By finding the perimeter of the inscribed and that of the circumscribed regular polygon of 393 216 (i.e.
If there are no redundant members in the frame there will be only two members abutting at the point of support, for these two members will be sufficient to balance the reaction, whatever its direction may be; we can therefore draw two triangles, each having as one side the reaction YX, and having the two other sides parallel to these two members; each of these triangles will represent a polygon of forces in equilibrium at the point of support.
The funicular or link polygon has its vertices on the lines of action of the given forces, and its sides respectively parallel to the lines drawn from 0 in the force-diagram; in particular, the two sides meeting in any vertex are respectively parallel to the lines drawn from 0 to the ends of that side of the force-polygon which represents the corresponding force.
Internally it is a polygon of sixteen unequal sides, and the cupola is supported by sixteen ribs, springing from the same number of columns.
Taking the circumference as intermediate between the perimeters of the inscribed and the circumscribed regular n-gons, he showed that, the radius of the circle being given and the perimeter of some particular circumscribed regular polygon obtainable, the perimeter of the circumscribed regular polygon of double the number of sides could be calculated; that the like was true of the inscribed polygons; and that consequently a means was thus afforded of approximating to the circumference of the circle.
Further, at any one of the centres of load let PL represent the magnitude and direction of the gross load, and Pa, Pb the two resistances by which the piece to which that load is applied is supported; then wifl those three lines be respectively the diagonal and sides of a parallelogram; or, what is the same thing, they will be equal to the three sides of a triangleS and they must be in the same plane, although the sides of the polygon of resistances may be in different planes.
A single known force in a polygon determines the direction of all the others, as these must all correspond with arrows pointing the same way round the polygon.
For longer bridges the funicular polygon affords a method of determining maximum bending moments which is perhaps more convenient.
In considering its properties, the load at each centre of load is to be held to include the resistances of those joints which are not comprehended in the partial polygon of resistances, to which the theorem of 7 will then apply in every respect.
In the polygon of loads the direction of a load sustained by parallel resistances traverses the point O-i i Since the relation discussed in 7 was enunciated by Rankine, an enormous development has taken place in the subject of Graphic Statics, the first comprehensive textbook on the subject being Die Graphische Statik by K.