if at 0° under a pressure _p_ = 1 (for example, under the atmospheric pressure), the volume (for instance, a litre) of a gas or vapour he taken as 1, and therefore _v_ and _b_ be expressed by the same units as _p_ and _a_. The deviations from both the laws of Mariotte and Gay-Lussac are expressed by the above equation. Thus, for hydrogen _a_ must be taken as infinitely small, and _b_ = 0·0009, judging by the data for 1,000 and 2,500 metres pressure (Note 28). For other permanent gases, for which (Note 28) I showed (about 1870) from Regnault's and Natterer's data, a decrement of _pv_, followed by an increment, which was confirmed (about 1880) by fresh determinations made by Amagat, this phenomena may be expressed in definite magnitudes of _a_ and _b_ (although van der Waals' formula is not applicable in the case of very small pressures) with sufficient accuracy for contemporary requirements. It is evident that van der Waals' formula can also express the difference of the coefficients of expansion of gases with a change of pressure, and according to the methods of determination (Note 26). Besides this, van der Waals' formula shows that at temperatures above 273(8_a_/27_b_-1) only one actual volume (gaseous) is possible, whilst at lower temperatures, by varying the pressure, three different volumes--liquid, gaseous, and partly liquid, partly saturated-vaporous--are possible. It is evident that the above temperature is the absolute boiling point--that is (_tc_) = 273(8_a_/27_b_-1). It is found under the condition that all three possible volumes (the three roots of van der Waals' cubic equation) are then similar and equal (_vc_ = 3_b_). The pressure in this case (_pc_) = _a_/(27_b_^2). These ratios between the constants _a_ and _b_ and the conditions of _critical state_--_i.e._ (_tc_) and (_pc_)--give the possibility of determining the one magnitude from the other. Thus for ether (Note 29), (_tc_) = 193°, (_tp_) = 40, hence _a_ = 0·0307, _b_ = 0·00533, and (_vc_) = 0·016. That mass of ether which at a pressure of one atmosphere at 0° occupies one volume--for instance, a litre--occupies, according to the above-mentioned condition, this critical volume. And as the density of the vapour of ether compared with hydrogen = 37, and a litre of hydrogen at 0° and under the atmospheric pressure weighs 0·0896 gram, then a litre of ether vapour weighs 3·32 grams; therefore, in a critical state (at 193° and 40 atmospheres) 3·32 grams occupy 0·016 litre, or 16 c.c.; therefore 1 gram occupies a volume of about 5 c.c., and the weight of 1 c.c. of ether will then be 0·21. According to the investigations of Ramsay and Young (1887),