DR WALLACE ON A FUNCTIONAL EQUATION. 



663 



AT'B'IF, be a section of the arch and the materials of which it is composed. 

 Assuming the roadway EF for the horizontal axis of 

 the curve, and CB'D', a perpendicular to EF through 

 B, the crown of the arch, as the vertical axis; 

 let CQ = .27, and P'Q = / be co-ordinates at any 

 point P' of the curve : let a' denote CB' the thick- 

 ness at the crown, which is the parameter of the 

 curve ; and let a, a constant line, be its modulus 

 (see art. 31). 



The equation of the curve is 



/ = |^jg^+e-^j (Art. 28). 



y 



Let ABH be a catenary whose parameter BC =« 

 is equal to the modulus of the equilibrated arch 

 A'B'H'; let the two curves have the same horizontal 

 axis EF, and their vertical axes CD', CD, in the 

 same straight line ; and let x = CQ, and y = PQ, 

 be co-ordinates of the catenary at any point P. Its 

 equation was found to be 



y^ 2"i ^""^^ / ■ 

 From these equations, it appears (as has been shewn, art. 37), that 



y : ^ = a' : a . 



By this property, the ordinates of the curve of equilibration may be found 

 from those of the catenary ; and for these, there are given in this memoir tables 

 sufficiently extensive, and more than sufficiently accurate, for all practical pur- 

 poses. 



Before we can employ the tables, however, the numeral value of a, the pa- 

 rameter of the catenary must be known. Produce A'E, H'F, the ordinates of the 

 equilibrated arch, until they meet the catenary in A and H. 



Put CE (half the span of the arch) = «„ ; A'E (the height of the roadway 

 above the base of the arch) = y\ ; AE (the distance of the extremity of the cate- 

 nary from the roadway) = y^. Because a' •.y'^ = a\y^, and that a' and ?/„ are 

 given, the ratio of a to y^ is given : hence, if a be found, y^ , the ordinate of the 



catenary, will be known. Again, because ^=— =/(-^), the tabular value 

 of the ordinate in a catenary whose modulus is unity, the amplitude being 



X if 



-2- ; therefore, the value of ^ may be found nearly by the table, just as x was 



