1903.] Thermal Relations of Energy of Magnetisation. 



237 



the integral being taken along the path. If (A, B, C) is a function 

 of (a, /?, y), that is if the magnetism is in part thoroughly per- 

 manent, and in part induced without hysteresis, so that the operation 

 is reversible, this work must vanish for a complete cycle ; otherwise 

 energy would inevitably be created either in the direct path or else in 

 the reversed one of the complete system of which Sr is a part. Thus 

 the negation of perpetual motion in that case demands that 



Adoc + Bd/3 + Cdy = dcf>, 



where <f> is a function of (a, ft, y), involving only even powers, and 

 practically quadratic for small fields. Its coefficients are then the six 

 magnetic constants for general aeolotropic material, no rotational 

 quality in the magnetisation being thus allowable by the doctrine of 

 energy. But if there is hysteresis, so that the cycle is not rever- 

 sible, 



-St ^(Ada + Bdfl + Cdy), 



or in vector product form - St J f dp, represents negative mechanical 

 work done, or energy degraded, in the cycle. 



In addition to this energy concerned with attraction, the external 

 field expends energy in polarising or orientating the individual 

 molecules against the internal forces of the medium, of aggregate 

 amount 



S T j (adA + fldB + ydV). 



In any case, whatever the hysteresis, the sum of this second part 

 and the first reversed is integrable independently of the path, giving 



8t I Aa + B/3 + Cy | , 



namely, the change in the total energy in the element, thus vanishing 

 for a cycle which restores things to their original state, as it ought 

 to do The latter part is purely internal, and of merely thermal 

 value as in §5. The former part represents the averaged waste of 

 direct mechanical energy in moving the iron armature through the 

 cycle, and accounts for the heat thus evolved. It is the expression of 

 Warburg and of Ewing for magneto-hysteretic waste of mechanical 

 energy in driving electric engines ; for a portion of a cycle it repre- 

 sents work partly degraded and partly stored magnetically. 



7. Reverting to § 5, we may profitably illustrate by working out into 

 detail a suggestion of Lord Rayleigh (he. cit.). Consider a ring-coil of 

 n turns with a flexible open core of soft iron of length / and cross- 

 section S, whose flat ends are bent round until they face each other at 

 a distance small compared with the diameter of section. We can 

 apply Hopkinson's theory of the open magnetic circuit to trace the 



