514 



Although tliis calculation (given as an example, not as a proof) 

 is applicable only to moderate and small degrees of cloudiness, 

 because a cloudiness 10 cannot be represented by circles, we may 

 conclude from this example that it is permissible to substitute a 

 linear for a surface-cloudiness of which it is impossible to give a 

 final detlnition because a given proportion of white and blue in a 

 plane can be represented in very different ways. 



3. In the foUov/ing calculations we assume the clouds to have 

 the shape of an ellipse rotating about a vertical axis and that these 

 clouds, if average values derived from many observations are used, 

 may be considered as small so that the tangents, drawn to the 

 ellipse in a vertical plane, may be regarded as parallel. The well 

 known fact that the dimensions of all objects (mountains, constel- 

 lations, sun, moon) when seen near the horizon appear strongly 

 exaggerated, is taken into account by introducing a physiological 

 factor f iff). 



The relation between true and apparent cloudiness in a point at 

 a zenith's distance (p then becomes : 



Ws = Wfiq) \/l -\- P tang V/), (1) 



where h denotes the proportion between the vertical and horizontal 

 axes of the ellipsoid i. e. greater than unit}^ when pointed upwards, 

 smaller than unity when oblate. 



We further assume that an object at the fiorizon is seen twice 

 as large as in the zenith, and as /(r/) must satisfy the condition 

 that / = 1 for </=: 0, the simplest forms which this function can 

 be given are : 



f\ztz\ -j- sill (f) , )\ =r 2 — COS '^(p , /g =: 2 — COS (f. 



Of these functions the first corresponds with the greatest, the last 

 with the smallest augmentation for values of (f near 45°. 



The value of the angle /?, the zenith's distance where, appai'ently, 

 the blue disappears out of the sky and the perspective cloudbank 

 begins, is then determined by the formula : 



W.fldi) . Vl~-\- ¥ tang '^ = 1 (2) 



and the sum of the cloudiness in the arc fp ^0 to fp .-= /i becomes : 



p 

 IW= W I f{fp) . V\ + ¥ taruf (p dff 



(3) 



The apparent cloudiness corresponding to the true cloudiness W 

 is then : 



W,=zl-2/jt i^—IW) (4) 



