Conditions which preclude the existence of critical solutions of an ordinary differential system

where yn?+ andf,?1 (x, yn) are by definition yi and fi(x, y,) respectively. By a critical solution of (S) through a point P: (xo, ylo, * * *, y.o) in R* is meant a solution W(x): w1(x), * * *, w,n(x) which passes through P and is such that for any other solution Y(x): y1(x), * * *, yn(x) which passes through P and exists in R* with W(x) over an interval I: (xo, xo +a), (a' > 0), either wk(x) 1 by E. Kamke [2] and further extended jointly by W. M. Whyburn and the author [3 ]. The system (S) is a special case of more general systems considered by these writers and it will have critical solutions if it satisfies the conditions imposed by them. In the following it will be shown that under other conditions it is impossible for (S) to have critical solutions. Some related facts will also be established. By the phrase "solution to the right of P" shall be meant a solution existing over an interval x0 0. Similarly, "solution to the left of P" shall designate a solution existing over xo -a 0. Throughout the ensuing discussion whenever the subscript i appears it is to be understood that i ranges over the set (1, * .. , n). The theory which shall be developed will be based on the prevalence