RECIPROCAL FIGURES, FRAMES, AND DIAGRAMS OF FORCES. 13 



If there are n holes in the frame, so that s' points lie on the circumference of 

 the frame or on those of the holes, and s x points lie in the interior, the degree 

 of stiffness will be 



— p = Sj + 3n . 



If a plane frame be a projection of a polyhedron of /faces, each of m sides, and 

 enclosing a space n times connected, then 



to/ = 2e 

 e — s — f=2n — 4 

 2s — e =p + 3 , 



whence 



P — 5 — 4:71 + ( 1 ) 



1 \ TO/ 



If all the faces are quadrilaterals m = 4 and p = 5 — 4w, or a plane frame which 

 is the projection of a closed polyhedron with quadrilateral faces, has one degree 

 of freedom if the polyhedron is simply connected, as in the case of the projec- 

 tion of the solid bounded by six quadrilaterals, but if the polyhedron be doubly 

 connected, the frame formed by its plane projection will have three degrees of 

 stiffness. (See Diagram II.) 



Theorem.— It every one of a system of points in a plane is in equilibrium 

 under the action of tensions and pressures acting along the lines joining the 

 points, then if we substitute for each point a small smooth ring through 

 which smooth thin rods of indefinite length corresponding to the lines are 

 compelled to pass, then, if to each rod be applied a couple in the plane, whose 

 moment is equal to the product of the length of the rod between the points 

 multiplied by the tension or pressure in the former case, and tends to turn the 

 rod in the positive or the negative direction, according as the force was a tension 

 or a pressure, then every one of the system of rings will be in equilibrium. For 

 each ring is acted on by a system of forces equal to the tensions and pressures 

 in the former case, each to each, the whole system being turned round a right 

 angle, and therefore the equilibrium of each point is undisturbed. 



Theorem. — In any system of points in equilibrium in a plane under the 

 action of repulsions and attractions, the sum of the products of each attraction 

 multiplied by the distance of the points between which it acts, is equal to the 

 sum of the products of the repulsions multiplied each by the distance of the 

 points between which it acts. 



For since each point is in equilibrium under the action of a system of attrac- 

 tions and repulsions in one plane, it will remain in equilibrium if the system 

 of forces is turned through a right angle in the positive direction. If this opera- 

 tion is performed on the systems of forces acting on all the points, then at the 

 extremities of each line joining two points we have two equal forces at right 



VOL. XXVI. PART I. D 



