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Hence, if (x, y) be the co-ordinates of P, as w* varies continuously from - oo to +00, P will describe continuously the right-hand branch A'AA" of the rectangular hyperbola, whose

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semi-axis-major is OA = a, in the direction indicated by the arrow-heads in Fig. 11.

If P be the point corresponding to u, Pr, Pr+1 the points corresponding to rujn and (r + l)«/«, and U the area AOP agreeing in sign with w, then, exactly as before,

* Adopting an astronomical terra, we may call u the hyperbolic excentric anomaly of P. u plays in the theory of the hyperbola, in general, the same part as the eccentric angle in the theory of the ellipse.

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= <iJ{ cosh. r»/» sinh. (r + \)ujn - siuh.?-!i/?icosli.(r + \)ujn}, = a* sinh. ujn.


U = \ashn sinh. ujn,
= |«2ML(sinh. u/n)l(ujn),
= la\ by § 29, (3').

Hence, if the area AOP = U, and u = 2U/«2, then, x and y being the co-ordinates of P, we might give the following geometric definitions of cosh u, sinh u, &c. :— cosh u = xja, sinh u = yja, tanh u = y/x, coth u = x/y, &c.

It will now bo apparent that the hyperbolic functions are connected in the same way with one half of a rectangular hyperbola, as the circular functions are with the circle. It is from this relation that they get their name.

Wo know, from elementary geometrical considerations, that the area 8 is the product of ^a? into the number of radians in the angle AOP. It therefore follows from (3) that the variable 6 introduced above is simply the number of radians in the angle AOP. Our demonstration did not, however, rest upon this fact, but merely on the functional equation cos -6 + sin 20 = 1. This is an interesting point, because it shows us that we might have introduced the functions cos0 and sin0 by the definitions cos0 = J {Exp (iff)

+ Exp(-t'0)}, sin 9 = ~..{Exp(iff) -Exp(-£<?)}; and then, by means of the

above reasoning, have deduced the property which is made the basis for their geometrical definition. When this point of view is taken, the theory of the circular and hyperbolic functions attains great analytical symmetry; for it becomes merely a branch of the general theory of the exponential function as defined in § 18.

When we attempt to get for u a connection with the arc AP, like that which subsists in the case of the circle, the parallel ceases to mu on the same elementary line. To understand its nature in this respect we must resort to the theory of Elliptic Integrals.

§ 31.] Expression of Real Hyperbolic Functions in terms of Real Circular Functions.


Since the range of the variation of cosh u when u varies from - oo to + oo is tho same as the range of sec 6 when 6 varies from - It to + 5t, it follows that, if we restrict 6 and « to have the same sign, there is always one and only one value of u between - oo and + oo and of 6 between - ir and + ir such that cosh u = sec 0 (1).

If we determine 6 in this way, we have sinh u = ± v/(coshs« - 1), = ± v/(sec20-l); hence, bearing in mind the understanding as to sign, we have sinh u = tan 6 (2).

From these we deduce

e" = cosh u + sinh u,

= sec 0 + tan 0; u = log (sec 6 + tan ti), = log tan {\ir+\6) (3).

Also, as may be easily verified,

tanh £« = tan \Ba (4).

When 6 is connected with u by any of the four equivalent equations just given, it is called the Gudermannian * of «, and we write 6 = gd u.

* This name was invented by Cayley in honour of the German mathematician Gudermann (1798-1852), to whom the introduction of the hyperbolic functions into modern analytical practice is largely due. The origin of the functions goes back to Mercator's discovery of the logarithmic quadrature of the hyperbola, and Dcmoivre's deduction therefrom (see p. 282). According to Houel, F. C. Mayer, a contemporary of Demoivre's, was the first to give shape to the analogy between the hyperbolic and tho circular functions. The notation cosh. sinh. seems to be a contraction of coshyp. and sinhyp., proposed by Lambert, who worked out the hyperbolic trigonometry in considerable detail, and gave a short numerical table. Many of the hyperbolic formulae were independently deduced by William Wallace (Professor of Mathematics in Edinburgh from 1819 to 1838) from the geometrical properties of the rectangular hyperbola, in a little known memoir entitled New Scries for the Quadrature of Conic Sections and the Computation of Logarithms (Trans. R.S.E., vol. vi., 1812). For further historical information, see Giinther, Die Lchrc von den gctcbhnlichen und verallgcmeinerlcn Hypcrbelfunktionen (Halle, 1881); also, Bcitriigc zur Qcschichte dcr Neueren MaOtemalik (Programmschrift, Ansbach, 1881).


It is easy to give a geometrical form to the relation between 6 and u. If, in Fig. 11, a circle be described about 0 with a as radius, and from M a tangent be drawn to touch this circle in Q (above or below OX according as u is positive or negative), then, since MQ- = OM2- OQ2=2?-a2 = !/-', we have ncoshw = x=asecQOM. Therefore QOM = 0, and we have j/ = MQ = « tan 9. From this relation many interesting geometrical results arise which it would be out of place to pursue hero. We may refer the reader who desires further information regarding this and other parts of the theory of the hyperbolic functions to the following authorities:—Greenhill, Differential and Integral Calculus (Macmillan, 1886), and also an important tract entitled A Chapter in the Integral Calculus (Hodgson, London, 1888); Laisant, "Essai sur les Fotictions by perboliques," Mini, de la Soc. Phys. et Nat. dc Bordeaux, 1875; Heis, Die Hypcrbolischcn Funetionen (Halle, 1875). Tables of the functions have been calculated by Gudermann, Theorie der Potential- odcr Cyclischhyptrbolischcn Funetionen (Berlin, 1833); and by Gronau (Dantzig, 1863). See also Cayley, Quarterly Journal of Mathematics, vol. xx.

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