ON THE TRIGONOMETRY OF THE PARABOLA. 



73 



Section II. 



IV. An expression for the length of a curve in terms of a perpendicular/? 

 let fall from a fixed point on a tangent to it, and making the angle d with a line 

 passing through the given point or pole, is found in most elementary works, 

 namely s=^pdd+i. In the following figure, 



jo=ST, e=VST, «=PT. 



Fig.l. 



Let n(>rt . 6) denote the length of the arc of a parabola whose parameter 

 is 4»i, measured from the vertex to a point at which the tangent to the arc 

 is inclined to the ordinate of that point to the axis by the angle 6. When 

 m=l, the symbol becomes 11(0). 



In the parabola whose equation is i/'^4;}nx, the focus S is taken as the 

 pole, and therefore p:=m sec d : while PT, or t^m sec d tan d. 



The arc of a parabola, measured from the vertex, may therefore be ex- 

 pressed by the formula 



n(»j . 6)=m sec tan 6 + m I sec 6 dd. 



The difference between the arc and its subtangent t may be called the 

 tangential difference. 



For brevity, and for a reason which will presently be shown, the distance 

 between the focus and the vertex of a parabola will be called its modulus. 

 Hence the parameter of a parabola is equal to four times its modulus. 



V. Let n(m . lo), U(m . ^), U(m . x) denote three parabolic arcs VD, VB 

 VC, measured from the vertex V of the parabola. Let, moreover, to, tj), and 

 X be conjugate amplitudes. Then 



n(»i. w)=»itan oj sec w+wifsec wrfw 



n(»j .0) =wjtan0 sec^ + mj sec^i 



>. 



(V) 



n(OT.x)=»i tan^secx+^j sec^^^x 



Whence, since Jsecwrfw =Jsec^c?^ + ("360 x^^X' because w, ^, and x are 

 conjugate amplitudes, we get, after some reductions, 



n(»j. (!>)— n(m.^) — II(»J.x)=2»^tanwtan0tanx. . . (8) 



