352 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
[520 
CL ry a 
For the real curve a extends from 7. through 0 to — , viz. 
a — /3 ° II 
cr = -—gives outcrop in plane z = 0, 
a= 0 „ umbilicar centre in plane y = 0, 
ct = - „ evolute-node in plane oo . 
It is to be noticed that the order of magnitude of the terms in the table is 
x , 
27 
7 — a 7 — a 
J— 1| 
-7 
/3-ry’ a -(3 
a -7« 
5 
— a 
2a — 37a 
O ’ 7 — a ’ 7 — a ’ (7 — a) 2 
, - 00 , 
so that the values - - g -—, 0, -7^- which belong to the real curve are contiguous; this 
CL— ¡3 
is as it should be. Several of the preceding investigations conducted by means of the 
quantities £, g, % 1} rj 1 might have been conducted more simply by means of the formulie 
involving cr. 
The Eight Cuspidal Conics. Art. Nos. 61 to 71. 
61. The centro-surface is the envelope of the quadric 
a-x- 
; + 
hhf 
+ 
C 2 Z 2 
—1 = 0. 
(a 2 +f) 2 (& 2 +f) 2 (c 2 + £) 2 
Hence it has a cuspidal curve given by means of this equation and the first and 
second derived equations 
era- 
; + 
by- 
+ ■ 
C-Z- 
(a 2 + f) 3 (> + |) 3 (c 2 +^> 
a?x 2 
(® , + f) , + 0 2 + f)‘ + (c ! +i)‘ 
which equations determine by, c 2 z 2 in terms of £, viz. we have 
- /3y arm? — (a 2 + f ) 4 , 
- 7a by = (b~ + £)\ 
- <x(3 cV = (c 2 + PY: 
by 
c-z 2 
0, 
= 0, 
so that, comparing with the equations — (3y a 2 x 2 = (a 2 + £) 3 (a 2 + 77) &c. which give the 
centro-surface, we see that for the cuspidal curve £ = 77; or the cuspidal curve now in 
question arises from the eight lines on the ellipsoid, which lines are the envelope of 
the curves of curvature : it is clear that the curve is imaginary.