Studies on the Motion of Viscous Flows—Ill 



,- • • . A - B - C - 3D = ... 



B + C (a.) + D (CD) = -1 

 - B - C - C (CO) - 3D(oo) = +1 . 



This gives 



Consequently, 



B=-l, A=-l, C=D=0 



4^' = -[r .1 



sin 



U' = - ( 1 + ^) COS0 (1.46) 



1 - — ] sin 



y 



(iii) lim u' t - COS 6* : Applying the conditions of Eqs. (1.41), (1.42), and 

 (1.44) to £'0*3. (1.39) and (1.40), we get 



A + B + D = 

 A - B - C - 3D = 



- B - C - C (CO) - 3D (m) =: +1 . 



The last of these three conditions requires that C = 0, D = 0, and B = -l. But 

 with C = D = 0, the first two demand that A = and B = 0. Hence we conclude 

 that there is no solution which can satisfy all three conditions of Eqs. (1.41), 

 (1.42) and (1.44). 



(iv) Hm v' 7^ + sin^ : Applying the conditions of Eqs. (1.41), (1.42), and 

 (1.44) to Eqs. (1.39) and (1.40), we get 



A + B + D = ^ ,. ;^;- .;; '\i, :\,, . ■{ V; ; . • 



A - B - C - 3D = • . ■ , 



B + C (co) + D(oo ) = -1 



The last of these three necessary conditions requires that C = 0, D = 0, and 

 B = -1. But with C = D = 0, the first two demand that A = B = 0. Hence, in this 

 case also there is no solution which can satisfy all three conditions of Eqs. 

 (1.41), (1.42), and (1.44). 



The conclusion is that Eqs. (1.45) and (1.46) are the only solutions of the 

 harmonic equations which satisfy three out of the four given conditions. Depending 



541 



