﻿Potentials of Moving Charges. 377 



The equations for the potentials may be written 



□ (ic<&, F, G, H) = /3 /c(zc, u, r, ic) 



\ i / A a 2 + i' 2 H-«> 2 



where /c=1/a/ 1— 2 



It is well to point out that this mode of writing the 

 equations is slightly different from the customary mode 



where QF-M so that our F is d¥'. This deviation 

 c 



from usage is justified by the greater symmetry and homo- 

 geneity of form resulting. The equations for h (magnetic 

 intensity) and d (electric intensity) will on account of this 

 •change assume the forms 



h=- rot(F, G, H), 



(2) 

 It is evident that the operator □ is an invariant under 

 n Lorentz transformation. It will therefore follow that 

 (ic<&, F, Gr, H) is a four-vector, because ic(ic, u, v, iv) is a 

 four-vector. Therefore c 2< &8t — FSx — G8y — HSr which 

 represents the scalar product of the four-vectors (zc<l>, F, G,H) 

 and (ic8t } 8x. hy, 8z) is invariant under a Lorentz trans- 

 formation. Therefore, 



e 2 <P8t-F8x-G8y^R8z=c 2 &8t , --F'8x'--G'8y'--K'8z', 



where the dashes refer to a system of axes moving with 

 velocity v along the axis of x. 



But 8x= K (8x' + v8t'), 

 Sy = Sy, Sz = 8z', 



and bt = K ht' + =\ where k = l/.y/ 1 - % . 



Substituting and equating coefficients of 8x', 8y', Sz', and 

 ■8t' we have 



F'= K (F-v®), G' = G, H'=H, 



F 



and <&' = *(<I> — ^-Y 



