CORRELATION WITH RAINFALL. 67 
of the profits and losses of the preceding years. The “credit balance” 
in their books at the beginning of the year has only somewhat less 
importance than the income during the current year. 
Mathematical relation of rainfall and growth.—In order to formulate 
the relation between rainfall and tree-growth, an effort was made to 
construct a mathematical formula for calculating the annual growth of 
trees when the rainfall is known. Any such formula must perform 
three principal functions: (1) reduce the mean rainfall to the mean 
tree-growth; (2) provide a correction to offset the decreasing growth 
with increasing age of the tree; and (3) express the degree of conserva- 
tion by which the rain of any one year has an influence for several 
years. In a formula of universal application other factors will play 
a part, but for a limited group of trees in one locality they can be 
neglected. 
The first process, namely, the reduction of the mean rainfall to the 
mean tree-growth, is a division by 250. This is the general factor K 
in the formula below. The second part, namely, the correction for the 
age of the tree, was practically omitted in forming the curves shown, 
since judging by the Flagstaff curves its effect would be very slight in 
the interval under discussion. In long periods it is an immensely 
important correction and its effect should always be investigated. 
Over the short periods used in this rainfall discussion the decrease of 
annual growth with age may be regarded as linear and an approximate 
formula is 
Gs 1— c(n—y) 
G, 
Where G, represents growth in any year n; G, is growth in middle 
year of series y, and c is the rate of change per year, a constant which 
was 0.0043 in the last half century of the Flagstaff series. Over the 
whole interval from 1700 to 1900, in the first Flagstaff curve, the 
growth was approximately an inverse proportion to the square root of 
the time elapsed since the year 1690 and may be closely expressed in 
millimeters by the formula 
r.=__10 
V n — 1690 
T,, is here the mean tree-growth for the year n. If G be the mean size 
of rings, then the factor to be introduced in a general formula becomes 
10 
GVn—1690 
Character of the conservation term.—This factor has two important 
features: (1) in this arid climate it applies better as a coefficient than 
as an additive term, and (2) it gives a prominent place to “accumulated 
moisture” as commonly used in meteorology. 
