584 
ON THE BICURSAL SEXTIC. 
[624 
viz. excluding the fixed points A, B, C, the six intersections are C, F, G, and the three 
intersections by the line. Hence, of the six intersections, we have C, F, G independent 
of 6, or we have (x — 0, z = 0) a triple point, say I, corresponding to the three points 
G, F, G, viz. these are the points 
(X, g, v) = (0, 0, 1), (0, g, -/), (h, 0, -/), 
(G, F, G). 
The equation Q + 6R = 0 is v (cXv + dgv + dXg) = 0 ; viz. the line v = 0 meets 0 = 0 
in the three points A, B, H, and the conic cXv + dgv + 6Xg = 0 meets 0 = 0 in the 
points A, B, C, and three other points: hence, rejecting the points A, B, G, the six- 
points of intersection are the points A, B, H, and the three variable points of inter 
section by the conic ; or we have (y = 0, z =0) a triple point, say J, corresponding to 
the three points A, B, H, viz. these are the points 
(X, g, v) = (l, 0, 0), (0, 1, 0), (h, -g, 0), 
(A, B, H). 
To find the tangents at the triple point I, these are x + dz = 0, where 6 is to be 
successively determined by the conditions that the line aX + bg 4- Ov = 0 shall pass 
through the points C, F, G * ; viz. we thus have 
0 = 0, 
bg 
0 = + -j, fx+bgz= 0, 
$ = + ( -y , fx + ahz = 0, 
x = 0, the tangent corresponding to the point C, (0, 0, 1), 
» » » » F, (0, g, -/), 
G, (h, 0, -/). 
And similarly, at the triple point J, the tangents are y + 6z = 0, where 6 is to be 
successively determined by the conditions that the conic cvX + dvg + OXg = 0 shall pass 
through the point H, and shall touch the cubic at the points A, B; viz. we thus have 
y= 0, the tangent corresponding to the point H, (h, — g, 0), 
0 = 0, 
0 = %, fy+cgz = o, 
0-7' fy+ dllZ =°: 
A, (1, 0, 0), 
B, (0, 1, 0). 
The two last values of 6 are obtained by the consideration that the equations of 
the tangents to 0 = 0 at the points A, B respectively, are gg+fv = 0, hX+fv = 0, 
where X, g, v are current coordinates of a point on the tangent: it may be added 
that the equation of the tangent at the point G is hX + gg = 0. 
* Observe the somewhat altered form of the condition: 6 is to be determined so that the cubic 
(a\ + b/j.+6p) = 0 shall touch the cubic 0 = 0 at one of the points C, F, G: but, as the first-mentioned 
cubic breaks up, and the component curve a\ + bfi + 6v = 0 does not pass through any one of these points, 
this can only mean that 6 shall be so determined as that the line shall pass through one of these points, 
viz. that there shall be at the point, not a proper contact, but a double intersection, arising from a node 
of the cubic \fj.(a\ + bix + 6v) = 0. And the like case happens for the other triple point; viz. there the cubic 
v (cv\ + dv/x+= 0 is to touch the cubic B=0 at one of the points A, B, H; the component conic 
cvX + dv/ji. + 0X^=0 passes through the points A and B but not through H; hence the conditions for 6 are, 
that the conic shall touch the cubic at A or B, or that it shall pass through H.