156 PROBLEMS OF RELATIVE GROWTH 



As in each case only one specific combination of these two 

 independent variables will produce such a result, we must 

 suppose that selection has controlled the precise values to 

 secure this result, which obviously secures greater strength 

 than does one in which the whorls do not touch. 



(5) The commonest condition, however, which ensures 

 even greater constructional strength than the one preceding, 

 is produced by a further increase of antero-posterior growth- 

 ratio, which has as result the partial overlapping of each old 

 whorl by the whorls formed later. The degree of overlapping 

 will obviously vary with the precise value of the ratio (as well 

 as with other properties of the shell : see below). 



A B 



Fig. 74. • — Diagram illustrating the co-operation of two growth-ratios in 

 determining the form of the Molluscan shell. The two plane logarithmic- 

 spiral shells both have the outer margin of one whorl just touching the inner 



margin of the next. 



In A the ratio of the distance along a given radius from the centre of the shell to the margin of one 

 whorl to that of the succeeding whorl is 3-0 ; in B it is 2-0. If there were no lateral growth-ratio 

 (i.e. if the shells were uncoiled). B would be a more elongate cone than A; correspondingly, the lateral 

 growth-ratio in B must be higher than in A to cause contact of successive whorls. 



(6) Finally, in some cases the growth-rate of the slower- 

 growing edge of the mantle becomes negligible or even zero, 

 and the growth-ratio accordingly rises towards infinity. In 

 such a case each whorl is completely overlapped by all suc- 

 ceeding whorls. This condition is completely realized in 

 Nautilus pompilius, and nearly so in Nautilus umbilicatus. 



This median growth-gradient may be orientated either way 

 in respect of the main axis of the molluscan body. In Ammon- 

 ites, for instance, the high point (growth-centre) seems to have 

 been ventral, so that the shell curved upwards over the back. 

 In other cases it is dorsal. In flat-shelled Gastropods (e.g. 

 Planorbis) the orientation is complicated by the fact of torsion. 



