460 EEPORT— 1878. 



Hence, if v =/(<p) be the tan ^ential equation of a curve, the intrinsic equation 

 of its involute is 



-W!)"*}/^*-. 



or, since s is the length of the involute, 



ds 



y-r is the length of the given curve. 



Hence, if v=f(<f>) he the tangential equation of a curve, the length of the 

 curve is 



iif+I$- 1 }M>'***> (14) 



SECTION IP. 

 PEDALS. 



9. If we make the axis of y the initial line, it is evident that the polar equation 

 <he first positive pedal of the curve 



is p =/(</>) sin cp. (15) 



Hence the tangential equation of any curve is transformed into the polar equa- 

 tion of its first positive pedal by changing v into p and multiplying the function on 

 the right-hand side hy sin <p. 



Thus the tangential equation of the parabola is v = a tan cp. 



Hence its first positive pedal is 



p = a sin 2 <fi sec cp. 



10. The tangential equation of the evolute of v =f(<j>) is 



v=f(cp) cot cf>+f(cp)\ 

 Hence the polar equation of the first pedal of the evolute is 



p-^(A*)"in*). (1G) 



This also appears from equation (11). 



And the polar equation of the first pedal of the w th evolute is 



P»=(4)"- A * )8in * i (17) 



and by supposing n negative we have the first positive pedal of the n lh in- 

 volute. 



11. By reversing the reasoning in art. 9 we have tbe following theorem: — 



If p = F(cp) be the polar equation of a curve, the tangential equation of its first 

 negative pedal is 



F(fl) 

 sin cp' ( 18 ) 



Hence the tangential equation of the first negative pedal of a conic section 

 from any point in its plane is 



(a cos 2 <p + 2b sin cp cos cp + b sin *<p)v 2 sin 2 <p 

 + 2{g cos cp +f sin <p)v sin cp + c = o. 



12. The equation of the line whose envelope is the negative pedal is 



X sin <p + y cos cp — F(<£) = o. (19) 



Hence the negative pedal is the result of eliminating cp from the equations 



x = (d>) sin cp + F'(<p) cos cp, 

 y = ¥(cp) cos <p-F'(<£) sin cp. 



