228 Prof. D. N. Mallik on the 



while, also from (7) and (10) we get, since 



"da? ~dy ds 



h — c 2 v 2 A = c 2 £ - ^- (pw). 



Thus, on the whole, the motion in the general case depends 

 on an equation of the form 



Q£->v , )* = *>,y,*,0- 



7. The object of the present paper is to deduce the various 

 solutions that have been proposed of these equations, starting 

 from Poisson's solution of f = c\ 2 % (which is reproduced 

 here for convenience of reference), in a synthetic manner. 



8. We know that if f = c 2 v 2 fj we ma J put 



i / x sinh (cW) , 

 I = cosh (c*V)%+ c t v +> 



X, ijr being arbitrary functions of x s y, z. 

 Suppose, initially, 



where F, /are given functions of a, y, z. 



Then the solution of the above equation may be put in the 

 form 



where the arbitrary functions are replaced by known, 

 functions. For, obviously, 



?o = X=/ and &> = ±V 



(putting t = Q in the solution). 



In order to interpret this symbolic solution, let a, /3, y, 

 be a point (P) on a sphere of radius r, centre 0(#, y, z), and 

 consider the integral 



!• 



.(■l.«l,«S) 



where JS is an element of surface of the sphere, 



