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corresponding to 500 hectolitres per hectare. But the same substance which had entered into fermentation gave in six hours results corresponding to the enormous quantity of 8800 hectolitres per hectare in the twenty-four hours, or 77,440 gallons per acre.

Experiments were made also with various organic substances: sugar as well as wood-charcoal moistened gave no carbonic acid, and guano only a small quantity; linseed-meal produced a considerable quantity of carbonic acid.

It is generally assumed that the carbonic acid absorbed by plants is almost entirely derived from the respiration of animals, and a balance between these two kingdoms has been established which must be considered too absolute if we take into account the enormous quantity of carbonic acid assimilated every year by plants. These experiments show, Corenwinder thinks, that it is more conformable to nature to ascribe to the carbonic acid produced at the surface of the soil the greater part of the alimentation of vegetables.

Boussingault has examined the water of the Dead Sea. His principal object was to ascertain if nitrates were contained in it, as these had not been found by Gmelin. Boussingault thinks that nitrates have frequently been stated to be present where they did not exist, owing to the imperfection of the methods, or the impurity of the materials used in their detection. He had used the method with gold, but he found that it was not to be depended upon, owing to the impurity of the hydrochloric acid, which frequently contains nitrous compounds. By boiling the acid so as to drive off about a fourth, they are got rid of. The method used by Boussingault, consisted in adding to the water to be tested about an equal bulk of pure hydrochloric acid, colouring the mixture with a few drops of water rendered blue by tincture of indigo, and then boiling. If nitrates be present, the liquid becomes decolorized. This test is so sensitive that Boussingault could detect the presence of nitrates even in so small a quantity as 0-0000031 grm. in a cubic centimetre of water; and as this quantity may be the residue of the concentration of a litre, the test may be taken to be infinitely delicate. In examining for nitrates in sea-water or saline solutions, it is desirable to add an excess of hydrochloric acid.

The composition of the water Boussingault found to be as follows; the adjoined analysis by Moldenhauer is the most recent, with the exception of Boussingault's; that by Gmelin was made thirty years ago :—

* Annales de Chimie et de Physique, October 1856.

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The most remarkable thing about the water of the Dead Sea is its proportion of bromide of magnesium, which amounts to 3 or 4 kilogrammes in the cubic metre. If bromine were ever wanted in large quantity for industrial purposes, this would be the best source for it.

On comparing the densities of the water and the analyses of it executed at various times, discrepancies are observed which are not altogether to be accounted for on the supposition that they arise from differences in the analyses; and Boussingault supposes that the water varies, not only in the quantity, but in the nature and arrangement of the salts it contains. This is easily understood when it is known that the level of the water is two metres lower in the dry than in the rainy season. Boussingault thinks also that the liquid mass of the sea has not a homogeneous condition.

In connexion with this, some experiments made by Lieben* may be mentioned, the object of which was to ascertain if solutions originally homogeneous remain so throughout their whole mass, or whether they have a different degree of concentration at different heights after they have stood some time. A glass tube, 2 metres in length, and 2 or 3 centimetres in diameter, was filled with a solution of salt and suspended in a vertical position in a cellar for four months. On taking the strength of the solution at various heights, it was found to be perfectly the * Liebig's Annalen, January 1857.

Phil. Mag. S. 4. No. 89. Suppl. Vol. 13.

2 M

same throughout, and not to have varied from its original strength. A similar experiment made with a solution of sulphurous acid in water, gave results which showed that it remained quite homogeneous..

THE

LXVI. On the Persistent Appearance of the Lightning Flash. Saturday, June 20th, 1857. HE thunder-storm which passed over London last night, gave me an opportunity of repeating an observation on lightning which I had before made on four or five occasions. As the cause of the effect observed is, as far as I am concerned, only guessed at, I publish this notice that it may attract attention so that the effect may be verified; and if verified, perhaps find an exact explanation. Between 12 and 1 o'clock the storm was over London. Much of the lightning was within the clouds. At one moment the part of the heavens at which I was looking, directly above me, displayed the lightning flash in irregular lines: these formed in one part a large stretching across a space 60° or more in extent; beside which there were other lines of lightning not connected with this, but occurring at the same instant. These lines did not disappear instantaneously, but faded away, occupying, I think, a second or more before their extinction. As they faded, the form of the lines did not change, but their character did; for in place of an equal illumination at all parts, they assumed a granular character; i. e. parts near to each other were of different degrees of brightness, and the lines seemed to be dotted in with light. These enduring lines were not the mere effect of impressions on the eye; for, taught by former observations, I could distinguish them from such a result by directing the axis of the eye from the point directly regarded, to another part of the luminous map, and to another, and back to the first point, namely the angle on the, before the lines were quite gone. The form did not move with the eyes, but was fixed on the sky, and remained there, during its short existence, undisturbed in place. As I have already said, I have on different occasions noticed this phænomenon four or five times.

What is the cause of this effect? The most probable guess seems to me to be, that it is due to a highly exalted phosphorescent condition of the particles of cloud, along or through which, the electric discharge has passed; and perhaps an experimentalist might find means of realizing it in some degree. I believe that like luminous traces have been observed upon sugar and some other bodies when the electric discharge has been made over them.

M. F.

IT

LXVII. Analytical Solution of the Problem of Tactions.
By ARTHUR CAYLEY, Esq.*

T is well known that the eight circles, each of which touches three given circles, are determined as follows:-viz. considering any one in particular of the four axes of similitude of the given circles, and the perpendicular let fall from the radical centre (or centre of the orthotomic circle) of the give circles, there are two of the required tangent circles which have their centres upon the perpendicular, and pass through the points of intersection of the orthotomic circle and the axis of similitude, or in other words, the axis of similitude is a common chord or radical axis of the orthotomic circle and the two tangent circles. This suggests the choice of the radical centre for the origin of coordinates; and the resulting formulæ then take very simple forms, and the theorem is verified without difficulty.

Take then the centre of the orthotomic circle as the origin of coordinates, and let the radius of this circle be put equal to unity; then if (a, B), (a', B'), (a", B") are the coordinates of the centres of the given circles, the equations of these will be

x2+ y2+1−2a x-2ẞ y=0,

x2+ y2+1—2α'x—2B' y=0,

x2+ y2+1-2α"x—2ß"y=0;

and the radii of the circles will be va2+2-1, √a12+ß12—1, √ all2 + Bl12-1. It will be convenient to write

α

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where the three several signs are fixed once for all in a determinate manner. If, however, all the signs are reversed, the result is the same, so that the system is one of four (not of eight) different systems. The coordinates of a centre of similitude of the second and third circles are

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And forming the corresponding expressions for the coordinates of the centres of similitude of the third and first circles, and of the first and second circles, the three centres of similitude lie on a line which will be an axis of similitude: to find the equation, write

*Communicated by the Author.

A=By —B'y+B'y" —B"y' +B"y-By",
B=ya'―y'a + y'all -yla' +yla-ya",

C=aß'—a'ß+a'ß" - a"ß' +a"ß—aß",

▼=aß'y'' —aß'y' + a'ß"y-a'By" + a" By' — a"B'y,

values which, it will be observed, give

Aa + BB + Cy =V,

Aa' + BB'+Cy=V,

Aa" + BB"+Cy"=V.

Then the equation of the axis of similitude is found to be
Ax+By-V=0.

Whence also the equation of the perpendicular let fall from the radical centre upon the axis of similitude is

Bx-Ay=0.

It should therefore be possible to find two circles having their centres on the last-mentioned line and touching the three given circles. Take AO, BO as the coordinates of the centre of one of the two circles, and let r be its radius; the conditions of tangency are

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=

√ (A0—a')2 + (B0—ß')2±Y',

= √ (A0—a'')2 +(B0—ß")2±y",

where the sign has the same value in each expression. We have consequently

·(rFy)2= (A0—α)2 + (B0—ẞ)2;

or, observing that Aa+BB=V-Cy, and reducing,

r2 —(A2+B2) 02+2▼0−1=2y(±r+C0).

Forming the two analogous equations, the three equations will be satisfied if only

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which gives for the two values

(A2 + B2—C2)0=V± √ √2 — (A2+B2— C2);

and then r is determined linearly by the equation r = Co.

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