114 



THE CLIMATIC FACTOR AS ILLUSTRATED IN ARID AMERICA. 



all the years in a series from the start to and including the j'ear desired depart from the 

 mean. It may be expressed by a formula, thus 



A,, = {R,-M) + {R„-,-M)+-- ■ (i?i-M)=/?„+7?„_i+/?„_o+. • • R,-nM 



and conversely 



In this formula, i4 „ is the accumulated moisture for the nth year of a series of consecutive 

 years whose mean rainfall is M, Rn is the rainfall for that nth year, and 7?„_i is the rainfall 

 of the next preceding year, and so forth. 



Now the accumulated moisture curve for Prescott, when brought to proper scale, almost 

 coincides with the smoothed curve of tree growth: hence the relation of the smoothed 

 curve of rain (individual years vary too much) to the accumulation cm-ve represents very 

 successfully the temporary relation between rainfall and tree growth and it is only neces- 

 sary to change the annual rain in the same proportion to produce the tree growth, as 

 appears in figure 17. The smoothed curve of rain in this case consisted of successive or 

 overlapping 5-year means used in the place of the middle or third j'ear. For example, 

 the average rain of 1881 to 1885 was placed in 1883, the average of 1882 to 1886 was placed 

 in 1884, and so forth. Its formula appears thus: 



»S„ = \{R n-1-\-R n-\-\-R n-\-R „+\-\-R n+'l) 



The simple empirical formula for the tree growth, T, for the nth year of this series 

 thus was found to be: 



T. = K . ^^+^^- . R. 



On 



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

 moisture curve to proper scale. In actual numl:)ers this becomes 



„,..,, 1 0.90M + i^„ p ,. . , ^ 

 In (m mches) = - ,.- • • /?„ (in mches) 



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



Inches 1870 1880 1890 19U0 1910 

 30 



20 



10 





 Fio. 18. — Actual Rainfall Compared with Rainfall Calculated from Growth of Trees, Arizona. 



= Observed rain. = Rain calculated from trees. 



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 18, where the curve 

 has an average accuracy of 82 per cent for individual years. In producing this reversal 

 the following operations were performed : 



