240 Lord Rayleigh on the Behaviour of Iron and Steel 



to pass. If we reckon from the mean condition, we may- 

 express the relation between the extreme changes of magneti- 

 zation and force by the formula 



3'=a£' + /3£' 2 , (2) 



where a and /3 are constants, corresponding with the 6*4 and 

 6*4 x *8 of the example given above. But no such single 

 formula can express the relation for the rest of the cycle. 

 When «§ is diminishing from Jq = 1q / to «jp = — ffi, 



3=«£+y8£' 2 {l-Kl--§/#) 2 h 



but when § is increasing from <£)=—»£)' to § = §', 



3=a£+/3£' 2 {-i+-KH-£A§')n- 



These expressions coincide at the limits «£)=+»£)', but differ 

 at intermediate points. Since the force is supposed to be 

 periodic, we may conveniently write 



£ = $ cos ; 

 whence, putting also for brevity a' in place of ««§', $ in place 

 of /3<£)' 2 , we get 



3 = a! cos 0-f /3'{cos 6 + i sin 2 0} 

 from = to 0~7r, 



3 = a'cos0 + /3'{cos0-isin 2 0} 



from 6 = it to 6 = 27T. 



We have now to express 3 for the complete cycle in 

 Fourier's series proceeding by the sines and cosines of 6 and 

 its multiples. The part 



a 1 cos 6 -}• ft' cos 0, 



being the same in the two expressions, is already of the 

 required form. For the other part we get 



±isin 2 0=B 1 sin<9 + B 3 sin30 + B 5 sin50 + ...., . (3) 



where only odd terms appear, and B n is given by 



Bjl = 7ra(n 2 -4) ^ 



Thus 



3 = (a' + /30cos^ + ^[^sin6>-^sin3(9~^sin5(9-...|. (5) 



If the range of magnetization be very small, ft' vanishes, 

 and the influence of the iron upon the enveloping coil is 

 merely to increase its self-induction ; but if /3' be finite, the 

 matter is less simple. The terms in sin 30, sin 50, &c. indicate 

 that the response of the iron to a harmonic force is not even 



