ME. W. H. L. EUSSELL ON THE CALCULUS OE FUNCTIONS. 
271 
then since 
we have 
Hence 
1 - L CO^ = ^ + • ■ •> 
ao+^is"® +«2S''" + • . . 
+ + + . . . =^(g"*®). 
?.(sin^)=^{/g‘^+/g-^}, 
whence <p(x) can be determined. 
Similar treatment will of course apply to the equation 
Eja + Y,^a,x^ + Ega . + . . . 
Iq r„(«) + F„+i(«)a?2+ ... 
(p{x)—'F{cc), 
but the functional equation employed for its solution (when possible by this method) 
will be of a higher order. 
Let us, lastly, consider the equation 
JiFi(^)9(^'4/(a))=F(a), 
to find Ip, where is a known function of (x) not containing (a), and (a) varies 
independently of (x). 
Let \P(ci)=^, then and the equation becomes 
J;i'.(:r)?(:r/3)=F4.-(|3). 
Let 
<p{x^)=A,-\-A,x(i+A^oif^(i‘‘+ . . ., 
then we shall have 
then 
Hence 
AoJ' <Z^Fiir+A,J'<?.r . x'F(x ) . ^-\-A 2 ^\dxx^'¥{x) . /3^+ . . . 
=F^}--‘0+F^(,-'0 . p+W'^-'O • 1^+ . . . ; 
A.= 
EtJ/ ^(0) 
A,= 
F'v^-'O 
m= 
F^/ ’0 F^\J/ ’0 
F"r|/-'0 ^ 
F'"4/-*0 
^^^dxYjx'y^dx.xY^x' . ai^Fja? " 1 *2 ^^^dx.x^.F^.x 1-2.3 
I being any variable. 
Now suppose 
jp— 
and that we are able to express x(n) by a definite integral, so that 
Xin)=^fiv)mv)ydv, 
•7 
