202 
MATHEMATICS: I. A. RARNETT 
Proc. N. a. S. 
represents a group in function space with r = 0 as the identity parameter. 
The infinitesimal transformation is given by the equation 
'6u{x,t) 
dr 
= a{x). 
Furthermore, every invariant satisfying suitable restrictions of continuity 
and differentiability may be expressed as a functional of the invariant 
I[x,u] = 
u(x) f^u(^)d^ 
J a 
The transformations 
U{X) = TU{X) = F[XyUyT] 
form a group. The infinitesimal transformation is 
bn{x,T) 
dr 
and every invariant is a functional of^ 
u(x), 
u{x) 
I[x,u] = 
It is easily verified that the transformations 
u{x) = u{x)e~^ + {/ — e~^) j*^ u{^)d^ = F[x,u,t] 
form a group with r = 0 giving the identity transformation. The in- 
finitesimal transformation is defined by the integro-differential equation 
Every invariant is a functional of 
I[x,u] = u{x)Jy{k)d^ - { J/©^^}'- 
Consider the group defined by the infinitesimal transformation 
(>u{x,t) 
dr 
/27r 
sin (t(x - ^)u{^)d^ 
where a is an odd integer. It can be shown® that the finite group may 
be put in the form 
u{ = u{x) -{- —\e 
27rL 
+ ^-*(^-- _ 1) J^V"'^w(^)(i^j = FIx,u,t] 
Every invariant is a functional of the invariant 
