520] 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
327 
19. The surface consists of two sheets intersecting in a nodal curve connecting 
the outcrop with the urabilicar centre. As to the form of this curve there is a cusp 
at the outcrop; and the curve does not terminate at the umbilicar centre but, on 
passing it, from crunodal becomes acnodal (viz. there is no longer through the curve 
any real sheet of the surface): moreover the curve is not at the umbilicar centre 
o 
perpendicular to the plane of xz, and there is consequently on the opposite side of 
the plane a symmetrically situate branch of the curve, viz. the umbilicar centre is a 
node on the nodal curve. Completing the curve, the nodal curve consists of two 
distinct portions, one on the positive side of the plane of yz or minor-mean plane 
consisting of two cuspidal branches as shown in the figure; the other a symmetrically 
situate portion on the negative side of the minor-mean plane. 
Intersections of Evolute and Ellipse. 
20. Consider in the plane of xy the ellipse and evolute, 
First, these are satisfied by 
- Coordinates of Umbilicar centres in plane of xy (imaginary), 
viz. the equations respectively become 
= (_ _ 2)3 27 a. 3 j3 3 _ o 
7 7 7 
the first of which is a + /3 4- 7 = 0, and the second is (a 3 + /3 3 + 7 3 ) 3 — 27a 3 /3 3 7 3 = 0. But 
the equation a + /3 + 7 = 0 gives a 3 + ¡3 3 + 7 3 = 3a/3y, and the two equations are thus 
identically satisfied. Moreover the condition for a contact is at once found to be 
ft' 2 \_{a 2 x 2 + b 2 y 2 — 7 2 ) 2 + 9y 2 6 2 y 2 ] = a 2 [(a 2 « 2 + b 2 y 2 — f) 2 + 9y 2 a 2 # 2 ], 
or, what is the same thing, 
(a 2 — /3 2 ) (a 2 x 2 + b 2 y 2 — f 2 ) 2 + 9y 2 (ol 2 ci 2 x 2 — /3’ 2 b 2 y 2 ) = 0;