171 
1911-12.] On General Dynamics. 
q v q 2 , . . . .,q g are absent from the expression for H, and when, therefore, 
dH/dq v dH/dq 2 , . . . ., 0H /dq g are all zero (so that p v p 2 , . . ., p g are all zero, 
and p v p 2 , . . . ., p g are constants a v a 2 , . . . a g ), 
S = + a 2 q 2 + .... + a. g q g + j {2(pq) - H}dt, . . (8) 
where 2(pq) refers to the remaining co-ordinates, q g+1 , . . . ., q k , and H is 
expressed in terms of the ps (in this case a v a 2 , . . . ., a g , p g+1 , . . . ., p k ) and 
t, with the co-ordinates q g+1 , . . . ., q k . Thus S, determined by the com- 
plete integral of (14), § 1, has the form, with remaining constants 
<Xp a 2 , . . . ., ct k _ g , 
S = a l q 1 + a 2 q 2 + . . . . + a g q g + 4>(<^ +1 , ...., q k , t, a v a 2 , .... , a g , a v a 2 , ... . ct, k _ g ). 
(»>' 
If H is a constant, h, the function <f> has the form 
= — Jit + .... , q ki ttj, . . . . , Ti t cq, . ... , & k _g). . (9 ) 
If the zero of time be t — 0, we have 
< 10 > 
since 0S/0A is a constant, y say. 
In (9') the g constants, a g+1 , . . . . , a k , have been replaced by 
a v .... , a g , obtained by putting p 1 = a 1 , . . . . , p g = a g in (4) of § 1, and 
eliminating a g+1 , etc. 
As an example of the process of finding S and S' we take the case of 
a planet the co-ordinates of which at any instant are the radius vector r, 
the heliocentric longitude 0, and the co-latitude 6. Here H has a constant 
value h (which is negative for elliptic motion), and the potential energy 
is fim/v, where m is the mass of the planet. We easily find 
2h^m(r 2 + r 2 6 2 + r 2 sin 2 0 . 0 2 - 2-^-), . . . (11) 
or, with q 1 — r, p x = mr , q 2 = 0, p 2 = mr 2 0, q B — 0, p 3 — mr 2 sin 2 d . 0, 
2 h = — (p* + -L> a 2 + . 1 . , -pA - 2^ . . . . (11') 
mV 1 - r 2 sm 2 0 V r v ' 
The co-ordinate q B does not appear in this equation, and therefore we 
can write_p 3 =a, a constant. Thus by (16), § 1, we obtain, to a constant, 
But 
S = — ht + a0 + j m(r 2 + r 2 6 2 )dt = - lit + a0 + J mrdr 4- j mr 2 0d0. . (12) 
f 2 *f- x 2 Q 2 — 2( — — + — — - 
m r 
m 2 r 2 sin 2 0 
( 13 ) 
