974 
Journal of Agricultural Research voi. xxxi, no. 10 
ash was also analyzed for aluminum, calcium, iron, magnesium, phos¬ 
phorus, sodium, and sulphur. Computations have been made to 
acre yields on the basis of the number of plants per acre, 14,662. 
(The rows were 3.38 feet apart, and the plants averaged 10.55 inches 
apart in the row.) 
While the investigation was not planned as a physiological study 
of plant growth, nevertheless the data accumulated shows some¬ 
thing of the growth changes occurring in the sunflower crop. The 
curves given by the eight points of determination show, in the case 
of many of the crop constituents as analyzed, the general reverse 
curve characteristic of growth in annual plants and animals. Follow¬ 
ing the lead of Robertson, 3 particularly, the writers have utilized the 
general form of equation representing the course of an autocatalyzed 
monomolecular reaction in deriving a mathematical expression of 
the data from a growth standpoint. The results have seemed to the 
writers, rather suggestive as to the possibility of the constants of 
such growth equations serving a very useful purpose in crop studies. 
If the constants have the significance attached to them by Robertson, 
they should be of value in supplementing the data of final crop yield 
as customarily used in variety tests, etc. At least certain very 
striking differences appear in the constants as between species such 
as the sunflower and com, as indicated by the data at hand. It is 
with the thought of somewhat emphasizing the differences thus shown 
that the present paper is offered. 
THE AUTOCATALYTIC GROWTH CURVE 
A brief review of Robertson’s 4 presentation of the equations 
expressing the course of ai J ' tic monomolecular reaction 
will serve to bring out the DacKgrouna as to the significance to be 
attached to the constants of this sort of growth equation. Repre¬ 
senting the amount of material at the start of such a reaction by A, 
and the amount* * transformed at any later time by x, then the amount 
of the original material left at any time is A — x. The velocity of 
the reaction is accelerated (catalyzed) by a product of the reaction 
present in amount proportional to x. The velocity is also pro¬ 
portional to the amount of original substance left, that is, A — x. 
Hence the velocity at any moment is proportional to x(A — x). 
Designating time by t , the velocity is expressed as 
( 1 ) 
in which k is a specific constant. 
It is obvious from (1) that the velocity will increase to a maximum 
with time until x = A — x = iA, after which time the velocity will 
continually decrease. Equation (1), when integrated, gives the 
relation 
log-/_ x = Kit-U) 
( 2 ) 
in which K=kA, and U = t when x = A — x. Time is thus given a 
negative value preceding the point at which A is one-half trans¬ 
formed, and a positive value thereafter. 
3 Robertson, T. B. the chemical basis of growth and senescence. 389 p., illus. Philadelphia 
and London. 1923. 
* Robertson, T. B. Op. cit. 
