Is that branch of mechanical science which treats of the laws of equilibrium or repose; it is divided into three distinct sections, according as it may refer to solid, to fluid, or to gaseous matter. Hydrostatics and Aérostatics have already been treated of under their proper heads; and in the present article we shall confine our attention to the equilibrium of solid bodies, or to Geostatics, as this section may, with propriety, be termed.

If we could place a body in space so as that nothing might obstruct its motion, it would move on the application of the slightest pressure, and could then only be kept at rest by the application of antagonistic pressures, so that the doctrines of statics must relate to the manner in which pressures applied to a solid body keep that body at rest, or, as we say, in which they balance each other.

Our idea of pressure is obtained directly from the experience of resistance to muscular effort; and as our knowledge of the very existence of resistance is experimental, so our acquaintance with the laws of resistance must also be derived from experience, and cannot possibly be the result of mere intellectual ratiocination: the attempt to make statics a branch of pure logic must, in the very nature of things, be a vain one.

When a piece of solid matter is entirely detached from all other solid bodies, it, except in certain very peculiar circumstances, moves and continues to move until it come in contact with some other solid body, and all the repose of which we have any actual cognisance, exists among solid bodies in contact with each other.

On attempting to displace a body which is at rest, we experience three distinct kinds of resistance:—in the first place, on trying to lift it, we experience an opposition, as it were, in the object itself, and if we succeed in raising it we still feel that it tends to go down again; to this well-known and yet very wonderful quality, the name weight is given, and we recognise in it that very tendency in virtue of which the object, when let go, descends until it meet an obstacle. In the second place, on attempting to push the object against the body on which it is resting, we find that that body resists our utmost endeavours which never succeed unless by its displacement or by its fracture; this kind of resistance may, for the convenience of language, be called direct. And in the third place, on trying to push the object along the surface on which it is resting, we experience a resistance, not insuperable as before, but varying according to the nature of the surfaces and to other circumstances; this kind of resistance is commonly called friction, but as this word implies motion, it would be more accurate to use the expression resistance to friction or lateral resistance. When the surfaces are smooth and the substances hard, the friction is comparatively small, and, as is well-known, becomes less when grease or oil is applied.

EQUILIBRIUM OF TWO PRESSURES.

If we place an object O upon a smooth flat surface, and push it gently by means of a stick BA, we find that it remains at rest, unless the intensity of the pressure be sufficient to overcome the friction. If now another person press against it by means of a stick DC so as to prevent the motion, we conceive that the two pressures are of like intensity, or differ from each other only by the friction, while everyday experience shows that their directions must be exactly opposed to each other. The object O seems to be the means of communication between the two pressing bodies, and if the two sticks were applied end to end with pressures equally intense, there would also be equilibrium, so that we very readily admit the general proposition, "Two pressures equal in intensity and opposite in direction balance each other."

When the object O is hard, we do not readily perceive that any change is made upon it by the pressures at A and C, but if a soft body, such as a cork or a bit of elastic gum, be subjected to such pressures, a visible change is made in its form; this change increases as the pressures are augmented, so that it becomes a kind of index of the intensity of the pressure, and careful experiments show, that though often imperceptible in extent, analogous changes take place in even the hardest substances. When the pressure is removed, the body resumes its original shape more or less accurately, and with greater or less rapidity according to the nature of the material. With some substances, as lead, a permanent change of form results if the pressure be considerable; with others, as vulcanised caoutchouc, a considerable time elapses; while with the generality of what are called hard substances, the resumption of form is complete and almost instantaneous. The communication of pressure, then, from the point A to the point C, implies important and complicated changes in the form and position of the parts of the intervening body.

We have not, however, completely examined this experiment; for if one person press against the end B of the stick BA, while another presses against the end D of DC, the object O is almost invariably found to turn round: the slightest error in the direction of either of the rods is followed by a dislocation of the arrangement. The same thing takes place if two parties attempt to make trial of their strength by pushing the buttons of their foils against each other, and it is only by laying the rods between guides that we can prevent this inconvenience.

If we make two holes in the object O, one at A and the other at C, and having passed a cord through each, pull at the ends B and D of these cords, the object no longer gets displaced; but, on the contrary, the threads AB and CD always come into one straight line. The two pressures are still opposite in direction and equal in intensity.

These two modes of conducting this fundamental experiment furnish excellent examples of unstable and stable equilibrium, the contrast between them is instructive in another way. It appears at first sight, that there are two modes of urging a body in a given direction, one by pushing, the other by pulling; but a closer examination of the matter shows that pulling or traction is, in reality, pushing or pressure. No body can influence another body to move (setting aside gravitation and such influences as magnetic or electric attractions and repulsions), except by pressing upon it. We cannot draw an object to us by stretching out the hand unto it and withdrawing the arm again; no: the hand must be reached to and beyond some integral part of the object, the fingers must be bent round, and pressure must be applied to the further surface; we must take hold of it. Even when, as in the case of pulling a rope or seizing the end of a stick, we do not reach to a surface opposed to the direction of the intended effort, we grasp the object in the hand so as to occasion friction, and thus our ultimate resort is still to pressure. The perception of this truth leads to the use of hooks, eyes, loops, rings, ties, &c., which are all contrived for the very purpose of reaching beyond and pressing upon the further surface.

Vices and forceps, again, are constructed for pulling by friction; their jaws are pressed against the lateral surfaces of the object, and the friction thus occasioned offers resistance to the pulls.

Traction occasions complicated changes in the internal part of the bodies subjected to it. Thus, if a pin A driven firmly into the block B be drawn towards E by means of the hook CDE, the parts of the pin near C are compressed, but of the parts near the lower end a some are compressed and some are distended. The hook at C is compressed, the parts D on the one side of the line of traction are partly distended and partly bent, while the parts E in the line of strain seem to be merely distended; so that in this little arrangement we have compression, lateral flexure, and distension combined, and this combination is to be found in every case of traction.

When we pull by means of a chain, each link is strained, on the whole lengthened, and communicates its tension to the next, so that we have a series of examples of the equilibrium of two opposing pressures. So, when a rope is used, we imagine that each part is distended, and communicates the pressure to the adjoining part; but as we, in truth, know very little of the internal constitution of matter, this idea rather furnishes us with a convenient mode of speech, than with an explanation of the matter; it clearly points to the well-known fact that the length of the cord does not, unless its weight has to be considered, in any way influence the transmission of pressure from the one end to the other.

It is apparent that we cannot strain a rope by pulling at the one end unless the other end be held fast; and therefore, when two men pull at opposite ends of a rope, the rope is not strained by the exertions of two men, but only by the exertion of one man, and he the weaker of the two. This is very obvious, yet we have examples of its misapprehension. Thus, Otto von Guericke, when exhibiting publicly his famous experiment of the hemispheres, yoked eight horses to the one and eight horses to the other cup, in order to pull them asunder, thinking that he thus employed the strength of sixteen horses, whereas, had he fastened one of the hemispheres to the adjacent tree, the eight horses yoked to the other cup would have pulled quite as effectively as when pulling against the other teem.

The change which pressure causes in the form of a solid is well seen when the strain acts transversely. Thus, if a wire AB (fig. 4) be bent into the form of a screw, a slight pressure applied to the end is accompanied by a perceptible alteration in the length of the spiral. By using a thin wire with large and numerous convolutions, any degree of delicacy may be obtained in this kind of spring, so that we can construct a convenient indicator of pressure. Tension gauges of this kind are in common use; the spiral spring being generally inclosed in a tube, and the divisions to indicate the quantity of compression or distension are marked either along the side of the tube or on a style protruding from one end.

We are not entitled to assume, nor, indeed, is it strictly true, that the intensity of the pressure applied is proportional to the lengthening or shortening of the spiral spring, yet such spring gauges can be made truly to indicate the pressures, and they are the most convenient instruments for examining experimentally the laws of equilibrium. The unit of pressure is usually taken as the weight of some known heavy body, such as a pound, an ounce, or a grain; and this will answer sufficiently well our present purpose, although it be a fact that gravitation is not quite the same all over the world.

Let then a weight, say of one pound, be hung from the hook of a spring-balance, and let the protrusion of the style be marked; remove now the pound, substitute for it a mass somewhat heavier, and by paring or filing adjust it so as to bring the index exactly to the same mark, and you have another pound weight; in this way we may prepare as many weights as may be required. Hang on now two, three, four, and so on of these weights at once, marking the indication corresponding to each, and you have the spring-balance graduated to show pressures of one, two, three, etc., pounds. Having prepared several of such pressure gauges, we are ready to make experiments on the manner in which pressures balance each other.

On hooking two of these gauges together, directly or by the intervention of a cord, we obtain at once a confirmation of the law already stated, that two pressures balance each other when they are equal in intensity and opposite in direction.

EQUILIBRIUM OF THREE PRESSURES.

In order to examine the simplest case of the equilibrium of three pressures, we may knot three threads together, and attach the loose ends of these to the hooks of three pressure-gauges. On stretching the threads of this little arrangement, the first thing which offers itself to our notice is, that, sensibly, they are always in one plane; and we are convinced that, when three pressures acting on one point balance each other, this is always the case. Let (fig. 5) PA, PB, PC represent the three threads knotted together at P; in PA and PB take any points a and b, then if we join these by a fine line, as a silk fibre, the continuation of CP always cuts it; this is in accordance with the well-known property of plane surfaces. A very slight examination of a few cases shows us that the continuation of the direction of one of the pressures always passes between the other two, and also that the greater pressure has the lesser angle opposite to it. If the distances Pa, Pb be measured off proportional to the intensities of the pressures, it is found invariably that the continuation of the third line CP divides the line joining a and b into two equal parts. Any observer may convince himself of the truth of this law by bringing a strained system of three threads over a sheet of paper, by noting the arrangement and by making the appropriate delineation.

In order to be a general law, this statement must apply as well to the two pressures PB and PC as to PA and PB, wherefore if Pe be measured proportional to the tension of PC, be should be bisected by the continuation of AP, and ca by the continuation of BP; and this is in accordance with the truth that the three lines which join the corners of any trigon with the middles of the opposite sides all meet in one point. Having produced the line CPf until fg be equal to Pf, join ag and bg, then it can easily be shown that ag is equal to Pb, bg to Pa, and Pg to Pe, so that the intensities of three pressures which balance at a point are proportional to the three sides of a trigon drawn parallel to the three directions of the pressures; thus the tensions of the cords PA, PB, PC are proportional to Pa, ag, gP. Again, since the sides of a trigon are proportional to the sines of their opposite angles, and since the sine of an angle is equal to the sine of its supplement, it follows that, of three pressures which balance each other at a point, the intensities are proportional to the sines of the opposite angles. The same truth may be stated yet in another way. The two pressures Pa and Pb may be regarded as balancing or resisting the third pressure Pc; but a pressure represented by Pg would alone resist Pe, so that we may, as it were, regard the single pressure represented by Pg as equivalent to the two Pa and Pb; the pressure represented by Pg is called the resultant of Pa and Pb; and thus, if the two sides of a rhomboid represent in direction and in intensity two pressures acting on a point, the diagonal of the rhomboid represents the resultant of those pressures.

This theorem, in whichever way we may prefer to state it, is the foundation of all mechanical science. Various attempts have been made to show that it ought, necessarily, to be true, or, in other words, to deduce our knowledge of it from some fancied intuitive perception of what should be; but the authors of all such attempts proceed by assuming, stated in one form of words, the very law which, disguised in another form, they wish to establish. Experience, and experience alone, tells us that this law is, but why it is, is utterly beyond our comprehension.

If three pressures, not acting on one point, balance each other, they must act through the intervention of some solid body, and must produce strains in the parts of that body; for we have no example of pressures applied to one being resisted by other pressures applied to another and disconnected object. Let us then suppose that three points, Q, R, S, are connected by three slender rods, QR, RS, SQ, and let us endeavour to trace the effects of three pressures, QA, RB, SC, acting on these points and balancing each other.

In this arrangement, putting out of view the weights of the tie-rods, the point Q is kept in equilibrium by three pressures—one QA, one the strain on QR, and one the strain on QS; so, according to the previously observed law, the direction of QA must be in the plane QRS, and as the same may be said of RB, SC, it seems that, in such an arrangement as this, the directions of the three pressures must lie in one plane. Let us produce AQ and BR to meet in P, and draw the perpendiculars PT, PU, PV to the lines RS, SQ, QR. Then since PU is the sine of the angle AQS to the radius QP, while PV is the sine of AQR to the same radius, it follows that the strain on QR is to the strain on QS as PU is to PV, or denoting strain by a superscribed line,

\[ \frac{QR}{QS} = \frac{PU}{PV}. \] In the same way, by considering the three strains balancing at the point R, we find that

\[ \frac{RS}{QR} = \frac{PV}{PT} \]

wherefore by compounding the ratios we obtain

\[ \frac{RS}{QS} = \frac{PU}{PT} \]

Now this is what would ensue if the direction of the third pressure CS were to pass through P; and thus we conclude that when three pressures acting on three points connected by tie-rods balance each other, their directions must all tend to one point; and farther, it is very easy to show that the intensities of these pressures must be just as if they had acted directly on that point.

This proposition, which we have easily deduced from our first law in the case of a linear frame QRS, is also true of pressures applied to any three points in a solid body; but without having recourse to farther experiments, we are unable to show that it is so.

If, for example, the points Q, R, S be taken in a thin flat plate, we are unable to trace the manner in which the strains are transmitted from the one point to the other, and are thus without the means of pursuing the investigation. A few experiments, however, are enough to convince us that the three directions, AQ, BR, CS, converge to one point, and that the intensities of the pressures bear to each other the very same ratios that they would have had if the strings had been attached to that point.

If the plate have a forked form, as shown in fig. 9, the tendency is to lengthen the arms QR and RS, and to open the angle QRS, so that we have a cross or angular strain upon the arms of this, which is called a bent lever. On making trials with such a form as this, we find, as before, that the three pressures balance just as if they had been attached directly to their point of common intersection; and thus we may accept it as a general law, that when three pressures balance each other, their directions meet in one point, and are in one plane, and their intensities are proportional to the sides of a trigon drawn parallel to them.

From this law it follows that the resultant of two pressures is always less than their sum. If the pressure QA were made nearly equal to the sum of the other two RB, SC, the trigon formed with its sides proportional to these three would have the angle opposite to the representative of QA very obtuse, so that its supplement BPC would be a small angle, and the point P would necessarily be situated considerably beyond Q, in the direction QA, as shown in fig. 10. If the pressure QA were gradually augmented, so as to become almost equal to the sum of the others, P would move off to a very great distance, and the lines QA, RB, SC, would be nearly parallel; so that in the limiting case, when we may suppose these lines to have become absolutely parallel, the middle pressure must be equal to the sum of the other two.

From Q let fall the perpendiculars QT, QU upon the directions of the pressures RB, SC; then, since these are proportional to the sines of the angles QPB, QPC, they must also be proportional to the intensities of the pressures SC, RB; and as this proportion, viz., \( \frac{SC}{RB} = \frac{QT}{QU} \), holds true of every case up to the very limit of parallelism, we must conclude that it is true of parallel pressures also.

This brings us to the very well known doctrine of the lever. When a rigid body RQS, of any form whatever, is made to rest at one point, as Q, against an obstacle, and has pressures applied to it at two other points, it is called a lever, the resisting point being called the fulcrum, and the distances, as QR, QS, of the other two points from the fulcrum, being called the arms of the lever.

When the three points Q, R, S are in one straight line, the lever is said to be straight; and in that case, if the pressures SC, RB be parallel, the perpendiculars QU and QT are proportional to QS and QR; hence, when a straight lever is kept in equilibrium by weights hung on at its two extremities, these weights must be inversely proportional to the lengths of the arms; and when these lengths are equal, the weights also must be equal to each other.

This proposition concerning the equilibrium of three pressures applied to three points in a solid body, contains the essential principles of balances, steelyards, and other weighing machines; but it must be remarked, that such an equilibrium is hardly to be obtained in practice, even though it were desirable. The balance is always made to rest on the two ends of a knife edge, and thus there are at least four pressures acting on four points. (See the article Balance in Vol. IV.)

**EQUILIBRIUM OF FOUR PRESSURES.**

When four pressures, acting in the directions PA, PB, PC, PD, balance each other, we may seek to investigate their relations by finding the resultants of two of them, as PA, PB, which we shall suppose to be PG, and by then considering the equilibrium to subsist among the three, PG, PC, PD; but in following such a course we would be assuming that because PA, PB would be resisted by a pressure... equal and opposite to \( \overline{PG} \), the pressure \( \overline{PG} \) may, in all cases, be substituted for them. This, however, we cannot demonstrate logically; it is a mere surmise, and cannot be admitted as an established truth, until it shall have received a substantial confirmation.

On tying together four cords, and straining them by help of pressure-gauges, we observe, on varying the directions of the pulls, that they no longer necessarily lie all in one plane; and we readily notice that the continuation of any one of the directions passes within the solid angle or corner formed by the other three. If we now measure along \( PA \) and \( PB \) distances, \( Pa \), \( Pb \), proportional to the tensions, join the points \( a \), \( b \), by a fine thread, and mark \( f \) the middle of \( ab \), we find, on bringing the eye into the plane of the two other cords \( PC \), \( PD \), that \( f \) is in that plane. Thus, the line \( Pf \) must be the intersection of the two planes \( APB \), \( CPD \). Similarly, if this be really a law, on measuring \( Pe \), \( Pd \) proportional to the respective strains, and joining \( cd \), the middle, \( k \) of that line must be in the plane \( APB \), so that \( Pk \) must be the intersection of the same two planes, and must therefore be a continuation on \( fP \). In the same way, the line joining the middles of \( ad \) and \( bc \) ought to pass through \( P \), as also that joining the middles of \( ac \) and \( bd \). Now it is a well-known property of the tetrahedron, that the straight lines joining the middles of the opposite sides meet in one point, and bisect each other, so that this statical law, which we have observed, is not inconsistent with the truths of geometry.

The double of \( Pf \) represents in direction, and in intensity, the resultant of \( PA \) and \( PB \), and also represents a pressure which alone would resist \( PC \) and \( PD \), so that we have our surmise confirmed, and may safely conclude that, so far at least as statical effect is concerned, the resultant of two pressures may be substituted for them.

Having taken any point \( p \) in space, draw \( pa \) parallel and equal to \( PA \) of fig. 12, and \( ag \) parallel and equal to \( Pb \), then it is evident that \( pg \), if joined, would be parallel and equal to the double of \( Pf \); it would, therefore, represent the resultant of \( PA \) and \( PB \). Draw now \( ge \) parallel and equal to \( Pe \), and it is obvious that \( ep \) is also parallel and equal to \( Pd \), so that we have this general law, that when four pressures acting on one point are represented in direction and in intensity by four connected lines taken in order, they balance each other; and this proposition is true whether the connected lines \( pa \), \( ag \), \( ge \), \( ep \), be or be not all situated in one plane.

The pressure represented by \( pe \) may be regarded as the resultant of two represented by \( pg \), \( ge \); but \( pg \) itself represents the resultant of \( pa \), \( ag \); so that \( pe \) may be regarded as the resultant of the three pressures \( pa \), \( ag \), \( ge \). Complete now the rhomb or parallelopiped, of which \( pa \), \( pb \), \( pe \) are three concurring sides, and we see that, of three pressures, \( PA \), \( PB \), \( PC \), represented by those lines, the resultant, viz., the opposite of \( PD \), is represented by the diagonal \( pe \) of the solid.

**EQUILIBRIUM OF SEVERAL PRESSURES.**

These observations on the equilibrium of four pressures applied to one point, may be extended to that of any number; and we have this general law, that if there be any number of lines connected in the manner of the sides of a polygon, but not necessarily in one plane, pressures represented in intensity and in direction by these lines, taken in order, will balance each other when they all act on one point. The equilibrium of many pressures is, however, best investigated by help of the

**DECOMPOSITION AND RECOMPOSITION OF PRESSURES.**

We have seen that the resultant of two pressures may be substituted for those pressures, and so, conversely, we may put, instead of a single pressure, any two or more pressures of which it would be the resultant; this substitution is called, though with a little impropriety of language, the decomposition of the pressure. As, for the purpose of indicating the relative positions of points, we use the method of co-ordinates, so we employ, with great advantage, the same method in examining the conditions of equilibrium of a system of pressures. We decompose each pressure into three acting in the directions of the axes of co-ordinates, and thus carry on our investigations more readily.

For the convenience of description let us refer the positions of the points of a system to three planes—one horizontal; one vertical, passing from east to west; and one also vertical, and passing from north to south. Let us measure \( x \) northwards, \( y \) eastwards, and \( z \) upwards. The direction in which a pressure acts may be indicated by what astronomers call its zenith distance, that is the angle which it makes with the vertical line \( z \), and its azimuth or bearing, that is the angle which its projection upon the horizontal plane makes with the north line \( x \).

In the first place, we may regard the actual pressure \( p \) as the resultant of two imaginary pressures—one, which we shall denote by \( p(h) \), horizontally in the direction of the projection, and the other \( p(z) \) towards the zenith; and, in the second place, we may consider \( p(h) \) as the resultant of \( p(x) \) towards the north, and \( p(y) \) towards the east. The values of these, expressed in trigonometric language, are—

\[ \begin{align*} p(x) &= p \times \sin \text{zen. dist.} \times \cos \text{bearing} \\ p(y) &= p \times \sin \text{zen. dist.} \times \sin \text{bearing} \\ p(z) &= p \times \cos \text{zen. dist.} \end{align*} \]

By making this decomposition for each one of the pressures acting on a point, and taking the sums of all those acting in each direction, we obtain the three elements of the resultant, viz.—

\[ \Sigma p(x); \Sigma p(y); \Sigma p(z); \]

from which we can compute the bearing, the zenith distance, and the intensity of the resultant; and, if these pressures keep each other in check, we must have

\[ \Sigma p(x) = 0; \Sigma p(y) = 0; \Sigma p(z) = 0 \]

three equations which contain the conditions of equilibrium.

When the system under examination contains several points acted on by various pressures, the above equations must hold good of each separate point, as, otherwise, that point could not be at rest.

**FLEXIBLE POLYGON.**

When a number of points are connected by a succession of ties which bind each one to only two others, the polygon thus formed is flexible whenever the number of the points exceeds three. Let \( ABCDEF \) (fig. 14) be such a polygon acted on by pressures \( Aa, Bb, \ldots Ff \); the directions of which may or may not be all in one plane; and let \( AB, BC, \ldots \) represent the tensions on the various parts of the cord; then, since the point \( A \) is kept in equilibrium by the three tensions \( Aa, AB, AF \), the direction and intensity of \( Aa \) could be discovered if the tensions \( AB, AF \) were known; and hence the problem, "to find those pressures in virtue of which the parts of a flexible polygon may be..." If we decompose each of these strains in the directions \(x, y, z\), we have for the point A,

\[Aa(x) + AB(x) + AF(x) = 0;\]

and again, for the point B,

\[BB(x) + BA(x) + BC(x) = 0;\]

and so on for every other point in the series. Adding all these equations together, and observing that since \(BA\) and \(AB\), being representatives of the tension on the cord \(AB\), are equal and opposite to each, \(BA(x) + AB(x)\) is zero, we have

\[Aa(x) + BB(x) + c = 0;\]

and similarly,

\[Aa(y) + BB(y) + c = 0\]

\[Aa(z) + BB(z) + c = 0;\]

wherefore, the extraneous pressures would, applied to one point, produce equilibrium.

The same kind of demonstration may be applied to any system whatever, and we have this general proposition, that "the extraneous pressures which, when acting on any system, keep it in equilibrium, would balance each other if they acted on one point." The converse of this proposition is not true.

The most interesting case of a flexible polygon is that of a chain fixed at its two extremities, and carrying weights attached to it at intervals along its length. As the theory of this kind of polygon forms the foundation of the doctrine of chain-bridges, and of arches in general, it deserves our particular attention.

Let ABCDEF (fig. 15) represent a chord to which are attached strings Bb, Cc, Dd, &c., carrying weights; and let us examine the conditions of its equilibrium. Since the three tensions \(BA, BC, BB\) balance each other at B, we have

\[BA : BC : \sin bBC : \sin bBA : \csc bBA : \csc bBC;\]

and again, on account of the equilibrium at C,

\[CB : CD : \csc cCB : \csc cCD;\]

wherefore, compounding these ratios we have, since the angle \(eCB\) is the supplement of \(bBC\),

\[BA : CD : \csc bBA : \csc cCD;\]

and, as this reasoning can be continued, it follows that, in every such suspended chain, the tensions at the various parts are proportional to the cosecants of their inclinations to the vertical line, or the secants of their inclinations to the horizon.

Let the distance \(Cp\) be measured along \(eB\) to represent the intensity of the strain on \(BC\), and similarly \(Cq\) along \(CD\); also, having drawn a horizontal line through \(C\), draw the vertical lines \(pr, qs\) to meet it; then the strain \(Cp\) may be decomposed into \(or, rp\); and the strain \(Cq\) into \(cs, sq\). The two horizontal strains, \(Cr, Cs\), are equal to and balance each other; while the difference between the two vertical strains, \(rp, sq\), must be equal to that caused by the weight hung on at \(c\). Hence, if \(h\) denote the horizontal strain,

\[BA = h, \sec \text{incl. } AB; CB = h, \sec \text{incl. } BC.\]

wherefore the weights hung on any part of such a chain are equal to the horizontal tension multiplied by the difference between the tangents of the inclinations at its two extremities.

These are the well known principles which regulate the construction of bridges; they are most distinctly seen in the case of suspension-bridges, but they are not less essentially connected with stone-arches.

**SUSPENSION-BRIDGES.**

The essential parts of a suspension-bridge are—first, the chain attached to two well secured supports; second, the suspending-rods; and, third, the roadway. In general, the weight of the roadway far exceeds that of the other parts; so that, in a first examination of the matter, we may take it alone into consideration, and suppose that the weights of the chain and of the suspending-rods are so small as not to be worth notice. Let, then, \(B'ABC\), &c., represent the roadway of a chain-bridge suspended, at equal intervals \(B'A, AB, BC, &c.\), by rods \(B'b, Aa, Bb, Cc, &c.\), from the chain \(babc\); and let us suppose that \(a\) is the lowest point of the chain, the parts \(ab'\) and \(ab\) being equally inclined to the horizon.

Put \(h\) to represent the horizontal tension, \(r\) the weight of one foot of the road-way, and \(a\) the distance between the vertical roads, \(ar\) being thus the weight supported by each rod. Then, since the links \(ab'\) and \(ab\) are equally inclined, the tangent of the inclination must be \(\frac{ar}{2h}\). Draw \(aK\) parallel to the horizon, and we must have

\[aK : Kb :: 2h : ar,\] whence \(Kb = \frac{ar}{2h}.\)

Passing to the point \(b\), at which again there is an equilibrium of three pressures, and drawing the horizontal line \(bL\) we have

\[\tan Lbc - \tan Kab = \frac{ar}{h},\] but

\[\tan Kab = \frac{ar}{2h};\] wherefore,

\[\tan Lbc = \frac{3ar}{2h};\] and

\[Le = a \tan Lbc = \frac{3ar}{2h}.\]

Proceeding in this way, we easily show that the distances \(Kh, Le, Md, &c.\), must be as the successive odd numbers, 1, 3, 5, &c., and therefore the rises above the lowest point \(a\), must, at \(b, c, d, &c.\), be as the successive square numbers 1, 4, 9, and so on; wherefore, the length \(Nr\) of a rod situated at \(n\) intervals from \(aA\), would be given by the equation and thus it seems that the chain \( babc \) must be inscribed in a parabola.

If, however, we take into account the weights of the links of the chain, and the varying lengths and weights of the suspending-rods, we must obtain a form differing perceptibly from the parabola. Putting \( c \) for the weight of one foot of the chain, where it is horizontal, and \( i \) for the inclination at any other part, the strain there is \( h \sec i \), and, therefore, as the chain ought to be thickened to bear the greater strain, the weight of one foot of it should be \( c \sec i \), and consequently the weight of that portion of the chain which is vertically over one foot of the roadway must be \( c \sec i \), and thus, exclusive of the suspending-rods, the whole weight above a foot of the roadway is

\[ r + c \sec i^2, \text{ or } r + c + c \tan i^2. \]

The suspending-rods, having to carry equal weights all along, must be of uniform strength; and hence their own weights are proportional to their lengths, so that the weight of the suspending-rods may be assumed as proportional to the area \( AaDD \) of the elevation of the arch. Let, then, \( x \) be the horizontal distance measured along \( AH \), \( z \) the ordinate \( Pp \), reckoned from the horizontal line passing through \( a \), and let \( s \) be the weight of the suspending-rods, corresponding to one square foot of side-area: then the weight above a minute element, \( dx \), of the horizontal line is

\[ dz = (r + c + c \tan i^2 + sx) dx; \]

and this, according to the law already explained, is equal to the differential of the tangent of the inclination, multiplied by the horizontal strain; therefore

\[ h \cdot d \tan i = (r + c + c \tan i^2 + sx) dx \]

expresses the condition of the equilibrium in chain-bridges when none of the accidents of their formation is neglected.

The complete integration of this expression cannot be obtained in finite formulae, but we can develop the value of the ordinate \( z \) in a series containing the even powers of \( x \); thus, if we put

\[ z = Ax^3 + Bx^4 + cx^6 + Dx^8 + Ex^{10} + \text{&c.}; \]

and if, for shortness sake we make

\[ A = \frac{r + c}{2h}; \quad a = 4Ac; \]

\[ B = A \frac{a + s}{3 \cdot 4 \cdot h}; \quad C = B \frac{4a + s}{5 \cdot 6 \cdot h}; \]

\[ D = B \frac{3a^2 + 20as + s^2}{5 \cdot 6 \cdot 7 \cdot 8 \cdot h^2}; \]

\[ E = B \frac{496a^3 + 474a^2s + 84as^2 + s^3}{5 \cdot 6 \cdot 7 \cdot 8 \cdot 9 \cdot 10h^3}, \text{&c.} \]

In all practical cases, the fractions \( \frac{a}{h} \) and \( \frac{s}{h} \) are small, so that the coefficients \( B, C, D, \text{&c.} \) decrease rapidly in value, and for all the wants of the engineer, the above terms are amply sufficient.

The question, "What is the extreme limit to which a chain-bridge can be thrown?" is an interesting one, particularly in reference to proposals which have, from time to time, been made for throwing such bridges across our estuaries, across even the Bosphorus, or the Dardanelles. In order to answer this question definitely, we may remark that an unloaded chain, no part of which is thicker than the strain upon it requires, must be capable of spanning the utmost stretch which the material will allow. Let us, then, suppose a chain having its horizontal tension \( h \) as great as is compatible with the thickness of \( c \) pounds per foot at the lowest part, to be thickened towards each end in such a way as in every part to be just able to bear the strain; then \( c \sec i \) will be the weight per linear foot at any point in the curve, and the condition of equilibrium will be expressed by the differential equation

\[ h \cdot d \tan i = c \sec i^2 \cdot dx, \]

from which we at once obtain

\[ i = \frac{cr}{h}; \]

so that in this, which may be called the catenary of regulated strength, the inclination of the curve at the point \( p \) is proportional to the horizontal ordinate \( aP_1 \) and becomes \( 90^\circ \), when \( x = \frac{\pi h}{2c} \); and thus the utmost horizontal stretch of such a chain is

\[ 2X = \frac{h}{c} \times 3.1415926, \]

and even to attain this the vertical height would need to be infinite. By putting for \( \frac{h}{c} \) the modulus of strength, or the length of itself which a uniform chain can support, we would have the absolute limit of span for any given material.

The form of the catenary of regulated strength is given by the equations

\[ z = \frac{h}{c} \cdot \text{nep. log sec} \frac{cx}{h}, \]

\[ l = \frac{h}{c} \cdot \text{nep. log tan} \left( \frac{\pi}{4} + \frac{cx}{2h} \right), \]

in which \( l \) is the length of the curve reckoned from the lowest point \( a \) to the top of the ordinate \( z \).

When the chain is of uniform thickness all along, the form which it assumes is called the catenary. This curve, although of no great importance in regard to chain-bridges, has a close connection with the theory of equilibrated arches, and possesses besides certain very remarkable geometrical properties. The same notation remaining, its equation of condition is

\[ h \cdot d \tan i = c \sec i; \]

whence we obtain, on integrating,

\[ z = \frac{h}{c} \times \frac{1}{2} \left\{ \frac{cx}{h} + \frac{cx}{h} - \frac{h}{c} \right\}, \]

\[ l = \frac{h}{c} \times \frac{1}{2} \left\{ \frac{cx}{h} - \frac{cx}{h} \right\}, \]

\( e \) being the basis of the Napierian system of logarithms. The utmost horizontal distance which can be spanned by a chain of uniform thickness is 13254838, or about \( \frac{4}{3} \)ths of its modulus of strength.

The theory of stone-arches is almost a counterpart of that of chain-bridges; the difference consisting principally in this, that the load on the stone-arch is the quantity of material included between the surface of the roadway and the lower face of the arch-stones. Let AR (figure 18) represent the line of road over a bridge, apq the arch-stones; then the load upon the part ap of the intrados is the quantity of material included in the figure aAPp.

When the entire space is filled up and the arrangements made suitable to this condition, the arch is said to be equilibrated. The roadway is commonly almost level; but, in the general investigation, we shall suppose it sloped at an angle \( \alpha \) to the horizon. The point \( a \), as before, being made the origin of rectangular co-ordinates, we have

\[ P_p = Aa - x \tan \alpha + z \]

or putting \( v \) for the distance \( Aa \), and \( u \) for \( P_p \),

\[ u = v - x \tan \alpha + z, \]

and the condition of equilibrium is

\[ \frac{hd}{\tan i} = k \frac{d^2u}{dx^2} = u dx; \]

wherefore, if \( a \) be taken at that point of the curve where it is parallel to the roadway, we have, on integrating,

\[ u = \frac{v}{2} \left( e^{x/\sqrt{h}} + e^{-x/\sqrt{h}} \right), \]

which is the equation of a modified catenary.

It may be remarked, that the chords of this curve drawn parallel to the roadway are bisected by the vertical axis \( Aa \); and that, contrary to what has been supposed, the case of bridges with oblique roadways differs in no essential point from that of bridges with the roadway horizontal.

**EQUILIBRIUM OF PARALLEL PRESSURES.**

Parallel pressures can only balance each other by the intervention of some solid body, which may form a connection between the points to which the various pressures are applied. The simplest mode of viewing the subject is to imagine the points to be connected by straight rods, the lengths of which would, geometrically, determine the relative positions of the parts, and to investigate at each point of attachment the distribution of the pressures. But this method is inapplicable when all the points of attachment are situated in one plane; in such a case, we should have to assume some accessory points out of the plane so as to communicate firmness to the system.

If, in some plane which passes entirely to one side of the system of connected points, three points not in one straight line be chosen, the distances of any point in the system from these three arbitrary points will determine its position, and we may thus regard the rigidity of the system as obtained by means of inextensible rods connecting each point in it with each one of the three assumed points. Let, then, A, B, C (figure 19), be the three arbitrarily assumed points, and P one of the points in the system to which the pressure \( P_p \) is applied. These being connected by slender rods, the pressure \( P_p \) is resisted by the tensions of PA, BP, CP. In order to obtain the values of these tensions, let the direction \( P_p \) be continued to meet the plane ABC in \( p \), and through \( p \) lead planes parallel to BPC, CPA, APB; these, with the planes to which they are parallel, include a rhomb, of which \( P_p \) is the diagonal, so that if, for the moment, \( P_p \) be taken to represent the intensity of the pressure \( P_p \), the sides of that rhomb would represent the strains AP, BP, CP. To avoid confusion, this rhomb is not drawn in the figure, but only the plane passing through \( p \) parallel to BPC is shown, cutting PA in \( a \); \( Pa \) in this case represents the tension \( P_A \).

The rod PA conveys this strain to the point A, which is acted on by three pressures \( P_A, AB, AC \); and these not being in one plane, cannot be in equilibrium; a fourth pressure must therefore aid in establishing the balance. Through \( a \) draw \( aa \) parallel to \( Ap \); the pressure \( Pa \) acting at A may be decomposed into two, one parallel and proportional to \( aa \), another parallel and proportional to \( Pa \). That part represented by \( aa \) being exerted in the direction \( PA \) is resisted by the tensions of the rods \( AB, AC \); and the other part represented by \( Pa \), exerted in the direction of a line drawn through A parallel to \( P_p \), must be resisted by the other pressures, to which the system is subjected.

Since \( P_p : Pa :: PA : Pa \), and since the altitudes of the trigons BAC, BpC are evidently in the same ratio, we have—

\[ P_p : Pa :: ABC : BpC; \text{ and similarly} \]

\[ P_p : P_B :: ABC : CpA, \]

\[ P_p : P_C :: ABC : ApB, \]

so that the pressure \( P_p \) is distributed among the points A, B, C in shares proportional to the areas of the opposite trigons BpC, CpA, ApB.

If the same process be followed for each pressure, we obtain, on summing up all the shares which are resisted at each of the points A, B, C,

\[ \Sigma Pa = 0; \Sigma P_B = 0; \Sigma P_C = 0. \]

Since each pressure \( P_p \) is equal to the sum of the three \( Pa, PB, PC \), it follows, from the addition of the above equations, that the sum \( \Sigma P_p \) of all the parallel pressures must be zero; or, in other words, that the sum of those which act in the one direction must be equal to the sum of those which oppose them.

The trigons BAC and BPC having a common base BC, are proportional to their altitudes, and these altitudes again are proportional to the distances of the lines \( Aa' \) and \( P_p \) from BC, wherefore the tendency which the pressure \( P_p \) has to turn the system round BC as an axis, is just equal to that which \( Aa' \) would have, and thus the expression, \( \Sigma Aa' = 0 \) or \( \Sigma Pa = 0 \), may be interpreted as signifying that the tendencies of the several pressures to turn the system round any assumed axis BC must amount to zero; and it is easy to see that if, while \( \Sigma P_p \) the sum of the pressures themselves is zero, the sum of the tendencies round any two axes having different azimuths round the direction \( P_p \) be also zeroes, the system must be in equilibrium.

The above reasoning can, in strictness, only apply to a network or skeleton of straight rods connecting the points to which the pressures are applied; and seeing that we know nothing, and are not likely ever to know anything of the manner in which pressure is conveyed from one point to another of a solid body, it is only by inference that we can extend the doctrine to the general case of points anyhow connected. Experience, however, abundantly testifies that this law of equilibrium is universal.

This course of reasoning leads us to the remarkable conclusion that the pressure \( P_p \) exerts the same influence, so far as the ultimate resistances are concerned, at whatever point in the line of its direction it may act; but it is to be noticed, that the strains internal to the system of connecting-rods are altered by every change in the position of the point of attachment. Gravitation affords the best example of parallel pressures; each part of a body tends downwards, or has weight, and thus the mere resting of any object whatever upon a support implies the equilibrium of a multitude of conflicting pressures.

In order to examine this important subject minutely, let us consider the case of two material points, A and B, connected by a rigid straight rod AB.

If AB be divided at E, so that EB : EA :: A : B a pressure \(-(\bar{A} + \bar{B})\) applied at E would resist the two strains \(\bar{A}\) and \(\bar{B}\), and this in whatever position the rod AEB may be placed. E, in that case, is called the centre of gravity of the two weights A and B.

If we put \(x_A, x_B, x_E\) to represent the distances of A, B, E, from some assumed plane surface, we have

\[ (\bar{A} + \bar{B}) x_E = \bar{A} x_A + \bar{B} x_B. \]

Suppose now that there is a third point C, the weight of which is \(\bar{C}\), join CE and divide it at F, so that CF : FE :: \(\bar{A} + \bar{B} : \bar{C}\); then a pressure \(-(\bar{A} + \bar{B} + \bar{C})\) applied at the point F would resist \(\bar{A} + \bar{B}\) applied at E, and \(\bar{C}\) applied at C, that is, it would resist the three pressures \(\bar{A}, \bar{B}, \bar{C}\), applied at the points A, B, C.

Taking a fourth point D, which may or may not be in the same plane with A, B, C, join DF, and divide DF in the ratio of \(\bar{A} + \bar{B} + \bar{C} : \bar{D}\); the pressure \(-(\bar{A} + \bar{B} + \bar{C} + \bar{D})\) acting at G would balance the weights \(\bar{A}, \bar{B}, \bar{C}, \bar{D}\), placed at the several points A, B, C, D. In this case G is called the centre of gravity of the four weights \(\bar{A}, \bar{B}, \bar{C}, \bar{D}\); and in reference to any plane,

\[ (\bar{A} + \bar{B} + \bar{C} + \bar{D}) x_G = \bar{A} x_A + \bar{B} x_B + \bar{C} x_C + \bar{D} x_D. \]

As this line of argument can be continued to any extent, it follows, generally, that if \(x, y, z\) be the co-ordinates referred to three planes, the position of the centre of gravity G of any number of heavy points A, B, C, &c., is given by the equations

\[ x_G = \frac{\sum \bar{A} x_A}{\sum \bar{A}}, \quad y_G = \frac{\sum \bar{A} y_A}{\sum \bar{A}}, \quad z_G = \frac{\sum \bar{A} z_A}{\sum \bar{A}}. \]

These formulae, however, only apply to discrete points; in order to make them applicable to concrete matter we must extend the calculations to the infinity of infinitely minute parts of which we may conceive solid matter to be composed. For the purposes of this inquiry, we regard a solid body as composed of a multitude of slices, which may be rendered so numerous and so thin, that each one may, so to speak, be regarded as a mere surface. Such surfaces, again, we divide into elongated sections, which, when made excessively narrow, are almost lines; and these lines we again divide into multitudes of parts, which, from their smallness, we call points. The ideas of physical points, lines, surfaces, thus differ essentially from those which the geometer designates by the same names; and it is also to be noticed that this ideal decomposition of a solid body into an infinity of infinitely minute particles, does not show either the infinite divisibility of matter, or any other fact concerning its physical constitution.

The centre of gravity of a uniform physical straight line is, clearly, its middle. This becomes evident when we consider it as composed of equal parts symmetrically arranged from either end. The centre of gravity of each pair of such particles is at the middle point, therefore that middle point is the centre of gravity of the whole. But this mode of reasoning will not apply in more complex cases. Representing by \(w\) the whole weight of the thin rod AB, and by \(l\) its length, let us divide the whole length into a great number \(p\) of equal parts, of which \(Pp\) may represent one; the weight of this part \(Pp\) must be \(\frac{w}{p}\), and the tendency of this weight to turn the rod round A as a fulcrum must be more than if an equal weight were hung on at P; and less than if it were suspended at P; hence, if the number of parts in AP be \(n\), and therefore AP itself \(\frac{nl}{p}\), the effect of \(Pp\) in turning the rod round A is more than \(\frac{nwl}{pp}\) and less than \(\frac{(n+1)wl}{pp}\). As the same may be said of each of the other parts \(Pp\), it follows that the tendency of the whole weight to turn the rod round A as a fulcrum is more than \((0+1+2+3+\ldots+p-1)\frac{wl}{pp}\) and less than \((1+2+3+\ldots+p)\frac{wl}{pp}\); or, summing these series of natural numbers, more than \(\frac{p^2 - 1}{2p} wl\), but less than \(\frac{p^2 + 1}{2p} wl\); these limits may be otherwise written \(\left(1 - \frac{1}{p}\right)\frac{wl}{2}\) and \(\left(1 + \frac{1}{p}\right)\frac{wl}{2}\). By taking the number \(p\) enormously large, the value of the fraction \(\frac{1}{p}\) is made insignificant, and each of the limits approaches to \(\frac{1}{2} wl\), which thus takes the place of the sum \(\sum \bar{A} x_A\) of the general formula. Dividing this by \(w\), we find for the distance of the centre of gravity G from the end A, the value \(AG = \frac{1}{2} l\); that is, G is in the middle of AB. The ordinary operation of integrating is, virtually, this process in disguise. Thus, if we put \(AP = x\),

\(Pp = dx\), the weight of \(Pp\) is \(\frac{wdx}{l}\), and its effect in turning the lever round A (or its moment as it is called) \(\frac{wdx}{l}\);

the integral of this, viz., \(\frac{1}{l} \frac{1}{2} x^2\), represents the moment of AP, which, on putting \(l\) for \(x\) becomes \(\frac{1}{2} wl\) for the moment of the whole rod as before.

The position of the centre of gravity of a uniform flat trigonal surface may be found by an extension of the same method.

Let \(w\) be the weight of the trigon ABC; through A draw an axis \(AY\) parallel to BC, and suppose the surface to be divided into trapezoids by lines QR, QT drawn also parallel to BC. Put \(l\) for the length of the perpendicular AD, and suppose it divided into \(p\) equal parts \(n\) of which go to AP; then the weight of the trigon AQR is \(\frac{n^2}{p^2} w\), while that Centre of Gravity.

of Aqr is \((n+1)^2\) \(w\), so that the weight of the trapezoid QRrq is \(\frac{2n+1}{p}\) \(w\). The moment of this in reference to the axis Ay is more than \(\frac{n}{p} \cdot \frac{2n+1}{pp}\) \(wl\), and less than \(\frac{n+1}{p} \cdot \frac{2n+1}{pp}\) \(wl\); it is therefore between \((2n^2+n)\) \(wl\) and \((2n^2+3n+1)\) \(wl\). Giving to \(p\) all values from 0 to \(p-1\) successively, and summing, we find that the moment of the whole surface is between

\[ \left\{ \frac{2}{3} \left( 0^3 + 1^3 + \ldots + p^3 - 1 \right) + (0 + 1 + \ldots + p - 1) \right\} \frac{wl}{p^3} \]

or,

\[ \left\{ \frac{2}{3} \left( 0^3 + 1^3 + \ldots + p^3 - 1 \right) + 3 \left( 0 + 1 + \ldots + p - 1 \right) + p \right\} \frac{wl}{p^3}. \]

Therefore, on supposing \(p\) to be an enormously large number, the true value of the moment of the whole trigon is \(\frac{2}{3} wl\), and the distance of the centre of gravity from the axis Ay is two-thirds of the altitude AD. Now the centre of gravity must also be at the distance of two-thirds of the altitude drawn from C to AB; therefore, it is the intersection of the lines which join the corners with the middles of the opposite sides. The centre of gravity of the surface of a trigon coincides with that of three equal weights placed at its three corners.

Similarly it may be shown that the centre of gravity of a tetrahedron or triangular pyramid, is the intersection of the four lines which join the four corners with the centres of gravity of the opposite faces, and therefore coincides with that of four equal weights placed at the four corners.

Since all polygons may be decomposed into trigons, and all polyhedrons into tetrahedrons, we can easily find the centre of gravity of any rectilineal figure, or of any flat-faced solid, by imagining the weight of each of the component parts accumulated at its centre of gravity, and then finding the centre of gravity of these points. In the cases of surfaces bounded by curved lines, or of solids bounded by curved surfaces, the nature of the curve necessarily enters as an element of the computation. One or two examples may suffice to explain the procedure.

Let it be required to find the centre of gravity of a parabolic segment CAD.

Draw any chord QR parallel to CD, bisect CD in B, QR in P, and join BP, producing it to A; then, according to the property of the parabola, the line Ay drawn parallel to CD is a tangent to the curve; all chords parallel to CD are bisected by AB, and their lengths are in ratios sub-duplicate of the distances intercepted from A. Hence, if we put, as before, \(w\) for the whole weight or area, \(l\) for AB, \(x\) for AP, \(dx\) for PP, also \(b\) for CD, and \(a\) for the angle ABD, we have \(QR = b l^{-\frac{1}{2}} x^{\frac{1}{2}}\); therefore the area of the surface comprised between the proximate ordinates QR and qf is \(bl^{-\frac{1}{2}} \sin a \cdot x^{\frac{1}{2}} dx\), and its moment round the axis Ay is this, multiplied by \(x \sin a\) or \(dm = bl^{-\frac{1}{2}} \sin a \cdot x^{\frac{1}{2}} dx\).

On integrating these expressions, we find for the area QAR, \(\frac{2}{3} bl^{-\frac{1}{2}} \sin ax\); and for its moment \(\frac{2}{3} bl^{-\frac{1}{2}} \sin ax \cdot x^{\frac{1}{2}}\); Centre of Gravity.

so that, making \(x\) equal to the whole length \(l\), we have \(w = \frac{2}{3} bl \sin a\), while the whole moment is \(\frac{2}{3} bl^{\frac{3}{2}} \sin a\); and therefore the distance of the centre of gravity from the tangent Ay is \(\frac{3}{5} l \sin a\); therefore, if AG be made three-fifths parts of AB, the centre of gravity must lie in the ordinate drawn through G; but, as all these ordinates are bisected, their centres of gravity must be in AB, wherefore G itself is the centre of gravity of the parabolic segment.

As an example from among solids, we may take the segment of a sphere. Let OA \(= r\) be the radius of a sphere, and let it be proposed to find the centre of gravity of the segment having AP \(= x\) for its altitude. Making \(Pp = dx\) and leading planes PQ, \(pq\) perpendicular to AO, the solid comprised between these two planes is the differential of the solidity, or \(dw\), but the area of the circle described by PQ is \(\pi \cdot PQ^2\), or \(\pi \cdot AP \cdot PB = \pi (2rx - x^2)\); therefore \(dw = \pi (2rx - x^2) dx\), and the moment of this is \(dm = \pi (2rx^2 - x^3) dx\) round an axis passing through A.

Integrating these we have \(w = \pi \left( \frac{2}{3} rx^3 - \frac{1}{4} x^4 \right)\); therefore the distance of the centre of gravity from A is \(\frac{x}{4} \frac{3r-x}{3r-x}\), and its distance from P is \(\frac{x+4r-x}{4} \frac{3r-x}{3r-x}\). On making \(x=r\) we find that the distance of the centre of gravity of a hemisphere from the vertex is \(\frac{5}{8} r\), and from the centre \(\frac{3}{8} r\).

When a solid body is suspended by a thread, it takes such a position that its centre of gravity lies in the direction of the thread; hence, if we suspend an object first from one point and then from another, noting the direction each time, the intersection of these two directions will give us the position of the centre of gravity.

If, when a body is laid upon a flat surface, the vertical line drawn through the centre of gravity pass within the convex polygon formed by joining the points at which it touches, it rests securely. Any attempt to overturn it is accompanied by a rise in the position of the centre of gravity, and as soon as the disturbing influence ceases, the body returns to its former site. But if the under surface of the object be curved, the slightest influence causes a displacement; if the arrangement be such that the centre of gravity tend to rise, as is the case with a spherical segment laid on a flat surface, the tendency is to return again to the first position; in such a case the equilibrium is said to be stable; if, as in the case of a complete sphere or cylinder, the centre of gravity describe a horizontal line, the equilibrium is indifferent; and if, as when the centre of gravity lies above the centre of curvature, that point tend to come lower, the body falls still farther, and the equilibrium is said to be unstable.

In all statical computations we may proceed as if the whole weight of a body were concentrated at its centre of gravity; but whenever we have to consider motion this hypothesis must be abandoned, because the motion of the GENERAL CONDITION OF EQUILIBRIUM.

The equilibrium of a number of pressures, acting in various directions upon a solid body, may be investigated by decomposing each pressure into three, acting severally in directions parallel to the axes of co-ordinates \(x\), \(y\), \(z\), and considering that each set of components should, separately, be in equilibrium.

VIRTUAL VELOCITIES.

The whole doctrines of statics are summed up in one general proposition, from which we can most readily obtain the solution of any particular case. To this proposition the name, law of virtual velocities, has unfortunately been given; in its statical aspect it has nothing to do with velocity. As a thorough comprehension of its import almost amounts to a complete knowledge of this department of science, we shall endeavour to lay it clearly before the reader.

Let us suppose that the point \(P\) is kept in equilibrium by three ropes \(PA\), \(PB\), \(PC\), stretched each from a considerable distance. This being the case, let us imagine that, by some agent exterior to the system, the point \(P\) has been displaced to \(p\), the ropes now taking up the positions \(pa\), \(pb\), \(pc\). By this derangement some of the ropes may have been pulled forwards, some may have been let backwards. Let us suppose that three thin, straight wires are thrust, at right angles, one through each of the ropes, so as to form a trigon \(EDF\); then, since this trigon is evidently similar to any trigon formed by drawing lines parallel to the directions of the three ropes, each of its sides may be taken to represent the strain on the rope to which it is perpendicular; that is, \(EF\) may represent the tension of \(PA\), \(FD\) the tension of \(PB\), and \(DE\) the tension of \(PC\). With the derangement of the system the trigon \(EDF\) is displaced, being brought into the position \(edf\), such that the lines \(Dd\), \(Ee\), \(Ff\) are parallel and equal to \(Pp\). The distance between \(EF\) and \(ef\), measured perpendicularly, is, obviously, the distance by which the rope \(PA\) has been pulled forward; so the distance between \(FD\) and \(fd\) shows by how much the rope \(PB\) has been let backwards, and so of the third rope \(PC\). The area of the rhomboid \(Efde\) thus represents the product of the tension of the rope \(PA\) by the distance through which that tension has been overcome; that is, the quantity of work which has been gained upon it. In the same way the area \(FDdf\) represents the quantity of work lost by the letting down of the rope \(PB\), and \(DdeE\) the work lost by letting down the rope \(PC\). Now, if, from the pentagon \(EFfde\) we take away the trigon \(edf\), there remains the rhomboid \(Effe\); while if, from the same pentagon, we take away \(EDF\), the two rhomboids \(FDdf\), \(DdeE\) remains; wherefore the gain of work on the rope \(PA\) is exactly equal to the sum of the two losses on \(PB\) and \(PC\); on the whole, then, there is neither gain nor loss.

The same kind of reasoning can be applied to all combinations of pressures, and thus we obtain the following virtual general law, that "if any system of pressures in equilibrium be slightly disturbed, there is neither gain nor loss of work by the disturbance."

Strictly speaking, this proposition is only true of infinitesimally small disturbances, and this circumstance is intended to be pointed out by the use of the word virtual in the title, law of virtual velocities, which is given to this theorem.

Using the word work to mean the product of a pressure by the distance through which that pressure has been overcome, and representing it in general by the letter \(w\), let us suppose that by the performance of a certain quantity of work some system has been brought into its actual position, as represented by the co-ordinates \(x\), \(y\), \(z\). If now the system be subjected to a definite change, represented by the symbols \(\delta x\), \(\delta y\), \(\delta z\) of finite or actual differences, the quantities of work gained or lost on the several strains may be deduced from purely geometrical considerations, and, according to Taylor's well-known theorem, the change in the quantity of work is given by the expression

\[ \delta w = \frac{dw}{dx} \delta x + \frac{dw}{dy} \delta y + \frac{dw}{dz} \delta z \\ + \frac{d^2w}{dx^2} \delta x^2 + \frac{d^2w}{dy^2} \delta y^2 + \frac{d^2w}{dz^2} \delta z^2 \\ + \frac{d^3w}{dx^3} \delta x^3 + \frac{d^3w}{dy^3} \delta y^3 + \frac{d^3w}{dz^3} \delta z^3 + \text{etc.} \]

Now, the law enunciated above refers only to the first differential co-efficients; that is to say, for mere equilibrium we must have \(\frac{\partial w}{\partial x}\), \(\frac{\partial w}{\partial y}\), \(\frac{\partial w}{\partial z}\) each zero; and these conditions must hold good in reference to every separate point of the system.

The values of the second derivatives determine whether the equilibrium be stable, indifferent, or instable. When these derivatives have positive values the equilibrium is stable; it required the application of actual work to change the position of the system; and when the extraneous pressures cease to act, the system again returns to the position of equilibrium. When the higher derivatives also are zeroes, the equilibrium is indifferent; and, when the second derivatives are negative, the system, if once disturbed from the position of equilibrium, will tend to go still farther from that position.

This theorem, besides being of great value as enabling us readily to investigate the strains in any structure, is important as throwing a clear light upon the general action of machines as transmitters of force. As the machine contains in itself no source of power, it can only convey force, it cannot augment it; thus, although a crane enable one man to raise at once a load which he could only have lifted piece-meal, it does not enable him to do more work than he could have done in the same time with the load divided to suit his strength, it only enables him to perform the work more conveniently; nay, on account of the friction of the parts, it even reduces slightly the actual amount of work done. This consideration at once dispels all those dreams about perpetual motions which have emptied the purses of so many schemers, and, at times, even ruined their intellects.

(E.S.)