256 



the necessary and .sufficient conditions are: 



Qs u' (s) .s = f (x). 



dz " u' (t) t = f (y). 



at 



z (s, t) = u [u (s) + u(t)l. 

 The solution for the unknown function in (1), under the restrictions 

 named above is 



f (x) = A u (x), A = arbritrary constant, 



and for (2) is 



f (s) = A u (s), or as before; f (x) = A u (x). 

 It will be further noticed that if 



z [w, z (x, 3')] = synmiotric function, 

 t^ en 



f (x) + f (y) = f [z (x, y)], by Abel's theorem. 

 Wj prove the converse. Necessarily 

 z (x. y) = u [u (x) + u (y)]. 



z [w, z (x, y)] = u [u (w) + u\n (u (x) + u (y)) }■] = u [u (w) + u (x) + u (y)]. 

 which is a symmetric function. 

 Indiana VnivtrHily. 



