In the treatment of spur gears, we have considered them fundamentally as cylinders rolling upon each other (ordinary spur gears) or a cylinder rolling on the inner surface of a larger one (internal gears). We now come to consider cones of various diameters and relative proportions rolling together, as shown in Fig. 259. The surfaces of the cones represent their pitch circles in the same manner as the cylinders. While spur gears must have their shafts always parallel, bevel gears may be designed to run properly at any angle from parallel to 150 degrees. In Fig. 270 are shown several pairs of typical bevel gears with their shafts at different angles. Those of 90 degrees are the more common. The pair shown at 4

Fig. 271. Cross-Section of Pair of Bevel Gears.

are unusual but sometimes necessary, and operate quite as well as the others. In this case the larger gear is an internal bevel gear.

When two bevel gears of the same diameter and number of teeth run together, they are called miter gears, although this term is more likely to be applied to those whose shafts are at an angle of 90 degrees to each other.

Fig. 271 is a cross-section of a pair of bevel gears, and is designed to illustrate the principles applicable to the cutting of gears. The lines AA and BB are the center lines of the two shafts, their point of intersection being the apex of each of the cones representing the pitch surfaces. The line CC is parallel to AA and at a distance from it equal to half the pitch diameter of the larger gear. The line DD is parallel to the line BB, and distant from it one-half the pitch diameter of the smaller gear. Between the points of intersection of the lines AA and BB, and of CC and DD, the pitch line EE is drawn, giving the line of contact between the two cones. The outline of the cones is completed by the line FF. The outer and inner ends of the teeth are lines at right angles to the pitch lines; and upon the outer tooth lines the depths of the teeth above and below the pitch lines are set off; and the lines a and b, for the top and bottom of the teeth, are drawn radially from the common apex of the cones at c. The various dimensions of the teeth are laid off at the large or outer ends of the teeth, and are taken from Table X.

To facilitate the proper cutting of bevel gears, the drawing should give the face angle a and the cutting angle b for each gear, and the depth of the teeth at the outer end. The angles should be so expressed that a bevel protractor may be set against the hub of the gear, and its arm upon the face angle and cutting angle, to verify their correctness. When the shafts are at right angles, the sum of the edge angles will equal 90 degrees, and the sum of the face angles and edge angles will be equal.

The angles may be determined by this method, for the angle of the pitch line FF with the face of the hub (or the line DD parallel to it), we may consider as a right-angled triangle FFD. Divide the height by the base, and the quotient will be the natural tangent. From the table of tangents we get the angle in degrees and minutes.

Thus, suppose the base is 5 inches, and the height 2.5 inches, then 2.5÷5 = 0.5, which is the tangent. In the table of tangents we find the nearest number is .50004, whose corresponding angle is 26 degrees 34 minutes. The value of any angle expressed in degrees and minutes may be determined in the same manner if the base and height are known.