ON THE CENTRO-SURFACE OF AN ELLIPSOID.
[From the Transactions of the Cambridge Philosophical Society, vol. xn. Part i. (1873),
pp. 319—365. Read March 7, 1870.]
The Centro-surface 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 that 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 centro-surface 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 taken to be
X— + = 1; 2°, when the Absolute is the surface X 2 + Y 2 + Z 2 ■f W 2 = 0, and the
a 2 b 2 c 2
equation of the quadric surface is taken to be aX 2 + (3Y 2 4- yZ 2 + 8IF 2 = 0: in the first
of them by Salmon, Quart. Math. Jour. t. ii. pp. 217—222 (1858), and in the second by
Clebsch, Crelle, t. lxii. pp. 64—107 (1863): see also Salmon’s Solid Geometry, 2nd Ed.
1865, pp. 143, 402, &c. In the present Memoir, as shown 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 is further developed in various respects, and in particular
as regards the nodal curve upon the centro-surface: the distinction of real and
