ADDITIONS TO THE THEORY. SECOND APPLICATION 99 



It would be necessary, in order to obtain E 2 , to add to the segment 

 0\A 2 (Fig. 47), a section 



It amounts to the same thing if this segment is drawn parallel to 

 and to the left of A i} along AiA'i, and if a circular arc, of radius EI, 

 is drawn from A' 2 , to cut OA 2 precisely at the point O'\. 



Now if we join DA'\, it is seen that the triangles DA\A\ and 

 DA 2 B are " similar" triangles, since the side DA\ and DB, also 

 A\A'\ and BA 2 are respectively perpendicular and proportional each 

 to each. It may be deduced from this that the line DA\ is per- 



FIG. 48. 



pendicular to DA 2 . On the other hand, the vectors A<J) and DA'\ 

 being the resultants of the loss of potential in the motor and in the rest 

 of the circuit, evidently represent ZI W and ZI d , respectively, when Z 

 equals the total impedance of the circuit. The diagram composed of 

 the right lines A 2 D, DA\, A\A'\, A 2 O, is, therefore, nothing more 

 than the diagram which would have been obtained directly by the first 

 method. Q.E.D. 



Fig. 48 shows the same analysis for the case where I d is lagging 

 (I d being positive). The precision of the power-formula already given 

 can also be verified; for, in this formula, 



PZ= 2lw 



