DR WALLACE ON A FUNCTIONAL EQUATION. 655 



again is equal to Eev). Now, it was shewn that the lines eE, Vp, are equal; 

 therefore, e m is equal P Y or Q q, the inci-ement of CQ ; and E in to p Y, the in- 

 crement of PQ. 



It is a known property of a parabola, that the common intersection of a 

 tangent to the curve, and a perpendicular to the tangent from its focus, is in a 

 straight line touching the parabola at its vertex.* Hence it follows, that if a 

 parabola be described about F as a focus, with its vertex at V, so that YE touches 

 the curve at its vertex, the lines EN, E n, will touch that parabola at points N, n. 



Suppose, now, that the parabolic curve m N V is the edge of a mould of some 

 solid material, such as in practice is used for tracing the curve, and that a thread 

 is applied along that curved edge, beginning at its vertex V, and extending in- 

 definitely to some point in the cmwe, where it is fixed to the mould ; if the thread 

 be gradually unlapped from the curve, the extremity of the thread that leaves the 

 vertex V will generate a curve VHA, which will be the involute of the parabola. 

 The lines EH, e h, will be normals to this curve, and e m, the increment of the 

 normal ; but e m has been proved to be equal to Q, q, the increment of CQ, the 

 abscissa of the catenary ; therefore, that abscissa, and the normal EH, which be- 

 gin to be generated together, will always be equal. It has been shewn also that 

 E m, which is the increment of the line FE, is equal to p V, the increment of the 

 ordinate PQ of the catenary ; therefore, on the whole, we have this proposition. 



THEOREM (Figs. 7 and 8).— Let VN be a parabola (Fig. 8), of which F is the 

 focus, FL the axis, V the vertex, and VG a perpendicular to the axis at V. 

 Suppose a thread to be applied along the curve, with one end at V, and 

 the other fastened to the curve at some point indefinitely remote. Let 

 this thread be unwound from the curve, and kept tight, so that its extre- 

 mity V may describe a curve line VHI : this wiU be the involute of the 

 parabola. 



Take any point E in the line VG ; draw EF to the focus, and EH perpendicular 

 to EF ; meeting the involute in H. Assume C a given point, as an origin 

 in a straight line CD given in position (Fig. 7) ; in that line take CR equal 

 to FE, draw RP perpendicular to CR, and equal to EH : The point P will 

 be in a catenary, whose parameter CB is equal to FY in the parabola ; 

 and the arc BP of the curve, between the axis CB and P, is equal to the 

 straight line YE. 



49. This construction gives a perfectly distinct notion of the catenary : be- 

 sides, for a practical purpose, it is easy, requiring merely the correct construction 

 of a mould for making a parabolic curve. 



* See my Treatise on Conic Sections, Part i. proposition 14. 



