512 



of C'loiidiness", and, ii possible, to substitute a simplei- definition, 

 suitable for niatliematical treat ment, for this evidently complicated 

 conception. 



In lig-.(l ) the plane of the drawing 

 represents a plane ^oing through 

 the observer in and the zenith 

 in il7o ; the circles are projections 

 of spherical clouds upon this plane, 

 the diameter of the clouds is 

 denoted by d, their mutual distance 

 by A. 

 Fig. 1. The apparent proportion of white 



to white and blue in the sky will be different according to the part 

 of the arc q^, (f\ — (f\ v^+i — f/» considered ; obviously the apparent 

 cloudiness of the ?i^i' order is 



y«+l — <Pn 



As we have here to do with proportions, and angular values are 

 unsuitable for mathematical treatment, we substitute for the pro- 

 iection upon the sky -arc the projection upon the straight line 

 Mii Ml . . . Mn ; then the apparent cloudiness is defined as : 



il/„_l_l ?»„_|_i 4- Mn tin d cos {(fr,-\-l «,;+]) + COS [cfn + «n ) 



il/„+l Mn 



d 



Sn — 



Here 



A 



A 2 cos {rpn-\-l — «„_|_i) COS {fpn + «« ) 



denotes the true cloudiness W of the sky, the cloudiness 



being defined as a proportion between linear quantities. When the 

 spheres are not very large we can assume an average value for 

 n and it ~\- '^, and the apparent cloudiness of the n^^^ order becomes: 



S„ ^ W 



cos ffn COS an 



cos^ Cfn — sin"^ an 



or, as sin a„ = sm ao cos (fn 



COS^ (fn 



'] = 



Sn = W 



1 -\- n^ q- ^ h 



vn^q^ -{- cos" a,^ 



COS- «„ 



If the clouds are not large so that cos ^a = 1 may be assumed 

 as equal to unity, this expression becomes, owing to the relation 

 nq^ tang cp», 



