686 Prof. J. J. Thomson on the Magnetic Properties of 

 so that 



""~ ^Idt (kj' r dt dz' r dt y dx' dy''^'^ dy''^ "^ ^ dy' dz I r 

 dy/ d^ , d^ . f/'\l \ 



Now if the motion is periodic, the mean values of 775 y ? 

 z ~, and y -^ are evidently zero. 



Since 



dz 



ydi. 



dy 



1 / dz dy\ Id,, 



2V-dt-'tt)^2Tt^'^'^^ 

 Ifdz dy\ Id, . 



^dt--Aydt^'dtriit^y'^^ 



the mean values of v^ and —z~ are one-half the mean 

 ^ dt dt 



value of y 77 — -;±-. We shall proceed to find these values 



and also to show that the mean values of x ~f and y ^ 

 are zero. _ ^ ^^ '^^ 



The equations of motion are 



'''dt^^^^ w 



.5 = Y-H.|, ..... (2) 



™£ = Z + h4'. (3) 



The force whose components are X^ Y, Z is supposed to be 

 centi'al ; so that 



x" y y 



From (1) and (2) we see 



d f dx dy\ -p- dz 



/ dx du\ T-T dz 



\y-dt-'tt) = ^'^dt'^ 



"^dty dt 



dz 

 so that the mean value oi x -j- vanishes. Similarly, we may 



dz 

 show that the mean value oi y j vanishes. 



