132 Mr. Ivory on the figure requisite to maintain the equilibrium 
ellipsoid ; and farther, that an ellipsoid may be found that 
will fulfil all the conditions of the equilibrium, unless there 
be some cases in which the necessary relations between the 
figure and the given velocity of rotation, lead to equations 
that cannot be solved. 
6 . In order to apply the foregoing solution, it becomes ne- 
cessary to compute the integrals L, M, N ; or, in other words, 
to find the attractive forces of an ellipsoid upon a point in 
the surface. If we extend the problem generally to a point 
within or without the figure, it is attended with some diffi- 
culty ; and it is usual to deduce the latter case from the 
former, which is more easily solved. There is however a 
great analogy between the two cases ; or rather the distinc- 
tion between them may be dispensed with ; since the suppo- 
sition of a point within the figure is equivalent to that of a 
point in the surface, which is the extreme case of a point 
without the figure. In this view the problem admits of a 
general solution deducible, by a short analysis, from the 
transformations used in this Paper. 
Suppose that k, jp, k", represent the semi-axes of an ellip- 
soid ; and let x,y, z, respectively parallel to the axes, denote 
the three co-ordinates of a point in the surface of the figure. 
Farther let a, b, c , be the co-ordinates of an attracted point 
without the figure ; and conceive another ellipsoid, the sur- 
face of which passes through the attracted point, and which 
has its principal sections in the same planes with the principal 
sections of the given ellipsoid, and also the differences of the 
squares of its semi-axes h, h h ", equal to the differences of 
the squares of k, V, k" ; that is, h 12 — li = k'* — k 2 , and h" 2 — h 2 
z=k" 2 —k 2 . The equations of the two curve surfaces will thus 
be, 
