BASIS OF DETERMINING DIMENSIONS OF THE FRUSTUM 219 



of the tree, on the same basis as used in timber estimating to top 

 diameters, and that the vahies of the form factors tend to vary but httle 

 from a straight Hne, thus permitting the construction of curves of board- 

 foot volume with greater accuracy than when volumes are plotted 

 directly (§ 138). This advantage permits of constructing such tables 

 on the basis of fewer measurements of felled trees. 



181. Basis of Determining Dimensions of the Frustum. The top 

 diameter of the frustum is supposed to coincide with the top diameter 

 inside bark of the merchantable length of each tree class. The diam- 

 eter at its base, which is at stump height is arbitrarily fixed as equal 

 to D.B.H. outside bark. No pretense is made that this form factor 

 is a scientific basis for studying tree form. Actual D.I.B. at stump 

 may or may not coincide with D.B.H. outside bark. The base of the 

 cone must be correlated with D.B.H. rather than with stump diam- 

 eters (§ 175) and this assumption is satisfactory. 



Since the sides of a cone are straight, the upper diameters of each 

 " log," or standard length into which this frustum is divided, are 

 determined by proportion, to the nearest Yt inch. 



In calculating the volumes of the frustums of cones the determination of the 

 diameter at the top of each successive 16-foot log for cones of different top and 

 base dimensions is best per- 

 formed by plotting the form 

 of the cone on cross-section ^ 



paper, on which the vertical 

 scale shows diameters and the £] 

 horizontal scale shows heights g 

 in feet. Plot, first, D.I.B. "l 

 equals D.B.H. at zero or 

 stump height; next, top diam- 

 eter inside bark at the mer- 

 chantable height. Connect 

 these two points by a straight 

 line representing the side of Fig. 

 the frustum. The diameters 

 inside bark at top of each log 

 are then read at 16 feet, 32 



feet, etc., to the nearest ra inch. The log rule should be tabulated to show the 

 values for each ro inch. 



182. Character and Utility of Frustum Form Factors. That the frustum form 

 factor is a practical rather than a scientific basis of measurement is shown by the 

 following facts: The absolute form factor of the total contents of the bole (§ 175) 

 would be 0.5 when the tree has the form of a paraboloid. A truncated portion of 

 the bole, with the rapidly tapering top eliminated, when compared with a trun- 

 cated cone ha\'ing the same top diameter, represents the lower portion of a cone 

 of considerably greater height than that of the tree or paraboloid. 



For cone and paraboloid (or tree) of equal total height, the form factor of the 



tree, compared with the cone is - — or 1.50, since 0.5 and 0.33 are the respective 



8 16 24 32 40 48 56 64 

 Feet 



38. — Method of plotting a frustum from 

 which to determine the top diameters of the 

 logs which it contains. 



