This gives, for the equation of the radical axis, Now, to ascertain whether this touches the inscribed circle, we have, applying the condition of Chap. II. Art. 9, to investigate the value of {(b − c) (s − a) + (c − a) (s − b) + (a−b)(s−c)}, which is 0. Hence, the radical axis touches the inscribed circle, and therefore the inscribed and nine-point circles touch one another. Similarly, it may be proved that the nine-point circle touches each of the escribed circles. 19. The equation of the nine-point circle may be deduced by substituting the above values of λ, u, v, or (perhaps more neatly) by expressing the fact that the curve ua2 + vß2 + wy2 + 2u'By + 2v'ya+2w'aß = 0 passes through the middle points of the sides of the triangle, and combining the equations thus obtained with those investigated in Art. 14. The former gives Hence, the nine-point circle is represented by the equation a cos A. a2+b cos B. B2+ c cos C. y-aẞy-bya-caß = 0. COR. It hence appears that the nine-point circle passes through the points of intersection of the circumscribed and self-conjugate circles, or has a common radical axis with them. 20. We have investigated, in Art. 10, the equation of the pair of tangents drawn to the conic from a given point (f, g, h). If these two tangents be at right angles to one another, they may be regarded as the limiting form of a rectangular hyperbola, and must therefore satisfy the equation investigated in Art. 13. This, therefore, gives as the locus of the intersection of two tangents at right angles to one another Wg2+Vh2-2U'gh + Uh2 + Wf-2 V'hf+Vƒ2+ Ug2 - 2 W'fg + 2 (Uƒ2 + Ugh - V'fg – W'hf) cos A + 2 (V'g2 + Vhf - W'gh — U'fg) cos B +2(W'h2 + Wfg – U'hf - V'gh) cos C′ = 0. This may be shewn (see Art. 15) to represent a circle, as we know ought to be the case. This equation may also be expressed in the following If the conic be a parabola, then (see Art. 6) this breaks up into two factors, one of which is the line at infinity; and the other must represent the directrix, since that is the locus of the point of intersection of two tangents to a parabola at right angles to one another. The appearance of the line at infinity as a factor in the result in this case may be explained as follows: Every para bola touches the line at infinity, and this line also satisfies the algebraical condition of being perpendicular to any line whatever, since, whatever l, m, n may be, al+bm+cn−(bn+cm) cos A − (cl+an) cos B− (am+bl) cos C=0, identically. It therefore will form a part of the locus of the intersection of two tangents at right angles to one another, the two tangents being the line at infinity itself, and any other tangent whatever. The directrix of the parabola is therefore represented by the equation 21. To find the magnitude of the axes of the conic. Let a, B, y be the co-ordinates of the centre; and, for shortness' sake, put a-α=x, B-B=y, y-7=z. Then if r be the semi-diameter drawn from the centre to a, B, y, we have (see Art. 3, Chap. 1.) (a cos A. x2+b cos B. y2+c cos C. 2).......(1). Again, from the equation of the conic, 0 = $ (a, ß, y) = $ (a + x, B + y, y + z) = ☀ (a, B, y) + 2x (uã + w'ß + v'ỹ) + 2y (w ́a+vB+u'ŋ) + 2z (v'à + u'ß + wy) + $ (x, y, z). Now, by Art. 11 of the present chapter, ua+w'B+v'y _ w'a + vß + u'y_v'a+u'ß + wy = Also, ax+by+cz = a (a− a) + b (ß − ß) + c(v − y) = 0...(2); .. $ (x, y, z) = − & (α, B, y), or, ux2 + vy2+ wz2 + 2u'yz + 2v'zx + 2w'xy u, w', v' w', v, u' (2A)2 v', u', พ u, w', v', α Now the semi-axes are the greatest and least values of the semi-diameter. We have then to make 4A2 r2 = a cos A. x2 + b cos B. y2 + c cos C. z2............... ..(4) a maximum or minimum, x, y, z being connected by the relations (2) and (3). Multiply (2) by the indeterminate multiplier 2μ, (4) by λ, adding them to (3), differentiating, and equating to zero the coefficients of each differential, we get = 0 ux + w'y + v'z + λa cos A . x + μa = = 0 (5). v'x + u'y + wz + λc cos C. z + μc = 0] Multiplying these equations in order by x, y, z, and adding, we get |