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 u, 
to dr* 
h r* d v 
, 3 + i-T+ T=°» 
„2 
then, by multiplying all the terms by — and adding 1 to both sides, 
from which we deduce 
0 —^-+(>- t ) 2 =- 2 
The last equation is solved by the same assumptions as before, viz. 
dr e sin 9 
rdv 1 -he cos 9’ 
cos 9 -he 
1 + e cos 9 ’ 
but it must be recollected that in these formulas, a and e are both variable. 
By differentiating the expression of r, viz. 
we get. 
_ a (1 — e~) 1? 
r 1 + e cos 9 1 + e cos 9’ 
dr _ esin 9 . d9 2 dh cos9 . de 
r 1 + e cos 9 ' h 1 + e cos 9 ’ 
d v 
and by equating this expression to the value of — taken from the first formula, 
and reducing, we obtain, 
2 d ~h> 
e {dv — d&) sin & + cos 6 . de = (1 + ecos 6). 
It appears therefore that v — 6, or w, is a variable quantity; and the formulas 
that determine the elliptic orbit, and the variation of ar, are as follows: 
