Theory of Dynamo-electrical Machines. 129 



form the equation 



E = Ri, 



by the application of which the equation (II. a) passes into 



m =[M p+ M v -H^^¥ ■ (31) 



If we eliminate from this equation the factor i which occurs 

 on both sides, and then transfer the members on the right 

 side, which have no denominator, to the left, we have 



r a + iy b + ij b + i ' 



or written differently, 



1= *( p + AY* w -j±. «-^ r (32) 



a + i\ r b + i/U + pv + av 2 b + i U + pv + av 2 v y 



For the sake of abbreviation we will introduce the sign w with 

 signification 



(33) 



w= 



U + pv + av 2 ' 

 by which we obtain 



a + % 



(P + 7 . W— t rvw. . . . (34) 

 F b + ij b + i v y 



We can determine by this equation the strength i of the 

 current produced by the machine. If we multiply the equa- 

 tion by a + i and b + i, and arrange the members in powers 



of i, we get 



i 2 — {pic—- \vw— a — b)i— (pb + q)w + \aviv + ab = ; (35) 

 and by solving this quadratic equation we obtain 

 iz=±(pw — \vw — a—b) 

 ±:^\/( K pw—\vw—'a~—b) 2 + / i(pb + q)w—4:'kavw—£ab. (36) 



Of the signs before the root the top one is the only one ap- 

 plicable. For the other one would give negative values of i 

 for all values of v, which would bo opposed to the meaning of 

 the magnitude i defined by this equation. For this equation 

 is derived from equation (34), where i only occurs in the deno- 

 minators ; and these denominators must be understood to have 

 the meaning that the sign i in them signifies the absolute value 

 of the current-strength, and is therefore an essentially positive 

 magnitude. Using the upper sign we get positive values of 

 i for all the greater values of v, which must be looked upon as 

 valid; and only for small values of v which are below a 



Phil. Mag. S. 5. Vol. 17. No. 104. Feb. 1884. K 



