720 Proceedings of Royal Society of Edinburgh. [sess. 
shrinkage of the assemblage perpendicular to the plane HKK'H' 
implies elongation in lines parallel to FG, because the volume 
remains constant, and there is clearly neither elongation nor 
shrinkage perpendicular to the plane of the diagram. Now to fit 
the tactics of Baumhauer’s twinning by the knife, we must have 
no change of dimensions of the assemblage in the plane HKK'H'. 
Hence while the turning and shearing motions described above 
are taking place, there must be a continual elongation of the sub- 
stance of each oblate perpendicular to this plane, and shrinkage 
parallel to FG,* to just such an extent as to prevent the centre of 
each oblate from coming nearer to the plane HKK'H', but instead 
to cause all the centres to move in lines parallel to FG. The 
oblates are now no longer figures of revolution but are ellipsoids 
with three unequal axes : the shortest, vertical ; the longest, perpen- 
dicular to the plane of the diagram ; and the mean axis parallel to 
FG. To complete the process, proceed as follows : — 
§ 61. Turn the oblates farther on in the same direction (opposite to 
the motion of the hands of a watch, as that in which they were 
turned in § 60), and through the same angle; and while, in conse- 
quence, the assemblage of centres shears to the right, give to the 
substance of each oblate a gradual shrinkage perpendicular to the 
plane FIKK'H' and elongation parallel to the line FG, so as to 
cause the rightward shearing motion of the assemblage of centres 
to be still exactly parallel to the initial position of the line FG. 
The whole movement of which the first half has been described in 
§ 60, and the second half in § 61, constitutes exactly what is done 
in Baumhauer’s artificial twinning of an end portion of a prism of 
Iceland spar, by a knife applied at F, with its edge perpendicular 
to the plane of the diagram, and pressed against the edge FG of the 
obtuse angle between the two upper faces of the prism before and 
behind the plane of the diagram. 
* Perhaps the simplest way of looking at the affair is found by considering 
that the elliptic section of each ellipsoid in the plane HKK'H' must remain con- 
stant ; and so also must the horizontal and vertical axes of the elliptic section 
in the plane of the diagram. Hence, while the principal axes of the elliptic 
section turn in the manner described in §§ 60, 61, the ellipse itself must remain 
inscribed in a constant rectangle of vertical and horizontal sides in the plane of 
the diagram, while the third axis of the ellipsoid, which is perpendicular to the 
plane of the diagram, remains constant. 
