A theorem on order relations generated by totally positive kernels

This paper is a result of reflections on a theorem on 'sums of hyperbolae'$^1$ needed in mathematical geophysics. This theorem and several unexpected corollaries of it do not seem very plausible at first sight. The first proof of the theorem did not dispel this impression: the reasons why it was true remained obscure. The explanation was found in the properties of the Cauchy kernel $C(s,x)=1/(s+x)$. In this paper the original theorem on 'hyperbolae' is established as a particular case of a general result holding for a certain class $\mathbb G$ of totally positive kernels which contains $C(s,x)$.