MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 
97 
colour may change both in intensity and variability with age, much as variability in 
stature changes with children from birth to adult life. 
(c.) As a more or less natural result of (6) it follows that any group, male or 
female, having male relatives is more variable than the same group witln female 
relatives. Thus sires of colts are more variable than sires of fillies ; fillies half-sisters 
to colts are more variable than fillies half-sisters to fillies, &c. But out of the nine 
cases provided by our data there are three exceptions to the rule, and perhaps not 
much stress can be laid on it, at any rate in the above form. It would appear that 
males, relatives of males, are sensibly more variable than males relatives of females. 
The bay ranges are L3926 cr' and L4447 cr respectively, which indicates that the 
average a' is larger than cr. But if we treat the groups of females alone, we find for 
females with male relatives the bay range = L3694 cr, and for females with female 
relatives L3433 a', which indicates that the latter are more variable. The difference 
is, however, not very sensible. Possibly the rule is simply that extremes tend to 
produce their own sex, but our data are not sufficient for really definite conclusions on 
the point. 
In order that we may have a fair appreciation of the probable errors of the 
quantities involved and the weight that is to be laid upon their differences, I place 
here a table* of the probable errors of y, of £ — cr x /cr y and of r xy for typical cases. 
IV.—Table of Probable Errors. 
Relations. 
Vx 
Vv 
c 
u 
v! 
r xy 
Sire and Filly . . 
•0143 
•0170 
•0243 
•0363 
•0330 
•0288 
Grandsire and Colt 
•0143 
•0199 
•0237 
•0385 
•0319 
•0333 
Colt and Colt . 
(Whole siblings) 
•0186 
•0186 
— 
•0328 
•0328 
•0259 
Filly and Colt . 
(Half siblings) 
•0179 
•0185 
•0315 
•0335 
•0328 
•0363 
It will be seen from this table that the probable error in y is about 3 per cent., in 
£ about 2 to 4 per cent., in u about 2 to 2‘5 per cent., and the values of r about '03, 
growing somewhat larger as r grows smaller. The probable errors are thus some¬ 
what larger than those which we obtain by the old processes wdien the characters are 
capable of quantitative measurement, but they are not so large as to seriously affect 
the use of the new processes in biological investigations. As we have already 
indicated, the probable errors of the y’s may be roughly judged by Mr. Sheppard’s 
formula for the median (p. 95). 
It will be seen that the differences in the y’s and £’s of Table II., or the its of 
* I have to thank Mr. W. R. Macdonell for friendly aid in the somewhat laborious arithmetic- 
involved in calculating these probable errors. 
VOL. CXCV.—A. 
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