572 Scientific Proceedings, Royal Dublin Society. 
Notwithstanding this, however, we must look for a congruen 
extension of the assemblage in both directions from the nucleus, 
because the irregularity and incompatibility which would other- 
wise be found at the junction of the solid and fluid portions would, 
at any rate so long as the number of added layers is few, be more 
prejudicial to closeness of packing at this boundary than the modi- 
fication imposed by the difference of curvature referred to; indeed 
we reach the important conclusion that if the curvature is infini- 
tesimal as compared with the distances between contiguous centres, a 
large number of layers may be added before the departure from the 
most favourable curvature becomes sufficient to prevent congruent 
accretion. Transition is thus possible from molecular thinness 
to microscopic or even macroscopic dimensions. 
Now a slight straightening of the nucleus will, it is evident, 
favour closest-packing of layers added on the concave side by 
approximating their conditions to those of the favourably arranged 
nucleus in regard both to the distribution of the centres and the 
curvature of the layers. Such a straightening will also favour 
closest-packing of layers added on the convex side, so far as distri- 
bution of the centres is concerned, but not in regard to the 
curvature of the layers, the latter becoming still further removed 
from the most favourable curvature. It is clear, therefore, that 
in many cases layers added on either side of the bent nucleus will, 
in striving after closest-packing, exercise some force tending to 
straighten it, and that as fresh layers are added, this will increas- 
ingly be the case. And if the solidified nucleus yields to some 
extent to the strain thus put upon it without being ruptured, its 
original curvature will be flattened. And further, if more layers 
are added at one part than at another we shall have the alteration 
of curvature different at different places. 
If, for example, the number of layers added is greater as we 
pass from one end to the other of a band-shaped assemblage whose 
most favourable curvature for closest-packing is a cylindrical bend- 
ing about an axis transverse to the band, it is evident that the 
band will take the shape of a watch-spring spiral. 
If, when a certain amount of bending has taken place, the: 
solidified portion of the assemblage breaks rather than bend any 
more, it is evident that the gradual growth of an assemblage will 
