662 Dr. WHEWELL, ON THE INTRINSIC EQUATION OF 



part. The negative value of s goes on increasing till <p = Sir, at Z', when there is another cusp 

 Afterwards the arc becomes positive, and the curve returns to A, having deflected through iir*. 

 The curve is an epicycloid in which h = \a. 



12. Ao-ain, ii s = I sin 2^, s increases from A, where <p = 0, till !- = -n , when it is a maxi- 

 mum, and there is a cusp, Z, Fig. 8. After this the arc (from Z) is negative till cp = —, when 

 there is a second cusp, Z'. Then the arc is positive, till (p = — (at Z"). Then it is negative 



4 



till d) = - — (at Z"'). When (f> = 2Tr, the curve returns to A. 



4, 

 The curve is a hypocycloid in which b = ^a. 



d) d) , ■ ^ 



13. If s = Z sin 3- , s = Z sin i , s = / sin i , &c. 



3 4 5 



b . . , 



we have epicycloids in which - is respectively 



1, |, 2, &c. 

 The radius of the wheel in these latter cases is greater than that of the globe, and the curve is 

 deflected through more than a whole circumference before it comes to a cusp. Thus in the case 



g = / sin * , the curve deflects through 2 7r + ^ to come to a cusp. See Fig. 9. 



5 2 



14. In the same way, if we have 



s=lsin3(p, s = lsm4!(p, s = lsin5(p, &c. 



we have a series of hypocycloids, in which - is respectively 



1 3 4 



-, -, -, &c. 



3 s' 9' 



As m becomes larger and larger, - approaches more and more nearly to ^, but never attains 

 that magnitude. As is well known, for that supposition, the hypocycloid is a straight line. 



15. It is evident that the ordinary properties of epicycloids and hypocycloids, as to their 

 lengths, radii of curvature, involutes, evolutes, &c., all follow with great facility from the use of our 

 equation. Thus the length of the epicycloid from the vertex A to the cusp Z is had by making 



the ano-le - d) = -, which gives the length of that half of the curve = , and the 



" a + 2b^ 2 "• 



whole length — ^^ — • from cusp to cusp, the known values. 



" a 



Also the radius of curvature of the epicycloid 



ds i (a + b) b a ^ , , 1 



— = — ^ — cos d), the known value. 



d(p a + 2h a +2b^ 



• That there will be a cusp when s is a maximum, appears also by considering that in that case g^ = 0, tliat in, the radius of cur- 

 vature vanishes. 



