A Treatise on Analytical Geometry of Three Dimensions; Containing the Theory of Curve Surfaces and of Curves of Double Curvature

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. 1830 Excerpt: ...to the tangent plane at its extremity is x y z' m n 1 1 Tf+fy+7«zsa1' or-p+ py+ c-z =7-w The equation to a concentric sphere, radius (r), referred to the same axes is (Art. 2.) whichj by reduction, becomes (since as mz, y =nz') x y (m + n cos 7 + cos a)--+ (n + m cos y + cos p) + (1 + mcos a + wcos /3) =-7............(2). r z But since a principal diameter is perpendicular to the tangent plane at its extremity, this plane must coincide with a plane applied at the same point to a concentric sphere. Hence equations (1) and (2) are identical, wzr2 «r2...--73-= m + n Cos 7 + cos a,--= n + m cos 7 + cos p, (a' b' sin yf + (a c' sin a)2 + (b'c' sin/3)2 = (a6y + (ncy + (6c)'.................. (3) (a b'c'f l--2 costt cos/3 0057--cos2a--cos2/3--cos27 = (aftc)2................... (4) which are the three required results. Cor. I. If the conjugate diameters 2a', 26', 2c'3 are each equal to 2.R, then JJ = /-; also, since there are o only two equations viz. (3) and (4) to determine the angles of inclination a, /3, 7, of the conjugate diameters, therefore fliere may be an infinite number of systems of equal conjugate diameters; and their extremities all He in the intersection of the surface, and a concentric sphere, whose equation is Cor. 2. Of all systems of conjugate diameters of an ellipsoid, the principal diameters have their sum a minimum, and the equal diameters their sum a maximum. If possible, let there be a system, not mutually at right angles, whose sum is a minimum. Through any two, which are not at right angles, draw a plane; this will of course cut the surface in an ellipse, of which these two will be conjugate diameters, and their sum will be greater than that of the axes of the section; if therefore we join the two axes of the section to the remai...