593 



For uneven polynomia 7,, has to be given the same value as (9) 

 and then again the common zonal harmonic would result. As, 

 however, the quantities under consideration are essentially positive, 

 uneven functions can be left out of consideration. 



If the limits are go and 0, then the same reasoning holds; it is 

 then rational to put : 



U-in = C A" 2^2» e~^' U'2n = C ( A — 2)« R^". 



Putting 



2« 

 the polynomium assumes the form : 



U'2u = i?''" — n' i22"— -' -] ^^ R'^-i — . . . (—1)" n! . (13a) 



a! 



and 



00 

 jU^'in RdR = 2« . 7i! (ffn RdR 



n!n! 



~9r' 







In analogy with (12) the polynomium, by putting 



2»n.' 

 may be written also : 



U',n=-^--C TT. + '^C,- ^,...(-1)" . . (13/>) 



n! [n — \)! (n — 2).' 



This new function (13) seems to be the proper form of development 

 in the case of directed quantifies as wind-velocities, disregarding 

 direction ; it satisties the ditf. equations : 



dR' ^ ^ dR ^ 



d'U2n dL\ 



dR dR 



In applying this development, a simplilication may be obtained 

 by a change of scale- value: writing HR foi' R and putting 



1 



M"- 

 the second term with the coellicient A„ will disappear as 



U', = (R'—l). 

 Here ]\P denotes the moment of the second oi'der of the given 

 frequency- series. 



