Magnetic Induction in Iron and other Metals. 331 



not appear to have been suggested hitherto — that solutions 

 of an equation of the same form as Fourier's partial differ- 

 ential equation 



dv _ d?v 



dt~ K dx* W 



are capable of expressing the complex results obtained in the 

 numerous experiments which have been made on magnetic 

 induction in iron and other metals. 



Here are some of the results at which I have arrived in 

 the very limited intervals that I have at my disposal for such 

 researches. 



Induction by Continuous Increase of Field-Intensity . 

 It has been abundantly proved by Ewing and others that 

 the typical form of curve connecting field-intensity (H) and 

 the magnetic intensity (I) is of the nature shown in fig. 1 

 (p. 332) by the curves A, B, or C drawn with broken lines. 

 These curves all fulfil the terminal conditions : 



H = gives I = ; and -^ = 0. 



IT 



H=qo gives I finite, and -yn- =0. 



It is not difficult, of course, to find a variety of forms of 

 the function /(I, H)=0 fulfilling these terminal conditions. 

 I propose, however, to try to show that the functions which 

 are solutions of (1) are not only capable of satisfying the 

 terminal conditions, but also of expressing the complex 

 relationship of I and H, even when affected by strain, 

 temperature, &c. 



Let us begin by assuming an equation of the form 

 dl _ d?L 



dR.- p *de*> w 



where I and H have the usual meanings ; and p is taken as 

 a constant, whilst 6 is a quantity whose numerical value is 

 altered by strain, by change of temperature, and so on. 

 Instead of (2) we can write 



dR-dtf' [6) 



in which x = 6xp~K 



A solution of (3) periodic in H is 



I=- 1 F(a)e?al cos {2n(R— a) — a: Vnje-tj" .dn 



^Jo Jo 



(4) 



(cf. Lord Kelvin's Math. & Phys. Papers, vol. ii. pp. 63, 64), 



Z 2 



