MR. SYLVESTER ON FORMULA RELATING TO STURM’S THEOREM. 
457 
/ 7 / 
thus related, so that calling as before the roots of f, and 
the roots of p, we shall have in general 
^Qi+l 
L^l 
=fK»= 
^9t + l 
^9j+l 
X 
^9j+ 1 
_ ^9i ^92 * • • ^9£ ^9i+2 ’ * • ^9m _ 
_M92- 
M . 
9!_ 
_^9i + 2 \i+3"'^gm_ 
Consequently 
^9£+l ^9t + 2*”^9TO 
_’?! ??2 ••.^m-l_ 
^9i + l 
^1^2- ••^m-1 
X 
^9j + 2 
^2*”^m-l 
X&c.X 
r^9 ^ 
Hm 
fll ^2*”^m-l 
= 
^9£+l 
_ ^91^92* ••^9t_ 
X 
^9i + l 
J^H+2 ^9i+3 • • • ^9m- * _ 
^hi + 2 
X 
^^9i + 2 
_^9i+l ^9i+3''‘^9m-l _ 
X &C 
h. 
*lm 
h„h„ ...hg. 
L ?2 Hi _ 
9m 
Hence 
?e + l ?t +2 9 m 
^ 1^2 
•^m-1 
^9i+l ^9i+2‘”^9m 
,^9. ^92 --K J 
9t+l 
L^9t+2 ^9i+3"’^9m_ 
^ 9 i +2 
_^9» ^9i+l’”^^qm_ 
X 
X 
h„ 
Hm 
h„ hg...h„ . 
?i ?2 ?m — 1 
•^(^9i+] ^9!-l-2''*^9m)’ 
the ^ denoting the operation of taking the product of the squares of the differences 
of the quantities which this symbol governs. Hence the Bezoutian secondary to 
f and f of the {m—i— l)th degree in viz. — 
h„ h„ ...hg, 
H\ ?2 Hi 
^2 1 
kg hg ...kg. 
H\ 92 9t 
L ^9t+ 1 ^^9t+2 • • • ^9m J 
3 O 2 
becomes 
