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THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[AUGTTST , 



fereiice to three points, to tlie case of any number of points, inexten- 

 sihlv anil flexibly connected. Let the points be A, H, C, D, E, F, fig. 3; 

 then, if the whole system beat rest, each pair of contiguous points 

 will be at rest with respect to each other, and conserpiently will be 

 connected hy a straight line, and acted on by equal and opposite forces. 

 Bv combining, as before, the forces at B, C, D, and E, we obtain their 

 resultants P, Cj, R, S, and we observe that, in general, any number of 

 points may be kept in equilibrium by as many forces acting se|r.irately 

 on each. For the sake of greater clearness, let us, however, imagine 

 tliat two equal and opposite forces are made to act upon B, in the de- 

 ductions C C, C B, respectively ; then the system will be at rest as 

 before, and if we suj)|)ose the force C B to act at C, the point B will 

 be kept in equilibrium by three forces, B A, B C, BP. In the same 

 manner, bj' superimposirg equal and opposite forces at the points C 

 und D, each will be kept at rest by three receding forces, two of which 

 are always in the direction of the lines of connection. By calling the 

 forces which act along the lines of connection V, W, X, Y, Z, we have 

 therefore the following proportions: — 



sin. P C B : sin. ABC 

 sin. ABC: sin. A B P 

 sin. Q C D : sin. BCD 

 sin. BCD: sin. B C Q 

 sin. R D E : sin. C D E 

 sin. C D E : sin. C D R 

 sin. S E F : sin. D E F 

 sin. D E F : sin. D E ,S. 

 From these proportions the relation of any one force to any other may 

 be determined, and consequently any force may be represented in 

 terms of any other and the sines of the angles through which their 

 lines of direction respectively pass. For example, 



v = z 



sin. P B C 

 sin. D E^ 



, and P =r S 



siu. ABC 

 sin. D E F ■ 



If the original forces A B, C B, by the union of which the force P is 

 obtained, were equal, P B produced will bisect the angle ABC, and 

 the same is to be remarked of the forces Q, R, S ; consequently, by the 

 preceding proportions we have in this case, V ^ W = X ^ Y = Z. 

 Moreover, denoting by 2 a, 2 /3, 2 j, 2 5, the angles of the polygon, it 

 follows : — 



R 



cos. a . COS. $ • cos. -/ ; cos. 5. 



Figs. 4 and 



That is to say, the forces applied at the several angles of the poly- 

 gon are proportional to the cosines of the halves of those angles. Let 

 us now suppose that the lines A B and B C are equal to each other. 

 Through the points A, B, C, fig. 4, describe the circle A B C D, draw 

 the diameter B D, the arc A E, and E F at right angles to A B. Then 

 li D bisects the /ABC, and because B A D is a right angle (Euc. 

 J). 3L b. 3) :— 



B A ; B D : : B F : B E : : cos. o : rad. 



B A 

 .*. cos. a = 5-f;. Hence, as th« forces P, Q, R, S, are proportional to 

 B D 



cos. a, cos. 0, &c., if we suppose all the sides of the polygon to be 

 equal, it is evident they will be inversely, as the radius of the circle 

 passing through the points terminating the two contiguous sides. But 

 if we imagine the sides of the polygon to become indefinitely small, it 

 then assumes the form of a curve, and the circle becomes the osculat- 

 ing circle, or the circle of equal curvature. If, then, a flexible curve, 

 the two extremities of which are immoveably fixed, be acted on at 

 points equidistant from each otherby a number of normal forces, these 

 forces will be inversely as the radii cif curvature of the points of appli- 

 cation, and the forces developed in the direction of the curve will be 

 everywhere the same. If the normal forces be equal, the reciprocals 



of the radii of curvature w ill be equal, and therefore the radii of curva- 

 ture themselves ; consequently, in this case, the curve will be part of 

 a circle. 



If the normal force vary as the cube of the cosine of the angle 

 formed by the ordinate and tangent at any point, the curve is a para- 

 bola, as is jjroved by Ihe following investigation. 



Let P A R, fig. 5, be a parabola generated by the action of normal 

 forces, P T the tangent at the point P and N T, the subtangent. Let 

 A N = :r, N P = 7/, and/*, the principal parameter or latus I'ectum; 

 also call the radius of curvature R, and the normal force V. 



N P 



Then, cos. N P T = 5-=- 



But B P^ = N T- + K P-' = 4 AN2 + N P« 

 N P ?/ 



cos. N P T = 



Or since y- . 



cos.= N P T = 



V 4 A N= + N P- 

 p.r, cos. N P T = 

 p.r 



' v-1 .!•' + r 



4 X- + fx 



^/ix' +px 

 i.,+p 



Hence cos.- N P T a 



1 



^ X -\- p 

 But in the parabola R 



; or cos.3 N P T a 



{■ix'+p)% 



{■^ x-\-p)h_ 

 2 s/p 



Consequently V a -5 a eos.^ N P T 



Let « be the normal force at the vertex, and denote by <^ the 

 ^/ N P T : — hence, because at the vertex cos. t = 1> 



f : V : : 1 ; cos.' (p .■. v = r cos.' i^. 



Again, since in the catenary, R CC >* <(> denoting the angle 



° COS.- <p 



formed by the abscissa and tangent, it is seen at once, that when 

 V OC cos.= (p, the curve is a catenary. 



Figs. 6, 7, and S. 



Assuming the system of points A, B, C, &c. fig. 6, to be in equili- 

 brium, we sliall now imagine the connecting lines to become perfectly 

 rigid. It is evident that this supposition will not affect the equili- 

 brium, as it does not involve the addition or abstraction oi force, the 

 only agent by which equilibrium is preserved or destroyed. If then 

 the system was in equilibrium before, it will remain so now, and we 

 have consequently a rigid body acted upon by the forces V, P, Q, R, 



• Poinsot. Traite de Statique. 



