4 MR CLERK MAXWELL ON 



Professor Fleeming Jenkin, in a paper recently published by the Society, 

 has fully explained the application of the method to the most important cases 

 occurring in practice. 



In the present paper I propose, first, to consider plane diagrams of frames 

 and of forces in an elementary way, as a practical method of solving questions 

 about the stresses in actual frameworks, without the use of long calculations. 



I shall then discuss the subject in a theoretical point of view, and give a 

 method of denning reciprocal diagrams analytically, which is applicable to 

 figures either of two or of three dimensions. 



Lastly, I shall extend the method to the investigation of the state of stress 

 in a continuous body, and shall point out the nature of the function of stress 

 first discovered by the Astronomer Royal for stresses in two dimensions, extend- 

 ing the use of such functions to stresses in three dimensions. 



On Reciprocal Plane Rectilinear Figures. 



Definition. — Two plane rectilinear figures are reciprocal when they consist 

 of an equal number of straight lines, so that corresponding lines in the two 

 figures are at right angles, and corresponding lines which meet in a point in 

 the one figure form a closed polygon in the other. 



Note. — It is often convenient to turn one of the figures round in its own 

 plane 90°. Corresponding lines are then parallel to each other, and this is 

 sometimes more convenient in comparing the diagrams by the eye. 



Since every polygon in the one figure has three or more sides, every point in 

 the other figure must have three or more lines meeting in it. Since every line 

 in the one figure has two, and only two, extremities, every fine in the other figure 

 must be a side of two, and only two, polygons. If either of these figures be taken 

 to represent the pieces of a frame, the other will represent a system of forces 

 such that, these forces being applied as tensions or pressures along the correspond- 

 ing pieces of the frame, every point of the frame will be in equilibrium. 



The simplest example is that of a triangular frame without weight, ABC, 

 jointed at the angles, and acted on by three forces, P, Q, R, applied at the 

 angles. The directions of these three forces must meet in a point, if the frame 

 is in equilibrium. We shall denote the fines of the figure by capital letters, 

 and those of the reciprocal figure by the corresponding small letters ; we shall 

 denote points by the lines which meet in them, and polygons by the lines which 

 bound them. 



Here, then, are three lines, A, B, C, forming a triangle, and three other 

 lines, P, Q, R, drawn from the angles and meeting in a point. Of these forces 

 let that along P be given. Draw the first line p of the reciprocal diagram 

 parallel to P, and of a length representing, on any convenient scale, the force 

 along P. The forces along P, Q, R are in equilibrium, therefore, if from one 



