(356 DR WALLACE ON A FUNCTIONAL EQUATION. 



Theoretically, a single point in the catenary is all that is required to deter- 

 mine any number of pairs of co-ordinates : For, let x and f(x) be co-ordinates at 

 a given point of the curve, then 



will be another pair, which may he found from the former by a geometrical con- 

 struction ; and any number /(2.r), /(3,i') &c. from //=/(*') by the formulae of 

 Art. 27. Also, having given three of these four ordinates/(,y^), f{,x^, f{x^ + x^. 

 f{x^-x^, the fourth is obtained by the relation 



2/(^ J/W = « {/(*, + ^,) +/ (^0 - ^,) } • 



50. Returning to the parabola and catenary (Figs. 7 and 8) ; since the tri- 

 angle FE t^ = i FE . e m is the increment of the triangle F VE ; and the space 

 PQ q Y = PQ . Q '7 is the increment of the curvilineal space BPQC ; and, since 

 FE = PQ, and em= Q,q, therefore the triangle FEV is half the space BPQC, and 

 that space is ecjual to CB x arc PB. 



And because the triangles EVE, PQK are similar, EV : VF = PQ : QK. Now 

 EV = arc PB, and VF = BC ; therefore, in the catenary, the subtangent QK is a 

 fourth proportional to the arc PB, the parameter BC, and the ordinate PQ. 



51. At the points P, p, which are infinitely near, draw PO, po perpendiculars 

 to the tangents PK, j} k ; these will meet at 0, the centre of the drde of curva- 

 ture at P : and the angle contained by the normals OP, p will be equal to that 

 contained by the tangents KP, kp at their intersection ; but that angle is equal 

 to the angle EF e in the parabola, which again is equal to the angle made by the 

 lines EN, e n, tangents to the parabola at T, their intersection ; therefore, the iso- 

 sceles triangles PO p, ET m are similar, and 



E )n : Pp = ET : PO, that is, since Pp = Ee, Em : Ee= ET : PO ; 



Join FN, and because the triangle E e m is similar to EFV, which again is similar 

 to FEN {Conic sections), so that 



Ewi : Ee = EV : EF = EF : NF ; 



Therefore (since ultimately ET = EN), EN: PO=EN:NF: 



Hence PO, the radius of curvature of the catenary at P, is equal to the line NF 



in the parabola : Now F N= -p-y- = p„ ; hence it appears that the radius of cur- 

 vature at any point in a catenary is a third proportional to the parameter, and 

 an ordinate to the horizontal axis at that point. 



51. From the four preceding articles, we derive the following proposition : 



Theorem (Figs. 7 and 8). — Let VN be a parabola, of which V is the vertex, and 



