practical Instrument of Physical Research. 483 



Maxwell makes no such assumptions as these, and does not 

 show that they are on his theory true — as, indeed, he was not 

 called upon to do. It is of interest, then, to inquire whether 

 they are true on his theory. At a surface of discontinuity 

 in any physical quantity let the two regions bounded be 

 denoted by the suffixes a and b, and let us for brevity write 

 ~~\ a+b instead of [] B + [] 6 . Thus, for instance, Uv a will be a 

 unit normal pointing away from the region a, i. e. into the 

 region b, and [Uv] o+6 = 0, or Uv a =— Uv 6 . In place of the 

 above assuuiptions Maxwell's theory gives 



4ttC = VvH . (7) 



[VUvH] a+6 =0 (8) 



VvA=B = H + 4:7rI (9) 



[SUvB]« +6 =0 (10) 



It appears rather a formidable problem to deduce the 

 ordinary theory from these equations, and to do it directly by 

 Cartesians would require rather a bewildering array of symbols. 



Before proceeding to see what expressions the ordinary 

 theory gives for A and H in terms of I and C, it is convenient 

 to deduce from equation (10) the equation 



[VTJvA]. +4 =VUv. W, (11) 



where W is some scalar function of the position of a point. 

 Equation (10) may be written 



SU, o B„_ 6 =0, or SUv aV A o _ 6 =0. 

 /.by equation (1) above jSrf/? a A a _ 6 is zero for any closed 

 curve drawn on the surface. .*. \$dp a A a _ b has the same 

 value for any two reconcilable paths on the surface from one 

 definite point to another ; i. e. the resolved part of A a -b 

 parallel to the surface is the resolved part of yW parallel to 

 the surface, where W is some scalar. Thus 



U^VUvA- S =Uv tt VU^vW. 



Multiplying by Uv a we get equation (11). 



The ordinary theory gives the following equations. De- 

 fining A and Q, by the equations 



A =jjj^9, (12) 



XI = -jJJSlvi«fc, .... (13) 

 it will follow that 



A = A +j]JVIV'"fc, .... (11) 



H=- v ^+v A o; .... (15) 



