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ke« r . n P 0sltl0n V.—The lines of direction of three forces^ d pin 8 ea<; h other in equilibrio, or a solid, and the intensityimp t ® ndenc y °f one of them being given ; to find thensity and tendency of the other two.tlir-p 0 ^ 6 Figure 4.—When two of the angles formed by thethe tt imeS direction are less than two right angles. Leth e • lree directions be b p, b e, b g, and let the given intensityWl!- 1 ^* ne B E ’ ar *d i e t i ta tendency be from e towards b.Par n , B D er l ua i to the given intensity, and complete thecJi r a ®*°gram abcd. a b is the intensity in its own line ofi s ,, 10 . n B E> its tendency being from b towards p ; and B Ctend 6 lntensit y the force in its line of direction b g, itsthe f n<t y being from b towards g : for produce e b to h, since° rCe actin S * n the line e b presses the point b, then, byEli m ** * s tb e same thing, whether the force in the linein ^* ess the point b, or an equal force on the other side of b :then 1 - ^ raw the point b, and instead of the force pressingat { j Wnt B by a force at e, let the point b be drawn by a forcea re . ’ thus the point b will be drawn by three forces, whichb 6 g 1 J r ’j e< l u ilibri o by the last Proposition. Or if the point b had

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qu en . p and b g would have pressed these lines, and conse-tli e V three forces acting at P, G, h, would be all pressingatligj, • n *' B ’'t therefore appears, when three forces keep eachle Ss , ln equilibrio, and their lines of direction make two angles•hedi * an . tw ° right angles, that the force acting in the inter-r pl e hne will be contrary to those in the two extreme lines,ejq *° u §h this example only shows how to find the two'“term f orces when the intermediate force is given ; yet theI'eadi l 6( late force and one of the extreme forces may asbe Cai) ^ be found by having the other extreme force given:and ( S , e when one of the angles of a parallelogram is given,it ttl Position of a diagonal passing through that angle,as tv described as readily by having either of the sides^ aia gonal.

gf e „. Se Figure 5 —When any two angles of direction are

]^ r ^an two right angles.

angj e A B > e b, c b, be the three directions, whereof any twoand 0 Itla< ^ e by these lines are greater than two right angles,angu tlSf: quently the remaining one less than two rightfile S ' 4,et the given force act in e b ; produce e b through•flab;; os ' te angle t0 D > 30 as t0 divide it into two angles ;hie D lj to represent the intensity in e b, then by completingifit eri V a fi f 4°g ra m abcd, as before, ba will represent thePos ed y m ba, and bc in bc; and as the forces are sup-fite p Q -° act at the points A, e, c, they are either all drawingP r ' nf: B or all pressing it.

file ^P° s ^i° n VI.—Given, the di rections of four forces in‘Ute^f 6 plane, keeping a solid in equilibrio, and one of theP r0( , les ’ to find the intensities of the other three.

Ptod Uee Ce any two directions till they meet each other; also,Join the 116 °^ ler two directions till they meet each other ;°tce o J' Wo angular points ; then, by means of the givenN f Q[ ,the other two at the same point: then, because• e ftm f S act * n g at each point of concourse in the same right1( ! thi s i. be equal, and have opposite tendencies, the force. IBe acting at the other point of concourse will now beS;it n e r „ ler efore, find the two remaining intensities in the

awn by a force acting at E, the two forces acting in the

a

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as at the first point of concourse.th ecti on f Figure 6. —Let e a, f b, g C, he, be thei Sivp °* ^‘e forces that support the body abcd, and letn l: „r n f wce be in e a. “ ‘ ‘

Produce e a, f b, till they meetE lq Prod” P^°duce gc, h d, till they meet in q. Join iq,^'4 to p ; then let i k represent the given force,

tp arid ' ete *b e parallelogram iiclm. Make q p equal toibese^^aplete the parallelogram opqr; then will im

intensity in f b, o Q in GC, and R Q in u d.

Example II- Figure 7.—Let a b d be a lever witli threearms, ac, bc, DC, revolvable about c, as a fulcrum, sup-ported in the direction c O ; and let forces act at the extre-mities a, b. d, in given directions, a k, d e, e b, and keep it inequilibrio : it is required to find the proportion of the forces.Produce two of the directions till they meet ; also producethe other direction, and that of the prop, till they meet; jointhe two angular points, and proceed as in Example I., andfind the parallelograms hefg, and ituis: then lk is theforce acting at A, and m k that in the direction of the prop,h e, the force acting at b, and f e that at d. The tendenciesof these forces are thus distinguished : let the point b bedrawn towards e, then the line e b is in a state of tension ;and because the angles h e g and g e f are less than tworight angles, the force in the direction e d will also be in astate of tension, and the middle one, e k, in a state of com-pression. Again, because the angles lkh and mkn areless than two right angles, and because e k is in a state of'compression, k a is likewise in a state of compression, and themiddle one c k, is in a state of tension ; or the post, c o, onthe opposite side, is in a state of compression, acting on theother side of g.

It must be observed, when any force acts upon any pointof a solid body, that to draw on one side of the point is thesame as to press upon the other side, or to press upon one sideis the same as to draw upon the opposite ; therefore, as thepoint C is drawn by the force si k, the prop, c o, is compressedby the fulcrum at C. The arms CA , cb.cd, are supposedto be void of weight. If the forces acting at a, b, d, beweights, p, q, e, going over the pulleys s, t, u, all the lines,as, bt, du, will be in a state of tension.

Proposition VII.—Given, the direction of five forces inone plane, keeping a solid in equilibrio, and the intensitiesand tendencies of two of them, to find the intensities andtendencies of the rest.

Find a force equivalent to the two given forces ; thenunite this given force with the three remaining ones, and thedirections of four forces, with the intensity of one of them,will be given to find the rest, which may be found by the lastproblem.

Let a b c d f, Figure 8, be a lever, with four arms, F A,f b, f C, f d, revolvable about e, and let it be acted upon byfive forces, four of which act upon the arms at the pointsa, b, c, d, in lines of direction a q, b s, C k, d i, and the otherupon the centre at f, in the line of direction ff ; then theintensities and tendencies of the two forces acting in thedirections c k, and d i, are given ; the one from c to k, andthe other from i to d. Produce the two directions c k andD l to meet each other at G, and complete the parallelogramG K i H, as in Case 2, Problem II., and G H will be the direc-tion and quantity of the force equivalent to G I and g ic :then proceed, as in Problem VI., with the given force G Hnow found, and the three remaining directions, AQ, b s, pf,and complete the parallelograms iinop and qbst ; thenn p is the quantity that supports the point or axis f inthe direction sf, and SB that which supports B in thedirection be.

From this example it appears, that when the direction ofany number of forces is given, and all the intensities andtendencies but three, the intensities and directions of thesethree may be found by compounding any two of the givenforces, then uniting the force found with another of the givenforces, and again compounding these, and so on until all thegiven forces are compounded : then proceeding with the lastcompounded force and the three remaining directions, as inProblem.

Proposition VIII.—If there be two straight lines, a b, b c,