259. Corol. 1. The greatest stress is when the weight w is at the middle: for then the rectangle of the two halves, AC. ACABABAB2, is the greatest. And, from the middle point, the stress is less and less all the way to the extremities A and B, where it is nothing.

260. Corol. 2. The same thing will obtain from the weight of the beam itself, or from any other weight diffused equally all over it; the stress in this case being the half of the former. So that, in all structures, we should avoid as much as possible, placing weights or strains in the middle of beams.

261. Corol. 3. If w be the greatest weight that a beam can sustain at its middle point; and it be required to find the place where it will support any greater weight w; that point will be found by making, as w: w:: AB. AB, or AB2: BC or AC X (AB — AC)

AC

AB. AC

AC2.

PROPOSITION LI.

262. When a Beam is placed aslope, its Strength in that position, is to its Strength when Horizontal, to resist a Vertical Force, as the square of Radius is to the Square of the Cosine of the Elevation.

G

B

LET AB be the beam standing aslope, CF perp. to the horizon AFG; then CD is the vertical section of the beam, and CE, perp. to AB, is the transverse section, and is the same as when in the horizontal position. Now, the strength, in both positions, is as the section drawn. into the distance of its centre of gravity from the point c. But the sections, being of the same breadth, are as their depths, CD, CE; and the distances of the centres of gravity are as the same depths; therefore the strengths are as CD CD to CE. CE, or CD2 to CE2. But, by the similar triangles CDE, AFD, it is CD: CE :: AD: AF, as radius to the cosine of the elevation. Therefore the oblique strength is to the transverse strength, as AD2 to AF2, the square of radius to the square of the cosine of elevation.

263. Corol. 1. The strength of a beam increases from the horizontal position, where it is least, all the way as it revolves to the vertical position, where it is the greatest.

PRO

PROPOSITION LII.

264. When Beams stand Aslope, or Obliquely, and sustaining Weights, either at the Middle Points, or in any other Similar Situations, or Equally Diffused over their Lengths; the Strains upon them are Directly as the Weights, and the Lengths, and the Cosines of Elevation.

FOR, by the inclined plane, the weight is to the pressure on the plane, as AC to AF, as radius to the cosine of elevation: therefore the pressure is as the weight drawn into the cosine. of the elevation. Hence the stress will be as the length of the beam and this force; that is, as the weight x length x cosine of elevation.

265. Corol. 1. When the lengths and weights of beams are the same, the stress is as the cosine of elevation; and it is therefore the greatest when it lies horizontal.

266. Corol. 2. In all similar positions, and the weights varying as the lengths, or the beams uniform; then the stress varies as the squares of the lengths.

267. Corol. 3. When the weights are equal, on the oblique beam AB, and the horizontal one ac, and BC is vertical; the stress on both beams is equal. For, the length into the cosine of elevation is the same in both; or AB X COS. Á AĊ × radius.

268. Corol. 4. But if the weights on the beams vary as their lengths; then the stress will also vary in the same

ratio.

269. Corol. 5. And universally, the stress upon any point of an oblique beam, is as the rectangle of the segments of the beam, and the weight, and cosine of inclination, directly and the length inversely.

VOL. II.

P

FRO

PROPOSITION LIII.

270. When a Beam is to sustain any Weight, or Pressure, or Force, acting Laterally; then the Strength ought to be as the Stress upon it; that is, the Breadth multiplied by the Square of the Depth, or in similar sections, the Cube of the Diameter, in every place, ought to be proportional to the Length drawn into the Weight or Force acting on it. And the same is true of several Different Pieces of timber compared together.

FOR every several piece of timber or metal, as well as every part of the same, ought to have its strength proportioned to the weight, force, or pressure it is to support. And therefore the strength ought to be universally, or in every part as the stress upon it. But the strength is as the breadth into the square of the depth; and the stress is as the weight or force into the distance it acts at. Therefore these must be in constant ratio. This general property will give rise to the effect of different shapes in beams, according to particular circumstances; as in the following corollaries.

271. Corol. 1. If ABC be a horizontal beam, fixed at the end AC, and sustaining a weight at the other end B. And if the sections at all places be similar figures; and DE be the diameter at any place D; then

D

E

BD will be every where as DE3. So that, if ADB be a right line, then BEC will be cubic parabola. In which case of such a beam may be cut away, without any diminution of the strength. But if the beam be bounded by two parallel planes, perpendicular to the horizon; then BD will be as DE2; and then BEC will be the common parabola. In which case a 3d part of the beam may be thus cut away.

272. Corol. 2. But if a weight press uniformly on every part of AB; and the sections in all points, as D, be similar; then BD2 will be every where as DE3: and then EEC is the semicubical parabola.

A D

E

But, in this disposition of the weight, if the beam be bounded by parallel plains, perpendicular to the horizon; then BD will be always as DE; and BEC a right line, or ABC a wedge. So that then half the beam may be cut away, without diminution of strength.

C

273. Corol.

273. Corol. 3. If the beam AB be supported at both ends; and if it sustain a weight at any variable point D, or uniformly on

all parts of its length; and if all the sections be similar figures; then will the diameter DE3 be every where as the rectangle AD. DB. ·

But if it be bounded by two parallel planes, perpendicular to the horizon; then will DE2 be every where as the rectangle AD. DB, and the curve AEB an ellipsis.

274. Corol. 4. But if a weight be placed at any given point F, and all the sections be similar figures; then will AD be as DE3, and AG, BG be two cubic parabolas.

D F

B

E

G

But if the beam be bounded by two parallel planes, perpendicular to the horizon; then AD is as DE, and AG and BG are two common parabolas.

275. Scholium. The relative strengths of several sorts of wood, and of other bodies, as determined by Mr. Emerson, are as follow:

Red fir, Holly, Elder, Plane, Crabtree, Appletree

Beech, Cherrytree, Hazle

Lead

Alder, Asp, Birch, White fir, Willow

Fine freestone

87766

63

6

1

A cylindric rod of good clean fir, of 1 inch circumference, drawn lengthways, will bear at extremity 400 lbs; and a of fir, 2 inches diameter, will bear about 7 tons in that direction.

spear

A rod of good iron, of an inch circumference, will bear a stretch of near 3 tons weight.

A good hempen rope, of an inch circumference, will bear 1000 lbs at the most.

Hence Mr. Emerson concludes, that if a rod of fir, or of

P 2

iron,

iron, or a rope of d inches diameter, were to lift of the extreme weight; then

The fir would bear 84 d hundred weights.

The

rope

The iron

22 d ditto.

63 de tons.

Mr. Banks, an ingenious lecturer on mechanics, made many experiments on the strength of wood and metal; whence he concludes, that cast iron is from 3 to 4 times stronger than oak of equal dimensions; and from 5 to 6 times stronger than deal. And that bars of cast iron, an inch square, weighing 9 lbs. to the yard in length, supported at the extremities, bear on an average, a load of 970 lbs. laterally. And they bend about an inch before they break.

Many other experiments on the strength of different materials, and curious results deduced from them, may be seen in Dr. Gregory's and Mr. Emerson's Treatises on Mechanics, as well as some more própositions on the strength and stress of different bars.

ON THE CENTRES OF PERCUSSION, OSCILLATION, AND GYRATION.

r`a

276. THE CENTRE of PERCUSSION of a body, or a system of bodies, revolving about a point, or axis, is that point, which striking an immoveable object, the whole mass shall not incline to either side, but rest as it were in equilibrio, without acting on the centre of suspension.

277. The Centre of Oscillation is that point, in a body vibrating by its gravity, in which if any body be placed, or if the whole mass be collected, it will perform its vibrations in the same time, and with the same angular velocity, as the whole body, about the same point or axis of suspension.

278. The Centre of Gyration, is that point, in which if the whole mass be collected, the same angular velocity will be generated in the same time, by a given force acting at any place, as in the body or system itself.

279. The angular motion of a body, or system of bodies, is the motion of a line connecting any point and the centre or axis of motion; and is the same in all parts of the same revolving body. And in different unconnected bodies, each revolving about a centre, the angular velocity is as the absolute velocity directly, and as the distance from the centre inversely; so that, if their absolute velocities be as their radii or distances, the angular velocities will be equal.

PRO