  We investigate the spectral structure of the Neumann-Poincar\'e operator on
thin ellipsoids. Two types of thin ellipsoids are considered: long prolate
ellipsoids and flat oblate ellipsoids. We show that the totality of eigenvalues
of the Neumann-Poincar\'e operators on a sequence of the prolate spheroids is
densely distributed in the interval [0,1/2] as their eccentricities tend to 1,
namely, as they become longer. We then prove that eigenvalues of the
Neumann-Poincar\'e operators on the oblate ellipsoids are densely distributed
in the interval [-1/2, 1/2] as the ellipsoids become flatter. In particular,
this shows that even if there are at most finitely many negative eigenvalues on
the oblate ellipsoids, more and more negative eigenvalues appear as the
ellipsoids become flatter. We also show a similar spectral property for flat
three dimensional domains.
