FUNDAMENTAL FORMULATIONS OF ELECTRODYNAMICS. 215 



terms of a certain number of variables x lt x 2 , ... x t , which are known to be connected 

 with the co-ordinates q and their velocities q by a series of relations of the type 



M, = 



M s 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 X,, 

 functions of the time, and then to consider the variations of the integral 



P (L + 2X,M S ) dt 

 It, 



where the q'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 



d / v , 3MA v . 3M S 3L 



-=- 2X, -- * 2X, - - = 



dt \ . ' 3* / Bar, Sx 



for each variable x : these with the restricting equations will determine the x'x and X's 

 as functions of the co-ordinates q and the time. Then there is an equation of the 

 type 



d / 8MA 3M S _ 



dt \, ' Sq)~ '-fy : 



for the motion in each q co-ordinate. 



The latter equations only involve the Lagrangian function L through the quantities 

 X 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 



ZX^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 

 q's only and the other of the x's only. In this case the part of the integral required 

 in the above statement is only that part of it involving the q'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 



VOL. ccxx. A. 2 H 



