184 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



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

 distribution of temperature, and put 



Q=Ar 2 (1-3 cos 2 d) 



Q'= A ' (1-3 cos 2 6). 



JV = 



_ aGR, 2 2e sip a ^ l ^ q nno t m f A„f : dS 



H 2 



The functions E,F,H,J are now to be computed with 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 : 



V= aGR \l— 3 cos 2 6) I A | r 2 ^+ 2r (*»+.#) J 



^^ 2 G cos 0. sin j Ar( F+E) + A ' (F> + E<) \ 



in ^ T(i_3 cos 2 8) J Ar ('/^ + 2 (JT+ J) ) 



+ £( r ^~ 3 (jH ' +J,) ) I +G °° 8 ' * I Ar (jff+ J) + £ (fl/J/) I ] 



The actual computation, haviug 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 



r=R(l+ff) 



then is a small with respect to unity. If we introduce these quantities 

 in the above given equations and put 



r<W+2(E+F)=Rf(G), F+E=R ( p(<r); 



(I P 

 r~-S(E'+F')=Rf( ff ), F' + E'=R<p' (a); 



HIT 

 ^+2(ff+J)=ify(<r), B+J=R? r {G) 



r lfF -Z(n< + Ji) = R,y(o-), H'+J> = Rry (a); 



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

 obtain simple expressions for these functions. Primarily we iind that 

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



