1 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
reduction of some complexity) assumes ultimately a form which is very simple and 
remarkable; viz., writing 
P = a 2 + b 2 + c 2 , Q = b 2 c 2 + c 2 a 2 + a 2 b 2 , R = a 2 b 2 c 2 , 
the relation is 
(6P + 3 Qh + Ph 2 ) 
+ \ (3Q + 4Ph + 3h 2 ) 
+ h 2 (P 4- 3 h ) = 0; 
this is a (2, 2) correspondence between the two parameters A, h x ; the united values 
h x = h, are given by the equation 6 (R + Qh + Ph 2 + A 3 ) = 0, that is 
(a 2 + A) (b 2 + A) (c 2 + A) = 0; 
viz., the two points on the ellipsoid which have their common centre of curvature on 
the nodal curve are only situate on the same curve of curvature when this curve is 
a principal section of the ellipsoid. 
{Since the date of the foregoing communication, Prof. Cayley has found that the 
squared coordinates x 2 , y 2 , z 2 of a point on the nodal curve can be expressed as 
rational functions of a single variable parameter <r.}