436 
ON THE ATTRACTION OF AN ELLIPSOID. 
[75 
The substitution of the above values of P, Q, R (a somewhat tedious one which 
does not occur in the process actually made use of by Legendre) gives the very 
simple result, 
P*co dco dcf> 
© 
and the formula for the attraction becomes 
which is of the form A = 2 Ico 2 dco, where 
which last integral, taken between the limits cf> = 0 and = 27r, and multiplied by 
2a)' 2 dco, expresses the attraction of the portion of the ellipsoid included between two 
consecutive cones. The integration is evidently possible, but the actual performance of 
it is the great difficulty of Legendre’s process. The result, as before mentioned, is 
independent of the quantity k, or, what comes to the same thing, of the quantity S : 
assuming this property (an assumption which in fact resolves itself into the consideration 
of the ellipsoid for which the cone reduces itself to a straight line, as before explained), 
the integral is at once obtained by writing S = A where A represents the positive root 
of the equation 
l 2 a 2 m 2 b 2 
_i u 
co 2 + ZA <u 2 + mA a) 2 + n/\ 
Q = hnab (to 2 + nA), 
R = Inac (o> 2 + mA), 
values independent of cf>, or the value of I is found by multiplying the quantity under 
the integral sign by 27t : and hence we have 
. . . f co 2 (co 2 -f - ZA) 2 (<o 2 -f- mA)“ (<u 2 ■+■ ?iA)^ dco 
77 ( J l s a 2 (co 2 + mA) 2 (or + nA) 2 + m 3 b 2 (co 2 + nA) 2 (co 2 + ZA) 2 + n s c 2 (co 2 + nA) 2 (co 2 +1A) 2 ’ 
where of course A is to be considered as a function of co. By integrating from 
co = to, to co = co u , we have the attraction of the portion of the ellipsoid included 
between any two of the series of cones, and to obtain the attraction of the whole 
ellipsoid we must integrate 
by the equation 
where £ is determined as before