The industrial problem of the rectification of alcohols is based entirely upon the properties of volatile liquids, upon the laws of the maximum tensions of the vapors of these liquids, and upon the influence of temperature upon those different elements which find themselves in presence of each other in an alembic. If we desire to follow, in their least details, all the phenomena which succeed one another in a rectifying column, and which are connected with one another by a continuous chain of reciprocal influences, the problem becomes exceedingly complex. [Illustration: PICTET'S APPARATUS FOR THE RECTIFICATION OF ALCOHOL BY COLD.] In order that the new applications of the mechanical theory of heat may be readily understood, we shall divide this problem into a series of propositions, which we shall examine separately, and which collectively constitutes in its general features the methodical rectification of liquids. I. Knowing the maximum tensions of pure water and pure alcohol, can we calculate directly the tensions of the vapors of any mixture whatever of alcohol and water? Yes, we can calculate this tension by a general formula, provided we take into account the affinity of water for alcohol, which increases the value of the total latent heat of evaporation of the liquid. The results of the calculation are fully confirmed by experience. We thus establish the following laws: a. For any temperature whatever, the maximum tension of the vapors of a mixture of water and alcohol is always comprised between that of pure water and that of pure alcohol. b. The tension of the vapors of a mixture of water and alcohol approaches the tension of alcohol so much the nearer in proportion as the proof is higher; and, reciprocally, if water is in excess, the tension of the vapors approaches the tension of the vapors of water. c. The curves of the maximum tensions of vapors formed by all mixtures of alcohol and water are represented by the same general formula, one factor only of which is a function of the richness of the alcoholic solution. It results, then, from these laws that we may determine with the greatest exactness the richness of a solution containing alcohol and water, if we know the tension of the vapors that it gives off at a certain temperature. Such indications are confirmed by the centigrade alcoholmeter. We see likewise that, for these solutions of alcohol and water, the laws of Dalton are completely at fault, since the total pressure of the vapors is never equal to the sum of the tensions of the two liquids, water and alcohol. II. Being given a solution of water and alcohol, mixed in equal volumes, what will be the quality of the vapors emitted from it? In other terms, do the vapors which escape from a definite mixture of water and alcohol also contain volumes of vapor of water and alcohol in the same proportion as the liquids? We have discovered the following laws: