46() 
JMR. SYLVESTER ON FORMULiE RELATING TO STURMS IHEOREM. 
admissible by a well-known arithpietical theorem is — and the first number 
{ i+l){i + 2 )...{m~- 2 _ jggg other. It remains then only to 
1 .2 ...(to— « — 2) ’ 
prove the remaining step of the demonstration*. 
Art. (43.). To fix the ideas let m=10, i=5, and consider the expression 
^lo) “ (^5 + ^6 + ^1+^8+ K + ^'.o) {h + h+h+ 
+ ( ^5 + ^6 + ^7 + ^8+ ^9 + ^.o) (^1 • ^2 + ^1 ^’s + ^. A’4 + A'a + h ■ h + h . A-,) 
7(Ai Ag A3-I-A1 
Now suppose the six quantities h, h, A„ A3, A^, A,o to become respectively equal each to 
some one or another of the four quantities A„ A„ h, A„ as for instance, I shall suppose 
As =Ag=:A7^Ai 
Ag Ag A2 
Alo — A 3 . 
Then ^2 — (^ 3 — 1? 
and the formula of art. (41) becomes 
(3AM-2A^+Ai)-(3A?+2Al-fAl)(A,+Ab+A3+Ad 
-f - (3 Ai +2 Aa-f A3) ( Ai . A2+ Ai . A3+ Ai . A4-I- A2 . Aad- A4 . A4 + A3 . A4) 
— 7 (Ai.A 2 .A 3 -|-Ai.A 2 .A 4 -}-Ai A3 A4+A2.A3.A4) 
= 3 { (A? - A? . {K+h^k,-\-h) + Ai (A2 A3-f A2 A4-+- A3 . A4+ A4. A2 + A3+ Aq) 
-f 2 { A| — A1 . ('A,+A3+A4 -f A 2 ) + A2(A4 k,+k,h+h A 4 + A 2 Aq + Aq + Aq) 
^-(Aj— - A3 (Ai-|- A2-I-A4+ A3) -[-A3(Ai . A2 + A1. A4-I- A2.A44-A3 A1 + A2 + A4) 
— { Ag A3 A3 A4+AJ A2 A4-I-A1 Ag A3} 
=z-k,k, A3--2A1 Ag A4-3A, A3 k,~ 4 hh Aq 
= — Aj Ag 
^3^.fi;+"F:+T:+T: 
In the above investigation the quantities which with their repetitions make up 
the A’s system, are k„ A^, Ag, A3, appearing respectively 1, 2, 3, 4 times, that is to say 
repeated 0, 1, 2, 3 times ; 7 is 1 more than the sum of the repetitions 0+ 1 +2+3, and 
the numbers 1, 2, 3, 4 arise from subtracting from 7 the sums 1+2 + 3; 0+2 + 3; 
0-f 1 -ps ; 0+1+2 ; respectively, so that the remainders 1, 2,3, 4 denote respectively 
one more than the number of repetitions of A4, Aq, A., Aq, i. e. are the number of appear- 
* If this first step of the demonstration appear unsatisfactor)^ or subject to doubt, it may he dispensed with, 
and the result obtained in the succeeding article (the demonstration of which is wholly unexceptionable) being 
assumed, it may be proved that the formula there obtained on a particular hypothesis must be universally true, 
in precisely the same way and by aid of the same Lemma in and by aid of which the formula obtained in the 
Supplement to this section for the simplified quotients to-^ upon a like particular hypothesis is shown to be 
of universal application, i. e. by showing that otherwise a function of 2/-1 variables wmuld contain a function 
of 2i variables as a factor. 
