the surface of the sphere, and their apices at its centre. Hence, from

V

=

S base by L altitude,

since the surface and the bases, and the radius and the altitude coincide ultimately, we deduce

ART. 96.-Volume of an Ellipsoid. The volume of an ellipsoid is given by

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
6

=

(L long diameter)2 by L thick diameter.

If the two equal diameters are less than the third, the spheroid is said to be prolate, and

П V = L long diameter by (L broad diameter)?.

6

ART. 97.-Change of Volume. A volume may expand in three independent directions. Suppose that the several ratios of

linear expansion are

1+ a L expanded length = L original length,
1+BL expanded breadth = L original breadth,
1+ y 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.,

(1)

and

{(1+a) (1+B)(1 +7) - 1} L3 increment = L3 original vol. The reciprocal of (1) is

1

L3

(2)

(1 + a) (1 + B) (1 + y) L3 original vol. = L3 expanded vol.; (3)

and the rate of diminution.

1

1

( + a) (1 + ẞ) (1 + y)'

La decrement L3 expanded vol. (4)

===

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 + ẞ+y; of (3) as 1 of (4) as a + B+ y.

When, further, a = ß=y, the values of (2) and (4) become 3ɑ, 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 = 18.)

For a circular cylinder,

Take logs of the upper factors by themselves and of the lower factors by themselves, add each column, and subtract.

Ex. 2. A circular plate of lead, 2 inches in thickness and S inches in diameter, is converted without loss into spherical shot of the same density, and each of 05 inch radius. How many shot does it make?

As the density of the plate and of the shot is the same, we require to consider the volume of the materials only.

EXERCISE XIV.

1. Find the volume of a cone whose altitude is 2 feet, and the diameter of the base 1 foot 6 inches.

2. Find the surface of a sphere which is one yard in diameter.

3. Find the radius of the sphere the volume of which is equal to the sum of the volumes of two spheres, whose radii are 3 feet and 4 feet.

4. The area of the base of a cylinder is 2 square feet and its height 30 inches; find the height of a cylinder the solid content of which is three times as great, but whose diameter is only two thirds of the given one.

5. If the volume of the first of two cylinders is to that of the second as 10 to 8, and the height of the first is to that of the second as 3 to 4, and if the base of the first has an area of 165 square feet, what is the area of the base of the second?

6. Determine the volume of the earth, supposing its diameter to be 8,000 miles. How many masses of the size of the earth would make up one of the size of the sun, the diameter of which is 880,000 miles?

7. Two spheres, A and B, have for radii 9 feet and 40 feet; the superficial area of a third sphere Cis equal to the sum of the areas of A and B. Calculate the excess in cubic feet of the volume of C over the sum of the volumes of A and B.

8. A cone and hemisphere being supposed to have a common base and to lie at opposite sides of it; required, the ratio of the altitude of the cone to the radius of the hemisphere, in order that the volumes of the two solids should be equal.

9. Determine approximately the length of the radius of the sphere whose volume is 400 cubic feet.

10. A sphere and a cube have an equal amount of surface; what is the ratio of their volumes?

11. If a model is made on the linear scale of 1/40 inch to the foot; what is the scale for surface and for volume?

12. The altitude of a common cone equals the circumference of its base. Calculate the volume and the area of the whole surface of the cone, the radius of the base being 6 inches.

13. The interior of a building is in the form of a cylinder of 30 feet radius and 12 feet altitude, surmounted by a cone whose vertical angle is a right angle. How many cubic feet of air will it contain?

14. The long axis of a spheroid is 10 inches, and each short axis 6 inches. Find its volume.

CHAPTER THIRD.

KINEMATICAL.

SECTION XV.-TIME.

ART. 98.-Sidereal Units. The idea of time is fundamental. The general unit is appropriately denoted by T.

The standard of time adopted by all civilized nations is the time occupied by the earth in making one rotation about its axis. This interval is marked out by the successive transits of a particular star across the meridian of a place, and it is on that account denominated a sidereal day. The sidereal day is divided into 24 sidereal hours, the sidereal hour into 60 sidereal minutes, and the sidereal minute into 60 sidereal seconds. The sidereal units are used principally by astronomers.

ART. 99.—Mean Solar Units. By a year is meant the constant interval occupied by the earth in making a revolution round the sun; it is marked out by the sun leaving and returning to a certain position among the stars. The transit of the centre of the sun across the meridian of a place marks out an interval called the apparent solar day. This apparent day is not completely constant; its length goes through a cycle of small changes in the course of a year. Its average length for the course of a year is called the mean solar day. This interval of the mean solar day is measured out by clocks and chronometers, corrected by observation, on the part of astronomers, of sidereal time.