( 804 ) 



overcast sky (cloudiness ten) ought to be considered separately from 

 the other observations as they constitute climatological factors of 

 peculiar importance for tlie description of the climate (principally in 

 northerly latitudes). Moreover they are to be regarded rather as 

 discrete quantities, which do not show any continuous transition to 

 a cloudiness resp. of degree 1 or 9. 



The other degrees of cloudiness may then be regarded as obser- 

 vations of continuous quantities subject to the above mentioned 

 conditions. 



In this case we may easily cause all terms of the series (1) to 

 suit these conditions by simply multiplying the series by a factor 

 that vanishes for .r = =b 1 e.g. x^ — J, and then applying to the 

 new functions, which we shall call li, the same reasonings as sub a. 



The degree of the polynomia is then increased by two, so that 

 w^e have to start with /?,. 



The general expression becomes : 



i?„_|_2 = (<'■" — 1) R'n = ('«■" — 1) ['^"' + ^2 •^"'~" + • • • «/<] ' ** ^^"'^" 



= (a-^ — 1) [.I"' -f a^ .r«-2 -\- . . ttn—o], n odd. 



The result of this operation is evidently that the surface enclosed 

 by the curve, as determined by the first term of the series, is not 

 represented by a rectangle of base 2 and height 0.5 as in the case 

 of the Q-functions, but by a parabola of base 2 and height 0.75, 

 which makes again the surface equal to unity. 



By altei-uately asymmetrical and symmetrical deformations the shape 

 of this parabola is then altered by means of the next terms in such 

 a manner as to make it approach more and more to the frequency 

 curve corresponding to the given data. 



It may be noticed here that in the case of fixed limits, there is 

 no reason to choose for the origin of coordinates the point corre- 

 sponding to the arithmetical mean ; for logical and practical reasons 

 the point intermediate between the limits is then indicated. 



The condition, which has to be satisfied by the <7-coeff. of the 

 /^-function, and by which they are fully determined, is "now that: 



JEn-^o -'" ^f'V = I 7?',i A-'» i^v' — 1) d.v =0, m<:^n 



or 



(.v'^^+^E'ndx= (a;''^R'nd.v (8> 



The a-coeff. are calculated from the equations; 



