The previous eight papers in this series may be regarded as constituting the basic framework of the liquid theory developed from the fundamental postulates of the work. Once such a general framework of theory has been erected, practically every new development, no matter how minor it may be in itself, serves to corroborate or clarify some part of what has gone before, and in order to facilitate recognition of the manner in which the new information to be presented in the subsequent papers fits in with the previous findings it has seemed advisable to undertake a general review and appraisal of the results described in the preceding papers.

0rdinarily the investigator himself is not the beat person to appraise the results of an investigation, but the purpose of this particular appraisal is primarily explanatory rather than argumentative. The exposition of a new system of thought on any subject is usually difficult to follow, since it requires a certain amount of mental reorientation which is not easily accomplished, and it should therefore be helpful to have the author's own analysis of the work. On this basis the following are the salient features of the findings reported in this series of papers:

1. (a) The entire development is carried out on a theoretical basis and all mathematical expressions are derived from theory, consequently

(b) There are no unexplained terms and no arbitrary or adjustable constants in any of these expressions.

Every term and every constant has a definite physical meaning. Aside from the conversion constants which are required for the purpose of expressing the results in terms of whatever conventional system of units is being used, the only constants that enter into the equations relating the various properties are what we may call structural constants: relatively small integral or half-integral values which represent the actual numbers of the different kinds of physical units entering into the particular phenomena under consideration.

(c) These expressions contain no exponential functions other than simple 2nd, 3rd, and 4th power values, which are readily explained theoretically.

All properties which vary with distance, for example, follow the inverse square relation; the system contains no such things as inverse sixth or eighth power functions or terms with decimal exponents,

2. (a) All phases of the development are based on the __same__ general liquid theory, consequently

(b) The same general principles apply to all liquid properties.

Example: Paper I stated that the relation between temperature and any temperature-dependent property of a liquid molecule (a molecule which is individually at a temperature somewhere between the melting point and the critical temperature) is linear, Subsequent papers showed that this principle applies to volume, to viscosity, and to surface tension. The papers to follow will still further enlarge this list of properties for which the relation has been verified.

(c) The basic numerical values entering into the mathematical equations are identical.

Example: Wherever the liquid unit of temperature enters into the calculations, as it does frequently, this unit always has the same value, 510.20º C, irrespective of the nature of the other property involved in the relation under consideration.

(d) The equations representing different properties are closely related.

Example: Viscosity and surface tension are calculated by means of expressions which are essentially nothing more than modified forms of the same equation.

(e) Because of this close relation between the equations, the results obtained in each area serve as confirmation and support for the results obtained in related areas.

The close correlation between the calculated and experimental surface tension values, for instance, not only establishes the validity of the surface tension calculations but also supports the results obtained from the saw basic sources in the viscosity field, where direct experimental verification is more difficult because of the larger experimental uncertainties.

3. (a) The latitude for variability in the equations expressing the various relationships is very small, yet they produce results which agree with the experimental findings within the margin of uncertainty of the latter, consequently.

(b) There is a high degree of probability that these results are correct or nearly correct.

Since current scientific thinking runs along somewhat different lines, this point probably needs some elaboration. Present-day practice in developing mathematical expressions to represent physical properties is to start with some basic relation of a theoretical or semi-theoretical nature and then to modify this relation by additional terms and "adjustable constants" to secure better agreement with the experimental data. For instance the "equations of state" for gasses which have gone farther in this direction than any other physical expressions have as their foundation the general gas law PV = RT. To this base an increasing number and variety of additional terms and the constructors of the equations have added constants. The Beattie-Bridgeman equation has four adjustable constants the Benedict-Webb-Rubin equation has eight. The development of expressions to represent other physical properties is in general proceeding along similar lines.

If we regard the objective of this activity as the attainment of close agreement with the experimental values for the purpose of facilitating interpolation and extrapolation of the experimental results, the prevailing policy has been successful, since the correlation is usually increasingly better as the number of constants is increased. The fact that seems to be generally overlooked is that if the objective is to ascertain the correct relationships and numerical values, this program of adding more and more constants is definitely proceeding in the wrong direction. Every constant makes it easier for the equation to fit the experimental data, but in so doing it correspondingly decreases the probability that the products-the equation and the results it produces-are correct. This is an inescapable mathematical result of the increase in the number of possible variations of the experimental data, which will agree with the equation.

In order to make progress toward the correct answers it is essential to reduce rather than increase the adjustability of the equation, and the ultimate goal of a completely defined system cannot be reached until all terms and all constants that enter into the mathematical expression of the property in question are specifically determined by the basic structural constants of the molecules, and all latitude for adjustment is eliminated. As we move in this direction we must obviously keep the results of the calculations within the limits of experimental uncertainty, but so long as this requirement is met every additional restriction that can be placed on the quantities entering into the calculations increases the probability that the values obtained from these calculations are correct.

This is, of course, the hard road to follow. Unlike the conventional "more and better constants" approach, which follows a well-defined pattern and is practically certain to produce results of some kind most attempts to make progress toward the difficult goal of a more restrictive equation will inevitably end in nothing but frustration and disappointment. The preference for the currently popular easy route is therefore quite understandable) but here, as in so marry other lines of human endeavor, true forward progress can only be made in the hard way.

Let us then consider what has been accomplished in this liquid study in the way of progress toward the goal of a more completely defined set of equations; that is, equations in which the latitude for "adjustment" is minimized. The first and undoubtedly the most important of the steps that have been taken toward this goal was made possible by the previously mentioned theoretical deduction as to the linear relation between temperature and the temperature-dependent properties of the true liquid molecule. But the equations presented in these papers are not only limited by the requirement of linearity; each is still further restricted to a specific kind of a linear relation. Such linear expressions are defined by two values: the slope of the line and the temperature at which it intersects the zero axes. If either of these defining values can be related to some specific quantity or if the two can be related to each other, the range of variability is very greatly reduced and as long as the values calculated from these expressions still fall within the limits of the experimental uncertainty so that they __can__ be correct, the probability that they are correct is greatly increased. Limitations of this kind have been established in the preceding papers for each of the properties covered. The slope of the temperature volume relation is fixed by the unit temperature value, 510.20, the slope of the surface tension curve is a specific function of the molecular mass, while the slope of the fluidity curve is reciprocally related to the zero point temperature.

Although the general nature of the curves which express the relations between temperature and the liquid properties is thus established, a complete definition of each curve requires one more item of information: the coordinates of some point on the curve. When these can be unequivocally evaluated from some basic molecular constants such as the mass or the number of constituent atoms the ultimate objective will have been reached. In the meantime each additional restriction that can be placed on the values for the individual liquids is a step toward that objective.

The first move in this direction carried out during the present project was to develop a means of expressing the variable factor in each equation in a manner that would bring out the underlying cause of the variations between individual substances. In the case of viscosity, for example, it was found in the initial phase of the study that the slope of the fluidity-temperature curve is inversely proportional to the zero point temperature, To. This one value therefore defines the entire curve, but an intensive study of the To values applicable to different liquids failed to disclose any direct connection between the individual values and the basic constants of the molecular composition and structure. The scope of the investigation was therefore broadened and with the help of some deductions from the basic theory it was ultimately determined that the To values could be expressed in of the molecular mass plus a mass increment I. a quantity which did have characteristics that could be related to the molecular composition and structure.

Development of methods which would enable a direct determination of the values of I and __similar __quantities appearing in other equations has been found to be a difficult problem and considerable additional work will be necessary in this area. In the meantime, however, it has been possible to establish certain general characteristics of these values. and in this way the range of variability has been reduced to the point where evaluation by indirect means is feasible. The increment I applicable to fluidity has been found to be (1) an integer or zero, (2) a property of the individual atom, (3) related to and limited by the atomic weight, and (4) subject to modification in a systematic and regular way by the molecular structure, These items of information, each of which restricts the range of variability of the values produced by the viscosity equation and therefore represents some advance toward the objective of the work, have then been applied to a study of homologous series of compounds and some further restrictions of major importance have been derived from this study. In total these various items of information are sufficient to establish unique values for most liquids.

Every step in this long process of development which has been described is a move in the direction of restricting the amount of variability or possibility of "adjustment" and each step has therefore increased the probability that the mathematical expressions which have been developed are correct representations of the physical properties to which they correspond, and that the values obtained from the calculations based on these expressions are, within the degree of accuracy of the mathematical processes employed, the correct magnitudes of these properties.

4. (a) The liquid theory is merely one aspect of a broad general theory of the structure-of the universe, consequently

(b) The liquid equations are not only closely inter-related among themselves, as indicated in item 2. but are also closely related to similar expressions in non-liquid areas.

A striking example will be provided by the next paper in this series, which will present the results of a study of the effect of pressure on the melting point. Here it will be shown that the equation, which represents the pressure-melting point relation, is not merely similar to, but is identical with the equation previously developed for solid compressibility, which was described in a privately circulated paper that preceded this liquid series. We simply substitute T_{m} for the term V where it appears in the compressibility equation and change the numerical coefficient to conform to temperature units rather than volume units, and we then have the complete melting point equation. The values of the initial pressure applicable to the solid-liquid transition are somewhat different from those applicable to either the solid or the liquid, as might be expected, but they can be derived in a similar manner.

5. (a) Although the achievements of the new liquid theory, even in the incomplete form in which they are presented in the eight previous papers and summarized in the numbered statements preceding, add up to a very impressive total, they are actually the results of only a relatively small amount of development work applied to the basic assumptions: a single-handed effort almost infinitesimal in comparison with the collective amounts of time and energy that have been applied to the development of previous theories.

(b) It is therefore evident that the potentialities of the new theory are a long way from being fully exploited, and that further work toward development of the details of the theoretical structure can be expected to be very fruitful.

The three most serious obstacles that have hitherto stood in the way of a complete understanding of the physical and mathematical relationships between the various liquid properties are first, the large degree of uncertainty in many of the experimental values, second, the great variety of behavior exhibited by different liquids, and third, the existence of transitions, some evident, some not evident, in which the factors controlling the behavior of individual liquids are modified and the trend of the observed values is altered accordingly. Heretofore the only significant tests that could be applied to the experimental values were those of reproducibility and regularity. The currently accepted physical values listed in the handbooks and other reference volumes are generally considered to be reproducible but, as pointed out in previous papers, the variations between the results of competent investigators are normally much larger than each investigator's own estimates of uncertainty, and the possible margin of error is therefore considerably greater than is generally admitted. Furthermore, many measurements, particularly under extreme conditions where the theoretical significance is the greatest, have been made by only one method, which leaves us with an added degree of uncertainty as to the accuracy of the method itself. The criterion of regularity is based on the assumption that the values of the liquid properties change continuously and can therefore be represented by a-smooth curve (aside from definite transitions). This test enables its to eliminate minor irregularities in the measured values, but neither of the available criteria gives us any protection against systematic errors where only one method has been used or where results are available from only one investigator (as in much of Bridgman's work, for example), nor do they necessarily distinguish simple curves from compound curves containing one or more transitions.

As a result of the absence of any positive method of checking the accuracy of the experimental results the theorist has been practically at the mercy of the experimenter. In general it has been necessary for purposes of theory to assume the validity of the experimental work since the degree of correlation with the experimental values has been the only test which could be applied to theory. Furthermore, it has been necessary to assume, or at least it has in practice been assumed, that unless there is a discontinuity in some physical property or other definite evidence of a transition, the shape of the experimental curve indicates the true pattern of behavior of the liquid property under consideration: a very misleading assumption.

The findings of this work have in essence liberated the theorist from this complete dependence on the experimental values and, aside from their value as a forward base from which to make further advances in theory, their principal claim to merit rests upon this fact, which opens up a wide field of potential usefulness. It is __still__ necessary, at least for the present, to utilize the experimental results as the raw material to which the theoretical processes are applied, but it is no longer necessary to accept all experimental values as authentic or as accurate; we now have theoretical criteria which are capable of separating authentic and applicable values from erroneous and inapplicable results, and we have processes which are able to convert approximate and uncertain values into correct and accurate values.

In undertaking to apply the general liquid theories to a study of any specific property we must first identify the quantity to which the linear relation characteristic of the liquid applies. It is essential, for instance, to establish the fact that it is the fluidity rather than the viscosity which varies linearly with the temperature. We then compare the experimental data with this theoretical linear relation and invariably we find that a large proportion of the curves representing the values for different liquids are definitely linear or, if they approach the zero axis as in the case of fluidity, have a linear upper section and make the transition to zero at the lower end by way of a probability curve, as the fluidity or any other property of the liquid aggregate must do if the curve for each individual molecule, is linear. This experimentally verified linearity of a substantial proportion, usually the great majority, of the curves confirms the validity of the theoretical linear relation, since the probability that this could be a chance result is essentially zero.

Having established the linear relation as correct, we now go back to the nonconforming experimental results and examine the deviations from the theoretical linear curves. Aside from the minor deviations due to experimental errors, the most common situation that we find is that the experimental curve consists of two linear sections connected by a transition curve. Such second order transitions without discontinuities are often recognized in solids by collateral evidence of one kind or another, but rarely in liquids, and where they do exist in liquids without being recognized their effects are automatically absorbed into the adjustable constants of conventional equations, giving the behavior of the liquid properties involved an appearance of complexity which is not at all justified by the facts.

The foregoing procedure enables us to identify the basic elements of the curves, the linear sections, and to select the most accurate values of the points on these linear sections from the experimental data. All of this information is then studied to determine the nature and magnitude of the controlling factors, which define the exact position of each curve. In principle it should be possible to deduce the values of these factors directly from the fundamental theory, but at the present incomplete stage of the theoretical development this is not feasible. The theory does, however. give some definite clues to the correct answers and in the studies that have been made thus far it has always been possible to establish the factors for at least the majority of the organic liquids. The application of this procedure to the property of fluidity has already been described in a previous paragraph. Here we find that each curve is completely defined by the molecular mass plus an increment I, originating from increased effectiveness of some of the individual mass units under certain conditions. We further find that the value of this increment is constant for all of the members of each organic series beyond the first two or three. All that we need to know, therefore to calculate the viscosity of any normal paraffin (propane or beyond) at any temperature is that the value of I for the normal paraffin series is 6.

Again the probability principles confirm the validity of the conclusions which have been reached. The probability that mere chance would enable us to reproduce the experimental values of the viscosity of all of these normal paraffins by the use of a single integer in conjunction with the molecular mass, with close agreement in many cases and approximate agreement in practically all cases, is infinitesimal. We are therefore justified in concluding that the calculated values are essentially correct and that whatever divergence may exist between these and the experimental values is a measure of the experimental error.

When this procedure is repeated for several organic series and in each case the series value of the increment I is found to be integral, the same kind of reasoning justifies accepting this fact as a confirmation of the theoretical deduction that __all__ such values are integral. This finding, together with the previously established linearity of the curves, then eliminates the necessity of relying entirely on uncertain experimental values. As long as these experimental results are anywhere near the correct values, so that the controlling integral factors can be identified, the exact values can be computed.

It should be noted that the foregoing is possible only because all latitude for adjustment is eliminated. We cannot prove the validity of the conventional type of equation containing, one or more adjustable constants by probability mathematics, as an adjustable equation can be __made__ to fit any set of values, right or wrong, which conforms to the general pattern for which the equation was constructed. An equation, which is not adjustable, cannot be forced into a fit; it can agree with the experimental values within the margin of experimental uncertainty, only (1) by chance, or (2) because it correctly represents the physical facts. Under the circumstances which have been discussed it is possible to demonstrate that chance cannot be responsible.

Of course, a great many liquids are not members of any large and thoroughly studied homologous series comparable to the normal paraffins, and identification of the controlling factors for the various physical properties of these less gregarious liquids is a more difficult matter. There are, however, other useful relations similar to those within series, and there are also inter-relations between the factors applicable to different properties of the same liquid, from which considerable information can be extracted. Extension of the work to the additional liquids that can be reached by such means is going ahead as rapidly as the tine that can be allocated to this particular phase of the total project will permit, and steady forward progress is being made, not only in the organic division but also among the more complex liquid structures of the elements and inorganic compounds.

This rate of progress will no doubt be greatly accelerated as soon as the results of the original work are sufficiently well known to encourage other investigators to explore additional consequences of the new liquid theories. Some very broad fields, which have had to be excluded from this initial project in order to keep the work within reasonable limits, are wide open for attack by anyone who wishes to undertake the task. It should be relatively simple, for instance. to extend the relations developed for pure liquids to liquid mixtures. Solutions will present some more complicated problems, but the general principles involved are already fairly well defined-the concept of physical state as a property of the individual molecule is obviously the key to an understanding of the nature of solid-liquid solutions-and some very interesting and significant results can undoubtedly be attained in this field with a minimum of effort.

It should be understood, however, that the data, which it has been possible to include in the tabulations accompanying the text of the previous papers, does not by any means reflect the full amount of information already available. The surface tension values given in paper VIII, for example, are representative of the results obtained in detailed studies made on over 500 liquids. Similarly, the presentation of the mathematical relations developed for the various liquid properties has in most cases been confined to the "regular" liquids because of the necessity for economy of space, but the relations applicable to many of the compounds with special behavior characteristics, such as water and the alcohols, have been determined with the same accuracy as those of the regular liquids and are also available far publication. The case for the new theories will, of course, be strengthened considerably when it becomes possible to release this material in one way or another. It will be necessary, however, to limit the forthcoming papers in this present series to new subjects. leaving the more extended treatment of the previous subjects for future publication.