by a Magnet in a Rotating Conductor, 521 



Further, V 1 and V 2 being the values of the potential of free elec- 

 tricity at any two points of the surface of the conductor, it fol- 

 lows that 



Vj — V 2 = — 2wfyt(cos 7 t — cos y 2 ) . 



If these two points be connected by a stationary conducting 

 wire of considerable resistance, the latter will be traversed by a 

 current whose intensity will be proportional to the difference of 

 the values of the potential, and therefore to the difference of the 

 cosines*. Now, according to a formula given by Green, we 

 have in the case of the sphere, 



Vfl ~ 4ttR JJ r 3 ' 



where V denotes the value of V at the element d$ of the sphe- 

 rical surface, p the distance of the external point from the sphere's 

 centre, and r its distance from the element dS. The integra- 

 tion is to be extended, of course, to all points of the surface of 

 the sphere. In the case under consideration, 



*-- *e$N* 



5. The solution of the general problem is considerably sim- 

 plified by the assumption of a velocity of rotation so small that 

 the intensity of the induced currents is sufficiently weak to 

 justify our neglecting, in presence of the direct inductive action 

 of the magnet, the induction which takes place between the 

 several parts of the conductor*}*. By this assumption the ex- 

 pressions (11) are reduced to their first terms. For if n denote, 

 as before, the velocity of rotation, the effective electromotive 

 forces as well as the generated current-densities u, v, w will be 

 magnitudes of the order nk. Consequently the expressions 

 within brackets which, multiplied by k, constitute the second 

 terms in the formula? for L, M, N will also be of the order nk ; 

 whilst in the equations (15*) those portions of the expressions 

 nL, wM, nN which proceed from the terms in question will be 

 magnitudes of the order n 2 k 2 . Even when the velocity of rota- 

 tion is tolerably great, the latter terms may, without incurring 

 appreciable error, be neglected in presence of the terms of the 

 order nk, seeing that the value of the constant of induction k is 



* Weber's unipolar induction. 



f This simplifying assumption has in fact been tacitly made in all pre- 

 vious investigations in connexion with the subject. 



