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.