The compressible flow corresponding to a line doublet
Thus F(p) vanishes at p = 1, 2, and 3 and changes sign at each of these zeros which proves that (1.5) has at least three periodic solutions for small e. Since (1.2) is assured by A3 > 0 in (1.9) it remains only to show that f(x) satisfies (1.0) and (1.1). Since the A,are already fixed and since As > 0 and A0 > 0 it follows easily that if C is large enough then Cx = A3x6 — A2x4 + A^x2 — An has only one negative root, which is near x = 0 and will be designated by —x[, and only one positive root, which is large and will be designated by x1 . Clearly f(x) now satisfies (1.0) and (1.1) and the example is complete. With polynomials of higher degree the existence of more periodic solutions can be established. (The argument of Serbin fails in that he does not recognize the restricted range of the monotonicity of <f>, in Lefschetz's notation, as stated by Lefschetz [1, p. 193] and to which Serbin refers.)