viii Transactions of the Royal Society of South Africa. 
The form taken by Binet and Cauchy's well-known expansion of the 
year 1812 is that of a sum of products of pairs of determinants : the form 
of the expansion now given is that of an aggregate of single determinants. 
The relation between the two is explained and a historical remark added. 
" Notes on Radiation of Crystals," by J. Steph. van der Lingen. 
(a) Eadiation patterns of the transformation of magnesium hydroxide 
to magnesium oxide. The patterns show that the reflecting planes of the 
crystal are disturbed when water is driven off. The patterns show that 
the " spots" become drawn out into radial lines, and that these radial lines 
reflect the intensity of the X-ray spectrum. These observations show that 
Friedrich's two-dimensional grating may only be attributed to crystals under 
definite physical conditions. 
(b) Diamond tests by radiation patterns. 
The following stones were examined : 
" Made," " spotted " stone, " spotted rejection " stone, and an " inferior 
brown block " with a spot in it. The patterns show that a " spot " in a 
stone causes a discontinuity in the intensity of individual spots of the 
patterns, and that a fracture of the lattice causes a discontinuity of the spots 
so that they now represent irregular markings on the plate. An ideal 
diamond's pattern shows a uniform intensity in all the spots. 
(c) Bultfontein Apophyllite — (1) ideal, and (2) showing a cleavage 
crack along a cleavage plane. 
The flaw causes the spots of the " flawed " crystal to present a nebular 
appearance, whereas the ideal stone shows a uniform distribution of intensity 
in the elliptic spots. This represents a case of discontinuity in the lattice 
normal to the incident rays. 
(d) Serpentine, malachite and pseudomorph quartz. 
Serpentine shows a regular " radial line " pattern symmetrical to a line 
parallel to the threads of the crystal. This indicates that serpentine is not 
tricliuic unless every specimen examined was a " twin." 
Malachite shows three " lines " parallel to the threads and some minor 
radial lines normal to the former deviating slightly from the normal. 
Crocidolite : A long exposure shows that it is micro-crystalline and that 
the elementary units have a tendency to favour a direction parallel to the 
threads. 
(e) A square plate of iodine showed, after an exposure of about an hour, 
diifraction, a phenomenon similar to that described by Prof. Laub of 
Buenos Ay res. 
In this case the plate shows diagonal lines of zero intensity. 
" A Summary of the Distribution of the Genera of South African 
Flowering Plants (with special reference to the Flora of the Uitenhage and 
Port Elizabeth divisions)," by S. Schonland. 
This is to a large extent based on published data checked and enlarged, 
