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fig. 89 to be the normal form of such curves; which figure the student is recommended to examine carefully, for the values of the lines in connexion with it will be deduced from the geometry of it.

Lets be the pole, sx the prime radius, APQ the curve, PSX = 0, SP = r. Let xSP be increased by a small angle QSP do, then sq = f(0+ d0) = r+dr. From centre s, with radius SP = R, describe the small arc PR, subtending do.

=

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Let PQ, the element of the arc of the curve, be represented by ds;

... PQ2 = PR2+RQ2,

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Through the two points, P, Q, on the curve, let the straight line QPT be drawn; then, as the two points are infinitesimally near to each other, the line is a tangent, in accordance with the definition of a tangent given in the last Chapter; through P draw the normal PG, and through s draw TSG perpendicular to the radius vector SP, and sy perpendicular to the tangent PT. The lengths PT and PG are respectively called the Polar Tangent and the Polar Normal; so is called the Polar Subnormal; ST the Polar Subtangent; and sy, the perpendicular from the pole on the tangent, is symbolized by p. The value of these lines we proceed to determine.

Since tan PQR =

PR
RQ

, we have, from (10) and (11),

rdo tan PQR = dr

(13)

And since SPT = SQT + PSQ = PQR +d0; therefore, SPT and PQR being in general finite angles, and do being an infinitesimal angle, we must neglect de in the above equation, and thus SPT = PQR; and SPT is the angle contained between the curve and the radius vector;

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by reason of Preliminary Theorem I, and equation (12) above. Hence also the following values result:

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Similarly may the values of other lines be determined.

(18)

270.] The value of p may be put under another form which is often convenient. Let u be the reciprocal of the radius vec

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The value of p in (18) might also have been deduced as follows, from the expression for p in equation (44), Art. 219, viz. :

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271.] It is frequently necessary to express the geometrical quantities of Article 269 in terms of p and r.

By similar triangles PQR, SPY,

PRICE, VOL. I.

3 H

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272.] Examples illustrative of the preceding theory.

Ex. 1. The Spiral of Archimedes.

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Ex. 2. The circle, the pole being at the end of a diameter.

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= r cos 0 = x = the rectangular abscissa;

r do

Ex. 3. The logarithmic spiral; see fig. 86.

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which is a constant; and therefore the curve cuts all its radii

vectores at a constant angle; and accordingly it is called the Equiangular Spiral.

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and this is the equation to the equiangular spiral in terms of p and r, and wherein m is the sine of the constant angle contained between the radius vector and the curve.

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Ex. 5. To find the relation between p and r in the conics,

the focus being the pole.

The general equation in terms of r and is

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wherein 2a is the distance from the focus to the directrix.

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and the equation represents an ellipse, a parabola, or a hyperbola, according as e is less than, equal to, or greater than, unity. Hence the equation to the parabola is

p2 = ar.

SECTION 3.-Asymptotes to polar curves.

273.] Curves referred to polar coordinates admit of rectilinear and curvilinear asymptotes, in the same manner as those referred to rectangular coordinates. As curvilinear asymptotes however are of little use in determining the course of a curve, we shall say nothing of them in general, and shall describe only one remarkable species, viz. the asymptotic circle.

As a rectilinear asymptote is a tangent to a curve at an infinite distance, the formulæ of Art. 269 enable us to determine it.

If for any finite value of 0, say = a, r is infinite, then either the radius vector itself, or a line parallel to it, is an asymptote to the curve; and the polar subtangent, which is equal to d Ꮎ becomes in this case the perpendicular distance from the dr' pole on the tangent; thus, if the value of

2.2

to

r2 do

dr

r2 do
dr

corresponding

a and r∞, is finite, the line can be constructed; if

= 0, the radius vector itself is the asymptote; and if it is

equal to ∞, the asymptote, being at an infinite distance from the pole, cannot be constructed. An inspection of fig. 90 will render this plain; in which SP is the infinite radius vector, TL the asymptote, sT the value of when 0 = a, and r = ∞ .

r2 do
dr

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If has many values for which r is infinite, there may be many rectilinear asymptotes. Hence, to determine them, we must find what finite values of render r = ∞. If the polar subtangents, corresponding to such infinite values of r and finite values of 0, are finite, the curve has rectilinear asymptotes which may be constructed in the way explained above.

It is to be borne in mind that when

asymptote lies below the radius vector,

r2 do

dr

is positive, the

as in fig. 90; and if it is negative, the asymptote lies above it, as in fig. 91.

do

Or, in other words, according as 2 is positive or negative,

dr

so is the perpendicular on the asymptote to be drawn in consequentia or in antecedentia.

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