Propositions Relating

It follows that, since the secant of the quadrant is infinite,. when 4/ g. becomes equal to c, the de flection will be infinite, and the resistance of the co lumn will be overcome, however small distance b may be taken, provided , that it be of'ate magni eadetandainoe in this case = aa .8225 which is the utmost force that the co 'ee • luans: sill hear: and for a cylinder we find, by the aa same seasonings, -e- c = .616g m . If 6 be summed tosanisk yea &hal& llama in. ihAuryan.equi, gent s, expressing me amerence or mammon of the end of the beam and the direction of the force, which is also that of the middle of the sup posed curve, to the tangent of the extreme inclina tio• of the curve to its else* which will therefore the effect of a small lateral farce in weakening a beam or pillar, which is at the same time compres sed longitudinally by a much greater force; consi dering the parts on each side of the point, to which the lateral force is applied, as portions of two beams, bent in the manner here described, by a single force slightly inclined to the axis.

D. A bar fired at one end, and bent by

a trans. verse force applied to the other end, assumes initially the form of a cubic parabola, and the ddlection at The ordinate of a cubic parabola varying as x', its second fluxion varies as 6x (dx)!, or since the first fluxion of the abscise is constant, simply as the ab scise x, measured from the vertex of the .parabola, which must therefore be situated at the end to which the force is applied, and the abscise must coincide with the tangent of the bar. But if we begin from the other end, we must substitute e—x for x, and the second fluxion of the ordinate will be as 6 (e—x) (dx)', the first as and the fluent as Sex'—x', which, when x=e, becomes 2e', while it would have been Se' if the curvature had been uni form, and the second fluxion had been every where 6e(dx)'. Now the radius of curvature at the fix Maa ed end being 1 ef —' and the versed sine of a small portion of a circle being equal to this versed sine will be expressed by ; and two third* of Maathis, ' or ' will be the actual deflection.

Maa E. The depression of a bar, faxed horizontally at ease end, and supporting only its own weight, is Se* d =Soma ; ea being the height of the modulus of elm ticity.

The curvature here varies as the square of the distance from the end, because the strain is propor tional to the weight of the portion of the bar beyond any given point, and to the distance of its centre of gravity conjointly, that is, to (e —x) 4 (e—x), so that if the second fluxion at the fixed end be as will elsewhere be as (e (dx)' and the cor responding first fluxion being Ord; elsdx ledx, the fluents will be 1 e'x', and — +es" + or, when x = e, and (1 —1 4- I consequently the depression must be half the versed sine in the circle of greatest Maa curvature. Now the radius of curvature be

f 12h Maa comes here — the force being applied at the die- . 6tepee e : and since the weight of the bar is to that of the modulus of elasticity an the proportion of the 3 the versed sine for the ordinate e will be half maa of which is the actual depression.

F. The depression of the middle of a horizontal bar, fixed at both ends, and supporting its own weight only, is se.

The transverse force at each point of such a bar, resisted by the lateral adhesion, is as the distance x from the middle (Art. Barnoz, under Prop. L); but this force is proportional to the first fluxion of the strain or curvature, consequently the curvature itself must vary as the corrected fluent of ±xdr, taking here the negative sign, because the curvature diminishes as x increases; and the corrected fluent will be I since it must vanish when x= e; the first fluxion of the ordinate will then be I e'zir — 4 Aix, and the fluent 4 text or for the whole length 1 e, instead of or which would have been its value if the had been equal throughout. Now the strain at the middle is the difference of the opposite strains, produced by the forces acting on either side; and these are the half weight, acting at the mean distance * e, and the re. sistance of the support, which is equal to the same half weight, but acts at the distance 1 the differ ence being equivalent to the half weight, acting at the distance * e, so that the curvature at the middle is the same as if the bar were fixed there, and loose at the ends, that is, as in the last proposition, sub.

stituting 1 e for e, r = ; and the versed sine a thedistance e being — 8 or 16maa' 1 of this will be This demonstration may serve as an illus.

32maa tration of two modes of considering the effect of a strain, which have not been generally known, and which are capable of a very extensive application.

It follows that where a bar is equally loaded throughout its length, the curvature at the middle is half as great as if the whole weight were collected there, the strain derived from the resistance of the support remaining in that case uncompensated. The depression produced by the divided weight will be as great as by the single weight, since x 1 is to as 5 to 8. Mr Dupin found the proposition by many experiments, between # and ; and in a very good mean for representing these results.