RECIPROCAL FIGURES, FRAMES, AND DIAGRAMS OF FORCES. 39 



EXPLANATION OF THE DIAGRAMS (Plates I. II. III.). 



Diagrams I.a and 1.6 illustrate the necessity of the condition of the possibility of reciprocal 

 diagrams, that each line must be a side of two, and only two, polygons. Diagram I.a is a skeleton of 

 a frame such, that if the force along any one piece be given, the force along any other piece may be 

 determined. But the piece N forms a side of four triangles, NFH, NGI, NJL, and MM, so that if 

 there could be a reciprocal diagram, the line corresponding to N would have four extremities, which is 

 impossible. In this case we can draw a diagram of forces in which the forces H, I, J, and K are each 

 represented by two parallel lines. 



Diagrams II. a and II. 6 illustrate the case of a frame consisting of thirty-two pieces, meeting four 

 and four in sixteen points, and forming sixteen quadrilaterals. Diagram Il.a may be considered as a 

 plane projection of a polyhedron of double continuity, which we may describe as a quadrilateral frame 

 consisting of four quadrilateral rods, of which the ends are bevelled so as to fit exactly. The pro- 

 jection of this frame, considered as a plane frame, has three degrees of stiffness, so that three of the 

 forces may be arbitrarily assumed. 



In the reciprocal diagram II. 6 the lines are drawn by the method given at p. 7, so that each 

 line is perpendicular to the corresponding line in the other figure. To make the corresponding lines 

 parallel we have only to turn one of the figures round a right angle. 



Diagrams IILa and III. 6 illustrate the principle as applied to a bridge designed by Professor F. 

 Jenkin. The loads Qj Q 2 , &c, are placed on the upper series of joints, and R x R 2 , &c, on the lower 

 series. The diagram III. 6 gives the stresses due to both sets of loads, the vertical lines of loads being 

 different for the two series. 



Diagrams IV. a and IV.6 illustrate the application of Airy's Function to the construction of 

 diagrams of continuous stress. 



TV.a represents a cylinder exposed to pressure in a vertical and horizontal direction, and to 

 tension in directions inclined 45° to these. The lines marked a, b, c, &c, are lines of pressure, and 

 those marked o, p, q, are lines of tension. In this case the lines of pressure and tension are rectangular 

 hyperbolas, the pressure is always equal to the tension, and varies inversely as the square of the 

 distance between consecutive curves, or, what is the same thing, directly as the square of the distance 

 from the centre. 



IV.6 represents the reciprocal diagram corresponding to the upper quadrant of the former one. 

 The stress on any line in the first diagram is represented in magnitude and direction by the corres- 

 ponding line in the second diagram, the correspondence being ascertained by that of the corresponding 

 systems of lines a, 6, c, &c, and o, p, q, &c. 



"We may also consider IV.6 as a sector of a cylinder of 270°, exposed to pressure along the lines 



_ 2 



a, 6, c, and to tension along o, p, q, the magnitude of the stress being in this case r 3 . The upper 



quadrant of IV. a is in this case the reciprocal figure. This figure illustrates the tendency of any 

 strained body to be ruptured at a re-entering angle, for it is plain that at the angle the stress becomes 

 indefinitely great. 



In diagram IV. a — 



F = - r* cos 40 G = - r % cos 20 H = - r 2 sin 20 . 



4 2 2 



In diagram IV.6 — 



/=-/>^cos-<£ g = -p 5 cos -cj> h= -p* sin -<}>. 



Diagrams Y.a and V.6 illustrate Airy's theory of stress in beams. 



V.a is the beam supported at C and D by means of bent pieces clamped to the ends of the beam 

 at A and B, at such a distance from C and D, that the part of the beam between C and D is free from 

 the local effects of the pressures of the clamps at A and B. The beam is divided into six strata by 



