Convection Currents in the Atmosphere. 453 



K 



dm 



dz 2 



-x 2 n\-^xB = 





- g{bve- vz + ce~ mz ) + f(be~ VK + ce- me ) 





+ k{b{v 2 —-tf)e-» z + c(m 2 -\ 2 )e- mz }, . (22) 





,7W 

 A, A + — = cayibe-^ + eg-™*). . . (23) 











= o, 



, 2 -A. 2 













/* 





These five equations are evidently not independent, as 

 they have all been derived from the three equations of 

 motion and the equation of continuity. Any four of them 

 may therefore be used to find their solution. Consider 

 first the complementary functions, where II, W, A, and B 

 are all proportional to ^~, where //, is a constant. We find 

 that the last four lead to the determinantal equation 



2(o -kfJ + kX^ty 







—2co\ 



X 



which reduces to 



( - k/jL 2 + k\ 2 + iy) 2 (/J-\ 2 ) + 4rfi 2 G> 2 = 0. . . (24) 



In general kX 2 is small compared with co. The roots are 

 found to be approximately 



±<i + .>(^)', ±^. . . , 25 > 



We shall make A, the height of the free surface in the 

 undisturbed state, large compared with (h/co)*, but small 

 compared with 1/X. Thus the only admissible values of \x 

 are the small roots containing X as a factor and the large 

 roots that have a negative real part. They are then four in 

 number, but which of the large roots are retained will 

 depend on the sign of y—2co. Let these be /^ and /jl 2 , 

 and let the smaller ones be yu, 3 and /x 4 , which are equal 

 and opposite. Then we can write for the complementary 

 functions 



II = p^w + prfW+p-tew+piew, . . . {26) 



and for the particular integral we find 



n _ gabv<r vz gacme- mz 



y 2 - 7 2 ^/( 7 2 -4a> 2 ) m 2 - 7 2 A, 2 /( 7 2 -4a> 2 )** l ' 



As m is not equal to any of the values of //, when rotation 

 is taken into account, it will be unnecessary to treat the 



