420 Prof. Rowland on Spherical Waves of Light 



Let © m; P m , and N m be the components in the directions 0, 

 p, and « respectively. Then we have 



d . * d cos sin d 



7 = sin^ -7-, 



ar ap p a//> 



d! ^rf sin 2 (i 



— — cos 0-y- + — , 



a.a? ap p ayu/ 



@™ = — F m sin + R m cos 0, 



P m = F m cos # + Rm sin #, 



p sin rf/9 ? 

 p __ 1 dM w -i 



rm ~ r 2 ^ ' 

 N m =0, 



©«+i=0, 



_ J_ f gMg-i , sin 2 fl^M TO _i \ 

 1Nm+1 ~ psintfl dp 2 + p 2 rfp 2 /' 



The equation of motion of light becomes in this case 

 J_ (PM,-! _ ^ 2 M m _i sin 2 ^ 2 M m _i 

 V 2 eft 2 ~ dp 2 + p 2 dp 2 ' 



whence 



Now, in this case of symmetry we readily find 



■tio — WP dr ' 



and for convenience putting 



U = C w p*e^- v ^ 

 we readily find 



M 1 = JSi^==psm6/-T-^ sm ^~^ cos^ >. 



Putting 



— — rfpT' 



and omitting the constant term, we have 



