To apply the same principle in the second place to the brace AC, and the truss FE, let the straining force be supposed to be applied at E, and EK to denote its measure and direction. Complete the parallelogram Ef K g. Join f g, and through E draw It i parallel to f g. Then the truss FE being on the same side of h i as the straining force, EK, will be in a state of compression; and the brace AC being on the opposite side of the same line, will be in a state of extension as determined in the preceding case.
To apply the mechanical principle in the next place, to the brace AC or AB and the longitudinal timber DE, let the straining force be allowed to act at D, and let DE be its measure and direction. Com plete the parallelogram D k L 1, and join k 1; and through D draw m n parallel to the last mentioned line. Then the longitudinal timber DE being on the same side of m n as the straining force, it will be in a state of compression, and the brace AB or AC, as be fore determined, in a state of extension.
Fourthly, Let the parts now to be selected, be the longitudinal timber GH and the truss HE. Then if the straining force be applied at let HAI denote its measure and direction; and on it as a diagonal, let the parallelogram H o DI p be constructed, having its sides coincident with the directions of the timbers proposed; join o p. and through I-I draw q r parallel to it. Then since the truss Ell is on the same side of q r as the straining force, it will be in a state of compression; and the longitudinal timber All being on the opposite side of the same line, will be in a state of extension.
Hence it appears that the resultant of the various forces acting on the diagonal frame proposed will ope rate so as to extend the braces AB and AC and the lon gitudinal timber GH; but on the remaining parts of the frame, viz. the trusses GD, DF, HE, El", and the longitudinal timbers DE, BC, the effect will be to pro duce compression; agreeing with the experimental con clusion of Sir Robert Seppings, that the frame with this disposition of the braces ••comes more in contact by the pressure." Let us now endeavour to estimate the effect of a similar system of forces, on a system of framing whose braces and trusses are disposed in opposite directions to those of the preceding investigation. For this pur pose, let the first application of the lemma be to the longitudinal timber BC, and brace AC, Fig. 8, A be ing the fulcrum; and let the point C be that to which the straining force is applied. Suppose CI to be its measure and direction, and complete the parallelogram C a I b. Join a b, and through C draw d e parallel to that diagonal. Then since the brace AC is on the same side of d c as the straining force, it will be sub ject to compression, contrary to the effect produced in the former case. But the longitudinal timber BC, like GH in the former figure, will undergo extension. In the next place, let the straining force be supposed to be applied at E, in order to estimate its effects on the brace AC, and the truss FE; and let EK be its mea sure and direction. Complete the parallelogram Ef K g; join f g, and draw h I parallel to it through the point of application E. Then the brace AE being be low the line it i, will undergo compression as before; and the truss FE being above the same line will un dergo extension.
In the third place, let the straining force be applied at D to produce an effect on the brace BA and the longitudinal piece DE, and DL be its measure and direction. Complete the parallelogram of force D k L 1. Join k I, and through D draw m n parallel to k I. In this case, therefore, the brace DA being be low m n must undergo compression, and the longitudi nal timber DE being above the same line, must un dergo extension.
Fourthly, Let the straining force be applied at H, to estimate its effect on the truss Eli, and the longi tudinal timber GH, and let its measure and direction be HM. Complete the parallelogram of forces, H o
I\I p, having its sides in the axes of the timbers pro posed. Draw the diagonal o p, and parallel to it, through H, the line q r. Hence it appears- that the truss EH, being above the line q r, must undergo ex tension; and the longitudinal timber GH, being below the same timber, must undergo compression.
With this disposition of the timbers, therefore, it appears, that the forces operating on the frame will produce a compression of the braces B.:1, C.1, and of the. longitudinal timber GH; but on the remaining parts of the frame, viz. the trusses BD, DA, CE, EA, and the. longitudinal timbers DE, BC, the effect will be to pro duce compression, agreeing also with the experimen tal conclusion of Sir Robert Seppings, that on the ap plication of a straining force, the trusses an 1 middle longitudinal piece •• will be immediately disengaged and fall out." The preceding results may be conveniently arrang ed in the following table: The primary object of the diagonal framing is to prevent arching; and if we suppose Al', in both fi gures, to represent the natural line From which the arching proceeds towards both extremities, it is evi dent that it is the mechanical combination represent ed in Fig. 7. which can alone prevent it. For since A, in that figure by the hypothesis, is one of the neu tral points of the system, it may be regarded as fixed, and the tendency of arching being to depress the point If, C, and C,B, the effect on the braces AC and All will be precisely similar to the weights applied in the preceding investigation; that is, to produce extension, and which is effectually provided for by the fastenings. The effect, moreover, brought at the same time into action by the trusses, in consequence of the disturb ing force, is to resist, by the whole longitudinal strength of their fibres all tendency to alteration of form; so that the effect exerted to depress the point (2, is at once resisted by the fastenings appertaining to the brace AC, and to the longitudinal strength of the fibres of the truss proceeding from the unchangeable point F. The point E becoming, in this point of view, fixed, the action of the force which tends to de press the point II, in common with the point C, is re sisted by the fastenings of the longitudinal timber All, and by the longitudinal resistance of the fibres of the truss Ell; so that, provided the fastenings of the braces and of the upper longitudinal timber are suffi cient, and the abutments of the trusses and of the mid dle longitudinal timber are also proper, all tendency to arching will be resisted in proportion to the perfec tion of the materials, and the excellence of the work manship.
But by referring to the converse disposition of the braces, as represented in Fig. 8, it appears, from the preceding investigation, that the braces AC and AB are subject to congwes,5.ion. And since the point A is by the hypothesis, the neutral or fixed point, the ef fect of the compression of the brace AC must be to depress the point C, and thus to promote the tendency to arching. Nor is this tendency to lower the point C prevented by the action of the truss FE; since the point F being fixed by the supposition, the tendency to extension, which takes place in the truss must tend to lower the point E, and thus to promote the further declension of the point C. The point E being thus depressed, must add its effect to the extending force called into action in the truss EH, and thus produce a declension in the point H. Hence the whole effect of the disturbing force is to lower every part of the frame from C to II, and thus to promote the arching of the vessel. Ilence the superiority of the present system of diagonal framing becomes apparent, and the advantages derived from it are demonstrated by the small alteration of form which ships now undergo in the act of launching.