Transactions of the American Mathematical Society Volume 23-24

Transactions of the American Mathematical Society Volume 23-24

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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. 1922 edition. Excerpt: ...canonical form is given by (67) and (68) in which case it is identically self-dual and belongs to a linear complex; or else it is an harmonic congruence whose canonical form is given by (65), (66) and (69). 6. THE PROPERTIES AND CANONICAL FORM or CONGRUENCES WHICH POSSESS I II II I COINCIDENCES (dldz) AND (d, d, ) We consider finally the congruences which have the two coincidences (d 101;') and (d: d;). The conditions to be satisfied are ' E. J. Wilczynski, Brussels Paper, p. 28 and p. 43. If we multiply (75) by 5 and subtract (71) from the result, we find Using this relation, the six conditions reduce to the four given by (71), (72) and the two following: But e = 1/5 enters in such a way into equations (71), (72), (78) and (79) that the same set of equations results from using k =--1 as from using k = 1. So there is no loss of generality in assuming 12 = 1. Use the transformation group (3) to make (84) /= const., 0 = d' = 0; these relations will be left unaltered by the sub-group of (3) given by K _ C1 _ _ C2 _, #--I where 01 and C2 are arbitrary constants. From (84) and (82), it follows that 6 =--Z =--centsucents1; then by using transformations (3) we can choose a and B so as to make (85) d = m, These equations show that the developables of the congruence intercept isothermally conjugate _nets on both sheets of the focal surface. Subtracting (80) from (81), and using (11) and (85), one finds By a similar computation based on the two equations corresponding to (80) and (81) one obtains If we differentiate the second equation of (90) with respect to v, and the first equation with respect to u, and subtract one resulting equation from the other, we obtain If we perform the differentiations indicated in (91), and useshow more

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  • Paperback | 224 pages
  • 189 x 246 x 12mm | 408g
  • United States
  • English
  • black & white illustrations
  • 1236737652
  • 9781236737656