Transactions of the American Mathematical Society Volume 2 (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. 1901 Excerpt: ...be transformed into projective groups of the plane, a result Lie obtains after determining the canonical forms of the primitive groups, f This fact can, however, be established from the general properties of such groups, and its use leads to a new determination of these primitive groups, to show which is the object of this paper. A primitive group will be defined as a group which does not leave invariant a differential equation of the first order. J Such a group is at least three-parametric, as a two-parametric group possesses a differential invariant of the first order, J say, and therefore an invariant differential equation of the first order, f (J) = constant. 1 It will first be necessary to show that any group in two variables and of more than two parameters leaves invariant at least one differential equation, which is integral and algebraic in the derivatives dy/dx, (Pyjdx1, etc., involved. Let Xf = dfjdx + n df.dy indicate any one of the infinitesimal transformations of the group. If we "extend" these n times, and write yl, y2, yn for dy/dx, (Py/dx2, dny/dx," and p, q, px, p2, pn for dfjdx, dfjdy, dfjdy, Sfjdy., dfjdy, respectively, we easily find these extended transformations to be of the form 6- + Vq + lVx + ix)yY-+ + + (PnVn + Qn)P- Presented to the Society April 27, 1901. Received for publication Maroh 3, 1901. t S. Lie: Vorlesangen fiber continuierliche Qruppen, herausgegeben von Dr. Scheffkrs, p. 359. This book will hereafter be designated Continuierliche Gruppen. % Invariant differential equation denned in Continuierliche Grupptn, pp. 213-214, and in Differentialgleichungen mit bekannten infinitesimalen Transformationen, by S. Lie, p. 277. Continuierliche Gruppen, p. 228. i Continuierliche Gr...