Theory of Dynamo-electrical Machines. 55 



coil. This potential is proportional to the length of the por- 

 tion, and therefore approximately proportional to the fraction 



- ; and as in the course of the calculation a multiplication by 

 n 



n occurs, we acquire for E 2 an expression which is almost in- 

 dependent of n. 



In determining the value E 2 we are dealing with the poten- 

 tial of a portion on itself, and the potential of one portion on 

 another. With the exception of a member occurring in the 

 former, which is independent of the length, these two poten- 

 tials are proportional to the square of the length of a portion, 



and are therefore approximately proportional to the fraction -g; 



and as a multiplication with n is necessary to determine the 

 total action, a value of E 2 must result which is approximately 



proportional to the fraction — 



Hence, between the magnitudes E x and E 2 there is this differ- 

 ence — that by changing the number n of divisions into which 

 the rotating coil is divided, the former is but little, while the 

 latter is very considerably affected, and so that E 2 decreases 

 with an increase in n. Now as E 2 is negative, and thus repre- 

 sents a loss of electromotive force, it is advantageous to make 

 the number n as large as possible. 



After we have got expressions for the electromotive forces 

 Ei and E 2 acting in the entire circuit, in the equations (6) 

 and (7), by adding them we obtain an expression for the 

 whole electromotive force E, and thus arrive at the expression 



E = tt(W-W">-fm>. • . • • (8) 



§ 6. Work done by the Electromotive and the Ponder omotive 



Forces. 



In order to express the work which is done in unit time by 

 the electromotive force E previously determined, we must 

 multiply the latter by the strength of the current which 

 flows through the two halves of the rotating coil — that is, 

 by i. If we use, then, for E the previous expression, we get 



Ei=w(W / -W // )w-piV .... (9) 



This work may be compared with another expression for 

 work. The coil traversed by the current experiences from the 

 other parts of the system, to which the magnets also belong, 

 a ponder omotive force, the work of which we will also determine. 



If we consider a single portion of the rotating coil which is 



