Sentence Examples

  • Under the general heading "Geometry" occur the subheadings "Foundations," with the topics principles of geometry, non-Euclidean geometries, hyperspace, methods of analytical geometry; "Elementary Geometry," with the topics planimetry, stereometry, trigonometry, descriptive geometry; "Geometry of Conics and Quadrics," with the implied topics; "Algebraic Curves and Surfaces of Degree higher than the Second," with the implied topics; "Transformations and General Methods for Algebraic Configurations," with the topics collineation, duality, transformations, correspondence, groups of points on algebraic curves and surfaces, genus of curves and surfaces, enumerative geometry, connexes, complexes, congruences, higher elements in space, algebraic configurations in hyperspace; "Infinitesimal Geometry: applications of Differential and Integral Calculus to Geometry," with the topics kinematic geometry, curvature, rectification and quadrature, special transcendental curves and surfaces; "Differential Geometry: applications of Differential Equations to Geometry," with the topics curves on surfaces, minimal surfaces, surfaces determined by differential properties, conformal and other representation of surfaces on others, deformation of surfaces, orthogonal and isothermic surfaces.
  • Such a determinant is of importance in the theory of orthogonal substitution.
  • We can eliminate the quantities S l, E2, ��� In and obtain n relations AbXi = (2B 11 - Ab)'�k1 +2B21x2+2B31x3+���, AbX2 = 2B12x1+ (2B22 - Ab) x2 +2B32x3+..., and from these another equivalent set Abx1 = (2B11 - X1 +2B12X2+2B13X3+���, Abx2 = 2B21X1+(2B22 - Ab)X2+2B23X3+���, and now writing 2Bii - Ab 2Bik - aii, Ob = aik, Ob we have a transformation which is orthogonal, because EX 2 = Ex2 and the elements aii, a ik are functions of the 2n(n- I) independent quantities b.
  • We may therefore form an orthogonal transformation in association with every skew determinant which has its leading diagonal elements unity, for the Zn(n-I) quantities b are clearly arbitrary.
  • Similarly, for the order 3, we take 1 v Ab= -v 1 A =1 +x2 + 1, 2 + � - A 1 and the adjoint is 1+A v +A� -� +Av -v +A� 1+11 2 A +/-tv pt+AvA +�v 1 +1,2 leading to the orthogonal substitution Abx1 = (1 +A 2 - / 22 - v 2) X l +2(v+A�)X2 +2(/1 +Av)X3 1bx2 = 2(A� - v)Xl+(1 +�2 - A2 - v2)X2 / +2(Fiv+A)X3 Abx3 = 2(Av +�)X1 +2(/lv-A)X2+(1+v2-A2- (12)X3.

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