two parts into which this surface divides the system; then the locus of the consecutive intersections of these resultants is that curved line to which I have assigned the properties of equilibrium described in the preceding page. I wish now to correct this definition. To the properties assigned to this line it is necessary that at each of the points where it intersects contiguous surfaces of the component masses, the whole pressure upon those surfaces should be supposed to be applied. Now, according to the definition given of it, this supposition is not, except under certain circumstances, admissible. The resultant of the pressures upon each surface of contact is necessarily at some point or other a tangent to the locus of the intersections of the resultants, but it may be, and except in particular cases, will be, a tangent to it at a point other than that in which this line intersects the surface of contact itself. The point where the resultant intersects the dividing surface to which it corresponds, is that element in the theory on which the condition, “that one portion of the system shall not turn over upon the boundary of its surface of contact with the adjacent portion,” depends. I propose, therefore, in the following paper, to determine the line which is the locus of intersections of the consecutive resultants, with the corresponding imaginary surfaces of division, these surfaces being, here, supposed to be planes. This line I shall call the LINE of RESISTANCE, including as it does the points of application of the resultants of all the resistances of the surfaces of contact. The direction in which the resultant intersects two common surfaces of contact, is that on which the condition, “that these surfaces shall not slip upon one another,” depends; moreover this direction is a tangent to the line which is the locus of the intersections of the consecutive resultants, drawn from the point where the line of resistance cuts the surface of contact. The determination of this line is therefore also an important feature in the theory. I propose that it should retain the name before given to it of the LINE of PREssurE. One of these lines—the line of resistance—determining the point of application of the resultant of the pressures upon each of the surfaces of contact of the system, and the other—the line of pressure—the direction of that resultant, the determination of the two includes the whole theory of the equilibrium of the system. In its application to the theory of the arch there belong to the line of resistance all those properties treated of in my former paper which have reference to the condition “that the voussoirs shall not turn upon the angles of one another.” It follows, therefore, on the principles established in that paper that this line touches the intrados of the arch at certain points equidistant from the crown, called points of rupture, and that the position of these points, and, consequently, that of the point of application of the resultant of the pressures upon the key-stone, are subject to the condition that this resultant is a minimum; and this condition being supposed, all the circumstances which connect themselves with the equilibrium of the circular arch, as a complete segment, and a broken or gothic arch, subjected to any variety of loading, are discussed and determined in the eleventh section of the following paper. The condition, however, that the resultant pressure upon the key-stone is subjected in respect to the position of its point of application to the condition of a minimum, is dependant upon hypothetical qualities of the masonry. It supposes an unyielding material for the arch-stones, and a mathematical adjustment of their surfaces. These have no existence in practice. On the striking of the centers the arch invariably sinks at the crown, its voussoirs slightly opening there upon their lower edges, and thus pressing upon one another exclusively by their upper edges. Practically, the line of resistance then touches the extrados at the crown; whilst the condition of the minimum is satisfied by its contact with the intrados at the points of rupture in the haunches. This condition being assumed, all consideration of the yielding quality of the material of the arch or of its abutments is eliminated. It is thus discussed as a practical question in the twelfth section of this paper. 1. Let a continuous mass to which are applied certain forces of pressure, be supposed to be intersected by a plane whose equation is Let the sums of the forces impressed upon one of the parts and resolved in directions parallel to three rectangular axes, be respectively Mi, M., M., and the sums of their moments Ni, N., N. Let, moreover, the position of the plane be such that these forces are reducible to a single resultant, a condition determined by the equa If between the four preceding equations in which M., M., M., N., N, N, are functions of A, B, C, these three quantities A, B, C be eliminated, there will be obtained an equation in a y, z, which is that to a surface of which this is the characteristic property; that it includes all the points of intersection of the resultant force with its corresponding intersecting plane in every position, which, according to the assumed conditions, this last may be made to take up. This surface is the SURFACE OF RESISTANCE. If to the preceding conditions there be added this, that in each two consecutive positions of the intersecting plane the corresponding resultants shall intersect, the surface of resistance will resolve itself into a line, which is the LINE OF RESISTANCE. Differentiating on this hypothesis the equation III. in respect to A, B, C, we have From the elimination of A, B, and C, between the five equations The inclination of the resultant pressure to a perpendicular to the intersecting plane, in any of its positions, may be determined (see paper on Equation of Arch), independently of the line of pressure, from the 2. Let the mass be a PRISM whose axis is horizontal, and the forces applied to which are, its weight and certain pressures, P, whose directions are in planes perpendicular to its axis and inclined at angles p to axis of x, and whose points of application are uniformly distributed along lines on the surface of the prism parallel to its axis; all those pressures which are applied in each line being equal to one another. * It will be observed that the condition V. is included in these. Vol. VI. PART III. 3 O The relation of the forces which compose the equilibrium of the whole Prism, will then be the same with that of the forces impressed on any one of its sections perpendicular to the axis. Let CBD, (Fig. 1,) represent any one of these sections. Suppose the mass to be intersected in any direction parallel to its axis by a plane, and let N. N. be the intersection of this plane with the section CB of the mass. And first, let this intersecting plane in altering its position be supposed to remain always parallel to itself. equation I, the second of equations III, and the second of equations IV. These resolve themselves into the following:— * = C............................................................................................................... (1), {X. Psin op-sin 9 syl-y) do) z+} cose sy,”-y;”)d C+sin 9 JC(y1-yl) dO+2+ Pk cosop 2 y = − X P cos p + cos 6 s(y1-ys) do ... (2), *****ac.” “...” ac-(y-y) (2-c) in e-, (, , y) cose y = Tsport-................ (3). dC dC + (y – yo) cos B |