One-parameter families and nets of plane curves

Abstract

iff. r " it"', " a c + o c +ci = ac+oc-f-cu.Now every non-degenerate net of plane curves determines a unique system of form (4), for which the conditions (5) must of course be satisfied.On the other hand every system of form (4), for which the integrability conditions (5) are (5) 1911] * It is easy to show that an absolute semi-covariant must be a homogeneous function of degree zero of y, yu and y".Any such function C may be written as a function of the two ratios p/y, a/y and of the coefficients of (4) ; 0-/(j,J... I......).But, as in the argument for seminvariants, this reduoes to C=f(?,°y,A,B,C,..), whence follows easily the theorem as stated.* Neither of these functions can vanish identically, since A" and H must be different from zero.Otherwise the transformed net would be degenerate.

Locations

• Transactions of the American Mathematical Society - View - PDF