and cr1, cr2, as shown in the figure. It is evident that the two forces acting on each rod must be equal and opposite, i. e. Ap1=Bq2, Bq1=Cr2, cr1=Ap2. It will be remarked that the tendencies of the forces are to compress A B and ac, and to stretch BC, i. e. AB and AC sustain a thrust, and Bca strain (Art. 28). Referring to Art. 46; if the ends B and cof the rafters are connected by a beam вс (fig. 39), called a tie beam, they will constitute a triangular frame like that we have just considered; it can be easily shown that the tie beam is subject to a strain equal to the horizontal thrust of each rafter, i. e. equal to wXBC÷4AD (Ex. 225). Under these circumstances the roof will act on the walls merely by its weight, and each wall will, of course, support half the whole weight of the roof.

Ex. 227. If in fig. 40 the point o fall within the triangle, show that all the bars will be compressed or all stretched.

Ex. 228. Two rafters AB and ac are each 20 ft. long, their feet are tied by a wrought-iron rod BC whose length is 35 ft., and a weight of 1 ton is suspended from A; determine the strain it produces on the tie, the weight of the rafters, &c., being neglected. If the rod have a section of a quarter of a square inch, determine the weight that must be suspended at a to break it. Ans. (1) 2024 lbs. (2) 18,590 lbs.

Ex. 229. There is a roof whose pitch is 22° 30′, the rafters are 40 ft. long; the weight of each square foot of roofing is 18 lbs.; determine the diameter of the wrought-iron tie necessary to hold the feet of the principal rafters with safety, supposing them 10 ft. apart. Ans. 1·28 inches.

48. Note. The foregoing remarks as to the thrusts on the rafters and the strain on the tie beam, apply to the cases in which the joints are perfectly smooth: as this is never the case, the thrusts, &c., may not equal the calculated amount; but it is generally considered that reliance should never be placed on the resistance offered by a joint to the revolution of a rod round it. It will be instructive, however, to consider the case in which the rods and the joint at a (fig. 41) are perfectly rigid. Suppose two points, b and c, to be taken near to a, and joined by a rod bc; if this rod were inextensible, and if

FIG. 41.

there would be no horizontal thrust

on the wall, which, as before, would merely have to sup

port the weight of the roof.

If we suppose the rod be to be replaced by a metal

plate firmly fastened to the beams, as

FIG. 42.

d

resistance by the materials to crush

ing on the bolts, and to the tearing of the plate across a d. Hence, under all circumstances, the walls have to sustain the whole weight of the roof, and besides this, a horizontal thrust which will more nearly equal W×BC÷4 AD as the joint is less rigid.

CHAPTER IV.

THE FUNDAMENTAL THEOREMS OF STATICS.

49. Axioms. The following chapter contains demonstrations of the fundamental theorems of statics, so far as forces acting in one and the same plane are concerned. It may be well to invite the reader's attention to the order of proof adopted. In the first place the case of two forces and their resultant is fully discussed, together with the conditions of the equilibrium of three forces, and the case in which two forces do not have a resultant. In the next place, the results obtained for two forces are extended to any number of forces. Lastly, a peculiar property of parallel forces the possession of a 'centre' is proved. The demonstrations are of a very abstract character, and should be thoroughly mastered. Applications of several of the theorems have been already given in Chap. III., and many more will be found in the succeeding chapters. The demonstrations are based on certain assumed elementary principles or axioms. The assumption of these principles is, of course, not arbitrary, but justified by experience of the action of forces. The axioms are as follow:

Ax. 1. The line which represents the resultant of two forces acting on a point, falls within the angle made by the lines that represent those forces (See Art. 25).

Ax. 2. If two equal forces act on a point, the line that represents their resultant bisects the angle between the lines that represent those forces.

Ax. 3. If a force acts upon a body, it may be supposed to act indifferently at any point in the line of its action, provided that point is rigidly connected with the body.

Ax. 4. It is necessary and sufficient for the equilibrium of any system of forces, that one of them be equal and opposite to the resultant of all the rest.

Ax. 5. If a system of forces in equilibrium be imposed on or removed from any system of forces, it will not affect the equilibrium of that system, if it be in equilibrium, nor its resultant, if it have a resultant.

Proposition 3.

The principle of the parallelogram of forces (Art. 32) is true of the direction of the resultant of two equal forces.

Since AC equals AB it equals CD, therefore the angle CAD is equal to the angle ADC, but since CD is parallel to AB, the angle ADC is equal to the angle BAD, therefore the angle BAD equals the angle CAD, and the line A D bisects the angle PAQ; but the line of action of the resultant of P and Q bisects the angle PAQ (Ax. 2), therefore the resultant acts along A D. Q. E. D.

50. Remark. The following proposition may be regarded as the foundation of the science of statics; the demonstration generally seems obscure to readers who meet with it for the first time: this results from the somewhat unusual form of the proof; it may therefore be well to remark that the demonstration consists of two parts; in the first part it is shown that if the principle is true in two cases, viz. with regard to the pair of forces P and P1 and the pair P and P2, it must also hold good in a third case, viz. in regard to the pair of forces P and P1+P2; this part of the proof is purely hypothetical, as much so as in the case of a demonstration by reduction to an absurdity; the second part of the proof takes up the argument, but as a matter of fact the proposition is true in two certain cases; therefore it must be true in a third case, therefore in a fourth case, and so on.

Proposition 4.

The principle of the parallelogram of forces is true of the direction of the resultant of any two commensurable forces.

Let the force Pact on the point a along the line A B, and the forces P1 and P2 on the point a along the

FIG. 44.

P

D

C

B

E

F

line AC; take AB, AC, CD, P. respectively proportional to P, P1, and P2, and complete the parallelograms BC, ED, then is the figure BD a parallelogram; draw the diagonals AE, CF and AF, and suppose

the points C, D, E, F to be rigidly connected with A.

(a) The lines AB and Ac represent the forces P and P1; assume that their resultant acts along AE; then can P and P1 be replaced by their resultant acting at A along a E, and, since A and E are rigidly connected, by that resultant acting at E along AE (AX. 3); but this resultant acting at E can be replaced by its components acting at E, viz. by P1 along BE, and by palong CE; and these again, since cand Fare rigidly connected with E, by P1 acting at Falong B F, and P acting at calong CE.