# Papers by Bevan Baker in Proc. Edinburgh Math Soc.

We give titles and some introductory remarks for six of Bevan Braithwaite Baker's papers in the Proceedings of the Edinburgh Mathematical Society. We note that the date on which the paper was 'read' is the one on which Bevan Baker addressed a meeting of the Edinburgh Mathematical Society while the (always later) 'received' date is the one on which the paper was received for publication in the Proceedings.

**1. The Vibrations of a Particle about a Position of Equilibrium.**

By BEVAN B BAKER, Lecturer in Mathematics in the University of Edinburgh, and E B Ross, Professor of Mathematics in the Madras Christian College.

*Proc. Edinburgh Math. Soc.*

**39**(1921), 34-57.

Read 11th March 1921. Received 10th August 1921.

In obtaining a solution of the differential equations corresponding to the motion of a particle about a position of equilibrium, it is usual to express the displacements in terms of a series of periodic terms, each sine or cosine having for its coefficient a series of powers of small quantities. Korteweg has discussed the general form of such solutions, and, from the developments in series which he has obtained, has deduced certain features of interest. In particular, he has shown that, under certain circumstances, it is possible that certain vibrations of higher order, which are normally of small intensity compared with the principal vibrations, may acquire an abnormally large intensity. ... The solution in periodic series has the disadvantage of becoming unmanageable for certain values of the frequencies, and it is difficult to determine whether this is due to actual divergence or whether the trouble arises from the presence of a part increasing steadily with the time (secular term). The particular problem discussed, which was suggested by Prof E T Whittaker, has the advantage of being soluble not only in periodic series but also in terms of elliptic functions, and this second form of solution gives results where the series solution breaks down. This paper is chiefly concerned with a discussion of the conditions, obtained from the elliptic function solution, under which any valid solution of the problem exists. In a later paper it is hoped to apply the conditions so obtained to ascertain the cause of the divergence of the series solution.

In obtaining a solution of the differential equations corresponding to the motion of a particle about a position of equilibrium, it is usual to express the displacements in terms of a series of periodic terms, each sine or cosine having for its coefficient a series of powers of small quantities. Korteweg has discussed the general form of such solutions, and, from the developments in series which he has obtained, has deduced certain features of interest. In particular, he has shown that, under certain circumstances, it is possible that certain vibrations of higher order, which are normally of small intensity compared with the principal vibrations, may acquire an abnormally large intensity. ... The solution in periodic series has the disadvantage of becoming unmanageable for certain values of the frequencies, and it is difficult to determine whether this is due to actual divergence or whether the trouble arises from the presence of a part increasing steadily with the time (secular term). The particular problem discussed, which was suggested by Prof E T Whittaker, has the advantage of being soluble not only in periodic series but also in terms of elliptic functions, and this second form of solution gives results where the series solution breaks down. This paper is chiefly concerned with a discussion of the conditions, obtained from the elliptic function solution, under which any valid solution of the problem exists. In a later paper it is hoped to apply the conditions so obtained to ascertain the cause of the divergence of the series solution.

**2. On the Relation between Pincherle's Polynomials and the Hypergeometric Function.**

By BEVAN B BAKER, Lecturer in Mathematics in the University of Edinburgh.

*Proc. Edinburgh Math. Soc.*

**39**(1921), 58-62.

Read 13th May 1921. Received 20th June 1921.

As is to be expected from their definitions, Pincherle's polynomials have properties which are closely analogous to those of Legendre's and Gegenbauer's polynomials. The object of this present note is to exhibit this similarity by obtaining expressions for Pincherle's and Humbert's polynomials in terms of the hypergeometric function.

As is to be expected from their definitions, Pincherle's polynomials have properties which are closely analogous to those of Legendre's and Gegenbauer's polynomials. The object of this present note is to exhibit this similarity by obtaining expressions for Pincherle's and Humbert's polynomials in terms of the hypergeometric function.

**3. The Vibrations of a Particle about a Position of Equilibrium - Part II.**

**The Relation between the Elliptic Function and Series Solutions.**

By BEVAN B BAKER.

*Proc. Edinburgh Math. Soc.*

**40**(1922), 34-49.

Read 13th January 1922. Received 30th June 1922.

In a previous paper, entitled the "Vibrations of a Particle about a Position of Equilibrium," by the author in collaboration with Professor E B Ross (

In the present paper it is shown that the solution in terms of periodic series may be deduced from that in terms of elliptic functions by certain transformations. It is thus possible to discuss the cause of divergence of the periodic series and to define the region in which they are convergent. It is further shown that in the remaining region in which real solutions of the problem exist, the solution in terms of periodic series derived from the elliptic function solution takes a quite different form.

In a previous paper, entitled the "Vibrations of a Particle about a Position of Equilibrium," by the author in collaboration with Professor E B Ross (

*Proc. Edinburgh Math. Soc.***XXXIX**, 1921, pp. 34-57), a particular dynamical system having two degrees of freedom was chosen and solutions of the corresponding differential equations were obtained in terms of periodic series and also in terms of elliptic functions. It was shown that for certain values of the frequencies of the principal vibrations, the periodic series become divergent, whereas the elliptic function solution continues to give finite results.In the present paper it is shown that the solution in terms of periodic series may be deduced from that in terms of elliptic functions by certain transformations. It is thus possible to discuss the cause of divergence of the periodic series and to define the region in which they are convergent. It is further shown that in the remaining region in which real solutions of the problem exist, the solution in terms of periodic series derived from the elliptic function solution takes a quite different form.

**4. The Vibrations of a Particle about a Position of Equilibrium - Part III.**

**The Significance of the Divergence of the Series Solution.**

By Dr BEVAN B BAKER.

*Proc. Edinburgh Math. Soc.*

**41**(1923), 128-140.

Read 10th November 1922. Received 27th August 1923.

§ 1. In the two parts of this investigation previously published it has been shown that the solution in terms of elliptic functions represents the motion of the particular dynamical system under consideration throughout the whole range of values of $s$ and $g$ for which a real solution exists, except for those values for which $s = 2g$ and $k = 1$, but that, on the other hand, the series solution is convergent and represents the motion only so long as

[complicated inequality]

for values of $s$ and $g$ for which the sign of this inequality is reversed the trigonometric series representing the solution are divergent. It is of importance to investigate what discontinuities, if any, of the system correspond to values of $s$ and $g$ which lie on the boundary of the region of convergence; the present part is concerned primarily with showing that under such circumstances no discontinuity of the system exists, thus confirming the suggestions made in Part I., § 12.

§ 1. In the two parts of this investigation previously published it has been shown that the solution in terms of elliptic functions represents the motion of the particular dynamical system under consideration throughout the whole range of values of $s$ and $g$ for which a real solution exists, except for those values for which $s = 2g$ and $k = 1$, but that, on the other hand, the series solution is convergent and represents the motion only so long as

[complicated inequality]

for values of $s$ and $g$ for which the sign of this inequality is reversed the trigonometric series representing the solution are divergent. It is of importance to investigate what discontinuities, if any, of the system correspond to values of $s$ and $g$ which lie on the boundary of the region of convergence; the present part is concerned primarily with showing that under such circumstances no discontinuity of the system exists, thus confirming the suggestions made in Part I., § 12.

**5. The Vibrations of a Particle about a Position of Equilibrium - Part IV.**

**The Convergence of the Trigonometric Series of Dynamics.**

By Dr BEVAN B BAKER.

*Proc. Edinburgh Math. Soc.*

**41**(1923), 141-145.

Read 10th November 1922. Received 27th August 1923.

It has been shown in the previous three parts of this work, that the whole question of the convergence of the series solution, for the particular dynamical system under consideration, has turned upon the cubic equation

$4a^{2}x^{3} - (4a^{2} + s^{2})x^{2} + (a^{2} + 2sg)x - g^{2} = 0$

Before proceeding to generalise the results it will be shown how this cubic equation may be derived in a slightly different fashion. ... It has been shown by Whittaker that for any dynamical system in which the motion is of a type not far removed from a steady motion or an equilibrium-configuration, the equations of motion may be expressed in a general form applicable to all such cases. It has also been shown that the same general form may be applied to motion which is not of this character and in particular to motion such as that of the planets round the sun, or the moon round the earth.

It has been shown in the previous three parts of this work, that the whole question of the convergence of the series solution, for the particular dynamical system under consideration, has turned upon the cubic equation

$4a^{2}x^{3} - (4a^{2} + s^{2})x^{2} + (a^{2} + 2sg)x - g^{2} = 0$

Before proceeding to generalise the results it will be shown how this cubic equation may be derived in a slightly different fashion. ... It has been shown by Whittaker that for any dynamical system in which the motion is of a type not far removed from a steady motion or an equilibrium-configuration, the equations of motion may be expressed in a general form applicable to all such cases. It has also been shown that the same general form may be applied to motion which is not of this character and in particular to motion such as that of the planets round the sun, or the moon round the earth.

**6. An Extension of Heaviside's Operational Method of Solving Differential Equations.**

By Dr BEVAN B BAKER.

*Proc. Edinburgh Math. Soc.*

**42**(1924), 95-103.

Read 2nd May 1924. Received l0th September 1924.

Introductory.

An elegant symbolic method of solving differential equations was developed by Heaviside in his "Electrical Papers" and "Electromagnetic Theory," chiefly in connexion with problems concerning electric currents in net-works of wires. Attention has recently been called to the method by Bromwich, who applied it to a wider range of problems and gave an extension of Heaviside's formula; another generalisation of the formula has been obtained by Carson. In the present paper a formula is obtained which contains the formulae of Heaviside, Bromwich and Carson as particular cases, and whose form is such that it may be readily applied to physical problems.

Introductory.

An elegant symbolic method of solving differential equations was developed by Heaviside in his "Electrical Papers" and "Electromagnetic Theory," chiefly in connexion with problems concerning electric currents in net-works of wires. Attention has recently been called to the method by Bromwich, who applied it to a wider range of problems and gave an extension of Heaviside's formula; another generalisation of the formula has been obtained by Carson. In the present paper a formula is obtained which contains the formulae of Heaviside, Bromwich and Carson as particular cases, and whose form is such that it may be readily applied to physical problems.

Last Updated July 2020