બ Strength of train; and this is almost the only cafe of a fimple tranfverfe 54 Experi ments mad to afcertain it. 55 Their re ftrength of a lever tional to the area of the section. Experiments were made for difcovering the rcfiftances made by bodies to this kind of frain in the following manner: Two iron bars were difpofed horizontally at an inch diftance; a third hung perpendicularly between them, being fupported by a pin made of the fubftance to be examined. This pin was made of a prifmatic form, so as to fit exactly the holes in the three bars, which were made very exact, and of the fame fize and shape. A scale was fufpended at the lower end of the perpendicular bar, and loaded till it tore out that part of the pin which filled the middle hole. This weight was evi dently the measure of the lateral cohefion of two fections. The fide bars were made to grasp the middle bar pretty ftrongly between them, that there might be no distance impofed between the oppofite preffures. This would have combined the energy of a lever with the purely tranfverfe preffure. For the fame reason it was neceffary that the internal parts of the holes fhould be no smaller than the edges. Great irregularities occurred in our firft experiments from this cause, because the pins were fomewhat tighter within than at the edges; but when this was corrected they were extremely regular. We employed three fets of holes, viz. a circle, a fquare (which was occafionally made a rectangle whofe length was twice its breadth), and an equilateral triangle. We found in all our experiments the ftrength exactly proportional to the area of the section, and quite independent of its figure or pofition, and we found it confiderably above the direct cohefion; that is, it took confiderably more than twice the force to tear out this middle piece than to tear the pin afunder by a direct pull. A piece of fine freeftone required 205 pounds to pull it directly alunder, and 575 to break it in this way. The difference was very conftant in any one fubftance, but varied from ds to 4ds in different kinds of matter, being fmalleft in bodies of a fibrous texture. But indeed we could not make the trial on any bodies of confiderable cohesion, because they required fuch forces as our apparatus could not fupport. Chalk, clay baked in the fun, baked fugar, brick, and freeftone, were the strongest that we could examine. But the more common cafe, where the energy of a lever intervenes, demands a minute examination. Let DABC (fig. 5.n° 1.) be a vertical section of a prifmatic folid (that is, of equal fize throughout), projecting horizontally from a wall in which it is firmly fixed; and let a weight P be hung on it at B, or let any power P act at B in a direction perpendicular to AB. Suppofe the body of infuperable ftrength in every part except in the vertical fection DA, perpendicular to its length. It muft break in this fection only. Let the cohefion be uniform over the whole of this fection; that is, let each of the adjoining particles of the two parts cohere with an equal force f. There are two ways in which it may break. The part ABCD may fimply flide down along the furface of fracture, provided that the power acting at B is equal to the accumulated force which is exerted by every particle of the fection in the direction AD. But fuppofe this effectually prevented by fomething that fupports the point A. The action at P tends to make the body turn round A (or round a horizontal line paffing thro' A at right angles to AB) as round a joint. This it cannot do without feparating at the line DA. In this cafe the adjoining particles at D or at E will be feparated horizontally. But their cohefion refifts this feparation. In order, therefore, that the fracture may happen, the en erry or momentum of the power P, acting by means of the Strength of Let us therefore firft fuppofe, that in the inftant of frac- The reader who is not familiarly acquainted with this The accumulated energy therefore of the cohefion in the inftant of fracture is fdXd. Now this mult be equal or juft inferior to the energy of the power employed to break it. Let the length AB be called ; then Pis the correfponding energy of the power. This gives us ƒ d1⁄2 d=pl for the equation of equilibrium correfponding to the verti cal fection ADCB. Suppose now that the fracture is not permitted at DA, but at another fection a more remote from B. The body being prifmatic, all the vertical fections are equal; and therefore fd is the fame as before. But the energy of the power is by this means increafed, being now = Px Bo, inftead of Px BA: Hence we fee that when the prifmatic body is not infuperably ftrong in all its parts, but equally ftrong throughout, it must break clofe at the wall, where the ftrain or energy of the power is greatest. We fee, too, that a power which is just able to break it at the wall is unable to break it anywhere elfe; alfo an abfolute cohefion fd, which can withstand the powerp in the fection DA, will not withstand it in the fection, and will withftand more in the fection d' a'. This teaches us to diftinguish between abfolute and relative strength. The relative strength of a fection has a reference to the ftrain actually exerted on that fection. This relative ftrength is properly measured by the power which is juft able to balance or overcome it, when applied at its Strength of proper place. Now fince we had ƒdid=pl, we have proportion each fibre is extended. It feems most probable Strength of Materials. fd + d for the measure of the strength of the fection We fhall suppose this to be really the cafe. Now recollect that the extentions are proportional to the distances from A. Materials. DA, in relation to the power applied at B. the general law which we formerly faid was obferved in all moderate extenfions, viz. that the attractive forces exerted by the dilated particles were proportional to their dilatations. Suppofe now that the beam is fo much bent that the particles at D are exerting their utmost force, and that this fibre is just ready to break or actually breaks. It is plain that a total fracture muft immediately enfue; because the force which was fuperior to the full cohesion of the particle at D, and a certain portion of the cohesion of all the reft, will be more than fuperior to the full cohesion of the particle next within D, and a finaller portion of the cohefion of the remainder. In the folid is a rectangular beam, whose breadth is b, it is plain that all the vertical fections are equal, and that AG ord is the fame in all. Therefore the equation expreffing the equilibrium between the momentum of the external force and the accumulated momenta of cohesion will be pl = fdbx 4 d. The product db evidently expreffes the area of the fec- Now f's is a proper expreffion of the abfolute cohesion of Since the action of equable cohesion is fimilar to the ac- or Because the relative ftrength of a rectangular beam is Now let F represent, as before, the full force of the ex- The force exerted by a fibre whofe thickness is x is therefore-fxx but this force refifts the ftrain by acting d by means of the lever EA or x. Its energy or momentum is therefore ƒx2x and the accumulated momenta. of all the fibres in the line AE will be ƒX sum of d This, when x is taken equal to d, will exprefs the momentum of the whole fibres in the line AD. This, therefore, ÷ 43 is f, or fi d', or fdxd. Now fd expreffes the abfolute cohesion of the whole line AD. The accumulated Such are the more general refults of the mechanifm of momentum is therefore the fame as if the abfolute cohesion this tranfverfe ftrain, in the hypothefis that all the particles of the whole line were exerted at 4d of AD from A. hypothefis are exerting equal forces in the inftant of fracture. We are From thefe premifes it follows that the equation expref- The indebted for this doctrine to the celebrated Galileo; and it fing the equilibrium of the ftrain and cohetion is p/fdtrength "As thrice the was one of the firft fpecimens of the application of mathe-Xd; and hence we deduce the analogy, length is to the depth, fo is the abfolute cohesion to the relative principles. strength." 57 Aftert.in cd on ile clequal cobefion; 58 But that not conformable to nature. meters. matics to the fcience of nature. We have not included in the preceding investigation that action of the external force by which the folid is drawn fidewife, or tends to flide along the furface of fracture. We have fuppofed a particle E to be pulled only in the direction Ee, perpendicular to the section of fracture, by the action of the crooked lever BAE. Dut it is alfo pulled in the direction EA; and its reaction is in fome direction E, compounded off, by which it refifts being pulled outwards; and e, by which it refifts being pulled downwards. We are but imperfe&ly acquainted with the force e, and only know that their accumulated fum is equal to the force p: but in all important cafes which occur in practice, it is unneceffary to attend to this force; because it is fo fmall in comparison of the forces in the direction E e, as we eafily conclude from the usual smallness of AD in comparison of AB. The hypothesis of equal cohesion, exerted by all the parhypothefis ticles in the inftant of fracture, is not conformable to nature: for we know, that when a force is applied tranfverfely at B, the beam is bent downwards, becoming convex on the upper fide; that fide is therefore on the ftretch. ticles at D are farther removed from each other than those at E, and are therefore adually exerting greater cohesive for We cannot fay with certainty and precision in what ces. 'The par This equation and this proportion will equally apply to rectangular beams whole breadth is b; for we fhall then have pl/ b d x ÷ d. = We alfo fee that the relative ftrength is proportional to As this is a most important propofition, and the founda 4 59 afcertained un other Strength of in another point of view. will be equal to AD, and their cohefions will be reprefented Strength of by triangles like ADd; and the whole actual cohesion Materials, will be reprefented by a wedge whofe bafes are vertical planes, and which is equal to half of the parallelopiped AD XDdXaa, and will therefore be = fbd; and the distance AO of its centre of gravity from the horizontal line A A will be of A D. The momentum of cohesion of a joift-will therefore be fld × 3 d, or ƒ b d — d, as we have determined in the other way. す T T ན Σ 1. 2 T The beam reprefented in the figure is a triangular prifm. The pyramid 19`a a d is of the prism a a Dda a'. If we makes reprefent the furface of the triangle a Da, the pyramid is offs. The distance AO of its centre of gravity from the horizontal line A A' is of A D, or 1⁄2 d. Therefore the momentum of actual cohesion is }ƒs × 4 d, =ƒs} d; that is, it is the fame as if the full cohefon of all the fibres were accumulated at a point I whose distance from A is th of AD or d; or (that we may fee its value in every point of view it is th of the momentum of the full cohefion of all the fibres when accumulated at the point D, or acting at the distance d=A D. Fig. 5. 12. A is a perfpective view of a three fided beam Matc ials. projecting horizontally from a wall, and loaded with a weight 60 at B juft fufficient to break it. DABC is a vertical plane The fare through its higheft point D, in the direction of its length. propofition a Da is another vertical section perpendicular to AB. The prefented piece being fuppofed of infuperable strength everywhere except in the fection a Da, and the cohesion being allo fuppofed infuperable along the line a A a, it can break nowhere but in this fection, and by turning round a A a as round a hinge Make D d equal to AD, and let Dd represent the abfolute cohesion of the fibre at D, which abfolute cohesion we expressed by the fymbol f. Let a plane a da be made to pals through a and d, and let dda' be another cross fection. It is plain that the prifnatic folid contained between the two sections a D) a and ada' will reprefent the full cohesion of the whole fection of fracture; for we may conceive this prism as made up of lines fuch as F, equal and parallel to Dd, representing the abfolute cohelion of each particle fuch as F. The pyramidal folid d Da a, cut off by the plane daa, will reprefent the cohesions actually exerted by the different fibres in the inflant of fracture. For take any point E in the furface of fracture, and draw Ee parallel to AB, meeting the plane ada in e, and let ·e A È be a vertical plane. It is evident that Dd is to Ee E as AD to AE; and the fore (fince the forces exerted by the different fibres are as their extenfion, and their extenfion as their distances from the axis of fracture) Ee will represent the force actually exerted by the fibre in E, while D is exerting its full force D d. In like manner, the plane FFffexpreffes the cohesion exerted by all the fibres in the line FF, and fo on through the whole furface. Therefore the pyramid da a D expreffes the accumulated exertion of the whole furface of fracture. Farther, suppose the beam to be held perpendicular to the horizon with the end B uppermoft, and that the weight of the prism contained between the two fections a D a and a'da (now horizontal) is juft able to overcome the full cohefion of the section of fracture. The weight of the pyramid d Da a will also be just able to overcome the cohesions actually exerted by the different fibres in the, inftant of fracture, because the weight of each fibre, fuch as Ee, is juft fuperior to the cohesion actually exerted at E. Let o be the centre of gravity of the pyramidal folid, and draw o perpendicular to the plane a D a. The whole weight of the folid d Daa may be conceived as accumulated in the point o, and as acting on the point O, and it will have the fame tendency to feparate the two cohering furfaces as when each fibre is hanging by its refpective point. For this reason the point O may be called the centre of actual effort of the unequal forces of cohefion. The momentum therefore, or energy by which the cohering furfaces are feparated, will be properly measured by the weight of the folid d Daa multiplied by OA; and this product is equal to the product of the weight p multiplied by BA, or by / Thus fuppofe that the cohefion along the line AD only is confidered. The whole cohesion will be reprefented by a triangle A Dd. Dd reprefents f, and AD is d, and AD is x. Therefore A D d is ƒ d. The centre of gravity o of the triangle A D d is in the interfection of a line drawn from A to the middle of D d with a line drawn from d to the middle of AD; and therefore the line o O will make AO of AD. Therefore the actual momentum of cohesion is ƒ × ÷ d × } d, = ƒ •× d×÷d, = ƒ d×÷d, or equal to the 2 Σ T I I d abfolute coliefion acting by means of the lever If the 3 fection of fracture is a rectangle, as in a common joist, whose breadth a a is b, it is plain that all the vertical lines This is a very convenient way of conceiving the momentum of actual cohefion, by comparing it with the momentum of abfolute cohesion applied at the distance AD from the axis of fracture. The momentum of the absolute cohefion applied at D is to the momentum of actual cohesion in the inflant of fracture as AD to AI. Therefore the length of AI, or its proportion to AD, is a fort of index of the ftrength of the beam. We fhall call it the INDEX, and exprefs it by the fymbol i. Its value is easily obtained. The product of the absolute cohesion by AI must be equal to that of the actual cohesion by AO. Therefore fay, "as the prifmatic folid aa Dda a' is to the pyramidal folid a a D d, fo is AO to A I." We are affifted in this determination by a very convenient circumftance. In this hypothefis of the actual cohefions being as the diftances of the fibres from A, the point O is the centre of ofcillation or percuffion of the surface D a a turning round the axis a a: for the momentum of cohesion of the line FF is FFX FXEA=FFXE A2, because Ff is equal to E A. Now A O, by the nature of the centre of gravity, is equal to the fum of all these momenta divided by the pyramid a a D d; that is, by the fum of all the FFX Ff; that is, by the fum of all the FF×EA. fum of FFX EA' Therefore AO= which is just the fum of F F XEA' value of the distance of the centre of percuffion of the triangle a a D from A: (See ROTATION). Moreover, if G be the centre of gravity of the triangle a Da, we fhall have D A to G A as the abfolute cohefion to the fum of the cohefions actually exerted in the inftant of fracture; for, by the nature of this centre of gravity, A G is equal to fum of FFX EA and the fum of FFX AG is equal fum of F F to the fum of FFXE A. But the fum of all the lines FF is the triangle a Da, and the fum of all the FFX EA is the fum of all the rectangles F Fff; that is, the pyramid Daa. Therefore a prifm whose base is the triangle a Da, and whofe height is AG, is equal to the pyramid, or will exprefs the fum of the actual cohefions; and a prifm, whose bafe is the fame triangle, and whofe height is Dd or Da, expreffes the abfolute cohefion. Therefore DA is to GA as the abfolute cohesion to the fum of the actual cohefions. Therefore we have D A:GA=OA:IA. Therefore, whatever be the form of the beam, that is, whatever be the figure of its fection, find the centre of ofcillation O, and the centre of gravity G of this fection, Call 3 Strength of Call their diftances from the axis of fracture o and g. Then og Materials. thefis. - AI or i = curve Add' will be a common parabola, having AB for its Strength of , and the momentum of cohesion is fs X ADd will be d ADXDd; and in the cafe of a rectan 2, where s is the area of fracture. This index is easily determined in all the cafes which generally occur in practice. In a rectangular beam A I is +d of AD; in a cylinder (circular or elliptic) AI is ths of AD, &c. &c. ठ In this hypothesis, that the cohesion actually exerted by each fibre is as its extenfion, and that the extenfions of the fibres are as their distances from A (fig. 5. no 1.), it is plain that the forces exerted by the fibres D, E, &c. will be reprefented by the ordinates D d, Ee, &c. to a ftraight line A d. And we learn from the principles of ROTATION that the centre of percuffion O is in the ordinate which paffes through the centre of gravity of the triangle A D d, or (if we confider the whole fection having breadth as well as depth) through the centre of gravity of the folid bounded by the planes DA, dA; and we found that this point O was the centre of effort of the cohesions adually exerted in the inftant of fracture, and that I was the centre of an equal momentum, which would be produced if all the fibres were accumulated there and exerted their full cohesion. This confideration enables us to determine, with equal facility and neatnefs, the ftrength of a beam in any hypothefis of forces. The above hypothefis was introduced with a cautious limitation to moderate ftrains, which produced no permanent change of form, or no fett as the artists call it: and this fuffices for all purposes of practice, feeing that it would be imprudent to expose materials to more violent ftrains. But when we compare this theory with experiments in which the pieces are really broken, confiderable deviations may be expected, because it is very probable that in the vicinity of rupture the forces are no longer proportional to the extenfions. AD q+2 We may obferve here in general, that if the forces As far as we can judge from experience, no fimple algebraic Materials. We must now remark, that this correction of the Galilean hypothefis of equal forces was fuggefted by the bending which is obferved in all bodies which are ftrained transverfeHow the That no doubt may remain as to the juftnefs and comrelative pleteness of the theory, we muft fhow how the relative ly. Because they are bent, the fibres on the convex fide have ft,ength may be de- ftrength may be determined in any other hypothefis. There- been extended. We cannot fay in what proportion this obtains in the different fibres. Our moft diftinct notions of termined fore fuppofe that it has been established by experiment on by any o- any kind of folid matter, that the forces actually exerted in the internal equilibrium between the particles render it highther hypo- the inftant of fracture by the fibres at D, E, &c. are as the ly probable that their extenfion is proportional to their ordinates Dd, E é', &c. of any curve line A ed. . We are distance from that fibre which retains its former dimensions. supposed to know the form of this curve, and that of the But by whatever law this is regulated, we fee plainly that folid which is bounded by the vertical plane through AD, the actions of the ftretched fibres muft follow the proporand by the surface which paffes through this curve A edtions of some function of this distance, and that therefore perpendicularly to the length of the beam. We know the the relative ftrength of a beam is in all cafes susceptible of place of the centre of gravity of this curve farface or folid, mathematical determination. and can draw a line through it parallel to A B, and cutting the surface of fracture in fome point O. This point is alfo the centre of effort of all the cohefions actually exerted and the product of AO and of the folid which expreffes the actual cohesions will give the momentum of cohetion equivalent to the former ƒs 3. Or we may find an index fs A I, by making A I a fourth proportional to the full cohe. fion of the furface of fracture, to the accumulated actual cohefions, and to AO; and then ƒs Xi (AI) will be the momentum of cohefion; and we fhall Itill have I for the point in which all the fibres may be fuppofed to exert their full cohesion ƒ, and to produce a momentum of cohefion equal to the real momentum of the cohesions actually exerted, fsi and the relative strength of the beam will still be por 62 We also fee an intimate connection between the strain and Bernoulli's og the elastic curve. the curvature. This fuggefted to the celebrated James problem of Bernoulli the problem of the ELASTIC CURVE, i. e. the ; curve into which an extenfible rigid body will be bent by a tranfverse strain. His folution in the Ata Lipfie 1694 and 1695 is a very beautiful specimen of mathematical dif cuffion; and we recommend it to the perufal of the curious reader. He will find it very perfpicuously treated in the first volume of his works, publifhed after his death, where the wide steps which he had taken in his investigation are explained fo as to be eafily comprehended. His nephew Dan. Bernoulli has given an elegant abridgment in the Peterburg Memoirs tor 1729. The problem is too inalfo too intimately connected with our present subject to be tricate to be fully difcuffed in a work like ours; but it is entirely omitted. We mult content ourselves with fhowing the leading mechanical property of this curve, from which the mathematician may deduce all its geometrical properties. Thus, if the forces be as the fquares of the exten frons (ftill fuppofed to be as the distances from A), the VOL. XVIII. Part I. When a bar of uniform depth and breadth, and of a given length, is bent into an arch of a circle, the extenfion of the C outer Strength of outer fibres is proportional to the curvature; for, because for, because Materials. the curves formed by the inner and outer fides of the beam are fimilar, the circumferences are as the radii, and the ra63 Its leading dius of the inner circle is to the difference of the radii as the mechanical length of the inner circumference is to the difference of the property circumferences. The difference of the radii is the depth of defcribed. the beam, the difference of the circumferences is the extenfion of the outer fibres, and the inner circumference is fuppofed to be the primitive length of the beam. Now the fecond and third quantities of the above analogy, viz. the depth and length of the beam, are conftant quantities, as is also their product. Therefore the product of the inner radius and the extenfion of the outer fibre is also a constant quantity, and the whole extension of the outer fibre is inverfely as the radius of curvature, or is directly as the curvature of the beam. It is not a circle. The mathematical reader will readily fee, that into what ever curve the elastic bar is bent, the whole extenfion of the outer fibre is equal to the length of a fimilar curve, having the fame proportion to the thickness of the beam that the length of the beam has to the radius of curvature. Now let ADCB (fig. 5. n° 3.) be such a rod, of uniform breadth and thicknefs, firmly fixed in a vertical pofition, and bent into a curve AEFB by a weight W fufpended at B, and of such magnitude that the extremity B has its tangent perpendicular to the action of the weight, or parallel to the horizon. Suppose too that the extenfions are proportional to the extending forces. From any two points E and F draw the horizontal ordinates EG, FH. It is evident that the exterior fibres of the sections Ee and Ffare ftretched by forces which are in the proportion of EG to FH (thefe being the long arms of the levers, and the equal thickneffes Ee, Ff being the short arms). Therefore (by the hypothefis) their extenfions are in the fame proportion. But because the extenfions are proportional to fome fimilar functions of the distance from the axes of fracture E and F, the extension of any fibre in the fection E e is to the contemporaneous extenfion of the fimilarly fituated fibre in the fection Ff, as the extenfion of the exterior fibre in the fection Ee is to the extenfion of the exterior fibre in the fection Ff: therefore the whole extenfion of Ee is to the whole extenfion of Ff as EG to FH, and EG is to FH as the curvature in E to the curvature in F. Here let it be remarked, that this proportionality of the curvature to the extenfion of the fibres is not limited to the hypothesis of the proportionality of the extenfions to the extending forces. It follows from the extenfion in the dif ferent fections being as fome fimilar function of the distance from the axis of fracture; an affumption which cannot be refufed. This then is the fundamental property of the elaftic curve, from which its equation, or relation between the abfciffa and ordinate, may be deduced in, the usual forms, and all its other geometrical properties. These are foreign to our purpose; and we shall notice only fuch properties as have an immediate relation to the ftrain and ftrength of the dif ferent parts of a flexible body, and which in particular ferve to explain some difficulties in the valuable experiments of Mr Buffon on the Strength of Beams. We observe, in the first place, that the elastic curve cannot be a circle, but is gradually more incurvated as it recedes from the point of application B of the ftraining forces. At Bit has no curvature; and if the bar were extended beyond B there would be no curvature there. In like manner, when a beam is fupported at the ends and loaded in the middle, the curvature is greatest in the middle; but at the props, or beyond them, if the beam extend farther, there is no curvature. Therefore when a beam projecting 20 feet from a wall is bent to a certain curvature at the wall by a weight Strength of Material fufpended at the end, and a beam of the fame fize projecting 20 feet is bent to the very fame curvature at the wall by a greater weight at 10 feet diftance, the figure and the mechanical ftate of the beam in the vicinity of the wall is dif ferent in thefe two cafes, though the curvature at the very wall is the fame in both. In the first case every part of the beam is incurvated; in the fecond, all beyond the 10 feet is without curvature. In the first experiment the curvature at the diftance of five feet from the wall is ths of the curvature at the wall; in the fecond, the curvature at the This muft weaken fame place is but of that at the wall. the long beam in this whole interval of five feet, because the greater curvature is the refult of a greater extension of the fibres. 65 has a cer In the next place, we may remark, that there is a certain Every beam determinate curvature for every beam which cannot be ex- tain deterceeded without breaking it; for there is a certain fepara- minate cura tion of two adjoining particles that puts an end to their co- vature. hefion. A fibre can therefore be extended only a certain proportion of its length. The ultimate extenfion of the outer fibres must bear a certain determinate proportion to its length, and this proportion is the fame with that of the thickness (or what we have hitherto called the depth) to the radius of ultimate curvature, which is therefore determinate. 66 of uniform where the strain is A beam of uniform breadth and depth is therefore moft And when incurvated where the ftrain is greateft, and will break in breadth and the most incurvated part. But by changing its form, fo as depth is to make the ftrength of its different fections in the ratio moff incurof the ftrain, it is evident that the curvature may be the vated fame throughout, or may be made to vary according to any law. This is a remark worthy of the attention of the greatest. watchmaker. The most delicate problem in practical mechanics is fo to taper the balance-spring of a watch that its Hooke's wide and narrow vibrations may be ifochronous. principle ut tenfio fic vis is not fufficient when we take the inertia and motion of the spring itself into the account. The figure into which it bends and unbends has also an influence. Our readers will take notice that the artist aims at an accuracy which will not admit an error of 4th, and that Harrison and Arnold have actually attained it in feveral inftances. The taper of a fpring is at prefent a noftrum in the hands of each artift, and he is careful not to impart his fecret. Again, fince the depth of the beam is thus proportional to the radius of ultimate curvature, this ultimate or breaking curvature is inversely as the depth. It may be expressed by /. I 67 When a weight is hung on the end of a prismatic To what beam, the curvature is nearly as the weight and the length the curva directly, and as the breadth and the cube of the depth in- ture is proportional. b ď Let us fuppofe that versely; for the strength is=ƒ 31 |