58 



PROBLEMS OF RELATIVE GROWTH 



the pupal instar, to emerge at the final moult in its definite 

 shape and size. And as there are no further moults, it is 

 incapable of further growth or form-change. 



We are thus driven to suppose either that all the processes 

 connected with the organ's heterogonic growth are confined 

 to a very brief period, presumably just before and just after 



the moult from last 

 larval in star to pupa ; 

 or else that, although 

 the visible growth of 

 the organ is confined 

 to this short period, it 

 depends for its amount 

 on some substance 

 whose chemical ac- 

 cumulation during the 

 larval phase has had 

 a constant differential 

 growth -coefficient 

 relative to body- 

 weight (see e.g. Teiss- 

 ier, 1 93 1, for a confir- 

 mation of this latter 

 possibility). 



As we shall see later 

 in considering dimor- 

 phism (p. 68) the for- 

 mer hypothesis is the 

 more probable ; but in 

 any case we have, as 

 in deer's antlers, the 

 fact that the amount 

 of growth attained is 

 proportional to body- 

 size raised to a power, 

 the value of the power 



30 40 50607080 



Total length, mm. 



Fig. 35. — Relative growth of male mandibles 

 in three species of stag-beetles (Lucanidae). 



-f , Lucanus lunifer ; X , L. cervus ; 0, Cyclommatus laran- 

 dus. ' Total length ' is true total length for Cyclommatus ; for 

 the others it is represented by (elytron length + mandible 

 length). All the curves inflect at large absolute sizes (see 

 text) ; for the remainder of the curves k is about 1-6 for 

 L. lunifer, 2-3 for L. cervus, and nearly 2-0 for C. tarandus. 



being equivalent to 

 that of the constant differential growth-ratio in cases where 

 visible growth is continuous over long periods. Thus the true 

 growth-ratio of the organ during its short growth-period is far 

 more rapid than indicated by the value found for the ' growth- 

 coefficient ' by the method of comparing organ and body 

 at different absolute sizes. We are, in fact, again in the same 



