232 



THE RADULA OF HYALINIA. 



II. 



Variation in the Radula of Hy, helvetica. 



By a. E. boycott. 



(Read before the Society, February nth, 1914). 



The object of the present enquiry is to ascertain the extent to which 

 the radula of H. helvetica is subject to "spontaneous" variation. The 

 preceding paper {supra p. 214) has shown that the size factor must be 

 eliminated by using individuals of approximately the same dimensions: 

 it is also plain that the possible influence of race and locality must be 

 excluded by taking the material from a single restricted area. Some 

 of the snails already dealt with are suitable for our present purpose. 

 Strictly speaking, perhaps, all should be of precisely the same size, but 

 it is of some importance that the number of observations should be 

 numerous, and I propose to deal with the 66 specimens with major 

 diameters of from 7*0 to 7*9 mm. as if they formed a truly homogeneous 

 size group.^ There are also 23 specimens between 8'o and 8"8 mm. 

 in diameter which afford another series. The details for each speci 

 men are shewn in Tables I. and II. The method and terms used 

 have been already described {supra p. 215). 



The statistical methods of expression which are used can only be 

 very briefly explained." Suppose a series of five measurements gives 

 the results 8, 9, 10, 11, 12. Then the w^a// is 8.+ LtiP^+i_i + i2_^5^^jQ_ 

 Take the differences between each measurement and the mean 

 (2, I, o, I, 2); square them (4, i, o, i, 4) ; add the squares together 

 (10); divide the sum of the squares by the number of observations 

 -y*-=:2 ; take the square root of the result=:i"4i, which is the sta?idard 

 deviation. The coefficient of variation is the per centage of the standard 

 deviation on the mean, in this case ---* y^i-° = i4"i and is the measure 

 of the variability by means of which we can compare the variability of 

 one shell or organ with another. The process is simpler than it seems. 



Working out the detailed figures along^ these lines, we obtain the 

 following summary results for the 66 snails of the 7*0 — 7 ■9mm. group. 



1 This group is taken simply because it comprised the greatest num ber of larger snails. In 

 the particular locality shells over 8 mm. were not common, and those over 7 mm. were anatomi- 

 cally sexually mature. It would have b^en more convenient to have taken an S'otoS'gmm. 

 group. 



2 For a most lucid and non-mathematical exposition of this important subject see G. U. 

 Yule : An introduction to the theory of Statistics (London : Grififia cfc Co ., 1912). 



