40. _Mechanical Causes of the Geometrical Ground-Forms._--The great variety of ground-forms exhibited by the Radiolaria is of special interest, since in most instances their causes admit of recognition, and since they are so intimately related to each other that even in the remaining cases the assumption that they have arisen by purely mechanical _causæ efficientes_ seems justified. In this respect the first rank is taken by statical conditions, especially the indifferent or stable equilibrium of the whole organism, which floats freely in the water. With regard to these fundamental statical relations, three principal groups of ground-forms may be distinguished, pantostatic, polystatic, and monostatic. 41. _Pantostatic Ground-Forms._--By pantostatic or indifferently stable ground-forms are meant those in which the centre of gravity coincides with the centre of the body, so that they are in equilibrium in any given position. Strictly speaking, the only form which possesses perfectly indifferent equilibrium is the sphere, that being the only truly homaxon and perfectly regular form. Nevertheless, in a somewhat wider sense many Polyaxonia, especially the endospherical polyhedra with very numerous sides, may be {xx}included in this category. Such indifferently stable bodies are found among the SPUMELLARIA in many #Collodaria# and #Sphæroidea#, as well as in the Astrolophida among the ACANTHARIA. On the contrary they are entirely wanting among the NASSELLARIA and PHÆODARIA, since their central capsule constantly presents a main axis with a differentiated basal pole, and determines the position of stable equilibrium. 42. _Polystatic Ground-Forms._--Those ground forms are defined as polystatic or multistable in which the body is in equilibrium in several different positions (though not in an infinite number). The number of these positions is usually twice as many as that of the constant equal isopolar axes exhibited by the form. Hence the regular polyhedra have as many positions of equilibrium as they have angles or sides, the icosahedron twenty, dodecahedron twelve, octahedron eight, cube six, tetrahedron four. The isopolar monaxon ground-forms (lens, ellipsoid, cylinder) and the diplopyramidal ground forms (quadrilonchial and lentelliptical) have two positions of stable equilibrium, since the two poles of the vertical axis are equal and similar and the body is divided into equal halves by the equatorial plane. This is the case in many SPUMELLARIA (especially #Discoidea#, #Prunoidea#, and #Larcoidea#), as well as in the great majority of ACANTHARIA. Perhaps the same holds good also in certain NASSELLARIA (_e.g._, isopolar Tympanida) and PHÆODARIA (_e.g._, isopolar #Phæosphæria#), though here unistable equilibrium appears to be necessitated by the constant main axis of the central capsule and the differentiated basal pole of the main axis.