Electrodynamics and CrSmieu's Experiment. 329 



Hence t is only derivable from a potential function at places 

 where a vanishes. 



In this equation a is the actual electric force. If the 

 velocity is small compared with v, we may for an approxi- 

 mation put o- equal to the electric force in a state of rest. 



From the equations satisfied by t the value of the line- 

 integral of t is the same for all circuits that can be deformed 

 into one another without leaving the free aether, in which 

 alone we can be certain that the equations are satisfied. If 

 the charged body is not infinite in length, the circuit can be 

 reduced to zero, and hence the line-integral is zero. The line- 

 integral of Yya does not vanish, and thus the second term 

 cannot cancel the first. Hence there must be some magnetic 

 force around the moving body. 



If the assumption XI. is not true, the velocity of the aether 

 must, at a sufficient distance from the body, vary inversely 

 as the cube of the distance. The magnetic force just found 

 varies inversely as the square, and hence cannot be cancelled 

 out by any effect that a motion of the aether can produce. 

 Whether XI. be true or not, there must be some magnetic 

 force produced. 



If the body is infinitely long, the argument fails. If, 

 however, the infinitely long body be cut through at any point,, 

 however narrow the gap may be, the argument applies. 



9. If the body moves inside an infinite tube the results of 

 the last paragraph hold for the space inside the tube. In the 

 case of the space outside there is some difficulty, as we 

 cannot prove that the term t is unimportant for our purpose. 

 If, however, we take the tube to be of finite though consi- 

 derable length, and suppose the body to be at a distance from 

 the end, the argument applies; and the magnetic force outside, 

 which reduces to t , has its line-integral of value zero. Hence,, 

 if everything is symmetrical about the axis of the tube, Tq, 

 vanishes ; and in this case at least there is no magnetic force 

 outside the tube. 



10. If we suppose the tube not to be continuous, but to 

 consist, for example, of a gilt glass tube where the gilding is 

 separated into a number of narrow rings by transverse 

 scratches, it seems clear that, if unelectrified, it will produce 

 little or no effect on the field of electric force in the case 

 where a single charged body moves along the tube. There 

 will therefore be an external magnetic field. If a number of 

 such charged bodies separated by narrow gaps move along 

 the tube, there must be an external magnetic field. An 

 electrification of the tube sufficient to produce an electric 

 field equal but opposite to the mean of the nearly constant 



Phil Mag. S. 6. Vol. 1. No. 3. March 1901. Z 



