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which is continuous when x^<x<Xi. Let X(xo) be the least upper bound of values Xi for which the Riccati equation admits a continuous solution when #o<x<Xi . Then the inequality (1.4) holds when Xo<x^X(xo), and Petrov [2] has shown that if pi and pi are constants no better result is true. That is, if p\ and pi are constants and X(xo) is finite, then there exists a function v(x) satisfying (1.3) and (1.2) for which v(x) =y(x) a t a point arbitrarily close to but greater than X(xo). I t is the purpose of this paper to show that this last result is true without the restriction that pi and pi are constants. We prefer to state our results in terms of and make our proofs depend

The converse of a theorem of Tchaplygin on differential inequalities