MR. SYLVESTER ON FORMULA CONNECTED WITH STURM’S THEOREM. 483 
Art. (46.). Reverting now to the simplified Sturmian residues, since by the theory 
set out in the first Section these differ from the unsimplified complete residues 
required by the Sturmian method only in the circumstance of their being divested 
of factors, which are necessarily perfect squares and therefore essentially positive, 
these simplified Sturmians may of course be substituted for the complete Sturmians 
for the purposes of M. Sturm’s theorem. The leading coefficients in these simplified 
Sturmians, reckoning/'(^) as one of them, will be 
which it is easily seen, as remarked long ago by Mr. Cayley, are the successiv'e prin- 
cipal minor coaxal determinants of the matrix 
^0) ^1} ^2) 
°’lJ ^2) ^3 
^”2’ ^”3 
a 
2m— 2J 
where in general — ^i+^ 2 +---+^m 5 JiRd of course M. Hermite has improved 
upon this remark by observing, which is immediately obvious, that if we use a,, to 
denote, not the quantity above written, but — .-4— ^4- 
x—h^ x—h^ ' 'x — h, 
determinants of the above matrix will become respectively 
, the successive coaxal 
2_J_. V ^{KKK) 
x—h^^ \{x—h^){x—h^)\^ [x—h^)[x—h:^[x—h^)^ [x—h^)[x—h^) ...{x—h^)'^ 
that is to say, these successive coaxal determinants, when multiplied up hy fx, will 
become respectively 
^x-h,){x-K)...{x-h:)-, ^Whk){{x-h,){x-h,)...{x-h^)} ; 
that is to say, will represent the simplified Sturmian series given by my general 
formulee. M. Hermite further remarks, that the matrix formed after this rule will 
evidently be that which represents the determinant of the quadratic function (which 
may be treated as a generating function) 
in which, since only the squared differences of the terms in the {h) series finally 
remain in the successive coaxal determinants, we may write (x—h^), {x — h^) ...{x—h^) 
simultaneously in place of h^h^.-.h^ without affecting the result, consequently the 
generating function above may be replaced by the generating function 
2^3^. {m, + (^ - /l,)W2+ (j?— A,)^W3-{- . . . -b (^ — A,)”*-' 
