practical Instrument of Physical Research. 485 



Here, as usual, the element ds is taken twice, viz., once 

 for each region bounded, except in the expression 



where it is taken only once. 



This last expression requires transformation. If the surface 

 of discontinuity in B or YyA is not closed, we see by equation 

 (11) that the component of yW parallel to the surface is 

 zero at the edges ; L e., W is constant at the edges. Let W 

 be this constant value. In this case 



-V$ w VUi/ a vW^= - v §uYTJv a v(W-Y? )ds 



= v ^j*(W-Wo)yUv a vw^-Jw(W-W )^}[equation(l)]. 



The line integral is zero V W = W at the edges. The sur- 

 face integral gives 



_jJ( W -W ) Vl VU, o Vi« 1 &=jJ(W-W„)VU, ( ,vV"^ 



= -jj( w - w ») su n.v • v«* [••• v 2 « = o] = vJJ(W- w )8u^v« *». 



If the surface is a closed one we may regard any point on 

 it as the bounding curve, or we may proceed thus : — 



-§uYXJv a vWds= -jjyVvfavWJds [equation (2)] 



= -JjJVvw'vWifc = jJJY V (W vu)ds = JJW YTJ v aV uds. 

 Hence 



- V&VTJv a V W ds = V $W VUv B yuds = vjjWS TJvj/uds 

 by the same transformation as for 



V§(W-W )YUv a vuds. 

 Defining then the scalar w by the equation 



4tto= jj (W- W ) SUv v^-j^SUvA^ +ffiu$ V Afc, 

 we have 



4tt(A- yw) = 4ttJJj Yly^9 + jJjVHv^? 

 and 



JjjYHvw^=jJjwYvHJ5+^YHU^5 [equation (2)] 



= 47rjyjwCd? [equations (7), (8)]. 

 Hence 



A=Ao + jJJVlVH<fc + Vw> .... (16) 



so that Maxwell's theory differs from the ordinary theory 

 solely by having an arbitrary vector of the form yto in A. 



