DR WALLACE ON A FUNCTIONAL EQUATION. g47 



If a mtenarp ABU, and curve of equilibration A'B'H', have a common horizon- 

 tal axis EF, atid their vertical axes CD, CD', in the same straight line; and if an 

 ordinate PQ, P'Q, to each cu7've-, pass through the sams point in the horizontal axis, 

 then, 



1. (1) Straight lines draivn touching the curves at the tops of the ordinates, 

 shall intersect each other in the horizontal axis. 



2, (2) The tangents of the angles which the touching lines make with the hori- 

 zontal axis, shall have to each other the ratio of the parameters of the curves. 



38. These are entirely analogous to knovra properties of a circle and ellipse, 

 also of an equilateral hyperbola, and any other hyperbola ; and as an ellipse may 

 be constructed from a circle, and any hy^jerbola from an equilateral hyperbola, 

 when the ratio of the axes is given, so, in like manner, a curve of equilibration, 

 whose parameter and modulus are known, may be constructed from a catenary. 

 For, from what has been shewn, it is manifest that, supposing an equilibrated 

 curve and a catenary to have the same horizontal axis and their vertical axes on 

 a straight line, any ordinates of the two curves, which have the same amplitude, 

 will have to each other the constant ratio of the parameter of the curves. 



39. Deferring, then, for the present, the farther consideration of curves of 

 equilibration, having two parameters ; let ABH be a catenary (Fig. 5 or Fig. 6), 

 of which EF is the horizontal axis, CBD the vertical axis, CB the parameter, PQ 

 any ordinate corresponding to the amplitude CQ ; let PK touch the curve, and 

 meet the horizontal axis in K. 



Let the parameter BC — a, 

 the amplitude CQ = x, 

 the ordinate PQ = ^ =/ {x) , 

 the arc VB = z = Y{x), 



the space BCQP = s, 

 the angle PKQ = <^. 



In addition to the properties of the curve stated in art. 35, it has these ; 

 x^ and x^ being any two amplitudes. 



«/(*o + *-) =/(^=) / W + F {X,) ¥ (x;) ; (1) 



«/(^o - X,) =/(^J / W - F (^J F (;.,) ; (2) 



aF(xo+ x) = F {x^} fix) +f{x,) F {x) ; (3) 



aF(x^- x) = F (x^) f(x) -/(.r J F (z,) ; (4) 



area of space s = aF (x) ; (5) 



y =-f(x) = asec<p ; z = F (x) =a tan (p . (6) 



We may enunciate the formulae of art. 34, which give the values oif{n x) 

 and F (w x), as properties of the catenary, simply by assuming that c, the modulus 



