Schönflies’ simple definition of the nature of the structure is that every molecule is surrounded by the rest collectively in like manner, when likeness may be either identity or mirror-image resemblance. Von Fedorow finds the extra 165 types to be comprised in “double systems” consisting of two “analogous systems” which are the mirror images of each other. Barlow proceeds to find in how many ways the two mirror-image forms can be combined together, there being in general three distinct modes of duplication, including the insertion of one inside the other. He also shows that all homologous points in a structure of the type of one of these additional 165 point-systems together form one of the 65 Sohnckian point-systems, the structure being capable of the same rotations or translations, technically known as “coincidence movements” (movements which bring the structure to exhibit the same appearance as at first), as those which are characteristic of that point-system. This fascinating subject of mirror-image symmetry, and the optical activity connected with it, will be reverted to and the latter explained in Chapter XI. We have thus seen how satisfactorily the geometrical theory of the homogeneous partitioning of space has been worked out, and how admirably it agrees with our preliminary supposition that a crystal is a homogeneous structure. The fact that the 230 homogeneous point-systems all fall into and distribute themselves among the thirty-two classes of crystals, the symmetry of which has also now been fully established, affords undeniable proof that as regards this branch of the subject something like finality and clearness of vision has now been arrived at. CHAPTER X LAW OF VARIATION OF ANGLES IN ISOMORPHOUS SERIES. RELATIVE DIMENSIONS OF UNIT CELLS. FIXITY OF ATOMS IN CRYSTAL. We are now in a position to approach the conclusion of the long controversy as to the constancy or otherwise of crystal angles in the cases of greatest similarity, those of isomorphous substances, and to appreciate how the conflicting views of Haüy and Mitscherlich and their schools of thought have at length been reconciled. As the result of a comprehensive study, on the part of the author, of the sulphates and selenates of the rhombic series R_{2}S/SeO_{4}, and of the double sulphates and selenates of the monoclinic series R_{2}M(S/SeO_{4})_{2}.6H_{2}O, in which R represents the alkali metals, potassium, rubidium and cæsium, and in which M may be magnesium, zinc, iron, nickel, cobalt, manganese, copper or cadmium, four facts of prime significance have been definitely established.