Properties of the Electric Field. 161 



The electromotive intensity X parallel to the axis of x is 

 equal to the mean of dT/df. 



Hence 



X=±7r(w%(fw-hu) -vtigu-fv))} 



= (wfi-vy) ? • • ( 7 ) 



= Pw—<yv. ) 



Similarly, if Y and Z are the components parallel to the 

 axes of y and z, we have 



Z = otv—/3u, 



where a bar placed over any quantity indicates that the 

 mean value of the quantity is to be taken. 



Thus, when a system of tubes of electrostatic induction is 

 in motion, the electromotive intensity is at right angles both 

 to the resultant magnetic force and the mean velocity, and 

 is equal to the product of these two quantities into the sine of 

 the angle between them. 



From the preceding equations, we see that if we have a great 

 number of positive and negative tubes moving about, if 

 the positive tubes move in one direction and the negative 

 ones in the opposite, there will be a resultant magnetic force, 

 but if there are as many positive as negative tubes in each 

 unit of volume, there will be no resultant momentum ; and if 

 there are as many moving in one direction as the opposite, 

 there will be no resultant electromotive intensity, due to the 

 motion of these tubes. We see then that when the electro- 

 magnetic field is in a steady state, the motion of the tubes of 

 electrostatic induction in the field will be a kind of shearing 

 of the positive past the negative tubes, the positive tubes 

 moving at one direction, and the negative at an equal rate in 

 the opposite. When, however, the field is not in a steady 

 state, this ceases to be the case, and then the electromotive 

 forces due to induction are developed. 

 We get from equation (7), 



dZ dY d . . d N d t > ( du Q du du\ 

 dy-Tz = T X W + dy ^ + dz (wx) -VTx+Pdy ^ dl )' 



since 



^ + *? + ^y = o. 



dx dy dz 



If we suppose the magnetic lines of force to be moving with 

 the velocities u, v, w, the right-hand side of this equation 

 represents the rate of diminution of a. 



