235 
But since from the nature of the case both 7 and P are 
unaffected when 7 — ¢ is written for ¢, therefore b=0. Thus 
+, N, and P are determined; and it is easily seen that the 
variable portion in the expression for the tendency to break 
varies as the square of the distance from a point situated in 
the axis of x. 
The theory advanced in this paper applies to leaves whose 
structure is such that the marginal vigour of growth is the same 
at all points; and in the case of aquatic plants with floating 
leaves which may be otherwise organized, some modification of 
the same theory will bé necessary; it applies during the period 
between the first unrolling of the leaf at the surface of the 
water and the completion of its growth. ach lobe of a 
divided leaf must be treated to a separate calculation, and 
new lobes may be formed at those points where the tendency 
to break is the greatest. Growth is supposed to proceed steadily, 
while the leaf is submitted to a suitable external pressure; 
when violent pressures exist, as, for example, when the current 
is very rapid after heavy rains or otherwise, growth is probably 
checked for a time and fissures may be started, or the leaf- 
margin maintains its shape by the support of its interior which 
then resists the external pressure, until at a proper time growth 
pushes forward the margin again, and its form is matured in 
obedience to the above-investigated laws. 
It is further to be noted that the form of the leaf-margin 
remains the same, even if at different times the power of growth, 
the direct resistance due to the velocity of the current, and the 
marginal tension all vary, provided only that their proportional 
values remain unaltered. It is a tenable hypothesis and by no 
means improbable that, during much or most of the time when 
actual growth is taking place and when the velocity of the 
current is subject to many and various vicissitudes, the plant 
has the power of adapting its growing efforts to the circum- 
stances just necessary for its development, that is, in the no- 
