DIFFERENTIAL EQUATIONS OF THE n"' OrDER. 27 



T h core m IL 



If u' , u'-\ ..., ?«'"* be solutions of (42), their sum will also be a 

 solution of the same equation. 



r = n r = n 



For putting z = ft«"*, whence r„„, , = Oî«!;!,,, we 



r = I r = 1 



get by substitution in (42) 



r = n i=n 



r=\ J = u 



an equation, that is identically satisfied, as every sum within the brac- 

 kets is 0. 



In the following we suppose the coeflficients U to be constants and, 

 for the sake of distinctness, we put C in their places; thus (12) becomes 



S C^ .„-.. = V. 



1 = 



We will, however, previously treat the simpler equation 



i = n 



g az„_,,, = (43). 



! = 



The auxiliary equation (20) becomes here 



I - n 



g(— !)■ am"- =0 (44) 



1 = 



and has constant roots , which we represent by mj , m.^ , • . . , »"„ . 



Th eorem III. 

 If «V be any root of (44), z = Ç (y — jji^a*) is a solution of (43). 



For let z = (p (ij — m^x) , whence we have ;„-,-,, 



= ( — \)"-' m"-* (p^"^ (y ~m^œ) , which, when multiplied by (— 1)"" 

 = (_ 1)2'-» ^ reduces the left member of (43) to 



(p<"' (y — m,x) g (— !)'• a mp--, 



1 = 



that is to 0, as m,. is a root of (44), and therefore the sum vanishes. 



