FUNDAMENTAL FORMULATIONS OF ELECTRODYNAMICS. 
215 
terms of a certain number of variables ... x,,, which are known to be connected 
with the co-ordinates q and their velocities q by a series of relations of the type 
= 0 
being a function of the co-ordinates q, the velocities q, the variables x and the 
differential coefficients of these latter variables with respect to the time. For the 
sake of simplicity we shall restrict our statement to the case when the first differentials 
only appear. The usual method of procedure is to introduce a set of multipliers 
functions of the time, and then to consider the variations of the integral 
(L + 2A,M,) dt 
where the g’s and x’s undergo independent variations. The equations obtained for 
the vanishing of the variation are of two types. Firstly, there is an equation of the 
type 
for each variable x : these with the restricting equations will determine the x’s and X’s 
as functions of the co-ordinates q and the time. Then there is an equation of the 
type 
dt 
for the motion in each q co-ordinate. 
The latter equations only involve the Lagrangian function L through the quantities 
A and x which enter into it, and once these are determined the rest of the solution 
involves only the restricting conditions. In fact when once these multipliers and 
variables are determined and regarded as functions of the time only the motion in 
the q co-ordinates is completely determined by the condition that the integral 
I 2A,M, dt 
is stationary for independent variations of the co-ordinates q. It may even happen 
that the relations M involve the co-ordinates q and the variables x in such a way that 
it is possible to separate M into two terms, one of which is a function explicitly of the 
g’s only and the other of the ic’s only. In this case the part of the integral required 
in the above statement is only that part of it Involving the g’s and this is independent 
entirely of the co-ordinates x. 
This remark has an important bearing on a question which occurs in the sequel, and 
it shows that the existence of a variational form for the equations of motion does not 
2 H 
2A, 
0g / 
2A. 
dq 
- 0 
VOL. CCXX.-A. 
