666 Db. whewell, on the intrinsic equation of 



we shall have the figure in which a curve such as = p sin - , runs along the circumference of a 



circle. 



And in the same manner, by adding to the value of (p, in any of the other cases previously given, 



a term - , we have the equation to the pattern curve there considered, made to run round the 



circumference of a circle. 



Thus d) = sin - h — gives such a figure as Fig. 21 ; 



b a 



TT m + cos s s , „ „. 



(p = - — -r + — such a ngure as rig. 22, 



^ 2 {I + m cos sf a ^ " 



m being about -, as in Fig. 1 6. 



27. The radius of the circle round which the pattern runs is less than a. When (p has gone 

 through all its values, so that « = 2-jra, the curve has not been laid along the circumference of the 

 circle, but has, besides, followed all the sinuosities of the pattern. 



Of the Catenary and Tractrix. 



28. The intrinsic equations simplify the properties of these curves. 



Fig. 23. Let CO be any arc of a Catenary from C the lowest point; OS, CS, tangents, OV 

 vertical, meeting CS ; therefore OSV is the triangle of the forces which support the weight of 

 CO; and if O be the tension at C, expressed in length of the curve, 



s OV 



- = =tan OSV, and if OSV = 0, 



a SV ^ 



s = a tan (p, 



the equation to the catenary. 



29. For the Tractrix, let PT be the tangent, AT the fixed line, PN, perpendicular on 

 AT = or, tan NPT = p. Then PT = ic y/i + pS = c, a constant, by hypothesis. 



Hence w = . ; 3 ;-,; also (s being now CP,) — = - V 1 + p: 



y/l+f dp {\ + pT- dx 



ds cp 



Therefore - — 



dp 1 + p* 

 But if d) be the angle of deflexion, beginning when the curve is perpendicular to AT, p = tan (p ; 



therefore = sec^ d) = 1 + p^. 



d <p 



ds , sin m 



Hence -— =cp = c tan d) = c '- . 



d<p ^ cos(p 



This is the equation to the evolute. Integrating 



1 _• 



s = cl ; or cos = e~<^, 



cos (p 



the equation to the tractrix. 



