40 



Substituting these values of v and - in formula (v), we haye 



If in the above formula^ *\ == i^ = '3 = , we have. 



y^«i^i + «2^2 + «Bys F„,^„]a (^i). 



«1 + «2 + «S 



The employment of this, formula presupposes a knowledge of the 

 respective areas occupied by the different age-classes. It is, there- 

 fore, not adapted for the calculation of the mean age of a crop 

 composed of trees of various intermixed ages, but its special use is 

 for the determination of the mean age of several crops considered 

 together, or even of an entire working circle, and it is generally 

 employed for this purpose. It gives the same result as Formula 

 (v) when T is approximately the age at which the highest mean 

 annual increment occurs. 



It now remains to investigate a formula for finding out the 



mean age of the crop when we know only the numbers Mj, w^, Mj 



of trees included respectively in the several classes whose ages are 



^1, ^2, ^3 By the ordinary rule of aritlimetie for obtaining 



averages, we have the mean age of the crop 



r = "■y + »^y^ + «3y. Fo,^„i^ (,ii)_ 



«1 + »2 + «3 - ^ ' 



If this formula is to yield the same result as Formula (v), it is 

 necessary that the mean annual increment of all the diameter-- 



classes should be one arid the same. For let j^,, ^g, j^j = the 



mean annual increments of the average trees of the several classes 

 then 



''1 = Xi^i"); ^2 = Xsy2"2i «'s = ?C3^8«si 



and by formula (v) 



j_ Xiyi^'i + XiSfj^s + X3.ys% 



Xi «i + Xi »2 + Xs »3 ' 



Hence, in order that formula (vii) should at the same time be 

 true, we must have 



Xi = %2 = Xs 



For formula (vii) to give the same result as formula (vi), it 



