7^> Mr. Robertson on the Precession 
respect, and all other circumstances being the same, the forces 
themselves must be equal. Now at either of the equinoxes 
the other circumstances are exactly the same in the two figures. 
At the vernal equinox, for instance, the straight line SACB in 
Fig. g. must be in the plane of the equator, and therefore the 
plane DCF, perpendicular to AB, at this time must pass through 
the poles P, L. At the equinox, therefore, the straight line 
SACB, and the plane DCF in Fig. g. are justly represented' 
by ACB, and DMEL in Fig. 10. Hence the librating force 
T x ~~j~~ x ^ ^ * Z at the commencement of its action, at the 
equinox, applies to Fig. 10, and at its commencement it is 
equally efficacious to cause revolution about DE or ML. We 
are therefore enabled to compare the effect of the librating 
force, or the revolution it is capable of producing, at the 
equinox, about ML, with the diurnal revolution of the earth' 
about DE, in the following manner. 
It being admitted that each of the two forces, stated in the be- 
ginning of the article, produces the same angular velocity, then- 
— x x ci — Fx Z — ~Z, and therefore d x rnn x * ~ e — v. 
a 5 5 . a 
But if a constant force act for a given time i, and cause the 
body to move on which it acts, the velocity generated from 
the commencement of the motion is as the force. Conse- 
quently, as v denotes the force acting on a particle at A, during 
the given time /, and as the forces acting, on the other particles 
of the spheroid are proportional to their distances from the 
plane DMEL ; the angular velocity of A, acquired in the given 
time i, is also accurately expressed by v. If therefore the 
force j Z cease to act at the end of the given time i, the point 
A y as the spheroid is in free space, will afterwards revolve 
