583 
the apex of the conch in all arcuate and coiled forms is also very 
strongly marked and there is an obvious correlation between the 
close coiling of the young, the size of the umbilical perforation and 
the rate of increase of the outer or ventral side.- Thus in shells 
with large umbilical perforations, the first whorl increased slowly and 
was more nearly equal on the ventral and dorsal sides than in those 
with small perforations in which the outer sides or ventral increased 
much faster than the inner. ‘This is shown by the lines of growth 
whenever they are observable and by the distance apart of the sutures, 
both of these being much more widely separated on the venter than 
on the dorsum, and also by the extremely long and gibbous outline 
of the venter as compared with that of the dorsum. 
One can readily illustrate this by drawing a circle with lines radi- 
ating from the centre and then roughly projecting upon this back- 
ground the figure of any of the species given, allowing the centre 
of the radii to coincide with the centre of the umbilical perforation. 
It can then be easily seen, that as the whorl grows, if the umbilical 
perforation be small, the outer side has necessarily in keeping pace 
with the inner to describe a much larger arc in proportion than it 
does when the umbilical perforation is larger. This necessarily 
follows because the two sides, starting froma given place in the 
plot of the radii, are more nearly parallel in proportion as the per- 
foration is larger. Thus in shells with small perforations the increase 
of the ventro-dorsal diameters of the body is often much in excess 
of all other diameters and this preponderance in highly involute 
shells may be continued until near the end of the gerontic stage. 
The proportional increase in breadth of the growth bands of the 
venter as compared with those of the dorsum is a corollary of 
this proposition, or in other words the bands on the outer convex 
side necessarily have quicker growth than those of the dorsum, 
being built out farther in the same periods of time. 
The ananepionic substage, as a rule, has the lines of growth 
straight or with ventral crests broader in the median line than at 
any other part, but in the metanepionic or early paranepionic 
at latest the hyponomic sinus is introduced. While the bands of 
growth still remain broader on the venter in spite of this depres- 
sion on that side, there is after this stage a constant lagging 
behind of the central ventral surface due to the presence of the 
hyponome. 
Among Ammonoidea this is not the case except in the more gen- 
