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liberty to make n new conditions. Let these be that the coefficients of the differentials, viz. da\, rfa*, da„, be equal

to zero; whence we have


> (*>)

and between the equations (45), (46), and (49), which are 2n

in number, we may eliminate the (2» —1) quantities, oi, a^,

a„, Xi, X2, A„_!, and ultimately arrive at an equation between x and y only, which will be the required envelope to the family of curves.

In the particular case in which the general equation contains only two variable parameters, and one equation is given connecting them, the result assumes a form identical with that considered in Art. 168; and therefore, as is therein shewn, the ratio of the coefficients of the above differentials in the two differential equations is constant.

317.] Ex. 1. To find the envelope of a series of straight lines, such that the perpendiculars from the origin on them are equal to a given straight line c.

Let a be the angle which the perpendicular on any one of

the lines makes with the axis of x; then the equation to the

line is • •

x cos a 4- y sin a = c; (ou)

whence, differentiating with respect to a,

ycosa —xsina = 0; (51)

whence, squaring and adding (50) and (51), and bearing in mind the condition (cos af -f (sin a)2 = 1, we have

x2 + y2 = ca;

the equation to a circle, which is manifestly correct.

Ex. 2. A straight line of given length slides down between two rectangular axes; to find the envelope of the line in all positions.

Let c be the length of the line; a and b the intercepts of the axes of x and y by the line: then the equation to the line is

f + (52)

wherein a and b are connected by the equation

a2 + b2 = c2. (53)

Differentiating therefore (52) and (53) by making a and A to vary, we have

ada + bdb = 0; and therefore, by reason of the remark at the end of the last

Article, „.,»..

x y x y

a b a2 b% c2'

.-. a = x$c%, b = y*c*;

.-. a2 + b% = (xi + yi) c*;

.•. + y% = c*.

Which curve is drawn in fig. 45, and of which therefore the length of the tangent intercepted between the two rectangular axes is constant.

Ex. 3. To find the envelope of a series of concentric and coaxal ellipses, of which the area is constant.

g + f! = l, ab = c2;

... 4rfa+grf6 = 0, *? + ^ = 0;
a3 b3 a b

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••• *y = Y'

the equation to a hyperbola referred to its asymptotes as axes.

Ex. 4. To find the envelope of a system of straight lines whose equation is

5 + (54) a and b being related by the equation

J + i-L .55,

a? y # y

I m I m
.-. a = (/a?)*, i = (»iy)i;

••• (!)*+

The equation to a parabola, referred to two tangents as coordinate axes, the intercepts of which by the curve are I and m.

The geometry of the problem is represented in fig. 112: Oa = a, Ob = b, Ol = /, Om = m;- therefore Ml is the fixed line whose equation is (55), and of which a and b are current coordinates; and Ab is the varying line whose equation is (54). The formation of the curve is manifest from the figure.

Ex. 5. The centres of a series of equal circles are on a given straight line; it is required to find the envelope. Let the equation to the given straight line be x cos a + y sin a = p;

and x, y being the coordinates to the centre of, and f, ?j the current coordinates to, the circles of radius c; their equation is

(t-x)2 + (v-yT = c2; whence, differentiating, we have

cos a dx + sin a dy = 0, (£—x)dx + (t]—y)dy = 0;

f—x _ rj—y _ + c _ f cosa + Tjsina—(arcosa-f-y sin a) '' cos a sin a (cos a)2 + (sin a)2

= f cos a + r} sin a — p;
.•. £ cos a + ?j sin a = p+c;

the equation to two straight lines parallel to the given line, and at distances + c from it.

Ex. 6. From a given point on the circumference of a circle chords are drawn, and on these, as diameters, circles are described; it is required to find their envelope.

In fig. 124 let s be the given point in the circumference of the circle: from which let the chord Sq be drawn; and on so., as a diameter, let the circle s Pq be described; it is required to find the envelope of all circles described similarly to Spq.

Let Sp = r, Psa = 0, a\ = 2a, Qsa = ff;

.-. Sq = 2a cos ff;

and since Sp = Sqcospsq,

r = 2acos0'cos(0-0'); (57)

differentiating which with respect to 0', since Sp remains the same when ff varies, we have

0 = 2a{cos0'sin(0-0')-sin0'cos(0-0')}; .-. sin(0-20') = 0, and & =z%\ whereby (57) becomes,

r = 2a (cos ,

= a (1 +cos 0); which is the equation to the cardioid.

Ex. 7. Again, suppose that on the radii vectores of the cardioid, as diameters, circles are described as in the last example; and again, on the radii vectores of the envelope, as diameters, circles are described, and so on continually; it is required to prove that the envelope ultimately is a circle whose radius = 2 a.

Suppose in fig. 124 Sqa to be a cardioid, and the circle Spq to be described on Sq as a diameter; then, if Sp = r, Psa = 0,

QSA = 0-, & .

r = 2a (cosy) cos (0—0^.

Differentiating which with respect to 0', we have

0' { 0' & )

0 = 2acosy |sin(0—(f) cos-^- —cos(0 —0') sin^-|;

.-. 2a cos-g- sin (0 —J = 0, and .-. 0^=-^-;

and the equation to the envelope is,

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and if n = 00 , r = 2a; that is, the ultimate envelope is a circle whose centre is s and radius is 2 a, and which is dotted in the figure.

Section 3.—Reciprocal curves and the theory of reciprocation.

318.] We are now able to complete, so far as the scope of our work requires, the inquiry into poles and polars which was opened in Articles 226 and 227; and we shall also be able to investigate some matters of interest connected with that inquiry. I shall take the requisite equations in a homogeneous and symmetrical form, so that the results may have all the elegance which they are capable of. Let us suppose the curve, which in these articles is called the base-curve, to be an algebraical curve of the rath degree, and of the form (49), Art. 208; and to be represented by the equation

Let the pole be (f, r), f); then the equation of the first polar of this pole with respect to the curve (58) is

which we may, as in Art. 224, express by the equation pt = 0; and let the successive polars of the same pole be pi = O,... p„_i = 0, pn = 0. Let us for the present confine our attention to the first polar, viz. (59); this is evidently a homogeneous equation of the (» — l)th order with respect to x,y,z; and is the equation to the curve which passes through the n(n—1) points, real or imaginary, at which tangents drawn from the pole (£, f) meet the base-curve.

Let the pole move continuously along another curve, which I shall, as before, call the directrix; let the equation of this curve also be homogeneous and of the rth degree in terms of the coordinates £, 17, f; and let its equation be

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