Viscous Fluid between two Parallel Planes. 195 



each equal to minus the value of u, v, w for every point of 

 each boundary, we find the u, t?, it) of § 33. The solution of 

 our problem of § 32 is then completed by equations (31). To 

 do all this is a mere routine after an imaginary type solution 

 is provided as follows. 



38. To satisfy (21) assume 



= e tM+ ^ + ^ ) {He yy(w2+92) + Ke-^ /(m2+92) + L/(z/) + MF(?/)} . (49), 

 where H, K, L, M are arbitrary constants and /, F any two 

 particular solutions of 



i(« + mty)«r=/t ||p-(m 2 + ? >] . . (50). 

 This equation, if we put 



m/3lfju=y, and m 2 + q 2 + tco / fi = \ . . . (51), 

 becomes 



■ffi=fa + vw)<r ...... (52); 



which, integrated in ascending powers of (\ + tyy), gives two 

 particular solutions, which we may conveniently take for our 

 / and F, as follows : — 



rt^-i r~>+*73/) 3 , 7" 4 pi+^/) 6 7" g (*.+*yy) 9 , & r ) 



J\iJ) - 1 O o "I a K Q O U W « K 3 o + ^ c ' 



3.2 ' 6.5.3.2 9.8.6.5.3.2 



FM-X + ™ 7" 2 (^+^) 4 , Y" 4 (^7i/) 7 7' 6 (U^) 10 

 i F (y)- X + *W 473 + 7.6.4.3 10.9.7.6.4.3 +&C * 



39. These series are essentially convergent for all values of y. 

 Hence in (49) we have a solution continuous from y = to 

 y = b; and by its four arbitrary constants we can give any 



prescribed values to fy 9 and _, for y = and y — b. This 



dy 

 done, find p determinately by (24); and then integrate (25) 

 for w in an essentially convergent series of ascending powers of 

 \ + iyy, which is easily worked out, but need not be written 

 down at present, except in abstract as follows : — 



w =6l{J e <o>t+mx+qz) (54): 



where 



<W= H&(\ + ijy) + K% (X + cyy) + L&O + iyy) ) 



+ M&(\ + cyy) + p^"^) + Qe~^ m2+ ^ J ( ^ 



Here P and Q are the two fresh constants, due to the inte- 

 gration for w. By these we can give to W any prescribed 



02 



>(53). 



