348 Mr. R. Hargreaves on Radiation 



and for a typical current ixdTi=aidx, g being a small cross- 

 section. Now F depends on up-\-i x , and we may separate 

 the discontinuities by writing ~F=pyjr + F 0y and F^ then 

 satisfies 



v M4 + 4 + ^) 2f > + ^ v = - • (63) 



The condition ^f-rv~ + 2-j— =0 becomes, when — = 0. 

 V Dt ax dt 



F dx * dy dz dx 7 



which gives 2 -j— ° = 0. We have, then y 



=j*g pt(l-2/)rfT j+ |^24ForfT,.+J|[f2K-^F ]^. (64) 



In the first two terms each discontinuity is associated with 

 its own potential. In the statical case an element of potential 

 has the discontinuity in the numerator, and in the denomi- 

 nator a distance connecting two points 



With such a formula, for a denominator belonging to ele- 

 ments at (pcyz) , (x'y'z 1 ), the various sections contributed to 

 the numerator by the last term of (64) are 



p'tpix— ptpix + ptpij —p'2pi x ; 



and these are cancelled whether p and i x exist at one place, 

 or p only at one place and i' x at another. 



The form of denominator for the motional potential (to be 

 considered more fully later) is suggested by transforming 

 the wave-surface %x' 2 = V 2 £ 2 to a moving standpoint 

 X(x + ut) 2 = YH 2 , and treating this as a quadratic in t, viz., 



or (l-tp 2 )t-tpx+ V(i-2p 2 )2# 2 + (2^) 2 =0. 



If we call this %=0, and write R 2 = (1-Xp 2 )%x 2 + {2px) 2 \ 

 it is readily verified that R _1 /(%) satisfies the differential 



D /2 



equation =rvr 2 ^' , = V 2 V 2 ^ r ; and this solution replaces the 



