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previous Article will no longer be accurately true; but since the variation of p as a function of will be very slow, Y

T

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ρ

=

may still, for a first approximation, be considered constant from y = 0 toy a considerable value. Consequently both the equations (1) and (2) of Art. 25 may in our present case be considered as approximately true.

27. The case at which we have last arrived is exactly similar to that of Art. 23, which it is our object to investigate. For a portion ABGH of the lamina, bounded by a line HMG, similar to EQF,

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P,

may be considered as being retained in its actual position, by the tensions acting parallel to AU and BV, at every point of HG, exactly in the same manner as that in which we have supposed the lamina represented in the figure in p. 31, to be kept in its position by forces acting at each point of HG in that figure. Also it has been shewn (Art. 23,) that the curvature of any such line as EQF, varies very slowly with its distance from AB. Consequently the variation of the radius of curvature at Q, is extremely small, considered as a function y (AE). This being the case, it is manifest likewise (assuming the original system of tensions parallel to AB, to have been uniform)* that T (the tension of EF) will vary very slowly with AE; and that therefore function of y, may approximately be considered constant. Consequently we shall have in this case

T

P

as a

T
T' =
.y, nearly.
ρ

*This is not essential to the truth of our general conclusions.

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If the fissure be of considerable length, p will be extremely large, and this equation will hold approximately for large values of y, and if y be less than T' will be less than T.

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28. Hence then it appears, that if the fissure be such that the curvature of its sides is extremely small, the greatest tension at any point within the lines AU and BV, and not extremely remote from AB, will be in a direction parallel to AB; and that consequently, if any fissure were propagated through Q, by the tension there, it must necessarily be in a direction perpendicular to that line.

§. On the Formation of Systems of Fissures.

29. The result enunciated in the last Article is important, as shewing the impossibility of forming in succession parallel fissures not far distant from each other in a mass subjected to such tensions as we have supposed. Let us suppose, for instance, a fissure AB to have been formed in a lamina subjected to two systems of tensions, of which the directions are perpendicular to each other. The

B

Α'

B'

propagation of the fissure beyond A and B, may be conceived to have been prevented by a greater cohesive power of the lamina there, or by a diminished intensity of the tensions perpendicular to AB. Let us also suppose another fissure to commence at A', subsequently to the formation of AB, and not remote from it, from the increased intensity of the tensions perpendicular to AB. Its direction AE will be parallel to AB, but it cannot be propagated in that direction from E to F; for the tension at Q along EF (as above stated) will be greater than that in a direction perpendicular to it, and therefore if a fissure be formed at all through that point, it must be perpendicular to EF. Nor would the formation of a fissure from E to F be rendered the more possible by the existence of this fissure through Q perpendicular to AB;

for it will be immediately seen, that this latter fissure together with AB, would destroy all tension at Q, and would of course prevent the possibility of the formation of any other fissure through that point*.

30. Hence it follows, that in any system of parallel fissures which are not remote from each other, the fissures could not be formed in succession. It will be easy however to understand how, in the case above assumed, of two systems of tension perpendicular to each other, any number of parallel fissures may be formed simultaneously. Let AB, A'B' be two such fissures, and let GH be parallel to and equidistant from them.

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Now if the two fissures begin simultaneously at A and A', (the line AA' being perpendicular to the direction of propagation,) and be propagated with equal velocity, it is obvious that no point in the physical line GH will have any motion communicated to it by the relaxation of the portion of the lamina between the fissures. Hence, if the line GH were to become absolutely fixed, the formation of the fissures would not be affected; but in this case the portions of the lamina on opposite sides of GH might be regarded as two absolutely distinct laminæ, having that line for a common fixed boundary. Consequently it is as easy to understand the simultaneous formation of any number of parallel fissures, under the circumstances supposed, as that of a single fissure.

31. Let us assume the two systems of tension not to be perpendicular to each other, and suppose AB, A'B', two parallel fissures of which the directions are perpendicular to the maximum resultant tension. These fissures would not necessarily be continued parallel to each other.

It must be recollected that the impossibility here spoken of assumes the tensions not to be produced by impulsive forces acting on the mass, the intensity of these tensions being always supposed to increase continuously, till sufficient to produce the fissure, and not to acquire that requisite intensity instantaneously, as previously stated in the Introduction, p. 11.

For let YY' be parallel to the direction of one system, XX' (meeting the fissure A'B') to that of the other. The opening of A'B' will relax

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the tension along XX', while that along YY will not be affected. Consequently the ratio of the tensions at Q will not be the same as originally, when AB, A'B' began to be formed. The direction of propagation of the former will evidently deviate towards perpendicularity with YY', and that of the latter in the same manner more nearly to perpendicularity with XX'. They will not therefore in such case preserve their parallelism.

A finite time, however, will be necessary to produce the relaxation at Q, after the opening of A'B', and therefore if the distance between the fissures be not too small, and the velocity of propagation very great, as we have shewn it may be (Art. 13) AB may be propagated through Q before the relaxation is produced there, and the fissures might under such circumstances preserve, at least approximately, their parallelism.

32. It is evident, however, that in whatever manner a system of parallel fissures may be produced, that, after their formation, the only tension of the mass between them must be in a direction parallel to them. Consequently, should any other system be subsequently formed, it must necessarily be in a direction perpendicular to that of the first system. No two systems of parallel fissures, not perpendicular to each other, could be formed by causes similar to those of which we have been investigating the effects. It will appear also, as in Art. 30, that this second system must be of simultaneous origin.

33. From our assumptions respecting the variable cohesive power of the mass, it is manifest that different fissures might commence simultaneously at different points, and be propagated in opposite directions.

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Thus, suppose the fissure CD to commence at D, when AB and EF commence at A and E respectively. When the first of these arrives at C, as the two others arrive respectively at B and F, the further propagation of each of them may be prevented by the relaxation of the mass. Consequently a system of fissures might thus be formed similar to that represented in the above figure.

§. Application of the previous Propositions to a Mass of three dimensions.

34. These investigations have been applied immediately to the case of a thin lamina, to avoid the complexity which would necessarily have been introduced in their immediate application to a mass of three dimensions. The extension of the preceding propositions, however, to this latter case is sufficiently obvious to require little more than an ciation of the results, which may also serve as a summary of the most important of those at which we have arrived in this section.

A slight inspection of what has been advanced in Art. 15, will shew that the existence of a line of less resistance in a thin lamina, will have no effect on the propagation of a fissure in a direction perpendicular to it; and similarly, if we suppose any mass acted on by horizontal tensions, it is manifest that a horizontal plane of less resistance will have no effect on the verticality or horizontal direction of the vertical fissures resulting from such tensions. Consequently, the tensions being horizontal, the cohesive power of the mass may be sup

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