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the tendency of these forces to separate the contiguous particles on opposite sides of the elementary portion 3a, of the geometrical line AB, estimated by their tendency to give an opposite motion to these particles along any assigned line r PR, we must resolve the forces in the direction of that line. Let RPQ = 6; then will the sum of the resolved parts of our forces in the directions PR and Prbe

3a . F sin \, cos 8 + 8a of sin (N, - (3) cos (3–6)............... (A).

If the value of this expression, considered as a function of the independent variables N, and 6 be made a maximum, we shall manifestly obtain from the corresponding value of N, that angular direction of the line AB along which the two sets of tensions we are considering have the greatest tendency to form a fissure.

Differentiating the expression with regard to 6, we have 3a. Fsin N, sin 6 – 3a.fsin (N, - (3) sin (3–6) = 0. The left-hand side of this equation is the expression for the sum of the resolved parts of the forces 3r. Fsin N, and 3afsin (, — 3), perpendicular to the line PR. Consequently the equation expresses the

condition that PR must coincide with the direction of the resultant of the above forces.

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From the above equations we must determine and 6. If we put

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1 + u (c.4%) (c.1%) = 0.

- - - - - I From the inspection of these equations it is manifest that x = ~ is for

I putting this value for x, and therefore also T 3. for w, the two equa

tions will only be converted into each other. Substituting in the first

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4. Let \!, \, be the two values of N, given by this equation. Then, since the last term is – 1,

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and if 8, 9, be the values of 9 corresponding respectively to N, and No,. we have

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5. Since PR coincides, as shewn above, with the direction of the forces of tension acting on the element 3a of the line AB, the expression (A) is the value of that resultant. Consequently N. and No, which correspond to the maximum and minimum values of the quantity (A), determine the position of the line AB in which the resultant of the above-mentioned forces of tension is a maximum or minimum.

6. If the two systems of tensions be equal and perpendicular to each other, equation (1) becomes sin tr. cotol – 2 (1 + cost) coty—sin it = 0, and is satisfied independently of particular values of No. In this case,

therefore, there is no greater tendency to form a fissure in one direction than another. If F be greater than f, the equation becomes

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of which the two roots are 0 and 2, which shews that the greatest tendency is to form a fissure in a direction perpendicular to that of F.

7. The above investigation easily admits of generalization for any number of systems of parallel tensions superimposed upon each other. Let F denote, as before, the intensity of the tension in the direction from which 6 and N, are measured; f, f, &c. the tensions in directions making respectively angles (3, 3, &c. with the direction of the tension F. Then shall we have 8.r. Fsin locos 0+.f. sin(\!-3)cos(9–3)+.f. sin(\!-3.)cos(0–3)+&c.}= max.; and proceeding exactly in the same manner as in the previous investigation, and adopting an analogous notation, we shall manifestly obtain the following equations:

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- I and putting, for the same reason as before, a = --, we obtain

c)3 + 8,
2.

+ &c. = 0;

1 + pil (c, – 5,3)

‘. x + it, {c, s, + (c.”—s,’):-c, s, z*} + &c. = 0;
or - Łuic, s, + usc, s, +&c.}x'

+ {1 + ul(c.” – 8.”) + us (c.” – so) + &c.; x + ulcis, + usc, s, + &c. = 0; Vol. VI. PART I. C

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... cot V. – 2. cot. " = 0............(2).

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 3 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 eartremities 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 ul, us, &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 cir

cumstances we are supposing, will be determined by the whole tension

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