652 
PROFESSOR SYLVESTER ON THE REAL 
that point representing the family of forms is to be referred. But since the doubt can 
only arise when J is negative and D positive, and since by. hypothesis we have A= — yJD, 
we see that A takes the sign of [m ; and consequently the sign of M, when it becomes 
zero, is to be understood as following the sign of p, i. e. as positive when (m is 1 and 
negative when [h is —2. 
(84) The method above given for ascertaining the nature of the roots of a quintiq 
involves the use of only three criteria. It may be inquired how many would become 
needful in applying Sturm’s method. In the case of a cubic equation only the last of 
the two Sturmian criteria comes into use ; and it seems therefore desirable to ascer- 
tain whether all four of the Sturmian criteria applicable to that case are required, or 
whether a smaller number are sufficient. I speak of four criteria, inasmuch as the lead- 
ing terms fx and f'x cannot be considered as such, their signs being fixed; so that we 
are at liberty to consider them both positive. Suppose all six Sturmian functions to be 
written down, including fx (a function of x of the fifth degree) and f'x, and let us cha- 
racterize by the index (r, s\ any succession of signs of the leading coefficients which con- 
tain r continuations and s variations, and which therefore will correspond to the case of 
( r — s ) roots. 
The total number of cases to be considered are the sixteen following : 
(5, 0) 
+ 
+ 
+ 
+ 
+ 
+ 
f + 
+ 
+ 
+ 
+ 
— 
(4, 1) 
j + 
+ 
+ 
+ 
+ 
+ 

— 
i + 
+ 
— 
— 
— 
— 
( + 
+ 
+ 
+ 
- 
+ 
! + 
+ 
+ 
— 
+ 
+ 
(3, 2)- 
+ 
+ 
+ 
— 
— 
+ 
+ 
+ 
— 
+ 
+ 
+ 
+ 
+ 
— 
— 
+ 
+ 
+ 
+ 
— 
— 
— 
+ 
r + 
+ 
+ 
— 
+ 
— 
(2, 3) i 
! + 
! + 
+ 
+ 
— 
+ 
+ 
+ 
: 
l + 
+ 
- 
— 
+ 
— 
(i 5 4) 
+ 
+ 
- 
+ 
— 
+ 
the successions corresponding to the indices (2, 3), (1, 4) will become impossible, a$ 
corresponding to a negative number of real roots. An inspection of the eleven cases 
corresponding to the indices (5, 0), (4, 1), (3, 2) will show that no ternary combination 
of signs in the third, fourth, and sixth columns belongs to any of the three characters 
(5, 0), (4, 1), (3, 2) exclusively, and consequently all four signs must be used; and there- 
fore, if the method of Sturm is employed, four criteria are indispensable for determining, 
