ƒ 



595 



For F^ tlie development holds good -. 



U sin^" add = (f^ s«i-" <i — '2n *f.^ .sin-"''^ « | . . . 



(— l)"-i . 2>^--^ . » (» — 1) . . . 2 . (f„ (— 1)" 2" // ! r/o,,_^i 



where 



r/ J = I ilda <f\, = j f/ J s'ln a cos a da etc. 



Therefore, piitliiig 



S171 c( COS a da 



2"w! 



C sin-^'^^ a cos^"— 1 a and C = 



{2n)\ 



we find for the liinils - and 0: 



2 



2".n! 



F , rzz sin a cos a A" ff„ ^= cos 2na. 



(2?/) ! ' 



In the same manner : 



2- . 7i ! 



F^ = A"-i sm'"-i « co.s-"-i « = sin 2na 



(2«— 1)! 



2" 7i ' 



F^ = — ' — ' sin a A" sin^''-'^ a gos-"+^ a = cos (2« -f 1) rr 

 (2»)! 



2" . n ! 

 F, z=: — '—^ cos a A" sm"'"+i a cos''"-! « = sin (2h f 1) « * 

 (2/0! 



8. The solution of tlie second problem, as formnlated in §1, can 

 be simplified by putting- (^ — i^' = ^f in form. (1), i.e. by counting 

 the angular values not, as usual, from the North-direction, but from 

 JSfiiE; this has, of course, no influence on the sums of the velocities. 



It is, however, unfeasible to apply a simihir correction for the 

 components a and b of the resulting wind, and the problem to be 

 solved comes to the development in series-form of the expression: 



^^^' /-v v^ ,'2r /^2 Rcos6 = y 



~r R sin 6 z=. X . 



It appears from the. first of the communications cited in § 1 that, 

 in following the usual method of developing, difficulties are ex- 

 perienced which practicallv are unsurmountable. In the second com- 

 munication however, it w:^.s shown that the development (6) may 

 be extended to (he case of two variables x and y, and that such a 

 function can be developed in a series of poljnomia of the form: 



39 



Pioceeifings Royal Acad. Amsterdam. Vol. XV U. 



