the Molecule and Chemical Combination. 515 



arrangement of 8 electrons. This as Foppl (Proceedings of 

 the International Congress of Mathematics, Cambridge 1912, 

 vol. xi. p. 188) has shown is an arrangement when, the 

 8 electrons are arranged in two sets of 4 in parallel planes, 

 the electrons in each set being at the corners of a square, 

 one square being twisted through an angle of 45° relative to 

 the other so that the projection of the electrons on a plane 

 parallel to either square is a regular octagon. This arrange- 

 ment for 8 electrons shown in fig. 1 is analogous to that 

 for 4 or 6, for the tetrahedron may be regarded as two 



Fisr. 1. 



sets of 2 at right angles to each other and to the line joining 

 •their middle points; the projections of the electrons on a 

 plane at right angles to this line are at the corners of a 

 square, while the octahedron may be regarded as two sets 

 of 3 points, each set forming an equilateral triangle; the 

 planes of these triangles are parallel and the triangles are 

 twisted relatively to each other so that the projection of the 

 electrons on a plane parallel to either triangle is a regular 

 hexagon. Whereas all the plane faces of a cube are four- 

 sided, the twisted polyhedron has 8 triangular faces as well 

 as 2 four-sided ones, thus two such polyhedra could be placed 

 ■so as to have either 2, 3, or 4 corners in common. 



The stable arrangement for 5 electrons is when three are at 

 the corners of an equilateral triangle with its centre at the 

 central charge and the other two are at equal distances 

 on opposite sides of this triangle. With 7 electrons five are 

 at the corners of a pentagon whose plane passes through the 

 central charge and the other two are at equal distances 

 on opposite sides of this plane. 



The point in the tables to which I wish to direct special 

 attention is that, whether the electrons are arranged in one 

 plane or distributed over the surface of a sphere, whenever 

 the number of electrons exceeds a limit, which may vary 

 with the law of force, the positive charge required to keep 

 them in stable equilibrium will exceed the sum of the negative 

 charges on the electrons. Thus with the law of force just 

 assumed, if there are more than 5 electrons in the ring or 



