8 W. E. ALKINS — Morphogenesis of Brae hi op o da 



given by Day {loc. cit., Fig. 4) : it therefore appears likely that 

 Day took his length from the pedicle umbo to the anterior margin 

 — which would at once account for the fact that the rate of 

 decrease of the ratio during the growth of the shell does not agree 

 very closely with the rate which may (approximately) be deduced 



Since w= i"3S(/— i), we have : -^-=1/35'; 



i.e., the width increases at a constant rate, which is equal to 

 i«35 times the rate of increase of the length. 



ai w J * \ . J '35 



Also — = 1 -35^1— -j) = 1 -35— — 



dl P ' 



i.e., as the length increases, the ratio- — increases rapidly at 



b Length ^ ' 



first and then more and more slowly, the increase being very 

 small when the length is great. 



The relationship between depth and length is rather less 

 simple (Fig. 4). Over the whole range covered by the specimens 

 studied, the two dimensions are very well expressed by the relation : 

 D =0-6643/- |-oooi 1 / 2 -fo-ooo247/ 3 ; where D=depth, /=length, 

 in millimetres. The agreement between the experimental values 

 and the curve given by this equation is remarkably good (little 

 weight is to be attached to the last two or three points found, 

 on account of the relatively small number of such large specimens 

 in the series). 



Since D = o-6643/+o-ooi 1 1 2 -{-0-0002 47 1 3 , 



d~D 



—77- = o 66434-o-oo22/-)-o-ooo74i/ 2 ; 



i.e., the rate of increase of depth increases more and more rapidly 

 as the length increases. 



Further, — = 0-6643-}- o-ooi i/-f 0-000247/ 2 ; 



— - — -= o-ooii-j-o-000494/; 



dl 



i.e., the rate of increase of the ratio - — *— =- increases at a con- 

 Length 



stant rate as the length increases. 



It is therefore evident that the relation between depth and 

 length throughout growth is very different from that between 

 width and length. 



