1888 - 89 .] Sir W. Thomson on Constitution of Matter. 711 
eight of those tetrahedrons ; of which the twelve edges meeting in 
it lie in the lines of six rows of points which intersect in that 
point. 
(?) Best conditioned Tetrahedronal Grouping. No Obtuse Angles. 
— Instead of choosing our first two points and our first triangle at 
random, take any point 0 and its nearest neighbour on either side, 
P ; and its next-nearest neighbour Q on the side making the angle 
QOP acute. The two other angles of this triangle are obviously, as 
Eravais remarks, acute. The only other way of thus finding best 
conditioned triangles is by taking O’s other nearest neighbour, P', 
and its other next-nearest, Q'. The triangles Q'OP' and QOP are 
equal, homochirally similar, and oppositely oriented ; and thus we 
find the only other possible best conditioned triangular grouping. 
Every other triangle of the points in the same plane, having none 
of the points within its area, has, as Bravais remarks, an obtuse 
angle. Consider now the nearest parallel plane of points on one 
side of the plane of QOP. Let E and its congeners ( • black dots) 
be the orthogonal projections of its points on the plane of QOP. 
Let R' and its congeners ( ° white dots) be the projections of the 
points of the nearest parallel plane on the other side of QOP. 
These projections will be situated relatively to the triangle P'OQ' 
and its congeners as are the former projections ( • black dots) 
relatively to the triangle QOP. 
R being, of the projections on the plane of POQ of all the points 
of the two parallel planes, the one which lies within the area of the 
triangle QOP, we have in OPQR a best conditioned tetrahedronal 
grouping. OP'Q'R' is another and the only other best conditioned 
tetrahedronal grouping. It is a parallel pervert of OPQR [ see foot- 
note on § 45 a above]. Hence a homogeneous assemblage of single 
points is essentially free from monochiral anti-symmetry ; or it is 
dichirally symmetrical. 
(j) The tetrahedron found by taking, with 0, P, Q, any other 
point than R in the plane through it parallel to QOP, has an obtuse 
angle along one, or obtuse angles along two, of its three edges, OP, 
PQ, QO : and so with 0, P', Q', and any other point than R' in the 
other parallel plane. 
