A CURVE, AND ITS APPLICATION. 665 



Gtt m + cos s 



And the curve is = — . . [m = — . 



^ U {1+ m cos s)" \ 111 



TT TO + cos « ( 5\ 



1 he curve = - — — , [m = -\ 



^ 2 {I + m cos s)- \ 8/ 



8 TT 8 IT 



gives the angles of deflection = — - and - - ; that is, 55l° and 240°, which nearly resembles 

 the last, and may also be represented by Fig. l6. 



l6ir TO + cos s 



The curve may deflect through more than a circumference. Thus if cb 



L 



27 Q + m cos s)- ' 

 (to =-| , the greatest deflexions are - and — ; that is 60° positive, and 360° + 60° negative. 



Hence the curve at J and B, Fig. 17, is parallel, at both points making an angle of 6o with = 0. 



Such a curve has a loop; C being the place of minimum radius of curvature, the curve opens 

 both ways from C. 



Of Diminishing Running-pattern Curves. 



24. If d) = sin s^ a, where a is a quadrant, we shall have a sinuous curve; 



And if we make « = 1, \/3, \/5, •y/7, -y/S, \/ll, &c., 



we shall have a series of points of inflexion in the curve. And since these values of « have 

 for their differences 



y/3 - 1, ^/5 - y/S, y/n - -v/S, y/c, - y/i, ^\\ - ^9, &c. 



which are a decreasing series, it is evident that we shall have such a curve as Fig. 18, in which the 

 lengths of the curve between the points of inflexion, A A', A" A", A" A"', Sec constantly decrease. 



The same will be the case \i <p = p sin s^ a. 



If p be large enough, such curves will have loops, like those represented hy (p = p sin s. 



Thus if (p = — sin s^ a, we shall have a curve such as Fig. 19. (See Fig. 12, which 



27r 

 represents (p = — sin s, as to its general form). 



The lengths of the alternate loops will constantly diminish, and the whole curve will occupy a 

 triangular space, like a writing-master's flourish. 



25. We may have a similar flourish, liut unsymmetrical, by taking, instead of sin x'-'a, the 



TO + cos s'a 



expression : . 



(1 + TO cos s a)' 



This will give a figure like Fig. 20. 



Of Circularly-running Pattern Curves. 



s a 

 2G. If we take the equation (l) = p sin r + -, 



' b a 



