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The behaviour of air and other gases at low densities is a subject which presents peculiar difficulties to the experimenter, and highly discrepant results have been arrived at as to the relations between density and pressure. While Mendeleef and Siljerström have announced considerable deviations from Boyle’s law, Amagat finds that law verified in the case of air to the full degree of accuracy that the observations admit of. In principle Amagat’s method is very simple. The reservoir consists mainly of two nearly equal bulbs, situated one above the other and con­nected by a comparatively narrow passage. By the rise of mercury from a mark below the lower bulb to another on the connecting passage, the volume is altered in a known ratio which is nearly that of 2:1. The corresponding pressures are read with a specially constructed differential manometer. Of this the lower part which penetrates the mercury of the cistern is single. Near the top it divides into a U, widening at the level of the surface of the mercury into tubes of 2 centims. diameter. Higher up again these tubes re-unite and by means of a three-way tap can be con­nected either with an air-pump or with the upper bulb. Suitable taps are provided by which the two branches can be isolated from one another. During the observa­tions one branch is vacuous and the other communicates with the enclosed gas, so that the difference of levels represents the pressure. This difference is measured by a cathetometer. It is evident that when the pressure is very low the principal difficulty relates to the measurement of this quantity, and that the errors to be feared in respect to volume and temperature are of little importance. Amagat, hilly alive to this aspect of the matter, took extraordinary pains with the manometer and with the cathetometer by which it was read. An insidious error may enter from the refraction of the walls of the tubes through which the mercury surfaces are seen. But after all his precautions Amagat found that he could not count upon anything less than 1/100 millim., even in the means of several readings. It may be well to give his exact words (p. 494):—“Dans les expériences dont je donnerai plus loin les résultats numériques, les déterminations sont faites en général en alternant cinq fois les lectures sur chaque menisque; les lectures étaient faites au demi-centième, et les divergences dans les séries régulières oscillent ordinairement entre un centième et un centième et demi; en prenant la moyenne, il ne faut pas compter sur plus d’un centième; et cela, bien entendu, sans tenir compte des causes d’erreur indépendantes de la lecture cathétometrique . . . . . . Les résultats numériques consignés aux Tableaux que je vais donner maintenant sont eux-mêmes la moyenne de plusieurs expériences; car, outre que les lectures out été faites en général cinq fois en alternant, on est toujours, après avoir réduit le volume à moitié, revenu au volume primitif, puis au volume moitié: chaque expérience a done été faite aux moins deux fois, et sou vent trois et quatre.”

IV. On a new manometer, and on the law of the pressure of gases between 1⋅5 and 0⋅01 millimetres of mercury