Conway — Application of Quaternions to Electrical Theory. 3 



We have also SVi] = from the second equation, and thus we get the 



quaternion equations 



c"^ (e + 47r() = Vjj, 



- 17 - 47rc = Ve, 



so that as before we get 



(V - hc-^ d/dt) (e + hen) = - 47r (<3 - Ikt' i); 



or, if we introduce the biquaternion e = e - hc^i, 



Several modifications of this may be noticed. Let the result of changing h 

 into - /(, in a quantity like o- be denoted by &„. Then we have 



Also, KaKD = K{a:<T) = - iirKe, 



or (tD„ = - iire^. 



In like manner, (j^D = - Aire, 



where Z» = A - hc-^d/df. 



If we introduce a scalar potential ^j and a vector potential to, we have 



i = - V^J - w, 



J) = FVot, 

 where c"-jj = ^SVot. 



We thus get £ + hcit = - (V + hc-'dldi) (j) - hew) ; 



or, if we put ^j - 7icct = p, we have a- = - (jop. 



We may notice that in fi-ee aether 



a<f = - aaoV = o. 



and that if there is volume-density of electricity, 



a<' = - aao • p = - ^ttc, 



or (V + c--d-/df-) p = 47re. 



If we denote* Ijy [c'], the effect of replacing t in the value of e' at a point 

 p' by t - c-^T{p - p), we have 



p =\dv'\e'^T{p-py\ 

 where dv is a volume-element, and 



■w = c-'\ dv' [('] T{p - p'). 

 Then V=P~ hczs = j dv' [e'] T(p - p')"'. 



* This convenient notation is employed by Lorentz ; of. The Theory of Electrons, p. 19. 



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