336 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
[520 
37. The umbilicar centres or points (I) belong to values such as ^ = |i = — a 2 
which are the united values in the equation between (f, (fi), viz. writing herein = £ 
the equation becomes 
(I + a ~) (£ + b 2 ) (f + c 2 ) = 0, 
so that the united values are f = £ = — a 2 , - Z> 2 or - c 2 . (It may be remarked, that 
treating this cubic as a degenerate quartic, a united value would be £ = = oo, corre 
sponding to the umbilicar centres at infinity.) 
To a value such as £ = — a 2 there corresponds (not only the value = — a 2 , but 
also) a value & = - a? + , as it is easy to verify. And the outcrops or points (II) 
P — 7 
3 f3y 
belong to such values £ = — a 2 , ^ = — a 2 + ^ _ -. 
And the nodes of the evolute or points (III) belong to values such as £ = cob 2 + co 2 c 2 , 
^ = co 2 b 2 + coc 2 (co an imaginary cube root of unity) which, as it is easy to see, satisfy 
the relation between (£, £). But to complete the theory we require to have the values 
of 7], 97! and also the coordinates of the points on the centro-surface, and of the two 
points on the ellipsoid. 
38. 1 exhibit the results first for the umbilicar centres (imaginary), outcrops 
(imaginary), and nodes of the evolute (imaginary), in the plane # = 0; secondly for the 
real umbilicar centres in the plane y = 0 and for the real outcrops in the plane ¿ = 0. 
The formulae contain an expression il which is a symmetrical function of a, /3, 7 
(or a, b, c), viz. it is 
il = 0? — /3y = /3 2 — 7a = 7 2 — a/3 = \ (a 2 + /3 2 + 7 s ) = — (/3y + 7a + a/3). 
We have 
I. £ = - a 2 , v - 
X =0, 
Y 2 = — b 2 -, 
a ’ V (Umbilicar centre). 
x — 
by = - 
c 2 z 2 = — 
-a 2 ; fi = -a 2 , Vl = -a\ 
Zj=0, \ 
Y 2 = — b 2 - 
1 a ’ [ (Umbilicus). 
7 2 _ r 2 @ 
A - c a ,