172 Procrcdiiigs of the Royal Irish Academy. 



time t' on account of the velocity of propagation of the field being finite 

 If we denote the value of various quantities at the time f by a single acute 

 accent, we have t' satisfying the equation 



t' = t-v-'T{p-pO, (5) 



so that 



or 



'^ ^{l^iL-^S: /V{p- p:)r = T{p- p^Tip- p,') + ir^Sp:{p- p,')r. (6) 



From (5) by Lagrange's theorem 



F{t') = F{t) - v-^T{p - p,)F\t) + ^ 1 ^TXp - pOi^'(O) + etc. 

 Let us take for F(t') the integral* 



'dfqT-^p-p,), 

 where 5' is a quaternion function of t', and we get 



'dtYT-Xp-p/)=^'dtYT-Xp-p,') 



v-^ d v'^ 9- / \ 



dt' 

 On differentiating with respect to f, and substituting the value of — from 



above (6) we have , • • , 



T{p - pO + u^Sp^P - P) 



1-1 



= c^T-\p - p;) - u-'g 



U ' ( 6_ 



dt) '^^"^ ^'^ 3! \dt 



+ -! H7 1 5^(P - pO - ^ ( i7 ) ^^'(P - P') + etc. (7) 



By putting successively q = Ci and q = v'~e^pi we get expansions for the 

 retarded scalar and vector potentials of the charge c^ in descending powers 

 of ?/. 



In the case of the scalar potential, the first term of the series is 



c^T-\p-p^): 



this is the electrostatic potential, and the corresponding forces obey Xewton's 

 law of action and reaction, and therefore will disappear in the equations (1) 

 and (2) or (3) and (4). The second term vanishes and the third is 



.,,r'(|Jy(,-p,). 



X /? -^z 2 



* It is r.ot ditlicult to see tiiat this integral is a generalized form of the potentials of a uniform 

 spherical shell 



