Accurate solutions to the Graetz equation and to the similar equation for flow between two parallel plates are presented including the first ten or eleven eigenvalues and important derivatives. The first six eigenfunctions are also presented at intervals of 0.05 from y = 0 to y = 1. For one directional flow in a flat duct (between two parallel plates) the corresponding equation isThe corresponding equations for mass transfer by diffusion are usually written with concentration or partial pressure in place of temperature, and dsusivity in place of thermal diffusivity.The first published solution to Equation (3) was presented by Graetz (13, 14). The assumptions and boundary conditions made by Graetz are constant thermal diffusivity, constant tube wall temperature, temperature symmetrical about the axis, uniform temperature at tube inlet, Wly developed parabolic velocity profile at tube inlet, and negligible conduction in direction of flow.These assumptions correspond to the following boundary conditions:

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Equations for excess free energy and activity coefficients of binary, ternary, and quaternary liquid systems have been presented by several authors (1 to 7). These include the van Laar and Margules equations for binary systems extending to the five-suffix terms and for ternary systems extending to the four-suffix terms ( 6 ) . An extension of the four-suffix Margules equation to quaternary systems has also been made ( 3 , 4 ) , and a generalization of this equation to multicomponent systems is given here.The general expansion of the Margules equation to include two-, three-, and four-suffix terms and applied to a system containing n components results in which is simplified to n n-1 n -2

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