328 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
[520 
and substituting the foregoing values, this is 
a 3 + 8 s 
(a 2 — /3 2 ) (— 7 2 ) 2 + 97 2 
7 
zƱ a ^Æ = o 
7 
that is, 
-—~ (a 3 + /3 3 + 7 3 ) 2 4- 97a 2 /3 2 (a — /6) = 0, 
which, putting therein a + /3 = — 7, and a 3 + /3 3 + 7 3 = 3aySy, is also satisfied ; that is, the 
points in question are points of contact of the ellipse and e volute. 
21. Secondly, consider the values 
Coordinates of outcrops in plane of xy (real). 
Substituting in the equation of the ellipse, we have 
a (fi - 7) 3 + /3 (7 - a) 3 + 7 (a - /3) 3 = 0, 
which is 
(/3 - 7) (7 “ a ) ( a “ /3) ( a + /3 + 7) = 0, 
or the equation is satisfied identically: and substituting in the equation of the evolute, 
we have first 
a 3 (/3 — 7) 3 + /3 3 (7 — a) 3 + 7 3 (a — yS) 3 
7 (a-/3) 3 5 
o?x 3 + b 2 y 2 — 7 2 = 
which in virtue of a (/3 — 7) + /3 (7 — a) + 7 (a — /3) = 0 becomes 
+ 6y - y — 
a 2 # 2 + 6 2 y 2 — 7 2 = — 
3aff (/3 - 7) (7 - °0 
(a-/3) 2 
and then, completing the substitution, it is seen that the equation of the evolute is 
also satisfied. The points last considered are simple intersections, and we have thus the 
complete number (8+4, =12) of the intersections of the evolute and ellipse. 
22. We have a, 7 positive, /3 negative; whence a — ft is positive, ¡3 — 7 negative; 
ry — a (= a? + c 2 — 2b 2 ) is positive, and hence, the outcrops in the plane of xy are real; the 
umbilicar centres are imaginary for this plane, but real for the plane of zx, the coordinates 
being 
a? \