31 
Substituting for r in (1) we have 
A =b+mi ees 
aa 
which gives 
n(btmiy?=A.... 2... (2) 
WT 
If 6 be zero, then the equation simplifies into 
ni? = —— 
mar 
Since A and z are constants and mis also a constant for any 
one species on any one site we may replace the right hand 
member of the equation by the constant C. 
Hence we have 
Dib Ne: Soaks 3 Dredd PATE eG (3) 
This last equation says that the number of trees at any 
time multiplied by the square of the time gives a constant 
value. The simplified formula closely agrees with Wim- 
menauer’s now celebrated Scotch pine experiments. 
If the expansion of the crown be a linear function of time, 
then if any two points of the graph are known, the graph itself 
is known. From such a graph the number of trees per acre for 
any given year can readily be calculated by dividing the area 
of a single crown in square feet into the number of square feet 
in an acre. If, however, the graph passes through the origin 
then a good deal of calculation may be saved by merely cal- 
culating the trees per acre for any one year and then using 
formula (3) for the other years. 
For both oak and ash two points on the graph of the crown 
expansion were obtained. From young stands, presumably 
fully stocked, the following dimensions have been taken: 
TABLE XVI 
7 D.B.H. Height Crown Radius 
Species Inches Feet Feet 
Osi gs veaueee cd 1.2 18 1.4 
Aphis ohcinsle peaehaes 1.2 16 1.2 
