JIR. SYLVESTER ON FORMULA RELATING TO STURM’S THEOREM. 
4(>5 
we shall have 
^ +(_).-.-.^_.s„_,_3+(-)«-‘->k+i)S._,_j j’ 
where a, denotes in general the sum of the rth powers of the (^+I) quantities 
and denotes in general the sum of the products of the complementary {m~i—\) 
(juantities 
combined, r and r together. It will of course also be understood that ffo=i+l, so 
that o-o+l=Z+2. 
Art. (42.). To prove the correctness of this general determination of the form of 
’’m-i-ij let US suppose in general that Z+1 relations of equality spring up between the 
{m) quantities Z*,, ^'^e shall then easily obtain (N representing a certain nume- 
rical multiplier) 
±Q=N.^(Z-i 
hfH- 
/1 1 . Zt2 
P^m-i-V 
ki, k^...k^_i_^ being what the [k) system becomes when repetitions are excluded, and 
being respectively supposed to occur pj, ... times respectively, so that 
the fractional part of the right-hand member of the equation immediately above 
written will be readily seen to be equivalent to 
lo establish the correctness of the assumed form, we must be able, as in the parti- 
cular cases previously selected, to prove two things ; the one, and the more difficult 
thing to be proved is, that when the series of distinct quantities k^, k^, k^..,k^ become 
converted into groups of Zr, ; groups of Z-2...jW-,„_i_, groups of k,^_i_i, then that 
K K • • • ^0™ -*•_ 1 ) J 
or in other terms. 
becomes identical with 
^m-i-Z &C. l)S„j_j-_2. 
1 he other step to be made, and with which I shall commence, consists in showing that 
the number of terms in the expression last above written, considered as a function of 
(m— Z— 2)th degree of (Z-j-J) variables, is never greater than the entire number of 
ways in which (2-1-1) quantities out of m quantities may be equated to the remaining 
{m—i 1) quantities, viz. each of the first set respectively to all the same, or all dif- 
feient, or some the same and some different ; in short, in any manner each of the i -\- 1 
quantities with some one or another (without restriction against repetitions) of the 
remaining quantities. This latter number being in fact the number of ways 
in which — quantities may be combined (Z-l-l) together with repetitions 
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