

184 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



T have at first limited myself to the computation for the above given 

 distribution of temperature, and put 



Q=Ar' (1-3 cos- H) 



^^'=^3-' (1-3 cos-' ^). 



The functions U,F,E,J are now to be computed witli the help of ,| 

 this Q, and the corresponding E', F', H', and J' with the help of this Q'. 

 We first obtain the general expressions : 



F= ^^'(1-3 cos-' fi) U j r^'^^+ 2r {F+E) J 

 jV j /IF' _r^ -^, p^ ) -| 



iV = — "^^^^ G cos /^. sin h[ Ar{F-\-E)+^^'^{F'-^E') \ 



„ H ["(1—3 cos^ fl) { Ar fr^^ + 2 {H-\-J)^ 



+ 7/^^ -3 [B' + J') ) [ +0 cos^ H { Ar {H^rJ) -f ^(H'+J') | ] 



The actual computation, having due reference to the boundary con- 

 ditions, of the functions here introduced, gives results that are difficult 

 to be discussed. But this is simplified when we make use of the cir- 

 cumstance that the atmosphere fills a very thin shell in comparison with 

 the terrestrial sphere, wherefore the distances from the earth's surface 

 are all small in comparison with the earth's radius. If we put 



thtMi is o small with respect to unity. If we introduce these quantities 

 in the above given equations and put 



/^+2 {E+F)=Rf{cj), F+ E= Rcpia) ; 



r^-3(E' + F) = Rf{a), F'-\-E'=R(p' {a)r, 



r^+2{n+J) = R'g{<j), E+J=R'y{a) 



/-K-Z{H' + J') = R:'f/'(cT), H'-\-J' = Rv^v' (a): 



<lr ' ' 



then by restricting ourselves to the terms of the lowest order, we can 

 obtain simple exi)ressions for these functions. Primarily we Ui.d that 

 the functions/ and/', (p and cp', g and <j', y and y' are identical. 



