thus, if one wheel is twice the diameter (measured on the pitch-circle) of the other, it has twice as many teeth. If the teeth are properly shaped the linear velocity of the two wheels are equal, and the angular velocities, or speeds of rotation, are inversely proportional to the number of teeth and to the diameter. Thus the wheel that has twice as many teeth as the other will revolve just half as many times in a minute. The "pitch," or distance measured on an arc of the pitch-circle from the face of one tooth to the face of the next, consists of two parts-the "thickness " of the tooth and the " space "between it and the next tooth. The space is larger than the thickness by a small amount called the "backlash," which is allowed for imperfections of workmanship. In finely cut gears the backlash may be almost nothing. pl al FIG. 153. The length of a tooth in the direction of the radius of the. wheel is called the "depth," and this is divided into two parts: First, the "addendum," the height of the tooth above the pitch line; second, the "dedendum," "the depth below the pitch line, which is an amount equal to the addendum of the mating gear. The depth of the space is usually given a little "clearance" to allow for inaccuracies of workmanship, especially in cast gears. Referring to Fig. 153, pl, pl are the pitch-lines, al the addendum-line, rl the root-line or dedendum-line, cl the clearance-line, and b the back lash. The addendum and dedendum are usually made equal to each other. No. of teeth 3.1416 diam. of pitch-circle in inches circular pitch 3.1416 No. of teeth diametral pitch diam. Some writers use the term diametral pitch to mean No. of teeth circular pitch 3.1416 but the first definition is the more common and the more convenient. A wheel of 12 in. diam, at the pitch-circle, with 48 teeth is 48/12 = 4 diametral pitch, or simply 4 pitch, The circular pitch of the same 12 X 3.1416 3.1416 wheel is 4 = .7854, or = .7854 in. 48 which always brings out the diameter as a number with an inconvenient diam. = fraction if the pitch is in even inches or simple fractions of an inch. By the diametral-pitch system this inconvenience is avoided. The diameter may be in even inches or convenient fractions, and the number of teeth is usually an even multiple of the number of inches in the diameter. Diameter of Pitch-line of Wheels from 10 to 100 Teeth of 1 in. Circular Pitch. For diameter of wheels of any other pitch than 1 in., multiply the figures in the table by the pitch. Given the diameter and the pitch, to find the number of teeth. Divide the diameter by the pitch, look in the table under diameter for the figure nearest to the quotient, and the number of teeth will be found opposite. AUTHORITIES.-1. Sir Wm. Fairbairn. 2, 3. Clark, R. T. D.; "used by engineers in good practice." 4. Molesworth. 5, 6. Coleman Sellers: 5 for cast, 6 for cut wheels. 7, 8. Unwin. 9, 10. Leading American manufacturers of cut gears. The Chordal Pitch (erroneously called "true pitch" by some authors) is the length of a straight line or chord drawn from centre to centre of two adjacent teeth. The term is now but little used. 180° Chordal No. of teeth pitch of a wheel of 10 in. pitch diameter and 10 teeth, 10 x sin 18° = 3.0902 in. Circular pitch of same wheel = 3 1416. Chordal pitch is used with chain or sprocket wheels, to conform to the pitch of the chain. Chordal pitch diam. of pitch-circle X sine of Formulæ for Determining the Dimensions of Small Gears, (Brown & Sharpe Mfg. Co.) P diametral pitch, or the number of teeth to one inch of diameter of pitch-circle; a distance between the centres of the two wheels; b = number of teeth in both wheels; t = thickness of tooth or cutter on pitch-circle; 8 = addendum; D' working depth of tooth; = f amount added to depth of tooth for rounding the corners and for clearance; D'+ƒ whole depth of tooth; π = 3.1416. P' circular pitch, or the distance from the centre of one tooth to the centre of the next measured on the pitch-circle. The following proportions of gear wheels are recommended by Prot. Cole man Sellers. (Stevens Indicator, April, 1892.) Width of Teeth.-The width of the faces of teeth is generally made from 2 to 3 times the circular pitch - from 6.28 to 9.42 divided by the diametral pitch. There is no standard rule for width. The following sizes are given in a stock list of cut gears in "Grant's Gears:" Diameter pitch..... 3 Face, inches........ 3 and 4 22 134 and 2 14 and 1% 34 and 1 1⁄2 and 5% 34 8 1 116 2 21% 41% 6 3 4 5 6 9 12 16 20 Rules for Calculating the Speed of Gears and Pulleys.The relations of the size and speed of driving and driven gear wheels are the same as those of belt pulleys. In calculating for gears, multiply or divide by the diameter of the pitch-circle or by the number of teeth, as may be required. In calculating for pulleys, multiply or divide by their diameter in inches. If D= diam. of driving wheel, d = diam. of driven, R = revolutions per minute of driver, r = revs. per min. of driven. If N Rrd + D; r =RD÷d; D= drR; d= DR + r. To find the number of revolutions of the last wheel at the end of a train of spur-wheels, all of which are in a line and mesh into one another, when the revolutions of the first wheel and the number of teeth or the diameter of the first and last are given: Multiply the revolutions of the first wheel by its number of teeth or its diameter, and divide the product by the number of teeth or the diameter of the last wheel. To find the number of teeth in each wheel for a train of spur-wheels, each to have a given velocity: Multiply the number of revolutions of the driving-wheel by its number of teeth, and divide the product by the number of revolutions each wheel is to make. To find the number of revolutions of the last wheel in a train of wheels and pinions, when the revolutions of the first or driver, and the diameter, the teeth, or the circumference of all the drivers and pinions are given: Multiply the diameter, the circumference, or the number of teeth of all the driving-wheels together, and this continued product by the number of revolutions of the first wheel, and divide this product by the continued product of the diameter, the circumference, or the number of teeth of all the driven wheels, and the quotient will be the number of revolutions of the last wheel. EXAMPLE.-1. A train of wheels consists of four wheels each 12 in, diameter of pitch-circle, and three pinions 4, 4, and 3 in. diameter. The large wheels are the drivers, and the first makes 36 revs. per min. Required the speed of the last wheel. 2. What is the speed of the first large wheel if the pinions are the drivers, the 3-in. pinion being the first driver and making 36 revs. per min.? Milling Cutters for Interchangeable Gears.-The Pratt & Whitney Co. make a series of cutters for cutting epicycloidal teeth. The number of cutters to cut from a pinion of 12 teeth to a rack is 24 for each pitch coarser than 10. The Brown & Sharpe Mfg. Co. make a similar series, and also a series for involute teeth, in which eight cutters are made for each pitch, as follows: In order that the teeth of wheels and pinions may run together smoothly and with a constant relative velocity, it is necessary that their working faces shall be formed of certain curves, called odontoids. The essential property of these curves is that when two teeth are in contact the common normal to the tooth curves at their point of contact must pass through the pitch-point, or point of contact of the two pitch circles. Two such curves are in common use-the cyloid and the involute. The Cycloidal Tooth.-In Fig. 154 let PL and pl be the pitch-circles of two gear-wheels; GC and gc are two equal generating-circles, whose radii should be taken as not greater than one half of the radius of the smaller pitch-circle. If the circle gc be rolled to the left on the larger pitch-circle PL, the point O will describe an epicycloid, oefgh. If the other generatingcircle GC be rolled to the right on PL, the point will describe a hypocycloid oabed. These two curves, which are tangent at O, form the two parts of a tooth curve for a gear whose pitch-circle is PL. The upper part oh is called the face and the lower part od is called the flank, If the same circles be rolled on the other pitch-circle pl, they will describe the curve for a tooth of the gear pl, which will work properly with the tooth on PL. The cycloidal curves may be drawn without actually rolling the generating-circle, as follows: On the line PL, from O, step off and mark equal distances, as 1, 2, 3, 4, etc. From 1, 2, 3, etc., draw radial lines toward the centre of PL, and from 6, 7, 8, etc., draw radial lines from the same centre, but beyond PL. With the radius of the generating-circle, and with centres successively placed on these radial lines, draw ares of circles tangent to PL at 123, 678, etc. With the dividers set to one of the equal divisións, as O1 |