117 
1918-19.] On Hamilton’s Principle. 
integral does not involve explicitly either the co-ordinates or the velocities 
in the unmodified part of the system. 
5. So far our discussion has centred round the question of ignoration 
of co-ordinates, but the result obtained in paragraph 2 enables us to approach 
some of the most important results in the transformation theory associated 
with Hamilton’s form of the equations of motion. We first write 
H 1 2,Mr ~ L 
r — I 
for the Hamiltonian function, and then, if we regard the co-ordinates q r and 
the momenta p r as the independent variables, we see that the equations of 
motion of the dynamical system are of the Hamiltonian type 
8H _ dq r 9H _ _ dp r 
dp r dt ’ dq r dt 
these being the conditions that the integral 
- = ♦2 
is stationary, the integrand being regarded as a function of the 2 n 
independent variables q v q 2 , . . . q n , p v p 2 , • • • Pn and the time. 
Suppose we now write 
I bl + 2 Pr 
r= 1 
d^. 
dt 
then using S to denote variations in which the time is maintained constant 
we have 
-j n 
= jdLPrhr 
tit y-.-y 
It follows that 
r = 1 
taken round a closed curve in the 2^-dimensional space (q v q 2 , . . . q n , 
p v p 2 , . . . p n ) is an integral invariant of the dynamical system. 
Conversely, if 
