93 
<f> will have 
ttN 
L 
values by the permutation of the N variables, 
which can be formed upon 
And the number of different functions <J>, all of the same 
algebraic degree, of which no one is a value of another, is the 
number of groups equivalent to G. 
Theo. P. Let G be any group of L substitutions. 
Let 
tj a /3 7 $, 
P = x 3 ‘ • • • x N 
where 
“ 7 ft, P v 7, &c. ■ • • e 7 o, 
be a term such that it changes in its algebraic value by the 
operation MP performed on its subindices, where MG is 
any derived derangement of G, and such that no group 
equivalent to G gives the same algebraic function, 
<I» = Pi + P 2 + ‘ + Pi 
with G. The function d> has 
7 r 
N 
values, which are con= 
structed on G and on its — r — 
Theo. Q. If two equivalent groups G and G 1 of L substi- 
tutions give for a certain system of exponents of Xi x 2 ’ * x x the 
same algebraic function <!>! has not 
ttN 
L 
values. 
Theo. R. The number of groups equivalent to a group G 
being given, and any system of exponents of x x x z • • x N being 
selected, there is always an assignable number of algebraic 
functions, 
^ = ?i + Pa ~r + 
of the degree determined by that system, such that each 
function has - values, and that no function is a value of 
JL t 
another. 
1 . There are three equivalent functions, and no more, of the 
Gth degree, made with four letters, vdiich have each six values. 
