  We consider three fluid phases meeting at a line of common contact and study
the linear excesses per unit length of the contact line (the linear adsorptions
Lambda_i) of the fluid's components. In any plane perpendicular to the contact
line, the locus of choices for the otherwise arbitrary location of that line
that makes one of the linear adsorptions, say Lambda_2, vanish, is a
rectangular hyperbola. Two of the adsorptions, Lambda_2 and Lambda_3, then both
vanish when the contact line is chosen to pass through any of the intersections
of the two corresponding hyperbolas Lambda_2 = 0 and Lambda_3 = 0. There may be
two or four such real intersections. It is required, and is confirmed by
numerical examples, that a certain expression containing \Lambda_{1(2,3)}, the
adsorption of component 1 in a frame of reference in which the adsorptions
Lambda_2 and Lambda_3 are both 0, is independent of which of the two or four
intersections of Lambda_2 = 0 and Lambda_3 = 0 is chosen for the location of
the contact line. That is not true of Lambda_{1(2,3)} by itself; while the
adsorptions and the line tension together satisfy a linear analog of the Gibbs
adsorption equation, there are additional, not previously anticipated terms in
the relation that are required by the line tension's invariance to the
arbitrary choice of location of the contact line. The presence of the
additional terms is confirmed and their origin clarified in a mean-field
density-functional model. The additional terms vanish at a wetting transition,
where one of the contact angles goes to 0.
