Mr. Ivory on the Attractions 
lienee 
( = __ ■ 
l dk J (A 2 + e*)i (A 2 + c'*)± 
o 
S'ZtT 
x/< 
(*‘ + «*h (** + or 
the fluent to be taken so as to vanish when k is infinitely great; 
because Q decreases, when k increases, and the former quan- 
tity is infinitely small, when the latter is infinitely great. 
From this value of O, we get 
I I d CM C — dk 
e \ de j 
f d( V \ 
e' 1 de 1 I 
2 'ST x 
r x 
f 
f\* + e*)i (A 2 + e' 2 )| 
(A 2 + e 2 )| (A 2 + 
— dk 
1 / d Q~\ 2/ar 1 
“ T 1 IF J “ T • (A 2 + «*>£ (A 2 + *' 2 )i 
and from these it is easy to infer that 
“ T [lk) + V (if) + T ( TiH == 2<ar ></a 2 (A 2 + e 2 )| (A 2 + «'*)£ ' 
— rfA 
therefore, if M = ^ . kk’k" — the mass of the ellipsoid, the 
last formulas for A, B, C will become, by substitution, 
A = 3<7M x/ A 2 (A 2 + e 2 )i (A 2 + e' 2 )| 
B = 36 M x/ + e n| (A 2 + oi (7) 
C = gcM + e *)± (A 2 -f : 
all these different fluents are to be conceived, as beginning to 
increase when £ is infinitely great, and are to be extended till 
£ has decreased, so as to be equal to the least of the serai-axes 
of the ellipsoid. In the general case of the problem, the expres- 
sions that have been obtained transcend the limits of the or- 
dinary analysis ; and their integration requires the introduction 
of other quantities besides algebraic expressions and circular 
arcs and logarithms. They belong to the class of elliptical 
transcendants ; a branch of the mathematics which has been 
