not think that at the ordinary pressure a gas could completely fill a vessel with 20,000 times the original volume of the gas. The substance of comets is therefore a kind of very divided matter, with its molecules isolated and destitute of mutual elastic reaction.

It follows from the preceding that both the mass and the density of a comet are infinitely small, and without any hypothesis we may say that a sheet of common air of 1 millimetre in thickness, if transported into the region of a comet and illuminated by the sun, would be far more brilliant than the comet.

The mass of the earth, according to the average density given by Baily, may be reckoned at

6,000,000,000,000,000,000,000,000 kilogrammes ;

the matter of comets being assimilated above to air of which the density would be

45,000,000,000,000,000

times less than that of the ordinary air, this would lead us to assimilate it to the substance of the earth diminished to about

194,000,000,000,000,000,000,000

times less than its ordinary density. By this estimate a comet as large as the earth would only weigh 30,000 kilogrammes; this makes thirty tons of 1000 kilogrammes, or the weight of thirty cubic metres of water.-Comptes Rendus, February 23, 1857, p. 357.

ON AN APPARATUS TO DETERMINE THE SOLUBILITY OF SALTS AT HIGH TEMPERATURES. BY CHEV. CH. VON HAUER*.

Chev. Ch. von Hauer has invented an apparatus to determine exactly the solubility of salts at high temperatures, an operation of great difficulty and precarious exactitude under ordinary circumstances. Ch. von Hauer's apparatus (made by M. S. Markus), called by him "thermo-lysimètre," consists essentially of a strong vertical cylinder of copper, with a solid bottom, and a cover to be screwed on to the cylinder. In the side of the cylinder is an aperture closed by a cylindrical piece of metal, so that this may be turned without admitting the passage of air. This piece of metal supports in the interior of the cylinder a ring, into which may be inserted a small vessel of determined capacity, the opening of which is turned towards the bottom of the cylinder. A sufficient quantity of the salt, of which the solubility is to be found, is placed on the bottom; the cylinder is then filled with the dissolving liquid, screwed up so as to exclude the air, and heated to the required temperature. By turning the external cylindrical piece of metal, the ring and the small vessel supported by it are brought upwards, and by a special piece of mechanism the vessel is at the same time shut. When the whole is sufficiently refrigerated, the small vessel is taken out of the cylinder, and as its capacity is known, the proportion of salt dissolved by the liquid contained in it may be easily determined. This apparatus may be used with security even for temperatures far above 100° Centigrade.-Proc. Imp. Geol. Instit. Vienna, Feb. 19, 1856.

* Favoured by Count Marschall.

LONDON, EDINBURGH AND DUBLIN

PHILOSOPHICAL MAGAZINE

AND

JOURNAL OF SCIENCE.

[FOURTH SERIES.]

MAY 1857.

XLIV. On Equally Attracting Bodies. By Dr. T. A. HIRST*.

HOWE

[With a Plate.]

OWEVER great the practical difficulties may be which have to be overcome in order to calculate the attraction of a given body upon a given material point according to Newton's law of the inverse square of the distance, nothing is more evident than that innumerable other bodies exist, differing from each other in shape as well as situation in space, each of which would attract the given point in precisely the same manner as the given body. Taken in its widest sense, the problem to find all such equally attracting bodies is clearly indeterminate; by introducing certain restrictions, however, this problem, by becoming capable of solution, will lead us to the detection of certain groups of bodies possessing the property in question. The possible advantages arising from a determination, in an independent manner, of such groups of bodies will at once suggest themselves; at any rate they need not be cited in justification of any attempt to solve a problem which has undoubtedly sufficient inherent interest.

In confining ourselves for the present to one of the above groups of equally attracting bodies, we propose to examine

I. Equally attracting curves, i. e. bars or wires of infinitesimal thickness;

II. Equally attracting surfaces; and

III. Equally attracting solids;

to the first of which the present communication will be restricted.

* Communicated by the Author.

Phil. Mag. S. 4. Vol. 13. No. 87. May 1857.

Y

I. On equally attracting Curves.

To determine any point in space, we shall employ polar coordinates, the attracted point being always considered as the pole; and when a system of curves are referred to this point as pole, we shall, for the sake of brevity, call all points of such curves situated upon the same radius vector corresponding points; the arcs of these curves intercepted between the same two vectors we shall term corresponding arcs or elements, according as the same are of finite or infinitesimal length; and corresponding elements produced indefinitely, corresponding tangents. This being understood, the problem with which alone we shall at present occupy ourselves may be thus enunciated: To find all the curves whose elements attract the pole in the same manner as the corresponding elements of a given curve.

At first let the given attracting curve be in the plane of the attracted point, so that the equation of the former may be

r=f(0);

then, assuming the density of the attracting matter to be everywhere the same, the attraction of an element of the given curve upon a point situated at the pole will be proportional to

pendicular from the pole upon the tangent, it is well known that

p: r=r de: ds,

so that each of the above three expressions is equivalent to

The attraction of the corresponding element of any other curve

we easily conclude that the corresponding elements of two or more curves, and hence the curves themselves, will attract the pole equally, provided their corresponding tangents are equidistant from the pole. The curves or their corresponding elements may here, of course, attract the pole in the same direction or in opposite directions; practically no difficulty will arise in distinguishing between the two cases; and if we allow the term ' equally attracting' to em

brace both, we may at once conclude from the above theorem that all right lines equidistant from the pole, together with the circle which they envelope, constitute a system of equally attracting curves. This conclusion is merely a more general expression of a theorem long known, and communicated some years ago to the Cambridge and Dublin Mathematical Journal by Professor Joachimsthal.

The condition that the corresponding elements of the two r=f(0), r1=fi(0)

curves

may attract the pole equally, is also expressed by the differential equation

which must be fulfilled for all values of under consideration. This equation may be written thus,

where F(0) is an arbitrary function of 0. Integrating these equations, and introducing two arbitrary constants c and c1, we find

}

(5)

By addition and subtraction in accordance with equations (3), we thus deduce the following general equations of a pair of curves whose corresponding elements attract the pole equally:

r

1

u1 = -=ce

do

(6)

Before considering a few special cases of this general formula,

it will be well to examine more minutely its geometrical signification. Let P and Ρι be the variable radii vectores whose reciprocals are v and v, respectively; then c and c, being variable parameters, the equations (5) represent two systems of orthogonal curves. For if 7 and 7 be the angles through which a radius vector must be turned, in a positive direction, in order to become parallel to the corresponding tangents of a pair of curves from the systems (5), then

These expressions, being each independent of the variable parameters c and c1, show that the corresponding tangents of all curves belonging to the same system are parallel to each other. But according to (4),

hence

cot T cot T1 = -1,

that is, the corresponding tangents of any two curves belonging to different systems are perpendicular to each other, consequently each curve of the one system cuts every curve of the other orthogonally.

Again, by equation (3) we have, for the same values of 0,

so that

ρ is the harmonic mean between r and r1; in other words, every line through the pole is divided harmonically by the prime radius and the three curves, whose variable radii vectores we have represented by r, r, and p. On this account we may refer to the last curve as the intermediate curve with respect to the two equally attracting ones.

The equation of the tangent to the curve r, at the point r, 0, may be thus written :

where

and U (the reciprocal of the radius vector) are the vari ables. For when p=0, we have evidently

so that the right line represented by (7), inasmuch as it not only passes through the point r, 0, but also cuts the radius vector to