G is proportional to mass at constant composition (extensive) and G,is independent of mass at constant composition being dependent on composition only (intensive). By combining (11) and (14), MacDougall(6) derives the generalized Gihbs-Duhem equation in an alternate manner.
The ahove method is essentially that used by Guggenall texts do not heim' and by G i b b ~ . Admittedly, ~ present this method in a satisfactory form, hut such presentations are reflections upon the texts, and not upon the method. The prmf of Euler's theorem given ahove may be extended t o functions which are homogeneous of positive integral degree. If the degree of homogeneity is two, equation (1) still holds, but the f,'s are now homogeneous of degree one. Hence, using (3), fi
Literature Cited (1) GIBBS,J . W., "The Collected Worka of J. Willard Gihba." Vol. 1, Yale University Press, New Haven, Conn., 1948, n R7~ -. --( 2 ) GIBBS,J. W., ibid., p. 106. (3) GILMONT,R., "Thermodynamic Principles for Chemical Engineers," Prentice-Hall, Inc., Englewood Cliffs, N. J., 1959, p. 1 3 H 3 7 . (4) L ~ w r s , G . N., A N D RANDALL, M., "Thermodynamics," MeGraw-Hill Book Co.. Inc., New York, N. Y., 1923, p. 42. (5) MACD~UGALL, F. H.. "Thermodynamics sad Chemistry," John Wiley and Sans, Inc., New York, N. Y., 1939, p. 2.526. F . H., ibid., p. 28. ( 6 ) MACDOUGALL, ( 7 ) MILLER,F . H., "C&lculu~,"John Wiley and Sons, Inc., New York, N. Y., 1946, p. 153 exercise 27.
To the Editor: The total differential of any fimction f of r variables (xl, . . ., x.) i s df = I:fjdxi
where f,
= I: ( 5
wazi)zi
(4)
Introduction of (4) into (1) gives df =
r
(aji/az,)zidzi
[,J
This may he integrated as before, since the second partial derivatives df&rj are homogeneous of degree zero. The result is f ( z , , . . . ,&I = '/* I: (aj
