346 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
[520 
and for Bz (= z) we have 
c 2 (Bz) 2 = — 
1_ 
a/3 
— 9i2g 3 
~ (a- /3) 3 ’ 
3il 2 h ’ 
so that writing for greater simplicity, (a - /3) q x = - a/3nr, the formulae become 
2Bx 
x 
% 
V 
3a 
y — a 
3/3 
ft — y 
nr, 
cBz = 
(— a/3nr)% 
nV 3 
52. This shows that there is at the outcrop a cusp, the cuspidal tangent being 
in the plane of xy. It appears moreover that this tangent coincides with the tangent 
of the evolute. In fact, from the equation (ax) 3 (by) 3 - 7 2 = 0 of the evolute we have 
(ax) 3 , dx (by) 3 , dy __ Q 
x y 
or substituting for (x, y) their values at the outcrop, 
/3 (7 ~ a ) dx a (/3 - 7) dy _ Q . 
7*(a-/3) ¿c y 3 (a-/3) y 
that is, 
/3(7-a)^ + *(/3-7)^ = 0, 
which is satisfied by the foregoing values of — , and —, and the two tangents there- 
x y 
fore coincide. 
We have 
which in virtue of 
4{(&r) 2 +(%) 2 } 
,.A2i _ - (/3 (7 ~ a ) + «(/3 ~ 7) 
7 (a - £) 3 
b 2 
is 
a 2 a (/3 - 7) + 6 2 /3 (7 — a) + c 2 7 (a — /3) = 3a/37, 
4 ¡(fa)* + (8y« = !3a/3 - ' (a ■- 
(observe 3a/3 — c 2 (a — /3), = — c 2 (7 — a) — 3a 2 a, is negative) 
- 9OT 2 a 2 /3 2 
a 2 b 2 ( a — /3) 
, 2 ft»