412 
ON CUBIC CONES AND CURVES. 
[351 
26. If T— 0, the equation becomes 
or reducing, this is 
(* 2 + ^+l) 3 -^ 2 (S- + 1)2 = 0; 
1C* - 1) <* + *)(* +2)}» «0, 
that is the six roots are 1, — — 2, each twice: and the four tangents form therefore 
a harmonic pencil, which is the geometrical interpretation of the condition T = 0. 
(^•2 _j_ _|_ ]\3 
27. The function - ^ is constantly positive and it has three equal mi 
minima 
if 1 — is positive and less than unity, that is, if S and 64$ 3 — T 2 are each of 
values corresponding to the last-mentioned values 1, — — 2 of S-, viz. this minimum 
value is = . Hence we see that the equation in S will have its six roots all real 
T 2 
64$ 3 
them positive: but when these conditions are not satisfied the six roots are imaginary: 
the limiting case 1 — =1 or T = 0 gives, as already mentioned, the three roots 
1, — —2, each twice. 
28. The quantities a, b, c, d which determine the four tangents may be all real, 
or two real and two imaginary, or all four imaginary; but the imaginary values 
appear as usual as a conjugate pair or conjugate pairs; and this being so, it is easy 
to see that in general if Sr be real the quantities a, b, c, d are all real or else all 
imaginary; but if S^ is imaginary then a, b, c, d are two of them real, two imaginary: 
in fact if a, b are real and c and d are conjugate imaginaries 7 + Si, then we have 
for one of the six values of S, 
_ (a — b) . 2 Si 
— (a — 7 — Si) (b — 7 + Si) ’ 
which is in general imaginary. 
29. But, as might have been foreseen, the limiting values Sr = 1, — —2, are an 
exception, viz. for these values a, b, c, d may be two of them real the other two 
imaginary: in fact the last-mentioned value of S- is real and = ~ > = 2, if 
(a — 7) (b — 7) + S 2 = 0, that is ab + 7 2 + S 2 = 7 (a + b), or, as the condition may also be 
written, 
2ab + 2 (7 + Si) (7 — Si) = (7 + Si 4- 7 — Si) (a -I- b), 
that is 2 (ab + cd) = (a + b)(c + d), or if a, b, c, d form a harmonic system.