X. HYPERBOLIC RATIOS.

46. Definitions of the Hyperbolic Ratios. An important class of exponentials, which because of their relation to the equilateral hyperbola are called the hyperbolic sine, cosine, tangent, cotangent, secant and cosecant, and are symbolized by the abbreviations sinh, cosh, tanh, coth, sech, csch, are defined by the following identities:

48. Geometrical Construction for Hyperbolic Ratios. For the representation of the hyperbolic ratios the equilateral hyperbola is employed. Its equation in Cartesian co-ordinates is

Let OX, OY be its axes, OJ an asymptote, P any point on the curve, x and y its co-ordinates, AQB the quadrant of

a circle with centre at the origin and radius a, NQ a tangent to the circle from the foot of the ordinate y, RS a tangent to the hyperbola parallel to the chord PA, a, B the co-ordinates of Q, the arc-ratio of the angle AOQ.

It is obvious from its definition that cosh u has 1 for its smallest and for its largest value corresponding to u=0 and ∞ respectively, and if the variations of x / a be confined to the right hand branch of the hyperbola its range of values is likewise between 1 and +∞; hence we may

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 В to OP, a sech u OM, intercept of tangent at P upon OX,

=

a csch u= LO,

(intercept of tangent at P upon OY).

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 ù — sech u) · =a sinh a / (cosh? t −1)= a csch a.

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 0 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 formula:

* By Cayley, Elliptic Functions, p. 56, where the equation of definition is u = In tan (+ ().

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.

cg u. cg v

(3). cg (u+v)=1+ sg u.sg v'

(4). Ifi=√=1, {gd({ gd u)= — u,

or more briefly

(gd)

2

u = -u. (Prof. Haskell.)

52. To Prove Limit [(sinh u) /u]=1, when u=0. 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

λα

Let x represent the true speed of P at a given instant within the interval considered, Ax, -8, λx. +8′ the speeds at its beginning and end respectively; then

x x。 − d < x = x < λ x。 + 8′;

λ χο

and if the interval ť

ť

t

t be made to decrease in such a way

that 8 and 8' simultaneously approach zero, the three mem