280 
PROFESSOR K. PEARSOX ON THE MATHEMATICAL THEORY 
Table XVII.—Motion of Bright Line. 
1. 
2. 
3. 
3-2. 
1-3. 
2-1. 
11-2 
5 ■ 6561 
5-5027 
13-2709 
12-0942 
16-1136 
11-1019 
rz 
+ 4-7424 
+ -5452 
+ 5-5761 
+ 21-0453 
+ 16-2877 
-20-4072 
+ -9755 
± -9361 
± 3 - 5060 
+ 3-0502 
+ 4-6909 
±2-6827 
r-i 
160-3883 
152-1762 
456-8883 
583-8991 
693-2790 
569-4060 
-1243 
-0018 
-0133 
-2504 
-0634 
-3044 
-0000 
-0000 
-0000 
-0000 
-0000 
-0000 
/52 
5-0135 
5-0256 
2-5942 
3-9919 
2-6701 
4-6918 
+ -1450 
+ -1450 
+ -1450 
± -1450 
+ -1450 
±-1450 
d 
-2193 
-0247 
-3020 
-6432 
-7219 
-5706 
+ -0862 
+ -0851 
+ -1321 
±-1261 
+ -1456 
+ -1088 
Sk. 
-0922 
-0105 
-0829 
-1850 
-1798 
-1713 
+ -0363 
± -0363 
+ -0363 
+-0363 
+ -0363 
± -0363 
-3524 
-0422 
-1153 
-5004 
-2518 
-5517 
+ -0725 
± -0725 
± -0725 
±-0725 
+ -0725 
+ -0725 
K 
-3-6541 
-4-0460 
-8514 
- 1-2327 
-8501 
-2-3266 
+ -2901 
+ -2901 
± -2901 
+ -2901 
+ -2901 
± -2901 
N.B.—The units of this table are | centim. on the recording strip—not on the observation strip; 
these are the units of our grouping in Table XV. In most of my previous tables the unit has been taken 
as 1 centim. of the recording strip. The ^ centim. is retained here, as it will be required in the plotted 
diagrams as unit of grouping. 
Now let US examilie these results, rememheriug that on the basis of a random 
sampling the odds against a quantity exceeding its supposed value by twice, thrice, 
four, five times its probable error, are 10 to 1, 49 to 1, 332 to 1, 2700 to 1 respectively, 
in round numbers. 
In the first place, /xg differs from zero by 4 to 6 times the probable error in 
(l), (3-2), (1-3), and (2-1). Further, d differs from zero by 2‘5 to 5 times the 
probable error in the same cases, and the skewness also by 2'5 to 5 times its j)robable 
error, vdiffers from zero by 3 to nearly 8 times its probable error in the same 
four distributions. I consider that it is really impossible to look upon these distri¬ 
butions as random samplings from symmetrical material. On the other hand, (2) 
and (3) or the absolute personal equation of Dr. Macdoxell and Dr. Lee might well 
be symmetrical distributions. Do they, however, fulfil the conditions for normality 
ySg = 3, K = 0 ? The deviations of /Sg from 3 are in the two cases 2‘0256 and 
2’4058, or nearly 14 and 2‘8 times the probable errors respectively. Further, their 
values of K differ from zero by nearly 14 and 2'9 times their probable errors. Thus 
the odds are enormous against Dr. Macdoxell’s judgment being a random sampling 
from a normal distribution of errors, and are about 300 to 1 against Dr. Lee’s being such! 
Of the other distributions the odds are enormously against normal distribution in 
cases (1), (3-2), and (2-1). They are less marked in (1-3), oMy differing from 
3 and Kfrom 0 by about 2‘1 and 2'9, their probable errors respectively—but this case 
has already been excluded from normality on account of its sensible skewness. 
