Fundamental Concepts of Electrical Theory. 407 



where the ordinary notation for Jacobians is used. It is- 

 easy to verify that these expressions give 



BH y 1/BE. \ 1 



(2) 



B-£ By B^ 



in accordance with the usual electromagnetic theory *► 

 They also give 



H f = 5B I -iE„ etc (3) 



c c y 



We shall define E x , E y , E z as the components of the 

 electric intensity E, H x , H y , H^ as the components of the 

 magnetic intensity H ; the quantity p then represents the 

 volume density of electricity, and pv x , pv y , pv z the components 

 of the convection current. It is convenient also to regard 

 the vector v as a velocity. If we use the customary vector 

 notation, the relation between E, H, and v may be written 

 in the form 



H = *[vE] (4) 



This is the fundamental relation which Sir Joseph Thomson 

 takes as the starting point of his theory. The constant c, 

 which has been introduced to bring the equations to the 

 familiar form, will be called the velocity of light. 



It is important to notice that the three functions X, Y, 

 and Z satisfy the partial differential equations 



bx x^ bx^ BX 



ot 0% y oy B- 



bY x BY^ BY j BY n 



ot d# By B* 



BZ^ JZ, BZ BZ A 



B^ 0# ay * B* 



* When Z = l we have p = 0, pv=0, and our expressions ^ specify 

 an electromagnetic field in free aether. This case has heen discussed 

 in a previous paper, ' Messenger of Mathematics/ Nov. 1915. The con- 

 ditions p = Q, pv = are also satisfied when Z=/(X, Y); but we can 

 generally choose two functions X , Y such that 



tf(X ,Y )=/(X,Y)d(X,Y). 



Consequently there is no loss of generality in putting Z = l. 



