OF THE ROOTS OF AN EQUATION AND ITS COEFFICIENTS. 
in which the law of succession is obvious. Hence 
>= 6.5) =6' 55 
2,= eae S; ze : ps 
5. OMS, =6 4-8, 
3, =o MM 8, =; hes, 
_ AAS Ss 2 PsPs. 
ays ey eee PsP 
p= (a—B), p= 3 {(a—B)*(a—y)"(B—y}} 5 Ge. 
6 
where 
503 
(47), 
These results agree with SYLVEsTER’s, and lead to the same conclusion, viz., 
that the number of imaginary roots depends on the number of variations of signs 
of the functions (e.g., the sextic), 
bore Anny AN, Ae, Ais, has 
and the values of these functions beginning with A, are respectively proportional 
tO Po, Ps, Ps5 Ps» De- 
The form A’, (42), for successive Sturm’s functions is interesting, and -I 
believe new; as also the form (39). 
