For the generalisation to apply absolutely it is essential that the interchangeable elements shall belong strictly to the same family group of the periodic classification of Mendeleéff. Potassium, rubidium, and cæsium fulfil this condition absolutely, and so the law of progression of the crystal properties with the atomic weight of the interchangeable elements applies rigidly to their salts. Now there are two bases, the metal thallium and the complex radicle group ammonium NH_{4}, which are not thus related to the group of three alkali metals just mentioned, but which are yet capable of replacing those metals isomorphously in their crystals without more change of angle or of structural constants than is provoked by the replacement of potassium by cæsium; and often indeed the amount of change has been singularly like the lesser amount observed when rubidium has been interchanged for potassium. But although this is so, the directions of the changes are irregular, being sometimes the same as when rubidium or cæsium is introduced, and sometimes contrariwise, and in the case of thallium there are also striking optical differences, the thallium salts being exceptionally highly refractive. Still, morphologically the ammonium and thallium salts may legitimately be included in the same isomorphous series with the salts of potassium, rubidium, and cæsium, and a somewhat wider interpretation has to be given to the term “isomorphism” in order to admit these cases. To distinguish the inner group formed by family analogues, that is, the more exclusive group obeying the law of progression according to the atomic weight, the term “eutropic” is employed. Thus the “isomorphous series” of rhombic sulphates, selenates, permanganates, and perchlorates, and the monoclinic series of double sulphates and double selenates, comprise the potassium, rubidium, cæsium, thallium and ammonium salts and double salts of sulphuric, selenic, permanganic, and perchloric acids, while the inner more exclusive “eutropic series,” following the law absolutely, comprises in each case only the salts containing the family analogues, potassium, rubidium, and cæsium. In this beautiful manner has the controversy between the schools of Haüy and Mitscherlich now been settled, the interesting law described in this chapter having definitely laid down the true nature and limitations of isomorphism, while at the same time absolutely proving as a law of nature the constancy and specific character of the crystal angles of every definitely chemically constituted substance. CHAPTER XI THE EXPLANATION OF POLYMORPHISM, AND THE RELATION BETWEEN ENANTIOMORPHISM AND OPTICAL ACTIVITY.