2. Much of the variation in specific gravity at breast height may be simply the 

 result of sampling at different distances from the apex. 



3. Regression coefficients depend on the past climate, especially for estimating 

 specific gravity of increment over the bole surface . 



An analytical expression for this function can be derived from the following assumptions: 



1. One component of specific gravity at distance (T) from the apex is inversely re- 

 lated to the concentration of auxin per unit area of cambium (C) (perhaps through the 

 mediation of other substances present in proportion to the auxin transported from the apex) . 



2. A second additive component represents effects proportional to the distance from 

 the apex. An example of this latter type is the effect of wind stresses acting through a 

 moment-arm proportional to the length of the stem (T) above the point of measurement. 

 Thus 



S = a + b/C + cT (1) 



3. The tree bole is approximately a paraboloid. For D-p = diameter at distance T 

 from apex: 



= gT^ (2) 



Since concentration would vary inversely proportional to cambial area, for a given 

 vertical dimension, it is inversely proportional to the horizontal dimension. Hence 



C = k/D = I t'^ (3) 



because circumference is proportional to diameter. 



Taking differentials of (1) and (3) 



6S = -b C"^dC + c 6T (4) 

 dC = T'^'/^dT (5) 



and, substituting (3) and (5) in (4): 



dS = -b T) (~ t""^^^) 6T + c6T 

 k^ 2g 



6S = ^ t"2 aT + cdT. (6) 



Integrating: / dS W T ^ + c) 6T 



gives S = ^ T^ + cT + a (7) 



4k o 



5 



