370 



Prof. (r. H. Darwin. On an Apparatus for [Dec. 15 



Now let cos n (9x) be the observed value of a function whose 

 true value is cos n 0, and suppose that x, the error of 0, has a fre- 

 quency / (x) then the mean value of the function deduced from many 

 observations will be 



p+oo /+oo 



f(x) cos n(0x) dx+\ f(x)dx. 



J -00 . J 00 



In our case /(a?) is the ordinate of, the frequency curve whose 

 abscissa is x. 



Let OQ = fc, QB = a, QB' = 6, OA = a+h, OA' = b + h ; then 



r 



= 



^o 



pfl+A 



(a + h x 



J a 



| (6 + ^- 



)cosn(0 x) dx 



