59Ó 



and, by (4) : 



Un"" dx — - e-^' dx = - [/jr. 



— OS 00 



The series (6), proposed by Bkuns ') and Charijkr '^), is in matlie- 

 niatics known as Hermitk's function and might, if applied to 

 analysis of frequencies, be called the 7,, function, as proposed by Bruns. 



It is the most appropriate form for quantities as atmospheric and 

 watertemperatures, barometi-ic heights etc., moving between un- 

 certain limits, and also for wind-observations if generalized for 

 application to functions of two variables. 



In either of the cases considered above the terms of even and 

 uneven power are separated automatically because 

 +1 +00 



I A-2"+i dx = and | .r2n+i e-^^ dx = . 

 ~i - 00 



If, however, the limits are 1 and or 00 and 0, then such a 

 separation does not take place and we must either maintain the 

 complete poljnomium or consider both cases separately. 



4. Considering the even polynomia separately for the limits 1 

 and 0, every polynomium 1/2,1 contains only n constants and the 

 development (5) takes the form: 



X 1 



j f72„ .r-" dx = ,r2« (f^ — 2ji x-^" -•-' ^2 + 2' '« ('« — !) ■^•""~"* Uz— ( (7) 







(— l)"-i 2«-i n (n — 1) . .. 2 x^ q>„ (- 1)" 2" . n! </)„+^ 



where 



X X X 



f/ J — I U-in dx , (f.^ = I (f^ Xdx . . . (fn-\-\ = j ipn ^vdx 



Ü 



X X 



j L'^2. d.v — (— 1)" 2» . n! I (fn xdx . 



Putting 



X dx 



1) Wahrscheinlichkeitsrechnung und Kollektivmasslehre. 1906. 

 ~) Researches into the theory of Probability. (Conim. from the Astron. Observ, 

 Lund.). 1906. 



