796 JOURNAL OF FORESTRY 



combined, while those of intermediate size are apphcable either to 

 combinations of I and B or II and A, and are so designated. This 

 combination of scales at first seems complicated, but gives great flexibil- 

 ity to the graph and yields values over a very large range of sizes, while 

 obviating errors from abnormally acute intersections. 



SMALIAX FORMULA USED FOR ENTIRE TREES 



The Smalian formula already described is sometimes used for obtain- 

 ing the volume of entire trees with a single computation, based on 

 measurements at equal intervals along its stem. The formula becomes 



Y={b,-^2b.^2b. + 2b, . . . . 2^"-^ + ^,,)^, 



where V is the volume of the tree in cubic feet, b^, b^, . . . . 6„, etc., 



the basal areas at the first, second and ;/th sections, and L 



the length (or average length) of section or log. lid^.d., rf,„ 



etc.. are diameters at the successive sections, this ma}' be written 



4 X 144 

 An alinement chart for this equation involves no new principles, but 

 certain difficulties as to arrangement and scale are encountered. It 

 necessitates first the successive addition of an unknown and varying 

 number of d values, followed by the multiplication of the sum by the 

 expression involving L. The addition must obviously be handled by 

 the parallel line form of chart and the multiplication by the r form. 

 On account of difficulties with the scales two forms of this chart have 

 been prepared, each of which has certain advantages one over the 

 other. 



Figure 14 has a series of 9 parallel axes for performing the addition. 

 From what has been said in connection with figure 7, it is evident that 

 three axes are necessary for the addition of two variables, 5 for 3 

 variables, 7 for 4 variables, etc. A seven-log tree with its 8 dififerent 

 values for d would therefore require 15 axes. These would be ex- 

 ceedingly confusing, however, and in the plate presented they have been 

 so combined as to reduce them to the number shown. The principle on 

 which this combination is effected is that in figure 7 : C and D can be 

 made to coincide by making the intervals between A, B, C, D, and E 

 equal. There must then be two series of graduations on the combined 

 CD axis. If this principle is adopted, it is evident that each new 

 variable to be added requires but one additional axis instead of two. 



The only complication in working out this system of axes is con- 



