4i6 ON INCREASE IN SIZE [pt. iii 



in Drosophila. Janisch treats the S-shaped growth-curve as the re- 

 ciprocal of a catenary exponential curve. These presentations have 

 the advantage that they do not prejudge the issue from a physico- 

 chemical angle. 



Exception to Robertson's views has been taken on quite other 

 grounds by Snell, who points out that Robertson's equation 



^ = K^Ax - K^x" 

 dt ^ ^ 



and Crozier's modification of it 



^ = (r^ + K,x) {A-x) 



(where x is the concentration of the end-product at time t, A the 

 concentration of the substrate at time t, and K^ and K^ the velocity 

 constants of the forward and reverse reactions respectively) do not 

 take into account the fact that the system concerned is not a closed 

 one. All the time the embryo is growing it is also eating, i.e. 

 absorbing nutritive material, and in addition it is giving out 

 waste products. Accordingly these equations derived from the 

 law of mass action as we know it in the inorganic world do 

 not allow for the effect of increasing size on the concentration 

 of the reagents involved in growth. The equations hold true 

 only on the condition that the volume occupied by the resulting 

 substances remains constant, and since a growing organism is con- 

 stantly increasing in -volume, this condition is not met. "When a 

 chemical process is carried out in the laboratory," said Snell, "the 

 reagents are ordinarily dissolved in water or some similar solvent, 

 and the volume of the solvent is kept constant throughout the whole 

 process. To make the conditions of a laboratory process comparable 

 to those involved in the synthesis of new protoplasm the volume of 

 the solvent would have to be increased as fast as the amount of the 

 end-products is increased. As the solids of new protoplasm are 

 formed, they do not stay in the same little parcel of liquid occupied 

 by the old, rather they cause the liquids to expand with them. Hence 

 the volume occupied by the end-products of growth is proportional 

 to their amount, and the concentration of these products, instead of 

 increasing, remains constant. This is a very important difference, 

 for it is on the concentration and not the amount of reagents that 

 reaction velocity depends." Thus the equations of Robertson and 



