Vol. 6, 1920 
MATHEMATICS: 1. A. BARNETT 
201 
sidered by G. Kowalewski,^ although, so far as the writer knows, no at- 
tempt has been made to discover the invariants of the group. 
Let M or u(^) be a real continuous function defined on the interval 
aS by and r a real variable ranging over the interval | r — t© ] ^c. Con- 
sider also a functional operation F[x,u,z] which, for every element {x, u, r) 
of the set defined by 
a^x<b, max|zi(|) - Uo(^)\Sa, \r - To\^c 
determines a real number. If, furthermore, F is a contiguous functional^ 
of its arguments, then for each r, one may regard F as a transformation 
taking the continuous function u into another continuous function u 
of the variable x, 
u{x) = F[x,u,t] (1) 
As usual, the transformations (1) are said to form a group if the product 
of every two transformations of the set is in the set, and if, furthermore, 
the set contains the identity transformation and for every transformation 
the corresponding inverse transformation. 
An infinitesimal transformation is defined by the equation 
^ = fM, (2) 
where 
f[x,u] = ^ F[x,u,t] , = 
To denoting the parameter giving the identity transformation of (1). If 
one supposes that the functionals have suitable properties of continuity 
and differentiability, 2 one can show that equation (2) determines a one- 
parameter continuous group of transformations. 
By a method exactly like that employed by Lie it can be shown that the 
functionals G{u) which are finite invariants {finite is here used in contra- 
distinction from diferential) of the group (1) may be found from the solu- 
tions of the Stieltjes integral equation 
(^J[x,u]dx<plx,u] = 0 (3) 
»/ Or 
where the <p is defined by 
^^u(x)dx<plx,u] = ^[u]t 
and <i> is the Frechet differentiaP of G. It can be shown that if the func- 
tional / has the properties already referred to, solutions of (3) exist. ^ 
2. Examples. — ^The set of transformations 
u{x) = u{x) -f a{x)r = F[x,UyT] 
where a{x) is an arbitrary but specified continuous function evidently 
