AND DIAGRAMS OF FORCES. 163 



in one diagram, and draw one diagram of the frame and a separate diagram 

 of the forces. By this method we shall not only avoid confusion, but we shall 

 greatly simplify mechanical calculations, by reducing them to operations with 

 the parallel ruler, in which no useless lines are drawn, but every line repre- 

 sents an actual force. 



A Diagram of Forces is a figure, every line of which represents in mag- 

 nitude and direction the force acting along a piece of the frame. 



To express the relation between the diagram of the frame and the dia- 

 gram of forces, the lines of the frame should each be indicated by a symbol, 

 and the corresponding lines of the diagram of forces should be indicated by 

 the same symbol, accented if necessary. 



We have supposed the corresponding lines to be parallel, and it is neces- 

 sary that they should be parallel when the frame is not in one plane ; but 

 if all the pieces of the frame are parallel to one plane, we may turn one of 

 the diagrams round a right angle, and then every line will be perpendicular 

 to the corresponding line. 



If any number of lines meet at the same point in the frame, the corre- 

 sponding lines in the diagram of forces form a closed polygon. 



It is possible, in certain cases, to draw the diagram of forces so that if 

 any number of lines meet in a point in the diagram of forces, the corre- 

 sponding lines in the frame form a closed polygon. 



In such cases, the two diagrams are said to be reciprocal in the sense in 

 which we use it in this paper. If either diagram be taken as representing the 

 frame, the lines of the other diagram will represent a system of forces which, 

 if applied along the corresponding pieces of the frame, will keep it in equi- 

 librium. 



The properties of the " triangle " and " polygon " of forces have been long 

 known, and a " diagram " of forces has been used in the case of the " funi- 

 cular polygon," but I am not aware of any more general statement of the 

 method of drawing diagrams of forces before Professor Rankine applied it to 

 frames, roofs, &c., in his Applied Mechanics, p. 137, &c. The "polyhedron of 

 forces," or the proposition that forces acting on a point perpendicular and pro- 

 portional to the areas of the faces of a polyhedron are in equilibrium, has, I 

 believe, been enunciated independently at various times, but the application of 

 this principle to the construction of a diagram of forces in three dimensions 

 was first made by Professor Rankine in the Philosophical Magazine, Feb. 1864. 



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