( 688 ) 



4. The calculation of the five charateristic constants of a wind 

 distribution amounts in one respect to the integration of (2), in 

 another respect to the means applied in this integration to a given 

 set of observations. 



The integration of (2) takes place by the introduction of rectangular 

 coordinates : 



x ■=. R sin 6 , y = R cos 6, 



where the element RdRdd is replaced by the element dxdy, whilst 

 the limits which were oo and for R, 2jz and for 6, now become 

 oo and — oo. 



Then the expression (2) under the sign of the integral is multiplied 

 successively by 



&, y* **i V* and asy. 



If we then put : 



R cos (a — ft) = a, x = x' sin ft A^ y' cos ft, 



R sin (a — ft) = b, y = as' cos ft — y' sin ft, 



the variables x' and y' can be separated and the integration can be 

 done; in this way we find for the determination of the five quantities 



to be obtained the five equations : 



M x = a cos ft — b sin ft, My = a sin ft -f- b cos ft 



cos** 3 siTt^ 3 



M x * = A 4- a* cos* ft A- b* sin* ft — ab sin 2ft 



2h* 2A" 



sin* ft cos* ft ( (o\ 



M* = A + a* sin* ft A- b* cos* ft A- ab sin 2ft ( ' l d ) 



y 2h* 2h l * 



2M xy = (^~ ^\ sin 2ft At (a* - b*) sin 2ft + ab cos 2ft 



out of which, ou account of (1) 



M x •=. R cos a, My = R sin a 

 M x * + M y * - [ K M x f + {M y )*] = M* + M" 

 M x * - My* - [(M r y — (My)*] =(M*- M'*)cos2ft 

 2M xy — 2M x M y — (M* — M'*) sin 2ft 



