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 6 be the longitude of the node of the orbit of m (Venus), and <p its 
inclination: the orbit of ml (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 — 6 is the 
distance of m from the node. Let r be the true radius vector of m : then 
x' — r'. cos v' 
y = r'. sin v 1 
x = r {cos (v — 0) . cos 6 — sin {v — 6). sin 6 . cos p} 
y =r {cos {v — 6). sin 6 + sin {v — 6). cos 6 . cos <p} 
z = r. sin {v — 0) . sin <p 
Substituting these, the expression for R becomes 
712 T C 1 
-jr < cos {v' — 0 ). cos {v — 0) + cos <p . sin ( v' — 0 ). sin (v — 0) > 
m 
— 2 dr ^cos (t/ — 0). cos {v — 0) + cos <p . sin (v' — 0). sin (u — 0)^ + r~ 
in which it must be remarked that r and v, when expressed in terms of t , will 
not involve the constants 6 and <p. This may be changed into 
m 
r 
^ ^ cos (v —v) — sin 2 . cos {v 1 — v) + sin 2 cos {v r + v — 2 0) j> 
m 
d 2 —2dr . cos(t/— v) + r 3 + 2r , r.sin 2 cos (d — v)—2dr. sin 2 -^cos(t/ + t>—2 0) ^ 
or. 
711 t . , . _ m _ 
r 2 C0 ^ V ' { r n — 2 r' r. cos (t/ — v) + r 2 } 
+ sin 2 -^- {cos {v 1 —v )—cos 2 6} . 77” 
7n dr 
2 dr . cos {d— v) + r 2 }" 
711 d 1 
~J 
•a, the longitude of the perihelion of m, must be measured in the same manner. 
