A2A Mr. A. Cayley on the Partitions of a Close. 
periments. When the solution obtained by employing a large 
quantity of hydrochloric acid and zine is diluted with about 
thrice its volume of water, it has a dirty yellowish colour, and 
for the moment is tolerably transparent ; in the course of a few 
minutes it becomes turbid and loses its transparency, owing to 
a finely-divided grecnish-grey precipitate which forms after a 
while ; on the addition of more water it is deposited, so that the 
liquor may be decanted. An assay with hydrochloric acid and 
tin shows this deposit to be hydrous dianic acid; for it is dis- 
solved to the characteristic sapphire-blue fluid when treated in 
the way so often referred to. If this blue solution is boiled for 
a few minutes with zine, the dianic acid also is thrown down 
with the tin; the precipitate is deposited with readiness in light 
grey flakes on the zine, which is coated with spongy tin, 
and the supernatant fluid is transparent and colourless, On 
filtering the portion containing the flakes and boiling the 
acid collected with hydrochloric ‘acid and tinfoil, the blue solu- 
tion is obtained again on the addition of a little water. ‘Thus 
the behaviour of the acid of the Tammela dianite and that of 
the acid of samarskite from the Iimen Mountains is strictly iden- 
tical. The behaviour of zine and tin therefore with reference 
to the solutions of dianic acid in question, is, to an extent not 
to be anticipitated, entirely different, it may be said antagonistic. 
Those who wish to repeat the investigations described would 
do well to adhere to the quantities mentioned by me, or to em- 
ploy them proportionally; for without this precaution it 1s possible 
that the properties may not be as distinctly brought out as they 
will be re Spa 8 to them. 
—_————_—_— ————— Wo ———S=S=————. 
LXIIL. On the Partitions ut Closes, iis A. hee LEY, Esq.* 
ie F, 8, E denote the number of faces, summits, and edges of 
a polyhedron, then, by Kuler’s w eH-known theorem, 
F+S=E+2 
and if we imagine the polvhedron ah on the plane of any 
oue face in such manner that the projections of all the summits 
not belonging to the face fall within the face, then we have a 
partitioned polygon, m which, if P denote the number of com- 
ponent polygons, or, say, the number of parts, F=P+41, or we 
have | | 
P+S=E--1, 
where 8 is the number of summits and E the number of edges 
of the plane figure. I retain for convenience the word edge, as 
having a different initial letter from swmmt. 
* Communicated by the Author, 
