474 
A DISCUSSION OF THE STUEMIAN CONSTANTS 
[290 
In the particular case where the third constant vanishes, or 
b 2 — ac = 0, 
we have 
— a 2 d 2 + Qabcd — 4 ac 3 — 4 b 3 d + 3 6 2 c 2 
= — (ad — be) 2 + 4 (6 2 — ac) (c 2 — bd) 
= — (ad — be) 2 , is negative; 
hence, regarding the evanescent term as being at pleasure positive or negative, we 
have in each case a combination of signs corresponding to one real root. 
The general result (which is well known) is, that there are three real roots or 
one real root according as 
— a 2 d 2 + Qabcd — 4<ac 3 — 46 3 d + 36 2 c 2 
is positive or negative. 
For the quartic equation 
the Sturmian constants are 
(a, b, c, d, e)(æ, 1) 4 = 0, 
if, as usual, 
a, a, b 2 — ac, 3aJ + 2 (6 2 — ac) /, I 3 —27J 2 , 
I — ae — 46d + 3c 2 , 
J = ace — ad 2 — b 2 e + 2bcd — c 3 . 
If the signs of the 
constants, that is, 
of the functions for 
+ oo, are 
then the signs of the 
functions for - 00 
are 
+ + + + + 
0 
+ - + - + 
4 
+ + - + + 
2 
+ + 
2 
+ + + - + 
2 
+ — + + + 
2 
+ + — — + 
2 
+ + + 
2 
+ + + + - 
1 
+ — 4- — — 
3 
+ + — + — 
3 
+ 
1 
+ + + — — 
1 
i 
+ 
+ 
i 
+ 
3 
+ + — — — 
1 
4~ — — + — 
3 1 
4 real roots, 
no real root. 
2 real roots. 
— 2, cannot occur. 
2 real roots. 
The non-existing combination of signs is 
I 3 — 27 J 2 =-, 
3aJ + 2 (6 2 — ac) I = +, 
b 2 — ac = —. 
To show a posteriori that this case cannot occur, write 
^ = a 2 d — 3abc + 26 3 , 
X = 3a J + 2 (6 2 — ac) I,