The curved surface is obtained by multiplying half the slant side by the periphery of the

base.

If the base be not circular, but any regular rectilinear figure, the solid is a regular pyramid, and the rule as to volume applies, substituting parallelopipedon for cylinder, i.e.

Volume area of base x height.

=

Fig. 20.

Each side is a triangle and its area can be calculated separately, and hence the surface obtained.

EXAMPLES.

1. Find the weight of a boiler plate, 10 ft. 6 in. by 4 ft. 9 in., weighing 35 lbs. per sq. ft. Ans. 15 cwts. 65 lbs.

2. A triangular piece of plate, sides 3 ft. 6 in., 4 ft., and 4 ft. 6 in. respectively, weighs I cwt. Determine the weight per sq. ft. Ans. 17 lbs.

3. The cross-section of a canal is a trapezoid. Width at top 42 ft., at bottom 28 ft., depth 10 ft.

Find its area.

Ans. 350 sq. ft.

4. The sides of a field are AB 150, BC 42, CD 130, DE 72, and EA 50, all in yds. Also BE is 1571, and BD 156 yds. There are no re-entrant angles. Find its area. Ans. 2.4 acres. 5. A boiler contains 530 tubes, 2 in. internal diameter. Find area for draught through them. Ans. 21.87 sq. ft.

6. Each tube in the preceding is 7 ft. long between tube plates, and 3 in. external diameter. Find the total heating Ans. 2915 sq. ft.

surface.

7. The tube plates in (6) are in. thick, and each tube projects in. outside each plate. The tubes are of wrought-iron, weight of a cubic inch lb. Find the weight of one tube. Ans. 27 lbs.

8. A boiler is 10 ft. diameter, 12 ft. long; the water surface is at of the height. Find the volume of the steam space.

Ans. 185 c. ft.

9. Find how many sq. yds. of non-conducting covering material will be required to cover the cylindrical portion, and one end, of the preceding boiler. Ans. 506.

10. The circular shell plates being 1 in. thick, and the end plates; material mild steel, weighing the same as wroughtiron. Find the total weight of the shell, not taking laps or holes for furnaces into account. Ans. 8.65 tons.

II. Find the weight of a cast-iron hexagonal column, 12 ft. high, 8 in. outside across. the corners, circular hole 5 in. diameter. Weight, lb. to 1 c. in. Ans. 788 lbs.

12. Construct a parabola on an 8-in. base, 3 in. high. Ist, symmetrical; 2d, with greatest height at 5 in. from one end. 13. Find the area of the preceding curves; and also by Simpson's rule.

14. An elliptical door is 18 in. by 16 in.; iron; mean thickness in. Find its weight.

by calculation, Ans. 16 sq. in.

material castAns. 42 lbs.

15. A hollow steel propeller shaft is 10 ft. long over all; flange at each end 23 in. diameter, 4 in. thick; external diameter of plain part 14 in., diameter of hole 9 in. Find its weight. Ans. 3593 lbs.

16. The total depth of a cast-iron beam is 12 ins. The top flange is 6" by 3", the bottom 9′′ by 1′′, and the web is " thick. Find its weight per foot length.

Ans. 55 lbs.

PART I

THE PRINCIPLE OF WORK

CHAPTER I

MOTION-SLIDING, TURNING, AND SCREW PAIRS

WHEN two bodies shift their relative positions they are said to be in motion relatively to each other. Thus, a train shifts its position relatively to the earth; then the earth and the train are in relative motion.

The usual mode of expression in the preceding case would be to say that the train was in motion, but a little consideration shows that this is not a full statement. To a spectator on the earth, the train alters in position while the earth does not. Therefore he says the train moves. But to a traveller in the train it is the earth that moves while the train is still; so that, to be consistent, he should say the earth is moving. But actually he would still say that the train moved. If we examine more closely, we see that the reason of this clearly is that in each case the earth is tacitly treated as if it were a fixed body, which for all ordinary purposes it may be assumed to be. But now when we come to examine larger motions, as those of the heavenly bodies, we know that the earth is not treated at all as a fixed body, but is in motion relative to the sun and to all the other bodies. Questions relative to the solar system are sometimes treated as if the sun were fixed; but this again will not do when we have to consider the motions of the so-called fixed stars.

We see then that there is in nature no such a thing as an actual fixed body. And so we cannot speak of

C