Mathematical Problems 



85 



area very soon, and the dorsal part continues and gives rise to 

 the tail. Could not the mathematician tell us exactly, given 



a 



d' 



Bl 



Fig. 39. An attempt to explain graphically the origin of the true archenteron in 

 the Anuran egg by cleavage as the direct action of the same forces which 

 produce the true archenteron in Amphioxus by invagination. a-af a line 

 which represents a zone of mergence of small and large segments (the effect 

 of cell division in the horizontal plane) at an early stage of segmentation [cf. 

 Morgan and Ume Tsuda, Q. J. M. 8. Vol. xxxv, PL 24, Fig. 111]. 6-6' the 

 zone at a later stage, c-c'-c" the zone at a still later stage after gastrulation 

 has begun. The zone has progressed superficially until after passing the 

 equator; then at one point it has left the surface and passed inwards and 

 backwards c". d-d'-d" the zone still later. It is now about to leave the 

 surface at all points and to complete the ventral lip of the blastopore. The 

 split caused by its ingression below the surface becomes archenteron as soon 

 as the sides of the cleft separate. The "lips" formed by the ingression, 

 become ipso facto the secondary growth centre, because, whereas before the 

 turning in, any increase in bulk of the superficial layer (i.e. the blastula wall) 

 leads to mere increase in the size of the radially symmetrical form, after the 

 ingression any increase in size must on account of the folding or doubling of 

 the superficial layer on itself bring about a conversion of the radial to the 

 cylindrical form and thus initiate growth in length (deuterogenesis). This 

 must be true of all gastrulae. 



a force such as postulated, what the distribution of the inert 

 or impermeable material must be to produce this form? The 



