38 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 51 



This expression for V, as well as those that must obtain for F, 

 for <p and for p respectively, when we write these out as functions 

 of p, X and r, all possess the properties that Laplace has demon- 

 strated for all functions of this kind that have definite real values 

 for constant r and for all values of X, from o° to 360 and of p from 

 — 1 to + 1. That is to say, since this latter condition (the having 

 a definite real value) is evidently fulfilled in the earth's atmosphere 

 as to temperature r, density p, and the function ip whose differential 

 coefficients are the component velocities, therefore, in accordance 

 with the Laplacian theorem just referred to, we may similarly assume 

 for V, for F, for <p, and for p, respectively such expressions as the 

 function 



a P° + P P' + y P" ■ ■ ■ • + v P {n) + . . . . 



in which, as we pass from one to another of these four functions, 

 V, F, <p and p, there occur: 



(1) Those coefficients a, /?, y . . . u which only vary with r 

 (2) the constant numbers that enter into P°, P 1 , P 2 , P n , as defined 

 in the next following paragraph. 



In general P n is defined by the partial differential equation 



d { (1 - p n ) 



dp J \ dk" 



= — — - + T 2 + n (« + 1) P (n) . 



op 1 — p" 



and from this it follows explicitly that 



P (n) = 5 n ° X^ + (A' n sin / + B' n cos /) X (1 ~^^ . d ^- 



n dp 



(1 - u 2 Y /2 a' Y (2) 



+ . . . . (A} sin * X +5 n * cos i X) K - ^ . —^- + .... 



n (n — 1) (w — i + 1) dp 1 



where B n °, A n ' , B n ' are constant numbers and 



X („) = n _ n(n-l) 2 n(n-.l)(n~2)(n-3) ^ _ etc 

 2(2»-l) 2.4. (2» -1) (2 » -3)' 



Since the development of each function, V, F, <p, p, in the form 



aP° + PP' + etc., 



