equilateral hyperbola, this is the same as saying that the two points at infinity on any circle lie each on the polar of the other with respect to the curve. Given five points.-We have shown (Ex. 12, p. 283) how by the ruler alone we may determine as many other points of the curve as we please. We may also find the polar of any given point with regard to the curve; for by the help of the same Example we can perform the construction of Ex. 2, Art. 149. Hence too we can find the pole of any line, and therefore also the centre. Five tangents. We may either reciprocate the constructions of Ex. 12, p. 283, or reduce this question to the last by Art. 266. Four points and a tangent.—We have already given one method of solving this question, p. 280. As the problem admits of two solutions, of course we cannot expect a construction by the ruler only. We may therefore apply Carnot's theorem (Art. 314), Ac. Ac'. Ba. Ba'. Cb. Cb' : = Ab. Ab'. Bc. Bc'. Ca. Ca'. Let the four points a, a', b, b' be given, and let AB be a tangent, the points c, c' will coincide, and the equation just given determines the ratio Ac2: Bc2, everything else in the equation being known. This question may also be reduced, if we please, to those which follow; for given four points, there are (Art. 318) three points whose polars are given; having also then a tangent, we can find three other tangents immediately, and thus have four points and four tangents. Four tangents and a point.-This is either reduced to the last by reciprocation, or by the method just described; for given four tangents, there are three points whose polars are given (p. 134). Three points and two tangents.—It is a particular case of Art. 337 that the two points where any line meets a conic, and where it meets two of its tangents, belong to a system in involution of which the point where the line meets the chord of contact is one of the foci. If, therefore, the line joining two of the fixed points a, b, be cut by the two tangents in the points A, B, the chord of contact of those tangents passes through one or other of the fixed points F, F', the foci of the system (a, b, A, B), (see Art. 264). In like manner the chord of contact must pass through one or other of two fixed points G, G' on the line joining the given points a, c. The chord must therefore be one or other of the four lines, FG, FG', F'G, F'G'; the problem, therefore, has four solutions. Two points and three tangents.-The triangle formed by the three chords of contact has its vertices resting one on each of the three given tangents; and by the last case the sides pass each through a fixed point on the line joining the two given points: therefore this triangle can be constructed. To be given two points or two tangents to a conic is a particular case of being given that the conic has double contact with a given conic. For the problem to describe a conic having double contact with a given one, and touching three lines, or else passing through three points, see p. 283. Having double contact with two, and passing through a given point, or touching a given line, see p. 237. Having double contact with a given one, and touching three other such conics, see p. 257. We have already alluded (p. 252) to the problem, "to describe a conic through four points to touch a given conic." Let the required conic be S+kS', which is to touch S". Then the polar of the point of contact, with regard to S", is the tangent at the point, and is also its polar for SkS', and therefore passes through the intersection of the polars with regard to S and S'. Now let it be required to find the locus of a point such that its polars, with regard to S, S', S", should meet in a point. If §, 7, be the current co-ordinates, we have to eliminate these between the equations of the three polars, a curve of the third degree, whose intersections with S" give the six solutions sought. If S, S', S" all pass through the same two points A, B, the locus reduces to a line and a conic: for the line joining those points must be a factor in the locus, since the polar of any point C on that line must pass through D, the fourth harmonic to A, B, C. If S, S', S" represent circles, the equation just written represents the circle cutting all those at right angles. The locus will also break up into a line and conic, if one of the quantities S' be a perfect square L2; since L will then be a factor in the locus. Hence we can describe a conic to touch a given conic S at two given points (S, L), and also touching S"; for the intersection of the locus with S" determines the points of contact with S" of conics of the form S + L2. THE END. PUBLISHED BY Messrs. 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