254 Mr. W. Williams on the Relation of Dimensions 



are mutually at right angles, and Z is at right angles to both, 

 being the intersection of the electric and magnetic equi- 

 potential surfaces. Let this relation between the directions 

 of the axes of reference and the displacements hold for every 

 point of the medium, so that the axes constitute an instanta- 

 neous system at every point. In passing therefore from 

 point to point in the medium, and for different epochs at the 

 same point, the axes and the displacements preserve their 

 relative directions, while their absolute direction in space in 

 general alters. 



Fi<r. 5. 



Let AO' be a closed electric circuit, and BO a correspond- 

 ing closed magnetic circuit, both being circles in planes at 

 right angles to each other. Taking instantaneous axes as 

 above, every element of the circuit AO' is "&x, and of the 

 circuit BO is "dy, while an element of the intersection of the 

 planes of the circuits is ~dz. The length of the circuit AO' is 

 2d#, and of the circuit BO, 2d?/, while the surfaces of the 

 circuits are ultimately 2(B^B£)> and S(dy"dz). We have 

 therefore : — 



1. Circulation H=2(Hd#) = C. 



2. Circuitation E = 2(EBa?)=E. 



3. Surface-integral of D = 2(D(ty32) = £. 



4. Surface-integral of B = 2 (Hftadz) = m. 



To express these dimensionally, we have to neglect the 

 summation 2, and substitute for d#, ~fty, ~ftz respectively X, 

 Y, Z. The relations then become : — 



1. D =JcE. 



2. C =eT~\ 



3. e =D(YZ). 



4. C =DT- X . 



5. E =E(X)=C w = mT- 1 = B(XZ)T- 1 , 



6. B =fiB. 



