288 



Prof. Karl Pearson. 



Proceeding we found 



e = 0-00699 r = 0' 17958 



m x — —0-8774 a x — 4*8109 



m 2 — —0-9430 a 2 = 5'1705 



The negative values of m x and m 2 snow us that the theoretical 

 curve has changed from its usual form to a U- sna P e( l figure. The 

 range given is b = a x + a 2 — 9-981.4, instead of the actual 10. 

 The distance d between mean and mode 



a 2 — a x 



— — • = 2'0022. 



r 



Thus the start of the range is 4-8270-4-8109 = 0*016, instead of 

 0, and it runs to 9*998, instead of 10. We conclude accordingly 

 that if the range of possible cloudiness had been quite unknown 

 a priori, it would have been closely given by theory. 



The modal frequency y was found to be 50*7505. 



Thus the theoretical equation to the frequency is — 



/ x \-°' 8774 / x \-°' 9430 



y = 50-7505 1+ (1 — - , 



u \ 4-8109/ \ 5-1705/ 



the origin being at 4*8270. 



The modal value now corresponding to a minimum and not to a 

 maximum as usual, the name " mode " ceases to be appropriate.* 

 The observations and the above curve are given in the accompanying 

 diagram, and it will be seen that there is a complete transforma- 

 tion of the usual frequency distribution to fit the altered state 

 of affairs. With the asymptotic character of the curve, it is 

 impossible to compare ordinates as giving the frequencies between 

 and 1, and 9 and 10. Accordingly the areas of the curve between 

 0*016 and 0*5, and between 9*5 and 9*998 were taken as the true 

 measure of the frequencies of the degrees and 10. These were 

 obtained by means of the following formulas : — 



A 1 = 



nfi n 2 n > fxA 1 -^ f 1 n 2 I x-\ n 2 Q 2 + l) fxA 2 \ 

 UW X ( Wl + «,)«,+». X U j 1 1 - n j2-n l \ b r 2(3-n x ) \ b ) J ' 



A 3 = 



( h , nfi n 2 n 2 / x 2 V~ n 2 J 1 n x ( x 2 \ n x (n x + 1) / x 2 \ 2 \ 

 {V ° } XT) I l-nj2-n 2 \TrY{^-^ \T) J ' 



where 



n x = — m X) n 2 — — m 2 , 



* The name antimode is now convenient. 



