For a crossed belt, as in A and C, fig. 109, L depends solely on c and on r1 + r2. Now c is constant because the axes are parallel; therefore the sum of the radii of the pitch-circles connected in every position of the belt is to be constant. That condition is fulfilled by a pair of continuous cones generated by the revolution of two straight lines inclined opposite ways to their respective axes at equal angles.
For an uncrossed belt, the quantity L in equation (32 B) is to be made constant. The exact fulfilment of this condition requires the solution of a transcendental equation; but it may be fulfilled with accuracy sufficient for practical purposes by using, instead of (32 B) the following approximate equation:—
L nearly = 2c + π (r1 + r2) + (r1 − r2)2 / c.
(33)
The following is the most convenient practical rule for the application of this equation:—
Let the speed-cones be equal and similar conoids, as in B, fig. 109, but with their large and small ends turned opposite ways. Let r1 be the radius of the large end of each, r2 that of the small end, r0 that of the middle; and let v be the sagitta, measured perpendicular to the axes, of the arc by whose revolution each of the conoids is generated, or, in other words, the bulging of the conoids in the middle of their length. Then
v = r0 − (r1 + r2) / 2 = (r1 − r2)2 / 2πc.
(34)
2π = 6.2832; but 6 may be used in most practical cases without sensible error.
The radii at the middle and end being thus determined, make the generating curve an arc either of a circle or of a parabola.