Compressible Fluid moving in Incompressible Liquid. 143 

 Substituting in ( L9) we get 



u= --i (a;si ""^- /3 "' eos " 0,) + iv//^ 

 + ^| i (-iy-'sin.^'- f )''7 



x [l + iZi2(«a cos n0' + p„' sin n0') 1 



+ S -- — — (*„ cos >?e + /3„ sin we) 2 ( — IV" 1 

 ^n(n+l)...(n + s-l)b*~ l . 



00 



+ (/S„cos»€ — a„sin ?ie) 2 ( — l) s_1 



n(n + l)...(n-M— 1) &*" 1 ,.. ,1 



X v /- ^ 7 J cos 5 <9'-e 



(5—1)! c s K } \ 



x Fl + ^i2K' cos nd' + n ' sin n^)] 

 n(n+l)...(n + «— I)*-- 1 , 



oo 



H-(/3nC0S?*e — a /; sin ?ie) 2 ( — l) 8 " 1 



; = 1 



n(n+l)...(w + 5-1) // 



^■sin,(fl'-e)] 



- K s in(^- e )-J-^cos(^-e). 



Z7T<" J7T6' a£ 



. . . (20) 



The two values of 9ft as given by (17) and (20) must be 

 the same ; so that we can equate the terms independent of 

 0' in either expression as well as the coefficients of the 

 sines and cosines of multiples of 6' . The terms independent 



of 6' simply verify the known result that . =2irb-j-. 

 Equating the coefficients of cos 6', we have 



v*/ = Z? (Bi cos 2e ~ " i sin 2e) + 5? % (ai oos 2e + Bi sin 2e) : 



