gg4 DR WALLACE ON A FUNCTIONAL EQUATION. 



found in the last problem ; and thence, — the amplitude of the function/(^ j ■ 



Now X is known, therefore a becomes known; and « = —4^, the ordinate of the 



catenary is knoMTi. 



Besides this way of finding y^ by the table, there are two direct methods given 



y v' 

 in art. 42. From the first of these, considering that — ~-^ , we have 



'^ = (Nep. log 10) X Com. log ■^""'"^ ^ """ ^ . 



Now, a' and /„, and Nep. log 10 = 2.3025851, are given; therefore ^ is 

 given, and x^ is also given ; therefore a is given. And since y„= -r j/'o ' therefore 



i/^, either ordinate of the catenary, at the end of the roadway, is given. 



By the second method, putting (p to denote the angle which a straight line 

 touching the subsidiary catenary at the top of either of its extreme ordinates p^ , 

 makes with the horizontal axis, we find that angle, and thence a and y^ by these 

 formulae (to which logarithms are particularly applicable), 



cosrf> = ^ = ^; f^ = 2.3025851. Com. log (l'^'L(i^!±M I; 



X a , 



In this way, by either method, we determine the catenary whose parameter 

 is the modulus of the equilibrated arch ; and then, the ordinates of the latter by 

 those of the former. 



Example. — Find co-ordinates of an equilibrated arch A'B'H' (Fig. 9), having 

 given its span A'H' = 100 feet ; its height B' D' = 40 feet ; the thickness at the 

 crown B' C = 6 feet ; and therefore A'E the height of the roadway above the base 

 of the arch = 46 feet.* 



In this case, 



CE=^„=50, A'E=y,=46, B'C=a' = 6. 



Calculation hy the first formvla : 



J yt - «") = V 2080 = 45.6070170, 



V(y/-^")+.y'„ =15.267836. 

 d 



The common logarithm of this nvunber is 1.1837775; 



* These are nearly the dimensions of the middle arch of Blackfriars' Bridge, London. 



