7 48 
MR. A. McAULAY OX THE MATHEMATICAL 
simple a theory for the explanation of all hysteresis phenomena, yet I believe it could 
be made to account for nearly all the known facts. # But, at present, even if this 
simple assumption were made, we are very much in the dark as to how I 0 varies with 
H and ' V F, and are compelled to fall back on such pure conjectures as are illustrated in 
the foot-nore. To mention only one thing—nearly all the detailed experiments on 
hysteresis deal only with variations of H parallel to itself. 
73. Thus it is useless to attempt a satisfactory theory of hysteresis at present, 
though we can see vaguely how, perhaps, in the future it may be made to fit into the 
present theory. 
But these considerations show that we must be very cautious in discussing results 
which depend upon the form of l in terms of H, for they imply that we are very 
ignorant of this form, even when we know how I varies with H under assio-ned 
circumstances. Thus, for instance, in equation (30), § 55, we can learn little of the 
true meaning of <j>' 0 so far as it depends upon H. The rest of ft in this equation, 
however, being independent of the form of l, gives us information of no doubtful 
character. 
C. On the Strains accompanying these Stresses. 
74. The object of the present paper is to discuss the general theory of electro¬ 
magnetism. It is not proposed, therefore, to deal more than is absolutely necessary 
in particular problems. A word, however, must be said as to a certain class of 
problems connected with the stresses just investigated. 
After the question of hysteresis has been settled in some such way as just 
indicated, it will be possible to discuss in detail the exact form of l' Q of equation (29), 
§55 above. To do this, the data on which to argue will generally be the strains 
which accompany electromagnetic phenomena. This necessitates the consideration of 
such strains. 
75. It must not be supposed that these strains will bear the same relation to the 
stresses as strains bear to the ordinary stresses considered in the mathematical theory 
of elasticity. From equations (26), (27), § 50 above, we see that in the case of 
equilibrium (no external stress) 
D.'V'W - F = f A', F, = [f UV]„ + 6 . 
* By suitably choosing the form of /i in terms of I 0 , and the four functions now to be introduced. 
Let | |, [a>], {«.'} be three positive scalar functions of Tw, and let Qw be a vector function of the form 
T«; function (Uto)—not in general linear—such that StoQw is always negative. The form of Q, like that 
of fi, is a function of I 0 . Let H be the present and h any previous value of H. Then assume that 
i 0 = QN where N = | H | H | H [h] e~ jf Tt?h T (cth + UHTdhj, 
the lower limit of the first integral sign being, strictly, the value of H at an indefinitely remote epoch, 
but practically at a time determined by the exponential. I give this merely to show in what sort of 
way we may suppose I 0 to depend on the history of.the body. 
