Let us for simplicity take for origin the centre of the circle, the point of contact with which we are seeking, that is to say, let us take a = 0, b = 0, then if A and B be the co-ordinates of the centre of Σ, the sought circle, we have seen in the last article, that they fulfil the relations But if x and y be the co-ordinates of the point of contact of Σ with S, we have from similar triangles for Now if in the equation of any right line we substitute mx, my x and y, the result will evidently be the same as if we multiplied the whole equation by m and subtract (m − 1) times the absolute term. Hence, remembering that the absolute term in SS' is (Art. 110) No′2 – p2 - a'2 – b'2, the result of making the above substitutions for A and B in (S- S') = 2R (r – r') is -a Ꭱ 1 · (S - S') + = · (a'2 + b'2 + p2 − p12) = 2R (r − r), (R + r) (S− S′ ) = R { (r − r′ )2 – a'2 – b′2 }. Similarly (R+r) (S − S′′) = R {(r— r'')2 — a′′2 — b′′2 } . Eliminating R, the point of contact is determined as one of the intersections of the circle S with the right line 126. To complete the geometrical solution of the problem it is necessary to show how to construct the line whose equation has been just found. It obviously passes through the radical centre of the circles; and a second point on it is found as follows. Write at full length for SS' (Art. 110), and the equation is 2a'x + 2b'y + r22 — p2 — a'2 — b′2 12 a'2 + b2 − ( r — r' )2 2a"x+2b"y +"2 — p2 — a′′2 — b′′2 Add 1 to both sides of the equation, and we have et 4 showing that the above line passes through the intersection of But the first of these lines (Art. 118) is the chord of common a'x + by + (r' − r) r = 0, a′′x + b'y + (r" − r) r = 0 is the pole of the axis of similitude of the three circles, with regard to the circle S. Hence we obtain the following construction: Drawing any of the four axes of similitude of the three S" circles, take its pole with re spect to each circle, and join (a, b; a', b'; a", b′′), the circle through a, a', a" will and that through b, b′, b′′ will S P C R be another. Repeating this process with the other three axes of similitude, we can determine the other six touching circles. 127. It is useful to show how the preceding results may be derived without algebraical calculations. (1.) By Art. 123 the lines ab, a'b', a"b" meet in a point, viz., the centre of similitude of the circles aa'a", bb'b”. (2.) In like manner a'a", b'b" intersect in S, the centre of similitude of C', C". (3.) Hence (Art. 121) the transverse lines a'b', a"b" intersect on the radical axis of C', C". So again a"b", ab, intersect on the radical axis of C", C. Therefore the point R (the centre of simi litude of aaa", bb'b") 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 S' and 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 cos a = =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, 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 which will enable us to determine the five quantities, &c. 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 x, 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 the new F, F' = Ax'2 + Bx'y' + Cy'2 + Dx' + Ey' + 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, we 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. |