Vol. 6, 1920 MATHEMATICS: 1. A. BARNETT 203 
I[x,u] = u(x) + - f^''e''MOd^'^]^e~''''-e'''''P''e-^^^^ 
Some interesting special invariants are obtained when the particular 
invariant functionals 
p''e~'''''Ilx,u]dx and J^''|lb,ti] j^^^t; 
are taken in which cases one obtains, respectively, the invariants 
I^J^^'cos a^^i^ci^j^ _|_ [^J^%in p\^{x)dx. 
This group is a special instance of the orthogonal group in function space^ 
defined by the infinitesimal transformation 
Or ^ 
where K{Xy^) is a skew-symmetric kernel; i.e., K{x,^) = -K{^,x). The 
invariants of this group are quite complicated and will not be given. 
As a last example, let a group have the last mentioned infinitesimal 
transformation where the given kernel function K(x,^) is continuous and 
symmetric. The finite transformations may be put in the form^ 
^ - 6 
u{x) = u{x) + ^<^«W(/" - 1) I <Pni^)u{^)d^ =F[x,u,Lr]^ 
^ a 
where the constants X„ are the characteristic numbers of the kernel K{x,^) 
and the functions <^„(^), are the corresponding normed orthogonal set of 
characteristic functions. Every invariant is a functional of 
I[x,u] = m{x) + 
"5l'P„w| r r 'Pn{iMi)di\ ^ [ 
If the particular invariant functional 
/^I[x,u](ph(x)dx, k=^l, 
a 
is taken, one obtains the invariants 
One could now try to extend some other results of the theory of con- 
tinuous groups to functions both for one parameter and for several para- 
meters. For example, it would be interesting to know what the invariant 
manifolds of a given group are; also the applications to partial differential 
equations in infinitely many variables, which the writer is now engaged 
