Magnetic Induction in Iron and other Metals. 339 



cover an exceedingly large number of periods. The changes 



in I become then truly cyclic. 



For if in (9) we put, in place of H, H + ^ Z, and ultimately 



n 



make Z = x> , we get for (9) the simplified expression 

 I= SfA^e-^cos (2nK-x \A0 + B n e-* v " . sin (2nH-# \/n)}. (10) 



All the other terms in the right-hand member of (9) have 

 vanished in the limit. 



The gradual transition from the value of I expressed by 

 (7), through the values expressed by (9) to the final con- 

 dition given by (10), is well exhibited in E wing's experi- 

 ments [cf. fig. 154, Ewing's ' Magnetic Induction in Iron'). 



We can write (10) in the form 



r=co f 



= 2 U r 



r=l «■ 



6 . COS 



+ B. 



Vg- 





(ii) 



where R is the " period " o£ H when r=l. 



It will be observed that the right-hand member of (11) 

 does not vanish, in general, when H is any whole multiple 

 of the period R. Hence, in general, there will be residual 

 magnetism under the conditions expressed by (11). 



To fix our ideas let us take a particular case. 



Fig. 5. 



When a- = in (11) let us assume that the graph repre- 

 senting the relationship of I to H is of the form shown in 



fig. 5. 



