396 ELECTROSTATICS AND MAGNETISM. [PT. II. CH. IX. 



vanishes, since the value of X* at both limits is the same. The 

 integral 



may be integrated by parts, giving 



Of this the integrated part vanishes, since, as found by Warburg 

 and Ewing, after the cycle has been once traversed 3 returns to 

 the same value on traversing the complete cycle. We thus find 

 that in taking the particle through the whole cycle of magnetic 

 operations, and leaving it in its original state, we have done a 

 quantity of work, which is equal, not to zero, but to 



the integral being taken around a closed loop. Each term of the 

 integral must of course be obtained from a separate loop. The 

 whole energy dissipated in the body is 



Yd + Zd3\ dr. 



Of course the general theory is so complicated that it is not even to 

 be assumed that when we have carried the magnetization through 

 a closed cycle in one point of the body we have done so at all 

 points. In practice we can calculate the dissipation only in the 

 case of a uniformly polarized body, where A, B, C are the same at 

 all points of the body and in the direction of the force. The cycle 

 is then the same for all points, and the energy dissipated is 

 equal to 



vol. of body x -: I j 



477-J 



The integral 



is evidently the area of the hysteresis-loop. This area is inde- 

 pendent of the time of description of the cycle. In the case of 

 viscous hysteresis there is an additional dissipation which depends 

 in a complicated manner on the rate of description of the cycle. 



