172 Proceedings of the Royal Society of Edinburgh. [Sess. 
The value which this gives for r 2 when 0 is constantly 
suggests putting (K = a constant) 
2t— + , r 2 6 2 — - a2 + 1 — . 
\m r J r 2 ’ m 2 r ' 2 sin 2 # r 2 
These give 
mf = ± m v / 2^ — + ^ , mr 2 # = ± rn x / K 2 - — a — — . 
V \?n r / r l V m 2 sin^ 
at once 
• (1^) 
. (15) 
We have thus obtained for mr a function of r, and for mr 2 6 a function 
of 0, while the value of m(r 2 -\-r 2 0 2 ) is left unaltered. The value of the 
constant K 2 thus introduced is settled by the conditions of the problem. In 
point of fact mK is by the second of (15) the angular momentum of the 
planet about the sun. Of course, K 2 >a 2 /m 2 sin 2 #. 
Equation (12) becomes by (15) (ambiguities of sign understood) 
s = - ht ^ + m l dr J + 7) - 5 + m /V k2 ^S - (i6) 
which gives the value of S reduced to quadratures. The three necessary 
constants involved in S are h, a, K 2 . Partial differentiation of this 
expression with respect to h, gives the time t taken by the system to pass 
from the configuration at time t = 0, to that at the instant under considera- 
tion : the partial derivatives with regard to a and K 2 give the path. 
The usual process is to write (11) in the form 
asy 
+ 
2i 
asy 
dq 2 J ' c[ 2 sin 2 g 2 \dqj 
+ 
-2m(h+^) = 0. 
h 
■ (17) 
To solve this differential equation the variables are separated by splitting 
it into the three, 
'2m(^h + — = 0, 
- K 2 = 0, 
(18) 
which are then solved separately. It is easy to see that the result of 
integrating 
as 
as 
as 
3S 
/S — - — dt + =—dq 1 + - ~dq 2 + — dq. 
a t 
s 2i ^2 
a <h 
(19) 
agrees with (16), but the process is not quite so direct and clear. The only 
difference, however, is that in the first process the terms —ht + a(p come as 
a matter of course ; for the others mr, mr 2 0 are determined in the first 
process, and their equivalents dS/dq v 0S /dq 2 in the second. The ambiguities 
of sign which are common to both methods correspond of course to different 
conditions of the problem, and must be carefully considered when applica- 
tions are worked out in detail. 
