1911-12.] 
On General Dynamics. 
169 
where the expression on the right denotes the partial derivative of H' with 
respect to pi taken after the values of the qs in H' have been supposed 
replaced by their expressions as given in (4). Subtracting (25) from (26) 
we obtain 
2 
3H' fdq 3 
0H'\ 
dfj 
Zqi 
dt 
3 H ' d Pi d Pr 
This equation suggests a function S' of the momenta p v p 2 , . 
a v a 2 , . . . ., a k such that 
as' as' as' 
(Zi — ~ — 3 q<2 — -s i 
3 Pi d P 2 
dp k 
( 8 ) 
P k , t, 
(9) 
With these values of the qs the left-hand side of (8) vanishes. If, moreover, 
S' be such that 
( 10 ) 
— = - H', 
dt 
where H' is supposed expressed in terms of the ps, the as, and t, 9H'/ dp { 
taken on this supposition is the quantity which appears in (8) as (0H'/3p,), 
and we have 
_aH_^ as'_^as'_a^ (n) 
dpi 0 Pi 3 t dt dpi dt ’ 
so that the right-hand side of (8) vanishes. If, then, we replace the qs in 
H, as usually expressed, by the values given in (9) we obtain the partial 
differential equation 
as' 
dt 
-H 
V 13 Vv 
as' as' as' . 
Pkl — , — 3 • • • - 3 ^— 3 ^ 
0. 
( 12 ) 
d Pi 
The function S'(p v p 2 , . . . ., p k , t, a v a 2 , a k ) referred to above is 
the complete integral of this equation, and fulfils all the conditions which 
result from the canonical equations. 
As in § 2 (for S) it can be shown that the partial derivatives of S' 
with respect to the as are all constants if these constants are distinct. 
4. Relations of the two forms of the principal function. Calculation 
of the functions in different cases. Derivation of the modified Lagrangian 
function for ignoration of co-ordinates . — An equation similar to (15), § 1, 
holds for S' also. For we have 
dS' as' 
dt dt 
so that 
as'. 
H + S(?p) 
T 
S' = / { H + %(q p)}dt. 
( 1 ) 
( 2 ) 
Taken along with (15), § 1, this gives 
!(S + S'): 
~(P'i +/"/)> 
(3) 
