of the Fishery Board for Scotland. 



201 



really decreasing or not is therefore simple. It is necessary to calculate 

 the average -size at first-maturity and that of the total mature, care being 

 taken that the specimens are not specially selected and are yet representa- 

 tive for the area in question. The changes in the averages, and the 

 changes in the relative distance between them, will show the change in the 

 number of mature specimens, and thence in the total number of fish on 

 that area. The investigations should be repeated for several years in 

 succession in order to avoid the possible yearly fluctuation, and again 

 some years later. 



Variability and the Method of Treating Variations. 



This complex subject has been dealt with recently by so many different 

 writers and in so many different ways that it is impossible to give a com- 

 plete digest of all the views concerning the method and its usefulness 

 within the reasonable compass demanded by this paper. The mathe- 

 matical aspect of the study has now assumed a definite form, with regard 

 to its more elementary portions at any rate, and the introduction of 

 shorter modes of working by Pearson and Duncker has greatly reduced 

 the old labour and drudgery. This aspect of the method is described by 

 Davenport,* and more fully by Duncker. f 



So far as the biological aspect of the method is concerned, the study of 

 variations is passing through a transitional period. For this aspect 

 one must turn to Galton,J Weldon,§ and Heincke.l! 



In the following pages the endeavour will be made to maintain the 

 biological aspect constantly in the foreground, and to display the 

 mathematical method as merely an instrument and aid, though a power- 

 ful one. The higher developments of the method, the determination of 

 curves, and the correlation of different structures, are not shown here 

 because the material is not very well suited for them. Attention is 

 therefore directed to the simpler and more elementary portions, which in 

 the present condition of the study are by far the most important. 



The necessary foundations for the study are of course the facts of 

 variation and the characters that vary. Here, at the very beginning of 

 the study, we find that its use with regard to fishery research has diverged 

 widely from that in other departments. In the latter one or two 

 characters only have been taken, such as the number of fin-rays, of glands, 

 the height or breadth of various parts, and so on ; their manner of varying 

 has been displayed, and sometimes the correlation of one part with another. 

 There are certain noteworthy exceptions, such as the works of Weldon on 

 the Crustacea and of Bumpus on the Sparrow, and it is more than probable, 

 now that the newness of the method has worn off, that more characters in 

 each animal will be examined in future. Heincke has set the example for 

 this in fishery research, and one can readily understand the reason. We 

 do not wish to know how this or that character varies by itself ; we wish 

 to know how many vary in relation to one another ; in short, how the 

 organism varies. For this purpose a large number of characters has to 

 be examined, and fewer specimens can be taken than was formerly 

 the case. 



In fishery research the method has been employed for the separation of 

 groups of individuals from one another. Before entering upon the 

 research it has been impossible hitherto to foretell how many individuals 



*Davenport, C. B. : " Statistical Methods," 1899 ; Chapman & Hall, London. 

 fDuncker, G. : "Die Methode der Variationsstatistik, Arch, fur Entwick, B. VIII., 1899. 

 +Galt<.n, F. : "Natural Inheritance," 1889 ; also Proc. Roy. Soc, Vol. LXI., 1897. 

 §Weiaon, W. F. R. : " Remarks on Variation in Animals and Plants, Proc. Roy. Soc, 

 Vol. LVIT., 1895 ; also Presidential Address to the British Association, 1898. 



HHeincke, F. : " Natnrgeschichte des Herings," Abhandl. d. Dent. Seefisch, V., 1898. 



