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XI. — On the Practical Application of Reciprocal Figures to the Calculation of 

 Strains on Framework. By Professor Fleeming Jenkin. (Plates XVII. to 

 XXII.) 



(Read 15th March 1869.) 



The theory of reciprocal figures used as diagrams of forces was first com- 

 pletely stated by Professor T. Clerk Maxwell, in a paper published in the 

 " Philosophical Magazine," April 1864. The following definition of reciprocal 

 plane figures, and their application to statics, are there given as follows : — 



" Two plane figures are reciprocal when they consist of an equal number of 

 lines, so that corresponding lines in the two figures are parallel, and correspond- 

 ing lines which converge to a point in one figure form a closed polygon in the 

 other." 



" If forces represented in magnitude by two lines of a figure be made to act 

 between the extremities of the corresponding lines of the reciprocal figure, then 

 the points of the reciprocal figure will all be in equilibrium under the action of 

 these forces." 



The demonstration of this statement is given. The conditions under 

 which stresses are determinate, and some examples of reciprocal figures, are 

 also given in the paper, which leaves nothing to be desired by the mathe- 

 matician. 



Few engineers would, however, suspect that the two paragraphs quoted put 

 at their disposal a remarkably simple and accurate method of calculating the 

 stresses in framework; and the author's attention was drawn to the method 

 chiefly by the circumstance that it was independently discovered by a practical 

 draughtsman, Mr Taylor, working in the office of the well-known contractor, Mr 

 J. B. Cochrane. The object of the present paper is to explain how the principles 

 above enunciated are to be applied to the calculation of the stresses in roofs and 

 bridges of the usual forms. 



The construction of a reciprocal figure for any frame requires the exercise of 

 a little discrimination, and the method employed can be best explained by 

 examples; those frames only being considered which are so braced as to be 

 stiff, but have not more members than is sufficient for this purpose. 



The simplest example is that of a triangle loaded in the middle, and supported 

 at the two ends. 



At each point three forces are acting. Thus we have at point 1 the upward 



VOL. XXV. PART II. 5 X 



