98 BRIGGS: THE LIVING PLANT AS A PHYSICAL SYSTEM 



3200 years, are summarized graphically in figure 3. The dotted 

 line in this figure connects the 100 -year means, through which a 

 smoothed graph has been drawn. If we assume the latter to 

 represent the relation between the thickness of the rings and the 

 age of the tree, we see that after the trees have reached an age 



dv 

 of about 1200 years, the ring thickness — is a linear function of 



tit 



the time, or 



<t=-at + b (9) 



at 



Integrating this equation and evaluating the constants from 

 figure 3, we have as the relation between the stump radius of the 

 tree in millimeters and the time in years, starting with trees 1200 

 years old, 



r = - 0.00005* 2 + O.S3t + 1670 (10) 



From 1200 to 3200 years then the radius does not increase pro- 

 portionally to the time, but is subject to a negative correction 

 term varying as the square of the elapsed time. 



The straight line portion of Huntington's graph (fig. 3) would 

 if extrapolated cut the axis of abscissae at about 9000 years. 

 It is more probable that the graph approaches the axis of ab- 

 scissae asymptotically, since the straight line relationship would 

 lead to a shrinkage after 9000 years. The relationship ex- 

 pressed in equation (10) must therefore be restricted practically 

 to the period covered by the observations. 



It is perhaps of more interest to -consider the rate of growth of 

 these old trees. In this connection let us determine how the 

 tree would increase in diameter if a uniform quantity of woody 

 tissue were laid down each year. We may for this purpose as- 

 sume the stem of the tree to have the form of a right cone with 

 a base radius r and height h, and we will also assume that the 

 height of the cone increases in proportion to the radius. We 

 may then look upon the growth each year as consisting of a thin 

 layer or shell of thickness dr wrapped closely about the cone. 



The volume V of the trunk of this idealized tree at any time 

 would be 



