Higher Geometry; An Introduction to Advanced Methods in Analytic Geometry (Paperback)

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. 1922 Excerpt: ... by aid of (2) and (3), is v + vdx3 + v3dxg + vtdxf = 0. (8) Consider now in order the previous cases. I. If xi satisfy a single equation (6), we have By comparison of (8) and (9) we have pf, .= r--, which shows CXi that vi are the coordinates of the tangent to = 0 at the point xi. II. If xi satisfy the two equations (7), we have A comparison with (8) gives pvi=s +--i which shows that CX" CXr vi passes through the line of intersection of the tangent planes to f1l = 0 and 2 = 0 and hence is tangent to the curve denned by the two surfaces. III. If the points xi are discrete points, we may say that each plane of the extent is tangent to the point, through which it passes, thus extending the use of the word "tangent" in a manner which will be useful later. Summing up, we say: A two-dimensional extent of planes consists of planes which are tangent either to a surface or to a curve or to a point. The theorem has reference, of course, only to the neighborhood of a plane of the extent. The entire extent may have the same nature throughout or different natures in different portions. 89. Change of coordinates. A tetrahedron of reference and a set of coordinates xi having been chosen, consider any four planes not meeting in a point the equations of which are Vl + V + V3 + A =, the coefficients being subject to the single condition that their determinant a shall not vanish. We assert that if we place px( = a, ., + a, A + aax2 + aitxt, (2) then x are the coordinates of the point xi referred to the tetrahedron formed by the four planes (1). The proof runs along the same lines as that of the corresponding theorem in the plane ( 29) and will accordingly not be given. It is also easy to show that by the same change of the tetrahedron of refer...