368 



take in OE produced a point P such that OP shall be to OE 

 as p to c. Then, if a right line drawn from P touch the focal 

 ellipse in the point T, and if the angle OPT be denoted by Q, 

 it will be found that 



cos B = ; cos ^. 



p 



Suppose that the point P moves to p, when (p is changed into 

 <p -\- d(j». Then it may be easily shown that the attraction of 

 dM upon the point C is proportional to the interval Pp mul- 

 tiplied by the cosine of 9. In this form the differential of the 

 attraction is immediately integrable. For if from p we draw 

 a right line pt touching the ellipse in t, and if s denote the 

 difference between the tangent PT and the elliptic arc ET, 

 while s + ds denotes the difference between the tangent ^^^ and 

 the arc E#, it will appear, by a lemma which I have fre- 

 quently had occasion to use (see the Proceedings of the Aca- 

 demy, vol. ii. p. 508), that ds is equal to Pp multiplied by 

 cos 6. The integral, beginning when = 90°, or p = c, is 

 therefore proportional to s. At the other limit we have <p zz 0, 

 and p=b, which determines the extreme position of the point 

 T. The difference between the tangent PT and the elliptic 

 arc ET, corresponding to this position, is to be multiplied by 

 a certain function of the semiaxes, in order to get the whole 

 attraction of the ellipsoid on the point C. 



" When the attracted point is at B, the extremity of the 

 mean axis, we proceed exactly as before ; but instead of the 

 focal ellipse we make use of the focal hyperbola, whose plane 

 is at right angles to OB. Putting now 



p = ^/b^ _ (62 _ c') cos V, p' = \/6" + {a" - 6^) cos V, 



and calling E the extremity of the primary axis of the hyper- 

 bola, we take in O E a point P (which will lie between O and 

 E) such that OP shall be to OE as p to b. Then, drawing 



