668 Dh. whewell, on the intrinsic equation of 



dx a?b'' tan (p . sec^ (f) a^ b'- sin <p 



d^~ ~ {a' + hHan' <p)i ~ ~ (a'cos^tp+b" sia' (p)i' 



. :,ds 1 „ ds a'b' , . . . 



And— = - -: — -. Hence ■— - = -— — - — . „ ,^. , the intrinsic equation to the ellipse. 



da! iin (p d<p {a'' cos'' <p + b' sm'' (p)i ^ '^ 



TT , :,. ^ • «'&' 



Hence the radius of curvature is 



(o^ cos^ (p + b' sin* 1^)* ' 



When (f) = 0, this radius is — ; (^ beginning at the extremity of the major axis, which was the 



supposition made. 



The intrinsic equation to the evolute of the ellipse is 



a'b' b' 



{d'cos'cp +6^ sin' 0)8 a 

 if s' begin from a cusp, where d) = 0. 



33. Given the intrinsic equation, to find the equation to rectangular coordinates. 

 Let the coordinates .v, y, be in the positions = 0, (p = — . 



Then it is evident that x = fds . cos (p,y = fds . &in (p: 



and the equation being given, these coordinates are found by integration. 



Thus in the cycloid, s = a sin (p. Hence ds = a cos (p . d(p ; 



w = fa cos'* (p . d(p, y = fa sin (p cos (pd(p. Hence, integrating, 



a? = - sm cos (p + -<p, y = - sin^ (p : the equation to the cycloid. 



34. In the running-pattern curves (Art. 17, &c.) of which the equation is 



d(p ds I 



m = m sin s, we have -—■ = m cos s = -v/(m* - (jy\ : = / . 



^ ds ^ ^ ^ ' d<p s/{m^ - ^') 



Hence .v = f '°' t>d<p ^ f sin 0rf0 



If these could be integrated, we could find the dimensions of the loops in Figures 11, 12, 13. 



There is one case for which f ., ^ ^„, taken from d) = to d) = »i is = 0. In this case the 



■'\/(m'-(p^) ^ ^ 



curve neither runs forward as in Fig. U, nor backward as in Fig. 12, but is simply two loops. Fig. 24. 



35. The following proposition, enunciated by John Bernoulli and proved by Euler, may 

 easily be proved by means of the intrinsic equation. 



If JB be any curve, AB' its involute beginning from A, B A the involute of AS beginning 

 from C, A B" the involute of A' B' beginning from A' ; and so on alternately and indefinitely : 

 the successive involutes approach indefinitely to the form of the common cycloid, provided the tan- 

 gents at A and B in the original curve are perpendicular to each other. 



(The proof of this is here omitted, being included in the proof of the extended propositions 

 given in the Additional Note.) 



