1.1. Introduction.

In order to account for the Title that has been given to this Volume a few words of explanation will be necessary. The printing of the work was commenced in 1857. I had then only the intention of going through a revision of the principles of the different departments of pure and applied mathematics, thinking that the time was come when such revision was necessary as a preparation for extending farther the application of mathematical reasoning to physical questions. The extension I had principally in view had reference to the existing state of the science of Hydrodynamics, that is, to the processes of reasoning proper for the determination of the motion and pressure of fluids, which, as is known, requires an order of differential equations the solutions of which differ altogether from those of equations appropriate to the dynamics of rigid bodies. I had remarked that although by the labours of Lagrange, Laplace, and others, great success had attended the applications of differential equations containing in the final stage of the analysis only two variables, the whole of Physical Astronomy is, in fact, an instance of such application, the case was far different with respect to the applications of equations containing three or more variables. Here there was nothing but perplexity and uncertainty. After having laboured many years to overcome the difficulties in which this department of applied mathematics is involved, and to discover the necessary principles on which the reasoning must be made to depend, I purposed adding to the discussion of the principles of the other subjects, some new and special considerations respecting those of Hydrodynamics. The work, as thus projected, was entitled "Notes on the Principles of Pure and Applied Mathematics," the intention being to intimate by the word "Notes" that it would contain no regular treatment of the different mathematical subjects, but only such arguments and discussions as might tend to elucidate fundamental principles.

After repeated efforts to prosecute this undertaking, I was compelled by the pressure of my occupations at the Cambridge Observatory, to desist from it in 1859, when 112 pages had been printed. I had not, however, the least intention of abandoning it. The very great advances that were being made in physics by experiment and observation rendered it every day more necessary that some one should meet the demand for theoretical investigation which the establishment of facts and laws had created. For I hold it to be indisputable that physical science is incomplete till experimental inductions have been accounted for theoretically. Also the completion of a physical theory especially demands mathematical reasoning, and can be accomplished by no other means. When, according to the best judgment I could form respecting the applications which the results of my hydrodynamical researches were capable of, I seemed to see that no one was as well able as myself to undertake this necessary part in science, I gave up (in 1861) my position at the Observatory, under the conviction, which I expressed at the time, that I could do more for the honour of my University and the advancement of science by devoting myself to theoretical investigations, than by continuing to take and reduce astronomical observations after having been thus occupied during twenty-five years. The publication of this work will enable the cultivators of science to judge whether in coming to this determination I acted wisely. Personally I have not for a moment regretted the course I took; for although it has been attended with inconveniences arising from the sacrifice of income, I felt that what I could best do, and no one else seemed capable of undertaking, it was my duty to do.

It should, farther, be stated that after quitting the Observatory, and before I entered upon my theoretical labours, I considered that I was under the obligation to complete the publication of the meridian observations taken during my superintendence of that Institution. This work occupied me till the end of 1864, and thus it is only since the beginning of 1865 I have been able to give undivided attention to the composition of the present volume. In April 1867, as soon as I was prepared to furnish copy for the press, the printing was resumed, after I had received assurance that I might expect assistance from the Press Syndics in defraying the expense of completing the work. In the mean while I had convinced myself that the hydrodynamical theorems which I had succeeded in demonstrating, admitted of being applied in theoretical investigations of the laws of all the different modes of physical force, that is, in theories of light, heat, molecular attraction, gravity, electricity, galvanism, and magnetism. It may well be conceived that it required no little intellectual effort to think out and keep in mind the bearings and applications of so extensive a physical theory, and probably, therefore, I shall be judged to have acted prudently in at once producing, while I felt I had the ability to do so, the results of my researches, although they thus appear in a somewhat crude form, and in a work which in the first instance was simply designated as "Notes." Had I waited to give them a more formal publication, I might not, at my time of life, have been able to accomplish my purpose. As it is, I have succeeded in laying a foundation of theoretical physics, which, although it has many imperfections, as I am fully aware, and requires both correction and extension, will not, I venture to say, be superseded. In order to embrace in the Title page the second part of the work, the original Title has been altered to the following: "Notes on the Principles of pure and applied Calculation; and Applications of mathematical principles to Theories of the Physical Forces."

The foregoing explanations will serve to shew how it has come to pass that this work consists of two distinct parts, and takes in a very wide range of subjects, so far as regards their mathematical principles. In the first part, the reasoning rests on definitions and self-evident axioms, and although the processes by which the reasoning is to be conducted are subjects for enquiry, it is presumed that there can be no question as to the character and signification of definitions that are truly such. The first part is not immediately subservient to the second excepting so far as results obtained in it are applied in the latter. In the second part the mathematical reasoning rests on hypotheses. It does not concern me to enquire whether these hypotheses are accepted, inasmuch as they are merely put upon trial. They are proved to be true if they are capable of explaining all phenomena, and if they are contradicted by a single one they are proved to be false. From this general statement it will appear that in both portions of the work the principles and processes of mathematical reasoning are the matters of fundamental importance.

There are two general results of the arguments contained in the first part which may be here announced, one of them relating to pure calculation, and the other to applied calculation.
(1) All pure calculation consists of direct and reverse processes applied to the fundamental ideas of number and ratio.
(2) "All reasoning upon concrete quantities is nothing but the application of the principles and processes of abstract calculation to the definitions of the qualities of those quantities."

Having made these preliminary general remarks I shall proceed to advert to the different subjects in the order in which they occur in the body of the work, for the purpose of pointing out any demonstrated results, or general views, which may be regarded as accessions to scientific knowledge. I may as well say, at once that the work throughout lays claim to originality, consisting mainly of results of independent thought and investigation on points chiefly of a fundamental character. The first part is especially directed towards the clearing up of difficulties which are still to be met with both in the pure and the applied departments of mathematics. Some of these had engaged my attention from the very beginning of my mathematical career, and I now publish the results of my most recent thoughts upon them. I take occasion to state also that the commencements of the Physical Theories which are contained in the second part of the volume were published from time to time in the Transactions of the Cambridge Philosophical Society, and in the Philosophical Magazine. They are now given in the most advanced stages to which my efforts have availed to bring them, and being, as here exhibited, the result of long and mature consideration, they are, I believe, free from faults which, perhaps, were unavoidable in first attempts to solve problems of so much novelty and difficulty.

In the treatment of the different subjects I have not sought to systematise excepting so far as regards the order in which they are taken. The order that I have adopted, as arising out of the fundamental ideas of space, time, matter, and force, is, I believe, the only one that is logically correct.

All that is said in pages 4-20 on the principles of general arithmetic rests on the fundamental ideas of number and ratio. As we can predicate of a ratio that it is greater or less than another ratio, it follows that ratio is essentially quantity. But it is quantity independent of the magnitudes which are the antecedent and the consequent of the ratio. Hence there may be the same ratio of two sets of antecedents and consequents, and the denomination of one set is not necessarily the same as that of the other. This constitutes proportion. Proportion, or equality of ratios, is a fundamental conception of the human understanding, bound up with its power of reasoning on quantity. Hence it cannot itself be an induction from such reasoning. The Elements of Euclid are remarkable for the non-recognition of the definition of proportion as the foundation of quantitative reasoning. The fifth definition of Book V. is a monument of the ingenuity with which the Greek mind evaded the admission of proportion as a fundamental idea. By arguing from the definition of proportion, I have shewn (in page 13) that Euclid's fifth definition may be demonstrated as if it were a proposition, so that it cannot in any true sense be called a definition. It is high time that the method of teaching general arithmetic by the fifth Book of Euclid should be discontinued, the logic of the method not being defensible.

In Peacock's Algebra, mention is made of "the principle of the permanence of equivalent forms." The word "principle" is here used where "law" would have been more appropriate. For it is certain that the permanence of equivalent forms is not a self-evident property, nor did it become known by intuition, but was rather a gradual induction from processes of reasoning, the exact steps of which it might be difficult to trace historically, but which nevertheless actually led to the knowledge of the law. In the arguments which I have adduced in pages 15-20 I have endeavoured to shew how the law of the permanence of equivalent forms was, or might have been, arrived at inductively.

In the rapid review of the principles of Algebra contained in pp. 21-28, the point of chief importance is the distinction between general arithmetic and algebra proper. In the former certain general rules of operation are established by reasoning involving considerations respecting the relative magnitudes concerned; in the other these rules are simply adopted, and at the same time are applied without respect to relative magnitude. In order to make the reasoning good in that case the signs and are attached to the literal symbols. The use of these signs in the strictly algebraic sense is comparatively recent. It was imperfectly apprehended by Vieta, who first used letters as general designations of known quantities. The rules of signs were, I believe, first systematically laid down by our countryman Oughtred. Regarded in its consequences the discovery of the algebraic use of + and is perhaps the most fruitful that was ever made. For my part I have never ceased to wonder how it was effected. But the discovery being made, the rationale of the rules of signs is simple enough. In pp. 22-24 I have strictly deduced the rules for algebraic addition, subtraction, multiplication, and division, on the single principle of making these operations by the use of the signs independent of the relative magnitudes of the quantities represented by the letters. This principle is necessary and sufficient for demonstrating the rules of signs in all cases. As far as I am aware this demonstration had never been given before.

In p. 25 I have remarked that algebraic impossible quantities necessarily arise out of algebraic negative quantities; the former equally with the latter being indispensable for making algebra an instrument of general reasoning on quantity. It would be extremely illogical for any one to object to impossible quantities in algebra without first objecting to negative quantities.

The rules of the arithmetic of indices are demonstrated in pp. 25-27, on the principle that all modes of expressing quantity with as near an approach to continuity of value as we please must be included in a system of general arithmetic. It is then shewn that an algebraic generalisation of these rules gives rise to negative and impossible indices, just as negative and impossible algebraic expressions resulted from the analogous generalisation of the rules of ordinary arithmetic.

In p. 28 I have proposed using the mark $\equiv$ to signify that the two sides of an equality are identical in value for all values whatever of the literal symbols, the usual mark = being employed only in cases of equality for particular values of an unknown quantity, or particular forms of an unknown function. The former mark contributes greatly towards distinctness in reasoning relating to analytical principles, and I have accordingly used it systematically in the subsequent part of the work.

The Calculus of Functions (p. 37) is regarded as a generalisation of algebra analogous to the algebraic generalisation of arithmetic. In the latter, theorems are obtained that are true for all values of the literal symbols; in the other the theorems are equally applicable to all forms of the functions. Under the head of the "Calculus of Functions of one Variable" I have given a proof of Taylor's Theorem (p. 40), which is in fact a generalisation of all algebraic expansions of $f(x + h)$ proceeding according to integral powers of $h$, involving at the same time a general expression for the remainder term. As the function and this expansion of it are identical quantities, the sign $\equiv$ is put between them. The coefficients of $h, h^{2}$, etc. in the expansion contain as factors the derived functions $f'(x), f''(x)$, etc. It is important to remark that the Calculus of Functions does not involve the consideration of indefinitely small quantities, and that the derived functions just mentioned are all obtainable by rules that may be established on algebraic principles.

It is nevertheless true that by the consideration of indefinitely small quantities the Differential Calculus is deducible from the Calculus of Functions. The possibility of making this deduction depends on that faculty of the human intellect by which, as already remarked, it conceives of ratio as independent of the magnitudes compared, which, the ratio remaining the same, may be as small as we please, or as large as we please. This is Newton's foundation in Section I. of Book I. of a calculation which is virtually the same as the differential calculus. Having fully treated of the derivation of the differential calculus from the calculus of functions in pp. 47-49, I have occasion here to add only the following remark.

In p. 47 I have shewn that the ratio of the excess of $f(x + h)$ above $f(x - h)$ to the excess of $x + h$ above $x - h$, that is, the ratio of a finite increment of the function to the corresponding h finite increment of the variable, is equal to $f'(x) + f''''(x)h^{2/6}+$ etc., in which there are no terms involving $f''(x)$, etc. Usually in treatises on the Differential Calculus the expression for the same ratio, in consequence of making it apply to a position at the beginning instead of at the middle of the increments, has $f''(x) h$ in the second term. As far as regards the principles of the differential calculus, the logic of the foregoing expression is much more exact than that of the one generally given, because it shews that the limit of the ratio of the increment of the function to that of the variable is equal to the first derived function whatever be the value of $f''(x)$, even if this second derived function and the succeeding ones should be infinitely great. When the expression for that ratio has a term containing $f''(x)h$, it is by no means evident that that term vanishes on supposing $h$ to be indefinitely small, if at the same time the value of $x$ makes $f''(x)$ indefinitely great. For this reason, in applications of the differential calculus to concrete quantities, when an expression for a first derived function is to be obtained by a consideration of indefinitely small increments, the only logical course is to compare the increment $f(x + h) - f(x - h)$ with $2h$; which, in fact, may always be done. This rule should be attended to in finding the differentials of the area and the arc of any curve, and in all similar instances. It has been adopted in the present work (as, I believe, had not been done in any other) both in geometrical applications and in dynamical applications.

The differential calculus as applied to a function of two variables is analogously derived from the calculus of functions of two variables. In the course of making this deduction I have expressed, for the sake of distinctness, the partial differentials with respect to $x$ and $y$ of a function $u$ of $x$ and $y$ by the respective symbols $d_{x} u$ and $d_{y} u$. This notation is particularly applicable where every differential coefficient, whether partial or complete, is regarded as the ultimate ratio of two indefinitely small increments. I might have employed it with advantage in my hydrodynamical researches; but on the whole I have thought it best to adopt the rules of notation stated in p. 174.

Under the head of "the principles of geometry,", I have discussed Euclid's definition of parallel straight lines and its relation to Axiom XII. These points, as is well known, have been very much litigated. I think I have correctly traced the origin of all the difficulty to what I have already spoken of as the non-recognition in the Elements of Euclid of our perceptions of equality, and equality of ratios, as the foundation of all quantitative reasoning. This foundation being admitted, there should be no difficulty in accepting as the definition of parallel straight lines, that "they are equally inclined, towards the same parts, to the same straight line." Equality is here predicated just as when a right angle is defined by the equality of adjacent angles. Euclid's definition, that parallel straight lines do not meet when produced ever so far both ways, is objectionable for the reason that it does not appeal to our perception of equality. Moreover, if the proposed definition be adopted, the property of not meeting is a logical sequence from Prop. XVI. of Book I.; for, supposing the lines to meet, a triangle would be formed, and the exterior angle would be greater than the interior angle, which is contrary to the definition. In p. 64 I have shewn that by means of the same definition Axiom XII. may be proved as a proposition.

Another instance of a definition in Euclid being such as to admit of being proved, is presented by Def. XI. of Book III., which asserts that "similar segments of circles are those which contain equal angles." This is in no sense a definition, because it is not self-evident, nor does it appeal to our perception of proportion. Def. I. of Book VI., inasmuch as it rests on equality of ratios is strictly a definition of similarity of form, but applies only to rectilinear forms. By adopting a definition which involves only the perception of equality of ratios, and applies equally to curvilinear and rectilinear figures, I have proved that "similar segments of circles contain equal angles."

In p. 70 I maintain that the proportionalities asserted in Propositions I. and XXXIII. of Book VI. are seen at once by an unaided exercise of the reasoning faculty, and cannot be made more evident by the complex reasoning founded on Def. V. of Book V. The use made of that definition in proving the two Propositions is no evidence that it is a necessary one.

The object of the discussion commencing in page 70 and ending in page 88 is to shew that by the application of abstract calculation all relations of space are deducible from geometrical definitions, and from a few elementary Propositions the evidence for which rests on an appeal to our primary conceptions of space. This argument was, in fact, required for proving that the generalisation announced in page viii is inclusive even of the relations of pure space. In page 82 I have been careful to intimate that the discussion was solely intended to elucidate the fundamental principles on which calculation is applied in geometry, and not to inculcate a mode of teaching geometry different from that usually adopted. At the same time I have taken occasion to point out a distinction, which appears not to have been generally recognised, between geometrical reasoning, and analytical reasoning applied to geometry. The former is reasoning respecting the relations of lines, areas, and forms, necessarily conducted by means of diagrams, on which account it is properly called "geometrical reasoning." But it involves no measures of lines and angles, and in that respect is essentially distinct from analytical reasoning, in which such measures are indispensable. By many minds geometrical reasoning is more readily apprehended than analytical, and on that account it is better fitted than the latter to be a general instrument of education. Regarded, however, as a method of reasoning on relations of space, it is incomplete, because it gives no means of calculating such relations. The method of analytical geometry, on the contrary, is not only capable, as I have endeavoured to shew by the argument above referred to, of proving all geometrical theorems, but also, by the intervention of the measures of lines and angles, of calculating all geometrical relations. In short, analytical geometry is the most perfect form of reasoning applied to space.

[Note. I quite assent to the propriety of that strict maintenance of the distinction between geometrical reasoning and analytical geometry which is characteristic of the Cambridge system of mathematical examinations; but I am wholly unable to see that this is a ground for the exclusion of analytical geometry to the extent enjoined by the recently adopted scheme for the examinations. According to the schedule the examiners have no opportunity, during the first three days of the examination, of testing a candidate's knowledge of the application of algebra to geometry, and it is consequently possible to obtain a mathematical honour without knowing even the elementary equations of a straight line and a circle.]

In page 90 I have employed the terms "Plane Astronomy" as being in common use; but I now think that "Spherical Astronomy" would have been more appropriate, inasmuch as applied calculation in the department of Astronomy which those terms designate consists mainly in finding relations between the arcs and angles of spherical triangles. The arcs are such only as are measured by astronomical instruments, either directly, or by the intervention of time. The element of time makes a distinction between the astronomical problems of this class and problems of pure geometry. The purpose of the notes in pages 90-96 on the science of Time is to shew how measurements of the uniform flow of time, and determinations of epochs, are effected by astronomical observation, and depend on the assumption of the uniformity of the earth's rotation about its axis. In page 91 I say, "there is no reason to doubt the fact that this rotation is perfectly uniform." But in page 127 I have admitted the possibility of a gradual retardation resulting from the moon's attraction of the tidal waters. This inconsistency is attributable to the circumstance that the reasons adduced in p. 127 for the latter view became known in the interval from 1859 to 1867, during which the printing of the work was suspended after it had proceeded to p. 112.

The simple and satisfactory explanation of the Aberration of Light given in pages 97 and 98 was first proposed by me in a communication to the Phil. Mag. for January 1852, after attempts made in 1845 and 1846 with only partial success. That Article was followed by another in the Phil. Mag. for June 1855 referring more especially to the effect of aberration on the apparent places of planets. The explanation wholly turns on the facts that instrumental direction is determined by the passage of the light from an object through two points rigidly connected with the instrument, and that, by reason of the relative velocity of the earth and light, the straight line joining the points is not coincident with the direction in which the light travels. One of the points is necessarily the optical centre of the object-glass of the Telescope. Although this explanation has now been published a considerable time, it has not yet found its way into the elementary Treatises on Astronomy, which continue to give nothing more than vague illustrations of the dependence of the phenomenon on the relative motion of the earth and light. This being the case, I take the opportunity to say, in order to draw attention to what is essential in the explanation, that if the cause of the aberration of light were set as a question in an examination, any answer which did not make mention of the optical centre of the object-glass would not deserve a single mark.

Under the head of the Principles of the Statics of rigid bodies (pp. 98-104), I have shewn that Lagrange's beautiful proof of the general equation of Virtual Velocities, after the correction at one part of it of a logical fault (p. 102), rests (1) on the fundamental property of a rigid body according to which the same effect is produced by a given force in a given direction along a straight line at whatever point of the line it be applied; and (2) on the definition of statical equilibrium. These are the fundamental principles of Statics, whatever be the mode of treatment of statical problems.

In stating the principles of Hydrostatics, a fluid is defined (1) by its property of pressing, and (2) by that of easy separability of parts. The second of these definitions has been adopted on account of its having important applications in Hydrodynamics, as will be subsequently mentioned. The law of the equality of pressure in all directions from a given fluid element is rigidly deduced from these two definitions.

In the statement of the principles of the Dynamics of solid bodies in motion, I have adhered to the terms which came into use at and after the Newtonian epoch of dynamical science, although I should be willing to admit that they might in some respects be improved upon. But whatever terms be adopted, all reasoning respecting velocity, accelerative force, momentum, and moving force, is founded on certain elementary facts which have become known exclusively by observation and experiment. These fundamental facts are the following:-
(1) in uniform velocity equal spaces are described in equal times;
(2) a constant force adds equal velocities in equal times;
(3) the velocity added by a constant force in the direction in which it acts is independent of the magnitude and direction of the acquired velocity;
(4) the momentum is given if the product of the mass and the velocity be given;
(5) the moving force is given if the product of the mass and the accelerative force be given.
It is especially worthy of remark that although these facts were not discoverable by any process of reasoning, it is possible by reasoning to ascertain the function that the space is of the time in the case of variable velocity, and the functions that the velocity and space are of the time in the case of a variable accelerative force. Since in these cases functions are to be found, it follows from the principles of abstract calculation that we must for that purpose obtain differential equations. The processes by which these are deduced by the intervention of the facts (1), (2), and (3), are fully detailed in pages 109-117. In this investigation Taylor's Theorem has been used in the manner indicated in page xiii.

In the Notes on Physical Astronomy commencing in page 119, I have, in the first place, adverted to the essential distinction which exists between the labours of Kepler and those of Newton in this department of science. This distinction, which holds no place in Comte's system of philosophy, is constantly maintained in Whewell's History and Philosophy of the Inductive Sciences. I select the following passage from the History: "Kepler's laws were merely formal rules, governing the celestial motions according to the relations of space, time, and number; Newton's was a causal law, referring these motions to mechanical reasons. It is no doubt conceivable that future discoveries may both extend and farther explain Newton's doctrines;- may make gravitation a case of some wider law, and disclose something of the mode in which it operates; questions with which Newton himself struggled." In accordance with these views I have noticed that Kepler's observations and calculations do not involve the consideration of force, and that the laws he discovered were really only problems for solution. Newton solved these problems by having found the means of calculating the effects of variable forces. This was his greatest discovery. By calculations made on the hypothesis that the force of gravity acts according to the law of the inverse square, Newton gave dynamical reasons for Kepler's laws, which may also be called causative reasons, in-as-much as whatever causes is force, or power, as we know from personal experience and consciousness. The principle which is thus applied to physical astronomy I have extended in a subsequent part of this work to all quantitative laws whatever. I have maintained that all such laws, as discovered by observation and experiment, are so many propositions, which admit of à priori demonstration by calculations of the effects of force, founded on appropriate hypotheses. This, in short, is Theory.

In making the remarks contained in pages 120-124 I was under the impression that the first evidence obtained by Newton for the law of gravity was derived from comparing the deflection of the moon from a tangent to the orbit in a given time with the descent of a falling body at the earth's surface at the same time, and that he did not have recourse to Kepler's laws for that purpose. This, at least, might have been the course taken. But on consulting Whewell's History of the Inductive Sciences, I find that the inference of the law of gravity from the sesquiplicate ratio of the periodic times to the mean distances, as given in Cor. 6 of Prop. IV., Lib. I., and the converse inference of the sesquiplicate ratio from the law, preceded historically those computations relative to the law of action of the Earth's gravity on the moon, which Newton finally made after obtaining a corrected value of the earth's radius.

A discussion of considerable length (contained in pages 128-152) is devoted to the determination of the physical significance of the occurrence, in the developments of radius-vector and latitude, of terms which increase indefinitely with the time. The consideration of this peculiarity of the Problem of Three Bodies falls especially within the scope of the present work, inasmuch as it is a question to be settled only by pure reasoning, and points of principle are involved in the application of the reasoning. As this question had not received the attention it deserves, and as I could be certain that the clearing up of the obscurity surrounding it demanded nothing but reasoning from the given conditions of the problem, and would, if effected, be an important addition to physical astronomy, I felt strongly impelled to make the attempt, although my researches had previously been much more directed to the applications of partial differential equations than to those of differential equations between two variables. My first attempts were far from being successful, and it was not till after repeated and varied efforts that I at length ascertained the origin and meaning of the terms of indefinite increase. As the decision of this point is necessary for completing the solution of the Problem of Three Bodies, I thought it might be regarded of sufficient interest to justify giving some historical details respecting the steps by which it was arrived at.

My attention was first drawn to this question by a paragraph in Mr Airy's Lunar Theory (Mathematical Tracts, art. 44*, p. 32, 3rd Ed.), where it is asserted that the form of the assumption for the reciprocal of the radius-vector, viz. $u = a [1 + e \cos (c\theta - a)]$, "is in no degree left to our choice." It is then shewn how that form may be obtained by assuming for $u$ the general value $a (1 + w)$; but the principle on which this assumption is made is not explained. My first researches were directed towards finding out a method of integrating the equations by which the above form of $u$ and the value of the factor $e$ should be evolved by the usual rules of integration without making any previous assumption. Having, as I supposed, discovered such a method, I offered to the Cambridge Philosophical Society a communication entitled "Proofs of two new Theorems relating to the Moon's orbit," respecting which an unfavourable report was made to the Council, and not without reason; for it was a premature production, and had in it much that was insufficiently developed, or entirely erroneous. The paper, however, contained the important differential equation at the bottom of page 145 of this Volume, arrived at, it is true, by imperfect reasoning, and also the deduction from it of the equation $e^{2} = 1 - \Large\frac{Ch^{2}}{\mu^{2}}\normalsize + \large\frac{1}{2}\normalsize m^{2}$ which is equivalent to one near the top of page 147. This last equation, for reasons I shall presently mention, forms an essential part of the solution of the Problem of Three Bodies.

In this first essay I obtained the above mentioned differential equation without distinctly shewing that it involves the hypothesis of a mean orbit. This defect is supplied by the argument contained in pages 142-145, where the equation is arrived at by suppressing terms containing explicitly the longitude of the disturbing body, which process is equivalent to making that hypothesis. Also, as is proved in pages 146 and 147, the integral of the equation completely answers the purpose of obtaining the proper form of $u$, and an approximation to the value of $c$, without any previous assumption relative to that form. But it is important to remark that the deduction of these results wholly depends on the antecedent hypothesis of a mean orbit, which hypothesis is, in fact, involved in every process made use of for getting rid of terms of indefinite increase. This is the case in the method adopted in Pratt's Mechanical Philosophy, where the substitution of $u - b$ for be $\cos (\theta - a)$ seems like employing for the purpose a species of legerdemain, until it be understood that by this step the hypothesis of a mean orbit is first introduced. The same remark applies to the method already referred to as having been proposed by Mr Airy, which in principle is the same as that just mentioned.

It also appears that the differential equation in p. 145 is the same that would be obtained on the supposition that the body is acted upon by the force $\Large\frac{\mu}{r^{2}}\normalsize - \Large\frac{m'r}{2a'^{3}}$ tending to a fixed centre, and consequently, as in Newton's Section IX., the radius-vector is equal to that of an ellipse described by the action of a force tending to the focus and varying inversely as the square of the distance, the ellipse revolving at the same time uniformly about its focus. This was my Theorem 1., to which, after the explanation that it applies only to the Moon's mean orbit, there is nothing to object. (See the discussion of this case in pages 149-151.)

But Theorem 11. was wholly erroneous, being deduced from the foregoing value of $e^{2}$ by arguing on grounds which cannot be sustained that $\mu^{2} = Ch^{2}$ and consequently that $e^{2}= \large\frac{1}{2}\normalsize m^{2}$, $m$ being the ratio of the Moon's periodic time to the Earth's. I ought to have inferred from that value, as is shewn by the course of reasoning concluded in page 148, that $\large\frac{1}{2}\normalsize m^{2}$ is an inferior limit to the square of the eccentricity of the mean orbit.

Having published the two Theorems in the Philosophical Magazine for April 1854, in the June Number of that year I invited Professor Adams, who was one of the reporters on my paper, to discuss with me its merits. Accordingly, in a letter addressed to the Editors of that Journal, contained in the July Number, he gave in detail the reasons of his disapproval of the new theorems. These reasons, I now willingly admit, proved that I had no right to conclude from my arguments that $Ch^{2} = \mu^{2}$ and hence that the eccentricity of the Moon's orbit is $\Large\frac{m}{√2}\normalsize$. It was also justly urged that the same theorem, applied to the orbit of Titan, was contradicted by the actual eccentricity. Some of the objections, which depended on my not carrying the approximations far enough, are met by the more complete investigation contained in this Volume. Professor Adams took no notice of the equation $e^{2} = 1 - \Large\frac{Ch^{2}}{\mu^{2}}\normalsize + \large\frac{1}{2}\normalsize m^{2}$, which, as I thought, should have saved my views from unqualified condemnation.

In my reply in the August Number I said much in the heat of controversy that had better not have been said, and some things also that were untrue. Still I claim the merit of having seen that the question respecting the meaning of terms of indefinite increase was of so much importance, that till it was settled the gravitational theory of the motions of the heavenly bodies was incomplete. Professor Adams was precluded from adopting this view by having taken up exclusively the position, that the solution of the differential equations obtained by introducing the factors usually called $c$ and $g$ is "the true and the only true solution," because it contains the proper number of arbitrary constants and satisfies the equations. On the other hand I argued, but not as logically as I might have done, that under those circumstances "the constants $a, e, \epsilon, w$ are not necessarily [i.e. absolutely] arbitrary and independent of each other," inasmuch as the solution is limited by the introduction of the factors $c$ and $g$, and is therefore not the complete, or most general, solution of the given equations. The same argument, put in the form which longer consideration of the subject has led me to adopt, may be stated as follows.

What is done by the insertion of the factors $e$ and $g$ is to impose arbitrarily the condition that there shall be a mean orbit, that is, an orbit in which the longitude shall oscillate about that in a revolving ellipse, and the latitude about that in a fixed plane. That this is the case is demonstrated by conversely deducing the values of $c$ and $g$, as is done in pages 147 and 152, from equations not containing explicitly the longitude of the disturbing body, and, therefore, by arguing on the hypothesis of a mean orbit. The fact that the differential equations can be satisfied after introducing these factors, is the proper proof that a mean orbit is possible. The analytical circumstance that before the introduction of the factors the integration leads to terms of indefinite increase proves that there is not always, or necessarily, a mean orbit. If the differential equations could be exactly integrated, the integrals would contain the same number of arbitrary constants as the limited solution under discussion, but they would embrace non-periodic motion as well as periodic, and the constants introduced by the integration would be absolutely arbitrary. Hence the constants of the limited solution, although the same in number, cannot be in the same degree arbitrary, but must have been subjected to limitations by the process which limited the solution. On this point I have given the following direct evidence.

The equation (A) in page 139 is a first integral of the given differential equations, obtained by usual processes, and its right-hand side contains exclusively the terms involving the disturbing force. If in this side the elliptic values of a first approximation be substituted for $r$ and $\theta$, all its terms will contain $e$ as a factor. The case would be the same if the approximate values of $r$ and $\theta$ belonging to a revolving ellipse were substituted. Hence it appears, since $e^{2}$ was put for $1 - \Large\frac{Ch^{2}}{\mu^{2}}\normalsize + \large\frac{1}{2}\normalsize m^{2}$, that if $1 - \Large\frac{Ch^{2}}{\mu^{2}}\normalsize + \large\frac{1}{2}\normalsize m^{2}$, the equation (4) becomes

          $\Large\frac{dr^2}{dt^2}\normalsize + \Large\frac{1}{C} (\frac{\mu}{r}\normalsize -C\Large)\normalsize^2$;

that is, the equation of a circle of radius $\Large\frac{\mu}{C}\normalsize$. Now this orbit could not possibly be described so long as there is a disturbing force, and we are thus driven to the conclusion that if there be a disturbing force, $e$ cannot be zero, but must have a certain limiting value. Thus it is shewn that in this solution one of the arbitrary constants is subject to limitation. I believe I may say it was after discovering that $e$ and the disturbing force vanish together, that I fell, not unnaturally, into the error of supposing that $e^{2}$ must contain the disturbing force as a factor. By subsequent research I found that this inference is not necessary.

The conclusion that $e$ has a limiting value ought, I think, to arrest the attention of mathematicians engaged on the theory of gravitation. It had not been previously arrived at, because the differential equation (A) of the first order, which is intermediate to the given differential equations of the second order and their complete integrals, has been singularly overlooked by investigators in the theory of gravity. The determination of the limiting value will be presently adverted to.

The foregoing narrative will have sufficiently indicated the grounds of the divergence of my views from those of Professor Adams. When I found that our discussion had not settled the question as to the signification of terms of indefinite increase, I pursued the enquiry in a series of communications contained in the Numbers of the Philosophical Magazine for December 1854, and February, March, and May 1855, which will, at least, attest the diligence with which I laboured to get at the truth of this question. These investigations, which referred mainly to the Moon's orbit, were followed by a more elaborate paper on the Problem of Three Bodies, read before the Royal Society on May 22, 1856, and printed in their Transactions (1856, p. 525). This treatment of the problem applies more especially to the Planetary Theory.

The method of solution I adopted relative to the Moon's orbit is characterised by successive approximations both to the mean orbit and the actual orbit, proceeding pari passu. The former approximations are made on the principle of omitting terms containing explicitly the longitude of the disturbing body, which is the same as the principle of omitting in the Planetary Theory periodic variations of short period in the investigation of secular variations. The solution of the problem of three bodies in the Philosophical Transactions is a direct determination of the actual orbit only, peculiar in the respect that by making use of the equation (4) the approximations are evolved without any initial supposition as to the form of solution. The expressions for the radius-vector and longitude are the same as those obtained by Laplace. I may as well state here that I had no intention in my researches in physical astronomy to furnish formulae for the calculation of Tables. I have perfect confidence in the principles on which those that have been used for this purpose have been investigated. My concern was solely with the logical deduction of consequences from the analysis which, although they do not affect the calculation of Tables, are important as regards the general theory of gravitation.

In pages 128-152, I have collected from the above-mentioned papers, all the arguments which, after mature consideration, I judged to be valid, (1) for explaining the nature of terms of indefinite increase; (2) for determining the limiting value of the constant $e$.

On the first point, I have come to the conclusion that by terms admitting of indefinite increase, the analysis indicates that in the general problem of three bodies, the motion is not necessarily periodic, or stable, and that the motion of a particular planet, or satellite, is proved to be stable by finding, after calculating on the hypothesis of a mean orbit, that the resulting solution is expressible in a series of convergent terms. This conclusion is, however, more especially applicable to the Minor Planets, because they are not embraced by the known general theorems which prove that the stability of the motions of the larger planets is secured by the smallness of the eccentricities and the inclinations.

With respect to the other point, by the approximations to the actual orbit and to the mean orbit, and by determining certain relations between their arbitrary constants, I have been finally conducted to the equation $e^{2} = e_{0}^{2} + \large\frac{1}{2}\normalsize m^{2}$ at the top of p. 148, which, however, was obtained on the supposition that both $e$ and $m$ are small quantities. Since $e_{0}^{2}$ is an arbitrary constant necessarily positive, this equation shews that if $e^{2} = 0$, we have $e_{0}^{2} = 0$ and $m^{2} = 0$, the last result agreeing with that mentioned in p. xxvi. The equation proves also that $e^{2}$ may have different arbitrary values, but all greater than the limiting value $\large\frac{1}{2}\normalsize m^{2}$.

In page 141 I have obtained the value of $e_{1}$, the eccentricity of the mean orbit, which is, in fact, what is called the mean eccentricity, being independent of all particular values of the longitude of the disturbing body. It is shewn also that $e^{2} = e_{1}^{2}$, if $e$ and $m$ be small. Hence it may be inferred from the foregoing limit to the value of $e^{2}$, that $\Large\frac{m}{√2}\normalsize$ is an inferior limit to the mean eccentricity. This theorem, which may, I think, be regarded as an interesting addition to the theory of gravitation, has been arrived at by patiently investigating the meaning of an unexplained peculiarity of the analysis, in perfect confidence that an explanation was possible, and could not fail to add something to our theoretical knowledge. It should, however, be noticed that the theorem is true only for the problem of three bodies. I have not attempted to extend the reasoning to the case of the mutual attractions of a greater number.

In the Notes on the Dynamics of the Motion of a Rigid System, there are three points to which I think it worth while to direct attention here.

(1) In page 153 I have endeavoured to state D'Alembert's Principle in such manner that its truth may rest on a simple appeal to our conception of an equality. It has already been remarked (p. xiv.) that a principle or definition which satisfies this condition is proper for being made the basis of quantitative reasoning.

(2) After deducing the general equation of Vis Viva by means of D'Alembert's Principle and the Principle of Virtual Velocities, I have remarked that there is impropriety in speaking of the principle of the conservation of Vis Viva, as expressed by that equation. For since the equation is a general formula obtained by analytical reasoning from those two principles, it is properly the expression of a law, it being the special office of analysis to deduce laws from principles and definitions. The distinction will not appear unimportant when it is considered that the law of Vis Viva has been relied upon by some mathematicians as if it were a principle of necessary and universal application, whereas the applicability of a law is determined and limited by the principles from which it is derived. To speak of the principle of Virtual Velocities is not in the same manner incorrect, because, for the reasons stated in page 102, the general equation of Virtual Velocities rests only on the fundamental principles of Statics, and may be regarded as the expression of a single principle substituted for them.

(3) All problems in the Dynamics of Rigid Bodies admit of being solved by means of the six equations given in page 157. When the known values of the impressed moving forces for a particular instance have been introduced into these equations, the solution of the problem is a mere matter of reasoning conducted according to the rules of abstract analysis. All circumstances whatever of the motion are necessarily embraced by this reasoning. I have been induced to make these remarks because it is usual to solve problems of this class by the initial consideration of angular motions about rectangular axes. This method is, no doubt, correct in principle, and is generally more convenient and elegant than that of directly integrating the differential equations. But it should be borne in mind that the latter method is comprehensive of every other, and that all the equations involving angular motions about rectangular axes are deducible from the integrations.

To illustrate these points I have attacked the problem of the motion of a slender hoop, by first adapting the six general equations to the particular instance, and then integrating for the case in which the hoop has a uniform angular motion about an axis through its centre perpendicular to its plane. I have emphasised at the top of page 164 the inference that "when a hoop rolls uniformly on a horizontal plane, it maintains a constant inclination to the plane and describes a circle," in consequence of having noticed that in the usual mode of solving the problem, this inference, being regarded as self-evident, has not been deduced by reasoning. Nothing that can be proved ought to be taken for granted.

The mathematical theory of Foucault's Pendulum Experiment is prefaced by a remark which may serve to explain why this problem had not been mathematically solved before attention was drawn to it by experiment. By reason of the earth's rotation about its axis, there is relatively to any given position an equal motion of rotation of all points rigidly connected with the earth about a parallel axis passing through that position. This circumstance ought in strictness to be taken into account, when it is required to refer motions, such as oscillations due to the action of gravity, to directions fixed with respect to the earth. This, it seems, no mathematician had thought of doing.

In the subjects that have hitherto been mentioned, I have succeeded, I think, in shewing that in some few particulars they admitted of additions to, or improvements upon, the processes of reasoning that had been applied to them by my predecessors and contemporaries in mathematical science. But in the subject of Hydrodynamics, (which occupies the large portion of this work extending from page 170 to page 316), I found the reasoning to be altogether in a very unsatisfactory state. After accepting the fundamental definitions on which the propositions of Hydrodynamics are usually made to rest, I discovered that methods of reasoning had been employed which were, for the most part, either faulty or defective. The following statement relates to an instance of the prevalence of a faulty method of reasoning.

My first contribution to the science of Hydrodynamics was a paper "On the theory of the small vibratory motions of elastic fluids," read before the Cambridge Philosophical Society on March 30, 1829, and printed in Vol. III. of the Transactions. That paper contains the first instance, I believe, of the determination of rate of propagation by differentiation, the principle of which method is insisted upon in pages 189 and 190 of the present work. At the very commencement of my scientific efforts I was unable to assent to Lagrange's method of determining rate of propagation, although it appears to have been accepted without hesitation by eminent mathematicians, and continues to this day to hold a place in elementary treatises. I perceive, however, that Mr Airy in art. 24 of his recently published work On Sound and Atmospheric Vibrations, has employed a method equivalent to that of differentiation, and I have reason to say that other mathematicians have now discarded Lagrange's method. But no one except myself seems to have discerned that as that method determines by arbitrary conditions a quantity that is not arbitrary, it involves a violation of principle. This, from my point of view, is a very important consideration; because if principle has been violated in so simple a matter, what security is there that the same thing has not been done in the more advanced and more difficult parts of the subject? My researches have led me to conclude that this has actually taken place.

The evidence on which I assert that reasoning has been usually employed in Hydrodynamics which is defective in principle, and requires to be supplemented, is in part given by the solution of Example I., beginning in page 193. Without any departure from the ordinary mode of reasoning the conclusion is there arrived at that the same portion of the fluid may be at rest and in motion at the same instant. When I first published this reductio ad absurdum, Professor Stokes attempted to meet it, (as I have mentioned in page 196), by saying that the analysis indicated something like a breaker or bore, forgetting, so it seems to me, that as breakers and bores are possible natural phenomena due to special circumstances, they cannot be included in an investigation which takes no account of those circumstances, which, besides, is found to lead to an impossibility, or to what is per se a contradiction. I have adverted also (p. 196) to similar views advanced by Mr Airy in a communication which by his own admission "does not consist of strict mathematical reasoning, but of analogies and conjectures." It will suffice for pointing out the character of these surmises to refer to the passage in the communication in which Mr Airy speaks of "the probable sensational indications" of the physical phenomenon "interruption of continuity of particles of air," such as a hiss, a buzz, &c. Admitting the possible applicability of these conceptions under circumstances which were not taken into account in the antecedent investigation of the differential equation, I deny altogether that the analysis in the present case indicates any interruption of continuity of the particles, inasmuch as, according to its strict meaning, after the above-mentioned contradiction is consummated, the motion goes on just as smoothly as before; which is only another phase of the absurdity. Since, therefore, strict mathematical reasoning, which neither of these two mathematicians has controverted, has shewn that the differential equation on which their views are founded leads to a reductio ad absurdum, it follows by necessary logical sequence that the equation is a false one, and that analogies and conjectures relating to it are misapplied.

That same equation is discussed by Mr Earnshaw in a paper On the Mathematical Theory of Sound, contained in the Philosophical Transactions for 1860, p. 133. At the time of the publication of his paper the author was well aware of the argument by which I had concluded that the equation is an impossible one. In the course of the discussion there occurs the singular assertion that a wave, after assuming the form of a bore, "will force its way in violation of our equations." Now the only interpretation that can possibly be given to this sentence is, that Mr Earnshaw conceives he is justified in supplying by his imagination what the equations fail to indicate, whereas it is unquestionable that we can know nothing about what the wave does except by direct indications of the equations. For the foregoing reasons I think I may say that Mr Earnshaw has applied a false method of reasoning to a false equation. It is not surprising that his views are approved of by Mr Airy (Treatise on Sound, p. 48) and by Professor Stokes (Phil. Trans. for 1868, p. 448), since they are the same in principle as those which had been previously advocated by themselves. But Mr Earnshaw in the sentence above quoted has divulged the mental process by which the existence of a bore, &c. is inferred, and has shewn that it involves an exercise of the imaginative faculty.

[Note. In a Lecture on "The Position and Prospects of Physical Science" delivered by Professor Tait of Edinburgh, on November 7, 1860, mention is made of the "meagre development" of Hydrodynamics, and the whole subject is represented as having been "till lately in a very backward state." Two "very great improvements" are then said to have been very recently made. On one of these I shall have to speak afterwards; the other was considered to be effected by Mr Earnshaw's paper on Sound. The author of the Lecture had evidently not comprehended the arguments by which I had proved that the equation on which Mr Earnshaw relied was vitiated by defect of principle, nor the reasoning by which already in 1851 and 1852 I had succeeded in supplying what was wanting in the received principles of Hydrodynamics.]

The contradiction above discussed is not the only one that results from reasoning founded upon the principles of Hydrodynamics as usually accepted. The solution of Example II. in page 197 leads to another contradiction. Perhaps the evidence in this instance may be made more distinct by remarking, that in the integral $\sigma r = f(r - at)$ in page 198, the arbitrary function may be discontinuous in such manner that the values of $\sigma$ in two spaces separated by any indefinitely thin partition transverse to the direction of the motion may be expressed by different algebraic functions, if only the condensations immediately contiguous to the partition be equal on the opposite sides. This is a legitimate consequence of the fundamental property of easy separability of parts. It hence follows that the uniform propagation of a solitary wave either of condensation or rarefaction from a centre is possible. But in that case the condensation necessarily varies inversely as the square of the distance $r$, whereas the formula shews that it varies simply as the inverse of that distance. This is so direct a contradiction that the principles on which the reasoning was based must be pronounced to be either false or insufficient.

On the publication of Mr Airy's Treatise on Sound and Atmospheric Vibrations (in 1868), I naturally looked for some attempt to meet the two foregoing arguments, which I had urged as long since as 1849 in the Numbers of the Phil. Mag. for February and April of that year. I found that by giving only an approximate solution of an equation which is equivalent to that in page 194 of this work, of which Poisson's exact integral is well known, Mr Airy has avoided the direct consideration of the first argument. He refers, however, in art. 34 to a "conjectural" change of the character of the wave as "suggested by Mr Earnshaw." On this point I have already said enough. The exact integral logically treated leaves no room for conjecture.

With respect to the second argument, I have to remark that Mr Airy has admitted (in Art. 31) the possibility of the transmission of a solitary wave; but in the discussion (in Art. 50) of the symmetrical divergent wave in air of three dimensions, no notice is taken of the law of the variation of the condensation inversely as the distance from the centre (the condensation itself not being once mentioned), although the law is readily deducible from the solution. Accordingly no attempt is made to shew how the uniform transmission of a solitary condensed or rarefied wave, as resulting from the principle of the discontinuity of the arbitrary function, can be reconciled with the existence of that law; so that the consideration of the second argument is omitted. Thus a reader of Mr Airy's book might be led to suppose that the received principles of Hydrodynamics are not liable to the objections which I have urged, which, to say the least, are not such as can be overlooked. To keep difficulties out of sight is not likely to advance philosophy.

It may surprise the non-mathematical world to find that on a question the premises of which are not in dispute, mathematicians are not agreed as to the process of reasoning. This, in fact, ought not to be; for in such a case it is certain that some rule of logic has been violated either on one side, or on both. In the present instance the fault does not lie with me. By strict reasoning from the received principles of Hydrodynamics I have been led to contradictions, and have consequently concluded, according to an undoubted rule of logic, that the principles require reforming. My opponents, without contesting either the reasoning or the contradictions, will not accept the logical consequence. Rather than do this they have recourse to conjectures and to imagined analogies and probabilities. This sort of unreasonableness is no new thing in the annals of science. There have been epochs before in which argument has availed nothing against prejudgment founded upon error of long standing.

I will not do the mathematical contemporaries whose names I have had occasion to mention the injustice to think that they will impute to me any other motive in making the foregoing remarks than an honest contention for scientific truth. There are special reasons for insisting on the arguments by which I have inferred the insufficiency of the received principles of Hydrodynamics. The whole future of theoretical philosophy, as this work, I think, will shew, turns upon this point. As the great discovery of the Newtonian epoch of theoretical science was the method of calculating the motion of a single particle under the action of given forces, so the existing state of theoretical physics demands a knowledge of the method of calculating the motion and pressure, under given conditions of force, of a congeries of particles in juxtaposition. It was in researches for this object that I encountered the failure of the usual principles of Hydro-dynamics, and subsequently discovered what was required for making them good. Respecting this last question, on account of its importance, a few subsidiary remarks may be appropriately made here.

Having no reason to suppose that the commonly received principles of Hydrodynamics were not true, I concluded, by a rule of logic, that the aforesaid arguments only proved them to be insufficient, and I took the course of endeavouring to supply what was wanting. By slow degrees I arrived at the conviction that a new equation of geometrical continuity was required. The investigation of such an equation in pages 174 and 175 is founded on Axiom II. in page 174, which is a perfectly intelligible expression of a principle of geometrical continuity necessarily applicable to the motion of a fluid mass, if only such motion admit of being analytically calculated. On this account the axiom must be granted. The equation (1) in page 175 is investigated so as to secure that the above principle, viz. that the directions of motion in a given element are normals to a continuous surface, shall apply to all the elements of the fluid at all times, and is, therefore, a veritable equation of continuity. This name has been improperly given to the equation (2) in page 177, which only secures that the quantity of matter shall be always the same. It should be noticed. that the continuity here considered is purely geometrical, and, therefore, distinct from that mentioned in p. 181, according to which the direction of the motion of a given particle is determined by dynamical conditions to be so far continuous that it cannot change per saltum.

The new general equation (1) introduces two variables $\phi$ and $\lambda$, for determining which in addition to the other variables there are a sufficient number of equations, as is shewn in p. 179. Also since $\lambda (d\phi) = udx + vdy + wdz$, the same equation enables us to reason generally, without being restricted to the cases in which the right-hand side of that equality is an exact differential, which are usually treated by the intervention of the function $\phi$, or, as it is called in Mr Airy's work, the characteristic function $F$. To make the logic of analytical Hydrodynamics good, it is absolutely necessary to be able to argue independently of that restriction; which the new equation gives the means of doing.

By so arguing I have, first, shewn (Prop. VII., p. 186) that the abstract supposition of the integrability of $udx + vdy + wdz$ conducts to rectilinear motion, and then inferred from the reasoning in pages 193-200, and from the solution of Prop. XI. (p. 201), that the straight line along which the motion takes place is an axis relative to the condensation, and to transverse motion, and that both the direct and the transverse motions are vibratory. As these results are obtained antecedently to any supposed case of disturbance of the fluid, it is concluded, on principles carefully stated in pages 180 and 188, that they indicate, with respect to the mutual action of its parts, certain laws depending only on the relations of its properties to time and space, and, therefore, not arbitrary, which laws may yet coexist with the results of arbitrarily imposed disturbances, in a manner analogous to the co-existence of the particular solution of a differential equation with the complete integral (see p. 200). These principles are similar to that which is adverted to (p. xxxi.) in connection with the determination of rate of propagation. Considering how long mathematicians persisted in determining rate of propagation by Lagrange's method, notwithstanding its violation of principle, I have, perhaps, little reason to expect that the above-mentioned arguments will be readily apprehended.

By the reasoning under Prop. XI. it is shewn that the rate of propagation along the axis of the motion is the same quantity α for every point of any given wave, and consequently that the waves undergo no alteration by propagation. This with regard to future applications is a very important result. The analytical expression for $\kappa$, terms of the second order being neglected, is $(1 + e\lambda^{2}/\pi^{2})^{1/2}$ as found in p. 206, which, since $e$ is always positive, is greater than unity. Thus the rate of propagation, as deduced exclusively on hydrodynamical principles, is greater than the constant $a$. Also this rate is independent of the maximum condensation of the waves; but without determining the value of $e\lambda^{2}/\pi^{2}$ there is no reason to assert that it is independent of $\lambda$ their breadth. It is necessary to find that value in order to calculate theoretically the velocity of sound.

For a long time I thought I had succeeded in solving this question in a communication to the Phil. Mag. for February, 1853, having relied too much on an accidental numerical coincidence. But eventually I became convinced, by the expression in p. 289 which had been obtained by Sir W R Hamilton and Professor Stokes for the values of $f$ corresponding to large values of $r$ in the series (20), p. 210, that I had used erroneous values of that function. (See Camb. Phil. Trans. Vol. IX., p. 182.) I then made another attempt, in the Phil. Mag. for May, 1865, employing this time the values of $f$ given by the above-mentioned expression. The value of which resulted is the same as that obtained in p. 224 of this volume by the argument commencing in p. 216, which, however, makes no use of that expression, the values of belonging only to points immediately contiguous to the axis of the motion. This last is the best solution I have been able to give of a very difficult problem, of which, possibly, a simpler or a truer one may still be discoverable. The velocity of sound deduced from it exceeds the experimental value by 17.5 feet. Perhaps the difference may be owing to the hypothesis of perfect fluidity, which cannot be supposed to be exactly satisfied by the atmosphere, especially near the earth's surface.

It is unnecessary to add anything here to the reasons I have adduced in pages 225 and 317 of this work, and elsewhere, for concluding that the velocity of sound is not increased by the developments of heat and cold accompanying the condensations and rarefactions of a wave. I may, however, state that my difficulty in apprehending Laplace's theory was long anterior to the investigations which led me to the inference that the excess of the velocity above the value a might be accounted for hydrodynamically. The same kind of difficulty must, I think, have induced Poisson to abandon Laplace's a priori views, and to substitute for them the bare hypothesis, that the increments of temperature by the development of heat are at all points of a wave instantaneously and exactly proportional to the increments of density. The advocates of the usual theory are bound to shew in what manner this entirely gratuitous hypothesis can be connected with experiments made on air in closed spaces.

The two examples, the solutions of which on the received principles of Hydrodynamics led, as before stated, to contradictions, are solved in pages 243-254 in accordance with the reformed principles. No contradictions are met with in this method, which conducts to the important results, (1) that plane-waves, or waves limited by a prismatic tube, whether they are large or small, are transmitted to any distance without alteration, either as to condensation, or velocity, or rate of propagation; (2) that a solitary condensed or rarefied wave can be propagated uniformly from a centre, the condensation and velocity varying inversely as the square of the distance from the centre. In both cases the discontinuity of the condensation, and by consequence that of the motion, is considered to be determined and limited by the fundamental property of easy separability of parts, as explained in page 248. It results, farther, from the new principles that the limited method of treating hydrodynamical problems employed by Laplace, and since extensively followed, is defective in principle. There can, I think, be no doubt that the method of commencing the reasoning by obtaining general equations on general principles, as adopted by Euler, Lagrange, and Poisson, is logically exact, and in other respects far preferable.

[The question has been recently raised as to whether a fluid which when at rest presses proportionally to its density, retains this property when in motion. That it does so is simply an intelligible hypothesis, the truth of which can neither be proved nor disproved by à priori reasoning. Already a presumption has been established that the hypothesis is true, at least quam proxime, by comparison of results deduced from it mathematically with facts of experience; such results, for instance, as those relating to vibratory motions. Utterly absurd results obtained from such an hypothesis do not prove that the hypothesis is untrue, but that some fault has been committed in the reasoning.]

The solution of Example Iv. in pages 200-272 consists of a lengthened discussion of the problem of the motion of a ball-pendulum and the surrounding air, embracing both the application of the appropriate analysis, and a comparison of the results with experimental facts. In page 272 I have stated that in my first attempts to solve this problem, I erroneously supposed that the prolongations of the radii of the vibrating sphere were lines of motion of the fluid. Here again I relied too much on a numerical coincidence, viz. that of the result obtained on this hypothesis with Bessel's experimental correction of the coefficient of buoyancy. Subsequently I was confirmed in the error by a misapplication of the general law of rectilinearity, which, as stated in page xxxvii, I deduced from the new general equation, and which I supposed to be applicable to the motion impressed by the moving sphere. These views are corrected in the present volume in pages 256-259 (see particularly the note in page 259), and the differential equations obtained for solving the problem, viz, the equations (33) and (34) in page 258, are identical with those employed in Poisson's solution, with the exception of having $\kappa^{2}a^{2}$ in the place of $a^{2}$. This difference has arisen from the circumstance that all the antecedent reasoning takes account of the indications of the general equation (1), which was clearly the only correct course of investigation, the truth of that equation being supposed to be admitted. On the ground of this admission I am entitled to say that my solution is more exact, and rests on truer principles, than any that had been given previously.

After effecting the above solution I have inferred, what I believe had not been noticed by other mathematicians, that a vibrating sphere causes no actual transfer of fluid in the direction of its impulses, just as much flowing backwards at each instant as it urges forwards. (I convinced myself of the reality of a backward flow by the experiments mentioned in page 272). Conversely it is shewn by the solution of Example vi, that when plane-waves are incident on a smooth sphere at rest, as much fluid passes at each instant a transverse plane through the centre of the sphere as would have passed a plane in the same position if the sphere had been away. These results, which I arrived. at only after extricating myself from misconception and error, are applied in a very important manner in the part of the work devoted to physical theories. It seems to be not uncommonly the case, that those who possess the power of carrying on independent research, and trouble themselves with exercising it, fall into error before they succeed in advancing truth.

In pages 267-271 I have entered into experimental details with the view of accounting for the difference between Bessel's correction of the coefficient of the buoyancy of a vibrating sphere, which is very nearly 2, and the theoretical value, which is 1.5. The result of the enquiry is, that the difference is mainly to be attributed to the effect of the resistance of the air to the motion of the wire or rod by which the ball was suspended. The solution of Example v, a problem which, as far as I know, had not been before discussed, gives the means of calculating the resistance of the air to the vibrations of a slender cylindrical rod.

The object of the solution of Example vi is to calculate the distribution of condensation about the surface of a smooth fixed sphere, when a series of plane-waves are incident upon it, and considerations are adduced in pages 288-296 relative to the way in which the distribution is modified by transverse action, or lateral divergence, of the incident waves. In the solution of Example VII. (p. 296) like considerations are applied to the case of waves incident on a moveable sphere, and an attempt is, besides, made in pages 298-306 to extend the reasoning so as to include terms of the second order. The result of chief importance is, that when only terms of the first order are taken account of, the motion of the sphere is simply vibratory, but when the calculation includes terms of the second order, the vibrations are found to be accompanied by a permanent motion of translation of the sphere. This conclusion, and the inferences and Corollaries contained in pages 307-312, have important bearings on some of the subsequent physical theories.

It must, however, be stated that on two points of much difficulty, the effect of lateral divergence, and the translatory action due to terms of the second order, the solutions of Examples VI. and VII. are neither complete nor accurate. While the work was going through the press, I discovered a more exact mode of treating Example VII., which is the more important problem of the two, and this improved solution, as far as regards terms of the first order, is given in pages 422 and 423, with reference to its application in a theory of the Dispersion of Light. The more complete solution, inclusive of all small quantities of the second order, is taken up at page 441, and concluded in page 452, under the head of "The Theory of Heat," the analytical determination of the motion of translation forming a necessary part of that theory. In this new solution some of the difficulties of the problem are overcome, but others remain, as, especially, that mentioned in page 453 relative to finding expressions for the constants $H_{1}$ and $H_{2}$. The determination of these functions would, it seems to me, require expressions to be obtained, to the second order of small quantities, for the velocity and condensation at all points of the fluid, whereas the investigation to that order of small quantities which I have given is restricted to points on the surface of the sphere. This generalisation of the solution I have left to be undertaken by more skilful analysts who may feel sufficient confidence in the antecedent reasoning to be induced to carry it on. It may, however, be here stated that from considerations entered into in the solution of Proposition XVII, I am of opinion that it would be allowable to suppose $udx + vdy + wdz$ to be an exact differential, although the motions would not be wholly vibratory, and that from the first approximation obtained on that supposition it would be legitimate to proceed to the second by the usual rules of approximating.

At the end of the solution of Prop. XVII the remarkable conclusion is arrived at that if $udx + vdy + wdz$ be an exact differential to terms of the second order, the total dynamical action of simultaneous disturbances of the fluid, so far as regards the production of permanent motions of translation, is the sum of the effects. that would be produced by the disturbances acting separately.

Under Proposition XVIII a demonstration is given of the coexistence of steady motions. This law had not, I think, been noticed till I drew attention to it. It is an essential element in some of the subsequent physical theories. The solution of Example VIII serves to determine the dynamical action either of a single steady motion on a small sphere, or that of two or more steady motions acting upon it simultaneously. These results. also receive important physical application.

__________

I have now gone through all the particulars in the first portion of the work which I thought it desirable to advert to in this Introduction. As to the Physical Theories constituting the remaining portion, the new investigations and new explanations of phenomena which they contain are so many and various, that it would be tedious, and occupy too much space, to speak of them here in detail. I can only refer the reader to the Table of Contents and hope that on all the physical subjects there indicated sufficient explanations will be found in the body of the work. What I propose to do in the remainder of the Introduction is, to sketch in few words the leading principles of the several Theories of the Physical Forces, and to take occasion at the same time to state some facts and circumstances relating to theoretical physics, which have come under my notice during a long course of devotion to scientific pursuits, and which seem to me to be proper for illustrating the modern progress and existing state of Natural Philosophy. My object in recording the facts and reminiscences I shall have occasion to mention, will be to shew that a great deal of misapprehension has prevailed respecting the true principles of physical enquiry, and to endeavour to correct it, with the view of gaining a hearing for the method of philosophy advocated in this volume.

The Theory of Light, contained in pp. 320-436, rests on hypotheses of two kinds, one relating to the qualities of the aether, or fluid medium, in which light is supposed to be generated and transmitted, and the other to the qualities of the ultimate constituents of the visible and tangible substances by the intervention of which phenomena of light are either originally produced, or are modified.

The hypothesis respecting the aether is simply that it is a continuous elastic medium, perfectly fluid, and that it presses proportionally to its density. Out of this hypothesis, by sheer mathematical reasoning, I have extracted explanations of twenty different classes of phenomena of light, namely, those enumerated in pp. 321-354, which are all such as have no particular relations to the qualities of visible and tangible substances. Among these are the more notable phenomena of rectilinear and uniform propagation, of composition and colour, of interferences, and of polarisation. It might have been supposed that to have to account for the transmission of light all the distance from the fixed stars without its undergoing any change of character, would have put in peril the hypothesis of a continuous fluid. But the mathematical reasoning above mentioned gives results completely accordant with this fact. There is just reason, I think, to say that the number and variety of the explanations of phenomena deduced by strict reasoning from this simple hypothesis establish a very strong presumption of its truth.

But my mathematical contemporaries will not allow of the very reasonable hypothesis of a continuous fluid medium. This is to be accounted for, in part, by the anterior refusal to admit the logical consequence on which I ground the necessity for reforming the principles of hydrodynamics, and, as matter of course, the non-acceptance of the reformed principles, on which, in fact, the explanations which attest the reality of such a medium depend. The opposition is, however, mainly due, I believe, to another cause, with which certain historical details are connected, which, as being illustrative of the course of scientific opinion on this subject, I shall now proceed to give.

To Mr Airy is due the great merit of introducing by his Professorial Lectures the Undulatory Theory of Light as a subject of study in the University of Cambridge. I had the advantage of attending the lectures, and, from the first, felt no hesitation in accepting that theory in preference to the theory of emission, which still held its ground. In 1831 Mr Airy published the substance of his Lectures as part of a volume of "Mathematical Tracts," and gave therein an able exposition of the merits of the Undulatory Theory, accompanied by a fair statement of its difficulties and defects. In the Preface he distinguishes between "the geometrical part" of the theory, which is considered to be certain, and "the mechanical part" which is conceived to be far from certain. This distinction I have difficulty in comprehending, a physical theory, according to my view, being altogether mechanical, as having necessarily relation to force. My conclusion on reading Mr Airy's Treatise rather was, that the theory was satisfactory so far as it was strictly undulatory, that is, rested on hydrodynamical principles, and that the difficulties begin as soon as the phenomena of light are referred to the vibrations of discrete particles of the aether. After this modification is introduced into the theory it ought to be called oscillatory rather than undulatory, the latter word applying to a wave, or a congeries of particles in vibration. I was quite confirmed in the above conclusion by what is said at the end of the Treatise in Arts. 182 and 183 (editions of 1831 and 1842), where it is admitted that the oscillatory theory does not distinguish between common light and elliptically polarized light, although they are proved by facts to be distinguished by difference of qualities. In consequence of this contradiction by fact, it follows, by an acknowledged rule of philosophy, that the oscillatory (not the undulatory) theory of light must be given up. I say this with the more confidence from having proved that the undulatory theory, placed on a hydrodynamical basis, does make the proper distinction between the two kinds of light.

That the oscillatory theory is incapable of distinguishing between these lights is only made more manifest by Mr Airy's attempt to escape from the conclusion. To do this he assumes that the transverse vibrations are subject periodically to sudden transitions from one series to another accompanied by changes of direction; but as it is not pretended that these changes are deducible from the antecedent hypotheses of the theory, and as no attempt is made to account for them dynamically, the assumption can only be regarded as a gratuitous personal conception. The advocacy of similar ideas by Professor Stokes (Camb. Phil. Trans. Vol. IX. p. 414), does not in any degree help us to conceive of a cause for the transition from one series of vibrations to another. I am not aware that such views have been adopted by continental mathematicians.

When in 1837 I commenced Professorial Lectures on Physical Optics in continuation of those of Mr Airy, I judged it right to point out the failure of the oscillatory theory, and to endeavour to place the undulatory theory on a more extended basis of hydrodynamical principles. I was blamed at the time for going against the current of scientific opinion. But what else could I do? Whatever views others might hold, I felt that I could not disregard the consequences of the above-mentioned application of a rule of philosophy. All that has occurred relative to the Theory of Light in the last thirty years has only convinced me that I was right in the course I took, which will also, I think, be found to be fully vindicated by the success with which the Theory is treated on hydrodynamical principles in this Volume. Professor Stokes, when he succeeded me in lecturing on Optics, recurred to the oscillatory hypothesis. I must here be permitted to express the opinion that the adoption of a different course might have contributed towards forming at Cambridge an independent school of philosophy on principles such as those which Newton inaugurated, which in recent times have been widely departed from both in England and on the Continent.

When Fresnel first ventured to make the hypothesis of the transverse vibrations of discrete particles, he stated that he did so on account of "the incomplete notions respecting the vibrations of elastic fluids that had been given by the calculations of geometers." (Mémoires de l'Institut, Vol. VII. p. 53). Had it been known in his time that transverse vibrations were deducible by calculations properly applied to a continuous elastic fluid he might, perhaps, not have had recourse to this method. As it has happened, that hypothesis, together with the isotropic constitution of the aether, imagined by Cauchy, has obtained a very firm footing in the theoretical science of the present day. I think, however, that this remark applies in less degree to the mathematicians of France than to those of other countries. It is well known that Poisson did not accept these views. A very eminent French geometer, in the course of a conversation I had with him at the Cambridge Observatory, only said of Cauchy, "Il ne conclut rien." It is by British mathematicians especially that these hypotheses have been unreservedly adopted and extensively applied. It does not, however, appear, as far as regards the Theory of Light, that the success in this line of research has been proportionate to the magnitude of the efforts. I say this on the authority of Professor Stokes's elaborate and candid Report on Double Refraction in the British Association Report for 1862. After giving an account of the profound analytical processes applied to that question by several eminent mathematicians, and of the use made of Green's very comprehensive principle, he expresses the opinion, that "the true dynamical theory of double refraction has yet to be found." I think it must be allowed that from my point of view there is reason to say, that the failure thus acknowledged, which, in truth, is apparent from the whole tenor of the Report, is attributable to the radical vice of an oscillatory theory.

The foregoing statements may sufficiently indicate the chief cause that has operated to prevent the acceptance of the hypothesis of a continuous aether. The contrary hypothesis of a discrete isotropic constitution of the medium, which was invented by Cauchy to account for the polarisation of light by transverse vibrations, obtained such extensive recognition, that mathematicians, influenced by authority and current scientific opinion in greater degree, perhaps, than they are themselves aware, are unwilling to surrender it, although, as above stated, it has failed to explain phenomena, and is actually contradicted by fact. It will thus be seen that I have been thrown into opposition to my scientific contemporaries, first, by maintaining the consequences of applying a rule of logic (p. xxxvi), and, again, by contending for the strict application of a rule of philosophy. I cannot forbear saying that under these circumstances the opposition on their part is unreasonable, and that, in my opinion, it very much resembles the opposition in former times of the Aristotelians to Galileo, or that of the Cartesians to Newton. History in this respect seems to repeat itself. Cauchy's isotropic constitution of the aether is relied upon in the theory of light, in the same manner as the vortices of Descartes were relied upon for a theory of gravitation, and what Newton said of the latter hypothesis, "multis premitur difficultatibus," is equally true of the other. I hold myself justified in thus strenuously contesting the two points above mentioned, inasmuch as they are like those strategic positions in warfare by gaining or losing which all is gained or lost. If the rules of a strictly philosophic method be not maintained, philosophy will become just what those who happen to have a scientific reputation may choose to make it, which, I believe, is the case with respect to much that is so called in the present day.

In page 354 it is stated that the explanation of the phenomena of diffraction is incomplete, owing to mathematical difficulties not overcome relative to lateral divergence, which, as mentioned in page 292, I have left for the consideration of future investigators. Poisson regarded the problem of the propagation of a line of light ("une ligne de la lumière") as one of great physical importance. (I remember to have heard this said by the late Mr Hopkins; but I have not myself met with the expression of this opinion in Poisson's writings.). The possibility of such propagation appears to be proved by the considerations entered into in pages 290 and 291, the object of which is to shew that composite direct and transverse vibrations contained within a cylindrical space of very small transverse section might be transmitted to any distance without lateral divergence; but they do not determine the law of the diminution of the density towards the cylindrical boundary. The general determination of lateral diminution of condensation under given circumstances, is a desideratum with respect to the complete explanation of other physical phenomena as well as diffraction. There is nothing, however, in these views opposed to the method in which problems of diffraction are usually treated on the undulatory hypothesis.

The explanations in pages 362-436 of phenomena of light which depend on its relations to visible and tangible substances are prefaced by certain hypotheses respecting the qualities of the ultimate constituents of the substances. These constituents are supposed to be inert spherical atoms, extremely minute, and of different but constant magnitudes. Excepting the spherical form, the qualities are those which were assigned to the ultimate parts of bodies by Newton, and regarded by him as "the foundation of all philosophy." According to hypothesis v, no other kind of force is admissible than the pressure of the aether, and the reaction to that pressure due to the constancy of form of the atoms. Hence the aether at rest is everywhere of the same density. I wish here to draw particular attention to the circumstance that in the explanations of phenomena of light, and in all the subsequent theories of the physical forces, no other hypotheses than these, and the former ones relating to the aether, are either admitted or required.

Although the evidence for the reality of the aether and its supposed qualities, given by the explanations of the first class of phenomena of light, adds much to the confidence with which those of the second class may be attempted, the latter explanations do not admit of the same degree of certainty as the others, on account of the greater complexity of the problems, and our defective knowledge of their precise mathematical conditions. The theory of Dispersion is given in pages 362-375, and again in pages 422-427, after introducing the correction spoken of in page xliii. The results by the two investigations differ very little, shewing that numerical comparisons, in the case of this problem, afford scarcely any test of the exactness of the formula.

The Theory of Double Refraction on the undulatory hypothesis is briefly given in pages 375-383. It accounts satisfactorily for the fact that "one of the rays of a doubly-refracting medium, if propagated in a principal plane, is subject to the ordinary law of refraction". In the Report on Double Refraction before referred to Professor Stokes admits that "this simple law" is not accounted for on the principles of the oscillatory theory. It appears also from the same Report that on these principles inconclusive results are obtained as to the direction of the transverse vibrations of a polarized ray relative to the plane of polarisation. The theory I have given determines without ambiguity that the direction is perpendicular to the plane of polarisation. These particulars are here mentioned because, while they confirm the assertion in page xlix, that the oscillatory theory has failed, they shew that the proposed undulatory theory of double refraction is entitled to consideration.

The theories of reflection and refraction at the surfaces of transparent bodies are given at great length in pages 383-415. In page 411 it is found that the direction of the transverse motion in a polarised ray is unequivocally determined to be perpendicular to the plane of polarisation, as was inferred from the theory of double refraction.

The hypotheses respecting the qualities of the ultimate constituents of bodies have been as little accepted by my scientific contemporaries as those relating to the aether. For instance, in the Phil. Mag. for July 1865, Professor W Thomson has expressed an opinion decidedly adverse to "finite atoms," and in the Number for July 1867, p. 15, has not hesitated to pronounce views admitted by Newton relative to the qualities of atoms to be "monstrous." As I have already said (p. viii.), I need not concern myself about a mere opinion, however strongly expressed, respecting my hypotheses; but I am entitled to ask for a fair consideration of the mathematical reasoning founded upon them, and of the results to which it leads. These results alone determine whether the hypotheses are true or false. All the explanations of phenomena in this Volume (the phenomena of light of the first class being excepted) depend on the hypothesis of finite atoms, the reality of which, when the number, variety, and consistency of the explanations are taken into account, can scarcely be regarded as doubtful.

Professor Thomson not only rejects Newton's atom, but puts another in its place. He considers that results obtained by M Helmholtz in an elaborate mathematical investigation [This is the other "great improvement" in Hydrodynamics referred to in the note in p. xxxiii.] respecting vortex-motion (see Phil. Mag. vol. 33, p. 485) indicate motion of such "an absolutely unalterable quality" as to suggest the idea that "vortex-rings are the only true atoms." From my point of view I can readily grant that investigations of this kind, regarded only as solutions of hydrodynamical questions, may admit of important physical applications. I have, in fact, given the solution, although by a very different process, of a problem of vortex-motion, which I had occasion to apply in the theory of galvanic force. But I cannot see that there is any reason for putting "the Helmholtz atom" in the place of Newton's foundation of all philosophy.

The Theory of Heat in pp. 436-462 answers the question, What is heat?, by means of mathematical reasoning applied to the aether of the same kind as that which applied to the air enables us to answer the question, What is sound? The perceived effects are produced in the two cases by vibrations obeying the same laws, but acting under different circumstances. Heat, accordingly, is not a mode of motion only, as has been recently said, but essentially a mode of force. Light is also a mode of force, the dynamical action which produces it being that of the transverse vibrations accompanying the direct vibrations which are productive of heat. For this reason I include light in the number of the physical forces.

With respect to the mathematical part of the theory it may be stated that the reasoning contained in pp. 441-452 is much more complete and satisfactory than any I had previously given. The principal result is the expression in p. 452 for the constant acceleration of an atom acted upon by aetherial vibrations, the investigation of which takes account of all terms of the second order, and therefore embraces both vibratory motions and permanent motions of translation of the atom. The general theory of the dynamical action by which repulsive and attractive forces result from vibrations of the aether, depends on this formula. But the information it gives is imperfect because, as the functions that $H_{1}$ and $H_{2}$ are of $m$ and $\lambda$ have not been determined, the values of the expression for different values of these quantities cannot be calculated. It can, however, be shewn that caloric repulsion corresponds to waves of the smallest order, and that these waves keep the atoms asunder in such manner that collision between them is impossible.

In the Theory of Molecular Attraction, in pp. 462-468, the attractive effect is supposed to be produced by waves of a new order resulting from the composition of all the waves from a vast number of atoms constituting a molecule. The values of $m$ and $\lambda$ resulting from the composition are assumed to be such as make the above-mentioned expression negative; but the theory is not sufficiently complete to determine the values for which the expression changes sign.

The theory of atomic and molecular forces is followed by an investigation in pp. 469-485 of the relation between pressure and density in gaseous, liquid, and solid substances, (particularly with reference to the state of the interior of the earth), together with some considerations respecting the different degrees of elasticity of different gases.

The Theory of the Force of Gravity, in pp. 486-505, depends on the same expression for the acceleration of an atom as that applying to the forces of Heat and Molecular Attraction; but while in the case of the latter the excursion of a particle of the aether may be supposed to be small compared to the diameter of the atom, for waves producing the force of gravity the excursions of the aetherial particles must be large compared to the diameter of any atom. For large values of $\lambda$ it appears that $H_{1} = 1$; but since the function that $H_{2}$ is of $m$ and $\lambda$ is not ascertained, the theory is incomplete. Nevertheless several inferences in accordance with the known laws of gravity are deducible from antecedent hydrodynamical theorems.

For a long time there has prevailed in the scientific world a persuasion that it is unphilosophical to enquire into the modus operandi of gravity. I think, however, it may be inferred from the passage quoted in p. xix. that the author of the History of the Inductive Sciences did not altogether share in this opinion. Not long since Faraday called attention to the views held by Newton on this question, and proposed speculations of his own as to the conservation of force and mode of action of gravity, which, however, he has not succeeded in making very intelligible. (Phil. Mag. for April, 1857, p. 225.) Faraday's ideas were combated by Professor Brücke of Vienna, who, in arguing for the actio in distans, introduces abstract considerations respecting "the laws of thought," such as German philosophers not infrequently bring to bear on physical subjects (Phil. Mag. for February, 1858, p. 81). I have discussed Newton's views in p. 359. It would have been a fatal objection to my general physical theory if it had not been capable of giving some account of the nature of the force of gravity.

So far the aether has been supposed to act on atoms by means of undulations, whether the effect be vibratory or translatory. In the three remaining physical forces the motions of translation are produced by variations of condensation accompanying steady motions. The mathematical theory of this action on atoms, which is given as the solution of Example VIII. p. 313, is very much simpler than that of the action of vibrations. It is necessary, however, to account for the existence of the steady motions. Here I wish it to be particularly noticed that this has been done, not by any new hypothesis, but by what may be called a vera causa; if the other hypotheses be admitted. It is proved in pp. 544-548, that whenever there is from any cause a regular gradation of density in a considerable portion of any given substance, the motion of the earth relative to the aether produces secondary aetherial streams, in consequence of the occupation of space by the substance of the atoms. These streams are steady because the operation producing them is steady, and to their action on the individual atoms the theory attributes the attractions and repulsions in Electricity, Galvanism, and Magnetism, the distinctions between the three kinds of force depending on the circumstances under which the gradations of density are produced. In a sphere the density of which is a function of the distance from the centre the secondary streams are neutralised.

In the Theory of Electric Force, in pp. 505-555, the internal gradation of density results from a disturbance by friction of the atoms constituting a very thin superficial stratum of the substance. The law of variation of the density of this stratum in the state of equilibrium is discussed in p. 466 under the head of Molecular Attraction. A large proportion of the theory of electricity, extending from p. 507 to p. 544, is concerned with the circumstances under which this equilibrium is disturbed, and new states of equilibrium of more or less persistence are induced, and with the explanations of electrical phenomena connected with these changes of condition. In this part of the theory it is supposed that attraction-waves and repulsion-waves intermediate to the waves of molecular attraction and gravity-waves are concerned in determining the state of the superficial strata, but not in causing electrical attractions and repulsions, which are attributed solely to the secondary streams due to the interior gradation of density.

In The Theory of Galvanic Force, in pages 555-604, consideration is first given to the relation between the electric state and galvanism. It is admitted that electricity not differing from that generated by friction is produced by chemical affinity, or action, between two substances, one a fluid, and the other a solid, and that the interior gradation of density thence arising originates secondary streams, as in ordinary electricity, but distinct in character in the following respect. The galvanic currents, it is supposed (p. 598), result from an unlimited number of elementary circular currents, analogous to the elementary magnetic currents of Ampère, but altogether aetherial, and subject to hydrodynamical laws. These resultants, after being conducted into a rheophore, are what are usually called galvanic currents. The investigation in pages 563-569, already referred to, shews that the current along the rheophore must fulfil the condition of vortex-motion, but it does not account for the fact that the whirl is always dextrorsum. The explanation of this circumstance would probably require a knowledge of the particular mode of generation of the elementary currents.

The above principles, together with the law of the coexistence of steady motions, are applied in explanations of various galvanic phenomena, for experimental details respecting which, as well as respecting those of electricity, reference is made to the excellent Treatises on Physics by M Jamin and M Ganot, and to the large Treatise on Electricity by M De La Rive.

The Theory of Magnetic Force, in pages 604-676, embraces a large number of explanations of the phenomena of ordinary magnetism, as well as of those of Terrestrial and Cosmical Magnetism. With respect to all these explanations it may be said that they depend upon principles and hypotheses the same in kind as those already enunciated, the only distinguishing circumstances being the conditions which determine the interior gradations of density. It is assumed that a bar of iron is susceptible of gradations of density in the direction of its length, with more or less persistency, in virtue of its peculiar atomic constitution, and independently of such states of the superficial strata as those which maintain the gradation of density in electrified bodies. The same supposition is made to account for the diamagnetism of a bar of bismuth, only the gradation of density is temporary, and in the transverse direction. The proper magnetism of the Earth is attributed to the mean effect of the asymmetry of the materials of which it is composed relative to its equatorial plane. The diurnal and annual variations of terrestrial magnetism are considered to be due for the most part to gradations of the density of the atmosphere caused by solar heat. The Moon, and, in some degree, the Sun, generate magnetic streams by the variation of density of the atmosphere due to unequal gravitational attraction of its different parts. The Sun's proper magnetism, and its periodical variations, are in like manner produced by unequal attractions of different parts of the solar atmosphere by the Planets.

This theory of Magnetism is incomplete as far as regards the generation of galvanic currents by magnetic currents, as mentioned in pages 636-638. The reason is, that we are at present unacquainted with the exact conditions under which the elementary circular currents, which by their composition produce galvanic currents, are hydrodynamically generated. The difficulty is, therefore, the same as that before mentioned with respect to galvanism.

The proposed theory of Terrestrial and Cosmical Magnetism agrees in a remarkable manner with results obtained by General Sabine from appropriate discussions of magnetic observations taken at British Colonial Observatories, and at various other geo-graphical positions. In the treatment of this part of the subject I have derived great assistance from Walker's Adams-Prize Essay (cited in p. 645 and subsequently), which is a good specimen of the way in which theory can be aided by a systematic exhibition of the past history and actual state of a particular branch of experimental science. For the facts of ordinary magnetism I have referred to the works already mentioned, and to Faraday's experimental Researches in Electricity.

__________

In writing this long Introduction I have had two objects in view. First, I wished to indicate, by what is said on the contents of the first part of the work, the importance of a strictly logical method of reasoning in pure and applied mathematics, with respect both to their being studied for educational purposes, and to their applications in the higher branches of physics. Again, in what relates to the second part, I have endeavoured to convey some idea of the existing state of theoretical physics, as well as to give an account of the accessions to this department of knowledge which I claim to have made by my scientific researches as digested and corrected in this Volume. On the state of physical science much misconception has prevailed in the minds of most persons, from not sufficiently discriminating between the experimental and the theoretical departments, language which correctly describes the great progress made in the former, being taken to apply to the whole of the science. Certainly the advances made in recent years in experimental physics have been wonderful. I can bear personal testimony to the skill and discernment with which the experiments have been made, and the clear and intelligible manner in which they are described, by the extensive use I have made of them in the composition of this work, many of the experiments being such as I have never witnessed. During the same time, how-ever, theoretical philosophy arrived at little that was certain either as to the principles or the results. This being the case it is not to be wondered at that experimentalists began to think that theirs is the only essential part of physics, and that mathematical theories might be dispensed with. This, however, is not possible. Experiments are a necessary foundation of physical science; mathematical reasoning is equally necessary for making it completely science. The existence of a "Correlation of the physical forces" might be generally inferred from experiment alone. But the determination of their particular mutual relations can be accomplished only by mathematics. Hoc opus, hic labor est. This labour I have undertaken, and the results of my endeavours, whatever may be their value, are now given to the world. The conclusion my theoretical researches point to is, that the physical forces are mutually related because they are all modes of pressure of one and the same medium, which has the property of pressing proportion-ally to its density just as the air does.

It is a point of wisdom to know how much one does know. I have been very careful to mark in these researches the limits to which I think I have gone securely, and to indicate, for the sake of future investigators, what I have failed to accomplish. Much, I know, remains to be done, and, very probably, much that I suppose I have succeeded in, will require to be modified or corrected. But still an impartial survey of all that is here produced relative to the Theories of the Physical Forces, must, I think, lead to the conclusion that the right method of philosophy has been employed. This is a great point gained. For in this case all future corrections and extensions of the applications of the theory will be accessions to scientific truth. To use an expression which occurs in the Exploratio Philosophica of the late Professor Grote, "its fruitfulness is its correctibility." Some may think that I have deferred too much to Newton's authority. I do not feel that I have need of authority; but I have a distinct perception that no method of philosophy can be trustworthy which disregards the rules and principles laid down in Newton's Principia.

The method of philosophy adopted in this work, inasmuch as it accounts for laws by dynamical causes, is directly opposed to that of Comte, which rests satisfied with the knowledge of laws. It is also opposed to systems of philosophy which deduce explanations of phenomena from general laws, such as a law of Vis Viva, or that which is called the "Conservation of Energy." I do not believe that human intelligence is capable of doing this. The contrary method of reaching general laws by means of mathematical reasoning founded on necessary hypotheses, has conducted to a meaning of Conservation of Energy not requiring to be qualified by any "dissipation of energy." From considerations like those entered into in page 468 it follows that the Sun's heat, and the heat of masses in general, are stable quantities, oscillating it may be, like the planetary motions, about mean values, but never permanently changing, so long as the Upholder of the universe conserves the force of the aether and the qualities of the atoms. There is no law of destructibility; but the same Will that conserves, can in a moment destroy.

In the philosophy I advocate there is nothing speculative. Speculation, as I understand it, consists of personal conceptions the truth of which does not admit of being tested by mathematical reasoning; whereas theory, properly so called, seeks to arrive at results comparable with experience, by means of mathematical reasoning applied to universal hypotheses intelligible from sensation and experience.

After the foregoing statements I am entitled, I think, to found upon the contents of the theoretical portion of this work the claim that I have done for physical science in this day what Newton did in his. To say this may appear presumptuous, but is not really so, when it is understood that the claim refers exclusively to points of reasoning. If I should be proved to be wrong by other reasoning, I shall be glad to acknowledge it, being persuaded that whatever tends towards right reasoning is a gain for humanity. The point I most insist upon is the rectification I have given to the principles of hydrodynamics, the consequences thence arising as to the calculation of the effects of fluid pressure having, as I have already said, the same relation to general physics, as Newton's mathematical principles to Physical Astronomy. I am far from expecting that this claim will be readily admitted, and therefore, presuming that I may be called upon to maintain it, I make the following statement, in order to limit as much as possible the area of discussion. I shall decline to discuss the principles of hydrodynamics with any one who does not previously concede that the reasons I have urged prove the received principles to be insufficient. Neither will I discuss the theory of light with any physicist who does not admit that the oscillatory theory is contradicted by fact. There is no occasion to dispute about the hypotheses of my physical theories, since I am only bound to maintain the reasonings based upon them. These conditions are laid down because they seem to me to be adapted to bring to an issue the question respecting the right method of philosophy. It is much against my inclination that I am in a position of antagonism towards my compatriots in matters of science, and that I have to assert my own merits. It will be seen that the contention is about principles of fundamental importance. Nothing but the feeling of responsibility naturally accompanying the consciousness of ability to deal with such principles has induced me to adopt and to persevere in this course.

It may be proper to explain here why I have contributed nothing in theoretical physics to the Transactions of the Royal Society. This has happened, first, because I thought the Philosophical Magazine a better vehicle of communication while my views were in a transition state, and then, as I received from none of my mathematical contemporaries any expression of assent to them, I was desirous of giving the opportunity for discussion which is afforded by publication in that Journal. About two years ago I drew up for presentation to the Royal Society a long paper giving most of my views on theoretical subjects; but finding that it necessarily contained much that would be included in this publication, and might be therein treated more conveniently and completely, I refrained from presenting it.

I have only, farther, to say that in the composition of this work I have all along had in mind the mathematical studies in the University of Cambridge, to the promotion of which the discussion of principles which is contained in the first part may con-tribute something. The subjects of Heat, Electricity, and Magnetism having, by the recently adopted scheme, been admitted into the mathematical examinations, it seemed desirable that they should be presented, at least to the higher class of students, not merely as collections of facts and laws, but as capable of being brought within the domain of theory, and that in this respect the Cambridge examinations should take the lead. It is hoped that the contents of the second part of this volume may in some degree answer this purpose. It was with this object in view that the physical theories have been treated in greater detail than I had at first intended, especially the theory of Magnetism.

CAMBRIDGE,

February 3, 1869.

2.1. Introduction.

The contents of this work are devoted almost exclusively to discussing the principles and the reasoning appropriate to the theoretical department of Natural Philosophy, and the mutual relation between this and the experimental department. The discoveries in recent times of new facts and physical laws by experimental means have been so remarkable and abundant, and have given rise to so much speculation, that there seemed to be reason to apprehend that the part of philosophy which is properly theoretical might be either set aside or wholly misunderstood. The purpose of this Essay is to endeavour to counteract this tendency.

The method of theoretical philosophy adopted in this work, as well as in my larger one on the Principles of Mathematics and Physics, published in 1869, is for the most part the same as that which is indicated by Newton in Book III. of the Principia. In both publications this method has been largely applied in the explanation of physical phenomena and laws, advantage having been taken for that purpose of the modern advances that have been made in physical determinations by experiment, and in abstract mathematics. The arguments employed in its application (which, in order that they may be more generally understood, are exhibited in the present work with as little reference as possible to analytical formulae) have conducted to explanations so numerous, and of such different kinds, as apparently to justify the conclusion that this System of Philosophy is true both as to its principles and as to the reasoning it demands; in short, that it is the true method of theoretical philosophy. With the view of assisting to form a judgment on its essential character, and on the necessity arising therefrom for adopting it, I propose to make in this Introduction some brief remarks on the antecedent history and actual state of the theoretical department of Physics.

All theoretical investigation is carried on by means of calculation, and the calculation is not simply algebraic, but consists essentially of the formation, according to given conditions, of differential equations, and the solutions of them by the rules of analysis. Newton is therefore to be regarded as the founder of theoretical philosophy, having in his Principia, by reasoning of which there had been no previous example, virtually formed and solved the differential equations which are necessary for calculating the motions of the Moon, the Earth, and the Planets with their satellites. And because these calculations rested on the hypothesis of an attractive force varying with distance according to the law of the inverse square, the motions, by being thus calculated, were referred to an operative cause. Kepler's laws, as determined by observation, are merely formal relations of time and space, and, like all quantitative laws, are deducible by calculation founded on the hypothesis of a producing cause, being, in fact, problems demanding such a solution. Newton solved these problems by his theory of gravitating force. It is evident that a step like this, inasmuch as it establishes the reality of causative action, is an advance of physical science beyond the mere knowledge of formal laws.

The truth of the hypotheses on which the theory of universal gravitation rests is not capable of demonstration by calculation alone, nor by observation alone, but by a combination of both. Such experiments as those of Galileo for determining the laws of the vertical descent of bodies acted upon by the earth's gravity, and those which prove that the vertical acceleration of a projectile by the same force is independent of its motion in a curvilinear path and of the rate of such motion, were absolutely necessary for the discovery of the fundamental laws on which, as hypotheses, all calculation relative to the motions produced by accelerative forces must rest. These hypotheses do not admit of à priori demonstration except on physical principles of a higher grade than those now under consideration. (See Art. 24 of the Essay.)

But after admitting on experimental grounds the above-mentioned laws respecting the action of constant forces, it is provable by abstract reasoning that an accelerative force, whether variable or constant, estimated in a given direction, is quantitatively denoted by the second differential coefficient of the function which expresses the distance at any time of the accelerated particle from a fixed plane perpendicular to that direction. (The demonstration as given in pp. 109-113 of The Principles of Mathematics and Physics is conducted by means of Taylor's Theorem, which is legitimately employed for this purpose on the general principle that abstract calculation is comprehensive of all concrete physical relations.) On this symbolic expression for accelerative force, which, as said above, could only be arrived at by experiment and abstract calculation combined, the whole of Physical Astronomy depends, together with every department of applied science which requires the calculation of motions produced by accelerative forces. The discovery of the calculation necessary for this purpose characterises in a special manner the Newtonian epoch of Natural Philosophy.

Besides the above-mentioned general hypotheses, or foundations, of all calculation relating to accelerative forces, there are particular hypotheses which are distinctive of the class of questions to which the calculation is applied. Such are the hypotheses of Physical Astronomy, discussed in Art. 10 of the Essay, respecting the mathematical deductions from which a few historical notices, illustrative of the principles of theoretical reasoning, may here be introduced. After forming the appropriate differential equations on those hypotheses, it only remained to integrate them, and to apply the integrals, according to the abstract rules of analysis. Now it appears that Newton published only an imperfect calculation of the motion of the apse of the moon's orbit, and that his successors, after trying to complete, by means of the analytical solution of the problem, what he began, failed at first in the attempt, and were disposed to conclude that the assumed law of gravity was not exactly true. Eventually, however, it was found that the reasoning was at fault, and the difficulty was overcome by an extension of Newton's method of treatment of the problem, conducted strictly according to the rules of successive approximations.

But it is to be observed that in order to apply that method it was necessary to get rid of certain terms in the analytical solution which increase indefinitely with the time. Now since it may be laid down as universally true that inferences strictly deduced by reasoning from premises must have some significance relative to the premises, and as these terms were so deduced, it is necessary, in order to complete the mathematical theory, to ascertain their origin and meaning, and the rationale of the process for getting rid of them. To these questions I devoted much consideration, not having met with any satisfactory answers to them, and came at length to the conclusion that every process by which the terms of indefinite increase are avoided introduces arbitrarily, without altering the number of arbitrary constants, the condition that there shall be a mean orbit; so that those terms prove that there may be perturbed motions which do not fulfil that condition. Accordingly the Problem of Physical Astronomy is not the general Problem of Three Bodies, but the more restricted one of disturbances producing oscillations relatively to a mean orbit. Assuming, as is allowable, that the mean orbit is an ellipse the elements of which can be calculated from observational data, it follows that the mean eccentricities of the orbits of the Moon and Planets must have inferior limits. For the mean eccentricity could not be zero unless the orbit was always an exact circle, which under the actual circumstances of the attracting bodies is impossible. This argument may be considered to be necessary for completing the deductions that can be drawn by abstract reasoning from the hypotheses of Physical Astronomy.

The foregoing remarks may suffice to shew in what way it is necessary to combine abstract reasoning with the results of observation for the formation of a theory in any other department of Physics.

Newton attempted to solve the Problem of the Precession of the Equinoxes; but it is evident from the mode in which he made the attempt that he had not recognized D'Alembert's Principle. This principle is indispensable for calculating the accelerated motions of masses, when they cannot be treated as if they were elementary particles, and is as legitimately applied to fluid as to solid masses. In fact, it is not so much a principle as an axiom, which admitted of being enunciated after the discovery was made that an accelerative force is quantitatively expressed by the second differential co-efficient of linear space regarded as a function of the time.

All the calculations relative to accelerative forces referred to in the foregoing remarks depend on differential equations which are reducible to a single one containing only two variables. For calculating the accelerated motions of a fluid, differential equations are required which in no case can contain fewer than three variables; on which account the analytical reasoning is of a higher order than that of the other class of problems, and is attended in its applications with peculiar difficulties. For the science of analytical hydrodynamics we are mainly indebted to the researches of Euler, Lagrange, and Poisson. Laplace, although he surpassed his contemporaries in theoretical deductions by means of the ordinary class of differential equations, effected but little relatively to the applications of partial differential equations. I take occasion here to recommend mathematical students not to adopt the method of treatment of hydrodynamical problems employed by Laplace, and also by some English mathematicians; which, as not introducing a symbol (called by Lagrange and Poisson) to indicate the existence and variability of "surfaces of displacement," keeps out of sight an essential principle of Hydrodynamics.

Newton gave a solution of the problem of the velocity of sound on the assumption (which was legitimate) that the aerial particles vibrate according to the law expressed by an harmonic function. He obtained a value of the velocity less by one-sixth than that given by experiment; and modern analytical methods have conducted to the same result, because, in fact, they do not differ in principle from that of Newton, so far as relates to the determination of rate of propagation.

I think it unnecessary to give here the particulars of the well-known theories proposed by Laplace and Poisson to account for the excess of the experimental above the Newtonian value of the velocity of sound: it will suffice to remark that their views concur in ascribing to the agency of the heat and cold developed by the condensation and rarefaction of the aerial particles in vibration, an increment of accelerative force having a constant ratio to the force due, apart from any such agency, to the actual variation from point to point of space of the density of the fluid, its temperature being invariable.

Long since I called in question the legitimacy of inferring the amount, or even the reality, of such instantaneous changes of the effective accelerative force in aerial vibrations from experiments made on air in closed spaces. The accession, or diminution, of the temperature of the air operated upon in these experiments is apparently to be attributed to the circumstance that the heat-radiants, or cold-radiants, generated by the sudden compressions or dilatations of the air are prevented from being propagated into surrounding space by the solid boundaries of the containing vessel; the consequence being that by successive reflections at the interior surface they are made to traverse the enclosed air repeatedly, and thus, by altering the amount of heat-waves emanating from the individual atoms (see Art. 60), produce a change of temperature. According to these views the change is wholly due to the air being enclosed. But whether or not this explanation be true, the circumstances of the compression and dilatation of air in closed vessels differ from those of the condensations and rarefactions of aerial vibrations in free space in such manner as to give no experimental evidence of the effects of the development of heat and cold by the latter.

Since, however, the determination of the velocity of sound by Newton's method is a matter of reasoning, it is open to enquiry whether erroneous or imperfect reasoning may not account for the discrepancy between the calculated and observed values. I propose, therefore, to give here a brief account of the argument from which I have concluded in the Principles of Mathematics and Physics, as well as in other publications, that the reasoning commonly employed in this investigation is faulty.

After forming an exact differential equation applicable to motion of an elastic fluid which at any given time is a function of the distance from a fixed plane to which its directions are perpendicular, a particular integral is obtainable which exactly satisfies the equation. By giving to the arbitrary function the form of an harmonic function, whereby the differential equation is still satisfied, it may be proved by strict mathematical reasoning that "at the same distance from the fixed plane and at the same moment the velocity of the fluid may be zero and yet have its maximum value". Such a result is a contradiction per se, indicating, according to a known rule of logic, either fault or defect in the premises, or fault in the reasoning. No fault in the reasoning being discoverable, it follows of necessity that the premises of Hydrodynamics require to be rectified. Although my mathematical contemporaries do not deny the reality of the above contradiction, they have hitherto refused to admit the necessity of the logical consequence. It is by such disregard of the rules of reasoning that error is introduced and perpetuated.

The way in which I have proposed to rectify the principles of Hydrodynamics is explained in Arts. 18 and 19 of the Essay. I have maintained that in addition to the two general hydrodynamical equations depending on the principle of constancy of mass and D'Alembert's principle, a third is necessary for expressing analytically the condition that "surfaces of displacement" may be drawn through all the elements at all times. This equation gives expression to a principle of geometrical continuity to which the motion, if it admits of being calculated, must be subject. It might reasonably be urged that an additional general equation must modify essentially the whole of the reasoning in analytical Hydrodynamics, and the modes of its applications. That this is actually the case will appear from the following statement of results for obtaining which that equation is indispensable.

(1) It is proved independently of particular conditions as to the manner of putting an elastic fluid in motion, that vibratory motions, expressed by harmonic functions, take place simultaneously in directions parallel and transverse to an axis, such motion being due exclusively to the elasticity and inertia of the fluid. On this result the Undulatory Theory of Light, including the theory of polarisation, wholly depends.

(2) The velocity of sound obtained by means of the same result is found to differ very little from the observed value. (Principles of Mathematics, &c.)

(3) When the analysis which gives the above-mentioned spontaneous vibratory motions is extended to terms of the second order, it can be proved that undulations incident on small spheres not only cause them to vibrate, but produce also accelerated motions of translation either in the direction of the propagation of the undulations or in the opposite direction. The translatory motion is attributable to the distribution of condensation about the surface of the sphere, as modified by the inertia of the vibrating fluid; on which account it is necessary to proceed to second-order terms. On this deduction from the reformed principles of Hydrodynamics the theory of attractive and repulsive forces depends.

(4) By means of the third general equation it may be proved that steady streams can flow along, and in the vicinity of, an axis of any curved form and any length, if the courses be in spirals about the axis, and if they complete a circuit. This result of the analytical reasoning is the foundation of the hydrodynamical theory of galvanism. The phenomena of magnetism are also referable to steady motions of the aether, but the courses of magnetic streams are not subject to the condition of being spiral.

(5) When the motion of an elastic fluid is central, and a function of the distance from the centre, analysis shews that the condensation varies inversely as the distance; whereas if the propagation of a solitary wave of condensation from a centre be possible (as has been assumed), the condensation must vary inversely as the square of the distance. This difficulty I pointed out long ago, although I have only recently discovered the solution of it. By taking into account the spontaneous vibratory motions deduced, as stated above, from the rectified principles of Hydrodynamics, it appears that the generation of a solitary wave of condensation or rarefaction is not possible, inasmuch as, by reason of the elasticity and inertia of the fluid, an impulse not vibratory instantaneously excites in it alternations of condensation and rarefaction. [This point is discussed at the end of an article on "Attraction by vibrations of the Air" in the Philosophical Magazine for April 1871.]

The foregoing statement of the results deducible by means of the third general equation may be considered to demonstrate the necessity there was, as regards the advancement of theoretical physics, for thus completing the principles of hydrodynamics. The proof of the new equation is, in fact, an essential part of the analytical reasoning by which the motion of an element which moves in juxtaposition with other elements is to be calculated. Newton discovered the mathematical reasoning proper for calculating the accelerated motion of a single particle: the existing state of theoretical physics demanded the discovery of the reasoning proper for calculating the motions of a congeries of particles constituting a fluid. I claim to have taken a necessary step towards meeting this demand, by completing the mathematical principles of Hydrodynamics.

Having by the preceding considerations indicated the character of the addition to the received mathematical principles of theoretical science which is proposed and maintained in the Essay, and having shewn by historical notices that such addition is consistent with, and was demanded by, the antecedents of experimental and theoretical physics, I have fulfilled the main purpose of this Introduction. There are, however, some remaining particulars which appear to me to call for remark or explanation.

It will, perhaps, be noticed that in this Introduction, and throughout the Essay, I have made very little reference to the productions of contemporary mathematicians who have written on the same subjects. The reason for this may be readily given. Although these subjects have engaged the attention of many physicists, I can point to no one who has treated of them theoretically, in the sense in which I understand theory. What I mean by this assertion will be perceived at once by referring to the views expressed in Art. 113 respecting Gauss's supposed proof of the law of the inverse square in magnetic action. The proof rests on hypotheses which, as not being intelligible from sensation and experience, do not conform to the Newtonian Rule of Philosophy. I cannot see that any knowledge is gained by conclusions drawn from hypotheses that are themselves unexplainable, or unintelligible. Such hypotheses Newton referred to by saying "somnia confingenda non sunt," and again, towards the end of the Principia, "hypotheses non fingo." These expressions refer to arbitrary, as distinguished from necessary, hypotheses, inasmuch as the first occurs under the very Rule of Philosophy which contains, respecting the qualities of the ultimate constituents of bodies, hypotheses which Newton pronounces to be "the foundation of all philosophy."

Not having felt any difficulty as to the admissibility of these hypotheses, and those relating to the existence and qualities of the aether, as foundations of reasoning whereby their physical reality might either be disproved or established, I have devoted much time and labour to physical researches conducted by mathematical reasoning resting on this basis, and I cannot in the least degree understand why there should exist as there does exist among physicists of the present day an antecedent objection to entertaining this method of philosophy. Prejudgment is allowed to stand in the way of giving consideration to the principles of a philosophy which commended itself to the minds of Newton and Locke, and consequently explanations of phenomena mathematically deduced from them are disregarded.

Empirical formulae derived from a large number of experimental data, such as the formulae by which Gauss succeeded in approximately expressing certain of the laws of terrestrial magnetism, are sometimes improperly spoken of as if they were theoretical. They are important as embracing in one view facts of the same kind, and indicating relations of groups of facts; also they may assist in forming a theory which is truly such; but they differ altogether from inferences deduced by mathematical reasoning from à priori principles. The distinction will be at once perceived by the contrast between Gauss's theory of terrestrial magnetism and that proposed in Arts. 120-149 of the Essay.

Neither can Fourier's analytical theory of heat be called theory in the proper sense of the word. For it consists of deductions, by appropriate analysis, of formulae expressing laws of heat, from certain experimental data; but professedly contains nothing respecting the intrinsic nature of heat. The Theory of Heat expounded in Arts. 47-62 of the Essay will be seen to be something very different as respects both principles and reasoning. The value of Fourier's work consists in its grouping phenomena of heat under formulated laws, so that the phenomena are all theoretically explained as soon as the experimental data from which the laws were deduced have been accounted for by reasoning founded upon the à priori principles of the Newtonian Philosophy.

The theory proposed by Ampère to account for magnetic attractions and repulsions by an agency analogous to that of galvanic currents in coils, is so far properly theoretical that, if experimentally verified, it would contribute something towards understanding the intrinsic nature of magnetism. But, as I have intimated in page 98, there is reason to question whether that theory be not contradicted by experiment; and it is, moreover, to be said that all the magnetic phenomena which are considered to be accounted for by referring magnetic force to galvanic agency, can be explained by the hydro-dynamical theory of magnetism, and that according to the principles of this theory magnetic currents cannot be composed of galvanic currents.

A method of philosophy wholly different from, if not inconsistent with, that I have been advocating, is adopted by some physicists of the present day. It consists in deducing explanations of phenomena from general laws (improperly called principles), such as the law of Vis viva, and that which is called "the conservation of energy." With respect to the law of Vis viva, we know that it is capable of being expressed by a formula arrived at by mathematical reasoning, and that the reasoning consists in forming according to dynamical principles, and integrating, general differential equations comprehensive of all particular cases. By being thus formulated the law of Vis viva has become matter of science, and it does not appear that it could have been made such by any other process. Being demonstrated by this inductive reasoning, it may be applied deductively in the solutions of particular problems.

Any general law which is similarly applied in accounting for physical phenomena, requires to be analogously arrived at by inductive reasoning, before it can be legitimately and definitely so applied. I fail to see that this has been done with respect to the above-mentioned law of the conservation of energy. I even venture to say that no such law admits of being demonstratively established excepting by reasoning on the basis of the Newtonian principles of philosophy. The results I have arrived at, by the adoption of those principles, respecting the qualities and agency of a universal aether, are clearly adapted to giving reasons for the "correlation" and "transformation" of forces, considered to be indicated by experiment, and for the conservation of uniformity in the total energy of the universe.

I take occasion to state here that I have adhered to the term "Hydrodynamics" in preference to "Hydrokinetics," not merely because it is established by long usage, but chiefly because it may be taken to imply, although not expressly significant of motion, that the department of science it designates treats of motions as necessarily having relation to force. The word "motion" has been used by some eminent physicists with so little reference to the essential distinction between its signification and that of "force," that, as I am able to testify, others have thence been led to infer that motion is capable per se of producing motion; whereas it is a fundamental axiom of natural philosophy that motion cannot be generated by motion itself any more than by the negation of motion. The following is an instance of disregard of the distinction between force and motion. The boring of a cannon, which may well be called a tour de force, is known to produce heat; and hence, taking into account the law of the mechanical equivalence of heat established by experiment, it might with reason be affirmed that heat is a mode of force. Instead of which, from this and like experiments the inference is drawn that heat is "a mode of motion," an expression implying that motion is per se operative. In order not to give occasion to such misconception of the quality of motion, I have avoided substituting for "Hydrodynamics" a term not well adapted by its etymology to make a distinction between force and motion.

The foregoing particulars relative to the philosophical views and productions of recent date will serve to indicate on what grounds I said that I could refer to no one who had treated of the physical forces theoretically, that is, in accordance with the principles of Newton's "foundation of philosophy." Modern physical science is characterised by great and successful efforts, to extract laws of greater or less generality from the results of observation and experiment, and it is worthy of remark that the experimental demonstration of a law is generally accompanied by some speculation on the part of the experimenter as to its cause. Such speculations, although they do not supersede the necessity for a mathematical theory, are of assistance in carrying on experimental research, and may be regarded as testifying that experimentalists themselves look upon physical science as something more than the mere determination of laws. But the demonstration of physical laws by mathematical reasoning founded on necessary and intelligible hypotheses, in other words, the prosecution of the Newtonian Philosophy of Causation, has scarcely been attempted since Newton's time either in England or on the Continent. In this respect the work I published in 1869 on the Principles of Mathematics and Physics stands alone, and has placed me in a kind of isolation relatively to my scientific contemporaries. Still I maintain that its contents, so far as they are true, give a legitimate extension to the course of theoretical philosophy that Newton began; and more than this, it may be asserted that works of this kind, devoted exclusively to doing what is to be done by reasoning, are absolutely necessary for advancing and completing physical science, inasmuch as for this end that which can be effected by reasoning alone is just as necessary as that which can be effected by experiment alone.

I am well aware that in so large an undertaking errors of detail, if not of principle, may have been admitted, and that there is also much which has been very imperfectly accomplished. In fact, various errors have been corrected, and improvements indicated, in the present work, which was written chiefly for the purpose of forming a kind of appendix to the larger one, and thereby making it more trustworthy and complete. With time and strength at disposal I should be able to make many more improvements.

Notwithstanding errors and imperfections, the contents of the two works will, I think, suffice for establishing eventually a claim to having initiated an advance in the application of mathematical reasoning to physics, having the same relation to the existing state of physical science as Newton's new application of mathematics had to the science of his day, and adapted in like manner to inaugurate a new scientific epoch.

In concluding this Introduction I propose to say a few words on the bearing which the philosophical views I have advocated may possibly have on the mathematical studies of the University of Cambridge. I have argued that the superstructure of physical science is raised by two essentially different means, by experiment and by mathematical reasoning, and that for making it complete, it is absolutely necessary to employ them in combination. The latter of these means has long been amply promoted at Cambridge by the facilities there afforded for acquiring an accurate and extensive knowledge of the different departments of pure and applied mathematics, while for an equally long period there was great deficiency as to the means available for becoming acquainted by experiment with physical facts and laws. Happily this want is now being supplied by the erection of new buildings, and the provision of appropriate apparatus, for oral and practical instruction in experimental physics. It may be that as far as regards instruction merely by instruments and experiments students at Cambridge will not have greater advantages than those at other educational institutions; but because there is no place in the world at which mathematics are more efficiently and extensively studied, probably at no other place could the study of theoretical physics, as requiring the union of mathematical reasoning with experimental research, be more effectively prosecuted.

Under these circumstances it is to be desired that the studies of the University should be directed more than formerly towards physical subjects, and that questions on such subjects should be introduced in greater degree into the examinations for mathematical honours. It may also be hoped that a proportion of Cambridge mathematicians who have had this direction given to their studies will not be content with merely acquiring a knowledge of physics, but will be induced to employ the mathematical and experimental means at their command in making efforts to extend the boundaries of physical science.

But as respects any endeavours to make advances in physical knowledge by mathematical theory, Newton's "foundation of all philosophy," the claims of which to recognition I have so long urged, and now urge again (perhaps for the last time), is indispensable, and cannot without detriment to scientific truth be superseded by any other.

CAMBRIDGE,

June 6, 1873.

3.1. Preface.

Within the last forty years several works on the Cambridge Mathematical Studies have been produced by members of the University who had taken an active part in improving and extending them either by teaching or writing, and were in other ways well qualified to form a correct opinion respecting their character and tendency. These works were specially called for in consequence of various changes which it had been thought desirable to make in the system of University instruction, in respect both to the modes of teaching and examining, and the subjects taught. In the year 1837 a work was published by Whewell entitled "On the principles of English University Education," and in 1845 another of like character was produced by the same author, the full title of which is, "Of a Liberal Education in general, with particular reference to the leading studies of the University of Cambridge." As might be expected from the titles, so much of these works as is devoted to mathematical studies treats of them mainly as educational means; and accordingly very much is said about the superiority in this respect of reasoning by geometry above reasoning by analytical symbols, and no account is taken of analytics as an instrument of physical research. In 1833 Professor Sedgwick published "A Discourse on the Studies of the University," which in 1850 was re-published with Additions consisting of a very long Preface and Appendices, which, in fact, constitute by far the greatest part of the volume. The occasional remarks, contained in the latter publication, relative to mathematical studies, I consider to be peculiarly apposite, especially as regards what the author calls "The Newtonian System of Philosophy," and its relation to results obtained solely by experiment or observation. These views, however, refer only to Physical Astronomy. At a date somewhat later a pamphlet entitled "Remarks on the Mathematical Teaching of the University of Cambridge" was produced by Hopkins, who, by his experience both as a private tutor and as a cultivator of applied mathematics, was eminently qualified to speak on the subjects he took in hand. The pamphlet bears no date, but as it is stated on the Title-page that the author was at the time President of the British Association, it must have been circulated in 1853 or 1854. The matters it treats of are almost exclusively restricted to the means of teaching mathematics by private tuition, and by the Lectures of College Tutors and University Professors, and on these points valuable suggestions are thrown out, which were afterwards to a considerable extent carried into effect. The suggestions had special reference to the Report of the Cambridge University Commission issued in 1852.

Since the last-mentioned date various changes have been made in the Scheme of Examination of Candidates for Mathematical Honours, the most important of which are those which were confirmed by Grace of the Senate on June 2, 1868, the principal effect of which was to sanction the introduction into the Examination of a wider range of questions in experimental and theoretical physics. Taking this circumstance into account, together with the limited character, as shewn by the foregoing statements, of the discussions which the Cambridge Mathematical Studies have hitherto under-gone, I thought the time was come when they might with advantage be brought again under review, both as regards the principles and the reasoning adopted in the treatment of the several subjects, and as to their relation to modern advances in physical science. I may claim, I think, to have some pretensions for entering upon this undertaking, from having had to do with the study of mathematics as pupil, lecturer, examiner, or professor, during more than half a century, and having spent much time and thought on independent mathematical and physical researches.

Accordingly the contents of this publication consist for the most part of arguments relating to the principles and processes of reasoning that are legitimate and necessary both in applied mathematics and physics, with prospective reference to the fundamental hypo-theses of the Newtonian System of Philosophy, and to the methods of deducing from them by mathematical reasoning results that might be compared with modern experiments and observation. The questions considered are generally such as involve points of difficulty which, as having respect to principle, require to be cleared up before farther advances can be securely made, and the arguments employed to meet the difficulties are exhibited in as much detail as was practicable in a publication like the present; but generally the reader is referred for the details of arguments to my work on The Principles of Mathematics and Physics published in 1869, and to the supplementary Essay on the Mathematical Principles of Physics, published in 1873. I have had occasion also to cite for the same purpose communications in the Philosophical Magazine, especially some written after the publication of the former work. It will be proper to mention here that the first of the two above-mentioned works is uniformly cited in the Remarks, for the sake of brevity, as 'Principles' with numbers added indicating pages, and the other as 'Essay' with indications of articles.

In publishing the present work I have also had a motive of a personal kind, which I beg leave to take this opportunity for stating. In February of next year I shall have completed the fortieth year of my tenure of the Plumian Professorship. In the ordinary course of nature I can expect to be able to continue but a short time longer the discharge of its duties, and I really think that the time is come when in respect to lecturing and examining, which are more suited to younger men, I may ask to be allowed to take the place of Professor Emeritus. I make this suggestion with the less hesitation because I feel that I am still able to devote my time to writing and publishing, and may hope thereby to contribute something both to the advancement of Natural Philosophy, and to giving a proper direction to the Mathematical Studies of this University. My plea is (1) that at the present time Natural Philosophy stands in no greater need, whether as regards the teaching of it, or its advancement, than that of being placed on the basis of the principles insisted upon by Newton in the Third Book of the Principia; and (2) that hitherto no one besides myself has undertaken to supply this need. (See what I have urged in Conclusions below) It is hoped that this brief publication may suffice to give the Authorities of the University and Members of the Senate the means of forming an opinion on these two points, and may induce them to determine, if it should be in their power, whether my being engaged in the way I propose in advancing Experimental Philosophy as distinct from Experimental Physics, may be taken to be, under existing circumstances, a discharge in full of the duties of the Professorship. I certainly think that in any new arrangements that the University may make respecting the Professoriate, it would be well to provide, if practicable, that a Professor who for a long time has performed the prescribed duties of his office, should be allowed to devote himself wholly to original research, after giving reasonable evidence of his willingness and ability to do so.

CAMBRIDGE,

November 1875.

3.2. Conclusions.

I have now completed an ungracious task, imposed upon me by the necessity of having to find fault. It is with reluctance that I have taken up a position antagonistic to the philosophical views of contemporaries who, I have reason to think, are very eminent as physicists and mathematicians. I felt compelled to do this from having observed that the Newtonian principles of theoretical philosophy are at this time either neglected, or strenuously opposed, that true theory is in danger of being overwhelmed by speculations such as those I have had occasion to advert to in this work, and that this questionable phase of philosophy is beginning to make its way in the Cambridge system of Mathematics. The contention between me and my opponents is the old one of experiment versus theory. Flamsteed thought that the laws of the Moon's motions might be extracted from his observations apart from Newton's theoretical calculations. It was Faraday's failing to attempt to theorise without mathematics, in consequence of which he was led to entertain the idea that the force of gravitation exists only so far as it has matter to act upon. Dr Tyndall does not hesitate to maintain that imagination is entitled to take a part in physical research, not being aware, apparently, that although such aid may with advantage have been invoked in conducting experiments, in theory that is made truly such by being based on information given immediately by the senses, there is no place for imagination. The divergence between my philosophical views and those of physicists who agree with the authors of "Unseen Universe" has an origin of the same kind. While I endeavour to account for experimental facts by mathematical reasoning founded, as just stated, on primordial sensible entities, they seek to make a foundation for theory out of experiment. This explains the occurrence in their writings of such terms as "work", "ergal", "reversible engine", "manufactured article", &c., used in such manner as to shew that their method of philosophy is rather an exercise of the perceptive, than of the reasoning, faculty. I regard with great admiration the ingenuity of the contrivances and fertility of resource whereby modern experimenters perform the necessary part of ascertaining facts and formulating laws (the more so, perhaps, because I am conscious that this is not my department), but they are under complete misapprehension if they expect that what may be effected by experiment can supersede the part that consists of mathematical reasoning founded on the Newtonian principles of philosophy. The old adage, Ne sutor ultra crepidam, applies even here. The parts are essentially distinct, but one is as necessary as the other for making up complete physical science.

The two works I have so often referred to in the course of these Remarks under the designations, 'The Principles' and 'The Essay', were composed chiefly for the purpose of bringing more into notice, and maintaining, the principles of philosophy which Newton considered to be necessary and fundamental, and extending the application of them as far as might be practicable. I am well aware that the larger work contains numerous faults and imperfections, as was to be expected in a first attempt to accomplish so extensive and complex an undertaking. If I were now to re-write that Volume I should make many corrections, and curtail much that has only personal or temporary relation to the main object, and I should take occasion to add some results of physical investigations which I have contributed to the Philosophical Magazine since the Volume was published. My object being, as I have just said, to carry out Newton's principles of theoretical philosophy, the work as to character, whatever may be the merits of the execution, holds the same place relatively to modern physical science as Newton's Principia did to the science of his day. The same ground is not occupied by any existing Treatises on Natural Philosophy or Physics, however useful and complete they may be in other respects, and I think, therefore, that it is not without reason that I feel a desire to be released from the more immediate duties of my Professorship, in order that I might have complete leisure for preparing a new edition of the portion of this work that embraces Physics, with the view of producing a Treatise on the Theoretical Principles of Experimental Philosophy as exact in point of reasoning, and as free from defects, as in the existing state of physical science may be possible. I should hope by this means not only to add something to the scientific reputation of the University, but also to compose a work which, as not demanding any mathematics of a more transcendental character than the applications of partial differential equations, might be thought fit to be put into the hands of Candidates for Mathematical Honours. Remembering, however, the historical precedent furnished by the fact that the Vortices of Descartes held their ground a long time against Newton's Principia, I do not feel confident that this view will be assented to by my scientific contemporaries.

4.1. Preface.

I began in the year 1843 a Course of Lectures on Practical Astronomy and Astronomical Instruments, having been at that time seven years Director of the Cambridge Observatory. Lectures on these subjects had already been given by Dr Peacock as Lowndean Professor, which he discontinued on the understanding that I would undertake to carry them on. The circumstances that an Astronomical Observatory so well appointed as that of Cambridge was near at hand, and was provided with various instruments of first-rate quality, appeared to me to give facilities for lecturing on Practical Astronomy which ought to be taken advantage of; and accordingly I commenced lecturing in the above-named year, having previously produced a 'Syllabus' of the subjects of the lectures, and procured a considerable number of wooden models, which, together with some apparatus that had been collected by Professor Peacock, I made use of in conducting and illustrating the Lectures. I had also the advantage of having at command several portable instruments pertaining to the Observatory. At the end of the Syllabus a list of formulae applicable to the reduction of astronomical observations was introduced, accompanied by brief demonstrations. Also in giving the lectures orally, I adopted the plan of exhibiting in ink-writing on large sheets of white paper descriptions of the instruments, and investigations of formulae, with the requisite illustrations by Figures, all being of such size as to be readily seen by the students from their seats. I had recourse to this method of lecturing partly because I thought it would serve to convey adequate information with little expenditure of the students' time, and partly because I was unable to meet with a text-book on this department of Astronomy which I could regard as sufficiently accurate and complete.

The contents of the Syllabus, together with the explanations inscribed on the above mentioned papers, have formed the ground-work of the present publication. In fact, as far as regards the divisions of the subjects treated of under the head of each instrument, and the order of their treatment, the Syllabus has been closely followed; but in the course of writing the Treatise, the composition of the Lectures, as originally conceived, has been in various respects modified and added to. The subjects which make up the additional matter are the following:-

A method of correcting the errors of a transit-instrument for deviation of the pivots from the cylindrical form; a detailed description of the construction and applications of the collimating eye-piece; the chronographical method of registering transit-observations; a discussion respecting personal equation in taking eye-and-ear transits; an experimental investigation of the effect of the flexure of a Mural Circle on the mean of its Microscope-readings; a description of the construction of the new Transit-Circle of the Cambridge Observatory and the mode of using it in taking observations (inserted with the consent of Professor Adams); the method employed at Greenwich for making observations of the Moon with an Altazimuth Instrument (not previously introduced, as far as I am aware, in an Elementary Treatise); and the process of taking observations with the Greenwich Reflex Zenith Tube.

The treatment of these subjects has given to the work an extension much beyond what I at first contemplated, but at the same time its usefulness as an astronomical manual may be considered to be much increased by being thus made to exhibit with a great degree of completeness the actual state of observational astronomy.

Although the instruments of the Cambridge Observatory, and processes of observation I adopted in the use of them, have been more especially described, and the Treatise consequently partakes somewhat of a local and personal character, I may venture, I think, to say that as having been written after twenty-five years of continuous labours in astronomical observations and calculations, and containing what may have occurred to me in the course of that experience as contributory to the advancement or improvement of practical astronomy, it will be found of some general utility as respects the work carried on in an Astronomical Observatory.

A particular account has been given of the investigation of formulae for the calculation of the mean places and annual variations of the fixed stars, and care has been taken to derive the numerical coefficients from the most accurate data procurable. The values assigned to the coefficients were calculated for the year 1879, and generally may be used without important error for epochs less than ten years before and after that date.

I take this opportunity for saying a few words respecting Practical Astronomy considered as a subject included in the course of the Mathematical Studies of the University of Cambridge. It seems to me that the tendency of our mathematical instruction and examinations has been of late years to promote the acquisition of a knowledge of formal relations of symbols and the power of readily producing them, apart from a distinct exercise of the reasoning faculty. Now as far as regards Practical Astronomy and Astronomical Instruments, it may be asserted that it would be impossible for the student to make himself acquainted with this department of science without understanding the reasons of the mutual relations of the different parts of the instruments, and the processes of observation by which the intended purposes are effected. He must be able to see how the many ingenious mechanical contrivances which the wants of astronomy have called forth, contribute to facility and precision in making and recording observations, and although this accomplishment may not demand a very high order of intelligence, it is still a mental exercise of much educational value, inasmuch as it is altogether unlike any process of reasoning by abstract symbols, and may serve as a corrective of the effect of too exclusive an attention to reasoning of that kind. The amount of mathematical knowledge which a complete understanding of Practical Astronomy requires is not more than what every candidate for Mathematical Honours is expected to possess, and accordingly the subject may be regarded as being within the reach of candidates of all degrees of proficiency. For these reasons I think it is much to be desired that Lectures on Practical Astronomy should always form a part of University teaching, and that the subject should at the same time receive due recognition in the Senate House Examination.

I am of opinion that a Professor of Astronomy, or Lecturer, who may undertake to lecture in this University on Practical Astronomy, if he should recommend selected portions of a Treatise like the present for the students' reading, and give only a few Lectures, would be able to convey sufficient instruction to candidates for Mathematical Honours, without requiring a disproportionate expenditure of time on this one branch of only one of the many subjects that engage their attention. The Lectures might be given partly in the Astronomical Lecture-Room in the Museum Buildings, to take advantage of the collections of instruments and models contained either in the Cabinet of the Plumian Professor's private room or in the Astronomical Apparatus Room, and partly at the Observatory for the purpose of pointing out the construction and mounting of the several instruments, and the methods of taking observations.

The contents of this Volume to page 337 are limited to the consideration of what is done with fixed instruments in a fixed observatory, and as having, consequently, exclusive relation to the foundations of exact Astronomical Science, might be appropriately separated from all other parts of Practical Astronomy, and form a distinct Volume. But because in the before mentioned Syllabus additional subjects were inserted which might be ranged under the heads of Observations with Transportable Instruments, and Miscellaneous Astronomical Information not given in the previous part of the Treatise, which also in delivering the Lectures formed the concluding portion of the Course, I have included them in the last two Sections of the present Volume, although on account of failing health and strength, I have not been able to treat of them so fully and in as much detail as the subjects comprised in the fundamental portion. It will, however, I think, be found that The Theodolite and The Sextant, two portable instruments of essential use in the Sciences respectively of Geography and Navigation which are of so much national importance at the present time, have been adequately handled. Also due consideration has been given to the methods recently employed for determining the Solar Parallax, and, in particular, inferences deduced from the observations of the Transit of Venus across the Sun's disk, on December 8, 1874, have been brought into notice.

Cambridge,
8 May 1879.