AND DIAGRAMS OF FORCES. 175 



functions of s, variables, and *, pieces may be removed from the frame without 

 rendering it loose. 



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

 ference of the frame or on those of the holes, and s 1 points lie in the interior, 

 the degree of stiffness will be 



p = s x + 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 



mf= 2e, 

 e s f=2n 4, 



whence p = 5 4n + ( 1 -- e. 



If all the faces are quadrilaterals m = 4 and p = 5 4n, 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 pro- 

 jection 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. If 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 pres- 

 sures 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 



