of Unipolar Induction. 413 



plane makes with the horizontal plane. The cosine of this 

 angle is evidently equal to 



\/R?-\-r 2 + ZRrcosu 



\/W+TC 2 + r 2 + 2Rr cosm" 



The velocity of the element of circuit will be then 



v \/W + r 2 + 2Rr cos u ; 



the angle which the direction of movement makes with the 

 line of junction between the element of circuit and that of the 

 magnet will be 90°, and consequently its sine = 1 ; the dis- 

 tance between the two elements will then amount to 



VH 2 + E 2 + r 2 + 2Rrcosw. 



Lastly, the cosine of the angle between the vertical element 

 As and the normal to the plane passing through the direction 

 of movement and the element of the magnet will be 



\/R 2 + r 2 + 2Rrcosw 



VH 2 + JR 2 + r 2 + 2RrcosM 



M 



If now we introduce these values for -%, (3, and i/r into the 



formula (6), drawn from the mechanical theory of heat, we 

 obtain for the inductive action of the element of the magnet 

 M.rdu, situated at c, upon the element of circuit As, when both 

 are in rotation in the same direction and with the same angular 

 velocity, the following expression : — 



Mrv ( R 2 + r 2 + 2Rr cos u) Asdu 



(H 2 + K 2 + r 2 + 2Rrcosw)* 



Multiplying by 2 the integral between the limits and it 

 of this expression, we obtain the sum of the whole inductive 

 action upon the element of circuit As of the magnets situated 

 upon the periphery of the magnet. But 



C u =" (R 2 + r 2 + 2Rr cos u)du _ C u ~ 

 J_ (K 2 + R 2 + r 2 + 2Rrcos^)|-J_ 



-H 2 



Jm = 



die 



(H 2 +R 2 + r 2 + 2Rrcosw)i 

 du 

 (H 2 + R 2 + r 2 + 2Rrcosw)*' 



it u 



If we remember that cos u — 1 — 2 sin 2 — , and if we put - = (f>, 



and, lastly, ^ — - = sin ®, we obtain for the two 



J '[H 2 + (R + r) 2 ]* 



