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the contiguous beds, of that enormous hydrostatic pressure, which the process of injection of an extensive horizontal bed would necessarily call into action. That the toadstone of Derbyshire is not an injected bed, admits, I think, of the most indubitable proof from observation; and if the interstratification of the whinsill of the north, with comparatively thin beds of limestone and shale, be as regular as it is represented to be, I should have no hesitation in coming to the same conclusion with respect to that bed, for the reasons which have been heretofore mentioned, (Art. 77).

In the preceding investigations, I have spoken of the law of parallelism only as recognized in phenomena of faults, mineral veins, &c., comprized within narrow boundaries as compared with those to which it has been attempted to extend it, in the theory of Elie de Beaumont. It is very possible, however, that the physical causes to which I have referred this law, may have had a far more extensive operation than that I have ventured to assign to them. The parallelism of two mountain chains might thus be accounted for as simply as that of two neighbouring anticlinal lines; but it is obvious, that the more remote they should be from each other, the less would be the probability of the fissures to which our theory would refer them, belonging to the same system, and the less satisfactory would our solution become.

I have been anxious to avoid, for the present, any speculations respecting the interior constitution of our globe, beyond what is comprized in the simple assumptions on which these investigations have been founded; we may, however, include in those assumptions, the hypothesis of the elevatory forces having acted in different cases at different depths. The application of our theory, alluded to in the preceding paragraph, would perhaps require the hypothesis of these forces having acted at a much greater depth in such instances, than in those where the resulting phenomena are on a much smaller scale; and we may observe, that if the formation of the fissures should commence very far beneath the surface, it is extremely probable that very few would become complete fissures (see Art. 39), or would ever reach

nearly to the surface, in comparison with those which would do so in cases where these fissures should originate at a much smaller depth. The complete fissures would consequently be distant from each other and very large, and all the phenomena of elevation resulting from them might be expected to be of proportionate magnitude. I have no intention, however, of insisting on this extended application of our theory, but merely to indicate its possible extension (should established geological facts appear hereafter to require it) to account for phenomena on a much larger scale than those to which I have considered it essential to refer in the preceding investigations.

ST PETER'S COLLEGE,

May 4, 1835.

W. HOPKINS.

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Investigation of the Equation to Fresnel's Wave Surface. By
ARCHIBALD SMITH, ESQ., Trinity College, Cambridge.

[Read May 18, 1835.]

"THE mathematical difficulties under which the beautiful and interesting theory of Fresnel has hitherto laboured are well known, and have been regarded as almost insuperable. He tells us in his Memoir (see the Memoirs of the Royal Academy of Sciences of Paris, tom. vii. p. 136.) that the calculations by which he assured himself of the truth of his construction for finding the surface of the wave were so tedious and embarrassing, that he was obliged to omit them altogether. A direct demonstration has since been supplied by M. Ampère (Annales de Chimie et de Physique, tom. xxxix. p. 113.); but his solution is excessively complicated and difficult." A geometrical demonstration of considerable simplicity has been given by Mr M Cullagh in a paper in the xvrth Volume of the Transactions of the Royal Irish Academy, from which the preceding paragraph has been quoted.

The difficulties which were experienced in this problem arose from two causes of the same nature:―want of symmetry in the fundamental equations, and the use of the essentially unsymmetrical method of differential coefficients. By putting the fundamental equations of Fresnel under a symmetrical form, and by the use of the Method of Multipliers as it is employed in the Mécanique Analytique, the eliminations may be effected without difficulty.

To render what follows more intelligible, and to show in what it differs from the other methods, I shall give the fundamental equations

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of Fresnel and Ampère, and state shortly the steps by which they are obtained.

In the Memoir of Fresnel referred to above it is shewn, that if a section of the "surface of elasticity", whose equation is

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the greatest and least radii vectores of the section will be the values of v, derived from the equation

(a2 -- v2) (c2 − v2) n2 + (b2 − v2) (c2 − v2) m2 + (a2 − v2) (b2 — v2) = 0 ... (3),

and that if a plane be taken, parallel to the former and whose distance from the origin is one of the values of v, this plane, whose equation is

z=mx+ny + v √ 1 + m2 + n2

.....

(4)

will be a tangent to the wave surface.

To deduce the equation to the surface we may solve (3) to find v, and substitute this value in (4), which will give the equation

(≈ — m x − ny)2 = 1 { (c2 + b2) m2 + (a2 + c2) n2 + a2 + b2

± √/[(c2 − b2) m2 + (a2 − c2) n2 + (a2 − b3)]o — 4 (c2 — b3) (a2 — c2) m2 n2 } .

And if we differentiate this equation first with regard to m and then n, and eliminate m and n between the three equations, we shall obtain the equation to the wave surface. This is the method which M. Ampère employed with success.

Instead of eliminating v at first, we may differentiate (4), considering v as a function of m and n determined by equation (3), we shall thus obtain the equations

(a2 − v3) (c2 — v2) n2 + (b2 — v2) (c2 − v2) m2 + (a2 − v2) (b2 — v2) = 0 ... (1),

(x — mx — ny)2 = v2 (1 + m2 + n2)

(2),

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{(1 + n2) (a23 − v3) + (1 + m2) (b2 − v3) + (m2 +n°) (c2 − v2)}

(3),

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dv

dn

{(1 + n3) (a2 − v2) + (1 + m2) (b2 − v2) + (m2 + no) (c2 − v2)}

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Between these six equations the five quantities m, n, v,

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be eliminated, and the resulting equation will be that of the wave surface. These are the equations given by Fresnel, but he was not successful in effecting the requisite eliminations.

The fundamental equations may be put under a symmetrical form by the introduction of an additional symbol.

If form and n we substitute respectively and

1, m, n, connected by the equation

12 + m2 + n2 = 1,

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n

m

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n'

and suppose

we shall, instead of (3) and (4), have the three equations

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Differentiate these equations with regard to l, m, n, and v.

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