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



tion. On applying the proper criterion it readily appears that 

 the locus of this equation is an hyperbola. For the purpose of 

 a graphical representation of the continuous relation between E 

 and E ; , let us assume E. W / equal to 225 ; minimum E will then 

 equal 30 units. 



Solving equation (2) for E, and substituting in the result suc- 

 cessive different values of E as the independent variable, we 

 obtain corresponding values of E, which satisfy the equation. 

 Each assumed value of E gives two plus values of E r The 

 hyperbola I is thus plotted, making the assumed values of E 

 abscissas and the corresponding ones of E, ordinates. E / is 

 considered essentially positive. 



By a comparison with the most general equation of a conic 

 section, the constants of this particular one may be deter- 

 mined. 



Thus independently of the assumed value of EW ; , e 2 =4± \/8 

 and e=l"0824, the other value of e making n imaginary. 



Also cos a=db- and a=22° 30', the minus value of the cosine 

 e 



not being admissible because then m would become negative. 

 But all values of E are positive, and hence m cannot be neg- 

 ative. 



Since the cosine of the angle between the transverse axis of 

 an hyperbola and its asymptotes is the reciprocal of e, it follows 

 that the asymptotes of this curve are the axis of X and the 

 diagonal OGr, the angle between them being 45°. The charac- 

 ter of the curve is thus entirely determined without knowing 

 the value of RW / . 



Assuming now RW / equal to 225, the coordinates of the 

 focus are rn = 32*96 and n = 13 - 65. 



It is evident also from the properties of the hyperbola that 

 m and n in this case equal A and B respectively. The perpen- 

 dicular distance of the directrix from the origin is 30*45. 



A number of inferences can be drawn directly from the curve 

 by inspection. It is seen that E is a minimum when it is double 

 E r At this point the hyperbola is tangent to an ordinate ; the 

 point of tangency will travel along the line OF, away from the 

 origin or toward it according as the assumed value of RW / 

 increases or diminishes. OF is therefore the line of minimum E 

 and its equation is E y "=-J E. 



The electrical efficiency at any point of the curve, being the 

 quotient of E y by E, is the tangent of the angle which a line 

 drawn from the point to the origin makes with the X-axis. 

 The efficiency at T is the tangent of the angle TOE, or one- 

 half. The electrical efficiency increases from zero, when E, 

 is zero and E is infinite, to unity, when E, equals E and both 

 are infinite ; the angle, whose tangent is the measure of the efn- 



