A CURVE, AND ITS APPLICATION. 661 



9. In the Epicycloid, if we take Newton's construction (Princip. B. iv. Sect. 10, Prop. 49), 

 Fig. 6, CB the radius of the globe, VB the diameter of the wheel when the describing point is at 

 P, E its center, we have, by Newton's proposition, only measuring the arc from A the vertex, 

 instead of Z the cusp, the arc at P perpendicular to the chord BP ; and 



CB : 2CE :: chord VF : arc AP. 



Let a be the radius of the globe, b the radius of the wheel : 6 the angle DCB, through which 

 the wheel has rolled upon the globe. Then (PQ being a diameter) by the mode of description, arc 



BQ = BD. Therefore angle BEQ = ^ : therefore chord VP = 2fc sin — ; and 



b 2h 



o/ IX oi • °^ > i (a + b) b . aO 



a : 2(a+b) :: 2 6 sin — :s; whence s = —^ —sin — . 



26 a 26 



But VBP = — , and DCB = 6. Hence BP makes with CA an angle = — + 9. And since the 

 2o ab 



curve at A is perpendicular to CD, and at P, to BP, it is evident that if (p be the angle through 



which the curve has deflected at P, d) = h 9 = ^ 9. 



^ 26 26 



26 

 Hence 9 = <p; and the intrinsic equation to the epicycloid is 



o + 26 



4 (a + 6) 6 



s = sm 



:<P- 



a + 2b 



This may coincide with any curve of which the equation is 



s = I sin r>i(p, where m is less than i. 



- , . a , 4 (a + 6) 6 



In this case, m = , I = ^— ; 



a + 20 a 



, 6 1 - m , 2(l+»n)6 



whence - = , I = — ^^ — . 



a 2m m 



10. In the same manner we shall find that the intrinsic equation to the hypocycloid is 



4 (o - 6) 6 . 



-sin 



.^- 



a a — 2b 



And this may coincide with any curve of which the equation is 



s = I sin m <p, where m is greater than 1 , by making 

 6 TO - 1 2 (1 + wi) 6 



n 2»» ' m 



11. It is evident from the equation s = Isinm^, that the curves rcjjrcscntcd by that equation 

 will be of such forms as are seen to result from the epicycloidal mode of description. Thus the 



equation « = ;sin- gives a curve in which « continues to increase from A, where (p = o, till 



f/> = TT, after which it decreases. Hence there will be a cusp when the curve has deflected 

 through two right angles, as at Z, Fig. 7. After this point the curve goes on in an identical 

 inverted course, till ^ = 27r, as at A', when s - 0, the negative part having destroyed the positive 

 Vol. VIII. Part V. 4Q 



