studies on the Motion of Viscous Flows — III 



where Ylj and Y2j, l ^ j ^8 are constants determined by the boundary condi- 

 tions, while the constants KllkS K13k^ M14,,J, Dl^^ Tl2y,i, and MIO^S 1 <j < 8, 

 1 < k < 00 are obtained by using the recurrence relations arising from the gov- 

 erning differential equations. These are explicitly defined, and the manner of 

 obtaining them is described in detail in Ref. 2. On applying the boundary condi- 

 tions to the general solutions in Eqs. (2.74), (2.75), (2.76), and (2.77), we obtain 

 the following expressions for the stream functions: 



^0 = -te^ 



e'^^-l) 



•(-"-!)] Sin 



(2.78) 



V'l = -4^0 + 



e 'c / „cs v^ Kll^ ^ 



L 



k= 1 



K13 



^ k 



— s I sin 



e c 2e 





L 



M14,, 



^ sin 20 ; 



(2.79) 



^2 = ( E Olk^'^'M si"^ + ( E I^^^sl^-i ) sin 29 ; 



(2.80) 



where 



Kllk = E YljKlV . K13i, - Z YljKia, 



j = i 



j = i 



MIO, 



^ Yl.MlO^i , M14, - X; YI3MI4, 



j = i 



j = 1 



Dlk = E Y2jDl,i , D2^ = j] Y2.D2,i 



j = 1 



j = i 



(2.81) 



The constants Ylj and Y2j in Eqs. (2.81) are known values and Y2g is by defini- 

 tion taken as unity. Adding Eqs. (2.78), (2.79), and (2.80), we obtain 



^ = 



e 'c / pCs 

 2 



E 



Kll 



_ gK _ g e 



- K13^ \ " 



,k-l 



e 'c / i„cs 



" MIO, 



k= 1 " k= 1 



k 1. ,o.cs 



— S - e 





E D2,. 



k-l 



sin 261. (2.82) 



As shown in Ref. 2, Eq. (2.82) can be rewritten in the following form: 



4j = [F(s) + G(s) cos 61] s^ sin^ , (2.83) 



573 



