A CURVE, AND ITS APPLICATION. 663 



16. Again, for the evolute of the epicycloid, let, in Fig. fi, ZO = s be the arc of the evolute. 

 Therefore 



, i(a +b)b a 



s = ~ cos (h. 



ra + 26 a + 2b^ 



x> .. .. rr u '„ " , 1" , ■Jra + 26 



Hut at Z, where « = 0, d) = _ (6 = - . 



a + 9.b^ 2 '^ 2 a 



And the deflexion of the evolute beginning from Z and going to is the excess of this value 

 of (p above the value at P, because at every point the evolute is perpendicular to the curve. 



Therefore if d)' be the deflection of s, (h'=~ . (h ; and — - — d) = - ? (h' 



2 n '^ a + 2b^ 2 a + 2b'^' 



, i {a + b) b a , 



1 herefore s = — sin (h . This is an epicycloid similar to the first ; for the 



ra + 26 a + 2b ' 



., , 4(o'+6')6' . a' ,, .~b' b , o' a 



equation agrees with s = ; sin —. r (h, if — = - , and - = . 



^ ^ a a'+2b'^ a a a a + 2b 



Of Ru7ining-2i(ittern Curves. 



17. By Running-pattern Curves I mean curves in which a certain form of curve is repeated 

 over and over again in the progress of the whole curve. For example, let d) = sin « ; as s increases 



from to infinity, it becomes successively 0, — , tt, — , 2 tt, , &c., and the corresponding values 



of (p are 0, 1, 0, — ], 0, 1, &c. : and the curve is evidently a sinuous curve, as represented in Fig. 

 10, in which the same form is constantly repeated every time that s goes through the value 2 tt. 



The greatest angle which the curve makes with the original direction is 1 and - 1 ; that is, 

 the angle of which the arc = 1, to the one side and to the other. 



18. If ^ = rre . sin s, we shall in like manner have a sinuous curve in which the greatest 

 angles of deflexion to one side and the other are = m. 



If (p = — sin s, these deflexions become right angles, and the curve is as represented in Fig. 1 1. 



19. 11 (p = TT sin s, the curve from « = to s = — is of the form CA, Fig. 12 ; J being 

 behind C. For in this case, — — = . Hence the radius of curvature is - at C, where .« = 0, 



d(p TT cos .<( TT 



and increases to A, wliere it is infinite. The evolute is of the form BD, and has for its asymp- 

 tote the line AJE, perpendicular to the original direction. And hence the general form of the curve 

 CA is manifest. At A there will be a point of inflexion ; and after A the curve will be repeated 

 in inverse position, as AC', and then continued reversely from C' to A', and so on, as in tlie Figure. 



20. If ^ = 27r sin 4-, it will l)e seen, in like manner, that the curve from » = to ,v =- 

 is of the form CA, Fig. 1.3, and by the repetition of this, we have the curve as represented. 



21. The pattern in the above curves is symmetrical with regard to a line transverse to the 

 line d) = 0. But we may have patterns which arc not thus syiiinietrical. 



sin a? , rfw m+coso! 



Let y = , whence ~— = ; r, . (m< 1). 



1 + m cos a? d,v ( 1 + ?n cos toy 



id 2 



