- It is obvious that both of the quantities ξ(x1, y1,) η(x1, y1,) cannot vanish identically, else we should have no transformation. We may assume then that ξ(x1, y1) does not vanish. It is clear then that cannot, in general, vanish; for if it should vanish identically for all values of x and y, φ(variant);(x, y, a) would be free from a and hence x would not be transformed at all, i. e. ξ(x1, y1) would be identically zero; but the latter is contrary to hypothesis.
- For example, a solution of the equation w(a)-1=0 would be such a value of a that would reduce the functions X and Y to functions of x1 and y1 alone.
- See Lie—Vorlesungen über Differentialgleichungen mit'bekannten infinitesimalen Transformationen, bearbeitet und herausgegeben von Dr. Georg Scheffers, Leipzig, 1891, pp. 38 et seq.
The American Mathematical Monthly
Volume 5, 1898 - Issue 1