charged ion in a magnetic field. 51 



origin at the centre of force and the direction of the magnetic 

 force as the axis of Z, our equations are 



dx _ k 2 X + k>y Y 



~dt~ Jc* + ky 2 '*'' 



dy = k 2 Y-k y X 



dt k 3 + ky 2 K h 



dz_ k 2 Z+y 2 Z = Z 



dt k 3 + ky 2 k ^ '* 



X x 

 Since ^ = - , we get from (1) and (2) 



k (ydx — xdy) = <y {ydy + xdx), 

 or if x = p cos cf>, y= p sin </> 



— kp 2 dcp = ypdp, 



or p = Ce v . 



From 1, 2, and 3 we get 



xdx + ydy k 2 zdz 

 x 2 + y 2 k 2 + y 2 z 2 



2k 2 



or a? + tf=G'P++ (4), 



hence z = C'e'^*'* . 



Thus the path of the particle is a spiral traced on the sur- 

 face (4). 



The relative importance of the three components of the 

 velocity depends upon the value of H/k. Now k is the reciprocal 

 of the velocity acquired by the ion under unit electric force ; if we 

 call this velocity v the relative importance of the three com- 

 ponents of the velocity depends upon the value of Hv a ; if this is 

 large, the ions follow the lines of magnetic force ; if it is small, 

 they follow the lines of electric force, while in intermediate cases 

 they pursue a spiral path. Thus if we keep the magnetic force 

 constant and consider ions which move with different speeds 

 under unit potential gradient, the more quickly moving ions may 

 travel along lines of magnetic force, while the more slowly moving 

 ones may travel along spirals. In the discharge of electricity 

 through gases, whenever the velocity of the ions has been measured, 

 the velocity of the negative ion has always been found to be 



4—2 



