202 
MR. IVORY ON THE THEORY 
By integrating, 
r 2 dv — hdty/Ju, 
h? = V + 2 ifr 2 (VR - 
** = V+2 f^^^+^dsy. 
(3) 
r 2 dv 
- when t — 0. 
the constant h 0 being equalto dt 
Further, let the first of the equations (A) multiplied by y be subtracted from 
the second multiplied by x ; then 
d.(xdy-ydx) R 
x 
R 
d^y- 
pdf- dy 
and, by converting the quantities in this equation into functions of r, X, s, 
( 
1 +s~ ' dt 
d\ x 
t \/p) 
dt Vp 
<7R 
d\ ‘ 
v~ 
and by multiplying both sides by 2 . ——„ ,d\, 
d. 
0 
d\ \2 
l+f * dtVp) 
and, by integrating, 
= 2 
d R 
1 + s 2 d A 
d /.: 
r~ d X 
1 + s 2 
= h! dt 
] 
h ' 2 = K + \f nr? • d ^ cl} ^ | 
> 
(4) 
I 
h 0 ' being a constant. 
The equations that have been investigated, which are only three, the first 
and second being one equation in two different forms, are sufficient for deter¬ 
mining the place of a planet at any proposed instant of time, whether it revolves 
solely by the central force of the sun, or is disturbed by the irregular attrac¬ 
tions of the other bodies of the system. The second and third equations 
ascertain the form and magnitude of the orbit in its proper plane, and the 
place of the planet; the fourth equation enables us to find the angle in which 
