360 Mr. Megb Nad Saha on the 



In three- dimensions this equation represents a sphere, but 

 in four-dimensions this represents a spherical cylinder having 

 infinite extension along the time-axis. The equation shows 

 that the electron is at rest. 



We shall now write down the equation of a spherical 

 electron moving with a uniform velocity (u b u 2 , u 3 ). 



In three-dimensions, the equation of a circular cylinder 

 having the line 



#— a'o = y-.vo _ z— ZQ 



I m n 



as the axis is given by the equation 



-[l(x-x )4-m(y-~y )+n(z-z )Y = r\ 



Similarly, in four-dimensions, since the axis of motion 

 is given by 



x — hiq __ y—y _ z—z _ l—l 

 iw 1 iw 2t iw 3 iw± ' 



therefore the equation of the cylinder having this as axis is 



^_. 1 , )2 +(7y _ 2/o)2+U _, ())2+(/ _/ o) 2 



+ [wi(x-Xo) + W2{y-yo) +w 2 (z-Zo) + W4(l-lo)] 2 = r 2 . 



That this is so can easily be observed by introducing a 

 Lorentz-transformation in which the line of motion is the 

 new time-axis, and the velocity is equivalent to the moment 

 of transformation. Then if (£, 77, f, v) be the new coordinates, 

 we have 



^(oc-xtf+iy-ytf+iz-zy+il-ltf 

 and 



i[w^(x — x Q )+ w 2 (y—y ) + w s (z-Zo) -f w 4 (Z — Z )] =v — v . 

 .*. the equation of the electron becomes 



We shall now calculate the potential-four-vector due to 

 the motion of electron at an external point (a, b, c, \). 

 We have 



*-±{N( Po{w)dxdydzdl 



"j}Jm*-ay+(y--by + (z-cy+Q-\yy {Wa) 



the integration being extended over the whole world- space 

 enclosed by the electron. 



