28 
Mr. Ivory on the Method of computing 
when the attracted point is in the surface then r =r p, and the 
preceding expressions for s and (~) will become 
«-t- 1 
5 ~f' { 2P 2 ( 1 — *)}—.*. 
(§) = 7 T'f' {2p'(i-7)}~ r -dm + {n + i) . (r-p) . 
f-f- % • dm 
observing that the second term on the right-hand side of the 
latter formula is to be valued on the supposition of r — p = o: 
therefore, by combining the two formulas, we shall get 
(%) — ^ •*=(» +i )•('•— -dm. (F) 
When n is equal to unit or greater than unit, it is plain that 
the quantity under the sign of integration in equation (F) will 
have a finite value at both the limits ; and therefore, on ac- 
count of the vanishing factor (r — p), that side of the equation 
will be equal to nothing. Consequently by putting a for p, 
which is permitted (because s is of the order a), we shall get 
f . s = o : whence it follows that the value of the 
\ dr 1 2 a 
function . V will depend only upon the sphere 
whose radius is p ; since the part of that function which is 
derived from the difference between the spheroid and sphere 
has been proved to be equal to nothing; which is in other 
words the theorem of Laplace. 
The demonstration we have just gone through is drawn 
from the same considerations as that contained in the Mecanique 
Celeste, from which it does not differ so much in spirit as in 
the manner of stating the reasoning. It must therefore be 
admitted that, when the exponent of the law of attraction is 
