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the radius vector produced backwarks: if this be not attended to, the curve will want branches or spires, and will appear to be discontinuous. Some authors neglect the negative values of r, and trace the spiral only with the positive values of the radius vector ; that this is an incomplete mode of tracing the curve may easily be seen by transferring the equation from polar to rectilinear co-ordinates, when it will be found that, according to the principles of interpretation used for the latter, the tracing of the curve from its rectilinear equation will give more branches than that from the polar equation. The remark which was made regarding the interpretation of the symbols in rectilinear co-ordinates applies equally to polar : there is no necessity for interpreting all the symbols which arise in our operations, but we gain much in the generality of our formula when we do interpret them, and we should sacrifice many advantages by not doing so*. Ex. (1) Let the equation to the curve be

r = a cos 0 + b, a>b. When 0 = 0, y = a + b, a maximum. From 0 = 0 to 0 = cos (-), which is an angle in the second quadrant, » is positive and continually diminishing till when 6 = cos-- (-) it is equal to 0, and therefore the curve passes through the pole cutting the axis at an angle whose cosine is

From A = cos-1 to 0 = 1, r is negative and increasing, and being measured on the radius vector produced backwards it traces out the portion OEB (fig. 42) of the curve; and when A = 1, r = - (a - b) = OB.

* It has been usual among writers on this subject to neglect the negative values of , and so to deprive the curves of their due allowance of branches : a marked instance of this may be seen in the spiral of Archimedes, which, as usually traced, appears shorn of one half of its length. Professor De Morgan is, so far as I know, the only writer who has insisted on the interpretation of negative values of r. See his Diff. Calc. p. 342.

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From 0 = 7 to 0 = cos-- (-) in the third quadrant m is still negative and diminishing, and traces out the portion BFO of the curve. When 0 = cos-- (-9), r = 0, and the curve passes

I a again through the pole, cutting the axis at the same angle as before, but measured in the opposite direction.

From 0 = cos-(-) in the third quadrant to 6 = 27, p is positive and increasing, till it again reaches the maximum value a + b or 04, after tracing the portion OGHA of the curve. On increasing the values of 0 the same values of r recur, showing that the curve is complete; and it is obviously unnecessary to give to negative values, since these will give the same values for r as the positive values 271 – 0 have done.

When a = b the smaller oval OEBF vanishes, and the point O is a cusp; the curve then becomes the common cardioid.

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(2) Let r= a sin 30 be the equation to the curve.

r = 0 when 30 = N7; that is for 0 = 0, 0 = 0 =<, 6=, 0 = ", B

When 0 = 27 or upwards the same series of values again recur. The curve therefore passes six times through the pole, and as r never becomes infinite, it must consist of six equal loops arranged symmetrically round that point. A little consideration will show that the form of the curve is that given in fig. 49.

This curve belongs to a class represented by the general equation r = a sin mo, the properties of which have been very elaborately treated of by the Abbé Grandi, in a paper in the Philosophical Transactions for 1723, and in a book called rather quaintly Flores Geometrici. From a fanciful notion that these curves resembled the petals of roses, he gave them

the Abbes of which by the se

the name of “Rhodoneæ," and endeavoured to trace analogies between them and the flowers after which he had named them. The first paragraph of his paper in the Philosophical Transactions will give an idea of his way of treating the subject : “ Suos Geometria hortos habet in quibus, æmula (an potius magistra ?) naturæ, ludere solet, sua ipsius manu flores elegantissimos serens irrigans enutriens ; quorum contemplatione cultores suos quandoque recreat ac summa voluptate perfundit.” (3) Let the curve be

70 = a (sin 20 – sin 8) = a sin 0 (2 cos 0 – 1), no is equal to 0 when sin 0 = 0) and cos 0 = 1, or when

o 7

5 7 0 = 0, 0 = 1, 0 = "

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The values of 7 recur when 0 = 27; and as r never becomes infinite, it appears that there are four loops arranged round the pole, one pair being smaller than the other.

From 0 = 0 to 0 = }, 1 is positive.

From 0 = to 0 = 77, 7 is negative as 2 cos 0 - 1 is negative, and sin is positive.

. 5T From 0 = 7 to 0 = *, - is positive, since both factors are negative.

From 0 = * to 0 = 27, » is negative.
The form of the curve is easily seen to be that in (fig. 50).
(1) Let the equation to the curve be

go = a (tan 0 – 1) (fig. 48). When 0 = 0, y = – (1 = OB if OA be measured in the positive direction.

From 0 = 0 to 0 = 171, r is negative and decreasing, and traces out the portion BDO of the curve.

When 0 = 1, r = 0), and the curve passes through the pole, cutting the axis at an angle of 45°.

From 0 = 47 to = ! , r is positive and increasing, and traces out the portion OEL.

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When 0 = 1, r = 0. To see whether this corresponds to an asymptote we must find me

Now

Now 10 = a (1 + tan 0); therefore hd do _ a' (tan 0 – 1) a (sin - cos 0)

dra (1 + tan 8) (sin?0 + cos2 O) when 0 = 1. Therefore AL drawn perpendicular to the axis at a distance OA = a is an asymptote to the curve.

From 0 = = = , r is negative and diminishing, and it traces out the portion KHACO: the prolongation AK of AL being an asymptote to this branch.

57 When 0 = 2, r = 0, and the curve again passes through the pole, cutting the axis at an angle of 45°.

From 0 = 5 to 9 = , " is positive, and traces out the portion OFN; and when 0=17, p = 0, and it is seen as before that a line BN perpendicular to the axis is an asymptote.

From 0 = ** to 0 = 27, r is negative and diminishing, and traces out the portion MGB.

When 0 = 27 the curve joins on to the first portion, and is therefore complete. It is obviously unnecessary to consider negative values of 0 as they are included in what has already been done.

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(5) Let the equation to the curve be

sin 30
= a

cos 0.
The form of this curve is given in fig. 46.

CHAPTER XII.

ON THE CURVATURE OF CURVED LINES.

Sect. 1. Radius of Curvature.

When the curve is referred to rectangular co-ordinates, if p be the radius of curvature

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If 0, a quantity of which both x and y are functional, be taken as the independent variable,

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