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p = r sin ,
dp = dr sin 0 + r cos o do; or, multiplying by ds and observing that sin p ds = rdo, and cos o ds
dp ds = rd0dr + rdpdr = rdr (do + dp) = rdrdy. But we know that, p denoting the radius of curvature at P, ds hence
Chord of Curvature through the Pole. 156. Let the radius vector PS, produced if necessary, meet the osculating circle at P in a point p; and let Pp = 9. Then it is plain that
9 = 2p sino
Radius of Curvature in terms of r and 0. 157.* We have
dr and therefore, considering do constant,
dr2 - rd2r
+1 röd0' + door
rodo? + dr?
* This method of investigating the expression for p is given in the Cambridge Mathematical Journal for November, 1840.
(9dO® + dra) Hence
do (2dr2 + réd0– rd’r)
dr goz + 2
d0 d02 The expression for p may be written, in a form perhaps rather less difficult to be remembered,
d0 (dr? + ds” - rd’r)
Evolutes of Polar Curves. 158. Let P' be the centre of curvature at any point P of a curve AB referred to polar coordinates. The line PP' will,
as we know, be a tangent to the locus of P'. From S draw SY' at right angles to PP and join SP'. Let SP' = "', SY' = p.
rdr Then, since
rdr we see that
(2). Between these two equations and the relation between p and q deducible from the equation to the involute, we may eliminate p and r, and thus obtain an equation between pand i which will determine the nature of the evolute. Conversely,
having given the equation to the evolute, or an equation
på = p2 - a?: then
dp and therefore, from (1),
gu? = p. Also, from (2) and the proposed equation,
p'? = go!? = a*, it follows that the evolute is a circle of which the radius is a and of which the centre is at the pole.
Asymptotes. 159. An asymptote is a tangent to a curve, at a point infinitely distant, which passes within a finite distance from the origin of coordinates. Let then
f(0, p) = 0 be the polar equation to a curve. Assume p = 00, and obtain from this equation any corresponding value of 0, if there be any such. . Then ascertain whether the corresponding value of the expression do
for the subtangent is finite, or infinite. If it be finite, dr a condition necessary for the existence of an asymptote, there will be an asymptote corresponding to the value assigned to 0. The asymptote will be constructed in the following manner. First draw the indefinite line SP inclined at the proper angle to the fixed line SX: from S draw ST at right angles to SP
d0 and equal to the value of g? for the particular value of 0:
dr through T draw a line parallel to SP; this line will be an asymptote. If there be several values of 0, when p = 0, which
: d0 make guz finite, there will be several asymptotes.
Ex. 1. To find whether the curve r = has an asymptote.
0 Assume r = 0: then 0 = 0. In this case
= a. dr
Hence there is an asymptote parallel to the line SX. Through S draw ST = a, and draw TP parallel to SX; TP will be the
required asymptote. Had the value of v been negative instead of positive, we ought to have taken ST", at right angles to SX, equal to a, and drawn a line through T', parallel to Sx, for the asymptote. The positive direction of 0 is indicated by the arrow; we have considered only the positive values.
If Ꮎ 1,0 =
1,0 ļa. Hence the curve has two asymptotes.
Through S draw Pp, P'p', inclined at angles, each of a circular measure unity, on opposite sides of SX. Draw ST = a at right angles to Pp, and ST"= } a at right angles to P'p. Through
T, T', draw KL, K'L', parallel respectively to Pp, P'p. Then
Asymptotic Circles. 160. If, for any finite value a of p, o becomes infinite, the curve will have an asymptotic circle, that is, a circle to which, as @ keeps increasing, the curve continually approaches without ever actually meeting. Ex. 1. Take the curve
02 then, when 0 = 0, r = a: when 0 has any finite value greater
A than unity, r is greater than a. Hence the circle of which y = a is the equation, is an interior asymptote.
Ex. 2. In the case of the curve
r = a is the equation to an exterior asymptotic circle.
Conditions for the Concavity and Convexity of the Curve towards
the Pole and for Points of Inflection. 161. When the curve is concave or convex towards the pole in the neighbourhood of any point, it is easy to see that p inor decreases respectively as