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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.

Let s be the pole, sx the prime radius, Apq the curve, Psx = 0, 8P = r. Let Xsp be increased by a small angle Qsp = dd, then Sq = f(0 + dd) = r + dr. From centre s, with radius Sp = R, describe the small arc Pr, subtending dd.

.-. Pr = rdd, (10) Rq = dr. (11) Let Pq, the element of the arc of the curve, be represented

hJd*> ... PQ* - PR3 + RQ*,

= drMrW. (12)

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 = —, we have, from (10) and (11).


tan Pqr = —j—. (13) ar

And since Spt = Sqt + Psq = Pqr 4- dd; therefore, Spt and Pqr being in general finite angles, and dd being an infinitesimal angle, we must neglect dO in the above equation, and thus Spt = Pqr; and Spt is the angle contained between the curve and the radius vector;


.-. tan Spt =; {14)

sin Spt cos Spt 1 rdO = dr = Ts' (15) by reason of Preliminary Theorem I, and equation (12) above. Hence also the following values result:

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FY = SP COS SPY = -j— = (r2 —»2)*.


Similarly may the values of other lines be determined.

270.] The value of p may be put under another form which is often convenient. Let u be the reciprocal of the radius vector, so that u = -; then r


du = r;


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,. r* dO r

Also p = —j— = -;

* {(log^+l}*

which may be written in the form,

p = tnr; (25)

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.

Ex. 4. The lituus.

_ a _ adO _ rdd

r~ -g-i' •'• - --20'

dr —rdd + ds

1 20 (1 + 408)*

r*d0 2a*r

.-. p = —j— = . (26)

ds (r* + 4a*)*

Ex. 5. To find the relation between p and r in the conies, the focus being the pole.

The general equation in terms of r and 0 is


r = =;

l + e cos 0

wherein 2 a is the distance from the focus to the directrix. Taking formula (19),

l + e cos 6 du — sin 6

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'' p2 ae 4a2e*

_ J l-e» ,

aer 4a2e2'

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

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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 formulae of Art. 269 enable us to determine it.

If for «ay finite value of 6, say 6 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


r* j-, becomes in this case the perpendicular distance from the

. r^dO pole on the tangent; thus, if the value of ^ , corresponding

to 0 = a and r = oo, is finite, the line can be constructed; if r*d6 •.

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

equal to oo, 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 6 = a, and r = ac .

If 6 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 0 render r = oo. If the polar subtangents, corresponding to such infinite values of r and finite values of d, are finite, the curve has rectilinear asymptotes which may be constructed in the way explained above.

r'2 dd

It is to be borne in mind that when is positive, the

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


Or, in other words, according as r2 -j- is positive or negative,


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

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