764 Sir Oliver Lodge on 



them generally is not easy ; though it appears physically 

 necessary that 



x must develop into a series of circular functions of odd 

 multiples of pt, while 



y develops into a similar series with even multiples of the 

 same quantity. 



The series may be stopped at any stage by ceasing to 

 provide resonating circuits and permitting radiation and 

 dissipation of energy. 



To prove the above equations — remembering that they only 

 have that simple form when tuned circuits are provided for 

 every intended frequency — write the number of lines of force 

 from stator to rotor as N ; and from rotor to stator as N' ; 

 then the E.M.F. generated is dN/dt or dWjdt respectively. 



mx. 



But N = wi#, and N' = 



and w = M cos|i, i. e. is a sinuous function of 



time as the armature revolves. 



Hence, if the rotor circuit resistance is R, while that of 

 the stator is R', and if the conditions of § (6) are satisfied, 



E 1 dN 1 d . . M d , iN 



^ ,== t=t = tt— tt — r>~r vny) = -77^7 (y cos pt). 

 R R dt Udt y JJ Rdt yj rJ 



Similarly, if ?/ is the stimulating current initially supplied 

 to the stator, the current in it is 



M d , 



So they are proved. I have not yet succeeded in solving 

 them generally, but it is not difficult to proceed by stages. 



Working out of the Equations. 



To deal with these equations easily, take them piecemeal, 

 and apply them to say four provided branch circuits beside 

 the original exciting circuit of stator. Two of these tuned 

 circuits are connected to rotor, with resistances R 3 and R 3 , 

 two with the stator, with resistances R 2 an( l ^4- The latter 

 circuit involves useful dissipation by radiation as well as all 

 hysteresis and ohmic losses. 



The initial current in stator being y , the current first 

 generated in rotor is 





