Let V denote the volume of the instrument immersed (i.e. of liquid displaced) when the surface of the liquid in which the hydrometer floats coincides with the lowest division of the scale, A the area of the transverse section of the stem, l the length of a scale division, n the number of divisions on the stem, and W the weight of the instrument. Suppose the successive divisions of the scale to be numbered 0, 1, 2 ... n starting with the lowest, and let w0, W1, w2 ... w_n be the weights of unit volume of the liquids in which the hydrometer sinks to the divisions 0, 1, 2 ... n respectively. Then, by the principle of Archimedes, W = Vw0; or w0 = W/V. Also W = (V + lA)w1; or w1 = W/(V + lA), w_p = W/(V + plA), and w_n= W/(V + nlA), or the densities of the several liquids vary inversely as the respective volumes of the instrument immersed in them; and, since the divisions of the scale correspond to equal increments of volume immersed, it follows that the densities of the several liquids in which the instrument sinks to the successive divisions form a harmonic series. If V = NlA then N expresses the ratio of the volume of the instrument up to the zero of the scale to that of one of the scale-divisions. If we suppose the lower part of the instrument replaced by a uniform bar of the same sectional area as the stem and of volume V, the indications of the instrument will be in no respect altered, and the bottom of the bar will be at a distance of N scale-divisions below the zero of the scale. In this case we have w_p = W/(N + p)lA; or the density of the liquid varies inversely as N + p, that is, as the whole number of scale-divisions between the bottom of the tube and the plane of flotation. If we wish the successive divisions of the scale to correspond to equal increments in the density of the corresponding liquids, then the volumes of the instrument, measured up to the successive divisions of the scale, must form a series in harmonical progression, the lengths of the divisions increasing as we go up the stem.