OF EREORS OF JUDGMENT AND ON THE PERSONAL EQUATION. 
265 
Table VIII.—Distribution of the 500 Experimental Lines. 
Magnitude in -J inches. 
Frequency. 
Magnitude in J inches. 
Frequency. 
3-00 
.5 
5-75 
44 
3-25 
7 
6-00 
37 
3-50 
4 
6-25 
22 
3-75 
12 
6-50 
18 
4-00 
20 
6-75 
13 
4-25 
28 
7-00 
9 
4-50 
30 
7-25 
13 
4-75 
48 
7-50 
0 
5-00 
.^8 
7-75 
3 
5-25 
64 
8-00 
4 
5-50 
58 
8-25 
3 
Here the frequency corresponding to any magnitude m in half-inches denotes all 
the lines whose lengths fall between rn — ’125 and m -f '125 half-inches. The 
lengths of the experimental lines had before this grouping been read off to the nearest 
of an inch. 
From this frequency we found ;— 
niu = mean value of =5-3165 half-inches. 
cr„ = standard deviation of u = -9513 half-inch. 
Vu = (T„jin„ = -1789. 
Now let x'q = distance of experimental point of Ihsection from real midpoint of 
line, positive if it fall to the right, and = distance from left-hand terminal of line 
to experimental point of bisection in the case of the (/th observer. Let us write 
X'y = X ,Ja, then X'^ is the ratio error which we liad previously dealt with, and 
X^ = xjii. 
Clearly x’,, = x,^ — hi, 
and if m, denote the mean value of a variant 3, we at once find : 
Now 
Wx, = «Lx'„ + -S'! 
o-x, = o-x-,, J 
— Xy - '5 = Xrj/U - '5. 
(xii.) 
1 renting in the usual way variations as differentials, whose squares and products 
may be neglected, we liave :— 
. 
VOL. CXCVIII.—A. 
SXb, = Su 
‘lyiu 
2 M 
