130 
[444 
444. 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
[From the Proceedings of the London Mathematical Society, vol. ill. (1869—1871), 
pp. 16—18.] 
The President [Prof. Cayley] gave an account of his investigations on the centro- 
surface of an ellipsoid (locus of the centres of curvature of the ellipsoid). The surface 
has been studied by Dr Salmon, and also by Prof. Clebsch, but in particular the theory 
of the nodal curve on the surface admits of further development. The position of a 
point on the ellipsoid is determined by means of the parameters, or elliptic coordinates, 
h, k\ viz., if as usual a, h, c are the semi-axes, and if X, Y, Z are the coordinates of 
the point in question, then 
_X^ _F>_ = 
n 2 + h b 2 4- h c 2 + h 
X 2 _Yf Z 2 
a 2 + k b 2 4- k c 2 -f k ’ 
and hence 
— /3yX 2 = a 2 (a 2 + h) (a 2 + k), 
— yaY 2 = b 2 (b 2 4- h) (b 2 + k), 
— a/3Z 2 = c 2 (c 2 4- h) (c 2 4- k), 
if for shortness 
a — b 2 — c 2 , /3 = c 2 — a 2 , 7 = a 2 — 6 2 , (a 4- ft + 7 = 0). 
This being so, the coordinates of the point of intersection of the normal at (X, Y, Z) 
by the normal at the consecutive point of the curve of curvature 
X 2 F 2 X 2 =1 
a 2 4- k + b 2 4- k^ c 2 4- k