48 THRUST OF A GRANULAR SUBSTANCE. As increases from zero, the thrust Q also increases from zero; and when 0+e=π, or the plane EM is parallel to the talus DF, Q=&wla2cos2e, the same as for liquid of density w cos2e. But this implies that the wall DE is surcharged to an infinite height; but if surcharged to a finite height b, then when the plane of cleavage EN meets the horizontal level surface in N, W=wl{(a+b)2tan 0 — b2cot a}, and the corresponding value of Q will become a maximum for a value of depending not only on e, but also on the ratio of b to a. The determination of this maximum value must be deferred; but now it is important to notice that P and Q are not equal, the difference between them being taken up by the frictional resistance of the ground BE. *32. The Thrust due to an Aggregation of Cylindrical Particles or of Spherules. An exact Theory of Earth Pressure can be constructed if we suppose the substance which is held up by a retaining wall to be composed of individual particles or atoms of cylindrical form, such as canisters, pipes, barrels, or cylindrical projectiles, regularly stacked; or else to be composed of spherules, such as lead shot, billiard balls, or spherical shot and shell, piled in regular order, as common formerly in forts and arsenals. It will be necessary to begin by supposing that the lowest layer of cylinders or spheres is imbedded in the ground; as otherwise a wedging action takes place, due to the slightest variation of level, and the problem is to a certain extent indeterminate, as in the preceding article on Earth Pressure. AGGREGATION OF CYLINDERS. 49 Now in the case of cylindrical bodies, regularly packed as close as possible (fig. 22), the slope of the talus DF is 60°; and if EN is drawn through E the foot of the retaining wall DE parallel to the talus DF, the thrust between the cylinders across the plane EN will also make an angle of 60° with the horizon; so that considering the equilibrium of DENF, of weight W, the thrust Q on the retaining wall DE is given by Q= W cot 60°. A Fig. 22. Also, if w denotes the apparent heaviness of the substance, measured in lb/ft3, W = {wl (h2 — b2)cot 60° = wl(3a2+ab)cot 60°, so that Q=wl({a2+ab)cot260° = {wl({a2+ab), the same as the hydrostatic thrust of liquid, of heaviness w, on the portion DE of a vertical wall, of which the top edge D is submerged to a depth b in the liquid. If the cylindrical particles are of diameter d, and composed of solid metal of density p, then since the triangular prism formed by the axes of three adjacent cylinders is of cross section 1/3d2, of which only the area d2 is occupied by solid metal, therefore so that pd2=1/3wd2, or w/p=π/3; the same as for liquid of density 3p. 50 THRUST OF AGGREGATION Now, suppose the cylinders in the lowest layer are imbedded in the ground, and regularly separated so as to be at a distance x from axis to axis; the slope a of the talus DF is given by and COS α= x/d, Q= W cot a = wla2+ab)cot2a; but now w, the apparent heaviness of the substance, is When >d/3, vertical planes of contact come into existence; and alternate vertical columns descend, so that w/p=πd/x. The thrust Q will become very large when the cylinders are nearly in square order. The thrust Q is theoretically infinite when the cylinders are in square order; but this arrangement being unstable, a seismic rearrangement takes place, and the original triangular order is regained, except that the talus now appears stepped. Suppose the wall AB or DE to yield horizontally a slight distance; the cylinders in ABC or DENF will roll and wedge down along planes of cleavage BC or EN. The lowest layer of cylinders being imbedded, no further motion is possible; but if they were free to roll sideways, a molecular rearrangement would take place, and the cylinders would appear wedged against the walls AB and DE in close order, except along two planes of cleavage. OF CYLINDERS AND SPHERES. 51 *33. When the substance is composed of spherules or spherical atoms, we suppose the lowest horizontal stratum is embedded in the ground and arranged (i.) in square order: (ii.) in triangular order. In (i.) the spheres in the talus DF are seen in triangular order, in a plane having the slope a of the face of a regular octahedron; and therefore while the thrust across the parallel plane EN is also inclined at an angle a to the horizon; so that Q= W cot a =wl (4a2+ab)cot2a = {wl({a2+ab), the same as for liquid of density w. In (ii) the internal arrangement is essentially the same as in (i.), but now the talus DF may show the spheres arranged, either (ii., a) in triangular order, or (ii., b) in square order. In case (ii., a) the slope a of the talus DF is the slope of a face of a regular tetrahedron on a horizontal base, so that COS α= 198 sin a= 3/2, tan a=2√2; while the reaction across the plane EN, parallel to DF, will be inclined to the horizon at the angle ẞ, the slope of the edge of the regular tetahedron, so that In case (ii., b) the values of a and B are inverted; so that, in each case, (ii., a) and (ii., b), Q= W cot a cot ẞ=1wl({a2+ab), the same as for liquid of heaviness w. 52 THRUST EXERTED BY AN If p denotes the density, real or apparent, of a single spherule, while w denotes the apparent density of an aggregation of a large number of spherules, we shall find For if we suppose the horizontal layers in square order, and we take a volume consisting of a very large number n3 of spheres, standing on a square base whose side is of length nd, then the height will be 2nd, and the volume/2n3d3; while the volume occupied by the no spheres will be n3d; and therefore So also if the horizontal layers are in triangular order, the length of the volume being nd, the breadth will be 3nd, and the height /6nd; so that the volume will be/2n3d3, as before. When the number of spheres is limited, the effect of the irregularity of the arrangement on the outside of the volume makes itself felt. Thus 1000 spheres, each one inch in diameter, can be packed in cubical order in a cubical box, the interior of which is 10 inches long each way; but other arrangements are possible by which a larger number of spheres can be packed in the box; the discovery of these arrangements is left as an exercise for the student. (Cosmos, Sep. 1887.) This problem of the packing of spheres is known of old as that of "the thirsty intelligent raven"; the story is given by Pliny, Plutarch, and Ælian; it is quoted by Mr. W. Walton in the Q. J. M., vol. ix., p. 79, in the following form as due to Leslie Ellis :— |