734 Dr. W. F. G. Swarm on the Longitudinal 



Now, confining ourselves to the case in which the electron 

 is moving along the axis of X with velocity v, we have 



x -i_/3U\ tt -i-W v-i-W 



r=0 r=0 r=0 



These expressions are of course the values of the so-called 

 longitudinal and transverse masses, the last two being the two 

 transverse masses, which are of course equal. Now although 

 when p — v, q = 0, r = 0, V is zero, it does not follow 



BV 



that — — is also zero. Again, though each of the quantities 



U, V, W is zero when p = q = r = 0, it does not follow that 

 the derivatives are zero also. Of course from symmetry, 

 when p = q = r = 0, all three masses are the same. 



Let us now proceed to the deduction of the expressions 

 for the masses : to do this it is necessary to find the general 

 expression for the momentum of an electron moving along 

 any line. Take axes of £, 77, f, not coincident with those of 

 x, y, z, and let the electron move along the axis of f with 

 velocity co. Let a, /3, 7 be the magnetic vector due to the 

 motion ; then the kinetic energy per unit volume is 



The resultant momentum of the electron per unit volume is 

 BT If B* .3/3 3y\ 



<jO) 47TV o&> OW 0(0/ 



Since if /, g, h are the components of the setherial displace- 

 ment 



a = 0, ft = — 47r/ia), y = 4:7rgco, 

 therefore 



0&> 0'J) 0« 



and 



0(0 



where P is the component of the setherial displacement 

 resolved perpendicular to the line of motion of the electron. 

 The total momentum is 47rjjJ PWf d^df, the integral being 

 taken throughout all space. The rest of the analysis depends 

 on the shape and nature of the electron. If we take the 



