On a generalization of Synge’s criterion for sufficient stability of plane parallel flows

say 0 ^ X g 1.Let qt = <p{ (X), = < (X), (i = 1, 2, ■ • • , n), where <p{ , i/\ have continuous derivatives on A and (<, <) traces out a simple closed curve c, as X traverses A. This induces the relationships Q, = <£, (X), P{ = SF, (X).Because the transformation is canonical, ($, (X), ^ (X)) also traces out a simple closed curve C, in the (Q, , P<) plane as X traverses A, andNow the mistake is made of using Stokes' theorem J"cxdy = J/x dx dy to complete the "proof".But for this formula to hold, C must be positively oriented.Now in the case at hand and c, have the orientations which are given to them by the mapping from A, and these are not necessarily all the same (examine the Example).We could use Stokes' theorem in the form JJ dqt dpi = (-1)" £ Pi dqt and jj dQ{ dPi = (-1)" (J) Pi dQ< , S i C i iS % C i where