42 



JorifXAL OF TIIK 



drels. On that account he was led to undertake the very 

 serious labor of computing the depths of key for a number 

 of arches after the theory proposed, and to compare with 

 the results given by some of the empirical formulas. 



The results are shown graphically in the figure, the line 

 through the small circles (which is nearly straight) being- 

 plotted from the values given in the table above. 



20 



40 



60 80 100 



SPAN IN FEET 

 RISE=-i-SPAN 



120 



140 



160 



The depths of key proposed by the following authors are 

 given by the ordinates to the various lines for the corre- 

 sponding spans given by the horizontal lines : Trantwine 

 (line T), Croizette-Desnoyers (line C-D), Perouet (line P), 

 SchefHer, by interpolation from his tables (line S, dotted) 

 and Dejardin (line D). 



These results refer to materials of only average strength 

 (second-class masonry for the Trantwine line) and vary very 

 greatly; thus for an arch of i6o feet span and thirty-two 

 feet rise Trantwine gives a depth of key of 4.3 feet, 

 whereas Dejardin requires eight feet and then increases the 

 radial length of joint according to the secant of the incli- 

 nation of the joint to the vertical as we approach the abut- 

 ment. 



■ The theoretical depths of key agree more nearly with 

 those of Dejardin and Scheffler than with any of the others, 

 though it is in excess for the smaller spans over any of the 

 empirical results, as should naturally be expected. 



