Page:Climatic Cycles and Tree-Growth - 1919.djvu/85

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 conservation 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

${\frac {G_{n}}{G_{y}}}=1-c(n-y)$

Where G_n represents growth in any year n; G_y 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

${T_{n}}={\frac {10}{\sqrt {n-1690}}}$

T_n 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

${\frac {10}{G{\sqrt {n-1690}}}}$

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.