  We analytically derive transition probabilities for four-neutrino
oscillations in matter. The time evolution operator giving the neutrino
oscillations is expressed by the finite sum of terms up to the third power of
the Hamiltonian in a matrix form, using the Cayley-Hamilton theorem. The result
of computation for the probabilities in some mass patterns tells us that it is
realistically difficult to observe the resonance between one of three active
neutrinos and the fourth (sterile) neutrino near the earth, even if the fourth
neutrino exists.
