238 Mr. Ivory on the figure requisite to maintain the equilibrium 
To this expression we must join the equation of the surface 
of the ellipsoid that passes through the attracted point, viz. 
a 2 6 2 . c 2 
A* A 2 + e 2 « A 4 -f e ' 2 ” 1 ’ 
by means of which h, the limit of x in the several integrals, 
is to be determined. When the attracted point is in the 
surface of the given ellipsoid, it is plain that h — k; and the 
limit of x is, therefore, one of the semi-axes. 
Thus an expression of V(r) has been found, that is ge- 
neral for all positions of the attracted point ; nothing more 
being requisite than to determine the limit of x in every par- 
ticular case. The several integrals are closely connected 
with one another ; they are in forms well known to geo- 
meters, and susceptible of many transformations ; but, in a 
general solution, it seems most simple to leave the expression 
as it is above exhibited. 
But although a general expression of V(r) has been found, 
yet it does not immediately make known the attractive forces 
acting upon a point. These forces, estimated in directions 
parallel to the co-ordinates, are represented by the partial 
fluxions of V(r) relatively to the co-ordinates ; but, in per- 
forming the operations, it must be observed that x is a function 
of r, and consequently of the co-ordinates. Thus the attrac- 
tions of the ellipsoid, respectively parallel to a, b, c, are equal 
d . V (r) 
d . V (r) 
dx 
d a 
d x 
da 
d.V(r) 
d.V(r) 
dx 
db 
d x 
‘ db 
d.V(r) 
d . V (r) 
dx 
d c 
d x 
d c 
But, according to the foregoing value of V (r), 
d.V(r) 
d x 
f M 
s/(x* + t*) (a* + e ' 4 
1 
+ i + 
b* 
x 2 -f- e i 
