434 
ON THE ATTRACTION OF AN ELLIPSOID. 
[75 
is that generated by the revolution of the curve round the line CM. The surface of 
the fourth order being once described for any particular value of co, the cone corre 
sponding to any one of the series of similar, similarly situated, and concentric ellipsoids 
is at once determined by means of the intersection of the ellipsoid in question with 
the surface of the fourth order. It is clear too that there is always one of these 
ellipsoids which has a double contact with the surface of the fourth order, viz. at the 
points where this ellipsoid is intersected by the normal to the confocal ellipsoid 
through the attracted point; thus there is always an ellipsoid for which the cone 
corresponding to a given value of co reduces itself to a straight line. 
Consider the attracting ellipsoid, which for distinction may be termed the ellipsoid 
8, and the two cones C, C, which correspond to the values co, co — dco of the variable 
parameter. Legendre shows that the attraction of the portion of the ellipsoid S 
included between the two cones C, C' is independent of the quantity Jc, which determines 
the magnitude of the ellipsoid: that is, if there be any other ellipsoid T similarly situated 
and concentric to and with the ellipsoid 8, and two cones B, B', which for the 
ellipsoid T correspond to the same values co, co — clco of the variable parameter; then the 
attraction of the portion of the ellipsoid 8, included between the two cones C, C, is 
equal to the attraction of the portion of the ellipsoid T included between the two 
cones B and B'. By taking for the ellipsoid T the ellipsoid for which the cone B 
reduces itself to a straight line, the aperture of the cone B' is indefinitely small, and 
the attraction of the portion of the ellipsoid T included within the cone B' is at once 
determined; and thus the attraction of the portion of the ellipsoid S included between 
the cones C, C' is obtained in a finite form. Hence the attraction of the portion 
of the ellipsoid 8 included between any two cones C t , C n corresponding to the values 
co,, co t/ of the variable parameter, is expressed by means of a single integral, and by 
attraction of the whole 
extending the integration from co = 0 to 
ellipsoid is obtained in the form of a single integral readily reducible to that given 
by the ordinary solutions. It is clear too that the attraction of the portion of the 
ellipsoid 8 included between any two cones C t , C /t , is equal to that of the portion 
of the ellipsoid T included between the corresponding cones B / and B /t . Hence also, 
assuming for the ellipsoid T, that for which the cone D„ reduces itself to a straight 
line, and supposing that the cones C / and B / coincide with the circumscribing cones, 
the attraction of the portion of the ellipsoid S exterior to the cone C u is equal to 
the attraction of the entire ellipsoid T. More generally, the attraction of the 
portion of the ellipsoid 8 included between the cones C / and C // is equal to the 
attraction of the shell included between the surfaces of the two ellipsoids, for which 
the cones B / and B u respectively reduce themselves to straight lines. 
§ 2. Proceeding to the analytical solution, and resuming the equation of the 
ellipsoid 
lx 2 + my 2 + nz 2 — 2 (lax + mby + ncz) + 8 = 0, 
and- that of the cone 
(lx 2 + my 2 + nz 2 ) 8 — (lax + mby + ncz) 2 + co 2 (x 2 + y 2 + z 2 ) = 0 ;