Problems concerning characterization of eigenvalues of some linear and homogenous differential systems by the Pellew and Southwell method of conjugate eigenfunctions in the domain of hydrodynamic instability are discussed and a general mathematical framework described. In this general survey we look back on and rewrite this work almost in exactly the way it evolved out of a few naive looking calculations in hydrodynamic instability. We show in the process the close relationship that exists between mathematical analysis and its applications with due credit to intuition as the main source of mathematical activity.
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