672 Mr. G. A. Schott on Radiation from Moving Systems 



Using the well-known properties of spherical harmonics 

 we get from (16)... (18), on the average, 



y- 4U r^ i ^ i j 



~ 3(C 2 -U 2 ) i R+ 10 Aso ' U2 j ' j 

 5C\/C 2 -U 2 



2IP 



1_U 2 /C 2 ~ ' 15(C 2 -U 2 ) A20 * R2 * * ( 25) 



. _ 4U 2 r— i — ~i 

 w= 3Tc^^{ R+ lo A W- • • • (26) 



It is noteworthy that inequalities of distribution o£ the 

 second order alone have influence on the total radiation and 

 radiation pressure. Such inequalities can only arise in a 

 non-uniform field and must be due to collisions. 



§ 15. In averaging (19)... (22) we must bear in mind that, 

 in the diagram of § 12, the poles P, Q are to be kept fixed, 

 while A takes up every position on the unit sphere. We 

 replace do) by its value sin a dad% and integrate for ^ from 

 to 2-7T, and for a from to it. In the case of (19) and (21) 

 no further transformation is necessary, for T' 2 , P /2 are 

 functions of a alone, which in fact replaces 6 in equations 

 (14), (15). We get 



^=4^(l^U7cT- {^ + A 2 o.5 2 + A 40 .R 4 +...}. (27) 

 §i.' = (28) 



The last equation obviously follows by symmetry. 



In the case of (20), (22) it is convenient to transform 

 from A to Q as pole, because T' 2 , P /2 are given as functions 

 of d and not of a. We replace dco by sin dO ch/r, and in (23) 

 substitute for a, % their values in terms of 0, i/r. Since 

 T /2 , P' 2 do not involve yjr, we require the integrals 



-LpM^ for (20), and - 1 - f^F cos 2f<fyr for (22). 



These are functions of 6 alone ; write 

 -j r2jr t=oo 



«r- F<ty=2QPi(«os0). 



