These equations show that the displacements in the three rectangular directions are, to the extent to which we have carried the approximation, independent of each other.

We have supposed the masses of the molecules to be all equal; but if the medium be composed of two fluids uniformly mixed, and if the masses of the molecules of one fluid be all equal to m, and of the other all equal to m', the equations (2.), which we have just obtained, will still be of the same form; because each of the sums Σ may then be divided into two parts, one of which parts, multiplied by m, will embrace the molecules of one of the fluids, while the other part multiplied by m', will comprehend the molecules of the other fluid; and the molecules of each fluid may be conceived to be arranged in the manner which we have supposed. In this way the equations may be extended to the case of any compound medium in which the elementary media are uniformly mingled. In another communication I propose to deduce the inte grals of the equations (2.), and point out the extent of their application. I am, Gentlemen, yours, &c.

Evesham, Feb. 9, 1836. JOHN TOVEY. P.S. There are three typographical errors in my last paper. At page 9, line 12, for x read Ax; page 10, line 3, for d read ▲ 24; and line 22, same page, for s22 read

5/2 3.4

LVI. An Abstract of a Memoir on Physical Geology; with a further Exposition of certain Points connected with the Subject. By W. HOPKINS, Esq., M.A., F.G.S., of St. Peter's College, Cambridge.

[Continued from p. 236.]

II. HAVING now reduced the determination of the hori

zontal directions of the fissures produced in the elevated mass to that of the fissure which would be produced in a plane lamina every point of which is subjected to known tensions, we may proceed with this latter problem. Our first object is to determine the direction in which the tensions have

the greatest tendency to cause a fissure to begin at any proposed point. To give all requisite generality to the investigation, let us suppose there to be any number of these tensons, and let F, fi,f2, &c. denote their respective intensities at any proposed point, B1, B2, &c. the angles which their directions make with that of F;

Σμ cos 2 β

2

= μ1 cos 2ß1 + μ2 cos 2 ß2 +, &c. the angle which the required direction makes with that of F; we shall have for the determination of,

This equation will determine the direction in which the tensions have the greatest tendency to cause a fissure to begin at any assigned point, but when its formation has begun, it is obvious that the state of tension in its immediate vicinity must be altered, and that the tensions thus modified may not have a tendency to continue the fissure in the same direction as that in which it was the tendency of the original tensions to make it begin. I have shown†, however, that with our hypothesis as to the mode of action of the elevatory force (see p. 234) the above equation will be very approximately applicable to the action of these modified as well as to that of the original tensions.

The actual direction in which the fissure will be formed will not in all cases depend solely upon this tendency of the tensions, but partly also on the constitution of the elevated mass. If, however, its cohesive power be perfectly uniform, it is manifest that this direction will be determined by the tensions alone, or will coincide with that given by the above equation. It will appear also that this is equally true in certain other cases; when it is not so, the effect of any peculiar constitution of the elevated mass must be investigated. I shall now proceed with these points.

Let us still confine our attention to a simple lamina of uniform thickness. Its cohesive power at any proposed point may be estimated in exactly the same manner as the intensity of the tension at that point. Let the point of the lamina be designated by P, and draw through it, in any direction in the plane of the lamina, a straight line whose length is unity. Then conceive two equal and opposite forces (f,f) acting

• See Memoir, p. 18. + Memoir, pp. 20, 21. Third Series. Vol. 8. No. 47. April 1836.

Memoir, p. 13.

2 F

uniformly along this line perpendicular to it, and in the plane of the lamina, on the contiguous particles situated respectively on opposite sides of the line, thus tending to form a fissure along it. The cohesive power opposes this tendency, and if it be uniform along the line just mentioned it will be measured by that value off which is just sufficient to overcome it. If the cohesive power along this line be variable, ƒ will manifestly not be a measure of it with reference to the single point P. In such case we must conceive the cohesive power to be equal (for the unit of length) at every point of the line to that at P, and then that value of f (which we may designate by II) which would, under such circumstances, just overcome the cohesive power, may be taken as a measure of it at the point P, when estimated in the direction perpendicular to the above line through that point.

In the first place let us suppose the value of II the same for every direction of this line; then is it manifest that the direction in which a fissure may be formed immediately at the point P cannot be determined in any degree by the cohesive power, since its value is the same for every direction through P. The same conclusion will clearly apply to every point where the value of II is independent of angular direction, and equally so whether I be the same or different for different points, i. e. whether the cohesive power be uniform or variable, so long as its variation depends solely on the position of the point P; or, in mathematical language, the above conclusion will hold whenever II is a function only of the coordinates of P. In such case then, the fissure will be formed through P in that direction in which the tensions there have the greatest tendency to form it, and our equation will be as strictly applicable for the determination of this direction as if the lamina were perfectly homogeneous. We shall be able shortly to extend still further the conditions under which this equation will be similarly applicable.

It is easy to extend the above reasoning from a lamina to the general elevated mass.

If, however, the value of II be different for different angular positions of our line of a unit of length through P, (as, for instance, when a laminated or jointed structure prevails in the mass, or any accidental line of less resistance passes through the proposed point,) it is manifest that the direction of the fissure there will depend on the tensions and this variable value of II conjointly, and the equation above given will no longer suffice generally for its determination. The case of laminated or jointed masses I professedly exclude from these investigations, since their lines of dislocation will necessarily

be principally determined by their peculiar structure, and will therefore be in great measure independent of the causes whose effects I am investigating. The case however of the existence of partial and irregular lines of less resistance, regarded as modifying, and not as principal causes, comes within the sphere of our investigations. We may now proceed to this point.

Recurring again to the simple case of a lamina, it is easily shown that if a fissure in its continuous propagation through consecutive points meet a line of less resistance, it will be propagated across it without change of direction, or along it, according as a certain condition is or is not satisfied, this condition depending on the angle at which the fissure meets the line of less resistance, and the cohesive power along that line estimated in a direction perpendicular to it. If this angle be a right angle the condition is necessarily satisfied, as it must be also if the angle do not deviate much from a right angle, unless the cohesive power just mentioned be extremely small, so that in such cases the line of less resistance will have no effect on the direction of the fissure. If the angle just mentioned deviate too much from a right angle, the fissure will be propagated along the line of less resistance; but I have shown that when this ceases to be the case it will almost immediately resume the direction determined by our equation, so that if these lines of less resistance exist only partially and irregularly, and be of limited extent, they will only produce partial deviations in the direction of the fissure, without very materially affecting its general bearing. This reasoning again is easily extended to the general mass.

We shall now be able to arrive (as intimated above) at another and important condition respecting the constitution of the elevated mass, with which our equation will be strictly applicable to determine the direction of a fissure. If a single tension act at a point of a lamina, it is easily shown‡ (and in fact is in itself sufficiently obvious,) that the resuiting fissure will be perpendicular to the direction of the tension, the cohesive power being such as above shown (p. 274) to be consistent with the strict application of our equation. In like manner it may be easily conceived, that since all the tensions act in the planes of their respective lamina, whatever their horizontal directions may be, the resulting fissure, whatever may be its horizontal direction, must necessarily (independently of perturbing causes,) lie in a plane perpendicular to each Jamina at the points where it intersects it. Hence, then, it Memoir, p. 14.

Memoir, p. 24.

+ Memoir, p. 23.

follows that however small the cohesion may be between two successive laminæ or strata, this will produce no effect on the position of the fissure. In such case then its horizontal direction will still be accurately determined by our equation. This is important, because in a stratified mass the cohesion between different beds must probably be often much less than that between the constituent particles of each bed. The same conclusion will hold with respect to any accidental planes of less resistance which do not deviate too much from horizontality; but if they be vertical, or nearly so, they will produce the accidental and partial deviations which have already been noticed.

In forming a judgement of the probable extent of these planes of less resistance, we must be careful not to be too much influenced by the impressions produced by the examination of a disturbed district, since we are now speaking of the existence of these planes in the undisturbed mass. I would also observe, that we are only concerned with this kind of discontinuity in the cohesive power, so far as it depends on local and irregular, and not on general causes, since, as already stated, I exclude those cases in which any regularly jointed or laminate structure may be supposed to have existed in the mass previously to its elevation. Now as far as the planes we are speaking of might be caused by accidental circumstances in the constitution or deposition of the mass, it would seem necessary to suppose them irregular in position and partial in extent; in which case, as we have seen, partial deviations only would be produced by them in the vertical or horizontal directions of the fissure.

It appears then from what has preceded, that the equation above given will accurately determine the direction of a fissure at any proposed point, produced by tensions such as we have supposed, not only in a homogeneous mass, but also in a mass in which there may be any number of planes of less resistance, provided they do not deviate too much from horizontality, and notwithstanding any variation in the cohesive power of the mass depending on the difference of position of one point and another. From the interpretation of the equation, it appears that the fissure (or rather its intersection with a horizontal plane) will in general be rectilinear only in the particular case in which the ratios,, &c. are the same

for every point through which it passes, supposing the directions of the tensions at one point respectively parallel to those at another. There is, however, one important exception, viz.