Notes

¹ The hyperbolic functions are not so named on account of any analogy with what are termed Elliptic Functions. “The elliptic integrals, and thence the elliptic functions, derive their name from the early attempts of mathematicians at the rectiﬁcation of the ellipse. …To a certain extent this is a disadvantage; … because we employ the name hyperbolic function to denote $cosh u, sinh u$ , etc., by analogy with which the elliptic functions would be merely the circular functions $cos φ, sin φ$ , etc. …” (Greenhill, Elliptic Functions, p. 175.)

²This angle was called by Gudermann the longitude of $u$ , and denoted by $l u$ . His inverse symbol was $L$ ; thus $u = L (l u)$ . (Crelle’s Journal, vol. 6, 1830.) Lambert, who introduced the angle $θ$ , named it the transcendent angle. (Hist. de l’acad. roy. de Berlin, 1761). Hoüel (Nouvelles Annales, vol. 3, 1864) called it the hyperbolic amplitude of $u$ , and wrote it $amh u$ , in analogy with the amplitude of an elliptic function, as shown in Prob. 62. Cayley (Elliptic Functions, 1876) made the usage uniform by attaching to the angle the name of the mathematician who had used it extensively in tabulation and in the theory of elliptic functions of modulus unity.

³The relation $gd u = am u, (mod. 1)$ , led Hoüel to name the function $gd u$ , the hyperbolic amplitude of $u$ , and to write it $amh u$ (see note, Art. 22). In this connection Cayley expressed the functions $tanh u$ , $sech u$ , $sinh u$ in the form $sin gd u$ , $cos gd u$ , $tan gd u$ , and wrote them $sg u$ , $cg u$ , $tg u$ , to correspond with the abbreviations $sn u$ , $cn u$ , $dn u$ for $sin am u$ , $cos am u$ , $tan am u$ . Thus $tanh u = sg u = sn u, (mod. 1)$ ; etc. It is well to note that neither the elliptic nor the hyperbolic functions received their names on account of the relation existing between them in a special case. (See foot-note, p. 40.2)

⁴Forms 7–12 are preferable to the respective logarithmic expressions (Art. 19), on account of the close analogy with the circular forms, and also because they involve functions that are directly tabulated. This advantage appears more clearly in 13–20.

⁵For ${tanh}^{- 1} (. 5)$ interpolate between $tanh (. 54) = . 4930$ , $tanh (. 56) = . 5080$ (see tables, pp. §, §); and similarly for ${tanh}^{- 1} (. 3333)$ .

⁶The use of vectors in electrical theory is shown in Bedell and Crehore’s Alternating Currents, Chaps, XIV–XX (ﬁrst published in 1892). The advantage of introducing the complex measures of such vectors into the diﬀerential equations is shown by Steinmetz, Proc. Elec. Congress, 1893; while the additional convenience of expressing the solution in hyperbolic functions of these complex numbers is exempliﬁed by Kennelly, Proc. American Institute Electrical Engineers, April 1895. (See below, Art. 37.)

⁷It is to be borne in mind that the symbols cosh, sinh, here stand for algebraic operators which convert one number into another; or which, in the language of vector-analysis, change one vector into another, by stretching and turning.

⁸The generalized hyperbolic functions usually present themselves in Mathematical Physics as the solution of the diﬀerential equation $\frac{d^{2} φ}{d u^{2}} = m^{2} φ$ , where $φ$ , $m$ , $u$ are complex numbers, the measures of vector quantities. (See Art. 37.)

⁹“If two $r$ th-degree functions of a single variable be equal for more than $r$ values of the variable, then they are equal for all values of the variable, and are algebraically identical.”

¹⁰This method of generalization is sometimes called the principle of the “permanence of equivalence of forms.” It is not, however, strictly speaking, a “principle,” but a method; for, the validity of the generalization has to be demonstrated, for any particular form, by means of the principle of the algebraic identity of polynomials enunciated in the preceding foot-note. (See Annals of Mathematics, Vol. 6, p. 81.)

¹¹The symbol $exp u$ stands for “exponential function of u,” which is identical with $e^{u}$ when $u$ is real.

¹²In this table $. 7$ is written for $\frac{1}{2} \sqrt{2}, = . 707 \dots$ .

¹³Such a chart is given by Kennelly, Proc. A. I. E. E., April 1895, and is used by him to obtain the numerical values of $cosh (x + i y)$ , $sinh (x + i y)$ , which present themselves as the measures of certain vector quantities in the theory of alternating currents. (See Art. 37.) The chart is constructed for values of $x$ and of $y$ between 0 and 1.2; but it is available for all values of $y$ , on account of the periodicity of the functions.

¹⁴See a similar problem in Chap. 1, Art. 7.

¹⁵ For the theory of this form of arch, see “Arch” in the Encyclopædia Britannica.

¹⁶This curve is used in Schiele’s anti-friction pivot (Minchin’s Statics, Vol. I, p. 242); and in the theory of the skew circular arch, the horizontal projection of the joints being a tractory. (See “Arch,” Encyclopædia Britannica.) The equation $φ = gd \frac{t}{c}$ furnishes a convenient method of plotting the curve.

¹⁷Jones, Trigonometry (Ithaca, 1890), p. 185.

¹⁸Merriman, Mechanics of Materials (New York, 1895), pp. 70–77, 267–269.

¹⁹See references in foot-note Art. 27.

²⁰Chapter V, Art. 8.

²¹Kennelly denotes these constants by $r$ , $l$ , $c$ , $g$ . Steinmetz writes $s$ for $ω L$ , $κ$ for $ω C$ , $θ$ for $G$ , and he uses $C$ for current.

²²Thomson and Tait, Natural Philosophy, Vol. I. p. 40; Rayleigh, Theory of Sound, Vol. I. p. 20; Bedell and Crehore, Alternating Currents, p. 214.

²³ In electrical theory the symbol $j$ is used, instead of $i$ , for $\sqrt{- 1}$ .

²⁴Bedell and Crehore, Alternating Currents, p. 181. The sign of $d x$ is changed, because $x$ is measured from the receiving end. The coeﬃcient of leakage, $G$ , is usually taken zero, but is here retained for generality and symmetry.

²⁵These relations have the advantage of not involving the time. Steinmetz derives them from ﬁrst principles without using the variable $t$ . For instance, he regards $R + j ω L$ as a generalized resistance-coeﬃcient, which, when applied to $i$ , gives an E.M.F., part of which is in phase with $i$ , and part in quadrature with $i$ . Kennelly calls $R + j ω L$ the conductor impedance; and $G + j ω C$ the dielectric admittance; the reciprocal of which is the dielectric impedance.

²⁶The complex constants $m$ , $m_{1}$ are written $z, y$ by Kennelly; and the variable length $x$ is written $L_{2}$ . Steinmetz writes $v$ for $m$ .

²⁷See Art. 14, Probs. 28–30; and Art. 27, foot-note.

²⁸Art. 30, footnote.

²⁹See Table II.

³⁰The data for this example are taken from Kennelly’s article (l. c., p. 38).

³¹Gudermann in Crelle’s Journal, vols. 6–9, 1831–2 (published separately under the title Theorie der hyperbolischen Functionen, Berlin, 1833). Glaisher in Cambridge Phil. Trans., vol. 13, 1881. Geipel and Kilgour’s Electrical Handbook.

Go Back