528 ON INCREASE IN SIZE [pt. m 



modified the temperature characteristic for the remainder of develop- 

 ment. This it could not be expected to do on the Robertsonian 

 view, for where there is only one velocity constant, as in his 

 presentation, its temperature characteristic must always be the same, 

 and changing the temperature would only multiply the time co- 

 ordinate by some constant, and drag out or compress the time 

 taken to complete the autocatalytic curve. Within one cycle the 

 temperature characteristics should be the same for all partial 

 developmental periods as well as for the whole. This, however, 

 was found not to be the case by Bliss for Drosophila, by Brown 

 for the cladoceran Pseudosida bidentata, and by Titschak for the 

 clothes-moth Tineola biselliela. If an animal is allowed to develop 

 for 50 per cent, of the total normal time at 15° C, and is then 

 transferred to a temperature of 25° C, it may take less or more time 

 to finish its development than would be predicted on the basis of 

 the fact that it has still 50 per cent, of its normal time to go 

 at 15°. Brown expressed the differences as per cent, gain or loss in 

 time under such treatment, and found that they were, though small, 

 quite significant statistically. This means that the growth equation 

 of Robertson, , 



= Kx (a — x) 

 dt 



(where a is the initial amount of growth-forming substance, x the 

 amount formed after time t and K a constant), cannot hold, and 

 must give place to an equation with two velocity constants. Crozier 

 has suggested one, the differential form of which is 



~={K, + K,x) [A - x), 



where Kj^ is the velocity constant proper to the reaction A^x, but 

 in the absence of the catalytic effect of x, while K^ is the velocity 

 constant associated with the process when x is functioning as a 

 catalyst. The point of inflection will then be 



The integral form of the differential equation is 



/ = 



, A {K^x + K^) 



K^ + K^A = K^{A- x) 



