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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. 1905 Excerpt: ...djdv = vp. The coefficients have geometrical interpretations, for if iv is the intercept on the normal by the plane x = 0, N-, . = x/lvi = i (X-a)/(v--a). Hence p2-(tf, + N2 + N3-Xir) p + NN-l = 0. If /(X) = (X-a)(X-6)(X-c), /' (X)// (X) = (X-a)1 + (X-6)' + (X-c)S the differential equation of lines of curvature can be written in the form dX2-(X-v) f (v)/f(v) + 3-X/v ddv + dvf (X)/f(v) = 0. This has been integrated when f is quadratic but not when, as in the general case, / is cubic. 6-c e--a a-6' showing that the intersections of the normal with the planes of reference and perpendicular central plane form a range of constant cross ratio. The line element is given by., " dX2 abcdu? 4ds' =-f--As in the general Kummer surface, the asymptotic curves have an envelope consisting of the singular conics, and a cusp locus which reduces to isolated nodes. Hence the elliptic and hyperbolic regions of the Wave surface are separated by the circles of contact of the four tropes and the four nodes. It is easy to see on which side of these boundaries the curves lie and that the hyperbolic regions consist of four detached portions, each bounded by one circle and one node (fig. 13, p. 104). A complete asymptotic curve consists of four branches, one in each portion, and each branch touches the circle at one point and has a cusp at the node. There are two elliptic regions, namely the entire inner sheet, and the outer sheet bounded by the four circles. 68. Painvin's Complex. The quadratic complex of which the wave surface is the singular surface is Painvin's complex of lines through which the tangent planes to a quadric are at right angles (cf. the generation of the harmonic complex, p. 97, 58). Let the quadric be /a + y//3 + /7=l. then the complex, with curre...
| ISBN-13 | 9781153111966 |
|---|---|
| ISBN-10 | 1153111969 |
| Weight | 0.42 Pounds |
| Dimensions | 9.00 x 6.00 x 0.29 In |
| List Price | $19.99 |
| Format | Paperback |
|---|---|
| Pages | 122 pages |
| Publisher | |
| Published On | 2010-01-01 |
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