(4) Rantsheff endeavoured to classify the elements in their periodic relations by a system dependent on solid geometry. He communicated this mode of expression to the Russian Chemical Society, but his communication, which is apparently not void of interest, has not yet appeared in print. (5) _By algebraic formulæ_: for example, E. J. Mills (1886) endeavours to express all the atomic weights by the logarithmic function A = 15(_n_ - 0·9375_t_), in which the variables _n_ and _t_ are whole numbers. For instance, for oxygen _n_ = 2, _t_ = 1; hence A = 15·94; for antimony _n_ = 9, _t_ = 0; whence A = 120, and so on. _n_ varies from 1 to 16 and _t_ from 0 to 59. The analogues are hardly distinguishable by this method: thus for chlorine the magnitudes of _n_ and _t_ are 3 and 7; for bromine 6 and 6; for iodine 9 and 9; for potassium 3 and 14; for rubidium 6 and 18; for cæsium 9 and 20; but a certain regularity seems to be shown. (6) A more natural method of expressing the dependence of the properties of elements on their atomic weights is obtained by _trigonometrical functions_, because this dependence is periodic like the functions of trigonometrical lines, and therefore Ridberg in Sweden (Lund, 1885) and F. Flavitzky in Russia (Kazan, 1887) have adopted a similar method of expression, which must be considered as worthy of being worked out, although it does not express the absence of intermediate elements--for instance, between magnesium and aluminium, which is essentially the most important part of the matter. (7) The investigations of B. N. Tchitchérin (1888, _Journal of the Russian Physical and Chemical Society_) form the first effort in the latter direction. He carefully studied the alkali metals, and discovered the following simple relation between their atomic volumes: they can all be expressed by A(2 - 0·0428A_n_), where A is the atomic weight and _n_ = 1 for lithium and sodium, 4/8 for potassium, 3/8 for rubidium, and 2/8 for cæsium. If _n_ always = 1, then the volume of the atom would become zero at A = 46-2/3, and would reach its maximum when A = 23-1/3, and the density increases with the growth of A. In order to explain the variation of _n_, and the relation of the atomic weights of the alkali metals to those of the other elements, as also the atomicity itself, Tchitchérin supposes all atoms to be built up of a primary matter; he considers the relation of the central to the peripheric mass, and, guided by mechanical principles, deduces many of the properties of the atoms from the reaction of the internal and peripheric parts of each atom. This endeavour offers many interesting points, but it admits the