The methods of the second class, which are processes of fractional crystallisation, depend on the differences in solubility which are observed in analogous compounds in passing from one member of the group to another. The value of these methods, as opposed to the methods depending on differences in basic strength, was clearly shown by Auer von Welsbach, who in 1885 succeeded in resolving Mosander’s ‘Didymium’ into two new elements, praseodymium and neodymium, by fractional crystallisation of the ammonium double nitrates; since that date, much attention has been devoted to the task of finding rare earth compounds which will lend themselves to such processes. The method is extremely laborious, and may involve several thousand recrystallisations, in consequence of the generally very slight differences of solubility, and the ease with which the rare earth compounds, being almost always isomorphous with one another, form mixed crystals. Whilst the method of fractional crystallisation has come into general use for the separation of one element from another only within the last thirty years, processes for the separation of the cerium group from the yttrium elements, depending on differences of solubility, have long been known and used. The most important of these, the double sulphate method, depends on the fact that the potassium double sulphates of the cerium metals are almost insoluble, whilst those of the terbium group are sparingly, and of the yttrium group readily soluble in a concentrated solution of potassium sulphate. The cerium elements may be thus completely removed from a solution of mixed salts by addition of a crust of potassium sulphate crystals, or of an hot concentrated solution of the same reagent. In other cases, _e.g._ in the double carbonate and double oxalate processes, separation is effected by taking advantage of the greater tendency to the formation of double salts possessed by the yttrium metals. In effecting a separation of closely related bodies by fractional processes, in which a large number of repetitions of the same operation are necessary, only the most careful and systematic procedure can avoid much waste of valuable material; in these processes, the object of the chemist is to obtain pure end fractions, whilst keeping the middle fractions as small as possible. One method of procedure generally adopted is illustrated in Fig. 7, which represents a fractional crystallisation of a mixture of four or five substances, α, β, ... φ; the separations being usually conducted in such a way that subgroups of three, four or five elements are first obtained, these being then further fractionated to obtain the pure elements. In the diagram, crops of crystals are represented by crosses, the mother-liquors by circles; for the sake of illustration, the process is made to appear as simple as possible. [Illustration: FIG. 7]