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 



Where G n represents growth in any year n; G v 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 



10 



T = 



J. n 



Vn - 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 



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. 



