\ XXVIII XLI] I '/// n>,i '//////, clxxxix 



of the day, we ha\v n<> hesitation in interpreting 2 na covering the range of 

 cloudiness from 1'5 to 2-5 tenths, but we have doubt* in supposing the subrange*! 

 and 10 to extend from - o to +0'5 and +9'5 to -f 10'5, which would signify 

 that the heavens could be more than quite clear of cloud, and mon- than wholly 

 enveloped in cloud. We are forced by considerations other than the fn-<ju<-n( \ 

 data to place our asymptotes for the actual {/-curve of frequency at and 10. 

 so that the total range is, as it should be physically, 10. But this involves the 

 terminal ranges being only one-half the other ranges, or the assumption that 

 to 0*5 and 9'5 to 10 are what the observers would record as and 10. To such 

 changes in the terminal subranges attention must be paid not only in calculating 

 the abruptness-coefficients, but in selecting appropriate positions for the asymptote 

 or asymptotes. 



After these warnings we may return to curve (i). Supposing ni = Ar/ii',?f = Arnj',... 

 n 6 = Nn & ' and putting x=l, 2, 3, 4, 5, we obtain five relations between the six 

 constants A, E, G, D, E and q and the Jive relative frequencies n/, n/, n 3 ', w/ and n 6 '. 

 q is then given a series of values between and TO; in this series it wsvs found 

 needful to make the intervals smaller for the range 0*0 to O'l, where q proceeds by 

 hundredths, than for the range O'l to TO, where q proceeds by tenths. We have 

 now got an auxiliary curve, which leaves q arbitrary, but gives the correct areas 

 for the first five equal subranges. How is q to be determined? After much con- 

 sideration, it was settled to determine q by making the sixth subrange frequency 

 (w 6 = Nil*) equal in the auxiliary curve to the value given by the data. Since 

 A, B, C, D and E arc known in terms of /, w a ', n 3 ', w 4 ', n 5 ' and q, we have 



NC' = q C 6 n 6 ' - qCiiit + qOzKs - 9 CW + 7 CV"i' (ii), 



where the constants g C 5 , 9 (7 4 , q C 3 , q C Zt q Ci are functions of q only. They are pro- 

 vided in Table XXXVIII. On th.e machine by continuous process it is fairly easy 

 to find the value of ?? 6 ' corresponding to any given q. We proceed therefore by trial 

 and error, and having found two values of q, one of which makes nj lie above and 

 the other below the true data value, we proceed to find a suitable q by linear inter- 

 polation. This is the purpose of Table XXXVIII, i.e. it is used to select a suitable q. 



The next four tajbles (XXXIX) give (when multiplied by ^Vand with due regard 

 to sign) the first four moments about x = 1, 



AV/^i", NnW, NnJri' and JV^W, 



(with due regard to sign) of the first element of the auxiliary curve, and they are 

 taken to represent the corresponding moments of the observed frequency on the 

 first subrange. But having ascertained the contributions to the total moments of 

 this first subrange wherein the asymptote lies, we can find tin- corrections for the 

 crude moments of the rest of the curve by the usual formulae for an abrupt, but non- 

 asymptotic, terminal *. This process would involve abruptness-coefficients in terms of 

 w 2 , n a , 4, n 6 , ?? 8 , but for each value of q, ?> 6 is known in terms of HI, n t , n 3 , 4 and w 8 . 



* For the actual nmnncr in which this lias been carried out see Fairman and Pearson, 

 Vol. xn. pp. 251253, and O. E. Fenrse, ibid. Vol. XX A . pp. 315319. 



