the radius vector revolves in the negative direction, the same three loops will be traced out.

From this and the former example it appears that in all curves whose equations are of the form

r = a sin no,

the curve consists of n loops if n is an odd number, and of 2n loops if n is an even number.

Ex. 3. Trace the curve whose equation is

= 0,

is never greater than a; and r = 0, when 0 =

= = 2nπ.

therefore

dr

do

...

=

0, when = ′′, = = 3π, = = (2n + 1) π.

...

Hence the radius vector is zero, when = 0, and attains a maximum value a, when = 7; whence it decreases, becoming 0, when 02π, until it reaches a minimum =

a, when 0 = 3π;

47,

after which it increases, passing through zero, when ✪ = and becomes a, when = 5; wherefore the curve is that drawn in fig. 97.

Ex. 4. Trace the curve whose equation is

r2 = a2 {(tan 0)2-1}.

r = ± a {(tan ()2 — 1 } § ;

therefore r cannot be constructed whenever (tan 0)2 is less than 1. Also as r is affected with ±, the pole is the centre of the curve.

Hence there are two asymptotes perpendicular to the prime radius, at distancesa, from the pole.

The curve therefore is that delineated in fig. 98.

Ex. 5. Trace the curve whose equation is

Also r∞, when 0 = + 1; therefore we must find r2 in order to determine the asymptote.

do

dr

therefore are inclined at +1 to the prime radius, and the per

a

pendicular distances from the pole on them are 7.

2

Also, when = ∞, r = a, and therefore the circle whose radius is a is asymptotic, and is an interior asymptotic circle, bceauser is greater than a.

=

=

As the radius vector revolves in the positive direction, r = 0, when = 0, and is negative, and negatively increases, until 01, when r∞; and changes too, approaching to the rectilinear asymptote, receding on one side of it, and returning on the other; after which, as 0 increases, r decreases, until it attains its least value a, when 0 = ∞.

r=

Again, as r revolves in a negative direction, it must be measured backwards from 0 to 1, at which latter angle - ∞, and then changes its sign to +; that is, the branches of the curve have approached the rectilinear asymptote, and cut it at infinity; and as increases, r continually decreases and approaches to the asymptotic circle, of which the radius is a. See fig. 99, in which the dotted branches indicate the parts due to the negative revolution of r.

if is positive, r is positive, since the arc is greater than its sine. And since for all values of 0 in the first and second quadrants sin is positive, and for values in the third and fourth quadrants sin is negative, therefore in the first and second quadrants r is greater than a, and in the third and fourth r is less than a.

=

And when = 0, sin 0 0; and therefore when = 0, r = ∞; hence, to determine the corresponding polar subtangent, we have

When

= ∞, r = a; therefore there is an asymptotic circle,

whose radius is a. Hence we tabulate as follows:

It appears then that the curve starts from infinity, as delineated in fig. 100, and periodically, when = п, = 2π, = ..., passes through the points A and B, which are the extremities of the diameter of the circle whose centre is the pole and whose radius is a; to which circle the curve continually approaches, being outside in the first and second quadrants, and inside in the third and fourth. There is then this peculiarity, that the curve on the outside is gradually becoming nearer and nearer to the circle, and the curve on the inside is receding from the diameter as ✪ increases and approaching to coincidence with the circle.

CHAPTER XII.

ON THE CURVATURE OF PLANE CURVES.

SECTION 1.-Curves referred to rectangular coordinates.

280.] IMAGINE a tangent to be drawn at a point in a plane curve, which is such that the curve lies entirely on one side of the tangent; then the curve is said to be convex towards that side of space on which the tangent lies, and concave towards the other side; such is our definition of concavity and convexity; and on such a conception were investigated in Chapter X the analytical criteria for determining the direction of curvature. Let us moreover suppose that at the point of the curve under consideration there is no discontinuity, or indeterminateness of derived-functions; then, as the curve deviates from the tangent line, such a deviation may be greater or less, or, in other words, the curve may be more or less bent; herein then we have a new affection, viz. the amount of bending or of curvature, as it is called the nature of which we propose to examine in the present Chapter.

:

And to consider it from another point of view; an infinitesimal element of the curve commencing from a given point being straight, it is in its length coincident with the tangent line at that point; and the next element being inclined at an angle to the former one deviates from the tangent. Now let the two consecutive elements be of equal lengths, and from the extremity of the second let a perpendicular be drawn to the tangent as this perpendicular is longer or shorter, so will the deviation be greater or less, and the curve will be more or less bent.

These terms however are but relative; and accordingly it is necessary to fix on some standard with which to compare such amount of bending, and to investigate some means by which the comparison may be made.

The circle naturally suggests itself for a standard; whatever its curvature or bending is, it is the same at all points of the same circle and in different circles, as the radius changes,

: