A c c u R A T E and internally consistent thermodynamic property tables are necessary for efficient design and anyalysis of modem engineering and scientific equipment. To construct these tables it is often convenient to fit experimental data with functional relations which accurately represent the data and are compatible with theory. Although other methods may be used for the determination of thermodynamic properties, the use of such functional relations is becoming common because of the ease with which complicated expressions may be evaluated on digital computers. The simultaneous representation of various thermodynamic data, such as P-p-T, specific heats, virial coefficients, Joule-Thomson coefficients, velocity of sound, etc., is also becoming important to the determination of more accurate and thermodynamically consistent property representation. The ability to constrain a function readily to a state point or to a certain behaviour at that point is a desirable feature in the fitting of thermodynamic equations to data.
The object of this paper is to describe mathematical methods by which the above features can be incorporated into a least squares fit. For completeness and continuity, the development of the normal least squares relations are included.
Least Squares without Constraints
Suppose N experimental data points Yn, X.I, X.2 ..... Xng (n = I, 2 ..... IV) of the true physical relation ~(21, 2'2 .... ZK), have been obtained. The dependent variable is denoted by ~ and the independent variables by Zl, X2 ..... Zx. However, in many instances the distinction between dependent and independent variables has no physical significance. We wish to approximate this