of the Equinoxes. 
*7& 
■t & 
t- we are to put — x a • — u*, and instead of X we are to put 
X. These substitutions being made, we have x — x 
a* 
a z — e 2 
— 5 a* — u 
x a — u x — r~ 
X for the sun’s disturbing force on the 
-ellipse HKNG, tending to turn the ellipse about an axis 
passing through C and perpendicular to the plane PELO. 
This expression for the force being multiplied by u, gives the 
fluxion of the force on that part of the spheroid between the 
ellipses PELQ, HKNG ; and the fluent of this, when u be- 
X. Consequently the 
t . d mn a 1 — e z 
comes equal to a is — x — x — t- x 
1 a 4 or 
8a* 
~ 
4« 
double of this, viz. — x mn x a 2 — e 2 xpX expresses the sun’s 
disturbing force on the whole spheroid. Hence if Z = yX, 
which expresses the solid content of the spheroid, the force on 
■the whole spheroid is y x — x — e * x Z. Let this be called 
the librating force or pressure, or the force causing libration. 
1 7. It is evident, from the manner in which the librating 
pressure is calculated, that the whole of the disturbing force 
is occasioned by the protuberance of the spheroid above the 
greatest inscribed sphere. For if PELQ were a sphere, as VI 
(Fig. 7. ) is parallel to the diameter DF, and AC perpendicular 
to it, the straight line VI (3. III. ) would be bisected in R ; and 
therefore the disturbing forces, above and below AC would 
exactly counteract one another. 
Let DCF (Fig. 9.) denote a plane perpendicular to the 
straight line SC, then it is evident that the librating pressure 
tends to move the earth about that diameter of the equator, 
which is the common section of the equator and the plane 
DCF. For the sake of precision hereafter let this diameter 
of the equator be called the axis of libration. 
L 
MDCCCVII. 
