520] ON THE CENTRO-SURFACE OF AN ELLIPSOID, 
321 
or, what is the same thing, 
, = x(l + |), ,-r(l + £). 1 + 1). 
Suppose now that the normal meets the consecutive normal, or normal at the 
point X + dX, Y+dY, Z+dZ; and let x, y, z belong to the point of intersection of 
the two normals; we must have 
0 = rfx(l + ^) + ^X, 
0 = dr(l + ±) + ? d x, 
0 = dZ (l+^ + ?d\, 
which determine the direction of the consecutive point; the equations in fact give 
0 = 
dX, 
dY, 
dZ, 
or, what is the same thing, 
dX 
X 
a? ’ 
a? 
dY 
Y 
b 2 ’ 
6 2 
dZ 
£ 
c 2 ’ 
c 2 
dX, 
X 
dY, 
Y 
dZ, 
Z 
curve 
of 
therefore be satisfied by taking for X + dX, Y+dY, Z+dZ, the coordinates of the 
consecutive point along either of the curves of curvature,—say along that which is 
the intersection with the surface 
Y 2 Z*_ _ 
a? + b 2 + r\ c 2 + rj 
9. To verify this, observe that we then have 
XdX YdY ZdZ _ 
a 2 + ¿2 + C 2 ~ U > 
XdX YdY ZdZ 
a 2 + 97 + b 2 + rj **" c 2 + 77 ’ 
or, what is the same thing, 
XdX : YdY : ZdZ=a*(a* + 77) a : & 2 (& 2 + t?)0 : c 2 (c 2 + V )r 
C. VIII. 
41