]\rATHEMATICAL CONTEIBUTIONS TO THE THEORY OF EVOLUTION. 297 
Thus the fecundity of a mare was described by one of these 16 letters. Here the 
centre of the j group, for example, is ’6, and it covers all fecundities from *56 to 'GS. 
Thus midway between y and h we are at about the mean fecundity. 
The more recent Stud-Books, vols. 12 to 17, were taken as containing more 
complete details and, what is more important, less in-and-in breeding, although as we 
shall see, this is still an important factor. These volumes cover 30 and more years 
of English'^ stud life. From these 30 years’ records upwards of 5000 mares, who had 
been covered upwards of four times, had their fecundity ascertained. The process 
w’as a very laborious one, as each mare had generally to be sought for in several 
volumes, and the records in each volume are not continuous, but overlap by quite 
arbitrary numbers of years. Further, great care had to be taken to identify each 
mare properly, as the same name is very frequently repeated, and the like difliculty 
occurs, though to a lesser extent, in the case of sires. A card was then written, 
giving the name of the mare and those of her sire, her dam, and her dam’s sire. 
Upon this card the letter indicating her fecundity was placed. A card alphabet of 
mares was thus formed, consisting, in the first place, of about 3000 entries. This 
alphabet was again gone through and the fecundity of the dams of the mares inserted 
on the cards till there were about 2500 cases known of mare and dam. The dams 
were partly found from the existing series, but it was also largely necessary to work 
out fresh cases. Lastly, the cards were gone through and the fecundiU of the grand- 
dams entered in upwards of 1000 cases. This forms the first series of cards. 
In the next place a card index was formed of all the sires serving during these 
30 years. This contained upwards of 1000 cards. On these cards the sire’s sire was 
entered, and the fecundity of all the mares contained in the first or mare alphabet 
was now' taken off and placed on the card of the mare’s sire. Thus the card of each 
sire had the letters a, h, c, d, e, &c., upon it, and a frequency distribution was formed 
on the card of each sire for the fecundity of his daughters. 
The same thing was done for the sires’ sires ; only here recourse had again to be 
had to the stud-books to obtain the fecundity of the daughters of the more ancient 
sires. Finally, a sire-alphabet was obtained which gave the average fertility of the 
daughters of a sire and of the daughters of his sire, or his half-sisters. On these 
cards was also placed the number of mares upon which each average was based. 
These two card-alphabets, the mare and sire alphabets, form the “ dressed ” 
material upon wdiich all the subsequent calculations were based. 
(14.) At this point it seems desirable to insist somewhat on the many causes which 
tend to make the fecundity of mares, as thus determined, to a considerable extent 
fictitious. Many of these were only apparent to me as I became more and more 
familiar with the material. 
* IrisE mares were excluded except where, for pedigree purposes, it was necessary to deal witli them. 
Many Irish mares were further included when it came to the valuation of the fertility of mares duo to 
a given sire. 
2 q 
VOL. CXCII.—A. 
