•. V = fyda = 20 fada - «fa'da

x1dx

segment = (6a—2x) +2 = 7 (3d—2x) z', io d= 2.

sphere. d3, by making x = d.

(3.) Find the volume of the paraboloid.

Here y2 = 4mx, is the equation to the generating parabola

But y2x volume of a cylinder, whose base = = y2 and height:

1

.. volume of paraboloid = || volume of circumscribing cylinder.

(4.) Find the content of the prolate spheroid formed by the revolution of a semi-ellipse round its major axis.

(5.) Find the content of the oblate spheroid formed by the revolution of s semi-ellipse round its minor axis.

Hence prolate spheroid : oblate spheroid :: ab2 : a2b :;b: a

.. oblate spheroid : sphere on minor axis::ab:¿1

MECHANICS.

DEFINITIONS AND FUNDAMENTAL NOTIONS.

1. Mechanics is the science which treats of the laws of rest and motion of bodies, whether solid or fluid, and is usually divided into the four following branches:

(1.) Statics, which treats of the laws of forces in equilibrium.

(2.) Dynamics, which treats of the laws of motion of solid bodies.
(3.) Hydrostatics, of the laws of the equilibrium of fluid bodies.
(4.) Hydrodynamics, of the laws of motion of fluid bodies.

2. Force or power is the cause which produces, or tends to produce, motion in a body, or which changes, or tends to change, motion.

3. A body is a portion of matter limited in every direction, and is therefore of a determinate form and volume.

4. All bodies have a tendency to fall to the earth; and the force which they exert in consequence of this tendency is called their weight.

5. When forces are applied simultaneously to a body, and produce rest, they balance each other, or destroy each other's effects; and therefore such forces are said to be in equilibrium.

6. The measure of a force, in statics, is the weight which that force would support.

7 The quantity of matter of a body is proportional to its weight.

8. The density of a body is measured by the quantity of matter contained in a given space.

9. Gravity is that force by which a body endeavours to fall downwards. 10. Specific gravity is the relation of the weights of different bodies o equal magnitude, and is therefore proportional to the density of the body.

STATICS.

THE COMPOsition and EQUILIBRIUM OF FORCES ACTING ON A

MATERIAL PARTICLE.

11. Def. The resultant of any number of forces is that singie torce which is equally effective with, or equivalent to, all the forces, and these forces are termed component or constituent forces.

РХОР. І.

12. To find the resultant of a given number of forces acting on a pariale in the same straight line.

The resultant of two or more forces acting on a particle in the same direc tion is equal to their sum, and acts in the same direction; but the resultant of two forces acting in opposite directions is equal to their difference, and acts in the direction of the greater component. Also, if several forces act in one direction, and others in a contrary direction, the resultant of all these forces will be equal to the excess of the sum of the forces acting in one direction, above the sum of those acting in the contrary direction, and it will act in the direction of the greater of these sums.

PROP. II.

13. To find the resultant of two forces acting on a particle not in the same straight line.

1. To find the direction of the resultant of two forces acting on a point. When the forces are equal, it is obvious that the direction of the resultant will bisect the angle between the directions of the forces; or if the two forces be represented in magnitude and direction by two lines drawn from the point where they act, the diagonal of the rhombus described on these equal lines will be the direction of the resultant.

Assuming that the diagonal of a parallelogram described on the two lines representing the forces in magnitude and direction is the direction of the resultant; then if p, p, be any two unequal forces, and p, p, also two unequal forces, we can prove that the direction of the resultant of the two forces p and pp, is the diagonal of the parallelogram whose adjacent sides are p and p1+P2.

Let A be the point on which two forces p and p, act; AB, AC, their directions and proportional to them in magnitude. Complete the parallelogram BC, and draw the diagonal AD; then, by hypothesis, the resultant of p and p, acts in the direction of AD.

Again, produce AC to E, and take CE a fourth proportional to p1, P2, and AC; that is, make p1. P2:: AC: CE. Now, since the point of application of a force may be transferred to any point of its direction, without disturbing the equilibrium, so long as the two points of application are invariably connected, we may suppose the force p, to act at A or C, and therefore the forces p, p, p, in the lines AB, AC, CE, are the same as p and P+P2, in the lines AB and AE.

Now replace p and p, by their resultant, and transfer its point of application from A to D; then resolve this force at D into two, parallel to AB and AC; these resolved parts must evidently be p and p1, where p acts in the direction DF and p, in the direction DG. Transfer these two forces p to C and p, to G; but by the hypothesis p and p, acting at C have a resultant in the direction CG; let, therefore, p and p, be replaced by their resultant, and transfer its point of application to G. But p, acts at G, and therefore by this process we have, without disturbing the equilibrium, removed the forces p and and p1 +Pa, which acted at A to the point G; hence the resultant of p and

P1 +P2 acts in the direction of the diagonal AG, provided our assumption is correct. Now the hypothesis is correct for equal forces as p, p, and therefore it is true for forces p, 2p; consequently for p, 3p, and thus it is true for p, mp. Again, if it be true for p, mp, and p, mp, it is is also true for 2p, mp; also for 3p, mp, and thus it is true for np, mp, where n and m are positive integers.

D

E

K

We have now to show that the proposition is true for incommensurable forces. Let AB, AC represent two such forces, and complete the parallelogram BC. Then if their resultant do not act along AD, suppose it to act along AE, and draw EF parallel to BD. Divide AB into a number of equal parts, each less than DE; divide CD into F parts equal to these, and let G be the last point B of division of the former, which will obviously fall between B and F. Draw GK parallel to BD; then two forces represented by AC, AG, have a resultant in the direction AK, because they are commensurable; but this is nearer to AG than the resultant of the forces represented by AC, AB, which is absurd, since AB is greater than AG. In the same manner we may show that every direction besides AD leads to an absurdity, and therefore the resultant must act in the direction AD, whether the forces be commensurable or incommensurable.

2. To find the magnitude of the resultant.

Let AB, AC be the direction of the given forces, AD that of their resultant; take AE in the prolongation of DA, and of such a length as to represent the magnitude of the resultant; then the forces represented by AB, AC, AE balance each other. Complete the parallelogram BE, and therefore AF is in the same straight line with AC, since the forces AB, AC, AE, balance each other; hence FD is a parallelogram, and therefore AD=FB = AE; that is, the resultant is represented in magnitude as well as in direction by the diagonal of the parallelogram.*

F

B

Cor. 1. The forces in the directions AB, AC, AD, are respectively proportional to the lines AB, AC, AD, and in these directions.

Cor. 2. The two oblique forces AB, AC, are equivalent to the single direct force AD, which may be compounded of these two, by drawing the diagonal of the parallelogram. Or, they are equivalent to the double of AE drawn to the middle of the line BC.

B

And thus any force may be compounded of two or more other forces; which is the meaning of the expression, composition of forces.

Example.-Suppose it were required to compound the three forces AB, AC, AD; or to find the direction and quantity of one single force, which shall be equivalent to, and have the same effect as if a body at A were acted on by three forces in the direction AB, AC, AD,

D

E

The preceding demonstration of the parallelogram of forces is due to M. Duchayla, and is exceedingly simple and beautiful. Analytical demonstrations of this fundamental property have been given by Laplace, Pontécoulant, Poisson, and others; but want of room prevents us from giving them here.

and proportional to these three lines. First, reduce the two, AC, AD, to one AE, by completing the parallelogram ADEC. Then reduce the two, AE, AB, to one AF, by the parallelogram AEFB. So shall the single force AF be the direction, and as the quantity, which shall of itself produce the same effect, as if all the three, AB, AC, AD, acted together.

Cor. 3. Any single direct force AD may be resolved into two oblique forces, whose quantities and directions are AB, AC, having the same effect, by describing any parallelogram whose diagonal may be AD; and this is called the resolution of forces. So the force AD may be resolved into the two, AB, AC, by the parallelogram ABCD; or into the two AE, AF, by the parallelogram

E

AEDF; and so on for any other two. And each of these may be resolved again into as many others as we please.

PROP. 111.

14. If three forces, A, B, C, acting together, keep one another in equilibrio, they will be proportional to the three sides DE, CE, CD, of a triangle, which are drawn parallel to the directions of the forces AD, DB, CD.

Produce AD, BD, and draw CF, CE, parallel to then. Then the force in CD is equivalent to the two AD, BD, by the supposition; but the force CD is equivalent to the two, ED and CE or FD; therefore, if CD represent the force C, ED will represent its opposite force A, and CE or FD its opposite force B; consequently, the three forces A, B, C, are proportional to DE, CE, CD, the three lines parallel to the directions in which they act.

B

Cor. 1. Because the three sides CD, CE, DE, are proportional to the sines of their opposite angles E, I,C, therefore, the three forces, when in equilibric, are proportional to the sines of the angles of the triangle made of their lines of direction; namely, each force proportional to the sine of the angle made by the directions of the other two.

Cor. 2. The three forces, acting against, and keeping one another in equilibrio, are also proportional to the sides of a triangle made by drawing lines either perpendicular to the directions of the forces, or forming any given angle with those directions. For, such a triangle is always similar to the former, which is made by drawing lines parallel to the directions; and therefore their sides are in the same proportion to one another.

Cor. 3. If any number of forces be kept in equilibrio by their actions against one another, they may be all reduced to two equal and opposite ones. For, by Cor. 2, Prop. II., any two of the forces may be reduced to one force acting in the same plane; then this last force and another may be likewise reduced to another force acting in their plane: and so on, till at last they be all reduced to the action of only two opposite forces, which will be equal, as well as opposite, because the who'e are in equilibrio by the supposition.