462 
MR. SYLVESTER ON FORMULAE RELATING TO STURM S THEOREM. 
o^fx, and k to denote h~-x\ the equation 
•4~^l ^2 ^3 ^4 ^6 ky.T3=-'2kq^.kqJ^{k^^ kq^ kq^ kq^ kq^ 
-~^iK K K K K K) X MiK K K K. K K )) ; 
and Tj will vanish whenever more than three relations of equality exist between the 
A-’s, for then each term in hath of the two sums in the right-hand member of the 
equation above written will separately vanish ; and of course three relations of 
equality between the same are sufficient to make all the terms in the first of these 
sums vanish. This relationship between the different A-’s corresponding to a multipli- 
city 3 may arise in different ways ; the multiplicity 3 may be divided into 3 units 
corresponding to 3 pairs of equal roots, or into 2 and 1 corresponding one set of 3 
equal roots, and a second set of 2 equal roots, or may be taken ‘^en bloc,” which 
corresponds to the case of one set of 4 equal roots. I shall make the first of these 
suppositions, which will sufficiently well answer our purpose in the case before us. 
Thus I shall suppose \~k^ k^=k^ k^^-k^, 
then, as above remarked, kq^ kq_^ kq^ kq) =0 for all values of q-, qe q^, and therefore 
also 2 A,, kq^ kq^ kq^ k q^ kq^ becomcs 
Ai Ag Ag (A1.A2.A3 + 2 A7.A1 Ag-j-Ai A3 + Ag A3), 
and vanishes, except for the cases where qiq^q^q^ represent respectively, 
q, the index"\ or 4, q^ the index 2 or 5, q^ the index 3 or 6 , and q, the index /• 
Hence 2 A,, A,, A,^ kql{kq^ kq^ k,^ AJ = 2 ^Ai k, k, kZ{k, h h 
and consequently becomes 
±8^(A. A2 A3 ky) X {Ai A2 k,^ 2 ky{k, A2-I-A1 A3+ A2 A3)}. 
Hence we are able to predict that the general expression for our r in the case befoi e 
us will be 
r3=+2{^(A,A,^A,^AJX 
+ (^,.+A,,+AJ(A,,.A,^+A,^.A,3-l-A,,Av-fA,^.A,3+A,^.Av-bA,,.A^..) 
K ^%+K K K K+^^2 
For in the first place, the fact that the r vanishes when more than three relations 
of equality exist between the A’s, proves that we may assume of the form 
2 { UK K K K) xKK K K K ; K K K} ^ 
the semicolon (;) separating the A’s into two groups, in respect of each of svhicn 
severally ?) is a symmetrical form. But if in the expression last above written for r, 
we make k^=k^ k^^k^ A 3 — Ag, 
it becomes , , , . , ^ 
(A^+AI+A1)-(A1+A1+A1)(Ai+A2+A3+Av) 
+ (Ai+A 2 H-A 3 )(Ai Ag+^'i A 3 +A 2 A 3 +A 1 A 7 -I-A 2 A^H-A'g Ay) 
— 4 (Ai A2 A3-I-A1 Ay A7+A1 Ay A7-I-A2 A3 A7) 
+8^(Ai A2 A3 ky) 
