BY SUEFACES OF A HYPOTHETICAL CLASS INCLUDING THE ELLIPSOID. 
0 
10. The preceding articles contain all that is essential. But it may be as well to 
deduce the expression for the potential, on an exterior point, of the finite homogeneous 
shell 
Let I, ri, ^ be the coordinates of the attracted point. The expression (E.), art. 8, is 
a function of h, and through h a function of k ; for A, at the lower limit of the integral, 
is a function of |, tj, determined by the equation (art. 3) 
f(^, /i, /t)=0 ( 10 .) 
(1 assume, for simplicity, that is independent of 7r.) We have then to integrate (E.) 
with respect to from to Now putting F(/i) for the integral in (E.), we have 
and, between the limits 77, A", this gives 
J^'F(70(77:=FF(7i'')-7:'F(7f)-j;::'7:F(/0f7A^ 
where A", A' are the values of A corresponding to A", k\ and given by the relation 
dh 1 
f(|, 4 A, A)=0. The actual value of F(A) is ( and therefore F(7i)=— ; 
hence the required potential is 
w+iwl’ 
where k, in the last integral, is the function of A determined by the equation (10.). 
11. To verify this in the case of the ellipsoid, we have 
y 
■-k-, 
here =^(7i), and Q|-l-4^^=0, whence w= 4 ; therefore 
•v/y( 7 i)=£”*^*^^ ^ =i^cd-\-A){If-FA){c‘^-\-A^^*. 
Also A^ = CO. Let us take 7:'=0, F=l, 7q = 0, so that the formula (P.) will give the 
potential, on an external point (|, ??, t), of the homogeneous solid ellipsoid 
Then ■>^(I)^)=abc ; and A, k being now connected by the equation 
« 2 y2 
^ ^ I ^ — h 
d^ + h^d^ + h'c^ + h ’ 
we have 7i'= oo, and A''= the positive root of this equation when 7:=1. The expression 
(P.) then becomes 
fi 
— '!rabc\ - 
Jh" 
2 ^2 
+ - 
2 + A^ Z.2 + /7 ‘ C^ + h 
■1 ]dh 
((a2 + A)(62 + ^)(c" + -^))^ ’ 
which is the well-known value of the potential. 
* The arbitrarj" constant, which might he introduced, would disappear in the result. 
C 
MDCCCLX. 
