And the same being true of every other element of the machine, we have

which is a general expression for the coefficient of equable motion in the case supposed. The value of A in equation (371) is evidently represented by

282. To determine the pressure upon the point of contact of any two elements of a machine moving with an accelerated or retarded motion.

Let p, be taken to represent the resistance upon the point of contact of the first clement with the second, p, that upon the point of contact of the second element of the machine with the third, and so on. Then by equation (370), observing that, P, and p, representing pressures applied to the same element, 2 is to be taken in this case only in respect to that element, so that it is represented by I,, whilst A is in this case represented by we have, neglecting friction,

f

a

b1

απ

Substituting the value of ƒ from equation (371), and solving in respect to P1,"

where the value of A is determined by equation (373), and that of Ew by equation (372). Proceeding similarly in respect to the second element, and observing that the impressed pressures upon that element are p, and p, we

have

f representing the additional velocity per second of the b. point of application of P1, which evidently equals f a1 Substituting, therefore, the value of ƒ from equation (371) as before,

Substituting the value of p, from equation (374), and solving in respect to p1, we have

(2) AL (PAP)

Σωλ

And proceeding similarly in respect to the other points of contact, the pressure upon each may be determined. It is evident, that by assuming values of A and B in equations (370) and (371) to represent the coefficients of the moduli in respect to the several elements of the machine, and to the whole machine, the influence of friction might, by similar steps, have been included in the result.

PART IV.

THEORY OF THE STABILITY OF STRUCTURES.

GENERAL CONDITIONS OF THE STABILITY OF A STRUCTURE OF UNCEMETED STONES.*

A STRUCTURE may yield, under the pressures to which it is subjected, either by the slipping of certain of its surfaces of contact upon one another, or by their turning over upon the edges of one another; and these two conditions involve the whole question of its stability.

a

THE LINE OF RESISTANCE.

6

283. Let a structure MNLK, composed of a single row of uncemented stones of any forms, and placed under any given circumstances of pressure, be conceived to be intersected by any geometrical surface 1 2, and let the resultant a A of all the pressures which act upon one of the parts MN21, into which this intersecting surface divides the structure, be imagined to be taken. Conceive, then, this intersecting surface to change its form and position so as to coincide in succession with all the common surfaces of contact 8 4, 5 6, 7 8, 9 10, of the stones which compose the structure: and let bB, cC, dD, eE be the re

Lind of Resista

Line of Pressure

10

Extracted from a memoir on the Theory of the Arch by the author of this work in the first volume of the "Theoretical and Practical Treatise on Bridges," by Professor Hosking and Mr. Hann of King's College, published by Mr. Weale. These general conditions of the equilibrium of a system of bodies in contact were first discussed by the author in the fifth and sixth volumes of the "Cam bridge Philosophical Transactions."

sultants, similarly taken with aA, which correspond to these several planes of intersection.

In each such position of the intersecting surface, the resultant spoken of having its direction produced, will intersect that surface either within the mass of the structure, or, when that surface is imagined to be produced, without it. If it intersect it without the mass of the structure, then the whole pressure upon one of the parts, acting in the direction of this resultant, will cause that part to turn over upon the edge of its common surface of contact with the other part; if it intersect it within the mass of the structure, it will not.

Thus, for instance, if the direction of the resultant of the forces acting upon the part NM 1 2 had been a'A', not intersecting the surface of contact 1 2 within the mass of the structure, but when imagined to be produced beyond it to a'; then the whole pressure upon this part acting in a'A' would have caused it to turn upon the edge 2 of the surface of contact 1 2; and similarly if the resultant had been in a" A", then it would have caused the mass to revolve upon the edge 1. The resultant having the direction aA, the mass will not be made to revolve on either edge of the surface of contact 1 2.

Thus the condition that no two parts of the mass should be made, by the insistent pressures, to turn over upon the edge of their common surface of contact, is involved in this other, that the direction of the resultant, taken in respect to every position of the intersecting surface, shall intersect that surface actually within the mass of the structure.

If the intersecting surface be imagined to take up an infinite number of different positions, 1 2, 3 4, 5 6, &c., and the intersections with it, a, b, c, d, &c., of the directions of all the corresponding resultants be found, then the curved line abcdef, joining these points of intersection, may with propriety be called the LINE OF RESISTANCE, the resisting points of the resultant pressures upon the contiguous surfaces lying all in that line.

This line can be completely determined by the methods of analysis, in respect to a structure of any given geometrical form, having its parts in contact by surfaces also of given geometrical forms. And, conversely, the form of this line being assumed, and the direction which it shall have through any proposed structure, the geometrical form of that structure may be determined, subject to these conditions; or lastly, certain conditions being assumed, both as it regards the form of the structure and its line of resistance, all that is

necessary to the existence of these assumed conditions may be found. Let the structure ABCD have for its line of re

E

B

P

sistance the line PQ. Now it is clear that if this line cut the surface MN of any section of the mass in a point n without the surface of the mass, then the resultant of the pressures upon the mass CMN will act through n, and cause this portion of the mass to revolve about the nearest point N of the intersection of the surface of

section MN with the surface of the structure.

Thus, then, it is a condition of the equilibrium that the line of resistance shall intersect the common surface of contact of each two contiguous portions of the structure actually within the mass of the structure; or, in other words, that it shall actually go through each joint of the structure, avoiding none: this condition being necessary, that no two portions of the structure may revolve on the edges of their common surface of contact.

THE LINE OF PRESSURE.

284. But besides the condition that no two parts of the structure should turn upon the edges of their common surfaces of contact, which condition is involved in the determination of the LINE OF RESISTANCE, there is a second condition necessary to the stability of the structure. Its surfaces of contact must no where slip upon one another. That this condition may obtain, the resultant corresponding to each surface of contact must have its direction within certain limits. These limits are defined by the surface of a right cone (Art. 139.), having the normal to the common surface of contact at the above-mentioned point of intersection of the resultant) for its axis, and having for its vertical angle twice that whose tangent is the co-efficient of friction of the surfaces. If the direction of the resultant be within this cone, the surfaces of contact will not slip upon one another; if it be without it, they will.

Thus, then, the directions of the consecutive resultants in