926 
Journal of Agricultural Research 
Vol. XXXI, No. 10 
regrouping and averaging. We have once more an apparent prece¬ 
dent in algebra, for if we have equations between x ana 3 and y and z 
such as z = ax + b and y = cz + d, we may with perfect propriety sub¬ 
stitute the value of z from the first equation in the second and obtain, 
as the equation between x and y , 
y = c (ax + b) + d = acx + bc + d 
But again, let us examine the actual result with our empirical data. 
The curve in Figure 5 is thus worked out. For example, from 
Figure 1 we find that the average height of 4-inch trees is 34 feet. 
From Figure 4 we find that the average volume of 34-foot trees is 
1.6 cubic feet. We may therefore plot 1.6 cubic feet as the average 
volume of 4-inch trees. In a similar way we may find values for 
volumes corresponding to as many successive values for diameter as 
HEIGHT - FEET 
Fig. 4— Curve of volume over height, for same data 
we please. Since the values are derived from two curves they will 
themselves inevitably fall on a smooth curve. 
The best test of the accuracy of this curve is to see how well it fits 
the data when the latter are handled in the straightforward manner. 
The same material has therefore been sorted into 1-inch diameter 
classes and the average value for each class obtained. The resulting 
values are plotted as small crosses in Figure 5. It is clear that the 
derived curve is entirely inadequate. Even at the lower end the 
percentage differences between the plots and the curve are very con¬ 
siderable (over 30 per cent), while at the upper end the discrepancies 
in both position and trend are gross. 
Nor is this accidental. With three variables under consideration 
six curves are possible (height over diameter, diameter over height, 
height over volume, volume over height, volume over diameter, 
diameter over volume). All six of these have been drawn both 
directly from the data and by the indirect method just described. 
