great number of physical investigations, but also in the province of chemistry, where instances of the passage of substances from a gaseous to a liquid state are so common, and where the very processes of dissociation, decomposition, and combination must be identified with a change of physical state of the participating substances, which has been elaborated by Gibbs, Lavenig, and others. For a _given quantity_ (weight, mass) _of a definite substance_, its state is expressed by three variables--volume _v_, pressure (elasticity, tension) _p_, and temperature _t_. Although the compressibility--[_i.e._, _d(v)_/_d(p)_]--of liquids is small, still it is clearly expressed, and varies not only with the nature of liquids but also with their pressure and temperature (at _tc_ the compressibility of liquids is very considerable). Although gases, according to Mariotte's law, with small variations of pressure, are uniformly compressed, nevertheless the dependence of their volume _v_ on _t_ and _p_ is very complex. This also applies to the coefficient of expansion [= _d(v)_/_d(t)_, or _d(p)_/_d(t)_], which also varies with _t_ and _p_, both for gases (_see_ Note 26), and for liquids (at _tc_ it is very considerable, and often exceeds that of gases, 0·00367). Hence, the _equation of condition_ must include three variables, _v_, _p_, and _t_. For a so-called perfect (ideal) gas, or for inconsiderable variations of density, the elementary expression _pv_ = _R_[Greek: a](1 + [Greek: a]_t_), or _pv_ = _R_(273 + _t_) should be accepted, where _R_ is a constant varying with the mass and nature of a gas, as expressing this dependence, because it includes in itself the laws of Gay-Lussac and Mariotte, for at a constant pressure the volume varies proportionally to 1 + [Greek: a]_t_, and when _t_ is constant the product of _tv_ is constant. In its simplest form the equation may be expressed thus: _pv_ = _RT_; where _T_ denotes what is termed the absolute temperature, or the ordinary temperature + 273--that is, _T_ = _t_ + 273. Starting from the supposition of the existence of an attraction or internal pressure (expressed by _a_) proportional to the square of the density (or inversely proportional to the square of the volume), and of the existence of a real volume or diminished length of path (expressed by _b_) for each gaseous molecule, van der Waals gives for gases the following more complex equation of condition:-- (_p_ + _a_/_v_^2)(_v_-_b_) = 1 + 0·00367_t_;