CORRELATION WITH RAINFALL. 69 



which the trees had exhibited in the preceding few years. But this 

 average growth bore the same relation to the average or smoothed 

 rainfall that the accumulated moisture bore to the smoothed rainfall. 

 Hence the ratio between accumulated moisture and smoothed rainfall 

 gave at once the required ratio between smoothed tree-growth and 

 smoothed rainfall. These relations are shown in figures 16 and 17. 



Accumulated moisture is simply the algebraic sum of the amounts 

 by which all the years in a series from the start to and including the 

 year desired depart from the mean. It may be expressed by a formula, 

 thus 



A n =(R l -M) + (R z -M)+. . ..(R n -M)=R l +R z +Rs+.. .R n -nM 

 and conversely 



In this formula A n is the accumluated moisture for the nth year of a 

 series of consecutive years whose mean rainfall is M. 



The simple empirical formula for the tree-growth T n for the nth 

 year of this series thus was found to be: 



T cM+dA n p 



* = A - o - Kn 



On 



in which c and d are small constants found advantageous in reducing 

 the accumulated moisture curve to proper scale. S n is the reading 

 of the smoothed rainfall curve and the term cM+d A n is the accumu- 

 lated moisture expressed in values above a base-line instead of depar- 

 tures from a mean. In actual numbers this becomes 



m / i x 1 0.90 M + l A n D ,. - , x 



T n (m inches) = . - ^ - . R n (m inches) 



o n 



The mean value of the rainfall M is 17.1 inches. The application of 

 this formula in calculating tree-growth at Prescott from the rainfall is 

 shown in figure 18. 



The reversal of the process in order to ascertain rainfall from tree- 

 growth seems to be fully as accurate over this limited period, and its 

 result is shown in figure 19, where the curve has an average accuracy 

 of 82 per cent for individual years. In producing this reversal the 

 following operations were performed : 



1. A 5-year smoothed curve was made of the tree-growth. This 



,, 0.90M+ \A n . , , 



gives us the term - - in the reversed tormula 



p _ _ &n _ m 



n ~- 



250 



2. This term is multiplied by 1,000 and 3.6M subtracted, leaving 

 A n in inches. 



