it i ih ai 4 % FOR DEIERPEOPLBE FOR EDVCATION FOR SCIENCE LIBRARY OF THE AMERICAN MUSEUM OF NATURAL HISTORY KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN -- TE AMSTERDAM -:- PROCEEDINGS: OF TEE = OC HONsOF SCIENCES 5.66( A : VOLUME XXII — (28> PART) — = (610) — a JOHANNES MULLER :—: AMSTERDAM = : SEPTEMBER 1920 : AH-QAYUZ Ob Varel. 15 (Translated from: Verslagen van de Gewone Vergaderingen der Wis- en Natuurkundige Afdeeling Dl. XXVII and XXVIII). ON ENS. en ce en EN en eer 688 er ie, Oad. Ale rade re ee LS KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN TE AMSTERDAM. ime CEE DINGS VOLUME XXII N°. 6. President: Prof. H. A. LORENTZ. Secretary: Prof. P. ZEEMAN. (Translated from: “Verslag van de gewone vergaderingen der Wis- en Natuurkundige Afdeeling,” Vol. XXVII and XXVIII). CONTENTS, JAN DE VRIES: “On an involution among the rays of space”, p. 478. JAN DE VRIES: “On an involution among the rays of space, which is determined by two REYE congruences”, p. 482. G. SCHAAKE: “On an involution among the rays of space”. (Communicated by Prof. JAN DE VRIES), p. 488. é JAN DE VRIES: “On an involution among the rays of space, which is determined by a bilinear congruence of twisted elliptical quartics”, p. 493. A. W. K. DE JONG: “The trimorphism of allocinnamic acid”. (Communicated by Prof. P. VAN ROMBURGH), p. 497. A. W. K. DE JONG: “The Truxillic Acids”. (Communicated by Prof. P. VAN ROMBURGH), p. 509. P. ZEEMAN and Miss A. SNETHLAGE: “The Propagation of Light in Moving, Transparent, Solid Substances. Il. Measurements on the FIZEAU-Effect in Quartz”, p. 512. (With one plate). M. J. SMIT: “On some nitro-derivatives of dimethylaniline”. (Communicated by Prof. P. VAN ROMBURGH), p. 523. P. H. J. HOENEN S. J.: “Pressure- and temperature-coefficients, volume- and heat-effects in bivariant systems”. (Communicated by Prof. F. A. H. SCHREINEMAKERS), p. 526. P. H. J. HOENEN S. J.: “Extension of the law of Braun”. (Communicated by Prof. F. A. H. SCHREI- NEMAKERS), p. 531. D. COSTER: “On the rings of connecting-electrons in BRAGG’s model of the diamondcrystal” (Com- municated by Prof. H. A. LORENTZ), p. 536. F. A. H. SCHREINEMAKERS: “In-, mono- and divariant equilibria’, XX, p. 542. W. KAPTEYN: “On a formula of SYLVESTER”, p. 555. A. J. P. VAN DEN BROEK: “About the influence of radio-active elements on the development”. Communicated by Prof. H. ZWAARDEMAKER), p. 563. (With one plate), B. VON KeréwjartTé: “Ueber die endlichen topologischen Gruppen der Kugelfliche’. (Communi- cated by Prof. L. E. J. BROUWER), p. 568. NIL RATAN DHAR: “Catalysis. Part. VII. Notes on Catalysis in heterogeneous systems”. (Com- municated by Prof. ERNST COHEN), p. 570. NiL RATAN DHAR: “Notes on Cobaltammines”. (Communicated by Prof. ERNST COHEN), p. 576, C. H. VAN OS: “An involution of pairs of points and an involution of pairs of rays in space”. (Communicated by Prof. JAN DE VRIES), p. 580. J. A. SCHOUTEN and D. J. STRUIK: “On n-tuple orthogonal systems of n—I-dimensional manifolds in a general manifold of n dimensions”. (Communicated by Prof, J. CARDINAAL), p. 504, L. RUTTEN: “On Foraminifera-bearing Rocks from the Basin of the Lorentz River (Southwest Dutch New-Guinea)”, p. 606. H. GROOT: “On the Effective Temperature of the Sun” (2nd Communication). (Communicated by Prof. H. W. JULIUS), p. 615, C. B. Biezeno: “Graphical determination of the moments of transition of an elastically supported, statically undeterminate beam”. 1. (Communicated by Prof, J. CARDINAAL), p. 621. Proceedings Royal Acad. Amsterdam. Vol. X XI Mathematics. — “On an involution among the rays of space”. By Prof. Jan pe Vrins. (Communicated at the meeting of September 29, 1918). 1. Among the rays of space four arbitrary plane pencils of rays determine an involution of pairs of rays; each pair consists of the two transversals ¢,t’ of four rays a,b,c,d appertaining one to each of the four pencils. The pencil (a) we shall also denote by (4,«); A is the vertex, « the plane of (a). Similarly the other pencils are denoted by (B,8), (Cy), (D,9). A straight line ¢ determines four rays a,6,c,d, which, in general, have still another transversal ¢’. If «,b‚c,d appertain to a quadratic regulus, then each line of the complementary, regulus is conjugated to every other line of this complementary regulus. If ¢ describes the pencil (Yv), ¢’ engenders a ruled surface which we shall denote by (¢’)", where « is the degree of this surface. By the rays of (77) the four pencils (7), (0), (c), (d) are rendered projective. 2. We now suppose that in some way a projective correspondence is established between these pencils and consider the ruled surface T engendered by the transversals 4! of four corresponding rays. From A the pencils of rays (6), (c), (d) are projected by three projective pencils of planes; hence three transversals of corresponding rays 6,c,d exist which intersect the corresponding ray a at A, so that 4,B,C,D are triple points of 7. Similarly «,g,y,d are triple tangent planes, since they contain three lines /. The intersection of 7’ and « thus consists of three straight lines and a curve. which has a triple point at A. Since every ray a contains the points of transit of a pair of transversals ft’, the said curve is of the fifth degree. Hence T'is a ruled surface of the eighth degree. Hight tangents of the curve «°‚ which 7’ has in common with a, pass through A; it follows from this that 7’ contains eight rays t, which coincide each with its corresponding ray ¢’. 3. If ¢ is made to describe the pencil of lines (7x), the ruled surface 7’ breaks up into the pencil (¢) and a ruled surface (¢’)’. Hence the transformation (t,t) converts a plane pencil of lines into a ruled surface of the seventh degree. 479 On the line e8 the pencils (a), (6), as soon as they have become projective, determine two projective point-ranges. One of the united points (coincidences) thereof is the point «grt; through the other passes a line ¢’. Besides this ray ¢’ the line « 8 also meets the three rays ¢’, which lie in @ and the three rays ¢’ in 8. By « the surface (¢’)’ is intersected along a curve «* witha triple point at A. Through every point of intersection of @* and r passes a line # which coincides with its conjugated line ¢. Hence the double- rays of the involution (t, t') constitute a line-complex of the fourth order. 4. In order to make sure of this by another reasoning we con- sider the threefold infinity of ruled quadrics (Bed). Through three points, arbitrarily chosen on the lines eg, ay, ad, there passes one ruled surface (hed). Hence the system is linear (comp/ex) and inter- sects « along a complex of conics @?. The conies which touch a ray a at a point B, constitute a system of single infinity. The ruled surfaces of which they are the curves of transit, also constitute a single infinity; the base-curve o* is at & tangent to a. Every chord t of eg“ passing through F lies on a ruled surface (bed), which is at R tangent to the ray AR; the two transversals of the quadruplet a,b,c,d thus coincide in ¢. Hence the cubic cone which projects o* from R, consists of double-rays t= t?’. In addition to these the point A carries a plane pencil of double- rays, lying in the plane a. In order to understand this we observe that every line ¢ of a belongs to a pair of lines of the complex {a’}. Let S be the point of intersection of ¢ and the second line of this pair. Since AS is at S tangent to the corresponding hy perboloid (bed), t represents the two transversals of a ray-quadruplet a, b,c, d and is therefore a double-ray of the involution (t, ¢’). In this way we find again that the double-rays constitute a line- complex of order four. Simultaneously it has become evident that this complex has four principal planes a,3,7,0 and, by analogy, four principal points ag, CD. 5. We now consider three rays 6,c,d meeting a line t4, which passes through A. Each line ¢ which intersects 6, c,d, meets in « a certain ray a and is therefore conjugated to ¢4. Hence every ray t4 corresponds to oo! rays t’; otherwise: the rays of the sheaf of lines |A\ are singular. By the rays /4 a trilinear correspondence is established between the pencils (b),(c),(d); in fact, two arbitrary rays b,c determine a transversal ¢4, which defines in its turn the corresponding ray d. 32 480 In any arbitrary plane pencil of rays (/) a trilinear correspondence is similarly originated. Each of the three double-rays (coincidences) is a transversal of three corresponding rays b,c,d, and therefore a line ¢’ of a ruled surface (bed), which passes through A and is con- seqnently conjugated to a ray t4. Hence the rays which in the correspondence (¢, t') are conjugated to the rays through A form a cubic complex. Similarly every ray ¢, (in the plane a) is also conjugated to a regulus; so the rays of the plane a, denoted as a whole by [vl], too are singular. 6. We now consider a ray ¢ of the sheaf of lines which has D* = apy for its vertex; let d be the ray of (D,d) which meets t. To ¢ now are conjugated all the rays ¢’ of the plane pencil which has D* for its centre and is situated in the plane D* d. Hence this pencil consists of singular rays which are each conjugated to every ray of the pencil. The sheaf [ D*] contains o’ of such singular pencils of rays, the planes of which pass through the line AA*. By an analogous reasoning it is found that the rays in the plane de = ABC are singular and constitute o* plane pencils conjugated to each other and having their centres on the line dd”. Hence the involution (é, 7’) contains eight sheaves and eight planes of singular rays. The ray AB meets two definite rays c,d, but al/ rays a, 6, and is therefore conjugated to all transversals of these rays c, d. Similarly the ray ag is conjugated to oo’ rays ¢’. Thus there are twelve principal rays, each of which is conjugated to all rays of a bilinear congruence. 7. The hyperboloids (bed) which pass through . Hence the order of = is fwe. The number of lines of © in a plane W is found by conjugating to a ray w of JW the line ¢, which passes through S and meets the same rays a and 0, and to ¢ the line w’, which meets the same lines c and d as ¢. The correspondence (2v,w’) in W is again birational and of degree four, and so has six double rays. Here there is no line 2 coinciding with a line ¢; the class of = is therefore sav. Hence to a sheaf is conjugated a congruence (5,6). To the four lines SA* ete. and to the transversals through S which meet the six pairs (AB, yd) correspond plane pencils of lines. Hence each congruence = contains ten singular pencils of lines. In the same way it is proved that to a plane field of rays V corresponds a congruence ® (6,5). If S, or V, contains one or more principal rays, this reduces the order of ©, or ®, which reduction is easy to calculate in each case. 6. Two complexes |t} (vide $7 of Prof. pu Vrins’ communication), which are conjugated to special linear complexes of lines ¢, with axes / and m, have a congruence C (49,49) in common, which we are going to investigate. In the first place C contains the congruence A conjugated to the bilinear congruence Z which has 7 and m for its directrices. Now L (4,1) has with a congruence 2 (6,5) eleven rays in common just as with a (5,6), from which it follows that A is a congruence (11, 11). Moreover each |} contains as double rays the lines of the sheaves A, Bb, C, D, those of the planes a, 3, 7, d, and the lines of the con- gruence (3,3), constituted by the singular reguli. For, in fact, the line / meets two generators of every singular regulus, each generator corresponding to the entire regulus. In the intersection of the two complexes corresponding to / and m, each of the nine above-mentioned complexes is to be reckoned for four. Together they therefore count for a congruence (28, 28). Furthermore each |/{ contains as single rays the lines of the four sheaves A“, etc.. those of the four planes «*‚ ete. and the lines of the six bilinear congruences (AL, yd), ete. (§ 4). These form together a congruence (10, 10). By the foregoing investigation the congruence (49,49) is com- pletely accounted for and the completeness of the discovered system of singular rays is controlled. 492 7. By virtue of the HarpneN theorem the congruences >, which are conjugated to two sheaves P and Q have 61 rays in common. To these belongs in the first place the line ¢’ corresponding to the linet — 20) Each congruence has among its generators the principal rays AB etc. (6 in number), ag etc. (6), AA* ete. (4). For, in fact, to each of the twelve former lines corresponds a bilinear congruence, to the latter four a sheaf and all these have one ray which passes through an arbitrary point. This accounts for 16 of the common rays. The remaining 44 are found as follows. According to § 1 there exist reguli (bed) which pass through A and also through P and Q. As follows from $ 5 of Prof. pw Vries’ communication a regulus (bed) through A is conjugated to a ray passing through A. Hence the two congruences = have in common five rays passing through A and as many passing through B, C and D resp. Furthermore there are six surfaces (bcd) which are tangent to @ and pass through P and Q. A surface (bed) which is tangent toa, is conjugated to a line lying in e. Hence the two congruences 2 bave in common six rays lying in « and six in B,y and d each. So in this way we find indeed 4 «5+ 4 « 6= 44 additional common rays. The investigation of the intersection of two congruences ® is quite analogous. Slightly different is the case of the common rays of the congruences = and ® which are conjugated to a sheaf P and a plane of rays V, which by virtue of the Hanpsen theorem have 60 rays in common. To these common rays belong the principal rays AB (6), «B ete. (6), 12 lines in all. There are six surfaces (be dl) passing through A and Pand simultane- ously tangent to WV. This furnishes six lines through A, common to = and ®. The same holds for B, C and D, so we have 24 lines in all. The six surfaces (bed), which are tangent to a, pass through P and are also tangent to V, furnish six common rays lying in a. Just as many we find in the planes 8, y and d, together 24. By the foregoing enumeration the 60 common rays are indeed accounted for. This § constitutes a proof of the completeness of the system of principal rays. Mathematics. — “On an involution among the rays of space, which is determined by a bilinear congruence of twisted elliptical quartics’. By Prof. JAN pr Vries. (Communicated at the meeting of March 29, 1919). $ 1. The twisted quarties (first species) which intersect each of two fixed curves of the same species at eight points form a bilinear congruence *). An arbitrary line ¢ is bisecant to one of these curves o*; at its point of transit, P, through a fixed plane « the line is intersected by one other bisecant ¢’ of the said 9*. The lines ¢ and t constitute one pair of an mvolution of rays which will be investi- gated in the sequel. Every bisecant 6 of the fixed curve * is singular with respect to the congruence [o*]. For, in fact, this congruence is generated by two systems, (8°) and (7), of quadries of which the fixed curves g' and y* are the base-curves; the hyperboloid p*, which contains 6, is intersected by the surfaces of the system (y°) along curves o' each having the line 4 for a chord. The second line 6* which this hyperboloid sends through the point be, is a common chord to the same curves o*. Hence the bisecants of the base-curves p* and y' are not singular with respect to the involution (¢, ¢’). § 2. The congruence [o*] however contains more singular bisecants s; on each line s the systems of quadrics determine the same invo- lution; the lines s constitute a line-congruence (7,3) *). By the involution determined on s by (3?) and (#°), these systems of quadries are rendered projective. The curves 6*, which are gene- rated by two homologous hyperboloids, lie on a quartic surface, which contains the line s. With a plane 6 through s this surface nas a curve o*? in common, the locus of the point-couples which the curves of have in common with 5 outside s. The involution of these point-couples is central je. the chords ¢’ which carry the couples, meet at a point S of 50°. The chords / of any ot which meet s constitute a regulus; the hyperboloids containing these reguli constitute a system, of which s and o® form the base. Since o* has two points hj Vide my communication: “A bilinear congruence of quartic twisted curves of the first species”. (These Proceedings vol. XIV p. 255). 2) Le, p. 257. 33 Proceedings Royal Acad. Amsterdam, Vol. XXII 494 in common with s, the centre S of the involution will, if ois made to revolve about s, twice occupy a position on the line s at each complete revolution. Hence the chords ¢’ meeting s at a point P constitute a cubic cone with s for a double edge. The singular lines s are therefore also singular with respect to the involution (t, t'). § 3. We shall now consider the locus W of the curves o*, each of which furnishes one ray of the plane pencil (Zr) by one of its chords. The curves og* which intersect 8* at B, lie on the hyper- boloid 7g? which passes through B. This hyperboloid intersects r along a conic, on which the curves op* determine an involution /*; the corresponding directing curve is of the third class. Hence the pencil (4) contains three chords of curves o* and from this it follows that the base-curves 3* and y* are triple curves on the surface ¥, The locus of the point-couples Q,Q’ at which the rays ¢ of the pencil (Zr) are twice intersected by curves of is a curve t° with a triple point 7, which curve contains the points of transit B, and Cr (kk: = 1, 2,3, 4) of the base-curves 8* and y*.’) A hyperboloid 3? has, in addition to the four points B, three point-couples Q,Q’ in common with t°. Hence the rays ¢ establish a (3,3)-correspondence between the quadrics of the systems (8*) and (y?); in consequence the locus W is a surface of degree twelve. Let (Z,4) be another pencil of lines, 4° the corresponding curve (analogous to t°). Of the points of intersection of 2° and ¥1? 8 x 3= 24 coincide with the points Bz, C.; the remaining 36 form 18 couples of points, each couple common to a curve of and one of its chords. It follows from this that the bisecants of the curves v* of each of which a given pencil of lines contains one chord, constitute a line- complex of order 18. .§ 4. The curve of passing through a point P, assumed in «@, is projected from P by a cubic cone, the edges s* of which are suugular rays of the involution (é,¢’)..Hach edge is conjugated to every other generator of this cone and is therefore at the same time a ray of coincidence. The curve t° determined by a pencil of lines (Zr) has five points P in common with the line ar; hence each of these points furnishes one singular ray s* in the pencil. The rays s* therefore constitute a complex of the fifth order. In general a line ¢, in the plane a is chord to one of. All chords of this v* which meet 4, are in the involution (¢,t’) conjugated to t.. To this singular ray t, correspond therefore the rays of a quadric regulus and the rays of the two cubic cones which project o* from its points of intersection with ¢,. 1) Le. p. 256. 495 We shall now consider a pencil of lines (7,1). The curve 0 which has a ray ¢ of this pencil for one of its chords, has six chords in «; the points of intersection Q of these six lines ¢, with tr we conjugate to the point P=(t, a). To the pencil (Q, @) corresponds (§ 3) a line-complex of order 18, constituted by the bisecants of the curves of which meet the rays of (Q, a) twice; this complex has 18 rays ¢ which belong to (7,r), so that to Q are conjugated 18 points P. Since P thus coincides 24 times with Q, (7,r) contains 24 chords of curves of, each of which chords meets one of the chords of the same of, which lie in «. The curve t°, corresponding to (Tr) (§ 3) determines on ear five points P,; the curve o* passing through one of these points conjugates three rays ¢, to the ray 7'P,; hence ZP, is to be counted thrice in the above-mentioned group of 24 rays 4. It follows from this that by the transformation (tt) a plane pencil of lines is transformed into a combination of jive cubic cones and a regulus of degree nine. This regulus (¢’)*? has the line ar for its directrix. The ruled surface on which it lies has, in addition to the line at and the five rays TP,, three more lines ¢’ in common with the plane vr of the pencil (7,7). A confirmation of the foregoing result may be obtained as follows. In the plane r each curve of determines four points Zi; the chords u= Rk, Rk, and v=(R,, R‚) are reciprocally conjugated by a quadratic transformation’). If w describes the pencil (7,7), v envelops a conic; the point of intersection of w and v therefore will thrice reach a position on the line ar. Hence there are three rays t of the pencil, of which the corresponding ray ¢ lies in t (and does not coincide with 2). § 5. A sheaf of lines with vertex M is transformed by (t, t’) into a congruence. A curve of of which one chord ¢ belongs to (M)\, has two chords w passing through the point N. To the point of intersection, P, of ¢ and « we conjugate the points of transit, Q, and @Q,, of the chords u, and u,, Similarly to each point Q correspond two points P. If P moves along a line ¢ describes a plane pencil; in the complex {uj{'*, determined by this pencil, w will then describe a cone of degree 18, Q a curve a'*. Hence P and Q are correlated in a (2,2) correspondence of degree 18. Since this correspondence in general contains 22 coincidences, the order of the congruence which corresponds to [|M | is 22. The curves o* which possess a chord passing through J/, consti- tute a surface of degree five. Hence in « there lies a curve «* each 1) Vide my communication: “A quadruple involution in the plane and a triple involution connected with it.” (These Proceedings vol. XIII, p. 86). 33* 496 point of which radiates a cubic cone of singular rays s*. Since these rays are at the sume time coincidences ¢=1', the image of the sheaf [7] has at M a singular point of order five. Let u be an arbitrary plane, ® the plane through M and au. To each ray ¢ of (MZ, P) correspond in u six chords of the particular curve of which meets ¢ twice; their points of intersection Q with au we conjugate to the point of intersection, P, of ¢ and au. The line-complex fu}* which is conjugated to the pencil (Q,u) has 18 rays ¢ which belong to the pencil (J/, >) and therefore determines 18 points P. Since Q coincides 24 times with P, u contains an equal number of rays ¢’ which are in (ft!) conjugated to rays of [M]. Hence the class of the congruence which constitutes the image of a sheaf of lines is 24; it is a congruence (22, 24). The total of the rays [u] of a plane is transformed into a con gruence of which the order evidently is 24. In order to find its class we have to bear in mind that a plane @ can only contain such rays ¢’ as pass through the point of intersection, Q, of the planes a,u and &. The cone with vertex Q of the complex {2}* which corresponds to the pencil (Q, «), breaks up in the pencil (Q,u), the cubic cone which projects the curve of passing through Q, and a cone of degree 14. The last mentioned cone contains the additional chords which are sent through Q by the curves of meeting rays of (Q,u) twice. The three rays ¢’ in ® which are furnished by the cubic cone, are each conjugated to each of the three generators lying in w, and are therefore to be counted thrice. It follows from this that the class of the congruence is 23. So the image of the total of lines of a plane is a line-congruence (24, 28). Chemistry. — “The trimorphism of allocinnamie acid.’ By Dr. A. W. K. pe Jone, Buitenzorg. (Communicated by Prof. P. van Rompuren). (Communicated at the meeting of March 29, 1919). As has already been pointed out in a previous communication, the formation of the same double acid of normal- and allo-cinnamic acid with the different forms of allo-cinnamic acid is in conflict with the view that these forms are chemical isomerides. Stropse, the most zealous exponent of this view, has undertaken, in conjunction with ScHönNBurG'), a detailed investigation in which, according to them, it is clearly shown that the allocinnamie acids are chemical isomerides. Their first series of experiments?) with the two modifications melting at 68° and 42° respectively led to the following conclusion: “Aus den in diesem Abschnitte beschriebenen 70 Einzelversuchen geht hervor, dass die stabile 68° - Säure und die metastabile 42° - Säure bei Abwesenheit von Keimen unverändert umzukrystallisieren sind; die erstere aber mit Sicherheit nur dann, wenn bei dem Lösungsakte und bei den späteren Vorgängen die Temperatur ihres Schmelzpunktes bzw. des durch das Lösungsmittel erniedrigten Schmelzpunktes nicht erreicht wird, wenn also ein Schmelzen der 68°- Sáurekrystalle vermieden wird. Tritt dieses ein, so kann 42° - Sáure als Krystallisationsproduct auftreten. Hiernach bewahren also die beiden Säuren in ihren Lösungen bei genau be- kannten Bedingungen ihre Individualität. Die Lösungen beider Säuren sind trotz der gleichen Lichtabsorption und trotz der gleichen Leit- fähigkeit doch verschieden. Die Allozimtsäure (68°) und die Isozimt- sáure (42°) sind zwei chemisch isomere Verbindungen”. On page 233 of the same communication this conclusion is extended to the case of the acid melting at 58°. It appeared to me to be not impossible that the above mentioned investigators had been led astray by the presence of crystal nuclei which, as is only too well known, play a prominent part in the 1) Annalen, 402, 187 (1914). *) loc. cit. p. 199. 498 case of these acid forms. This was the more probable, since it is clear from their communication that they had formed no adequate idea of the nuclei in question. Although they mention the “beim Einfüllen der Lösung etwa an den Wandungen entstandenen Keime” ’), they neglect to give sufficient attention to the nuclei which can occur in the solution, and to those which are notoriously present in the air. Before describing how the experiments of SroBBr and SCHÖNBURG were repeated, it is desirable to discuss the considerations which form the basis of the experimental method adopted. By ‘nuclei’ are to be understood molecular complexes which remain over from the solid state after solution, and which can be formed in the liquid as a preliminary to crystallisation. According to the solvent used, nuclei and single molecules, or nuclei, double molecules and single molecules can oecur in the solution. If a solution of one of the forms of allocinnamic acid is prepared, a complete or an incomplete dissociation into single molecules results, according to the temperature and the concentration. The higher the temperature and the smaller the concentration, the more complete is the dissociation. It is thus possible to prepare two kinds of solutions, namely, those which contain only single mole- cules, and those which, in addition, contain also nuclei. In solvents in which double molecules can occur, a third kind of solution is also possible containing single and double molecules, while a solution with nuclei may also at the same time contain single and double molecules. While within the solutions equilibria between the different kinds of molecules are established, for which, of course, a longer time is required according as the molecules are more strongly held together, the concentration of the solution is greater, and the temperature lower; equilibria are also established above the solutions between the nuclei, the double-molecules and the single molecules. These equilibria are dependent on the composition of the solution. It is now sufficiently known from experimental investigation, that the atmospheric nuclei of the forms of allocinnamic acid are very persistent. It follows therefore from this that the equilibrium in the air lags behind variations in the solution. As a consequence nuclei are often still present in the air when the solution consists of single molecules only. 1) loc. cit. p. 198. 499 In the experiments it is therefore especially necessary to exclude the dangerous air-nuclei, and, at the same time, to give the solutions sufficient opportunity to reach the equilibrium condition. The experiments of SroBBE and ScHöNBure were repeated with due regard to these considerations in the following manner. The solutions were prepared some time before the distillation and were kept during this time in the dark. At the same time the solutions were several times transferred to another flask. During this process the air nuclei were got rid of by transferring the solution first of all to a small flask which was filled to the brim, blowing away the air above it, and then pouring it into the new flask. Before use any nuclei still remaining in the air above the solution were removed by filling outside the laboratory a small measuring cylinder to the brim with the solution and blowing away the air over it. The solution was then poured from the cylinder through a glass funnel into a distillation flask (50 e.e. to 100 e.c.). The latter was closed by means of a cork through which passed a glass tube reaching to the middle of the bulb of the flask. A plug of cotton wool was inserted into the glass tube, while a larger plug was tied round the tube externally, fitting into the neck of the flask above the side-tube. Both the flask and the tube with the plugs were heated for several hours beforehand in a steam-heated air oven. In fitting them together care was taken that the fingers did not come into contact with any interior part. In order to exelude the possibility of accidental inoculation, a rapid current of air saturated with the solvent was drawn through ihe flask for about five minutes. The air was led in through the tube, so that any atmospherie nuclei present could be carried off through the side-tube. The cork was now raised a little, and, by means of a copper wire, which had been heated red-hot, the larger of the cotton wool plugs was pushed below the side-tube of the flask. The latter was then closed again by the cork. Since Srosppe and Scnönrsure observed that the melting of the 68°- and the 58°-acid, which takes place in petroleum ether and in water at 50° and 40°—42° respectively, must be avoided, as this gives rise to the formation of 42°-acid, it was necessary to drive off the solvent at a temperature not exceeding 35°. At this tempera- ture they found no transformation of the two higher melting forms into the 42°-acid. When the concentration of the solutions was small, the distillation was generally effected in a partial vacuum, while with more concentrated solutions the solvent was evaporated by means of a 500 current of dry air at ordinary temperatures (25°—380°). After the distillation and after the removal of the remainder of the solvent by a dry air current, the flask was closed and placed in ice, where- upon after a longer or shorter time crystallisation began. The preparation of the allocinnamic acid by subjecting an aqueous solution of sodium or potassium cinnamate (containing about 1 °/, acid) to light was somewhat modified, so that a more rapid trans- formation was attained, and at the same time the unaltered cinnamic acid could easily be used again. Flat tin-plate vessels were used. In these the solution was set out in the daylight, and the water lost by evaporation was made up daily. After about eight to fourteen days, exposure the solutions were worked up. In order to separate out the alloeinnamic acid the solution was evaporated to about one tenth of its volume and acidified with sulphuric acid when cold. After cooling, the normal cinnamic acid was filtered off, washed, and immediately after drying was used again for the preparation of a new solution. If this acid melts on the water-bath, then allocinnamic acid is still present. This can be extracted by means of hot ligroin. The filtrate was neutralised with alkali, and the solution then evaporated, until crystals began to form. After cooling sulphuric acid was added, which caused the allocinnamic acid to separate out as an oil. This is dissolved in ligroin, and the solution is allowed to erystallise quietly after ‘seeding’ with the 68°- or the 58°-acid as required. These forms erystallise in large crystals, which even by their appearance and also by their melting points are easily distinguished from the crystals of the double acid of normal and alloeinnamie acids, which occurs only in small quantity. In this way almost perfectly colourless crystals are obtained at first which, after recry- stallising once from petroleum ether solution, are quite pure. Repetition of the above procedure yields crystals with a pale yellow colour. These may be decolourised in alcoholic solution by means of animal charcoal. The water solution can afterwards be extracted with ether and yields a further small quantity of impure allocinnamic acid. The transformation of the 68°-acid into the 58 acid is easily brought about by boiling the erystals with a little water for a quarter of an hour. The flask is then closed by means of cotton- wool, and the boiling continued. On cooling the alloeinnamie acid separates out in oily drops which are transformed into the 42°-acid in ice. If the solution with the oily drops is “seeded” with a trace 501 of the 58°-acid, beautiful needle crystals of this acid are formed on standing. Experiment gave the following result: Solutions of the 58°-acid and the 68°-acid in petroleum ether, saturated at 25° (about 0.26 erm. and 0.17 grm. in 5 c.c. respectively) were allowed to stand for eight days in the dark at 25° to 30°. Without the previous presence of crystals the solutions gave, on evaporation of the solvent at the ordinary temperature by means ot a current of air, a residue which in ice was transformed into the 42°-acid. After having stood for eight days in the dark, less concentrated solutions of the two acids gave always the 42°-acid. This 42°-acid remained unchanged during the whole period of observation, about one month. If, however, the solvent was distilled off immediately after the preparation of the solution, it was found impossible to effect the transformation even of a solution containing only 0.05 grm. of either of the two acid forms in 5 ee, into the 42°-acid. Solutions in petroleum ether which were kept for eight days before distillation in contact with crystals of one or other of the two higher melting forms, gave, either during or immediately after the distillation of the solvent, crystals of the acid from which the solution was made. LieBERMANN and TrucksAss *) succeeded frequently in excluding nuclei of the higher melting acids by filtering the petroleum ether solution and afterwards heating it on a water bath at 35°. In six out of ten experiments with 68°-acid the transformation into the 42°-acid was effected. The same result was obtained with the 58°-acid in two out of four cases. Experiments were carried out to ascertain if it were not perhaps possible to remove the nuclei more quickly than by a complete dissociation at ordinary temperature. The same procedure was adopted as before to exclude atmospheric nuclei. Heating at 35° was however, omitted. It was found that when solutions, almost satu rated at the ordinary temperature and prepared immediately before- hand, were filtered through cotton-wool, ordinary filterpaper, or even through quantitative filterpaper, they yielded the original acids. If the solutions, even those containing crystals, were filtered after standing for twenty four hours, a residue was frequently, though not always, obtained, which yielded crystals of the 42°-acid. The reason for this may be ascribed to an alteration in the size ij Ber. 43, 411 (1910). 502 and also in the number of the nuclei, or possibly in both causes together. The transformation of the nuclei of the 68°- and the 58°- acid in ethyl ether and benzene solutions, saturated at 25°, without crystals, did not take place even after they had stood in the dark for fourteen days. This is very probably due to the great concentration of the solutions. At 25° about 4.4 germs. of 58°-acid is soluble in 1.6 grms. of ethylether, and about 4.6 germs. in 2.3 grms. of benzene. The 68°-acid dissolves to the extent of about 6.6 grms. in 3.2 germs. of ethylether and about + grms. in 3 grms. of benzene. An ether solution containing 2 grms. of 68°-acid in 5 ¢.c. and a benzene solution containing 2.4 grms. of acid in 3.6 grms. of solvent, gave a residue after standing for eight days in the dark which yielded the 42°-acid on crystallisation. After two months, standing in the dark a solution containing 2 grms. of the 58°-acid in 1.6 grms. of ether and a benzene solution with 1.8 grms. of acid in 2.5 grms. of solvent gave also -a residue which crystallised out as 42°-acid. An attempt to remove the nuclei from ethylether and benzene solutions, saturated at 25°, by filtration through cotton wool or filter paper was not successful, even-when the solutions had been kept for more than ten days free from crystals. From the foregoing it appears that the transition of the 58°-acid and the 68°-acid into the 42°-form in solution can take place inde- pendently of the melting of these forms, and that at 25°—30° they can form solutions which, apart from differences of concentration, are identical, provided that the nuclei are afforded an opportunity for transformation, and that atmospheric nuclei are excluded. When, however, the concentration of the solution is high, as may be the case with ether and benzene solutions, then it is not possible to break up the nuclei, or to remove them by filtration. In this case there exists most probably an equilibrium between the nuclei and the other molecules. One of the principal arguments of SroBBr and ScnHönBurG for the chemical isomerism of these acid forms is thus rendered ineffective, while the results are in complete agreement with the assumption of the trimorphism of the allo-cinnamic acids. In connection with what Stops and ScHöNBurG have communicated regarding the transformation of the 42°-acid and the 58°-acid into the 68°-acid at —14° (ice and salt), it was of importance to investi- gate if the same change also took place in solution. Various solvents were used. The most important results were obtained with water, so that these may be detailed first. An experiment was made with 68°-acid which had been freed 503 from 68°-acid nuclei by boiling with water. The solution was evapo- rated down until the acid separated out as an oil at ordinary temperature. This solution gives crystals of 42°-acid on cooling to O°. The presence of a small quantity of liquid acid indicates at once the occurrence of undesired inoculation. Small quantities of this solution were introduced into the flasks with the cotton wool plugs. After filling and drawing air through the flask, the solution was again boiled. After cooling the flasks were placed in a freezing mixture. The temperature of the mixture was in some experiments about —10°, in others about —16°. After twenty-four hours these tem- peratures were 0° and 5° respectively. (The mixtures were kept in a box packed with hay). The cooling was continued until transformation had taken place. This point is easily recognised from the more copious crystallisation and also from the form of the crystals. After the ice had melted, the flask was opened, the tube with the plug withdrawn, and the solution carefully poured out so that the crystals as far as possible remained in the flask. The flask was then closed in the usual way, and the few drops of water were removed by means of a stream of dry air at the ordinary temperature. The melting point was then determined. It was found that, whenever the initial temperature of the mixture was —10°, the 58°-acid was always formed, while, when the initial temperature was —16°, the 58°-acid was formed in nearly as many cases as the 68°-acid. (58°-acid in five experiments and 68°-acid in seven). There is apparently a range of temperature within which the 58°-acid is formed, while at lower temperatures the 68°-acid is obtained. The transformation at —10° sometimes requires several days; at —16° it is complete after a few hours. In this way a method is given by which one or other of these acid forms may be prepared. Experiments with a solution in petroleum ether of low boiling-point were carried out as follows. A dilute solution of the 68°-acid in petroleum ether was prepared and freed from nuclei at the ordinary temperature. Portions of this solution were introduced into several flasks through which a rapid stream of air was drawn. After the cotton-wool plug had been pushed below the side tube, a large portion of the solvent was distilled off. The solution was then cooled. It was found that, both when the initial temperature was —10° and when it was —16°, the 58°-acid was obtained in several cases, but ’ generally the 68°-acid was formed. 504 Solutions in benzene prepared in a similar way showed at —10° a transformation only in exceptional cases. In these cases the 68°- acid was obtained. At —16° after twenty four hours the 68°-acid was always formed. Further experiments were made by adding a few drops of the solvent to erystals of the 42°-, 58°-, and the 68°-acid contained in the flasks with the cotton wool plugs, in such a way that crystals still remained in the solution. The solvent was introduced into the flasks through the glass tube with the cotton wool plug, the part of the tube projecting beyond the cork having been previously heated in order to prevent infection. It was found that the 58°- and the 68°-acid were unchanged. The 42°-acid was, however, transformed, the same changes being observed as with the solutions. STOBBE and ScHöNBURG*) assert that the 42°-acid and also the 58°-acid in the solid state are transformed into the 68°-acid on cooling in ice and salt. As the results obtained by me would seem to cast doubt on the correctness of this assertion, the solid substances, after having been carefully dried, were cooled for six days in ice and salt in the flasks with the cotton wool plugs. In the case of the 42°-acid the drying was effected by heating the flask to 80°—90°. It was observed that none of the three acid forms was altered by cooling. As Sroppe and ScHönBurG always worked with capillary tubes, a small quantity of the dry acid was introduced into a capillary from the flask. After six days cooling this also showed no change in the melting point. The solid substances are thus unchanged by cooling in this way. If, however, the 42°-acid is moist, transformation can take place. To this fact the changes of this acid observed by the above investigators are probably due. It is difficult to understand the transformation of 58°-acid into 68°-acid on cooling, as observed by Srospe and ScHönBurG, unless one assumes that they used only ten capillaries for the cooling of the acid, in which after determined intervals of time the melting- point was taken. In that case the possibility is always present that 42°-acid is formed by the melting, as this takes place more easily in capillaries than would appear from the results of these investiga- tors (p. 239). From the 42°-acid in a moist state the 68°-acid would then be formed. As a further result of the investigation may be deduced, that there is a great difference between double molecules and nuclei. The transformations which have been described take place only in solu- 1) loc. cit, p. 218, 236. 505 tions of suitable concentration, while even a chloroform solution which contained 35.4°/, by weight of alloeinnamie acid, and in which, therefore, double molecules were certainly present, no trans- formation was observed after six days’ cooling. In my opinion the explanation of this is that only solutions containing nuclei can be transformed, and that there exists only one kind of double molecule. Van DER Waars’ supposition that double molecules consist merely in the temporary association of the single molecules, is thus rendered more probable. Further investigation is required to clear up this point. In the nuclei we have thus a determinate arrangement of the molecules. They represent the smallest particles of the substance in the solid condition, the simplest nucleus consisting of two mole- cules. It is not necessary to assume that the molecules are united at the carboxy! groups in order to explain the existence of different isomerides, since, as | have found, coumarine, which has no carboxy] group in the molecule, occurs also in a metastable form. It appears to me more probable that the reason for the occurrence of isomerides must be sought in the double bond. I hope shortly to return to this point. STOBBE and Scudnpure (p. 200) consider the occurrence of “Lösungs- gemische’ of the 42°- and the 68°-acid as an argument for the isomerism of these acids. It is clear from what has been said above, that the nuclei in the solution play an important part. The authors have however, interpreted their experiments in a different sense. They supposed that they had found that “In jedem Hinzelfalle als Verdampfungsriickstand 68°-Säure erhalten wird nach einem Gesamt- zusatz von 2.9-—4.1 Proz. 68°-Säure, d.h. waren weniger als 2.9 Proz. 68°-Säure zur 42°-Säurelósung zugesetzt, so schmolz der Ver- dampfungsrückstand bei 42°; betrug der Zusatz mehr als 4.1° Proz. 68°-Säure, so zeigte der Verdampfungsriickstand den Schmelzp. 68°”. They therefore conclude that the distillation residues are to be regarded as mixtures of the 42°-acid and the 68°-acid. On p. 204 they state: “Wenn, wie oben gezeigt worden, die Lösung von der 68°- Säure verschieden von der Lösung der 42°-Säure ist, und wenn nach Zusatz von wenig 68°-Siurelésung zur 42°-Säurelösung ein bei 42° schmelzender Verdampfungsriickstand erhalten wird, so kann dieser nicht reine 42°-Siure (fest) sein. Es muss vielmehr ein Gemisch der beiden isomeren Sáuren sein’. It is then assumed that solid solutions of these forms exist, and that the erystals are mixed erystals of the 42°-acid and the 68°-acid. It is obvious from what has been said above that this hypothesis is devoid of foundation. Also on theoretical grounds, from the point of view of the authors, this hypothesis is untenable. 506 If the 68°-acid and the 42°-acid are chemically different, the “seeding” power of the former must be a property resident in the molecule. It is therefore impossible to explain why a nucleus of 68°-acid or a trace of this substance is capable in a short time of converting a large quantity of 42°-acid, while, when the molecules of the 68°- acid were distributed in a regular manner among those of the 42°- acid, as must be the case with mixed erystals, no transformation took place until 2.9°/, of 68°-acid was present. Fusion experiments have also led these investigators to assume the existence of solid solutions or mixed erystals. On p. 213 they write: “Hin Teil der eben besprochenen, bei 42° schmelzenden Erstarrungsproducte bleibt jahrelang unverändert, ein anderer Teil verwandelt sich bei Zimmertemperatur, zuweilen schon nach Minuten oder Stunden ohne erkennbare Ursache in 68°-Säure. Diese erstarr- ten Schmelzen sind also unter einander nicht gleich; sie sind ebenso wie die aus den Lösungsgemischen erhaltenen Verdampfungsriick- stände, feste Lösungen oder Mischkristalle mit wechselnden Anteilen 68°-Säure und +42°-Saure'). It is not altogether improbable that these transformations could be brought about by one or more atmos- pherie nuclei which had not been broken up, in cases where no care had been taken to ensure the removal of these. A single nucleus remaining in the melt is sufficient to cause transformations of this kind. On the assumption of the trimorphic nature of allocinnamie acid it might be expected that, when the different forms were melted, the dissociation into single molecules would be more complete according as the time of heating is longer and the temperature higher. This was confirmed by SroBBr and Scndnsure for the 58°-acid (p. 239) and the 68°-acid (p. 211). In these experiments only 5—7 mer. was introduced into each capillary. Experiments with 10—50 mer. of the 68°-acid in larger capillaries showed that even heating for twenty-five minutes at 70° was not sufficient, even in a single case, to bring about a permanent change into the 42°-acid, while on heating 5—7 mgr. of the acid for ten minutes the transformation was effected in four out of ten experiments. The same thing has already been stated by other observers, namely, that large quantities are more difficultly transformable than small quantities. melt, is certainly by no means always correct. Nuclei are molecule-complexes, and the molecules of the nuclei in different parts of the melt with nuclei are not necessarily uniformly distributed. On the other hand it-is precisely in the case of a solid solution or of mixed crystals that we have a uniform distribution of the molecules among themselves. 507 The presence of nuclei affords an easy explanation of this. Let us suppose, for example, that the dissociation of the 68°-acid is allowed to proceed so far that there are now only two nuclei remaining in 20 mgr. of the substance. Suppose also that the substance may now be divided into four equal parts. In two parts there are now at the most one nucleus each; in the other two parts there is no nucleus. In this way we have, using portions of 5 mgr., 50°/, of the substance transformed into 42 -acid under the applied conditions of temperature and heating. If the substance had not been subdivided, then the 20 mgr. with the two nuclei would have been transformed, either immediately or after several hours, into 68°-acid.*) It is now easy to see that, if the probability of transformation for 5 mgr. is 50°/, it is 25°/, for 10 mer., 12.5 °/, for 20 mer, 6.25 °/, for 40 mer., etc, that is, the probability of complete (ransformation with a given temperature and time of heating becomes smaller and smaller as greater quantities of substance are used. For each experiment STOBBE and SCHöNBURG heated only ten tubes, so that the figures obtained by them are certainly not to be used as mean values. They found, for example, on heating 5—7 mgr. of 68°-acid. for 10 minutes at 70° that four tubes out of the ten were transformed. (In the case that one part melted at 42° and another at 68°, complete transformation was not obtained). At 100° also only four of the ten showed a transformation into the 42°-acid. At 70° the mean value was probably somewhat lower. If we assume that the mean for ten minutes’ heating at 70° was 40°/,, then the probability of the conversion of 0.05 gr, in ten minutes is only 5°/,, that is, the conversion should take place in one experiment out of twenty. In order to investigate if perhaps atmospheric nuclei were in part responsible for the difficulty of the transformation, quantities of the 68°-acid were melted in U-tubes. One limb of these U-tubes was provided with a plug of cottonwool. Through this limb a rapid current of air at 70° was drawn during the time the U-tube was being heated in a water-bath at 70°, in order to drive out the air nuclei through the other limb. After the heating the other limb was closed by means of a cotton wool plug. Working in this way | was as unsuccessful as Sroppe and Scsonsore in transforming 0.05 gr. of 68°-acid permanently into 42°-acid by heating for twenty minutes at 70°. ') This is the reason for the phenomenon, observed by Stopse and Scuönsura, that in the capillary tubes some portions melted at 42° and others at 68° or 580, 508 In this experiment it frequently happened that the melt first erystal- lised to the 42°-acid and afterwards was transformed back into the 68°-acid. This was also noticed by SroBBr and ScnönBure. This phenomenon is therefore not due to atmospheric nuclei, but to nuclei in the liquid. Although for a permanent transformation into the +2°-acid a complete absence of 68°-acid nuclei is essential, it is, of course, to be understood that a single nucleus of this acid, remaining over on melting the acid, is not necessarily sufficient to cause an immediate transformation of the melt. This may only take place after the lapse of several hours. Moreover, the nuclei vary in size, and the effect produced by the smallest, which is probably built up of only two molecules of allocinnamiec acid, is presumably less effective and less rapid than that of nuclei consisting of several molecules. From the foregoing it appears that the arguments used by STOBBE and ScHONBURG in support of the chemical isomerism of the allocin- namie acids are fallacious, while the experimental results obtained admit of satisfactory explanation on the assumption of the trimorphism of these acids. hort summary of the results ained. Short y of tl lts obtained 1. An experimental method was worked out and applied by which it is possible to avoid inoculation, to exclude atmospheric nuclei, and to bring about the complete dissociation of those nuclei present in the liquid. 2. With this method it was shown that dilute solutions of the 58°-acid and the 68°-acid give, after removal of the solvent by distil- lation at ordinary temperature, a residue which is transformed into the 42°-acid. 3. When the concentration of the solutions of the 53°-acid and the 68°-acid is great, as may occur with ether and benzene as solvents, these acids are not transformed into the 42°-acid at ordinary temperatures. 4. The solutions may be transformed by cooling in ice and salt. In this way an aqueous solution containing no nuclei of the 58°- acid or the 68°-acid gives at —10° the 58°-acid and at —16° the 58°-acid or the 68°-acid. 5. In the solid state the acid forms appear to be unaltered after cooling for six days in ice and salt. 6. The arguments of SroBBe and SCcHÖNBURG in support of the chemical isomerism of these acids are shown to be fallacious. All the results obtained are completely explicable on the assumption that the alloeinnamie acids are trimorphous. Buitenzorg, January 1919. Chemistry. — A. W. K. de Jone. “The Truzillic Acids’. (Communi- cated by Prof. van ROMBURGH.) (Communicated at the meeting of May 3, 1919). The separation of truailhe acids from cinnamic acid. Since mixtures of cinnamic acid and the truxillie acids are obtained by the action of light on the salts of the former, it was necessary to be in possession of a good method of separation. Although various attempts have been made to carry out the separation in the wet way, these have so far failed to yield quantitative results. This is partly due to the increased solubility of the truxillie acids in presence of other truxillie acids and especially of cinnamic acid; and partly to the smallness of the quantity of the truxillie acids compared with the cinnamic acid present. With petro- leum ether, for example, undoubtedly one of the best solvents for the pur- pose, no quantitative separation is obtained, since 6-cocaic acid and d-truxillie acid arevery appreciably soluble in proportion to the amount of cinnamic acid present. The same is true in a less degree of the other acids. The attempt was also made to effect the separation by means of the acid potassium salt, which is diffieultly soluble in alcohol. This, however, appeared to be impracticable, since some of the traxillie acids were also precipitated to some extent. For the present there remains only the sublimation method. This, however, with RrBer’s apparatus proceeds very slowly. For this reason the sublimation was carried out at ordinary pressure in a current of air at 130°. The substance was placed in a little boat, which in its turn was placed in a glass tube. The whole was heated in a sand bath at 130°. Sublimation was continued until the weight of the residue became constant. Separation of the trucillic acids from each other. The acids were dissolved in the calculated quantity of N/,, potassium hydroxide solution on beating. To the solution anhydrous calcium chloride was added, 1.5 grm. for each 10 c.c. of solution. After twenty- four hours the precipitate, which may contain the calcium salts of B-, d- and e-truxillie acid, was filtered off and washed with calcium chloride solution (1.5 grm. per 10 ¢.c.). The acids in the filtrate 34 Proceedings Royal Acad. Amsterdam. Vol XXII. 510 were separated by means of hydrochloric acid and ether and then weighed. They were once more dissolved in the calculated quantity of N/,, potassium hydroxide, and calcium chloride (1.5 grm. for each 10 e.e.) again added. After twenty-four hours the precipitate was filtered and washed with a little calcium chloride solution (1.5 grm. in 10 ¢.c.). The precipitate is added to that first obtained. Separation of B-, d-, and e-truxillic acids. The calcium salts are treated with hydrochloric acid and ether, and the acids dissolved in the calculated quantity of N/10 potassium hydroxide. Twice the volume of water is then added, and as much N/10 barium chloride solution as was used of N/10 potassium hydroxide. After twenty- four hours the precipitate is filtered and washed with water. It consists of the barium salts of 8- and e-truxillie acid. The acids are extracted from the filtrate by means of hydrochloric acid and ether. They are redissolved in N/10 potassium hydroxide. Twice the volume of water is added, and as much N/10 barium chloride as was used of N/10 potassium hydroxide. In this way a little more 8- and «- truxillie acids are obtained as barium salts. The filtrate now obtained yields d-truxillie acid with hydrochleric acid and ether, which, if necessary, can be purified by recrystallisation from boiling water. The precipitated barium salts are boiled with water, cooled, and filtered. Hydrochloric acid is added to the filtrate. If a precipitate is formed, the above treatment is repeated until no precipitate is obtained. The filtrates yield e-truxillie acid on treatment with hydro- chlorid acid and ether. This may be purified, if necessary, by recrystallisation from boiling water. The undissolved barium salt gives B-truxillie acid with hydrochloric acid and ether. Separation of a-, y-trurillic acids and p-cocaic acid. To the filtrate from the precipitated calcium salts 8.5 grms of anhydrous calcium chloride per 10 c.c. is added. The precipitate is filtered after twenty- four hours and washed with a solution of calcium chloride prepared by dissolving in water as much calcium chloride in grams as there are c.c.’s of water. The acids are extracted from the filtrate, and these are subjected to a similar procedure in order to separate a small quantity of p-cocaic acid as calcium salt. The precipitated calcium salt gives g-cocaic acid, when treated with hydrochloric acid and ether. This may be recrystallised from boiling water if necessary. The filtrate from the precipitated calcium salt gives a- and y- truxillie acid with hydrochloric acid and ether. In order to separate these the acid mixture is boiled with water (25 c.e. per 0.1 grm.) with a reflux condenser for half an hour and is then filtered hot. 511 The residue consists of «-truxillie acid. On cooling the filtrate yields y-truxillie acid which, if necessary, may be recrystallised from boiling water. In order to test the effectiveness of the method of separation a mixture of the six truxillie acids was subjected to the treatment above described with the following result. i | ; | | Melting point Quantity used |Ouantty found) ating point | AER Feet water. g-truxillic acid 0.119 0.086 270° — ER 0.100 0.096 2029 204° = y - = 0.134 0.099 200° — 215° 220° — 226° P 5 5 0.106 0.132 gummy 1729 — 1749 € - - 0.078 0.079 208° — 220° 230° B-cocaic , 0.106 0.120 165° — 175° 189° — 190° Total. . 0.643 0.612 | . For the sum of the «- and the y-acid 0.224 grm. was found. The method is therefore sufficient for the detection of the truxillic acids in presence of each other. If there are only two truxillic acids in the mixture an almost quantitative separation may be effected. From the above separation several properties of the truxillie acids may be noted. The following may be added. p-cocaic acid‘) forms with cinnamic acid a well crystallised double avid with equal proportions of the components. This is obtained by boiling a petroleum ether solution of einnamic acid, saturated at the ordinary temperature, with a little g-cocaic acid until the latter is dissolved (0.1 grm g-cocaie acid in 500 c¢.c.). On cooling the double acid separates out, frequently only after several days, in long needles, which melt at 139°. The filtrate gives a fresh quantity of double acid whenever 0.1 grm. of each of the acids is dissolved in it by boiling. The composition is determined by sublimation at 130°— 140°. The solubility of y-truxillie acid in chloroform is increased in a remarkable degree by the presence of 8-cocaic acid. The ammonium salts of the truxillic acids slowly lose their ammonia when their aqueous solutions are evaporated on a water bath and are transformed into the free acids. The ammonium salt of cinnamic acid also possesses this property. ') The acid (m.p. 190°) formerly separated from the acids derived from the coca alkaloids appears to be (-cocaic acid. 34* Physics. — “The Propagation of Light in Moving, Transparent, Solid Substances. IL. Measurements on the Fizwav-fject in Quartz’. By Prof. P. Zeeman and Miss A. SNETHLAGE. (Communicated at the meeting of May 3, 1919) 1. In communication | the apparatus has been described that has proved suitable for the investigation of the Fizeau effect in solid substances. We have now carried out experiments with quartz, which was traversed by beams of light in the direction of the optical axis. We were led to the choice of this substance by the consideration that in general, crystals are the most homogeneous bodies that we know, and the scattering of light in a crystal must be exceedingly slight on account of its regular structure *). It appeared to us later that the best optical glass, for our purpose, can be compared in some respects with quartz, in others it is even preferable. In some series of experiments 10 quartz rods were used, supplied by the firm of Srere and Reuter, with endplanes normal to the optical axis, and of the dimensions 10 Xx 1.5 1.5 em. Later on four similar rods supplied by the firm of A. Hrreer, Ltd. were added to them. For a series of experiments the rods were joined together to form a column of a length of 100 em.; in a second series of 140 em. They were placed one behind another in a groove which was milled in a wooden beam, fastened to the driving apparatus by means of four solid screws. The different rods are separated from each other by rubber dises with round apertures of a diameter of about 13 mm. Each quartz rod rests in a groove 13 mm. deep, and is pressed down by two brass plates, fastened with screws in the upper surface of the wooden beam, a thin piece of cork being placed under the plates. The space remaining at the ends of the groove is filled up by a piece of brass tubing. Solid brass plates, which clasp the beam, shut off the ends of the groove. ‘) Lorentz. Théories statistiques en thermodynamique, p. 42. 513 2. In order to place the quartz rods in the groove, we proceeded in the following way. After the interference-bands had been produced with great distinctness, and the beam had been placed on the apparatus, ene quartz rod was put in the groove, and if necessary the inter- ference-lines were made distinct anew. It was then ascertained which of the four positions obtained by rotating the rod round its longitudinal axis, gives bands that change least, when the machine is made to assume different positions. Then the second rod is placed behind the first, likewise in four positions ete., till all the rods have been arranged. In order to prevent reflected light from entering the interferometer, each of the rods is placed in a somewhat sloping position by putting a piece of thin cardboard at one end. The rods are put in one by one. After each addition it is tried, whether the correct position has been obtained. We may still remark in this connection that the glass cylinders with which we have made experiments (see the following commu- nication) have been manufactured so exceedingly well by the firm of Zeiss, that on rotation about the longitudinal axis in a cylindrical groove there does not appear an appreciable change of the inter- ference bands. Hence the optical control becomes a great deal simpler than for quartz. The interference bands finally photographed through the quartz column are decidedly less distinct than the interference bands that are observed when the column has been removed. The lines have become slightly diffuse. This is not the case when the glass cylinders of Zriss have been introduced. The diameter amounted to 25 mm. with a length of 20 em. As there were used six cylin- ders, there were twelve reflecting planes for a total length of glass of 120 cm. In the experiments with the quartz column of 140 em. length the number of reflecting planes amounted to twenty-eight. Though this great number of reflecting planes must have an unfavourable influence on the distinctness of the system of fringes, yet it was beyond all doubt that it was not owing to this cause that the quartz column had a more unfavourable influence than the glass column. We might still have eliminated the reflections on the interfaces by introducing a liquid of the mean index of refraction of quartz be- tween the successive rods. The complication of the apparatus, which would ensue from this, and the unfavourable experience which we had with moving liquids, made us resolve to put up with the reflections. 4. As source of light a 12 Amperes are-lamp was used, the light of which was made sufficiently monochromatic by means of filters. 514 Experiments were carried out with three different colours, the effective wave-lengths of which amounted to 6510, 5380, 4750 ALU. 4. When white light traverses the apparatus, we easily distinguish the central band. Its centre is the point whose displacement we should wish to measure in an experiment witb white light. Also for incident monochromatic light we can speak of the centre of the central band. It is the point that remains fixed when the interference lines rotate, or become narrower or wider through any cause that does not depend on the Fizeau effect. The position of the centre can be determined by means of the horizontal and movable vertical cross- wires in the telescope, by subjecting the interference bands to some modification with the compensator, thus causing the centre to be observed clearly. When the centre has been determined, the movable vertical wire is displaced over a few bands, so that this wire can have no disturbing influence on the measurement on the photo. A series of photos is then taken on one photographic plate, in which the directions of the movement alternated. The observed effect is derived from the displacement of the centre. Of course plates on which a notable rotation of the interference bands has occurred, are rejected. 5. The following table may serve as an example of the results obtained by measurement of a plate taken with: Green light 4 = 5380 A.W. Maximum | Length of | Effect reduced bomen velocity | column | Observed [to 1 m. of quartz aa vale ee in cm. | of quartz | effect jand max. velo- fate P per sec. in cm. city 10 meters P 48 750 100 92 123 137 183 118 157 152 99 132 125 167 The effects are given in thousandths of the distance of the fringes. Altogether photos have been taken on eleven plates with green light 4 = 5380 A.U. In all fifty-one values have been obtained in this way 515 for the observed effect, but not the same number of photos have been taken on all plates as on Nr. 48. The velocities used ranged between 750 and 950 em./see. the length of the quartz column being 100 em. for nine plates, and 140 em. for two. The obtained values may now be used in two ways to derive a final result from them. For each plate a mean value can be derived, and the arithmetical mean may be taken of the eleven values thus obtained. In this way: 0,146 += 0,012, as final result of the effect reduced to a velocity of 1000 cm./sec. and a length of quartz of 100 cm., the mean error being recorded after the + sign. Another way in which the values can be combined is by taking the arithmetical mean of the fifty-one values. Thus we find: 0,148 + 0,006. From formula (4), which was given in our communication I, and will be proved presently, follows for the theoretical value of the effect : 0,143 6. With red light 2 — 6510 A. U. twenty-seven values have been obtained for the effect on six plates. To eight of them corresponds a quartz column of 140 em., to nineteen one of 100 cm. The velocities range between 750 and 960 em./sec. The result, when the mean values of the different plates are combined, is: 0,123 + 0.014. The arithmetical mean of the twenty-seven separate values yields: 0,125 + 0.007. The calculation gives for the expected effect: 0.115. 7. The results with violet light 2 == 4750 A. U. should be received with some diffidence, as it appeared afterwards that the violet filter transmitted some red light, which had not been detected at first. Hence it is possible that this cause slightly vitiated the later series. It must be said, however, that no trace of change could be ascer- tained in the values of the later series. 516 When the eight mean values of the different plates are combined, the result becomes: 0,156 + 0.008 The arithmetical mean of the thirty-one separate values differs very little from this: 0,156 0.007 The theoretical value is 0.166. 8. We collect the results in the following table. a A Am 4750 0.156 + 0.007 0.166 0.156 + 0,008 5380 0.148 + 0.006 0.143 0.148 + 0.012 6510 0.125 + 0 007 0.115 0.123 + 0.014 The observed displacement of the bands is indicated under A. The mean error has been calculated in two ways, as was discussed above. The second values are those derived from the average of the mean values of the individual plates. Under A, the theoretical value is given calculated by the aid of the data for the index of refraction for the ordinary ray in quartz, taken from Kontrauscn’s data. It is not to be denied that taking the particular difficulties of the experiments into consideration, the agreement between theory and observation is very satisfactory. The change of the effect with wave-length as well as the absolute value of the effect are represented very well. In the discussion of the experiments with glass, for which the dispersion is greater than for quartz, we shall have an opportunity to point out the very pronounced influence of the dispersion term. 9. The formula for the optical effect. We consider two of the rays which bring about the interference phenomenon, and which have passed over opposite paths. We shall denote quantities which refer to the first ray, by one accent, and those belonging to the second ray by a double accent. Each of the paths traversed, consists of three parts: 1 a path 1 in the air, 2 a path 2 in the quartz column, 3 a path 3 in the air. P. ZEEMAN and A. SNETHLAGE: “The Propagation of Light in Moving Transparent Solid Substances. II. Measurements of the Fizeau-Effect in Quartz”. Proceedings Royal Acad. Amsterdam. Vol. XXII. 517 The times expressed in seconds, which the light requires to pass over each of the parts we call RODE nnen bern ep GER If the quartz is at rest, of course ¢,— 1", ete. If, however, the quartz moves with the velocity w, the time required to traverse the quartz column (length /) in the direction from 1 to 2 = = iy reat. erker oD) c 1 c w A NED er in which the difference between the velocity of the light in quartz, and of that of the column itself must be taken into account. While the light is passing through the quartz, the quartz moves on, hence i, is changed by an amount: lw 1 = = (2) C w C ema? We get for the ray in the opposed direction: l “= (3) c w u Wg and for the other quantity : lw 1 (4) planken in Ap u u For the first ray the entire difference of time becomes, therefore: l lw 1 En : w ¢ w : c ' Is TR and for the other 518 When we now consider that: wu + eee bre (7) B Dee crd vore ie) and 4 duw u an te ee eee rau ree (2) we find, after substitution of this in the formulae (5) and (6), and after subtraction for the entire phase difference of the two rays that: — nn Ae i ) nh nd or, on reversal of the direetion of motion, an optical effect: — ns 4 ue 9) =— 2 = lee shige Wnts Ss ear ( With regard to the dispersion term it is still noteworthy that in Fizrau's experiment with water the light is transferred from : 2 W OE 5 standing water to moving: water, and or must be written in the c/u 4 w formulae instead of —. Cc 10. Derivation of formula (9) from the theory of relativity. After we had communicated formula (9) to Prof. Lorentz, he had the kindness to give us a derivation strictly from the theory of relativity, which will follow here. Let 2’, ¢’ be a system of coordinates, in which the rod AB is at rest; length of the rod /. Light motion on the lefthand side of A: a \ acorn! (¢ —= + ph) REA ca (JI) c a! a, cos n° (« ——-+ P's) v On the righthand side of B: a, cos n c= +P) A MSE eet (4) ¢ v’ velocity of propagation belonging to n’. We easily find: In the rod: 1 1 weert) gone ar) 519 Through the relativity transformation: mar bet , tai NO (4) c and ' Li U b ' nest deb gtt c we may pass to a system, in which the rod moves with a velocity : be ee atten) re eer Eft (BD) a From (2) and (3) we derive a, cos |» (« — =) - | and a, cos E (« — = + n' p | . (6) c c Tad) eenn a Te ee as dak CD) The phase difference between (1) and (2), ie. the change of phase brought about by the presence of the rod, is given by (3) in angular measure, and this same difference of phase still exists between the expressions (6). Expressed in wave-lengths or periods, it is: A 0, if the equilibrium is stable. The equation (5) is thus established. Oa 0: We may remark that the separate sums, as a6 Ie = = AREN QV AVy are not zero except in the special case when — = : 3. We shall now attempt to establish the general equation (2). We assume that we have a liquid phase consisting of one mol of solvent and «,y,z,... mols of the dissolved substances. At pressure p and temperature 7’ the solution is saturated with respect to these sub- stance. We have thus m-++1 components in as many phases and have therefore two degrees of freedom at our disposal. The equilibrium conditions are (for the notation see above): Die oN AN nec dz rme ee Mn IE (6) dz 0 TR The expressions on the left-hand side of these n equations are again functions of z,y,z,... p, and 7. The last two we consider as independent variables. If we differentiate, first with respect to p alone and then with respect to 7’ alone, we obtain the two sets of equations : WZ 02 OF dy OF dz 2, + Lj ll EAT == — AV; dz? Op dady Op 0x0z Op 0°Z 0e 07Z dy 0'Z dz 7% drdy Op iz dy” Op aE On0z Op "4 y sxe (6a) 0°Z da 02 Oy 0727 0z Ordz Op dydz Op | Oz? Op 1 534 0°Z Ox A 0°Z dy fg 0?Z dz i _& dx? OT OxdyOT Owdz OF °° Te 17 0:Z Oz 0°Z Oy 0?Z dz Q, OudyOT | Oy? OT | dedzdT |T BEE C7 0°Z Ow 0°Z dy 0?Z de Ms Oxdz OF’ Oydz OT dz? OT Gs TET : , OV Qe We have again written AV, for nn ete., and — for Lv oH ——Y7,, ele. Dm 0: If we multiply the first of the equations (6a) by = the second Oy by — =, TE ete. and add together the 2n equations, we obtain an equation the right hand side of which is: A 4 Ox Oy etc, the first of equations (66) by Do’ the second by an’ P p The left hand side of the resultant equation is zero. This may be shown as follows. Each term of the left hand side contains one of dz O Uno UM OE = iat etc. from the equations (6a). Let us consider Pp Op : : Ow the terms which contain one of these unknowns, e.g., ae In the sum- mation these terms are contributed (1) by the first terms of the equations (6a) and by no other terms of these equations, (2) by the complete left hand side of the first equation (66), which was multi- 0 plied throughout by = and by no other equation of (65). P Om The terms involving = therefore : Ie Ox OZ Oy OZ Oz 0°Z 07 dc?’ or dwdy’ OT drde’ OZ0« OZ0y 0°Z dz Oa? 07" dady 07" dxdz OT" ” . from (6a) and from (60), 0 all terms being multiplied by ar B From the structure of equations (6a) and (65) it appears that the 535 Ox sum of the factors by which 55 is multiplied is zero, and that the Pp same holds for each of the “unknowns”. The left hand side of the resultant equation is therefore shown to be zero. We have then as a result: Oz Qzx Ox Oy Qy dy 5 Xe + Ve de kK 44+ 40,4 ..... ==, EL ror that is, equation (2) results. This is the equation which we set out to establish as an extension of Braun’s law. Note J. It is not necessary that the solvent should be a pure substance. It may be a mixture of different substances of which, however, none occurs in the solid state. With this assumption the above method of proof remains exactly the same, and the validity of the result is unaffected. The quantities Q, etc., have, of course in general different values when the “solvent” is differently constituted. Note [/J. In the above treatment we have nowhere made use ot any explicit relation connecting Z and the composition. It follows from this that the results are valid both for constant and for reacting components. The only assumption made was that the components were independent in the sense of the phase theory. Note III. For the general case we can give a demonstration on the lines of that given for the simple case of three components. It would then be seen that we must deal with a state of stable equi- librium. Since the proof is more involved than that given above we do not reproduce it here. In a later communication dealing with a more general problem another proof will be found. Katwijk a. d. Rijn, August 1919. Physics. — “On the rings of connecting-electrons in Braaa’s model of the diamonderystal” By D. Coster. (Communicated by Prof. H. A. Lorentz). (Communicated at the meeting of October 25, 1919). The beautiful investigations of the two Braces‘) have given us a clear insight in the structure of the diamonderystal. As is known according to these investigators the structure of this crystal may be represented by the following scheme: a set of cubes, where the C- atoms are situated in the corners and in the centres of the side- planes; in which another set of identical cubes, which may be obtained from the first by translating it parallel to itself in the direction of one of the cube-diagonals over a quarter of this diagonal (see fig. 1, where only those atoms are represented, which are situated within a fundamental cube). If we assume, that the valency of the atoms also have a principal meaning in the crystal, this system is of a perfect symmetry. Every C-atom namely has in its neigh- bourhood four other atoms at the same distance and symmetrically situated. (The lines which join each atom with its + neighbour- atoms form the diagonals of a cube). In this way the four valencies of the C-atoms are satisfied. Now we may assume, that the “bonds” between the atoms are formed by rings of electrons as it is the case in Bour’s model of the hydrogen-molecule. DrByx and SCHERRER °) for instance suggest a model, where each carbon-atom sbould part with four electrons, one for each valency, for which consequently two electrons should be available. These should revolve about the connecting-axis of two nuclei in a plane perpendicular to this axis and half-way the distance between the nuclei. So the nucleus itself should still retain two electrons and behave at a distance as a four- fold charge. If once we have admitted, that the “bonds” are formed by rings of electrons, from the point of view of symmetry there is much to be said in favour of this model’). Degije and ScHERRER however arrive at the conclusion, that such a model is inconsistent with the experimental data of the two 1) Proc. Roy. Soc. Londen (1914) A 89, p. 277. See also: Brace. X-rays and crystalstructure. 2) Phys. Z. S. (1918) XIX, p. 476. 8) Of course many difficulties yet remain, e.g.: how is the direction of rotation in the orbits. We can also say but little about form and magnitude of the orbit. 537 Braces (and also with the data, they have obtained with their own method of erystal-photography). In my opinion however they neglect an important element in their reasoning and in this state of things nothing can be said about the existence or non-existence of such rings of electrons on account of data about scattering of Röntgen- rays. This | hope to prove in the following. To this purpose | intend to follow the clear method in which Brace has treated the subject. We consider the octahedronplanes (the planes (111) in the usual notation), which contain the C-atoms, e.g. the plane A, B,C, F,G,H (see fig. 1), a second plane contains D, a third £. All these planes contain an equal number of atoms, their mutual distance is alternately + d and ? d, as represented by fig. 2. If we only regard the reflec- tion by the planes a, accord- ing to the ordinary suppo- sitions we shall have a maximum intensity in the reflected beam for 2d sin p = nA, here 7 has the values 1, 2, 3 etc. Regarding also the planes a’ we see that the spectrum of the 2"¢ order (n = 2) disappears, because the planes a’ give half a wave- length phase-difference with the planes a. For the same reason the spectrum of the 6% order would disappear. Fig. 1. The Braces have observed with the use of Rh.-K-rays spectra as far as and including that of the 5 order; of the spectrum of the 2°¢ order nothing could be detected. This very result has given them one of their strongest arguments in favour of the erystalmodel they suggested. With the model of Dugum b------------—- ‘ and Scnerrer it is another case. In i= the usual way they assume, that the Der See wen scattering is only caused by the electrons Lissa zi and may be calculated in the classical ei : 5 : 3 Fig. 2 and 3. manner. In their caleulations they 538 suppose, that the connecting-electrons may be placed in their common centre of gravity. The octahedron-planes are situated as represented by fig. 3. In 6 and 6’ we now have the connecting-electrons, in 6 three times as many as in 0’. In this ease the nucleus-electrons give also no contribution to the spectrum of 2rd order, the connecting- electrons however should give an intensive spectrum; whereas, as has been said before, the experiment does not give the slightest in- dication of it, therefore DeBijk and ScHeRRER reject this crystalmodel. Regarding however a definite octahedron-plane (for instance that with positive indices 111), we see, that only 4 of the orbits of the connecting-electrons coincide with those planes (i.e. those belonging to 6 fig. 3). The other orbits form angles of about 70° with these planes. From the following calculation it may be concluded that it is not admissible to assume, as in fact is done by DreBijr and SCHERRER, that the electrons of these orbits always remain in the same octahedron-plane. For the sake of simplicity we assume the connecting-electrons moving uniformly in a circular orbit. Suppose 66 (fig. 4) to be the considered octahedron-plane, cc the plane of ie the orbit, both perpendicular to the plane of the paper. The l different phases of the beams b b reflected in the ordinary way | by the electrons of the plane bb are only determined by the c distance h of the electron to 6 6. Fig. 4. To caleulate the total reflected beam we are to multiply the separate beam from each electron by he Ee (p is the complement of ich the phasefactor e °°, where x= the angle of incidence). If we assume the electrons distributed at random in their orbits, then the probability that an electron is at a distance h > h- dh, is dh a VIT Therefore the total amplitude of the reflected beam is to be multiplied by (1) +1 : 0 eh de =. oe — e—ilxcos» day —= J, (lx) oh ies . (2) a Vla? It —l —7 here J, is the BessrrraN function of order zero. 539 4 a sing Taking into consideration, that « = , we find, if Brace’s relation 2d sin p= nh is satisfied, 2 rn Deden /(6) As is known the function J, is real for real values of the argument and oscillates between decreasing positive and negative limits and so behaves like a “damped” sinefunction. Here this means, that the phase-difference between the resultant beam and a beam reflected by the plane 64 is zero or 180°. The absolute value of (2) is always less than 1 except for the argument 0; the motion of the electrons therefore implies a decreasing of the intensity of the reflected Rönt- genbeam. The experiment requires, that the spectrum of the second order by reflection from 6 and 6’ disappears. This happens strictly if TUe REEL TER) since the plane 6 contains thrice as many electrons as 6’. The smallest value of /, which satisfies (4) and (3) for n= 2 is 0,258 d. If we assume according to Brace d= 0,203.10 8 ¢.m., then /—0,524.10-§ ¢.m., which should give for the radius of the orbit of the electron »=0,56.10—-§ c.m., which value cannot be excluded for being impossible '). Here it is of importance that the relation (4) holds independently of the wave-length of the Röntgenrays. Now | do not intend to attach high value to this calculation of the radius of the orbit. Firstly because my supposition (uniform circular motion of the electrons) is too schematic, secondly it is not probable, that Desye and Scenerrer should have been able to ascertain an intensity which should remain for instance below a 100" of that of the spectrum of the first order. This gives in the above case for r all values between about 0,52 and 0,62.10—8 and also between 0,70 and 0,81.10-® e‚m. Greater values of 7 are a priori improbable. Now the question arises if the existence or non-existence of the rings of connecting-electrons yet may be proved in the manner suggested by Dupre and Scuerrer. The spectra of higher order obtained by reflection from the octahedron-plane are not adapted for the purpose. Thus the spectrum of the 6" order should give a difference between the model with the connecting-electrons and that without. ') If we only take account of the change of the two nuclei concerned vas a fourfold charge) and neglect all the disturbances, then according to Bour an orbit of one quantum and two electrons has a radius of about 0,75 10-8 ¢.m. 540 First however the intensity decreases in general with the order of the spectrum’); secondly the intensity which we should expect according to (2) is very small, because J, (lx) is again negative for nm = 6 for the considered value of 7 (about 0,55.10—8 e.m.). Now it is interesting to consider the reflection by the other erystal- planes. Here we shall follow the method also used by DeBrw and SCHERRER. When we have a regular erystal, then the intensity of a beam retlected according to the relation of Brace is proportional to the square of the so-called structure-factor®) S, which is given by Sa Sane Teh Pn eh) vo eee Here A, is proportional to the amplitude of the beam radiated by the zt" centre of the fundamental cube, pn grin are the ordinates of this centre in the cube, whose edge is 1; h,h, h, are the indices of the considered crystalplane. These may have a common divisor. If they are for instance 024, then the spectrum of the Ard order of the plane 012 in the ordinary notation is meant. For Brace’s crystalmodel this factor is: Be. „ir Gelet) Na Relea SE Sade cy DerByr and ScHERRER assume that the connecting-ring scatters in the same way as the nucleus-electrons. Also for their model we may put all A,’s=1. Therefore they obtain: iS Oale) iS athe ths) i= at) +e 14e 4 — Spe (tee leah cana. gen Cal) eee (07) im (hj +h i = (hath) iz (hath ie gi all ar otk sal) Taking into consideration the position of the orbits of the ring- electrons in the above-given way, we get for the structure factor: i= (yp he bh BE) S=2lite? Et Oe | i= (hath i= (hgh itho) | rede jee E iz (hy +h) ix (hath ix (hs+hy) } | ‘des Taig ed a ak Here 7 and x have the same signification as in (2) and (3); the indices at the different magnitudes / refer to the four different angles 1) See e.g. Braga. Proc. Roy. Soc. A. 89, p. 279, fig. 2. 2) See D. and Scr. Phys. Z. S. (1916), p. 279. For the meaning of this factor see: Marx. Handb. d. Rad. Bd. V. p. 581. 541 which the orbits of the electrons can make with the crystallografic plane under discussion. 2) The annexed table gives =. calculated‘) for the three cases; in the last case once for a value r = 0,56.10—8 and once for r =0,81.10 Sc.m. Here the ratio between the numbers standing in the same column is only of importance. We have to remark that the spectra (002) and (024) disappear independently of the assumed value of r. Only to make also the spectrum (222) disappear we are bound to certain limits in the choice of r. Indices. | Br. D. and Sch. r =0.56.10 * |r—0.81.10" (11) eee Ae 2.9 5.8 (002) | 0 0 0 0 (022) | 36 4 0.61 7.8 (13) | 18 0.34 1.64 3.55 (222) OR NLG 0 0.038 (004) BB te dle ed SAAD 9.0 (133) 18 2 2.1 2042000 (024) ost Hildo 0 0 In calculating this table no account has been taken of the different factors’) that strongly affect the intensity of the expected spectra (mostly those of higher order). Because as yet all is quite uncertain and the foregoing speculations are very schematic, I thought it unnecessary to involve them in the calculations. The table however shows that especially the numbers of the fourth column do not more contradict the experimental data than those of the first®). From which we may conclude that for the present it will not be possible to draw a conclusion from the experimental data concerning the existence or non-existence of the connecting-rings. Perhaps here the study of the erystals of homologous elements (Si,Ge)*) may bring a decision. 4) The first two columns are taken from D. and Scu. 4) e.g. LORENTZ- and Depise- factor, see Marx Handbuch V, p. 581 af. 5) See Brace |.c. and Degije and Scuerker l.c. *) Si seems to behave completely as diamond, cf. Desise and Sen. Phys. Z. S. (1916) p. 282. With Ge the number of connecting-electrons is already small compared with that of the nucleus-electrons. 36 Proceedings Royal Acad. Amsterdam. Vol XXII. Chemistry. — ‘“J/n-, mono- and divariant equilibria’. XX. By Prof. ScHREINEMAKERS. (Communicated at the meeting of November 29, 1919). Equilibria of n components in n phases, in which the quantity of one of the components approaches to zero; the influence of a new substance on an invariant (P or T) equilibrium. In the communications XVI, XVII and XVIII we have seen that a region is two-leafed in the vicinity of a turning-line and one-leafed in the vicinity of a limit-line [e.g. curve ab or ed in fig. 1 (XVI)]. We shall consider the latter case more in detail. We take the equilibrium H=— #, + F,...+ HK of n components in 2 phases under constant pressure. This equilibrium is (Comm. XVII) monovariant (P); viz. it has one freedom under constant pressure. The equations (2) and (3) (XVII) are true for this equilibrium; on change of one of the variables e.g. of x, this equilibrium traces in the P,7-diagram a straight line parallel to the Z-axis. In the vicinity of a limit-line of a region e.g. in the vicinity of curve ab or cd in fig. 1 (XVI), the quantity of one of the com- ponents approaches to zero. When this is the case with the component X, viz. with that component, the quantities of which are indicated in the different phases by w,a,...a,, then in (2) and (3) (XVII): : OZ, OZ, OZ n Ox, , On, ; ay Òzn become infinitely large, viz in Z, the term 2, /og x, is found, in Z, the term w, log a, ete. Now we write: Z,=Z, + RY x, log x, D= IRI le « o (Ì) Herein Z,’ Z,’... and their differential quotients remain always finite also for z, — 0, 2,0... It follows from (4): Oy VE care = + RT (1 + loge) Ou, Ow, ; Wipe (2) 07, 07," ry == + RTL + log «,) Oz, Oa, ete. The m equations (2) (XVII) now pass into: fae || 543 0Z,' 0Z,' AR Ee es BS = | 2, Oy, VAR 2 Ze — RT zE Ya .-.-- == Oz, ~~ Oy, ete. The first series of the equations (3) (XVII) passes into: 0Z,' 0Z,' Slog ae ted logian wat, KG SEE (84) Ox, Ox, The following series of the equations (3) (XVII) become: dZ,' 0Z,' OZ, (5) SS SS = +0000 == = vo dy, Oy, un etc. It follows from (4): x, 0Z,' 0Z,' RT log == UE OER 4 (6) OO, RT log —= 1 Oz, Ox, or =n By SMG, oo eos =o s (U) in which u,‚u,... are defined by (6). For values infinitely small of z,2,... the ratios between LE, L,--- En are consequently defined by (7). Now we give the increments: d7’.x,2,... dy,... dy, etc, to the variables 727, 2,...4,y,-.-. etc, in which we put z,=— 0 == : Now it follows from (3): 0Z,' H dT + RT a, + y,¢d— + oib & oe = — dK oy, (8) nr 0Z,' pd B Elendes dk | Ya ete. in which the sign d indicates that we have to differentiate according to all variables. Now we add the m equations (8) after having multiplied the first by 4,, the second by 2, ete. Then we obtain, when we use the relations which follow from (5): ; 0Z,' 2(AH) dT + RT Z (A 2) 4 zona) ) +...=— ZS (3) dK (9) Ji Now we define 2,À,... in such a way that they satisfy the n—1 equations (10) 36* Dt + a, =0 (A= ) Y, ae A5 Ys dp Dr 000 OD =| dn Yn = 0 (10) ADE ae EZ dte th SE Aln Zn =S) etc. By this the 2—1 ratios between the coefficients 2, 2, .. . are defined. As Po (CEO == 25 OR ae Gn ae 6 oo oo SE Onan | NDL eee AE | GD the ratio B(aw): 2(2H) is also defined. Now it follows from (9) *): RT J (A) ON ne (4 (2H) The value of dT’ in (12) depends on. 2(%z), consequently on the n increments 2, 2,...an. We may express them, however, in one of those increments e.g. in 2,. With the aid of (7) we obtain then: RT «, (Ap) Deis (wel (dT)p = (13) wherein : SAP) Att He Aas ee ees Eeen OL) When we take the equilibrium H— /,+ F,+...+F, of n components in 7 phases at constant temperature, then it is mono- variant (7Z’). In the same way as above we find now: RT Z (Az) RT a, = (Ap) adr =S == 15 SS SA SOY) © Herein 2, 2,... have again the values, which are defined by (10) Z (Ae) has also the same value of (11) viz.: 25 (2 x) == À, x, of A, Wai 005.0 4 + an En while Pe (l@) 23 (2 Vi a VA + ds Ve San tes eerie dn Va | X(4u) has again the same value asin (14). In the previous considerations it is assumed that the quantity of the component X in the equilibrium H= F4 H+... + Fh of mn components in 2 phases is very small. When, however, this quantity becomes zero, then ME passes into an equilibrium of n—1 components in ” phases. This is monovariant and is represented in the P,7-diagram by a curve. Under constant pressure it is inva- riant (P), at constant temperature invariant (7’). In this invariant (P or T) equilibrium between the phases F,... Fn may occur a 1) For another deduction see F. A. H. Scurememaxkers, Die heterogenen Gleich- gewichte von H. W. Baxuuis Roozesoom. III. 289. 545 reaction; the quantities À,...2 of the phases participating in this reaction are defined by (10). The change in entropy occurring with this reaction 2(/H) is defined by (11), the change in volume 2(4 V) is defined by (16). Some of the coefficients 2,...4n are positive, other ones are negative. As long as we do not assume for this a definite rule, we may arbitrarily interchange positive and negative. We assume the following: The coefficients of the phases, which occur with a reaction, are taken positive; the coefficients of the phases which disappear with the reaction, are taken negative. Now (4) is the algebraical sum of the quantities of the phases which participate in the reaction, of course this is zero. X(4y) is the algebraical sum of the quantity of the component Y which participates in the reaction; this is also zero. The same is true for the other components. As the component X does not occur in the invariant (P or 7) equilibrium, (A«) has, therefore, another meaning. When we add, however, a little of the component X to this equilibrium, then it is divided between the » phases; this division is defined by (7), so that z,...z» and consequently also 2 (Ar) are defined. Now we imagine a reaction in the invariant (Por 7’) equilibrium; 2,...4, represent, therefore, the quantities of the phases participating in the reaction. When those phases would contain the quantities E‚--- An Of the new component, then 2(2r) would be the algebraical sum of the quantity of the component X, which participates in this reaction. For this reason we shall call 2 (Ac) “the fictitious quantity of reaction of the component X”. Now we take a point on the limit-curve of a region, e.g. point h on the limit-eurve ab in fig. 1. (XVI). This limit-curve represents an equilibrium of »—1 components (viz. the components Y, Z, U.) in 7 phases, consequently a monovariant equilibrium. In the point h itself PT y,y,..2,2,..ete. have definite values; the same is true for the ratios of 4,...4n which are defined by (10) Now we adda little of the component X, this is divided over the n phases; this division is defined by (7). For a delinite value of e.g. a, the ratios Ear): LAH) and Er): (AV) are also defined. In accordance with (12) and (15) we know consequently also (/T)p and (dP) 7. When (/T)p is positive, then the region Mis situated at the right of the point h; we enter then the region, just as in fig. 1 (XVI) starting from A in the direction Al. When (dr is negative, then the region M-is situated below 546 point A; then we enter the region, just as in fig. 1 (XVI) starting from h in the direction hm. Consequently the region H is situated at the right and below the point h. The direction of curve ab itself is defined in every point by: OPDE) NS SIA (17) AT KAD It follows from our assumption over the sign of (d7’)p and (dP)r that we have assumed 2(2r): 2(4H) to be negative and 2 (Az) : 2(AV) also to be negative. Then it follows from (17) that curve ab must be a curve, rising with the temperature, in the vicinity of point h, as is also drawn in fig. 1 (XVI). In fig. 3 (XVI) abchd represents a limit-curve which has a maxi- mum of pressure in 6 and a maximum of temperature in c. It follows with the aid of (17) from the direction of branch ab that E(,H) and Z(AV) have the same sign; we now choose the signs of a,...4, in such a way that both are positive. Then it follows from the direction of the branches be and cd with the aid of (17), which signs 2(2H) and 2(AV ) must have on those branches. Then we have: on branch ab Z(AH)>0 z(aV) >0 in 6 E(AH)=0 (AMV S30 on branch bc Z(ÀH)0 in C Z(AH)<0 (AL 1) 0) on branch cd 2(2H)<0 Z(AV) 0, then it follows from (12) and (15) that the region must be situated at the left of and above branch ab, at the right of and above branch be, at the right of and below branch ed. Then we bave fig.5 (XVI). [As it is apparent from the position of the letters, the printer has turned this figure; for this reason the reader has to place it in such a way that the tangent is horizontal in 6 and vertical again in c}. We may assume also that 2(2r) is positive in the one part of the curve, negative in another part. We assume that ZX (dz) is 547 positive in part abf of curve abed fig. 3 (XVI) and negative in the part fed. Then it follows from (12) and (15) that the region must be situated as is drawn in fig. 3 (XVI) viz. that a part a fe of this region must be situated outside the limit-line and that this region must have a turning line ef. It appears from the following that this point f must be a point of the turning-line. In this point 2(2e)— 0. As in this point also the equations (10) are valid, a phase reaction 2,#, +... ar #,=0 may oecur between the » phases of the equilibrium H= F, +...+ F,, in which an infinitely small quantity of the component X occurs now also. ; Consequently when ina definite point f of curve abcd ZY (Ar) = 0, then f is a common point of turning- and limit-line; later we shall see that f is a point of contact. When (42) changes in sign in f, then f is a terminating point of the turning-line as in fig. 3 (XVI); when however (Ax) does not change its sign in f, then f is not a terminating point, but the curve proceeds further. From (12), (15) and (17) follows the relation: : dP (dP)r : (aT) p -— (55) ze oen nem ks) d1 Ti 0 5 yaar dP The index =O in the second part of (18) indicates that TP C is true for the limit-curve, in which the component X is missing. In order to comprehend the meaning of (18), we imagine the P,T-eurve of the limit-equilibrium, to be drawn in which the com- ponent X does not occur, therefore. For this we take the curves ab and ed in the figures 1, 2 and 4 (XVI) and curve abed in the figures 3 and 5 (XVI). [We have to place again the latter figure in the right position |. We shall call the branches on which the pressure increases with increase of 7 the “ascending” branches, e.g. the branches « b and ed in the figures 1, 2, 3, 4 and 5 (XVI). A branch like eg. be in figs. 3 and 5 (XVI), on which the pressure decreases at increase of 7, is called a “descending” branch. : ENS she ; On an ascending branch (=) iE positive, then it follows from (18) that (/P?)7 and (dT)p have opposite signs. When (/7’)p is positive and consequently (d/’)y negative, then the region is situated at the right and below the branch; this is the case with respect to branch ab in the figs. 1, 2 and 4 (XVI) and with respect to branch ed in the 5 548 figs. 2, 4 and 5 (XVI). When (dT’)p is negative and (dP)r consequently positive, then the region is situated at tbe left and above the branch; this is the case with respect to branch ab in figs. 3 and 5 (XVI), and with respect to branch cd in the figs. 1 and 3 (XVI). Consequently we find: A region is situated always at the right and below or at the left and above the ascending branch of its limit-curve. : dP d On the descending branch of a limit-curve ( ) is negative. It de follows from (18) that then (dP)r and (dT)p have the same sign. When both are positive, then the region is situated, therefore, at the right and above the branch. When both are negative, then it is situated at the left and below the branch. In fig. 5 (XVI) the region is situated at the right and above branch 6c; in fig. 3 (XVI) the region is situated at the right and above the part bf, and at the left and below the part fc of branch 5 c. Consequently we find: a region is situated at the right and below, or at the left and above the ascending branch of its limit-curve; it is situated at the right and above, or at the left and below the descending branch of its limit-curve. In Communication XI on: Equilibria in ternary systems, we have already deduced this same property for a special case viz. for the ternary region # + LG, in which ¥ represents a binary compound, with respect to its binary limit curve #+L-+G. Now it appears that this is true in general for each arbitrary region with respect to all its limit-curves. We may express the results obtained above also in another way. The equilibrium H= F,+...+ F,0f2—1 components in n phases is monovariant or invariant (P or 7’). When we add a little of a new substance X, then a new equilibrium #'— FF", +...+ F'n may arise. Herein the invariable phases have the same composition as in HH; the variable phases (which of course not all need to contain the new substance X) differ still only very little from those in #. We now may put the question: how must the temperature change under constant P or: how must the pressure change at constant 7’ in order that in both cases the equilibrium £ passes into EH’. It is clear that both questions are only another form of the questions, treated above: how must the temperature be changed 549 under constant P and the pressure at constant 7’ in order to pass from a limit-curve into the corresponding region. We take the equilibrium F=L+ F,+ F,+... of n—1 com- ponents in m phases (or of 2 components in 2 + 1 phases). Herein fF, F,... represent solid substances of invariable composition and L a liquid. On addition of a new substance X this occurs then only in the liquid. When in this equilibrium Z at constant 7’ and under constant P there occurs the reaction: ; ALE Alo IE PV a De S02 peo (19) then E(Àz)—2, 2, when viz. a represents the concentration of the new substance in the liqnid. When we put A, =1, then 2(AH) and L(V) are the increases of entropy and volume, when one quantity of liquid is formed at the phase-reaction. We represent them by AH and AV. (12) and (15) pass now into: ze and (AP) =. REZ) When we represent by AW the quantity of heat which is to be added in order to form with the reaction one quantity of liquid, then (20) passes into: (dT)p = — redt as TRIE 8 NT See l= Ry Reaction (19) may represent the common melting of the solid substances PF, F,...; this is the case when the reaction is of the form: VIRE IRE RR Le NEEN Eer (22) and when 4, 4, are positive. When the reaction is of the form: (d7)p = — (21) AT AR, Hated ag beta Po! ee ame . (23) in which we take also positive all coefficients, then it represents the conversion of the liquids 7, #,... into #,... when simultaneously liquid is formed. Now we assume that heat is to be added at the formation of liquid from solid substances, consequently at melting in accordance with (22) and at conversion in accordance with (23); then AW is positive; the change in volume at melting or conversion may be as well positive as negative. Now it follows from (21): The common melting- or conversion temperature of one or more substances is lowered by addition of a new substance; the common melting- or conversion-pressure of one or more sub- stances is; 550 raised by a new substance, when the volume increases on melting or conversion ; lowered, when the volume decreases on melting or conversion. This increase and decrease are at first approximation proportional to the quantity of the new substance. When we apply those rules to the melting of a simple substance, then follows the known rule of the decrease of melting or freezing point; the first formula (21) is then the known formula of Raovnt- VAN "T Horr. We may apply the previous deductions also when we substitute in (19) the liquid ZL by a gas G. In general AV is then positive and approximately equal to the volume V of the gas; by this we may give another form to the second formula (21) viz. GPP Pri NE BON (20) We may deduce the previous rules also in the following way. We take the equilibrium H= LFF... in which the new substance X is not yet present under constant pressure; then it is invariant (P) and it consists at a definite temperature, which we shall call 7. When we assume that reaction (22) takes place from left to right at addition of heat, then it follows: Pir Me ai J SO (L) (HEE CE ee cy towards lower 7 | towards higher 7. Consequently the equilibrium (D= F,+ F,+... consists at temperatures lower than 7,. When we add the new substance X, then # passes into #’ = L’+ F,+ F,+..., in which JZ, differs from ZL; this equilibrium £” exists at a temperature 7” which differs from 7. When we take away the liquid Z’ from H’, then it passes into FE, 4 F,+..., consequently in the equilibrium (Z) discussed above; as this exists at lower temperatures than 7,, it follows 7” < 7%. On addition of the new substance the common melting-point must fall, therefore. From reaction (23) we find the same for the common point of conversion. When we take at constant temperature the equilibrium E=L4+F,+ F,+..., in which the new substance is not yet present, it is invariant (7Z’); then it exists under a definite pressure P,. 551 When reaction (22) takes place with increase of volume from left to right, then follows: jp ain aa at ke Sb AWSS 0 (Z) [ie IED) aos aya towards higher P | towards lower P The equilibrium (Z)=— F,+ F,+.... exists, therefore, under pressures. larger than P,. Hence is follows that the equilibrium 2’ occurs also under a pressure higher than P,. Consequently when at common melting increase of volume takes place, then the melting pressure rises. From reaction (23) the same follows for the common point of conversion. When we assume that AV < 0, then it follows that on addition of the new substance the pressure of melting or conversion falls. Now we take the equilibrium: TN Oi DR SPOE A A RE of n —1 (or 7) components in m (or n +1) phases. Again L, L, represent liquids, #, F, solid phases of constant composition. Formerly *) we have called the temperature at which this equilibrium occurs under constant pressure the “Schichtungstemperatur”; we may call it also the stratification-temperature. We write the reaction occurring in this equilibrium: ZT EE iy Vial Oe NEEN du FF, +o, Et... ==) Sa (2,5) We may distinguish at this reaction the 2 main types: By oe, oe ea, Ged, Ee tg Fo Fs. (26) AL, +--+ oF, tud. Ap Ep Hd... +g Fy.--. (27) in which we take all coefficients positive. In (26) the solid substances may be wanting on the right side, in (27) on the right or on the left side. Experimental examples of both types are known’). In order to express the difference between the two reactions we shall say: in (26) all liquids are situated in reaction-conjunction, in (27) two or more are situated in reaction-opposition *). jk, A. H. ScHrermagers, die heterogenen Gleichgewichte von H. W. Baxuuis RoozeBoom ILI? 108. 2, fF. A. H. SCHREINFMAKERS ibid. III? 106 —113, 193—203. *) In order to prevent confusion, see the following. In the books III) and III? mentioned above phases are many times spoken of which are situated in the diagram in conjunction or opposition. When we call this situation diagram conjunction and diagram opposition, then it appears that reaction-conjunction corresponds with dia- gram opposition and reaction-opposition with diagram-conjunction, 552 When we add a new substance, then this divides itself between the liquids; its concentrations #,2,... are defined by (7). [It is apparent that the w’s in (7) have quite another meaning as in (25), (26) and (27)). For reaction (26) 2 (2e) = 2,2, + Aa,#, +..., in which occur only the 2’s, not the w’s. As the 2’s are all positive, 2(2r) is also positive. With this we assume that heat must be added, in order that reaction (26) takes place from left to right, so that also 2(4H) is positive. The sign of 2(2V), however, is indefinite. [It is apparent that in (AH) and Z(AV) the w’s of (26) occur also]. Now it follows from (12) and (15) _ PED apd ED) Oa . 6 20 » (ZE) Hence it follows: when we have an invariant (Por 7’) equilibrium with 2 or more liquids, which are situated all in reaction-conjunction and when we add a new substance, then: under constant P the stratification temperature is lowered ; at constant 7’ the stratification-pressure is raised when the volume increases at the formation of the liquids; lowered when the volume decreases at the formation of liquids. For reaction (27) is Ba (AV) ec yt et —i1,« so that E(4r) may be as well positive as negative. This depends on the partition of the new substance X between the different liquids. In order to illustrate this further we consider a definite case, viz. the equilibrium i Ib TE OEE vene SPUN rots o (8) between the 7 components YZ... N. Consequently in this equili- brium all components, excepted Y, occur as solid phases. As there are, therefore, n—1 solid and 2 liquid phases, it is invariant (P or T). Now we represent the reaction by: Melee debe ZE i Tp EN + Anti N=0 . (80) so that Z(Ae)=A,a,-+4,2,. For the definition of the relation between 4, and A, we take from (10) the equation 2() 7)=0. As the substance Y occurs only in the two liquids, it follows: 23 (Cl) = 2) de 2,95 SSO go oo oo (GI) Hence it appears that A, and 2, have opposite signs, so that reaction (30) belongs to type (27). We write it in the form: AZ ee a lin silk. (PE) We have put, therefore 2,1, consequently 2, is positive; of course one or more of the coefficients 4,.... may be negative. = 1), Big oee 553 Further we assume that Z, is the liquid, which is formed on addition of heat. {When this should be the case with Z,, then we should have placed £, in the left part of (82) ]. Now we have: Tay) =—y,—4,y, and Dax) = a, — Ar. za (EE). BRA ME (EE) Hence it follows: and (85) Herein AW is the heat, wanted for forming one quantity of the liquid £,; AV is the inerease of volume occurring at this formation, which can be as well positive as negative. Now we shall mean by partition-coefficient of a substance: the concentration of that substance in the liquid, which is formed on addition of heat, divided by the concentration of that substance in the other liquid. x,:2, is consequently the partition-coefficient of the new substance, y,:7, that of the component, which does not occur as solid phase. Consequently we find: when in an invariant (P or 7’) equilibrium with 2 liquids only components occur as solid phases, then both liquids are situated in reaction-opposition. The stratification-temperature under constant P by addition of a new substance: is elevated (lowered) when the partition-coefficient of the new substance is smaller (larger) than that of the component which does not occur as solid phase’). We may deduce from (35, similar rules for the influence of a new substance on the change in pressure at constant temperature. We may also give a more simple form to (34) and (35). We have viz. expressed the concentrations of the components in the liquids in such a way that each liquid contains in all one molecule. We may, however, also mean by concentration the quantity of the ') For some examples of the influence of a third substance on binary equilibria see F. A. H. Scureinemakens, Die heterogenen Gleichgewichte von H. W. BaKuHUIS Roozegoom III? 160, 554 components when the liquid contains one molecule of the component which does not occur as solid phase, consequently in our case of the component Y. As, therefore, y, and y, become — 1, (34) and (35) pass into: RT? (va) AW Len RT (2; 7) (a7) p = — and (dP)r = I (36) Now we find: the stratification-temperature is raised (lowered) under constant P on addition of a new substance, when the concentration of the new substance in the liquid, which is formed on addition of heat, is smaller (larger) than its concentration in the other liquid. The first formula (86) has been deduced formerly for equilibria with two’) and more’) components. To be continued. Leiden, Inorg. Chem. Lab. 1) F. A. H. ScHREINEMAKERS. Zeitschr. f. Phys. Chem. 25, 320 (1898). 2) H. A. Lorentz ibid. 25. 332 (1898). Mathematics. — “On a formula of Sytvestir’. By Prof. W. KAPTEYN. (Communicated at the meeting of November 29, 1919). In his paper “On the partition of numbers” Quart. Journ. of Math. I (1857) p. 141—152, Sy.vesrer has given a general formula for the number of solutions in integers (zero included) of the equation Oo ba, SE SEES Eo doa (Rr Don ou cee oan (1) where nm and a are given integers. Applying this formula, which is given without proof, to a partic- ular example, I found a fractional number. Of course this result is absurd. I therefore tried to construct a proof and found, as will be shown hereafter, that SYLveEsSTER’s formula wants a slight correction. If the fraction 1 1 D(z) (lea) (l—2m) . . (1 @) is developed in ascending powers of 2, it is evident that the coef- ficient of 2 gives exactly the number of solutions in integers of the equation (1). We therefore proceed to reduce this fraction to its partial fractions and to develop every one of these in ascending powers of z. The denominator being a compound quantity, the first thing wanted is to determine its different factors. Let 1—a" = 0 denote the equation containing all the prime roots ver) : of the equation 1—r" = 0, then we have k 1—a" —— 7 ladi, . ° . . e e » (3) i=1 ————_ where d,,d,,...d.(d,=1,d,—=m) represent the different divisors of m. To prove this theorem let m=p%q?.. ¥, p,q,...t being prime numbers, then the divisors of m are the several terms of the contin- 2 Bp ) (143r)(14 £0). (04 209), 1 1 1 One of these being ued product EE gan. we know that the number of prime roots of the corresponding equation 1—«d = 0 is 556 rea D013) The number of the prime roots corresponding to all the different divisors of m is therefore 1 a ip 1 B Re ie De Dd q/ 1 TN A 5 Eee Sese. .Ò =m. t 1 Now these prime roots being all different, they must satisfy an equation of degree m, which, because every one of these roots is also a root of 1—«v" =O, must coincide with 1—a” — 0. To illustrate this theorem, put m= 20 — 2? . 5, then the divisors are Us A 45 Oo MO, ZO and the factors corresponding to the prime roots l—a, Ita. 14-27, 1ta4 27+ 2*. 1—a?— oaf, l—a?+ a4 ada! or l—a, 1—2?, lef, 1e), 1—2z, 1, The continued product of these factors is evidently 1—«?°, or 6 la? = (ledi). ii Developing in the same way the several factors of @ (x), we may write Ti To 2 7 DOMUS US) re (ISEB Te 5 ar 0 (4) where the quantities «;, ranged according to ascending magnitude, represent the different divisors of a,,a,..a,, and 7; the numbers of the divisors «;. We may remark here that ¢,—=1 and r, =7. If, for instance a, +20e,+ 52, + 100, 4+ 202,=n is the given equation, we have ® (x) = (le) (la?) (1 —#) (1 — 2°) (1— 2”) where La = lr Nay = Nang lee EE IPS tlw oo log? 5 lw lg a , la lea a Nee Og NG 10 30 5 ll hence @ (x) =(i—z)® (le) (et) (la) (lS =<")? ASN) or or 1 1 Proceeding now to determine the partial fractions of Ae x we know by Cavcar’s formula that lie 1 1 EEOC EN EEE DE) (vO) where the double parentheses denote that the residues must be taken for all the roots of ®(z)=0, viz. for all the roots of the equations 124 = 0, ls =O dee 0. By developing the factor 1 2 gt we get immediately for the required coefficient of 2” a] 8 pra. oe an od BLL = 5 W, on REKEN en where 1 Toes Hi (lazy (Lae)? (Let em) en or, restoring the original form of d(z) 1 lee UL t ert (l—zn) (Lea) (lee) ( (1--2)) where the residue is to be taken for all the roots of the equation OME eee (6) Representing one of these roots by o and putting Olea the preceding value of W,, takes this form 7 gn pnt t pe Sb aes a at (1—o% e—% 1) (l—o% lem Aad) (ij AG 7 ) ((6)) pent (20) wherein the summation must be extended to all the roots @ of the equation (6). The term W, may be further developed, for the corresponding equation (6) 1—-z=0 or 1—z=0 shows that the only root is g==1. Therefore W, reduces to N ent t a E (l—e at) (1—e CA) eb (l—e ")(() Proceedings Royal Acad. Amsterdam. Vol. XXII. 558 : : : ; ly This residue is the coefficient of Fu in eae (GO = C=) (Noch), Ln da) i= : Now (a) JB 2 152 di lg (le) =lge — bed af at 1 ry in 2/2 4/4 !6 as may be shown by integrating 1 ee. B B TB - = 1 12 Sc =< P=, mod z Qa Pi 7 Aj Th Beh 6 IE where B =t, B, = zh B, =d between the limits o and z Substituting the values of lg (l—e%), we obtain Be IBS, 0 1 vt = = 6 Wp Be onl 42 2 a,4,-..4,¢ where pnt ds and s;—af Haf 4... af hence W, is the coefficient of #—*t in the product 1 1 ( 1 1 1 el | DÛ de =D DP de pe ie Ae ANRA u 2 T By 2 Tp ig 1 t? : (1-3 ker a fy te cae EE 2880 Applying the preceding to compute the number of solutions of the equation 2, toer, + 102, + 20e, ==. we first observe that the different divisors of 1, 2, 5, 10, 20 are ard Uc OMEN Vesa nino be enone wv 5, 4 10, 5 10 20 thus D («) = (le) (la)? (Le) (le)? (1 — a7")? (Lr?) 559 The number of values a; being six, we have six different terms Wa, In this case, having Ina 455, a, == 10,4, == 20 530, s,=170642, »v=n+19 we obtain oe i ee eu! 2 ESOC es MO or ai - 48000 For IW, the equation (6) ] E + 762? + 1901 n? + 17366 + 41930 “a or U sss). shows, that also in this case there is only one root v—=— 1. There- fore W, reduces to a = (—1)" ent t TC ele (Ie 9) (1—e-19) (Ie) (() 1 or (—1)" multiplied by the coefficient of in nt—Iq(1 He!) —lg(1—e- 2) —lg(1 He) —Ig(1 —e—1') — lg(1—e—20), NE Developing the logarithms in this expression by means of the equation (a) and lg (L+e -) = lg (1—e- 27) — lg (l—e—?) = | ee BINAS B (0) Re mm me we get 2 hbe En EEE IOO ene Tes Hence ge (15»* — 7274) 7 48000 or (CD (15 2 4 570n 4- 4687 1). * 48 00 ; For W, the equation (6) is 1— z*=0 or : Le 24—0 37* 560 hence o-” ent t i, = Sy <= (1— ge~')(1 —9%e—*')(1—o%e 5')(1 —e*%e—101) (1 9?%e~ 204 ((¢)) where the summation must be extended to both the roots o of N the equation 1+ z273=0O Writing 9? =— 1, we obtain Ee” ent t Wi Zi ee re IgA Fe 09) (12209) (9) 1 0" l = Den == DT 80° (1—o)? 160 Denoting by >, the sum of the pt" powers of the roots, and observing that e*—1, we know D2) == Oh =S (, == 2, 0} and generally, & being an integer number Zar, Zug), Zire, Zuppa 0. Therefore W, i 160 Dan 1 which gives different values for different values of 7. According to n=4p, 4p+1, 4p4+2, 4p+3 we get respectively the four values Wa fo, i, 6 1) STE ’ ? ’ For W, the equation (6) gives 1—a2'=0 or lte+e7+a*4+ aet*—0 thus w dz >S oe” ent t TT Se ee SS OEE) Putting nt—lg (1—ge—')—lg (1—97e —2')—Iq (L—e—5 tlg (1—e— 10 *) lq (1—e— 204) Ke —e 6 and reducing by means of the equations (a) and (6) 20? 35 40? 525 D= e = t 4 5 + - e x ete lo 1—o’ 2 2(1—e)? 2(1— 0’)? 24 E 1000 (1—e) (1—0?) # we get = 9 60? WRS 5 n° + gele n+ 200: (1—@) (1— 0”) re = rn 525 340 + 9807 — 420° — ae] 2 Gens 561 or 25 n* + 35n + Ge Dee nea OET Lol ; 2000 (1—e) (1—9’) Eyres = o-" (20 + 607) 2000 ~ (Ie) de _ 1 yer B4o_~ 989" — 429" — 11494) 2000 (1 —e)(1 —e7) With the same notations as in the preceding case, we obtain Se SS EES ee De and generally ee ete Las pea) According to the values n= dp, 5p+1, 5p+ 2, 5p+ 3, 5p4+4 we find Er zele == Zo" (1 tete = Tek Ge 1 Bs eae L4n42— Dant3 — Dani) = (1. 0, t =a =) „e "(2e + 69’) 1 EME 5 „2 "(B4g + 9807 — 429*— 114%) 1 a (eo) 07); 25 + 32 Zine + 162 Dy,43 + 352 Dan) = (2 Zan43 ap é Zana — 6 Ean) == (= 6, 0, 0, 2, 4) (— 588 Dan — 8 Zan + (2069) AN 16.481, 176) or} Do and therefore l = 3 N \ ng Bl s 18000 (24n? + 840n + 6300) (1, 0, 1, il ) l + ——. , 3, 0, 0, —1,.— 2 48000 48n ( ) 1 _48 F—_ ___ (269, 4, —16, —81, — 176). 48000 5 In the same way we obtain, according to n==10p, 10p+1, 10p42,... 10p +9 I W,,=— 120 n + 1800) (—1, 0, 1, 8, 3, 1, 0, —1, —3, — 3 ze 45000 ° de, )( , 4 . 60(5, 16, 21, 29, 19, —5, — 16, 21, 29, 19) 48000 and according to 562 n=20p, 20p+1,... 20p419 1200 == (—5, —8, —5, —7, —5, 3,0, 3 Ms ty a £8000 ° ? OAN SKO NON RO OD — 7). From the preceding formulae we may deduce the several results Wor pO (Gre > HE) To illustrate this, taking n =10p + 7, we obtain (n* + 76n? + 1901n* + 17366n + 419302, 1 48000 W, =i lors Wp LAB) 1 Wenn 600) W, 00 24n? + 840n + 6300 ) taal AE) We = gon aor 1207) 800K) ] * 48000 { en >= 3600 ) and finally = W. We 48000 =p + 1)(p + 2) p® + 22p + 9). That this value is an integer may he easily seen by writing = W..=3P(p + 1)(p -+ 2) (10 p + 22) + 3(p + 1) (p + 2). Comparing the general formula of SYLVESTER ——— (nf + 76n* + 1910n? + 17516n + 43329) o” ent t (lg e-% 4) (l—e eat) . (l-e7e 7) (0) with the formula (7), it is evident that in his formula @” ought to be replaced by or. This makes no difference in the values W, and W,. In the example treated by Syuvesrer ot 2,3, 4, Hoe, t 6x, =n the values of W,, W,, W, and W, however want correction. VP) a Physiology. — “About the influence of radio-active elements on the development’. By Prof. A. J. P. van pen Brork. (Communi- cated by Prof. H. ZWAARDEMAKER). (Communicated at the meeting of November 29, 1919). One of the elements, composing the living protoplasma, potassium, is radio-active. The investigations of ZWAARDEMAKER and his pupils about the signification of potassium in the organism have proved that, chiefly by streaming experiments of the isolated frog-heart, potassium can be substituted by an aequiradio-active quantity of any other radio-active element. On account of this ZWAARDEMAKER concludes‘): ‘die Radioaktivitat und keine andere Eigenschaft der sich gegenseitig vertretenden Atome erfüllt die für die Automatie notwendige Bedingung” (Le. pag. 49). Next to this substitute ZWAARDEMAKER has fixed the attention on a second fact, viz. an antagonism between different groups of radio- active elements. The antagonism is expressed in the following scheme: Uranium Potassium | Thorium Rubidium > _ Radium (Casium) = Ioniam Niton (Kmanation) The uranium substitutes the potassium in certain experiments; but the elements together neutralize each others’ effect. These investigations raise the question if it were possible to sub- stitute the potassium during the development by another radio-active element. | tried to obtain an answer on this question by experimental investigation. I will give a short account of the experiments taken and of the results which [ obtained. The experiments were taken with frog-eggs and carried out in the following way. After the fecundation (in the laboratory) the egglump was parted ') H Zwaarpemaker, Die Bedeutung des Kaliums im Organismus. Pllügers’ Archiv. Bd. 173. 564 immediately in equal quantities, these are placed in liquids containing potassium or in which different quantities of uranium-salt had been dissolved. There is practically no potassium found in the Utrecht water, thus, as much care as possible was taken, to bring up the uranium- tadpoles, with food containing no potassium, while the tadpoles that had been put in the liquid containing potassium, got as much ordinary (animal) food, as was possible. Rice, boiled in distilled water was given as food without potassium. In the first year the uranium-tad poles were brought up in glass-bowls; to prevent the dissolving of potassium from the glass, the tadpoles were brought up in quartz bowls in the second and third years of the experiments. I. A first series of experiments consisted in adding to the water a certain quantity of uranium-nitrate UO,(NO,),. The potassium which was present in the eggs was compensated by giving 44 mgr. uranium-nitrate pro liter; moreover another 12,5, 25 mer. (altogether 17 and 29} mgr.) and 50 mgr. pro liter. At the same time as the tadpoles in these liquids, others were brought up in ordinary water, with piscidine*) and rice. The eggs were laid on April 14", On May 29* the piscidine- tadpoles are long 11—12 m.m.’) and have hind-limbs; on June 6'% there are tadpoles of 14.2 m.m., which have hind- and front limbs; then metamorphosis and tailreduction regularly follow. On June 26" only a few tadpoles of 8—12 m.m. remain, these are not yet meta- morphosed. The rice-tadpoles develop far more slowly, and it now appears that at the same time the stage of development of the tadpoles differs considerably. On June 26" I found tadpoles of 6—12 m.m.; then the development slowly continues; on August 6% 1 found the first complete metamorphosis, (the tail has disappeared) the length being 15 m.m. On October 10% another tail-reduction takes place. Some tadpoles do not metamorphose, they become quite big animals, viz. 16—17 m.m., with only short hind-limbs. As to the uranium-tadpoles they remain backwards and develop far more slowly, which are the most striking characteristics. The following table informs us about the size. From this table it appears, that the development takes place considerably more slowly than with the animals under control, also the sizes are smaller. Although the differences seem little, the tad- 1) Piscidine is a preparation containing dried and powdered fish. 2) In these and all following measurements the length must be considered as taken from the top of the head to the beginning of the tail. A. J. P. v. d. BROEK: “On the influence of radio-active elements on the development”. v8 Ro. 26 dliet JITeL Gun Jz On this, plate are shown the minimum- and maximum-length of tadpoles from different solutions of uraniumsalt at various data. The fourth row from above are tadpoles fed with rice, the fifth (last) row are tadpoles fed with piscidine. 565 en | 41/, m.gr. uranium-| 17 m.gr. uranium- | 29!/, m.gr. uranium- nitrate pro L. nitrate pro L. nitrate pro L. June 26th | 5.5—10 m.m. 5— 9.8 m.m. 5 —7.5 m.m. July 10th 6 —11 m.m. 5-10 mm. 4.5—9 m.m. July 30th 7 —11 mm. 6— 9.5 m.m. 5 —10 m.m. August 14th 9.5 m.m. August 24th 10.5—11.5 m.m. September 3 12 m.m. ) front-limbs (10.2 mm. front-limbs Cae Konin September 14th | 11 m.m.§ reduction | 8.4 m.m.9 reduction | 8.5—9.6m.m. (frontl.) November 27th 10 m.m. 11 m.m. 8—11.8 m.m. poles are all the smaller in proportion as the quantity uranium- nitrate is greater. The following gives a more detailed account. a. 41/, mgr. uranyl-nitrate pro Liter. On June 26th and July 10th the measurements are as mentioned; the biggest tadpoles only have an indication of hind limbs. On July-20th the biggest tadpoles have hind limbs which do not lie any more straight along the tail, but which are abducted. On August 14th it was the first time that front limbs broke through with one tadpole of 9.5 mm ; on Sept 3rd. and Sept. 14th the first tail-reductions were observed. On November 27th the last tadpole of 10 min. with small hind limbs, showing signs of diminishing vitality, was killed and fixed. 6. 17 mgr. uranyl nitrate pro Liter. From April 14th till July 10th the growth makes very little progress, the maximum length only being 10 mm ; tiny little points of hind limbs are present, and it is not before July 30th that one tadpole has hind limbs in abduction; on September 3rd. the first tail-reduction is observed. On November 27th the last living tadpole has a length of 11 mm. c. 29'/ mgr. uranyl-nitrate pro Liter. The growth.still goes more slowly than the preceding ones. On July 80th I see tadpoles with points of hind limbs. The beginning of tail-reduction was observed for the first time on September 7th; after that regularly; the size of the tadpoles then being 9,6—11,5 mm. On November 27th the last two tadpoles of 8.5 and 11 mm. were fixed. Il. In different periods of the development tadpoles were taken out of the uraniam solutions and put into ordinary water. It then appeared that almost immediately the development took place much more quickly than with the tadpoles that remained in the uranium- solution; it must be well understood that the piscidine-tadpoles grew more quickly than the rice-tadpoles. 566 Ill. A number of tadpoles being in a young stage of development (on April 30%) were put into a liquid, containing a quantity of uranium-salt and an aequiradio-active quantity of potassinm. In this liquid the tadpoles grew almost as quickly as the animals under control; on July 15 I already found one tadpole with front- and hind limbs; several other tadpoles soon get them as well; the smallest ones are 8—9 m.m. So these tadpoles grew more quickly than the uranium-tadpoles. It may be said, that as a general result of the experiments made, tadpoles in a medium containing a radio-active substance antagonistic to potassium, grow and metamorphose less quickly than in a medium only containing potassium. The food given to the tadpoles, is not the cause of this lingering, for in regard to the tadpoles in ordinary water, fed with rice, the uranium-tadpoles also remain backward in development. The first question that can be put is, if the tadpoles have taken uranium next to or instead of potassium. This question cannot yet be affirmed. From an investigation volun- tarily undertaken by Prof. Rincer, it appeared that no uranium could be demonstrated in the tadpoles. In another respect this experiment had a very important result though the percentage of potassium found in the tadpoles fed with pisci- dine, was in two cases 0.76 and 0.82 °/, from the dried tadpoles. The ura- nium tadpoles had evidently taken the minimal potassium quantity and even 0.91 °/, kalium was found. The potassium-perceniage was only 0 49°/, of those tadpoles, which had been in water containing 175 mer. UO, (NO,), and 50 mgr. KCl pro Liter (being aequiradio-active quantities). It here makes the impression that the presence of = aequiradio-active quantities of the antagonistic substances has thrown obstacles in the way of taking in potassium. Quite remarkable it is though that the concerned tadpoles should only stay little behind in growth to those, which had been normally brought up. *) The remaining backward in growth might be imputed to a possible poisoning caused by the uranium salt. In 1919 I have made a single experiment on this subject. On May 6'® I had five bowls containing 4 L. water each, and resp. 0, 2*/,, 5, 7'/, and 10 mer. uranium-salt; every single bowl contained 75 tadpoles as well; on June 19" resp. 51, 38, 24, 20 and 20 tadpoles 1) Prof. Ringer fixes the attention on the fact, that the chemical investigation, viz. the investigation of the radio activity of the dried tadpoles, took place with so small a quantity of the dried tadpoles, that it is desirable this investigation should be repeated, namely about the absence of uranium. 567 were still alive in these bowls; it thus seems as if the tadpoles die sooner and in greater number, while being in higher concentration. This result does not agree with Huirscn’s’*) results, who supposes that the quickest development takes place in the concentration, nearest to the concentration in which life is impossible. In this case one should expect a quicker development in the higher concentrations than in the lower-ones, but this has not been proved. Moreover Hirsca has every time extended his investigation over a very short period (7 days) which does not seem desirable, if one takes into consideration the great variability in the development of tadpoles.. Altbough apparently a poisoning in the solutions with a greater quantity of uranium-salt does not seem impossible, the fact that many tadpoles develop, and that they live quietly on, in much stronger uranium solutions, and the failing of uranium in the body, might plead against the poisoning of the uranium-salt as a cause of the more slow development. Microscopic investigation of series sections from some uranium- tadpoles, in comparison with normal tadpoles of the same size, has not yet shown differences in structure or in degree of development of certain organs, which, should be of importance for the growth. 4) E. Hiascu, Die biologische Wirkung einiger Salze. Zool. Jahrbücher. Band 34. Mathematics. — ‘Ueber die endlichen topologischen Gruppen der Kugelfliiche’. By B. von KerfkJArtó. (Communicated by Prof. L. E. J. Brotwer). (Communicated at the meeting of November 29, 1919). Die vorliegende Arbeit gibt eine neue Herleitung des Resultates, dass die endlichen topologischen Transformationsgruppen mit invarianter Indikatrix der Kugelfläche mit den Gruppen der regulären Körper identisch sind, was nach dem Brouwrrschen Grundsatz *), laut dessen die topologischen Gruppen mit den konformen homöomorph sind, aus dem die konformen Transformationsgruppen der Kugelfläche betreffenden, bekannten Satze folgt. Wir betrachten eine Gruppe G von n topologischen, die Indikatrix erhaltenden Transformationen der Kugelfläche in sich. Eine willkürliche Transformation ¢ von Gist nach dem Rotationssatz *) eine v-periodische Drehung, die also zwei Fixpunkte P und Q hat; die Anzahl der mit P bei G äquivalenten Punkte ist ~. Wir verbinden P mit Q durch einen Weg 6, der seine bei den Potenzen von ¢ entstehenden Bilder ausser in P und Q nicht trifft. Sei von Paus R der erste solche — Punkt von 6, dass PR eines seiner bei G entstehenden Bilder ausser- halb P trifft. Wenn R= Q, so ist G mit der zyklischen Rotations- gruppe 1, ¢, ¢?,.... # + identisch. Wenn aber Rk ~ Q, so kann auf dem Bogen PR höchstens ein mit A aquivalenter Punkt A’ liegen. — Wenn auf PR kein mit R äquivalenter Punkt liegt, so ist R bei Za einer Transformation von G invariant, sodass der Bogen PR ein zwei nichtäquivalente Fixpunkte von G verbindender, seine Bilder ausserhalb der Endpunkte nicht treffender Bogen c ist. Wenn aber — auf PR ein mit AR äquivalenter Punkt A’ liegt, der bei keiner Transformation von G ausser der Identität invariant ist, so betrachte IE man das System der Bilder des Bogens A’ von 6; es besteht aus einander nicht treffenden Jordanschen Kurven, da & zu genau zwei solehen Bogen gehört; ferner ist dieses System bei G invariant. 1) Diese Proceedings XXI, S. 1143 (29. Marz 1919). 2) Math. Ann. Bd. 80, S. 36. 569 Sei y eine der genannten Kurven; da das innere, d. h. keinen Bild- punkt von P enthaltende Gebiet von y bei jeder es invariant lassenden Transformation von G einer Potenz derselben Drehung unterworfen ist, so kann man /’ mit dem im Innern von y existierenden einzigen Fixpunkt S (der nicht mit Q zusammenfallen kann) durch einen seine Bilder nicht treffenden Weg verbinden, welcher mit dem Bogen PR’ von 6 zusammen einen seine Bilder ausserhalb der Fixpunkte von G nicht treffenden und zwei Fixpunkte verbindenden Weg c bildet. Die Bilder von c zerlegen die Kugelflache in Elemente; falls eines dieser bei einer Transformation von G invariant, also einer Drehung unterworfen ist, so kann man einen Fixpunkt seiner Grenze mit dem in seinem Innern liegenden einzigen Fixpunkt 7’ durch einen seine Bilder nicht treffenden Weg d verbinden. Die sämtlichen Bilder von e und d ergeben zusammen ein bei G invariantes System H von folgender Beschaffenheit: 1. H zerlegt die Kugelflache in Elemente, von denen je zwei äquivalent sind und jedes nur bei der Identitat invariant ist; 2. jeder Fixpunkt von G liegt auf H; 3. jeder Punkt von H, der zu mehr als zwei Bogen von H gehört, ist ein Fixpunkt von G. Die Anzahl der Elemente, in welche H die Kugelfläche zerlegt, ist nm; die Anzahl der nicht äquivalenten Fixpunkte ist 3, ihre gesamte Anzahl ist also, wenn »,,r,,v, ihre Multiplizitäten be- 1 1 1 zeichnen, gleich 7 (—+ ): die Anzahl der Kanten jedes hi >] Yv 1 ‘3 3 Elementes ist 4, also die gesamte Anzahl der Kanten 2». Mithin besteht nach dem Eurerschen Polyedersatz die Formel n 1 2 n+ S>——2In=2, oder S—=—1+—, Dv. Dv. n 1 t woraus sich die bekannten Lösungen ergeben '). Mittels der gleichen Methode werde ich die Brouwurschen Resultate *) in bezug auf die endlichen Gruppen von topologischen Transformationen des Torus herleiten. 1) Krein, Vorlesungen über das Ikosaeder, S. 119. 4) C. R. t. 168, S. 845 (28. April 1919). Chemistry. — “Catalysis” — Part VII — Notes on Cutalysis m heterogeneous systems. By Nm Ravas Daar (Communicated by Prof. Easer Comms). (Communicated af the meeting of November 29, 1919). 1. Ithas been known for a long time that violet ehromie chloride is practically imsolable im water, bat im presence of reducing agents solution takes place due to the transformation into the soluble modification. Anhydrous ferrie sulphate dissolves slowly im water at the ordi Gary temperature, im other words, it may be said to have a small velocity of solutioa. 1 have found that reducing agents hike stannous chlemde, ferrous sulphate, salphurous acid ete, markedly accelerate the velocity of soluttiom of ferrie sulphate im water. Im the ease of chrome chloride, it is assumed that the reducing agemit first reduces the imsoluble chromie chloride to chromous chloride, and the orginal chromous chloride is transformed into soluble chromie chloride. The newly formed chromous chloride them acts om the mseoeluble chromic chloride as before The diffieuliy of am explanation like this is that we assume thaf the reducing agent acts om the solid chromie chloride and reduces it; from oor experience im heterogemeous systems, we know how diffi- ealt it is to redme a solid substance rapidly with a solutiom of a reducing agemt. . Moreover, im the ease of ferrie sulphate we do not know two varieties of the salt as im the ease of chromic chloride Tt is dificult to assume that ferrous salphaie would reduce ferrie sulphate, for we kmow that a mixture of ferrie and ferrous salts eam exist unchanged for am indefinite period im absence of oxygen. So we cam say with Osrwarp (Grundlimiem der anorganischen Chemie, Leipzig 1900; A. Frspnar’s trans. 603, 1902) that a sufficient explanaiiom of these actions is still wanting 2. The action of mitrie acid om the metals generally, is somewhat complex, because the maim reactiom is complicated by side or con carremt, amd by consecutive reactions. These again depend not only upon the particular metal under consideration, but also om the 571 concentration of the acid, the temperature and the concentration of the products of the reaction accumulating in the solution. Mitton (Compt. rend. 1842, 14, 904) and Verey (Phil. Trans. 1891 A, 182, 279) have shown that metals like copper, silver, mercury and bismuth have no action on cold dilute nitric acid unless a trace of nitrous acid is present. The nitrous acid may be present in the nitric acid as an impurity; it may be formed by the incipient decomposition of nitric acid when it is warmed. According to Ve.ry, therefore, the dissolution of copper in nitric acid proceeding: Cu + 3 HNO, = Cu (NO), + HNO, + HO, is a resultant of a series of consecutive reactions: Cu + 4 HNO, = Cu (NO), + 2 H,O + 2 NO; followed by Cu(NO,), + 2 HNO, = Cu (NO,), + 2 HNO,. The small trace of nitrous acid thus acts as a catalytic agent; nitrous acid is continuously produced and continuously decomposed according to the following equilibrium: 3 HNO, = HNO, + 2 NO + H,0. Similar results have been obtained by Ray (Trans. Chem. Soe. 1911, 99, 1012) in the case of mercury and by Sranspiz (J. Soe. Chem. Ind. 1913, 32, 311) in the case of silver. Now Mirror (loc. cit.) and Verer (loc. cit.) have pointed out that the presence of ferrous sulphate, “which removes the nitrous acid as fast as it might be formed” serves to prevent the chemical change between nitric acid and the metals. But I have observed that ferrous sulphate exerts an accelerating influence on the complete dissolution of copper in 20°/, nitrie acid at 18°. This result being different from those of previous investigators, I thought it worth while to observe the effect of both ferrous and ferric salts and various other substances on the complete dissolution of copper in excess of 20°/, nitric acid. Equal lengths of copper wire of uniform sectional area were placed into test tubes and covered with an excess of 20° , nitric acid. The mean temperature of the experiments was 18° and the tubes containing equal volumes of nitrie acid and equal weights of copper wire were kept at rest. Weighed quantities of the solid substances used were added at the beginning of the experiments. The whole of the copper wire dissolved in about 30 minutes and the exact time of dissolution was noted. In order to get exactly comparable results one test tube was always set apart for a blank parallel experiment. : It has been found that the following substances exert an accelera- 572 ting effect on the complete dissolution of copper in 20°/, nitric acid : ferrous sulphate, ferrous chloride, ferric sulphate, ferric chloride, ferric nitrate, lead sulphate, lead nitrate, lead acetate, copper nitrate, copper chloride, barium nitrate, thallium nitrate, lithium nitrate, sodium nitrite, manganese chloride, chromic chloride, arsenious oxide, strychnine sulphate, ethylene bromide, carbon tetrachloride, hexachlorobenzene, phthalic anhydride ete. On the other hand, the following substances have a retarding effect: hydrogen peroxide, potassium chlorate, potassium permanganate, chromic acid, sodium nitrate, ammonium nitrate, manganese nitrate, thorium nitrate, sodium sulphite, titanic acid, molybdic acid, ammo- nium persulphate, mangenese sulphate, cobalt chloride, copper acetate, copper sulphate, calcium nitrate, tartarie acid, ether, urea, acetic anhydride, benzoic anhydride ete. In a foregoing paper of this series (Trans. Chem. Soc. 1917, 111, 707) 1 have shown that sulphuric acid in small concentration is an accelerator, whilst in large concentrations it is a retarder in the oxidation of oxalic acid by chromic acid. Similar results have been obtained in the action of nitric acid on copper. The following sub- stances in very small concentration exert a slight accelerating effect, whilst in large concentrations they have retarding effect: Zine chloride, nickel chloride, cobalt nitrate, aluminium nitrate, potassium chloride, strontium nitrate, cadmium nitrate, magnesium chloride ete. When the concentration is very small, the effect of potassium nitrate, uranium nitrate, citric acid, potassium dichromate ete. is practically nil, but in concentrated solutions they are all retarders. The effect of monochloracetic acid is very peculiar. In small con- centrations, it is a feeble accelerator and in concentrated solutions it has a retarding effect, which instead of increasing, decreases with increase of concentration. A similar phenomenon has already been observed in the case of the oxidation of formic acid by chromic acid in presence of manganese chloride (loc. cit. p. 726). It is practically impossible to give a complete explanation of these results, they being so diverse. Ferrous sulphate and ferrous chloride behave as marked accelerators. It would appear that the acid nucleus in this particular instance, plays no part. A part of the ferrous ion reduces the nitric acid to nitric oxide and passes into the ferric state. The nitric oxide dissolves in the ferrous salt solution forming the unstable substance FeNO°°. The dissolved nitric oxide then reduces a part of the nitric acid according to the following equation: HNO, + 2 NO + H,0 23 HNO. 573 It is quite possible that some nitrous acid is produced by the direct reduction of nitric acid by ferrous ions. The formation of nitrous acid either by the direct reduction of nitric acid by ferrous salts or by the indirect reduction through the intervention of nitric oxide is proved by the following experiment. If nitric acid of the strength used in this research be taken in a test tube and a crystal of ferrous ammonium sulphate or ferrous sulphate be added to it, almost immediately the crystal is covered with the deep brown FeNO°° ion and a little nitric oxide also escapes. If urea crystals are now added, they are immediately oxidized with the evolution of carbon dioxide and nitrogen, indicating the presence of nitrous acid. So in presence of ferrous salts, nitrous acid, which is the active substance in the action of nitric acid on copper, is formed when we have an excess of nitrie acid. This explains the accelerating influence of ferrous salts in the complete dissolution of copper in 20°/, nitric acid. As a matter of fact, the accelerating effect of ferrous salts is slightly greater than the accelerating effect of sodium nitrite on the dissolution of copper in nitric acid. The greater the concentration of the ferrous salt, the greater is the acceleration. Ferric sulphate, ferric nitrate and ferric chloride exert a marked accelerating effect, though their activity is slightly less than that of sodium nitrite and the accelerating effect is proportional to the concentration of the ferrie salt. It would appear that the acid nucleus in this ease also, plays no part. The explanation of this activation seems to lie in the reduction of ferric salts by the nitric oxide which is a product of the chemical change between nitric acid and copper. The ferrous salt, which may thus be formed, will reduce a part of the nitrie acid into nitrous acid, which activates the action of nitric acid on copper. It seems plausible that a part of the ferric salt would be reduced to the ferrous state by the metallic copper. It is well known that when a solution of a ferric salt is shaken with metallic copper, the ferric salt is partly reduced to the ferrous state and the copper is oxidized to the cupric salt and an equilibrium is set up: — 2 FeCl, + Cu Z 2 FeCl, + CuCl, The ferrous salt thus formed reduces the nitric acid to nitrous acid, which accelerates the action of nitric acid on copper. In a similar way the accelerating effect of arsenious oxide, strych- nine sulphate, phthalic anhydride ete. may be explained on the basis of the formation of nitrous acid by the action of these reducing agents on the nitric acid. The retarding effect of the oxidizing agents like, H,O,, KMuO,, 38 Proceedings Royal Acad. Amsterdam Vol. XXIL. 574 H,Cr,0,, KCIO,, (NH,),5,O, ete. and of the reducing agents likeurea, sodium sulphite ete. is certainly due to the destruction of the nitrous acid as soon as it is formed. When the experiment was performed in such a condition so as to cover the copper wire with solid urea, the reaction became very slow, but it did not stop altogether. It is very difficult to explain the difference in the behaviour of the nitrates on the oxidation of copper by nitric acid. Lithium nitrate is an accelerator, whilst sodium and potassium nitrates are retarders; from analogy we should expect calcium nitrate to have an accele- rating effect, but as a matter of fact, both calcium and strontium nitrates are retarders, whilst barium nitrate is an accelerator. Rennie and Cook (Trans. chem. Soc. 1911, 99, 1035) have found that the accelerating or retarding effects of the nitrates of K, Rb, Cs were functions of the temperature and of the concentration of the acid. Hienry (Amer. chem. Jour. 17, 18 (1895)) has shown that both NO, and N,O, are the products of the reaction between copper and nitric acid. Evidently in the solution, we should consider the following equilibria : SHINO} == NO USEINO SoHE Ont eee ee een NO; HO HNO“ ENO, ee ENE) SNOR EKO =O NOM INOl. = ane) Lewis and Epear (J. Amer. chem. Soc. 1911, 33, 292) have shown that in equilibrium (1) there is a change in the equilibrium constant with the concentration of nitric acid. It seems probable that nitrates may affect one or more of these equilibria and change the concen- tration of nitrous acid, which being the activating agent. In this connection, it is interesting to observe that several reac- tions, in which nitric acid is the oxidizing agent, are autocataly tic. As for example, the actions of nitric acid on metals like Copper, Silver, Bismuth, Mercury ete., on starch, on sugar, on arsenious oxide, on hydrogen iodide (Ecksräpr, Zeit. anorg. Chem. 1901, 29, 51), on nitric oxide (Lewis and Epaar, loc. cit.) etc. become more pronounced as the chemical change proceeds. The explanation is not far to seek. The nitrous acid is the active substance and its concentration and hence the reaction velocity increase with the progress of the chemical change. In all these cases I have found that the chemical change becomes more rapid when a nitrite is added at the commencement of the reaction. It has been observed that the chemical change between nitric acid and copper may be practically stopped by agitating vigorously the 575 tube containing copper and nitric acid, because the nitrous acid cannot accumulate round the copper. Summary and Conclusion: 1. The velocity of solution of anhydrous ferric sulphate can be increased by the presence of sulphurous acid, stannous chloride, ferrous sulphate ete. No satisfactory explanation of reactions of this type is forthcoming. 2. The action of nitric acid (20°/,) on copper has been studied in the presence of various substances and it has been observed that when the nitric acid is in excess and the whole of the copper is made to dissolve, ferrous and ferric salts exert a marked accele- rating effect. In the light of the present investigation, the view hitherto aecepted as regards the part played by ferrous salts in des- troying nitrous acid, has to be modified. As a matter of fact, it has been proved that nitrous acid, which is the active substance in this reaction, is formed by the action of nitrie acid on ferrous salts. Oxidizing agents like H,O,, KMnO,, H,Cr,O, ete. destroy the nitrous acid and hence retard the change. Out of the 56 substances, the effect of which was investigated, 22 act as accelerators and 22 exert a retarding influence in all concentrations; whilst 8 of them are slight accelerators in small concentrations and are retarders in concentrated solutions. Four of these 56 substances have been found to be neutral in small and retarders in large concentrations. My best thanks are due to the van ’r Horr Fund Committee for a grant for this research. Chemical Laboratory, Muir Central College, Allahabad, India. 38% OO en Chemistry. — “Notes on Cobaltammines’. By Ni Ratan Duar. (Communicated by Prof. Erxst Coues). (Communicated at the meeting of November 29. 1919). In two previons investigations (Zeit. Anorg. Chem. 1913, 80, 43; 84, 224) | had occasion to study certain proporties of the cobalt- ammines. This note is the result of the continuation of my previous work. 1. Let us pof ip slollo wane series of compounds _ ‘ 3 NH), (NH), (NH), > 1 CI E / [Co (NH,),] Cl, , | 0 aa CL, [co plea Ne [eo Ae | K | Co 50) el and K, [Co (NO Det: For the preparation of NN _ (NA), NHJ. _ (NED, Er NO, je [ee nis BL [ee SO.) ql ie jee on the general meihod of procedure is to mix a cobalt salt, ammonium chloride, ammonium hydroxide and a nitrite; by this a complex cobaltous compound is formed which is turned into the stable cobaltic compound by oxidation. The amount of a certain compound which will be formed, depends on the concentration of the reacting substances and on the solnbility of the resulting complex compound. If the concentration of the nitrite in the solution is large in comparison with the concentrations of ammonium hydroxide and ammonium chloride. we should expect that several (NO,) groups would enter the complex. It bas been known from a long time that aquopentammine salts can be converted into the corresponding hexammine salts by heating the aquo compound with ammonia in a sealed tube or in a bottle under pressure. | ad u: I found that if | Co ** | Cl, is warmed with a dilute solution (NO,) (NH), den which of sodium or potassium nitrite, we get mainly | co could be purified by reerystallisation. NH), In a similar way croceo cobalt chloride | co (NO,), Ja can be ond hast) enn —— Te a a ye Venn 517 (NH). - - - - - - 4 converted into | co ] by warming it with a dilute solution of (NO). v ; NH a nitrite, whilst | co al can be converted into K [co aM (NH), by warming | co ] with a concentrated solution of potassium (NO), nitrite, ammonia escaping from the solution. N I tried to prepare the compound K, ‚| Co ae (NH). (NO), } which is still anknown, by warming K| co, Al with potassium nitrite, but was unsuccessful. en __ On the other hand, one can convert K| Co Seale into | co 0 and | co Se nto | co San Cl by warming the compound in question with a mixture of ammonium chloride and ammonium hydroxide. In all these cases, ammonium salts are used along with ammonium hydroxide, and their function is to suppress the ionisation of the base and form undissociated NH,OH, which is in equilibrium with NH,. The NH, then enters into the complex molecule. 2. If a fairly concentrated solution of aqnopentammine cobaltie chloride [corn a)s 4 oe is left, it slowly gives a precipitate of the eorresponding purpureo salt [co ea | Cl, NAL), SE) [co Cl) Je + H,0 =| Co (1.0) Je This is a case of equilibrium in solution and the purpureo salt being much less soluble comes out as a precipitate. If we start with a solution of purpureo cobalt chloride | co pen Je. and add ammonium hydroxide and warm the mixture, we get the (NH), method of preparation of the aqno salt. I find that the ammonium hydroxide has only a catalytic effect on the hydrolysis of the purpureo salt into the aquopentammine salt. A solution of the purpureo chloride takes up a molecule of water aquopentammine salt [co ‘Je. in solution, and this is the usual 578 and passes into the aquo salt very slowly even at the ordinary temperature. This hydrolysis is markedly accelerated by the presence of hydroxyl (OH’) ions. The greater the concentration of the hydroxide ions, the greater is the acceleration. The study of the reaction velocity of this hydrolysis may serve as a means of determining the concen- tration of hydroxide ions in a dilute solution of a base. Thus if we make a solution of the purpureo salt and add a few drops of a dilute solution of potassium hydroxide, the purple colour changes and becomes rose in a few minutes; but with a weak base like ammonium hydroxide the colour change takes a long time. This explanation may be true in the case of bydrolysis with the corre- sponding compounds of chromium and platinum. There is evidence to show that in some other cases of hydrolysis by alkali, the action of the hydroxide ions is catalytic. The decomposition of sodium chloracetate by alkali is a case in point (Senter, Trans Chem. Soc. 1907, 91, 473). One can get the hydroxides of the cobaltammines in solution by treating the corresponding halide with moist silveroxide : (NH), re. OE, | co an Cl + AgOH = | co Arn OH + AgCl The solution slowly decomposes even at the ordinary temperatures. The hydroxides of the other members of this series can also be prepared by this double decomposition. These hydroxides turn phenol- phtalein pink and electric conductivity measurements show that they are strong bases of the type of sodium hydroxide. But one cannot prepare the hydroxide from purpureo cobalt chloride | co i Cl, by double decomposition with silver oxide. The explanation becomes simple on the light of the catalytic effect of hydroxide ions on the hydrolysis of purpureo salts into the aquo compounds. The hydroxide ions set free by the double decom- position act catalytically on the purpureo salt and actually one gets nto (OH),, which is stable in alkaline solution, (compare URBAIN et SÉNÉCHAL, Chimie des com- plexes, p. 280, “Les sels purpureo ne donnent pas une réaction de ce genre’). the aquopentammine hydroxide | co Summary and Conclusion. 1. The principle of the preparation of the cobaltammines is guided by the law of mass action and thus depends on the concentration 579 of the reacting substances. One can substitute a nitro (NO,) group in a compound by the group (NH,) on warming it with a mixture of ammonium hydroxide and a ammonium salt and on the other hand, NH, is replaced by NO, when the salt is warmed with a nitrite solution. NH), (NH), 2. C CI BOAC Cl, . [ee Cl | al 2 [eo uo) This hydrolysis reaction is catalytically accelerated by the presence of OH’ ions and the velocity is proportional to the concentration of hydroxide ions. (NH) (NH,) | Co ** | (OH),, | Co ~~ | OH! ete: [exo | [© wo, are strong bases and can be prepared in solution. The base obtained from the purpureo cobalt chloride is the aquopentammine hydroxide (NH,) C OH) | 0 H,O ( )s Chemical Laboratory, Muir Central College, Allahabad, India. Mathematics. —‘‘An involution of pairs of points and an involution of pairs of rays in space.” By Dr. C. H. van Os. (Commu- nicated by Prof. Jan pu Vrins.) (Communicated at the meeting of September 29, 1918). § 1. Introduction. By several authors involutions have been treated, consisting of groups of points in the plane or in space. On the contrary involutions, consisting of groups of straight lines, do not seem to have been considered. In the following such an involution will be investigated. This involution is derived with the help of an involution of pairs of points, which is itself again connected with a certain bilinear congruence of twisted cubics. The congruence in question [y*] is formed by all the curves 9 which pass through two given points A, and A,, and have three given straight lines a,, a, and a, as bisecants. These curves are the moveable intersections of the quadratic surfaces out of two given pencils (07,,) and (9’,,). The base-curve of the pencil (0’,,) consists of the lines a, and a, and the common transversals 6,,, and 6,,, which we can draw through the points A, and A, to these straight lines; that of the pencil (o°,,) consists of the lines a, and a, and their common transversals 6,,, and 6,,, passing through A, and 4, *) 145 Through a point P passes one 9° of the congruence; if we asso- ciate to P the point P’, which on the curve g° is harmonically separated from P by the points A, and A,, we get an involution of pairs of points (P, P’). A straight line ¢ is chord of one gy’; let P and Q be its support- ing points. Through the involution just found there are associated to the points P and Q two points P’ and Q’. If we now associate the line ¢’ connecting the points P’ and Q’, to t‚ we get an involu- tion of pairs of rays (t, t’). § 2. Deyenerate 0° of the involution. We shall show that the 1) This congruence [c°] has been investigated by M. StuyVAERT (Etude de quelques surfuces algébriques engendrées par des courbes du second et du troisième ordre Dissertation inaugurale. Gand 1902) and by J. pr Vries (Bilineaire congruenties van kubische. ruimtekrommen. Proefschrift, Utrecht 1917). 581 congruence [g*] contains seven systems of o' curves g’, each of which is degenerated into a conic £* and a straight line d. In the first place the conic 4? can pass through the points A, and A, and therefore lie in a plane 2 through the straight line A, A,. Such a plane intersects the lines a,,a, and a, in three points A,, A, and 4,, which together with the points A, and A, define one conic £?. The ruled surface yw? formed by the common transversals of the lines a,, a, and a,, intersects this conic £’ besides in the points A,, A, and A, in one more point D; the transversal d passing through D, forms with 4? a degenerate g°. The surface w’ is intersected by the line A, A, in two points B, and B,; the generatrices 6, and 5, of yw? passing through these points, form each with the line 4, A, a degenerate 4? of the system just considered. The transversal d, which completes the degenerate ?, formed by the lines A,A, and 6, to a gy’, is apparently no-other than the line b,. The three lines b,, A, A, and 6, form therefore together a degenerate 9’. [t has just appeared that to every conic &’ there belongs a definite transversal d; is the reverse also the case? In order to examine this we remark that the line A, A, is twice a component of a degenerate &*, and is therefore nodal line of the surface formed by these conies 47. A plane a through A, A, intersects this surface along the nodal line and along a conie k?; it is therefore of order four. A transversal d intersects this surface besides in the lines a,, a, and a, in one point D and so forms together with one conic £? a degenerate Q’. $ 3. In order to get a second series of degenerate 9’, we draw the transversal 6,,, mentioned in § 1 and bring throngh the point A, and the line a, a plane a,,. This plane intersects the transversal b,,, in a point D,, the lines a, and a, in two points C, and C, The points A,, D,, C, and C, determine a pencil of conics each of which forms with the line 5 . B 8 ia, à degenerate o°. As we can take one of the transversals 6,,,, boss, Orsa Onas Osis instead of the transversal 5,,,, we get in all sir pencils of conics degenerated in this way. Each of the corresponding pencils of conics contains three pairs of lines; for the pencil lying in the plane e,, they are the pairs (A, D,, C,C,) (A, C,, D,C,) and (A,C,, D,C,). Each of these pairs forms with the transversal 6,,, a @’, which has degenerated into three straight lines. Lying in the plane «,, the line A,C, intersects the line a, and is therefore the transversal 6 e E N . 7 Ì = aay; in the same way the line A,C, is the 582 same as the transversal b,,,. The curve (0,,,, A,D,, C,C,) belongs evidently only to the pencil of degenerate o° which contain the line bis As component; the curves (bs, Oss: D,C,) and (6,,,, Oos DC) belong each to fwo pencils of degenerate 0°. There are therefore six curves of the first and as many of the second kind. Hence together with the curve (b,, A,A,, 6,) the congruence [9°] contains thirteen @° which have degenerated into three straight lines. § 4. Singular points and bisecants of the congruence. The points of the three lines a,, a, and a, are singular points of the congruence. Let us consider for instance a point A, of the line a,. The surface ’,, through A, intersects an arbitrary surface p’,, along a curve 0°, which passes through the point A,. Through A, passes therefore a pencil of curves 09’; all these curves pass also through the second point of intersection of the surface g’,, witb the line a,. Also the points of the transversals bij, are singular points; for each of these transversals is component of a pencil of degenerate 0°. The straight lines through the points A, and A, are singular bisecants; for through any point of such a straight line there passes one oe’ and as this passes also through the points A, and A,, it has that straight line as bisecant. In the second place the straight lines in the planes a, a, ds a> Cas @, brought through the points A, and A, and the lines a,, a, and a, are singular bisecants. For each of these planes contains a pencil of conics k°, each of which is a component of a degenerate oe’, and a straight line in such a plane is bisecant of all these conics. In the third place the generatrices g,, of the surfaces y,,, which eross the lines a, and a,, are singular bisecants of the congruence. Such a line g,, is intersected by the surfaces g’,, in the pairs of points of a quadratic involution and the two points of such a pair are every time the supporting points of a curve @°. As the surfaces _@’,, pass through the lines a,, a,, 6,,, and 6,,,, the lines g,, are the transversals of the lines 6,,, and 3,,,. In the same way the transversals g,, of the lines 6,,, and 0,,, and the transversals g,, of the lines 6,,, and 6,,, are singular bise- cants of the congruence. The singular bisecants form therefore two sheaves, six fields and three bilinear congruences. a § 5. Pairs of points on a degenerate 9’. o’. We now pass on to the consideration of the involution (P, P') and examine first what becomes of this correspondence, if the points P and /” lie on a degenerate o@°. 583 With a view to this we remark, that the four barmonical points P,A,, P', A, of a curve g° from each of its chords s are projected by four harmonical planes. This must remain the case, if we let the e° degenerate into a conic &? and a straight line d. In the degeneration considered in $ 2, the points 4, and A, lie both on the conic £°. The following two cases are now possible: 1. The point P lies also on the conic 4°. If we take as chord s a common secant of the conic £? and the line d, we see that also the point P' lies on #? and is harmonically separated from P by A, and A,. 2. The point P lies on the line d. If we take the chord s in the same way, we see that the point P’ lies on &? and by 4, and A, is harmonically separated from the point of intersection D of the two components £* and d. To the point D', which is harmonically separated from D, all the points of the line d are therefore associated; for the rest there belongs to every point P of the degenerate 0° one definite other point P’. In the degeneration considered in § 3, the point A, lies on the line d, the point A, on the conic £° (or inversely). Two cases are again possible: 1. The point P lies on the conic £°. If we again take as chord s a secant of &? and d, we see that the point P’ lies also on the conic k* and is harmonically separated from P by the points A, and D. 2. The point P lies on the line d. If we take as chord s a straight line in the plane of the conie k°, we see that the point P lies also on the line d and is harmonically separated from P by the points A, and D. To each point of this degenerate 0’ belongs consequently a definite other point. If P coincides with D, the point P' does the same. If the g° is degenerated into three straight lines, considerations of the same kind hold good. § 6. Singular points of the involution (P, P’). On every non dege- nerate g*® the points A, and A, are associated to themselves; it appears from the preceding § that this is also the case for the dege- nerate y*. These points are therefore not singular points of the invo- lution. On the contrary the points of the lines a,,a, and a, are singular points. Let us consider e.g. a point A, of the line a,. In order to find the point A,’ associated to A, on a curve @° passing through A,, we must bring through the bisecant a, of this curve 9° a plane which by the planes a,, and a,, is harmonically separated from the plane whieh touches the curve gy’ in the point A, and 584 passes through the line a,; this plane intersects the curve vy? in the point A’, in question. As the plane (4’,, a,) passes through the line «,, it is a tangent plane of the surface v’,, which contains the considered curve g?. Now the tangent planes of a ruled surface in the points of a gene- ratrix are projectively associated to the points of contact; the point of contact B, of the plane (A’,,@) is therefore harmonically separa- ted from the point A, by the points of contact of the planes a,, and a,,. As these two planes pass through the lines 6,,, and 6,,,, their points of contact B, and B,, are the intersections of these transversals with the line a,. If the surface ,,? describes the pencil (p,,’), the plane (4’, a), which touches the surface g’,, in the point B,, describes a pencil which is projectively associated to the pencil (g’,,). The figure produced by these projective pencils is a surface of the third order. To these planes of contact belong the planes a,, and a,, each of which is at the same time part of a degenerate surface @?,,; conse- quently these planes belong to the product and the rest is a plane. The figure produced by two projective pencils passes through the base-curves of these pencils. The plane just found contains therefore the line a, as this is the case with neither of the planes a@,, and «,,. It must also pass through the point B, as the intersection of a curve ~’,, with its tangent plane in the point B consists besides of the line a,, of a generatrix through the point By. The locus of the points A’, which are associated to the point A, on the different curves g’* laid through the point A,, belongs to the intersection of the plane (4,,a,) with the surface y’,,, on which all these curves @° are situated and which passes through A,. This intersection consists besides of the line a, of a straight line 2; this is the locus in question. This line 2 passes through the point of intersection of the plane (B,,a,) with the line a,. Evidently this point of intersection is pro- jectively associated to the point B,, therefore also to the point A, The same must hold good for the intersection of the line 2 with the line a, If the point A, describes the line a,, the intersections of the line 2 with the lines a, and a, describe two projective sequences of points. Consequently the line 4 describes a guwadratic surface w’, the locus of all the points associated to the points of the line a,. To each of the lines a, and a, belongs a similar surface w’. § 7. The points of the transversals },,, etc. are not singular points of the involution. For from the construction given in § 5 it follows 585 that to every point of such a transversal a definite other point of the same transversal is associated, no matter of which degenerate o* we consider the transversal to be a component. From $ 5 follows further that on each degenerate g° of the first series there lies one singular point D’. We shall determine the locus of these singular points. It appeared in § 2 that to this series belongs a ov? consisting of the straight line A,A, and the two transversals 6, and 6, of the four lines A, 4,, a,, a, and a,. AS we can combine each of the two transversals with the line 4,4, to a degenerate conic £°, there lie on this conie “ve singular points D', and D',. A plane = through the line A,A, contains one conic &? and con- sequently it intersects the locus in question, besides in the points D', and D',, in one more point D’; the locus is therefore a twisted cubic o°. A point D’ is associated to a straight line d, which intersects the three lines a,, a, and a,. To the points associated to D’ belong therefore three points which lie on the three lines mentioned; con- sequently the point D’ must lie on the three surfaces w? found in the preceding §. All these three surfaces pass therefore through the curve o°. § 8. If a point P describes a straight line Z, the point P’ asso- ciated to P describes a curve (/). As the line / has two points in common with each of the three surfaces w?, the curve (/) has two points in common with each of the three lines a,, a, and a,. A surface ,,’ intersects the line / in two points and contains both the points of the curve (/) associated to this line, so that in all this surface y,,? has stv points in common with the curve (é). For this reason (/) is a twisted cubic. In general the line / and the curve (l)? have no points in common, for as a rule no two associated points of the involution (P, 2’) lie on /; for the rest this involution has only a finite number of coin- and the points D, found in $ 5, etc. intersect the corresponding planes cidences, viz. the points A, and A, in which the transversals 4,,, a,, ete. As a rule therefore the line / does not contain any coinci- dences either. If a point P deseribes a plane WV, the point P' associated to P, describes a surface (V). In order to find the order of this surface, we draw in the plane V a straight line /. The curve (/)* associated to this line /, intersects the plane V in three points, each of which is associated to a point of £. The line / intersects therefore the locus 586 of the pairs of points (P, P) lying in the plane V, in three points; consequently this locus is a curve of order three. The plane V con- taining as a rule no coincidences, this curve is the complete inter- section of V with the surface (V), which for this reason is a surface of order three. The surface (V)* contains the lines a,, a, and a,, for each point of one of these lines is associated to a line }, which cuts the plane V in one point. In the same way the surface (V)* passes through the curve 6’. Let Q be a point of the plane V, / a straight line of V passing through Q. This line contains three points P, associated to a point P' in V. If we connect these points P' with Q and associate these lines of connection to the line /, we get in the pencil of the rays through Q a correspondence (3, 3) with siz coincidences. These must originate from the rays (P, P') passing through Q and each of these rays furnishes two coincidences, as the correspondence (P, P’) is involutory. Through Q pass therefore three rays PP’ which lie in plane V and accordingly the lines PP’ as a rule form a cubic line complex. § 9. Singular straight lines of the involution (t, t’). We now proceed to the consideration of the involution (¢, ¢’) and first investigate its singular rays. The line A, A, is bisecant of all the curves 9%. On an arbitrary curve @° each of the two supporting points A, and A, coincides with its associated point; in this case the line A, A, is associated to itself. According to § 7 the line A, A, is a component of one degenerate e® and as such contains two singular points D,’ and D,'; to these points correspond all points of the two transversals 6, and 5, of the lines A, A,, a,, a, and a,. If we consider the points D,' and D,' as supporting points, there is associated to the line A, A, a belinear congruence of rays which has the lines 6, and 5, as directrices. If we consider one of the points D,' and D,' and one arbitrary other point of the line A, A, as supporting points, we find that to the line A, A, there are moreover associated two fields of rays lying in the planes which connect the lines 6, and 6, with the line A, A. Also the line a, is bisecant of all curves 9’. The supporting points EF, are each time the two points of intersection of the line a, with a surface g?,,. The points 4’, and F”’, associated to these, lie on the generatrice 4 and u of the surface w’ corresponding to /, and F,. Through each pair of points (£,, /,) pass o'* curves g°; the cor- 587 responding points #,’ and #, describe apparently two projective sequences of points. Moreover the pairs of points (#,, F) form an involution on the line a,; the pairs of generatrices (2, u) form there- fore also an involution. Consequently the pairs of points (£,', F,') form an involution on the surface w? and the lines connecting associated points of this involution are the rays associated to the line a, We shall first demonstrate that each generatrix v of the surface w* which belongs to the same system with the lines a, and a,, contains one pair of points (#,', F,'). With a view to this we remark that two points #,’ and F,' are situated on the same curve 9°; Q this curve intersects the surface w° besides in the supporting points of the bisecants a, and a,. The congruence [g*] being bilinear, each line wv belongs as bisecant to one o°; the corresponding supporting points are the points in question /,’ and F,’. Through a point Z, of the surface w? there pass two rays of the congruence in question, viz. the line connecting Z,’ with its associ- ated point /,’, and the line v passing through the point £,’; con- sequently the order of this congruence is two. A tangent plane of the surface w* contains one line v and one line 2. The straight line u, associated to the line A, cuts this tangent plane in a point Ff,’ and the line connecting this point with the associated point #,’ is a ray of the congruence in consideration, which together with the line v lies in this tangent plane. For this reason the class of the congruence is two as well. Analogous considerations hold good for the lines a, and a,. Con- sequently to each of the lines a,, a, and a, theré corresponds a congruence (2,2). 4 § 10. A straight line / through the point A, is bisecant of oo! curves o°. The point A, corresponds to itself; the locus of the points P’ corresponding to the points P of the line / is according to § 8 a curve (/)*. This passes through the point A,; for when P gets into A,, P’ coincides with P. The rays associated to the line / project the curve (/)? from the point A, and form therefore a quadratic cone. The same holds good for a straight line through the point A,. A straight line / in the plane «,, is bisecant of oo! conics k?. Let E and F be the points of intersection of the line / with such a conic. The points £’ and /’, associated to these points / and F, lie according to § 5 also on the conic 4? and the straight line 2’ F’’ is associated to the line /. The locus of the points 4’ and /” is a conic k?, for the line / 588 has one point in common with the line a,. To this point corresponds a line 4, so that the curve (/)* which corresponds to the line /, must degenerate into this line 2 and into a conic 47, the locus of the pairs of points (E’, FF’). These pairs of points form an involution on the conic £7; the line e’#’’ passes therefore through a fixed point, so that to the line 2 a plane pencil of the plane «,, is associated. The same holds good for a straight line in one of the planes a Ong, U hon Gay GUNG! Ge 25? According to § 4 each transversal g,, of the lines 6,,, and 6,,, contains an involution of pairs of points (G,H) which are each time the supporting points of a curve o°. The associated points G’ and A’ lie on the curve (/,, which through the involution (P, P’) is associated to the line g,,. The pairs of points (G’,H’) form an involution on this line with two coincidences and the lines G’H’ determine a _ quadratic ruled surface, associated to the singular line q,,. In the same way there corresponds to each of the lines g,, and Js, A quadratic ruled surface. The straight lines which are associated to all the lines g,,, form together a line complex, the order of which we shall determine later on. $ 11. It appeared in § 5 that on each degenerate ge’ of the first system lies one singular point D’ which is associated to all the points of the line d. A bisecant / of this o° through the point D’ corres- ponds therefore to a plane pencil which projects the line d from the point which is associated to the second supporting point of the bisecant. These bisecants 7 form two plane pencils, which both have the point D’ as base point; the first lies in the plane of the conic 4’, the second projects the line d from the point D’. The plane of the conic 4? passing through the line 4,A,, the bisecants / of the first kind are the common secants of the line A, A, and of the locus o* of the points D’. As A,A, and o* have two points D', and D', in common, their common secants form a con- gruence (1,3). A plane JV intersects the curve o* in three points; through each of these points passes one bisecant / of the second kind lying in the plane WV; these biseeants form consequently a congruence of class three. From a point P the curve 6? is projected by a cubic cone K°. The planes which project the corresponding lines d from P, envelop a quadratic cone of which the tangent planes are projectively asso- 589 ciated to the generatrices of the cone A*; it happens jive times that such a plane passes through the corresponding straight line, so that this line is a bisecant / of the seeond kind passing through P. Hence the order of the congruence formed by these bisecants is five. To each ray / of one of the congruences (1,3) and (5, 3) corres- ponds a plane pencil of straight lines / which project a line d from a point of the corresponding conic £*. For the lines / of the second kind this point coincides with D’, so that the congruence (5,3) is transformed into itself; for those of the first kind it is an arbitrary point of the conic £*. A plane V intersects the conics 4? in the points of a curve cf that has a node in the intersection of the plane V ‘with the line A A,, and the lines d in the points of a conic c°. Between the points of the curves c* and c’ there evidently exists a correspondence (1, 2). The three points of intersection of these curves lying outside the intersections of the plane V with the lines a,, a, and a, and with the two transversels 6, and 6, of the four lines 4,A,, a, a, and a,, are points D, hence coincidences of this correspondence. The lines connecting associated points of this correspondence, in other words the rays /' lying in the plane JV’, envelop therefore a curve of class five. The rays U corresponding to the rays | of the congruence (1,3) form consequently a line complea of order five. The degenerate curves 0° of the second series, found in § 3, do not contain any singular points. § 12. Coincidences. A line A produces a coincidence if its sup- porting points P and Q coincide with their associated points 1’ and Q’. The involution (P, P’) has a finite number of coincidences, viz. the points A,, A, and the six points D found in §5, in which the trans- versals 4,,, ete. cut the corresponding planes a,, etc. The line A,A, and the lines connecting the points A, and A, with the points D are therefore rays of coincidence. Let us further consider a line / through the intersection D, of the line 4,,, with the plane «,,. This line is bisecant of a degene- rate y® formed by the line b,,, and a conic £* in the plane «,,; in the point D, this conic touches the plane brought through the lines Land be of which the supporting point P lies on the line 6,,,, the supporting point © on the eonie k?, to approach D,, we get such a straight line /. The point P/ associated to P lies on the line 6,,, and is barmoni- 39 For if we cause the two supporting points of a bisecant PQ Proceedings Royal Acad. Amsterdam. Vol. XXII. 590 cally separated from P by the points A, and D,; it approaches therefore also to D, and in such a way that lim. PD: P'D, = —1. In the same way the point Q on the conic 4? approaches to the point D,. From this it is easily seen, that in the limit the line P'Q' coincides with PQ so that the line / is a ray of coincidence. Consequently the straight lines through these six points D are also rays of coincidence. A line ¢ is also a ray of coincidence, if P’ coincides with Q and Q' with P, so that the supporting points P and Q are associated to each other in the involution (P, P’). According to § 8 these rays form a cubic complex. § 13. When a straight line ¢ describes a plane pencil, the associ- ated ray ¢' describes a ruled surface R, of which we shall determine the order. Each ray is bisecant of one curve 9°; the locus of the supporting points is a curve c; this has a node in the base point B of the plane pencil, for on the two rays ¢ connecting B with the two other points of intersection of the v* passing through £4 one of the two supporting points gets into B. Hence the curve c is of order four. The curve c* has one point in common with each of the three lines a,, a, and a,; for if a ray ¢ intersects one of these lines, one of the two supporting points gets into the point of intersection. Through the involution (P,P') a curve (/)* is associated to a line l, hence to a curve of order four, in general one of order twelve. The curve of has one point in common with each of the straight lines a,, a, and a, and to each of these points a line 2 is associated, so that moreover a curve 9° is associated to the curve 9°. The pairs of supporting points form on the curve c* an involution with sir coincidences; these are the points of contact of the six tangents which can be drawn from the node B at the curve c’*. The pairs of points of the curve 9’, associated to them, form there- fore also an involution with six coincidences. The lines connecting associated points of this involution form consequently a ruled surface of order siz, which is the surface R. We can also determine the order of & by trying to tind the number of points of intersection of this surface with the line a,. With a view to this we remark that to the line a, a surface w’ is associated, so that whenever one of the supporting points of a ray ¢ lies on this surface w?, one of the supporting points of the associated ray t’ lies on the line a,. The surface w? passes through the lines the curve cf intersects this surface besides in the points a, and a,; 591 it has in common with the lines a, and a,, in sie more points, so that the plane pencil in consideration contains sir rays ¢ of which one of the supporting points lies on the surface w*; consequently there are sir rays ?¢’ intersecting the line a, In the third place we can determine the order of A by trying to find the number of intersections with the line 4,A,. For this purpose we remark that a ray ¢’ intersecting the line A,A,, if it is not a singular ray, must be bisecant of a conic £*. The two sup- porting points are associated to two points of the same conic, so that also the associated ray ¢ intersects the line A,4,. The plane pencil contains one ray ¢ intersecting the line A,A,; the associated ray ¢’ rests also on the line A,A,. According to § 11 there is a complex of order five consisting of rays ¢ associated to singular rays # which form a congruence (1,3) and each of which intersects the line A,A,. The plane pencil contains 5 rays of this complex, hence the surface R five rays ¢’ of the (1,3). In all the line A,A, is intersected by six rays ¢’, so that the surface FR is of order sie. § 14. We can now also determine the order of the line complex associated to the congruence of the singular rays g,, found in § 10. A singular ray 7’, intersecting the line 6,,,, is bisecant of a degenerate 0° consisting of the line 6,,, and a conic £° of the plane @,,, passing through the point of intersection of this plane with the line ¢’. The supporting points of the associated ray ¢ lie also on tbe line 6,,, and on the conic £°. Now the plane pencil considered in the preceding § contains one ray ¢, which intersects the line 0,,,; hence there is one ray ?¢’, which intersects the line 0,,,. The other five generatrices of the ruled surface f° intersecting the line 6,,, must be singular rays, therefore lines g,,. The plane pencil contains five rays associated to rays g,,; consequently these rays form a complex of order five. § 15. To a sheaf of rays corresponds a congruence [!]. In order to determine order and class of this congruence [t’], we take the base point B of the sheaf on the line A, A, It has been found already in § 13 that to a ray ¢ intersecting the line A,A, a ray ¢’ is associated also intersecting A,A,. We shall now show that the rays ¢ and ?¢’ intersect the line A,A, in the same point. Let /* be the conie which has the line ¢ as bisecant, P and Q the corresponding supporting points, P’ and Q’ the points associated 39* 592 to these. Through a linear transformation of the plane a of the conic 4? we can transform the points A, and A, into the circle points at infinity. If S be an arbitrary point of the conic k?, the straight lines SP and SP’ will be harmonically separated by SA, and SA,,. hence they will be perpendicular to each other after the transformation, so that PP’ is a diameter of the circle 4?, the same as the line QQ’. The chords PQ and P’Q’ are therefore parallel and consequently intersect on the line A,A,. To an arbitrary ray ¢ through the point B corresponds, therefore a ray through the same point, so that to the congruence {t’| there belongs in the first place the sheaf itself. To the line A,A, corresponds a bilinear congruence of rays, also belonging to the congruence [¢’|, besides two fields of rays. Through the point B passes a cubic cone of singular rays of the congruence (1,3) considered in § 11. To each of these rays corres- ponds a plane pencil which projects a line d from a point Q’ of the corresponding conic &*. The point Q’ is associated to the second point of intersection Q of the ray with the conic £?. The cubic cone mentioned has the line A,A, as nodal generatrix. The two generatrices coinciding with A,A, belong to the two dege- nerate conics k? consisting of the line A,A, and of one of the two lines 6,,6,; hence the two leaves of the cone A *, which pass through the line A,A,, touch at the planes of these degenerate conics conse- quently they also touch the two leaves both passing through A,4,, of the surface of order four, found in § 2, described by the conics k?; the line A,A, belongs therefore siv times to the intersection of the cone K* with this surface. The rest of the intersection consists of the curve o* projected by the cone KX? and of the locus t° of the points Q. The cone A®* has three points in common with each of the lines a,, a, and a, lying on the quartic surface mentioned ; the curve 6? having these lines as bisecants, two of these points lie every time on the curve o*, while the third must lie on the curve t’. It is further easily found that the curves o* and zr’ lying on one and the same cubic cone, have tree points in common. In general through the involution (P.P’), to a cubic curve a curve of order nine is associated. However the curve t° having one point in common with each of the lines a,, a, and a,, three straight lines 2 belong to this associated curve and as it has three points in common with the curve o and for this reason contains three singular points D’, three lines d belong to it. The complete locus of the points Q' is therefore a Curve r‚*. The rays in question, associated to the generatrices of the cone 593 K*, project the lines d from the corresponding points Q’ of the t,". In the same way as in § 11 we should therefore find that these rays form a congruence (5,3). But it happens three times that the point Q’ coincides with the point D and hence lies on the line d; these points are associated to the three points of intersection of the curves o* and t°, for in them the point Q coincides with the point D’. In this case all rays through the point Q’ intersect the line d. Accord- ingly, from the congruence (5,3), which we should find in general, three sheaves are split off and we only find a congruence (2, 8). To the sheaf of rays through a point 5 of the line A,A, are associated one sheaf, two fields of rays, one bilinear congruence and one congruence (2,3). In general there corresponds therefore to a sheaf of rays a congruence (4, 6). § 16. To a field of rays corresponds also a certain congruence. In order to investigate this, we consider the rays lying in a plane a through the line A, A, A non singular ray of this field is bisecant of a conic &? in this plane z, hence associated to another bisecant of this conic. To the congruence in question belongs therefore in the first place the field of rays itself. To the line A, A, in the plane z correspond a bilinear congruence of rays and two fields of rays. To an arbitrary straight line through the point A, corresponds a quadratic cone with the point A, as vertex. This intersects the plane a along two straight lines. The sheaf of the rays through the point A, belongs therefore also to the congruence in question and each of these rays must be counted twice, because it is associated to two rays of the plane a. The same holds good for the sheaf of the rays through the point A, The plane zr intersects the curve 0° besides in the points D', and D', in one more point; through this point passes a plane pencil of singular rays of the congruence (1,3). To each of these rays corres- ponds a plane pencil, which projects the line d, belonging to the conie £°, from a point of this conic; hence to the plane pencil mentioned corresponds the congruence of the lines resting on 4? and d. As these have a point D in common, those lines of intersection form a congruence (1,2). A field of rays is therefore transformed into a congruence (6, 6). Mathematics. — “On n-tuple orthogonal systems of n—1-dimensional manifolds in a general manifold of n dimensions.” By Prof. J. A. Scuouren and D. J. Srrurk. (Communicated by Prof. J. CARDINAAL). (Communicated at the meeting of June 28, 1919). I. 1. Notations’). A p-dimensional manifold may be denoted by V,, a p-dimensional euclidean’) manifold by Ze. A, may also denote an infinitesimal region, determined by p independent directions, in the vicinity of a point of V,. As original variables in a V,, we use the systems a and y,..., with the corresponding covariant and contravariant vectors ex = Va, ey 9 eee aera beeen! ed (Jt) Sj — Vils): which satisfy the conditions: ’ eN. = &? 3; e2.e,—e8) 7 ’ Bae | : 5 -— TT 2 Sj - Sj — Oj H Sj.Sj = 0; 0 when AZu eN eu = mai) Aro RC) “—=(—1l) 2 when A=wu 0 when jk 85. Sz’ = * when j=khk. The fundamental tensor of this V, may be written ’g: di, mop li Uv eers U, lS otd 1,...,n WE LZ Hyp eey= Z Phe ey’ = J gjrnsjsn— = pks; sy’. (3) dy ps do Jk Jk We will choose the aequiscalar V,_4 belonging to a and 17 in different ways according to the circumstances. 2. Normal and V-creating fields. In a manifold V, may be given 1) Wor the notations used in this communication see also: J. A. SCHOUTEN, Die direkte Analysis zur neueren Relativitätstheorie, Verh. der Kon. Akad. v. Wetenschappen XII, 6 (1918), here further cited as A. R. 4 2) We will call a manifold euclidean when its Riemann-Christoffel affinor K is zero. Compare A. R. p. 58. ; 595 an f,-tield by the p-vector-field of the simple p-vectors’) „v. Then any system of simple (n— p)-vectors, perfectly perpendicular to „v, determines an R,—)-field. When the field ,_,w is everywhere per- fectly perpendicular to a system of oo" 7 manifolds Vj,qg<>p, the R,-direction of JV, in each point lies in the region of ,v. In the case that ,_,w is not also perpendicular to a system of o”—@—! manifolds V,4; we call ,_,w Vynormal and ,v V‚-creating. When ,v=vV, ... Vp, then ‚v is V,-creating, if the p equations: AD ee etic. RE) have n—gq independent solutions. This condition is sufficient too, Sn the V, can always be given as intersections of n—q systems Ce Cea Chine, en, C, ake variable parameters. It is necessary that qr > p might be obtained, which would be satisfied by all solutions of (4). But this system certainly has more than ”— r independent solutions. Hence the condition (7) is necessary and sufficient. AS: ZUM (V vi) vj —(V «vy PVE eV); (Vii WI) vi a (8) (7) is equivalent to h A simple p-vectov can be written as an alternating product of p non-ideal vectors and can be represented by a part of a definite Fe, with definite volume, but of indefinite shape, together with a hyper-screw-direction, 596 io VEER = VA i == ie We (Wa ig) Oes ca (6) or also to pul WE VERS SUG oo Bus Gore (YY) hence to NG Rl eae ee ees (A) In this equation the auxiliary vectors v and w occur no more. It is the required condition that the ,v-field may be V-creating. As: we -{V 1 (vi vj} = we- (a. V) (a! (Vi — vj) ®) = = (a. V)iwea? (vi — v,)}) —(a.V) we fat (vi vj} —= = (11) = (vi > vy)? V Wi, (A) is equivalent to (vi — vj) 2 V >We), LT Ge cae SV (12) and as } Ln —p Vn pW = AW We (QV) Wie Witten Wimpy eo (13) k also to DVCAM, ne) on aks ol te ere) (B) can be deduced from (A) without returning to the auxiliary vectors v; and wz. We can show also independently of (A) the necessity of (B). For, when ,_,w is V,-normal, we always have an) n—pW = À (VA) 50 6 (7 fn—p)} Re (14) in which 2 is a function of the place. Hence ni SDN de so Wife) 6 vere er (LB) from which (B) is a direct result, because every v is 1 V fs. When p=n—1, we see from (B), or clearer from (15), that V — w is a simple bivecior. From this we may deduce the following theorem. When a field w is V_{-normal and w is interpreted as 1) The forms (10) and (12) of the condition are identical with those occurring in E. von Weser, Vorlesungen über das Pfaffsche Problem (TEUBNER, 1900) page 99 and 100. 3) (A) and (B) were already given without proof in J. A. ScHoureN, Over het aantal graden van vrijheid van het geadetisch meebewegende assenstelsel. Versl. der Kon. Ak. v. Wet. 27 (18) 16—22. 3) The differentiating effect of a differential operator extends to the first coming closing bracket. 4) This term is zero, because wj,Lv; and LV, 597 the vector of velocity of a streaming liquid, in which case the component of rotation of the movement (with respect to a geodesically moving system) is given by V — w‚ in this rotation every point ot the R, 21 V — w remains unaltered.') Indeed, dr!1(V — w) is a vector in the plane of V — w. In the same way we can prove that, if ,v is V,-creating and thus ,_,wW is V,-normal, the equations exist: | n—qW WA py =i, Qi, Sor OAT EEND (A!) Lv? Van vd. . Oo . . e 5 5 (B) in which ‚„v represents a g-vector in Va and „—,w an n—gq-vector LV, For p=n—1 we see from (B) that for a V,-normal vector- field w the component of Vw in V, is a tensor. 3. Canonical congruences. A field of unit-vectors i, determines a congruence’), u,—*xi,! Vi, is the vector of curvature of the curves of this congruence and the modulus w, = (U,), is the geodesic curvature. As (V ray i, = 4 Vv (in ° i,) = 0, . 0 . : 0 . (16) the second ideal factor of Vi, does not contain an index n. Hence 2 Ji, consists of two parts, a part h in the B, Li, and a part 16: i, V th, = zi, U: 2 vl,=h zi in Un, . NTS ITD (17) 9 In general h is the sum of a tensor *h and a bivector jh. If i is a unit-vector in one of the m—1 mutually perpendicular principal directions of *h, we have Se SNL rjg hol To Pt, a eee a (LS) and as VS in == th + 5 (iron + ont), . . . (19) we have ANA NA Dd a Hidde oh oe 20) ) For Rn this is observed by A. SommerreLp. Geometrischer Beweis des Dupis'schen Theorems und seiner Umkehrung, Jahresberichte der Deutsch. Math. Ver. 6 (99) 123—198, p. 128. 4) In A. K. p. 38 et seq. the word “Hyperkongruenz” is used. For the sake of simplicity we will use here the word congruence, in harmony with among others Moor and Levi-Civira. 598 or (C2 7th A ETE 6 0 bo 2 oo (2115) for TEE Te (DE) From (21) follows: in! (xV = in — 2 1) =e Wi helen Saad & (23) or, when X, are the covariantive coordinates of V ~i, and 7, those of in, in coordinates: 0 Ühahosd 6.6 oo Dore 0 tna, Un a Xa, En A Yara, « Xa, ae Ja, a, —= 0. (24) a X == eke qe —A nay, ony Jae, tn Ja, a, This equation of degree n—1 in / is called by Rrcor the algebraic characteristic equation of the congruence i,. *) Since the tensor *h has, as is known, n—1 real principal directions, the equation (24) has n—1 real roots.?) When all roots are distinct (that is when no two roots are equal in all points of V,, which does not exclude that they may be equal in some manifolds of less than n dimensions), we see from (21) that to a definite root 4; belongs the direction: SOC typ = data" ies 0 ve ove (8) Two directions belonging to distinct roots are mutual perpendicular, because with regard to (20): Zins We == ais (WY LE es Aptis a, HA so (PP) A p-fold root determines a region fi, perfectly perpendicular to the regions of the other roots, and in this region we may choose p arbitrary mutually perpendicular directions as principal directions. In every case we can indicate to the given direction i, in every point n—1 mutually perpendicular principal directions that join to the congruence i, n—1 mutually perpendicular congruences ij, j=1,2, ...,m—I1. Riccr calls these congruences the orthogonal canonical congruences belonging to in. *) 1) G. Ricci. Dei sistemi di congruenze ortogonali in una varieta qualunque, Memorie R. Acc. Lincei Ser. V 2 (95) 276—322, p. 301. 2) For a direct proof see e.g. G. Ricci, Sui sistemi di integrali independenti di una equatione lineare ed omogenea a derivate parziali di 1° ordine, Ann. di Mat. Ser. II 15 (87/g.) 127—159, p. 134. 3) G. Rreer. Dei sistemi, p. 302. For the sake of brevity we will speak here of canonical congruences. See also G. Riccr and T. Levi Crvira, Méthodes de calcul differéntiel absolu, Math. Ann. 54 (Ol) 125—201; J. E. Wricut, Invariants of quadratic differential forms. Cambridge Tracts N°. 9 (08), p. 73. 599 For the sealars 4 and u from (20) follows: figs fale? Wise 0 6 o 6 6 5 6 (20) NaS NW See eon et a ale TEE) and from (18) for *h: Ien hi = Aj ij ij, sne ate EEE (29) J and OOMEN) or Dei BH) In the special case that in is V‚_—-normal, (B) gives: ine ONNIE (32) hence ‚h — 0. Instead of (17) then the equation holds: Ti, es gta bh cant aee) By means of the idea of geodesic alteration, that is alteration with respect to a geodesically moving system, a simple geometrical interpretation can be given to the canonical directions. In conse- quence of (17) and (30): prin Uti aver Wii ZN Sin wl ike (34) Now ziz1 Vin is the geodesic increment of i, when moved in the field along ig pro unit of length, and so i;i,? Vi, is the projection of this specific increment on the j-direction, i.e. the specific geodesic rotation in the m —j direction. Hence when „Bz is the bivector of specific geodesic rotation of the system i,,...,i, when moved in the k-direction : in! V tc = „Br! in A hy Pa Sent Oee rt ANNE (35) then i;1,2 Vin is the 7j-component of „Br: in ij 2 „Br = ij ik ? V iy, i] F k. . . . . . (36) So the nj-component of the rotation „Br is equal to the nk-compo- nent of the rotation „By. ) When i, is V,—;-normal, we get in consequence of (831) and (82): ATA Wee (Dm ae ar (37) Thus the nj-component of „Bz is zero when j # k, or: the geodesic h Rrcor, Dei sistemi, p. 303, gives another geometrical interpretation, in which he makes no useof the idea geodesically moving. Then however it is necessary to lay the Vn in a euclidean space of more than 7 dimensions. 600 rotation of i, when moved in the f#-direction occurs in the nk-plane. When we define the principal directions of curvature in a point of the V,-; as the directions i, in which the geodesic rotation of i, occurs in the nk-plane’), we can conclude that the canonical congruences are the principal directions of curvature of the V‚ a LAn. Still a consequence of (29) and (83) is zij! V in = 7; ij, Hells 9 dlg NSS) equation equivalent to (37), which may also be considered as definit- ing equation of the principal directions of curvature of the V,,_; 1 i,. 4. The second fundamental tensor of the V,—, Lin. In order to make clearer the signification of *h, we choose the V, 4 Li, as aequiscalar regions of the original variable 2’n and n—1 arbitrary systems of VV, through i, as aequiscalar regions of the original variables a’, 2 = a,,...@,—1. Then the directions of e, and e,’ lay in the Vl i,, while @, and ea have the direction of i,: . a ij Sh SG VY G8 Ön eg n En n LOE 5) Then the contravariant 2u-characteristie number of 7h is: TPP == HGn? SHOES Wid Fr With 6 va (0) or, because i, Le and | es, and: VASES) D=Ggooso Gein o oo (EN) also: "Y= — tinen* Ver tire? Ver 1 =— (ese 2 Vetere? Ve) = Ze we 1 da dat ) cS == ae + Coal Zen ‘= 0 fou In the same way the covariant characteristic number of *h is: 1) This definition is the natural extension of the definition of the lines of curvature on a Vs, in Rg as the lines, along which the normals form a developable surface. ESO Oy 2 8 Or OLE Win Gro 2 Vinh = Ì S=— OF Ge PW de EV = Zen 2e, ig 1 (94a Òaa 1 ade da; . . (48) n dE n i aie A Ss || Se a) an |= a) A Qe, \ Oat Ow) Zen \Qatn Aan a : =—(e'o . V) 0p =FH(in - V) Giz. DE ow Hence *h is the second fundamental tensor of the Va Lin’). When i, is geodesic without being V,,-;-normal, then we have u, =O and Vi, lies totally in the region 1i,: wi Baalen 70 (4) When i, is normal too, Vi, is symmetrical : VAi Ne ae ee (4D) In this latter ease choosing 2°" as the length measured from a definite WV, Li, along the curves of the congruence i,, we get: ea, = Ea, Sin - - - . « « « (46) 5. Mutually orthogonal V,y-systems through a given congruence when the canonical congruences are singly determined. When given an i,, it is required to choose the original variables y',..-y"—1 in such a way that the corresponding aequiscalar Va pass through 1, and that the vectors s=Vy,j=1,..., n—-1, are mutually perpendicular. Hence the system of equations: heal == oar spe a aT) Citic SY (iO aes HP San) ae Soa ine wena) — alt) must allow for every value of 7 m —2 independent solutions. The necessary and sufficient condition is, according to (7): in) Vers! Vin=arsr Honing - . » « (49) in which «, and «, are arbitrary coefficients. Since in LS, and accordingly : in) V sp =(V sz)! in = V (Sp. in) -— (Vin)! sr = — (Vi)! sz, . (50) (49) is equivalent to: ') Compare Biancai-Lukat, Vorlesungen über Differentialgeometrie (1899) p. 601, form. (7). The principal directions of curvature may also be defined as the principal directions of the second fundamental tensor. So Brancut, p. 609, 618. 602 HOW ih) eS Ee Cis a ae (HN) This equation however is of the shape (20) and hence each of the desired vectors sz forms one of the canonical congruences belonging to i,. At first we consider the case that the n—1 roots of (24) are all different. In this case every vector 8; must be equally directed with a definite i: 1 ig = OkSk=— Si? - ol or (62) Ok The n—1 canonical congruences belonging to i, must therefore all be V,—1-normal. In order that this may be so, i, has to satisfy certain conditions that may be obtained as follows. Application of V to (31) gives: (Tij)! (Vin)! ie + (Vint (V ~ in) 1 ij {VT il? ir = 09%, (53) and transvection with in: int (Vit (V in)? ie + int (Vip) (V — in)t ij Hij ir in? V(Vin)=0.(64) In consequence of (46) and (51) (V~i,) 1 iz contains only iz and i,. Further i; is V,-1-normal, so that according to (B): int (V ij)? ik =iz 1 (V i;)! bon Se (55) Hence (54) is equivalent to: — iz! (V ~— in)! (Vi)! ij—ij} (Vi)! (Vi)! iz aa i; ik i, 3 V7(V~i,)=0. (56) If now 1,...,n—1 tn an = Dn Dn == = ij ij eere c z 5 (57) J we have a quantity: 4 Sn — an Dr b, An. : . . . . : : : . (58) which when transvected twice with an arbitrary affinor of second degree gives the component of this affinor in the A, 1 Li, Intro- ducing the tensor: 4 De WLA SE oe GD) we get from (56): iin? *p = 0, Wc le li oe atthe Ca) Hence the first condition is that the tensor *p has the same principal directions as *h. Since on account of (19): 4 4 En 2 (in . V) (U xe in) = Sn 2 (in . V) *h AF Xl Uns. 0 5 (60) und on account of (19) and (30): 603 4 4 6.2 27 (9 ~ in)! (V in) = 2 *h! Ph + xn un + gn? 2 Th! ‚h, (61) we have: 4 Pp Enne Vb 2 *hi sh) — 27h th (62) Since *h! *h has the same principal directions as *h, we may express the first condition also in another way, viz. that 4 gn 2 {(n- V) *h—2 Th! oh} has the same principal directions as *h: iin? {Gn V) *h—2 Fn! MOF EE 1, 2,..,n—-1.| (C) (C) ean also directly be found when we start from (30) and reason in the same way as we did when deriving (C’). In order to get a second condition we apply V to (30) and after that we transvect with i,. This gives: (Gv. V) ij} ie Ph + YG - V) ix} ij 2 7h + iin? Gd. V)*h=0 . (63) or, according to (29): Miki? Vij + 47 ij)? Vin Heijin? (iu. V)"*h=O0 . . (64) or: (AA) iris? Vi; +xijiz? (vu. V)*h=90. . . . (65) The vectors ij, i, and iy being all V,,-1-normal and mutually per- pendicular, so that iki? VG? Vij= gi? Vi =—iki;? Vij= 9 | (66) =a Vip Vil NE VA or: Oe LAE NOY. (67) the equation (65) is equivalent to ic? i. VI =O, 7B Allin hk =D eg nd (D) Since (Jij) | i; and in consequence (i,.V) V (ij. ij) is zero, so that: (MEAN Ip TET MGE also the principal directions of *h are singly determined. The tensor zdsij1 7 *h being the geodesic differential of *h, when moved over ds in the direction of i;, the second condition (D) expresses that by an infinitesimal translation in a direction perpendicular to i, and perpendicular to m 9 De 9 rotation ern III ,; Pav MWe BV „v are the same for all possible charges of the beam. For if we consider the two triangles 4,’ B III, and 4,’ B, Il, 1 (the latter of which is supposed to bear upon an arbitrary charge differing from the one given), in these two affined figures the points Pa B and Bae B, Pur, „IV and Pir, Iv are homologous points. x x From this it can immediately be derived, that also the points Ei and Pir, ul, P iv aN and Pav ‚v are corresponding points, so that the lines connecting them must pass through the pole of affinity, the point at infinity of the straight lines J. p) —— . ri =D ¥ * +, Pi, u, and Pu, il, as well as P Iv ‚Vv and LAAN wv lie therefore perpendicularly above each other. From this follows the theorem referred to in the beginning of this §: The situation of the centres of rotation Ly Nyt te B, Pir, „IV, x P 1v vy, relative to the lines /, is quite independent of the charge of the beam; it is exclusively connected with the stiffness of the beam and that of its supports. 8. Beam on four points of support. When we have once made ourselves familiar with the line of thought, developed in the preceding paragraphs, it is rational to try and find a solution for the beam on four points of support according to the following program. 1. Cnt the beam at the last point of support but one, and construct the situation of the point D in two ways. First by determining the reaction yp of the beam CD, freely supported at its extremities, and secondly by drawing for the beam ABCD the link-polygon belonging to Mc=0O. In this way two points 628 .. and .,D,*) appear, which do not coincide, unless in reality there is no moment of transition above the point of support C. 2. Construct then in a similar way two points ‚D and UD on the supposition that the moment of transition Mc is one metreton. 3. Prove, that the series of points yD and nD) arising on intro- duction of various moments of transition J/¢—y metreton, are similar. Then the double point D of these series, to be constructed by the help of oD, oD and ,D, ,D, will indicate the real situation of the last point of support. 4. Starting from this point D construct the link-polygon in question DIX VIII VILCVIV IV B II IIA: | | : / | ! / of U 4 fae | { 7 o \ fi / | | a | (CO eo A B YI £ VOA 1 ' 9. In fig. 2 the working program, developed to this end, has been put into execution on the supposition that each of the fields AB, BC and CD of the beam is charged in the middle by a force of one ton. First the construction given in $$ 3—7 bas been executed for the beam ABC, which besides by the two forces of one ton on each of the fields AB and BC is supposed to be charged at its extremity C by a force of } ton (originating from the charge of the last field CD). 1) By the indices O and . added to the letters D, is indicated that the moment of transition in C is zero and that the moment of transition in B has the right value belonging to the supposition Mc = 0. 629 If we cut this beam at B the whole extra charge acts on the spring under C, so that in the determination of the points A >, B , C only the point C appears to have an extra 00 00 00 00 descent. Without any diffienlty with the help of the link-polygons Palin Retain CnandeALtoillaone 2 oro Nn Cathe points C and C 00 00 00 10 10 00 10 ean then be constructed, which together with the point C and C 00 10 determine the point C through which the beam ABC must pass at „0 its extremity C, when besides the given charges it must bear in C a force of 4 ton. On the supposition M‚/==0 the side 12” u is Vo eN Q can now be prolonged as far as lvirr. After that from Ca segment C, C'=c .0 0 .0 must be drawn in downward direction in order to make it possible to draw the side C' oVIII oD. In this way, however, the point oD „0 is determined. 10. It is far more difficult to find the point C, through oi which the beam ASC, considered as a whole, must pass if in C a moment of transition of one metreton is applied. For, when the connection of the beam above B is broken, this moment will, in opposition to the force just applied in C, besides on C also exert its influence on the point B. The place of the point B is taken by a point B, which lies 7 00 01 p : 7 (10 é u Y higher. In the same way the point C' lies a distance 15 below C, o1 00 because the couple of unity acting on the field BC as well as that, . : u : acting on the field CD, gives an extra descent — to the spring under C. 4 In case the moment of transition in B is supposed to be zero we have therefore to do with the link-polygon A'oj, [Io1, B, 01 oVs, oVI, C, of which the two last sides now deviate and cut a segment 01 of known length from /r. ') By a second index, placed to the right or to the left of a letter, the value of the moment of transition in the second point of support to the right or to the left is indicated, etc. 630 If in Ba moment of one segmentton is introduced, the point C is serment 01 t replaced by the point G lying 7 higher, while tbe construction of the elastic link-polygon A’,, II, UL, B, IV, ‚Vo, Vs C removes u u the point C to C. ol u Instead of the earlier fixed centres of rotation Pry 9 BIV: AA Wp 5 i 5 BY >! 1] 8 . = 1 Wp Voy other points Pira Ivy: Np Vil SIVA wyply ing perpen dicularly above them, appear; of these points for the present only the last is of importance. For when the double point C of the series Cand Cis constructed, 1 xe 1 xl also the point C” is known, through which the side V, VI must pass. al fle cal But this side must also contain the point PAs Vas it is therefore determined. Consequently also the sides NL; UG. WWI UL and ‚VEL oD can be drawn, so that now BD) is determined. *) The construction of the points „D and ,D conjugated to the points ND) and mid just found, does not present any difficulties. 11. According to the outline given in $ 8 we must now investigate whether the series of points yD and yD: which appear on the introduction of various moments of transition 1/c— y metre-ton in the way described above, are similar. To that purpose we consider in the first place the centres of rotation Bis us Pi, „IV PAW, Vi just mentioned, belonging to the moment of transition Mvo == l metre-ton. These centres of rotation 1) Strictly speaking the construction of the link-polygon A11, 1u, B... C, 11 11 mentioned in this 8 and drawn in fig. 2 for completeness’ sake, is superfluous. ORG) For it serves exclusively for the determination of the ratio wee which only 01 11 depends on the horizontal situation of the centres of rotation Pir hey LER LV sal sn EO AES Al aC : 01 11 : ete. which corresponds to that of Bnr, Vly P ivy ave etc. CC therefore he ol 11 10 put equal to the ratio ze already found. 631 lie perpendicularly above the centres of rotation Piro ine en IN P Ivo „Vo On & straight line through A’,,, determined by the point Ze 8 i lying © above B. L 00 As on introduction of the other moments of transition Mc=y meterton there appear points B, defined by B, B=y. B, B, it is Oy 00 Oy 00 01 evident, that the centres of rotation mentioned, undergo vertical displacements, which are proportional to these moments. Especially at the introduction of Mc = y metreton the ; 4 : : segment P'_1vo Vo P IV, iy will be equal to y times the segment Ua / P x!V9 xVo P „IVi Vr: On account of the law of superposition, on which the whole problem is founded, the descent C C of the point C will increase 5) cal in direct ratio to the value y of the moment of transition Mc. The distance of the point C" to the point C can therefore be y 0 put equal to: y .(C" C—C C). ott oil a) onl The lines (EAN aie C") connect therefore corresponding points x Vy 3 e of two similar series of points; they pass through one point. As VI, .VI, can be linearly expressed in C" C and P’ rv, ‚v‚, ay 0 ¥ 8 P’ iv Vix the series of points .VI, is also similar to the series yx C, so that the lines .VIy, ‚VII, too have a fixed centre of rotation P ‚vj But then also the sides VII VIII and VIII D have „VI VII sy) Ly y y fixed centres of rotation P vu ‚VII and P* vil ‚VIT The series of points „D is therefore similar to the series C", C, y LY P’ tv, „v‚--- which in their turn are similar to the series ,D, for yxy J which holds good: yD oD=y XD oD. Hence the series of points D and D are also similar. Their double point D at finite distance is the extreme point of the link-polygon in question for the beam on four points of support. Now that this double point is known, the construction of the whole link-polygon no longer presents any difficulty. 632 For by the centre of rotation B Vi VAL the side DIX VIII is determined, by the centre of rotation P vir Vii the side VIII VII C", by the centre of rotation Pvr vit the side VILC VI. If we furthermore draw C' VI in the first place the side VIV and in the second place the centre of rotation P'iy y, hence also the point Piv v, lying perpendicularly above it, througb which VIV must pass, are fixed. By means of Piyy we also find the line A'oo Pyyy, on which the centre of rotation of all the other sides must lie. Now the link-polygon in question DIX VIII VIICVIVIV BIA can be completed. KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN TE AMSTERDAM. PROCEEDINGS VOLUME XXII Nes, 7 and 8. President: Prof. H. A. LORENTZ. Secretary: Prof. P. ZEEMAN. (Translated from: “Verslag van de gewone vergaderingen der Wis- en Natuurkundige Afdeeling,” Vols. XXVII and XXVIII). CONTENTS. JAN DE VRIES: “Quadratic involutions among the rays of space”, p. 634. JAN DE VRIES: “A Congruence of Conics”, p. 641. W. VAN DER WOUDE: “On a Quartic Curve of Genus Two in which an Infinity of Configurations ” of DESARGUES can be Inscribed”. (Communicated by Prof. J. C. KLUYVER), p. 645. J. WOLFF: “On the quasi-uniform convergence”. (Communicated by Prof. L. E. J. BROUWER), p. 650. J. WOLFF: “Series of analytical functions”. (Communicated by Prof. L. E. J. BROUWER), p. 656. A. J. DEN HOLLANDER and F. E. VAN HAEFTEN: “On the Nitration-Products of p-Dichlor-Benzene”. (Communicated by Prof. A. F. HOLLEMAN), p. 661. EuG. DuBOIS: “The Quantitative Relations of the Nervous System Determined by the Mechanism of the Neurone”, p. 665. B. P. HAALMEIJER: “Note on linear homogeneous sets of points’. (Communicated by Prof. L. E. J. BROUWER), p. 681. J. A. SCHOUTEN and D. J. STRUIK: “On n-uple orthogonal systems of n—1-dimensional manifolds in a general manifold of n dimensions”. (Communicated by Prof. J. CARDINAAL), p. 684. J. WANNER: “Ueber einige palaeozoische Seeigelstacheln (Timorocidaris gen. nov. und Bolboporites Pander)” (Communicated by Prof. G. A. F. MOLENGRAAFF), p 696. (With one plaat). v A. DE KLEIJN and W. STORM VAN LEEUWEN: “Concerning Vestibular Eye-reflexes. II. The Genesis „of cold-water nystagmus in rabbits”. (Communicated by Prof. R. MAGNUS), p. 713. FERNAND MEUNIER: “Quelques insectes de l'Aquitanien DE ROTT, Sept-Monts (Prusse rhénane)”. (Communicated by Prof. K. MARTIN), p. 727. (With two plates). J. F. VAN BEMMELEN: “The wing-design of Chaerocampinae”, p. 738. A. A. HIJMANS VAN DEN BERGH and P. MULLER: “On Serum-lipochrome”. (First part), p. 748. P. VAN ROMBURGH: “The unsaturated alcohol of the essential oil of freshly fermented tea-leaves”, p. 758. R. BRINKMAN and Miss E. VAN DAM: “A method for the determination of the ion concentration in ultra filtrates and other protein free solutions”. (Communicated by Prof. H. J. HAMBURGER), p. 762. H. A. BROUWER: “On the Crustal Movements in the region of the curving rows of Islands in the Eastern part of the East-Indian Archipelago”. (Communicated by Prof. G, A. F. MOLENGRAAFF), p. 772. 42 Proceedings Royal Acad. Amsterdam. Vol. XXII. Mathematics. — “Quadratic involutions among the rays of space.” By Prof. JAN De Veres. (Communicated at the meeting of December 28, 1918). In a communication which is to be found in part Vol. XXII, p. 478 of these Proceedings I have dealt with an involution, the pairs of which eonsist of the transversals to quadruplets of straight lines belonging one to each of four given arbitrary plane pencils of rays. In the sequel [ shall consider a few involutions related to the above mentioned. 1. In the first place we assume two plane pencils of lines (A, a) = (a), (B, 8) = (6) and a quadrie regulus (c)? te. one set of generators of an hyperboloid I. An arbitrary line ¢ meets one ray a, one ray 6 and two rays c. If we conjugate to ¢ the second transversal # of these four lines, a quadratic involution among the rays of space is thereby detined. If ¢ describes a plane pencil, an involution is thereby determined in (c)*, the pairs of which correspond projectively to the rays of the pencils (a) and (6). Now consider the more general case where a quadratic involution in (c)? is brought into a projective correspondence to the pencils (a) and (6) in an arbitrary way. The transversals 7, ¢’ of the quadru- plets of rays a, 6,c,c’ will constitute a ruled surface, the order of which we shall determine by an investigation after the number of lines ¢ which rest on the line of intersection of the planes « and ~. On the line @f the projective pencils (a), (6) determine two projective point-ranges. Through each of the two united points (coin- cidences) passes a line £ The remaining rays ¢ which meet «ag, lie in « or in 8. On the intersection of I? and a the points of transit of the pairs c,c’ constitute an involution; the joins of the pairs of this involution form a pencil (Ce), which is projective to the point-range cut out on «-by the pencil (6) and therefore also projective to the line-pencil which projects this point-range from C. Since each of the two united- rays (coincidences), rests on four corresponding rays a, 6,c,c’ there are in « (and in 8 too) two rays of (4). Hence the ruled surface (t) is of degree six. The plane « intersects (¢?) still along an additional curve a‘, which 635 must needs have a double point at A, as an arbitrary ray a is met by two transversals ¢,?¢’ only. Since the line AB outside A and B meets two lines ¢*) and therefore at A has two points in common with (@)°, it is necessary that A, and B too, is a double point of the ruled surface. The curve @* has six tangents passing through A; hence (£)° con- tains six united-rays (double rays) of the involution (¢, ¢’). The transversals of the pairs a,6 form a quadratic line-complex ; for, in an arbitrary plane (a) and (b) determine two projective point- ranges and the joins of corresponding points envelop a conic. This complex has four rays in common with the second regulus (set of generators) (y)? of FT“. Each of these tour rays meets two corre- sponding rays a,b and at the same time the rays c,c’ conjugated thereto. Hence the ruled surface (£° has four lines in common with the hyperboloid I’. 2 If ¢ is caused to describe the pencil (7’,r) the ruled surface (D° breaks up into this pencil (¢) and a-ruled surface (t’)’. Thus the transformation (t, t’) converts a pencil into a ruled surface of degree five. Of the two united-points of the projective point-ranges on «8 one now lies at apr; through the other passes a ray ¢’. Thus in « (and in 2) there lie again two rays ¢’. The remainder of the intersection of 4’) and « is a nodal «° with double point at A. Each point of intersection of «° and r is the transit of a ray ¢’ which coincides with its conjugated ray ¢. Hence the double rays of the involution (t,t) form a cubic complex. A confirmation of this enunciation can be obtained as follows. With 7? (t’)’ has four rays y in common (§ 1) and in addition thereto a twisted curve y°. At a point of intersection, C, of y° and tT a ray ¢ is intersected by the corresponding ray ¢t’; hence C lies on a double ray t= and the second line of (c)* resting on this double ray meets rt in a point C’, which must lie also on y*. Thus the six points of transit of y’ lie in pairs on three double rays belonging to (Tr). 3. A ray tú through A is intersected by a ray 6 and by two rays c,c’ of (c)*. Bach ray 4 which meets 6,c and ec’ intersects on a a certain ray a and is therefore conjugated to ¢4; hence the ray tg is singular. ') Lying in the united-planes (coincidences) of the projective pencils of planes which project (a) and (hb) from AB. 42% 636 The tangent plane of the hyperboloid (6,c,c’) at A intersects a along a line « which touches (6, c,c’). The transversals of the four rays a,b,c,c’ therefore coincide. Hence every ray ta is also to be regarded as a double ray; thus the cubic complex of double rays has principal points at A and B and, accordingly, « and B as principal planes. It follows from the above, that the Snakes of lines A and B and the planes a and 8 consist of singular rays of the inwolution (t, t’). Together with A a ray 6 determines a pencil (¢4) and thereby at the same time a quadratic involution J? among the rays of the | regulus (c)’. Now, let there be given in the plane 4 a pencil (/) with vertex L; then each point of 82 determines, by means of /?, an involution 7? on the conie (17, 2). Through Z therefore passes a ray Ll joining the points of transit of two rays c, ce’, which in combination with 6 determine a transversal ¢4. If this ray / is conjugated to the ray U, which meets 6, a projectivity is established in (J). Each of the two united-rays is then a ray € which is conjugated to a ray ty. It follows from this that the reguli (¢’)? which are conjugated to the singular rays t4, constitute a quadratic line-complex. Three other quadratic complexes {t’}? correspond to the sheaf of lines [tp] and to the plane systems of rays [t,] and [4]. The pencil (7,1) contains two rays of each of these complexes; accordingly A, B,@, and B each carry two rays tf of the ruled surface (¢’)° into which (¢) is transformed by the involution (¢, 4’). Thus it appears again that (¢’)? has A and B as double points, a and B as double tangent planes. The ray AB meets two definite rays c‚c’‚ but all rays a and 6. To t=AB therefore are conjugated all the rays of the bilinear congruence which has ¢ and c’ as directrices’). Similarly {ag is conjugated to a7 rays ¢’. Thus the involution (¢, ¢’) has two principal rays, AB and af. 4. The lines of the regulus (y)? too are principal rays, for a line y meets two definite rays a and 6, but all rays c; each transversal, t’, of a and 6 rests on two rays c and is therefore conjugated to = The involution (¢, t’) has still other singular rays. If the point of intersection, S, of two rays a and c lies in the plane o passing through two rays 6 and c’, then the pencil (S, 5) consists of rays s 1) The congruence [t’], conjugated to AB belongs to the intersection of the line-complexes which correspond to the sheaves [¢4] and [tg]. 637 each conjugated to all the other, hence of singular rays. Now a plane. 6 is intersected at two points S by the conic @, which I? has in common with @; every plane tangent to £'* therefore contains two pencils (s). In any arbitrary plane lie two points S, and therefore two rays s; through an arbitrary point pass two planes o and consequently four rays s. Since a second system of singular rays is obtained by interchanging a and 5 in the foregoing reasoning the pencils of singular rays form two congruences (4,2). The vertices of the pencils (s) lie on the conics a? and g°, their planes envelop the hyperboloid J”’. 5. In order to obtain another involution among the rays of space, we consider two reguli (c)? and (dj, of the hyperboloids I’? and A? respectively. Any two rays c,c’ determine in combination with any two rays d,d’ a pair of transversals (¢, t’) constituting one pair of the involution which will here be considered. Now suppose that on T° an involution (c,¢’) be given which in some way is projectively related to an involution (d, d’) assumed on A’. The transversals of the pairs d,d’ form a linear line-complex, for, in a plane 4 the points of transit D, D’ of these pairs determine an involution on the transit (conic) of A*, so that the joins of the point-couples D, D’ form a pencil. This complex contains two lines / of the second regulus of I’. There are therefore two transversals of pairs d,d’ which meet all the rays c. In addition to these two an arbitrary ray c meets the two transversals of the pairs in (c)’ and (d)? which are determined by c. Hence the transversals ¢, ¢’ of the pairs c,c’ and d,d’ form a ruled surface of degree four, denoted by (#)*. Evidently (f° contains also two rays of the second set of genera- tors, (d)? of A’. 6. Thus to the rays ¢ of a pencil (7) t) corresponds a ruled surface (t’)?, which contains two lines y and two lines d. This surface meets the intersection gf of I and A’ at 12 points, eight of which lie on the last mentioned four lines; the remaining four carry each one ray c and one ray d intersecting + at two points which are collinear with 7’. This statement may be corroborated as follows. Through each point of v* pass a line ce and a line d, Their points of transit, C and PD, through r determine two point-ranges related by a 2,2- correspondence on the curves of transit y* and Jd’ of I? and A. 638 The lines 7’C and TD are therefore reciprocally conjugated in a correspondence (4,4). Of the 8 united-rays of this correspondence four pass through the points of intersection of 7? and d?; the remaining four each meet a pair c,d the point of intersection of which lies on ef and therefore carries a ray ¢’ conjugated to a ray ¢. In addition to the two lines y already mentioned the ruled sur- face (t’)* has a twisted quartic y* in common with £*. This curve intersects t at four points, which are two and two collinear with T (§ 2). It follows from this that the double rays of the imvolution (t, t’) form a quadratic complex. The single directrix of (¢’)* lies in +t, the double one passes through 7. 7. The rays of the reguli (y)’ and (0)? are evidently (§ 4) princi- pal rays of (t,t’). To each of these rays a bilinear line-congruence is conjugated having two lines c or two lines d for directrices. As each line c acts as directrix to two congruences (1,1), there emanate two pencils (¢’) from each of its points. The congruences (1,1) cor- responding to the principal rays therefore constitute two quadratic complexes. In a similar way as in $ 4 we find a congruence of singular rays. Of the intersection e* of the byperboloids I and A?’ each point is the vertex of a pencil (S, 0) consisting of rays s which are each conjugated to all the others, hence singular. For, in fact, the plane o through the lines y and 0, which are concurrent at $, intersects o* still in the additional points C of y, D of 0 and Z. Evidently CH belongs to (c)’, DE to (d)*. Each ray of (S,o) meets two rays c’,d' at S and intersects the lines c= CEH and d= DE; therefore (S,o) consists of reciprocally conjugated rays s of the involution (¢, t’). Since the vertices of the pencils S lie on of and the planes o envelop a developable of the fourth class, the pencils of singular rays form a congruence (4,4). 8. Any three rays c of a cubic regulus (c)* determine in combi- nation with each ray a of a pencil (A,@) two transversals, which form a pair of an involution of rays in space. By the rays of a pencil (¢) the rays of (c)° are ordered in an /?, the sets of which are projectively correlated to the rays a. To begin with we again suppose that this correspondence is established in an arbitrary manner; then the transversals ¢, ¢’ of the quadruplets of rays constitute a ruled surface which will here be investigated. 639 On the nodal curve y°, along which the ruled cubic I* is inter- sected by the plane «, the triplets of rays c determine an /*. The conics joining two sets of this /* with the double point D and another point B of y° have in addition to these points two points B', B" in common, not lying on y*. The sets of the /* are therefore cut out on y° by the system of conics with basal points D, B, B', B". Only the pair of lines DB, B'B" furnishes a set consisting of three collinear points. It appears from this that the plane « contains one line of the ruled surface (é), for the line t= B' B" does not rest on the three rays c of a triplet only, but also on the ray a conjugated thereto. Through A passes similarly one ray of (¢). Since a is still inter- sected by two additional transversals ¢, t', the ruled surface (t) is of the fourth degree. The remaining curve a? which (é* bas in common with @, sends four tangents through A. Hence (¢)* contains four double rays of the involution (tf). If ¢ is caused to describe a pencil (7’,r) then (¢)* breaks up into (4) and a cubic regulus (¢')’. Now again «a contains one of the rays t'; the points of transit of the remaining lines ¢' constitute a conic «@, which passes through A and intersects t on the double rays which belong to the pencil. Hence the double rays of the involution (t,t) form a quadratic complex. 9. Let a, be the particular ray of (A, a) which is intersected by the single directrix e of (c)’. Every line ¢’ which rests on a, is in (4, t’) conjugated to e. To the line t= e therefore correspond all the rays of a special linear complex. Similarly the double ray d of (c)° is conjugated to all the rays of the special linear complex having the ray ag which rests on d for its axis. In this involution (¢,?’) also the rays ¢4 through A are singular and each conjugated to the rays of a regulus having three lines c for its directrices and containing the lines d and e. Similarly the rays 4, lying in the plane a, are singular too and each correlated to the rays of a regulus which contains d and e. Now consider the system of the hyperboloids (//), which are each determined by three lines c. The specimens whieh pass through a given point P arrange the lines ¢ into the sets of a cubic involution of the second order. The involutions /?, which thus belong to the points P,P’, P", have one set in common; the hyperboloids /7 therefore form a complex (triply infinite system). The hyperboloids 640 corresponding to the rays ¢4 and therefore passing through A then constitute a met (twofold infinite system) all the specimens where- of have the lines d,e and the transversal 4, through A of d and e in common. Through a point P therefore passes a single infinity of hyperboloids and these still have the transversal through P of d and e in common. Hence the lines ¢’ through P which are conjugated to the rays of the sheaf [A] form a pencil in the plane (P4,). 10. There are still other singular rays. Each plane « through e contains two lines c. In « lies a pencil of rays ¢, which has the point of intersection # of e and ae for its vertex; these rays are singular, since they rest at # on a third line c and are therefore all conjugated to each other. The sheaf |E]| is therefore composed of pencils of singular rays. The plane d passing through d and aq contains a linec,; through each point D of d pass two lines c, hence oo! lines ¢, which rest at the same time on c, and ag. It follows from this that the plane of rays |d] is composed of op* pencils of singular rays. These have their vertices on the line d. 11. Lastly we consider a ruled surface I'* with a double curve o*. The linear complex which can be laid through five generators ec of I* contains all the lines c. The four rays c which rest on a line ¢ meet besides the line ¢’, which by the complex is conjugated to ¢. The involution (4,4) then consists of the pairs of conjugated directrices of a linear complex; its double rays are the rays of this complex. Another well-known involution (¢,¢’) is originated by the pairs of reciprocal polar lines of a hyperboloid. Its double rays are the two sets of generators of the hyperboloid. Mathematics. — “A Congruence of Conics’. By Prof. Jan px Vers. (Communicated at the meeting of January 31, 1920). 1. We shall suppose, that a trilinear correspondence’) exists between the ranges of points (A,), (A), (A,) lying on the crossing straight lines a,, a,, «,. Through each triplet of corresponding points A,, A,, A, let a conic 2? be passed which intersects the fixed conic 8? twice. The congruence [27], arising in this way, will be examined more closely; it passes into a congruence of circles, if 8? becomes the imaginary circle at infinity. 2. Pairs or LINES. In four different ways 2? can degenerate into a pair of straight lines. 1. One of the lines, g, rests on a,, d,, a, the other, h, lies in the plane 8 of 8°. If we keep the point A, fixed, A, and A, describe projective ranges, so that g,,—=A,A, describes a quadratic scroll. There are therefore two lines g,, resting on a,; the two supporting points 4’, will be associated to A,. Each point A’, belongs to one point A, ; for the transversal through A’, of a, and a, determines two points A,,4A,, hence one point A,. Three times A, coincides with A’,; there are therefore three lines g,,,, each containing a group A,, A,, A,. Each line /,,, in B, intersecting g,,,, forms together with this line a pair of lines belonging to the congruence. To group 1 belong accordingly three systems, each consisting of a fixed straight line and a ray of a plane pencil. 2. One of the lines, g,,, rests on a, and a,, the other lies in @. To the intersection A,* of a, with B a seroll (g,,*) is associated, which intersects 8 in a conic y,,”. Hach ray of the plane pencil (A,*, 8) intersects on y,,’ two lines g,,, and forms with each of them a pair of lines. Group 2 contains therefore three systems, each con- sisting of a ray of a plane pencil and a straight line of a quadratic seroll. 3. Let us denote the point of a, associated to A,*, A,*, by A,**. Sach line through A,** resting on A,* A,*, forms with the latter a ') B, Srurm, Die Lehre von den geometrischen Verwandtschaften, 1, 320, 642 pair of lines. Also here we find three systems, each consisting of a fixed line and a ray of a plane pencil. 4. The line g rests on a,, a, and 8’; the line A cuts a, and 8? Through the point B of B? passes one transversal g,, =A, 4,; the corresponding point A, determines the plane of 2? and in this: way the point B’ of 6’; h, =A, B’ forms with g,, the pair of lines. We find therefore three systems of pairs of lines in group 4. Let us consider the correspondence (5,5’). Any ray h, of the plane pencil (B'A,) is eut by two rays g,, of the scroll corresponding to A,; the transversal through B’ of a, and a, is associated to a definite point of a,, and intersects the corresponding ray A, in B’. Hence the ruled surface of the pairs of lines g,, which we have associated to the rays h,, intersects the plane (B’a,) along a cubic passing through 4’. But in this plane lies a line g,, connecting the points A,,4, in (B'a,). The ruled surface (g,,) is therefore of order four; it intersects 8? besides in B’ in seven points B, which in the correspondence in question are associated to B’. Each of the eight coincidences is the double point D, of a pair of lines; the locus of D, is for this reason a twisted curve of order eight, d,°. The lines g,, form a ruled surface of order four with nodal lines dy, A, and directrix B. To each point A, are associated four points D,, while to a point D, there corresponds one point A,. From this follows, that the order of the ruled surface (,) with director lines a, and d,°, is twelve. 3. ORDER AND cLass. With a view to defining the order of the congruence, we consider the conics 4? through a point P in 8. To them belong in the first place the three pairs of lines of group 1, each formed by one of the lines g,,, together with the line through P and the point (g,,,, 8). Further the six pairs of lines of group 2, defined by the three rays PAz. As each of these three rays belongs to two pairs, we come to the conclusion, that the order of [47] is nine. A plane through an arbitrary line & intersects a, and a, in the points A,,A,, and a, in a point A’,, which we associate to the point A, corresponding to A,,A,. Of the scroll (y,,) defined by A,, two lines rest on £; hence two points A’, are associated to A,. As A', coincides three times with A,, three planes A,A,4, pass through k, which is consequently a chord of three conics 47. The class of [27] is therefore three. 4. SINGULAR CHORDS. According to a well known property of the 643 trilinear correspondence there are two neutral pairs A,”, 4,”, which form a group with any point A,. The line A,”4,” is therefore a singular chord. One of the conics 4? consists of this chord and the line in 8 resting on it and on a,. From this follows, that the locus of the À* which pass through A,” and A,", is a cubic dimonoid, containing a. The conical points of the six dimonoids can be indicated by Ar, "A,, Ay", "A,, A,", "A,; in this order the six neutral chords are each time defined by two successive symbols. They form a hexagon, inscribed in @,, a,, ds. To the singular chords belong apparently also the three lines 9:2, and the three lines A;* A;* in g. Also the three lines a; are singular. For each plane through a, contains the conic determined by the intersections with a, and a, Let us consider the intersection of the surface %,, formed by these conics, with the plane 8. To this belongs the conic 3’; the rest consists of straight lines. On a, rest two lines g,,; their intersections with 8 determine together with the point A,* two straight lines belonging to WU, The line A,* A,* is cut by a line A, A, of the scroll correspond- ing to A,*; it lies} therefore on 2,, as well as the line 4,* A,*. Each of the three lines g,,, forms a pair of lines with a straight line in 8 through A,*. The intersection of AU, with 8 is therefore of order nie. The locus of the conics 47 which intersect a, twice, is accordingly a surface U? with a sevenfold line a,, containing the lines a,, a, and the conic p’. 5. SINGULAR POINTS. All points Ax of the lines aj are singular. A straight line £ through a point A, is intersected by two lines ss is therefore a chord of two # passing through A. The planes of the 4* through A, envelop consequently a quadratic cone; from this follows, that through any point of 3’ two of these, 4? pass. Hence the locus of the 2? through A, is a surface (A‚)* with double curve g* and conical point A, Also the points B of 8° are singular. Through two points B, b’ 2. hence Pp? counts three times in the locus ® of the 2? pass three 4 through ZB. Moreover 6 and B have in common the three lines through B meeting the lines g,,,, and the lines joining B and the points Az*, which have to be counted twice. We conclude from this, that B is a surface of order fifteen with threefold curve # and three nodal lines aj; the point B is twelvefold. 644 6. SURFACE OF THE CONICS RESTING ON A GIVEN LINE J. Let us consider the intersection of this surface with the plane 8. To this 8’ belongs jifteen times. Further three rays h,,, which intersect / and each of which forms with one of the straight lines g,,, a 2’. Also the three lines joining the points Az* with the point (/,8) and each belonging to two pairs of lines. Then the two rays of the plane pencil (4;*, 8), each forming a 2’ with a straight line A; A resting on 2; in all six rays. Finally the three lines A;*A;*, each of which belongs to a A of which the second component is the ray through 4,,** intersecting /. The complete intersection is there- fore of order 48. The surface in question is accordingly a A** with fourfold lines di, As, Aj, fifteenfold curve 8? and three double conics 27; these are the conics which have / for a chord and therefore intersect it twice. Besides the lines mentioned lying in the plane 8, 4 contains the three lines g,,,, two lines g,,, two lines g,, and two lines g,,, all crossing the line ¢; further two lines g,,, two lines g,, and two lines g,,, intersecting /; then three lines resting on /, successively directed to the three points A;**; finally 3 X 16 pairs of lines, the components of which each contain one point of 8°; in 3 x 4 of them the line gj; and in 3 >< 12 the line / rests on /. Mathematics. — “On a Quartic Curve of Genus Two in which an Infinity of Configurations of Dusarcurs can be Inscribed.” By Prof. W. van per Wovpr. (Communicated by Prof. J.C. KLUYVER). (Communicated at the meeting of January 25, 1919). In an article entitled “The quartic Curve and its Inscribed Con- Jigurations’ H. Bateman’) comes to the conclusion that there exist quartic curves of genus two in which an infinity of configurations (10,, 10,) of Desaraurs can be inscribed. BATEMAN makes only mention of the existence of these curves without entering more deeply into their properties. Starting from considerations quite different from those of Bateman, I wish to indicate in this paper, what condition is sufficient for a uninodal quartic curve y, being circumscribed to an infinity of these configurations. It will appear that each point of y, is part of one of these configurations, and that we can construct each of them from one of its points if we consider y, as being given. I shall first mention a few known properties of an arbitrary uninodal quartic. 1. Let for the present y, represent a quartie which has a double point in O and is for the rest arbitrary. We denote the tangents at O by z and y, their equations are c=U and y= 0; each of these two lines meets y, in One more point; the line joining these points is represented by z= 0. We can then represent y, by: 7, = wy (@° + may + y? + 27) + 2 (av? + bay + cay? + dy*)=0. (1) Out of OU we can draw 6 tangents to y, if we do not count the two at O; the points of contact are the intersections situated out of O, of y, with the first polar curve of O, represented by a 2 wyz + ant + bay + cay’ 4+- dy? = 0. Hence Ya + Nrs 0 indicates a pencil of quartics all of which possess a double point in O and touch « and y in that point; the other base points are h American Journal of Mathematics (36). H. Bareman. The quartic curve and ils inscribed configurations. 646 the points of contact of the 6 tangents drawn out of O to y,, and the + points of intersection of z with y,. If we now put r=—-—], we choose out of the pencil a curve which has degenerated into the two lines w and y and a conic } with the equation B= x* may 4+ y’?— 2? =0. 8 is called “the conic of Brrtint’. On the cone B are situated the points of contact of the 6 tangents drawn out of O to y,, and the two points of intersection of y, with z. If we further draw through O an arbitrary line y—l«=0, its points of intersection with y, are found from Use? (1 + ml + 1?) + 27} + we (a 4 bl 4 cl? + dl’) — 0, those with @ from a? (1 4+ ml + [)— 27? =0, from which appears at once: Any line | through O intersects y, besides in O in 2 more points which are harmonically separated by the points of intersection of l with B. 2. The curve in question is now obtained by putting in (I) the coefficient m equal to zero. The geometrical significance of this is the following : The curve y, considered is cut by the line z joining the two other points in which y, ts intersected by its tangents at the node, in two more points, harmonically separated by the former two. For convenience’ sake I shall speak in the future of the “harmo- nical uninodal curve y,”. Its equation is: Y, = wy (a? + y? + 27) + 2 (ar? + bx*y + cry? 4+ dy*)=0. (1) If we now put 1=@ 1+¢ g=(1+C)a* + (1—C)y? + 22 |«( ih me) tele nd and en 4a i ime Ad EO in which C is an arbitrary constant, we get: re OER “CO. 2 5. 647 For this reason we can produce y, as the locus of the intersections of corresponding curves of 2 projective pencils of conics: Pa dw=0 Ne eten te (9) and (!1—C) a? + (1+ C)y?— 2aay = 0, . . . . (A) where the pencil (4) consists of an involution of rays with O as centre. 3. We will now first give a geometrical interpretation of the way in which the projectivity between (3) and (4) has been fixed. By putting 4=/, we choose an arbitrary conic from (3) inter- secting z in the points A and A’ defined by: (1+ C) aw + 2a,ay + (I—C) y? = 0. We put moreover (LC) x* + 2a,ey + (L—C) yr =(1—C) (ype) Ype), so that : 1+ C=(1—C) pp’ and 2/4, =(1—C)(p + p’). The points B and B’, harmonically separated from A and A’ by the points of intersection of z and 8 are then respectively found from : ce Di) and « — p'y= 0 Hence the pair of rays which project the points B and B’ out of O, have for equation: C= == or (LC) 2? + (14 C) y? — 2),cy = 0. For this reason the projectivity between (B) and (4) is fived in this way: a pair of corresponding conics of (3) and (4) always intersect z in pairs of points (A, A’) and (B, B’) so that A and B and also A’ and B’ are harmonically separated by B. 4. Concerning the pencil (3) we remark that both p and yw are harmonically circumseribed to p, i.e. circumscribed to an infinity of polar triangles of B, as appears directly from this that one of the simultaneous invariants, — generally called @ —, formed out of the coefficients of 3 and p (or ws), becomes zero. From this follows: 1. each conic of (3) is harmonically circumscribed to B; 2. the base points of the pencil (3) form a polar quadrangle of B, ie. a quadrangle of which each side passes through the pole of the opposite side with regard to B. 648 We call the base points of (3) S,,, Sas S,,, S,,, the join of S,, and S,, is called s,,, that of S,, and S,, bes,,..To the pencil (3) belongs a conic which has degenerated into (s,,, s,,); let z be inter- sected by the first of these lines in A,,, by the second in A,,; the point of z harmonically separated from A,, by &, is B,,; also A, and B,, are harmonically separated by 8. The conic (pair of lines) of (4) corresponding to this degenerate conic of (3), is therefore formed by OB,, and OB,,. Let the intersection of OB,, with s,, be called §S,,, that with s,, be 7,,; S,, and 7’, are points of y,. Now O is the pole of z with regard to 8; OB,, is therefore the polar line of A,,; for this reason the pole of s,, lies on OB. We knew already that the latter point also lies on s,,; hence S,, is the pole of s,, with regard to 3. In this way we find that the pole of each side of the quadrangle S,,S,,S,,.S,, lies on y,. Now the four corners of a polar quadrangle form with the six poles of the sides a configuration (10,, 10,) of Drsaraums; hence all the corners of this configuration lie on y,. It is noteworthy that the points Sj;, Sm Sri always lie on a line Sim Of which Sj, is the pole. Each of these lines has a fourth point of intersection with y,; if we choose e.g. s,,, it cuts y, besides in Ss Sus Soo in one more point 7’, which also lies on OS. By giving to the equation of y, the form ¥,=2eyp + {(2—O) 27+ (14 0O)y7}w=0-... (5) we have been able to show that in it a Cf. (10,,10,) of Desarcuns is inscribed. But the equation (5) also contains the entirely arbitrary constant 7; by varying this we shall find an infinity of pencils (3) and (4) and an infinity of configurations. Hence: In y an infinity of configurations (10,,10,) of Desareuus can be inscribed ; each configuration is self-polar with regard to 3. 5. In (5) p and w are functions of x,y,z and C. Let P(a',y/, z’) be an arbitrary point of y,, so that VAN (o Wine) = 0; Let us then determine C, so that ODE a OSU We can find two values of C’ satisfying this condition; then also according to (5) OE a5 AN = de If we therefore consider C as a variable parameter, each point of y is twice a base point of a pencil (3). On the other hand it is 649 clear from what precedes that every point e.g. S,,, can only be part of one configuration, which we can easily construct, starting from that point. With a view to this we join S,, O; the intersection of this line with y, is 7,,; we draw s,,, the polar line of S,, with regard to B, which cuts y, besides in 7,, in S,,, S,, and S,,; the polar lines of these 3 points cut y, in the 6 other points of the configu- ration and 7, 7,,, 7,,. Yet each point, e.g. S,,, though belonging to only one configuration, is twice a base point of the pencil (3); for we can as well produce this pencil and also y by starting from a pencil (3) with the base points S,,,.S,,,.S,,,.S,, as from one with the base points SS. S,,, Sas Any point of a harmonical curve y, belongs to one configuration (10,,10,) of Dersarcurs; if we consider y, as given we can easily construct these configurations from one of their points. Moreover it has appeared from the way in which we produced y, that: Lf a configuration (10,,10,) of Desaraurs and a point O are given, we can produce a harmonical curve y, circumscribed to this configuration, with its node in O, while the conic of Burtini connected to it coincides with the come relative to which the configuration is self-conjugated. 43 Proceedings Royal Acad. Amsterdam. Vol. XXII. Mathematics. — “On the quasi-uniform convergence”. By Prot. J. Worrr. (Communicated by Prof. L. E. J. Brouwer). (Communicated at the meeting of February 22, 1919). I. We consider within an interval a> N such that for each w of the interval a number n, of terms of the series can be determined between N and N', the sum of which, Su), differs less than e from f(«). This theorem constitutes, it is true, a complete solution of the problem: to replace the ordinary uniform convergence by another condition which is not only sufficient, but also necessary for the continuity of f(«). However the quasi-uniform convergence can again be replaced by a wider condition by which a slight extension of ArzeLa’s theorem is obtained. We shall namely prove the following theorem : 1. If the series is quasi-uniformly convergent at the points of a set E which is everywhere dense within the interval a N such that for every « of FE an index n, can be determined (V << nz << N') such that | f(@)—Sn,(@)| Ze Now choose an arbitrary point « of the interval. In consequence of the convergence of the series a number MN can be found such that, s denoting an arbitrary positive number: AO en (D) | COU Oe UIN eo Se (i) For every n between N and the number V' conjugated to te, N, we can now in consequence of the continuity of S,(#) determine 1) Mem. R. Acc. Bologna 1899. Boren, Legons sur les Fonctions de variables rééelles. 651 an interval (w—d,r + d) such that for every point § within it the inequality SON sO holds good. Since the number of these indices» is finite, an interval TI exists, with centre z such that for any & within / the relation (2) is satisfied by any nm between N and N’. Thus, if § is a point of E lying within /,and if for » is chosen the index n: corresponding to £, in the first place the relations (1) and (2) are satisfied and besides Sn. (RACLETTE PS a. (8) It follows from (1), (2), and (8) that | F(e) —F(O) |< Hence we have f(@=limf ©, where § coincides successively Er with all the points of the everywhere dense set /. If «' is a point of / not belonging to #, then f(@)—F(e) | =lin| F@ —7O| Se Hereby the continuity of f(x) is established. 2. In connection with Arzrra’s theorem it appears thus that the quasi-uniform convergence at the points of a set £ everywhere dense within a Sr <5 involves quasi-uniform convergence throughout the whole interval. 3. From the fore-going it may be easily concluded that the quasi-uniform convergence in ARZELA'’s theorem can be replaced by the following criterion: “for every ¢, NV there exists an N’ > N and a set of points B(e, NV), belonging to the first category of Barre, such that for every « of the interval not belonging to B (e, N) an index nm, (Ner and N everywhere dense in the interval (0,1). Let the function y= S,(w) be represented by the following line: from the right to the left first the zig-zag A,b,A,b,....AnB,, then the segment of B,A,41 to the point of intersection, C,, with the line P= ips last the line CP. Evidently S,(e) is continuous in the interval O oo wards S,(7) coincides with f(x) for a thus situated point. Hence the series S,(x) + {S,(c)—S,(x)} + -..+ {S,(z)—S,—1(z)} +... converges in the interval O< x<1 to f(x) and all the terms are continuous within O<2<1. This convergence is quasi-uniform. In order to make this clear we choose e and N arbitrarily. Since the set (P;) is everywhere dense within (0,1), we can now choose a finite number of points P,,,Pn,,-.- Pi, between O and 1 of which the indices are >> N and which divide the interval into &-+ 1 segments all < «. Now, let « be an arbitrary value between O and 1. If (x) coincides with one of the & values Sn (2), then | f (v) — Sn, =O Ge: If f(z) does not coincide with any of these values, then Sn, (2) = Vp, ded A ey, i Since 0 < f(r) <1 one of the & indices satisfies LF @) — Sn, (@) | Se Hereby the quasi-uniform character of the convergence in the interval O< «<1 is established. At 0, however, /(#) does not assume a limiting value, but oscillates between O and 1. UL. 7. Let flo), fr)... be functions which are continuous within the interval a > n, such that |S, («,)—f(#,)| <€,, hence there exists an index n, satisfying Sd) ul Se, Thus pursuing we find that there is a partial sequence of functions Sn(@), Sn(@),... which at a converges to the value u and for a N exists for which Sn, (a) lies between ZP; and P44. Thus we find » indices. Since the functions S,,(x), S,,(z)....S,,_,(2) are continuous at a,a number d, can be found such that for z—a < d,, On; lies between P; and P14, where 7=0,1,....p—1. Also a number d, can be found such that for «—a< d, f(w) lies between P, and P,, for, M and m are the maximum and the minimum of f(x) at a. Let d N, since the given sequence in consequence of the continuity of f(«) converges quasi- uniformly in this interval. Hereby the theorem is established. It is evident that in this theorem the words “to every number k between M and N” may be replaced by: “to every number k of a set which is everywhere dense in the interval m, M”. Mathematics. — “Series of analytical functions’. By Prof. J. Worrr. (Communicated by Prof. L. E. J. BROUWER). (Communicated at the meeting of September 29, 1918). Oscoop’s theorem: “Jf the series f, + f,+..., all the terms whereof are analytical functions within a region T of the complex plane, is convergent at the points of a set (B) everywhere dense in T, and if besides, | fi Hf, H.+ fn | SG at every point of T (G being a constant), the series converges everywhere in T and there represents an analytical function” *) has been again demonstrated by ARZELA ®). Viraut*) and Porter’) have extended the theorem by proving that it is sufficient if only the set (@) of the points where the series converges has an internal point of 7 for a limiting point. Of the thus extended theorem a simple demonstration shall be given in the sequel. 1. To this end we suppose that the f; are analytical in 7’, that in T everywhere |S, |< G for every n, G being a constant, and that the series is convergent at the points ò,, 8,,...., having the internal point z, of 7’ for limiting point, and we shall prove that the series converges uniformly in every region lying with its boun- dary within 7’. Now describe a circle (f) with centre z, and radius R, lying in T. Let 8; be a point of (5) inside a circle (4 R) with centre z,, and let f(8;) denote the sum of the series at (8), S,(8;) the sum of n terms, then 1 ('S(@dt Tes S,(t)dt Su) 221 t—B; andes (2) Aad tim f t—B; (R) (R) If 8, denotes another point of (8) inside (3 R), then 4G |Sn(Bi )—S,.(8x) | << 7 |B; Fes Bel ’ 1) W. F. Oseoop. Functions defined by infinite series. Annals of Mathematics, Series 2, Vol. 3, Oct. 1901, p. 26. 3) V. C. Arzera. Annals of Mathematics, Series 2, Vol. 5, 1904, p. 51. 3) G. Virau. Sopra le serie di funzioni analitiche. Annali di Matematica, Serie 32, tomo 10, 1904, p. 65. 4) M. B. Porter. Annals of Math. Series 2, Vol. 6, 1904—5, p. 45 and p. 190. for every 7, so that | (Bi )—F(Bx)| Se Bi Br: ’ whence it follows that 7(3;) tends to a limit f(z) as B; tends to z,. Let » denote an arbitrary positive number. For all sufficiently small values of |8;—z,| we then have: FE) SEN SE: 2G Also | S,(3:) — S, (z.) | <5 |8i—z,| for every n, so that for all sufficiently small values of |8;—2,| the relation SG) Sule) <5 holds, where n is arbitrary. Now choose a 6; satisfying these two conditions, then from a certain m onwards we have: SE) FEN CT. It follows thence that from this value of m onwards we have continually : Sz) (Z0)| Sn, from which we conclude to the convergence of the series at z, The sum there is f(z). At the same time it has become evident that lim f(Bi) = f(20) i=Z, 2. Provisorily we consider n constant. Then for |z—z,|<4R we have: Sn —S, 0 A a (z,) in ne) ein ) ze Wn(2) ’ Ze Sn Zz —S,(z where lim w,(z) — 90. The function nd Jobat ‚) 220 2—2 is analytical inside 0 © U (4), its absolute value being less than me If m now is made to increase infinitely, this function tends to a limiting value at the points @; and therefore, according to § 1, also at z,. At z, it has 658 for every 7 the value S,'(z,). It follows thence that S,'(z,) tends to a limit f@(z,). Similarly Si(z)—Sn(2,) = S (zo) SAG) = 21 + Dz), (ez)? where lim d‚(2) =0. The first term of the right-hand side is analytical Jd SP(z) » ss : ; AG inside (472), its absolute value being less than Te since the absolute value of the coefficient of (z—z,)* in the development 4 oo G : Sn(Z) = Sne) + Daz(e—z,)*, is less than BE As n increases infi- 1 nitely this function tends to a limit at the points 9; and therefore at z, too, from which it follows that a limit lim S",(z,) = f(z.) noo exists. Thus pursuing we find that for every & a limit lim S@(z) = f Pz). n= exists. 3. For an arbitrary z inside (42) we have: Ceol? aap S,(2) = Sz.) zi (z—2,) S (2) ie 0000 air a Sn) (z9) ar EG (1) For a fixed value of z the terms of this series, if n increases infinitely, tend to those of the series (2 =) f (2) =d) zE (ez) f® (z,) zi 500 SF kl Fe.) +... (2) ee Sm fe) which represents a function, analytical in (4), since i Cr HA | = lea \/N0, \ JNO, Cl Cl in which the structure of the intermediate products was every time determined. When we try to separate the three components out of the crude nitration-product of JUNGFLEISCH, we succeed by means of very prolonged continued fractionated crystallisation from alcohol in obtaining the compounds [ and Il in pure state, but not Ill. We get at last a pretty considerable quantity of a cauliflowery mass, which is no longer liable to further separation by crystalli- ij A. 196, 221 (1879). 3) Atti (5) 22 Il, 632 (1913). 664 sation, and which — as appears from the treatment with alcoholic ammonia applied to it as given above — consists chiefly of I, further of a little I], and pretty much III. NH, Cl From the obtained compounds NO, NNO, and é as the \ JNO, Cl Cl corresponding dichlordinitro benzenes can be easily regained through diazotation. A full deseription of this investigation will shortly appear in the Recueil. ; November 1919. Org. chem. lab. of the-Univ. of Amsterdam. Physiology. — “The Quantitative Relations of the Nervous System Determined by the Mechanism of the Neurone.” By Prof. Eve. Durois. (Communicated at the*meeting of December 27, 1919). For animal species with the same organisation of the nervous system (homoneurie species) the quantity of the neurone varies in the same way, relative to the body quantity, as that of the brain’), ie. proportional to P°5 or Pb. The establishment of this fact may certainly be considered as a confirmation of the validity of our present views of the constitution of the nervous system, which, on solid grounds, was considered, also in its highest forms, as entirely consisting of series of structural elements, highly specialized cells, the neurones, which conduct processes of stimulation (impulsions) in definite directions, and preserve conditions of stimulation (impressions), directly or indirectly dependent on the sense organs and the muscles, and which are, accordingly, in mechanic dependence on the body. The other constituent parts of the nervous system were thought to be non-essential: they were supposed to serve only as a support and protection to the neurones. From the parallelism of the relation of the total quantity of the brain with that of the neurone, including the medullary sheath and the neurilemma, it also appears that the same significance must be !) We refer here to a ratio of two quantities expressed in the same unit with regard to a ratio of two other quantities which may be expressed in another unit, provided it be the same for these latter two. For two animal species whose body weights and brain weights are P and P,, and W and W, and whose volumes of homologous neurones and their ganglion cells are N and Nj, C and (Cj, the following equations may be pul: (= z C 4 EAN NEN i LENG : 5) — C: an P, — N, as we as P, — Actually it appeared that: ; (ON D000: Ol) ter cand (5 ) = a The specific weight of the components of the nervous system (about equal to E = . EH, N 1 that of the body) lies, indeed, so little above 1 that, when the ratio of the volumes is taken, instead of that of the weights of these components to the body weight, the exponents are only reduced to 0.2754 and 0.5508 (instead of 0.27 and 0.55). 44 Proceedings Royal Acad, Amsterdam. Vol. XXII. 666 attached to that intermediate substance of the essential elementary components of the brain as to the coverings of the neuraxone. That these could not be entirely inactive, had been understood long ago. Just as the propagation of the process of stimulation in the axone has nothing in common with the conduction of the electrie current through a wire, the comparison of the medullary sheath to the insulator of an electric cable undoubtedly represents this living sub- stance as being much more passive than it is in reality, though it is true that it does not directly take part in the propagation of the impulses. Nor could the simple function of giving support and pro- tection to the neurones be assigned with conviction to the neuroglia. No more does the equality of the volume of the medullary sheath with that of the axone, which Donatpson and Hokn') established for all classes of Vertebrates, fit in with the view that the former would only have the same significance as the insulating covering of the conductor in the cable. Taking this into consideration and in view of what physiological experiments have taught, the nervous system, hence the neurone, appears more and more in the light of a mechanism; though a stringent proof could not yet be furnished, chiefly because it is only in the last few years that a relation has been found between dimension and function of the nervous system. Thus doubt could still be entertained, chiefly as regards the organ of the brain, of the rational significance of the determined quantitative relations, which beyond any doubt point to the existence of a (as yet unknown) mechanical relation. For — thus the reasoning ran — part of the brain must as “the organ of the mind” be as independent of the size of the body as the psychical processes that are enacted there; is it then possible to assume that the body mechanically determines the total quantity of the brain? ; But the psychical processes, certainly, are not independent of the quantity of the brain and its parts. For Man is not only psychically superior to all the animals, he is also distinguished by the great- est true relative brain quantity, and by the extraordinary develop- ment, particularly of that part of the brain which performs the highest functions. We also meet with great differences in the well- calculated relative brain quantity (determined by the cephalisation 1) H. H. Donatpson and G. H. Hoke, On the Areas of the Axis Cylinder and Medullary Sheath as seen in Cross Sections of the Spinal Nerves of Vertebrates. Journal of Comparative Neurology and Psychology. Vol. XV. Chicago 1905, p. 1—16. 667 of the nervous system) and in the relative development of the parts of the brain of higher and lower function between different animal species, apparently in accordance with their psychical capacities. With equal body weight as the Chimpanzee, Man has indeed three and a half times the brain weight of this most human-like animal species; the Chimpanzee, in its turn, has twice as much brain as would possess a Macaque, obtaining the same body weight, ten times as much brain as a Mouse or a Rat of the same body weight. Besides, the different Mammals differ very considerably with regard to the proportion of the more highly to the lower organized parts of the brain. It is beyond our power to estimate the amount of the psychical differences between the animal species, but these psychical differences are connected, so far as we can see, with the well- calculated relative brain quantity and the relative development of more highly organized parts of the brain. Though, in view of those facts, we cannot reasonably assume the existence of essential, qualitative differences between the species, either in one or in the other respect, yet there is some difficulty in conceiving that even in case of the greatest quantitative differences, essentially equal ‘‘psychical powers’, only differing in degree, cor- respond to this qualitative similarity of the nervous system. They think that a certain quantity of the brain, though it cannot be anatomically separated from the rest, would be mechanically inde- pendent of the body, and specially set apart for the intellect. This view formulated in 1885 by Manoovrier’), though with a great deal of reservation and for want of something better, was refuted by Lapricgue?) in 1907. The latter demonstrated that an equal brain quantity 2 can, indeed, be calculated every time for two psychically equivalent species, but not for three or more. Between the Lion and the Puma the calculation of the “brain quantity for the psychical functions” gives a value four times greater than between the Puma and the Cat. Yet these members of the Cat-tribe may be assumed to’ have quite the same organization of the brain. Ll can now add a few more trios of other genera to these three species of my paper of 1897. In the Dog-tribe, the value of 7 found between the Wolf and the 1) L. Manouveien, Sur l'interprétation de la quantité dans l'encéphale. Mémoires de la Société d'Anthropologie de Paris. 1885. 2e Série, Tome 3me, p. 316 et seq. 5 L. Laricgue in Bulletins et Mémoires de la Société d'Anthropologie de Paris 1908, (Séance du 2 Mai 1907), p. 256 et seq. 44% 668 Jackal is three times that found between the Jackal and the Fennec. Likewise in Rodentia, the calculation of 2 between the large Malayan Squirrel and the Common Squirrel gives double the value obtained by comparison of this European species with the small Hudson Squirrel; between the Brown and the Black Rat z is found more than three times as much as between the Black Rat and the House Mouse '). Nor can such a quantity, which would only serve for psychical functions and be independent of the mechanism of the body, be reasonally .assumed to exist in the human brain. The hypothesis under consideration, which lacks anatomical or physiological foun- dation, must, therefore, be relinquished. This can be stringently proved from the mechanism of the neurone. If there is no room for a non-mechanically determined quantity in the neurone, then this cannot either be the case in the complex of neurones, the nervous system. The existence of fixed quantitative relations between the neurone and the body, and between the parts of the neurone inter se left hardly any room for doubt that these relations are determined mechanically; a closer consideration of them gives complete certainty on this head. 1) Let i be the hypothetical brain quantity (the weight), set apart for the “intellect”, m a constant for the influence of the “masse organique” (Manouvrrer) on the quantity (weight) of the brain, -P and P, the body weights of two com- pared species, one of which is greater than the other, E and & their brain weigths, then on the supposition, H=mP +7 and L, =>mP, +1 from which BD. ATS P—P, P(E—E)) and 1= PP, . In grams the weights of P and E are, for Canis lupus 37000 and 189, for Canis aureus 10000 and 73 (these two according to L. Lapicgue in Bulletin du Muséum d’histoire naturelle. Paris 1912. N°. 1, p. 4), for Canis zerda 1500 and 25 (according to B. Krarr, in Sitzungsberichte der Gesellschaft naturforschender Freunde, Berlin 1918, p. 37), for Sciurus bicolor, Sciurus vulgaris, and Sciurus hudsonicus 1400 and 12, 323 and 6.1, 159 and 4.1, for Mus norvegicus, Mus rattus and Mus musculus 448 and 2.36, 200 and 1.59, 21 and 0.43 (according to my records in Zeitschrift fiir Morphologie und Anthropologie, 1914, p. 327, and my Paper in the Verhandelingen of this Academy of 1897), for Felis leo, Felis concolor and Felis domestica 119500 and 219, 44000 and 137.5, 3300 and 31 (according to records in the same Paper). a 669 It has appeared in the first place that the volume of homologous, at the same time analogous ganglion cells (functioning in the same way) varies proportional as the power 0,27 or 5/;s of the body weight. As it further appears from the found equality of the relations of different kinds of neurones and the brain, that the same mechanism must hold for all the neurones, we can study it by means of the neurones with peripheral nerve fibers, which are most accessible to investigation. In animal species having the same form the length of homologous nerve fibers varies necessarily in direct ratio to the length of the body, ie. to P°33 or P's, in animal species of dissi- milar form the nerve lengths vary with a greater or smaller power of the body weight. In case of dissimilarity as well as in case of similarity in form the volumes of homologous, at the same time analogous neurones are however, found varying in proportion to P° or P%>. Hence the cross section of the nerve fiber must vary in inverse ratio to its length. As the nerve fiber constitutes by far the greater part of the volume of the neurone, the variation of the cross section may be put proportional as Pe? or P* for animal species having the same form, but different sizes. Man and the Mouse are not species of similar form; the Mouse has relatively much shorter limbs, nevertheless homologous neurones which function in the same way, such as the motor neurones for the finger muscles, can very well be compared, which appears from the careful measurements by Irvine Harpesty*). He found the nerve fiber of these neurones on an average 35 mm. long in full-grown mice, and the homologous nerve fiber of a man weighing 72 kg., i.e. 3600 times the weight of the mouse, 800 mm. long. These lengths are to each other as 22.86:1, i.e. as the power 0.3821 of the body weights, whereas in case of conformity the proportion would be as the power 0.33 of the body weights or 15.32: 1. The length of the nerve fiber of man ought then to be no more than 536 mm. From Harpesty’s measurements of the diameters of the axones, the nerve fiber of man appears, however, to be thinner in exactly the same ratio as it is longer. The area of the cross section varies proportionally as the power 0.1693 of the body weight, instead of as the power 0.22 in case of similarity. Thus it is found that the volumes of these dissimilar neurones are exactly in the ratio of the power 0.55 or %/ of the body weights, and the square ') Ixving Hanpesty, Observations on the Medulla Spinalis of the Elephant with some Comparative Studies of the Intumescentia Cervicalis and the Neurones of the Columna Anterior. Journal of Comparative Neurology, Vol. XII 1902, p. 171—172. 670 of the volumes of the nerve cells, as would be the case when the neurones were similar in form. In other cases of dissimilarity in form the comparative length of homologous nerve fibers in the large species is smaller than in the case of conformity. But with all differences of the relative length and section of the nerve fibers the relation between the volumes of the neurones and their ganglion cells yet remains the same. We always find: (= oN C, mi N, and the volume of homologous ganglion cells always varies. in pro- portion to the power 0.27 or 9/8 of the body weight: CPN o-r) If this power were 0.33 or 6/18, its meaning would be clear at once. For the movements of animal species of the same form, but of different sizes, are slower, and the muscular contractions more prolonged, in proportion to the greater length of the body, the result being that large and small animals move along equally rapidly. The Tiger, e.g., moves at the same speed as the Cat, but with slower steps. It would be natural to conclude to a corresponding variation in the volume of the ganglion cells supplying the stimulation energy. D P Ns The proportion found departs little from (=) , but the deviation 1 is constant, and has, accordingly, a rational significance. This opinion is supported by the fact that the area of the retina varies in the \5 same way, in proportion to (=) is The comparison of the receptive 1 cells of the retina with the ganglion cells of the brain is certainly reasonable, on account of the origin of this membrane as a bulging out of the primitive cerebral vesicle; the retina is actually a complex of neurones. But then it follows that the area of the retina must vary to the same degree as the nerve cell vo/wme. In animal species of different sizes the impressions of the retina (images) vary with its area; those of the nerve cells of the spinal cord and the brain with the volumes of the cells. In the retina the area of the cross section and the volume of the receptive elements must vary to the same degree *). 1) The available data are not sufficient to allow us to judge about the variation of the area of the cross section of the receptive retina-elements with the body weight of homoneuric animal species. GisA ALEXANDER ScHärer (Pflügers Archiv. 671 How can we account for this constant deviation from the simple proportion between the cell-volume and the length of the body? We are put on the right track of this explanation by a dissimi- larity of the cell in relation to its nerve fiber. We found that in similar animal species of different sizes, the area of the cross section of the nerve fiber varies in proportion to Po22 or Ps With uni- formity of the ganglion cell from which the nerve fiber proceeds, this section would have to vary with the °/3 power of the volume of this cell, so that the cell volume itself would then have to vary proportionally to P°3 or P's, In reality the cell volume increases and decreases proportionally to Ps Whence this unconformity in animal species of similar form ? The answer to this question is supplied by the closer examination of the cell volume. Only part of it, the plasm, is in direct relation to the nerve fiber; the axis cylinder arises in the plasm, its structure proceeds in it, passing by the nucleus. It has already been known for some time, especially with regard to the large ganglion cells, that the size of the cell increases more than the size of the nucleus, hence the cytoplasm still more so *). The existence of a definite quantitative relation can here again be found by means of a power-equation. When the volumes of a large and a small cell C and C,, and those of the nuclei A and K, are known, the value of the power & for the relation between the two relative volumes can be calculated by means of the equation: In table I I have collected the diameters of a number of nerve cells and their nuclei, borrowed from the works of GauswePr Levi (1906 and 1908)'). Of course only between cells having the same Bd. 119 (1907), p. 574) gives 5.14 micra for the diameter in the Hare, and 4.6 micra for that in the Rabbit. The relative cross section 1.248 is here proportional to the °/;, power of the relalive body weight. ') Giuseppe Levi, Studi sulla grandezza delle cellule. I. Ricerche comparative sulla grandezza delle cellule dei Mammiferi. Archivio [taliano di Anatomia e di Embriologia. Vol. V. Firenze 1906, p. 291—358. Cf. on the relation of nucleus and plasm of the ganglion cells, the tables and graphic figures, particularly the table of the spinal ganglion cells, p. 330, and the two figures XIX, and also the conclusions formulated p. 354. Further data in: G. Levi, | gangli cerebrospinali. Studi di Istologia comparata e di Istogenesi. Supplemento al Vol. VII dell’ Archivio Italiano di Anatomia e di Embriologia. Firenze 1908. 392 pp., 60 Tavole. (These papers will be referred to as 1906 and 1908). TABLE I. 672 Diameter of ganglion cells and their nuclei, according to he measurements of GiusEePPE Levi (1906 and 1908), and linear dimension of the plasm of these cells corresponding to it. (Micra) OO © al HD OO a WO NH Oo ee o FP BY LO 16. 17. 18. 19. 20. 21. 22. Animal species . Bos taurus . Bos taurus . Tragulus kanchil . Lepus cuniculus . Cavia cobaia . Cavia cobaia . Mus decumanus . Mus musculus . Arvicola arvalis . Felis domestica . Felis domestica . Python (species) . Varanus arenarius . Seps chalcides . Bos taurus Mus musculus Canis familiaris Canis vulpes Bos taurus Tragulus kanchil Canis familiaris Putorius putorius Larg. in gln.sp.cocc. I) | Ganglion cervicale su-/ | Kind of the ganglion) cell Largest in spinal ganglia id. id. id id. id. id. id. id. id id. id. id id id. id. id. Largest in ganglion spinale cerv. V | Largest in ganglia sp. id. id. Largest cell. radic.ant. medull.spin.intum. | lumb. id. id. Cells of Purkinje in} cerebellum id. id. Larg. pyramidal cells of grey cortex | id. id. perius n. sympathici id. id. Mean diameter of the cell | 104.3 ter of the nucleus Mean diame- ter of the plasm Mean diame- 32.2 23.2 13.8 16.3 20.1 14 14.2 1.8 24.7 11 Reference to the page of G. Levi's papers 1908. 200 1906.330 ” ” 1908.200 ” ” 1906 .330 1908.200 ” ” 1906 .330 1908.200 673 shape can we derive the accurate relation of the volumes from the proportion between the lengths. To have a good chance in this respect | compared allied animal species, wherever obtainable, or at least those in which detinite homologous cells may be considered as similarin form. Besides spinal ganglion cells were particularly chosen for the calculations, on account of their regular, round shape, which as such leads more to conformity. It is self-evident, that cells were compared which differed as much as possible in size; thus individual deviations, which would affect regular relations, as are supposed to exist between the volumes of the nucleus, the plasm, and the cell, are minimized and recede into the background. The linear dimension of the plasm (equal to the cube root of its volume) was taken as the difference between the diameter of the cell and that of the nucleus. An exponent d can be calculated giving the relation between the volumes of the plasm YD and D, and the volumes of the cells C 1 C, first column of figures in Table Il are found. Most of these approach very closely 1.2 or 6/5. Old (6) P 0.27 d Besides it was found that el = Cv so that el ae Thus the value of the plasm-exponent 2 (= 0.27 d) can be calculated d D and C,, in the equation ( ) — D then the values recorded in the 1 Ja Neen JD} from the equation =) = — with the value of d calculated for lee D, every pair of cells investigated. The results of these calculations are given in the second column of figures of Table IL. Most of them differ but little from 1/3. Not being acquainted with the details of the data made use of in these calculations, we could not be certain beforehand, however guided in their choice by the principles stated, that they were indeed serviceable. In the case of Tragulus e.g. there seems to have been something wrong with them. But finding for a fair number of cell- couples, chosen on those principles, such a striking conformity in the results of the calculations, we feel justified in admitting the real existence of regular proportions. As was already stated couples of cells of similar form had to be chosen for the calculation to enable us to derive the volume from the diameter; but the general validity found earlier for the relation of volume between the ganglion cell and body, and considerations in connection with what follows, leave no room for doubt, that also between cells dissimilar in form, but functionally equal, the 15. 674 TABLE II. — Calculated values of the exponents d, A (= 0.27 d) and & for the variation of the plasm-volume D with the cell-volume C and with the body-weight P, and of the nucleus-volume K with the cell-volume C Animal species . Bos taurus (1) and Mus musculus (8) . Bos taurus (2) and Mus | musculus (8) . Bos taurus (2) and Tra- | gulus kanchil (3) . Lepus cuniculus (4) and | Mus decumanus (7) . Lepus cuniculus (4) and Mus musculus (8) . Mus decumanus (7) and Mus musculus (8) . Cavia cobaia (5) andAr- vicola arvalis (9) vicola arvalis (9) . Felis domestica (10 and 11, gln.cerv.Vand cocc.1) . Python (species) (12) and Seps chalcides (14) . Varanus arenarius (13) . Cavia cobaia (6) and Ar- | and Seps chalcides (14) | . Bos taurus (15) and Mus musculus (16) Canis vulpes (18) . Bos taurus (19) and Tra- gulus kanchil (20) Canis-familiaris (21) and Putorius putorius (22) . Canis familiaris(17) and | Mean of 13 comparisons(without N° 3 and N°, 14) ; ze aia |) ali lk te Kind of the Sanelton (c jep (2 =p (Gp | Ci Di | P, D, C, Ki | From Gangl. spin. 1.198 0.3327 | 0.5348 ie fel 1.203 | 0.3342 0.5268 id. id. 1.070 | 0.2971 0.7864 | id. id. 1.202 0.3338 0.5987 id Ad, 1.206 0.3351 0.6143 ia id 1.210 0.3362 0.6288 fel, = 1.216 0.3378 0.6703 ids) sid’ 1.259 0.3497 0.6025 ii 1.123 0.3119 0.6466 id. id. 1.187 0.3296 0.5892 idd 1.203 0.3341 0.5386 | Rad. ant. med. spin. 1.195 0.3320 0.6555 Purkinje cerebell. 1.199 0.3330 0.6651 Gr. pyram. cerebr. | 1.422 0.3950 0.6071 | Ganglion cervic. sup. n. sympath. 1.248 0.3466 0.6523 1.204 0.3344 0.6095 Pe 675 relation of volume of the plasm would remain the same. We arrive therefore at the following statement: The plasm volume of the nerve cell varies in proportion to the cube root of the body weight, te. to the mean linear dimension of the body. Hence we are justified in considering the plasm volume of the nerve cell as determined dynamically. The mean linear dimension of the body varies in inverse ratio to the rapidity of the movements, and in direct ratio to the duration of the muscle contractions (com- pare the Cat and the Tiger), because the muscular force, which is determined by the transverse section of the muscles, and the body weight are in this proportion. The calculated values of the exponent 4, in the equation for the variation of the nucleus-volume with the cell-volume, are given in the third column of Table IT. It appears that most of these values, and their mean, are slightly below 2/3, which value £ ought to have, if the variation of the nucleus volume were proportional to the surface area of the cell. We may assume 0.6 or 3/5 for the real value of &. This lies exactly halfway between that for proportio- nality with the area of surface of the cell and with the area of surface of the nucleus, that is the outer and the inner surface of the plasm. This leads to the conclusion that the regulation of the metabolism of the plasm of the ganglion cell must be attributed to the nucleus. Apparently we are, therefore, justified in considering the nucleus as the assimilator and dissimilator of the plasm — by catalysis or enzyme action, — and as the process of stimulation in the nerve fiber undoubtedly proceeds from (or ends in) the plasm, which is closely connected with the axone, the name of neurokinete may be applied to the nucleus. *) Further, it can be deduced from the value found for Z, that the volume of the nucleus varies proportionally as P16 op Pile, and the square of the volume of the nucleus as the volume of the plasm of the cell. Hence these two are in the same relation to each !) The action of the same enzymes can give rise both to synthesis and analysis. P. Scuterrerpecker (Muskeln und Muskelkerne, Leipzig 1909, p. 150 et seq.) found that in the rabbit the relative nuclear volume of the red muscle fiber, which is rich in muscle haemoglobin, is much greater than that of the while muscle fiber, which is poor in muscle haemoglobin. In the red Soleus the relative nuclear volume is 2'/, times greater than in the white Gastrocnemius, which consists of muscle fibers of similar form, and which acts in conjunction with the Soleus. This, too, points to a catalytic relation between the nuclear quantity and the rapidity of the metabolism (oxygen consumption) of the cell. 676° other as the volume of the nerve cell to that of its nerve fiber. This is of great significance in the mechanism of the neurone. Through the found proportionality of the volume of the plasm with P's it now becomes clear that in animal species of the same form, but of different sizes, the area of the cross section of the axis cylinder (and also of the nerve fiber) is in a relation of uniformity with the plasm of the nerve cell from which it proceeds. Hence this area varies as the 2/3 power of the plasm-volume, ie. by P% or Po, The volumes of homologous nerve fibers of animal species having the same shape, the lengths of which are proportional as P*%h or P33, must therefore vary proportionally as Ph or P055, And as the ganglion cel! constitutes only a very small part of the volume of the neurone (in the above described motor neurone of the finger muscles of man, e.g., the nerve fiber has 870 times larger volume than the ganglion cell), we may also say that the volume of the neurone, hence also the complex of neurones, which we call the brain, varies by the power %/ or 0.55 of the body weight.) Thus the rational character of this apparently incomprehensible power is clearly shown. At the same time, the mechanism of the neurone becomes more distinct. Also for species of different shape, as Man and the Mouse, we saw the volame of homologous, also analogous ganglion cells (functioning in the same way) vary proportionally as P°2* or Phs, and we may, therefore, assume that the volume of the plasm of these cells varies proportionally as P0 or P's, hence as the mean linear dimension (for uniform species, as every homologous linear dimension) of the body. The proportionality with the mean linear dimension of the body is, indeed, a necessary condition for the cooperation of ail the neurones in the nervous system, the nerve fibers of which inter se differ greatly in length. Consequently the relations of the elements of the neurocyte must be valid both for species of different shape and for species having the same shape. The existence of these fixed relations of volume of the 1) The degree of accuracy of the data does not allow us to ascertain whether the volume of the neurone or only that of the nerve fiber varies proportionally as the square of the volume of the ganglion cell. Hence it may very well be that the ganglion cell, which besides being the station, is also the road for the impul- sions, at least as regards its plasm, must also be included in the proportional section of the nerve fiber. Then the volume of the neurone is at least almost exactly proportional as P or PS, ti 677 neurone and its parts leave, indeed, hardly any room for doubt of the perfectly mechanical character of its arrangement. Though it is not possible to demonstrate this in details, because it is not yet fully known what takes place in the nerve fiber during the trans- mission of the stimulation process, the impulsions, from and to the ganglion cell, yet our present view of the nature of this mechanism ean be tested by what we know about this arrangement. What is transmitted in the nerve fiber as impulsions is, beyond doubt, a process of dissimilation, and it is highly probable that the colloids, contained in the living substance, play an important rôle. The plasm of the ganglion cell and the axone possess ‘“Spumoid- struktur’ (Rhumbler), is an emulsive foam mixture, consisting of two liquid phases. The denser of these colloid substances forms the walls of the minute spumoid compartments (“Schaumkämmerchen” of Rhumbler); so semipermeable membranes or osmotic films can act selectively on the ions liberated by the disaggregation of the mole- cules. Thus the anions diffuse in centrifugal or centripetal direction from one spumoid minute compartment to another *). Material particles move in any case from one end of the nerve fiber to the other. Not the same particles: the movement is trans- mitted from one spumoid compartment to another lying in front of it, as it were from one transverse layer to another, but the work performed thus must be equal to that of a transverse layer of equal mass which moved from one end of the nerve fiber to the other. But the process of stimulation is not transmitted in this way from one end to the other in the whole nerve fiber; this takes only place in the axone. We see the axone taking its central origin or termi- nating centrally iz the plasm of the neurocyte, with gradual transi- tion; it is enveloped with myeline only at some distance from this ; the medullary sheath terminates at the muscle fiber, and the motor nerve end-plate is but a plate-like or antler-like extension of the axone. It also appears that at the peripheral ends of the afferent nerve fibers the axone is the real conductor. Though the medullary 1) Cf: W. Neanst, Zur Theorie der elektrischen Reizung. Nachrichten von der Kön. Gesellschaft der Wissenschaften zu Göttingen. (Mathem.-physik. Klasse). 1599, p. 104—108. — Max Verworn, Allgemeine Physiologie. Sechste Auflage. Jena 1915, p. 134 et seq., 319 ete. — O. Bürscuur, Untersuchungen über mikros- kopische Schéiume und das Protoplasma. Leipzig 1592. Hans Heup, Beiträge zur Struktur der Nervenzellen und ihrer Wovrtsätze. (Zweite Abhandlung). Archiv fir Anatomie und Entwickelungsgeschichte. Leipzig 1897, p. 204—289. — L. tnumeten, Das Protoplasma als pliysikalisches System. Ergebnisse der Physiologie. (Asner und Spino). Jalirgang XIV. Wiesbaden 1914, p. 484—617. 678 sheath has undoubtedly not the entirely passive significance of the insulator in the electric cable (just as the axone cannot be compared to the wire), it nevertheless certainly does not take part directly in the transmission of the process of stimulation. The axone, certainly, is directly involved in this transmission. With the varying size of the animal species the volume of the axone always varies proportionally to half the square of the volume of the ganglion cell. The length 7 and the area of the cross section q of the axone (like that of the nerve fiber) indeed vary (as was discussed above with regard to the nerve fiber *)) in inverse ratio to each other, so that their product /q remains the same. Hence the product /q varies proportionally as 4 C?. Let us consider in this connection the propagation of the process of stimulation in the nerve fiber. On discharge of the ganglion cell (we shall confine ourselves to the efferent neurone; what follows holds inversely for the afferent neurone) potential energy of some form must certainly be consumed to supply motive energy in the nerve fiber. Let us suppose this to be performed by a layer of anions (or other material particles) placed in the cross sectional plane of the axone. It leaves the ganglion cell with a velocity v, travels the whole path to the other end of the axone, and assumes there a state of rest. This layer, whose mass is proportional to g, must have possessed a potential energy in the ganglion cell proportional to /q, and obtained a kinetic energy, equal to this, proportional to } qv’. We also found /g proportional to 4 C?, from which follows that Y v is proportional to an As for animal species having the same form q is proportional as P°22 or P*hs and C as Poi or Phs, we find v proportional as P16 or Ps. This is the same ratio in which the nucleus volume increases with increasing body weight. The velocity of the metabolism of the cytoplasm and the velocity of the process of dissimilation im the axone varies proportionally with the increase of the nucleus volume; we must therefore consider the movement in question as having begun in the cytoplasm with a velocity which 1) Comp. my communication “On the Relation between the Quantities of the Brain, the Neurone and its Parts, and the Size of the Body. These Proceedings Vol. XX, p. 1828—1337. The areas of the cross sections of analogous nerve fibers of Man and the Mouse referred to above, are based on direct measurements of the diameters of the axones by Irvine Harpesty. 1 assumed double the cross- seclional areas of the axones for the nerve fibers, in virtue of researches by Donatpson and Hoke and others. 679 is determined by the volume of the nucleus. The name of neurokinete may, indeed, very appropriately be given to the nucleus. We may onee more point out the analogy to catalysis or enzyme action. In dissimilar species (and neurones) the section of the axone varies proportionally as a smaller power of the body weight than 0.22 or 4/;3 (between the Monse and Man 0,1693) or as a greater power of the body weight. Therefore the velocity v, which a layer of anions (or other material particles) of the axone obtains in the ganglion cell, must in one case be greater (between the Mouse and Man v varies in the proportion of P°19%1), in the other case smaller than with species (and neurones) of equal form. The kinetic energy imparted to this layer by the ganglion cell, whose volume varies in the same relation with the body weight for species of dissimilar form as for species of equal form, continues to vary in the same way. For, the mass of the layer, determined by the cross section of the axone, varies iversely as the length of the uxone (i.e. between the Mouse and Man in the ratio of 05821), and the square of its velocity (between the Mouse and Man in the ratio of P8862) varies inversely as the cross section, hence directly as the length. The energy remains proportional to 4 qv? and to /g. In a previous communication I discussed the relation between the velocity of the propagation of the stimulation process, and the area of the eross section of the nerve fiber and the axone.') It is self- evident that per unit of length in the same time the stimulation process performs more work in relation to the cross section of the axone. It corresponds morphologically that the joint area of the parts of the walls of the spumoid compartments that are placed transversely increases with the cross section of the axone, for in the same degree more anions diffuse in the unit of time. 7) Thus our present views on the nature of the mechanism of the transmission of the stimulation process in the nerve fiber find full confirmation in the quantitative relations of the nervous system. ” 1) “The Significance of the Size of the Neurone and its Parts”. These Proc. Vol. XXI, (1918) p. 711—729. *) In connection with this the chronaxy, which expresses the time during which a nerve elaborates an electric stimulation, so that it reacts to it, is slight for thick nerve fibers, great for thin nerve fibers: L. Lapicgue et R. Leaenpre in Comptes rendus de l'Académie des Sciences, Paris 1913 Tome 157, p. 1163—1166. L. Laricove et R. Leoenore in Bulletin du Muséam d'histoire naturelle. Paris 1914, N°. 4, p. 248—252. — Cf. on chronaxy also J. K. A. Werrueim SALOMONSON, especially in “Nederlandsch Tijdschrift voor Geneeskunde, Jaargang 1919. 2e. Helft N°. 15. rn 680 Besides it appears that it is the nucleus of the ganglion cell, the neurokinete, that controls the mechanism of the neurone, of the nervous system, and indeed that of the entire animal organism. The sixth power of the volume of the nucleus varies then proportionally as the body weight, or the square of the nucleus-volume proportionally as the length of the body; thus also the sixth power and the square of the velocity of the movement of the anions in the cytoplasm and the axone. ; Airy *) knew only one instance in physical science in which the sixth power came really into application: if the velocity of a sea- current (or a river) be doubled, it will carry stones, pushing them on along the bottom, of sixty-four times the weight of those (having the same shape) carried before. In the animate world the sixth power plays a very important part. SUTHERLAND *) found that the body weights of all bird species are proportional to the sixth power of their brooding-times; those of mutually allied mammals to the sixth power of their times of gestation; thus the body lengths of the adult animals being proportional to the squares of these times. In all these cases, just as in the case treated here, we have to do with movement of, or relative to bodies having the same form. Hence that cytoplastic metabolism, like this functional metabolism of the neurone, is mechanically determined, and the general oceur- rence of quantitative relations between nucleus and plasm *) is only the consequence of the mechanical relation in the dependence of these cell-elements on each other. 1) G. B. Amy in Minutes of Proceedings of the Institution of Civil Engineers. Vol. XXIII. Session 1863—64. London 1864, p. 227. 2) ALEXANDER SUTHERLAND, Some Quantitative Laws of Incubation and Gestation. Proceedings of the Royal Society of Victoria. Vol. VII, (New Series). 1895, p. 270—286. 3) Compare the researches and studies of: J. J. Gurassimow in Zeitschrift für allgemeine Physiologie. Bd. 1. 1992, p. 220—258, — Tu. Boveri in Verhandlun- gen der Physikalisch-medicinischen Gesellschaft zu Würzburg. Neue Folge. Band 35. 1903, p. 67—88, — R. Hertwie in Biologisches Centralblatt. Bd. 23, 1903, p. 49—62 and 108—119, — Auge. Pürrer, Vergleichende Physiologie. Jena 1911, p. 32 et seq. Mathematics. — “Note on linear homogeneous sets of points’. By Dr. B. P. Haatmerer. (Communicated by Prof. L. B. J. Brouwer). (Communicated at the meeting of October 25, 1919). We shall call a linear set of points « homogeneous in the interval AB, if its subset, interior to an arbitrary sub-interval, allows of a uniformly continuous one-one representation on the subset of ar interior to 4B *). If the set z is everywhere dense in the interval 4B*®), each of these representations determines a continuous one-one correspondence between the entire linesegments. As will be shown, we may in this case, assume the correspondences, postulated for a homogeneous set of points, to leave relations of order unaltered. Let CD be a sub-interval of AB (possibly identical to AB) and E a point between C and D. We consider the following possibilities : 1. For every system of points C, D, and ZE the representation of the interval CD on CE leaves relations of order unaltered. 2. This is not the case. First case. Suppose a representation of 4B on FH has to be effected (order from left to right A, F,H, B). According to the assumption both AB and FH can be represented on AH with unaltered relations of order, hence AL on FH in the same way. Second case. The assumption postulates the existence of an interval CD which can be represented on its sub-interval CE with inverted relations of order. Considering this representation is a continuous one-one correspondence between entire linesegments, it follows from the Depekinp axiom that a point P exists (not necessarily belonging to the set 2), which corresponds to itself. This however establishes the possibility of representing the part of a interior to CD on itself with inversion of order-relations. It follows that the part of zr interior to an arbitrary sub-interval of AB, possesses this same ') An analogous definition has been given by Hausporrr for ordered sets, Grundz. der Mengenlehre p. 173. Wor linear sets of points Brouwer has introduced the following more extensive definition: a linear set of points + is homogeneous in the interval AB if for each couple PQ of its points interior to AB, there exists a continuous one-one transformation of the interval AB in itself, such that zr passes into itself and the point P into the point Q. These Proceedings XX, p. 1194. *) Which obviously is the case if + has any points inside AB. 45 Proceedings Royal Acad. Amsterdam. Vol. XXII. 682 property, hence all correspondences, postulated for the homogeneous set or, can be effected in such a way as to leave relations of order unaltered. We now formulate the following theorem: The linear continuum cannot be composed of two homogeneous sets of points, possessing the same geometric type. Our demonstration is to be an indirect one. Let the open line- segment AB consist of two sets of points a and z’ of the kind mentioned. These sets a and 2’ possess the same geometric type, that is there exists uniformly continuous one-one correspondence between them. Evidently « and a’ are both everywhere dense on AB. To begin with, we assume that this correspondence inverts rela- tions of order. Then a can be divided into two subsets 2, and zr, such that every point of a, is situated on the left, and every point of zr, on the right of the corresponding point of a’. Besides, every point of 2, lies on the left of every point of a,. Hence, as a, + 1, is everywhere dense, the DuprKIND axiom postulates the existence of a separating point A. This point & however can belong to neither a, nor z,. For instance let us assume A to be a point of a,, then it is situated on the left of the corresponding point of z' and the continuity of the correspondence makes that this is also the case for all points of a inside a certain finite neighbourhood of R, which means a contradiction. Hence A belongs to 7’, but this also leads to a contradiction as the fact that A is situated either on the left or on the right of its corresponding point cannot be made to agree with the circumstance that all points of 2’ on the left (right) of R are also situated on the left (right) of their corresponding points. We now come to the second possibility, namely that the corres- pondence between a and a’ leaves relations of order unaltered. We distinguish two cases: 1. The set a contains both points situated on the left, and points situated on the right of the corresponding points or a’. 2. All points of am lie on the same side of the corresponding points. First case. Let the point P, of z be situated on the left of its corres- ponding point P and P, on the right of Re The subset of a between P, and P,, including the endpoints shall be called 2,. Let ‚mT, be the subset of mr, consisting of those points, which, together with all points of a, situated more to the left, precede their corres- 683 ponding points*), and let A be the last limiting point of ‚a, on the right hand side. Then the assumption that A precedes its cor- responding point, as well as the assumption according to which R follows on its corresponding point, leads immediately to a contra- diction (we here consider the transformation of the entire segment AB in itself, which is determined by the correspondence between a and 2’). Hence the point R must correspond to itself, but this is out of the question, both if A belongs to 2 or to a’. Second case. All points of a lie on the left of the corresponding points. Let the points P. and P. of 2’ correspond to P, and P, of x respectively and let the order from left to right be P,, P., 12, P.. Of course, such a system of points can always be found. We choose a point C of a’ on the left of P, and we subject a! to a uniformly continuous one-one transformation in itself, such that P passes into C and P. remains in its place. A transformation of this kind can certainly be found as a! is homogeneous. Let 2" be the transformed set, then a uniformly continuous one-one correspond- ence exists between 2” and a, such that 2" contains both points preceding and points coming after the corresponding points, and the reasoning used for the jist case can now be applied. To Prof. L. E J. Brouwer I am indebted for several remarks turned to advantage in the preceding note. bj “Precede” here stands for “are situated on the left of”. Mathematics. — “On n-uple orthogonal systems of n—1-dimensional manifolds in a general manifold of n dimensions.” By Prof. J. A. Scnourrn and D. J. Srrum. (Communicated by Prof. J. CARDINAAT). (Communicated at the meeting of October 25, 1919). ie 7. Dupin’s theorem and an inversion. From theorem I we conclude that Durin’s theorem also holds for a general manifold: The Va of an n-uple orthogonal system intersect along the lines of curvature. This theorem may be inverted in the following way: When n—1 mutually orthogonal V,,-1-systems, determined by the congruences 1,...,14n—1 perpendicular to them, intersect along a con- gruence in, and when we can choose the arrangement of the first congruences in sucha way that the congruence 1, in each V,, 441i, ..,1n—1 is a congruence of lines of curvature for the V,,_; being the inter- section of this Vr with the V, 11%, k=1,....,n—1, then 1, is perpendicular to a V,,-,-system, orthogonal to the n—l given systems, and i,,...,14, are the congruences of the lines of curvature for each of the n systems. Proof. When the fundamental tensor °g of the V, is written: lil it has wee eo sal nao (7E) then the ideal factor a can be decomposed as follows: VEE EN ee Rent . (0S) in which a’ contains but iz, :.., in, @ but i,,..., 14-4. 'g—a a —bb—.... is the fundamental tensor of the Vz L i,,...,i,—-, and the geodesic differentiation of a vector v, which is wholly situated in this WV, 74, is determined by. the equation: VE EAT AN EEND IE EEEN AE ey Hence for ij, we have: in! Viz =in1 V (ie. a)a in? Vlir.a)a Hin! V (iz. a) a’ = ye aaa tt WAC EN Ms iy © ‚ (75) 685 According to the supposition in is a congruence of lines of curvature for the Vp; being Liz in the considered Vz, so that according to (38): cots Wissing vele servo le ue (UO) in which oz is a still unknown coefficient. Hence we conclude from (76): opl zin! Vir@kint 2 umij, . . . … « (77) j in which gz; are still unknown coefficients. So it is supposed that it must be possible to arrange i,,...,in_) in such a way that the equation (77) is satisfied in the same time for all values =1,...,n—1. Since iz iv = 0, kb =1,... 25m, AE) we find by application of in. V: vin A pi Thott 2 WT 8 ver bet eon ot UE) For £ 3, then however it is no longer the only condition. 9. Rreer’s conditions. Be i, again V,_,-normal. Then we can choose an original variable y” and vectors s„ and s’„, so that:' : 1 , In = On Sn = — Sn Ot Oe vei On BPL (89) On By means of this equation we can eliminate in from (C) and (D) and substitute s, for it. Since: (in : V) (V =A in) => (in : V) {On V sn + 3 (V On) Sn + 4 Sn Von}, . (90) we have: 4 4 gn? (in. V) (V — in) = gn 2 { (in -V On) V sn + On ini VV sn + (91) + 4(V on) int V su + b (in? V sn) V on}, or, since: WV on == V (sn. Sn) % = — XOn° (VSn)* Sn —=— On Un -+ X%On Sn! (Von) sn, (92) also: 4 4 En? (in. VY) (VY — in) = En? § (in. Von) V Sn + On? sn! VV Sn—xttn Un}. (93) Since on account of (31) and (69): bir? {2x7 (TV ~ in)! Vin =ijik? unum, . . . (94) the condition (C') gets the shape: iig }%On? Sp} VV Sn — 2 un that == 0) Saws (C,) Since: 1 1 Va=—Vh+(V—) In, artes st) Listes (95) we further have, in connection with (30) and (33): ij ir 2 Vsn = 0, . . Oe ey ur oF i e (96) from which by application of (ix.V) may be concluded: ] 1 3 — (i; 1 Vig) 4 (V in) Lig + — (itt V ix) t (V in) * ij Hij ici: ? VV sn—0. (97) Gn In h R. vy. LittentHar, Ueber die Bedingung, unter der eine Flaichenschar einem dreifach orthogonalen Wlächensystem angehört. Math. Annalen 44 (94), 449—457. 4) W.G. L. Wierinaa, Over drievoudig orthogonale oppervlakkensystemen. Diss. Groningen, (18) 59 pp., see p. 13. 5) See note bj of next page. 688 (Vi,)! i, containing but i; and i, on account of (38), we find in connection with (67): ij inv? VI sn =O, j AE, heals kl, hill pean — 5) (D,) This equation (D,) can be decomposed into: ieu Wend, . Eu NCD) iS Wv) wreede en (C8) or: and: PW WIED ete Sos a (©) 4 When K is the Riemann-Caristorret-affincr of Vn, (99) can be written: °) 4 AVEN IES Der aten Sa 3 5 (OO) or a B SOHN Ker ot keneden (UD!) The equations (C,), (D,), (D';) and (100) are deduced by Riccr.®) (n—1) (n—2) (n—8) Loe) The number of the equations (D',) is ‚the number : n—1 —2) (n—3 of the equations (D") is Cae 5 eer 8) mutate not only j and &, but also & and 7‘). (D,') contains third, (D,") only first differential quotients of y”. The conditions (D,") vanish identically, when the characteristic , because we may per- 4 numbers /4£j7n of K vanish. Since in a space of constant RifmMann- curvature K,: 4 K=2K,(a~b)(a—b) 5). . . . . . (101) the equation holds: 4 ND EG too pe oa) (LOE) so that the condition (D,") is an identity in such a space, and hence also in a euclidean space. Thus (D,) reduces in this ease to (D',). For 1) (Cs) can also be deduced from (84) in an analogous: way as (Dj). 2) Comp. A. R. page 59. 5) G. Rreer, Dei sistemi etc, p. 314. Here the equations (C3) and (Dj) are lettered (Aj) and (Bj). G. Rreer, Sui sistemi, p. 151. *) Compare the observations of Riccr on occasion of a paper of Dracu, Comptes Rendus 125 (97) 598—601 and 810—811. 5) Compare Cf. Brancur-Lukar, Ist. german edition, p. 574. 689 euclidean space the condition (D',) has been given by Dargoux ') *). The 4 ~ characteristic numbers (lljn) of K vanish too, when the Vs Li, are geodesic. 10. Lé&vy’s, Caytry’s and DarBoux’s conditions. Differentiating the relation: Deonet AO ie Ul Rha CTOS) we get V7 Ie => (V Gn) Sn + On WS nee sie 2 ors (104) Differentiating again, we get: VV in = (VV Gn) Sn + (V Sn) 1 a(V Gn) a + (V on) V sn + On VV Sn, (105) and. from this and (104) we have for VV on: 2x “VV on = On (VV in) 1 in — Gn? (VV sn)! in + (V on) (WV on). (106) Since: (VV in)! in=V {(V in) 1 in} — (V in) 1 a (Vin) 1 a=~*h 1 *h-(un. un) in in,(107) we get, in connection with (92): 4 4 Én 2 VV On —= — On ‘ht *h— x0n’ En 2 (VV Sn) 1 Sn ++ 2 On Un Un. (108) In connection with (C,) this equation gives a new shape to the first condition : 1) G. Darpoux, Legons sur les systèmes orthogonaux et les coördonnées curvi- lignes I (98), p. 130, form. (35). 2) As a simple example for the application of (C,) and (D‘;) for euclidean space, we can take the system u = Y, (y!)-++...-+ Yn (y"), in which y!,..., yn are Cartesian coordinates. To calculate gaz etc. it is necessary to find a system of m—1 Vn—1 which determines in the V;,—1 U = const. a system of coordinates ea,... Then Kea, . Ca, — Yaya etc. For this purpose we must try to find »—1 independent solutions of the differential equations : TRAAN eco apn! Se ey aaa Sh Gi) le 5 Oy? dy 5 RO For the calculation compare e.g. WiIeRINGA, Diss. p. 21 and seq. Then we can see that the condition (D’)) is identically satisfied, so that only LartenrnaL’s con- dition (Cj) remains, which can be written in this case: 1 ] 1 iy Vark verd 0, Y/Y /" —2 Yi PY —2 ¥/'? Ye Ye — 2 A or Yi Yi"—2 Yi’? = AYi’ + B, in which A and B are constants. This result has been deduced for n —=3 by Street, and for a general n by DAgBOUX in another way as has been done here. Comp. Darsoux, Legons sur les systémes orthogonaux etc., p. 140 and 141. 690 4 1j in? VV On = 2 “On® ij iz ? {sn 1 (V ae V) Sn} = KOn ij in? (in 1 Ki in) (109) or 4 i; ix 2 VV on=xon in i; iz in 4K. (C,) Thus for a V,, for which the characteristic numbers (nkjn) of 4 K vanish, this first condition can be written: li; ie Oo WW Op == 0. | (C",) This equation expresses that the tensor VVon has the same prin- cipal directions as *h. The geometrical signification of o, is that this quantity is proportional to the infinitesimal distance between succeeding V1 Lin measured along in. In space of constant RimMany-curvature AK, we have, in connection with (101): ii? fin! kt in} = — Ko ij de? (°8 — in in) —0, . . (110) from which we conclude that in this manifold the first condition has the shape (C,), hence also in euclidean space. In this latter case the condition is deduced for n=8 by Levy’), CayLey’), DARBOUX®), and for general values of n by Darpovx f). Thus the necessary and sufficient conditions for manifolds of constant RreMANN-curvature are (C,') and (D,’). 11. Wertearten’s condition. We will try to find a shape of the conditions that only depends oni, and no more on ij, j=1,2,.. ‚nd. When a tensor, whose principal directions do not coincide with those of *h, be transvected once with *h, an affinor arises whose alternating part is certainly not annihilated. Thus the condition that the principal directions coincide, is that the alternating part of the first transvection with *h vanishes. Hence (109) is equivalent to: 4 4 Bg? {(V in)! (VV on) — on (V in) In 7K inf} =0, . . (111) in which B may indicate that the bivector-part has to be taken. 1) M. Levy, Mémoire etc., p. 170. 2) A. CAYLEY, Sur la condition pour qu'une famille de surfaces fasse partie d'un système orthogonal, Comptes Rendus 75 (72), a series of articles. 5) G. Darsoux, Sur l'équation du troisième ordre dont dépend le problème des surfaces orthogonales. Comptes Rendus 76 (73) 41—45, 83—86. See also e.g. Brancur-Luxatr 1st. german edition. 4) G. Darpoux, Legons sur les systèmes orthogonaux etc. p. 128. His formula (32) is our formula (C’,). 691 Since: V ' (V in) a (Von | — (VV in)! WV On + a(V in) 1 (a 5 V) V On, (112) we have: 4 Vl(Vin) 1 V oni =BY | (Vin)! Von} = vis K2in,Vor—B (Vin)! VV on,(113) so that (111) is equivalent to: 4 14 4 gs? |- Vv (Vin) 1 V on} + 2 K2 in V On—6, B(V in) in? K! is [=0. (114) Since in a space of constant Rimmany-curvature on account of (92) and (101): Ay 8 4 4 4 g? K? In WV On = — On £2 K? in Un = — 20, K, 8? in — Ur = 0, (115) the condition for such a manifold is, on account of (110), that the component of V — {(Vin)! Von} in the region Jin vanishes. On account however of Strokes’ law‘), we have for each vector v: fra zaf tt Oo 0d zee (1811) s 5 in which s is a closed curve and *fdo the bivector of the surface- element of any surface o bounded by this curve. From this we conclude that in a space of constant Rimmann-curvature we can also give as first condition that the linear integral of the vector (Vin)! Von along each closed curve in a Va Lin vanishes. This condition is the only one for V,. Foran R, it has been first indicated by WeEINGARTEN *) and Ricci *) has observed on occasion of WeINGARTEN’s paper that the condition holds also for a V, of constant Rimmany-curvature. From the above-mentioned we see that the condition, but no more as the only one, holds also for manifolds of constant Rimmann-curvature, for which n > 3. 1) Comp. A. R., page 37 and 61. 2) Weincarten, Ueber die Bedingung, unter welcher eine Wlächenfamilie einem orthogonalen Flächensystem angehört. Crelle 83 (77), 1—12. *) G. Rreer, Della equazione di condizione dei parametri dei sistemi di superficie, che appartengono ad un sistema triplo orlogonale. Rendiconti Acc. Lincei Ser. V, Ill, (94) 93—96. Ricct observes for the case n=3 that Wrincarren’s theorem remains also 4 valid, when K has the shape: 4 == i (a — b) (a ~ b) + » (1, 1,) (1, SH) when v is an arbitrary coefficient. This however holds also for general values of ». 692 12. Mutually orthogonal V,—1-systems through a given congruence, the canonical congruences being not singly determined. When the roots of (24) are not all different, these roots determine in general q mutually perpendicular regions R,,,... Rp, Within the region A, every set of pz mutually perpendicular directions satisfies the canonical conditions. The equations (47—51) teach us that if must be possible to choose the canonical directions in each of the regions A, in such a way that they are V,_;-normal, when through i, there shall pass n—1 mutually orthogonal V,,—1-systems. Thus the conditions (C’) and (D), depending on (55) resp. (67), i.e. of the being WV, {-normal of al/ canonical congruences, will no more remain valid without any restriction. VG ail, 6 0 6 rg are the unit-p-vectors belonging to the regions I@pvein oo Ry the equations: ind Vye=0. (117) pit Vy2=0 a=l,…, B—1, BH Ung must be satisfied by pz independent solutions. On account of (B) we thus have: (in phere vaak vate pd? Vm pJd=0. … . « (118) and from this we conclude: psa OVA pe Oh ne oee Gero ol roer (ID) AART Bo. (EO) in which i; belongs to another region than it and ij, and for the rest the choice is arbitrary, provided k /. (119) has entirely the same form as (55) and from (120) follows for the special. case that i;, iz, iv each belong to different regions: Ve EO WA SGD eva tte con, ol (ZI) an equation of the same form, and deduced in the same way as (67). The equations (C’) and (D) only remain valid under the above- mentioned restricting conditions. They are besides no longer sufficient. A supplementary condition will be found in the following way: The equation (65) shows: (4% — Aj) hiv? vy ij + xisin iS V *h= 0, (122) (A — 25) Wi? Vi; + xi; viz? V *h — 0. valid for the case that i, and i/ belong to the same region and i; to another one. Then, subtracting the equations (122) one from the other we conclude, in connection with (121): 693 OE UO waa ==) ee iel eels lele yy di re ome le (E) Under the mentioned conditions the equations (C’), (D) and (£) are not only necessary, but also sufficient. In fact, from (£) may be coneluded, in connection with (122), since 27, = 2), that V i; is symmetrical in / and &, when / and & belong to the same region, but j and / do not. From (D) we conclude, in analogical way as we have explained in the first part of this paper, that V ij is symmetrical in 7 and k, when / and & belong to different regions, different from j. (C”) tells that V ij is symmetrical in » and 4, when k differs from j. Hence these three conditions are sufficient to show that Yi; is symmetrical in the region 1i;, and thus that i; is V,,-1-normal. When we call’) p,,p,...pq the multiplicity of the roots of the algebraic characteristic equation (24), the number of equations (C’) is the sum of the two-factorial products of the numbers ° Py» Pa ---Pg, and the number of the equations (D) is thrice the sum of the three-factorial products of these numbers. The number of the equations (£) is equal to the sum of the products . + Pk of the form pz pr (mae = 1). 13. Simplifications for the case that the given congruence is V,,-1-normai. When i, is V,—;-normal, (C) passes into (C,) or (C,), being valid for the case that i; and iz belong to different regions. (D) can also be brought into the form (D,) and is then valid for the case that ij, 1, and iy belong to different regions. From (97) follows for the case that i, and i; belong to the same region and i; to another: ieee ire, Vasu Onset a ey (128) This equation can also be written in the form: 4 VARS CA OE oe med Nd) which has a formal analogy to (D,"), but which is valid under different conditions. But the increment of the vector i,, when ') (0) is the equation (C) of Ricci, Dei sistemi, page 312, but deduced from th, and not from V — in *) Compare Ricci, Dei sistemi, p. 312. 4) (1) is (Cy) of Paces, Dei sistemi, p. 314. 694 geodesically moved along the boundary of the surface-element do, is:7) 4 Dison = Cole el 6 5 sa 6 oo (lees) So (H,) demands this increment to remain in the region formed by i, and ij. *) Thus we have obtained the following theorem: Ill. A system of co VWV, in a Vn, whose second fundamental tensor, apart from determined V,,r 3 too. 1) A. R. p. 64. *) An analogous geometrical interpretation can also be given to condition (D,”). 5) E. Corron. Sur une généralisation du problème de la représentation conforme aux variétés à trois dimensions, Comptes Rendus 125 (97) 225—228, compare also E. Corroy, Annales de Toulouse 1 (99) 385—488, Chap. III. — 695 15. Addendum. In this paper the product i.i=x of the system R,°') is used. x can be found from the dualities existing in the orthogonal group, on which the identifications used in the system &,,° are founded. Now in investigations on differential geometry these identifications (e.g. of i, and ii) are practically not used. In this case it is convenient to substitute x by + 1, then x vanishes in all formulae, and the calculation grows much easier. It has however to be noted, that taking +1 for x it is no longer permitted to make use of the identifications founded on the dualities of the orthogonal group. 1) J A. ScHoureN, On the direct analyses of the linear quantities etc., These Proceedings 21 (17) 327—341; Die Zahlensysteme der geometrischen Gröszen, Nieuw Archief (20) 141—156. Palaeontology. — “Ueber einige palaeozoische Seeigelstacheln (Timorocidaris gen. nov. und Bolboporites Pander)’. By Prof. J. Wanner. (Communicated by Prof. G. A. F. MOLENGRAAFF). (Communicated at the meeting of January 31, 1920). I. Timorocidaris gen. nov. Die bis jetzt bekannten palaeozoischen Seeigelstacheln zeichnen sich im Vergleich zu den meso- und känozoischen durch eine bemer- kenswerte Einförmigkeit aus. “As far as known, spines are very uniform in character within the species in the Palaeozoic, cases of marked deviation such ‘as occur in some Cidaridae being almost unknown in these older types” sagt Jackson *). Seeigelstacheln aus den permischen Ablagerungen der Insel Timor zeigen, dass dieser Satz fiir das jiingste Palaeozoikum nicht mehr als zutreffend gelten kann. Aber auch ausserhalb der einzelnen Art herrscht hier eine betrachtliche Mannigfaltigkeit der Typen, ähnlich wie bei den Krinoiden, Blastoiden und Korallen, die allerdings einen noch weit grösseren Reichtum an neuen und eigentümlichen Formen aufzuweisen haben. Die Modifikationen, die bei den permischen Stacheln von Timor zu beobachten sind, fallen zum Teil in den Rahmen der Abänderungen, wie sie manche Cidariden zeigen. Wie dort erscheinen Gestalt und Skulptur der Stacheln innerhalb der gleichen Art in verschiedener Weise modifiziert, und wir wissen hauptsächlich durch Beobachtungen an rezenten Cidariden, dass diese Modifikationen im wesentlichen mit der Position der Stacheln an der Schale zusammenhängen. Fremdartiger ist die Modifikation des Gelenkes, das hier in einer Ausbildung erscheint, wie sie bisher noch bei keinem Seeigelstachel beobachtet worden ist. Am bemerkenswertesten ist in dieser Hinsicht eine Seeigelart, deren Stacheln bei Basleo, der bekannten reichhaltigen Fundstatte permischer Versteinerungen auf Timor, zu den häufigsten und auffallendsten Fossilien gehören. Sie sind von mir und der Expedition MoreNGRAAFF’s in mehreren Tausenden von Exemplaren gesammelt 1) Jackson, R. T., Phylogeny of the Echini with a revision of palaeozoic species. Mem. of the Boston Soc. of Nat. Hist. Vol. 7, p. 78, Boston 1912. 697 worden, während von den hierzu gehérigen Asseln bis jetzt keine Spur entdeckt werden konnte. Ich schlage fiir dieselben den Namen Timorocidaris sphaeracantha vor. Wie Tafelfig. la—c zeigt, handelt es sich zumeist um eigentiim- liche, knopfahnliche Stacheln mit einem annähernd halbkugelförmigen Körper, der auf der hemisphaeroidalen Oberseite gekörnt, auf der flachen Unterseite, die im allgemeinen senkrecht zur Längsachse des Stachels steht, glatt ist Auf der Unterseite wachst ein kürzerer oder längerer Stiel oder Hals heraus, der unten durch drei Flachen spitz zugeschnitten wird Das Bemerkenswerte ist nun, dass diese drei Flächen als Gelenkfacetten mit ausgesprochenem Krinoiden-Charakter ausgebildet sind. Jede Fläche (Tatelfig. 2) besteht aus einer segment- artigen Ligamentfläche und einer im Umriss dreiseitigen Muskel- fläche, die von der ersteren durch ein Querriff getrennt wird. Die Ligamentfläche liegt oben (distal) und zeichnet sich durch eine deutliche schlitzformige Ligamentgrube aus, die Muskelfläche unten (proximal) und wird durch eine fast oder ganz bis an das Querriff reichende Medianfurche halbiert. Dieser Typus zeigt nun in der Gestalt, in der Skulptur und auch im Gelenk mannigfaltige Abänderungen, von denen hier nur die wichtigsten kurz besprochen werden sollen. Bei der Betrachtung der Gestalt der Stacheln fällt am meisten auf, dass bei vielen (Fig. 5a-b) der Körper im Querschnitt nicht kreisrund, sondern an einer oder an mehreren Stellen seitlich mehr oder weniger abgestutzt ist. Das trifft für ca. */, aller vorliegenden Stacheln zu. Die Körnelung der Oberseite ist bei diesen im Umriss häufig dreiseitigen, manchmal auch vierseitigen oder polygonalen Stacheln auf den sphaeroidalen Teil der Körperoberfläche beschränkt und auf den seitlichen abgestutzten Flachen entweder unvollkommen oder garnicht ausgebildet. Die meisten Stacheln haben sich also in ihrem freien Wachstum gegenseitig behindert und dürften somit eine fast geschlossene Decke, einen wahren Panzer über der Schale ge- bildet haben, vergleichbar mit dem Stachelpanzer, den der bekannte lebende Colobocentrotus Mertensit trägt. Diese Stacheln waren unbe- weglich oder in ihrer Beweglichkeit zum mindesten sehr beschrankt. Andere Stacheln (Fig. 6 und 12) sind mehr oder weniger birn- oder keulenfórmig; bei manchen (Fig. 7) tritt über dem halbkugel- fórmigen Körper scharf von diesem abgesetzt ein zweiter ähnlich gestalteter auf; bei wieder anderen wächst der halbkugelfórmige Kórper am distalen Ende zu einem spitzen Kegel aus. Die Mannigfaltigkeit der Skulptur kommt am augenfalligsten durch das Auftreten von gekérnten und völlig glatten Stacheln (lig. 3 und 14) 46 Proceedings Royal Acad. Amsterdam. Vol. XXII. 698 zum Ausdruck. Von 1768 näher untersuchten Stacheln erwiesen sich 1546 oder 87,4 °/, als gekörnt und 222 oder 12,6 °/, als glatt. Dass diese beiden Modifikationen keine verschiedenen Arten oder Varie- (aten repräsentieren, ist bei ihrer völligen Uebereinstimmung in allen übrigen Merkmalen als sicher anzunehmen. Es kann sich demnach bei den viel weniger häufigen glatten Stacheln, die mit den gekörnten auch durch Uebergänge verbunden sind, nur um eine durch die Position der Stacheln an der Schale bedingte Modifikation handeln. DöperLeiN *) hat gezeigt, dass bei der Mehrzahl der Cidariden auf den dem Bucealfelde zunächst stehenden Stacheln die Körnelung ihrer Oberfläche ganz allgemein mehr zuriicktritt, sodass sie häufig ganz elatt werden. Es ist somit sehr wahrscheinlich, dass auch bei Timorocidaris die glatten Stacheln vorwiegend in der Umgebung des Buccalfeldes auftraten. Eine andere skulpturelle Modifikation (Fig. 10ab) zeigt auf der Oberseite des Körpers anstatt einzelner Körner oder Pusteln ein unregelmässiges Netzwerk von Leisten; die rundliche oder verlängerte, grubenartige Vertiefungen umschliessen. Diese seltene Modifikation ist durch zahlreiche Uebergänge mit dem gekörnten Ty pus verbunden. Zu diesen Uebergangsformen gehört u.a. der in Fig. 13 abgebildete, auch gestaltlich modifizierte Stachel, der auch bemerkenswert ist, weil die. Pusteln, kurzen Leisten und Vertiefungen der Oberseite auf der hohen becherartigen Seitenwand des Körpers von verschieden starken und langen Längsleisten und Furchen abgelöst werden. Aehnliche, durch unregelmässige zellenartige Vertiefungen aus- gezeichnete Skulpturen finden sich bei Seeigelstacheln nur selten, so z. Bsp. bei ,,Cidaris” scrobiculata Braun aus den St. Cassianer- schichten. Man vergleiche insbesondere das von Barner®) in Fig. 339 auf Taf. XI abgebildete Exemplar, von dem gesagt wird: ,,The surface... is covered with small deep pits irregularly distributed and having a granular border apparently of fused pustules’. Aus diesen Worten geht klar hervor, dass nach Baruer’s Meinung die Wande, welche die Vertiefungen umgeben, durch eine Verschmelzung von Pusteln gebildet worden sind. Man kann jedoch auch annehmen, dass umgekehrt die Pusteln durch eine Auflösung der Wände in Pusteln entstanden sind und somit die zellige Skulptur die primäre und die körnige die sekundäre ist. Diese Auffassung dürfte vielleicht deshalb vorzuziehen sein, weil die zellige Skulptur bei keinem späteren Seeigelstachel mehr auftritt, wohl aber bei älteren Seeigeln, so 1) Döperrei, L., Die Japanischen Seeigel. I. Teil, 1887, p. 34. *) Barner, F. A., Triassic Echinoderms of Bakony, Budapest 1909, p. 183. 699 bei dem permischen 7%morocidaris und, wie wir unten sehen werden, sogar schon bei Stacheln aus dem Unter-Silur. Hine dritte sehr häufige Abänderung der Skulptur (Fig. 6 und 12) kommt durch eine mehr oder weniger ausgesprochene Anordnung der Körner in parallele, gerade oder gebogene Querreihen zustande. Von den verschiedenen Modifikationen des Gelenkes schliesst sich eine sehr enge an die Stacheln mit drei krinoidenähnlichen Gelenk- facetten, die weitaus am häufigsten sind, an. Es sind Stacheln mit nur zwei krinoidenähnlichen Facetten. Die Lage und Beschaffenheit dieser Facetten ist genau dieselbe wie bei den dreifacettigen Stacheln. An der Stelle der fehlenden dritten Facette verlängert sich die Aussenseite des kreisrunden Stieles geradlinig nach unten. Diese Modifikation ist nieht allzu häufig und in ihrer Bedeutung neben- sächlich. Interessanter ist eine weitere Abänderung. Von 2422 genauer untersuchten Stacheln zeichnen sich 2150 oder 88.7 °/, durch krinoiden- ähnliche Gelenkfacetten aus; die übrigen besitzen am unteren Ende des Stachelkopfes eine konkave, mehr oder weniger tiefe Aushöhlung ähnlich wie ein normaler Seeigelstachel. Die Aushöhlung ist entweder ziemlich regelmässig schüssel- oder trichterförmig (Fig. 46) oder in unregelmässiger Weise von einigen Furchen und Wiilsten durch- zogen (Fig. 55) im Umriss gewöhnlich kreisrund, gelegentlich gerundet dreiseitig und hierdureh an den Umriss der drei Facetten der häufigsten Stacheln erinnernd. Der stielförmige Hals selbst ist wie bei den Stacheln mit krinoidenähnlichen Gelenkfacetten bei verschiedenen Individuen kürzer (Fig. 44) oder länger (Fig. 13); er kann auch ganz fehlen (Fig. 55), sodass die konkave Gelenkfläche nur von einem niedrigen Wall umgeben wird, der sie von der übrigen Unterseite des Stachels trennt. Die Modifikation mit einfachem konkaven Gelenk ist durch Ueberginge mit der durch drei krinoidenähnlichen Facetten aus- gezeichneten verbunden. Diese Uebergangsformen (Fig. 8 und 9 a—c) zeigen, dass das einfache Gelenk morphologisch nicht den drei krinoidendlinlichen Facetten zusammen, sondern nur einer einzigen entspricht. Es kann daher das konkave Gelenk nur aus einer krinoidenähnlichen Facette bei gleichzeitiger Reduktion der beiden andern oder, wenn man umgekehrt eine Entstehung des krinoiden- artigen Gelenkes aus dem konkaven annimmt, aus der konkaven Gelenkflache nur eine krinoidenähnliche Facette hervorgegangen sein, wáhrend fiir die beiden übrigen eine Neubildung anzunehmen ist. Wie ist nun das Zusammenvorkommen dieser beiden Gelenkty pen bei ein und derselben Art zu verstehen? Zunächst mag bemerkt werden, dass die Annalime, dass die Stacheln mit krinoidenähnliehem Gelenk einerseits und mit konkavem Gelenk andererseits verschie- 46% 700 denen Arten oder Varietäten angehören, als héchst unwahrscheinlich beiseite gelassen werden kann. Die Uebereinstimmung dieser beiden Gelenktypen in allen ihren übrigen Merkmalen, der Befund, dass auch die Abänderungen in der Gestalt und Skulptur bei beiden Gelenk- typen genau dieselben sind, und schliesslich das schon erwähnte gelegentliche Vorkommen von Stacheln, die alle Uebergange von dem einen Gelenktypus zum andern zeigen, sprechen bestimmt gegen eine solche Auffassung. Die verschiedenartige Ausbildung des Gelenkes bei Timorocidaris lässt sich mit dem Zusammenvorkommen von gekerbten und glatten Hauptwarzen und dementsprechend mit dem Zusammenvorkommen von Stacheln mit gekerbiem und glattem Stachelkopf bei vielen fossilen Cidarisarten und bei manchen rezenten Arten von Plegioct- darts, Tylocidaris, Dorocidaris und Leiocidaris vergleichen. Nun ist nach DöpurLErN *) bei einer Reihe von Plegiocidarisarten, die sich durch gekerbte Hauptwarzen auszeichnen, das Vorkommen einer mehr oder weniger grossen Zahl von ungekerbten Hauptwarzen auf die obere Schalenhalfte beschränkt und auch von Leiocidaris, einer durch vorwiegend glatte Hauptwarzen ausgezeichneten Gattung, wird angegeben, dass es zumeist die dem Apicalfelde zunächst stehenden Hauptwarzen sind, die gekerbt sind. Man könnte daher daran denken, dass auch bei Timorocidaris die verschiedenen Gelenkmodifikationen mit der Stellung der Stacheln an der Schale in Beziehung zu bringen sind. Diese Auffassung scheint gestützt zu werden durch das Haufig- keitsverhältnis der beiden Gelenkmodifikationen. Denn von 2422 näher untersuchten Exemplaren besitzen 2150 oder 88,7 °/, krinoi- denahnliche Gelenkflachen und 272 oder 11,3°/, einfache konkave Gelenkflachen. Diese Zahlen stimmen anffallend genau mit denjenigen überein, die das Verhältnis zwischen gekörnten und glatten Stacheln bezeichnen. Gleichwohl ist diese Auffassung nicht haltbar, wie sich leicht feststellen lässt, wenn man die gekörnten und glatten Stacheln auf die Beschaffenheit ihres Gelenkes gesondert untersucht. Dabei ergibt sich nämlich, dass unter den gekörnten Stacheln 87,3 °/, und unter den glatten 87,8°/, durch krinoidenähnliche Facetten und der Rest, also 12,7 bezw. 12,2°/, durch konkave Gelenkflächen ausge- zeichnet sind. Unter den glatten Stacheln aus der Umgebung des Buccalfeldes befindet sich also ein genau ebenso grosser Prozentsatz von Stacheln mit konkaver Gelenkfläche, wie unter den Stacheln der übrigen Schalenzonen. Auch unter allen übrigen Modifikationen der 7%morocidarisstacheln 1) Je. p. 42 und 43. 701 ist keine einzige ausfindig zu machen, bei der nur einer der beiden Gelenktypen auftreten wiirde. Es bleibt somit als wahrscheinlichste Annahme übrig, dass bei der gleichen Zimorocidarisart in allen Schalenzonen unter einer überwiegenden Zahl von Stacheln mit krinoidenartigem Gelenk ein gewisser Prozentsatz von Stacheln mit konkavem Gelenk mehr oder weniger unregelmässig zerstreut auftritt. Die Art der Gelenkung ist labil. Schwieriger ist es, zu ermitteln, welche Gelenkmodifikation wir als die primäre und welche als die sekundäre anzusehen haben. Ist das krinoidenähnliche Gelenk aus dem einfachen konkaven hervorgegangen oder umgekehrt? Timorocidaris darf zweifellos als ein in sehr eigentümlicher Weise spezialisierter Typus gelten. Das Vorkommen einer überwiegenden Zall seitlich abgestutzter Stacheln weist auf eine sehr geringe Beweglich- keit, wenn nicht Unbeweglichkeit dieser Stacheln hin. Diese Erschein- ung dürfte wohl kaum als eine ursprüngliche gelten können, sie dürfte als eine Anpassung an die Lebensbedingungen auf einem Riff aufzufassen sein. Da nun auch die krinoidenähnliche Ausbildung des Gelenkes eine freie allseitige Beweglichkeit der Stacheln ausschliesst — je nach der Zahl der Facetten können sich die Stacheln in zwei oder drei Richtungen bewegen — so liegt es nahe, in der krinoi- denäbnlichen Ausbildung des Gelenkes und dem engen Zusammen- schluss der Stacheln einen ursächlichen Zusammenhang zu sehen und anzunehmen, dass das krinoidenartige Gelenk aus einem einfachen konkaven entstanden ist, wobei die allseitige Beweglichkeit zugunsten einer besseren Verfestigungsmöglichkeit der schweren massiven Stacheln mit der Schale aufgegeben wurde. Wenn diese Auffassung die richtige ist, dann möchte man allerdings erwarten; dass die seitlich abgestutzten, garnicht oder nur äusserst wenig, beweglichen Stacheln auch diejenigen sind, bei denen vorzugsweise das krinoidenartige Gelenk auftritt und dass umgekehrt die kreis- runden, in ihrem freiem Wachstum nach keiner Seite hin behinderten Stacheln, deren Gestalt auf keine Beschränkung der Beweglichkeit schliessen lässt, diejenigen sind, die hauptsächlich konkave Gelenkung zeigen. Wenn man das vorliegende Material darauf hin prüft, so ergibt sich, dass solche Beziehungen in keiner Weise bestehen. Bei den kreisrunden Stacheln sind beide Gelenk- bildungen ebenso häufig wie bei den seitlieh abgestutzten. So bietet das Material selbst allerdings keine Stütze fiir die Auffassung, dass die Ausbildung des krinoidenihnlichen Gelenkes bei Timorocidaris zu dem engen Zusammenschluss der Stacheln in Beziehung zu 702 bringen ist. Auch beim lebenden Colobocentrotus ist es trotz des Zusammenschlusses der Stacheln zu einer panzerartigen Decke zu einer von der normalen konkaven abweichenden Ausbildung der Gelenkflache nicht gekommen. Zum weiteren Vergleich mag nochmals die Kerbung der Haupt- warzen herangezogen werden, der DöperIEIN in seinem anregenden Werke über die Japanischen Seeigel ein besonderes Kapitel gewidmet hat. Er sagt"): „Die Frage, ob die Cidariden init glatten oder die mit gekerbten Hauptwarzen den urspriinglichen Zustand darstellen, lässt sich nicht mit Sicherheit beantworten’, halt es jedoch für sehr wahrscheinlich, ,,dass unabhängig voneinander auf verschiedenen Linien aus Formen mit gekerbten Warzen solche mit ungekerbten allmahlich sich herausgebildet haben”, oder m. a. W., ,,dass die Kerbung der Hauptwarzen ein Charakter ist, der bei den Cidariden auf verschiedenen voneinander unabhängigen Entwickelungslinien allmählich verloren gegangen ist’. Die kompliziertere Gelenkver- bindung wäre somit hier als der ursprüngliche Zustand, die einfachere als der spätere anzusehen. Ob wir diese Erfahrung auf die ver- schiedenen Ausbildungen des Timorocidarisgelenkes übertragen dürfen, lässt sich zur Zeit wohl nicht entscheiden. Immerhin scheint der Schluss, dass auch bei Zimorocidaris die krinoidenartige Ausbildung des Gelenkes die primäre und die einfache konkave die sekundäre ist, eine gewisse und vielleicht sogar ebenso grosse Berechtigung zu besitzen wie die umgekehrte Annahme. Wie dem aber auch sein mag, das Bemerkenswerte bleibt jeden- falls das Vorkommen eines Gelenktypus bei palaeozoischen Echiniden, der mit demjenigen gewisser Krinoiden vollkommen übereinstimmt. Es ist das ein neuer Konvergenzfall, der in die Reihe derjenigen Erschei- nungen gestellt werden kann, die Eimer als ,,unabhangige Ent wicklungs- gleichheit” oder ,, Homéogenesis”’ bezeichnet. Denn der gleiche Charakter gelangt hier bei ganz verschiedenen Gruppen selbstandig zur Aus- bildung. Die Art der Verbindung und Befestigung der 7?morocidarisstacheln mit der Schale ergibt sich aus der Beschaffenheit des Stachelkopfes und der Gelenkfacetten. Die nach unten zugespitzte Form des mit den Gelenkfacetten besetzten Stachelkopfes und das Vorhandensein eines Querriffes in jeder Facette, das ein Widerlager erfordert, lässt darauf schliessen, dass der seitlich facettierte Stachelkopf nicht auf, sondern im dem Warzenkopf gesessen hat, wie es Textfig. 1 sche- matisch veranschaulicht. 1) lep. 37, 38: 703 Die zu diesen Stacheln gehörigen Warzenköpfe müssen demnach eine der Form des Stachelkopfes annähernd entsprechende Vertiefung fig. 7. Fig. 1. Schematischer Längsschnitt durch zwei Stacheln von Timorocidaris sphaeracantha Wann. X 2. Links ein Stachel mit krinoidenartigen Gelenkfacetten, Gh, Gelenkhécker. Gp, Gelenk. einfachem konkaven Gelenk. rechts ein Stachel mit H, Warzenhof. M, Muskelfläche mit der unteren Muskelschicht. I, Ligamentfläche mit der oberen Muskelschicht. R, Querriff. besessen haben und auch sehr gross gewesen sein. Zur Anheftung der Muskeln dienten ausser den Muskelflächen wahrscheinlich auch die Fig.2. Fig. 2. Schematischer Längsschnitt durch eine Stachelbasis S und den Gelenkhöcker Gh eines normalen Seeigelstachels. M,, Mo, innerer Muskelmantel. B, Vereinfacht nach dusserer und Bindegewebesubstanz. Lupwie -Hamann, Seeigel in Bronn’s Klassen und Ordnungen des Tierreichs. Ligamentflachen. Nachdem jetzt von HAMANN') u.a. bei den Arm- gliedern rezenter Krinoiden auch die dorsalen Fasern als echte Mus- kelfasern gedeutet werden, scheint mir diese Annahme die zutref- fendste zu sein. Es wäre demnach eine innere (untere) und äussere (obere) Muskelschicht und somit eine, allerdings nur äussere Aehn- lichkeit mit der Muskulatur eines normalen Seeigelstachels vorhan- den, wo diese bekanntlich gleich- falls aus einer äusseren und einer inneren Schicht besteht, die den Gelenkkopf Mantel umgibt. (Siehe Textfig. 2). wie ein doppelter Auch bei der durch eine konkave Gelenkfläche ausgezeichneten Stachelmodifikation von Timorocidaris muss die Gelenk(läche zur ') Brons's Klassen und Ordnungen des Tier-Reichs. Il. Bd. 3 Abtlg. Echino- dermen, p. 1463. Leipzig 1905. 704 Anheftung der Muskulatur gedient haben. Denn ausserhalb des flachen Randes, der die Gelenkfläche umgibt, ist kein Platz für die Befesti- gung von Muskeln, wie die Oberflächenbeschaffenheit des Halses zeigt. Dementsprechend ist auch über diesem Rande keine Spur eines Ringes vorhanden. Diese konkave Gelenkfläche lässt sich daher nicht ohne weiteres vergleichen mit der Gelenkpfanne am unteren Ende eines normalen Seeigelstachels, wo sich die Stachelmuskulatur bekanntlich zwischen dem Ring und dem unteren Rande, der die vertiefte Gelenkfläche umgibt, anheftet. Der glatte Rand, der bei Timorocidaris die konkave Gelenkfläche umgibt, entspricht daher dem Ringe und nicht dem unteren Rande des normalen Seeigel- stachels; er entspricht ferner dem oberen Rande der Ligamentfläche bei den Timorocidarisstacheln mit krinoidenähnlichen Facetten, wie die schon oben besprochenen Uebergangsformen zwischen den beiden Gelenkmodifikationen von Tunorocidaris zeigen. Die gleiche Gelenkausbildung wie bei der durch eine konkave Gelenkfläche ausgezeichneten Stachelmodifikation von Zimorocidaris treffen wir u.a. bei den Stacheln von Bothriocidaris und auch bei mesozoischen Cidariden noch gelegentlich an. So sagt Quenstupr *) von den Stacheln von Cidaris elegans aus dem weissen Jura: ,,Hin- zelne Individuen (Taf. 62 Fig. 8) haben keine Spur eines Halsringes”’, und von Cidaris coronatus: ,,Besonders hervorzuheben sind die Gelenkgruben ohne Gelenkkopf, bloss mit scharfem Rande”. Auch bei Cidaris marginatus, florigemma u.a. kommen nach QuENsTEDT Stacheln vor, die des Gelenkkopfes ermangeln. Die starke Ausbreitung der glatten Unterseite der meisten 7imoro- cidarisstacheln lässt auf grosse Warzenhöfe schliessen. Diese Warzen- höfe dürften zumeist ineinander geflossen und die Serobicularringe, soweit solche überhaupt vorhanden waren, vielfach unterbrochen gewesen sein, da die meisten Stacheln sich gegenseitig berührten und zum Teil dicht aneinander geschlossen waren. Dazu kommt noch die schon erwähnte grosse Ausbildung der Hauptwarzen. Das sind alles Merkmale, durch die sich nach Döperrein?) die heute lebenden Cidariden in jugendlichem Alter auszeichnen; sie können deshalb als primitive Merkmale aufgefasst werden. Auch die knopf- formige Gestalt der Stacheln ist ein primitives Charaktermerkmal, wie die Ontogenie der Cidariden zeigt. „A young spine of a cidarid is short, broad, and distally rounded and reminds one of the character of the spines of Colobocentrotus”’, sagt Jackson. *) Als weitere Merk- 1) Quenstept, F. A., Petrefaktenkunde Deutschlands. Echiniden, p. 41 und 50. 705 male, die wahrscheinlicb als urspriingliche anzusehen sind, sind sehliesslich noch die eigentümliche Ausbildung des Gelenkes, insbe- sondere auch das Fehlen eines Ringes und das gelegentliche Auftreten einer netzartigen Skulptur an Stelle der Körnelung zu nennen. Il. Bolboporites Pander. Die Pimorocidarisstacheln geben uns in mehrfacher Hinsicht den Sehliissel für das Verständnis der von Panprr') schon 1880 unter dem Namen Bolboporites beschriebenen eigentiimlichen Fossilien aus dem Unter-Silur von Russland, deren wahre Natur bis jetzt nicht richtig erkannt worden ist. Die Panprr’schen Bolboporiten®) (Fig. 14 und 15) sind bekanntlich mehr oder weniger kreiselförmige, halbkugelförmige oder stark abge- plattet-kugelige, aus spätigem Kalk bestehende massive Körper. Die äussere Oberfläche dieser Körper ist mit eigentiimlichen zelligen, unregelmässig eckigen oder abgerundeten, nur wenig tiefen Griibchen versehen, die umso länger und breiter werden, je mehr sie sich der Unterseite des Körpers nahern. Die Unterseite ist glatt und zeigt in der Mitte oder noch häufiger dem Rande genähert eine kleine Grube, zuweilen auch noch einige weitere unregelmässige Hindriicke. Nach Panper stehen die Bolboporiten am nächsten den Dactyloporen also den früher für Foraminiferen gebaltenen Kalkalgen. Bronn 1848°) bezeichnete Bolboporites als ,,Bryozoorum foss. gen. Cala- moporae affine’. Die meisten späteren Autoren haben diese Gattung zu den tabulaten Korallen, zumeist in die Nahe von Favosites gestellt, so Minne Epwarps und Harmer 1852 *), pr Fromenrer, 1858 °), BicuwaLp 1860°), Zire, 18797) und Quunstrpr 18815), letzterer allerdings mit grosser Reserve, da diese tabulate Koralle ‚aus spätigem Kalk besteht, wie wir es bei Stacheln von Echinodermen zu finden gewohnt sind.” Bréccer 1882") hielt die Stellung dieses rätselhaften Gebildes h Panper, Beiträge zur Geognosie Russlands 1830, p. 107. 2) Die besten Beschreibungen und Abbildungen finden sich bei PAnper (l.c), Eicuwatp (Lethaea rossica) und Quensrept (Petrefaktenkunde Deutschlands, Echiniden). 5) Bronn, H. G., Index palaeontologicus, p. 170. 4) Mitne Epwarns and J. Harmer, Monogr. des Polypiers fossiles des terrains paléozoiques, p. 246. 5) pe Fromenvet, E., Introduction a |’étude des Polypiers fossiles, p. 269. *) EronwaLp, Lethaea rossica, I, p. 495. 1) Zuren, K. A, Handbuch der Paliontologie I. Bd. 2. Lfg. p. 236. *) Quenstept, F. A., Petrefaktenkunde Deutschlands |. Abtlg. 6. Bd. Die töhren- und Sternkorallen, p. 58. % Broager, W.C, Die silurischen Etagen 2 und 8 im Kristianiagebiet und auf Eker, p. 43 706 fir noch ganz unermittelt. Erst Linpsrröm ist 18831) als erster mit Bestimmtheit für die Echinodermen-Natur dieses Fossils ein- getreten, indem er sagt: „There can be no doubt left, that the fossils commonly named Bolboporites are neither corals nor bryozoa, but, as is evidently shown by their intimate structure, parts of the skeleton of some Echinodermatous animal, possibly some unknown startish, amongst the recent ones of which blunt loosely affixed spines of nearly the same appearence often occur’. 1888 spricht LiNDsTROM ®), worauf mich Herr G. Horm in freundlicher Weise aufmerksam machte, nochmals von Bolboporites als „Fragments of some unknown Eechinoderm, as shown by its intimate, cha- racteristie structure’. Auch Jarker und von WornrMANN 1899%) sind sich über die Echinodermennatur dieses Fossils im Klaren gewesen. vON WOErHRMANN war jedoch merkwiirdigerweise geneigt anzunehmen, dass die Bolboporiten im Innern der Theca zweier Chiroerinusarten ihren Platz hatten, wozu JarkKer bemerkt: „Ich weiss nicht recht, welehen Platz und welche Funktion ein solcher massiger Körper im Innern der Theca gehabt haben soll. In Betracht könnte wohl nur die Möglichkeit kommen, dass diese Körper ur- spriinglich sehr porös als innere Madreporenfilter funktionierten. Die hufeisenformige Narbe, die sich bei ilnen auf der einen flachen Seite findet, entspricht etwa in Form und Grösse der äusseren Oeffnung des primären Steinkanales und könnte diesem also innen angesessen haben als Hingangsoeffnung in den Filter”. Seitdem scheint dieses interessante und merkwiirdige Fossil ganz in Vergessenheit geraten zu sein. Es wird in keinem neueren Lehrbuche der Palaeozoologie erwähnt. Da die Bolboporiten in den Sammlungen weit verbeitet sind, kann es sich nicht um sebr selten vorkommende Fossilien handeln. Nach Bröacer sind sie im Kristianiagebiet im oberen Teil des Expansus-Schiefers „überall ganz haufig’. Massenhaft kommen sie indes, wie mir Herr G. Horm in Stockholm auf meine Anfrage in freundlicher Weise mitgeteilt hat, nach seinen Erfahrungen wohl nirgends vor. Die Bolboporiten liegen überall in Unter-Silur. Sie sind besonders in der Gegend von St. Petersburg bei Zarskoje, Pulkowa, Ropscha, am Lynnofluss bei Koltschanovo (Kalkschanovo) und an anderen Stellen im Orthocerenkalk und im oberen Teil des Glaukonitkalk- 1) LinpsrröMm, G., Index to the generic names applied to the corals of the palaeozoic formations. Bihang till K. Svenska Vet.-Akad. Handl. Bd. 8, Nr. 9, p. 7. 2) LinpsrröMm, G., List of the fossil faunas of Sweden. I. Cambrian and Lower Silurian, p. 10. Stockholm 1888. 5) JAEKEL, O., Stammesgeschichte der Pelmatozoen, p. 246. Fussnote. 707 steins verbreitet'). In Schweden finden sie sich nach Linpstrém (1888, le) im „Lower Gray Orthoceratite Limestone” und im „Chasmops Limestone’, im Kristianiagebiet, wie erwähnt, im oberen Teil des vorwiegend kalkigen „„Expansus-Schiefers”. Auf Irland scheinen sie in der Landschaft Waterford im höchsten Teile der Etage 2 der „Tramore Limestone Series” vorzukommen *®). Ich glaube nun, diese Fossilien mit Bestimmtheit als Seeigel- stacheln ansprechen zu können. Unter den Merkmalen der Gattung Bolboporites ist die zellige Beschaffenheit der äusseren Oberfläche zweifellos dasjenige, das früher für die Deutung dieses Fossils als tabulate Koralle oder Bryozoe den Ausschlag gegeben hat. Quensrepr und besonders LinpstréM haben erkannt, dass eine solche Deutung an der spätigen Natur der Bolboporiten ohne weiteres scheitern muss. Eine Erklärung für die eigentümliche Oberflachenstruktur haben sie indes nicht geben können. Durch das Auftreten einer ganz ähnlichen Skulptur bei manchen Stacheln von Pimorocidaris sphaeracantha und bei den schon oben erwähnten Stacheln von ,,Cidaris” scrobiculata Braun ist das Vorkommen zelliger Skulpturen auch bei Seeigelstacheln erwiesen und steht also mit der Deutung der Bolboporiten als Seeigelstacheln nicht im Widerspruch. Der hauptsächlichste, aber unwesentliche Unterschied, der sich zwischen der Skulptur der Bolboporiten und derjenigen der genannten permischen und triadischen Stacheln ausfindig machen lässt, ist der, dass bei Bolboporites die Wande, welche die Vertiefungen umgeben, ganz glatt sind und abgesehen von den Stellen, wo sich die Wände benachbarter Zellen vereinigen, keine Körner oder Pusteln erkennen lassen. Das würde fiir die oben geäusserte Auffassung sprechen, dass die Netzskulptur ein ursprünglicher Zustand ist, und dass aus ihr die Körnelung durch Auflósung der Wande in Körner hervorgegangen ist. Vielleicht tritt die zellige Skulptur bei den ältesten Seeigelstacheln überhaupt häufiger auf, als es heute den Anschein hat. Es sei hier kurz auf die Fossilreste hingewiesen, die Eicnwatp*) als Seeigelstacheln beschrieben und abgebildet hat mit dem Bemerken, dass sie wabrscheinlich zu seinem Bothriocidaris globulus gehören. Zwar hat später Fr. Scumipr *) ') Scumipt, F., On the Silurian strata of the Baltic provinces of Russia. 1882. Quart. Journ. of the Geol. Soc. Vol. XX XVIII, p. 519. *) Reep, F.R. C., The lower palaeozoic bedded rocks of county Waterford. Quart. Journ. Geol. Soc. Vol. LV, p. 732. 1899. ®) ErcHwatvp, Lethaea rossica, Bd. | p. 655, Tab. XXXII, fig. 23 a, b. Stuttgart 1860. *) Scumipt, Fr, Ueber einige neue und wenig bekannte baltisch-silurische 708 gezeigt, dass die Stacheln von Bothriocidaris globulus eine ganz andere Beschaffenheit besitzen, und dass die von HErcnwaLp als Stacheln zu dieser Art citierten Stiicke von Pulkowa nichts damit zu tun haben. Ob sie jedoch, wie Scumipt meint, eher als kleine Bryozoen anzuselhen sind, scheint mir noch recht zweifelhaft zu sein. Nach der ausführlichen Beschreibung und den Abbildungen Ercawarp’s scheinen alle Merkmale für einen Seeigelstachel mit zelliger Oberflächenstruktur zu sprechen. Leider fehlt mir das Material, um diese Frage entgiltig zu entscheiden. Fiir das Verständnis der Bolboporitenskulptur als Seeigelstachel- skulptur ist ferner von Belang, dass die einzelnen Zellen umso länger und grésser werden, je mehr sie sich der Unterseite nähern, eine Erscheinung, die, wie oben gezeigt wurde, in ganz abnlicher Weise bei einigen Stacheln von Zimorocidaris auftritt; ferner, dass „die Zellen häufig’”’, wie schon Panper bemerkt, ,,in einer gewissen Ordnung aneinandergereiht erscheinen, indem sie in konzentrischen einander berührenden Kreisen liegen’, eine Anordnung, zu der auch die Körner bei vielen Seeigelstacheln neigen. Schliesslich ist auch die glatte Beschaffenheit der Unterseite ein Merkmal, das, wie die glatte, flache Unterseite der 7%morocidaris- Stacheln zeigt, gleichfalls für einen Seeigelstachel spricht. So lässt sich also die Oberfläche der Bolboporiten nach allen ihren Eigen- tümlichkeiten als Seeigelstacheloberfläche auffassen. Noeh wichtiger ist das Vorhandensein einer echten Gelenkfläche auf der Unterseite der Bolboporiten. Sie ist von den meisten Autoren zwar bemerkt, aber nur als ,,Grube” oder „Vertiefung” angesprochen worden. Selbst von einem so ausgezeichneten Beobachter wie Qovensrepr, der übrigens, wie alle übrigen Forscher mit Ausnahme von Panper die Bolboporiten mit ihrer Spitze nach unten abbildet, wird sie mit der Bemerkung abgetan: „Die Oberseite hat eine grosse, schwer zu reinigende Grube”. Der richtigen Deutung am nächsten ist anch hier schon Panper gekommen, indem er sagt, dass die Grube daraufhin zu weisen scheint, dass die Unterseite der Bolboporiten „vielleicht auf einem Stiele getragen wurde”. An mehreren gut erhaltenen Exemplaren (Fig. 14 a—c) des Bonner Museums vom Flusse Lynno bei Koltschanovo (Gouvernement St. Petersburg) auf die mich Herr Prof. Srsmmann in freundlicher Weise aufmerksam gemacht hat, zeigt diese Grube folgende Be- schaffenheit: Im Umriss ist sie verlängert elliptisch, in der Mitte jedoch eingeschniirt, sodass sie in zwei mehr oder weniger gleiche Petrefacten. Mém. de l’Acad. imp. des Sciences de St.-Pétersbourg. VII. sér. t. XXI N°. 11, p. 41. 1874. 709 Hälften zerfällt. Soweit die Stücke im Querschnitt nicht vollkommen kreisrund sind, ist die längere Achse der Grube zu dem grössten Querdurchmesser des Stachels annähernd parallel. Die Grube ist in der Regel von einem niedrigen Wall umgeben, ähnlich wie die konkave Gelenkfläche solcher Z?morocidaris-Stacheln, bei denen der Hals und der Stiel sehr stark verkürzt ist. (Vel. Fig. 4a, 6). Dieser Wall kann demnach als ein stark verkürzter oder noch unvollkommen entwickelter stielförmiger Hals aufgefasst werden. Auf einer Seite wird er da, wo er eingeschniirt ist, von einer schlitzartigen Furche unterbrochen; auf der entgegengesetzten Seite, die bei einer excen- trischen Lage der Grube zugleich diejenige ist, die dem Rande der Unterseite genäbert ist, zieht sich von der Einschnürung des Walles ein schwacher Riicken in die Tiefe der Grube hinab, ohne jedoch den Schlitz zu erreichen. Bei anderen Exemplaren (Fig. 15) felilt der Wall, und der Schlitz liegt in der Tiefe der Grube. Es ist selbstverstandlich, dass eine so ausgesprochen bilateral symmetrische Ausbildung des Gelenkes nur eine Bewegung in zwei diametral entgegengesetzten Richtungen erlaubte. Die merkwiirdige Beschaffenheit dieser Vertiefung konnte in der Tat kaum fiir einen Seeigelstachel sprechen, solange eine von der normalen wesentlich abweichende Ausbildung der Gelenkfläche von keinem Seeigel bekannt war. Zwar hat schon Scnuurze 1866 *) seine Gattung Nenocidaris auf die abweichende Bildung der Gelenkfläche dieser in Eifeler Mittel-Devon vorkommenden Stacheln gegründet. Bei NXenocidaris zeigt sich statt der knopfförmigen Verdickung der Basis eine concave perforierte Gelenkfläche, jedoch ist dieselbe nicht gleichmassig eingesenkt, sondern stark ausgekerbt, sodass der Stachel sattelartig auf dem ihm entsprechenden Tuberkel aufruht”. Diese Gelenkbildung ist jedoch bei weitem nicht so aberrant wie diejenige von Bolboporites. Sie scheint mir zwischen dem Gelenktypus, wie ihn die Timorocidarisstacheln mit konkavem Gelenk zeigen, und dem- jenigen der normalen Seeigelstacheln zu stehen. Die Timorocidarisstacheln zeigen nun zum erstenmal, dass bei palae- ozoischen Seeigeln auch andere, von der normalen stark abweichende Gelenkbildungen möglich sind. Es liegt somit kein Grund mehr vor, der gegen die Deutung der Bolboporitengrube als Gelenkgrube sprechen könnte. Die richtige Deutung dieser Grube wurde vielleicht auch durch ihre wenig konstante Lage erschwert. Die Grube liegt nämlich bald in der Mitte der Unterseite, bald mehr oder weniger excen- trisch dem Rande genáhert. Bei genauerer Betrachtung zeigt sich ') Senuurze, Monographie der Echinodermen des Kifler Kalkes, p. 14. 710 jedoch, dass die Lage der Grube anf der Unterseite keineswegs eine willkiirliche ist. In der Mitte liegt sie bei den mehr oder weniger radialsymmetrischen Bolboporiten (B. semiglobosa und B. mitralis), exzentrisch stets bei den hornférmig gekriimmten (B. wncmata) und zwar so, dass sie sich stets nach derjenigen Richtung verschiebt, nach der sich die distale Spitze des Bolboporiten kriimmt. Verbindet man die distale Spitze mit der Gelenkfläche durch eine Gerade, so steht diese letztere mehr oder weniger senkrecht auf der durch die Peripherie des Stachels gelegten Ebene. Parper hat unter den ihm vorliegenden Bolboporiten auf Grund der äusseren Gestalt der Körper und der Grösse der Zellen vier Formen (B. semiglobosa, triangularis, uncinata, mitralis) unterschieden. EicnwarD vereinigte diese in einer einzigen Art (B. mitralis). Dass er damit das Richtige getroffen hat, dürfte jetzt kaum mehr zweifel- haft sein, nachdem wir glauben, den Nachweis erbracht zu haben, dass es sich bei den Bolboporiten um Seeigelstacheln handelt. Es liegt jetzt nahe, die Panper’schen Formen als Stachelmodifikationen aufzufassen, die am gleichen Individuum in verschiedenen Schalen- zonen auftraten und anzunehmen, dass der Panper’sche B. mitralis vielleicht vorwiegend auf die Umgebung des Apicalfeldes, die semiglobose Form auf die Umgebung des Bucecalfeldes beschränkt war, während die beiden übrigen Formen (B. uncinata und triangu- laris) Stacheln der dazwischen liegenden Schalenzonen sind. Dass sich, wie PaNper sagte, „nicht viele Vebergänge von der einen Form zur andern finden lassen’, steht mit der Deutung dieser Formen als verschiedene Modifikationen derselben Stachelart nicht in Wider- spruch. Der Ercuwarp’sche B. stellifer dürfte hingegen einer von B. mitralis verschiedenen Art angehören. Die Tatsache, dass bis jetzt noch nie eine Assel der Bolboporiten- schale gefunden wurde, kann selbstverständlieh nicht als Einwand gegen die Deutung der Bolboporiten als Seeigelstacheln vorgebracht werden. Das Gleiche ist, wie oben bemerkt, bei den Z?%morocidaris- und vielen anderen Seeigelstacheln der Fall. Es sei nur an Xeno- cidaris aus dem Mittel-Devon der Hifel und an die zahlreichen Stacheln aus der oberen Trias von St. Cassian und vom Bakony erinnert. Zusammenfassung. Als wesentlichste Ergebnisse der vorangehenden Ausfiihrungen sind hervorzuheben : Seeigelstacheln sind im Palaeozoikum in einer grésseren Mannig- 711 faltigkeit der Typen vertreten, als man dies bisher annehmen konnte. Auch innerhalb der Art treten im Palaeozoikum schon dieselben mannigfaltigen Modifikationen wie bei manchen späteren Cidariden auf. Zu diesen Abänderungen der Gestalt und Skulptur gesellt sich bei Timorocidaris sphaeracantha gen. nov. et sp. nov. aus dem Perm von Timor eine sehr bemerkenswerte Moditikation des Gelenkes. Bei den meisten Stachelindividuen dieses neuen Typus besteht das Gelenk aus drei Facetten von ausgesprochenem Krinoiden-Charakter, eine Ausbildung, wie sie bisher von keinem andern Seeigel bekannt geworden ist; bei anderen, weniger häufigen Individuen ist eine einfache konkave Gelenkflache vorhanden, die nicht mit der Gelenk- pfanne am unteren Ende eines normalen Seeigelstachels verglichen werden kann. Die unter dem Namen Bolboporites beschriebenen Fossilien aus dem Unter-Silur von Russland nnd Skandinavien sind als Seeigel- stacheln zu deuten. Es- sind somit die ältesten Echinidenreste, die wir kennen. Die auffallende zellige Oberflachenskulptur der Bolbo- poriten steht mit dieser Deutung durchaus im Einklang. Sie ist als altertümliche, auch noch in der Trias vorkommende Stachelskulptur anzusehen. Die Grube auf der Unterseite der Bolboporiten ist eine echte Gelenkgrube und als ein weiterer neuer Typus der Gelenk- bildung bemerkenswert. Bei den palaeozoischen Seeigelstacheln kommen somit verschiedene Gelenkbildungen vor. Als solche sind zu nennen: 1. Das Bolbo- poritengelenk; 2. das Krinoidengelenk (bei Pimorocidaris); 3. das konkave Gelenk bei Stacheln ohne Ring und ohne Verdickung des Stachelkopfes (T'imorocidaris, Bothriocidaris u.a.); 4. das Xenocida- risgelenk; 5. das konkave (normale) Gelenk mit oder ohne Kerbung des unteren Randes an Stacheln mit verdicktem Stachelkopf und mit Ring (Archaeocidaris ua). Für die beiden ersten Typen ist bezeichnend, dass sie nur eine beschränkte Beweglichkeit in wenigen Richtungen, für die übrigen, dass sie eine allseitige Beweg- lichkeit gestatten. Die Gelenkbildung war somit bei den palaeo- zoischen Seeigeln noch nicht so konsolidiert wie das bei den späteren Seeigeln der Fall ist. Die Natur hat urspriinglich auf ver- schiedene Weise versucht, die Stacheln mit der Schale zu verbinden, aber doch unfähig, die Art der Gelenke unbegrenzt abzuändern, hat sie in zwei verschiedenen Tiergruppen die gleiche Form der Gelenktláchen hervorgebracht. 712 TAFELERKLARUNG. Fig. 1—13. Timorocidaris sphaeracantha gen. nov. et spec. nov. aus dem Perm von Basleo, Insel Timor. , Fig. 1. Häufigster Stacheltypus mit gekörnter Oberfläche und krinoidenartigen Gelenkfacetten. a. Von der Seite. 6. Von der Unterseite. Nat. Gr. c. Skulptur der Körperoberfläche > 4. Wig. 2. Gelenkfacette <5. Fig. 8. Glatter Stachel mit ausgehéhlter Unterseite und verkürztem Hals. a. Von der Seite. Nat. Gr. b. Von der Unterseite >< 2!/,. Fig. 4. Stachel mit konkaver Gelenkfläche. a. Von der Seite. Nat. Gr. b. Von der Unterseite >< 21/.9 Fig. 5. Seillich abgestutzter Stachel. Nat. Gr. a. Von der Seite. b. Von der Unterseite. Fig. 6. Keulenförmiger Stachel. Körner in parallelen Querreihen angeordnet. Von der Seite. Nat. Gr. Fig. 7. Fast glatter Stachel mit eingeschntirlem Körper. Von der Seite. Nat. Gr. Fig. 8. Mittlere Partie eines Stachels von der Unterseite mit dem Gelenk: x 4 Zeigt vom Beschauer abgewandt eine grosse konkave Gelenkfläche, dem Beschauer zugewandt zwei unvollkommen krinoidenähnliche Faceften. Fig. 9. Stachelkopf. X4. a. Von der Unterseite. Zeigt eine grosse konkave Gelenkflache und zwei seitliche, unvollkommen krinoidenähnliche Facetten. b und c. Letztere von der Seite (spiegelbildlich) gesehen. Fig. 10. Stachel mit zelliger Oberflächenskulptur. a. Von der Seite. N. Gr. b. Von oben X 21/9. Fig. 11. Kleiner glatter Stachel mit dickem Hals. Fig. 12. Birnförmiger Stachel mit + parallelen Körnerreihen. Fig. 13. Stachel mit seitlich stark verlängert-zelliger Oberflächenskulptur. Fig. 14—15. Bolboporites mitralis Pander aus dem russischen Unter-Silur. Fig. 14. Vom Lynnofluss bei Koltschanovo. a. Von der Seite. 5. Von der Unterseite. Nat. Gr. c. Gelenk auf der Unterseite X 6. Fig. 15. Von Pulkowa. a. Von der Seite. b. Von der Unterseite. Nat. Gr. c. Gelenk auf der Unterseite X 6. Die Originale zu den Figuren 6 und 13 befinden sich in der Sammlung Moteneraarr’s in der Technischen Hochschule Delft, alle übrigen im geolog. pal. Museum der Universität Bonn. WANNER: Ueber einige palaeozoische Seeigelstacheln. 10a *2% Timorocidaris und Bolboporites. MELIOT VINE VAI Loe AMATENRDAM Physiology. — “Concerning Vestibular Eye-rejleves. 11. The Genesis of cold-water nystagmus in rabbits’: By Dr. A. pe Kuryn and Dr. W. Storm van LEEUWEN. (Communicated by Prof. R. Magnus). (Communicated at the meeting of January 31, 1920). For an explanation of cold-water nystagmus we may have recourse to two theories. Barany’s theory is founded on the assumption of a stream of endolymph in one or more semicircular canals, brought about by local cooling of the labyrinthwall. This will cause also the endolymph, present there, to cool down and to flow off to the lowermost part of the semicircular canal. The ensuing lymph stream stimulates the sensory epithelium of the ampulla. In case the head of the animal is in a position in which the ampulla lies higher than the cooled part of the semicircular canal, the stream will be ampullofugal; if the reverse be the case an ampullopetal stream will result. The nystagmus elicited by each stream is of an opposite character. Bartets holds that by douching of the meatus with cold water the labyrinth would be eliminated, so that the nystagmus provoked would be like the spontaneous nystagmus after unilateral extirpation of the labyrinth. A warm water flow would be like stimulation of the N. vestibularis on the same side. In a previous paper, issued from this institute, we have disproved BARTELS’s theory’). Moreover, it has already been contended by many other researchers. lt was first of all pointed out that, if Barreis’s conception were correct, a cold-water nystagmus could not possibly be elicited from the unimpaired ear after unilateral extirpation of the labyrinth. Hover?) has phrased it so well: “dieses tatsächliche Auftreten eines rotatorischen Nystagmus nach der operierten Seite ware nach Barrnus’ 1) A. pe Kiewsn and W. Storm van Leeuwen. Ueber vestibuläre Augenreflexe I. Ueber die Entstehungsursache des kalorischen Nystagmus, nach Versuchen an Katzen und Kaninchen, Graefe’s Arch. 5 Bd. 94 316, 1917. A. pe Krog and W. Srorm van Leeuwen. Over vestibulaire oogreflexen | Mededeeling. Kon. Acad. van Wetensch., Amsterdam. Wis- en Nat. Afd. Versl. Deel XXVI, 381, 1917. 4) J. Hover. Untersuchungen über den calorischen Kaltwassernystagmus. Monatsehr. f. Ohrenheilk. (1912) S. 1313. 47 Proceedings Royal Acad. Amsterdam. Vol. XXII. 714 Theorie, wie er ja selbst zugesteht, total unmöglich, weil eben das operierte Labyrinth fehlt und also nicht überwiegen kann über das gesunde, welches durch die kalte Ausspülung gelahmt werden soll; es sollte also nach Barrens in so einem Fall gar kein Nystagmus auftreten, was aber den klinischen Tatsachen vollständig wider- spricht” (S. 1817 und 1318). This argument, however, is not valid. Brcnterew’s'!) wellknown experiments have shown us that when we extirpate a labyrinth and remove the other after some days, a nystagmus will occur again in the direction’) of the labyrinth that was removed first. So, if the cold-water-nystagmus were resulting from extirpation of the labyrinth on the douched side, we might also expect, some days after unilateral extirpation, on douching the unimpaired ear, a nystagmus towards the extirpated side. Indeed, Barters*) himself has suggested this interpretation. Another argument put forward by Barrers®) against Barany’s theory, we do not quite understand. In a rabbit, with one octavus cut through, a cold-water or a warm water flow into the meatus of the unimpaired ear could provoke a nystagmus only towards the unimpaired ear. This finding of Barrers’s is not explained by Barany’s theory nor even by that of Barrers. Neither were we ever confronted with this case in a prolonged series of experiments’). It is difficult to say what may have led to Barrers’s abnormal experience. It would be better perhaps in similar experiments to perform an extirpation of the labyrinth than a section of the octavus, since the latter operation may be attended with lesions of the central nerve-system. Another cogent argument against the theory of Barrers, put forward also by Barrers himself, is that experimenters succeeded, by provo- king a caloric nystagmus with various positions of the head in space, in obtaining now a nystagmus towards the non-douched ear, now again towards the douched one. This, indeed, is the main argument that turns up repeatedly in the literature. Still, it cannot be adduced against Barrers’s theory without also considering that, when examining 1) W. BecutEREW. Ergebnisse der Durchschneidung des N. acusticus nebst Erör- terung der Bedeutung der semizirkulären Kanäle für das Körpergleichgewicht. Pflüg. Arch. Bd. 30. (1883) S. 312. 2) In speaking about a nystagmus in a certain direction we always mean a nyst. with the quick component in that direction. 3) M. Barrens. Ueber die vom Ohrapparat ausgelösten Augenbewegungen (Ophtalmostatik). Klin. Monatsbl. f. Augenh. Jhrg. 50. (1912) S. 200. 4) Discussion Verh. d. Otol. Gesellsch. Frankfurt. (1911) S. 214. 5) See F. Qurx. Ein Fall von translabyrintharisch operiertem Tumor acusticus. Verh. d. Otol. Gesellsch. Hannover (1912) S. 252 715 the calorie nystagmus, with various positions of the head in space, tonic reflexes of the eye-muscles may occur: the so-called compen- satory eye-positions, which alter the position of the eye in the orbita. Therefore, it must be ascertained beforehand whether or no the spontaneous nystagmus occurring after unilateral extirpation of the labyrinth, alters its direction with different positions of the bead in space. Such experiments have been carried out, for aught we know, only by Kuso‘). They will be briefly discussed here: Kuso severed one octavus. He does not tell us how he did it, nor whether he tried to ascertain by a subsequent control section if the process was successful. It would seem from the protocols that this is highly doubtful. Six of the experiments are reported in detail, of which a short description follows here: Experiment 1, +, and 5 will not receive consideration, because in them the nystagmus was not examined with different positions of the head. Experiment 2. In this experiment a nystagmus appeared with the quick component towards the operated side, after section of the right octavus had been performed. The nystagmus consequent on unilateral extirpation of the labyrinth, however, turns towards the unimpaired ear. Kuso adds only: “Diese Bewegungen bleiben unverändert, wenn man die Körperlage des Tieres ändert.” Experiment 3. Left acusticus cut through. Subsequent vertical nystagmus-move- ments. After a couple of hours perfectly horizontal nystagmus witb the quick component on the operated side towards the nose. Just as with the vertical nystagmus this direction is the same for any position of the animal. A flow of cold water into the right meatus is of no influence. After the semi-circular canal of the right ear has been laid bare, the experimenter states: ‘Nach Hinspritzen von kaltem Wasser ändert sich die Richtung und es tritt eine rückweise Bewegung nach der Nase hin auf der operierten (linken) Seite auf.” This, however, was also the existing direction and opposite to the one we can look for in the case of cold-water nystagmus from the right ear. The vertical nystagmus also points to an imperfect section. Experiment 6. Section of left acusticus. First vertical, afterwards horizontal nystagmus (on the left with the quick component towards the nose.) 1) Kuso Iso. Ueber die vom N. acuslicus ausgelösten Augenbewegungen (besonders bei thermalen Reizungen.) Pflüúg. Arch. 114. (1906) S. 143. 167. 47* 716 On the right a cold-water flow: Reversion of the nystagmus. In ventral position right eye with quick component towards the nose. In other position same direction. Here, then, in caloric examination no influence on the direction of the nystagmus through change of the position of the head in space. This, no doubt, is anomalous. Compensatory eye-positions are no longer distinct. This again indi- cates the deficiency of the experiment. Repeated application of cold water in the right ear yields on the right a nystagmus with the quick component towards that ear. This nystagmus is not affected by the position of the head in space. The appearance of a nystagmus towards the douched ear on cold-water flow is the reverse of what is normally observed, and also the reverse of what was seen after the first washing. The imperfection of the experiment is also seen in the absence of any influence of the position of the head in space. In our first communication it has been shown that in cats the spontaneous nystagmus after unilateral extirpation of the labyrinth, with different positions of the head in space, varies in nature and frequency, but not in direction. In our investigation of the cold- water nystagmus in normal animals and in animals after unilateral extirpation of the labyrinth, on the contrary, a considerable differ- ence in the direction of the nystagmus with different positions of the head in space, has been demonstrated. It also appeared from subsequent experiments with rabbits that with them the case was fundamentally the same. Slight variations in the direction of the spontaneous nystagmus after unilateral extirpation of the labyrinth, however, do manifest themselves here, when the position of the head is varied, in consequence of the compensatory eye-positions, to be discussed later on, whereby the place of insertion of the eye-muscles in the orbita is altered. In the first communication evi- dence was also adduced to show that BArrrrs’s conception of the origin of the caloric nystagmus cannot be correct. In the present investigation we purpose to ascertain whether additional data can be collected to support the theory of BARANY, who ascribes the calorie nystagmus to endolymph-streams. There are plenty of indications in the literature; to our know- ledge an extensive experimental investigation has not been perfor- med as yet. Doubtless, the first question that arises is, whether douching of the meatus with cold-, resp. warm-water through the tympanum will engender such cooling down, resp. warming of the labyrinth- wall that endolymph streams are possible. 5 717 The result of a similar investigation carried on') together with Prof. Maenus, was published in GrarFm’s Archiv., and led to the following conclusion : “Bei Katzen, bei denen die Sympathicusbahnen zum Auge durch das Mittelohr verlaufen, tritt bei Ausspritzen des äusseren Gehör- ganges mit kaltem Wasser eine Sympathicuslähmung am Auge auf, die sich vor allem im Vortreten der Nickhaut äussert. Sie beruht auf einer Kalteparese der genannten Bahnen. Dadurch ist der Beweis geliefert dass beim Auslösen des kalorischen Nystagmus mit kaltem Wasser die Wand des Mittelohres über dem Labyrinth sich nach- weisbar abkiihlt.” We now pass on to report the results of our new experiments on the cold-water nystagmus in rabbits. Our reason for selecting rabbits, while our previous experiments were chiefly carried out with cats, is the following: First, in rabbits we seldom meet with rotatory nystagmus, of which the direction is always difficult to indicate. It is encountered in cats. The principal reason, however, is that in our experimenta- tion we made use of an inquiry into the compensatory eye-positions, which have been carefully determined for the rabbit in conjunction with v. p. Hoevr*), but are difficult of determination for the cat. Technique of our method. A rabbit was suspended on an operation-board, and the head fixed firmly in a Czermak-clamp. Now in order to be able to bring the animal in any given position in space, the following contrivance was made (Fig. 1). The operation-board p-q-r-s is fixed to a wooden frame P-Q-R-S in such a way that the board p-q-r-s can rotate on the axis U-T, while the frame P-Q-R-S is again fixed to a second frame A-B-C-D, so that both P-Q-R-S and p-q-r-s can rotate on the axis V-W. A protractor is attached to P-Q-R-S, as well as to A-B-C-D, so that the degree of the rotation can be noted exactly in every direction. Now when the animal has been tied to the board in ventral position, a rotation on the axis V-W causes the animal to rotate on its bi-temporal axis. When moving the board round the axis U-T the animal turns on its occipito-caudal axis. When finally 1) A. pe Krein und R. Maanus. Sympathicuslähmung durch Abkühlung des Mittelohres beim Ausspritzen des Gehörganges der Katze mit kaltem Wasser. Graefe's Archiv Bd. 96. (1918) 5, 368. 4) J. v. p. Hoeve und A. pe Kreis. Tonische Labyrinthreflexe auf die Augen. Pflúg. Arch. Bd, 169. (1917) S. 241. ~ 718 the board p-q-r-s is first revolved 90° about U-T, so that the animal is in lateral position, and when in this position the board is turned Fig. 1. about the axis V-W, the animal will revolve about its dorso-ventral axis. A combination of rotations round the axes U-T and V-W enables us to bring the animal in any given position in space. In _ all of them the direction of the nystagmus consequent on a cold- water flow could be determined. In the following expositions the rotations in the different directions are described : Rotation I. Animal in ventral position, mouth-fissure horizontal. Rotation of the animal on its bi-temporal axis. Direction of rotation: head down, tail up. Rotation U. Animal in ventral position, mouthfissure horizontal. Rotation of the animal on its occipito-caudal axis. Direction of rotation: douched ear downwards. | Rotation UI. Animal in lateral position with irrigated ear downwards, mouth- fissure vertical. Direction of rotation: head down, tail up. In these experiments the direction of the nystagmus consequent on a cold-water irrigation, was determined 37 times for every rotation of 360°. The first determination was always made at the normal position of that rotation; so e.g. at rotation I: the animal in ventral position, mouth-fissure horizontal. After this, while the ear was constantly being douched, the animal was moved every time 10° in the given direction and the direction of the nystagmus was noted. At the 37% determination the animal had come round again 749 to its original position. Then the last determination served for a eontrol-estimation. A short interval after every rotation of 10° was required before each reading, to preclude the possibility of a nystagmus, resp. deviation brought about by the rotation itself. The irrigation of the right meatus took place from a height of 1,5 m., the cold-water used was of a temperature of + 12° C. For every position in space, after it had continued for some time, the direction of the nystagmus was valued and the direction of the rapid component was marked down. (Figs 2, 3, and 4 not corrected). This method does not yield perfectly reliable data; for a correct determination of the direction one might resort to cinematographic photos from which to decide on the direction. However, this was impracticable for a large number of determinations. Still, from what follows here we may infer that our method of valuation of the direction of the nystagmus yielded useful results. In figs. 2—4 — = Direction of the quick component of the nyst. towards the nose == ” np > 52 » », towards the temporal t= » 5 9 A » », upwards relativeto the orbita. == ys 35 Pb 95 55 » » downwards. Fig. 2—4 (not corrected) gives the mean of 5 experiments. Now, however, the question rises: what influence is exerted on the nystagmus by the above-mentioned tonic eye-reflexes, occurring in the eye-muscles (compensatory eye-positions) with different positions of the head in space. On p. 246 of V. p. Horvw’s research, mentioned above, a curve is given of the rotatory movements. With the aid of this curve the directions of the caloric nystagmus found, were now corrected as follows We assume that douching with cold water, with the head in normal position, engenders an absolutely horizontal nystagmus with the quick component towards the nose. Now, when the position of the head changes from the normal into another position in space, so that a rotatory movement of the eyes ensues, e.g. of 45° with the upper cornea-pole towards the temporal, the insertion-points of the eye-muscles, notably of the Mm. internus and externus, will also be changed by this rotatory movement, and the same contrac- tions and relaxations of these two eye-museles, which caused with the normal position an horizontal nystagmus, will bring about a nystagmus of quite a different direction, viz. about 45° anteriorly upwards. So for instance if an horizontal nystagmus appears at the normal 2 Mp Se. ine ae ne pes SIZE HKG, SS Z NS a amy ix N ' + 4) Ss ) | ee 721 position and at another position of the head in space, with a rotatory Rotation III (no correction). Fig. 4. movement of 45° with the upper cornea-pole towards the temporal, a nystagmus of 75° anteriorly upwards, the correction is 75°—45°=30°. The corrected direction, therefore, is that direction of the nystagmus that would be found, if the eyes were only under the influence of the labyrinth-stimulant consequent on the douching, and if there were no compensatory eye-posttions. Figs 2—4 illustrate our results before and after correction. RESULTS. As stated above, it had already been detected by BARANY, Hornr, and others that the direction of the nystagmus in man varies with different position of the head in space. This result was borne out by our experience. When examining a rabbit, first in ventral position and subsequently with its head hanging downwards, we found a difference of 180° in the direction of the nystagmus. 722 At first we supposed that, when e.g. the nystagmus of the left eye on douching the left meatus was directed anteriorly upwards in ventral position, and posteriorly-downwards with the head down, there would be an intermediate position in which there would be no nystagmus at all. In other words, if the nystagmus in ventral position is Owing to an ampullo-fugal stream in the horizontal semi- circular canal, and the nystagmus with the head down to an ampullo-petal stream, there would be no difference in the level of ampulla and of that portion of the semicircular canal that is cooled down by the douche and the nystagmus would consequently not appear. This proved not to be the case. True, in this reasoning the possibility has been eliminated of an influence of the cold water on the lymph-streams in the vertical semi-circular canals. Considering that, although also the vertical canals may come into play, the horizontal canals are on account of their anatomic loca- tion, most exposed to the influence of the cold water, it could be anticipated on the ground of Barany’s theory that in the transition from ampullo-fugal to ampullo-petal stream in the horizontal canals, there would exist a short zone in which, with a slight variation in the position of the head, a marked change in the direction of the nystagmus would manifest itself abruptly. The critical point at which neither ampullo-fugal, nor ampullo-petal streams occur in the horizontal canals, so that only streams in the vertical canals can exert an influence here, receives a full discussion below. Now when looking at the corrected figures, which illustrate the mean result of our experiments with the several rotations, the following observations can be made: a. Rotation I. Douche of the right ear. Observation right eye. With the animal in ventral position the nystagmus is anteriorly upwards. At 20° (i.e. head 20° below the Fig. 5. 723 horizontal plane) the direction is still the same; at 30° a slight deviation begins; at 50° it is much more pronounced. At 80° the direction of the nystagmus deviates as much as 135° from the initial position and at 100° the change of direction of the nystagmus of 180° has been completed. Something like this occurs between the position of 170° and 270°. Observation left eye. Fundamentally the same as right eye. b. Rotation Ml. Douche of right ear. Observation left eye: A sudden change in the direction of the nystagmus takes place here between 140° and 150°. While the nystagmus at 140° moves posteriorly-upwards, at 150° it is already anteriorly upwards. A similar marked change of direction is observed between 310° and 320°, the direction being respectively anteriorly downwards and downwards. Observation right eye: Here it is less easy to say where the change of direction takes place. Presumably also between 140° and 150° and between 300° and 330°. That in this case the curve differs from all the others may be explained by the fact that the process of the experiments averagely represented by this curve, was very irregular in two out of five cases, which could not but be of great influence on the mean curve. There was no such irregularity with the left eye of these animals (which was examined on another day). ce. Rotation LIL. Douche of the right ear. Observation left eye. Very great change of direction is found between 50° and 70° and a second change between 210° and 230°. Observation right eye. Very great change between 40° and 60° and a second between 220° and 240°. After the above facts had been ascertained, the critical point was Fig. 6. 724 determined for the various rotations, i.e. the point at which the horizontal semicircular canal has reached its optimal horizontality and consequently no or hardly any streams can exist in this canal after douching the meatus. This determination was performed with the aid of a model in wax!) formerly made of the semicircular canals of a rabbit, which contrivance was arranged, after the indi- cations of DE BurLer and Kostpr?), so as to afford an exact imitation of their natural position in the rabbit’s skull. This was to the following effect: With the animal in ventral position with horizontal mouth-fissure (Fig. 5) the level of the ampulla of the horizontal semicircular canal is higher than the canal itself, so that an ampullo-fugal endo- lymph-stream will occur on a cold-water douche of the meatus. With rotation I the horizontal canal is approximately horizontal at 40° (Fig. 6). With rotation II the horizontal canal is approximately horizontal at 150° (Fig. 7). With rotation III the horizontal canal is approximately horizontal at 57° (Fig. 8). In figures 2—4 these points are indicated with crosses. At a glance it may be seen that a marked change in the nystagmus occurs at the very place where the horizontal canal is approximately horizontal. When taking into account the considerable individual variations in the position of the semicircular canals in various animals of the same species, and when also considering the fact that our results are based upon the observation of tive different animals, while the correction for the compensatory eye positions as well as the data Fig. 7. ') H. M. pe Burver and A. pe Kreun. Ueber den Stand der Otolithen- membranen beim Kaninchen. Pfliig. Arch. Bd. 163. (1916) S. 321. *) H. M. pe Burver and J. J. J. Kosrer. Zur Bestimmung des Standes der Bogengänge und der acustica im Kaninchenschädel. Arch. f. Anatomie und Physiologie. Anatomische Abteilung. (1916) 59. 725 from the model in wax, refer to an animal that does not belong to this series, a striking resemblance can be stated between the changes of direction observed and those that could be anticipated with reference to the model. The fact, however, is that with none of the rotations I—III does the horizontal semicircular canal attain horizontality. Dr BurLer and Kosrer's researches showed that the right horizontal semicircular canal is approximately horizontal when the animal turns from the ventral position about 30° round the bi-temporal axis with the head down and at the same time round the fronto-occipital axis about 7° to 8° with the left eye downwards. We examined different animals in this position, from which it appeared that in most cases the nystagmus had not disappeared alto- gether and could neither be made to disappear by applying different Variations in the rotation round the said axes. We observed, how- ever, that the nystagmus-movements are very small in this position. Only in two cases could the nystagmus be made to disappear completely, viz. with a rotation about the bi-temporal axis of 37° in the one and 30° in the other rabbit and combined with a rota- tion about the fronto-occipital axis of 5° in both animals. This urges us to conclude that the horizontal semicircular canal plays a principal part in caloric stimulation, that, however, in most cases also the vertical canals exert some, though a small, influence. This influence, however, was not such as to enable us to make an accurate analysis of it from the curves. For a positive solution of the problem it would be necessary to determine in one and the same rabbit the nystagmus in various positions of the head in space, as well as the compensatory eye- positions in the said positions and finally through microscopic exa- mination of the labyrinth, to determine accurately the position of the semi-circular canals in that animal, after the method of Dr 3urLET and Koster. | Ji Wig. 8. 726 SUMMARY. 1. The above experimental results lend support to the theory of Barany of the origin of cold-water nystagmus. The theory of BARTELS, on the other hand, conflicts with these results. 2. In the genesis of cold-water nystagmus the cooling down of the horizontal semi-circular canal plays the principal part; however, in the majority of cases some influence (though little) is also to be assigned to the vertical semicircular canals. 9. Harlier inquiries by Magnus and pr Kiryn have demonstrated a distinct cooling of the labyrinth-walls in cats, on douching the meatus with cold water. 4. Compensatory eye-positions should be taken into account when cold-water nystagmus with various positions of the head in space is observed. Palaeontology. — ‘“‘Quelques insectes de I’ Aquitanien pr Rorr, Sept-Monts (Prusse rhénane).” By Dr. FerNAND Meunter. (Communicated by Prof. K. Martin). (Communicated at the meeting of January 31, 1920). La faunule entomologique décrite dans ce travail est assez variée. Elle fait suite a des travaux antérieurs, elle signale de nouvelles formes, complete ou rectifie, s'il y a lieu, les observations de Hnypun ou formule quelques remarques relatives aux anciennes descriptions de Germar. Dans le monde des Coléoptères, relatons des empreintes bien con- servées: Anomala tumulata Herp., beau Melolonthidae et Stenus seribai Herp., gracieux petit Staphilinidae. Une aile de Trichoptère ou Phryganien appartient au nouveau genre Ulmeriella. Parmi les insectes métaboles mentionnons la présence, a Rorr, d’intéressants hyménopteres Apides des genres Andrena et Eucera et de minuscules Terebrantia des genres Bracon et Cryptus. D'autres métaboles ne sont pas moins curieux à connaître. Citons d’abord l’empreinte et la contre- empreinte d'un frêle Mycetophilide, ou diptêre fungicole Macquart, Boletina philhydra Heryp., espece si soigneusement décrite par le paléontologiste rhénan; ensuite, un Empide, Empis melia Heryo., dont le dessin du réseau des veines des ailes (nervures) manque d’exactitude et nécessite un complément de diagnose. Si les Bibionides sont fréquents sur les schistes pu Rorr, en revanche, leur état de conservation est souvent loin d'être parfaite. Bien des formes de Germar et de HeypeN resteront vraisemblablement toujours problématiques ou pour le moins douteuses. En effet, plu- sieurs des descriptions de ces paléontologistes manquent de précision et leurs dessins sont souvent imparfaits ou fantaisistes! Protomyia veterana Herp. est une espèce bien critère, par sa petite taille et l'ensemble de ses caracteres morphologiques. Bibio heydeni n. sp. (B. pannosus? Heyp.) et Bibio germari n. sp. (B. lignarius? GurM.) sont de si bonnes espèces, pr Rorr, qu'il est possible de les étudier irés rigoureusement et de donner de bons dessins de leurs caractères les plus saillants. Les espèces de Gurmar, signalées dans son travail de 1537, ne sont données ici qu'à titre de curiosité, examen des insectes fossiles étant encore a cette époque tout-à-fait rudimentaire. On sait 728 que leur étude n’a commence a être basée sur des données rigou- reuses, et n’a pris un réel essor, que depuis les remarquables travaux de feu S. H. SCUDDER. Description des especes. 1. Nevroptera. Trichoptera. Genre Ulmeriella nov. gen. U. bauckhorni n. sp. Fig. 1. Dans des travaux antérieurs,*) j'ai décrit Phryganea ulmeri du Sannoisien d’Aix, en Provence, et Phryganea elegantula, de lAqui- tanien pe Rorr*®). La collection de Monsieur Bauckuorn, de Siegburg, renferme l’empreinte et la contre-empreinte d’un autre Trichoptere, a curieuse morphologie de la veination (nervation) des ailes. *) La nouvelle espèce, représentée seulement par une aile, mesure dix millimètres de longueur et 3 millimètres de largeur. Nervure sous-costale anastomosée aux trois quarts de la longueur du bord antérieur de Vaile, Radius simple, puis offrant deux fourches; son secteur sortant au delà du milieu de la longueur de Vaile, fourche de ce secteur plus longue que la premiere fourche du radius; nervure médiane d’abord simple, à la base de laile, ensuite longuement fourchue: la branche supérieure de cette fourche l'est aussi, l'infé- rieure est simple. Trois nervures cubitales simples et deux *) nervures anales qui le sont aussi. 1) Entomolog. Mitteil. Bd. VIL NO. 10—12. S. 198—200 u. 3 Fig.; Berlin 1918. 8) Jahrb. d. preuss. geol. Landesanstalt. Bd. XX XIX. S. 143. Taf. 10, fig. 1. Berlin 1918—19. 8) Le manque de réticulation du champ alaire et la conservation du fossile empêchent de décider avec quel genre de trichoptère le nouveau type de Rorr a le plus de rapports phylogéniques. *) Cette partie de l’aile est un peu altérée par la fossilisation. 729 2. Coleoptera. Staphylinidae. Genre Stenus, Latr. Stenus scribai*) Heyp. (Palaeontograph. Bd. XV, S. 137; Taf. 22, fig. 13). Ce Staphylien est une bonne espèce. Il a six millimetres de longueur. Tête arrondie, assez aplatie, moins large que le thorax, qui est aussi long que large. Elytres du tiers de la longueur de l’abdomen, ce dernier organe est composé de six segments. Fémurs renflés en massue, amincis à la base; tibias cylindriques, assez robustes. Les antennes, les articles tarsaux et les ailes postérieures ne sont pas représentés sur le schiste. Coll. BAvcKHorN. 1 spécimen. Melolonthidae. Rutelini. Genre Anomala Samouell. Anomala tumulata HeyYDEN. (Palaeontographica Bd. XV, S. 140; Taf. 23; fig. 18—19). Fig. 2. Fig. 3. Cette espèce a déjà été assez bien décrite par Hrypen. Je complete ici la diagnose en l’accompagnant d'une reproduction phototypique, plus précise que celle de l'auteur allemand. Tête petite et aussi large que le thorax. Antennes assez longues 1) Dans „Verhandelingen der K. Akademie van Wetenschappen van Amsterdam,” p. 3. 1917, (du tiré à part) ce mot est erronément écrit comme Scribei. Cette espèce est dédiée 4 feu Scriba. 48 Proceedings Royal Acad. Amsterdam Vol. XXII. 730 et composées de six articles: !e scape cylindrique et plus long que les autres articles suivants réunis, qui sont plus larges que longs; le dernier article sub-ovoide. Thorax (il devait étre convexe) distincte- ment plus large que long; scutellum minuscule; élytres *) recouvrant les segments de abdomen, ovoïdes et ornés d'un sillon, très distinct, longeant parallèlement, a peu de distance, leur bord antérieur. Pattes robustes, fémurs assez dilatés et un peu plus longs que les tibias; articles tarsaux antérieurs composés de 4 articles; le 1°” environ aussi long que les deux suivants réunis, le 4® plus long que le troisième; ongles des tarses courts, un peu robustes. Cavités des banches bien développées. Longueur du corps 6 mm. Empreinte et contre-empreinte Coll. BAUCKHORN. 3. Hymenoptera. Apidae. Les Apides sont rarement conservés sur les schistes aquitaniens du Rhin. V. HeypeN a signalé naguere deux espèces, assez frustes, Apis dormitans et Anthophora effosa. J'ai déerit, en 19157), Apis oligocaenica du même gisement, dont il m’a été possible de donner tous les détails de la veination des ailes antérieures. L’espece signalée brièvement, ci-dessous, me semble devoir se ranger avec les Andrénides du genre Andrena. On sait que chez les Halictes, le dernier segment dorsal de l’abdomen est orné d’un sillon longitudinal, très caractéris- tique, chez toutes les espèces de ce genre. Genre Andrena Fabr. Andrena tertiaria n.sp. c-Tête un peu plus large que le thorax. Antennes robustes, insérées en dessous du milieu de la face et composées de treize articles: le scape assez long, le funicule cylindrique et formé d’articles environ aussi longs que larges; le dernier article des antennes paraissant assez conique; mandibules robustes, larges, et échancrées a l’extrémité. Mésothorax convexe, scutellum semilunaire. Abdomen ovoide, a premier segment plus développé que les suivants; le dernier assez conique au bout. Epines des tibias tres appréciables; métatarse postérieur plus long que les articles 2—5 pris ensemble. Ailes aussi longues que labdomen, mais a veination très éffacée *) sur le schiste. 1) Ils étaient lisses, très vraisemblablement. *) Zeitschrift d. deutschen geol. Gesellschaft. Bd. 67S. 210 Taf. 21 Fig. 4; Berlin. 8) Elle devra être décrite après l’examen de spécimens, en meilleur état de conservation. 731 Longueur du corps 3 mm. Observation: Au dire de Menge, le genre Andrena a été observe dans lambre de la Baltique; je ne l’ai jamais rencontré parmi plusieurs milliers d’inclusions d’insectes . du succin: Genre Eucera Latreille. Eucera mortua u. sp. Wig. 4. Parmi les Anthophorides fossiles, on ne connait que quelques formes tertiaires du genre Anthophora de l'Aquitanien de Rorr. Von Herpen signale Anthophora effosa (Palaeontographica, Bd. X, S. 76; Taf. 10, Fig. 10). La description de cette espèce est peu précise. De Corent (France), feu E. Ousrarer donne la diagnose de Anthophora gaudryi. O. Herer et d'autres paléontologistes citent plusieurs espèces des gisements d’Oeningen et de Radoboy. Le genre Anthophora a été observé dans le sucein du Samland. A ma connaissance, le genre Kucera Latreille n'a jamais été remarqué sur les plaquettes de Rorr. Kucera mortua est une des plus récentes trouvailles de M. l’Ingénieur BAucKHORN, de Siegburg. Longueur de l'insecte 7 mm., longueur de laile 6 mm., largeur 3 mm. L'insecte est fortement écrasé sur le schiste bitumineux. Toutefois, les caractères de la veination des ailes, des pattes et des organes copulateurs sont si bien conservés qu'il est aisé de ranger, avec certitude, cette nouvelle forme d’apide dans le genre Eucera. 2-Téte robuste et paraissant aussi large que le thorax, qui était vraisemblablement enti¢rement ponctué. Pattes courtes et a tibias bien élargis pour la récolte du pollen et ornés, a leur extré- mité postérieure, de calcars très distinets; articles tarsaux robustes, surtout le métatarse, qui est environ aussi long que les articles deux a cing pris ensemble; ongles des tarses robustes, et parais- sant unidentés. Abdomen ovoide; organes copulateurs saillants et bifides à Vextrémité'). On sait que chez les Anthophora et les Eucera, les armures copulatrices des 4 fournissent de bons caractéres spécifiques pour le démembrement des espèces affines. ') La fossilisation empéche de décrive le détail de leur structure morphologique. 45” 732 Ailes antérieures offrant une cellule radiale et deux cellules cubi- tales dont la deuxième recoit les deux nervures recurrentes. Ailes postérieures non distinctes. Terebrantia. Braconidae. Genre Bracon Fabr. Bracon rottensis, Meun.: Zeitschr. d. deutsch. Geol. Gesellsch. Bd. 67, S. 224—225, Taf. XXVII, fig. 2; Berlin 1915. Fig. 5. Q-Antennes assez longues, articles cylindriques et environ 3 fois aussi longs que larges. Téte un peu plus large que le thorax et tant soit peu aplatie. Scutellum du thorax bien développé. Abdomen ovoide, les stylets de la tariere plus longs que cet organe. Pattes assez robustes (elles sont peu indiquées sur le schiste). Pour les autres caractères, voir la diagnose de 1915. Coll. BauckHorn de Siegburg. Observation. Ce Braconide s’observe, assez fréquemment, sur les plaquettes pe Rorr. La ponctuation du thorax semble avoir été comme chagrinée. Cry ptidae. Genre Cryptus Fabr. Cryptus sepultus n. sp. On ne connait que peu les Cryptides fossiles. Oswarp Herr signale une espèce douteuse des schistes d’Oeningen; MeNGE mentionne, sans les decrire, des Téré- brants de ce genre de l'ambre de la Baltique; Cyartes Bruers a observé des Cryptines sur les plaquettes miocéniques de Florissant. Je n'en ai pas remarquê dans l’ambre sicilien ni dans le Copal subfossile de Zanzibar. L’espece de la collection BauckHorn pourrait être mieux conservée; elle se classe cependant rigoureusement avec les Cryptus. Longueur de insecte 5 mm. @-Tête arrondie et aussi large que le thorax. Antennes cylindri- Fig. 6. 733 ques’) et paraissant être ornés d’articles rapprochés, comme c'est le eas chez les Cryptus GRAVENHORST. Dos du mésothorax et du méta- thorax gibbeux; ailes antérieures a nervation caractéristique des Cryptus, avec stigma très distinet et cellule radiale divisée; pas de cellule aréolaire? Ailes postérieures peu visibles. Abdomen composé de sept segments: le premier assez long, formant pétiole, un peu renflé apres sa base, le deuxième segment cupuliforme; la tarière, qui est tigelliforme, sort du cinquieme segment ventral; elle a environ la longueur des segments précédents, non compris le pétiole. Les fémurs et les tibias sont robustes, les articles tarsaux un peu grêles. 4. Diptera. Empidae. Genre Empis, LiNNÉ. Empis melia, Herpen. ‘Palaeontographica, Bd. XVII, S. 259— 260; Taf. 45, fig. 27). L’ambre renferme une intéressante faunule de dipteres de la famille des Empidae, notamment des Empis et des Rhamphomyia. Ils doivent être rares sur les schistes de Rorr car von HEYDEN ne déerit de ce gisement que Empis melia, espece qui n'a que 2'/, lignes de longueur. Le fossile, mentionné ici, a 10 millimetres de long, une longueur alaire de 8 mm. et une largeur de 3 millimetres. Je le considere comme la @ de cette espèce, l'exemplaire signalé par v. Heyprn étant vraisemblablement le ¢. On sait que ches les Empis, les mâles ont la taille beaucoup moins grande que chez les femelles. Le thorax et abdomen sont robustes. Les ailes offrent la veination si caractéristique des Empis mais imparfaitement figurée par von Hevpex. Pattes postérieures vigoureuses et courtement ciliées: les fémurs et les tibias d’égale longueur; métatarses environ aussi longs que les articles 2—5 réunis; le deuxième article a peu près aussi long que les articles trois et quatre pris ensemble, le cinquième plus court que le quatrieme; ongles des tarses paraissant grêles. Coll. Bavcknorn. Empreinte et contre-empreinte. Mycetophilidae. Genre Boletina Staeger. Boletina philhydra Hevp. Palaeontographica Bd. XVII, 5. 246; Taf. 44, fig. 11.) ) Elles sont trop altérées par la fossilisalion pour décrire le détail de leur structure. 734 2-Tête un peu aplatie et un peu plus large que le thorax. Ocelles indistinets. Antennes dépassant notablement la longueur du Fig. 7. thorax; les deux premiers articles et ceux de l'extrémité peu appré- ciables*), les autres articles cylindriques et un tiers plus longs que larges. Palpes non représentés sur le schiste. Thorax un peu gibbeux et orné, aux côtés latéraux, de rares cils, écusson garni au bout de deux cils, assez longs. Le thorax devait Être pourvu de trois bandes ou fascies de teinte plus sombre que le restant du thorax. Ailes plus longues que l'abdomen. Nervule assistante réunie au bord costal un peu avant le dessus de lextrémité de la cellule humérale. Bord costal alaire peu prolongé apres le cubitus (radius sec. Comstock and Needham). Pétiole de la fourche discoidale (médiane) assez long, fourche posticale (Cubitale) distinctement plus longue que la discoidale: Les deux nervures anales sont peu accusées. Abdomen de six? segments, finement ornés de cils courts et munis, a Vextrémité de chaque segment, d’ une large bande de teinte sombre; bout de l’abdomen (oviducte) assez effilé*). Parties externes des tibias ornées de rares cils espacés; calcars assez longs, surtout les postérieurs. Articles tarsaux de la troisième paire de pattes longs, métatarse de cette paire plus long que les articles 2— 5 réunis. Coll. BauckHorn. Empreinte et contre-empreinte. ? inconnu. Bibionidae. Genre Bibio Linné. Bibio germari n. sp. Bibio lignarius? Gurmar. Bibio lignarius? HwypeN. L’espece décrite par GERMAR, comme B. lignarius, est très problé- matique. Il en est de même de la fig. 23 des „Insecta carbonum 1) On ne peut compter exactement le nombre de leurs articles. *) Les lamelles ne sont pas représentées sur le schiste. 735 fossilium” et de celle de Hryprn ,,Palaeontographica Bd. VIIT.S..14 Taf. I fig. 4”. La nouvelle forme, dont la diagnose suit, est représentée Fig. 8. par lempreinte et la contre-empreinte, d'une conservation remarquable. Ce Bibionide mesure 12 mm. de longueur, l’aile a 9 mm. de long et 3'/, mm. de large. 2-Tete assez grande, orbiculaire et un pen moins large que le thorax. Yenx bien séparés sur le front. Pipette robuste et ornée de cils courts. Cou tres appréciable. Thorax assez gibbeux. Abdomen largement ovoïde, de sept segments, dont les côtés sont trés distinctement garnis de poils courts; dernier segment échaneré a la partie centrale ; lamelles de loviducte eylindriques. Ailes assez larges (elles devaient être assez enfumées*), la sous-costale plus rapprochée de la nervure radiale que du bord costal; Je secteur du radius, qui part de la radiale avant le milieu de la longueur de l'aile, n’atteint pas l’apex de cet organe. Une petite nervule transversale velie le secteur du radius a la médiane. Cette dernière longuement fourchue; fourche de la nervure cubitale partant a peu de distance de la base de l’aile qui est pourvue de „Flügellappen’” ou lobes alaires. L’aile paraît avoir deux nervures anales. Les pattes, peu représentées sur le schiste, sont ornées de courts cils. Coll. BAUvCKHORN. 4 Ineonnu. s1bio heydeni n.sp. Cette espèce correspond peut-être a B, pannosus, forme de Bibionidae tres imparfaitement déerite et figurée par GumMmar. L’exemplaire de la collection Bavucknorn, d'une conservation remarquable, permet d'en donner une diagnose plus précise. —, ') Elles sont très foncées chez B, infumatus Meun. et aussi plus longues et plus larges. 736 ¢. Ce Bibionide a 10 mm. de longueur, l’aile mesure 10 mm. de long et 3 mm. de large. Les ailes sont un peu enfumées; la sous costale courtparalélement à la costale et se réunit a cette dernière avant le milieu de la longueur de l'aile. Le radius s’anastomosant aussi au bord costal, a peu de distance de la sous-costale; secteur du radius un peu convexe et n’atteignant pas lapex de Vaile; une nervule transversale oblique, dirigée vers le bout de |’aile, réunit la nervure radiale a son secteur; nervure médiane fourchue, la nervure cubitale a la base de la fourche rapprochée de la base de Vaile; une nervule transversale oblique. dirigée vers la base de l’aile, relie la base de la fourche médiane à la branche supérieure de la fourche cubitale; il y avait proba blement 2 nervures anales rapprochées. Abdomen cylindrique, assez large (il a 3 mm.), de sept segments bien arrondis aux bords latéraux et paraissant ne pas étre ornés de cils, comme c’est le cas chez B. germari; Lamelles de l’oviducte petites, cylindriques. gd Inconnu. Protomyia veterana Hyp. (Meun.) (Palaeontographica Bd. XIV, S. 25, Taf. 8, fig. 4). Par sa petite taille et sa forme trapue, cette espece est bien reconnaissable. Longueur de l’insecte 4'/, mm., l’aile mesure 5 mm. de long et 2 mm. de large. 9-Téte arrondie et aussi large que le thorax. Yeux bien saillants; abdomen ovoide et composé de sept segments. Ailes notablement plus longues que abdomen, assez larges; nervure sous-costale ana- stomosée au bord costal, un peu au dela du milieu de sa longueur. La distance entre Sc. et Ra. plus courte que celle entre Ra. et Ra,. (secteur du radius). Ce dernier n’atteignant pas Vapex de Vaile. Pétiole de la fourche médiane environ aussi long, que la nervule unissant le secteur du radius a la nervure médiane (discoïdale). Fourche cubitale distinctement plus longue que la médiane. Dessous du dernier segment ventral comme incisé au centre. Pattes assez robustes. Coll. BAvCKHORN. 1 spécimen g Inconnu. — Quelques types de GERMAR. — L’Institut paléontologique de Université de Bonn possède quelques types du paléoentomologiste de Halle. EF. MEUNIER: Quelques insectes de 'Aquitanien de Rott, Sept Monts (Prusse-rhénane). GE Fig, 5 ¢rdinand Bastin, phot roceedings Royal Acad Amsterdam Vol. XXII HELIOTYP B, VAN Lures AMUTERDAM ad _2 = ss... eS a ekeren en TO ee a —— ——— Si 2 EF. MEUNIER: Quelques insectes de l'Aquitanien de Rott, Sept Monts (Prusse rhénane). Prijs Ferdinand Bastin, phot Proceedings Royal Acad. Amsterdam Vol. XXII NELIOTYP B, VAN LEKIE, AMBTLIEDAM i rd} 737 Ce sont les espèces suivantes: Buprestis carbonum, B. major, Ydsolophus insignis, Prionites umbrinus, Tenebrio effosus, Saperda lata, Silpha striatum, Alydus pristinus, Bibio xylophilus, Locusta exstincta. Sous Vinfluence des actions chimiques prolongées et de Pair, ces fossiles sont devenus trop frustes pour les décrire et pour en donner de bonnes reproductions phototy piques. EXPLIGATION DES FIGURES. 1) (Texte). Fig. 1. Aile antérieure de Ulmeriella bauckhorni nov. gen. n. sp. Fig. 2. Antenne de Anomala tumulata Heyd. Fig. 3. Articles tarsaux de ce Melolonthidae. Fig. 4. Aile de Eucera mortua n. sp. Fig. 5. Antenne de Bracon rottensis Meun. Fig. 6. Abdomen de Cryptus sepultus n. sp. Fig. 7. Aile de Boletina philhydra v. Heyd. (Meun.) Fig. 8. Aile de Bibio germari n. sp. EXPLICATION DES PLANCHES. *) Fig. 1. Ulmeriella bauckhorni nov. gen. n. sp. Fig. 2. Stenus scribai Heyden. Fig. 3. Anomala tumulata Heyden. Fig. 4. Andrena tertiaria n. sp. Fig. 5. Eucera mortua n. sp. Fig. 6. Bracon rottensis Meun. 2 Fig. 7. Cryptus sepultus n. sp. Fig. 8. Empis melia Heyden. 2 Fig. 9. Boletina philhydra n. sp. Fig. 10. Bibio germari n. sp. Fig. 11. Bibio heydeni n. sp. Fig. 12. Protomyia veterana Heyden. 1) Elles ont été faites par Mme |. Meunier. *) Les clichés ont élé exécutés, avec soin, par mon ami M,F. Bastin d'Anvers. Zoology. — “The wing-design of Chaerocampinae’. By Prof. J. F. VAN BEMMELEN. (Communicated at the meeting of October 25, 1919). In a monograph, which is now being published as a supplement to Zeitschrift für Wissenschaftliche Insectenbiologie von Car. SCHRÖDER in Husum, and of which I received the first part a few months ago, Dr. P. Denso, the author of Palaearctic Sphingides in Skitz’ Macrolepidoptera, begins a description of the lepidopterous hybrids that have hitherto got known, with considerations on the wing- design of the species of Celerio. On page 1 he says about this: “Thorough investigations and theoretical considerations, which it would lead me too far astray to reconsider here, clearly show that the markings (and hues) of all Celerio-moths may easily and without constraint be derived from a primitive form, which only very slightly deviates from the pattern still found in the oldest species of Celerio, viz. zygophylli O., or likewise in lineata, when we only abstract from the white striation of the wing-veins. It must be mentioned here, that the original design of the species of Ce/erio is nearly related to that of the more closely-connected species of Pergesa”. I deplore that Denso did ‘not think fit to publish in detail his “thorough investigations and theoretical considerations” on the phy- logenetic interrelations between the different species of Celerio. For now we are obliged to deduce the grounds for his assertion “that zygophyll and lineata have to be considered as the (phylogene- tically) oldest species’ from a few remarks, which must be picked up here and there in the course of his paper. Such being the case, I prefer first to expose my Own views inde- pendently of Denso’s considerations and afterwards to discuss his deductions. In my eyes the only way to acquire a trustworthy insight into the wing-design of Celerio-species, is to compare it with that of other genera of Sphingids, especially Chaerocampinae. When keeping this course, it becomes evident that their colour-pattern is a highly modified variation of the general ground-design of Heterocera-wings, due to reduction and obliteration of the general primitive set of seven transverse bars, by the influence of the V-diagonal-motive (this being the name which in my foregoing paper on the wing- pattern of Saturnidae [ gave to the system of linear markings running 739 obliquely across the wing from tip to root). Consequently in my opinion the most original pattern must be looked for in those Chaerocam- pinae that show the fewest traces of this influence of the V-diagonal on the transverse bars. Now it is evident that this does not at all occur in zygophylli and lineata, but on the contrary in Pergesa (Deilephila, Metopsilus) porcellus, and better still in Berutana (Metop- silus) syriaca. In this latter the forewing shows a set of transverse bars which remarkably agree with that of Smerinthus populi, though they do not to any notable degree pass over upon the hindwing. The V-diagonal is only very slightly indicated at the apex of the wing in the shape of the foremost external triagonal spot, which extends from the wing-tip along the front-border, and shows the form of a dark-brown threesided blotch, growing fainter and of lighter hue from before backward. It remains separated from the convex blotch along the external margin by a narrow space, which is occupied by the well-known oblique white apical stripe, that is seen in so many different forms of Lepidoptera. In the same way the posterior triagonal spot is well-developed, but remains separated from the anterior one by three internervural Spaces, containing only faint traces of dark marginal spots. Bar Il is complete and well-marked, III on the contrary hardly visible, IV is a broad dark band, imperceptibly passing at its external side into the area where III] would have occurred, had it been visible, but very sharply traced at its internal border. “V is rather sharp, but does not reach the back-margin of the wing. VI is just indicated by a faint trace, VII on the contrary is invisible. On the upper side of the hindwing a broad marginal seam and an obscuration of the root-field are the only traces of the pattern. On the inferior surface the common heterocerous pattern occurs, viz. a design which is the same for front- and backwing, and betrays clear traces of reduction, when it is compared to that of the upper-side. For it only consists of a well-defined marginal range of (coalesced) spots, inwardly bordered by an irregular zig-zag-line, and moreover of the bars Il and III, represented by brown lines on a lighter ground. The wing-design of porcellus may rather easily be derived from that of syriaca, and this deduction presents a certain amount of probability, as the complete set of seven transverse bars is clearly discernible along the front-margin of the forewing. Three of these bars: the outward or distal ones (1, Il and IIL) reach the hind-margin. When I call this pattern an original one, this expression should not be taken in the absolute sense generally connected with it. 740 Precisely in the case of porcellus, it can be proved in a very striking way, that this would be inappropriate. For not only the pattern of the seven transverse bars is repre- sented on the fore-wing, but that of the V-diagonal as well. Or ex- pressing it in other words, we may assert that the pattern of porcellus could be obtained by combination of that of Smerinthus populi with that of H/penor gall and euphorbiae, of course under omission or reduction of certain parts of each. The best proof for this assertion can be given by superadding the wing-patterns of the above-mentioned species to that of porcellus, or, otherwise, by marking with a darker hue those elements of foreign wing-designs in the porcellus-pattern that can be discovered in it. Bar I, otherwise called the marginal seam, shows in porcellus the usual type of an irregularly indentated, wine-red streak, which is characteristic of Chaerocampinae. It begins at the wing-tip with the above-mentioned oblique white stripe, which likewise is of so frequent occurrence among Sphingides, and can be considered as the outmost fragment of the V-diagonal. Bar II begins with a tolerably distinct, rather dark blotch, in the wine-red streak along the front-margin, but gets much fainter as soon as it enters the yellow-brown central area of the wing, which it traverses in a well-marked inward curve. Bar III likewise begins at the front-margin with a double-blotch, but becomes a single band when entering the yellow area, and at the same time gets into contact with the discoidal spot, which itself may be considered as a remnant of Bar IV. Furtheron B. III runs parallel to II, both being dislocated a little in the direction of the wing-root. V and VI are represented by a pair of small, greenish-brown stripes in the red field, VII can confusedly be traced in the brownish root-area. On the upper side of the hindwing the only point of similarity with the forewing is formed by the wine-red marginal seam, but the underside once more proves, that also in porcellus well-marked remnants of the transverse bars occur in the shape of dark spots and stripes on the light-yellow and rose-red fond. As in most other cases these are especially well-marked along the front-margin. At the underside the similarity between fore- and hindwing is again much more pronounced than on the opposite surface, the design on the firstnamed wing being more reduced than on the last, especially as regards the root-field. The same red, violet and greenish-golden-brown hues that decorate 741 porcellus, are found back in elpenor, a great superficial similarity resulting from this, which finds its expression in the popular names. But in tbe pattern an important difference prevails, for in elpenor the traces of transverse bars along the front-margin are almost completely absent, while on the contrary the V-diagonal-design is strongly expressed, though in factit reaches the back-margin of the wing in its more distal part, and therefore deviates in a lower degree from the original transverse direction of the primary bars than is the case with other Chaerocampinae. Fig. 1. (after Denso). Denso pays special attention to this vicariating relation between the two parts of the back-margin of different species of Celerio, into which it is divided by the above-mentioned oblique line, which, starting from the wing-tip, forms the outer border of the light middle- field. He arranges these species in a series, beginning with lineata, where the meeting-point of this line with the back-margin lies farthest towards the proximal side, and ending with ntcaea, which in its more distal position of this point more or less agrees with elpenor. As far as 1 feel able to understand his views, he seems, for the just-mentioned reason, to consider lineata as more original than nicaen. According to my conviction the relation between these two species is precisely the opposite one. To me it just seems remarkable that Denso, when speaking of another detail of the wing-design, which he remarked in a few 742 specimens of nicaea only, comes to a conclusion that exactly agrees with my views. For Denso considers the occurrence of a dark line over the middle part of the wing, which appears from time to time (called by him fa, and running parallel to his median bar am) as an atavistic phenomenon. Now this line can scarcely be anything else than Bar III of porcellus, and therefore in my opinion may really be considered as the reappearance of an element of the original design. In truth this unexplainable confusion and contradiction in his views can be remarked in different passages of Drnso’s contentions: e.g. when he says in describing the /meata-design : “my investigations led me to assume, that /ineata and certain specimens of zygo- phylli show a design, that very nearly approaches tbe original Celerio-pattern. In fact, when drawing the contours of the lineata- design, they completely include the elements of the pattern of all remaining species of Celerio, these latter therefore appearing to be due to the more or less far-reaching reduction of the original design. This may be demonstrated by Fig. 2”. Fig. 2. (Copied after Denso). Now in this figure we remark, how lineata, besides showing the V-diagonal, possesses only a single vestige of a bar (1) along the external wing-margin, and therefore next to nothing of the original design, while in ewphorbiae on the contrary the remains of at least four transversal bars are present along the front-margin, though in truth only in the shape of isolated blotches. Were we obliged to share Denso’s views, we should have to assume that self-colour is 743 the more primitive condition of wing-coloration, all patterns taking their origin from it by dissociation of the homogeneous hue into spots and bars. This really seems Denso’s opinion, notwithstanding a few lines before he asserts: “As to the underside of the wings, we also here find, that progression in phylogenetic development always goes hand in hand with an increasing loss of elements of the pattern. Lineata is richest in details, galli? less so, zygophylli the same, while euphorbiae and nicaea show the fewest components of the pattern”. I see no need here to remonstrate that this assertion can as well be applied to the upper side of forms like C. lineata and D. elpenor, in comparison respectively with C. eupkorbiae and D. porcellus. Neither can I agree with Dunso’s contentions (p.5) about the ‘‘manifes- ny | iy Fig. 3. (after Denso). tation of atavistie characters’. He writes: “Very often we remark in pure species, e.g. gallii or euphorbiae, a dark, indistinct line, starting at the wing-tip near to the transverse bar p, and running parallel to the distal border, across the marginal field. In most cases this line is rather short, and disappears nearly halfway between apex and hind-corner; rarely it attains this corner and there joins with the bar p. It takes exactly the same course as does the distal bordering of bar p in lineata. Without doubt we here meet with an atavistic feature; it is nothing but the old borderline of the p-bar. In vain therefore should we look out for it in the lineata- group, while in zygophylli it will only occur on rare occasions and in a weak condition, as the regression of p has only just begun. Gallit often shows this line, euphorbiae more rarely, nicaea extremely 744 seldom, which can be easily understood, as the two latter, being relatively young species, have already long since lost this border- line of p”. Comparing the wing-designs of gallii, zygophylli and livornica (as figured in Seitz, comp. vol. II Taf. 41. d.) I come to the con- clusion that in the second of these species the line in question, fa, is present without exception, but that it has been dislocated a little towards the internal side, and moreover that this line is also present in Pergesa oldenhamii and japonica, and likewise in Celerio bois- duvalii and minor, though its shade may differ in saturation. For the rest I cannot well understand, why precisely this line should be of special atavistic importance; though on the other side it is of course beyond discussion that it has originated by the coalescence of the external row of spots (Bar I), which runs parallel to and in the immediate neighbourhood of the external margin of the wings. Nor am | able to see why there still should exist differences in the degree of atavism between the several transverse striae, which, according to Denso occur from time to time as variations in the different species. Denso himself seems inclined to accept this difference, for he says: “In contrast to the line fa, which forms a feature restricted to forms within the limits of the genus Ce/erto, another atavistic line goes back to a far wider plan, viz. to elements of design also appearing in the genus Pergesa. I mean a dark line fa,, which, beginning at the costal spot mc,, runs along the costal zone ac and parallel to the proximal margin of p, towards the posterior wing- border. Very often this line forms a connecting link between the spots mc, and mce,. It only occurs in specimens, where the tendency to dissolution of the costal zone into separate costal-spots shows itself, or in which this dissolution has already been achieved, e.g. zygophylli, vespertilio, the euphorbiae-group and nicaea. Never on the contrary does it appear in Aippophaes, gallii and lineata”. Judging from vespertilio, when compared with askoldensis and mellus, the line in discussion must be the one I designed as Bar III, but which here must have blended with IV, traces of this line being present not only in some, but in all specimens of dahli and euphorbiae, near to the posterior margin of the wing. In accordance with these remarks, it is self-evident that my views about the wing-markings of the ewphorbiae-group are absolutely in contradiction with those of Denso. For this author says: “In C. euphorbiae L. the process of dissolution of the original Celerio-design has proceeded very far already”. 745 I feel convinced that in this case a process of dissolution is out of the question, but that quite on the contrary we can still discover the last traces of the transversal bars along the front-margin of the wing, in the shape of isolated spots, the posterior and distal part of the wing meanwhile remaining under the dominion of the J’- diagonal-pattern. Yet I am willing to admit, that the reduction of the transverse rows of spots to three or four irregular blotches along the front-margin (called by Denso costal-spots) and the gradual diminution in size of these spots towards the wing-tip undoubtedly are in connection with the course of the V-diagonal, and that the entire set of these three or four blotches responds to the dark anterior marginal field of C. lineata. This latter area however I consider as a blending of those four blotches, i.e. as partial self- coloration, leading to uniformity of hue of the whole anterior marginal field. The justification for this way of regarding the question, I see in conditions as found in C. gallii, where the blotches, though in connection with each other, in such a way that the front border of the wing is entirely and uniformly dark-coloured, yet are perfectly distinct in their original extension by the occurrence of arcuate incisions from the side of the light diagonal middle V-bar. That Denso looks at this condition from an opposite point of view is revealed by his expression: “Galli possesses a broad costal margin, in which the (light) groundcolour has intruded, (the italics are mine), especially from three points of the middle-area am’’. The same considerations can be applied to the dark triangular area, which forms the postero-external border of the light diagonal bar, and which Denso calls p. When speaking of C. euphorbiae, he remarks about this bar: “The proximal limit of the transversal bar p, in its hinder part, which touches the back margin of the wing, has been removed towards the posterior wing-angle”’. According to my view, it has remained at its original place. Though his remarks about gallii are restricted to the words: “The bar p is broader than in euphorbiae. Its terminal point P is situated more towards the base of the wing”, he declares in a preceding passage: “Starting from the distal border” (of the light median area) “the marginal coloration of al increases in extension at the cost of p, and moreover am broadens along the posterior wing-margin, thus causing the proximal limit of p to stand more perpendicularly to that margin’. Also in this regard therefore, Dunso's views are diametrically opposed to mine. And yet I could see a possibility that Dunso’s view of the matter 49 Proceedings Royal Acad. Amsterdam. Vol XXII. -4 46 might after all prove right. For this would be the case, when we had to surmise, that in gall, and still more in euphorbiae, the presence of the spots along the anterior wing-margin was due to reversion of the archaic pattern, i.e. to atavism. We then should be obliged to imagine that in the pattern of lineata, the uniform dark anterior region of the diagonal-pattern, itself derived from the coalescence of the anterior parts of the original seven transversal rows of spots, had again been solved into a certain number of free blotches. The fact that this number is lower than seven, renders some probalility to the supposition that we have here to do with a secondary dissociation of an originally coherent longitudinal bar along the entire anterior wing-border. But according to my view the primary cause of this dissociation may be seen in the hereditary presence of the tendency to the formation of isolated marginal spots, belonging to the ancient pattern of transverse rows of maculae, which is common to all Heterocera. When trying to analyse in this same way the complicated pattern of the upper-side of the forewings of Deilephila (Daphnis) nerit, we come to the conclusion that without constraint derivatives of all the seven transversal bands can be recognized in the alternately dark and light areas along the anterior wing-margin, but that only one of them, viz. V, runs on unbroken to the posterior margin, VI nearly doing as much, as it only becomes crossed by the white external seam of the dark root-field. The disturbances in the rest of the transverse bars may for the greater part be attributed to the well- known influence of the V- and the A-diagonal-design. The first manifests itself in the same manner as in euphorbrae, gallü etc, but in nerw only fragments of the light median bar of the remaining Deilephilas can be discovered. In the first place we remark the light apical marking, strongly contrasting to the extremely dark anterior segment of bar I. Then comes the white curved stripe in the middle of the posterior margin, abutting towards the median side against a peculiarly dark hinder part of a transverse bar (probably a fragment of IV) and which in its forward zig-zag-course gets twice abruptly broken. I presume that this characteristic white zig-zag-line represents part of the distal border of the triangular light central part, which broadens towards the hind margin and is so characteristic of Chaerocampinae. For the rest this light central field is only represented next to the root-field by its most proximal part running along the posterior wing-border. This part narrows and describes a convex curve, thereby passing into the area of bar VI, and reaching the anterior margin. In the same way the light colour-party at the external border of V 747 advances distally towards the anterior wing-margin, and so comes_ in contact with the fore-end of a still lighter bar, which begins in the area of III at the said margin, but takes such a sinuous course in a postero-external direction, that the dominion of III so to say curves up to that of Il. A similar feature can likewise be observed in another Sphingid, whose forewing-pattern agrees with that of nerd? in a remarkable number of points, viz. Dillina tiliae. Here the feature in discussion is seen in the anterior part of the external border-line of the dark central field, by which the forewing is so characteristically divided into a proximal and a distal light area, and which itself is broken up (either completely or nearly so) into a larger anterior and a smaller posterior portion by a constriction along the course of the second cubital vein. This constriction corresponds in position and character to the above-mentioned white zig-zag-line of nervi. That this explanation of the forewing-pattern of nerü is well founded, becomes especially evident when we compare it to that of nearly-related species, e.g. Aypothous (Moorn, Lepidoptera Ceylon, Pl. 83; Cramer, Pap. Exot. III pl. 285 D; Seitz, X 63a), layardi (Moore, Pl. 81; Seitz 63a‘), protrudens (Novara Exp. Zool. Il, 2, Taf. LXXVI, 7; Seitz X, 63b®), angustans (Nov. Exp. Zool. Bd. II 2), placida (Seitz, X 63a‘). But as the most remarkable patterns in regard to this feature I consider those of omissa and its congeners (miskini, anceus, sericeus, cunera), because here parts of the nerii-pattern are so to say pro- jected on that of Smerinthus populi, the latter appearing as if it were visible by transparency beneath the first. Groningen, October 1919. 49* Physiology. — “On Serum-lipochrome’. (First part). By Prof. A. A. HijMANS VAN DEN Beren and Dr. P. Murren. (Communicated at the meeting at December 27, 1919). In a previous investigation') I gave evidence to show that the normal human blood-serum contains two pigments: bilirubin and a lipochrome. Prior to this, opinions about the materials that yield the colour of normal serum, were contradictory and confused. The French clinician GirBeRT e.g. believed that the colour of the human bloodserum was due exclusively to bile-pigment and that it never - contained lutein (lipochrome). The Italian researcher Zoya, on the other hand, asserted that bilirubin is never present in the serum of normal man, but that the yellow colour is owing to lutein. We suspect these clashing opinions to have arisen from unsuitable methods of separating the pigments. It is especially the extraction of a protein- rich fluid like blood-serum, by shaking with ether and similar solvents, that yields unsatisfactory and differing results. When, however, we precipitate the serum by an appropriate amount of alcohol most of the bilirubin will pass over into this fluid, while from the ensuing protein-precipitate the lipochrome can be readily extracted with ether. In this way we are enabled to separate both pigments from the serum. After having watched the fate of bilirubin under various circumstances*), an inquiry on lipochrome naturally suggested itself to us. Yellow pigments, which for the present may conveniently be termed “lipochromes”, have until recently been investigated chiefly by botanists*). Srokges*) and SorBy®) discovered that in the green parts of plants besides chlorophyl numerous yellow pigments are to be found. Prior to their findings carotin had already been separated from 1) HiuMANs VAN DEN Beren u. Snapper. Deutsch. Arch. f. klin. Mediz. 110, 540, 1913. 2) HiuyMANS vAN DEN Beren. Der Gallenfarbstoff im Blute. Leiden 1914. 5) For the literature see T. Tammes. Flora 87, 205, 1900, and C. v. WisseLinen, Flora 107, 371, 1915. *) Srokes. Proc. Roy. Soc. 18, 144, 1864. 5) Sorsy. Ibid, 21, 442, 1873. 749 Daucus carota, while ArNnaup showed in 18857) that a yellow pigment in the green parts of plants is identical with the carotin from carrots. ARNAUD made an extensive study of carotin and esta- blished that it is an unsaturated autoxydable carbohydrate. His analysis and further inquiries produced the empiric formula C,,H,,. Ever since many inquiries into these pigments have been undertaken. Latterly they have received Wu.rsrätrer’s *) attention. He established in accordance with the assumption of previous inquirers that plants contain different pigments which — as Boropin*) had observed — may be divided into two large groups. The pigments of the first group, to which carotin belongs, are rather easy to dissolve in benzene, hardly so in alcohol. The second group is represented by xantho- phyll, which can readily be dissolved in alcohol, less readily in benzene. Either of these substances could be obtained in pure, erystal- line condition. The elementary analysis, the determinations of mole- cular weight, and the analysis of iodine-addition products yielded the formula C,,H,, for carotin and C,,H,,O, for xanthophyll. WitisTarrer also corroborated that the two carbohydrates are highly unsaturated and autoxydable. They are very sensitive to acids, but are not attacked by alkali. Also in animal produets, particularly in the egg-yolk, in the serum of animals and men, former observers have found lipochrome pigments (KRUKENBERG, THupicum, SCHUNCK, Kürne) and have published interesting communications about them. They usually term them luteins. To Witistatter and his co-workers we are indebted for considerable advance in this respect. It appeared that also the animal lipochromes or carotinoids may be divided into two groups according to their relative solubility in benzene and alcohol. Winistarter’s pupil Escuer managed to separate pure carotin from the corpus luteum of the cow. WursrÄärrer, in conjunction with Escner, has obtained lutein from the egg-yolk, and established that it is quite identical with the xanthophyll from plants, with a difference only in the melting-point. Some years ago a series of papers appeared from the American Algricultural-Chemist Patmer*). We shall frequently refer to this work, but it may be expedient to state here what we deem to be the chief result of Parmer’s work. The American researcher comes 1) A. Anmnaup. C. R. Ac. Sc. 100, 751, 1885, and 102, 1119, 1886. %) Wittstitrern u. Srou, Untersuch. über Chlorophyll. Berlin 19138. 5) Quoted from Wittstitten. *) Patmen, The Journ. of biolog. Chem. 1915—1919. 750 e.g. to the conclusion that the yellow pigment of the body-fat, milk- fat, and blood-serum of the cow is identical with carotin, whereas the yellow colouring matter of the egg-yolk, body-fat and blood- serum of fowls corresponds with xanthophyll. He also demonstrated that these pigments in animals are of alimentary origin. Finally that in the cow’s intestine carotin is resorbed well nigh exclusively, whereas in the fowl’s intestinal canal only xanthophyll is resorbed almost to the exclusion of other pigments. Our prolonged investigation of lipochrome in the serum of the human blood led us to study some of its qualities more in detail. As already stated in the paper referred to above’), we had observed that the lipochrome pigments behave differently in man and in the cow towards ethylalcohol. When we precipitate cow’s serum with 2 vol. of alcohol and when centrifugalizing the preci- pitate, the lipochrome can be extracted from it with ether. The cow’s pigment, then, is nearly soluble in 64 perc. alcohol. When we submit human serum to the same process, we generally fail to extract pigment with ether from the protein precipitate; it can be obtained when we precipitate 1 vol. of human serum by an equal volume of alcohol. It appears then that the human lipochrome is often soluble in 64°/,, but invariably insoluble in 48 perc. of alcohol. This different behaviour of lipocbromes towards 64°/, alcohol we purposed to examine. First of all we ascertained in which of the two groups of caro- tinoids, as established by Wuttistarrer, the pigments from carrots, ege-yolk, fowl’s serum, cow’s serum and human serum, have to be placed. The materials to be examined are treated with 96 °/, ethylalcohol ; subsequently with petroleum-ether. By adding an appropriate quantity of water all the lipochrome passes over to the petroleum-ether, which floats on the surface as a limpid, gold-yellow layer. This layer is pipetted off. It contains besides the pigment, also fats, cholesterin and presumably still other substances. The fats are removed by saponification, the cholesterin is precipitated by digitonin. What remains is an incompletely purified solution of lipochrome in petroleum-ether. When adding to this fluid methylaleohol (90°/, or stronger) the pigment will pass over to the lower methylaleohol layer, if we have to do with xanthophyll, to the benzene layer in the case of carotin. Following WittstArtrer’s example we always used methyl-alcohol for this process; ethylalcohol proved to be ‘) Deutsch. Arch. f. klin. Mediz., loc. cit. 751 unserviceable. Neither did we deem it suitable to distinguish between the two groups of carotinoids by their spectroscopic properties. The same holds in Tswerr’s method. He filters solutions of the pigment through a column of calcium-carbonate. The carotin will then pass through without being adsorbed, while the xanthophyll is left behind. Tswett applies this method to separate the different sorts of xantho- phyll, that according to him exist. However, since we only look for a separation between the two main groups, we have confined ourselves to the method of distribution between methylalcohol and benzene. It was apparent from our results, as Parmer had already shown, that cow’s serum contained only carotin *) egg-yolk and fowl’s serum only xanthophyll. Human serum yielded results varying with the individual from which it was drawn. It usually contained a mixture of ecarotin and xanthophyll, carotin mostly preponderating. In only one case xanthophyll predominated slightly; in a few cases — very rare though — the amounts of xanthophyll and carotin were nearly equal. Not unfrequently did we find that along with carotin there was only very little xanthophyll. In order to determine the solubility in 64 perc. aleohol we have extracted carrots, blood-serum and egg-yolk with ether after treat- ment with aleobol. The ether was pipetted off and evaporated to dryness in fraction- ating flasks, in vacuo, at room-temperature (or gentle heating on a waterbath). Subsequently 64 perc. ethylalcohol was added in each of the flasks and shaken up rapidly. The colour of the fluid was taken for the index of solubility. It then appeared: Colour of 64 pere. alcohol. Carrots. . . . . +—+4++4+ + Beeyvolke ae fend Fowl’s serum. . + + + Cow’s serum. . . faint Human serum . . + + or + + + This makes it clear that whereas carotin obtained from carrots is easily soluble in 64 perc. ethylaleohol, cow’s serum is almost insoluble. The pigment from human serum (a mixture of carotin and xan- thophyll, carotin most) is sparingly soluble. h Parmer rightly observes that also some very small quantities of xanthophyll may occur, which will come forth only when working with large quantities of serum. 752 What strikes us most is the different behaviour of the pigment of egg-yolk and of fowl’s serum (xanthophyll), the former being insoluble, the latter readily soluble. These properties are no doubt partly due to the presence of sub- stances accompanying the pigments. When purifying the egg-yolk- xanthophyll, which is almost insoluble in 64 pere. alcohol, by saponi- fying and removing the fats in the ether-solution, the solubility increases. Something like it, but in a smaller degree, was witnessed in cow’s serum. From this it is evident that some properties of lipochromes are markedly influenced by the presence of other sub- stances. We have to keep this in mind when studying the lipochromes of blood-serum and other human products, since in this case it is impossible to examine them in a pure state. This will be easily understood when considering that in clinical inquiries the investigator has seldom more than a few cubic centimeters at his disposal, while Wirrsrärrer and Escurr used 6000 eggs to prepare 2.6 grms. of pure yolkpigment and Escuer required 10.000 cow-ovaries to produce 0.45 grm. of carotin. As has been stated above our first investigation showed us that in order to prepare lipochrome from bloodserum it is necessary to precipitate it with alcohol and after this to extract the precipitate with ether. Extraction by shaking the serum with different solvents yielded varying and generally bad results. KrvKENBERG had also noticed that lipochrome can be extracted from cow’s serum only with amylacohol. He insists that other means of extraction such as chloroform, ether, methyl-, ethylalcohol are not suitable. A more extensive inquiry in this direction revealed that no trace of pigment could ever be obtained with petroleum ether from cow’s serum, human serum or fowl’s serum. With ether we most often obtained no pigment from these three sera, at other times only little. This result does not quite tally with the experience of PArmrr, who also gave his attention to this point. He records that from cow’s blood the pigment can never be extracted with ether, from fowl’s blood always. To what this difference is due, we have not been able to make out. Anyhow, it is certain that 8 specimens of fowl’s serum, examined by us, did not yield a pigment even after being rapidly shaken with pure ether; whereas from two other specimens, treated in a similar way, a rather considerable amount of pigment could be obtained. When shaking cow’s serum with amylalcohol, a trace of pigment 753 is transmitted to the aleohol. At the same time, however, the lower- most layer is far more decolourized than the amount of pigment, which has passed over to the amylalcohol, conld lead us to expect. Chloroform does not extract pigment from cow’s serum. On the other hand the pigment can be easily extracted from egg- yolk with ether. From which it may be concluded, that in fowl’s blood-serum pigment occurs in a condition or in a combination different from the pigment in the egg. To benzene again the egg-yolk yields nothing. When we boil the yolk, a considerable amount can be extracted with benzene. From carrots a pigment can be readily obtained with ether, with henzene and likewise with alcohol. Pigment may be extracted from finely ground maize with ether. More easily when the maize is boiled with alcohol. It is well to watch the behaviour of cow’s serum towards ether and benzene. By shaking it with one of the two solvents, the pigment cannot be liberated. When, however, we add ethylaleohol to the serum and then ether, and subsequently separate it by a small. quantity of water, the pigment will pass quantitatively to the upper (ether) layer. We have pointed out before that this behaviour of cow’s serum had also struck Parver. He believed (although we could not corroborate it) that fowl’s serum invariably yields its pigment to ether. This induced Parmer to assume that fowl’s lipochrome occurs in free state in tne serum, whereas in cow’s serum the lipochrome is present in combi- nation with albumen. Of this compound termed by him caroto- albumen he has endeavoured to establish some properties. We doubt whether the pigment occurs in cow’s serum in combi- nation with albumen. If this were so, we might expect that the pigment could be set free not only with alcohol, but also with other solvents that denature protein, so that it could be extracted with ether. However, when salting out the albumen of the serum with ammo- nium-sulphate and subsequently treating it with ether or benzene, the pigment will not be taken up by it. Neither is this the case when extracting with ether or benzene after precipitating it by boiling. Still the protein, as may be expected, is denatured more by boiling than by precipitation with alcohol, since the latter reaction is initially reversible, the former is not. There is one more reason for our assertion that the “liberating” action of alcohol on lipochrome, which renders it fit to be extracted, resis upon something else then the decomposition of the protein-mole- cule. Whereas e.g. the denaturing of protein by alcohol is compara- 754 tively a slow process, the action on serum, which renders lipochrome accessible for ether or benzene, manifests itself instantly. Lipochrome, then, is freed during the first phase of the action of alcohol on the protein (precipitation) and prior to the second phase (denaturation). I doubt, therefore, whether the liberation of lipochrome by alcohol is due to a decomposition of a supposed albuminous compound. That this alcoholic action may occur, without any question about the destruction of an albuminous compound — as in lipochrome solutions which are free from protein — is borne out by the following experiment. A concentrated solution of carotin is prepared by extracting finely rubbed carrots with a mixture of alcohol and ether. The ether is removed, after which a gold-yellow, alcoholic solution of carotin is left behind. When this solution is diluted with water, so that the alcohol-content in the mixture is very low, the carotin will not be precipitated, nevertheless the solution keeps clear. By evaporation in vacuo (if required with gentle heating in the waterbath) the rests of alcohol, still left behind, are removed as much as possible. The carotin does not precipitate then either, but a solution remains in which no solid particles are visible. It passes through the filter unchanged. Solutions of a higher concentration are opalescent, those of lower concentration are clear. The investigation Prof. Kruyt kindly performed confirms that the carotin in this solution is in a colloidal condition. It appears, then, that by this procedure we are able to solve carotin in water in a colloidal state, although under ordinary circumstances it is insoluble in water, as e.g. is also the case with mastic, cholesterin and many lipoids. However, an attempt to extract this colloidal solution with ether fails. Even a two hours’ rapid shaking in the shaking apparatus does not enable us to transfer the slightest trace of yellow pigment to the ether or the benzene. But as soon as we add to the mixture a small amount of alcohol, e.g. some drops to 5 em®* of colloidal carotin solution and 3 ecm° of ether, the pigment will pass over instantly and quantitatively into the upper layer, whereas the lowermost layer is completely decolourized and generally becomes rather more opalescent. The best result is achieved, when first some drops of alcohol are added to the solution and after this the ether. A similar result is obtained when, before carrying out these experiments, the fats and the cholesterin are removed from the alcoholic carotin solution respectively by saponification and by digitonin. 755 Similarly to the carrot-carotin, the carotin from cow’s serum and human serum, and the xanthophyll from fowl’s serum and egg-yolk ean be obtained in an aqueous, colloidal solution. The latter are more opalescent than the carotin solution from carrots, the egg-yolk solution most of all. Nevertheless they pass through the filter unaltered, and the opalescence may be largely diminished by removal of the fats and of the cholesterin. t When extracting these aqueous colloidal solutions with ether, no trace of pigment passes over to the ether — as has been described for the carotin from Daucus carota. After the addition of a small amount of alcohol, the pigment will again pass over quantitatively into the ether layer. It appears, therefore, that also in aqueous colloidal protein-free solutions alcohol plays a liberating influence upon lipochrome. Precisely the same influence is exerted by a small quantity of alkali: if to an aqueous carotin-solution ether is added, (which by itself does not extract any pigment from it) and subsequently some drops of 10 pere. NaOH, and this solution is shaken rapidly, all the pigment will pass over into the ether. We feel greatly indebted to Prof. Krurr for investigating for us these colloidal solutions and for discussing the problem with us. He stated that the solutions presented TynDALL’s phenomenon, while under the ultramicroscope small particles are visible performing the Brownian movements. He also found that, besides by alcohol and 10 pere. NaOH, also all sorts of other salts, especially the bivalent and the trivalent metals exert the same liberating influence on lipochrome. For instance when we add to the colloidal carotin-solution some drops of aluminium-sol, and subsequently shake with ether, the lowermost layer will immediately be completely decolourized. After having searched in vain for a similar phenomenon in the literature to which we had access, we learnt that Wirrsrärrer had observed the same in aqueous colloidal solutions of chtorophyl. He prepared them — as we did the analogous lipochrome-solutions — by adding a large quantity of water to an alcoholic chlorophyl- solution and subsequently evaporating the alcohol in vacuo. He did not succeed now in extracting the chlorophyl from the aqueous solution with ether. Still the green pigment passed directly over into the ether, when he had added a small quantum of an electrolyte to the aqueous solution. WitisrArrer did not ascertain whether some drops of alcohol had any “liberating” effect. He accounts for the action of the salt by assuming that the dispersed chlorophy|- particles cannot be reached by the ether. The addition of the 756 electrolyte results in salting out the colloid whereby the particles are at first so small that they are invisible to the unaided eye and that the fluid seems to be perfectly clear. Gradually the flakes become visible. But even in the phase in which they are still invisible, the extremely small particles are accessible to the ether and are dissolved in it. This interpretation also undoubtedly holds in the phenomenon observed by us with the lipochrome, when alkali or Al-sol, are added. However, it naturally does not account for the ‘‘liberating”’ action of some drops of alcohol, as in that case there is no question of precipitation. The alcohol, if added in adequate quantity, would rather change the colloidal solution into a true solution. We are unable to account for this phenomenon, but we believe it to be analogous to another reaction previously observed by us. In studying the bile-pigments we had detected that bilirubin, as found in bile and in the blood-serum of patients with obstructive icterus, is directly and completely capable of combining with diazonium salts. However, when the same reaction is performed in the blood-serum of patients suffering from what was formerly called hematogenous icterus, the reaction is retarded and incomplete. On addition of a small quantum of alcohol the reaction takes place directly and completely. We cannot but assume that the bilirubin in the serum in the case of obstructive icterus, and in the bile from the gall-bladder, is in a different condition from the bilirubin in the serum of patients with hematogenous icterus. In the latter case it would seem that the bilirubin particles cannot get in touch with the diazonium- solution, except through the action of small quanta of alcohol. This behaviour of bilirabin in the serum from hematogenous icterus is similar to that of lipochrome in cow’s serum. The resemblance in the behaviour of an aqueous colloidal solution of carotin to that of the native cow’s serum containing lipochrome, suggests the idea that, also in the sernm the carotin occurs in a colloidal and analogous condition. However, this seems not to be the case, since in the aqueous colloidal solution the lipochrome is precipitated by the above-named substances (NaOH, Al-sol, etc), so that it can be extracted with ether. On the other hand, when adding these reagents to the native serum, the pigment will not pass over to the ether. Another remarkable difference between the native serum, con- taining lipochrome and the artificial colloidal solution, is the action of light. Barlier researchers detected already that carotin from 757 Daucus carota (subsequent researches proved the same to hold good for xanthophyll) exposed to the sunlight, is decoloured by the ab- sorption of oxygen and after some time is completely decolourized. With our impure or incompletely purified solutions of pigments in alcohol and ether our results varied with the nature of the lipo- chrome and the solvent. After a close exposure to the quartzlamp the aqueous solutions are almost completely decolourized after 15—90 minutes, the interval varying with the nature of the lipochrome and of the solvent. Contrariwise the native substances (egg-yolk, carrots, cows serum) are not decolourized under similar circumstances. It is evident, therefore, that with regard to sensitiveness to light, there is a difference between the native solutions of lipochromes and the colloidal aqueous solutions. Chemistry. — “The unsaturated alcohol of the essential oil of freshly fermented tea-leaves.” By Prof. P. van Rompouren. (Communicated at the meeting of May 31, 1919). In 1895 in collaboration with my assistant at that time, Mr. C. E. J. Loumann, I investigated the ethereal oil from freshly fermented tea‘), a small quantity of which we were successful in preparing with the cooperation of several tea-planters. The yield of this ethereal oil is extremely small, fifteen kilograms of the fresh leaves giving only one c.c. We were able at that time to detect the presence in the oil of an unsaturated alcohol (b.p. 153°—155°) of the composition C,H,,0O, evidently a hexylene alcohol. From this by oxidation an acid could be obtained, smelling like rancid butter, the calcium salt of which gave on analysis a result which indicated the presence of butyric acid. Lack of material prevented us from determining whether the acid formed was the normal or the iso-butyric acid. Later, shortly before my departure from Java, | had the opportunity of obtaining a larger quantity of the ethereal tea-oil (about 120 ec), which enabled me to resume the research and to investigate more in detail whether by the oxidation of the unsaturated alcohol one or other of the butyric acids is really formed. A knowledge of the nature of the acid is of course of primary importance for the elucidation of the structure of this acid. After treatment with alkali in order to saponify the methyl sali- cylate*) (the presence of which we had demonstrated in 1896) and - other esters*) possibly present, the crude oil was fractionated several times. The largest fractions boiled between 154° and 156° and between 156° and 158°. These were mixed and distilled in vacuo, the principal fraction boiled at 75°—80° at 28—30 mm. pressure. The sp. gr. at 15° was 0.8465; npw 1.43756. Elementary analysis gave 71.17°/, C. and 12.74°/, H. The formula C,H,,O requires 71.91°/, C. and 12.10°/, H. 1) Verslag omtrent den staat van ’s Lands Plantentuin te Buitenzorg for the year 1895, p. 119. *) The same for the year 1896, p. 168. 8) The salicylic acid isolated was not odourless. The smell resembled that of phenyl! acetic acid. 759 The liquid was now treated with anhydrous sodium sulphate and again distilled in vacno. Further analysis of the product gave, however, no better results. (71.08 °/, C. and 12.59 °/, H). The unsaturated tea-alcohol forms with avidity an addition compound with bromine, as was previously shown. The quantity of bromine added. however, was smaller than is to be expected from a substance of the formula C,H,,O, being only 87°/, of that amount. Two fresh determinations gave the following results: I. 1.017 grm. of the alcohol in chloroform solution ‘cooled in ice-water add 1.363 grm. bromine. II. 0.529 grm. add 0.707 grm. : From these result it appears that only 83.2°/, and 83.5 °/, respectively of the calculated quantity of bromine is added. As I suspected that the unsaturated alcohol perhaps contained a hexyl alcohol as impurity, | attempted to purify a larger quantity of the bromine addition compound *) from this by heating in vacuo at 100°. A subsequent treatment with zine dust should give the hexylene alcohol in a pure state. Since, however, the bromine addition product gave hydrobromie acid, I was unable to carry out this intention. Treatment of the unsaturated alcohol with phenylisocyanate gave no crystallised product. On the other hand a-naphthylisocyanate gave an «-naphthylurethane (m. p. 76°), the melting point of which could be raised to 80° after repeated recrystallisation from petroleum ether. On treatment with phthalic anhydride, the tea alcohol gave a liquid acid ester of which the silver salt melted at 140°. Oxidation of the tea alcohol with potassium permanganate in neutral as well as in alkaline sodium carbonate solution, proceeds very smoothly. About 3 ¢.c. of acid were obtained from 11.5 grm. on treatment with 50 grm. potassium permanganate in 4 °/, solution, This acid, as before, had a smell resembling that of butyric acid. On distillation of the acid, however, the principal fraction, besides a small first fraction in which formic acid could be detected, was a liquid boiling between 125° and 145°, while the residue in the flask consisted of a liquid of higher boiling point with a smell of perspiration. The principal fraction, on redistillation, gave a liquid of boiling point 140°—145°; which on being boiled with water and calcium carbonate was transformed into a calcium salt which was found on analysis to contain 21.2°/, Ca. This result in conjunction with the boiling point of the acid obtained, show that the latter ') This does not solidify in liquid ammonia 760 consists of propionic acid, the calcium salt of which contains 21.5 °/, Ca. On heating the ammonium salt an amide with a melting point of 78° was obtained which, on mixing with propion-amide, produced no depression of the melting point. The hexylene alcohol obtained from tea-oil might therefore be hexene-3-ol-6 of the formula CH,. CH, CH: CH. CH,. CH,OH. A hexylene alcohol has been obtained by H. WarBauM *) from Japanese peppermint oil to which after investigation he attributes the structure of a g-y-hexenol. This alcohol is presumably the same as that extracted from tea oil. On oxidation with potassium permanganate the 8-y-hexenol gives propionic acid as principal product. With chromic acid a hexylene acid is obtained. The «-naphthylurethane prepared from the alcohol melts at 80°, while the melting point of the silverssalt of the acid phthalic acid ester melts at 126°. On treatment with bromine only 70°/, of the quantity required by theory is absorbed. It is true that the melting points of WarBaum’s silver salt and my own do not agree, but the other properties of the tea-alcohol justify the assumption that the latter to a great extent consists of B-y-hexenol. I am, however, for the moment unable to explain why, on oxidation with potassium permanganate, an acid was obtained previously, the calcium salt of which contained only 18.6 °/, of calcium. The acid on that occasion was not distilled, as the quantity available was too small, and may have contained, for example, hexylic acid, by which the calcium content of the propionic acid formed would be lowered. Finally it may be possible that the heating of the crude oil with alkali in order to remove the methyl salicylate, has caused a shifting of the double bond. This research, as well as the investigation of the other constituents of the tea-oil, is being continued ’”). Postscript. Since the above paper was communicated, the firm Messrs. SCHIMMEL and Co. of Leipsic sent me at my request a small quan- tity of the unsaturated alcohol prepared from Japanese peppermint oil, for which I desire herewith to express my thanks. The «-naph- 1) Journ. f. prakt. Chemie, 96, 254 (1917). 8) The ethereal oil of tea was some time ago the subject of an investigation by Dr. Deuss (Mededeelingen van het Proefstation voor Thee, XLII, 21, 1917). This research merely confirmed our own observation that the oil contained an unsatu- rated alcohol together with methyl salicylate. 761 thylurethane obtained from this (m.p.80°) when mixed with that prepared from the tea-oil caused no alteration of the melting point. The acid phthalic acid ester prepared from it, gave a silver salt- melting at 128°. By recrystallisation from alcohol the melting point could be raised to 134° (not sharp). With the silver salt prepared from the tea-alcohol it gave a mixture melting at 138°. The assumption is thus justified that the unsaturated alcohol prepared by me from the tea-oil is identical with the 8-y-hexenol, that is, with hexene-3-ol-6. Utrecht. Org. Chem. Laboratory of the University. 50 Proceedings Royal Acad. Amsterdam, Vol. XXII. Physiology. — “A method for the determination of the ion con- centration in ultra filtrates and other protein free solutions”. By Dr. R. BRINKMAN and Miss E. van Dam. (Communicated by Prof. HAMBURGER). (Communicated at the meeting of October 25, 1919). A. Determination of the concentration of free calcium ions. With regard to the biological actions of salts the actions of ions claim the first consideration. It is therefore desirable that we have at our disposal a method by which the ion concentrations are measured. Up to this only the concentration of the free H'-ions have been measured directly; the concentrations of other, also physiologically important ions were not measured at all or determined only indirectly by calculation. The concentration chain method can be applied only with great difficulty to the physiologically important metals owing to the disturbances brought about by the liberation of gas. Drucker !) has offered a method in which Ba-amalgam was used as an electrode. An analogous method can perhaps be worked out for the alkali metals. Such determinations have, however, not been made as yet. As an example of a case where it is necessary to know the ion concentration, we can point to the state in which the calcium occurs in the blood. It occurs there namely in three forms: as Ca’ ion, as undissociated calcium salt (Ca (HCO,),) and as colloidal caleium- protein compound. More or less 25 °/, of the total quantity of cal- cium occurs in the latter state. According to Rona and TAKAHAsHI *) the ion concentration of the calcium in the serum is determined by the equation [Ca ].[HCO,'] Nes) KonkanWaverase) [LH] For the serum which has the physiological [H-] and carbonic acid tension, this means a [Ca] of 20—25 mgr. per L. Of the more or less 100 mgr. per L. of calcium which occurs in the serum, therefore, only */, part is present in the ion form. We learn from 1) Zeitschr. für Elektrochemie 19, 804 (1913). *) Biochem. Zeitschr. 49 p. 390. 763 the equation that this concentration of Ca ions is not directly dependent upon the total quantity of calcium; the concentration of the physiologically most important part of the plasma calcium is thus not governed by the amount of calcium salts present, but by the concentration of the hydrogen and bicarbonate ions. By means of the method offered by us it is now possible in a simple way to measure directly the Ca: ion concentration. In prin- ciple the method can equally well be applied to other ions. We started with the determination of the concentration of Car ions, because the results of the determination can in this case very easily be controlled by calculation. I. General principle of the Method. A few technical remarks. If in a binary electrolyte the concentration of the anion = C4, that of the cation = C} and that of the undissociated salt = C,, then, according to the law of mass action, the following relation exists ca . Ck=k . cn, where k is a constant. If the electrolyte is only slightly soluble the salt is practically completely dissociated and the concentration of the undissociated part may be neglected. If now the solubility of the slightly soluble salt = A, then C4 = C,= A, and the product C4. C, = A’ has a constant value (solubility product). If this product and the concentration of one of the ions is known, the concentration of the other can therefore be calculated. Supposing that the solution has a concentration of Ca” ions = Cn then the concentration of the C,O, ions which can exist free beside > —, if P represents the solution pro- 5 duet of CaC,O,. If now still more C,O, be added, then the CaC,O, will be precipitated or will remain in supersaturated solution. If the formation of a supersaturated solution can be avoided, then it will be noticed, that, upon the gradual addition of C,O," ions to the solution containing Ca ions, at a certain moment a slight tur- bidity due to CaC,O, results. At this stage the concentration of C,0, ions has become so strong that the solubility product is just exceeded. The C,O, ion concentration is then known, and also the these Ca” ions, maximally = — solution product and the Ca” ion concentration can thus be calculated. Vice versa, if we start with a known [Ca] we are able to deter- mine the valne and constancy of the solubility product. Where this method is used therefore it is necessary to observe bo 764 how great the [C,0,"] is while only the merest sign of turbidity due to CaC,O, can be detected. In general it can be done in the following way: A series of small tubes, each containing 1 e.c. of a known CaCl, solution, was taken, and to the tubes in succession quantities of oxalate solution increasing gradually with each new tube were added with a capillary pipet divided into tenthousandths of c.c. The tubes were then left to themselves for from */, to 1 hour and consequently it was observed in which tube the first sign of turbidity due to CaC,0, appeared. It is clear that the formation of supersaturated CaC,O, solutions has to be avoided. In cases where the solutions held other salts besides (e.g. RINGER solution, ultra filtrate) we have never noticed supersaturation. As a matter of fact supersaturation occurred in the case of pure solutions of CaC,O,. This can be avoided by setting to work in the follow- ing way: With a capillary pipet the desired quantities of a, say 0.05, N. strong oxalate solution is brought into the dry tubes. In a waterbath the tubes are evaporated down to dryness. After this the liquid con- taining the Ca” is introduced into the tubes. In this way is prevented the formation of already supersaturated solutions. For the determination of the calcium ion concentration it is moreover necessary to use tubes that are well closed with ground glass stoppers. This is necessary to keep the water free of carbonic acid or to keep a fixed carbonic acid tension constant. It is necessary for the judging of the appearance or non-appear- ance of the CaC,O, precipitate that the tubes should be cleaned as thoroughly as possible; this can be done in the usual way (chromic acid, ABEGG’s steaming process etc.). The best way for viewing the tubes is in a box with a slit in the bottom from which the light falls through the solution. Care should be taken that the light does not fall on the eye of the observer. The Tyndall phenomenon makes it possible to appreciate the slight- est turbidity. Should the solution before the experiment already evince a slight opalescence (not due to CaC,O,) as is sometimes the case with serum and ultra filtrate, it is advisable to view the solu- tions by red light. The wavelength of this light being too great to cause refraction the opalescence is not apparent. The temperature during the experiment must of course remain constant. It is therefore best to work in a waterbath of constant temperature. The results obtained by the above method can be controlled in 765 another way still, viz. by measuring the electrical conductivity of the solutions. If to a solution containing C-- and Cl’ ions, C, 0," and Na’ ions be added, then the product Ca X C,O, cannot exceed the square of the solubility of CaC,0O,. If too many C, 0, ions have been added, then undissociated CaC,O, must be formed. How much CaC,O, will be formed, if the product is exceeded by a fixed quantity of C, O,? To a binary electrolyte with a solubility A, a salt which has an anion in common with the first is added in a concentration x. Through this the solubility of the first salt is changed to A’. The total concentration of the anion then amounts to A’+ 2, that of the kation to A’. The solubility product is therefore ; A‘(A’+ x), and because this is constant we have: A' (A! + 2) = A? or A — wt WAA Ee? 5 8 The quantity of undissociated salt which results when x Mol salt that has 1 ion in common with the first is added, therefore is: =~ se VAR dE A culls SII Lice 4 Ae maar ena (| 5 (1) if A represents the solubility of the first salt. We have now e.g. 5c.c. of an aqueous solution of CaCl, .6aq., free of COs, con- taining per litre 0.56 millimol Ca: and (2 Cl’). To this there is added several times successively 0.0050 ce. of a 0.05 N solution of Na,C,O,. After every addition the conduetivity is measured. The Na,C,;0, may here be added in solution, for here the solution may be supersaturated. By means of the first method the value now found for the solubility product is 0.055. From this it follows that a C,O, concentration of a magnitude 1 millimol corresponds to the 0.56 millimol Ca. Upon every addition of 0.0050 c.c. 0.05 N. Na,0,0, to 5 ce, of a solution of CaCl, 6 aq. the C,0, concentration increases by 0.25 mm. After 4 additions therefore the solubility product is reached. What is the relation between the total concentrations of ions during these additions? For the first addition the total ion concentration is 0,56 millimol Ca” + 0,56 m.m. (2 Cl”) = 1,12 m.m. After the first addition of 0.025 mm. NasC,0, 0,56 Ca + 0,56 (2 CI”) + 0,025 40, + 0,025 (2 Na”) = 1,17 m.m. Thus the total ion concentration after the 2nd addition is 1.22 m.m., after the Srd 1.27 mm. and after the 4th 1.32 m.m. Upon the 5th addition the solubility product is exceeded. According to the deduced formula (1) the amount of undis- sociated CaC,0, formed = 0,025 + V4 « 0,055 +4- 0.025? VY 90,055 0 ONO A Ke , “ The total ion concentration becomes thus after the 5th addition 766 1.32 m.m. + 0.025 C,0,” + 0.025 (2 Na”)—0.0115 Ca\—0.0115 €,0,” = 1.347 m.m. The total concentration of ions therefore does not increase by 0.05 mm but only by 0.027 mm. With the 6th addition we get a value for the undissociated CaC,O, of: 0,050 + V4 < 2000 c. Ohm 5 cc. CaCl, 6 ag.t0.010 cc.Na,C,0, 8.200 >< 2000c.Ohm | 0.252 2000 c. Ohm 5 cc. CaCl, 6 aq. +0.015 cc. Na;C,0,| 7.929 2000 c. Ohm | 0.271 > 2000 c. Ohm 5 cc. Calls 6 ag.t0.020 cc.NagC,0,, 7.696 & 2000c.Ohm | 0.233 2000 c. Ohm 5 cc. CaCl, 6 aq.+-0.025 cc. Na,C,0,) 7.600 * 2000c.Ohm | 0.096 2000 c. Ohm see. CaCl, 6 aq.+0.030 cc. Na,C,0, 7.500 X 2000c.Ohm | 0.100 > 2000 c. Ohm 5 cc. CaCl. 6 ag.+0.035 ce. NajC,0, 7.410 > 2000 c. Ohm | 0.090 2000 c. Ohm Subsequently more determinations of a similar kind were made by us, which always gave a result of 0.053—0.58, — a mean of 0.055 — for the solubility product. Here it must be remarked still that it cannot be expected that the solubility product will just have been reached at the end of an addition; the mean value, therefore, has to be taken. There is still the possibility that the decrease in resistance came about because the oxalate added in such large quantities did practically not dissociate completely; the way in which the decrease would take place then would not be such a sudden one. To control this the same quantities of oxalate were added to 5 c.c. of distilled water; the conductivity kept increasing proportionally to the quan- tities added. We have now found by two methods which are independent of each other the constant value of 0.055 m.m. for the solubility product, when Ca” and C,O," ions are added together. The solubility of CaC,O0, has been found by Konurauscn to be 4.35. 10 5 Mol per L. (18°); the solubility product calculated from this is 0.0019 m.m. per L. and this is much below the value found by us. 5 Konrrauscn measured the conductivity of a saturated solution of CaC,O0,; he therefore did not start out from the individual ions. Herz u. Mens!) found by adding together the ions a value of 0.034 gram per L. for the.solubility of CaC,O, from which follows 1) Ber. 36. 4, p. 3717. 1 769 a solubility product of 0.054. This product thus agrees perfectly with ours. The determination of Herz v. Muns and our own determinations, by two methods, show thus conclusively that we have to reckon with a solubility product of 0.055. UI. Determination of the concentration of Ca ions in solutions which hold other salts besides. 1. The concentration of Ca’ ions of 0.02°/, CaCl, 6aq. in 0.5°/, NaCl. For the system CaCl, = Ca” + 2 Cl’ the following also holds: CaCl, <2 K Ca” Cl: K can be found if the degree of ionisation z of a given CaCl, solution is known. For Ca(NOs), 0,1 %/, (= 6m.M. per L.) « is 0,67 }). We have therefore [Ga(NO3)2] — « [Ga(NOs),] = K a Ca” a? (NO3)* or because [CGaNO,] = [Ca ] = [NO3] = 6 m.M. per L. 1 — 0,67 = KX 0,673 (NOs)? 1 — 0,67 = K X 0,675 X 0,036 K = 30. This is therefore the dissociation constant for Ca(NO3); that for CaCl will differ very slightly from it. For 0,02°/, CaCl,6 aq. or 0,91 millimol per L., in 0,5°/, NaCl also holds: [CaCl,] = K Ga Cl?. The conc. of CY is given by the dissociation of 0,5°/) NaCl. Here « = 0,82 (Osmot. Druck u. lonenlehre, p. 53); [Cl] thus becomes 7 m.M. In addition to this there is still [CI of 0,91 m.M. CaCl,, + half of which we may consider to be dissociated without committing a large error. The total [Cl’] then becomes +8m.M. Thus CaCl, = K (0,91 — CaCl) 0,064. K=30. CaCl, = 0,60 m.m. Of 0,91 m.m. CaCl, thus 0,06 m.m. is not dissociated while 0,31 m.m. is disso- ciated. The solution therefore contains 12,4 mgr. free Ca” per L. Experimentally it appears that an oxalate concentration of 0.18 m.m. is necessary before turbidity results in a solution of NaCl 0.5 °/, + CaCl, 6 aq. 0.02 °/,. From this follows a [Ca] of — 0.055 : 0.18 = 0.30 m.m., or 12 mgr. per L. This determination thus is perfectly in correspondence with the calculation. 2. Determination of the concentration of Calcium ions in physio- logical salt solutions. In a solution of the composition: NaCl 0.7°/,, NaHCO, + 0.18°/,, KC} 0.02°/, and CaCl, 6 aq. 0.040°/,, with a certain carbonic acid ') Osmot. Druck u. lonenlehre. I, p. 53. 770 tension which was not exactly known, the concentration of hydrogen ions was 0.3.10~7 (determined with neutral red after SÖRENsEN) and the concentration of bicarbonate ions 0.02 N (determined by titration with 0.01 N.HCl and methyl-orange). From this follows for the concentration of Calcium Oe UO CH IT = F [Ca] 0,08 20 mg. per L Experimentally a CaC,C, turbidity resulted with a concentration 0.1 millimol oxalate. From this follows a [Ca ] of 0,055: 0,1 = 0,55 m.M. = 22 mgr. per L. In a similar solution in which the [H’] however was 0.45 . 107 and the [HCO,']=0,02N., the CaC,0O, milkiness was seen with [(C,0,*] = 0,07 m.m. Thus: [Ca**] = 0.055 : 0.07 = 0,8 m.m. = 82 mgr. per L. 0,45 . 107 From ti leulati Cau 350 rom the calculation [Ca] 0,02 m.m. = 80 mgr. per L. 3. Determination of the concentration of calcium in ultra filtrate. Human serum was centrifuged for 2 hours in ultra filters after pe Waarp *). CO, was passed through the ultra filtrate until (H]=0.3.10 7. (This was ensured by comparing the colour of neutral red in the ultra filtrate with neutral red in a phosphate mixture, according to SÖRENsEN, which had an CH) == (0810) A precipitate of CaC,O, occurred with a [C,O,"] of 0.4 m.m. per L. From this follows a [Ca] of 0.55 m.m. or 22 mgr. [Ca | ions per L., as has also been made probable by Takauasui and Rona. An attempt to apply these measurements directly in serum often fails because the turbidity due to CaC,O, is very much less evident, and the opalescence which normally occurs so often in serum is a drawback. In the limited number of instances where we could notice a definite turning point the same concentrations of Ca” ions as in ultra filtrate were found. As a rule, however, it is necessary where serum determinations have to be made to make ultra filtrate, which, after pu WAARD, is very simple. 1) Arch. Néérl. de phys, 2 530 (1918). 771 Summary. A simple method is described by which it is possible to measure the concentration of Ca’ ions in a solution of a mixture of salts e.g. ultra filtrate. The method is based upon the following principle: To a solution containing Ca” there are added so many C,O," ions till the solubility product of Ca,C,O, is just reached. The juncture at which so many C,0, ions are added that this product is just exceeded, is ascertained by the appearance of a slight milkiness due to CaC,O,. It does not matter whether in the mixture of salts there are present other ions still that can give a precipitate with oxalate. It is only necessary that CaC,O, should be the most insoluble sub- stance which can result in the solution. The method is correct to 2—3 mgr Ca’ per L. The value of the solubility product was tested by the measurement of the electrical conductivity of the solution. The principle of the method can likewise be applied for the determination of other ions. The only condition is the disposal of a reagent that gives a salt which is very slightly soluble with the ion whose concentration has to be measured. Physiological laboratory of the University of Groningen. September 1919. Geology. — “On the Crustal Movements in the region of the curving rows of Islands in the Kastern part of the East-Indian Archipelago”. By Prof. H. A. Brouwer. (Communicated by Prof. G. A. F. MOrLENGRAAFF). (Communicated at the meeting of November 25, 1916). Of late years various explorers *) have pointed out a resemblance in the tectonic structure of the curving rows of islands of the Moluccas and that of many curving chains of Alpine structure. Large overthrusts formed down to the miocene, have been discovered in various islands and the zone, characterised by overthrusts, is bordered on the outside by a region, in which the mesozoic and tertiary deposits are slightly or more intensely folded, but no overthrusts occur. Actual facts seem to indicate that in the curving rows of islands of the Moluceas may be distinguished : 1. A zone characterised by overthrusts (Timor-Ceram row of islands). 2. A marginal zone without overthrust-tectonie (Sula-islands— Misool, Western New Guinea south of the Mac Cluer bay and probably also the Kei-islands). 3. An inner zone with the young active volcanoes. 4. A zone lying between 1 and 2 of older voleanic rocks (North coast of Netherlands-Timor, Wetter, Ambon, peninsula of Huamual in South-Western Ceram and Amblau). We will now pass in review the features of these zones. General situation and origin. If the sea-level in the East-Indian archipelago were to subside 200 m., Sumatra, Java and Borneo would form one mass of land with the peninsula of Cambodja and Siam, just as Australia with the Aru-islands, the vast tract now occupied by the shallow Arafura-sea G. A. F. MoLENGRAAFF. Folded mountain chains, overthrust sheets and block- faulted mountains in the Hast Indian Archipelago. Compte Rendu du Xlle congrès géol. internat. Toronto 1913, p. 689. H. A. Brouwer. On the Tectonics of the Eastern Moluccas. Proc. Kon. Ak. v. W. Amsterdam. Vol. XIX. N® 2, p. 242—248. 773 and the bay of Carpentaria, New-Guinea and the islands of Misool, Waigeu, Batanta, Salawati, west of it. Between those two landmasses lies an area in which deep sea- basins alternate with upheaved islands. The region of the curving rows of islands (the Timor-Ceram row and that of the young active volcanoes) considered by us, presents an aspect similar to that which parts of the geo-synclinal of the Mediterranean region must have presented in some part of the mesozoic period. In the Jurassic period several geo-anticlines were formed in the latter region, which divided the original geo-syncline into a number of secondary geo-synclines and in connection with the parallelism between the direction of the (more recent) alpine mountain ranges and the axes of these mesozoic geo-synclinals, Haug‘) thinks it legi- timate to assume that the formation of these mesozoic geo-synclines is due to beginning mountain-building movements. MoLENGRAAEF *) assumes on the ground of different features of the curving rows of upheaved islands of the Moluccas and of the adjacent deep sea-basins, that these islands have originated in the same way. The outlying position of the Tenimber islands. If we imagine the islands to the east of Timor (Letti, Moa, Lakor, Luang, Sermata and Babber) joined by a curve to the islands south- east of Ceram (Drie Gebroeders, Kur, Téor, Kasiwui, Gorong and Ceram Laut) the islands of the Tenimber group will be seen to lie outside this curve. This curve is e.g. also found on map N° 1 of Verserk’s Molukken Verslag*), on which the Tenimber islands and the Kei-islands are lying outside his “belt of older rocks”. Now it is striking, that im the Sahulbank, which constitutes the submarine continuation of the Australian block — 1%. e. the “ Vorland” 1) E. Haua. Traité de Géologie. II, p. 1127. 3) G. A. F. Motencraarr. On recent crustal movements in the island of Timor and their bearing on the geological history of the East-Indian Archipelago. Proc. Kon. Ak. v. Wetensch. Amsterdam. June 1912, 3) RB. D. M. Verserk. Molukken Verslag. Jaarb. v. h. Mijnwezen 1908. Wetensch. Ged. Allas. Kaart |. See also: A. WicHmaNnN. Gesteine von Kisser, Jaarb. v. h. Mijnw. 1887, p. 120 and Samml. des Geol. Reichsmus. in Leiden. (The curving row of islands, separating the Banda Sea from the Arafura Sea is also here represented as a mountain range). Ibid. Der Wawani auf Amboina und seine angeblichen Ausbriiche. UL Tijdschr. Kon. Ned. Aardr. Gen. XVI. 1899, p. 109. K. Martin. Die Kei Inseln und ihr Verhältniss zur australisch-asiatischen Grenz- linie, Tijdschr. Kon, Ned. Aardr. Gen. VII, 1890, p. 241 ff. 174 against which the overthrust mountain chain is pushed up — a depression occurs just opposite the Tenimber islands. We know that the shape of the folds of several mountain-chains is influenced by the resistance of the “Vorland”. This also holds for the folds to which the formation of the uplifted curving rows of islands and the alternating deep ocean-basins have been aseribed above, and then we can compare the bending of the Timor-Ceram curve near the Tenimber-islands with the pushing forward of the Penninic overthrust sheets of the Alps in the lower parts of the hereynian mountains, against which they were pushed up (as between Mont Blane and the Aar-massif). Behind the parts of greatest resistance of the “Vorland”, the tectonic axes at a deeper level, and the islands at the surface will rise higher; this need not be, but may be, the reason why the Tenimber islands are not uplifted so high above the sea-level, as Timor is. The Tenimber islands have been considered by us‘) to belong to the overthrust mountain-range, and if the mountains on the South Coast of Timor, characterised by an imbricated structure with a uniform dip to the north-north-west, are autochtonous 7), the over- thrust mountain-range must in all likelihood also have been bent at the site of the Tenimber islands. The outlying position of the Kei-islands. The Kei-islands are like the Tenimber-islands situated opposite a depression in the region covered by the shallow Arafura Sea, and their outlying position can be explained in a similar way. Along the north coast of Groot-Kei the terraces of miocene limestone are surrounded by a younger and lower (probably quaternary) coral- terrace, while the terraces of miocene limestone in the southern part of the island are found down to the sea level. This points to an intenser uplift of the northern part of the island in post-tertiary time and this may, just as the outlying position, point to the persistence of crustal movements similar to those which gave rise to the overthrusts of the Timor-Ceram row of islands. The northern part of the island, namely, lies just opposite a protruding point of the depression in the region covered by the Arafura sea, and opposite this more resistant part of the ‘“Vorland” the tectonic axes at a deeper level and the islands at the surface will be more elevated. 1) H. A. BROUWER. loc. cit. 3) G. A. F. MoLENGRAAFF. Folded mountain chains ete, loc. cit., p. 691. Halmahe ra Fig. 1, 775 as N.e w Guinea re 5 re ie Si De 32 a, oi = ks) Co => ow 9 i) & 8 ay RR EE / ~ a e S fee" Bye ae f iS 7 Ae …e 1 a x ° 6 9 1 5 3 E/1 IN ag ld 2 O\!l £ % = Cey es \ 3 = ~ e 3 S & < 3 3 & al -s Obi 1X" P4 Sx a Ah ¢ 4 y hs The youngest crustal movements in the curving row of islands of the eastern Indian Archipelago. —----- the two geoanticlines, rising and moving towards the *Vorland’’, approximate limit of the *Vorland”. 776 The large island of Jamdena of the Tenimber-group consists — at all events in part — of mesozoic rocks. As regards the Kei-islands we found there only in one spot of small extent near the east coast amidst limestones of eocene age mica sandstone and ferriferous rocks, strongly resembling mesozoic rocks. *) The eocene in. Groot-Kei is not folded intensely; the miocene is not folded at all.?) More to the west the strata seem to be folded more intensely, for in a new island near Ut (Klein-Kei-group) contorted, about vertical strata of probably eocene limestone were found. The tectonic relation of the above-mentioned mica sandstone and ferriferous rocks to the widely spread tertiary limestones and marls in Groot-Kei, has not been explained yet, neither is it possible yet to fix the eastern limit, once reached by the overthrusts of the Timor-Ceram curve near the Kei-islands. Perhaps here also the overthrust mountain-range has already made an outward bend; the old-miocene of Groot-Kei however lies about horizontal. The Aru-islands. These islands form a small elevation inside and near the border of the tract which we consider to be the “Vorland”. They may perhaps also be considered as bulges similar to those which are elsewhere believed to result from the pressure to which the most exposed part of the “Vorland” is subjected. On the occurrence of rocks older than Permian rocks. In the Western Alps the central parts of the chain are formed by a series of autochtonous massifs (Mercantour, Montblanc, Aar Massif and others) which belong to the ancient hercynian mountains. Part of the overthrust sheets was pushed over these massifs and deposited north of them. For Australia the equivalents of the hereynian folding of Europe are known, but nothing is known for certain in this respect for the curving rows of islands under consideration. In the large island of Timor, which has been pretty well explored, we have no certainty about older rocks than those of Permian age, and MoLENGRAAFF says about this island: ““The Fatu sheet is like the Tethyssheet, composed of rocks ranging in age from Permian to Eocene and probably to Miocene”. Indeed, so-called “old slate rocks’ occur in numerous islands. 1) H. A. Brouwer. Geologische verkenningen in de oostelijke Molukken. Verh. Geol. Mijnb. Gen. 1916, p. 47. *) R. D. M. VerBreK. Molukken Verslag. loc. cit., p. 501. 177 VERBEEK supposes*) that Archean as well as old-Palaeozoie rocks occur among them. But, for reasons, which have been expounded in other papers we feel justified in assuming that these rocks — in part at least — are of younger date. *) When confining ourselves to the island of Timor, the available data seem to bear out that the older massifs, constituting the base of the Tethysgeosynelinal, have not, at all events not here, been raised sufficiently by the folding process to be denuded through the erosion, to which that portion of the overthrust mountain range that had been lifted up above the sea-level, was exposed for a long time. This must have been the case also in an earlier stage in the region of the Alps, when in the middle-mesozoic period the Tethys- geosyncline was divided into different geosynclines and geoanticlines, with partial emersion of the latter. Prolongation of the curve west of Timor and west of Ceram. The island of Roti may be considered as a direct continuation of Timor; we find there rocks of the same kind and various facts point to a similarity in the tectonic structure®). Similar rocks are also found in Savu, but Sumba presents a totally different structure; not a vestige of the intense miocene foldings is found here. That the prolongation of the overthrust mountain-range does not proceed over Sumba is not surprising in connection with the contour of the “Vorland”. South of Timor the limit of the Australian block bends southward, so that Sumba lies further from the ‘‘Vorland” and consequently assimilates itself more to the more northern row of the Sunda islands. With respect to the prolongation of the curve west of Ceram Martin believes that vast overthrusts possibly also occur in Buru‘). The elliptical “belt of ancient rocks” indicated by VrrBerK on Plate | in his Molukken Verslag, diverges from Buru in south- RD. M. Verpeex. Molukken Verslag, loc. cit, p. 738. Verslagen der Afd. Natuurk. DI. XXV, 1916/17. 5 Comp. H. A. Brouwer. Geologisch Overzicht van het oostelijk gedeelte van den Oost-Indischen archipel. Jaarboek Mijnwezen in Ned.-Indië. 1917. Verh. Il, p. 33—35. Devonian rocks with Spirifer Verneuili occur in Celebes (H. A. Brouwer. Devonische afzettingen in den Ol. archipel. De Ingenieur. 29 Nov. 1919). *) H. A. Brouwer. Voorloopig Overzicht der geologie van het eiland Roti. Tijdschr. Kon. Ned. Aardr. Gen. XXI. 1914, p. 611. OL G. A. . Morgraraarr. Verslag betreffende de wenschelijkheid etc. Tijdschr Kon, Ned. Aardr. Gen. XXXL. 1914, p. 369 ff. 51 Proceedings Royal Acad. Amsterdam. Vol XXII. 778 western direction, but we cannot find sufficient evidence to look in this direction for the continuation of the Timor-Ceram row of islands. Hotz’) reports the occurrence of rocks in the western part of the eastern peninsula of Celebes, which show a great resemblance to rocks, widely spread in Buru (Martrin’s Buru-limestones) while also the tectonic structure becomes more complicate than that of the eastern part of the east arm, where, as in the Sula islands, simpler tectonic relations prevail. This, however, does not convince us even- tually of a prolongation of our overthrust mountain-range. Curve with the young Volcanoes. The young volcanoes of the Banda Sea.are joined by VrrBrEK by an ellipse of which only one half embraces volcanoes, no volcanoes being known on the northern half between Banda and the Gs Api, north of Wetter. This ellipse runs concentrically with VrrBEEE’s elliptical “belt of older rocks’. In my opinion, we may as well assume that the volcanic islands rest upon a submarine ridge, which forms the continuation of the rows of islands to which Sumbawa and Flores belong, and which bends round considerably past Banda in the direction of the Siboga ridge with the Schildpad- and Luci- para islands and the G& Api to the north of Wetter. On this sup- position the Banda Sea would be encircled by two ridges, running concentrically wide apart, but the inner ridge bending sharply towards its termination. Additionally we are able to record here, that between the Timor- Ceram row and the row of the young volcanoes, another zone seems to exist with a certain autonomy. We mean a zone of older voleanic rocks, having many features in common and occurring near the north coast of Dutch-Timor, in Wetter, in Ambon and in the peninsula of Huamual in South-West-Ceram. Then a very consider- able portion of this zone would be covered by the sea. Among these voleanic rocks are serpentine breccias and serpentine conglomerates, tuffs, rhyolites, and andesites. Peculiar andesitic to basaltic rocks with glassy crusts, reminding us of the “pillowy lava” of Mullion Island and the upper-Devonian ‘“Wulstdiabase” of the Westerwald occur in all localities. Their typical structure is indicative of submarine origin; the origin of such structures was observed by ANDERSON?) where the lava of the new volcano Matavanu in Savaii 5 W. Horz. Vorläufige Mitteilungen über geologische Beobachtungen in Ost- Celebes. Zeitschr. d a. geol. Ges. LXV. 1913. Monatsber. N°. 6, S. 329. 2) Tempest ANDERSON. Volcanic craters and explosions. The Geogr. Journ. Febr. 19127 p. 129; 5 779 (Samoa islands) reaches the sea, and also for the rocks of Mullion Island, which occur together with sediments with radiolaria TeALr ') assumes a submarine origin. Comparisons with the Alps. Although the geology of the region under discussion is as yet known only in broad outlines, it is permissible to conclude from the results of the inquiries of the last few years that the crustal movements bear some resemblance to those by which other curving alpine mountain ranges were built up, to witness the known over- thrusts in an outward direction everywhere in the Timor-Ceram curve and the adaptation of the folds to the shapes of the ““Vorland”. Additional data that are being collected, prove this resemblance to be beyond dispute. We know that the folded eurves of mountains of the Mediter- ranean region correspond to the geosynclinals accumulated by bathyal sediments in the mesozoic and in the beginning of the tertiary period. The jurassic and the cretaceous deposits reach a considerable thickness there, their horizontal extent is very large, fossils of the neritic zone are rare; all these characteristics are wanting in the generally little disturbed deposits of the same age outside the region of the alpine mountains. For the sake of comparison we point to the striking resemblance of the triassic to the jurassic and perhaps even younger deposits of the deep-sea, covering a vast extent in islands of the Timor-Ceram curve (Roti, Timor, Buru) which are situated far from each other, while different reasons justify the assumption that in that time an open sea connected the region of the East-Indian archipelago, the Himalaya and the Alps’). The in- vestigation of the permian fauna of Timor also teaches us that the Tethys geosynclinal extended already in permian time from the Mediterranean Sea to the region of our Archipelago and a conform- able succession of perm and trias seems to be the rule. The fact that permian deposits are as yet known only in the southern islands of the Timor-Ceram row of islands, goes to show that in that time the sea covered a smaller area in the eastern part of our Archipelago than in mesozoic time. In the Mediterranean region the hereynian crustal movements were no longer distinctly perceptible already towards the end of h J. J. H. Tea. On greenstones associated with radiolarian chert. Trans-Royal Geol. Soc of Cornwall 1894. *) G. A. PF, Morenoraarr. L'expédition néerlandaise à Timor en 1910—1912, Arch. Néerl, des Sciences exactes et nat. 1915, p. 395 seqg. 780 the permian and in the triassic period this movement does not recur. What we do observe at the site of the future intensive tertiary folds, is the formation of geosynclines, in which tbe bathyal trias is deposited. In the jurassic period different geosynclines and geoanti- clines were formed whose course has been reconstrued by Haue *) with the aid of stratigraphical data and by removing the deposits of the overthrust sheets to their original site. In the formation of these geoanticlines some parts may rise above the sea-level, which will cause rows of islands and also (under favourable circumstances) coralreefs to be formed, such as we know now in the eastern part of the East Indian archipelago. Have (loc. cit. p. 1126) says of the géanticlinal brianconnais: “La zone axiale du Brianconnais et la nappe supérieure des Préalpes, qui a sa racine dans son prolonge- ment, sont caractérisées par un Lias corralligene ou tout au moins zoogene, faisant quelquefois défaut, par des couches a Mytilus, représentant le groupe Oolitique inférieur, et par du Tithonique coralligène. Ces formations néritiques indiquent la présence d’une crete sous-marine, voire d'un chapelet diles, correspondant a un nouveau géantielinal”. In the cretaceous period intensive crustal movements took place in most of the geanticlines, from which resulted partial upheaval above the sea-level, as is borne out by lacunae in the series of cretaceous deposits. Already in old-tertiary time real mountain ranges in the geographical sense were formed, while chiefly in the neogene the high mountain ranges arose, such as the Alps and the Himalaya. We do not purpose to make a reconstruction of the aspect of the Tethys-geosyncline, as it was, during the mesozoie period, in the region of the East-Indian Archipelago. Such a reconstruction must be incomplete, since a considerable portion of the region is covered by the sea, so that our knowledge of it is little as yet. The Alpine geologist will in this respect always have the advantage not only in that the structure in the deep erosion valleys is much more denuded, but also because several continuous parts of the mountain range can be compared with each other. On the other hand Arcanp®) has already pointed out, that the study of the rows of islands of Eastern Asia and Oceania teaches us what the condition may have been of Alpine mountain ranges with a similar distribution of land and water in earlier periods. We can compare the curving rows of islands of the Moluccas with the con- 1) E. Hava. Traité de Géologie. II, p. 1125. : 2) E. ARGAND. Sur l’are des Alpes occidentales. Eclogae Geol. Helv. Vol. XIV. 1916, p. 179. 781 dition of the Western Alps in their development in the Jurassic period, as described by ArGanp'). Also here we see two geanticlines and a “Vorland” separated from each other by geosynclines. In the liias the formation of the geosynclines and geanticlines is more accentuated, which continues down to the middle-jurassic, the gean- tielines above the sealevel having disappeared. In the Upper-Jura this is followed by a moderate submersion, after which in cretaceous times the intense crustal movements begin, which reach their maxi- mum in the tertiary period. The overthrust sheets moved in the direction of the “Vorland” and eventually were pushed over it; the sea-basins of the anticlines are moving down gradually and at last disappear altogether. Oscillations, such as occurred in the jurassic period in the Alps and to which we have alluded above, are also known to us in the curving rows of islands in the Moluceas. The formation of the overthrusts was followed by a long period of denudation, then a submersion and deposition of sediments, which was followed again by upheaval above the sea-level. SUMMARY. The outwardly directed overthrusts to be observed everywhere in the Timor-Ceram curve mark the action of a tangential pressure, which caused the sediments, deposited in this region in mesozoic and tertiary until the beginning of miocene time, to be pushed in the direction of the “Vorland” and to be raised above the sea-level. The subsequent submersion may be accounted for by a temporary decrease of the intensity of the tangential pressure. The character- istics of the now appearing rows of rising islands and of the alternating sea-basins point to a recurrence of the crustal movements and do not clash with the assumption that these movements occur again in the direction of the ‘‘Vorland” and that consequently the rows of the uplifted islands indicate the spots where at greater depths the folding process continues with a tendency to form overthrusts. In this connection we refer once more to the outlying position of the Kei-, and Tenimber-Islands opposite the depressions of the “Vor- land” and the stronger uplift of the northern part of Groot-Kei. As the movements proceed the uplift of the rows of islands (with alternate intervals of temporary subsidence through decrease of the intensity of the tangential forces) will be accompanied by a shifting ') E. Ancasp. La formation des Alpes occidentales. Eclogae Geol. Hely. Vol. XIV, 1916. Pl. 3. 782 in the direction of the ‘‘Vorland’’, the sea-basins will narrow and eventually the masses of the present rows of islands will be deposited on the site of the present Australian continent, a stage which e.g. was reached long before in the Alps. The bend in the inner curve Le. that of the active volcanoes, as assumed by us, will widen and lengthen in consequence of the outward pressure in all directions. The same holds for the Timor-Ceram curve. In conclusion we will compare the way in which the voleanic rocks of the inner curve of islands occur with that of the voicanic rocks encountered at the inner side of the Timor-Ceram row of islands. A very considerable portion of the produets of the young vol- canoes is now deposited under the sea and we saw that part of the older voleanie rocks alluded to, evince characteristics indicative of a similar formation. Tbe inner curve, less elevated than the outer one will rise higher above the sea-level as the crustal movements are prolonged. When the voleanie deposits, which at this day are still lying far below the sea, will be lifted up above the sea-level, they will perhaps have been folded already by the same crustal movements and will already have been uplifted or overthrust. When these deposits become visible at the coast, erosion has for a long period already been affecting the voleanie cones and the voleanic products lying far inland; they may even have disappeared completely through erosion. It appears, therefore, that the volcanic rocks will occur in the inner row of islands in the same way as now in the outer row. KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN TE AMSTERDAM. PROCEEDINGS VOLUME XXII Nes, 9 and 10. President: Prof. H. A. LORENTZ. Secretary: Prof. P. ZEEMAN. (Translated from: “Verslag van de gewone vergaderingen der Wis- en Natuurkundige Afdeeling,” Vol. XXVIII). CONTENTS. A. SCHOENFLIES: “Zur Axiomatik der Mengenlehre”. (Communicated by Prof. L. E. J. BROUWER), p. 784. L. E. J. BROUWER: “Ueber eineindeutige, stetige Transformationen von Flächen in sich”. (Sechste Mitteilung), p. 811. F. M. JAEGER: “On the Symmetry of the Röntgenpatterns Obtained by means of Systems Composed of Crystalline Lamellae, and on the Structure of Pseudo-Symmetrical Crystals”,p.815.( With one plate). L. EERLAND and W. STORM VAN LEEUWEN: “On Adsorption of Poisons by Constituents of the Animal Body. I. The adsorbent power of serum and brainsubstance for Cocain”. (Communicated by Prof. R. MAGNUS), p. 831. A. D. FOKKER: “The Contributions from the Polarization and Magnetization Electrons to the Electric Current”. (Communicated by Prof. H. A. LORENTZ), p. 850. N. G. W. H. BEEGER: “Ueber die Zerlegungsgezetze fiir die Primideale eines beliebigen algebraischen Zahik6rpers im Körper der /-ten Einheitswurzeln”. (Communicated by Prof. W. KAPTEYN). p. 873. A. SMITS: “The Electromotive Behaviour of Aluminium” I. (Communicated by Prof. H. A. LORENTZ), p. 876. ARNAUD DENJOY: “Sur les ensembles clairsemés”. (Communicated by Prof. L. E. J. BROUWER), p. 882. FERNAND MEUNIER: “Quelques insectes de l'Aquitainen DE ROTT, Sept-Monts (Prusse rhénane)”. (Communicated by Prof. K. MARTIN), p. 891. (With one plate). M. W. BEIJERINCK: “Chemosynthesls at denitrification with sulfur as source of energy”, p. 899. K. W. RuTGers: “Complexes of Plane Cubics with Four Base Points”. (Communicated by Prof, JAN DE VRIES), p. 909. W. STORM VAN LEEUWEN and Miss C. VAN DEN BROEKE: “Experimental Influence on the Sensitivity of various Animals and of Surviving Organs to Poisons”. (Part 1). (Communicated by Prof. R. MAGNUS), p. 913. W. STORM VAN LEEUWEN and Miss VAN DER MADE: “Experimental Influence on the Sensitivity of various Animals and of Surviving Organs to Poisons”. (Part Il), (Communicated by Prof. R, MAGNUS), p. 927. JAN DE VRIES: “A Congruence of Orthogonal Hyperbolas”, p. 943. R. MAGNUS and A. DE KLEIJN: “On Optic “Stellreflexe” in the Dog and in the Cat”, p. 948, C. PH. SLUITER: “Rhythmical Skin-growth and Skin-design in Amphibians and Reptiles”, p. 954, J. DE HAAN and K. J. PERINGA: “On the genetic relation between lymphocytes and granulated leucocytes”. (Communicated by Prof. H. J. HAMBURGER), p. 962. Erratum, p. 974. 52 Proceedings Royal Acad. Amsterdam. Vol. X XIL Mathematics — “Zur Amomatik der Mengenlehre’. By Prof. A. SCHOENFLIES, Frankfurt a. M. (Communicated by Prof. L. B. J. BROUWER). (Communicated at the meetings of February 28 and March 27, 1920). Die Hilbertsche Grundlegung der Geometrie darf für alle analogen Untersuchungen als vorbildlich gelten. Zwei ihrer Eigenschaften sind es, auf die es hier ankommt. Erstens wird von allen sprachlichen Definitionen der Objecte, mit denen sie operiert, wie Punkt, Gerade, zwischen u.s.w. abgesehen; nur ihre gegenseitigen Beziehungen und deren Grundgesetze werden axiomatisch an die Spitze gestellt *). Zweitens werden die Axiome in verschiedene Gruppen gewisser Higenart und Tragweite gespalten (die des Schneidens und Verbin- dens, die Axiome der Ordnung, der Kongruenz u.s.w.), und es ist eine wesentliche Aufgabe des axiomatischen Aufbaues, zu prüfen, bis zu welchen Resultaten eine einzelne oder mehrere dieser Gruppen fiir sich fiihren. Die gleiche Behandlung eignet sich fiir die Mengen- lehre. Von sprachlicher Hinfiihrung der Begriffe Menge, Bereich u.s.w. ist daher ebenso abzusehen, wie von der des Punktes oder Raumes. Ebenso kann man hier gewisse Axiomgruppen unterscheiden, die Axiome der Aequivalenz, die Axiome der Ordnung u.s.w. und kann die gleichen Fragen stellen, wie im Gebiet der Geometrie. Dies soll im Folgenden geschehen, und zwar fiir denjenigen Teil, der nur mit der Aequivalenz der Mengen, der Mengenteilung und Men- genverbindung, sowie der Mengenvergleichung operiert. Will man die Probleme der Mengenlehre einer derartigen Behand- lung unterwerfen, so ist es oberstes Erfordernis, die Begriffe der endlichen und der unendlichen Menge auf einer Grundlage einzu- führen, die nur die ebengenannten Fundamentalbegriffe benutzt. Solehe Definitionen sind ja in der Dedekindschen Begriffsbestimmung vorhanden: Eine Menge JM heisst unendlich, wenn es eine (ächte) Teilmenge M’ von M giebt, die aequivalent M ist; sie heisst endlich, 1) Der Euklidische Aufbau beginnt noch mit den Worten: Ein Punkt ist, was keine Teile hat. Eine Linie ist eine Länge ohne Breite usw. In dem Verzicht auf alle solchen sprachlichen Begriffsbestimmungen liegt einer der wesentlichen Hilbertschen, und durch ihn modern gewordenen Gedanken. Die Mengenlehre hat sich ihm bisher nicht erschlossen. 785 wenn es eine solche Teilmenge nicht giebt. Sie haben daher den alleinigen Ausgangspunkt zu bilden. Die historische Entwiekelung der Mengenlehre ist freilich wesent- lich anders vor sich gegangen. Während vorstehend die wnendliche Menge als das logisch posztiv bestimmte Object erscheint, und die endliche Menge als ihr logisches Gegenteil, ist die historische Ent- wiekelung umgekehrt von den endlichen Mengen als woh!bekannten mathematischen Objecten ausgegangen, und hat die unendlichen Mengen als Gegensatz der endlichen Mengen eingefiihrt. Der so benutzte Begriff der endlichen Mengen gehört aber bereits einem Gebiet an, das sich nicht mehr ausschliesslich an die Aequivalenz- beziehungen anschliesst. Der historisch überkommene Begriff der endlichen Menge ruht ja überhaupt nicht auf axiomatischer Grund- lage. Mag man ihn sprachlich oder empirisch oder anschaulich auffassen, er war im wesentlichen an der Hand des Zahlbegriffs entstanden und ruht jedenfalls auf Voraussetzungen, in die auch die Ordnung als Grundbegriff eingeht. Diese gehort aber bereits einer Begriffsgruppe an, von der hier abzusehen ist. So laufen in der historischen Entwickelung der Mengenlehre zwei wesentlich verschie- dene Bestimmungen der endlichen und unendlichen Mengen unver- mittelt neben einander her und erschweren infolgedessen die Frage nach dem, was den einzelnen Sätzen axiomatisch zu Grunde liegt. Auch insofern ist eine Klarung des Sachverhalts wiinschenswert. Das Resultat erweist sich in zwei Punkten als durchaus eigenartig. Die Vergleichung der Mengen beziiglich ihres Grössencharacters ist nämlich nichts, was dem Mengenbegriff allein eigentümlich ist; sie betrifft allgemeiner alle Objecte, für die man das Ganze und den Bestandteil unterscheiden kann. Die Axiomatik, die hier entwickelt wird, ist also richtiger eme Awiomatik der Gröszenlehre, und zwar in dem besonderen Fall, dass es auch Grössen von wend hichem Character giebt. Dies bedingt, dass die Mlemente der Mengen im Folgenden gar nicht benutzt werden; immer nur bilden die an sich möglichen Beziehungen zwischen den Ganzen und ibren Teilen den Gegenstand der Untersuchung. Deven auf axiomatischer Grundlage ruhende, umfassende Krörterung bildet den eigentlichen’ Inhalt der Arbeit. Ich habe aber doeh die gewohnten Mengenbezeichnungen beibebalten. Für die Mlemente der Mengen wird erst am Schluss eine auf den Begriff der Teilmenge sich stützende Winfiihrungs- mögliehkeit gezeigt. Sie erscheinen als solehe Teilmengen, die selbst niebt weiter in Teilmengen zerlegbar sind (gleichsam als die Atome). Kine zweite Kigenart der Untersuchung betrifft die logischien Not- wendigkeiten, die dev axiomatische Aufbau dieses besondern Gebietes 52” 786 verlangt. Ausser den selbstverstandlichen axiomatischen Festsetzungen über die Regeln, nach denen man mit den Begriffen der Aequivalenz, der Teilmengen usw. zu operieren hat, treten auch noch Annahmen ~auf, die man wohl nicht erwarten mag. Bei ihrer Einführung handelt es sich aber — und darin besteht die genannte Higenart — weniger um spezifisch mathematische Notwendigkeiten, als vielmehr um rem logische, also um Festsetzungen, die deshalb nötig sind, weil man ohne sie — um welches wissenschaftliche Gebiet es sich handeln mag - aus den in Frage stehenden Voraussetzungen Schlüsse über- haupt nicht ableiten kann. Hin allgemeiner Grundsatz der Logik lautet: E mere negativis nihil sequitur; d.h. aus lauter negativen Prämissen kann eine Folgerung nicht gezogen werden. Aus den Sätzen kein A ist ein B, kein ® ist ein © lässt sich in der Tat eine Beziehung zwischen % und € nicht ent- nehmen; und ebensowenig gestatten die Sätze kein ®B ist ein U, kein © ist ein U eine Beziehung zwischen B und €). Gerade solche Prämissen sind es aber, die uns bei den mengentheoretischen Problemen mehrfach begegnen, und deshalb der Einführung einer zwischen 4 und € oder zwischen DB und © vorhandenen Beziehung den Stempel der axio- matischen Notwendigkeit aufdrücken. $ 1. Die Aequivalenz. Die mathematischen Objecte, von denen im Folgenden die Rede sein wird, heissen Mengen (Teilmengen, Verbindungsmengen). Alle sollen denselben Aequivalenzbeziehungen gehorchen, die wir als Axiome der Aequivalenz (~) einfiihren. Sie lauten: Sind M, NV, P verschiedene Mengen, so gilt I Aus M= N folgt N~ M. Ik Aus) Mie NE unde P folet iP: 1) Aus den Vordersätzen U ist nicht BV, U ist nicht © kann freilich in gewissen Fallen doch eine positive Folgerung gezogen werden und zwar für Xl selbst. Nämlich dann, wenn man eine zwischen 8 und © bestehende positive Beziehung kennt. Aus den Sätzen: Das Dreieck D ist nicht spitzwinklig und Das Dreieck © ist nicht stumpfwinklig folgt, dass D rechtwinklig ist. Hier liegen nämlich nur scheinbar ausschliesslich negative Prämissen. vor; zu ihnen kommt als positive der Satz: Jedes Dreieck ist entweder spilzwinklig oder stumpfwinklig oder rechtwinklig. Vgl. auch die Anmer- kung auf S. 839. 787 Der Aequivalenzbegriff hat also sowohl kommutativen, wie auch assoziativen Character. Aus diesen Axiomen folgt: 1. Aus M~ N und N nicht ~ P folgt M nicht ~ P. Denn ware If ~ P, so würde daraus in Verbindung mit N ~ M gemäss I weiter V ~ P folgen, im Gegensatz zur Voraussetzung. Die Axiome | u. II zeigen, dass sie die Ausdehnung auf den Fall zulassen, dass M und N dieselbe Menge bedeuten. Wir fügen also als weiteres Axiom hinzu Ill. Es ist M= M. § 2. Teilmengen und Verbindungsmengen. Ist M’ Teilmenge von J, so soll dies durch M't M bezeichnet werden. Wir nehmen durchweg an, dass M' von M verschieden ist, und nennen insofern M’ auch ächte oder eigentliche Teilmenge von M. Für die Teilmengen sollen folgende Axiome gelten (Aviome der Teilmengen) : I. Aus M’tM und M’’ tM’ folgt M’’ tM. II. Jede Teilmenge M’ von M bestimmt eindeutig eine zweite Teil- menge M/, von M, die ihre Komplementürmenge bezüglich M heisst. UL Die Komplementärmenge von M, ist wiederum M'. Wir dürfen daher folgende Bezeichnungen einführen. Wir schreiben M,k M' resp. M'k M, und setzen dem gemäss (III) in die Form ll’. Aus M,k M' folgt Mk M,. Fiir die Beziehung von M, und MW’ zur Menge M selbst schreiben wir M=(M,, M')=(M', M,), und sagen, dass M in die Teilmengen M’ und M, zerfällt. Zusam- menfassend kénnen wir also sagen: Aus M’tM folgt M,tM, Mk M’, MkM, M=(M’, Mm). Seien nun M und MN zwei Mengen, so können bezüglich ihrer Teilmengen zwei Fille eintreten. Entweder gibt es für M und N identische Teilmengen, oder es giebt keine solchen Teilmengen. In diesem Fall nennen wir die Mengen fremd zu einander, oder kurz fremd, und schreiben Mf N resp. Nf M. Fiir fremde Mengen gilt der Satz: 1. Sind M und WN fremde Mengen, so ist auch jede Teilmenge von M zu jeder Teilmenge von N fremd; d. h. 785 Aus MN, M’tM, N’ tN folgt M’ f N’. Waren nämlich die Teilmengen MW’ und WN’ nicht fremd, und ist P eine in beiden enthaltene Teilmenge, so hatte man 10 AD UE ET obi JPG I NGE und daher gemäss I auch PtM und Pt N, im Widerspruch mit der Voraussetzung. la. Der Satz gilt auch so, dass M’ zu N selbst, und ebenso NV’ zu M fremd ist. Der Beweis ist derselbe. Wir stellen weiter folgende Axiome auf: IV. Die beiden Komplementirmengen M’ und M, einer Menge M sind fremde Mengen; d. h. Aus M,k M’ folet M, f M’. Diese Beziehung soll aber auch umgekehrt gelten ; zu diesem Zweck führen wir folgendes weitere Axiom ein (Aviom der Verbindungs- mengen). V. Zwei fremde Mengen N und P bestimmen eine und nur eine Menge M, deren Komplementdrmengen sie sind; d. h. Aus Nf P folgt Ni M, Pt M und NEP. Die Axiome IV und V lassen sich also auch so auffassen, dass die Beziehungen N&P und NFP gleichwertig sind. Wir nennen die Menge (NV, P) die Verbindungsmenge von N und P. Es folgt noch 2. Die Mengen MN und P sind von ihrer Verbindungsmenge M = (N, P) verschieden. Denn da sie nach V Komplementärmengen von M sind, so ist jede eine ächte Teilmenge von M. Die Menge (N,P) hat ausser N und P gemäss Axiom I auch jede Teilmenge N’ und P’ zu Teilmengen. Damit sind aber, wie wir durch ein weiteres Axiom festsetzen, nicht ihre sämtlichen Teilmengen erschöpft. Gemäss Satz (1) und (la) ist auch N’ zu P’ fremd, ebenso N’ zu P und WN zu P’; nach Axiom V giebt es daher je eine Menge CNG ACNE) Bun AEN Ge): Für sie setzen wir nun fest: VI. Ist M = (N,P) so sind auch die Mengen. (OSL (UNG LO) (UNG 124) Teilmengen von M; es ist aber auch jede von N, N’ P, P’ ver- schiedene Teilmenge von dieser Form. Wir folgern hieraus den Satz: 789 3. Ist M= CN, P) und ist die Menge Q fremd zu N und fremd zu P, so ist sie auch fremd zu M; d.h. Aus Qf N und QfP folgt Qf(N. P). Ware nämlich die Menge Q nicht fremd zu M, so gabe es für sie und JM eine identische Teilmenge; d.h. es gabe eine Teilmenge Q’, die gemäss VI eine der Formen Mo INS JPL TP, (UNG IBN (A IP) KNLS) haben miisste. Diese Teilmenge Q’ hätte also jedenfalls N oder P oder eine Teilmenge von N oder P als Teilmenge; d.h. es gabe eine Teilmenge Q" von Q', die mit N oder P oder einer Teilmenge von N oder P identisch wäre. Nun ist aber nach I Q" auch Teilmenge von Q, und damit ergiebt sich ein Widerspruch gegen die Voraus- setzung Der Satz (3) lässt sich auch in die Form setzen: 3a. Ist die Menge Q nicht fremd zur Menge (JN, P) aber fremd zu MN, so ist sie nicht fremd zu P. Will man den Begriff der Verbindungsmenge auf mehr als zwei Mengen ausdehnen, so hat man ein neues Axiom nötig. Es ist jedoch fiir das Folgende nicht erforderlich dies näber auszuführen. § 3. Die Verknüpfung der Mengen. Die verschiedenen Beziehungen, die zwischen zwei Mengen M und .V Platz greifen können, sind aus der folgenden von CANTOR angegebenen Aufzählung aller Möglichkeiten ersichtlich, die unsern Ausgangspunkt abgeben soll: a. Es giebt ein M’ ~ N, und ein N’ ~ M. b. Es giebt kein M, ~ JN, aber ein N’ ~ M. c. Es giebt ein M’ ~ N, aber kein NM, ~ M. d. Es giebt kein M, ~ N, und kein NV, ~ M’). Wir wollen diese vier Beziehungen durch MaN, MbN, MeN, MdN...... (A) darstellen. Man erkennt zunächst unmittelbar: 1. Die Beziehungen (a) (4) (c) (d) schliessen einander gegenseitig aus. 2. Die Beziehungen Ma N und Na M, ebenso Md Nund Nd M sind identisch. Die Beziehung 176 N ist identisch mit Ne M. Wir erörtern sofort, welche dieser Beziehungen sich auf den Fall ausdehnen lassen, dass M/ und N dieselbe Menge bedeuten. Es findet sich 1) Die Anwendung oberer und unterer Indizes bei den Teilmengenim positiven und negativen Fall soll im Allgemeinen zur Erleichterung der Auffassung beibehalten werden. 790 2a. Die Beziehungen Mb M und Me M sind widerspruchsvoll. Sie fordern nämlich das gleichzeitige Bestehen von M’ — M und kein M, — M. Dagegen sind die Beziehungen Ma M und Md M widerspruchsfrei. Uebrigens lässt sich dies auch als unmittelbare Folge von (1) und (2) auffassen. Sei P eine weitere Menge, so besteht zwischen N und P eben- falls eine der Beziehungen NINE NBSRR MINE END aero sc RK) und es entsteht die Frage, welche Folgerung sich für die Mengen M und P einstellt, wenn man eine Beziehung der Reihe A mit einer Beziehung der Reihe 5 kombiniert. Diese Aufgabe lässt sich ohne Hinfiihrung neuer Axiome nicht erledigen. Ein erstes, das den Begriff der Teilmenge mit dem der Aequivalenz verbindet, sei das folgende: I. Aus den Relationen M'tM, MN lassen sich die Relationen N't M, N'— M' folgern; d.h. Ist M~ N, so bedingt eine jede Teilmenge M’ von M die Existenz einer Teilmenge N’ von N, die zu M’ aequivalent ist. Vielleicht mag man erwarten, dass die Menge N’ als diejenige wolbestimmte Menge eingeführt wird, die der Menge J/’ gemúss der zwischen M und N bestehenden Aequivalenz entspricht. Aber dies ist fiir den hier vorgenommenen Aufbau — jedenfalls an dieser Stelle — weder möglich noch nötig. Es genügt, die Mwistenz einer Menge N’ zu fordern; welches diese Menge ist, darf ganz offen bleiben. Es hängt dies damit zusammen, dass die Aequivalenz M— N in ihrer besondern Eigenart hier nicht in Frage kommt; nur die Relationen, die die Higenart des Aequivalenzbegriffs kennzeichnen, und fiir zwei Mengen und ihre Teilmengen bestehen, werden in Betracht gezogen. Einen Teil der oben gestellten Frage hat bekanntlich schon Cantor selbst beantwortet; man zeigt leicht 3. Aus Ma N und NaP folgt Ma P. Aus Ma N und Nb P folgt Mb P. Aus Ma WN und Nc P folgt Mc P. Aus Mb N und NOP folgt Md P. Aus Mc N und Nc P folgt Mc P’). Sw) oP CN he 1) Diese Tatsachen entsprechen bekanntlich dem Umstand, dass wenn man den Fallen a,b,c die Beziehungen ,gleich”, ,kleiner”, grészer’ zuweist, die für 791 Die Beweise sind natürlich ausschliesslich auf die in a, b,c, d ent- haltenen Beziehungen zu stiitzen. Ein Beispiel mége zeigen, wie sie sich fiihren lassen. Um aus den Relationen MobN und Nb P weiter Mb P zu folgern, haben wir von kein M, — N, ein N’ — M kein Ne eine iN, auszugehen, und daraus die Beziehungen kein Mi =P, em PM abzuleiten. Wir beweisen zunächst den zweiten Teil. Wegen P’ ~ N giebt es nach / eine Teilmenge P" ~ N’, und aus N’ ~ M folgt nun P" ~ M. Die Richtigkeit der ersten Behauptung erweisen wir indirect. Ware nämlich ein M/ ~ P,so folgte gemäss / aus V’ ~ M wiederum die Existenz einer Menge N" von WV’, fiir die WV" ~ MW’ sein miisste, und aus MENG = weiter Nis a0 has während kein NV, ~ P sein kann. Es bleibt noch übrig, das gleichzeitige Bestehen der Beziehungen 116 N und Ne P zu untersuchen, sowie die Kombination von M/ d N mit einer der Beziehungen Nar IND) SEL IN GIES INIGLTE, Hier gilt zunächst, dass aus 1/6 N und Nc P eine bestimmte Beziehung zwischen M und P nicht folgt; d.h. 8. Mit M5 N und Ne P ist jede der vier Beziehungen Ma P, Mb P, Mc P, Md P verträglich. Der Beweiss darf unterbleiben. Nur sei bemerkt dass dies dem realen Tatbestand entspricht, dessen axiomatische Grundlegung hier in Frage steht *). Wir gehen nun zu dem Rest unseres Problems über und prüfen zunáchst die Kombination von VRON ATION AG RENE NC) Die Frage lautet auch hier, ob die Beziehungen (a) eine bestimmte Beziehung zwischen M und P bedingen und eventuell welche. Hier liegt der in der Einleitung genannte Fall vor, dass es sich um lauter negative Prámissen handelt. Diese Prämissen sind diese Beziehungen geltenden assoziativen Gesetze erfüllt sind (z, B. aus a = b und b=c folgt a=c usw.) lj Für Mächtigkeiten würden die Relationen mp bestehen; sie bedingen keine Grössenbeziehung zwischen m und p. 792 kein M, - N, kein NM, — M, 8 RE Ten VP, MAN Aus ihnen lässt sich auf directem Wege über die Beziehung von M zu P nichts schliessen. Teilweise gelingt es allerdings auf indirec- tem Wege; in einzelnen Fallen kommt nämlich dadurch zu den obigen Prämissen eine neue Tatsache hinzu, die positiver Natur ist. Um die Untersuchung durchzuführen, hat man nämlich zu prüfen, ob die Annahme einer der Beziehungen Mia PAMGENMeEP MAP. 4 IRO) auf Grund der bisherigen axiomatischen Festsetzungen einen Wider- spruch mit dem gleichzeitigen Bestehen der Beziehungen @ bedingt, und zwar kommen naturgemäss hier nur die Axiome von § 1, das obige Axiom § 3, / und der obige Satz 2 in Frage. Diese Prüfung haben wir ausführlich vorzunehmen *). Zunächst sieht man leicht, dass die Beziehungen Mb P und ebenso Mc P als Folgen von (a) auszuschliessen sind. Wegen Satz (2) kann man nämlich die. Beziehungen (a) auch in die Form Pd N und NdM setzen, und müsste daher als Folgerung von (a) auch Pb M oder Pc M erhalten. Aber M5 P und P6 M, und ebenso Me P und Pe M sind nach Satz (2) nicht identisch, womit die Behauptung erwiesen ist °). 1) In den Math. Ann. Bd. 72, S. 551 (1912) ist diese Untersuchung schon teil- weise durchgefiihrt worden, 2) Die logische Eigenart des oben behandelten Problems entspricht also nicht ganz dem in der Einleitung genannten Tatbestand. Es lautet nämlich genauer so: Welche von vier möglichen Beziehungen wird durch die dem Problem eigen- tümlichen nur negativen Prämissen ausgeschlossen? Bei der Annahme, Mb N oder Mc WN seien die Folgen dieser negativen Prämissen, wird von selbst eine neue Tatsache eingeftihrt; die Symmetrie der Beziehungen Md N und Nd P bezüglich Mund P steht nämlich im Gegensatz zu der Unsymmetrie der Folgerungen MbP oder McP fiir Mund P. Und daher ergab sich oben ein Resultat. Die Annahme, MaP oder MdP seien die Folgen der negativen Prämissen, liefert dagegen eine solche neue Tatsache nicht; es ergiebt sich daher, wie das obige weiter zeigt, ein Resultat nicht. Allgemeiner gesprochen: Wenn die Prämissen: 9 ist nicht Bund B ist nicht € sich auch in die Form setzen lassen € ist nicht ® und ® ist nicht 9, so kann damit nur eine solche Beziehung zwischen U und € vereinbar sein, die zugleich die ndmliche Beziehung zwischen © und Y bedeutel. Eine genauere Analyse des hiermit mehrfach besprochenen logischen Problems von Seiten der Logiker wäre sehr erwünscht. Das letzte Wort soll mit dem vorstehenden nicht gesprochen sein. 793 Es ist weiter zu untersuchen, ob sich die Beziehung IR DER NE EM ch | erase) als Folge von (a) einstellen kann. Hier ist ein Resultat, das dies unmöglieh macht, nicht erhältlich. Die Beziebung Ma P bedeutet nämlich im Wd) on JL leen ba deeg (67) Die Verbindung mit («) liefert gemäss § 1 die weiteren Relationen kein VV, — P’, kein N, — AV’. Genauer bedeutet dies: Es giebt eine Teilmenge P’, der keine Teilmenge von .V aequivalent ist, und es giebt auch eine Teilmenge MW’, der keine Teilmenge von MN aequivalent ist. Dies stellt aber einen Widerspruch zu («') oder zu (y') nicht dar. Es soll noch eine zweite Prüfung vorgenommen werden; wir haben auch den assoziativen Character der Beziehungsregeln in Betracht zu ziehen. Ist Ma P das Resultat von («), so heisst dies, dass das gleichzeitige Bestehen von MaN, NdP, MaP nicht widerspruchsvoll sein darf. Nun sollen aber zwei von diesen Beziehungen stets eine dritte bedingen, und daraus folgt, dass aus Ma P und Pd WN wieder Md N und aus Nd M und Ma P wieder Nd P folgen muss. Es ist nun die Frage, ob diese Regeln einen wider- spruchslosen Character haben. Dies ist in der Tat der Fall. Man sieht es am einfachsten daraus, dass man die assoziativen Gesetze, die die Beziehungen (a) und (d) mit einander verbinden, wenn man noch Satz (3) beachtet, in die einfache Form (aa) =(dd)=a, (ad) =(da)=d setzen kann; sie sind das genaue Analogen zu den Vorzeichenregeln EIZ ti HI) (WI) deren assoziativer Gesamtcharacter feststeht. Wir haben endlich noch die Beziehung MAREN EAR ve en tte Uv) als mógliche Folge der Beziehungen (a) zu erörtern. Sie bedeutet kenen kenen oe es re (0!) Hier zeigt sich zunächst, dass sich aus ihr und den Relationen («') weitere directe Folgerungen überhaupt nicht entnehmen lassen, da sie jetzt saml und sonders negativer Natur sind. Wir prüfen auch hier noch den assoziativen Gesamtcharacter. Ist Md [ das Resultat von Md N und Nd P, so bedingt es jetzt, dass 794 aus Md P und Pd N wieder Md N und aus Md M und Md P wieder Md P folgt ; hier aber ist der widerspruchsfreie Character evident. Also folgt: 9. Mit den Beziehungen MdN und NdP kann sowol die Beziehung MaP, wie MdP zugleich bestehen. Keine der beiden Annahmen y und d führt also auf einen Wider- spruch mit den in (a@’) enthaltenen Prämissen; wir können daher auf diesem Wege nicht zu einem Resultat über die vorliegende Frage gelangen. Man muss daher in der Tat die Folgerung, die sich aus MdN und NdP ergiebt, aviomatisch einführen ; naturgemäss so, wie es durch den realen Tatbestand der Mengenlehre gefordert wird. Ihn aufzubauen ist ja einer der Zwecke dieser Darstellung. Wir setzen daher fest (Aviom der Verkniipfung) U. Aus Md N und Nd P folgt Md P. Hieraus erhalten wir nun leicht die Antwort auf die noch aus- stehenden Verknüpfungen für die Beziehungen (A) und (B). Zunächst beweist man 10. Aus Mb N und NdP folgt Md P. 10a. Aus Mc N und NdP folgt Mc P. Fiir den Beweis von (10) haben wir auszugehen von kein M, — N, ein N’ = M, kein VN, - P, kein P, — N, und daraus die Beziehung Mb P, also kein M, — P, ein P'- M abzuleiten. Wir folgern zunächst, dass eine Beziehung M" — P unmöglich ist. Aus V’ ~ M würde nämlich auf Grund dieser Annahme die Existenz einer Teilmenge NV” folgen, fiir die N" = M"=P wäre, im Widerspruch zu kein NV, ~ P. Damit ist die Beziehung kein M, ~ P erwiesen. Es ist jetzt noch zu zeigen, dass es ein P’~ M giebt. Ware dies nicht der Fall, so bestände auf Grund des vorstehenden jetzt die Beziehung kein UM, — P, kein P, — M, also die Relation Md P, und zusammen mit der vorausgesetzten Beziehung Pd N folgte gemäss Axiom II die Beziehung Md N, im Widerspruch zu M6 N. Damit ist der Beweis wieder geliefert. Ebenso wird der Beweis für Me N und Nd P geführt, was einer ausführlichen Darstellung nicht bedarf. 795 Wir haben schliesslich noch die Kombination von Ma N und Nd P zu erörtern. Wir folgern zunächst, dass diese beiden Relationen an sich nur die Folge MdP gestatten. Wir haben auszugehen von M’ = N, N’ = M und kene Va ave keine = NV, und zeigen zunächst, dass hiermit nur kein M, — P, kein P, — M, verträglich sind. Gabe es nämlich eine Menge MM" ~ P, so folgerte man wie oben eine Relation INU = i ac J) im Widerspruch mit der Voraussetzung: kein NV, ~ P; ebenso folgt die Unmöglichkeit einer Beziehung P" ~ M. Es kann also an sich nur die Relation MdP bestehen. Wiederum ist noch der assoziative Character des Resultats zu prüfen. Diese Prüfung führt hier auf einen Widerspruch. Aus MdP und NdP würde nämlich gemäss dem Axiom IL Md N folgen, im Widerspruch mit der Annahme Ma N. Das gleichzeitige Bestehen von Ma N und Nd P führt also auf einen Widerspruch; d.h. 11. Die Beziehungen Ma N und Nd P können nicht zugleich bestehen. Dagegen sei ausdrücklich festgestellt, dass die Sätze (10) und (10a) einen solchen Widerspruch nicht herbeiführen. Denn gemäss (2) ist Mb P mit Pe M identisch, und die beiden Beziehungen Pe M und MEN sind, wie wir oben unter (8) erwähnten, mit jeder der vier an sich möglichen Beziehungen zwischen N und P verträglich. Damit ist unsere Untersuchung abgeschlossen; sie zeigt zugleich die Widerspruchslosigkeit des Axioms 1. Wir ziehen aus ihm zunächst noch eine Folgerung; nämlich die, dass der Satz (11) auch in der Weise gilt, dass er das gleichzeitige Bestehen von Ma M und Md N, sowie von Ma N und Nd N ausschliesst. Aus Ma M folgt MW’ ~ M und hieraus gemäss § 3, | M" — M'=— M, und daher besteht auch die Relation M'a M; diese kann aber nach Satz (11) nicht mit Md N zugleich bestehen. 796 Weiter folgt aus Ma N zunächst Vis NAE NIE also auch N' ~ M'~ N, während dagegen Nd N besagt, dass kein N,~ N ist. Also 1la. Die Beziehungen MaM und Md N, ebenso Ma N und Nd N schliessen eimnander aus. Es ergiebt sich damit das folgende Schlussresultat. Mit den Be- ziehungen MdN und NdP erscheint sowol die Folgerung Ma P, wie auch die Folgerung Md P vertriglich. Wird die Relation Md P axiomatisch als Folgerung eingeführt, so bedingt dies, dass die Beziehungen Ma N und NdP nicht zugleich bestehen können; würde man dagegen die Beziehung Ma P axiomatisch als Folgerung einführen, so ergiebt sich ein derartiges Resultat nicht. Trotzdem erfordert der Aufbau der Mengen- lehre die Einführung der Folgerung Md P. Auf die Deutungs- möglichkeit der axiomatischen Annahme Ma P komme ich in § 7 zurück. Für die Beziehungen (a), (b),c,d gelten noch die folgenden beson- deren Sätze: 12. Aus den Relationen MaN, MbN, MeN, MdN und M- i, Ne N folgt auch Ma N, MEN, mc N, Md N und MaN. MON, MeR, MAN Für den Beweis mag ein Beispiel genügen. Werde von MbN und M= MN ausgegangen, so heisst dies IN’ — M, jedes M, nicht — A. Wir erhalten daher, falls M, ~ ©, ist, gemäss $ 1 sofort N’ — MX, jedes M, nicht — A, womit die Behauptung erwiesen ist. 13. Aus M’t M folgt M'a M oder M’6M; dh. Für jede Teil- menge M’ gilt entweder M’ a M oder Mb M. Es giebt nämlich eine Teilmenge von M, die aequivalent M’ ist, nämlich J’ selbst, und daher ist die Beziehung (c) und (d) ausge- schlossen. 797 14. Aus M’tM und MON folet Wb N; d.h. Besteht die Beziehung M5 N, so besteht für jede Teilmenge M’ von M die Beziehung JM’ 6 N. Man hat nämlich gemäss (13) und nach Voraussetzung. M’ aM oder M’b M und Mb N, und damit gemäss Satz (4) und (6) die Behauptung. 15. Aus MtM, M’tM’, M''b M folgt M'’5 M; d.h. Sind M' und MM" Teilmengen von M, für die die Beziehung M" 6 M' gilt, so ist auch M"b M*). Man hat nämlich wieder zugleich (nach 13) M’bM und M aM oder M'b M und folgert daraus wie eben M"5 M. § 4. Endliche und unendliche Mengen. Nach § 3, Satz (1) und (2) sind Ma M und Md M die beiden einzigen der Beziehungen (a), (4), (c),(d), die eine Menge zu sich selbst haben kann; wir definiren nun: 1. Kine Menge heisst wn- endlich, wenn die Beziehung M a M besteht; sie heisst endlich, wenn M dM gilt. Man hat also im ersten oder zweiten Fall ein M’ — M; kein M, — M, und damit die Dedekindsche Begriffsbestimmung. Wir folgern zunächst: 2. Aus MaM oder MdM und M— ®M folgt MaM und MAM. Dies ist eine unmittelbare Folge von $ 3, (12). Fúr endliche und unendliche Mengen bestehen gewisse Sonder- sdize; diese sollen jetzt abgeleitet werden. Das Haupttheorem lautet: 3. Fiir unendliche Mengen können nur die Beziehungen (a), (0), (c) bestehen; für endliche Mengen nur (b), (©), (d). Der Beweis ergiebt sich unmittelbar aus den in $ 3 abgeleiteten Resultaten. Sind nämlich M und N unendliche Mengen, und würde die Be- ziehung Md N bestehen, so hätte man Ma M und Md N, und dies verstésst gegen den Satz (11a) von § 3. Ebenso, wenn M und MN endliche Mengen sind, so hätte man. falls sie die Beziehung Ma N gestatten, Na M und Md M, und auch dies verstösst gegen Satz (11a) von § 3. Damit ist der Saiz (3) bewiesen. Er giebt zugleich den inneren ') Dieser Satz berfihrt sich inhaltlich mit dem Satz 25 in Zermelos Grund- lagen (Math. Ann. 65, S. 271). 798 Grund fiir die im Satz (11) von § 3 enthaltene Unvereinbarkeit von Ma N und Nd P. Denn unserm Satz (3) gemäss besagt Ma N, dass M und XN unendliche Mengen sind, und N d P, dass N und P endliche Mengen sind. Beides schliesst sich aber aus. 4. Fiir jede Teilmenge einer endlichen Menge besteht die Beziehung Mose dhe: Aus Md M und M’t M folgt MW’ dM. Gemäss Satz (13) von § 3 gilt nämlich für jede Menge M und eine Teilmenge J’ von ihr M' a M oder M’ 6 M. Hierzu kommt, da M eine endliche Menge ist, Md M. Diese Be- ziehung kann aber nach Satz (11) von §3 mit W a M nicht zugleich bestehen ; also muss es J/’ 6 M sein. Die weiteren noch abzuleitenden Sätze machen die Einführung eines neuen Axioms nötig, und zwar eines Awioms über die Aequi- valenz von Verbindungsmengen. Es lautet: [Au SUSSEN SD) RENE Lat ENDE) EE d.h. werden in der Verbindungsmenge (N, P) die Mengen N und P durch die zu ihnen aequwalenten zu einander fremden Mengen N und P ersetzt, so ist die neue Menge der urspriinglichen aequivalent. Das Axiom gilt gemäss § 1, III auch für den Fall, dass nur eine Menge durch eine aequivalente ersetzt wird, d.h. 5 As Wil UNG 12), AN = ote Sty JP roller GN, 1) (OR) 5 Wir beweisen nun der Reihe nach folgende Sätze: 6. Jede Teilmenge einer endlichen Menge ist selbst eine endliche Menge; d.h. Aus Md M, M’tM folgt M’d M’. Ware nämlich M’ eine unendliche Menge, so müsste eine Beziehung M"— M' bestehen. Setzt man nun M=(M',M), so ist gemäss § 2, VI auch M" — (M", M,) eine Teilmenge von M, und aus Satz (5) folgte M" = M; was einen Widerspruch gegen MJM darstellt. 7. Ast M eine endliche, N eine unendliche Menge, so kann nur die Beziehung MON bestehen*);d.h. Aus MAM und NaN folgt MON. ') Es liegt nahe, Satz 5) als Axiom hinzustellen, und das Axiom als Folge. Der Beweis hätte aber die sachlich überflüssige Annahme N f P notig. 2) Auf diesen Satz wurde ich vor längerer Zeit von Herrn H. Haun auf- merksam gemacht. TSS) Der Beweis wird so gefiihrt, dass die Unvereinbarkeit der Voraus- setzungen mit MaN, MeN, MdN gezeigt wird. Würde zunächst die Beziehung Ma N bestehen, so hätte man M'—= N; und demgemäss erhielte man aus der Annahme MaN nach $ 3 Satz 12 weiter auch Ma M’ resp. M’ a M, was aber, da M endliche Menge ist, gegen Satz (4) verstösst, Ware zweitens MeN in Kraft, so folete daraus M' — N,und nun hieraus und aus NaN weiter Wa’, was wiederum einen Widerspruch zum Satz (6) darstellt. Endlich ist auch die Beziehung MdN unméglich. Dennaus MaM folgt zunächst INN hieraus und aus NaN und der angenommenen Relation MdN folgte dann weiter NaN’ und Md N? resp. N’d M. Die Beziehungen NaN’ und N’dM sind aber gemäss $ 3 Satz (11) nicht zugleich möglich. Also gilt in der Tat die Beziehung MON. 8. Ist M eine unendliche Menge, so ist auch die Verbindungs- menge (M, N) eine unendliche Menge. Der Beweis ist eine unmittelbare Folge des Axioms I. Denn aus M’ — M folgt (M,N) — (M’, N) und damit ist der Satz, da (J/’, N) Teilmenge von (M7, N) ist, bewiesen. 9. Eine Menge ist unendlich, wenn sie eine unendliche Teilmenge hat. Ist nämlich M’ diese Teilmenge, so ist M= (M', M‚) und daher gemäss Satz (8) auch M eine unendliche Menge. Man kann diesen Satz auch noch so formulieren: 9’. Eine Menge ist endlich, wenn jede ihrer Teilmengen endlich ist. 10. [st M eine endliche Menge, so ist stets Mb (M,N); d.h. Aus Md M folgt Mb (M,N). Es ist nämlich M Teilmenge von (VW, NV). Ist nun (M/, N) endlich, so folgt der Satz aus (6), ist aber (7, NV) unendlich, so folgt er aus (7). Zur Ableitung weiterer Sätze bedürfen wir neuer Axiome. Das Axiom | besagt, dass die Verbindungsmengen aequivalenter Mengen selbst aequivalent sind; wir haben jetzt noch zwei Axiome nötig, die die Nichtaeguivalenz der Verbindungsmengen nicht aequivalenter Mengen betreffen. 53 Proceedings Royal Acad. Amsterdam. Vol. XXII. 800 II. Sind M und N fremde Mengen, ist M, Teilmenge von M und N, Teilmenge von N, und ist M, nicht == M, N, nicht~ N, so folgt daraus die Beziehung (M,, N,) nicht ~ (M,N); d. h. Aus MfN, M,tM, N,tN, M, nicht — M, N, nicht N folgt (M,, N,) nicht — (M, N). Dieses Axiom soll für alle Mengen gelten. Für endliche Mengen reicht es aber noch nicht aus, und werde durch das folgende ersetzt und ergänzt: UI. Sind M und N fremde und zugleich endliche Mengen, und ist M, Teilmenge von M, so soll stets(M,, N) nicht ~ (M, N) sein; d.h. Aus MfN, MdM, NdN, M,tM folgt (M,,N) nicht — (M,N). Für unendliebe Mengen braucht dieses Axiom bekanntlich nicht erfüllt zu sein. Auch die Voraussetzungen dieser Axiome besitzen durchaus den in der Einleitung genannten logischen Sondercharacter; sie sind sdmtlich negativer Natur, soweit es sich um die hier allein in Frage stehenden Aequivalenzbeziehungen handelt. Man könnte freilich annehmen, dass in diesem Fall ein indirectes Beweisvertahren zum Ziele fiihren werde; die Annahme (M,, N,) — (WM, N) resp. (,, VN) — (M, N) ist ja von positivem Character. Aber diese Vermutung triigt. Die Aequivalenz von Verbindungsmengen ist nämlich keineswegs nur so möglich, dass M,~ M und N, — N, ist sondern auch auf andere Weise; und daher kann aus der angenommenen Aequivalenzbeziehung ein Widerspruch mit den Voraussetzungen M, ncht — M, N, nicht — N nicht abgeleitet worden. Die negative Fassung unserer Axiome stellt uns zunächst vor die Aufgabe, die bestimmte Beziehung (a), (6), (c),(d) aufzufinden, die zwischen (J, NV) und den Mengen (JZ, N,) und (JZ, N) besteht. Für das Axiom II kann es erst im nächsten Paragraphen geschehen; fiir das Axiom III soll es hier folgen. Da (1, NV) Teilmenge von (MS, N) ist, so kann nach Satz 13 von $ 3 nur die Beziehung (a) oder (6) realisirt sein. Aber der Fall (a) d. h. (U, N) a(M, N) ist unmöglich. Jede Teilmenge von (M,, N) hat nämlich nach § 2, VI eine der Formen M,, M,, N,N,, (M,,N), (MN), (MN), wo M, eine Teilmenge von M, ist. Keine von ihnen kann aber zu (M, N) aequivalent sein. Da nämlich M und N endliche Mengen sind, so hat man für sie gemäss (10) die Relationen 801 Mb (M, N) und Nb (M, N). Gemäss Satz (4) hat man weiter M,bM, M‚bM, N,bN und damit folgt die Behauptung nach Satz (6) von § 3 bereits fir M,, M,, N, N,. Für die drei Verbindungsmengen folgt sie aus den Axiomen selbst; es ist ja, da MJ und WN endliche Mengen sind, M, micht — M, M, nicht — M, N, nicht — N und damit ist in der Tat die behauptete Nichtaequivalenz eine Folge von (II) und (III). Also 11. Für endliche (und fremde) Mengen M und N gilt die Beziehung (M,, N) b(M, N). 12. Die Verbindungsmenge zweier endlichen Mengen ist selbst endlich; d.h. Aus Md M und Nd N folgt (M‚, N) d(M, N). Wir haben nachzuweisen, dass die Beziehung (M‚ N) a(M, N) ausgeschlossen ist. Nun hat jede Teilmenge von (M, NV) wieder eine der Formen M, M,, N, N,, (M‚N), (U, N), (4. N,) und wir beweisen, genau wie eben (vgl. auch § 5, 2), dass keine dieser Mengen zu (VM, N) aequivalent ist. Damit ist der Satz bewiesen. § 5. Das Aequivalenzproblem. Die wichtigste Aufgabe, die zu behandeln ist, betrifft den Nach- weis, dass die Mengen M und WN aequivalent sind, talls für sie die Beziehung Ma N oder Md N besteht; also der Satz (Aequivalenzsatz) 1. Aus Ma N oder Md N folgt M— N. Ehe der Beweis gefiilrt wird, sollen die Aequivalenz-Relationen vorangestellt werden, die sich aus den vorstehenden Paragraphen unmittelbar ergeben : 2. Aus Mb N und MeN folgt M nicht — N. Ware nämlich 1 ~ MN, so hätte man auch (§ 3, 12) NbN oder Ne N, was aber gemäss § 3, 3 widerspruchsvoll ist. Hieraus folgt unmittel- bar weiter 3. Mit M=— N ist nur Ma N oder Md N vertriglich. Die Umkehrung dieses Satzes 3 ist es, die den eigentlichen Aequi- 53 802 valenzsatz (1) bildet. Ist er bewiesen, so folgt endlich noch, als Umkehrung von (2) 4. Aus M nicht — N folgt MbN oder McN. Man kann diese vier Sätze auch folgendermassen zusammenfassen: Die Beziehungen (a) und (d) sind hinreichende und notwendige Be- dingungen fiir die Aequivalenz, (b) und (c) ebenso fiir die Nicht- aequivalenz. Als Folge von (4) ergiebt sich, was in $ 3 und 4 noch offen bleiben musste, 5. Aus M‚tM und M, nicht~ M folgt M,6 M. d.h. Besteht fiir die Teilmenge M, von M die Beziehung M, nicht~ M, so gilt M, b M. Denn nach (4) gilt 17,6 M oder M,c M; nach Satz (13) von § 3 nur M,aM oder M,6 M, also gilt M,b M. Hine Anwendung hiervon giebt auch Antwort auf die beziiglich des Axioms II in § 4 gestellte Frage. Es folgt jetzt 6. Sind M, und N, Teilmengen von M und N, undist M, nicht ~ M,N, nicht ~ N, so folgt daraus stets (M,, N,) 6 (M, N). Wir gehen nun zum Satz (1) über und beweisen zunächst den ersten Teil, also den eigentlichen Bernsteinschen Aequivalenzsatz. Sein Beweis folgt aus dem Axiom II von § 4 über die Nichtaequi- valenz der Verbindungsmengen. Aus der Voraussetzung Ma N folgt zunächst ein M’ — N, ein N’ = M. Ware nun M nicht ~ N, so hätte man nach § 1, 3 SM mcht — M’, N’ nicht — N. Mit M und NV sind aber auch J’ und N’ fremde Mengen (§ 2, 1); sie bestimmen daher eine Menge (M’, N’), und für sie müsste gemäss Axiom II nunmehr (M’, N’) nicht — (iM, N) folgen. Andererseits folgt aber aus den beiden ersten Relationen unmittelbar nach § 4, I (M'‚ N') = (M.N) und damit ergiebt sich ein Widerspruch. Damit ist der Beweis bereits geliefert Freilich beruht der Beweis auf einer gewissen Voraussetzung, die noch zu erörtern ist. Wir operieren mit der Verbindungsmenge von M und N und haben deshalb die Voraussetzung nötig, dass M und N fremde Mengen sind. Sind sie es nicht, so wird man am ein- 803 fachsten so vorgehen, dass man folgendes neue Axiom zu Grunde legt: *) I. Sind M und N keine fremden Mengen, so giebt es stets zwei tinen aequivalente, zu einander fremde Mengen ® und ® ; so dass also end — Ne nd Ik ft Gemäss § 3, 12 besteht auch für sie die Beziehung M a M, und auf sie lässt sich daher der obige Beweis übertragen. Aus MM ~ N folgt dann auch WM ~ N. Es handelt sich nun noch um den gleichen Nachweis fiir die Beziehung Md N. Ehe ich dazu übergehe, erinnere ich daran, dass die Eigenart der Beziehung M/d N in der Cantorschen Theorie offen geblieben war; für das durch sie bedingte Verhältnis von Jf zu N hatte sich ein Resultat nicht ableiten lassen. Das darf nicht Wunder nehmen; das hierin enthaltene Problem stellt nämlich wieder ein logisch unlösbares Problem, und damit eine illusorische Aufgabe dar. Wir haben ja als Prämissen zunächst nur die Aussagen kein M, ~ N, kein N, ~ M. Dazu kommen, da M und N endliche Mengen sind, kein M, ~ M, kein N,= N, also lauter Aussagen von negativem Character. Selbst der Weg des indirecten Beweises ändert daran in diesem Fall nichts; denn man müsste noch die Annahme M nicht ~ N hinzufügen. Nun wäre es ja möglich, dass die für den Beweis einzig in Frage kommenden Axiome II und III der Nichtaequivalenz von § 4 die Prämissen positiv beeinflussen könnten; aber auch das ist nicht der Fall. Denn diese Axiome lauten ja in ihrem Schlussteil übereinstimmend (M,, N,) nicht ~ (M, N). Wir müssen also von Prämissen ausgehen, die samt und sonders negativ sind, und kommen zu dem Schluss, dass sich die Aequivalenz M — N im Fall endlicher Mengen ohne eine nochmalige neue axiomatische Festsetzung nicht folgern lässt. Das so gewonnene Resultat lásst sich auch in seiner allgemeinen Bedeutung leicht ver- stehen. Es láuft dem Tatbestand parallel, der uns aus der allge- meinen Theorie der endlichen Zallgrössen geläutig ist. Dort muss die Festsetzung, wann zwei Grössen als gleich gelten sollen, erst frei — 1) Es entspricht dem von Zermero in seinen Grundlagen (Math. Ann. 65) enthaltenen Theorem 19. 804 natürlich zweekgemäss — geformt werden, ehe man die Frage, ob zwei gegebene Grössen als gleich zu gelten haben, in Betracht ziehen kann. Man denke z. B. an die Weierstrassische Theorie der Irratio- nalzahlen; sie setzt bekanntlich die Gleichheit zweier Zahlen « und b so fest, dass jeder Bestandteil von a kleiner ist als 6 und jeder Bestandteil von 6 kleiner als a. Eine solche axiomatische Festsetzung erweist sich also auch im Gebiet der endlichen Mengen, wenn man sie, wie hier, ausschliesslich auf die Mengenbeziehungen, d.h. auf die Nichtaequivalenz von Menge und Teilmenge griindet, als eine Notwendigkeit. Es fragt sich nur, welche Festsetzung man zweckmassig zu Grunde legt. Beachtet man, dass es sich im Grunde um eine Axiomatik der Grössenlehre handelt, so liegt offenbar nichts näher, als die eben genannte Definition zu benutzen, und dies soll in der Tat geschehen. Wir setzen also fest (Aviom der Aquivalenz endlicher Mengen) U. Zwei endliche Mengen M und N sind aequivalent, wenn für gede Teilmenge M' und N’ die Beziehung M’ b N resp. N’ 6 M besteht; d.h. Aus MdM, NdN, M’bN, N’6M furjedes M’, N’ folgt M~ N. Hieraus lässt sich der Satz, dass aus Md N auch M~ N folgt, unmittelbar folgern. Ehe wir dazu übergehen, wollen wir noch die Berechtigung unseres Axioms und seine Stellung im gesamten Aufbau näher erörtern. Wir wollen zunächst nachweisen, dass von den vier Beziehungen MaN, MbN, MeN, Md N nur die letzte mit dem Axiom verträglich ist. Aus Ma N folgt ein M’ ~ N; gemäss unserm Axiom ist aber für jedes M’ M'b N und man erhielte also V6 N, was aber nach § 3,3 widerspruchs- voll ist. Aus 6 N folgt Cin IY =~ slp was analog zur Relation Mb M führt, die ebenfalls widerspruchsvoll ist. Endlich folgt aus Me N genau wie eben die widerspruchsvolle Relation NV 6 N. Unser Axiom kann also in der Tat nur mit der Beziehung M d NV verträglich sein. Dies ist aber auch wirklich der Fall. Die Folge- rungen, die sich aus M’bN und Md N, aus N’b M und Md MN 805 ergeben, lauten gemäss § 3, 9, dass für jedes M’ und N’ M’bM und N'5N ist; sie entsprechen der Endlichkeit von M/ und MN und stellen die in $ 4,4 gefundene Eigenschaft der endlichen Mengen dar. Zusammenfassend folgt also: Das Axiom II ist nur für endliche Mengen realisiert, und überdies weder im Fall M5 N, noch Me N; damit ist aber der Beweis seiner Berechtigung geliefert. Hs ist für die endlichen Mengen und ihre Aequivalenz characteristisch. Der Beweis des Aequivalenzsatzes ergiebt sich nun tolgendermassen. Gemäss § 4, Satz 4 ist für jedes M’ und N’ M’6M und N’b N; ferner gilt nach Voraussetzung Md N und Nd M, und hieraus folgt nach $ 3, 9 sofort M’b N und N'b5 M und nunmehr nach unserm Axiom M~N. § 6. Sdtze über Verbindungsmengen. Seien M und JN einerseits, und M und N andrerseits fremde Mengen. Zwischen M und I”, sowie zwischen N und ® besteht je eine der Beziehungen Mam, MbM, Mem, MdM und NaN, NOR, NeR, NAR. Es ist die Frage, welche Beziehung fiir (M,N) und QM, N) resultiert, wenn wir irgend eine Beziehung der ersten Zeile mit einer Beziehung der zweiten Zeile kombinieren. Wir beweisen zunächst folgende Sätze 1. Aus MaM und Na® folgt (MZ, N)a (MN, N). 2. Aus MbM und NAR folgt (M, N)b AM, N). 3. Aus McM und NeR folgt (M, N) cM, N). 4. Aus MdM und Nd® folgt (M, N) d(mM, N). 5. Aus Ma®M und NdM® folgt (M, NV) a (M,N). Die Beweise von Satz (1), (4), (5) lassen sich folgendermassen zusammenfassen. Die Voraussetzungen lauten gemeinsam M= M und N~®™®, woraus gemäss Axiom I von 64 (M, N) Gd (M, N) 806 folgt. Im Fall (1) und (5) sind nun M und ® nach $4, Satz 3 unendliche Mengen, also gilt dies nach §4,8 auch von (JZ, N) und ON, 2) und daher ergiebt sich wieder (ML, N) a (MM, N). Im Fall (4) sind dagegen M, N, MN, NM endliche Mengen, also auch (64,12) (MZ, N) und (9R,N) und daher ist (M,N) d (NR, Ny. Wir beweisen nun den Satz (2)'). Dazu gehen wir von den Relationen MbM und NR aus, also von den Beziehungen kein M,~ wt MM! — M, kein NS NE und erhalten zunächst (M', W’) T(M, N) Wir folgern nun aus den gegebenen Relationen Mb ® und NON mittels M — M’ und N~ N’ weiter MEM und MN oder aber ($ 5, 2) M' nicht ~ M, MN! nicht — N und daraus endlich, gemäss Satz (6) von § 5 (ON, NW’) 6 AN, M) oder (IM, N) b (N,N). In derselben Weise beweist man den Satz 3. Ein letzter Satz, der sich ableiten lässt, lautet: 6. Ist M eine endliche Menge, so folgt aus MbM und NdN (M,N) (M,N). 1) Geht man zu Mächtigkeiten über, so bezieht sich der obige Satz auf den Fall, dass m,< m, und n, v) also, differs only unappreciably from that of a rhombic crystal. The mineral is strongly birefringent (about: 0,038), with negative character. In first instance now, the Rönreenpattern of a single lamella (d=0.32 mm.) was obtained, as reproduced in fig. 1 of Plate I; a stereographical projection of this beautiful and rich diffraction- image is given, moreover, in the diagram 1 of the text. It manifests the ordinary bilateral symmetry of the monoclinic crystals parallel to {001} or {100}; on more detailed examination, three directions may, moreover, be clearly discerned in it, which include angles of almost 60° with each other, and which are closely related to the hexagonal “radiation-figure’ of this mzca-species. obtained by pres- sure with a sharp object. The direction of the optical axial plane may also be discerned in it without much difficulty ; it is indicated by a row of numerous smaller spots, situated perpendicularly to the plane of symmetry of the diffraction-pattern. Fig. 1. Stereographical projection of the Rönraenpaltern of a single Muscovite-lamella, parallel to {O01}. 54* 818 $ 3. In the second place we investigated the behaviour of: four preparations, these being dextro-, respectively laevogyratory muca- piles consisting of muscovite-lamellae crossing at 45° or 60°. The composing lamellae were cut from a muscovite-crystal in such a way, that their longer side was parallel to the optical axial plane of the mineral, their shorter edge thus being parallel to its. plane of erystallographical symmetry. The central part of the complexes-com- posed of hexagonally arranged lamellae, manifested in convergent polarized light between crossed nichols the almost perfect axial image of a uniaxial crystal endowed with circular polarisation ; in the interference-image of the dextro- or laevogyratory complexes built up by lamellae crossing under 45°, there appeared only a single dark beam interrupted in the central part of the image,:while also the coloured rings showed a somewhat elliptical distortion with a slight spiral constriction in the immediate vicinity of the dark beam. For the rest, the optical properties of the preparation appeared to vary quite continuously in all azimuths, being almost the same in all directions. The Ronrexnpatterns obtained are reproduced. in fig. 2 and 3 of Plate I, in the right position with respect to that Fig. 2a. Stereographical Projection of the normal Diffraction-pattern of a dextro or laevogyratory Complex of Muscovite-lamellae, crossing at 60°. Fig. 2b. Stereographical Projection of the abnormal Diffraction-pattern of a dextro- or laevogyratory Complex of Muscovite-lamellae, crossing at 60°. Fig. 3a. Stereographical Projection of the normal Diffraction-image of a dextro- or laevogyratory Complex of Muscovite-lamellae. crossing at 45°. 820 Fig. 35. Stereographical Projection of the abnormal Diffraction-image of a dextro- or laevogyratory Complex of Muscovite-lamellae, crossing at 45°. of fig. 1 of this plate, while the text-figures 2a and 26, respectively 3a and 35, represent stereographical projections, immediately relating to these diffraction-images. In all experiments the time of exposure of the photographic plates was two hours. Secondly, we can remark, that the normal RöNtGEN-images of fig. 2a and 3a show a perfect hexagonal, respectively octogonal sym- metry, evidently consisting of a pattern repeated siz, respectively eight times, the structure of which is, however, in the hexagonal image clearly diferent from that in the octogonal image, although the absorption of the RönrceNrays in these very thin lamellae plays only an insignificant part. Evidently the character of the composing patterns is here dependent in some way or other on the special way in which the secondary waves, emerging from the upper lamella, are modified by their passage through the next following lamella; and from this experience it seems, that this influence varies with the magnitude of the crossing-angle of two subsequent lamellae’). 1) On the modifications of a primary RöNTGeN-pattern, if a secondary ray of it passes through a second and identically orientated crystalplate of the same sub- 821 The character of the whole pattern as that of an original figure repeated regularly a number of times, equal to the number of lamellae contained in a full turn of 360° (here, therefore 6 or 8), — was observed in all cases of normal diffraction-images ; it can be considered as the normal character of the diffraction-patterns of such complex systems of lamellae. Basing ourselves upon the experience gained in these and other cases, we may, therefore, safely enunciate as a general rule: [7 the central part of a regular complex of crystalline lamellae, cut perpendicular to a plane of symmetry of the crystals, and crossing at angles 22 a=—, be radiated through by Rontexnrays, then the normal dif- n fraction-pattern thus obtained, will exhibit an axis of n-fold sym- metry, showing, therefore the image of an original pattern repeated n-tumes. The diffraction-image of the dextro- and laevogyratory com- plexes of this kind are always identical. § 4. From what has been said, it must be concluded directly, that pseudo-tetragonal, circularly polarizing crystals can not be con- sidered as built up in the way supposed by Marrarp, namely, if they do not consist of a substance, the molecules of which are them- selves endowed with rotatory power. For it may be easily foreseen, that even in the case where the composing lamellae possessed no symmetry-plane whatever, the final diffraction-image will at least show an axis of octogonal symmetry, the eight planes of symmetry in fig. 3a then having disappeared. In this most general case of lamellae crossing at 45°, therefore, the pattern should all the same show an octogonal symmetry-axis, which, however, is impossible in crystallography, and which, in agreement with this fact, wasnever found by us in any diffraction-image of real or apparent tetragonal crystals. The RonxtceNnpatierns of any optically-inactive, pseudo-tetragonal erys- tal-species'), or those of optically-inactive, pseudo-tetragonal crystals stance, conf. the paper of R. Grocker, Ann. der Physik, (4) 47, 337, (1915). We have now started the systematical investigation of the phenomenon stated in the above, according to which the special character of the diffraction-image of such crossed lamellae varies with the angle 9, at which subsequent lamellae cross. From the fact, that the text figures 2—7 are drawn on the same scale as fig. 1, it will immediately be clear, that there can be not a mere superposition of images here, as e.g. a considerable number of the outer spots of fig, 1 have completely disappeared, even in so simple a case as that of fig. 4. ') Conf. the pattern of strychnine-sulphate, in: E‚ M. Jaranr, Lectwres on the Principle of Symmetry and ils Application in all natural Sciences, 2nd Edition, Amsterdam, (1920), p. 194, 195. However, in this case the molecules of the sub- stance have a rotatory power in solution also. 822 such as polassium-ferrocyanide, always manifested in their most complete and undisturbed form an axis of fourfold symmetry at the highest, and the same appeared to be the case with all true tetra- gonal crystals hitherto investigated. But in such complexes of lamellae, an axis of fourfold symmetry of the diffraction-image results only, when the composing subsequent lamellae include an angle of 90°, instead of 45°, as could be demonstrated e.g. by the pattern repro- duced in fig. 4, obtained with a system of muscovite-lamellae, care- fully arranged at 90°. Fig. 4. Stereographical Projection of the Rönrtaen-pattern of a Compiex of Muscovite-lamellae crossing at 90°. It must be concluded, therefore, that, if pseudo-tetragonal crystals be of the nature of polysynthetical intergrowths at all, the composing lamellae cannot cross at other angles than 90°. But from the mathe- matical theory of optical superposition’) it follows necessarily, — and the early experiments of NöRREMBERG and others are in full agreement with this conclusion, — that such systems of lamellae !) In 1906, at my request, professor LoreNrz was kind enough to develop once more the theory of the optical phenomena in systems of regularly piled-up lamellae. His results agree, although not quantitatively, yet in their principal features with those obtained by Marrarp and others. 823 crossing under angles of 90° will never manifest an optical rotation. The supposition made by Marrarp is, therefore, only allowable for pseudo-tetragonal crystals without optical rotation, and there is no possibility to explain the special behaviour of real pseudo-tetra- gonal crystals endowed with rotatory power in this way, at least in those cases, where the molecules of the crystallised substance do not possess a molecular rotation of their own. It will be necessary to look for a special explanation in all cases concerning this kind of objects, as e.g. in that of the pseudo-tetragonal ethylenediamine- sulphate, ete. § 5. On closer examination of the original photographic plates of the patterns obtained with the hexagonal and octogonal complexes, which correspond with the projection-figures 2a and 3a, it became evident that, although the situation of the spots on the plates completely agreed with that of the normal images in fig. 2a and 38a, yet a distinct and rigorously determined abnormal distribution of their intensities was present, in such a way, that equivalent spots in the images did not possess the same intensity. Especially in the immediate vicinity of the centre, where very intensive spots were situated, the said phenomenon manifested itself most clearly. A more detailed study taught us, that this distribution of the intensities in the two images was, as drawn in the figures 25 and 35, i, e. symmetrical with respect to only a single plane. By special experiments it could be proved, that these anomalies did not depend on the position of the preparation with respect to the plane of the anti- cathode, or generally, to that of the luminous source: for on turning the preparation from its original position through 45° e.g., the plane of symmetry £ in the images appeared to have turned also through the same angle on the new photograms. The cause of the said abnormalities must, therefore, be ascribed to the preparations them- selves; and the close analogy of these anomalies with those formerly observed by us in real pseudo-symmetrical crystals, must be obvious, as also in those cases we observed a bilateral symmetry of the pattern, instead of the expected one, as was e.g. demonstrated with rac. triethylenediamine-cobalti-bromide and other preparations. The chief difference between these cases is, that in the preparations formerly studied, a number of spots were lacking altogether, their intensities being reduced to zero. Thus the bilateral symmetry of the patterns came there to expression in a higher degree, than was the case in our photograms which were obtained with objects, composed of a much smaller number of superimposed lamellae. 824 That it must be special properties of the preparations, that are the cause of such anomalies, becomes also evident from the fact, that the pseudo-symmetrical substances showing them, under favourable circumstances may occur in such well developed individuals, as to give perfectly wndisturbed RÖNrerN-patterns: thus e.g. with potassiwm- ferroeyanide in most cases certainly only bilaterally symmetrical images were obtained’), but occasionally there were found also perfect tetragonal patterns. And while we obtained an only bilaterally symmetrical ROnTGEN-pattern with an apparently irreproachable individual of benitoite *) cut perpendicular to its optical axis, RiNNw *) afterwards was able to obtain a quite normal trigonal diffraction- image of the same mineral. Moreover, it was found that the direction of the single plane of symmetry in the RöNrGeNpattern was completely analogous in the two cases studied in the above: its situation being in that of the hevagonal complex, as well as in that of the octogonal one, coinciding with the bisector of one of the angles of two’ subsequent lamellae of the mica-piles. As the optical and microscopical investigation of the preparations did not reveal any abnormality in these directions, the only possible conclusion was, that the cause of this phenomenon must be attributed to some peculiarity in the lamellar arrangement. In the case of the preparations with lamellae crossing at 60°, the explanation of the phenomenon may be given in the simplest way as follows. The preparation of such mica-piles was hitherto executed only with the purpose of demonstrating the optical effects of such com- plexes: the apparent uniaxiality and the rotation of the plane of polarisation of the incident rays. Because of the fact that the optical orientation of each lamella does not differ appreciably from that of a rhombic crystal cut perpendicularly to its first bisector, it could be considered hitherto of no interest to the prepairer of such mica- piles, whether he piled up these lamellae in the same position as they were cut from the original crystal, or whether he turned them accidentally through 180° about an axis perpendicular to the plane of the lamella. For the final optical effect of the preparation will not be affected in the slightest degree by this turning. However, such a change of right and left, of the anterior or posterior part 1) The plane of symmetry being in these and other cases often parallel to the direction of the composing lamellae, apntrary to what was observed here. 2) F. M. JAEGER and H. Haga, Proceed. Acad. of Sciences Amsterdam. 17, 1204 (1915), 3) F. Rinne, Miner. Centralblatt, (1919), p. 193. 825 of the lamellae, etc., is by no means indifferent any longer, if the distribution of the intensities of the spots in the diffraction-image by RÖNTGEN-rays be considered. For the »wscovite-crystal has under all circumstances a true “monoclinic” molecular structure; the inten- sities of the diffraction-spots of a lamella parallel to {001} e.g., will, therefore, always be different to the left or to the right of the optical axial plane, while they will appear the same to the right or to the left of the plane of crystallographical symmetry. It must thus be evident, that the interchange of the two sides of a lamella in the way mentioned above, must really be of influence with respect to the special symmetry, which will be manifested in the distribution of the intensities of the diffraction-spots, as they will appear in the photographical image of the lamellar complex as a whole. If the subsequent lamellae of a hexade are numbered 1 to 6, while the longer side of the lamellae, — as was really the case with our preparations, — is parallel to the direction of the optical axial plane in each lamella, then, if in the piling-up of the lamellae at angles of 60° first the lamellae 1 to 5 be taken in their right position, but N°. 6 be turned now through 180° in its own plane, the thus obtained hexade will give a diffraction-image, in which the intensities of the spots will be no longer distributed symmetrically with respect to six planes of symmetry, but in which there is only a single plane of this kind, exactly bisecting the angle between the superimposed pairs of lamellae: (1—4+4) and (2—5), and, therefore, being perpendicular to the pair: (3—6). By means of schematic figures, in which the distribution of the intensities of the spots, as effected by a single lamella is taken into account, it is possible to deduce systematically the general symmetry-character of the final distribution of intensities in the diffraction-image resulting from the complete hexade. Undoubtedly such a reversion of-a lamella will have accidentally occurred during the preparation of the mica-piles considered, just because there was no need for the prepairer to draw special atten- tion to avoid such a reversion, and because with regard to his aim he was quite free to fix the subsequent lamellae in those positions, in which they accidentally were presented to him. Of course, there is a fair chance also, that during his work, he turned two or three lamellae in the way described; and it is necessary, therefore, also to consider the consequences of this for the final character of the diffraction-image, if all possible combinations of lamellae be in this way taken into account. In the case of such hexades, it appears unnecessary, however, to consider any combinations with a number 826 of reversed lamellae greater than three: for it will be evident, that an accidental reversion of four lamellae, for example, will have’ the same effect as the turning through 180° of two lamellae, of five the same as if one were reversed, etc. These cases are, there- fore, already contained amongst the possibilities formerly deduced in turning one, two, or three lamellae respectively. A closer examination now taught us, that in the case of six muscovite-lamellae crossing at 60°, three kinds of diffraction-images might be produced: with respect to the intensities of the spots normal patterns (VV); or such as are symmetrical with respect to a single plane bisecting the angle between two subsequent lamellae (diagonally-symmetrical; D); or finally such of the same symmetry, but in which the symmetry-plane now coincides with the direction of the lamellae themselves (lamellar-symmetrical; Z). If one of the six lamellae be turned, there are siv possibilities; if two be reversed, fifteen cases must be considered; and if three lamellae be turned through 180°, twenty possible combinations must be accounted for. In the first mentioned six cases only images with the bilateral sym- metry D appear to be possible, as we found it just now in the case of fig. 26; in the case of two reversed lamellae, we may find three combinations of pure hexagonal, normal symmetry, siz combi- nations of diagonally-symmetrical character D, and siz of lamellar symmetry L. In the last mentioned case of three reversed lamellae, we may find two possible combinations of normal character, here, however, not with hexagonal, but with trigonal symmetry; and eighteen combinations, corresponding to diagonally symmetrical diffraction-images D: A Review of the Possible Types of Intensity-Distribution in the Diffraction-Patterns, |. Obtained by means of Mica-Complexes with Lamellae Crossing at 60°. If one lamella If two lamellae | If three lamellae be turned: be turned: be turned: Number of possible combinations: 6 5 20 Normal images: 0. 3 (hexag.). 2 (trigon.). Asymmetrical Images: 0. 0. 0. Diag. symm. Images: 6. 6. 18. Lamell. symm. Images: 0. 6. 0. From this it appears, that a mica-complex piled-up arbitrarily and without special care, with lamellae crossing alt 60°, will 827 never produce a completely asymmetrical diffraction-pattern; and that there is an appreciably fair chance that the symmetry of it will be diagonally-symmetrical, as found in the case of fig. 2; it is no wonder, that we just now met with ts symmetry in the case of the preparation investigated in the above, In the same way it is possible to deduce the possibilities to be expected, if the composing lamellae cross at 45°. However, because in such mica-piles there are always lamellae present per- pendicular to, or coinciding with the geometrical symmetry-plane in one of the eight lamellae, the case of fig. 36 will never result from the reversion of a single lamella but only a lamellar symmetry L of the intensities can be produced thereby. A general review of the possible cases can be given as follows *): A Review of the Possible Types of Intensity-Distribution in the Diffraction-Patterns, Obtained by Mica-Complexes with Lamellae Crossing at 45°. | If one lamella Iftwolamellae If three lamel-| If four lamel- | | be turned: be turned: | lae be turned:| lae be turned: | | Number of possible | Combinations: 8 28 56 10 Normal images: 0. 4 (octogon.). 0. 6 (octogon.). Asymmetrical Images: 0. 0. 16. 0. Diag. Symm. Images: 0. 16. 0. 48, Lamell. Symm. Images: 8. 8. 40. 16. If only two lamellae be turned, there is an appreciable chance of a diagonally-symmetrical image, as found in fig: 36; but if four lamellae be accidentally reversed, this chance is extremely great. For the rest, there are about equal probabilities for the bilaterally- symmetrical images D and L, both of which were observed formerly in the case of natural pseudo-symmetrical crystals. § 6. A number of mica-piles were, moreover, prepared, in which the right orientation of the muscovite-lamellae was rigorously checked by comparison with their true position in a single muscovite-crystal *). First a dextro-, and a laevogyratory combination, in which the ) My assistant Dr. A. Simex was kind enough to check the number of these possible combinations systematically. | wish to express my best thanks to him here once more for the trouble he has given himself in this matter. 2) This erystal my colleague Prof. Bonnema most kindly gave me for this purpose from the mineralogical collection of the University. 828 lamellae were crossed at 120°, while attention was given to prevent a rotation of them through 180° about an axis perpendicular to the plane of cleavage. In these and the following preparations, the longer sides of the lamellae were always parallel to the geometrical plane of symmetry of the muscovite-crystal, contrary to what occurred in the mica-piles studied before. A normal image with a trigonal axis and three planes of symmetry passing through it, could be expected here beforehand. Because of the not wholly irreproachable material available, the patterns obtained were not suited for photo- graphical reproduction; but notwithstanding this, it was possible to confirm the exactness of this prediction completely. A schematical projection of these patterns, which also in this case appeared to be identical for the dextro-, and laevogyratory complexes, is reproduced in fig. 5. Fig. 5. Stereographical Projection (schematical) of the Réntcenpattern of dextro and laevogyratory Mica-piles, with Lamellae crossing at 120°. Finally in fig. 6 the stereograpbical projection is reproduced (schematically) of two diffraction-images, obtained by two different muscovite-piles. In the first complex the lamellae crossed at 60°, and a rotation of the lamellae was carefully prevented; in the second preparation, however, the subsequent Jamellae included angles 829 of 120°, but each following lamella was turned with respect to the “preceding through 180° about an axis perpendicular to its plane of cleavage. It will be easily understood that in this way the symmetry of the intensity-distribution in the two patterns must be essentially the same; it is remarkable, moreover, that also the patterns themselves appeared identical, notwithstanding the fact, that the sequence of Fig. 6. Stereographical Projection (schematical) of the Rönreenpatterns of two Mica-piles, the Lamellae of which crossed at angles of 60° and 120° respectively, with a partial reversion of some of them. subsequent lamellae was different in the two mica-piles: both dif- fraction-images show a senary axis and. six planes of symmetry passing through it. § 7. Regarding the results obtained in the above, hardly any doubt can remain as to the principal justification of our former view, according to which the observed abnormalities in the R6nreunpatterns of pseudo-symmetrical uniaxial crystals are in reality caused by a simple reversion of the position of the composing lamellae. Rotations of this kind may, for example, occur in some cases of tvin-formation between those lamellae, if the axis of rotation or twinning be only no real symmetry-element of the crystallographical structure of the : 830 lamellae; at best it may be an axis of pseudo-symmetry of this structure. It is, therefore, by no means improbable, that finally submicroscopical twinformation between the lamellar units, composing the pseudo-symmetrical crystal, has to be considered as the primary cause of the anomalies formerly observed in the Rönraer-diffraction- images of such crystals. But then the question arises, why such a twinning-process happens oftener in one principal direction of intergrowth, than in the other equivalent ones? This question must arise, however, because in such crystals one has to deal not with a relatively small number of super-imposed lamellae in each direction, but with an extremely great number of them. It might be supposed that there were special influences during the growth of the crystal from its mother-liquor, which caused such a directing and preferential action in this respect: but it is at the moment difficult to guess, of what nature those influences really may be. Perhaps a factor of some importance therein may have been the /eat-effect during the crystallisation, which causes convection-, and concentration-currents to appear in the environing liquor, corresponding in their turn to greater or smaller changes of the viscosity of the solution in those directions. It is well known, that the degree of viscosity of a medium plays an important rôle with respect to the occurrence of twins, and generally in such a way, that an increase of the viscosity appears most favourable to the occurrence of twin-formations. It is not improbable that influences of this kind may in the end appear to favour also the twinning of the very thin submicroscopical lamellae, of which the crystal is built up in one special direction. Perhaps systematicul investigations on the phenomena of erystal- lisation of such pseudo-symmetrical erystals under variable, but well-determined external circumstances, may in not too distant a future bring us better evidence on this subject. Laboratory for Inorganic and Physical Chemistry of the University. Groningen, April 1920. Physiology. — “On Adsorption of Poisons by Constituents of the Animal Body. 1. The adsorbent power of serum and brain- substance for Cocain”. By L. BERLAND and W. Srorm van Leeuwen. (Communicated by Prof. R. Magnus). (Communicated at the meeting of January 31, 1920). In a previous paper’) STORM VAN LEEUWEN has shown that in the sernm and the tissues of rabbits there are substances capable of inactivating pilocarpin. At the same time he was able to demon- strate that this does not happen by destroying pilocarpin, but through a physical adsorption of pilocarpin by certain components of the serum, whose nature could not be determined thus far. From quan- titative investigations it also became evident that this physical adsorp- tion proceeds according to the same laws that hold for the adsorption of dyes by animal charcoal. In the paper alluded to just now, STORM VAN LEEUWEN has already pointed out that the adsorption of pilocarpin by rabbit’s serum is not the only case of the kind, since many facts, described in the literature, render it highly probable that many similar adsorptions appear in the animal body. We know, for instance, that many poisons such as digitalis, atropin, cocain, strychnin ete. may be rendered inactive by animal tissue. This inactivation is commonly conceived to be a decomposition of the poison; we, however, believe that in many of those cases adsorption comes into play. True, in numerous cases poisons in the body are inactivated chemically, but we believe that this chemical action is in many cases preceded by a physical adsorption. The reason why we attach great importance to the question whether poisons are rendered inactive along the chemical path, or through adsorption, is that the great difference in the sensitivity of various individuals to poisons that bring about a very quick, acute poisoning process, can be accounted for by an adsorption, not by a chemical process. The following example may serve to illustrate this: h W. Storm van Leeuwen. Sur l'existence dans le corps des animaux de substances fixant les alcaloides. Arch. Neerl, de Physiol. Tome 2 p. 650 1918. 55 Proceedings Royal Acad. Amsterdam. Vol. XXII, 832 It is well known that some people are less sensitive to the poisoning action of cocain than others. According to HarcHer and EGGLEsTON *) cases are known in which 16 mgr. and 20 mgr. given subcutaneously were fatal, whereas in other cases 1.25 grms of cocain given subcutaneously had no effect whatever. HATCHER and EGGLESTON have proved conclusively that cocain, novocain and many other local anaesthetics become inactive very soon after being in- jected into an animal, while they have also demonstrated that various tissues, above all the liver, are able to decompose these poisons chemically. This, indeed was no novel experience, for Bier already found, when experimenting with rabbits, that cocain that has for some time been in contact with animal tissue, has thereby become less active, while Sano*) had come to the same conclusion for cocain with respect to brain-substance. Bier and Sano believed that this inactivation was caused by chemical decomposition. HATCHER aud Heeieston’s assertion that the liver can decompose cocain to a large extent, is incontestible. Still, this decomposing process cannot be so quick as to afford an explanation for the large differences in the sensitivity of different people. When after an in- jection of a few milligrammes of cocain the patient shows after a short time (a few minutes) serious symptoms of intoxication, the reason can not be that the cocain in his body is not decomposed quickly enough, for this decomposition cannot be so quick even with normal individuals. This, in fact, has also been pointed out by HATCHER and Keerrston themselves. Now it would seem to us that the abnormal sensitivity of some individuals to cocain might be ex- plained as follows: When cocain is administered to a normal man or animal it will be used: A in those places (i.a. the central and peripheral nervous system) where it exerts an influence. B in other places (i.a. free chemoreceptors distributed in the blood). The sensitivity of a special individual to cocain will then be largeiy determined by the ratio between the number of the places of adsorp- tion referred to under A and B. *) 1) C. Eeereston and R. Harcuer. A further contribution to the pharmacology of the local anaesthetics. Journ. Pharm. and exp. Therap. vol. XIII. p. 433. 1919. 2) Torata Sano. Ueber die Entgiftung von Strychnin und Kokain durch das Riickenmark. Ein Beitrag zur physiologischen Differenzierung der einzelnen Rückenmarks-abschnitte. Pfliigers Arch. Bd. 120; p. 367. 1907. Torata Sano. Ueber das entgiftende Vermögen einzelner Gehirnabschnitte gegen- über dem Strychnin. Pflügers. Arch. Bd. 124, p. 369. 1908. 3) The places of adsorption sub A may be termed “dominant chemoreceptors”, those sub B “secondary chemoreceptors.” 833 In order to confirm this hypothesis it must first of all be ascer- tained whether the places mentioned sub B (i.e. the secondary chemoreceptors) really exist in the body. In this paper we shall endeavour to settle this question with regard to cocain. As already stated the researches of Brier, Sano, Harcurr and Eeeieston, and others had already brought to light that cocain can be inactivated by animal tissue. It lay with us to show that this inactivation takes place through physical adsorption. We had to proceed as follows: 1. We had to ascertain the action of a cocain solution of known strength on a special organ. | 2. We had to show that the cocain solution became less active after the addition of animal tissue. 3. We had to demonstrate that the cocain was not decomposed in the less active mixture, so that all the active cocain could again be extracted from the mixture. The effect of cocain upon the nervus Ischiadicus of the frog was taken as the index for cocain-action. We applied Zorn’s *) method °), of which we give a brief description (see: Fig. 1). The nerve of a-nerve-muscle preparation is led through a small ebonite basin, which is to hold the cocain (and other liquids); on either side of the place where the nerve is in contact with the local anaesthetic, electrodes can be applied, which communicate with the secondary coil of an inductorium. By the aid of Pohl’s commutator the nerve can be stimulated alternately by EH’ and KE". First the position of the secondary coil is determined (to be read from S) at which the muscle can just be stimulated from KE’ as well as from E". Subsequently the liquid with the local anaesthetic is put in the basin, and after this we investigate how strong the solution must be in order to make the muscle after a certain time irresponsive to the stimulus from the electrode E’. The stimulus from E” must retain its effect upon the muscle to make sure that during the expe- riment the excitability of the muscle itself is not diminished. We invariably experimented with a gastrocnemius-ischiadicus preparation of Rana esculenta. Due care was taken to keep the room-tempera- ture constant. We made sure beforehand that the liquids used for h Zorn. Beiträge zur Pharmacologie der Mischnarcose. Il. Zeitschr. f. exp. Path. und Ther. Bd. 12, p. 529 1913. 1 Cf W. Storm VAN LEEUWEN. Physiologische waardebepalingen van geneesmiddelen. Wolters, Groningen, 1919. 55” 834 the cocain solution were in themselves indifferent to the nerve. This proved to be the case for 0.6 °/, Ringer, for 0.9 °/, Ringer, for serum as well as for an emulsion of brain-substance. Experiment 1. The liquid used was: 0.2 ee. hydrochloras cocain (5 °/,) + 4 ¢.c. 0.9°/, Ringer’s fluid, Le. a solution of '/, °/, cocain in 0.9°/, Ringer. We found: Accumulator 2 volts. | Reading taken ofthe Stimulation at: | inductorium with Control (E”) stimulation at (E’) 3.00 h. 1.96 1.96 3.02 1.96 1.96 3.04 1.92 1.96 3.06 1.92 1.96 3.08 1.90 1.96 3.10 1.90 1.96 3.12 1.88 1.96 3.14 1.86 1.96 3.16 en 86, 1.96 3.18 1.86 1.96 3.20 1.86 1.96 3.24 1.82 1.96 3.26 1.82 1.96 3.28 1.80 1.96 3.30 1.80 1.96 3.32 1.78 1.96 3.34 1.78 1.96 3.36 1.74 1.96 3.38 1.64 1.96 3.40 1.40 1.96 3.42 1.38 1.96 3.44 1.34 1.96 3.46 1.2 1.96 (muscle still responsive). No contraction at the strongest current. MME KEEN | Fig. 1. Apparatus after Zorn (borrowed from a communication by Zorn). So it appeared that the nerve had become irresponsive after 48 minutes by the effect of '/,°/, cocain solution. The process of the experiment will be seen from the curve in Fig. 2. Reading taken of the inductorium minutes Fig. 2. Effect of 1/,°/, hydrochloras cocain upon the nervus ischiadicus of a muscle-nerve preparation of Rana esculenta. Abcisga: Time in minutes. Ordinate: Stimulus required to make the muscle contract through indirect stimulation. The same experiment was repeated several times to the following effect : 836 Erp. 2. */,°/, cocain solution nerve irresponsive after 43 min. pe ake oh On ER 55 5 sb EN NE DCE hl 5 5 5 yt AAN es ” 5. “fe */, Ld Te) LE > 2 43 LE) ” 6 whe ae LE) ” EE) ” ” 42 ” 99 le “fos Wp ” ” LE 2 ” 45 ” 2, 8. We he ” ” ” ” EE) 41 ” »? 9. “Ya de ” ” LE) ” ” 42 ” dd 10. “eo Wb ” id ” ”? ” 43 EE) It follows, then, that on an average the nerve is irresponsive in */,°/, cocain solution in 48 minutes. We now proceeded to ascertain the adsorbent power of human bloodserum. Exp. 11. The liquid consisted of: 0,1 ec. 5°/, cocain solution + 1,9 ce. of serum, i.e. a concentration of */,°/, cocain in serum. In this case the muscle remained normally responsive for a whole hour, so that the effect of 5 mgr. cocain is eliminated by 2 ce. of human serum. The process of this experiment will be seen from the curve in Fig. 3. Reading taken of the inductorium 60 minutes Fig. 3. Effect of 1/,0/, cocain in human serum on the nervus ischiadicus of a muscle nerve preparation of Rana esculenta. After this Exp. 11 was repeated with dog’s serum. Exp. 12. The liquid used was 0.1 ec. 5°/, cocain + 2 ce. of dog’s serum, that is about */, °/, cocain in serum: (see table p. 837. Here also the inhibiting influence of the serum can be seen distinctly. A similar result was obtained in exp. 13 with cat’s serum and in 837 exp. 14 with rabbits serum. Hereafter we endeavoured to detach the cocain from the serum. To this end we used the liquid of exp. 12 and 13 to the following effect: elen | 4 | 1.9 1.9 6 1.9 1.9 8 1.9 1.9 10 1.9 1.9 12 1.9 1.9 14 1.9 1.9 16 1.9 1.9 18 1.9 1.9 20 1.9 1.9 22 1.9 1.9 24 1.9 1.9 26 1.9 1.9 28 1.9 1.9 30 1.86 1.9 32 1.86 1.9 34 1.86 1.9 36 1.86 1.9 38 1.86 1.9 40 1.84 1.9 42 1.84 1.9 44 1.82 1.9 46 1.8 1.9 48 1.8 1.9 50 1.8 1.9 52 1.8 1.9 54 1.8 1.9 56 1.8 1.9 58 1.8 1.9 60 1.8 1.9 838 To 14 ec. of the liquid (serum + cocain) was added 1} times the volume of alcohol 96°/, + 2 drops of HCL. This was centrifuga- lized and filtered, the filtrate was turbid. The precipitate was subse- quently washed with alcohol and part of the alcohol was evaporated down in vacuo. After this the solution was acidified and shaken out twice with ether. The ether extract was then acidulated with */,, N. HCL to get an aqueous cocain solution. This solution was again neutralized with bicarbonas natricus. With this liquid the experiment was repeated. Exp. 15. We used the liquid of exp. 12 after extracting it with alcohol, the amount of cocain was calculated at about '/, °/,. Stimulation after: 2 min. 4 So after 40 minutes the Reading taken of the induct. (E’) 1.96 1.96 1.9 1.74 1.68 1.68 1.68 1.1 (no longer any contraction). Control (E”) 1,96 1.96 1.96 1.96 1,96 1.96 1.96 1.96 1.96 1.96 1.96 1.96 1.96 1.96 1.96 1.96 1.96 1.96 1.96 1.96 nerve was anaesthetic, from which it 839 appears that through the treatment with acid and alcohol all the cocain adsorbed by the serum was detached. (Normal value for ‘/,°/, cocain is 43 minutes). Exp. 16. The liquid used is that of exp. 13 treated with alcohol and acid. Here also we found that after 40 minutes the muscle had lost its contractility, so that the result coincided with that of expe- riment 15. In the following experiments we used a stronger solution of cocain, Mize o/c /. COCR: 3 0 Exp. 17. The liquid is 0.4 ce. 5°/, cocain + 4 ce. Ringer 0.9 °/,, consequently */, °/, cocain hydrochloricum. 18 08 a6 Reading taken of the inductorium 0.4 40 50 60 minutes Fig. 4. Effect of 1/,°/, cocain hydrochloricum solution in Ringer’s fluid (0.6 9/5) on the nervus ischiadicus of a muscle nerve preparation of Rana esculenta. The result of this experiment is represented by the curve in Fig. 4. After 28 minutes the nerve was irresponsive, while it appeared that, through stimulation with electrode E", the muscle itself had remained responsive. Two other experiments yielded the same results. Exp. 18. Liquid '/,°/, cocain; nerve irresponsive after 30 min. Exp. 19. Liquid '/,°/, cocain; nerve irresponsive after 30 minutes. Average time in which the nerve becomes irresponsive with '/, °/, 840 cocain: 29} min. When serum was added the adsorptive action revealed itself again distinctly. Exp. 20. 0.4 ce. 5°/, cocain + 4.5 ce. rabbit’s serum, which is equal to ca. '/, °/, cocain hydrochloricum in serum. Stimulation after : Reading taken ae Control (E’’) 2 min. 1.8 1.8 4 1.8 1.8 6 1.8 1.8 8 1.8 1.8 10 1.8 1.8 12 1.8 1.8 14 1.8 1.8 16 1.8 1.8 1.8 1.8 42 1.8 1.8 44 1.78 1.8 46 1.78 1.8 48 1.78 1.8 50 1.78 1.8 52 1.78 1.8 54 1.76 1.8 56 1.76 1.8 58 1.76 1.8 60 1.76 1.8 It wil be seen that we found distinet inhibition by serum also in this experiment, for after an hour the conductibility of the nerve had diminished only slightly. This experiment was repeated (exp. 21), which again showed no anaesthesia of the nerve. Subsequently we inquired into the action of 1 °/, cocain. Eep. 22. Liquid: 0.8 ee. 5°/, coeain solution + 4.22 ce. Ringer 0.9°/, equal to 1°/, cocain hydrochloricum in Ringer’s solution. 841 eee Stimulation after: | Reading RED of Control 2 min. 1,95 1.9 4 1.9 1.9 6 1.88 1.9 8 1.84 1.9 10 1.8 1.9 12 Let 1.9 14 1.6 1.9 16 1.4 1.9 18 183 1.9 20 1.2 1.9 22 (irresponsive) 1.9 Nerve does not respond any more after 22 min. Erp. 23 1°/, cocain irresponsive after 18 min. Pap. 24 „ ;; 55 menen Exp. 25 „ 3 3 pa oO me. From which we see that after about 20 minutes the conductibility of the nerve is eliminated by 1°/, cocain hydrochloricum. In experiment 26 and 27 we ascertained the influence of serum on the 1°/, cocain solution. Exp. 26. Liquid: 1°/, cocain hydrochloricum in rabbit’s serum: Stimulation after: Readings taken of Control (E”) 2 1.8 1.8 4 | 1.8 1.8 6 | 1.78 1.8 8 | 1,78 1.8 10 1.74 1.8 12 | 1.74 1.8 14 1.74 | 1.8 16 1.72 ? 1.8 18 | 1.66 1.8 842 (Table continued). Stimulation after: Sanat aa EB) Control (E”) 20 1.64 1.8 22 1.6 1.8 24 1.56 1.8 26 15 1.8 28 1.46 1.8 30 1.4 1.8 32 1.2 1.8 34 1.16 1.8 36 1.1 1.8 38 — 1.8 After 38 minutes the nerve appears to be no longer responsive, so there must be distinet inhibition. Exp. 27. Liquid: 0.8 ec. 5°/, cocain solution + 4.2 cc. of cavia’s serum i.e. to 1°/, cocain hydrochloricum in cavia’s serum. Stimulation after: Ree EIEN Control (E”) 2 1.9 1.9 4 1.9 1.9 6 1.9 1.9 8 1.9 1.9 10 1.9 1.9 12 1.9 1.9 14 1.9 1.9 16 1.8 1.9 18 1.8 1.9 20 1.8 1.9 22 1.76 1.9 24 ie) 1.9 843 (Table continued). Stimulating after: | EE Control (E”) 26 | 1.6 1.9 28 | 1.52 1.9 30 1.46 1.9 32 1.4 1.9 34 1.4 1.9 36 1.34 1.9 38 1.26 1.9 40 152 1.9 42 1 1.9 44 = The result of this exp. is similar to that of exp. 26, viz. only after 44 minutes irresponsiveness of the nerve. The liquid of experiment 27 was treated with alcohol and acid as in experiment 15, and was used in experiment 28 (the cocain content was calculated at 1°/,). After 22 minutes the nerve was no longer responsive from which it appeared that (compare the results of experiments 23, 24, 25) through extraction with alcohol the cocain had been detached. We now considered the question whether the behaviour of brain- substance toward cocain is similar to that of serum. To 5 grms of rabbit’s brains was added 10 ce. of a 2°/, solution of cocain in Ringer 0.6°),. After standing for 30 minutes at room-temperature it was centrifugalized and the supernatant fluid was examined. A control experiment was made on 5 grammes of brains and 10 cc. of Ringer without cocain. The latter liquid proved to be indifferent to the nerve. Erp. 29. Liquid: 5 grms of rabbit's brain-substance and 10 ec. 2°/, cocain; contains 1,33 °/, cocain (see Table Exp. 27). So it appears that after 50 min. the nerve has become anaesthetic. Since in the normal experiments with 1°/, cocain anaesthesia appears after 20 minutes, we must conclude that also brain-substance inhi- bits cocain. Exp. 30. Repetition of experiment 29 but with cat’s brains, 844 Ra a Stimulation after: Reading taken of Control 2 1.9 1.9 4 1.9 1.9 6 1.9 1.9 8 1.9 1.9 10 1.9 1.9 12 1.9 1.9 14 1.9 1.9 16 1.8 1.9 18 1.7 1.9 20 1.7 ILS) 22 1.7 1.9 24 ed 1.9 26 1.7 1.9 28 oa 1.9 30 1.7 1.9 32 1.7 1.9 34 1.7 1.9 36 1.6 1.9 38 1.5 1.9 40 135 1.9 42 155 1.9 44 1.4 1.9 46 1.26 1.9 48 1.2 1.9 50 — 1.9 52 — 1.9 845 | Reading taken of | Stimulation after: | Em Control 2 | 1.8 1.8 4 1.8 1.8 6 1.8 1.8 8 1.8 1.8 10 1.8 1.8 12 1.8 1.8 14 1.8 1.8 16 1.8 1.8 18 1.8 1.8 20 1.8 1.8 22 1.7 1.8 24 1.6 1.8 26 | ho) 1.8 28 1.5 1.8 30 1.5 1.8 32 1.4 1.8 34 1233 1.8 36 1.3 1.8 38 (m3 1.8 40 183 1.8 42 1.3 1.8 44 13 1.8 46 1.3 1.8 48 1.3 1 8 50 1.3 1.8 52 2 1.8 54 1.1 1.8 55 == 1.8 From which we see that the nerve is irresponsive after 55 minutes. Here then there is also adsorption. In order to prove that the cocain is not decomposed, but adsorbed physically, brain-substance and cocain is treated with alcohol and acid as in experiment 15. 846 Exp. 31. Liquid: brain-substance + cocain after treatment with hydrochloric acid and alcohol, computed at 1 °/, cocain hydro- chloricum. Stimulation after: Reading taken of Control 2 1.8 1.8 4 1.8 1.8 6 1.7 1.8 8 1.68 1.8 10 1.6 1.8 12 1.5 1.8 14 1.44 1.8 16 1.4 1.8 18 1.4 1.8 20 1.3 1.8 22 1.1 1.8 24 — 1.8 Here, then, the cocain action manifests itself again, for the nerve is irresponsive after 24 minutes, so that no cocain has been decom- posed by the brain-substance. In order to show that from brain-substance, after extraction with hydrochloric acid and aleohol, no materials are abstracted which, of themselves, are deleterious to the nerve, so that thereby in ex- periment 31 the cocain action might have been intensified, we under- took a control exp. 32, in which a liquid was added to the nerve that was composed of 5 grms. of cat’s brains and 10 c.c. RINGER 0.6 °/, and then extracted with hydrochloric acid and alcohol. This liquid again proved to be indifferent to the nerve, because within an hour the responsiveness had not diminished. Exp. 33. This experiment is a repetition of exp. 31. Liquid: Cat's brain-substance and cocain-solution equal to 1 °/, cocain hydrochloricum. After 54 minutes the nerve is irresponsive, which again shows that the cocain action is inhibited by brain-substance. Exp. 34. The liquid of exp. 33 was again treated with alcohol and hydrochlorie acid. 847 Stimulation after: | Reading laken of Control 2 min. | 1.9 1.9 4 | 1.8 1.9 6 1.7 1.9 8 1.68 1.9 10 1.66 1.9 12 1.64 1.9 14 1.64 1.9 16 1.62 1.9 18 1.6 1.9 20 1.56 1.9 22 | 1.4 1.9 24 1.2 1.9 26 1.1 1.9 28 — 1.9 We see from this that the cocain has again been detached. Of the ecocain thus obtained. Dr. Le Heux determined the melting point, which was 96,6° (uncorrected), which again proves that the cocain has not been decomposed (not even partially), but that only a physical adsorption has taken place. (Normal melting point of cocain hydrochl. 98°). Since it had now become evident that brain-substance is capable of adsorbing cocain, we ascertained whether one of the familiar brain lipoids viz. lecithin’), could also exert this action. Erp. 35. Liquid: 1 ee. 5°, lecithin solution +1} c.c. aqua distillata + 24 ee. Rincer (1.2 °/,) without cocain. In this experi- ment the responsiveness of the nerve had hardly changed, from which we see that lecithin of itself does not injure the nerve. Erp. 36. 1 ec. 5°/, lecithin solution + '/, ec. aq. dest. + 1 ce. 5°/, cocain + 2'/, ce. Ringer (1.2°/,), that is 1°/, cocain hydro- chlorieum in 1°/, lecithin. ') The lecithin was supplied by Menck. Proceedings Royal Acad. Amsterdam, Vol. XXII. 848 Stimulation after: | Ee Control (E”) 2 min. | 1.9 1.9 4 1.9 1.9 6 1.9 1.9 8 1.9 1.9 10 1.9 1.9 12 1.9 1.9 14 1.9 | 1.9 16 1.9 1.9 18 1.9 1.9 20 1.9 1.9 22 1.9 1.9 24 1.9 1.9 26 1.9 1.9 28 1.9 1.9 30 1.9 1.9 32 1.9 1.9 34 1.8 1.9 36 | 1.72 1.9 38 1.6 1.9 40 1.5 1.9 42 1.3 1.9 44 1.12 1.9 46 = 1.9 The nerve is irresponsive after 46 minutes. Exp. 36. Liquid: 3 ce. 2°/, lecithin and 2 ce. Ringer (1.8°/, Wen 1 ce. cocain solution 5°/, is equal to 0.83°/, cocain hydrochloricum in a 1°/, lecithin solution. Result: After 62 minutes the nerve is still responsive. From experiments 35 and 36 it appears then that 50 mers of lecithin can inhibit the action of 50 mgrms of cocain considerably. Zep. 37. Here we examined the influence of an ether extract of dried cats brains. Of itself this extract is indifferent to the nerve, which after 60 minutes is still normally responsive. 849 jep. 38. Liquid: 0.8 ee. 5°/, cocain solution and 4.2 ec. extract of dried cat’s brains, thus containing 1°/, cocain. | Reading taken of inductorium (E’) Stimulation after: Control (E”) | Dati 1.8 1.8 4 | == == 46 1.8 1.8 48 1.7 1.8 50 | 1.7 1.8 52 1.68 1.8 54 1.6 1.8 56 | 1.5 1.8 58 | 1.42 1.8 60 | 1.3 1.8 62 1.1 1.8 64 cs 1.8 Result: This extract proves to possess distinct inhibiting power, since only after 64 minutes the nerve becomes irresponsive (normally after 22 minutes). CONCLUSIONS. 1. Our experiments produced evidence for our assertion that the action of cocain can be considerably inhibited by the addition of: a. the serum of man, dog, rabbit and cavia; 4. the brain-substance of rabbit and cat; c. ether-extract of dried cat’s brains; d. \ecithin. He's and SANo’s experiments are hereby supported and extended. 2. This inhibition of cocain, is not brought about by a chemical decomposition of the cocain but by a physical adsorption; for, through extraction with hydrochloric acid and aleohol of a mixture with a reduced cocain action, all the cocain can be restored, which has still retained its activity. The melting point of this cocain also lies very near to normal values. 3. Serum, brain-substance and lecithin are of themselves not deleterious to the frog’s nerve, nor when these materials (in control experiments) were extracted with hydrochloric acid and alcohol. 56% Physics. — “The Contributions from the Polarization and Magne- tization Elections to the Electric Current’. By Dr. A. D. Fokker. (Communicated by Prof. H. A. Lorentz). (Communicated at the meeting of Juve 27, 1919). 1. An important point in the theory of electrons is how to evaluate the electric current proceeding from the electrons which in their movements are bound to the atoms of matter. We require it for establishing the equations of the electromagnetic field in ponderable matter, and we know that it is responsible for the effects of polarization and magnetization. Consider a stream of moving neutral atoms, and imagine them as consisting of a positive nucleus and one accompanying electron. The beavy nuclei will contain the centres of mass of the atoms their motion therefore will be identified with the motion of matter in bulk. The accompanying electrons will move round the nuclei or in their immediate neighbourhood. Now the stream of positive nuclei will form an electric current, and the stream of electrons of course will constitute another. For a great part these two currents will cancel one another, but not completely, as they would, if both motions were the same: the resulting current is clearly what arises from the intra-atomical motions of the bound electrons. Obviously we shall know this current if, given the motion of a stream of particles, we can find the variation effected by displacing them slightly from their tracks, for it is by such small displacements that the motion of the electrons may be found from the motions of the nuclei. Our problem thus presents itself as a variation problem. M. Born has told us*) that the idea to put it thus is due to Hermann Minkowski. He has developed it after Minkowskr's death and compared his deductions with MinkowsKrs posthumous notes. I venture to offer to the Academy a novel development of the same idea, which might claim a great simplicity and might be more exact in some points. In addition, a new second order contribution of the bound electrons is arrived at, which has been neglected until now, so far as I know (§§ 9 and 11). 1) H. Minkowski—M. Born, Mine Ableitung der Grundgleichungen fiir die elektromagnetischen Vorgdnge in bewegten Kérpern vom Standpunkte der Elektronentheorie, Math. Ann. 68, p. 526, 1910. 851 The Varianonal Displacements. 2. We consider a field of streaming discrete particles, the velo- cities being continuous funetions of the coordinates and the time. We imagine a picture in a four-dimensional space-time-extension designing the motion-trails of the particles indicating their positions in successive instants. Now the displacements will consist of a shift in space and a shift in time, and we shall define these shifts with the aid of a field of a four-fold vector 7“, the components of which: rr, 13), being space-components and 7@) being the time-compo- nent, will be continuous functions of the coordinates and time «4 (ad . 4). Mathematically, we define the shifts as the one-membered infini- tesimal transformation group determined by the functions r¢ (a = 1.. 4), with parameter 0: or 4 ct Agard 40S (ret. 1 Ox This will be clearer if we explain the nature of the r¢. If the variational parameter increases by an amount d@, then the particles are supposed as suffering an additional shift given by 72 dO (a = 1, 2, 8, 4), the values of +7 being taken such as they are in the momentary point-instant occupied by the particle. Leaving out second order terms with 6%, we at once see that the first approximation of the total shift will be 6 re, (a =de, 4), and proceeding to second order terms we obviously get 6 A ne — Í « 0 Ara A at = Ora + ws = (Oe 6 ae Ore ra {+ > (c) 3 — 7c vd dy, axe where now the values of 7 and their derivatives have been taken in the point-instants of the particle’s undisturbed motion. The Variation of the Stream. 3. The following conception of the stream components will greatly facilitate our deductions. Let N, a continuous function of space-time-coordinates, represent the density of the particles’ distribution through space. At the instant 852 v4, take an element of volume dV, situated at the point 2, 2®), xv), It will contain NdV particles. We assume that NdV is still a great number, notwithstanding d@V being physically infinitesimal. Now, in our four-dimensional picture, consider the trails of these NdV particles, run through during an interval of time de). These trails will cover an element of space-time-extension of magnitude dVda@). In the direction of the coordinate X* the components will in the aggregate amount to Nd Vda. It will readily be seen that the streamcomponent in the direction of Xe is the aggregate of the X*-components of the four-dimensional trails, run through by the particles per unit of volume per unit of time: NdV dae dat N DA = N= = Nue == il, 4, 8,4) dV da =e dr or Ue (a = Sh So So) ) We shall put wd, w®, w) for the components of the velocity: de da, dede, da)/da. The fourth component equals unity: wt) = de/dx®, and accordingly the fourth streamcomponent Nw) is the number of particles per unit of volume. It is obvious that the equation of continuity must be satisfied by these streamcomponents : __, ONw! = (b) Sb 0 By the displacements the components will change to Nwt + dNwe + 4d? Nwt, where the first variation 0 Nw is proportional to 6 and the second variation J? Nw will contain the second order terms with 67. It may be anticipated that the first variation will account for by far the greater part of the effects of polarization, whereas the second variation mainly gives the effects of magnetization. 4. We proceed to the evaluation of the first variation. Here we may consistently neglect 6’. The displacements will have changed the aggregate of the Xc- components of the trails under consideration: each dx* passes into OO F ae vb, dat 4 = (b) so that the aggregate becomes N dV {dae + & (b) dx? dgre Ox On the other hand the four-dimensional extension covered by the 853 trails has changed too: we find its magnitude by the aid of the JACOBIAN determinant: jd (ee + Azo) (ze + Ae) | ve Ox! 0(x? + Ax) d(x) + Ax) V dz) = : dV dic) Y de ) dze Ow? | Be) | | Fi 7 Ore Ore | a5 dze Ox? f = | eee AV dO lee Slee lav dew = | ar JE ge zl 2 (6) TD wv We must divide by this, and so when we follow the displacement, we may state a change of the streamcomponent into 0 pt 0 a) Nus + A Nue =| Nut + = (b) Not |.| 1— = (0) 0 — |. Ox? Ow é But this is not the thing we want. This value is found in the point-instant «*-++ Aw, after the displacement. We require the varia- tion of the stream which we get if we stick to one and the same point-instant «* both when the particles are shifted and when they are not. It is clear that the shifted particles which will by the displacement get to our point, had their starting-points elsewhere, in a point-instant which may be found if in the formula for Ava we change 4 into —6. So we have to correct the above expression by accounting for this different starting-point: instead of Nw we are to take ONwt Nwt — XD (b Orb, Oa! and we get ONw* Nw } JSNwe == Nw" | 0 (b) EE! Orb dra Del Or! — Nw 0 Fi t+ Nw 6— |. zb ub Oar? Availing ourselves of the equation of continuity, we may put our result in the symmetrical form: 0 dNw! = Z (6) Ora Nw! Orb Nw}. Ox! 854 This formula is also given by Born. It may be found, without deduction however, in a paper by Lorentz’). 5. The second variation is easily found, without calculation, by submitting the first variation in turn fo the operation which we had to apply to Not in order to find dNw*. So we get without difficulty Oe d.d Nwt = & (b) 5, {Ora dNw? — Orb dNwt}, v 0 Oae It is, however, important to remark that this formula implies the accurate definitions of the displacements as given in § 2. This can be verified by a direct deduction, following throughout the same line of argument as in the case of the first variation. We may refrain from reproducing the calculus, but it will be good to point out, that one has to develop the Jacobian with the required exactness up to the terms with 67, and, above all, that at the last step to be taken one has to be careful to choose the right starting-point from where the displacements will carry the particles to the point under consideration, viz., 0 0 JS Nwt = & (be) TE jor [Ort Na red [OraNwe-ôre Nw] °. Ore wt — Grt + 10E (Ore, Òze and not «*—Awt, as we might be tempted to take. Next, we have to give an interpretation of our mathematical result in physical terms such as polarization and magnetization. The Simultaneous Displacements. 6.1. Before turning to the physical interpretation we must look closer into the nature of our displacement vector r¢ and the under- lying assumptions. We are to assume, that the trails of the electrons can be found from the trails of the nuclei with the aid of the vectors 7% in the indicated manner. First of all, we have taken these to be continuous functions of the coordinates. This implies that neighbouring atoms are supposed as having their electrons at similar distances in similar directions, the positions of the electrons relative to the nuclei varying but extremely slowly from one atom to the next. Of course this will not 1) H. A Lorentz, Haminton’s Principle in Einstein's theory of Gravitation, Proc. R. Ac. of Amsterdam, 19, p. 751, 1915. 855 exactly or even nearly exactly correspond to reality. But we can commit no essential error by assumiug the atoms as behaving in such a continuous way. Secondly. we must observe, that the only reality we are concerned with is the aggregate of trails of nuclei and electrons, and that the choice of the vectors r* is entirely arbitrary provided they furnish us with the right motion of the electrons relative to the nuclei. Obviously the choice can be made in a great many different ways. Sometimes it will be suitable to choose the 7* such that the time- component 7 vanishes in all points where matter is in a station- ary State. We need not specify a particular choice. 6.2. As yet the displacements considered have been accompanied by a shift in time. In view of the physical interpretation of the formulae obtained, it will however be necessary to realize the simultaneous positions of the electrons relative to the nuclei. Now, in a first approximation, we find the electron belonging to the nucleus, which at the instant 2@) is in the point 2, x®, x), shifted to the point ci) + Ord), wl?) + Or), a) + Gri) at the instant a4) + Orl4) . Thus we see that its position at the time «© will be given by alt) + 9G), 2® + (2), 29) + 08), where oe — Art — wt Ord) : For an obvious reason gt — 0. _ Next, to obtain the second approximation, consider the nucleus at the instant a ( : Or) Ort) a4) — Ort) — & (c) 44 6? re ——— we Ord) ——_ 3, ( de dae when its coordinates are k { _ _Ori4) Ord | dwt ma wa Ord) — wt > (c) | 4 Or — we Ord) — —+- 4 A? rrd, | Ou! Owe { ~ da) This line implies the preceding as a special case, for a =d. Then the displacements of the electron will be dr Ore Ort + 40% & (c) pe 08 D (c) we Le = so that its actual position will be given by dw D i One Or(4) za + Art — wt Grit) + 4 — Or Dld + S (6) (4 OP ref — — wt ]}] — Owe Owe 2 Ore Or) — ()? 74) we | ——- — we —— }} ; Oe Oue Taking «=4, this formula yields the instant z®, for w= 1, and all terms vanish except the first. So we see that for a= 1, 2, 3 it gives the simultaneous displacements. We can simplify considerably. Writing 0 d ; = (c) we = dze _ der > Pen inet «2 Or g (c) en OE qe) we get for the simultaneous displacements : dor Owe dod veeel 2 — = (0) os + LEWoe—, (62) da\4) Oue For ~w=4 we have s4)=0. 6.3. Let us inquire what will be the polarization of matter, viz. the electrical moment per unit of volume. The electrical moment of one atom being es*, where e is the charge of an electron, the answer is, in a first approximation, that the polarization has components Ne st, (Gale): Proceeding more carefully, we must take some closed surface, a sphere, say, sum up the electrical moments of the atoms within and divide by the volume. But what about the border atoms, which are intersected by the sphere? Must we leave them out, or must we reckon them as lying within the sphere? The difference will be of second order only, but it does make a difference. A similar question has been raised by Lorentz in his Theory of Hlectrons (note 53). Lorunrz decides himself to leave out the inter- sected atoms, and this is certainly right when we restrict ourselves to the first order terms, neglecting 6%. But here we retain 6’. Fortunately, our calculus leads us to the answer: it will show a correction to be made to the same effect as establishing the rule: the atoms are to be reckoned as lying within the surface, whenever more than half of the line joining nucleus and electron lies within the surface. This is a quite satisfactory rule. Thus the polarization is: 0 Nest sc dze 6.4. The magnetic momentum of an atom has the components Nest — } 2 (Cc) (6.3) 857 1 dst dst el st sb ; 2c dat dat Hence the components of the magnetization are ds? Isa cmb — 1 Ne (« en sb en ) (6.4) de da) It is possible this ought to be corrected in the same way as shown for the polarization. The correction, however would be of the third order and contain 6*; and this we drop throughout our investigation. For this same reason we are justified in replacing s* by o“ in the expression for the magnetization. Interpretation of the Variation of the Stream. 7. If e be the charge carried by an electron, then the current carried by the electrons is eNwt + ed Nwt + Led? Nw%, (a = 1,2, 3, 4). Adding the current carried by the nuclei, viz. —eNw*, we get for the resulting current: ed Nwt + 4 ed? Nw, Our results indicate that this can be written as a divergency of a skew-symmetrical tensor 7’: drab ed Nw" + ted? Nwt = & (b) ape vu where 7’ is givan by 0 Tab — e6(r2 Nw! —r> Nwt) + 4 e673 rt & (c) ane (ro Nwt—re Nw?) — Ue — rb > (re Nwe— pe Nut) 2 we and Pah oe We shall see what this tensor contains. First writing Tb = (94*Nw? 0! Nw?) | € 0 0 + } Or wt a> (o! Nur x. 0! Nw) 4 Ort wb B (ot Nwe oe Nw") + Oz’ Owe d 0 4 } o 0 (ot Nuc — 0" Nw) } ob Dj (o” Nwe e° Nw"), On’ One we can arrange terms in such a way as to get i dot Owe doa òNeotoe Tao — wb Ne ot Ort > Ive <= = oF Ve + 1E oe SS 1yb> 2 org te OE | Ò xe ue do? Ow? do! dNeobge — wtNe| 9 —14rl4) S | we — e LS Os + swt —— ioe ii ue NS Ome Ò ae ane Ow? Owe do do ——1 Neot 0° — Neo’ So Nel ot we —-- 04 we : g N Q ke Oat at ane We recognize the simultaneous displacements (6.2), and find 0 Nests¢ 0 Nes? se fab = wh) Nese tt — wa }Nesh —1 S Ome 02‘ LN ES Ow? LANE Owe LN ds a) == est se SS es sc - 5 (1 — Owe cS da:\4) da 8. Taking 5 —= 4, some terms vanish, and we get 0 Nest se Oare : Remembering what has been found about the polarization in (6.3), we at once see that 74 (a= 1, 2,3) are the components of the polarization. Thus the polarization is no 4-dimensional vector: its components are the space-time-components of a tensor. When neither a nor 6 have the value 4, then the part of 7 containing the polarization : 0 Nests¢ | — wt Owe Tt = Nest — 4 5 (c) 0 Nes? sc Ox? is nothing else but a component of the well known RÖNTGEN-vector, which in three-dimensional analysis is written [p.w], where p and ware the three-dimensional polarization and velocity vectors. We see that in our tensor the components of polarization are always accom- panied by the components of the corresponding RontTeEuN-vector. we {Nest — 4E Nest — 4 3 9. In another part of Te (a44, bd), viz. ds? dst emu) = 4 Ne| st — sb ' da\4) de we recognize the components of magnetization. The remaining part however: On — 4 Nest (c) se 5 vb GE - + 4 Net 3 (Os. & Ow c indicates the existence of a new effect. It is of the second order and 859 therefore has been neglected by Lorentz’) and by CUNNINGHAM °*). Born does not separate it from the magnetization. But we can imagine an experiment (see below) where this effect will manifest itself apart from magnetism. So we shall keep these terms apart. Here the quadratic electric moments of the atoms appear: est sb, the same quantities which occur in recent papers of Drgijr and HorrsMark on the broadening of spectral lines from luminous gases under increased pressures.*) Half the sum of these quantities per unit of volume we shall call the electrical extension of matter, unless a better name be proposed. If an atom contains more than one electron, then we can have an electrical extension without polari- zation. We denote it by Kab — 1 Nest sb. and the corresponding part of the tensor can be written Owe dwt kab — — ZS (e) | Kee — — Ke 5 (°) dare : Owe 10. In order to review the results reached thus far, let us gather them in a scheme, and let as for convenience’ sake use rectangular coordinates 2, 7,2; ¢ for the time, and three-dimensional notations for the (three-dimensional) vectors of polarization, magnetization, and velo- city: p,m(m,—m”*, etc.) and w. In addition, write *K for the three-dimensional extension tensor, and for the new vector k: k= —[(K. V) wi, where (*K.7) is an operator having vector properties. Thus k, = k??, etc. Then the contents of the tensor 7’ are: —>b | em: + ke + [p.w]z -emy—ky —[p.wly pe zat —CiMz k, -— [p.w|- ema + Ky + [p.w |. py. cm, + ky, + [p.W], -cmz — ke — [p.w]x pz — ps == py =p Applying the formula for the current from the bound electrons: 1) Eneyelopaedie der Mathem. Wissenschaften. 4) The Principle of Relativity, Camb. Univ. Press. 5 P. Desye, Das molekulare elektrische Meld in Gasen, Phys. Ztschr. 20 p. 160, 1919. J, Hotvsmank, Ueber die Verbreiterung von Spektrallinien, ib. p. 162, ’ See also P. Desiue, Die Van ven Waausschen Kohdsionskrdfle, Phys. Zschr. 21, p. 178, 1920. | 860 0 Tab due : and putting it in the right hand members of the equations of the field, we get for the fundamental equations for moving non-conduct- ing media, in three-dimensional vector notation: > (0) rot B — eae 4 Mr Ln [p.w] ate C CG C C and div E = — div p. These are Lorentz’ equations with the addition of rotk to the current. We see a polarization current p, a RONTGENCurrent rot [p.w]| and the current of magnetization rot cm. A proposed Experiment. 11. Let us inquire further into the nature of the second order current rot k. Referring to the definition : ke (OS Vo ll, we see that it is an effect due to the non-uniformity of motion in matter where the atomical charges lie outside one another. If these charges bad fixed positions, i.e. if the electrons were rigidly fixed between the nuclei and if they therefore could be said to have exactly the motion of matter in bulk (i.e. of the nuclei, or rather motions inter- polated between the nuclei) then our calculus indicates, that there would be no current resulting from the charges: the streams of positive and negative particles cancelling each other. But in this case, the motion of matter being non-uniform, the electrons clearly would turn round the nuclei in an absolute sense, and the atoms would have a magnetic momentum. It is the part of k to counterbalance this slight magnetization, it then equals cm with opposite sign. On the other hand, in case the electrons, instead of being rigidly fixed in the frame of the nuclei, always kept the same distance and in the same direction from the nuclei, not turning round in the rotating motion of matter, then k comes into play, not being balanced by a slight magnetization; so an induction field will be produced. It should be possible to keep the electrons in the same direction from the nuclei by applying an electric field and maintaining a constant polarization. A rotatory motion then should produce an induction. We must be careful, however, to separate this from the 861 Ronreen-effect, by eliminating the latter. This might be done in the following way: Take a sphere of insulating material, which is mounted to perform rotatory oscillations round a vertical axis. Surround its equator by a circuit fixed in space. Apply an electric field of constant horizontal direction, and the oscillations of the sphere must induce an oscilla- ting current in the circuit. The effect will be small, but it should be detectable with the aid of the modern detectors of radiotelegraphy. It will be proportional to the square of the electric field applied. It might be pointed out that a comparison of the effect with the produced polarization, would provide us with means to determine the number of electrons per atom, which are involved in the pola- rization, because, for a given polarization, the displacement 8 of the electrons is inversely proportional to the number ” of displaced electrons per atom, and so the effect of k per electron is inversely to n°. Materials with the same di-electric constant should show the effect to a degree inversely proportional to the number of polarizing electrons per atom. Spontaneous Electric Polarization of Moving Magnets. 12. Though we have used in the title of this paper tne deno- minations “Polarization and Magnetization Electrons’, yet it is well known that it is impossible to make a rigorous distinction between the two. For even though there may be in some cases electrons which only produce polarization and no magnetization, there can be no electron which gives rise to a magnetization and never produces polarization. In fact, whenever magnetized matter moves in a direction per- pendicular to the magnetization, then it shows a polarization at right angles both to magnetization and motion. The explanation runs as follows. A magnetic atom contains elec- trons sweeping round the nueleus, in circles, say, with uniform velocity, under the actions of electromagnetic forces. When the atom acquires a motion in the plane of the circling electrons, then the forces are modified in a way given by the theory of electrons and of relativity. The effect of this alteration of the forces will be that the orbit is no longer a circle, and becomes an ellipse, and that the velocity changes in such a way that the electrons during a longer time stay in one part of the ellipse near an end of the long axis than in the other. This clearly results into a polarization. 862 We shall call this the polarization of moving magnetism. It explains why no current is set up in a moving magnet on account of a motion perpendicular to its own internal induetion field, so that with sliding contacts no current can be taken off. Thus, e.g., if we take a circular spring, the two ends pressing together, we can put a long magnet into it. Suppose that we can draw the magnet across the ring, the ends of the spring giving way and making a sliding contact: there will arise no current in the ring if we do it. Again, this polarization is responsible for the electric force set up in a homogeneous magnetic field if the magnets producing the latter acquire a uniform motion at right angles to the field. The magnetic field may remain stationary and homogeneous: neverthe- less an electric force will be induced by the motion of the magnets. Afterwards these problems will be treated more adequately when we shall have explained the character of our deductions from the relativity point of view (see below § 20). Then we shall also define a distinction between the di-electric pola- rization which is independent in itself, and the polarization of moving magnetism. The Invariancy of the Results. 13. Thus far we did not want to refer to a single theorem of the theory of relativity to deduce our results. Nevertheless they possess the property of complete invariancy, not only in EiNstwIN- Minkowski’s theory of restricted relativity, but also in EiNsTEIN’s theory of general relativity. We proceed to show this. This theory ascribes to a four-dimensional track the length ds: ds* = = (ab) gap dat da), if dart (a=1..4) define the increments of the coordinates and time. The determinant of the gas is called gy, and its minors divided by g are denoted get. What is the character of Nw«? Remembering the definition ($ 3 :) Vg NdV dae Vg dV dal) ’ we notice that NdV is a number, dz“ is a contravariant vector and WgdV de constitutes a scalar. Thus Nw° is a contravariant vector multiplied by Yq. 6r* is a contravariant vector too, and so Ort Nw — Or? Nw is an skew-symmetrical contravariant tensor, multiplied by g. (This is sometimes called a volume-tensor or a tensor-density, after Wry). Nwe= Then we know that 863 0 d Nut = = OT {Ora Nw — Ort d Nw} is the contravariant vector-divergency of this tensor, multiplied by Vg, and thus of the same nature as Nw itself. In like manner the second variation Om d' Nw = = (b) AE {Ora JNwt — Or’ SNw%} x is a contravariant vector multiplied by p/q again. It follows that our results are in complete accordance with relativity theory in the most general sense, and we are justified in applying any theorem of that theory. Having thus recognized the true character of our tensor, we shall henceforth write Yg 7 instead of 7’? Va Te) = eGre Nw? — eOr Nwt + $ & Ore d Nw — Or d Nw%}. This will cause no confusion. We must further keep in mind that 2* is no four-dimensional vector, but w* da/ds is. We shall not introduce a new notation for this velocity vector. The General Covariant Equations for the Field. 14. The covariant tensor of the field can be written as the rotation of the potential vector ga: = (GER WS ile); (14.1) From these we get the contravariant components: Fab = & (ed) gee 94 fear and the fundamental equations of the theory of electrons are ò : rr (EM Os (14.2) where o is the density of the electric charges, and ev" is a contra- Jab a variant vector multiplied by Vg. From the relations (14.1) arises another equation. Multiply by the contravariant fourth rank tensor 4d¢¢/)/g, and contract twice. Here deed ig 1 whenever the figures abcd constitute an even permu- tation of 1234, and in other cases vanishes. Then we get the conjugate tensor/,@” *): "al > (ed) 1 fabled f. a — * a0c6 fet = Eed) — dated fg ') In order to get the covariant conjugate tensor components /*ay from the contravariant fed, multiply in the same way by the covariant tensor Ig Vg Dabed - Oabed = jabed), Proceedings Royal Acad. Amsterdam. Vol XXII. 864 If now we write EO 5 Was) =0, (14.3) this must be an identity in virtue of (14.1). The Muinkowskian force acting on a moving charge e has the covariant components: fa =e = oe = 1 fat These equations are supposed to hold within the finest structure of matter. To obtain the equations of matter in bulk, we take the mean over a small region, containing a great many atoms. We define S fas Vg det). ded pas IFS Vg ded dls) SWVgded..de® ” fYgdelt).. dal)’ It is readily seen that still 1 SS EH) GFE GE Ng: The mean of the convection current gv’, as produced by the bound electrons, we have just found, and so the equations for non- conducting matter are: == 0 ag Va PO: (14.41) In conducting matter, the current from the conduction electrons Vgl* must be added in the right hand member. The other equations oo. 0 =) Veh = =O. | > (6) en Wa Fy) =0 (14.42) Now, we could try a solution #«’— 7, and add a solution E of the equations Ore “Va Fat) — 0 (14.51) and =O, CARENS = (0) "Wa T,"), (14.52) By’ and 7,°° being the conjugate tensors a Ee and Tet. Then Fab Tab 4 Fab is a solution of equations (14.41) and (14.42). We shall call 7% the internal, and Ze the external field. Separation of the Polarization and the Magnetization Tensor. 15.1. It has been remarked, that in our tensor Vg 7 the - 865 polarization and magnetization are intimately interwoven. Indeed, it is not always easy to separate them. We shall assume that our system of reference is such that ga vanishes for a— 1, 2,3. This means, that in our system of reference the velocity of beams of light in opposite directions is the same. This also implies that whenever the first three contravariant components of velocity vanish, also the first three covariant components vanish. Then we may in stationary points of matter (or in an arbitrary ‘point after changing variables in a way that renders it stationary) separate 7’*) into two tensors, one consisting of the (a4)-components, the other components vanishing, and vice versa. These teusors we may call the polarization and magnetization tensors. It should be understood, that a change of variables, or a motion of matter, cannot leave half of the components zero, as they are in stationary points: the magnetization tensor, e.g., completes itself with polarization components. Write „Tet for 7 in a stationary point, and separate: „Tab = ,Mab 4 „Pb, so that Py bi ne 0 ‚Met (=) RT aps Lou nf lee elt 0 0 0 and 0 0 agree »P (=) yO Ont IE 0 0 „Ds pe Aad wapen We could have taken the covariant components ot the tensor T,, and the separation would have had the same effect. This is due to the fact that g,4 vanishes for a= 1, 2,3. If this is not the case, then we have first to change variables to make them vanish and afterwards make the separation. 15.2. We shall now briefly indicate what becomes of the constitutive relations between polarization and electric force, magnetization and magnetic force, and between the conduction current and electric foree. We proceed in a quite formal way First, to find the generalization of the equation P = (e—1) EB, we form from the field-tensor a force-vector 4%: 57* 866 datt) Fa 2 (6) wo Fab, ds and from the polarization tensor we form a vector Pa: dat) Rab) wy Pa, 8 and the required generalization will be Pa — — (e—]) Fe, Secondly, to generalize the relation B — uH, or rather ul u we proceed in a similar manner. From the conjugate field tensor we form a vector Ga: M = B, da(4) ds and from the conjugate magnetization tensor a vector Q,: da(4) ds Ga = = (6) we) Faas, Qn ===" (6) The generalized relation is wb May. On ge Ga. u The current of the free electrons is partly a convection current, partly a conduction current. The latter will be the component of the four-dimensional vector-density Yg/* in a direction perpendicular to the four-dimensional velocity vector. The conduction vector thus is: ENE: ee (en iT. 8 Je [a — we This can be put otherwise, if we first form a skew-symmetrical tensor da) Me ass flaw) — [> wt ds and afterwards from this tensor form a vector again: dar Ja (b) wy Tet, ds The equation for the conduction current must be Jeu — ) Fe, We notice that in the common equation J=oE, 5 — Ay. 16.1. Now take the contravariant tensor Pe? and form its con- jugate : Prat = > (ed) $ V9 dasca Pd. 867 then we get the conjugate tensor with covariant components VgP™ Jey Wap: ople En VgP** VYyP Vgkh™ Peat (=) VoP™ —VY9P" VoP" a VYgP” —VoP" —VYgP" By multiplying by the velocity vector and contracting, dx) S = (6) wb Peo, we get a vector. This vector clearly vanishes in a stationary point, because wi, w®), wd, and „Pes vanish, and it therefore always vanishes. Thus we conclude that we shall always have 0 = w?) gP** — w®) V¥gP** + YoP*, (16.1) and similar relations for cyclic permutations of the figures 123. It is thus confirmed that where w/gPe* (a= 1, 2,3) are polarization components, the other components of this tensor consist of compo- nents of the corresponding RONTGEN-vector. 16.2. Apply a similar reasoning to the magnetization tensor. Multiply by the velocity vector and contract: ) dix) we Mab — >(b) ds ds This will be a vector vanishing in stationary points, since 2,, w‚, w,, and ,J/*4 vanish. Therefore it will always vanish, and we shall have da = (be) Jbc wy, Mab, 0=w, MW? + w, MP Lw MY. (cycl. 123). (16.2) Here we meet the polarization of moving magnetism, Vg M%*, in terms of Met. We know from §§ 8, 9 that W/g Met must contain, besides the components of the magnetization and of £¢, the components of the RöxraeN-vector corresponding to the polarization of moving mag- netism also. This will afford us means completely to express the polarization of moving magnetism in terms of the magnetization and k of moving matter (§ 19). Comparison with Other Theories. 17. In constructing the polarization tensor Einstein, following Minkowski, starts from the vector /” defined in §15.2, and he puts for his tensor *) h Die formale Grundlage der allgemeinen Relativildtstheorie, Berl, Sitz, 41, p. 1065, 1914. 868 dax —— { Paw — Pb wa}. s In order to show that this is the same as our tensor Pe, take a special case, a=1, b—= 2 eg., and write in full dal) de)? —— | Pa wb— Pow — | ds ds wl?) (w, PE + w, PY + w, Pi) — — wb (w, PE + wy, P?*? + w, P?*)). We can rearrange: (do) Pace) ds P? (wD w, + ww, + w) w, -+ wt) w,) + + w, (wh) P?? + wl) PES + wl) PA) + w, WD PP u?) PH wt) P?1)) . and now we remark that the latter two bracket forms vanish in virtue of (16.1), for de) ds 1 de T V4 ds (wD PE? +. w@) PB Ll) PII) = (w(t) Jr wf?) Ln + w3) Peso) = 0. As da\*))? St a) om, = IE ds the required identity is shown to exist. In the same way it can be shown that the magnetization tensor or rather its conjugate in the form dal) ds | Qa wy — Qo wa} agrees with our Mss. 18. Let us make the simplifying assumption of the absence of gravitation. Then the ya, and ge? have the values: Neier Om OO ihe Oe Ou 80 ORO RKO Ee ONO dab (=) OM OMIM 80: EEE) tae Onell Oleg ce) OO O° One Ome OMmaly/c7 If A and p denote the common vector and scalar potentials, then the components g, are A, A, A. and — cp. The components of the field are 1) In order to avoid imaginaries, we shall everywhere in Vg take |g]. B = B, cEx — B- Bz cEy Fat (—) B, — Bz cEz —cE; — cE, —1cE B — B, — E‚/c — Bz B. — E,/e a) B, —B — E-/e E‚/c E‚/e E./c The equations for the field are (14.41) 0 0 3 (6) —- (Vg Fe’) = DB) (Ng Pet + Vg Me), dx? Ox? and we have, if P is the principal di-electric polarization : [Pw]. — [Pw], Pz =, P Zz ) Vege eS [Pw] [Pw] Py (18.1) Dee A «el Ds and em-+k-+[n.w]. —cem,—k,—[n.w]y na VaMe(=) —Cm-— k-—[n.w]- emz+k+[0.W |x Hy (18.2) em,+k,+|0.W], —cemg—ky—[n.W Ja: Nz — 0x — Il, Th where n denotes the (electric) polarization of moving magnetism. For the conjugate tensor of the field we have EEN B,/c — E‚ E B,/c NE ; a —B,/e —B/e — B,/c We see that the equations (14.42) amount to crotE + B=0, (18.31) div B= 0, (18.32) From the equations of the field we see that div E = — div (P + n), (18.41) and erot B —E = rot (om + k + |n.w] + [P.w]) + [P +n]. (18.42) These are the equations we have met in §10. Only we had not yet separated p= P-+ Dd there. 870 19. Let us solve n in terms of mandk. Referring to the equation of § 16.2 we must notice that MD == = Wa Dj = Vijn Û SN U Zn De =d and we get c? nr = Wy (em- + ke + [n- w]-) — wz (em, + ky + [n- wlj) or en=([w.(cm+ k + [n. w)) J. (19.1) From this it is easily seen that (a. w) = 0, and as [w.[n.w]] = wn — w(n. Ww), we get w. (cm +k) | zy; = Vg AM *4 == DEE a.So. (19.2) w ¢? (2 = == C and 5 z 5 . (c k Vonne Ee AC eo (19.3) ’ w? ‘ w? I= GU — — (a c? In this form our result for the magnetization tensor can be readily compared with the corresponding formulae of Born '). He also points out the existence of the vector n and states that it is the magnetic analogon to the ROnteun-vector. We see that the factor 1/(1—w?/c’) disturbs the analogy. The difference in the appreciation of the result is this that Born (apart from not separating k) takes the whole of the components VgM**, wgM* and gM" to be the components of magnetization and seems not to have become aware of the fact that they contain the Rénreen-vector components belonging to n as well as the magnetization components proper. Born emphasizes the complete symmetry of his electric and mag- netic equations and certainly one can enjoy the mathematical beauty of the formulae thus written. It would, however, be erroneous to believe that the difference from Lorentz’ equations is more than a difference in form. Our investigation shows that the physical contents of Born’s equations is no other than what has been expressed by LORENTZ. Action of Polarization of Moving Magnetism. 20. Let us illustrate some effects of n by considering a long 1) Le. form. 39 and 39’, pp. 546 and 547. 871 magnet moving at right angles to its magnetization. We shall follow the distinction of “internal” and “external” field at the end of § 14. The effect of this electric polarization n, called into existence by the motion of magnetized matter, is to produce an internal electric field (18.41): P= hh This could be expected to act on free electrons, present in the magnet, and cause a conduction current. But these electrons are carried along with matter and therefore are moving with velocity w through the internal magnetic field where the induction vector is (see § 18.42): cB — cm + k + [n.w] and, where the external field may be neglected *), they consequently are subjected to the Newtonian force (E+ = [w . B]). This expression vanishes according to the formulae of $$ 16.2 and 19, so that the free electrons moving along with the magnet are not driven sideways. Therefore it is impossible with sliding contacts at the magnet’s sides to get a current from it, and the experiment with the long magnet drawn across a circular spring is explained. (§ 12). On the other hand, if we cut the magnet at right angles to the magnetization, and take out an infinitely thin lamella, so that a thin wire might be kept in the same place while the magnet is drawn across, then the “external” field in this split will simply be the continuation of the internal field. and the free electrons in the wire, not sharing the motion of the magnet, will be subjected to the electric force E only, so that an induction current will be set up in the wire. Thus we see that it is the polarization of moving magnetism that accounts for the inductive force, when a magnetic pole moves across a wire, in a case where the magnetic-field is homogeneous and stationary. Conclusive Remarks. 21. In conclusion we may remark that the result of the first variation is wholly incorporated in the polarization tensor. The ') Suppose the magnetization as being homogeneous, and the free poles of the magnet as being at infinite distance. 872 greater part of the result of the second variation is represented in the magnetization tensor. Consider once more the complete polarization (6.3 and 6.2) : neler tor | zegel +4265 OEE eS e 5 Owe ; oh dal) 5 9 Owe Owe Here Neot is the term, by far the most important, which results from the first variation. It is difficult to tell in a few words. which part from the second order terms is exactly the polarization of moving magnetism. If the 7 are so chosen that 7%) vanishes in stationary points, then we can say that the greater part of dp 8 I Neo ge BING NO da == ON clo" figures in the polarization tensor. A small fraction of it (in as much as JN is no scalar) appears, however, in the magnetization tensor, together with do ; Owe — 1 Ne Grl4) TD ee 2 oO SUE as the polarization of moving magnetism. But we refrain from entering into detail here. Mathematics. — “Ueber die Zerlegungsgesetze fiir die Primideale eines beliebigen algebraischen Zahlkörpers im Körper der ten Hinheitswurzeln.” By Dr. N. G. W. H. Brraer. (Com- municated by Prof. W. Kapreyy). (Communicated at the meeting of March 20, 1920). Im letzten Hefte der “Mathematischen Zeitschrift” *) hat Herr T. Retta die Zerlegungsgesetze für die Primideale eines beliebigen algebraischen Zahlkörpers im Körper der (ten Einheitswurzeln dargestellt. / war dabei eine Primzahl. Im Folgenden werde ich zeigen dasz seine Methoden auch in dem Falle benutzt werden können wenn man statt des letztgenannten Körpers, den Körper der J-ten Einheitswurzeln nimmt. Man musz dann seinen Betrach- tungen einige hinzufügen. Ani Es sei / eine ungerade Primzahl; ¢ =e Yk ein Körper der mit k{(& einen Unterkörper vom Grade 7n=—=ali#—! gemein hat, wo a ein Teiler von /—1 bedeutet. Der aus & und 4 (&) zusammengesetzte Körper (f, $) ist vom Relativgrad = über £ und relativ-zyklisch. Wir 7 setzen zur Abkiirzung p statt p(l). In (&, $6) gelten folgende Zerle- gungsgesetze : 1. Ist p eine von / verschiedene Primzahl, p ein in p aufgehendes Primideal von & von Grade /. Gehört p (mod /") zum Exponenten /, und ist ff’ das kleinste gemeinschaftliche Vielfache von f und /, so zerfallt p in (4,5) in z’ Primideale vom Relativgrade /”, wenn Pp Le m 2. Ist tl ein in / aufgehendes Primideal von & von Grade f und /= la, (a,!)=1; d der gröszte gemeinschaftliche Teiler von e und (lj; n die gröszte ganze Zahl für welche eine Kongruenz 1 == at (mod (e+) besteht, wobei « eine ganze Zahl von 4 bedeutet; d, der gröszte gemeinschaftliche Teiler von ” und p(/t#), so ist m ein Teiler 1) Band 5. S. 11. 874 von d, und d, ein Teiler von d. Setzt man d= f’d, = f’z’m so gilt in (4,5) die Zerlegung: 9 == (ee Seniesa; Nu(&)=1" . Der Beweis für 1. ist derselbe wie fiir den Satz. 1. des Herrn Renta, wenn man darin Zin /# und /—1 in p ändert. Beweis fiir 2. 22 3 IW’ 8 a Wir setzen erst £ zusammen uit (el) zu einen Körper 4. BAE . gl Dieser ist relativ-zyklisch vom Relativ-grade —— zu 4. Der Relativ- a grad ist nicht teilbar durch /. Ist also in 4,: [= (f EL so hat das Primideal & keine Verzweigungsgruppe, weil der Grad dieser Gruppe eine Potenz von / sein musz und zugleich ein Teiler i) VON Benet a Hieraus ergibt sich weiter dasz g’ prim zu / ist, da die höchste Potenz von /, durch welche g’ teilbar ist, dem Grade der Verzwei- gungsgruppe gleich ist. Man sieht leicht ein, dasz der Beweis des Herrn Rerra auch hier seine Gültigkeit behält wenn man darin wiederum 7 in und A in p (ll), ändert. Es ergibt sich dann die Beziehung: ae, d woraus folgt dasz e’ nicht durch 7 teilbar ist, und weiter e’ =1 Dann hat man in &, die Zerlegung gefunden: OCT (=P) @ NAD eers Ww Nun haben 4, und & (ps ) den gemeinschaftlichen Unterkörper Oni ani Ce) Wir setzen 4, zusammen mit dem Körper (me) einem neuen Körper #,. Dieser Körper ist relativ-zyklisch von Rela- tivgrade / in Bezug auf 4. Und wir würden ebenso den Körper 4, bekommen haben wenn wir gleich % mit dem zuletzt-genannten Kreiskörper zusammengesetzt hätten. Ai Ist & die den Körper #, bestimmende Zahl und Z7= Ann stellen die Zahlen 1) Weser „Lehrbuch d. Algebra” Il. S. 664 u, s. w. SO 875 i yl ae eee a Or (eae I eine Basis von &, dar, wenn g der Grad van &, ist. Die relativen Substitutionen von &, in bezug auf &, haben die Form (7: 7°). Dazu gehört das Element an Ge) Es ist hieraus ersichtlich dasz € durch das Primideal |, = (1—Z) teilbar ist. Und weil es ein Ideal von 4, ist, ist es also teilbar durch £' wenn dieses Ideal in &, auf / teilbar ist. Dann ist auch der Relativ discrimant von &, in bezug auf &, durch {" teilbar. Also ist €” ein ambiges Primideal*) und &! ein Primideal von #,. Es ist daher in 4,: e= eid Rit is her eal aie Snr PN (2) und DE == In derselben Weise findet man, indem man wiederum 4, zusammen- od 8 (ee) : oe APSA ABTS. ft setzt mit &\e zu einem Körper &,, dasz im Körper 4, die Zerlegung OUA Rd RENS) gilt. Wenn man das Verfahren fortsetzt so findet man aus (1), (2), GA den Beweis des zu erweisenden Satzes. Schlieszlich bemerke ich dasz die Sätze ihre Gültigkeit behalten für m= 1. 1) BACHMANN, “Allgemeine Arithmetik der Zahlenkörper’”, S. 450. 3) Hitpert. “Bericht über die Th. d. a. Zahlkörper. Jahresb. d. D. M. V. Band IV. Satz. 95. Chemistry. — “The Electromotwe Behaviour of Aluminium.” 1. By Prof. A. Smrrs. (Communicated by Prof. H. A. Lorentz.). (Communicated at the meeting of February 28, 1920). 1. Jntroduction. As early as 1914') we began to consider the behaviour of aluminium from the point of view offered by the new theory of the electromotive equilibria. As regards its electromotive behaviour aluminium is a most in- teresting metal. It has generally not been inserted in the electromotive series, because no certainty has been attained as yet about its place. In alkaline solutions aluminium precipitates the zinc, but it does not do so in neutral or acid solutions. To this is added the very remarkable fact that the amalgamated alumininm does precipitate the zine from neutral solutions, and acts with violent decom- position on water, that it further rapidly oxidizes when exposed to the air, and exhibits a character that indicates that aluminium in this condition must be placed directly after the metals of the alkaline earths, thus: Mg—Al—Mn—Zn. In connection with this the conclusion was obvious that commer- cial aluminium is in a noble, less active condition, or in other words that it is in a state of passivity. This was decidedly a step in the right direction, but an explanation of the behaviour of aluminium had not yet been given. Most handbooks and publications state that commercial alumi- nium is covered with a coat of oxide, and that its passivity is owing to this. Also the anodic polarisation of aluminium has made the peculiar character of this metal evident. It was found before, that when an Al,(SO,),-solution was used, the density of the current, i/o, on anodic polarisation continually decreased, whereas the electric potential rose, which may be seen from the following table, which has already been published before.*) Here the potential has been measured with respect to another aluminium rod as auxiliary electrode. 1) Smits, Aten, These Proc. 22, 1133 (1914). 8) SMITS, ATEN, Loc. cit. 877 Al-electrode in 1/2 N Al2(SO4)3 i/o - anode 0.8 | + 2.56 | 0.53 + 3.48 | 0.46 + 3.84 0.36 + 4.13 It has been tried to account for this phenomenon by assuming the formation of an AI,O,-layer with great resistance, which sup- position is, however, hardly tenable, for when the above-mentioned phenomenon presents itself, the aluminium-anode is perfectly bright. Besides when the current is reversed, the resistance has entirely disappeared. When the tension is increased, there is actually formed a coat of Al,O,, Al(OH), or of a basic salt. Then the density of the current is practically reduced to zero, but when the current is reversed, the potential of the aluminium-electrode is considerably smaller. When the anode potential is carried up very high, e.g. to 200— 500 V., the potential is reduced to from '/,, to '/,, on reversal of the current. This property, the so-called valve-action, is used to transform an alternating current into a continuous one. With high current-densities the electric valve-action stops under ordinary circumstances owing to rise of temperature. Fiscuer*), therefore, used as anode an aluminium tube, through which water flowed, and in this way he succeeded in getting coats of oxide of a thickness ot some tenths of millimeters. The most extensive researches on the valve-action of aluminium have been performed by Scnurze*). He assumes, that every newly- 1) Zeitschr. f. phys. Chem. 48, 177. 1904. *) Ann. der Phys. 21, 929. 1906. Ply, voe 548! 1907: » ¥ - 23, 226. 1907. - ‘ » 24, 43, 1907. » 5 = 2010008; . , . 28, 787. 1909, See 34, 667, 1911. s = . 41, 598. 1913. Zeitschr. f. Elektrochem, 20, 307. 1914. . " 9 20, 592. 1914 878 formed aluminium-surface is immediately covered with a solid, not porous layer of oxide of molecular thickness. This layer, indeed, insulates, but according to him it can be pierced by the anions of the salt-solutions or by the O”-ion on anodic polarisation, and the oxygen formed then combines with the metal to Al,O,. The porous oxide layer offers an ever increasing resistance with increasing thickness, and at last the anions are almost exclusively discharged at the layer of oxide, and only very few succeed in traversing this layer, and reaching the metal, which he tries to prove by the fact that the quantity of generated oxygen is 96°/, of the quantity of electricity transmitted. When with a certain thickness of layer a definite potential gradient has been reached, sparking commences, which puts a stop to the increase of tension and the thickening of the oxide layer. This maximum tension is greatly dependent on the nature and the con- centration of the anions; when this concentration increases, the maximum tension diminishes. It is remarkable that, when the current is reversed, no current passes below a certain potential, and that this minimum potential of the cathode is then many times smaller than the anodic-minimum-potential. Also the cathodic minimum potential depends greatly on the nature of the ions. SCHULZE, and before him Taytor and INeus ') and Gurue ®), thought that they could find the explanation of this peculiar phenomenon by assigning to the gas layer that is formed in the pores of the Al, (OH),, the property of allowing the anions to pass less easily than the cations. It is clear that this explanation is not entirely satisfactory, the more so because there are still a great many other exceedingly remarkable phenomena on which it does not throw any light. Two of them may be mentioned here, first the phenomenon that amal- gamated aluminium does not show valve-action, and secondly that a chlor-ion concentration in the electrolyte of 0.2°/, renders the valve-action quite impossible. 2. When aluminium is considered from the point of view of the theory of the electromotive equilibria, the conclusion is readily reached that this theory is able to account for the above-mentioned remarkable behaviour of aluminium by means of the same principles as the polarisation-phenomena in the other metals. In the first place it may be pointed out that it can easily be 3) Phil. Rev. 15, 327, (1902). 879 demonstrated that it is erroneous to assert that commercial alumi- nium is covered with a coating of oxide. It was shown before that when the bottom of a vessel with an Al, (SO), solution is covered with a layer of mercury, and when through the solution an alumi- nium rod is immersed in the mercury layer, the aluminium rod immediately assumes the potential of the mercury, from which follows that the aluminium rod was not covered with an insulating layer of Al,O,, but was in direct contact with the mercury ‘). Now that this fact has been established, and the initial condition is uncovered metal, it is clear that it must be explained why on anodic polarisation the potential of the metal becomes so strongly positive already with very small current densities that the tension of separation of the oxygen is reached. lt is seen that here the same question presents itself as in the case of anodic polarisation of other inert metals. It was pointed out before that in the first place the most essential, the primary phenomenon, should be explained viz. the change of the potential in noble direction; the oxygen, separation and the subsequent oxide formation are secondary phenomena. The strong ennobling of the potential of aluminium on anodie polarisation must be explained by this, that while the withdrawal of electrons from the metal which is represented by 36, is immediately followed by aluminium-ions going into sulution Al. { Al, because this heterogeneous equilibrium is instantaneously established, the homogeneous reaction Al, > Als +3 6. proceeds with very small velocity, so that the metal becomes poorer in ions and electrons. In consequence of this the potential of the metal becomes less negative or more positive, aS appears from the equation : 0,058 Ly == log : — 2,8 B (ML) because in this case Ly, becomes smaller. This phenomenon is, therefore, primary, and if the metal is inert, ') Swits, ATEN. I.c. Proceedings Royal Acad. Amsterdam Vol. XXII. 880 as it is here, the potential of separation of the oxygen will soon be reached, and oxygen generation will set in, which under certain circumstances may lead to the formation of an adherent coating of oxide or hydroxide round the metal. Of course this coating gives rise to a certain resistance, which may rise to considerable amounts with increasing thickness. The assumption, however, that the resi- stance of such a coating should be different for different directions of current is not justifiable, so that there can be no doubt that the sudden decrease of the resistance on reversal of the current, must be owing to some other cause. So far there is no reason to doubt that oxygen and hydrogen are negative catalysts for the setting in of the internal metal equilibrium. Hence the slight quantities of oxygen absorbed by the metal on anodic polarisation have a greatly retarding effect (Fe, Co, Ni). Most probably this is likewise the case for aluminium, and to this it will have to be attributed that such a strong anodic polari- sation has been observed in aluminium. Accordingly this fact leads to the assumption that the metal is disturbed here to a great extent, i.e. that the metal surface becomes very poor in ions and electrons, or in other words, that the metal passes at its surface into a state which agrees with a metalloid in this that it possesses an exceedingly small electric conductivity. On this strong anodie disturbance the aluminium surface becomes, therefore, a metal coating of great resistance, and this coating is in its turn surrounded by another of Al, O,. As the study of the phenomenon of polarisation in other metals has taught, the disturbance that has arisen by anodic polarisation, stops immediately through reversal of the current. This behaviour must be explained by the fact that hydrogen, just as oxygen, though in a different degree, is a negative catalyst for the establishment of the internal metal equilibrium, and can yet apparently act positively catalytically, when it separates on a metal surface that has previ- ously absorbed oxygen, as both negative catalysts then disappear amidst formation of water. The small quantity of oxygen absorbed then enhances the disturbance of the aluminium on anodic polari- sation; hence the removal of this negative catalyst will immediately stop the disturbance, and the strongly metastable state of the alu- minium surface will be transformed with great velocity in the direction of the state of internal equilibrium. This transformation changes the metal coating of great resistance suddenly into another of smaller resistance, which must, therefore, be attributed to the great velocity of the reaction: 881 Al, Al” 436, Hence the resistance that remains when aluminium is made from anode to cathode, is chiefly the resistance of the coating of Al, O,. In the earlier paper cited here a great resistance was simply assigned to the solid solution of oxygen in aluminium, and too little stress was laid on the fact that the strong disturbance of the aluminium may iead to the formation of an aluminium surface that is very poor in ions and electrons. Since 1914 researches on the electromotive behaviour of aluminium have been made in my laboratory, first by Miss Riw1in, and after- wards continued by Mr. pr Gruyrer, both as regards commercial aluminium and amalgamated aluminium. Working with very pure commercial aluminium, the polarisation and the curves of activation in different aluminium-salt solutions, have been determined, in which again, just as with iron, the strongly positively catalytic influence of halogen-ions came to light. A closer examination of the valve action is in progress, and also the thermic and electromotive investigation of the system mercury- aluminium. In connection with this investigation there are other thermic and electromotive investigations being made on systems mercury-metal, as mercury-magnesium, mereury-tin ete. As a pecu- liarity it may be mentioned here that, as was already found by De Leeuw®) in tin, mercury exerts an accelerating influence on enantiotropic conversions, so that, where a point of transition in the pure metal is not, or hardly, to be observed, this is generally very clearly seen on addition of a little mercury. | A good example of this is furnished by aluminium, for which a point of transition at + 580° was found with great clearness from the investigations of amalgams rich in aluminium. Also with a view to this the investigation mercury-metal with other important metals will be continued. General Anorg. Chemical Laboratory of the University. Amsterdam, Jan. 30, 1920. » 1) These Proc. 55” Mathematics. — “Sur les ensembles clairsemés.” By Prof. ARNAUD Densoy. (Communicated by Prof. L. E. J. Brouwer). (Communicated at the meeting of March 27, 1920). Selon une définition que j'ai proposée, (Journal de Math. pures et appliquées, 1916), je dis qu'un ensemble est clairseme quand il est non dense sur tout ensemble parfait. Soit £ un ensemble queleonque, Z, l'ensemble des points de ZE qui sont limites a Z. Soit a un nombre ordinal queleonque. Si « est de première espèce, soit F, l'ensemble des points de ZE, ; qui sont limites a #,_,. Si « est de seconde espèce, soit Z, l'ensemble des points communs a tous les #,, de rang inférieur a «. Chacun des H, contient tous les ensembles suivants. Je dis que tous les Z, sont nuls ou coincident à partir d'un certain rang de «. En effet £,4, est l'ensemble commun a Z, et a son dérivé E’,. Done l'ensemble £’, contenant Mp, contient tous les ensembles HE, d'indiees 2 supérieurs a «. Comme E’, est fermé, EL’, contient tous les ensembles £’; si A >a. Done, dapres un théoreme connu, il existe un rang 8 tel que H's > H’s, si Bp’ <8, et tel que Bie Ei, ,) =. -. Ey élant situé dans EH’, pour 2>>«, E’; est situé dans le dérivé H", de E’,. Done, si E’3 n'est pas nul, W's est parfait, puisqu’il coincide avec un ensemble E's, contenu dans son dérivé W's Dans ce cas, Pipi, situé sur W's et ayant pour dérivé HH’; lui-même, Ez, est partout dense sur “’3. H244, que nous désignons par Ff, est dense en lui-même et a pour dérivé Pensemble parfait 4”; ou P. Si E's est nul, Hz a un nombre limité de points, ou est nul. En tous cas, Easy, est nul. Soit P, un ensemble parfait sur lequel # est partout dense, et H Pensemble commun a P, et a #. H est dans E, et, de proche en proche, dans Z, quelque soit «, done dans He+1, donc, le dérivé de H, soit P,, est dans P, dérivé de Hay. Si done Es, est nul, E est non dense sur tout ensemble parfait. Si Ea n'est pas nul, soit G l'ensemble des points de / qui ne font pas partie de #. l'ensemble G, est contenu dans ZE, quelque soit «. Done, Gan est dans Bay, done dans #, mais Gap, est aussi dans G. Comme G est distinct de #, G244 est nul. Done. G est chuirseme. Tout ensemble est done la réunion d'un ensemble dense en lui-méme 883 et d'un ensemble clairsemé, proposition dont on trouvera une autre démonstration dans le mémoire rappelé plus haut. Il nous sera commode, avant d’aller pour loin, de considérer la famille d’ensembles fermés A, ainsi définie. Si « est de première espèce, KA, est identique a A’,;. Si a est de seconde espèce, K, est l’ensemble commun a tous les ensembles K,, d’indice a’ infé- rieur a «. Nous désignons la totalité de l'espace par K,, et # facultativement par £,. Je dis que Ex est l'ensemble commun à E et à K,,. Pour a—=1, K, est le dérivé de EZ, ensemble identique a Z, et EB, est bien l'ensemble commun a £ et a K,. Supposons la propo- sition vraie pour @’ 9 pour toute valeur de p, Végalité étant réalisée pour au moins une valeur de p. Done, au moins un point M; de £; est dans la suite MM»). D'ailleurs Es contient Es, done MP, quelque soit p. Done, MN est un point limite de Es Mais ceci est impossible, puisque /M;) contiendrait N et que, par hypothèse /(Ms) ne contient aucun point de £’s. La con- dition est done suffisante. Soit A un ensemble fermé. Supposons d’abord que H n’ait pas EE nd denn dc de 885 de point commun avec K, — £’. Alors, il n’y a évidemment qu’un nombre fini d’ensembles /(J/) contenant à leur intérieur au moins un point de H. On voit en effet comme ci-dessus, que si ces points étaient en infinité, chacun de leurs points limites serait sur H, puisqu’il n’y a qu'un nombre limité d’ensembles /(J/) dont le rayon surpasse un nombre positif donné. Si done H est distinct de Z/ dérivé de £, nous aboutissons a une contradiction. Plus généralement, st ensemble fermé H est situ sur Kz et sil na pas de point commun avec K,44, il n'ewiste qu'un nombre limité d'ensembles I(M) contenant à leur intérieur au moins un point de H. En effet, si 8 lel Soit Cn 1’ le rayon de cette sphere. Entourons M, d'une sphere /'; de rayon r;. L'ensemble M, n’étant pas clairsemé, il y a des points de Vespace intérieurs à une infinité de sphères /',, Pour chacun de ces points .V, la série «, f,(N—WM,) est divergente comme ayant une infinité de termes supérieurs 4 1 en valeur absolue. La condition est suffisante. En effet, si EW est clairsemé, nous pouvons autour de M, déerire une sphère /, de centre M, et de rayon 7, telle que tout point de l'espace ne soit intérieur qu’à un 886 nombre fini de spheres /,. Soit, hors de la sphere de centre 0 et de rayon r,, u, le maximum de | f, (M) |. uw, existe, puisque par hypothèse | 7,(J/)| est borné a l’extérieur de toute sphere ayant son centre a lorigine. Soit @, un nombre quelconque inférieur en module a = La série @nfn(M—M,) converge en tout point Nn Un M, n’ayant qu’un nombre limité de termes supérieurs en M, comme n’ayant qu’ ombre limité de t ss A , . 1 module aux termes de même rangs de la série —. n On montre aisément que la série a, f, (M—WM,) converge unifor- mément sur tout ensemble fermé H sans points communs avec E’ ou plus généralement sur tout ensemble fermé contenu dans K, et ayant aucun point commun avec Kop. En effet, il n'y a qu'un nombre limité d’ensembles /, contenant des points d’an tel ensemble H. Done, a partir d'un certain rang N, le n° terme de la série et 1 4 est inférieur à — en tous les points de H, quelque soit n > N. n Supposons que f,(M) soit la somme d'une série Un. 1 (M1) se Un.2 (M) +... + Un.p (I) lfede ie uniformément convergente et à termes bornés (chacum séparément) a Textérieur de toute sphere ayant son centre a origine. Alors, a l'exté- rieur d'une telle sphere ayant le rayon r, défini plus haut, les sommes Un .p (M) + Un pti (M) H.H Un.g (M) sont, indépendamment de p, de q et de M, bornées en module par un même nombre 4, .(en particulier, avee g=p, | Un.» ML) | < 2»). 1 ; Soit a, un nombre de module inférieur a ae Je dis qu’en ajoutant Nn An par colonnes les séries a, fy, (M—M,), nous obtenons une série 1, (M) +, (M) + . . +, (M) + convergente en tout point M. En effet, on a: Wp (M) = & Up (M—M,) =F a, U2» (M—M,) AF oo + An Un.p (M— M,) + . La série w, (JZ) est convergente puisque, M n’étant intérieur qu’a un nombre limité de spheres /,, la série w, (J) n’a qu'un nombre limité de termes supérieurs en valeur absolue a l'inverse du carré de leur rang. Soit e un nombre positif. Nous voulons prouver que, M étant choisi, il est possible de déterminer MN, de facon que | w,41 (JZ) + HH wg MD) | Se quelque soit p > N,, et quelque soit g. = 887 Cette relation s’écrit: mpg m=pdg m=p+q a ZE uim(M—M,)+ 4a, 3 ven(M—M,)+..4-an 2 Unn(M—Mn)-+ ... Ie. m=p-+-1 m==p-+1 m=p+1 Nous allons même montrer que l'on peut résoudre par p > N, inégalité m=, +9 m=p+9 | ay, = tin (M—M,) | Foes =F | Gy Um (M—M,) | + nen (1) m==p--1 m=p-+1 Nous divisons les termes de la série du premier membre de (1) en trois catégories. 1° M étant intérieur & un nombre limité (ou nul) de sphères IM), soient M,,, M,,,..., Mn, les centres de ces spheres. Puisque les séries poe E oo oo Zun.p(M Ms), Bun.p (MM) Bun, (MZ —M,) pl p=1 pl sont convergentes au point J/, nous pouvons déterminer AN, de facon que, si p > N,, Un p+ (MM) 45 Uni pte (M—M.,,) + Soor ns pa (M— — Mn. 2) Gr DE Ea ha: 9 : npt pour 7=1,2,...,, quelque soit g. Les termes leen; na M-M,)| n=p-++-1 an ont alors une somme inférieure a —. (St) 76. 3 , 2 1 2° Soit V’ un entier supérieur à — La série 2 — a une somme € N'+1 2 ee a „€ ie inférieure Et donc An: Tous les termes de la série (1) de rangs supérieurs a MN’ et différents des ni, ont done une somme inféri- IG eure a —. 3 Mp 3° La série © u‚m(M) étant uniformément convergente pour n mel fixe et M variable avec dist. OM >> rr, nous pouvons déterminer un nombre Ns, tel que, si m=p-+-q p>Ns,,onal 2B un.n (M) | <= | An |. m=p+1 Donnons a n les valeurs 1, 2,. N’ Lae des n;, les No, ont une valeur maximum N,. Gare n Ani, M est extérieur a la sphere J, de centre M, et de rayon 7, Si mpg p> N, ‚onal ay Fin m (M— —M,) | Tr m=prh 888 Done, la somme des termes correspondants a n 4 nj, n< N’ est eye ae : eo 1 e n° & inférieure, si p > N,, a= B—=>-._< 9,n? 9 6 3 grand des deux nombres N, et N,, la condition p >> N, entraîne: | wpa (MZ) + .... + wore (M)| Ze, quelque soit g. La série w,(JZ) est done partout convergente. . Done, si -N, est le plus Application. p(n) étant une fonction positive de Tentier n, jamais croissante, la série p (1) sn AO + yp (2)sm2A0+...4 y(n)snnd+..., est convergente quelque soit 6. Soit /(@) sa somme. 6 f(0) tend vers 0 avec @. Si la série n p(n) est divergente, f (9) n'est pas som- 1 mable et |/(@)| croit indéfiniment avec —. Soient 0, une suite de 6| valeurs de 4 situées sur le segment (— z, Jz) et y formant un ensemble clairsemé queleonque. Il existe alors une suite de nombres positifs wy tels que, st \en| Z wr, la série p (1) Lam sin (O—O,,) + ..-. + P(n) Dan sin n(O—G4,) +... ml m=1 est convergente quelque soit 6. Soit (A) sa somme. J'ai défini sous le nom de totalisation un procédé d’intégration de certaines fonctions non sommables. La première condition remplie par les fonctions totalisables, — savoir que l'ensemble H des points d'un ensemble parfait P au voisinage desquels la fonction est non sommable sur P, H est non dense sur P, — cette condition est remplie par toutes les fonctions limites de fonctions continues, puisque, celles-ci étant ponctuellement discontinues, l'ensemble A des points de P au voisinage desquels lune d’elles est non bornée sur P, K est non dense sur P. K contient évidemment H. A toute fonction limite de fonctions continues, on peut donc faire cor- respondre une suite d’ensembles parfaits P,. P,,.., Pz, .., correspon- dants aux divers nombres ordinaux des classes let II. Par définition, si a est de première espèce, P, est le noyau parfait de ensemble fermé constitué par les points de P‚ 4 au voisinage desquels f est non sommable sur P,-1. Si a est de seconde espèce, P, est le plus grand ensemble parfait commun a tous les P, quand «