Theory of Dynamo-electrical Machines. 123 



§ 10. Modification which the foregoing Results undergo with 

 rapid Rotation. 



It has been assumed in what precedes that the magnetism 

 evoked by the combined action of the fixed electromagnet and 

 of the current in the coil on the rotating iron core is the same 

 as in that at rest. When the rotation is slow this is approxi- 

 mately the case ; but with a more rapid rotation it follows, 

 from the inertia which the iron has in reference to changes in 

 its magnetic condition, that the strength and position in space of 

 the poles in the rotating core are somewhat different from what 

 they are when at rest. So far as I know, this circumstance has 

 not hitherto been determined in dynamo-electrical machines. 



As to the position of the poles, that is the direction of the 

 magnetic axis, the angle <jf> in § 7 which the magnetic axis of 

 the stationary iron core forms with the opposite direction of 

 the axis of the fixed electromagnet, that is defined by the equa- 

 tions (15). If the iron core rotates rapidly, it can be assumed 

 that its magnetic axis is thereby displaced in the direction of 

 the rotation by a small angle which is proportional to the 

 velocity of rotation. Hence if <// is the angle which the 

 altered direction of the iron core forms with the opposite 

 direction of the axis of the electromagnet, we may put 



<J>' = <l> + ev, (21) 



in which e is a small constant. 



The strength of the poles must be somewhat less in a rota- 

 ting than in a stationary iron core. No considerable error is 

 made, if it be assumed that the changed magnetic moment is 

 as great as the change in the components of the original mag- 

 netic moment which fall in the altered direction, and which 

 we have called P, and defined by equation (14). Hence if P' 

 is the magnetic moment in the rotating iron core, we may put 



P^Pcoseu (22) 



If we decompose this magnetic moment P', as we have 

 previously done with the magnetic moments P, into the two 

 components which fall in the opposite direction of the axis of 

 the fixed electromagnet, and the direction at right angles 

 thereto, and which may be called P/ and P 2 X , we have to put 



P/ = P ; cos <£/ = P cos ev . cos ((/> -f ev), 

 P 2 ' = P x sin 4> f = P cos ev. . sin (cf> + ev). 



If we develop these expressions in powers of ei\ and from the 

 smallness of the coefficient e neglect the members of ev which 



