APPENDIX. 



METHOD OF DERIVING EQUATIONS. 



In plotting the various points to a natural scale (i. e., the sluinkage or a given 

 mechanical property vs. specific gravity) it was found that in roany cases tbey 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 tlirough 

 the origin, i. e., that a piece of wood of zero weight or specific gi-avity should have 

 zero strength, it was found that in practically every case a curve of the form f=fG'n 

 (where / is the strength value, G the specific gravity, and p and n 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/=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 / against log G on ordinary cross-section paper and find the straight line 

 which best averages the points ; then n and log ]) 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 n 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, or 4. 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 jp 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. 



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