will be greater with flatter than with steeper pitch cones. Only in mitres will the same cutters serve for the two wheels in engagement.

But the selection of cutters is not always made strictly on this basis, when pinions with low numbered teeth are in question. It is impossible to shape the flanks of any bevel wheel teeth with absolute accuracy by rotary cutters, because the section of the cutter is constant, and that of the tooth varies continually from one end to the other. Several courses are open. One is to select cutters strictly to correspond with the larger diameters. Then the curves will not be correct elsewhere, until at the small ends of the teeth there will often be a difference equal to that of three or four diametrical pitches. Another is to select cutters for a pitch situated about one-third of the length of the teeth away from the large ends, and so average the difference between the large and small ends of the teeth, neither being then exactly correct. Or two cuts can be taken down each side, instead of one, the second cut on each side being adopted specially to complete the smaller ends of the teeth. Or, what is generally done in wheels of small dimensions, the file is used at and near the small ends, following after the cutting, to reduce the widths of the tooth points towards the small ends.

Generally the first operation in cutting the teeth is to remove the bulk of the metal with a central or 66 stocking" cut, which goes down to the bottom of the teeth. This operation may be frequently omitted in the smaller gears after the settings of the wheel and cutter have been settled exactly by trial. The first cut when taken is done with the cutter set centrally with the axes of the blank. If the cutter happens to be of exactly the same thickness as the spaces between the teeth at the small end that determines the width of these spaces

at once. But when, as more often happens it is thinner, trial has to be made to get the exact width of the space correctly. The wheel blank has to be rolled by rotating the shaft of the dividing worm by hand, and the tooth faces to right and left of a space are cut by trial until the tooth thickness is right at the small ends of the teeth. If now the cutter is shifted

laterally, first to right and then to left, just to come in contact with the small ends, and cuts taken right through, the widths of the teeth spaces and the tooth thickness at the large ends should be right. It is in checking these dimensions that the value of gauges comes in more than in cutting spur gears. When all is right the amount of rolling over is noted and also the lateral displacement of the cutter, and one side of all the teeth is then finished and afterwards their other sides.

When these data are once obtained it is not necessary in small gears to take a central cut, but the first cut can be utilised to finish all one side, and the second cut all the other sides.

Though the tooth thicknesses at the small and large ends are thus secured, the tooth cones are not then right at all sections, simply because the cutter produces uniform cones from end to end. What happens is this. The wheel blank is set on its arbor at such an angle that the bottoms of the teeth spaces are cut to the correct angle. If the cutter is of the correct section at the large ends, the teeth cannot be tapered properly at the points, but will spread too much from the pitch line outwards. It is this excess which is filed off tapering down from the small ends to nothing at the large. Or the same result can be secured by rolling the gears after the first cut has been taken and removing more off the upper parts. This is a compromise, since it leaves the large ends rather thin. The steeper the angle of the pinion the more apparent is this result, so that in some cases a cutter having a flatter radius but narrower is substituted.

This is a general statement of the production of the teeth of bevel gears by means of rotary cutters, which is the most common method. For other methods, see Bevel Gear Generating Machines, Bevel Gear Planers.

Fig. 168, Plate VIII., illustrates the usual or standard type of wheel-cutting machine for bevels, spurs, and straight-toothed worm wheels. The blank is carried in a dividing headstock which is traversed along its bed to suit the different radii of gears to be cut. The large size of the dividing wheel will be noticed, because it

is essential in order to reduce errors in pitching. The worm, which is hidden behind the bed, can be thrown out of gear to permit of turning the work around by hand when making rough adjustments. On a bed at right angles to that just named, the cutter head is carried on a traversing bracket, and can be swivelled from the vertical and horizontal with suitable graduations for cutting bevels or worms. The head has a vertical slide and screw for adjusting the height of the cutter. The spindle is geardriven. Change gears are provided for varying the feed, and the self-acting feed motion has automatic throw-out. A set of 33 change wheels is provided for cutting all numbers up to 100, and every even number up to 200 teeth. Samples of the work done on the machine are seen lying around.

Fig. 170.-Crown Bevel Gear.

Fig. 169, Plate VIII., is a machine designed for cutting bevel gears specially, and one which has long been established in favour in the shops. The feature of the machine is, that although a rotary cutter is used, its spindle has a reciprocating movement as well, which by means of an eccentric decreases in amount as it approaches a certain point, so that it cuts the tapered form of tooth by virtue of this reciprocation.

The headstock is carried on the top of a hollow casting. It has self-acting traverse and automatic stop motion, so that the machine is brought to a stand at the termination of each cut, remaining thus until started on a fresh tooth. The work table is carried on a knee with longitudinal and transverse slides, the former having an index by means of which the axis of the work can be set exactly under the centre of the cutter. The table has a universal head with

hollow spindle; it can be set to any angle. Dividing is through worm and worm wheel.

Bevel Gear Generating Machines.—In these machines there is no former used, as in the planer type, nor rotary cutters having sectional shapes the reverse of those of the teeth, but the mechanism of generation is embodied in the

Fig. 172.-Evolver or Roll Cone of Bilgram Machine. design of the machine. Few of these have been made, but their numbers may be expected to increase. There are five or six successful types, the Bilgram, the Robey-Smith, the Warren, the Monneret, and the Beale, and each differs in important elements of design from the others. A brief notice of the principles embodied in each follow:

In general, the essential principle which lies at the basis of all machines of this type may be stated thus. The generating tool, or tooth represents in spur gears a rack tooth having straight flanks set at an angle, and the machines are so designed that this cuts all teeth of all

Fig. 173.-Wedge-Shaped Cutting Tool used in Bilgram Machine.

wheels of the same pitch, as though the blank were a plastic body being rolled in the same relation to the cutter as the gears would take relatively to a rack. But as in strictness rack teeth do not exist in bevels, the real basis is an absolute crown bevel wheel, Fig. 170, or one in which the pitch planes do not form a cone, but an absolutely horizontal plane, and the tooth