484 MR. SYLVESTER ON FORMULA CONNECTED WITH STURM’S THEOREM. 
the corresponding matrix to which becomes 
^^5 ^05 ^15 
^05 ^IJ ^2J 
6 , 
where 6^ denotes 't{x—a')\ and 2 ^ 3 ^=^ 
form 
3 ? 
Hence every simplified residue is of the 
25 15 
fU 
6-2 . . • 
f’xX < 
^2 ^3 •••^r + 1 
••^2i— 1 ^ 
> X •< 
0 K 1 
-'0 
% k 
K ^r+1 •••^2r-i J 
The residue in question will be of the degree m—r—2 in x, and consequently we 
have, according to the notation antecedently used for the syzvgetic equations 
■'I ^2 •••'"r 
, ^2 ^3 • • • ^r + 1 
L+i — • S 
r 0 ^0 ^1 
^0 
• • ^j-+2 ^ 
Elegant and valuable for certain purposes as are these formulae for and they 
are aflfected with the disadvantage of being expressed by means of formulae of a 
much higher degree in the variable x than really appertains to them, the paiadox 
(if it may be termed such) being explained by the circumstance of the coefficients 
of all the powers of x above the right degree being made up of terms which mutualh 
destroy one another. Upon the face of the formulae, and r,. which are in fact 
only of the degrees r+ 1 , and r respectively in x would appear to be of the degree 
l_j_ 3 _j- 5 _j_.„-}-( 2 r— 1 ), i. e. of the degree 
Art. (47-) • I important remark, which does not appear to have 
occurred immediately to my friend M. Hermite when he communicated to me the 
above most interesting results, that in fact, by virtue of the law of inertia for quadratic 
forms, we may dispense with any identification of the successive coaxal determinants 
of the matrix to the generating function 
{W,-l-/ZiW2“l"^d*^3"l~ ••• 
