When V is not defined as L3, we have some number instead of 1. This rate may be written as an equivalence, 1 V = L long by L broad by L thick. For example, in a rectangular slab the number of feet in the thickness is the same as the number of cubic feet per square foot of surface. In the case of a rectangular rod, the number of square inches in the cross section is the same as the number of cubic inches per inch of length. When the parallelepiped is not rectangular, let the length and breadth be inclined at an angle of 0°, and the thickness be inclined to the base at an angle of 4°, then sin 0 sin V = L long by L broad by L thick. EXAMPLES. Ex. 1. Given the rate of exchange 25-22 francs per £, and that 4.54 litres are equivalent to 1 gallon; deduce the relation of francs per litre to £ per gallon. 4.54 litres = 1 gallon; .. dividing the one equivalence by the other, Ex. 2. A schoolroom is 40 ft. long, 26 ft. 6 in. broad, and 19 ft. 3 in. high. If 80 cubic feet of space and 8 square feet of floor must be provided for each scholar, what is the maximum number of scholars which the room can provide for? Hence the greatest number of scholars provided for is 132. EXERCISE XIII. 1. Express the litre in terms of the cubic foot. 2. An imperial gallon is 277 274 cubic inches; a Winchester bushel 2150 42 cubic inches; how many Winchester bushels are equal to 100 imperial bushels? 3. Reduce 85 litres to gallons, and 54 gallons to litres. 4. By how much does a quart exceed a litre ? 5. Reduce 234 hectolitres to bushels, and 432 bushels to hectolitres. 6. What is 1.05 dollars per bushel in terms of shilling per quarter? 7. What is one shilling and sixpence a gallon in francs per litre, taking exchange at 25 francs per £? 8. A gas stove burns 7 cubic feet per hour, and the cost of the gas is 4s. per 1,000 cubic feet. Find the hourly cost. 9. A rectangular block of stone is as broad as it is long, and contains a cubic feet. If it were as broad as it is high the bulk would be b cubic feet. Find the length. 10. If a wall, 42 feet long, 10 feet high, and 2 feet thick, contains 12,800 bricks, how many bricks of the same kind will be required for a wall 112 feet long, 6 feet high, and 2 feet thick? 11. If 81 gallons of water will fill a cistern 4 feet 4 inches long, 2 feet 8 inches broad, and 1 foot 1 inch deep, how many cubic inches are contained in a pint? 12. The price of wheat is 36s. 6d. per quarter; express it in terms of francs per hectolitre, the rate of exchange being 25 30 francs = £. 13. The price of brandy is 52 centimes per litre ; express it in pence per gallon, the rate of exchange being 25 80 francs = £. 14. By taking the decimetre as equal to four inches, what percentage of error is introduced, first, in linear measure; second, in square measure; third, in cubic measure. Take the legal equivalent as the true length of the metre? 15. Find the length of the side of a cubical vessel that shall contain twice as much as one whose side is 8 inches. SECTION XIV.—VOLUME, Continued. ART. 94.-Volume of a Cylinder, a Cone, a Pyramid. The volume of a cylinder is given by 1 V = S base by L altitude, where V denotes L3 and S denotes L2. In the case of a cone the altitude is variable, but it can be shown that the average altitude is of the greatest altitude. Hence VS base by L greatest altitude. In a pyramid the altitudes vary in the same manner as in a cone. Hence V = S base by L greatest altitude. ART. 95.-Volume of a Sphere. A sphere may be considered as made up of a great number of pyramids having their bases on the surface of the sphere, and their apices at its centre. Hence, from VS base by L altitude, and 4 S surface = since the surface and the bases, and the radius and the altitude coincide ultimately, we deduce for which, approximations are 11 V 21 (L diameter)3, = ART. 96.-Volume of an Ellipsoid. The volume of an ellipsoid is given by π 6 V L long diameter by L broad diameter by L thick diameter. = In the case of a spheroid, two diameters are equal. If the two equal diameters are each greater than the third, the spheroid is said to be oblate, and Π V = (L long diameter)" by L thick diameter. If the two equal diameters are less than the third, the spheroid is said to be prolate, and π 6 V = L long diameter by (L broad diameter)2. A volume may expand in Suppose that the several ratios of ART. 97. Change of Volume. three independent directions. linear expansion are 1+a L expanded length = L original length, 1+BL expanded breadth = L original breadth, 1+7 L expanded depth = L original depth; G then, if the included angles remain constant, (1 + a) (1 + ẞ) (1 + y)L3 expanded vol. = L3 original vol., and {(1 + a) (1 + ẞ) (1+ y) −1} L3 increment = L3 original vol. The reciprocal of (1) is 1 (1) (2) 3 L3 original vol. L3 expanded vol.; (3) (1 + a) (1 + ẞ) (1 + y) and the rate of diminution = a-ẞ-y; and If a, ẞ, y is each a small fraction, the value of (1) may be taken as 1+ a + ẞ+y; of (2) as a + B +y; of (3) as 1of (4) as a +B+y. When, further, a = ß=y, the values of (2) and (4) become 3a, that is, 3 times the value of the linear rate of expansion. EXAMPLES. Ex. 1. Calculate the volume of a granite monument, consisting of a right cylindrical shaft 8 feet high, surmounted by a right circular cone 5 feet high, the common radius of the cone and cylinder being 2 feet. (Take = {{}.) For a circular cylinder, |