440 Mr. W. H. L. Russell on the Calculus of Symbols. 

 this may be written 



d 



or 



X^)= 



d d 



={l+aeda.-2+a2(e^^-2)2+. . . .}F(a?) 



To make this solution complete, we must add a complementary ftmction, 

 and we have 



«= 1 + r 



+ F(a;)+gF , - +a^F 



Jo 



As an example of this, put F(^)=a?, and the series becomes 

 iPi — / ■ 4- 4- — +• • ■ 



dv e-«2(l4.ae-2*V + a'e-4*^' + . . .) 



As a second example, we will take the equation 



This equation may be written 



(e 4-3e^^ + 2)/(^)=F(a?), 



(e'^o. _l)(e^-a._2)/(^)=F(a?). 



Now let 



(e 5 -2M^)=x(a.), 

 and the functional equation resolves itself into the two, 



and 



which are known forms. 



