22 Proceedings of the Royal Irish Academy. 
where j/^ is a giveii function ; tlien whatever be the function F, 
where, for shortness, cb = cf3(cc) = ,., 
x}/ {x) 
Thus, for instance, let 
A ^ • 111 ^ 
And since — = 1 — , s = 
z cc \ — X 
Hence ^ -^(^y^' 
also 
5. Thus, if we are given the functional equation 
^(iT^)^"^W' 
if ^W = y, e-^^^'^y = ay, 
as before, - hx^Dy = log «y ; 
I 
.-. <^(ii;) = Ca hx, 
6. In the same Paper it is shown that 
Hence the functional equation 
cji [x'') = atji {x) 
is equivalent to 
log r . X log X . Dy = log a .y; 
log a C dx log log X 
.'. log y = r-^ — = log » . ^ + C; 
log rjx log ^ log r 
or <^(a;) = C{logx) log^ 
This example, by putting (ji{x) ^ ij/ (log ^i;), would have given 
ij/ (r log x) = aij/ (log x), 
and thus be reduced to (2), above. 
