APPENDIX B. 375 



and the ratio of the electromotive forces (a) to the electromotive 

 forces (b] is equal to the ratio of n to n f . In order to establish 

 these propositions, consider the arrangement shown in Fig. 7, 

 namely, a squirrel-cage rotor placed in a uniform magnetic field h. 

 The radial component of this uniform field at the surface of the 

 rotor is equivalent to a harmonically distributed flux. Further- 

 more, suppose that the uniform field h pulsates in value in 

 accordance with the equation 



h = Hsm2Trnt. (i) 



The motion of the rotor induces electromotive forces in the regions 

 AA f , and the pulsation of the field produces electromotive forces 

 in the regions BB' '. Con- 

 sider two pairs of opposite 

 rotor rods // and qq t as 

 shown in Fig. 8, the angle /3 



having any given value. The 



, . . , 



above propositions are estab- 



lished if we can show that the 

 electromotive forces induced 

 in the rods pp by pulsation 



Fig. 8. 



are njn' times as great in 



value and in time quadrature with the electromotive forces which 

 are induced in the rods qq by motion. Let r be the radius 

 of the rotor and / its length parallel to the motor shaft. 



The radial component of h at the rotor rods qq is equal to 

 h cos /3, and the velocity of the rods qq is equal to 2irn r r. 

 Therefore the electromotive force which is induced in each of the 

 rods q by motion is equal to 2irn r r x / X h cos /3, or, substi- 

 tuting the value of h from equation (i), we have 



sin 2irnt. (ii) 



The component of h which is perpendicular to the plane of 

 the turn of wire which is formed by the rotor rods // is equal 



