94 THE BOTANICAL MAGAZINE. IToi. xxxiy. tfo. 403. 



true radius of the tree ; R, theoretical value of the radial curve ; d, 

 deviation). From this table we see that the modified formula is 

 quite suitable arid its validity can also be shown in the two other 

 cases, p varying from 20 (No. I) to 10 (No. III). (Art. Jap., p. (150), 

 Table 2 1 , Fig. 2 2 ) 



If R=\^— t R 2 =^r~or ~R 2 =pX. ~R 2 is equal to the area A of a 



circle with radius R, so that A =pX, and as p is a constant in each indivi- 

 dual tree, A is proportional to X or the age. If, therefore, we accept 

 this formula, we may say in general, that the area in cross section 

 of wood produced every year or every several years is constant. Now, 



p 



we may transform the formula again into R= V PX > where ~ = P. 



The curve represented by R= V px or R 2 — PX is of the parabolic 

 type, so that we can say that the radial curves of growth in thickness 

 of these trees follow, in the main, the parabolic type. 



We will turn to examine other wild-grown trees (No.- IV, V, VIII). 

 The central part through nearly 50 annual rings shows very slow 

 thickening growth which is followed by rings of far greater thickness. 

 These trees, therefore, show quite the different type from the above- 

 mentioned planted trees. Though the amount of wood produced in 

 the young stage is small, we can not neglect it, as it occupies the 

 growth of a considerable duration of age. 



Now, we turn to the general consideration of growth curves. 

 If the rate of growth of succeeding years increases gradually, the 

 curve will be concave (type a), if it is constant, the curve will be 

 straight (type b), while if it decreases, the curve will be convex 

 (type c). There is a formula which may represent these three types 

 of curves, viz. y 2 —px r . If r=l, y 2 =px or y— ^ px > which is a convex 

 curve (c) ; if r=2, y 2 =px 2 or y— Vp x > which is a straight line (b) ; 

 and if r=3, y 2 =px s or y= Vp T | } which represents a concave curve (c). 

 Thus we can apply this formula for all cases of growth-curves by 

 choosing the value of r correspondingly. 



The radial curves of these wild-grown trees are not simple, but 



^of complex nature, consisting of three types of curves (Art. Jap., p. 



(152), Table 3 1 ). An example of the application of the formula will 



be shown in table II (X, age in years ; r, true radii of the tree ; R, 



1. A, No. I; B, No. III. 



2. Graphical illustration of Table 2. Full line, true radial curve of the tree ; 

 broken line, theoretical line denoted by the formula. 



