EQUATIONS OF ELECTRON MOTION 169 



where the functions .t„(/) represent a natural motion, the ^„(t) are small 

 functions, and Ati and A/2 are small parameters. 



It follows from the results of Section VI that we have (to within terms 

 of the second order in small quantities)'* 



AW = At2L[x,(t2), ••• ,a-3(/2),/2] 



- A/iL[xi(/i), ••• ,a-3(/i),/i] 



dL 



dx' 



Uti) 



n=l \_OXnJt=-t2 71=1 



AhL[xr{h), ••• ,x[{h),to\ 



- AtiUxiih), ■■■ ,xz{li),ti] 

 3 



+ Z Wnit-^Ui-i) - 7r„(/i)^„(/i)]. 

 74 = 1 



Let us write 



(A:v„)2 = .T„(/2 + A/o) + Ui2 + A/o) - :v„(/2) = Uh) + x'M a/2, 



(A.r„)i = Xnih + A/i) + Uli + A/i) - .Y„(/i) = Uk) + .t'„(/i)A/i, 



so that (Axi)2, (A.V2)2, (Ax's)2 are the coordinate differences of the terminal 

 points of the varied and unvaried curves, and similarly (A.ri)i, (A:r2)i, (Ax3)i 

 are the coordinate differences of the initial points. Then we have the 

 formula 



AW = (iMk), ■■■] - Z 7r„(/2)/„(/2) j A/2 



- f LU-i(/i), • • •] - S 7r„(/i)xl(<i) j A/i 



3 



+ S [7r„(/2)(Ax„)2 - 7r„(/i)(A.v„)i], 



n=l 



which, by equation (13), can be written in the form 



AH' = -H[xM, • • •] A/2 + HMh), • • •] A/i 



3 



+ E Wn{k){AXn)2 - 7r„(/i)(A.T„)l]. (24) 



n=l 



Now, the integration in (23) being taken over a natural motion of the 

 particle, the value of W depends upon the initial instant, the initial coordi- 



1^ This is also the sense in which the following equations are to be understood. 



