406 Prof. E. Edlund on the Theory 



of motion of the jacket, cos <£ will be unity. If the jacket 

 moves with the velocity h } the work done in unit time by 

 the action of the pole upon the current dz will be equal to 



Mr(I-i)hdz 



In a similar manner, we obtain for the north pole 



-Mr(l-i)hdz 

 ^ + {l + zf^ 



By integration of these expressions between z = and z = l, 

 we have for the whole jacket 



— {l - i)h \w^i~w^^y * * () 



Multiplying this expression by A, we obtain the quantity of 

 heat corresponding to the amount of work in question. 

 Consequently we have from the formulae (2) and (4), 



t Wl-fl=2M(I-2)7t.-( - 1 - ; ^-rl- 



If we remove the battery from the circuit so that 1 = 0, the 

 preceding equation nevertheless continues applicable if the 

 jacket is maintained in rotation by an external mechanical 

 force so that h does not become equal to zero. In this form 

 the equation shows that the square of the intensity of the in- 

 duced current multiplied by the resistance of the circuit is 

 equal to the mechanical work consumed in the rotation of the 

 jacket. If this work is zero, the intensity of the current will 

 also be zero. 



If we suppose that the jacket consists only of an element 

 of the circuit As, we shall obtain, by equating the expressions 

 (2) and (3), after dividing the first by A and putting I equal 

 to zero, the following law for the induction in an element of 

 the circuit in rotation round a magnetic pole : — 



•2 M • • A7 A 



i'm= -2 i sin a cos (palls 



or 



M 



im= -^-sinacos (j>hAs (5) 



Consequently, when an element of the circuit is in rotation 

 round a magnetic pole, the induced current which results is 

 proportional to the magnetic intensity at the place where the 

 element is, multiplied by the sine of the angle which the ele- 



