68

Strength of loaded point is pulled down, and the fpace through which Materials, it is drawn may be called the DEFLECtion. This may be confidered as the fub-tenfe of the angle of contact, or as the Deflection. verfed fine of the arch into which the beam is bent, and is therefore as the curvature when the length of the arches is given (the flexure being moderate), and as the square of the length of the arch when the curvature is given. The deflection therefore is as the curvature and as the fquare of

69

The theo

the laws of

tion.

the length of the arch jointly; that is, as

3 13 av fb dз

31w fbd3 fb d3×12, or as The deflection from the primitive shape is therefore as the bending weight and the cube of the length directly, and as the breadth and cube of the depth inversely. In beams juft ready to break, the curvature is as the rems refult- depth inverfely, and the deflection is as the fquare of the ing from this fubject length divided by the depth; for the ultimate curvature at afford the the breaking part is the fame whatever is the length; and fineft me in this cafe the deflection is as the fquare of the length. thods of exWe have been the more particular in our confideration amining of this fubject, because the resulting theorems afford us the corpufcular fineft methods of examining the laws of corpufcular action, that is, for discovering the variation of the force of cohefion by a change of diftance. It is true it is not the atomical law, or HYLARCHIC PRINCIPLE as it may juftly be called, which is thus made acceffible, but the specific law of the particles of the substance or kind of matter under examination. But even this is a very great point; and coincidences in this refpe&t among the different kinds of matter are of great moment. We may thus learn the nature of the corpufcular action of different fubftances, and perhaps approach to a discovery of the mechanism of chemical affini ties. For that chemical actions are infenfible cafes of local motion is undeniable, and local motion is the province of mechanical difcuffion; nay, we fee that these hidden changes are produced by mechanical forces in many important cafes, for we fee them promoted or prevented by means purely mechanical. The converfion of bodies into elaftic vapour by heat can at all times be prevented by a fufficient external preffure. A ftrong folution of Glauber's falt will congeal in an inftant by agitation, giving out its latent heat; and it will remain fluid for ever, and return its latent heat in a close vetfel which it completely fills. Even water will by fuch treatment freeze in an inftant by agitation, or remain fluid for ever by confinement. We know that heat is produced or extricated by friction, that certain compounds of gold or filver with faline matters explode with irrefiftible violence by the fmalleft preffure or agitation. Such facts should roufe the mathematical philofo. pher, and excite him to follow out the conjectures of the illuftrious Newton, encouraged by the ingenious attempts of Bofcovich; and the proper beginning of this ftudy is to attend to the laws of attraction and repulfion exerted by the particles of cohering bodies, discoverable by experiments made on their actual extenfions and compreffions. The experiments of fimple extenfions and compreffions are quite infufficient, because the total stretching of a wire is so fmall a quantity, that the mistake of the rooth part of an inch occafions an irregularity which deranges any progreffion fo as to make it ufelefs. But by the bending of bodies, a diftenfion of sth of an inch may be eafily magnified in the deflection of the fpring ten thousand times. We know that the investigation is intricate and difficult, but not beyond the reach of our present mathematical attainments; and it will give very fine opportunities of employing all the addrefs of analyfis. In the last century and the beginning of the present this was a fufficient excitement to the firft ge

Materials.

niufes of Europe. The cycloid, the catenaria, the claftic Strength of curve, the velaria, the caustics, were reckoned an abundant recompenfe for much fudy; and James Bernoulli requested, as an honourable monument, that the logarithmic fpiral might be inscribed on his tombstone. The reward for the ftudy to which we now prefume to incite the mathematicians is the almost unlimited extenfion of natural science, important in every particular branch. To go no further than our prefent fubject, a great deal of important practical knowledge refpecting the ftrength of bodies is derived from the fingle obfervation, that in the moderate extenfions which happen before the parts are overstrained the forces are nearly in the proportion of the extenfions or separations of the particles. To return to our subject.

70

James Bernoulli in his fecound differtation on the elaftic Bernoulli

calls in

curve, calls in queftion this law, and accommodates his in- queftion veftigation to any hypothefis concerning the relation of the this law, forces and extenfions. He rclates fome experiments of lute ftrings where the relation was confiderably different. Strings of three feet long, 2, 4, 6, 8, 10 pds.

Stretched by

Were lengthened

9, 17, 23, 27, 30 lines. But this is a moft exceptionable form of the experiment. The ftrings were twisted, and the mechanism of the extenfions is here exceedingly complicated, combined with compreffions and with transverse twists, &c. We made experiments on fine flips of the gum caoutchouc, and on the juice of the berries of the white bryony, of which a fingle grain will draw to a thread of two feet long, and again return into a perfectly round sphere. We measured the diameter of the thread by a microscope with a micrometer, and thus could tell in every ftate of extenfion the proportional number,of particles in the fections. found, that though the whole range in which the distance of the particles was changed in the proportion of 13 to I, the extenfions did not fenfibly deviate from the proportion of the forces. The fame thing was observed in the caoutchouc as long as it perfectly recovered its first dimenfions. And it is on the authority of thefe experiments that we prefume to announce this as a law of nature.

We

Dr Hooke.

Dr Robert Hooke was undoubtedly the first who attend. Which was ed to this fubject, and affumed this as a law of nature. med by Mariottè indeed was the firft who exprefsly used it for determining the ftrength of beams: this he did about the 1679, correcting the fimple theory of Galileo. Leibnitz indeed, in his differtation in the A&a Eruditorum 1684 de Refiftentia Solidorum, introduces this confideration, and wishes to be confidered as the discoverer; and he is always acknowledged as fuch by the Bernoullis and others who adhered to his peculiar doctrines. But Marriottè had published the doctrine in the molt exprefs terms long before ; and Bulfinger, in the Comment. Petropol. 1729, completely vindicates his claim. But Hooke was unquestionably the difcoverer of this law. It made the foundation of his theory of springs, announced to the Royal Society about the year 1661, and read in 1666. On this occafion he mentions many things on the ftrength of bodies as quite familiar to his thoughts, which are immediate deductions from this principle; and among these all the facts which John Bernoulli so vauntingly adduces in fupport of Leibnitz's finical dogmas about the force of bodies in motion; a doctrine which Hooke might have claimed as his own, had he not perceived its frivolous inanity.

72

Though

corrected

not proper

But even with this firit correction of Marriottè, the mechanism of tranfverfe ftrain is not fully nor juftly explain-by Maried. The force acting in the direction BP (fig. 5. n° 1.), and otte it does bending the body ABCD, not only ftretches the fibres on the fide oppofite to the axis of fracture, but compreffes they explain fide AB, which becomes concave by the ftrain. Indeed it nifm of cannot do the one without doing the other: For in order tranfverfs to strain,

C 2

the mecha

1

Materials

com

quite accidental, and is not strictly true in any body. In Strength of most bodies which have any confiderable firmnefs, the Materials. preffions made by any external force are not fo great as the dilatations which the fame force would produce; that is, the repulfions which are excited by any fuppofed degree of compreffion are greater than the attractions excited by the fame degree of dilatation. Hence it will generally follow, that the angle dAD is less than the angles A 4, and the ordinates D d, Ee, &c. are less than the corresponding or dinates 4, E‹, &c.

Strength of to ftretch the fibres at D, there must be some fulcrum, fome fupport, on which the virtual lever BAD may prefs, that it may tear asunder the stretched fibres. This fulcrum must sustain both the preffure arifing from the cohesion of the diftended fibres, and alfo the action of the external force, which immediately tends to caufe the prominent part of the beam to flide along the section DA. Let BAD (fig. 5. n° 1.) be confidered as a crooked lever, of which A is the fulcrum. Let an external force be applied at B in the direction BP, and let a force equal to the accumulated cohesion of AD be applied at O in the direction oppofite to AB, that is, perpendicular to AO; and let thefe two forces be fuppofed to balance' each other by the intervention of the lever. In the first place, the force at O must be to the force at B as AB to AÒ: Therefore, if we make AK equal and oppofite to AO, and AL equal and oppofite to AB, the common principles of mechanics inform us that the fulcrum A is affected in the fame manner as if the two forces AK and AL were immediately applied to it, the force AK being equal to the weight P, and AL equal to the accumulated cohefion actually exerted in the inftant of fracture. The fulcrum is therefore really preffed in the direction AM, the diagonal of the parallelogram, and it must refift in the direction and with the force MA; and this power of refiftance, this fupport, must be furnished by the repulfive forces exerted by thofe particles only which are in a state of actual compreffion. The force AK, which is equal to the external force P, must be refifted in the direction KA by the lateral cohesion of the whole particles between D and A (the particle D is not only drawn forward but downward). This prevents the part CDAB from fliding down along the fection DA.

73

As is fully verified by experi

ment.

If the frac

This is fully verified by experiment. If we attempt to break a long flip of cork, or any fuch very compreffible body, we always obferve it to bulge out on the concave fide before it cracks on the other fide. If it is a body of fibrous or foliated texture, it feldom fails fplintering off, on the concave fide; and in many cafes this splintering is very deep, even reaching half way through the piece. In hard and granulated bodies, such as a piece of freeftone, chalk, dry clay, fugar, and the like, we generally fee a confiderable fplinter or fhiver fly off from the hollow fide. ture be flowly made by a force at B gradually augmented, the formation of the fplinter is very diftinctly seen. It forms a triangular piece like a I b, which generally breaks in the middle. We doubt not but that attentive obfervation would show that the direction of the crack on each fide of I is not very different from the direction AM and its correfpondent on the other fide. This is by no means a circumftance of idle curiofity, but intimately connected with the mechanism of cohesion. Confequen- Let us see what confequences refult from this ftate of the ces refulting from case respecting the ftrength of bodies. Let DA KC (fig. 6.) the ftate represent a vertical section of a prifm of compreffible materials, such as a piece of timber. Suppose it loaded with a weight P hung at its extremity. Suppofe it of fuch a conftitution that all the fibres in AD are in a ftate of dilatation, while those in A▲ are in a state of compreffion. In the inftant of fracture the particles at D and E are with-held by forces D d, Ee, and the particles at a and E repel, refift, or fupport, with forces. AJ, E‚§.

74

of the cafe.

Some line, fuch as de AS, will limit all these ordinates, which reprefent the forces actually exerted in the inftant of fracture, If the forces are as the extenfions and compreffions, as we have great reafon to believe, de A and A ‹ ♪ will be two ftraight lines, They will form one ftraight line d As, if the forces which refift a certain dilatation are equal to the forces which refift an equal compreffion. But this is

But whatever be the nature of the line d A &, we are certain of this, that the whole area AD d is equal to the whole area. A 4♪: for as the force at B is gradually increased, and the parts between A and D are more extended, and greater cohelive forces are excited, there is always fuch a degree of repulfive forces excited in the particles between A and 4 that the one fet precisely balances the other. The force at B, acting perpendicularly to AB, has no tendency to push the whole piece clofer on the part next the wall or to pull it away. The fum of the attractive and repulfive forces actually excited muft therefore be equal. Thefe fums are reprefented by the two triangular areas, which are therefore equal.

The greater we suppose the repulfive forces correfponding to any degree of compreffion, in comparison with the attractive forces correfponding to the fame degree of extenfion, the smaller will A ▲ be in comparison of AD. In piece of cork or fponge, A ▲ may chance to be equal to AD, or even to exceed it; but in a piece of marble, A a will perhaps be very small in comparison of AD.

a

75

the com

Now it is evident that the repulfive forces excited be- An impor tween A and ▲ have no fhare in preventing the fracture, tant confe They rather contribute to it, by furnishing a fulcrum to quence of the lever, by whofe energy the cohesion of the particles in preflibility AD is overcome. Hence we fee an important confequence of body of the compreffibility of the body. Its power of refifting fully prothis tranfverfe- ftrain is diminished by it, and fo much the ved. more diminished as the stuff is more compreffible.

This is fully verified by fome very curious experiments made by Du Hamel. He took 16 bars of willow 2 feet long and an inch square, and fupporting them by props under the ends, he broke them by weights hung on the middle. He broke 4 of them by weights of 40, 41, 47, and 52 pounds: the mean is 45. He then cut 4 of them d through on the upper fide, and filled up the cut with a thin piece of harder wood ftuck in pretty tight. These were broken by 48, 54, 55, and 52 pounds; the mean of which is 51. He cut other four through, and they were broken by 47, 49, 50, 46; the mean of which is 48. The remaining four were cut ds; and their mean ftrength was 42.

Another set of his experiments is ftill more remarkable. Six battens of willow 36 inches long and 1 square were broken by 525 pounds at a medium.

Six bars were cut d through, and the cut filled with a wedge of hard wood stuck in with a little force: these broke with 551.

Six bars were cut half through, and the cut was filled in the fame manner: they broke with 5.42.

Six bars were cut 4ths through: thefe broke with 530 A batten cut ths through, and loaded till nearly broken, was unloaded, and the wedge taken out of the cut. A thicker wedge was put in tight, fo as to make the batten ftraight again by filling up the space left by the compref fion of the wood: this batten broke with 577 pounds.

From this it is plain that more than 3ds of the thickness (perhaps nearly 4ths) contributed nothing to the ftrength. The point A is the centre of fracture in this cafe; and in order to eftimate the ftrength of the piece, we may fup

pose

78 The

Atrength of a piece de.

beam was

21

;

in the second (of attractive forces proportional to the extenfions) it was fbd2 ; and in the third (equal 31 attractions and repulfions proportional to the extenfions and fbd2 fb da compreffions) it was or more generally 67, ml m expreffes the unknown proportion between the attractions and repulfions correfponding to an equal extenfion and compreffion.

where

Hence we derive a piece of useful information, which is confirmed by unexcepted experience, that the ftrength pends chief. of a piece depends chiefly on its depth, that is, on that dily on its menfion which is in the direction of the ftrain. A bar of depth, timber of one inch in breadth and two inches in depth is four times as ftrong as a bar of only one inch deep, and it is twice as ftrong as a bar two inches broad and one deep; that is, a joift or lever is always strongest when laid on its edge.

19

And therefore a -choice in the manner

in which

the cohe.

fion is op the ftrain.

There is therefore a choice in the manner in which the cohesion is opposed to the ftrain. The general aim must be to put the centre of effort I as far from the fulcrum or the neutral point A as poffible, fo as to give the greateft energy or momentum to the cohefion. Thus if a triangular bar projecting from a wall is loaded with a weight at its extremity, it will bear thrice as much when one of the fides is uppermost as when it is undermoft. The bar of fig. 5. no 2. would be three times as ftong if the fide AB were upper80 moft and the edge DC undermoft. The strongHence it follows that the strongest joift that can be cut eft joift has not the out of a round tree is not the one which has the greatest greatest quantity of timber in it, but fuch that the product of its quantity of breadth by the fquare of its depth fhall be the greateft poffible. Let ABCD (fig. 7.) be the section of this joift in fcribed in the circle, AB being the breadth and AD the depth. Since it is a rectangular fection, the diagonal BD is a diameter of the circle, and BAD is a right angled tri. angle. Let BD be called a, and BA be called x; then AD is = √ a2 — x2. Now we must have ABX AD, or x X a2 · x2, or a2x — ×3, x3, a maximum. Its fluxion 32x must be made = 0, or a2 = 3 ×2, or x2 = If therefore we make DE DB, and draw EC perpendicular to BD, it will cut the circumference in the point C, which determines the depth BC and the breadth CD.

timber.

8 r A hollow

उ

a2

3

Because BD: BC= CD: CE, we have the area of the fection BC CD BD.CE. Therefore the different sections having the fame diagonal BD are proportional to their heights CE. Therefore the fection BCDA is lefs than the fection Bc Da, whose four fides are equal. The joift fo fhaped, therefore, is both ftronger, lighter, and cheaper.

The strength of ABCD is to that of a Bc D as 10,000 tube strong- to 9186, and the weight and expence as 10,000 to 10,607; er than a fo that ABCD is preferable to a Bc D in the proportion of hollow rod 10,607 to 9186, or nearly 115 to 100. containing

the fame From the fame principles it follows that a hollow tube is quantity of ftronger than a folid rod containing the fame quantity of matter, matter. Let fig. 8. reprefent the fection of a cylindric tube, of which AF and BE are the exterior and interior

diameters and C the centre. Draw BD perpendicular to Strength of BC, and join DC. Then, becaufe BDCDCB, Mareriale. BD is the radius of a circle containing the fame quantity of matter with the ring. If we eftimate the ftrength by the first hypothefis, it is evident that the ftrength of the tube will be to that of the folid cylinder, whofe radius is BD, as BDX AC to BD2 x BD; that is, as AC to BD: for BD expreffes the cohefion of the ring or the circle, and AC and BD are equal to the distances of the centres of effort (the fame with the centres of gravity) of the ring and

circle from the axis of fracture.

The proportion of these ftrengths will be different in the other hypothefes, and is not eafily expreffed by a general formula; but in both it is still more in favour of the ring or hollow tube.

The following very fimple folution will be readily understood by the intelligent reader. Let O be the centre of ofcillation of the exterior circle, o the centre of oscillation of the inner circle, and w the centre of ofcillation of the ring included between them. Let M be the quantity of ring included between them. furface of the exterior circle, m that of the inner circle, and that of the ring. M•FO — m• Fo We have Fw=

the ftrength of the ring =

5

छ

ƒXFw, and the ftrength of

2

the fame quantity of matter in the form of a folid cylinder is fx BD; fo that the ftrength of the ring is to that of the folid rod of equal weight as Fw to BD, or nearly as FC to BD. This will eafily appear by recollecting that fum of p.rz FO is = (fee ROTATION), and that the mom. FC

mentum of cohesion is fm FC Fa_fm Fo for the inner circle, &c.

2 FC

2

Emerfon has given a very inaccurate approximation to this value in his Mechanics, 4to.

82

This property of hollow tubes is accompanied also with And more greater ftiffness; and the fuperiority in ftrength and ftiffnefs ftiff. is fo much the greater as the furrounding fhell is thinner in proportion to its diameter.

83

forming the

bones, &c.

Here we see the admirable wisdom of the Author of Hence the nature in forming the bones of animal limbs hollow. The wifdom of bones of the arms and legs have to perform the office of le- God in bones of the arms and legs have to perform the office of le- God in vers, and are thus opposed to very great transverse strains. By this form they become incomparably ftronger and ftiffer, hollow. and give more room for the infertion of muscles, while they are lighter and therefore more agile; and the fame Wildom has made use of this hollow for other valuable purposes of the animal economy. In like manner the quills in the wings of birds acquire by their thinnefs the very great ftrength which is neceffary, while they are fo light as to give fufficient buoyancy to the animal in the rare medium in which it mult live and fly about. The ftalks of many plants, fuch as all the graffes, and many reeds, are in like manner hollow, and thus poffefs an extraordinary ftrength. Our best engineers now begin to imitate nature by making many parts of their machines hollow, fuch as their axles of cast iron, &c.; and the ingenious Mr Ramsden now makes the axes and framings of his great aftronomical inftruments in the fame manner.

In the fuppofition of homogeneous texture, it is plain that the fracture happens as foon as the particles at D are This is a feparated beyond their utmost limit of cohesion. This determined quantity, and the piece bends till this degree of It follows, extenfion is produced in the outermoft fibre. that the smaller we fuppofe the distance between A and D, the

Materials.

84

The momentum of cohesion must be equal to this in Strength of every hypothefis.

Having now confidered in fufficient detail the circumftances which affect the ftrength of any section of a folid body that is ftrained tranfverfely, it is neceffary to take notice of fome of the chief modifications of the ftrain itself. We fhall confider only those that occur most frequently in our conftructions.

The ftrain depends on the external force, and also on the lever by which it a&s.

Materials.

87

It is evidently of importance, that fince the ftrain is ex- The ftrain erted in any section by means of the cohesion of the parts lepends on intervening between the fection under confideration and the the exterpoint of application of the external force, the body must be nal force,. able in all these intervening parts to propagate or excite the ftrain in the remote fection. In every part it must be able to refift the strain excited in that part. It fhould therefore be equally ftrong; and it is ufelefs to have any part ftronger, because the piece will nevertheless break where it is not ftronger throughout; and it is ufelefs to make it ftronger (relatively to its ftrain) in any part, for it will nevertheless equally fail in the part that is too weak.

Strength of the greater will be the curvature which the beam will acquire before it breaks. Greater depth therefore makes a beam not only stronger but also ftiffer. But if the parallel fibres can flide on each other, both the ftrength and the ftiffness will be diminished. Therefore if, inftead of one Fig. 6. beam D 4 KC, we suppose two, DABC and A ▲ KB, not cohering, each of them will bend, and the extenfion of the How a fibres AB of the under beam will not hinder the comprefftrong fion of the adjoining fibres AB of the upper beam. The compound beam may two together therefore will not be more than twice as be formed. ftrong as one of them (fuppofing DA = A ▲) instead of being four times as ftrong; and they will bend as much as either of them alone would bend by half the load. This may be prevented, if it were poffible to unite the two beams all along the feam AB, fo that the one shall not slide on the other. This may be done in small works, by gluing them together with a cement as strong as the natural lateral cohefion of the fibres. If this cannot be done (as it cannot in large works), the fliding is prevented by JOGGLING the beams together; that is, by cutting down feveral rectangular notches in the upper fide of the lower beam, and making fimilar notches in the under fide of the upper beam, and filling up the fquare spaces with pieces of very hard wood firmly driven in, as reprefented in fig. 9. Some employ iron bolts by way of joggles. But when the joggle is much harder than the wood into which it is driven, it is very apt to work loose, by widening the hole into which it is lodged. The fame thing is fometimes done by scarfing the one upon the other, as reprefented in fig. 9. (n° 2.); but this waftes more timber, and is not fo ftrong, because the mutual hooks which this method forms on each beam are very apt to tear each other up. By one or other of these methods, or fomething fimilar, may a compound beam be formed, of any depth, which will be almoft as ftiff and strong as an entire piece.

85

How ftrength may be

On the other hand, we may combine ftrength with pliableness, by compofing our beam of several thin planks laid on each other, till they make a proper depth, and leaving combined them at full liberty to flide on each other. with pliaIt is in this blenefs. manner that coach-fprings are formed, as is represented in fig. 10. In this affcmblage there muft be no joggles nor bolts of any kind put through the planks or plates; for this would hinder their m tual fliding. They must be kept together by straps which furround them, or by fomething equivalent.

$6

Maxims of conftruction.

The preceding obfervations fhow the propriety of fome maxims of construction, which the artists have derived from long experience.

Thus, if a mortife is to be cut out of a piece which is exposed to a cross ftrain, it fhould be cut out from that fide which becomes concave by the train, as in fig. 11. but by no means as in fig. 12.

If a piece is to be ftrengthened by the addition of ane. ther, the added piece must be joined to the fide which grows convex by the strain, as in fig. 13. and 14.

Before we go any farther, it will be convenient to recal the reader's attention to the analogy between the ftrain on a beam projecting from a wall and loaded at the extremity, and a beam fupported at both ends and loaded in fome intermediate point. It is fufficient on this occafion to read attentively what is delivered in the article Roof, no 19. We learn there that the train on the middle point C (fig. 14. of the prefent article) of a rectangular beam AB, fupported on props at A and B, is the fame as if the part CA projected from a wall, and were loaded with the half of the weight W fufpended at A. The momentum of the strain

is therefore WX AB, = WX AB=p, or

Suppose then, in the first place, that the ftrain arifes from a weight fufpended at one extremity, while the other end is firmly fixed in a wall. Suppofing also the crofs fections to be all rectangular, there are feveral ways of fhaping the beam so that it fhall be equally ftrong throughout. Thus it may be equally deep in every part, the upper and under furfaces being horizontal planes. The condition will be fulfilled by making all the horizontal fections triangles, as in fig. 15. The two fides are vertical planes meeting in an edge at the extremity L.. For the equation expreffing the balance of ftrain and ftrength is pl=fbd2. Therefore fince d' is the fame throughout, and alfo p, we must have fb, and b (the breadth AD or any fection ABCD) must be proportional to (or AL', which it evidently is.

Or, if the beam be of uniform breadth, we must have d everywhere proportional to. This will be obtained by making the depths the ordinates of a common parabola, of which L is the vertex and the length is the axis. The upper or under fide may be a ftraight line, as in fig. 16. or the middle line may be straight, and then both upper and under furfaces will be curved. It is almost indifferent what is the fhape of the upper and under furfaces, provided the distances between them in every part be as the ordinates of a common parabola.

3

Or, if the fections are all fimilar, fuch as circles, fquares, or any other fiinilar polygons, we must have d3 or b3 proportional to, and the depths or breadths must be as the ordinates of a cubical parabola..

88

which it

It is evident that these are also the proper forms for a And on the` lever moveable round a fulcrum, and acted on by a force at form of the the extremity. The force comes in the place of the weight levers byfufpended in the cafes already confidered; and as fuch levers acts. always are connected with another arm, we readily fee that both arms fhould be fashioned in the fame manner. Thus in fig. 15. the piece of timber may be fuppofed a kind of fteelyard, moveable round a horizontal axis OP, in the frort of the wall, and having the two weights P and in equili- ` brio. The ftrain occationed by each at the fection in which the axis OP is placed must be the fame, and cach arm OL and Oa must be equally ftrong in all its parts. The lon gitudinal fections of each arm must be a triangle, a common parabola, or a cubic parabola, according to the conditions previously given.

λ

4