360 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
[520 
and the order of the nodal curve is thus =24: two of the equations in fact are 
a, 
b 
= 0, 
b, c 
a, b, 
c 
a, c 
a', b', 
c' 
a', c', d' 
which are surfaces of the orders 4, 6; or the nodal curve is a complete intersection 
4x6. By the results above obtained as to the nodal curve, it appears that the two 
surfaces must have an ordinary contact at each of the 16 umbilicar centres, and a 
stationary or singular contact at each of 48 outcrops. 
i 
79. The derivation of the centro-surface from the surface 
a 2 x 2 ( b 2 y 2 t 
af+% + W+~% + 
c 2 z 2 
C 2 +lr 
-1- £ — m = 0 
requires to be further explained. The surface, say F= 0, is a quadric surface depending 
on the two parameters f, m; the axes coincide in direction with those of the ellipsoid, 
and their relative magnitudes are as 
- Va 2 + P : l VP + P : - Vc 2 + P, 
a b c 
viz. these are as the axes of the confocal surface 
x 2 
a 2 + £ + 
y 2 z 2 
b 2 + %* c 2 + P 
-1 = 0 
divided by a, b, c respectively; to fix the absolute magnitudes 
equation may be written 
x 2 + y 2 + z 2 — m — 
y 2 
b 2 + £ 
+ 
observe that the 
viz. the surface F= 0 is a surface through the spheroconic which is the intersection 
of the confocal surface by the arbitrary sphere x 2 + y 2 + z 2 — m= 0; but, while the 
surface is hereby and by the preceding condition as to the axes completely determined, 
the geometrical significance is anything but clear. 
80. Considering then the quadric surface F = 0, depending on the parameters £, m; 
suppose that m remains constant while £ alone varies; we have thus three consecutive 
surfaces V = 0, V' = 0, V" = 0; and these I say intersect in a point of the centro-surface ; 
the point in question will depend on the two parameters (£, m), and if these vary 
simultaneously we have the whole system of points on the centro-surface; but if only 
one of them varies, the other being constant, we have a curve on the centro-surface. 
The three equations may be replaced by V= 0, S^F=0, Sf 2 F=0; of which the first 
alone contains m ; and it thus appears that if m be the variable parameter, the equations 
of the curve are ^F=0, ^ 2 F=0, viz. the curve is then the quadriquadric curve which 
is the concomitant centro-curve of the curve of curvature for the parameter £. But if