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the necessary and sufficient conditions are: 



5s u' (s) s = f (x). 



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



at 



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

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

 named abov'e is 



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



and for (2) is 



f fs) = / u (s). or as before; f (x) = / u (x). 

 It will be further noticed that if 



z [w, z (x, y)] = sj'mnictric function, 

 ^ t! e.i 



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

 Wo prove the converse. Necessarily 

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



z [w, z (x, y)] = u [u (w) + u -ju (u (x) + u (y)) \] = u [u (w) + u (x) + u {y)\, 

 which is a symmetric function. 

 Indidna Unii'crsily. 



