the Attractions of Spheroids of every Description . 
8i 
If we make \d — ■■■■ -~-f . — — the integral in the last ex- 
pression will be transformed into 
i_ f* x % . dx % i i [ dF\ 
V ef J j i +*'•**!* V e S * ' ldA ' 
therefore the value of this term is 
2 ?r . i j dF ^ 2v . b z 
""" *.T l ~ 3 ' 
And, by proceeding in a manner entirely analogous, it may 
be shewn that the remaining term multiplied by c 2 , is equal to 
27! , C 1 I I dF \ 2 7! . c 1 
~~ vtr ' ” i^'J r* 
If M denote the mass of the ellipsoid, then M = ~ . kk'k" 
= “7= ; and ~= = : therefore by collecting all the 
parts of V, into one sum, we have 
V — 3M F _ 3M | I, jl J_ ffi iJLfflV 
“ 2k * A ztf • 1 i /(i+A*).(i+*'*) T * [dx! O A ' [dxq J 
3M . 
+ 
2 k 3 
k Z -k Z 
•T (^) + 
3 M.C 1 
2k 3 
a , 
k ll7 -—k* 
- 1 — ) 
A' [(/*•/' 
/2 
= A . 
F = /* ^ ( from x = o,toi = i). 
J 
The case of an oblate elliptical spheroid of revolution cor- 
responds to the supposition of k'— k" or a = a': but in taking 
the partial fluxions of F we must attend to the peculiarity 
that takes place when x = a': for in general dF = dx 
(^) dx ' ; and hence when a™ A', dF = e[~) .dx: now when 
x = x', F = -i- . arc. tan. x ; consequently -j- (j|) = 7 (f ) 
* Mec. Cel. Liv. 3e, No. 3. 
M 
MDCCCXII. 
