Part 34

Setting 1·625 on C to 2 on D, it is seen that 80 (driver) and 65 (driven) are possible wheels.

COMPOUND GEARS.—When wheels so found are of inconvenient size, a compound train is used, consisting (usually) of two drivers and two driven wheels, the product of the two former and the product of the two latter being in the same ratio as the simple wheels. Thus with 60 and 40 as drivers, and 65 and 30 as driven, we have, (60 × 40)/(65 × 30) = (2400)/(1950) = (2)/(1·625) as before.

With the slide set as above, values convenient for splitting up into suitable wheels are readily obtainable. Thus, (1600)/(1300); (2400)/(1950); (4000)/(3250); (4800)/(3900) are a few suggestive values which may be readily factorised.

SLIDE RULES FOR SCREW-CUTTING CALCULATIONS.—Special circular and straight slide rules for screw-cutting gear calculations have long been employed. For compound gears these usually entail the use of six scales, two on each of the two slides and two on the stock. The upper scale on the stock may be a scale of threads per inch to be cut, the adjacent scale (on the upper slide) a scale of threads per inch in the guide screw. Setting the guide screw-graduation to the threads to be cut, the lower slide is adjusted until a convenient pair of drivers is found in coincidence on the central pair of scales, while a pair of driven wheels are in coincidence on the two lower scales.

Some years ago, a slide rule was introduced by which compound gears could be obtained with a single slide. Assuming the set of wheels usually provided—20 to 120 teeth advancing by 5 teeth—the products of 20 × 25, 20 × 30, etc., up to 115 × 120 were calculated. These products were laid out along each of the two lower scales. The upper scales were a scale of threads per inch to be cut and a scale of the threads per inch of various guide screws. Setting the guide screw-graduation to the threads to be cut, any coinciding graduations on the lower scales gave the required pairs of drivers and driven wheels.

FRACTIONAL PITCH CALCULATIONS.—The author has long advocated the use of the slide rule for determining the wheels necessary for cutting fractional pitch threads, and it is gratifying to find its value in this connection is now being appreciated. For the best results a good 20 in. rule is desirable, but with care very close approximations can be found with an accurate 10 in. rule. In any case a magnifying cursor or a hand reading-glass is of great assistance.

EX.—Find wheels to cut a thread of 0·70909 in. pitch; guide screw, 2 threads per inch.

To 0·70909 on D, set 0·5 (guide screw pitch in inches) on C. To make this setting as accurately as possible, the method described on page 112 may be used. Set 10 on C to about 91 on D, and note that the interval 77–78 on C represents 0·91 of the interval 70–71 on D. Set the cursor to 78 on C and bring 5 to the cursor. The slide is then set so that 5 on C agrees with 7·091 on D.

Inspection of the two scales shows various coinciding factors in the ratio required. The most accurate is seen to be (55 on C)/(78 on D). These values may be split up into (55 × 50)/(65 × 60) to form a suitable compound train of gears.

GAUGE POINTS AND SIGNS ON SLIDE RULES.

Many slide rules have the sign (Prod.)/(−1) at the right-hand end of the D scale, while on the left is (Quot.)/(+1.) It is somewhat unfortunate that these signs refer to rules for determining the number of digits in products and quotients, which are used to a considerable extent on the Continent, and conflict with those used in this country. By the Continental method the number of digits in a product is equal to the sum of the digits in the two factors, if the result is obtained on the LEFT of the first factor; but if the result is found on the RIGHT of the first factor, it is equal to this sum − 1. The sign (Prod.)/(−1) the right-hand end of the D scale provides a visible reminder of this rule.

Similarly for division:—The number of digits in a quotient is equal to the number of the digits in the dividend, minus those in the divisor, if the quotient appears on the RIGHT of the dividend, and to this difference + 1, if the quotient appears on the LEFT of the dividend. The sign (Quot.)/(+1) at the left-hand end of the D scale provides a visible reminder of this rule.

The sign

+ⵏ– ⟵ⵏ⟶ –ⵏ+

found at both ends of the A scale is of general application but of questionable utility. It is assumed to represent a fraction, the vertical line indicating the position of the decimal point. If the number 455 is to be dealt with in a multiplication on the lower scales, we may suppose the decimal point moved two places to the left, giving 4·55, a value which can be actually found on the scale. If we use this value, then to the number of digits in this result, as many must be added as the number of places (two in this case) by which the decimal point was moved. If the point is moved to the right, the number of places must be subtracted. Similarly, in division, if the decimal point in the divisor is moved n places to the left, then n places must be subtracted at the end of the operation; while if the point is moved through n places to the right, then n places must be added. The sign referred to, which, of course, applies to all scales, completely indicates these processes and is submitted as a reminder of the procedure to be followed by those using the method described.

The signs π, c, c′, and M are explained in the Section on “Gauge Points,” p. 53.

On some rules additional signs are found on the D scale. One, locating the value (180 × 60)/(π) = 3437·74 and hence giving the number of minutes in a radian, is marked ρ′. Another, representing the value (180 × 60 × 60)/(π) = 206265, and hence giving the number of seconds in a radian is marked ρ″. A third point, marked ρ_{˶}, placed at the value (200 × 100 × 100)/(π) = 636620, is used when the newer graduation of the circle is employed.

These gauge points are useful when converting angles into circular measure, or vice versa, and also for determining the functions of small angles.

A gauge point is sometimes marked at 1146 on the A and B scales. This is known as the “Gunner’s Mark,” and is used in artillery calculations involving angles of less than 20°, when, for the purpose in view, the tangent and circular measure of the angle may be regarded as equal. For this constant, the angle is taken in minutes, the auxiliary base in feet, and the base in yards. The auxiliary base in feet on B is set to the angle in minutes on A when over 1146 on B is the base in yards on A. The value (1)/(1146) = (π × 3)/(180 × 60).

TABLES AND DATA.

MENSURATION FORMULAE.

Area of a parallelogram = base × height.

Area of rhombus = ½ product of the diagonals.

Area of a triangle = ½ base × perpendicular height.

Area of equilateral triangle = square of side × 0·433.

Area of trapezium = ½ sum of two parallel sides × height.

Area of any right-lined figure of four or more unequal sides is found by dividing it into triangles, finding area of each and adding together.

Area of regular polygon = (1) length of one side × number of sides × radius of inscribed circle; or (2) the sum of the triangular areas into which the figures may be divided.