A method has been described for obtaining the density of air quickly without resort to measurement of the prevailing atmospheric conditions and with accuracy sufficient to enable the correction for the buoyancy of air on a weighing to be made to better than one part per million. The method involves the weighing in air in quick succession of two objects, a hollow cylinder of stainless steel (density 0.5 g./cm.('3)) and a solid counterweight of equal mass and area in the form of a stainless steel tube (density 8.11 g./cm.('3)). The densities of these objects have been obtained from the volume as determined by high-precision hydrostatic weighing and by measurements of the dimensions and from the mass as determined by weighing in an actual vacuum ("vacuum weighing bottle"). The density of air obtained by weighing this "absolute density cylinder" and counterweight has been compared to the density calculated from measurements of the atmosphere, made at the same time in a constant temperature room ((+OR-)0.075(DEGREES)) and with high-precision instruments, of the barometric pressure and temperature and gravimetric determination of water and carbon dioxide contents. The density of air determined by the cylinder was 3.4 parts per thousand greater than that calculated from the prevailing conditions. Possible explanations for the discrepancy have been examined with particular reference to the oxygen content of the atmosphere; the discrepancy is left unexplained but it is recommended that for the high-precision measurement of mass that the density of air be measured with a cylinder such as the one described;A method has been proposed for the high-precision determination of the oxygen content of the atmosphere;Modifications of the Mettler single-pan balances have been described to increase capacity and sensitivity and eliminate electrostatic effects;It has been proposed to replace the International Kilogram with a kilogram mass of a corrosion-resistant stainless steel the surface of which has been rendered nonpolar by chemical treatment and to subdivide the kilogram in an actual vacuum using a set of weights constructed on the basis of the Fibonacci sequence. Fabrication of such a set of weights has been started. A mathematical theory has been developed for the calibration of this set in which advantage is taken of the numerous possible cross checks on the experimental work afforded by the characteristics of the Fibonacci numbers.