Rectilineal Motion of Viscous Fluid. 189 



g(U + ..) + (U + .)g + ,|<U + .) + «£=,V«(U + «)-£ +,*V 



du /T _ x dv dv dv „ 9 <fy> t ( /o\ 



dt • J dx dy dz r fy 



dw ' /Tr -cdw ■ dw dw 2 rfy? 



dt K ' dx dy dz r dz J 



d 2 d 2 ^ 2 

 where V 2 denotes the " Laplacian " -=-$ + -^ 4- -=-$ 



28. If we have u = 0, v = 0, w=0; p = C— #cos Iy; the 

 four equations are satisfied identically ; except the first of (2), 

 which becomes 



dV d 2 U ■ . T , Q . 



¥ = ^ + ^ SmI (3) * 



This is reduced to 



dv _ d 2 v ,.-. 



di'^df W ' 



if we put 



U^ + ^sinl//..^ 2 -*/ 2 ) . . . (5). 



For terminal conditions (the bounding planes supposed to be 



y = and y = b), we may have 



v=F(t) wheny = 0\ (Q) 



v=%(t) „ y = bi ' ' Kh 



where F and § denote arbitrary functions. These equations 

 (4) and (6) show (what was found forty -two years ago by 

 Stokes) that the diffusion of velocity in parallel layers, provided 

 it is exactly in parallel layers, through a viscous fluid, follows 

 Fourier's law of the " linear " diffusion of heat through a 

 homogeneous solid. Now, towards answering the highly 

 important and interesting question which Stokes raised, — Is 

 this laminar motion unstable in some cases ? — go back to (1) 

 and (2), and in them suppose u, v, w to be each infinitely 

 small : (1) is unchanged ; (2), with U eliminated by (5), 

 become 



where 



c=gsml/fj, (10) 



