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. 1901 Excerpt: ...them as before, then the vectors of the four real umbilics (if such exist) must admit of being thus expressed: LXXVIII... + tfrA: F = abc (gJJX + MTA); LXXIX... 0: /Ffx = abc (gUp + Al; and thus we see anew, that an hyperboloid with one sheet has (as is well known) no real umbilic, because for that surface the product ...
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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. 1901 Excerpt: ...them as before, then the vectors of the four real umbilics (if such exist) must admit of being thus expressed: LXXVIII... + tfrA: F = abc (gJJX + MTA); LXXIX... 0: /Ffx = abc (gUp + Al; and thus we see anew, that an hyperboloid with one sheet has (as is well known) no real umbilic, because for that surface the product abc of the semiaxes is imaginary; or because it has no real tangent plane parallel to either of its two real planes of circular section. (30.) Of whatever species the surface may be, the three umbilicar rectors (23.), of whicli only one at most can be real, with the particular signs there given, but which have the forms of lines in the three principal planes, must be conceived, in virtue of their expressions LV. LVI. LVII., to terminate on an imaginary right line, of which the vector equation is, -a(e'+l) /--Tb(/ + Saa) c(e'-l) LXXX...p = i--j-L-y-1, + J /; a + a V aa a--a e being a scalar variable, which receives the three values, -Saa', + 1, and-1, when p comes to coincide with a, o, and u" respectively. And such an imaginary right line, which is easily proved to satisfy, for all values of the variable e', both the rectangular and the bifocal forms of the equation of the surface (e), or to be (in an imaginary sense) wholly contained upon that surface, may be called an Umbilicar Generatrix. (31.) There are in general eight such generatrices of any central surface of the seoond order, whereof each connects three umbilics, in the three principal planes, tico passing through each of the twelve umbilicar points (23.); and because en disappears from the square of the expression LXXX. for p, which square reduces itself to the following, LXXXI... P =-P2e' + e + Sao) =-6-21V, they may be said to be the eight generating ...
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