Referring to Fig. 1, let B equal one-half the length of the bow, L equal one-half the length of the rigid handle section, B1= B-L equal the length of the active bending portion of each limb, S equal one-half the length of the string, H equal the distance from middle of bow to middle of line connecting bow tips, P equal distance from arrow nock to middle of line connecting bow tips, D=H+P equal length of draw, Y equal one-half length of line connecting bow tips, A equal angle between the line connecting bow tip and point O on the line representing the position of the undeflected limb. (The point O is located at a distance of 3B1/4 from the tip of the bow.) Let E equal angle made by string with the line connecting the bow tips, f equal the static force at each bow tip in a direction tangent to its path, T equal static tension in the string, F equal static drawing force, Y1=Y-B1/4-L and N equal the distance along the path made by the bow tip during the draw.
It may be shown that the path made by the bow tip is part of a cardioid. The portion of the cardioid traversed by the bow tip is almost a perfect arc of a circle whose radius is 3B1/4 and whose center is located at a distance of 3B1/4 from the tip of the undeflected bow. Since the equations which are to be obtained are based on the assumption that the bow tips travel along arcs of circles having radii of 3B1/4, the accuracy of this assumption will now be investigated.