APPENDIX. 
METHOD OF DERIVING EQUATIONS. 
In plotting the various points to a natural scale (i. e., the shrinkage or a given 
mechanical property vs. specific gravity) it was found that in many cases they arranged 
themselves in the form of a curve, or if their trend was along a straight line, this line 
would not pass through the origin. Assuming that the function should pass through 
the origin, i. e., that a piece of wood of zero weight or specific gravity should have 
zero strength, it was found that in practically every case a curve of the form f=pG” 
(where f is the strength value, G the specific gravity, and p and vn are constants) 
would fit the points quite well. This equation is the general equation of the parabola 
of the nth degree passing through the origin. 
In order to simplify the determination of the proper values for the constants p and n 
the equation was transformed into the logarithmic form, log f=log p+n log G. This 
equation represents a straight line having its slope equal to n and its intercept on the 
y axis equal to log p. Consequently, to find the constants p and n it is only necessary 
to plot log f against log G on ordinary cross-section paper and find the straight line 
which best averages the points; then nm and log p are determined from the slope and 
intercept of this line. 
To find the straight line which best averages the points in the logarithmic plot the 
following plan was adopted: 
By successive trials the parallel lines BB and CC (see fig. 9) were so located that 25 
per cent of the points were above BB and 25 per cent were below CC and at the same 
time the vertical distance between the two was a minimum. ‘Two lines (not shown 
on the figure) were then drawn as follows: Both parallel to BB and CC, one bisecting 
the distance between them and the other in such a position that 50 per cent of the 
points were on each side of it. AA was then drawn midway between these two lines 
and assumed to be the line which best averages the points and best represents the 
relation between specific gravity and the strength property in question. This method, 
as can readily be seen, is very likely to produce values of m such that the resulting 
equations can not be handled without the use of logarithms. As the slope of the 
lines could in most cases be varied through a considerable angle without appreciably 
affecting the distance between the lines BB and CC, the slope was so taken that 
n would be a fraction with the denominator 1, 2,3, or4. The solution of the equation 
is then possible by using the rules for the extraction of square and cube roots. When- 
ever it happened that more than one direction of the lines BB and CC fulfilled the 
conditions outlined above, preference was given to that slope which would simplify 
the form of the equation. The constant p was changed at the same time, so that the 
_ new line A‘A! passed as nearly through the center of gravity of the points as possible. 
The analytical process known as the ‘‘method of least squares’’ can be applied to 
determining the mathematical relations between two properties of a substance as 
ascertained from experimental results. This method was used in one or two instances 
to determine the specific gravity strength relations; but it was found that the long 
and refined computations essential to the application of this method to so large a 
number of tests was not justified by the added accuracy of the final determinations. 
Especially is this true since it is desirable to obtain n to the nearest 0.125 only, and 
since undue refinement in the value of the constant p is unnecessary in view of the 
fact that there is a considerable variation of actual results from the values given by 
any equation which may be derived. 
11 
