542 
MATHEMATICS: S. D. ZELDIN 
Proc. N. a. S. 
the group G has q ( < r) exceptional infinitesimal transformations (Lie 
calls them "ausgezeichnete"), which for simplicity are taken to be 
Then 
(Xi, Xj) = 0 {i = 1,2, ... r q;j = r + l,...r + q). 
The purpose of this paper is to find the conditions to be imposed on the 
group G which would make the structural constants c^v _|_ i , ... c^v + ^ all 
zero for i, j = 1, 2, . . . r q. 
Since the adjoint of G has the same structure as G itself, it follows that 
r 
{Ei, Ej) = 2^ ^iJkEk ih j = 1,2, ... r). 
1 
Therefore we may say that if G contains q exceptional infinitesimal trans- 
formations, there exists another group, say G\ with r essential parameters 
having the same structure as the group G. 
We shall denote the operators of the adjoint of G' by the symbols 
r r 
Mk = 2^" = 1, 2, . . ., r) 
1 1 
The condition we impose on the adjoint of G' is that it shall have one 
invariant spread. From that follows that the nullity of the matrix formed 
by the coefficients of the r differential operators is equal to one, i. e., at 
least one of the minors of the determinant of the coefficients of order 
f — 1 is not zero. But every minor of that determinant is also a minor of 
the determinant formed by the coefficients of the differential operators 
of the adjoint of G, therefore in this larger determinant there will be at 
least one non-vanishing minor of order r—1. Furthermore, for ai. . . 
a^. . .aj. ^ q assigned, the symbolic equations 
rr + q r + q v 
^ih ] = 0 
have the following g + 1 independent solutions : 
^1 = ai, . . . /?, = a,, + 1 = . . . = /3, + g = 0 (1) 
^1 = . . . = = 0, + 1 = 1, + 2 = . ■ . = + 0 (2) 
iSi = = /5, + 1 = 0, + 2 = 1, + 3 = • • • = + g = 0 (3) 
/3i = = + g_ 1 = 0, /3, + g = 1 (g + 1) 
That shows that there will be in the large determinant at least one non- 
vanishing minor of order r — q—\. Therefore, the nullity of the matrix 
formed by the coefficients of the operators of the adjoint of G is equal 
to q -\- \, which means that the adjoint of G leaves invariant g + 1 
