CoNWAV — Applicatiim of Quatcrniom to Electrical Theory. 5 



which becomes, on transforming by Green's theorem, 



I d(ildt . dv = (87r)"'c J (rdvcTo. 



The scalar part of this equation is Poynting's theorem, and the vector part 

 expresses the theorem tliat the rate of change of electromagnetic momentum 

 is balanced by the Maxwellian stresses over the surface. If the space contains 

 electricity, we will have an additional term 



-PcJ(e„ffo + ,76,) dv, 



of which the scalar part is the activity of the electric forces, and the vector 

 part is the resultant of the mechanical forces. By the aid of the theorem, 

 which is not difficult to verify, that the expression VpVcjD^cT, has the same 

 value, whether B, acts on p or not, we can prove a similar theorem about 

 angular momentum. 



The integral Jj' (j-dvdt has some interesting properties. The first variation 

 olith ll2S(jl<jdvdt or - 2 jj Sad^^dvdt, which is equal to 



-2jjSaD,S^dvdt, 



since aB^ = in free space. This latter integral can be replaced by a triple 

 integral on separating Bg into its parts A and hc-'d/dt, and applying Green's 

 theorem to the former, and an integration with respect to t to the latter. 

 Hence the first variation vanishes with suitable boundary conditions. It will 

 he found, on taking the real part of jj a-dvdt, that we reproduce the Principle 

 of Least Action as formulated by Larmor,* and, on taking the imaginary part, 

 we find that Sjj Stridudt = 0, a theorem which is perhaps new. 



We give here a comparison of our results with the formulae of electro- 

 statics. In the left-hand column we have the latter formulae, and in the right 

 the general formulae or analogues. 



£ = - Vj}. (Definition of potential.) <t = - (JoP- 

 Va = - 47re. (Poisson's equation.) (ja = - 4:we. 

 V=-S.cT{p-py\ p=2[e']r(p-p')"'- 

 - (8n-)"' £'. (Energy per unit- volume.) - (Stt)"' ado- 

 £ = 47r Uve. (At surface of conductor.) o- = ^tt UvB. 



e = 2nUve + V J Tdv'e'T(p - p')-\ <T = 2TrUve + aJ '-^'^^V [e'] T{p - pY. 

 (Integral equation at surface 

 of conductor.) 



* Liunior ; Aether and Matter, p. S2. 



