The theory of the lamellar eutectoid reaction by volume diffusion has been extended to account for solute segregation within the product phases and the effect of dilute third element additions. It has been demonstrated for symmetric binary systems that the segregation can account for 10% or more of the free energy stored in the product phases and can lead correspondingly to a predicted lamellar spacings appreciably greater than those obtained when segregation is neglected. This segregation is relatively high in the systems Cu-In, Cu-Be and Ag-Cd and may account for the fact that a secondary coarse-grained pearlitic reaction follows the initial fine grained one in these systems. The binary theory has been-used to analyse the available data for the eutectoid reaction in the Cu-Al and Fe-C systems and satisfactory agreement is obtained. The theory for ternary systems, while complete in principle, proves to be. intractable in all but the simplest version of the solution thermodynamics. It is concluded that the effect of the third element on the binary eutectoid reaction is mainly through its effect on the phase diagram, the ternary cross effects in the diffusion matrix tending to cancel out. Any additional element which lowers the eutectoid temperature will retard the reaction. / Thesis / Doctor of Philosophy (PhD)