1892.] Mathematical Theory of Electro-magnetism. 401 



of the theory, The establishment of general results, and The detailed ex- 

 amination of these results. 



The groundwork of the theory, though not the longest of these, 

 calls for most attention here. It is divided into two parts — Funda- 

 mental assumptions and Preliminary dynamical and thermodynamical 

 considerations. I do not propose to give here a resume of the different 

 parts, but to call attention to certain prominent features. 



The two most important of the fundamental assumptions are, 

 perhaps, first, that in all cases 4tt0 = VyH, which I take to be one 

 of the most characteristic features, if not the most characteristic, of 

 Maxwell's theory ; and, secondly, that the modified Lagrangian 

 function per unit volume, though of course it contains H, does not 

 contain any term involving magnetic moment per unit volume 

 or magnetic induction. Neither of these assumptions seems to 

 be at variance with Maxwell's, and, as hinted, the first is taken 

 up mainly because it is a fundamental feature in his theory. 

 From the first it follows that C must obey the laws of incom- 

 pressibility, and . this naturally leads to the assumption that D 

 also invariably obeys those laws. The second leads to very important 

 consequences, which I believe have not before been traced, and which 

 I wish to call attention to here. Though not put quite in this form 

 below, they amount to this, that H vZ, where I is the modified La- 

 grangian function per unit volume of the standard position of matter, 

 obeys the law of incompressibility, that round every circuit there is 

 an electromotive force equal to the rate of decrease of the surface 

 integral of 4tt h vZ through the circuit, and that H vZ— H/4?r appears in 

 subsequent equations in such a manner as to compel us to identify it 

 with the magnetic moment per unit volume.* It is clear then that 

 47r H vZ is, according to the present theory, the magnetic induction. 

 As the theory is developed below, it is convenient to define B as 

 equal to 4tt h vZ, and call B the magnetic induction, leaving the justi- 

 fication till we examine the detailed consequences of the theory. It 

 is well to insist on this result here, as it does not appear obvious in 

 the work below, but only comes out when a general review of a great 

 part of the paper is made. To put the matter in the form of a 

 proposition : 



If the two fundamental assumptions are made (1) that 4 7rC = VvH 

 and (2) that I, the Lagrangian function per unit volume, can be expressed 

 in terms involving H, but independent of magnetic induction and of 

 magnetic moment per unit volume, then the magnetic induction must be 

 = 477- H y Z. 



* Strictly speaking, the last clause should be modified by the condition "if the 

 present position be taken as the standard position." This, however, is only an 

 accident of the particular form of enunciation, which at the present stage is 

 unavoidable. 



