The angle KHMi is the angle of rotation required, and from the spherical triangle KHMi we have
T^-uAi cos MiK - cos HK . cos HMi cos KHMi = : ==== . sin HK . sin
(substituting from (a) and reducing) ; where iU 2 = (AD +
= 2r 2 2 + (n + r 2 )2 + 2 J'2r, (n + r 2 ) 2 - 27? ; for AD =
(This method may be readily extended to groups of spheres, in which two of radius r% are each in contact with g spheres of radius r\, whose centres form a regular polygon ; and s spheres of radius r are placed in contact to form a regular polygon of s solid angles.)
For the arrangement to be possible, we must have DU < ?2, i.e., <s/(n + r. 2 )' 2 - 2?y 2 < r 2 , i.e., r 2 < i\.
It is of interest to consider the possible effects of change of tem- perature upon the volume of such a configuration as we have described. In discussing this subject in the case of silver iodide, it was suggested that the volume of the atoms of iodine possessed a higher coefficient of expansion than that of the atoms of silver ; from this analogy and that of sulphur and selenium as compared with copper, we may plausibly suppose that the oxygen of cuprite has a larger coefficient than the copper. If this view be admitted as having some likelihood, important consequences follow from it ; to examine these let us consider the effects of a rise of temperature upon the configuration ; the first result will be an expansion of atomic volumes, but since according to supposition, the atoms of oxygen enlarge at a higher rate than those of copper, the disparity in size of the two kinds of atoms will be increased, but if the centres of the oxygen atoms be regarded as fixed in space, this will bring about an increase in the divergence of the atoms of copper from another, and thus an increase in the length of the diagonal of the copper square. So far all these changes are in the direction of ex- pansion, but it has next to be remembered that an increase in the length of the copper diagonal does not necessarily involve an increase in the total volume of the configuration, because this diagonal may rotate in a direction which leads towards the configuration of the limiting case 2, which is that of minimum volume.
Expressed crudely it may be said that room is furnished for the increased volume of the oxygen by utilising the space which lies in the
