the penetration of the second cube through the brass which forms the frame of the first. The prisms must be connected at top and bottom by two similar and equal flanges, the edges of which must lie between the diagonal of the cube and the extreme corner of the flange. If too far from the diagonal, connexion with the prisms would cease; if too close to the diagonal, the second cube could not pass between the flanges. The geometrical conception of a cube and the material solid representing such a conception to the sight and touch are different things. It is easy to conceive how one geometrical cube can pass through another of equal dimension by a square opening whose diagonals are perpendicular to the sides of the first cube. The triangular prisms left by this opening would be equal; but they would touch by mere lines, and could not be represented by a continuous solid material substance. They would necessarily be distinct and separate. If the second cube were passed through the first parallel to the diagonal of the square on one of its faces, two triangular prisms would be cut off each distant from the other by the side of the cube. The sides of the base of each prism p would be manifestly pa (1-1), and the long sides a (2-1). But these prisms would again be unconnected. They could be connected by triangular flanges having knife-edges, and equally inclined planes terminating at the corners of the cube. These knife-edges must manifestly be equal to the side of the cube; and as the sliding cube on each flange has its side perpendicular to the flange, the two flanges must have their edges a little distant from the diagonal to which each is parallel. The interval secures junctions of the flange with the two prisms. The thickness t of flange downwards must be also secured in order that the flange holds its place. The inclination of the face of the flange will depend upon these two quantities. The relations between a the distance of the knife-edge from the corner of the cube, t the thickness of the flange at the points of junction with the prisms, and the angle of inclination of the inner face of the flange to the face of the cube can be easily found. As the sliding cube must have one of its sides always perpendicular to the face of the flange, the following equation must subsist :— a=(a−t) cos 0+[p√2 — (x−‡a)] sin 0. As the least thickness of the flange parallel to the long side of the prism may be represented by From (2) the relation between 2 and 0 gives for 0, (1) (2) From (1) the value of 0 which makes t a maximum can be found by the usual methods, 2t=a(√2—1) tan 0—a (sec 0—1), a(√2—1) a sin e 2 dt 2 = de cos Ꮎ a cos3 0-2(2-1) cos 0 sin 0-2a sin2 0 cos 0 When dt2 dt do =0, sin 0= √2—1, this, substituted in the value for shows that the latter must be negative; hence d029 = sin 0-2-1 gives for t a maximum, or 0=24° 28' would give the greatest thickness for t. Between this and zero the thickness would give a smaller inclination and also a different value for x. In the model x has been chosen between the α two extreme values2 x This value of a would allow greater values of t and than in the model, but they have been both determined by making the greatest thickness of the flange at its extreme end equal The value of ≈ numerically is a. ·60355 nearly. The side of the cube is very nearly 1.92 inch, and the above results agree with the measured values of a, p, 0, and very nearly with that of t. It thus follows that a material solid cube can be so constructed as to allow of a cube of the same dimensions passing through it by an aperture cut in the former without separating the remaining portions. As crystals are known to be penetrated by others of similar shape, this problem may possibly illustrate questions connected with the study of isomorphous groups of the cubical type which are frequently known to present the appearance of interpenetration. XIX. On the Liquefaction of Gases. My occupied ever since the year 1883, Y researches concerning the liquefaction of gases, with have been published in various scientific periodicals in the Polish, French, and German languages, viz., in the publications of the Academy of Sciences of Cracow (in Polish), in the Bulletin International of the same Academy (in French and German), in the Annals of the Academy of Sciences of Vienna, and in Wiedemann's Annalen der Physik und Chemie and in his Beiblätter, as well as in the Comptes Rendus. Though I suppose that my labours are sufficiently known to the scientific world, yet there are motives which lead me to ask the Editors of the Philosophical Magazine to insert the following summary of the more important results of my experiments. : Firstly, because my researches appeared irregularly in different scientific papers, as they proceeded; such as wished to become acquainted with them being obliged to look them up in all the papers I have mentioned. Secondly, because of the experiments and public lectures of Prof. James Dewar, concerning the liquefaction of large quantities of oxygen and air. In several cases Prof. Dewar merely repeated my experiments for instance, as regards the absorption spectrum and the colour of liquefied oxygen. In these cases he confirmed the observations I have made, and mentioned the results of my work in the manner usually received in the scientific world. But in his last experiments and lectures respecting the liquefaction of considerable quantities of oxygen and air and their employment as cooling agents, Prof. Dewar has thought fit not to make any mention of my labours in the same field, which had been published several years before Prof. Dewar went over them again. In the number for June 1890 of the Bulletin International de l'Académie de Cracovie, I have described an apparatus serving to liquefy a greater quantity of oxygen or air in a steel cylinder, from which it can be poured out into an open glass vessel, and used as a frigorific agent. It is entitled "K. Olszewski. Transvasement de l'oxygène liquide ;" and a brief report on the subject is contained in the Beiblätter of Wiedemann, vol. xv. p. 29, under the title "K. Olszewski. Ueber das Giessen des flüssigen Sauerstoffs." That my labours should have thus been passed * Communicated by the Author, over in silence is all the more astonishing, because as soon as the above-mentioned Bulletin was printed I sent a proof of it to Prof. Dewar; I also forwarded him proofs of my other researches, knowing that they interest him. The apparatus I constructed and described works very well and can be used without danger, so that in October of the same year (1890) I was enabled to obtain 100 cub. centim. of liquid oxygen in the presence of an audience consisting of over 100 students. In the following year, during the Congress of Polish naturalists and physicians in Cracow (July 1891) I obtained 200 cub. centim. of liquid oxygen in the presence of a good many physicists, and showed its peculiar properties; as, e. g., its bluish colour and its absorption spectrum. Subsequently, without having altered my apparatus in any way, I got about 200 cub. centim. of liquid air and used it as a frigorific agent in order to liquefy hydrogen. The construction of my apparatus is very simple, and it can easily be enlarged by using a steel cylinder of the capacity of 300, 400, 500, or more cubic centimetres. The only reason that I have never hitherto employed a steel cylinder of greater capacity than 200 cub. centim., is the circumstance that the quantity of oxygen or air which can be liquefied in this cylinder was quite sufficient for my experiments. After these remarks, I shall now give a summary of the more important results of my former labours concerning the liquefaction and solidification of gases, and then describe the apparatus I constructed, which serves to obtain great quantities of liquefied oxygen and air; also stating my experiments made in order to liquefy hydrogen, by using large quantities of liquid oxygen or liquid air as frigorific agents. Summary of the Results of my former Experiments. In 1883, and for several years following, I liquefied the gases in a strong glass tube, about 30 centim. in length, 14-18 millim. in diameter within, with walls from 3 to 4 millim. thick. Oxygen, nitrogen, atmospheric air, carbon monoxide, nitric oxide, and methane, submitted to the influence of cold in the tube by means of liquid ethylene, boiling in vacuo at a temperature of 150° C., were easily liquefied under a pressure not beyond 50 atm. As my experiments proceeded, I published their results in the periodicals I have mentioned; and a detailed description of the apparatus I used in my experiments is contained in Wiedemann's Annalen der Physik und Chemie, 1887, vol. xxxi. p. 58, under the title "K. Ölszewski. Ueber die Dichte des flüssigen Methans sowie des |