of the Equinoxes. 77 
with the uniform angular velocity v ; but by the doctrine of 
constant forces, the angle described by A, during the action 
a * 
r av r-r • 1 . V d Cl 
of — Z/ is equal to — == — x mn x - 
20 . Let AB ( Fig. 6. ) represent that diameter of the equator 
about which the librating force begins to cause revolution at 
the equinox. Let G be the centre of the earth, and in GB let 
GV be taken equal to — , or — x mn x — ■•. Let w denote the 
angular diurnal velocity of the earth about its axis DC : and 
in GC let GW be taken equal to w. The points V, W being 
joined, let TGS be drawn parallel to YW, and by article 6, 
TS is the axis about which the earth will now revolve, in con- 
sequence of the diurnal revolution being combined with the 
libration about AB. From T, a pole of this axis, let TK be 
drawn perpendicular to DG. Then, by article g, GW : GV : : 
GK : KT. But as the angle DGT is extremely small, GK 
may be considered as equal to the radius, and the arc DT as. 
equal to its sine. Consequently, using the notation already 
d' 
specified, and considering a as radius, zu : — x mn x 
ad a *_ 
— x mn x — j 
2W CT 
— the angular velocity caused by the librating, 
force. Our next object is to find the value of d in known 
terms. 
2 -i. If t be put for the time of the earth’s diurnal revolution 
round its axis, and T be put for the earth’s annual revolution 
round the sun, then ~ is equal to the centripetal force on a 
body revolving at the equator in the time t, with the velocity 
w; and', using the notation of article n , is equal to the 
centripetal force- of the sun on the eartln By the doctrine of 
centripetal forces therefore 
_M isa z _ _ SC _ a | axM 
sch ' T : : T» : anc sc ? v v- 
