622 Scientific Proceedings, Royal Dublin Society. 
frequency or constancy of occurrence are presented which are incom- } 
patible with fortuitousness. Where, for example, all or most of the © 
crystals of a substance which are found at a given spot are 
twinned it is evident that some definite predisposing cause, and — 
not mere accidental inability to attain a more stable equilibrium, 
must lie at the root of the phenomenon. 
That there are many instances in which the twinning of — 
crystals is an accidental occurrence may be inferred from the 
observations of Lehmann, which show that the greater the viscosity 
of a solution the greater is the branching (Verzweigung) of the 
crystallization, some of which leads to twin production. 
If the form of an assemblage which gives closest-packing has 
enantiomorphous symmetry, and consequently admits of two kinds, 
right-handed and left-handed, both of which are packed equally — 
close, it is evident that accidental twinning of these two kinds 
together may be produced in'the way suggested, by rapid change 
of state; and not only so, but that this kind of twinning is likely to 
occur wherever solidification of the mass originates at more than one 
nucleus. For this will stereotype arrangements of both kinds, if 
such happen to be present at the different centres, and will conse- 
quently prevent either arrangement from extending its boundaries 
to absorb the entire assemblage, however much the principle of 
closest-packing may favour such a proceeding. 
It may be remarked at this juncture that, except in one class 
of cases, the nature of the surface separating the individuals forming 
a twin assemblage is manifestly completely determined by two factors, 
(1) the relative situation of the nuclei, or centres, from which the 
individuals respectively commence to grow; (2) the relative rates 
of growth, at every moment, of juxtaposed faces, one of each indi- 
vidual, which meet in the separating surface. Any symmetrical 
arrangement of the juxtaposed faces, and consequently any 
symmetry of form of the separating surface will depend on the 
relative orientation of the individuals. 
The cases which furnish the exceptions, in which an additional 
factor comes in, are those in which two faces thus meeting are in 
the same plane. For it is evident that the separating surface can 
1 Zeitschr. fiir Kryst. I., p. 484. 
