98 Carhart — Direct and Counter Electromotive Forces. 



its asymptotes are equal, that is, AH equals DR for instance, 

 when the counter B. M. F. passes through the series of values 

 represented by the Ordinates DR, CQ, TS, BQ, AR in succes- 

 sion, the current will be proportional to AR, BQ, TS, CQ, 

 DR in succession. With the condition that the motor absorbs 

 a constant amount of energy per second, the current is a maxi- 

 mum when the counter E. M. F. and the electrical efficiency 

 are a minimum : and it diminishes continuously as the effi- 

 ciency increases. 



If now the the constant chosen is not the rate at which 

 energy is absorbed by the motor, but the electrical energy con- 

 tributed by the generator per second, we then have the equa- 

 tion 



CE= E ( E - E /) _ W; or E 2 -EE =W . (3) 



This is again an equation of the second degree and its locus is 

 the hyperbola II. Its axis makes an angle of 157° 30' with the 

 axis of X and its asymptotes are OG- and the axis of Y. Its 

 eccentricity is 2'613, the second value obtained from the equa- 

 tion e 2 =4rb v/8, the other value being the eccentricity of hyper- 

 bola I. 



In this case the two machines and the conductor joining 

 them form a system the only condition laid down being the 

 constancy of RW. Hence the relation between B and E, may 

 be found either when the motor is driven by the current and 

 furnishes counter E. M. F., or when it is driven by mechanical 

 means so as to contribute to the direct E. M. F. If the motor 

 is actuated by the current then E has its minimum value when 

 E^O, or at the point where the hyperbola crosses the axis of 

 X; but if the motor dynamo contributes E. M. F. in the same 

 direction as that of the other machine, then E may diminish 

 still further, even to zero in the limit. In this case RW repre- 

 sents still the energy furnished by the generator, but not the 

 total electrical activity per second. The part of the hyperbola 

 lying below the axis of X corresponds to values of E and E y 

 obtained when the motor is driven mechanically so as to fur- 

 nish direct E. M. F. 



In the first curve any assumed value of E greater than the 

 minimum gives rise to two values of E„ both positive. In the 

 equation of the second curve the substitution of a positive 

 value of E / gives two values of E, one positive and the other neg- 

 ative, indicating another branch to the curve. This other 

 branch represents the relation between E and E / when the part 

 played by each machine is reversed, the motor becoming the 

 generator, and the generator becoming the motor by being 

 driven electricallv. 



