A CURVE, AND ITS APPLICATION. 667 



30. It appears in this investigation, that the evolute of the tractrix is the catenary, a well- 

 known property. 



General Properties of the Intrinsic Equation. 



31. Given the equation of a curve to rectangular coordinates tv, y, to find the intrinsic 

 equation. 



Let y = / (r) : hence, /' («) = ^ = — ^ being for y. 



d.v tan ^ ° ^ 



Hence -v is known in terms of tan (ft. Let x = F (tan (b), 



d X 

 Then ~~ = F (tan (b) x sec" (b. 



dtp ' ' 



A 1 d^ 



Also -— = cosec (h. 



dx ^ 



Hence -— = F' (tan <h) . sec= (b . cosec (h = ^^ JlL. 



d(p T T T sin^.cos^d) 



32. Examples. 1. The Common Parabola. 



, /a dy I 



■ = 4 o.r ; \/ _ = -— = ; hence x = a tan- (b. 



* /^ rf.r tan m ' 



f- 



X dx tan (p 



dx 2nsin0 



Hence - — =ia tan &>. se&(b = -. 



d<p ^ ^ cos'0 



, , rfs 1 ,^ rfs 2 a 



And -— = -, . Hence 



dx sin <p d(p cos'rf) 



2 a 



2 a 



Hence the equation to the evolute of the parahola is s = . 



' ' cos'0 



2 a 

 The length of the parabola may be found by integrating — - — . 



cos (p 



2. The Semicuhical Parabola. 



dy 2 a^ 1 , 8«tan'd) 



v' = a x\ — = - . — J = ; hence x = . 



dx 3 .r* tancp 27 



dx 8a ,ds l ds 8a sinrf» 



- — = — tan'd) .sec'rf); and — = ; ; whence -— - = — . r—; 



dcj) I) -r -r ^^ ^,^^ ,ifp ,, cos'0 



tlie intrinsic equation to the curve. 



8 « I 1 I 



Integrating, we have i = — { — 1 >. 



27 (cos'0 J 



3. Thi' Ellipse. 



l> ,—- ■ dy h a> ' IT *' 



y = - V «' - y, = — . - , = . Hence m = . = . 



a rf.t a y/a'-al' tan<p y/a' + b'tan' (p 



