338 



Mr. J. Buchanan on the Theory of 



Hence a closed curve, like fig. 4, Q, can be represented by 

 periodic functions of a continuously increasing variable. It 

 is in this manner that I propose to form the expression for 

 the relation of I to H when the latter is varied cyclically. 

 There is nothing artificial in this ; for although by continuous 

 increase of H from zero we can reach a given intensity of 

 magnetization, we cannot return along the same path. As 

 experiment amply shows, the effect of commencing to de- 

 crease H, is to begin the establishment of I as a periodic 

 function of H. The part played by H in the analysis here, 

 as in experiment, is similar to that of " time " in heat 

 problems. Negative values of H are not admissible here, 

 since the right-hand members of (6) and (7) have the value 

 zero when H is negative. But although in the analysis H, 

 like " time," cannot become negative, the functions whicb 

 express the effect of cyclical variations of H can, and do, 

 assume negative as well as positive values. 



Beginning with equation (6) again, let us assume the 

 following values for F(«) : 



F(a) = aa from a=0 to ^ = 7^ ; 

 F(a) = c from a=h 2 to a = h 2 ; 



n=ca 



¥(a) = % ( A B cos 2noc + ~B n sin 2na) from a—h 2 to oc—cc . 

 The integration of (6) now gives 



: -5s(«fl 



ax 



V4(H-A X ) 

 V4H 



4(11-*!) 



•H-Ai-e 4H n/H 



9. r V4(H-* 2 ) »=oo ._ _ 



+J1 \ € s\dy+ 2 JA n e-* Vw cos (2riE-xSn) 



V4(H— *!> 



»=1 



+ B H e-* Vn sin (2wH-a? y/n) \ 



V4(H-* 2 



_^= 2 ( A » cos 2nH + B n sin 2nE ) \ c °s 2^ • e~ y2 . dy 

 s/ir n= i Jo V 



2 n: 



-4- 2(A M sin2»H-B n cos2nH)l 



V7r«=i ^o 



Let us next suppose that H varies continuously, so as to 



V 4 (H-* 2 ) 2 



sing,. e~y\dy. . . (9) 



