304 Sir J. J. Thomson on a Theory of 



between AB and A' B' is 



1 ire 2 dr 1 ire 2 dr 



Thus the energy in one half of the cone is equal to 



J" 



2 co r- ~ 2 



And the energy in the double cone is 7re 2 /coa. This is greater 

 than if the lines of force were uniformly distributed round 

 the corpuscle in the proportion of 2ir to co. 



Energy in a tube moving with uniform velocity. 



Suppose the tube is moving with a velocity u in a direction 

 making an angle with the axis of the tube. The velocity 

 of the tube at right angles to itself is u sin : there is 

 therefore a magnetic force at right angles to the tube equal 

 to B sin 6 u. The energy due to this magnetic field is equal to 



1 RVsin g fl 



<5tt c 2 



per unit volume. Thus the energy due to the magnetic 



9 

 11 



field is -J sin 2 6 times the energy due to the electrostatic field. 



The expression for the electrostatic energy obtained in the 

 last paragraph requires modification, as when the lines of 

 electrostatic force are moving they suffer a compression 



parallel to the direction of motion equal to \ / 1 -, 



which we shall denote by q ; we can easily show that this 



/ . cos 2 0\f 

 contraction diminishes the solid angle to q i sin 2 6-\ ^— I co, 



and since the energy is inversely proportional to the solid 

 angle, the total energy, electrostatic and magnetic, is equal to 



tt e 2 \ -, u 2 . „ - } 



/.«/,, cos- 0\u- ale 2 J 



co qy sin- v-\ 2 — I 



The kinetic energy thus depends on 0, and the motion will 

 not be steady unless 6 has the value which makes the 



kinetic energy a maximum, i. <?., unless 6= —, so that the 



