A.1"-1C1 



39 



,9U, 



au 



ns6[ — 2 + 5 — i + ... ] - nsy[VQ + &¥-]_ + . . . ] 



at 



aT 

 aii 



ax 



aii^ an. 



+ 0[--2 + 6 — -i + . . . ] = nseA [U. + 67t + . . . J 



- (1 + a sinT )cos nsy 



(9) 



a V av 



ns6[— 2 + 6 — i + ... ] + nsy[U^ + 6 U-, + ...] 



aT aT u X 



,an^ an. 



+ e[~^ + 6 ^ + .., ] = nseA [V^ + 6V, + ...] (10) 



ay ay o i 



au 



ax 



au. 

 o + 5 ^ + ... + 



9V^ av 9H^ 



o + 6 ™-i + ... = -6[— 2 



9y 9y aT 



aH 



1 



+ 6 -— i + .. . ]. 



9x oy oy St; a-r (n) 



If we regroup each of these equations so as to combine 



the coefficients of each po'/er of 6 ? we have, upon retaining 

 terms In 6° and 6 only: 



r 



y[u^^ + v^, 1 + v^ - £[v^__ + v__ - u^_„ - u_,„J 



ox oy 



oxxx oxyy oxxy 



oyyy 



+ (1 + a sin T)sln nsy f + -^ V 



1 . J 



U. 



1 



r ^ " OXT "^oyT 

 + V, - U. 



T 1 xvv 



+ y^ui^ + Viy] 



+ Vt - e[VT_^ + Vt.„_ - U,„„„ - U,,„„J ^6 +... =0 (12) 



Ixxx T ixyy ixxy lyyy j 



aH, 



-nsyV + 9 —~- - nseAU^ + (1 + a sin t:)cos nsy 



au. 



+ i ns .— 



aT 



- nsyVi + © — i - nseAU-j^ U6 + ... = (I3) 



1 nsyU^ + © — 2 - ns eAV_ l+'<ns — ^ + nsU. + © _-i-nseAV^>6+,« = 



O rl-,7- '9t '' 



9y 



9y 



~9Uo 9^0 1 /^^l ^^^1 ^^0^- 



ax ay I ax ay ^ aT 



& + . , . = 0. 



ah) 



(15) 



^ax ay J ^ 



Setting each of the coefficients of b equal to zero we 

 have as the zero order equations for (12) and (l5) 



