664 db. whewell, on the intrinsic equation of 



If X, y, be ordinary coordinates, these equations represent a curve sinuous, but each siriris 

 not symmetrical. The angles at which the curve cuts the axis are alternately those of which the 

 tangents are 



1 1 



I + m \—m 

 Hence the descending side is more inclined than the ascending. 



We shall obviously have a curve nearly resembling this, if we take the intrinsic equation 



VfL "i* COS J? 



<p = r^ ; which differs from the former by putting <p for tan (p, and s for x. 



The curve will be a sinuous curve, inclined to the original line d) = 0, at maximum angles 



<b = , d) = , on one side, and on the other, when cos s = ± l. And if a be the arc in 



' I + m ^ 1 - m 



the first quadrant for which cos a = m, (p = when s = w - a, tt + a, 3 tt - a, 3 ir + a, &c. ; and 



the curve will be as represented in Fig. 14. 



For example, if m =-, the curve deflects alternately on the positive side, so that the angle of 



3 3 



deflexion is -, and on the negative side, so that the angle is -; that is, the angles are respectively 



43" and 86" nearly. 



We have, in this case, - — = 



d(p {2m' -1 + mcoss) sins 



This is the radius of curvature, which becomes infinite when s = 0, tt, 2 tt, &c. ; that is, at J, 

 A', A", &c., when there are points of inflexion. 



m + cos s 



22. If we have <h - p . r- 



(1 + jre cos s) 



P P 



we shall still have a sinuous curve, and the greatest deflexions will be , and - 



1 + m 1 — m 



■K 1 



Thus if /) = — , and m = - , the greatest angles are 



- -, and -2. — ; that is 60" and - 180". 

 3 2 2 



Hence the curve will be of the form represented in Fig. 15, making at A an angle of ikf with 

 the line 0=0, and at B, where cos s = — I, the curve being parallel to = 0, but in the opposite 

 direction. 



The radius of curvature is infinite at A and at B, and has a minimum value at some intermediate 

 point, nearer to B. 



23. It is easy to construct running patterns curves of this kind which have any given angles 

 for their extreme deflexions. Thus let Fig. l6 represent a pattern curve which sinuates between the 

 angles 60° one way and 3 x go" = 270° the other. Then, 



■K p Sir p I + m 9 7 Sir 



3 1 + m 2 l-m 1-m 2 II '^ II 



