OF THE PERTURBATIONS OF THE PLANETS. 
207 
side of which is the arc X — N in the immovable plane, and the remaining side 
is the latitude having s for its tangent: wherefore we have 
tan (X — N) = tan (v — P) cos i, 
s = tan i sin (X — N). 
The first of these equations enables us to compute X when v is given, and con¬ 
versely ; by means of the second, the latitude is found. The practical calcula¬ 
tions are much facilitated by expressing the quantities sought in converging 
serieses : but the discussion of these points is beside our present purpose. 
4. We now proceed to investigate the effect of the disturbing force of the 
planet P' in altering the orbit of P. For this purpose we have the equations 
(3) and (4), viz. 
r 2 dv = hdt 
r 2 _ 
1 — a . dX = h' dt ^/(M ; 
of which the first is the expression of the small area described round the sun 
by the planet in the time d t, and the other is the projection of that area upon 
the immovable plane of xy. Wherefore, if i denote the angle of inclination 
which the plane passing through the sun and the radii vectores r and r + dr, 
has to the plane of x y, we shall have 
t* 
. i +s *' rix h ' 
cos i — --= ~r '• 
r J a v h 
and, as h! and h vary incessantly by the action of the disturbing forces, it 
follows that the momentary plane in which the planet moves is continually 
changing its inclination to the fixed plane. Let i 1 be the value of i when t — 0 ; 
then cos i' = -r°; and, by the formulas (3) and (4), we shall have, 
h' 2 = h 0 2 cos 2 1 + 2f r 2 
JR dx 
• ~dx ' 1 + s® 
h " 2 = h 2 - U 2 = h 2 sin 2 i' + 2 \fr 2 . 
s 2 dx d R \ 
rr? + Ts ds ) ■ 
