520] 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
319 
And in like manner k = const, gives the series of curves of curvature one of which 
is the section by the plane Z — 0, or ellipse semi-axes a, b; say this is the major-mean 
series. In particular k= — c 2 gives the ellipse just referred to; and k= — b 2 , or say 
k — — b 2 + e, gives the remaining portions of the ellipse semi-axes a, c; viz. these are 
two portions each extending from an umbilicus above the plane of xy, through the 
extremity of the semi-axis c, to an umbilicus above the plane of xy. 
The ellipse last referred to may be called the umbilicar section, the other two 
principal sections being the major-mean section and the minor-mean section respectively. 
In the limiting case h = k= — b 2 , we have the umbilici, viz. these are given by 
X 2 
a 2 
1 
P’ 
a 
P' 
The two series of curves of curvature cover the whole real surface of the ellipsoid; 
so that at any real point thereof we have £ = h, r) = k, or else f = k, r) = h, where h, k 
are negative real values lying within the foregoing limits — a 2 , — b 2 and — b 2 , — c 2 
respectively. But observe that f, rj taken separately may each extend between the 
limits — a 2 , — c 2 . 
6. Suppose % = r], the equation in v will have equal roots, or the condition is 
(P — X 2 —Y 2 — Z 2 ) 2 = 4 {Q- (b 2 + c 2 ) X 2 - (c 2 + a 2 ) Y 2 - (a 2 + b 2 ) Z 2 ), 
viz. this surface by its intersection with the ellipsoid determines the envelope of the 
curves of curvature. This envelope is in fact a system of eight imaginary lines, four 
of them belonging to one of the systems of right lines on the ellipsoid, the other four 
to the other of the systems of right lines. For in the values of X 2 , Y 2 , Z 2 writing 
V — we find 
± \Z_/3y^ = a 2 +£ 
y 
± V - 7 a -^ = & 2 + £. 
y 
+ V — a/3 - = c 2 -I- f, 
c 
or representing for shortness the left-hand functions by + X', + Y', + Z', the eight lines 
are 
tt" + £ — 
X' 
= X' 
= -X' 
= -X' 
b 2 + Z = 
Y' 
-- Y' 
= Y' 
= - r 
c 2 + £ = 
Z' 
= -Z' 
= -Z' 
= Z' 
a 2 + % = 
-X' 
= X' 
= X' 
= -X' 
b 2 + g = 
Y' 
= — Y' 
= Y' 
= - V 
c 2 + f = 
Z' 
= Z' 
= -Z' 
= -z\