266 
ON THE THEORY OF INVOLUTION IN GEOMETRY. 
[40 
which are still equations of considerable generality. If now </> = 0 and U e is a function 
of m + n + p + ... of the order 0, the quantity a {0 d ~ g A (0)} reduces itself to the single 
term of Ue which contains the product mnp.... Hence, if 
U e = [a+m + n+p 0] 
in which afterwards a. = r — m — n — p— ... we have the formula (A). Again, if cj> = 0 +1, 
and Ue +1 is a function of m + n+p... of the order 0, the sum a {0 e t 1- v A (<£)} vanishes; 
whence writing U g+1 = [m + n +p ... — 0, 0\ we have the formula (B). Similarly, if in the 
second formula <fr = 0+l, and U e+1 is a function of m + n+p ... of the degree 0, then 
{(<9 + 1 -g) O d -9A(0+l)}, 
reduces itself to the term which contains mn ... +np ... + mp ... + &c.; whence, if 
U e+1 = [m + n+p+ ... — 0, 0], 
we have the formula (C).