;.n-ioi 38 



The equations become 

 ns6 -^ - nsy V + © ^^i = nseAU - (1 + a sin t;)cos nsy (5) 



dx ax 



ns6 M + nsy U + Q ^ = ns e AV (6) 



^ d-z 9y 



and (3«^) becomes 



^ ^M = , ^^ , (7) 



9x ay a^ 



Attempts to solve equations (5) to (5) in closed form 



were unsuccessful, I7e therefore resorted to seeking solutions 



by a perturbation expansion in the parameter 6. 



Let P 



U = Uq + 6U-L + b^Ug + ... 



V rr Vp + 6V-j_ + 6^V2 + ... 



H = Hq + 6H^ + b^E^ + ... . 



Our formal procedure is to regard the coefficients U^, U, etc, ^ 

 as coefficients in a power series in 6, 



Let us substitute the expansions into equations ik) ^ 

 (5), (6) and (7). We have 



+ y [u^x + 6U;lx + ••• ^ V "" ^^ly ■" -'-^ 



-f Vq -I- 6V^ + ... = eCV^xxx + ^^Ixxx -^ ••• 



■*' ^oxyy + ^^Ixyy + •♦• " ^oxxy " ^^Ixxy ' '•' ' 



■ Vyy " ^^lyyy -••.]- (l + « sin 'r)sin nsy (8) 



