of Unipolar Induction. 409 



right angle, by cos (ACa). It will be necessary, finally, to 

 multiply this product by the cosine of the angle made by the 

 direction of movement with the normal to the plane aCP, 

 i. e. by cos (bCGt). But the plane GCB being perpendicular 

 to the plane 6CB, the dihedral angle B is a right angle. We 

 obtain consequently from the spherical triangle, 



BGb : cos (6CG) = cos (GCB) : cos (6CB). 



The law of induction (5), deduced from the mechanical 

 theory of heat, obtains thus the following form : — 



im= ^cos (oOA) cos (GCB) cos (JOB) . A As. . (5b) 



r 



We will now transform formula (6) in a similar manner. 



As before, let aC denote the element of the circuit As, and 

 let bC indicate the direction of movement or the direction of 

 the current of translation produced by the motion of the ele- 

 ment of the circuit. This current must be multiplied by 

 sin {bCT). or, what comes to the same thing, by cos (6CB). 

 The magnetic pole tends to conduct it into the normal to the 

 plane passing through CP and bC, i. e. in the direction FC. 

 To obtain the component in the direction aC of the element 

 of the circuit it will be necessary to multiply, finally, by 

 cos (aCF). The expression for the induction thus becomes 



^=cos(aCF)cos(&CB)AA* (a) 



In the spherical triangle aAF, the dihedral angle at A is a 

 right angle ; we obtain therefore 



cos (aCF)= cos (aCA) cos (FCA). 



Introducing into the formula (a) these values of cos (aCF), 

 and recollecting moreover that cos (FCA) = cos (GCB), we 

 have the desired formula of induction, viz. 



M 



im= ^cos (aCA) cos (GCB) cos (bGB\kAs, . (5b) 



r 



a formula identical with the formula (5b). 



The formula (6) established by me for the calculation of 

 the unipolar induction in an element of the circuit moving in 

 a magnetic field ought therefore to give correct results. 



It is evident that the magnitude of the induction depends 

 on the relative movement between the magnetic pole and the 

 element of the circuit. But the relative movement is not 

 altered if we give to the pole and to the element equal and 

 parallel velocities in the same direction. Having regard to 



Phil. Mag. S. 5. Vol. 24. No. 150. Nov. 1887. 2 E 



