546 THE NATURE OF THE GROWTH RATE 



Robertson has called attention to the correspondence between 

 growth of organisms and autocatalysis. He has shown that the rate 

 of growth may also be expressed by the differential equation 



— = kx {i — x) 

 dt 



in which x is the size of the organism at time /; a is the final size of 

 the organism; and ^ is a constant. When integrated, the above equa- 

 tion becomes 



log -^ = Kit - ti) 

 a — X 



where K = ak, /i is the time at which the organism has reached half 



its final size; i.e., the time when x — -. With the assistance of tables 



2 



which Robertson (1910-15) has prepared, the constants and the theo- 

 retical values of x are easily obtained from observational data. The 

 value of a was taken as 114 cm. and h as 47.4 days. Table I shows 

 the computations made and the several values of K. With the 

 average value of K the values of x were calculated by the formula and 

 are given in the fourth column of Table I. The agreement between 

 the two is good except for the first three measurements. Fig. 1 shows 

 the curve obtained from the calculated values together with the 

 observed values of mean length at weekly intervals during the grow- 

 ing season. 



This growth rate is of considerable physiological interest if we com- 

 pare it with the growth rates of organisms mentioned elsewhere in 

 the literature. Attention may be called to the fact that we are here 

 dealing with the mean growth of a number of separate shoots on a 

 small population of trees. 



A word concerning the heterogeneity of the population may be 

 useful. On the final day of measurements the shoots ranged in length 

 from 69 to 171 cm. with an average of 113 cm. The standard devi- 

 ation of the measurements was 21.14 cm. This indicates that we are 

 dealing with a random selection of material and not one which was 

 so closely selected that it was not representative of the shoots of such 

 trees. 



