W. R. Macdonell 195 
Now — ^„ = -7= e- 5^ dw : 
therefore, if H = — i= e~^*', 
V27r 
we have - Bn, = NHMi and In, = tnJ{NH,). 
Similarly - S??, = i^^^sS/^s and t,„ = ^nJ{NH,). 
Also 1ti^h^Rh^h^ = ^n^n,Rn,nJN'^H\H3, and it is shown in Professor Pearson's 
memoir " On the correlation of characters not quantitatively measurable " {Phil. 
Trans. Vol. 195, A., p. 11), that 
^n,- , ^ «3 - 
N ' 
therefore = ^3^,^ , S^a, = , 
The probable error of cr is accordingly •674492^, 
We have next to find the probable error in p^, that is, the probable error of the 
mean. 
Pi = hio; 
Therefore 1%, = h\ X\ + + 2h,al,thRah, \ 
also Ai + ^3 = - ; 
(T 
ha- 
Bhs = — — 7] 
cr' 
Substituting for '^,„%h,R<Th, from the first equation, and remembering that 
7] = a- {hi + hi), we have finally, after a few reductions, 
and the probable error = •67449Sp,. 
