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 
and conversely 
R,=M+A,—A,i41 
In this formula A, 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 7, for the nth 
year of this series thus was found to be: 
in which c and d are small constants found advantageous in reducing 
the accumulated moisture curve to proper scale. S, is the reading 
of the smoothed rainfall curve and the term cM-+d A, is the accumu- 
lated moisture expressed in values above a base-line instead of depar- 
tures from a mean. In actual numbers this becomes 
ee 1 0.90 M+iA ze, 
x ig Ee iy je bade cSt Pat . n 
(in inches) 550 os (in inches) 
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 
1 
0.90M +34, in the reversed formula 
250 
8, 
R,=————. . T, 
0.90M+7A, 
250 
2. This term is multiplied by 1,000 and 3.6M subtracted, leaving 
A, in inches. 
gives us the term 
