[734 
735] 
111 
735. 
NOTE ON THE THEORY OF APSIDAL SURFACES. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvi. (1879), 
pp. 109—112.] 
I obtain in the present Note a system of formulae which lead very simply to 
the known theorem, that the apsidals of reciprocal surfaces are reciprocal; or, what is 
the same thing, that the reciprocal of the apsidal of a given surface is the apsidal 
of its reciprocal; the surfaces are referred to the same axes, and by the reciprocal is 
meant the reciprocal surface in regard to a sphere radius unity, having for its centre 
unction a determinate point, say the origin; and it is this same point which is used in the 
construction of the apsidal surfaces. The apsidal of a given surface is constructed as 
follows; considering the section by any plane through the fixed point, and in this 
section the apsidal radii from the fixed point (that is, the radii which meet the curve 
at right angles), then drawing a line through the fixed point at right angles to the 
plane, and on this line measuring off from the fixed point distances equal to the 
apsidal radii respectively, the locus of the extremities of these distances is the apsidal 
surface. We have the surface, its reciprocal, the apsidal of the surface, the apsidal of 
the reciprocal; and I take 
(x, y, z), (x, y, z), (X, Y, Z), (.X', Y', Z') 
for the coordinates of corresponding points on the four surfaces respectively. 
The condition of reciprocity gives xx' + yy' + zz' — 1 = 0, and (the equations being 
¿7=0, U' = 0) x', y', z' proportional to d x U, d y U, dJJ, and x, y, z proportional to 
cl X ’U', dy’TJ', d Z ’U'; or, what is the same thing, we must have 
x'dx + ydy + z'dz = 0 and xdx'+ ydy + zdz = 0; 
one of these is implied in the other, as appears at once by differentiating the equation 
xx' + yy + zz' — 1 = 0.