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approximately. Consequently, if the normal make an angle : – m with
#: + p sin m = 0,
Again, let us suppose another physical line exactly similar and equal to the former, with its extremities fixed to two other points in lines through A and B respectively, and perpendicular to AB, and so that the two lines shall be in contact, when not acted on by any force. When the force p acts in exactly the same manner on both, they will assume exactly similar positions, and those elements of the two lines respectively which were in contact when the lines were straight, will remain so when they have assumed their curvilinear form, and will be in exactly the same relative positions with respect to each other, as if the lines had been united into one previously to their becoming curved. Whence it follows, that there can be no more action between these lines when united, as we have just supposed, than if they were perfectly independent, and therefore the tension of each must remain the same as if this independence existed. If we conceive any number of lines to be united in a similar manner, so as to form a lamina, the same conclusion will apply to each.
25. Let us now take then a rectangular lamina ABGH, which we may conceive to be formed in this manner, and which we will suppose to be brought into the position represented in the annexed figure, by the force p acting perpendicularly to AB, and in the plane of the lamina. EF represents a physical line originally parallel to AB; and PM another originally straight and parallel to AH, and therefore, still evidently remaining so, though in a different position, in the curved form of the lamina. Let a be the original distance of PM from AH, which will be approximately = AP, or EQ; then will &r be the original width of the element PM; and if AE, or PQ = y, 3 y will be the width of EQ.F. Also, if T denote the tension of the lamina at Q, (estimated as in Art. 2.), in the direction of a tangent to EQF at
that point, it is evident that T. 3 y will equal the tension above denoted by T. Therefore the force produced by this tension in the direction of
the normal to EQF at Q, will _ T . 3r. 89, acting on the element - p common to the two physical lines PM and EQF at Q.
Now it is manifest, that the tension T, and f , will remain unaltered
so long as the position of every element of the lamina remains so, whatever be the forces by which it is kept in that position. The action of p will be the same at any point Q in PM as at P, since Q and P are similarly situated points in EQF and APB, and by hypothesis this force acts in the same manner upon each physical line, similar to APB. Consequently, the whole force on PM = p. PM. 3a. Let us suppose this force instead of acting on each element of PM, to be applied entirely at its extremity M. If this be done to every such line as PM, and the lamina be sensibly inextensible in the direction of these
lines, the position will remain undisturbed, and the normal force T 3.r. Öy, p
at Q will not be altered. Hence, if T denote the tension of the lamina at Q in the direction PM, and therefore T3a; the tension of PM at that point, and n the angle which the normal there to EQF makes with PM, we shall have for the conditions of equilibrium of the element common to PM and EQF,
- - - T . In the case we are considering, + is a function of a alone, and therep
since T = 0, when y = 0. This is subject to the condition T". 34 = force at M = q. PM. 8a, or T = p. PM.
The second equation gives
26. If instead of supposing the lamina inextensible in the direction PM, we suppose it capable of small extension in that direction as well as in that parallel to AB, and still assume it to be acted on by forces applied at each point of HMG, so as to keep that extreme boundary in the same position as before, the physical line EF will assume a position differing in a small degree from its former one. Since the angle m will still be very small, we shall still have T = const. nearly. The curvature at Q will no longer be the same as that at M, and p will therefore be a function of y, as well as of w. Consequently equation (1) of the previous Article will no longer be accurately true; but since the variation of p as a function of y will be very slow, f may still, for a first
approximation, be considered constant from y = 0 to y = 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,
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 p, 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 f aS a
function of y, may approximately be considered constant. Consequently We shall have in this case , T To = 7-9. nearly.
* This is not essential to the truth of our general conclusions.
Vol. VI. PART I. E
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 p, T' will be less than T.
28. Hence then it appears, that if the fissure be such that the curvature of its sides is eatremely 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
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;