159 
Communications made to the Society : 
(1) On the Centrosurface of an Ellipsoid. 
By Prof. Cayury. 
The centrosurface of any given surface is the locus of the 
centres of curvature of the given surface—or say it is the locus 
of the intersections of consecutive normals, (the normals which 
intersect the normal at any particular point of the surface being 
those at the consecutive points along the two curves of curvature 
respectively which pass through the point on the surface). The 
terms, normal, centre of curvature, curve of curvature, may be 
understood in their ordinary sense or in the generalised sense 
referring to the case where the Absolute (instead of being the 
imaginary circle at infinity) is any quadric surface whatever: 
viz. the normal at any point of a surface is here the line joining 
the point with the pole of the tangent plane in respect of the 
quadric surface called the Absolute; and of course the centre 
of curvature and curve of curvature refer to the normal as just 
defined. 
The question of the centrosurface of a quadric surface has 
been considered in the two points of view, viz. 1°, when the 
terms “normal” &c. are used in the ordinary sense, and the 
equation of the quadric surface (assumed to be an ellipsoid) is 
2 2 2 
taken to be*, + S, i 
face X*+Y*+Z*+ W?=0, and the equation of the quadric 
surface is taken to be aX? + BY? +yZ?+5W?=0 :—in the first 
of them by Salmon, Quart. Math. Jour. t. 1. pp. 217-222 
(1858), and in the second by Clebsch, Crelle, t. 62, pp. 64-107 
(1863). See also Salmon’s Solid Geometry, 2nd Ed. 1865, pp. 143, 
402, &c. In the present memoir, as shewn by the title, the 
quadric surface is taken to be an ellipsoid; and the question is 
considered exclusively from the first point of view: the theory 
=1. 2’, when the Absolute is the sur- 
is further developed in various respects, and in particular as 
13—2 
