1887.] On the Theory of the Alternate Current Dynamo. 169 



Let I be the total induction in the magnet core, and let at time t I 

 be distributed into I' through A x ', I" through A/' and V" as a waste 

 field to the neighbouring poles. 



The line integral of magnetic force from the pole to either adjacent 

 pole is I'" /Jc, where h is a constant. 



We have first to determine I', I", I'", in terms of x and y. 



Take the line integral of magnetic force in three ways through the 

 magnets, and respectively through area A 1 ', through area A/', and 

 across between the adjacent poles — 



/ I \ I" 

 ^\Ay ^ r ^ l ~A T ' = ^ 7rmx ~^ 7rn y-> 



/ 1 \ r" 



whence I' + r' + I'" = I 



When t, x, and y are given, this would suffice to determine I by means 

 of the known properties of the material of the magnets as represented 

 by the function f. We will, however, consider two extreme cases 

 between which other cases will lie. 



First. — Suppose that the intensity of induction in the magnet cores 

 is small, so that Z 2 /(I/A 2 ) may be neglected, the iron being very far 

 from saturation. We have — 



r ~ r = ^{"»(Ai'-V)«+»(Ai'+.V)y} 



= |™(^]Cos-^ + 6 3 cos — + ...)x 



We see that the coefficient of self-induction 7 in general contains 

 terms in cos (47r£/T). 



Second. — In actual work it would be nearer the truth to suppose 

 that the magnetising current x is so great that the induction I may 

 be regarded as constant, and the quantity Z 2 /(I/A. 2 ) as considerable. 

 But as small changes in I imply very great changes in Z 2 /(T/A 2 ), its 

 value cannot be regarded as known. We have then — 



