310 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



When t=A (go) sin (nt+A), then e, b, c, are to be sought in expres- 

 sions of the following form : 



£=E (go) sin (nt+X), 

 b=cp (go) cos (nt+X), 

 e=ip (go) sin (nt + X) t 



wherefore the last of equations (iVa) becomes 



~ ^ . s 1 I dimsm go) , . ) „ 



whilst the first two give 



dE ™2cos go 

 RT dGo sin go 



*?" nS 1—4 cos 2 go 



4,=- 



dl^ 



RT.dco 



2cosgl>- 



E 



sin go 



nS 



1—4 COS 2 GO 



These latter values substituted in the preceding equation lead to a 

 relation between E and A only, or between e and r. It will be con- 

 venient for the further computation to introduce an auxiliary function, 



<P(Go)=^jr<p(oo) sin (go) 



1 



(I — 4 cos 2 go) @GOz=- 



1 d(Es\n 2 Go} 



sin go 



dco 



tfS 2 

 BT, 



E = ^— — f (p (go) sin go (4 sin 2 go— 3) di 

 sin' 5 go J 



S 2 1 i d<P ,2 cos go E { 



T u { ^sin cj ( dco sin ca sin©) 



(11.) 



then 



sin go sin g? 

 If we assume $ to have the following form : 



<P (co)=cos G9(a t sin oo-\- a 3 sin 3 oj+a 5 sin 5 GL)+ . . . ) 

 E (oo)=bi sin co+b^ sin 3 Go-\-b 5 sin 5 go-\- . . . 



_4«i— 3a 3 _4a 3 — 3a 5 

 0i=&i, o 3 — = , 05 — y , 



Let the temperature amplitude diminish from the equator to the pole 

 according to the cosine of the latitude or 



and for brevity put 



A (go) = G sin (go) 

 S 2 



k—n 2 



h"J' 



