Manchester Memoirs, Vol. Ixv. (192 1), No. 9 3 



the form of the table, is high — + 09598. The equations of 

 regression are : — 



(a) W = 0015 + i'24i L, 



with a standard error of + 0*301 ; 



(b) L =0-515 + 0742 W, 



with the standard error ± 0*232. 

 The regressions are throughout assumed to be linear. 



(c) Correlation of Length and Thickness. Table III sets 

 out the data for a correlation of length and thickness. The 

 correlation is again high, though not so high as in the case 

 of the relationship between length and width ; the coefficient 

 of correlation has the value + 09254, and the equations of 

 regression are : — 



(a) L =1505 + 1 147 D, 



with the standard error + 0*314; 



(b) D = - 0*480 + 0747 L, 



with the standard error + 0*253. 

 L = Length ; | 

 W = width ; >in millimetres.] 



D = thickness ; J 



(d) Correlation of Width and Thickness. In Table IV are 

 given the data for the correlation of width and thickness. In 



this case the value of the coefficient of correlation \- 0*9340 — 



is intermediate between the values which it possesses in the 

 case of the other two pairs of dimensions. The three 

 coefficients are thus in the same order as they were found 

 to be in the case of Sph. lacustre. The equations of regression 

 are : — 



(a) W= 1*55 + 1498 D, 



with the standard error ±0*383; 



(b) D = - 0*333 + 0*583 W, 



with the standard error +0*239. 



(e) Distribution of the Ratios 7 =- and — : — 



v J J Length Length " 



Data, shewing the distribution of the ratios . r- and 



' s length 



-; = — are set out in Tables V and VI. The rangfe over 



length fe 



which the ratios vary is very similar to that found for Sph. 



lacustre, though the dispersion is rather greater. The mean 



. r ratio, too, is somewhat higher than in the case of (he 



length fe 



latter species — 1*242 as compared with 1*196. 



