EQl'ATIONS OF ELECTRON MOTION 



165 



By a well known theorem concerning the differentiation of definite 

 integrals with respect to parameters," we have 



di 



de 



a/ 





+ <^'n7„-r 



F{t, X + e(^, • • • , z' + Oo)') dt, 



d(x -\-e<p) ^ d(x' + e/)_ 

 = -F[l\ + h , x(Ti + h) + e<^CA + /i), • • • , z'(T, + k) + do^'iT, + /,)], 



= F[To + /2 , x(To + /,) + ec^(r2 + /2), • • • , z'CA + /o) + dc^'iTo + /2)I. 



The formulae for dl/drj and dl/dd are similar to that for dl/de, and need 

 not be written down. 



In particular, if [dI/de]o, etc. denote the values of the derivatives at the 

 point e — 7] = 6 = ti = t2 = 0, we have 





a.r 



dx' 





dy 



dy'_ 



F{t, X, ■•• ,z') di, 

 F(t, X, • • • , s') dt, 



Fit, x,---,z') dt. 



(18) 



ado = I. r ^2 "^ "' ^' 



[i]o ^ ^^^^' '^^^^^' ■ • ■ ' ''^^'^^' 



The first three of equations (18) can be transformed to advantage, as 

 follows. Integrating by parts, we obtain the formula 



£V i, F(/, .,..., z')J^ = [,^,^(,, ,,...,, J 





(i/aa;' 



, 2') ^/, 



dt 



and similar formulae for the integrals 



and 



f \' ^, F{t, x,---,z') dt. 

 J Ti dz 



" The theorem is given, often in the form of two separate theorems, in most works on 

 Advanced Calculus and the Theory of Functions of Real Variables. See the bibhography. 



