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surface of the mass, the tension of it may become so far relaxed that the further formation of the others shall cease. We may therefore suppose it highly probable that the number of fissures formed in the inferior parts of the elevated mass, will be considerably greater than the number which reach the surface.
40. The phenomena, then, to which our investigation at present extends, may be represented as in the annexed diagram, a few of the fissures being complete ones, or running up to the external surface of
the mass, and the others being incomplete ones, or rising to different heights, without reaching the surface.
41. If we recur to what has been previously advanced respecting the depths of veins, (Introd. II. p.), we shall see the importance of the fact established above, that the formation of fissures produced by the causes we have supposed must necessarily begin in some lower portion, and not at the upper surface of the mass, where it might perhaps at first sight be supposed more probable that they would begin.
42. We may also see, in what has been above stated, one cause of the inclination or hade of a fissure. (See Introd. II. K.)
43. In the case we have been considering, the whole tendency of the elevatory force, acting with perfect uniformity, will be, as we have before remarked, to produce longitudinal fissures; and a vertical section of the elevated mass parallel to the general axis of elevation, will be bounded above and below by straight horizontal lines. If, however, we now conceive this force to act with greater intensity at particular points along the general line of elevation, the section just mentioned will present such an appearance as represented in the annexed diagram,
in which the line ABC, previously to the elevation, was horizontal. In such case we shall have longitudinal extension, (equal to the difference between the line ABC and the dotted line AC), which, if sufficiently great, will necessarily produce transverse fissures, similar to the longi
44. We may represent to ourselves this more intense action at particular points, by conceiving an additional force superimposed on a uniform force producing the general elevation independently of the irregularities resulting from this partial action. It is manifest therefore that the tension perpendicular to the line of elevation will result from the sum of these forces, while the longitudinal tension will be produced by the superimposed force alone. The former will therefore, when the partial force is not great, be much the greatest; and we may consequently conclude, that the longitudinal fissures may in such case be formed first, during the continuous though rapid increase of intensity in the elevatory forces, according to the assumption we have made respecting them, (Art. 12.); and when this system is once formed (the fissures in it not being remote from each other), the transverse system must necessarily be approximately perpendicular to it, whether it be formed at the next instant, or at any succeeding epoch, and notwithstanding any irregularity in the forces producing it, provided they do not act impulsively. In this manner it is easy to understand the formation of a transverse system of fissures approximating to the law of parallelism, though resulting from forces which, acting partially, and under other circumstances, would produce the most irregular phenomena.
45. If however this more intense action at particular points be sufficiently great, and exactly simultaneous with that of the general elevatory force, it may modify materially the position of the longitudinal fissures. To determine the nature of this modification, we must consider the directions of the tensions which would be produced by an elevatory force, acting solely in the vicinity of any proposed point of a mass; because such tensions superimposed upon those produced by a force acting uniformly along the whole range, will be very nearly equivalent to the tensions produced by the simultaneous action of two forces such as those just mentioned.
46. For the greater simplicity, we may take a cone as the approximate type of the partial elevation we have to consider.
Let A'C'B' represent this cone, CD its axis. Then if we assume the physical line Ap C to be equally extended, and AD to be its original length, we have
The original length of A'p : A'p : A' D : A'C', and therefore, - - r , A'D The original length of Ap = 4 p. AFC,
mp being parallel to DC". Consequently, the distance of the physical point p from the axis of the cone, will not be altered by the elevation; and since the same holds for every physical point in the circumference of the horizontal circle whose radius is pn, there can be no tension at any point of the physical line forming that circumference, in the direction of its tangent at that point. This is consistent with our assumption of the equable extension of every part of the line A'C', which will therefore be true”. Similarly, if we conceive the whole mass AA'B'B to be formed by the superposition of similar conical shells, it is easily seen that the same result will hold for every horizontal circle concentric about the axis of the cone. Hence it follows, that if any vertical plane be drawn through the axis of the cone, there will be no tension at any point of the mass in this plane in a direction perpendicular to it. The tension will be entirely in the plane, and parallel to the slant side of the cone.
If, then, a fissure which should pass through any proposed point P, were formed according to the greatest tendency of the tensions of the unbroken mass to form it, it would manifestly coincide with the surface of an inverted cone, whose base would be the circle of which the radius is pn, and whose axis would coincide with that of the elevated cone. If p should coincide with C, an orifice would be formed along the axis C"C; and if we consider that the force will act, according to our hypothesis, with the greatest intensity at C, it seems highly probable that the first dislocation will usually take place along, or very near to that axis. For the greater distinctness, suppose this to be the case.
47. The instant this has occurred, the conditions of the problem will be entirely altered. The force at C maintaining every such line as A'C' and B'C' in its state of tension, being now destroyed, the extremities of those lines at C" will separate from each other by the contraction of A'C' and B'C'; and the same will be true for every similar pair of lines. An extension of the orifice at C will thus be produced, and consequently a tension of the mass contiguous to it in the direction of a tangent to a horizontal section of it, while the tension in the direction of such lines as C"A" will be entirely destroyed near to C, and much lessened at lower points. The whole tension therefore in the upper part of the mass, will be in the directions of the tangents of horizontal circles concentric about the axis; and the tendency to form a fissure there, will be entirely in a vertical plane passing through the axis of the cone. It is easily seen also that the tension at the vertex will be greater than in any other part. Consequently, if fissures be formed under these circumstances, they will commence at the vertex, and be in positions such as that just mentioned.
* Suppose a tension T to exist along the physical line forming the circumference of the
circle whose radius is pn. This would produce a force #. acting at p in the direction p n,
the resolved part of which in the direction p C would increase the tension of A'p. In such case the extension of A'C' would be greatest at A', and our assumption of the uniform extension of that line would not be true.
48. Let us now suppose the elevatory force to act with additional intensity beneath the point C of the annexed diagram, (which represents a horizontal section,) so as to superimpose on the general elevation
a conical one, having its apex at C. In addition to the tension (F) acting at any point P within the bounds of the cone, and in the direction perpendicular to the general axis of elevation, we shall also have another tension (f) acting at P, in the direction PQ' perpendicular to CP, (taking the case of Art. 47.) and the tendency of these tensions will be to form a fissure deviating from perpendicularity with PQ, in a degree depending on the relative intensities of f and F. Consequently, a fissure APB will deviate from parallelism with the line of general elevation, approximating towards C in the manner above
form, be more nearly spherical, without any such rupture at C, as Wol. VI. PART I. G