VI. Second Memoir on the Intrinsic Equation of a Curve and its Application. 

 By W. Whewell, D.D., Master of Trinity College, Cambridge. 



[Read April 15, 1850.] 



38. The method of deducing properties of curves which I employed in a former memoir 

 on the Intrinsic Equation of Curves* is capable of many other applications. I the more readily 

 proceed to point out some of these, in consequence of remarks which have been made to me 

 upon the former memoir. 



Scruples have been entertained by some of my readers as to whether we rightly suppose 

 the portion of curve, added after reaching a cusp, to be negative. It has been said that every 

 cusp may be conceived to be the remnant left by a loop when the breadth of the loop vanishes ; 

 and as in a looped curve the increment of the length of the arc could nowhere become negative, 

 it ought not to do so in the ultimate form of the looped curve. 



39- In order to see whether this remark is generally true, let the intrinsic equation to a 



curve be s = ad> + b sin 0. Therefore — = a + b cos <b ; and if b < a, it is evident that the 



T T d(p T 



radius of curvature will vary between the limits a + b and a — b, but will never become or 



negative ; and the curve will a looped curve. 



But let us begin by considering a < b. When a = 0, we have a = b sin (p, the equation to 



ds 

 a cycloid. And since — = a + b cos (p, it is evident that the curve is that which is produced 



by taking the involute of a cycloid, adding a to the describing radius. In fig. 1, if CD be a 



semi-cycloid, of which D is the vertex, and b the length, and if the string OQ unwrapping 



from OD, produce the equal semi-cycloid DB, then if OQ be produced to P, so that QP = a, 



ds 

 P will describe a curve of which the equation is — — = a + b cos <p, the curve proposed. 



It is evident that we have a cusp when coscj)= ——, as at Z; and at Z'. The curve 

 consists of sinuses alternately larger and smaller, as in the figure. 



40. The base EC of the semicycloid is = — . If this be = a, it is evident that the 



7T 



• Namely, an equation between s, the length of the curve, and $, the angle of deviation from the original direction. 



