Viscous Fluid between two Parallel Planes. 193 



whence, by (22), 



i.[mx+(n-m(it)tf + qz] 



Using this in (25), and putting 



t0= f e l[wt( ' l "'"^ )y+ ^ C37) 



we find 



^ = ^ K + (ra - m g^ + ^ ] W- K + ( ^g ) , + g2] . (38), 



which, integrated, gives W. 



Having thus found v and w, we find u by (1), as follows: — 



u, (»-"»#)" + 8" (39) . 



35. Realizing, by adding type-solutions for ±c and + w, 

 with proper values of C, we arrive at a complete real type- 

 solution with, for v, the following — in which K denotes an 

 arbitrary constant : 



6 -^[m2+»2+? 2 -»m/3*+ii»2|82<2] ^ 



<8e * K { V + '(.-,^« + y»- riD [™* + (»-^«)y + j*] 



This gives, when £ = 0, 



+K . sin , N ,. 1X 



V=—s r, r. sin ny nna (mx + qz) . . . (41), 



m 2 + n' + g 2 ^ cos ^ * y v '' 



which fulfils (28) if we make 



n = iiry/b (42); 



and allows us, by proper summation for all values of i from 1 

 to co , and summation or integration with reference to m and 

 g, with properly determined values of K, after the manner of 

 Fourier, to give any arbitrarily assigned value to v t=Q for 

 every value of x, y, z, 



from x= — co to ^=-hoo,^ 



„ y=0 „ y=b, I . . . (43). 



„ z=—cc „ e=+oo.J 



The same summation and integration applied to (40) gives 

 v for all values of t, x, y, z ; and then by (38), (37), (39) we 

 find corresponding determinate values of w and u. 



36. To give now an arbitrary initial value, w , to the 

 Phil. Mag. S. 5. Vol. 24. No. 147. August 1887. 



