  Correlation functions of the two-dimensional Ising model on the periodic
lattice can be expressed in terms of form factors - matrix elements of the spin
operator in the basis of common eigenstates of the transfer matrix and
translation operator. Free-fermion structure of the model implies that any
multiparticle form factor is given by the pfaffian of a matrix constructed from
the two-particle ones. Crossed two-particle form factors can be obtained by
inverting a block of the matrix of linear transformation induced on fermions by
the spin conjugation. We show that the corresponding matrix is of elliptic
Cauchy type and use this observation to solve the inversion problem explicitly.
Non-crossed two-particle form factors are then obtained using theta functional
interpolation formulas. This gives a new simple proof of the factorized
formulas for periodic Ising form factors, conjectured by A. Bugrij and one of
the authors.
