290] 
473 
290. 
A DISCUSSION OF THE STURMIAN CONSTANTS FOR CUBIC 
AND QUARTIC EQUATIONS. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. IV. (1861), 
pp. 7-12.] 
For the cubic equation 
(a, b, c, d) (x, l) 3 = 0, 
the Sturmian Constants (or leading coefficients of the Sturmian functions) are 
a, a, b~ — ac, — a 2 d 2 + 6abcd — 4ac 3 — 4>b 3 d + Sb 2 c 2 . 
If the signs of 
the constants, 
that is, of the 
functions for 
+ oo, are 
+ + + + 
4- + — + 
+ + + - 
+ + - - 
then the signs 
of the func 
tions for -oo 
are 
- + - + 
— + + + 
— + + - 
three real roots, 
case cannot occur. 
one real root. 
The second case would give a loss of variations of sign in passing from oo to — oo , 
which is inconsistent with Sturm’s theorem. To show d posteriori that the case cannot 
occur, we may form the identical equation 
(a?d — 3abc + 2b 3 ) 2 = — a 2 (— a?d 2 + 6abcd — 4ac 3 — 4ib 3 d + 36 2 c 2 ) + 4 (b 2 — ac) 3 , 
and, this being so, then in the case in question, the right-hand side would consist of 
two terms, each of them negative, while the left-hand side is essentially positive. 
C. IV. 60