410 
ON CUBIC CONES AND CURVES. 
[351 
and thence 
M = 64>S 3 - = 64Æ 8 {k (k + 4) 3 - (k 2 + 6k + 6) 2 }, 
= 6№ (- 8k - 36), 
= - 256& 8 (2k + 9). 
It may be right to 
remark that from the value k = 
-4(1 -If 
1-21 + 4Z 2 
we deduce 
k + 4 = 
4Z (1 + 1 + P) 
1-2Z+4Z 2 5 
and that thence 
if 
— 2 (1 — 20Z 3 — 8Z 6 ) 
kr + № + 6= (l-2t + 4ty ■ 
S=cPS', T=a 6 T, 
rjl-2 JI'S 
S 3 = F 3 
S' = -1 + P, 
T = 1 - 20Z 3 - 8Z 6 , 
23. The equation 
4 (1 - If k 
1 -27 + 4Z 2 “ l - 1 • 
4 (Z — l) 3 
4Z 2 -2Z+ 1 
or as it may also be written 
7. 16(Z-1) 3 
16(Z-i) 2 +3’ 
gives without difficulty 
and 
k + | = 
i6(;-tf+27(i-j) 
16(i-J)= + 3 
A: + f — 
16 (Z + !) 2 
16 (Z — -|) 2 + 3 ' 
24. Hence treating Z, & as coordinates, we see that the locus is a cubic curve, 
viz. a hyperbolism of the ellipse, having a centre (Newton’s species 62), the coordinates 
of the centre being Z = k = — f, and the equation of the asymptote being 
k +1 = Z — i, (that is the asymptote passes through the centre and is inclined at an 
angle =45° to the axis of Z). The centre is of course an inflexion, the equation of 
the tangent at this point is & + f = 9(Z — £), and for the other two inflexions we have 
Z = l, k = 0, and Z = — I, k = — f, the tangents at the two inflexions respectively being 
k = 0 and k — — ^, that is the tangents at the inflexions are parallel to the axis of Z. 
The curve consists of a single branch lying below the asymptote for large negative 
values of Z, k, crossing the asymptote at the centre and lying above it for large 
positive values of k, l. For each value of Z there is consequently a single value of k 
and reciprocally; and Z, k pass together from — go to + oo. There are certain critical 
values of k and Z, the meaning of which will appear from the following article.