H. S. Carhart — Direct and Counter Electromotive Forces. 95 



Aet. VIII. — Relation between Direct and Counter Electromotive 

 Forces represented by an Hyperbola ; by H. S. Carhaet. 



In the usual discussions respecting the relations between the 

 electromotive force (E. M. F.) of the generator, the counter E. 

 M. F. resulting from the electromagnetic reactions taking place 

 in the motor, and the rate at which energy is absorbed by the 

 latter in the electrical transmission of power, it is implicitly 

 assumed that E is constant. Thus the equation, representing the 

 division of the whole electrical energy spent in the circuit in 

 unit time into heat and mechanical work, viz : 



CE=C 2 R + W, 



is differentiated so as to obtain the first differential coefficient 



of W, with respect to C, R and E being constants. Hence 



dW 



-^^E — 2CR=0 for a maximum, the second differential co- 

 d\j 



E 

 efficient being negative. C is therefore equal to -^, or the 



current is reduced to half the value it would have with the 

 motor at rest, by the reduction of the effective E. M. F. to 

 one-half. 



If, however, we take the equation expressing the electrical 

 energy absorbed by the motor per second as the product of the 

 counter E. M. F. and the current, we have, as is well known, 



E '( E R ^ E ' ) =W / or E,(E-E,)=RW, . . . (1) 



From this it appears that if R and W / are constants, E/E, — E,) 

 is also a constant. But when the product of two factors is a 

 constant, their sum is a minimum when they are equal to each 

 other. It follows that the sum of E, and E-E„ or E, is a 

 minimum, when E,=E — E /t or when E^-JE. By substitu- 

 tion in equation (1) E / =. N /RW / . 



With an assumed amount of work spent upon the motor per sec- 

 ond and a given resistance R, E has a minimum value equal to 

 twice E r This corresponds with Jacobi's law of maximum rate 

 of working, or greatest electrical activity and constant E. M. F. 

 Evidently the conditions of greatest activity and constant E. M. 

 F. of the generator are identical with those of constant energy 

 absorbed by the motor and minimum E. M. F. of the generator. 



Equation (1) gives 



E^-EE^-RW, . . . . (2) 



an equation of the second degree and therefore of a conic sec- 



