456] 
303 
456. 
NOTE ON THE DISCRIMINANT OF A BINARY QUANTIC. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. x. (1870), p. 23.] 
It is well known that the discriminant of a binary quantic {a, h, c, d,...'§t, 1)” is 
of the form 
Ma + N¥, 
but it is further to be remarked that if 6 = 0, then the form is 
a {Ma + Ac 3 ), 
if 6 = 0, c = 0, the form is 
a 2 (Ma + Nd 4 ), 
and so on, until only the lowest two coefficients are not put = 0. Or, what is the 
same thing, if in the discriminant of the original function we put a = 0, then the 
discriminant divides by 6 2 ; if 6 = 0, the discriminant divides by a, and, omitting this 
factor, if we then write a = 0, it divides by c 3 ; if 6 = 0, c = 0, the discriminant divides 
by a 2 , and omitting this factor, if we then write a = 0, it divides by d 4 ] and so on, 
until as before. 
Thus if 6 = 0, the discriminant of (a, 0, c, 
this factor it is 
a 2 e 3 
d, e\t, l) 4 , divides by a, and omitting 
— 18 ac 2 e 2 
+ 54 acd 2 e 
- 27 ad 4 
+ 81 c 4 e 
- 54 c 3 d 2 
which for a = 0 has the factor c 3 ; if 6 = 0, c = 0, the discriminant of (a, 0, 0, d, e$t, l) 4 
has the factor a?, and omitting this factor it is 
ae 2 
- 27 d 4 , 
which for a = 0 has the factor d 4 \ the series of theorems here terminates, since the 
lowest two coefficients d, e are not to be put = 0.