206 
MR. IVORY ON THE THEORY 
n t + s — sr, the true anomaly being v-tv. The equation may be put in this 
form, 
, i /» \/1 — e*.dv .,*/I — e* . s\n(v — zz) 
n t -j- £ — ts = /—r -7-, — ex ---- 
-f e cos (u — zr) 
1 + e cos (u — w) 
and, if Ave assume 
. \/l — e 3 . sin (v — zr) cos (v — zr) -f- e 
sm U 1 + c cos {v — zr) ’ C0S U 1 + ecos(u — zr) ’ 
Ave shall find, 
u 
«/\ — g 2 . d V ' 
1 + e cos (v — zj) " 
so that Ave readily arrive at these results, 
nt z — vs — u — e sin u, 
a [1 — e") t 
r — -—;-7-\ = a (1 — e cos u), 
1 + e cos [y — zt) a n 
v — zs u /l 4- e 
tan —= tan ¥ X \J 
These last are the formulas that occur in the solution of Kepler’s problem, 
the arc u being the anomaly of the eccentric. Having found the expression of 
the eccentric anomaly in terms of the mean anomaly from the first of the for¬ 
mulas, we thence deduce the true anomaly v — vs, and the radius vector r, for 
any proposed instant of time. The analytical solution of these questions is 
omitted ; the sole intention of treating here of the motion of a planet circu¬ 
lating by the central force of the sun, being to elucidate the investigations that 
are to folloAV respecting the orbit of a disturbed planet. 
The purposes of astronomy require further that the motion of the planet in 
its orbit be connected Avith the longitudes and latitudes estimated Avith regard 
to the immovable plane of xy. The orbit being supposed to intersect the im¬ 
movable plane, and the angle of inclination being represented by i, let N stand 
for the longitude of the ascending node, and P for the place of the same node 
in the plane of the orbit and reckoned from the same origin Avith the true 
motion v : then v — P, or the distance of the planet from the node in the 
plane of the orbit, is the hypothenuse of a right-angled spherical triangle, one 
