( 803 ) 



ill calculating uQn in (3), we have to deal with the unnecessary 

 factor kn . 



However the relation (5), where: 



_ (2«)/ 



2« . n! n! 

 SO that : 



2" . n! n! 



«•' = T2^)r^" (*^> 



is useful in deriving from the well known properties of the 

 P-functions those of the Q-functions. 



They satisfy Legendre's equation as well as the zonal harmonics : 



(•^■' - 1) -y^ + 2.1- -^ _ n {n f 1) Q, = 0. 

 dx^ ax 



The recurrent formula becomes : 



and 



QnJ^X — xQn + 77 Qn-\ = 0, 



(2?i-j-l)(2n — 1) 



n! d>'(x- — ]y' _ , 



Qn = -^— (6a) 



22"+i n! n! n! n! 



Hence, we find: 



a-^ — 1 0,, Qn dx — ~ \ Pn Fn dx = 



J"'"" k"-J P(2/i + l) (2n+l)/(2w)/ 



and for An ■ 



An = a 



n{n-l) n(n-l)(n-2)(n -3) "] 



ƒ*" - 2:(2n-^) ^"-^- + 2.4.(2n-l)(2.":^ ^*"~^ " '''' J ^^^ 



6. (r/?;ön u = for ^ = ± 1. 



The case discussed sub a, where nothing is supposed to be known 

 concerning the function to be developed, will seldom occur in practice 

 and, as all adaptation is due to the accomodating power of the 

 ^-constants the application would, in such a case, necessitate the 

 calculation of many terms and, therefore, hardly be profitable. 



Now, in dealing with observations of the degree of cloudiness, 

 the case presents itself, that a curve has to be found, which is 

 characterized by the limiting values mentioned above. 



The observations of serene sky (cloudiness zero) and of an entirely 



