204 
MR. IVORY ON THE THEORY 
by multiplying all the terms by 
and adding 1 to both sides, 
A* 
a 
dr 3 
r- d v‘ 
+ fl “ 
A 2 
a ’ 
and by introducing the new quantity e 2 , 
e 9 - = 1 
(l-* 2 ) 
This equation is solved by assuming 
dr e sin 9 
r dv 1 + e cos 9 ’ 
cos $ + e 
1 + <?cos 9 ’ 
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 6, let the second of the formulas be differentiated, and equate 
dv • • 
— to the like value in the first formula ; then, 
dv = dd ; and v — vs = 0. 
The nature of the orbit is therefore determined by these two equations, 
dr e sin ( v — w) 
r dv 1 + e cos (u — -ot)’ 
a ( 1 —e 2 ) 
J' • 
1 + e cos (u — ct) ' 
% % d v 
the first of which shows that the two conditions ^ = 0, and sin (v — w) = 0, 
must take place at the same time ; so that vs is the place of the planet when 
its distance from the sun is a minimum = a (1 — e), or a maximum = a (1 + 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?; — zs the 
true anomaly, that is, the angular distance from the perihelion or aphelion ; 
