54 PARTICULAR CASES OF INDUCTION. 



are proportional to the same function <j>(I) of the intensity. The 



efficiency 



_ 



P ~ 



is about equal to the ratio of the velocities of the two machines, 

 and the useful work in unit time is expressed by 



E'l 



As the strength I is a function of n - ri, the maximum useful 

 effect can only be determined if we know the form of the function <. 



558. The preceding considerations only apply rigorously to the 

 case of a uniform current like that obtained with a machine such 

 as Faraday's disc. In a periodical machine, for instance, a frame 

 turning with a uniform motion about a vertical axis under the 

 influence of the terrestrial field, the phenomena are more com- 

 plicated. 



If the machine is used as an electromotor, the ends of the 

 wire being joined by sliding contacts with an external conductor 

 in such a way that the coefficient of mutual induction of the two 

 parts of the circuit may be neglected, the current should satisfy 

 equation (518) 



in which L is the coefficient of self-induction of the whole circuit, 

 R the resistance, and Q the flow of terrestrial force across the 

 frame. If S is the surface of the frame, H the horizontal com- 

 ponent of the terrestrial field, and T the period of rotation, we 

 may write 



Q = HS sin27r-. 

 / 



The electromotive force is sinusoidal, like that which we investi- 

 gated in (535). When the permanent regime has been established, 

 the current is itself periodic, and the work necessary to keep up 

 the motion corresponds altogether to the heat disengaged in the 

 circuit, which for unit time gives 



* H2S2 



