This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1893 Excerpt: ...When 6 is defined as a function of u by the relation tan 0 = sinh u (Art. 48) it is called the Gudermannian of - and as written gd u. Sin 6, cos 6 and tan 0 are then regarded as functions of u and are written sg u, eg u and tg u. 51. Agenda. From the definitions of the Gudermannian functions prove the formulae: By ...
Read More
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1893 Excerpt: ...When 6 is defined as a function of u by the relation tan 0 = sinh u (Art. 48) it is called the Gudermannian of - and as written gd u. Sin 6, cos 6 and tan 0 are then regarded as functions of u and are written sg u, eg u and tg u. 51. Agenda. From the definitions of the Gudermannian functions prove the formulae: By Cayley, Elliptic Functions, p. 56, where the equation of definition is! = lntan (1 JT + J Q). Since tan (i IT + 1 6)-' + si" and sinh-' tan 6-In ' + si" (Art. 56), the cos () COS () equivalence of the two definitions is obvious. The name is given in honor of Gudermann, who first studied these functions. 'f-(' and the average speed of P during the whole interval is x'--x _ "'--6" f--i t'--f Let k x0 represent the true speed of P at a given instant within the interval considered, A x0--8, X x0-f-8' the speeds at its beginning and end respectively; then x'--x X x0--8-j, _-A xa + 8'; and if the interval t'--t be made to decrease in such a way that 8 and 8' simultaneously approach zero, the three members of this inequality approach a common value, their limit; that is, and in particular when = limit Keu--if-") limit (sinh?) u = o I 2 u ) u==o' U ) Q. E. D. (Cf. Art. 43.) 53. Area of a Hyperbolic Sector. Let the perpendicular / be dropped from any point P, of the equilateral hyperbola x--y = a, upon its asymptote, meeting the latter in S, and let OS = s. The following properties of this hyperbola are well known and are proved in elementary works on conic sections: sp = a' j 2, x+y = s /1, x--y=PV 2, __ y = s-p) IV 2. But a a a' hence, assuming x / a--cosh it and y / a = sinh u, (4). Deduce these formulae also geometrically from the constructions of Arts. 48, 53, assuming for the definitions of sinh u and cosh u the ra...
Read Less
