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take in OE produced a point P such that OP shall be to OE 
asp toc. Then, if a right line drawn from P touch the focal 
ellipse in the point T, and if the angle OPT be denoted by @, 
it will be found that 
Ve — & 
cos § = —— 
p 
—— OSs ¢. 
Suppose that the point P moves to p, when ¢ is changed into 
~ +d. 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 @. 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 ¢, and if s denote the 
difference between the tangent PT and the elliptic are ET, 
while s + ds denotes the difference between the tangent pt and 
the are Eé, 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 #. The integral, beginning when # = 90°, or p =¢, is 
therefore proportional to s. At the other limit we have ¢ = 0, 
and p= b, which determines the extreme position of the point 
T. The difference between the tangent PT and the elliptic 
are 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 =V 8 — (B — c*) cos *, p =V0 + (a& — B*) cos *6, 
and calling E the extremity of the primary axis of the hyper- 
bola, we take in OE a point P (which will lie between O and 
E) such that OP shall be to OE as p tod. Then, drawing 
