300. Proceedings of the Royal Irish Academy. 
those by which they remain mutually connected. In fact, we have now 
presented a cell with four quadrate segments, mutually bonded toge- 
gether by the inner angles, the whole as yet forming but one common 
cavity, each of these four segments, except as to their being so united 
by one of their angles, offering all the same characteristics as the original 
single quadrate cell (fig. 4). A further degree of complexity seems now 
to arise. Taking origin from each of the emarginations at the centre of 
each of the sides of all the four quadrate segments, into which the 
original cell has become subdivided (as well from those bounded by the 
lines of incision as those external), a secondary incision takes place, 
proceeding likewise in a direction perpendicular to the sides, and pro- 
gressing until each of these too almost reaches the centre. Like the inci- 
sions of the primary cell, they also are somewhat wide, rounded below, 
and with a somewhat undulate outline, due to precisely the same circum- 
stance, that at the middle of all the sides of each and every of the new qua- 
drate segments (the inner as well as outer), a minute emargination pre- 
sents itself, and the angles being rounded, with a slight concavity between 
these and the central emargination. Pari passu with the formation of 
these further incisions of each of the now new quadrate segments of the 
segments of the original quadrate cell, they, too, in their turn become 
increased in size, so as to equal their predecessors, as these equalled the 
original cell, whilst they, likewise, remain bonded together by the inner 
angles, the incisions not proceeding so far as to effect a complete sepa- 
ration. There is thus now a highly complex form produced : sixteen 
quadrate segments have resulted from the formation of the primary and 
secondary sets of incisions, and these are combined into one tahlet by five 
points of mutual union ; that is to say, the single median point of union 
of the group of four primary segments, the result of the first segmentation 
of the original quadrate cell, and the four points of union of four sets of 
four segments of the primary segments, now having each attained the 
original dimensions. The original quadrate cell has now become a 
quadrate tablet of sixteen times its superficial dimensions, composed of 
sixteen compartments, divided off by deep incisions, three from each 
side of the tablet ; each of those proceeding from the middle of each side 
to the centre, is, of course, cruciform, whilst the two others, reach- 
ing only to the point of union of each of the secondary sets of seg- 
ments, are simple; but all the cavities of all the compartments of 
the tablet maintain, of course, a common intercommunication at the 
before-mentioned five points of junction (fig. 6.) Such an example, in- 
deed, as this calls to mind a door key of many wards ^ if so homely an 
illustration be allowable. 
Nor does this appear to ba all : judging form Reinsch's figures^* (see 
our figures 5 and 6, after Reinsch), and from just a single mention made 
in his explanation of the plate, a further apparently singular character- 
istic appertains to this most curiously complex form. It would appear 
Op, cit. i. ii. fig. I., k, m. 
