by a Magnet in a Rotating Conductor, 509 



and the velocity with which the conductor rotates remain unal- 

 tered. The electromotive force thus induced in any element of 

 the conductor depends partly upon the relative motion of the 

 element with respect to the given external magnetism, partly 

 upon its motion relative to the system of currents existing in 

 every other part of the conductor and fixed in space, and partly 

 also upon the distribution of free electricity which ensues on the 

 surface and also, as will be seen, on the interior of the conduc- 

 tor. Let u, b, to be the components, referred to three stationary 

 rectangular coordinate axes, of the velocity with which an element 

 at x, y, z within the conductor moves ; further, let u } v, w be 

 the components, referred to the same axes, of the intensity of 

 the current at the point x,y,z ; and similarly let u\ v } , w denote 

 the components of the current in any other element of the con- 

 ductor at x J , y', z'. It will then follow from Weber's law* that 

 the electromotive force induced in the conductor-element, moving 

 with the component velocities u, b, to, by the current-element 

 u', v ! , u>', existing in the space-element dx ! dy' dz\ is expressed by 

 the formula 



— - — ^— ( cos e — ~- cos $ cos 3' J dx 1 dy 1 dz\ 



wherein 



r= \/( X <- X Y + {yl-yY-\-(z'-zf 



denotes the distance between the elements of the conductor and 

 of the current ; 



the density of the current in the point at, y ! , z f ; 



OT = >/u 2 + b 2 + fo 2 



the absolute velocity of the moving conductor-element; e, 3, 

 and 3' respectively the angles between the directions of i' and &, 

 of -cr and r, and of i 1 and r ; and lastly, 



155370. 10 6 " 



the constant of induction referred to millimetres and seconds as 

 units. Accordingly 



* Electrodynamische Maassbestimmungen, 1st memoir, p. 339; and 4th, 

 p. 268. In the latter memoir the intensities of currents are estimated by 

 electro-magnetic units of measures, whilst electromotive forces are referred 

 to absolute mechanical units. 



