Leaf Surface of Cucumis sativus. • 113 
are forced to the conclusion that r itself is a variable depending in magnitude 
on the external conditions. 
Theoretically, then, the equation of increase in total area with time 
should be of the form 
A n = ae (ri + ™ + r s +..;+m) ) (n) 
where r v r 2) r 3 » &c., are rates characteristic of successive days according to 
the total amount of light for each day. In practice, however, the whole 
problem is complicated by the changing weather conditions, so that we have 
no means of determining the values of r from the complex results. In 
spite, however, of the fact that the values of r for successive days are unknown, 
it is certain that there is a general trend of increase in the value of r from 
December 21 to June 21, and after this date by analogy the value of r 
should again fall. By assuming that r falls off according to a linear law 
and applying successive decreasing values of r in a compound interest 
formula, a series of values for the area will be obtained, which when plotted 
give a curve of S form, such as is the complete curve of leaf area, as shown 
by Kreusler. 
By a simple modification of the compound interest formula the curve of 
S form is thus produced. 
Now it is the resemblance of this curve to that of an autocatalytic 
reaction 1 curve which led Brailsford Robertson to apply, with great 
success in certain cases, the formula of an autocatalytic reaction to growth 
phenomena. Both Robertson’s fundamental assumption and his conclusion, 
however, have been called into question by Enriques, who has shown that 
the form of function used by Robertson is limited in its application to 
S- curves in which the point of inflexion is not far removed from the half 
period of the reaction. Enriques, moreover, shows that the function used by 
Robertson is only a special case of a general function derived by integration 
of a differential equation of form 
clx 9 
-7- = a + bx + cx 1 + dx 3 + . . . . 
a t 
To quote his words : ‘So ist es moglich zu sagen dass die benutzte Formel 
in keiner Weise eine privilegierte Stellung in bezug auf die Nachahmung 
zwischen den anderen von demselben Typus besitzt,’ &c. Evidence has been 
brought forward in this paper to show that, at least in the early stages of 
growth, the increase in area of the leaf surface conforms with a compound 
interest law, and it has been indicated how an extension of the conception of 
this law will account for the form of the complete curve of growth of surface 
area. The compound interest law, even in its extended form, is only an 
empirical expression of the course of the phenomenon of surface growth, 
1 When speaking of an autocatalytic reaction the term is employed in the following sense, 
namely, a reaction in which one of the products acts as the catalyst, and in which the material 
catalysed continually decreases in mass as the reaction proceeds. 
I 
