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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.)
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.
5. 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
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.
Wol, WI. PART I. F
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 ABBA" 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
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 eartension 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 p, p, and q, will be inclined, as in the figure. Hence it immediately appears that the difference between the lengths p,q, and as 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 Ap 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.
Hence, then, we may conclude that the eatension of the physical line ACB, under the circumstances supposed, will be at least equal, and generally greater, than that of any similar line in the higher portions of the uplifted mass. It seems also probable, that in cases occurring in nature the eatensibility will be less in the lower portion of the elevated mass (at least to a certain depth) than in that which constitutes its upper surface.
Now the tendency of any horizontal portion of the mass to separate, so as to form a vertical fissure, will vary directly as the extension, and inversely as the extensibility. We may therefore safely conclude, that when a mass has been elevated as above supposed, the greatest tendency to rupture will not be in its upper portion; and consequently, that if any fissure be produced, whether by a gradual increase of the horizontal tension, or by any more sudden impulsive action on the mass in its state of tension, such fissure will not commence at the surface, but at some lower part of the mass.
37. It appears, from what has been proved in the previous Section, that if we suppose the fissure produced solely by the tensions to which the mass is subjected, the plane in which it will lie will be perpendicular to the direction of the single system of tensions which, in this case, act upon the mass, and will consequently decline as much from a vertical plane as that direction deviates from horizontality. According to the hypothesis we have made, however, of the force acting on the elevated mass through the medium of an elastic vapour, this vapour will necessarily ascend into the fissure, and exert a fluid pressure on its sides, in a direction perpendicular to them, and of which the intensity may bear a considerable ratio to that of the tension. To form a rough estimate of this intensity, let r be the radius of the circle which shall most nearly coincide with the curve ACB (Fig. p. 41), p the pressure of the fluid on a unit of surface, T the intensity of the tension (supposed uniform) of the elevated mass estimated as in the previous section, and b the thickness of the mass. Then the whole tension exerted on a portion of the mass included between two vertical planes perpendicular to the axis of elevation, at a distance unity from each other, will = b T, and we shall therefore have
The value of r, according to the same rough approximation, will be nearly = #, which will always be very large; but as b also is probably large, p may bear a very considerable ratio to T.
Here then we have the case which has been anticipated in the investigation of Art. 20; and it appears that the action of this force p will greatly tend to increase the effect of any local causes in producing partial deviations in the plane of the fissure from a vertical plane, but that it will not alter generally its position when considered with reference to its whole extent.
38. Again, with respect to the comparative width of the fissure at different depths, it is manifest, taking the case of the Fig. p. 41, where the extension of each lamina is the same, that if the mass, when relieved from its tension by the rupture, return to its original horizontal length, the width of the fissure will be the same throughout its whole depth; and in the case of the Fig. p. 42, the same conclusion might be considered as very approximately true under the same hypothesis. If, however, the different laminae, which I have supposed to have different powers of cohesion, have also different degrees of elasticity, this difference may materially affect any approximation to this uniformity of width. It seems probable, however, that the mean width (at least within certain limits) will rather increase than decrease with the depth.
39. Any number of these fissures might thus be formed simultaneously, (Art. 30.); and this simultaneous formation would be very much facilitated by the action of the pressure p in the interior of the fissure. If it be supposed, however, that partial causes prevent the commencement of the formation of each fissure at the same instant, exactly equal forces will not be exerted in the production of each, and consequently they will not be propagated with the same velocity. Some therefore will reach the exterior surface sooner than others; and when a certain number have thus been formed from the lower to the upper