ON THE CALCULATION OF THE EQUILIBRIUM AND STIFFNESS OF FRAMES. 599 



equations of extensions, and e equations of elasticity ; so that the solution is 

 always determinate. 



The following method enables us to avoid unnecessary complexity by treating 

 separately all pieces which are additional to those required for making the frame 

 stiff, and by proving the identity in form between the equations of forces and 

 those of extensions by means of the principle of work. 



On the Stiffness of Frames. 



Geometrical definition of a Frame. A frame is a system of lines connecting 

 a number of points. 



A stiff frame is one in which the distance between any two points cannot 

 be altered without altering the length of one or more of the connecting lines 

 of the frame. 



A frame of s points in space requires in general 3s 6 connecting lines to 

 render it stiff. In those cases in which stiffness can be produced with a smaller 

 number of lines, certain conditions must be fulfilled, rendering the case one of 

 a maximum or minimum value of one or more of its lines. The stiffness of 

 such frames is of an inferior order, as a small disturbing force may produce 

 a displacement infinite in comparison with itself. 



A frame of s points in a plane requires in general 2s 3 connecting lines to 

 render it stiff. 



A frame of s points in a line requires s 1 connecting lines. 



A frame may be either simply stiff, or it may be self-strained by the intro- 

 duction of additional connecting lines having tensions or pressures along them. 



In a frame which is simply stiff, the forces in each connecting line arising 

 from the application of a force of pressure or tension between any two points 

 of the frame may be calculated either by equations of forces, or by drawing 

 diagrams of forces according to known methods. 



In general, the lines of connexion in one part of the frame may be affected 

 by the action of this force, while those in other parts of the frame may not 

 be so affected. 



Elasticity and Extensibility of a connecting piece. 



Let e be the extension produced in a piece by tension-unity acting in it, 

 then e may be called its extensibility. Its elasticity, that is, the force required 



