602 Proceedings of the Royal Irish Academy, 
Ey putting in (2) F(z - <^z) for F{%)j we have the result 
F{x) = D-'^rLl)F{x-<p{x)), 
which we observe is independent of 2, or of the equation (1). For 
instance, let <^ {x) = x^. 
(7) 
2. Several curious results as to infinite series follow from Lagrange^s 
theorem: thus, if z = x + y(j>{z), and we write (j> = cli{x), (the operand 
is 1) 
let ;^ = 1, then 
we thus obtain the n*^ power of ^this series, viz. if <^ be any function 
of 
thus let (f> = e*, by reducing and putting h = ye", 
Again, let s = + tf", so that 
n = 1 + + -J— + &c. 
Qg» I = e'^j but (2) e» = i)-ine*, and ^ = ^1 • 
if n = -l. 
