EXAMPLE VII.-The cross area of a cylinder is 19635 square inches. What is its diameter in feet and inches?

Or, by using the fraction (22), and dividing the 22 and the 1963.5 by 2, and then dividing numerator and denominator by 11, and multiplying by the 7, we get ".

./624-75 = 25" . d = 50 inches nearly (as before).

Note. It is often of importance in such questions to get the answer quickly and near enough to the truth without going through an elaborate series of multiplications and divisions by long numbers (such as multiplying or dividing by 3'1416). We consequently recommend the above method of cancelling by easy stages. If the student should happen to have a logarithm book, and know how to use the tables, he will find that he can work such questions very easily and quickly; but he must remember that logarithm books are not usually permitted in written examinations.

EXAMPLE VIII.-Steam is admitted into the cylinder of a steam engine whose diameter is 2 feet 4 inches, at a pressure of 50 lbs. per square inch. What is the total pressure on the piston, if the diameter of the piston-rod on the side upon which the steam is admitted be 3 inches?

In this question, the first thing to be done is to find the nett area upon which the steam acts; in other words, we have to subtract the cross area of the piston-rod from the cross area of the cylinder.

.. 50 (lbs.) × 608-68 (square inches) = 30,434 lbs. total pressure.

5. To Find the Surface Area or Envelope of a Sphere. DEFINITION.--A sphere is a perfectly round body, every point on the surface of which is equidistant from the centre.

RULE.--The surface of a sphere is equal to the concave surface

of the circumscribing cylinder; consequently,

Multiply the square of the diameter by 3*1416, or by

22

7

EXAMPLE IX.-A spherical cast-iron ball, 8 inches diameter, is used as a counter-weight to a reversing lever, and, in order to keep it bright and clean, it is desired to electro-plate it with nickel plating, which costs 5s. per square foot. What will be the cost?

6. To Find the Circumference of an Ellipse.

DEFINITION.—An ellipse is a plane curve formed by cutting a right circular cylinder by a plane inclined to the axis, which plane does not meet the base of the cylinder.

RULE.-Multiply π or (3°1416) by half the sum of the two dia

The superficial area or envelope of an elliptical cylinder is × 7, where l is the length of the axis of the

therefore =π

cylinder.

(d1 + d2

2

EXAMPLE X.-What is the area in square feet of the elliptical manhole doors of the s.s. St. Rognvald, whose longer diameter is 16 inches and shorter diameter is 12 inches?

LECTURE II.-QUESTIONS.

1. A furnace door is of this shape and size,

H

2'

(2 feet long, I foot high at ends, and 1 foot 3 inches high in middle). Find its area in square feet. Ans. 2.25 sq. ft.

I

2. Find the area in square inches, by Simpson's rule, of the diagram illustrated at p. 114. Ans. 1050 sq. ins. (taking pressures marked as

inches).

...

3. Referring to the diagram at p. 114, let the length of the figure be 2 feet, and the various ordinates, 65, 60, 46, . 17, represent inches. Find the area of the figure in square feet. Ans. 7.25 sq. ft. 4. A cylinder is 53" diameter. Find the circumference. Ans. 166.5 inches.

5. What is the girth of a circular steam pipe whose diameter is 10"? Ans. 32'98 inches.

6. The girth of a circular furnace tube, measured by a string, is found to be 3' 6". What is its radius? Ans. 6.69 inches.

7. The inside diameter of a furnace tube is 3' 11", and the thickness of the metal is ". Find the outside circumference. Ans. 10'08 feet, or Io' of" full.

8. The funnel of a steamship is 5 feet diameter, and is made of three plates in girth; the lap of each plate is 1 inch. Find the width of each plate. Ans. 5'38 feet.

9. The outside diameter of the flange of a cylinder-cover is 6 feet; the pitch between the bolt-holes is to be 63 inches. How many bolts will be required if the pitch-circle be 2 inches from the outside edge of the flange? Plot this out to scale. Ans. 33 bolts.

10. What is the area of the rubbing surface in a cylinder, the diameter of the piston being 52 inches and the stroke 3 feet. Ans. 40 84 sq. ft. II. How many square feet of iron plate are there in a ship's circular funnel, 4' 6" diameter and 26 long? Ans. 367.5.

12. The diameter of a cylinder is 25.5 inches. Ans. 510 7 square inches.

13. The area of a cylinder is 2002'96 square inches. Ans. 25 inches.

What is its area?

What is its radius ?

14. The cylindrical boilers of the s.s. St. Rognvald are each, say, 15 feet internal diameter and 10' 3" long inside. Find the total pressure on the back end and on the barrel of the boiler with 90 lbs. pressure of steam per square inch. Ans. 2,290,226.4 lbs. on back end, 6,259,952 on barrel.

15. The diameters of the high and low pressure cylinders of the s.s. St. Rognvald (Lecture XX.) are 36" and 70". The diameters of the piston-rods are 6" below and 44" above the pistons in each case. What is the effective area for the steam-pressure to act on, in each case? Ans. H.P. cyl. 984'68 square inches below and 1001'97 above; L.P. cyl. 3815'28 square inches below, and 3832 56 above.

16. An elliptical manhole is 164" longer diameter, and 12" shorter diameter. What is the circumference and area of the whole in square feet? The wrought-iron strengthening ring, which is flush all round with the manhole on the inside, is 4" broad. What is its outer circumference and its superficial area? Ans. Oc=3'73 ft., area=1'08 sq. ft. Outer Ooo= 5·825 ft. superficial area=1'6 sq. ft.

02

LECTURE III.

CONTENTS.-Table of Weights and Measures-Table of Specific Gravities of Metals, Coals, Woods, &c.-Volumes of a Rectangular Solid or a Tank, Parallelopiped, Prism, Cylinder, Cone, Frustrum, and a Sphere -Surface of a Cone and a Frustrum-Examples.

HAVING explained and worked out a series of examples on Linear and Superficial or square measure in the two first lectures, we are now prepared to deal with the capacities or volumes or contents of solids, and their weights. We, therefore, begin by giving

of the most useful and common tables of weights and measures used by British mechanical engineers.

B.-TABLE OF SPECIFIC GRAVITIES.

RULE 1.-To find the weight of a CUBIC INCH of any of the substances mentioned in this Table, multiply its specific gravity by 036, the weight of a cubic inch of fresh water.

RULE 2.-To find the weight of a CUBIC FOOT of any of the

* This number is used for finding the displacement of a vessel in seawater or the weight of sea-water ballast.

Specific gravity is the ratio of the weight of a given bulk of a given substance to the weight of the same bulk of pure water, at a standard temperature (62° Fahr. in Great Britain). The above table of specific gravities gives only rough averages, and, therefore, must not be adopted for scientific or very exact calculations.