432 
[75 
75. 
ON THE ATTRACTION OF AN ELLIPSOID. 
[From the Cambridge and Dublin Mathematical Journal, vol. iv. (1849), pp. 50—65.] 
Part I.—On Legendre’s Solution of the Problem of the Attraction of an 
Ellipsoid on an External Point. 
I propose in the following paper to give an outline of Legendre’s investigation 
of the attraction of an ellipsoid upon an exterior point, [“ Mémoire sur les Intégrales 
Doubles,” Paris, Mem. Acad. Sc. for 1788, published 1791, pp. 454—486], one of the 
earliest and (notwithstanding its complexity) most elegant solutions of the problem. 
It will be convenient to begin by considering some of the geometrical properties of a 
system of cones made use of in the investigation. 
§ 1. The equation of the ellipsoid referred to axes parallel to the principal axes, 
and passing through the attracted point, may be written under the form 
l (x — a) 1 + to (y — b) 2 + n(z — cf — k = 0, 
(where are the semiaxes, and a, b, c are the coordinates of the 
attracted point referred to the principal axes). Or putting la? + mb 2 + nc 2 — k = 8, this 
equation becomes 
lx 2 + my 2 + nz 2 — 2 {lax + mby + ncz) + 8 = 0. 
The cones in question are those which have the same axes and directions of 
circular section as the cone having its vertex in the attracted point and circumscribed 
about the ellipsoid. The equation of the system of cones (containing the arbitrary para 
meter ro) is 
(lx 2 + my 2 + nz 2 ) S — (lax + mby + ncz) 2 + or (x 2 + y 2 + z 2 ) = 0 ; 
or as it may also be written, 
(to 2 + l% — l 2 a 2 ) x 2 + (or + mb - m 2 b 2 ) y 2 + (o> 2 + nl — n 2 c 2 ) z 2 - 2mnbcyz — 2nlcazx - 2lmabxy = 0.