40] 
ON THE THEORY OF INVOLUTION IN GEOMETRY. 
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supposing the coefficients of all the equations to be of the order unity, the order of the 
result, free from extraneous factors, may be shown to be 
\r — m, 6] +...—2 {[r — m — n, 6]+ ...} +3 {[r — m — n—p, 0] + ...} -&c. 
= mn... + mp... +np... + &c 
(C), 
(the equality of which will be presently proved) a result which agrees with that deduced 
from the theory of symmetrical functions; but I am not in possession of any mode of 
directly obtaining the final result in this its most simplified form. My method, which 
it is not necessary to explain here more particularly, leads me to the formation of a set 
of functions 
P,Q, X,Y, z, 
6 in number, such that Z divides F, this quotient divides X, and so on until we have 
a certain quotient which divides P, and this quotient equated to zero is the result of 
the elimination freed from extraneous factors. It only remains to demonstrate the 
.formulae (A), (B), and (C). Suppose in general that (k) denotes the sum of all the 
terms of the form m a n b ..., which can be formed with a given combination of k letters 
out of the (f) letters m, n, p ...; and let 2 (k) denote the sum of all the series (k) 
obtained by taking all the possible different combinations of k letters. It is evident 
that 2 (k) is a multiple of (</>), {(cp) denoting of course the sum of all the terms 
m a n b ..., m, n... being any letters whatever out of the series m, n, p...). Let g be the 
number of exponents a, b, ..., then (0) contains [<f)] g terms, also (k) contains [k^ terms, 
and the number of terms such as (k) in the sum 2 (k) is [</>]^-* 4- [(/> — Hence 
evidently 
or, what comes to the same thing, 
Let A be an indeterminate coefficient, a a summatory sign referring to different 
systems of exponents ; then 
ZaA(4>-k) = r [ ±^A(<fi), 
or, giving to k the values 1, 2 ...(f), multiplying each equation by an arbitrary coeffi 
cient, and adding, putting also for shortness aA ((f) — k) — U^-k, we have 
whence in particular, 
...= a {(F-M (</>)}, 
2- 22 P*_ 2 + ... = cr {(0 - g) 0(</>)}, 
c. 
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