220 

 Then 



Mr. A. W. Flux on the 



—rlij —mi, n x will be the direction-cosines of B'E'. 

 n )) » 



» )i » 



I, — m, n 



in . n 



EG. 



If we refer C to spherical coordinates with the radius 

 through as polar axis, cf>, yjr being those coordinates, 



f j = r sin cf) cos ^, tj 1 = r sin <£ sin ^ 

 5i=— r-f rcos(2>= — 2r sin 2 ^ 



r<i 2 I. 



0-) 



The normal at C has direction-cosines, 



sin (f> cos yfr, sin <£ sin i/r, cos <£. 



Z' =— Z + 2U sin cos yfr , 



m'= — m + 2TJ sin sin ^, 



n' = — n + 2U cos j> ; 

 where 



U = Z sin $ cos i/r + m sin (/> sin ^ +ncos<£ 



= l<j) cos ty + m<\> sin ^r + nfl — ^ J to the 2nd order; 



V ~—l +2n^>cosi|r +2<£ 2 cos^[Zcos-»Jr + msinT/r],"j 



m!=— m + 2n(f) sin ^r +2<^> 2 sini|r[Zcos'^-fmsin'^], 

 n' = n +2^[Zcos^r + msm^r]— 2n<£ 2 . 



We have also 



h=l/p, m 1 =m/fi 9 n x ^^/^—l + ri*/^ 



h'=l'lto m^m'liJL, ni ' = s/~j#-l + n '*/iJL. 

 Whence 



and 



1 If 2n /7 . 2rc 2 J9 



2 r 3^ 2 "1 ~i 



i i r 2 



n fZ= n\}~ n W> cos ^ + W> sin ^) + 2<£ 2 



4 -i 



+ ^(^cos^ + w^sini/r) 2 . j 



> (iii.) 



