With the classical theory used as a guide, general conditions are derived for a linear field theory in order that it meet the requirements of quantum theory and relativity theory. If one does not demand that the mass be a c-number, the spin is required to be the sum of two commuting operators, one of which describes the spin in the constant mass case. Three expressions for the mass operator consistent with the general requirements are obtained, and these are shown to lead to a new description of particles of spin 0, 1/2, 1, 3/2, 2. These expressions are equivalent to adding extra terms to the Dirac or Kemmer-Duffin equations. Apart from numerical constants, the rest-energy eigenvalues, charges, spins and decay mechanisms of these particles are unambiguous consequences of the theory. (Author).