Properties of the Electric Field, 159 



both the direction of motion of the tube and the magnetic 

 force produced by it. 



From equation (3) we see that 



dZ dY du , da /dj3 , dy\ , /dv dw\ „du du 

 or, since 



we have 



^ + ^? + ^ = o 



dx dy dz ' 



dZ dY d . s , d , v , d . x du n du du ,.. 

 dy~Tz = d-J M)+ Ty^ + dz^-^-^d-y^Tz- < 4 > 



The right-hand side of this equation is equal to — -7-, the 



rate of diminution in the number of tubes of magnetic force 

 parallel to the axis of x. 

 Hence, since 



§(Xdx + Ydy + Zdz) 



taken round a closed circuit is equal to 



jj{'(f-f)-(f-f)-(S-f)}* 



taken over any surface entirely surrounded by the circuit, if 

 I, m, n are the direction-cosines of the normal to the surface, 

 we see by (4) that the line-integral of the electromotive force 

 taken round a closed circuit is equal to the rate of diminution 

 of the number of lines of magnetic force passing through the 

 circuit. 



Collecting these results, we see that a tube of electrostatic 

 induction when in motion produces (1) a magnetic force at 

 right angles to the tube and the direction of motion, (2) a 

 momentum at right angles to the tube and the magnetic force 

 produced by it, (3) an electromotive intensity at right angles 

 to the direction of motion of the tube and the magnetic force 

 produced by it. 



The momentum and the electromotive intensity are thus 

 both in the plane containing the direction of the tube and its 

 velocity ; the first of these is at right angles to the tube, the 

 second to the velocity. 



We have hitherto only considered the case of one tube, or 

 rather of a set of tubes, moving with a common velocity. We 

 can, however, without difficulty extend these results to the 



