1888-89.] Sir W. Thomson on Constitution of Matter. 709 
each of these triangles a point of the assemblage is found. ISTo 
point of the assemblage is to he found elsewhere in the same plane. 
Fig. 1 shows a homogeneous distribution of points in a plane. In 
the diagram they are joined by lines, determinately chosen accord- 
ing to § (i), so that all the angles of triangles formed by them are 
acute. Closely related to this triangular arrangement are three 
others. One of these is obtained by omitting PQ and its parallels 
and taking instead the other diagonal OD of the parallelogram 
QOPD and drawing parallels to it through all the points. The two 
others are obtained similarly by omitting OQ and taking instead 
the other diagonal PE of the parallelogram QPOE; and by omitting 
OP and taking instead the other diagonal QF of the parallelogram 
PQOF. 
(g) All the points of the assemblage lie in equidistant planes 
parallel to the plane of (f); similarly placed at the angles of tri- 
angles equal, similar, and similarly oriented to the triangles of (/). 
The distance between each of these planes, and the next plane to it, 
is easily proved to be equal to the reciprocal of the product of twice 
the area of the triangle into the number of points per unit volume. 
vol. xvi. 23/11/89 2 z 
