282 PRINCIPLES OF EMBRYOLOGY 



It is worth while just to glance at the main varieties of these. 



(i) The 'monomolecular' formula is derived from the idea that the 

 growing substance is formed from a store of some substance a which is 

 gradually used up. This gives the relation 



^ k(a — tv), 

 at 



whence i^ = 4^ - ^^"'')' 



when h is a constant depending on the initial amount of a. 



When time is plotted horizontally, this gives a curve which is convex 

 upwards and which approaches the upper hmit a. Rather few growing 

 systems behave in this way. r ^■rr ^ 



(2) The 'logistic' formula can be deduced from a number of difierent 

 assumptions. For instance, if the substance lu is formed from a precursor 

 a as before, but if the rate of formation is also increased m proportion to 

 the amount o(w already present (i.e. if the reaction is 'autocatalytic ), we 



shall have 



dw J f \ 



dt 

 Or, if we suppose that the rate of formation per unit mass decreases in 

 proportion as w is formed, we shall have 



1 dw , s 



w at 

 where p and q are constant, which amounts to the same thing. 



The differential equation can be solved to give an expression of the 

 form 



a 



^ = r— ^ 



I + he- 



This when plotted gives a sigmoid curve, which eventuaUy approaches 

 the limit a. 



(3) Another assumption is that^ -^ decreases as time goes on, but in a 



different way, being proportional to log w. This is the 'Gompertz 



equation : 



1 dw ^ 

 __ =p\ogw-q. 



tv dt 

 This gives a solution of the form 



which is again a sigmoid curve which approaches the upper limit a. 



