On General Dynamics. 
165 
1911-12.] 
Here a v a 2 , . . . ., a k , b v b 2 , . . . ., b k are constants of values to be assigned 
by the initial circumstances. For example, the as may be the initial 
co-ordinates, and the 0s the initial velocities, or the initial momenta. 
Differentiation of (1) with respect to t gives q v q 2 , ..... q k . From 
these values and the expression of T in terms of q v q 2 , . . . ., q k , namely, 
T = i(4ij^i 2 + ^ Ai 2 5'i^2 + • • • • + u qiq k 
+ ^ 22 ^ 2 2 + ^A 23 5 ' 2 ^ + + 2A 2k q 2 q k 
+ + A kk q k 2 
+ Aj^i + A 2 q 2 4- .... + A k q k + A, . (2) 
where the As are functions of the qs and t, we find p v p 2 , . . . ., p k 
( = dT/dq v 0T /dq 2 , . . . ., 0T /dq k ) in the form 
Pi=9i( a v a 2 > • • • •> a k, K b 2 , . . . ., b k , ty 
P2=9z( a v a v • • • - «*> K b v • •••» b k , 0 ^ ^ 
Pk — 9k( a i> a 2 , . . . a 7c , 0 1? ft 2 , . . . ., ^ 
Now by means of (1) we can express any k constants, let us say the 
6s, in terms of q v q 2 , q k , t, and the remaining constants, thus 
obtaining by substitution in (3) 
Pi — ^2’ ' " 
. ., q k , t, a v a 2 , . . 
. a k y 
P2 = ^ 2 (^ 1 ’ ?2’ • 1 
. ., q k , t , a v a 2 , . . 
. . a k ) 
- • • • (1) 
Pk = &M V 9.2* • ■ 
■ • 9?c, t, a v a 2 , . , 
■ ■ ; <h). 
The canonical equations are 
of the type 
P = 
0H . _0H 
dq * ^ dp 
(5) 
where, however, H is supposed to be expressed as the function of the' ps, 
the qs, and t, which we obtain by finding q v q 2 , . . . ., q k , in terms of the p s, 
the qs, and t, from the equations 
0T 0T 
Pk = 
0T 
b q k 
(6) 
derived from (2), and substituting in (2) and in 2( pq ). 
From the ps as expressed in (4) we find for any p 
d P_ b P, , , 
+ dp a + dp 
+ + di 
which by the canonical equations (5) can be written 
(D 
0H + 0H dp + 0H dp f + 0H dp _ _ dp 
dq dp 1 dq % dp 2 dq 2 dp k dq k dt' 
