D.

To find the contents in cubic feet of a cylinder of diameter _d_ in inches and length _l_ in feet.

Find area in feet as before, and multiply by the length.

If dimensions are all in inches or feet, set the mark _c_ (= 1·128) on C to diameter on D and over length on B, read cubic contents on A.

To find the area of an ellipse.

Set 205 on C to 161 on D; bring cursor to length of major axis on C, 1 on C to cursor, and under length of minor axis on C read area on D.

EX.—Find the area of an ellipse the major and minor axes of which are 16 in. and 12 in. in length respectively.

Set 205 on C to 161 on D; bring cursor to 16 on C, 1 on C to cursor, and under 12 on C read 150·8 in. on D.

To find the surface of spheres.

Set 3·1416 on B to R.H. or L.H. index of A, and over diameter on D read by the aid of the cursor, the convex surface on B.

To find the cubic contents of spheres.

Set 1·91 on B to diameter on A, and over diameter on C read cubic contents on A.

WEIGHTS OF METALS.

To find the weight in lb. per lineal foot of square bars of metal.

Set index of B to weight of 12 cubic inches of the metal (_i.e._, one lineal foot, 1 square inch in section) on A, and over the side of the square in inches on C read weight in lb. on A.

EX.—Find the weight per foot length of 4½in. square wrought-iron bars.

Set middle index of B to 3·33 on A, and over 4½ on C read 67·5 lb. on A.

(N.B.—For other metals use the corresponding constant in column (2), below).

To find the weight in lb. per lineal foot of round bars.

Set R.H. or L.H. index of B to weight of 12 cylindrical inches of the metal on A (column (4), below), and opposite the diameter of the bar in inches on C, read weight in lb. per lineal foot on A.

EX.—Find the weight of 1 lineal foot of 2 in. round cast steel.

Set L.H. index of B to 2·68 on A, and over 2 on C read 10·7 lb. on A.

To find the weight of flat bars in lb. per lineal foot.

Set the breadth in inches on C to (1)/(weight of 12 cub. in.) of the metal (column (3), below) on D, and above the thickness on D read weight in lb. per lineal foot on C.

EX.—Find the weight per lineal foot of bar steel, 4½in. wide and ⅝in. thick.

Set 4·5 on C to 0·294 on D, and over 0·625 on D read 9·56 lb. per lineal foot on C.

To find the weight per square foot of sheet metal, set the weight per cubic foot of the metal (col. 1) on C to 12 on D, and

────────────┬──────────┬──────────┬──────────┬─────────── │ (1) │(2) │ (3) │ (4) │Weight in │Weight of │ (1)/(Wt. │ Weight of Metals. │ lb. per │12 cubic │of 12 cub.│ 12 │cubic ft. │in. │ in.) │cylindrical │ │ │ │ in. ────────────┼──────────┼──────────┼──────────┼─────────── Wrought iron│480 │3·33 │0·300 │2·62 Cast iron │450 │3·125 │0·320 │2·45 Cast steel │490 │3·40 │0·294 │2·68 Copper │550 │3·82 │0·262 │3·00 Aluminium │168 │1·166 │0·085 │0·915 Brass │520 │3·61 │0·277 │2·83 Lead │710 │4·93 │0·203 │3·87 Tin │462 │3·21 │0·312 │2·52 Zinc (cast) │430 │2·98 │0·335 │2·34 „ (sheet) │450 │3·125 │0·320 │2·45 ────────────┴──────────┴──────────┴──────────┴───────────

above the thickness of the plate in inches on D read weight in lb. per square foot on C.

EX.—Find the weight in lb. per square foot of aluminium sheet ⅜in. thick.

Set 168 on C to 12 on D, and over 0·375 on D read 5·25 lb. on C.

To find the weight of pipes in lb. per lineal foot.

Set mean diameter of the pipe in inches (_i.e._, internal diameter _plus_ the thickness, or external diameter _minus_ the thickness) on C to the constant given below on D, and over the thickness on D read weight in lb. per lineal foot on C.

┌────────────┬─────────────────────┬─────────────────────┐ │ Metals. │ Constant for Pipes. │Constant for Spheres.│ ├────────────┼─────────────────────┼─────────────────────┤ │Wrought iron│ 0·0955 │ 6·87 │ │Cast iron │ 0·1020 │ 7·35 │ │Steel │ 0·0936 │ 6·73 │ │Brass │ 0·0882 │ 6·35 │ │Copper │ 0·0834 │ 6·00 │ │Lead │ 0·0646 │ 4·65 │ └────────────┴─────────────────────┴─────────────────────┘

EX.—Find the weight per foot of cast-iron piping 4 in. internal diameter and ½in. thick.

Set 4·5 on C to 0·102 on D, and over 0·5 on D read 22·1 lb. on C, the required weight.

To find the weight in lb. of spheres or balls, given the diameter in inches. (W = 0·5236_d_^3 × wt. of 1 cub. in. of material).

Set the constant for spheres (given above) on B to diameter in inches on A, and over diameter on C read weight in lb. on A.

EX.—Find the weight of a cast-iron ball 7½in. in diameter.

Set 7·35 on B to 7·5 on A, and over 7·5 on C read 57·7 lb. on A.

To find diameter in inches of a sphere of given weight.

Set the cursor to the given weight in lb. on A, and move the slide until the same number is found on C under the cursor that is simultaneously found on A over the constant for the sphere on B.

EX.—Find diameter in inches of a sphere of cast-iron to weigh 7½lb.

Setting the cursor to 7·5 on A, and moving the slide, it is found that when 3·8 on C falls under the cursor, 3·8 on A is simultaneously found over 7·35 on B. The required diameter is therefore 3·8 in.

The rules for cubes and cube roots (page 40) should be kept in view in solving the last two examples.

FALLING BODIES.

To find velocity in feet per second of a falling body, given the time of fall in seconds.

Set index on C to time of fall on D, and under 32·2 on C read velocity in feet per second on D.

To find velocity in feet per second, given distance fallen through in feet.

Set 1 on C to distance fallen through on A, and under 64·4 on B read velocity in feet per second on D.

EX.—Find velocity acquired by falling through 14 ft.

Set (R.H.) index of C to 14 on A, and under 64·4 on B read 30 ft. per second on D.

To find distance fallen through in feet in a given time.

Set index of C to time in seconds on D, and over 16·1 on B read distance fallen through in feet on A.

CENTRIFUGAL FORCE.

To find the centrifugal force of a revolving mass in lb.

Set 2940 on B to revolutions per minute on D; bring cursor to weight in lb. on B; index of B to cursor, and over radius in feet on B read centrifugal force in lb. on A.

To find the centrifugal stress in lb. per square inch, in rims of revolving wheels of cast iron.

Set 61·3 on C to the mean diameter of the wheel in feet on D, and over revolutions per minute on C read stress per square inch on A.

EX.—Find the stress per square inch in a cast-iron fly-wheel rim 8 ft. in diameter and running at 120 revolutions per minute.

Set 61·3 on C to 8 on D, and over 120 on C read 245 lb. per square inch on A.

THE STEAM ENGINE.

Given the stroke and number of revolutions per minute, to find the piston speed.

Set stroke in inches on C to 6 on D, and over number of revolutions on D read piston speed in feet per minute on C.

To find cubic feet of steam in a cylinder at cut-off, given diameter of cylinder and period of admission in inches.

Set 2200 on B to cylinder diameter on D, and over period of admission on B read cubic feet of steam on A.

EX.—Cylinder diameter 26 in., stroke 40 in., cut-off at ⅝ of stroke. Find cubic feet of steam used (theoretically) per stroke.

Set 2200 on B to 26 on D, and over 40 × ⅝ or 25 in. on B, read 7·68 cub. ft. on A, as the number of cubic feet of steam used per stroke.

Given the diameter of a cylinder in inches, and the pressure in lb. per square inch, to find the load on the piston in tons.

Set pressure in lb. per square inch on B to 2852 on A, and over cylinder diameter in inches on D read load on piston in tons on B.

EX.—Steam pressure 180 lb. per square inch; cylinder diameter, 42 in. Find load in tons on piston.

Set 180 on B to 2852 on A, and over 42 on D read 111 tons, the gross load, on B.

Given admission period and absolute initial pressure of steam in a cylinder, to find the pressure at various points in the expansion period (isothermal expansion).

Invert the slide and set the admission period, in inches, on Ɔ to the initial pressure on D; then under any point in the expansion stroke on Ɔ find the corresponding pressure on D.

EX.—Admission period 12 in., stroke 42 in., initial pressure 80 lb. per square inch. Find pressure at successive fifths of the expansion period.

Set 12 on Ɔ to 80 on D, and opposite 18, 24, 30, 36 and 42 in. of the whole stroke on Ɔ find the corresponding pressures on D:—53·3, 40, 32, 26·6 and 22·8 lb. per square inch.

To find the mean pressure constant for isothermally expanding steam, given the cut-off as a fraction of the stroke.

Find the logarithm of the ratio of the expansion _r_, by the method previously explained (page 46). Prefix the characteristic and to the number thus obtained, on D, set 1 on C. Then under 2·302 on C read _x_ on D. To _x_ + 1 on D set _r_ on C, and under index of C read mean pressure constant on D. The latter, multiplied by the initial pressure, gives the mean forward pressure throughout the stroke. (N.B.—Common log. × 2·302 = hyperbolic log.)

EX.—Find the mean pressure constant for a cut-off of ¼th, or a ratio of expansion of 4.

Set (L.H.) index of C to 4 on D, and on the reverse side of the slide read 0·602 on the logarithmic scale. The characteristic = 0; hence to 0·602 on D set (R.H.) index of C, and under 2·302 on C read 1·384 on D. Add 1, and to 2·384 thus obtained on D set _r_ (= 4) on C, and under 1 on C read 0·596, the mean pressure constant required.

Mean pressure constants for the most usual degrees of cut-off are given below:—

Cut-off in fractions of stroke Mean pressure constant ¾ 0·968 ⁷⁄₁₀ 0·952 ⅔ 0·934 ⅝ 0·919 ⅗ 0·913 ½ 0·846 ⅖ 0·766 ⅜ 0·750 ⅓ 0·699 ³⁄₁₀ 0·664 ¼ 0·596 ⅕ 0·522 ⅙ 0·465 ⅐ 0·421 ⅛ 0·385 ⅑ 0·355 ⅒ 0·330 ¹⁄₁₁ 0·309 ¹⁄₁₂ 0·290 ¹⁄₁₃ 0·274 ¹⁄₁₄ 0·260 ¹⁄₁₅ 0·247 ¹⁄₁₆ 0·236

To find mean pressure:—Set 1 on C to constant on D, and under initial pressure on C read mean pressure on D.

Given the absolute initial pressure, length of stroke, and admission period, to find the absolute pressure at any point in the expansion period, it being assumed that the steam expands adiabatically. (P_{2} = (P_{1})/(R^{¹⁰⁄₉}) in which P_{1} = initial pressure and P_{2} the pressure corresponding to a ratio of expansion R.)

Set L.H. index of C to ratio of expansion on D, and read on the back of the slide the decimal of the logarithm. Add the characteristic, and to the number thus obtained on D set 9 on C, and read off the value found on D under the index of C. Set this number on the logarithmic scale to the index mark, in the opening on the back of the rule, and under L.H. index of C read the value of R^{¹⁰⁄₉} on D. The initial pressure divided by this value gives the corresponding pressure due to the expansion.

EX.—Absolute initial pressure 120 lb. per square inch; stroke, 4 ft.; cut-off ¼. Find the respective pressures when ½ and ¾ths of the stroke have been completed.

In the first case R = 2. Therefore setting the L.H. index of C to 2 on D, we find the decimal of the logarithm on the back of the slide to be 0·301. The characteristic is 0, so placing 9 on C to 0·301 on D, we read 0·334 as the value under the R.H. index of C. (N.B.—In locating the decimal point it is to be observed that the log. of R has been multiplied by 10, in accordance with the terms of the above expression.) Setting this number on the logarithmic scale to the back index, the value of R^{¹⁰⁄₉} is found on D, under the L.H. index of C, to be 2·16. Setting 120 on C to this value, it is found that the pressure at ½ stroke, read on C over the R.H. index of D, is 55·5 lb. per square inch. In a similar manner, the pressure when ¾ths of the stroke is completed is found to be 35·4 lb. per square inch.

For other conditions of expanding steam, or for gas or air, the method of procedure is similar to the above.

To find the horse-power of an engine, having given the mean _effective_ pressure, the cylinder diameter, stroke, and number of revolutions per minute.

To cylinder diameter on D set 145 on C; bring cursor to stroke in feet on B, 1 on B to cursor, cursor to number of revolutions on B, 1 on B to cursor, and over mean effective pressure on B find horse-power on A.

(N.B.—If stroke is in inches, use 502 in place of 145 given above.)

EX.—Find the indicated horse-power, given cylinder diameter 27 in., mean effective pressure 38 lb. per square inch, stroke 32 in., revolutions 57 per minute.

Set 502 on C to 27 on D, bring cursor to 32 on B, 1 on B to cursor, cursor to 57 on B, 1 on B to cursor, and over 38 on B read 200 I.H.P. on A.

To determine the horse-power of a compound engine, invert the slide and set the diameter of the _high_-pressure cylinder on Ɔ to the cut-off in that cylinder on A. Use the number then found on A over the diameter of the _low_-pressure cylinder on Ɔ as the cut-off in that cylinder, working with the same pressure and piston speed, and calculate the horse-power as for a single cylinder.

To find the cylinder ratio in compound engines, invert the slide and set index of Ɔ to diameter of the low-pressure cylinder on D. Then over the diameter of the high-pressure cylinder on C, read cylinder ratio on A.

EX.—Diameter of high-pressure cylinder 7¾in., low-pressure 15 in. Find cylinder ratio.

Set index on Ɔ to 15 on D, and over 7·75 on Ɔ read 3·75, the required ratio, on A.

The cylinder ratios of triple or quadruple-expansion engines may be similarly determined.

EX.—In a quadruple-expansion engine, the cylinders are 18, 26, 37, and 54 inches in diameter. Find the respective ratios of the high, first intermediate, and second intermediate cylinders to the low-pressure.

Set (R.H.) index of Ɔ to 54 on D, and over 18, 26, and 37 on Ɔ read 9, 4·31, and 2·13, the required ratios, on A.

Given the mean effective pressures in lb. per square inch in each of the three cylinders of a triple-expansion engine, the I.H.P. to be developed in each cylinder, and the piston speed, to find the respective cylinder diameters.

Set 42,000 on B to piston speed on A; bring cursor to mean effective pressure in low-pressure cylinder on B, index of B to cursor, and under I.H.P. on A read low-pressure cylinder diameter on C. To find the diameters of the high-pressure and intermediate-pressure cylinders, invert the slide and place the mean pressure in the low-pressure cylinder on ᗺ to the diameter of that cylinder on D. Then under the respective mean pressures on ᗺ read corresponding cylinder diameters on