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cause is the true one, we must shew that, supposing all local and partial causes with which we are acquainted to be removed, it would produce effects strictly in harmony with those laws to which the actual phenomena are observed to approximate. The most obvious cause of deviation in our phenomena from strict geometrical laws, is irregularity in the intensity of the elevatory forces, and in the constitution of the masses on which they are supposed to act. Abstracting these sources of uncertainty, we have before us a definite problem, viz., to determine the nature of the effects produced by a general elevatory force acting at any assigned depth on extended portions of the superficial crust of the earth, and with sufficient intensity to produce in it dislocations and sensible elevations. To this simple and definite form the problem may be reduced; and at least a correctly approximate solution of it must necessarily be obtained by some means or other, before we can pronounce on the adequacy of the assigned cause to produce the observed effects. The complete solution of the problem presents many difficulties, which, however, are avoided by restricting ourselves to a first approacimation, which will amply suffice for all practical applications of our results. This approximate solution is what I have now to offer; and I may be allowed to observe, that those who may object to the mathematical resources of which I have availed myself, are at least bound to offer a solution equally conclusive and available by some method more adapted to the general reader. A slight examination however of the problem will suffice to shew that it can admit of no accurate solution independently of reasoning too intricate to be clearly embodied in any language but that of mathematical analysis.
The hypotheses from which I set out, with respect to the action of the elevatory force, are, I conceive, as simple as the nature of the subject can admit of. I assume this force to act under portions of the earth's crust of considerable extent at any assignable depth, either with uniform intensity at every point, or in some cases with a somewhat greater intensity at particular points; as for instance, at points along the line of maximum elevation of an elevated range, or at other points where the actual phenomena seem to indicate a more than ordinary energy of this subterranean action. I suppose this elevatory force, whatever may be its origin, to act upon the lower surface of the uplifted mass through the medium of some fluid, which may be conceived to be an elastic vapour, or in other cases a mass of matter in a state of fusion from heat. Every geologist, I conceive, who admits the action of elevatory forces at all, will be disposed to admit the legitimacy of these assumptions.
The first effect of our elevatory force, will of course be to raise the mass under which it acts, and to place it in a state of eatension, and consequently of tension. The increase of intensity in the elevatory force might be so rapid as to give it the character of an impulsive force, in which case it would be impossible to calculate the dislocating effects of it. This intensity and that of the consequent tensions will therefore be always assumed to increase continuously, till the tension becomes sufficient to rupture the mass, thus producing fissures and dislocations, the nature and position of which it will be the first object of our investigation to determine. These will depend partly on the elevatory force, and partly on the resistance opposed to its action by the cohesive power of the mass. Our hypotheses respecting the constitution of the elevated mass, are by no means restricted to that of perfect homogeneity; on the contrary, it will be seen that its cohesive power may vary in general, according to any continuous law ; and moreover, that this power, in descending along any vertical line, may vary according to any discontinuous law, so that the truth of our general results will be independent, for example, of any want of cohesion between contiguous horizontal beds of a stratified portion of the mass. Vertical or nearly vertical planes, however, along which the cohesion is much less than in the mass immediately on either side of them, may produce considerable modifications in the phenomena resulting from the action of an elevatory force. The existence of joints for instance, or planes of cleavage in the elevated mass, supposing the regularly jointed or slaty structure to prevail in it previously to its elevation, might affect in a most important degree, the character of these phenomena. To a mass thus constituted, these investigations must not be considered as generally applicable. Vertical or highly inclined planes of less resistance, will only be assumed to exist partially and irregularly in the elevated IIla SS.
With these hypotheses then respecting the nature of the elevatory force, and the constitution of the elevated mass, I shall proceed in the next section to investigate the directions in which fissures will be formed in it when subjected to given internal tensions sufficiently great to overcome the cohesive power which binds together its component particles. These tensions, so far as this investigation is concerned, may either be supposed to be produced by external forces causing an extension of the mass, or by such as prevent that contraction of it which might be conceived to result from the loss of moisture or of temperature. It must be understood however that these internal forces are quite distinct from that sort of molecular action on which any kind of laminated or crystalline arrangement of the component particles may depend.
1. THE simplest form of the mass in which we have to consider the formation of fissures, is obviously that of a thin lamina. The investigations therefore of this section will be applied directly to this case, from which the results applicable to a mass of three dimensions are immediately deducible. It will appear that its cohesive power may vary according to any continuous law.
§. Lamina subjected to one System of Tensions.
2. Suppose the lamina acted on by external forces, which shall place it in a state of tension, such that the direction of the tension at every point shall be parallel to a given line CD*. Let AB be any
proposed line in the lamina; P any point in this line. Also assume F to be the tension at P, estimated by the force which the tension at that point would produce, if it acted uniformly on a line of which the length should be unity, and which should be perpendicular to CD, the common direction of the forces of tension. Then if we take Pp a small and given element of the line AB, and draw PQ parallel to CD, and pm perpendicular to PQ, the force of tension on Pop in the direction PQ will be measured by F. pm, or F. Pp. sin N, (BPQ = \!); or the tendency of the forces of tension to separate the particles which are contiguous, but on opposite sides of the geometrical line Pp, by causing them to move parallel to CD, will be measured by
* The reference will always be made to the figure in the same page, unless stated to the contrary.
The tendency to separate the particles at P, by causing them to move in a direction making an angle 6 with CD or PQ, will be estimated by
This is greatest when 6 = 0, and N = 3. as of course it ought to be,
A B being then perpendicular to CD.
5. Lamina acted on by two or more Systems of Tensions—Direction in which their tendency to produce a fissure is greatest.
3. Let us next suppose a second system of parallel tensions superimposed on the former, their common direction making an angle (3 with that of the first system. Let PQ, PQ, in the following figure, be the directions of the tensions acting on the element 3a, of the line AB at
P, and therefore QPQ = 3; and let the intensities of these tensions (estimated as in Art. 2.) be represented by F and f. Then if QPB = \,, and therefore QPB = Q, - 3, we shall have the forces 3r. F. sin \,, and 3a ..f. sin (, — 3) acting on the element 3a; ; and to find