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posed to vary continuously or discontinuously along any vertical line, and, as explained in Art. 8, it may vary according to any continuous law in any horizontal lamina of the mass. The same assumptions are made respecting the continuous but rapid increase of the tensions, as in Art. 12.

I. If this mass be acted on by a single system of horizontal parallel tensions, a fissure beginning at any point will be propagated in a vertical plane perpendicular to the direction of the system. (Art. 2).

II. If the mass be subjected to any number of systems of parallel tensions, the fissure will be propagated through any point in a direction perpendicular to the maximum resultant tension at that point, at the instant the fissure reaches it, (Art. 12.) the horizontal direction being determined by equation (2), (Art. 7). If the ratios of the tensions at each point at the instant of propagation through it be the same, the fissure will, in general, be formed in one vertical plane. (Art. 14.)

III. If there be only two systems of horizontal tensions, and these be perpendicular to each other, the fissure will lie in one vertical plane perpendicular to the direction of the system of the greatest intensity, whatever be the ratio of the tensions at each point in the two systems, provided the tension at each point always remain the greatest in the same system. (Arts. 6, 14.)

IV. Each fissure under the conditions assumed, will be propagated with extreme velocity. (Art. 13.)

V. The tendency of the tensions to propagate the fissure in one particular direction rather than in any other, or the permanence of the permanent direction of cleavage, depends on the rapidity with which the magnitude of the resultant tension, estimated in a particular direction, decreases as that direction deviates from that of the maximum resultant tension; or generally, on the ratio which the maximum bears to the minimum resultant tension, which is perpendicular to it. (Art. 18.)

VI. If in addition to a system of horizontal tensions, there be also a force acting on the opposite sides of the fissure, perpendicularly to its direction, and tending to increase its width*, the permanence of direction in the progressive formation of the fissure will be diminished, but the permanent direction will remain the same as if there were no other force than the system of horizontal tensions, i. e. if the direction in which the propagation of the fissure is taking place be disturbed by any partial cause, it will still constantly tend again to perpendicularity with the directions of the system of tensions; but this tendency will be less than if the force always acting perpendicularly to the fissure did not exist. (Art. 20.) Consequently, deviations from the permanent direction of cleavage will, in the case we have supposed, be greater than if the sides of the fissure were not subjected to the action of this last-mentioned force.

VII. If there be no tension acting on the mass, and a fissure be formed solely by this force, acting perpendicularly to its sides, the fissure will be propagated in the plane in which it begins to be formed, if the cohesive power of the mass vary according to any continuous law. There will be, however, but little permanence in its direction, so that if it be turned from its original direction by planes of less resistance, there will be little tendency to resume that direction, and the fissure may thus assume any form of irregular curvature. (Art. 20.)

VIII. If a fissure commence at, or in the course of its progressive formation meet, a partial plane of less resistance at an acute angle, it will, under certain conditions, be propagated along it; but when from any cause this ceases to be the case, the fissure will almost immediately resume a direction parallel to its original one, supposing it produced by tensions, which, independently of the existence of planes of less resistance, would produce rectilinear fissures. (Arts. 17, 18.)

IX. If the mass be subjected to two systems of parallel tensions, of which the directions are perpendicular to each other, two systems of

* This will be the case if the fissure be filled with any kind of fluid subjected to a great pressure from some external cause.

parallel fissures may be produced, of which the directions will be perpendicular to each other. No two systems of parallel fissures could be thus formed, of which the directions should not be perpendicular to each other. (Art. 32.)

X. If the fissures in either of these systems be near to each other, they could not be formed by such tensions as we have been considering, in succession. They must be formed simultaneously in each system. One system, however, might be formed at any time subsequently to the other. (Art. 30, 32.)

SECTION II.

35.

LET us now proceed to apply the results obtained in the last section to the actual case of a portion of the earth's crust, under the hypotheses respecting the action of the elevatory forces and the cohesive power of the mass, which have been already stated, (Introd. p. 11, and Art. 12.) And, first, let us suppose, for the greater simplicity, the surface of the mass acted on to be of indefinite length, and bounded laterally by two parallel lines. If we first suppose the elevatory force to be uniform, it is manifest that the extension, and therefore the tension, will be entirely in a direction perpendicular to the length; so that its whole tendency will be to produce longitudinal fissures, or such as are parallel to the axis of elevation.

§. Formation of Longitudinal Fissures-Their Position and Width— Complete and Incomplete Fissures.

36. Let the annexed diagram represent a transverse section of the elevated mass, and let us suppose it symmetrical with respect to the line CC', and also that the mass below the horizontal line AB remains

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perfectly undisturbed. The cavity ACBD, containing the fluid through the medium of which the elevatory force is supposed to act on the lower surface of the elevated mass, (see p. 10), may either be supposed to have existed previously to the action of the elevatory forces, or to have been partly produced by them.

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If we suppose the mass not to become compressed, and the disturbance not to extend beyond the vertical lines AA', BB, it is manifest that the lengths of the lines ACB, A'C'B' will be equal; and since their original lengths were so, their extension will be the same.

It is evident, however, that the force required to elevate the mass ABB'A' will be much greater than that just necessary to overcome its weight, on account of the forces called into action at the extremities of the elevated mass, and that some degree of compression of the mass will consequently exist, which will render the vertical line

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CC shorter than its original length. It is also evident that the disturbance of the upper part of the mass will extend laterally beyond the verticals through A and B, as above represented.

The compression of CC' will clearly make the curvature of A'C'B less than that of ACB, and will consequently render its extension less than it would otherwise be. The greater extent of lateral disturbance in the upper portion will also produce the same effect. For let us suppose the portion A'p of the upper curve exactly similar, and equal in length to p C', then is it easily seen (assuming the extension of A'B' to be uniform throughout) that the line joining the physical point p, and its undisturbed position will be vertical, while similar lines for PP and q will be inclined, as in the figure. Hence it immediately appears that the difference between the lengths P. and aß will be less in this case than if p and q were in the verticals through A and B respectively. We may therefore infer that the same will hold generally, since the condition of the similarity of A'p and pC will be approximately satisfied when the tangents at A' and C' are parallel, and the curvature small, as we may here assume it to be.

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