where ƒ (Art. 95.) represents the additional velocity actually acquired per second by the driving point of the machine, if P1 and P, be constant quantities, or, if not, the additional velocity which would be acquired in any given second, if these pressures retained, throughout that second, the values which they had at its commencement.

281. To determine the coefficient of equable motion. Zwx represents the sum of the weights of all the moving elements of the machine, each being multiplied by the ratio of its velocity to that of the driving point, which sum has been called (Art. 151.) the coefficient of equable motion. If the motion of each element of the machine takes place about a fixed axis, and a1, a2, a,, &c. represent the perpendiculars from their several axes upon the directions in which they receive the driving pressures of the elements which precede them in the series, and b1, b2, b1, &c. the similar perpendiculars upon the tangents to their common surfaces at the points where they drive those that follow them; then, while the first driving point describes the small space AS,, the point of contact of the pth and p+1th elements of the series will be made (Art. 234.) to describe a space represented by

so that the angular velocity of the pth element will be represented by

ap

and the space described by a particle situated at distance p from the axis of that element by

and the ratio à of this space to that described by the driving point of the machine will be represented by

The sum Σxx' will therefore be represented in respect to this one element by

Or if I, represent the moment of inertia of the element, and the weight of each cubic unit of its mass, that portion of the value of Σx2 which depends upon this element will be represented by

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 (361) 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 element with the second, p, that upon P2

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 p1 representing pressures applied to the same element, Σwx2 is to be taken in this case only in respect to

that element, so that it is represented by μ,I1, whilst A is

Substituting the value of ƒ from equation (371), and solving

where the value of A is determined by equation (373), and that of Ewx2 by equation (372). Proceeding similarly in respect to the second element, and observing that the impressed pressures upon that element are Pi and P2 we have

f representing the additional velocity per second of the point of application of p1, which evidently equals f. Substi

απ

tuting, therefore, the value of ƒ from equation (371) as before,

Substituting the value of p, from equation (374), and solving in respect to p2, we have

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.

406

PART IV.

THE THEORY OF THE STABILITY OF
STRUCTURES.

GENERAL CONDITIONS OF THE STABILITY OF A STRUCTURE OF UNCEMENTED 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.

THE LINE OF RESISTANCE.

283. Let a structure MNLK, composed of a single row of

5

Line of Resistanc

Link of Pressure

K

b

M

a

10

6

8

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 aA 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

* 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 "Cambridge Philosophical Transactions."

:

contact 3 4, 5 6, 7 8, 9 10, of the stones which compose the structure and let bB, cC, dD, eE be the resultants, 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 12 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 12; 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.