Algebra Volume 2; an elementary text book for the higher classes of secondary schools and for colleges

by George Chrystal

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. 1889 ... Cos x Cos yi-Sin x Sin yi = cos x cosh y-i sin x sinh y, we have u = cos x cosh y, v =-sin x sinh y (2); and therefore Ma/cos x vsjain. x=l (3); «2/cosh 2y + i/smh 'y = 1 (4). iv Fio. 12. In order to avoid repetition of the values u and v, arising from the periodicity of cos x and sin x, we confine z, in the first instance, to lie between the axis of y and a parallel UCGCU to this axis at a distance from it equal to Tt (Fig. 12). If we draw a series of parallels to the y-axis within this strip, we see, from equation (3), that to each of these will belong half of a hyperbola in the w-plane (Fig. 13), having its foci at the fixed points F and G, which are such that OF = OG = 1. Thus, for example, if in the 2-plane FP = ir and FQ = then to the parallels LPL, NQN correspond the two halves LPL, NQN of a hyperbola whose transverse axis is PQ = V2. To the parallel MAM, which bisects the strip, corresponds the axis of v (which may be regarded as that hyperbola of the confocal system which has its transverse axis equal to 0); and to the parallels KFK and UGU, which bound the strip, correspond the parts KFK and UGU of the M-axis, each regarded as a double line (flat hyperbola). Again, if we draw parallels to the z-axis across the strip, to each of these will correspond one of the halves of an ellipse belonging to a confocal system having F and G for common foci. Thus to BRDSC and BRDSC equidistant from the z-axis correspond the two halves BRDSC and BKDSC of the same ellipse whose semi-axes are cosh y and sinh y. In particular, to FPAQG on the x-axis itself corresponds the double line (flat ellipse) FPAQG. Thus, to the whole of the first parallel strip between KOK and UU corresponds uniquely the whole of the w-plane. Hence, if we confine ourselves to this strip,...

108 pages, Paperback

Published January 1, 2012