166 Proceedings of the Royal Society of Edinburgh. [Sess. 
But by direct differentiation of H, regarded, as stated above, as a function 
of the ps, the qs, and t , we obtain 
8H 0H 0^ ^ 0H 0p 2 + ^dRdp^fdR^ ^ 
dq + 9Pi dq dp 2 dq dp k dq \dq / 
The q in this equation corresponds to the p of the last, and the dH/dq in 
brackets on the right is the partial derivative of H with respect to q, taken 
after the values of the ps in H have been replaced by their expressions in 
terms of the qs and t as given in (4). Subtracting now (8) from (9), 
we get 
0H/0p2_0p\ . . . no) 
d Pl \dq dqj dp 2 \dq dqj dp k \dq dqj \dq ) df 
There are of course k such equations. 
The left side of (10) at once suggests the idea of a function S of the 
co-ordinates, q v q 2 , ... ., q k , the time t, and k constants (either the as, 
say, or k other constants c v c 2 , . . . ., c k , related to the as) such that 
as 
as 
dq 1 3 q 2 
as 
dq,’ 
. ( 11 ) 
for the left side of (10) will then vanish identically, since the quantities 
0H /dp v 0H /dp 2 , 0H /dp k , which are the velocities q v q 2 , ... ., q k , are supposed 
to be finite, and d 2 S/dq 1 dq 2 = d 2 S/dq 2 dq 1 , etc. 
If, moreover, S be such that 
( 12 ) 
where H is now supposed to be expressed in terms of q v q 2 , ... ., q k . , t , and 
the constants a v a 2 , ... ., a k , then dH/dq taken on this supposition is the 
quantity denoted by (0H/0g) in (10), and we have, dropping the brackets, 
_sh a as aas a^ 
a q a q a t a t a q a t 3 
so that the right-hand side of (10) also vanishes. 
Taking then H in its usual form H(p 1 , p 2 , . . . ., p k , t, q v q 2 , . . . ., q k ) 
and replacing the ps by their values, the partial derivatives of S with 
respect to the qs, we obtain as the necessary and sufficient condition fulfilled 
by the function S, the differential equation 
S +H 
as as 
dqi dq 2 
9S 
k* 
t, q v q 2 , . . . ., q^ = 0. 
(14) 
The function S(g p q 2 , . . . ., q k , t, a v a 2 , . . . ., a 7c ), including as it does 
k arbitrary constants, is the “ complete integral ” of this equation, and fulfils 
