► 
of the Equinoxes, 
% 
Hence, as R?7 or -j expresses the number of particles in Rf 
or v\, it follows that the force of all the particles in RI, to 
turn the ellipse, is — x RC xRI x x RC x RI 2 . In the 
same way it may be proved that the force of all the particles 
in VR, to turn the ellipse, is ~ x RC x RV 1 . But the force ol 
the particles in RI tends to turn the ellipse upwards in the 
direction FAD, and the force of the particles in RV tends to 
turn it downwards in the contrary direction DAF. The force 
of all the particles in VI, therefore, to turn the ellipse, is 
— x RC x RV 2 —RF. But as TV is half the sum of RV, RI, it 
2 a 
follows that RT is half their difference, and therefore RV*— 
RI*= VI x 2RT. Consequently the force of the particles in 
VI, to turn the ellipse, is ™ x RC x VI x 2RT = — x RC x VI 
x RT. 
13. Let c = CG, b = CD,/ = GH, g = CH, y = CR, and 
x — CT. Then by similar triangles, c:f::x:y = ^; and 
c : g : : x : ^ = RT. Also, by the property of the ellipse, 
C : b* : : GTxTK : TV 2 : : 7+7 x 7 ~^x : TV* : : 6*— x' : TV*. 
Consequently TV =yv / c“ — x\ and VI = — >/ x a , and 
the force of the particles in VI, to turn the ellipse, is ™- xjy x 
— V 6'* — x 2 x — = zhd ^f- \/ c z ~x z , by putting ~ for y. The 
fluxion of this force is therefore f . \/ c*—x z xj) = - hd ^ gx 
ac 3 J ac 3 
s/ C — , 
■X 2 X 
jf zbdffgx* X 
c ac 4 
V c* — x*, as — =y. The fluent of 
this expression may be found in the following manner. 
14. The fluxion of ixc s — x 2 j 2 is x x r a — afl 2 -j- J- x x x 
c' — x* i 2 x — 2,xx = — 3/ c* — x* x x* x=x xc* 
