400] 
and, as mentioned above, 
GIVEN PENCIL OF SIX LINES. 
113 
Q 10 = c s R 3 (— T 2 + 64$ 3 ). 
i.he just-mentioned value of Q 10 should, I think, admit of being established a priori, 
and if this be so, then the substitution of the values of S and T in terms of r, 2 f1, 
V, c®, cR, would be the easiest way of arriving at the before-mentioned expression of 
Qio i n terms of these same quantities. The calculation by which this expression was 
arrived at, is however not without interest, and it will be as well to indicate the 
mode in which it was effected. 
Calculation of Q 10 . 
We have to find the discriminant of 
c [(a, h, k, b\x, yff + 32l 3 cc?y 3 . 
Consider for a moment the more general form P 2 + 4Q 3 , then to find the discriminant, 
we have to eliminate between the equations 
these are satisfied by the system P = 0, Q 2 = 0, and it follows that if R be the resultant 
of the equations P = 0, Q = 0, then the discriminant in question contains the factor 
R 2 . For the other factor we may reduce the system to 
dP dQ dP dQ 
dx dy dy dx 
Now writing Q = 2Ixy, these equations become 
P ——h 48l 3 x 2 y 3 = 0, 
dx 
the resultant of which is = l 3 into resultant of the system 
dP 
pTL + 48^y = 0, 
dx 
C. VI. 
15