114 Proceedings of the Eoyal Society of Edinburgh. [Sess. 
therefrom the variations of the velocities : this makes the velocity variations 
dependent on the variations of the co-ordinates. We can, however, formu- 
late the principle in such a way that the velocities and co-ordinates may 
all be treated as independent variables in forming the variation, latitude 
being allowed for the ultimate relations which must hold between them by 
the introduction of a number of undetermined multipliers. We form the 
variation of the integral 
f tz Ldt 
Jtx 
wherein the function L is considered as a function of the 2 n variables 
q v q 2 , . . . q n , q v q t , . . . q n , these being, however, subject to n equations 
of the type 
r=l, 
n . 
The usual method is to' introduce ^-arbitrary functions of the time 
Xj, X 2 , . . • A», then to express the conditions that the integral 
is stationary when the 2?i-quantities q v q 2 , . . . q n , q u q 2 , . . . q n are all 
independently variable, and finally to choose the functions A r so as to make 
the solutions of the derived equations satisfy the conditions which 
necessitated their introduction. The equations obtained for the vanishing 
of the variation are of the type 
3 L dX r _ q 
dq r dt 
which are equivalent to the ordinary Lagrangian equations of motion for 
the system. The undetermined functions introduced are seen to be the 
momenta corresponding to the different co-ordinates ; denoting these by 
p v p 2 , . . . p n respectively, we see that our result is equivalent to the 
statement that the equations of motion of the system can be derived by 
varying the integral 
2 'Mr + 2 Tr 
dqr 
dt 
dt 
wherein the co-ordinates q r and the velocities q r are all independent, and 
the momenta p r are functions of the time. 
3. The result derived in the last paragraph enables us to proceed 
immediately to the question of the ignoration of some of the co-ordinates 
of the system, We have obtained the integral of an explicit function of 
2 n independent variables whose variation vanishes when the motion of the 
