298 PROFESSOR K. PEARSON AND MR. L. N. G. FILON 



Further, we have the following BERNOULLI number series for G (r, v), where 



tan <f> = v/r : 



log {e~^ G (r, v )} = log y(27r/r) + (r + 1) log cos <f> + v<f> 



(2s 



COS 



To find ?/, Sx, the mean frequency, we have only to put x = avjr in (i.), and we 

 have 



log y, = log y + (r + 2) log cos <j> + v<f> 



= log n log a log [e~ 1 "" G (r, v)} + (> + 2) log cos <f> -\- v<f> 

 = log n log a log v /(27r/? 1 ) + log cos <b x, 



where x stands for the summation in (cxxxiv.). Hence 



^ = ~v i^ e ~ Xco8< ^ ( cxxxv -)> 



or, 



* ..... (cxxxvi.)- 



As typical constants we require the probable errors of the mean, the standard 

 deviation, the skewness, and the mean frequency. It is clear that these will require 

 us first to find 2,,, 2,., , 2,,, and R,, (( , R,,,, R,,,, R, m , R,,,, R m , R_ a . 



We shall only indicate briefly the steps towards finding the integrals of the second 

 differentials of log y. 



log y = log i/o in log U 



^ (log?/) v 1 



' T T+7-eay " " 



^-(log?/) _ 2 f v.v/a m 2m "I 



rf* 2 " " 2 I ~ {1 + (/.) 2 } 2 ~" 1 + (a/a)* + (1 + ^/ 2 ) 2 J 



= (2/a-) { v sin cos 3 6 m cos' 2 + 2m cos 4 



-- e-v 



" ir/2 



whence, remembering that 2m = r + 2, and integrating the first integral by parts, 

 we find 



