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From a great number of well arranged experiments, on the strength of iron beams and tubes, it follows that they, may be safely reduced in strength from the middle towards the extremities in the ratio indicated by theory.

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Let AB be a beam supported at its extremities E and F, and put F'equal to the necessary strength at the

Then, the necessary strength at D= Fx

middle of the beam. A C2 — C D2

A C2

The tensile force of wrought iron is to its compressive force as 2

to 1.

Hence, the plate on the upper side of hollow wrought-iron tubes should contain an area twice as great as the plate on the under side.

Strength of Cast-Iron Pillars.

The breaking weight of solid cylindrical cast-iron pillars.

In solid pillars, with their ends rounded, and moveable,

Breaking weight in tons = 14.9 ×

d36

717

(1)

In solid pillars, with their ends flat, and incapable of motion,

Breaking weight in tons

= 44-16 x

d3'6 217

(2)*

where l is the length of pillar in feet, and d the diameter in inches. In hollow pillars of cast-iron, where D, d are the external, internal diameters, and 7 the length; both ends of the pillar were moveable.

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In hollow cast-iron beams, whose ends were flat and firmly fixed,

Breaking weight in tons = 44-3

D36d36
71-7

Of three cylindrical pillars of steel, wrought and cast iron, and wood, all of the same length and diameter, the first having its ends

* Formula (1) was obtained from the mean result of eighteen pillars, varying in length from 121 times the diameter down to 15 times. The formula (2) was derived from eleven pillars, with flat ends, varying in length from 78 to 25 times the diameter.

rounded, the second with one end round and the other end flat, and the third with both ends flat, the strengths are as 1, 2, and 3.

These formula and results were obtained from experiments on pillars, varying in length from 121 times the diameter down to 15 times.

Effects of Temperature upon the Strength of Cast-Iron.

The strength of cast-iron is not reduced when its temperature is raised to 600°, which is nearly that of melting lead; and it does not differ very widely whatever the temperature may be, provided the bar be not heated so as to be red hot.

EXAMPLE.

Find the strength of a hollow cylindrical cast-iron pillar, 14 feet long, 6-2 inches external diameter, and 4·1 inches internal; the pillar being flat and well supported at the ends.

6-236712-22 and 4·196 160-7

141788.801

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Comparative Strength of Long Pillars.

If the strength of cast-iron pillars be 1000, then wrought-iron will be 1745, cast-steel 2518, Dantzic oak 108.8, and red deal 78.5. The strength of similar pillars is as the square of their linear dimensions.

Resistance to Torsion.

Let = length of prism from the fixed end to the point of application of the lever used to twist it.

T= radius of prism, if round.

b, d

=

breadth and thickness, if rectangular.

W the weight acting by means of the lever to twist the prism.

L= length of the lever to which the weight Wis applied. 0= angle of torsion.

R

resistance to torsion at the time of fracture. C constant for each species of body.

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For a rectangle,

3 L l W (b2 + d3) = C ́0 b3 d3 and 3 W L √ b2 + d2 = R ¿a3 ♫o

The Ultimate Resistance of a Cast-iron Beam to Torsion.

In a cylinder, WL= 51055 r3.
In a square prism, WL=7660 ď3.

In a rectangular prism, W L = 10834

All the dimensions are taken in inches.

b2 d2

√ b2 + d2.

Strength of Ropes.

The cohesion of hempen fibres is 6400 lbs., for every square inch

of section.

Breaking weight in tons =

circumference squared

4

the circumference being measured in inches.

Ex-Find the breaking weight of a rope 6 inches in circum

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PROCESSES FOR STAINING WOODS.

Mahogany Color (Dark).-Boil lb. of madder and 2 oz. of logwood in a gallon of water; then brush the wood well over with the hot liquid. When dry, go over the whole with a solution of 2 drachms of pearlash in a quart of water.

Mahogany Color (Light)-Brush over the surface with diluted nitrous acid, and when dry apply the following, with a soft brush: Dragon's blood, 4 oz.; common soda, 1 oz.; spirit of wine, 3 pints. Let it stand in a warm place, shake it frequently, and then strain. Repeat the application until the proper color is obtained. To Stain Maple a Mahogany Color.-Dragon's blood, alkanet, oz.; aloes, 1 dr.; spirit of wine, 16 oz. Apply it with a sponge or brush.

oz.;

Rosewood.-Boil 8 oz. of logwood in 3 pints of water until reduced to half; apply it, boiling hot, two or three times, letting it dry between each. Afterwards put in the streaks, with a camel's hair pencil, dipped in a solution of copperas and verdigris in a decoction of logwood.

Ebony-Wash the wood repeatedly with a solution of sulphate. of iron; let it dry, then apply a hot decoction of logwood and nutgalls for two or three times. When dry, wipe it with a wet sponge; and when dry again, polish with linseed oil.

Rd.-1. Take a pound of Brazil wood and mix it with a gallon of stale urine. Pour over the wood while boiling hot. Before it dries it should be laid over with alum water. 2. A fine red may also be obtained by a solution of dragon's blood in spirits of wine.

Yellow-Nitric acid, lightly diluted, will produce a fine yellow on wood. Sometimes, if the wood is not in proper condition, it will create a brown. Care must be taken that the acid used be not too strong, or it will render the wood nearly black.

Blue-Take of alum 4 parts; water 85 parts. Boil.

Purple.-To produce this color, take of logwood 11 parts; alum

3 parts; water 29 parts. Boil.

Mahogany.-1. Linseed oil 2 pounds; alkanet 3 ounces. Heat them together and macerate for six hours, then add resin 2 ounces; beeswax 2 ounces. Boiled oil may be advantageously used instead

of the linseed oil.

2. Brazil-wood (ground); water sufficient; add a little alum and potash. Boil.

3. Logwood 1 part; water 8 parts. Make a decoction, and apply it to the wood; when dry, give it two or three coats of the following varnish: dragon's blood 1 part; spirits of wine 20 parts. Mix.

To take Stains out of Mahogany.-Spirits of salts 6 parts; salt of lemons 1 part. Mix, then drop a little on the stains, and rub them until they disappear.

To Stain Musical Instruments-Crimson: Boil one pound of ground Brazil wed in three quarts of water for an hour; strain it, and add half an ounce of cochineal; boil it again for half an hour gently, and it will be fit for use.

Purple: Boil a pound of chip logwood in three quarts of water for an hour; then add four ounces of alum.

LOGARITHMS.

Logarithms literally signify ratios of numbers; hence Logarithmic Tables may be various, but those in common use for the facilitating of arithmetical operations generally are of the following corresponding progressions, viz.:

Arithmetical, 0, 1. 2. 3. &c., or series of logarithms.
Geometrical, 1. 10, 100, 1000, &c., or ratio of numbers.

And thus it may be perceived, that if the log of 10 be 1, the log. of any number less than 10 must consist wholly of decimals, because increasing by a decimal ratio. Again; if the log. of 100

be 2, the log. of any intermediate number between 10 and 100 must be 1, with so many decimals annexed; and in like manner, the log. of any intermediate number between 100 and 1000, must be 2, with decimals annexed proportionally, as before.

Application and Utility of Common Logarithmic Tables,

The whole numbers of the series of logarithms, as 1, 2, 3, &c., are called the indices, or characteristics of the logarithm, and which must be added to the logarithm obtained by the table, in proportion to the number of figures contained in the given sum. Thus suppose the logarithm be required for a sum of only two figures, the index is 1; if of three figures, the index is 2; and if of four figures, the index is 3, &c.; being always a number less by unity than the number of figures the given sum contains.

EXAMPLES.

The index of 8 is 0, because it is less than 10.
The index of 80 is 1, because it is less than 100.

The index of 800 is 2. because it is less than 1000.

The index of 8000 is 3, because it is less than 10,000, &c.

The index of a decimal is always the number which denotes the significant figure from the decimal point, and is marked with the sign, thus, —, to distinguish it from a whole number

The index of 32549 is

the first decimal.

The index of 032549 is the second decimal.

EXAMPLES.

-1, because the first significant figure is

2, because the first significant figure is

The index of 0032549 is-3, because the first significant figure is the third decimal, &c., of any other sum.

If the given sum for which the logarithm is required contains or consists of both integers and decimals, the index is determined by the integer part, without having any regard to the other.

1. To find the logarithm of any whole number under 100.

Look for the number under N in the first page of any Logarithmic Table; then immediately on the right of it is the logarithm required, with its proper index. Thus the log. of 64 is 1·806180, and the log of 72 is 1.857332.

2 To find the logarithm of any number between 100 and 1000, or any sum not exceeding 4 figures.

Find the first three figures in the left-hand column of the page under N, in which the number is situated, and the fourth figure, at the top or bottom of the page; then the logarithm directly under the fourth figure, and in a line with the three figures in the column on the left, with its proper index, is the logarithm required. Thus, the log. of 450 is 2 653213, and the log. of 7464 is 3-872972. Or, the log. of 378.5 is 2.578066, and that of 7854 is-1.895091.

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