6. To Find the Volume of a Frustrum of a Pyramid or of a Cone.

RULE.-To the areas of the two ends add the square root of their product; multiply the sum by the height of the frustrum, and divide the product by 3.

Volume of frustrum of right cone = [ "(r," + r,') + √π(r," + r.”) ] 1⁄2

3

Or, if we know the height, h1, of the complete cone, as well as the height, h,, of the top cone cut away, then the difference between the volume of the complete cone and that of the smaller part cut away leaves the volume of the frustrum.

7. To Find the Volume of a Sphere.

RULE-Take of the area of the circumscribing circle, and multiply this by the radius.

For the purpose of having a combined ready reference to the different formule for the various rules given in the preceding lectures, we now arrange them in the form of tables, adopting the same notation throughout.

LECTURE III.-QUESTIONS.

1. Find the weights of a cubic inch and of a cubic foot of (1) cast iron; wrought iron; (3) steel, mild; (4) steel, tempered. Ans. (1) 255; 443'75; 277; 481 25; (3) 277; 48125; (4) 28; 490'6 lbs.

2. Find the weight of a cubic inch and of a cubic foot of (1) copper, cast; (2) copper, hard wire; (3) brass, cast; (4) lead; (5) mercury. Ans. (1) ·3; 537 5; (2) 31; 550; (3) 3; 525; (4) 41; 712'5; (5) 49; 848.75 lbs.

3. Find the weight of a cubic foot; also how many cubic feet make up a ton weight in the case of (1) Scotch, (2) Welsh, (3) Newcastle, (4) Lancashire coals. Ans. (1) 79; 28'4; (2) 81 ; 27·5; (3) 78; 28·8 ; (4) 79.5; 28′2. 4. Find the weight of a cubic foot of steam at atmospheric pressure from the Specific Gravity Table. Ans. 038 lb.

5. How many cubic inches and cubic feet are there in a rectangular box which is 2 feet long, 18 inches wide, and I foot deep? Ans. 5184 cubic inches; or 3 cubic feet.

6. What is the capacity of a rectangular tank 3' deep, 15' long, 27" wide? Ans. 101 25 cubic feet.

7. A tank 4 feet square in horizontal section is sounded by a rod, and there is found to be a depth of 3' 3" of water in it. How many pounds and gallons of fresh water are there in the tank? Ans. 3250 lbs.; 325 gallons.

8. You are supplied with 100 gallons of oil. What will be its weight, and its bulk in cubic feet? Ans. 920 lbs.; 16 cubic feet.

9. A wrought-iron cap for the cover of a marine crank-shaft bearing is 16" by 121" by 3". Find its weight. Ans. 194 lbs.

10. What weight of coal in tons will a bunker hold which is 18' by 5'6" by 12', allowing 45 cubic feet to the ton for coals stored in this way? Ans. 26'4 tons.

11. How many cubic feet of steam will be required per minute for a steam-engine cylinder 60" diameter; steam being cut off at 1' of stroke when the engine is making 90 strokes per minute? Ans. 1767.

12. In the last question, suppose the engine develops 60 horse-power with steam supplied at 50 lbs. pressure, which weighs 12 lb. per cubic foot. What weight of steam would be required per hour per horse-power? Ans. 212 lbs.

13. Find how many cubic feet of steam will exhaust from a steam-engine cylinder per hour, and also the weight of this steam at atmospheric pres sure (see Sp. Gr. Table), if the diameter of the cylinder be 60", stroke 6 feet when the engine makes 90 strokes per minute. Ans. 636,174 and 24,175.

14. Find the weight of a wrought-iron shaft of the dimensions and shape given in Example VII. of this lecture. Ans. 3 tons 14 cwt. 24 lbs.

15. Find the weight of a circular plate of wrought iron 3 feet in diameter and " thick. Ans. 1945 lbs.

16. A feed-pump is 4" diameter and 14" stroke, but it is only full each stroke. Find the number of gallons and weight of water which it pumps into a boiler when making 72 strokes per minute. Ans. 34'36 gallons per minute; or, 343'6 lbs.

17. Find the weight of the brass liner to an air-pump which is 24" internal diameter, 24" long, and ğ" thick. Ans. 3512 lbs.

18. Find the weight of the rim of a cast-iron fly-wheel of rectangular parallelogram section outside diameter 6', inside diameter 4' 6", breadth 9". Ans. 4116.6 lbs.

19. Find the total weight of the crank-shaft of the s.s. City of Rome. There are three cranks of the shape and dimensions given in the following

<------ 45 ------>

gure, and they are made of Whitworth's compressed steel, the specific gravity of which may be reckoned at 78. Sketch each separate part in your exercise-book before your calculation of the volume thereof, and sum up the whole in a neat table. Ans. Weight reported by owners to be 63 tons, weight by careful calculation from this figure 62 tons.

20. Find the weight of a spherical cast-iron ball (used as a safety-valve balance weight) which is 10" inches in diameter. Ans. 1314 lbs.

LECTURE IV.

CONTENTS.-Temperature-Thermometers-Pyrometers.

It is necessary, at the very outset of this section of our subject, to clearly understand what is meant by the different expressions : 1. The temperature of a body.

2. The quantity of heat in a body, and the unit of heat. 3. The capacity for heat, and the specific heat of a body.

Temperature. The temperature of a body is its thermal state considered with reference to its power of communicating heat to other bodies (MAXWELL).

Two bodies are said to be at the same temperature if, when placed in thermal communication, there is no tendency to the transfer of heat between them; if, however, one of them loses heat, and the other gains heat, that body which gives out heat is said to have a higher temperature than that which receives heat.

Temperature, therefore, indicates the quality or condition of the heat in bodies, and is capable of greater or less intensity according to circumstances. It is measured by Thermometers and Pyrometers.

Thermometry.-Thermometry is the method of ascertaining temperatures, or the intensities of heat. The action of thermometers is based on the change of volume to which bodies are subject with a change of temperature. Air, water, spirit, and mercurial thermometers are severally used under different circumstances, but the mercurial thermometer is the one most commonly employed by engineers. The mercurial thermometer consists of a stem or tube of glass, formed at one end into a bulb, to contain the mercury which expands into the tube (see left-hand figure). If the stem be of uniform bore, the expansion of the mercury being practically equal for equal increments of temperature, it follows that, if the scale be uniformly graduated, the divisions will indicate equal increments of temperature.

It was early ascertained that the freezing and the boiling points of water at the normal pressure of the atmosphere (viz., 14.7 lbs. on the square inch) were constant temperatures, and advantage is taken of this physical property in order to graduate thermometers. The interval between these two fixed temperatures is