167 
1911-12.] 
On General Dynamics. 
k conditions of the form expressed in (10) which result from the canonical 
equations. If the k constants in S are not “ distinct,” being connected 
by a relation or relations, S will still be a function of q v q 2 , . . . , t, and a 
smaller number of distinct constants, which satisfies (10), and now repre- 
sents a family of solutions. 
It will be noticed that 
dS 0S V 0S . K/ 
T< = S + V = {n) 
H = L, 
(15> 
where L is what is sometimes called the Lagrangian function or the 
“ kinetic potential.” S is Hamilton’s “principal function.” It may be 
determined by the equation 
s- fWp4)-H}dt, 
(16) 
or otherwise, as will be noticed in the sequel. 
2. Proof that the partial derivatives of the complete integral with 
respect to the constants are themselves constants . — We calculate the 
time-rate of variation of the partial derivative of S with respect to 
any one of the k distinct constants, say %. We have 
0 0S 
d as 
dt da. 
dt dttj 
/, i»v 
^ /0H a 0S\ 
V 9 * 3 q t da) 
dt daj ^ 
J \?Pi % 
(i> 
where 0H /dp if derived from H expressed in terms of the ps, the qs, and 
t, is put for q i} by the canonical equation for f. Now (1) can be written 
d 0S 0H ^ 0H dp, 
Y Oil 
dt daj daj ^ dp { da^ 
0, 
( 2 ) 
for the as only enter into H by the replacement of the ps by the 
corresponding partial derivatives of S. We have therefore by (2) 
as 
da. 
(3) 
a constant. If cq be an initial co-ordinate, Cj is — b if bj be the correspond- 
ing initial momentum. 
There are k such equations, and these are the finite equations of motion. 
For they enable the k co-ordinates q v q 2 , .... , q k to be expressed in terms 
of t and the 2 k constants a v oq 
a k , b v b 2 , 
K as in (1), § 1. 
If, however, H = /q a constant, so that H — h is a first integral of the 
equations of motion, then h becomes one of the constants in the expression 
of S, which has now the form given by 
S= -ht + W(q v q 2 , ... ., q ki a v a 2 
3 a k - 13 h), 
(D 
