PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 2 67 
Now whether the sample be large enough or not seems to depend on the just 
determined value of the. probable error, and in Professor Weldon’s case the probable 
error is so small, as compared with the value determined for Galton’s function, 
that I think v T e may safely draw conclusions from his results. 
Taking the case of shrimps, we have for the most reliable determination of r, that 
for total length of carapace and length of post-spinous portion # :— 
71. 
7'. 
■p.e. of r. 
Plymouth .... 
1000 
•81 
•0057 
Southport .... 
800 
•85 
•0050 
Thus the difference between the r s is not very large, but still between five 
and six times the probable error ('0075) of their difference. 
Taking two cases from Professor Weldon’s results for crabs, t with r’s of con¬ 
siderably different order, we have :— 
71. 
r. 
p.e. of r. 
Breadth, frontal, and . ... f 
Naples 
1000 
•29 
•0187 
R. antero-lateral margin . . . \ 
Plymouth 
1000 
•24 
•0196 
R. antero-lateral margin, and . f 
Naples 
1000 
•60 
•0117 
L. dentary margin.\ 
• 
Plymouth 
1000 
•70 
•0089 
With these probable errors the identity of the first pair of r’s is unlikely ; the 
identity of the second excessively improbable. 
The conclusions therefore to be drawn from our results are these :—The samples 
taken were sufficiently large to determine r with close practical accuracy. Hence, 
therefore, unless there were large errors of measurement, or in the determination of r, 
the evidence of these observations is against the constancy of Gallon’s function for 
local races of the same species. If the differences in the values of r be attributable 
not to deviation in the sample from the mean, but to experimental error or to 
methods of calculation, then it would appear that the methods adopted or the 
measurements are not sufficiently close to supply an answer to the problem proposed, 
it being an essential condition of the requisite observations that the experimental, or the 
arithmetic error shall be less than the probable error of the sample. It seems to me 
extremely improbable that the divergence should be due to errors of measurement, 
and Professor Weldon’s papers, I venture to think, illustrate not the constancy of 
'* ‘ Roy. Soc. Proc.,’ vol. 51. p. 2, 1892. 
f ‘ Roy. Soc. Proc.,’ vol. 54, p. 327, 1893. 
2 m 2 
