300 
Proceedings of the Royal Society of Edinburgh. [Sess. 
XIX.— On Systems of Partial Differential Equations and the Trans- 
formation of Spherical Harmonics. By H. Bateman, Ph.D. 
(' Communicated by Professor Whittaker.) 
(MS. received July 1, 1916. Read July 3, 1916.) 
§ 1. It is well known that Laplace’s equation V 2 Y = 0 possesses certain 
classes of solutions of the form Y = F(X, Y) where F satisfies a partial 
differential equation with X and Y as independent variables and X and Y 
are real functions of x, y, and z. Such solutions have been called binary 
potentials .* They are of some interest in the problem of finding cases in 
which the equations of motion of an electron in a steady magnetic field 
are readily integrable.f There are also classes of solutions of the same 
type, except that X and Y are complex quantities and the coefficients in 
the partial differential equation for F may also be complex. In particular, 
there are solutions of the form Y = Y/(X) where / is an arbitrary function 
with continuous second derivative.! 
Passing on to the case of four independent variables, we find that the 
equation of wave-motion 
9 2 y aw aw_i^ aw 
a^ 2 + a y 2 + dz 2 ~ c 2 dt 2 ’ ' ' ‘ • \ / 
possesses classes of solutions of the form Y = F(X, Y, Z) where F satisfies 
a partial differential equation § and X, Y, Z are real functions of x, y, z, t. 
It also possesses solutions of a similar type in which X, Y, and Z are not 
necessarily real, and in particular it possesses solutions of type Y = ZF(X, Y) 
where F is an arbitrary function of its two arguments. || 
* Of. Y. Volterra, Ann. Sc. Normale di Pisa (1883). T. Levi Civita, Turin Memoirs , 
t. 49 (1899), p. 105, classifies the binary potentials. 
f Of. C. Stormer, Gomptes Rendus, Paris, t. 146 (1908), pp. 462, 623. 
1 These solutions have been discussed by Jacobi, Forsyth, Levi Civita, and the author. 
See the author’s Electrical and Optical Wave Motion , pp. 114, 136, 153. Also Annals of 
Mathematics , vol. xiv (1912), p. 51. Solutions of type V = ZF(X, Y), where F satisfies a 
partial differential equation, have been discussed by Amaldi, Rend. Palermo , t. 16 (1902), 
pp. 1-45. See also Hantzschel, Reduction der Potentialgleichung. 
§ These solutions are of interest in the problem of transforming a special type of electro- 
magnetic field into an electrostatic field. One solution of this problem is derived by 
setting up a Lorentzian transformation between X, Y, Z, T and x, y , z, t. As far as I know , 
a solution of this problem is not necessarily associated with a set of functions of type 
X, Y, Z ; for instance, I have not yet succeeded in determining the special type of electro- 
magnetic field associated with the functions X = x cos ut — y sin «£, Y=x sin at A y cos cot, 
Z=z — vt , where v and u are constants. 
|| Messenger of Mathematics (1914), p. 164. Solutions of type Y = WF(X, Y, Z), where F 
satisfies a partial differential equation, have been discussed in a paper by the author, Gambr. 
Phil. Trans. (1910), vol. xxi, p. 257. 
