694 Proceedings of Royal Society of Edinburgh. [sess. 
taken from Bravais’ doctrine of homogeneous assemblages, which 
we may look upon as the grammar of molecular construction. 
Space-Periodic Partitioning (§§ 3-13). 
§ 3. Given a homogeneous assemblage of points : let it be required 
to partition all space accordingly. The thing to be done is concisely 
defined in the second sentence of § 6 below. 
§4. The problem is clearly indeterminate. Here is a solution 
which has obvious relation to Brewster’s kaleidoscope and the 
corresponding doctrine of electric images, and which may be import- 
ant in respect to Yortex Theory for a crystal or ether. From P, a 
point of the given assemblage, draw a line, PH, of any length in any 
direction, provided only that H is not a point of the assemblage of 
P’s. Do the same relatively to every other of the P-assemblage. 
We thus have a homogeneous assemblage of double points, PH. 
Let Q be any point in space, and let S denote summation for all 
the PH’s. Let <£( D) be a function which decreases as D increases 
from 0 to oo . The equation 
2ft(QP)-«KQN)] = 0, 
expresses a locus for Q which partitions space periodically, and 
divides each periodic portion into two cells containing respectively 
an H and a P. Every cell containing an H is a parallel pervert 
(footnote on § 45a below) of every cell containing a P. That this 
is true we see by drawing any straight line to equal distances in 
opposite directions through the point midway between H and P. 
Its ends are similarly related, one of them to all the H’s ; the other 
to all the P’s. 
§ 5. Here is a perfectly general solution. Around any one of the 
points P describe a closed surface S, of which the greatest distance 
from P is less than that of P’s nearest neighbour. Describe an 
equal, homochirally similar, and same- ways oriented surface around 
every other point P. Hone of these surfaces cuts or touches any 
other. Expand all of them simultaneously, equally, and without 
altering shape or orientation, till one of them touches another. All 
corresponding pairs of the surfaces touch simultaneously at corre- 
sponding points. Continue the expansion, annulling in each case 
the mutually enclosed portions of the expanding surfaces, and sub- 
