(147.) By the earth’s attraction, all the particles which compose the mass of a body are solicited by equal forces in parallel directions downwards. If these component particles were placed in mere juxtaposition, without any mechanical connection, the force impressed on any one of them could in nowise affect the others, and the mass would, in such a case, be contemplated as an aggregation of small particles of matter, each urged by an independent force. But the bodies which are the subjects of investigation in mechanical science are not found in this state. Solid bodies are coherent masses, the particles of which are firmly bound together, so that any force which affects one, being modified according to circumstances, will be transmitted through the whole body. Liquids accommodate themselves to the shape of the surfaces on which they rest, and forces affecting any one part are transmitted to others, in a manner depending on the peculiar properties of this class of bodies.

As all bodies, which are subjects of mechanical enquiry, on the surface of the earth, must be continually influenced by terrestrial gravity, it is desirable to obtain some easy and summary method of estimating the effect of this force. To consider it, as is unavoidable in the first instance, the combined action of an infinite number of equal and parallel forces soliciting the elementary molecules downwards, would be attended with manifest inconvenience. An infinite number of forces, and an infinite subdivision of the mass, would form parts of every mechanical problem.

To overcome this difficulty, and to obtain all the ease and simplicity which can be desired in elementary investigations, it is only necessary to determine some force, whose single effect shall be equivalent to the combined effects of the gravitation of all the molecules of the body. If this can be accomplished, that single force might be introduced into all problems to represent the whole effect of the earth’s attraction, and no regard need be had to any particles of the body, except that on which this force acts.

(148.) To discover such a force, if it exist, we shall first enquire what properties must necessarily characterise it. Let A B, fig. 37., be a solid body placed near the surface of the earth. Its particles are all solicited downwards, in the directions represented by the arrows. Now, if there be any single force equivalent to these combined effects, two properties may be at once assigned to it: 1. It must be presented downwards, in the common direction of those forces to which it is mechanically equivalent; and, 2. it must be equal in intensity to their sum, or, what is the same, to the force with which the whole mass would descend. We shall then suppose it to have this intensity, and to have the direction of the arrow D E. Now, if the single force, in the direction D E, be equivalent to all the separate attractions which affect the particles, we may suppose all these attractions removed, and the body A B influenced only by a single attraction, acting in the direction D E. This being admitted, it follows that if the body be placed upon a prop, immediately under the direction of the line D E, or be suspended from a fixed point immediately above its direction, it will remain motionless. For the whole attracting force in the direction D E will, in the one case, press the body on the prop, and, in the other case, will give tension to the cord, rod, or whatever other means of suspension be used.

(149.) But suppose the body were suspended from some point P, not in the direction of the line D E. Let P C be the direction of the thread by which the body is suspended. Its whole weight, according to the supposition which we have adopted, must then act in the direction C E. Taking C F to represent the weight; it may be considered as mechanically equivalent to two forces (74), C I and C H. Of these C H, acting directly from the point P, merely produces pressure upon it, and gives tension to the cord P C; but C I, acting at right angles to C P, produces motion round P as a centre, and in the direction C I, towards a vertical line P G, drawn through the point P. If the body A B had been on the other side of the line P G, it would have moved in like manner towards it, and therefore in the direction contrary to its present motion.

Hence we must infer, that when the body is suspended from a fixed point, it cannot remain at rest, if that fixed point be not placed in the direction of the line D E; and, on the other hand, that if the fixed point be in the direction of that line, it cannot move. A practical test is thus suggested, by which the line D E may be at once discovered. Let a thread be attached to any point of the body, and let it be suspended by this thread from a hook or other fixed point. The direction of the thread, when the body becomes quiescent, will be that of a single force equivalent to the gravitation of all the component parts of the mass.

(150.) An enquiry is here suggested: does the direction of the equivalent force thus determined depend on the position of the body with respect to the surface of the earth, and how is the direction of the equivalent force affected by a change in that position? This question may be at once solved if the body be suspended by different points, and the directions which the suspending thread takes in each case relatively to the figure and dimensions of the body examined.

The body being suspended in this manner from any point, let a small hole be bored through it, in the exact direction of the thread, so that if the thread were continued below the point where it is attached to the body, it would pass through this hole. The body being successively suspended by several different points on its surface, let as many small holes be bored through it in the same manner. If the body be then cut through, so as to discover the directions which the several holes have taken, they will be all found to cross each other at one point within the body; or the same fact may be discovered thus: a thin wire, which nearly fills the holes being passed through any one of them, it will be found to intercept the passage of a similar wire through any other.

This singular fact teaches us, what indeed can be proved by mathematical reasoning without experiment, that there is one point in every body through which the single force, which is equivalent to the gravitation of all its particles, must pass, in whatever position the body be placed. This point is called the centre of gravity.

(151.) In whatever situation a body may be placed, the centre of gravity will have a tendency to descend in the direction of a line perpendicular to the horizon, and which is called the line of direction of the weight. If the body be altogether free and unrestricted by any resistance or impediment, the centre of gravity will actually descend in this direction, and all the other points of the body will move with the same velocity in parallel directions, so that during its fall the position of the parts of the body, with respect to the ground, will be unaltered. But if the body, as is most usual, be subject to some resistance or restraint, it will either remain unmoved, its weight being expended in exciting pressure on the restraining points or surfaces, or it will move in a direction and with a velocity depending on the circumstances which restrain it.

In order to determine these effects, to predict the pressure produced by the weight if the body be quiescent, or the mixed effects of motion and pressure, if it be not so, it is necessary in all cases to be able to assign the place of the centre of gravity. When the magnitude and figure of the body, and the density of the matter which occupies its dimensions, are known, the place of the centre of gravity can be determined with the greatest precision by mathematical calculation. The process by which this is accomplished, however, is not of a sufficiently elementary nature to be properly introduced into this treatise. To render it intelligible would require the aid of some of the most advanced analytical principles; and even to express the position of the point in question, except in very particular instances, would be impossible, without the aid of peculiar symbols.

(152.) There are certain particular forms of body in which, when they are uniformly dense, the place of the centre of gravity can be easily assigned, and proved by reasoning, which is generally intelligible; but in all cases whatever, this point may be easily determined by experiment.

(153.) If a body uniformly dense have such a shape that a point may be found on either side of which in all directions around it the materials of the body are similarly distributed, that point will obviously be the centre of gravity. For if it be supported, the gravitation of the particles on one side drawing them downwards, is resisted by an effect of exactly the same kind and of equal amount on the opposite side, and so the body remains balanced on the point.

The most remarkable body of this kind is a globe, the centre of which is evidently its centre of gravity.

A figure, such as fig. 38., called an oblate spheroid, has its centre of gravity at its centre, C. Such is the figure of the earth. The same may be observed of the elliptical solid, fig. 39., which is called a prolate spheroid.

A cube, and some other regular solids, bounded by plane surfaces, have a point within them, such as above described, and which is, therefore, their centre of gravity. Such are fig. 40.

A straight wand of uniform thickness has its centre of gravity at the centre of its length; and a cylindrical body has its centre of gravity in its centre, at the middle of its length or axis. Such is the point C, fig. 41.

A flat plate of any uniform substance, and which has in every part an equal thickness, has its centre of gravity at the middle of its thickness, and under a point of its surface, which is to be determined by its shape. If it be circular or elliptical, this point is its centre. If it have any regular form, bounded by straight edges, it is that point which is equally distant from its several angles, as C in fig. 42.

(154.) There are some cases in which, although the place of the centre of gravity is not so obvious as in the examples just given, still it may be discovered without any mathematical process, which is not easily understood. Suppose A B C, fig. 43., to be a flat triangular plate of uniform thickness and density. Let it be imagined to be divided into narrow bars, by lines parallel to the side A C, as represented in the figure. Draw B D from the angle B to the middle point D of the side A C. It is not difficult to perceive, that B D will divide equally all the bars into which the triangle is conceived to be divided. Now if the flat triangular plate A B C be placed in a horizontal position on a straight edge coinciding with the line B D, it will be balanced: for the bars parallel to A C will be severally balanced by the edge immediately under their middle point; since that middle point is the centre of gravity of each bar. Since, then, the triangle is balanced on the edge, the centre of gravity must be somewhere immediately over it, and must, therefore, be within the plate at some point under the line B D.

The same reasoning will prove that the centre of gravity of the plate is under the line A E, drawn from the angle A to the middle point E of the side B C. To perceive this, it is only necessary to consider the triangle divided into bars parallel to B C, and thence to show that it will be balanced on an edge placed under A E. Since then the centre of gravity of the plate is under the line B D, and also under A E, it must be under the point G, at which these lines cross each other; and it is accordingly at a depth beneath G, equal to half the thickness of the plate.

This may be experimentally verified by taking a piece of tin or card, and cutting it into a triangular form. The point G being found by drawing B D and A E, which divide two sides equally, it will be balanced if placed upon the point of a pin at G.

The centre of gravity of a triangle being thus determined, we shall be able to find the position of the centre of gravity of any plate of uniform thickness and density which is bounded by straight edges, as will be shown hereafter. (173.)

(155.) The centre of gravity is not always included within the volume of the body, that is, it is not enclosed by its surfaces. Numerous examples of this can be produced. If a piece of wire be bent into any form, the centre of gravity will rarely be in the wire. Suppose it be brought to the form of a ring. In that case, the centre of gravity of the wire will be the centre of the circle, a point not forming any part of the wire itself: nevertheless this point may be proved to have the characteristic property of the centre of gravity; for if the ring be suspended by any point, the centre of the ring must always settle itself under the point of suspension. If this centre could be supposed to be connected with the ring by very fine threads, whose weight would be insignificant, and which might be united by a knot or otherwise at the centre, the ring would be balanced upon a point placed under the knot.

In like manner, if the wire be formed into an ellipse, or any other curve similarly arranged round a centre point, that point will be its centre of gravity.

(156.) To find the centre of gravity experimentally, the method described in (149, 150) may be used. In this case two points of suspension will be sufficient to determine it; for the directions of the suspending cord being continued through the body, will cross each other at the centre of gravity. These directions may also be found by placing the body on a sharp point, and adjusting it so as to be balanced upon it. In this case a line drawn through the body directly upwards from the point will pass through the centre of gravity, and therefore two such lines must cross at that point.

(157.) If the body have two flat parallel surfaces like sheet metal, stiff paper, card, board, &c., the centre of gravity may be found by balancing the body in two positions on an horizontal straight edge. The point where the lines marked by the edge cross each other will be immediately under the centre of gravity. This may be verified by showing that the body will be balanced on a point thus placed, or that if it be suspended, the point thus determined will always come under the point of suspension.

The position of the centre of gravity of such bodies may also be found by placing the body on an horizontal table having a straight edge. The body being moved beyond the edge until it is in that position in which the slightest disturbance will cause it to fall, the centre of gravity will then be immediately over the edge. This being done in two positions, the centre of gravity will be determined as before.

(158.) It has been already stated, that when the body is perfectly free, the centre of gravity must necessarily move downwards, in a direction perpendicular to an horizontal plane. When the body is not free, the circumstances which restrain it generally permit the centre of gravity to move in certain directions, but obstruct its motion in others. Thus if a body be suspended from a fixed point by a flexible cord, the centre of gravity is free to move in every direction except those which would carry it farther from the point of suspension than the length of the cord. Hence if we conceive a globe or sphere to surround the point of suspension on every side to a distance equal to that of the centre of gravity from the point of suspension, when the cord is fully stretched, the centre of gravity will be at liberty to move in every direction within this sphere.

There are an infinite variety of circumstances under which the motion of a body may be restrained, and in which a most important and useful class of mechanical problems originate. Before we notice others, we shall, however, examine that which has just been described more particularly.

Let P, fig. 44., be the point of suspension, and C the centre of gravity, and suppose the body so placed that C shall be within the sphere already described. The cord will therefore be slackened, and in this state the body will be free. The centre of gravity will therefore descend in the perpendicular direction until the cord becomes fully extended; the tension will then prevent its further motion in the perpendicular direction. The downward force must now be considered as the diagonal of a parallelogram, and equivalent to two forces C D and C E, in the directions of the sides, as already explained in (149). The force C D will bring the centre of gravity into the direction P F, perpendicularly under the point of suspension. Since the force of gravity acts continually on C in its approach to P F, it will move towards that line with accelerated speed, and when it has arrived there it will have acquired a force to which no obstruction is immediately opposed, and consequently by its inertia it retains this force, and moves beyond P F on the other side. But when the point C gets into the line P F, it is in the lowest possible position; for it is at the lowest point of the sphere which limits its motion. When it passes to the other side of P F, it must therefore begin to ascend, and the force of gravity, which, in the former case, accelerated its descent, will now for the same reason, and with equal energy, oppose its ascent. This will be easily understood. Let C′ be any point which it may have attained in ascending; C′ G′, the force of gravity, is now equivalent to C′ D′ and C′ E′. The latter as before produces tension; but the former C′ D′ is in a direction immediately opposed to the motion, and therefore retards it. This retardation will continue until all the motion acquired by the body in its descent from the first position has been destroyed, and then it will begin to return to P F, and so it will continue to vibrate from the one side to the other until the friction on the point P, and the resistance of the air, gradually deprive it of its motion, and bring it to a state of rest in the direction P F.

But for the effects of friction and atmospheric resistance, the body would continue for ever to oscillate equally from side to side of the line P F.

(159.) The phenomenon just developed is only an example of an extensive class. Whenever the circumstances which restrain the body are of such a nature that the centre of gravity is prevented from descending below a certain level, but not, on the other hand, restrained from rising above it, the body will remain at rest if the centre of gravity be placed at the lowest limit of its level; any disturbance will cause it to oscillate around this state, and it cannot return to a state of rest until friction or some other cause have deprived it of the motion communicated by the disturbing force.

(160.) Under the circumstances which we have just described, the body could not maintain itself in a state of rest in any position except that in which the centre of gravity is, at the lowest point of the space in which it is free to move. This, however, is not always the case. Suppose it were suspended by an inflexible rod instead of a flexible string; the centre of gravity would then not only be prevented from receding from the point of suspension, but also from approaching it; in fact, it would be always kept at the same distance from it. Thus, instead of being capable of moving anywhere within the sphere, it is now capable of moving on its surface only. The reasoning used in the last case may also be applied here, to prove that when the centre of gravity is on either side of the perpendicular P F, it will fall towards P F and oscillate, and that if it be placed in the line P F, it will remain in equilibrium. But in this case there is another position, in which the centre of gravity may be placed so as to produce equilibrium. If it be placed at the highest point of the sphere in which it moves, the whole force acting on it will then be directed on the point of suspension, perpendicularly downwards, and will be entirely expended in producing pressure on that point; consequently, the body will in this case be in equilibrium. But this state of equilibrium is of a character very different from that in which the centre of gravity was at the lowest part of the sphere. In the present case any displacement, however slight, of the centre of gravity, will carry it to a lower level, and the force of gravity will then prevent its return to its former state, and will impel it downwards until it attain the lowest point of the sphere, and round that point it will oscillate.

(161.) The two states of equilibrium which have been just noticed, are called stable and instable equilibrium. The character of the former is, that any disturbance of the state produces oscillation about it; but any disturbance of the latter state produces a total overthrow, and finally causes oscillation around the state of stable equilibrium.

Let A B, fig. 45., be an elliptical board resting on its edge on an horizontal plane. In the position here represented, the extremity P of the lesser axis being the point of support, the board is in stable equilibrium; for any motion on either side must cause the centre of gravity C to ascend in the directions C O, and oscillation will ensue. If, however, it rest upon the smaller end, as in fig. 46., the position would still be a state of equilibrium, because the centre of gravity is directly above the point of support; but it would be instable equilibrium, because the slightest displacement of the centre of gravity would cause it to descend.

Thus an egg or a lemon may be balanced on the end, but the least disturbance will overthrow it. On the contrary, it will easily rest on the side, and any disturbance will produce oscillation.

(162.) When the circumstances under which the body is placed allow the centre of gravity to move only in an horizontal line, the body is in a state which may be called neutral equilibrium. The slightest force will move the centre of gravity, but will neither produce oscillation nor overthrow the body, as in the last two cases.

An example of this state is furnished by a cylinder placed upon an horizontal plane. As the cylinder is rolled upon the plane, the centre of gravity C, fig. 47., moves in a line parallel to the plane A B, and distant from it by the radius of the cylinder. The body will thus rest indifferently in any position, because the line of direction always falls upon a point P at which the body rests upon the plane.

If the plane were inclined, as in fig. 48., a body might be so shaped, that while it would roll the centre of gravity would move horizontally. In this case the body would rest indifferently on any part of the plane, as if it were horizontal, provided the friction be sufficient to prevent the body from sliding down the plane.

If the centre of gravity of a cylinder happen not to coincide with its centre by reason of the want of uniformity in the materials of which it is composed, it will not be in a state of neutral equilibrium on an horizontal plane, as in fig. 47. In this case let G, fig. 49., be the centre of gravity. In the position here represented, where the centre of gravity is immediately below the centre C, the state will be stable equilibrium, because a motion on either side would cause the centre of gravity to ascend; but in fig. 50., where G is immediately above C, the state is instable equilibrium, because a motion on either side would cause G to descend, and the body would turn into the position fig. 49.

(163.) A cylinder of this kind will, under certain circumstances, roll up an inclined plane. Let A B, fig. 51., be the inclined plane, and let the cylinder be so placed that the line of direction from G shall be above the point P at which the cylinder rests upon the plane. The whole weight of the body acting in the direction G D will obviously cause the cylinder to roll towards A, provided the friction be sufficient to prevent sliding; but although the cylinder in this case ascends, the centre of gravity G really descends.

When G is so placed that the line of direction G D shall fall on the point P, the cylinder will be in equilibrium, because its weight acts upon the point on which it rests. There are two cases represented in fig. 52. and fig. 53., in which G takes this position. Fig. 52. represents the state of stable, and fig. 53. of instable equilibrium.

(164.) When a body is placed upon a base, its stability depends upon the position of the line of direction and the height of the centre of gravity above the base. If the line of direction fall within the base, the body will stand firm; if it fall on the edge of the base, it will be in a state in which the slightest force will overthrow it on that side at which the line of direction falls; and if the line of direction fall without the base, the body must turn over that edge which is nearest to the line of direction.

In fig. 54. and fig. 55., the line of direction G P falls within the base, and it is obvious that the body will stand firm; for any attempt to turn it over either edge would cause the centre of gravity to ascend. But in fig. 56. the line of direction falls upon the edge, and if the body be turned over, the centre of gravity immediately commences to descend. Until it be turned over, however, the centre of gravity is supported by the edge.

In fig. 57. the line of direction falls outside the base, the centre of gravity has a tendency to descend from G towards A, and the body will accordingly fall in that direction.

(165.) When the line of direction falls within the base, bodies will always stand firm, but not with the same degree of stability. In general, the stability depends on the height through which the centre of gravity must be elevated before the body can be overthrown. The greater this height is, the greater in the same proportion will be the stability.

Let B A C, fig. 58., be a pyramid, the centre of gravity being at G. To turn this over the edge B, the centre of gravity; must be carried over the arch G E, and must therefore be raised through the height H E. If, however, the pyramid were taller relatively to its base, as in fig. 59., the height H E would be proportionally less; and if the base were very small in reference to the height, as in fig. 60., the height H E would be very small, and a slight force would throw it over the edge B.

It is obvious that the same observations may be applied to all figures whatever, the conclusions just deduced depending only on the distance of the line of direction from the edge of the base, and the height of the centre of gravity above it.

(166.) Hence we may perceive the principle on which the stability of loaded carriages depends. When the load is placed at a considerable elevation above the wheels, the centre of gravity is elevated, and the carriage becomes proportionally insecure. In coaches for the conveyance of passengers, the luggage is therefore sometimes placed below the body of the coach; light parcels of large bulk may be placed on the top with impunity.

When the centre of gravity of a carriage is much elevated, there is considerable danger of overthrow, if a corner be turned sharply and with a rapid pace; for the centrifugal force then acting on the centre of gravity will easily raise it through the small height which is necessary to turn the carriage over the external wheels (142).

(167.) The same waggon will have greater stability when loaded with a heavy substance which occupies a small space, such as metal, than when it carries the same weight of a lighter substance, such as hay; because the centre of gravity in the latter case will be much more elevated.

If a large table be placed upon a single leg in its centre, it will be impracticable to make it stand firm; but if the pillar on which it rests terminate in a tripod, it will have the same stability as if it had three legs attached to the points directly over the places where the feet of the tripod rest.

(168.) When a solid body is supported by more points than one, it is not necessary for its stability that the line of direction should fall on one of those points. If there be only two points of support, the line of direction must fall between them. The body is in this case supported as effectually as if it rested on an edge coinciding with a straight line drawn from one point of support to the other. If there be three points of support, which are not ranged in the same straight line, the body will be supported in the same manner as it would be by a base coinciding with the triangle formed by straight lines joining the three points of support. In the same manner, whatever be the number of points on which the body may rest, its virtual base will be found by supposing straight lines drawn, joining the several points successively. When the line of direction falls within this base, the body will always stand firm, and otherwise not. The degree of stability is determined in the same manner as if the base were a continued surface.

(169.) Necessity and experience teach an animal to adapt its postures and motions to the position of the centre of gravity of his body. When a man stands, the line of direction of his weight must fall within the base formed by his feet. If A B, C D, fig. 61., be the feet, this base is the space A B D C. It is evident, that the more his toes are turned outwards, the more contracted the base will be in the direction E F, and the more liable he will be to fall backwards or forwards. Also, the closer his feet are together, the more contracted the base will be in the direction G H, and the more liable he will be to fall towards either side.

When a man walks, the legs are alternately lifted from the ground, and the centre of gravity is either unsupported or thrown from the one side to the other. The body is also thrown a little forward, in order that the tendency of the centre of gravity to fall in the direction of the toes may assist the muscular action in propelling the body. This forward inclination of the body increases with the speed of the motion.

But for the flexibility of the knee-joint the labour of walking would be much greater than it is; for the centre of gravity would be more elevated by each step. The line of motion of the centre of gravity in walking is represented by fig. 62., and deviates but little from a regular horizontal line, so that the elevation of the centre of gravity is subject to very slight variation. But if there were no knee-joint, as when a man has wooden legs, the centre of gravity would move as in fig. 63., so that at each step the weight of the body would be lifted through a considerable height, and therefore the labour of walking would be much increased.

If a man stand on one leg, the line of direction of his weight must fall within the space on which his foot treads. The smallness of this space, compared with the height of the centre of gravity, accounts for the difficulty of this feat.

The position of the centre of gravity of the body changes with the posture and position of the limbs. If the arm be extended from one side, the centre of gravity is brought nearer to that side than it was when the arm hung perpendicularly. When dancers, standing on one leg, extend the other at right angles to it, they must incline the body in the direction opposite to that in which the leg is extended, in order to bring the centre of gravity over the foot which supports them.

When a porter carries a load, his position must be regulated by the centre of gravity of his body and the load taken together. If he bore the load on his back, the line of direction would pass beyond his heels, and he would fall backwards. To bring the centre of gravity over his feet he accordingly leans forward, fig. 64.

If a nurse carry a child in her arms, she leans back for a like reason.

When a load is carried on the head, the bearer stands upright, that the centre of gravity may be over his feet.

In ascending a hill, we appear to incline forward; and in descending, to lean backward, but in truth, we are standing upright with respect to a level plane. This is necessary to keep the line of direction between the feet, as is evident from fig. 65.

A person sitting on a chair which has no back cannot rise from it without either stooping forward to bring the centre of gravity over the feet, or drawing back the feet to bring them under the centre of gravity.

A quadruped never raises both feet on the same side simultaneously, for the centre of gravity would then be unsupported. Let A B C D, fig. 66., be the feet. The base on which it stands is A B C D, and the centre of gravity is nearly over the point O, where the diagonals cross each other. The legs A and C being raised together, the centre of gravity is supported by the legs B and D, since it falls between them; and when B and D are raised it is, in like manner, supported by the feet A and C. The centre of gravity, however, is often unsupported for a moment; for the leg B is raised from the ground before A comes to it, as is plain from observing the track of a horse’s feet, the mark of A being upon or before that of B. In the more rapid paces of all animals the centre of gravity is at intervals unsupported.

The feats of rope-dancers are experiments on the management of the centre of gravity. The evolutions of the performer are found to be facilitated by holding in his hand a heavy pole. His security in this case depends, not on the centre of gravity of his body, but on that of his body and the pole taken together. This point is near the centre of the pole, so that, in fact, he may be said to hold in his hands the point on the position of which the facility of his feats depends. Without the aid of the pole the centre of gravity would be within the trunk of the body, and its position could not be adapted to circumstances with the same ease and rapidity.

(170.) The centre of gravity of a mass of fluid is that point which would have the properties which have been proved to belong to the centre of gravity of a solid, if the fluid were solidified without changing in any respect the quantity or arrangement of its parts. This point also possesses other properties, in reference to fluids, which will be investigated in Hydrostatics and Pneumatics.

(171.) The centre of gravity of two bodies separated from one another, is that point which would possess the properties ascribed to the centre of gravity, if the two bodies were united by an inflexible line, the weight of which might be neglected. To find this point mathematically is a very simple problem. Let A and B, fig. 67., be the two bodies, and a and b their centres of gravity. Draw the right line a b, and divide it at C, in such a manner that a C shall have the same proportion to b C as the mass of the body B has to the mass of the body A.

This may easily be verified experimentally. Let A and B be two bodies, whose weight is considerable, in comparison with that of the rod a b, which joins them. Let a fine silken string, with its ends attached to them, be hung upon a pin; and on the same pin let a plumb-line be suspended. In whatever position the bodies may be hung, it will be observed that the plumb-line will cross the rod a b at the same point, and that point will divide the line a b into parts a C and b C, which are in the proportion of the mass of B to the mass of A.

(172.) The centre of gravity of three separate bodies is defined in the same manner as that of two, and may be found by first determining the centre of gravity of two; and then supposing their masses concentrated at that point, so as to form one body, and finding the centre of gravity of that and the third.

In the same manner the centre of gravity of any number of bodies may be determined.

(173.) If a plate of uniform thickness be bounded by straight edges, its centre of gravity may be found by dividing it into triangles by diagonal lines, as in fig. 68., and having determined by (154) the centres of gravity of the several triangles, the centre of gravity of the whole plate will be their common centre of gravity, found as above.

(174.) Although the centre of gravity takes its name from the familiar properties which it has in reference to detached bodies of inconsiderable magnitude, placed on or near the surface of the earth, yet it possesses properties of a much more general and not less important nature. One of the most remarkable of these is, that the centre of gravity of any number of separate bodies is never affected by the mutual attraction, impact, or other influence which the bodies may transmit from one to another. This is a necessary consequence of the equality of action and reaction explained in Chapter IV. For if A and B, fig. 67., attract each other, and change their places to A′ and B′, the space a a′ will have to b b′ the same proportion as B has to A, and therefore by what has just been proved (171) the same proportion as a C has to b C. It follows, that the remainders a′ C and b′ C will be in the proportion of B to A, and that C will continue to be the centre of gravity of the bodies after they have approached by their mutual attraction.

Suppose, for example, that A and B were 12lbs. and 8lbs. respectively, and that a b were 40 feet. The point C must (171) divide a b into two parts, in the proportion of 8 to 12, or of 2 to 3. Hence it is obvious that a C will be 16 feet, and b C 24 feet. Now, suppose that A and B attract each other, and that A approaches B through two feet. Then B must approach A through three feet. Their distances from C will now be 14 feet and 21 feet, which, being in the proportion of B to A, the point C will still be their centre of gravity.

Hence it follows, that if a system of bodies, placed at rest, be permitted to obey their mutual attractions, although the bodies will thereby be severally moved, yet their common centre of gravity must remain quiescent.

(175.) When one of two bodies is moving in a straight line, the other being at rest, their common centre of gravity must move in a parallel straight line. Let A and B, fig. 69., be the centres of gravity of the bodies, and let A move from A to a, B remaining at rest. Draw the lines A B and a B. In every position which the body B assumes during its motion, the centre of gravity C divides the line joining them into parts A C, B C, which are in the proportion of the mass B to the mass A. Now, suppose any number of lines drawn from B to the line A a; a parallel C c to A a through C divides all these lines in the same proportion; and therefore, while the body A moves from A to a, the common centre of gravity moves from C to c.

If both the bodies A and B moved uniformly in straight lines, the centre of gravity would have a motion compounded (74) of the two motions with which it would be affected, if each moved while the other remained at rest. In the same manner, if there were three bodies, each moving uniformly in a straight line, their common centre of gravity would have a motion compounded of that motion which it would have if one remained at rest while the other two moved, and that which the motion of the first would give it if the last two remained at rest; and in the same manner it may be proved, that when any number of bodies move each in a straight line, their common centre of gravity will have a motion compounded of the motions which it receives from the bodies severally.

It may happen that the several motions which the centre of gravity receives from the bodies of the system will neutralise each other; and this does, in fact, take place for such motions as are the consequences of the mutual action of the bodies upon one another.

(176.) If a system of bodies be not under the immediate influence of any forces, and their mutual attraction be conceived to be suspended, they must severally be either at rest or in uniform rectilinear motion in virtue of their inertia. Hence, their common centre of gravity must also be either at rest or in uniform rectilinear motion. Now, if we suppose their mutual attractions to take effect, the state of their common centre of gravity will not be changed, but the bodies will severally receive motions compounded of their previous uniform rectilinear motions and those which result from their mutual attractions. The combined effects will cause each body to revolve in an orbit round the common centre of gravity, or will precipitate it towards that point. But still that point will maintain its former state undisturbed.

This constitutes one of the general laws of mechanical science, and is of great importance in physical astronomy. It is known by the title “the conservation of the motion of the centre of gravity.”

(177.) The solar system is an instance of the class of phenomena to which we have just referred. All the motions of the bodies which compose it can be traced to certain uniform rectilinear motions, received from some former impulse, or from a force whose action has been suspended, and those motions which necessarily result from the principle of gravitation. But we shall not here insist further on this subject, which more properly belongs to another department of the science.

(178.) If a solid body suffer an impact in the direction of a line passing through its centre of gravity, all the particles of the body will be driven forward with the same velocity in lines parallel to the direction of the impact, and the whole force of the motion will be equal to that of the impact. The common velocity of the parts of the body will in this case be determined by the principles explained in Chapter IV. The impelling force being equally distributed among all the parts, the velocity will be found by dividing the numerical value of that force by the number expressing the mass.

If any number of impacts be given simultaneously to different points of a body, a certain complex motion will generally ensue. The mass will have a relative motion round the centre of gravity as if it were fixed, while that point will move forward uniformly in a straight line, carrying the body with it. The relative motion of the mass round the centre of gravity may be found by considering the centre of gravity as a fixed point, round which the mass is free to move, and then determining the motion which the applied forces would produce. This motion being supposed to continue uninterrupted, let all the forces be imagined to be applied in their proper directions and quantities to the centre of gravity. By the principles for the composition of force they will be mechanically equivalent to a single force through that point. In the direction of this single force the centre of gravity will move and have the same velocity as if the whole mass were there concentrated and received the impelling forces.

(179.) These general properties, which are entirely independent of gravity, render the “centre of gravity” an inadequate title for this important point. Some physical writers have, consequently, called it the “centre of inertia.” The “centre of gravity,” however, is the name by which it is still generally designated.