The mixture is dissolved up, and allowed to crystallise; the crystals are filtered off, the filtrate concentrated, and a second crop obtained; this is repeated until five or six crops of crystals have been obtained. These, with the mother-liquor, constitute series A. The first fraction is now recrystallised; it yields a crop of crystals, fraction 1 of series B, and a mother-liquor, which is added to fraction 2 of series A, as indicated by the dotted arrow and circle; on recrystallisation of this mixture, a crop of crystals, fraction 2 of series B, is obtained, together with a mother-liquor, which is recrystallised with fraction 3 of series A. In this way, by continued repetition, series are obtained, of which each contains one fraction more than its predecessor; the least soluble constituent is thus concentrated in the fractions represented on the left of the diagram, whilst the most soluble accumulates in the mother-liquors. After a greater or smaller number of series have been traversed, according to the differences in solubility, the end fractions in each series will be pure. These are no longer fractionated, and the number of fractions in each series begins to diminish, as shown on the diagram. The middle fractions will contain the compounds of intermediate solubility; these may be separated by further fractionation on the same lines, or may perhaps be better treated by a different or modified process. In a modification of the method, each fraction of series A is recrystallised separately, yielding a crop of crystals, and a mother-liquor; series B is then built up by adding to the crystals from fraction 2 the mother-liquor from fraction 1, to the crystals from fraction 3 the mother-liquor from fraction 2, and so on; the fractions in this series are then recrystallised separately, and the third series built up by the similar combination of the crystals and mother-liquors. Similar systematic methods of procedure must be adopted in working out any method of fractional separation; it can at once be seen that where, as in the rare earth group, only small variations in properties exist, much time and care must be expended, if pure products are required. Since the development of the methods of spectrum analysis, the difficulty of testing the efficiency of a method of separation, and of examining the purity of the products obtained, has been greatly lessened. The only reliable test at the disposal of the earlier chemists was the determination of the equivalent weight, which still constitutes an important check on the modern methods. Some account of the methods available for the control of the methods of separation is essential in a general account of the rare earths; but before describing these, it will be convenient to give a short description of the methods used in the extraction of the elements from the rare earth minerals. EXTRACTION OF THE RARE EARTHS FROM MINERALS