and the co-ordinates of Q being a, ß, also and finally, since OBK is similar to ONP and a2=y• LO,* Hence if a be made the denominator in each of the hyperbolic ratios, their numerators will be six straight lines, drawn from O, A, B, or P, which may be indicated thus: = a sinh u NP, perpendicular distance of P from OX, a cosh u = ON, distance from centre to foot of NP, a tanh u = AH, distance along a tangent from A to OP, a coth u= BK, distance along a tangent to the conjugate hyperbola from B to OP, OM, intercept of tangent at P upon O.X, (intercept of tangent at P upon OY). a sech u a csch u = LO, This construction gives pertinence to the name ratio as * Obtained by writing the equation of the tangent MP and finding its intercept on OY; or thus, OL OM=NP | MN, that is, OL-=(OM. y) | (x − OM) = a (sech u sinh u) / (cosh a — sech u)=a sinh u / (cosh u−1): = a csch u. applied to the six analogues of the goniometric ratios. Compare these with the constructions of Art. 42. 49. Agenda. Properties of the Equilateral Hyperbola. Prove the following propositions concerning the equilateral hyperbola. (Fig. 25 of Art. 48.) (1). The tangent to the hyperbola at P passes through M, the foot of the ordinate to Q. (2). The locus of I, the intersection of the tangents NQ and MP, is AJ the common tangent to the hyperbola and circle. (3). The line OIV bisects the angle and the area OAVP and intersects the hyperbola at its point of tangency with RS. (4). A straight line through P and Q passes through the left vertex of the hyperbola and is parallel to OV. (5). The angle APN= one-half the angle QON. 50. The Gudermannian. When is defined as a function of u by the relation tan 6 = sinh u (Art. 48) it is called the Gudermannian of u* and is written gd u. Sin 0, cos and tan are then regarded as functions of u and are written sg u, cg u and tg u. 51. Agenda. From the definitions of the Gudermannian functions prove the formulæ: * By Cayley, Elliptic Functions, p. 56, where the equation of definition is u = In tan (}π + }(). and sinh−1 tan 0 = ln I + sin 0 (Art. 56), the equivalence of the two definitions is obvious. The name is given in honor of Gudermann, who first studied these functions. 52. To Prove Limit [(sinh u)/u] = 1, when uo. In the construction of Art. 23 suppose that, during the interval of time t-t, P moves over the distance x'— x, Q over the distance u'-u. Then speed being expressed as the ratio of distance passed over to time-interval, the speed of Q is and the average speed of P during the whole interval is Let Ax, represent the true speed of P at a given instant within the interval considered, λ x,-8, Ax. +8' the speeds at its beginning and end respectively; then and if the interval ť t'be made to decrease in such a way that 8 and 8 simultaneously approach zero, the three mem bers of this inequality approach a common value, their limit; that is, x' bu u' Ax, when tt. λ =- xo, when t't; In the important case when u′ = — u, and uo, since u=o requires that x=1, the expression last written be 53. Area of a Hyperbolic Sector. Let the perpendicular p be dropped from any point P, of the equilateral hyperbola x2-ya, upon its asymptote, meeting the Assuming these equations as known, lay off upon the asymptote OS, OS1, OS2,..., OS = a/√2, S1, S2, ..., S, such that these lengths are in geometrical progression, and let the corresponding perpendiculars upon the asymptote be AS, PS1, PS2, ..., PS=Po, P1, P2, ..., p. Then if Ρ So, S1, S2, be the common ratio of the successive terms s, we have If now (a, o), (x,, Y1), (X2 Y2), . . ., (x, y) be the co-ordinates of A, P1, P„, . . ., P, the area of the triangle OPP, is (x ̧1⁄2 — x,y), and by virtue of the relations 2 3 we have but Thus the triangles OAP, to one another in area and OAPP.... Pis n |
