1888 - 89 .] Sir W. Thomson on Constitution of Matter. 695 
stituting the portion of fixed surface traced, or left behind, by the 
expanding line of mutual intersection. This portion of surface we 
shall call (after my brother, Professor James Thomson) an interface. 
Follow the same rule when another, another, and another contact 
takes place. When the borders of two of the growing interfaces 
thus traced meet and begin to intersect, annul their projecting por- 
tions, so that the intersection and wdiat is left of the expansion of 
its previous border now constitute the boundary of the interface. 
Continue the process until fresh growing intersections of interfaces 
are formed, and the ends of these growing intersections meet, and at 
last nothing is left of the expanded original surfaces, and therefore 
nothing of space is left unenclosed by the cells — polyhedrons of 
interfaces — thus constructed. 
§ 6. The interfaces formed in § 5 are generally curved, but, as we 
shall see (§ 7), may be plane, and are so in particular cases of special 
interest. In every case each cell contains one, and only one, of the 
P’s ; there is no interstitial space between them ; they are all equal, 
homochirally similar, and con-orientational. 
§ 7. If the initiating surface, S, of § 5 is a polyhedron of plane 
facets, the periodic partition to which it leads is in polyhedrons of 
plane facets. So it is also if the initiating surface is any ellipsoid 
with P for centre. 
§ 8. Let S be a sphere. The partitional polyhedron, to which it 
leads, is the dodekahedron obtained by drawing planes through the 
middle points of the lines between P and its twelve next-neighbours, 
perpendicular to these lines. 
§ 9. If S is an ellipsoid similar to and con-orientational with that 
determined in § 47 below, the partitional polyhedron to which it 
leads is the rhomboidal dodekahedron to which the rhombic dodeka- 
hedron of § 21 below is converted by the homogeneous strain of 
§ 46. In this case the whole number of contacts of the expanding 
surfaces (§ 5) is twelve, and they all take place simultaneously. 
§ 10. If the assemblage becomes equilateral, the partitional dode- 
kahedrons of §§ 8, 9 become, each of them, the rhombic dodeka- 
hedron of § 21. 
§ 11. If S is an ellipsoid, having conjugate diameters along lines 
from P to other three points of the assemblage, and of magnitudes 
proportional to the distances from P to the nearest points in these 
