FIGURES TO THE CALCULATION OF STRAINS ON FRAMEWORK. 447 



forces, now explained as reciprocal figures, over the ordinary methods of calcula- 

 tion used by engineers. 



The graphic method of calculation hitherto employed has been to draw a 

 separate polygon of forces for each joint. To do this it was necessary at each 

 joint to start afresh, setting out the known forces, and from them determining 

 the unknown forces. In thus continually measuring and setting out new lines 

 considerably greater accumulations of error, and more frequent errors, are pro- 

 bable, than when each line when determined by an intersection is used where it 

 lies, and if the successive polygons are drawn to any considerable scale, they lap 

 over one another on the drawing in an awkward and complex manner. Moreover, 

 it is impossible to complete the diagram of the reciprocal figure without taking 

 every line into account ; and the closing of the diagram by the final line is an 

 almost perfect check on the accuracy of the delineation. 



When compared with algebraic methods, the simplicity and rapidity of exe- 

 cution of the graphic method is very striking ; and algebraic methods applied to 

 frames such as the Warren girders, in which there are numerous similar pieces, 

 are found to result in frequent clerical errors, owing to the cumbrous notation 

 which is necessary, and especially owing to the necessary distinction between 

 odd and even diagonals. If this is the case when the loads are uniform or 

 symmetrical, the advantage is much more strikingly in favour of the graphic 

 method when the loads are not symmetrical, and when they are inclined, as in 

 fig. 13, or in such cases as the framed arch and suspension bridge. In fine, the 

 diagram once drawn acts as a sort of graphic formula for the strain on every part 

 of the bridge or roof, and it is a formula which can hardly be misapplied. 



In conclusion, the author begs to acknowledge with thanks the assistance of 

 his students, Mr T. H. Cunningham and Mr A. M'Culloch, in preparing the 

 diagrams ; and takes the opportunity of repeating, that the merit of discovering 

 the method is entirely due to Professor Maxwell and Mr Taylor, the object of 

 the present paper being to put the theory in such a form as should be intelligible 

 to the engineer and mechanician. 



