VERHANDELINGEN DER KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN EERSTE SECTIE (Wiskunde - Natuurkunde - Scheikunde - Kristallenleer - Sterrenkunde - Weerkunde en Ingenieurswetenschappen.) DEEL XI Met 7 platen en één kaart. AMSTERDAM — JOHANNES MULLER Juni 1913. LA Li We 4 Pa [28 California Academy of ne RECEIVED BY EXCHANGE Yv ¢ a ‘ 4 à b a f \ 1 A ‘ * ce | xs a | 20 4 ON ny ‘ HA: jt \ vite à \ x > ‘A { RURALE . Ld ’ > sh r k en - D L | , er, VERHANDELINGEN fa, DER KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN ERS SCEE (Wiskunde - Natuurkunde - Scheikunde - Kristallenleer - Sterrenkunde - Weerkunde en Ingenieurswetenschappen.) DEEL XI Met 7 platen en één kaart. AMSTERDAM — JOHANNES MULLER Junt 1913: Gedrukt bij Jon. ENscHEDÉ EN Zonen. — Haarlem. i oo a en) . INHOUD. Miss A. Boore Storr. Geometrical deduction of semiregular from regular polytopes and space fillings (With 3 plates). M. H. van Beresteyn. Getijconstanten voor plaatsen langs de kusten en benedenrivieren in Nederland, berekend uit de waterstanden van het jaar 1906. (Met één kaart). P. H. ScrHoure. Analytical treatment of the polytopes regularly derived from the regular polytopes. (Section I). (With one plate). H. Douritu. Theoretische en experimenteele onderzoekingen over par- tieele racemie. (Met één plaat). P. H. Scuours. Analytical treatment of the polytopes regularly derived from the regular polytopes. (Sections IT, HII, IV). (With one plate). B. P. Moors. Etude sur les formules (spécialement de Gauss) servant à calculer des valeurs approximatives d’une intégrale définie. (Avec une planche). 5, line 12 from top replace ,,edges’’ 29 8 9 15 I 8 22 AR ARS oe 7a a Re by ,,limits 4,” ie ja hare, 19x 227, 227 Geometrical deduction of semiregular from regular polytopes and space fillings BY bins. A BOOLE STOET: Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam. (EERSTE SECTIE.) DEEL XI. N°. 1. (With 3 plates). AMSTERDAM, JOHANNES MULLER. 1910. Geometrical deduction of semiregular from regular polytopes and space fillings BY Mrs. A. BOOLE STOTT. Introduction. 1. The object of this memoir is to give a method by which bodies having a certain kind of semiregularity may be derived from regular bodies in an Euclidean space of any number of dimensions; and space fillings of the former from space fillings of the latter. These space fillings or nets for threedimensional space have been given in a paper entitled ,,Sulle reti di poliedri regolari e semire- golari e sulle corrispondenti reti correlative” by Mr. A. ANDREINI D, who deduced them by means of the angles of the different poly- hedra. Photographs prepared for the stereoscope, taken from that paper, representing the various semiregular space fillings were sent to me by Prof. Scnourg to whom I desire to record here my thanks for the generous help he has given me during the whole course of this investigation. Thesc photographs suggested a method by which at once the semiregular bodies and the manner in which they combine to fill fourdimensional space could be derived from regular polytopes and nets in that space. It will be seen that this method can be applied to spaces of any other number of dimensions. The semiregularity considered here is that in which there 1s one kind of vertex and one length of edge ?), and the symbols used *) Memorie della Società italiana della Scienze (detta dei XL), serie 38a, tomo XIV. *) So the greater part of the forms called semiregular here will have a degree of regularity less than + in the scale of Mr. E. L. Eure. . A 1* À GEOMETRICAL DEDUCTION OF SEMIREGULAR ETC. for the polyhedra of this description, almost the same as those given by ANDREINI, are as follows: T, C, O, D, T indicate the five regular polyhedra, using the initial letters of their ordinary names Tetrahedron, Cube, Octahe- dron, Dodecahedron, Icosahedron; and if y», indicates a regular polygon with 2 vertices we have | FT truncated. T, san ee eee limited by 4p, and 4%, ‘C= y» Css Vie ai ee ae 7 / 6pz n Spas AOL Orbn CRE Re 7 7 816 „Ops, A Disana ETAPE DL TER 1275 ” 2073, Ai M £5 ia Le Sa oe ee eee 1 1 20D, 7 12p;, COO and 0 im egulibnom..s. m7 0e 7 2 674 „En LDSN A Dep EE EE CR EUR 7 1 127 nu 2Ups, RCO — combination of rhombic D, Cand-O.... » uv : 18p, CHE AD " 7 Br BEET PARA 7 1 12ps, 30p, ” 20p3, (GO =S truncated) MOOIE Bin ee on en ee 2 uv. Op, 12p4 n° Boe (ID = / DELE Ze a 2 7 F " n 12pio, 80p, ” 20pe. Moreover we want: Ps, P,,.. for threedimensional triangular prisms, square prisms (cubes), etc. | Po, Po,... for fourdimensional prisms on a cube, an octahedron,.… as base, P (8;3) or simply (3 ; 3) for a prismotope *) with two groups of threedimensional prisms ?, as limiting bodies, P (6;8) or simply (6; 8) for a prismotope having for limiting bodies six octagonal and eight hexagonal prisms, etc. 2. The transformation of the regular into the semiregular bodies and space fillings can be carried out by means of two inverse ope- rations which may be called expansion and contraction. In order to deine these operations conveniently, the vertices, edges, faces, limiting bodies, of a regular polytope are called 1) By Tr we indicate the solid limited by 30 lozenges in planes through the edges of [or D normal to the lines joining the centre to the midpoint of each edge. *) According to custom the word “truncated” is used here, though this body and the next one cannot be derived from the CO and the JD by truncation. *) This body is also a ,simplotope” as the describing polygons (placed here in planes perfectly normal to each other) are triangles (compare Scnoure’s „Mehrdimensionale Geometrie”, vol. II, p. 128). In general a prismotope is generated in the following way: Let Sp and Sq be two spaces of p and q dimensions having only one point in com- mon; let P be a polytope in Sp, Q a polytope in Sy. Now move Sp with P in it parallel to itself, so as to make any vertex of P describe all the points of Q. Then P generates the prismotope. Here we have to deal only with the case of two planes (p=q=2); by the symbol (6;8) we will indicate the polytope limited by eight hexagonal and six octagonal prisms obtained in the indicated manner if we start from a hexagon and an octagon situated in two planes perfectly normal to each other. GEOMETRICAL DEDUCTION OF SEMIREGULAR ETC. 5 its limits (7) and are denoted respectively by the symbols /,, 4, Bik I. The operations of expansion and contraction. Definition of expansion. 3. Let O be the centre of a regular polytope and 47, M,, M... the centres of its limits /, /, Z,... The operation of expansion e, consists in moving the limits /, to equal distances away from O each in the direction of the line O M, which joins O to its centre, the limits /, remaining parallel to their original positions, retaining their original size, and being moved over such a distance that the two new positions of any vertex, which was common to two adjacent edges in the original polytope, shall be separated by a length equal to an edge. The polytope determined by the new positions of the limits /, will have the kind of semiregularity described above. The limits i, are said to be the subject of expansion or briefly the subject; and the new polytope is denoted by the symbol of the original regular polytope preceded by the symbol e,. A few particular cases, in 2, 3 and 4 dimensions, will now be examined. TEE Lvamples of the e, expansion. 4. Here the edges (/,) are the subject. It is evident that this operation applied to any regular polygon changes it into a regular polygon having the same length of edge and twice as many sides. In Fig. la a square is changed into an octagon by the application of the e, expansion 1). Fig. 14 shews the e, expansion of a cube. The real movement of any edge AB is in the direction of the line OM, but that move- ‘ment may be resolved into two. Thus instead of moving 46 directly to the position 4,2, it might have been moved to 4 B or to 4°B" and then to 4,B,. If the movements of all the edges be thus resolved the result is the same as if the faces AC, AD... (Fig. le) of the original cube had been first transformed into octa- gons by an e, expansion of each in its own plane, and then moved *) In these drawings the thick lines represent edges of regular polytopes in their original or in new positions, the thin lines edges introduced by expansion. 6 GEOMETRICAL DEDUCTION OF SEMIREGULAR ETC. away from the centre O until the edges 4’B’ and 4°B”, which correspond to an edge AB of the cube, are coincident and become the common edge of two octagons (transformed squares). It is to be noticed that as each vertex of a cube is common to three edges (three members of the subject) it takes three new positions, which, owing to the regularity of the cube, are the three vertices of an equilateral triangle. ‘Thus the faces of the cube have been expan- ded into octagons and the vertices into triangles. Fig. 2a shews the e, expansion of a tetrahedron. Each face is — changed into a hexagon, each vertex into.a triangle. Here again a vertex of the tetrahedron is common to three members of the sub-. ject; ‘the ‘result is 2 72. Fig. 26 shows the same expansion of an octahedron. Each face is changed into a hexagon; but each vertex into a square because in an octahedron each vertex belongs to four edges (four members of the subject); the result is a 70. From these examples it is easy to find the e, expansion of an icosahedron and of a dodecahedron. 5. This investigation leads to the determination of the e, expan- sion applied to the fourdimensional polytopes. For instance in the C, each cube is transformed (in its own space) by the e, expansion and becomes a /C (Figs. 14 and 2c). These transformed cubes must be so adjusted that an edge which was in the C, common to three cubes 1) is, in its new position, common to three transformed cubes. Again each vertex in a Cs is common to four edges and must take four new positions which are the four vertices of a regular tetrahedron. Thus the vertex of the C, is expanded into a tetrahedron, which is said to be of vertex import. This tetra- hedron might have been determined in another way; for four cubes meet in a vertex of a Cj and in each the vertex is changed into a triangle; therefore a vertex of Cg is replaced by a body limited by four triangles 1. e. a tetrahedron. The two kinds of limiting body of the new Dolto e, Ce are shewn in Fig. 2c; in Fig. 2d are shewn the limiting bodies of e, G. In Cg, where six edges meet in a vertex, the e, expansion changes each tetrahedron into a 47 (Fig. 2a) and each vertex into an octahedron (of vertex import) whose vertices are the six new positions of a vertex of the C,, Again in C, eight edges meet in a vertex, so that the e, expan- *) In order to facilitate the application of the operation of expansion it is desirable to have at hand a table of incidences; this is provided on Table III. GEOMETRICAL DEDUCTION OF SEMIREGULAR ETC. 7 sion here changes each octahedron into a ¢O (Fig. 26) and each vertex into a cube (of vertex import) whose vertices are the eight new positions taken by a vertex of Cy. In a similar manner the e, expansions of C,, and Coop may be determined. 6. ule. These examples lead to the general rule for the e, expansion of a regular fourdimensional polytope P. The limiting bodies of P are transformed by the e, expansion and the vertices expanded into regular polyhedra each having as many vertices as there are edges meeting in a vertex of P. Examples of the e, expansion. 7. As the faces are the subject in this expansion there can be no application to a single polygon in twodimensional space. The e, expansion of a cube, an ACO, is shewn in figure 34; there are three groups of faces: 1** : squares corresponding to the faces of the original cube Dr i kn Wen ede halt . ji 8" : triangles . Ree TR VERLIES Te > Dl In this expansion of any regular polyhedron the faces of the first group are like those of the original polyhedron; the faces of the second group are always squares, since they are determined by the two new positions of an edge of the original polyhedron; those of the third group are triangles, squares or pentagons according as a vertex of the original polyhedron belongs to three, four or five faces. | As the cube and the octahedron are reciprocal bodies, the num- ber of vertices lying in a face of one being equal to the number of faces meeting in a vertex of the other, it follows that the e, expansion of the octahedron is also an RCO (Fig. 36). Again the tetrahedron is self reciprocal, the number of vertices lying im a face being equal to the number of faces meeting in a vertex; so in the e, expansion the faces of vertex import are, like the faces of the tetrahedron, equilateral triangles (Fig. 4). The e, expansion of the icosahedron and dodecahedron, which are reciprocal bodies, is an A/D. 8. The e, expansion of the C transforms each cube into an RCO and, as in the C each face is common to two cubes, so those faces in the RCO which are faces of the cubes in new posi- tions must now be common to two BCO. In the CG each edge belongs to three faces, so in the new polytope each edge takes 8 GEOMETRICAL DEDUCTION OF SEMIREGULAR ETC. three new positions which are the three parallel edges of a right prism on an equilateral triangular base. This manner of determining the prism (expanded edge) bears the most direct relation to the particular expansion under consideration, namely that in which the faces are the subject; but it could have been determined otherwise. ‘Thus in a © three cubes meet in an edge and as each is changed into an 2CO, its edges are changed into squares, so that instead of three coincident edges there are now three squares, the side faces of a right prism. Again in the C, each vertex belongs to six faces and therefore must assume six positions. From this it 1s evident that the body taking the place in the new polytope of the vertex in the © has six vertices and it remains to determine its faces. In figure 5 are represented, in their true relative positions as far as threedimensional space will allow, two of the four RCO and two of the four P, which have taken the place of the four cubes and the four edges meeting in a vertex of the Cs. It shews that each ACO supplies a triangular face and each prism a trian- gular face — all equilateral — to the body that takes the place of the vertex of the Cs. This body therefore is a regular octa- hedron, four of whose faces are in contact with RCO and four with P. The new polytope then, e, C, is limited by 8 RCO, 32 P, of edge import, 16 O of vertex import. 9. Rule. The rule for the e, expansion of a regular fourdimen- sional polytope P may be stated thus: The limiting bodies of P are transformed by the e, expansion. The edges are expanded into prisms each having as many edges parallel to the axis as there are faces meeting in an edge of P. The vertices are expanded into bodies having two groups of faces, one kind of edge, and as many vertices as there are faces meeting in a vertex of P. One group of faces is supplied by the bases of the prisms of edge import and of these the number is equal to the number of edges meeting in a vertex of P; the other is supplied by the expanded vertices of the transformed limiting bodies, of which the number is equal to the number of limiting bodies meeting in. a Vertex ot... 7 a ° Kivamples of the e, expansion. 10. Here the limiting bodies are the subject; and it is at once evident that this expansion applied to reciprocal fourdimensional GEOMETRICAL DEDUCTION OF SEMIREGULAR ETC. 9 bodies, e. g. to G and Og, also to Cog and Gp; must produce the same result; while applied to a self reciprocal form it pro- duces a polytope whose limiting bodies of vertex import are like the original limiting bodies and of the same number, and whose limiting bodies of face import are of the same number and kind as those of edge import. Thus as in the © each face belongs to two, each edge to three, and each vertex to four cubes, it follows that in the expansion each face takes two, each edge three, and each vertex four posi- tions. The e, C, is therefore limited by 8 cubes of body import (cubes of the original ©), 24 P, of face import, 32 P, of edge import, and 16 tetrahedra of vertex import (Fig. 6a). In the C,, each face belongs to two, each edge to four, each vertex to eight tetrahedra, so in the expansion each face takes two, each edge four, and each vertex eight positions and the e, ©, is limited by 16 tetrahedra of body import, 382 P, of face import, 24 P, of edge import; and 8 cubes of vertex import (Fig. 64). These two polytopes are alike except that the zmpords are reciprocal. 11. Generally there are four groups of limiting bodies: 1*: polyhedra of body import like the limiting bodies of the original cell, 2": prisms of face import defined by their bases (two positions of each face of the original cell), 3™: prisms of edge import defined by their edges parallel to the axis (as many positions of an edge as there are bodies meeting in an edge of the original cell), 4'": polyhedra of vertex import having as many vertices as there are bodies meeting in a vertex of the original cell. So in e;C, there are 10 7, 20 P, ; in Co, there areas O, 1922. _ This expansion of a C,, and a Cop (reciprocal cells) can easily be determined. 12. ule. The rule for the e, expansion of a regular polytope P of fourdimensional space is as follows: The limiting bodies of P are moved apart (untransformed). The faces are replaced by prisms whose bases are parallel posi- tions of a face of P. The edges are replaced by prisms each having as many edges parallel to the axis as there are limiting bodies meeting in an edge of P. Hach vertex is replaced by a regular polyhedron, the number of whose vertices is equal to the number of limiting bodies meeting in a vertex of P. 10 GEOMETRICAL DEDUCTION OF SEMIREGULAR ETC. Generalization. 13. The foregoing result may be generalized thus. If any set of limits e, be the subject of expansion in a regular polytope P, in a space of # dimensions the polytope P,’ detined by the new positions of the members of the subject has for its limits 7, ,: 1°: a group consisting of the limits Z,_, of P, transformed by the e expansion (e, Z,_,), 2": a group of vertex import, each member of the group being determined by its vertices, the number of which is equal to the number of limits 7, meeting in a vertex of P, and having one kind of edge. This polytope is regular in the e, and the e,_, expan- sions. These two groups are the principle ones. 3": there are besides various kinds of prisms. Those of edge import (l-import) are determined by the new positions of an edge of P, and the number of these positions is equal to the number of limits Z. meeting in an edge of P,. The prisms of face import (2-import) are determined each by the new positions of a face of P,, and the number of these is equal to the number of limits J, meeting in a face of P, and so on. The whole series of prisms is as follows: l-import, 2-import, .....—J-import. Combination of operations. 14. The expansions described above have been applied to regular bodies according to the definition given on page 5, transforming them into bodies possessing a particular kind of semiregularity. The question now arises: can these semiregular bodies be trans- formed by the application of any further expansion without having lost the kind of semiregularity defined above? It is evident in the first place that a movement of all the edges or of all the faces would produce bodies with edges of different lenghts. But an inspection of the transformed bodies in three- dimensional space (Figs. 16, 2q@ and 26) shows that in each of the polyhedra /C, 47 and 40 there are two groups of faces, each of which taken alone defines the polyhedron: one group corres- ponds to the faces (expanded), the other to the vertices (expanded) of the original polyhedron, and these two groups differ as to a particular characteristic. The members of the first group are in contact with members of the same group; the members of the second are separated by at least the length of an edge from members of their own group. As GEOMETRICAL DEDUCTION OF SEMIREGULAR ETC, 11 the operation of expansion applied to a set of limits has the effect of separating any two adjacent members, it follows that the first group can, the second group cannot, be made the subject of ex- pansion. For instance in e, C (Fig. 7a) the triangles cannot be moved away from the centre without increasing the length of the edges joining them, but the octagons niay be moved away from the centre until the edge 4B common to two has assumed two new positions AB, A'B" which are the opposite sides of a square. The new positions of the octagons define a polyhedron having the required kind of Beele ) 15. This double operation may be denoted by the symbol e, e, C where it is understood that the faces forming the subject of the Ca expansion are only those which have en the place of faces in the original cube. Similarly the interpretation of the sym- bele er C is that the e, expansion is applied to a cube and that the subject of further expansion is composed of those faces which have taken the place of edges in the original cube. This is shewn in Fig. 74 where the group of 12 squares (corresponding to the edges of the original cube) form the subject of expansion. ‘These two figures 7a and 76 show that | EN C= 100 and it is evident that the order in which the operations are applied to any regular polyhedron is indifferent, for the two operations could have been carried out simultaneously. In Fig. 7e is shewn the result of the double operation e, e, O applied to an octahedron. This is also a 7¢CO. If the double operation be’ applied to a 7 and an 2 the result in both cases will be a ¢/D. This body and the ¢CO are incapable of further expansion. 16. Thus it appears that three expansions can be applied to the cube, octahedron, dodecahedron, icosahedron, namely re, €, e‚ €. But more can be done with the tetrahedron owing to the fact that it is self reciprocal. Fig. 7d and 7e show respectively the result of the e,e, and the e,e, expansion applied to a tetrahedron, and the result in both cases Is a ¢O which can be further expanded into a {CO (Fig. 7c). Thus the self reciprocity of the tetrahedron allows an expansion which cannot be carried out in the other four polyhedra. The *) Here the group of octagons may be called the ,,independent” variable, while the triangles, which are transformed into hexagons, are the dependent” ones. 12 GEOMETRICAL DEDUCTION OF SEMIREGULAR ETC. combination of operations may be applied in the same way to the cells of fourdimensional space as one or two examples will show. 17.. Case e,e, Cz. — The e, operation applied to a C, produces a polytope limited by 8 RCO, 32 P,, 16 O (Fig. 5). The symbol ee Cg directs that the new subject of expansion comprising those limiting bodies in e, Cg which correspond to edges of ©, i. e. the 32 P,, shall, themselves unchanged, be carried away from the centre (of the e, C,). These P, in their new positions define the polytope sought. This movement changes the RCO and the O. Each RCO was derived from a cube by the e, expansion; the new expansion e, carries out the group of 12 squares (corresponding to the edges of the cube), thereby producing a 7CO (Fig. 76). Im order to determine the change in the octahedron of vertex import it 1s only necessary to observe that four of its faces (those in contact with bases of P;) are still in contact with them and are only changed in position, while the other four (those which were in contact with RCO) are changed into hexagons in contact with ¢CO. Thus the octahedron is changed into a 47. The effect on a single octahedron is the same as if its alternate faces had been made the subject of expan- sion (Fig. 8). 18. Case e,e, Cg. — The result of applying the e, operation to a C, is a polytope limited by cubes (original cubes of the C,), P, of face import, P, of edge import, and tetrahedra of vertex import (Fig. Ga). The symbol e, directs that the square prisms of face import shall be moved away from the centre of e, Cy, they them- selves remaining unchanged except in position. These in their new positions define the new polytope and it only remains to determine in what manner their movement has modified the remaining limi- ting bodies of the e, Cz. This can be seen at once in a drawing. In figure 9a are shewn seven limiting bodies of the e, Cg; one isa cube of the original C,, after having been separated by the es movement from the adjacent cubes; three are cubes of face import interposed by the same movement between the cubes of the CG; three are ?, of edge import, their bases being faces of a tetra- hedron of vertex import. The symbol e, directs that the cubes of face import are to be moved out. The result is shewn in figure 96; the original cube is changed into an RCO, the P, into a P, and the 7 into a 47. It is necessary to bear in mind that only one limiting body of any polytope can be in threedimensional space at a time, and in representing several at once in it there must be either distortion of the limiting bodies or separation of faces and edges which ac- GEOMETRICAL DEDUCTION OF SEMIREGULAR ETC. 13. tually coincide. Moreover the direction of the real movement cannot be represented; but valid conclusions may be drawn from diagrams such as these, if the mind always distinguishes between the actual and the apparent relation of parts. | These two examples suffice to show how the result of the com- bination of operations may be applied to the fourdimensional cells. There are seven expansions of each: ETE EERE LE a en) but owing to the reciprocity of some of the figures these are not all different. Thus it appears that in any expansion a set of limits, which define the body and which is such that each member is in contact with other members of the same set, may be made the subject of expansion. Definition of contraction. 19. In each of the expansions e,, e,, e,... the resulting semi- regular polytope may be reduced to the regular one from which it was derived, by an inverse operation which may be called con- traction. Here the limits which formed the subject of the expansion are moved towards the centre and brought back to their original positions. The direct operation separates the members of the subject; the inverse operation brings them again into contact, annihilating the edges introduced by expansion. In both positions they define the polytope of which they are the limits. The conditions necessary to the inverse operation are: 1%, the limits forming the subject must define the polytope; 2%, no two members of the subject can be in contact before contraction. The polytopes of vertex import always satisfy these conditions and can be made the subject of contraction. ‘The symbol ¢ is used to denote contraction. ‘The import of the limits forming the subject is shown by means of subscripts, as in expansion. Eramples of contraction. 20. The inverse operation will be made clear by one or two examples. In figure 10 the square 4 B C D has been expanded by the e, operation; the edges of vertex import in the resulting octagon have been made the subject of the inverse operation, that is, they have been moved nearer to the centre so far that the edges of the original square are annihilated, and the final result is the square 14 GEOMETRICAL DEDUCTION OF SEMIREGULAR ETC. HL F G H, denoted by the symbol ¢ e, S where 8 is the square A BD; ar In figure 104 is shown a cube transformed by the e, operation, ie. an e, C; the triangles of vertex import are brought nearer to the centre by the c, operation and the result is a CO whose sym- bol is now ce, C. Again, the ¢CO may be considered in two ways. It may be deduced from either the octahedron or the cube (compare Figs. 7a and 74), so it may be denoted by e, e, C or e, ez O. Though the identity of these results may be expressed in the form of an equation: e‚ eg C= e,e, O, it must still be borne in mind that the imports are different. Let each of these symbols be preceded by c,. What are the results? If the {CO has been derived from the cube, the hexagons are of vertex import; if, on the other hand, it has been derived from the octahedron, the octagons are of vertex import. Thus the symbol ce, ez C indicates that the hexagons, and the symbol ¢)e,é@, 0 that the octagons, are the subject of the inverse operation whence c, ee, C= ¢O (Fig. Te), ep neg O = tC (Fig. Ta): But the octagons correspond to the faces of the cube and the hexagons to the faces of the octahedron, so that ce, e, C= ce, e, O, Ge EN. 21. An example will show the combination of inverse operations. The ¢CO derived from a cube (Fig. lla) may be reduced to an octahedron by moving the squares and the hexagons nearer to the centre; the ¢CO derived from an octahedron (Fig. 114) may be reduced to a cube by moving the squares and the octagons nearer to the centre. These operations are denoted respectively by the equations Co Ci Cen Oy ECO CO 22. In figure 5 are shown, the limiting bodies of an e,G. If the octahedra of vertex import be made the subject of the inverse operation, the following changes will take place: each #3, sepa- rating two neighbouring octahedra, is reduced to two coincident triangles. This annihilates the edges of the prism parallel to the axis. But these are the edges of the original @ in the new positions due to expansion and if these be annihilated each RCO will be reduced to an octahedron. ‘Thus the new body is a Cy, eight of whose limiting bodies are compressed RCO, while sixteen are of vertex import in the expansion e, Cs. As in the enumeration of the polytopes and the nets given in the three Tables only the c, appears, c, has been replaced by c. GEOMETRICAL DEDUCTION OF SEMIREGULAR ETC. 15 Partial operations. 93. It has been seen that in both expansion and contraction it is a necessary condition that the subject of operation shall define the polytope both before and after the movement. In expansion, each member of the subject must be in contact with other members. In contraction, each member must be separated from the other members by at least the length of an edge. It sometimes happens that one of these conditions is satisfied by a group consisting of the alternate members of a set of limits. Such a group may then be made the subject of expansion or contraction. If the members be in contact, they may be made the subject of expansion; if they be not in contact, they may be made the subject of contraction. | 24. Thus, an octahedron 1s defined by a group of four alter- nate triangles, but each of these triangles is in contact with the other three, so that these four may be made the subject of expan- sion. ‘This partial operation, which changes the octahedron into a truncated tetrahedron, is denoted by the symbol $e, O. So Le, O — {1 Again, a CO whose symbol is ¢ eC is defined by a group of four alternate triangles. Hach of these is separated from the others by the length of an edge. This group may therefore be made the subject of the c operation, which changes the CO into a 7. So fqq¢e,C= T. It may be remarked that the partial. contraction $c, can never take place without a previous complete contraction c,. | 25. The corresponding case in fourdimensional space is expressed by the symbol +cce Cs. This indicates that first, the edges of the C, are made the subject of expansion; second, the sixteen tetrahedra of vertex import are made the subject of contraction; third, a group of eight alternate tetrahedra are made the subject of still further contraction. This last partial movement changes the cubes of the C% into tetrahedra and anmihilates eight of the tetrahedra of vertex import, thus changing the C into a Cg, eight of whose limiting tetrahedra are derived from the limiting cubes of the Cs, the remaining eight being of vertex import. So Beg Gh Ce Oi Og These examples suffice to show in what manner and under what conditions the partial operations may be applied. 16 GEOMETRICAL DEDUCTION OF SEMIREGULAR ETC. Il. Application to space fillings. Expansion applied to the nets. 26. A space filling or net in any space #, may be considered as a polytope with an infinite number of limiting spaces of z dimen- sions in a space #,,, of one dimension higher. ') According to this view the operations of expansion and contraction and their com- binations may be applied to nets; but the fact that the net is a particular case of a polytope modifies the manner in which the operation is to be applied. Expansion has been defined as a movement of any set of limits away from the centre of a polytope. This movement in general separates the members of the subject. In a polytope in #, with an infinite number of z—1-dimensional limits (a net) the centre is at an infinite distance in a direction normal to the space S, of the net and no movement away from the centre can separate the limits forming the subject, in other words can expand the net. Now it has been shewn that the real movement taking place in an expansion may be resolved into two, one of which transforms the limits each in its own space and the other adjusts those transformed limits. In this way the operation can be applied to the special case under consideration. Thus if the e, expansion be applied to a net of squares (Fig. 12) they are transformed into overlapping octagons and then the octagons must be moved apart until an edge which was common to two squares becomes common to two octagons. | This adjustment leaves a gap A, A, A, A, (vertex gap) between the octagons corresponding to the vertex 4 common to four squares. ‘Thus the transformed net of squares is composed of two constituents, octagons corresponding to the squares, and squares corresponding to vertices of the original net. 27. In threedimensional space there is only one regular space filling 1. e. the net WC of cubes. The net V(O, 7) of octahedra and tetrahedra is semiregular. If the e, expansion be applied to a net of cubes each cube is trans- formed into a 4C. These will overlap and must be moved apart until an edge which was common to four cubes becomes common to four 7C (Fig. 13) By this adjustment octahedral gaps (vertex gaps) are left at the vertices. So the net e‚ VC is formed of ¢C and O. In order to determine the octahedra it is necessary to observe that as a vertex of the original net belongs to six edges, 1. e. ‘) See the quoted paper of ANDREINI, art. 47. PPT ES GEOMETRICAL DEDUCTION OF SEMIREGULAR ETC. 17 six members of the subject, each vertex takes six new positions, forming the six vertices of an octahedron (see rule, art. 6) whose eight faces are supplied by the expanded vertices of the eight cubes meeting in a vertex of the original net. 28. The application to fourdimensional space is simple. a For instance if the e, expansion be applied to a net of C, each Cg is changed into an e, Cs (Fig. 2c), two adjacent ones having a ¢C in common. As a vertex in the net C, belongs to eight edges (eight members of the subject) each vertex takes eight new posi- tions which are the eight vertices of a Cig. The limiting bodies of this Ci may be identified as follows. In the net ( each vertex is surrounded by 16 members. Each vertex of a C is changed by expansion into a tetrahedron, so that the vertex gap in the net is surrounded by 16 tetrahedra, the limiting bodies of a Cy. Thus by the e, expansion a net of Cs has been converted into a net e, VC, of two constituents, e, Cg and Cg, in which two adjacent e, CG have a ¢C in common, while an e,C, and a QC, have a tetrahedron in common. 29. Again the e, expansion may be applied to a plane net. In this case the constituents of the net are moved apart until an edge assumes two positions, the opposite sides of a square, and the vertex gap is a polygon with as many vertices as there were constituents meeting in a point in the original net; figure 14 (a and 6) snews this with regard to a net of triangles. If the e, operation be applied to a net of squares, it moves apart the squares and the result is again a net of squares; but they are not all of the same kind, some being the squares of the original net, some of edge import, others of vertex import (Fig. 15). From this simple example it may be seen that the e, expansion applied to a net of measure polytopes in z-dimensional space produces again a net of measure polytopes; but the latter is composed of constituents with different imports, and the subject of any further expansion must be suitably chosen. For instance if the e, e, expansion be applied to a net of squares the subject of the e, expansion comprises only those squares of edge import intro- duced by the e, expansion in a net of squares (Fig. 154). The result is that the squares of the subject remain unchanged except in position. Those of vertex import and those corresponding to the squares of the original net are changed into octagons of different imports. The corresponding double expansion of the net of triangles is shewn in figure 1 4c. | 30. If the e, expansion be applied to a net of cubes each cube Verhand. Kon. Acad. v. Wetensch. (1° Sectie) DI, XI. À 9 18 GEOMETRICAL DEDUCTION OF SEMIREGULAR ETC. is changed into an RCO. Four of these are shewn in Fig. 16 after having been adjusted so that a face which was common . two cubes: de common to two ÆÀCO.. à This adjustment leaves edge gaps and vertex gaps. As an edge belongs to four and a vertex to twelve faces ca bers of the subject) Ene edge gap is defined by four new parallel positions of an edge and the vertex gap by twelve new positions of a vertex. ‘Therefore the first is filled by a square prism (a cube) and the second by a CO. In the CO the triangles are supplied by triangular faces of the eight RCO (expanded cubes) and the squares by the bases of the six prisms (expanded edges) surrounding the gap. ‘Thus the net of cubes is changed by the e, transformation into a net e, VC with the three constituents RCO, Cand CO (A. 20) 4. The e, expansion may be applied to a net MO,7) of O and T by taking either the group of O or the group of 7’ as inde- pendent variable, and the faces of that group as subject. Whichever group is chosen, its faces in their original position define the net N(O,T), in their final position the new net. Thus if the e, expansion be applied to the O each O is changed into an RCO (Fig. 36) whose triangular faces are in contact with the untransformed tetrahedra. The vertices of each O are now changed into squares (Fig. 36) and as six octahedra meet in a vertex of M(O,7) the vertex gap is a cube. Thus the new net e,MO,7) has three constituents ROO; 0, 7 (Pigs Vay, In figure 18 is shewn the result e, MO, T) = = e, MOT) of the e expansion applied either to the octahedra or to the tetrahedra of the net (O;7Y. | 31. In fourdimensional space an example is given of the es expansion e, MNC, Hach GC, is changed into an e, C, limited by 24 RCO, 96 P,, 24 CO (see rule, art. 9 and Fig. 197).4 The ACO are transformed octahedra, the P, are expanded edges, and the CO expanded vertices. When the transformed C,, are adjusted so that. an octahedron which in the regular net is common to two C, is changed into an RCO common to two e Cy, thea are edge gaps and vertex gaps. In order to facilitate the determination of these gaps it will be well to state clearly the manner in which the three kinds of limi- ting bodies are mutually arranged in the e Cy. ‘) This means Fig. 20 in Anprerni’s memoir quoted in art. 1. In order to facilitate comparison a table of threedimensional nets is given on plate III. *) Here and in the following figures + means “principal” constituent, while a, @, etc. stand for the pelytopes filling the vertex gap, the edge gap, etc. GEOMETRICAL DEDUCTION OF SEMIREGULAR ETC. 19 A shaded. face 4,B,C, common to two RCO (in Fig. 19) is the new position of a face ABC common to two octahedra in G,; A,B,, AB, A,B; are. three new positions of an edge AB of the C,,, and the two positions 4,B,, 4,B, in the RCO are identical with the two positions 4,B,, 4,8, in the prism. Again, the vertices of the CO are the 12 positions taken by a vertex 4 of the C, of which four. 4, 4, 4, A; are identical with four 4, 4, A, A; in the £CO. In the net of C,, an edge is common to four and a vertex to’ 82 faces (members of the subject), so that the edge gap is defined by four positions of an edge and the vertex gap by 32 positions of a vertex. The limiting bodies surrounding these two gaps may be found in the following manner. Four C,, meet in an edge and eight in a vertex of the net C, In each, the edge is changed into a P; and the vertex into a CO. Thus among the limiting bodies. surrounding the edge and vertex gaps there must be four P; in parallel positions in the former and 8 CO in the latter. Now in the original net two adjacent C,, let us say M & JN, have a common octahedron, or it may be said that two octahedra, limiting bodies of two adjacent C5,, coincide. So in the transformed net two adjacent e,C,, have an RCO (transformed octahedron) in common; or it may be said that two ACO, limiting bodies of two adjacent e,C,, M & N, coincide. Thus the RCO (Fig. 197) represents two coincident limiting bodies, one belonging to #7 and the other to MN. In each the face (4, B,, 4,8.) 1s In contact with a P, and these two P; can have no other point in common, or else the polytopes 47 and MN, having already one common limiting body, an ECO, would coincide. Thus two adjacent P; surrounding the edge gap have a square face in common. It remains now to seek a polytope which satisfies the following conditions. It must be determined by four parallel positions of an edge and have amongst its limiting bodies four parallel P; of which any adjacent two have a square face in common. A fourdimensional prism on a tetrahedral base is the only body which satisfies these conditions, so that the limiting bodies are 4 Ps, ee (hig. 199). Kach of the tetrahedra is determined by its vertices 1. e. four positions assumed by the end point of an edge of the net C, and is therefore of vertex import. As 16 edges meet in a vertex of the net G,,, there are 16 of these tetrahedra surrounding the vertex gap. The limiting bodies of the polytope which must fill the vertex 20. GEOMETRICAL DEDUCTION OF SEMIREGULAR ETC. gap are therefore 16 tetrahedra and 8 CO (Fig. 198). The regu- lar net of C,, has thus been transformed into one of three constituents: (1) es (limited by 2CO, Ps, CO), Fig. 197 (2) a Fig. 198 (3): ca G (limited by SCO, 167), Fig. 192. In this net ee polytopes (7) have an RCO, a (7) and a (@) have a Ps, a (7) and an (æ&) have a CO, and an (&) and a © have a 7’ in common. The ez expansion applied to a block of cubes. 32. The figure 20 shews the result es VC clearly. It has already been remarked that this expansion leads to a block of cubes of different kinds, some having face import (a), some edge import (0), and some vertex import (c). | In figure 21 is shewn the result of the operation ee; VC; the cubes corresponding to those of the original net are changed into (0; the cubes of edge import (subject of the second operation e,) remain cubes; those of face and vertex import are changed respectively into P, and RCO (A. 22). The ez expansion applied to a net of Cg. 33. Each (C,, Is expanded according to the rule and produces a polytope limited by 7’, P,, P,, C (Fig. 227). When these are adjusted, so that tetrahedra which were common to two (ig are common: to two es Cg, there are face, edge, and vertex gaps; these are defined respectively by three parallel positions of a face, 12 parallel positions of an edge, and 96 positions of a vertex; since in the WC, a face is common to three, an edge to 12, and a vertex to 96 tetrahedra (members of the subject). It remains only to determine the limiting bodies surrounding these gaps. 34. In order to find those of the face gap the three new parallel positions of the face ABC are represented by the triangles 4, B, C,, A, B,C, As By Orie. It follows from the definition of expansion that the lines 4, 4,, aida, As Andee are normal to the face 4BC and equal to an edge. ‘Thus the face gap is surrounded by two groups of three P,; one group consists of the Ps: 4, B, C, ABC, A, B, C, As Ba G,, Az Bz C; A, B,C, of face import and the other of 4, 4,4,6,B,8;; b, B, Bs C, C, C3, CC Cs A, A, Ag of edge import. GEOMETRICAL DEDUCTION OF SEMIREGULAR ETC. 21 The members of each group are in triangular contact with mem- bers of the same and in square contact with members of the other group. This polytope, called a simplotope, is a special case of a group of polytopes called prismotopes '). Two kinds of limiting bodies surrounding the edge gap have now been found, 1. e. square prisms due to the transformed Cs (Fig. 227) and P; due to the expanded face (Fig. 22); there are six of the former and eight of the latter, since six C,, and eight faces meet in an edge of NO. As the axes of these 14 prisms are parallel, the body must be a fourdimensional prism whose base is a CO of vertex import (since its vertices are the 12 positions taken by the end point of an edge). The vertex gap is surrounded by cubes (7) i CO (2), and there are 24 of each since 24 CG and 24 edges meet in a vertex of NO. | Thus there are four constituents in the new net es NC: es Ce, prismotope (3; 3), Peo; and a polytope e, Ci limited by 24 C, 24 CO. The manner in which these different bodies are in contact is indicated by the imports in the drawings and by the vertical lines. 35. Two examples are given in order to show how a second operation may be applied to the result of a single expansion (Figs. 24 & 25). | Let it be desired to apply the e, expansion to the net obtained above. Here those constituents taking the place of edges in the original VC; are the subject and must be moved unchanged into new positions. Thus the edge gap in the new net is like that in the ez expansion (compare Figs. 226 & 24). Moreover those limiting bodies of edge import in the transformed C,, and in the prismotope (face gap) must also remain unchanged (compare the parts 7 and y of Fig. 22 and Fig. 24). The tetrahedra (Fig. 227) are transformed by the e, expansion into ¢Z7' (Fig. 247). A careful examination of the manner in which the P, of face import and the cube of vertex import in the same polytope (7) are in contact with the tetrahedra will show in what manner they must be changed (see Fig. 247). From these may be traced the changes in the face gap (y) and vertex gap (4). 36. If it be desired to apply the e, expansion to e, VC, the *) Compare the foot note *) in art. 1. 22 GEOMETRICAL DEDUCTION OF SEMIREGULAR ETC. face gap remains unchanged (Figs zet and 25y), as well as the | limiting body of face import in the es Cg: (7). The tetrahedron (Fig. 227) is changed by the e, expansion se a CO (Fig. 257) and again the manner in which the other limiting bodies of this polytope are affected by the change can be traced — by an examination of the manner in which they are connected with the tetrahedra. ay The changes.in the edge and vertex gaps can also he traced (compare Figs. 22 and 25). The polytope of vertex import in Fig. 25 Is “ed as it is limited by 48 semiregular polyhedra of the same kind. The e, expansion. 37. The e, expansion applied to a net of C,, Gp ar sepa- rates the adjacent constituents by a distance equal to an edge. Thus two neighbouring members of à block are separated by a fourdimensional prism whose two opposite bases are the two limi- ting bodies that coincided in the reguiar net. The net of C, so treated results in another net of © of different imports. The net of Cg transformed by the e, expansion leads to the following result. The C,, are separated, so that instead of two having a tetrahedron in common they are separated by a distance equal to an edge. | In other words the tetrahedron common to two dad Che has assumed two parallel positions, the bases of a fourdimensional prism (Fig. 264). The side limiting bodies of this fourdimensional prism are four (of face import). As three C, meet in a face in the net of Ci, each face must assume three positions which define a prismo- tope (3; 3) (Fig. 267). Again six Cg meet in an edge of the net, therefore each edge takes six positions, 1. e. the new positions are the side edges ot a fourdimensional prism on an octahedral base (6). It may be seen by (7), (2), (y) and (@) that only one of these four polytopes pos- sesses a limiting body with vertex import, 1. e. the one filling the edge gap (8), so that the vertex gap is surrounded by octahedra, and as in the net of C, there are 24 edges meeting in a vertex it follows that 24 octahedra surround the vertex gap; that is, it is a Cy. This new net evidently may also be obtained by applying the e, expansion to the net VO. GEOMETRICAL DEDUCTION OF SEMIREGULAR ETC. 23 38. The foregoing investigation leads to the following conclusion as to the nets of fourdimensional space. | If the edges are the subject there are only vertex gaps. If the faces are the subject there are edge and vertex gaps. If the limiting bodies are the Ben, there are face, edge, and vertex gaps. If the constituents are the rie there are body, face, edge, and vertex gaps. The vertex gaps are filled by polytopes determined by their vertices. Their limiting bodies are regular or semiregular polyhedra. The edge gaps are filled by fourdimensional prisms determined by edges parallel to their axes. ‘Their bases are either regular or semiregular polyhedra and their other limiting bodies are prisms. The face gaps are filled by prismotopes determined by parallel positions of a face and are limited by two groups of prisms. The body gaps are filled by fourdimensional prisms determined by two parallel positions of a regular or semiregular polyhedron. Contraction applied to the nets. 39. One or two examples will suffice to shew the application of this process to the nets. If in the net e, WO,7) (Fig. 18) (A. 24) the CO corresponding to the vertices of the original octahedra be made the subject of contraction, the #0 are reduced to CO, the #7' to O, while the CO remain unchanged. Thus ce, (0,7) denotes a net composed BO and CO (A. 18). 40. In the net e, VC, (Fig. 19) the polytopes filling the vertex gap (x) may be made the subject of contraction, when the following changes take place. The polytope & remains unchanged except in position; the prism (2 is reduced to a tetrahedron common to two of the polytopes a; the CO of 7 remain unchanged while the #CO are reduced to cubes. Thus the net of three constituents is re- duced to one of two constituents, one limited by 8 CO and 16 7, the other by 24C and 24 CO. Tables. 41. The chief results of this memoir are tabulated in the Tables I and IT. Table I gives the 48 polytopes of expansion (the regular polytopes included) and the 42 polytopes of contraction. The first set has 24 GEOMETRICAL DEDUCTION OF SEMIREGULAR ETC. been numbered from 1 to 48; if p stands for any number, p' of the second set is obtained by application of the operation ¢(= cp) to p of the first set. The first set consists of 39 different polytopes; the second set contains only eight new ones. Table IT gives the 48 nets of expansion (the regular nets included) and of the nets of contraction only the seven new ones, so altogether 39 + 7 1. e. 46 fourdimensional nets. Table III gives the nets of threedimensional space and a table of incidences. DOT UI OI of Ww bl df: | Limiting bodies & import expansion | body face | edge Expansion. vertex TABLE OF POLYTOPES IN vo © x Symbol of Limiting bodies & import ver- tex expansion pe body face | edge Expansion. Ciao ej C190 € C120 | e3 Chao #2220 €13 C120 €303 C120 €025 Cao Contraction. | ce Cs ces C3 ces Cz Ce es C3 Cee Cz Ceye3 Cs Ce C63 Cy ce Ci20 ces Ciao ces Chao ce ea Ciao ce, es Ciao cese3 Ciao ce, ea 20 -| Symbol of expansion | tee 5 B [Limiting bodies & import | €508 Coo €023 C5oo ce Ci ceo C16 ces Chg cees cees Cie ceses C16 0) C63 Cis ces Ooo ces C00 ces Coo ceye C0 ce, es Oooo cezes C00 ce 503 Coo body | face |edge Expansion. ver- tex TABLE OF NEUES INES / IT. me Gaps ced , en oenen | PE session | et | expen | Ee tuent | body | face | edge | vertex IEEE vertex A | arent vortex Expansion. Expansion. Expansion. 1 NC, (cha eee = = 17 NC Ge 33 NC GR | 2 e NC, ABS | = = Ge 18 e,NC,, el = Ss ter 34 BREN aL Sj =) = (ei 3 e, NC, e, Ca — — Po Cas 19 e,NC,, Cl — | — Po € Cis 35 e,NC,, Al = — Pap ce, C, 4 e, NC, eC, — |(4;4)| Po ce, C, 20 e,NC,, el — |(3;3)| Pco er Cie 36 e NC, oC (SHS GE 5 e,NC, G | Pe |(4354)| Pe C, 21 e NC; Cl Pr 853) Po C,, | = 31137 e,NC,, Cul Po (853) Pr Chol == 6 e,e, NC, e,e, Cy = = Po ei Cie 22 GANGA Ne er Cl Pe e‚e, Cio 38 GANA AEC == SD eC, T e‚e‚NC, e,e,C, — |(43;8)| Po e, Cy, 23 @,e,NC,,] e2C,.| — |(3;6)| Pao 2, Can 39 e,e,NC,,|| e,e,€,,| — |(3;6)| Po e, Cg 8 e,e,NC, eG | Pre |(4;8)| Pe e,Cs |= 11] 24 ee NC, e‚ Ciel Prrl8:6)) Po e,C,, | = 43] 40 e,e,NC,, €,C,,| Pig \(336)| Pr eG, | =27 9 e,e, NC, e,€, Cy — |(434)| Pio ce, e, Cy 25 @,e,NQ,,|| eeGs| — |(8:3)! Pic | cee, 41 @,e,NC,,|| eel) — (859) Pir) cee, 10 e,e,NC, QG | Prco |(434)| Paco e, Cy 26 e,e,NC,,|| @,€,6| Poo |(3 33)| Prco e,€,, | = 42) 42 e,e,NC,, € Ca |Prco|G ;3)| Pco e‚C | —26 ial e,e,NC, eC. | Po |(834)| Pie é,€, |= 8} 27 e,e,NC,,|| eG) — |\(6;3)) Pio e,e,C,, | = 40] 43 e,e,NC,, eC| Po |(633)| Per e Cie | = 24 12} e,e,e,NG, eee, CG | — |(4;8)} Pio e,C,, 28| e,e,e,NC,q|| eee, Ciel Pr (8:6) Pi ee; Ca AA} eee,NC.,leee CG, | — (356) Pr eet 13 | eee, NC, ee: C% | Pico | (4: 8) | Prco e,€,C, | = 15] 29) e,e,e,NC,, | ae Ciel Pro |(8:6) Procol Ca 45| e,e,e,NC,, | ee Ce | Pico |(8:6)|Pcol eo | =31 14 | eee NC, ee, Cy | Pic |(8;8)| Pic e‚e, Ca 30| eeeNCll ¢¢,C,.| Por |(636)| Pio €,e,C,, | = 46/46] e‚ee,NC,ll e,e,C,. | Pio ((6;6)| Por) ee,G | =30 15 | e,e,e,NC, e‚e, Cs | Parco | (834)! Pico e,e,C, | =18] 31 eeletNGE| ee Ciel Pcol(6;3)| Pico | e162 C2, | = 45/47] e,ee.NC,.| e,e,C,, |Prcol(633)| Pol Ge 16 |e,e,e,e,NC, |laee CG | Pico |(8:8)| Pico | 020 C 32 le eee, NG | eee, Cl Pro 6: 6) Pico l'a ee, GC, | = 48] 48 BOEG eee, | Pico |(65 6)| Pool eee, Cie | = 32 Contraction. Contraction. 49 ce, NC, ce, C, — — — Cy. 51 | ce,NC,,|| ceGel — | — | — e, Cie | 50 cee, NC, || ce,e,C, — — — e Ce 52 ce,e,NC,,|| ce,e,C,.| — | — — AGG 53 ce,e,NC,,|| ce,e,€C,,| — (353) — e, Cr 54 ce,e,NC,,!| ce,e,C,,) — | — — | ce,e,C,, 55 | ce,e,e,NC,,|\ce,e,e,C,,| — |(3;3)| — ce, Cy, TABLE OF NETS IN S,. IT. ke) n ls n — 2 Gaps ‘x oa Gaps ag 8% 5 E & Hie € a es BAT 55 = Bad es 5 a | Symbol | SE BE | Symbol | SE | aa = 8 | face| edge (vertex 5 5 E 8 Al face | edge |vertex 4 À [e] A le) a S |A 5 L] nel Cc 12 MOT) || 0,7 e NC 6 4 | e NCOI) || 10,27 e NC ‘ eg N(O,T) | RCO, T ex NC } } 23 lere MOD) || (COLL exe NC| at N(O,T)| 10,10 ae NC 5 |3@N(O,T)| (LT es NC || RCO Q4bis |e,ees NC {CO 15 ce NC| CO C. C 120 Y 16 | Cells meeting in a body .. facen edge .. vertex.. face... edge .. vertex . edge... vertex.. MA BOOLE STOTT: “Geometrical deduction of semiregular from regular polytopes and space fillings." PDE MMA 24 Hig7 a Verhand. Kon. Akad. v. Wetensch. (1£ Sectie) DI. AT. qu | — Mh jy je 77} a b je RQ i | | A | | N 53) ESS Ÿ 111] In AS (ff | Verhand. Kon. Akad. Wetensch. (Sectie). DIAL. MS A.BOOLE STOTT: “Geometrical deduction of semiregular from regular polytopes and space fillings’ fal ——* PLIIT. 1 Verhand. Kon. Akad. v-Wetensch.(1° Sectie)DLAT. Fig: 22. PA ile (es | oee 16 ï 24 8 \/ à | à | 3 3 3 | | oe VN A À ) X} / | If Ci os | : 6 8 5 nd E | 5 on eS Fig: 25. zl im i sl a « 24 24 Jie 24 16 5 Ô A A 5 = Y : En A a : : 5 d PAL k | | ; A f Fig 24. 8 : | i pi D À | | 4 | +. s 2 ZN ed Fig: 20 | ze Be N7 24 re 24 4 gory BEREKEND wr ee WATERSTANDEN VAN HET JAAR 1906, _ Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam. . oe Na LS pe Zig EERSTE SECTIE.) en EO “490 Sac > DEEL XI. N°. 2. | ee Ss - (Met één kaart). on Ten à i . ’ AMSTERDAM, a é ; aes. 4 a: JOHANNES MULLER. | an Februari 1911. i GÉTIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN EN BENEDENRIVIEREN IN NEDERLAND BEREREND UIT DE WATERSTANDEN VAN HET JAAR 1906. M. H. VAN BERESTEYN. Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam. (EERSTE SECTIE.) DEEL XI. N°. 2. (Met één kaart). AMSTERDAM, JOHANNES MULLER, LONT. LITERATUUR. G. H. Darwin. Scientific Papers. Volume I Oceanic Tides (1907). Prof. Dr. C. BôRGEN. Die harmonische Analyse der Gezeiten- beobachtungen (1885). | Ableitung der harmonischen Konstanten der Gezeiten aus drei täglichen Wasserstandsablesungen zu bestimmten Stunden, nebst Bearbeitung dreijähriger Beobachtungen zu Kameron. (Methode von Dr. VAN DER Stok). Annalen der Hydrografie und Maritimen Meteorologie Heft X, XI 1903. M. Lévy. Lecons sur la Théorie des Marées (1898). Pa. Harr. Des Marées. Prof. Dr. H. G. van pr Sanpe BAKHUYZEN. Over de getijden te Helder, IJmuiden en Hoek van Holland. Verhandelingen Kon. Ac. v. Wet. 26 Jan. 1895. Dr. J. P. van per Srox. Studiën over getijden in den Indischen Archipel. I en Il. Tijdschrift K. I. v. I. afdeeling Ned.-Indië. ¥390/91., 1891/92. Etudes des Phénomènes de Marée sur les côtes néerlandaises. oo FT: Kon. Ned. Met. Inst. n°. 90. ee NOR re s | | . = ee J er] L 4 + . alr he A “ ay 2 . kid “ + a ER ed ' à = ” = ; A 0 + a ¥ 1 2 ‘ , . ' à + | : 1 » f L . oe ‘ . ’ | big g 1 . 5 : “A 3 si ‘ à a >. * De eenige publicatie’s waarin getijconstanten van plaatsen aan de Nederlandsche kusten vermeld worden, zijn de hiervoren onder „Literatuur’” genoemde verhandelingen van Prof. Dr. H. G. van DE SANDE BAKHUYZEN en Dr. J. P. vAN DER Stok. In de eerste zijn nagenoeg de constanten van alle elementaire getijden, waarin de getijkromme ontbonden wordt, gegeven; terwijl in de laatste (n°. D) over een tijdperk van 18 jaar de constanten van de getijden der zonnegroep en het halfdaagsche Maangetij M, uit de waarnemingen van waterstanden te 2—8—14—20 uur voor Katwijk, Harlingen en Urk, berekend zijn. Waar dus voor betrekkelijk weinig plaatsen de constanten bekend waren, was het wel van belang deze voor meerdere havens te berekenen en de bestaande gegevens aan te vullen. Niet alleen voor het practische doel: het samenstellen van getij- tafels, welke aangeven den tijd en hoogte van hoog- en laagwater voor komende jaren of wel de voorspelling van de waterstanden voor de uren van den dag. Hierin wordt trouwens, dank zij den overheerschenden invloed van het halfdaagsche Maangetij M, zeer voldoende voorzien door de „getijtafels”” bewerkt bij den Algemeenen Dienst van den Waterstaat volgens de methode van den Oud- tloofdingenieur-Directeur van den Waterstaat H. E. pr Bruyn. Maar ook uit een theoretisch oogpunt is de kennis van deze constanten van groot gewicht. De vorm toch van onze kusten is zoodanig, dat talrijke vraagstukken over de voortplanting en inter- ferentie van golven zich daarbij voordoen en dus de uitkomsten der theoretische oplossing van een gegeven vraagstuk met die der waarneming vergeleken kunnen worden. Bovendien kunnen wanneer de constanten over meerdere jaren bekend zijn systematische afwijkingen opgespoord worden, zooals die, welke voorkomen in de A ,,constante” van het halfdaagsche Maangetij 4/,. Deze is nl. niet constant maar verandert met de lengte van den klimmenden knoop der maansbaan, zoodat de af- wijkingen van het gemiddelde over 18 jaar van deze ,,constante” 6 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN NZ duidelijk te voorschijn komen bij de groote amplitude van MZ, in de zuidelijke kustplaatsen van ons land. Deze omstandigheid, reeds vermeld in „Ebbe und Fluth” door Hueco Lenz, doet zien dat de reductie coefficient fm, om uit de constante J/,, uit een waarnemingsgroep gedurende eene zekere periode berekend, de amplituden voor een bepaald jaar te bepalen, onjuiste uitkomsten kan geven. Om dus de constanten der getijbewegingen zoo nauwkeurig mogelijk te verkrijgen is het wel noodzakelijk deze over meerdere jaren te berekenen en zal eene geregelde publicatie van deze geenszins nutteloos zijn. Integendeel, meer in bijzonderheden leeren kennen, de eigenaardigheden, en de veranderingen die de getijbeweging langs onze kusten heeft of ondergaat. Hoewel aanvankelijk het voornemen bestond de analyse’s der getijkrommen te verrichten uit waarnemingen op 24 uur per dag gedurende een jaar, — op deze wijze zijn berekend de constanten in de genoemde verhandeling van Prof. v. D. SANDE BAKHUYZEN — bleek na eenige proefberekeningen, dat voor de meeste getijden zelfs de kleinere nagenoeg dezelfde uitkomsten verkregen worden, wanneer men de waarnemingen van de waterstanden op de 8 equidistante uren 2—5—8&—11—14—17—20—23 gebruikt. Deze proefberekeningen zijn verricht voor de drie getijkrommen te Hansweert, Brouwershaven en Delfzijl van het jaar 1900 en waarvan de uitkomsten verzameld zijn in bijlage 1. Zoowel voor de waarnemingen op de 24 uren 0— 23 als voor die.op de genoemde 8 uren zijn de rangschikkingen der water- standen geschied ‘naar de methode van Darwin. (Sc. P. Vol. D. Voor de laatste groep van waterstanden ondergingen de reductie- factoren van de amplituden en de correctie’s aan de phase eene kleine verandering (zie n°. 22). Voor flansweert zijn op deze methode voor de 8 uur waar- nemingen ook bepaald de combinatiegetijden MS en 2 MS. Boven- dien op eene andere wijze, omschreven in n°. 15. De uitkomsten daar mede verkregen leidden er toe de rangschikking naar de Darwinsche methode van deze getijden als ook van 24M niet meer te volgen voor de overige plaatsen. De constanten, die nu bepaald zijn, uit de twee waarnemings- groepen 2—8—14—20....(1) en 5—11—17—28....(11) waar deze bekend waren zijn: man Er A; 4 AR ele he", EREN GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 7 Aya Sy » 9 | cos (ko — 60°), P, Ky, Ko, T, Suoss Mo, M, en M, en ha : | sin (Aso Wet 60°), ky, #, kh’, ki, Kesut.2.3.49 kings Ana king: In aanmerking moet genomen worden, dat de periode van één jaar onvoldoende is voor deze constanten en men minstens 4 jaar waarnemingen hebben moet om de storende getijden in voldoende mate te elimineeren. Wat het getij 47, betreft, maakt men door dit te bepalen uit een der beide groepen eene geringe doch constante fout (zie n°. 16) en is bovendien de constante J/, aan eene periodieke verandering onderhevig zooals boven reeds is gezegd. De constanten bepaald uit de waterstanden op de 8 uren 2—5—8—11—14—17—20—23 (I) zijn die van de getijden, genoemd in onderstaande staat en ontleend aan Darwin Sc. P. I. A, — gemiddelde waterstand : Spoed. | S, B se bys per mid. zonneuur. 5 2 (y — ») UE A yy = 14.9589314 K, y/ = 15.0410686 Ky y == 930. 0821372 Ja 27 —3y =a (od. 9989814 Rk 2 y —" = 30.0410686 a, y = 00410666 Se 2 y koren Lore ‘sig 3 4 00.296205 Di, 4 y — 0.1642744 — 14.492052] — 28.9841049 at 43 4764508 — 57.9682084 M, 6y—o — 86,9523126 M, S8(y—o) 115.9364168 N Poe Le 28 .4397296 29 .5284788 u 28.5125830 ; 2y—o¢tw—2y= 29.4556254 0 y—2¢ = 13.9430356 8 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. Spoed. | | Q y—3T+0 13°.3986609 per mid. zonneuur. J An Pi Ro 15 .58544338 | MS 4y—20—2y 58.984104 (ral 2MS ®2y—Act wy 27.9682084 2 SM *y+2c0—A4y 31.0158958 s Mm g — w = 0.5443747 Mf 2e — 1.0980880 MSf 2(¢—y7) = 1.0158958 en de waarde van # cos (4,, — 120°). = 15°.0410686 = hoeksnelheid der aardrotatie = 0°.5490165 = gemidd. maansbeweging | = 0°.0410686 = gemidd. zonsbeweging = 0°.0046418 = gemidd. beweging maansperigeum ES Ar Voor de beteekenis der hierboven door letters aangeduide getijden moge verwezen worden naar de werken over de Harmonische Analyse der getijden; in het bijzonder naar de verhandeling van Dr. J. P. van per Stok: De Harmonische Analyse der getijden in het Tijdschrift van het Kon. Inst. v. Ing. Afd. Ned.-Indië 1890/91. Behalve de numerieke opgave der constanten (bijlagen 2, 3, 4) is in dit stuk alleen een korte opgave der uit verschillende bronnen samengestelde wijze van berekening gegeven en alleen wat op deze betrekking heeft. Vervolgens een kaart (bylage 5) waarop de af- standen der peilschalen bij de verschillende plaatsen in Æ. M. zijn aangegeven. De cijfers, die deze afstanden aangeven, zijn loodrecht op de verbindingslijn van twee punten geplaatst. GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ 9 1. Voor de volgende plaatsen zijn getijconstanten bepaald, voor de cursief gedrukte plaatsen werd eene volledige analyse uitgevoerd; voor de overige zijn alleen de getijden, vermeld in n°. 2, en de maangetijden M,, M, en Me berekend. À p Af ORE TEE NE 2 6:20 ole 02:20 MARNE 2): 0.22 51.4 —0.11 OI CORRE 0.24 51.4 —0.09 INE 0.26 51.3 —-0.07 Hansweert. cr. 0.27 DA: am 0e 06 A 00a eee. oe 0.24 51.5 —0.09 Wemeldinge... 0:27 Bio A= Oo PerIRBC ee 0:26 DÉC 007 Brouwershaven... 0.26 51.7 —0.07 Brüinisse.: 57 Nue 0.27 51.7 —0.06 Steenbergsche Vliet. 0.29 51.7 —0.04 Willemstad........ 0.30 Bs? sede Wier chile NS Mr mete 0.31 51.7 —0.02 Wallemsdorpic cae. xg! 54.8. —-0.02 Mond der Donge... 0.32 51.8 —0.01 ‘s-Gravendeel . ..... OST EU SEE Ent Dordrecht: +. see 0.31 51.8 —0.02 Alblasserdam...... DES 51.9 —0.02 Puttershoek ....... Gel GOO Spijkenisse tn 0.29 51.9 —0.04 Hellevoetsluis. . .... 0.28 DANS 0.00 Hoek van Holland. . Oy 52.0 .—0.06 Maassluis. acte. 0.28 51.9 -—0.05 Vlaardingen... es 0:29 51.9 —0.04 hotterdam: Fo Sea 0.30 51.9 ——0.03 Krimpen a/d. Lek.. 0-81 51.9 —0.02 Streefkerk 1020 O52 i eee Schoonhoven....... 0233 520 0.00 Wares Wilkie re an 0.34 52.0 0.01 Scheveningen. +, 0.28 52.1 —0.05 KOPER Sees an aes 0.28 bns Elmanden vanen: 780 52.5 —0.08 ÉLIRE A RS 0532 bee 0.01 VIBRO el Zee 0-34 bo DOr Pakhuient "#22 0.35 DT 0.02 Oranyesluizens 2 0.33 52.4 0.00 10 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. À p At Nienke ee. 0".36 52°. 0".04 BLOED NIET 0.39 Dn 0.06 US Wha ya SAND 0.87 52:47 0.04 Schekland = 2:3... 0.39 52 #7 0.06 Kraggenburg ...... 0.40 Deed 0.07 EE Tr EE 0.38 SP 005 ee 0.35 52.9 0.02 Elmdeloupen,:… ... 0.36 52.9 0.03 TPPRAG ON NME nn 0.36 Roce 0.03 opel. eu 0.30 53.9 0.03 POPRADID. sed 0.41 53.3 0.08 a dun 0.46 Tere Oris Nieuw-Statenzijl . . 0.48 53.38 0,15 A == Lengte der plaats in uren oostelijk van Greenwich. p — N. breedte der plaats. | Af = verschil in uren tusschen den plaatselijken tijd en den tijd, dien het uurwerk aangeeft. De waterstanden te 2 en 8 uur voor- en namiddag zijn ontleend aan de: ,,Verzamelingstabellen der waterhoogten”’ volgens de bladen der registreerende peilschalen voor het jaar 1906 bewerkt door den Algemeenen Dienst van den Waterstaat; die van 5 en 11 uur voor- en namiddag eveneens aan bovengenoemde bladen. Voor Ostende zijn de waterstanden bepaald uit: Diagrammes des Variations de niveau de la mer observées à l’extrémité de l’Estacade Est du chenal d'entrée au port pendant l’année 1906. (Ministère des Finances et des Travaux publics.) Met uitzondering van Hansweert, waar de registreerende peil- schaal van 4 Maart tot 19 April ontbrak, kwamen geene belangrijke onderbrekingen van de waterstanden voor. Waar somtijds door een of andere stoornis van den getijmeter waterstanden ontbraken, zijn - zi} gegist in overeenstemming met waterstanden van naburige plaatsen of wel, werd eene getijkromme geconstrueerd uit eenige bekende standen en aldus de ontbrekende bepaald. Als begintijdstip werd aangenomen 1 Januari 1906 te 12 uur ‘smiddags en de uren gerekend van 0...28. Zoo dat 2—5—8 —11 namiddag — 2—5—8—11 2—d—8—1] voormiddag — 14—17—20—28. De periode der waarnemingen: 1 Jan. 1906 0"—4 Jan. 1907 0°. ML Soe) eu a + welt ré PU PE dins ee ae de dé Doré Là GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 11 De tid, die de uurwerken der getijmeters aangeven, is aange- nomen te zijn die, welke bepaald wordt door den meridiaan 0°.33 oostelijk van Greenwich, en welke ongeveer overeenstemt met Amsterdamsche tijd. Dit is met alle het geval met uitzondering van Ostende en Hellevoetsluis. In Ostende wijst het uurwerk Greenwich tijd, in Hellevoetsluis plaatselijke tijd aan, en zijn voor de reductie op Amsterdamsche tijd de correcties aan de uit de waarneming afgeleide constante” Æ aangebracht (n°. 24), en daarom zijn de constanten # onmid- dellijk onderling vergelijkbaar. Een getijtafel voorspeld met gebruikmaking van achterstaande constanten #, geeft dus de tijdstippen van hoog- en laagwater in Amsterdamsche tijd. 2. De getijden van korte periode: Dyes pe hen en die van lange periode Sai, 2, 35 à werden bepaald volgens eene methode, ontleend aan en samen- gesteld uit de aangehaalde werken van G. H. Darwin, pp. 221—287. Brot Dr. C.. Borgen; J.P. VAN DER STOK: 3. Hen korte opgave van de betrekkingen tusschen de maand- gemiddelden der waterstanden té 2—5—8—11—14—17—20—23 uur en de componenten der bovengenoemde getijden moge hier- onder volgen. Men vindt deze voor de 2—8—14—20 uur water- standen terug in het aangehaalde werk van van DER Stok, terwijl voor de 5—11—17—23 uur waterstanden deze betrekkingen eenvoudig zijn af te leiden wit de voor deze getijden gegeven spoed of verandering per middelbaar zonneuur. Voor de bepaling der maandgemiddelden der waterstanden werd het jaar verdeeld in 12 maanden van 80 dagen (vie Darwin p.224) n.l. Maand aanvangende Maand aanvangende 0 1 Januari VI 3(2) Juli I 31 à VII 2(1) Augustus FT 3(2) Maart VIII 1(31 Aug.) September il 2(1) April IX 2(1) October IV 3(2) Mei X 1(3: oet. ) November V 21) Juni XI 2(1) December. De cijfers in () zijn de begindata voor schrikkeljaren. 12 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. Voor iedere maand m (m — 0,1...XI) wordt berekend de ge-. middelde som der waterstanden te 2—5 —8—11—14—17—20—23 uur, welke gemiddelde sommen respectievelijk zijn voor te stellen door: Jp. 25,8, 11, 14, 17, 20, 23 m en waarmede de volgende combinatie’s gevormd worden. ss — 4 [ ln 7 — he sh wee 20 am | a hn J 50. oc (1) SR [ ( oy: = A jas ey + Hn Jel | hn ne aa ae hin IE an + 4 sags hep = 4 [ Die pen h, RES i, Sh | ek bi: 4 | (4, sr > + Bn EE (4,7 == 4 ] | Um Fit LI (A m > + ee =e Gas == ba) | | Uit de 12 waarden 4," (m — 0,1, XVe Tie ann de gewone wijze te berekenen voor beide groepen van combinatie’s (1) en (2) de volgende uitdrukkingen: (Zie voor de wijze van berekening: Darwin, Sc. P. I. pp. 54—55. BôRGEN, Harmonische Analyse der Gezeitenbeobachtungen 1885 prAa0e. a) dj eae += Î ee | eo a yy += D [ aes | | | ie a | Hage | Ai aeons PA is | Om 80°. | > is sin A at COS ge Ag — | . COS aie B ee 6 We | ge | 30m | B === COS < = pon L ps | > | 60°.m | a IV —4 COS pu 4 LAN 30m 12,84 a TRE GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 13 resp. te voorzien van de letters / en r, al naarmate de 2—8—14—20 uur of wel de 5—11—17—23 uur waterstanden gebezigd zijn. 4. Correctie’s. Aan de combinatie’s 4 "1% of aan de daarmede berekende waarden behoeven geen correctie’s aangebracht te worden. De maximum invloed toch, op 4,7" bedraagt voor een storend getij met amplitude — /7. Be DOS —.0,. H sin 12n¢,, Nh, = + Saba. (a == 30). TI SUR Ie, | en wordt voor de halfdaagsche getijden, waarvoor a, ongeveer 30° is, zeer gering. | Voor de enkeldaagsche getijden (¢,, ongeveer 15°), kan AZ, relatief groot worden, b. v. voor het maansdeclinatie getij O vindt men Af, = 0.0528 O (O = amplitude) Op onze kusten is dit getij ongeveer 10 cM. Bovendien heeft deze invloed een periode van ongeveer een */, jaar in de combi- natie’s A", terwijl die van de getijden P en K, eene jaarlijksche periode in dezelfde combinatie’s vertoonen en is dus een correctie niet aangebracht. De maximum invloed op 4," voor een zeker getij (77, ¢,) 1s / H sin 12nc, Aln SS eee 080 CSST, n Sin LAC, en wordt dus zeer gering zoowel voor halfdaagsche als voor één- daagsche getijden. De maximum invloed op de combinatie 4," is voor een zeker I mn getij (M,o,) ies H sin l2n¢,, Rm LA cos 0e, 3.0, == Co ET n Sin 12, zoodat alleen de invloed in aanmerking komt van halfdaagsche getijden en is van deze alleen de invloed van het halfdaagsche Maangetij 17, berekend. (Zie Darwin Sc. P. I. p. 227, v. p. Srox Etudes des Phénomènes de Marées enz. p. 6). De correctie aan de combinatie 4," is voor te stellen door: 14 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. A byl! = — G, Hsin [1e — e, + 29.12 0, + 240, (80m + 2)] A, h, i” = — C6, H sin [la — ¢, + 29.12 or, + 240, (30m + &)] waarin €, — phase van het getij op 1 Januari 0". en voor m æ 0 0 I 0 IT ] III 2 IV 2 V 3 m VI Vil VIII IX X XI Voor M, worden deze uitdrukkingen: m=) 0.0156 M, sn (825 — &%) I Il Ul IV V VI Vil VITE IX X XI sin (S14—E8,,) sin (TRE) 810 (ODE) sin (230 — eno) sin (219 — Eno) sin (208 — Eng) SMT eee) sin (100 SE sim (124 — 8,9) MAE TD = Ego) SOS En CS CO à o = & Co Ir A m9 hes 0.0156 M, sin ( 52 — Eno) sin ( Al — €,5) sin ( En sin (353 — Eng) sin (SLT sin (806 — é€no) sin (270 — En) sin (259 — Eno) sin (247 — Eg) sin. (211 EEN sin (200 —=%,,) sin (189% Men kan nu aan ieder der 12 waarden van 4." bovenstaande m correctie’s aanbrengen (zie v. D. SToK a. w.) of wel aan de daar- mede berekende uitdrukkingen: 4," Bis" am (zie Darwin Sc. P. Lt ape) Door de analyse van A, 4, lijke hoek in getallenwaarde is bepaald vindt men: mg Cm Urn m2 A A A III waarin de six en cos der verander- = — 0.0055 M, sin (En. + 159) = + 0.0055 My cos (8,2 + 162) =; D Farc mets Beas Bk Eek Aff A A A, TER at + 0.011617 (eo) — 0.0104 Morse an do — 0.0021 Ms SAE re — 0.0044 M, sin (Eno + 62) GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 15 Amg 4,8" = — 0.0116 My cos (Eng + 179) Ane B, Nr + 0.0154 M, C08 (Emo + 76) Ang A, = +. 0.0021 M, cos (Em + 27) Moe = 00044 A COS (Eng + 65) In alle deze uitdrukkingen is MZ, de amplitude van het getij M, voor de periode waarover de waarnemingen genomen zijn. Men kan deze amplitude voor die periode berekenen of wel uit de eenmaal bekende ,,constante”” M, ver met een reductie- factor (fig), bepalen. Evenzoo de phase ¢,,. op den 1° dag der waarnemingen onmid- delijk uit deze afgeleid of wel uit de constante #,, herleid op den stand van het fictieve hemellichaan op den 1 dag der waar- nemingen te 0 uur. 5. In aansluiting van de uitdrukkingen in n°. 3, zij nu gesteld de op eene der beide wijzen voor M, gecorrigeerde waarde van: TEER ag ET A Gr ONE ke S, den PAIE RA RE ea Dn DE 6. De betrekkingen tusschen de combinatie’s 4 van de water- standen te 2—S8-—14—20 uur, (index 7) en de componenten der getijden bovengenoemd, zijn dan Oi COS S, (ka — 30° RE | san = (A bj | sin > p | — | — 15° FE, | tr ) ENTRE Po \ COS on Ai sin | 2K, = (ea — 45°) 1 == (Ay br) | cos Cm == SQ COS (Aig — 60°) Cit — as SON COS si (e, — 45°) ; | we Er 10) | F, Fe DE — sin | sin 16 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. K. a (8x > 90°) DEE | sin M COS | S, F (Epe Pela = Aj ek) B == sin en eindelijk de gemiddelde waterstand 4, uit: an = A,. 7. Voor de combinatie’s met de waterstanden te 5—11—17—23 uur (index 7): a, = COS | Si (he Geit 15°) Oes | SIM | eS AP Sy | sin | 2P Fe (Ep — 60 ) | f.= AY + BY = | cos Cr = A + B — sin ze | — 90° F, | (Ex ) EU Ir [les d= A diy, — Sy sin (ko — 60°). CLONE lt sin ff ft zo EN @—45) LE (e, — F4 1 1 DIE COS | | — COS Cr COS | x . 7 | (€,. — 180°) GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 17 COS (Esap —P- 15°) |(w=l,2,3,4) san Evy, Ae Ann == p lijn Bis pee en de gemiddelde waterstand 4, i à S. De in de uitdrukkingen van 5 en 6 voorkomend factor Ee Pp is de verkleining die de amplitude van het getij ondergaat door de gemiddelde som over 30 dagen te nemen. Zij is te berekenen uit: 1 n 12 3 En Kee (y — 0°.0410686). B 30 sn 12 XD X y Men vindt: log F, = 0.00478 log F, = 0.01939 log F, = 0.04419 log F, = 0.08000 Met de factoren /, behooren nu de uit de waarnemingen af- p 8 geleide amplituden (waar noodig) vermenigvuldigd te worden om de juiste waarden te verkrijgen. 9. Uit 6 en 7 kunnen nu afgeleid worden de betrekkingen tusschen de componenten en de combinatie’s der waterstanden op de 8 equidistante uren FE Ae wl: + (Ary; ap Ay, ) == Ay + (Ary: — Arvr) = Sz cos (hoy — 120°). an Har = bry + by, = Cin = COS Led 9: COS (deg 590.5) | SIN COS 8, (kg — 60°) din CP Verhand. Kon. Akad. v. Wetensch. (1e Sectie) Dl. XI. B SIN DO 18 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. ler Fe) = ‚sin 4 P —— cos 22°.5 Spat as fy et In). cos CHRIS PUS EE sin x, 4 cos 22°.5 — (€,4— 67°.5) i (d, +d.) = COS ca san De Une cn 27 | Den (é 2499 | es UI + Die 1 COS j CHE yy) Mee sin 2 R ve (e FR 75°) oa ns D, = à ane | Cian fe DE Se | sin 2K 2 (ero 90" Gye ek Die Es À COS BA Je BTE sin | 28 Fe (Eze ee He 1. Zan p A +- Ae sn COS 10. Berekening van de getijden der M groep, de combinatie- getijden MS, 2 MS, 28M en MSF. Wat de berekening van de getijden 4, 4 betreft, deze is ver- richt volgens de methode van v. p. Srok (zie v. D. SToK a. w. Tyds. K. I. v. I. 1891/92. Böreen Ann. der Hydr. 1908? Daartoe werden de waterstanden op één der uren 2—5....23 gerangschikt volgens M ,,uren”. Deze „uren’’ — in zulk een uur doorloopt de fictieve maan een boog van 15° — zijn te berekenen volgens de uitdrukking : 15° (r + a) = 154 — (© DL ee waarin nt nt 4 a = à = > GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 19 Tr — een der M ,,uren” 0... .23. 4 — eene waarde die tusschen —+ 0.5 en — 0.5 getijuur varieert. { — een der § uren 2-—5....258. y, — gemiddelde beweging van zon en maan. B aantaksdagen: naden. 1 dag (1°: dag = 0). Voor een constant S uur ¢ kunnen nu voor iederen dag van het jaar (2 — 0...364) de daarmede overeenkomende M ,,uren”’ bepaald worden. Waren in plaats van voor ieder 8 uur de met het 8 uur 12 correspondeerende J/ ,,uren’’ 7 berekend en de waterstanden ge- rangschikt volgens deze 7’s dan komt deze methode neer op die van Darwin. (Zie Sc. P. I. p. 216), toegepast op enkele uren. De gemiddelde som der onder het uur 7 van (1) gerangschikte waterstanden te { uur S tijd is voor te stellen door: 1 _ EE M, cos [15° p(t + a) — Eng) HR LEE ORNE stad wanneer > het aantal der onder het uur 7 voorkomende water- standen is; en daar g over de # waarnemingen gelijkmatig verdeeld zal zijn kan men (2) schrijven M, | =? Fe COS Bip eee dean Wiad KE Irae EEE" p De factoren #, waarmede de door de berekening gevonden p ) vermenigvuldigd moeten worden, zijn te bepalen M amplituden ( — GF uit : | 1 1 #20.55 X 15°» FOR ADD Soar Men vindt .0084 log F, = 0.00149 OLS tog = 0) 00008 ‚0315 log fF, = 0.01348 .0570 log F, = 0.02408 .1850 log F, = 0.05492 .2586 log F, = 0.09982 ppb esr sl Soh se ee ail ae Pel 20 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 11. Door de rangschikking der waterstanden volgens M ,,uren”, welke uren alleen bepaald worden voor een constant S uur # door 7 en (¢—y) volgens (1), worden tevens, behalve de getijden der M serie, die getijden opgenomen, waarvan de verandering per etmaal van het argument een veelvoud van (¢—y) is; dus die getijden die na eene semi-lunaire periode dezelfde phase ten opzichte van S, innemen n.l. de combinatie getijden \ 2 SM MS 2 MS MSf 12. De invloed van deze getijden op de gemiddelde som der waterstanden onder een zeker 7 ,,uur’’ en die op onze kusten een belangrijke rol in de getijbeweging spelen, kan op deze wijze bepaald worden : De spoed van het getij 2 SM is: Cam ye dn = 2 X 15° + 2 (a) waaruit volgt de invloed op den waterstand te ¢ uur van den (dae (das Sn) (2 SM) cos [30° 4 4-2 (x —y) LA 24 & 2 on In verband met (1) kan deze uitdrukking geschreven worden in den vorm (2 SM) cos {30° (7 + a) — 6) waarin €, == 60° ¢ — ev, en dus de invloed op de gemiddelde som 4} der onder het M uur + gerangschikte waterstanden — cos [80° 7 — Easy). bo De spoed van het getij MS (MS, o,s, Ens) 18: De invloed dus op den 2°" dag te 4 ae (MSN) COS 160° t— 2 (¢ =a y) { — 24 DI 2 (a B: y) Pers Ems): GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 21 Na substitutie van (1) in dit argument wordt dit: (MS) cos (30° (T + 2) — En) Ems == Ems — 60° 4 en de invloed op de gemiddelde som fae (MS) F, COs (30° oe E ms) Op overeenkomstige wijze vindt men. voor den invloed van 2 MS (2 MS, Cong, Eos) waarvan de spoed Tons > 2 Eg) op de gemiddelde som 4. 2 MS 4 Kindelyk voor het getij MSf (MSS, a, Ens) waarin à / / COs (60° PS UE a Eons — Ems a= 60° t. O msf — 2 (a es y) de invloed op de gemiddelde som 40 MSf a de ‘ SS Ms COS 130° PSE He Emsf — nine Emsf 13. Recapituleerende, vindt men voor de gemiddelde som 4®, wanneer 4, — de gemiddelde waterstand en de invloed van storende getijden met uitzondering van het halfdaagsche zonnetij #8, buiten rekening latende, onderstaande uitdrukking : DA, + B, cos (80° t— A 0 MM. + 5 à … COS (EST EE Er i. 2 SM a = 7 ze COS (50° Pine cree 2 + F, cos (30° + — €.) EMS Oe CA) COS (6 0° Lin de bd F, M Sf +- U De COS (3 0° (has À of) 2 22 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 14. Nu zijn achtereenvolgens onder de M ,,uren” 7 — 0...23 afzonderlijk gerangschikt de waterstanden te /— 2—5—8.. .23 uur. en dus bepaald de gemiddelde sommen : 19 Z® 7 700 Ca AS GE La AO zen Zan 703) T T La Tu Daarmede kunnen de volgende combinatie’s gevormd worden: 2 1 HE HUP HAD HUP + 4) HU HUP HAD) — GO + 40) HY = HO AD) + GO 4 49) Hr = HOP HAD — HY + 20) | Gemakkelijk is nu na te gaan, dat, wanneer men in de uit- drukking van 2@ in n°. 18 achtereenvolgens ¢ = 2, 14, 8, 20, enz. stelt, in aanmerking nemende de uitdrukkingen van € = g 30°¢ + € in n°. 12, men vindt: ; EA H = A4, 2 Er cos. (15° pt ene p 29 M ( à COS (3 0°r — En m nt ee Eos F He — Scones BON MS : a = COS (30° ra Eye) Ens md er 60° 9 MS / / aie a cos (60° Cre Ems) E Ms TT ons ale 60° MSf | | + 2 = cos (30° T — En) à Enr = \Epiep en 17 Ee WT, Ko HA, + ren (15° p 7 — Ep) D CSM) | et — cos (30° 7 — € ae) Eon = PU Eon 7 Vi ») GETIJCONSTANTEN VOOR PLAATSEN LANG HY — S, Sin (hea — 60°) + Ge cos (30° T — €) Gite 2 se ie cos (60° T — eo) €: of EE 2 cos (30° 7 — Ep) € a E msf CE S DE KUSTEN ENZ. 23 Ens TE 150° sae fms == 150° ck. 15. Door de analyse van de 24 waarden H, op de gewone wijze de componenten van de #° orde (4, men ten slotte: voor de groep der 4 uren (2—8—14—20) A, = gemiddelde waterstand a —— iz COS M, F. En a= 4 ro — sin ee = COS COS M, (281) (5 9 F4 Vi ma Le = — sin sin ie — COS M; TR FER F Hod 3 M, F En 4 Dir sin Ae cos Me F. €06 6 B sin AY —— COS Ms vg E118 De Sin B) bepalende, vindt (1 20° TAN Exam) 24 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. en: en S, cos (ko — 60°) AS — COS COS (MS) EN FE, Cer 0 ) ae 3! (60 rors aan D — sin | sin no COS (2MS) : E! F (Eons + 60 ) B — sin en voor de groep der waterstanden te 5—11—17—23 uur A,” = gemiddelde waterstand. Ir FAT 43,868 = cos M, 5,1,6,8 À ( mA, 3, 4, 6, 8 N Fis, h, 6,8 Ek 4 Bises = Ls A = | cos cos M, (2 SM) PRE F Ema | — F (120 ESA aan) 2 2 Wi sin sin en Ag, Siri Ge ow A COS gl | cos (MS) (MS) =O = NO 7 (ere CE 150 ) + CAMES (150 TR Eno oe | sim sin rn COS “| (MS) : ENT ra (Eo ms + | 50 ) { i — sin En uit de combinatie der beide groepen 7 en r voor de compo- nenten der getijden van de 47 serie: GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 25 Fr + AM = Lo == Ep (p = lig 2e 3, 4, 6, 8) le LM + Br) =" Lain voor die der combinatiegetijden : BOE AE) — cos 7 (2 SM) 8 CR Cr = E20 ) 4(BY — BIJ) = | sin | 4 ca! ie B) et a | cos 7 ) hae 60°) 1 ss Fu Je Bo ae: | sin 4 (47 + Be) AE COS (SF) o Er ae. (Esr = 60") LCA — B) = sin 1 (4, + Be) = : zi D cos Se. AS Es 4 (BY ae à Le | son = Uit het bovenstaande blijkt, dat uit de enkele rangschikking der waterstanden volgens M/ „uren” de voornaamste combinatiegetiden bepaald kunnen worden bij geschikte keuze der waarnemingstijd- stippen. Vallen deze op 4 equidistante uren van een dag, dan kan alleen het getij 2 4/8 berekend worden, terwijl 2 $M niet van M, ~ kan bevrijd worden en evenmin MSf van MS en omgekeerd, na een der beide getijden 2 SM of M, en MSf en MS bekend is. 16. Correcte’s. De componenten 4 en B uit de M rangschikking der water- standen verkregen, kunnen met uitzondering van 4,, 5,, gebruikt worden voor de bepaling der getijden vermeld in n°. 10 zonder eenige correctie voor storende getijden aan te brengen. Het grootste van deze — #8, — is uit den aard der methode geëlimineerd. Voor de overige WV, ZL, O is de invloed onderzocht COS VE AL: door de waarden en 94 gi in de kolommen 7 — 0...23 in te in 26 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. schrijven en aldus te bepalen den invloed op 4. Hoewel deze in- vloed van eenige beteekenis kan zijn op ééne der 24 waarden A, blijkt alleen dat voor MN eene correctie aan 44, Bg behoort aange- bracht te worden, zoowel voor de groep der 2—8—14—20 uur, als voor die der waterstanden te 5—11—-17—23 uur en dus eveneens voor de combinatie der beide groepen nl. KA, == 0.1 AN sie, le AB ==. 0117, Name 144%) f, = reductiefactor; zie Darwin Se. P. I p. 46. Wanneer voor de bepaling van M, de 4 uren 2—8— 14-20: of wel die te 5—11—17—23 gebrukt worden dan zou men aan dit getij correctie’s moeten aanbrengen voor 2847. Daar dit in verhouding tot #7, zeer klein blijkt te zijn kunnen de correctie’s in plaats van aan de componenten 4, en #&, onmid- dellijk aan de amplitude 2’ en phase €, aangebracht worden: A Pr = as Jing (28M) cos (Es == Era = 60°) 7 ZM) | AE + [M . L SUN (ee aa Bp Sl 60°) m2 (bovenste teekens voor de bepaling uit 2— 8—14—20 uur, onderste voor die uit 5—11—17—238 uur.) Door fo — 1 te stellen in verband met de kleine waarde van vt 28M en in aanmerking te nemen, dat de som van het astronomisch gedeelte der argumenten — 27 en dus €, + €,,,, — "constante, kunnen deze correctie’s als constant beschouwd worden over ver- schillende jaren. Men vindt dus (4Z,, &) steeds te groot of te klein uit de 4 waterstanden per dag op tijdintervallen van 6 uur. Voor de combinatie getijden zijn evenmin correctie’s aan de componenten 4, B aan te brengen. Het grootste getij dat hier invloed kan uitoefenen, het getij M, is door de verschillen te bepalen van de gemiddelde som der onder hetzelfde 17 ,uur” T gerangschikte waterstanden op verschillende uren, nagenoeg geëlimineerd. Daar echter onder hetzelfde uur 7 niet de juiste waarde van de 7 ordinaat der 7 sinusoide is geplaatst, maar die welke tusschen T + 0.5 en 7 — 0.5 liggen, kan de eliminatie niet volkomen zijn bij een betrekkelijk gering aantal waarnemingen. Sit Men hl | Te a GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 27 17. De combinatiegetijden 28M, MS, 2MS, MSF, zijn dus berekend zonder eenige correctie aan de componenten aan te brengen. De afwijkingen, die de amplitude en phase van MS op deze wijze berekend vertoonden, met die, verkregen uit 24 en 8 waterstanden per dag op de methode van Darwin voor de 3 plaatsen Hansweert, Brouwershaven en Delfzijl 1900 uitgevoerd, deden eene nog niet geëlimineerde invloed vermoeden bij toepassing der laatste methode. Inderdaad blijkt er aan de componenten van M/S op de methode van Darwin verkregen eene niet onbelangrijke correctie voor M, noodzakelijk te zijn. Op de volgende wijze kan deze bepaald worden: De „spoed” van MS per middelbaar S uur is: Gy 2g ek) = 60° — 2 (x — y). Zoodat de bepaling der met het 8 uur 12 van een zekeren dag à overeenkomende MS „uur 7 (zie Darwin p. 237) kan verricht worden naar de uitdrukking LN 60° (7 + a) = 60° 12 — 2 (a — y). 12 — 2 (© — y) 241. (C= 49) en is de waterstand op het S uur 12 — 7 van dienzelfden dag, die op het MS „uur” TT (zie n°. 22) dus: 60° (T° + a) = 60°. 12 — 60° £ — 9 (¢ — y). 12 — — 2.24 (a — y)e....(2) De verandering van het argument van M, per middelbaar S uur is — 30° — 2(¢ — y). . .(3) zoodat de invloed van dit getij op den 7 dag (1° dag = 0) te (12 — #) uur S tid, wanneer f= de amplitude Eno — phase op den 1 dag der waarnemingen te 0 uur bedraagt : cys SO er ES Go LA Se) = R cos [[60° 12 — 60° — Be — Ei 12 — 2 on | ROPE 2 (ey) Fe nal | of volgens (2) en (3) | T= Foor (80° Looe fash Zijn nu de waterstanden op de 24 uren van Le dag ol volgens de methode van Darwin in de MS „uren” gerangsch dan varieert in de kolom 7’, « van + 0.5 tot — 0.5 en £ — 11 tot +12 {12 —¢#=0 — 23} bij een Boo, me | nemingen. | standen onder het MS „uur” 7’ ak ae . 1 nA OFS pe 24 » R cos (60° gd + a) + (60°— x) é — € en spé | | ‘MERS ay) "A ats » af El Ven AGO ao SO Ta) I» cos|60°r—e,,. +1}, nn od ; TA F, 24 sin (60° — Ono) 1, (F, = reductiefactor, zie n°. 21) en op de componenten Hp _ Ay = 1 COS ne AT 60° 7 By ee sin Te) Va 2% it SIN 24 COO ‘ley ae { | 1 0 iD DA sin Go na) i], X | Eng Sees 1 (60 = Ta) sin Opgemerkt moge worden dat na substitutie van ¢,,, voor Oyo 1 (5) men den invloed van het getij MS verkrijgt en daarmede — ie reductiefactor À, (zie Darwin p. 240). | en de correctie ©,,, aan de phase € MS RNL i nl. Fry = F, 24 si (OO Sala | | sin 24 (60°— or) Yo Or. Fra —- Fi (60° TS Cae GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 29 en daar fF, = en (zie n°. 21) Jp) Pop MenPdp@ 4 ms gin 1°.5 p sin ZA X T°.5 08 bk Dt PE Overeenkomstig die, in eene hieronder gevonden uitdrukking. fn. 22). 18. Op dezelfde wijze kan de invloed van M, op de compo- nenten van MS bepaald worden bij toepassing der methode van Darwin op de 8 waterstanden te 2—5—8—11—14—17—20—23 uur voor de berekening van MS. Men neme slechts in aanmerking dat de uurreeks van een dag niet is 12 — # voor é— — Il .... +12 maar 12 — (10 — 3 g) voor g=0....7 en dus in (4) te stellen (= 10 —3 g. De gemiddelde invloed voor M, wordt dan: 1 2 0.5 5 eS if cos {60°(¢ + &) + (60 — ono) (10 — 39) —émo} da q1=0a 0.5 of : 1 sin 8 (60° — o,s) °/, ooo OO 2 joa ADs N F, sin (60° poe Cina) is oe iM tf Ema ls ( ) Ting) en de gemiddelde invloed op de componenten 4,, BP, De COS 1 sin 8 (60° — sue) 9 m2 R 5 So À 60° — 5 |. F, 8 sin (60° — oo) 8/5 lea + ‘a ( Tin) Os sin 30 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. Substitueert men hierin voor a: &,, dan verkrijgt men de reductie die de amplitude en de verandering die de phase ¢ onder- gaat, bij toepassing der methode van Darwin, wanneer de 8 ge- hanne waterstanden gebruikt worden. De correctie’s aan de componenten van MS aan te en Nay voor M, zijn dan AA=—1, AB=—4i, De getallenwaarde van ¢,,. invoerende vindt men ten slotte bij de Darwinsche rangschikking van MS voor 24 uren, de correctie’s: nds OOST AM core CAT [log coef’. = 8.49737] A B, = —0.0814 fr M, sin (eus — 15°.5) n2 voor 8 uren: A A, = —0.0847 fo My cos (eo + 15°.5) | [log coef’. = 8.54078 | | A B, = —0.0347 fs M, sin (eo + 15°.5) 19. Deze correctie wordt niet vermeld in het handboek van Darwin. Toch kan, waar M, groot is, de invloed aanzienlijk zijn en is de bepaling van MS uit waarnemingen gedurende één jaar onvoldoende. B. v. werd gevonden voor Hansweert 1900 zonder correctie uit 24 uur waarnemingen 369 dagen 4, == 80°.9 MS — 4.43 eM. hel ne e > Di — 109°.2 = 6.53 EE) BE „volgens nel eae == en uit 24 uur waarnemingen (met correctie) #,,, = 198.0 MS 2.97 cM. Bee, 4 53 193.0 2-80 Be 7 volgens n°. 15 2011 5.250 dus onderling veel beter overeenstemmende waarden. 20. De overige getijden van korte periode, waar deze berekend zijn, werden bepaald volgens de methode van DARWIN (zie DARWIN pp. 216venv.) ia GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ 51 Daartoe werden voor de verschillende getijden de met het zonne- uur 12 van iederen dag van het jaar correspondeerende getijuren Tr—0....28 bepaald, zóó, dat voor eenen bepaalden dag 4 het murb vonnetijd —:het getijuur (rt 0.5). Dit kan geschieden volgens de uitdrukking : 15° p(r + 0.5) — 12 & + 2e X 2 waarin ¢ de ,spoed”” van het getij per middelbaar zonneuur, p = 1, 2 enz., afhangende van de orde van het getij. Zie verder voor de berekening dezer uren 7: BöreeN Ann. der Hydrografie 1903 Heft X p. 444. v. D. Srok. Studiën over getijden in den Indischen Archipel IT. Tijdschrift van het K. I. v. I. Afdeeling Ned. Indië 1891/92. 21. Bij de oorspronkelijke methode voor de scheiding der ver- schillende getijden, werden de waarnemingen op de 24 uren van een dag zoodanig gerangschikt, dat een waterstand op een zeker uur van een bepaalden dag geplaatst werd in de kolom van een der met dat S uur overeenkomende 24 getijuren 7, binnen de grenzen van een 1}, getijuur. | De waterstand te ¢ uur S tijd werd dus geacht te zijn waar- genomen op het getijuur: 7 + « (« van + 0.5 tot — 0.5). Zie Darwin pp. 48 e. v. Lévy Theorie des Marées pp. 81 e. v. _BöreeN Die harmonische Analyse ‘der Gezeitenbeobachtungen pp. 42 e. v. Over een groot aantal waarnemingen kan men nu aannemen dat de afwijkingen van het juiste getijuur 7 gelijkmatig verdeeld zijn tusschen de grenzen —+ 0.5 en — 0.5 uur en wordt dus de gemiddelde waarde der functie: y = Rcos (nr — €) T +05 D =f R cos (nar —e dr ae DNS ea se [2% RK cos (nt — €). “Is we De juiste amplitude ondergaat hierdoor een verkleining bepaald ee, ñ door den factor # = site waarmede de uit de waarnemingen sin 1,7 afgeleide amplitude moet vermenigvuldigd worden. 32 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. n == 15° p stellende, vindt men voor: p =1 log F, = 0.00124 2 log F, = 0.00498 3 log F3 = 0.01122 A log F, = 0.02008 6 log K, = 0.04561 8 log Fy = 0.08250 (Börarn, Die h. A. der Gez. p. 48.) Voor alle getijden van dezelfde orde p, (p — 1 voor enkel daagsche == 2 _.,,.. half-daagsche eux zijn deze factoren constant terwijl door deze gelijkmatige verdeeling der afwijkingen tusschen + 0.5 en — 0.5 uur, van het juiste getijuur geene correctie aan de phase ¢ behoeft aangebracht te worden. 22. Deze gelijkmatige verdeeling der afwijkingen van het juiste getijuur 7 heeft niet plaats bij de methode van Darwin, waarbij zooals boven reeds is vermeld, alleen het S uur 12 binnen de grenzen van een half uur overeenstemt met een zeker getyuur 7. Voor de overige 24 uur wordt dan aangenomen, dat het getijuur T + r samenvalt met het S’ uur 12 — r van zekeren dag. Gedurende een dag wordt de ,,spoed’’ van het getij gelijkgesteld aan die van den middelbaren zon. Men maakt daardoor een zekere fout want is de spoed van het getij per middelbaar zonneuur dan is Bs | 1 Suur = (1 — 8) getijuur Sie Te D =; 1 2 | en zijn de waarnemingen niet verricht op het getijuur: | maar op het uur: Voor waarnemingen op 24 uren van een dag neemt 7 de waarden aan van —12.,..-+ 11 en is de grootste positieve afwijking van het juiste getijuur + (0.5 + 12 8) getijuur de grootste negatieve afwijking — (0.5 ++ 11 8) getimar. = “2 r di e 3 / zE ed SO ee » De TE LEN RS GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ 33 Nu worden de waterstanden zóó gerangschikt, dat men in een zelfde kolom plaatst, die, welke op het getijuur (r + r) =7 = constant, kunnen geacht te zijn waargenomen. Over een groot aantal waarnemingen verkrijgt bij constante 7’ Tr alle waarden tusschen Ce) 3 PU ke » — 12...+11 ers oe » 0.59... + 0.5 en is de gemiddelde waarde van de functie R cos (nt’ — r) voor 7 =7 + 4 + rd, met bovengenoemde variaties van g en r te stellen: ee deers +0.5 ae 5h = rz R cos|n(r + a+ rp) — «| da y= —11 es sin sin 24 2 ne 2 2 ue R cos \n (r + 5) — el n - ni? 7 24 sin ee Om dus de juiste amplitude en de juiste phase ¢ te vinden moet men de eerste vermenigvuldigen met den a 6 214 ne factor Here ue (1) je 5 ; x wie sin = sin 24 en aan de berekende phase de correctie | nf DN. £ OP tee eet Vere alk ee RE aanbrengen. Zijn de gemiddelde waarden der ordinaten y voor eene zekere rangschikking op bovengenoemde methode bepaald uit de water- standen op de & equidistante uren 2—5...23 dan kunnen op analoge wijze / en ©), bepaald worden. Correspondeert voor een bepaalden dag 7 het S uur 12 met het getijuur 7, dan komt de waterstand op een der 8 uren 2—5...23 of 12 —(3g + 1), ¢=—4....-+ 8) overeen met het juiste getijuur : Ba denten et (Sgt le en wordt in dat geval de Verhand. Kon. Akad. v. Wetensch. (1e Sectie) Dl, XI. Bes 34 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. grootste positieve afwijking — (0.5 + 10 2) getijuur dl negatieve ,, — (0.5 +11) „ De gemiddelde waarde der functie Reos nT +—a+(39 + wordt dan, over een groot aantal waarnemingen waarbij el "+ u varieert van 0.5 tot — 0.5, g de waarden — 4 .... + 3 aanneemt, r constant blijft aa hea 0.5 —~ > | ee dd q=—4ea — 0.5 RD 8 nf? ‘ sins Sin 8 do es Sal of : X — cos \n (r Or 2 8 sin Be ‘ zoodat de berekende amplitude moet vermenigvuldigd worden met n yy ane eae ae 8 sin 3 3 ey sin” sin 8 X 3 "Ê ae vy be en men aan de gevonden phase eene correctie moet aanbrengen van: ~ (4) Stelt men in (1), (2), (3) en (4) 2 = 15° p, dan wordt IN te: en GE p sin 24 DC 19 P £ OP = Eh pp Fie et TD DMS " sin 7°.5 p sin 8 X 3 BX 1.59 O9 = Ain we. In de onderstaande staat zijn voor eenige getijden berekend yy log F en © GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 35 log FSP EC) log FY 9 M, 0.00205 + 0.25 0.00204 — 0.25 M, 0.00825 + 0.51 0.00821 — 0.51 a, 0.01860 + 0.76 0.01849 — 0.76 BEE 0.03320 + 1.02 0.03301 — 1.02 M, 0.07544 + 1.52 0.07504 — 1.52 M, 0.13617 + 2.08 0.13544 — 2.68 N 0.01272 + 0.78 0.01261 — 0.78 L 0.00568 0.24 0.00567 — 0.24 y 0.01200 + 0.74 0.01190 — 0.74 A 0.00591 + 0.27 0.00590 — 0.27 O 0.00478 —+ 0.53 0.00472 — 0.53 Q 0.00940 + 0.80 0.00928 — 0.80 ee 0.00232 — 0.29 0.00231 + 0.29 MS 0.02330 + 0.51 0.02326 — 0.51 2 MS 0.01814 + 1.02 0.01795 — 1.02 2 SM 0.00824 — 0.51 0.00820 + 0.51 De op deze wijze gevonden reductiefactoren verschillen weinig van die ,door Darwin (Sc. P. I. p. 240) opgegeven en bepaald na constructie eener frequentie kromme der afwijkingen van 7 door de uitdrukking : 1200 LA DIX ADP sin 1°.5 p sin 24 K p XK 1.9 B Deze factor kan ook verkregen worden door de evaluatie van 4 den dubbelintegraal : p= 1218 #=+0.5 1 ne / fees nr Had r) — el da dr p= lg ¢=—0.5 na substitutie van 15°» = ». 23. De getijden van lange periode Mm Mf zijn bepaald geheel overeenkomstig de methode beschreven in Darwin Sc. P. I. p. 244. Bovendien is op deze wijze eveneens het getij MSf berekend. Met de dagelijksche sommen van de waterstanden te 2—5...23 uur, kan gemakkelijk nagegaan worden, dat de invloed op de gemiddelde som onder het uur 7 van eene groepeering, bedraagt voor: BL97 36 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. (Mm) C IL Mm: os |6°.8 — 6, + 15° 7} log Fi, = 0.00098 Mf: cos (13°.7 — ef + 30° 7) log F, + 0.00375 ñ | MSF: 2 cos (192.7 — en, + 80° 7) log F, = 0.00324 terwijl de invloed van 47, op de gemiddelde som der voor MS/ gerangschikte waterstanden onder het uur 7 kan gesteld worden : — 0.0384 fing M, cos (— 2°.3 Hema = 30° 7) waaruit volgt de correctie voor: A: AA==0.0384 fing Ma cos (— 2°.3 + Eno) (log coef’ = 8.58394) B: A B= — 0.0384 fing M, sin (— 2.38 + Eng) M, = constante i ( Jing = factor voor de reductie tot Amplitude ) Eng = phase op den 1° dag der waarnemingen te 0 uur 24. In de nummers 2—23 zijn aangegeven de wijzen waarop de componenten 4 en B der verschillende getijden kunnen bepaald worden. De amplituden £ en de pnasen ¢ op den 1 dag te 0 uur vindt men dan door: R=/) A+ B? EO . en ten slotte de constanten H en # uit: R 7. k= Vi dude a jj | | Voor de berekening van — en W, + vu zie de tabellen van BörGeN. Die harm. An. der Gez. pp. 61—67 en 35—37. GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 37 De constanten #, zooals reeds in | is vermeld, hebben betrek- king op den Amsterdamschen tijd, zoodat alleen voor Ostende en Hellevoetsluis reductie’s noodig waren. Om de juiste verachteringen van de golven met de fictieve sterren te verkrijgen, moeten de volgende correctie’s aan de hierachter volgende #s aangebracht worden + Ales waarin Af, voor alle plaatsen behalve Ostende en Hellevoetsluis de waarde van A4 in | heeft; terwijl voor Ostende Alde Hellevoetsluis — — 0.05 en « de „spoed”’ van de fictieve ster. (zie de opgave in 1). ‘s-Gravenhage, Mei 1909. ha an) A ¢ y 7a A a A (A RE AND et ne, } Di ggd À.» v ay ie LA | , 1e i. ia " \ | | ' + | | LE | CT GE UT STE À of GRILLE NES DEAN DEN +. et 1 | | Pur, BEE en NN | Mau » . ET berekend. it de ci SGR "24 (0—1.,..23) en 271 Me oe ; > a Ae £ GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 41 1900. Plaats: | Hansweert | Hansweert | Hansweert Aantal waar- 94 | 8 nemingen per dag: H k FER k Va k ELI. | En | @M) | (ge) | CM) | (er) | CM) | (er) | A, —6.13 —6.20 | S, 1.50 323.2 1,55 325.4 S, 48.14 128.8 48.29 | 128.7 P 3.57 9.6 3.06 8.4 K, 7.38 23.7 7.38 25.8 | ne 14.54 124.4 14.30 125.7 T 5.08) be 01.3 ro OGD R 0.44 | 251.5 0.66 236.3 Sa, 4.96 | 206.1 | 4.94 | 207.5 Sa, Zale 208:9 2e lee 206.8 Sa, 2.18 204.0 2,18 200.8 Sa, pe SR OL we 2.47 54.0 M, 0.72 35.4 0.74 38.4 0.67 16.8 M, 186.89 69.4 | 187.04 69.3 LE 187.24 69.3 M, 0.57 98.3 0.49 101.0 0.73 162.9 M, 5.05 133.0 4.48 183.8 4.76 124.1 M, 6.05 16.5 6.68 16953 6.77 166.6 M, 3.56 130.5 2299 114.4 2.96 152.1 N 31.84 42.0 31.68 41.4 E 20:95 86.3 20.85 86.1 y 14.84 40.1 14.50 39.8 À 405 81.6 7.09 18.4 O 10.65 200.8 OET MAD B, Q 4.04 153.2 4.17 154.4 a AD 4a 1.14 156.4 | aa sh n MS 297 193.0 2.89 193.0 3.28 201.1 uw of 2 MS 17.40 KOI 16.73 14h es, 17.68 170.2 2 SM 6.04 26.8 OR 359.2 MK tack 13.4 2 MK 2.41 234.8 MN 2.34 85.4 Mm 2.23 214.6 2.04 216.9 Mf 1.48 1498 1:51 té » C | a | Msf EU NN RE | a 42 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ, 1900. Plaats: | Brouwershaven | Brouwershaven Aantal waar- | 94 8 nemingen per dag: : H k H k GET. ; (c.M.) (gr.) (c.M.) (gr.) A, —15.93 —15 ,92° St 1.42 318.9 1:02 302-3 S, Oi A? bp as | 27.40 12136 P 3.46 34908 3.39 350.3 K, 1720 4,9 ARE | 6:2 K, 7.50 149.5 Tod 119.8 P 0.98 115.6 119 109.2 Kk 0.48 196.5 0.67 200.2 Su, 1:40 23 à FEAT 7.14 2125 Sa, 2.86 165,:9 2.88 165.6 Sa, 2.27 234.8 2.11 210 Sa, 2.54 90.0 2,48 90.0 M, 0.91 941.9 1.18 56.9 M, 110.99 66.7 110.72 66.6 M, 0.54 6.7 0.34 178.8 M, 12.49 129.0 12.92 127.8 M, 5.55 11529 6.32 113:0 M, 140 159.4 2:01 210.2 \ Toate 01.8 18.98 LEA I 13.02 88.8 19.26 87.4 y 8.10 40.9 8.49 42.0 A 4.18 38.9 3.95 87.6 O 10.57 189.5 10.79 188.3 Q 4,58 144.9 LT 144.4 J 1.02 11275 1:22 143.6 US 8.05 184.1 8.02 18E rt mz of 2 MS 9.26 189.8 10755 184.3 2 SM 4.83 30.1 onde 16.2 Vm 2.89 210.4 2.14 207.9 Mf 1.15 163.9 1.18 Ee ds 0.63 12% se 0.70 145.2 1.45 148.8 ue GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 43 1900. Plaats: Delfzijl | Delfzijl Aantal waar- 94 | 8 nemingen per dag: Ef k H k GET iJ: (c.M.) (gr.) | (c.M.) | (gr.) A, — 9.90 —10.13 5 1.60 53.4 1.49 53.4 Si 29,85 26.4 29.07 26.3 P 2:29 28:9 DE 29,6 UG 1:29 33.7 1.39 33.3 K, 8.40 20:58 8.22 29.9 F 0.95 TOR 0.91 Lo R 0.38 38.5 0:75 20.7 Su, 1299 204.0 12.54 204.5 Sa, = £92 102.4 1.88 105.6 Sa, 3.84 155.9 4.03 156.1 Sa, 6.41 46.3 6.59 41.3 M, 0.60 85.8 0.55 15.38 M, iy ae SLi A 122.91 11.2 M, 0.03 PA i | 0.40 941.6 M, ASE 114.9 14.21 114:2 M. 6.37 303.3 6.20 305.9 M, YI 140.7 1.26 194.4 N 20.19 288.0 20.08 287.8 iB 13.48 336.4 13.43 336.4 y 9.41 289,5 JA 289.8 À 5.90 315,6 5.66 Ole O 9.63 Dt 8.90 DO Q 3.51 184.8 ee) 183.8 A 0.33 349.0 0.40 895.9 MS Tel 1674 7.93 189.3 um of 2 MS 12.63 ze 12240 46.1 2 SM 4.60 295.38 2.40 181.2 Mm 4,80 215.6 4.84 206.6 Mf 2.37 169.7 2,11 169.0 5 : 2,99 132.1 MSf 1 ET 134.5 955 164 0 = | nan ; Le ie : th ‘ ET ted Je 4 : r À À ' 4e ae ie berekend uit de Waterstanden te Pin J 46 GETICONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 1906. Plaats | Ostende Wielingen | Neuzen Hansweert H k H k H H k GETIJ (c.M.) | (gr.) (e.M.) (er (c.M.) (gr. (c.M.) | (gr.) A, | 314.48 — 17.47 — 11.52 Ë 1.168 CA 0:37 oid 0.67 210,6 1.54 DD 0.92 3.5 S, cos (ks,—60°) |-- 54.63 + 38.40 + 28.03 + 18.79 P d'A 318.4 2.59 350.0 2,82 05 3.26 16.7 K, 6.62 OE 5.84 358.7 6.60 19.5 1.00 25.2 K, 15:91 62.8 12:93 15.8 MA gn 12230 1258 7 212 COL 5.81 70.3 OS 88.7 3.29 15.0 Sa 1.83 25077 6.64 255.7 1.02 264.8 5.94 210.3 Sa, 4.15 522 50 188.9 4,01 192.8 5.81 223,4 Sa. 1:20 245.9 0.98 290.3 1 en BALL 1-45 212.6 Sa, 1.88 295.8 2703 230.5 1.52 Dit 2.05 239.1 | —— Es eS eS = 7 M, 176.55 16.5 160.49 BO 179.59 59.1 18979 68.8 M, 10.60 | 342.4 14.99 51.4 Jago | 94.4 5.62 108.2 W, 6.32 | 385.6 | 18.14 07 904 9.76 | 84.2 | SA GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 47 1906. Plaats: Vlissingen | Veere Wemeldinge | Zierikzee H k IT k H k | H ki GETIJ. (c.M.) (gr.) | (c.M.) (gr. (c.M.) (er.) | (e.M.) (gr.) A, — 17.84 — 14.04 —8.27 — 14.12 S, 0.78 314.8 0.64 325.0 0.60 323.0 0.88 326.6 S, cos (ks,—60°) [+ 36.40 + 20.89 + 10.71 + 16.01 Pr 2.41 352.3 2,55 356.2 3.67 12.8 2.83 3.3 K, 6.73 6.4 6.68 359.2 (PE XI 22.5 1.21 15.5 K, 12.66 93.9 9:90 108.0 10.22 140.6 8.58 127.4 i 3.15 68.8 3.01 88.1 2.69 115.8 8.27 100.5 Sa, 1:29 249.0 1.35 249 29 7.68 241.7 8.02 244.0 Sa, 4.04 189.6 4.23 16921 3.85 £9332 4,40 201.0 Sa, 1.08 350.4 2.00 Load 2.80 10.0 2.87 353.3 Sa, 2.28 226,9 2.02 261.0 2.24 284.8 1.59 258.6 M, 168.73 43:81 .182;97 56.8 | 150.59 16.2 | 136.66 66.4 M, 12.76 63.0 10.16 961 5.98 199:8 5.44 126.0 M, 12.47 46.6 9:31 6927 gots Ie LA abs B18 | 48 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. ESP ei a eer. à An er 1906. Plaats: Brouwershaven Bruinisse Steenbergsche Vliet Willemstad H k H h H k H k GETIJ. J (M) | (er) | (0M) | (erde | (eM) Mer) | eee (er) A, 2e 1447 — 10.02 — 9.49 + 6.69 S, 0.50 | 320.9 0:60 STA 1.38 7.4 0.43 53.9 S, cos (ks,—60°) [+ 11.55 + 7.33 — 0.46 — 9.52 P 3.20 0.6 3.55 10.4 3.51 17.5 Bend 18.5 K, 7.23 9.5 1.73 19.8 7.20 24.7 6.68 29.0 R 6.66 | 135.5 9.17 | 133.6 5.94 | 165.9 4.81 | 192.6 T SA JBA 2.69 | 128.6 1:27: | 10636 1.84 | 168.2 Sa 8.06 | 248.6 7.39 | 258.5 7.92 | 258.6 7.86 | 288.3 Sa. 4.68 | 198.4 4.54 | 211.9 0.83 | 183.3 6.48 | 298.6 Sa CN CR ae 2.47 0.0 9,99 11.4 3.40 5.3 Sa, 2.13 | 284.2 2.07 | 286.8 2.77 | 305.3 3.84 | 52007 M, 113.31 69.9 | 135.50 79.9 | 123.87 92.4 | 95.38 | 115.5 M, 12.25 | 196.9 8.06 | 193.5 | 10.00 | 202.1 | 12.43 | 190.1 M, 7.79 97.3 20 +) DAT 2.96 | 208.2 2,81 | 2041 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 49 1906. Plaats: | Moerdijk Mond der Donge Willemsdorp Hellevoetsluis H k H k H k H k T a | (c.M.) (er) | (cM) | (gr) | (M) (gr) | (M. (gr) A, + 21.42 + 22.40 + 15.34 — 4.60 S, 0.80 69.2 0.76 84.7 0.60 36.3 0.86 321.6 S, cos(ks,—60°) |— 14.89 foot — 15.20 + 2.36 iP 2.56 29.0 2.24 67.2 3.00 32.0 2.89 5.6 K, 6.60 34.9 4.67 56.0 6.06 41.8 6.72 1269 K, 5.86 208.3 Del 259.8 5.95 210.9 5.84 149.3 a 1.54 189.4 1-02 347.5 1.80 190.9 Lod 115:3 Sa, 8.03 303.2 25.42 44.7 9.65 317.2 1.54 259.4 Sa, 7.44 230.9 10.90 47.9 St 231.1 5.43 209:9 Sa, 4.10 4.2 4.68 234.4 3.61 353.0 2:39 357.8 Sa, 5229 325.6 8.07 250-9 5.00 325.0 2.67 303.8 M, OAT 133.3 54-67 178.9 89.08 136.6 88.25 89.4 M, 10.94 218.6 6.49 307.6 WIED 220.5 14.07 145.2 M, 0.85 153.7 2.42 131.8 16 171.2 4.63 104.9 Verhand. Kon. Akad. v. Wetensch. (1e Sectie) Dl. XI. B 4 50 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ, 1906. Plaats: Spijkenisse Puttershoek ‘s-Gravendeel Dordrecht H k IT k I k H k GET. | CH) | Gr) | CM) (er) | (c.M.) (er) | (M) | (gr.) A, + 11.67 + 11.31° 125.22 + 35.63 S, 0.35 201.9 0.93 20.0 1.61 233.8 0.76 84.7 S, cos (ks,—60°) | — 7.43 Zld — 15.68 — 14.98 2 2.15 24.2 2.19 41.8 2.81 Zhe 2.72 46.6 K, D.67 23.0 D.68 37.8 4.87 219.7 5.34 42.2 K, 4.57 190.5 Daf 215-6 6.07 216.8 5.47 229.5 id 1:95 162.3 1.88 196.6 1.15 184.0 LA 214.3 Sa 1.83 291.2 9.21 221.2 12.64 324.4 12.92 336.7 Sa 1:29 229.2 9.16 234.7 +0 241.1 10.50 241.2 Sa, 3.33 300: 3,07 306.0 3.48 308 .D 4.64 350.5 Sa, 4.23 323.9 6.02 330.4 6.68 399.0 1.62 338.3 M 67.83 120.4 74.07 144.8 17.64 147.0 69.84 159.0 M, 10.66 1911 11:96 233 .5 12.22 213 4 12.61 251.4 M, 3.28:| 164.1 1.36 | 186.1 0.65 | 188.0 0.84 | 193.9 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 51 1906. ats: Alblasserdam Vreeswijk Schoonhoven | Streef kerk H k H LE H k H k GETIJ. (c.M.) (gr.) (eM) (gr.) | (c.M.) (er.) (c.M.) (gr.) de + 32.95 + 190.94 + 73.43 + 45.94 S, 0.41 41.8 0.58 13.8 0.24 38.3 Oa 42. S, cos(ks,—60°) |— 13.90 — 0.46 — 8.33 — 11.40 P el 44.7 0.64 108.2 1795 58. 2:05 B K, 5.47 39.4 £350 82.6 4.14 44,9 5.07 43.5 K. 4,94 226.6 1227 327.5 3.85 25972 AAD 240 "à 1.44 207.6 0.99 280.9 jen 284.5 1.45 220 Sa, St 333,9 tot 13,0 30.10 0.8 ksa 346. Sa 10.30 240.0 31.58 248.7 16222 pies 6.99 259 Sa 401 349.0 13.67 20112 Sn 347.2 6.48 318 Sa, 1.34 22352 95.92 20.9 16221 353.4 10.59 345 M, 65.20 157.3 8.10 268.5 39.88 188.5 51.40 171 M, 11.28 3449 211 A1 .4 10755 298.0 11.68 PRE M, 1.46 208.7 Ong 92310 1.41 50.4 0.44 Bld 52 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 1906. Plaats: | Krimpen | Rotterdam Vlaardingen | Maassluis H k H k H k | H k GETIJ. (c.M.) (gr.) (c.M.) (gr.) (c.M.) (gr.) | (c.M.) | (gr.) A, + 34.19 + 19.45 | + 11.25 + 0.89 =. 0.56 38.3 0.43 60.1 0.40 331.3 0.52 304.8 & «cos (ks, 60) | 12-93 — 9.80 t= pont — #00 P 2.63 41.8 2,14 21:98 2.68 19.4 2.42 9:2 K, 5.29 95.1 Deel Pis el 5.68 15.6 6.05 6.3 Ki 4.97 224.6 4.61 198. 4.19 188.3 4.38 171.8 T 1.67 | 203.8 | 2.01 | 178.7 | 1.99 | 157.2 | | )aneeeeee Sa, 6.30 344.9 902 314.5 1.68 2984 1-44 281.2 Sa, 10.61 240.7 8.56 233.0 To 229% 6.54 2918 Sa, 4.51 348.8 3.16 304.6 3.53 352.0 3.22 354.9 Sa, 1.3) 337.5 5.35 330.7 4.26 329.2 3.44 314.2 M, 61161 153.5 64.53 133.4 65.07 528 68.05 98.1 M, 10.50 245.9 9.93 Pl WAR 10.67 181.5 137% 162.7 M, 1.70 208 . 4 3.30 1854 3.80 1529 4.51 122.4 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ 93 1906. Plaats: | Hoek van Holland Scheveningen | Katwijk | IJmuiden H k | fie k IT k IT k pet ls. | (c.M.) | wr) | CM) | (er) | (a) (er) | (M) (er) En AAE A, — 11.59 ET ERO 16.57 | NER Ss, (07 |) SA £191 300-6 bs OS sa | os S, cos (ks,—60°) | + 6.74 + 3.04 — 0.66 8.29 p 166 sal ease 4 die || sieg 3598 [4 5.301 8387 K, 1.51 | 355.7 7.22 | 354.8 woe See |e i Tete ab K, ae | eee Dior last + SPORTS QC act T ER SAR Po Mer De BED OAD LI ven Ie Pa d pees Sa, 8.78 | 259.7 | 10.50 | 268.5 | 10.57 | 264.6 | 11.46 | 251.8 So, 5.51 | 204.8 | 4.86 | 208.7 5.26 | 203.3 | 6.95 | 222.0 Sa 2.07 | 353.1 | 2.18 | 355.8 | 4.98 | 359.8 | 1.64 | 298.8 Sa, 2.27 | 275.8 | 2.76 | 282.7 7.24 | 351.3 | 2.11.| 225.2 M, 77.51 fa 0e as | eared tee 54") 998.8 1 66.35 115 0 M, Wieder |) 19608" Mio ie en res | Eur de Ie besa M, 5.62 T0061 88.28 ARE PE EDS re ON 94 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ, d — >” 1906. Plaats: Helder | Vlieland Enkhuizen Oranjesluizen : H k H k H k H k GE TI. (c.M.) | (gr.) (c.M. (gr.) (c.M.) (gr.) (c.M.) (gr.) Ao — 11.12 — 12.73 — 5.69 — 1.75 ze AE : _______| S, 1.05 306.6 1.02 307.0 0.34 265.9 1.44 | 164.6 S, cos (ks,—60°) |— 15.03 — 10.19 + 1.19 + 2.16 52 2.70 345.0 2.31 355.8 0.84 87.4 1.45 119.2 f 5.09 6.0 5.40 14.8 is’ ES | 2.16 140.5 ; 5.27 233 .4 3.84 288 .4 1.02 18.7 5 101.4 1.82 215.4 2.57 231.1 0.29 321.8 0.46 125.1 Sa, 11.96 259.9 12.41 249.8 6:55 262.4 1.42 241.1 Sa, 5.52 225.0 4.79 215.6 5.48 233.4 2.98 257.6 Sa, 2.97 345.6 2.78 338.3 3.45 355.2 . 4.02 0.9 Sa, 2.26 250.6 1.26 241.9 5.26 320.6 8.33 338.1 M, 53.01 169.6 63.99 233.7 11.47 294.1 13.88 13.5 M, 9.40 190.1 5,10 325.9 2.21 338.9 1.61 271.1 M, 5.34 | :299.9 2.72 29.9 1.05 | 227.0 0.06 53.5 : GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 55 1906. en Plaats: Nijkerk | Elburg Kraggenburg Schokland k H k D H GET. (c.M.) (er) | (c.M.) (er) | (c.M.) (er) | (cM.) (er A, —9,91 + 1.20 + 6.12 | + 5.45 S, POF 117.6 0.65 15.1 0.68 350.0 0.43 SIA ib, cos (ks, — 60°) | + 2.49 + 2.12° + 1.69 + 1.25 P 1122 98.6 0.78 88.6 0.69 16.7 0.66 109.3 K, 2,95 116.4 2.69 122.6 SO 118.5 2,20 189 i. 1.15 63.0 0.78 48.0 0.91 36.2 0.59 40.5 F 0.30 973 0.07 197.9 0.28 308.3 OCT 97.6 Sa, 1140 30.5 ».61 300.7 5.74 M2: 4 5.99 21:71 Sa, 4,50 224.9 6.60 238.1 7.30 239.6 6.59 234.4 Sa, 4.78 B27 5.43 5.6 5:07 3.8 5.85 5.9 Sa, 9.98 Boot 6.70% 321.4 7.76 3120 1:95 316.5 M, 13.40 47 10.21 989 6:35 D 4 5.99 ns M, 1.99 260.2 T 301 286.3 0.70 294.1 0.78 305.2 M, 0.85 59.0 1.38 12.4 1.88 8.3 142 22): 56 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 1906. Plaats: | Urk Lemmer | Stavoren | Hindeloopen. | H k k FT k REE ASA MEM) @r) | (eM) (er) | (Cr) (gr) A, 3:07 +5.97 + 0.66 S, 0.29 | 282.0 | 0.68 | 303.0 0.60 | 320.6 | 0.85 S, cos (ks,—60°) | + 1.47 EAU LAS LH p 0.69 | 103.9 0.69 97.2 1.00 34.6 1.18 K, RENNEN 1 TAN (088 2.09 50.6 2.71 À 0.70 28.8 0.68 21.2 1.335) 191920 1.95 T 0.28 | 315.5 0.49 | 318.8 0.55 | 279.7 0.7% Sa, 6.44 | 264.9 8.82 | 269.0 8 96 | 266.1 | 10.22 Sa, 6.02 | 230.7 100) Pen 5.97 | 230.1 6.42 Sa, 4.94 8.2 4.48 | 354.4 2.09 | 359.8 3.88 Sa 6.55 | 316.9 5.53 | 303.5 4,76 | 303.2 4.49 M, 6.74 | 346.4 4.34 | 342.0 | 21.14 | 245.4 | 96.41 M, 1464) Sie dS SIET 3.45 | 325.5 2.87 M, 0.74 | 315.3 0.76 | 335.1 1.28 | 142.7 1.50 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 57 1906. Plaats: Harlingen | Roptazijl Zoutkamp Delfzijl H k H | k H k H k BE. GETIJ (c.M.) | (gr.) (c.M.) | (gr.) (c.M.) (gr.) (c.M.) (gr.) +, — 4.23 | — 1.85 + 0.01° — 10.68 S, 1.08 314.3 1.02 327.0 2.73 2.3 1.66 22.8 S, cos (ks,—60°) | — 0.61 + 0.07 + 10.39 + 26.23 à 210 14.5 E95 11.6 2.76 36.4 3.31 38.1 nok, 4.81 31.4 4.78 281 6.22 37.7 6.62 41.2 K, 3.63 331.4 3.65 336.0 8.01 16.4 10.31 39.3 ij 05 285.7 1.16 278.8 2.06 313.4 2.27 344.3 Sa, Loo Dee? 11.93 261.8 14.82 278.2 10.75 240.9 Sa, Det 228.1 6.26 231.2 9.24 233.8 6.24 244.9 Sa, 3.10 349.3 3.93 353.5 4.54 324.7 6.87 348.3 Sa, 3.85 285.6 4.00 282.7 97 294.6 6.20 306.3 M, 56.32 261.6 61.50 264.4 96.09 ARS TION 324.1 M, 3.82 Ets0 4.08 oves OM 44.3 12.59 1579 M, 3.49 183.4 3.25 194.1 3.06 212.3 9.54 312 .4 58 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 1906. Plaats: Nieuw-Statenzijl ie | k S, 4.58 357.4 S, cos (ks,—60°) +- 24.89 P 3.31 56.6 K, 8.21 18.4 K, dad 69.6 T 4.20 262.1 Sa, 14.53 296.2 Sa 16:29 251 9 Sa, rea) 1.6 Sa, 10.74 31935 M, 105.87 354.0 M, 7.49 214.1 M, 2.41 350.0 ue de STA st GETICON berekend uit de Vaterstanden ln 23 _uur 60 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 1906. Plaats: | Ostende Neuzen | Hansweert Vlissingen FT k Jad k H k H k Dr (c.M.) (gr.) | (c.M.) (gr.) (c.M.) (gr.) | (c.M.) (gr.) A, + 317.38 — 10.68 — 0.79 — 17.20 S, : i We 3.8 3.53 302.3 1.39 333.0 0.84 307.4 S, sin (ks,—60°) | — 1.55 + 37.68 + 43.77 + 27.81 P 3.20 348.7 2.16 341.4 2.92 0.4 3.31 337.7 K, 6.07 356.2 8.11 1558 tbe 22.3 6.83 5.5 B 15.18 (or 14.68 11279 15.15 129.3 13.66 103.6 fl 3.36 75.5 3.16 93.2 3.38 132.4 3.10 91.3 Sa 8.94 231.2 7.10 264.5 8.58 283.9 1.2 251.8 Sa, 5.40 1974 3.89 210.4 4.61 231.2 4.12 209.6 Sa 0.75 399.2 100 321.7 0.87 334.1 1.30 349.3 Su, 0.66 218.0 1:97 255.5 Lay 211.2 1.03 193.6 181.86 Pio. Lid 282 56.0 | 183.38 65.91 41075 41.2 M, 11.43 357.0 8.44 122.2 5.67 155.5 10.43 89.0 M, 6.06 316.1 6.64 131.5 9.61 189.2 6.03 64.2 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 61 1906. Plaats: | Wemeldinge Zierikzee Brouwershaven Willemstad H k EN k Jul k Va k GETIJ | 4 (c.M.) (gr.) (c.M.) (gr.) | (c.M.) (gr.) | (c.M.) (gr.) A, es rte 1258 RSC AT 04 5 1.24 341.9 1.20 331.6 1.35 290.6 0.78 39.3 S, sin(ks,—60°) [+ 34.90 + 29,37 + 94,88 + 19.39 P 2.89 346.3 2.83 343.9 5.85 353.0 2.51 22.8 x 7.59 21.2 7.31 12.5 7.19 78 5.56 6.1 i, 12.33 131.8 959$ 2128.3 8.96 126.6 6.71 172.6 T at 112.9 3.34 99.7 2.80 109.1 +51 146.3 Sa, 7.27 248.9 THT 248.6 7.74 255.3 9.00 290.7 Sa, 3.56 192.9 3.32 200.7 3.78 208.5 5.86 20.1 Sa, 1.61 12.1 2,76 352.3 2,33 343.0 1.48 2.6 Sa, da | 987:1 OVAA | 97687 1.88 | 280.2 2.99 | 335.5 M, 14317 | 74.1 | 130.66 | 63.7 | 106.71 | 67.7 |: 92.20 | : 117.7 M, 8.37 198.9 6.71 155.0 LT 135.4 14.00 | 183.3 M. 7.28 217.1 3.26 183.9 4.08 139.7 6.50 | 249.8 62 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 1906. Plaats: Hellevoetsluis Rotterdam Hoek van Holland GET IJ | (c.M.) {gr.) (c.M.) | (gr.) | (c.M.) (gr.) A, — 3.86 + 19.51 — 10.70 S, : 4:00 D | 0.48 817 1/32 292,4 S, sin (ks,—60°) [+ 20.17 + 10.38 + 17.20 P 2.75 0.0 208 28.1 3.41 338.9 K, 6.80 195 5.63 20.8 1:82 357.0 K 6.70 143.8 3.95 198.6 6.53 13514 ‘à 2.83 tt 27 1738 MOT 2.43 120.5 Sa, 8.04 | 264.1 8778.1 3144 8.19 | 254.9 Sa, 5.09 21128 128 93052 A STS Weg Sa, 2.51 390.0 3261 349.9 2.03 $45.7 Sa 2,45 291.8 4.56 324.8 2.00 263.9 M, 82.63 89.6 64.43 136.2 72.86 Fe er M, 15.28 | 149.6 10.67 200:3 16.74 19275 M, HAE | Tom 302 226.9 1.56 14.0 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 63 1906. Plaats: Helder Vlieland Delfzijl k H if ae Let Lo | ey | a je | oy Les ee) | (eam) | Gr (eat) | Gr) S, sin(ks,—60°) | + 1.28 — 14.35 — 13.87 94.5 00 TBS i) on er) = 2.67 346.0 6791 39. (ae) K, 5.46 | 359.3 ee ep de) OD © ©) (SU) bo (SU) E> a GO (=>) e ©) di — iw) (@ 2) =] 3.91 242.0 r 0.70 189.9 11.63 290.7 3.12 49. 4 6987 50 55.21 2614 127-98 325.7 4.58 =] =] — bd i D Qo 4 ©9 =] QD M, 13.35 | 183.3 — a =~] us =] © de) bo e 2.00 148.4 2.80 21.4 A, — 10.92 — 12.47 ORN RER EUR ae — 3.99 S, 0.96 | 312.6 0.62 1-07 | 348.1 1.42 74.5 he Sie IS RES Te va ARE RES, Tie at) - 2 ey TEN. 4 © LE 7 AN van 1906. | | berekend uit de GETIJCONST er ctanden te 2—5 —&—11—1 47-17 BO 28 UE 47 EA = Verhand. Kon. Akad. v. Wetensch. (Je Sectie) Dl. XI. et: “+ OPMERKINGEN. De constanten van Hansweert van de S serie zijn berekend na schatting van de ontbrekende gemiddelden. Zie n°. 1. De overige, uitgezonderd het getij y, waarvoor eene correctie wegens JV, en ASF, waarvoor de bekende correctie wegens M, is aangebracht, zijn zonder correctie’s voor storende getijden berekend. Evenwel met inacht- neming van de voorschriften voor den aanvang en het einde van een bepaalde periode (zie Darwin Se. P. I p. 243). Bij MSf zijn de uitkomsten volgens n°. 15 onder die, volgens de methode van Darwin verkregen, geplaatst. | GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 67 1906. Plaats: | Ostende | Neuzen | Hansweert | Vlissingen H k H k H k GET IJ | in | | (c.M.) (gr | (c.M.) 5) | CM) (gr.) | (c.M.) (gr.) À, + 317.58 — 11.10 — 1.16 — 17.52 A 0.72 321.1 1:59 305.3 1.00 339.2 0579 308. S, 54.65 68.4 46.96 113.4 47.63 126.8 45.80 de Beos (ks, —120°) | + 0.20 — 0.42 — 0.37 — 0.32 P 2.89 339.2 2.56 391.3 2.88 10.4 2.70 339.8 K, 6.26 309.3 1.24 14.6 7:56 24.4 6541 at K, 16.81 67.4 14.76 PET 13.72 EMD 13.14 pj E 3.03 Ee 3.44 90.6 2292 104.0 3.35 78.8 R 0.54 88.8 0.33 237.8 1.60 287.6 0.74 284.8 Sa, 8.39 230.5 tsa 264.6 8.72 244 20 (imal 250 Sa, 4.77 TAO 4.05 20151 50 226.8 4.02 199 Sa, 0.58 248.7 +67 332.3 0.86 299.8 +19 349 Sa, 1527 220. ek 265.0 Fete 224.3 1-60 ie 286 7 EE = | En en M, 1:02 69.0 1.58 10.8 1.05 92.8 1.33 61 M, 179.60 PT PERRET BA 60 [166752 67.4 | 167.89 42 M, 1.46 85.4 1530 ES 2.68 156.5 NED 124 M, 10.98 350.7 BG 107.6 Het 132.0 11.38 Wi M, Gell |) 6242 7.54 103.0 6.63 168.5 EG 52 M, 2 Die 262.0 2700 72.4 3.98 148.3 3.20 26 N s0r80R |) ony x6 gode IN sG 29.70 44.5 24.10 23. E hen he DD 14.58 18.8 16.87 101 19:49 46. y 992 321.0 9:85. | 1994 12200 31.2 SUE 300. À 5.24 55.4 4.54 90.2 4.60 12 O ged 183.3 10440 201.7 12.69 197.6 10.60 19 Q 4.57 12546 4.54 138.2 5.14 140.8 ele 132.5 k 0.08 359.0 0.87 152 0.86 187.3 0.33 228.6 MS 1.49 2008 peda tes 2.09 206.0 6.89 122 4: wu of 2 MS 10.73 252.4 1346 167.4 17.63 167.7 1109 153.5 2 SM 3-07 242 6 5.30 859.2 De 356.8 8.92 aoe Mm 4.72 246.9 4.66 272.0 5.65 Zod sl 4.77 260.1 Mf Boils 62.8 ASIE O0 2 LE 198 DEMO T2 MSf .68 63.8 OU RER 4.79 19.4 SEEN Naas | 1.98 21.4 DCE TRA 5.37 1.3 SAC AE 68 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 1906. Plaats: Wemeldinge Zierikzee Brouwershaven Willemstad k H GE RIJ. (ed | (CM. A, — 7.88 toon" See 4 6080 S, 0.98 | 327.8 1.06 | 325.9 0.91 | 301.4 0.57 37.4 Si 36.51 | 132.9 | 33.46 | 121.4 | 27.45 | 195.1 | 20 S, cos (ks,—120°) | —0.39 —0.30 —0.66 Gel P 2.93 3.9 2.58 | 353.3 3.17 | 356.2 2.85 23.0 K, 7.44 ai .5 7.18 13.9 7.15 8.5 6.29 29.5 K, 11.24 | 135.8 9.95 | 127.9 7.79 | L 150,2 5.67 | 180.9 T BAT.) 1 3.29 100.0 a 18 88.9 1.63 | 156.1 R 0.54 13.5 0.05 45.9 0.98 | 291.6 0.42 77.9 Sa, 7.47 | 244.8 1.19 | 947.9 7.89 | 251.9 8.43 | 289.6 Sa, Bie.) 18547 3.86 | 200.9 4.91 | 202.9 6.15 | 224.5 Sa, 1.96 10.9 2.01 | Lx 2.20 | 352.8 2.44 4.5 Sa, 2.30 | 285.7 1.98 | 269 2.01 282.3 3.36 | 328.2 M, 1.43 83.1 1.20 79. 1.39 75.5 1.00 90.1 M, 146.85 15.2 | 133.62 65. 109.99 68.8 | 93.78 | 116.8 M, 1.02 186.1 Ost lt 1746 0.58 | 163.3 0.77 | 229% M, 7.16 199.3 5.88 142.1 12.67 131.0 13.19 186.5 — M, 4.42 | 197.2 3.02.; 102.6 5.58 | 111.5 1148 | 2040 M, 1251 10127 2.64 95.6 0.41 185.5 1.42 ak 0 N 93.97 | 58.0 | 21.65 4 16.79 49.5 14.03 98.4 L 12.94 18.7 11.34 q 9.56 79.0 8.89 |° 1946 y 9.49 17.2 7.65 5.96 20.2 6.28 63.3 A 5.06 | 111.9 3.91 10 3.21 | 101.8 3.12 | 163.3 0 10.55 | 202.6 | 10.52 | 196.8 9.22 95.6 9.15 (AR QO 5.18 | 148.4 5.04 | 148.1 5.05 | 137.6 4.54 10416978 J 0.45 | 195.1 0.27 128 0.32 | “121.4 0.47 | 316.8 MS 4.32 | 9239.3 3.71 196. 7.81 185.4 7.11 NAS u of 2 MS 19.54.) | BAH 10.75 | 180.0 9.64 | 187.3 8.99 | 2297.9 2 SM 4.53 10.5 4.31 9. 3.92 18.6 2.19 46.6 Mm 4.58 | 268.8 5.53 | 267.4 5.20 | 261.0 5.53. | 275.9 Mf 4.94 48.4 4.95 50.0 4.57 50.0 5.67 46.7 MSf ( 3.59 42.8 3.31 34.1 3.18 38.5 4.51 42.8 es | 3.78 23,9 3.89 14.9 3,3 15.1 4.53 28.0 A s i GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 69 1906. 4 Plaats: Hellevoetsluis Rotterdam Hoek van Holland IJmuiden 5 | eee Ya EAN Re | k HT k H k H k | GET TJ. | (c.M.) (gr.) | (c.M.) (gr.) (c.M.) (gr.) | (e.M.) (gr.) aR . A, — 4.23 + 19.48 Lite 2 15.01 S, | 0.99 351.4 0.39 48.5 1370 297 .6 FO 301. S, 20.30 144.8 14.28 193.5 18.48 128.6 16729 180. BEDS (ke, 120°) | — 0.37 — 0.03 — 0.47 LEE P 2240 Dro 2.49 50.1 3.00 366.21 Es 99920 K, 6.75 1250 5.61 22.4 1.12 355.9 Gone 354.7 K, 6.26 146.3 4,28 198.6 5.50 191% 4.47 18332 T 2:25 11277 1:69 170.4 1 UO; le MO 1.47 146.6 R 0.61 14.5 0.34 101.9 0.82 338.0 0.69 36.4 Sa, 7.78 261.8 8.90 314.5 8.49 208.1 10673 D538 ok Sa, i260 - 21038 8.08 39314 5.09 210 :1 6.18 De Sa, 2.44 353.8 3.66 348.5 2.05 349.4 iste 294.1 Sa, 2155 298.0 495 328.0 2:12 280.2 2.84 22163 | . M, 1.01 14.6 1067 96.6 1.35 67.6 138 65.3 M, 89.44 89.5 64:87} 184.8 15:28 12.54 65.10 115.8 M. 0% | 1095 |. 0.40 | 19.6 | 0148 | 980.7 |, 0.49 # 271.0 M, 14.67 147.5 10.24 206.3 16.99 150.7 18.18 1545 M, 210% 109.9 BAD 206.7 Seo rip lee | 4.37 rAd M, 1:22 163.5 0.43 189.6 1.40 ast 2.66 265. N 13.29 11230 9.46 118.4 10.84 5278 915 101.6 ie 8:88 98.7 DU 146.0 1:10 82.4 1:19 112,7 y iy 24 2912 ad 82.3 4.46 21.8 A 3d 65.2 À Puts Halal 1.46 141.9 2.60 11873 2.26 148.9 102 198.6 8:29 pe a NE TE 249 187.0 PA? 187 Q 4.86 146.8 3.98 160.8 4.94 BPs HN 4.89 134 0.88 412 0.36 59 O71 EG 0.46 166 MS 8 99 20172 6485 2614 10:91 182:9 10.10 eal LE of 2 MS 0.92 207.8 7.76 209.9 7.68 201 53 8.06 219 2SM 2.82 Bi 2D 1:55 19:90 2.48 BAT 59 42 Mm DAT 267.8 6.49 24128 5.66 262.0 6.20 Jo Mf 5.30 | 65:2 4,18 45.9 Lio 4129 Si) Dene MSf 2.86 Zoek 4,45 40.2 2.98 = NS, 3.01 | 40.2 ei HUE SLP 4.13 ADE? 3.08 18.3 a2 ls | 13.4 10 GETIJCONSTANTEN VOOR PLAATSEN LANGS DE KUSTEN ENZ. 1906. Plaats: is Helder Vlieland Harlingen Delfzijl H k H k H HAE GETIJ. ick I [.) (er) (e.M.) (gr.) (c.M.) (gr.) (c.M.) (gr.) A, =f 11502 — 12.60 | ae 102 : se ; | | n S, 1.08: | : 310.5 0.88 | 322.4 1.16 | -\ 3812 1.66 oe pj S, 15.08 | 235.1 | 16.60 | 292.1 | 14.36 | 327.6 | | Bouma 32.14 S, cos (ks,—120°) | — 0.10 — 0.14 | — 0.12 + 0.06 À LE KEN | EN P 2.70 | 345.6 2.3 357.3 2.20 19.3 2.88 28.0 À K, 5.36 2.9 5.15 12.5 4.89 30.6 6.64 40.0 K, 4.58 | 287.1 4.75 | 294.5 4.61 | 3825.0 9.20 32.4 4 i EN 2 1.89 | 9444 1.29 | 292.6 1.66 | 339.5 K 0.63 | 62.8 0:68 | OL 0.55 | 177.9 0.63 | 302.9 Sa, 11.80 | 255.7 12.84 250.2 11.82 260.6 10.65 241.3 Sa, 5.20 | 219.6 5.14 | 91820 5.97 | 229.5 6.10 | 244.7 Sa, 2 #01) | 341.3 2.77 | 336.4 3.70 | 349.3 6.92 | 349.2 Sa, 2.69 | 248.9 1.26 | 243.3 3.62 | 289.9 5.82 | 299.3 M, 1.01 80.3 0.91 101.7 0.93 | 169.4 0.76 | 136.0 M, 4.56 | 170.0 | 64.64 | 231.8 | 55.79 | 261.7 M0 OS M, 0.54 | 249.3 0.44 | 353.9 0.20 54.5 0.69 | 153.2 M, 11.36 | 187.1 2.81 | 308.0 3.64 50.0 | 13-2100 M, 6.39 | 293.1 4.82 32.5 2.62 | 170.9 5.42 | 326.0 M, 1.34 | 340.6 1.41 | 296.0 8.07 | 315.5 0.21 | 152.3 \ 8.79) | 158.38 | ! 10.27 |) Aidan 9.58 | 254.0 | 18.99 | 3078 i bale: | 156-9 5.20 | 994.9 5.48 | 958.5 9.09 | 323.8 y 3.42 | 106.8 379.1 0 3.93 | 204.2 7.23 | 269.6 À 231) | 186.6 2.05 | 257.8 2.01 | 246.5 4.50 | 353.2 0 8.27 | 196.4 6.35 | 210.9 6.20 |") Zak 9.08 | 241.4 0 3.13 | 142.5 3.04 | 156.0 2.70. | 170.8 3.39 | 178.0 J 0.41 20517 0.45 19129 0.34 189.7 0.70 252.4 _ MS 6.43 | 247.4 1.82 44.0 2.93 | 128.4 7480 TERRE u of 2MS 5.*0 | 262.0 7.25 | 344.9 6.29 | 358.1 | 12.56 57.9 2 SM lan 107 :4 D. DE OR Be 0.56 | 218.7 1.96 | 276.2 Mm 6.51 | 261.6 6.23 | 264.3 ets LD 5.94 | 279.3 Mf 4.60 59.9 1.15 49,5 ». 76 DOS 5:22 37.8 Msf ( 1.98 47.6 2.79 54.5 5.22 55.2 2.69 | 108.4 zi (2.44 8.1 2.39 28.2 5.17 37.6 134 51.1 MH VAN BERESIEUN. ,, Getyconstan ten voor plaatsen langs de kusten enz. in Nederland. : e Terschellingerbant: “a en ( © 7 Onanjesluizen A we ©, iS) ae “Ostende % Se Vlieland A Je “so pe ze "+ y 8 & Ô $# Ropdax ijl » HEC Harlingen / „600 4 Jane „> a ce 30.00 APES N Haales ip Es UD 4400 of Prises / AG + Stavoren | | SE LE — 6 Streefkerie == ©" s) =e > = ke ro \Sr 1% Far IE Ib don KS D ak ujhenissee = ct Gorinchem Her wynen Ÿ kes = SE Ly ets CR TS SN Oe Puttershise ay sas Lei À) © ses So < vantL à 4 Cx tt) ae i th Gia pee 2) SS 3 & Brouwershavè WEE ond GRR © = derDonge M 5 Hind % 500 ara 20 as 5 Hille Tt atts \ Je re. i) $ = Zieriksche_ En SE Stenbergache vliet wv RS a2 =< \ iS Ry &, 4 ki / Veere EEN \ EN G en Kas ms DRS LR \ = Ee Wemeldinge \ 5 4 er Vlissingen e — / Hansweert \. 700 50 ; 2 ~ West Hinder 1250 © deg 00] Oe SRD Wandelaar : LT 7 Wielingen =~ So x î Ho 4 ey, he il Nieuw Statenzijl 4 Afstand | If: tusschen wk M Ne c oordxee 200 Hoek v. Holland 00 Maassluis Le ri di 7.65 aardingen 11.05 Rotterdam z Krunnen eee Str, Za pee of ker|c D eef kerk: 8.10 Schoonhoven 21.40 ee wyk Des C wlemborg Noordzee ee 17.50 Spujkenisse i 19.20 Puttershoek 7.10 Dordrecht A id 13.50 ee Renu 9.25 lerwynen 10.65 Zalt-Bonunel Krünpen 60 Alblasserdant Ari Dordrecht ees fi 19 sGravendeel A 5 if illentsdorpt pene Moerdyk | | Térhand.Kon.Akad.v.Wetensch{(I£Sectie) DL XT N22. JBijtel lit. PI Mulder umpin Leiden ik 5 oe À A É LES de E (Section EThe simplex) + 00 Te) J a P. H. SCHOUTE. 3 _ Verhandelingen der Koninklijke itelae van Wetenschappen te Amsterdam. | Ee (EERSTE SECTIE). : | NE Dn DEEL XI. N°. 8. (With one plate and three tables). AMSTERDAM, | 7 es JOHANNES MULLER. | RÉ November 1911. | ee > Pa ACT med herken Analytical treatment of the polytopes regularly derived {rom the regular polytopes. BY PH SG EG Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam. (EERSTE SECTIE). DEEL XI. N°. 3. AMSTERDAM, JOHANNES MULLER. 1911. Analytical treatment of the polytopes regularly derived from the regular polytopes. BY PH. SCHOUTE: INTRODUCTION. In a memoir recently published by this Academy (Verhandelingen, vol. XI, n°. 1) M. A. Booze Storr has given the geometrical treat- ment of the polytopes regularly derived from the regular polytopes of polydimensional space. In these pages I wish to complete her beautiful considerations by giving the analytical counterpart ?). The basis of this analytical counterpart is the fact that the coor- dinates of the vertices of the tetrahedron may be represented by one of the symbols (1, 0, 0, 0) and +[1,1, 1}, while those of the ver- tices of cube, octahedron and icosahedron can be put in the forms [1,1, 1], [1, 0, 0] and[1 + //5, 2, 0]: 2 respectively. The meaning of these symbols will be explained later on. This paper is divided into five sections. In the first, concerned with the offspring of the regular simplex, we will meet chiefly the amplifications of the symbol (1,0,0,0) of the tetrahedron. The second and the third, dealing in the same manner with the measure polytope and the cross polytope, will bring us chiefly amplifications of the symbols [1, 1, 1] and |1, 0,0] of cube and octahedron. The fourth will deal with the half measure polytopes and allied forms represented by amplifications of the symbol }[1, 1,1]. Finally in the fifth section about the extra regular polyhedra and polytopes we will have to use the symbol of the icosahedron. *) I had the great advantage of reading the original manuscript to Mrs. Srorr; the ensuing d.scussion — I acknowledge this with thankfulness — has led to a simplification of the proofs of several of the theorems. 1* À ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY Section |: POLYTOPES DEDUCED FROM THE SIMPLEX. A. The symbol of coordinates. 1. In a preceding paper (Nieuw Archief voor Wiskunde, vol. IX, p. 133) I found that the distance 7 between two points P, P’, the barycentric coordinates of which — with respect to a regular simplex S(a + 1).of space 8, — are Hi, bo,» +s Hin va aNd 24, (Zo, OS is represented, with the length of the edge of the simplex of coor- dinates as unit, by the simple formula We insert here a much simpler deduction of this formula; of this deduction fig. 1 gives a geometrical represensation for the particular case 2 — 2 of the plane. a4 Let © 2, — 1 represent the space S,, determining in a space of a= operation #,,,, on the axes of a given system of rectangular coor- dinates with origin O, the points 4,, 4,,..., 4,4, at positive distances unity from O. Let P and P’ with the orthogonal coordinates #,, m,,..., man 44 chek and #4, m5,..., My», be any two points of this #,,; then, accor- ding to the expression for the distance in rectangular coordinates, n +1 we have PP? => (m, —m;) and, as the points lie in #,, the EN nn nil condiuonseem— 1 S ms AG Mle == Al ae Now let us consider the normal distance coordinates U, and u, G—= 1, 2,..., 2 +1) of P and P with respect to them simplex S’(z + 1) with the vertices 4, 4,,..., 4,4, in 8; then from similar rectangular triangles we deduce immediately the relation where 4 is the height of the regular simplex. But, as the barycentric coordinates are normal distance coordinates measured by the corres- ponding height of the simplex and all these heights are equal in the regular simplex, we find for the barycentric coordinates yz, and y’, FE BE LL; U ; i — IN; , u', =? Me l / | A : 4 4 DERIVED FROM THE REGULAR POLYTOPES. 5 and therefore n+i = 2 a U) "| Ww But here PP’ is expressed in O4, as unit. By taking the edge A, A, = OA, |/ 2 as unit we find as above ~ n +1 AE FE 2 À (Hu The formula 1) enables us to find an answer to the following question, now forming our starting point: “Under what circumstances will the series of points obtained by giving to the set of barycentric coordinates 2, 2,,..., æ&,:, & determinate set of values taken in all possible permutations form the vertices of a polytope all the edges of which have the same length, say the length of the edge of the simplex of coordinates?” The very simple answer is given by the theorem: Turorem I. “If the x + 1 values a, a,,..., 4,41 satisfying the n +1 relation Z a, — 1 are arranged in decreasing order, so that we have i=1 Oper Ogio D A ee RN the De dy — 4,4 Of any two Ha values must be either one or zero.’ Proof. According to 1) the square of the distance PQ between any two vertices P, Q of the set is a sum of squares; from this it is evident that in order to make the distance a minimum we have to select two points P, Q which are transformed into each other by interchanging only one pair of coordinate values, say 4, and a. But then the square of PQ is $ [(a, — aye)? + (ay — a} |= (a, —a,)*, and therefore PQ itself is a,—a,. Now this diffe- rence becomes a minimum, if a, and a, are unequal adjacent values. As this minimum distance must be an edge, the condition that all the edges are to have the length unity implies that the difference between any two different adjacent values must be one. 2. As the condition stated in theorem I depends upon the differences of the corresponding barycentric coordinates 2; and a’; we may drop the conditions À #, — 1, Xa’; = 1 by allowing these coordinates either to increase or to diminish a// of them by the same amount, so as to make e. g. either the smallest or the greatest of the ~ + 1 values equal to zero. So in order to avoid fractions 6 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY we will indicate the system of points (21, 14,4, — +, — 14) of S, for which the difference of any pair adjacent values is unity by (4, 3, 2, 1, 0). Indeed, it is easy to return from (4, 3, 2, 100) where the sum of the values is 10 and therefore too large by 9, | to the real coordinate values by subtracting ? from each; so this important simplification — of a temporary character — can do no harm. : We indicate the entire system of points obtained by taking the values 4, 3,2, 1,0 in all possible orders of succession by putting these values in decreasing order between round brackets; this sym- bol (43 2 1 0) will be called the “zero symbol” in distinction with the symbol of true coordinates. 3. The simplification alluded to above allows us to indicate in a very transparent manner the sets of values furnishing the polytopes with one kind of vertex and one length of edge (equal to the edge of the simplex of coordinates) found by M. Storr. So we have for n == 2,3,4,5 successivily in the symbols explained in the memoir Grew TON: DERIVED FROM THE REGULAR POLYTOPHS. (9) 6 2 29 9 = (OIELVE) #& * (9)$ 2212 (OTE) x ‘ (9) 9 222 —(O11887) | (9) g *a%' =(QOLEEN) … ‘ (g),$ fotolog —(OOIEE) « | (G) 9 2 T2 — (OIGEV) x (G) ica Va — = (¢) g *a Alo (OT3EE) igs @—— (che a ..—= (OTIEE) (chy 2 — (00188) OF= (OTE) x 09 = (OLIS) « en (OC) + € € € (g) 6 72 °a = (OLISS8) « (9) 5 2% = (001888) (g)g 2%% =(o1gesp) (9) 2% — —(9)$ 2" (9) 2 =(OLITIZ@ x “ (9) 9 9 = (OLLI) = (OLIL8S) ‘ (9) 9 2 = (000LTS) (ONSEN HOTLEL) 2 (0).8'2 == CO000TS) (9)9 % —=(Q00IE) ‘ (9) 22— (GOO1T I) x (9) 52.729 — (GOLICS) « ‘ (9)8 I= (000011) (g)s 22 (00018) ‘ MS —(000001) G zr (g) & — = (g)$ ‘2 22 — (G) gy fo 29 = (OLS) (G) gf Sa Voo = (N0TSS) x (q)y* =(OTIIS) » (c)y% = (0011 (Bs = (00018) p= 4 Li = (0018) E— = (OUD e= U M= == (11) gu € 6 € € € 6 ‘ (Eg =(g)9 9=(OLILD (g) gy 129 — = (¢) 9 29 — (OOTLD (¢) $ 29 = (000TT) (chy = (00001) OG == Ree i = (0001) = (OO) 8 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY In general a form obtained in this way may present itself in two different positions with respect to the simplex $'( + 1) of coor- dinates. So we may write e. g. for (21100) also (0—1 —1—2—2), or, if we invert the sign of all the coordinates and indicate that we have done so by putting the sign minus before the brackets, — — (01122), i.e. — (22110); so (21100) = — (22110) and likewise (22110) = — (21100). Really the symbols with the values satisfying the condition X +, — 1 corresponding to (21100) and (01122) are _ (g 2 2— 3 — 3) and (— 10011) representing, if we omit the brackets, two points P, P’ situated symmetrically to each other with respect to the centre of gravity (4444+) of the simplex 8 (5) of coordinates. So (22110) is the form (21100) in opposite orientation ; in the equation (22110) == — (21100) the difference in orientation is indicated by the sign minus. The forms the symbols of which are not affected by the “inversion” mentioned are marked by an asterisk; as they do not alter as a whole when they are put into the opposite position they possess central symmetry 1). 4. The results obtained show that the geometrical method followed by M. Srorr and the analytical method developed here cover exactly the same ground, 1. e. that they lead up to the same system of forms. Nevertheless we should jump to a wrong conclusion, if we deduced from this coincidence of results that by either of the methods all the possible forms with one kind of vertex and one length of edge have been found. We show this by remarking that the combina- tion of the two zero symbols (1000) and (1110), or of the proper values (1000) and (4 4— 4), of Zand — 7 gives us the vertices | of the cube, which implies that all the forms deduced from the cube by M*. Srorr can be represented by couples of symbols in bary- centric coordinates as derived from the tetrahedron 2). *) For the deduction of the e and c symbols from the symbols of coordinates and reversely compare the part D of this section. The first of the tables added at the end of this memoir is destined to put on record for n = 3, 4, 5 the different polyhedra and polytopes deduced from the simplex with their principal properties. Of this table the first column contains Mrs. Srorr’s symbol, the second the symbol of coordinates and the third the value by which the coordinates have to be diminished in order to find the true coordinate values for which = ai = 1. The following columns will be explained farther on. ‘) So the most complicated form e, e, C—1CO can be represented by (+3 8—V —1—1 —5—1 i ae Wane 4 ee Ee SNe: i 4 ANAL PEER Tye TRE ri if 1’ and 3’ stand for 1/2 and 3 v/2. : | LI : | DERIVED FROM THE REGULAR POLYTOPES. 9 5. We finish the first part of this section by mentioning a theorem already proved in the paper quoted in art. 1, as this theorem will be very useful in future. It is: “Any two spaces 5,4, #,_, containing together the vertices of a regular simplex $'(x + 1) of 4, are perfectly normal to each other.” This theorem is an immediate consequence of the property that any two edges without common end point determine a regular tetra- hedron and are therefore at right angles to each other. For this implies that, if S,_, contains the vertices 4,, 4.,..., 4, and S,,_, the vertices 4,14, 4,,,,..., 4,14, each of the 4 — 1 independant Imes 4, A, (/ = 2, 3,...,4) of S,, is normal to each of the » — A independent lines 4,,, 4,4. (m= 2, 8,...,2—A+1) of &_, 4). B. 7e characteristic numbers. 6. We will now explain how the characteristic numbers of the vertices, edges, faces, limiting bodies, etc. can be deduced from the symbol of coordinates. The larger u is, the more elaborate the process becomes. So, in order to divide the difficulties, we will begin by treating the cases n —4 and ~=5 at full length by means of an easy method, working from two different sides, the vertex side and the limiting polyhedron (u = 4) or limiting polytope (x = 5) side, of the series _ vertex, edge, face, etc. Afterwards we will show a more direct way leading immediately to the knowledge of the forms and the numbers of the different kinds of limits (7), of p dimensions. In the cases # — 4 and x — 5 of the four and the five charac- teristic numbers we determine for itself the first two and the last two, using the law of Euler for x — 4 as a check and for 2 = 5 as a means of finding the lacking middle number of the faces. The number of vertices is easily found in all possible cases. If all the x + 1 digits of the symbol of coordinates of the polytope in S, are different it is (2 + 1)! This number (4 + 1)! must be divided by 2! for any two, by 3! for any three, by 4! for any four digits being equal, etc. The number of edges can be calculated as soon as we know how many edges pass through each vertex. For the product of this latter number by the number of the vertices indicates how many times *) Compare the theorem of art. 30 (on p. 42 of the first volume of my textbook, “Mehrdimensionale Geometrie’) where the “two groups of lines through 0” may be replaced by “one group of lines through O (here A,) and an other group of lines through O° (here 4,,4).” 10 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY an edge passes through a vertex; so the total number of edges is half this product. Now the number of edges passing through a vertex is equal to the number of vertices lying at distance unity from the chosen vertex, and this number is easily determined, as will be shown by examples for x == 4 and x = 5 separately. In order to be able to find the number of the lmiting bodies (x = 4) and that of the limiting polytopes (x = 5) we prove in general the following theorem : Tagorem IT. „The non vanishing coefficients c; of the coordinates #2, in the equation e, 2, + ec æ, + ... =p of a limiting space 8,4 of the polytope deduced from the regular simplex S(#-- 1) of S, must all be equal to each other”. Proof: The linear equation e, a, + co æ + . .. =p represents, as far as the vertices of the polytope are concerned, more than one equation, if the coefficients ¢, are different. We show this by a simple example. If in the case (32110) of S, we start from the | equation 2a, + «, — p and we try to determine the vertices of the polytope for which the expression 2 #, + x, becomes either a maxi- mum or a minimum we find.the maximum 8 for a2, = 3, @ = 2 and the minimum 1 for a, = 0, à, = 1. So for values of p situated between 8 and 1 the space 2 a, + #,—=~p intersects the polytope, while it contains a limiting face only — and not a limiting body — for the extreme values 8 and 1 of p, as each of the couples of equations v, = 3, a, — 2 and a, = 0, a, — 1 determines a plane; of these planes the first contains the triangle 2, = 3, #, — 2 and a3, a, @= (110), the second the hexagon x, = 0, a, = 1 and 35, a; Ge From this example can be deduced generally that the equation Ci di À Cds +... =p represents / different equations, as far as the vertices of the polytope are concerned, if the non vanishing coefficients c‚ admit together % different values. The theorem is not reversible, i. e. not every linear equation with equal coefficients ¢, represents for the maximum or the minimum value of p a limiting space #,_, of the polytope in S,. So in the case of the simplex (10000) of 8, only the five spaces a, = 0 bear limiting tetrahedra of #(5), while the ten spaces 2, + a, = 0 bear faces (100), the ten spaces a, + a, + a,==0 bear edges Cio ett. In order to find the number of the faces (2 — 4) and that of the limiting bodies (7 — 5) we determine the form of the limiting bodies (2 — 4) and that of the limiting polytopes (x = 5). For the number of faces (7 — 4) is half the sum of the numbers of the faces of all the limiting bodies, and the number of limiting bodies DERIVED FROM THE REGULAR POLYTOPES. 11 (x = 5) is half the sum of the numbers of the limiting bodies of the limiting polytopes. 7. We now treat at full length the example (32110) mentioned above. The number of vertices is 5! divided by 2! 1. e. 120:2 — 60. The number of edges passing through each vertex is five, for in $2110 each of the five brackets indicates two coordinates with difference unity the interchanging of which furnishes a new vertex joined by an edge to the “pattern vertex’, 1. e. to the point with the coor- 60 dinates 3, 2, 1, 1, 0. So the number of edges is ee — Vo In the case of this polytope we have to consider successively the equations. Be Cu = De OR mer Le OP ee A, GE Crdi CT nt Deeds Os dé NO did D 2, dl in: di OF Oe a di À Each of the two equations placed on the same line is a conse- quence of the other; for in treating the polytope (32110) we have to suppose that the true coordinate values of any point have been increased by such a common amount as to make the sum of the coordinates equal to seven. a). Both the equations à, + 2 + 23 + x, — 7 and a, = 0 give no 2.2... — (3211), & e. for. the coordinates #25, #3, 4, we. can take any permutation of 3,2, 1, L. But if we subtract a unit from all the coordinates and write 7; — — land a, a, a3, v, — (2100), whereby the constant sem 7 of the coordinates is changed into 2, we see (compare the table in art. 3) that the obtained form is a ¢7'?). this, 77 presents itself five times, as the subscript 5 in 2, — 0 may be any of the five numbers 1, 2, 3, 4, 5. 6). In the case a, + # + #, — 6 which implies x, a = 1 we have to combine the two systems 2, 4, #3 — (321) and a, a; — (10); i. e. we have to combine the system %, #,, #3 — (321) with each or the two possibilities z, — 1,2, — 0 and z, — 0, a, = l giving ) That (2100) is a polyhedron with 12 vertices, 18 edges, 8 faces limited by 4 regular hexagons and 4 equilateral triangles is immediately found by treating the symbol (2100) in the manner taught here. 12 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY — two regular hexagons in parallel planes, or we have to combine the system @,,2; — (10) with each of the six possibilities 321, 312, 231, 213,182, 123 for 2,23 contained in 4, 4,23 — (eee giving six parallel edges of the same length. So in order to prove that the result is a hexagonal prism P, we have only to show yet that the planes of the two hexagons are normal to the six edges. But this follows from the theorem in art. 5, as the planes of the hexagons are parallel to the plane x, — 0, x; = 0, 1. e. to the face” A, A, A; of the simplex of coordinates, while the six edges are parallel to the line — 0, x, — 0,2, = 0, 1. e. to. the opposiie edge A, 4; of that simplex, The hexagonal prism P, obtained in this manner occurs ten times, as for the subscripts 4 and 5 in the equation 2, + x, — 1 we can take any combination of the five numbers 1, 2, 3, 4, 5 by two. c). For a, ae 2, — 5 we have either — 3, #, == SO 2, = 3 and in both cases 23, 2, a; — (110). So we find here ten prisms P3. d). Finally «2, — 3 gives 2, @3, a, a, — (2110); so we taie five CO. All in all we have got the limiting polyhedra 5 #0: 10210 2 eee so their number is 30. The number of the faces is easily found. As the numbers of faces of. #7, Pe, Ps, CO are respectively 8, 8, 5, 14 weiden 4065 X8 H- 10 X8 +10 X 5 E 5 X 14) = 120. So the final result (e, 4, f, 7) !) is (60, 150, 120, 30), in accor- dance with the law of Euler. Remark. In the case of the simplex (1000) of S, we would have to consider the equations a) SAT V4 + U —- U == dy — JE or Vz — 0 > Hir Ly op Te a Cc) ie YA +- U = — Il or V2 = Vs, + Ur == is | Rise Hell HO ot pd gee containing as we remarked already in art. 6 — respectively a limiting tetrahedron, a face, an edge, a vertex of the simplex # (5). Therefore in the expressive language of M. Srorr the limiting polyhedra of (32110) are distinguished, as to their orientation, as *)eThis is the general symbol I used for S, in my textbook; here e, k, f, r stand for “Ecxe, Kante, FLicue, Raum,” i.e. for vertex, edge, face, limiting body. DERIVED FROM THE REGULAR POLYTOPES. 13 5 4T of body import, ROLE ut ce RES Pr edge RO Be Ont venter 8. We add an other example, this time of a polytope in #,, and choose (432110), showing all possible particularities. 1) The number of vertices is 6! divided by 2! i.e. 720:2 = 360. The number of edges passing through each vertex is six, for in ND ete ines ate Ae |, each of the six brackets indicates two coordinates with difference RS 00 6 unity. So the number of edges is — SO: Here we have to consider the equations BE Bi a ene dee OE Pe Oe Oren ag td Or zr CM. rdt OR Me ae ae OR. aay OPB de DRAC Oe ope Re — 4e a). The equation 2, — 0 gives a, æ,, #3, @%, æ, = (43211), or — if we diminish all the coordinates by one — we find in 2, = — 1 tne polytope 2, #, 22, 21, #5 == (82100), 1, e. an ee, S(G), occur- ring six times ?). 0). For æ + 2, = 1 we have the two possibilities z; — 1, 2, = 0 and. 2, — 0, 7, — 1, combined with 2, a, £3, 2, — (4321), ‘which may be reduced to ay, 2, #3, æ, == (3210) by subtracting unity from all the coordinates. So we find a rectangular fourdimensional prism Po with ¢O as base, occurring fifteen times. c. Here we have to combine the two systems ay, x, #3 — (432), onl Dd @ — (110). So we get a polytape with 6 x 3 — [8 vertices arranged in six equilateral triangles in planes parallel to me planeur .==.0 7, == 0,2, ==. 0) contaming the ince AA tot | I *) The fourth and the sixth columns of Table I contain the characteristic numbers and the limiting elements of the highest number of dimensions of the new polytopes. The meaning of the small subscripts in column four and of the fractions in column five will be explained in part G of this section. *) The characteristic numbers of this form — compare Table I — can be deduced in the manner indicated in art. 7; see farther under a’). In the memoir quoted of Mrs. Storr the regular simplex of space S, is indicated by the symbol C,; we prefer to use here S(5), as this allows us to discriminate between the regular simplex S(8) of space S, and the measure polytope C, of S,, etc. 14 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY the simplex of coordinates and in three regular hexagons in planes parallel to the plane 2, = 0, 2, == 0, x, = 0 containing the opposite face A, A, A, of that simplex. As these faces are perpendicular to _ each other according to the theorem of art 5, we find a regular prismotope (3; 6), occurring twenty times. d). For a, + x, — 7 we have to combine each of the possibilities _ a, =A, 2, == 3 and #, — 3,2% — 4 with a Bi B — 1 ae So we find fifteen prisms Peo. | e). The equation x, — 4 gives 2, 43, %, &, 4% = (82110). So here we find six & e $ (5). So the limiting polytopes are Ge, 2.8 (5); 15 Po, 20 (356), 15 Peo Mene SM their number is 62. Of these the 6 e,e, (5) are of polytope 1m- port, the 15 P,, of polyhedron import, the 20 (3; 6) of face import, the 15 Peo of edge import and the 6 e,e3 8(5) of vertex import. In order to find the number of the lmiting polyhedra we enu- merate the limiting polyhedra of the limiting polytopes. a’). The determination of the limiting polyhedra of the six poly- : topes (32100) runs parallel to the investigation of (32110) in the preceding article, as we remarked already in the last footnote. So we find in the same order of succession and with the import with respect to the simplex $(5) of its space: 5/0 of polyhedron _ import, 10 P, of edge import and 5 #7’ of vertex import, i. e. twenty limiting polyhedra. 6’). The prism ?,, is limited by two ¢O and fourteen prisms, viz. six P, (or cubes) and eight P,, i.e. by stateen polyhedra. c). The prismotope (3 ; 6) is limited by ze prisms, six 2, and three Pe. d). The prism Peo is limited by two CO and fourteen prisms, viz. six P, (or cubes) and eight P,, 1.e. by séateen polyhedra. e). The polytope (32110) of the preceding article is limited by thirty polyhedra. So the number of polyhedra is + (6 X 20 + 15 X 16 + 20 X 9 E 15 X 16+ 6 X 30) == AEM According to the law of Euler the number of faces is 1080 + 480 + 2 — (360 + 62) = 1140. So the resulting symbol of characteristic numbers is (360, 1080, 1140, 480, 62). 9. The direct method alluded to in the beginning of art. 6 rests DERIVED FROM THE REGULAR POLYTOPES. 15 on the division of the limiting elements (7), of p dimensions accor- ding to their symbol into different groups; as introduction we explain what this means by considermg the edges of the polytope (432110) treated in the preceding article. The edges are obtained by joining two points which pass into each other by interchanging in the symbol of coordinates two digits with difference unity. So we have edges with four different symbols, viz. (43), (32), (21), (10), if by the symbol (p, g) we indicate any edge the end points of which pass into each other by interchanging two coordinates with the numerical values » and 9. It is easy to find their numbers. To that end we calculate first the numbers of edges of different symbol passing through a deter- minate vertex, e.g. through the pattern vertex 4, 8, 2, 1, 1, 0, which point will be indicated by ?. Through ? passes only one edge (43) and one edge (32), as the symbol of coordinates contains only one 4, one 3 and one 2; but two edges (21) and two edges (10) concur in P, as the symbol of coordinates contains two digits 1. So we have 1 edge (43) + 1 edge (32) + 2 edges (24) + 2 edges (10) through P. Now the number of vertices of the polytope being 360, the numbers indicating how many times each of the four edges of different symbol passes through any vertex would be 360, 360, 720, 720; as each edge bears two vertices we find 180 edges (43) + 180 edges (32) + 360 edges (21) + 360 edes (10), Le. once more altogether 1080 edges. _ In order to be able to extend the method of deduction of edges with different symbols to limits (7), of p dimensions we are obliged to introduce some new terms, So we call (43), (32), (21), (10) the unevtended symbols of the four groups of edges found above and designate as the eavtended symbols of these groups respectively (43) (2) DD (0), (4) (82) DD 0), (4) (8) 2D) (1) (0), (4) (3) (2) 1) (10), containing also the remaining four digits, each placed in brackets. Moreover we call the constituents (48), (2), (1), (1), (0) of the extended symbol (43) (2) (1) (1) (0) the sy//ables of that symbol and exclude from our considerations the “petrified’’ syllables with two or more equal digits as (22), (111), etc., where the influence of the interchange of the digits is nihil. 16 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 10. We now prove the general theorem: Tasore III. “We Bh all the groups of d-dimensional limiting. polytopes (P), with different symbols of any given #-dimensional polytope (P), derived from the simplex S’(z + 1) of #',, if we split up the # + 1 digits of the pattern vertex in all possible ways into n—d-—+t-1 groups of adjacent digits and consider these groups, each of them placed in brackets, as the syllables of the extended symbol.” We remark that the extended symbols of the four groups of edges of the preceding article satisfy the conditions of the theorem. We represent the 2 — d + 1 different syllables by (..)",(..)”, ‚Crdi, where Ai, Ao, ...,4,,,32 Indicate the numbers ot ae digits, so that we have A, LEZ EL... + Ara =H moreover in order to fix the ideas we suppose that the coordinate values of (..)"* correspond to the coordinates z,,4,, . . .æ,, those of (..)* to: the coordinates: a i mp 45, on Me OR Proof. If we put it short the three moments of the proof are: a) As petrified syllables have been excluded we obtain by proceeding according to the indications of the theorem a d-dimensional polytope P,, the vertices of which are vertices of P,,. 6) As the digits of the syllables are adjacent digits of the symbol of (P),, the (P), obtained is a limiting polytope of (P),,. c) As the system of equations representing any limiting polytope (P), of (P), occurs under all the systems of equations corresponding to the limiting (P), of (P), furnished by the theorem, we obtain by means of the theorem all the limiting polytopes (P), of (P),. ‘We consider each of these three parts for itself. a) The polytope obtained is a (P),. By the exclusion of petrified syllables we are sure that any syllable (..)* with # digits allows the vertex, the coordinates of which are the 2 + 1 digits of the symbol of (P),,, to coincide successively with all the vertices of a determinate # — 1-dimensional polytope (P),_ 4 situated in a space 8,_, parallel to a limiting space S,_, of the simplex Á'(x 1) of coordinates. So in the case of the a — d + Ì syllables(, .)"", (..), ...,(. rd +1 under consideration the poly- tope obtained will be a prismotope, the constituents of which are polytopes (P),,4, where & is successively A, Aa, . . … Ap — a 45 Situated in spaces parallel to the limiting spaces 8, _4 — (4, 4, . . ., Ar), Ser Nen PE. PL de à. the simplex of coordinate which spaces are by two normal to one another according to art. 5. This prismotope which may be represented by the symbol (Pj4_4; Pias «5 Pr, 33,9 18 a polytope (P),. For its number of dimen- 4 DERIVED FROM THE REGULAR POLYTOPES. ET sions is the sum of the numbers 4, — 1,4,—1,..., 4, ¢4,—1 of dimensions of the constituents, 1. e. the sum of the numbers hy, hy, ..., Ap ara diminished by the number of the constituents, i.e. 2 + 1 diminished by »—d-+- 1, Le. d. _ We pass from the extended symbol of a (P), formed according to the prescriptions of the theorem to the unextended symbol by omitting the syllables containing only one digit. So the unextended symbol contains only syllables with two and more than two digits. If all the syllables of the unextended symbol bear two digits, the polytope (P), is a measure polytope; if this symbol contains only one syllable with more than two digits, the polytope (/), is a prism, may be of higher rank; if the symbol contains at least two syllables of more than two digits the polytope (P), is a prismotope, may be of higher rank, im the restricted sense of the word. This explains how we have to interprete the result found above that all the limits of (P),, are prismotopes. In the particular case of the limits (2),_, of the highest number of dimensions, where we have to split up the digits of the pattern vertex of (P), into two groups, we find, if + l is split up into & and u — # + 1, the result (Pias Por), which is a non specialized +) polytope (P),_, for 4 = 1 and # ==, a prism on a non specialized polytope (?),_, as base for &=2 and &£=xn—1, and a prismotope in the narrower sense for all the intermediate values; 1. 0. w. of the limits (/), 4 represented elsewhere (Proceedings of the Academy of Amsterdam, vol. XIII, p. 484) in connexion with the notion of zmport by go, ga, + Ona the forms 9, and g,_, are non specialized polytopes, the forms g, and ÿ,_ are prisms and the forms 4, g3,... 9,3 are prismotopes. 6). The (P), obtained is a limiting body of (P),. A polytope (P),, the vertices of which are vertices of (P),, 1s a limiting body of (P), — and not a section of it ——, if we can indicate 2—d limiting spaces #,_, of (P), containing it. Now, according to the manner in which (P), is obtained, the coordinates of its vertices satisfy the »—d-- 1 equations En ope =. = Op =D, Vini he Phe ee in En rar le Uy, Pr A + Uy, of Ie 0 + se tte —- Uy, ne k + ka a ELC if.y, is the sum of the first 4, digits of the pattern vertex, p, the sum of the next #, digits, », of the then next 43 digits, etc. These n — d + 1 equations, only connected by the relation holding for *) Here “non specialized” means: according to the mode of generation neither prism nor prismotope. About this last form art. 13 will give more particulars. Verhand. Kon. Akad. v. Wetensch. (4ste Sectie) Dl. XI. C2 18 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY | n +1 all points of S,, that the expression Xe, is equal to the sum of all = il the digits of (P),, form a system of # — 4 mutually independent equations, representing therefore, in accordance with the result of the first part of the proof, a space S,, bearing the (P), found above. If we write this system of equations in the form: ke, + koo ky +k, Bs St, ps BE Pag à Ye =1 aay . + Kn—d =P, + Pa +. + Ppa I a i it is evident that each of the equations represents an #—1-dimen- sional limiting space of (P),, the constant of the right hand member being a maximum. As we remarked already the crux of the proof of di part lies in the true interpretation of the expression “adjacent digits”. It cannot be replaced by the condition that all the syllables should be formed according to the first theorem. We show this by means of two simple cases concerned with the determination of faces of threedimensional polyhedra. In the case of the (2110) = CO the hexagon (210) (1) is no face but a section, likewise in the case of the (1100) = O the square (10) (10) is no face but a section. In both cases the syllables satisfy the conditions of the first theorein ; but the impossibility of putting the syllables behind each other so as to obtain the order of succession of the digits of the pattern vertex implies the impossibility of finding an equation where the constant that is equal to the sum of some of the coordinates is either a maximum or a minimum. Under the five polyhedra 7, O, (7, CO, 10 which can be represented by a symbol with four digits (compare the small table of art. 3 for 2 — 3) the O and CO are the only ones with sections y, and », with sides equal to the edges; at the same time they are the only ones with four edges through each vertex. c). By means of the theorem we obtain all the limits (P), of (P),. It is always possible to represent any limit (P), of (P), by x — d equations of spaces #,_, containing x — 1-dimensional limits (7), _4 of (P),; as the vertices of this (P), are also vertices of (P),, this system of equations will be in accordance with the symbol of (P),, i.e. this system must be included into the set of systems of equa- tions provided by the theorem. 1. We apply the theorem III to an other fivedimensional form (321100), showing at the same time how we can determine the numbers of a//Z the different limits. DERIVED FROM THE REGULAR POLYTOPES. 19 Vertices. There is only one kind of vertex (8) (2) (1) (1) (0) (0). According to the rule given in art. 6 the number of vertices is 6! divided by 2°, i.e. 180. Edges. There are three groups of diese represented in extended and in unextended !) symbols by (32) (1) (1) (0) (0) = (82), (8) (21) (1) (0) (0) = (21), (3) (2) (1) (10) (0) = (10). We indicate a new method of determining the numbers of these edge groups. In the case of (10) the coordinates corresponding to the two digits between the same brackets can be #,, x, where 7, 4 is any combination of the subscripts 1, 2,3, 4, 5, 6 by two, giving (6), = oa en - chosen the four remaining ones can be assigned anyhow to the four digits (3), (2), (1), (0), giving 4! = 24 possibilities. So the number of edges (10) is (6),.4! — 360. In the case of (21) the number 360 must be divided by 2 on account of the two equal syllables (0), (0), in the case of (32) this number must be divided by 2° on account of the two pairs of equal syllables (1), (1) and (0), (0). So we have — 15 possibilities; these two coordinates having been 90 edges (32) +-- 180 edges (21) 360: edges. (10) 1. e. altogether 630 edges. _ Faces. There are six groups of faces, represented in extended and in unextended symbols by (321) (1) 0) (0) = (821) = pe, (82) (1) (10) (0) = (82) (10) = 7, (3) (211) (0) (0) = (211) = psy, B) (21) (10) (0) = (21) (10) = p,, RELECO) (0) == (PEO) == HOME =e OUPS pe Taken in the order of succession of the rows the numbers of these polygons are mai, 60: (6), (4), 22 ee ACCES SEP RO D IRO, (6h00, Le. we find 300 pz + 360 p, + 60 ps = 720 faces. Limiting bodies. There are seven groups of limiting bodies, viz.: ") As we have seen in the preceding article the unextended symbols are deduced from the extended ones by omitting the syllables of one digit. DE Ad 20 ANALYTICAL TREATMEMT OF THE POLYTOPES REGULARLY (3211) (0) (0) — (3211) = #7, (821) (10) (0) = (321) (10) = B,, (32) (110) (0) — (32) (110) = Py, (82) (1) (100) = (32) (100) = Ps, (3) (2110) (0) = (2110) = CO, (3) (21) (100) = (21) (100) = Ps, (3) (2) (1100) — (1100) = 0, the numbers of which are respectively (6,:..21:9 TD (CLONE 00 LOE ESE (6), 43) = OU, AO): ST EE AO os NN (Gi OR | 1.16: 15 47" 30 (0 + CO) + 60 P, + 180 P, = 315 limiting bodies. Limiting polytopes. There are four groups of limiting polytopes, viz. : (32110) (0) — (82110) =e, e, 45 B, 21) 100) COEN (32):(1100) =P, (8) (21100) RFO, the numbers of which are. é (6); = 6 , (6k—20 , (6,—15 , (6);=—6. So we find | 6 e, e3 5 + 20 (6; 3) + 15 Po + 6e, S; = 47 limiting polytopes and the characteristic numbers are (180, 630, 720, 315, 47), in accordance with the law of Euler. 12. Though the introduction of the extended symbols has enabled us to simplify the theoretical considerations it cannot be denied that the unextended symbols are better fit for practical use. Therefore we insert here a corresponding version of theorem III, but to that end we have to enter first into a distinction of the digits of the syllables of the unextended symbols. We will distinguish the digits contained in any of these syllables into end digits and middle digits, the first and the last digits and the digits equal to these being the end digits, the remaining ones — if there are some — the middle digits. So in (3210) there are two middle digits 2 and 1, in (2110) there are two equal middle digits 1, while in (2210), (2100) there is only one middle digit and in (1000), (1100) none. Now we can repeat theorem III in the new form: TagoreM I. “We obtain a (P), the vertices of which are vertices *) Compare the small table unter art. 3. DERIVED FROM THE REGULAR POLYTOPES. 21 of the given polytope (P),, if we fix either the values of — d coordinates and allow the remaining d + l to interchange their values, or the values of # — d — 1 coordinates and split up the remaining d + 2 into two groups of interchangeable ones, or the values of #— d— 2 coordinates and split up the remaining d + 3 into three groups of interchangeable ones, etc., this process winding up for # < 2d in a symbol with x — d + 1 and for x > 2(d— 1) in a symbol with d groups.” “This (2), will be limiting polytope of (P),, if: 1°. each syllable of the unextended symbol with middle digits exhausts these digits of the symbol of (P),,, 2°. no two syllables without middle digits have the same end digits.” Proof. The first part of the new theorem is a consequence of this that in the different cases communicated the corresponding extended symbol is always consisting of u — d + 1 syllables, 1. e. of & syllables with more than one digit and # — d — 4 + 1 syl- lables with only one digit for #4 — 1, 2,...,d; so it 1s equivalent to part a) of the proof of theorem III. The second part of the new theorem is equivalent to part 6) of the proof of theorem IT; for the only cases in which it is impossible to put the syllables of the extended symbol behind one another so as to obtain the order of succession of the pattern vertex are the two excluded by the two items 1° and 2°, 1. e. 1° that a syllable with middle digits does not exhausts these digits and 2° that two syllables without middle digits do have the same end digits. Finally the part c) of the proof of theorem III can be repeated here. By means of theorem III’ we find e.g. in the case of the (P), represented by (5443322210) the following 58 different kinds of limiting (6: (8443322), — (544332) (21), (544332) (10), — (54433) (221), (54433) (210), — (5443) (3222), (5443) (322) (21), (5443) (322) (10), (5443) (32) (221), (5443) (32) (210), (5443) (2221), (5443) (2210), — (544) (33222), (544) (3322) (21), (544) (3322) (10), (544 (832) (221), (544) (332) (210), (544) (82221), (544) (3222) (10), (544) (322) (210), (544) (32) (2210), (544) (22210), — (54) (433222), (54) (43322) (21), (54) (43322) (10), (54) (4332) (221), (54) (4332) . (210), (54) (433) (2221), (54) (433) (2210), (54) (43) (32221), (54) (43) (3222) (10), (54) (43) (322) (210), (54) (43) (32) (22.10), (54) (43) (22210), (54) (832221), (54) (83222) (10), (54) (8322) (210), (54) (332) (2210), (54) (822210) — (4433222), — (443322) (21), (443322) (10), — (44332) (221), (44332) (210), — (4433) (2221), (4433) (2210), — (443) (82221), (443) (8222) (10), (443) (822) 22 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY (210), (443) (32) (2210), (443) (22210), — (4332221), — (433222) (10), — (43322) (210), — (4332) (2210), — (433) (22210), — (43) (322210), — (3322210). 13. We will insert a few remarks about the character of the limiting (P); obtained. | In the case (5443322) of one syllable we find a non specialized form (5221100) which will prove to be an es e, S(7) in M. Srorr’s language. In the cases (5443832) (21) and (544332)(10) we find right prisms on (322110) = e, e, S(6) as base. In the case (54433) (221) we find a prismotope the constituents of which are a (21100) = e, S(5) and a (110) =p; So this (P), can be generated in the following way. Consider a space S, and a plane Sj, perfectly normal to each other. Take in #, an e, S'(5), in S, a p,, and let P be a definite vertex of the former, Qa definite vertex of the latter. Now move either e, S'(5) parallel to itself in such a way that P coincides successively with all the points inside ,, or p,; parallel to itself in such a way that Q coincides” successively with all the points inside e, £(5). Then the (2); can be considered as the locus either of the e, S(6) in the first case or of the P, in the second; its vertices are given in the first case by the three positions of e, S(5) in which P coincides with one of the vertices of »,, in the second by the thirty positions of 93 in which Q coincides with one of the vertices of e, £(5). We represent it by the symbol je, 8 (5); 3}. In the case (54433) (210) we find an le, 8 (5); 6}. In the three cases (5443) (8222), (5443) (2221), (5443) (2210) we find successively (CO; 7), (CO; 7), (CO; tT). In the cases (5443) (322) (21), (5443) (822) (10), (5443) (82) (221) we have to deal with right prisms on a (CO; 3) as base, whilst (9443) (32) (210) is a right prism. on a (CO; 6) as base. These prisms may also be represented by the symbols (CO; 3; 2) and (CO;6;2) as prismotopes of the second rank. But a prismotope proper of the second rank is the (P); represented by (544) (822)(210), which may be represented as such by the symbol (3;3,;6). To generate it we have to start from three planes &,, &,, @, two by two perfectly normal to one another, and to place in @ and a, equilateral triangles and in 4, a regular hexagon; then the (P), 1s obtained by the parallel motion of the hexagon in such a way that a definite vertex of that hexagon coincides successively with all the points inside the fourdimensional prismotope (3; 3) determined by the two triangles. DERIVED FROM THE REGULAR POLYTOPES. 23 The (54) (43) (32) (2210) is a prism of the third rank on a £7’ as base; it may also be considered as a prismotope (C; 47). If in the case of (P), we deduce the limits (P),_, we find them in the order of succession g,_4, Jy—2,--+Jo Of polytope import to vertex import, when, in proceeding from left to right we take in the first syllable as many digits as possible and keep in it the first digit as long as possible. his principle has been followed throughout in the enumeration of the limiting (P), of the given (P),, as well as in the sixth columm of Table I. In the notation of art. 10 (page 17 in the middle) a limit (P),_; represented as to its import by g,, is a prismotope (P,; P_n). 14. It is worth noticing that m space S, the series of limiting elements may include the series of the measure polytope 47, for n even up to the polytope Mn of Sw for x odd up to the polytope My et of 8, nyt. SO, for x — 2m ie, 1 the (P),,,., represented. by EE mec, . Cy, S (2m 2) = (2m ET, 2m, 2m — 1,..., 3, 2, 1, 0) admits as limiting element (P),,,, the d/,,,, with the symbol (2m + 1, 2m) (2m — 1, 2m — 2). RO CET Ne On the other hand, amongst the polytopes tMemselves, no measure polytope occurs and of the cross polytopes only the octahedron presents itself. We prove that this must be so, for each of the two series separately. Measure polytopes. The number of vertices of (5443322210) 1s aa which can be written in the form 31 on (LD SO as to be able to generalize it for any (2), as (2 + 1)! al ble AE: where a,6,c,.../ are arranged in decreasing order and their sum is # + 1. Now this form is a product of binomial coefficients (2 + 1), (#—a-1),@—a—b6b-+] and there is only one possibility under which this product contains no factors different from two and is therefore a power of two, me. we the case mi 1 == 2") à = 2) i ot avis Or ip 24 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY But this case corresponds to the semplex S(2”) of space Sp, Cross polytopes. The cross polytope is characterized by the pro- perty of having all its diagonals of the same length (—]/ 2 times an edge) and passing through the same point. So in order to represent a cross polytope the symbol of coordinates of (2), can contain end digits only, for the supposition of free different digits as in (210) leads inevitably to three different distances. Let us suppose the two end digits are 1 and 0. Then we have to take in at least two of each in order to create the possibility of inter- changing two pairs of digits; this gives us the octahedron, the diagonals of which are the joins of the pairs of vertices repre- sented by 1100) 1010) 1001 0011 | ? 0101 |? 0110 |? and pass therefore through the point $,4,4,4. Finally in the n —1 n —1 cases (1100 ... 0) and (11... 100) we have to deal also with polytopes admitting only diagonals — ]//2 times an edge, but here these diagonals do not pass through the same point (centre). For in the case of (11000) the centre is the point all the coor- dinates of which are # and this point lies not on the diagonal joining the points 11000 and 00110, etc. C. Zxlension number and truncation integers and fractions. 15. “How can all the new polytopes (adc ...) found analyti- cally be deduced geometrically from the regular simplex?” As we remarked in the introduction the new polytopes have been discovered geometrically by M. A. Boor Srorr; we will consider her method thoroughly under J. Here we wish to indicate first that the answer to this question can also be given by the theorem: Turorem IV. “The new polytopes, all with edges of length unity, can be found by means of a regular extension of the regular sim- plex of coordinates followed by a regular truncation, either at the vertices alone, or at the vertices and the edges, or at the vertices, edges and faces, etc.” Proof. This theorem is an immediate consequence of that given in art. 6 (theorem IT) about the equality of the non vanishing coefficients ¢, of the coordinates a; in the equation c, 7, + co + ...==p of a limiting space 8,_, of the new polytope deduced from the simplex S'(2 + 1) of #8, So in treating in art. 7 the example (32110) we found that the limiting spaces zn Salt ter ah DERIVED FROM THE REGULAR POLYTOPES. 25 di = 3 containing a limiting CO, WV mi V2 cy 5 >) EE) ” P3, Li + Vy + v3 0 >) >) D Pes 4 Al di —- Lo = = v3 + Lg. == it >) 22 2 tl are respectively parallel to the spaces 5 Ly — | containing a vertex, By > By == zt an edge, nm id à a face, M da em rl ; , limiting body of the regular simplex, while they are normal to the line joining the centre O of the simplex to the centre of that limiting element. Moreover it is evident that all the spaces of the same group, say dr + & + @, = 6, have the same distance from the corresponding P@aces ZA 7, + +, — 1, ete 16. The meaning of the expression “extension number” is clear by itself: an extension to an amount € transforms the simplex SO (x + 1) with edge unity of #, into a simplex S©) (u + 1) of edge €. But we have i define beforehand what we will understand e.g. by a truncation À. If we split up the 2-1 vertices of the extended seer SC) (n + 1) into two groups of # ag l and x —4 pouits (see fig. 2, where the case 2 = 6, 4 = 3 is represented), forming the vertices of regular simplexes S©)(4 + 1), S(©) (u — 4) lying in spaces #,, 8,1, and we cut S()(z-+- 1) by any space S,,__, at the same time parallel to these spaces S,, #,_,_4, 1. e. normal to the line joining the centres 47, 41’ of S()(4— 1), SC) (un —#) in a certain point O, any edge PQ joining a vertex P of S@) (4-1) to a vertex Q of 8) (un —k) will be cut in a certain point & for which Ze and therefore independent from the choice of the vertices P, Q. This ratio is the “truncation fraction”’ of SCE) (n + 1) at the limiting SO) (4-1) by the truncating space and its ‘à the ratio = is equal to QR EURE complement Ee to unity is the “truncation fraction’’ of $(S) (u + 1) at the limiting SCS) (u — #) opposite to S()(4— 1) by the same space. But, if we like, we can use the term “truncation number” for the number of units contained in the segment PA or QR, accor- ding to the truncation being performed at the side of P or of Q. As the number of units of the denominator of the truncation fraction, 26 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY i.e. the number of units of the edge of the extended simplex, is the extension number ¢, the truncation numbers 7, which — as e itself — will prove to be always integer, are simply the nume- rators of the truncation fractions with the extension number € as common denominator. 17. If we indicate the truncation numbers corresponding succes- sively to a truncation at a vertex, an edge, a face,... by Tat Mis Vases os the TheOLem Mos: Tavorem V. “Let (mo, »4, m,,..., 0) be the zero symbol of the polytope; then the sum m= Xm, of the digits is the extension number €, and the truncation numbers 7), 74, T»,... are repre- sented by the forms 22 To = m — Mo, Ta = M — My — Mas To = M — My — M4 — Hae EE By the extension € the simplex of coordinates 8 (7 + 1) Eg (1 00: . .0) of 8, is changed into the concentric simplex S©) (e Jl pere ln (¢ 00...0) with edges ¢. Then the space #,_, represented in the latter case by 2, — 0 contains a limit (/),_, of the considered polytope (P), pee a part of the limiting simplex 8£© (4) = een (£ 00...0) of (¢ 00...0), at which limit of the Mighest order of an this £() rar 1) is not slieed off. If now we go back to true coordinate values the last digits in the two symbols of (P), and #()(2 +1) must still be the same, which will be the case, 1f we have to He from nought in both cases the same =m; — E— | amount, 1.6. af and — — are equal, i. e. if we have mia n +1 € — DM; = mM. From what we remarked in the preceding article it follows that the truncation ian of the extended simplex #8 (x + 1) at any limiting S“’? (4-4-1) can be derived from the mutual position of three parallel spaces #',_, normal to the line joining the centre of S°? (4-1) to the centre of the opposite S°” (x — #); of these three spaces one passes through 8” (4 + 1), an other through the opposite 8S” (2 — 7), whilst the third is the truncating space lying between these two. For, if as in the preceding article P is any vertex of 8 (4 + D, Q any vertex of 8" (u — #) and Z the point of intersection of PQ with the third of these three parallel spaces, which may be represented by &®,,_,, S®,_,, 8%, 4, according to definition _ DERIVED FROM THE REGULAR POLYTOPES. 21 | ae GPT the truncation fraction 7, 1s the ratio PO and now we have the theorem: “lf a line ‘cuts any three parallel spaces SP 82, S?,_, of 8, represented by the three equations b, a, + 6,%,-+-...... Ben — ¢, @ = I, 2, 3)-in the points ?, Q; A, we have For this is obviously true for the edge 4, 4,,, of the simplex of coordinates, the values of x, for the three points of intersection with this line being determined by the relations 4, a, = c¢,, (¢= 1, 2, 3); therefore it is true for any transversal, according to a well known theorem, already used implicitly in the preceding article, the ratio in question being the same for all possible transversals. Now in the case under consideration the spaces SP, S®,,_4, S% ,_, may be represented by the equations à, 4- 2, +... a,44= 4, (¢—1,2,3), where c, and c, are the maximum and minimum values of the left hand member with respect to the vertices of the n extended simplex (#% 00....0), while c; is the maximum value of the same expression with respect to the vertices of (uu, mm, m3, . . . , 0). So we have OE Utr tet oe Tee 2 — (My my FH m,) 1 es n giving for the truncation fraction the result and therefore 7, == m — (m + my, + ... + m,). so we find in the case of the (2), of art. 12 represented by 5922210) m= 20, 5 == Re epee Ta 10: ee Ga Te Ti For x — 3,4,5 the extension number and the truncation numbers are indicated in ‘lable I, the seventh column containing the exten- sion number £, the eighth column giving what may be called the “truncation symbol”. So in the case of ee, e, S(6) the extension number is 11, the truncation symbol is 7,4, 2,1 where these numbers represent successively the values of Ty, 74, Ts, Ta; SO we find mentioned here a truncation „5 at the vertices, 54, at the edges, 7, at the faces and ,4 at the limiting bodies. 28 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY D. Expansion and contraction symbols. 18. If we compare the symbols containing the operators e, and c of expansion and contraction introduced by M. Srorr for the offspring of the 7’— S(4) in #, and the S(5) in S, with the zero symbol of these polyhedra and polytopes, we remark that all these cases underlie certain general laws, up to now of an empirical character. By proving these laws we will promote them to theorems, the first of which can be stated as follows: Turorem VI. “The expansion e,, (4 = 1; 2, 38,..., de to the S (w+. 1) of 8, changes the symbol of coordinates (1 00... 0) of that simplex into an other zero symbol which can be spite by adding a unit. to the first £ + 1 digits.” Indeed this gives (compare the small table 2 — 4 under art. 3): e, (5) — (21000), e, 5) = (21100), es 85) — (21110). Proof. The operation of expansion e, consists in moving the limiting S(4 + 1) of £® (u + 1) to equal distances away from the centre O of SP (x + 1), each A4 + 1) moving in the direction of the line OM joining O to its centre 47, these S(4 + 1) “remai- ning parallel to their original position, retaining their original size and being moved over such a distance that the two new positions of any vertex which was common to two adjacent limits (Z),, in the original S“ (x + 1) shall be separated by the length of an edge”. 5) Now let us consider (fig. 3) the plane through OM and any vertex À of the S(A-+ 1) of which M is the centre; then, on account of the regularity of $P (2 + 1), the angle AMO is a right one. This plane will also contain the new position AM’ of AM. What we have to do now is this: We select from the symbol of coordinates (1 00 .. 0) the vertices of any limiting S(4-+ D), calculate the coordinates of 47, deduce from the coordinates of O and 47 those of 47° on the supposition that OM : OM = À is known. Then we have to determine the coordinates of 4° by adding to the coordinates of the vertex A chosen arbitrarily among the k + 1 vertices of the S (4 + 1) the differences of the correspon- ding coordinates of J/ and 47. Finally we have to determine A by the stated condition that two new positions of the same vertex A of S“(~-+- 1) shall be separated by the length of an edge, *) Compare p. 5 of the memoir quoted of Mrs, Srorr. : DERIVED FROM THE REGULAR POLYTOPES. 29 or — which comes to the same — by the condition that the coordinates of A’ satisfy the law stated in theorem I, that the difference of any two different adjacent values must be unity. We now set to work and select for the # +- 1 vertices the ver- tices 4,, 4.,..., 4,4, of SY (m+ 1) and for A of fig. 3 the vertex A. According to this choice the coordinates of M are hi | Sle in RARE TR Orr his = Hy ya 0. So the coordinates of the three points O, 4, M satisfy the equations Ne Rt A SSL opr ee Vy 4.43 but then these relations hold for any point of the plane OAM, as the # — 2 equations represent a plane. As moreover AA = MM' = (A — 1) OM, or in the notation of vector analysis Aa ANS MS (A — 1) (M — 0), we have successively for the mentioned coordinates in true values and for the mentioned differences of coordinates: di Lo —L3— TE — dy + 1 Vier Ui + BE Saat zer Hen 4 1 L 1 0 “Een DE se 1 1 i k+l ki ° 1 1 | 1 1 1 “meld k+l n+1 | k+1 n+1 n+l ' 1 it | 1 Be TN = = = sn: 4 Ay DG +1 To) | (A DG a nl Ca Mate A A 1 | 0 0 / 1 1 1 1 SIMS a 146-1) (ri) Cee te ati . ° À FE Ì f h li ] f ! 1.6. the difference RE the coordinates z,,, and e+, of A is either unity or zero. But if we make it zero we get.A 1, Le. we find back the original SP (u 4-1). So for the expanded DIE Al à 43 polytope we have to take HI NS l or A # 2, giving for 4 7 the coordinates 30 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY ars A, D= ral, Uy po = eee = My yy HO RS Le. the symbol (2 11. .1 00..0), what was to be proved. __ If by the operation e, the limits S®(£—+ 1) of SO (+ 1) are moved away from the centre O to a distance equal to À times the original distance, the extended simplex £®( 41), the limits SOD (A1) of which will contain these 8% (44 1) in their new positions, will be a simplex 8()(z-+ 1), 1.0. w. A=#+2 isthe extension number of the new polytope. ‘his comes true, for accor- ding a theorem V the sum #-- 2 of the digits of the zero symbol n—k (2 ne ..1 00...0) is the extension number. So we find by the way: | | TaxoreM VII. “In the expansion e, the limits 8S(4-+ 1) are moved away from the centre to a distance equal to 4-++ 2 times the original distance.” Remark. We may express the influence of the operation e, on the symbol (1 00..0) of the simplex 8“? (u + 1) presenting only one unit interval between the first and the second digit by saying that it creates a second unit interval between the # + 1* and the 4 + 2" digit. This remark which holds also with respect to the symbol of true coordinates will be of use in the following articles. 19. Trrorem VIJL “The influence of any number of expansions en er, Ons. of SY (a 4-1) on its zero symbol (1 00. . .0) is found by adding the influences of each of the expansions taken sepa- rately’. Indeed this gives (compare the small table # — 4 under art. 3): é, € A (5) = (32100), e, es S (5) = (32110), me SO ei Ces OLD) = (49200 Proof. We begin to prove the theorem for the case of two operations of expansion only. It is stipulated expressly by M. Srorr that in the succession of two operations of expansion the subject of the second is to be what its original subject has become under the influence of the first. So in the case e,e, 7’ of the tetrahedron (fig. 4") the original triangular subject of e, is transformed by e, into a hexagon (fig. 4°) and now the hexagon is moved out, in the case ee, 7' the linear subject of e, is transformed by e, into a square (fig. 4°) and now the DERIVED FROM THE REGULAR POLYTOPES. 31 square is moved out; in both cases the result (fig. 4%) is the same, 40. In general, for # => /, in the case e, e, 8 (x + 1) the subject S (4 + 1) of e, is transformed by e, into an e, S (4 + 1) and now this e,S“(4-+- 1) is moved away from the centre, while in the case e,e, S (un + 1) the subject $P (7 + 1) of e, is trans- formed by e, into an #—1-dimensional polytope of the import / corresponding to S(Z+-1) which polytope is moved away from O as a whole. Now it is evident that the geometrical condition “that the two new positions of a vertex shall be separated by the length of an edge” makes the distance over which the second motion of any of these two pairs has to take place equal to the distance described in the first motion of the other pair; i. e. if SO(ZH TI) is a limiting element of S°(4-+- 1) and À is a vertex of that S (7+ 1), the segments described by 4 in transforming SD (a + 1) into the two polytopes e,e, Slu + 1) and e,e, Sn + 1) are the two pairs of sides of a parallelogram leading from A to the opposite vertex 4’. In other words: we find the true coordi- nates of 4’ by adding to the coordinates of 4 the variations cor- responding to the motions due to each of the operations e, and e, taken separately. Taking for S® (4% + 1) the simplex 4, 4, ... 4,,4 for SY (7+ 1) the simplex 4, 4... 4,,, and for 4 the point 4, we have to n vary the coordinates 1, 0,0,...0 of 4 so as to admit two more unit intervals, one between the # + 1% and the # + 2%, an other between the 7-+ 1% and the 7+ 2” digit. If then afterwards we l kl n—k pass to the zero symbol we get (8 22 .. 2 11 .. 1 00 .. 0), what was to be proved. | Now we have still to add that the proof for the composition of three and more operations of expansion runs entirely on the same lines. In the case of three operations we will have to compose three displacements according to the rule of the diagonal of the parallelopipedon, in the case of more we will have to use the extension of this rule to parallelotopes. To this geometrical compo- sition of motions always corresponds the arithmetical addition of the symbol influences, where the order of succession is irrelevant: this arithmetical addition leads to the creation of new unit inter- vals independently. So the general rule is proved. The preceding developments lead to a new theorem, viz: Turorem IX. “The operation e, can still be applied to any poly- tope deduced from the simplex in the zero symbol of which the hk + 1% and the # + 2” digit are equal.” 22 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY This theorem enables us to find immediately the expansion sym- bol of a polytope with given zero symbol. We show this by the example (5443322210) of art. 12. In (5443322210) five unit intervals occur, viz. 1f we en the p'" digit by 4, between (4,d,), (da, d,), (ds, de), (ds, do), (do, dro). Of these the first corresponds to the original unit interval of ie EPO simplex (1 00...0), whilst the others are introduced by the expan- sion Has es 61; 15e. SO we find ese, e eg 59 (T0). Reversely it is quite as easy to find back the zero symbol of ee, 6103 9 (10). As there are to be four unit intervals more than the original one the zero symbol begins by 5 and 4, and has to show a unit interval behind the third, the fifth, the eighth, the ninth digit, etc. 20. It is obvious that the system of expansion operations cannot lead to a zero symbol with two or more equal largest digits. So the system of the expansion forms is not complete as to the total number of possible forms. But the scope of this incompleteness is not so large as we might think at first. For, if the zero symbol winds up in two or more zeros, the inversion indicated in art. 3 will bring about a new zero symbol with more than one largest digit. Nevertheless, after this extension of the system of expansion forms, still the forms with a zero symbol containing two or more largest digits and two or more zeros are lacking. So it was desirable to have at hand a new geometrical operation leading to forms with a zero symbol containing more than one largest digit. This now is given us by M. Srorr in the operation of contraction; but before we show this we may devote a single word to the introduction of different kinds of contraction. The subject of the operation ec of contraction of an expansion form in #, is always a group of limiting elements of the same import and of the highest order of dimensions available; so we designate the contraction ¢ as a co, a &, a €, etc. according to the subject elements being of vertex import, of edge import, of face import, ete. Moreover these limits of the same import can be sub- ject of contraction, when and only when all their vertices form together exactly all the vertices of the expansion form, each vertex taken once; in this case any two of these limits are still separated from each other by the distance of an edge at least and now the operation of contraction consists merely in this that all these limits undergo a parallel displacement, of the same amount, towards the DERIVED FROM THE REGULAR POLYTOPES. 39 centre O of the expansion form, by which any of these limits gets a vertex or some vertices in common with some of the other ones. We illustrate this by the example of fig. 4. Here the results can be tabulated as follows: BEL — O der Ge l'impossible) — TT ¢, cie, ee À get DER QUE = CA 7 | G6; 6 — —e,T Gel =d In this small table the negative sign indicates the inverse orien- tation; the impossibility of ae, 7 and ce, 7' is caused by the fact that the polygons, in the first case of edge and in the second of face import, forming the subject of contraction, have already a vertex or an edge in common. But we can also account for the impossibility of e, e, 7’ and ce, 7 — and for other similar results — by remarking that the contraction c, undoes the expansion e, and that it can be applied, when and only when the expansion form has been obtained by applying amongst the different expansions the operation e,. So c, is the only contrac- tion operation which we have to introduce in order to be able to deduce all the forms with a symbol satisfying the law of theorem I. As we will use henceforth exclusively the operation c,, the sub- script of the ¢ can be omitted. 21, We now prove the general theorem: Frrorem X. “By applying the contraction c to any expansion form the largest digit of the zero symbol of this form is diminished by one’. Proof. The groups of polytopes of vertex import of the expansion form represented. by the zero symbol (a + 1, a,6,c,.., 0), where Weste... is found by putting sal, leaving (a, 6, c,.., 0) for the other coordinates. By diminishing a +- 1 by one we get an other form with the zero symbol (a, a, 6,c,..,0) possessing also polytopes of vertex import represented by (a, 6,c¢,.., 0). So the polytopes 4 of vertex import of the second form are congruent to and equally orientated with the corresponding polytopes of the first, but they lie in spaces 2, —= 4 nearer to the centre than 2, = a-+ 1. For, if p is the extension number of the original form (a + 1, a, 6, ¢,.., 0) of #,, and therefore y—1 that of the new form (a, a, 6,c,.., 0), the true coordinate values of +, corresponding to the values a + 1 of the first and « of the second zero symbol are a + 1—* = C 3 Verhand. Kon. Akad. v. Wetensch, (4ste Sectie) Dl, XI. 34 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY i and a —— ae as the true coordinates of O are ET cases, the distance to O is diminished by in both EEn ET ME Moreover it is evident that any two of these polytopes 4 of the first form, e.g. those lying in the spaces m — +1, x, a Ph are separated by the right prism with the base polytopes Bi GR VEE a » gs Ds oe 2 0 == CORRE dy == , Rent Mae OP Las Lg, eere —: (0, eran while the corresponding two g, of the second form are in contact with each other by the 7 — 2-dimensional polytope Li =, Da OT WN By combining the theorems IX and X we can find the symbol in operators ¢ and e, of any contraction form, i.e. of any form the zero symbol of which contains two or more largest digits. To that end we have. 1°. to pass to the corresponding PE form by add one to the first digit, 2°. to treat the zero symbol of this expansion from according to the rule deduced from theorem IX, 3°. to put e before the obtained result. In the following we give some examples of the deduction of c and e, symbols from zero symbols, in connexion with the three possibilities which may present themselves, if we consider the two different zero symbols of a form without central symmetry, according to the appearance of the contraction symbol; they are (4332210) = —(4822110),i.e. eee S0(1)=— e ee 8D), (4432210) = — (4322100), ,, ce,¢,¢,e, 8° (N=— 62,6 20000 (3332100) = — (3321000), ,, ce,e,e, 8 (7) =— ce EON Remark. According to the developments of the preceding article the contraction c, always cancels the expansion e,; so we can deduce from the theorems VI and IX that the operation ¢, can only be applied to expansion forms in the zero symbol of which the 4 + 1* and the # +2" digit are unequal and that the zero symbol of the new form is found by subtraction of a unit from the first — DERIVED FROM THE REGULAR POLYTOPES. : 39 i +1 digits of the zero symbol of the given form. Of this general result theorem X considers the special case # zero. Now if we apply the contraction c,) — € to the simplex of coor- ÉTAT dinates (100...0) itself we find the point with the zero symbol DHL (000...0) i.e. the centre O. This result is geometrically evident: if we bring the vertices nearer to the centre so as to annihilate the separating edges the result is a single point. In this point of view the inverse operation e, can be considered as corresponding to the generation of the simplex starting from a point. Remark. By introducing the operation e, the contraction symbol e can be shunted out. So, if SP (u + 1) represents the point which is to become the SP (z + 1) by applying the operation e,, we can replace ce,e, $P (x + 1) by ee SP (x + 1), but this implies that we write ee,e, Sy? (x + 1) for ee, SY (x + 1). This new notation will prove to be preferable in the case of the nets (see under E the art. 30 at the end of page 57). E. Mets of polytopes. 22. As to recent literature about space fillings or nets we may mention A. Anprerni’s “Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative’? (Roma, 1905), two papers of mine (“Fourdimensional nets and their sections by spaces” and “The sections of the net of measure polytopes 47, of space Sp, with a space Sp, _, normal to a diagonal’, Proceedings of Amsterdam, vol. X, pp. 536, 688) and the memoir of M. Srorr quoted several times. We exclude what may be called a prismatic net, 1. e. a net in #, obtained by prismatizing a net of S,,_, in a new direction, and divide the remaining wziform nets derived from the simplex into two groups 1): 1°. pure nets with only one (central symmetric) constituent and 2°. mved nets either with one non central sym- metrie constituent in two opposite positions or with constituents of different kind. If we restrict ourselves to the plane the first group consists of the hexagon net only, while the second is represented e. g. by the triangle net and the net of hexagons and triangles; if we proceed to ordinary space the first group contains the 40 net only, while the second is represented e. g. by the net of 7’ and ©. *) This division — of no fundamental importance in itself — is introduced here, merely in order to smooth the way leading to the analytical representation of the nets. 3% 36 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY It is our aim to unearth in the following articles a// the nets of simplex extraction possible in space S, from » = 2 to a= 5 included. This task, concerned with new material, breaks up into several parts. Jirst we will have to deduce general characteristic | properties of the analytical symbols which are to represent the nets. Secondly we will derive a simple rule solving the question under what circumstances the symbols obtained do represents possible nets. Thirdly, application of this simple rule will lead to the knowledge of all the possible nets and to a tabularization of them. Finally we will pass in review the tabulated nets and devote some words to an other method by which at least a part of these results can be obtained. 23. Theoretically speaking a net can be determined analytically in two different ways, either as a whole or decomposed into its constituent polytopes. So we will try to find either one symbol of coordinates, representing all the vertices of the net at a time, or in the case of pure nets one pair, in the case of mixed nets several pairs of symbols, each pair consisting of a symbol repre- senting all the vertices of any constituent and an other symbol from which can be deduced all the centres of the repetitions of that constituent in the same orientation occurring in the net. In order to blow life into this theoretical skeleton — forming as it were a kind of working hypothesis — we consider the generally known and simple case of the zet of triangles in the plane. If we start (fig. 5) from a triangle 4, 4, 4, = p;®, i. e. with sides unity, and complete the three sides produced to three systems of equidistant parallel lines, the distance of any two adjacent parallel lines being the height of triangle #6), we get the net V(p,). From this generation it is at once evident that with respect to the original p®, as triangle of coordinates all the vertices of the net can be represented by the coordinate symbol (a,, a,, 43), where a,,@ = 1,2,38) are any three integers for which Za nn (a, a, ap), La, == 1 is the net symbol of N{(p;), under the condition — stated that a, are three integers. In this ever so simple case the round brackets may be omitted, for the faculty of taking for di, 45, 43 any set of three integers with a sum unity includes that of interchanging the three digits. The net JV(p;) consists of two sets of triangles, triangles ps corresponding in orientation with A, 4, A, and triangles p™; of opposite orientation. If we consider only one of these two sets of triangles and of these triangles only one of the three sets of homologous Neem ii ta dze 4 ae DERIVED FROM THE REGULAR POLYTOPES. aT vertices we get all the vertices of the net and each vertex once. In other words: the system of the centres of either of the two sets of triangles is equipollent to the system of vertices of the net, 1. e. if we move all the vertices of the net in the direction 4,O over that distance it passes into the system of the centres of the triangles corresponding in orientation with 4, 4, 4,, whilst we get the system of the centres of the other set of triangles by a motion over the same distance in opposite direction. So, as the three coordinates of any vertex of the net are found by adding to the coordinates 1, 0, 0 of À, three integers with a sum zero, and the true coordinates of the centres O and O, are 1, À, 5 and — 5e 2,2 the symbols GE 4,6, + 4, 6; + 4) and (4, — 4, 6, + À, 6, + 5) represent the centres of the two sets of hal under “le condition that the three 4, are integers with sum zero. In both cases the three integers b, with sum zero indicate what is to be added to the coordinates of any centre of each set in order to obtain the whole set; as we call the two sets of centres the “frames” of the two kinds of triangles, we call the system of differences b,, b,, b, the “frame coordinates” and (0,, bs, 43), 26, — 0 the “frame symbol” of both sets of triangles. Recapitulating we find the following result for M (p3): NÉS MINOR wat rates CRT A ay EE Set of triangles { Symbol of constituent 4, 4 4, ...... lr ed Frame. en On ey EI Other set of | { Symbol 5, 0, VEA de 4,2,2, triangles Rreme ren 4,6,+ 2,6,+ 2), 26= 0. Frame symbols een “as CR MDN DEA LE Here O, O, O, represents a central triangle oppositely orientated to A, 4, 4. We may still remark that the second frame may be written in the more symmetrical form (4, + id 6,+ 2, 6, + 5) 20, = — 1, or if one likes (4 — 1, 6, — 4, 6,— 4), 24; de But it is much simpler here to doctor the met intorthe repetitions of the two triangular constituents by introducing a new symbol still, the symbol (4, + 1, 4, + 0, 4; +0), obtained by addition of the corresponding digits of the frame symbol and the symbol of the constituent 4, 4, A}, the heavy round brackets meaning that only the parts of the digits written in heavy type are to interchange places, whilst the arbitrary integers 4, satisfy the condition © 4, — 0. For each system of values of the 4, satis- fying the condition stated the symbol represents a definite triangular constituent of the set to which 4, 4, 4, belongs; so by this symbol 38 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY the net V(y,) is decomposed into the different constituents of the set of triangle 4, 4, 43 In the same way the symbol ( + 1, 4, +1, 0, + 0) characterizes the other set of triangles under the condition > 4, = — 1. 4) 24. In the first place we remark that the net of triangles admits _ a net symbol with only integer digits and we examine now to what extent this property is a general one. If we choose for simplex of coordinates S® (u + 1) a simplex with respect to which a definite polytope of the net — let us call it the central polytope (P)° of the net — can be represented by its zero symbol and we restrict ourselves to the cases in which the constituents of the net are exclusively forms derived from the simplex, we can easily prove that the coordinates of all the vertices of the net must be integer. To that end we call any polytope of 29 the net ‘orientated”” with respect to the simplex of coordinates, if a translational motion of the polytope which brings its centre into coincidence with the centre of that simplex gives it a position in which it is represented by its zero symbol or by the reverse; this definition enables us to state the following lemma: “If two polytopes of the net in 8S, have a limiting # — 1- dimensional polytope in common, they are both orientated with respect to the simplex of coordinates, as soon as this is the case with one of them”. We remark — in order to prove this lemma first — that, if two polytopes derived from SP (x + 1) have a limiting u — 1- dimensional polytope in common, this limit has eter with respect to both the same import or its import with respect to the one is complementary to that with respect to the other. For, according to the last two lines of art. 13, any two limits (/),_,, represente das to their imports by 9, and g,, are prismotopes (?,; P,_,_;) and (Ps P_i), and these prismotopes cannot coincide, unless we have either # =f or k —n—#—1. | This remark leads to a proof of the lemma in the following way. Let (P)% and (P); be the two given polytopes and (P),_, their common 2 — 1-dimensional limit ae in the space S®%. Let the S (a 1), from which (?)" can be derived by means of the operations e. and c, be our simplex of coordinates; then PW is not only orientated with respect to that simplex but also concentric with it. *) As soon as the idea of splitting up the digits of the symbol into two parts, an unmovable part and a permutable one, had presented itself, the analytical deduction of the nets of simplex extraction was within grasp. DERIVED FROM THE REGULAR POLYTOPES. 39 Let (P’)? and (P’) represent farthermore the two 1) polytopes con- gruent to (P)) and concentric to (P)*% which admit of a zero symbol with respect to the simplex of coordinates. Then we have only to prove that either (P), or (PY is equipollent to (?)?. Now from the fact that (2)! and (P)? have (P),_, in common it follows that (PY and (PY must admit a set of limits congruent and therefore of the same or of complementary import with (/),_, of (P)%; so one of these limits of (P)/ — say (P’),_, — and one of these limits of (P’)? — say (P”),_1 — must lie in spaces 8”,_, and 8”,_, parallel to S%’, and on both sides at the same distance from the meno oleh iy -Of these-spaces: and; 9", -, let 8; be that one which lies on opposite sides with respect to O with Si". Then it will be possible to bring (2), into coincidence with (P)! by means of a translational motion; for, if by such a motion the limit (P’),_, is brought into coincidence with (P),_,, the polytopes will coincide, as this is the case not only with the limits mentioned but also with their centres. So (P)? is orientated with respect to the simplex of coordinates. From the lemma to the theorem in view we have only to take one step more. The lemma immediately shows that, if the net in S, consists exclusively of polytopes derived from the simplex, all the polytopes are orientated with respect to the simplex of coor- dinates, as soon as this is the case with one of them; for we can always consider any two polytopes of the net as the first and the last of a series of polytopes any two adjacent ones of which are in x —- l-dimensional contact. So with respect to the simplex from which the central polytope (P)° has been derived all the polytopes of the net are orientated. But this includes that by passing from any vertex of the net to an adjacent one the coordinates change by integers and as we can reach any vertex of the net by means of a set of these motions — starting from a determinate vertex of (P) — the coordinates of any vertex of the net must be integers. So we have shown now that the property of admitting vertices with integer coordinates only belongs to all the nets, the polytopes of which are exclusively of simplex extraction. This very general result brings us in contact with the two following questions : a). Can the result be expressed by saying that any net with the assigned property of its constituents admits a zet symbol with integer coordinates only ? This question must be answered negatively. We cannot pass to 1 : = a . ej. . . ) In the particular case of a central symmetric (P)? these two positions coincide, etc. AQ ANALYTICAL TREATMENT OF 'THE POLYTOPES REGULARLY this new version, unless we prove that each net of the icn . Wa * kind does admit a net symbol; as soon as such a net admits a net symbol we can choose our system of coordinates in such a manner that this net symbol contains integer coordinates only. We take position with respect to this point De supposing beforehand that each net of the assigned kind admits a net symbol, which _ brings us under the obligation to prove afterwards that this is so. 6). Are there Rae nets not satisfying the condition that all the constituents are of simplex extraction ? oe dispose of this question by pointing to three plane nets, vig. 2, N(p3 Pi, Po) Of triangles, squares and hexagons (fig. 6), — = N (pi, Pos Pio) Of squares, hexagons and dodecagons (fig. 7), 3°. N(p,,p) of triangles and dodecagons (fig. 8), which must undeniably be considered as simplex nets, as they can be derived from the three generally known plane nets VV (ps), WV (pg), NV (p3, pe) by means of the e-operations. If W (p,; p,;) represents a net with the polygonic constituents p,,y,,p, of which p, is of face, p, of edge, py, of vertex import, these deductions are indicated by the equations ey N (ps) = N (D3; Pas Pe); 2 LV (ps) = N (Po; Pas Ps); ee NAP) = N (ps: Di: Pin Cy Ca AD N (Di Vis Pee C0, ea N (Ps) == MP De): As these three nets contain constituents not deducable from the simplex of the plane, the triangle, by means of the operations e, and c, they must form exception to the general rule about the net symbol with integer coordinates only; for, in the coordinates with respect to the simplex, only the polytopes derived from the simplex can be represented by a symbol with integer coordinates only. On account of the property of the three plane nets mentioned — to admit at the same time constituents derivable and constituents not derivable from the simplex — we call them ‘‘hybridous’’. In order to be able to deduce general results from the simple law of integers found above we discard provisionally the three hybribous plane nets and all the hybridous nets that space and hyperspace may contain, considering only the nets we call simplex nets “proper”; meanwhile we promise to come back to these exceptional cases, after having secured. the general rule alluded to in art. 22 and the main results to which it leads (see art. 34). 25. In the second place we remark that the two sets of triangles of the net V(p;) admit the same frame symbol with integer coordinate DERIVED FROM THE REGULAR POLYTOPES. 4] values only. We show that this property too is a general property i. e. that all the different sets of constituents of any simplex net proper admit the same frame symbol with integer coordinates. In diseussing the number of the regular polyhedra in ordinary space the plane nets /V( ys), N(p,), N(p) appear as polyhedra with an infinite number of faces, unyielding as to this that their faces remain in the same plane instead of bending round in three dimensions. Of these regular polyhedra with an infinite number of faces the centre is at infinity in the common direction of the normals to their plane in the space of three dimensions which is supposed to contain them and the anallagmatic 1) rotations and reflections of the regular polyhedra proper pass into transiations and reflections in the case of N(p3), V(p,), V(p,). In the same way each net of &, may be considered as an z + 1-dimensional poly- tope with an infinite number of limits (2), which instead of ben- ding round in 8, ,, fill a space #,. On account of this each net must be transformable in itself by a translational motion which brings a constituent polytope of the net into comeidence with any repetition of that constituent in the same orientation. By means of this property we prove now the following general theorem: Theorem XI. “Any possible simplex net proper admits a net symbol and for all the different sets of constituents the same frame symbol. Moreover the frames of all the possible nets of #, are similar to each other.” a) We show first that al the different frames of a net are equipollent. Let (P) and (Q) with the centres C, and C, be any two polytopes of different kinds of a simplex net proper having at least one vertex / in common. Let (/) be any polytope of this net equipollent to (P) and let (Q), 7”, C°, C”, represent the new positions of (Q), V, CC, after a translational motion which brings (P) into coincidence with (2°) and therefore the net with itself. Then (Q’), V’, C’,, C’, are respectively a polytope of the net equipollent to (Q), a vertex of the net homologous to V, the centre of (P’), the centre of (Q’). From this we derive the equipollency of the three lines VV’, C,C', CAMES CHE", onde’: C’, are mutually equipollent as they are both equipollent to VV’. So all the different frames of a net are equipollent, i.e. each of these frames can be brought into coin- cidence with any other of them by means of a translational motion. ) These rotations and reflections which transform a polytope in itself will be studied in part G. 42 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 6) In the second place we prove that each net admits a frame symbol, and that the frames of all possible nets are similar. If a rectilinear translational motion of the net over a distance d — brings any polytope (P) of it into coincidence with its repetition (P)® and therefore the net with itself, a rectilinear translational motion _ of the net over a p times larger distance pd in the same direction, p being any integer, will bring (P) into coincidence with an other of _ its repetitions (P)” and therefore also the net with itself. This is self evident if we consider the motion over pd as the result of p motions d in the same direction executed one after another. In other terms: the frame of any set of constituents of any simplex net proper must be characterized by the property of containing the point C™ determined by the vector equation CO” = p.CC™ as soon as it con- tains the centres C and C® and » is any integer. Now let d,, d,,..., d,, with the condition Xd, = 0 represent the frame coordinates of C with respect to any centre C, of the frame, and let us consider di, ds, ...,4d,, — 1. e. all these integers, ¢, ,, alone excepted —, as the rectangular coordinates of a point V lying in an other space 8", bea- ring the system of coordinates O (X,, X5,...,X,,). Then to each point C, CD, C of the frame correspond points /, Vv, V™ of S”,, and the vector equation CC=p. CCP includes the vector equation VV =p, VV, 1. e. there is a correspondence one to one between the centres C of the frame and the images / in S",, the points V in S",, being characterized by the property of having integer coordinates @,, @,...,@,. But ad/ the points / with integer coordinates form evidently the vertices of a net of measure polytopes with edge unity; so the system of images V is either the total system of vertices of this net of measure polytopes or a portion of it, containing always the origin O corresponding to the centre C, and partaking of the geometrical property of containing the point VP, determined by the vector equation VV =p.VV’ if it contains V, WV and p is integer. In this form it is immediately evident — in connection with the equivalence of the different coordinates — that the portion can only be a system of vertices, the coordinates of which are integers admitting a common factor 7, i. e. the set of vertices of a net of measure polytopes with edge 7. So we have shown now that the system of points V of 8”, must be (r4,, 7d,,...,76,), where the # quantities 4, are arbitrary integers, whilst 7 is a definite integer. rom this result it follows immediately that the frame of the centres C admits the frame symbol (Os. rb, , rb. , DDR) T0 020 à MS ets ve 0 0 Lf) the arbitrary integers 6, satisfying the condition 24, — 0. The DERIVED FROM THE REGULAR POLYTOPES. | 43 quantity 7, which is the same for the different frames of the same net, may vary from net to net. We call it the period of the net. All the simplex nets proper have similar frames, as their images of points V are similar. 1) If, as in art. 1, our space #, is the space Er 1 lying in a space of operation S,,,, and deter- i=l ; mining on the axes of a given system O(X,, X,,...,X,44) of coordinates equal intersepts O4, we can say that the nets of &,, the vertices of which admit with respect to the simplex of coor- dinates 4, 4, ... A, ‚4 integer coordinates only, always admit frames projecting themselves normally on any of the #- dimensional spaces 8’, of coordinates of #,,, as sets of vertices of systems of measure polytopes of that S’,,. c) In the third place we prove that each net of S,, admits a nel symbol. By combining the zero symbol (13 Gens Gye OY of the-"eentral polytope with the frame symbol (rb; 7b,,...,76,, 76, 44), 2 6, = 0 of the net we obtain the symbol aay her ee ee PRET sone N) where the 9, and r are given integers, whilst for the 5, we can take any system of integers with sum zero. As this symbol contains the coordinates of a//) the vertices of the net, it is the wet symbol. If we write this symbol in the form (7h, + 1,76, + do. .... 76, + a, Tn za + 0) we have got a symbol representing the net decomposed into the repetitions of the central polytope. ) We remark already here that later on cases will present themselves which are at variance with this simple result. We will treat these cases — and explain why they appear as exceptions — as soon as they turn up. *) This is only true, if each vertex of the net is also ne of a repetition of the central polytope in the same orientation. So, if the net contains a non central symme- trie constituent in two opposite orientations and in each vertex only one of these two differently orientated constituents concurs, the net symbol corresponding to one of these constituents as central polytope would only contain half the number of vertices of the net and would have to be completed by a second symbol giving the other half. In that particular case the system of vertices breaks up into two equivalent parts P and Q with the property that the net is equipollent to itself for any two points of the same half as homologous but congruent with opposite orientation to itself for any two vertices of different halves as homologous. This particularity presents itself in the plane in the case of the net of triangles and dodecagons (fig. 8), already discarded above for an other reason. Here we exclude, also provisionally, all the eventually possible nets where this particularity of the division of the system of vertices into two equivalent systems might present itself. 44 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY At first sight it may seem that the introduction of the common factor 7, by means of which the frame is only enlarged but not changed in form, is of no avail, as the scale of the diagrams is of no importance whatever. But then one overlooks the fact that the, frame is enlarged, while the central polytope (P)° remains unaltered. So in the ease of the central triangle 4, 4, 4, in the plane: if we take r= 1 we have to deal with the net WV(y,) of fig. 5, whilst the supposition *==2 gives the vertices of the net VV (ps3, pe) by means of the triangle 4, 4, 4 and its equally orientated repetitions (fig. 9). This simple example shows in the first place the influence of the period #. But on the other hand it gives a glimpse of the fact that with a given central polytope not all integer values of > lead to existing nets. So the supposition 7 — 3 brings already the central triangle 4, 4, 4; and its repetitions too far apart. +) 26. We pursue our investigation in the direction of the last sentence of the preceding article, entering into details about the relationship between the period 7 and the largest digit g, of the zero symbol (g,, 92,-- … Jn, 9) of the central polytope (P)°. If we call any repetition (P) of the central polytope (P)° corres- ponding with it in orientation adjacent to it, if the distance between their centres C, and C is as small as possible *), 1. e. if the coordinates of C can be deduced from the equal coordinates PSE 24: ‘ atl tion and subtraction of only oze time 7, we find: “The central polytope and one of its adjacent repetitions overlap for r << g,, whilst for r — 9, they are iz contact and ion free from each other”. _ Of these three cases of relationship between 7 and 7, we consider first the case. r — g,, then the two cases r Ag, at a time. Case r — gi. The two adjacent polytopes represented by of C by altering only ove pair of coordinates by addi- UE >, Garry J, 0), lade en Ys>- NE > 9) have all the vertices ‘) Application of the case »=8 to the triangle 4, A, A, gives one of the two sets of triangles of the net of fig. 8, already discarded for two different reasons. *) This is the case if the image V of C (compare the preceding article under b) lies on an axis OX; at distance r from O, bes acne se DERIVED FROM THE REGULAR POLYTOPES. 45 dj Gi en 0, (ass ise . > &n41) = (Yar 3. : +» Gn) in common; this is immediately evident, if for +, and x, we take in the first symbol the digits g, and 0, in the second + + 0 and — 7 + g,—=—r-r. In general these common vertices define a polytope of ~—2 dimensions situated in the space $,_ for which 4 = Qh, 2 = 0, 1.e. the two polytopes are in contact with each other by a limit (/),_,; as any common vertex of the two polytopes lies at equal distances (radius of the circumscribed spherical space) from the centres ( and C, this common (/),_, lies in the space 8, _, normally bisecting ( C, 1. e. this (7),_, of contact has the midpoint M/ of CC for centre, 1. e. the contact by the (7),_, 1s external. But in ieerceceptional ease, in thescas 42, 95 == Gg... = On = | n—1 of the central symmetrie polytope (2 11...10), the common limit (/),-» shrinks together into a single point, the midpoint of G C, as in that case (Gs, g3,..., 4) becomes a petrified syllable. At any rate, for 7 — 9, the net is med, as the central polytope and one of it adjacent repetitions are zot in contact by a limit (Z),,_4. 29; If for brevity we represent ae by g the coordinates of G, and nt C are Cure g Cee RAS TEE LEA SNE ee ee RL OU AEN, So, according to formula 1) of art. 1, the distance CC is equal to the period 7; this result will be useful in the treatment of the next case. Case rg, Let us start from the case 7 = 9, treated above and vary r. As the relation C, C—=r holds always, this variation of r implies a variation of C, C, the effect of a translational motion of the repetition (P) of the central polytope (P)° in the direction CoC if r increases, in the opposite direction CQ, if r decreases. In the first case when CC is enlarged, the polytopes which were either in (/),_,-contact or in point contact, will become free from each other. In the second case when C, C diminishes the midpoint 47 of the new CC will lie inside both polytopes, i. e. the polytopes will overlap. So the theorem is proved. As we cannot use overlapping polytopes we have to discard all the cases r << 4, 1. e. we have to consider 9, as an inferior limit of r. But if the net symbol — as we suppose — contains a// the | vertices, there is also a superior limit. For in the case r — y, + 4 A6 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY the distance of the two limits (/),_, or of the two vertices, which coincided for rg, has now become # and this distance may not surpass unity. So we have also to discard all the cases » > g, + 1. So the result is that we can only use the values 7 = g, andr = 9, +1, or inversely: the only values of the largest digit g, of the zero symbol of the central polytopes are r and r—1. Now as any polytope of the net can be promoted to central polytope we have in general: - Tarorem XII. “Any possible net with period 7 contains only constituents with zero symbols having for largest digit g, either _ r—l or r. Two adjacent repetitions of a constituent for which g\—7—1 are free from each other, whilst two adjacent repeti- tions of a constituent for which g,=—=r are in contact, in general See el 1 by a limit (J), but in the particular case (211 ... 10) by a point.” 27. But now unexpectedly a difficulty presents itself. In the case gr any two adjacent repetitions of a definitely orientated consti- tuent are in (/),_,-contact or in point contact, in the case g, =r—1 these two repetitions are free from each other. In both cases we need other constituents to fill up gaps, in other words all the nets are mived. But this result is at variance with the existence of the net MN(po) in the plane, of the net V(¢O) in space. So we have to look out for a way out of this difficulty. his way will present itself immediately, if we examine how to find the other constituents of a net, the central polytope and the period of which are given. Let the zero symbol of the central polytope (P)° of a net with B r be represented once more by (94,95, —, 9 On e have either g, =r or g, =7—1. Then we tan ask by what processes we can deduce from the symbol (rb, + qu, rh + Qe, QC tds EE representing the net decomposed into the repetitions of the central polytope, other constituents. There are two of these processes com- pleting each other in this sense that the first can be used in the ease 7, — 0, the second im theveaseag, are 1°. In the case of the zero symbol (44, a, ...,¢,—1, 0 taming more than one zero we can write the decomposing symbol (rb, Quirbs Eos © rh a A ras "bn + 0,78 p14 +0), © 3, =0 in the form + etree DERIVED FROM THE REGULAR POLYTOPES. 47 (74, Mdr Qu: + 704 rh 0, rb ark) zn) 25,0 by allowing 7 units to pass from the -unmovable part 76, ‚4 of the digit #b, 4 + 9 to the permutable part; for by that variation we alter only the grouping of, the vertices of the net to vertices of polytopes but not the total system of vertices of the net. If now we write 4 for 4,,,—1 and put the permutable digit r fore- most we get (74, + 7,76, + 1,76. +... rs 1 + qua, 7b, +9), 24, = — 1, bringing to the fore the constituent with the zero symbol (7, 4, gs... Gn — 41 0). : 5; peru thie Gase of the zero symbol (9,955... 9, 241; 1; 0) con- taining only one zero an application of the same process leads from (14, a0 NM: rby ne D... ro, 1 # Yn-1> rb, na 1 > 10,44 TP 0), Xb; == to (Cr zi 2, rb, BIE (A ’ rb, se 2 SEREN AE + fen rb, + iy 2 D. en | end therefore to the constituent (r; 91, gay. …-, Yn —45 1), the zero symbol of which is (r — 1, — Ll, ga _— 1, ..., gna — 1, 9). In order to obtain this zero symbol we can write the decomposing symbol in the form (74, +1+r—1,744+1+q—-1,...,7,4+1 + Y4,-,—1, 76, + Ì +0), 24, = — 1 and pass to an other sum YX, — (u + 1) of all the digits by omitting the unit of the unmovable part of the digits. So, if we take notice only of the zero symbols of the constituents deduced by means of the two processes, we can word these pro- cesses as follows: 1°. “If the zero symbol of the given constituent contains more than one zero, we can replace one of these zeros by 7’. _ 2°. “If the zero symbol contains only one zero, we can replace this zero by r and diminish all the digits by unity afterwards”. We now come back to the difficulty about the pure nets stated above. To that end we have to ask under what circumstances one of the two processes leads back to the original constituent; therefore we repeat that: the first deduces RUE ae OOG eg es 0, 0), ,, second „ (r—1,9,—1,92—1,...,Gn—4— 1,0) „ (G43 Qos.Qn_1 1,0). 48 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY As a polytope with a zero symbol with 4— 1] zeros cannot be a repetition of a polytope with a zero symbol with / zeros, the first process does not suit our aim; but the second may do so under the conditions Pret re In NT 1 = 2,42 — | — 3» vn le ee i.e. in the case of the central polytope (7 — 1, r— 2, ...,2,1, 0), 1. e. if we have in S, the case (2,#—1,...,2,1,0)withr = It is indeed easy to prove that the particular case of the reap- pearance of the original constituent presents itself, when and only when we have r — # + 1 and g, — x. For, according to the law of theorem I, 7, == exacts that the zero symbol contains no two equal digits and under this circumstance the substitution of 2 + 1 for zero followed by the diminution of all the digits by umity reproduces the original zero symbol. In art. 30 (page 57 at the top) it will be shown that the suppositions 7 = » + 1, 2 lead to the unique self space filler of &,. But now that the manner in which we have to account for the existing pure simplex nets is secured we have to revise our notion of ‘constituent of the same kind”, if we will keep the analytical theory developed just now in touch with the geometrical facts. According to that theory we are obliged to say that the plane net ÂV(p;) contains three different groups of hexagons, though geometrically all the hexagons are equipollent to each other and therefore of the same Aid. For im the case z= 2 the suppositions r=an+1, g\—n give rise to the net with the decompocme symbol (34 +2, 34,11, 38%, L 0), ©4,=0, corresponding (fig. 10) to the set of hexagons a with a heavy lined circuit, whilst the net contains two other groups of hexagons which admit alternately thick and thin sides, one group à where the horizontal thick side is below, an other group c where the hori- zontal thick side is above. So, though we keep saying that the hexagon is a self plane filler, we will consider N(pe) from an analytical point of view as admitting three different groups of hexagons, using here henceforward the more precise term of “group of constituents’ in order to indicate a “set of equipollent poly- topes, the vertices of which form together all the vertices of the net, each vertex taken once”. Only under this extension of our former kind of constituents by our now introduced group of con- stituents the theorems XI and XII are generally true. If we follow the interpretation of the net M(¢O) as a net with one kind of DERIVED FROM THE REGULAR POLYTOPES. 49 constituent only, we get a frame !) dissimilar to that of other nets, with any two adjacent repetitions of the unique constituent in contact by a limit (/),_,, Le. a face here; these “exceptions” disappear, if we adhere to the analytical idea, according to which N(tO) admits four groups of constituents. ?) 28. If we indicate by o, the number of the digits of the zero symbol of the central polytope (P)° leaving the remainder y when divided by 7, i. e. if p, represents in general the number of the digits p of the zero symbol, but in the particular case of p, the sum of the numbers of the digits zero and the digits / (the latter being absent in the case g, — 7 — 1 of the original zero symbol), we have the: Tarorem XIII “The two operations stated above which may lead to new constituents of the same net do not affect the circwlar order meeerecession of the terms of the séries 0,_,, be, ---» 1» Po- This series with the sum 2 + 1 will be called “partition cycle of n + 1, Semen 0 thé fet and be represented by (0,4; 0,0, . - ., 01, Pon This theorem is self evident. For the first of the two processes does not affect the series at all, whilst the second transforms it into D De, «=> Pi We apply the two processes to an example in order to show the circular permutation of the partition and suppose to that end that in space À; there zs a net with the period 4 admitting -the con- stituent (3222100). Then application of the two processes gives successively | | Partition cycle (3222100) 1312 (4322210) | 1312 (4482221) = (3321110) 2131 (4832111) — (3221000) 1213 (4322100) 1213 (4432210) 1218 (4443221) = (3332110) 3121 (4833211) = (3222100) 1312 Here every new symbol in the first column is derived by the first process from the one in the line immediately above it which con- *) In art. 39 the system of the centres of all the 40 of N(tO) will prove to form the vertices of a net of rhombic dodecahedra, which latter net is not of simplex extraction. *) In order to avoid misunderstanding we stipulate expressly that it is not our inten- tion to replace the notion of kind of constituent by that of group, but that we wish to stick to the notion of kind of constituent, complemented by that of group as soon as the partition cycle (see the next article) is a power cycle (see page 57). Verhand. Kon, Akad. v. Wetensch. (1ste Sectie) Dl. XI. C4 50 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY tains a zero, whilst by the second process the symbols of the second column are deduced from those of the first column placed on the same line. In the third column we find the partition cycle 1312 proceeding one step to the right at every application of the second process. if the case considered is that of an existing net — which question does not yet interest us here —, this net must admit seven different constituents, 1. e. three pairs of oppositely orientated ones (3222100) , (3321110) (3221000) , (3332110) (4322100) , (4432210) and the central symmetric (4322210). This example leads us to a general rule about the nome of groups of constituents any net is to have; we state it in the form of: Tarorem XIV. “In general the number of kinds of constituents of a net of 8, as ge dean the case r— Part The proof of the general case runs as follows. The zero symbol (Ga, Yos ++ +; Ins Yn» 9) of the central polytope we start with passes either into (7, 94, Go, . . ., Gn-1, 0) or into @ —1,9,—1, qo — 1, .; {na — 1, 0) according to g, being either zero or one. So, in continuing the application of the two processes of art. 27, at each step the digit 7, moves one place to the right and comes back to its original place after # 1 moves. Moreover it reappears there with its original value g,. For the increase by 7 at the jump from the rear to the front is exactly counterbalanced by the loss of a unit every time when of two unequal adjacent digits the right hand one jumps to the fore, this loss occurring exactly r times; indeed, in the circular permutation of the digits from the left to the right — executed for simplicity for a moment without increasing or decreasing — the zero at the end has to be replaced successively by 1, by 2, by r — 1 and finally in the case g, = r — 1 by zero, in the case Mr by r. So after 2 + 1 moves the original zero symbol recurs and the total process has come to a close. 1) In the exceptional case » = 1 we find only g = 1, i.e. the zero symbol of any constituent can only contain units and zeros. So we can SCSI n—1 start with the simplex (1 00...0) and find successively (11 00...0), n=? LME RE (LE 00%. .0), etc. But when we have to pass from (liz i ele 0) 1 1 . . ) The process may come to a close sooner. Compare for this exception page 57. DERIVED FROM THE REGULAR POLYTOPES. 51 ees Was _to the next symbol we find by means of the second process (00...0), which falls out. So we only get 2 kinds of constituents for 7 = 1. 29. We come now to the general rule about simplex nets proper; it can be stated in the following form: Taxorem XV. “To every possible cyclical partition ‘of # +1 corresponds a definite simplex net proper of S,,.” In order to prove the theorem for the general case with » +1 and the particular case with x groups of constituents we first of all determine a list containing these different groups of constituents, to be derived from the partition cycle. Then we select from this list a definite polytope (P), of a definite group and show that this (P), is in contact by any of its limits (/)\” with one and only one other polytope (P), of the list, whilst the list contains no polytope overlapping (P),. Case r > 1. We start from the partition cycle ,(0,_4, Pp—os- - «Pt» Po)n and deduce from it the net symbol Pp 1 Pr—2 ep PO (ra; ra, +r—1, Nig eek Phi ‚ra, +1, ra; + 0): (a Ay see, » In44) i.e. the symbol with o,._, digits congruent to 7 — 1 mod. r, 03 digits congruent to 7 — 2 mod. 7, etc., the different quotients Ms &, - .., Ara Of the division of these digits by 7 having a sum Xa, — 0, whilst the sum of the remainders r — 1,7 —2,...,0, r—1 Me > 76, may be represented by ‘£,- -4=1 If we write this symbol in the form Pr_1 Pr—9 PA PO (ra, + r — 1, rime ‚ra, +1, ra, + 0) 2 PAC VASE ie) and permutate only the remainders 7—1, r—2,...,0, the net is decomposed into the group of constituents to which the central polytope belongs; but we can have this rather complicated symbol in our mind quite as well if we simplify it by omission of the unmovable parts of the digits. So the first line of the following list repeats the group of constituents to which the central polytope belongs, while the other lines give all the other groups of consti- tuents, deduced from the “central group” in the manner and order of succession of the preceding article. 4% Pr Pr—3 Po Py1 ( Qr—2, Qr—38,..,r, rl) Oran) Ao + El pr 1 Pro Pr 3 bgn Ge ; (Url, 2w—2, 2w—3,.., 7, rl) | pra (nt) | ho + (wn E2— 0, ,)r te Ppa Pros Po 1 (Q7—l1, 2r—2, 2r—38,.., r, r7—1) —n ky + nr 59 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY Pa; k Pep ie hho | PER PREP EG SUC à 0 A 1 Pr—4 Pr—2 Py | = oet Sie EIN 0e eet hor. Po Pr Pz 4 En pigs ky + pot PO Py—1 Pa | oor og T—1 , sy Zo ee ky +(e For each of these groups of constituents have been indicated in a second column the value of Xa,, in a third column the value r—1 of 4 = 210. Moreover, in order to point out the regularity of Ak | the process by means of the variation of these two sums, the use of the zero symbol has been sacrificed for a moment, 1.e. the diminution of the digits by unity every time as the last zero is replaced by 7 (exacted by the second of the two processes of the preceding article) is not executed here, which implies that a digit h jumping to the fore becomes A+ r. Here at each step Xa diminishes by a unit and # increases by 7. 3) But in the selection of a definite polytope (P), of the list we return to the zero symbol and suppose 1) that the cyclical permutation of the partition cycle from which (P), has been derived begins by 0,4 and winds up in b,, 2) that », of the », zeros have been replaced by r, 3) that the equation of the space S,,_, containing (/)®°, is ij *) This relation also holds when we pass from the last group of constituents to the first, when we diminish all the a; by unity at the transition. From this point of view we can introduce the notion of “cycle of constituents”. hart de Oe aen Aad ee DERIVED FROM THE REGULAR POLYTOPES. 53 determined by making the sum a + #, +- ... ++, maximum, the > digits which are to make that sum maximum consisting of 0, times 7, 6;_, times r—l, etc. and p, of the p,, digits r—/--m. Under these circumstances the polytope (P), is represented by the symbol Pr PLA Pm Pm—1 PO Pr_A PEA Pi—Pr eee. em, r—l--m—1,.., rd, r—/—1,.., 1, 0 ), (a, URSS An +1) a,b while this symbol passes into that of (/)%:% by the introduction of intermediate brackets between the digits a,r—+r—/-—+m and aardr lm, i.e. (D%:4 is represented by Py» Pr Pu re Pm—Pu Pm—A es PO Pr 4 Py44 Py Py, (r,r—l,..,r—l-+-m) (r—l-+m, r—l-+m—l1,..,r—l,r—l—l,..., 1, 0) (a, a, dy) (ay +4, 449, +++ An +1) under the condition that of the two parts of this symbol the first refers fo the coordinates z,, 2, ...,z, and the second to #,,,,4,:2,...,2,11. The determination of a second polytope (2), of the list containing (Ut! as limit must be guided by the remark that in each of the two parts of the symbol of (/%% considered for itself we may transfer the same amount from the unmovable parts of the digits to the permutable ones. But in order to obtain a symbol satisfying the law of theorem I, when the intermediate brackets are omitted, we have moreover to select these two amounts in such a way as to obtain a set of permutable parts containing «+ 1 integers distributed over 7 different ones, succeeding one another with dif- ferences unity. So we can esther diminish all the digits 7,7 — 1,..., r — lm included between the first pair of brackets by 7, coun- terbalancing: this by increasing a,a,,...,a, by unity, or — which comes to the same — increase all the digits r — /+ m, r—l—+m—1,...,0 included between the second pair of brackets by r, counterbalancing this by diminishing @,,4, G19, ---,@n44 by unity; that these two results differ in form only can be shown by remarking that the first passes into the second if we increase all the digits 0, —1,..,—/+m,r—/+m,r—/+m—1,..,0 by r, counterbalancing it by diminishing a,+1,a,+1,..., Gr Lot Gao, ---, 2,1, by unity. So we find one and always one second polytope (P), with (/%:% as limit, represented by the symbol : Pr Py4A Pre Pme Ver | Po Pr Pi+4 Py Py oe i- .—l+m,r—l+m,r—i+m—l,. ., r—l,r—l/—I]1,.., 1, 0) (@,+41,4,+4,...,ay +4, ay+1,ay +9, ... an +4) 54 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY We now pase to the determination of the centres G,,@,,@, of (P),, (P),, OD! which are collinear, as G, @, and Gi G', are normal — in the centre G, to the space S, 4, bearing (24, in order to prove that G, lies between G, and G,, 1. e. that (P), and (P), lie on different sides of that space 8,4. If we consider the #,,, of these three points we find for CES rp a en | ro, Fr piace - Pm —T door =D Hora lr FF ft Bera oe | — 0p 2 8p Eu : Se mme. Ÿ Ra, - 1 | (0m — Pye) PEM) Jr Pol? Dr Oral ver Pt | Now in comparing the three values a, 2, 2%” of zt we can omit the common part ra,,,. But then if we write y,,, for Vr za TA, 4, it is evident that we have 99 < 7% 9 < y@. For gy” y, y are arithmetic means, 4“ of a series S of positive integers 1,2,...,7—/Z-+m, each of them taken a certain number of times, y of an other series of integers consisting of 8 and of positive numbers 7 —/-+-m, r—/l-+m+1, ....,r—l, r equal to or larger than the largest of 8, y” of a third series of integers consisting of S and negative numbers. So G, and G, lie on different sides of the space S,,_, bearing (/)\, i. e. the system of polytopes contained in the list admits no holes, every limit (/),_, of an arbitrarily chosen polytope P, being covered by an other polytope P,. We have still to show that no two polytopes of the net can overlap. We do so by simply remarking that the vertices of the polytopes of any group of constituents form together the total system of vertices of the symbol derived from the partition cycle }) (see above at the beginning of the treatment of the case r > 1 under consideration), as each of the groups of constituents of the list has been deduced from that symbol according to the processes of art. 27. For — while overlapping of polytopes of the same group *) This fact can also be put on duty in the proof about the position of two polytopes with common (/),4 ou different sides of that limit, DERIVED FROM THE REGULAR POLYTOPES. 55 is already excluded (art. 26) — this remark excludes overlapping of any two polytopes, as we can derive from it that not a single vertex can lie inside any polytope of any group of constituents. Case r= 1. In this case the enumeration of the zero symbols om —_ ee ir 00 20) 2.2, CET 00. 0) . AT. 10) n n —1 p n—p +1 n of the # groups of constituents is much simpler. Moreover the polytopes of the first group and those of the last admit only one kind of limits (/),_, viz. simplexes, those of any other group only n two, limits (/),_, with respect to {1 00 .. 0) of the lowest and of the highest import. Here overlapping is also excluded, as can be shown by means of the same remark used above. Here the polytope (P), can be represented by fad, att, ....:. AE LD a Le Ona 0), its limit (/){“®) lying in the space #, _, with the equation &, — 4, +1 by (as + D) (a +1, ...... NPA RE ME AREA RO , 4,1 +0), the second polytope (P), of the list containing also this limit by (4+1+0,a+1,...... ma Oeser. Ara 0). For the rest the proof can be copied from that given above. Now that theorem XV has been proved we go back to the polytopes (2), and (P), in contact with each other by a common limit (/),_4 in order to indicate a relation between the import of that common limit with respect to (P), and (P), on one hand and the places of the groups of constituents, to which (P), and (P), belong, in the list of polytopes of the general case r > 1 on the other. To that end we indicate by G,, G,,..., G,; GC, ‚4 successively the kinds of polytopes represented by the first, the second, ... the last but one, the last line of the list of polytopes and — as on page 17 — by the symbols 9, 91,92, -- + 5% —1 in relation to any #- dimensional polytope limits (7),_4 of vertex import, edge import, face import, ..., the highest import of that polytope. Then we find: Tagorem XVI. “If two polytopes of the net, (P), of group G, and (P), of group G,_,, are in (/),_, contact, the common limit Bot ONE), and alge, tor (P}. The proof of this theorem lies in the remark that (P),, according to the subscript ie nS ee Sn ele ee a A DOD 56 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY of the symbol representing it, gives for Xa a value surpassing the corresponding sum for (P), by 7, on account of the units added to the digits a, a», ... a. As Xa diminishes by a unit if we go down one line in the list, our (P), belongs to group G,_, if (P), belongs to group G,. Now we know that the sum of the » definite coordinates is maximum for (P), and minimum for (P), in the space je the common limit, which proves that this limit sa mode (FP), and 29, 10 NEE | | By means vat this theorem we can indicate the group to which belong the polytopes touching a given polytope along its limits. of a given import; if (P), belongs to group G, and it has limits g,, it is touched along these limits by polytopes (P), belonging to PEOMD,. Eh a | The theorem also holds for the case 7 — 1, where the zero symbols of the successive groups @, G5, ..., G,, ..., G, ate n n — 1 Ne nee Lies À n (100... 0), (1100 .: 0), ..; (1b... 1400 20), CC There we can state it in this form: “Any polytope (P), of a net for which 7 — 1 is touched along its limits of vertex import by polytopes of the preceding, along its limits of highest import by polytopes of the following group”. 30. We now apply the theorem XV to the cases 7 = 2, 3, 4, 5 and put the results on record in the second table added at the end of this memoir. | First one word about the general plan of this table. Horizontally it is divided mito four parts, corresponding successively to the cases ~ n — 2,3,4,5. Vertically it breaks up into seven columns with the first five of which we are concerned here. The first column, indi- | cating the rank number of the net, enables us to individualize _ | each net by a very short symbol, consisting of the value of # in italian figures, bearing at the right a roman rank index, 2,,, in- dicating the net of hexagons in the plane. The second column gives the value of the period + from 1 to 2 + 1 upward. The third column contains the partition cycle, represented by that permutation in which the first digit is as small as possible. The fourth column brings the net symbol corresponding to that eyclical permutation of the partition cycle, whilst the fifth is concerned with the zero sym- bols of the different groups of constituents. With respect to these columns — the others will be explained in part G — we have to insert a few remarks. ‘ F DERIVED FROM THE REGULAR POLYTOPES. D7 In the cases 2,7, Sy, Avrr 9x, Where the partition cycle consists of #1 units, we find back the self space fillers of simplex extraction, pe — (210), /0 = (8210), etc. The net symbol of these self space fillers is characterized by the property that its » + 1 digits, when divided by # +1, leave all possible remainders n,n—1l1,+:.,1,0, each remainder once. In the case of the partition 2,2 of the net 3,,, an other particu- larity presents itself: in the process of formation of new zero symbols we fall back at the second step on the original symbol (1100), (2110), (2211) — (1100). This is due to the fact that the partition cycle consists of (two) equal parts. So this particularity repeats itself in the cases 5,5, 5, and 5, with the partition cycles (3, 3), (2, 2, 2) and (1,2, 1, 2), in general if we have # + 1 = uv and the partition cycle consists of the v digits &, 4», ... a,, this set of v digits being repeated in the same order of succession so as to have w sets. In the latter case where the partition cycle is said to be “a cycle of powerv”, Es v the self space fillers, which present themselves for v= x -+-1,w== 1). We point out two other particularities occurring for the first time in S;. The two partition cycles 1, 2, 3 and 1, 3, 2 of which the second written in the form 3, 2, 1 is the inversion of the first, have been inscribed both as 5,,, as these two nets, differing only in orientation with respect to the simplex of coordinates, are essentially the same. On the other hand the two nets 5,, and 5, are essentially, different, though the four digits of the partition cycle are two times 2 and two times 1 for both. The fifth column forms the principal part of the table. As to the number of different constituents of a net in #, this column is sub- divided into x + 1 small ones. In the first of these 2 + 1 small columns is placed the central polytope; on each horizontal line the polytope mentioned in a following small column is deduced by the two processes of art. 27 from that in the immediately preceding one. For brevity we have only inscribed the geometrically different forms, using from 2 — 4 upward the symbol e, explained at the end of art. 21 and indicating the orientation by means of the signs. we find only —= w constituents of different form; it includes *) If we wish to indicate the number of constituents of different form and orientation we can complete theorem XIV by saying that this number is n for r —1 and et Vv if the partition cycle is a cycle of power v, 58 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY We now come back to the particularity of the nets 5%", 5y° hinted at above. We see now at a glance that these two nets are one and the same, the polytope of the p“ small column of the one being equal but oppositely orientated to the polytope of the 7 — y" small column: of the other (» — 1,2, ..., 6). So in each the six constituents present themselves in only one of the two possible orientations, which implies that none of them can be central sym- metric, as in the range of the # + 1 different constituents of a net of #, in the order of succession obtained by a regular appli- cation of the processes of art. 27 adjacent polytopes of a central symmetric one differ in orientation only. Or otherwise: two opposite _ limits (/),_1 of a central symmetric constituent are covered by two congruent but oppositely orientated polytopes, i.e. if we project on the line CC joining the centre © of any polytope of the net to the centre C’ of any limit (7),, of this polytope all the poly- topes of the net the centres of which he on that line, the projection of any central symmetric polytope with its centre on CC acts as a “turn table” with respect to that projection. 31. The simple rule of theorem XV enables us to extend the list of nets to any value of z we like. So we would find for # = 6 and x — 7 respectively the 17 and the 29 cases represented as to their roman rank index, their partition cycle and the character of their constituents in the following small tables, where the three subdivisions of each last column give successively the number of central symmetric constituents, the number of the asymmetric con- stituents occurring in pairs and the number of asymmetric con- stituents occurring in one orientation only. WO; I | 6 VII Rib He XI a | TI] 16116 VIII} 223116 XIV} 111221116 Il 25/16 TX{1114]1/6 ell Liziahi6 fal 3411/6 X|1123l | |7 XVII 1111121146 VILLSII 6 X1/121 3/116 XVII aaa) VIJL 24 | i Xi 2224116 | DERIVED FROM THE REGULAR POLYTOPES. 59 AT; Il S[16 XI] 1115 | XXII 112131 | 8 ri! 17] |8 XII| 1124] | 8 XXII] 11222126 nl 261216 XII 1214/26 XXII 12122126 Iv 35] |s XIV} 1133]2/6 XXIV} 111113} |8 vl 4412/2 XV 1313] 4 XXV| 11112212'6 VI/116)2 6 XVI 1223] 8 XXVI 111212] (8 vili2sl | 8, XVII] 1232) 8 XXVI 112112/22 vinli34l | 8 XVIII 2222/2 XX VIII 11111 12/26 Ix|22al2i6l | XIX|11114L216 XXIX]11111111]1 X]23312/6 | XX[11123 8 Under x = 6 no cases of a power partition cycle (except 6x, the self space filler) present themselves, as x-+-1 is prime here. wane, wood besides 7,3. stll 7 A5, Tey Wik? Mae) ev With vo 4. Instead of pushing this general investigation any further we will give here the generalizations of the three nets of the plane to space 4, Tarorem XVII. “In space S, the central symmetric polytope with the zero symbol (x, ~—1, n— 2, ..., 1, 0), represented also by the expansion symbol e, ee, ... €, 9 e, 1 Alu +- 1), is the only self space filler of simplex extraction. This unique geometric constituent of the net presents itself in 2-+-1 different groups with the property that the vertices of the constituents of each group form the vertices of the net, each vertex taken once, in other words: that no two constituents of the same group have a vertex in common. In this “cycle of constituents’? (compare the footnote of art. 29) formed by these groups G, G,, ..., G, any polytope of the group G, is touched along its limits g) of vertex import by (2+ 1), polytopes of group G,_,, along its limits g, of edge import by (Jl), polytopes of group G,_», ete. So, in order to perform the task of colouring the polytopes of this net im such a way that any two polytopes bearing the same colour are free from each other (a polydimensional bud of the renowned shrub “map colouring’) it will be necessary and sufficient to have at hand nl different paints, one for the polytopes of each group”. “The n-dimensional angle of the self space filler of S, is oe right ones”, 60 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY What is said above about contact of constituents of different groups by a limit (/),_, is a mere application ') of theorem XVI. As the net of measure polytopes 47, of S, shows, the #-dimen- sional space round any point contains 2” right angles, if the a-dimensional angle of 47, is called a right one. As all the #-dimen- sional angles of the self space filler are equal and x-+1 of these polytopes concur in a vertex of the net, the n-dimensional angle of n the self space filler is aT right angles. Turorem XVIII. “The net of 8, with the period unity admits Re Oren napa f, the constituents (11-.. 1 00 .. 0), (w= 1, 2, .-.,4) consists ; 5 FE of 3 constituents in both orientations for z even and of a constituents in both orieutations and one central symmetric consti- tuent for 2 odd” Trrorem XIX. “The net of 8, with the partition cycle 1, a n EN es sch admits the # + 1 constituents ie 00 … 09,-(22- 2 1 O0 en ie ñ DEEE p—=1,2,..,n— 1 and (11 .. 10) consisting of 9 constituents In both orientations and one central symmetric constituent for x even n +1 5 3 : : re) | and of aS constituents in both orientations for x odd These theorems immediately follow by specializing the general results. We give them here expressis verbis as we will indicate later on an other deduction of them. ?) 32. A survey of the results for # — 2,3,...,7 suggests one or two general remarks. The first can be stated in the form of: Trrorem XX. “Every simplex polytope partakes in the formation of two nets. This is true without any reserve for the central sym- metric constituents, it is also true for each of the two different positions of an asymmetric constituent.” *) It is an easy task to demonstrate theorem XVII by itself by showing that the image points of the centre of the central polytope with respect to the spaces S, —4 bearing the limits (/),,4 as mirrors form the centres of the polytopes in n— 1-dimen- sional contact with the central polytope. We consider this verification as a useful exer- cise, even in the special case n = 3 of ordinary space. *) Though we do not wish to push the general investigation any further we still mention the following theorem: “The net of So”, with the power partition cycle 2” is built up of two central sym- metic constituents only”. =e Sais le hiérss °s DERIVED FROM THE REGULAR POLYTOPES. | 61 Let us take the polytope (221000) of S;. This polytope can belong to a net the period r of which is either 2 or 3. In the first case we find 4221000) which can be reduced to ,(100000) by going two steps backward; in the second case we have (221000) which passes into (211100) also by going two steps backward. Or, let us go back to the constituent Pr Pr Po Pr PRE PIS Er M DE DE A TES Mage DA RD). used in art. 29 in order to make the proof as general as possible. This constituent can belong to two nets, one with the period 7, an other with the period # + 1; the two partition cycles of these nets are Bras Pres + +25 Pos ras s+ +s Pints Pt) | rrt(Ors Oras Oras +++» Pos Aras +++» Pt44s Pr Pr) and may be reduced to ARE Pr Re Ades PO) r t(Ors Oras Oras + + +5 Rigas Pi— rs Pia + + +5 Pis Po) In the case of an asymmetrie constituent it may happen as we have seen that a definitely orientated one occurs in two different nets, if we consider as different two nets as Bn Be which are each others reversions. So under this point of view the two positions of 221000) occur together in three different nets. But the statement of the theorem about each of the two positions of an asymmetric constituent holds under any point of view. A second remark refers to the expansion symbols used in the table. In order to bring the two different orientations of the asymmetric constituents into evidence we have introduced the expansion symbols provided with the negative sign. But the law of succession of the different constituents of each net proceeding in the list from column to column would have been much more evident if we had stuck to expansion symbols without sign. ‘Then the order of succession in the case of net 5, would have been e,,‚e,, e5, e3, e, leading to the sup- position that in general at each step the index of each e increases by unity, an illusion which is already destroyed by the series Cor Colis C1L2 C263, 6361, €. At any rate this second remark places us before the question by which rule the expansion symbols of the constituents of a net can be deduced from the partition cycle. The answer may be given in the form of: Turorrm XXI. “The constituents of the net of #, corresponding to the partition cycle ,(0,_:,0,_2, +++») are found by applying to the system of digits consisting of the series 62 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY il; pr Er ea ee EE , 2 — Lo preceded by the repetition of all its terms after having subtracted n + 1 from each (of which repetition the negative terms with an absolute value surpassing 2 may be omitted) a number of xz times — the process of increasing all the digits by unity and throwing out — (as soon as they appear) digits surpassing z — 1 and (afterwards when the process is finished) all the negative digits. Then the n + 1 rows obtained represent the indices of the e operations to n +1 be applied to (00 ... 0) in order to obtain the expansion symbols of the constituents”. Before proving this general rule we elucidate its meaning by applying it to an example, for which we choose the case 5,42. Here the series is — 1,0, 38 which has to be proceeded by — 3. So, if we indicate in heavy type the figures which are to be kept, the operation is 3 4 + GOOD = © giving € 63, Cy 4 Ens C1 Cas Cp Cz C3, 1 C3 Cy, Caen for the six expansion symbols of 5 y;’. As we have e, e, e, — — e, & e, and & e, = — & &3 this series is the same as that inscribed in the table. | The proof of this general theorem splits up into three parts. In the first we show that the top row corresponds to the constituent Rd in Ed for which (r— l,r — 2, ..., 1,0) is the zero symbol. In the second we explain that the addition of a unit to all the digits corresponds to what happens to the digits in the processes of art. 27 but for the transplantation of the digit at the end to the begin- ning. In the third we will be concerned with the influence of that transplantation. The first and the second parts are mere consequences of theorem Py—1 Pr—2 P1 PO IX. In the case of the zero symbol ,(r — 1, r— 2, the unit intervals present themselves behind the digits of rank Ors Pr +- Pr—2> Aldo kes st + 1 a Po and this proves in connection with theorem IX the first part. Moreover the circular permutation over one digit to the right hap- DERIVED FROM THE REGULAR POLYTOPES. 65 pening at each step of the two processes of art. 27 (see the example following theorem XIII) changes the ranks 4% + 1 and kh +2 of two adjacent digits into 4+ 2 and 4+ 3, 1e — according to theorem IX — the operation e,,, is still to be applied or has already been performed on the new constituent according to the operation e, being still to be applied or having already been performed on the original constituent; i.e. if e, occurs in the e-symbol of the original constituent, e,,, must occur in the e-symbol of the new one, what proves the second part. In the third part we have to consider all the possible cases of the transplantation of a digit from the end to the beginning; these cases, four in number, are the following: (r —1,.., 1, 0) becomes (r — 1, r — 2,..,0) … loss of e, 4, gain of e,, oe 0,0) se ay rel, 0) gant of e,, DO > (by Fe, 0). JOSS OF Br, Seen i Wy. Fee. rev yO). neither loss mor gam, So we find the two rules: 1°. If e,_, appears in the symbol of the original constituent it falls out in the next one, though an other e,_, may be introduced (if e, . was contained also in the original symbol). 2°. If the number of the operation factors e, 1s 7 — 1 the symbol €) appears in the next constituent. But this is also the effect of the operation indicated in the theorem, the first rule being a consequence of the omission of the digits surpassing #7 — 1, the second being deducable from the repetition of the series — i, 6,4—1, o,,-+,.—l, ete. If, in order to add still one word about the second rule, G, 4, G,, G,., indicate three constituents, consecutive in the sense of the theorem, and the e-symbol of G, bears only 7 — 1 expansion factors, then the e-symbol of G,_, contains # — 1 and therefore also — (x + 1) + (x — 1) = — 2, before the negative digits have been omitted; this — 2 becomes — 1 for G, and O0 for G44. 33. The theorem XXI enables us to show how the “principal” net of S,, Le. the net with the period » — 1 always inscribed first, can be transformed successively into all the other ones. The result for S; is given in the following table, in the left half in the symbols to be applied to the different constituents of V(7’, 0), in the right half by the results of this application. 64 Ver- Constituents tex I e) eps @y4 “gap — () 1 e. u | eee | 2 — e, Ill 2 — 0 e ] 02 ] Ep €» IV 2 0 OS) ee 12 2 Dee V 12 02 01 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY Ver- Constituents tex 7 O | —7 | gap fie PTI NR | ra —i7 | —T B CO O CO O LT, tO | —tT CO CO vi 4102 | —tT tO — tT CO | EP | tO | tO 10 — | According to this table the principal net V(7, O) can be transformed into the net 3,, either by applying to O and — 7'the operations e, and e, or by applying to 7' and: O the operations e, and e,; these two trans- formations are of the same kind; as they pass into each other by inter- changing the two sets of tetrahedra and at the same time the two sets of four non adjacent faces of each octahedron in contact with them. Whilst each of the two nets 3,,, and 3, can be deduced in one way only, there are three manners of deduction of net 3,7; of these the first stands by itself and the second and the third pass into each other by the indicated interchange of the two sets of tetrahedra, etc. The table for 8, is the following Constituents re: Cs = 0 i ] 2 3 2 3| — 3| — 0 10 PTS 3 On Oa 12. 23 3 Pa 03 1 13 DS 23 3 12.023) 13 sea OP On is: Qe oy Ol 12912231. 03 apn De. Obs Ver- tex Sap C3 Co (1 Ca Cp es Cy Ca ep 4 €, €3 eo C5 Cy Co | Co & C3 Co Cy Ca | €4 © €3 | Co 4 Co | Co Co ep ey eo © Eyres eo ey ep E3 GONE Cues Ces — ee: es Co &4 Co Cy Cy C3 — 0) 64 C3 Cy C2€3 a | €061 E63 Constituents e, es perd Fot C1 Cy |-— Ep C4 — ee) —& e, CN Ep € Cs |—€p Ey Co Boer Sgt tues — €y€1€s| — Co & NAS ei Cy |p yes — 0,0, | Cy ee Cp €4 0x 02 — Cy Ey 60 Cy &y C2 Cy & Cy €3 Cy GB En bi eo — Cx, — Cpe 63 Co &4 Ca C3lC0 C1 C2 C3 Ver- tex Ca _ gap — € | — € — 6 ep Co €3 és". — En Bal —€@ — Ep € ep €3 — C1 — € & Cy Ea CNA — 00163] — E Cg — Ep C2 CA ey Cy Cy C3 €, Co —€ 1,63} Cp 43 — 5 €1 Cal — Cy & C3 ep C4 el Cnty eo, Cy ee: aes a Cy Cy Cp Ca) = DERIVED FROM THE REGULAR POLYTOPES. 65 In the following table for S; we give only the indices of the e-symbols which are to be applied to the constituent of the prin- cipal net in order to obtain all the other ones. Constituents Ver- Constituents Ver- tex tex eden 4 reap) IOT 28 4 gap Il va 0M1| 2) 3) 4 12) 023) 134) 24) 03) 014 BABA ll. 0 14} 02] 013) 124} 23) 034 “ar 23) 4/—| 0) 1" | 84) 04 01) 019] 128) 284 DELE pa pot (8 123) 234) 34) 04] 01] 012 BRS Al |: 6) 1) 2 13} 024] 13] 024) 13] 024 | 1/02/18) 24) 3/04 ai 24 03] 014) 12} 023] 134 V{-4| 0/01/ 12/23] 34 se 124) 28) 034| 14 02] 018 112/93] 34) al ol 01 | 284) 34) 04 01] 019] 123 | 2103114) 2 03/14 { 23] 34] 14) 02] 013) 124 vis 113/24] 3/04] 1/02] *| 134 24 03] 014) 12) 023 | 34| 4! 001/19] 23 12310234) 134) 024) 0130124 3/04/ 1/02/13] 24 124) 0230134) 124) 023/0134 VP }14| 2/03/14} 2103 | XI{ 134) 024) 013/0124) 1230234 23/34) Al 0/01/12 | 234) 034) 014) 012/0123/1234 VII 24] 3/04] 1/02/18 1234) 234) 034) 014) 01210123 XII 1234/0234/0134/0124/0123 We only remark here that the number of ways in which the principal net can be transformed into any other one is equal to the number of different cyclical permutations of the partition symbol of the latter, if we make allowance for the fact that two of these ways may be essentially the same as they pass into each other by interchanging the different positions of the constituents without central symmetry, etc. 34. In the outset of this paragraph (art. 22) we have excluded prismatic nets, restricting ourselves to uniform ones; moreover we have disregarded 1°. all cases in which not all the constituents are of simplex extraction (the hybridous nets of art. 24, 6) and 2°. the nets with two systems of vertices (art. 25, c). Now that our general considerations about simplex nets are come to a close we wish to add a few words about these two exceptional groups of nets. Hybridous nets. In order not to become too circumstantial we only mention the decomposing. symbols of the three plane hybridous nets indicated in art. 24. They are (in the notation of art. 24): Verhand. Kon: Akad. v. Wetensch. (1*t¢ Sectie) Dl. XI. C5 66 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY (4,-+V3)+1, an (11/3) HO, 4(1+V3)+0), Za—0, N (p33 Pas De) (a 43) HAB IO, a(1+v3)+4v3 +1, mi --v3)+4V84+)22 N( pes Pa: ps). (al 4-48 +2, a +43) +1 as (1 + 41/3) +0), Za, = 0. N (pes pas Pa). (4 A+V3)+2, ao (1 4-3) LL 25 (1 HV) +0), Zaj=0, the a; different from each other with respect to mod. 8. _ (a,(2+V3)+1, 40 (2 HVB) +0, a +V3)+0), Za;—0, N(P35—; Pas). - | ((@ +3V3)(2 + V3) + 0, W+ 3V8)(2 + V8) + Le (&L4V3)@ VB ED Za ee In space we find two hybridous nets. If N(4; B) represents a net with the polyhedric constituents 4, B, the first being of body, the second of vertex import, these two nets and their generation are indicated by the equations | eN(T, 0) = N(T, RCO; C), ae N(T, 0) = NT, #00; tC), the stroke under O referring to this that the expansions are to be applied to O. Here we even abstain from mentioning decomposing symbols. Which prospect opens hyperspace for the hunting up of hybridous simplex nets? Very probably none at all. For the most powerful instru- ment in the plane, the operation e,, is quite ineffective in ordinary space already, whilst the two hybridous nets of that space are due to the special character of the octahedron as simplex polyhedron. Nets with two kind of vertices. Neither is it probable that hyperspace contains nets with a constituent occurring in such a manner in two different orientations that any vertex of the net only belongs to one polytope of one of the two sets; for in S, and in 8} the only nets admitting this particularity are precisely hybridous nets, the net NV (ps3, Pr) with respect to pz, the net e, V (7, O) with respect to 7’ and the net e,e,V(Z, O) with respect to 47. 35. We finish this paragraph by mentioning other generations of the nets #, and »,, of the theorems XVIII and XIX. “If we start from a simplex S(z + 1) of S, and complete the n + 1 spaces S,,_, bearing the 7 — 1-dimensional limits $'(x)® to n —+- 1 systems of equidistant parallel spaces #',_,, the distance between SRG OT A DER BEN Vele DERIVED FROM THE REGULAR POLYTOPES. 67 any two adjacent parallel spaces S,,_, being either the height of S(x + 1)° or twice that height, we get either the net 2, or the net a.” “If we intersect a net of measure polytopes J/,,,, of space 8,., by a space 8, normal to a diagonal of a measure polytope and we make that S, to pass either through a vertex or through the cen- tre of an edge of that polytope we generate either a net x, or a net 27. In order to obtain nets #, and »,, with length of edge unity we must start in the first case from a net W (M377), in the second case from a net W(#,K°?).” The first generation is easily proved, if we consider the cases of the triangle net V(p;) and the triangle and hexagon net N(p3, 7%) of the plane and the cases of the net (7, O) and the net V(7, ¢7’) of threedimensional space first. But the second generation, used already in two different papers, 4) has this great advantage that it furnishes at the same time an easy method of deducing the character of the different constituents. We only trace this method here, as the different constituents have been found otherwise already. The generation itself shows that all the constituents are sections of the measure polytope 47,,, by a space S,, normal to a diagonal. In the first of the two papers quoted just now is demonstrated that “the section of #7,,, by a space 8, normal to a diagonal can always be regarded as a part of that space 8, enclosed by two definite, concentric, oppositely orientated, regular simplexes S(n +1) of that space”, i.e. that this section is a “regularly truncated regular simplex’’. Moreover the second of the two papers indicates how to find the amount of these truncations, whilst finally the theorem V, or rather its inversion, teaches how to deduce the zero symbol from the truncation numbers. F. Polarity. 36. If we polarize one of the regular or one of the Archimedian semiregular polyhedra with respect to any concentric sphere, 1. e. if we replace that polyhedron characterized by its vertices by the polyhedron included by the polar planes of these vertices with respect to that sphere, we pass from a body with one kind of vertex and edges of the same length to a body with one kind of face and equal dihedral angles. We suppose the simple laws of this “inversion” to be known; so we state only that the lines bearing the edges of the ") Proceedings of the Academy of Amsterdam, vol. X, pp. 485 and 688. Pa J a 68 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY new body are the reciprocal polars of the lines bearing the edges of the original one, that the vertices of the new body are the poles of the planes bearing the faces of the original one, etc. This definition of reciprocal polyhedra can be extended imme- diately to space #,, where we have to use a concentric spherical — space (with o”~* points) as polarisator. If this concentric spherical space is the circumscribed one, the limiting spaces S,,_, of the new polytope pass through the corresponding vertices of the original one and are normal in these points to the lines joming these points to the centre. We use this most simple disposition in order to show that the property of having one length of edge is transformed into that of the equality of the dispatial angles. To that end we consider (fig. 11) the plane determined by any edge 4B and the centre O of the original polytope and remark that the polar spaces S,,_, of A and B project themselves onto that plane in the lines a and 6, in 4 and B normal to OA and OB respectively; so the space of intersection #',_, of these two spaces #,_, projects itself in the point C common to a and 5 and the angle ACB is the dispatial angle between the two spaces #,_,; but this angle is the supple- ment of the angle 4OB which is constant, OA = OB and AB being constant. By applying this inversion to any semiregular polytope of simplex extraction the characteristic number symbol of it is inverted too. So the symbol (15, 60, 80, 45, 12) of ce, S(6) — see the table — passes into (12, 45, 80, 60, 15). 4) If, in inverting a definite polytope of simplex descent in 8,,, we assume as polarisator the imaginary spherical space for which the vertices of the simplex from which the polytope was derived admit as polar spaces 8,_; the opposite limiting spaces S,,_, of that simplex, and (a, a, .... 4,11) is the coordinate symbol of the *) It is a very good exercise to deduce the limiting bodies of the reciprocal polytopes of S, by polarizing the properties of the edges passing through the vertices of the ori- ginal simplex polytopes. So, if Le, eg e, stands for “the limiting bodies of the reci- procal polytope of e e9 eg S (5)”, if T (13, 1944, 241444) indicates a tetrahedron of which one vertex bears three equal edges, one two equal and one unequal edges, two three different edges, if Pliwo;a means pyramid on a deltoid base, P?,,, double py- ramid on an isosceles triangle as base, Rh rhombohedron, the results to be obtained are represented by the equations Le, = 20 T (15, 3514), Les ez = L (— e eg) = 60 PA acitoiar Le; — 20 Rh, Lee, = 10 P23, Le, Co te 60 Ti (15, 1, 1; 243444) Lee, Co es 30 1 le (4544): DERIVED FROM THE REGULAR POLYTOPES. 69 polytope in true value coordinates, this symbol also represents all the limiting spaces #,_, of the new polytope in space coordi- nates, i.e. that these spaces S,,_, are represented by the equations b, @ H-bom +... LG, jar = 0, where bo, &,...,8,,1 Stands for any permutation of the z + 1 digits a, +) Finally it is easy to see in what manner the process of trun- cation is transformed by inversion. As we have no intention of studying the new system of semiregular polytopes for itself, it may suffice here to remark that truncation at a limit (/),, which implies the determination of the intersection of a definite space 8, _, with the limits (Z),,, passing through that (/),, 1s transformed into the assumption of a point in the line joining the centre of a limit (/),_,-1 Of the new polytope to the centre O of that polytope, which implies that this point is joined to all the limits (/), 5 of that (2),—p—1 by new limits (7),_,_, replacing the chosen one, etc. 37. We now prove the theorem: Tusorem XXII. “Any polytope (P), of simplex descent in S, has the proporty that the vertices /; adjacent to any arbitrary vertex V lie in the same space #,_, normal to the line joming that vertex V to the centre O of the polytope. The system of the spaces S,,_, corresponding in this way to the different vertices 77 of (P), include an other polytope (P),, the reciprocal polar of (P),, with respect to a certain spherical space with O as centre’. In order to prove the theorem we consider the polytope with the zero symbol (4, a, ... 4,4) and in connection with it the linear expression Gy di À da. si Oren This expression assumes the value dy + do +... + ha for the pattern vertex 77 and the same value diminished by unity for each of the points WV, adjacent to 7. For we pass from the ( pattern vertex V to any vertex WV, adjacent to it by making two Lt digits p and p—l interchange places and by this process the sum n +1 PH (p—l)} contained in Xa, is replaced by p(y—l)+(p—l) p | 1 = 2p°—2p. So the coordinates of the points 7; adjacent to the ") Compare “Nieuw Archief voor Wiskunde’, vol IX, p. 188—141. 70 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY n+1 n +1 pattern vertex satisfy the equation Sy “= Sa 1; as this equa- 1 tion represents a space #,_, normal to the line from the centre of the coordinate simplex to the pattern vertex 1), the first part of the theorem is proved. From the regularity of the considered polytope it can be deduced that the distance OP from the centre O to the space 84e ontaining the vertices 7; adjacent to any vertex V does not change with that vertex. So all the vertices of the considered polytope are trans- formed into the spaces #,,, containing their adjacent vertices by means of an inversion with respect to the spherical space with O as centre and |/ OP. OP’ as radius. 38. If we use the symbol 8, (z-+-1) introduced in art. 21 we have: Tnrorrm XXIII. “The two polytopes Cn Oy Cy ee Cp es Oy So (u + ID, Cy Cy Cy ... Er Cy Cp Sy (RD are equal and concentric, but of opposite orientation, if and only if we have generally att{=b+sSet+r=.... =rtess4bSt+d=a—l Por a= a,b = b,c) NES se tope in which the two given ones coincide is central symmetric, if and only if we have aid +s= cree en under which conditions there may be an unpaired middle expansion pes, tor odd”. 2 This theorem gives in analytical form the results published in a joint paper of M Srorr and myself ?), already quoted on page 17, as far as the simplex offspring is concerned; for the supposition that the reciprocal polytopes 4 and 4’ mentioned in art. 3 of that paper are eo S)(w+-1)® and e,_, 8% (n +-1)P, i. e two concentric and equal simplexes #(2+-1)% of opposite orientation, specializes the general results found there to the simplex theorem stated just now here. To prove the latter analytically we have only to write out the result of the operations e,e,e,...e,e.e, and eee. ..e,ese, *) Compare “Nieuw Archief voor Wiskunde”, vol. IX, p. 140, remark I. *) Reciprocity in connection with semiregular polytopes and nets, “Proceedings of the Academy of Amsterdam”, September, 1910, _ DERIVED FROM THE REGULAR POLYTOPES. 71 on S (2+ 1) and to investigate under what circumstances the zero symbol of the one is the inversion of that of the other. If each of the two products Been ets a a Gy yi. Car Ope Oras" s Ope Cy Ori”, where we have Mee VEE KNO OO, OU TE bears / factors, the two results are represented by a +1 b—a c—b S—r t—s n—t and the same expression in which the a, 4, c, .. r,s, ¢ are dashed. So the conditions are atl=a—t,6—a=t—s,c—b=s'—Yr,. .58—r=e—b,t—s=b)—an—t=a+]l, giving immediately ati=b+s—=c+r—=... =r+¢e=s4+0=ttd=r—l. So the first part is proved and the second is deduced from this by suppression of the dashes. In this second part the unpaired middle expansion e,_, occurs, if and only if both # and # are odd. 2 It is an easy task to return to the e and ec symbols referring n to the simplex (1 00 .. 0); to that end we have to omit the e symbol and to add ¢ to any expansion form, where e, is lacking. In doing so we arrive for 2 = 3, 4, 5 by means of the first part of the theorem to all the cases, as & e3 S'(5) — — e, e, S(5), of equal and concentric polytopes of opposite orientation mentioned in the table, and by means of the second part to all the cases, as ce, (6), of central symmetry. 39. In the joint paper of M Srorr and myself quoted in the preceding article, the notion of reciprocal polytopes has been exten- ded to that of reciprocal nets by considering a net of #, as a polytope with an infinite number of limits (7), in S,,,. In this case the centre of the circumscribed spherical space of the polytope lies at infinity in the direction of the normal to the space $, bea- ring the space filling, from which it ensues that the poles of the limits (2), coincide with the centres of these polytopes. So one obtains a net reciprocal to a given one by considering the centres 12 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY of the polytopes of the given net as vertices (see art. 51 of An- DREINI'S memoir, quoted in art. 22). Under what circumstances polarization of a simplex net leads to an other simplex net? The answer to this question is: “this only happens in the plane with the nets N(p3), Nr), WV (ps3, pe) and the for two reasons discarded net (ps3, mo)’. For, if other- wise the net contains two or more different constituents the reci- procal net will contain two or more differents kinds of vertices, and if the net is formed by one constituent only and this self space filler is partially regular the vertices of the new net will be partially regular. So the only possible case of two reciprocal simplex nets is that of the pair V(p;) and W(pe) in the plane, the centres of the two sets of triangles of N(p;) being the vertices of an (y), the centres of the hexagons of V(») being the vertices of an (y). In the treatise “Sulle reti, ecc.” quoted once more above M. ANDREINI has indicated how to draw up a complete list of all the reci- procal nets of threedimensional space ; in this research he comes to the remarkable result (art. 59) that the rhombic dodecahedron and some other less regular polyhedra into which this semiregular polyhedron of the second kind can be decomposed form the constituents of the different reciprocal nets. If we restrict ourselves to the cases concerned with nets of simplex extraction this result is that the constituent of the reciprocal net of N (Tf, O) is the rhombic dodecahedron 2D, NT, tT) » the rhombohedron (+ AD), N(O, CO) , a double pyramid on a square (+ RD), NGT, tO, CO) „ a pyramid on a lozenge (+ RD), N (10) „ à tetrahedron limited by four equal isosceles triangles (ol, RD). What corresponds to this remarkable result in space #,? It goes without saying that this question deserves an answer. But that ans- wer can only be fragmentary, unless we surpass the limits between which we wish to confine ourselves in this paper. So all we can do now is to express the hope that we may be able to give a complete answer to that question in a new paper of its own. Only we cannot retain the remark that the constituent of the reciprocal net of the net corresponding to the undivided partition # + 1 of n + 1 and that of the reciprocal net of the net corresponding to the partition of # + 1 consisting of units only are very interesting . polytopes, worthy of study for their own sake. DERIVED FROM THE REGULAR POLYTOPES. 13 G. Symmetry, considerations of the theory of groups, regularity. 40. We begin by determining the spaces S,,_, of symmetry which may be indicated by 87,_, and we consider to that end successively the case of the simplex S(z + 1) of S, and that of any polytope (P), deduced from that simplex S'(x + 1) by the operations of expansion and contraction. | Case of the simplex. The vertices of S(z + 1) lying outside a space of symmetry Sy,_, of this S(x + 1) occur in couples. Now there must be at least one of these couples, as Sy,,, cannot contain all the vertices of S(z + 1), and on the other hand there cannot be more than one of these couples, as S(z-+ 1) does not admit parallel edges. So any space Sy,_, must bisect orthogonally one edge of S(z-+- 1), i.e. the number of spaces 87,4 is Ja (n + 1). It is not at all difficult to indicate the equations of the 4x (u + 1) spaces Sy, , of (1 00..0). For the space S,_, bisecting normally the edge 4, 4, joining the points 4, and A, with the coordinates dre), 0) and e= 1; #57, — 0) 1s’ represented -by the equation: #,: — a. Case of the polytope (P), deduced from the simplex. It goes without saying that the + x (x 1) spaces of symmetry a, = 2, of S(n 1) are at the same time spaces Sy,_, for any polytope (P), derived from that S(z-+-1) by the operations e and c, and that any two limits of that (P), which are each others mirror images with respect to any of these Sy,_, are of the same import. So the only question is, if the polytope (P), can possess a space of sym- metry which is no Sy, , for the S(z-+1) from which the (P), has been derived. To answer this question we suppose there 7s such a space Sy, , and we examine the consequences to which this supposition leads. According to this supposition (P), is its own mirror image with respect to that definite Sy,,_,, which may be represented by the symbol Sy,,_,, whilst the mirror image of the simplex S'(z + 1) from which (P), has been deduced is an other simplex S’ (u + 1) concentric to S'(2 + 1). But then the figure consisting of (P), on one hand and the two simplexes S(#—+-1), S°(2 + 1) on the other is symmetric with respect to Sy, 4; so it must be possible to de- duce (P), by the same set of expansion operations from the new simplex (Jl). From this we can draw two conclusions, one with respect to the two simplexes, an other with respect to (P),. It we can deduce the same polytope (P), from two different simplexes, these simplexes must be concentric and oppositely orientated; 1 we 14 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY can do so by means of the same set of expansions, (P), must be central symmetric. So we have to solve first the new question if the simplex S(z-++-1) admits a space S,,_, reflecting it into a con- centric simplex S'(z-+-1) oppositely orientated to S@-+-1). We once more suppose that there is such a space S,,_, and we examine the consequences of this supposition. Let 4,4,....4, be the given simplex and 4,'4,’.... A, the concentric simplex of opposite orien- tation, the common centre O being at the same time centre of the n+.1 segments 4,4;, (¢ = 1, 2,...,2-+1) Then the image of 4, wit respect to #,_, must be either 4’, or one of the other verti- ces of S' (nl), say 4°, We consider these two different cases each for itself. If 4, is the mirror image of 4,, the space S,,_, is normal to the line 4,0 joining in S(z-+ i), if produced, the vertex 4, with the centre 47, of the opposite limit S(#), which line 4,47, may be called a “first transversal’? of S (a+ 1); this Sz) with the ver- tices 4, 4,...4,,1 18 contained in a space S,_, parallel to Sn whilst the mirror image of it is the limit S’(z) of S (u + 1) oppo- site to A, with the vertices A, 4’s...., 4,44. 90, as a mio these S'(2) and JS’ (x) have to be at the same time equipollent and opposi- tely orientated to each other, equipollent as reflections of figures lying in spaces #,_, and S',_4 parallel to the mirror S,,_,, oppo- sitely orientated as corresponding parts of the oppositely orientated simplexes S'(7<+ 1) and S' (+1). This is impossible for 2 > 2, e. g. two triangles lying in parallel planes cannot be equipollent and oppositely orientated at the same time. Now the case „ = 1, meaning- less in itself, leads to two conciding simplexes, i. e. to a point of symmetry of the simplex of the linear domain, the line segment. So the casen=—=?2 of the triangle 4, 4, 4, with the lines through O parallel to the sides is the only remaining one. If A’, (fig. 12) is the mirror image of 4,, we consider the tri- angle A, À, 4, with a right angle in 4,, as the centre O of A, 4’, is at equal distance from the three vertices; if M,, is the centre of 4, 4», the line MO, parallel to 4, 4’,, passes, if produced, through the centre of the limit $(x— 1) of S'(x + 1) containing 43, 4,..., 4,44 as vertices and may therefore be called a “second transversal”? of S'(x + 1). Now the mirror 8, , bisects 4 4, orthogonally and is therefore normal to the second transversal 47,4, 0, while the limits S(u — 1) with the vertices 4, 4,,...,4,,, and S'(#—1) with the vertices A3A,...A,.4, lie in parallel spaces #,_, and #°,_.. Here too these limits have to be at the same time equipollent and oppositely orientated to each other, which is impossible for 7 — 1 > 2. So we find here DERIVED FROM THE REGULAR POLYTOPES. 19 two cases, the case » — 2 found above and the case x= 3 of the tetrahedron 4, 4, 4, A, with the planes through O parallel to a pair of opposite edges. So we have proved the general theorem: Nn Tasorem XXIV. “The simplex (1 00 .. 0) of S, and the poly- topes deduced from it by expansion and contraction admit + n(r +1) spaces Sy,_, of symmetry, the spaces 2 = >,. Moreover in the plane the e (3) admits the three new axes of symmetry 2, = ÿ of the hexagon, whilst in space the ce, 7'= O, e, T— CO, e,e,T =t0 admit the new planes of symmetry 2; + 2, — x, + x, of the octa- hedron”’. 41. We now prove the following theorem *): Trrorem XXV. “The order of the group of anallagmatic dis- placements of the simplex S(z~-+ 1) of #, and of the polytopes deduced from it by expansion and contraction is $(a-— 1)!” “The order of the extended group of anallagmatic displacements of these polytopes, reflexions with respect to spaces Sy,_, of sym- metry included, is (x + 1)! In this extended group the first group of order $(z-+-1)! forms a perfect subgroup”. COTE 2==2 and 2 — 3 these general results have to be Com- pleted in the generally known way”. The simplest proof of this theorem is connected with the remark that reflexion of the polytopes with respect to any space Sy, _, corresponds to the interchanging of any pair of vertices of the sim- plex. So the order of the group of reflexions (and anallagmatic displacements) is equal to the number of permutations of the 2 + 1 vertices of S(u +1), Le. (w+ 1)!, and the group of the anallag- matic displacements is of an order half as large, i.e. of order }iv#-+-1)! For the cases 7 —2? and n—8 we refer to F. Krer's “Vor- lesungen über das Ikosaeder’’ (Leipsic, ‘Teubner, 1884). 42. The manner in which the polytopes considered here have been derived from the simplex is a guarantee that all the vertices are of the same kind and all the edges have the same length. But this is all that can be asserted; so e.g. the polyhedron #7 has two kinds -of edges, edges common to two hexagons lying in planes including a definite acute angle and edges common to a hexagon and a triangle lying in planes including the obtuse supplementary angle. So in judging of the regularity we have to look at the edges from two different points of view; we must not only take into account the length but also consider angles on or faces through the edges, etc. *) Compare Report of the British Association, 1894, p. 563. 76 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY In his dissertation — which is about to appear — M. E. L. Erre has created an artificial system !) according to which it is possible to count the degree of regularity of the partially regular polytopes deduced from the regular polytopes by regular truncation. In this system the regularity of such a polytope is expressed by a fraction, the denominator of which is equal to the number of dimensions, while each group of limiting elements as vertices, edges, faces, etc. may contribute a unit to the numerator. With the exception of the group of vertices”) every group of limiting elements has this unit subdivided into two halves, one half for equality of form, the other half for equality of position with respect to the surroundings; moreover only successive contributions count, begimning at the vertices. So in the case of #7 the contributions of vertices and edges are 1, + and l+3 ; if aan the degree of regularity 1 Sg ee and this is the case with all the Archimedian semiregular Bee except CO and ZD, where the dihedral angles on the edges are equal and the degree of regu- J EE eee larity is ie eee | Of the two halves corresponding to equality of form and to equality of position with respect to the surroundings the first needs no expla- nation, while the second may seem rather difficult to grasp. But this second half also will become clear, if we indicate it as follows. Equality of vertices means that the figures formed by the systems of edges concurring in the different vertices (vertex polyangles) are congruent, equality of edges means that the edges have the same length (first +) and that the figures formed by the systems of intersecting lines of the faces passing through the different edges with spaces S,_, normal to the edges (edge polyangles) are congruent (second À, equality of faces means that the faces are congruent (first +) and that the figures formed by the systems of intersecting lines of the limiting threedimensional spaces passing through the faces with spaces #,_, normal to the faces (face polyangles) are congruent (second 3), etc. So we will be able to determine the regularity fraction of a given polytope derived from the simplex in the scale of M. Exrs, if we have found the different subgroups of each of the limiting ‘) We can only give a glimpse of the system here. For more particulars we must refer to the dissertation written in English. *) If we count from the other side (see the next page) we must say: “with the excep- tion of the group of limits (l)n—1”, etc. DERIVED FROM THE REGULAR POLYTOPES. tt elements (2), (Us. (U) 4. So this research is closely related with theorem III of art. 10 which enables us to find the subgroups of the same system of limiting elements /, characterized by different symbols, the more so as we have the theorem : Tasorem XXVI. “Any two limiting elements of the same group (7), belong to the same subgroup or to different subgroups, in the sense of the scale of regularity, according to their zero symbols being equal or different, if we consider two different zero symbols of a central symmetrie polytope as being equal when they pass into each other by inversion.” This theorem is nearly self evident. A rigid proof of it can be based on the consideration of the limits (/),_, passing through the (Dy. So in the case of the form (321100) treated in art. 11 the different unextended edge symbols (32), (21), (10) correspond to subgroups of edges with different positions in relation to the sur- roundings. For, if we consider the four groups (82110), (321)(100), (32) (1100), (21100) of limiting polytopes it is immediately evident that the second group distinguishes (10) from the others, that the fourth group distinguishes (32) from the others, whilst the third group alone shows already that no two of the three subgroups of edges can be equal. We remarked above that we count the contributions to the numerator of the regularity fraction beginning at the vertices and taking in only successive contributions. But the case may present itself that a polytope derived from the simplex shows also some regularity at the side of the limiting elements (/),,_, of the highest number of dimensions. We then indicate two fractions of regularity, one for each side, as will be shown in an example in the next article. The fifth column of Table I contains the regularity fraction of the different forms obtained in the cases x — 3, 4, 5, only counted from the vertex side. In the fourth column the subscripts indicate the numbers of the different subgroups of each limiting element (/),. 43. We elucidate the theory by applying it to several examples: a). Example (321100). Here we find three different groups of edges. So the vertices contribute 1, the edges contribute 4 to the 2 1 numerator and the fraction is À EE = oe 6). Example (110000). This form has only one kind of edge (10) but two subgroups (110) and (100) of triangular faces. So we find ep et : es 9 : 18 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY c). Evample (111000), This central symmetric form has one kind of edge (10), one kind of face (110) = -— (100), but two sub- groups (1110) = — (1000) = 7' and (1100) = O of limiting bodies and once more one kind of limiting polytopes (11100) = — (11000). 3 1 So we fi Zi o we find en : Remark. The degree of regularity of the polytopes of S,, found 1+4 | ; 3 here is at least ———= — — and therefore for xz — 3 at least + ñ In So the Archimedian polyhedra of the stereometry are semiregular in the right sense of the word, if we take semiregular to mean that the degree of regularity is } at least but less than unity. 44. As the scale used for the determination of the regularity is independent from the number of vertices, edges, faces, etc. of the polytope, the same method may be applied to nets of polytopes, by considering a net in #, as as polytope limited by an infinite number of limits (2), in #,,,. This new application depends only on the problem how to determine the different kinds of vertices, edges,‘ faces; ete: ofthe: met: All the nets considered here have vertices of the same kind and edges of the same length. So for a net in $, the fraction of regu- 1 TURN HET ad 1) number of cases in which a constituent of the nets admits two or more ad shaped faces we have only the choice between 9 larity is at least So in the most frequent aA and 2 5 D of which the first value corresponds to the case of only one kind of edge, the second to that of two or more differents kinds of edges. In order to make the determination of the fraction of regularity of the nets in S, and S; as easy as possible we enumerate in Table IIT the different limits (/),, (/3, (2), of the nets in S, and the different limits (/);, (/),, (2)3, (2), of the nets in #5. In the part corresponding to #— 4 we find under the seven headings I, IE, .., VIT the subdivisions 4, 3, 2 standing for (7), (2);, (Z),, in the part corresponding to # = 5 likewise under I, If,..,XIT the subdivisions 5, 4, 3, 2 standing for (/), (Us, (2), (2). These limits are indicated in abridged notation: under 5 the symbols 1, ce,, ce,, etc. denote S(6), ceyS(6), ce,S(6), etc.; under 4 the symbols 1, ce, etc. signify S(5), ce,S(5), etc. ‘The results of Table IIT are inscribed in Table IT in the sixth column — Pe ae pins mais a É DERIVED FROM THE REGULAR POLYTOPES. 79 under the headings (/), (2h, ...,(7);; so the number 11 on the lines of the nets 5,;*, 5,” under (/), indicates that in these equal nets of space S; the constituents admit together eleven differently shaped limits (2), What is taken from Table III — and what is self evident — is incribed in small type. The other numbers — inscribed in heavy type —, of which only two correspond to faces, have been found separately. We treat here two of these cases in detail. Case An. Here the constituent (11000) has only one kind of edge. Does this imply that the zet has only one kind of edge? The example of the net 3, where the CO admits also only one kind of edge, whilst Anprurnt rightly mentions the fact (see his treatise , p. 32 under n°. 21) that of the five edges concurring in a vertex one is common to 247 and 240 and each of the four others to tT, 10, CO, must prevent us from jumping too rashly to this con- clusion. So we investigate this point and examine if, e.g. in the case of the constituent (21100) with two kinds of edges, (21)100 and 21(10)0 these two edges are different with respect to the net or not. So we enumerate first the different limits (/), to which the vertex 21100 is common. They are (oe gist ) (2 Es ) (2, Lens SUR La, 2) PROD] geen pale © 1 Hat nd 2) ( ) ( ) ( + 2) ( fm, fh jet jt | 2 + vo 2, 1 —2+3,—24+2,—2+ Oan t 8,948, — 242 2,—2+3,—24+3,—2+2, rae 4 sr yet 4, — 0. (2, — 2+3,—4+4, a Starting from (21100) we have indicated in this list of ten polytopes first the two polytopes deduced from (21100) by varying the form of one of the digits 1, 1, 0, 0, then the only polytope obtained by varying two of the digits, etc., see the curved brackets and the numbers 1, 2, 3, 4 at the right. As we can augment all the interchangeable parts by the same integer provided that we diminish all the unmovable parts by the same amount, we find in this manner all the polytopes to which the chosen vertex 21100 is common, though we leave the first digit 2 alone. If we denote the ten polytopessor thelist by (Dr CP),,...%,CP io 80 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY we find that the edge (21) 100 is common to (P),(P)2,(P)s, (Ph, Ps and the edge 21(10)0 to (P),, (P)3, AP) (Pz, (P)3. So both char are common to 36, Ce, és. But this fact is not yet decisive, as the possibility exists that the grouping of the sets of five polytopes around the edges (21) and (10) is different. In order to decide this point we draw up the following table of threedimensional contact, where 1, 2,..., 10 stand for (P),, (P),,..., (Po and contact by a prism is indicated by a small asterisk. | ads 3 4 5 & 6 7 | 8 | 9 a 2 | 1 1 | 9 1* | 1 3 | 2 4 ide AS 285 NON soe eye DR PS a ec 6 | 6*| 6*| 6* 6* | -6* |} 6* | 6 4 8 7 7 à soer dl ie A € A, 7 972 POT PTO ee ge | O* 10% | | | ee: This table shows that if we arrange each of the two sets of five polytopes as follows in three groups Hinge aa rarer Beke PEG fee P3, P ie Eds LT, Pe each polytope is in bodily contact with the polytopes of the other groups of its horizontal row, whilst two polytopes in the same column are equal. So there is no difference whatever in the threedimen- sional contact, 1.e. there is only one kind of edge Case 5%. Here the point 321000 is common to the 17 polytopes De ten Ey Zhan 1. OO cer ee. (3, poe 1 ,—3+8,...0...: Of zee. (9; Bidt 1 ,—3+5—3+3 0 )3 a RU NAN: À LÉ I ,—3+3, -——3 +53, —3 +3)" CSM LP 2 ,—3+4,—3+3,—3+3,—3-+8) 1 a....3—3+6,—814 3138 ee,...(0,—3+9,—3+4,—3+3,—3+3,—6+6)/8 ce, 02e. (d, —3-+9,—3+4,—3+3,—6+6,—56-+ 6) 3 —¢é.,.(6,—38-+5,—3+4,—6+6,—6+6,—6-+6)/ 1 DERIVED FROM THE REGULAR POLYTOPES. 81 In this list we have availed ourselves of the occurrence of the three zeros in order to represent the 17 polytopes by nine symbols. So the second line represents three different polytopes which can be obtained by putting seccessively — 3 + 9 under each of the three zeros. For each line the number at the right indicates how many different polytopes through the point 321000 correspond to that line. This list shows that we may find here two kinds of edges though we have three groups, edges (32) 1000 common to 9 polytopes, edges 3(21)000 common to 16 polytopes, edges 32(10)00 common to 9 polytopes, as in the first and the last group the sets of 9 polytopes are both 4 e,¢,, 8 ce ee, e, e, By investigating the fourdimensional contact between the nine polytopes of each set can be found whether the edges (32) and (10) belong to the same kind or not. From the numbers of differently shaped limits the fraction of regularity has been deduced; it is given in the last column of Table - IT. 45. We finish this part of our memoir concerned with the off- spring of the simplex by a remark about what may be called the “circumpolytope’ of a net. This polytope, which has for vertices the vertices of the net joined by edges to any arbitrarily chosen vertex of the net, is by its form a criterion for the regularity of the net. If the net admits one kind of edge the circumpolytope must admit one kind of vertex, etc. This circumpolytope is in the cases of the threedimensional nets: a OO: Sy -.. a prismoid limited by two equilateral and six equal isosceles triangles, 3n7--- à prismoid limited by two squares and eight equal isosceles triangles, Sy ... à pyramid on a rectangular base, 3, ... a tetrahedron limited by four equal isosceles triangles ; of these five polyhedra only the fourth has vertices of two diffe- rent kinds. The theories developed here enable us to find the circum- polytopes corresponding to the different nets of simplex extraction in S, and S;. But instead of deducing these polytopes here we conclude by the following general problem, for the proof of which we refer to the dissertation of M'. Kirn: SE M re - SL 4 Eet PE te nr 3 sare: erf Ade À 2 À ah , gage à EX : MISE Ne Re + + Tee dots ee ; ne ia 5 ‘ i 4 pe : oe $ on F À war 8 FE à EE br. je 10 i Ne” = a ¢ À ee | Sr : Pee Soe | ä ; a © . ca $ ; © S ml ES Cr RS D vi je « $33 : fab) . met en das DST pe : + 8 1e A) oe a à ñ rs ' ue fe © . 2 Rd yi ; = St a. ï ME on En pan! Fe ON en 3 2 Bee on” Mo F nt I “eu D e 4, d 1 à last 1 7, line DC) SAR OA. PH.SCHOUTE, Analytical treatment of the polytopes regularly derived from the regular polytopes. Ad GB Z CV \ ee RN PUM, Fig 7. xx AY XX WA, 4 WA ES Fig 9. is Verhand.Kan. Akad v. Wetensch. 1° Sectie, DLAT N°3. JBytel ith. PJ Milde mon L ED LIST OF POLYTOPES DEDUCED FROM THE SIMPLEX. Table I. D= 4 4 6 | 4 DO (1000) | 0 CAN) | Pa = = 1 e S(4)— tT (2100) | 2 OEE) i De = D. : 1 ey S(4)= CO] *(2110) | 3 (12, 24, 14,) 2 Ds p 2 AA es qe, SLO | SRO | 3 (24, 36, 143) | Ds en ONE ce e S(4)= 0 | *(1100) | 4 COR a8") Pa eg NET >| 1 cs I= A 5 10 10 5 S (5) (10000) | 0 ey Mh ET 1 iP — Ok 5 1 1 1 Es 2 l e 8 (5) (21000) | 2 CPU 20s = 805 a0 3 (7 D = p Sala e> 8 (5) (21100) | 2 80: 90-807 205) 3 00 — i 0 Ang sers DES gene i a 3 8 3 Ezi ION en DOC 10, 8, | 7 P p ip SE I cs ec S(5) | (82100) 60, 120, 80, 20, î 10 es P, iT BIL ea e, e3 S (5) (82110) | & 60, 150, 120, 30,) 3 tT P. P. CO Heat e» es 8 (5) ee 8(5) | (32210) | 3 GO 150, 120, 30, à CO P, Pp, 7 Pole hey et e es 9(5) (43210) | À (120, 240, 150, 30, : 10 P. D. 10 el à ce 86) | (11000) | À (10, 30, 80; 10) 3 0 Be : 7 le : ce, ¢y S(5)| *(22100) | 4 30, 60, 40, 10,) 4 tT = = iT bale sel Dd 6 15 20 ae 6 8 (6) (LO0000) | OIC ser 015; 2 208 15, 26) 0 8 (5) = | = 1 e, S(6 (210000) BN (80h 75,7 80,2 45, 125) | e 8(5)| - | 8(5) | 3] 1 e, 86) (211000) 271 60; 240, 290; 135," 27.) 123, 8, SO) — — Pe || ee SOL ex S(6 211100) | # | (60, 270, 420, 255, 47,) |Z, ON D Pp ce SOME Saeed ca 8(6) | *(211110) Zi (30, 120, 210, 180, 62,) | 2 S(O) ey ELN '8(5) |6| 4,3,2,1 cs e,‚ €» S(6 (821000) 101207 300; 290, 135, 278) a CUS (D |e Pr IDEE er ¢ 8 (6) 821100) | 1 | (180, 630, 720, 315, 41) | aes — | (:8) | Po ss) neue ee, S(6) (821110) | F | (120, 420, 570, 380, 62) | a SO) Pp 1-65 8) er e, (5) | 8 | 5,3,2,1 | =e e, S(6) ex ey (6 822100) (180, 540, 570, 255, 47,) ce SH - | (838) | Pin | ene SOINS | 5e ee eze, 8(6) | *(822T10) | 8 | (180, 720, 900, 420, 62) |H Oe NE ETON EON Ci cs. ex e4 S(6 (822210) | # | (120, 420, 570, 380, 62,) | 3; a See Pre 5 e, S(5) |1O| 7,5,3,1 | =e, e, S(6) e, Cy ex 8(6 452100) | # | (860, 000, 810, 916, 41) llaees@)l — | @:6) | 2 | ee 8G) (10) su oe a ee, (6 (432110) || (860, 1080, 1140, 480, 62.) |-%| a 286) Po | (6:8) | Pao ee SOIT 7,42) 1 | ee, e, 806) RER 86) | *(482210) | 41] (360, 1080, 1080, 420, 62) || ee S(5)| Pr | (6:6 NEO en el bn u 8O | (488210) | 2 | (60 1080, 1140, 480, 62) || ee 8(5)| Pao | (6:8) | P ae 8(5) |13| 9,6,3,1 | —e ee 86) 5 Je f LS 9 14 ec on : re aye ste 0 2 es É LA] > > == 2 ù ec “4 a Caen en En on ate de wa a Peebos Loa ee, es 5 5) |15| 10, 6, 8,1 es. zn 8 al nee EEn 2 EGY EERE =" 5 (5 1 oe ao (6) conne 6 | (20, 90, 120, 60, 12,) | 5 | —ce SH — — = Ceres jee ad c.s Ge. 6,.5'(6) (221000) | & | ( 60, 150, 140, 60, 12) || ca & SG — — — HO aay rl ee, €3 8(6) | *(221100) | & | ( 90, 360, 420, 180, 32) | 2 e, 8(5)| — — = e, S(5) | 6] 4, 2,1 c. 8 ce ee, 8(6) | *(832100) Ht O80) 450 00 2180 82.) |e eee SOI — = = a So) GRE i LIST OF NETS DEDUCED FROM THE SIMPLEX. Table IT. n= 2 CONONONONONG: 1[8 | (a Es de) | (100) = pz (110) = —p, | | 1 1| 1 Jil, 2 (Qa, + 1, 2a, , 2a ) (100) = p, (2110) == 7 (110) SSS #5 1 3{1,1,1 | (Ba, + 2, Bag + 1, 8a, ) | (210) = “pe | : | n —= 3 1 |: (ae 5 CE NCA) (MOOD) = 72 QUO) == (0) (110) = — 7 | Ta he ake 2) 2 |1, 3 (2a, + 1, 2a, , 243 , Za) | O0) = 7: C100) ee | (2210) = — #7 | (1110) = — T | Wa Mele ay 2 12,2 (Qa, = 12a, 4- I 2a, , 2a,) | (1100) = * O NO) == Of) | | Il Bay 2 Olle ke 22 (8a, 2,82 10843 ‚ 344) | (2100) — #7 (3210) — * #0 (2210) = — #7 (MALO) = COM JN S2 A33 AN did (4a, + 8, 4a, + 2, 4a, + 1, 4a,) (3210) — * 40 | Mt) Si n — 4 115 (a, Ay os 0 (1) (10000) = e | (OO Oppen | (11100) = —e, (11110) = — e, We EP ij 2 2/1, 4 (2a, + 1, 2a, , 243 , 244 , 245 (10000) = + | (21000) = ee, OO ENEN (22210) = — ee, HALO) WE a) AS) 212,3 (2a, 1, 24, 15245 ‚ 2a, EA | (11000) — e COO) RENES | (22110) = — @ 63 (11106) = — e, (CAMMY) es “= ee 5 | JMS a) SiS | (8a, + 2, 3a, + 1, 3a, ‚da, ‚ ds) | É 21000) = ee, | (82100) = ee, & | (88210) = — ae eg | (22210) = — ee, (21110) = * qe, | Wy Pall Shi ey) 8) sil, 2, 2 (Ba, + 2, 3a, + 1, 3a, + 1, 8a, , da, (21100) = &e } (82110) = Qn Gs | COO ENE (82210) =— ene, e, | (22110) = — é @ Ma) ai) Bh] 5 8 ADL Al al (4a, + 3, 4a, + 2, das + 1, 4a, ‚da, (82100) re "ee, (43:21 0) = 5, ey C5 Es (SS LONEN Ge Gy ELO = a. Gs (32110) = ERGE = Nee Bh) BB by Hil, Wy il, ak al (5a, 4, bay 3, 5a, + 2, 5a, + 1, 54, (43210) = * e, €, € 5 Le ey) a 221 ae D= 1 [6 (a, Tr 5 GR ua, NU , ){(100000) = e, (110000) = e, (111000) = * e, (111100) =—e, (111110) =—e, | TY TES PRICES Q 11,5 (2a, + 1, 2a, , 243 ed ‚Zas ‚ 2a) (100000) = e, | 210000) = ee, (221000) = Gee (222100) =—e, & (222210) = —e,e, (111110) = —e, WE Bt 2 19, 4 (2a, + 1, 2a, + 1, 2a, , 2a, , Zas , 2ag) (110000) = e, | 21600) ==" en es (221100 = * @ € (222110) =— @e, (111100) = —e, (ALTO) Ee 1/1) 2) 4) 5) 4 ZIS (2a, Il 2e 1, 2a3 1, 2a, » Jas 2de) (11000) =e, (211100) = ee, (221110) == ee 1) 1) 2) 3] 4] 2 sil, 1, 4 (Ba, + 2, 3a, + 1, 8a, ‚ day ‚ 34; , 34)(210000) = ee, 3.25000) SMEER LS (332100 OR CED ANT (2222110) —— ene; (ODO) EEEN Wi 2 385)) 40) saa oH, Bs, 33 (Ba, 2, 3a, 1, 8a, 1h Be ‚dd , 34)(211000) = € & 22100) = "ere: (832110) eee 222100) = — e, €3 322210) See ey \(221110) =— ene, GX MEN ANAL 6 ol, 3, 2 Gr an ele ard Bee RG , 8&)(211100) = ez (821110) ==" ee, Er (221000) = ee (3822100) "eee 332210) =—e ee, (222110) =— ae, 1) 3| 3] 6/11 6 9112, 122 2 Ga SP 07 Le ESS , 84) (221100) = *e, e, (BOSE ere; | 18 BANEN ANS RI? ANNI Il (da, + 3, Aa, + 2, 4a, + 1, Aa, ‚da, ‚4a,)(821000) = ege, ey ASN ONSEN EEE (443210 = ee 6, 63 (888210) Set B) 322210) eee, (821110) = eee, INR SSA ONES NS Ali 1 2202 (da, 3, da, 2, das 1, da, + 1, 4a; , dag) |(321100) = eee ets ol Cy el Cn Cx (332100) = eee (433210) ere e, ep|(802210) = — sE Ea (322 MORE ve, JSS ZEAL teh it Ac, 2) 1. 2 (da, + 3, Aa, + 2, 4a, + 2, Aa, + 1, 4a, , 4ag) (322100) = e, ee, | 0) — 08 eze, (832110) = —e, ee, | LOR PSE tl (5a, 4, Bas 3, Das 2, 0a, + 1, 5a; ‚ Sag) (432100) = ege, eye, (54 ne Kee; € 03 Cal 148210) = — e, e, & €34 (433210) = 1 Oy Cg 4482210) = Ge. CE 10) = aaawc|l1|8|383|6|8|4 Gil AE le de Ala TGZ Sp 2 Oi SP Bs (Wek A 0) (543210) —*e,e, ey es ey ee hel eo ee Ko col nop po} loc 5 te, ol re Sel a ae ci apo cu = oro ole le oe le ble ole ple ph el con cn cof nolo LIMITS OF CONSTITUENTS OF NETS. Table III. 5 4 3 2 5 4 3 2 5 4 3 2 5 4 2 D 4 3 2 5 4 3 2 1 1 Jp Pa ] gl Ps Gey Ce (0) Pa Ce» Ger O Ds e Cj tle 7 ey e> CO Pa Ger GIE O Gi e tT’ Ps Cs J T Da es @5 if Pa € €» 1 11 Pa €, €3 Pr JB Pa Ce» Gener SCE REIN OG is ey CO (Ges) |) LEE Bleep Ca), Or tO Ds Cy C5 ce, O Ps | es De Je Po Chap eer JE OG Gy RENES JE | (33) (8: 3) ae, |(6:3)| 4 | ea Jas Jor | (3; 3) | Puy GENE) 6 | É VII VIII IX Xx XI XII 5 4 3 2 5 4 3 2 5 es 2 dD 4 3 2 5 4 3 2 5 4 3 2 BO GG) CO Ds Cres ete» tO Ps Cres ln Cue, uw Pa Ge VE Co ar Heep ex eneen LO Pa (Er Cp €s ule nea) LO Pa Gn 10 LE Da enk Le Ps ens ENEN 0e AOF) Be Pa Eeen (859) Ba Pa le; 90304 (8:6) Jen Pa Dio Te Pe (3;3) | O GN Gi iP Po [ace] Po Ps Ye Per Pe Pe [cel Per Ps Ye (6; 6) P, Len € 6563 fr C5 eg Co CO genen | ee Gey Cie, Cp tT" (8 ; 6) Je Oe O | (6; 6) Po Bi | De IB HO) (GCAO) GO | ez JP co 72) 4 | iy co | | (3 ; 3) | ei €3 H. DUTILH 4 Sa Chem. Docts. — rhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam. ee (EERSTE SECTIE). | | Fee Fe DEEL XI. N°. 4. | ge AMSTERDAM, JOHANNES MULLER. a Februari 1912. j THEORETISGHE EN EXPERIMENTELLE ONDERZOEKINGEN — OVER PARTIEELE RACEMIE Ee BN B PE mn th Chem. Docts. AMSTERDAM, JOHANNES MÜLLER. For? Theoretische en Experimenteele Onderzoekingen over Partiéele Racemie DOOR He DEE EEE + Chem. Docts. VOORWOORD: Den 17° Januari 1910 overleed te Utrecht mijn vriend Herman Dur. Kort te voren had Prof. van Romgurex in de Koninklijke Akademie van Wetenschappen te Amsterdam een mededeeling aan- geboden, waarin Durirx voorloopig wees op enkele resultaten van een onderzoek, waarmede hij sinds enkele jaren bezig was en waarop hij weldra tot doctor in de scheikunde hoopte te promoveeren als einde van een studietijd, die met „cum laude” afgelegd candidaats- en doctoraalexamen, zijn leermeesters en zijn studiemakkers zooveel van hem deed verwachten. Het onderzoek was zoo goed als afgesloten en de hoofdstukken IT en III, ongeveer in den vorm, waarin ze hier volgen, geschreven. Daar Durrm en ik sinds jaren gewoon waren dagelijks onze werkzaamheden samen te bespreken, kende ik, althans in hoofdtrekken, zijn opvattingen en toen na zijn dood allen, die hem kenden, gevoelden, dat de vrucht van zooveel arbeid niet voor de wetenschap verloren mocht gaan, verzocht Prof. van Rom- BURGH mij te trachten uit Durrun’s journaal, zijn aanteekeningen en mijn herinnering het ongeschreven gedeelte aan te vullen. Ik heb dat gaarne gedaan. Maar nu het voltooid is gevoel ik sterk, en acht mij verplicht dat hier voorop te stellen, hoe enorm veel beter deze verhandeling geweest zou zijn, indien hij zelf haar geschreven had. Moge, wie het hier volgende leest, en zich een denkbeeld over Durum wil vormen, daarmede rekening houden. 7 ve 4 VOORWOORD. De bedoeling van den gestorvene was een hoofdstuk vooraf te doen gaan over de splitsing van racematen in het algemeen en daarvan een aan eigen, oorspronkelijk inzicht getoetst overzicht te geven. Helaas was daarvan nog niets uitgewerkt op papier gebracht en kende ik zijn inzichten niet voldoende om ook maar eenigszins zijn bedoeling met dat hoofdstuk naar waarde weer te geven. Ik heb er dan ook geheel van afgezien. Wat de positieve conclusies betreft ben ik vrij zeker Durrrs inzichten aangaande de strychnine-tartraten weergegeven te hebben. Ten opzichte der zure brucine-tartraten hebben wij wel vaak daar- over gesproken, zooals ik het hier heb neergeschreven, maar ik durf niet absoluut zeker zeggen of hij, indien hij de zaken zelf neer- geschreven had, er zich evenzoo over uitgesproken zou hebben. _ Moge ten slotte het geheel zoo zijn geworden, dat het de na- gedachtenis van dezen betreurden vriend ter eere zij. H. R. KROM September 1910. HOOFDSTUK I. INLEIDING. Alvorens de theoretische kwesties te bespreken, die het hier volgende onderzoek zullen beheerschen, komt het ons gewenscht voor een en ander vooraf te doen gaan aangaande de soort stoffen, wier eigenschappen hier bestudeerd zijn en de bedoelingen , die wij met dit onderzoek hadden. De partiëele racematen zijn verbindingen waaraan weinig aandacht zou zijn besteed, indien ze niet op merkwaardige wijze een hinder- paal waren geweest bij de bereiding van andere, zeer belangrijke verbindingen. Hun optreden is nl. beletsel tot de uitvoerbaarheid van een der methoden, door Pastrur ontdekt, om racematen, ver- bindingen resp. mengsels van optische antipoden, in hun compo- nenten te splitsen. Twee verbindingen, wier moleculairconfiguraties slechts voor een deel spiegelbeelden, voor een ander deel congruente vormen zijn (die nochtans niet met hun spiegelbeeld tot bedekking zijn te brengen), zoodat dus de geheele moleculairconfiguraties noch spiegelbeeiden noch congruente vormen zijn, — vertoonen verschil in al hunne physische eigenschappen. Van dat feit maakte Pasreur gebruik bij een zijner splitsingsmethodes. Men kwam echter tot de ervaring, dat deze methode faalde, zoodra, behalve die stoffen met half symetrische constitutie, ook nog hun verbinding optrad. Heeft men zoo b.v. een racemaat 4, 4, (een racemisch zuur b.v.) en laat men daarop inwerken een optisch actieve stof B,, die in staat is met A een verbinding aan te gaan (een optisch actieve base dus b.v), dan kunnen zich 4, B, en A, B, vormen, welke stoffen noch identieke, noch spiegelbeeld-isomere moleculairconfigu- raties hebben. Deze lichamen hebben verschillende oplosbaarheid, zoodat bij partiëele stolling dus in het algemeen de minst oplosbare zal uitkristalliseeren; door splitsing der 42 verbinding zal dan een optisch actieve 4 verkregen zijn. Deze methode faalt natuurlijk in het algemeen als behalve de verbindingen 4B ook nog een verbinding van het type 4, 4,2 B, 6 THEORETISCHE EN EXPERIMENTEELE ONDERZOEKINGEN ENZ. uit de oplossing uit kan kristalliseeren. Aangezien deze verbinding opgebouwd te denken is uit een racemisch gedeelte 4, 4, en een optisch-actief gedeelte (2)B heeft men zulk een verbinding een partieel racemaat genoemd, een naam over welks juistheid overigens te twisten valt. Met het oog op de belangrijkheid, zoowel van louter chemisch als van biologisch standpunt, die het vraagstuk der racemaatsplitsing ongetwijfeld toegekend moet worden, als ook om het interessante van het vraagstuk uit physisch chemisch oogpunt, komt het er op aan de stabiliteitscondities dezer partiëele racematen te kennen. In de onderzochte voorbeelden bleek er nl., evenals bij de dubbelzouten en de racematen vaak voorkomt, een temperatuur te bestaan, waar- boven, resp. waarbeneden, het partiëele racemaat bij zich instellend evenwicht in zijn componenten uiteen valt; deze temperatuur geeft dan de grens aan, waar de methode van Pasteur toe te passen is. De eerste onderzoekers der partiëele racemie (wij noemen LADENBURG) hebben dan ook dadelijk het gewicht der kennis van deze overgangs- punten ingezien en er experimenteel naar gezocht. Hun resultaten worden verder in deze verhandeling uitvoerig besproken. Op deze onderzoekingen is nu spoedig van theoretische zijde een kritiek van principieelen aard gekomen, nl. van H. W. Barauis RoozreBoom, wiens verhandelingen evenzeer hieronder een nadere bespreking zal geworden. De uitkomsten van LADENBURG’s experi- menten verschilden van de door zijn theorie geeischte waarden niet meer dan hij proeffouten zijner methode acht, terwijl van geen andere zijde experimenteel materiaal aangebracht werd. Wij hebben ons daarom ten doel gesteld LADENBURG's experimenten in eenige gevallen te controleeren en zoomogelijk de verhoudingen in een of meer systemen met zoo groot mogelijke zekerheid bloot te leggen. Allereerst komt het er echter op aan scherp in te zien, wat hier eigenlijk aan de hand is, wat de physisch chemische functies van zulk een overgangspunt zijn, hoe en aan welke functies de kennis omtrent de ligging van zulk een overgangspunt nagegaan kan worden; in het bizonder zullen deze conclusies dan op systemen met racemaat resp. partiëel-racemaatvorming moeten toegepast worden (Hoofdstuk IT). Dan rijst de vraag naar een scherpe begripsstelling voor de partiéele raceme, waarbij de historie van het ontstaan dezer bestreden term niet kan gemist worden (Hoofdstuk IID). Daarna zullen wij de uit- komsten onzer onderzoekingen mededeelen (Hoofdstuk IV en V) en ten slotte de resultaten overzien. HOOFDSTUK IE OVER OVERGANGSPUNTEN. Het is een overbekend ‘feit, dat latere onderzoekers na PASTEUR meermalen te vergeefs getracht hebben, om druivenzuur via het natrium-ammoniumzout langs den weg der spontane kristallisatie in d- en /-wijnsteenzuur te scheiden. Zoo verkreeg Sranper D in 1878 aanvankelijk steeds goed ontwikkelde kristallen uit het monokliene stelsel, waaraan geen hemiëdrische vlakken konden worden waar- genomen en wier oplossing optisch inactief was. Eerst uit de moeder- loog scheidden zich de rhombische kristallen der beide wijnsteenzure zouten van de formule C, H, O, Na NH. Aag af. Srarpen kon dit verschijnsel niet verklaren. Reeds tevoren waren dergelijke waar- nemingen gedaan. Zoo gaf Mrrsenrrrien *) in 1842 op, dat hij door vermengen der oplossing van neutraal druivenzuur-natrium met eene oplossing van iets meer dan één molecuul neutraal kalium- racemaat een in het trikliene stelsel kristalliseerend kalium-natrium racemaat heeft verkregen, dat verschilde van het Seignettezout. Fresenius ©) vond verder, dat ook zonder overmaat van kalium- racemaat dit trikliene zout met vier moleculen kristalwater, beneden 8° echter met 3 moleculen water wordt verkregen, en Dutrrs +) beschrijft zelfs een dergelijk zout met Ad ag, terwijl daartegenover staat, dat volgens Pasteur, wat door RammxrsBerG bevestigd is geworden, in dergelijke omstandigheden eene splitsing van het druivenzuur onder afscheiding van de dubbelzouten van d-wijnsteen- zuur, naast die van /-wijnsteenzuur optreedt. Scaccut °) heeft het vraagstuk opgelost door waar te nemen, dat, wanneer de kristallisatie bij verhoogde temperatuur plaats heeft, *) Ber. d. D. chem. Ges. 11, 1752 (1878). *) Pogg. Ann. 57, 484 (1842). *) Lieb. Ann. 53, 230 (1845). *) Pogg. Ann. 81, 304 (1850). *) Rend. dell’ Acad. di Napoli 1865, 250. 8 THEORETISCHE EN EXPERIMENTEELE ONDERZOEKINGEN zich het dubbelracemaat (C, H, O, Na NH). 2 Hy O afzet, terwijl bij gewone temperatuur echter de beide tartraten C, HW, O, Na NH, 4H, O uit de oplossing kristalliseeren. In eene serie van uitgebreide onderzoekingen gelukte het later aan Wyrougorr !) te constateeren, dat de beslissende temperatuur in dit geval 28° bedraagt, zoodat, wanneer oververzadigingsver- schijnselen vermeden worden, boven 28° het racemaat, beneden 28° de beide tartraten zich uit de oplossing afzetten. Die temperatuur van 28° draagt in dit systeem den naam van overgangstemperatuur. Het zal wenschelijk zijn ter behandeling van het verschijnsel der partiéele racemie in dit hoofdstuk en de volgende, de begrippen overgangspunt en overgangstemperatuur van phasen- theoretisch standpunt nader te bespreken. Overgangspunten zijn door Baxuuts Roozesoom *) gedefinieerd als punten, waar in stelsels van 2 componenten (x + 2) phasen kunnen coexisteeren. Zij zijn gekenmerkt door de samenkomst van (w + 2) curven, die elk op zichzelf het monovariante evenwicht van (7 + 1) phasen onderlmg aangeven. Het eenvoudigste geval van een overgangspunt doet zich voor in één-component-stelsels als het bekende tripelpunt, waar, bij Fig.1 een bepaalde temperatuur en een bepaalden druk, vast, vloei- stof, en gas naast elkaar in nonvariant evenwicht kunnen bestaan. Overal, waar in systemen oo van één component drie pha- sen kunnen coexisteeren, treft men zulke overgangspunten aan. Er kunnen zich in der- gelijke systemen dus nog andere overgangspunten voor- doen, n.l. het evenwichtspunt voor ÿ, Dp L: Si 5, G ; S, 5 A enz. 3) + Dat inderdaad dergelijke punten overgangspunten zijn, dat er dus ") Bull. Soc. Chim. 41, 210 (1884). *) Zeitschr. f. physik. Chem. 2, 474, (1888). ) Met S worden vaste phasen aangeduid; L en G geven de vloeistof- resp. gas- phase aan. ‘) Verg. Baxnuis Roozenoom, Die heterogenen Gleichgew. enz. Ien Heft, Braunsch- weig (1901). pele B ‘ aS eer eee OVER PARTIBELE RACEMIE. 9 een overgang (misschien beter omzetting) plaatsgrijpt bij warmte- toevoer of onttrekking van warmte, blijkt wel hieruit, dat men in het geval van het gewone tripelpunt fig. 1, by constant volumen uitgaande van een punt der lijn 7 waar de vaste phase naast de gasphase kan bestaan, bij toevoer van warmte, deze lijn in de rich- ting naar O zal volgen; in O gekomen treedt nu naast de phase S+-G, de vloeistofphase Z op. Gaat de smelting met uitzetting gepaard, zooals in fig. 1 wordt verondersteld, en zal de druk con- stant blijven, dan moet een deel der gasphase verdwijnen, derhalve S-+ G@— L. Omgekeerd, door onttrekking van warmte in O grijpt plaats de reactie Z— S + G. lets dergelijks doet zich voor in de andere tripelpunten, die in stelsels van één component kunnen optreden; hunne beteekenis is aldus samen te vatten : Tripelpunten, d. z. overgangspunten in stelsels van één component, zijn de combinaties van een bepaalden druk met een bepaalde temperatuur, waarbij drie phasen van één component kunnen coexisteeren. In dergelijke punten komen drie curven van mono- variant evenwicht samen, overeenkomende met de drie systemen van twee phasen, die uit de drie phasen kunnen worden gevormd. Bij toe- of afvoer van warmte heeft in die punten door het verdwijnen van één der phasen eene omzetting plaats, waaraan alle phasen deelnemen. In de eene richting verdwijnt steeds ééne be- paalde phase, voor welke het punt dan ook in waarheid het overgangspunt is; in tegengestelde richting daarentegen verdwijnt één der beide andere phasen, afhankelijk van beider hoeveelheid en van het volumen van het systeem. Het tripelpunt is eene overgangstemperatuur in die richting, waarin slechts ééne curve is gelegen, en voor die phase, wier bestaansgebied tusschen beide andere curven zich uitstrekt. Meer samengesteld worden de verschijnselen daar, waar in binaire stelsels het quadrupelpunt als overgangspunt optreedt. Het quadrupel- punt ontstaat door de samenkomst van twee evenwichtscurven voor drie phasen van twee stoffen. In dat punt zijn dus 4 phasen met elkaar in evenwicht, zoodat dit punt tevens het snijpunt moet zijn voor nog 2 andere curven van monovariant evenwicht tusschen drie phasen onderling. Beginnen wij ook hier weer met het eenvoudigste geval, dan wordt onze aandacht gevraagd voor de coëxistentie van de phasen S, Sn £ G, wier bestaan naast elkaar alleen mogelijk is in het 1. g. eutecticum onder dampdruk. Daar men in binaire stelsels, in tegenstelling met unaire, drie vrijheidsgraden heeft n.l. de tempe- ratuur 4, den druk », en de concentratie >, kunnen hier het over- 10 THEORETISCHE EN EXPERIMENTEELE ONDERZOEKINGEN gangspunt en de curven, die er in samenkomen slechts als eene projectie, b.v. op een p/-vlak gegeven worden, zoodat in die projectie de curven van monovariant evenwicht (drie phasenlijnen) geen inzicht kunnen geven omtrent de verandering van æ met p en £. Eene dergelijke projectie geeft fig. 2. Gaan wij uit van een punt der driephasenlijn IT waarop de beide vaste componenten met hun gemeenschappelijken damp in evenwicht zijn, dan zal deze lijn bij warmtetoevoer in de richting naar O worden doorloopen. In O treedt de phase Z op, doordat A en B smelten tot Z. Stel, deze smelting gaat met uitzetting gepaard; de transformatie in O is dan (v-constant) #, + SJ GS L O is minimum temperatuur voor de phase Z. Bij warmte- onttrekking gaat men omge- keerd van elk der curven JV, T en JL op ZIT over; daarbij verdwijnt Z en ontstaat één der phasen #,, Sp, of G. _ In tegengestelde richting zal, afhankelijk van het volu- men, dat. het systeem in- neemt, S,,° Sp 2010 dwijnen, waardoor men van HI op 7, JI ot Tews gaat. Het punt O is dus weer overgangspunt voor de phase Z bij warmteonttrekking onder constanten druk. In fig. 2 is echter niet de eenige mogelijke situatie van het overgangspunt voor S,, Sz, Z en G geteekend. Ik volsta met op te merken, dat een curve als // ook terugloopend kan zijn; in dat geval heeft de omzetting S, + S,—> Z onder contractie plaats en daardoor wordt de transtormatie in O: S, +. Sp L + G. Als dit zoo is, dan kan men uit O naar hoogere en naar lagere temperatuur telkens op 2 curven overgaan, al naarmate de eene of de andere phase verdwijnt. O is dan niet een overgangstemperatuur voor één enkele phase,-maar wel in elke richting voor een systeem van 2 phasen. Andere quadrupelpunten kunnen in binaire stelsels ontstaan door de coexistentie van b.v. S, JB L, Ly; 8, 6 L, Los Sa 19455 G enz Vatten wij deze beschouwing over quadrupelpunten samen, dan resulteert: Hen quadrupelpunt geeft in binaire stelsels de eenige waarden van p, ¢ en æ aan, waarbij 4 phasen uit twee componenten OVER PARTIÈELE RACEMIE. Li in evenwicht kunnen zijn; in dat punt komen 4 curven samen overeenkomstig de 4 systemen van mono-variant evenwicht, die mogelijk zijn. Warmteonttrekking of warmtetoevoer heeft in een quadrupelpunt eene omzetting tengevolge, waaraan alle aanwezige phasen deelnemen. Is één der phasen verbruikt, dan gaat men uit het overgangspunt op één der curven over. Nu kan de vergelijking, die de omzetting in het overgangspunt uitdrukt aan beide zijden twee phasen, of één en drie phasen bevatten. In het eerste geval kan in beide richtingen één van twee phasen verdwijnen in het laatste geval verdwijnt in de eene richting één bepaalde phase, in de andere één der drie overige; welke, hangt van het volumen . van het systeem af. | In dit laatste geval is het quadrupelpunt weer een waar over- gangspunt voor die ééne phase. Ingewikkelder nog zijn de verschijnselen in ternaire systemen. Hier geeft het overgangspunt de waarden van druk, temperatuur en concentraties aan, waarbij 5 phasen naast elkaar in evenwicht kunnen bestaan. Daar ik nu bij het hier volgend onderzoek uit- sluitend oplosbaarheidsverschijnselen heb bestudeerd van partieel- racemische verbindingen, welke eene volkomen analogie moeten vertoonen met de dubbelzouten uit de anorganische en de organische chemie, zooals b.v. astrakaniet en calcium-koperacetaat, zal ik hier alleen het quintupelpunt beschouwen, waarbij evenwicht heerscht tusschen de beide vaste enkelzouten, de oplossing en de gasphase. Konden wij in binaire stelsels het quadrupelpunt en de curven, die in dat punt elkaar ontmoeten geven als eene »{ projectie der 4 curven voor mono-varlant evenwicht met hun snijpunt, zooals die op het p {ez oppervlak in binaire stelsels zijn gelegen, thans krijgen wij één variabele meer. Onafhankelijk veranderlijk zijn nu niet alleen p, ¢ en +, maar p, t, æ en y, waarin met # en y bedoeld zijn de concentraties aan elk der beide enkelzouten. Hunne evenwichtsvoorwaarden zouden op dezelfde wijze voortgaande dus in de ruimte slechts met behulp van 4 onderling loodrechte assen voor te stellen zijn, waartoe men de 4 dimensionale meetkunde zou moeten te hulp roepen. Echter zijn de 5 curven van mono-variant evenwicht, die thans in het overgangspunt samen komen, ook in eene pf figuur te demonstreeren, waarbij men dan te bedenken heeft, dat langs die curven twee concentraties voortdurend veranderen. Al naar gelang nu een dubbelzout naast oplossing zijn bestaansgebied naar hoogere temperaturen of naar lagere uitstrekt, wordt de voorstelling van het quintupelpunt door fig. 3 of door fig. 4 weergegeven. 12 THEORETISCHE EN EXPERIMENTEELE ONDERZOEKINGEN Duiden wi) de enkelzouten met 4 en B, het dubbelzout met Das de oplossing met Z en de gasphase met G aan. Het is nu mogelijk, dat aan weerszijden van het overgangspunt twee en drie, of één en vier curven zijn gelegen. Na hetgeen voorafgegaan is bij de binaire stelsels zal het duidelijk zijn, dat alleen in het laatste geval het quintupelpunt een overgangspunt voor ééne enkele phase nl. voor D kan zijn. Fig. 8 heeft betrekking op astrakaniet, (-pipecolinebitartraat, hydrochinaldinebitartraat en dergelijke dubbelverbindingen, die eerst bij temperaturen, hooger dan het overgangspunt gelegen, naast één „hunner bestanddeelen, oplossing en gas stabiel zijn. Wij zullen nu in de eerste plaats de curven om het punt O beschouwen. De lijn JZ geeft het evenwicht aan tusschen de beide enkelzouten, de oplossing en de damp. Warmtetoevoer bij constanten druk voert ons van de lijn JZ in het veld tusschen 7 en JV, waarin de aan beide curven gemeenzame phasen i.c. 4, B en G zijn gelegen. Derhalve LS AJ B+ G. In het veld links van curve Z bestaan naast elkaar óf 4+ B+-L, óf d44+-G-+-L, of BGL, in wier gebieden men komt al naarmate bij de warmteonttrekking onder constanten druk de omzetting 4 + B + G— Z het verdwijnen van G, B of A ten gevolge heeft. Dit wordt beheerscht zoowel door het totaal volumen van het systeem, als door de onderlinge verhouding in de quantiteiten der componenten. De lijn // is de aaneenschakeling van evenwichten in het phasen- complex bestaande uit beide enkelzouten, dubbelzout en gas. Druk- verlaging bij constante temperatuur of temperatuursverhooging bij OVER PARTIBELE RACEMIE. — 13 constanten druk voert ons in het veld van B DG en 4D G, der- halve wordt de omzetting aangegeven door 4+ Bz D-+-G. De lijn V geeft het evenwicht aan tusschen de beide enkelzouten, dubbelzout en oplossing. Hierbij is de omzetting door temperatuurs- verandering 4+ Be DL. De lijn Z7 is de evenwichtscurve voor één der enkelzouten, dubbel- zout, oplossing en damp. Hier is de omzetting Z + 4— D+G, wat weer af te leiden is uit de phasen, die in de velden links en rechts van deze curve kunnen bestaan. Evenzoo is curve 7/77 de lijn, die de evenwichten aangeeft tus- schen het andere der beide enkelzouten, dubbelzout, oplossing en gas. De omzetting wordt hierop voorgesteld door L> B+ DG. Wat is nu de transformatie in OP In het quintupelpunt zijn naast elkaar in evenwicht 4, B, D, L en G; bij warmtetoevoer (v-constant) zal men overgaan op één der curven /7, LI of V en dus zal B, A of G verdwijnen. De omzetting zal dus zijn: 44 B4 GDL en zij zal ten einde zijn gekomen, als 4, B of G is opgeteerd en men van uit O op één der curven, aan de rechter zijde daarvan, zal kunnen overgaan. Omgekeerd zal by afkoeling D of Z verdwijnen en daardoor gaat men òf op de curven J òf ZV over. Hieruit volgt, dat O niet een overgangstemperatuur voor één enkele phase is; wel in de ééne richting voor een systeem van twee phasen en in de andere richting voor een systeem van drie phasen. Beschouwt men de systemen, die alleen tusschen / en // en tusschen VY en Z// kunnen bestaan, dan vindt men 4 BG resp. DL BDL en DLG. Voor 4 BG is O een maximumtemperatuur, voor 4 + Z daar- entegen een minimumtemperatuur. Voor het dubbelzout D is echter het punt O geen minimum- temperatuur, immers 2) kan naast zijne samenstellende bestanddeelen A en B met de gasphase G (curve ///) ook bij lagere temperaturen bestaan. Geheel anders is het verloop der curven van mono-variant even- wicht in fig. 4, die het gedrag van caliumkoperacetaat, neutrale druivenzure strychnine, zure druivenzure brucine enz. weergeeft. Thans heeft men in het temperatuursgebied, beneden O gelegen, 2 curven nl. voor de evenwichten 4 D LG en BDLG, terwijl zich van O uit naar hoogere temperaturen de curve voor de phasen AB LG uitstrekt. De curve /V is geheel in overeenstemming en vergelijkbaar met 14 THEORETISCHE EN EXPERIMENTEELE ONDERZOEKINGEN de gelijknamige curve in fig. 3. Curve V geeft de evenwichten der drie zouten naast hunne smelt bij verschillende temperaturen en drukken. Zij is in het geval van het caliumkoperacetaat .experi- menteel bepaald en terugloopend gevonden, doordat de omzetting _ D A+ BL, welke bij constanten druk door temperatuurs- verhooging plaats grijpt, eene contractie tengevolge heeft. Bij de bovengenoemde druivenzure zouten is zij niet langs proefondervinde- lijken weg vastgesteld, echter is daar haar verloop zeer waarschijnlijk eveneens in overeenstemming met fig. 4. In deze figuur loopen dus van O uit 4 curven naar lagere temperatuur en ééne naar hoogere. Daardoor wordt in O de trans- b" hb” b b formatie uitgedrukt door het symbool D> 4+ B LG, wat tot het onmiddellijke gevolg leidt, dat het punt O in waarheid eene overgangstemperatuur voor het dubbelzout is en wel de maximum- temperatuur, waarbij dit zout kan bestaan. Uit bovenstaande volgt: het quintupelpunt, zooals dat zich bij waterhoudende dubbelzouten in het invariante evenwicht 4 B DLG voordoet, 1s slechts een overgangspunt voor het dubbelzout, wanneer dit hooger gehydrateerd is dan de beide componenten samen, en zijne omzetting in de beide componenten en oplossing onder con- tractie plaats heeft. In alle overige gevallen kan het dubbelzout bij temperaturen zoowel hooger als lager dan die van het quintupel- punt bestaan. OVER PARTIÉELE RACEMIE. 15 Gaan wij nu in plaats van p en f als assen te gebruiken, over op een diagram, waarin æ en 7, (de concentraties van elk der beide enkelzouten in eene constante hoeveelheid van de 3 compo- nents samen) als coördinaten zijn gekozen, dan krijgen we in overeenstemming met de figuren 3 en 4 de figuren 5 en 6 4). Deze figuren gelden voor constanten druk en zijn eene aaneen- schakeling van doorsneden, bij een bepaalde temperatuur, van het ptegy oppervlak, dat de evenwichtscondities in ternaire stelsels aangeeft. Legt men de temperatuur vast, dan zal men voor die constante p en ¢ op genoemd oppervlak een lijn krijgen als voor- b’ gesteld is door acd in figuur 5. Het punt a geeft de oplosbaarheid *) Er zÿ hier de volgende opmerking ingelascht. Sinds Durin's dood is het derde deel der Heterogene Gleichgewichte van de hand van Prof. ScnREINEMAKERS verschenen. Daarin is in volkomen algemeenheid nagegaan welke gevallen mogelijk zijn en daarbij zijn meer mogelijkheden voor den dag gekomen, dan hier behandeld zijn. De twee typen der fig. 3 en 4 zijn dan ook niet de eenig denkbare en aan de fig. 5 en 6 moet nog een derde toegevoegd, die men bij SonREINEMAKERS l.c. pag. 164 als Fig. 76 vindt aangegeven. Of de behandeling van dat type in dit verband van belang zou zijn, wil ik hier in het midden laten. Maar in elk geval scheen mij een uitbreiding van dit hoofdstuk (mede in overleg met de commissie uit de Akademie) niet gewenscht en bepaal ik mij hier dus tot de verwijzing naar genoemd werk. Kort voor Duriin’s dood heeft hij met Prof. ScnRrrEINEMAKERsS overleg gepleegd over deze zaken en waarschijnlijk heeft slechts zijn onverwachte dood hem belet dit hoofdstuk in overeenstemming met die besprekingen te wijzigen. Kruvr. 16 THEORETISCHE EN EXPERIMENTEELE ONDERZOEKINGEN in eene constante hoeveelheid oplossing van den component 4 aan bij de temperatuur ¢ en 1 atmosfeerdruk. Op dezelfde wijze wijst het punt 4 de oplosbaarheid van de component B onder dezelfde _ omstandigheden aan. De lijn ae stelt nu de verandering voor van de oplosbaarheid van A bij toevoeging van B aan de oplossing. - Ditzelfde geldt voor de lijn 4e, wat betreft de oplosbaarheid van B in oplossingen, die reeds 4 bevatten. Eindelijk geeft punt c het gehalte aan 4 en B aan van de met beide zouten verzadigde oplossing. Kan echter bij eene temperatuur f, (zie fig. 5) het dubbelzout — onontleed naast zijne verzadigde oplossing bestaan, dan zal de op- losbaarheidsisotherm voor die temperatuur uit drie takken gevormd worden. Op. de lijn a” f" is de component 4 „Bodenkörper”, op den tak fe” de dubbelverbinding, op de lijn e” 6” de component B. d'” geeft de oplosbaarheid der dubbelverbinding aan, als we aannemen, dat J eene aequimoleculaire verbinding van 4 en B is. Bij temperaturen hooger dan #, blijft voorloopig de isotherm uit drie takken bestaan. Hoe hooger echter de temperatuur wordt, des te kleiner wordt de oplosbaarheidseurve van de dubbelverbinding en des te meer breiden zich die der enkelzouten uit, totdat bij eene temperatuur # de oplosbaarheidscurve der dubbelverbinding tot een enkel punt O, is ineengekrompen. In dit punt O, en by die temperatuur 4, kan dus blijkbaar eene oplossing tegelijk ver- zadigd zijn aan A, B en D. Wij hebben hier dan vier phasen van drie componenten, de druk is vastgelegd en dus is het evenwicht nonvariant. — Dit punt O, is strikt genomen niet hetzelfde als het punt O in fig. 4; het komt overeen met het snijpunt van curve V in die figuur met de lijn voor p — 1 atm., welk snijpunt in temperatuur zeer dicht bij O gelegen zal zijn. 4) De oplossing, wier gehalte aan 4 en B door de coördinaten van het punt O, in fig. 5 wordt bepaald, kan derhalve verkregen worden, door bij de temperatuur ¢, hetzij 4<+ D, hetzij B + D, hetzij A+B tezamen tot verzadigde vloeistof op te lossen. — Merkwaardig is nog de isotherm der temperatuur Z. Hier snijden de lijnen a’ d" en d”e” elkaar juist op de lijn, die den assenhoek midden doordeelt, zoodat de oplossing, die door d” wordt aan- gegeven, verkregen kan worden, door bij 7, het oplosmiddel aan D of aan DH 4 te verzadigen. Tot deze temperatuur toe kan het dubbelzout verzadigde oplossingen geven, waarin de verhouding der ‘) Men zou misschien, ter vermijding van misverstand een punt als O, in fig. 5 en fig. 6 het overgangspunt bij 1 atmosfeer druk kunnen noemen. ein Vka anti OVER PARTIÈELE RACEMIE. ry gehalten aan 4 en B met de samenstelling van D overeenkomt (zooals d bij #, enz.). Boven die temperatuur is D niet meer naast zijne oplossing stabiel, maar lost op onder gelijktijdige afzetting van de minst oplosbare zijner componenten. ') — Verzadigt men der- halve bij temperaturen boven 4, het oplosmiddel met het dubbelzout, dan krijgt men niet eene oplossing, wier innerlijke samenstelling met een punt der lijn OD overeenkomt, maar eene oplossing, waarvan het gehalte aan 4 en aan B wordt weergegeven door een punt der lijn d O,. Alle punten der lijn O, 4 geven dus bij ver- schillende temperaturen de aan d + D verzadigde oplossingen aan. Ditzelfde geldt voor de lijn O, B met betrekking tot de aan B + D verzadigde oplossingen. Hindelijk stellen de punten der lijn O, Z de by verschillende temperaturen aan 4 + B verzadigde oplossingen voor. Vertragingsverschijnselen uitgesloten, zijn dus in de figuren 5 en 6, alleen de getrokken lijnen realiseerbaar. Heeft echter de vorming van het dubbelzout uit zijne beide componenten, resp. zijne ontleding, onder afzetting van de minst oplosbare zijner componenten naast zijne waterige oplossing bij temperaturen, waar die oplossing niet meer stabiel is, uiterst langzaam plaats, dan zal het duidelijk zijn, dat dus ook de verlengingen van de lijn #O,, m. a. w. punten als Berend. en de voortzetting van Od”-d.7. dé puntend , d enz. te verwezenlijken zijn. Dit doet zich bij partiëele racematen als b.v. druivenzure strych- nine inderdaad somtijds voor. Im het temperatuurgebied, waar het partiëele racemaat ,,Bodenkorper’ kan zijn van oplossingen met wisselende samenstellingen ?), kunnen nochtans door vertraging in de vorming der dubbelverbinding uit hare componenten de oplos- baarheidscurven af", ad en be”, be“ resp. tot hunne snij- punten ec’ en c° worden gerealiseerd en evenzoo kunnen punten der lijnen d D (fig. 5) en Od’ (fig. 6) worden bepaald, hoewel deze eigenlijk moesten gelegen zijn op de lijnen d O,. Deze figuren ondergaan geen groote verandering, indien men in plaats van op eene constante hoeveelheid oplossing, + en y betrekt op eene constante hoeveelheid oplosmiddel, als men tenminste, zooals bij de strychnine- en brucinezouten der beide wijnsteenzuren en hunne dubbelverbindingen (1. c. de partiëele racematen), te doen heeft met verbindingen, die eene kleine oplosbaarheid in water bezitten en men hun gedrag in een dergelijk oplosmiddel bestudeert. *) Hierbij is natuurlijk ondersteld, dat het snijpunt der oplosbaarheidstakken ligt aan de zijde van de concentratieas der meest oplosbare componente, zooals in normale gevallen ook steeds het geval is. *) Men vergelijke de isothermen voor de temperaturen ¢, en f, in fig. 5. Verhand. Kon. Akad. v. Wetensch. (ste Sectie) Dl, XI. D 2 18 THEORETISCHE EN EXPERIMENTEELE ONDERZOBKINGEN Ten slotte Zi] nog in verband met fig. 5 en 6 opgemerkt, dat het temperatuursgebied gelegen tusschen het overgangspunt en de hoogste resp. laagste temperatuur, waarbij het dubbelzout naast zijne verzadigde oplossing kan bestaan, (welk gebied in fig. 5 resp. fig. 6 is gelegen tusschen de temperaturen #4 en #), den naam heeft verkregen van overgangstraject. 1) Partiëel-racemische verbindingen als drnivenduresatrychninet die zich naast hunne verzadigde waterige oplossing van zekere tempera- tuur af kunnen splitsen in d-wijnsteenzure-strychnine + /-wijnsteen- zure-strychnine en wier componenten in alle physische eigenschappen verschillen, zijn volkomen vergelijkbaar in de stabiliteitsverschijnselen naast hunne verzadigde oplossingen in water, met anorganische dubbelzouten als het astrakaniet, Va, Mg (S 0). 4 H, O, dat uiteen kan vallen in Va, SO,. 10 H, O en MgSO, TH, O onder opname van water *). Eenvoudiger worden de oplosbaarheidscurven, wanneer men te doen heeft met eigenlijke racematen als natrium ammonium- racemaat, dat zich beneden 27° splitst in d-natrium ammonium tartraat + /-natrium ammonium tartraat. Hier zijn de componenten, behalve in kristalvorm en in hun gedrag ten opzichte van het ge- polariseerde licht, in alle physische eigenschappen en dus ook in hunne oplosbaarheid in hetzelfde medium aan elkaar gelijk. Dit heeft tot gevolg, dat wanneer men hunne oplosbaarheidscurven in een a-y diagram teekent, deze geheel symmetrisch ten opzichte van de lijn, die de samenstelling van het racemaat aangeeft, zullen zijn. Dit is in de figuren 7 en 8 weergegeven. *) „Umwandlungsintervall”. Verg. Mrveruorren. Zeitschr. f. phys. Chemie 5, 97 (1891). *) Vgl. Baxuuis Roozesoom, Zeitschr. f. phys. Chem. 28, 494 (1899). OVER PARTIÉELE RACEMIE. 19 Deze figuren staan in dezelfde betrekking tot elkaar als fig. 5 en fig. 6. In figuur 7 is het geval voorgesteld, dat het racemaat !) bij hoogere temperaturen zich naast zijne verzadigde oplossing in zijne beide componenten splitst, terwijl in fig. S is ondersteld, dat het racemaat juist eerst bij hoogere temperaturen 1 evenwicht han zijn met zuivere racemaat-oplossingen. Im beide figuren snijden de oplosbaarheidscurven van de d en de /-verbinding in het gebied, waar het conglomeraat (d.1. 1 + ZL) bestendig is, elkaar steeds op de lijn OD, die weer aequimoleculaire hoeveelheden van D en Z in de oplossing aangeeft. Door toevoeging van D aan de verzadigde oplossing van Z wordt de oplosbaarheid van dit lichaam op gelijke wijze veranderd als bij toevoeging van Z aan de verzadigde D-op- — lossing de oplosbaarheid van dit zout gewijzigd wordt. Dit heeft tot gevolg, dat de geheele figuur verkregen kan worden door spiegeling van de helft x O0 D op de lijn OD. Alle (D + £) oplossingen liggen dus op de. lijn OD. Bij de temperatuur #, wordt in beide figuren het racemaat naast zijne verzadigde oplossing bestendig. De overigens nog geheel meta- stabiele oplosbaarheidscurve van het racemaat gaat hier dus door O, en daarom stelt O weer de samenstelling eener oplossing voor, die verkregen kan worden door by #, het oplosmiddel hetzij met D + Z, met D+ LR, of met Z +R te verzadigen. Bi de temperaturen 4, 4 enz. kan het racemaat met oplossingen van verschillende gehalte aan 4 en B in evenwicht zijn en deze samenstellingen zijn begrensd bij elke temperatuur door twee punten, resp. der lijnen O, 4 en O, B. De oplosbaarheidscurve van het racemaat is ook weer geheel symmetrisch ten opzichte van de lijn OD, daar de invloed van tevoren opgelost D of Z op de oplosbaar- heid van & dezelfde zal zijn. Het essentieele verschil in oplosbaar- heidsverschijnselen van eigenlijke racematen met partieel-racemische verbindingen en alle andere dubbelzouten is dus hierin gelegen, dat de oplosbaarheidscurven der racematen volkomen symmetrisch zullen zijn ten opzichte van de lijn, die met hunne samenstelling overeenkomt (1. c. de lijn OD), terwijl de oplosbaarheidscurven der laatsten asymmetrisch moeten verloopen. Daardoor kon bij racematen een ontledingstraject niet bestaan; daarentegen moet dit bij partiëele racematen en alle andere dubbelverbindingen steeds voorkomen. Hoewel door LADENBURrG en zijne leerlingen tal van oplosbaarheids- bepalingen bij partieel-racemische verbindingen zijn uitgevoerd, met *) Wij zullen in het vervolg de beide componenten kortweg door D en L en de verbinding van beide door R aangeven. DX a 20 THEORETISCHE EN EXPERIMENTEELE ONDERZOEKINGEN de bedoeling om daardoor het overgangspunt vast te stellen, meen ik, dat zij dit laatste niet voldoende in het oog hebben gehouden. Zij hebben steeds het overgangspunt van het partiëele racemaat allereerst geconstateerd als de temperatuur, bij welke de oplosbaar- heid van de heide componenten samen gelijk werd aan die van het racemaat alleen. Dit is juist, wanneer men met de eigenlijke race- maten te doen heeft, (wat in het bovenstaande is aangetoond). Betreft het onderzoek echter partieel-racemische dubbelverbindingen, dan moet men figuur 5 of figuur 6 beschouwen, en dan blijkt daaruit, dat zij dus punten als 4”, d’, d’, d met de punten als ce”, c’, O,, c hebben vergeleken. Weliswaar hebben zij, wanneer zij meenden het overgangspunt te hebben gevonden, de oplosbaarheid van BJ D en van + Z bij die temperatuur eveneens bepaald, en deze dan gelijk aan die van #& en van D + Z bevonden, zoodat dus (R-+ D), (HL) en (DHL) in oplosbaarheid aan elkaar gelijk waren. Dit is de voorwaarde voor het overgangspunt; maar een louter toeval blijft de gelijkheid der oplosbaarheid van Z aan die der splitsingsproducten bij die temperatuur en principieel onjuist is het, het overgangspunt te gaan zoeken, door de snijding te bepalen van de lijnen, die de oplosbaarheid van Z en van D + Z, in hare afhankelijkheid van de temperatuur, aangeven. Nog een andere voorstellings- wijze van de verhouding van dabbelverbindingen t.o.v. hunne splitsingsproducten in een oplos- middel is mogelijk. 1) Men kan nl. als abscissen de temperatuur uitzettende, als ordinaten de totaal-oplosbaarheid der mono- variante systemen uitzetten, d.1. de hoeveelheid opgeloste stof der verschillende systemen, waarin twee vaste phasen optreden. Geeft men dan ook nog de oplosbaarheidslijn voor de in ons geval (partieel racemische) dubbelverbinding, dan zullen ook in deze voor- stelling twee merkwaardige punten optreden, (zie fig. 9) nl. 4, ook hier de snijding van R+-L, R+ D en D-+-L curve en bovenste grens van het overgangstraject, en B, snijpunt van À en R+Z tak en onderste grens van dat traject. Formeel is deze voorstellings- tot aal conc. Fig. 9. 1 r7 : | FA RET . = : ps - ) Zie b.v. Meyernorrer, Gleichgewichte der Stereomeren, Leipzig—Berlin 1906, S. 48. May le OVER PARTIEELE RACEMIE. 21 wijze juist en kunnen inderdaad de punten 4 en B beschouwd worden als de snijpunten dezer zoo gemakkelijk te bepalen lijnen. Maar toch zal hier, bij vergelijking met fig. 5 een groot bezwaar aan den dag treden. ‘Trekt men in fig. 5 een lijn die loodrecht staat op de lijn OD dan hebben alle oplossingen, die door de punten van die lijn worden voorgesteld, dezelfde totaal-concentratie, zooals gemakkelijk in te zien is. Wanneer nu eens de punten zooals f”” en e”, punten dus der RAL resp. R+ D curve, eener zelfde isotherm, op zulk een loodlijn op OD lagen, dan zou by de betreffende temperatuur in fig. 9 de R+ LZ en K+ D tak samenvallen. Men kan dus bi deze methode van Mryrruorrer snijpunten hebben zonder dat die een iden- tiek worden van phasen beduiden, zooals in de punten 4 en B wel het geval is. En aan dit bizonder voorbeeld zal men het algemeene bezwaar begrijpen, dat deze voorstellingswijze haar waarde kan doen verliezen: hoe meer de richting der racemaattakken in de isotherm van fig. 5 (eigenlijk van verbindingslijnen zooals e” 7) den 45°-stand nadert, zooveel kleiner zal het verschil der bedoelde oplosbaarheden zijn en dus zooveel dichter zullen de curven À + D en LL in fig. 9 bij elkander komen te liggen, en derhalve zooveel moeilijker zal het snijpunt A te bepalen zijn. Eenzelfde bezwaar kan m.m. voor B optreden, vaak voor beiden gelijktijdig. LADENBuRG en Docror hebben de samenstellingen hunner aan (D+ L) verzadigde oplossingen bepaald door polarimetrische analyse. Zij bepaalden eerst het specifieke draaiingsvermogen van elk hunner stoffen, dus van D, Zen À in hunne afhankelijkheid van de concentratie. Daarna werden de afdampingsresidus der verzadigde (D + Z) oplossingen bij verschillende temperaturen op hun draatings- vermogen onderzocht in oplossingen, waarvan de concentratie nauw- keurig bekend was. De aldus gemeten draatingshoeken &, werden nu aldus gesplitst 4 —= 4, J- @,, waarin &, en @,, de draaiings- hoeken voorstellen, die D en Z elk op zich zelf in de oplossing van het zoutmengsel zouden veroorzaken. De concentratie aan D werd nu p, gesteld, die aan Z p, en die aan (D+ L) p; dan is p =p; Jp. Voorts zij / de lengte der polarimeterbuis en s het spec. gew. der oplossing. 442 LA Bale Le] 2, 2.8 s 1 Piao Derhalve *) 0 100 100? *) Voor [+] moet men feitelijk [2], lezen (natriumlicht), Ter voorkoming van ver- warring is deze D in de formules weggelaten, _ 22 THEORETISCHE EN EXPERIMENTEELE ONDERZOEKINGEN oe = hg Pa + [ml ne . . | I 5 [4,] in zijne verandering met de condens wordt voorbe door [aj] —— Aj Bp, evenzoo geldt voor [g,| (u) =— 4, + Ba = Ar B WH) In deze beide vergelijkingen zijn de #s en de B’s bepaald door een uitgebreide serie bepalingen. banden men nu in vergelijking I bovenstaande waarden voor [z,| en [@,] dan krijgt men: 100 & Ls = — Ap, + Bp?, — 4, (p— pi) + B, (p — Hal Men komt dus tot eene vierkantsvergelyking in p,, waaruit deze en derhalve ook p, is op te lossen. Deze rekenwize is evenwel niet als de juiste te aanvaarden, want oplossingen van (D-+ Z) zijn homogene phasen, waarin krc de wet van de massawerking een evenwicht moet worden aan- genomen tusschen D, Z en &, welk evenwicht bij eene bepaalde temperatuur door de concentraties is bepaald. Wanneer men dus den afgelezen draaiingshoek eenvoudig beschouwt als samengesteld uit de draaimg der D-moleeulen met die der Z-moleculen in het opgeloste mengsel, dan begaat men een fout, door niet de draaiing, die de /-moleculen veroorzaken, tevens in rekening te brengen. Is derhalve de totaal-concentratie p, de partiaal-concentratie voor de D-moleculen y, en die voor de Z-moleculen p,, dan komt aan de /?-moleculen eene partiaal-concentratie (p — pj — p,) toe en dan worden de bovenstaande vergelijkingen als volgt gewijzigd: ra CA _ [alma ds [a] a € ls [|W pip 100 a6 100 Sree ice ad C4 10 Oc RES = [ca] Pa + La lp + mill) (D zr hi ca *) Verg. G. Doctor, Inaug. Diss. Breslau, p. 62 [1899.] OVER PARTIEELE RACEMIE. 23 Voegt men hier weer in [æ,|, [æ,] en [ar] in hunne afhankelijk- heid van de concentratie der oplossing dan krijgt men opnieuw eene vierkantsvergelijking, ditmaal echter met fwee onbekenden nl. y, en p,, waarvan één enkele polarimeter-bepaling niet de oplossing kan geven. Bij de berekening der polarimetrische analyse-methode van LADENBURG en Docror en de verbeterde rekenwijze, die ik daarop liet volgen, is als vereenvoudiging bovendien nog ondersteld, dat in oplossingen, waarin D-, Z- en /-moleculen naast elkaar voor- komen, deze op de grootte van elkaars optisch draaiingsvermogen geen invloed zullen uitoefenen en dat op zulke oplossingen nog de tevoren experimenteel bepaalde formules voor het specifieke draaiings- vermogen van elk der zouten op zich zelf in hunne afhankelijkheid van de concentratie der oplossing mogen worden toegepast. Of dit juist is, moet nog bovendien worden nagegaan. Het zal echter duidelijk zijn, dat de weg, dien LapEnBure en Docror gevolgd hebben, om de innerlijke samenstellingen hunner verzadigde (D + /)- oplossingen te leeren kennen, naar alle waarschijnlijkheid niet de juiste is geweest. Het was dus, vooral waar hunne uitkomsten in sommige gevallen verrassend en a priori onwaarschijnlijk waren te noemen, wenschelijk eene andere methode te bedenken, wier resul- taten met die van LADENBURG en Docror konden worden verge- leken, en die minder aan bedenkingen onderhevig is. Welke die methode geweest is, zal in het experimenteele gedeelte van dit proefschrift nader worden uiteengezet. In dit hoofdstuk werd de overgangstemperatuur van dubbel- verbindingen in ternaire stelsels eenigszins uitvoerig besproken, omdat slechts met de kennis daarvan het gedrag van partieel- racemische lichamen behoorlijk kan worden bestudeerd. Ik kan er thans toe overgaan, in het volgende hoofdstuk een litteratuur over- zicht te geven van de partiëele racemie, waarbij naar volledigheid gestreefd is, en daarin tegelijk wijzen op de theoretische fouten, in de conclusies, uit vorige onderzoekingen getrokken. HOOFDSTUK HE. DE PARTIEELE RACEMIE. HistoriscH OVERZICHT. Hoewel de splitsing van racemische zuren met behulp van actieve basen, en die van racemische basen door actieve zuren in tal van gevallen gelukt is, zijn er toch een zeker aantal gevallen bekend, waarbij pogingen, om langs dezen weg eene scheiding tot stand te brengen, mislukten. Zeer dikwijls werden die gevallen in de litte- ratuur niet vermeld en riep men, waar de eene actieve stof geen resultaat gaf, eene andere te hulp, om het vooropgestelde doel te bereiken. Naar de oorzaak, waardoor de splitsing mislukte, werd niet gezocht; of, zoo dit al gedaan werd, was eene verkeerde ver- klaring dikwijls het gevolg daarvan. Zoo vermeldt Prcrer !), dat eene poging, om het appelzuur, uit fumaarzuur verkregen, door middel van cinchonine in zijne optische antipoden te scheiden, niet gelukte. Door gefractioneerde kristalli- satie was het niet mogelijk, twee verschillende lichamen te krijgen. De waterige oplossing gaf bij indampen, steeds hetzelfde cinchonine- zout, dat een standvastig smelttraject (135°—140°) bezat en by ontleding met ammonia telkens inactief zuur terugleverde. 4 Prerer trachtte dit verschijnsel te verklaren, door aan te nemen, dat dit appelzuur een analogon zou zijn van het antiwijnsteenzuur. Het is duidelijk, dat de structuur van het appelzuur een dergelijke isomeer niet toelaat. | BREMER ?) constateerde hetzelfde verschijnsel toen hij zijn, door reductie uit druivenzuur verkregen, appelzuur met cinchonine wilde splitsen. Het gelukte hem ten slotte toch eene scheiding tot stand *) Ber. d. d. chem. Ges. 14, 2648 (1881), *) Ibid. 18, 351 (1880), | THEORETISCHE EN EXPERIMENTEELE ONDERZOEKINGEN ENZ, 25 te brengen, door de oplossing met /-appelzuur-cinchonine te enten. Hoogst merkwaardig is het, dat daarbij niet /-appelzuur-cinchonine uitkristalliseerde. Integendeel, d-appelzuur-cinchonine zette zich uit de oplossing af. Intusschen heeft Krannicu !) later opnieuw het gedrag van het appelzuur ten opzichte van cinchonine bestudeerd en is daarbij tot eene conclusie gekomen, die afwijkt van de gevolgtrekking, welke uit Prerer’s proeven is te maken. Hij bracht eene heete oplossing van zuur-appelzuur-cinchonine op het waterbad tot kristallisatie door deze oplossing met kristallen van d-appelzuur-cinchonine te enten. Terstond scheidde zich eene groote hoeveelheid in kristallen af. De uitgekristalliseerde verbinding bleek bij ontleding met animonia en verwijdering van het neergeslagen cinchonine, een zout van het rechtsdraaiende appelzuur te zijn geweest, want de aldus verkregen oplossingen draaiden het polarisatievlak naar rechts. Hieruit meende Krannicu het besluit te mogen trekken, dat g-appelzuur-cinchonine niet bestaat. Ik meen tegen deze conclusie te mogen aanvoeren, dat KRANNICH op deze wijze mogelijk de oplossing door te enten heeft gedwongen, de verbinding van het d-zuur af te zetten, terwijl toch misschien het partieel-racemische lichaam onder omstandigheden van evenwicht het lichaam is, dat bij de temperatuur van het waterbad de naast de oplossing stabiele phase is. Voor het mislukken der seheiding zonder enting, geeft hij echter geene verklaring. Het is nu de groote .verdienste van LADENBURG geweest, dat hij dit proces nader heeft bestudeerd, en de oorzaak heeft kunnen aanwijzen waardoor splitsingen langs dezen weg vaak een ongunstig resultaat kunnen opleveren. Toen hij het B-pipecoline door middel van het bitartraat dezer base trachtte te splitsen ?) in zijne beide actieve componenten, liet hij dit zout op het waterbad uitkristalliseeren. Uit de afgescheiden kristallen werd de base vrij gemaakt; zij bleek echter inactief te zijn. Hij kwam nu op het denkbeeld, het bitartraat zich bij lagere temperatuur te laten afscheiden en zegt hierover ®): ,, Bei den ersten Versuchen erwies sich die Base als optisch inaktiv, weil die Kris- tallisation auf dem Wasserbade geschah, und diese Temperatur zu hoch war, d.h. die Umwandlungstemperatur niedriger liegt, als diejenige, bei welcher die Ausscheidung der Kristalle erfolgte.” *) Inaug. Diss. Breslau, 1901. *) Ber. d. d. Chem. Ges, 27, 75 (1894). ie ec. 26 THEORETISCHE EN EXPERIMENTEELE ONDERZOEKINGEN Hij verklaart dus het negatieve resultaat zijner scheiding door aan te nemen, dat zich bij hoogere temperatuur uit de oplossing eene verbinding van het inactieve pipecoline met d-wijnsteenzuur _afzet, eene verbinding derhalve van twee lichamen, die in configu- ratie slechts voor een deel elkaars spiegelbeeld zijn nl. van d-£- pipecoline-d-bitartraat met de overeenkomstige verbinding van Zf2- pipecoline. | | Eene dergelijke verbinding is, indien zij zich bij hoogere of bij lagere temperatuur naast hare verzadigde oplossing in hare zooeven genoemde bestanddeelen splitst, een volledig analogon van de door Van ‘rt Horr en zijne medewerkers bestudeerde anorganische dubbel- zouten. Zij bezit dan naast oplossing eene overgangstemperatuur, zooals die in Hoofdstuk IT besproken is, een overgangstraject en ten slotte een temperatuursgebied, waarbij zij naast hare verzadigde oplossing onontleed kan bestaan. Volgens deze theorie ontstaat er een nauw verband tusschen de spontane splitsing door kristallisatie (de natrium-ammoniumtartraten) en die met behulp van optisch actieve (zoutvormende) stoffen. Het gelukken van beide hangt af, of men beneden het overgangspunt, resp. het overgangstraject, de kristallisatie tot stand doet komen; dit traject is, (zooals we in Hoofdstuk IT gezien hebben) voor de eigenlijke racematen nul, daar bij hen beide componenten gelijke oplosbaarheid bezitten; vandaar, dat hier werken beneden het overgangspunt voldoende is. De ongelijkheid van alle physische eigenschappen van lichamen met gedeeltelijke gelijke spiegelbeeld-configuratié, dus ook van hunne oplosbaarheid, is de oorzaak, dat bij hen een overgangs¢raject ontstaat. De verschillende oplosbaarheid der actieve componenten biedt hier echter juist het groote voordeel, dat in de methode is gelegen. Ferwijl immers by de methode der spontane kristallisatie de beide actieve splitsingsproducten in gelijke hoeveelheden zijn uitgekristal- liseerd en men de kristallen nauwlettend op grond van hunne verschillend gelegen hemiëdrische vlakken moet uitzoeken, treft men nu het groote voordeel aan, dat de minst oplosbare component zich het eerst en het rijkelijkst afzet, zoodat uit de eerste fractie der kristallisatie meestal het eene der beide bestanddeelen van de oor- spronkelijke inactieve verbinding zich laat afzonderen. Nu kunnen zich bij partiëele racemie twee gevallen voordoen : 1°. de dubbelverbinding (p 7) strekt haar bestaansgehied naast oplossing naar Aoogere temperaturen uit, zoodat het overgangspunt de laagste temperatuur is, waarbij zij naast oplossing kan bestaan ; 2° het bestaansgebied breidt zich naast oplossing naar Jagere temperaturen uit, in welk geval het overgangspunt de maaimum q << OVER PARTIÈELE RACEMIE. 27 temperatuur aangeeft, waarbij de dubbelverbinding naast oplossing op den bodem kan voorkomen. 5) Het eerste geval doet zich voor in het 7-@-pipecoline-bitartraat 2) en het 2-tetrahydrochinaldine-bitartraat. %) Het tweede geval heeft tot voorbeeld druivenzure-strychnine # en brucine-biracemaat °). Al deze voorbeelden zijn door LADENBURG en zijne medewerkers gevonden. Reeds vóór LapunBure’s onderzoekingen in het gebied der par- tiëele racemie, had Emm Frscurr®) nagegaan, of lichamen, wier structuurformules gedeeltelijk elkaars spiegelbeeld zijn, zooals bijv. d-gluconzuur COOH ~~ en /-mannonzuur COOH | | | H COH H COH An Hl ae ae HO ie jee: bes neiging vertoonen, om zich met elkaar tot verbindingen te ver- eenigen. Hij vond, dat dit noch bij de zooeven genoemde zuren noch by de calciumzouten van /-mannon- en d-gluconzuur het geval was, want uit een mengsel van gelijke moleculaire hoeveelheden der genoemde stoffen, in water opgelost, scheidde zich in beide gevallen slechts één der stoffen uit de oplossing af. Ook Ascuan 7) vond, dat kamferzuur en iso-kamferzuur zich niet vereenigen en ditzelfde constateerde LIEBERMANN ®) bij kaneelzuur- dibromide en allo-kaneelzuur-dibromide. Fischer besluit daaruit, dat er derhalve geen neiging bestaat tot vorming van dubbelverbindingen, waarin de beide samenstellende bestanddeelen in configuratie gedeeltelijk elkaars spiegelbeelden zijn. Hij voerde voor dergelijke lichamen den naam „partieel-racemische verbindingen” in. *) De verschijnselen worden hier bij één bepaalden druk beschouwd. *) Ber. d. d. Chem. Ges. 36, 1649 (1903) Inaug. Diss. Breslau 1903. 2) Ibid. 41: 7966 C1908) NEED 3) 18908, à) Ibid. 31, 1969 (1898) en 32, 50 (1899) Inaug. Diss. Breslau 1899. ei) Ibid. 40, 2279 (1907) Inaug. Diss. Breslau 1905. 5) Ibid. 27, 3225 e. v. (1894). à Ibid. 27, 2001 (1894). *) Ibid. 27, 2045 (1894). Z 28 THEORETISCHE EN EXPERIMENTEELE ONDERZOEKINGEN Deze naam is door LADENBuRG !) overgenomen voor de verbin- dingen, die hij verkregen had uit een racemisch zuur met eene actieve base resp. eene racemische base met een optisch actief zuur, en die dus feitelijk tot eene geheel andere klasse van lichamen dan de door Frscuer oorspronkelijk met dien naam aangeduide verbin- dingen behooren. Sprekende over het chininezout van het racemische pyrowijnsteenzuur, zegt hij: 7) „Es ist also selbst eine racemische Verbindung, und zwar findet hier, da in den Componenten die Base jeweils dieselbe ist, die Säuren aber Spiegelbilder sind, das statt, was man zweckmässig als „partielle Racemie’”’ bezeichnen kann; d. h. es ist eine Verbindung zweier Körper, die nur theil- weise Spiegelbilder sind, sodass also durch die Verbindung eine theilweise Aufhebung der optischen Activität stattfindet, und der racemische Körper noch optische Activität besitzt”. Definieert men de partiëele racemie als het verschijnsel, waarbij twee stoffen, die in configuratie gedeeltelijk elkaars spiegelbeeld zijn, zich verbinden, dan is tegen deze benaming in vergelijking met de eigenlijke racemie, die de vorming van verbindingen met volkomen spiegelbeeld-configuratie bedoelt, niet veel te zeggen. Toch heb ik van den aanvang af tegen deze benaming een bezwaar gevoeld. Wie toch met de partiëele racemie niet bekend is, zal meenen, dat eene partieel-racemische verbinding een lichaam is, dat bestaat uit een inactief racemisch deel, het zuur, en een deel, dat optische activiteit bezit (de base); zoo zal men zich, krachtens de benaming, het molecuul druivenzure-strychnine denken opgebouwd uit druivenzuur en strychnine, terwijl toch het gedrag van dit lichaam naast zijne waterige oplossing er op wijst, dat het is op- gebouwd uit d-wijnsteenzure-strychnine + /-winsteenzure-strychnine. Stelt men het molecuul aldus voor inenen any eLink /-wijnsteenzuur- | strychnine’ komt blijkbaar de binding in het dubbelzout ‘niet tot stand tus- schen actieve base en racemisch zuur, maar wel tusschen de beide biactieve zouten, i.e strychnine-d-tartraat en strychnine-/-tartraat. Daar echter deze naam voor de verbindingen in quaestie langzamer- hand burgerrecht heeft verkregen, komt het mij niet wenschelijk voor er eene andere benaming tegenover te stellen. Het leek mij evenwel niet ongewenscht, deze objectie tegen de gebruikelijk geworden nomenclatuur te maken. Alvorens er toe over te gaan, de afzonderlijke gevallen, waarbij partiëele racemie met zekerheid geconstateerd is, te bespreken, wil ik *) Ber. d. d. Chem. Ges. 31, 938 (1898). gh Neos bes OVER PARTIËÈELE RACEMIE. 99 nog een enkel woord zeggen over den strijd, die jaren lang tusschen LapenpurG en Emin Fiscumr geduurd heeft over het bestaan der partiëele racemie, en die eerst in den allerlaatsten tijd beslecht is. Ik heb reeds vermeld, dat de term „partiëele racemie” van E. Fiscaer afkomstig is en dat deze de uitdrukking wenschte toe te passen op mogelijke verbindingen, als bijv. van Z-mannonzuur met d-gluconzuur. Het is nu aan Fiscuer niet mogen gelukken, verbindingen van dergelijke lichamen af te scheiden en zoo komt hij er toe, in zijn uitvoerige mededeeling betreffende zijne onderzoekingen over amino- zuren, polypeptiden en proteïnen) op pagina 572 te zeggen, dat het in de meeste gevallen niet gelukt, om isomeren, die gedeeltelijk elkaars antipoden zijn, door eenvoudige kristallisatie te scheiden. Het niet-gelukken dier scheiding meent hij daaraan te moeten toeschrijven, dat in de meeste gevallen de isomeren wegens hunne groote gelij- kenis meugkristallen zullen vormen en dan schijnt hem geen bezwaar tegen het gebruiken van zijn term ,,partiéele racemie’’ voor meng- kristallen, die 50 in van beide half-antipodische isomeren bevatten. Een jaar later is E. Fiscuer opnieuw op dit onderwerp terug- gekomen ?) en heeft op de verwarring, die door het gebruik van zijne uitdrukking ,,partiéele racemie’ voor de door LADENBURG ontdekte dubbelzouten, ontstaan is, een helder licht laten vallen. Het is nl. nooit zijne bedoeling geweest, de dubbelzouten van LADENBURG voor mengkristallen te houden. Deze beschouwt hij evenzeer als echte verbindingen als alle andere, die goed gedefinieerd zijn. Maar in vele gevallen, waarbij ,,partiéele racemie” in de be- teekenis van FrscHeRr, (verbinding tusschen bijv. de beide isomere broomisocapronyl-/-asparaginen) zou kunnen optreden, bleek meng- kristal-vorming op te treden en wiet het ontstaan van verbindingen tusschen de bedoelde isomeren. Nochtans is ook hier de mogelijk- heid, dat „verbinding’’ tusschen de isomeren optreedt, volgens hem geenszins uitgesloten. Fiscuer wijst echter ten slotte op de weinig scherpe definitie, die LADENBURG van de partiëele racemie geeft. Volgens zijne mee- ning is ook een lichaam van het type als de door Pasreur ontdekte verbinding van d-ammoniumbitartraat met /-ammoniumbimalaat een partiëel-racemaat. °) *) Ber. d. d. Chem. Ges. 39, 530 (1906). *) Ibid. 40, 943 (1907) noot 4. *) Het schijnt dat er eenige onzekerheid bestaat, of ook dit lichaam inderdaad eene verbinding dan wel een isomorph mengsel is. Verg. W. Meyernorrer „Gleichgewichte der Stereomeren” Leipzig 1906 p. 62. 30 THEORETISCHE EN EXPERIMENTEELE ONDERZOEKINGEN Hiermede is de strijd Frscuer-LADeNBure tot een eind gekomen en blijkt hij dus louter te berusten op eene ongeoorloofde toepas- sing van Frscner’s uitdrukking „partiëele racemie” door LADENBURG. Wij kunnen nu de gevallen, waarin het verschijnsel, dat ons bezighoudt, geconstateerd is, kortelings nagaan en de methoden mededeelen, volgens welke in al die gevallen met beslistheid kon worden bewezen, dat men inderdaad met verbindingen te doen had. Tevens zullen de voorbeelden, waarin het overgangspunt van deze verbindingen werd vastgesteld, worden opgegeven en tevens zal worden uiteengezet, langs welken weg die punten werden bepaald. LapexBurG heeft tot nu toe over de partiëele racemie een achttal mededeelingen gepubliceerd, waarvan de laatste, tijdens mijn onder- zoek verschenen, eene samenvatting geeft van de resultaten, die in de voorafgegane onderzoekingen zijn verkregen. In zijne eerste mededeeling +) vermeldt LapeNBvre, dat het in- actieve pyrowijnsteenzuur niet met behulp van chinine en het inactieve (-oxyboterzuur niet met strychnine te splitsen is; de zouten die in beide gevallen uit de oplossingen uitkristalliseeren leveren bij verwijdering van de actieve base het inactieve zuur in den vorm van het ammoniumzout weer terug. Er moest nu worden aangetoond, dat dit chinine- en dit strychnine- zout werkelijke verbindingen waren en niet beide een aequimoleculair mengsel van d-pyrowinsteenzure-chinine met /-pyrowinsteenzure- chinine resp. de overeenkomstige (2-oxyboterzure-strychninezouten waren. Daarom werd eerst aangetoond, dat, wanneer bij verschillende temperaturen de kristallisatie plaats vond, steeds een zout ontstond van een zuur, dat optisch inactief was. Het isoleeren van het zuur uit de afgescheiden kristalmassa had steeds op dezelfde wijze plaats en wel zóó, dat de kans op auto-racemisatie ervan, uitgesloten mocht worden geacht. Met oplossingen in absoluten alcohol kreeg men bij 0°, 18°, 30° steeds dezelfde verbinding. Dit feit pleitte krachtig voor de werke- lijke ,,verbindings’’-natuur van het lichaam ; immers, mocht het al toevallig mogelijk zijn, dat bij een der genoemde temperaturen de oplosbaarheid der beide half-antipoden gelijk was, onaannemelijk is het, dat hij twee zoo totaal verschillende lichamen als dergelijke isomeren plegen te zijn, de oplosbaarheid van beide over een tem- peratuurstraject van 0°—30° steeds gelijk zou blijven, waardoor zij *) LapensurG en Herz, Ber. d. d. Chem. Ges. 31, 524 (1898). er OVER PARTIËELE RACEMIE. eee uit oplossingen, waarin zij in aequimoleculaire verhouding voorko- men, zich ook in diezelfde verhouding kristallijn zouden kunnen afzetten. Nu werden de chininezouten van d-, /- en 7-pyrowijnsteenzuur bereid. Het zout van het /-zuur kon niet zuiver verkregen worden, daar dit zuur slechts te bereiden was met een draaiingsvermogen van ?/, van het overeenkomstige d-zuur. Het d-pyrowijnsteenzure-chinine smolt bij 169°—171°, het zout uit het z-zuur bij 174°—175°, dus eenige graden hooger, een feit dat LADENBURG ten gunste van het bestaan eener dubbelverbinding aanvoert. Het is echter duidelijk, dat eene dergelijke dubbelver- binding evengoed lager als hooger dan hare componenten kan smelten. De oplosbaarheid van het d-zure zout bleek grooter dan die van het g-zure zout. Dit levert een argument voor eene verbinding, Immers, wanneer het zout, dat zuur blijkt te bevatten, een mengsel was, dan zou krachtens de onderzoekingen van Rüporrr 4) de oplosbaarheid van dit mengsel grooter moeten zijn dan van eene enkele component afzonderlijk. In de 2° mededeeling *) wordt ver- meld, dat men het /-zure zout zuiverder heeft verkregen en de op- losbaarheid ervan in alcohol heeft bepaald. Deze is zeer veel grooter dan die van d-zuur zout. Hiermede had LaApeNBura de dubbelzout- natuur van het 2-pyrowijnsteenzure-chinine bewezen en het eerste geval van partiëele racemie met zekerheid aangetoond. De volgende verhandelingen leveren nieuwe gevallen en tegelijk wordt hierin telkens gepoogd de overgangstemperatuur der nieuw- ontdekte partiëel racemische verbindingen vast te stellen. Zoo bestudeert LADENBuRG in zijne derde verhandeling %) met Doctor het neutrale strychnineracemaat. Hier zijn de componenten d-wijnsteenzure-strychnine, /-wijnsteen- zure-strychnine en druivenzure-strychnine gemakkelijk in zuiveren toestand te bereiden. De smeltpunten, het kristalwatergehalte en de dichtheden worden van de 3 zouten bepaald en van allen verschillend gevonden. Tegelijk worden bi twee temperaturen (20° en 40°), later by meer temperaturen, de oplosbaarheden bepaald en steeds verschil- lend gevonden. Het verschil in smeltpunt van een aequimoleculair mengsel en van het dubbelzout wordt vastgesteld, en men vindt, dat bij toevoeging van de voor de vorming van het dubbelzout ‘) Pogg. Ann. 148, 456 en 555. *) Zie pag. 27. le: 32 THEORETISCHE EN EXPERIMENTEELE ONDERZOEKINGEN uit zijne componenten vereischte hoeveelheid water, het aanvankelijk vochtige mengsel, na eenige dagen in een afgesloten fleschje be- waard te zijn, kurkdroog is geworden. De stof vertoont dan het smeltpunt van de dubbelverbinding. Hiermede was bewezen, dat het lichaam geen mengsel was en verder werd getracht de overgangstemperatuur in dit geval te bepalen. Dilatometerproeven voerden hierbij niet tot de gewenschte hit. komst, daarentegen wordt in de 4° mededeeling bericht, dat oplos- baarheidsbepalingen het overgangspunt op 30° deden vinden en deze temperatuur wordt later door tensimeterbepalingen op 29.5° gecor- rigeerd. Ik zal op dit onderzoek hier niet A ingaan, daar er in het experimenteele gedeelte van dit proefschrift, gelegenheid genoeg zal bestaan om LADENBURG en Docror’s uitkomsten kritisch te be- spreken en ze met de door mij gedane waarnemingen over deze zelfde zouten te vergelijken. LapenBure’s 5° mededeeling brengt als nieuw geval van par- tiëele racemie het (-pipecoline-bitartraat. Deze stof heeft, zooals wij op pag. 2 zagen, historisch belang, daar bij haar het eerste geval van partiëele racemie werd opgemerkt. De eigenschappen en de bestaansvoorwaarden van het racemaat naast zijne waterige oplossing zijn hier echter nader bestudeerd. Het racemaat blijkt ook hier weer, in zijne physische eigen- schappen, geheel te verschillen van een aequimoleculair (d + /) mengsel. Daartoe zijn de smeltpunten, dichtheden en kristalwater- gehalten der drie stoffen in kwestie bepaald. Waar we hier stoffen hebben, die misschien normale smelt- verschijnselen vertoonen, (in tegenstelling bye met de brucine- en strychninetartraten, die alle onder ontleding smelten) lijkt het mij 145 hier niet ongepast, op te merken, dat het bewijs, dat het partiëele racemaat werkelijk ve een chemisch individu en geen aequimole- culair mengsel van zijne samenstellende bestanddeelen is, ook kan geleverd worden (en dan met alle zekerheid) door de bepaling van de smeltlijn van het binaire stelsel d-[- -pipecoline- -d-tartraat en /-(2-pipecoline-d-tartraat. Is het racemaat nl. eene werkelijke verbinding, dan zal zich dit vertoonen in den vorm der smeltlijn en hare gedaante ongeveer zijn als in fig. 9ézs. Op dergelijke wijze is, zooals ik kort voor de samenstelling van Fig. 9bis. mert OC E OVER PARTTÜELE RACEMIE. 33 dit hoofdstuk heb bemerkt, door Finpuay en Hickmaxs !) het karakter der 7-amandelzure-/-mentholester als dubbelverbinding met volledige klaarheid blootgelegd. ‘Tegelijk blijkt uit hunne onderzoe- kingen, dat deze ester van het racemische amandelzuur in zijne smelt tot een aanmerkelijk bedrag is gedissocieerd in zijne beide componenten. _ Ook de overgangstemperatuur liet zich met behulp van den tensi- meter volgens Brrmer-Frowern vaststellen op 39.5°. Ik heb reeds vroeger vermeld, dat men hier een partiëel racemaat heeft, welks bestaansgebied zich naar hoogere temperaturen uitstrekt. Nog een geval van partiëele racemie vonden LADENBURG en FiscHeR (6° mede- deeling) *) in het zure druivenzure brucine. Deze onderzoeking wijkt, ook in hare fouten, geenszins af van de vorige. Door oplosbaarheids- bepalingen werd de overgangstemperatuur op 44° bepaald. Langs tensimetrischen weg kon dit punt niet nader gecontroleerd worden, daar de omzetting zonder waterafsplitsing volgens LADENBURG en Fiscrer plaats heeft. | Het laatste *) door Laprnpure en zijne leerlingen bestudeerde geval betreft het tetra-hydrochinaldine-bitartraat. Deze base is reeds door LapexBurG in 1894 #) met behulp van het gewone wijnsteen- zuur gesplitst en op die wijze is de rechtsdraaiende base in zuiveren toestand bereid. Bij herhaling van deze proeven verkreeg men een base, die een kleiner draatingsvermogen bezat dan het aanvankelijk geisoleerde d-tetrahydrochinaldine. Dit deed een geval van partiëele racemie vermoeden. Niet zonder moeite gelukte het nu, een zout te bereiden, welks base geheel vrij van optische activiteit was, nl. door vermenging van zeer geconcentreerde wijnsteenzuur oplossing, die op 60°—63° verhit was, met racemisch hydrochinaldine van diezelfde temperatuur. Onder deze omstandigheden kreeg men eene kristallijne brij, waarvan de kristallen bleken racemische base te bevatten. Waarschijnlijk waren de grootere moeilijkheden van de bereiding van het partiëele racemaat hier gelegen in het lage smeltpunt van het zout nl. 72°—73° (d 90°—91°, 7 62°— 63°) en de zeer ge- makkelijke hydrolytische splitsing ervan. Deze laatste eigenschap maakte eene vaststelling van de overgangs- temperatuur door oplosbaarheidsbepalingen onmogelijk; niettemin *) Journ. Chem. Soc. 91*, 905 (1907). a) Ze pag. 27. *) Te meded. zie pag. 27. *) Ber. d. d. chem. Ges. 27, p. 76, 1894. Verhand. Kon, Akad. v. Wetensch. (5e Sectie) Dl. XI. Did 34 THEORETISCHE EN EXPERIMENTEELE ONDERZOEKING EN gelukte het, haar op ca 59° vast te stellen langs den weg der spanningsevenwichten in den tensimeter. Ook hier is het racemaat naar hoogere temperaturen stabiel naast zijne oplossing. LapexgurG stelt zich ten slotte de vraag, hoe eigenlijk hier het proces plaats grijpt en zegt: 1) „Man kann allerdings annehmen, dass beim Zusammenbringen von d-Weinsäure mit @-Hydrochinaldine zunächst das partielle racemische Salz entsteht, welches sich aber, da es sich jenseits der Umwandlungstemperatur befindet, alsbald in die Einzelsalze zer- legt. Diese Zerlegung wird um so langsamer vor sich gehen, je mehr man sich der Umwandlungstemperatur nähert, und es wäre daher immerhin möglich, dass das geringe Drehungsvermögen der bei höherer Temperatur auskrystallisierenden Salze in dieser Weise erklärt werden könnte. Wahrscheinlich finde ich es aber nicht. Plausibler ist die Annahme, dass die Löslichkeiten der beiden Einzelsalze bei höherer Temperatur einander näher kommen, so dass das auskrystallisierende Salz bei höherer T'emperatur mehr von der leichter löslichen Komponente enthält und dadurch weniger aktiv ist. Leider haben die Eigenschaften der Salze nicht gestattet, Lôslich- keitsbestimmungen zu machen und eine Entscheidung zwischen beiden Môglichkeiten zu treffen.” Daar bij deze zouten hydrolytische splitsing kan optreden, is hier moeilijk te beoordeelen, wat er bij de reactie tusschen d-wijnsteen- zuur en het racemische hydrochinaldine gebeurt. Zien we echter voor een oogenblik hier van af, dan meen ik, dat fig. 5 blz. 14 hoofdstuk [IT in staat is, de verklaring te geven. Men gaat uit van de racemische base en d-wijnsteenzuur in waterige oplossing. De samenstelling wordt dus aangegeven door een zeker punt der lijn OC. Heeft nu de kristallisatie bij een constante tem- peratuur plaats en ligt deze onder het overgangspunt 1. c. beneden het bestaansgebied rn partiëel racemische verbinding, naast oplossing van hare samenstelling, dan behoeven we slechts na te gaan, waar de lijn OC de isotherm van de temperatuur, waarbij het uitkris- talliseeren plaats heeft, snijdt. Dit snijpunt is natuurlijk gelegen op de oplosbaarheidslijn der minst oplosbare componente ?). By afkoeling tot de kristallisatie-temperatuur zal zich dus eerst deze afscheiden, hierbij verandert de oplossing in samenstelling en wel *) Ber. d. d. chem. Ges. 41, p. 969 (1908). *) Indien tenminste de fig. 9er de verhoudingen weergeeft, wat in an normale ge- vallen inderdaad zoo is. OVER PARTIÉELE RACEMIE. 35 zóó, dat ze rijker wordt aan de meest oplosbare componente. De afscheiding van de eerste gaat voort, tot men gekomen is in punt P (fig. Mer), waar zich nu beide componenten naast elkaar uit de waterige oplossing afzetten. De ligging van dit laatste punt P is nu beslissend voor de activiteit der uit de kristallen te isoleeren base. Ligt P dicht bij de lin OC, dan is de afgescheiden base zeer zwak draaiend, is echter DP groot, wat samenhangt met het verschil in oplosbaarheid der beide componenten bij de kristallisatie- temperatuur, dan zal de base grooter draaiend Q' vermogen vertoonen. Oplosbaarheidsbepalingen Fis: Oter) zijn hier echter niet verricht en dus laat zich hier de theorie niet met het experiment vergelijken. Zeer onlangs!) heeft LADENBURG in een samenvattend artikel zijne onderzoekingen over de partiëele racemie bijeengevoegd. Behalve, dat eenige experimenteele bizonderheden worden mede- gedeeld, die in het volgende hoofdstuk ter sprake zullen moeten komen, levert dit artikel niets nieuws op. Van belang is echter het slotwoord, waar hy in korte trekken het proces der splitsing van racematen door zoutvorming nogmaals schetst. Ik wil hieruit het volgende citeeren: „Früher hatte man geglaubt, dass, wenn man zu einem Racemkörper eine optisch active Substanz hinzufüge, die mit jenem eine Verbindung bilden könne, vorher eine Spaltung des Racemkörpers stattfinde. Eine solche Annahme ist aber durch Nichts zu begründen, widerspricht im Gegentheil einer Reihe von bekannten analogen Thatsachen. So bildet die Traubensäure mit den Alkalien und vielen anderen Metallen wohl definirte Salze, aus denen man die Traubensäure auch wieder regeneriren kann. Warum sollte nun durch Zusatz von z. B. Cinchonin plötzlich d- und /-weinsaures Salz entstehen? Dazu müsste eine Hinwirkung eines optisch-activen Körpers auf einem Racemkörper angenommen werden, die durchaus unverständ- lich und unphysikalisch wäre. ?) Durch den Nachweis der Existenz partiell-racemischer Salze wird nun Alles klar. Es bildet sich zunächst stets die Verbindung des Racemkörpers mit der aktiven Substanz, welche eventuell, wenn innerhalb ihres Existenzgebietes gearbeitet wird, isolirt werden kann, B *) Lieb. Ann. 364 p. 227—271 (09). *) Die Auffassung von Lanporr (Optisches Drehungsvermögen 2 Aufl. S. 60, 86 u.s.w.) ist nur zu verstehen, wenn man annimmt, dass Racemkörper in Lösung nicht existiren, was aber zweifellos irrthümlich ist. 3x 36 THEORETISCHE EN EXPERIMENTEELE ONDERZOEKINGEN im anderen Fall aber in die beiden Einzelsalze zerfällt, die, weil sie nur teilweise Spiegelbilder sind, ganz verschiedene Higenschaften — haben, zur Trennung und Isolirung der activen Componenten mit grossem Vortheil benutzt werden können.” Ik kan mij met dezen gedachtengang niet vereenigen en stel mij de verschijnselen aldus voor: Wanneer men eene druivenzuur- oplossing heeft bereid, dan bevindt zich in die oplossing niet het druivenzuur als zoodanig; maar er bestaat in die oplossing, omdat zij eene homogene phase is, wier innerlijke toestand beheerscht wordt door de wet van de massawerking, een evenwicht tusschen inactieve druivenzuur-moleculen en moleculen van d- en /-wijnsteen- zuur. Uit de bepalingen van de vriespuntsverlaging en kookpunts- verhooging van druivenzuuroplossingen is gebleken, dat hier dit evenwicht zeer naar de zijde der actieve componenten is verschoven. 4) Neutraliseert men nu eene dergelijke oplossing met bijv. strych- nine, dan ontstaan in de oplossing de strychninezouten, zoowel van d- en /-wijnsteenzuur als van druivenzuur, en weer zal zich volgens de wet van GuLpBERG en Waaar een evenwicht instellen; ditmaal echter tusschen d-tartraat, /-tartraat en partiëel racemaat. Bij afkoeling zal dit evenwicht verschuiven en wel in den zin van partiëele racemaatvorming, indien de dissociatie van dit lichaam in zijne componenten een endotherm proces 1s. Komt men nu op een temperatuur, waarbij het partiëele race- maat naast eene oplossing van zijne samenstelling Kan bestaan en is aan de voorwaarden voor de kristallisatie voldaan, dan zal ?) zich ook dit zout uit de oplossing afzetten en niets ates dan dit zout. Ligt de temperatuur van kristallisatie niet in het bestaansgebied van het partiéele racemaat, dan gebeurt wat op pag. 16 e. v. is geschetst. Strekt het partiëele racemaat zijn bestaansgebied naast oplossing naar hoogere temperaturen uit, dan is wat gebeuren zal, gemak- kelijk uit fig. 6 pag. 15 te begrijpen. Twijfel aan het bestaan van partiëele racematen en van eigen- lijke racematen in oplossing of in vloeibaren toestand behoeft niet te bestaan. Er is steeds een evenwicht tusschen racemaat resp. partiëel racemaat en zijne splitsingsproducten. ") Prof. SCHREINEMAKERS maakte mij er attent op dat hij |Zeitschr. f. phys. Chem. 33, 74 (1900)| tot eenzelfde conclusie is gekomen op grond van het gedrag van het stelsel water-phenol-wijnsteenzuur (resp. druivenzuur). KRUYT. : ‘) Men bedenke echter, dat hier, waar het verschijnselen uit de organische chemie geldt, groote kans is op vertragingen, zoodat wel het racemaat boven zijne overgangs- temperatuur uit de oplossing kan kristalliseeren en anderzijds (d + 1) zich in het race- maatgebied niet tot racemaat verbinden. OVER PARTIBELE RACEMIE. on Ik wil thans nog kort de aandacht vestigen op de overige gevallen van partiéele racemie, die men in de litteratuur verspreid vindt en zal deze in chronologische orde vermelden. G. Gonpscumipt') geeft op, dat het hem niet gelukt is, het tetrahydropapaverine, dat hij trachtte in het bitartraat om te zetten, om aldus een splitsing ervan te bewerkstelligen, in zijne compo- nenten uiteen te doen vallen. Hij verkreeg met wijnsteenzuur steeds het neutrale zout daarvan, dat steeds inactieve basis bleek te be- vatten en schrijft nu de mislukking der splitsing toe aan de om- standigheid, dat hier het bitartraat niet kon worden geisoleerd. Na hetgeen voorafgegaan is, eischt het wel geen betoog, dat hij d-wijnsteenzure r-tetrahydropapaverine in handen heeft gehad. Ook Porr en Pracney ?) konden in dit geval geen zuur tartraat van genoemde basis bereiden. Zij zagen echter terstond in, dat men hier met een geval van partiëele racemie te doen heeft en verwijzen naar LApeNBure’s beschouwingen daaromtrent. Kreerne %) bericht zelfs de vondst van isomere partiëel-racemische verbindingen in het geval van r-hydrindamine met broomcampher- sulfonzuur, chloorcamphersulfonzuur en cis-7-camphaanzuur, welke isomerie, hoe interessant ook, wij niet nader zullen beschouwen. Ongeveer terzelfder tijd deelde Bac *) mede, dat het wijnsteen- zure zout van phenyl-g-picolylalkine bij regeneratie van de base, deze in haren racemischen vorm blijkt te bevatten, wat dus als een nieuw geval van partiëele racemie mag worden beschouwd. Van meer belang zijn de beschouwingen ontwikkeld door Krrring en Hunrer 5) naar aanleiding van gevallen van partiéele racemie bij de tartraten van pheno-g-aminocycloheptaan en van hydrindamine. In de bedoelde publicatie vinden we naar aanleiding van de moge- lijke, zouten, die gevormd kunnen worden uit d-wijnsteenzuur en | dB B’ ook dit zout zou men partiëel racemisch kunnen noemen, immers het is gevormd uit een tweebasisch zuur, welks eene zuurfunctie is geneutraliseerd door de d-base, de andere door de /-base. Zulk een zout zou niet eene partiëel racemische verbinding in den ge- wonen zin zijn, het is nl. geen verbinding van twee lichamen, die in configuratie gedeeltelijk elkaars spiegelbeeld zijn. Dat is het (7, 4) pheno-g-aminoeyeloheptaan o.a. genoemd het zout d 4 ") G. Gorpscumipr, Monatshefte 19, 321 ('98). *) Pope en Peacuey, J. Chem. Soc. 73, 902 ('98) en Zeitschr, f. Krist. 31, ('99), *) Kippine, J. Chem. Soc. 77, 861 (00). *) Bacu, Ber. d. d. chem. Ges. 34, 2237 (01). *) KipriG en Hunter J, Chem, Soc, 81, 576 (02). 38 THEORETISCHE EN EXPERIMENTEELE ONDERZOEKINGEN echter weer wel, als het zout bimoleculair (a4) is, het kan à | dan nl. eene verbinding zijn van d 4 dB LB | en of nu de normale tartraten van de d-base en van de Z-base zich misschien vereenigen tot eene verbinding, die identiek is met die, welke men verkrijgt door achtereenvolgens op d-wijnsteenzuur eerst één molecuul d-base te laten inwerken en daarna één molecuul /-base, is echter niet onderzocht. | mg Partiéele racemie werd geconstateerd bij het (d-/) hydrindamine bitartraat, daarentegen niet bi] het zure wijnsteenzure zout van pheno-g-aminocycloheptaan, van welke base de constitutie zeer veel overeenkomst met die van g-hydrindamine (d. 1. pheno-g-aminocyclo- pentaan) vertoont. Wat echter in deze publicatie ons het meest interesseert, zijn de beschouwingen, die KrPPING aangaande partiéele racemie ontwikkelt en die vrijwel lijnrecht staan tegenover die van LADENBURG. Hij bespreekt allereerst LApENBure’s definitie van partiéel race- mische verbindingen en wijst er op, dat deze auteur hen soms als verbindingen van gedeeltelijke antipoden beschouwt, soms als actieve zouten van een werkelijk racemisch zuur of racemische base. Dit is dus hetzelfde bezwaar, dat boven reeds door mij tegen den term ,,partiéele racemie” is aangevoerd. Nu schijnt Kipprne verder racematen uitsluitend als „kristallijne vereenigingen van d- en /-iso- meren te beschouwen, die geen ander bestaan dan in dien vorm” bezitten, terwijl juist LADENBURG een voorvechter is voor de vloei- bare racematen en de racematen in oplossing. | Karrine denkt zich dus bij het smelten of oplossen van een race- maat totale dissociatie; LADENBURG daarentegen absoluut geene. Het zal wel niet noodig zijn te betoogen, dat de waarheid in het midden ligt en er in oplossing of smelt steeds racemaat-moleculen aanwezig moeten zijn in eene hoeveelheid, afhankelijk van het even- wicht, dat zich in die homogene phase volgens de massawerkings- wet moet instellen. Hoe toch zouden anders racematen zich uit oplossing of smelt in vasten toestand kunnen afzetten, indien hunne moleculen niet reeds daarin aanwezig waren? Bi het smeltpunt van een racemaat of in diens verzadigde op- lossing zijn toch de moleculaire thermodynamische potentialen in het eerste geval van de moleculen van het vaste racemaat en die van het vloeibare aan elkaar gelijk, in het tweede geval die van de moleculen van het vaste racemaat en van die in oplossing. In beide gevallen heeft j,,,, een bepaalde waarde, dus moet OVER PARTIÈELE RACEMIE. 39 ook Kn. Of Un. die zelfde waarde bezitten. Hieruit vloeit de noodzakelijkheid van het bestaan der racemaat-moleculen in smelt en oplossing voort. — Kiprine gaat ten slotte nog na, welke gevallen zich kunnen voordoen, indien een uitwendig gecompenseerd tweebasisch zuur (d, !) A geneutraliseerd wordt door een actieve basis dB. | | dB | dB en of als het zuur één basisch is d 4 dB en / A dB. dB dB | 1B en 71 IB verschillen in physische eigenschappen en zijn te scheiden door gefractioneerde kristallatie, wat vaak, hoewel niet altijd, het geval is. 2°. Zij kunnen zich bij de afscheiding uit de vloeistof (Krpprne’s standpunt) vereenigen in kristallografischen zin en eene stof leveren, die in kristalvorm en in overige physische eigenschappen verschilt van beide componenten. 3°. Zij kunnen mogelijkerwijze in gelijke hoeveelheden „naast elkaar’ uit de oplossing afgezet worden als zuiver mengsel. 4°, Zij kunnen mengkristallen geven, in zekeren zin vergelijkbaar met de pseudoracemie, zooals die door Kipprne en Porg is ontdekt, indien daartoe bij de componenten voldoende kristallografische verwantschap bestaat. Evenzoo kan een actief tweebasisch zuur met een racemische In oplossing kunnen dan gevormd worden 2 zouten nl. d 4 dB dB? Nu zijn de volgende 4 gevallen mogelijk: 1°. 74 LA base in oplossing de volgende zouten doen ontstaan: “) dA fe dB | (/B b C Rp en OAN DE Mogelijkerwijze kan zich het zout *) als bepaalde verbinding uit de oplossing afzetten, indien dat niet zoo is, dan kunnen de onder ’) en ©) genoemde zouten zich gedragen op één der vier reeds aan- gegeven wijzen. Krireine zegt nu, dat in alle genoemde gevallen, behalve in dat van gefractioneerde kristallisatie (geval 1°) en in het onder “ geclassificeerde geval het afgescheiden zout overeenkomt met LADEN- BURG's definitie van partieel-racemische verbinding, „unless the meaning of „Verbindung” !) he interpreted as a crystallographic union, the result of which is to give a product differing from at least one of its components in crystalline form, and consequently in other properties; if this limitation be not made the term ,,parti- ally racemie” would include a number of salts of different types ) Cf. Lapensura’s verhandelingen op pag. 27 en 28 besproken, 40 THEORETISCHE EN EXPERIMENTEELE ONDERZOEKINGEN in much the same way, as did at one time the term ,racemie’ (Krepina and Porr Trans. 1897 71, 989). Hier is slechts aan toe te voegen, dat LADENBURG, hoewel niet altijd even consequent in zijne terminologie, met partiëele racematen niet anders bedoeld heeft dan dubbelzouten of dubbelverbindingen van zouten of andere lichamen, die slechts gedeeltelijk elkaars optische antipoden zijn. Dus wel ,,crystallographic unions” maar in stoechiometrische verhouding, lichamen dus, die als dubbel- verbindingen, zoowel in vasten als in vloeibaren vorm (hoewel daar gedeeltelijk geplitst) voorkomen. | Zeer juist is aan het slot van dit artikel de opmerking, dat het karakter eener partieel-racemische verbinding als zoodanig kan uit- gemaakt worden volgens dezelfde methode, die leidt tot identificatie van eigenlijke racematen nl. door bij gegeven temperatuur oplos- singen te bereiden van het vermeende partiëele racemaat alleen, van hem zelf met de eene zijner splitsingsproducten en van hem met de andere zijner componenten. Zijn die oplossingen verschillend, dan is het lichaam eene werkelijke dubbelverbinding en geen aequi- moleculair mengsel, zijn die drie oplossingen daarentegen onderling gelijk, dan is het lichaam een mengsel zijner beide componenten. Ware dit feit aan LADENBURG en zijne leerlingen, die met hem de partiëele racemie hebben bestudeerd, bekend geweest, dan zouden zij zich vele pogingen, om bewijzen aan te brengen voor het ware karakter als dubbelverbinding hunner partiëele racematen, hebben kunnen besparen. Niet onvermeld mag in dit overzicht blijven, dat MEyErHorrer +) in zijne ,,Stereochemische Notizen” nog eens ten overvloede op de analogie tusschen partiëele racematen en anorganische dubbelzouten de aandacht heeft gevestigd. Het schijnt intusschen, dat noch het uitnemende artikel van Bakuurs RoozreBooMm®), noch deze publicatie tot degenen, die zich met de studie van ons onderwerp hebben beziggehouden, behoorlijk is doorgedrongen. De latere publicaties uit LADENBURG’s school toch geven nog evenzeer blijk van een onvolledig inzicht, als de vroegere van LADENBURG en Doctor. Veel uitvoeriger nog wordt de aandacht geschonken aan de par- tiëele racemie in MeisernorreR’s zeer lezenswaardig boekje „Die Gleichgewichte der Stereomeren’’ waarin ook tevens op de fouten en onwaarschijnlijkheden in de uitkomsten van vorige onderzoekers wordt gewezen. In het bizonder wordt daarin ook de partiëele racemie van het druivenzure strychnine ter sprake gebracht. *) Ber. d. d. chem. Ges. $7, 2604 (1904). 2), Ac blz. 18; y OVER PARTIEELE RACEMIE. 41 Ook in verschillende leer- en handboeken heeft thans het begrip partiëele racemie ingang gevondén. Zoo wordt zoowel het ,,Lehr- buch der Stereochemie” van A. Werner‘) als in den 2° druk van het „Lehrbuch der Organischen Chemie” van Victor Meyer en P. JacoBson ?) de vorming van dubbelverbindingen bij gedeelte- lijke optische antipoden besproken en men vindt vooral in het laatste boek een vrij volledige litteratuuropgave. De opmerking echter, dat de „tot nu toe vastgestelde gevallen van partiëele racemie uitsluitend betrekking hebben op zouten” is niet meer als „up to date” te beschouwen. Een merkwaardig geval van partiëele racemie is door Mc. Kenzin ®) gevonden in den / menthol-ester van het racemische amandelzuur. Deze is, voor zoover mij bekend, het eerste voorbeeld van verbin- dingen, die in configuratie gedeeltelijk elkaars spiegelbeeld zijn, bij esters. Zijn bestaan als „verbinding” is door A. FinprAy en mej. Hick- MANS *) bewezen door een bepaling der volledige smeltlijn van het binaire stelsel d-amandelzure /-menthol-ester + /-amandelzure- /- menthol-ester. Hierin komt de partiëel racemische ester als ware verbinding door het bezit eener eigen smeltlijn te voorschijn, wel is waar met een vlakken top, wat op hoogen graad van dissociatie in zijne beide biactieve bestanddeelen in de vloed wijst. Ken tweede publicatie over dezen zelfden merkwaardigen ester ver- scheen zeer korten tijd geleden van deze zelfde auteurs, waarin de stabiliteitsverschijnselen naast zijne verzadigde oplossing in 80% alcohol bij’ verschillende temperaturen is bestudeerd. De aard van dit onderzoek is in sterke mate vergelijkbaar met dat van LADENBURG en Doctor, al moet erkend worden, dat de Engelsche onderzoekers een beter inzicht in de zaak hebben. Gemeten zijn oplosbaarheden van Z,D,R, (R+ D) en (R + ZL) bij 3 temperaturen nl. 35°, 25° en 10° in het genoemde oplosmiddel. De oplosbaarheid wordt in een oplosbaarheidsdiagram, dat be- trokken 1s op een constante hoeveelheid oplosmiddel met tot assen een D en Z as, waarop de hoeveelheden opgelost D en Z worden afgezet. Om in dit diagram ook de oplosbaarheid van (2 + D) en (+ Z) te kunnen aangeven, (die van À ligt natuurlijk *) A. Werner, Lehrbuch der Stereochemie, p. 81. Jena 1904. *) Victor Meyer und P. JacoBson, Lehrbuch der Organischen Chemie 1er Band 2te Auflage, p. 105. Leipzig 1907. *) J. ch. Soc. Trans 85, 383 (1904). 4 Idem. 91, 30341907): + Idem. 95, 1386 (1909). 42 THEORETISCHE EN EXPERIMENTEELE ONDERZOEKINGEN ENZ. steeds op de lijn van gelijke samenstellingen D en Z) was eene analyse dier oplossingen noodig, welke FiypLay en mej. Hickmans hebben bewerkstelligd op dezelfde wijze als LADENBURG en Doctor hunne (D + L) oplossingen hebben geanalyseerd nl. door den draaiingshoek der afdampingsresidu’s hunner oplossingen te meten en uit dezen hoek in verband met het bekende draaïingsvermogen van Z en D, het gehalte aan elk van deze in de oplossingen uit te rekenen. De juistheid dezer methode is aan bedenking onder- hevig, zooals boven (cf. pag. 21 e.v.) reeds besproken is. Aan- gestipt zij nog, dat de door Finpray en Hickmans telkens uit de vijf door hen bepaalde punten geconstrueerde isothermen een onaannemelijk verloop vertoonen, en daarmee eene sterke indicatie leveren, dat de gevolgde analyse-methode onbetrouwbaar is. Niet onvermeld mag worden gelaten, dat dit onderzoek met de menthol-_ amandelzure esters tot onderwerp, sterke verwantschap vertoont met mijne onderzoekingen over de strychnine tartraten, waardoor ik mij genoodzaakt heb gezien tot het doen eener voorloopige mededeeling 4) om de gelijktijdigheid van mijne experimenten met die der Engelsche onderzoekers ter kennis te brengen. | Ik meen hiermede de gevallen van partiëele racemie, voor zoover _ ik die in de litteratuur heb kunnen vinden, voldoende te hebben — besproken en zal thans overgaan tot het experimenteele gedeelte dezer verhandeling, waarin ik verslag zal doen van mijn experi- menteele werk, en tevens mijne waarnemingen en de door mij gevolgde methode van onderzoek aan die der overige onderzoekers in dit gebied zal toetsen. | ") Versl. Kon. Academie 18", 329 (1909). HOOFDSTUK IV. EXPERIMENTEELE ONDERZOEKINGEN OVER DE STRYCHNINE TARTRATEN. 1. INLEIDING. a. Bereiding der wtgangsprodukten. Als grondstoffen werden gebruikt : d-wijnsteenzuur, uit den handel; druivenzuur, afkomstig van KarrBAuM en strychnine van Zimmer & C°. en van Merck. Het Zwijnsteenzuur werd bereid uit het druivenzuur door middel van cinchonine. De drie stoffen voor het onderzoek vereischt: het strychnine /- resp. d-tartraat en het strychnine 7-tartraat werden verkregen volgens het voorschrift van Docror. !) h. Het kristalwater. Het kristalwatergehalte werd op de gebruikelijke wijze bepaald door een afgewogen hoeveelheid in een droogstoof op 110° tot constant gewicht te verwarmen. De volgende uitkomsten worden daarbij verkregen: BABE l-tartraat. d-tartraat. r-tartraat. hydraat 0.4789 1.1516 0.6928 1.0102 0.9330 1.5926 na verwarming 0.4407 1.0596 0.5932 0.8649 0.8314 1.4236 | dus water 0.0382 | 0.0920 0.0996 0.1453 0.1016 0.1690 dr. |. water 1.998 1.98 14-316 14.384 10.89 10.61 di 4. 3.99aq 8.95aq 1.6 C6 5.5 5D dus 4aq AT 5*|,aq *) Ik moet hier helaas zeer onvolledig zijn. Omtrent deze (uit den aard der zaak dagelijks weerkeerende) bereidingen liet Durin geen enkele aanteekening achter. Kleine, door hem ingevoerde wijzigingen, blijven mogelijk. Kruvr. 4,4 THEORETISCHE EN EXPERIMENTEELE ONDERZOEKINGEN Vergelijkt men deze uitkomsten met die van andere onderzoekers, dan komt men tot de volgende tabel (alle gemiddelde der series). PACD ala see l-tartraat. d-tartraat. r-tartraat. 1 H, 0 | mol. aq. {-°/, H, O | mol. ag. | /, HO | mols Deore: 1:20 3%, 13.37 7 12.47 67 PASTEUR nee. 7.8 nr 14.45 1e DUT A heee Le 1-99 4 14.38 Kai. 10,75 ij Men ziet dat voor d- en /-tartraat de uitkomsten van Pasteur en de onze overeenstemmen, die van Doctor afwijken. c. Let specifiek gewicht. Specifiek gewicht werd zoowel van de anhydrische als van de gehydrateerde zouten bepaald. Het geschiedde door middel van gewone fleschjes-pycnometers met nauwkeurig ingeslepen stop. Als vloeistof werd watervrije toluol gebruikt, terwijl de lucht verwij- derd werd door evacuatie aan de luchtpomp gedurende eenige minuten. De bepalingen geschiedden bij 25°.0. De uitkomsten zijn nu: | TABEL anhydrisch d-tartraat | anhydrisch Ztartraat | anhydrisch racemaat *) 1:430 1.428 1.428 1382242382 1.384 12368 gemidd. 1.429 gemidd. 1.382 gemdd. 1.885 Docror®) 1.43218 Docror 1.34050 Docror 1.36653 cy UC: *) Ann. d.-Ch. et d. Ph. [3] 38. *) De bepalingen van Doctor zijn bij 20° geschied. ‘) Deze getallen ontbraken in Durin's journaal. Daar juist hunne herbepaling van gewicht is, werden deze spec. gew. bepalingen op mijn verzoek door een leerling van Prof. van Rompurcu, den heer C. F. van Duin, chem. stud., verricht. Hem zij daarvoor hier dank gebracht. Kruyt. OVER PARTIËELE RACEMIE. 45 mgee gehydrateerd d-tartraat | gehydrateerd /-tartraat | gehydrateerd racemaat 901 1.390 1.386 1.388 3794870 gemdd. 1.391 gemidd. 1.387 gemidd. 1.372 Docror 1.548 Docror 1.60802 Doctor 1.46968 Ook hier bestaat dus een vrij groot verschil met Docror’s cijfers, vooral voor de gehydrateerde zouten. d. De Smeltpunten. Doctor geeft l. c. de volgende smeltpunten op racemants ses oe 222° detartraat. cate a 220° | LR NT RE ety, 942° terwijl een aequimoleculair mengsel van d- en /-tartraat bij 283— 236° smolt. Er rees nu de vraag: zijn dit de smeltpunten der hydraten of der anhydrische stoffen? A priori is het laatste wel het waarschijnlijkst, daar bij deze hooge temperatuur de waterdampspanning natuurlijk een zeer aanzienlijke is en daardoor de bepaling van het smeltpunt (als dat al realiseerbaar is) groote voorzorgen eischt. Bij de eerste bepalingen met het racemaat-hydraat in open capil- lairen bleek bij gewoon opwarmen nu al terstond, dat de hydraten zich ontleden, dat men bij onscherp te bepalen temperaturen gedeelte- lijke smelting te zien kreeg van de kristallen, die door waterverlies dof werden. Bovendien had bij deze langzame verhitting bruinkleuring plaats. Brengt men de capillairen echter in H,SO,, dat reeds hoog verhit is (+ 220°), dan heeft terstond een uitkoken van het kristal- water plaats en vervolgens een smelten. Bij de hydraten van de enkel-tartraten werd het volgende waargenomen. Brengt men het d-hydraat in een bad van 220° dan heeft eerst een moment geheel smelting plaats, onmiddelijk gevolgd door koken en hernieuwde stolling, die bij 227 à 228° door blijvende smelting wordt gevolgd. Daar er dus smelting van het hydraat was opgetreden, werd door toegesmolten capillairen in een telkens hooger verhit bad te dom- pelen, dit smeltpunt opgezocht. Geen smelting had plaats in baden van 120°, 140° resp. 160°, wel in een van 180°, niet bij 175°, wel bij 177.5° eveneens bij 174.5°. Het verschijnsel komt dus niet geheel regelmatig voor. Er werden nu capillairen gebruikt, die vlak boven de vaste stof waren toegesmolten; daarbij bleek dat totale AG THEORETISCHE EN EXPERIMENTEELE ONDERZOEKINGEN smelting zelfs bij 170° mogelijk is, bij 155° gedeeltelijke smelting tot een troebele vloeistof plaats had en bij temperaturen beneden 150° practisch geen vervloeiing plaats had. Wij zullen de zaak dus zoo moeten opvatten: het hydraat heeft een smeltpunt omstreeks 177°, maar slechts metastabiel, daar de stabiele toestand daar anhydride-vloeistof en damp is. Een nadere bestudeering der mono- varlante evenwichten wordt echter onmogelijk gemaakt door de betrekkelijk spoedig optredende ontleding, die uit geelkleuring der smelt blijkt. Bij racemaat en /-tartraat werden in toegesmolten buisjes analoge verschijnselen waargenomen. Het was intusschen genoegzaam ge- — bleken, dat de smeltpunten boven 200° aan de anhydrische stoffen toekomen. Bij de bepaling van de smeltpunten der gebruikte praepa- raten werd nu gevonden TABEL 4. Domm <= Docror anhydrisch d-tartraat........ 227°—228° 228° ENE 246 | 249° 22 3°— 224° 922° een zeer bevredigende overeenstemming met de bepalingen uit LADENBuRG’s laboratorium. Er moet nu intusschen op een hoogst be- langrijk feit de aandacht gevestigd worden, een feit waarvan de beteekenis aan LADENBURG c.s. ontsnapt schijnt te zijn. Een aequimole- culatr mengsel der anhydrische tartraten smelt bij 233°—236°, het anhydrische racemaat daarentegen bij 222°. Met anhydrische race- maat is dus geen mengsel der tartraten maar een chemisch individu. Deze conclusie wordt bevestigd door de bepalingen der s. g. in onze Fig. 10. vorige $ besproken: Met s.g. van het anhy- drische racemaat is met hetzelfde als dat hetwelk: men voor een mengsel der enkeltartraten berekent. Wij kunnen ons derhalve het smeltdiagram der anhydrische zouten voorstellen, zooals in fig. 10 geschetst. Wellicht ook is het anhydry- sche racemaat zimmer stabiel naast vloeistof. De beteekenis dezer conclusie zal ons in de volgende $ blijken. CS > OVER PARTIÈELE RACEMIE. À e. De onderzoekingen met den tensimeter van Bremer-lrowein. Docror heeft het overgangspunt in het hier besproken systeem ook tensimetrisch trachten te bepalen. Hij bracht daartoe in den eenen bol van den tensimeter racemaat, waaraan eenig water onttrokken was. Aannemend nu dat deze reactie zich afspeelt: 2 [(r-tartraat) 64 ag | (d-tartraat) 7 ag + (/-tartraat) 31/, ag 4-21/, H,0 zal dus aan deze zijde van den tensimeter een vier phasen systeem, bestaande uit 7, d en /tartraathydraat en gas ontstaan; de druk van dit monovariante systeem zal nu bij de temperatuur van het over- gangspunt gelijk moeten zijn aan dien van het monovariante systeem, dat in de andere bol van den tensimeter zich bevindt nl. d- en /-tartraat, met hun verzadigde vloeistof en damp. M. a. w. de temperatuur, waar het spanningsverschil 0 is, 1s die van het over- gangspunt, omdat in het overgangspunt de dampdruk-curven der monovarlante systemen elkaar snijden. (Zie Hoofdstuk 11). Zulke verhoudingen blijken nu intusschen in het onderzochte systeem absoluut niet op te treden. Onttrekt men aan 7-tartraat hydraat water dan splitst het zich niet in de twee enkeltartraat hydraten maar er ontstaat r-tartraat anhydride! In het eerste bolletje is dus heelemaal geen monovariant, maar een bivariant evenwicht, de dampspanning is door de temperatuur niet bepaald, van een 0 worden in het overgangspunt als criterium is dus geen sprake. Onbegrijpelijkerwijze heeft Docror echter een precies kloppende uitkomst gekregen, hetgeen niet slechts op bovenstaande theoreti- schen grond verwonderlijk is, maar ook om een experimenteel- technische reden. Zijn geheele onderzoek is nl. op een middag tusschen 2 uur en 5.45 verricht, alle dampspannings-evenwichten stelden zich bij temperatuursveranderingen binnen enkele minuten in, hetgeen volkomen in strijd is met onze ondervinding en die van anderen, die ons over hun ervaring met den differentiaal tensi- meter inlichtten. Wij hebben dan ook maanden er aan besteed deze proef van Docror te reproduceeren, maar steeds zonder eenig succes. Vreezend, dat zulks aan eigen onbekwaamheid of gebrek in de instrumenten zou liggen, bepaalden wij het overgangspunt van Va, SO, 10 ag, vonden dit evenwel met zeer normaal functioneerden tensimeter tusschen 32° en 33°. De conclusie waartoe wij komen 1s derhalve deze: de bepaling van het overgangspunt is onmogelijk Hu den door Docror beschreven tensimetrischen weg. AS THEORETISCHE EN EXPERIMENTEELE ONDERZOEKINGEN Il. DE OPrLOSBAARHEIDSBEPALINGEN. Wij hebben er boven op gewezen, dat de conclusies waar LADEN- BuRG en Doctor toe gekomen zijn, in strijd zijn met de theoretische verwachtingen en wij hebben dan ook reeds op fouten in hun proef- methoden gewezen (Cf. pag. 19 e. v.). Het kwam er dus op aan vast te stellen op een wijze, die aan geen bedenking onderhevig is, -hoe de zaken zich in dit systeem verhouden en daarom werd besloten de oplosbaarheids-isothermen bij verschillende temperaturen te bepalen, door na te gaan hoe de oplosbaarheid van D resp. L verandert als men Z resp. D aan de oplossing toevoegt. De uitvoering dezer bepalingen geschiedde aldus: Voor de bepaling der oplosbaarheid der enkele stoffen in water werd in een fleschje de vaste stof met water geschud, door het fleschje aan een horizontale, roteerende as, in een thermostaat aan- gebracht, rond te draaien. Na eenigen tijd werd in een pipet, die van onder met een buisje was verbonden, waarin zich een watten prop in bevond, ongeveer 5 c.c.m. heldere vloeistof opgevangen en overgebracht in een klein glaasje met vlakgeslepen rand, waarop een vlakgeslepen deksel-plaatje werd gelegd. Door weging werd de hoeveelheid oplossing bepaald, waarna het glaasje ongedekt in een droogstoof werd gebracht en op 110° de inhoud tot constant ge- wicht ingedampt. Door eenige uren later eenzelfde proef te herhalen werd vastgesteld na hoeveel tijd de oplossing verzadigd was. Op grond daarvan werd steeds minstens 2 dagen geschud. De bepaling der oplosbaarheid van de eene stof in onverzadigde oplossing van de andere werd als volgt uitgevoerd. Wil men b.v. de oplosbaarheid van d-tartraat bepalen in oplossingen, die /-tartraat bevatten, dan bereidt men zich eerst een hoeveelheid verzadigde /-tartraat-oplossmg en zuigt die door watten van het vaste zout af. Brengt men nu in een fleschje, waarin zich reeds vast d-tartraat be- vindt, een bekende hoeveelheid water en een bekende hoeveelheid der verzadigde /-oplossing toe. Pipeteert men nu, nadat de verzadiging aan d-tartraat bewerkstelligd is, wat heldere oplossing af, dan is uit te rekenen, hoeveel /-tartraat zich in die hoeveelheid bevindt en uit het gewicht der droge stof, die na indampen terugblijft (en natuurlijk d- + /-tartraat is) vindt men gemakkelijk de oplos- baarheid van het d-tartraat. Heeft men op deze wijze de verschillende takken der isotherm bepaald, dan kan men de ligging der snijpunten, zooals die door graphische interpolatie gevonden zijn nog verifieeren door water OVER PARTIÈELE RACEMIE. 49 met twee Bodenkörper te schudden. Men vindt dan door indampen de som der beide oplosbaarheden. De meetkundige plaats van alle oplossingen, die een zekere totaal oplosbaarheid van de twee bestand- deelen representeeren, is een rechte lijn die de punten verbindt, welke op elk der consentratie-assen die totaal oplosbaarheid aangeven. Het door interpolatie gevonden snijpunt moet dus op deze lijn liggen. Uit Tabel 5 ziet men de uitkomsten voor de isotherm van 40.0°. 4) _ Hierbij zij het volgende opgemerkt. Als concentratie is steeds opgegeven het aantal m.G. op 5000 m.G. water. Uit proef 17 blijkt, dat het racemaat, hoewel niet stabiel naast oplossing, toch vrij bestendig is: de 1° en 2° bepaling geschiedden na 2 dagen, de 3° en 4° na 3 dagen, de 5° en 6° na 12 uur schudden. Voorts ziet men in de graphische weergave dezer uitkomsten (fig. 11) dat het punt voor vloeistof verzadigd aan d- en /-tartraat gevonden door graphische intra- (resp. extra-) polatie bevredigend overeen- stemt met de zooeven genoemde meetkundige plaats. Bij proef 9 is het voorts gelukt een punt op het metastabiele verlengde der d-verzadigingslijn te bepalen. ‘) Alle temperaturen zijn gecontroleerd met een door de Phys. Techn. Reichsanst. te Charlottenburg geijkte thermometer, verdeeld in 0.1°. Verhand. Kon. Akad. v. Wetensch. (4ste Sectie) Dl. XI. D 4 50 THEORETISCHE EN EXPERIMENTEELE ONDERZOEKINGEN ABB 15: Isotherm van 40°.0. 2S Bij indampen van nl re Gemiddeld Vaste Ë = Bs ie EEn IND a ¢ phase. 3 3 Gr. Gr. L 2 | | STEN d © ° | oplossing.|droogrest. | mel 124-9799 | 0.1708 0 177.8 | 5.0132 | 0.1694 0 174.9 | 5.0038 | 0.1692 0 175.0 6 een 5.0044 | 0.1690 0 174.8 5.0016 | 0.1694 0 175.3 5.0028 | 0.1716 0 177.6 A Ae a 4.9872 | 0.1114 | 114.2 0 5.0046 | 0.1120 | 114.5 0 4.9932 | 0.1118 | 114.5 0 114.5 0 4.9810 | 0.1118 | 114.8 0 3 | 1 | 35.44 | 4.9986 | 0.1308 | 99.0 35.4 5.0050 | 0.1306 | 98.6 35.4 98.8 35.4 A sa lis 0168 | 0 TAA 59.3 5.0198 | 0.1482 | 92.8 59.3 92.5 59.3 10 l 71.44 | 5.0120 | 0.1560 89.2 nlet | 5.0148 | 0.1568 | 89.9 | 71.4 89.6 | A4 io eat wi BO ba | 6.0184 | 0 TELE 89.5 5.0126 | 0.1696 | 85.6 89.5 85.5 89.5 13 | 7 | 108.04 | 5.0216 | 0.1888 | 82.0° | 108.0 5.0308 | 0.1852 | 83.1 | 108.0 5.0190 | 0.1846 | 82.9 | 108.0 82.8 | 108.0 5.0046 | 0.1844 | 88.8 | 108.0 5 | a | 23.01 | 5.0118 | 0.1716 | 23.0 | 154.3 5.0188 | 0.1790 | 230 | 154.4 |, 28-0 | 1544 7 | 4 | 46.91 | 5.0196 | 0.1770 | 46.9 Me 5.0108 | 01766 462 mle | 46.2 136.5 6 | a | 57.92 | 5.0174 | 0.1796 | 57.9 | 1977 5.0190 | 01790 | 57,9 | 127.0 |} 57.9 | 1274 9 a) 87.87 | 5.0104 | 01904 dl are 1/9850 | 0.1900 | 87.3 | 1111 |, 87-8 | 1107 EE SS nd 140 Water | 5-0246°/| 0.1990 196.9 5.0174 | 0.1930 200.0 on 5.0306 | 0.1930 199.5 5.0280 | 0.1938 200.4 hike tid: 5.0192 | 0.1874 193.91 5.0228 | 0.1876 194.0 7 5.0140 | 0.1843 190.8) ioe 5.0200 | 0.1836 189.8% a 5.0070 | 0.1853 199.9 : 5 0210 | 0.1854 191.7) OVER PARTIËÈELE RACEMIE. TABEL 6. Isotherm van 25°.0. 5 | Bij indampen van Samenstelling der Gemiddeld mir 5 © opl. in m.G. Vaste) £ = Gr? P | N°. A à phase. 2 6 Gr: Gr. | : 3 © er = © © | oplossing.| droogrest. Sieg © | Water. | 5.0193 |. 0.1123 0 114.6 5.0100 | 0.1126 0 114.9 7 ube 10.0170 | 0.2234 0 Hk sine 10.0150 | 0.2239 0 114.4 sl Eik 4.9934 | 0.0712 | 72.3 0 4.9921 | 0.0711 | 72.2 0 ze ‘ 10.0080 | 0.1439 | 73.0 0 2. 10.0012 | 0.1434 | 72.7 0 ee 23.0 4 | 5.0061 | 0.0857 | 64.1 23.0 5.0000 | 0.0847 | 63.2 23.0 4.9997 | 0.0838 | 62.2 93.0 | 63.0 | 23.0 4.9917 | 0.0889 | 62.5 93.0 Pepi |» 46 9d | 50056 | 0.0996 | 55.3 46.2 5.0162 | 0.1005 | 55.8 AG 2 5.0046 | 0.0996 | 55.3 46.2 | 55.4 46.2 5.0093 | 0.0995 | 55.1 16 2 Bere 4. 69.74 | 5.0190 |. o 152 |: 47,9 69.7 5.0040 | 0.1150 | 47.9 69.7 5.0152 | 0.1156 | 48.3 69.7 48.0 69.7 501640 1155 |. 48.1 69.7 NR ite 50120 LOTTI 0146 | 109.9 5.0002 | 0.1148 | 146 | 102.9 14.6 | 103.2 B | a | 29.21 | 5.0027 | 0.1176 | 29.2 91.2 Fa te 5.0100 | 0.1186 | 29.2 | 92.0 9.2 JL. BE | 43.91 | 50130 | 0.12138 | 43.9 80.1 510168 | 0.1918 +) 04349 80.0 5.0130 | 0.1210 | 43.9 79.8 | 43.9 80.0 5.0160 | 0.1214 | 43.9 80.1 | Ne) ee _ 9 |d+1| Water | 5.0303 | 0.1229 125.2 5.0107 | 0.1225 195.3 ae 5.0124 | 0.1222 194.9 2 5.0031 | 0.1223 125.3 rtl rtl À RASE à id: 5.0140 | 0.1176 120.1 5.0040 | 0.1172 119.9 120.1 5.0100 | 0.1176 120.2 r + d r + d Be geer dd. 5.0024 | 0.1180 120.8 | 5.0200 | 0.1180 120.4 120.5 5.0030 | 0.1176 120.4 | Je 7 Ds zal 5.0060 | 0.1209 123.7 5.0136 | 0.1213 124.0 | aus 5.0130 | 0.1208 123.5 | see 5.0140 | 0.1204 123.0 52 THEORETISCHE EN EXPERIMENTEELE ONDERZOEKINGEN In tabel 6 vindt men de uitkomsten der geheel overeenkomstig uitgevoerde bepaling voor de isotherm van 25°. Men ziet hoe door de bepaling 1, 2, 7, 6, 8, 3, 5 en 4 de takken voor de oplosbaarheid van /- en d-tartraat zijn bepaald en door bepaling 9 de ligging van het snijpunt bevestigd is; dat ver- volgens door de oplosbaarheid van 7 + / te bepalen is vastgesteld in welk punt der /-tak het snijpunt met de racemaattak gelegen is; en dat vervolgens eenzelfde punt op de d-tak is gevonden voor r+ d. By deze bepalingen 9, 10 en 11 is natuurlijk weer van diezelfde meetkundige-plaats-methode gebruik gemaakt als by de isotherm van 40° is aangegeven. Eindelijk is de oplosbaarheid van racemaat in water bepaald. Het daardoor gevonden punt moet natuurlijk op het metastabiele gedeelte der 7-oplosbaarheidstak liggen. Ziet men de graphische voorstelling (fig. 11) aan, dan blijkt terstond dat wij 4 25° im het overgangstraject zijn. De tak der r-oplosbaarheid snijdt de lijn, die de assenhoek middendoor deelt nog niet, het racemaat is naast een oplossing, die / en d aequimoleculair bevat, niet bestendig. Ten overvloede is door een proef dit ook nog eens direct bewezen. | | Een hoeveelheid der afgepipeteerde heldere vloeistof, die aan race- maat verzadigd was (van proef 13 afkomstig), werd gedurende eenige dagen met een weinig /-tartraat geschud; uit onze graphische voor- stelling kunnen wij aflezen wat er gebeuren moet: aangezien de. oplossing oververzadigd is t.o.v. /tartraat, zal dit uitkristalliseeren en de vloeistofsamenstelling zich dus bewegen in de richting der met een pijl voorziene loodlijn. Inderdaad bleek de droogrest van 123.5 teruggegaan te zijn op 116.4, zie onderstaande Tabel. De proeven werden gedaan na resp. 1, 2, 3 en 8 dagen schudden. FA Bin N°. | Gr. Oplossing | Droogrest Samenstelling Gemiddeld 15 5.0102 0.1140 116.4 | FN 5.0086 0.1140 116.4 5.0056 0.1132 11860 4.9964 0.1144 117.1 | Ten einde eveneens te bewijzen, dat de aan d + JZ verzadigde vloeistof (N°. 9) in labielen toestand was t.o.v. het racemaat, werd als proef N°. 14 deze vloeistof twee dagen met 7 geschud, waarbij eveneens een terugloopen werd geconstateerd. Zie Tabel 8. 150 à (15) N > f. PROD 735 « ; 5) 5 Ÿ 3 DR a ee 50 100 150 200 OVER PARTIÉELE RACEMIE. 53 Pe poe. N°. | Gr. Oplossing Droogrest Samenstelling | Gemiddeld 14 5.0120 0.1212 123.9 ‘ 5.0080 0.1214 124,9 123.6 5.0146 0.1202 122.8 Ten einde vast te stellen bij welke temperatuur het overgangs- traject zijn laagste begrenzing had, werd nu een isotherm bij aan- zienlijk lagere temperatuur bepaald nl. bij 16°. Tabel 9 geeft een overzicht van de uitkomsten, die in fig. 11 weer graphisch zijn weergegeven. Men ziet nu terstond in dat wij hier bij 16° zog steeds in het overgangstraject zijn; de r-oplossing is nog steeds labiel. Bij deze tabel dient het volgende aangeteekend te worden. Het principe is natuurlijk weer hetzelfde als bij de isotherm van 25°: eerst de d- en /-oplosbaarheidstakken bepalen en hun snijpunt con- troleeren, dan de snijpunten bepalen met de racemaat-oplosbaar- heidslijn (7 + / en r +d). Dat bleek nu echter minder eenvoudig te gaan. Schudt men nl. racemaat en d met water, dan heeft men aanvankelijk en nog gedurende een betrekkelijk langen tijd (ten- gevolge van het langzaam voortschrijdende oplossen) racemaat naast water. Aangezien wij nog in het overgangstraject zijn is het rac. onder deze omstandigheden labiel en zal zich in d en / splitsen. Het aanwezige vaste d-zout zal dat proces, bijwijze van entings- materiaal, bevorderen. Zoodoende zal al spoedig zooveel 7 gesplitst zijn, dat er vast / op den bodem ligt. Met zuiver water als oplos- singsvloeistof werden dan ook zeer onregelmatige uitkomsten ver- kregen. Ten einde nu deze racemaatsplitsing te voorkomen werden geheel of gedeeltelijk verzadigde oplossingen der enkeltartraten als schud-vloeistoffen gebruikt. Met zuike oplosmiddelen zijn de proeven 5 en 6 uitgevoerd, die, zooals men ziet, nu zeer regelmatige uit- komsten gegeven hebben. Nochtans lijken ons de uitkomsten voor de oplosbaarheid van r + d niet geheel vertrouwbaar en schijnt het ons onmogelijk een volkomen correcte methode te vinden voor de bepaling van de oplosbaarheid van het racemaat naast zijn meest oplosbaar splitsingsproduct. Gelukkig is het echter juist de oplosbaar- heid van racemaat met het minst oplosbare splitsingsproduct waaruit wij het einde van het overgangstraject moeten leeren kennen. 24 THEORETISCHE. EN EXPERIMENTEELE ONDERZOEKINGEN TABEL 9. Isotherm van 16°. Samenstelling der = Bij indampen van : vie E 5 RES in m.G. | N°. EH ‘à © : os Home) roi | 2 | d Water 5.008 0.0868 0 88 :2 | 5.012 | 0.0873 0 88.6 | + 5.002 0.0876 0 Sa: | 5.003 0: 082024) Sea 88.5 2 l id 4.9886 | 0.0563 DAL 0 | 4.9804 | 0.0564 51.2 0 Hae 4.9500 | 0.0566 57.8 0 5 5.0000 | 0.0572 ay es) 0 11 l 17.8d 4.996 0.0654 48.5 178 48.8 4.998 0.0660 A0 soy Ge, . In l 59.1 d 4.997 0.0768 49.3 21.4 4.996 0.0768 492.6 35.7 49 5 5.002 OO 42.5 oD. 1 ‘ 5.001 0.0772 42.5 DO 13 l Deed 4,962 0.0880 36.6 53: ¢ | 4.961 0.0881 36.6 53.1 97 0 4.958 0.0887 at. 53.7 ; 4.963 0.0886 Bi D9.1 8 d dell 5.007 0.0862 hi eae A0 ) 11.5 5.000 | 0.0868 | 11.5 Ac Er 9 d 4.11 5.000 0.0904 34.1 57.4 5.007 0.0902 34.1 DAS 24 7 5.002 0.0906 34.7 Di 7 5.007 0.0914 SAUT 58.5 10 d 46.41 4.965 0.0937 46.4 49.8 4.950 0.0929 46.4 49,9 46.4 4.950 0.0950 46.4 Ad? . 4.966 050955 46.4 49:25 3 | d+l Water 5.006 0.0960 OW ca. 5.011 0.0966 98.3 98.1 5.010 0.0962 98.3 | rtd r + d Beel 8.009" Yos 89.9 5.004 0.0882 SET | 89 9 4.996 0.0882 89.9 | En 4,935 0.0874 90:1 r+] aie 6er ia 5.006 | 0.0912 92.8 5.005 0.0910 92.8 | 99 5.004 | 0.0915 92.8 | oF 4.999 020917 03.4 r r 4 r Water 1.962 0.0928 95:58 4.952 0.0934 95.4 4,961 0.0936 671 95.6 4.963 0.0932 95.6 à 5.004 0.0928 94.5 4,005 0.0930 94.5 OVER PARTIÉELE RACEMIE. 55 LAB ET 10. Isortherm van 7°.35. Samenstelling der 3 3 By indampen van opl. in m.G. Gemiddeld Vaste| £ © ere dh ee giek N°. a ‘à phase. 3 8 Gr. Gr. Ë | ; o & oplossing. | droogrest. | 2 d Water | 3.9882 | 0.0550 6 69.9 4.4532 | 0.0617 0 70.2 | + HE 4.9483 | 0.0697 0 71.4 || 4.9644 | 0.0698 0 71.3 1 l id. 4.9731 | 0.0467 | 47.4 0 £9618 | 0.0465 | (47.3 0 I * 4.9278 | 0.0465 | 47.6 0 4.9544 | 0.0465 | 47.4 0 5 l 26.3 d | 5.0248 | 0.0628! 37.0 26.3 |) a4 26.3 5.0080 | 0 0628 37.2 Lie Li Hs EU 5 1 39.5d | 5.0021 | 0.072 | 33.9 39.5 ne 4.9938 | 0.0714 | 34.0 39.5 34.0 gen 4 1 13.14 | 5.0474 | 0.0558] 42.6 Bet ; 4.8934 | 0.0532 | 41.8 13.1 22 | 18.1 7 d 10.02 | 4.9863 | 0.0708 | 10.0 62.0 |) 3 4.9850 | 0.0709 | 10.0 62.1 |; 10.0 62.1 8 d 20.01 | 4.9538 | 0.0714] 20.0 53.1 a re 4.9430 | 0.0716 | 20.0 53.5 20.0 58.3 9 d 80 11 | 429640") 20-0040 19304 46.2 ; 4.9612 | 0.0740 | 30.1 156 | 20-1 | 45.9 14 |a-+i| Water | 4.9617 | 0.0745 76.2 4.9808 | 0.0744 75.8 75 & 14bis 4.975 0.0736 Zoe t 7 4.996 | 0.0738 75.0 r + d r + d nt er id. 5.0164 | 0.0743 75.2 50 5.0270 | 0.0742 74.9 De rtl r+l fad ap ie J id. 4.8998 | 0.0701 72.6 4.9582 | 0.0699 71.5 12bis id. 4.950 | 0.0688 70.4 4.968 | 0.0685 69.0 ae 12ter id. 5.032 0.0684 68.9 ke 4.948 | 0.0674 69.0 5.022 | 0.0682 68.8 4.983 | 0.0679 cd À fe ie epe id 4.942 | 0.0675 69.2 BATS 4.946 | 0.0678 69.5 Ster id. 4.940 | 0.0688 70.6 nt 4.959 | 0.0691 70.7 4.927 | 0.0686 70.6 4.948 | 0.0689 70.6 56 THEORETISCHE EN EXPERIMENTEELE ONDERZOEKINGEN Om het einde van het overgangstraject te vinden moest dus naar nog lagere temperatuur gegaan worden, welke proeven in een koude Januarimaand werden uitgevoerd. De thermostaat werd daarom op een binnenplaats in de open lucht geplaatst en was door die plaat- sing buiten den wind (die het vlammetje zou uitblazen) en het zonlicht. Zoodoende kon een reeks proeven gedaan worden bij 7°.35, waarvan de uitkomsten in Tabel 10 zijn aangegeven. Men zal uit de tabel en uit fig. 11, die de uitkomsten weer- geeft, zien dat wij thans inderdaad het einde van het overgangs- traject bereikt hebben. De oplosbaarheid van het racemaat (70.1) is binnen de proeffouten gelijk geworden aan die van racemaat + / tartraat (70.0); de lijn, die de assenhoek midden doordeelt, gaat juist door het snijpunt der 7 en 7 oplosbaarheidtakken. Toch diende nog gecontroleerd of inderdaad bij een iets hoogere „temperatuur de twee juist genoemde oplosbaarheden genoegzaam verschilden om met eenige zekerheid te kunnen zeggen, dat hier inderdaad het einde van het overgangstraject gevonden is. Daartoe werd nog bij 8.°9 de oplosbaarheid van r en van r + / bepaald. Tabel 11, die deze uitkomsten opsomt, toont aan dat daar inder- daad een verschil valt te constateeren, dat de proeffouten verre te boven gaat. TAB EME Oplosbaarheden bij 8°.9. | ke ; ‘ | | a Vaste Oplossngs Bij indampen van Oplosbaar Gemiddeld een | GE | Gr.Oploss.|Gr. droogr. Se | 15 AME water 51002 0075 14.8 | | 4.766 | 0.0710 | 75.6 “59 4.979 . | 0.0735 14.9 4.950 0.0736 15.5 25 end 4.958 0.0712 72.8 | 4.961 | 0.0714 | 73.0 79.3 5.098 0.0714 da: 4.994 0.0714 12.5 Wij mogen derhalve als conclusie zeggen, dat het overgangstraject zich naar lagere temperaturen tot (afgerond) 74° witstrekt. OVER PARTIÉELE RACEMIE. 5 Door enkele proeven wenschten wij nog te constateeren of het door LApeENBurG en Docror (l.e.) aangegeven punt van ongeveer 30° als overgangspunt, d. i. bovenste grens van het overgangstraject, inderdaad juist was. Het zekerst was dat na te gaan, wanneer men althans niet weer geheele isothermen wilde bepalen, uit het gelijk- worden der oplosbaarheden van 7 + / en d +7. Bij 25° waren deze waarden resp. 120.1 en 125.2 en waren wij dus beslist beneden de overgangstemperatuur. Bij 80° vonden wij uitkomsten, weergegeven in Tabel 12. ABEL 12 Oplosbaarheden bij 30°. i} indan ee dean an Oplosbaar: | Gemiddeld phase heid Gr. oplossing Gr. droogrest Mene / 5.011 0.1362 139.7 5.010 0.1371 140.7 | 110,6 5.010 0.1372 140.7 | 5.009 0.1375 141.4 | ae hal 5.018 0.1372 140.5 5.018 0.1372 140.5 | 141.0 4.964 0.1368 141.7 5.008 0.1376 141.3 | Daar dus de waarde voor d + 7 niet meer hooger is dan die voor > + / schijnen wij juist even boven het overgangspunt te zijn; daar de afwijking niet veel van de proeffout verschilt mogen wij dit punt slechts op 29°—30° vaststellen. 58 THEORETISCHE EN EXPERIMENTEELE ONDERZOEKINGEN Ter contrôle werden intusschen nog de in Tabel 13 aangegeven bepalingen bij 32° verricht. TABEL 13. Oplosbaarheden bij 32°. Bij indampen van ro en D EPS LEE Fe es Gemidd Pias. (Gr. oplossing Gr. droogrest : i rey 5.009 0.1234 1263 | 5.007 0.1234 126.3 4 5.006 0.1238 126.8 26,4 5.004 0.1235 126.5 ga 130975010 mede 152.8 | 5.020 0.1484 159.3 152,8 5018 | 0.1492 LBB et Deze uitkomsten bevestigen het voorafgaande, zooals zonder meer duidelijk zal zijn. | VUE Een opmerking zij gemaakt over de waarde der door LADEN- BURG en Docror gegeven cijfers voor de oplosbaarheid. Op het eerste gezicht ziet men al dat aan hun methode deze twee fouten kleven: 1°. werd na 3 uur reeds tot analyseeren der oplossing overgegaan; 2°. werd daarbij niet van een watten filter gebruik gemaakt, terwijl men toch hier nimmer een volkomen bezinken der vaste stof bewerkstelligen kan. Is het feit dat L. en D. tot theoretisch onmogelijke uitkomsten gekomen zijn reeds een bewijs tegen hun methode, Tabel 14, die eenige uitkomsten bevat van die onderzoekers en de onze bij zelfde temperaturen, toont genoeg- zaam, dat zij nimmer de eindevenwichten bereikt hebben. Eenige malen vindt men niet gelijke temperaturen vergeleken, maar dan zal men zien, dat de door L. en D. gevonden oplosbaarheden, hoewel bij een hoogere temperatuur bepaald toch nog een geringere waarde aanwijzen dan bij onze proeven gevonden werd. > OVER PARTIEELE RACEMIE. 59 TABEL 14. Je Oplosbaarheden volgens ae este vie Bemerkingen © | phase. | LADENBURG | arena 3 ; en Docror | 49°| 4 = Je 187.4 ps | De r oplosbaarheid bij 42° vol- . Be eed i AQ° oh raw 199.2 | ieee ee as die volgens 80° df 128.8 141.0 SE RS de at 12607 140.6 Ted HZ 74.0 15.5 vase Lae Hs at loch die van L. en D. r 69.4 70.1 || fételÿk bij 7° + O°. Md 97.5 Fe | 16° id. en 98.1 _ Men merke. op dat D’s bepa- 19° x 94.9 Rize lingen hier 3° lager werden uit- : gevoerd dan by L. en D. Deed. -— 95.6 Zooals men ziet zijn de waarnemingen van LADENBURG en Docror aanzienlijk veel te laag, behalve bij 7°, waar het verschui inder- daad gering is. Het spreekt vanzelf dat men dan ook beter doet uit deze cijfers van LADENBURG en Doctor geen conclusies te trekken. Meryeruorrer heeft op eenige afwijkingen in die cijfers op- merkzaam gemaakt +), welke op omzettingen in de bestanddeelen zouden wijzen. Wij vermoedden, dat deze afwijkingen, gezien het voorafgaande, wel aan de onjuistheid der cijfers zouden zijn toe te schrijven. „Doch müssen diese Verhältnisse noch emgehender geprüft werden”, schrijft Mrvyernorrer, die dus terecht ook niet op dit getallenmateriaal wil afgaan. Wij hebben nu één dier afwijkingen nagegaan: M&yrruorrer zegt van de cijfers van L. en D.: „Auch weist die Löslichkeit des /-Tartrats zwischen 27 und 30° eine Unregelmässigkeit und hernach eine Richtungsände- d-Loslichkeit dt grösser ist als unterhalb 27°. Dies deutet auf eine Veränderung des /-Tartrats zwischen 27 und 30° hin..... | Wij hebben nu de oplosbaarheidslijn van het /-tartraat van 25° tot 30° bepaald. Tabel 15 geeft daarvan de uitkomsten weer. rung auf, in dem merkwürdigerweise oberhalb 30° ") Gleichgewichte der Stereoisomeren (1. c.) pag. 49 voetnoot. 60 THEORETISCHE EN EXPERIMENTEELE ONDERZOEKINGEN Men ziet daaruit dat van een dergelijke bizonderheid geen sprake d-Lôslichkeit” ARR EL de oplosbaarheid lineair met de temperatuur verandert. is, dat, binnen de proeffout constant is, dat dus pd TABEL 15: Oplosbaarheidscurve van het /-tartraat van 25° bis 30°. mn Oplosbaar- EEEN d-Lôslichk.” | 2 5 _ heid STE Bij indampen van Gr. oplossing (Gr. droogrest | 25°| 4.993 0.0712 72.3 4.992 0.0711 72.2 À 10.008 0.1439 13.0 hao 10.001 0.1434 72.1 26 26°| 4.997 0.0738 15.0 | 4.982 0.0740 10.4: oa ee ea i 27) 5.005 0.0768 .| 77.9 era 4.997 0.0768 78.0 on | 28°| 4.997 0.0792 80.5 2.2 4.999 0.0788 80.1 : | 4.998 0.0782 79.5 i 4992 | 0.0792 | $0.6 | à 29°| 4.951 0.0800 82.1 | À Bik 4.954 0.0804 $25 Ate 30°} 4.950 0.0827 85.0 || | 2.8 4.949 0.0829 85.2 ail ARE Ten einde ook op dat gebied een inzicht te krijgen omtrent de waarde van L. en D’s. cijfers, hebben wij nog eenige bepalingen van het optisch draatingsvermogen aan verzadigde oplossingen verricht. In een polarimeter (volgens LaNporr, driedeelig gezichtsveld, fabr. Scumipt & Harnscu) werd eerst onderzocht de bij 30° aan d en / verzadigde oplossing, van welke wij (zie Tabel 12°) ge- vonden hadden, dat de concentratie 141.0 per 5000 m.G. oplos- middel was d.1. 2.82 gr. per 100 gr. Gevonden draaiing — 2,50°. OVER PARTIBELE RACEMIE. 61 Nu kan men uit de cijfers van L. en D. terugrekenen welke draaiing zij afgelezen hebben, indien zij hunne bij 30° aan d en / verzadigde oplossingen gebezigd hebben. Zij geven nl. op: Gevonden conc. 2.575 Gr. op 100 Gr. water. nr 9 deet el Ss Gr -0p. 100 Gr. water. LA Mle te open ER © Dit is geconcludeerd uit pla = — 20.6073 + 0.9367 X 1.313 dus == — 19.38° [ap], = — 31.3634 + 1.8564 X 1.262 dus = — 29.65° [a | d +1 CE 49,03° zoodat de afgelezen draaiing « (indien eenzelfde buislengte is ge- bruikt) aldus is te vinden ee 1002 Re AU 100 z M DENON waaruit volgt @ == — 2,52° merkwaardigerwijze (practisch) hetzelfde getal als wij vonden. Hieruit (aangenomen dat de fout niet in de polarimeterbepa- lingen gelegen is) zou men moeten concludeeren, dat L. en D. en wij dezelfde verzadigde vloeistoffen in handen gehad hebben en de fout bij hen in hun analyse-methode gelegen is, waarover wij boven al gesproken hebben. Een toetsing der bovengebruikte LApENBuRrG-Docrorscum-formule van het /-zout leverde een zeer onbevredigend resultaat. De op- lossing, die bij 25° aan /-zout verzadigd was, en bij analyse bleek te bevatten 72.6 m.G. per 5000 m.G. water en dus 1452 Gr. per 100 Gr. water, leverde een draaiing a van — 0.93°. Volgens bedoelde formule was hare concentratie negatief! Ook andere be- palingen leverden geen overeenstemming. ‘) De door L. en D. gegeven concentratie. 62 THEORETISCHE EN EXPERIMENTEELE ONDERZOEKINGEN Samenvatting. Overzien wij de onderzoekingen boven beschreven omtrent de strychninetartraten dan komen wij tot de volgende conclusies: Uit de onderzoekingen aangaande smeltpunt en spec. gew. der strychninetartraten blijkt, dat het druivenzuurstrychnine anhydrisch bestaanbaar is en dat dientengevolge de tensimeterbepaling van het overgangspunt zooals door LADENBURG en Doctor pues onmo- gelijk tot juiste resultaten kan voeren. | Door een reeks oplosbaarheidsbepalingen, die zoo ingericht waren, dat de hoeveelheden der. componenten rechtstreeks bekend waren, werden isothermen bepaald en vastgesteld, dat er geheel overeen- komstig de door BaAkuuis Roozrsoom gestelde verwachtingen, een overgangstraject in dit systeem optreedt en wel tusschen 71 en 30°. Met proeven gedoeumenteerde bezwaren werden tegen alle onder- zoekingsmethoden van LapeNBure en Docror aangevoerd. Ook een door Meyrernorrer uit hun data afgeleid vermoeden werd door experimenten opgeheven. HOOFDSTUK V. EXPERIMENTEELE ONDERZOEKINGEN OVER DE ZURE BRUCINE-TARTRATEN. 1. INLEIDING. a. Door de in het vorige hoofdstuk beschreven onderzoekingen is feitelijk de juistheid van RoozrBoom’s critiek op de onderzoekingen van LADENBURG geheel gerechtvaardigd. Het nader nagaan van een tweede systeem behoefde dus niet zóó uitgevoerd te worden, dat ook hier een volledig isothermen-net werd bepaald. Slechts kwam het‘ er op aan na te gaan of in dit systeem, waar het stabiliteits- gebied van het partiëele-racemaat zich naar /oogere temperaturen uitstrekt (cf. fig. 6), ook een overgangstraject zou zijn aan te toonen en daarmede de LaApenNBurasche voorstelling ook in dit systeem afgewezen zou kunnen worden. De uitkomsten zullen echter in nog veel hooger mate de onvoldoendheid van het onderzoek van Lapensure en Frscur ') aanwijzen. b. Bereidingswijze der mtgangsproducten. Omtrent de wijnsteen- en druivenzuur praeparaten zij naar het vorige hoofdstuk verwezen. De gebruikte drucine was een praepa- raat afkomstig van Zimmer & C°. 6. Bereiding van het zure d-wijnsteenzure brucinezout. Er werd afgewogen 1.000 gr. wijnsteenzuur, die, na oplossing, met brucine werd geneutraliseerd tot lakmoespapier niet meer rood werd; daarna werd de overmatige brucine uit de heete oplossing *) Ber. der deutsch. chem. Ges. 40, 2281 (1907). 64 THEORETISCHE EN EXPERIMENTEELE ONDERZOEKINGEN afgefiltreerd en wederom 1.000 gr. wijnsteenzuur toegevoegd. Uit de aldus verkregen + 150 ccM. kristalliseerde eerst in koud water, dan in ijs het zure zout als fijne naaldjes (langer dan bij het over- eenkomstige d-zout) uit. Ze werden op den zuigtrechter van de moederloog gescheiden. Bereiding van het zure l-wijnsteenzure brucine. De bereiding van dit zout geschiedde op overeenkomstige wijze: het zout zette zich als kleine, straalsgewijze zich tot wratten samen- hoopende kristallen op bodem en wanden van het bekerglas af. Het afzuigen werd bemoeilijkt door de brijige consistentie der moeder- loog, daarom werd deze verwijderd door de kristallen tusschen filtreerpapier zooveel mogelijk te drogen. Daarna bleven ze eenige dagen aan de lucht liggen. Bereiding van het zure druivenzure brucine. Ook hier geschiedde de bereiding op gelijke wijze. Slechts werd bij verschillende bereidingen de kristallisatie bij verschillende tem- peraturen uitgevoerd. Eenmaal werd ook afgewogen: 2.000 gr. druivenzuur en 9.890 brucine, welke hoeveelheden elkander juist moesten neutraliseeren indien de brucine anhydrisch was. Dat bleek ook hier weer inderdaad het geval. c. Analyse der proefstoffen. Daar een eerste reeks oplosbaarheidsbepalingen voor ons onver- klaarbare uitkomsten had gegeven, hebben wij, ter vaststelling dat geen onregelmatigheden in onze proefstoffen oorzaak daarvan waren, deze aan verschillende analysemethoden onderworpen. TA BBL: Elementair Analyse Stikstof bepaling (Dumas) gev. ber. gev. ber. PER O0 7 59.56 I. 5.43 5.14 a PatieG. 1.5 5.88 2, 5.44 5.14 BNR EEN 59.56 /-zout | H 6.00 5 88 5.5 5.14 0.14 Or r-zZ0 ut niet verricht D. OVER PARTIBELE RACEMIE. 65 Allereerst werden elementair analyses en stikstofbepalingen gedaan. (Zie Tabel 16). Deze gaven normale uitkomsten, maar vormen geen criteria ervoor of we nu inderdaad de zure zouten zuiver in handen hebben. De procentische elementairsamenstelling der zewtrale en zure zouten verschilt te weinig, dat een verontreiniging van eenige procenten der eerste in de laatste aanleiding zou geven tot grootere afwijkingen dan de normale proeffouten dezer methoden zijn. Een titrimetrische analyse bleek intusschen de gewenschte zeker- heid te verschaffen. Titreert men nl. de zure zouten met ongeveer _0.01 n. KOM en rosolzuur als indicator, dan heeft de kleuromslag plaats als het neutrale brucine-kaliumzout is gevormd. De vloeistof blijft dan ook helder tot den kleuromslag, terwijl bij gebruik van b.v. phenolphtaleine de troebeling door brucine, door KOM afge- scheiden, reeds optreedt vóór nog kleuromslag heeft plaats ge- vonden. Een koolzuurvrije KOH-oplossing werd gesteld op barnsteenzuur en bleek 0.01163 n. I e.e.m. dezer loog correspondeert dus met 1163 1000 B. en C. werd een andere oplossing gebruikt, waarbij 1 c.c.m. met 649 mgr. correspondeerde. De tabellen 17 bevatten de uitkomsten : X 5.445 mgr. — 633 mgr. winsteenzuurzout. Bij de series ARE LUE A. Tilratie zuur d-brucine tartraat. Afgewogen 0.5291 Gr. anhydr. zout, opgelost in 100 c.c.m. water. mgr. zout eem oplg eenn. KOH | ———— gevonden | berekend 25.00 | 20.97 | à Het (eenmaal om- © gekristalliseerde) zout eN € 29 « O 25.00 | 20.88, 3 1324 132.3 | is dus als zuiver te Bei Pinal | 8 beschouwen. | | le, | Verhand. Kon. Akad. v. Wetensch. (ste Sectie) Dl. XI. D 66 THEORETISCHE EN EXPERIMENTEELE ONDERZOEKINGEN B. tratie zuur l-brucine tartraat. Afgewogen 0.6070 gehydr. zout, opgelost tot 100 c.c.m. mgr. zout e.c.m, ople Brem O 17 gevonden | berekend ie A | OE heee = Het (eenmaal om- 5 50 gekristalliseerde) zout LOO Oe 08) Ve. 51.92 52,5 RE |= = is dus als zuiver te 20 00 Jos = 2 beschouwen. C. Vitratie zuur druivenzuur brucine. Afgewogen 0.5324 anhydr. zout, opgelost tot 100 c.c.m. mer. zout e.e.m. opl. | c.c.m. KOM ee gevonden | berekend 10.00 oe = 52.2 53.2 10.00 | 8.06)” Ook het racemaat = is dus het zuivere zure 20.00 16.20] 8 zout. = | 1051 106,1 20.00 16.20 | = Ep Ten slotte werd nog nagegaan of het zure bestanddeel van het racemaat inderdaad optisch inactief was. Daartoe werd uit een zout- hoeveelheid (die volgens straks te vermelden proeven 11.8°/, kristal- water bevatte) met ammonia de brucine gepraecipiteerd. Daar de oplossing echter met ZZNO, zich nog rood kleurde en dus niettegen- staande de overmaat ammonia nog brucine bevatte, werd zij tot droog toe ingedampt, het residu weer opgelost, gefiltreerd en met loodacetaat gepraecipiteerd. Het loodracemaat werd afgezogen, uitgewasschen en gesuspendeerd in lauw water, waarna in de op het waterbad - geplaatste oplossing zwavelwaterstof werd geleid, terwijl voor her- haald omroeren werd zorggedragen. Loodsulfide zette zich grof af, het werd afgefiltreerd, weer met water aangeslibd en ZZ, 8 werd Na RE Ie A1 étés hydraat 0.7945 | 0.8532] 0.4598) 0.7624) 0.9819) 1.1611] 1.6069) 1.0370) 1.1587 na verwarming| 0.7925 | 0.8518] 0.3964) 0.6566; 0.8510) 1.0050] 1.4758) 0.9134) 1.0213 dus water 00920 | 0.0024} 0.0634) 0.1058) 0.1339) 0.1561] 0.1311) 0.1236) 0.1374 di. OVER PARTIÈELE RACEMIE. 67 weer ingeleid tot het filtraat niet meer zuur reageerde. Zoodoende werd het zuur zuiver en voor polarimetrisch onderzoek geschikt verkregen. Zoowel een 10°/, oplossing van het zuur als die van de er uit bereide ammoniumzouten was volkomen inachef. d. Het kristalwater. De bepalingen van het kristalwater geschiedden als bij de strych- nine-tartraten beschreven is. De uitkomsten vindt men in Tabel 18. LABEL 8: d-tartraat. | l-tartraat. racemaat. °l, water 0 Or 1155 (8 11385" (13.595 SL AP 84> ed. BA TA EO 85 dei; anhydr.| anhydr.| 4.84aq| 4.86ag| 4.74ag| 4.7Oagf 4.06aq) 4.08aq| 4.07aq dus anhydr. 5 aq 4 aq LADENBURG en Frscur (l.e.) vonden voor de enkel tartraten de- zelfde waarden, voor het rac. daarentegen 24 ag. Wanneer men nu bedenkt, dat hierboven vermeld is hoe ons zout een optisch inactief zuur bevat (juist dit 11.8 °/, kristalwater bevattende zout is voor bepaling gebruikt!) en dat een inactief mengsel tot 24 ag zou voeren, dan rijst er wel zeer ernstige bedenking tegen het praeparaat waarvan L. en |. het kristalwatergehalte bepaalden. Aan het einde van dit hoofdstuk komen wij hier nog nader op terug. e. Het specifiek gewicht. Voor de bepalinesmethode van het sp. gew. wordt weder naar 8 p- 8 het vorige hoofdstuk verwezen. De uitkomsten zijn samenecvat in Tabel 19 en gelden (ook die D van Fiscxz) voor 25°, bt 68 THEORETISCHE EN EXPERIMENTEELE ONDERZOEKINGEN LAURE PO; anhydrisch d-tartraat | anhydrisch /-tartraat | anhydrisch racemaat 1.492 1.492. 1.493 | 1.455 1.452 1.457 1.422 1.421 gem. 1.492 gem. 1.455 gem. 1.422 Fiscaz 1.30967 Fiscar, 1.14972 Fiscaz 1.26029 gehydrateerd d-tartraat | gehydrateerd /-tartraat| gehydrateerd racemaat 1.434 1.436 1.436 bestaat niet 1.437 1.448 1.444 1.449 gem. 1.436 gem. 1.443 Frscaz 1.74633 Fiscaz 1.63962 Terloops zij hier opgemerkt, dat het racemaat de eigenaardigheid vertoont zich met het spec. lichtere water tot een specifiek zwaarder hydraat te verbinden. Aangezien deze eigenaardigheid zeldzaam is en voorzoover ons bekend nimmer bij een stof met zoo hoog spec. gew. (1.4) geconstateerd, hebben wij het verschijnsel door een dilatometerproef gecontroleerd. In de peer van een dilatometer werd 3.5697 gr. anhydrisch racemaat gebracht, en de dilatometer op de gebruikelijke wijze met toluol gevuld door de 0.8 mM. wijde capillair. Door middel van een zweepcapillair werd nu 0.3 c.c.m. water in de peer gebracht. Deze waterdruppel viel niet naar onderen maar bleef aan den wand hangen. Nadat de capillair van boven was dicht- getrokken werd de stand van den meniscus bij 25°.00 aangeteekend. Vervolgens werd de druppel met de anhydrische stof in aanraking gebracht. De meniscus bleek nu 5 c.m. gedaald, 18 uur later zelfs 7 c.m. De hydratatie had dus onder merkbare contractie plaats gehad. e. De smeltpunten. De algemeene opmerkingen over de smeltpunten der strychnine tartraten gemaakt gelden ook geheel voor de brucine zouten. Van ie ths. ARS © i 6 OVER PARTIÈRLE RACEMIE. 69 de hydraten waren weer geen scherpe smeltpunten te bepalen en de feitelijke bepalingen geschiedden dus weer aan de anhydrische zouten. Gevonden werd: BARRE 1,205 Doria Frscur d-zout 290! 246° F2 249° 3989 Pers a? 240° J. De tensimeterbepalingen. Fiscut, meenende dat de omzetting in het overgangspunt aan- gegeven wordt door een vergelijking: 2 rac. 24 ag. @ d + 1 5 ag. dat er dus geen vloeistofphase bi] gevormd wordt, heeft er van afgezien tensimeterproeven te verrichten. Nu ons gebleken was, dat de reactie, die zich eventueel in een overgangspunt af zou moeten spelen, Brat. Arade ded are D HO was, hebben wij getracht tensimetrisch dit punt op te sporen. Tevergeefs echter; nu zou dat aan dezelfde oorzaak kunnen worden toegeschreven als by de stryehninetartraten. Immers uit de Tabel 19 blijkt ten duidelijkste, dat het anhydrische racemaat geen mengsel der enkeltartraten is, dat het racemaat zich dus ook hier by water- onttrekking niet noodzakelijk in de enkeltartraten behoeft te splitsen. Maar uit het hier volgende onderzoek der oplosbaarheden zal blijken, dat er nog een andere reden is en wel de meest klemmende, die maar mogelijk 1s, nl. dat er hoogstwaarschijnlijk heelemaal geen dergelijk overgangspunt in dit systeem optreedt. 70 THEORETISCHE EN EXPERIMENTEELE ONDERZOEKINGEN Il. De OPLOSBAARHEIDSBEPALINGEN. Zooals boven gezegd lag het niet in onze bedoeling een volledig stel oplosbaarheidsbepalingen te verrichten van het isothermennet im dit systeem. Met het oog op de groote verschillen in oplosbaarheid der enkeltar- traten bestond de hoop dat, wanneer men bepalingen op de MEYERHOFFERSCHE manier (Vel. pag. 20 e.v.) verrichtte, de lijnen voor de evenwich- ten - met - twee - Bodenkorper voldoende ver uiteen zouden liggen om met scherpte Fig. 12. conclusies daaruit te kunnen | trekken. Daarom hebben wij bij verschillende temperaturen achtereenvolgens de totaal-oplosbaar- heden bepaald van resp. d, /, r, d + J, r + ¢ en rd Wij herinneren nu (Vel. fig. 12) dat in het overgangspunt de oplosbaarheden van 4 + 7, r + / en r + d gelijk zullen moeten worden, terwijl in het einde van het overgangstraject de x + A oplosbaarheid gelijk aan die van 7 wordt. TABEL 21. Oplosbaarheden bij 20°. Bij indampen van Vaste Concen- Ge- NS 8 Opmerking. phase.| Aantal | Gr. Gr. tratie. | middeld. dagen geschud. oplossing.| droogrest. 47 d +1 1 4,997 0.0688 69.8 | 69.6 4.994 | 0.0684| 69.4 || 48 r+l 1 4.994 0.0794 80.8 | 80.6 5.003 | 0.0792| 80.4 || 49 r + d if 5.052 0.0690 692 | IL 69.3 5.061 | 0.0694| 69.5 || 50 ose 1 5.011 | 0.0682 | 69.0 |} | | | 69.0 | 5.010 0.0682 | 69.0 32 33 19 21 22 03 d+l r + d EN OVER PARTIBELE RACEMIE. TABEL 22. Oplosbaarheden bij 25°. Bij indampen van dagen gesch ud. oplossing. bo wol t + 4.855 À NANA EE hwo BE CORPO PR NOU OUR OU OPENED E OORT Gr. „Je „974 „9749 „992 „965 „999 „0025 „1638 „9815 „010 „001 „8816 9205 ‚2923 . 9526 .9508 .9624 .026 .056 .061 .005 „010 „004 „010 „018 „0076 „9775 „3077 2919 „915 „008 „002 „9526 . 9024 „645 „964 „144 „O1 “ooo .023 .9490 .9582 . 9666 .9548 004 .224 .008 .924 083 „057 Gr. 0.0494 0.0495 0.0436 0.0496 499 496 0.0899 0.0897 0.0852 0.0894 0.0914 912 913 925 560 848 946 997 895 895 898 893 879 879 866 864 859 851 TAL 850 856 866 864 1068 1058 „0535 „0994 „0438 „0215 „1025 . 1026 858 858 854 855 886 159 907 888 874 882 71 RE Concen- tratie. WONOROW HOVEN WR RO OU DORE OH DV MIO DO DO HO Où UI DO Soe Poe. Poe. 4 a en ed ————— nnen ns Ge- middeld. 50.4 92 Opmerking. De bij 35° verza- digde oplossing gebruikt. De bij 20° verza- digde oplossing gebruikt. Zie 19. Zie 19. Zie 19. Zie 54. 35 36 15 94 58 16 55 did 56 57 d+1 rl Vaste phase. — = En À B E L 23. es Oplosbaarheid bij 35°. Bij indampen van Aantal dagen ie has Gr. oplossing. droogrest. 4.987 4.968 5.085 5.076 5.0727 5.0708 5.015 5.018 5.022 5.027 5.055 5.038 5.019 5.0270 5.0226 5.011 5.0260 9.0200 5.008 5.003 Gr. 0.0625 0.0624 0.1600 0.1600 0.1474 0.1476 0.1468 0.1468 0.1385 0.1389 0.1372 0.1368 0.1484 0.1486 0.1486 0.1483 0642 „1428 „1326 1632 1628 „1610 ‚1449 05377 0782 „1465 1472 1458 1462 1192 1283 So ó © SMOS SS em Concen- | tratie. _ | Opmerking. © middeld. ean ed LA: oe Son us: > L #0 -Æ L * ars À 152 OVER PARTIBELE RACEMIE. TABEL 24. Oplosbaarheid bij 44°. Bij indampen van Concen- Ge- Vaste : 4 merking. phase.| Na Gr. | Gr. tratie. | middeld. à : schudtijd. | oplossing.| droogrest. d EX 4.968 0.0778 19.5 | 19.5 4.959 | 0.0777 | 79.6 i l LD. 5.013 0.2251 233 939 3.302 0.1458 | 231 r 18 Uur 5.037 0.2444 | 255 5.052 0.2451 | 255 1 Cur 5050 0.1494 220 2 Uur 5.068 0.2109 217 4 Uur 5.066 0.2030 | 209 15 Min 5.021 0.2113 220 aor. 5.042 02120414 219 3. Uur 5.032 0.2339 | 244 d+l| 1D 5.060 0.2751 287 5095900272 71128) 5 5.031 0.2674 | 280 0 5.023 | 0.2656 | 279 28 1 5.022 0.2374 | 248 4.794 0.2270 | 248 4.501 OQ: 2152 249 5.024 0.2376 248 1 War 5.065 0:453er 254 2 Uur 5.046 0.2509 | 261 : 4 Uur 5.048 0.2539 268 2D: 5.028 0. 26341-2216 5.033 0.2635 | 276 eeb heet D: 5.048 0.2642 276 5.048 0.2663 278 3D. 2,020 0.1062 AU LD: 5.0764 | 0.2689 280 - 5.0928 | 0.2686 279 9D. 5.0254 | 0.2665 280 1 Uur 5.061 0.2976 313 15 Min 5.037 0.2956 305 pen 4.816 0.2825 Si r+d| 3 D. 5.043 0.1642 168 5.037 0.1648 169 1 5.033 0.1632 168 5.015 0.1622 167 5.027 01660 ne ATA. 5.018 0.1657 171 1 Uur LAS Le Ot 210 Jur 4.750 0187 206 5 Uur 5.018 0.1964 | 204 15 Min 5.035 0.2169 | 225 oa DOTE 0.2187 225 14 THEORETISCHE EN EXPERIMENTEELE ONDERZO TABEL 25. Oplosbaarheid bij 50° Bij indampen van 8 Vaste phase. Na Gr. schudtijd. | oplossing. 49 d 1 D. 4,999 : 4.989 43 l de. 5115 5.068 23 r SD, 5.0798 5.0512 AD: 5.0468 5.0572 2g di) 5.0168 4.9963 40 ED 5.044 5.032 41 5.001 hd 44 1 Var 5.022 5035 2 Une 4,821 4,913 3 Uur 5.079 HAUTS 45 t mr 54054 5.058 2 ar 5.021 4,797 46 4 Uur 5H 1050 5.041 67—68 20 Min 5 Ona 5.020 BBA % 3.493 5.319 69 |d+1| 16 Utr 9.116 DS FA D À 3.643 4.126 eet.) 3D 5.058 5.067 AD: 5.0595 5.0338 29 DD! 5.080 5.014 71 15 Min 5.194 EDS. b.062 6 Uur 4,008 20 Uur 4,108 4.156 26 |r+d)| 3 D. 4,910 5 013 4 D. 4,989 4,981 30 5e, 4,940 4.663 70 20 Min 5.061 3 2 5.032 1 War 5.025 Gr. Be 0.0910 0.0908 0.3147 0.3122 0.2021 0.2008 0.2016 0.2018 0.2010. 0.2000 0.2469 0.2464 0.2524 0.2537 0.2400 0.2396 0.2276 071961 0.2383 0.2379 0.2786 0.2769 0.2603 0.2476 0.2891 0.2888 „2846 „2844 . 1966 . 1866 „2669 1550 „2563 „2912 3529 .3821 .3198 .3453 .3407 .4336 A274 „3246 „2972 „3007 „2139 2156 ‚2159 2138 2126 „2002 „2631 Zoi 0.2530 .3036 | Concen- Ge- | tratie. middeld. EN NL eh EKINGEN — ik Opmerking. a Vervolg van 23. Oe | TR art is — CMS » See . . + PE Ps > De oplossing ge ag bruikt, die bij 44° — verzadigd was. Te 918 Vervolg van 25. Vervolg van 26. OVER PARTIBELE RACEMIE. 15 De Tabellen 21, 22, 23, 24 en 25 vatten de uitkomsten dezer bepalingen samen. | Men ziet, dat deze tabellen een geheel ander aanzien hebben dan die bij de strychnine tartraten. Terwijl daar na 1 à 2 X 24 uur de uitkomsten zóó standvastig waren, dat van het telkens herhaald opgeven der schudtijden is afgezien, veranderen hier de uitkomsten op zonderlinge en (men lette op paralelbepalingen, met zelfde schud- tijden) onregelmatige wijze. Dat is tenminste het geval met die proeven waarin racemaat, hetzij alleen, hetzij met d of /, als Bodenkörper optreedt. De oplosbaarheden van d, / en vrijwel ook die van d + 7 verloopen normaal. Volgens LADENBURG en HIscHr zou het racemaat boven 44° stabiel zijn, onze tabellen laten zien, dat hoe hooger de temperatuur wordt, des te onregelmatiger wor- den alle oplosbaarheidsbepalingen, waarbij met / geschud wordt; tot aan 50° krijgt men uit deze tabellen al zeer weinig den indruk, dat met een stabiel lichaam oplosbaarheidsbepalingen worden uit- gevoerd. Integendeel, er blijkt een ontleding plaats te hebben, die met verhooging van temperatuur sneller voortschrijdt. Het zijn vooral de bepalingen, waar wij tenslotte toe gekomen zijn, die met volg- nummers boven de 44, die deze meening bevestigen. Gaat men b.v. eens na wat er gebeurt met een oplossing van r in water van 50°. Wij ontleenen dan aan Tabel 25 de volgende gemiddelden. TABEL 26. Oplosbaarheid van 7 bij 50°. Ne: Na tijd. Opl. Andere Serie. 67—68 20 Min. 300 De 298 45 PLU 291 247 (No. 44) 2 Uur 273 a ee ee Sede AD CREER) 40 TA 257 23 3 D. 207 4 D. 208 gD: 209 Men ziet daaruit, dat na enkele minuten een oplosbaarheid wordt bereikt, welke van af dat oogenblik voortdurend afneemt. Dit 1s toch zeker niet het gedrag van een stabiel lichaam. Nu zou men 76 THEORETISCHE EN EXPERIMENTEELE ONDERZOEKINGEN deze proef nog kunnen verklaren met de veronderstelling, dat nochtans de opvatting van L. en F. van een overgangspunt by 44° juist is, maar dat deze 7 oplosbaarheid bij 50° er een is binnen het over- gangstraject, waarmede dus ons oorspronkelijk doelwit zou zijn ge- troffen. Wij gelooven echter niet, dat deze uitlegging juist is, de proeven pleiten er veelmeer voor, dat de verhoudingen in het stelsel der zure brucine tartraten met water een geheel andere is. Werpt men een blik in Tabel 24, dan ziet men, dat het bestaan van een overgangspunt bij 44° niet te rijmen is met onze oplosbaarheids- bepalingen bij die temperatuur. Welke cijfers men ook als de juiste wil nemen, de maxima na enkele minuten, dan wel de ge- tallen, die men na dagen schudden vindt, geen enkele combinatie toont, dat bij die temperatuur Oplosbaarh. „+; == Oplosbaarh. , , , = Oplosbaarh. ,. , 4. De getallen van d + / en r, die wet gelijk behoeven te worden, lijken nu en dan wel wat op elkander (Vgl. proef 37 en 8) en men begrijpt dan ook wel, dat Frscur de foutieve praemisse hier met een onjuiste interpretatie van het experiment gecombineerd heeft. Dat hij voor r + / ook een getal in die buurt vond is ook te begrijpen (Vgl. proef 13), maar hoe hij op kan geven, dat ook voor > + d een waarde in die buurt optreedt (Vgl. de proeven 10, 14, 61 en 65) is weer even onbegrijpelijk als de tensimeterbepa- lingen van Docror zijn! Trouwens de eigenaardige gang in de waarden, die wij vonden voor 7 + / en r +d by en boven 44°, duidt op de instabiliteit van het systeem. Wanneer wij dus tot de conclusie komen, dat LADENBURG en Fiscai, onjuiste conclusies omtrent het onderhavige systeem gesteld hebben op een uiterst zwak feitenmateriaal, dan dringt de vraag zich natuurlijk aan ons op, hoe wij ons de verhoudingen in dit systeem dan wel te denken hebben. Allereerst moeten wij dan vaststellen, dat blijkens in de inleiding van dit hoofdstuk vermelde proeven de zure druivenzure brucine ongetwijfeld een chemisch individu is, maar dat zij, althans beneden 50°, nimmer stabiel naast oplossingen is. Dat zij het alsnog bij hoogere temperaturen zou worden is natuurlijk niet uitgesloten, maar waarschijnlijk is dat niet. Zet men nl. de gevonden waarden voor oplosbaarheden, hoe ruw men daarbij ook middelen moet, uit (zie fig. 13), dan met men toch nog wel een tendenz in die lijnen, maar zeker geen "M 300 200 100 OVER PARTIÉELE RACEMIE. 11 van + + / eu r +d om elkaar te snijden naar hoogere tempera- turen. Ons lijkt dus het meest waarschijnlijk, dat het zure druiven- zure zout een zeer onbestendige verbinding is. Hoe men zich nu het eigenaardige verloop der oplosbaarheden met den tijd moet verklaren, hoezeer wij onze aandacht daaraan gegeven hebben, durven wij daar geen positieve uitspraak over te doen. Een voor de handliggende onderstelling zou zijn, dat het onbe- stendige racemaat zich telkens in de enkeltartraten splitst; dan zou men echter steeds op d + / oplosbaarheden moeten uitkomen, als 20 30 40 Fig. 13. er tenminste voldoende Bodenkörper is, wat bij onze proeven inder- daad het geval was. Daar nu die waarden niet gevonden zijn, moet de verklaring elders gezocht worden en dan schijnt het ons zeer waarschijnlijk, dat het gecompliceerde verloop der oplosbaar- heidsproeven daarin te zoeken is, dat de normale brucine tartraten hier tevens een rol spelen. Er is echter te weinig omtrent deze stoffen bekend (oplosbaarheid), dan dat men dit met zekerheid zou kunnen zeggen, maar hun bestaan opent zeker in dit systeem de mogelijkheid tot complicaties als wij in bovenstaande tabellen aan- treffen. 20 18 THEORETISCHE EN EXPERIMENTERLE ONDERZOEKINGEN Samenvatting. Overzien wij de resultaten van de bovenbeschreven onderzoekingen omtrent de zure brucine tartraten, dan komen wij tot de volgende conclusies. j | De physische constanten (spec. gew., smeltpunt) der zure brucine tartraten zijn anders, dan door LapeNBurG en Fiscar opgegeven, eveneens het kristalwatergehalte der druivenzure verbinding. De bestaanbaarheid van een anhydrisch druivenzuur zout belet echter _ weer tensimetrisch onderzoek. De oplosbaarheidsbepalingen door Fiscur verricht laten zich niet reproduceeren. Integendeel leeren nauwkeurige bepalingen, waarbij na verschillende tijden analyses zijn verricht, dat in de met racemaat geschudde oplossingen een ontleding plaats heeft, ook boven 44°. bij die temperatuur ligt zeker niet het overgangspunt, want de oplosbaarheden van 7 + d en r + / worden er absoluut niet gelijk. Kortom, de verhoudingen in dit systeem zijn van totaal anderen aard dan door LADENBURG c.s. vermoed wordt. Naar alle waar- schijnlijkheid is de partiëel racemische verbinding steeds een labiel hehaam en zijn slechts de enkeltartraten naast oplossing bestendig. OVER PARTIBELE RACEMIE. 79 SLOTBESCHOUWING. Zooals in de inleiding dezer verhandeling vooropgesteld werd, was de bedoeling dezer onderzoekingen na te gaan of RoozrBoow’s beschouwingen over de systemen, waarin partieel racemische verbin- dingen optreden, juist zijn; een dergelijk onderzoek scheen gewenscht, omdat LADENBURG nog steeds de aangevochten voorstellingswijze gehandhaafd heeft. ‘Tegenover RoozrBoom’s theorie stond al het experimenteele mate- riaal uit LADENBURG’s laboratorium. Dat materiaal is nu aan een nauwgezette contrôle onderworpen in twee gevallen. Bij het nawerken van het strychnine-tartraten-systeem kwamen al dadelijk zoowel numerieke als principieele afwijkingen voor den dag, terwijl een breeder opgezet onderzoek der isothermen, o.1. de eenige geheel „einwandsfreie’”” onderzoekingsmethode, absoluut de opvatting van foozeboom bevestigde. Im het systeem der brucinetartraten zijn de verhoudingen gecompliceerder, onze onderzoekingen hieven echter in elk geval de waarde van dit Ladenburgsche materiaal geheel op, daar de verhoudingen in dit systeem zeker geheel anders zijn, dan ze door zijn leerling Frscur bepaald zijn. Daarbij komt, dat wij op algemeene fouten in de onderzoekingsmethoden wezen, be- zwaren, die ook gelden voor de overige, niet nagewerkte systemen. Onze verwachting, aan de (trouwens theoretisch onaanvechtbare) opvatting van RoozrBoom een experimenteel bewijs terzijde te stellen en daarmede tegelijk het materiaal, dat ermede in tegenspraak scheen, te kunnen: kritiseeren, mag dus naar wij meenen wel als vervuld beschouwd worden. uc a ar polytopes. 7 co rs sits der Koninklijke Akademie van Wetenschappen te Amsterdam. EERSTE SE 5 8 | DEEL XI N°. 5. : (WITH ONE PLATE). Sa ee NEEE 3 AMSTERDAM , JOHANNES MÜLLER. : + April LOTS. Analytical treatment of the polytopes regularly derived from the regular polytopes. (Sections I, If, IV). BY Pr SCÉAOUTE: Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam. (EERSTE SECTIE). DEEL XI N°. 5. (WITH ONE PLATE). AMSTERDAM , JOHANNES MU LEER: 1913: ge DRE ee | KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN TE AMSTERDAM. Verzoeke het stereoscoopplaatje, behoorende bij de verhandeling van den Heer P. H. SCHOUTE: ,Analytical treatment of the polytopes regularly described from the regular polytopes.” Section II, I, IV, (Verhandelingen le Sectie, Deel XI, N°. 5) te vervangen door het hierbijgaande. Please replace the stereoscopic plate belonging to the memoir of Prof. P. H. SCHOUTE: „Analytical treatment of the polytopes regularly described from the regular polytopes.” Section II, IL, IV. (Verhandelingen fe Sectie, Dl. XI N°. 5) by the enclosed one. Analytical treatment of the polytopes revularly derived from the regular polytopes. (Sections II, IE IV). BY PEL SCEAOEPEE: Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam. (EERSTE SECTIE). DEEL XI N°. 5. (WITH ONE PLATE). D Ee D AMSTERDAM, JOHANNES MULLER. PUS: Analytical treatment of the polytopes regularly derived from the regular polytopes. Section Il: PorLYroPEs AND NETS DERIVED FROM THE MEASURE POLYTOPE. A. The symbol of coordinates. 46. The distance r between two points P, P’, the ordinary rectangular coordinates of which are (4, [42,...,%, and (24, Us... ., Ky is represented by the formula Pu (ABA IE tn Re ener ele 2). Now we repeat here the question of art. 1: “Under what circumstances will the series of points obtained by giving to the set of coordinates (4, 42,..-., 4%, a determinate set of values taken in all possible permutations form the vertices of a polytope all the edges of which have the same length, say unity ?” The answer is nearly the same as that given in art. 1: “If the » values a,, 45,...,4, are arranged in decreasing order, so that we have B es Be Or ed ss a GS the difference a, — a, ‚4 of any two adjacent values must be either Er OT ZOO. The proof runs on the same lines as that given in art. 1. The geometrical result can be stated in the following general form: “Under the conditions stated, the polytope fie vertices of which are represented by the symbol (des 35 Mag teams Os is the same as that obtained im the first section for 7 — 1 and a, — Gj 4 either one or zero. It is a derivative of the regular simplex the vertices of which determine on the a axes OX, of coordinates positive segments O4,, (4 = 1,2,...,n), of the same ° n lenoth Xe. 1 À ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY This simple result, in close connection with the new deduction of formula 1), shows us that we shall have to enlarge the scope of our symbol of coordinates in order to find something new. 47. We remember that the symbols [4, +, 4] and [4 V2, 0, 0] represent the coordinates of the vertices of cube and octahedron with edge unity, if the sguare brackets indicate that all the per- mutations of the values they include must be taken, each value being affected successively either by the positive or by the negative sign. Moreovor +[4,4,4] and — +f[4, 4,4] can represent in the same way the two tetrahedra, the vertices of which form together the vertices of the cube [+, 4, 1], if by the coefficient + we indi- cate the vertices with an even, by the coefficient — + the vertices with an odd number of negative coordinates. In connection with this we amplify the question of art. 1 as follows: “Under what circumstances will the symbols [ler Ais LS alle Sei Gon BE eee represent the vertices of polytopes in S,, all the edges of which have the same length, say unity?” The answer to this question runs as follows: Turorem XXVIII. “If the values a,, 4, ...,a, are arranged in decreasing order, a, being the smallest non vanishing one, and if U, App 1 represent any couple of adjacent unequal ones, we must have inthe case of thé frstisymbols seen either p= na, har Ar Me or PAV det ne |? in the case of the second symbol + +[a;; a, ...,a, Sis ae aye a ¢ 1 “a p= 0,0, = a, EVE en le Proof. The part of the proof concerned with the common value 4 V2 of the difference a, — a ‚4 of two unequal adjacent digits is the same as that given in art. l. So we have to add only a few. words about the values of a, in the case of the first and of 4,4 and a, in the case of the second symbol. Symbol Ta, as, ...,a,]. In the supposition a, — an ‚1 = + WV2 the length of the edge of the polytope is unity. Therefore the distance 2a, between the points D / weve) = v4 = U d'y == di, Vs =a Aa, ng a GPa Vi == —~ Ay, by = as VEN ee which are transformed into each other by inverting the sign of DERIVED FROM THE REGULAR POLYTOPES. 5 a,, must be unity, which gives a, — À, unless P and Q coincide which happens for a, — 0. So in the case p — x we have a, = 4. In the case py << u we consider the points Bd dd US a iy a OA nd a passing into each other by interchanging x, and a@,. The distance a, V2 between these points is unity for a, — 4 V2. Symbol + 4[a, 4, ...,a,]. Here a, differs from zero; for the supposition a, — 0 is incompatible with the division of the vertices represented by the symbol [a,,a5,...,a,] into the two groups + 4[u, 4, ...a,], the inversion of the sign of zero having no effect whatever. Here the point eeen ES ARE Pare dE: must be considered in combination with the points Q ES Vv —— Ceca Vy —— dy , V2 — Ups; BET D De sag Oy i A RE corresponding with it as to the coordinates 23, #4, ... æ, +4, as these points Q and Z are the nearest ones to P obtainable either by interchanging two digits or by inverting the signs of two digits. Now we have under these circumstances POS laga ee AG, ani) frem which ensues PQ < PR. So we must have PQ — 0) PR = 1, DU, =d rk MB. 48. In the case of the first symbol | a, a, ..., a,]we are confronted with two possibilities, as we have to choose between a, =} and a, — 0, 1. e. between a group containing the measure polytope [4, 4, ... 4] and an other group containing the cross polytope [4V2,0,...0]. Do the two regions lying on different sides of the limiting demarcation line cover the same ground as the group of the measure polytope on one side and the group of the cross polytope on the other? The answer to this question depends on the manner of deduction of these two groups. If we follow closely the geometrical manner of deduction developed by M. Srorr the contraction forms derived from the measure polytope do possess coordinate symbols winding up in zero, whilst on the other hand the form derived from the cross polytope by means of a set of expansions under which e,_, occurs are represented by coordinate symbols containing no zero. These two exceptional facts which ~ seat rees LE) Les: LA ~ DD DO eee Ye if f a hi) me: 6 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY prove the close relationship between the progeniture of the two patriarchs, cube and octahedron, can be extended so as to make the two families quite zdentical with each other; to that end we have only to derive from each of the two, cube and octahedron, ad/ the expansion and contraction forms, the number of which amounts in #, to 2" — 1. This important fact, which will be proved later on, enables us to treat in the second and third sections the forms with the symbols [a, a, ...,4] and [a,, a, . . ., 0] successively, without being obliged to postpone the study of the corresponding nets built up by atin’ of both groups. In order to avoid fractions we will multiply. the digits by two in this section and the next one; under this circumstance the last digit is unity or zero, the difference a, — a, ‚4 of two unequal adjacent digits is V2 and the symbol represents a polytope with edge 2. Moreover in order to simplify the symbols we will write p for 1+pV2 and put if possible V2 outside the brackets, substituting e. g. [11100]V2 for [V2,V 2, V2, 0, 0]. 49. For = 2,3,4,5 we have successively in the symbols explained in the memoir of M. Srorr: 4) N= [li] =p, | [I] =e p= Ps | | {10} 2= ce pa = Pa n= 9 111]= C 1] = Cy C=; OO [100172 — ce, C= O BA = 71047 Ue, C=O [110112 = ce, C = CO [2101172 = ce, eg C=tO NRA El | = Cs [2211] = eC} Cy C [211011 2—= ce ej Cs el id V1] = e1 Cs [2’ M Sa = 1 63 Cs jLllojy2= ce Cs [2110] 72 == Ce 63 Cs Ag be lj = ex Cs [2 el le Co 3 O3 [11001772 == Ce, Cs [2100] LL Ce C3 Cs eal ah | 1] = C3 0. (ove ca = C1 Cy 63 Cs [10001172 — Ces, Cs [8210] 1/2 = ce € es Cs ni A — Col RTE Cis [22100172 ce e3 Cio PAL = € Co L'ANPE I= Cy C4 Co [11110]1/2—= ce Cro [21100] ~2= Ce) ey O19 ‘igs Ue es — Cy Cro 12” A HE © j= e3 Cy, Co [1110012 ce Co [21000172 — Ce ey Cio Li M à it JN = C3 Co Bioorg al Co (1 Co [110007 Ie C3 Cio [33210] 2 (441 Cy Ca Co AT ei Co r3’2"2/ |’ |= ey € Ex Cy [10000172 — EE Co [32210] 2= came, Co vo 9 4 1] — 41 (4) Co | Ex Ae 1 i 1 — 41 Ca C4, Cho [22210] Là — 1 22 > Cio BEA 10] Vis Ce ez a Cio gelet 1) = € 63 Cio (82 RARES 23 Cy Cho [22110172 — Beate: [32100] Vis ce; e3 €, Cio TINI = & eg Co 43 UI 1e 6, 6 0, Co [21110] 1/2 = ceres Cio | [48210] 112 = ce egeg 4 Cr *) For the deduction of the e and c symbols from the symbol of coordinates compare the part D of this section; here p, means: p, turned 45° about the centre. In Table IV added at the end of this memoir are put on record for n= 8, 4, 5, the different polyhedra and polytopes deduced from the measure polytope. Of this table the first column contains the symbols of deduction of the polytope from measure polytope and cross polytope — with the first of which we are concerned in this section only — and the third the symbol of coordinates, The second and the following columns will be explained farther on, AM ie" start from the equation 2a, DERIVED FROM THE REGULAR POLYTOPES. | 7 Here we have [1100]V 2 = C®,,,[1000]V 2 = C®,,, [10000]V2 = Cg. Remark. If we invert the sign of all the coordinates of a vertex // of the polytope we get the coordinates of an other vertex 7 of that polytope for which the centre of the segment PP’ is the origin of coordinates O. So, all the forms derived analytically from the measure polytope admit central symmetry, as the geometrical deduction by means of the operations e and c requires it. B. The characteristic numbers. 50. In the case of the simplex the direct method for the deter- mination of the characteristic numbers proceeding regularly from vertices to edges, from edges to faces, etc. was preceded by an easier method fulfilling the exigencies of the cases n = 4 and x = 5, working from both sides, the vertex side and the side of the limi- ting element of the highest number of dimensions; in this case of the measure polytope we will do likewise. 4) Here also the number of vertices is easily found. If all the x digits of the symbol of coordinates are different it is 2”. »!; of the two factors 2" and xz! of this product the first is due to the power of choosing arbitrarily the signs of the x digits, whilst the second corresponds to the power of permutating them. This product must be divided by 2! for any two, by 3! for any three digits being equal, ete. In order to be able to find the number of the limiting bodies (2 = 4) and that of the limiting polytopes (x — 5) we have to prove here the 7 Prroreu XXIX. “The non vanishing coefficients c; of the coor- Ginates a, in the equation cz, 4-8 Fr. .—=p of a limiting space #,_, of the polytope deduced from the measure polytope of #, must all of them have the same absolute value.” ~The difference between this theorem and the corresponding one for the simplex (theorem II of art. 6) lies in the addition of the word “absolute”, therefore printed in italics. This amplification is necessary here, in connection with the power of assigning to each of the x digits of the coordinate symbol either the positive or the negative sign. But the proof runs quite in the same lines. If in the case of the polytope [1 + 2V2, 1+ V2, 14+ V2, 1] we 2, =p and try to determine the *) The treatment of the offspring of the measure polytope with which we are con- cerned now — and of that of the cross polytope which comes next — will be copied as much as possible from Section I, 8 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY vertices of the polytope for which the expression 2a, — x, becomes either a maximum or à minimum we find the maximum 3 +- 5V 2 for m—=1+9V2, ~=—(1+V2) and the minimum — (38-+-5V 2) of the same absolute value for 4, = —(1 + 2V 2), m —1 + V2. So, for values of p between 3 + 5V2 and — (3 + 5V2) the space 2a, — #, — p intersects the polytope, whilst it cannot contain a limiting body but at most a limiting face only for the extreme values + (8 + 5V 2) of p, as each of the two couples of equations a@=112V2, a= —(14+V2) and 1 = —(1 +2V2), æ, = 1 + V2 determines a plane. Here too, as far as the vertices of © the polytope are concerned, any linear equation ¢ a, +-¢,a,-+...=p represents # different equations if the non vanishing coefficients ¢; admit Æ different absolute values. Here too the theorem is not reversible. As to the theory of the determination of the number of faces (u — 4) and the number of limiting bodies (7 = 5) compare the end of art. 6. Remark. In accordance with the central symmetry of the polytope [a,,@,...,@,| any two parallel spaces S,_,, represented by the equations a, +, La, +...— dp and lying therefore on different sides at the same distance from the origin, bear either both or none of them a limit (/),_, of the polytope. So, in the determination of the limits (/),_, we can restrict ourselves here to the equations eta, +a +... = maximum. 51. We now treat at full length two examples, one in &, and one in Ss. Example [1 + 2y2, 1+ y2, 1+ y2, IJ). The number of vertices is 2 4! divided bye Po, CA 2 == The number of the edges passing through each vertex is five. For the pattern vertex Lh WL ee ve > is adjacent to the five vertices | + V2,1+2V2 ,1+ Ve 1 RCE V2 ,1+ EN Ke ] | 1+ 2V2, ] , i+ V2 ,1+V2 j 1+9V2, 1 + Ve | 1+ve | 1+2V2, 1 + VALUE l + ARE | *) In vol. XI of the „Wiskundige Opgaven” we have recently treated the polytope [(1+3V2,1+2)V2,1+4)2, 1] and its projections on its four kinds of axes (pro- blem 78) and deduced the symbol of characteristic numbers of the polytope [1 + (n—1) V2, 1+ (n—2))V2,..., 1+ 12,1] of Sn (problem 80). For the latter point compare also my paper „On the characteristic numbers of the polytopes e, e,... e, » &, 4 Sin-+1) and es... &, 9 &, 4 M, of space S,’ (Mathematical congress, Cambridge, August 1912). DERIVED FROM THE REGULAR POLYTOPES. 9) which may be indicated by the brackets and the negative sign after 1 in the symbol TD ERR en PLD Te Are ee AE Aa Ee een Oo 92 So the number of edges is ee ol. In order to find spaces which may contain limiting bodies we have to consider the equations FA NN Noes ear DNA, OI ie Ee = 2 + 8V2, Cee Pride ley =) AM, PRE eier das My a Oy eae AE VS) a). The equation a, = 1 + 2V2 gives us for the other coordi- nates the system represented by a., #3, 2,=[1 + V2,1+V2, 1], 1.e. an e, C. This ¢C presents itself 2. 4 times, as in the equation + #,=1-+2V2 the sign may be either positive or negative (factor 2), while the index # may be any of the four indices 1, 2, 3, 4 (factor 4). 6). The condition a Jam — 2 + 8V2 gives a,, a, = (1 + 2V2, 1+ V2) and 3, 2, = [1 + V2, 1], i.e. we have for the coor- dinates in their natural order of succession da, Po, #3, U = (1 + 9V 9,1 + WOII + V2, 1] representing an octagonal prism P, with end planes parallel to OLX; X,) and edges normal to these planes parallel to the lines Pi | # == constant in‘ OLX, Zo); this P, occurs 2°. 6 times, as we dispose in + a, + à, = 2 + 38V2 over two couples of signs (factor 2°) and the pair of indices 7, 7 stands for any of the combinations of the four indices by two (factor 6). c) In the supposition v, + a + 2} — 3 + 4 V2 we find in the same way Li, Lo, ®, & = (1+ 2V2,14+V2,14+ V2) [1], i.e. a triangular prism P3 occurring 2°. 4 times. d) Finally for w= 4(1 + V2) we get di ; Lo, ds , Vy == (1 - Veo I + VAL | + KA ee which — compare the last result of art. 46 — is a CO, occur- ring 2* times. 10 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY So, all in all we have got the limiting bodies SLE AT ne ES CO: SO fen number is 80. As the numbers of faces of ¢ C, Ps, P3, CO are respectively 14, 10, 5, 14, the total number of faces is 4 (8 X 14 + 24 X 10 + 32 X 5 + 16 X° 14) = 368 So the final result is (192, 480, 368, 80), in accordance with the law of Euler. Remark. Im the case of the measure polytope (4 of #, repre- sented by [1, 1, 1, 1] the spaces represented by FA MARNE EEE | D) Mei = CNT ap Bd d) .... a te ++, = 4 contain respectively a limiting cube, a face, an edge, a vertex of Cs. So we find here in the case of the chosen example 8 ¢C of body import, SAP. face a patie oes S 16004 vertexca = 02. Evample [1 +32, 1+2y2,1+2y2, 14+7%, I. The number of vertices is 2°. 5!: 2! — 82. 120: 2 = 1920. The number of edges passing through each vertex is six, as can be derived from the symbol ME oe PI ce UE | 1+3V2,14+2V2,14+2V2,1+V2, 1(—, mn containing five brackets and the negative sign after 1. So the 1920 6 aa none 0 number of edges 1s In this case the limiting polytopes can only lie in spaces #, with equations of the form a) de = dre On — 25 V2, 6) nn — 8 + 7 V2, OOS se a ly Lors =4+8V2, On «Se Oy Ae EEE ER corresponding respectively to DERIVED FROM THE REGULAR POLYTOPES. Hi a)..2 . 5polytopes(1+3V2)[1+2V2, 149V2, 14V2, 1], 6)..27.10 „ (H3V2, 1H2V2)[1+2V 2, 14V2, 1], c)..2%.10 , (1+8V9, 1+2V2, 142V2)[1+V2,. 1], d)..2*, 5 , (+8V2,1+2V2, 14+2V2, IHV) [IN PE. » (1+3V2,1+2V2, 1+2V2, 14V2, 1), Of these groups of polytopes the first, of polytope import, can be studied by itself; it proves to be a form with the characteristic numbers (192, 354, 248, 56), an me, C&. The second group consists of prisms on [1 +2V2, 1+-.V2,1]=+#CO as base, the third group of prismotopes (3 ; 8), the fourth group of prisms on (1+ 3V2,14+2V2,1+2V2, 1+V2)= CO as base. According to art. 46 the fifth group, of vertex import, contains forms e, €; S(5). So we find 10 & e, CG + 40 P ico + 80 (8; 3) + 80 Peg + 32 a e SD) = ; — 242 polytopes, Bas ee, Co, Pos (85.9), Pos a es 9.0) admit respectively 56, 28, 11, 16, 30 limiting bodies $(10 K 56+ 40 & 28 + BOX 11 + 80 X 16 + 32 X 30) = — 2400 polyhedra. So, according to the law of Euler, the number of faces is 6000, and the final result a (1920, 5760, 6000, 2400, 242). 4) 93. We pass now to the more direct method going straight on from vertices to limits with the highest number of dimensions, and apply it to the second example [1+3y2, 14+2y2, 1 +212, 1+y2, 1 of the preceding article. But in order to make the symbols less clumsy and thereby the method more manageable we represent once more 1 +pV2 by 7’. The number of vertices was and remains 1920. According to the symbols the edges split up into four groups, viz. (3 2), (2 1), (Ll B, [1]. Here (32°) means that any deter- minate pair of coordinates each affected by a given sign take the interchangeable values 3’ and 2’, the other coordinates retaining the same values; whilst [1] means that any determinate coordinate takes successively the values +-1 and — 1, the other coordinates remaining unaltered. *) The fourth and the sixth column of Table IV contain the characteristic numbers and the limiting elements of the highest number of dimensions. The meaning of the second column, of the small subscripts in column four and ofthe fraction in column five, will be explained later on. 12 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY It is easy to calculate the numbers of edges of each group. Through the pattern point with the coordinates 3’, 2’ 2’ 1”, 1 pass — on account of the two digits 2° — fwo edges (3° 2) and (2:10 and one edge (1’ 1) and 1]. So there are in toto 1920 edges (3 2’), 1920 edges (2’ 1’), 960 edges (1° 1), 960 edges [1 |, 1.e. 5760 edges. Remark. We may notice that [1] with one digit only is equi- valent, as to the representation of edges, to (3’ 2’), (2 1), (L° 1) with two digits. This difference is explained by the different cha- _ racter of the symbols: the digits between square brackets have given absolute values, whilst the digits between round brackets satisfy a linear equation, the sum of the digits being constant. This diffe- rence will repeat itself throughout the whole section; so [1° 1] is a face, an octagon, and (3° 2 2’) is a face, a triangle, etc. By applying the notions of “unextended”” and “extended” symbols, of the ‘syllables’ of these symbols, etc., given for the offspring of the simplex in art. 9, to the group of polytopes deduced from the measure polytope we easily extend this direct method to faces. According to the symbols the faces split up into eight groups, viz: the triangles (3 2’ 2’) and (2’ 2’ 1’), the squares (3 2’) (2’ 1), (3° 2)(1 J), (B 2’) [1], (2°.1')[1], the hexagon (2’1'1) and the octagon [1 1]. In the pattern vertex P concur one of each of the two groups of triangles, one octagon and — on account of the two digits 2’ — two of each of the four groups of squares, two hexagons. So we find © 2 triangles _ 8 squares , 2 hexagons , 1 octagon 1920 | a _- ETE 5 ae =) — 1280 triangles + 3840 squares + 640 hexagons 240 octagons, Een 0000" faces. According to the symbols the limiting bodies split up into nine groups: (3' 2’ 2'1', (3221 (822 (829 211), 6 2) aa (BEE (2°91), 2 Va er Le. taken in the same order of succession, of DOE Post Tae ap, ON Pies C Woe” ANT à ecard ES CC} So we find through P CO + 8 Ps RP, 22 OOR CRM and therefore in toto DERIVED FROM THE REGULAR POLYTOPES. 15 COR Sr 0 eae bis ETES FCO Ni es lin en 1920 (45 ieee Saas Gob Ds 48 = 160 CO +960 Pz + 820 P, + 480 C+ 240 P, + 16047+ 804C0 i.e. 2400 limiting polyhedra. According to the symbols the limiting polytopes split up into five Beaups viz. (322 ED (322 1117, (32 2) 11 (8 22 TN, [22 1’ 1], i. e., taken in the same order of succession, of e, e, S(5), Pos (3 > 8), Picos Ci Co Cs. So we find through P ex € (5) + Peo + (358) + 2 Pico Fe Ce and therefore in toto p Co C2 S(5) : ig (3 5 5) 2 Ee ey Co Ca ve 0 60 24 24 + 96 Ae PO ae — 32 ¢, e, S(5) + 80 Peo + 80 (3 ; 8) + 40 Pico + 10 Cy Cy Ce i.e. the same 242 polytopes found in the preceding article. 54. If we exclude once more the “petrified” syllables (11), (111), etc. introduced in art. 9 we can state the: Tagorem XXX. “We obtain the extended symbols of all the groups of d-dimensional limits (2), with different symbol of any given #-dimensional polytope (P), derived from the measure polytope 47, of space S,, if we split up the x digits of the pattern vertex in all possible ways, either into x —d or into un — d + | groups of adjacent digits, place all these groups with exception of the last one of the second case between round and this last one between square brackets, and consider these bracketed groups as the syllables of the extended symbol.” Proof. As in art. 10 we represent the ~ — d different syllables . : 5 [ge = at: . a Mi, Round, bracketssby.¢: OE (ar), fe aw) ee, an thelist case we have the relation 4, + 4,... + 4, — 4 — 2, whilst addition of the syllable [..] with 4 digits leads in the second case to the condition Aj + As + ..-+4,_,+4# =x. In both cases we suppose in order to fix the ideas that to (..)" correspond the coor- dinates! tas ays yey do (22) the coordiates a; res tes #,+%,, ete. and in the second case to[. . |* the coordinates a, ,- , 3, Ben Sh fetes Ur Here too the proof splits up into three parts. As the first case can be deduced from the second by supposing 4 = 0, we indicate the alterations which the three parts of the proof of art. 10 have to undergo for the second case only. 14 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY a). The polytope obtained is a (P),. By the exclusion of petrified syllables we are sure here too that any syllable (..)* with # digits allows the vertex, the coordinates of which are the x digits of the symbol of (P),,, to coincide successively with all the vertices of a definite 4 — 1-dimensional polytope (P),_, situated’ in a space #,_, determining equal segments on / of the z axes OX;. Moreover the unique syllable [.. |“ with 4” digits allows that vertex to coincide successively with all the vertices of a definite Æ'-dimensional polytope (P), situated in a space 8, parallel to the space of coordinates &' containing the # axes OX,, where « is successively-2 — # +1, »—# +-2,...,. The spaces bearing these xn —d-+-1 polytopes (P),, (4 = &, Aj, ... Ana), and are by two normal to each other. For (P),, lies in the space #8, = OVX, Ao. A), (P),, hes in the space 8, = OLX, +1 Xn, 10 . App) etc. and now the spaces S,, S,,..., S,,_,, Sy form a set Of coordinate spaces containing together all the axes OX, once, 1. e. they are by two perfectly normal to each other. So, as two spaces lying in spaces perfectly normal to each other are themselves perfectly normal to each other, the spaces bearing the 2 — 4 + 1 polytopes found above partake by two of that property. So the polytope under consideration is a prismotope with #— d + 1 constituents and this prismotope is a (/),; for its number of dimensions is the sum of the numbers #4, — 1, Al, 4,_,— 1), Ze Ga dimensions of the constituents, i. e. the sum of the numbers ky, Ko, ... Ay—q diminished by 7 — 4, 1e. x diminished by nd, i.e. d. 6). The (P)q obtained is a limit of (P),. | According to the manner in which (P), is obtained the coordi- nates of its vertices satisfy the #—4 mutually independent equations Bit dote. ey == Pis By ea Oy ee if p, 1s the sum of the first 4 digits of the pattern vertex, p the sum of the next #, digits, etc. As in art. 10 these equations can be written in the form k, k,+k, k, + ke, + … + nd LE EN sree is eee Re Zap, =P ps. Lae =H= Mh + he. eee i=! i=1 i=1 representing #—d limiting spaces #8, , of (P),, as each of the right hand members is a maximum. For the rest of this part we eter to art. 10: | c) By means of the theorem we obtain all the limits (P), of (P),. For this part compare also art. 10. ns DERIVED FROM THE REGULAR POLYTOPES. 15 55. We apply the notion of end digits and middle digits of the syllables, introduced in art. 12, to the syllables in round brackets occurring in the symbols of the polytopes deduced from the measure polytope, in order to be able to repeat theorem XXX, in a version connected with the more practical unextended symbols, in the following form: Tarorem XXX‘. “We obtain the unextended symbol of a pee tope (P), the vertices of which are vertices of the given (P),, i we put the lowest # digits of the pattern vertex between square brackets, where 4 takes successively one of the values 0, 1, 2,..., d, and place before it, of the x — # remaining digits, between round brackets either one group of d — # + Ll interchangeable digits, or two groups containing together d — # + 2 interchange- able digits, or three groups containing together d — # + 3 inter- changeable digits, etc., this process winding up where the total number of groups is 2 —- d + # for # < 2d—£#- 1 and d for n > 2d—k— 1”. “This (P), will be a Hanne polytope of (P),, if the syllables between round brackets satisfy the two following conditions: 1°. each syllable with middle digits exhausts these digits of the symbol of (P),, 20, no two syllables without middle digits have the same end digits” The proof of this new version can be deduced fron the articles MO 1-2 and 54: By means of theorem XXX’ we deduce the limits (P), of the polytope (Pho represented by the symbol [5443 322211], ily shown!) — the (P) of art. 12 represen ted by (5443322210) is the limit gy of vertex import. If we put together the different (?), for which the # has the same value we find for £—0O the 58 polytopes given in art. 12 and for Elsen.) 6 -emecessively/proups: ons D Ier 1e. in toto 120 polytopes. If for brevity the last syllable — between square brackets — 1s sy at the head of each SE, these are 1 OI O1 be simplified be passing to parallel axes with the point 1, 1,..., 1 as origin, i.e. by subtracting a unit from all the coordinates. If we then bea = aia that according to art. 1 we have to divide the coordinate values by “2 if we pass to barycentric coor- dinates on account of the new unit of length, we find (5443322210). From this relation between a polytope deduced from the measure polytope and its polytope of vertex import can be deduced generally that the number of these polytopes in Sn, the measure polytope itself included, is C + 2N + 1, where C and N represent the numbers of Central symmetric and of non central symmetric polytopes in Sn—1 of simplex extraction, the simplex itself included. 16 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY k = 1, last syllable [0] (544339), — (54433) (21) — (5443) (322), (5443) (32) (21), (5443) (221), — (544) (3322), (544) (332) (21), (544) (3222), (544) (822) (21), (544) (32) (221), (544) (2221), — (54) (43322), (54) (4332) (21), (54) (433) (221), (54) (43) (3222), (54) (43) (332) (21), (54) (43) (32) (221), (54) (33222), (54) (3322) (21), (54) (832) (221), (54) (32221), — (443322), — (44332) (21), — (4433) (221), — (448) (3222), (443) (322) (21), (443) (82) (221), — (433222), — (43322) (21), — (4332) (221), — (433) (2221), —(43) (32221), — (332221) hk == 2, last syllable [107 | (54433), — (5443) (32), — (544) (322), (544) (322), — (54) (4332), (54) (43) (322), — (44332), — (443) (822), — (43322), — (48) (3222), — (33229) k = 3, last syllable [210] (5443), — (544) (82), — (54) (433), (54) (43) (82), — (4433), — (443) (32), — (4382), — (43) (822), — (3322) | k =A, last syllable [2210] (544), — (54) (43), — (443), — (488), — (48) (32), — (332) Æ — 5, last syllable [22210] (54), — (43) k = 6, only syllable [322210]. We remark, that in general the # of the theorem indicates how many of the axes of the rectangular system of coordinates are parallel to the space 4 bearing the (P),. For d = 2-— Caem we determine the limits of the highest number of dimensions, the Æ is at the same time the index of the symbol g, indicating the import. For comparison we put side by side in the next table the different g, of the polytope (Pho just treated and those of its polytope of vertex import (5443322210) ...... Jo (544332221)[0]..... Gad (SAAB SIRO ee tees Js (54433222)[10]..... Ja | (64433222)(10). 2. 7 (5443322)[210]..... gs | -(0448322)Q10)..... I (544332)[2210]..... gs | (544332)(2210)..... Is (54433)[22210]..... Js | (54433)(22210)..... VA (5448)[322210]. .. Je | (5443)(322210)..... 3 (544)[3322210]..... 91 | (844)(83B22210) 000% 7 (54)[43322210]..... ga | (84) (43822200)25 et 7 [443322210]..... Jo (4483822210)... 9 _DERIVED FROM THE REGULAR POLYTOPES. PT From the examples given in the art. 51 and 52 it is clear that in the enumeration of the limits of the highest number of dimen- sions we proceed from # — # —- 1 to # — 0; this principle has been followed too in column five of Table IV. C. Æxlension number and truncation integers and fractions. 56. Taroreu XXXI. “The new polytopes, all with half edges of length unity, can be found by means of a regular extension of the measure polytope followed by a regular truncation, either at the vertices alone, or at the vertices and the edges, or at the vertices, edges and faces, etc.” This theorem is an immediate consequence of that given in art. 50 (theorem XXIX) about the equality of the absolute value of the non vanishing coefficients ¢, of the coordinates x, in the equation + oa, Hoe, +...—>p of a limiting space 8, _, of the polytope. As to the proof we can refer to art. 15. | The extension number is always equal to the largest digit of the symbol of coordinates. So, if in the case [2° 1° 1] of CO of three- dimensional space the cube [1 1 1] with edge 2 is extended to the cube [2'2'2’] with edge 2 (1 + 2V 2) it is precisely large enough to enable us to deduce [2’1'1] from it by truncation; for the limit of face import lies in the space + x, = 2’. Likewise in the case [V2, V2, 0,0] of C, in S,, which symbol winds up in zero, we have to extend the eightcell [1111] to [V2,V2,V2,V2] by multiplying its linear dimensions by V 2, etc, The manner in which the amount of truncation is measured most easily can be explained as follows. If the measure polytope 7 EE — M,® —1[11...1] of S, with centre O is extended to M, °° } En lee. wel, £ being the extension number, and this extended M,°? is truncated at a #-dimensional limit M,°°? with centre 47 by a space S,_, normal to OM cutting in PR any edge PQ of à FE TM M,®% one end point P of which belongs to M,°®, then PO is considered as the “truncation fraction”. Now, as we will prove immediately, PR is always a multiple of V2 with half the edge of M,® as unit, whether the symbol of coordinates of the polytope deduced from 47,%° by truncation terminates in unity or in zero; so, in the relation PR — 9 V2? the multiplicator g which is integer may be called the “truncation integer”. So the truncation Verh. Kon. Akad. v. Wetensch. 4° Sectie Dl XI No. 5. E 2 18 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY gV2 fraction is irrational if the symbol of coordinates of the polytope winds up in 1 and rational if the last digit of that symbol Is zero. | 57. If we indicate the truncation numbers corresponding succes- sivily to a truncation at a vertex, an edge, a face,... by To, Ti To,... and p' stands once more for 1 + pWV2 we have: Tarorem XXXII. “If [m'o, m4, ms, Mn 1] is the symbol of coordinates of a polytope deduced from the measure polytope 4, of #, — where m, 4 stands for either 1 or 0 — the truncation numbers Tj Fy. Past. ate n —1 n — 2 n — 3 To—=2 My— 2 M;, Ta—=(n—1)m—X2m,,T—=(n—2) Mg DE Mise =0 i=0 i=0 a 2 Proof. Here »', is the extension number. Now, if we wish to calculate 7, and we take for the vertices P and Q of the extented measure polytope [my Mo; +. Mo] the points #5, #5, 00 and — my, Mo,..., mo differing in the sign of +, only, we have to apply the theorem of page 27 (art. 17) with respect to the equation ot+a,t...te,_,=«, (=1, 2, 3), where c, is determmed by the condition that this space is to contain successively the points P, Q and the pattern vertex mo, m1, m',,..., m,_, of the polytope under consideration. So we find n—k—1 a (ah) = (2 — 4 Da à EM i=0 and therefore n—k—1 4 / n — À) mo — Em, PR ( ) 0 ES i POSTE 2 mo But, as 2%, is PQ, the numerator is PR. As the rational part n—k—1 of (x — k)m’, is equal to that of X m’;, viz. x — for m,_,=1 Hil and zero for #',_1 — 0, this numerator is a multiple of V2, as we have stated at the end of the preceding article. So we find n— k —1 Ty = (u — hk) mo — Zm,, as the theorem has it. p=) In the case of the polytope P,, represented by [5'4’4'3'8'2'2 21 1) and in the case of [5443322210] we get T) = 24, T, = 19, 7. DERIVED FROM THE REGULAR POLYTOPES. 19 But the Poe number of the first polytope is 1 + 5V 2, that of the second is Remark. In he enn of the method of measuring the amount of truncation introduced for the simplex to the measure polytope we experience that the truncation fraction may become an improper fraction. This means that the point of intersection R of the truncating space #,_, with the edge PQ lies on PQ produced at the side of Q. If we wish to avoid this inconvenience we can determine the amount of truncation in the following new way. If O is once more the centre of the polytope and 47 the centre of the limit 47,2 of the extended measure polytope /,°? at which the truncation is to take place, whilst the truncating space #,_, normal at OM cuts OM in P, we may consider ie as measure for the amount of truncation. Then we find PM a — jm Em, OM, bele Bg na) from which it ensues that the new truncation fraction is deduced from the old one by multiplication by 5 5 7 But instead of alterimg our method of measuring the amount of truncation we prefer to put up with the inconvenience indicated. So in Table IV the truncation numbers are indicated, after the extension number where g = 1 + 4 V2? and g’ = 4 V 2, according to the original system in column seven. D. Mepansion and contraction symbols. 58. We now prove the theorem: Taxorem XXXII. “The expansion e,, (4 = 1, 2, 8,...,n— 1), applied to the measure polytope M,® of #, changes the symbol of coordinates [1, 1,..., 1] of that polytope into an other symbol which can be obtained by adding W2 to the first » — # digits. Proof. The operation of expansion e, is performed by imparting to all the limits M,® of M,® a translational motion, to equal distances away from the centre O of M,®, each A, moving in the direction of the line OM joining O to its centre #7, these 7, remaining equipollent to their original position, the motion being Qe 90 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY extended over such a distance that the two new positions of any vertex which was common to two adjacent M,® shall be separated by the length 2 of an edge. Now if we move the limit M,® for which we have k U FU ==... Cr Pa Rn 1 in the manner described in the direction of the line joining O to its centre MZ, for which ULg SS. HD ES Un k +1 EN ANT VTT. =d to a A times larger distance from O we get a new position of this M,® characterized by EES == Lo —. A leans Uy —k+1) Un —k+29° “5 D = se = a lt Te in which it is a limit M,® of the new polytope AA TN me Be and according to the last ten lines of art. 48 this polytope belan to the progenies of M,® if we have A — 1+ V2. So the rea n—k ica. ae em which proves the theorem, and we find by the way: Trrorem XXXIII. “In the expansion e, the limits 17 of M, are moved away from the centre to a distance always equal to 1 + V2 times the original distance.” This comes true, for 1 + V2 is the first digit of the aba of coordinates of the new polytope and, as we found in art. 56, this first digit represents the extension number. As the distance OM was V (a — #) it becomes (1 + V 2)V (u — A). Remark. We may express the influence of the operation e, on k the symbol [11...1] without interval between the digits by saying that it creates an interval V2 between the x + 4” and the nk + 1* digit. 59. Tarorem XXXIV. “The influence of any number of expan- n SIONS “C,,€;, E35. Of M OO svn ao bbe i .1] is found by adding together the lue of each of the expansions taken separately.” Proof. We begin by combining two expansions only. In the succession of two expansions the subject of the second is to be what its original subject has become under the influence of the first. So in the case e,e, C of the cube C (fig. 13%) the B Ser ci 7 @ DERIVED FROM THE REGULAR POLYTOPES. 21 original subject of e, (the square) is transformed by e, into an octagon (fig. 13°) and now the octagon is moved out, in the case e €, C the linear subject of e, (the edge) is transformed by e, into a square (fig. 13°) and now this square is moved out; in both cases the result (fig. 13") is the same, a ¢CO. In general, for 4 > /, in the case e,e, M,® the subject M,@ of e, is transformed by e, into an M,®, while in the case e,e, M,® the subject M,® of e, is transformed by e, into an 2 — 1-dimensional limit g, of the import /. Here also the geometrical condition “that the two new positions of any vertex shall be separated by the length of an edge” makes the distance over which the second motion of any of these pairs has to take place equal to the distance described in the first motion of the other pair; i.e. if M,® is a limit of the limit 7° of M, and A is a vertex of that M,®, the segments described by A in transforming 17, into the two polytopes e, e, M,® and e,e, M, are the two pairs of sides, with the length V 2(2—4) and V 2(xn—/), of a rectangle leading from A to the opposite vertex 4”. So we find the coordinates of 4’ by adding to the coordinates of 4 the variations corresponding to the motions due to each of the opera- tions e,,e, taken separately. So, in the case of three or more ex- pansions we will have to use the extension of this rule to parallel- opipeda and parallelotopes; to this geometrical composition of motions always corresponds the arithmetical addition of influences. So the general rule is proved. By the way we still find the theorem: Tasorem XXXV. “The operation e, can still be applied to any expansion form deduced from J/,® in the symbol of coordinates of which the 2 — 4” and the x» — # + 1% digit, 1. e. the 4” and the # +- 1 digit counted from the end, are equal” This theorem enables us to find immediately the expansion symbols of an expansion form deduced from 47, with given coordinate symbol. We show this by the example [5443322211] of art. 55. In [5 44388222 1'1 | five intervals occur, viz, if we represent the py” digit from the end by d, between (d,, d,), (do, da), (ds, dé), fd, =); (dy; dig). So: we find: eee 07-80 M, 60. By means of the operations e, we can deduce from MZ, all the possible polytopes the square bracketed symbol of coordinates of which winds up in a unit. If we wish to deduce from M,® also all the forms with a square bracketed symbol ending in zero — which _ is a desideratum as to the treatment of the nets — we have to introduce the operation c of contraction. The subject of this contraction | 22 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY is the group of limits (/),_, of vertex import, sometimes denoted by go, the vertices of which form exactly all the vertices of the expansion form, each vertex taken once, and now the operation c consists in this: all these limits undergo a translational motion, of the same amount, towards the centre O of the expansion form, by which any of these limits gets a vertex or some vertices in common with some of the others. By this contraction the edges of the expansion form parallel to the axes of coordinates are annihilated. We have now the general theorem: Trrorem XXXVI. “By applying the contraction c to any expansion form all the digits of the symbol of coordinates of this form are diminished by a unit”. This theorem, which shows that the preceding one still holds for contraction forms deduced from 47%, is almost self evident. So, as the motion of the limit g, lying in that part of 8, where all the coordinates are positive takes place in the direction of the line making in that part of $, equal angles whith the x axes, all the coordinates of the pattern vertex diminish by the same amount, and this process has to go on untill the smallest of the digits disappears. For then we once more obtain a polytope the symbol of coordinates of which satisfies the laws of the first part of theorem XXVIII (art 47). Remark. By combining the theorems XX XV and XXXVI we can find the symbol in the operators c and e, of any form deduced from M,®. But this process can be simplified by introducing the opera- tion e which transforms the centre O of 47,% considered as an infinitesimal measure polytope 47,9 into 4/,%. Then the contraction symbol c can be shunted out by substituting e, e, ... en M, for Ce, e)...€, M,, but this implies that we replace e, e,. . .e, M,® by € ¢,@..-@, M,°. This remark will be useful in part F of the next section (compare theorem LIII). B. Nets of potytopes. 61. The theory of the nets derived from M,® is based entirely on the consideration of the most simple of these nets, the net N (M,) of the measure polytope itself. So we begin by the analytical representation of that net WV (M,®). By means of the symbol [2a, + 1, 2a,+1,..., 24, +1] the net of J," is decomposed into its measure polytopes, if &, 4», . … 4, are arbitrary integers and the heavy square brackets mean that im order to obtain a definite M,® of the net we have to permutate and to DERIVED FROM THE REGULAR POLYTOPES. _ 29 take with either of the two signs the units printed in heavy type only. Of the #7” brought to the fore by this symbol itself the Pire isthe point 2a,, 2a,,.:., 24; So [2a,-2a&,..., 2a, | may be called the “frame” of the net, and this symbol may be written quite as well with round or even without brackets, as the faculty of taking for the a, all possible integer values includes permutation and changing of signs. 62. If we consider the net NM(M,®) as a polytope! of 8,4, with an infinite number of limits (7), which instead of bending round in 8,,, fills #,, we can apply to this polytope the expansions €;,@,...,e, and the contraction ce, either separately or in possible combination ; in this simple way the measure polytope nets e, A(J/,), e, N(M,), etc. have been determined by M. Storr. We introduce the corresponding analytical considerations by the following: Tasorem XXXVII. “Let any expansion or expansion and con- traction form (P), of M,® be represented by the symbol of coor- dinates/[4,, a);..., 2,1, @,|. Let MC” be the measure polytope with edge 2a concentric and coaxial to this (P), and W(4,°) the net of measure polytopes to which the JZ,°” belongs. Let us suppose in each of the oo” measure polytopes of this net a concentric polytope equipollent to (P),. Then the vertices of all the oo” polytopes obtained in this manner cannot form together the vertices of a net, if a differs from a, and from a, + 1.” This theorem of a negative tendency can be proved thus. If we call two (P), “adjacent” if the measure polytopes M,°” concentric to them have this position with respect to each other, 1. e. if these M,®® are in M,_,°% contact, and we consider the limits (é),,_, of the highest import of any two adjacent (P), deduced from the common 47, _,°" of the two 47,2" concentric with these (P),, we see at once that these limits ÿ,_, coincide for a = a,, whilst they are at edge distance from each other and form therefore the end polytopes of a prism for a — a, + 1. In all other cases two adjacent (P), are either too near to each other or too far apart. What we shall have to show farther is this that the vertices of the oo” polytopes (P), do form together the vertices of a net in each of the cases a — à, and a — a, + 1. We prepare the general proof of this assertion by indicating by the special case of the threedimensional net of truncated cubes [1 + V2, 1 + WV2, 1] included in larger cubes J/;°”, where a = 2 + V 2, how the other constituents are to be found. ‘This will give us occasion to introduce *) Compare art. 39. 24 ANALYTICAL TREATMENT OF THE POLITOPES REGULARLY some new geometrical terms by the use of which the expression of general laws will be simplified. In fig. 14 is represented in heavy lines one of the /C with centre O and an eighth part of the 47,%° surrounding it, viz. that part ‘lying in the octant of the positive coordinates taken in the directions OV, OV’,, OV;. Now we make to correspond to the different limiting elements of the surrounding cube the limiting elements of the ¢C into which the first are transformed if the #C is deduced from the surrounding cube by truncation at vertices, edges and faces. So the triangle ABC of vertex import corresponds to the vertex V, the edge 44° (or the face of edge import which replaces it in an other case) corresponds to the edge VW,, the octagonal face B’ BCC’... corresponds to the face W, VW. Then by reflecting the triangle ABC into the three faces of 47,°° through the corresponding vertex VY as mirrors and by dealing in the same way with the edge 44° with respect to the two faces through the corresponding edge V/W, and with the face B BCG... with respect to the corresponding face WVW, we get successively the eight triangular faces of an ACO with V, the four upright edges of a P, with V,, the two end planes of a Ps with /, as centre. We simplify these expressions by saving that “multiplication” of the triangle 4BC round V, of the edge 44’ round VW, of the face B'BCC... round WV W generates the indicated polyhedra RCO, P, = C, Pa. In fig. 14 have been represented in ordinary lines the ACO generated by the triangle 4BC, the three cubes generated by the edges 44’, BB’, CC’ and the three P, generated by the faces B'BCC'.., C'CAA'.., A'ABB'.. From this diagram it is clear that the indicated ACO, C, Ps fill up the interstitial space between the 4C, i.e. that the net bearing in ANDREINT’s memoir the number 22 exists; we facilitate the inspection of this diagram by adding a stereoscopic representation of it. }) The deduction of the coordinate symbols of the new constituents RCO, C, Ps from those of the ¢C and its surrounding cube shows us, what we have to do in general in order to obtain the coordinate symbols of the new constituents. We begin with RCO obtained by multiplying the triangle 4BC round VY. In order to get the representation of the triangle 4BC with respect to the original axes we have to replace the square brackets of the symbol [1 + V2,1+V2,1] of ¢@ by round ones. In order to represent that triangle with respect to new axes *) The effect is enhanced if we place it so, as to have the small arrow at the left. DERIVED FROM THE REGULAR POLYTOPES. 25 VV, VV, VV; we have to replace the digits of (1 + W2,1 + W2, 1) by their complements to a = 2 + V2, giving (1,1,1 + V2), Le. (1 + V2,1,1). In order to multiply the last triangle round the new origin V we have to return to square brackets. So [1+ V2,1,1] is the symbol of the new constituent RCO. We repeat that the digits of this new symbol are the complements to a=2 + V2 of the digits of the “groundform”’ ¢C taken in inversed order. In the case of the edge AA’ and the cube derived from it we have to assume V,, the centre of the cube, as new origin, and eV, V; V,, Vi V's as new axes. Thereby a, —= [1], 2,=1+ VQ, Ml V2 is transformed into 2;—=([1], 7: = 1,#:— 1; so by muluplication we get — [1], 2, 7==|[1;1] or shorter | i}, (1, 1}, which in this special case may be combined to 2, 2’, a’3= gay AE or shorter [1, 1,1], the cube. Finally the face A'ABB'...represented by &,, 2, = [1 + V 2,1], @,=1-+ V2 passes by multiplication into a,, 2, =[1+ V2, 1], #3, — [1] or shorter [1 + V2, 1][1]. So if we arrange the constituents in the order 43, 4, 91,9 of decreasing import we get g3 =([1+V2,14+V2, 1] jo=([1 + V2, LEE ee, es rie The ee the first and the last being semiregular polyhedra deduced from the cube, whilst the intermediate ones appear as prisms. We remark that the pairs of syllables of the symbols of 7, and 9, can be derived from the symbols of 7, and 9, by taking for g, the last two digits of g3 and the last digit of gj, for g, the last digit of g, and the last two digits of g. Now it is obvious that in the general case of the polytope (P), of S,, represented by [a ,a@,...,a,_1,a,] the introduced multipli- cation of the limits of dre import, which multiplication can be performed for any value of the constant a, leads in general to n + 1 constituents g,,9,_4,.., 41, Jos represented by Jo nn lid 5 ay 9 Us Sree Meteen fo leit: telle 3 per, Gy 49 a, | Le enn ns [as » As , (477 Site! ‘eis; of ewe Mo lee (© he 13 2 TX ne D = [aes » Uy. MES NE RES ec ETES Ge AE staak: ESO, ‚Ads, A—a, | Jo = [a—a,,, a—a,,_, AA OE ADE: ‚ d—Ay, A—a, | >) 26 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY where g, is given, go is obtained by subtracting the digits of g, from a and taking the differences in inverted order, while the two syllables of 7,_, are got by taking the last nk digits of 9, and the last # digits of g for #—1,2,...,n—1. 63. We now prove the following problem of positive tendency completing the preceding one. Trrorem XXXVIII. ‘In either of the two cases a — a, and a — a; + 1 the vertices of the oo” polytopes (P), of the preceding theorem do form together the vertices of a net. The constituents of this net are obtained by means of the algorithm developed at the end of the preceding article.” We march in the direction of the proof of this general theorem: 1°. by deducing from the symbol of coordinates of the given groundform (?), the symbol representing all the repetitions of this polytope and therefore all the vertices of the system, 2°. by deriving from this new symbol the symbols of the polytopes different from the groundform the vertices of which belong to the system (which set of new constituents will prove to be equivalent to that obtained above by the geometrical multiplication introduced above), 3°. by showing that the system of polytopes obtained in this way fills space, i.e. that there is neither overlapping, nor hole. Symbol of the total system of vertices. The symbol of a definite repetition of the groundform is [26,4 + di, bra Hose 26; 44+ Ars 26,4¢+ 4, where 6,, 6,,..., 6,4, 6, 18 a definite set of arbitrarily cme integers. So this symbol represents the total system of vertices, if the 4; denote all possible sets of integers. From the symbol 7 we deduce the frame symbol [25,45 Wasa 20,-4.0, bal RCE F representing the system of vertices of a net of measure polytopes M,°, one of which has the origin as vertex and the x spaces D 0, (¢ = I, 2,222) tis spaces: Presumptive new constituents. 'The most general transformation by which the total system of vertices 7') passes into itself consists in a transport of ya units from the permutable to the unmovable part of 2,, the # quantities p, being integer. But this process is limited by the restriction that in the case of a new constituent sought the permutable parts placed within the same pair of square brackets have to satisfy the conditions of theorem XXVIII, from DERIVED FROM THE REGULAR POLYTOPES. 21 which it ensues that the extent of the restriction depends on the number of syllables which the symbol of any constituent may contain. This number is evidently two at most. For the process can only afford besides the original minimum digit a, one new minimum digit, viz. zero in the case a — a, and unity in the case a — a, + 1. So we have to hunt up only new constituents the symbols of which are either monosyllabic or composed of two syllables. If we take 4/7 the p, equal to one we find [28 + lha+a,—a, (24+ Data—a,..., (26,4 = la = asa a, (26, Bi 1) a ai u, LES a), or, if we replace negative permutable parts by the positive ones of the same absolute value, rearrange these positive parts according to decreasing order and substitute for brevity (© for 28 +1, [Braad B'ata-a,_s...,8, 1a+a-@,8,a+a-a;].7) winding up in zero for a=a, and: in unity for a— a, + 1. So we find the repetitions of the new constituent 9, of the last list of the preceding article. This form g, and the given form g, we started from are the only constituents of measure polytope descent proper. If we transform the first 4% digits of 7' by the transport of a units from the permutable parts to the unmovable ones and put each of the two sets of digits, the set of the / transformed ones and the set of the x—/ untransformed ones, between square brackets, we get after rearranging, if (©, still replaces 26, + 1 and 2; is substituted for 25, [Prada Pia +d—d,,...,8B;ia+a—ai] [Bra + Aras Pire 0 + Unser Orr A + Ons Ba +]. .77) revealing the new constituent | A—Ay, » AA Aar arr Oera © © 09 Ars GJ; a prismotope (P,; P,_,) with the constituents (P), and (P),_, repre- sented by each of the two syllables of the symbol taken separately ; if the digits of the second syllable correspond to the coordinates 1, #o,.. +, &»_, and those of the first syllable to æ,_;,4, 4, 149, .., n> this prismotope is the constituent g,_, of the last list of the preceding article. In the latter case the different positions of (P),_, are parallel to OX Hee Xe.) those ot (P),t0. OX, A zel) So we find again all the new constituents obtained formerly by geometrical multiplication. 28 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY No overlapping and no hole. By a translational motion in the direction of one of the axes over a distance 2a the system of vertices 7’) is transformed in itself; so, if the central measure poly- n tope [«,a,..., a] 1s filled exactly by the set of constituents found above, these constituents form a net By a reflection in one of the n spaces a,—=0, (—1,2,...,2), the system 7’) also is transformed in itself; so, if the part of the central measure polytope 44,2" containing the points with positive coordinates only is filled exactly, the constituents form a net. We indicate this part of the central measure polytope by the symbol 47,7. We now prove the following lemma: | „Let (P),” be a constituent lying partially within 44,79 and (Pÿ any of its limits lying partially within 4/9. Then the set of polytopes obtained above always contains one and only one polytope (P),’ having with (P), the limit (P%% in common; this (P),’ lies with respect to (P), on the opposite side of (Py? 7.” The condition that (P),” hes at least partially within 47,9 is fulfilled, if we consider that repetition of the chosen constituent the coordinates of the centre of which admit the values + a and zero only. We find, if all the coordinates are zero the groundform contained in 7’), if all the coordinates are + a a polytope contained in 7”), if some coordinates are —+ a and the other ones zero a polytope contained in 7”). Now the first case, of the groundform, and the second case, of all coordinates — + a, are included in the third case, as we get them by putting 4=—0 and #4 —». So we can choose for (P)," the polvtope lada A, a HAA aten dry, - +s Oa U V4 9 Uy, eee e/we le ’ dy Vers Up: 499 es 6 6). an Th where the +, placed under the two syllables indicate the coordinates to which the two sets of digits refer, and occupy ourselves with the question how to get a limit (/),_, of this prismotope. Now in general the limits (/),_4 of the prismotope (?,; P,_,) present them- selves in two groups, viz. if (P),_, 1s any limit (/),_, of (P), and (Pra any limit (Da Of (P), ,, Im the two forms (Pl and (P,; P,_, 1). So D, we have to consider the two different cases : NE a, b : , Pte +a : *) For a limit (P) i lying at least partially within M none of the coordinates == y may assume values equal to or surpassing + a for all the vertices of that limit; therefore in the first case (P, 4; P,;,) we have to place between round brackets a certain number s, of the largest digits |a + a— aj| where æ — aj is taken with the reversed Sign, 1. €. ap, @,_4,+++, y 44 taken in inverted order. DERIVED FROM THE REGULAR POLYTOPES. 29 [a+ 4-0; a+A-Qr-s,_1 MO ata, | (aj 8, +1? hy — 3,493 inden »4x) UA Poe ras em > Dis, Cis, +1 Ves, +2 ee es Pp Pa Sikes. Sopra | ; n Vista» pose et DÉS NON RASE D ds | (aa > Uy+9 IQ RON yek= D Orte) [di ee > Ae+s,+29 EN EE a, | PP a ne Ro AR AD Ne which two limits (/),_, admit as centres the points ks, ST n—k Rae A bai. te Ot ae k Sa N—k—S2 P TT ae RE DO CU 4 and 4 being determined by the relations k k+s, Sie, , Si, , i=k—s,+1 is=k+1 showing that we have 0 < ¢;< a for 1 — 1,2. So the centres of these two (/),_, le on the boundary of the measure polytope M,*® and therefore the (/),,_, themselves le partially within that measure polytope. Now for each of the two cases there is only one constituent passing through the chosen limit (/), 4, viz. (EEA U a+, 1; ARE a+a-ay | ater As, +29 es My 4, Un Ce .., Lys, Tps, +1 ’ Vi — 9,42 >...) Uy [a+ A-Ax 4s, a+, Ay ++ + aad | [apar Apts, +29 +%>4y,_4, Un Vv , do CR Vr+s, Vets, +1 3 Vets, +2 CHROME) Vr So, all we have to do yet is to investigate the position of the centres. If we indicate these points by the letters G,, G,,, G,,, G a? D, ? Gay, and we remark that for these five points we have ab, ? USS. Uh Tps Tps, — + —dp Uh Urga Dors) Cets, +4 = Cets,42 St dy; we find the following list of coordinates Piste Di ut NPA NE ee dea 6 a a 0 0 Gy, a 0 0 0 | Gs, a a a () cane a La 0 0 Gi a a ly 0 According to this list of the two triples (G,, G,, Ga), (Gas Go, Go) of collinear points G,, lies between G,, G,,, and G between ab, ab, 30 ANALYTICAL TREATMENT OF THE POLYTOPESSREGULARLY G,, G,, So the proof of the lemma is given. So neither of the two systems of constituents can admit holes. In order to show that no two polytopes of any of the two systems can overlap we remark that by means of the symbols 7), 7"), 17) any polytope of the chosen system can be promoted to central polytope, which shows that not a single vertex can lie inside that polytope. | So we have proved completely now the theorem under consi- deration. 64. We now formulate the manner of deduction of all the measure polytope nets as follows : | THEoREM XXXIX. “Let G —[a,a,...,a, _1,a,] be the symbol of coordinates of the “groundform” of the net. Deduce from it the symbol O =[a—4a,, a—a,,_4....,@—4d,, a— a | of the “opposite form”, where a is either a, or a, + 1. Derive from these two symbols G, O the mixed symbol 7, of the “intermediate forms” represented by (as Ty 2450 a eek [a — Ann U Anr 4 — My, a — Q |, of the two syllables of which the first contains the last / digits of G, the second the last z — 4 digits of O. Then G, the forms 7,, A=n—l, n—2,..., 2,1), O are respectively the constituents Ds Un Asrai AOP AERDEN “The frame of the constituent 9, _,; 18 ORE a, Pise d,... Eur a, Bee, Bt a, Bi A,.. shee a, 2’, a |, where we have B, — 26, and 8’, = 25, + 1, the 4, being integer and the digits of the first syllable being related to the odd, those of the second syllable being related to the even multiples of a”. “If (e,c), etc. indicates a net with an expansion groundform and a contraction. opposite form, the theorem includes the four cases: a, == NS a aR EEN (etc) an RCE GR SRE (e, e), Dy, = SCENE ee (eo) a, = 0 Sine: (c, e).” In this theorem the deduction of the intermediate constituents differs slightly from that given in the preceding article, the two me- thods passing into each other by interchanging # and # — #, and the two syllables. In the new form the succession of the different constituents is a more regular one, as the following examples prove. DERIVED FROM THE REGULAR POLYTOPES. ol Example 1. The two nets with [54433222 11] as ground- form admit the constituents : BDA BS YUI moetn 44/88 2991 gist Ae OET EL or EERE NE EAM Te [IT] EEN OMZ | gn. UBB 9 9/0 ll LT) ee 2 er Pea TOV 8 8 BRT 1| (2 11) D 2 21 2 ROWO ge 9/9 11] eee Pe ee OW Se ere RE sen SZ ONWEL [seau PTT) [582 S110 le. TH [888 er PI MP 43332410 lg. TIAS 382 DT Tele Mer [6483892711 01V2 ly. oc: [5 4" 8’ 3’ 3'2/9'1'1'1) form admit the constituents: Gao + - [5443322210]V 2 heard FAS Seek bOI? gs) (A 4B S22 1017 8 1453922210) 2, FLOW 2 | wee Ve 483.2 2 2102 Be B22 2104V 2... 120]Y 2: graal 1), [8 8222402 eee el 21085 PAL EON Pe ge he VA oe BAO pe) (222101 2-5; (221 TON Zur gene EV 11 112 22 10e gen (2210 | 2 82211012 4 gn. [8 MO Vie 1224012 BeOS (83221102 ass pa 8 272 AT PED et |[101V2 , [8383221102 a 0188 B22 EE HALO 2 Rue re. [5433322110]V2 | go. . .[5'4'3'3'3'2/2/1'1'1] The nets of measure polytope extraction of the spaces S;, 5, S; are put on record in the Tables V and VI. The first column of these tables is concerned with the “name” of the net; it contains the system of operators e, and c which are to precede the general symbot V(47,”) in order to obtain the symbol of the net. This system of operators is in close connection with the consideration of the net of S,, as a simple polytope of 8, ,,; for a= a, it is equal to the system of operators characterizing the groundform, for a=a,-+ 1 it consists of latter system completed by e,. So of the three parts into which each of the three cases 2 — 3, 2 — 4,2 —5 has been subdivided, the first contains the nets (e, c), the second the nets (e,e), the third the nets (c, c). Therefore the question rises where the nets (c,e) are to be found. The algorithm indicated in our last theorem immediately shows 92 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY that by interchanging the two extreme forms with one another the intermediate constituents return in inverted order of succession. This remark suggests an answer to the question raised just now. By taking the constituents 9,,9,_41,...,41,90 contained in the second, third,...,2 + 1%, » + 2% column of the same horizontal line corresponding to a certain net in reversed order of succession we get the constituents 9,9 „—4,---, 94, Jo of a net bearing in general an other name, the operators occurring in which are inscri- bed in the x + 3% column; this net with constituents with com- plementary import is essentially the same as the original one. So by inverting the order of succession of the imports the three groups (e, c), (e,e), (c‚c) pass into (c,e), (e, e), (ec, c), in other words the first group furnishes the group (c, e), whilst each of the other groups passes into itself. We have used this fact, to which we shall have to come back in part / of this section, in order to simplify the Tables V and VI. So on one hand the nets (c, e) have been omitted totally, whilst on the other the number of lines of the groups (e, e) and (c,c) have been diminished by writing down the nets in a trans- parent systematical order and omitting at any time the net appearing already in inverted order under the preceding ones. !) In the column under the heading p. some particularities of the nets have been inscribed. By 7. we have indicated that the net is regular, by s.p. that itis “semiperiodic’’, 1. e. that the two extreme forms are the same which implies the equality of any two consti- tuents with complementary import. The other columns will be explained later on. A survey of the results contained in the tables suggests the following remarks: a). There is a great difference in character between the consti- tuents of a simplex net proper on one hand and those of a measure polytope net. All the constituents of a simplex net proper are expan- sion and contraction forms of the simplex, whilst we found just now that in a measure polytope net in general only two of the consti- tuents, the groundform and the opposite form, are expansion and extraction forms of the measure polytope.*) *) The cases ce, N(C,), ce, N(C,), etc. do not figure in the first third part of Table II contained in the memoir of Mrs. Storr, as they appear already as expansion forms under either M(C,,) or N(C,,). In order to spare room we have omitted in Table VI the column containing the name of the net taken in inversed order. For the upper and middle part it is always the symbol before M, under 4, to which e, has been added, for the last part it is that symbol itself. *) Compare for the prisms and prismotopes entering here my paper: “On the cha- racteristic numbers of the prismotope”’, Proceedings of Amsterdam, vol. XIV, p. 424. DERIVED FROM THE REGULAR POLYTOPES. | 39 This difference in character implies a difference in the number of different positions a constituent of definite form may admit. In the case of a simplex net proper this number is {wo in general and only ove if the form is central symmetric. In the case of a measure polytope net this number is ove for the two extreme constituents, whilst the intermediate form J, generally occurs in a number of different positions indicated by half the number of limits J/, of M,®, i.e. in 2"—"~* (n), different positions. In the case of the simplex net we have considered as kind of constituent any polytope of the net with egwpollent repetitions ; when the partition cycle was a power cycle we have even been obliged to split up a kind of constituent into several groups, in order to keep the analytical treatment in contact with the geometrical facts. On account of the extreme transparency of the measure polytope nets we can allow ourselves to be less exacting and extend the notion of constituent here by admitting that the 2”~"~' (x), different positions of the intermediate form 7, introduced above belong to the same constituent. 6). In order to be able to indicate the number of different con- stituents according to the new point of view we fall back ‘on the different cases (e,c), (e, €), (e, c), (ce, e) mentioned at the end of the last theorem. By generalizing the results of the two examples given above one finds immediately that the required number is in general n — p —+- 1, where p indicates the number of e’s contained in the symbol. But this general number # — p + I 1s still to be considered as a maximum, 1. e. under circumstances the number of constituents may become less. This decrease can be due to two different causes. If in the first place in one of the two groups (e, c), (c, c) of a net in #, the expansion operator with the largest index is e,, where hk for | d D 7 apo ytope an eat for a net, if the symbol of the polytope or that of the groundform of the net contains no zero, 2°. if the net admits a constituent 9, _:. For in both cases there are at least two kinds of edges: in the first case the edges [1], in the second case the erect edges of the prisms 9, 1 differ in character from the remaining ones. The results about regularity have been indicated in the Tables IV, V, VI. In Table IV the regularity fraction is contained in column 5, whilst the subscripts in column 4 give the different groups of limits (/),. In Tables V and VI in the cases 7 — 4 and 2 — 5 the last column contains the regularity fraction, the last but one +) the „different groups of limits (/),, whilst the part. 2er of Vahle*V contains two columns more, one indicating the number of the ANDREINI diagram of the net, the other indicating the particularities of the edges passing through a vertex (see ANDREINI’s list, page 30—32 of the memoir quoted in art. 22). Section II]: PoLyroprs AND NETS DERIVED FROM THE CROSS POLYTOPE. Asv Phe symbol of coordinates. 72. In this section which is so closely related to the immediately preceding one that it may be considered as a mere supplement of the latter we have to start fron m the cross polytope Cn” of 8, repre- sented by the symbol [100 ... ET V2 and to remember that we are to prove by and by that there is no difference whatever between the offspring of this cross polytope and that of the measure polytope Pits: lor Se For n — 3, 4, 5 we have successively in the symbols of M’. Srorr:?) ') The numbers of the different groups of limits (/), for k>1 have been found in the manner indicated for the simplex in Table III, but we have judged it of no impor- tance to insert an analogous table for the measure polytope. *) For the deduction of the e and c symbels from the symbols of coordinates compare part D of this section. In Table IV second column are inscribed the e and c symbols of the polytopes deduced from the cross polytope corresponding to the symbols of coordinates of the third column. 42 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY hd. [100112 = O NIS: GSR LT) F21011/2 =e, O=10 | PAIS ae OD dE Ve =D Bere n= À. MODO 2 C,,| (5210121 =P ee, Oe [2 2 L 01 == MOI =. e, Ol (2VEH = tee 0, (10012 == Cal eae = [21102 = e Col [2 IT — ea 3 Ce | [L110] p72 = ce (yg SEE = BEE == a [3'2"11] = eee LH Ce ie DO = ce n =D BSD p= . 18221008 = mme [222102 — 20002 = a Oa [2 LT = ee [11000] 12 = ce Caz | [LT LTS L100 2 = e Oa li2 LILI] = ee aol P1100) 72 = ce Coo LLL (2110112 = ee, 143210112 = eee (yo lit 11101 v2 = ce Col 320000 f1'1 111) =) OL PAT PEN EEE Er Oan ERAAN = 60, Cap |[2 2 LINE [3 2 100112 =e, e Co (3 2 L'L'1] — 1 3 6, Cag | [2.2 100] 1/2 = ceren Ong | [LAT LT [3 21101172 =e, ez Cao 13221 11 = E303E, Coy | [2 2 1101 2 = ce, eg Cao VUIL = AVI —eae Ogg | [43211] ey eg yey Cog (VIII: © = ceren Cg [SS B. The characteristic numbers. 73. From the preceding section concerned with the measure polytope _ can be gathered the symbols with the characteristic numbers of the polytopes deduced from the cross polytope, the symbols of coor- dinates of which wind up in a unit, as these polytopes also belong to the offspring proper of the measure polytope. So we have only to add a couple of examples about polytopes, the symbols of coor- dinates of which end in zero. Example {2110}, method working from two sides 3). The number of vertices is 2°. Al divided by 2!, i. e. 8. 24:2 == 90: The number of the edges passing through the pattern vertex is six, for this vertex is united by edges to the vertices: - 20 2041; 201—1, EL 205 eA ie 210—1. 6.6 Ro: So the number of edges 1s In order to find spaces containing limiting bodies we consider successively the equations: tat, Lntaes, +e, tots, te The equations + a, = 2 give 8 forms | 1 1 0], i.e. 8 CO of vertex import. *) In the two examples we omit the common factor // 2. DERIVED FROM THE REGULAR POLYTOPES. 43 The equations + x, + x, — 3 give 24 forms (21) [10], i.e. 24 P, of edge import. ie equations: 2 + 4, — dirgive: b6 forms (21-10) 1e, 16-00 of body import. So, we find 24 CO and 24 C,1.e. 48 polyhedra, and therefore + (24 X 14 + 24 K 6) = 240 faces. So the result is (96, 288, 240, 48) in accordance with the law of Euler. Erample [32110], direct method. The number of vertices is 2*. 5! divided by 2!, 1. e. 1920 :2 == 960. The edges split up into three groups (32), (21), (10). Through the pattern vertex pass: owe edge (32), two edges (21) — on account of the two digits | — and four edges (10) — on account of the two digits 1 and of the facuity to make the last digit to correspond either to + x; or to — x So there are in toto 480 edges (32), 960 edges (21), 1920 edges (10), Le. 3360 edges. | The faces split up into six groups, viz. the triangles (211) and (110), the squares (32) (10), (21)(10) and [10] and the hexagon , (821). In the pattern vertex concur: I 5° one triangle (211), {two triangles (110), on account of + a, four squares (32) (10), on account of the two digits 1 and of + a;, BRE Se eB DEU EN asta: oss) de any OR ot ae ete ae two ,, FD, fe ws foo hexagons. (321), „ So we find: 060 3 ue ae 10 ee ir 2 se) | j — 960 triangles + 2400 squares + 320 hexagons, 1. e. 3680 faces. The limiting bodies split up into the seven groups: (3211) = #7, (321) (10) = Po, (32) (110) = Py, (2110) = CO, (32) [10] = (21) [10] = P,, [110] = CO. *) In the case [10] the difference between + «5 and — x; has no effect, on account of the square brackets. A4 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY Of these seven polyhedra concur, on account of the reasons given above, in the pattern vertex in the indicated order: 147, 4 P,, Spe ae DE 2 P,, 2 P,, 1 CO. So we find: GAP 4 P 2 P 060 (35+ ag ig + a as =) — 80 #7 240 CO + 320 P, + 480 P, + 820 Py, Le. 1440 limiting polyhedra. Finally the limiting polytopes split up into four groups: (32110), (821)[10], (82)[110], [2110] and so we find: 32 ee; 8 (5), 80 (6;4), 40 Pog, AU eeen i.e. 162 limiting polyhedra. So the result is (960, 3360, 3680, 1440, 162) in accordance with the law of Euler. With respect to the import we have still to add that we pass to the complementary import, if a polytope of the measure polytope family is regarded as a polytope of cross polytope descent. So in the first of the two examples where the cross polytope import has been indicated the result is complementary to that registered in Table IV read from left to right. C. Zrlension number and truncation integers and fractions. 14. Tarorem XLY. ‘The new polytopes, all with edges of length unity, can be found by means of a regular extension of the cross polytope followed by a regular truncation, either at the vertices alone, or at the vertices and the edges, or at the vertices, edges and faces, etc.” For the proof we refer to the art. 15 and 56. Here the limit (7),_, of the highest import, i.e. g,,, corresponds to the equation a, +-#,-+ ... + x, — constant. So the extension number is the sum of the digits of the new polytope divided by the sum of the digits of the cross polytope, i.e. by V 2. “Sethe extension number of [3'3'2'1'1] is 5+9V2 divided is V2, 1. €, 942V We can stick here to the method of measuring the amount of the different truncations on the edges. But we must point out a DERIVED FROM THE REGULAR POLYTOPES. 45 difficulty underlying this method. So, in the case of truncation of an octahedron (fig. 16) at the edge BC, it makes a difference whether we choose BA or BC’ as the edge on which we deter- mine the amount of truncation. For if we move the truncating plane (through BC normal to OM, where J/ is the midpoint of BC) parallel to itself untill it passes through O it contains the other extremity 4 of the edge BA, while it bisects the edge BC’. This difficulty can be overcome by stipulating that the edge to be chosen may not contain a vertex opposite to one of the vertices of the limit at which the truncation takes place. But this implies always that we measure quite as well on the line 470 joining the centre of that limit to the centre of the polytope. So if the trun- ee AR cating space cuts 470 in ? the amount of truncation is WO Now the complement oes of this quantity can be deduced immediately from the symbol of coordinates [a,, &,,..., a,] of the cross polytope form considered. If we suppose that the truncation takes place at the limit n — 1 n (Dj, of the corresponding extended cross polytope [1,0,..., 0|a, 1 lying in the space represented by z, + 2 +...—+ a#, = constant it is immediately evident that is equal to the quotient MO k of the sum Xa, of the first # digits of the symbol of the trun- 1 cated polytope by the corresponding sum of the extended cross k n PO 2 a; polytope, i.e. by Xa, So from ii = we deduce: ‘ a Al PA M P srl amount of truncation = —— — “'—, MO LÉ La 1 We illustrate this theory by the example [33211] for which we have determined above the extension number. Here we find moreover 5 5 : Ya, =44+6V2, Se,=—384+3V2, Se,—2+V2,4,=1 2 3 . and therefore A Or 3 + 3V2 2+ V2 1 5+9V2 ° 5+9V2 ° 54+9V2 * 5+9V2 46 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY as the amount of truncation at (4), (/4,, (2), , (/)3. As these numbers rd + 8V2), 5 aap (lB +4V2), Be ENE Eton 5) 137 137 are rather bé we only put on record in Table IV the results relating to the cross polytope forms proper, where the i MP denominator and the numerator of the fraction WO are both integer multiples of V2. Here the result 9 | 6, 3, 1 correspon- ding to [83210]V2 expresses that the amount of truncation at ah. Keel (Bo, (2)4, La is respectively 3° 3° 9° D. Z£ypansion and contraction symbols. 15. What we have to prove here is: ‘TnroreM XLVI „The expansion e,, (k—=l, 2, 3,..,2—2), applied to the cross polytope Cy of S, changes the symbol of coordinates nt [100...0]V2 of that enr polytope by addition of V2 to the NA first Æ + 1 digits into BTI .1 00..0]V2, whilst in the case of en 1 Where application of this rule would give a symbol without zero we have to add unity instead of V2 to all the digits, qe nl BEA | FES Pr or We treat the cases /< »—1 and # —» — 1 separately. Case k > 7, in the case e,¢, C,, the sue SE + 1) of e, is transformed by e, into the form e, S{4 + 1) of the same number of dimensions, while in the case e,e, © the_ subject S(Z-- 1) of e, is transformed by e, into an 2 — | -di- mensional limit g, of import Z Here also the geometrical condi- tion: “that the two new positions of any vertex shall be separated by the length of an edge” leads to the ordinary composition of the motions of the centre accord ng to the rule of the parallelogram in the case of two expansions, etc. By the way we find: Tasorem XLIX. “The operation e, can still be applied to any polytope deduced from Cy» in the symbol of coordinates of which the # + 1% and the 4+ 2” digit are equal.” We indicate by means of this theorem the expansion symbol of DERIVED FROM THE REGULAR POLYTOPES, 49 the example [54433222 11] of art. 55, considered as a descen- 9 ment of | L 00..01 Of the five intervals V2, indicated by (4, 4), (da, di), (ds, de), (ds, do), (d,, dio) the first corresponds to the original interval of the symbol of coordinates of Cf? whilst according to the theorem the others result from the four operations e,, e‚, €, es. But as the symbol winds up in a unit instead of a zero we have to add e. So we find eee es eo On”. 17. By means of the operations e, we can deduce from C,@ all the possible polytopes the square bracketed symbols of coordinates of which are characterized by the fact that there is an interval V 2 between the first and the second digits. If we wish to deduce from C,\ also polytopes with square bracketed symbols the two digits di, d, of which are equal we have to follow M. Srorr by intro- ducing the operation ¢ of contraction, the subject of which is the group of limits (/),_, of vertex import. With respect to this opera- tion we can prove the theorem: Prrorem L. “By applying the contraction ec to any expansion form deduced from C the largest digit of the symbol of coordi- nates of this form is diminished by V2.” Proof. Here we have to consider the two cases of the symbol of coordinates, winding up either in 1 or in 0. Case [1 +(a+1)V2, ltaV2, 1+6V2,...,1]. — If we replace 1+ (a+-1)V2 by 1-+aV2 the limit g, represented by ay—=lIta@tlV2, aaan == (la VIV.) passes into | nl Lave, btn TV a LEV ane Le. that limit (/),_, moves parallel to the axis OX, towards the centre O over a distance W2. Evidently application of this process to all the limits g, corresponds to a substitution of 1 + aV 2 for the digit 1 L(a+1)V2 within the square brackets. Evidently any two adjacent limits represented. originally by DATE EAN dM D) ml +(a LV, a,a;.-.@,=U -eVv2,1+bvVe2,...,), which were separated by the right prism vita DW la), 2,.:..,2,—(1+#+6V2,...,1), pass into the two limits Verh. Kon. Akad. v. Wetensch. 1e Sectie Dl. XI No. 5. E 4 50 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY m=l+aVe, mia =O +eV 2,14 bY 25.5, @=ltaV2, aye. esn == (0 aM LE Veren which are in contact with each other by the 7 — 2-dimensional polytope | edel n—=1+aV2, p= HaV 2, as ai ., 2 = OV M) Case [a 41, a,b,...,0]V2. — Here we have to consider the influence of the replacing of a+ 1 by a. The proof runs exactly in the same lines. Remark. By combining the theorems XLVIII and XLIX we can find the symbols in ¢ and e, of any form deduced from GŸ. But this process can be simplified by introducing the operation e which transforms the centre O of Cf considered as an infinitesimal cross polytope G,\” into CG. Then the contraction symbol ec can be shunted out by substituting e, e,...e,, Gi? for Ce, €...€, CC, but this implies that. we replace ee. Enon by e,e,e,...e, Gy”. This remark — corresponding literally to that of art. 60 — will also be useful in part # of this section. . Meanwhile we Aave shown now that any coordinate symbol be- tween square brackets satisfying the laws of the first part of theorem XXVIII (art. 47) can be interpreted both ways, either as a form deduced from the measure polytope or as a descendent from the cross polytope. So we have proved the following theorem already stated implicitly in art: 48: Trrorem LI. “The families of polytopes deduced from the two patriarchs, measure polytope and cross polytope, are identical.” B. Nets of polytopes. 78. In accordance with the last theorem the net of measure polytopes N(M/) can also be considered as a net N (ce Co) of polytopes ce, _, C,. So the nets put on record for x= 3, 4, 5 can be transcribed as nets of cross polytope descent. But instead of doing this we point out a particularity of the case » — 4. For x — 4 both the half measure polytopes + $[1,1, 1,1] are cells C and in relation with this fact we find a new fourdimensional net of regular polytopes, 1. e. #, possesses besides the measure polytope net exceptionally a cross polytope net too. If we suppose that the net WV (J/,°) be composed of alternate white and black polytopes, so that two J/,° with a common ,'” differ in colour, and that each white J/,° is truncated at one set of eight vertices, so as to retain a +-4[1,1, 1,1], whilst DERIVED FROM THE REGULAR POLYTOPES. © 61 each black M,® is truncated in the same way so as to retain a — +[1,1,1,1], the interstitial spaces between these two sets. of inclined CV?) can be filled up by a third set of erect C,(2 2), and we obtain a fourdimensional net formed by three equally numerous groups of cells C,(22) with the property that all the polytopes of the same group are equipollent. Moreover we can transform the net N(M,) of alternate white and black polytopes into a net of regular cells C,® by decomposing each white VM, into eight mutually congruent pyramids with the centre of the polytope as common vertex and the eight limiting cubes ‘of the polytope as bases, and uniting each of these white pyramids to the black measure polytope with which it is in contact by its base D. Now what concerns us here is that by treating the new regular net W(C,) in the same way in which the net N(M,) has been treated we find several new _ fourdimensional nets; for these nets the reader may compare Table Il of the memoir of M, Storr quoted several times ©). Remark. In art. 64 we have seen that with respect to measure polytope nets any net (c, e) is also a net (e,c). This particularity does not present itself for the nets deduced from V(C,,). So here we will have to distinguish four cases °), 1. e. (e, ¢), (e, e), (c, c) and moreover (c, €). 79. We have seen that the vertices of the net N(M,®) can be represented by the symbol [2 a, + 1,2 as +- 1,24,+1,2a,+ 1] where the a, are arbitrary integers. By considering the point 2; — 1, (¢ = 1, 2, 3, 4), as the new origin of parallel axes and omitting the square brackets we get for the coordinates of these: vertices MO Ld From this we deduce that the vertices of the net V(C,(2v 2)) can be represented by the same coordinate values under addition of the 4 condition that 2 a, has a defined character of parity. If we choose 1 fy the condition “2 a; is even” we get for the three sets of (,,(2V-2) 1 < the coordinate symbols: 1) Compare p. 242 of vol. IL of my textbook “Mehrdimensionale Geometrie” or Proceedings of the Academy of Amsterdam, vol. X, p. 536, 537. *) In the part of that Table concerned with the nets deduced from N(C36) the Pr of the line with the number 28 ought to find a place in the same column in the line with the number 27. Moreover we can add in the last column of the line 29 that this net is equal to that of line 47. The fact that several nets of this part are equal to nets deduced from cell Coy will be explained in part F of this section. *) In (e,c), etc. the first letter is related to C6, the second to Ca. AF 52 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY ae: [2a,++ 2,2 a3 + 0,24, + 0,24, + 0], XEe,even, MEE: 4, 4f2a,+1+1,2a+1+1,2¢,+1-+1,2¢,+1-+ 1], >) , —4[2a,+1+1,2¢,+1+1,2¢,+1+1,2¢,+1-+]1], „ even. Of these three sets I represents the erect group, while IT and II form the two inclined groups. If we wish to represent analytically the fourdimensional nets derived from WV(C,) we have to start from the three symbols I, IL, HI, and to study the influence of the operations e,, c. As to the repre- sentation of all the vertices of these new nets by coordinate symbols these influences can be split up into two inadequate parts; of these the first deals with the variation in form of any C;,, of each of the three groups, whilst the second is concerned with the variation of the distance of any two C,. We treat each of these two parts for itself, a) Variation in shape. We know the influence of the operations c,, ¢ on the coordinate symbol [2000] of the central C,(272) of the erect group and from this we can deduce the corresponding influences on the C22) of each of the inclined groups by means of the transformations of coordinates by which [2 0 9 0] passes into 4{1111] and —{if1111] The formulae corresponding to the first transformation are 244 = % + LH A3 Hr Wa D Ye = Li do 3 Vs, 23 == di — do + Bz &% Zyn V4 V3 + Vy by changing the sign of y, we get formulae corresponding to the second transformation. In the following small table we put on record the result of the first transformation: D ORO Br Prien to a Lis Bog a “4 9 2 0] Pots Es wih Tg Sea a 4/2111] and —+j11'1', pe — 1 ACC a) [6 4 2 0] Zale Ge a bee Ge 1 1} and — +13 +72, 3 +72, l', v2—1] D PI... aL 2/2, 9 9 0] and —3[4+y2, 242, 2+p2, 2] en EL OE 1... 6 1-22, 4 2 01 and —3[6+ )2, 441772, 2+7/2 12] (® SY aie poy 07? 2200) WEN eS oe a EAS ie Pie aye 0" 200 01 pe and, Ne AE ATEA —3[531 1] Deere IO a. as a's [2 + 27/2, 2 0 0] and — }3{[1'l’1 1ljp2 cey e,[1'1'1'1] ve SES — 413+ 9172, b 1 Tand 23) Sey ee Cente LUE GE Ce us. we —3([5+4+ 272,38 1 1] and —}$ (5+ 2, 3+yp2,1+)72, 14/12] DERIVED FROM THE REGULAR POLYTOPES. 53 6) Variation in distance. We account for the variation of the distance of any two sixteencells due to the extension of these cells by multiplying the immovable parts of the digits of the three sym- bols of coordinates given above for the three groups of sixteencells by a certain constant. This constant is the extension number itself when the operation e, is lacking, 1.e. in the two general cases (e, c) and (c,c) of nets deduced from V(C,,); in the remaining general cases (e,e) and (c,e) we have to add V 2 to that multiplier in order to create room for the intermediate prisms with 2V 2 as height. As we start from [2000] the extension number is half the sum of the digits. So we find for the multiplier the values given in the following table | (e,c) | (2, @) | (¢, ¢) | (c, €) e,...1+ V2 ces... V2 ‘8 ee... .8+ V2 Ce ceren. V2 4 nes... A+ V2 GED cesen. 3d V2 ..142V2 e,¢,...1+8V2 Ces woe C630, SVR . 6 een ue OF WIE eas ceresen. 5d V2 84-2V 2) eee...8+8V2| ceeg...2+2V2| ceyese,...2+3V2 ..442V 2) ejese...4+8V2) ceyes...8+2V2]| ce,ee,...38+3V2 612V 2) e,e,ee,...618V 2] ce,ee,...5+2V2 | ceeese,...5+8V2 SO. By means of the preceding developments we can find the three net symbols for all the different nets deduced from V(Cg). But this work can be reduced by the remark that it will do to use only the net symbol of the erect group in the cases of the seven nets 1, @,, es, €, 5, Ce,, ces, ce, e,, while we want these of two groups only for the eight nets ez, e‚ e3, 5&3, 263, Ces, CE, C3, Ces, ce, e es, and all the three symbols in the remaining cases where e, occurs. The proof of this assertion is based on the following theorem, where we distinguish the three sets of cases just indicated as the set without e, and e,, the set with e, and without e,, and the set with e,: TaxoreM LIT. “Any of the three net symbols represents all the vertices of the net in the set without e,and e,, two thirds of all the vertices in the set with e; and without e,, one third of all the vertices in the set with e,”’. This theorem is an immediate consequence of the following lemma: “Any limiting tetrahedron of the net W(C,) is common to two Ae belonging to different groups, any limiting triangle is common to three Cj, no two of which belong to the same group”. 54 ANALYTICAL TREATMENT OF THE POLITOPES REGULARLY The first part of this lemma is evident by itself. As to the second part related to a face we state that the angle formed by the two spaces of adjacent tetrahedra ABCD, ABCD of Cg at the common face ABO is 120° (see my paper: “On the angles of the regular polytopes, etc”, Amer. Journ. of Math., vol. XXXI, p. 307), from which it ensues that any face is common to three Cg; as any two of these three Cg have a limiting tetrahedron in common they belong to different groups, etc. | The lemma just proved immediately shows the truth of the theorem. If, after having driven asunder the cells C2) of the net N(C,;) so as to create room for the extension recorded above, the extended C,, receive the shape exacted by the character of the net under consideration by means of a regular truncation, the contact of the cells — belonging to different groups — by faces will remain uninfluenced if the operations e;, e, do not yet present themselves, the truncations being then restricted either to the ver- tices alone or to vertices and edges; so, as any vertex of the net …_ belongs at least to one face and each face belongs to three poly- topes of the set without e,,e,, one of each group, each vertex of the net must be contained in each of the three net symbols of any case of that set. So in this case the net itself can be represented by any of the. . three symbols, which includes that the constituents furnished by one symbol are identical with those furnished by each of the two others, though constituents of polytope and body import of one symbol may become under certain circumstances constituents respec- tively of vertex and edge import of an other. | Now the state of affairs changes as soon as e, makes its appea- rance. This operation still preserves the contact by limiting bodies of body import between cells belonging to different groups, but it annililates at the same time face contact between limiting bodies of body import of the same cell. So here the limiting bodies of body import of any constituent have been split up into two sets P and Q dividing the vertices equally between them, in such a way that any two of these limits which were in face contact before belong to different sets. So here the arrangement of the three groups 4, B, C of constituents is such that any constituent . of group 4 is in body contact by its set of limits P with consti- tuents of group £, by its set of limits Q with constituents of group C. So each of the three net symbols contains all the verti- ces of one group and only half the number of vertices of each of the two other groups, Le. 2 of the total amount. DERIVED FROM THE REGULAR POLYTOPES. 55 Finally, in the set with e,, two cells — belonging to different groups — cannot have a vertex in common; so here each net symbol represents only + of the system of En We now indicate schematically how we can determine all the constituents of the different nets of Oi. To that end we have 1°. to deduce from the preceding ee the net symbols necessary in every case, 2°. to calculate the coordinates of the centres of the different constituents, by multiplying the coordinates of a vertex, of the midpoint of an edge, of the centre of a face and of the centre of a limiting body of [2, 0, 0, 0] by the extension number, 3°. to determine the vertices contained in the net srmbeld. lying at the same minimum distance from these centres. As we shall have to consider the “extended’’ vertex, midpoint of edge, centre of face, or centre of limiting body mentioned sub 2° as new origin of parallel axes of coordinates in order to be able to obtain the simplest representation of the sets of vertices menti- oned sub 3° we will denote this extended point henceforth by O'. Of each of the three sets we will treat some examples, of the first e, es, (Cg) and ce, e, NC), of the second e, es MC) and ce, es MC), of the third ee, MCx), ey es ese, MG) and ce, es eze, NC. Afterwards we will put on record the coordinate symbols of all the constituents in Table VII. Sl. Case e,e,N(O,,). Net symbol 4, [l2a,+6, 12a +4,12 a + 2, 12a, + O], Z a, even. 1 Here the constituent of polytope import is [6, 4, 2, 0] = e e Cie. There are no constituents of body and face import as the opera- tions e, and e, do not present themselves. So we have only to determine the polytopes of edge and vertex import. | Edge gap prism. By extension the centre 1, 1, 0,0 of the edge (2,0)00 of [2, 0, 0,0] becomes 6, 6, 0, 0. By putting in the net symbol a; = 0, (: = 1, 2, 8, 4), we find among others the vertices (6, 4) [2,0] and by putting a, —a,—1, a,—a,—0, and taking the movable digits 6,4 with the negative sign we find also the vertices (6, 8) [2,0]; with respect to the new axes with the point 6,6,0,0 as new origin O' these two groups of vertices can be represented together by the symbol [2,0] [2,0]. So we find a measure polytope C, which is to be interpreted here as a prism on a- cube, ‚Pe: 56 ANALYTICAL TREATMENT OF THE POLITOPES REGULARLY Vertex gap polytope. By extension of the vertex 2,0,0,0 of [2,0, 0,0] we get 12,0,0,0 as new origin Q. By substituting a, 0, @= 1,2, 3,4), n the first place and a, — 2, a, — 0, (¢= 2, 8,4), in the second (with he movable digit 6 taken negatively) we put in evidence the two sets of vertices 6[4, 2, 0] and 16/4, 2,0], i.e. with respect to O' the vertices [6][4,2,0] contained in the net symbol. But this symbol still contains other vertices lying at the same minimum distance 2V 14 from O', 1e. all the vertices represented with respect to that point by [6, 4, 2, 0] and no other. So we find e.g. the point 4, 6, 2, 0, with the coordinates 16,6,2, 0 with respect to the original axes, by considering the vertices 12 a, + 4, 12a,—6, 12a +2, 124, and putting a;—a,—1 and a—4—0, etc. So the result is” that the constituent of vertex import is a [6,4,2,0]—=e es Co and therefore identical with the constituents of polytope import. Case ce,e, N(C,;). Net symbol 4 [10 a + 4,104, +4, 104, +2, 104, +0], Xa; even. 1 Here the constituent of polytope import is [4, 4, 2, 0] == ce, e, Cg. As in the preceding case of ee MC) the constituents of body and of face import are lacking. Moreover by the contraction the original edge and therefore also the constituent ?, of edge import is annihilated, i.e. P‚ is reduced to its base C. We verify this analytically as follows. By extension of the midpoint 1,1, 0,0 of the edge (2,0) 0,0 of [2,0,0,0] we get 5,5,0,0 as new origin 0. Now the vertices at minimum distance from O' contained in the net symbol are found by putting a, = 0,(¢= 1, 2,3, 4), giving 4,4[2,0], and a—a—1, a,=a,—0 (with the:two di taken negatively) giving 6,6[2,0], i.e. with respect to O' the two squares 1,1[2,0] and —1,—1[2,0] forming two opposite faces of a cube with O' as centre. Finally we remark that the contraction c does not affect the con- stituent of vertex import. This is easily verified by determining the vertices at minimum distance from the point O' with the coordi- nates 10,0,0,0 presenting itself here. 82. Cuse e,e;NW(C,,). As the operation e, presents itself here we have to find besides the constituent [2’1'1'1]V2—=e,e3 Gg of polytope import those of face, of edge and of vertex import, and in order to be able to gather all the vertices of these constituents we have to use two of the three net symbols. But we prefer to Cure OPE land. 5 ve xj DERIVED FROM THE REGULAR POLYTOPES. 51 investigate how far we can proceed in this way by using the first net symbol only. This much more complicated symbol is EÉ(2 + 1/2)en +4 +172, 4(2 41/2 )ao +2412, 4(2 +1/ Daz +2 BAL + a, +21, Xa, being even. We abridge it into the following form, clear by itself: | A | [44+-V2,21+V2,24V2, V2], (8H AV 2) a, 43,41, 2a; even, 4 where a,, as, as, 4, preceded by the common factor 8 4 4V 2 repre- sents the immovable part. Face gap prismotope. By extension the centre 2, 5, 2, 0 of the face (2,0,0) 0 of [2,0,0-,0] passes into the new origin O' with the coordinates 4 (2 + V2), 4 (2 +WV2), 4(2+ V2),0. By supposing the four a, of the net symbol to disappear we get inter alia the set of vertices (4+V2, 2H V2, 2H V2)[V2], Le. a Ps. These are the only vertices contained in the net symbol above mentioned lying at minimum distance 4 V3 from 0’, but as we shall see immediately the two other net symbols contain other ver- tices partaking of this property. However, in order to sharpen our analytic tools, we leave these other net symbols alone for a moment and try to deduce these lacking vertices from the simple properties of the prismotope with two regular generating polygons in planes perfectly normal to each other. By means of the ?, just found we know that one of these polygons is a triangle, and the character of the other polygon can be deduced from its circum- radius. For the relation p,* + 0° = p° between the circumradit 04, Ro, p of the two generating polygons and the ‘prismotope itself Sivés, as we have p= 4V 3 and 9, 2V 6, gs V 6, ic. the second polygon is also a triangle and the prismotope a (3 ; 3). We have therefore only to find a third position of the first triangle, the two end planes of P; containing already two positions, and this third position can be found by remarking that the centres of these three equipollent triangles are the vertices of an equilateral triangle with O' as centre. So, if p,g,r,s are the coordinates of the centre of this third position we have that the triangle with the three vertices +V2, B4V2, $4V2, VO SNe BS y/o tae P > q » 2 = S must admit E(2HV2), 4@+V2), 424V2), 0 as centre. From this it ensues that we have p=g=r—=i+2V2è,s—0, 58 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY furnishing (4 J-2V2, 2+ 2V2, 2 + 2V2), 0 for the third po- sition of the first triangle '). Indeed the part of the second net symbol corresponding to [4 +- IVS Dib; 01e. | 4 3 [4+ 27/2, 2,2,0], (44+ 2172) 2e +1, 2a,-+1, 2a,+!, 2a,+1, Sa odd, 4 gives for d == 4, = a, = 0 and a,=-— J the set of vertices represented by | (412V2—0, 419V2—2, 4-1 2V 2-8) 4 oy oe ie. (41 2V2, 2+2V2, 2+ 2V 2) 0. Edge gap prism. By extension the centre 1, 1, 0, 0 of the edge (2, 0)0, 0 of [2, 0, 0, 0] gives 2(2 24/9) 22 VR for the coordinates of O'. By reducing the first net symbol to this point as new origin we get een ae PE Bt 2, 24+12,9+12,12), (4462/72) 2a, = 1, a, 1, Ba ta Ze | An By putting a = 0, (— 1, 2, 3,4), and taking the permutable digits in the indicated order and with the positive sign we find the vertex —V 2, —(2+V 2), 2+V 2, V2 lying at minimum distance 2V(4-+-2V 2) from O. As this distance is smaller than 4 + V2 we are obliged, in order to find all the vertices contained in that symbol lying at that distance from ©”, to put a; = a, — 0 and to take either a, = a, — 0 or a, =a, = 1" So we find the 22 tices [2 + V2,V2][2 +V2,V 2], where the + refers to the first syllable corresponding to the coordinates 2,, 2, only. Now we have furthermore to examine the other two net symbols. For O' as origin the second net symbol is | MES: D et SK) a) oe Ee Be ONE ee - , (41912) 9m, 2a, 2a 1/2 1, :S4todd; —#{4+ V2, 2+12, 2+12, v2! SPIGA) Manta Ae te Le = *) Until now we have only used implicitly the condition that the planes of the generating polygons are perfectly normal to each other, in the equation py” + po? = p2. As the plane 7; + a) + 3 —0, x4=0 is parallel to those of the first triangle, the plane x; — #3 — 3 perfectly normal to it must be parallel to thcse of the second. We verify this by the following table of the nine vertices of the prismotope lat 12,94 19,94 V2, V2 94 2,442,942, Waat 2,242,442, V2 (44+ WI 94 12,94 V9 V9 24 V2,4+ 172,94 V2, —12 DES 172,94 12,44 V2,—V2 |, PAH 2,29 2,9+ 919 O0:12+9 2 4199 941979 0 (249172, 24912, 44.212, 0 the three rows forming the positions of the first triangle and the three columns (of sets of coordinates) those of the second. So for the triangle of the first column we have Lj Lo — 2, Lo — Xz, etc. | By continuing this research it can be verified, that each of the three net symbols con- tains the six vertices of a P3 with two positions of the first triangle, i,e. two rows of the table of the nine vertices, as end planes, _ DERIVED FROM THE REGULAR POLYTOPKS. 59 the two sets of permutable digits having to be combined with the same set of immovable ones. Here we find only vertices lying at a greater distance from O’, unless we take a, — a, — 0. So we get for @3, 4 —(0,— 1) by means of the upper half of the symbol the 16 new vertices [2, 0][2 (1 + V2), 0], by means of the lower the 16 vertices z,, v7, =4[2+V2,V2],#,,2,=—4[24+V2,V 2] already contained in the set $[2+ V2, V2] [2+V2, V2] deduced from the first symbol. From this may be deduced that the two halves of the third symbol will furnish the two sets [2,0] [2(b-+-V 2),0) and 4, 2 = 1/2 + V2, V2], aa — 4/24 V2, V2] So the result is a polytope with 48 vertices represented by the combination of the two symbols $[2 +V2,V2][2 + V2, V2] and {2,0][2(1 + V2), 0]. It proves to be a P. For, by applying on the ¢C represented by the symbol [V2][2 + V2,2+V2,V2] the transformation Ba =H | vy + a= ga V 2 | BVR 2, —m—=yV?)| we get, 1/2 + V2, V2] [24+ V2, V2] for the 32 vertices [V21[2 = V21[2 +V2,V2] and [2,0][2 (1 + V2), 0] for the remaing 16 vertices [V2][V2][2 + V2,2+ V2]. Vertex gap polytope. By extension the vertex 2,0,0,0 of | 2,0,0,0| gives 4(2-+V2),0,0,0 for O’. With respect to this origin the first net symbol is [4+V2,21+V2,24+V2,V 2], (8+4V 2)a,—1,4,,a3,a,, Za;even, | 1 which can be reduced to [ALV2,21V2,24LV2 VEL (SB HAV2) aa aa Da,0dd. 1 By taking in this last symbol a, &,, 43, a, —[1,0,0,0] and putting the digit 4 + W2 always where the 1 stands with the opposite sign of it, we get the 192 vertices [443V2,2+V2,2+V2,V2] lying at the minimum distance 4 (1 + V 2) from 0’. With respect to the same origin OQ’ the second net symbol is re ME vd: hid pa; 2 ty, 22, vi the immovable part of which can be reduced to (4+ 2V2)2a,11,2a,+-1, Zag del, 24,11, Sa even, i, ores be ee > odd, 60 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY By considering the three groups of cases ds (= 1,0 3, 4) —, 44, 43, A3, Ay == (= bi 15:0, UD ai din Le Ae 3,4), and adding to the immovable parts the permutable ones taken in any order, generally affected by the sign which tends to decrease the absolute value of the coordinate but — in connection with the negative sign before the lower half of the symbol which exacts an odd number of negative permutable digits — with exception of the smallest of these digits V 2 the sign of which is to be chosen inversely so as to /rcrease the absolute value of the coordinate, we get by the upper half the 96 new vertices [4-+ 2V 2,2 + 2V2, 2+ 2V2, 0] and by the lower the 96 vertices $/ 44-3 VSD Oey ee 24+V 2, V 2], obtained above. So the result is a polytope with 258 vertices repre- - sented by the combination of the symbols [4+38V2,24+V2,24+V2,V 2], [4+2V2,24+2V 2, 24+2V 2, 0]. As we will prove in section V this polytope with the characteristic numbers (288, 576, 336, 48) limited by 487C is ce,e, Cy. Case ce,e,M(C,,). Besides [1'1'11]V 2 = ce,e; CG, we have to look out for the face gap filling and the polytope of vertex import, the edge gap filling being reduced by contraction to the base poly- hedron of the prism occurring in the case of e, e; V(C,,). Face gap prismotope. Here we get for the new origin O° the coordinates 2 (2 + 2V 2), 2(2 + 2V2), 2(2 +2V2),0,as2+2V2 is the extension number. So the first and the second net symbol are : Renesas ETEN 4 [2 lams 2, 2 + 2, V2, py 2], (4 + 47/2) a, — à, Go — 5, a, — à, ay, > Xdi even, 1 (2-219 9 0 Det ee RS. STUNT à ie AT A AO L a | L > > > J IE aL NE alt 5 yet pe ot pa pe, pap CD Ra RG LE Pm FE MFT, Des odd By taking in the first symbol a, —0,(:— 1,2, 3,4), we find . 2—py2 2—p2 —4—yp2 : as ; the vertices ( Ee ; en : 5 5 [2] lying at minimum dis- tance 4V3 from O/, i.e. a Ps; by substituting in the upper half of the second symbol a, — 0, (¢ = 1, 2, 8),a,—= — 1 we get moreover 2142 2132 — 4199 | 8 en L a ) 0, the third triangle of the pris- 3 3 3 | motope [3;3] to be found. DERIVED FROM THE REGULAR POLYTOPES. 61 Vertex gap polytope. The new origin is 2 (2 + 2V 2), 0, 0, 0 and the first and second net symbol become, in the shortest form possible , sah ot EA a NS = NE ee EES 4 EED, VD, TAART Lot due cn BE poe odd, 1 B 2 ad ED 2, 21/2, 1-2; de , (242172) 2a,+ 1, Zag +1, 2a3 +1, 2a, +], > even. Putting into the first symbol 4,, a, 43, a, — [ 1, 0,0, 0} and com- bining with the a, differing from zero one of the two digits 2 + V2 taken with the sign tending to decrease the absolute value of the coordinate we get the 192 vertices [2+3V2,2+V2,V2,V 2]. Putting into the upper half of the second symbol a4,= 0,(:—1,2,3,4), we find moreover the 96 vertices [2-+2V2,212V2,2V2, 0]. So the result is a polytope with 288 vertices which will prove later on to admit the characteristic numbers (288, 864, 720, 144) and to be e, Ci. 83. Case e,e,N(C,,). Here the extension number is 3 + V2. So we have to reduce the three net symbols 4 [4,2,0,0],(6+2V2) a, ds PE ‚Xa;even, 1 h 113,3,1,1 |,(3+ V 2)2a,+1,2a,+1,2a,+1,2a,+ 1, 2a,odd, —4/3,3,1,1], (8+ V2)2a,+-1,2a,+1,2a,11,2¢,+1,2a, even Al for the constituents of ae face, edge, vertex import to the new origins (3 + V2) 1 2) à 1,4,4, (8 4-V2) 2,2 2,2,0 2,0, (83 +V2)1,1,0,0, (3 + V2) 2,0,0,0 respectively, the constituent of polytope import being [4, 2, 0,0]— e, Ci. Body gap prism. The three net symbols become h [4,2,0,0],(3 + V 2)2a,—4,2a,—4,2a,—H, Pa ZA: even, aa ope ea V 2)24a,+4,2a, 4,2a,+4,2a,+ Lda, odd, —4/3,3,1,1], (8 +V 2)2a,+4,24,+4,2a,+1,2a,+14 ‚za even. 62 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY By making the a, to disappear the first and the third!) N V2 1-2 —38—p2 =e symbol give the sets of vertices (P= ; 5 Reet Gas re ae = fas eee each of which naan to a (2100), i.e. to a #7. So the result is a P,;, all the vertices of” the second symbol lying at larger distance from 0° than the cireum- radius Wi3 of this P,;. Face gap prismotope. Here the three net symbols are 4 [4,2,0,0 |, (3 <5 V2) 2a,—4, Zas, 243—#, 24 2 24 EVER 1[3,3,1,1], (3 + W2) tartte Bart + Baat d Pal, Sa, oi —1[3,3,1,1], (34 V 2) 2a,4-4, 24,4, 2a,+4, 2e +1, Sa, even. 1 2 By taking in the first symbol a, De 0: = 1,2, 3, 4) RE second a, —0, (4—1,2,38)}, a,=— |, in the had (i = 1,2,3,4), we get the three hexagons FVG Sey ee ee (QLIVI. “yg ese ye Q+4V2, AVR So the result is a [6; 31. Edge gap prism. Now the three net symbols become 4 [4,2,0,0], (8 + V 2) 2a,—1, 2a,—1, 2a,° „Zas , 2a, even, 1 4 $/3,3,1,1],(8+V2)2a, „Zes , 2a;+-1, 2a, +1, da, odd, | 1 za AT 3 vs n —113,8,1,1], (8 + V2) 2a, , 243 , 2a3+1, 2a, +1, Xa; even. 1, ARE By taking in the first symbol a; — a, — 0 and either a, =a, 0. 1, in the second 4, —4;—= 0. “ahd az, a = in the third a, = a, = 0 and either a, == a, = or and by combining with the not disappearing immovable digits the greater permutable ones, generally affected by the sign tending to decrease the absolute value of the coordinate but — on account of the sign before $[3,3;1,1] of the second and the third symbol — oF a, =a, = ‘) That one of the three symbols must remain inactive in the generation of the body gap prism is an immediate consequence of the lemma of art. 80. DERIVED FROM THE REGULAR POLYTOPES. 05 with exception of one of the permutable units, we get successively the three quadruples of vertices lying at minimum distance V6 from O'. These 12 points form the vertices of a prism P, with octahedral base; each of the three quadruples just found lies in a plane passing through the axis of the prism and consists of a pair of opposite vertices of each of the two limiting octahedra. The equations of the three planes are @,—=0,%,=— 0 —,a4,+2,—0,7,—=%,—,7+2,—0,2,+ 4, = 0. So the axis of the prism is -represented:‘by 2 = 0, 2, = 0, Bit Zo 0. Moreover it is easily verified that the three quadrangles are rectangles with sides 2V2 and 4. As we can unite the second and third symbols the Pg can be represented by the two symbols Lil +V2,—1+V2]0,0 and (1, —1[V2, V2}. Vertex gap polytope. Finally the three net symbols are, in the simplest form, Ee OON Sa tig een ap ic a coda, 1 4 $[8,3,1,1], B+ V2) 2q-F1, 24+, 24,1, 20,41, Za, even, Es 4, —4/3,3,1,1], (Sa V2) Bids Zal, 2a3+ 1, Zat 1, Za; odd. Ay: taking for 4, 45,45, a, inthe first symbol [1,050 0 "in the second either 0,0,0,0 or (—1,—1,0,0) or —1,—1,—1,—l, in the third either (—1,0,0,0) or (—l, —l, —1, 0), and by assigning to the permutable digits the sign which decreases the absolute value of the coordinate, we find the three sets of 48 points represented by the symbols [2+12, 2,0, 0], 4(2+12,2+12, 2, 12), — 412 +12, 2 +12,12, 21, which can be reduced to | [2 + 2V/2, 2,0,0], [2 +V2,2+V2,V2,V 2]. These 144 points prove to be the vertices of the polytope e, C,, with _ the characteristic numbers (144, 576, 672, 240). e, e, ee, MC). Extension number 6-+-3V 2, three net symbols 64 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 4 , > dj even, [6 + p2,4+p2,2+12,V2] , (12462) a ‚ ay iS ee 1 CH, 4, 2.2 ern a —}[6+ p/2,4+72,2+72,12] » (64-32) 2a +1, Zag 4-1, Zag +1, ut pn : 4 aa vase vale 6 0903) Lad 24 ea which are to be reduced to the new origins, indicated in the pre- ceding example e,e, MC). But in the case of the body gap we will mention only the first net symbol and the lower part of the third, which lead to the desired result. Body gap prism. We tind 4 (64472, 4-112, 21 1/2, VAL (64 31/2) Pa de De tm za; even, 4 1[6+ 1/2, 44772, 21172, 12], (6 + 37/2) 2a, + 3, 2a, +4, 2a3 HE, 2a,+ 4, Ta; even, 1 giving by means of the suppositions of the preceding example the prism Po, the two bases of which are (3—1V2,1—1V2, —1—1V2, —3—1V2), (84+-V2,1+14V2,—1+4V2, —-3 LIV2). Face gap prismotope. Here we have 4 [6 p72,4-4- 2,24 72,172] , (OLB) ij 26, 2 95, =, ES 1 eye) A = 9.00) | JU DW Re EB 13,44172,2-- papa Oe aa ee sk odd, (6 + 2y2, 4 5 2 5 0! 5 end = T ok ret 2, Ade, eve, ver OBR) Zar t fa, tE Aj RER giving by means of the suitable substitutions easily found succes- . sively (2—V2, —V2, —2—V2) [V2], Q+V2, V2,—2+V2)— V2, (52 DES 2 )—2V2, Q+V2, V2, —2+V2) vo: ( 2 Bee 2 AZM ee which can be combined to representing together a prismotope [6 ; 6]. DERIVED FROM THE REGULAR POLYTOPES. 65 Edge gap prism. We get ; SR RAR EE We RES CR PB 4 ee een Beb a) my — |, Dae DE du, Our even, a ie yp, EN D 2, 42. 8 Ve, il ey) a 8 e+ we 4128-412, aj 4 na (6+ 31/2) 2a, ‚ Lo , Zag +1, 24, +], Za; odd, 4 1 4 a (6+ 31/2) 2a, , 2a, » 2a, +1, 2a; + 1, Fai even, i giving by means of the suitable substitutions (24+ 2V2, 2V2] Pousada 1 [ 2 à 0| 4(2+3V2, W2], (2+ V2, V2] $[24+2V2,2V 2], [ 2 : 0] 4[2+3V2, V2], 4(2+ V2, V2] 4[2+2V2,2V 2], which can be combined into a (2+ BP, 3-2] (2+12,12]—, [2, 0] [2+ 37-2, 1-2] —, 42 +12, 1-2.) (2 +92, 2), representing together the 96 vertices of a P,co. For the transformation ndr Lada V2) Bz = y VR 71 v4 (sap Vee fe La — di = Yi V2 gives immediately Yo—=|V 21-5 Hoan = (4 + V2,2+V2,V 2). Vertex gap polytope. Finally we have to deal with 4 (6+ pa,4+p2,2-p2,p2), (124672) a, as ee sd > Sa; odd, 1 eye A su YO] on ( 64 8/2) 2a, H- 1, Lao + 1, 2a. 1524, 1, Sa; even, En Dt 0E 4 | ep a eo poppe ee tee) a aor ee ee giving by adequate substitutions (6+5V2,4+ V2,2+ V2, V2], $(6+38V2,4138V2,2+3V2, VZ], 4[(6+4V2,442V2,2+2V2,2V2], —1/643V2,4+13V2,2+3V2, V2], —4[(6+4V2,44+ 2V2,242V2,2V2], i (64512, 4412, 2412, 1/2] [6+31/2, 4431/2, 24312, 1/21, (6-4+4)2, 4421/2, 24212, 212), representing together the 1152 vertices of the polytope e, es ez Cy. Verh. Kon, Akad. v. Wetensch. 1° Sectie DI. XI No. 5. E 5 66 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY Case ce,e,e,e,N(C,,). Extension number 5 + 3V 2, three net symbols | te Pr ton 4 [4H V2AHWV22 2, V2 } (0+6/2) a ‚ Ag > Ag » & veen 1 BAE, 93) eee Fea HB VSH IVe le gy? ee RER Rte Fe para HF La FL Ba FL Re FL, 2 Soe which are to be reduced to the new origins, to be formed according to the indications of the preceding example. Here the polytope of edge import is lacking. In the case of the body gap we mention only the first net symbol and the lower part of the third, which lead to the desired result. | Body gap prism. We find To + À, [4 +172, 4772, 2172, WZ ], (5H 3172) 2a, — +, 2ag — E, 2a, — +, 2a,—3, Ya; even, 1 PL Er ES Per DS PERDRE HIB, 3-+p2, 1-Ly/2, 1412], (54-302) 2a, +5, Bag LE, 2a, 4, 2a, FA, Sa; even, il giving by means of the substitutions a, — 0, (: — 1, 2, 3,4), the prism ?,,, the two bases of which are Ee pein a eee LE 4 de D FT RON PRET esse EYE Sie ae DT Ds 2 2 Face gap prismotope. Here we have REE 4 [4+ V2,4+)2,2+12, p2 ], (5-32) 2a, — 2, 2a, — 2, 2ay— 2, ta, ,Zajeven, | 1 tol whe TE ee 7 s+ 12,3-4+12,1+12,1-4p2I (5 31/2) 2a, + 4, Zag JH, 2ag+ 4, 2a,+1, zl odd, el yay dot (OE 88) ce Be WS ve LAW va (5 +32) 2a, + 4, Lag + 4, Lag JF, 2a,+ Zan | tol nl ler | > Ot OX giving by means of the suitable substitutions easily found BV EVR BVZ, 212, =a ee ( à BO CA g+Ve2 EEV CA CN re which can be telescoped into DERIVED FROM THE REGULAR POLYTOPES. 67 (2—V2,2—V2Q, ede : @+V2,24+V2,—44 V2(V2]—.¢,3,-PRV2I, representing together the vertices of a prismotope [6; 3]. Vertex gap polytope. Here we find finally de SENTE Se 4 ae} -72,4+772,29+ 72, p2 1 (104-672) a, a ag; ‚Zei odd, por 3 1 1 2 ’ 2 ’ 4 pi 2 anna ape Ga aa ee giving — as ze ene to do — quite the same result as in the preceding example. 84. Probably after all the indications contained in the treatment of several special cases Table VII would be quite clear by itself but for the first column of the part corresponding to the second extreme polytope and the last column but one; so we have to add a few words about these two columns. }) In the two special cases treated in art. 81 the vertex polytopes proved to be polytopes all the vertices of which can be represented by one symbol, i. e. polytopes of measure polytope extraction, viz. Ce, €, €; Cz == &, € Cig. But in the five cases studied in the art“. 82, 83 we had to deal with vertex polytopes the vertices of which cannot be represented by one symbol only, i. e. with forms which do not belong to the measure polytope family. These forms were said to be derivable from the cell C,, by applying respectively the sets of operations ce, &, €, 63, 1e Now in part F of this section will be shown, not only that a// the forms appearing here as vertex polytopes — whether their vertices are represented by one, two or three symbols — can be deduced from cell C,, by applying the operations e, and €, but also by which set of operations any required result is to be obtained. This set of operations is indicated for all possible cases in the first column of the part of Table VII corresponding to g,. So im the second case of art. 82 we found ex Cy; but as the general theory (compare Theorem LV) demands ce e3 Cy, which is equal to e,C,,, we have inscribed ce, e; Cy. ?) The remark of the second foot note: of art. 78 — that several nets deduced from MC,,) are equal to nets deduced from MC) — *) The very last column will be explained later on. *) The deduction of the symbols contained in the Table by applying the operations e, and c to the cell Cy, i.e. to |1, 1, 0, 0] 12, will be given in the last section of this memoir. F y. o* ~ 68 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY must now be generalized to this: “Every net deduced from (Gg) can at the same time be deduced from MCC)” Now the last column but one indicates the name of the corresponding C-net. So we have eze, WC) = @ € MC), ete. We must remember that the symbols given in Table VIT have to be completed by applying the transformations indicated in art. 79. Moreover we fix our attention on the particular form in which the symbols of each constituent appear. Every prismotope g, 1s decomposed as to its vertices into two or three fourdimensional prisms, one of which degenerates in some cases into a regular polygon ; of the fourdimensional prisms g;, g, the first is determined by its two bases, whilst the latter appears as prismotope (4; 4) or as a combination of prismotopes, etc. F. Polarity. 85. In art. 67 we remarked that in 8, any polytope derived by means of the operations e, with or without c from the measure polytope JZ, can also be derived from the cross polytope Cn. In art. 77 we stated this result in the form of theorem LI after having demonstrated it by showing that the /ofa/ set of symbols of coor- dinates of the group derived from Cn is equal to that of the group derived from M,. We have to come back to this result once more here, in order to indicate how it depends on the laws of reciprocity and what is the general relation between the two symbols of expan- sion operations of the same polytope deduced from 7, on one hand and from Cn on the other, which couples of symbols have been given for # = 3, 4, 5 (compare the foot note in art. 72) in the first and the second column of Table IV. It goes without saying that the dependence between theorem LI and the laws of reciprocity merely consists in this that the polar reciprocal polytope of a regular polytope À of S, with respect to a concentric spherical space is an other regular polytope 4° and that in this polarity the vertices, edges, faces, etc. of the one cor- respond to the limits (7), 4, (2), 0, (7), 2, etc. of the other. So we have still only to deduce the relation between the two symbols of the same polytope. This task can be performed by comparing the first two columns of Table IV with each other and by generalizing for an arbitrary 2 the outcome of this comparison. So for bee) TE ee B pe pe immediately deduce from Table EV: the following general laws: DERIVED FROM THE REGULAR POLYTOPES. 69 Ca Cy re bs eC; Er M, RE, Cn—t-1 Cn—s—1 PR Cn =-bA | En 4 Con | Clay Cp + - + Cs O4 M, No Ens es Cn p14 On—a- Cyn | Ca Cp Om te és C, M,, a Cn—t-1 Cn—s-A said En—b—A Cn—a-1 Cn Cn | COg Gy... CC M, = C Ort Pn—s-1 + > + On—bA “n-a-1 Cyn : The proof of these general laws can be based on the remark that each pair of polytopes forming the two members of any of these four equations admits the same symbol of coordinates; if # is the number of the symbols e,, e,, ...,e,,e, these symbols of coor- dinates are successively: n—t—1 t—s b—a “ \ f1+¢+)V2,1 +4V2,1+¢—DV3...., pes L 1 n—t—1 t—-s b—a ee PA eee EEK Be 0) n—t t—s pice Siete Soe a Pay ETL eens a 1 I t—s b—a C4 Here ee en By introducing the operation e, corresponding to the generation of the regular polytopes starting from a point and representing this point for 47, by Py, for Cn by Py we can unite these four general laws in: | TusoreM LIL “The two polytopes P 2, 0,0 De CO Oro iO Oe Oped Tae Er en wan Ge bu Ctl are equal !) if and only if we have generally a+ = Es Set r=] Srtd Hs Ht+d=a—1.” 86. The influence of theorem LIIT on the results laid down in art.” 65 and 66 is evident. By polarizing an expansion or contraction form derived from the cross polytope Cy» of S,, with respect to a concentric spherical space (with oo”! points) as polarisator we get a new polytope admitting one kind of limit (/), ‚ and equal dispacial angles?), to which corres- ponds the inverted symbol of characteristic numbers of the original polytope, ete. *) This theorem gives for M, and Cyn what theorem XXIII contains about the two differently orientated positions of the simplex; it holds not only for M, and Cyn, n being general, but also for the polytopes C9 and Coo of S4 and in the same way there exists a theorem analogous to theorem XXIII for the cell Co4 of S4 in its two different positions with respect to the system of coordinates. We shall have to come back to this point in the fifth section of this memoir. *) Compare for Si the foot note of art. 65. 70 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY Trworem LIV. “Any polytope (P), of cross polytope descent in S, has the property that the vertices 7; adjacent to any arbitrary vertex V lie in the same space #,_, normal to the line joining this vertex V to the centre O of the polytope. The system of the spaces S,_, corresponding in this way to the different vertices V of (P), include an other polytope (P),, the reciprocal polar of (P), with respect to a certain concentric spherical space. But in the case of the cross polytope itself these spaces pass through the centre.” This theorem is a mere transcription of theorem XL. 87. If we apply the general relations of polarity, which have led us in art. 67 to theorem XLI, to the particular case of the polarly related nets ÂV(C,;) and WV(C,,) of 8, we get: Turorem LV. “If the sets of operations # and 4” are comple- mentary to each other, i. e. if 4” contains the operations e,_,, comple- mentary to the operations e, of Wand no other, we have EN Cy)=cH'e, MC), He, M Cy) = Ee, M Cor), CEN( Cie) =ch MC,), cHe, NC) = BMC)” An analytical proof of this theorem would require a more ample knowledge of the net symbols of the nets deduced from ÆV(C,,) than we have at our disposal, after having nearly finished the third part of this memoir. We will therefore invert the order of ideas, 1. e. we will content ourselves here by giving the analytical form of the geometric facts and use theorem LV and the last column but one of Table VIT based on it in the last section of this memoir dealing with the extra regular polytopes, to facilitate and control the deduction of the polytopes and nets, deduced from C,,. There we shall have occasion to apply the same principle to the polarly related polytopes Coo and Ci. 5) 88. ‘The connection between Cs, Cig, C according to which the Cy can be split up with respect to its vertices into a © and a Cr?) and with repect to its limiting spaces into a C22) and a C; leads to connections between the polytopes and the nets which cannot be deduced from polarity only. So we find: Cy, = ce, Cy (= ce, Cig), à Cy = Cy Ca Cg and N Gy, i= 68, N( Cy) = ce, M Ce) = ce, MC). But there is still a more striking coincidence to be indicated, viz. that the nets e, MC) and e, e, W(C,,) are respectively equal to the nets ) We defer the investigation of the reciprocal nets of those given in Table VII to the paper announced in the foot note of art. 68. DERIVED FROM THE REGULAR POLYTOPES. 71 ce, ez (Cx) and ce, e, ex N(C;), the constituent C, forming at the same time the g, of the former couple and the g, of the latter. We shall have occasion to profit by this coincidence in the next article. G. Symmetry, considerations of the theory of groups, regularity. 89. On account of the fact that the offspring of the cross polytope is identical with that of the measure polytope, the theorems XLII and XLIII may be applied to any form of cross polytope descent. So we have only to add a few lines with respect to the regularity and for the same reason this task has to be performed with respect to the nets deduced from M(C,,) only. If we individualize the 31 nets of Table VIT by an MV bearing the rank number as subscript we can say that the nets W,, Ns, Wo, are regular and that the degree of regularity of the nets Ns, Ns with two equal extreme constituents is known, as these nets are at the same time measure polytope nets. As moreover each of the 26 remaining nets admits faces of at least two different shapes, the _ degree of regularity of each of these nets is either 4 or 3), according - to its having only one kind or more than one kind of edge. But now it is immediately clear that any net admitting a constituent g3 has at least two different kinds of edges, as the erect edges of the fourdimensional 3, characterized by the property that the same coordinates of the two end points differ by unity, cannot be at the same time edges of the groundform in any of its three orientations. So we have still to treat the twelve cases V,, N,, Ne, No, Ne, Mis, Mo, Moss Vor, Noos Nos, Vo, forming two different Groups, one of the nets Mis, Vig, Vo, with groundforms admitting only one kind of edge and one containing the other nine not characterized by this property. Now we can decide the question with respect to any of the nets of these two groups with the least amount of trouble by means of the following general problem, where G is the groundform given in Table VII, P the pattern vertex obtained by omitting the square brackets of G, whilst Q, and Q’, represent the vertices of the net adjacent to P, of which Q; are and Q'; are not vertices of G: “Determine the repetitions r of G (in its three orientations) with P as vertex and examine whether or not all the vertices Q, and Q’; are vertices of the same number of these repetitions > (Grincluded)*. The first case must present itself for the three nets Mig, Vig, Wor. For the groundform of each of these nets admits one kind of edge and its repetitions containing P are grouped regularly round /; | i wl wl tole ole ia ot 1 72 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY so these repetitions must be arranged in the same manner round every edge. But this decides that the arrangement of a// the con- stituents round constituent, viz every edge is the same, as there is only one other Jo. So the degree of regularity of Mig, Vo, Nar 18 2. In all the nine cases of the second group there are two or more different kinds of edge and the degree of regularity is =>. From tliese cases we treat a couple of examples. Krvample ei W(Cjs). All the repetitions of the ner are repre- sented by the system of the three symbols [ 6a, +4, 6a, +2, Gas +0, 6a, 0], Ya; even L[Ga +3 +8, Gas + 3-++3, Gas LIST, Ga, +3 HH), Ya, odd}. — 1 [60 +3+3, 6a,+3-+5, 6a,+3-+1, 6a,+3-+1], Xa, even So the repetitions 7 with 4, 2, 0, 0 as vertex are: pa which may be adjacent vertex of 74, 8, 4 only. of Ee 1S Keample e; N(C, into VA 72> ee [24+-V2,V2,V So two kinds of Een [ 4, 2, 0, O} [6 Ae =. 0, 0]. TI 3 + 1, 83 — 1, — 6 +3 +53, 3-8) “et epee ees ARE 386 4 [ 3 +1 desk SE: 3—3]... 4 LI 1 3-—1,—6+3+46,—6+3-+5]...4 indicated by the symbols 7, 7, .., 4. Now the 2, 4, 0, 0 is vertex of the six repetitions and 4, 0, 2, 0 So there are two kinds of edges and the degree re 10: | o.Hwe telescope [ ppt Qu, ppt @, pps Qs, ppit Qi] Jia. (D) Pas Pa Pa, PA the repetitions of the groundform 2, V2] can be represented by ln Sl Re 2 2, V2 , V2 , V2 LCI) ar 20 5 2a,” 2a, 1 1-22, LE | eee Se! 1 } t= h LS EEE zj OT 21/2) 2a,+-1, 24 +1, 241, tal, Za: odd, EB 1 beo et EEE rvi +212) 2a El, 2a,--1, Zaal, 242 ‚Zj even. So the repetitions r with 2 + V2, V2, V2, V2 as vertex are only 9 | V2, J 9 BA V2 i 22 +-1—172, 1+ 24v2— 1+] EN 1+2p/2—(1 +72), Le +R Now V2,2+V2,V2,V2 is vertex of both, whilst on the other hand 2+ V2,V2, V2, —V2 is vertex of the first only. Von PAA : , € 1] i 3 edges, degree of regularity À. The very last column of Table VII contains the results. DERIVED FROM THE REGULAR POLYTOPES. 13 Section IV: POLYTOPES AND NETS DERIVED FROM THE HALF MEASURE POLYTOPE. A. The symbol -of coordinates. 90. Several times we have commemorated the fact that the eight vertices of a cube can be split up into. two groups of four points, the vertices of two regular tetrahedra, and that with respect to the cube the vertices of each group may be said to be non adjacent, 1.e. not connected by an edge of the cube — see e. g. the introduction of section I and the foot note of art. 4.1) Also that the sixteen vertices of an eightcell can be split up into two groups of eight non adjacent points, the vertices of two regular sixteencells (compare e.g. art. 78). So in general the 2” vertices of the measure polytope 47, of space #, can be split up into two groups Of 2-7 non adjacent points, but the polytopes of which these groups of points are the vertices are not regular for n >> 4. So in the case 2 —5 there are two different kinds of limits (/),, viz. cells C4 forming what remains of the limiting eightcells of A7; and simplexes (5) replacing the vanished vertices of 47. In relation with their generation we call these new polytopes half measure polytopes and we investigate in this section these polytopes and the nets which can be derived from them. In the cases [111] and [1111] of cube and eightcell we have represented the two half measure polytopes by the symbols + [111 | and +4/1111] respectively. Likewise we indicate by +4[11...1] the two half measure polytopes into which 47, can be decomposed n according to the vertices, where + 4[11...1 ] includes all the vertices nt of which an evez number of coordinates is negative and —1[11...1] all the vertices of which an odd number of coordinates is negative. These symbols immediately reveal a difference in character between the half measure polytopes of S,,, and S,,,., which we will represent for short by ZM, and HM,,.,. In the case of ZM, the polytope admits a centre of symmetry, as the reversion of the signs of all the coordinates of any vertex furnishes an other vertex of the same group; on. the contrary in the case of HM,,,,, every vertex is *) The result mentioned contains a numerical error; it ought to be replaced by GC oy), OO) 4a aye), rn see „Wiskundige Opgaven”, vol. XI, problem 96. 74. ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY opposite to the simplex replacing the opposite vertex of the measure polytope. So in this respect MM, presents analogy to measure polytope and cross polytope, whilst AZJZ,,,, imitates the simplex. We still remark that the case ~== 2 is exceptional in this sense that the corresponding MM, is a line, 1. e. a diagonal of the square, instead of a form of two dimensions; as we shall see this remark is essential in the theory of the nets derived from the half measure polytopes. 91. It is easy to prove that the half measure polytopes partake of the two properties characterizing the semiregular polytopes con- sidered in the preceding sections, i.e. that all the vertices are of the same kind and all the edges of the same length, here 2V2. Indeed we already solved incidentally in art. 47 the more general question : “Under what circumstances will the symbols Aer dico represent the vertices of polytopes in 8,, all the edges of which have the same length, say 2V2”? For the length 2V/2 of the edges the solution takes the form of Tarorem LVI. “The symbol + 4[a@, a5,..., 4] for which Oy ge len a represents a polytope admitting the required properties under the conditions: 4,_, — 4, = 1 and the difference between any two unequal adjacent digits equal to 2”. So. we find in 8; the two forms 1[111], 4[311), | our » +[1111], [3311], HBI), 1[531M, » À5 „ eight , $/{11111],4[33311], 41383111], LISE 1155311],4[533811], 1153111], 4[ 75311], etc., which are represented in the following table by other sym- bols referring to 7’, Cj and MM; these symbols will be explained later on. !) ÈS À" 4 ia | | | == Ne 1131 1] == 17 =e ie NE [IJ CG AM) HUIS ce eee : [8311] =e, Cg =e HMS 4[5811] = cee, Og =e, ce; HM) ‘) We remark here that the symbols e before HM,, are related to the limits of M,. DERIVED FROM THE REGULAR POLYTOPES. 19 | 1f11111]= HM, 1[83311]— e HM, DETTE 1[81111]—e, HM; BBI é 4 AM. dB Fibs € à HJK BT Fee) ay 5311] —=eee, HM, Lo bo} bo} non ot Sr Art I Ol We introduce for these forms and for the corresponding ones in spaces of a higher number of dimensions the collective “half measure polytope descendent”, which we abreviate to Ampd. B. Te characteristic numbers. 92. It is not difficult to determine the characteristie numbers of AM, for a general ». For, if a, and a’, denote the numbers of limits (7), of 47, and ZM, respectively, we have the relations Nee EN | | d'A on do d'a — A as } 43 = 43 +42), % ee, = 4(2)s do, ay = a, +6 (See = Ay - 1 = (2), a | where at the right the numbers are arranged in two columns of which the first contains old, the second new limits. Indeed the process transforming M, into HM, — which may be called an alternate truncation — destroys half the number of vertices, all the edges, all the faces, and maintains all the other limits (/}, (Da, nar of M, but in an altered shape, bringing new sets of edges, faces, limiting bodies, ete. into existence. Now each face of M, produces an edge of AHM,, each limiting body of M, — becoming a 7° — produces four triangular faces of HM, and finally in general any set of p + 1 vertices of M, adjacent to a vertex destroyed produces a regular simplex 8 (pl) forming a limit Ds of HM,, for p=4,5,...,2-——1. This accounts for all the relations given above. Now, as the characteristic numbers of M, are given by the equation Gr ee age EE we find for ZM,: 16 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY do = 2" Ee as roa (2) ee Æ A (2) DE Be (2), 2 ee re 33195. RE REA Neither is it difficult to prove that the characteristic numbers a, satisfy the law of Euler. To that end we go back to the relations given above and transform the Eulerian expression ddr ee Re El Maal nn ie JH (— 71)" "a, — [ay — 4ay + + «(as — Ws +. +(— I Wald of which two sums between square brackets the first contains the _ contributions of the first column (old elements) and the second of the second (new peas Now we add to each of these two sums between square brackets À a, — a, + a,. So we get La, — a, Ha, —...+(— 1" a] Edd 20, Ade ve (OP See sen OM! But as we have — a, + 2a, — daz = Ja eh + @h — Ga the second sum disappears, as it is equal to bal — (eh HH DW = al — 1”. So we find that the Eulerian expression of MM, is equal to that of À, and has therefore the value 2 for x odd and the value 0 for x» even, ete. We give here the results up to # — 8. They are m= 5 ...( 16, 80: 160, «ease n—6...( 82, 240, 640, 640, 252, 44), n=1...( 64, 672, 2240, 2800, 1624, 532, 78), n=8... (128, 1792, 7168, 10752, 8288, 4032, 1136, 144). In the outset we remarked that MM; admits two kinds of limits /),, viz. cells C4 and simplexes #5). Here we remember that in general for u > 4 the 77%, is bounded by two kinds of limits lus, viz. limits ZM, 4 forming what remains of the limits M,,_, of M. and limits S(z) replacing the vanished vertices of JZ,. It will be useful to call the ZM, _, the “original”, the S(x) the “trun- cation’ limits. 93. In the cases of the offspring of simplex, measure polytope, and cross polytope we have used two different methods for the determination of the characteristic numbers, one fulfilling the exi- gencies for < 6 as far as these numbers only are concerned, an vise: 2 bee 4 DERIVED FROM THE REGULAR POLYTOPES. 77 other giving for # > 5 not only the characteristic numbers but also the numbers of any limit of any kind; here we will do likewise. So in the case of the polytopes connected with ZM; in the manner indicated in theorem LVI we have to determine: 1°. the number of vertices according to general principles, 2°. the number of edges concurring in any vertex and thereby the total number of edges, 3°. the limiting polytopes (/),, which limits reveal at the same time the limiting bodies (Z)s, 4°. the number of faces (by means of Euler’s rule). But before applying this method to a definite example we give some further explanation with respect to the equations of the four- dimensional spaces containing the limits (/),,., of the Ampd. deduced from HM, in S,, as this will save us trouble in the exposition of the direct method. If $[a,a,...a,] is the symbol of coordinates, where the digits have been arranged in diminishing order, we consider the vertices represented by CASS re De NEN aie ene hy) Uy, Lo. ., Vy Ep +1 Up 499 es Vy lying in the space 8,_, represented by the equation fae ne se ET Evidently these vertices will determine a limit (/),-_4 of the po- lytope, if (a, @,...a,) and L{a,,14,,2...a,] represent polytopes (P),.1 and (P),_, respectively, this (/), 4 being then a prismotope which may be denoted by (P,_,; P,_,). Now (a a. ..a,) always represents a (P),_,, unless all the digits a,a,,...,@, are equal, in which ease (4, a,...a,) is a petrified syllable. On the other hand + |@y11%49 ..a,] always represents a (P),_,, unless we have either p—x—2, or p==2—1; for, as we remarked already p=nu—2 gives the syllable [11], i.e. a line segment instead of a square, and p = x — 1 gives a vertex only instead of an edge. To this we have to add a few words only about the extreme cases p— | and p—». For p— 1 we find the polytope with the coordinate symbol [as 43 ...a,] lying in a space S,,_, repre- sented bye ¢,— a; it cam be deduced rom 20 2 Lor gk the result is different for 2 even and x odd, the polytope having as HM, itself a ‘centre of symmetry in the first case and two different limits, either a vertex and an (/), 4, or two differently shaped (/),_4, opposite to each other in the second. Or otherwise, 18 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY as follows. For # even the diagonals of JZ, split up into two groups of non adjacent ones, of those bearing vertices belonging also to HM, and of those bearing vertices cut off by the alternate truncation leading from #, to ÆM,; the 2"? diagonals of the first group are normal to two limits of vertex import 1) in the considered polytope, whilst the 2"~° diagonals of the second are normal to two limits which may be called of truncation import as they are derived from truncation limits of ZM, in passing to the polytope under consideration. For x odd there is only one group of diagonals of JZ,,, each of which bears only one vertex of HM,; so each of these diagonals is normal to two differently shaped limits of the polytope, to one limit of vertex and to one limit of truncation import. But in the two cases, of » even and n odd, we have to deal with the two equations X + #,—= Xa; and X + 2, — Xa; — 2, the last digit a, — 1 having to be taken — with the positive sign for limits of vertex import, with the negative sign for limits of truncation import. After this introduction we treat a definite example. 94. Case TL [53311]. The number of vertices is 2* times 5! divided by 2°, i. e. 480. The vertices adjacent to the pattern vertex 5 33 1 1 are 688 83511 | 53181 Bost ih 153113 D393—1—1 which may be indicated by the brackets and the negative sign after the two units in the symbol CT co co id HE D So seven edges concur in any vertex, 1. e. the total number of edges is half the product of 480 and 7 , 1e. 1680. Now we have to pass to the limiting iS PoE The spaces 8, represented by + 7, — 5 give 26) — Ohm 11331 2 of polytope import. *) Also the import of the different limits (1), 4 of HM,, will be considered in relation with the limits (1),_4 of M,. So the equations + x; = a, will give limits of (/),, 4 import, the equations + x, + x, = a, + ag will give limits of (1), » import, ete., this series ending in gener: il in limits of body and limits of vertex import, as no ee or face of M, partakes in the limitation of HM, 4 > 4 , q | ; DERIVED FROM THE REGULAR POLYTOPES. 19: The spaces #, represented by + x; + a2, — 5 + 8 give 2°.(5) = 40 limits (53) 4[3 1 1] of body import. The spaces 8, represented by X + x, = 13 give 2* limits (5 3311) of vertex import, where (538 11) —(42200). 5) The spaces S, represented by X + a, = 11 give 2“ limits(5331—1) introduced by the alternate truncation. So the limiting polytopes are 10e Cr 40 Pnt Lee 8(5) = 16 ee, 8(5), 1.6. 82 in toto. Now from the list of limiting bodies Berne Oe ee 2 EE IE AR D AVR ES À Cx DOs. re D GO EE PO Oo Cj CS ON ar bi, ee ey TOP DECO of the four different limiting polytopes we can deduce that our polytope is limited by (IO 8+ 16 5) 0, $7010 X 16 +40 K 2 E 16 5) 7, LUHOEX 5 + 16 X 5) CO, (40 4+ 16K 10 E IG & 10) Ps, L(40 X 4 +16 X 10) Ze ed by 720 polyhedra, viz. 80 O, 160 #7, 80 CO, 240 P,; 160 P,. Now finally, according to Euler’s rule, the number of faces is 1840. So the result is | (480, 1680, 1840, 720, 82). This example shows that the method explained #s sufficient for S;, as far as the characteristic numbers themselves are concerned. But if we want to extend our knowledge of these Ampd. — in relation with the difficulty of realising their lopsided form — by determining the numbers of the different kinds of limits the method is insufficient even in S; and has to be completed, in one sense or other, with respect to the different kinds of edges and of faces. We shall see that the direct method, which will be explained in the next article, furnishes this complement at least expense. 95. Here once more the direct method in view is based on the *) This (42200) with edges 2/2 is similar to (21100) with edges )2, i.e. to es S(5). Likewise (5331-1) leads by (64420) to (32210) or (32110), i.e. to ej e3 S(5) . / 80 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY — distinction of the different kinds of limits (7), by what we have. called formerly “unextended’’ symbols. If we take care to exclude always the petrified syllables we can formulate the method in: Tarorem LVIi. “We obtain the unextended symbol of a poly- tope (P), the vertices of which are vertices of the given Ampd. of S, by applying to the x digits of the symbol of coordinates la a ...a, ;a,] of this polytope one of the three following processes: 1°. Take the last digit a, first with the posifive and afterwards with the zegative sign, and place for both cases between pairs of round brackets either one group of d+ 1 digits, or two groups containing together d + 2 digits, or three groups containing together d + 3 digits, etc., omitting the digits not included. 2°. Place before 4[11] of the remaining digits. 4,0. 9am between pairs of round brackets either one group of d digits, or two groups containing together d + 1 digits, etc., omitting the digits not included — and the syllable with one digit for d= 1. 3°. Place before Ta, va On 49+ * -G,1@,), where re 4,...,d successively, between pairs of round brackets either one group of d—A-+-1 of the remaining x — # digits, or two groups containing together d — £ + 2 of these digits, etc., omitting the digits not included — and the syllable with one digit for d = #.” “In each of these cases the (P), obtained will be a Zimiting poly- tope of Ampd., if the: syllables between round brackets satisfy the two following conditions: a) each syllable with middle digits exhausts these digits of the symbol of the given Mmpd., 6) no two syllables without middle digits have the same end digits.” The proof of this theorem, forming an adaption of theorem XXX to the special character of the Ampd., embodied in the + before the square brackets of their symbol, can be copied from that of theorem XXX and theorem XXX’. We apply it to two definite examples, one in S(5), the other INT. Case } [55311]. — If we place before a vertical stroke the limits deduced from 55311 and after it the different ones furnished by 5531 — I, we get Or 63) Gls, 4 fig DBS. nie 8) LIL CARS 6091) (Gall), DH, 113111} | (531—1), enero 05 1 des (5531 — 1), We an DERIVED FROM THE REGULAR POLYTOPES. 81 where the small subscripts at the right indicate the number of limits concurring in any vertex '). So we find through any vertex five edges, two #3, twO 21, SIX De, one. Pa, five fZ’, four ¢0O, one (55311) — ce, e, S(5), two (5531— De == 0,104 AB) two [5311] ce, e, Cg, and this gives in a transparent way in toto .480 a EMS EN Gy pe APTE RC RES oa AEA EM omen in RENE hese Nob AEEA IDG taks art NEE al te co ott 1.6. 1200 Ce. .480 2.480 6.480 ae = 320 m3 RTE re 240 p, ; = = 480 pg he coe EO () 5.480 4.480 : = EU P, fae — 200 42 ig BOO. ee AEON 480 2.480 2.480 om 16 ce, e S(5), en eere ET Kees mn 42 (U). So the result 1s (480, 1200, 1040, 360, 42) in accordance with the law of Euler. Case 1 [755311]. — | In the same way we find here the table: (O1 (75), (53), (B Da, $[ 11], 21755, (15) (53), (75) (31),, (553), (631), (811), | G1—D, BOLLE), (53) 11], 1553), G55 (D. (75) (53D;, (75)(311),, (6581); (T5) GEST) 631-2P, (6311), (755) 4 1[11},(75)(63)4[11},(653)4[11}, 21811} (75531), (755)(311),, (75) (531)),, (65311), (Sit Ds Ga) (ool by (7553) £[11],, (75) 4[811}, 4[5311), (5531—1), | NDS, (165) 4811, (15) 4/5811), [55801], | (F65812 T0), So we find through any vertex seven edges, three tp, ten 2. Sms one CO, five 47, six Ps, eight P,, two C, four #0, two (32110), one (22100), two (32100), four Pr, four Po, one Peo, one (3;3), two (3;6), two [5811], one (322100), one (p3; 67), two Piss, one H[ 55311], two (432110); *) So (531) is to bear the subscript 4, as the 5 may be related either to ao or to xz and the 1 either to a4 or to #5; so (31— —1) is to admit the subscript 2, as the three digits may apply either to + x3, + 24, — az or to + 23, — 24, + x5, etc. Verh. Kon. Akad. v. Wetensch. 1e Sectie Dl. XI N°. 5. E 6 82 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY as the number of vertices is 2°.6! divided by 2°, i.e. 5760, we get in toto / 60 fi = er nn Ur ie. 20160(7), 5760 105760 : 6.5760 RMC 1 ~=14400p,— — 51607 „ 25920), 57 5.5760 6.5 x6 (OD Aen dof i — 240077, PO Gr gute 9 12 6 .5760 .5760 38402, 14400, 4.5760 | of — 960/0....................... , 14880(Z)s, 2.5760 576 + = 1924 8, SB), an = 192 005), 2.5760 4.5760 Go 12e 85), Ga — = 960 Pr, 4.5760 5760 ie =480 Po 57 = 240 Poo, 5760 2.5760 WNR 640(3 ; TER 640(3 ; 6), 2.5760 open 120ce es CPE ore Geertie Me le Ve telle delete elfen hande he 5 3656(/),, 5760 5760 2.5760 5760 192, = OOF verte me 2.5760 . By moving this face away from the centre O to a distance À times as large its centre is transported to the point æ Aa (¢ = 1,:2,. 3, 0 9), 88, SES ROUES So the new position of the face leads to a new polytope d[Aa, Aa,,...Aa,_3,...]. As the length 2V 2 of the sides of this face is maintained and the length of the edge (Aa, À a, 4.4) is 2AV 2 if a, and 4,,, are unequal, we only can arrive for AA 1 at a polytope all the edges of which have the same length 2V2 if all the’ digits a,, @2,/.., 4, 3 are equal, 2 sim ne RIRES n—3 n—2 n—3 EENES A, 4[88. 8111), ASS ate CAN In these cases the face becomes A+2 A+2 1 — 4 D= NOL DB (TE Al : ) Og) ete 5) di — dA, à F = Je : L,== 3A, ne 75 len 2 À 2) BDA, 3 . = à | furnishing for the edge (a,,_», “n-a) of the new polytope the four symbols DE AN gale: : | ni), Kn: El Gal ace De So, if € represents either 2 or 0 we have in these four cases dn OEI, RAE An so the values of A different from unity are respectively _ À, gi i ¢ | 2 4 D 2 2 Dn ? of which the integer values are the only available ones. So the nN NS face Eon, can be applied to 1111..1] giving [44.4220] nt N — and to 1133. = |] giving [66. 64207, 1. e. in both available cases measure ld forms deducible from 47,%. Therefore we can disregard altogether the expansion of the AZmpd. according to their own DERIVED FROM THE REGULAR POLYTOPES. 89 faces and take into consideration the expansions e,, (4—2,3,...n—2), of the 17, only. We now pass to the contraction. À motion of the limits of vertex import of &|a, a ... da 4, 1. e. of (a, a... aa a,) towards the centre gives (a, — À, dy —2À,...,4, 1 —2À,a,— A). So the only new form we can get is (a, —1,a,—1,...,a,.—1,0,0), Le. a form deducible from M,®, etc. 100. We conclude this part by proving the following theorems, which will be useful in the next: Turorem LX. “The limits of truncation import of Cry Cp © + Cra Ch, HM, (1?) are Cha Cha + + Chey A C1 S(n) 22)” v p—1 According to the preceding theorem we have Nn —k, Ky—ky_4 4 EM, = tw + 1, 2% — 1, ee de) hky—ky Ie EL ARMES A e hy hy Ky 4 i t So the limits of truncation import are n Ky, Ky — Ky 4 hig—ky ky—1 i Ce ea eee a en n—k heeht kh ky—1 p CODEN Bp! AIDE or reversed hyt kjky k,-k, 4 Bep — (20 + 2, 2p ,2n—2,..... ALDE OORD Dre "pra, Ao EN e, _, ISnK21à), fey—1 CA Key 41 Ch, —I (m)( Taxorem LXIV. “The number /, of the units figuring in the he | symbol of coordinates t[a,a,_,.... 11..1] of an /wpd. in S, indicates how many limits of truncation import pass through any vertex.’ n—k,, oo hiy—ky a The number of vertices of }| 2p + 1, 29 —1,...,383..3, 11. .1], eae NN Wer ghy hj A respectively of its limits (2 + 1, 2p ET ne 2 of truncation import is represented by IN dpi n | D ÉESD: = ETE (nk) (kp Ap A) ... (aA)! fa! (nk) Oep AL eeh) So the 2’! limits of truncation import admit together a number of vertices equal to #, times that of the Ampd. itself. 90 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY E. Nets of polytopes. 101. Let us consider the net W(4,7) and suppose that it is composed of alternate white and black 47,%, so that any two 17, with a common limiting M,®, differ in colour. Let us imagine that each white M,® is split up into an inscribed positive HM, (= + 4[11..1) and 2" pyramids on regular sim- plexes Sz) the vertex edges of which have a length 2 and meet at right angles, and that in the same way each black 77, is split up into an inscribed negative AW, (= —4[11..1}) and 27 pyramids. Then it is clear that a space filling of S,, is formed by three groups of polytopes, two groups of HM,, i.e. a group of positive ones and a group of negative ones, and one group of n—1 cross polytopes [200..0], each of which has for centre a vertex of the net V(J/,7) not belonging to an ZM, and is generated by the addition of 2” of the equal pyramids. This net, which may be represented by the symbol W(+ AMW,, Cr,), forms our starting point here. It is our aim to deduce from this simple net several other ones the constituents of which are forms derived from the regular polytopes and Ampd., partaking with each other of the proper- ties of admitting one kind of vertices and one length of edge, by considering in the application of the expansion operations either the two sets of half measure polytopes as independent and the set of cross polytopes as dependent variables, or reversely. Any HM, of the original net V(+ HM,, Cr,) is limited by HM,,, of (2),-, import and by simplexes S(z) of truncation import; by each ZM, , it is in contact with an MM, of the other kind, by each S(z) with a Cr,. We now follow two polytopes HZM,, Cr, in Ax) contact through any group of expansion operations leading to a new net, by which operations MM, and its S(z) pass into (P), and (Q),_, and likewise Cr, and its S(z) into (P), and (Q),-1. Then it is evident that (Q),_, and (Q),, must coincide, as the application of the operation e, with respect to the group of Cr, origin on one hand and the group of HM, origin on the other would lead to a net with two different kinds of vertices, those of the group of Cr, origin and those of the group of HM, origin. This coincidence dominates the mpd. nets, as it creates a very close relation between the two chief constituents. If we denote by the symbol e, MM, the separation of the two groups of HM, from each other by the intercalation of prisms on their original DERIVED FROM THE REGULAR POLYTOPES. Di limits, the relation between the two chief constituents of an Ampd. net can be thrown into the following form: Tasorem LXV. “In the Ampd. nets the constituent of ZM, origin unequivocally determines that of Cr, origin and vice versa. If the former 1s Or, Chgs Er, 4 Se, HM,, the latter is represented by e;, 1 CA CRC | ne das Op 4 1 ko 1 k, _ 1 We divide the proof of this theorem in two parts. In the first part we suppose Æ, different from x, in the second we trace the influence of the occurrence of e, HM, Let the set of operations to be applied to the Cr,,, in order to obtain a polytope able to form an mpd. net with On, Chee Ck, 4 ek, HM, be represented by e e,,...e Ors, Then according to the q—1 results obtained in the preceding section the limiting S(z)(?v2) of Or, is transformed into Ors, Og er à Ue, Sr), whilst on the other hand the #{(2)21?) of HM, is transformed into ES En 1 EA Vads ME LR S(n)ev 2), As the negative sign of the second symbol is accounted for by the position of the two polytopes at different sides of the common limit deduced from S(x) the coincidence requires that we have log Ss hy Anta? I k's = Bs == +, he kga mn tem 3h kK = Ks aos 1e as the theorem states. We now suppose that the operation e, is added to the set of e, expansions to be applied to the ZM,, 1. e. that we drive the two groups of ZM, apart by prisms. Then the enlargement of the edev Hof. the triangle’ CH Er (Me kB) Stormiea br the centres ©, H,, H_ of any triplet of constituents of different kind in mutually (/),_, contact, eaused by the intercalation of the prism implies enlargement of the two other sides, as the triangle must remain similar to itself. This enlargement of CH, and CH_ cannot be effected by the application of the operation e, between the two coustituents of different form (see pag. 90); so it must be caused by application of the operation e,,_, to the polytopes of Cr, origin. In other words: the theorem to be proved aiso holds for the case that e,, occurs under the operations e, to be applied to the ZM, groups. Moreover from theorem LXIV we deduce: Turorem LXVI. “The totality of the vertices of any /Ampd. net can always be represented by means of one net symbol, viz. that corresponding to the constituent of Cr, origin.” We still remark that the number of Ampd. nets in #8, is 2”"~*. 92 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY For we can start either from MM, as it is, or from one of the (n—1), forms e, HM, or from one of the (z—1), forms e, e,, HM, etc., giving altogether 1+ (nl) +(w—D, +... ... + (al) == (1 HI 2" possibilities. These nets must all be new for u > 4, if they prove to exist. On the other hand a preparatory study of the cases 7 = 3 and x — 4 will show that » = 3 furnishes nothing new, whilst n == 4 produces four new cases only. 102. mpd. nets in 8; — If we interprete the net of Zand O as N(+ HM, Cr:) the four cases we meet here are RETE 2 DEE D. oop ke Res bern UE nettes CL Me OUT Aret Mee CE or in other form A AE RE Mig aen 1213... RCO . CNE AL LU) LEP TC ORAN 24 | Ape LI LOO UC Here the third constituents CO, C, {C are polyhedra filling gaps, whilst the numbers 12, 24, 19, 23 refer to the stereoscopie diagrams of ANDREINI. Compare also Table III of M. Srorr’s memoir. Let us pass now to the deduction of the coordinate symbols of these four nets. To that end we have to start in the first case from a 7 and an O in face contact — and in the other cases from what these polyhedra have become — and to calculate by means of the distance of their centres the periodie term which is to figure in the symbol. We therefore elucidate the mutual position of the two polyhedra in face contact in fig. 19, in projection on to a plane normal to one of the three diameters of the O group. But for clearness’ sake we have represented in each of the four cases the 7’ and the O — or what they have become — lying apart; in order to re-establish the real state we have to move the 7’ parallel to itself so as to bring the invisible shadowed face of 7’ indicated by dotted lines in contact with the visible shadowed face of O, 1. e. 4 B into coincidence with AB. As we want only the net symbol with respect to the group of O, the origin of the system O(Y YZ) of coordinates has been chosen in the centre of the O of the diagram. The simple diagrams of fig. 19 show an easier way leading to the knowledge of the periodic term of the net symbol. Indeed, im each of the four cases the O — or what it has become — is in DERIVED FROM THE REGULAR POLYTOPES. 93 contact by the edge 4B with an other polvhedron congruent to it. In other words: if the coordinates #,7 of the centre M of AB are p, the centres of the O group are represented by the frame 3 [2a p,2a,p,2a;p] under the conditions a, as, az integer and Xa, even, 4 1. e. 29 is the period. of the net. So, as the p has in the four cases successively the values 1,3,1+-V 2,3 + V2 we find for the four net symbols under the stated conditions BT Ba +2 , de 2as + 01, Zhed 6 ay + 4 6a,+ 2 Gas + 01, D a le res ip Lys ES a ABBA Da HAVE et ZB BD as +12 Though we pursue the study of these threedimensional nets merely from a didactic point of view it is not necessary to deduce from these net symbols of the O group the net symbols of the two 7’ groups. All we want is to show how the third constituents CO, C,tC can be found. Therefore we give here the net symbols of the two 7’ groups in the form: Pe ee, 2m EL 4, PER EE à Cat Ot Gael, ne Zee ck 3(2a, + 1)+8, 3 (Zag +1)+41, 3(2a, + 1)+1), oy + ala +1)0+p2)+1, Ca + D0 +y2)4+1, (Zag +1) 0+12)-+1), 4.24 3[2q +1)(8--V2)-+8, (a EIB HV) 1, (Re + DEVA) 1), where the double sign refers to the two groups + HM; and the conditions about the a, and their sum remain the same. As the polyhedra, of the O group remain in contact by faces with those of the two 7' groups and by edges with each other we have only to look out for new polyhedra filling vertex gaps which make their appearance in the second, third and fourth cases on account of the truncation of the polyhedra of the O group at the vertices. Though all the vertices of these new constituents are con- tained in the net, the second and the fourth cases show that it may happen that some of the faces of these new bodies have to be furnished by the polyhedra of the Z’ groups. At any rate we have to determine the new constituent by starting from an octahedron vertex and deducing from the net symbol the vertices at minimum distance from that point. We treat further each of the four cases by itself. Case (O, 7). — In this case there is no third constituent. Never- theless we deduce from the net symbol of group O given above that the vertices of all the COrepresented by[2a,+ 2,2a,+2,2a,+ 0], 3 Xa, odd, are vertices of the net. But these CO are no constituents 1 of the net; for the centre of the CO corresponding to any set of 94 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 3 integers a, satisfying the condition & a; odd is the point 2 a, 2 ay, 2 as, 1 and for “a, odd this centre itself is a vertex of the net, i. €. 1 these CO overlap. Case (tO, {T). — As we have p = 3 the point 2, 0, 0, originally common to the central O and an other in vertex contact with it, is carried away from the origin to thrice the distance and arrives at 6, 0, 0. So with respect to this centre of a new constituent as new origin the original net symbol becomes| 6 (a,—1)+-4, 6a,+-2, 6a,+-1}, Ya; even, i.e. [6a, + 4, 6 a, + 2, 6 a; +0], Xa; odd. Now the supposition a, ——1l,a;, —a;—=0 gives the square — 2[ 2,0 ]and so the six suppositions a, &, 43 — [100] give the six squares of the [2,2,0], i.e. of the CO. The eight triangles of this CO are fur- nished by #7, four of each group. So by putting a, = 0, a, = — 1, ax = — 1 in the net symbol of the group of positive ¢Z’ we get 4 {3+3,—3-+1,—3-+ 1], i.e. reduced to the new origin 6, 0, 0 the symmetrical form 4 [— 3 +8, —3+1,—3-+ 1], the triangle (0,—2,— 2) of which is a face of the CO found above. Case (RCO, T). — Here we have p — 1 + V2 and the centre of the new constituent becomes 2(1 + V2), 0, 0. So the net symbol with respect to that new origin is 3 2 +12)q,+2+72, 20 +12) +12, Ul +12) a, +121, Ya odd. 1 Here the six suppositions a,, &, ag = [100] give the six limiting squares of the cube [V2, V2, V2]. Case (tCO, tT’). — Herep = 3 + V 2 and therefore 2 (3 + V 2), 0, 0 is the new origin, leading to the new form 3 [2403 --V2)q, +42, 2(3-+ 12) a, + 2412, 2 (3+ 12) a, + 1/2] x a; odd. 1 of the net symbol. Here the same suppositions give the six limiting octagons of the #C represented by [2 + V2, 2+V2, V2]. By putting a, — 0, a, = — 1, a, = — 1 in the net symbol of the group of positive #7’ we get here or with respect to the new origin 4[—8+V2)+3,—6+V2+1,—-@6+V2)+ Ih the triangle (— V 2, —2—V 2,— 2—V 2) of which isa face of the 40. ftemark. Vhe p introduced above is not to be confounded with the extension number of the octahedron group which according DERIVED FROM THE REGULAR POLYTOPES. 95 to the rule connected with the sum of the digits would be 1, 3, 148V2,3 + 8V2 in the four cases. 103. The four cases of Ampd. nets in #, considered above agree in this that the third constituent is the contraction form of the constituent of octahedron origin. Indeed the contraction forms of O, tO, RCOp {CO are respectively a vertex, CO, C, tC. This fact is too general to be accidental, we will show why it mwst be so. Therefore we recur to theorem LXVI. As all the vertices of the net figure in the net symbol of the octahedron group — which implies as we already remarked that all the vertices of the new constituent are contained in the net symbol —, the faces which that new constituent has in common with the adjacent polyhedra of the octahedron group must define that new polyhedron. Now in the original net (O, 1) any vertex V is a point of concurrence of six O, the centres of which are the opposite vertices 77; of the six edges of the net of cubes from which (0, 7) has been deduced. So the six faces of contact of the new constituent with the six polyhedra of octahedron origin lie in planes normal to the lines OV, in the centres of these faces, lying at equal distance from O. These simple considerations lead to three possibilities compatible with the condition that the new constituent must admit vertices of the same kind and edges of the same length: either the new constituent is equal to the constituent of octahedron origin, or the new one is the contraction form of the other, or the other is the contraction form of the new one. But the first and the last sup- positions are to be rejected. For the first would bring equality between the two kinds of limits of the constituent of tetrahedron origin which have been called original limits and limits of trun- cation import, whilst the last is inadmissible as the constituent of octahedron origin is no contraction form. We now prove that the preceding result holds for any Ampd. net in ©, If once for all we distinguish for short the constituent of HM, origin as the first and that of Cr, origin as the second we can extend theorem LXV by proving: Tarorem LXVII “Any Ampd. net has three different con- stituents, none of which is a prism. The third is the contrac- tion form of the second. So, if the first is Chey Chg + Ce, 4 ky HM, pt Elp Cr,, the third sex Za Cr,. Im this form of the statement p and therefore the second OA Ck a» » ER IS ce EREN Saeed as ky—1 hk —1 Roy each of the three unequivocally determines the two others.” 96 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY Proof. In the original W(AW,, Cr,) any two Cr, in contact are either in edge contact, or in vertex contact, in other words the contact of the highest order between two Cr, of the net is edge contact. Now this contact of the highest order can only be annihi- lated by separation of the Cr, i.e. by applying the expansion e, to them. As this operation is excluded (as leading to a net with two kinds of vertices) the edge contact between the polytopes of Cr, origin is maintained, though it is changed in character by the operations e,, 1 Ee Ve > > > | 9 ZA fi mi PCA CEA CA + + Ce y A Cl S(n)(?v2), So the limit (7), 4 of Pa hh oe) DERIVED FROM THE REGULAR POLYTOPES. 97 highest import of the third constituent is the contraction form of the corresponding limit of the second constituent, 1. e. the third constituent itself is the contraction form of the second. Or shorter still: by the reversion of the symbols the transition from (a, a,...4, 2 1—1) to (a, as. . 4-2 1 1) manifests itself by the diminution of the jrs¢ digit by 2, 1.e. by the operation of contraction, leading to the result mentioned in the theorem. Remark. There is a characteristic difference between the three groups of nets — ® the simplex nets, ® the measure polytope nets, the half measure polytope nets — as to the character of the con- stituents. As we have seen in the preceding sections the simplex nets admit exclusively principal constituents, 1.e. neither prisms nor prismotopes, whilst the measure polytope nets admit only /wo principal constituents with exception of the original net of measure polytopes. Now in the case of the Ampd. net we always find three principal constituents with exception of the original net (AZM, Cr,); as soon as two of the three constituents become equal to each other we fall back on a measure polytope net. This only happens for n >> 8 in #,, as we shall see in the next article. 104. Hmpd. nets in S;,. Here we have to examine the eight cases: RS Pa EA DME ot PRN Oo ee a ese LLM), se CoCr, RS ONL res COM, ep OR ONLINE": enter Mr SEEN EON On it PRES CM OOP ANNAE ese, TIM 5, ey CaF, BRA ar eet nen en AM esse Cr NS ares Cae, Ty One enh Of these eight cases only four are new. The first is MC), the three equal groups of Cg being the groups of + HM/,, — HM, Cri. The second case is e‚ MC); as es HM, =e, Cr, we find only two principal constituents. The third case is ce, MC); as e3 HA, = ce, Cr, the third constituent is equal to the first. Finally the fifth case is ce, ea MCG); as e e3 HM, = ce, e, Cr,, here also the third constituent is equal to the first. In the four remaining cases the three chief constituents are different; so these cases are new. We represent them in the following small table e, HM,, Ca Cr,, ée, Cr, Pr], 20+)772) a; (2-+1V2) p2., 12 4 wa), ver Men Oa: Chats Cees Che, Perles B 2) DIN NDA baal > ese FEM, ee Crise en Cry, Pr), USE) a ER RE EL RER Bree, AM yo, Cher Cs Chg 60, en esra, Pr (DEEE) GI EVA ME ETAPE oe eel, enumerating the quadruplets of constituents and in condensed form the net symbols; in latter symbols the immovable parts of the digits are placed before the square brackets, whilst the sum of the four integers a; is always even. Verh. Kon. Akad. v. Wetensch. 1e Sectie Dl. XI No. 5. Ed 98 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY In order to get a better insight into the constitution of the fourdi- | mensional. Zmpd. nets we tabulate the contact between the different . constituents. To that end we introduce first a short notation with respect to the nets themselves and to their constituents and the threedimensional limits of these. We denote the Ampd. nets in 8, by the collective symbol NM, and distinguish them mutually from each other by putting before that symbol the system of expansion operations applied to the second constituent C7,; so the four nets found above are ez NH,,e, e NH, es e3 NH, e ee; NH). Moré- over we indicate the four constituents of each net, 1.e. the three principal ones taken in the order of succession assumed in theorem LXVII and the prism, by 4, B, C, D and we represent their different limits (2), by means of subscripts in connexion with their import; so 43, 4,, 4) will represent the limits of body, trancation, vertex import of 4, whilst B,(4—= 8,2, 1,0) and (4 Sn will represent the limits of (7), import of B and of (7), import of C, and D;, D,, D, will stand for the bases of D and the upright limits (7), of that prism which correspond to the faces of face import and of vertex import of the bases. So we find the following small table, where the numbers under the columns show how many (7); of each kind each polytope admits: ee 8 P, | 4CO | 4 Net A,.| A, | Ay | À | Bo | Bl BOI COUNTER ENEN TT |Z |B PE an | i | AT | O0 | EP, Py PRO ON PCO an mé 7 | T | CO | T | co | Ps | PS VONN CNRS tT 16 ry tle |S 407 SN AP of OL, PEN PA | 8 8 Siete Ge ee a4 | 8 | 32 | 8 | 2008 This table shows that the contact between the four different constituents is the same in the four nets, i.e. that we have in general A» = Je A, es DE 21, = Ce D == Dn BP, = Ch; CG — De whilst B is in contact by its limits B, of edge import with other polytopes B, this transformed edge contact being preserved. So the different threedimensional limits cover each other two by two. The contact between the different constituents can also be deduced from the following small table in which we repeat the constituents of the net in an other form: Bet a] jie B Any D es NH, |[1111]V2|(1111]V2] 4/1111] AMI V2 es » -|[1111], |[2111), | [SSS TET Cz C3 29 Rocke ike PP) RAA 2) 29 [3111] 9 [111] Re ” ee » |[2211], |[8'2'1'1] ,, l.(5311 SI DERIVED FROM THE REGULAR POLYTOPES. 99 So from this table we deduce 4,=— D; by remarking that the digits of the first syllable of D are the last three digits of the unique syllable of 4; in order to facilitate comparison of 4 and D we have reversed the order of 4, B, C. So we find 4, = B; (or rather 4, — — B,) as we get the same form by placing the four digits of 4 between round brackets after having taken the last unit with the negative sign and by placing the digits of B, multiplied by V2, between round brackets; ete 105. Before passing to the case x — 5 we will put the last two small tables of the preceding article on duty as to the general results they may suggest for 2 > 4. We begin by fixing our attention on the extreme case of the relation between the two constituents 4 and C, being governed in the case e; NH, by a vertex only. Here: 4,, the limit of vertex import of A, is still a vertex; so we have to accept for Cthe polytope deduced from Cr, which admits as limit G, of body import a vertex and this is the eightcell ce; Cr, The same remark holds for e, VH, already, i.e. for the third of the four cases treated in art. 108. But the first of our two tables, 1e. the table of contacts, suggests a remark of much wider scope. We deduce it from the fact that each constituent with three kinds of limits (2), is in contact with the three others, whilst the only one with four different kinds of limits (2); is in contact with the three others and with itself. This fact suggests that in space #, we will want in all ~ different constituents 4, B, C,..., of which B only admits at most x different limits (/),_, and all the others at most # — 1. We have used this suggestion as working hypothesis and found by its help the sixteen hmpd. nets of S;; this was an easy task: as theorem LX VIT gives the three principal constituents 4, B, Cand the prism D can be deduced from them, the table of contacts shows immediately which limits (2, remain uncovered and these limits reveal the character of the fifth constituent. }) ‘But there is an other method of deducing the new constituent, much more capable of being extended to S,,, viz the determination of their coordinate syinbols by transformation of the net symbol to *) It may seem in accordance with this suggestion that in the cases e, NH, and e, NH, of S, we have found no fourth constituent i.e. no prism, though they require the operation e, with respect to the two groups of HM, of different orientation, driving these groups asunder. But this not appearing of the prism is rather due to the fact that two adjacent HM, of different orientation are in contact by an edge only instead of by a face, so that the separation intercalates a square instead of a prism. 7% 100 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY new origins. We introduce this method by remarking that the addition of the second syllable [1]V2 of the symbol of the prism D.in the last table of the preceding article has a deeper meaning than might be supposed: in this form the coordinate symbol of D is derived from the net symbol, and by examining how this process runs in S, we easily hit upon its generalization for #,, if necess- ary by the assistance of the knowledge of the fifth constituent in S. found in the manner described above. So we indicate for any net in S, how the coordinate symbol of the constituents can be derived from the net symbol. | In fig. 20 we represent by O (4, X, 43 X,) the system of coordinates and by the shaded pentagon with the axis of symmetry OM a fourth part of the section of the plane O(X, X,) with the central polytope B. Then OP, is the “period” p of the net and the point P, of OX, lying at twice that distance from O is the centre of an adjacent polytope C filling a vertex gap, whilst P, with the coordinates 2p, 2”, 0,0 is the centre of an other polytope £ in contact with the central one by a polyhedron of edge import. Moreover P, is the point 2p, 2p, 29,0 and P, the point all the coordinates of which are 2p; of these P, corresponds in character with P,, and P, with O and ?,. So the midpoint Q, of OP, must be the centre of a polytope in threedimensional contact of body import with the two polytopes B with the centres O and P,, 1e. of a polytope 4. On the other hand the midpoint Q, of OP; must be the centre of the prism interposed between the two polytopes A of different orientation with the centres Q,, latter point being the image of Q, with respect to the space 2, — 0 as mirror, ‚as these polytopes are derived from the two M/M, of the original net, (AM,, Cr,) which were in body contact in that space wv, = 0. In this manner we find in general for all the cases in S, for the coordinates of the centres of the adjacent polytopes 2p, 0: 050 an” the creme „Ds O53 Dos re ee core | Pes D Os ND ee NN OENE PEI ENE DES END) 2 2p whilst the upright edges of the prism D are parallel to the axis OX,. Now we consider the case e, e,e, NH, in order to show how the process runs. Here we have p — 5 + V2, whilst the central B is represented by [6 +V2,4 + V2,24V2,V2]. So we obtain C, 4, D successivey as follows: DERIVED FROM THE REGULAR POLYTOPES. 101 6+ V2,44+V2,2+V2,V2 10 + 2V2 , 0 0 ed —(4+ V2),4+V2,2+V2,V2 furnishing the polytope [4 +V2,44-V 2,21 V2,V2],ie.C; 6+V2 ,44+V2,2+V2, V2 Be ee ae ee M oye Me Sues 1 OR Marah oreo tr Me pe — 5 leading to the polytope — }[ 5811), i.e. 4; 6+V2 ,A4AV2,2HV2,[W2] 6+V2 ,54+V2,54+V2, 0 1 we Eee ae giving finally the polytope 1[311][V2] , ie. D. This will be clear, if we only add one word about the factor + before the symbols of A and D, viz. that we want this factor in order to have symbols representing polytopes with one kind of vertex and one length of edge. subtr. subtr. 106. Hmpd. nets in S;.— We have determined the sixteen /mpd. nets of S; by means of the two methods given in outline in the preceding article. The results of the first method are put on record in Table X. This table is divided by vertical lines into eight parts; of these the first contains the symbol of the nets, the last two their consti- tuents and the five others the limits (/, of each of the five constituents 4, B, C, D, F. In the construction of this table we started from theorem LXVIT enabling us to register in the last part but one in the columns with the superscripts 4, B, C the character of the three principal constituents and to add under D, in the cases where e, appears amongst the expansion symbols of the net, the prisms on the polytopes of polytope import of 4 as bases. After having finished this task we have inscribed in the columns with the headings 4,, 4,,...D,, Dy the limits (4, of these consti- tuents 4, B,C, D, taken from the tables given in the preceding sections of this memoir; this will be clear if we add the remark that the notation D,, D),, D, for the limits (/), of D differing from the bases D, has been chosen in accordance with the consideration of these bases as deduced from MM, This second task having been performed we can formulate the contact between the constituents A, B,C, D; we find generally: A, = A, (i e, 18 absent) and 4, — D, (if e, is present), A, = B,, Ay = C,, Bs = D, , By = B, By = Co, C; = Do. 102 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY So A3, B,, C,, D, remain uncovered, 1. e. have still to be covered by limits (7), of #. We represent these limits (7), of # by M,, B,, £,, £,, indicating by the subscripts the constituents with which they are in (/), contact and repeat these limits in the column with the headings L,, B,, Ho, Hy. Finally from these limits we deduce the constituent 77 itself, see the last column of the seventh part of the table. We remark that this fifth constituent is a prismotope, a two components of which are ZM; (or e, HM/3) and p, (or pj); presents itself if and only if either es, or e,, or both oa are present. In applying the second method to #8; we have to tend the MCP of fig. 20 with the broken line O P, P, P; P, of edges leading from O to the opposite vertex P, into an MP” with OP, P, P, P, P; as corresponding broken line of edges from O to the opposite vertex P. If we represent the midpoints of OP;, OP,, OP; respectively by Qs, Q,, Q we find for the new origins leading to the consti- tuents C, 4, D, # the points P,, Q;, Q,, Q, with the coordinates Fe Op Ds VE p p, Ps es #, 0 | P> VIE P » Oy : 0 2p, 0, dE U ‚0 | So in the case e, eze, WH,, 1. e. [3'2'1'1'1]V 2 withy=5+V2 the constituents 4, D), # are obtained by the three processes BEND, AY IEM oo V2 BV 2) BV! DEMI He ee TE go OO sae 6+V2,4+V2,24V2,24V2, [V2] B+V2,5+V2,51+V2 51/72. Le Le PORC a : — SUDLT. ee ee Tk A: ED, 1 bc. V 2] O+V2, 5M 2; 5+ VR, > AE À subtr. 0 POELE TARN on v?] giving respectively $[53311], 4[3311][1]V2, 4[811][11v2. The results obtained in this way are collected in Table XI. To this we have only to- add a few remarks. The processes used just now show clearly why the syllables y[ 3311] and $[311] of D and # must correspond in digits with Le Sok DERIVED FROM THE REGULAR POLYTOPES. 108 the last digits of $[53311], the symbol of 4, and likewise why the other syllables [1]V2 and [1’1]V2 must correspond in the same manner with either of the symbols [3'2’1’1’1] and [22111] of B and C. Also why D must be a prism and 7 a prismotope, in connexion with the faculty of inverting the signs of V2 in the case of D, and of 2+ V2 and V2 in me case of 7, these in- version having no influence whatever on the distance of the vertices obtained of the new origin which is to be the centre of the gap filling polytope. Moreover the processes themselves indicate under which circum- stances the prism J and the prismotope £/ present themselves. If the symbol of 4 winds up in zero the second syllable of the symbol of D is [0], 1. e, the prism is lacking; but we know from theorem XXXV that the last digit of the symbol of B vs zero, if the operation e, has not been applied to B. Likewise, if the last two digits of the symbol of B are zero, the second syllable of F is [0,0], ï.e. there is no prismotope #, and the last two digits of the symbol of B are zero, if neither e; nor e, has been applied to B. Finally it is evident why we Be add a fourth process to the three considered ones and subtract 5+-V2, 54-V2, 0, 0, 0. For then we would get 1, —1,[2+V2,2+V2, V2], leading to L[11][1"1'1]V2, ie. — as 1[11] is an edge instead of a face — to a limiting body and not to a limit (/),. 107. Hmpd. nets in S,. — It is easy to see how the processes of the preceding article must be extended to #,, as the algorithm always remains the same and the number of the subtractions has to be augmented until only three digits of the subtrahend differ from zero. So, if we indicate by A” the constituent obtained by n—k k the subtraction of pp..p 00..0 we can formulate the general result in the following theorem : Tasorem LXVIIT. — “In any net deduced from N/Z, we find, besides the three principal constituents 4, B, C always present, under certain circumstances one or more prismotopes 4“ for 4 = 1, 2, .,2-—8, which may be called accidental constituents. The pris- motope A“ presents itself if — and only if — one or more of the expansions e,_,,e6,_141,6,_1 have contributed to the trans- formation of Cr, into B; the two syllables of its symbol are the last x —& de ol A between square brackets preceded by + and the last # digits of B between square brackets.” 104 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY So we find in the case B—15443322211|V2 of So: C = [444839921102 A =43[9755588111] 49 — ,[755533111] [12 A® = „(55588111 [11], AO Th 5 881 A) ER AD: [538111] ITS IR, AP a= STR RA r ET UMR 46 — TBAT) [so Rot. AD = LIT TS see | By applying this theorem we find immediately the sixteen nets of S5, as they have been registered in the eighth part of Table X with the heading ‘constituents in an other notation”. Moreover Table XI gives the corresponding results for the 32 nets of 44. F. Polarity. 108. By polarizing an #-dimensional Ampd. with respect to a concentric spherical space (with oc”! points) as polarisator we get a new polytope admitting one kind of limit (/),_, and equal dispacial angles, to which corresponds the inversed symbol of characteristic numbers of the original polytope. Moreover, if La, &,...,4,_1,4,] is the coordinate symbol of the original hmpd., this symbol also represents the limiting spaces S,,_, of the new polytope in space coordinates. The fact that there is no Ampd. proper in 8, and S, implies the corresponding fact with respect to the new forms. So, if by the subscript s is indicated that space coordinates are meant, we have: Pee) (4564 Aj 11311] = (8, 18,12) = 7 with py: ramids on the faces, 1[1111],—(16,32, 24,8) =, 1[3311],—(24,96,120,48)=%, with pyramids on the cubes, etc. 109. Tarorem LXIX. “Any Mmpd. in $, has the property that the vertices V; adjacent to any arbitrary vertex 7 lie in the same space S,,_, normal to the line joining this vertex V to the centre O of the polytope. The system of the spaces S,,_, corres- ponding in this way to the different vertices of the Ampd. include an other polytope, the reciprocal polar of the original polytope with respect to a certain concentric spherical space, unless the chosen DERIVED FROM THE REGULAR POLYTOPES. 105 hmpd. be the cross polytope MM, of S, in which case all the spaces 8; pass through the centre.” The simple geometrical proof of this theorem can be copied from that of theorem XL (see art. 66). 110. We have to add a single word about the reciprocation of the Ampd. nets. The results obtained here run parallel to those of art. 68. In general the system of vertices found by polarizing an Ampd. net is the combination of several groups of limits MCP of the measure polytopes of the net MMC”), p being the period. These groups are formed by the centres of the constituents B, C, A, DIES I Re PACE | i for B the even vertices of N(47,@2P)), represented by [2pa,,..-, 2pan], Ya; even, 1 VW ni „odd yy „ ERN 7 7 7 Sj odd, 1 NA vy centres of the M, of N(M,,@P)), LÉ MAR 7 y w limiting M,—a of the My, of N(M,G@r)), „ AR) 7 7 ore) 7 NG Sa TAD 7 1 2 Te KE F ; AA) <7 7 TEL 7 M. 3 RS i 7 7 In the case of the net NH, itself only the first and the third group are present; so in #, we find then the net MC). In all other cases we have to deal with at least three groups, the first three. As we already remarked in art. 68 an other paper, also destined to complement art. 39, will contain more ample develop- ments about these reciprocal nets. G. Symmetry, considerations of the theory of groups, regularity. 111. We first determine the spaces of symmetry Sy,,_, of HM, itself and afterwards those of any /mpd. derived from it. Case of HM,. — We have to investigate here how the reasoning which led us to the spaces of symmetry of the measure polytope is affected by the alternate truncation. In the case of 47, we found two possibilities under which the space #,_;, bisecting orthogonally the join 4, 4, of two vertices A,, As is a space Sy, _, of the polytope, i.e. that 4, 4, is either an edge or the diagonal of a face; in the first case we got the » spaces 2, = 0, in the second the (x — 1) spaces 2, + #,—=90. Now on the one hand it is immediately evident that the alternate trun- 106 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY cation behaves itself differently with respect to these two groups of spaces: it destroys the symmetry property of the first and pre- serves that of the second. But on the other hand we have to examine whether the alternate truncation does not enervate the force of the argument by means of which we excluded the cases that 4, 4, was a diagonal of a limiting WM, of the 47, for # > 2, i. e. that the projections of the two regular simplexes $(%) of the vertices of Jf, adjacent to A, and to A, on the space normal to A, A, are of opposite orientation. Indeed this argumentation has to be revised, as the two simplexes S(/) disappear altogether by applying the truncation and are replaced as groups of vertices of HM, adjacent to 4, and to 4, by the two sets of 4 #4 (4—1) ver- tices of JZ, lying in the following layers 8,_, normal to 4, 4. But the two polytopes‘) determined by these groups of vertices are neither central symmetric and maintain the property of the diffe- rently orientated projections, unless they coincide in the space S,_, normally bisecting 4, 4, for 4 = 4. So any space orthogonally bisec- ting a diagonal of a limiting sixteencell of HMM, is an Sy, , and therefore MM, also admits two groups of spaces Sy,_,, the spaces Et 2, — 0 and the spaces 2, + a += be, — 0. Lhe ae of the former is always #(2—1), whilst that of the latter is 42 (nl) (n—2) (n—-3) for » > 4 and four for » = 4. Case of the hmpd. derived from HM,. — From the structure of the Mmpd. it is immediately evident that a space S,_, is an Sy, for an Ampd. if and only if it is an Sy,_, for the HM, from which the /mpd. has been dirived. So we have proved the THEOREM LXX. “Any /mpd. of S, admits two groups of spaces ASy,_,, viz. the m (n—l) spaces zx, + 4, — 0 and the 2 (n—1) (n—2) (n—3) spaces: 47 tr RE 112. From theorem XLIIT we deduce: Taeorem LXXI “The order of the group of anallagmatic displacements of HM, and of the Ampd. derived from it is 2" a! for 2 > 4”. “The order of the extended group of anallagmatic displacements of these polytopes, reflexions with respect to spaces Sy, ; included, is 27 xl. In this extended group the first group of order 2"! z! forms a perfect subgroup”’. 1) Compare for these polytoyes: “The sections of the measure polytope Mn of space Sp, with a central space Sp, 4 perpendicular to a diagonal”, Proceedings of Amster: dam, vol. X, p. 495. DERIVED FROM THE REGULAR POLYTOPES. 107 The proof of this theorem is to be based on the remark that the order of the group must be half of that of theorem XLHI on account of the alternate truncation. 113. As to the application of Eurr’s scale of regularity we have to use theorem XLIV. We illustrate this, sticking to the original scale, by the following examples. a). Evample {HN Here we find one kind of edge, one kind of face, but two kinds of lmiting tetrahedra, viz. tetrahedra of body import and tetrahedra of truncation import. So the contribu- tions to the numerator are 1 from each of the three groups of vertices, edges, faces, and + from the limiting bodies. So the frac- eee aot RAM HI, 6). Bwample 4[553111]. Here we find three different groups of tion is À edges (5, 3),(3, 1), 4{1, 1]. So the fraction is a —=1 c). Evample N CH, Cr). This simple net admits one kind of edge, one kind of BL but two kinds of limiting tetrahedra, as a tetrahedron of body import of MM; is common to four MM, a tetrahedron of truncation import to two HM; and one (Cr:. So we 3 Ti if find ——# rae! da). a ample e‚e, NH,. Here we have to deal with three groups of constituents represented with their frames in the table . (82100012... (2p, » 2P9 ‚ Pa » Pa ‚ 2Ps ‚ Ups ) 5, Speven, Dn 2122100012. (3p; ‚ 2Pa > 23 “20: ie pe 2e ) 5, Sp odd, D Ball on Vp, ob Nog, ee So through the vertex 6, 4, 2, 0, 0, 0 pass | 6, L 422, 0, 0, Cea B, [10+4,10+6, 2, 0, 0, OF End B, (1044, dere Ore EN: OA G —1[.5+1, 5—1,5—3, 5—5, 5—5, 5—5]..... 4 1] 541, 5—1,5—8,(—5+5, 5—5, 5—05)]..24,, 4, 4, —3[ 5+1, 5—1,5—8,(—5+5, —5+9, 5—5)]..4,, 4, 4, IT 5+1, 5—1,5—3, —5+5, —5+5,—5+5]..... 42 Now the edge (64)2000 belongs to all these polytopes with exception of ©, 6(42)000 belongs to all with exception of P,, whilst 64(20)00 belongs to seven only. So we find three kinds of edges and the fraction is =i: , Oy ee 4 pase ur. oth _ As a rule the ne L Av + La derived from it — | GE | Groningen, December, 1912. a . wl el zt : ot pet ee oe ALP RTL LA og a. Vl AT wey PH SCHOUTE, Analytical treatment of the polvtopes regularly derived from the regular polytopes. Fig 16. Verhand. Kon. Akad.r-Wetensch. 1° Sectie, DIA N° 5. ACER an EE ne me ho me tt me M. le ERIS ar NS LIST OF POLYTOPES DEDUCED FROM MEASURE POLYTOPE AND CROSS POLYTOPE. Table IV. D= 6 12 8 C| = ce, O [ weet] MS 12h 265) 1 Ds = == 1 C=. 1C| = ce, es O PAT (24, 36, 14) 4 Ds == Ps ile 1 e, C= RCO | = es O pe D a (244 48, 263) il D pa Ps LE eee pa Bret 100 | = ee O eea (48, 723 263) a De Ph Pe JS GERO CONS ce, O [110] V2 (12, 24, 14.) 2 Pa — Pa igen | 1 el ce, C= Ol = O [100] V2 (rules) 1 ze = Ds Wap oT 1 ceC— 10| = e, O [210] V2 (24, 36, 14) 4 Pr — De Pye Sh oT gel Ds Ul 8 24. 32 16 G |= ces : (ELL LE BDA EK 1 C = = = 1 eG |= certes Gon Lel alde 64, 128, 85, 24,) 3 tC = == Gp Pel Oh || = CG Op || [WV LA] ( 96, 288, 248, 56, 3 RCO — 12 (0) he Ddl es OC | = ex GN NL) ( 64, 192, 208, 80, 3 C P, 1 1 88) DE I HAG | == CAGE Ce lk 192, 884, 248, 56, 3 ¢CO ae P, tT OY |e Be I Ry CS | = CG CN ll scl Lt | (192, 480, 868, 80, 3 AC JEN 12 CO 0 WE ay Oe dl Se OL || = ec Oren ale ellen (192, 480, 368; 80, 3 RCO 72; IE. i Dek GB ei ee O, | = oneness Lala) 384, 768, 464, 80, 3 100 pj 12. 10 SAGE ce, 0, | = ce Ce | [1110] V2 (32; 96, 885124.) 1 CO os = 7 ey 1 B ce, O,| = G,= ce On | [11001 V2 ( 24, 96, 96, 24) 1 O — — O IAE 5 ah Qe lial ces Cx | = QC, | [10007 V2 (Sn ATS AI 1 = =: = fh Wee Sel | ce, C,| = Ce, € Ce | [2210] V2 ( 96, 192, 120, 24 3 10 = = tT Oe Gell Ba, Ce es (68 || == @, Oe | (2110) V2 ( 96, 288, 240, 48, 3 CO P, — CO Oan ARN Zee ce ex Cz | = e Os | [2100] V2 ( 48, 120, 96724.) 3 (0) = = ijn Oh Wk Ds Bs dl 3 || Ce € ex Op | = ee Ci | [8210] V2 (192, 384: 240, 483 a 10 12, — 10 OISE CRIE: Neb 10 40 80 80 32 Ol = dr Oa EE Er EO EOD Ent Di — — = 1 On | = Well Al ES a 160, 400, 400, 200, 42) | # OC, = — = SOE IE It RO ceren CEA All OL EU] 320, 1280, 1520, 680, 122) | 5 AC = == Pa ce, S(5) | 1° | 2,1 es Co | = ces € Oy | [111 11] |:( 820, 1440, 2160, 1240, 202,) | 33, CCR (CE Pe OAN || momen CnC e, Cal [1'1 111] | ( 160, 640, 1040, 800, 242,) | # COPA UE) EP S(5) | 1 4, 3,9, L HAG | = ce cie Lee ay 640, 1600, 1520, 680, 122) | 3 oh ES eS Ie. CASE 2 | aoe er Col = ce ese, G, | [2'2'1'1'1 | 960, 3360, 3760, 1560, 202,) | 4% CNE Oe =" EI IA GSM 91 ener == Esten Goele Oe a 640, 2240, 2960, 1600, 242.) | 53, Cen DE SE), GS (ONE 2 | MORS ee Co | — 0666.) /22 111) 960, 2880, 2960, 1240, 202,) | & 2 Cl 8) PE emcees So) ulus Del ese, Oo | = ee Oy | [2'1'1’1 1] | ( 960, 3840, 4720, 2080, 242) | 53, ey OP en UE Leo 2, Sp) IEI ena 2.1 ONE PA A a J Pa 640, 2240, 2880, 1520, 242,) | -# DCE lhe 2? ea SE) | eae 5, 3,1 Eenes Go | = rererese Oy | 13320101] | (1920, 4800, 4240, 1560, 202) |E | aaah — (8:3) Pr ae SE) 3 | 63,1 ra On er eg ex Co | [32211] | (1920, 5760, 6000, 2400, 2425) | À Onl CE) APS AED pB. DAS Aton = er C3 €y Coa | [3 2'1'1'1 | | (1920, 5760, 5760, 2160, 242) | À PRECIEZE ESO) KSO GO. Onl eee Cp | [ 8'2'1'1 1] | (1920, 5760, 5920, 2320, 242) | # RO DEA: OL OSH MI ONG I ex C2 €3 Cy, Col = eee, Co | [43211] | (8840, 9600, 8160, 26404, 242,) | À er ee Cg co Lr AE ANTON CN AI CRC ces Co | [11110] V2 80, 3207 400, 200 dede Rl ne EDE lil 4 | 3, CC ce, Cy, | [11100] V2 80, 480, 640, 280, 42) | À CONO — — CES) NTI 2 A St 2 Cex Oro | ce, Cy, | [11000] V2 [( 40, 240, 400, 240, 42) | 4 CONC — — ec Si(D) |e ane 3,274 2] 1 ce, Go | = CA OOOOT VAAR Keen Zom Bon BOMS _ SON 49727000 CRC ceres On | [22210] V2-| ( 320, 800, 720, 280; 42) | 33 CC = eb) lS 3,1 TL || ti CAE |= ce €; Oy | [22110] V2 | ( 480, 1920, 2160, 840; 122) | 33 CE CE CA 8 -— € 8#(5) | 2 | 4,2,1 6 | 4, ce &, Co | = es Cy | [21110] V2 | ( 320, 1440, 2160, 1200, 162) | & ce, GO Pro (4:33) — ENSOR 5, Seo: Gez Cs Cio | = ce, es Chy | [22100] V2 160, 480, 560, 280, 42) | À EDG — + ENH 2 5,3, 1 Bolk Sy ces e, Oo | = es Oy | [21100] V2 240, 1200, 1520, 640, 825) | 33 ce Cs Po tay | OF ECE Oe TE 7h aby. Geren Onl = & Cy | [21000] V2 |( 80, 280, 400, 240, 42) | 3 CCC =S = — ASS) | Ok" ND Belle ER al cerenes Col — cue Coy | [33210] V2 960, 2400, 2160; 840; 122) | 43] eme — (453) — e%e8(5)| 8° | 6,3,1 9 | 6, ce, ese, Oo | = ese On | [32210] V2 960, 2880, 2960, 1200, 162) | À ceren Pie (Us ESS NN 7,4, Ll RER NDE Ceres e, Cap | — € e3 Cp | [32110] V2 960, 3360, 3680, 1440, 162,) | À Ce CN CPE MAT OC) NEC OISE SOMS il a ce; eze, Co | = € & Op | [82100] V2 480, 1440, 1520, 640, 82) | 3 Geen Gs aPe A ME O2 ce, es €3 en Cio | = Cy €203 Cop | [43210] V2 | (1920, 4800, 4160, 1440, 162,) | Br | ce ee G& Po (456) — «ee; S(5) | 4 | 10,6,3,1 | 10] 6 MEASURE POLYTOPE NETS IN 8, AND 8, Table V. CAUSES Fees Pa ser hade had CD D= 4 = Mp Je N Jo : P: Or, (Ce Os De |p DE == (0) 7. 1 1 1 i (Gi [111] == 0 [J 0 0] V2=0 Ce C3 2 2 2 à 17 43 (0, 2103), 1, (Aleag) NER 700 [1][10]) V2 = P, | [110] V2=CO | cee, ) | he SE ANR Lan) ee, | [211] — 160. RARES EN LÉ NVAO ce, Co €3 3 3 3 3 15 (c, 240038), 23 (Ce, Cons, Lore), Va (2feogs, toog) «| = So. TT mn eso pes SO rele le lek til ae | ATI] — 0) HAES | DEE LN — RCO Nee 3 3 4 | 3 | 22 1, (e, Wig, tag), 25 (Cs Pins 7C03,), 23 (Whe, blag, E03) eee | [2/11] == EO NOT AIEE io RIEL AO ORN REN CNE TDA 3 2 | 245 | 9, (pis, Drs 10mg), 23 (Page ÉCOgs» ECO%,) ce, | ALO VAG | | [100] V2=0 | Ces | | 1 2 2 | + | 18 | 83 (205, 0) Ge, e, | [210] V2 = 40 Padova Oe Ve rl Weeen De LNA | aera, ton) N= À Ur 93 Ja I Yo Pp. Or Ch (3 On Ms ee Ate pat fa é, (ELITE) = 0 BEOOR Crete, QE ees Ie Can AS eben 1] — @ [LOOF V2 = Po [1100] VANGEN CAE, LOIR ws Wey |e Pale), 7 MoN = Cre Ce AVEIRO Vy 1110] V2 — ce, CO 219 /4|4|: ee | [2211] = Gi \ OEREN 12100] W2— ce, e, Ce» C3 Cy 2)/4)4) 31] 5 e es | [2111] — 2 © ARLON VERS 777) MOINE = Peo [2 dl 10] V2 = ce, en ORG Ge aSa Delen POE SAN ez es |[2 111] NEC B PON VAL Dd G BOND LE 221 0! V2 = ce 6 ce, Cy €, 2.13 5 | AN eee; | [3211] — 2 Ce [11] [1.0] V2 = (pe; pi) BNO) 5 210] V2 = ce, eres | Ce; ee, e,|8.p.| 2 | 8 | 5 | 4 | 5} ENE =a NDS BDU me Ie GI) © =) é, a ea laide = es ME A een Pye = ae SC ay = za Tele a = ee | [1111] =; BEN) ER ED = (7h87) 0 He} ELA] = D pet] == 2, es Cy Gas | 2k |e 2 eee Se a exe, | [2211] = (iG Fee ls EL == (P5391) EAN) == Pico [2111] =a A Cy Ex Ch 2 Atal Del | e ese | [2111] RENES BEANS BEE = (Pa; #) BREE = 2; [2111] — C1 Ca er C3 Cy SO Sn ANNE: a eee [3211] 440 NNS BENE = (ps; Ps) 41e) = D; LS STE] ENEN een ese | sp. | 2 | 8 | 5 | 3B || | LION ce, [1 0 0 0] V2 = ce, = Cg [ces 21219 Ce | RIMIEDIDIA A2 — ces Ok [1100] V2 = ce, = G, [ces gat ee el ce, e [2210] V2 =ee, e, [2 100] V2 = ces eg CC €3 ZBB RAN: ces e | [2110] V2—= ee, eg [10] V2[10] VS mn) = CG [2 MONET ER Nice re, SN We SON 52 cees ea | [8210] V2 = ce, ec [lO] W2[l0} V2= (pis pi) = G [3210] V2 = ce, e& €3 GEen eg Ier 2 22 2) | jee el ol (= NETS OF MEASURE POLYTOPE DESCENT IN S;. Table VI. Is In 3 Js J I Jo P ( Or Ox Os fps Tete tt pe M; 7. 1 ele Wel Wesley = des [10000] v2 = - ce, M, |) 2) 1 2 mii Gi POO] WPS 2a [11000] v2 = Ces » lee 1 CARRE CE 11] (100) V2 = (wm; 0) RON [11100] V2 = Ce» à m9 1 a OPA) = u» PIPO C2) — M, | [11] [110] V = (,; CO) DOME", MAO VS cer » 2 1 ON NPA er € » PS OOO EP [21000] V2 = Ce Cy, » Tee 4 Gay PR TN =S Gy Oss, [1’1] [100] V2 = (ps; 0) (NOON Pe [211007 V2 = Ces Cy » a 2 4 HOE || LILO WA = NOS ALO) V2 = (#0; p,) [1’1} [110] V2= (7; CO) HU) ee 1 rs [21110] V2= ce, 1 | 2 | 1 ANNEE ENGE [11][100) V2 = (7; 0) PNO OT VAREN [22100112 = Tee ., l 9 1 Ge NPA] ee » [11 1] [10] V2 = (RCO; pi) [11][110] V2 = (~; CO) Hd V2 08, ae [22110] V2 = cue» hee 1 a LA Ce, MAT ON Cr) =U, AN 210142 = (pe; 40) BENN ROIS || 2 1 eee [3 3 2 1 1] = x Cy Cy » [IL 100] V2 = (#3; 0) R200) Veo ee. [21001 = centes ere, ] 2 L AO MERE as [2'1'1] [10] V2 = (#00; p,) [111] [110] V2 = (pq; CO) EON [32110] V2= cee, ,, Teale? 1 DAC | (|e) 23 I ER à [VI (10] V2 = (éC; p) [1’1] [210] V 2 = (pe; #0) LNV end [32210] W2= ce Ene » TR 4 CONN OI T pa 26. 5 [1/11] [10] V2 = (ROO; p‚) [11] [210] V2 = (a; #0) ANNEN ae [33210] V2 = ee , oy D) 1 AGN ACS PNR. [21/1] [10] V2 = (C0; p,). 1’1} [210] V2 = (a, ; 20) ENE Va Bren 148210] V2 == ce, eene » 12 4 CA Da a pes ARE NE OM PEP as ©) ONE EM LINE M; | r. | | 1 i G54) Wey a PDE LELLI NOD) EE eal ps; 0) BS Sn GV IME jj C4 LINE | 1 ENEN [HN ea ae 1 i) Pe Elle eel Wile CO) BN SCG) NANA 2 RU Gan Le RE 4 en NO Wel ae RES Lo SR Eren Sz) BLP Sl AS Os 5 ee 4 GO| OW Chanel eae) =P, [111] [11] =e; wa) MA Et 2000) RARES RNA ANNE Chen SRE) 1 aac n= 2) Gs 4 | OR 2% Sa tee) LOL feels ic) LN CL pes 12 IE =S ee, sanne 1 ENGEN NE AIN at 5 || (RATATAT ae MEN CO 7) RLG 072 CO) BRE FS es € sp. | 1 | 2 4 HEE Ne all lal S — en EE P. one PORT CEGO) Pap SEN) Be ESSO MIES EREN iL 2 i ner Ne RE ET PET =P [211] [1'1] (CO; po) E00 BET a FSE TEEN Len i €4 Cy C3 Cy C5 (PAC Si Qin Teall 61 €503 Ci, ’, LS Neve [een RAR SOR) ple (sa = (y, ;,400) 1] (324101) ec [432 V1] = arret Gede db |) ee | 4 ger || NAO AY ce, M; [10000] V2 = ce, M. 1 | LN 2h | À ce, | [11100] V2 = Ce» 5, OO ONNA CC 55 1 | 1 | 2 | 4 CC, Cy [22210] VD = CCCs 5, (21000! VA Ces Cy | | 2 | | 1 ce, es | [22110] V2 — ces e [10] V2[100] V2 = (m; 0) [21100] V2 ce» e Ee | 1 ca e | (20110) V2 — cer e [110] V2[10] V2 =(CO;p,) . | [10] V2[110] V2 = (y,; CO) [21110] V2— eye, Brard al 2 1 ce, es | [22100] V2 = Ce » [22100] V2 = BNG Soon | 2 | 1 CE, Cy Cy [33210] VO GD; G One [10] V2[100] V2 =(m; 0) [32100] V2= een es & | 2 | L CC; Cy Cy [32210] VASTE 6; » [210] V2[10] V2 = (20; m,) [10] V2[110] V2 =(m; CO) [32110] Vad cence, 1 | 2 | 1. ces €, eze, | [43210] V2 = ce, ea 63, ,, [210] V2[10] V2 = (0; p,) [10] V2[210] V2 = (ym; 60) [43210] V2 = Ce, ee Staal 2 | 1 Bw x + 1 ee € cB 5 e | € 6 el € 7 GTA Cp Cg 8] ee ele & ey 10 € & a 11 Co) Ea Ce 9 12 Carl Cz 18 GEGE NE Gr 14 CA ei €3 15 Co Cy Cy € Cg 16] @ @ es ey] a €5 C3 17 ce, ce, 18 Ce» ce 19 cez CA 20 GE, Col ce, Cy 2 1 ce, €3| ce eg 22 Ces C3] Ceres 23] ce; € e3|Cé tes 24 ces 9 np. n, 25 ce; ey ce, 26 Cy Cy ces | 9 7 27 ce, Cy Ces 26 oe 28 ce, Cy 4] Ce Cy 29 ce cel 30| cac] cel Bl fee, ey e, ey | Ce; Cy Cy © 1 torche remt vo tercko meed tenen rears tre nat 22 | nat trs echte torche riche reste vo tie nebineh rentes tenten vente reine telnr tenten w ww CCE CCE RE renten to < ren = wen w ow ae wens whic Dre wo vo KS) rw ww =f three refine ww wlan ter ie ee Ihe pepe roe réfute Sty eo << -- a vo ww a ee gemeen uee et ee relo refrein elen a ee Se Ei mr tree ww = = cherche two Ie -=- ww rene wo terzi vefsersko hee whens ete ID to rw ww ww weenskes tech nen (CET ET (G ; (6; 6 (6;: CROSS POLYTOPE NETS (IN efron en << << << co SS ww echec shout Sos ww << ww ww [=] re K CRS zo a K ©] efron < SENS PRIE U ee w oelroseleo w ua semccho exaceto cenecho | UC CCS Neve ket Keke SSS oefroteleo ecfroscko cehmecko acte RE eee 2 V2 ie ace wro celeste ofnie EN zero slede cedeo Sail cho i +2 secte arl vefeocef eefrosc euro il oshoorat colracicho CO CEE u wo << COTCES EN zE 3 ze 3 ke celeste zele << << ARR Che ie wlroex|ro — oslo PUN) to ww exleoas|es D DL V2) (2) 0 INE Va 42) VAA )[2V2 JE Vv 2)[ \ )[2 V En wh st En rn Ar So vo vo to 29 kad „De Wo ,2+2V2, vo ew 2 19 vo co oo ER nd a Ed Ee ee www to to to wo wo to 0 9 wr ww w RO LO 19 to u to wwe we ww vw ET 9 L9 19 22 vo 2+ V2 2+3V2, Om VV 9 D 9 at V2 219V2 9 9 2 2 9L9V29 v2 3V2 9 D} Of at V2, V2, 3V2, 2V2, ew ww pasty eS Ww ww LL ©] ST wo to L ee www www to Table VII. ces Ce, C4 Ce Ey ce, ey Cl Eg Ey ce, Ca Ey ce, eg Cy ce Co Ez ey ly Cy € een ey Co Ey €, Co Ez Ey = €1 € a Co Ca = wR wl rol no HMPD. IN 8, 8, 85. Table VIII. oO Faces Symbol of en rs ASE coordinates Characteristic numbers. | _ Ps | m | ES „Dj 7 (4, 6,4) 4| 3 | sn Limiting polyhedra. 1[811]=— #7 (12, 18, 8) al 1 WT | 5 | P. | nn | a | PF. | es Bele or, (8, 24, 32, 16) 32/12 16. 8 | | BRE ,_., (48, 120, 96, 24) 64) 4 3214 sil 1614 | Rire, [8111]—= ce, , (32, 96, 88, 24) 64| 6| 24) 3 16) 2 83| | | [531 Ice ez , (96, 192, 120, 24) 32) 1| 24] 1| 64/4 | 116121 812 Limiting polytopes. | | | | on A ONE TE) SUT) eat ly (16, 80, 160, 120, 26) | 160/30 12030 | | Onl | S(5)| 5 47383811) (160, 560, 640, 280, 42) | 480) 9 160.6] 80 2]803 1209} | a 3 | ce, S(5)|1 en | 1138 111) (160, 720, 880, 360, 42) | 640/12] 240 6} | |120 8/80/38] 80! 3 80/6) | hen? | olene omer AiG) Janse ( 80, 400, 720, 480, 82) | 480/18 | 240112| | | 24012 240118 | | | GP, 4 S(5)|1 es 4 1[55311] (480, 1200, 1040, 360, 42) | 320) 2] 240) 2 | 4806 | 80| 112005] | 804|ce e „ |2 URO nec a 1[5 83811] (480, 1680, 1840, 720, 82) | 800) 5| 720} 6] 3204 {sop [240/ 3] 1604] 802] 1604 e IE Er eee 1 [5 3 1 11] (320, 1120, 1280, 560, 82) | 640) 6] 480) 611603 | 160) 2 160 3] 80/3} 803] 803 Boen LZ ie Peer 118 4(7531]] (960, 2400, 2080, 720, 82) | 320] 1] 960) 4] 8005] | (160! F102" | 12403 1160/4 ler nr nec nee sere HMPD. IN 4. Table IX. Faces. Limiting polyhedra. Limiting polytopes (P),. Limiting polytopes (P),. HAM, oe Characteristic numbers. 2 Ps | Pa | as | 2 | g | Ps | g, | za | GB | Ps 5 | 10 | og | He | ep RP gan | ee IL kad | 649 GA | = 12 Ee 60; ete Lao ts SRE = oe 5 LNE LTS ARL ON GAO, 640, 252, 44) 6.40/60 | | | 640/s0 | | | | : | | ICE Dö) | 4[11111]| S(6) € » |1(833311]]( 480, 2160, 3200, 2080, 636, 76) ppeoils 64081 960) 8] 4506 | | | 64016 | es leren e A5) ce, 5) 1(33311] | ce, S(6) Crees e3 » |1(3833111]|( 640, 3840, 5920, 3520, 876, 76) SUED 1440 9 1120, 7| 9609] 960! 9 | 4809 | | GC € BBE > ce, [eer $[8 38111) CES € A 3(331111]]( 480, 3360, 7360, 6240, 1996, 236) | 4480.28) 2580/24 1920/16 | 4806} 384048 | | | | le: Ge ce » |2Pr| Po (3 ; 3) 4(81111)| 18313011] Oe 6 Cs e, » |4[811111]|( 192, 1440, 4000, 4800, 2344, 296)] 256040 | 1440/30 192040 | [2880190 | | 2 1 IES BRT (3 ; 3) ALA) (81)1H111]|(8314[111 S(6) e, zeg „ J$[(555311]]( 1920, 5760, 6560, 3520, 876, 2560) 4] 1440, 3) 2560/8} 480) 1 | | 960) 3 | 1600/10 | 4180 6 CIC TC Dr CHE ER he 45531 11 | Cee » €, Co exe » [4[553311]f( 2880, 12960, 18240, 10560, 2636, 236 1680) S| 8640/12] 19204 | 96021 5760 12 | | 960! af 960/1[1920| 8 | Gi 5 (Hm Oya een || Po LL 1G:3)|(8;:614[53311] (55 3)4[811]] ca e „ GES € es, 4 331 1J|( 1920, 9600, 16800, 12480, 3656, 296)] 7680/12] 7200/15] 19206] 1440, 3} 960,3 | 672021 1440! 9 1920/12 | eis: Om 3 » ce „| Pr |2Po | 4 P| (3; 8)| (85 6)f 4[38 331 1)| (53) 4[3311]/ (533) 4(311) he ws ej €, Cy es 4[553111)]( 1920, 7680, 10720, 6720, 1996, 236)f4450| 7| 4320) 9] 1920/6} 960! 2 | |2880| 9 | 1440) 9} 48013} 960) 6 | OG op Cras Cie » | Ce » |2Pr| Pr (3 ; 3) $(53111)] 16530111] cae „ Cy C3 ene, ${5 3311 1)]( 1920, 10560, 16960, 11040, 3016, 296)] 8: 013 5640/18 1440, 3] 9601313760 18 | 1440 6 | 144019 | | Zee, €; € ce, 4 Py | 3 Po (3 ; 3) 4(33111)) (53) 4[3111]/(5 33) 4(1 11) Gm CCE: ee, n +(531111]|( 960, 5280, 10720, 9120, 3016, 296)f 5760/18 | 4320/18 640 4] 288012 4800 30 | | 480) 6 96012 | 2, U; eren es / 4 1],63)4{1111] 63814111] Chee Br om exe C4 4/7 7531 1)]( 5760, 17280, 19680, 10560, 2636, 236) 3840| 2] 10080) 7] 5760/6 | 4800! 5 | |1440| 3 | 2880] 6] 1440/6] ce , lee „ OO L 1} (7175) 4[311]ice, ee „ REG ENGEN 4[755311][( 5760, 20160, 25920, 14880, 3656, 296) 5700) 3} 1440010 5760/6 | | [5760 6] 14402 2400) 5| 4801 3840, 8] 960/4)2ce¢ ec » lee » GE || WAG 4 1}, (75) 4(5 311)}(7 5 5)$[3 11] SG 5 CAC Co €4 5 1[753311])|( 5760, 23040, 31200, 17760, 4136, 296) 9600! 5} 1584011] 57606 | | 9601|6720) 7 | [2400! 5! 960 | 6720/14 | Je lee OON ee 4 (5 3311]| (75)h[8-311])/(75 8)a(B11]]| ec "ele €3 4 AG =m 75311 1)/( 3840, 15360, 21760, 13440, 3496, 296)1 7680) 6) 1152012) 25604 1920) 2 5760, 9] 14403] 960) 3} 9603]1920) 6) 4803] Zee, „ Clare: Gis ee » |6 Pir} 2Pco| Pio} (6; 3) 4(58111)} (75) 4[(81119)(7 5 3)4{111] EG on OA 5 GLACE 15311][(11520, 34560, 38400, 19200, 4136, 296)| 3840, 1] 23040) S| 11520/6 | 3840! 2] 1440/1 | 1920! 2 | | 9600}10 | 2400/5 J2ce e , lac „eee „lee » 16 Lrl3Po (6; 3)] (6; 6)f4(75311]| O7N4F[5311}|O75)4[311]] ee, eve „Ve, ey ey ey HMPD. NETS IN #;. Table X. CONSTITUENTS. CONSTITUENTS in an other notation. Nets | A, A, 4,| 4 | B | Bl B |Z B | G CG G G | D, D, | D, | D, | Zl A IN Te || A | B ___ @ | base of D De 0 1 B A bi ae ED, B AZ Gi. SE) =| = 865) = | = = = |] f= pS] S|) =S HM, Crs Crs — = = [10000] 2 |4f11111] = = a | eee aro; SEON G ce, 86) — | — où Den a. Oh — |[11000] 2 |f21000) BST = = Gy 5p || EE Ga || | Gay & »|— | — |Po OG, || CQ,» =i) = Ce, -- Œ GES Ce on — — BEANO KOOI > [33111] — — Gs Gi a | Pr S(5) a AE Oa, cen » SOIN AC | me EN GS [eee pe a ME ces » = (Psp) |(11110] , |f21110) , |,,[81111] = 1[111)(10] 2 eee on So | = 5)| Pp | (4; 3 Pe Pen ee eal eet oe ces » Copa lp El te) | ee De ey» ce, » (EIL (Psa) [11111 V2 | 1111) V2|,911119 2119p )v2e,01Np ve Gis ov CE Co» Gl |CE Cy US(O)| GE en fee, SD) — | — Br | — AG Center | CEE ee — — [22100] 2 [82100] 2 |,[55311]| — — € E3 » CE eg» | Perl € » 1&3 » | — |(4; 6) Poo & » | € » | — | (433) yy NT Nl CSN GRE) ey Cy» als » ce es » | EE (2; p.) |(22110] „ [32110] , |,[53311]| = 2(811110]2 ae en En ces, e » | Pir | (43 6) Pe eg „| ea » | Po | (4; 8) Grint | er Ge lePin|| Pr Po | = (4; 6) (458) Pe Cy es» ae » | ene » [BBL fp) [1111] V2] (21111) V2], (83311) {8 3112, (811) IV? Ep &3 » | CE » Co eg » | Pr Gun BE Dl NER Ron AA all — BA By. 55 = — | — | —|?P;1(4:38) — | — BE CCS Ceres = (Psp) |(22210] 2 [82210] 2 |,[53111 — 11 1){1 0] 2 BA nl OA > Gun = || CA & » | Poo|(4s8) Pac) Ca es » | ea » | Pr] — | cert » | ce Cg] Pr | Pool Pr | — |(4:8) — | Pr ae ee, » case, » |H(8111]| (23a) (nav | (27711) V2 | (83111) [8 1 Vel, (12 IL Wve ae. Ce ets le XB) es» | Pr 1838) Pic | cep eq » Oli) == ee GREE AEP ler) eal! Be ERR ENEN | 2 ces e;. 5 lle) AAA 1 Yee 2 0 NE arm’ Crees, a3 IGEnEanen (eye es > | Lil eea » Meet, | — (4: 6) Pro ee ole es a) |-—t (45 8) een — ee | Perl; 6) 4; 3) — En C3 Cy 3s MEN CD Ca 3s COC Ce — (47; py) [383210] 2 |[43210] 2 5311] BIE ag ial Oj) Er Gayer oo | AC a5 A IE 5 ee » | Po l(4; 6) Preol 3 » CA » | Pir |(4338)) ee » eee Cy, | Pip| Pol Per | — | (436) (4:38) Pr | eze » | CoC » ce Ce, » |F(5311]| C2; op) |(22111]V21(82111]V2| [5-5 311] 2}, (8 1 1][1 1]V2 HTC Ce Nc. | Pin (856) Lic | ee, ne, EPONE SN eel a » | Perl Pir| | Pir Bs OIB: B La | eue » | Been | caue.,|,(8811]|67;#)1(22111)., |f32114) 53311] Sab yaa) Gre mol CE 5 203 » | Pr GU €303 » | Pco| (8; 3) Pico lee & &3 » er » | Pr | — Lace » | ce Pr | Po) Pr | Pr |(S; 8) — Py mue m | Clg es» | €¢p eye, » | (S111) (Psp) |[22212) , [322710] | ies 171) | [(111]01] C3 Cy » [CCl >, |C Ces » | Pin ae » | ace » | Po l(B5 6) Pol ee e3 » Ve Ca » | Prl(85 3e, pe » lee » | Perl Pio | Perl Pr | (8 3 6N (85 8) Pir [ey ea eres » lame Ca » [CC Cp Cy » |,(5311]| (Z;pe) |L [43211] , |,([75311]|,{5311]{1] BI) , 10 16 40 | 16 32 |80| 80 |40| 10 | 32 |80| 80 10 2 Gelk Slide Ace (eee lle | | | SE confier nn We tt na } 1 A a a = M WI APR ce eo HMPD. NETS IN &. Table XI. CONSTITUENTS. Nets (0) B À A A® Ae NH = Heo OOOO Be pla eh ay == ae 2 HA OO OO OO EERE = = ae: Co MO ON PEOP ie eral at ty — = en Br EON IE 00 Eeen er = — HATE | EO Oh 2 Cee AA TO AT ar Oil sve boule tal. EN == Re = C5 (Te Th A VAE A nan ern eat) ak le shy A = — CE 22000) 2132 100 0)-2. |, [ibis 5.8 1 1] — — _ 6,3 » |[221100] BO IE TES AIRE — — Ces OO) 2 ena Tk ROn et RSA == SM) ONE ee [Th a AV nt RE SE TO eel ay esse Tl JA NVA — — C2 C3 RÉ On 222 1007) 2 | fos B LIN — == LM] NO 07e? C3 C4 MO EE KUn 5 BS TE = RSE Oy) 2 — C2 € DS An 21 ANDA ES ER BRON LE ELST — = eyes (22272 0 2202101820 EE | — AAC AO) MON ATEN) OND C3 € UE eA ONO TL PE A RÉ ENS EU = ART CO EE ELLE enal EAU Ms EL EMA En — C1 ey es [832100] 2 |[432100] 2 |,,[775311] — — 1[811][100] 2 Cy Ca €4 (BEOOR ene — esn EO 2 — | C1 Cs C5 PIL VAST 1 2) is 65.8 1 lal 58 TAN V2 — — C1 C3 Cy (SSE Oi ATEA KON CSE = eee NO ela ee Oil ei C3 Cs eM IEI 0 NTS AU A OPA CU Etre A RU — (EV Thy 2 ei Cy es PANNES) MANN) foes oro elan Ears ste MEI) ses) Sen) MAN — Co C3 C4 3.020 0) 224832 10) 20 RA bl aly = MSIE (Ol) SE oe Cy C3 Cs LVNL TASTEN LOR 16 oo A Sen] — Benthe ene san 001668 ane enne rn — 3 41e FNB ae To LT A AN ee An en AL Re ER Cy Cy Cy Cy [443210] 2 |[543210] 2 [9753811] = Fs TOON es IU Oy) 0103036 » | (3/8211 1] V2) [43211 1)V2],, [775 811]|4[75 31 1) [1]V2 = Picea hy) Ci Cy Cy Cr ALAN SG ea APE ES Ne SOE —: ER deu No) MMS Run eel es SIP Sn) te 88 Le) NAS aah PAL ye Gen NB SE a eee Nea As ee EEND PE C1 Cy Cs C4 Cs [4’4'3'2'1’ ies DAS TT le ben TOME S HET es levees lala (Tl) SR Meet A) eee pr er Mrs EE | té. om ef ang re mate mnt Dar A ete . a . - : - y 7 Cah, Sama] L SPY ee À st SAT ENT : : } ; : NT." x ¢ h ik > drf x ours ps Lit ILES € (SPÉCIALEMENT DE de DR MOORD, Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam. (EERSTE SECTIE). DEEL XI N°. 6. Stee (Avec une planche.) | | 2 JA 4 | RTS er i À ENS “Raed | REN E NE EE —E————— rex > EE es Co me ie 5 3 PRE : AMSTERDAM, | Re JOHANNES MÜLLER. EE Juni 1913. 5 SE re EAN NEE 3 e ÉTUDE SUR LES FORMULES (SPÉCIALEMENT DE GAUSS) SERVANT A CALCULER DES VALEURS APPROXIMATIVES D'UNE INTEGRALE DEFINIE PAR Beed MOED: Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam. (EERSTE SECTIE). DEEL XI N°. 6. > SEE OSS AMSTERDAM , JOHANNES MULLER. Tors, Valeur approximative d'une intégrale définie. b $ 1. La plupart du temps la valeur d’une intégrale | p (+) dx ne peut pas, par intégration, être déterminée d’une façon simple; souvent même la fonction placée sous le signe intégral a une forme si compliquée, qu'il est extrêmement difficile et même parfois tout à fait impossible de l’intégrer. Généralement on ne peut non plus, sans des calculs embarrassants, exécuter le calcul approximatif de l'intégrale en développant sa fonction dans une série. Par contre, on peut toujours aisément, et d'habitude assez vite, et cela avec toute Vexactitude désirable, trouver, au moyen de formules dites (approximation, une valeur approximative de presque toutes les intégrales. ; La valeur de l’intégrale | y (vx) dx, entre les limites a et à, peut, dans le sens géométrique, être représentée par Vaire de la surface plane A 4 B B'!) limitée par l’axe X, les ordonnées qui appartiennent aux abscisses O' 4 =a et 0 B =6 et la courbe dont l’équation est Si maintenant la fonction sub (1) peut être représentée par une série continue, infinie ou non, de la forme DL KR) PKR RO, at ete Pere (2) alors on peut toujours, comme nous le verrons ci-dessous, déve- lopper des formules par lesquelles il est donné de calculer des valeurs approvimatives de l'intégrale définie, sans qu’il soit néces- saire d’intégrer ou de connaître la série elle-même, et, dans ce dernier cas, malgré que les formules d’approximation en question se basent sur cette série. *) Voir la figure à la fin de cet exposé; les axes des coordonnées sont considérés comme perpendiculaires l’un sur l’autre. ( À ÉTUDE SUR LES FORMULES Si la série est convergente, elle peut être représentée, avec un degré illimité d’exactitude, par la série fine: y=f@—K, Kot ee KS ee ee qui représente l’équation d’une courbe parabolique passant par z points de la ligne sub (1). Dans (3), on peut attribuer à z une valeur aussi grande que l’on voudra. Nous attribuerons, dans cet exposé, à z une grandeur telle que les deux lignes sub (3) et (1) coincident à extrêmement peu de chose près, de sorte que laire de la figure limitée par la ligne sub (3) a assez bien la même grandeur, que celle limitée par la ligne sub (1). Dans ce cas, la première aire peut prendre la place de la seconde. § 2. Les séries, dont il est question sub (3), peuvent être divi- sées en deux classes. Dans la première classe se ramènent les séries dans lesquelles les coefficients A représentent wuigwement des valeurs connues et dans lesquelles dans les 2 premiers, éventuellement dans les 2% premiers termes, aucun K west égal à zéro, c’est-à-dire qu'aucun des termes avec À, jusque Æ,_, inclusivement, éventuellement jusque Æ,, _; inclusivement, ne manque 1). Toutes les autres séries, nous les considérons comme appartenant à la seconde classe. $ 3. Alors que, en rapport avec la convergence, 1l suffit, au point de vue théorique, que la série sub (3) converge, afin d’en pouvoir calculer des valeurs approximatives d’une intégrale définie, la pratique exige que la série converge assez fort pour qu'un nombre borné de termes (u) puissent donner une approximation suffisante. L’approximation doit donc, lorsque le nombre des termes de la série augmente ou du moins lorsque cette augmentation est quelque peu considérable, s’améliorer d’une façon constante. Par exemple, dans la série ve a? a po go (a 4 2)? t (at SP Lada et ) Ce n’est que dans le § 7 qu'il apparaîtra que par n on entend ici le nombre de termes dune série dont les coefficients, chacun en particulier, sont, pour la déduction d’ une formule d’approximation, assimilés à zéro. OT (SPECIALEMENT DE GAUSS) ETC. après substitution des valeurs de & — 1 et a — 10.1 et calcul des termes, les s/m premiers termes semblent être assez convergents; cependant ce qu'ils fournissent ne ressemble aucunement au résultat qu'on doit obtenir, lequel n'apparaît que lorsqu'on prend au moins douze termes. — Des séries dont il faudra prendre plus d’une douzaine de termes afin d’en pouvoir calculer une valeur suffisamment approximative conviennent moins à la déduction de formules d’approximation, parce qu'elles exigeraient des calculs trop étendus. Dans cet exposé, nous admettrons toujours que les séries sur lesquelles se basent les formules d’approximation en question, con- versent suffisamment apres un nombre relativement petit de termes. SECTION | Formules d'approximation lorsqu'on peut substituer à la fonction sous le signe intégral une série de la première classe. Aire exacte d'une figure plane limitée par une courbe parabolique dont l’équation peut étre représentée par une série de la première classe. $ 4 Sans nuire à la généralité du problème que nous traitons dans cette section, nous pouvons admettre que l’origine des ab- scisses coïncide avec la première ordonnée de la figure. A ce titre, l’axe O' Y” (voir la figure à la fin de cet exposé) est déplacé du point O' au point 4’; par là, à égard de la ligne A Y’, considérée comme Vaxe des y, l’équation de la vraie ligne limite de la figure sub (1) devient et celle de la courbe parabolique sub (3) g= LE het Le +. +L, a + Let. bt Be RE ne EE PDT to Tt ER ARD) Nous représenterons laire de la figure limitée par la courbe, dont il est question sub (5), par Z et nous l’appellerons l'aire exacte de la figure, parce que z, ayant une valeur aussi grande qu’on le veut, la différence entre Z et Zaire vraie, c’est-à-dire l’aire parfaite de la figure limitée par la ligne sub (4) est inférieure, en valeur absolue, à toute grandeur donnée si petite qu'on veut. ÉTUDE SUR LES re » + K Aire approximative de > Jigure et erreur de Papproain § 5. Etant donné, que de la vraie ligne limite y — ne, (4), ‚on ne connait les de que de n de soit j ee de la figure À ABB par tn hind Bagot Bate heb Bales OÙ #1, Yor Ys, - Yn représentent les ordonnées des n pone co de la vraie ligne limite de la figure et hy, Bs PA Ms nombres arbitraires. En Lorsque Verreur de l’approximation de J, sub (7), c'est-à-di | la différence entre laire exacte Z et l'aire approximative he | représentée par /# alors |; EC Ee x ys A hdmi der ARE ++ (8) a Q Hi $ 6. Si les valeurs des ordonnées des x ‚points connus, cate 5 lées de y= f(@), sub (4), sont transportées dans (7), alors not s pouvons, en vertu de (5), établir: LR, Lo Ha + Lait. KL ef LENS ae ee +h Bi +R, \ Ly La, Lott Lig te ETS oh td ae AP OREERES 21d, + Rs Ly + Ly234+ Lodi + Li AT EL af +... LL, jae + he EEE 105 + li, Lo PU Ly By, ae Let. dr Lisi U a Ligh nee sat OE CE : 5 Linda tee +h = ZE RHB Hal Rot tar RL. Cette équation étant soustraite de celle sub (6), donne, en vertu KE de (8), KE BE CA Holl + a8 Re +. ot Rs] Ly OY et (SPÉCIALEMENT DE GAUSS) ETC. | I=1,4+ F=1f,42|-j—@? R + ah, +a Rs +... + RE De IT a REP (10) où p doit être remplacé successivement par 0, 1, 2, 3,...(2—1). $ 7. Dans (9) se présentent, outre les grandeurs tout à fait inconnues Z, deux groupes chacun de # grandeurs, à savoir: un. groupe de x grandeurs æ et un groupe. de # grandeurs #, donc en tout 2% grandeurs. Aux grandeurs 2 et & que l’on veut sup- poser connues, peuvent s’attribuer des valeurs arbitraires, alors que les grandeurs inconnues peuvent se calculer, au moyen des équations que Von obtient en égalant dans (9) autant de coefficients de Z à 0, que contient le nombre des grandeurs inconnues de @ et L. Ainsi dans (9) nous pouvons, ¢adépendamment des valeurs inconnues de L, éliminer à notre gré 0, 1, 2, etc. jusqu’ av plus 2n termes de #5) | $ 8. L'expression dans le dernier membre de (10) donne laire exacte de toute figure, qui est limitée par une courbe parabolique, dont l’équation se trouve sub (5); elle est valable pour toutes les valeurs attribuées à x, æ et &. Cependant, quoique une formule dans laquelle 2 et & sont arbitraires, soit mathématiquement exacte, elle ne convient pourtant pas pour en obtenir une valeur utilisable de /,. Notamment, nous ne connaissons aucune des grandeurs Z et nous sommes donc obligés de négliger tous les termes de #7, sub (9), pour autant qu’ils ne sont pas égalés à 0, qui pris ensemble peuvent constituer une valeur considérable. Par conséquent on doit tâcher de développer pour / des formules, dans lesquelles les termes de 7, sub (9), aussi bien quant à leur nombre que quant à leur grandeur, sont réduits à un minimum; car apparemment dans (10) Z, differera moins de Z à mesure que FH, sub (9), devient plus petit. Quoique nous puissions éliminer chaque terme de chaque groupe de x termes arbitraires de # qui se suivent ou non, il convient que nous prenons à cet effet de préférence les termes dont les grandeurs Z sont affectés du plus petit indice, parce que, la série sub (5) étant convergente, les termes affectés de l’indice le plus petit ont ordinairement une valeur considérablement plus grande que ceux affectés d’un indice plus grand. A ce titre, pour la déter- *) Dans chacun de ces cas, il se forme un groupe spécial de formules d’approxima- tion dont quelques-unes, qui sont connues sous le nom de leurs auteurs, seront indiquées dans le § 10, 8 ÉTUDE SUR LES FORMULES e . : / à . mination des valeurs de æ et Zl, nous égalerons toujours, dans cet exposé, à zéro les coefficients des termes affectés de l’indice le plus petit dans (9). Si par conséquent dans (9) les 2x coefficients de Zo jusque Lo, inclusivement sont chacun en particulier égalés à zéro, si des 2x équations qui en résultent on déduit les 2 valeurs de @ et les x valeurs de ZX, et si on transporte ces valeurs dans (7) et (10), alors on obtient pour (10) une expression pour J dans laquelle les termes de # avec Lo jusque Z,,, ne se présentent plus et on obtient donc pour (7) une expression pour J, qui est la plus exacte qui puisse être déduite de (9). Si l’on suppose, par exemple, le cas où l’on veut, de a ordon- nées, déterminer une valeur approximative d’une intégrale, alors on trouve les x valeurs de Z, sub (7), qui pour * ordonnées, con- duisent à la formule la plus exacte, au moyen des 2% équations qu'on obtient en égalant à 0 chacun en particulier les coefficients de Zo jusque Lo, inclusivement. On a de la sorte (pour p— 0, D AT RD SI Rite Bek) EE D d'A R,+ Vy Rot V2 Rd er tar mer vy, FREE ? yr Rt lt aad... a . ge Pe Rat Re ee gear à Avant de poursuivre, nous faisons remarquer qu’à chaque @,, qui satisfait aux équations sub (11), s'attache en même temps une valeur (1 —a@,) 3). Pour le démontrer on prend des equations sub a1) les (£ +1) premières, on les multiplie en numéro d'ordre par les coefficients binomiaux du 4°" degré affectés de signes alternativement + et — et additionne le tout. Alors on obtient A | Bye | Ames rt 4 aa ne oe | ( 1) Ls a | : Á nk LE: LAN Nr à LT - T9 LT: + (—1) 2,1 + ) Il en résulte immédiatement que les ordonnées doivent se placer deux 4 deux à égale distance des deux côtés de la ligne qui, au milieu de la base de la figure, est perpendiculaire à cette base. Le nombre des ordonnées doit donc toujours être pair; il est vrai deux d’entre elles, notamment les deux du milieu, peuvent coïncider et, dans as, le nombre des ordonnées à calculer est impair. Voyez l’alinéa final du § 26. a 1 (SPÉCIALEMENT DE GAUSS) ETC. 9 +R, | — —W oe ie En B 9 Rh so (—1)*2,,"| LE DS pu it A k Sere ar RHONE: Dans le premier membre de cette équation, les formes avec les- quelles 2,, R,, L,R, ont été multipliés sont égales aux 4° puissances de (1—x,), (l—e), (1—a3).. .(1—za,,); le second mem- PRET jg bre est égal à — — , notamment égal au terme initial de la série hl des #°™°° différences de la série harmonique 1, +, 4,... 3) Par conséquent on obtient Ra a) HB) JH Bla)’ Er Cette équation est valable pour 4=0, 1, 2,...(2#—1). On obtient de nouveau, après substitution de ces valeurs pour #, les 1) Si l’on forme d’une série de nombres arbitraires Arlon Aars: Agee An les séries des le, 2e,...ke différences, alors on obtient pour le terme initial des ke diffé- rences, la formule jee abet) k k(k—1) ke A ASAT 45: ETS ER mi om Ai 1 1 FRA < Si l’on y suppose Aj = 1, 49 — 9 A3 = FE AUS a alors, en renversant l’ordre, on a ak k k (k —1) 1 t A es ke | ae 5 at de, Kern Re a Ie Ee + (12) Par une soustraction effective, on obtient dans le cas de la série harmonique ie 1 il 1 1 1 serie donnée AT CON ans A Boites. oe de 1 1 1 le différences a. AD “ — 9 3 ; EC ; = ae ne ei tits) eea ofgepen ele tellelle (ej effe 2e différences Pan Le Biro Hol Mate mete Wt oat 5 PAR A OO LO NR LS RRO! es 18) (8) sise ee BS. GO +, Oe Dear Oe, ee TO ke ow ewe, Oe, ies ass ale Oe en) "MR OS 00 04e, One: Dee eve es. es ele Par induction, on décide que la série des ke différences peut être représentée par k k} k kh (—1) Gas monk Never La justesse de cette induction apparaît, quand on déduit de ces ke différences suppo- sées les (kh + 1e, On obtient alors immédiatement la même chose que quand on sub- stitue (k + 1) à kh. En simplifiant, on obtient ar nd tie alge TE En égalant cette expression < celle sub (12) et en divisant par (— 1)" on a HAN ce Re en k+1 10 ÉTUDE SUR LES FORMULES équations sub (11), seulement chaque +, est remplacé par (l1—a,), ce qu'il fallait démontrer. Apparemment cette déduction, où il est supposé que la fonction sous le signe intégral peut être remplacée par une série de la pre- mière classe, ne peut être poursuivie lorsque cette fonction ne peut être représentée par une série complète. Voir $ 23, 3° phrase. Développement des formules d'approvimation lorsque l'axe des ordonnées est placé au milieu de la figure. $ 9, L’équation de la courbe limite de la figure étant géné- ralement donnée par rapport à la ligne qui, comme axe des y, est élevée perpendiculairement au milieu O de la figure, nous supposerons désormais dans cette section, que l’axe 4° Y est placé au point O au lieu du point 4, de sorte que l’équation sub (4) de la vraie ligne limite de la figure, par rapport à la ligne O Y comme axe des y, devient et l’équation sub (5) de la courbe parabolique devient y = A+ Ara + Ao + Aga +... + À, a... (14) Si nous représentons les coordonnées d’un point, à gauche de Paxe Y, par (—x,,7_,) et celles d’un point qui, à droite, se trouve à la même distance de l’axe Y que le point précédent, par (@,, 74), alors il faut, puisque z est ici toujours pair, remplacer z par 22, de sorte que nous avons Y_»p= AoA, Ana; — Ae Joen Aad A et Van AT Apt Ae gt Agee .-. PASSE AN done Vr VY +p __ ; Ay HAHA + Aoi. a ge et / est représenté par 1 A ‘his | Ay + Ar + Aya! Aga? +... + Age pt | dae; _ par conséquent IAEA EE AHP ADA dt PEA ok (16) (SPÉCIALEMENT DE GAUSS) ETC. 11 On peut désigner une valeur approximative de J, c’est-à-dire /,, par ene ae ok Zn tes a Ym 4 mi GELE 4 Fe aaa (17) OÙ (Yar Fa Ya +712), etc. représentent des couples d’ordon- nées des 2m points connus de la vraie ligne limite de la figure sub (13) et. #,, A, ....#,, des nombres arbitraires. Si l’on représente Verreur de l’approximation, c’est-à-dire la différence entre l’aire exacte Z sub (16) et laire approximative /, sub (17) par Z, alors on a ban bre ET RE L'ART EST PER el (18) En transportant dans (17) les demi-sommes calculées de (15) des 2m ordonnées connues, on obtient 1 = Thy en Ad IE AD + ee Aen Ed EE nn en D 0 ir + Ro AE At zin Star se Ay, 9 en. = Se (a li B a hy = dj? Li, ER OB nn Ban hn) Ao: Cette équation soustraite de (16) donne, en vertu de (18) 1 5 2} 9 2 BE OP Ry + ay? Ry ae? B+... a, B)} dy, (19) ou il faut remplacer successivement p par 0, 1, 2, 8,...2—l. Au sujet des formules d'approvimation les plus connues, spécialement celles de Gauss. § 10. Des expressions pour 4, et #, développées dans le $ précédent on peut déduire les principales formules d’approximation connues, ainsi que les fautes qui y appartiennent comme, par exemple, les formules d’approximation selon Nuwron-Corns, SrIRLING, EULER, MacLauriN, Gauss, Carisrorrer, LoBaArro et Hermrre-'cHEBICHEFE. Toutes les formules mentionnées ci-dessus excepté celle selon Gauss pour un nombre pair d’ordonnées et celles selon Hermrre-Tcur- BICHEFF, s’obtiennent en attribuant certaines valeurs fixées d’avance respectivement à toutes les abcisses a, #5, #3, .. . Bm seulement d’entre elles et en calculant les valeurs correspondantes de Ry, Ros. B, de (19) en y assimilant à zéro autant de coeffi- cients de 4 que W comporte d’inconnues @ et 2. Dans le développement des formules d’approximation selon Her- ou à une partie 12 ÉTUDE SUR LES FORMULES Mite-Tonesicaerr, les coefficients des premiers m termes de Z (19) sont égalés à 0, dans ces coefficients les Æ sont considérés comme égaux entre eux, c’est-à-dire qu’on prend 2,—À, = R; =...—= B, = a et au moyen des m équations qui se sont formées de la sorte, on détermine les valeurs de @ 3). Des formules mentionnées ci-dessus, nous ne discuterons que celles de Gauss. Les formules @approximation de Gauss. (Pp $ 11. Gauss traite deux cas ?). 1° Dans le premier cas, aucune des 2m grandeurs mz et À ne sont considérées comme connues, de sorte que le nombre de coeffi- cients de 4, c’est-à-dire le nombre de termes de Z qui dans (19) A Z 7 \ Pd 21 NN peuvent être égalés à zéro, est de 2»; le nombre des ordonnées à calculer est, dans ce cas, pair. On a alors pour déterminer 2 et & les 2» équations suivantes R,+ B Yel Ak Br Li À, ac © Rs + Dz Bs + A RS le On Bn 5 G ay Ry LE B Tes LE ds Ry + TEE + | hae dy t G if , 4n—2 4m—2 4m—2 4m—2 eee (1-)4m—2 VY hiya, Bs 4-3 Ee ONE eT li, Bet 2) j Si Pon résoud de (20) +? et À, on obtient les carrés des abscis- SES £1, Las. LE, pour lesquelles les ordonnées y, #45 Va, doivent être calculées de (13) et pour 2 les valeurs qui doivent être substituées dans (17) pour obtenir la formule pour /,, exprimées dans 4, Gag, en À cet effet on suppose dans les $$ 26 et 27 (où la solution de æ et À d’équations comme celles sub (20) est donnée en général) gd), k= 2,0 =O, metre NE et 28: PA x)” et dans $ Ae DN i Bi Brae By = 2S. ae "Sn, alors on trouve que les carrés 2,7, #,..-.@, de (20) sont les racines de Péquation (voir (71)) (29% Ne Ur Cahen A 5 or as SS 15 ee = 0 où, en vertu de (74) ~ re] = oe ‘) Toutes les formules citées dans ce § sont développées dans notre ouvrage: Valeur approximative d'une intégrale définie. Paris. Gautruser-Vittars. 1905. ‘) Voyez Cart Frixpricn Gauss Werke. Dritter Band. p. 165 et suiv. Heraus- gegeben von der Künigl. Gesellschaft der Wissenschaften zu Göttingen. 1866. (SPÉCIALEMENT DE GAUSS) ETC. 13 2 ge 2m— 1 | a De La “ ’ 5 (2) } mt m—1. .2m—3 Sd Fer as (2) 2 AE PAR PE „m3 .2m—d Ss (4) 3 et Am F D Ss 7 Fh aa (21) 2 3 20) Sm—2 ? m—1 (à) ml Im +3 m—2. | — (1) + ’ Els De : m 2m +1 | Les valeurs de 2, s’obtiennent de (68) après substitution de d= 0, etc. (voir plus nn on obtient Ai _ Wm 127 6 seach foe fae is a ee sG ye me Re m—1.p D ET iP a Sie eo ee (22) L’erreur de Z, sub (17) est représentée, en vertu de (19), où tous les termes avec Ao (p= 0), jusque Ayo (== 2m—1) inclu- sivement sont égaux à 0, par | 6 jee (a, ih el He Re" Rs + .. +» hoen Fil Aint ee ee tn By WA gm tor + et ainsi de suite. De (21) on peut facilement déduire, voir (56), D) Sip Ups | 2 So DER Áo Sip Ty : 8 =S 8 2? 3.p 3 EN ME aN ORR) ee (23) ane jee, sert sue, je) “e/g, tes 2 D zap >< Sin—2 FE an 3.p Ly ’ A9: en 2 m—1.p TT “m—1 Ss es Vy 2°. Dans le second cas, une grandeur +? est préalablement con- sidérée comme connue, c-a-d. 4, — 0, de sorte que les deux ordon- nées médianes y_, et y,,, coincidant dans l’axe des y, ne forment qu’wxe seule ordonnée à calculer et que, par conséquent, le nombre d’or- données à calculer est impair. Les autres (#7 —1) grandeurs + et les m grandeurs 2, forment le nombre d’inconnues à déterminer, c’est pour cette raison que dans (19) il n’y a que (2 »—1) coefli- cients de 4 qui peuvent être égalés à zéro. On a te ed Ur HE SEPT SC et 5, pet rk ed etl, ml re 5, ER (2) Al Am—8 ; À ss gp? m3 | D. Ti 5) 9 Am—5 F4 rs Ad 3 Ss ew) Be a | un eere Bin. Pr ER 23 Ds, I 3 À pe 4 tandis que les valeurs de /% s'obtiennent des deux formules vantes, notamment la valeur de #, de la formule | ] ws oe Im 1\2m—4 ae meden Pas Le rois REESE Pli T= 4 2) 19m 32) 5° EDS, EE HA 2 = pe m—-1., 8 et les autres valeurs de Z de la formule A MES 16 ies S 5 ln mar) iE (Oe bren a R,— = ik = pi TOME mtn p +(— Ke Zend Pi : ya died de ze ef Me >. |! L + » CES Ie À L'erreur de lexpression pour Z, de (17) est, en vertu de (9) KE ln 1 HE he Pe représentée par ee ij | PK: BP te 17 hm—2 vin? hm—2 AVES EN fe ee By À 4 i. + Lil se Ge a + 2, rp zn R,,) dins + tae D" tr, (ws hes + Bs ie |. » R.,,)| Aum + et ainsi de suite. Ensuite on trouve facilement (voir § 22) (SPÉCIALEMENT DE GAUSS) ETC. 15 D) ES, Sy Vy 2 à 2 Hop Do. ue Ae ? 2 Sp = 83 Bh pee ORAN EME (27) 2 Dea. Dier Sh. re p Vy ? 2 Sin—2 TOK SA SEB 1:7) Vy ° Gauss donne les valeurs numériques de a et ZX, dont il est question sub (17), pour 1 a 7 ordonnées inclusivement, en 16 décimales exactes (voyez note 2, p. 12). M. R. Rapav a encore calculé ces valeurs pour 8, 9 et 10 ordonnées en 10 décimales (Voyez R. Rapav, tude sur les formules d'approximation qui servent à cal- culer la valeur numérique d'une intégrale définie, dans le Journal de Mathématiques pures et appliquées. 3° série. Tome sixième, 1880). Dans la Table 4 à la fin de ce travail-ci, on les trouve pour 2 jusque 10 ordonnées en 16 décimales. $ 12. De toutes les formules d’approximation, celles de Gauss donnent les résultats les plus exacts, pourvu qu’on puisse substituer une série de la première classe à la fonction sous le signe de l'intégrale en question; elles exigent cependant, même lorsqu'on applique un petit nombre d’ordonnées, des calculs extrèmement longs, où peuvent aisément se glisser des fautes de calcul 3). Gauss suppose qu’en appliquant ses formules, les longueurs des abscisses seront toujours exprimées en 16 décimales, puisque ce nest que lorsque 2 compte un nombre suffisant de décimales, que l’on peut être assuré qu’en appliquant sa méthode les premiers 2m, eventuellement les premiers (2m—1) termes de erreur / dispa- raissent de (19). Si on applique les formules de Gauss et si, dans les calculs des y de p(@), f(e) ou ¥(z) sub (1), (4) ou (13), les longueurs des abscisses sont exprimées en moins de 16 décimales et si pour # on admet cependant la même valeur, que celle qui s'offre dans les formules de Gauss (Table 4), alors il se peut aisé- ment que les coefficients de quelques-uns des termes avec 4, jus- que A,,, 5 inelusivement ne soient pas assez proches de zéro, auquel cas l’inexactitude de /, serait sensiblement plus grande qu’on ne le déduirait de la formule. *) Loparro (Calcul intégral, p. 442) dit à ce sujet: „La méthode d’approximation d’après Gauss présente, quant au degré d’exactitude, des avantages évidents et est pré- férable à celle de Newron et de Cores. Il est seulement regrettable que cette méthode plus correcte comporte des calculs assez compliqués, qui rendent assez difficile son appli- cation.” 16 ÉTUDE SUR LES FORMULES S'il y a neuf ordonnées ou plus, il sera probablement nécessaire, en appliquant la méthode de Gauss, d'employer parfois pour x plus de seize décimales, auquel cas le calcul de ces formules exige un temps extrêmement long. $ 13. Si l’on calcule les valeurs des coefficients de 4 de (19) pour siz ordonnées p.e., on obtient lorsqu'on prend dans la Table 4 les abscisses seulement en 2 décimales exactes, c’est-à-dire lorsqu'on établit di 0. FRS De Vi 9a.” 6b, ani TS I=1,+ #=0.2339 5696 7286 3455 (y_1—+ 744) + +-0.1803 8078 6524 0693 (y_2+-719) + 0.0856 6224 6189 5852 (y_3+t-y43)— —0.0005 3714 4, —0.0002 3546 4, —0.0000 8191 4 — — 0.0000 2468 4, —0.0000 0686 4,,—0.0000 0173 Ais — — 0.0000 0039 A,,— 0.0000 0008 4,,— 0.0000 00015 4,,— ete. (25) Si l’on prend 2 en 5 décimales exactes, alors on a BTO 2, == 0.383060" Et “2, == 40020 et T=T,+ F=0.2339 5696 7286. 3455 (yad) + +-0.1803 8078 6524 0693 (y_s+-y4.)+ 0.0856 6224 6189 5852(y ya) H +.0.0000 0183 4039 4, + 0.0000 0057 3819 4, + + 0.0000 0014 7821 4, + 0.0000 0003 7069 4 + +-0.0000 0000 9386 4,4 0.0000 0009 2415 4, + 4-0.0000 0007 3335 4, 0.0000 0003 6299 4 + 0.0000) 0001, ABS AE eht DO RER (29) Pour z en 10 décimales, on trouve 1 —=0.1193 0959 30, z,=— 03306 0469 32 et 2, 04662 SAN en [=I,+ F=0.23889 5696 7286 3455 (y, y) +-0.1803 8078 6524 0693 (y_y+-y4s) + 0.0856 6224 6189 5852 (y_3+-y44) + +-0.0000 0000 0012 7996 4, + 0.0000 0000 0001 9693 4, + + 0.0000 0000 0000 3218 4, + 0.0000 0000 0000 0511 4, + | 0.0000 0000 0000 0122 45 + 0.0000. 0009 0097 4940 4, + +-0.0000 0007 2760 5382 A, + 0.0000 0003 6157 2569 Ais + + 0.0000 0001 4316 9888 4,,— ete, 27.7) (30) ) Le coefficient de A,, (qui en comparaison du coefficient du terme immédiatement antérieur est remarquablement grand) est à peu près égal au coefficient de A,, dans la formule pour sia ordonnées de la Table A. Voyez aussi les formules sub (30), (31), (32) et (33). (SPÉCIALEMENT DE GAUSS) ETC. 17 Pour > en 16 décimales, on trouve dx ==0:1193 0959 5041 5985: , 2, = 0:3306 0469 3233 1328 et æ = 0.4662 3475 7101 5760 et T= 1, + B= 0.2339 5696 7286 3455 (y_, 4 y_,) + + 0.1803 S078 6524 0693 (y_2 + 445) + 0.0856 6224 6189 5852 (y + 74s) + 0.0000 0009 0097 4927 4 + 0.0000 0007 2760 5509 4,, + 0.0000 0003 6157 2582 4,, + +. 0.0000 0001 4316 9388 Ais + etc........ (31) Formules @approaimation dans lesquelles les longueurs des abscisses selon Gauss sont exprimées en moins de seize décimales. § 14. Voici comment on obtient des formules d’approximation qui, avec un nombre égal d’ordonnées et un nombre égal de déci- males pour a2, sont plus exactes que celles qui ont été développées dans le § précédent. On prend de la Table 4, pour chaque formule en particulier, æ en nombre égal, inférieur à seize décimales exactes, et on cal- cule, suivant ces longueurs arrondies, les valeurs de ZR de (22), éventuellement de (25) et (26), l'expression pour Z, de (17) et enfin Pexpression pour / de (19). | De cette manière on trouve, pour un nombre égal d’ordonnées: et pour les mêmes longueurs des abscisses que celles dont on s’est servi respectivement dans (28), (29) et (30) les formules suivantes: Pour m — 0.12 , æ —0.33 et a — 0.47: I= I, + E— 0.2324 6285 8338 5646 (y_, + y.4) + + 0.1855 3350 9700 1764 (y_ + 442) 0.0820 0363 1961 2591 (y_3 + 99) — — 0.0000 1633 6188 4 — 0.0000 0870 1796 4, — — 0.0000 0317 1460 4,, — 0.0000 0089 5102 A, — — 0.0000 0020 7914 A, — 0.0000 0003 9648 Ay, — 10000 0000 6491 ee ote... re (32) Pour 2, — 0.11931 , a —0.33060 et 2, — 0.46623 TI, + B= 0.2339 5631 2171 0420 (ya + y4) + + 0.1803 7353 2778 4742 (y_ sy) -+ 0.0856 7015 5055 4838 (y + y13) + -+ 0.0000 0000 4306 4, — 0.0000 0000 3825 4, + + 0.0000 0000 1833 4,3 + 0.0000 0009 0771 4, — — 0.0000 0007 2975 A, + 0.0000 0003 6220 4x + SF -AEDOONEOO0 AA oto RER AU TEE (33) Pour 44° =0: 14030950, 305.25) — =P aa O46 9. 32 etre. == 0.4662 3475 71: Verh. Kon. Akad. v. Wetensch, 4e Sectie DI. XI No. 6. F 2 + 0.1803 8078 6572 5009 (y_o + y) + 0.0856 6224 6198 4350 [= I, + B 0.2889 5696 7227 0741 (ya +74 en +. 0.0000 0000 0600 0140 4, — 0.0000 0000 0000 0016 . — 0.0000 0000 0000 0016 4,, = 0.0000 0009 0097 onde a 0000 0001 4316 9388 A, i Sse ee hes + En hi Ee formules sub (28), 29 et an aves § 15. Pour He, numériquement, la plus grande exac des formules du $ précédent, lorsque les calculs pour J, ont lan étendue, en comparaison de celles du $13, nous inscrivon: dessous, pour chacune de ces sept formules, le résultat du ca JA À ES : Pe 2 hg 1 d’une valeur approximative de l’intégrale | —, c’est-à-dire du_ 8€. SFU . A De 5 4 ë ‘EURE rithme zéperien de °/,, valeur qui, en 24 décimales, est exacte- “ment équivalente à | 1h 0. 1177 8303 5656 3834 5453 8794. Attendu que l’axe des y est placé au milieu de la figure, nous Fe a substituons, pour la commodité, dans l’intégrale ci-dessus 8.5 ke: à a, de sorte qu'on obtient à ie dx t ik Ale 2 ke ee e pe ne Me Seb Z a ef Mm c'est-à-dire pour siz ordonnées, ou m= 3: 1 5 pour e= : HVA 5. ES YA 8. BEN 1 een 3 2 1 2 BETES MTI Er DE os Wieck: >) » LdU “YY too 2 ee ase Be le On obtient, lorsque les abscisses sont données en deux ikan er = ae, notamment lorsque z—0.12, x, — 0.33 et z,=0-47, de es de 54 mule RU. sub (28) LONT SI 92% Bu (52) L,= 0: 1177 8303 9683352 oe (35) Lorsque les abscisses sont données en 5 décimales, on obtient — de la formule : (SPÉCIALEMENT DE GAUSS) ETC. 19 sub (29) L210) IPS OR RM ee St (36) (83) I,=0. 1177 8308 5656 3820.......... (37) Si les abscisses sont prises en 10 décimales, alors on obtient de la formule sub (30) Ei 0. 1177, SOS CAO SCA Lie Et (38) (A) 1,—0. 1177 8303 5656 3834 5445 91.. (39) _ Si les abscisses sont exprimées en 16 décimales, on obtient de la formule sub (31) Z—0.1177 8303 5656 3834 5446 29.. (40) Il apparaît, chaque fois, pour six. ordonnées : en comparant (35) avec (36), que le résultat pour + en 2 déci- males, suivant la formule (32), sera plus exact que celui pour + en 5 décimales, suivant la formule de la Table 2; en comparant (37) avec (38), que le résultat pour 2 en 5 déci- males, suivant la formule (33), sera plus exact que celui pour a en 10 décimales, suivant la formule de la Table susdite. Etc. Sur le nombre de décimales, dans lesquelles il convient d'exprimer x et A. $ 16. Dans la Table 4 nous avons, à la suite de Gauss, aussi pour l’application d’un petit nombre d’ordonnées, exprimé les abs- cisses en seize décimales, Ce nombre est pris arbitrairement; il doit toujours être relativement grand. Mais, dans les deux $$ précédents, il est apparu que, même lorsque la série sub (14) converge assez fortement, on peut, en appliquant six ordonnées, se contenter de moins de seize décimales pour @, parce que, dans le cas où la convergence ‘n’est pas trop forte, l’erreur de l’approximation reste assez constante, soit qu'on prenne pour @ dix ou seize et plus de décimales +). Si, par exemple, on compare la formule (31) où a a seize déci- males, avec celle sub (34), où a n’est donné qu’en dix décimales, il apparaît que, dans les deux formules — qui chacune sont desti- nées pour six ordonnées — les coefficients des 4 homonymes, de As jusque 4,4 inclusivement, sont semblables jusqu’à la 15° déci- male ou plus. Dans (34), les termes avec 4, Ag et 4, se présen- 1) En cas de faible convergence, on pourra probablement pour six ordonnées se con- tenter d’encore moins de dix décimales. DE 90 ETUDE SUR LES FORMULES tent aussi il est vrai, tandis que dans (31) ils ne paraissent plus; mais ces termes ont dans (34) des coefficients numériques si petits, que leur valeur totale n’a pas d’importance en comparaison de la valeur totale des termes avec A, Au et A, dont les coefficients numériques sont respectivement plus de 6, 45 et 22 millions de fois plus grands que ceux des termes avec 4, A, et Ai, et ce n’est que dans les séries qui convergent extrêmement fort que les relations a = et si 12 “444 16 est question ici. On obtient alors aussi, comme il résulte de (39) et (40), pour l'intégrale donnée, selon les deux formules (31) et (34) une même approximation, concordante jusque dans vingt déci- males. Par conséquent, pour des séries qui ne sont pas très forte- ment convergentes et, à plus forte raison, pour des séries faiblement convergentes, l’exactitude maximum, pour autant que celle-ci puisse être atteinte avec six ordonnées, s’obtiendra déjà, et cela à une minime différence près, si 2 est donné en dix décimales au maxi- mum, lorsque la formule pour /, est établie conformément au $ 14. Le surplus d'au moins six décimales, que Gauss propose d’em- ployer Zoujours, est donc souvent sans utilité, parce que, malgré la grande augmentation des calculs, l’exactitude du résultat n’en est pas appréciablement accrue. La chose devient cependant tout autre, quand il s’agit de séries très fortement convergentes. Alors en effet la valeur totale des termes affectés des plus petits indices, peut avoir une influence appréciable sur la dernière décimale dans laquelle Z, doit être exprimé. Dans ce cas, en appliquant la méthode de Gauss, on doit, lorsque le nombre des abscisses est assez considérable, exprimer les abscisses en plus de dix, peut-être en seize ou plus de décimales. Si l’on veut donc appliquer dans tous les cas les formules de la Table A, pour chaque fonction dont on ne sait pas d’avance si elle peut être exprimée par une série faiblement ou fortement convergente, 1l faut, pour toute sûreté, que pour chaque fonction, chaque # soit exprimé dans un grand nombre de décimales, mais alors encore il y a lieu de se demander si seize décimales pour + sont bien toujours suffi- santes si on met en compte, par exemple plus de neuf ordonnées. Le fait, qu’on ne peut pas d'avance juger définitivement com- peuvent se rapprocher de celles dont il bien de décimales il faut prendre pour + pour réduire les termes avec A, Jusqu'à 4,,_, inclusivement, de telle façon que leur valeur totale ne puisse plus avoir d'influence appréciable sur la valeur de /,, ce fait n’est pas un des moindres inconvénients de la méthode de Gauss et on est par conséquent, pour toute sûreté, obligé en (SPÉCIALEMENT DE GAUSS) ETC. | appliquant cette méthode, de prendre pour z un nombre plus grand de décimales que peut-être il n’est nécessaire. D'ailleurs dix décimales pour + donnent déjà lieu à des calculs très longs et très ennuyeux. C’est pour ces différentes raisons, que dans la Table B, placée à la suite de cet exposé, on n’a pris pour 2 que deux décimales, par là aucun chiffre n'est de trop et cependant les termes dans (19) avec A, jusque 4,, 2 inclusivement sont absolument sans influence, tandis que les termes homonymes avec 4,, etc. sont plus petits que ceux qui appartiennent, entre autre, aux formules selon Newron-Corms et MacLaurin pour un même nombre d’ordonnées. On ne cherche pas alors l’exactitude dans un grand nombre de déci- males pour æ, mais dans un grand nombre d’ordonnées, qui sont faciles à calculer. Les nombreuses applications de formules des Tables 4 et B m'ont prouvé d’une façon évidente, que le calcul des formuies pour m ordonnées de la Table 4 prenait beaucoup plus de temps que le calcul de formules pour (2#—1) ordonnées de la Table B, alors que, en comparant l’exactitude relative des deux Tables, il appa- rait encore que les formules pour (Qm—l) ordonnées de la Table A sont notablement moins exactes que celles pour le double d’or- données moins une, c’est-à-dire pour (4wm—8) ordonnées de la Table B. $ 17. Les valeurs de À sub (22), (25) et (26) consistent égale- ment de groupes, formés de suites infinies de décimales, qui ne peuvent donc être inscrites qu’en nombre restreint dans les formules pour /,. Le fait de ne mettre en compte qu'un nombre restreint de déci- males pour #, ne produit d’ailleurs aucune erreur, si À est ex- primé dans un nombre tellement grand de décimales que même si on en prenait davantage encore cela ne pourrait avoir aucune influence appréciable sur le calcul du résultat. C’est pourquoi, dans la Table B, les valeurs pour & sont indiquées par un nombre de décimales plus grand qu'il ne sera jamais nécessaire; cependant ce plus grand nombre de décimales ne complique pas les calculs, parce que la personne qui fait ursage des formules de la table B est elle-même juge du nombre de décimales qu'il lui convient, pour toute sûreté, de prendre dans chaque cas particulier, en rapport avec le degré d’exactitude qu’elle veut atteindre. Du nombre d'ordonnées a appliquer. $ 18. La formule (19) indique — conformément à ce qui se 29 ÉTUDE SUR LES FORMULES trouve dans la 2° phrase du $ 14 — que si toutes les m gran- deurs +, donc aussi leurs carrés, sont admises d’avance comme connues, le premier terme de #, dont le coefficient numérique n’est pas égal à 0, sera 5 (4 Me LE (m7 Si, En pele Ry +a" Rs +. wot mi R,)| Ass 2m + 1 indifféremment si +, est égal ou plus grand que zéro. Il en résulte que, soit que les deux ordonnées médianes coïncident dans l’axe des y, soit qu’elles soient à quelque distance de cet axe, dans les deux cas l’expression algébrique pour le premier terme de l'erreur #, sub (19), donc le rang de l’erreur de l’approximation, est la même; seule la grandeur, c’est-a-dire la valeur du coefficient numérique de ce terme diffèrera, étant une fonction des valeurs de >. J'ai fait expressément dans ce but un grand nombre de calculs de formules pour deux jusque dix ordonnées inclusivement et pour 2 et plus de décimales pour 2 et j'ai constaté que la différence dans la grandeur de la faute entre l’approximation pour (2m— 1) et celle pour 2m ordonnées est généralement tout à fait insigni- fiante. +) Si les longueurs d’abscisses sont données en deux décimales cor- rigées, comme il est indiqué dans le § suivant, alors l’erreur en question pour (2% —1) ordonnées est même quelquefois un peu plus petite que celle pour le nombre pair suivant; jamais cependant il ne me parut que le calcul plus long pour 2m ordonnées au lieu de (2m—1) présentàt des avantages suffisants. C’est pourquoi j'ai cal- culé pour la Table B seulement des formules pour (2m —L) ordon- nées, comme STIRLING l’a fait aussi (LoBarro, Calcul Intégral, p. 411). Correction des lonqueurs @abscisses de Gauss, lorsqu'elles sont cæprimées en deux décimales seulement. $ 19. Lorsqu'on applique les abscisses de Gauss (pourvu qu elles ed 2 7 KJ dr AC soient exprimées dans un nombrè suffisant de décimales) pour le calcul de la valeur de /,, alors les premiers 2m termes de l'erreur FE, sub (19), sont chacun en particulier égaux à 0. Ceci n’a plus lieu lorsqu'on a posé comme condition, que toutes les abscisses ‘) A cet égard, il convient de faire remarquer, que lorsqu'il s’agit de mesurer un nombre impair d'ordonnées, les deux ordonnées médianes, qui dans ce cas coïncident, sont toujours placées exactement, ce qui, lorsqu'il y a un nombre pair d’ordonnées n’est le cas pour aucune d’entre elles. (SPÉCIALEMENT DE GAUSS) ETC. 23 seront indiquées dans seulement deux décimales, comme il est ques- tion dans $ 16, alors les grandeurs w de (19) ne sont pas déduites des équations égalées à 0, mais elles sont déterminées d’avance et les # grandeurs Z# sont alors les seules inconnues qui restent à Pe determiner. Suivant (19), on trouve alors pour le premier terme de |’erreur # qui appartient à ce groupe d’abscisses réduites à 2 décimales (que nous nommons le premier groupe d’abscisses) A" — a” Re Roda Bs iik sas a DT tie) Ay, Ti p. Ao pols La valeur de (2, notamment re == hi D?" bros” (aj B + pe R, + ige Be + Ves + - om i. peut le plus souvent être réduite, en allongeant ou en raccourcis- sant d’un ou de deux centièmes de l’unité de longueur, une ou plusieurs des longueurs du premier groupe de longueurs d’abscisses, de telle manière qu'il en résulte un nouveau groupe d’abscisses (le deuxième groupe) dans lequel le coefficient du terme avec 4,,, devient un peu plus petit que celui du premier groupe. Il ne faut pas perdre de vue ici, que cet allongement ou ce raccourcissement ne peut pas comporter plus de deux centièmes de l’unité de longueur, alors que, en cas d’allongement ou de rac- courcissement plus considérable, quelques abscisses pourraient s’écar- ter trop des abscisses de Gauss et que par là les coefficients avec An» ete. pourraient sensiblement augmenter en valeur. Si nous représentons les longueurs des abscisses arrondies et non corrigées (ainsi celles du premier groupe) respectivement pas z,, PEN. uma celesxdu deuxième groupe par! ia Gao). ae, Ct si nous remplacons +, par 2, + ad et si d = 0.01 et u, — l ou 2, alors on peut écrire, suivant (19) yee eee ese R= 1. zi ad RH (2,-+-2,0) Bs (ez) PR, +. Rene = 4d) hig ~ ( ae ad) R, + Ha Va (23-a3d) Rs AR? Er dr L ( Rn Te ne Let She. Mere sp) We ae eer ter © 6. New wiee ye = ene ese, © ses ©, Jer he) 29) te) en an atgahetien nere tp et STe te tp ie) ns; le A» on ie Had" Ry (eof etd)?" RH (eafad) Rod. Heard)" Rn DAP Si nous introduisons les notations suivantes S,— la somme des m carrés (2, + ad), (2+ &0),.. (oe, Hud), S,— la somme des produits deux a deux de ces m carrés, S,— la somme des produits trois à trois de ces m carrés, et ainsi de suite, S,—le produit (continu) de ces m carrés; 1 ai 24 ETUDE SUR LES FORMULES © : D." alors ces carrés (+ æ&d) etc. sont les racines de léquation du 2m°% degré \ ie >, î = | FZ ie eel een” Bnn (2 + ad)?" — 8, (2 + ad)? 4 8, (ead? (IE = SUR ER à Si nous additionnons les équations sub (41) après les avoir mul- tipliées successivement, en commençant par la dernière, par 1, —A,, +. §,, —&,...(— 1)” So, nous obtenons l'égalité suivante: 1 1 ran: 1 RS Le mr OT RSS RCD, US Pour pouvoir déterminer maintenant les valeurs de &,, @, &3,.... æ,, qui rendent 6 le plus petit possible, nous at: dans (42) S, par la somme de tous les carrés (z,-+ 4,0)’, S, par la somme des produits deux à deux de tous ces carrés, et nee de suite, #, par le produit (continu) de ces carrés, nous développons les puissances et les produits et négligeons tous les termes dans lesquels se présentent des puissances du 2°" et plus haut degré de x, et de d; alors s'établit une équation dans laquelle les gran- déurs z,, élevées seulement au premier degré, sont les seules inconnues. k Après l'introduction encore des notations suivantes: Bij == la somme- des: carrés, 2,725", vu oke rentes S,, == la somme des produits deux à deux de ces mêmes (m—-1) carres, : S,, == la somme des produits trois à trois de ces mêmes (m— 1) carrés, et ainsi de suite, S14, == le produit (continu) de ces mêmes (m—1) carrés; et S, == Ja somnfe de ‘tous ‘les: carrés; 21127 dn ne S, == la somme de leurs produits deux a deux, S; == la somme de leurs produits trois à trois, et ainsi de suite S, == leur produit (continu), (42) devient (LL Am 1 2n—2 1 a m net) or tect ee San: GY (IT SN EN Nien en 5 ei en S) a Rs JI Sr ee — 220 A Sars PE Sz PE ITS ee gd + se © 0 « e 0e 0, © 9 9% 06 © 0 9 & @ + es a ee es Bos 8 6 8 6 let a, Bi ere Tors lo Le Ve SR ee ee ee 6) » 1 2m? 1 2M—! 1 MA = tal GS, gg byt + So PA DSi AN Valable pour 2m, c’est-à-dire pour un nombre pair d’ordonnées. Pour (2m—1), c'est-à-dire pour un nombre impair d’ordonnées, lorsque z, — 0, on trouve la formule (SPÉCIALEMENT DE GAUSS) ETC. 25 2 — — nt @ En sr “1 rer ae : =e D. 4 ye 5 chy sy eae a TNA LE Iml + 1) + LG MPa la NS 17 an SG AE Dn ae Sas hr" aes Lye Se ee Ÿ| a — En een à OA ars: — 22,0 ne) bi DO aor ec 00 0 ae eee). ant An | + D Sn ni me D En = MIN... (44) : mme application des formules du récédent, nous Ÿ 20,: Co pplicat des f les d P lent posons le cas de #—3, c'est-à-dire qu’il faut calculer 2» — 6 7 F rn . ordonnées. De la Table 4 nous obtenons alors pour /e premier groupe de longueurs d’abscisses ze Ds | ES CE avs 0k et de (19) pour le premier terme de l’erreur Z, faisant partie de ce groupe BAP GRR LE 28h; af Rl Ar 0.0000 1633-6188 Aes tandis que, d’après (43), on trouve pour le premier terme de l'erreur, appartenant au second groupe de longueurs d’abscisses LÉO Sek Ga ER Sh ES CD SN ADS, Tere Rd DR Pre tad EHS 1de — [— 0.0000 1633 6188 — — 0.0000 2177 4424 u, < + 0.0000 2592 0664 a — — 0.0000 3565 5694 Ay; expression, qui devient la plus petite pour & — 1, #,=—0 et a; —— |, de sorte que les abscisses du second groupe sont Pi OIB Sa Da NE Re ee (45) En effet, on trouve pour ces longueurs de w I= + F=0.2466 1426 3490 (y_a +-y41) + 4-0.1569 3803 5364 (y_.+-y,5)+ 0.09644770 1146 (y_34+-¥43)— — 0.0000 0314 7065 4, + 0.0000 0313 4520 4, + + 0.0000 0214 2798 4,0 + 0.0000 0094 1791 Ay + GOO ARENDA et. (46) Si on compare cette formule à celle sub (32) on constate, que dans (46) non seulement le premier, mais aussi quelques-uns des termes de l’erreur # qui suivent immédiatement le premier terme sont plus petits que les termes homonymes dans (32). af” Leg vi 4 + . HE DR ae De (46) on trouve pour une ane: approximative | de ri ité 4 [ de ; at | Fe En | : 49.5 +2 ; 1 ; a ie ik à #6 I, = 01104 8303 S667 A Se contre L = 0NTT Geesa eB er kde (32), voir 35). ordonnées à calculer sera ‘Ib: (2m—1)= 5, tandis qu’ Cl a aussi exprimé en deux décimales, et que maintenant aussi ash De dans ce cas, la Table 4 donne pour le premier groupe ded gueurs d’abscisses B 4=0 , 4=027 et 4—045 et on obtient, d’après (44), pour le premier terme de l'erreur, qui fait partie du second groupe | Lt a) — 4G) + 2) +44 es Be | — 0.022, 11 55 De a), — 0.022, in CoS Ga) 1428 = [0.0000 1983 + + 0.0000 23625 a, — — 0.0000 57825 a ] Ay; expression qui devient la plus petite pour &æ, —— 1] et zon | par conséquent il faut prendre Dy 02 Oet 7, —— 0, 45, „an } Abstraction faite du signe, on trouve ici pour le premier terme = uf de lerreur # de la formule pour cinq ordonnées, 0.0000 0336 ae 3 et contre 0.0000 1988 4, pour celui qui appartient à la formule pour — a les longueurs d’abscisses en deux décimales empruntées à la Table 4. zn 8 La formule pour 7 pour les longueurs d’abscisses corrigées est | I= I, + E—0.2688 7768 7681 ya + B +0.2398 7744 5928 (y_s+-y40) + 0.1256 8371 0282 (yap) —0.0000 0336 4, +0.0000 0133 4, +-0.0000 0251 A+ = = +-0.0000 0140 4, -+0.0000 0056 4y,4+-0.0000 0019 Agtete. (47) NM De (47) on trouve pour une valeur approximative de Vintégrale [ da ” I= 0) D177 20 S65 IA pour cg ordonnées, tout comme plus haut pour six ordonnées. 1) Les formules pour Z, de la Table B ont été déduites de la facon ieee dans ce §-ci et dans le $ précédent. ') Voir la remarque dans la dernière phrase du § 18. (SPÉCIALEMENT DE GAUSS) ETC. 27 Ze . . . . . Determination de l'exactitude d'une valeur approximative. $ 21. Dans le calcul de la valeur approximative d'une intégrale définie, il est généralement nécessaire de savoir jusqu’à quelle figure (chiffre ou zéro) cette valeur est exacte. À cet effet, on peut recourir à deux formules, qui dans la Table 4 (ou dans la Table B) se suivent immédiatement. Nous représentons ces formules, pour les distinguer l’une de l’autre, par 1, et Z,; la dernière étant arrangée pour plus d'ordonnées que la première, est par conséquent plus exacte que l’autre, de sorte que, à la série ininterrompue de figures égales, comptées à partir du premier chiffre, que les résultats calculés d’après les deux formules pour Z, et Z, ont en commun, on peut conclure à une valeur pour I, qui est exacte jusqu’à la dernière de ces figures égales. Cette détermination de l’exactitude exige il est vrai le calcul de deux formules avec des ordonnées toutes différentes. C’est d’ailleurs la la voie qu’on doit suivre, lorsqu'on fait usage des formules de Gauss; pour les formules de la Table B, on peut exécuter un cal- cul plus commode. Notamment il est plus simple de calculer pour /, une des for- mules de la Table B et de rendre plus faciles les calculs pour /, en employant pour /, les mêmes ordonnées que pour Z, en adjoignant deux nouvelles. Si nous admettons que, dans la formule à développer pour /,, on attribue aux (m—1) abscisses z,, 2, %,..-@m-,, les mémes valeurs que dans la formule pour Z,, il faut alors encore détermi- ner les valeurs des m grandeurs 2 et l’abscisse inconnue 2,,, donc en tout (m-—+-1) inconnues. A cet effet nous pouvons dans (19) égaler à zéro les coefficients des premiers (m 4 1) termes de 77, chacun en particulier, et, des équations ainsi formées, résoudre les (+ 1) inconnues qui en même temps satisfont à ces (# + 1) équations. Nous avons 7 Rw en eea Ed Pi A = 2 he + Be iit == tas Fe = By? B — + „Te fie este, où be eee ver, else en ele rvanik ele deu (spite, 0. 0. je «ns © sn eene Le , 2m | 2m ZIN yee it eN 1 1\2m d'A Ti are Lo Lo V2 Lt. + CC + ln Lt, —— Tee) . Sidans (66)det (BO Tous -posons dE = 0 nis MES S| 1 \2p : 5 Ae anaes AE ' et wv, =d)”, on trouve de (66) le même groupe d'équations, que celui sub (48) et par conséquent pour ces équations vaut, d’apres (68) l'égalité PE RER AE AD AE Ve dE; RP: tende j tee ra pare 28 ETUDE SUR LES FORMULES 4 2m—2 1 2n—4 1 2m—6 el Re Iml 2) , SL re i rte) A en re Re 1.p p een CARE tie à So pd. -(— 1)" 48, mp et d’après la première équation sub (72) l'égalité er HD nm DP Sn EPT. ir bee 1 +(— DS La dernière égalité devient (voir (5 6), après substitution de. ‘s DE Oe | Di S, = ; + se m = Si. me ie ? etc. ce ey We a Mince = MECS, nt. ml ot ae tee 3 | 6 m-1. itt Si: mlm den 0 n—A.meL Bd war a 3 | d’où 1 PERS 1 4 B md yn Si. mana) SS eme Co + a en) m—1.m MAT à CPS 1 Di 4 Dn © za) # meee omar th ST LE en ek Gr REE be) PENN (= ys ‘ a En appliquant cette formule au cas, par exemple, de m = 4, lorsqu’ auparavant on pose (voir (45)) | 50.18 4, va Sno ete HD on trouve 2 =: Vy —=—— =. Cette valeur négative pour 2,’ montre, que s’il faut attribuer a 4, & et wv; les valeurs que nous venons de mentionner, il est impossible d’y joindre une quatrième qui puisse, avec les trois précédentes, satisfaire aux (#7 + 1) —5 équations sub (48). Pour rester maintenant aussi près que possible de la valeur cal- culée de x, on peut poser æ, — 0. Cette valeur de >, jointe aux longueurs prescrites des autres abscisses, par conséquent dx = O18 4, ps ORDRE rg ORO TEL conduite à la formule auxiliaire suivante, 1, = 0.2414 1627 5088 (y_, + ya) +- +-0.1584 6732 4625 (y_»+-y45) + 0.0960 7580 3242 (y_ tye) + + 0,0080 $119 4089 y, tandis que, sub (46) on a trouvé pour J, T, = 0.2466 1426 3490 (y_4 + ya) + + 0.1569 3803 5364 (y_.+-y4.)+ 0.0964 4770 1146 (y_3+-y43) St nous entendons, comme Sririine 4) par ,,correction’’ ou ,,corr.” ) STIRLING. Methodus Differentialis. Londres 1730. (SPÉCIALEMENT DE GAUSS) ETC. | 29 la différence entre la valeur approximative de lintégrale calculée pour le cas de 2m ou de (2m—1) ordonnées et la valeur plus exactement approximative, que l’on obtient en adjoignant au nombre des ordonnées deux ordonnées nouvelles, alors corr. == /,— J, et donc, dans le cas envisagé plus haut, corr.: = — 0.0051 9798 8402 (y_1 + ya) + ad 0015 2998 9261 (4-00. 0003 7189 7904 (y_sH-4 3) H- OOS EEN MNSD COMORES NE (50) De (46) on a trouvé pour une valeur approximative de | 10 Ass TEA De (50) on trouve corr.: = — 0.0000 0000 0000 53, d’où l’on peut conclure que la valeur trouvée pour Z, est exacte jusqu’à la onzième décimale inclusivement, ce qui est d’accord avec l’énoncé de la valeur approximative de ladite intégrale dans le $ 15. Les corrections qui ont été admises dans la Table B ont le sens donné ci-dessus à ,,corr.”: et ont été développées d’une façon iden- tique à celle que nous venons d'indiquer. Prenons encore la formule pour 5 ordonnées de cette Table. A la fin du $ 20, on a trouvé de cette formule pour une valeur da ST bz J =; 171 8808580974 tandis que l’on trouve de la formule pour ,,corr.”’ corr.: = — 0. 0000 0000 0001 3 d'où il résulte que la valeur calculée de /, est exacte jusqu’à la onzième décimale inclusivement et est plus exactement représentée par Li = 0 VERES3 035-0056" Calcul des valeurs de S,, S p.q? 718 Ee approximative de l’intégrale [ $ 22. Les longueurs d’abscisses ge exprimées seulement en deux déeimales et dont nous représentons les carrés par 2 2 2 2 CRE B QUE EEN UN (51) étant connues, il est facile, pour de petites valeurs de m, de cal- culer assez rapidement les valeurs correspondantes des grandeurs D, Sp. , ete., au moyen des sommes des produits deux a deux, en à trois, etc. desdits carrés; mais lorsque m est grand, ces calculs deviennent extrêmement compliqués à cause du grand nombre 30 ÉTUDE SUR LES FORMULES de produits, qui alors doivent être déterminés. Cependant, on peut donner aux expressions pour #,, etc. une autre forme, qui est quel- quefois plus commode. 1°. Nous représentons en général la somme des pt" puissanses des carrés sub (51), à l’exception des carrés + et a,?, par 27, et la somme des produits p à p de ces mêmes (m—2) carrés par Sr, alors les carrés sub (51), sauf les carrés z,* et 2°, sont les racines de léquation | fe) de NE © ne Oe Dans (52), les facteurs (e?—2z,°) et (2° — 2,7) ne se présentent pas; pour (52) on peut écrire aussi PAVE Bagi ee a gee” = er OR la dérivée de cette dernière équation est FE) = (m— DS (m—3B) Sr ITH dalle Sas MEO a eee (53) Le rapport entre f(z) et f(z) s’exprime par 1) en Al 1 1 Lina | M D CD 23 m fz) en Te donc ft) 2 2 —2, ae Aan TOR FANS D'ESSAI CES 2 En 2 2 SERRE de Et Se) Ho: dans Ja dernière équation les termes ~——, et ~——, ne se pré- —2 — 3, q 7 RCA sentent pas. Si l’on divise f(z’) par (2—2,7) on obtient Te’) 2\m—3 2\m—4. ÿ | 2\m—5 9 ae SE (z ) Ta MO (2 ) + Ss (2 ) me: 9 1 2 2 oe 2 — At gir ln Si dans cette dernière équation on remplace z,? successivement par chacune des (m—38) autres racines, on aura en additionnant tous les résultats, l’équation suivante, qui est identique à celle sub (53) vaN 2\n—3 | 4 2\m—A SG) = (m — 2) (2 — |(m — 2) St.g.r— Ha Ca | [A Y, 1 2 9 he tr — Sir Br Stair = NE 1 ) J.A. Serret. Cours d’Algébre Supérieure I. p. 111 et 377—379. (SPÉCIALEMENT DE GAUSS) ETC. 31 La comparaison de cette expression pour /'(2) avec celle sub (53), fournit les relations que nous donnons ici: [ar 3) Si .g.r == (mM PA 2) Sy, q. RE ’ (mm — 4) Soo. = (m — 2) Dla te a pr Ee q. FES A ee f (m — 5) 55,4, = (m — 2) Dogue ZE. r So gar À Por: St, ar Ear > et ainsi de suite. Après quelques réductions on obtient les équations suivantes, qui ont été données, pour la première fois, par Nrwron: Si qr 25 ee ; 7 | 2 D, one Bes Si 4 r — ’ ee Dotan es roa ies ne ’ mt: .(b4) ee Sn US Le Ses Sn ears Suisses tel) Eerd. 2°. Les carrés sub (51) sont les racines de léquation (2) = (2 — § (2Y tt... L(—D"S, —=0... (55) Si l’on isole des carrés sub (51) le carré x,” et qu’on mette Sla somme des carrés sub (51), excepté le carré +”, S,,—la somme des produits deux à deux de ces mêmes (m—l) carrés, et ainsi de suite. alors les (w—l) carrés restants sont les racines de l’équation (as MIS Si, CORRE + etc. En multipliant cette dernière équation avec (a — x), de. puisque ce produit est identique à l’équation f(z) sub (55), | (g + Ie terme du produit sera égal au (g + a terme de a par conséquent on a en général oh, ors m.p =) m—A. (ape De ee Vs shoes ane (5 6) 39 ÉTUDE SUR LES FORMULES SECTION II. Formules d'approximation lorsque la fonction sous le signe intégral ne peut pas être remplacé par une série de la première classe. $ 23. Les formules des Tables 4 et B ont été établies dans la supposition que la fonction sous le signe intégral peut être repré-_ senté par une série de la première classe. Si ces formules sont appliquées à une intégrale dont la fonction ne peut pas être rem- placée par une série de puissances complète, les calculs n’attein- dront pas à la plus grande exactitude accessible. Si auparavant on découvre une lacune dans la série, on peut en tenir compte dans le cas où l’on veut connaître les abscisses les plus avantageuses pour l'intégrale, dont on veut calculer une valeur approximative. Pour la question qui nous occupera dans cette Section, nous introduirons (voir l'alinéa final du $ 8) de nouveau les lignes 4‘ X et A'Y' comme axes des coordonnées et nous considérerons les inté- grales de a” F(x) dx et se de entre les limites 0 et 1. Les at be lettres d et 4 représentent des nombres entiers plus grands que 1. Dans ces deux fonctions, on voit immédiatement : Dans la première, que la ligne représentée par l'équation y=" Fe) passe par le point d’intersection A4! des axes A'X et A'Y' et que par conséquent dans (5) Z—0, et, dans la deuxième, que, dans la série, qui peut remplacer te a il manque plusieurs termes, 3 a + be et qu’ainsi ces deux séries n’appartiennent pas à celles de la première classe. § 24. 1°. Lorsque dans #° F(a), la fonction F(x) elle-même peut être représentée par la série de la première classe FX) = a5 + da + a2? + aa’ + ete., alors la valeur de liitégrale de #'F(x)dz, entre les limites 0 et 1, est 1 ARS 1 1 1 [= a” F(a) dx = 75% + te Ge aram su a, + etc. . 1000 0 Si Von représente les coordonnées des xz points connus de la (SPÉCIALEMENT DE GAUSS) ETC. 39 vraie ligne limite de la figure par (@,, 1), (42, yo), (#3, #3) … (a, Yn) alors une aire approximative de la figure est 1 == Bate Betr LeVert 17 ee oe (58) et J=/,4+ #. Si l’on substitue dans (58) les 2 ordonnées calculées, alors on obtient T= R; (gat Ha, an + agar? + azar? + ete.) + + Boats + 42 + A, ty? + ayy" ** + ete.) + + Bs (ay 03° + a, af + a, 2 + agg + ete.) + Sere. aile) wee sa 6 valve! l'a eres vo Mon sip ea Vets cer ae Oe) le le eee ues sa. stee a5 Li, (@ zen ae ds Bas © a ay ae aie a: Pree a etc.) mer on À (aor? Ti, == gn Rh, + gee de + NE te + we, FTP Le). La dernière équation soustraite de (57) procure l’équation EN I _ » dtp d+p 1+] \ E> lagen i ae? By op ag Bg en ey ey) dans laquelle il faut poser successivement » —0, 1, 2, etc. Si dans la dernière expression on égale à zéro chacun des 2 premiers termes, alors les valeurs de > de > a 5 a | ai! | | rt 1 d'A Fi >, Rh, da Rs ere Vy, ER — d+ | d+1 , HA ,d+1 p , d+1 4 1 1? ay he +a 2 Vie +- sol >, dr —— d+2 De c € € 4 ait R, } 2 ele, | ar tee Be + PME | bee bi aS FES HNE (59) d+2n—1 , d+2n— | 1 d+2n—1 p d+2n—1 Merit | AA Kya, hy +2: Rat. + Vy Bn | donnent les longueurs d’abcisses pour les formules d’approximation les plus exactes pour ordonnées. Si dans (66) et (69) nous posons 1 : C—0,£—1,0—n et #,— gp On trouve facilement que dans (59) les grandeurs z sont les racines de l’équation a” — Sat + ST SSL... HS, — 0 tandis Qu'en posant dans (72) et (13) r=dd-r tl; v=”, ASL, Den nn D= on ebtiert n d+n ] A ed ne S, —- nie Sige) Ju 8 aire rs se detre els se, ee ee nee. eis Victen 18 Verhand. Kon. Akad. v. Wetensch. 4e Sectie. Dl. XI. No. 6. irs 34 ETUDE SUR LES FORMULES CONS A EEN REA da TON 1 d+] N a EE kn *Dy_4. É n dta-+1 On trouve, par exemple, pour d=1 et n— 4 les grandeurs # de l’équation suivante at —8,.a°+ 8a — Sa + 8, — 0 où S,= 4.32 — 20 S,= 2.2.3 — 4% et kit — 126 done ot — 20 a 1. 3 a? 197+ =0 d’où 2, = 0.1397 ra alae are oats a ae (60) a — 0.7231 5699 a — 0.9428 9580 Remarquons ici que la série pour (we) a été supposée de la première classe, ce qui n’est pas toujours facile à constater, il s’en faut, et si, parmi les premiers termes de cette série, il en manquait un ou plusieurs, alors les valeurs sub (60) ne désignent non plus les longueurs d’abscisses les plus favorables. Pour ¢d— 1 et u = 20m tore 2 —= 0.2123 4054 2, 0.5905 2314 EEN (61) et 2091144200 Pour d= 1 et n=: P == 0.3090) 5108 et 2 — 0.8448 4897 ef. pour d= let." 2. Remplacons [expression y = /, dans laquelle on suppose a + be |, par l’équation suivante (SPÉCIALEMENT DE GAUSS) ETC. 39 dans laquelle, des signes doubles, les signes supérieurs vont ensemble, ainsi que les signes inférieurs. 1 EE js Ae 4 Es Nous posons = Go» ae a U ida, RE “ox Beten da (63) devient ya + a + a,.2 + as. + etc. Si les abscisses des x points connus de la vraie ligne limite de ia heure sont 27,4; #3; +. .dn, alors Li = By (ao + ae + aay” + az + ete.) + + By (dy + «af + aa + az ae" + etc.) + + Bs (ay + a, v3" + aya” + a," + etc.) + ay fi, (ao 3 ay ne = do des ein a: a ae etc.) TE =( R, 4- fy + Barid B) + i ly ey Had ee. 2e de he de + (a, By + 2 Ro + a; R; Sets By Lig) | — et ainsi de suite. | i La dernière égalité ayant été soustraite d 1e] he \ dernière égalité ayant été soustraite de FT ARTE notamment 1 to ag Spee EL LIRE cree, donne, pour le calcul des x longueurs d’abscisses, les équations suivantes Be Ce CE OR pe eee a," Ky + to Be + Ga Berle el an Bey, = k+l De - Spay zit KE 188 a" Ry + By Ly En nm a By, = 2k +1 vus = sine ue 8) OPE w en a” er, er ce; re, © eee once Se 6 mn eo) en ae Jel” eee "ve, (ss le? ee ler Dean "bn eee) Ole Ore Ow ee See) ne ee È — der 5 = Ge 2 — Ye { 2k— Vie j Pe a le ee tie a tr een POON as =. (22 — IA + ] Pour 4—=%et:#—3, par exemple, om-trouve des longueurs d’abseisses de léquation (a) — 8, (a7) + 5, (2°) — 83 = 0 où KF 3% 36 ÉTUDE SUR LES FORMULES | 2 Fy ch pe QU =d PST es eee Ba, We St Dy ne nt Le a DN tn) et Sz 3.4.77 =337> par conséquent æ = 0.2386 1920 | ce 2 == 066 12 OKs 4s ree (64) et 23 — 0.9324 6953 | ee SS Conclusion. ae de pe . Ar + SR 4 A au + ke 2 § 25. Les distances des pieds des trois ordonnées pour (2m — 1) —=3 de la Table 4, calculées à partir du point 4', comportent =—=0.5—0.8872 9833 — 0.1127 0167 De 0.5 GO : et 23 — 0.5 + 0.3872 9833 — 0.8872 9833 € ~ Si Pon compare ces distances avec les longueurs des abscisses = sub (61) et (64) et ces longueurs-ci entre elles, on constate, que _ on Ee toute lacune dans le commencement de la série, qui peut rempla- cer la fonction sous le signe intégral, peut avoir une très grande influence sur les longueurs des abscisses les plus favorables. C’est ainsi que, par exemple, pour x — 3, l’abscisse à, dans (64) est plus de deux fois aussi grande que celle dans (65). Si l’on compare ensuite les 9»— 4 longueurs d’abscisses sub (60) avec celles correspondantes de la Table 4, celles-ci étant comptées à partir du point 4', notamment 3 a = 0.0694 3184, 2 — 0.3300 0948, as — 0.6699 9052, et a, —0.9305 6816, on voit, que lorsqu'on applique les longueurs d’après Gauss à des fonctions dont les séries ne sont pas de la première classe, les 14 ou 15 dernières des 16 décimales, dans lesquelles æ est exprimé dans la Table 4 n’ont absolument aucune valeur. La comparaison des longueurs d’abscisses sub (62) avec celles correspondantes de la Table 4 donne lieu à une remarque analogue. De ces différents faits on peut déduire, que les formules de la Table A ne peuvent être employées avec avantage, que pour un nombre restreint de fonctions, notamment pour celles dont on sait avec certitude qu’elles peuvent être remplacées par des séries de ‘(SPÉCIALEMENT DE GAUSS) ETC. 37 la première classe; pour toutes les autres il faudra parvenir à Pexactitude dans le calcul de la valeur de l’intégrale en prenant un nombre considérable d’ordonnées. Il en résulte, en rapport avec les observations dans les §§ 12, 16 et 21, qu’en général, pour le calcul de la valeur approximative . d’une intégrale .définie, les formules de la Table B méritent d’être préférées à celles de la Table 4. SC THOM RE Développement de formules sur lesquelles reposent celles destinées au calcul d'une intégrale définie. $ 26. Des (cv) équations *) ; 7 star op TR, + ODS SE da LR HEC qe Bory ep y= ? (66) 0 8: Re ol" Oy Oy (vaio et 8 8 eter se. C10... 8. Of eh eerie Te” ets ae uses © esa ot pts sie vs fe ey amie eo 9) oF 86)" 18: e+u—1)k-+d , (Ctv—1)k+d , (etu-—Mk+d > c+v—1)k+d Es ays TR ay rd Rot gs’ TER gt... La Ball où e, v, d et & représentent des nombres positifs arbitraires et les (cv) grandeurs w, de mème les nombres 2 sont connus et les nombres À sont inconnus, déterminer la valeur d’un ZX arbitraire ®). Solution. Remarquons que toutes les (c +) grandeurs + se présentent tout à fait de la même façon dans les équations sub (66). C’est pourquoi il ne faut pas résoudre immédiatement chaque LR séparément, mais il faut tout d’abord tacher de déterminer les coefficients de l’équation dont les (cv) grandeurs 2" sont les racines. C’est à cet effet que nous introduisons les notations suivantes: S,,—"la}somme des puissances mm. 44 à l'exception des”, Sy» == la somme des produits deux à deux de ces mêmes (¢-+-v—1) puissances, S;,== la somme des produits trois à trois de ces mêmes (c+v—1) puissances, et ainsi de suite. Sy 49 —le produit (continu) de ces mêmes (c+v— 1) puissances. *) Ici la somme (c + v) pourrait être remplacée par une seule lettre; cependant il est préférable, à cause de la clarté et en rapport avec la solution des équations sub (69) et avec les formules qui doivent en être déduites, de garder la somme (c+ v). *) Voir Mémoires de l'Académie de Berlin. Recherches par M. DE LA GRANGE. 1775, pag. 183 et 1792, pag. 247. 38 ÉTUDE SUR LES FORMULES L k Les puissances @,",...a@,,,", à l'exception de z,/, racines de l’équation | Ik (C+v—2)k „(cv —3)x cr ze Tee iy Ses + 82 p-@ ea So4 yt p=. sont alors les Nous représentons aussi le premier membre de cette équation par _ * té k I ky (ook k Ik PAT a cen Pp (2) = (2° — 2) 2) ND 0 NO S dans ce produit, il est entendu que le facteur (e"—z,") ne se pré- sente pas. | Pour résoudre les équations sub (66), nous les additionnons, après les avoir multipliées, la dernière par + 1, l’avant-dernière par —#S,,, la précédente par -+- S,,,, et ainsi de suite, et enfin la première par (—1) T8, 4 Si l’on réunit les termes avec le même ZR, on obtient (nn (e+v—1)k+d , (C+u—2)k+d (c+u—3)k+d ___ __y\etv—-t PRE hy, EA 8,24 +02. 24 + ( 1) : Sis vy CARE Mg ri == à VA l = Le el EA aen 1)k+d —S, ver" Moke +8,27 3)k+d EU Se ef 4 (C+v—1)k+d , (c+v—2)k+d , (etu—3)k+d c+v—1 DA sae +R; (es — V4 pts +823 se Healy Scute EN as sje a en © a + je ee a © a aa se €. lw 2 0e u ne se tes le ete lere Fer ae ve su eten se en arseen 78 xg SEN ee (c+tv-1)k+d__Q £ (c+v—2)k+d | ; (c+v—3)k+d __ __ 1 \c+v—1 A ee +hoslten Sateen 4 Does RP 1) Sos Very = en cHv—1 te Qu ore : Were + So.) . (PRE er ave te Cr Slik | Le premier membre de cette équation peut être également repré- senté par Le. Pp (az, = Fi. Pp (2,°) Bo” = fis. Pp (23°) Ae we se nn “Pp CI ue) Bn où, en vertu de la supposition au sujet de g,(2") sub (67), les coefficients de toutes les grandeurs &, excepté le coefficient de la grandeur #,,, sont égaux à zéro, par conséquent on obtient, après avoir divisé par le coefficient de Z,: | / c+v—1 be Uociy_1 ne : UBE Son Oje STe ts + Ge) " Sv. PTT p (C+Hv-—1)htd > (C+u—2)k +d c+u—3)k+d cu di Uy rr ES, nS ) Be je +(—1) Seta eee Dans les applications, que dans cet exposé, nous faisons du groupe d'équations sub (66) on a toujours 4<#% notamment 2,—=0 (voir la note a la p.- 8) (SPÉCIALEMENT DE GAUSS) ETC. 39 1 ‘ | 1° R+ Lye Pot. t gr R,+...+ GA fig y=Up 5 … k+d k+d k+d f , k : aT? R+ 2 TS Rot... + pT | ie Bte, $ 2k+d ok 1 7 ú 2k soe 6 aX, uy R+ Vs au Rote. + U, AE Pik Vov jens Berv=Uo 2 ( 9) en 8 Oe. (en en Ve € elle, Ce sut de el seule ee eeen se © 0: a € ne ps le wet an ©) a a. UO he se" de jee en pile: © 21) Hd 2u—1)k ae ne Rige Detd Bt 7 Core De A tes DU Pl te EREN dans lesquelles les (e+ 2v) grandeurs #, de même que les c gran- deurs 24, æ, @3,...v sont connues et dans lesquelles les autres v grandeurs a, ainsi que toutes les (c +) grandeurs # sont incon- nues, résoudre les (e + 2v) inconnues. Solution. Du $ précédent est apparu comment on peut trouver les valeurs de Æ au moyen des premières (e + v) équations de (69), quand toutes les grandeurs a, 2, %3,...% 4, sont connues. Pour pouvoir exprimer ici À, dans les données, il nous importe done d’abord de définir les v grandeurs inconnues 2. _ À cet effet, nous introduisons les notations suivantes: #,—la somme de toutes les (e+ v) grandeurs a”, &°… tors, S,—la somme de leurs produits deux à deux, (70) et ainsi de suite, | :, —=leurs produit (continu) | S c Les (cv) grandeurs (a) sont alors les racines de l’équation gere 5 EE LE, gere, ETS, ,—0...(71) Le premier membre de cette égalité peut être également nn par fe) où (a! — a," (a — 2!) da) — 24") 9) Qu’on prenne maintenant, des (e+ 2v) équations, sub (69), (e+-v-++1) qui se suivent immédiatement (ce qui peut se faire de v manières) et qu’on les additionne, après les avoir multipliées, la dernière par — — 1, l’avant-dernière par — $,, celle qui précède par + &, et ainsi de suite, et. enfin la première par (— 1)°*’S,_,.- Réunissant les termes avec le même A, on obtient | | Po an ee Saf (ctvu—1)k +8; Pi (e+v—2)k “eg Pass an | : (2) Re no — 82 (ctyu—1)k + Sra, (e+u—2)k eS Iets a | A a4 À VU + Re eset Sgt ap an =H Bl ver OL) Shy on [ee Kee Sne te) "a ete ren lant ey penta, ms BEPA et sn en Mes) Te. ee eet ie. eiser elst ie, ss Va tale a sm et ele ese 16 IB are lr TRS Aer Dk SS ae Ae +v—2)k + (— PSs, | 22 Ti Uctytz FA Si : Cot yor = Ss 2 ni: on zin == RES CH" U. a) Le mie de cette expression avec celle pour dp (x } sub (67) est désigné par fak) = (a* — 2). 0, (xt). 40 ÉTUDE SUR LES FORMULES Le premier membre de cette égalité peut être | également repré- senté par | : se B Raft fader fad) + Baas! Jelte Route Ja Puisque, dans cette expression, à cause de la supposition qui a été aa faite au sujet de f(z"), tous les termes sont égaux à zéro, nous on | u Ucty+z ore Duties = A hae pri ee == ees, ctu: 1, = 0. Pour z on peut poser successivement les valeurs 0, 1, 2,.. (v— 1) de sorte qu’on obtient les v équations suivantes: Ue — Ue HOU 02 — 8. Uc+ry—Z +... Steet tein = Ue+y+1 TS ee eat ae Ucty— 2 Se (DS c+v* U == Usps Ot Uervrt À Dodo en begr ee DES ou, = En Uorou À. HD Sopa DS # Etablissons encore S, — la somme des e grandeurs connues 2", 2", 2 ",...@,", S. — la somme de leurs produits deux à deux; etc. et S,=— la somme des » grandeurs inconnues 2,14", Gero sedens S:—la somme de leurs produits deux à deux; et ainsi de suite. Si nous substituons ensuite, dans (72) a #,, SS, Ss, etc. ses expressions dans S,, Ss, 838, 193 95, 8;,.,.8,, alors nous obtenons, après quelque réduction DE oge —St-Wevar Sologne Satis Te 4, >. — Moon Serve Sotern 3 Stern Heet (ID m4 SE id (te+y—2 Te Nu loin ae Sz opus +...--(—1)’. PAL LS, per ose + se ee » ee es © @, 06 p.18 jp. 8 ee): as © a © ae eve ee due) ee an ea ea lu Co D eee =i Dia, Su Ue Se Us Ne 3 is ad lo |S, Fe | Woo — Stor +S» Uoty—4 ——S3-Ue4y—9 esin), UH | = ae (eye mal (UE +S» Wes y—o = Ng Ue 3 an U, US, ae: aed ev — St Ure +8 Uc+y3 TS Ue vt inte ek U 1 „eee ess ee e 0e ee se + es + © ee a ss se ° es raa see a ee a ne FOO ter rc GO Get BS OS Phare ide Vv see 9 6 0 . © + vs ee 6 e ete. © in fe. ee Col sue ere" se we eee eene en ee | bet 2y—4 S4 Uoron2 Solos 3 — Seller 1 500 + (— 1 Ne Var ae ns Weren Sy. Ube +.9y-—3-t- So. p41 9y—4—Sg-Ue4-2y-5-- eee + (—1 ASE. S, + = ma | U, +23 S; Ue 49-4" SoU p4.9y-5-—S3 Waem stal +(—1)° Urs, oa) S, ——- TGD ero SU Soil tety—3 — Saber teen A A (SPÉCIALEMENT DE GAUSS) ELC. Al De ce groupe de v équations linéaires, on peut résoudre les v grandeurs §,, S,,... S,, qui en sont les seules inconnues. Les v inconnues +, sont alors les racines de l’équation SEN + Sa) STE HIS, = 0. Dans les équations sub (69), toutes les lettres w représentent des nombres arbitraires. Cependant dans chacune des formules d’approxi- mation, qui seront développées dans ce travail, il existe un certain rapport entre ces nombres. Si l’on tient compte de ce rapport, alors, comme il apparaîtra dans le prochain $, la relation des gran- deurs #' devient beaucoup plus simple. $ 28. Des v équations 1 | i 1 | 1 | B Ee A jé ee CDS Br | —A —2A RE SRE fe | I I a ] Ds 1 | D TG are JA SA Le se Re ) : (v Da v > | i En | R Il ie E rt(v—] JA 00 er (02 1 En r-+-(v—3)A AK Seer ) (4 DA AT eN +(—1) Ee hae ED ——; B= 0; | dans lesquelles 7, A et v sont connus et dans lesquelles on peut attribuer à ££) une valeur arbitraire, résoudre les v grandeurs incon- Men, Sn: Us, bah Solution. On pose (—1) Z,—$@, et on transfère les termes avec B, du premier au second membre, alors on a: B ne Rg ode lead es r—A f, ZA By, | r—3A By p(t Aa DE Vesta EN Ne ONE bon fes die mr y lB, i eta | B, r By jeg’ MT RL ou TBE A Pi MERE me Sh, Il En MGS ASA CS ae wy aL Ve. ee ee amen seroma) alte sheet Tej ta sole re mie we ee Le feux SPA NO CR IG „ale tie ey nl le, Sev merten Ke Gein: oise rd ker ley kee Bee 8 ets Ole es rie r+(v—2)a By oer at Mi ; 1 (2 En Ein 1 LE EE Les fractions ey sont proportionnelles aux déterminants. 0 Po de la matrice des coefficients en remplaçant la pene colonne: 8 colonne des valeurs connues et les signes négatifs par ae Alors a Ri correspond Kp | Il i i Il 1 if PATA 'r—(t—-1)A rr—(t+lhar—(tt na Lard | ] if 1 rr—A r—-(t—2)A r+-a r—tar—(F+ 1a ro DA a ee se fe ee ae, me Vel © mle ow ballet ate eier eme fe als ee ee et fe ee ea er NN CR A : ’ 1 1 1 1 on rdw BA rL(v—3)a | r—(t—v)A ro DA le 6 Ie I TES Fee AG ey Pr, Avec 5 0 | oA 1 l 1 es pA p= OA PE DA ae ee r—0A Es a oa Tl Len I ju 1 pr A rlr DA Er ITA ES EN r+(v—2)a 7 —(0—8)A rl) r—lt—(0— 1] a ro DA (i 1 % | 5 rlr En Qu'on soustraie, dans le premier déterminant, la (¢-- D" co- lonne et, dans le deuxième déterminant, la (4— 1)°"° colonne, de toutes les autres colonnes; les éléments deviennent alors des frac- tions, celles qui se trouvent dans la même colonne ont toutes les numérateurs égaux, celles qui se trouvent dans la même ligne _ horizontale ont toutes un facteur commun dans le dénominateur. __ On peut ainsi, du premier déterminant, isoler le facteur Gd DO ee EERDE (SPÉCIALEMENT DE GAUSS) ETC. 43 et, du deuxième déterminant, le facteur EE pl [ry —ta].[r—(¢—1)A].. .[r —(— 0 + Ja] Dans ces deux cas, le déterminant restant est le méme. Le rapport des grandeurs B, et B,,, est par conséquent égal à celui des facteurs isolés. Après simplification, 1l reste Ber NN Si dans cette expression on remplace successivement ¢ par 0, 1, 2, 83,...(v— 1), alors on obtient Vie AN v | B == a a 5 B > bordeel 1 Bi eee belies Oe Va) ee 27 9 Bi re 9 i Al 4 | PA vu — 2 prier en EE er ES ei 3 r+ (0 — 3)A 3 8 | oe dd ae NE eee | vl | — 9 vel eee | DNS ee SE 4 Uv TABLE A. Nombre d’ordonnées. [SL ~ 0. 10. & = 0,2886 7513 4594 8129. [= 5 ( nae ee) E = 0,0055 5556 4, + 0,0016 5259 4, +..... == d = 0,3872 9833 4620 7414. Li 0,4 71 + 0,27 (ya + 942) H = 0,0008 6714 4 + 0,0001 6278 AH æi— 0,1699 9052 1792 4281, do = 0,4805 6815 5797 0263.: L—0,3 3260 7257 7431 2731 (y_ B= 0,0000 2268 4, +. ie 1+ 94) 40,1739 2742 2568 7269 DES re) 20; do = 01,2692 3465 5052 8415, #3 = 0,4530 8992 2969 3320. T, = 0,284 y, + 0,2393 1433 5249 6832 (y_» + y42) + 0,1184 6344 2528 0945 (y_3 + 94) BE — 0,0000 0143 1549 4,,+-..... EN 1193 0959 3041 5985, & = 0,3806 0469 3233 1323, æ3 = 0,4662 3475 7101 5760. (nl) Lo 0,2029 2257 5688 6986, #3 = 0,38707 6559 2799 6972, EN 5895 6171 3793. Di 0,0917 1732 1247 § 8249, Ly = 0,2627 6620 4958 1645, da = 0,3983 3323 8706 8134, 2, = 0,4801 4492 8248 7681. 2 = 0, do = 0,1621 2671 1701 9045, #3 = 0,3066 8571 6350 2952, ®, = 0,4180 1555 3663 3179, | @ = 0,4840 8011 9753 8131. = 00 0744 3716 9490 8156, 9769 7064 6236, = 0.3397 0478 4149 5122, a = 0,4325 3168 3344 4923, 2s = 0,4869 5326 4258 5859. I, = 0,2089 7959 1836 7347 y al 0.1909 1502 5252 = 0,2339 5696 7286 8455 (y_, + yn) + 0,1803 8078 6524 0693 (y_. + ya.) + + 0,0856 6224 6189 5852 (y_3 + 443). BL = 0.0000 0009 0097 ie = ats 5595 ys 7 Ia) + + 0,1398 5269 5744 6383 (y_, + gas) + 0,0647 4248 3084 4348 (y_, + 44). B =0.0000 0000 5660 4,, +..... 1 —0.1813 4189 1689 1810 (y +- 0,1111 9051 7226 6872 (y HB = 0,0000 0000 0355 4 +... Hy) + 0.1568 5332 2938 9436 (y_» + 942) + 3 t+ 943) + 0,0506 1426 8145 1881 (y_, + ya). 1,=0.1651 1967 7500 6324 y, + 0.1561 7353 8520 0001 (y_5 Hy) + + 0,1303 0534 8201 4678 (y_s + 443) + 0,0903 2408 0347 4287 (y_, + yu) + + 0.0406 3719 4180 7872 (y_, + 445). E = 0,0000 0000 0022 Ais + Th =D AGN “2357 3764 a Dee ya) + 0,1346 3335 9654 9982 (y_» + ys) + + 0,1095 4818 1257 9910 (y_ + y43) + 0,0747 + 0.0333 3567 2154 3441 (y_s + 745). B — 0,0000 0000 0001 3950 4 +..... 2567 4575 2908 (ya + y44) + TABLE B. Nombre d’ordonnées à mesurer. a, — 0), T, = 0,4521 1483 6730 2213 y, + 0,2739 4258 1634 8893 (y_» + yy). 3. 2, — 0539. HF = —0,0001 1500 4, + 0,0003 0427 A, + 0,0001 4080 4,--..... TiO 05 corr: = — 11,5055 8842 8665 y, — 0,0037 8473 3036 (y_. + yy.) + 5,7565 7894 7368 (y_3-+- 743). DE 23 —10 01 DO 020; I, = 0,2688 7768 7681 1064 8453 y, + 0,2398 7744 5927 5115 0315 (y_» + y49) + 0,1256 5371 0231 9352 5459 (y_3 + y). 5 2, = 0.45 E = — 0,0000 0335 7143 4, + 0,0000 0133 1728 4 + 0,0000 0250 8965 Aij + 0,0000 0140 1848 4, + 0,0000 0055 4, +... mil gj 00) corr: = 0,0009 8087 5340 y, — 0.0010 0915 1447 (y_» + v4) + 0,0012 9362 7501 (y_3 +443) — 0,0007 7496 3725 (y_, + y_,). ile ON UE Mn ION — 05 J, = 0,2097 5447 0980 6416 9640 0461 y, + 0,1824 7432 9387 6422 8742 9068 (y_0 + yo) + ie LV dO 2, OAN + 0,1350 7607 6969 2986 6560 7711 (y_3 +743) + 0,0745 7235 8152 7381 9876 2990 (y_, + y). B =0,0000 0005 5192 4, + 0,0000 0019 5652 A, + 0,0000 0014 0693 4» + 0,0000 0007 1315 4, + 0,0000 0003 4% + UO —10 14; : | == 0,31, WOR + 0,1744 8969 5686 6362 9922 8778 4231 (Y_» = 749) +. 0,1485 6536 6367 4303 3946 3273 5653 (Y_3 7.) + 0. Lo == OAS. +. 0,0747 0874 5459 9309 2982 1885 8212 (y_, 7) + 0,0468 3556 5209 0569 5989 0544 1020 (y 7) F = 0,0000 0000 5188 4,5 + 0,0000 0003 2005 Aÿ 0,0000 0002 5372 4, +... .. OO 1,— 0,0828 1895 6609 8793 2657 4313 4813 8734 3572 y, + a= 0,25, 14 =—=0.37,| + 0.2074 9069 2760 1711 3256 6405 2835 9730 1815 (y_s Fy) + 11. = 0.44, = 0.49.| + 0.2624 6970 4997 3907 5235 9300 5662 0412 8537 (y_a + ya) + + 0.1829 1472 5915 8427 5919 8224 1585 0777 1593 (y 4 Lys) + +. 0.1287 3949 7353 4044 9347 3558 1538 7965 5414 (y_,+-y,,) + + 0.0577 4746 5753 4322 0925 3884 8750 3645 5497 (y_ Eye). EB =0,0000 0098 1552 4, +..... B. P. Moors. Etude sur les formules (spécialement de Gauss) servant a calculer des valeurs aproximatives d’une intégrale définie. Verhand. Kon. Akad. v. Wetensch. (le Sectie) Dl. XI. N°. 6. e tcénces. LIBRARY ni it il | GEDRUKT BIJ —o JOH. ENSCHEDE EN ZONEN o— HAARLEM