litude of aa'a", bbb") must be the radical centre of the circles C, C', C".

(4.) In like manner, since a'b', a"b" pass through a centre of similitude of aa'a", bb'b"; therefore (Art. 121) a'a", b'b" meet on the radical axis of these two circles. So again the points Sand S" must lie on the same radical axis; therefore SS'S", the axis of similitude of the circles C, C', C", is the radical axis of the circles aa'a", bbb".

(5.) Since a"b" passes through the centre of similitude of aa'a", bb'b", therefore (Art. 121) the tangents to these circles where it meets them intersect on the radical axis SS'S". But this point of intersection must plainly be the pole of a"b" with regard to the circle C". Now since the pole of a"b" lies on SS'S", therefore (Art. 96) the pole of SS'S" with regard to C" lies on a"b". Hence a"b" is constructed by joining the radical centre to the pole of SS'S" with regard to C".

(6.) Since the centre of similitude of two circles is on the line joining their centres, and the radical axis is perpendicular to that line, we learn (as in Art. 124) that the line joining the centres of aa'a", bb'b" passes through R, and is perpendicular to SS'S".

Ex. To describe a circle cutting three given circles at given angles.

By the help of (Ex. 5, Art. 117) this is reduced to the problem of the present article; or else the three equations

R22Rr cosa = S, R22 Rr cosẞ = S',

R2 - 2Rr" cos y = S",

may be discussed directly as in Art. 124.

CHAPTER X.

PROPERTIES COMMON TO ALL CURVES OF THe second degree, DEDUCED FROM THE GENERAL EQUATION.

128. THE most general form of the equation of the second degree is Ax2 + Bxy + Cy2 + Dx + Ey + F = 0,

where A, B, C, D, E, F are all constants.

The nature of the curve represented by this equation will

vary

according to the particular values of these constants. Thus we saw (Chap. v.), that in some cases this equation might represent two right lines, and (Chap. vi.) that for other values of the constants it might represent a circle. It is our object in this chapter to classify the different curves which can be represented by equations of the general form just written, and to obtain some of the properties which are common to them all.*

Five relations between the coefficients are sufficient to determine a curve of the second degree. It is true that the general equation contains six constants, but it is plain that the nature of the curve does not depend on the absolute magnitude of these coefficients, since, if we multiply or divide the equation by any constant, it will still represent the same curve. We may, therefore, divide the equation by F, so as to make the absolute term 1, and there will then remain but five constants to be determined.

=

Thus, for example, a conic section can be described through five points. Substituting in the equation the co-ordinates of each point (xy) through which the curve must pass, we obtain five relations between the coefficients, viz.,

A

&c.

which will enable us to determine the five quantities,

F'

129. We shall in this chapter often have occasion to use the method of transformation of co-ordinates; and it will be useful to find what the general equation becomes when transformed to parallel axes through a new origin (x'y'). We form the new equation by substituting x + x' for a, and y + y' for y (Art. 8), and we get

A(x+x')2+B(x+x') (y+y') + C(y+y')2+D(x+x') + E(y+y') + F = 0.

* We shall prove hereafter, that the section made by any plane in a cone standing on a circular base is a curve of the second degree, and, conversely, that there is no curve of the second degree which may not be considered as a conic section. It was in this point of view that these curves were first examined by geometers. We mention the property here, because we shall often find it convenient to use the terms "conic section" or "conic," instead of the longer appellation," curve of the second degree."

Arranging this equation according to the powers of the variables, we find that the coefficients of x2, xy, and y2, will be, as before, A, B, C; that

F' = Ax22 + Bx'y' + Cy'2 + Dx' + Ey' + F.

the new F, Hence, if the equation of a curve of the second degree be transformed to parallel axes through a new origin, the coefficients of the highest powers of the variables will remain unchanged, while the new absolute term will be the result of substituting in the original equation the co-ordinates of the new origin.*

130. Every right line must meet a curve of the second degree in two real, coincident, or imaginary points.

Let us first consider the case of lines which pass through the origin. The truth of the proposition will then easily appear by transformation to polar co-ordinates. If the angle between the axes be w, then for a line making angles a, ß, with the axes, we saw (Art. 12) that x sin w = p sina, y sin w = P sin ẞ, or, as we shall write for shortness, x = mp, y = np. Making these substitutions in the general equation, we have, to determine the length of the radius vector to either of the points where the line (whose equation obviously is my = nx) meets the curve, the quadratic,

(Am2 + Bmn + Cn2) p2 + (Dm + En) p + F = 0.

Since this equation always gives two values for p, wè see, as in Art. 81, that every line through the origin will meet the curve in two real, coincident, or imaginary points.

The case of a line not passing through the origin is reduced to the former, by transferring the origin to any point on the line. The equation will then become

Ax2 + Bxy + Cy2 + D'x + E'y + F' = 0 ;

where D', E', F' have the values found in the last article, and the distances from the new origin of the points where any line through it meets the curve, are the two roots of a quadratic equation, precisely similar in form to that already given.

This is equally true for equations of any degree, as can be proved in like manner.

دم

131. The next articles will be occupied with a discussion of the different forms assumed by the quadratic just found for according to the different values we may give the ratio m:n. The reader will better understand the method we pursue if he bear in mind the following elementary principles. Suppose that we have to discuss any quadratic,

ap2 + bp + c =

its solution may be written in either of the following equivalent forms,

the latter being the form in which the solution would have presented itself had we divided the given equation by p2, and solved it for the reciprocal of p.

1. If we have c = 0, the quadratic is divisible by p, and one of

its roots is p = 0, the other being

-

b

α

If we had not only c = 0,

but also b = 0, then the quadratic would be divisible by p2, and both its roots would = 0.

II. If we have a = 0, then one of the roots of the equation is p=. For if we had written the equation

same thing may be seen by making a = 0 in the general form of the solution. If not only a = 0, but also b = 0, both the roots

= ∞.

III. Ifb = 0, the roots of the quadratic are equal with opposite signs.

Iv. If we have b2 = 4ac, the two roots are equal, and we may

roots of the quadratic are real; if b' be less than 4ac, the roots are

imaginary.

R

132. Let us now apply these principles to the equation which determines the points where the line (my = nx) meets the curve,

viz.

(Am2 + Bmn + Cn2) p2 + (Dm + En) p + F = 0.

=

1. Let F 0. In this case one of the values of o is = P 0, or the origin is one of the points where the line meets the curve (see also Art. 79). The other value is

ρ

If, however, we have not only F = 0, but also the line be drawn in such a direction that Dm + En = 0, then the second value of is also = 0: the line (mynx) meets the curve in two coincident points at the origin, or, in other words, is a tangent at the origin. Multiplying by p the equation Dm + En = 0, and remembering that mp = x, np = y, we find the equation of the tangent at the origin, viz. Dx + Ey = 0.

transform the equation to parallel axes through this point, and find the tangent at it. Ans. 9x 5y O referred to the new axes, or 9 (x − 1) — 5 (y − 1) = 0

referred to the old.

133. To find the equation of the tangent at any point x'y' on the curve.

Transform to parallel axes through x'y', and (Art. 129) F' will vanish, since x'y' is on the curve. The equation of the tangent will then be D'x + E'y = 0 referred to the new axes, or D'(x − x') + E' (y - y') = 0 referred to the old. Write for D' and E' the values found in Art. 129, and the equation of the tangent

is (2Ax' + By + D) (x − x') + (Bx' + 2Cy' + E) (y − y) = 0, which may be written in a simpler form by adding to both sides the identity

2 Ax2 + 2Bx'y' + 2Cy2 + 2Dx' + 2Ey' + 2F = 0, when the equation of the tangent becomes

(2Ax' + By + D) x + (Bx' + 2Cy' + E) y + Dx' + Ey' + 2F

= 0.