Natural systems of trajectories generating families of Lamé

Abstract

(3) f2 +fy +S2 = F2 associated with (1). A set of sol surfaces of this sort will be termed a set of wave frollts or, more simpl)T, a wave set. The natural system being given, every surface determilles one of these wave sets. The phraseology employed is of eourse suggested by the optical interpretation of (1), ill which F(x, y, z) represents the index of refraction in a lloll-homogeneous isotropic mediula: the curves of the natural system ale then the possible paths of light; and a disturbance starting from any surfate will be propagated by means of a set of wave fronts. The question arises whether wave sets may be farnilies of Lame, that is, simply infillite families of surfaces which can form part of a triply-orthogonal system. Ordillarily this is not the case. The condition that (2) shall represent a Lame family is thatf satisfv the Darboux-Cayley equation of third order t