216 



Sing Be 

 r B n 



p F' (ct + r) dt 



Sing B5 

 cr 3 n 



F (ct + r) 



J- t' 



Sing B g 

 cr B n 



2-B F (ct + r) at 

 3 t 



(*«) 



As Jeans, we now take F, whicn is as yet an arbitrary function, to De sudh that it anfl all 

 its derivatives vanish except for zero arjument, for which particular value F becomes infinite in 

 such a way that its integral is unity. We also take t' sufficiently large so that, for all values 

 of r considered, the value of r - ct' is nsgativr;, and wc noti that since t" is positive, the 

 value of r + ct" is always positive. with these assumptions it is then seen firstly that the 

 right-hand side of (ai) is zero at both limits and seconflly that the intejrated parts of (*») (A5) 

 and («6) also vanish at both limits thus contributing nothing to the left-hand side cf (a7). 

 Finally the integrals on tne right-hanO sides jf (au) , (a5) md (a6) and jlso the similar integrals 

 of the last two terms in (j3) contribute to (ai) merely the values, at t(m>.' t = - r/c, of the factors 

 multiplying F (ct + r) . 



Collecting these terms together we then have 



Is 



3 n 



r c 3 n 3 t 



2 



7^ 



3_r 

 3 n 



2p d r 



1^ TZ 



3 t 



B n 



1 



-2 



Cos g d S 



t = - r/c 



c3 t 



^ 



Sin 



t = - r/c 



3g 

 Bn 



d s = 



Since the zero of time is arbitrary we nave only to replace t = 

 obtain equation (3) of the main paper. 



(A7) 



r/c by t = t ■ - r/c to 



Appendix B 



