Another relationship can be expressed as follows

(4)

Where Wt is the weight of the dowel as tested.
Dt is the deflection as tested of the dowel weighing Wt.
Dn is an arbitrarily selected deflection for which comparisons of weight are to be made
Wn is the weight of the dowel necessary to give deflection Dn.

The above relationships (3) & (4) hold true no matter what the diameter of the test dowel as long as it is constant over the entire length. This means that it is possible to take a group of dowels of assorted weights, diameter and material, weigh them, measure their deflection in a static bending machine and by using the above equations either obtain the spine rating number or the relative weights required for a fixed selection deflection.

Equation No. 3 was checked as follows. Roy Case made a number of dowels, determined their weight and deflection and computed "N" the spine rating number. He then reduced the diameters of all the dowels by a considerable amount. The new weights and deflection were then determined and a new rating number computed. The new rating numbers checked the original numbers exactly in all cases.

In order to illustrate the practical use of equations No. 3 and 4, Table No. 1 was prepared showing the properties of various materials. The dowels were selected from a bunch made up by Roy Case and were tested in his static bending machine.

The table shows the kind of material, its source, weight of the test dowel in grains, test deflection in inches with a distance of 28 inches between supports and a two pound weight suspended from the center, the diameter of the dowel and the spine rating number. Nearly all the diameters were alike. This is just a happenstance and is not necessary to obtain the rating number. The rating number was divided by 1000 so as to give smaller figures. Note that the dowels are all listed in the order of their rating number. The last two columns of the table show the computed weights and diameters of the dowels for a deflection of 1.2" with a two pound load.

The table will show that there is not a great deal of difference between the first nine specimens listed. Also note the great variation between different specimens of the same kind of wood. Due to this variation the table of course does not indicate definitely which is the best kind of material to use in arrows. For instance the specimen of Norway pine tested may have just happened to have a low rating whereas another sample might be as good as the best Port Orford Cedar or Spruce. However in a general way the table does show that the three woods Port Orford Cedar, Norway Pine and Spruce make the best arrows. The fact that experience has always shown these woods to be the best arrow material would tend to prove that the method given above for rating materials for arrows is correct.