255 



A Theorem on Addition Formulae. 



I'.v Leslie MacDill. 



The theorem stated here is a corollary of a general theorem on a certain 

 class of functional equations, whose theory has not been completed at the 

 time of writing. 



Abel has shown that if a function, ((> (x, y), has the property: 



^ [z, (p (x, y)] is a symmetrical function of x, y, and z; then there exists 

 another function such that: 



f (x) + f (y) = f [0 (x, y)]. 



The corollary mentioned proves the converse of this theorem, and shows 

 further, that a necessary and sufficient condition for the solution of an addi- 

 tion formula in the form: 



f (x) + f (y) = f [z (x, y)], 

 w'.iere z (x, y) is supposed given as a known function of x and y, is that the 

 ratio: 



dy 



shall assume the form of the ratio of a function of x alone, to a function of y 

 alone, both of which functions have an indefinite integral, possessing each aq 

 inverse function, viz: 



5x u' (x) 



dy 



Furthermore, if we designate the inverse function by the bar, 

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

 is another necessary and sufficient restriction on the function z (x, y). 

 If the equation be given in the form: 



(2) z[f (x),f (y)] = f (x + y), 



