Fig. 50.
3.Given Depth of Shaft and Angle of Dip to find where it outcrops.—Then AC = AB × natural tangent of angle ABC. Or by scale and protractor by inspection.
4.Given Depth of Shaft AB and Dip of Vein Angle CB to find Distance BC between Bottom of Shaft and Outcrop.—BC = AC × natural sine ACB. Or by scale and protractor by inspection.
One inch of rain = 22,680 gallons, or 102·35 tons of water per acre.
Co-efficientof friction between ordinary leather belting and cast-iron pulleys or drums = ·423. Ultimate strength of ordinary leather belting = 3086 lb. per square inch. Belts vary from ³∕₁₆ in. to ¼in. thick, average ⁷∕₃₂ in.
To calculate the power of single leather belts, the following formula may be used: Let HP = actual horse-power. W = width of belt. F = driving force. T = working tension from 70 to 150 lb. V = velocity of belt in feet per minute.
Example:—A 10-inch belt running 2500 feet per minute, what horse-power will it transmit? Assuming the working tension to be 100 lb.,
Nystrom gives this rule:—HP =V × F⁄₅₅₀. V = velocity of belt in feet per second. F = force in pounds transmitted by belt.
The first rule gives good practical results where there is no great inequality in the diameter of the pulleys.
Double Beltstransmit 1½ times as much as single belts.
A = covered area of driven pulley in square inches.
V = speed of belt in feet per minute.
H = indicated horse-power.
W = width in inches.
H =AV⁄66,000A =66,000 H⁄V. W = A/L, where L = length of belt on driven pulley in inches.
Another authority simply says H = {70 to 80} ×WV⁄33,000.
And a third says W =36,000 H⁄6VL, where L is here in feet.
Evan Leigh’s rule is W =66,000 x IHP⁄L x V.
L = length of arc of contact upon smaller pulley in inches.
V = velocity of rim in feet per minute.
A belttransmits its motion solely through frictional contact with the surfaces of the pulley. The lower side of the belt should be made the driving side when possible, as the arc of contact is thereby increased by the sagging of the following side. Increase of power will be obtained by increasing the size of pulleys, the same ratio being retained. Wide belts are less effective per unit of sectional area than narrow belts. Long belts are more effective than short ones. The proportion between the diameters of two pulleys working together should not exceed six to one. Convexity of pulleys to receive belt = ½ inch per foot wide. Thewidth of pulley should equal 1·2 times width of belt.
Belts have been employed running over 5000 feet per minute. Nothing, however, is gained by running belts much over 4000 feet per minute. About 3500 feet per minute for main belts agrees with good practice; lathe belts from 1500 to 2000 feet per minute. The life of a belt may be prolonged and its driving powers increased by keeping it in good working order. To ensure this it should be dressed on the back with castor oil every few weeks, more or less according the dryness of theatmospherein which it works.
The weight of a cubic foot of any material is its specific gravity multiplied by 62·425, or the weight of a cubic foot of water in pounds. To find the specific gravity of a stone, divide its weight in air by loss of weight in water of temperature of 60° F. = specific gravity.
Thus:
Then:
293·7⁄113·6= 2·59 = Specific gravity of quartz.
One ton of quartz when solid occupies 13 cubic feet, but when broken, about 20. Rocks when solid, as compared to the same when broken, usually increase in volume in the ratio of 1 to 1·5 or 1 to 1·18, the increase depending on size and form of fragments.
A dwt. of gold in a cwt. of ore = 1 oz. of gold per ton of ore.
For approximate calculation a grain of gold = two pence, and a dwt., four shillings.
In the following table of the chemical elements the standard of sp. gr. is hydrogen for the gaseous elements (hydrogen, oxygen, &c.) and water for the others.
The figures indicating the proportions by weight in which the elements unite with one another are called the combining or atomic weights, because they represent the relative weights of the atoms of the different elements. Since hydrogen is the lightest element, it is taken as the standard, and its combining or atomic weight = 1.
To find the proportional parts by weight of the elements of any substance whose chemical formula is known:
Rule.—Multiply together the equivalent and the exponent of each element of the compound; the product will be the proportion by weight of that element in the substance.
Example:—Find the proportional weights of the elements of Alcohol C₂H₆O.
Of every 46 lb. of Alcohol, 6 lb. will be H; 16, O; 24, C.
To find the proportions byvolume, divide by the specific gravity.
The following are the formulæ for the conversion of degrees of one scale to those of another:—
Pounds of water evaporated by 1 lb. of fuel as follows:—
= Sign of equality, denoting that quantities so connected are equal to one another; thus, 3 feet = 1 yard.
+ Sign of addition, signifyingplusor more; thus, 4 + 3 = 7.
- Sign of subtraction, signifyingminusor less; thus, 4-3 = 1.
× Sign of multiplication, signifying multiplied by or into; thus, 4 × 3 = 12.
÷ Sign of division, signifying divided by; thus, 4 ÷ 2 = 2.
{} () [] Brackets, denoting that the quantities between them are to be treated as one quantity; thus, 5 {3(4 + 2)-6(3-2) = 5 (18-6) = 60.
Letters are used to shorten or simplify a formula. Supposing we wish to express length × breadth × depth, we may put the initial letters only, thus,l×b×d, or, as is usual when algebraical symbols are employed, leave out the sign × between the factors, and write the formulalbd.
When division is to be expressed in simple form, the divisor is written under the dividend; thus (x+y) ÷z= (x+y) /z
° ’ ” are signs used to express certain angles in degrees, minutes, and seconds; thus 25 degrees 4 minutes 21 seconds would be expressed 25° 4’ 21”.
√ This sign is called the radical sign, and placed before a quantity indicates that some root of it is to be taken, and a small figure placed over the sign, called the exponent of the root, shows what root is to be extracted.
Thus ²√aor √ameans the square root ofa∛a” cube ”∜a” fourth ”
ρ This sign is used to denote the force of gravity at any given latitude.
π The Greek letter pi is invariably used to denote 3·14159,that is, the ratio borne by the diameter of a circle to its circumference.
When the figure 2 is affixed to any number, as diameter² or 12², the number is to be squared, as 12 × 12 = 144, the square; and with ³ affixed, the number is to be cubed—i.e., multiplied twice by itself, as 6³ = 6 × 6 × 6 = 316, the cube of 6.
12lines=1 inch.12inches=1 foot.3feet=1 yard.6feet=1 fathom.16½feet=1 pole.220yards=1 furlong.}8furlongs1760yards=1 statute mile.5280feet6086feet=1 naut. mile.7·92inches=1 link.}100links66feet=1 chain.22yards
144sq. inches=1 sq. foot.9sq. feet=1 sq. yard.}30¼sq. yards=1 sq. rod or pole.272¼feet40rods=1 sq. rood.}4roods160rods4840yards=1 acre.43560feet10sq. chains=1 acre.1hectare=2·471 acres.640acres=1 sq. mile.30sq. acres=1 yard of land.Sq. ins.× 0·007=square foot nearly.Sq. yds.× 0·00021=acres nearly.113·0977sq. ins.=1 circular foot.183·46circular ins.=1 square foot.
1728cubic inches=1 cubic foot.27cubic feet=1 cubic yard.}40cub. ft. of rough, or=1 ton or load.50cub. ft. of hwn. tmbr.}128cub. ft. of timber=1 cord of wood.
16drachms=1 ounce.16ounces=1 lb.14lb.=1 stone.28lb.=1 qr. cwt.112lb.=1 cwt.20cwt=1 ton.lbs.× 0·009=cwt. nearly.lbs.× 0·00045=tons.7000grains=1 lb. avdp.437½grains=1 oz.
Gallon (C) = 8 pints (O); 1 pint = 20 fluid ounces (oz. weight of water). Ounce (f ℥) = 8 drachms (f ʒ) = 480 minims (♏) = 720 drops (gtt.).
One wine glass = 4 tablespoonfuls = 16 tablespoonfuls = 2 ounces.
Symbols.—f. or fl. fluid; s.s. one half; a.a. for each. Thus f℥ss. ½ a fluid ounce.
Apothecaries’ weight, formerly used for dispensing medicines, superseded in 1864. 20 grains = 1 scruple; 3 scruples = 1 drachm; 8 drachms = 1 ounce; 12 ounces = 1 lb. (troy).
Gramme15·432349 grams troy.Décagramme (= 10 grammes)5·6438 drachms av.Hectogramme (= 100 grammes)3·527 oz. av.Kilogramme (= 1000 grammes)2·204621 lbs. av., or2·679227 lbs. troy.Quintal (= 100 kilogrammes)220·462 lbs. av.Tonne (= 1000 kilogrammes)2204·621 lbs. av.Decigramme (= ¹∕₁₀th of a gramme)1·5432 grain.Centigramme (= ¹∕₁₀₀th of a gramme)0·15432 grain.Milligramme (= ¹∕₁₀₀₀th of a gramme)0·015432 grain.
Mètre3·2808992 feet.Décamètre (= 10 mètres)32·808992 feet.Hectomètre (= 100 mètres)328·08992 feet.Kilomètre (= 1000 mètres)1093·633 yards.Myriamètre (= 10,000 mètres)6·2138 miles.Decimètre (= ¹∕₁₀th of a mètre)3·937079 inches.Centimètre (= ¹∕₁₀₀th of a mètre)0·39371 inch.Millimètre (= ¹∕₁₀₀₀th of a mètre)0·03937 inch.
Centiare (= 1 square mètre)1·196033 square yard.Are(= 100 square mètres)0·098845 rood.Hectare (= 10,000 square mètres)2·471143 acres.
Litre(= 1 décimétre cube)1·760773pint (61·027 cubic inches).Décalitre(= 10 litres)2·2009668gallons.Hectolitre(= 100 litres)22·009668”Kilolitre(= 1000 litres)220·09668”Décilitre(= ¹∕₁₀th of a litre)·17607pint.Centilitre(= ¹∕₁₀₀th of a litre)·017607pint.