a Horizontal Layer of Fluid. 537 



The modes of greatest instability are those for which s is 

 smallest, that is equal to 7r/f, and 



M =i+Jh <*> 



For a two-dimensional disturbance we may make m = 

 and l = 27rj\, where X is the wave-length along x. The X of 

 maximum instability is thus approximately 



X=2£ , . (2$) 



Again, if l = m — 27r/\, as for square cells, 



x=V*-fc ( 29 ) 



greater than before in the ratio ^/2 : 1. 



We have considered especially the cases where k is 

 relatively small and relatively large. Intermediate cases 

 would need to be dealt with by a numerical solution of (23). 



When zc is known in the form 



w = ~S\ J e ilx e im y$\nsz .e nt , .... (30) 



n being now a known function of /, m, s, u and v are at once 

 derived by means of (11) and (12). Thus 



il dw im dw /olN 



"= 72-7^2 T77- ■ ■ ' 0' U ) 



l' 2 -hm 2 dz ' l' 2 + nr dz ' 



The connexion between w and 6 is given by (15) or (16). 

 When jS is negative and n positive, 6 and w are of the same 



si s n - 



As an example in two dimensions of (30), (31), we might 

 have in real form 



w = W cos .?? . sin z . £ ?i *, .... (32) 



u = — W sin x . cos c. e nt , v = 0. . . . (33) 



Hiiherto we have supposed the fluid to be destitute 

 of viscosity. When we include viscosity, we must add 

 v(\/ 2 u, V 2 *>, V 2 ^) on the right of equations (1), (8), and 

 (11), v being the kinematic coefficient. Equations (12) and 

 (13) remain unaffected. And in (11) 



V l =iPAk > -P-» 1 (34) 



We have also to reconsider the boundary conditions at c = 

 and 2=f. We may still suppose = and ic = ; but for a 

 further condition we should probably prefer dw/dz = Q, 

 corresponding to a fixed solid wall. But this entails much 

 complication, and we may content ourselves with the 



