BOWS AND ARROWS — KLOPSTEG 577 



could not be improved. The}- must have felt certain that any attempts 

 to improve their product were predestined to failure. I have recollec- 

 tions of pre-1930 bowyers speaking with pride, if not boastfully, of 

 their ability in selecting yew wood for making bows par excellence. 

 But no matter how singularly excellent the quality of the wood they 

 selected, or how well it was seasoned, even the best of their bows 

 required a high initial angle of trajectory for the arrow to hit the 

 target at 100 yards. This did not contribute to high scores, nothwith- 

 standing their derogation of the higher velocity and flatter trajectory 

 of the new bows, not attainable with theirs. 



It may interest the reader to follow the major steps by which 

 the improvements were achieved. Our point of departure was the 

 longbow, as used in this country prior to the early 1930's w^hich was 

 the English pattern modified with the rigid grip. 



Mechanics, that section of physics w^hich deals with static and 

 dynamic forces, with kinematics, and with the properties of materials 

 subjected to stresses, shows that when a elastic beam is bent, it 

 is under tension which causes stretching on the convex side and under 

 compression, causing shortening on the concave side. Somewhere be- 

 tween there is a geometric "layer" of zero thickness which neither 

 stretches nor sliortens as the beam is bent. At this "neutral" layer 

 the shearing force between the stretched and the compressed sections 

 is a maximum, and this diminishes to zero as w^e move outward, at 

 right angles, to the surfaces of the beam. 



To illustrate this, consider the limb of a longbow (fig. 4). A force 

 Fg applied to the tip through the string causes the limb to bend in a 

 curve which depends on the force, and on the shape, dimensions, and 

 elastic properties of the limb. At any section AB within some finite 

 radius of curvature, the tensile force is Ft and the compressive force 

 Fc. These forces increase from zero to maximum values as we go 

 outward from the neutral layer. The bending moment at the section 

 is the summation of the tensile and compressive forces over the ele- 

 ments of area on each side of the neutral layer, giving a resultant 

 tensile and compressive force, respectively ; the sum of each of these 

 resultant forces, multiplied by the distance of its point of applica- 

 tion from the neutral axis of the section, is the bending moment at 

 the section. Tliis is equal to the moment represented by the force 

 along the string multiplied by the perpendicular distance of the sec- 

 tion AB from the string: FsXcIab. 



Figure 5 represents a section AB of the limb in figure 4, of a typical 

 longbow. Line CD through the center of mass of the section is the 

 neutral axis. The neutral axes of all the sections defuie the neutral 

 layer; conversely, the neutral layer contains all the neutral axes of all 

 possible sections. In the section shown, the distance from the neutral 

 axis to the outer surface of the back of the limb is less than the cor- 



