would detain us too long to describe, and in consequence of which the system Fig. 36 tends towards the form represented at Fig. 36, where the two faces abc and a'b'c are each occupied by a plane film. This form is completely attained at the moment when the inferior summit of the frame emerges from the liquid, but a change speedily occurs, and the system takes the form of Fig. 26. Although this change is very rapid, we may yet, by proper attention and by repetition of the experiment, observe how it is produced the two films which occupied (Fig. 36) the faces abc and a'b'c rise towards the interior of the frame by turning around the solid edges ab and a'b', and at the same time there is developed from the inferior summit a quadrilateral, at first very small, but which increases until its superior summit attains the centre of the frame, and which then constitutes the inferior quadrilateral of the definitive system; at the same time the summits ƒ and g of the curvilinear quadrilateral sfge ascend by a certain quantity, this quadrilateral shrinks, its edges become straight, and it finally forms the superior quad Fig. 37 : rilateral of the same definitive system. Fig. 37 represents the phenomenon in course of formation, at the moment when the quadrilateral, which is extending, has acquired half of its ultimate height. It will be easily conceived from this drawing how the four other quadrilaterals of Fig. 26 are generated. In order that all these phenomena should, with almost entire certainty, be produced, it is necessary that the frame should be withdrawn quite vertically; it is further necessary that this frame should be well constructed, that the iron wires which compose it should be of the least possible thickness, and, above all, that at the summits of the octahedron they should unite in the neatest manner, at least on the side which faces the interior of the frame; when this is not the case, the laminar figures obtained are often irregular. It should be added that the frame in question sometimes gives, when it is a little inclined in being withdrawn from the liquid, a system wholly different from that of Fig. 26, regular like it, but formed of curved films. This second system contains in the middle an hexagonal film, placed parallel to two faces of the octahedron, and having its sides slightly re-entering (§ 20;) these sides are attached to the summits of the above two faces by triangular films, and to the edges of these same faces by trapezoidal films; moreover, the summits of the film in question are attached by triangular films to the other solid edges. The different examples which I have given in detail will suffice to make it understood how laminar systems are generated, and to show that theory can render an account of all the particulars which this generation presents. § 29. In § 19, of the 5th series, I stated the laws which govern the laminar systems of polyhedral frames, when these systems are formed. Of these laws, which are five in number, three have already been discussed, (§§ 16 to 23 :) they relate to the number of films terminating at the same liquid edge, and the equality of the angles between these films; the number of liquid edges terminating at the same liquid point, and the equality of their angles; lastly, the formation of a film proceeding from each solid edge. Let us now consider the two other laws. One of these rests upon the fact that if care be taken that there shall be no bubble of air at the surface of the liquid of the vessel before the frame is plunged therein, the laminar system will present no space closed on all sides by films, and that hence all the films will be in contact by their two faces with the ambient air. In effect, if, while the frame is being withdrawn, the system, before it undergoes the rapid modification which gives to it its final arrangement, contained a space closed on all sides by films, this space must have originated during, and have increased in proportion to the elevation of the frame; now this is impossible, for the air which must fill it would have had no passage by which it could have entered; for the same reason the system, at this period of its generation, could present no space closed in part by films and in part by the surface of the liquid; in fine, when the rapid modification is effected, the film or films which then ascend into the system finding no space of the second kind to complete its laminar closure, the complete system will necessarily satisfy the law in question. The last law is the following: when the conditions of other laws can be fulfilled by plane surfaces, the films take that form; when this cannot be, all the films or several of them become more or less curved, but always in such manner as to constitute surfaces of mean curvature null. The first part of this law is a matter rather of evidence than demonstration; the plane being the most simple surface, nature, which always proceeds by the least complicated methods, will not unnecessarily give curved forms to the films. The second part of the law follows from the first, and I have already made application of it in what I have previously stated respecting the laminar systems of prismatic frames. It is proper that a reason should be here given for the special fact that the laminar system of the regular octahedron is formed of curved films when it is obtained with oil in the alcoholic liquid. We have seen (§ 16) that in this mode of producing laminar systems the thickness of small remaining masses has great influence. Now, in the octahedral frame, when, by the gradual withdrawal of oil, the point has been reached at which the system is spontaneously modified, since the masses of junction have still a considerable thickness, and the oil which composes them accumulates, in the definitive system, around the points where four liquid edges would terminate, so as to form masses much thicker than the edges in question, and since, in fine, several of these edges are sufficiently short for the masses which occupy their extremities to be in commanication of curvature with one another, it will be readily conceived that there must result from thence an influence on the form of the edges and of the films; and it cannot be doubted, that if we might, without occasioning the rupture of the system, sufficiently reduce the thickness of the masses in question, all the films would become plane. § 30. To return to the systems of the prismatic frames: Besides the facts already stated, these systems have presented others equally curious, which I here propose to notice. The system obtained with the pentagonal frame of Fig. 30 is, as was seen. (§ 20,) composed of films obviously plane. Now, if we consider the oblique films which proceed from two homologous sides of the two bases to unite at one of the sides of the central pentagonal film, and recollect that these two oblique films must form between them an angle of 120°, it will be evidently seen that, for bases of given dimensions, an augmentation in the height of the prism involves a diminution in the extent of the central pentagonal film, and that there is a limit of height beyond which the existence of this film is impossible. It will be found, without difficuly, that the limit in question corresponds to the case in which the ratio between the height of the prism and the diameter of the circle which might be inscribed at the base would be equal to V3, that is to say, to 1.732. It might naturally be asked, What becomes of the laminar system when his limit is overpassed? In order to know this, I had a frame constructed m which the height was about 24 times the diameter of the inscribed circle, and yielded me a singular result. When it is withdrawn from th glyceric liquid. as the lateral edges are of ordinary iron wire, all the lateral faces are at first ce cupied by plane films, and after complete emergence, a plane film is formed al-o in the inferior base, then ascends among the others, forming a pentagon wh.cn continues decreasing, everything occurring as with the frame of Fig. 30; but the pentagonal film decreases much more rapidly, then disappears, and at the instant the system undergoes an abrupt change, assuming an odd arrangement which it would be difficult to represent clearly in perspective, but of which I shall still attempt to give an idea. On the two bases respectively rest two identical assemblages, composed of five curved films; one of these assemblages is Fig. 38 represented in projection on the plane of the base by Fig. 38 It will be seen that there are in each of them a pentagonal film, two quadrangular and two triangular films. These two assemblages are connected with each other by films which proceed from the five lateral edges of the prism, and by two other intermediate films, whose direction is also in the length of the frame, and which proceed from the liquid edges a b and b c of one of these assemblages to terminate at the homologous liquid edges of the other. There is no need of remarking that the same thing would occur if, instead of overpassing the indicated limit, we should simply attain it-that is to say, if we gave to the prism the height which would precisely correspond to the annulling of the pentagonal film; in effect there would then be ten liquid edges terminating at the centre of the system, in the same liquid point, and consequently equili brium would be unstable. § 31. Although, in the laminar systems of the prisms with a greater number of sides, the oblique films must be considerably curved, it appeared to me probable that there would also be, for each of these prisms, a limit of height beyond which the system would no longer comprise a central polygonal film, and that this limit would differ little from that pertaining to the pentagonal prism. To ascertain this, I tried first the hexagonal prism, with a frame whose height was also about 2 times the diameter of the circle which might be inscribed at the base. Now, to my great surprise, a central hexagonal film was still formed, though much smaller than with the frame of § 20; but the system had undergone a modification which rendered possible the existence of this film. The points of the solid lateral edges from which proceeded the oblique liquid edges (§ 26) were situated much further from the summits of the bases, so that the arrangement was very nearly such as if, in reality, the frame had been shortened. In this arrangement, therefore, the films proceeding from the sides of the two bases remain, to a sufficently great distance from the latter, adherent to the solid lateral edges, whence it follows that the assemblage should be considered as an imperfect laminar system, resulting from a conflict between the tendency of the films to occupy the lateral faces of the prism, and the kind of traction which these films undergo on the part of the hexagonal film which ascends among them. I say imperfect, because the films which proceed from all the sides of the same base are, in the parts which remain attached to the lateral solid edges, separated from one another and rendered independent by these edges. § 32. This becomes more evident with prisms the number of whose lateral faces exceeds six; then the angle of two adjacent lateral faces being superior to 120°, the films have more tendency to occupy all these faces, and the portions which remain attached to the lateral solid edges are, in effect, much more extended. For example, with an octagonal frame in which the ratio between the height and the diameter of the circle inscribed at the base is nearly the same as in the above frames, the central octagonal film, instead of being small, is, on the contrary, very great, and the two oblique liquid edges proceeding from any one of its summits attach themselves to the corresponding Solid lateral edge, at two points, of which the distance is but about the sixth of the length of that edge, and consequently a little less than the half of the diameter of the circle inscribed at the base. In this case, therefore, the films which proceed from two homologous sides of the bases in order to direct themselves towards the central octagonal film, only abandon the solid lateral edges on approaching the middle of the height of the frame, and until then they occupy, under a perceptibly plain form, the lateral faces of the prism. In the hexagonal frame of the preceding paragraph, the distance between the points where two liquid edges proceeding from one of the summits of the central film attach themselves to the same lateral solid edge is about double the diameter of the inscribed circle. In the octagonal frame, as we have just seen, it is somewhat smaller than the half of that diameter; in a heptagonal frame it is, as might be expected, intermediate between those two values, and equal, nearly, to three-fourths of the same diameter. I have tried, also, a decagonal frame, and in this the distance in question is but the sixth of the diameter. The facts just recited would constitute an exception to our law, according to which one film ought to rest on each of the solid edges of the frame, since, reckoning from either base to the points at which the oblique liquid edges arise, two films are attached to each of the lateral solid edges; but, as I have shown, the laminar systems in question are imperfect systems. These facts are not owing to the thickness of the metallic threads: they still present themselves when the lateral edges of the frames are of the finest iron wire; only, in this case, the separation of the points of attachment of the oblique liquid edges is a little greater. § 33 Having afterwards employed an octagonal frame of which the height was a third of the diameter of the inscribed circle, I found that although, according to the value before given for the separation of these points of attachment, all of them should have been at the summits of the prism, it was not so: the points in question were still at a certain distance from the summits, and their separation was not more than the sixth of the diameter of the inscribed circle; hence the octagonal film was still enlarged. The same effect was produced with a heptagonal prism of which the height was half the diameter of the inscribed circle that is to say, less than the separation of the points of attachment as previously estimated in regard to prisms of that number of sides. The same thing occurs in the hexagonal prism, since with a frame of this kind, (§ 26,) the height of which was but 1 of the diameter of the inscribed circle, the points of attachment of the oblique liquid edges were still found to be a small distance from the summits. § 34. In order to discover the cause of these last facts, let us consider an octagonal or heptagonal prismatic frame sufficiently high for the films which proceed from the sides of the bases to occupy under a plane form considerable portions of the lateral faces. At the places where these films quit the faces in question to direct themselves towards the sides of the central polygonal film, they are necessarily convex towards the exterior in the direction of their length; but in consequence of the necessity of a mean curvature null, it is requisite that at these same places they should be concave towards the exterior in the direction of their breadth. If, then, we conceive the frame to be traversed by two planes perpendicular to its axis and passing by the two series of points where arise, on the lateral solid edges, the oblique liquid edges, these two planes will cut the films by arcs concave towards the exterior, and if we imagine these arcs solidified, the equilibrium of the system will not be disturbed. According to this, if we constructed a frame having for its height the separation of the points of attachment of the oblique liquid edges on the same lateral solid edge, and gave to the wires which form the sides of the bases the curvature of the above arcs. it is clear that the laminar system realized in this frame would have its oblique liquid edges proceeding exactly from the summits; but with a frame of th height or of a less height and the sides of whose bases are straight, the condi tion relative to the transverse curvatures of the re-entering films, and conse quently to the form of equilibrium of these films, can evidently not be sati-fied unless the points of attachment of the oblique liquid edges be placed at a certain distance from the summits on the lateral solid edges. § 35. If, in the different systems which we have been studying, we compare the central polygonal films with one another, it will be seen that the curvature of their sides goes on increasing from the hexagonal film to the decagonal film, a circumstance which constitutes new facts to be added to those of § 20 in confirmation of the law relative to the angles under which the liquid edges terminate at the same liquid point. The transverse curvature (§ 34) of the oblique films which are directed towards the sides of the central polygonal film being connected with the curvature of these sides, it must be less in the heptagonal prism than in the octagonal, and still less in the hexagonal; it is because of the weakness of the curvature in question in the last prism, that this yields, when it has not too much height, a laminar system almost perfect, with its hexagonal film. § 36. In reflecting on the generation of laminar systems, the inquiry has struck me whether there might not be realized, at least in the hexagonal prism and with the frame of § 31, a system devoid of the central polygonal film by withdrawing the frame from the glyceric liquid in such manner that the axis of the prism should be horizontal instead of being vertical, as in the preceding experiments. I have consequently so arranged the fork as to be able to operate in that way, and with complete success: I have obtained, indeed, two different systems, according as the frame was withdrawn in such a way that two lateral edges emerged at the same time from the liquid, or first one and then simultaneously the two neighboring ones. These two systems are of the same kind with that which is realized in a pentagonal frame sufficiently high, (§ 30;) that is to say, they are composed of two assemblages of oblique curved films connected with one another by other films directed in the length of the prism. The projection of one of these assemblages on the plane of the base is represented in the Fig. 39 first mode by Fig. 39, in the second by Fig. 40. Fig. 40 a third system of still another kind, a system of which, notwithstanding its simplicity, it is quite difficult to give a clear idea either by description or drawing: it contains two curved hexagonal films, resting, respectively, by one of their sides, on one of those of the bases, and directing themselves obliquely towards the interior of the frame; the other sides of these hexagons have, as usual, a concave curvature; the curved sides of each of these same hexagons are connected with the corresponding sides of the neighboring base and with the homologous sides of the other hexagon by curved films; lastly, at the liquid edges which unite these last films two by two, terminate other films proceeding from the lateral edges of the frame. The sides of the bases on which rest the two hexagonal films pertain to the face of the prism which first emerged from the liquid. The heptagonal prism yields analogous results, with a frame having its dimensions in the same ratio; only, at first, the three systems are imperfect, in the sense that, in the two former, the films which proceed from the sides of the bases remain, to a certain distance from the summits, adherent to the lateral solid edges, and that, in the third, the films which pass from the curved sides of one of the heptagonal films to the homologous sides of the other are attached, for the greatest part of their length, to the lateral solid edges, presenting, throughout this extent, a form decidedly plane; moreover, by a new singularity the system furnished by the second method is unstable; when barely formed, it undergoes a spontaneous modification: the two assemblages situated near the bases become elongated, at first slowly, then more and more rapidly, reach one another, and immediately there appears the imperfect system with the hepta |