W being the breaking load in lb., and M being a constant multiplier representing the unit strength of the material used for the gearing, and being equivalent to the breaking strength in lb. of a cantilever of that material, 1 in. square, and 1 in. long; taken at 6,000 lb. in cast iron.

Since the curvatures of teeth vary much, and as t measured at the root will vary slightly with variations in diameters of the wheels, it is usual in the formula to reckon t as being equal to the thickness on the pitch line, because that is then an invariable proportion, 48, of the pitch; being less than 5 pitch to allow for flank clearance in cast gears. Also, to make allowance for wear, it is properly taken as less than this, or as suggested by Professor Unwin, -36 pitch. The pressure is only concentrated on the extreme end of the length L at certain periods, that is as the teeth are entering, and leaving gear. At all other times the stress is less. But in basing the formula on the first contingency the position of greatest stress is taken, otherwise the formule would not be reliable.

Excessive increase of breadth B may not tend directly to increase of strength. For here the inaccuracy of practice comes into play. The teeth may be and often are not at precise right angles with the wheel faces, or the wheels may not be hung precisely true on their shafts. For these reasons some formulæ assume that the load is concentrated on one corner only of the teeth, tending to break off a triangular prism, as at A, Fig. 93. Wheel teeth do often break in that fashion, showing that the load is concentrated on the corner. But they as often break nearly straight across, in a more or less jagged irregular fashion. Moreover, there is less excuse now for accepting this contingency, because machine-moulded gears are practically free from error; and in pattern gears only those of exceptional width should be very much tapered. Then they can, in many cases at least, be so hung on their shafts that the taper of one is in the reverse direction to that of its fellow. After wheels have become worn a little they come into perfect contact, so that inaccuracy of contact occurs only when they are new, and at maximum strength. When weakened, their contact is absolutely true.

The proportions between the pitch and breadth vary. But B is seldom less than twice

the pitch, or more than four times the pitch. The requirements of modern engineering are too exacting to permit of the indiscriminate use of formulæ such as these just noted. There is a vast deal of difference in the manner in which gears have to be driven. Some are driven slowly and steadily, others in a more irregular fashion, while in other cases very violent shock comes into play, stressing the material in a trying fashion. Formulæ are prepared, therefore, in which these influences are estimated, and embodied. The Unwin formula evolved for the strength of wheel teeth is

ratio of the pitch to the breadth of wheel face, and introduces a factor not taken account of when the stress is supposed to act only upon one corner of the tooth, or when the face width happens to be less than twice the pitch, which very seldom happens.

A rule taken from a standard American work on gearing which appears to be practically identical with this of Unwin's is as follows:-To determine the pitch for a castiron gear; multiply the force to be transmitted by the ratio of the pitch to the face width, extract the square root of the product, and multiply the result by 078 for violent shock, 07 for moderate shock, or 05 for little or no shock. The equation for moderate shock therefore stands :

p = '078 √ px?

Having to do with force in rapid motion other conditions come into play. Work increases directly as velocity, so that if the rim of a toothed wheel travels at the rate of 4 ft. in a second that velocity will represent twice the number of units of work which would be done by the same wheel under the same conditions of pressure moving at the rate of 2 ft. per second. Horse power is only a convenient expression to represent 33,000 foot pounds of work per minute. The HP., therefore, of wheels varies directly as their velocity; a wheel moving twice as fast as the same wheel under the same conditions of pressure will develop twice the HP. in the former case as in the latter. But the pressure stress on the teeth varies inversely as the velocity, an important point.

The Wilfred Lewis formula for determining the strength of spur gears assumes that the whole load is taken upon one tooth, and considers the tooth loaded as a cantilever. A factor which depends upon the form of the tooth is introduced into the formula; this factor has been determined by the selection of the weakest cross section of involute, cycloidal, and radial flank gears. The factor naturally varies with the number of teeth contained in a wheel. The formula is as follows::

W = spfy.

where W = load on tooth in lb.,

8 = safe working stress of material, p = circular pitch,

f-face of gear in inches,

y = factor.

General Joiner, or Universal Joiner A combination type of wood-working machine which is specially adapted to the requirements of shops where an extensive plant cannot be laid down. Fig. 94, Plate VI., shows an example which comprises the following:-A circular saw on a rising and falling spindle, to which cutter-blocks for tonguing, grooving, and rebating may be attached. Planing and moulding spindle taking work 12 in. wide by 4 in. thick. Band saw with pulleys 24 in. diameter, sawing up to 9 in. deep. Circular moulding apparatus, with vertical spindle capable of working mouldings up to 4 in. deep. A tenoning apparatus with cutters for cutting complete tenons at one operation. A special table may also be provided for slot-mortising and boring. It will be seen that all the operations in joinery can be done on this machine, with the added convenience that the parts are located close together, so that one man may finish a piece of work rapidly. Alternatively, another helper could be doing some operation, such as bandsawing, as a preliminary to further work. The machine is driven primarily from the pulleys at the end of the frame on the extreme left. Generating Circle.-See Gears. Generating Machines. See Bevel Gear and Spur Gear Generating Machines. Generating Stations. See Central Central Stations.

Geometrical Mean. The geometrical mean of two quantities is the square root of their product. Thus the geometrical mean of 4 and 9 is 6; for 4 × 9 = 36, and √36 = 6; 9 is as many times greater than 6 as 6 is greater than 4. Stated generally, a: m :: m : b or ab = m2, where a and b are the two numbers, and m the mean. Geometrical Progression. A geometrical progression is a series of numbers increasing or decreasing by a common ratio or constant factor. 1, 3, 9, 27 is an increasing series; 3,,, a decreasing series. The common ratio (which is found by dividing any term by the one preceding it) is 3 in the first example, and in the second.

3

84

Algebraically, a geometrical progression is stated, a, ar, ar2, ar3, and so on, where a is the first term and r the common ratio. Since the index of the 3rd term is 2, of the 4th term 3,

German Silver.-Also called nickel silver, is an alloy of copper, zinc, and nickel. The proportions vary, one of the best alloys being produced by 4 parts copper, 2 zinc, and 2 nickel ; another alloy is obtained from 6 copper, 3 zinc, and 1 nickel. Spoons and forks, pots, dish covers, and bar fittings are largely made of German silver, its whiteness, and toughness, and the facility with which it takes a polish making it highly valuable for these purposes. It is frequently electroplated, and this is desirable in the case of articles where acids would act on the copper present, and produce verdigris. It is employed extensively in electrical work, for resistance wires.

Gib.-A shouldered strip of metal used as a backing for a cotter, to prevent opening out of a strap by the friction of the cotter. Examples may be noted in Figs. 57 and 58, Vol. IV. The term is also applied to Adjusting Strips. A gib-headed key is thus distinguished from one having no head, the gib being required when the key can only be drawn out by its head.

18

Gimbal Joint-See Universal Joint. Gimlet.-A boring tool for wood, used chiefly for holes for screws. It is not employed in larger sizes than ordinary shell bits, a medium size being in. or in. diameter. Being turned by its handle it does not bore so quickly as a bit operated by a brace, and consequently the latter is always preferred, unless, as is sometimes the case, a gimlet is more convenient. Gimlets are made in twist and in shell form, and less frequently in a combination of the two, and also with a twist like an auger bit, but the most popular form has a comparatively slight twist, and its full diameter is some distance back from the point. In all cases gimlets are provided with screw points.

Girard Turbine.-A wheel of the impulse type, in which the water is under atmospheric

pressure only, and has free deviation, or freedom to pursue its own course after leaving the guides. The development of this design was due to M. Girard, a French engineer. The ventilation of the buckets was introduced by him, and by preventing the water from filling the buckets, the chance of the wheel working by reaction was avoided. These turbines are made in radial outward flow, inward flow, and axial types. They have the advantage that the high speeds necessary in the reaction types when small quantities of water are available, are not re

to 1,000 feet. Girard turbines with vertical shafts are seldom made now, being mostly superseded by mixed flow turbines for low and medium falls.

The illustration, Fig. 95, is that of a Girard horizontal shaft, outward flow turbine, by Messrs Carrick & Ritchie, of Edinburgh. Fig. 96 is the 19 inch size in sectional views. The diameter is measured on the inner circumference where the water is received. A indicates the guides or vanes, в the buckets, c is the regulating gate actuated by a rack and pinion D.

quired; but relatively large wheels and moderate rates of revolution are available; for the water can be admitted to all the buckets or to a few only-partial injection-without affecting the efficiency. This is a feature of value in cases where the quantity of water varies in different seasons. This turbine is not suited for very low falls, 50 feet being considered the practical minimum for horizontal types, though some with vertical shafts are made for lesser heights.

Turbines with vertical shafts are only used for comparatively low falls, but those with horizontal shafts are suitable for heads up

Fig. 95 shows this turbine connected directly to a dynamo for electric lighting, the lights being run direct from the plant without an accumulator. It is therefore fitted with a governor to control the speed. The water from the main pipe enters the box casting, the inner side of which is closed by a cast-iron lid. The guide ports are formed in the lower part of this box, and the openings are closed by the slide c just now mentioned. The spindle for operating the rack passes out through a stuffing box, and is fitted with a quadrant gearing into a rack which forms an extension of the piston rod of