supposing that all the quantities under the double sign of integration are expressed in terms of the two variables y and z, and that is made equal to 1, after the integrations. This transformation is correct; and the introducing of the new integral is important, as it leads to detecting the fault of the investigation. M. Poisson assumes that the differentials of the new integral X" are all infinitely small, so long as is infinitely small: in which case, X" being itself infinitely small, it may be rejected, and the same conclusion will be obtained as in the first investigation. Now were the denominators in the successive differentials of X" always finite quantities, the assumption of M. Poisson would be allowable; but as both the numerators and denominators begin to vary from zero, it is not impossible that, while the first increase to a finite magnitude, and the others to some small quantity B, the quotients may pass through every gradation of quantity; their values may be infinitely small, or finite, or infinitely great. This point must therefore be examined before any just conclusion can be drawn.

=

Let uf(0, 4), u' =ƒ (0', '); then u'-u: put also g1-a; then

1 — a2 = 2 g − g3

-

1−2 ap+a2 = 2 (1 − p)−2 g (1 − p)+g3.

These values being substituted, the resulting expression of X" will be,

pycz (2g—g2). u'—u. sin e' d' d↓

าม

x" = SoSo (2 (1 − p) — 2 g (1−p) + g°

Now 1-p is a small quantity depending upon the values assigned to y and z; and g is a small quantity quite independent of any other; we may therefore suppose that g is equal to 1-p, or less than it, and even infinitely less than it. Now, c being any positive number less than, if we reject quantities of the second order in the last formula, the result may be thus written :

which is obviously the limit to which the expression of X" continually tends as values decreasing indefinitely are assigned to y, z, g. Since g may be considered infinitely less than is always

1-p, the ultimate value of the factor

2 g

(2(1-p))

zero: but distinctions must be made with regard to the other factor. When the value of

u-u

3

(2(1-p))-c

is either always finite or infinitely small, all the differentials of X" will be infinitely small as assumed by M. Poisson, and the equation (2.) will be proved. But the same equation will not be proved if the limit of the same factor be either infinitely great, or if it be a quantity that cannot be generally determined, and of which it cannot be said that it is either finite, or infinitely great.

If c = 1, the factor in question will be,

u-u

√2(1-p);

which has a finite value when u, or ƒ (0, 4) is a finite function of cos, sin cos y, sin sin. This readily follows from the usual transformation of such expressions. The same factor will be infinitely small when u'-u is divisible by (1-p)", n being any positive integer. In all these cases the equation (2.) is demonstrated. If n = 1, or if u'-u be divisible by 1-p, we fall upon the instance particularized by Laplace in the eleventh book of the Mécan. Céleste.

But if u, or f(0, 4), be not a finite function of cos, sin cos, sin sin, no determinate value can be assigned to the factor

u-u

(2(1—p)) —

by any transformation; and in all such cases the equation (2.) is not demonstrated.

Lagrange has considered this subject in the 15th cahier of the Journal de l'Ecole Polytechnique. His investigation possesses all the exactness and clearness and elegance which distinguish the writings of this geometer. But the success of his analysis demands that f(',') shall be a finite function of cos', sin sin V, sin ' cos'. For such functions his process leads to a strict demonstration of the equation (2.): for functions of a different description, the algebraic operations fail, and no other conclusion can be drawn except that the equation (2.) is not demonstrated. It is very remarkable that the illustrious author has not noticed a distinction so obvious and necessary.

July 26, 1836.

* Théorie de la Chaleur, p. 213.

JAMES IVORY.

XXXV. On the probable Cause of certain Optical Properties observed by Sir David Brewster in Crystals of Chabasie. By JAMES F.W. JOHNSTON, A. M., F.R.S.E., F.G.S., &c., Professor of Chemistry and Mineralogy in the University of Durham.*

A

T the Meeting of the British Association in Edinburgh, Sir David Brewster brought under the consideration of the Chemical section, some very interesting observations on the variations which the doubly refracting power is seen to undergo in different portions of the same crystal of certain varieties of chabasie. An abstract of this paper has since appeared in the Fourth Report of the Association, p. 575; but the leading fact is more precisely stated in the account of the Meeting published in the Literary Gazette (for 1834,) No. 952, p. 690. The double refraction in these crystals, which is positive in the rhomboidal nucleus or centre of the crystal, was seen "to diminish in succeeding layers from a positive state till it disappeared altogether; beyond this neutral line it became negative, and again gradually increased +." This observation was brought forward partly with the view of illustrating, as it does very beautifully, the importance of the optical character of minerals in throwing light upon the structure, composition, and mode of formation of such of them as occur in a crystalline state; but partly also to show that chemical analysis is liable to lead to error by treating as simple minerals what are in reality only aggregates of different substances deposited in successive layers around a common nucleus.

That the layers which exhibit the difference of optical properties mentioned by Sir David, have a different composition, is highly probable. A series of optical changes is produced on some substances, as on glauberite and topaz, by the elevation of temperature; or, as on unannealed glass, by sudden cooling it is possible, therefore, that the deposition of successive layers of the same chemical constitution under a pressure or temperature constantly varying, might produce also phænomena analogous to those observed in the present case. There is not much apparent probability, however, that any such variations actually took place in the circumstances under which the crystals of chabasie were formed. That change of atomic arrangement which gives rise to the interesting phanomenon of dimorphism must necessarily, we may suppose, give to the two forms very different optical properties: but * Communicated by the Author.

These phænomena are also detailed, and an explanation attempted, in a paper by Sir David Brewster on another subject, inserted in the Philosophical Transactions for 1830, part i. pp. 93, 94.

here the form of all the layers is the same; so that this mode of accounting for the appearances must also be rejected. Sir David Brewster indeed, speaking of the form of the crystal at the neutral line or line of no double refraction, says: At this period the form of the crystal would be a cube:" but he speaks here only in reference to the absence of the doubly refracting structure which in a single atom or group of atoms of a simple substance would indicate a cube; but which, in a mass, such as the smallest of these layers, is quite compatible with a congeries of rhomboids.

This transition from a positive to a negative state of double refraction may be explained on the principle that the molecules of two substances possessed of opposite optical properties may neutralize each other if the relative positions of the optical axes and the number of molecules of each be rightly adjusted. Let the proper material of a crystal be possessed of a positive double refraction, such as the nucleus in the present case indicates; and let a second substance having a negative index deposit itself during crystallization, either in the form of distinct layers or mixed with the molecules of the first as fluids mix, forming only part of the mass of such layers; let also the quantity so mixed increase, or the distance of the layers decrease, as we depart from the central nucleus, and let the optical axes of the two substances be parallel; we shall have a crystal possessed of properties analogous to those observed in chabasie. The positive energy of the nucleus will gradually decrease till it disappear in a neutral line, and from this neutral line the refraction will again increase with a negative sign.

But the second substance must not only be negatively double-refractive, but also isomorphous with the first or capable of replacing it during crystallization without affecting the form of the crystal. Now among the substances contained by chabasie in large quantity, is silica; and this substance is not only negatively double-refractive, but appears also to be isomorphous with chabasie. The form of this mineral as determined by Mr. William Phillips from cleavage planes, is an obtuse rhomboid of 94° 46'; while that of quartz is a similar rhomboid of 94° 15'. These two forms are as nearly identical as isomorphous bodies generally present themselves in nature: we may therefore consider them as capable of replacing each other.

Suppose then that during the crystallization of the chabasie a slight excess of silica is present in the solution and deposits itself in occasional layers or otherwise as above described, and that the quantity so deposited increases with the size of the

crystal; the form of the crystal, its transparency, apparent homogeneity, &c. will be unaffected; its optical properties alone will present a change, and one exactly such as we are considering*.

But if silica be isomorphous with pure chabasie, and be the cause of the phænomena in question, the quantity found in the mineral should be in some degree variable. Granting the positive nucleus of pure chabasie to have a fixed composition associated with a constant form, analysis ought to show that the quantity of silica present in some crystals is greater than in others. Now chemists have recognised two varieties of chabasie; one from Aussig in Bohemia, and from Fassa, analysed by Hoffman, and from Faroe analysed by Arfvredson; represented by the formula,

C

NS2+3 AS2+6 Aq,

K

and another from Nova Scotia, analysed by Hoffman, from Gustafsberg, by Berzelius, and from Kilmacolm, by Mr. Connell, represented by the formula,

C

NS+3 AS +6 Aq.

K

The only difference between these two formulæ is that the latter contains one half more silica in the first member than the former does. This circumstance is entirely in accordance with the view presented in this paper. If the first of the two formulæ be the correct one, the presence of the additional silica in the second should produce a perceptible change in the optical properties. It is impossible to say, however, that even in that formula the quantity of silica indicated may not be greater than the pure mineral may contain. The fact that silica is isomorphous with chabasie renders any formula for this substance exceedingly doubtful. Crystals from the same locality may not always contain the same quantity of silica; it is probably the paucity of analyses only that prevents us from knowing of crystals from one and the same locality to which each formula would apply. Sir David Brewster has not stated from what locality his crystals were obtained † ; but the analysis of the actual specimen, and not the locality, would be required to enable us to say which formula would apply to it. Were they

In the paper already referred to, Phil. Trans. 1830, p. 94, Sir David Brewster supposes the change to be due to the presence of a foreign body. + One of the specimens I have since learned is from the Giant's Causeway.