204 
MR. IVORY ON THE THEORY 
by multiplying all the terms by 
and adding 1 to both sides. 
and by introducing the new quantity e 1 , 
This equation is solved by assuming 
dr e sin 3 
r dv 1 + e cos 3 ’ 
r _ cos 3 + e 
a e 1 + ecosS ’ 
the arc 0 remaining indeterminate. For, if the assumed quantities be sub¬ 
stituted, the equation will be verified, and the arc 0 will be eliminated. In 
order to determine 0, let the second of the formulas be differentiated, and equate 
— to the like value in the first formula; then, 
r 
dv — d0-, and v — vs = 0. 
The nature of the orbit is therefore determined by these two equations, 
dr e sin (u — ot) 
r dv 1 + e cos (v — ct) 5 
a (l - e 2 ) 
- - • 
1 + e cos (d — ■&) * 
the first of which shows that the two conditions ^ = 0, and sin (v ~ m) = 0, 
must take place at the same time; so that w is the place of the planet when 
its distance from the sun is a minimum = a (1 — e), or a maximum =«(!+ e): 
and the second proves that the orbit of the planet is an ellipse having the sun 
in one focus; a .being the mean distance; e the eccentricity; and y — sr the 
true anomaly, that is, the angular distance from the perihelion or aphelion; 
