DR WALLACE ON A FUNCTIONAL EQUATION. 



667 



function f{x) in our catenary tables, so that, \i f{x~-h), f{x), f{x + h) be three 

 ordinates whose amplitudes have a common difference h, then 



2f(x)/(h-)=f{x + h)+/(z-h). 



By this formula we may interpolate an ordinate between any two (except the 

 last two), the formula for bisection being 



/w= 



fW 



It wUl be best to use the tabular amplitudes. Thus, to interpolate an ordinate 

 between /(2.60) and /(2.65), the difference of whose amplitudes is .05, we have 

 /(A)=/(.05) = 1.00125,/(.f-A)=/(2.60) = 6.76901,/(a; + A) = /(2.65) = 7.1]234: 

 these numbers substituted in the formula give 



3.55617 + 3.38455 



f (x)=f (2.625)=- 



= 6.93203. 



1.00125 

 These numbers reduced to feet, give 



X =2.625 X 18.343585=48.152 feet, ^ = 6.93203 x 6=41.592 feet. 

 It has been found that the angle (f>, which lines touching the curve at the 

 extreme ordinates of the catenary make with the horizontal axis, is 82° 30' 19" ; 

 and, (f)' denoting the like angle in the arch, we have a : «'= tan (p : tan </>' (art. 37), 

 therefore (p'=68° 5' 22". Professor Robison, in his Essay on Arch in the Enci/- 

 clo2>cedia Britannica, has taken this arch as an example from Hutton's Essay on 

 Bridges ; and he says, " It is by no means deficient in gracefulness, and is abun- 

 dantly roomy for the passage of craft ; so that no objection can be offered against 

 its being adopted on accoimt of its mechanical excellency." The reader may, 

 however, form his own opinion as to these qualities from the subjoined diagram, 

 which represents a vertical section of the arch along its road- way, constructed by 

 a scale from the table. 



Fig. 10. 



I believe enough has been done in this memoir to enable engineers properly 

 instructed in mathematics, to construct arches having the form of equilibrated 

 curves. The requisite tables now foUow. 



