In a note at the end of his paper {\it Recherches sur les fractions
continues}, Stieltjes gave a necessary and sufficient condition when a
continued fraction is represented by a meromorphic function. This result is
related to the study of compact Jacobi matrices. We indicate how this notion
was developped and used since Stieltjes, with special attention to the results
by M. G. Krein. We also pay attention to the perturbation of a constant Jacobi
matrix by a compact Jacobi matrix, work which basically started with Blumenthal
in 1889 and which now is known as the theory for the class M(a,b)