210 
MR. IVORY ON THE THEORY 
which are different from the corresponding equations in the last section in no 
respect, except that here h and a are both variable, whereas in the other case 
they were both constant. Treating these equations exactly as before, we first 
get by exterminating d t ^ 
A 2 
dr- 
r l d v- 
^--+- = 0 ; 
r a r 1 a 
then, by multiplying all the terms by — and adding 1 to both sides, 
/r d r 9 
a 
Jtn. . fi _ j :\ 2 _ 1 _ h * 
r 2 dv 2 ^ \ L a )~ 
a ’ 
from which we deduce 
* 2 =i- a . 
The last equation is solved by the same assumptions as before, viz. 
dr e sin 9 
rdv 1 + e cos d’ 
r cos 0 + e 
a e ^ 1 + e cos 0 ’ 
but it must be recollected that in these formulas, a and e are both variable. 
By differentiating the expression of r, viz. 
a ( 1 — e 2 ) h 9 
r 1 + e cos 0 1 + e cos 0’ 
we get, 
dr esin0.rf0 2 dh cos 0 . d e 
r 1 + e cos 0 h I + e cos 0 ’ 
d t 
and by equating this expression to the value of — taken from the first formula, 
and reducing, we obtain, 
e {d v — d 0) sin 0 + cos 6 . de ■= (1 + ecos 0). 
It appears therefore that v — 0, or zt, is a variable quantity; and the formulas 
that determine the elliptic orbit, and the variation of nr, are as follows : 
