IN THE MOTIONS OF THE EARTH AND VENUS. 
71 
Section 2. 
On the abridgement which the development admits of, and the notation which it 
permits us to use. 
4. Let 0 be the longitude of the node of the orbit of m (V enus), and <p its 
inclination : the orbit of m! (the Earth) being supposed to coincide with the 
plane of xy. Let v, the longitude of m, be measured* by adding the angular 
distance of m from its node to the longitude of the node. Then v — 0 is the 
distance of m from the node. Let r be the true radius vector of m : then 
sd = r'. cos d 
?/ = d. sin v' 
x —r {cos (v — 0 ) . cos 0 — sin (v — 0) . sin 0 . cos <p } 
y = r {cos (y — 0). sin 0 + sin {y — 0) . cos 0 . cos <p} 
z = r . sin (v — 0) . sin <p 
Substituting these, the expression for R becomes 
^sr cos (d — 0) . cos (v — 0) + cos <p . sin (d — 0) . sin (v — 0) j> 
m 
y' — 2 dr ^cos {d — 0) . cos (u — 0) + cos <p . sin ( v ' — 0) . sin (u — 0)^ + r 2 ^ 
in which it must be remarked that r and v, when expressed in terms of t, will 
not involve the constants 0 and <p. This may be changed into 
m d 
cos {d —v) — sin 2 . cos {d — v) + sin 2 cos (d + v — 2 0) 
m 
V' r ,2 — < lr'r. cos (d — v) + d + Qr' r. sin 2 cos (d — v)— 2 dr . sin 2 ~ cos (t/ + y — 2 0) 
or. 
mr . . . 
-i-COS (d — v) — 
m 
\/ {f 2 — 2 r' r . cos {d — v) + r 2 } 
+ sin 2 -§- {cos (t/ — i>)— cos (d-\-v ~ 2 0} . | 
4 (. {r' 2 — 
mr 1 r 
{ r n — 2 r'r . cos {d— v ) + r 2 } 1 
* the longitude of the perihelion of m, must be measured in the same manner. 
