Electric Waves round a Large Sphere. 523 



In particular, on the surface of the obstacle, where r = a, 



1 C1T-l2 



sin 2 



i sin- v ^oo -r, /n , 2iy "i 



ka 2 n= i 





(17) 



TA« values o/'R M , <£ n , % n . 



It has been shown by Lorenz* that, if z— m is of higher 

 order than z* and positive, 



R„(z)=z/(z>-m*)i "I (lg) 



<j> n (z) = (2 2 — ra 2 )* + m sin " *»i/s — ^mr, J 



and that if z—m is of lower order than z%, and positive or 

 negative, 



R.(r) = m*3-V-*{«<-« + m-.-.^(-i)( 2 iy ' 

 (m-c) ! , ,./ 24\i , 1 



^(^)=| + B„(,-m)-^ 2 ^^- 2 +... I 



where, in the latter formula, R n , R n ' correspond to z = m. 

 7r(s) denotes Gauss' it function, and is identical with T(s + 1). 

 An extension of the first result, required for diffraction 

 problems in which a second approximation is desired, has 

 been given by the writer f. 



If n + \ = m = z sin «, (20) 



thus defining a subsidiary angle a, 



R*0)= sec a +-^2 sec 3 «+ 7^sec 5 a+ ... 

 where 



\ 3 =— i, \ 5 =i(27-24m 2 ),X 7 = 1 i 6 -(1160m 2 -1125-40m 4 ), 

 and 



s + Z.\ s + i +{s + 2) 3 \ s +2 + 2m 2 s.s + l.s + 2.\ s +m i s.s 2 -±.\ s _ 2 =0. 



• - (21) 



Loc ctt 

 t Phil. Ma?. Dec. 1907. 



