consider the effect of the existence of lines of less resistance in the lamina, in which case the continuity above assumed will no longer exist along these lines.

Let DE be a line of this description, along which the cohesive power estimated in a direction perpendicular to it II', that of the lamina near to DE being = II. Also let R, acting in the direction

B

R

PR, be as before, the resultant at the time t and at the point P, of the general systems of tensions impressed upon the lamina; and let Ri denote the tension along PR perpendicular to DE at the time t Then if

it is manifest that the fissure will begin to be formed along the line DE, rather than in a direction perpendicular to R, in which it would be formed in the absence of a line of less resistance*.

16. Let us now suppose this line to terminate at D and E. When the fissure has been propagated to those points, its progress will be arrested till the tension R and that superimposed just beyond the extremities of the fissure, and before denoted by 4, (Art. 11), produce a resultant tension greater than the cohesive power II. The direction in which the fissure will be then immediately continued, will not be known, being unknown; but without staying to enquire what this may be, we may observe, that the fissure must very soon in its pro

It is assumed in the above condition, that if the fissure be formed along DE, the particles on opposite sides of the fissure in separating would move in lines perpendicular to DE. This would be only approximately true.

gressive formation, arrive at a point at which R will be very nearly equal to the cohesive power, since that force by hypothesis increases rapidly with t, (Art. 12)}, and where, consequently, the direction of the fissure must necessarily be very approximately that determined by equation (2), as explained in Art. 12. Hence then we may conclude, that under the hypotheses we are taking, whatever may be the direction first given to the fissure by any local cause, its subsequent direction will soon become independent of that cause.

17. If the fissure, instead of beginning at some point in a line of less resistance meet it, in its progressive formation, it will pass along it, or will cross it, according as a condition exactly similar to that given above (Art. 15), be satisfied or not. At the termination of this line, the fissure will soon resume the direction given to it by the general systems of tensions to which the lamina is subjected, as just explained. Such also will be the case at the point at which the line of less resistance, should it be a curved or broken line, may assume a direction in which the condition just referred to is no longer satisfied.

18. The condition given in Art. 15 gives us

The first of these ratios will in each particular case be a function of the angle RPR or EPB, the angle between the line of less resistance and the direction AB, (perpendicular to PR) in which the

general tensions tend to form the fissure, the value of the function decreasing as RPR or EPB increases from zero to a right angle,

since the resultant tension is a maximum in the direction PR, and a minimum in that perpendicular to PR. (Art. 5). Consequently, the greater the ratio which the former of these resultants bears to the latter, the more rapidly will R decrease while RPR increases, and the smaller will be the angle EPR, within which the above condition will be satisfied, and the narrower therefore will the angular limits, within which a line of less resistance must be situated, in order that it may cause a fissure proceeding in any assigned direction to deviate from its course. A line through P perpendicular to PR, may be fr fr termed a permanent line of cleavage. If the ratios F' F'

&c. be the

same at every point of the lamina, all such lines will be straight lines (Art. 14) and parallel to each other. A fissure will always have a tendency to resume this direction, when made by any partial cause to deviate from it, and will resume it {taking our assumptions respecting the impressed tensions, (Art. 12)} almost immediately after the cessation of such cause. It will be well to examine this tendency in a few parti

cular cases.

It may be considered as measuring what may be termed the permanence of the fissure's general direction.

19. Let there be two systems of tensions, the directions of which are perpendicular to each other, and of which the intensities are F and ƒ respectively, at any proposed point, when they become sufficient to form the fissure there. The greatest of these (F) will be the maximum, and f the minimum resultant tension, (Art. 6), and therefore the less is, the greater will be the permanence of the permanent direction, perpendicular to that of F. If ƒ = F, there will be no permanence in any particular direction. We have already seen (Art. 6), that there is, in fact, no greater tendency in this case to form a fissure in one direction than another.

20. Again, let us suppose in addition to the systems of tensions, of which the intensities are fi, fa, &c., and which have determinate directions, a force acting within the fissure perpendicularly to its direction, and with equal intensity on its opposite sides, exactly as a fluid would act when forcibly injected into a fissure formed in a solid mass. VOL. VI. PART I.

D

Let P'P be the fissure. It is manifest that this force (p) will produce a tension on the mass contiguous to the extremity of the fissure, in a direction Pp perpendicular to P'P, and must therefore tend to propagate the fissure along PP produced. Hence it will follow that such

P

R

P'

a force cannot affect the permanent direction of cleavage as determined by the tensions fi, f2, &c. alone. For, suppose PR the direction of the maximum resultant (R) of these tensions, it is manifest that the whole resultant tension (including that produced by p) immediately beyond the extremity P of the fissure, must be in a direction PR between Pp and PR; consequently, the direction of propagation from P will deviate from P'PN, and approximate more nearly to perpendicularity with PR', and therefore also with PR. For the same reason, the direction of its further propagation will approximate still more nearly to a line perpendicular to PR, till it coincide with it. The permanent direction will therefore be the same as if the force p did not exist.

If however p be large compared with R, it is manifest that the angle pPR' will be very small, and that the tendency to resume the permanent direction, when the fissure has been obliged by any partial cause to deviate from it, will be much less than if p were relatively smaller.

21. If the lamina be subjected to no tension, and the fissure be produced entirely by p, the tendency will be to propagate the fissure in the direction in which it may originally be formed. Suppose AP to be its original direction, but that from P, it follows a line PP

of less resistance; then if we suppose the force p not to act effectively in propagating the fissure, except near its extremity*, its action will not extend beyond the portion PP, of the fissure, and consequently

P

P

its tendency will be to propagate it in the direction of PP produced, after it has reached the termination of the line of less resistance. There will be no tendency, as in the former cases, to resume any particular direction.

§. Modification of the Tensions in the vicinity of a Fissure.

22. Let us now suppose a fissure to have been formed in the manner above described, and extending between two points in the lamina, where we may conceive its propagation to have been arrested either by an increased cohesive power, or by a diminution of intensity in the tensions. It is manifest that the state of tension in the vicinity of this fissure, will become entirely different from that which existed previously to its formation; and that the subsequent formation of any other fissure not very remote from the first, must therefore be influenced by the modification of the original tensions thus produced. It will now therefore be our object to examine this consequence of the existence of a fissure. For the greater simplicity, we may suppose it to be rectilinear. It will also suffice for our immediate purpose, to suppose the lamina subjected to two sets of tensions acting perpendicularly to each other, the direction of the fissure being perpendicular to that of the system of the greater intensity.

This will be true in the actual case to which it is intended to apply this part of the investigation.