1. Elementary Principles of the Theories of Electricity, Heat and Molecular Action: On Electricity (1833).
M Poisson in his Memoirs on Electricity, Magnetism and Molecular Actions, M Ampère in his 'Theorie des Phenomenes Electro-dynamiques,' and Fourier in his 'Theorie de la Chaleur,' have been the respective founders of the physical sciences considered in this treatise in a mathematical point of view. The subject of electricity (including what is called ordinary electricity, Voltaic actions and magnetism,) forming in itself a complete system, is the sole object of the first part of this work, the other subjects being reserved for the Second Part; and as the ordinary course of mathematical reading in the University is a sufficient preparation for the study of the branches of science here treated, it is hoped that the suggestion recently made by a distinguished member of the University, will be in some degree answered in the present Treatise. [The suggestion is in the Preface of William Whewell's A Treatise on Dynamics.]
As an acquaintance with the properties of the remarkable functions treated by Laplace in the Mécanique Céleste Book III is indispensable in investigations respecting electricity, instead of referring to that work I have here introduced them under the form of Preliminary Propositions; I have however followed a different route, making the functions which shall possess those properties, the objects of investigation; and have thus arrived at a more general class of functions (which are of great use in investigations relative to Latent Electricity) and also obtained several new and remarkable theorems with respect to Laplace's functions: it must be added that on referring to Crelle's Journal, I found that M Jacobi had anticipated me with a respect to few of the theorems alluded to.
It was natural to consider the manner in which electricity is disposed in bodies, previous to its becoming sensible by the action of electro-motive causes; this is the object of the second chapter, and I am not aware that it has been before made the subject of mathematical investigation.
It could answer no useful purpose to point out what is new in the remaining parts of the work; that will easily be recognised by those who are already acquainted with the subject, and those who are unacquainted would not benefit by the information; I shall only add that the sixth and seventh chapters contain the theories of Ampère on Voltaic actions, and Poisson on magnetism, with such modifications as seemed to simplify the processes employed by those writers.
I have to return my best thanks to Professor Cumming, for the facilities afforded me by the use of his apparatus, to confirm experimentally some of the results deduced in this work, from theoretical views.
Caius College, June, 1833.
The main view of the author of the work quoted, I learned, was to conduct his subject so as insensibly to lead the learner from pure algebraical theories to a knowledge of the principles on which the more advanced branches of analysis depend. To this advice from an excellent analyst I have adhered as well as I was able; but, in consideration of the recent progress of the "Theory of Equations," I felt it necessary to alter the plan, assuming however the propositions proved in the other work, to which therefore the reader will find several subsequent references.
I will now make a short statement of the plan adopted in the present work, premising that no treatise with exclusively the same object has been published of late, as far as I know, either at home or abroad. To collect and methodically digest the scattered elements of this theory, as far as its present advanced state imports, was attended with no inconsiderable difficulties; therefore, though an object of great utility has been, I hope, obtained by the composition of this work, it cannot be expected to be altogether faultless.
Before examining algebraical equations theoretically, it appeared necessary to convey a precise idea of the continuous nature of algebraic functions, and to show that their numerical magnitudes may be extended through every quantity from negative to positive infinity, notwithstanding the existence of certain maxima and minima values. This subject is discussed in a series of propositions, the more clearly to impress the reader with the steps of the reasoning. Having attained this object, the ordinary properties of equations relative to the existence, number, limits, and symmetrical relations of the roots, followed as easy consequences; on these deduced properties I have not much dilated, as they have been already ably treated in the work before referred to.
I have then given the theorems of both Sturm and Fourier relative to the discovery of the number of real and imaginary roots of an equation, the combination of which with the methods of approximation due to Newton and Lagrange conducts to the solution of all numerical equations of finite dimensions, except for imaginary roots, for the discovery of which I have employed a method deduced from recurring series. These numerical applications I have illustrated by examples in a later part of the work.
The formation of literal equations being understood, I have explained the logarithmic method for obtaining with rapidity the series which analytically represent the different roots and their functions; and have then shown how to effect some general and useful transformations of equations, and explained the algebraical solutions of the equations of inferior degrees, and the analytical meaning of the different surd parts which constitute the roots.
The theory of binomial equations is treated much in the manner of Lagrange, and the methods for the general resolution of equations are then discussed; and wherever useful applications to the kindred branches arose, I have supplied them in the form of Scholia, in order to preserve a proper arrangement of the subject more especially treated.
After giving several useful analytical results springing from the employment of the methods before given, I have passed on to discuss recurring series, which have been used from an early date for the solution of equations. I have then pointed out the useful extension made but left unproved by Fourier; I have supplied the proofs for that part which was correct, and substituted right theorems for those in which he has committed errors.
After then giving the various methods of approximation to the roots, and using all the appliances by which they may mutually assist each other, and thereby facilitate the numerical solution - on which occasion I have also considered several properties of continued fractions - I have, in conclusion, considered the properties of general classes of equations of finite and infinite dimensions, and shown in what cases the theory of the former may or may not be applicable to the latter.
All parts of the work I have taken care to illustrate with numerical or more general examples, and to draw such inferences as will be found useful in the higher branches of analysis. A glance over the table of contents, which I hope will form a useful epitome of the whole subject, will convey a more complete idea of the nature and extent of the matters here treated.
I have availed myself of an accidental delay in the publication of this Treatise, to revise, correct, and augment different parts, with the view of rendering the work as complete as possible.
February 3, 1838.