.S2„ . 2,n = /*-" a'?" 



(2m) 



i\ 2'» 



597 



JiHrQm-l 



1.3 ^ 1.3...(2m— 1)_ 



Although, tlierefore, in this case the f/ functions do not altogether 



vanish, still the form remains (he same as in (13'^) and (13') because, 



as appears from (17), I he polynoraium has the same value for all 



terms where n -\- m has (he same value so that e.g. the terms with 



-^4.0 -42.2 and .4o.4 



can be taken together. 



In order to investigate in how ftir a given collection of wind- 

 observations may be considered as a collection of two independent 

 quantities depending on chance, we have, therefore, in the first place 

 to calculate the constants a,b,i-i, A and // from the set of observations. 



In the second place the development (136) has to be applied to 

 the frequency-series of the wdnd- velocities, thereby taking for H 

 either h or k' so that the term U^ remains. 



A comparison between the A constants calculated in this way 

 with those determined according to (18) then gives an answer to 

 the question. 



9. By writing in (15) liR sin and JiE cos 6 for x and y^ multi* 

 plying by RdR and integrating with respect to R between the 

 limits GO and zero, we obtain a development representing the fre- 

 quencies of the directions independent of velocity. 



The even terms C/a» Jind Vo,n, or the product V^n U->m then give 

 rise to a series of terms of the type F^ (§ 7) all of which have the 

 factor cos- 2na in common. 



The even terras U^nJ,-! T^2.„-(-i, produced by the product of two 

 uneven terms have sin a cos « as a common factor and give rise to 

 terms with sin 2nn, according to the functions F^ in § 7. 



The uneven terms, analogous to F^ and F^, assume a simpler fornl, 

 namely : 



t^2n+i =^ K Sin a cos-" a hud Fon-j-i = K cos a snz'^" a 



and therefore give rise to terms with sin {2n-\-l) a sind cos {2n-{-l) a, 

 whereas all non-periodic terms vanish, except in the first term 

 with Ag. 



A comparison with the FouRiER-series thus produced and calculated 

 on the base of the five wind-constants with the FouuiER-series as 

 directly deduced from the observaiions of direction-frequencies, then 

 again gives an answer to the question. 



39» 



