96 Mr. Ivory on the figure requisite to maintain the equilibrium 
The integrations with respect to d R' should be executed from 
R' = o, to the value of R' at the surface of the body ; which 
cannot be done, because all the terms after the two first 
would be infinite. Conceive a sphere to be described about 
the origin of the co-ordinates with the radius r; then the 
whole value of the function V ( r) will be equal to its value 
with respect to the sphere added to its value with respect to 
the matter between the sphere and the surface of the body. 
The attracted point being in the surface of the sphere, the 
first part of the value of V ( r) will be equal to -LL x ^ ; and 
the second part will be found by integrating the foregoing 
expression, so that all the integrals shall vanish when R' = r. 
Thus we get, 
V(r) =AZ-r*+ffi (R' s — r*)dfidx' 
+ rff (R' — r) . C {l) dfi dir' 
+ r a JJ [Log. R' — Log. r) . d d -us' 
+ i ff (“ “ R 7 ) • C(3> d P 
+ r 
-f- &c. 
Now the integraf — f ffr^tT^ dii', taken between the limits 
^ — 1 , ^ — l ; and = 0,^=2^; is equal to — 2 7rr s . 
All the other parts of the above expression that contain r, are 
evanescent ; because we have generally, 
JJ& ] dd — 0. 
In order to prove this, it is to be observed that \n! and y 
are the cosines of the three sides of a spherical triangle, and 
37 — is the angle opposite to the side whose cosine is y : 
