I IO 
Gregory . — The Increase in Area of Leaves and 
where t. x represents the time elapsed since beginning of the reaction until the 
maximum velocity is attained. It is from a function of this type that the 
values of leaf dimensions in Table IV were calculated. In this case A repre- 
sents maximum length attained, represents length at any time, t since the 
unfolding of the leaf. It is clearly not necessary to know the exact time of 
unfolding, since the same error is introduced into t as into t v so that (t — t x ) 
is correctly known. This function is derived by integration from the 
dx 
differential equation — = Kx (a^x), which in our case represents the rate 
of growth. This rate will attain a maximum when x(a — x) is a maximum, 
i. e. when x = ~. The curve of the function (3) will therefore be S-shaped, 
2 
symmetrical about the point of inflexion. 
Enriques has shown that we may start from a differential equation of 
a more general form, namely, 
dx 
dt 
= a + bx + 
ex* 
( 4 ) 
( 5 ) 
and by integration derive a more general function : 
which will represent an S curve no longer symmetrical, but with the point 
A— B 
of inflexion corresponding with a value x — - . As B approaches A 
* 2 
in magnitude so the point of inflexion approaches the origin, so that 
A -\- x 
log A -x 
= Kt 
( 6 ) 
will represent a reaction with maximum initial velocity falling off with time. 
It is this modification which has been used to calculate the area of cotyle- 
dons ifi Table VI. The growth in area of individual leaves under artificial 
light thus falls into line with the growth in area under natural illumination. 
The increase in area of leaves under artificial light can, however, be repre- 
sented by a much simpler function of type : 
A = a + b log e t, (7) 
and it was from a function of this type that the curves of closest fit in 
Figs. 8 and 9 were constructed. 
Differentiating this function with respect to time we obtain 
d_A __b 
dt t ’ 
(«) 
so that the rate of increase in area falls off in time. This function (7), how- 
ever, goes on increasing to infinity, whereas the area of a single leaf reaches 
a maximum limit, so that the function is merely empirical, and holds only 
over a limited range of time. 
