168 
Proceedings of the Royal Society of Edinburgh. [Sess. 
where W denotes a function of the quantities within the brackets. We 
have p { = dW /dq { , and dS/dh = — t + dW/dh = const. The partial derivative 
dS/dh is thus constant ; it enables the time of passage from the initial 
configuration to that at the instant considered to be found. See § 4 below. 
3. Derivation of Hamilton s second differential equation . — It is only 
necessary to substitute in the canonical equations IF = — H to obtain as 
typical equations 
q - 
0H ' . = 0H ' 
dp ’ ^ dq 
( 1 ) 
in which q, p are related to H' as p, q are to H in (5), § 1. It might be 
inferred that a function S' of p v p 2 , . . . . , p k , t and the constants a v a 2 , 
. . . . , a which gives k equations of the form 
( 2 ) 
from which the integrals of the canonical equations can be deduced, may be 
found by the differential equation * 
as; tt,/— as; 
dt va^’ a 
as' 
• • • > 3— > Pv Pto • • • • 5 Pk, 
OPk 
( 3 ) 
A formal proof of this theorem may be constructed in precisely the 
same manner as (14), § 1, was established. By means of (3), § 1, the 
constants b v b 2 , . . . . , b k are expressed in terms of p v p 2 , . . . . , p k , t and 
the constants a v a 2 , .... , a k . Thus we get for a typical q (g,. say) 
Qi= E(Pp • • • • , Pk , t, a 2 , ... ., a k ). ... (4) 
From the qs as thus expressed we find 
dq i . dq i . 
qi = ^T 1 Pi + ?r L P2 + 
9Pl d P2 
d <li . , dq 
dqi 
qu + 
dt ’ 
which by the canonical equations (1) can be written 
air air a q { _ dq i 
dpi + j dq- dpj dt ' 
(5) 
(6) 
This equation corresponds to (8), § 1. 
Direct differentiation of IF regarded as a function of the ps, the qs, and 
t gives 
3H' -ySH'llwaH'X (7) 
°Pi \W 
* The duality represented by this equation and (14), § 1, is one of the first results of 
the contact transformation theory of dynamics. For the addition of {2S(pg) + H}d£ to 
{ %{pq) - H } dt gives the perfect differential %{pdq + qdp). Hence we get S' = J { %(pq) + H } dt, 
and so aS'/af = H, q = dS'/dp. 
