859 2(a/a)y' + (gla)y = 0 (6) where g is the gravitational constant. Claim: Assume that a(t) is a positive bounded above function. If a(t) satisfies one of the following conditions for all t > k 1) a' > 0 2) a' < 0 3) a' + r*a > 0, for some p > 1 then (6) is stable. Proof: Criteria 1 and 2 can be proved by Corollary 3 with M = 1, D = la 1 la, K = gla, and a(f) = max {-a'la, -4a'I a}. If a' > 0, then a(f) = -a'la. If a' ^ 0, then a(f) = -^a'la. This shows that /°°a (5) ds < c, which implies (6)

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... s stable. Criterion 3 can be obtained by Corollary 4 with A -0, M = 1, D + G = 2a'/a, # -£/0, and a(f) = r p , p > 1. The claim is then proved. Remark 7. Criteria 1 and 2 can also be seen in Hsu and Wu. 3 VI. Conclusion One stability criterion and two instability criteria for the firstorder linear time-varying system are given in this Note. These criteria are extensions and/or consolidations of the results in Refs. 3 and 4. These conditions, though not intuitive, can be checked easily for a given system. A general necessary and sufficient condition on stability is very difficult to derive. However, it is interesting to know if such conditions can be obtained in some forms of linear systems.