of the Fishery Board for Scotland. 



205 



the true average of the total number. The amount of probability can be 

 calculated from the following formula. The probably true value of the 

 standard deviation or 



P.o-. = 0-6745 cr, 



and dividing this by the square root of the number of individuals observed, 

 or s/ n, we obtain the probable error (r) of the observed average. The 

 true average for the total number of individuals in the group will then 

 probably lie between M±r, where M is the observed average. According 

 to Davenport, if we take three times r, then the probability is 19 to 1 

 that the true average will lie between M+_3r. If we take five times r, 

 then, according to Heincke, the probability is 1000 to 1 that the true 

 average lies between M±5r* Through this method, therefore, we are 

 given the means of testing and telling the amount of probability or 

 certainty in our results. 



The further developments of the mathematical theory — the plotting-out 

 and determination of the variations-curves, the correlation of variations 

 and of averages — are not employed in the present paper, and may be left 

 for future discussion. 



A few words should be said as to the meaning of the words "variation " 

 and "variability." According to the common usage in Biology, "varia- 

 tion " is a general and vague term, which might be applied to either 

 "variants "or "deviations" as defined above. For common usage this 

 does not much matter, and in the present paper the term is employed in 

 its general sense. But a good deal of confusion would arise in the 

 mathematical method if its meaning were not made more precise. In the 

 more restricted sense now employed, " variations " mean the observed 

 deviations expressed in terms of the standard deviation. 



The definition of the term " variability " is extremely difficult. It is 

 one of those words which imply potentiality or possibility, and is best 

 described by showing what it is not. Any character whatsoever presents 

 a series and range of variations, but variability is not the variations nor 

 the range of variations. These are the outward expressions or indications 

 of the variability, which is something potential and hmate. In one and 

 the same individual, again, the characters do not all remain constant and 

 unchanging throughout life, but alter with growth, and here, again, 

 variability is not the actually observed change, but the possibility of that 

 or any other change. A^ariability is, however, not indefinite, so far as we 

 know, but gives rise to variations which are limited within a certain 

 range. Variability is thus a general term which indicates the power 

 that an organism possesses of giving rise to variations. These variations 

 are particular observed objects more correctly spoken of by reference to a 

 particular observed average, and their standard deviation about that 

 average is a measure of the variability, and is thence called the/' index of 

 variability." 



The various forms of "variability" will be presently dealt with; 

 meanwhile it is of importance to show more closely the underlying 

 meauing and inter-relation of these conceptions. This subject has recently 

 been discussed by Professor Sedgwick, t who endeavours to make a generic 

 division of variations into two kinds — genetic and acquired. Genetic 

 variations are inherent in the constitution of the individual. They arise 

 in the germ in consequence of conjugation, and manifest themselves in 

 " habit, constitution, form, and structure." They have nothing to do 

 with the environmental conditions. On the other hand, acquired 



*Tlie "fluctuations of the averages" in the Tables are based upon the formula M+5r. 

 f Address to the Zoological Section of the British Association, Dover, 1899. 



