( 811 ) 



b. Given u=^0 for x = 0. 



In the same manner as the (^-series has been made to suit the 

 zero-condition of the function at the limits, the tf>series can be made 

 fit for the case that the function assumes the zero value for the lowest 

 limit by multiplication with x. This case presents itself e.g. for 

 frequencies of wind-velocity, the curve of which originates at the 

 zero-point as absolute calms do not occur. 



By this operation the degree of the polyuomia is increased by one 

 and we can write down at once the new T function from (16) by 

 multiplication with x and, at the same time, substituting n -[• 1 for n 

 except in the binomial factors which remain the same. 



The condition for the determination of the a-coëfF. is now : 



ƒ 



and the general expression : 



,, n (w+iy n(w— l)(n + iy 



From this evidently : 



1 dl\+x 

 7i-\-\ dx 



a similar relation as is shown by (11) between the Q and i? functions. 

 Hence, if we put : 



ƒ00 

 U T'n dx 







where : 



ƒ 00 /» 00 



e-^ T„+i T\ dx = I e-^ v» T^+i dx = 







«-^"'^"— J^ = ('i+1) I e-^x'^Sndx = {n\\)lnï 







so that: 



" 0/ {n^\)! n! \! n! (n-1)/ "^ V {n-\)! {n^^f ~"' ~^ * ^^^^ 

 If we call the series discussed here, the tp'„+i series, so that: 



