118 Proceedings of the Poyal Society of Edinburgh. [Sess. 
is an integral invariant in the above sense of a dynamical system, we 
must have 
=0 
so that, on account of the arbitrariness of the curve to which the 
invariantive property relates, 
d n n ( \ 
^ \prhr +vMr J 
r— 1 r=l 
is a complete differential of some function I of the variables (q v q 2 , . . . q n , 
Pi, p 2 , . . . p n ) which may also contain the time as a parameter. 
We have then 
n 
81 =^(p r Sq r + p r Sq r ) 
r = 1 
so that 
n n 
S H " 2 Mr) = 2 UPMr - fiPr) 
r — 1 r= 1 
and therefore using 
- H = I - 2 Pr4r 
r~ 1 
we see that the equations of motion of the system are of the Hamiltonian 
type 
8H . 0H _ . 
Wr Pr ’ ¥r ^ 
From this last result or by the same argument it is concluded that if a new 
set of variables (Q 1? Q 2 , . . . Q n , P 1? P 2 , . . . P w ), functions of (q v q 2 , . . . q n , 
p v p 2 , . . . p n , t), be chosen as co-ordinates for the dynamical system, and if 
r — 1 
is an integral invariant of the original system, then the new equations of 
motion are still of the type 
0K 
SQr 
-P 
r ? 
0K 
dl\ 
m i 
The transformations from the variables (p, q) to the variables (P, Q) are 
in general special to the problem considered, but they include all contact 
transformations, which are of a less special type. Let us consider the 
contact transformation defined by the relations 
O s = 0 s = 1, 2, . . . k 
, aw x' , ao s aw , ao s 
r = ^r+ Pr=-^—~ 
0Q r s=i ^Qr 0 q r 5—1 tyr 
