THE FIGURES OF EQUILIBRIUM OF A LIQUID MASS 287 



We may make an intcre.'^tinf? application of this principle. Among regular 

 polyhedrons, it has been seen that the tetrahedron, the cube and the octahedron 

 fl^rui^^h perfect laminar systems, and, in these three polyhedrons, the angle of 

 tAvo adjacent faces is, in efi'cct, less than 1209; there remain the dodecahedron 

 and the eikosahedron ; now, in the first of these the angle of two adjacent 

 faces is but 116° and a fraction, and consequently less than 120°, while in the 

 second it is 138° and a fraction ; it might be seen, therefore, that the dodecahe- 

 dron would give, like the preceding regular polyhedrons, a perfect laminar sys- 

 tem, but that such would not be the case with the eikosahedron, and this is cou- 

 lirmtd by experiment; with the eikosahedron, in whatever manner the frame be 

 withdrawn, none but plane films arc ever in nineteen of the faces, and the twen- 

 tieth is void. As to the system of the dodecahedron, there would be difficulty 

 in describing or drawing it. 



§ 39. In order to give more precision to these results, I would say : 1st. The 

 frame of any polyhedron in which the dihedral angles are inferior to 120° 

 gives a perfect laminar system ; such are the frames of the tetrahedron, the cube, 

 the octohcdron. the dodecahedron, the prisms whose luimber of lateral faces is 

 less than six, &c. There are, however, some rai"e exceptions, as that presented 

 by the frame of a quadrangular pyramid whose height does not much exceed, 

 the half of a side of the base. 2d. The frame of any polyhedron in which all 

 the dihedral angles are superior to 120° gives a laminar system null. Examples: 

 The frame of the regular eikosahedron, that of the assemblage of two hexagonal 

 pyramids, or one containing a still greater number of sides, united by their bases, 

 and such that at the edges of the commoir base the dihedral angles exceed 120°, 

 &c. Nevertheless, when all the dihedral angles but a little exceed 120°, we ob- 

 tain, in certain cases, a real laminar system ; this, for instance, occurs with the 

 frame of the polyhedron formed by cutting down the summits of a cube by 

 equilateral sections which join one another, so that there shall only be triangu- 

 lar faces and square faces. But in this polyhedron all the dihedral angles are 

 but about 125° ; moreover, the real and symmetric laminar system is produced 

 with difficulty, and only when the frame h withdrawn by a triangular face ; 

 when withdrawn by a square face it always yields a system null. 3d. A frame 

 of which the dihedral angles are some inferior and other superior to 120°, but 

 in which the faces which comprise between them these last angles are identical, 

 disposed in the same manner, and form a continuous and recurring succession, 

 gives an imperfect laminar system or a laminar system null : for instance, the 

 system is imperfect in the frames of prisms the number of whose sides exceeds 

 six, in that of a hexagonal pyramid when withdrawn from, the liquid by its 

 summit, &c.; it is null in that of a quadrangular pyramid whose height is less than 

 half the side of the base, in that of an hexagonal pyramid when withdrawn from the 

 liquid by its base, &c. 4th. When 'the frame falls within neither of the pre- 

 ceding categories, the laminar system is sometimes perfect, sometimes imperfect, 

 sometimes null. For example, we have a perfect laminar system with the frame 

 of a prism having for its base a lozenge of which the two obtuse angles exceed 

 or equal 120° ; we have, as has been seen, an imperfect system with the frame 

 of an hexagonal prism sufficiently tall when it is withdrawn with the axis ver- 

 tical ; with a frame representing the assemblage of two pentagonal pyramids 

 united by their bases and sufiiciently tall for the dihedral angles, in each of 

 them, to be less than 120°, while the dihedial angles which have for edges 

 those of the common base are greater than 120°, we have a system null when 

 it is withdrawn with the axis vertical, and an imperfect system with the axis 

 horizontal, &c. 



§ 40. Let us indicate some other curious modifications of our laminar sys- 

 tems. In the first place, we know that in the system of the cubic frame (Fig. 

 18) the central quadrangular lamina is parallel to two opposite fiices of the 

 cube, but, because of the symmetry of the frame, it is evidently indifferent, as 



