270 
ME. W. H. L. EUSSELL ON THE CALGULES OF FENCTIONa 
by making 
when ■\pi and Fj are known functions of (a?), may be reduced to the form 
<Pi(a + ^07+ = F(ir)4/(a + 
The second equation has already been considered ; it becomes, therefore, interesting 
to ascertain what equations are included under the form of the first equation, in other 
words, to consider what forms the algebraical expression can possibly take. 
The following are a few of them : — 
v' [x-^ 2 ac—b) 
x—a 
\/a+Vc.x ’ x[b + cx)^ {2a-\-bx + cx'^)x[b + cxy 
-v/ { a?® — 2bx + 5 ^— Aac 
a — cx^ 
Many others may, in like manner, be imagined; and the same methods, mutatis 
mutandis, apply to functional equations of the higher orders with variable coefficients. 
I now come to the consideration of equations involving definite integrals, when the 
equation contains an unknown function under the sign of definite integration. 
Let us take the equation 
J . FaW +r4W»“+nW-^" dxf{x)-Y{a,), 
where (p{oc) is an unknown function of [x) not containing (a), Fi(a), . . , F 5 (a) rational 
functions of (a) which is supposed to vary independently of (x), to determine (p(x). 
Suppose the equation can be written in the form 
where 
r f l-(x(«))^ , 1 — 
Jo v'l— — 2%a(l — 2a;2) + {xa)^ ' 1 — 2a(l— 2a;2) + «2j ^ 
X(x)—<p{x)\/l—x^. 
Let a7=sinp? and the equation reduces to 
or if 
we find 
>vL(a) = i d6. 
^ Jo l-2«cc 
2^u cos 6 + 1 —2a cos S + 
^}=2FW ; 
; cos Q + tx- 
= 2F(a). 
Suppose the solution of this equation be determined by the former investigations to be 
= /(«), 
then 
Assume 
^ (1 — 
Jo 1— 2acosd + a^ 
=/(«}. 
X sin p cos ^+<*52 cos 2^+«3 cos 3^+ . . 
