the Structure of the Electric Field, 313 



sets itself at right angles to the direction of motion of the 

 corpuscle; we shall now investigate what effect a magnetic 

 field will have on the direction of the axis of the tube of 

 force. We shall take the case when the magnetic field is 

 uniform and parallel to the axis of .v. Let I, m, n be the 

 direction cosines of the axis of the tube of force ; v, v, w the 

 components of the velocity of the corpuscle. If the angle of 

 the tube is small, the components of the magnetic force due 

 to the moving tube at a place where the electric force is It, 

 are respectively 



ll(mw — nv), R(nu—hv), R(lv—mii), 



and if a is the external magnetic force, the components of 

 the magnetic force at this part of the tube are respectively 



a + R.(mw— nv) f H(nu — hu), R(lv—mv), 



The energy per unit volume is therefore 



^ { (« + B(«w - ™)Y + &X™ - Iwf + r 2 o - muy \ , 



if the axis of the tube makes an angle 6 with the axis of .r, 

 and if the plane containing the axis and t ho axis of x makes 

 an angle <j> with the plane of ocu 



I = cos 0, m = sin 6 cos $, n = sin sin <£. 



AVhen things are in a steady state, if T is the energy per 

 unit volume, 



l T = o .' /T = o 

 dd ' <t<b ' 



The first of these is equivalent to 

 -r- Aicm — vn) + R.(lu + mv + nw){u sin #-|-cos 6{ ycos 6+ to sin 6)\ — 0. 



SID U T Y/3 9 



and the second to 



alu— (lu+mv + nu))(a+ R(mto—«t')) = <>. 



From these equations we get 



/ = 0, Iu-\-mc + nw=z0. 



The first of these shows that the axis of the tube sets itself at 

 right angles to the magnetic force, and the second that it is 

 also at right angles to the direction of motion of the corpuscle : 

 thus, where there is a magnetic field, the direction of the axis 

 of a uniformly moving tube is determinate. 



