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The equation to the locus of the extremity of the subtangent is evidently

22 go' =

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O being measured from a line 90° distant from the original axis as pe' is at right angles to r. If in a similar way we find the locus of the extremity of the subtangent of the curve d' = a?, and so on in succession, we shall have a series of spirals, the equations to which are



1. (n − 1)? the angle o in each case being measured from a line 90° distant from that in the preceding curve. (2) The equation to the hyperbolic spiral is


therefore the subtangent

du The locus of the extremity of the subtangent is evidently a circle, the radius of which is a: and as 0 = 0 makes r = co while the subtangent remains finite and equal to a, it appears that a line drawn parallel to the axis at a distance a is an asymptote. (3) The equation to the lituus is


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or W=



= a.

u 1 =

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then p = tan-'(-20), subtangent = 2 a0! ; and as 0 = 0 makes r = c and the subtangent = 0, it appears that the line from which is measured is an asymptote to the curve.

Also since ?0 = a* it appears that if a circle be described with radius r, the sector between the axis and the radius r is of constant area. (4) The equation to the Lemniscate is

po? = acos 2.

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Then p = tan-a, and is therefore constant ;

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p = r sin (tan-'a)

(1 + a") The subtangent = ra.

The locus of the extremity of the subtangent is the involute of the curve, the equation to it being

2. = ar = ace", and therefore a similar spiral.

Also if r; be the subnormal, that is, the portion of a perpendicular to the radius vector at the origin cut off by the normal, the locus of the extremity of r: is the evolute of the spiral, its equation being

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(6) The equation to the Cardioid is

r = a (1 – cos 6). If q' be a radius in the direction of r produced backwards,

r' = a {1 - cos ( + 7)} = a (1 + cos 0). Therefore r + p' = 2a, or the chords passing through the pole are of constant length. = tan 0; therefore

(7) Let the equation to the spiral be

qon = a" sin no.

tan o = tan nd; and p = ne. If o, be the value of Q corresponding to an angle 0 + TT; that is, to a tangent at the other extremity of the chord passing

tan o

= } 0.

through the origin, . = n (@++) and O. - = nt.

Therefore the angle between two tangents at the extremities of any chord passing through the origin is constant.

(8) Let the equation to a spiral be

0 (2ar – po)} = 1. Then when 0 = 0, (2ar – ??)= 0 and r = 0, r = 2a.

Therefore the circle, the radius of which is 2a, is an asymptote to the spiral. The pole also, for which p = 0, may also be considered as an asymptotic circle the radius of which is zero, as the curve makes an infinite number of revolutions before it reaches it. The same remark applies to the logarithmic spiral, and many other curves for which p is zero when is infinite.

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offers examples of both rectilinear and circular asymptotes.


aFor if 0 = + 1, r = 0, and as po?

the sub

dr tangent corresponding to 0 = +1 is + la, and there are therefore two rectilinear asymptotes inclined at angles + 1 and -1 to the axis.


= a


a? Also since p =

= a when = oo, the 02 - 1 circle whose radius is a is asymptotic to the spiral.

The form of the curve is given in fig. 28.

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By the Singular Points of Curves are usually meant those for which any of the differential coefficients of the one variable with respect to the other take the values o, co or We shall confine our attention to the first and second differential coefficients only; and of these the first is the more important.

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When 0, the curvé is at that point parallel to the

dx axis of v, and if the first differential coefficient which does not


vanish along with be of an even order, the ordinate is at

dx that point a maximum or a minimum. We shall not here consider any examples of such points, as the subject has been already sufliciently illustrated in Chap. vi.

dy dc When

= 0, and the abscissa is at that point dx a maximum or minimum.



dly WVhen = 0, the curve coincides at that point with a straight line; for as y = a x + b is the equation to a straight

dy line, it follows that for such a line = 0. The same result

d x2

dy may be deduced from the consideration that when 0,

dir? the radius of curvature is infinite, or the line at that point has no curvature, or is straight. If the first differential coefficient which does not vanish along with

be of an d x?

dy odd order, then is a maximum or minimum, and the

dx curve has a point of contrary flexure. Instead of finding what differential coefficient vanishes, it is generally more

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convenient to try whether change sign on substituting

d.r? in it values of w a little greater and a little less than that which makes it vanish. If it do change sign, the point is

dy one of contrary flexure, otherwise not. If


dx may

be a point of contrary flexure provided that it change sign for values of x a little greater or a little less than that which makes it infinite.


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If any values of x and y make

it is an indica

dr tion generally that the point in question is a multiple point, or that several branches of the curve pass through it. The

dy multiplicity may be of different kinds. 1st. If

is found

d.x by the usual method of evaluating vanishing fractions, to have several different possible values there are

as many branches of the curve cutting each other in one point. 2nd.

dy If is found to have two or more equal and possible

dx values, there are two or more branches of the curve touching each other in one point, which is called a point of osculation.

dy 3rd. If all the values of are found to be impossible, then

dc the point in question is an isolated or conjugate point, that is, one through which there passes no branch in the plane of the co-ordinate axes. In fact the point is that in which impossible branches of the curve meet the plane of the axes. With respect to the 2nd and 3rd class of multiple points a few more remarks are necessary.

dx equal values for a given value a of one of the variables, we find that for a value a +h the other variable is possible,

If when

has two


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