$3 
extensive Class of Spheroids . 
B (i) = JJf R' ,+2 . dR' . d?' . dW . 
In this expansion B (o) , in every case is equal to the mass of 
the spheroid: and with regard to the second term, Laplace 
has remarked that it may be made to disappear by fixing the 
origin of R', which is an arbitrary point, in the centre of gra- 
vity of the spheroid. To prove this, we have 
B (,) =fff R' 3 • dR ! . df . d* r ' . Q 0) : 
but dM = R' 1 . dR 1 . df . d-m' ; and R' . Q (l) = R' . y = p x Ry 
-j- V 1 — jM> a . COS. w x R' . V 1 — . COS. ■sr' \/ 1 — p* . sin. -ur 
x R' . V l — . sin. •or — p x x -{- sf l — [*," . cos. sr x y -f- 
y/i — [S . sin. tff x % ; where x, y, z denote as before the co- 
ordinates of the molecule dM : therefore, by substitution, 
B (l) = p xfx- dM + Vi jl/C,* . COS. US' x/y. dM 
+ — [S . sin. w x J z . dM. : 
now, if all the planes to which x,y, z, are perpendicular pass 
through the centre of gravity ; then, by the nature of that 
point, fx . dM = o ; Jy . dM = o ; fz . dM = o : therefore 
B (l) = o.f 
In the expression of none of the integrations can be 
executed in a general manner, excepting that relative to dR' : 
let R denote what R' becomes at the surface of the spheroid ; 
then 
B (i) = ~ . // R' +3 . df! . «V . Q f0 . 
8. When the attracted point is within the spheroid, the 
value of V will be represented by a series of the ascending 
powers of r : let 
* Mec. Cel. Liv. je. No. 9. 
f Ibid. Liv. No. iz. 
