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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. 1909 edition. Excerpt: ...of a point in space in terms of the parameters of the three quadrics which pass through the point. These parameters are called the elliptic coordinates of the point. It is evident that to each set of these Kirchhoff, Mechanik, p. 203. Leipsic, 1877. coordinates there correspond eight points in space, one in each of the eight compartments bounded by the coordinate planes. If one of the parameters mf in (8) be made constant, and the others up ut, where t k, be allowed to vary, these equations define in parametric form the surface, also defined by equation (1), in which u has this constant value ur The parametric curves Uj= const., mt= const. are the curves of intersection of the given quadric and the double system of quadrics corresponding to the parameters u and ut. If we put (9) a'1--ut = a, b2--M.= 6, c--M, = C u, --ui=u, t--u. = v, the equation of the surface becomes (10) Moreover, the quadrics which cut (10) in the parametric curves have the equations: 96. Fundamental quantities for central quadrics. By direct cal culation we find from (11), C '-L /( ) /(f) (14) where for the sake of brevity we have put (15) fe) = ia-e)(b-e)c-e). We derive also the following: (16) and (17) uv /( ) Since F and Z' are zero, the parametric curves are lines of curvature. And since the change of parameters (9) did not change the parametric curves, we have the theorem: The quadrics of a confocal system cut one another along lines of curvature, and the three surfaces through a point cut one another orthogonally at the point. This result is illustrated by fig. 22. From (14) and (17) we have (18) i =-J?, 1 =-J. Hence the ellipsoid and hyperboloid of two sheets have positive curvature at all points, whereas the curvature is negative at all points of...

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