Manchester Memoirs, Vol. Ixv. (192 1), No. 8 9 



abnormal in shape. In the case of each of these two shells, 



the position of which has been denoted by an asterisk in 



Tables II, III, and VII, an injury to the ventral margin at a 



relatively early stage of growth has caused the length to be 



considerably less than it would have been if growth had been 



normal throughout ; width and thickness are quite normal, 



and the consequence is that the two shells are conspicuous for 



, 11 i_- 1 1 r i_ • width , thickness 



abnormally high values of the ratios -: -r and — , -rr 



J length length 



which place them in the extreme lower right-hand corner of 



Table VII, at a considerable distance from the mass of the 



specimens. In view of this circumstance it has been thought 



advisable to recalculate the whole of the constants for the 



correlation of the ratios, with the omission of these two 



particular individuals ; for the series of one hundred and 



ninety-eight normal specimens, we have : — 



„ ' Width . 



Mean Le^th ratlo: - II955 ' 



Standard deviation : — 0*02626. 



Coefficient of variation : — 2*20. 



, T Thickness . r _ 



Mean — ■=. r ratio : — o*6sq6. 



Length Jy 



Standard deviation : — 0*02460. 



Coefficient of variation : — 3*73- 



Coefficient of correlation: h 0*419. 



Equations of regression :■ — 



(a) J - - go + 0*4474 | 



standard error ± 0*238 ; 

 (b) 3L = 0*190 + 0*3926 ^ ; 



standard error £ 0*223. 



Hence, the correlation between the two ratios studied 

 appears to cover little more than two-fifths of the whole 

 variation ; apart from this variation in the one ratio occurs 

 quite independently of variation in the other. 



Summary. 



Measurements of length, width, and thickness of a series 

 of two hundred Sphcerhim lacustre (Muller) from a small pond 

 near the Three Lows, N. Staffordshire, has shewn that : — 



1. The correlation of each pair of variables (length, width, 

 thickness) is high ; the coefficient of correlation exceeds nine- 

 tenths in everv case. 



