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



Squaring, introducing the standard deviations, and rearranging, we find 



^i -* ci w ) 4- / J ?*- ?s?Y s 2 = 1 1 4- cj iv 



yf ' " T 2 ^ \ 2* 7 / ij) 2n [ S W t; 



We must now evaluate Jp ^. This is easily shown from the BERNOULLI 

 number expansion for log F (p + 1) to be given by 



p 



where 





Thus we determine 



Expanding the expi-ession in brackets in inverse powers of p we find 



y, 49 28 248 1* 



Besult (Ixi.), however, with S and T calculated to 1/p 3 , gives a better value than 

 (Ixii.). 



To find the modal frequency error we must take the logarithmic differential 

 of (xlvi.) and proceed in the same way. We find almost at once 



2/ (l "7 P 

 Whence on squaring and completing the square of the factor of %> 2 , we find 



v- if 2T S ] 

 2, _ J_ J ! j_ L I , 



tf 2n\ J 



and 



Expanding as far as powers of 1/p 8 exclusive we obtain 



a very simple expression for the probable error of the modal frequency, y . 



