of the Equinoxes. 73 
plane DMEL, let v be the force of a particle at A. Let p = 
the area of a circle whose diameter is x. Then 4 pae = the 
area of the ellipse DMEL, and as a + x x a — x — Ql--*-~x % — 
rK’, a • : a 2 — x 2 : : 4^>a<? : ^ x a'— x* = the area of the ellipse 
FKGH. Again, a : x : : v : ~ = the force of a particle at r, 
and therefore x a ' x — a; 3 = the force of all the particles 
in the ellipse FKGH. Now as this force acts at r, by the pro- 
perty of the lever, the power of the ellipse FKGH, to turn 
the spheroid, either about DE or ML as an axis, is x 
a 2 x 2 — x* ; and the fluxion of this force is x a 2 x 2 x — ■ x 4 x. 
a 
The fluent of this, when x becomes equal to a , is ~ >L, - ^ e v ; and 
the double of this, for the force of the whole spheriod, is 
l 6 ^- — . Hence it is evident that if the force of each particle 
in the spheroid, to cause a revolution, be as its distance from 
the plane DMEL, the particles on one side of this plane hav- 
ing a tendency to cause a revolution in one direction, and the 
particles on the other side of the plane having an equal tendency 
to cause a revolution in the same direction, then the pressure 
with which the spheroid is urged to revolve, either about DE 
or ML, is as . This force is equal to y Z, if Z be put 
equal to ' g , the solid content of the spheroid. 
ig. As the librating force ~ x y x a 2 — e 2 x Z, ascertained 
in article 16, and the force y Z, obtained in the last article, 
are calculated on the same hypothesis, viz. that the force of a 
particle is as its distance from the plane DCF in Fig. g, or the 
plane DMEL in Fig. 10, if they produce equal angular velo- 
cities, the spheroids in the two figures being equal in every 
L 2 
