Publisher's Synopsis
This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1901 edition. Excerpt: ...in 407, (8.). For if b07, b', b% be the roots of the former cubio, and e0, c et the roots of the latter, inspection of those equations shows at once that we have generally, LXXV...-p = J ' ] V + V-2/'Saa' = /(, + 0, +?2 + Saa'); or LXXVI... o? = V = ' + V + = V + Ci + flJ = &o., where the semiaxes ff, in c% belong to the three confocals through any proposed point p. Making theD, LXXVII... a02 = a 6,1 = 0, c = cJ-&, we recover the expression assigned above, for the square of the length of an umbilicar semidiameter of an ellipsoid. (29.) For any central surface, the prinoiple (27.) shows that if X, p be, as in 405, (5.), &c, the two real cyclic normals, and if g be the real scalar associated with them as before, then the vectors of the four real umbilics (if such exist) must admit of being thus expressed: LXXVIII... tfr'X: /F = abc (/UX + /TA); LXXIX... fn: yFfi = abc (gUp + XT/u); and thus we see anew, that an hyperbohid 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 vectors (23.), of which only one at most can be real, with the particular signs there given, but whioh have the forms of lines in the three principal planes, must be conceived, in virtue of their expressions LV. LVI. LVIL, to terminate on an imaginary right line, of which the veotor equation is, -a(e'+l) /--ri + Soa') ce'-) i Jaaa... p =-----, ---/-----t--+ T; e' being a scalar variable, which receives the three values, -Saa', + 1, and-1, when p..."
