594 



6. In the same manner as in § 5 in the case of a directed quantity 

 in a plane, the development appropriate for quantities in space may 

 be found, e.g. for distances of stars, disregarding direction. 



The element of integration is then 4,n:RhlR, and the development 

 (11) holds f]?ood if in the left member /?-"+i is written instead ot 

 R^" and, at the same time for r/:^ 



oc 



(14) 



U2n R^dR 







so that 



r2„ =:- A" (fn aud q„ = Ci22"-hi e-R\ 



Jx 



Putting 



C =^ ~ 



U''^" becomes: 



« 2n+l » (2w+l)(2/i— 1) 



and 



Cu-^_n R'dR = ( - 1)" . 2" . nljffn RdR — (— 1)" 2»-i n!n! 







In applying this development a simplification may be obtained by 

 writing HR for R and putting : 



3 

 2M' 

 then A^ = 0, because 



U', = R^- V,. 



7. Although we ma}" expect a priori that the FoiRiER-series is 

 the most appropriate form of development for frequencies of directions 

 (disregarding velocity), it seems desirable in connection with the 

 foregoing to show that, following the same method, we, in fact, 

 come to this result. 



If 



U =: sin-'' a + fjj sin'-'^—- « -j- . . . cin , 

 then we may distinguish four different types of functions, namely : 



I^=z U F^=: U sin a cos a F^= U co.^ a en F^ = U sin a 



