STEADY MOTIONS OF EARTH S ATMOSPHERE ERMAN 39 



is only possible in one manner and since it always gives a converg- 

 ing series, therefore, each of the functions occurring in equation (B) 

 consists of a limited and, in fact, probably a small number of terms, 



aP°,pP', r P",etc, 



which altogether constitute a series progressing according to the 

 whole powers of 



p and (1 — fi 2 )* 



whose terms are multiplied into the sines and cosines of multiples 

 of X. Furthermore, since the terms of this nature, in equation (B), 

 resulting from the development of V and F contain respectively only 

 a well-known function of r, while, on the other hand, the terms aris- 

 ing from the differential quotients of <p contain the functions a, /?, y 

 of this same form and the constants A, B, C — which are the only 

 unknown quantities of the problem — therefore, these latter must 

 be determined by equating to zero each of the sums of known and 

 unknown terms that in equation (B) are multiplied by 



p q (l -p 2 )*siniX or p q {\ - pP)*cosiX. 



In order to practically execute the determination of the velocity 

 function <p, for a given temperature function F, we can easily con- 

 vert that form of the latter equation which results from the combi- 

 nation of equations (A*) and (B) into the equivalent differential 

 equation in r, p, and X whose specialization then leads directly to 

 the desired end. 



The two following relations between any two functions, $ and 

 <£', of the coordinates x, y, z are easily demonstrated 



00 00' 00 00' d$ dcf>' _ 00 d<p' 00 00' 1 - p 2 

 dx dx by dy Oz dz Or l)r dp dp r 2 



d<f> 00' 1 



+ 



2\,-2 



dX dX (1 - p*)r 



and 



old- 2 ) d ±\ (?*) 

 d2 ± + a V + #* = J_ dpi W/ r O 2 (r0 ) 



dx 2 dy 2 dz 2 dp 1 - a 2 dr 2 



vvuence it follows that equation (B) may be written thus: 



