weights w and w1, exert themselves in the directions of gravity, let the lines, in which they act, be produced upwards in a contrary direction, and let p e, be taken, of any convenient length, at pleasure, to represent the measure of the weight applied at P; then, on pe, as a diagonal, construct the parallelogram r a e b, and pa, will be the measure of the tension in the cord a P, while Pb, is the tension in the middle part p p1.

Now, when the cord is strained by the two weights w, and w1, acting at the points P, and p1, the middle portion p p1, is manifestly in the same condition as if it were fastened at P, and p1, under the same degree of tension; therefore, by proposition 2, whatever tension is produced at P, in the direction p p1, the same is produced at p1, in the direction pl p; hence the following construction :

Upon p1 P, the middle portion of the cord, set off p1 c, equal to pb, and through the point c, draw cf, parallel to p1 B, and meeting p1f, in f; complete the parallelogram p1 cƒd, and p1 d, will be the measure of the tension in the portion в p1; and p1 F, that of the weight w1, in the direction of gravity.

Make a R, equal to the sum of Pe, and p1ƒ, and produce a q, and в Q, beyond the point of intersection to D, and c; then, upon Q R, as a diagonal, describe the parallelogram QC RD; then is Q D, the measure of the tension in the cord a Q, and a c, the measure of that in BQ. But Q D, is equal to P a, and a c to p1 d, as will readily be perceived by those who are acquainted with the elements of geometry; therefore, the tension at a, and в, is precisely the same, whether it be produced by the two weights w, and w1, acting separately at the points P, and p1, or by the single weight w, equivalent to their sum, acting at the point a.

If another weight were added at a separate point in the cord, and the whole system left to the action of its own gravity, the result would still be the same; and if ever so many weights were applied at different points in the cord, the tension on the extreme portions would be precisely the same as if a single weight equivalent to their sum, were applied at the intersection of their directions. By this law, therefore, the tension in the chains of a suspension bridge can easily be found.

PROPOSITION 5.-If a heavy bar, of uniform figure and density throughout its length, be supported horizontally by two equal forces acting at its extremities in a vertical direction, or perpendicular to the horizon :

Fig. 4.

B

The cords, on which the forces act, will be equally stretched in all their parts, and each of the forces will sustain an equal portion of the weight of the bar. Let A B, fig. 4, be a straight inflexible rod or bar, of uniform figure and density, sup- O ported in a horizontal position by the two equal weights w, and w1, attached to the cords A pba, and pb в, passing over the pullies p, and p, in such a manner, that the portions b A, and b в, shall be perpendicular to the horizon, and consequently parallel to each other. Then, since the cords are no where attached to any object but at their extremities, and because the weights w, and w1, are equal between themselves, it follows, that the cords are equally stretched in all their parts, and each of them sustains one-half the weight of the bar a B, the tension on each cord being measured by the weight w, in whatever manner it may be expressed.

But if, instead of having two cords and two equal weights attached to the bar A B, we conceive one cord and one of the weights to be withdrawn, and in their stead, to have a solid immoveable prop or fulcrum, supporting the extremity в, as represented in fig. 5, while the other extremity

Fig. 5.

A, is supported by the weight and cord as before, then it is manifest, that half of the bar is supported by the fulcrum at B, and the other half by the weight w, the tension in every part of the cord being the same as in the preceding case.

B

If the direction of the cord should be inclined to the horizon, in an angle B A E, then the tension on the cord A E, is no longer represented by the weight w, nor is the pressure on the fulcrum B, equivalent to half the weight of the bar; for, in consequence of the oblique direction of the cord, there must be a force excited in the bar, in the direction of its length, so that some ob

stacle is required beyond the fulcrum, to prevent the bar from being urged in that direction by the weight w, acting obliquely at A, the other extremity. The bar is now sustained by three forces, namely, the pressure on the fulcrum at B, exerted vertically, the thrust on the obstacle beyond B, exerted horizontally, and the oblique force in the direction A E. From the fulcrum B, draw the straight line в D, perpendicular to A E, the direction in which the weight acts; then is B D, the leverage which produces the tension in the cord A E.

Fig. 6.

2

Let us now suppose that the weight of the bar is accumulated at its centre of gravity, as in fig. 6, where the weight is represented by w, suspended from the middle of the length. This is a

supposition perfectly admissible in mecha- A

nics, and is often resorted to for the pur

pose of simplifying the investigations, the absolute effect, in that case, being precisely the same as that which is produced by the bar in its natural state.

Then, since B D, is perpendicular to A E, the magnitude of the power, which, acting in the direction a E, sustains the system in a state of equilibrium, is found by the following proportion, viz:

But, by the principles of plane trigonometry, the perpendicular BD, is equal to A B x sin. L B A D, and, by the nature of the question, B G A B; therefore, by substitution, it is These then, are the

ABX W

PABX sin. L B A D

w cosec. L B A D.

principles by which the tensions in the different parts of the chain are to be calculated, and it has now to be shewn, that the tensions decrease from the base to the middle of the bridge, the strain being taken up in succession by the oblique suspending rods.

Let A B, fig. 7, be a bracket, abutting

against the wall E B, or what is the same
thing, let it be supposed to represent
half the platform or road-way of a sus-
pension bridge, and let a E,
taining chain, (it is of no consequence

be the sus

Fig. 7.

E

Ph D

whether it be considered as a straight line or a curve, for the

principle is the same on both suppositions); then it is obvious, that the platform A B, is sustained in the horizontal position by a tension in the chain A E, and a compression in the direction a B.

It is further evident, that in whatever manner the tension is transmitted to the point of support at E, it must always be of the same intensity at that point, while the weight of the platform remains the same; but the weight of the platform is constant, consequently the tension, on the point E, is constant also.

Now, let g h, ef, c d, and a b, be four oblique bars, attached to the chain at the points g, e, c, and a, and the platform at the points h, f, d, and b, and let these bars be brought to the proper degree of tension, in order to transmit the proportional weights of the platform to the chain at their respective points of junction. Then, if we suppose only one of them as g h, to act at the same time with the force in the direction of the chain, it is clear that the tension on g E, that part of the chain contained between the point of support, and the point where the bar is attached, must be equal to the tension in a g, the lower part of the chain, and of that in the bar g h; that is, the tensions in a g, and g h, are, together equal to the tension in g E, so that the strain on a g, being less than that on g E, it may be made of less dimensions.

Again, let another bar ef, be brought into action at the same time with gh, then the tension on ge, is equal to both the tensions on a e, and ef, so that ec, may be made of less dimensions than ge, the tensions on ec, ef, and gh, being, together, equal to the constant tension on g E.

If another bar cd, be brought into action at the same time with gh, ef, and a E, then the tension on a c, will be still more diminished, and of course, so may the dimensions, for the total constant strain is now transmitted through the lower part of the chain a c, and the three oblique bars c d, ef, and g h. Pursuing this mode of reasoning to any number of subsidiary bars, it will be found that the tension on the lower portions of the chain continually diminishes, a proportional part of it being taken up by each successive bar, so that at the centre of the bridge, where the position of the bars is reversed, the strain upon the chain is

evanescent.

This, then, is the principle of Mr. Dredge's bridge, in as far as the tapering of the chain is concerned; and a further advantage is gained by the obliquity of the suspending-bars, as well as by the curvature or deflection of the chains.

That part of the advantage which is gained by the obliquity of the suspending-rods, has a direct reference to the action of the lever; for let e p, fig. 7, be a perpendicular upon the platform A B, and suppose it to be attached to the same point of the chain as the oblique rod ef, then, since each of the suspending-rods, throughout the system, must support its own portion of the roadway, it is obvious that the portion в ƒ, between the fulcrum at B, and the point of attachment at f, will be more easily supported, by a force acting in the direction ef, than by the vertical force acting in the direction e p; not only because the oblique force acts at a greater distance from the fulcrum, but also in consequence of a thrust against the abutment at B, which does not obtain by the action of the vertical force.

With respect to the position of the suspending-bars, it has only to be observed, that the distance B D, must be fixed upon, according to the deflection of the chain and span of the arch, and the remaining part A D, is then divided into as many equal parts as there are links in the chain; so that, if the chain were a straight line, as A E, all the bars would be parallel to each other, making equal angles with the platform; but in consequence of the flexure of the chain, they deviate a little from the parallelism, in proportion to their respective distances from the points of suspension at E. This deviation, however, being a variable quantity, cannot be ascertained without applying the calculus of variables; but, in general, its quantity is so small, as to produce no appreciable effect upon the results. Indeed, the inventor erects all his bridges on the supposition of parallel bars; for, in this way, whatever deviation may occur from the results of theory, will always be found to lean to the side of safety.