The same remarks will apply to this equation as to equation (1).

Hence, then, when the directions of the different tensions to which a lamina is subjected, and the ratios of their intensities, are known, this equation will determine that position of the line AB passing through any proposed point P, in the direction of which there is the maximum or minimum tendency to cause a fissure to begin at that point. If ẞ be less than a right angle, it is manifest by inspection that the negative root will correspond to the former, and the positive root to the latter case.

8. The actual direction in which the fissure will begin to be formed at P, may, however, be different from that in which the tensions have the greatest tendency to form it; for if there be any particular line through that point, along which the cohesive power of the lamina is less than in any other, the fissure may begin to be formed in that direction, though it may not coincide with that of the maximum resultant tension. If however the cohesive power at the proposed point be equal in every direction, i. e. if it vary continuously in passing from one point to another, and not suddenly as at a line of less resistance, the direction in which the fissure will begin to be formed, will coincide with that of the maximum resultant tension determined by equation (2). This observation respecting the constitution of the mass to which the investigations of the previous articles are applicable, is important. The cohesive power may vary according to any continuous law, as was before stated. (Introd. p. 11).

Direction in which the Fissure will be continued.-Partial System of Tensions imposed on the Lamina about the extremities of the Fissure.— Direction of the Fissure not affected by it in the case proposed.

9. In the preceding investigations the tensions have not been considered necessarily sufficient to produce a fissure. Let us now suppose

their intensity to increase till the resultant tension becomes greater than the cohesive power at any proposed point P. A fissure will then begin to be formed in the direction determined by equation (2), in which the values of M1, M2, &c. express the ratios of the different tensions at P, at the instant the fissure begins to be formed there. Let us suppose the fissure AB to have been thus formed, and that the cohesive power

of the lamina beyond A and B is sufficient to prevent its further propagation, and let us then consider whether any modification of the tensions will be produced immediately beyond A and B, which may possibly influence the direction in which there will be the greatest tendency to continue the fissure.

10. Let GK be any physical line broken by the fissure. It is obvious that if it pass near the extremity of the fissure, its extension, and therefore its tension, will not be very much diminished; but since this tension is no longer counteracted at g and k by an equal and opposite tension, as in its unbroken state, it is manifest that the force exerted by each portion Gg, Kk, must produce an increased stress upon the portions of the lamina, immediately contiguous to and beyond the extremity of the fissure; and since a similar effect, differing only in degree, will be produced by each physical line broken by the fissure, it is possible that the intensity of the whole additional tension, thus partially superimposed upon the lamina, may be very considerable in comparison with the general tensions impressed upon it.

Now it is manifest, that the direction in which there is the greatest tendency to continue the fissure from A or B, under the circumstances we are supposing, will be determined by the whole tension

contiguous to those points, consisting of that superimposed as above described, as well as of that impressed generally on the lamina; and consequently, if we conceive the latter of these tensions, (and therefore also the former) to increase till the resultant tension is sufficient to overcome the cohesive power at A or B, the fissure will not necessarily be continued in the same direction, as if its continuation were independent of the partial system superimposed about its extremities.

It will be observed, however, that in the case just considered, in which the forces are not producing motion in the mass, the whole force exerted by gG, and kK, and similar lines is effective in producing the superimposed system of tension about the extremity of the fissure. We shall shew however, that such is not generally the case during the propagation of the fissure, if propagated in the manner we shall suppose it to be, and that consequently this force will have no material effect on the direction in which the fissure will be continued, and which will therefore be very approximately determined by equation (2).

11. For this purpose, let us suppose in the first place, any systems of tensions impressed on the lamina, of which the resultant tension (R) shall be less than the cohesive power (II), at any proposed point P; and let us then conceive subsequently superimposed on these another system of which the direction is different to that of R, and of which the intensity shall increase continuously with the time t, till the resultant of R and shall be equal to П, so that a fissure shall then begin to be formed at P. Its direction will evidently depend on R, and the value (), which shall have acquired at the instant the fissure commences. If R differ but little from II, 9, will be generally small*, and cannot (however the forces producing may subsequently act on the lamina), produce any material influence on the direction of the fissure, which will therefore, in such case, nearly coincide with the direction in which the tensions whose resultant is R may have the

* If the direction of coincided with that of R, the fissure would manifestly begin to be formed when R+, should = ПI, or 4,=ПI-R, which by hypothesis is small. If the angle between the directions of R and be not too near a right angle, it is equally manifest that must be small. In the actual case considered in the text, this angle obviously cannot be very considerable.

greatest tendency to form it, i. e. it will be nearly perpendicular to the direction of that resultant.

12. Let us now suppose P, to designate a point in the lamina, at which a fissure shall begin, and P. another point through which it shall be subsequently propagated; and let П,, П, denote the cohesive powers of the lamina at those points respectively, II, being the least. It has been already stated, (Introd. p. 11.) that in the case to which these investigations are to be applied, the intensity of the elevatory force, and therefore, of the tensions produced by them, will be assumed to increase continuously from the commencement of the action of this force, to the formation of the fissures; we shall here also make an additional assumption, viz., that this intensity shall increase rapidly, so that a very small time shall elapse between the commencement of the elevatory action, and the instant when the fissures shall begin to be formed. The tensions therefore to which our lamina is subjected, will be assumed to increase in the same manner. Let R denote the intensity of their resultant at the time t; then if t, be the time when the fissure begins at P1, R, must be equal to the cohesive power at P1 = ПI1. When the fissure is thus begun to be formed, the partial system of tensions described in Art. 9., will be superimposed about its extremities. Let , denote its intensity at the time t, and at any proposed point. As the fissure in its progressive formation approaches P, this force will be superimposed on the lamina there, in addition to the force R、 previously acting there, so that if t be the time when the fissure is first formed at P2, we must have at P2, the resultant of R, and of Now, if during the time t-t, R increases from R, or II, so that R nearly П,,, must be small at P2, and therefore can have but little influence on the direction of the fissure through that point, whatever be the direction of that tension, or the intensity it might acquire if the cohesive power at P, were sufficient to prevent the propagation of the fissure beyond that point (Art. 11.) In such case therefore the direction of the fissure will be at least very approximately determined by equation (2), p. 18, in which the values of do not include the tension,

=

29

= Π.

* This assumption is not absolutely necessary for the truth of the approximation we have to establish or for the proof of it. It renders however the approximation more accurate, and the proof much more simple.

but only the values F, f, f, &c. of the general tensions, at the instant when the fissure is propagated through the proposed point*.

13. Under the circumstances here supposed, the fissure will be propagated from P, to P, nearly in the time t-t, during which R, increases from II, to П. Consequently, if the difference between these latter quantities be not great, i. e. if the cohesive power do not vary rapidly; or if R (heretofore assumed to be the same at the same time at different points of the lamina) increase with rapidity, it follows that the velocity of propagation will be extremely great, becoming infinite, when the cohesive power, and the tension R, are accurately uniform throughout the lamina.

If R be not uniform, it is easy to see that reasoning similar to the above will hold equally true, with respect to the progressive formation of any fissure.

μ

14. The fissure will be propagated in a straight line, if the values of in equation (2) remain the same, i. e. if the ratios of the tensions at different points be the same at the instant the fissure is propagated through them. If these ratios be different for different points, the fissure will generally be curvilinear; there is, however, an important exception to this rule, when there are only two systems of tension, of which the directions are perpendicular to each other; for in this case it appears by Art. 6, that the direction of the fissure will always be perpendicular to that of the greater of these two tensions.

Effect

of Lines of Less Resistance on the Direction of a Fissure. Permanent Direction of Cleavage.

15. In the preceding articles, we have supposed the cohesive power of the lamina to vary according to some continuous law. Let us now

* When the cohesive power of the lamina is not sufficient to prevent the propagation of the fissure, the problem presented to us is no longer a statical one. In the case above considered, a small portion only of the extraneous forces producing the tension 4, is effective in causing an additional tension of the lamina before the formation of the fissure. The greater part is effective in communicating motion to those parts of the mass, the receding of which from each other causes the opening of the fissure. On the contrary, when the formation of the fissure is arrested, the whole of these forces is effective in producing this partial system of tensions.