108] 
CONNECTED WITH THE THEORY OF ATTRACTIONS. 
39 
The only obviously integrable case is that for which in the second formula q = 1; 
this gives 
f _ 2 Tr^ n fj 
J (x 2 +?/ {(x - af + {y- by + u 2 }^- 1 ~ r (in) (l 2 + u 2 f n ~ l (l 2 + e)*”“ 1 ' 
In the case of u = 0, we have, as before, when p 2 —f 2 and l 2 — fy are of opposite 
signs, e = 0, and therefore 1 + e = 1 ; but when p 2 —f 2 and l 2 — f 2 are of the same 
sign, the value before found for e gives 
1 +«={*■/.*+(p* -/.*) №■ -/.*))• 
Consider the image of the origin with respect to the sphere (x - a a ) 2 + (y - bj) 2 ... —f 2 , 
the coordinates of this image are 
Oj 
hi 
w-m 
and consequently, if p be the distance of this image from the point (a, b ...), we have 
=¿f W + ipt-fÔW-M; 
whence, by a simple reduction, 
or the values of the integral are 
P* ~fi and li—fy opposite signs, I = ^ - 7 —-¡r- 
J ° r(^w) l n ~ 2 {l 2 ~f 2 )’ 
p 2 —fi 2 and ly-fi 2 the same sign, I = ^ .. , 
J J 5 ’ T Qn) k n ~ 2 g n ~ 2 (l 2 ~f 2 )’ 
where p is the distance from the point (a, b...) of the image of the origin with respect 
to the sphere (x — a^ 2 + ... —fy = 0. 
1 + e: 
l ±PL 
Pf 2 ' 
Stone Buildings, August 6, 1850.