which is a special case of Lagrange's linear equation, and is easily solved as follows.
(a) Write down the corresponding subsidiary equations of La grange, namely, (b) Find any two independent solutions of these equations. The two simplest and most obvious solutions of (58) are those given in (Io) above, which are alternative forms of the adiabatic equation, and are also solutions of (57). (c) To find the most general solu tion, including all other possible solutions, make one of these ex pressions an arbitrary function of the other. This will be the most general form of characteristic equation consistent with (6) and (9). The most convenient form for most practical purposes, giv ing V explicitly as a function of P and T, is as follows, which is expressed in words by stating that P(V—b)/T must be constant along any adiabatic represented by P /Tn+i = constant. Thus (59) includes all possible forms of characteristic equation consistent with the adiabatic found experimentally for steam, and with the expression (6) for the intrinsic energy, which was as sumed as the basis of the equations first proposed for steam by the writer (Proc. R.S. 1900, p. 269). At that time none of the experimental evidence available, except that for the adiabatic, extended much beyond 200 lb. pressure and 200° C, and the state of knowledge did not justify going further than the first approxi mation represented by equation (42), in which the arbitrary function F was assumed to be of the simple form R/a—cP/T. This proved to be a very good approximation and amply suffi cient at moderate pressures or high superheats, but it appeared that higher powers of cP/T would be required at higher pressures, and that no equation of this type could represent the accepted theory of the critical state, as represented by the van der Waals equation.
While retaining the fundamental assumption (6) it would evi dently be possible to construct an equation of the van der Waals type, giving P as a cubic function of .r/(V —b), by replacing in (59) by i/(V —b)T., as follows, „..
but in the absence of accurate experimental data it was impossible to predict that this would be more satisfactory than (59) in terms of P, whereas it would certainly be much less convenient for practical calculations.
It has recently been found possible (Proc. R.S. Sept. 1928), to extend the experimental range for water and steam to 400° C and 4,000 lb. pressure, including the whole of the critical region. Results obtained for water, by the steady flow method described in the article CALORIMETRY, verify the thermodynamic equation there given for the total heat with extreme accuracy up to the critical point. Those for steam disagree materially on several fundamental points with the accepted theory of the critical state, and appear to show that an equation of the type (59) is capable of representing the critical phenomena with much greater accu racy than any equation of the van der Waals type. Since the points in question are of primary importance with respect to the relations between the liquid and vapour states, they are further discussed in the article VAPORIZATION, though they also afford a good illustration of the application of the laws of thermodynamics to experimental research.
As it would be impossible within the limits of this article to illustrate or explain adequately the applications which have been made of the principles of thermodynamics, it has been necessary to select such illustrations only as are required for reference in other articles, or could not be found elsewhere. For fuller details and explanations of the elements of the subject, the reader must refer to general treatises, such as Ewing's Thermodynamics for Engineers (Cambridge, 192o), Birtwistle's The Principles of Thermodynamics (2nd ed., Cambridge, 1927) or Preston's Theory of Heat (5th ed., 1928). One or two chapters on the subject are also generally included in treatises on the steam engine or other heat engines, such as those of Rankine, Ewing, or Perry. Of greater interest, especially from a historical point of view, are the original papers of Joule, Kelvin and Rankine. A more elaborate treatment of the subject will be found in many foreign treatises, such as those of Clausius, Zeuner, Duhem, Bertrand, Planck and others. (H. L. C.)