( 807 ) 



factor x^ — 1, a quantity which, by the definition of the limits, is 

 always negative. 



As well as the (^-functions, the /^-functions might be multiplied 

 by an arbitrary, constant factor, such that any peculiar development 

 becomes possible or also with a view of simplifying some expressions. 

 In our case e. g. kn might be chosen so that /? = 1 ; practically 

 however this would hardly affoid any advantage. 



c. Given ii = for x =^. zt 1. 



As has been remarked above, in working out observations of 

 cloudiness the case presents itself that the frequencies for the extreme 

 limits vanish ; if, however, we have to deal, not with the original 

 observations, but with average values as, e. g. daily means, the fre- 

 quencies of serene and overcast sky, although still of peculiar interest 

 for the knowledge of the climate, cannot be regarded as discrete 

 values because, owing to the operation of taking the means, a 

 continuous transition of these extreme values into the intermediate 

 values must be assumed. 



In this case, when the curves assume peculiar forms quite different 

 from the well known curves generally met with, we can take 

 care that the conditions for the extreme limits are bound to the first 

 term of the series whilst all other terms remain as they are in the 

 case discussed sub b. 



Now the first term must contain three constants, two for the 

 extreme values and one for the fixing of the area. 



In the expression 



«0 4- ^0'^ + Co^' (13) 



the constants must satisfy the three conditions 



"i = «0 + ^0 + ^'o 

 "o = «0 — ^0 + C„ 



hence -. 



4a„ =r 3 — (jfj + wj 



4c„ = 3 (m, + u,) - 3. 



The reasoning as well as the application then remain the same 

 as sub b ; again 



r 



^n+2-R'„,fZii? = 0, m different from n 



