Theory of Dynamo-electrical Machines, 49 



of being connected with the beginning of the following wind- 

 ing, or of the following group, is connected with its own 

 beginning. The rotating coil and the fixed circuit may ac- 

 cordingly be regarded as systems of closed conductors. In 

 respect of the magnetism of the masses of iron, we may, as is 

 well known, consider the electrodynamic action of magnets as 

 made up of innumerable small closed currents, and can deter- 

 mine their action. We are therefore only concerned with the 

 actions of closed currents on closed conductors, by w r hich the 

 discussion is greatly simplified, and at the same time rendered 

 more certain ; for the laws of induction which hold for closed 

 currents and conductors may be regarded as being firmly 

 established, while differences of opinion still prevail as to the 

 action exerted by unclosed portions of currents on unclosed 

 portions of conductors. 



The statement of the law of induction is greatly simplified by 

 applying the electrodynamic potential of the effective currents 

 on the conductors supposed to be traversed by the unit of cur- 

 rent. Given a system of closed currents which exist in the 

 conductors s, s^ s 2 , &c, and are of the strengths i, i^ i 2 , and, 

 further, a closed conductor which we suppose traversed by 

 unit current. Suppose, then, that els is an element of any 

 one of the conductors s, s 1? s 2 , & c -^ that da is an element of 

 the conductor a, while (sa) is the angle between the directions 

 of the two elements, and r their mutual distance, the electro- 

 dynamic potential W of the given system, on the current of 

 unit strength in the conductor <7, is defined by the equation 



^-ffi^iM^; (1) 



H\ 



in which one integral is to be extended to the series of con- 

 ductors s, 5 1? s 2 , &c, and the other to the conductor a. 



It is presupposed in this expression for W that the currents 

 are measured in electrodynamic or electromagnetic measure. 

 In order to apply electrostatic measure the expression must 

 be divided by K 2 , where K is the critical velocity of electricity 

 which is approximately equal to 30 quadrants of the meridian. 



If, now, any motion of the conductor, and at the same time 

 a change in the strengths of the currents i, i^ i 2 , &c, takes 

 place, the electromotive force e thereby induced in the con- 

 ductor is defined by the simple equation 



— f <» 



in which t is the time. 



Phil. Mag. S. 5. Vol. 17. No. 103. Jan. 1884. E 



