or, the difference of the squares upon the conjugate diameters is equal to the difference of the squares upon the axes. 184. Again, multiplying (1) and (2) together, and (3) by itself, then subtracting the results, and reducing, as in the article (135.), we have 1 Now 0' is the angle PCD between the conjugate diameters CP and CD; hence, drawing straight lines at the extremities of the conjugate diameters, parallel to those diameters, we have, from the above equation, the parallelogram PCDT the rectangle A' C B E, and hence the whole parallelogram thus inscribed in the figure is equal to the rectangle contained by the axes *. 185. Returning to article (182.), the equation to the curve, suppressing the accents on and y', as no longer necessary, is The theorems in articles 183 and 184 may be proved also in the following man ner: Referring the curve to its rectangular axes, as in art. (187.), let the co-ordinates of P be and y'; then the equation to CD is a2 y y b2 x x 0, and eliminating r يو = b2x2=- a2 b2) we have and y between this equation and that to the curve (a2 the co-ordinates C N and D N, independent of the sign √ — 1, with which they are both affected, Also the triangle PCD the trapezium PM ND+ the triangle DCN triangle PCM a − x) " + y' + x y − x' y ' — x' y − y′ x 1 = 2 we have the square upon Q V: the rectangle PV, V P':: the square upon CD: the square upon C P. 186. The equation to the tangent at any point Q (x′ y') is ayy' - b2 x x' — — a,2 b2. = 187. Let the curve be referred to its axes C A, C B, and let the co ordinates of P be x'y', then the equation to C P being y = b2 a2 equation to CD is y = tan. e'= x cot. = a2 y hence C D, or the diameter conjugate to C P, is parallel to the tangent at P. The equation to the conjugate diameter is the same as that to the tangent, omitting the last term a2 b2. 188. Let x' and y' be the rectangular co-ordinates of P; then from the equation a2 — b2 a2 - b2, we have b1o = a ̧2 — a2 + b2 — x' 2. + y22 — a2 + b2 = x12 + a2 + b2 a2a2e2 x12 — a2 = (e x' - a) (e x' + a) = r r'; e2 x' 2 That is, the square upon the conjugate diameter C D = the rectangle under the focal distances S P and H P. 189. If P F be drawn perpendicular from P upon the conjugate C D, (see the last figure but one,) we have the rectangle PF, CD = a b, (184-). Hence the rectangle P G, P F = the square on B C ; And the rectangle P G', P F = the square on A C; the square on C D. *If the distance CP u, and p the perpendicular from the centre on the tan gent, this equation is a2 b2 a2 + b2 SUPPLEMENTAL CHORDS. 190. Two straight lines drawn from a point on the curve to the extremities of a diameter are called supplemental chords; they are called principal supplemental chords if that diameter be the transverse axis. The equations to a pair of chords are y- y = α (xx') y+y' = a' (x + x'); as in (141.); hence the product of the tangents of the angles which a pair of supplemental chords makes with the transverse axis is constant; the converse is proved as in 141. 191. The angle between two supplemental chords is found from the expression tan. PQ P' = x'y-y' x 2 b And, if A R, A'R be principal supplemental chords drawn to any point R on the curve, tan. A RA' = 2 a b2 The angle AR A' is always acute, and diminishes from a right angle to 0; the supplemental angle A A' R' increases at the same time from a right angle to 180°; hence, the angle between the supplemental chords may be any angle between 0 and 180°. Chords may be drawn containing any angle between these limits, by describing on any diameter, except the axes, a segment of a circle containing the given angle, and then joining the extremities of the diameter with the point where the circle intersects the hyperbola. And therefore principal supplemental chords parallel to these may be drawn. 192. Conjugate diameters are parallel to supplemental chords (144.); and therefore they may be drawn containing any angle between 0 and 90o. 193. There are no equal conjugate diameters in the hyperbola, but in that particular curve where b = a, we have the equation a b12 = a9 b2 = 0; hence the conjugate diameters a, and b, are always equal to each other. The equation to this curve, called the equilateral hyperbola, is 194. We have now shown that most of the properties of the ellipse apply to the hyperbola with a very slight variation: there is, however, a whole class of theorems quite peculiar to the latter curve, and these arise from the curious form of the branches extending to an infinite distance; it appears from the equation tan. 0 . tan 0 = b? a2 in (180.), that as tan. recedes along the curve from the origin, the conjugate diameters for that point approach towards a certain line C E, fig. (179.), and finally at an infinite distance come indefinitely near to that line. We now proceed to show that the curve itself continually approaches to the same line C E, without ever actually coinciding with it. But as this species of line is not confined to the hyperbola, we shall state the theory generally. 195. Let CPP' be a curve whose equation has been reduced to the form For any value of x we can find from this last equation a corresponding с ordinate M Q, and by adding to M Q, we determine a point P in the x curve similarly we can determine any number of corresponding points (P', Q', &c.) in the curve and straight line. C Since decreases as a increases, the line P' Q' will be less than P Q, and the greater becomes, the smaller does the corresponding P' Q' become; so that when x is infinitely great, P' Q' is infinitely small, or the curve approaches indefinitely near to the line T B S, but yet never actually meets it: hence TBS is called an asymptote to the curve, from three Greek words signifying "never coinciding." The equation to the asymptote TB S is y = ax + b, or is the equation to the curve, with the exception of the term involving the inverse 196. The reasoning would have been as conclusive if there had been more inverse powers of x; and in general if the equation to a curve can be put into the form y = &c. + m x3 + n x2 + a x + b + d + + &c. Ꮖ 4 Then the equation to the curvilinear asymptote is y = &c. + m x3 + n x2 + a x + b Also the equation y = &c. + m x3 + n x2 + a x + b + much more asymptotic than the preceding equation, and hence arises a series of curves, each " more nearly coinciding" with the original curve. 197. Let us apply this method to lines of the second order, whose general equation is (75.) · √ { (b3 − 4 a c) x2 + 2 (b d − 2 a e) x + ď2 − 4 a ƒ} ± √ { m x2 + nx+p}, by substitution, b x + d m { 1 } {1 + 2 a b x + d 2 a n m x2 constant terms powers of x ± √m { x + } Now b2-4ac is negative in the ellipse, and therefore there is no locus to the above equation in this case; also if b2 - 4 a c = 0, the equation to the asymptote, found as above, will contain the term, and therefore will belong to a curvilinear asymptote; hence the hyperbola is the only one of the three curves which admits of a rectilinear asymptote. It appears from the sign, that there are two asymptotes, and that b x + d 2 a the diameter y = bisects them. Also these asymptotes we have pass through the centre; for giving to the value and these values of x and y are the co-ordinates of the centre (80.). 198. If the equation want either of the terms 2 or y2, a slight operation will enable us to express the equation in a series of inverse powers of yor; thus if the equation be b xy + cx2 + dy + ex+f= 0, |