268 
ME. W. H. L. EUSSELL ON THE CALCHLIJS OF FUNCTIONS. 
The general linear functional equation of the wth order with constant coefficients is 
••• -\-Ji<p-^{x)-{-1c(p{x)='F'{x\ 
where the subject -^x is supposed to be the same as in the preceding equations. 
Let x=x(z), and suppose; then ■^\x)—xm% ' 4 '^x=:-)Qn% &c., 
and the equation becomes 
+ ... +hcpx{mz)+k<pxiz)=Fx(z). 
Let z=-^, and the equation becomes 
This equation may be written 
or 
{ (l + a,£^) (l + a^s^) (l + . . . (l + } (px = F% ( J+;.) ’ 
+<4'Px(—^^ +&C. j 
+-+i^ +a3F%^^„+„+2^ + • • .j+&c. 
+ 4.&C 
where Ai, A2, A3, &c. are certain functions of a^, a^, 0^3, &c., and Cj, Ca, C3, &c. are 
arbitrary constants, from whence we at once obtain the value of <p(x). 
We now proceed to consider functional equations with variable coefficients. And first 
let the equation be 
p(x)-x(^)p\0^'^=F{x}, 
where x{x) and F(^) are known rational functions of (a). 
Let 
then 
x=u. 
a + bug 
M'j.+ i — 
c-\-eUg 
a + bug 
c + eug 
Suppose a solution of this equation of finite differences to be 
A + B^r 
^^—C+Wz 
