318 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
[520 
As usual, a is taken to be the greatest and c the least of the semi-axes; we have 
thus a, 7 each of them positive, and /3 negative, = — /3' where ¡3' is a positive quantity 
= a. + 7. A distinction arises in the sequel between the two cases a 2 + c 2 > 2b 2 and 
a 2 + c 2 < 2b 2 , but the two cases are not essentially different, and it is convenient to 
assume a 2 + c 2 > 26 2 , that is, a 2 — b 2 > b 2 — c 2 or 7 > a, say 7 — a. positive. The limiting 
case a 2 + c 2 = 26 2 or 7 = a requires special consideration. 
3. We have 
— /87 X 2 = a 2 (a 2 +(a 2 + 77), 
— 7a F 2 = 6 2 (6 2 + £) (6 2 + 77), 
— a/3 Z 2 = c 2 (c 2 + £) (c 2 + 77). 
It is in fact easy to verify that these values satisfy as well the equation of the 
ellipsoid as the assumed equations defining the elliptic coordinates £, 77. We may also 
obtain the relations 
X 2 + Y 2 + Z 2 = a 2 + b 2 + c 2 + £ + 77, 
a 2 X 2 + b 2 Y 2 + c 2 Z 2 = a 4 + 6 4 + c 4 + b 2 c 2 + c 2 a 2 + a 2 b 2 + (a 2 + b 2 + c 2 ) (f + 77) + £77. 
These, however, are obtained more readily from the equation in v, viz. the roots 
thereof being f, 77, we have 
-%- v = a 2 + b 2 + c 2 -X 2 -Y 2 -Z 2 , 
£77 = 6 2 c 2 4- c 2 a 2 + a 2 b 2 - (b 2 + c 2 ) X 2 — (c 2 + a 2 ) Y 2 — (a 2 + 5 2 ) F 2 , 
which lead at once to the relations in question. 
4. Considering f as constant, the locus of the point (X, Y, Z) is the intersection 
of the ellipsoid with the confocal ellipsoid 
Z 2 F 2 Z 2 
a 2 + f + b 2 + % + c 2 + %~ ; 
viz. this is one of the curves of curvature through the point; and similarly considering 
77 as constant, the locus of the point is the intersection of the ellipsoid with the con- 
focal ellipsoid 
X 2 F 2 Z 2 _ 
a 2 + 77 b 2 + 77 c 2 -f- 77 ’ 
viz. this is the other of the curves of curvature through the point. 
5. If instead of £ and 77 we write h and k, we may consider h as extending 
between the values — a 2 , — b 2 , and k as extending between the values — b 2 , — c 2 . 
h = const, will thus give the series of curves of curvature one of which is the 
section by the plane X = 0, or ellipse semi-axes b, c ; say this is the minor-mean 
series. In particular h= — a 2 gives the ellipse just referred to; and h = — b 2 , or say 
h = — b 2 - e, gives two detached portions of the ellipse semi-axes a, c; viz. each of these 
portions extends from an umbilicus above the plane of xy, through the extremity of 
the semi-axis a, to an umbilicus below the plane of xy.