654 



DR WALLACE ON A FUNCTIONAL EQUATION. 



shewed how it might be constructed by a parabola and hyperbola. Gregory's 

 construction is, however, complex,* and probably was never employed in de- 

 lineating a catenary. I propose here to shew how the curve may be actual- 

 ly generated from a parabola alone, and my analysis will not require the in- 

 tegral calculus, it being derived from the property investigated in art. 25, that 

 the increment of the curve at any point is always as the increment of the tangent 

 of the angle which a line touching the curve at that point makes with the hori- 

 zontal axis. 



•Fig. 7. Fig. 8. 



48. (Fig. 7.) Let ABP be a catenary, CQ its horizontal, CD its vertical axis, 

 and BC its parameter. From P, p, two points, comprehending between them an 

 infinitely small arc of the curve, draw ordinates PQ, x>q, and straight lines PK, 

 p k, touching the curve, and meeting CQ in K and k. Take a straight line VL, 

 terminated at V (Fig. 8) ; to this line draw VG, a perpendicular, and in VL take 

 VF, equal to BC, the parameter of the catenary. At the point F, make the 

 angles VFE, VF^, equal to the angles PKQ, pkq. 



By the nature of the catenary (art. 25), (see Figs. 7 and 8), 



arc P/> = BC (tan K-tan k) = FV (tan eFV-tan EFV) : 



But FV(taneFV-tanEFV) = eV-EV = Ee; 



therefore, E e, the increment of the line VE, is equal to Fp, the increment of the 

 arc BP. Now, by construction, the straight line VE, and the catenary arc BP, 

 must begin to be generated together ; therefore, they are always equal. 



In Fig. 8, draw EN perpendicular to FE, and entoF e, and produce FE to 

 meet n einm-, and, in Fig. 7, draw p Y parallel to KQ. The infinitely small right- 

 angled triangles PpY (Fig. 7), andeEm (Fig. 8), are similar, because the angle 

 ^ P Y is equal to the angle PKQ, that is, by construction, to the angle EFV, which 



* Philosophical Transactions, as quoted at art. 21. 



