1911-12.] On General Dynamics. 173 
We now find the function S' for the same problem. Starting from (2) 
of the present article, 
S'l/{H + S(gi>)}*, 
0 
we get for the same limits of integration 
Sg ht + Jq 1 dp 1 + Jq 2 dp 2 , .... (20) 
since, as we have seen, p 3 ( = mr 2 sin 2 0 . <j>) is a constant, denoted above by 
a. But by (14), since q x — r, p^mr, q 2 = 0, p 2 = mr 2 6, we have 
„2 
2 2m 2 /x m 2 K 2 
gl+ „2 O„.7 . = 0 > ^2 
p j 2 - 2 mh pp - 2 mh 
The first of these fives 
m 2 K 2 - p 2 2 
and the second 
Thus we obtain for S' 
J9.J 2 — 2 mh 
g 2 = sin -l ( ± 
{^±y ^ - ^(pi 2 - 2 «) } . • 
Jm 2 K 2 -p 2 s 
S ' = ht + IJ^Th { ^ ~ § (ft2 " *- mh) } + l dp > sin " ( 
• ( 21 ). 
• ( 22 ) 
• ( 22 ') 
*J m 2 K 2 — y 2 2 / ’ 
(23) 
In the usual mode of discussion we should begin with the differential 
equation, written down from (11), 
(p x 2 - 2 hm)i “ ) - 2$m 2 ^— +p 2 2 + — 01 — = 0, . 
*Pi 
which is then split into the two, 
'0S'\2 0S' 
0Pi 
• 9 0S 
sin^ — 
0y 2 
(ft 2 - - 2 *"” ! f^ + m2K2 = °’ £ - si 
• (24) 
■ (25) 
sin -1 ( -4 — - 
Jm 2 K 2 - p 2 2 / 
From this 0>Bj0p 1 is found by solving the quadratic, and with 0S '/dp 2 
and 0S'/0^ is used to form the expression on the right of 
7Q/ 3S' 7 . 0S' 7 . 0S' 7, 
db = — dp x + - — dp 2 + — dt, 
dpi 0p s 
dt 
(26) 
in which the variables are separated. Equation (23) is then found by 
integration. 
We obtain here a good example of the theorem stated in (7') above. 
From (16) we obtain 
1 
m 2 sin 2 # 5 
m 2 sin 2 # 
