354 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
[520 
65. In fact consider the two conics 
Va ax ± V'ft by + Vy 02 = V — afty, 
(1, 1, 1, -1,-1, — 1) (a Va ax, ± ft ^ft by, 7 Vy oz) 2 = 0; 
for the intersections with the plane y = 0 we have 
Va cm; + Vy cz = — afty, 
(a Va ax — 7 Vy az) 2 = 0 ; 
so that the two conics each meet the plane in question in the same two coincident 
points, that is, they each touch the plane y = 0 at the same point, viz. the point given 
by the equations 
Va gm? + Vy c.z = V — afty, 
CL\/cLax — y^ycz — 0\ 
. \JAJ 
-ft V-/3 
and the common tangent at this point is 
Va ax + \fy cz = V — afty, 
which is also the common tangent of the ellipse and evolute in the plane y — 0. 
66. It has been seen that the nodal curve meets each principal conic at four 
outcrops, which points are cusps of the nodal curve: it is to be further shown that 
the nodal curve meets each of the 8 cuspidal conics in four points (giving 32 new 
points, which may be called ‘outcrops,’ the 16 points heretofore so called being 
distinguished as the principal outcrops or 16 outcrops, and the points now in question 
as the 32 outcrops), which points are cusps of the nodal curve. 
In fact to obtain the intersections of the nodal curve with the 8 cuspidal conics, 
we must in the equation of the nodal curve, or (what is the same thing) in the 
expressions of £, rj in terms of a, write y = £. 
67. Putting for shortness, 
P) i ( 7 ~ g ) 2 °~(°'- !)(3o--2) 
i!o" + yCL 
and as before 
«7 (3c-2) 
<H)(1 + \/£) 3 = 1 - Vs, 
we have thus