QQ4: ^^ WALLACE ON A FUNCTIONAL EQUATION. 



found in the last problem ; and thence, — the amplitude of the f unction/ (—} • 



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



catenary is known. 



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



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



a a 



^ = (Nep. log 10) X Com. log ^^o+^ll^ln^l . 



Now, «' and /„ , and Nep. log 1 = 2.3025851, are given ; therefore — is 

 given, and <2?„ is also given ; therefore a is given. And since y^^—r yf^\ therefore 



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



By the second method, putting <^ to denote the angle which a straight line 

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

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

 formulae (to which logarithms are particularly applicable), 



cos 



<P = ^ = ^; ^ = 2.8025851 . Com. log ( j ^C^^^ + i^) 1 . 



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=a:„ = 50, A'E=y,=46, B'C=o'=6. 



Calculation by the Jirst formula : 



V ( j/'/ - a") = V 2080 = 45.6070170, 



V (/.'-«-)+/. ^15.267836. 

 a' 



The common logarithm of this number is 1.1837775; 



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



