224 Mr. A. W. Flux on the 



.'. To the first order, 



<f> cos yfr = <£j cos yfr ly (f> sin yfr = <f> l sin a/^. 

 And therefore to the second order, 



<f>l COS ^ = (f) COS ^ ?'(£ 2 , 



</>! sin yjr 1 =<fi sin i|r rc£ 2 . 



Similarly, we have 



h 



(f> 2 cos ^2 = ^>i cos-v/tjH- — </>i = (f>costy — 2~ <f> 2 y 



$2 sin -*|r 2 = <£i sin ^ -| <j> 1 2 = sin >/r— 2 — </> 2 . 



And, generally 



(f>k cos -*lr k =(f> cos y}r — k . -(f) 2 , j 



m 



J- • • 



$isin T/rj. = <^>sin yjr — k . — <£ 



(i-) 



Recalling the results of (ii.) section I., we have : — 



?2 = ^i + 2n! <£cos t|t 2 — 2^> 1 2 cos ^[^cosijr + m! sin^/r] 



= — I + AtkJ) cos yjr — 21(f) 2 -\- S(f) cos ijr\_l<f) cos yfr + ?n(j) sin -^] ; 



w 2 = — m + 47i<£sin -x/r — 2m</> 2 + 8</> sin ty [1(f) cos yjr + mcfrsmyjr'] ; 



1 — n 2 

 n 2 = n + 4:[l(f) cos i/r + m</> sin ^] — 8k</> 2 — 2 <£ 2 . 



Proceeding thus, we find that 



l h = -l + 2k n(f) cos f-k(k-l)l(f) 2 



+ 2P$ cos ijr [/<£ cos yfr + m<£ sin ^r] , 

 m]c = — ?7i + 2k n(f) sin ifr — k(k — l)m<£ 2 



+ 2k 2 (f) sin^r[Z$ cos yjr + nop sini/r], r ( n -) 

 nfc=n + 2A'[Z0 cos -*Jr + m(f) sin ^r] — 2P?i(/> 2 



1-i 



£ 2 



-&(&-l)±— ^4> 2 + 2 -[tycosf + wc/>sin^] 2 . 



