Kansas University Quarterly Volume 10

Kansas University Quarterly Volume 10

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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: ...that heading. If the transformation in the plane p is identical, the collineations are of type VII, and form a three-parameter group Ga(p). Dualistically, if the two-dimensional transformation through A is identical, the collineations are of type VII in space and form a three-parameter group of type VII, G8(A). Thus G6(Apl) contains a dualistic pair of three-parameter subgroups of type VII. If the one-dimensional transformations along the lin l and in the pencil of planes through l are both identical, the collineations are of type XII. Of the ooG collineations in Ge(Apl), oc4 satisfy these two conditions, and hence this group contains cx4 collineations of type XII. These constitute a four-parameter group of type XII. The constitution of this group will be discussed in the proper place in this series of papers. Theorem 3. The group Go(Apl) contains two five-parameter subgroups of type XIII, two three-parameter subgroups of type VII, and one four-parameter subgroup of type XII. B.--Analytic Verification. 1. The Six-parameter Group G6(Apl). Analytic expression for T.--Along the line l and in the pencil of planes through l the collineation T produces one-dimensional parabolic transformations whose constants we shall designate by mt and nt, respectively. Let h, k and g be three other constants determining T. Let the tetrahedron of reference (ABCD) be taken so that B is on l, C in the plane p, and D anywhere in space. The plane p is now the plane z=0; y = 0 passes through l; x = 0 passes through A; and w=0 is not specially related to the invariant figure. The collineation T is expressed by the following"equations: fr = 7 + nt. (!) ff=T+t7+-+ K (2) TT = 7 + mt7 + ( + ht)7 + T3 + (--2--H2 + g-(3) These equations may be thrown into the...show more

Product details

  • 12-17
  • Paperback | 90 pages
  • 189 x 246 x 5mm | 177g
  • Rarebooksclub.com
  • United States
  • English
  • black & white illustrations
  • 1236971442
  • 9781236971449