GRAPHICAL PROPERTIES OF THE CHARACTERISTICS. 



727 



The problem is always possible when the point C' is below the 

 curve that is to say, when we have <<(/ ). If this point is above 

 the curve at C", the right line of construction C"B" may meet the 

 curve at two points N and N', the former of which represents an 

 unstable equilibrium. Finally the problem is impossible if this right 

 line does not meet the curve. 



The same constructions give also a representation of the work. 



The total energy expended in unit time W = E 1 I, is equal to 

 n times the area of the rectangle PQ (Fig. 257). 



x 

 The ratio of the useful to the useless work is ; by taking 



AB X = , this ratio is equal to J- or * . The useful work 



Fig. 258. 



is then n times the area of the rectangle QM X , and the value of the 



. 



efficiency u is 





If the right line CB X be prolonged until it meets the curve at M', 

 CP' represents the current I which we should have with the re- 

 sistance R alone. To obtain the maximum useful work, the point M 

 of the curve must be determined such that the surface of the rect- 

 angle QM X is a maximum. 



1275. Analogous considerations enable us to represent the 

 action of dynamos, when the inductors are excited in the shunt. 



If I' is the current of the inductors, 2 = 1-1' the external current, 

 a the resistance of the ring, b that of the inductors, and putting 



<o 



