HIG Abn ht Kea aie is YY tls sith nt Hi iyi AN hi oe at} L HW, | (id FES Se = : = a ===) — _ en es Essen f H i % i 3 ' Hi i Î Hij 9 Phy A eit > aE / $ | HD Sr ee een SSS ae en —— i int i Sos LR b bas nt, dd Dj £9 ie Baia aR / ch dat ; ' Pith if tik ili t tilt € ne ~ \ ; = | sl f J g - AAs : f A rx \ We f? vot , ot é ° thi THE AME Bound A.M.N, fi BN i] : KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN -- TE AMSTERDAM -:- PROCEEDINGS OF THE Sie RON OF SCIENCES VOLUME Ix JOHANNES MULLER :—: AMSTERDAM EVA ROO: (Translated from: Verslagen van de Gewone Vergaderingen der Wis- en Natuurkundige Afdeeling van 26 Mei 1906 tot 26 April 1907. DI. XV.) KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN =- TE AMSTERDAM -:- PROCEEDINGS OF THE 5.c¢G@92 SEC TION OF SCIENCES VOLUME IX (= Pari) JOHANNES MULLER :—: AMSTERDAM : DECEMBER 1906 : , A0 04 OYE IP AD erf (Translated from: Verslagen van de Gewone Vergaderingen der Wis- en Natuurkundige Afdeeling van 26 Mei 1906 tot 24 November 1906. Dl. XV.) CONTENTS Proceedings of the Meeting of » » > » > » » » > > » > » > > ? > > <<>> May 26 June 30 September 29 October 27 November 24 » » Page KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN TE AMSTERDAM, PROCEEDINGS OF THE MEETING of Saturday May 26, 1906. DOGS (Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige Afdeeling van Zaterdag 26 Mei 1906, Dl. XV). CONTE EN ES: A. Smits: “On the introduction of the conception of the solubility of metal ions with electromotive equilibrium”. (Communicated by Prof. H. W. Bakuuis RoozeBoom), p. 2. A. Smits: “On the course of the P,7-curves for constant concentration for the equilibrium solid-fluid”. (Communicated by Prof. J. D. van DER Waars), p. 9. J. Morr van CHARANTE: “The formation of salicylic acid from sodium phenolate’. (Commu- nicated by Prof. A. P. N. FRANCHIMONT), p. 20. F. M. Jarcer: “On the erystal-forms of the 2,4 Dinitroaniline-derivatives, substituted in the NH,-group”. (Communicated by Prof. P. van RomBurGH), p. 23. F. M. JAEGER: “On a new case of formeanalogy and miscibility of position-isomeric benzene- derivatives, and on the crystalforms of the six Nitrodibromobenzenes”. (Communicated by Prof. A. F. HOrrLEMAN), p. 26. H. J. HAMBURGER and Svanre ARRHENIUS: “On the nature of precipitin-reaction”, p. 33. J. Stemx: “Observations of the total solar eclipse of August 30, 1905 at Tortosa (Spain)”. (Communicated by Prof. H. G. van DE SANDE BAKHUYZEN), p. 45. J. J. van Laar: “On the osmotic pressure of solutions of non-electrolytes, in connection with the deviations from the laws of ideal gases”. (Communicated by Prof. H. W. Baxuvis Rooze- BOOM), p. 53. Proceedings Royal Acad. Amsterdam. Vol. IX. (2) Chemistry. — “On the introduction of the conception of the solu- bility of metal ions with electromotive equilibrium.” By Dr. A. Smits. (Communicated by Prof. H. W. BaKmuis RoozEBoom). (Communicated in the meeting of April 27, 1906.) . If a bar of NaCl is placed in pure water or in a dilute solution, the NaCl-molecules will pass into the surrounding liquid, till an equilibrium has been established ; then the molecular thermodynamic potential of the NaCl in the bar has become equal to that of the Na Cl in the solution. As known, this equilibrium of saturation, represented by the equation: UNaCl = U'NaCl is characterized by the fact that per second an equal number of molecules pass from the bar into the solution, as from the solution into the bar. We shall call this equilibrium a purely chemical equilibrium. It is true that in solution the Na Cl-molecules split up partially into particles charged either with positive or negative electricity, which are in equilibrium with the unsplit molecules, but for the hetero- geneous equilibrium solid-liquid under consideration this is not of direct importance. If, however, we immerge a metal e. g. Zn into a solution of a salt of this metal, e.g. ZnSO,, we observe a phenomenon strongly deviating from the one just discussed, which according to our present ideas may be accounted for by the fact that a metal does not send out into the solution electrically neutral molecules as a salt, but exclusively ons with a positive charge. If the particles emitted by the bar of zine were electrically neutral, then the zine would continue to be dissolved till the molecular thermodynamic potential of. the zine in the bar of zine had become equal to that of the zine in the solution, in which case the equation: nn Bz would hold. This, however, not being the case, „and the emitted /n-particles being electro-positive, an equilibrium is reached long before the thermodynamic potential of the zinc-particles with the positive electric charge in the solution has become equal to that of the zine in the bar of zine with the negative electric charge. That in spite of this an equilibrium is possible, is due to the fact that an electrical phenomenon acts in conjunction with the chemical phenomenon. (3) The zine emitting positive Zn-ions, the surrounding solution becomes electro-positive, and the zine itself electro-negative. As known, this gives rise to the formation of a so-called electric double-layer in the bounding-layer between the metal and the electrolyte, consisting of positive Zn-ions on the side of the electrolyte and an equivalent amount of negative electricity or electrons in the metal. By the formation of this electric double-layer an electric potential difference between metal and electrolyte is brought about, which at first increases, but very soon becomes constant. This takes place when the potential difference has become great enough to prevent the further solution of the Zn-ions. In order to compute the potential difference between the metal and the solution, we shall apply the principle of the virtual dis- placement, as has been done before by Mr. van Laar.') If we have to do with a purely chemical equilibrium then with virtual displacement of this equilibrium the sum of the changes ot molecular potential will be = 0, which is expressed by the equa- tion of equilibrium: = (B,.dn,) = 0. If the equilibrium is a purely electrical equilibrium then with a virtual displacement of this equilibrium the sum of the changes of electric energy will be = 0. If however we have an equilibrium that is neither purely che- mical, nor purely electrical, but a combination of the two, as is the case with electromotive equilibrium, then with virtual displace- ment of this equilibrium, the sum of the changes of the molecular- potential + the sum of the changes of the electric energy will have to be = 0. a If we represent the mol. potential of the Zn-ions by we, in case of electromotive equilibrium, we know that Bik is much smaller than w., or the mol. potential of the zine in the bar of zinc. If we now suppose that a Zn-ion emitted by the zine virtually carries a quantity of electricity de from the metal towards the solu- tion, then this quantity of electricity being carried by a ponderable ; de quantity — when » = valency of the metal and ¢ = the charge pe of a univalent ion, the increase of the thermodynamic potential during this process will be equal to 1) Chem. Weekbl. N°. 41, 1905. 1* which increase is negative, because te, > Men In the virtual displacement of the quantity of electricity de from the metal towards the solution the change of the thermodyna- mic potential is not the only one that has taken place during this process. If we call the electric potential of the solution V, and that of the zinc V,,, we know that in the above case V, >V, and V. -—V»=A indicates the potential difference of the electrolyte and the metal. With the virtual displacement of the quantity of electricity de from the metal to the electrolyte this quantity has undergone an electrical potential increase A, and so the electric energy has increased with Ade. From the principle of virtual displacement follows that with electro- motive equilibrium + on ee i dae Ot EN VDE or ok ea te A (2) VE Now we know that the mol. therinodyn. potential of a substance may be split up as follows : wg Rw Tin where in diluted states of matter w may be called a function of the temperature alone. In non-diluted states however, w depends also somewhat on the concentration. If we now apply this splitting up also to equation (2), we get: de (Wen — Man) + RT ln C Ye SS = (3) where C represents the concentration of the Zn-ions in the electrolyte. If we now put: ah Uzn — pees hee we may say of this A that for diluted states of matter it will only in Ke nn EE _ Or — depend on the temperature, and will therefore be a constant at constant temperature. From equation (3), (4) follows EG ed FG _ A= ln See iy we Ma: Bsa, ee CO VE Y Mr. van Laar already pointed out that this equation, already derived by him in the same way is identical with that derived by dap ths oe P : - In —, in which therefore — stands instead of ve P Pp Nernst A —= K ay, J and p the “osmotic pressure” of the metal-ions in the solution. Rejecting the osmotic phenomenon as basis for the derivation of the different physico-chemical laws, we must, as an inevitable conse- quence of this, also abandon the osmotic idea “elektrolytische Lösungs- tension” introduced by Nernsr. The principal purpose of this paper is to prove that there is not any reason to look upon this as a disadvantage, for, when we seek the physical meaning of the quantity A’ in equation (5), it can be so simply and sharply defined, that when we take the theory of the thermodynamic poiential as foundation, we do not lose anything, but gain in every respect. In order to arrive at the physical meaning of the quantity A, we put for a moment . P represents the “elektrolytische Lésungstension” of the metal, Ce from which follows A=—0. From this follows that there is a theoretical possibility to give such a concentration to the metal-ions in a solution that when we immerge the corresponding metal in it, neither the metalnor the solution gets electrically charged. How we must imagine this condition is shown by equation (2), Let us put there A = 0, then follows from this for an arbitrary metal ao Un = Um or in words the molecular potential of the metal in the bar is equal to that of the metal-ions in the solution. So it appears that we have here to do with an equilibrium which is perfectly comparable with that between the NaCl in the bar NaCl, and the salt in the solution. (6) The only difference is this that the molecules of a salt in solution are neutrally electric, whereas the metal particles in solution are charged with positive electricity, hence the physical meaning of the equation gm = Um is simply this that in absence of a potential difference, per second an equal number of metal particles are dissolved as there are deposited. If we express this in the most current terms, we may say, that when C = K the metal-ions have reached their concentration of saturation, and that A, therefore represents the solubility of the metal-ions. To prevent confusion, it will be necessary to point out that the fact that the dissolved metal-particles in equilibrium with the solid metal have an electric charge, is attended by peculiarities which are met with in no other department. Thus it will appear presently that in every solution of copper- sulphate which is not extremely diluted, the concentration of the copper-ions is supersaturated with respect to copper. Yet such a copper-sulphate-solution is in a perfectly stable condition, because the copper-ions constitute a part of the following homogeneous equilibrium, CuSO, = Cu + SQ, which is perfectly stable as long as the solution is unsaturate or is just saturate with CuwSO,-molecules. If we now, however, insert a copper bar into the solution, the condition changes, because the Cu-ions which were at first only in equilibrium with the CwSO,-mols and with the SO,"-ions, must now also. get into equilibrium with the copper bar, and, the concentration of the Cu-ions with respect to copper being strongly supersaturate, the Cu-ions will immediately deposit on the copper, till the further depositing is prevented in consequence of the appearance of a double layer. We shall further see that in the most concentrated solution of a zine-salt the concentration of the zinc-ions always remains below the concentration of saturation, which appears immediately when we immerge a zine-bar into such a solution; the zine emits zine parti- cles with a positive charge into the solution, till the appearance of the electric double layer puts a stop to the phenomenon of solution. In order to find the values of K for different metals we make use of the observed potential difference with a definite value of C, Ce) We know the potential difference at 18° and with normal con- centration of the ions, i.e. when solutions of 1 gr. aeq. per liter of water are used. These potential differences are called electrode potentials, and will be denoted here by Ao. If we express the concentration in the most rational measure, viz. in the number of gr. molecules dissolved substance divided by the total number of gr. molecules, we may write for the concentration ober, eq: per: liter 1 in which » represents the valency of the metal. In this it has been further assumed, that the dissociation is total, and the association of the water molecules has not been taken into account. If we now write the equation for the electrode potential of an arbitrary metal, we get: RE IG A,=—ln VE il 55,9» +1 or RT A, = — ln K (55,5 v + 1) VE If we use ordinary logarithms for the calculation, we get: dard ly iy barns Oh Sere (56 epee Tree OS ae a If we now express F in electrical measure, then 0,000198 A, = ———— T log K (55,5 » + 1) Y end for == bee or 7291" 0,0578 6 avy Ae — log K (55,5 v + 1) p If we now calculate the quantity log K by means of this equation from the observed values of A,, we get the following. (See table p. 8). In the succession in which the metals are written down here, the value of A, decreases and with it the value of log A. For the metals down to Fe (Fe included) log K is greater than zero, so A greater than 1. Now we know that C for a solution is always smaller than 1; hence A will always be larger than C for the metals mentioned, and as A denotes the concentration of saturation of the metal-ions, (3) Values of log A at 18°. oa) Ao gah ee Do log E K: (+ 292) D| (448,77 ) Co — 0,045 — 1,805 2 Na‘ (» 2,54) ( 42,19 ) Ni: » 0,049 — 1,872 2 Ba: (» 2,54) ( - 42:92 2) Su < » 0,085 | <— 2,49 X 2 Sr: (» 2,49) ( 42,06 * 2) Pl: » 0,13 — 327 X 2 Ca (» 2,28) ( 38,42 * 2) JEL » 0,28 — 6,6 Mg: » 2,26 38,07 &K 2 Cu » 0,61 — 11,58 & 2 Al » 1,00 16,56 > 3 Biss: <<». (0:67 <— 1233 * 3 Mur » 0,80 4 Re) 8 Hos" » 1,03 —- 18,84 *& 2 Zn » 0,49 7,45 X 2 Ag » 1,05 — 19,92 Cd: » 014 4539 <2 Pd: » 1,07 — 19,03 X 2 Fe » 0,063 0,065 2 Tain “414 — 20,62 * 4 Th » 0,045 — 0,245x 2 Apps | » 1,36 — 26,27 X 3 the metal-ions will not yet have reached their concentration of saturation even in the most concentrated solutions of the corresponding metal-salts. Hence, when the corresponding metal is immerged, metal ions will be dissolved, in consequence of which the solution will be charged with positive and the metal with negative electricity. Theoretically the ease, in which A would always be smaller than C, can of course not occur. If log K is smaller than zero, so A smaller than 1, then the theoretical possibility is given to make the potential difference between the metal and the corresponding salt solution reverse its sign, which reversal of sign of course takes place through zero. Whether it will be possible to realize this, depends on the solubility of the salt. If we now take the metal copper as an example, we see that for this metal K has the very small value of 10-23. On account of this very small value of K, C is greater than A in nearly all copper- salt-solutions, or in other words the concentration of the Cu-ions is greater than the concentration of saturation. Hence copper-ions are deposited on a copper bar, when it is immerged, in consequence which the bar gets charged with positive, and the solution with negative electricity. But however small A may be, it will nearly always be possible to ') The values of A. between parentheses have been calculated from the quan- lity of heat, (9) make C smaller than XA. In a copper-salt-solution e.g. this can very easily be done, as is known, by addition of ACN, which in consequence of the formation of the complex-ions | Cu,(C.N),|", causes copper-ions to be extracted from the solution. The solution, which at first hada negative charge compared with the metal copper, loses this charge completely by the addition of ACN, and receives then a positive charge. In the above I think I have demonstrated the expediency of replacing the vague idea “elektrolytische Lösungs-tension’”’ by the sharply defined idea solubility of metal ions. Amsterdam, April 1906. Anorg. Chem. Lab. of the University. Physics. — “On the course of the P,T-curves for constant concentra- tion for the equilibrium solid-fluid.”” By Dr. A. Suits. (Commu- nicated by Prof. J. D. van per Waat1s.) (Communicated in the meeting of April 27, 1906). In connection with my recent investigations it seemed desirable to me to examine the hidden connection between the sublimation and melting-point curves for constant concentration, more particularly when the solid substance is a dissociable compound of two com- ponents. This investigation offered some difficulties, which 1, however, succeeded in solving by means of data furnished by a recent course of lectures giving by Prof. van per Waats. Though his results will be published afterwards, Prof. van DER WaALs allowed me, with a view to the investigations which are in progress, to use that part that was required for my purpose. In his papers published in 1903 in connection with the investi- gation on the system ether-anthraquinone’) vaN DER Waats also discussed the ZP, 7-lines for constant « for the equilibrium between solid-fluid *), and more particularly those for concentrations in the immediate neighbourhood of the points p and g, where saturated solutions reach their critical condition. Then it appeared that the particularity of the case involved also particularities for the P, 7-line, so that the course of the P, 7-line as it would be in the usual case, was not discussed. ') These Proc. VI p. 171 and p, 484 Zeitschr. f. phys. Chem. 51, 193 and 52, 987 (1905). *) These Proc. VI p. 230 and p. 357, (10) If we start from the differential equation in p‚r and 7’ derived by vAN DER Waats (Cont. IT, 112). 075 ji ry Vsp dp = (as — «f) (sez ptr 7 al) ae Fe we get from this for constant a that 1 1 a en SS ee: ge: oen I W, Te Vo ee oom, Sa en dT) yp Vi 2 rp If we now multiply numerator and denominator by a a8 will vf or prove necessary for simplifying the discussion, we get: Oba See ie. W ‘Sf ‚(dp Òvf” fi = == TTE ° . e . e e . (4) dT). Op . . Vsf Ov? In order to derive the course of the P, 7-lines from this equation, the loci must be indicated of the points for which the numerator, resp. the denominator = zero, and at the same time the sign of these quantities within and outside these loci must be ascertained. In the v, 2-fig. 1 the lines ab and cd denote the two connodal lines at a definite temperature. The line PsQs whose «=a, the concentration of the solid compound AB cuts these connodal lines and separates the w‚z-figure into two parts, which call for a separate discussion. If Ps denotes the concentration and the volume of the solid com- pound at a definite temperature, then the isobar VQRDD'R'QN of the pressure of Ps will cut the connodal lines in two points Q and Q', which points indicate the fluid phases coexisting with the solid substance AB, and therefore will represent a pair of nodes. } Mal fo dp E The points for which 5 Un 0 or — = =0 are situated where u ) the isobar has a vertical tangent, so in the points D and D' as VAN DER Waars') showed already before. In DD the isobar passes through the minimum pressure of the mixture whose r=wp, and so it has there an element in common with the isotherm of this concentration. In D' however, the isobar passes through the maxi- 1) These Proc. IV p. 455. A. SMITS. “On th y { y A. SMITS, “On the course of the P,T-curves for constant concentration for the equilibrium solid-fluid.” Proceedings Royal Acad, Amsterdam. Vol, IX (11) mum pressure of the mixture whose «= zp’, and will therefore have an element in common with the isotherm of the concentration «py. 2 As for the sign of —— we may remark that it is positive outside Ov? the points D and D' and negative inside them. The ordinary case being supposed in the diagram, viz. V; < Vy, we may draw two tangents to the above mentioned isobar from the point P, with the points of contact A and fh’, These points of con- tact now, indicate the points where the quantity V.,—= 0, as VAN DER Waars *) showed. This quantity is represented by the equation : dur Vs =: V, = V, meee ee CY —— ° ° e . 5 p= Wed), (5) and denotes the decrease of volume per molecular quantity when an infinitely small quantity of the solid phase passes into the coex- isting fluid phase at constant pressure and temperature. For the case that the coexisting phase is a vapour phase, Vr is negative, but this quantity can also be positive, and when the pres- sure is made to pass through all values, there is certainly once reversal of sign, for the case V; > V, even twice. To elucidate this Prof. vaN per Waars called attention to the geometrical meaning of Vr. Let us call the coordinates of the fluid phases Q coexisting with P, Vy and Xf and let us draw a tangent to the isobar in Q. Then P, P will be equal to Vr if P is the point where this tangent cuts the line drawn parallel to the axis of v through Z, If the point lies above P,, Vay is negative, and if /’ lies under ,, then Vy is positive. For the case that the tangent to the isobar passes through P,, which is the case for the points Rand L’, V‚s= 0. In this way it is very easy to see that for the points outside those for which Vs == 0, the value of V‚f is negative, and for the points within them, Vs is positive, but this latter holds only till the points D and D' have been reached, where V.r= a. Between D and D’, Vf is again negative. The transition from positive to negative takes therefore place through oo. As each of the lines of equal pressure furnishes points where 1) These Proc, VI, p. 234, (ELLE) 0° 5 De =0 and V.~=0, when connecting the corresponding points Vv we obtain loci of these points, indicated by lines. As, however, we simplify the discussion, as VAN DER WAALS has 0? shown, when we consider the quantity Vr instead of the quantity Vr, because this product can never become infinitely great and is yet zero when V.,,=0, the locus of the points where dp dof’ We know then too that this quantity on the left of the line ot the compound is negative outside this locus, and positive within it. V‚y=0 is given in fig. 1. 0? Further the locus of os = 0 is indicated, and we see that these v two lines intersect at the point where they pass through the line of the compound. In his lectures vAN DER Waats has lately proved in the following 2 =. : Ò way that this must necessarily be so: If we write for 5 a ges Of 2 (6) Vee — xf) ib . vf” Ov Ou we see that when this quantity —0O, and when at the same time To if": Ors Vs — ( Ves 7 or AT Dn I, too, had already arrived at the conclusion that in the left half of our diagram the two loci mentioned had interchanged places, by assuming that there existed a three-phase equilibrium also on the right, and by drawing the corresponding isobar M,Q,D, Rk, Ry DCN. It appears then that here the points AR, and A,’ lie within the points D, and D,', which points to a reversed situation (compared with 0? , 0? A Et alt) of meloen “&. Wr 20 andi Vans Ovf? Ov” has also given this graphical proof. ) on the right of the line As for the sign of the quantity ee vj (13) of the compound also there it is negative outside, positive inside the first mentioned locus. Before proceeding to my real subject, I shall, for the sake of completeness, first call attention to the fact that the spinodal curve, for which the equation: dw dw & 5 925 du? op any u . P Ov? == | i er ree iS holds, lies entirely ontside the locus = 0. Van DER Waats’) Ov? proved this in the following way : 2 2 On the spinodel curve rac and EE must both be positive, and so L U dw \? 07 Op also . As = — — is positive outside the line f ‚hi ee] Ov? 55 P or which dw Se = 0, the spinodal line will always have to lie outside the curve » 07w Ov? That the spinodal curve which coming from the left, runs between 2 2 0? —-= 0, cuts the line for Ee Ov p Ove Vsy=0 on the left of the line of the compound in two points g, and q, which will be discussed afterwards, follows from this, that 02 2 on the line of the compound En V‚p== 0 coincides with En 0 Òrf Òvf? 0 the lines —. Vp = 0 and : 2 EN. and that the line DE zs always lies within the spinodal line, whereas Uf on the right of the line of the compound 2 Ov7? . Vs = 0 lies within 2 When we start from the maximum temperature of sublimation, we get now v,7-lines which have been indicated by 7,, 7,, 7, and T, in fig. 1 for the equilibria between solid-fluid according to the equation *) ya’ O° en re EE | BE a (8) Ov? ¢ dif 1) loc cit. 2) These Proc. VI, p. 489, (14) The v,v-curve denoted by 7, relating to the maaimum temperature of sublimation, consists of two branches, which pass continuously into each other. The points of intersection with the connodal line ab indicate the vapour phases and those with the connodal line cd the liquid phases. In this way we get two pairs of fluid phases which can coexist with the solid compound at the same temperature. At the place where the two branches of the v‚z-line cut the locus 07y dv a Our Sr Ow With increase of temperature these branches draw nearer to each other, and when they would touch, intersection takes place; this is here supposed to take place for the v,z-line denoted by 7’. This point of intersection is the point q,, it lies therefore both on the 2 If we now proceed to higher temperatures, detachment takes place, and the v,2-figure consists of two separate branches, one of which, viz. the vapour branch is closed. This case is represented by the v,v-line 7, for which it is also assumed, that this temperature is the mdnimum-melting point of the compound, which follows from the fact that the liquid branch of the v‚z-line 7,, simultaneously cuts the connodal line cd and the line of the compound. | With rise of temperature the closed v,z-line contracts, and the corresponding liquid branch descends. The points of intersection of the closed vapour branch and the liquid branch with the connodal curves draw nearer and nearer to each other, and at a certain temperature the two branches will show contact. The closed vapour branch touches the connodal curve ab and the liquid branch the connodal curve cd. This is represented by the v‚z-figure 7’,, which represents the condition at the maximum-threephase-temperature, at which the points of contact on the connodal curves and the point for the solid substance must lie in one line. At higher temperature no three phase equilibrium is possible any longer, and both the closed vapour branch and the liquid branch have got detached from the connodal curves. The liquid branch descends lower and lower, and the closed branch contracts more and more, and vanishes as a point in qg,, where the upper branch of the 2 spinodal curve and on the curve Vri. b] w Ov ¢ spinodal curve and the curve V.¢ = 0 intersect. 2 If we now also indicate the locus of the points where ee Wy =0 f (15) the peculiarities of the course of the P7-lines may easily be derived by means of the foregoing. For the determination of the last mentioned locus, we start from the equation: de Wj = E EEn (52), | VOTE rie se 911 (0) The factor of V,- being naturally positive and (£,/), being always negative, Ws, can only be equal to zero in a point « where Vs; is ay Ov Further it is now easy to understand that at the same time with Vs the quantity Wp will become infinitely great, there where dy Ov, positive, so between the loci where V;,=0O and = 0. In order to avoid this complication vAN Der WAALS has 0? multiplied the quantity Ws by = as equation (4) shows; the obtained product never becomes infinitely great now. 0? If we multiply equation (9) by as we get: Uf dp def 0?) dp —, Wy = —— —— . Vip ee oe (1.0 Ov? f E = (EE) dv? s/ sie Ov,? ( 1) ( ) Now we know that the locus for aes Wsr = 9 will have to lie Uf 0? 0? between that for p . Vir= 0 and for ud = 5 as drawn in Ove? Ov? i am np fig. 1, which compels us to make ean Ws = 0 and ae ir == 0 In Ji intersect on the line of the compound. That this must really be so, is easily seen, when we bear in mind, 2 = 0 coincides do, that on the line of the compound the locus where 2 wy : 3 ; .Vsr == 0, from which in connection with with that where Ov? equation (10) it follows immediately that at the same point also 0? aoe Wy = 0. In this way we arrive at the conclusion, that the three loci Òvy? 0? 0?y 0? se hen ‚as Vof = 0 and ae Wp = 0 will intersect on the line Ov? Ov;? Ovs? (16 ) 2 0 of the compound, and that therefore the loci Te: Vey = 0 and uf op the line of the compound. By means of equation (10) we understand now easily that the . Wsy = 90 will interchange places on the left and the right of 0?» sign of the quantity us Ws, must be negative outside the locus of 0? a ss . Wer = 0, and positive within it. vf As connecting link for the transition to the P,7-lines we might discuss the V,7-lines ; for this purpose we should then have to make use of the following equation (Cont. II, 106) 0? = 02yp 0 0?w — Saar IT =e w Naf nen ee oe Ow oS! ike c e dup. dap e= va te Jap | = Oe 7 By taking w constant we derive from this ‚(dof — (Esf)v Jl aT == NED á ej (vs — wr 2 5 or dv? ; I shall, however, not enter into a discussion of the V 7-lines because it is to be seen even without this connecting link, what the course of the P,7-lines must be. 2 If now for simplication we call —— 07 ke VX and 5 Wy ae do Ov? and if we indicate what the signs are of these quantities in the different regions on the left and the right of the line of the com- pound, and where these quantities become = 0, we get the following : left right X, —X, — Ze NE Ee = ea *; oy 0 Snes ae oa) X, — Ayr Kl) Xe Ne Ppa EE es = X, + A, + # pi atlas 0 Fiat a ae a, De a XX, 7 (17) dp x T| — SS CS Pate aa erm NC dT zp ae we draw the 7, 7-line for a concentration on the left of the curve of the compound, we obtain a curve as given by GF'FD in fig. 2. As we have assumed in our diagram, that the vapour-tension of A is the greatest and of B the smallest, whereas that of A B is intermediate, we cut now that branch of the three phase line of the compound, which has a maximum. If now led by equation FY / L / / / FMN A / tons LAY, Ale, E / yea! 5 / aay AG isen LO 6, 7 / GE £ LEE ha Bt Lm Fig. 2. This intersection takes place in the points #” and #, about which it may be observed, that /” lies at a higher temperature than /. This situation can, however, also be reversed, and as appears from the diagram, the transition takes place at a concentration somewhat to the left of that of the compound. We see further, that the inter- mediate piece, which continuously joins the line of sublimation GF" to the melting-point curve /#' D, has a maximum and a minimum (points where A, =O), about which the isotherm teaches us, that, when we are not in the immediate neighbourhood of the critical state, they are very far apart and that the minimum lies at a negative pressure. Proceedings Royal Acad. Amsterdam. Vol. IX. (18 ) It is also noteworthy about this figure, that when following the P, T-line, starting with. the point G resp. with the point D, we first meet with a point, where the tangent is vertical, and (place where XY, == 0) after that with a point where it is horizontal. If we now consider a concentration on the right of the line of . the compound, the ZP, 7-line corresponding with this will cut the other continually ascending branch of the three phase line of the com- pound, and by means of equation (da) and the scheme for the reversal of sign of X, and X, preceding it, we obtain a curve as indicated by G, FF, D,. The situation of the loci X,=0 and X,=0 being different on the fight from that on the left, this P, T-line differs from that just discussed. When now, starting from the point G, resp. D,, we follow the P,7-line, we meet first with a point, where it is vertical, so we have just the reverse of the preceding case. About the situation of the points /,' and F, we may point out, that /' always lies at lower temperature than #,. The loci X,=0O and X,==0 intersecting on the line of the compound, the P,7-line for the concentration of the compound will have to give to a certain extent the transition-case between the two lines discussed. What happens when we approach the curve of the compound, we see immediately from fig. 1. The distance between the loci X, = 0 and X, =O becoming smaller and smaller, the points of contact of the vertical and horizontal tangents will draw nearer and nearer, which prepares us for what happens when we have arrived at the line of the compound. We see from the scheme for the signs of X, and X, that when the loci X,—0 and X,=0 have coincided, the signs of X, and X, reverse simultaneously, on account of which dp : : ee ie r( me retains the same sign, viz. remains positive. Combining this with what we know about the course of the P,7-lines somewhat to the right and the left of the curve of the compound we are led to the conclusion, that the ,7-line for the concentration of the compound will have two cusps, each formed by two branches with a common tangent. I have not been able to decide whether these points will be cusps of the first or the second kind. The former has been assumed in the diagram. It is further noteworthy for this P,7-line that, as VAN DER WAALS *) already demonstrated before, both the line of sublimation and the melting-point line must touch the three-phase line, so that the P, 7-line 1) Verslag 21 April 1897, 482. (19) for the concentration of the compound assumes a shape, as given bythe, hmesG Pa Pe Din fig..2 If it were possible to make the degree of association of the com- pound smaller and smaller, the points 4’ and #/, would move to lower pressure and higher temperature. Moreover these two points and the neighbouring point of intersection of the melting-point and sublimation branches would draw nearer and nearer to each other, till with perfect absence of dissociation these three points would have coincided. Another peculiarity will present itself for the case that we have a three-phase-line as described by me before, viz. with two maxima and one minimum *), for then there is a point where m7 == a") on this line, and then it is immediately to be seen that in consequence of the coinciding of the points /” and /’, we get for this concentration a P,T-line, as represented in fig. 3, which curve has the form of a loop. Amsterdam, April 1906. Anorg. Chem. Lab. of the University. 1) These Proc. VIII, p. 200. le oa 2) In this point the direction of the three-phase line is given by 7’— = : aT vq—vI ( 20 ) Chemistry. — “The formation of salicylic acid from sodium phenolate” By Dr. J. Mor van Caarants. (Communicated by Prof. A. P. N. FRANCHIMONT). (Communicated in the meeting of April 27, 1906). The communication from LoBry pr Bruyn and Tumstra read at the meeting of 28 May 1904 and their subsequent article in the Recueil 28 3885 induced me to make this research. Their theory, and particularly the proofs given in support do not satisfy me and as, in consequence of other work, I had formed an idea of the reaction I made some experiments in that direction. According to my idea, an additive product of sodium phenolate with sodium phenylearbonate, or what amounts to the same an additive product of two mols of sodium phenolate with one mol. of carbondioxide C,H,OC(ONa),OC,H, might be the substance which undergoes the intramolecular transformation to the salicylic acid OH derivative and then forms, dependent on the tem- perature, sodium salicylate and sodium phenolate or else phenol and basic sodium salicylate. This ÚONa view is supported by previous observations of various chemists and has been partially accepted OC, H, also by CraISEN '). As LoBry DE BRUYN and Tiymstra give no analytical figures in their paper it did not seem to me impossible that the phenolsodium- o-carboxylic acid obtained by them might be the substance formed by intramolecular transformation of my supposed additive product. vo GH ONa 6 I, therefore, took up their method of working, OH prepared sodium phenylearbonate in the usual manner, from sodium phenolate and carbon dioxide, CoH, ONa and heated this to 100° in a sealed tube for 100 Sons hours. On opening the tube considerable pressure was observed. This pressure was always fonnd when OC He the experiments were repeated. The gas liberated proved to consist entirely of carbondioxide and amounted to */,—!/, of that present in the sodium phenylearbonate. If we argue that the sodium phenyl- carbonate under these circumstances is partially resolved into carbon dioxide and sodium phenolate the latter compound ought to be present or else the splitting up might give carbon dioxide and my supposed 1) B. B. (1905) 38 p. 714. ( 21 ) intermediary product (C,H,O), C(ONa),. In the first case it is strange that during the cooling of the tube, which often was left for a few days, the carbon dioxide is not greatly reabsorbed. Those substances had now to be searched for in the product of the reaction. On treatment with ether a fair amount of phenol was extracted although moisture was as much as possible excluded. It was then brought into contact with cold, dry acetone, by which it was partially dissolved, but with evolution of gas and elevation of temperature. From the clear solution, petroleum ether precipitated a substance which, after having been redissolved and reprecipitated a few times in the same manner, formed small white needles containing acetone which efflo- resced on exposure to the air. On analysis, this compound proved to be sodium salicylate with one mol. of acetone. As an ebullioscopic determination in acetone, according to LANDSBERGER, did not give the expected molecular weight, sodium salicylate was dissolved in acetone and precipitated with petroleum ether and a quite identical product was obtained as proved both by analysis and determination of the molecular weight. Both products, after being dried at 100°, yielded no appreciable amount of salicylic ester when heated with methyl iodide. The amount of sodium salicylate obtained by heating sodium phenylearbonate in the manner indicated was, however, very trifling. I suspected that the evolution of gas noticed in the treatment with acetone, and which was identified as pure carbondioxide without any admixture, was caused by the presence of unchanged sodium phenylearbonate, so that, therefore, the reaction was not completed, and that the tube after being heated must still contain a mixture of unchanged sodium phenylearbonate, sodium phenolate, sodium sali- cylate and free phenol, besides the said additive product (C,H,O) 2 OH C(ONa), and the salicylic acid derivative possibly C,H, ONa formed from this. I now thought it of great NDA . 7 AL ‘ importance to first study the behaviour of acetone CONa with these substances as far as they are known. , OG He Sodium phenolate dissolves in boiling acetone, from which it erystallises on cooling in soft, almost white needles, several ¢.m. long, which contain one mol. of acetone. They lose this acetone, in vacuo, over sulphuric acid. At the ordinary temperature acetone dissolves only 0,1 °/,. Sodium phenylearbonate placed in carefully dried acetone gives off (22) carbon dioxide with a slight elevation of temperature. The quantity amounts to about */, of the carbon dioxide actually present,*at least if account is taken of the comparatively large solubility of that gas in acetone. The acetone, or if the mixture is extracted with ether, also the ether, contains a quantity of phenol corresponding with the total amount obtainable from the sodium phenylearbonate. The undis- solved mass consists of a mixture of neutral and acid sodium carbonate, nearly, or exactly in equivalent proportions. The decom- position of 3 C,H,OCOONa to 3 C,H,OH + CO, + NaHCO, + Na,CO, requires 2 mols. of water. As the experiments however, have been made in a specially constructed apparatus into which no moisture or moist air could enter, with extremely carefully dried acetone, we are bound to admit that this water has been generated by the acetone, and we may, therefore, expect a condensation product of the acetone which, however, could not be isolated, owing to the small quantities of materials used in the experiments. It seems strange that in this reaction the evolution of carbon dioxide is so extraordinarily violent. Sodium salicylate dissolves in acetone from which it erystallises, with or without addition of petroleum ether, in small needles, which may contain one mol. of acetone of crystallisation. In different deter- minations the acetone content was found to vary from one-half to a full molecule. At 16° it dissolves in about 21 parts of acetone. Disodium salicylate was prepared by adding an (95°/,) alcoholic solution of salicylic acid to a concentrated solution of sodium ethoxide in aleohol of the same strength. After a few moments it crystallises in delicate, white needles. By boiling with acetone in which it is entirely insoluble it may be freed from admixed mono- sodium salicylate. The behaviour of acetone with these substances now being known, the experiment of heating the sodium phenylearbonate for 100 hours was once more repeated, without giving, however any further results. A portion was treated with acetone in the same apparatus which had been used for the sodium phenylearbonate. A quantity of carbon dioxide was collected corresponding with an amount of unchanged sodium phenylearbonate representing 50—60°/, of the reaction- product. Another portion was extracted with ether and yielded about 20 "/, of phenol whilst, finally, a small amount of sodium salicylate was also found. The residue which had been extracted with ether and acetone contained sodium carbonate but no disodium-salicylate. It, however, contained phenol, probably from sodium phenolate. It seems strange there is such a large quantity of free phenol ( 23 ) in the heated sodium phenylearbonate, and as no disodium-salicylate has been found it cannot have been caused by the formation of that compound. I have not been able to find the looked for additive product ; perhaps it has been decomposed by acetone in the same manner as sodium phenylearbonate. The results obtained show in my opinion that the formation of salicylic acid from sodium phenylearbonate is not so simple as is generally imagined. A more detailed account of research will appear in the “Recueil”. Chemistry. — “On the crystal-forms of the 2,4-Dinitroaniline-deri- vatives, substituted in the NH,-group”. By Dr. F. M. Janaur. (Communicated by Prof. P. van RoMBurGH). (Communicated in the meeting of April 27, 1906). More than a year ago I made an investigation as to the form- relation of a series of position-isomerie Dinitroaniline-derivatives *). On that oceasion it was shown how these substances exhibit, from a erystallonomie point of view, a remarkable analogy which reveals clearly the morphotropous influence of the hydrocarbon-residues, substituted in the NH,-group. Among the compounds then investigated, there were already a few 1-2-4- Dinitroaniline-derivatives kindly presented to me by Messrs. VAN RompurcH and FRANCHIMONT. Through the agency of Prof. vAN RomBuren and Dr. A. Murper, I have now received a series of other derivatives of 2,4-Dinitroaniline which in the happiest manner complete my former publications. I wish to thank these gentlemen once more for their kindness. I will deseribe and illustrate all these derivatives in a more detailed article in the Zeits. f. Kryst. For the present I will merely give a survey of the results obtained, which have been collected in the annexed table. I have chosen such a form-symbolic, that the morphotropous rela- tion of the great majority of these substances is clearly shown. They all possess the same family-character which is shown in the values of the axial relations and the topic parameters. Only a few of these substances show no simple relationship with the other ones. 1) Jaeger, Ueber morphotropische Beziehungen bei den in der Amino-Gruppe substituierten Nitro-Anilinen; Zeits. f. Kryst. (1905). 40. 113—146, No. Name of the compound Ln 1 1-2-Nitro-Aniline. 2 4-4-Nitro-Aniline. 3 1-2-4-Dinitro-Aniline. 4 1-2-4-6-Trinitro-Aniline. 5 1-4-Nitro-Diethyl-A. 6 1-2-4-Dinitro-Methyl-A. 7 1-2-4-Dinitro-Ethyl-A. 8 {-2-4-Dinitro-Dimethyl-A. 9 1-2-4-6-Trinitro-Dimethyl-A. 10 1-2A-Dinitro-Methyl-Ethyl-A. de 1-2-4-Dinitro-Diethyl-A. 12 1-2-4-, + 1-3-4-Dinitro-Diethyl-A. (Double compound.) 13 4-2-4-6-Trinitro-Diethyl-A. 14 1-2-4-Dinitro-Ethyl-n-Propyl-A. 15 1-2-4-6-Trinitro-Ethyl-Isopropyl-A. 16 1-2-4-Divitro-Isopropyl-A. 17 1-2-4-Dinitro Dipropyl-A. 18 1-2-4-6-Trinitro-Dipropyl-A. 19 1-2-4-Dinitro-Isobutyl-A. 20 4-2-4-6-Trinitro-Isobutyl-A 21 1-2-4 -Dinitro-Diisobutyl-A. 22 1—-2-4-Dinitro-Allyl-A. 23 1-2 4-Dinitro-Methyl-Phenyl-A. 24 1-2-4-Dinitro Ethyl-Phenyl-A. 25 1-2-4- Dinitro-Benzyl-A. 26 1-2 4-Dinitro-Methyl-Benzyl-A. 27 1-2-4-Dinitro-Ethyl-Benzyl-A. 28 1-2-4-Dinitro-Phenyl-Benzyl-A. 29 4-2-4 6-Trinitro-Etbyl-Nitraniline. 30 4-2-4-6-Trinitro-Isopropyl-Nitraniline. 31 1-2-3-4-6-Tetranitro-Methyl-Nitraniline, *) On the isomorphism and the complete miscibility of this compound with p-Nitrosodiethylaniline Survey of th | Equiv. Vole. (in the solid Sym state.) 95.70 Rhoml 96.03 Mon. 113.30 Mon 129.39 Mon. 162.07 | Mom 125.24 Mon. 145.44 Tricl 492.41 Mon 950): 24 Mo 157.93 Tri 194.16 Mo 210.48 Mo 187.50 | Tri 204.44 M 219 87 M 250.00 M 183.09 M 201.53 M 189.71 M } ystallographically-investigated derivatives of 1—2—4—Dinitro-Aniline. y: Axial-Elements: Topic Parameters: mr} a:b: c = 1.3667 : n. a 20350 a se = 1.9826 : a Sey 1105607: mM. a se = 1.0342 : m. a e= 1.2286 : iC. a C=. 1 221 @ = 33°25}! 2 4.9154 : 4 eu) 424497 Me 1.2045 Mer 200102 = 83099! 1: 4.4585. 1: 1.4220; 1: 1 4088; 1: 1.5208; 1: 0.9894; — je) de) =I (= ~l 1: 1.0808. 9936 24: 1.3831. 4: 1.6639. au Wen eres OIS 14: 0 9462; 24) oO Tle dy. p= 88°10! y= 1504, a= 1546! g—= 92 += 68°57’ By Suber els ley 2 = 86939! gp = 86°28! 1907 9 @——199017! y= 11016)! zie == alls pyeal Sal etayheloy == 0.732524 30 3470. == 10747721 20.9124 ene — 1 02st 74: 0.9632 B AGE BAO" 7 ERD cr — 11154821: 126968 ; abi e— (04953-41210. 65865; abe: 6 — 10385" 1-5 0:8986 =de 086-242. 43276 ; nc 1125821 13087; te 137a 21°: 1,3645 ; Ker Se 15 1 0.963) sce —— at, O60. dis 1E DS 2 ese Proceedings (1905) p. 658, AOL BASSE“ v= a —118°43' 2 —104°33' y= 85°12}! Ameb = ¢ — 1.0191 24: 0.9246. nbr 13397201150 9055 ge 11940 24410! 102055! a:b:c — 0.7104 :1: 0.3591; P— 8534} P= 65°95)! = 116940! BSO! P= 1864! 78°23! p= 1°40! p= 84°! P = 64921} p= 86°23}! P= Beas! ES) p oi ) Oo t Bee) yb: o Lb a) ij Ay b Oo b 3 wb 6) T wy a) p 6) sy: wv 0) 7 b 6) w om U oo b 0) wb i) 7 pt 0 b o b i) U i) Lv a U} a gro WOR WER Lsa — O0 RD 6D Mn ‚8206 : 3 | ee bo “2 bo — Cyt oe „0871 = ==! S035. - % == 5 8455 4 ).0294 ; | Or . 3064 : 4 Se ee ier 5593 © 5 OLON MLS 1042 RRD 1870 : 8 — 7 .0686526 == 5) 94804 Se Ct a = ee) de) mn ~l no NG = er) | A0 eas „3676 : 5 = 1730 : 4 ‘9313 : 3. 6210 : 5. „8090 : 4. OAS DE „1556 2 6: dol OE ‚Ot : 4. 945 : DAZ: 4403 : ee DOE: „5184 : 0058 : 6940 : MO PARE „9890 : „1092; < „5960 : 4.6241 : 4871 50R EE Oost 1551: „7640 : „2913 : „2987 : 8119 : | 4351 : 7281 : : 5140 : 7946 : ! 6360 : 1197: 1782: § S024 : GO84 : Or ~ 6. „0880 „1583 ). 2493 „5106 „7631 5 1083 „2765 3466 „1849 ).0181 5.5890 „8650 „7401 „OS91 „3129 0310 „6856 ( 26 ) Crystallography. — “On a new case of form-analogy and misci- bility of position-isomeric benzene-derivatives, and on the crystal- forms of the six Nitrodibromobenzenes.” By Dr. F. M. Jaraer. (Communicated by Prof. A. F. HOLLEMAN.) (Communicated in the meeting of April 27, 1906). § 1. The following contains the investigation of the crystal-forms exhibited by the six position-isomerie Nitrodibromobenzenes, which may be expected from the usual structure-representations of benzene. It has been shown that, in this fully investigated series, there again exists a miscibility and a form-analogy between two of the six terms. The above compounds were kindly presented to me by Prof. HOLLEMAN, to whom I again express my thanks. This investigation is connected with that on the isomeric Dichloro- nitrobenzenes, which has also appeared in these proceedings (1905, p. 668). A. Nitro-2-3-Dibromobenzene. Structure: C,H,.(NO,) .Br .Br ; meltingpoint: 53° C. (1) (2) (3) The compound, which is very soluble in most organic solvents, Migs ol: erystallises best from ligroïn + ether in small, flat, pale sherry- coloured needles which generally possess very rudimentary terminating planes. Triclino-pinacoidatl. a: Oe — 1,4778 slik: 169513. A == 90730) OA B 11037 8110862: GE AOC te ( 27 ) The crystals, therefore, show a decided approach to the mono- clinic system ; on account, however, of their optical orientation, they can only be credited with a triclinic symmetry. The forms observed are: a = {100}, strongly predominant and very lustrous; 6 = {010}, smaller but yielding good reflexes ; c = {001}, narrower than a, but very lustrous; 0 = {111}, well developed and very lustrous; w = {111}, smaller but very distinct ; s = {111}, very narrow but readily measurable. The habit is elongated towards the b-axis with flattening towards {100}. Measured: Calculated : a:b = (100) : (010) =* 90°16"/,' a ae — (100) : (001) =* 69 23 = aio — (100): A11) =* 65 11 c:0o = (001) : A11) =* 75 477/, fe b:o = (010): 111) =* 36 6 au a:w = (100): (111) = 50 52 50°49! c:@ = (001): (111) = 56 52 56 43 b:w — O10): A11) —= 46 28 46 35 o:w = (111):(111) = 47138 47 297/, a:s =(100):411)= 4959 | 50 491/, b:s —(010): A11) — 45 48 db c:s =(00D): di == — 56 4 o:s = (111): (111)= 63 39 63 592/, Readily cleavable, parallel {100}. The extinction on {100} amounts to about 26'/, in regard to the b-axis; in convergent light a hyperbole is visible occupying an eccen- trie position. The sp. gr. of the crystals is 2,305 at 8°; the equivalent volume 121.47. B. Nitro-2-5-Dibromo-Benzene. Structure: C,H, . (NO). Bro). Bris); m.p.: 842,5. This compound has been previously studied erystallographically by G. Feis, (Zeits. f. Kryst. 82, 377). This paper, however, contains several errors, which render a renewed investigation desirable ; more- over, another choice of axial (coordinate) planes is required, which makes the erystals show more analogy with the other triclinic terms of this series. The crystals deposited from acctone + ligroin have the form of small plates flattened towards {001} (figs. 2 and 3). They are pale yellow and very lustrous. ( 28 ) Triclino-pinacoidal. Forms observed : a 3b C= 14909 1: 20204: A= 91° 81/,' B= 118°211/,' 90°27’ Ne strongly predominant and reflecting GE c= {001}, a= 90°57'/,' B = 113°21!/,' ideally +a — 100; sand = {101}, usually developed equally broad and also yielding sharp reflexes ; able ; m — {110}, 1 arge and lustrous ; very lustrous ; sometimes as broad as im. Broad flattened towards {001}. The approach to monoclinie sym- metry is also eb (1100); b == (Ol): Co DEE beam (O10): O27 (00). Cami 001); a n= (100) c:r = (001) r:m= (101) p:m= (113) r:b = (101) MP (101) : plain in this case. Measured : = 1) = (100) —*66 381/ (110) =*35 591/ (101) —*43 45 (110) — 75 46 : (110) = 53 33 (101) — 69 37 (110) = 65 20 (110) = 60 59 b = {010}, smaller, p = {113}, mostly narrow but readily measur- Calculated: 75°38?/,) 53 33/, 69 361/, 65 11 60 441/, 89 22 (29) Readily cleavable, parallel 77. The optical orientation is that of Fris, in which his forms {010}, {001} and {111} assume, respectively, in my project the symbols {O01}, {110} and {010}. It may be remarked that Fers has incor- rectly stated the structure and also the melting point. Moreover, his angles (111) : (100) and (111) : (010) appear to be >> 90°. Perhaps it is owing to this, that the agreement between the calculated and found values is with him so much more unfavourable than with me. I have never observed forms {552} and {15.15 . 4 The sp. gr. at 8° is 2,368; the equiv. volume: 118,66. Topical axes: x: : w = 5,2190 : 3,5005 : 7,0758. On comparing the said position-isomeric derivatives, one notices at once not the great similarity between the two compounds, which, although constituting a case of direct-isomorphism, still very closely resembles it. Nitro-2-3- Dibromobenzene. Nitro-2-5-Dibromobenzene. Triclino-pinacoidal. Triclino-pinacoidal. 20-6147 718 2 ROES G2: O56 14909 1 2.0214. A=90°30' B=110°37' C=90°16?/,' | A=91°31/,' B=113°21"/,' C=90°27' a=90°45?/,'B=110°36?/,'7y=89°59"/,' | a=90°574/,' B=113°21"/,’ y—90°2' 42 W: w= 5,2565 :3,5571:6,9409. | xy: wy: w = 5,2190 :3,5005 :7,0758. However: However: Forms: {100}, {010}, {001}, {111}, | Forms: — {100}, {010}, {001}, {101}, {1141} and {117} {110}, {113}. Cleavable parallel {100}. Cleavable parallel {110}. Habit tabular towards {100}. Habit tabular towards {001}. We, therefore, still notice such a difference in habit and cleava- bility that a direct isomorphism, in the ordinary meaning of the word, cannot be supposed to be present. There occurs here a case of isomorphotropism bordering on isomorphism. Notwithstanding that difference, both substances can form an interrupted series of mixed crystals, as has been proved by the determination of the binary melting point curve and also erystallo- graphically *). 3 The melting point of the 1-2-3-derivative (53°) is depressed by addition of the 1-2-5-derivative. The melting point line has also 1) The binary melting-curve possesses, — as proved by means of more a exact determination, — a eutectic point of 52° C. at 2%) of the higher melting com- ponent; therefore here the already published melting-diagram is eliminated. There is a hiatus in the series of mixed-crystals, from + 3% to circa 48° of the 1-2-3-deri- vative. I shall, however point out, that the possibility of such a hiatus thermody- namically can be proved, — even in the case of directly-isomorphous substances. (Added in the English translation). (30) not, as in the previously detected case of the two tribromotoluenes (Dissertation, Leyden 1908) a continuous form; the difference is caused by the lesser degree of form-analogy which these substances possess in proportion to that of the two said tribromotoluenes. The third example of miscibility, although partially —, and of form-analogy of position-isomeric benzene-derivatives*) is particularly interesting. Mixed erystals were obtained by me from solutions of both com- ponents in acetone + ether. They possess the form of fig. 1 and often exhibit the structure of a sand time-glass or they are formed of layers. With a larger quantity of the lower-melting derivative, long delicate needles were obtained which are not readily measurable. The melting points lie between + 75° and 843°; 1 will determine again more exactly the mixing limits. C. Nitro-2-4-Dibromobenzene. Structure: C,H; . (NO a. Broa). Bris); mp. 61°.6. Reerystallised from alcohol, the compound forms large erystals flattened towards « and elongated towards the c-axis. They are of a sulphur colour. Friclino-pinacoidal. Lbs =O de del GIS: Jm ko ule a= 97°36' B 11 352504e" BAAS 23 Gs SOES ni WOS. Forms observed: a = {100} predominant and very lustrous; b = {010} and c = {001}, equally broad, both strongly lustrous; p= {110}, narrow but readily measurable; 0 = {111}, large and yielding good reflexes. The compound has been measured previously, by Grorn and Boprwie (Berl. Berichte, 7, 1563). My results agree in the main with theirs; in the symbols adopted here, their a- and h-axes have changed places and the agreement with the other derivatives of the series is more Fig. 4. conspicuous. 1) The examples now known are 1-2-3-5-, and 1-2-4-6-Tribromobenzene ; 1-2-3-5-Tribromo-4-6-Dinitro- and 1-2-4-6-Tribromo-3-5-Dinitrotoluene; and 1-2-5-, and 1-2-3-Nitrodibromobenzene, partially miscible. a: b = (100): Gee (100) : be (O10) - D= (110) : C20 (OOH) a) — (111) Gap (O01): (31 ) Measured : Calculated : (010) =* 89°211/,' ai (001) —* 66 29'/, = (001) —* 82 461/,' = (100) —=* 46 36 an : (111) —* 48 42 = :(110) = 5143 (circa) 52° 1’ (110) = 100 29 (circa) 100 43' Cleavable towards {010}; Grorn and Boprwia did not find a distinct plane of cleavage. Spec. Gr. of the crystals = 2,356, at 8° C., the equiv. vol. = 119,27. Topic Axes: y:W:w = 5,2365 : 4,6304 : 5,4166. Although the analogy of this isomer with the two other triclino- isomers is plainly visible, the value of a: is here quite different. In accordance with this, the derivative melting at 84'/,° lowers the melting point of this substance. A mixture of 87°/, 1-2-4- and 13°/, 1-2-5-Nitrodibromobenzene melted at 56°. There seems, however, to be no question of an isomorphotropous mixing. D. Nitro-2-6-Dibromobenzene. Structure: C,H, (NO,)qay . Brey . Bro); m.p. 82°. Reerystallised from alcohol the compound generally forms elongated, brittle needles which are often flattened towards two parallel planes. Monoclino-prismatic. abi =O HHS 106257 B = 83°24’, Forms observed: 6 = {010}, strongly pre- dominant; g = {011} and o = {111} about equally strongly developed. The crystals are mostly flattened towards 5 with incli- nation towards the a-axis. (32) Measured : Calculated : q:q = (011) : (014) =* 63°437/,' os 0:0 = (111): A11) =* 47 52 == ( ( ) ong = 111) (DSE 72-2075 = 0%: ae == (111) : (011) =S AS ADE 45°42! q:b = (011) 2O1O)==F nS. 187: 98 181/) 6:6 ={O1OYE EMED == 66" 6 66 4 No distinct plane of cleavage is present. An optical investigation was quite impossible owing to the opaqueness of the crystals. Sp: Gr. == 2,211 fat 62 @-- he equiv. vol. ' 420500: Topic parameters: %: py: w = 4,0397 : 7,1147 : 4,4516. E. Nitro-3-5-Dibromobenzene. Structure: C,H, (NO,)a) . Bris). Br); m.p.: 104°,5. The compound has already been measured by Boprwia (Zeitschr. f. Kryst. 1. 590); my measurements quite agree with his. Monoclino-prismatic. Bopewie finds a:6:¢=0,5795:1 :0,2839, with B=56°12'. Forms: 110}, 1100}, {001} and {011}. I take B = 85°26’ and after exchanging the a-, and c-axis a0 6 = Ube: 104651; with the forms {011}, {001}, {201} and {211}. Completely cleavable towards {201}. Strong, negative double refraction. Sp. Gr. = 2,363 at 8° C.; equiv. vol. = 118,91. Topic axes: x: p:o = 4,3018 : 7,5761 : 3,6601. _ The great analogy in the relation a:b of this and of the previous substance is remarkable; also that of the value of angle (. F. Nitro-3-4-Dibromo-Benzene. Structure C,H, (NO) . Bris). Bray; m.p. 58° C. Has been measured by Grorn and Boprwie (Berl. Ber. 7.1563). Monoclino-prismatic. a:6=0,5773:1 with B= 78°31’. Forms {001}, {110} and {100}, tabular erystals. Completely cleavable towards {100}, distinctly so towards {010}. The optical axial plane is {010}; on a both optical axes (80°) are visible. I found the sp. gr. at 8°C. to be 2,354. The equivalent volume is therefore 119,34. (33 ) I have tried to find a meltingpoint-line of the already described type in the monoclinic derivatives in which some degree of form-analogy is noticeable. However, in none of the three binary mixtures this was the case; the lower melting point was /owered on addition of the component melting at the higher temperature, without formation of mixed erystals. For instance: A mixture of 82,3°/, 1-2-3- and 17,7°/, 1-3-5-Nitrodibromo-benzene melted at 48'/,° C. A mixture of 76,5°/, 1-2-6- and 23,5°/, 1-3-5-Nitrodibromo-benzene ate GS" /,°. C. A mixture of 90,5°/, 1-3-4- and 9,5°/, 1-2-6-Nitrodibromo-benzene at 54° C. Moreover, no mixed erystals could be obtained from mixed solutions. The slight form-analogy with the Notro-dichloro-benzenes *) investi- gated by me some time ago is rather remarkable. Nitro-2-3-Dichloro-Benzene (62° C. rhombic) and Nitro-2-6-Dichloro- Benzene (71° C. monoclinic) exhibit practically no form-analogy with the two Dibromo-compounds. There is also nothing in the Dichloro- derivatives corresponding with the isomorphotropous mixture of the 2-3- and 2-5-Dibromo-product. The sole derivatives of both series which might lead to the idea of a direct isomorphous substitution of two Cl- by two Br-atoms are the Nitro-3-5-Dihalogen-Benzenes (65° C. and 104°,5 C.); the melting point of the Dichloro-derivative is indeed elevated by an addition of the Dibromo-derivative. As a rule, the differences in the crystal-forms of the compounds of the brominated series are much less than those between the forms of the chlorinated derivatives — a fact closely connected with the much greater value which the molecular weight possesses in the Nitro-Dibromo-Benzenes than in the corresponding Chloro-derivatives. Zaandam, April 1906. Physiology. — “On the nature of precipitin-reaction.” By Prof. H. J. Hampurerr and Prof. SvANTR ARRHENIUS (Stockholm). (Communicated in the meeting of April 27, 1906). One of the most remarkable facts discovered during the last years in the biological department, is most certainly the phenomenon that when alien substance is brought into the bloodvessels the individual reacts upon it with the forming of an antibody. By injecting a 1) These Proc. VII, p. 668. Proceedings Royal Acad. Amsterdam. Vol. IX, (34) toxin into the bloodvessels, the result is, that this is bound and free antitoxin proceeds. Errricn explains this as follows. When a toxin is injected, there are most probably cells which contain a group of atoms able to bind that alien substance. Now Weicert has stated the biological law, that when anywhere in the body tissue is destroyed, the gap usually is filled up with overeompensation. So, it may be assumed, that when the cell looses free groups of atoms, so many of these new ones are formed, that they can have no more place on this cell and now come in free state in circulation. This group of atoms is the antitoxin corresponding to the toxin. As a special ease of this general pbenomenon the forming of precipitin is to be considered. When a calf is repeatedly injected with horseserum, which can be regarded as a toxic liquid for the calf, then after some time it appears that in the bloodserum of that calf an antitoxin is present. In taking some bloodserum from this calf and by adding this to the horseserum a sediment proceeds. This sediment is nothing else than the compound of the toxin of the horseserum with the anti-toxin that had its origin in the body of the calf. We are accustomed to call this antitoxin precipitin, and the toxin here present in the horseserum, and which gave cause to the proceeding of precipitin, precipitinogen substance. The compound of both is called precipitum. It is very remarkable that such a precipitate proceeds only, when the precipitin is brought in contact with its own precipitinogen sub- stance. Indeed by adding the designed calfserum containing preci- pitin, not to the horseserum but to the serum of another animal, no precipitate proceeds. In this we have also an expedient to state if in a liquid (e.g. an extract of blood stain) horseserum is present or not (UnLENHUTH, WASSERMANN inter alia). Meanwhile such a calfserum gives notwithstanding also a precipitum with serum of the ass related to the horse. To the same phenomenon the fact is to be brought, that when a rabbit has been injected with oxenserum, the serum taken from the rabbit does not only give a precipitate with oxenserum but also with that of the sheep and the goat, which are both related to the ox. Some time ago an expedient was given to distinguish also *) serum proteid from related species of animals by a quantitative way, and in connection with this a method ® was proposed to determine accu- 1) H. J. Hampurcer, Eine Methode zur Differenzirung von Eiweiss biologisch verwandter Thierspecies. Deutsche Med. Wochenschr. 1905, S. 212. 2) H. J. Hampurcer, Zur Untersuchung der quantitativen Verhältnisse bei der Pricipitinreaction. Folia haematologica. Il Jahrg. N°, 8. ( 35 ) rately the quantity of precipitate which is formed by the precipitin reaction. This method also permitted to investigate quite generally the conditions which rule the formation of precipitate from the two components. Immediately two facts had pushed themselves forward by a preli- minary study which were also stated in another way by EISENBERG *) and Asco.t’). 1. That when to a fixed quantity of calfserum *) (precipitin = antitoxin) increasing quantities of diluted horseserum (precipitinogen substance = toxin) were added, the quantity of precipitate increased, in order to decrease by further admixture of diluted horse serum. 2. that whatever may have been the proportion in which the two components were added to each other, the clear liquid delivered from precipitate always give a new precipitate with each of the components separately. This leads to the conclusion that here is question of an equilibrium reaction in the sense as it has been stated and explained for the first time by ARRHeNius and MapseEn £). This conclusion has become also the starting point of the now following researches of which the purpose was to investigate by quantitative way the principal conditions by which precipitin reaction is ruled. Methods of investigation. To a fixed quantity of calfserum ®) (precipitin = antitoxin) increas- ing quantities of diluted horse-serum (precipitinogen substance = 1) Etsenpere. Beiträge zur Kenntniss der specifischen Präcipitationsvorgänge Bulletin de l’Acad. d. Sciences de Cracovie. Class. d. Sciences Mathem. et nat. p. 289. *) Ascou. Zur Kenntnis der Präcipitinewirkung. Münchener Med. Wochenschr. XLIX Jahrg. S. 398. 3) They used sera of other animals. 4) ArrneNius und Mapsen. Physical chemistry to toxins and antitoxins. Fest- skrift ved indvielsen of Statens Serum Institut. Kjobenhavn 1902; Zeitschr. f. physik. Chemie 44, 1903, S. 7. In many treatises the authors have continued these investigations; compare e.g. sull : Arruenius. Die Anwendung der physikalischen Chemie auf die Serumtherapie. Vortrag gehalten im Kaiser]. Gesundheitsamt zu Berlin am 22 Sept. 1903. Arbeiten aus dem Kaiserl. Gesundheitsamt 20, 1903. ARRHENIUS. Die Anwend. der physik. Chemie auf die Serumtherapeutischen Fragen. ‘Festschrift f. Botrzmann 1904. Leipzig, J. A. BARrn. 5) To make it easy for the reader, we speak here only of calfserum and horse- serum. Compare the third note on this page. 3* ( 36 ) toxin) are added. There upon the mixtures are heated for one hour at 37° and then centrifugated in funnelshaped tubes of which the capillary neck was fused at the bottom. The in 100 equal volumes calibrated capillary portion contains 0.02 or 0.04 cc. The centri- fugating is continued till the volume of the precipitate has become constant *). ° Experiment with calf-horse serum. As it was of importance, at all events for the first series of proofs, to dispose of a great quantity of serum containing precipitin, a large animal was taken to be injected. Dr. M. H. J. P. Thomassen at Utrecht was so kind to inject at the Governement Veterinaryschool there, a large calf several times with fresh horse serum and to prepare the serum out of the blood drawn under asceptie precautions. The serum used for the following series of experiments was collected Nov. 28, 1905, sent to Groningen and there preserved in ice. On the day of the following experiment January 25, 1906, the liquid was still completely clear and free from lower organisms ; there was only on the bottom a thin layer of sediment, which naturally was carefully left behind at the removing of the liquid. ’) The horseserum used for the proof in question was fresh and 50 times diluted with a sterile NaCl-solution of 1°/). Each time two parallel proofs were taken as a control. The capillary portion of the funnel shaped tubes used for this experiment had a calibrated content of 0.04 ce. Bach division of the tubes thus corresponded to 0.0004 ce. To this series of experiments another was connected in which the quantity of diluted horseserum was constant, but increasing quan- tities of calfserum were used. From the first table it appears, that when to 1 ce. calfserum increasing quantities of diluted horseserum are added, the quantity of precipitate rises. When more horseserum is added as is the case in the second table, the quantity of precipitate descends. This appears from the following. 1) Compare Folia haematologica l.c, for further particulars of the method. 2) Fuller details of other proofs taken on other days with calf-horseserum, also of experiments with serum obtained by injecting rabbits with pig-, oxen-, sheep- and goat-serum will be communicated elsewhere. (37) TABLE I. os 1 cc of the mixture of 1 ce. Volume of the precipitate, after centri- The quantity calfserum (precipitin or | fugating for: of precipitate found in ‘ce. serum containing anti- of the mixtures calculated for toxin) +... cc. horse- the total quan- tity of the serum !/,, (precipitino- mixed compo- nents according gen or toxin containing to the last observation. serum. th.-$h. - 3h. -4h. - 3h.- 20m. - 15m. 0.04 3 CC. horseserum '/;, | 41 — 1/, — not to be measured accurately! 2 » » » 1—!/;,— » » » » » en » » 3— 3— 3— 3— 3— 3— 3 3.08 > » » 3— 3— 3— 3— 3 — 3 — 3 3.08 0.4 5 De? » » | 42 — 14 — 10 — 10 — 10 — 10 — 10 10.5 oe » » » | 42 — 14 — 10 — 10 — 10 — 10 — 10 10.5 EE » » » | 26 — 23 — 20 — 18 — 17 —17 — 17 18.4 Er » » | 26 — 23 — 20 — 18 —17 —17 — 17 18.4 0.2 5 » » » | 32 — 26 — 24 22 — 21 — 21 — A 23.1 » » | 832 26 = 24 — AS 23.1 0.13 » » » | 48 — 43 — 39 — 34 — 32 — 32 — 32 36.2 0.13 » » » | 48 — 43 — 39 — 34 — 32 — 32 — 32 36.2 0.15 » » » | 52 — 45 — 40 — 36 — 34 — 34 — 34 39.1 0.15 » » » | 50 — 45 — 40 — 36 — 34 — 34 — 34 39.1 018 » » » | 6 — 61 — 54 — 48 — 42 — 43 — 43 DO 0.18 » » » | 6 — 61 — 54 — 48 — 42 — 43 — 48 50.7 0.2 » » » | 65 — 62 — 55 — 49 — 45 — 45 — 45 54 0.2 » » » | 6 — 62 — 55 — 49 — 45 — 45 — 45 D4 0.25 » » » | 78 — 73 — 65 — 58 — 55 — 53 — 53 66.3 0525.) » » 78 — 73 — 65 — 58 — 54 — 53 — 53 66.3 Orsay » » | 8 — 80 — 70 — 62 — 58 — 57 — 57 74.1 0.3 » » » | 84 — 80 — 70 — 62 — 58 — 57 — 57 74.4 ( 38 ) So e. g. the quantity of precipitate when 0.3 cc. horse serum is added to 1 ce. calfserum, is 74.1 (table I). But when, as may be read in the second table 0.5 ec. horse serum is added to 0.9 ec. calfserum the precipitate has a volume TABLE II. 1 ce of the mixture of 0.5 cc horseserum feo cic Co calfserum. 0.1 cc calfserum, 0.1 » » 055) » 0.3 » » 0.5 » » 0.5 » » OMD » Oni» » 0.9 » » 0.9 » » 1.4 » » cli) » 185 » AD » ADD » AD) » 1.9 » » 1.9 » » Volume of the precipitate, after centrifu- The quantity of precipitate found in fee of the mixtures calculated for the total quantity of the mixed components, | according to the last observation. gating for th. - Jh. - ¢h. - $h. - $h.-20m.- 45m. 1 — 4 — not to be measured accurately 1 — — » » » » » Q— 2— 2— 2— 2— 2— 2! 1.6 2— 2— 2— 2— 2— 2 dalle alin 6—_ B— 5 — b— 5 — bD— bl 5 7— 5— 5— 5— 5— 5— 58] 5 95 — 81 — 67 — 58 — 52 — 50 — 50 | 80 of 51.6. If instead of 0.9 calfserum 1 ec. was used the quantity of horseserum ! would necessarily have amounted to 0,5 X 9 9 = 0,55 cc. So it appears that by the addition of 0.3 ec. horseserum to 1 ce. calfserum the precipitate amounts to 74.1 and by the addition of 0.55 cc. horseserum but to 37.3 !). This decrease must be attributed partly to the solubility of the precipitum in NaCl-solution, a solubility which is felt the more strongly as a greater quantity of diluted horseserum is added. (Compare also Fol. Haematol |. c.). So we see that the clear liquid above the precipitate contains, besides free precipitin and free precipitinogen substance, as has already been stated, also dissolved precipitate. These three substances must form a variable equilibrium, which according to the rule of GuLDBERG and Waaer is to be expressed by the following relation. Concentration of the free precipitinogen subst. > Concentr. of the precipitin —k, X Concentr. of the dissolved precipitate . . . . (I) in this 4, is the constant of reaction. Meanwnile it appears from the experiment, that a greater quantity of precipitate must be dissolved than corresponds with this equation, or to express it more clearly, than corresponds with the conception that the solubility of the precipitate in NaCl solution is the only fact by which the quantity of sediment decreases. To take away the difficulty, the hypothesis was made that still another portion of the precipitate forms a dissolvable compound with free precipitinogen substance (of horseserum) and that we have here a case analogical to the reaction of CaH,O, with CO,. As is known CaH,O, is precipitated by CO,, but by addition of more CO, the sediment of CaCO, decreases again, while CO, with CaCO, forms a dissolvable substance. As will soon be seen, a very satisfactory conformity between calculated and observed quantity of precipitate is obtained through this hypothesis, which could afterwards be experimentally affirmed. Let us now try, reckoning both with the solubility of the precipi- tate in NaCl-solution and with the forming of a dissolvable mixture of precipitate with precipitinogen substance, to precise more closely equation I. 1) The hyperbolic form of the precipitate curve with nereasing quantity of horse- serum may still appear from the following series of experiments taken on another day (Table (II). This series has not been used for the following calculation. ( 40 ) TABLE III. Zee of the mixture | Volume of the precipitate after centrifu- of 1cc calfserum +... cc horse- serum 1/,,. gating for: The quantity of precipitate found in 1 cc. of the mixtures calculated for the total quantity of the mixed components according to the last observation. th.—}h. — th. — $h. — 20m. — 15m. — 10m. 0.4 cc horseser. 1/,,| 38 — 38 28 — 24 — 9 — OA » » yl 2) 99 — 94° — 193" = 02» » > 66; = 54 WS) NN Ss U 0.2 » » ye 59) == 50 45 43 A == 0.3 » » y | 88° — 169 65 56 55 0.3) » » yl) co =. (ate) GS es GY es Sy, == 0.4 » » » | 98 — 76 10 = 162) —= 58) —— 0.4 » » » | 89 — 73 OSS 0.6 » » » | 84 — 62 SS ASW 0.6 » » Pd ND 53 — 47 — 43 — 0.7 » » » | 6 — 49 45 — 39 — 37 — OFT» » » | 66 — 49 45 — 39 — 37 — 08» » » | 61 — 45 40 — 38 — 33 — 0.8 » » » | 62 — 45 40 — 38 — 33 — 0.9 » > » | 4A — 30 265 —— 25 AS 0.9 » » » | 4 == to) 96 — PE 1 » » » | 244 — 17 1 Ss) aly = ley den Sp » el Lr SS 4 (0) 45 — 145 — 13 — 1092» D) » Y= OF ==. OF a OF 1 » » Q9— 9 Di Dames Peke 1.4 » D) » not to be measured 23 23 (25) Firstly we shall try to find an expression for the three substances occurring in the clear liquid which stands above the precipitate: for the free precipitinogen substance, for the free precipitin which it contains and for the quantity of dissolved precipitate. Firstly the quantity of free prcipitinogen substance. Let A be the total quantity of that substance used for an experiment. To determine how much of this is still present in the liquid in free state, it is to be determined how much is bound. Bound is: 1. a certain quantum to form the precipitate which is present in solid condition. If we set down as a rule that 1 mol. precipitum proceeds from 1 mol. precipitinogen substance and 1 mol. precipitin, then the wanted precipitinogen substance will be expressed by ZP, if the molecular quantity precipitate also amounts to P. 2. a quantity pV when p represents the percentage of the quantity of dissolved precipitate and V the total volume of the liquid. 3. a quantity necessary to form the compound of precipitate- and precipitinogen substance. Admitting that 1 mol. of this compound proceeds from 1 mol. precipitate and 1 mol. precipitinogen substance and then that y of this compound is present, then together 27 must be charged, while in each of the two components y mol. precipitinogen substance is present, so that the quantity of precipitinogen substance, which is left in free state, amounts to A—P—pV—2y. So when the volume of the liquid is V, the concentration of the free precipitinogen substance = A—P—pV—2y Se ee It is possible to calculate in the same way the concentration of the free precipitin. If B is the total quantity of precipitin, which is used for the experiment, then there is to be subtracted from this: 1st. a quantity P for the same reason as is given at the calcula- tion of the free precipitinogen substance (see above). 2nd, a quantity pV, likewise as explained there. 3rd, a quantity necessary to form the compound precipitate-preci- pitinogen substance. While in this compound but 1 mol. precipitin is present, only 1y is to be charged. So that the quantity of pre- cipitin which remains in free state, amounts to b—P—p V—y. While the volume of the liquid amounts to V, the concentration of the free precipitin is = ( 42 } B—P—pV—y - Sat ae EE) As for the concentration of dissolved precipitate in the third place, this must be expressed by pV roc Oe ee Ne (4) So the equation (1) becomes: A—P—pV—2y B—P—pV— V 5 y x Ond =) DE V Vi W or (A—P—pV—2y) (B—P—pV—y) =k, pv’. . . (5) Now one more equation, expressing the reaction according to which precipitate combines with preeipitinogen substance. This is to be written down as follows. Concentration free precipitinogen substance X concentr. dissolved precipitate = ko concentr. compound precipitinogen subst. — precipitate. A—P—pV—2: V P ul xP =f „ee V V V or (A—P—pV—2y)p=ky. .. . . (G6) By putting shortly P+-pV =P’ and by substituting iel value of y of equation (6) into equation (5) we obtain ee Ee De a (7) Pr 2p dn Ms eae In this equation are known: dst. A, the total quantity of precipitinogen substance (diluted horseserum added) ; gnd, $B, the total quantity of precipitin (oaltsairut) used ; 3rd, V7, the volume of the liquid resulting from the mixing of the two components ; 4th P, the quantity of solid precipitate directly observed. Unknown are: 1st. , the quantity in percentages of precipitate which is dissolved (so p represents the solubility of the precipitate) ; Jud, ke, the constant for reaction of the formation of precipitate; Bd, A, the constant for reaction of the formation of the com- pound precipitate-precipitinogen substance ; (43) 4, P’, this is however P+pV and therefore known as soon as p has become known. As equation (7) contains 3 unknown quantities three observations will be necessary to determine them. When we introduce then the so found values in the other experi- ments and calculating the quantity of precipitate, it appears that the calculated quantities correspond in very satisfactory way with those which are observed. Let us observe that to avoid superfluous zeros 1 ce calfserum (B) is taken = 100. While as appears from the experiments in the case in question 1 ce calfserum is equivalent to nearly '/, ce horseserum 1:50, 1 ce horseserum 1:50, that is A, obtains a value of 300. . 0,04 So, where in the first experiment ge horse serum was used 0,04 A obtains a value of ma SOA In the experiment, where on 1 ee. calfserum 0,3 ce. horseserum was used, with a value B == 100, A becomes 0,3 « 300 = 90. Let us now combine the two tables to one by calculating for the second table how much '/,, horseserum is used on 1 calfserwm.. We see that the comformity between the determined and caleu- lated precipitate (col. LIT and LV) is very satisfactory. The average of the discrepancy amounts to 1.3. | This result deserves our attention not only in view of the know- ledge of the precipitin reaction as such, but also from a more general point of view, this reaction belonging to the great group of the toxin-antitoxin reactions. Till now, in studying the last, we were obliged to deduce the equilibrium conditions from the toxins, that is to say by determining the toxie action which was left by the gradual saturation of the “toxin by increasing quantities of antitoxin, but with the precipitin- reaction the equilibrium conditions may be deduced from the quantity of the formed toxin-antitoxin compound. And not only that, but owing to the fact that the compound forms a precipitum, the quantity of this may be fixed in an accurate and direct way by simple measurement, thus without the aid of red blood corpuscles or of injecting-experiments in animals. So there is good reason to expect that a further study of the precipitin-reaction will facilitate too the insight in other toxin-anti- toxin reactions. (44) TABLE IV. ] fe HI IV Vv Acc. calfserum, B = 100. Used quantity, Used quantity Determined Calculated Difference of of volumes of the | volumes of the between horseserum horseserum precipitate precipitate III and IV. 1/54 (on Acc. expressed in 1 cc. of the | in 1 cc. of the calf serum). in the just mixtures. mixtures. accepted units 0.013 ce. 4A not to be measu- 0.2 red. 0.027 » 8 3 3.9 + 0.9 0.05 » 15 10 10.3 + 0.3 0.08 » 24 17 17.8 + 0.8 0.4 » 30 21 23.6 + 2.6 0:43 _» 39 32 29.7 — 2.3 074555) 45 34 34 0 OMS 54 43 40.1 — 2.9 0.2 » 60 45 43.9 — 1.1 0.25 » fe APE 52 52.1 0 0.266 » 79 51 53.6 + 2.6 0.24 » 88,3 55 57.1 + 2.1 0.3 » 90 57 57.5 + 0.5 0.33 » 100 59 58.9 — 0.1 0.385 » 115.4 55 57.4 + 2.4 0.457 » 137 50 51.3 + 41.3 UES vie 167 43 41.3 — 1.7 0.713 » 214 25 26.8 + 1.8 1 » 300 5 5.5 + 0.5 467. D 500 2 0 —2 Ld , RESUME, We may resume our results as follows. By mixing precipitin and precipitinogen substance (to compare resp. with antitoxin and toxin) an equilibrium reaction proceeds ( 45 ) obeying to the law of GuLpBerG and Waace. By this equilibrium reaction part of the precipitin molecules combines with the corre- sponding quantity of molecules precipitinogen substance, while by the side of this compound a certain quantity of each of the two components remains in free state. The compound is partly precipitated and partly remains dissolved. How much remains dissolved depends for the greater part on the salt solution which is present, for the sediment is soluble in Na Cl-solution. Besides this equilibrium reaction there is still another which consists in this, that part of the precipitate combines with free precipitinogen substance to a soluble compound. ‘This reaction too obeys the law of GurpBerG and Waacr. The case is to be compared with the precipitation of Ca(OH), by CO,. By excess of CO, a part of the resulting CaCO, is transformed in a soluble bicarbonate. So CaH,O, takes the function of the precipitin and CO, that of the precipitinogen substance. Astronomy. — “Observations of the total solar eclipse of August 30, 1905 at Tortosa (Spain).” By J. Stun S.J. (Communicated by Prof. H. G. VAN DE SANDE BAKHUYZEN.) At the invitation of Mr. R. Cirera S. J., director of the new “Observatorio del Ebro” I went to Tortosa towards the end of June 1905 in order to take part in the observation of the total solar eclipse. I was charged with making the measurements of the common chords of the sun and moon at the beginning and at the end of the eclipse and had also to determine the moments of the four contacts. The results might perhaps contribute to the correction of the relative places of the sun and moon. The determination of the co-ordinates was much facilitated by the circumstance that the signals of the three points Espina, Gordo and Montsia of the Spanish triangulation were visible at this place. The measurements of the angles with a theodolite yielded the following results : = 40°49' 13".43 ; Az1™ 58518 east of Greenwich. In these results the spheroidal shape of the earth is accurately taken into account. Later measurements made by Mr. J. Usacu gave the same results. Electric time-signals, directly telegraphed from the Madrid observatory, gave for the longitude: 1m 585.8 east of Greenwich. As the most probable value we have adopted 1m 585,5, ( 46 ) the mean value of the two determinations. As a test 30 other deter- minations of latitude have been made with an instrument temporarily adjusted for Talcott observations, from which I derived as mean value: p= 40°49' 14.8. The height above the sea-level is 55 meters. The instrument at my disposal for the eclipse observation was a new equatorial of Mamuar (Paris), 2™.40 focal length and 16 em, aperture, provided with an eye-piece with a double micrometer. I have determined the screw value of one of the two screws from 18 transits of circumpolar stars near the meridian. I found for it: R, = 60".3534 + 0".0117; the value of the other screw was determined by measuring the intervals by means of the first: aa OO0N 08 aie... The observatory possesses a good sidereal clock, the rate of which had been carefully determined during four months by means of star transits. In the night of 29—30 August, Mr. B. Berrory, a clever observer had observed 20 clock-stars, so that the accuracy of the determination of the clock-error left nothing to be desired. During the phase observations the object-glass was reduced to 25 mm. by means of a screen of pasteboard. The eye-piece with a power of 30 was provided with a blue glass. The observations of the chords were continued as long as was allowed by the field of view of the eye-piece, which was more than 20! in diameter. At my signal “top” the moments of the observations were noted by Mr. Brera, who was seated in front of a mean time standard clock, which before, during, and after, the observations was compared with the sidereal clock; another assistant recorded the micrometer readings. During the beginning and the end of the eclipse the sky in the neighbourhood of the sun was perfectly clear, so that I could per- form. the measurements of the chords undisturbed, although now and then I met with difficulties owing to irregularities in the rate of the driving clock. From some minutes before, until after, totality the sun was covered with light clouds, yet the moments of contact could be recorded with sufficient accuracy. In the derivation of the results I have taken the solar parallax — 8".80; for the rest I have borrowed the constants from the public- ation “Observatorio Astronomico de Madrid. Memoria sobre el eclipse total de Sol del dia 30 de Agosto de 1905”. They are: Mean radius of the sun &, = 15'59".63 (Auwers) J. be » „ moon 7, = 15'32".83 (KuEstNER and BATTERMANN) Parallax of the moon Rss OF 268 Car) OBSERVATIONS. hens First contact: {1 55 39 .1 (mean time of Greenwich.) Length of the chords (corrected for refraction) " 11 56" 28" „2 294.93 57 12 A 390.24 57 35 .2 437 22 58 20 .0 507.74 59 8 .2 566.98 59 38 .9 608.94 12 6 9.2 642.58 1 25 .0 721.69 2 49.9 798.82 418 3 876.43 457 .0 906.12 5 44 3 935.04 6 15 .9 959.75 6 53 .2 983.94 718 9 1004.93 8 1.2 1030 37 843 3 1052.59 9 23 3 1078.47 949 A 1096.89 — 10 16 A 4106.16 10 42 .2 1124.37 19.3 1138.90 1 96 A 4144.49 AA 56 .3 1160.37 12 4 .3 1178 82 h m Second contact: AGA £2 Third contact: 449 7 2 Length of the chords Li " 2 15 53 .0 1297.92 ( 48 ) Length of the chords ih’ #448 Wy Oe ala AV 4256 94 3 18 4.5 1232.27 18 25 .3 JONES 18 42 .5 1209. 51 19.43. :3 1193.25 19 38 .2 1181.49 20 45 .0 4157.42 219 5) 53 sO 21-28 <3 4117.78 ep Haas at) 1095.75 22 35 .3 1073.82 23; 4.41 1054.40 23 21 .3 1041 52 23 54 .3 1020.90 24 36 .0 993.28 DO a 975.01 25 35 .3 950.47 26. 2) 23 £20.28 26 29 .3 903 . 24 26 52 .3 880 81 27 13 3 863.90 27 36 .2 845.44 287 .6 819 14 28 43 .3 779.01 JORD 762.98 29 38 .6 726.38 30 2 .3 697 . 40 30 22 .3 677.17 30 52 .3 637.13 31 14 .8 610.37 31 40 .6 973.84 32 4.5 538 . 62 32 42 .6 480 78 33 3.3 437.21 33 13 .3 406.92 ho ms Bronakrtahsrctom tia cil 9 34 44 .7. ( 49 ) Right ascension of the sun, Aug. 30,125 M.T. Gr. ao = 158°10'44".24. Declination EE eet 2 da > oe Sas 19 Right ascension of the moon ,, ___,, - ac == 157°42'47".95 (HANSEN-NEWCOMB). Declination By ees Ss LT se dew F 5a. 348 (HANSEN-NEWCOMB). Each observation gives an equation of condition for the determin- ation of the corrections 4 of the elements of the sun and moon. Let these corrections be successively AR, Ar, Aap, Aag Ado, Ade Ax, then we obtain by comparing the observed distances and chords with those computed the following equations: (the coefficients have been rounded off to two decimals). EQUATIONS OF THE CHORDS. I. Observations after the first contact. Obs.—Comp. " Li +7.98 AR +7.97 Ar 47.14A¢ —3.20 A3 H1.67 Ar =H50.71 —10.36 GE GO BENSON PS 40, $5.88 5 °049 „HAAR HGe BT EERE Ae F380 4.3 | HOi, STR 12070 ERETON ev 14:09 43.56 5 —1.60 „ 40.81 „ 180.40 £4.78 NE a Ee ON “4.90... EE ts Ee 330 ABT 1D, LO OO AO EE AS © - 19/69 4 A0, 4059 5 A5 0-67 LOE Ee LE EEA NONO 1248, 12.06, —0.92 5, 40430 OGO MD seared. 296, 14.85 4, —0.82 008 Be LOI 06 HU , 4218, 44.76 „ —0.79 „ $0.36, +419.60 + 4.98 LAN , $240 , BB, —0.75 „ HOW ,, 443-42) — 4.49 TS 40, £1.68" , —0.73. „ On VAD GO 4.48 49.02 , $4.99, 44.57 „ 0.70 „ HOU „ H457 + 0.47 44.99 „ HB „ 44.53 „ —0.69 „ $0.30, -}17.85 + 4.46 44.94 ,- $1.90, HAB , 066 „ 4$0.99,, +45.73 + 9.82 44.89 4, 41.85, HAB „ 06 „ FOM, 41.49 — 0.97 44.85 , HU, HB ,, —0.62 „ 40.26, 442.97 + 0.87 44.82 , HB, HB „ —0.60 „ $0.2 ,, 446.44 + 4.68 1480 5 44.76 , HAB ,,. 0.50 4. FOD, 0:08 — 456 Ty Ae A3 HD Sp OBB ge eee NE LA FA EO eN SE EEN EAT ELE. 60) 2, HA Sy 058. Sea ST ey Th CANON HB 5, 0.55) ope HO 0 01 0.50 44.69 „ 14.65, HU „ O5 „ $0.22 ,, HAM + 2.06 Proceedings Royal Acad, Amsterdam, Vol, IX, if. Observations before the last contact. +1.46 Ar —0.99 Au —1 .05 „09 ( 50 ) 0.49 A5 10.16 Ar GO SO) eee ee Os et de AD OAN OO OO eet EDEN A ETE Die OO Bh e050 ee EO ED Nee EIO MA ORG SES REP EU RED eden OH Ee EO Git) a0 oe zn P0684 S06, ar E0270) eee 7 VEP A eae Ee Ne EE eik ate Ee Ne EONSO tn te Son Br 280289! 52° EO nd SEO en OSL EOSTA ones spe EOS EED 5) EOIGB ORE aOR ee ET RU SERT Oe EEE SRU EO DA SE DE Eee 02 ed E oe eee SER Ee Ee ie Gee On EAT OO IS os, ne Meso Ue Oe Obs.— Comp. n " = 5230) 110 — 5.30 + 0.97 5.84 IBH 130 OE = 859 AE — 8.19, eaf — 6.20 + 0.81 ERR — 2.58 + 4.99 = 5.07 ee =P 1:88 A00 LE BATT 0:07 = 9.75 A = 8.054 0:18 — 6.19 + 3.02 1.13 + OMS = £88 JAE —13.04 — 2.92 — 9.47 + 0.96 —12.56 — 1.80 —11.04 + 0.04 —10.50 + 0.87 53 40 SESAR —14.81 — 2.31 040 3:53 11.28 + 2.38 —14.62 — 0.37 —12.33 + 2.47 46.88 (= 448 —15.37 ++ 1.38 —18.89 — 1.29 —19.50 — 1.52 —18.51 + 2.74 O6 23°86 —38.87 —10.28 (51 ) Equations of the contacts I AR+ Ar + 0.903 Ago — 0.405 Ad-o = + 3'.78 IT AR— Ar — 0.9668 Aar > — 0.2007 Ado + + 0.0004 Ata_o — 0.0036 AaAd + 0.0091 A?&_o = — 6.52?) III AR— Ar + 0.3085 Aac-o — 0.9489 Ado + + 0.0104 A?ar-o + 0.0068 AaAd + 0.0012 A*d_o = + 4".02 IV AR+ Ar— 0.889 Aq—o + 0.485 Ade-o = — 11.18. A mere glance at the equations derived from the distances of the chords shows the impossibility to derive from them all the unknown quantities. On account of the proportionality of the coefficients we may use one single equation instead of the first 25 equations after the 18* contact; the same for the 35 others. In order to diminish the weight of the observations immediately after the first and before the last contact — when the chord is less sharply defined and varies rapidly — I have formed the two normal equations not according to the method of least squares but simply by addition. We obtain the following equations: 68.1(AR+ Ar) +56.2Aa-25.2Ad=+ 489".46 — 0.35(AR-Ar) — 12.9Ax —81.6(AR+ Ar)-+65.1Aa-31.6Ad=-+397".87-+0.24(AR-Ar)+12.8Az whence: AR + Ar=+4 1".05 — 0.015 Ad — 0.003 (AR — Ar) — 0.16 Az. Aa= + 7".428 + 0.465 Ad — 0.001 (AR — Ar) — 0.02 Aa. Neglecting the last terms, we find for the result from the equations derived from the length of the chords: AR 4 Ar= 41".05 — 0.015 Ado Aac-o = +7".428 + 0.465 Ad_o. From the equations of the 22d and 3rd contact we derive: Aa-O = + 1".793 + 0.464 Ad-O. Aaco = + 7'.18 + 0.667 (AR — Ar) Adio = — 1.43 + 1.437 (AR — Ar). And lastly the equations of the 1s* and 4 contact yield: Aa—o = + 8".35 + 0.468 Adi—o [AR + Ar = — 3".78] The latter result for AA + Ar, which differs entirely from that found above is little reliable. We can entirely account for it by assuming that the first contact has been observed too late and the last contact too early. It can hardly be doubted that the 1st contact 1) It is not allowed (as it is generally done) to neglect the quadratic terms in the equations of the 2"4 and 3rd contact, because the corrections Az and „©, as compared with the distance between the centre of the sun and that of the moon, (in this case 46") are too large. 4* (52) is recorded too late because the eclipse began earlier than was expected and in consequence took me by surprise. As an evidence that the time of last contact was given too late there is an instantaneous photograph of the sun (diameter = 10 em.) taken at the very moment when I gave the signal “top”. This plate shows a small impression on the limb of the sun. To enable me to compare the obtained results, Messrs. Ta. Wurr and J. D. Lucas kindly put at my disposal the results of their highly interesting observations of the 2°¢ and the 8" contact, made at Tortosa by means of sensitive selenium elements. (See for this Astron. Nachr. N°. 4071). They found: | beginning of totality 1° 16m 155,6 5 de 19 OEI which yield the following equations: AR — Ar — 0.9650 Aac-o — 0.2117 Adc-o + 0.0004 Atao — — 0.0039 Aa Ad + 0.0092 A'de-c = — 5'.73 AR — Ar + 0.3063 Aco — 0.9493 Adio + 0.0105 Atao + + 0.0069 Aa Ad 4+ 0.0012 A?h&_—O = + 4".10 whence end ms (4) Aq—o = + 67.42 + 0.653 (AR — Ar) Ado = — 1".76 + 1.404 (AR — Ar). When we subtract the two equations A from each other we get : Aa-O = + 7".238 + 0.465 Ad_o , which agrees exceedingly well with the result of the chord equations Aa = + 7".428 + 0.465 Ad; but it also appears that it is impossible to determine Aa, Ad and AR—Ar separately from the combination of the contact and chord equations. In the derivation of the final result we have accorded the same weight = 1 to the results of the chord measurements and to those of the contact determinations made by Wurr—Lucas, and the weight 4 to my observations of the 2nd and 3" contact. Thus we find, leaving out of account the first and the fourth contact : AR + Ar= + 1".07 — 0.02 (AR—Ar) Aq—o = + 6".66 + 0.66 (AR—Ar) Ad-O = — 1".65 + 1.42 (AR —Ar). The last column of the chord equations contains the deviations in the sense of observation — computation, which remain when we sub- stitute these numerical values. The mean error of the first 25 obser- vations (excluding the first) amounts to + 2."53; that of the last 35 (excluding the last) is + 2."24. (53) Chemistry. — “On the osmotic pressure of solutions of non-electro- bytes, in connection with the deviations from the laws of ideal gases” By J. J. van Laar. (Communicated by Prof. H. W. Bakuuts ROOzrBooM.) Communicated in the meeting of April 27, 1906). 1. By H. N. Morse and J. C. W. Frazer') very accurate experiments were recently made on the determination of the osmotic pressure of dilute sugar solutions in water. The solutions had a concentration up to f-normal, and as c is then about */,, [the association factor of the water is viz. at 18° C. about 1,65, so that in 1 L. of water about 55,6: 1,65 = 34 Gr.mol. of water (simple and complex molecules) are present], the difference between the exact expression — /log(1—.) and the approximate value 2 [formula (2)| is not yet appreciable. It is however not so with the difference between the molecular volume of the solution v = (1 — x) v, + av, (v supposed to be a linear function of #, about which more presently) and the molecular volume of the solvent v,, when v, (the molecular volume of the dissolved sugar) cannot be put equal to v,. We shall see that this difference for 1-normal solutions amounts to 19°/,, so that by means of the experiments we can very well ascertain, if we have to make use of v or of v,. And these have really taught us, that the osmotic pressures measured agree (and even with very great accuracy) with the calculated values, on/y when v, is put in the numerator, and not v. This harmonizes therefore perfectly with what I have repeatedly asserted since 1894 *). (What I have called above v, for the sake of symmetry, was formerly always indicated by v,). Not the molecular volume therefore of the whole solution, but the molecular volume of the solvent i the solution. And this deprives those of their last support, who in spite of all evidence (for not the dissolved substance, but the so/vent brings about that pressure) persist in trying to explain the osmotic pressure by a pressure of the molecules of the dissolved substance comparable with the gas pressure. If such a thing could be thought of, v should be taken into consideration and not v,, for the molecules of the dissolved substance move in the whole volume v and not in the volume v,, which is perfectly fictitious with regard 1) Amer. Chem.-Journ. 34, 1905, p. 1—99. See also the extensive abstract N? 274 in the Phys. Chem. Centralblatt [IL (1906). *) See inter alia my previous paper on this subject in These Proceedings, May 27, 1905, p. 49. (Some remarks on Dr. Px. Kounstamm’s last papers). (54) to the solution, which would be equal to v only when v, happened to be equal to 2. 2. In order to compare the results, found by Morsr and Frazer, more closely with those for the osmotic pressure already given by me in 1894, we shall return to its derivation for a moment, chiefly in order to ascertain on what limiting suppositions this formula holds. With equilibrium between the pure solvent (concentration 0, pressure p,) and the solvent in the solution (concentration c,‚ pres- sure p) [the dissolved substance is nowhere in equilibrium, for it is supposed that there is a membrane impenetrable to it] the molecular thermodynamic potentials must have the same value. Hence *): Ut, (@, p) = u, (9; po) Now in general : dZ A nnee when OE (ht lan De ee at MU 00 x es 0,= =; © being given by On, O—= | pdv — pv— RT Jn, . log Zn. For binary mixtures of normal substances we may now introduce the variable « and we obtain (=n, is now —1, so that the term with log Zn, vanishes), as may be supposed as known: when w is written for if pdv by way of abbreviation. This expression is perfectly accurate for the above mentioned mixtures. For the further calculation we now introduce the idea jdeal”’ mixtures. They are such as for which the im/fluence of the 2 Ww two components iter se may be neglected. Then a 0, and w Hij 2 v becomes a Znear function of wv. But also Den 0, so that v becomes U 1) The following derivation is only different in form from the cited one in these Proceedings. (55 ) also a linear function of z. We shall further demonstrate this in § 6, and show that in the case of such mixtures: a. wis a linear function of x b. Oss ” ” 2 a C. a » » EE) EE) 2 99 d. the heat of mixing is — 0, so that we may say: ideal mixtures are such for which the heat of mixing is practically = 0, or with which no appreciable contraction of volume takes place, when 1—w Gr.mol. of one component is mixed with « Gr.mol. of the second. The conditions a, 6, c and d are simultaneously fulfilled, when the critical pressures of the two components are by approximation of the same value. Ow 3. For w— rz ES may now write w,, as w=(Î —#) w, + Ou x TEN CK ae KL a Ov —"/,a Rn (Pe ED eee In the same way Rie ene and we get: tb, (%, p) = C, — w, + pr, + RT log (1 — 2) u, (op) = C, | al Po? mo) Ow OW i —— | Otherwise evidently w — En always when v, and w, are supposed to be dependent of the pressure. For else w, and v, would have another value at the pressure p than at the pressure p,. We must therefore also suppose that our liquids are imcompressible. But there is not the slightest objection to this supposition for ordinary liquids far from the critical temperature (and there is only question of such liquids in discussions on the osmotic pressure). Only when « draws near to 1, and so the osmotic pressure would approach to oo, v, (and so also w,) must no longer be supposed to be independent of p. By equating these two last equations, we get: pv, + RT log (1 — x) = pyr,, hence (56 ) RT RP ee RN), 1 the expression already derived by me in 1894. *) 1) Cf. Z. f. Ph. Ch. 15, 1894; Arch. Teyler 1898; Lehrbuch der math. Chemie, 1901; Arch. Teyler 1903; Chem. Weekbl. 1905, N°. 9; These Proceedings, June nl 905. In the original Dutch paper another note followed, which Mr. van Laar has replaced by the following in the English translation. A conversation with Dr. Kounstamm suggested the following observations to me. Dr. Konnstamm finds (These Proceedings, May 27, 1905) the quantity Te av in the denominator of the expression for 7. This is quite correct, and harmonizes perfectly with the general expression, which according to equation (1) on p. 54, would also have been found by me for non-linear variability of v. Then we should viz. have: ug = — RT log (1 J = ie == (@ SA | en 1_\p —— Q zi WwW vu Ind P\v U Ou Po (v)p, b 0g ( 9) Ou 5 ( 1/Po Ww dm : where, when calculating @ — x De by means of vaN perk Waats’ equation of state, 0 (v—d) , : q also a term — Bn — appears, in consequence of which p { v — x occurs & Ax in the first member. db da’ ae ; ee dv Now it is of no importance whether v is diminished by x ) or by 7 as © 0(v—b p ) 5) approaches to O both for small and for very large values of p. I therefore is obtained a correction term in the denominator, in connection with the size of the molecules of exactly the same value as Dr. Kounstamm. That this did not always clearly appear in my previous papers, is due to the fact that I then always introduced the 5 g U ; De: approximation v — & >= tj, which was perfectly justifiable for my purpose. 0a VERE NO an ee Dn vi — Yo A05) — etc, this is sufficiently accurate for prac- tical purposes. (for ideal mixtures, where v is a linear function of %, it Is of course qwite accurate). Yet in a so early paper as the one cited by K. of 1894 (Z. f. Ph. Ch. 15, p. 464) it is clearly to be seen that the result obtained by me agrees perfectly with his. For it says (line 4 from the top) that va’ (the index a’ is there always For as v— © used for the liquid Pa ae - But this is in the z-notation nothing but Nal hl Ox’ in the solution with the concentration «. The phrase occurring on page 466: “und niemals etwa va’ — b im Sinne etc.” refers there to the well-known attempts of Ewan and others. The same is the case with the phrase in the paper on non-diluted solutions in the Ch. Weekblad of June 7th 1905: “Ook heeft men getracht, etc.” (p. 5). 4G the physical meaning of which is: the molecular volume of the water (57) We repeat once more: this expression holds from #=0 to «= near 1, when the following conditions are satisfied : a. the solution is an ideal binary mixture of normal components; 6. the solution is practically @compressible. Then (2) represents the additional pressure on the solution, in order to repel the penetrating water (the so-called ‘osmotic’ pressure). As however in all the experiments made up to now water was the solvent, hence an anomalous substance, (2) must not be applied to solutions in water without reservation. It is, however, easy to show that the influence of the association does not play a part before the term with 2 (justas the influence of the two components inter se), so that in the above experiments, where 2? may undoubtedly be neglected (cf. $ 1), formula (2) may certainly be used. Let us, however, first reduce it to a form more practical for use. 4. Let us write (2) for this purpose: Re RT (@ + he mt...) (1 + Me A ve 2a) v, v, LE which is more than sufficient for solutions up to 1-normal. Let us further assume that c Gr. mol. are dissolved in 1000 Gr. H,O (called by Morse and Frazer “weight-normal solutions”), then : C c Aden Te when we put */,,c=c' (84 = 55,6: 1,65 is the number of Gr. mol. ERO ine t000 -Ge. at 18° GC; ef. §. 1). We find then: v or when we restrict ourselves to terms of the second degree with respect to c’: BTR We ol hs x= —e(l1— En ber leo): v, , 94 In this R= 82,13 (c.c.M., Atm), and v, =1001,4:34 cM® at r ET. 13°. For af we therefore find at 18° C.: v, RT _ 82,18 x 291,04 __ … = 340, 1001,4 Eieren hence ige = 23,87 ¢(1— 0,015 c) Atm. . . . . (25) We see from the calculation, as we already observed above, that ( 58 ) the influence of the association of the solvent is only appreciable in the term with c°. If water were a normal liquid, we should have had */,,, cimstead of */,, ¢ = 0,015 ¢. (4 c would then be = */,<'/;, .c)- Let us now consider what the last expression would have become for 2,,0, when not v, had occurred in the denominator, but the molecular volume of the solution v. When c Gr. mol. are dissolved in 1000 Gr. H,O, then the total volume will be (at 18°) 1001,4 +190c ccM. [For 1 Gr. mol. = 342,2 Gr. of melted sugar occupies a volume of 190 ccM. at 18° (density = 1,8)]. Altogether there are now 84 + c Gr. mol., hence the molecular volume of the solution will be: _ 100144 190¢ 1001,4 140,196 OR BES 2. a BAT 1 008 e= For v, we found however above: 1001,4 in uae so that the value of 2,,° with v in the denominator instead of w, would have become: v ’ 1 + 0,03 ¢ 53° == 23,87 c (1—0,015 c) DK 14 019: At es 1. 6. 1 + 0,015 Wis = 20/86 eee” An 1+ 0,19 c For (weight)normal solutions (¢ =1) we should therefore have found instead of 2,,° = 23,87 (1 — 0,015) = 23,51 Atm., z,,° == Ao zen == 20,36: At = 5 >< En Cker 5 m. Now Morsr and Frazer found 24,52 Atm., which is considerably nearer the theoretical value 23,51 Atm. (with v, in the denominator) than near the inaccurate expression with v in the denominator *). So it is out of the question that the molecules of the dissolved substance should exert a certain pressure comparable with the gas- pressure, for then the volume of the solution as such, viz. v, would have to be taken into account, and not the in that solution perfectly fictitious molecular volume of the solvent v,. 5. But there is more. We shall viz. derive the expression for the pressure which would be exerted by the dissolved molecules, 1) With 0,5-normal the two values would have been 11,85 and 10,98 Atm., whereas 12,08 Atm. has been found experimentally. (59) when they, according to the inaccurate interpretation of the osmotic pressure, could move free and undisturbed throughout the space of the solution. VAN DER WAALS’ equation of state, viz. PT. a gives for the rarefied gas-state: RTS» Tek died i b Gaden p=—|— —-—— |=—(1+---— }, v | v—b v v v v 1 when we again content ourselves with terms of the degree —. Dj} Let us now write: then where v now represents the volume, in which 1 Gr. mol. of the dissolved substance moves. This volume is however evidently (cf. also $4): 1001,4 + 190 c C ’ or ee lOUL, 4 (1 + 0,19 ©), so that we get: “A EL ¢ 1 yc TOE 019%) 1001,4 (1 + 0,19 ©) Y RT —— = is (c.f. § 4), and wi ‘= ———_-; or as = 23,87 is (c.f. §4), and with y 10014 1001,4 — ye — 23,87 Ans, Aa rae T MN ie ae. 6) and this is an altogether different expression from (2%). Not only is v, replaced by v (which gives rise to the factor 1 + 0,19 c), but we also find 1— y'c instead of 1 — 0,015 c. In this y' is different for every dissolved substance, dependent on the values of a and 3, whereas the coefficient 0,015 has the same value for all substances dissolved in water, independent of the nature of the dissolved substance (c.f. § 4). Also the coefficient 0,19 depends on the dissolved substance on its molecular volume). Moreover y' depends also on 7’ on account ( 60 ) of a: RT. Except with H,, where y is negative at the ordinary tempe- rature, y is everywhere positive. But at higher temperatures its value is reversed, and becomes negative. So, when comparing (2’) and (3), we see clearly, that it is out of the question that the so-called osmotic pressure should follow the gas laws. Only with ce =O this would be the case, but for all other values of c the deviation for the osmotic pressure is altogether different from that for the gas pressure. This is still more clearly pronounced, when we compare the original formulae. For the osmotic pressure viz. the equation Ld Al R RE mW edt log (1—a)) = a 2(1 + */, #7 +...) 1 A holds; for the gas pressure on the other hand: en Y D= 1 ——|, v v so that the deviations from the gas laws (at the ordinary tempe- ratures) are even in opposite sense from the deviations of the osmotic pressure for non-diluted solutions. In view of these facts it is in my opinion no longer possible to uphold the old conception of the osmotic pressure as arising in consequence of a pressure of the molecules of the dissolved substance comparable with the gas pressure. The molecules of the dissolved substance have nothing to do with the osmotic pressure except in so far as they reduce the water in the solutions to another state of concentration (less concentrated), which causes the pure water (concen- tration 1) to move towards the water in the solution (concentration 1—zx) in consequence of the dmpulse of diffusion. On account of ry this a current, of which the equivalent of pressure = — (— log (l-«)), Vv, arises in the transition layer near the semi-permeable membrane, which current can only be checked by a counterpressure on the solution of equal value: the so-called osmotic pressure. This is in my opinion the only correct interpretation of the osmotic pressure. As I already observed on former occasions, we might just as well speak of an “osmotic” temperature, when the impulse of diffusion is not checked by pressure on the solution, but by cooling it. For at different temperatures the temperature functions C, (ef $ 2) are no longer the same in the two members of Uy (2, i) ei (o, T,), ( 61 ) whereas the terms pv are now the same. In this case 7’ would have to be <7, because the temperature exerts an opposite influence on the change of u from the pressure. In consequence of the term RT log (1 — x), u, (x) will be < u, (0). u, . On, must therefore be increased. Now-— = v,, hence positive, whereas je: 0 (u, ake ; ate ne — (¢, + pr,), so negatwe. So the value of u, («), which is too small in consequence of 2, can again be made equal to that of u, (0), either by increase of pressure (“osmotic” pressure), or by lowering of the temperature (“osmotic” temperature). It would, however, be advisable to banish the idea ‘osmotie pressure” altogether from theoretical chemistry, and only speak of it, when such differences of pressure are actually met with in case of semi-permeable walls (cell-walls, and such like). 6. Appendix. Proof of some properties, mentioned in § 2. a. In a previous paper in these Proceedings (April 1905) I Ov 5 the perfectly accurate expression [equation (4), p. 651): & derived for db 1 (v—b)? da do de RT v de de — ret) JE Zo . db da ; SN With = 8 and eae oo Wa, in which 6 = 6, —b, and a= Wa, —WVa,, at av this becomes: 2af/a (v—b)? Ov 7 ETB Ye 07 ; 24/,(v—b)? | ~ RT o : 8 db And now we see at once, that this passes into 3 or = when BYVa == dun aya. : For then —— in the numerator becomes equal to ¢/, in the de- 4 dv db 0?v d*h nominator. But when ——-—,, then also ——- =O, Ow dx Ox? is a linear function of z. as 7, = 0, and v AT ( 62 ) [We above derived the condition BW/a=av from the general ae db ’ Ox would immediately follow from this by differentiation, and Tee it 0 expression for = If we knew this condition beforehand av Ox b. On p. 651 [equation (5)] of the paper cited the perfectly general expression : dv would not be necessary to start from the general expression for | 70 2 (av—By a)’ dx? vo eb RTR 070 was derived for ST which becomes therefore == 0, when again & 070 070 BY¥a=av. Now O= | pdv—pv=w—pv. And as EP and 5 are Ce 02 << will be = 0, in other words « U both =O when av= Ba, also is a linear function of 2. c. The heat of dilution. It is given by the formula ae Ë Oe 20) oT This is viz. the so-called differential heat of ‘inten per Gr. mol. m of the solvent when dn Gr. mol. solvent (« == —_) are added to m+n a solution consisting of m Gr. mol. dissolved substance and » Gr. mol. solvent. This becomes [see equation (1)]: oe ee 1 Ow Ov Ct 0’ dw Ov = 0, then w — «x — =w,; and v — « — will be = v,, when Ort ‘ 0a Ow +; = 0. But then Ly =0. qed. And hence also the total heat of mixing will be = 0, when z Gr. mol. of the 24 component are mixed with 1—wz Gr. mol. of the ‘1st component. Ve 2 d. The peculiarities mentioned in $2 under a, 6 and d, which ( 63 ) characterize the so-called zdeal mixtures, are therefore all satistied when Brad: This yields: 8 (Wa, + ze] =d [b, + x8], when it is permissible — for liquids far from the critical temperature — to replace v by 6. Hence we get: Blan ab. or (6, — b,) Wa, = (Va, — Va) d,, or also be1Z 6 0; Vas, hence Ya, = Ya, Cn 0 1 2 from which «we see, that the case of ideal mixtures occurs, when the critical pressures of the components have the same value. e. Finally Oa nr ar Ona: Ow? 5 EE b° i a . . . so we see that also ra will be a linear function of z, when b, Wa, = b, Wa, or p, =p, In this way also c of $ 2 has been proved. (June 21, 1906). KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN TE AMSTERDAM, PROCEEDINGS OF THE MEETING of Saturday June 30, 1906. (Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige Afdeeling van Zaterdag 30 Juni 1906, Dl. XV). € OS NE ENE Se L. E. J. Brouwer: “Polydimensional Vectordistributions”. (Communicated by Prof. D. J. KORTEWEG), p. 66. F. M. Jarcer: “On the fatty esters of Cholesterol and Phytosterol, and on the anisotropous liquid phases of the Cholesterol-derivatives” (Communicated by Prof. A. P. N. FRANCHIMONT), p. 78. F. M. Jancer: “Researches on the thermic and electric conductivity power of crystallised conductors” I. (Communicated by Prof. H. A. Lorentz), p. 89. H. W. Baxuvis RoozEesoom: “Three-phaselines in chloralalcoholate and anilinchydrochloride”, P99. H. Haca: On the polarisation of Röntgen rays”, p. 104. P. van Rompuren: “Triformin (Glyceryl triformate)”, p. 109. P. van RomBurem and W. van Dorssen: “On some derivatives of 1-3-5-hexatriene”, p‚ 111. L. E. J. Brouwer: “The foree field of the non-Euclidean spaces with negative curvature”. (Communicated by Prof. D. J. Korrewea), p. 116. A. PANNEKOEK: “The luminosity of stars of different types of spectrum”. (Communicated by Prof. H. G. VAN DE SANDE BAKHUYZEN), p. 134. Errata, p. 148. r Proceedings Royal Acad. Amsterdam. Vol. IX ( 66 ) Mathematics. — “Polydimensional Vectordistributions”’.') By L. E. J. Brouwer. (Communicated by Prof. D. J. KorreweG.) Let us call the plane space in which to operate S, ; we suppose in it a rectangular system of coordinates in which a C, represents a coordinatespace of p dimensions. Let a PX-distribution be given in S,; ie. let in each point of S, a p-dimensional system of vectors be given. By X,, ae, We understand the vector component parallel to C, indicated by the indices, whilst as positive sense is assumed the one corresponding to the indicatrix indicated by the sequence of the indices. By interchanging two of the indices the sense of the indicatrix changes, hence the sign of the vectorcomponent. Theorem I. The integral of PX in S, over an arbitrary curved bilateral closed S, is equal to the integral of 7+! Y over an arbitrary curved S,41, enclosed by S, as a boundary, in which Pt!Y is determined by ox ae a 00K Y AA = 72 48 +1 (eRe Pip (ey. (24 —— ER 5 a, TRE el De On, À TEN q eenn EI i where for each of the terms of the second member the indicatrix (ag, &g,- - « ag, Cg) has the same sense as (a, @,.-.@ p41). We call the vector Y the first derivative of PX. Proof. We suppose the limited space S,4 1 to be provided with curvilinear coordinates w,...%,4 1 determined as intersection of curved C's, i.e. curved coordinatespaces of p-dimensions. We suppose the system of curvilinear coordinates to be inside the boundary without singularities and the boundary with respect to those coordinates to be everywhere convex. The integral element of »+'!Y becomes when expressed in differen- tial quotients of PX: Oat eae ph! Dees du, du, ga 9 + B pHi} . . £ De De nn sE dU duty 4i- 5 5 k me ii n v Y Aj ee in el Sn A “+1 0x2, Oita ay Oupy et” thay Oup+ | 1) The Dutch original contains a few errors (see Erratum at the end of Ver- slagen 31 Juni 1906), which have been rectified in this translation. ( 67 ) We now unite all terms containing one of the components of BANE. Aronson We then find: Oty Ox, dx, a Slade nes Da x ; 2 : OXi25...p 8 Oi ae den de O41 . . . Oty +1 Ox, da’, Oup+1 Wie n 0ne Òup1 Òzpe Oz, Oz, Òu, Den ke x du, ax Á ; : hen ee AP ; du, - +» dur + Oee Òz te 0x, dx, Out Opp Opr + ...(n—p terms). If we add to these the following terms with the value 0: de, Òz, On, Ou, Oee Ou, Dos es j a du, .. … dup4i+ Ow, 0a, 0x, Men ier Oup41 Ow, Ow, de, du, da el iss Ou, 0X05... : g 5 + ae : du, ss dupe = On, On, Ow, Oe. ue ne Oy +... (p terms), the n-terms can be summed up as: OXj03 p Ou, Ow ) du aie dy) Aa hay eee ae Ou, u, du, u, du, du, OX103... p ae eg Fs ON ea a Ned eae ee ae | Ba. ae da PEN ( 68 ) Let us suppose this determinant to be developed according to the first column, let us then integrate partially each of the terms of the development according to the differential quotient of Xj93....), appearing in it; there will remain under the (p + 1)-fold integration sign p(p-+ 1) terms neutralizing each other two by two. Thus for instance: ar ah, Ox, 0 Xp Òu, Ot 41 Ou, 0Up41 | | On, 0.2) | Ou, Ou, du, - Atty +} | ‘ j | 0x, Up | Òu, Ou, | X93... | X193...p | da, Oz; Ou, Ou, and du se du : ; 7 be 0) o ph! . ’ Ow, dw, dup Ou, | 072, 0225 | Ou, Ou) 41 Ad Òu, 0,41 | as they transform themselves into one another by interchangement of two rows of the matrix-determinant. So the p-fold integral remains only, giving under the integration sign 0a, oy hese WS dur & ‚Sdu, | Ou, Ou, | 7a | . e X 123 sp | . . | . . | Ox, dx, | dy) 41 : dut Ul Ot 1 to be integrated over the boundary, whilst in a definite point of that boundary the ZB term of the first column gets the sign + when for the coordinate 2, the point lies on the positive side of the boundary. Let us now find the integral of Xj23,.., over the boundary and let us for the moment suppose ourselves on the part of it lying for all ( 69 ) ws on the positive side. The indicatrix is in the sense u, u,...u, 4) and if we integrate A23. successively over the components of the elements of boundary according to the curved C)’s we find: dw, 0, | du, . —— du, | 1 1 | Ou Òus, | ° : | > if X193 Pp | : | : | vee du ie du | 4 2 | Ux Pp Òus p | p Pp | where (e,114,.-.@)—(123...p(p+1)); so that we can write as well Oz, Ix, du adu Ou i Ou ; | 1 1 | |. ; . | Or, da, | duy—j Dil | | / 0 1 | Nga an . f X193....p | | | On Ox q=1,2..(p-F1) | | : dtgti.-- = 5 dua | | Oug 41 Ugh | | | 0x, du, = du du Ovy +1 Òu 41 | Ow, 0x, En Ue ne } > u, u, | . ort f Nans | : : : | 1 of; du dn du U a . . . ER Ed a] | Ou +1 Òu +1 $ | If we now move to other parts of the boundary we shall conti- nually see, where we pass a limit of projection with respect to one of the coordinates u, the projection of the indicatrix on the relative curved C, change in sense. So in an arbitrary point of the boundary the integral is found in the same way as on the entirely positive side; we shall tind only, that for each coordinate w, for which we are on the negative side, the corresponding term under the sign will have to be taken nega- tively, by which we shall have shown the equality of the p-fold (70) integral of PX over the boundary and the (p +1) fold integral of P+'!V over the bounded Sp +1. We can also imagine the scalar values of »X set off along the normal-S,—,’s. As such the integral over an arbitrary curved bilateral closed S,—, can be reduced to an (n — p + 1)-dimensional vector over a curved S,—»+41, bounded by S,_,. If again we set off the scalar values of that vector along its normal-S,—), the vector ?—1Z appears, which we shall call the second derivative of PX. For the component vectors of ?—'Z we find: 0X, a soy Banned = —_— emme pl Òz, q Sq op) le on The particularity may appear that one of the derivatives becomes 0. Mm vr . > . . Mar « If the first derivative of an _X is zero we shall speak of an 1X, en c > M vr if the second is zero of an m1. a U 5 7 +1 Da Theorem 2. The first derivative of a »X is a” pA, the second a sae in other words the process of the first derivation as well as that of the second applied twice in succession gives zero. The demonstration is simple analytically, but also geometrically the theorem is proved as follows: Find the integral of the first derivative of »X over a closed S, 41, then we can substitute for the addition given by an 5,41 element the integral of 7X along the bounding S, of that element. Along the entire S41 each element of those S, boundaries is counted twice with opposite indicatrix, so that the integral must vanish. The analogous property for the second derivative is apparent, when we evaluate the integral of the normalvector over a closed S,—,+41. By total derwative we shall understand the sum of the first and second derivatives and we shall represent the operation of total derivation by v. —n 0? Theorem 3. 7? = SN aa Eh h=l Proof. In the first place it is clear from theorem 2 that the vector vy’ is again a PX. Let us find its component Aro. The first derivative supplies the following terms Ci) q—n 7 OY rp „Pp ee Dr Dern, q g=p +1 where u=p Sees es Ue VCE OXo12.. ~~ I)(u+1)...p Yop = (+ sign for (ug 12... (u—1) (u+1)... p)—(q 1 …p) 0X12...p Oa, So LS DX ien = q12...(u—])(u+1)..p en De Oe, Òz, u=1 gp +1 (- sign for (ug 12...(u—1)(u+1)...p)=(q1 ~»)) g=n Ò°Xio..p hi 02,7 : g=p+ 1 The second derivative supplies the terms u=p ÒZis. (u—1 )(u+1)...p Ús 35 eee Da 02, u=1 (+ sign for (w12...(w—1)(w+1) … p) = (12 … p) or for (gul … (wl) WH) … p) = (q 12... »)) qn ON 7 a Sy qi2..(u—1)(u+!)...p where Oi u—1)(u+1)...p = — de, ee q=ptl ES Te (- sign for (w12...(w—1)(w+1)... p) = (12 -») . u=p —n 3 D { En = X12 (utp 0u,02g u=l g=p+l (- sign for (qul ..(w—I1)(w+1)..p) = (12.7) Wijd be OSE ig ul 0? ues GAD) The terms under the sign © S of 7, are annulled by those of 1 7, so that only h=n 0? Xa. Pp DE 02)? A= is left. Corollary. If a veetordistribution PV is given, then the vector- Ae et “Vd f distribution {---————., integrated over the entire space, has tor second derivative V. (if 4,7"—! expresses the surface of the n—I-sphere in S,). The theorem also holds for a distribution of sums of vectors of various numbers of dimensions, e.g. quaternions. We shall say that a vectordistribution has the potential property when its scalar values satisfy the demands of vanishing at infinity, which must be put to a scalar potential function in S,.1) And in the following we shall suppose that the vectordistribution from which we start possesses the potential property. Then holds good: Theorem 4. A vectordistribution V is determined by its total derivative of the second order. For, each of the scalar values of V is uniformly determined by the scalar values of V?V, from which it is derived by the operation i dv | linn — yr? Theorem 5. A vectordistribution is determined uniformly by its total derivative of the first order. For, from the first total derivative follows the second, from which according to the preceding theorem the vector itself. We shall say that a vectordistribution has the field property, if the scalar values of the total derivative of the first order satisfy the demands which must be put to an agens distribution of a scalar potential function in S,- And in the following we shall suppose that the vectordistribution under consideration possesses the field property. Then we have: Theorem 6. Each vectordistribution is to be regarded as a total 4) Generally the condition is put: the function must become infinitesimal of order n—2 with respect to the reciprocal value of the distance from the origin. We can, however, prove, that the being infinitesimal only is sufficient. (73) derivative, in other words each vectordistribution has a potential and that potential is uniformly determined by it. Proof. Let V be the given distribution, then WW ude Bel es is its potential. For V?7P=—VV, or V(VP)=VV, or VP=V. Farther follows out of the field property of V, that P i uniformly determined as 7-2 of VV, so as 7 of V. So P has clearly the potential property; it need, however, not have the field property. N.B. A distribution not to be regarded here, because it has not the field property, though it has the potential property, is e. g. the fictitious force field of a single agens point in S,. For, here we have not a potential vanishing at infinity and as such deter- mined uniformly. The magnetic field in S, has field property and also all the fields of a single agens point in S, and higher spaces. Let us call \1/V the first derivative of PV and \2/V the second; we can then break up 7 V into oa KTA OU St TP Py Ne ie se! a Ne pee \2/ V.dv a Keri Po V A Ee K: kn (n — 2) AET —t From the preceding follows immediately : and Ae bes En p Theorem 7. Each ,-,V has as potential a „ V. Each „4 V has p+ as potential a p V. ) We can indicate of the „+, V the elementary distribution, i.e. that Pp ; ae particular »4;V of which the arbitrary S, integral must be taken to P obtain the most general „+: V. P LoO7 . . For, the general p4,V is \2/ of the genera so it is the general S, integral of the ?/ of an isolated (p + 1)-dimensional vector, which, as is easily seen geometrically, consists of equal ?vectors in the surface of a Psphere with infinitesimal radius described round the point of the given isolated vector in the p41 of the vector. p+ l Ke P Loe ‘ In like manner the general,—1V is the \1/ of the general P—! V, (i) so it is the general S, integral of the \\/ of an isolated ?—! vector, consisting of equal Pvectors normal to the surface of an "~? sphere with infinitesimal radius described round the point of the given isolated vector in the A,~p+1, normal to that vector. From this follows: Theorem 8. The general ?V is an arbitrary integral of elemen- tary fields #, and 4£,, where Zale A en ene a ’ Ww ere iS) S Pao ) a EA ere » Z consists of the #—! vectors in the surface of an infinitesinal P-!sphere Sp, . . . . . . (A) Cavey p+l Bi jen h(n rr where , Y consists of the 7+! vectors normal n\n to the surface of an infinitesinal "~P—'sphere Spy. . . . (2) For the rest the fields #, and ZE, must be of a perfectly identical structure at finite distance from their origin; for two fields 4, and Z, with the same origin must be able to be summed up to an isolated Pvector in that point. We can call the spheres Sp, and Sp. with their indicatrices the elementary vortex systems Vo, and Vo;. A field is then uniformly determined by its elementary vortex systems and can be regarded as caused by those vortex systems. We shall now apply the theory to some examples. The force field in S,. The field E,. The elementary sphere Sp, becomes here two points lying quite close to each other, the vortex system Vo, passes into two equal and opposite scalar values placed in those two points. It : , COSp , furnishes a scalar potential ——— in which gp denotes the angle of the 7e radiusveetor with the S, of Vos, i.e. the line connecting the two points. The elementary field is the (first) derivative of the potential (the gradient); it is the field of an agens double point in two di- mensions. The field E,. The elementary sphere Sp, again consists of two points lying in close vicinity, the elementary vortex system Vo, has in those two points two equal and opposite planivectors. The plani- vector potential (determined by a scalar value) here again becomes cos ~ ; so the field itself is obtained by allowing all the vectors of r (75 ) a field EZ, to rotate 90°. As on the other hand it has to be of an identical structure-to /, outside the origin we may call the field KE, resp. H, “dual to itself”. In our space the field ZE, can be realized as that of a plane, infinitely long and narrow magnetic band with poles along the edges ; the field /, as that of two infinitely long parallel straight electric currents, close together and directed oppositely. The planivector (vortex) field in S,. The field E,. The elementary sphere Sp. is a circlet, the elementary vortex system Vo- a current along it. It furnishes a linevector sin f : : : potential aa vies directed along the circles which project themselves rT on the plane of Vo. as circles concentric to Vo, and where p is the angle of the radiusvector with the normal plane of Vo-. The field is the first derivative (rotation) of this potential. The field E,. The elementary sphere Sp, is again a circlet, the elementary vortex system Wo, assumes in the points of that circlet equal ‘vectors normal to it. The *V-potential consists of the *V’s normal to the potential vectors of a field /,; the field EZ, is thus obtained by taking the normal planes of all planivectors of a field £,. As on the other hand /, and ZE, are of the same identical structure outside the origin, we can say here again, that the field £, resp. Z, is dual to itself. So we can regard the vortex field in S, as caused by elementary circular currents of two kinds; two equal currents of a different kind cause vortex fields of equal structure, but one field is perfectly normal to the other. So if of a field the two generating systems of currents are identical, it consists of isosceles double-vortices. The force field in S,. The field HE. Vo. gives a double point, causing a scalar __» CSP potential oR where p is the angle of the radiusvector with the axis of the double point; the derivative (gradient) gives the wellknown field of an elementary magnet. The field E,. Vo, consists of equal planivectors normal to a small circular current. If we represent the planivector potential by the SU n = directed r linevector normal to it, we shall find for that linevector (76) along the circles, which project themselves on the plane of Vo, as circles concentric to Vos, and where p is the angle of the radius- vector with the normal on the circular current. The field ZE, is the second derivative of the planivector potential, i.e. the rotation of the normal linevector. According to what was derived before the field “, of a small circular current is outside the origin equal to the field £, of an elementary magnet normal to the current. In this way we have deduced the principle that an arbitrary force field can be regarded as generated by elementary magnets and elementary circuits. A finite continuous agglomeration of elementary magnets furnishes a system of finite magnets; a finite continuous agglomeration of elementary circuits furnishes a system of finite closed currents, i.e. of finite dimensions; the linear length of the separate currents may be infinite. Of course according to theorem 6 we can also construct the scalar potential out of that of single agens points the second derivative of the field), and the vector potential out of that of rectilinear 1 elements of current (perpendicular to TE < the first derivative of the big field), but the fictitious “field of a rectilinear element of current” has everywhere rotation, so it is the real field of a rather complicated distribution of current. A field having as its only current a rectilinear element of current, is not only physically but also mathematic- ally impossible. A field of a single agens point though physically perhaps equally impossible, is mathematically just possible in the Euclidean space in consequence of its infinite dimensions, as the field of a magnet of which one pole is removed at infinite distance. In hyperbolic space also the tield of a single agens point is possible for the same reason, but in elliptic and in spherical space being finite it has become as impossible as the field of a rectilinear element of current. The way in which ScuerinG (Göttinger Nachr. 1870, 1873; compare also Frusporr Diss. Göttingen 1873; Opitz Diss. Göttingen 1881) and Kiniine (Crelle’s Journ. 1885) construct the potential of elliptic space, starting from the supposition that as unity of field must be possible the field of a single agens point, leads to absurd consequences, to which Kiem (Vorlesungen über Nicht-Euklidische Geometrie) has referred, without, however, proposing an improvement. To construct the potential of the elliptic and spherical spaces nothing but the field of a double point must be assumed as unity of field, which would lead us too far in this paper but will be treated more in details in a following com- munication. With the foree field in S, the vortex field in S, dual to it has been treated at the same time. It is an integral of vortex fields as they run round the force lines of an elementary magnet and as they run round the induction lines of an elementary circuit. The force field in S, . The field E‚. Voz again gives a double point, which furnishes a __ C08 p scalar potential pean: where p is the angle between radiusvector and pn axis of the double point; its gradient gives what we might call the field of an elementary magnet in S,. The field E,. Vo, consists of equal planiveetors normal to a small *—?sphere Sp,. To find the planivector potential in a point P, we call the perpendicular to the S,—; in which Sp, is lying OL, and the plane LOP the “meridian plane” of P; we call p the angle LOP and OQ the perpendicular to OZ drawn in the meridian plane. We then see that all planivectors of Vo, have in common with that meridian plane the direction OZ, so they can be decomposed each into two components, one lying in the meridian plane and the other cutting that meridian plane at right angles. The latter components, when divided by the m— 2»¢ power of their distance to P, and placed in P, neutralize each other two by two; and the former consist of pairs of equal and opposite planivectors directed parallel to the meridian plane and at infinitely small distance from each other according to the direction OQ. These cause in P . : ae sin p : a planivector potential lying in the meridian plane =c——. The 5 = pT field ME, is of this potential the y = \?/, and outside the origin is identical to the field of an elementary magnet along OL. The force field in S, can be regarded as if caused 1st. by magnets, 2d, by vortex systems consisting of the plane vortices erected normal to a small ”—?sphere. We can also take as the cause the spheres themselves with their indicatrices and say that the field is formed by magnets and vortex spheres of m#—2 dimensions (as in S, the cause is found in the closed electric current instead of in the vortices round about it). Here also fields of a single plane vortex element are impossible. Yet we can speak of the fictitious “field of a single vortex” although (18) that really has a vortex i.e. a rotation vector everywhere in space. We can say namely: If of a force field in each point the divergence (a scalar) and the rotation (a planivector) are given, then it is the V of a potential: > div. dv rot. dv fk : ‘| NE + uf a this formula takes the field as an integral of fictitious fields of agens points and of single vortices. Crystallography. — “On the fatty esters of Cholesterol and Phytosterol, and on the andsotropous liquid phases of the Cholesterol-derwatives.” By Dr. F. M. Jancer. (Communicated by Prof. A. P. N. FRANCHIMONT.) (Communicated in the meeting of May 26, 1906). § 1. Several years ago I observed that phytosterol obtained from rape-seed-oil suffers an elevation of the melting point by a small addition of cholesterol. The small quantity of the first named sub- stance at my disposal and other circumstances prevented me from going further into the matter. My attention was again called to this subject by some very meritorious publications of BomER') on the meltingpoint-elevations of phytoterol by cholesterol and also of cholesterol-acetate by phyto- sterol-acetate. Apart from the fact that the crystallographic data from O. Miiccr led me to the conclusion, that there existed here an uninterrupted miscibility between heterosymmetric components, a further investigation of the binary meltingpoint-line of the two acetates appeared to me very desirable, as the ideas of Bömrr on this point are not always clear; this is all the more important, as we know that BomErR based on these melting point elevations a method for detecting the adulteration of animal with vegetable fats. My further object was to ascertain in how far the introduction of fatty acid-residues into the molecule of cholesterol would modify the behaviour of the esters in regard to the phenomenon of the optecally- anisotropous liquid phases, first noticed with the acetate, propionate and benzoate, with an increasing carbon-content of the acids. Finally I wished to ascertain whether there was question of a similar meltingpoint-elevation as with the acetates in the other terms of the series too. 1) Boer, Zeit. Nahr. u. Genussm. (1898), 21, 81; (1901). 865, 1070; the last paper (with Winter) contains a complete literature reference to which I refer. (79) $ 2. In the first place the esters of cholesterol and phytosterol had to be prepared. The cholesterol used, after being repeatedly recrystallised from absolute alcohol + ether, melted sharply at 149°.2. The phyto- sterol was prepared by MercK, by Hessk’s') method from Calabar fat, and also recrystallised. It melted at 137°. A microscopic test did not reveal in either specimen any inhomogeneous parts. First of all, I undertook the crystallographic investigation of the two substances. The result agrees completely with the data given by Müaere, to which I refer. I have not, up to the present, obtained any measu- rable crystals; on account of the optical properties, cholesterol can possess only triclinic, and phytosterol only monoclinic symmetry. Although an expert erystallographer will have no difficulty in microscopically distinguishing between the two substances, the crystals deposited from solvents are, however, so much alike that a less expe- rienced analyst may easily make a mistake. I, therefore, thought it of practical importance to find a better way for their identification with the microscope. This was found to be a very simple matter, if the crystals are allowed to form on the object-glass by fusion and solidification, instead of being deposited from solvents. Figs. 1 and 2 show the way in which the solidification of the two substances takes place. Fig. 1. Fig. 2. Cholesterol, Phytosierol, fused and then solidified. fused and solidified by cooling. Phytosterol crystallises in conglomerate spherolites. Between crossed nicols they exhibit a vivid display of colours and each of them is !) Hesse, Annal. der Chemie, 192, 175. (80) traversed by a dark cross, so that the whole conveys the impression of adjacent interference images of monaxial crystals, viewed perpen- dicularly to the axis and without circular polarisation. The charac- ter of the apparently simple crystals is optically negative. Cholesterol, however, presents a quite different image. When melted on an object-glass, the substance contracts and forms small droplets, which are scattered sporadically and, on solidification, look like little nug- gets with scaly edges, which mostly exhibit the white of the higher order. That the microscopical distinction in this manner is much safer than by Miiegr’s method, may be seen from fig. 3 where phytosterol and choleste- rol are represented as seen under the Fig. 3. microscope, after being crystallised Phytosterol and Cholesterol from from alcohol. A is cholesterol, B phy- 95%y Alcohol. tosterol. § 3. Of the fatty esters, I have prepared the acetates, propionates, butyrates and isobutyrates by heating the two alcohols with the pure acid-anhydride in a reflux apparatus. A two or three hours heating with a small flame, and in the case of the cholesterol, preferably in a dark room, gives a very good yield. When cold, the mass was freed from excess of acid by means of sodium hydrocarbonate, and then recrystallised from aleohol + ether, afterwards from ethyl acetate + ligroin, or a mixture of acetone and ligroin, until the melt- ingpoint was constant. Generally, I used equal parts by weight of the alcohol and the acidanhydride. The formiates, valerates, isovalerates, capronates, caprylates and caprinates were prepared by means of the pure anhydrous acids. These (valeric, caprylic and capric acids) were prepared synthetically by KanrBaum; the isovaleric acid and also the anhydrous formic acid were sold commercially as pure acids “KAHLBAUM’. Generally, a six hours heating of the alcohol with a little more than its own weight of the acid sufficed to obtain a fairly good yield. Owing, however, to the many recrystallisations required the loss in substance is much greater than with the above described method of preparing. Both series of esters crystallise well. The phytosterol-esters in soft, flexible, glittering scales; the formiate and the valerates present some difficulties in the crystallisation, as they obstinately retain a trace of ( 81 ) an adhesive by-product which it is difficult to remove. The choles- terol-esters give much nicer erystals; the formiate, acetate and ben- zoate have been measured macroscopically ; the other derivatives erystallise in delicate needles or very thin scaly crystals which are not - measurable; I hope yet to be able to obtain the butyrate in a measurable form *). In the case of the caprylate, the purification was much assisted by the great tendency of the product to erystallise, The purification of the capric ester was, however, much more diffi- cult; at last, this has also been obtained in a pure state even in beautiful, colourless, plate-shaped crystals, from boiling ligroïn *). The phytosterol-esters retain their white colour on exposure to the light; the cholesterol-esters gradually turn yellowish but may be bleached again by recrystallisation. The determination of the melting points, and in the case of the cholesterol-esters, also that of the transition-temperatures: solid — anisotropous-liquid, was always executed in such manner, that the thermometer was placed in the substance, whick entirely surrounded the mercury-reservoir. Not having at my disposal a thermostat, I have not used the graphic construction of the cooling-curve, in the determinations, but simply determined the temperature at which the new phases first occur when the outer bath gets gradually warmer. As regards the analysis of the esters, nothing or little can be learned from an elementary analysis in this case, where the formulae of cholesterol and phytosterol are still doubtful, and where the molecules contain from 28 to 37 carbon-atoms. I have therefore rested content with saponifying a small quantity of the esters with alcoholic potassium hydroxide, which each time liberated the cholesterol] or phytosterol with the known melting points. On acidifying the alkaline solution with hydrochloric acid, the fatty acids could be identified by their characteristic odour. The esters were called pure, when the melting points, and in the case of cholesterol-esters, both temperatures, remained constant on further recrystallisation. 1) I have even succeeded lately in obtaining the formiate in large transparent crystals from a mixture of ligroin, ethyl acetate and a little alcohol, 2) The crystals of the caprinate are long, flat needles. They form monoclinic individuals, which are elongated parallel to the b-axis, and flattened towards {001} The angle B is 88° a 89°; there are also the forms: {100} and ‘Ton: measured : (100) : (101) = + 20.°. The optic axial plane is {010}; inclined dispersion: p > v round the first bissectria. Negative double refraction. On {001} there is one optical axis visible about the limits of the field. The crystals are curved-planc. 6 Proceedings Royal Acad. Amsterdam. Vol. IX. (82) $ 4. I give in the following tables the temperatures observed ete.!) Next to my data are placed those of Bömrr as far as he has published them. The temperatures in [ | will be discussed more in detail later on. L FATTY ESTERS OF CHOLESTEROL. Chol. Formiate | — [+ 90°] 960,5 — 96°. | » Acetate — [BO A90} 1122.8 — | 14305 | » Propionate 93°:0 4079.2 oo SOENS | » -n-Butyrate 96°.4 | 107°.3 = 96° | 108° | » ITsobutyrate — — 1269.5 — — » -n=Valerate 91°.8 092 — — — » _ _Iso-valerate — [+ 1099) | 110°.6 — — » Capronate 910.2 100°.4 — — — » _ Caprylate — [+ 10409] | 10694 — — » _ Caprinte 820,2 90°.6 — — — » Benzoate 145°.5 17825 — 1469 | 178°.5 » _ Phtalate *) — — — = 1829.5 » Stearinate *) — — — 659 Benzoates and phthalates although not being fatty esters, have nevertheless been included. 1) According to ScnönBecK, Diss. Marburg. (1900). 2) According to Bömer loco cit. 3) According to Berruetor. It is as yet undecided, whether liquid crystals are | present here ; perhaps this case is analogous with that of the caprylate. | The temperatures in [] cannot be determined accurately; see text. § 5. Most striking with these remarkable substances are the splen- 1) It should be observed that in these substances three temperatures should be considered, namely 1. transition: solid — anisotropous-liquid ; 2. transition : aniso- tropous-liquid — isotropous-liquid ; 3. transition: solid — isotropous-liquid. This distinction has been retained, particularly on account of the cases of labile, liquid crystals discovered here. (83 ) did colour-phenomena observed during the cooling of the clear, isotropous, fused mass to its temperature of solidification, and also during the heating in the reverse way. These colour phenomena are caused by interference of the incident light, every time the turbid anisotropous liquid-phase occurs, or passes into the isotropous liquid. During this last transition we notice while stirring with the ther- mometer, the ‘‘oily slides” formerly described by Rerirzer, until the temperature ¢, has been exceeded. These colours also occur when the solid phase deposits from the anisotropous liquid, therefore below t,. The most brilliant, unrivalled violet and blue colour display is shown by the butyrate and normal valerate, also very well by the capronate and caprinate. The temperatures in | | ¢, answer to anisotropous liquid phases which are /abile in regard to the isotropous liquid, and which double- refracting liquids are, therefore, only realisable in undercooled fused material. Of this case, which is comparable with the monotropism, as distinguished by LEHMANN from the case of enantiotropous transfor- mations, the acetate is the only known example up to the present. Now the number of cases is increased by three, namely the formiate, the caprylate and without any doubt also the zsovalerate, to which I will refer presently. Cholesterol-formiate and caprylate melt therefore, perfectly sharply to a clear liquid at, respectively 961/,° and 106.°2. If, however, the clear liquid is suddenly cooled in cold water, one notices the appearance of the turbid, anisotropous, more-labile phase, accompanied by the said colour phenomena. The acetate in particular exhibits them with great splendour. It is quite possible that many organic compounds which are described as ‘melting sharply”, belong to this category and on being cooled suddenly possess a double-refracting liquid phase, even although this may last only a moment. The phenomenon of liquid crystals would then be more general than is usually believed. Prof. LEHMANN, to whom [ have forwarded a little of the cholesterol- esters, has been able to fully verify my observations. This investigator has, in addition, also found that cholesterol-caprinate may probably exhibit two anisotropous hquid phases. Although, personally, I never noticed more than one single phase, and Prof. LEHMANN’s determinations are only given provisionally, this case would certainly have to be regarded as one of the most remarkable phenomena which may be expected in a homogeneous body, particularly because the percep- tibility of those two phases implies that they would not be miscible in all proportions with each other. 6* SEL) § 6. The behaviour of cholesterol-isobutyrate is a very remarkable one. Microscopic and maeroscopic investigation shows absolutely nothing of an anisotropous liquid phase, not even on sudden cooling and this in spite of the fact that the normal butyrate gives the phenomenon with great splendour. This differently-behaving ester has been prepared from the same bulk of cholesterol as was used for preparing the other esters. The cause of the difference can, therefore, be found only in the structure of the fatty acid-residue, which contrary to that of the other esters, is branched. All this induced me, to prepare the analogous ester of isovalerie acid ; perhaps it might be shown also here that the branching of the earbon-chain of the acid destroys the phenomenon of the anisotro- pous liquid phase. At first I thought this was indeed the case, but a more accurate observation showed that in the rapid cooling there oceurs, if only for an indivisible moment, a /abile anisotropous liquid; the duration, however, is so short that, for a long time, I was in doubt whether this phase ought to be ealled stable or labile as in the ease of the formiate and caprylate! Even though the carbon- branching does not cause a total abrogation of the phenomenon of liquid crystals, the realisable traject appears to become so much smaller by that branching, that it almost approaches to zero, and the expected phase is, moreover, even still labile. From all this I think we may conclude, as has been stated more than once by otbers, that the occur- rence of the liquid phases is indeed a inherent property of the matter, which cannot be explained by the presence of foreign admix- tures etc. (TAMMANN C. 5.). § 7. We now give the melting points of the analogous phytosterol- esters which, with one exception, do not exhibit the phenomenon of the double-refracting liquids. As the phytosterols from different vege- table fats seem to differ from each other, and as BöMer does not mention any phytosterol esters from Calabar-fat in particular, I have indicated in the second column only the limits within which the melting points of the various esters prepared by him from diverse oils, vary. (See table following page.) From a comparison of the two tables it will be seen that the lowering of the melting point of phytosterol by the introduction of fatty acid- residues of increasing carbon-content, takes place much more rapidly than with cholesterol. On the other hand, the succession of the melting points of the acetate, propionate, butyrate and n-valerate is more regular than with the cholesterol-derivates. All phytosterol-esters share with phytosterol itself the great ten- (85 ) I]. FATTY ESTERS OF PHYTOSTEROL. | | | Limits according to BoMER: Phytosterol-Formiate | 1409 103°—113° Phytosterol-Acetate | 1290.1 123°—135° | Bintosterdl-Propionate | 105°.5 | 404°—116° | Phytosterol-Butyrate | 91°.2 | 859 90° | Phytostercl-Isobutyrate AG hee — Phytostervl-norm.-Valerate bn Odat ===" 0° == | | Phytosterol-Isovalerate 100°.1 == | dency to crystallise from the melted mass in sphaerolites; with an increasing carbon-content of the fatty acid-residue, these seem gene- rally to become smaller in circumference. The formiate crystallises particularly beautifully; this substance possesses, moreover, two solid modifications, as has been also stated by Prof. LeHMANN, who is of opinion that these two correspond with the two solid phases of the cholesterol-derivative. In the phytosterol- ester the sphaerolite-form is the more-labile one. On the other hand, when recrystallised from monobromonaphthalene or almond-oil, they form under the microscope well-formed needle- shaped crystals which, however, are always minute. Probably, we are dealing in all these cases with polymorphism. I have also often observed whimsical groroths and dendritics. A difficulty occurred in the determination of the melting point of the normal valerate. It melts, over a range of temperature at about 67°.1, but if the mass is allowed to cool until solidified, the ester fuses to a clear liquid when heated to 30°. This behaviour is quite analogous to that observed with a few glycerides of the higher fatty acids, for instance with P'rilaurin and Trimyristin by ScHEY. *) After half an hour the melting point had risen again to 53’/,° and after 24 hours to 67°. After 24 hours, small white sphaerolites had deposited in the previously coherent, scaly and slightly double-refrac- ting layer on the object glass, which exhibited the dark cross of the phytosterol. In order to explain this phenomenon, I think I must assume a dimorphism of the solid substance. Moreover, liquid crystals are formed here, as has also been observed by Prof. LEHMANN, 1) Scuey. Dissertatie, Leiden (1899) p. 51, 54, ( 86 ) According to Prof. LEHMANN, normal phytosterol-valerate forms very beautiful liquid erystals, which are analogous to those of chole- sterol-oleate ; like these they are not formed until the fused mass is undercooled. Consequently, the anisotropous liquid phase is here also labile in regard to the isotropous one. I do not think it at all improbable that the changes in the melting points observed by Scury with his higher tryglicerides also owe their origin to the occurrence of labile, double-refracting liquid phases. A further investigation is certainly desirable. § 8. We now arrive at the discussion of the mutual behaviour of both series of fatty esters in regard to each other. It has been sufficiently proved by Bömer that the meltingpoint- line of cholesterol and of phytosterol is a rising line. In connection with Müecer’s and my own crystal determinations we should have here indeed a gradual mixing between heterosymmetrie components ! In mixtures which contain about 3 parts of cholesterol to 1 part of phytosterol, the microscopical research appears to point to a new solid phase, which seems to erystallise in trigonal prisms. This com- pound (?) also oecurs with a larger proportion of cholesterol '). Whether we must conclude that there is a miscibility of this new kind of erystal with both components, or whether an eventual transformation in the solid mixing phases proceeds so slowly that a transition point in the meltingpoint-line escapes observation, cannot be decided at present. The matter is of more interest with the esters of both substances. According to Bémmr?) the formiates give a meltingpoint-line with a eutectic point; the acetates, however, a continuously rising melting point-line. The method of experimenting and the theoretical interpretation is, however, rather ambiguous, as BöMmrr prepares mixed solutions of the components, allows these to erystallise and determines the melt- ingpoint of the solid phase first deposited. By his statement of the proportion of the components in the solution used, he also gives an incomplete and confusing idea of the connection between the melting? point and the concentration. Although a rising of the binary meltingpoint-line may, of course, be ascertained in this manner quite as well as by other means — and Bömer’s merit certainly lies in the discovery of the fact 1) Compare Bömer, Z. f. Nahr. u. Gen. M. (1901) 546. 2) Boner, Z. f. Nahe u. Gen. Mitt. (1901) 1070. In connection with the aide: phism of the formiates, a mixing series with a blank is however very probable in this case. ( 87 ) itself — the determination of the binary meltingpoint-line must be reckoned faulty as soon as it is to render quantitative services, which is of importance for the analysis of butter; for if the meltingpoint- curve is accurately known, the quantity of phytosterol added may be calculated from the elevation of the melting point of the cholesterol acetate. [ have, therefore, now determined the binary melting point line in the proper manner. (Fig. 4). 700 _ 90 80 70.3% 60 50 42.4 30 20 10 0 Fig. 4. Cholesterol-, and Phytosterol-Acetate. Although the curve takes an upward course it still deviates con- siderably from the straight line which connects the two melting- points. As the course of the curve from 40 °/, cholesterol-acetate to 0°/, is nearly horizontal, it follows that the composition of mixtures can be verified by the melting-point, when the admixture of phytosterol in the animal fat does not exceed 60°/,. The results are the most accurate when the quantity of phytosterol-ester’) amounts to 2°/,—40°/,. In practice, this method is therefore applicable in most cases. The cholesterol-acetate used in these experiments melted at 112.°8; the phytosterol-acetate at 129.°2. A mixture of 90 °/, Chol. Acet. + 10 °/, Phyt. Acet. melts at 117° » » » 80 » » » +20 » » » » » 120.°5 » » » 733» » » Je 26.7» » » » » 4122.°5 » » 160» » » +40 » » » » » 1950 » » » 424» » » + 576» » D ». » 428° » » » 20 » » » +80 » » » De Di 120004 » » » 10 » » » +90 » » » » » 129,°9 1) It should be observed that although Bömer, in several parts of his paper, recommends the said method for qualitative purposes only, it is plain enough in other parts that he considers the process suitable for quantitative determinations in the case of small concentrations. In his interpretation of the melting point line this is, however not the case, for his experiments give no explanation as to the mixing proportion of the components in mixtures of definite observed melting point. Quantitative determinations are only rendered possible by a complete knowledge of the binary melting point line. When the concentration of cholesterol-acetate is 0,5 — 1°/,, the meltingpoint is practically not altered; when it is 20/, however, the amount is easy to determine. ( 88 ) Probably, a case of isomorphotropous relation occurs here with the acetates; both esters are, probably, monoclinic, although this is not quite certain for the cholesterol-ester. This is pseudotetragonal and according to Von ZrPHArovicH: monoclinic, with 8 = 73°38'; according to OBERMAYER: triclinic, with /?=106°17', a=90°20', y=90°6', while the axial relations are 1,85: 1: 1,75. The phytosterol-ester has been approximately measured microsco- pically by BrykrrcH and seems to possess a monoclinic or at least a triclinie symmetry with monoclinic limit-value. In my opinion both compounds are certainly „ot isomorphous. In any case it might be possible that even though a direct isomorphism does not exist in the two ester-series, there are other terms which exhibit isomor- photropous miscibility in an analogous manner, as found for the ace- tates by Bömer. I have extended the research so as to include the isovalerates ; the result however is negative and the case of the acetic esters seems to be the only one in this series. The following instance may be quoted : 31.8°/, cholesterol-butyrate + 68,2°/, phytosterol-butyrate indicate for ¢, 81° and for ¢, 83° etc. etc. For the formiates, the lowering had been already observed by Boer ; other esters, also those of the iso-acids behave in an analogous manner: at both sides of the melting-diagram occurs a lowering of the initial melting points. It is, however, highly probable that in some, perhaps in all cases, there exists an isod¢morphotropous mixing with a blank in the series of the mixed erystals. The anisotropous liquid phase of cholesterol-esters gives rise in this case to anisotropous liquid mixed crystals. I just wish to observe that for some of the lower-melting esters, such as the butyrate, capronate, caprinate, normal valerate, etc., the temperature ¢, for these mixed crystals may be brought to about 50° or 60° or lower and this creates an opportunity for studying liquid mixed erystals at such tempera- tures, which greatly facilitates microscopical experiments. In all probability, I shall shortly undertake such a study of these substances. Of theoretical importance is also the possibility, to which Prof. Baknuuts RoozeBoom called my attention, that in those substances where ¢, answers to the more-labile condition, the at first more labile liquid mixed crystals, on being mixed with a foreign substance, become, finally, stable in regard to the isotropous fused mass. Expe- riments with these preparations, in this sense, will be undertaken elsewhere. Perhaps, a study of the low-melting derivatives or else a similar study of the low-melting liquid mixed crystals by means of the wltra-microscope might yield something of importance. Zaandam, May 1906. ( 89 ) Physics. — “Researches on the thermic and electric conductivity + power of crystallised conductors.” 1. By Dr. F. M. JarGer. (Communicated by Prof. H. A. Lorentz). (Communicated in the meeting of May 26, 1906). 1. Of late years, it has been attempted from various sides to find, by theoretical means, a connection between the phenomena of the thermic and electric conductivity of metallic conductors, and this with the aid of the more and more advancing electron theory. In 1900 papers were published successively by P. Drupn’), J. J. Tomson *) and HE. Riecke*) and last year by H. A. Lorentz *). One of the remarkable results of these researches is this, that the said theory has brought to light that the quotient of the electric and thermic conductivity power of all metals, independent of their particular chemical nature, is a constant, directly proportional to the absolute temperature. When we assume that the electrons in such a metal can move freely with a velocity depending on the temperature, such as happens with the molecules in ideal gases and also that these electrons only strike against the much heavier metallic atoms, so that in other words, their mutual collision is neglected, whilst both kinds of particles are considered as perfectly elastic globes, the quotient of the thermic conductivity power 4 and the electric conductivity power 6 may be indeed represented by a constant, proportional to the absolute temperature 7’. The theories of Drupr and Lorentz only differ as to the ab- ; : 7 a \? solute value of the quotient; according to Drumr — = a) a O e À OMEN ; according to Lorentz — = aen T. In these expressions 4, 6 and 9) e é T have the above cited meaning, whilst « is a constant and e represents the electric charge of the electron. By means of a method originated by KoHLRAUSCH, JAEGER and DiesseLHorst have determined experimentally the values for — with 5 1) P. Drupe, Ann. Phys. (1900). 1. 566; 3. 369. 2, J. J. Tomson, Rapport du Congrès de physique Paris (1900). 3. 138. 3) E. Rrecke, Ann. Phys. Chem. (1898). 66. 353, 545, 1199; Ann. Phys, (1900). 2. 835. *) H. A. Lorentz, Proc. 1905, Vol. VII, p. 438, 585, 684. ( 90 ) various metals *). The agreement between theory and observation is in most cases quite satisfactory, only here and there, as in the case of bismuth ®, the difference is more considerable. From their meas- urements for silver at 15°, the value 47 DX 10° may be deduced in Id Al = 7 4 2 . € . C.G.S. units, for the expression — . (Compare Lorentz, loco cit. é p. 505); according to Drupe's formula: 38 > 10°. § 2. I hope, shortly, to furnish an experimental contribution towards these theories by means of a series of determinations of an analogous character, but more in particular with crystallised con- ductors, and in the different directions of those erystal-phases. If we take the most common case in which may be traced three mutual perpendicular, thermic and electric main directions in such erystals, the propounded theories render it fairly probable for all such conducting crystals that: A: = — = —, and therefore also: A: 2, : A= G; : Oy : Oz: Garm: In conducting crystals, the directions of a greater electric con- ductivity should, therefore, not only be those of a greater thermic conductivity, but, theoretically, the quotient of the electric main- conductivities should be numerically equal to that of the thermic main-conductivities. Up to the present but little is known of such data. The best investigated case is that of a sliglitly titaniferous Haemitate of 1) W. Jarcer und Diessernorsr, Berl. Sitz. Ber. (1899). 719 etc. Comp. Remneanuw, Ann. Phys. (1900) 2, 398. 2) With Al, Cu, Ag, Ni, Zn, the value of at 18° varies between 636 x 105 and 699 108; with Cd, Pb, Sn, Pt, Pd between 706 X 108 and 754 >< 108; with Fe between 802 and 832105, therefore already more. With bismuth Di at 18° — 962 Xx 108. Whilst in the case of the other metals mentioned the values of * at 100° and at 18° are in the average proportion of 1,3:1, with bismuth the proportion is only 1.12. In their experiments, Jaeger and Diesskrnorsr employed little rods, and bearing in mind the great tendency of bismuth to crystallise, their results with this metal cannot be taken as quite decisive, as the values of the electric and thermic conductivity power in the chief directions of crystallised bis- muth differ very considerably. ( 91 ) Swedish origin which has been investigated by H. BAckstrém and K. ANGsTRÖM *) as to its thermic and electric conductivity power. In this ditrigonal mineral, they found for the quotient of the thermic conductivity power in the direction of the chief axis (c) and in that perpendicular to it (a) at 50°: fo 4.19. de For the quotient of the electric resistances w at the same tempe- rature they found: jn 1.78, and, therefore: dts Dis: Wa Oc From this it follows that in the case of the said conductor, the theory agrees with the observations as to the relation between the conductivity powers only qualitatively, but not quantitatively, and — contrary to the usually occurring deviations — the proportion of the quantities 2 is smaller than that of the quantities o. JANNETTAZ’S empirical rule, according to which the conductivity for heat in crystals is greatest parallel to the directions of the more complete planes of cleavage, applies here only in so far as haematite which does not possess a distinct plane of cleavage, may still be separated best along the base {111} (Miner), that is to say parallel to the plane of the directions indicated above with a. § 3. In order to enrich somewhat our knowledge in this respect the plan was conceived to investigate in a series of determinations the thermic and electric conductivity-power of some higher and also of some lower-symmetrical crystalline conductors, and, if possible, of metals also. For the moment, I intend to determine the quotient of the conductivities in the different main directions, and afterwards perhaps to measure those conductivities themselves in an absolute degree. I. On the thermic and electric conductivities in crystallised Bismuth and in Haematite. Measurements of the thermic and electric conductivity of bismuth are already known. Marrevccr *) determined. the thermic conductivity, by the well- 1, H. Bäcksrröm and K. Anastrém, Ofvers. K. Vetensk. Akad. Férh. (1888). No. 8, 533; BAcxsrrés ibid. (1894), No. 10, 545. 2) Marrrvcer, Ann. Chim. et Phys. (3). 43. 467. (1855). (92) well-known method of INGeNHovsz, by measurement of the length of the melted off waxy layer which was put on the surface of cylindrical rods of bismuth, eut // and | to the main axis, whilst the one end was plunged into mercury heated at 150°. For the average value of the quotient of the main conductivities — perpen- dicular and normal to the main axis — he found the value 1,08. JANNPTTAZ’S rule applies in this case, because the complete cleavability of ditrigonal bismuth takes place along {111} (Mirrer), therefore, perpendicularly to the main axis. JANNETTAZ') has applied the SÉNARMONT method to bismuth. He states that in bismuth the ellipses have a great eccentricity but he did not take, however, exact measurements. A short time ago, Lownps*) has again applied the SÉNARMONT method to bismuth. He finds for the quotient of the demi-ellipsoidal axes 1.19 and, therefore for the quotient of the conduetivities 1.42. The last research is from Prrror’). By the SENARMonT method he finds as the axial quotient of the ellipses about 1.17 and conse- quently for the quotient of the conductivities | and // axis 1.368, which agrees fairly well with the figure found by Lownpbs. Secondly, Prrrot determined the said quotient by a method proposed by C. Soret, which had been previously recommended by Trourer®), namely, by measuring the time which elapses between the moments when two substances with known melting points 9, and #, placed at a given distance at different sides of a block of the substance under examination begin to melt. As indices were used; a-Naphtyla- mine (9 = 50° C.), o-Nitroaniline (9 == 66° C.), and Naphthalene ==) As the mean of all the observations, Perrot finds as the quotient of the main conductivities 1,3683, which agrees perfectly with his result obtained by SÉNARMONT's method. He, however, rightly observes that this concordance between the two results is quite an accidental one, and that the method of THOULET and Soret must not be considered to hold in all cases. The proof thereof has been given by CatrLer in a theoretical. paper ;°) the agreement is caused here by the accidental sma// value of a quotient ul 7 in which / represents the thickness of the little plate of bismuth 1) Jannerraz, Ann. de chim. phys. 29. 59. (1873). 2) L. Lownps, Phil. Magaz. V. 152. (1903). 3) L. Perrot, Archiv. d. Science phys. et nat. Généve (1904. (4). 18, 445. t) Trovrer, Ann. de Chim. Phys. (5). 26. 261, (1882) 5) C. Carrer, Archiv. de Scienc. phys. et nat. Genève (1904). (4). 18, 457. ( 93 ) and h and & the coefficients of external and internal conductivity. § 4. 1 have endeavoured to determine the quotient of the chief conductivities by the method proposed by W. Vorer. As is wellknown, this method is based on the measurement of the angle, formed by the two isotherms at the line of demarcation between two little plates which have been joined to an artificial twin, when the heat current proceeds along the line of demarcation. If 4, and 2, are the two chief conductivities of a plate of bismuth cut parallel to the crystallographic main axis, and if the angle which the two main directions form with the line of demarcation equals 45°, then according to a former formula’): À, Cheer é == 19 4) — Tome 2 § 5. The bismuth used was kindly furnished to me by Dr. F. L. Perror, to whom I again wish to express my hearty thanks. The prism investigated by me is the one which Dr. Prrror in his publications?) indicates with J/, and for which, according to a es iJ Aa A ‘ = SénaRMoNT’s method, he found for — the value 1,390. The prism °C given to Dr. van EvERDINGEN yielded in the same manner for ha a the value 1,408. Cc Two plates were cut parallel to the crystallographic axis, in two directions forming an angle of 90° and these were joined to twin plates with g = 45°. It soon appeared that in this case the Vorer method *) was attended by special difficulties which, as Prof. Voicr informed me, is generally the case with metals. First of all, it is difficult to find a coherent coating of elaidie acid + wax; generally the fused mixture on the polished surface forms droplets instead of congealing to an even layer. Secondly, the isotherms are generally curved and their form presents all kinds of irregularities, which are most likely caused by the great specific conductivity of the metals, in connection with the peculiarity just mentioned. On the advice of Prof. Vorer I first covered the metallic surface with a very thin coating of varnish ; this dissolves in the fused acid, and causes in many cases a better cohesion, but even this plan did not yield very good results. 1) These Proceedings. (1906). March p. 797. 2) p. 4, note 10. 3) Vorer, Géltinger Nachr. (1896). Heft 3, p. 1—16; ibid. (1897). Heft 2. 1—5 ( 94 ) However, at last, I succeeded in getting a satisfactory coating of the surface by substituting for white wax the ordinary, yellow bees-wax. This contains an adhesive substance probably derived from the honey, and, when mixed in the proper proportion with elaidie acid it yields the desired surface coating. I have also coated ') the bottoms of the plate and the sides, except those which stand _L on the line of demarcation with a thick layer of varnish mixed with mercury iodide and copper iodide. During the operation the heating was continued to incipient darkening (about 70’). The plates should have a rectangular or square form, as otherwise the isotherms generally become curved. It is further essential to heat rapidly and to raise the copper bolt to a fairly high temperature; the isotherms then possess a more straight form and give more constant values for «. I executed the measurements on the double object table of a LEHMANN’s crystallisation microscope on an object glass wrapped in thick washleather, to prevent the too rapid cooling and solidification of the coating. After numerous failures, I succeeded at last in obtaining a long series of constant values. As the mean of 30 observations, I found es — 22°12’ and therefore: P| —" — 1,489. 2 c § 6. The value now found is somewhat greater than that found by Prrror. I thought it would be interesting to find out in how far a similar deviation was present in other cases, and whether when compared with the results obtained by the methods of SÉNARMONT, JANETTAZ and ROENTGEN, it has always the same direction. In fact, the investigation of many minerals has shown me that all values obtained previously, are smadler than those obtained by the process described here. I was inclined at first to believe that these differences were still greater than those which are communicated here. Although a more extended research, including some plates kindly lent to me by Prof. Voier, showed that these differences are not so serious as I suspected, at first the deviation exists always im the same direction. For instance, I measured the angle e of a plate of an Apatite- crystal from Stillup in Tyrol and found this to be 17°. From the 1) Ricwarz’s method of experimenting (Naturw. Rundschau, 17, 478 (1902)) did not give sufficiently sharply defined isotherms and was therefore not applied. (95 ) de position of the isotherms it also follows that 2, > 2, so that ==. 1 “0 In a quartz-plate obtained from Prof. Vorer I found ¢ = 303°, | p 2 Ac ; : : ; therefore a 1,75. In a plate of Antimonite from Skikoku in Japan “a cut parallel to the plane {010}, : was found to be even much larger “a than 1,74, which value is deduced from the experiments of SENARMONT and JANNETTAZ as they find for the quotient of the demi ellipsoidal axes 1.32. For Apatite they find similarly 1,08, for quartz 1,73, whilst TucuscuMipT determined the heat-conductivity of the latter mineral according to WeBer’s method in absolute degree. His experiments . he give the value 1,646 for the quotient - ; “a The deviations are always such that if 2, >4, the values of the quotient * turn out to be larger when Voiet’s method is employed 3 instead that of DE SÉNARMONT. The method employed here is, however, so sound in principle, and is so much less liable to experimental errors, that it certainly deserves the preference over the other processes. Finally, a sample of Haematite from Elba was examined as to its conducting power. A plate eut parallel to the c-axis was found not to be homogeneous and to contain gas-bubbles. I repeatedly measured the angles ¢ of a beautifully polished preparation of Prof. Voier, and found fairly constantly 10°, whilst the position of the isotherms showed that 2: was again larger than 2, : } a For the Haematite we thus obtain the value: = 1,202. The ef value found by Bäcksrröm and ANGsrRÖM for their mineral with the aid of CHRISTIANSEN’s method was 1,12. In this case the deviation also occurs in the above sense. From the experiments communicated we find for the quotient 2. ne A A “a:% in both erystal phases, if by this is meant {—] : | —] the ; O Ja Cn), values : With Bismuth: eo 1,428. With Haematite : i 1,480. Xo ha ; In this my measurements of — are combined witb the best value c ( 96 ) 6, found by vaN EverDINGEN') with Prrror’s prism, namely — —1,68, Cc and with the value found by the Swedish investigators for haematite: WTSat 50 GC: 7. If there were a perfect concordance between theory and 3 7 Ka ; observation, we should have in beth cases — =1. The said values Xe 1,128 and 1,480 are, therefore, in a certain sense a measure for the extent of the divergence between the observation and the con- clusion which is rendered probable by the electron theory. In the first place it will be observed that the agreement is much better with bismuth than, with haematite. However, this may be expected if we consider that the theory has been proposed, in the first instance, for metallic conductors. The influence of the peculiar nature of the omide when compared with the true metal is shown very plainly in this case. The question may be raised whether, perhaps, there may be shown to exist some connection between the crystal structure and the chemical nature on one side, and the given values of “on the Cc other side. Such a connection would have some significance because it may be, probably, a guide for the detection of special factors situated in the crystalline structure, which stand in the way of a complete agrement of electron theory and observations. § 8. First of all, it must be observed that we are easily led to compare the structures of the two phases. Both substances inves- tigated erystallise ditrigonally and have an analogous axial quotient; for bismuth: a:c = 1:1,3035 (G. Rose); for haematite a:¢ = 1:1,3654 (Merrczer). In both substances, the habit is that of the rhomboid, which in each of them approaches very closely to the regular hexahedron. The characteristic angle @ is 87°34’ for bismuth for haematite 85°42’. Particularly in bismuth the pseudo-cubic construction is very distinct; the pianes of complete cleavage which answer the forms {111} and {144} approach by their combination the regular octahedron in a high degree. Although haematite. does not 1) van Everpincen, Archives Néerland. (1901) 371; Versl. Akad. v. Wet. (1895— 1900); Comm. Phys. Lab. Leiden, 19, 26, 37, 40 and 61. See Archiv. Néerl. p. 452; rods No. 1 and No. 5. (97) possess a perfect plane of cleavage, it may be cleaved in any case along {111} with testaceous plane of separation. It admits of no doubt that the elementary parallelepipeds of the two crystal structures are in both phases pseudo-cubic rhombohedral configurations and the question then rises in what proportion are the molecular dimensions of those cells in both crystals ? If, in all erystal-phases, we imagine the whole space divided into volume-units in such a manner that each of those, everywhere joined, mutually congruent, for instance cubic elements, just contains a single chemical molecule, it then follows that in different crystals the size of those volume elements is proportionate to a in which M represents the molecular weight of the substances and d the sp.gr. of the crystals. If, now, in each crystal phase the content of the elementary cells of the structure is supposed to be equal to this equivalent-volume = the dimensions of those cells will be reduced for all crystals to a same length unit, namely all to the length of a cubic-side belonging to the volume-element of a crystal phase, whose density is expressed by the same number as its molecular M weight ; then in that particular case a = 1: If we now calculate the dimensions of such an elementary parallelopiped of a Bravats ; M ; structure whose content equals the quotient a and whose sides are in proportion to the crystal parameters a@:6:c, the dimensions ¥%, w and w thus found will be the so-called topic parameters of the phase which, after having been introduced by Broker and MuTHMAnn independently of each other, have already rendered great services in the mutual comparison of chemically-different crystal-phases. In the particular case, that the elementary cells of the crystal-structure possess a rhombohedral form, as is the case with ditrigonal crystals, the parameters 4, Wp and w become equal to each other (= @). The relations applying in this case are MIRC SIN — 2 Oe ; ‚ with sin ——=——. sim? a.sin A 2 sin a If now these calculations are executed with the values holding here: Bi= 207,5; Fe,O, = 159,64; dp; = 9,851 (Perror); dre, 0, = 4,98, then | Proceedings Royal Acad. Amsterdam. Vol. IX. ( 98 ) Vg; = 21,064 and Vie, O03 — 32,06, and with the aid of the given relations and the values for @ and A we find for each phase: *) On; 2,7641 Oro, 811853 If now we just compare these values for the sides of the rhom- bohedral elementary cells of the crystal structure with those of the : daan ; quotients — in the two phases, they curiously enough show the Xe following relation: x x (=) : (“") =/00 se 807 == 4s32. He] Fes O3 ee) Bi Fa Bi Allowing for experimental errors, the agreement is all that can be desired: in the first term of the equation the value is exactly: 1.312, in the last term: 1,328. x In our case the quotient — may therefore be written for both Ke phases in the form: C.9*, in which C is a constant independent of the particular chemical nature of the phase. Instead of the relation 0,°: 0,7, perhaps 0,” sin a, : 0,” sin a, = 1.305 is still more satisfactory. These expressions, however, represent nothing else but the surface of the elementary mazes of the three chief planes of the trigonal molecule structure, for these are in our case squares whose flat axis =a. The quotient > in the two C phases should then be directly proportional to the reticular density of the main net-planes of Bravais’s structures. A choice between this and the above conception cannot yet be made, because a, and a, differ too little from 90°. Moreover, a further investigation of other crystals will show whether we have to do here with something more than a mere accidental agreement. Similar investigations also with lower-symmetric conductors are at this moment in process and will, I hope, be shortly the subject of further communications. Zaandam, May 1906. 1) For bismuth «= 87°:34' and A=87°40': for haematite u —=85°42' and A=86°0'. The angle A is the supplement of the right angle on the polar axes of the rhombohedral cells and z is the flat angle enclosed between the polar axes. (99) Chemistry. — “Three-phaselines in chloralalcoholate and aniline- hydrochloride’. By Prof. H. W. Baknuis Rooznsoom. It is now 20 years since the study of the dissociation pheno- mena of various solid compounds of water and gases enabled me to find experimentally the peculiar form of that three-phaseline which shows the connection between temperature and pressure for binary mixtures in which occurs a solid compound in presence of solution and vapour, The general significance of that line was deduced, thermodynamically, by vaN per Waars and the frequency of its occurrence was proved afterwards by the study of many other systems. That this three-phaseline is so frequently noticed in practice in the study of dissociable compounds is due to the circumstance that, in the majority of the most commonly occurring cases, the volatility of the two components or of one of them, is so small, that at the least dissociation of the compound both liquid and vapour occur in its presence. In the later investigations, which have led to a more complete survey of the many equilibria which are possible between solid liquid and gaseous phases, pressure measurements have been somewhat discarded. When, however, the survey as to the connec- tion of all these equilibria in binary mixtures got more and more completed and could be shown in a representation in space on three axes of concentration, temperature and pressure, the want was felt to determine for some equilibria, theoretically and also experi- mentally, the connection between temperature and pressure, in order to fill up the existing voids. Of late, the course and the connection of several p,f-lines, have been again studied by van per Waats, Smits and myself either qualitatively or qualitative-quantitatively. To the lines, which formerly had hardly been studied, belonged the equilibria lines which are followed, when, with a constant volume, the compound is exposed to change of temperature in presence of vapour only. They can be readily determined experimentally only when the volatility of the least volatile component is not too small. STORTENBEKER at One time made an attempt at this in his investigation of the compounds of iodine with chlorine, but did not succeed in obtaining satisfactory data. In the second place it was desirable to find some experimental confirmation for the peculiar form of the three-phaseline of a compound, recently deduced by Sirs for the case in which a Vh ( 100 ) minimum occurs in the pressure of the liquid mixtures of its components. Mr. Lrororp has now succeeded in giving experimental contributions in regard to both questions, by means of a series of very accurately conducted researches where chloralalcoholate and anilinehydrochloride occur as solid compounds. Solid compounds which yield two perceptibly volatile components (such as PCI, NH,.H,S, PH,.HCl, CO,.2 NH, ete.) have been investi- gated previously, but either merely as to their condition of dissociation in the gaseous form, or as to the equilibrium of solid in presence of gaseous mixtures of different concentration at constant temperature; but liquids occur only at higher pressures, so that the course of the three-phase lines had never been studied. These two compounds were selected because in their melting points neither temperature nor pressure were too high. Moreover, the diffe- rence in volatility of the two components in the first example (chloral + alcohol) was much smaller than in the second (aniline + hydrogen chloride). It was also safe to conclude from the data of both com- pounds that the liquid mixtures of their components would show a minimum pressure. ONE EE Zl. AET | | ME \ EES ARN SSE Fe ' En > EENS AR INT PSN Pee RN ae Geeta) ( 101 ) This last point was ascertained first of all by a determination of the boiling point lines, in which a maximum must occur. In both cases this was found to exist and to be situated at the side of the least volatile component, respectively chloral or aniline. The investigation of the three-phase lines showed first of all that these possess the expected form in which two maxima and one minimum of pressure occur. | In the first system (Fig 1) CFD is the three- -phase line, T and T, are the respective maxima for the vapour pressure of solutions with excess of either alcohol or chloral and saturated with chloralalco- holate; the minimum is situated very close to the melting point F. In the second system (anilinehydrochloride Fig. 2) the first maxi- mum, in presence of excess of HCI is situated at such an elevated REE ZADEN EAN LEER pies: | ET ( 102 ) pressure that this has not been determined, the second T, at a moderate pressure is situated at the side of the aniline. The minimum T, is situated at the same side and is removed further from the melting point than in Fig. 1. 7, minimum F' melting point p a oar elt 22.5 cM. t 7 199°2 The determination of these lines and also that of the equilibria- lines for compound + vapour or liquid + vapour which also occur in both figures can only take place on either side of point /’, for in measuring the pressures, we can only have in the apparatus a larger, or smaller, excess of either component. Moreover, it is possible to fill the apparatus with the compound in a dry and pure condition. In the case of the compounds employed, this was attained by preparing very pure crystals by repeated sublimation in vacuo. In the second example, the sublimation line ZG of aniline hydro- chloride was thus determined. On this line then follows the piece GF of the three-phase line, because beyond G, no vapour can exist which has the same composition as the compound, except in the presence of some excess of HCl, so that a little liquid is formed with a slight excess of aniline. If, however, the apparatus is properly filled with the compound so that there remains but little space for the vapour then he three-phase line G may be traced to very near the melting point /, where one passes on to the line F4, for the equilibrium of the fused compound with its vapour. We have here, therefore, the first experimental confirmation of the normal succession of the p,¢-lines when those are determined with a pure compound which dissociates more or less. Theoretically, the minimum 7, in the three-phaseline must be situated at the left of the terminal point G of the sublimationline. The difference here, although small, is yet perfectly distinct: qe G p 16 cM. 16.5 cM. ¢t 497° 198° In the case of chloralaleoholate the points 7, and G both coincide so nearly with # that this point is practically undistinguishable from the triple point of a non-dissociating compound, both Z/ and FA, or their metastable prolongation A’ appear to intersect in /”. Moreover, the investigation of the melting point line proved that chloralalco- holate in a melted condition is but little dissociated. ( 103 ) In both compounds the p,f-lines have also been determined with excess of chloral or aniline. A very small quantity of these suffices to cause the occurrence of liquid in presence of the compound at temperatures far below the melting point and we then move on the lowest branch of the three-phaseline. In the case of a slight excess of chloral (Fig. 1) this was followed from D over 7, to F, just a little below the melting point, and from there one passed on to the liquid-vapour line /',A,, which was situated a little above /’A. In the case of a slight excess of aniline the piece DT,7,GF, could be similarly followed (Fig. 2). In this occurred the minimum 7’, whilst the piece GH, coincided entirely with the corresponding part of GF, which had already been determined in the experiment with the pure compound. Just below / the compound had disappeared entirely and one passed on to the liquid-vapour line #,A4,, which, unlike that in Fig. 1, was situated below #4. If the excess of the component is very trifling, liquid is formed only at higher temperatures of the three-phaseline, and below this temperature a sublimationline is determined, with excess of the component in the vapour, which line must, therefore, be situated higher than the pure sublimationline. With chloralaleoholate a similar line BE (Fig 1) was determined, situated decidedly above LI’. At L, liquid occurred and a portion of the three-phaseline ZF was followed up to a point situated so closely to / that the liquid-vapourline, which was then followed, was situated scarcely above FA. The excess of chloral was, therefore, exceedingly small, but in spite of this, LH was situated distinctly above L/’. The position of BE depends, in a large measure, on the gas-volume above the solid compound, as this determines the extra pressure of the excess of the component, which is totally contained in the same; so long as no liquid occurs. It appeared, in fact, to be an extremely difficult matter to prepare chloralalcoholate in such a state of purity that it exhibited the lowest imaginable sublimationline LF, which meets the three-phaseline in #. Similar sublimation lines may also occur with mixtures containing excess: of alcohol. But also in this case, even with a very small excess of alcohol we shall retain liquid even at low temperatures and, therefore, obtain branch C7'F of the three-phaseline. Such hap- pens, for instance, always when we use crystals of the compound which have been crystallised from excess of alcohol. They then contain sufficient mother-liquor. ( 104 ) We then notice the peculiar phenomenon that the compound is apparently quite solid till close to the melting point and we find for the vapour pressure the curve CTF, whilst the superfused liquid gives the vapour pressureline A, which is situated much lower. Ramsay has found this previously without being able to give an explanation, as the situation of the three-phaseline was unknown at that period. In the case of anilinehydrochloride, it was not difficult, on account of the great volatility of HCl, to determine sublimationlines when an excess of this component was present. In Fig. 2 two such lines are determined BE and BE. From £, the three-phaseline was followed over the piece /,H, afterwards the liquid-vapourline #7, /,. From JL also successively HH and H/. With a still smailer excess of hydrogen chloride we should have stopped even nearer to /” on the three-phaseline. In the ease of chloralaleoholate we noticed also the phenomenon that a solid substance which dissociates after fusion may, when heated not too slowly, be heated above its meltingpoint, a case lately observed by Day and ALLEN on melting complex silicates, but which had also been noticed with the simply constituted chloralhydrate. An instance of the third type of a three-phaseline where the maximum and minimum have disappeared in the lower branch of the three-phase line has not been noticed as yet. The two types now found will, however, be noticed frequently with other dissociable compounds , such as those mentioned above, and therefore enable us to better understand the general behaviour of such substances. Physics. — “On the polarisation of Röntgen rays.’ By Prof. H. Haca. In vol. 204 of the Phil. Trans. Royal Soc. of London p. 467, 1905 BarkLA communicates experiments which he considers as a decisive proof that the rays emitted by a Réyrcen bulb are partially polarised, in agreement with a prediction of BLonpLor founded upon the way in which these rays are generated. In these experiments BARKLA examined the secondary rays emitted by air or by some solids: paper, aluminium, copper, tin, by means of the rate of discharge of electroscopes. In two directions perpen- dicular to one another and both of them perpendicular to the direction of the primary rays, he found a maximum and a minimum for the action of the secondary rays emitted by air, paper and aluminium. ( 105 ) The difference between the maximum and minimum amounted to about 20°/,. I had tried to examine the same question by asomewhat different method. A pencil of RéxTcEN rays passed through a tube in the direction of its axis, without touching the wall of the tube. A photo- graphic film, bent cylindrically, covered the inner wall of the tube in order to investigate whether the secondary rays emitied by the air particles showed a greater action in one direction than in another. I obtained a negative result and communicated this fact to BARKIA, who advised me to take carbon as a very strong radiator for secon- dary rays. I then made the following arrangement. ed —_ | | Let S, (fig. 1) be the front side of a thiek-walled leaden box, in which the RörrceN bulb is placed; S, and S, brass plates 10 « 10 e.m. large and 4 m.m. thick. Their distance is 15 c.m. and they are immovably fastened to the upper side of an iron beam. In the middle of these plates apertures of 12 m.m. diameter were made. A metal cylinder A is fastened to the back side of S,; a brass tube B provided with two rings A, and A, slides into it *). An ebonite disk / in which a carbon bar is fastened fits in tube B. This bar is 6 c.m. long and has a diameter of 14 m.m. At one end it has been turned off conically over a length of 2 ¢.m. 1) Fig. 1 and 2 are drawn at about half their real size. ( 106 ) The aperture in S, was closed by a disk of black paper; the back side of A was closed by a metal cover, which might be screwed off. The dimensions were chosen in such a way, that the boundary of the beam of RÖNTGEN rays, which passed through the apertures in S,, S,and S,, lay between the outer side of the carbon bar and the inner side of the tube B. The photographie film covering the inside of B was therefore protected against the direct RÖNTGEN rays. If we accept Barkua’s supposition on the way in which the secondary beams are generated in bodies of small atomic weight, and if the axis of the primary beam perfectly coincided with that of the carbon bar, then a total or partial polarisation of the RÖNTGEN rays would give rise to two maxima of photographic action on diametrically opposite parts of the film and between them two minima would be found. From the direction of the axis of the cathode rays the place of these maxima and minima might be deduced. A very easy method proved to exist for testing whether the primary beam passed symmetrically through the tube B or not. If namely the inner surface of cover D was coated by a photographic plate or film, which therefore is perpendicular to the axis of the carbon bar then we see after developing a sharply defined bright ring between the dark images of the carbon bar and of the ebonite disk. This ring could also be observed on the fluorescent screen — but in this case of course as a dark one, and the Rénrern bulb could easily be placed in such a way, that this ring was concentric with the images of the carbon bar and of the ebonite disk. This ring proved to be due to the rays that diverged from the anticathode but did not pass through the carbon bar perfectly parallel to the axis and left it again on the sides; these rays proved to be incapable of penetrating the ebonite, but were totally absorbed by this substance; when the ebonite disk was replaced by a carbon one, then the ring disappeared; it is therefore a very interesting instance of the selective absorption of RÖNrGeEN rays’). When in this way the symmetrical passage of the RÖNTGEN rays had been obtained, then the two maxima and minima never appeared, neither with short nor with long duration of the experiment, though a strong photographic action was often perceptible on the film. Such an action could for instance already be observed after one hour’s exposure, if an induction-coil of 30 cm. striking distance was used with a turbine interruptor. A storage battery of 65 volts was used; 1) Take for this experiment the above described arrangement, but a carbon bar of 1 cm. diameter and 4 cm. long. ( 107 ) the current strength amounted to 7 amperes; the RÖNTGEN bulb was “soft”. Sometimes I obtained one maximum only or an irregular action on the film, but this was only the case with an asymmetrie position of the apparatus. From these experiments we may deduce: 1 that the primary RONTGEN rays are polarised at the utmost only to a very slight amount, and 2"¢ that possibly an asymmetry in the arrangement caused the maxima and minima observed in the experiments of BARKLA, who did not observe at the same time in two diametrical opposite directions. With nearly the same arrangement I repeated BARrKrA’s experiments on the polarisation of secondary rays, which he has shown also by means of eleetroscopes and described Proc. Roy. Soc. Series A vol. 17, p. 247, 1906. K /\ 7 fre. 2 qs Let the arrow (fig. 2) indicate the direction of incidence of the RÖNTGEN rays on the carbon plate A large 8X8 em. and thick 12 mm. The secondary rays emitted by this plate could pass through the brass tube G, which was fastened to S,. This tube was 6 em. long and on the frontside it was provided with a brass plate with an aperture of 5 mm. It was placed within the leaden case at 8 em. distance from the middle of the carbon plate; leaden screens protected the tube against the direct action of the primary rays. In these experiments the above mentioned induction-coil was used with a ( 108 ) WEHNELT interruptor; the voltage of the battery amounted to 65 Volts and the current to 7 Amperes. A very good photo was obtained in 30 hours and it shows very clearly two maxima and two minima, the distance between the centra of the maxima is exactly half the inner circumference of the tube, and it may be deduced from their position that they are caused by the tertiary rays emitted by the conic surface of the carbon bar. | In this experiment the centre of the anticathode, the axis of the carbon bar and the centre of the carbon plate lay in one horizontal plane, and the axis of the cathode rays was in one vertical plane with the centre of the carbon plate; the axes of the primary and the secondary beams were perpendicular to one another. According to BarkLA’s supposition we must expect that with this arrangement the maximum of the action of the tertiary rays will be found in the horizontal plane above mentioned. In my experiment this sup- position really proved to be confirmed. In order to know what part of the photographic film lay in this plane, a small side-tube # was adjusted to the outside of cylinder A, and this tube # was placed in an horizontal position during the experiment. A metal tube with a narrow axial hole fitted in tube /, so that in the dark room, after taking away a small caoutchouc stopper which closed F, I could prick a small hole in the film with a long needle through this metal tube and through small apertures in the walls of A and B. This hole was found exactly in the middle of one of the maxima. So this experiment confirms by a photographic method exactly what BArKLA had found by means of his electroscopes and it proves that the secondary rays emitted by the carbon are polarised. In some of his experiments BARKLA pointed out the close agreement in character of primary and secondary RÖNTGEN rays; in my experi- ments also this agreement was proved by the radiogram obtained on the film placed in cover D. Not only did the secondary rays act on the film after having passed through the carbon bar of 6 cm., but also the bright ring was clearly to be seen, which proves that ebonite absorbs all secondary rays which have passed through carbon *). The ring was not so sharply defined as in the experiments with primary rays; this fact finds a natural explanation in the different size of the sources of the radiation: in the case of the primary rays the source is a very small part of the anticathode, in the case of the secondary rays it is the rather large part of the carbon plate which emits rays through the apertures in G and $S,. 1) The ring was perfectly concentric: the arrangement proved therefore to be exactly symmetrical. ( 109 ) This agreement makes it already very probable that the RÖNTGEN rays also consist in transversal vibrations; these experiments however yield a firmer proof for this thesis. If namely we accept the suppo- sition of BarkLa as to the way of generation of secondary rays in bodies with a small atomic weight, then it may easily be shown, that the supposition of a /ongitudinal vibration of the primary R6nTGEN rays would, in the experiment discussed here, lead to a maximum action of the tertiary rays in a vertical plane and not in an hori- zontal plane, as was the case. Groningen, Physical Laboratory of the University. Chemistry. — “Zriformin (Glyceryl triformate)’. By Prof. P. van ROMBURGH. Many years ago I was engaged in studying the action of oxalic acid on glycerol’) and then showed that in the preparation of formic acid by Lorin’s method diformin is produced as an intermediate product. Even then I made efforts to prepare triformin, which seemed to me of some importance as it is the most simple representative of the fats, by heating the diformin with anhydrous oxalic acid, but I was not successful at the time. Afterwards Lorin’) repeated these last experiments with very large quantities of anhydrous oxalic acid and stated that the formic acid content finally rises to 75°/,, but he does not mention any successful efforts to isolate the triformin. Since my first investigations, I have not ceased efforts to gain my object. I confirmed Lorin’s statements that on using very large quantities of anhydrous oxalic acid, the formic acid content of the residue may be increased and I thought that the desired product might be obtained after all by a prolonged action. Repeated efforts have not, however, had the desired result, although a formin with a high formic acid content was produced from which could be obtained, by fractional distillation in vacuo, a triformin still containing a few percent of the di-compound. I will only mention a few series of experiments which I made at Buitenzorg, first with Dr. Nanninca and afterwards with Dr. Lone. In the first, a product was obtained which had a sp.gr. 1.309 at 25°, and gave on titration 76.6°/, of formic acid, whilst pure triformin requires 78.4°/,. The deficiency points to the presence of fully 10°/, of diformin in the product obtained. 1) Compt. Rend. 93 (1881) 847. 2) Compt. Rend. 100 (1885) 282. ( 110 ) In the other, the diformin, was treated daily, during a month, with a large quantity of anhydrous oxalie acid, but even then the result was not more favourable. The difficulty in preparing large quantities of perfectly anhydrous oxalic acid coupled with the fact that carbon monoxide is formed in the ‘reaction, which necessitates a formation of water from the formic acid, satisfactorily explains the fact that the reaction does not proceed in the manner desired. A complete separation of di- and triformin cannot be effected in vacuo as the boiling points of the two compounds differ but little. I, therefore, had recourse to the action of anhydrous formic acid on diformin. I prepared the anhydrous acid by distilling the strong acid- with sulphuric acid in vacuo and subsequent treatment with anhydrous copper sulphate. Even now I did not succeed in preparing the triformin in a perfectly pure condition, for on titration it always gave values indicating the presence of some 10°/, of diformin. Afterwards, when 100°/, formic acid bad become a cheap com- mercial product, I repeated these experiments on the larger scale, but, although the percentage of diformin decreased, a pure triformin was not obtained. 5 I had also tried repeatedly to obtain a crystallised product by refrigeration but in vain until at last, by cooling a formin with high formic acid content in liquetied ammonia for a long time, I was fortunate enough to notice a small crystal being formed in the very viscous mass. By allowing the temperature to rise gradually and stirring all the while with a glass rod, I succeeded in almost completely solidifying the contents of the tube. If now the crystals are drained at O° and pressed at low temperature between filter paper and if the said process is then repeated a few times, we obtain, finally, a perfectly colourless product melting at 18°, which on being titrated gave the amount of formic acid required by triformin. The sp. gr. of the fused product at 18° is 1.320. dh MR. 35.22; calculated 35.32. The pure product when once fused, solidifies on cooling with great difficulty unless it is inoculated with a trace of the crystallised substance. On rapid crystallisation needles are obtained, on slow crystallisation large compact crystals are formed. In vacuo it may be distilled unaltered; the boiling point is 163° at 38™™, On distillation at the ordinary pressure it is but very slightly decomposed. The boiling point is then 266°. A product contaminated EEE) with diformin, however, cannot be distilled under those circum- stances, but is decomposed with evolution of carbon monoxide and dioxide and formation of allyl formate. If triformin is heated slowly a decided evolution of gas is noticed at 210° but in order to prolong this, the temperature must rise gradually. The gas evolved consists of about equal volumes of carbon monoxide and dioxide. The distillate contains as chief product allyl formate, some formic acid, and further, small quantities of allyl alcohol. In the flask a little glycerol is left *). Triformin is but slowly saponified in the cold by water in which it is insoluble, but on warming saponification takes place rapidly. Ammonia acts with formation of glycerol and formamide. With amines, substituted formamides are formed, which fact | communi- cated previously *). The properties described show that triformin, the simplest fat, differs considerably in its properties from the triglycerol esters of ‚the higher fatty acids. Chemistry. — “On some derivatives of 1-3-5-hevatrienc’. By Prof. P. van RompurGcu and Mr. W. van DorsseEn. In the meeting of Dec. 30 1905 it was communicated that, by heating the diformate of s-divinylglycol we had succeeded, in pre- paring a hydrocarbon of the composition C,H, to which we gave the formula: CH, == CH — CH = CH CH=CH Since then, this hydrocarbon has been prepared in a somewhat larger quantity, and after repeated distillation over metallic sodium, 50 grams could be fractionated in a LADENBURG flask in an atmosphere of carbon dioxide. The main portion now boiled between 77°—78°.5 (corr.; pressure 764.4 mm.). Sp. griss 0.749 ND13.5 1.4884 Again, a small quantity of a product with a higher sp. gr. anda larger index of refraction could be isolated. 1) This decomposition of triformin has induced me to study the behaviour Of the formates of different glycols and polyhydric alcohols on heating. Investigations have been in progress for some time in my laboratory. 2) Meeting 30 Sept. 1905. (112) In the first place the action of bromine on the hydrocarbon was studied. If to the hydrocarbon previously diluted with chloroform we add drop by drop, while agitating vigorously with a WirT stirrer, a solution of bromine in the same solvent, the temperature being — 10°, the bromine is absorbed instantly and as soon as one molecule has been taken up the liquid turns yellow when more is added. If at that point the addition of bromine is stopped and the chloroform distilled off in vacuo, a crystalline product is left saturated with an oily substance. By subjecting it to pressure and by recrystallisation from petroleum ether of low boiling point, fine colourless crystals are obtained which melt sharply at 85°.5—86° 1). A bromine determination according to Linpic gave 66.84°/,, C,H,Br, requiring 66.65°/,. | A second bromine additive product, namely, a tetrabromide was obtained by the action of bromine in chloroform solution at 0° in sunlight; towards the end, the bromine is but slowly absorbed. The chloroform is removed by distillation in vacuo and the product formed is recrystallised from methyl alcohol. The melting point lies at 114°—115° and does not alter by recrystallisation. Analysis showed that four atoms of bromine had been absorbed. Found: Br: 80.20. Calculated for C,H,Br, 79.99. A third bromine additive product was found for the first time in the bromine which had been used in the preparation of the hydro- carbon to retain any hexatriene carried over by the escaping gases. Afterwards it was prepared by adding 3 mols of bromine to the hydrocarbon diluted with 1 vol. of chloroform at 0° and then heating the mixture at 60° for 8 hours. The reaction is then not quite com- pleted and a mixture is obtained of tetra- and hexabromide from which the latter can be obtained, by means of ethyl acetate, as a substance melting at 163°.5—164". Found: Br. 85.76. Calculated for C,H,Br, 85.71. On closer investigation, the dibromide appeared to be identical with a bromide obtained by GRrINER') from s. divinyl glycol with phosphorus tribromide; of which he gives the melting point as 84°.5—85°. A product prepared according to Griner melted at 85°.5—86° and caused no lowering of the meltingpoint when added to the dibromide of the hydrocarbon. GrINer obtained, by addition of bromine to the dibromide prepared from his glycol, a tetrabromide melting at 112° together with a 1) Not at 89° as stated erroneously in the previous communication. ( 113 ) product melting at 108°—109°, which he considers to be a geome- trical isomer. On preparing ®) the tetrabromide according to Griner the sole product obtained was that melting at 112°, which proved identical with the tetrabromine additive product prepared from the hydro- carbon, as described above. For a mixture of these two bromides exhibited the same meltingpoint as the two substances separately. Prolonged action of bromine on the tetrabromide according to Griner, yielded, finally, the hexabromide melting at 168°—164°, which is identical with the one prepared from the hydrocarbon. The bromine derivatives described coupled with the results of GRrINER prove that our hydrocarbon has indeed the formula given above. According to THIELE’s views on conjugated double bonds we might have expected from the addition of two atoms of bromine to our hexatriene the formation of a substance with the formula CH,Br — CH = CH —CH=CH—CH,Br. . . (1) or CH,Br — CH = CH — CHBr —CH=CH,. . . (2) from the first of which, on subsequent addition of two bromine atoms the following tetrabromide would be formed. CH,Br — CHBr — HC = CH — CHBr— CH,Br. . . (3) As, however, the dibromide obtained is identical with that prepared from s. divinyl glycol, to which, on account of its mode of formation, we must attribute the formula CH, = CH — CHBr— CHBr— CH=CH,. . . (4) (unless, what seems not improbable considering certain facts observed, a bromide of the formula (1) or (2) should have really formed by an intramolecular displacement of atoms) the rule of THLE would not apply in this case of two conjugated systems. Experiments to regenerate the glycol from ‘the dibromide have not as yet led to satisfactory results, so that the last word in this matter has not yet been said. The investigation, however, is being continued, Meanwhile, it seems remarkable that only the first molecule of bromine is readily absorbed by a substance like this hexatriene, which contains the double bond three times. By means of the method of SABATIER and SENDERENS, hexatriene may be readily made to combine with 6 atoms of hydrogen. If its ') Ann. chim. phys. [6] 26. (1892) 381. 2) Investigations on a larger scale will have to decide whether an isomer, melt- ing at 108°, really occurs there as a byproduct which then exerts but a very slight influence on the melting point of the other product. 2 8 Proceedings Royal Acad. Amsterdam. Vol. IX. ( 144 ) vapour mixed with hydrogen is passed at 125°—130° over nickel reduced to a low temperature, the hydrogen is eagerly absorbed and a product with a lower boiling point is obtained, which, however, contains small quantities of unsaturated compounds (perhaps also cyclic ones). In order to remove these, the product was treated with bromine and after removal of the excess and further purification it was fractionated. As a main fraction, there was obtained a liquid boiling at 68°.5—69°.5 at 759.7 mm. Sp.-gr.‚e —:0,6907 npe ded: Although the boiling point agrees with that of the expected hexane the sp. gr. and the refraction differ still too much from the values found for hexane by Brian and by Eyxkman’). Therefore, the product obtained from hexatriene was shaken for some time with fresh portions of fuming sulphuric acid until this was no longer coloured. After this treatment were obtained one fraction of B. p. 69°—70°, Sp. gr.,, 0.6718 np,, 1.388250. and another of B. p. 69°.7—70°5, Sp. gr, 0.6720, np,, 1.88239. An n-hexane prepared in the laboratory, according to Briiu *) by Mr. Scurrinca gave the following values B. p. 69°, Sp. gr, 0.664 np,, 1.8792 whilst an n-hexane prepared, from diallyl according to SABATIER and SENDERENS, by Mr. SINNIGE gave B. p. 68.5°—70,° Sp. gr.,, 0.6716, np, 1.388211. It is, therefore, evident that the hexane obtained by SABATIER’s, and SENDERENS process still contains very small traces of impurities. There cannot, however, exist any doubt that 1-3-5-hexatriene absorbs 6 atoms of hydrogen with formation of normal hexane. Of greater importance, however, for the knowledge of the new hydrocarbon is the reduction by means of sodium and absolute alcohol. If, as a rule, unsaturated hydrocarbons are not likely to take up hydrogen under these circumstances, it becomes a different matter when a conjugated system is present. Now, in 1-3-5-hexatriene, two conjugated systems are found and we might therefore expect the occurrence of a 2-4-hexadiene : CH,—CH=CH—CH=CH—CH, 1) Briint (B.B. 27, (1894) 1066) finds Sp. gr..9 = 0.6603, np) = 1.3784,; Eyxman (R. 14, (1881) 187) Sp.gr44 = 0.6652 np, = 1.37725. 2) Ann. 200. 183. ( 115 ) or, of a 2-5-hexadiene : CH,—CH=CH—-CH,—CH=CH,,. The first, still having a conjugated system can again absorb two atoms of hydrogen and then yield hexene 3. CH,—CH,—CH—CH—CH,—CH, whilst the other one cannot be hydrogenated any further '). The results obtained seem to point out that both reactions have indeed taken place simultaneously, and that the final product of the hydrogenation is a mixture of hexadiene with hexene. 10 grams of 1-3-5-hexatriene were treated with 100 grams of boiling absolute aleohol and 15 grams of metallic sodium. After the sodium had dissolved, a current of steam was passed, which caused the ready separation of tbe hydrocarbon formed, which, however, still contained some alcohol. After redistillation, the hydrocarbon was washed with water, dried over calcium chloride and distilled over metallic sodium. At 75°.5 it commenced to boil and the temperature then slowly rose to 81°. The liquid was collected in two fractions. fraction I. B.p. 75°.5—78°, Sp.gr.,, 0.73826 np,, = 1.4532 pellet On a: my — , = 1.4665 These fractions were again united and once more treated with sodium and alcohol. But after purification and drying no liquid of constant boiling point was obtained, for it now commenced to boil at 72°.5, the temperature rising to 80°. The main fraction now possessed the following constants : B.p. 72°.5—74°, Sp.gr.,, 0.7146 np,, 1.4205 The fraction 75°—-80 gave np, 1.4351. An elementary analysis of the fraction boiling at 72°.5—74° gave the following result: Found Calculated for C,H,, Calculated for C,H,, C 87.06 Oat 85.6 H 13.32 12.3 14.4 The fraction investigated consists, therefore, probably of a mixture of C,H,, and C,H,,. The quantity collected was insufficient to effect another separation. We hope to be able to repeat these experiments on a larger scale as soon as we shall have again at our disposal a liberal supply of the very costly primary material. Utrecht, Org. Chem. Lab. University. 4) If CH,=CH—CH,—CH,—CH=CH, should be formed, this will not readily absorb more hydrogen either. 8* ( 116 ) Mathematics. — “The force field of the non-Euchdean spaces with negative curvature’. By Mr. L. E. J. Brouwer. (Commu- nicated by Prof. D. J. Kortrwse). A. The hyperbolic Sp,. I. Let us suppose a rectangular system of coordinates to be placed thus that ds = V A? du? + B? dv? + C° dw?, and let us assume a line- vector distribution X with components X,, X,, X», then the integral of X along a closed curve is equal to that of the planivector Y over an arbitrary surface bounded by it; here the components of Y are determined by: 1. ONK 0(X, C poro OA) os aN as BC Ow Ov For, if we assume on the bounded surface curvilinear coordinates € and 4, with respect to which the boundary is convex, the surface integral is dv Ow Av dw\ /0(X,B) 0(XwC) fs: Se ee |) a) in 05 soa On TOS Ow Ov Joining in this relation the terms containing X, C and adding and Be tana OC) 100 A subtracting Se Sa ° dE . on we Obtain : ONO rd 10 (Xx, C)™ Ow NEG oa rene eal Integrating this partially, the first term with respect to n, the second to §, we shall get uk X,Cdw along the boundary, giving with the integrals JE X, Bdv and i: Nu A du analogous to them the line integral of X along the boundary. In accordance with the terminology given before (see Procee- dings of this Meeting p. 66—78)') we call the planivector Y the first derivative of X. 1) The method given there derived from the indicatrix of a convex boundary that for the bounded space by front-position of a point of the interior ; and the method understood by the vector Xpgr... a vector with indicatrix opqgr.... We can however determine the indicatrix of the bounded space also by post-position of a point of the interior with respect to the indicatrix of the boundary; and moreover assign to the vector Xpgr... the indicatrix pqr...o. We then find: (117 ) Analogously we find quite simply as second derivative the scalar: a a oe in mi Se According to the usual way of expressing, the first derivative is the rotation vector and the second the divergency. . ly. - : si . II. If X is to be a 2X, i.e. a second derivative of a planivector =, we must have: 1 =; Zw A= AE ed. _ 2 Fe O) a ‚ etc BC Ow and it is easy to see that for this is necessary and sufficient A— Ill. If X is to be a oX, ie. a first derivative (gradient) of a scalar distribution g, we must have: Og dp Op xX; = es nnee ; 2. — - —————— ds Cow Adu Bov and it is easy to see, that to this end it will be necessary and sufficient that r= 6: IV. It is easy to indicate (comp. Scurrine, Göttinger Nachrichten, 1870) the »X, of which the divergency is an isolated scalar value in the origin. It is directed according to the radius vector and is equal to: 1 sinh?r when we put the space constant — 1 *). OX 2 Nn Ip er De aa} 45 - q a SH Ayan yy p+l Tp P+ OX... a 1% Z Ee P g Aje ape pl Or, = = ay o%, These last definitions include the well known divergency of a vector, and the gradient of a potential also as regards the sign; hence in the following we shall start from it and we have taken from this the extension to non-Euclidean spaces. 2) For another space constant we have but to substitute in the following formulae r — for r. R (118 ) It is the first derivative of a scalar distribution : — 14 coth r, and has in the origin an isolated divergency of 42. V. In fature we shall suppose that X has the field property and shall understand by it, that it vanishes at infinity in such a manner that in the direction of the radius vector it becomes of lower order 1 fae Gt than — and in the direction perpendicular to the radius vector of Y fe lower order than e—. For a (X this means that it is derived from a scalar distribution, having the potential property, i.e. the property of vanishing at infinity. Now the theorem of Gruen holds for two scalar distributions (comp. Fresporr, diss. Göttingen, 1873): Ow ; op [oxwo-foy vides fw do — fw yp dr IE \grad. p‚ grad. W} i.) ; If now g and w both vanish at infinity whilst at the same time lim. y we? = 0, then the surface integrals disappear, when we apply the theorem of GREEN to a sphere with infinite radius and fe vpe fp opde integrated over the whole space, is left. Let us now take an arbitrary potential function for p and — 1 +eothr for w, where r represents the distance to a point P taken arbitrarily, then these functions will satisfy the conditions of vanishing at infinity and lim. g pe” = 0, so that we find: 4X. Pp ={- 1 + cothr) 7’ . dt. So, if we put — 1+ cothr=F(r), we have: — = 2/7 oX wey) We Gye ick oa VI. We now see that there is no vector distribution with the field property, which has in finite nowhere rotation and nowhere diver- gency. For, such a vector distribution would have to have a potential, having nowhere rotation, but that potential would have to be every- where 0 according to the formula, so also its derived vector. ( 119 ) From this ensues: a vector field is determined uniformly by its rotation and its divergency. VII. So, if we can indicate elementary distributions of divergency and of rotation, the corresponding vector fields are elementary fields, i. e. the arbitrary vector field is an arbitrary space-integral of such fields. For such elementary fields we find thus analogously as in a Euclidean space (l.c. p. 74 seq.): 1. a field #,, of which the second derivative consists of two equal and opposite scalar values, close to each other. 2. a field £#,, of which the first derivative consists of equal planivectors in the points of a small circular current and perpendicular to that same current. At finite distance from their origin the fields #, and ZE, are here again of the same identical structure. ; VIII. To indicate the field 4, we take a system of spherical coordinates and the double point in the origin along the axis of the system. Then the field /, is the derivative of a potential: cos p It can be regarded as the sum of two fictitious “fields of a single agenspoint”, formed as a derivative of a potential — 1 + cothr, which have however in reality still complementary agens at infinity. IX. The field ZE, of a small circular current lying in the equator plane in the origin is outside the origin identical to the above field /,. Every line of force however, is now a closed vector circuit with a line integral of 4 along itself. We shall find of this field E, a planivector potential, lying in the meridian plane and independent of the azimuth. In order to find this in a point P with a radius vector r and spherical polar distance g we have but to divide the total current between the meridian plane of P and a following meridian plane with difference of azimuth d9, passing between P and the positive axis of revolution, by the element of the parallel circle through P over dd. For, if ds is an arbitrary line element through P in the meridian plane making with the direction of force an angle fF, if dh is the element of the parallel circle, 2 the above mentioned current and H the vector potential under consideration, we find: dE = dh. Xds sin F, ( 120 ) whilst the condition for H is: d (Hdh) = dh ds X sin F. 2 So we have but to take for H. ah To find = we integrate the current of force within the meridian zone through the spherical surface through ?. The force component perpen- coshr dicular to that spherical surface is 2 cos p Pat therefore : sinh r _coshr sf" cos p . sinhr dp . sinhr sin p dd = dd coth r . sin? p. sin h° if So: = 2 ba coshr dh sinhr sin pdd — sinh?r Ry | | | n &. X. From this ensues, that if two arbitrary vectors of strength unity are given in different points along whose connecting line we apply a third coshr vector = , the volume product of these three vectors, i.e. the Sin Ne ihe volume of the parallelepipedon having these vectors as edges taken with proper sign, represents the linevector potential according to the first (second) vector, caused by an elementary magnet with moment unity according to the second (first) vector. To find that volume product, we have first to transfer the two given vectors to a selfsame point of their connecting line, each one parallel to itself, i.e. in the plane which it determines with that connecting line, along which the transference is done, and maintaining the same angle with that connecting line. The volume product y(S,,S,) is a symmetric function of the two vectors unity of which we know that with integration of S, along a closed curve s, it represents the current of force of a magnet unity according to S, through s,, in other words the negative reciprocal energy of a magnet unity in the direction of S, and a magnetic scale with intensity unity within s,, in other words the force in the direction of S, by a magnetic scale with intensity unity within s,, in other words the force in the direction of S, by a current with intensity unity along s,. So we can regard y (S,, S,) as a force in the direction of S, by an element of current unity in aie direction of S,. With this we have found for the force of an element of current with intensity unity in the origin in the direction of the axis of the system of coordinates : (121 ) coshr —— sing, sinh? r directed perpendicular to the meridian plane. XI. For the fictitious field of an element of current (having mean- while everywhere current, i. e. rotation) introduced in this way we shall find a linevector potential V, everywhere “parallel” (see above under § X) to the element of current and the scalar value of which is a function of 7 only. Let us call that scalar value U, and let us regard a small elemen- tary rectangle in the meridian plane bounded by radii vectores from the origin and by circles round the origin, then the line integral of V round that rectangle is: 0 0 — antl om gy sinh rv dg} dr — a {U cos dr} dg. This must be equal to the current of force through the small rectangle: coshr , ——— sing. sinh r dg. dr, sinh?r from which we derive the following differential equation of U with respect to r: - U — a U sinh r} = coth r, the solution of which is: U = cosech r — } rsech? Sr + c. sech? br. Let us take c= 0, we shall then find as vector potential V of an element of current unity /: cosech r — 4 rsech? r= F, (r), directed parallel to EF. Let us now apply in an arbitrary point of space a vector G, then the vector V has the property that, when integrated in G along an elementary circuit whose plane is perpendicular to G, it indicates the force in the direction of G, caused by the element of current HE, or likewise the vector potential in the direction of WZ caused by an elementary magnet with intensity unity in the direction of G. - So, if we call of two vectors unity Wand F the potential x (Z, F), the symmetrie function F, (7, cos gy, where 7 represents the distance of the points of application of the two vectors and p their angle after parallel transference to a selfsame point of their connecting line, we know that this function x gives, by integration of e.g. # over (424) a closed curve e not only the negative energy of a magnetic scale with intensity unity bounded by e in the field of an element of current unity £, but also the component along F' of the vector potential caused by a current unity along e. From this ensues for the vector V of an element of current, that when the element of current is integrated to a closed current it becomes the vector potential of that current determined uniformly on account of its flux property. So really the vector potential of a ie Le. of a field of currents is obtained as an integral of the vectors V of the elements of current. XII. We can now write that in an arbitrary point: sl ay [WE pee, gut a ee where we first transfer in a parallel manner the vector elements of the integral to the point under consideration and then sum up. Let us now consider an arbitrary force field as if caused by its two derivatives (the magnets and currents), we can then represent to our- selves, that both derivatives, propagating themselves according to a function of the distance vanishing at infinity, generate the potential of the field. The field X is namely the total derivative of the potential : ae (r) dr + [roe An An The extinguishment of the scalar potential is greater than that of the vector potential ; for, the former becomes at great distances of order e—?", the latter of order re”. Farther the latter proves not to decrease continuously from o to O, but at the outset it passes quickly through O to negative, it then reaches a negative maximum and then according to an extinguishment ve—" it tends as a negative (1. e. directed oppositely to the generating element of current) vector to zero. XIII. The particularity found in Euclidean spaces, that 1 FE, (r) = F, (r) = —, is founded upon this, that in Euclidean spaces Ue 2 the operation of twice total derivation is found to be alike for scalar distributions and vector distributions of any dimensions (le. p. 70). Not so in non-Euclidean spaces; e.g. in the hyperbolic Sp, we find for the Y? of a scalar distribution w in an arbitrary point = ( 123 ) when choosing that point as centre of a system of RIEMANN normal coordinates V dx? + dy? + dz? x + y? a 2? (i e. a system such that ds = 1 vAn 0’ du du aa (5 1 Oy? + a) but as WV? of a vector distribution with components X, Y and Z, we find for the z-component Xyz: ME „=| the te ea) The hyperbolic Sp,. I. As first derivative Y of a vector distribution X we find a planivector determined by a scalar value: into Oz, A): 00GB) AB| ow ou ' As second derivative Z we find the scalar : 1 wee 5 Q(X, A) ) AB Ou Ov Fle 1 ° ° . : Il. If X is to be a 2X, i. e. a second derivative of a planivector with scalar value w we must have: Ow dw =e ; Kn Bov Adu to which end is necessary and sufficient: 7 = 0. DF 7 1 . - . . If X is to be a oX, i.e. a first derivative of a scalar p we must have: dy dy Ee ; XxX, = — i Adu Bov to which end is necessary and sufficient: Y= 0. N= EAP A HI. The 9X, of which the divergency is an isolated scalar value in the origin, becomes a vector distribution in the direction of the radius vector: 1 sinh r It is the first derivative of a sealar distribution : Leoth 4 r. ( 124 ) The divergency in the origin of this field is 27. The scalar distribution /coth}r has thus the potential property. (This was not the case for the field of a single agens point in the Euclidean Sp,). IV. In the following we presuppose again for the given vector distribution the field property (which remains equally defined for 2 and for 7 dimensions as for 3 dimensions); no vector field is possible that has nowhere rotation and nowhere divergency; so each vector field is determined by its rotation and its divergency and we have first of all for a gradient distribution : 7 0X Kay f mame ir dt, eee) | Xx x= Vf SPO. de BE V. For the field Z, of an agens double point we find the gradient of the potential : cos p ‘sinh TP It can be broken up into “fields of a single agens point’ formed as a derivative of a potential /coth } r. VI. Identical outside the origin to the above field Z, is the field E, of a double point of rotation, whose axis is perpendicular to the axis of the agens double point of the field £,. For that field Z, we find as scalar value of the planivector potential in a point P the total current of force between P and the axis of the agens double point, that is: sin p coth r. So if are given a vector unity V and a scalar unity S and if we apply along their connecting line a vector cothr, the volume product w of V, S and the vector along the connecting line is the scalar value of the planivector potential in S by a magnet unity in the direction of V. Of w we know that when summing up S out of a positive scalar unity S, and a negative S, it represents the current of force of a magnet unity in the direction of V passing between S, and S,, in other words the negative reciprocal energy of a magnet unity in the direction of V and a magnetic strip S, S, with intensity unity, in (125 ) other words the force in the direction of V by a couple of rotation S, =S, So we can regard w as the force in the direction of V by an isolated rotation in S. So that we must take as fictitious “force field of an element of rotation unity” coth r, directed perpendicularly to the radius vector. In reality, however, this force field has rotation everywhere in Sp, VII. Let us now find the scalar value U, function of r, which we must assign to a planivector potential, that the “field of an element of rotation unity” be its second derivative. We must have: dU — — == coth r. dr U = leosech r. And we find for an arbitrary 2X: i 1 \1/ 9X Py ay, l cosech r dr, 2 aly x= Wf Le AO dee ee CE) j 2n And an arbitrary vector field X is the total derivative of the potential \2/X 1/ X ih aa +f oe VIII. We may now wonder that here in Sp, we do not find F, and F, to be identical, as the two derivatives and the two potentials of a vectordistribution are perfectly dually related to each other in the hyperbolic Sp, as well as in the Euclidean Sp,. The difference, however, is in the principle of the field property, which postulates a vanishing at infinity for the scalar potential, not for the planivector potential; and from the preceding the latter appears not to vanish, so with the postulation of the field property the duality is broken. But on the other hand that postulation in Sp, lacks the reasonable basis which it possesses in spaces of more dimensions. For, when putting it we remember the condition that the total energy of a field may not become infinite. As soon as we have in the infinity of Sp» forces of order er, this furnishes in a spherical layer with thickness dr and infinite radius described round the origin as centre an energy of order e?r X &—lr dr = e—3)r dr; which for n > 3 would F, (r) dr. ( 126 ) give when integrated with respect to 7 an infinite energy at infinity of Spr. So for n Ze 3 are excluded hy the field property a vector distri- butions which ‘cannot have physical meaning. For n=2 however the postulation lacks its right of existence; more sense has the condition (equivalent for m > 2 to the field pro- perty) that for given rotation and divergeney the vector distribution must have a minimum energy. Under these conditions we shall once more consider the field and we shall find back there too the duality with regard to both derivatives and both potentials. IX. Let us consider first of all distributions with divergeney only and let us find the potential function giving a minimum energy for given \7’. We consider the hyperbolical Sp, as a conform “representation of a part of a Euclidean Sp, bounded by a circle; if we then apply in corresponding points of the representation the same potential, we retain equal energies and equal divergencies in corresponding plane elements. So the problem runs: Which potential gives within a given curve (in this case a circle) in the Euclidean Sp, under given divergency distribution a minimum energy ? According to the theorem of GREEN we have for this: 0 5 0.8 fez (G)ar= (=x. 2 ede fw. = 40 — fn gt bude, Òz so that, as V?du is 0 Hede within the boundary curve, the necessary and sufficient condition for the vanishing of the variation of the energy is: u =O, along the boundary curve. For the general vector distribution with divergency only in the hyperbolical Sp, we thus find under the condition of minimum energy also, that the potential at infinity must be 0. So we find it, just as under the postulation of the field property, composed of fields £,, cos p sinhr The lines of force of this field Z, have the equation, derived from a potential sin p coth r — ¢. Only a part of the lines of force (in the Euclidean plane all of them) form a loop; the other pass into infinity. None of the equi- potential lines, however, pass into infinity; they are closed and are all enclosed by the circle at infinity as the line of O-potential. (1275) » . 1 ° . » The same holds for the arbitrary oX; of the lines of force one part goes to infinity; the potential lines however are closed. X. If we now have to find the field with rotation only, giving for given rotation distribution a minimum energy, it follows from a consideration of the rotation as divergency of the normal vector, that the scalar value of the planivector potential at infinity must be 0, and the general eas composed of fields “/,, derived from a planivector Sp potential ——— (whilst we found under the postulation of the field sinh r property sin gp cothr). In contrast to higher hyperbolical spaces and to any Euclidean and elliptic spaces the fields Z, and M, cannot be summed up here to a single isolated vector. . . . . 1 a . . For this field #, and likewise for the arbitrary »X the lines of force (at the same time planivector potential lines) are clused curves. XI. We have now found cA. vf MW Lcoth & r dr, AD ax — 5 df Lcoth 4 r dr. : nT And from this ensues that also the general vector distribution X having under given rotation and divergency a minimum energy is equal to: es en CAPS EE DS Xdiv. + Xrot. = \V/ Jk a= L coth 4 r dv + \2/ { = lcoth 3 r dt. For, if V is an arbitrary distribution without divergency and without rotation in finite, it is derived from a scalar potential function, so it has (according to § VIII) no reciprocal energy with Aq; neither (as according to $IX all lines of force of X,;, are closed curves and a flux of exclusively closed vector tubes has no _ reciprocal energy with a gradient distribution) with X,.:,; so that the energy of Xaiv. + Xro.+ V is larger than that of Xa. + Xroz- So finally we have for the general vector distribution of minimum energy X: X pg vf = .lcoth 4 r dr. ( 128 ) C. The hyperbolic Spy. I. Let us suppose a system of rectangular coordinates, so that i= WA u,” Ines and let us suppose a linevector distribution X with components X,...X,, then the integral of X along a closed curve is equal to that of a planiveetor Y over an arbitrary surface bounded by it, in which the components of Y are determined by: Tyee ete) 0 (Xz, Ax, ) yes i: ; Y is the first derivative or rotation of X. Further the starting vector current of YX over a closed curved Spr—1 is equal to the integral of the scalar 7 over the bounded volume of that Sp,—1; here Y; Aya Dos Dir en se er n AS Tas Ay Or Z is the second derivative or divergency of X. => bee on 7 z U. If X is to be a 2X, i.e. a second derivative of a planivector =, we must have: DEE Ayo A Me) 91 7 1 12 In—1 KE A ee ee . AT 0x FE: a n The necessary and sufficient condition for this is: Z=V,; » Zee 1 Ne . . . a If YX is to be a oX, i.e. a first derivative of a scalar gy, we must have: L oy Ne —., A, 0@x The necessary and sufficient condition for this is: yi | Il. The 5X, which has as divergency an isolated scalar value in the origin (comp. Opitz., Diss. Göttingen, 1881), is directed along the radius vector, and if we put the space constant equal to 1 is equal to f sinhn—| r° It is the first derivative of a scalar distribution (129) a dr Nt sinh! p n (7); 5 and it has in the origin an isolated divergency of &, (if h,7"—! ex- presses the spherical surface of the Huclidean space Spr). IV. For two scalar distributions g and w the theorem of GREEN holds (comp. Opitz., Le.): a) fox - dO,-1 — [PpV"* wb . dt, = fw » 20,4 — fw Vp. dt Ov Ov (= SV oe SNE im) If at infinity g and w both become O whilst at the same time im pw err 0, then for an ”~'sphere with infinite radius the surface integrals dis- appear and we have left fy 4 Vw ; inn = [0 E ip PA integrated over the whole space. If here we take an arbitrary potential function for p and wy, (r) for yw, where r represents the distance to an arbitrarily chosen point P — these functions satisfying together the conditions of the formula — we have: kn Pp = fn (en prend If thus we postulate for the vector distributions under consideration the field property (which remains defined just as for Sp,) we have, if we put w,(r)— (7), for an arbitrary PRE EK ay [LE roar sie Tie ae eC from which we deduce (compare A $ VI) that there is no vector field which has in finite nowhere rotation nor divergency; so that a vector field is uniformly determined by its rotation and its divergency. V. So a vectorfield is an arbitrary integral of: 1. Fields 2, of which the second derivative consists of two equal and opposite scalar values close to each other. 2. Fields #,, of which the first derivative consists of planivectors distributed regularly in the points of a small "~*sphere and perpen- dicular to that ”—®sphere. 9 Proceedings Royal Acad. Amsterdam. Vol, IX. ( 130 ) At finite distance from their origin the fields Z, and £, are of identical structure. VI. In order to indicate the field Z, we assume a spherical system of coordinates') and the double point in the origin along the first axis of the system. Then the field ZE, is the derivative of a potential : COS & sinhn—1 The lines of force of this field run in the meridian plane. It can be regarded as the sum of two fictitious “fields of a single agens- point” constructed as derivative of a potential w,(7) to which, however, must be assigned still complementary agens at infinity. VU. The field MZ, of a small vortex--—2sphere according to the space perpendicular to the axis of the double point just considered is identical outside the origin to the field #,. Each line of force is now however a closed vector tube with a line integral /, along itself. We shall find for this field Z, a planivector potential H, lying in the meridian plane and dependent only on randg. It appears then simply that this Hf is a xe Let «¢ be an (n—2)-dimensional element in the n—2 coordinates existing besides 7 and g, then it defines for each rand p an element on the surface of an *—?-sphere of a size dh = ce sinh*—?r sin™—g, and for the entire Sp, what may be called a “meridian zone”. We then obtain for the current of force 2, passing inside a meridian zone between the axis of the system and a point P with coordinates 7 and g, if ds represents an arbitrary line element through P in the meridian plane under an angle f with the direction of force : d= =dh.X ds sin F, whilst we can easily find as necessary and sufficient condition for H: d (Hdh) = dh.ds. X sin F ; Dh so we have but to take = for H. L 1) By this we understand in Sp, a system which with the aid of a rectangular system of numbered axes determines a point by 1. 7, its distance to the origin, 2. @, the angle of the radius vector with Xj, 3. the angle of the projection of the radius vector on the coordinate space X)...Xn with Xs, 4. the angle of the projection of the last projection on the coordinate space X3...Xn with X3; etc. The plane through the Xj-direction and the radius vector we call the meri- dian plane. ( 1313 To find © we integrate the current of force inside the meridian zone passing through the ”—!spherical surface through P between the hr | for the axis of the system and P. As we have (n—1) cos p sinh ™r force component perpendicular to that spherical surface we find: ? coshr , ine == | (rn—1) cos p— sinh r dg .cé sinh "—®r sin"—*g = ce sin "Ip cothr. sinh ™r 0 = coshr jaf — sin &. dh sinh ™—|p VII. If thus are given in different points a line vector ZL unity and an "~2vector W unity and if we put along their con- cosh r ———, then the volume product w of L,W sinh™—|p and the vector along the connecting line is the ”-?vector potential in the direction of W caused by an elementary magnet with moment unity in the direction of ZL. We know of w(L,W) that with integration of W along a closed curved Sp,—2 Q it represents the current of force of a magnet unity in the direction of Z through Q, in other words the negative reci- procal energy of a magnet unity in the direction of Z and a magnetic *—!scale with intensity unity, bounded by Q, in other words the force in the direction of Z by a magnetic *—'scale bounded by Q, in other words the force in the direction of Z by a vortex system, regularly distributed over Q and perpendicular to Q. So we can regard y(L,W) as the force in the direction of L by a vortex unity, perpendicular to W. With this we have found for the force of a plane vortex with intensity unity in the origin: necting line a line vector cosh r - sin p‚, sinh "—|p directed parallel to the operating vortex element and perpendicular to the “meridian plane”, if now we understand by that plane the projecting plane on the vortex element; whilst p is here the angle of the radiusvector with the Sp,—2 perpendicular to the vortex element. IX. For the fictitious field of a vortex element in the origin intro- duced in this way (which meanwhile has vorticity every where in space) we shall find a planivector potential, directed everywhere “parallel” to the vortex element and of which the scalar value U is a function of r only. Let us suppose a point to be determined by its azimuth parallel Ox ( 132 ) to the vortex element and then farther in the Sp"—! of constant azi- muth by a system of spherical coordinates, of which we take the first axis in the ‘meridian plane” (see above under $ VIII), and in the plane of the vortex element, the second in the meridian plane perpendicular to the first, and the rest arbitrarily; let us understand meanwhile by p here the angle of the radius vector with the Sp,—s, perpendicular to the vortex element; let further ¢ be an (n—3)-dimen- sional element in the n—3 last coordinates, then this defines for each r and p an element on the surface of an ™—%sphere, of a size dk = ce sinh "—r cos "Sg. We then consider a small elementary rectangle in the meridian plane bounded by radii vectores out of the origin and circles about the origin and a Sp,—1 element consisting of the elements dé erected in each point of this small elementary rectangle. Applying to this Spn—1-element the reduction of an (n—2)-fold integral along the boundary to a (n—1)-fold integral over the volume according to the definition of second derivative, we find: 0 — — {U cos p . dr . ce sinh "—%r cos "—3g} dep — dp p p 0 DE {U sin p . sinh r dp. ce sinh "Ir cos "Ip dr = , cosh r = ce sinh “Sr cos "Sp . sinhr dg .dr.— -——— sin @. sinh "—|p cosh r sinh nr dU (n—2) U — EN sinh r — (n — 2) U cosh r = r cosh r dU nn U=— sinh nr The solution of this equation is: 1 —2(n—2)1 —381p ‚ cosh —2n—-2) bp Lf oot n—3ly dtr + (n—2)sinh nr OR Ii—s3 So we find as planivector potential V of a plane vortex: i i —_________ — —_______~____ | coth n—3ly. dir = F,(r), (n—2)sinh™—2r _ 2n3 cosh %An—2) ky directed parallel to that plane vortex. Let us now call Z the "—vector, perpendicular to the plane vortex, the field of which we have examined, and let us also set off the vector potential V as an"—®vector; let us then bring in an arbitrary point of space a line vector G; then the vector V has the property ( 133 ) that when integrated in G along a small curved closed Sp,» in a Spn—1 perpendicular to G, it indicates the force in the direction of G caused by the current element J, or also the vector potential in the direction of , caused by an elementary magnet with intensity unity in the direction of G. Let us now call the potential x(/, /) of two *—*vectors unity EL, F the symmetric function /,(r)cosg, where r represents the distance of the points of application of both vectors and p their angle after parallel transference to one and the same point of their con- necting line, then we know that this function x gives, when e.g. Fis integrated over a closed curved Sp,—2 which we shall call e, not only the negative energy of a magnetic "—'scale with intensity unity bounded by e in the field of a vortex unity perpendicular to / but also the component along /’ of the vector potential caused by a system of vortices about e with intensity unity. From this ensues again for the vector potential V of a vortex element, that when the vortex element is integrated to a system of vortices about a closed curved Sp,» it becomes the vector potential determined according to § VII of that vortex Spnr—2; so that the : . ly, . ° ° vector potential of an arbitrary »X is obtained as integral of the vectors V of its vortex elements, in other words: ae ee 1/7 9X “=S B) CG es ee n where for each point the vector elements of the integral are first brought over to that point parallel to themselves and there are summed up. X. So let us consider an arbitrary force field as if caused by its two derivatives (the magnets and the vortex systems), we can then imagine that both derivatives are propagated through the space according to a function of the distance vanishing at infinity, causing thereby the potential of the field. For, the field X is the total derivative of the potential: wy Xx wx ij A F, (0) de 4 C MS F, (r) de. The extinguishment of the scalar potential is the stronger, as it is at great distances of order e——lr, the vector potential only of eracr re— lr: ( 134 ) Astronomy. — “The luminosity of stars of different types of spectrum.” By Dr. A. PANNEKOEK. (Communicated by Prof. H. G. VAN DE SANDE BAKHUYZEN). The investigation of the spectra of stars which showed that, with a few exceptions, they can be arranged in a regular series, has led to the general opinion that they represent different stages of develop- ment gone through by each star successively. Voarr’s classification in three types is considered as a natural system because these types represent the hottest and earliest, the further advanced, and the coolest stage. This, however, does not hold for the subdivisions : the difference in aspect of the lines, the standard in this case, does not correspond to the different stages of development mentioned above. Much more artificial is the classification with letters, which PickERING has adopted in his Draper Catalogue; it arose from the practical want to classify the thousands of stellar spectra photographed with the objective prism. After we have allowed for the indistinctness of the spectra which, arising from insufficient dispersion and brightness, influenced this classification, the natural affinity between the spectra will appear and then this classification has the advantage over that of Vocer that the 2"¢ type is subdivided. The natural groups that can be distinguished are: class A: the great majority of the white stars (Sirius type), Voaer’s Ia; class B: the smaller number of those stars distinguished by the lines of helium, called Orion stars, Voarr’s Id. In the continuous series the latter ought to go before the first type and therefore they are sometimes called type 0. Class F forms the transition to the second type (Procyon); class G is the type of the sun and Capella (the E stars are the indistinct representatives of this class); class K contains the redder stars of the 2d type, which ap- proach to the 3¢ type, such as Arcturus (PrCKERING reckons among them the H and I as indistinct representatives). The 3d type is called in the Draper Catalogue class M. The continuity of the stellar spectra is still more evident in the classification given by Miss A. Maury. (Annals Harv. Coll. Obs. Bd. 28). Miss Maury arranges the larger number of the stellar spectra in 20 consecutive classes, and accepts groups intermediate to these. The classes I—IV are the Orion stars, VI—VIII constitute the first type, IX—XI the transition to the 2d type, XIII—XIV the 2d type itself such as the sun, XV corresponds to the redder Arcturus stars, XVII—XX constitute the third type. If we consider that from class I to II a group of lines is gradually falling out, namely the hydrogen lines of the other series, which are characteristic of the Wolf-Rayet (435 ) stars or the so-called fifth type stars (Voer 115), it is obvious that we must place these stars at the head of the series, as it has also been done by Miss Cannon in her investigation of the southern spectra (H. C.O. Ann. Bd. 28) *). Some of these stars show a relative intensity of the metallic lines different from that of the ordinary stellar spectra; Voeren and SCHEINER have found this before in @ Cygni and a Persei (Public. Potsdam Bd. 7, part 2). Maury found representatives of this group in almost all the classes from III to XIII, and classed them in a parallel series designated by IIIc—xXIIIc, in contradistinction to which the great majority are called a stars. According to the most widely spread opinion a star goes succes- sively through all these progressive stages of development. It com- mences as an extremely tenuous mass of gas which grows hotter by contraction, and after having reached a maximum temperature de- creases in temperature while the contraction goes on. before the maximum temperature is reached, there is a maximum emission of light; past the maximum temperature the brightness rapidly decreases owing to the joint causes: fall of temperature and decrease in volume. That the first type stars are hotter than the stars of the second type may be taken for certain on the strength of their white colour ; whether the maximum temperature occurs here or in the Orion stars is however uncertain. This development of a tenuous mass of gas into a dense and cold body, of which the temperature first increases and then decreases is in harmony with the laws of physics. In how far, however, the different spectral types correspond to the phases of this evolution is a mere hypothesis, a more or less probable conjecture; for an actual transition of a star from one type into the other has not yet been 1) According to Camppett’s results (Astronomy and Astrophysics XIII, p. 448), the characteristic lines of the Wolf-Rayet stars must be distinguished in two groups and according to the relative intensity of the two groups these stars must be arranged in a progressive series. One group consists of the first secondary series and the first line of the principal series of hydrogen: H@’ 5414, Hy' 4542, H3' 4201, principal line 4686); it is that group which in Maury’s classes I—III occurs as dark lines and vanishes and which in the classes towards the other side (class Oe—Ob Cannon) is together with the ordinary H lines more and more reversed into emission lines. The other group, which as compared with the hydrogen lines becomes gradually stronger from this point, consists of broad bands of unknown origin of which the middle portions according to Cannon’s measurements of yVelorum have the wavelengths 5807, 5692, 5594, 5470, 4654, 4443. The brightest band is 4654; its relative intensity as compared with the Hline 4689 gradually increases in the series: 4, 47, 5, 48, 42 (Camppe.t’s star numbers). ( 136 ) observed. The hypothesis may be indirectly tested by investigating the brightness of the stars. To answer to a development as sketched here the brightness of a star must first increase then decrease; the mean apparent brightness of stars, reduced to the same distances from our solar system must vary with the spectral class in such a way that a maximum is reached where the greatest brightness is found while the apparent brightness decreases in the following stages of development. § 2. For these investigations we cannot make use of directly mea- sured parallaxes as a general measure for the distance because of the small number that have been determined. Another measure will be found in the proper motions of the stars when we assume that the real linear velocity is the same for different spectral classes. In 1892 W. H. S. Monck applied this method to the Bradley-stars in the Draper Catalogue’). He found that the proper motions of the B stars were the smallest, then followed those of the A stars; much larger are the mean proper motions of the F stars’) which also con- siderably surpasses that of the G, H and K stars and that of the M stars. He thence concluded that these F stars (the 2% type stars which approach to the {st type) are nearest to us and therefore have a smaller radiating power than the more yellow and redder stars of the 2d type. ‘Researches on binary stars seem to establish that this is not due to smaller average mass and it would therefore appear, that these stars are of the dullest or least light-giving class — more so not only than the Arcturian stars but than those of the type of Antares or Betelgeux” (p. 878). This result does not agree with the current opinion that the G, K and M stars have successively developed from the F stars by contraction and cooling. It is, however, confirmed by a newly appeared investigation of EJNAR Herrzsprune: Zur Strahlung der Sterne’), where Maury’s classification of the spectra has been followed. He finds for the mean magnitude, reduced to the proper motion 0,01, the values given in the following table where I have added the corresponding proper motions belonging to the magnitude 4.0. Here also appears that for the magnitude 4,0 the proper motion is largest and hence the brightness smallest for the classes XII and 1) Astronomy and Astrophysics XI p. 874. 2) He constantly calls them incorrectly “Capellan stars” because in the Dr. Cat. Capella is called F, though this star properly belongs to the sun and the -G stars. 5) Zeitschrift fiir wissenschaftliche Photographie Bd. III. S. 429. ( 437 j Spectrum Magn. for P. M. for Maury | Draper C.| P. M. 0"01 | Magn. 4.0 RAL A B 431 | 0.012 VVI BÀ 7.25 | 0.045 VII—VIII A 8.05 _ | 0.065 Le F 9.06 0.103 X= i B= 11.93 | 0.279 KIRIK) GR 7-938 0.064 XV K | 9.38 0.419 WV XVI K—M Tia 0.057 CVA M 8.28 0.072 XIII that form the transition from F to G; for the later stages of development the brightness again increases. $ 3. A better measure than the proper motion for the mean distance of a group of stars is the parallactic motion. This investiga- tion was rendered easy by means of N° 9 of the “Publications of the astronomical Laboratory at Groningen”, where the components rt and v of the proper motion are computed with the further auxiliary quantities for all the Bradley-stars. Let + and v be the components of the proper motion at right angles with and in the direction of the antapex, A the spherical distance of the star-apex, then = v sina 1E sin? a is the parallactic motion for a group of stars, i.e. the velocity of the solar system divided by the mean distance of the group. The mean of the other component =. is, at a random distribution of the directions, equal to half the mean linear velocity divided by the distance. The mean magnitudes of the different groups are also different. Because we here especially wish to derive conclusions about the brightness, and as both the magnitude and the proper motion depend on the distance the computation was made after the reduction to 1) The Roman figures in italics in Maury’s classification designate the transition to one class higher. ( 138 ) magnitude 4.0; that is to say, we have imagined that every star was replaced by one which in velocity and in brightness was perfectly identical with the real one, but placed at such a distance that its apparent magnitude was 4.0. If the ratio in which we then increase the proper motion is idea 100.2 (m—4) we have = pv sind = pt 94.09 = ————— _ and mo= : ane > sin* À ig n In this computation we have used Maury’s classes as a basis. We have excluded 61 Cygni on account of its extraordinary great parallax, while instead of the whole group of Ursa Major (8 y de 5) we have taken only one star (e). In the following table are combined the results of the two computations. Spectrum | Typical fe mean | mean ie Fe 74.0 MAURY | Dr Cat. star | m aa = 1 I—III B e Orionis B sou OOO’, 0. ‘018 EN ‘007 7 013" IV—V B—A y Orionis 48 | 4.31 | 0.011 0.035 | 0.014 | 0.036 VI—VIII A Sirius 93 | 3.92] 0.040 | 0.054] 0.038 | 0.061 IX—XII F Procyon 94 14441 0.089 | 0.453 | 0.095 | 0.436 XIII—XIV G Capella 69 | 4.08} 0.44 0.457 je 0.1607)" 0499 XV K Arcturus | 101 3200 POB He OOS 0120702085 XVI—XX M Betelgeuze 61 3.85 | 0.049 | 0.068 | 0.050 | 0.061 In both the series of results the phenomenon found by Monck and HrrtzsprunG manifests itself clearly. I have not, however, used these numbers 140 and q4o, but have modified them first, because it was not until the computation was completed that I became ac- quainted with Herrzsprune’s remark that the above mentioned c stars show a very special behaviour; their proper motions and parallaxes are so much smaller than those of the a stars of the same classes that they must be considered as quite a separate group of much greater brillianey and lying at a much larger distance’). We have Ll, In his list of parallaxes Hertzsprune puts the question whether perhaps the bright southern star a Carinae (Canopus) belongs to the c stars; but he finds no indication for this except in its immeasurably small parallax and small proper motion. In her study of the southern spectra Miss Gannon has paid no regard Class | n | 4.0 | 14.0 Qt /q Saas oe ar | 7 m I 5) | 0.009 0.022 0.8 II | 15 | 005 009 alte Ill | 14 | 006 O15 | 0.8 IV | 18 014 | 023 Ae? IV | 16 016 | O44 O7, V | 11 009 042, 0.4 VI 16 030 068 | 0.9 VII | 30 040 086 0.9 VIII 4A 043 055 | 186 IX 95 050 064 | 1.6 x | 16 070 Ufa 0.8 XI | 22, 103 061 On | aul! 93 170 982, | At 9) XIII 18 297 346 | 47 XIV | 21 192 305 alee} XIV 90 077 025 6.2 XVA | 26 934 148 ON, DO 783 3D | 105 | 070 0) ONG: | 40 059 087 1 4 XVI 19 049 071 14 XVII 19 049 032 3.1 X VIII 16 050 O75 1 UE 7 057 078 1.5 to the difference between the a and the c stars. Yet all the same this question may be answered in the affirmative; on both spectrograms of this star occur- ring in her work, we see very distinctly the line 4053.8, which in Capella and Sirius is absent and which is a typical line for the c stars. Hence follows that « Carinae is indeed a c star. ( 140 ) therefore repeated the computation after exclusion of the c and the ac stars. The table (see p. 139) contains the results for all the classes of Matry separately ; class XV is divided into three subdivisions: XV A are those whose spectra agree with that of a Boötis, XV C are those which agree with the redder « Cassiopeiae, while XV B embraces all those that cannot with certainty be classed among one of the other two groups. The values for t49 and q4o differ very little from those of the preceding table. If we take the value of the velocity of the solar system — 4.2 earth’s distances from the sun, the q’s divided by 4.2 yield the mean parallax of stars of different spectral classes for the magnitude 4.0 (204). Reversely, we derive from the q’s the relative brightness of these stellar types, for which we have here taken the number which expresses how many times the brightness exceeds that of magnitude 4.0 when placed ata distance for which g = 0".10,~ hence with the parallax 0".024. Finally the last column 21/q contains the relation between the mean linear velocities of the group of stars and our solar system. In the following table we have combined these values in the same way as before. mmm Spectrum Typical L for Maury |Dr. Cat. star : “40 He “40 ie zg EE NEE RS Se | EL EN a | ae a a TI B = Orionis 32 0.0055} O 014 | 0.0033 | 51 08 IV—V B—A 7 Orionis 45 0.013 | 0.036 | 0.0086 Tel O7 VI—VIII A Sirius 87 0.040 | 0.063 | 0.015 2.5 1.3 IX —XIl F Procyon 86 0.401 | 0.144 | 0.034 0.50 | 1.4 XIH—XIV G Capella 59 0 182 | 0.224 | 0.053 0:20) 4126 XV K Arcturus | 101 0.120 | 0.096 | 0.023 aed 2.5 XVI—XX M Betelgeuze 61 0.050 | 0.061 | 0.015 Dd 1.6 $ 4. Conclusions from this table. The numbers of the last column are not constant but show a systematie variation. Hence the mean linear velocity is not constant for all kinds of stars but increases as further stages of development in the spectral series are reached. _ (Whether the decrease for the 3"¢ type, class M, is real must for the present be left out of consideration). That the linear speed of the Orion stars is small is known and appears moreover from the ( 141 ) radial velocities. While CAMPBELL found 19.9 kilometres for the velocity of the solar motion, and 34 kilometres for the mean velocity of all the stars, Frost and Apams derived from the radial velocities of 20 Orion stars measured by them, after having applied the correction for the solar motion: 7.0 kilometres as mean value’), hence for the actual mean speed in space 14 kilometres, whence follows the ratio 0.7 for 2r/g. Hence the Orion stars are the particularly slow ones and the Arcturian stars (class XV) are those which move with the greatest speed. § 5. When we look at the values of q4o or those of 249 or Loo, derived from them, we find, as we proceed in the series of development from the earliest Orion stars to the Capella or solar type G, that the brightness constantly decreases. That q was larger for the 2d type as a whole than for the first (the Orion stars included) has long been known; some time ago Kaprryn derived from the entire Bradley-Draper material that on an average the 2¢ type stars (F GK) are 2,7 times as near and hence 7 times as faint as the 1st type stars (A and B). This result perfectly agrees with the ordinary theory of evolution according to which the 2d type arises from the 1st type through contraction and cooling. A look at the subdivisions shows us first of all that the Orion stars greatly surpass the A stars in brightness, and also that among the Orion stars those which represent the earliest stage greatly surpass again in brightness those of the later stages. As compared with the solar type G the Sirius stars are 12 times, the stars which form the transition to the Orion stars 38 times and lastly the ¢ Orionis type 250 times as bright. This result is in good harmony with the hypothesis that one star goes successively through the different con- ditions. from class I to class XIV; we then must accept that the density becomes less as we come to the lower classes. Whether the temperature of the Orion stars is higher than that of the Sirius stars or lower cannot be derived from this result; even in the latter case it may be that the larger surface more than counterbalances the effect of smaller radiation. This must be decided by photometric measurements of the spectra. As the Wolf-Rayet stars follow next to class I, an investigation of their proper motion, promised by KaptrYN, will be of special interest. Past the G stars, the solar type of the series, the brightness again increases. The values obtained here for q confirm in this respect the results of Monck and Hurrrzsprune. 1) Publications Yerkes Observatory. Vol. II. p. 105. (A42) Against the evidence of the q’s only one objection can be made, namely that these classes K and M might have a proper motion in common with the sun, so that g would not be a good measure for the distance. A priori this objection is improbable but it may be tested by material, which, though otherwise of small value, may for this kind of investigations yield very valuable conclusions on this point, namely the directly measured parallaxes. Hnrrzsprune gives mean values of the measured parallaxes reduced to magnitude 0,0; by the side of these we have given the values for somewhat different groups derived from our 4.9: Observed 270.6 Derived from q 20 ESA 0".0255 (6) ES 0.021 ASN 01106 24) vy. 0 .054 VII— VIII 0 1:53.50) ViVi ORO iD Sal 0.226) (6) xe Xa Ord KISS 0.442 (2) XIV Ok beb) KSO RSS XV Obit (8) pa Orie XVI 0171 (3) XVI XX 01096 XVII—XVIII0 115 (8) In general HertzsPRUNG’s numbers are somewhat larger, this can be easily explained by the circumstance that many parallaxes measured in consequence of their large proper motions wil] probably be above the mean. It appears sufficiently clear from this, at any rate, that also the directly measured parallaxes markedly point at an increase of brightness past class XIV, and that there is not the least ground to assume for the other groups a motion in common with the sun. It is therefore beyond doubt that the K and M stars have a greater intrinsic brilliancy than the F and G stars. Monck derives from this fact that they have a greater radiating power, because about the same value for the masses is derived from the double stars. That the latter cannot be derived from the double stars will appear hereafter. Moreover Monck’s conclusion of the greater radiating power of the K and M stars is unacceptable. In incandescent bodies this radiating power depends on the temperature-of the radiating layers and of the atmospheric absorptions. With unimpaired radiance a greater amount of radiation is accompanied with bluer light (because the maximum of radiation is displaced towards the smaller wave- lengths) as both are caused by the higher temperature. The general absorption by an atmosphere is also largest for the smaller wave- lengths, so that when after absorption the percentage of the remain- ( 143) ing light is less, the colour of the radiated light will be redder. Therefore it is beyond doubt that a redder colour corresponds at any rate with a less degree of radiance per unit of surface. Then only one explanation remains: the K and M stars (the redder 24 type stars like Arcturus and the 3 type) possess on an average a much larger surface and volume than the other 24 type stars of the classes F and G. This result is at variance with the usual representation of stellar evolution according to which the redder K and later the M stars are developed from the yellow-white F and G stars by further contraction and cooling. § 6. A further examination of the constitution of these stars shows us that it is improbable that they should possess a very small density; the low temperature, the strongly absorbing vapours point to a stage of high condensation. These circumstances lead to expect greater (with regard to the F and G stars) rather than less density. From the larger volumes it then follows that the K and M stars have much larger masses than the F’s and G’s. This result is the more remarkable in connection with the conclusion derived above about their greater mean velocity. If the stars of our stellar system form a group in the sense that their velocities within the group ‘depend on their mutual attraction, we may expect that on an average the velocities will be the greater as the masses are smaller. No difficulty from this arises for the Orion stars with small speed, because the same circumstances which allow us to ascribe to them -a mass equal to that of the A, F and G stars, enable us likewise to ascribe to them a larger mass. The K stars which have both a greater mass and a greater velocity are characterized by this thesis as belonging to a separate group, which through whatever reason must originally have been endowed with a greater velocity. Arcturus with its immeasurably small parallax and large proper motion is therefore through its enormously great linear velocity and extraordinary luminosity an exaggerated type of this entire class, of which it is the brightest representative. Therefore it would be worth while to investigate separately the systematic motions of the K stars which hitherto have been classed without distinction with the F and G stars as 2™¢ type. If this result with regard to the greater masses of the K and M stars should not be confirmed, the only remaining possibility is the supposition that the density of these star is extremely small. In this case their masses might be equal to that of other stars and they may represent stages of evolution of the same bodies. Where ( 144 ) they ought to be placed in the series of evolution remains a riddle. There is a regular continuity in the series F—G—K—M; and accord- ing as we suppose the development to take place in one direction or in the other we find in the transition G—K either cooling accom- panied with expansion, or heating accompanied with contraction. The puzzling side of this hypothesis can also be expressed in the follow- ing way: while in the natural development of the celestial bodies, as we conceive it, the temperature has a maximum but the density continuously increases, the values obtained here would according to this interpretation point at a maximum density in the spectral classes F and G. In Vol. XI of Astronomy and Astrophysics Maunper has drawn attention to several circumstances, which indicate that the spectral type rather marks a difference in constitution than difference in the stage of development. ‘There seems to me but one way of recon- ciling all these different circumstances, viz.: to suppose that spectrum type does not primarily or usually denote epoch of stellar life, but rather a fundamental difference of chemical constitution” *). One of the most important of these facts is that the various stars of the Pleiades, which widely differ in brightness and, as they are lying at the same distance from the sun, also in actual volume show yet the same spectrum. The result found here confirms his supposition, One might feel inclined to look for a certain relation between these K and M stars and the c stars, which, according to HERTZSPRUNG, have also a much greater luminosity, hence either less density or greater mass than the similar a stars; and the more so as these c stars reach no further than class XIII. Yet to us this seems improbable ; the K stars are numerous, they constitute 20°/, of all the stars, while the cstars are rare. Moreover the spectra of all the K stars are with regard to the relative intensity of the metallic lines perfectly identical with the astars of preceding classes such as the sun and Capella. Therefore it as yet remains undecided to which other spectra we have to look for other phases in the K star lives and to which spectra for those in the ¢ star lives. The c stars, except a few, are all situated in or near the Milky Way : this characteristic feature they have in common with the Wolf-Rayet stars and also with the 4th type of Srocm (Vogel's HI), although these spectra have no lines in common which would suggest any relation between them. § 7. The constitution found here for the Arcturian stars among the third type stars may perhaps be tested by means of other 1) Stars of the first and second types of spectrum. p. 150. ( 145 ) data, namely by those derived from the double stars. The optically double stars cannot however teach us anything about the masses of the stars themselves as will appear from the following consideration (also occurring in “The Stars” by Nrewcoms). Let us suppose that a binary system is m times as near to us, while all its dimensions become 2 times as small, but that the density and the radiation remain the same. Then the mass will diminish in the propor- tion of n° to 1, the major axis of the orbit @ in the proportion of » to 1 and hence the time of revolution remains the same; the luminosity becomes n* times as small, therefore the apparent brightness remains the same as well as the apparent dimensions of the orbit, in other words: it will appear to us exactly as it was before. Hence the mass cannot be found independently of the distance. Let a be the angular semi-major axis, M/ the mass, P the time of revolution, d the density, 2 the radiating power, a the parallax and @ the radius of the spherical volume of the star, then ; 3 at 4 we shall have: a°M = a the mass Jf ‘is a constant value x 0°, the apparent brightness H is a constant X 2’0°2, Eliminating from this the parallax and the radius, we find Pr vak ee Er Thus from the known quantities: elements of orbit and brightness, we derive a relation between the physical quantities: density and radiating power, independently of the mathematical dimen- sions. This relation has been derived repeatedly. In the paper . 3 3 cited before Maunprr gives values for the density d = ec (3) 3 in the supposition of equal values of 2; he found for the Sirius stars (1s' type) 0,0211, for the solar stars (all of the 2°4 type) 0,3026, hence 14 times as large on an average; we can also say that when we assume the same density the radiating power of the Sirius stars would be 6 times as large; the exact expression would be that the quotient 2°/d* is 200 times as large for the Sirius stars as for the solar stars. In a different form the same calculation has been made by HertzspruNG by means of ArrkeN’s list of binary system elements *). By means of — 2,5 log H == m he introduces into his formula the stellar magnitudes ; if we put in the logarithmical form 1) Lick Observatory Bulletin Nr. 84. 10 Proceedings Royal Acad. Amsterdam. Vol. IX. ( 146 ) 3 log H + 4 log P — 6 log a = const. + 3 log 4 — 2 log d m—*/, log P+ dloga=m, then we have m, = const. — 2,5 log 4 + °/, log d. If we arrange the values of mm, after the spectra according to the Draper Catalogue (for the Southern stars taking Cannon; according to the brightest component « Centauri was reckoned to belong to class G), we find as mean values: Class A — 2.92 (9 stars — 4.60 to — 1.09) F —1.32(49 ,, — 3.61 ,, +0.14) ,- GandKE—049(11 ,, —1.60 ,, +1.28) The 3 stars of the type K (with H) give — 4.88 (y-Leonis), —1.05 and + 0,87, hence differ so widely that no valuable result is to be derived from them. To the extraordinarily high value for 2°/d* given by y Leonis attention has repeatedly been drawn, While for a great number of stars of the other classes the extreme values of m, differ by 3.5 magnitudes we find that y Leonis differs by 5 magnitudes from the mean of the two other values, that is to say : its radiating power is a hundred times as large, or its density is a thousand times as small as for these other stars. For the classes A and F we find that 2°/d? is 640 and 8 times respectively as large as for class G; conclusions about class K as a whole, such as are especially wanted here, cannot be derived from it. It may be that an investi- gation of binary systems with partially known orbit motion (for which we should require auxiliary hypotheses) would yield more results. About the mass itself, however, something may be derived from the spectroscopic binary systems. The elements derived from obser- vation a sind and P directly yield M sin *t; as it is improbable that there should be any relation between the type of spectrum and the angle between the orbit and the line of sight we may accept the mean of sin ®% to be equal for all groups. For systems of which only one component is visible, the element derived from observation contains another unknown quantity, viz. the relation 8 of the mass of the invisible to that of the visible star. If a is the semi major axis of the orbit of the visible star round the common centre of gravity, we have 3) 3 En (me PD “(EL B): It is not perfectly certain, of course, that on an average B is the same for all classes of spectrum; if this is not the case the M’s : a? sin % may behave somewhat different from the values of PE computed sin %. here. ( 147 ) Unfortunately, of the great number of spectroscopic double stars discovered as yet (in Lick Observatory Bulletin N°. 79 a number of 147 is given) the orbit elements of only very few are known, They give, arranged according to their spectra: Group II—IV (B) Group VI—VIII (A) Orion type Sirius type o Persei 0.61 3 Aurigae 0.56 1 Orionis 2.51 5 Ursae (3.41) 7) d Orionis 0.60 Algol 0.72 8 Lyrae 7.85 a Androm. 0.36 *) a Virginis 0.33 a, Gemin. 0.002 V Puppis 34.2 ; Group XII—XIV a (F—G) Group XII—XIV ae Solar type a Ursae min. 0.00001 a Aurigae 0.185 S Geminorum 0.0023 zy Draconis 0.120 74, Aquilae 0.0029 (W Sagittarii 0.005) d Cephei 0.0031 (X Sagittarii 0.001) t Pegasi 0.117 Group XV (K) 1 Pegasi 0.254 8 Herculis 0.061 Of the K stars only one representative occurs here, so neither this material offers anything that could help us to test the results obtained about this stellar type. But all the same, some remarkable conclusions may be derived from this table. It appears here that notwithstanding their small number the Orion stars evidently surpass the others in mass, while the Sirius stars seem also to have a some- what greater mass than the solar stars. Very striking, however, is the small mass of the ¢ stars approaching towards a. Hence the c stars combine a very great luminosity with a very small mass, and consequently their density must be excessively small. If it should be not merely accidental that the three regularly variable stars of short period, occurring in Maury, all happen to show c characteristics and a real connection should exist between this particularity of spectrum and the variability, we may reasonably include into the 1) In the case of £ Ursae a has been taken equal to the semi major axis of the relative orbit; hence this number is proportionally too large by an unknown number of times. 2) Assumed period 100 days, velocity in orbit 32.5 kilometres. ( 148 ) group W and X Sagittarii which also yield small values; as has been remarked, for the southern stars no distinction is made between the a and the c stars *). We may expect that within a few years our knowledge of the orbits of the spectroscopic double stars will have augmented consi- derably. Then it will be possible to derive conclusions like those found here from much more abundant material, and also to arrive at some certainty about the mean mass of the K stars. With regard to the latter our results show at any rate that in investigations on grouping of stars and stellar motions it will be necessary not to consider the 2rd type as one whole, but always to consider the F and G stars apart from the redder K stars. 1) In this connection may be mentioned that in 1891 the author thought he detected a variability of z Ursae minoris with a period of a little less than 4 days. The small amplitude and the great influence of biased opinions on estimations of brightness after ARGELANDER’s method in cases of short periods of almost a full number of days, made it impossible to obtain certainty in either a positive or a negative sense. C:AMPBELL’s discovery that it is a spectroscopic binary system with a period of 3423" 14m makes me think that it has not been wholly an illusion. PvR ak ATA. In the Proceedings of the Meeting of June, 1905, p. 81: line 7 from top, read: “cooled by conduction of heat”, EO ee tor ony eel, EN rend: Ee el NA ge In Plate V belonging to Communication N°. 83 from the physical laboratory at Leiden, Proceedings of the Meeting of February 1905, p. 502, the vacuum glass B’, has been drawn 18 em. too long. (August 21, 1906). KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN TE AMSTERDAM. PROCEEDINGS OF THE MEETING of Saturday September 29, 1906. (Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige Afdeeling van Zaterdag 29 September 1906, Dl. XV). GON TE EN PS; M. NieuwENHUIS-vON UEXKULL GÜLDENBAND: “On the harmful consequences of the secretion of sugar with some myrmecophilous plants”. (Communicated by Prof. J. W. Morr), p. 150. H. KaMmerLiNGH Onnes: “Methods and apparatus used in the cryogenic Laboratory at Leiden. X. How to obtain baths of constant and uniform temperature by means of liquid hydrogen”, p. 156. (With 3 plates); XI. “The purification of hydrogen for the cycle”, p.171.(With 1 plate); XII. “Cryostat especially for temperatures from — 252° to —259°”, p. 173. (With 1 plate); XIII. “The preparation of liquid air by means of the cascade process”, p. 177. (With 1 plate); XIV. “Preparation of pure hydrogen through distillation of less pure hydrogen”, p. 179. H. KAMERLINGH Onnes and C. A. CROMMELIN. “On the measurement of very low temperatures IX. Comparison of a thermo element constantin-steel with the hydrogen thermometer”. p. 180. H. KAMERLINGH Onnes and J. Cray: “On the measurement of very low temperatures X. Coefficient of Expansion of Jena glass and of platinum between + 16° and —182°, p. 199. XI. A comparison of the platinum resistance thermometer with the hydrogen thermometer, p. 207. XII. Comparison of the platinum resistance thermometer with the gold resistance thermometer”, p. 213. JAN DE Vries: “Quadratic complexes of revolution”, p. 217. J. K. A. WERTHEIM SALOMONSON: “A few remarks concerning the method of the true and false cases”. (Communicated by Prof. C. WINKLER), p. 222. J. J. van Laar: “The shape of the spinodal and plaitpoint curves for binary mixtures of normal substances. 4th Communication: The longitudinal plait”. (Communicated by Prof. H. A. LoRENTz), p. 226. (With 1 plate). 11 Proceedings Royal Acad. Amsterdam. Vol. IX. ( 150 ) Botany. — “On the harmful consequences of the secretion of sugar with some myrmecophilous plants.’ By Mrs. M. NmuweNnuis- von UBXKÜLL-GÜLDENBAND. Ph. D. (Communicated by Prof. J. W. Motz). (Communicated in the meeting of June 30, 1906). During my residence of about eight months at Buitenzorg in 1901 I occupied myself chiefly with an investigation of the structure and peculiarities of the sugar-secreting myrmecophilous plants. The results of these observations, extending over some 70 plants, are inconsistent with the opinion expressed by Derpino, Kerner, TRELEASE, BurckK and many others, that the extrafloral secretion of sugar by plants would serve to attract ants which in return would protect the plants against various harmful animals. For I was unable to observe in a single instance that the secretion of sugar is useful to the plant; on the other hand it appeared to me that the ants feed on the sugar, but that, instead of being useful at the same time, they injure the plant indirectly by introducing and rearing lice; moreover the extrafloral nectaries attract not only ants but also numbers of beetles, bugs, larvae, etc. and these are not content with the sugar alone, but at the same time eat the nectaries themselves and often consume the leaves and flowers to no small extent. In about one third of the plants, investigated with this purpose, the secretion of sugar in this way certainly does much harm; with another third the plants experience only little harm by attracting the undesirable visitors, while with the last third no indication at all could be found that by secreting sugar they were worse off than other plants. Of those that were indirectly injured by secreting sugar I here only mention a few examples out of the many which I shall consider more extensively elsewhere. Spathoglottis plicata Bl. is a common orchid in the Indian archi- pelago. In the environs of Buitenzorg it is e.g. found on the Salak, and it is used in the Botanical Garden to set off the beds in the orchid quarter. Its leaves (all basal leaves) have a length of as much as 1.20 M., according to Smrru, they are narrow, have a long point and are folded lengthwise; their inflorescence is erect, reaches a height of about 2 metres and bears at its extremity, in the axils of coloured bracts, a number of flowers, the colour of which varies from red violet to white. The bracts and perianth leaves have blunt, ( 151 ) thick and darker coloured points. On the inflorescences two kinds of ants always abound, one large and one small species. Even when the flower-buds are still closed the ants are already found on the bracts and no sooner are the flowers open than the ants also attack the perianth leaves. It appeared that sugar was secreted as a bait here. In order to prove this the flowers were placed for some time under a damp glass bell-jar; after a few hours by means of FEHLING’s reagent sugar could be proved to be present in the liquid secreted by the leaves at the exterior side. I could find no special organs for this secretion, however; probably the secretion is an internal one the product being brought out by the epiderm or the stomata. It was already known to De.pino that some orchids secrete sugar on the perianth; the remarkable point with the just mentioned Spathoglottis is that the ants have such an injurious influence on it. Whereas namely the small species remains on the flowers and is content with the sugar there secreted, the big species also descends to the basal-leaves and attacks these also, often to such an extent that only a skeleton of them remains. These harmful big ants are not expelled at all by the much more numerous small ones. It further appeared most clearly that the secretion of sugar was the reason indeed why such important organs as the leaves were eaten by the big species. The proof was namely afforded by those plants that had finished flowering and bore fruit; with these secretion of sugar took place no longer and the leaves, which were produced in this period, remained consequently uninjured. So it was the secretion of sugar during the flowering period which attracted the ants, while the leaves as such were no sufficient bait. A second instance of the great harm that may be caused to the plants themselves by the secretion of sugar, is seen with various tree- and shrublike Malvaceae. In the Botanical Garden stands an unnamed tree, a Malvacea from Indo-China. This not only has nectaries on the leaves and calyx, but also offers the ants a very suitable dwelling-place in the stipules, which occur in pairs and are bent towards each other. The spaces formed in this way are indeed inhabited by ants, but not by so many as might be expected. The reason is that in spite of the abundance of nectaries they find no sufficient food, since on these trees a species of bugs occurs which not only consume the secreted sugar but also eat the nectaries them- selves. These bugs moreover injure the leaves to such an extent that the tree suffers from it, as may be seen by a cursory examina- tion. The same may be stated of a tree named “Malvacea Karato” and of some other species of this family. 1E ( 152 ) In order to prove that the secretion of sugar by attracting harmful insects is indeed injurious to these trees it would still be necessary to show that they remain uninjured when the secretion of sugar does not take place. This proof is readily afforded by some other Malvaceae. Two shrublike Malvaceae of common occurrence in India, namely Hibiscus rosa sinensis L. and Hibiscus tiliaceus L. have nectaries on their leaves. They are not frequented by ants or other harmful insects, however, because in the nectaries, as far as my observations go, a fungus always occurs, which may be recognised already from the outside by its blaek colour. This fungus prevents the secretion of sugar, and the nectaries cease to have an attraction for insects which otherwise would be harmful to the plant. These shrubs by their healthy appearance contrast strongly with the above mentioned plants in the Malvaceae quarter, which are frequented by ants and other insects. On account of the circumstance that the extrafloral nectaries are found chiefly on and near the inflorescences, Burck proposed the hypothesis, that in some cases they would serve to attract ants into the neighbourhood of the flowers in order to protect these against bees and wasps, which would bore them and rob honey. But even with the plants investigated by him I cou!d find no confirmation of his hypothesis. Furst the nectaries only rarely occur on the inflores- cences exclusively; also the plants mentioned by him as proof as: Thunbergia grandiflora Roxb., Gmelina asiatica L., and Gmelina bracteata, Nycticalos macrosyphon and Nycticalos Thomsoni cannot serve as examples, since these plants also on their vegetative parts such as leaves and stems possess nectaries, which according to him are not present there or are not mentioned. In regard to the so-called “food-bodies”’ (Burck’sche Körperchen) on the calyx of Thunbergia grandiflora, it appeared to me that these are no ‘“food-bodies” at all, but ordinary sugar-secreting deformed hairs which I also found on the bracts, leaves and leaf-stalks of this plant. Further it appeared to me that the number of bored flowers stands in no relation to the number of nectaries occurring on the calyx, as should be the case according to BurckK. It is much more dependent on external factors, as e.g. the more or less free situation of the plants, the weather ete. As an example the creeper Bignonia Chamberlayni may be men- tioned. Of this plant on many days only 1,6 °/, of the fallen flowers appeared not to have been bored by Xylocopa coerulea, although numerous ants always occur on the nectaries of the calyx. ( 153 ) An example of the fact that the more or less free situation in fluences the number of perforations of the flowers is found in two species of the genus Faradaya, both having nectaries on the calyx and the leaves. With Faradaya papuana Scnerr., which stands in the Botanical Garden at Buitenzorg surrounded by many other richly flowering plants, the flowers are often perforated by a boring wasp; of the fallen flowers only 1 °/, was undamaged. This was different with another still unnamed species of the same genus which, as far as the nectaries were concerned, showed no difference with the former and grew at some distance from it in a less open site. Its branches hung partly to the ground and bore far fewer flowers than Faradaya papuana. Now of this three 19,3 °/, of the flowers remained unperforated. And in regard to the weather it appeared that the number of bored flowers closely depends on it. After a sunny day a much larger number of flowers had been bored the next morning than when rain had prevented the insects from flying out. This was e.g. very conspicuous with Zpomoea carnea Jacg., a shrub having nectaries as well on the leaves as on the calyx, the latter being bored by Vespa analis and two Xylocopas. Collected in the morning without regard to the weather of the preceding day 90°/, of the fallen flowers were bored; after rainy days 57 °/, of the flowers were damaged and after sunny days even 99,1 °/, were bored. From this appears most clearly how little value must be assigned to statistical data about the perforation of flowers and about their being eventually protected by ants if not at the same time all other circumstances which may influence the results have been taken into account. When trying to fix the part, either favourable or otherwise, played by insects with regard to a plant, one meets with greater difficulties in the tropics than e.g. in Middle Europe, because the vegetative period lasts so much longer. So one may meet an abundance of definite insects during one part of that period which are not found during another part. This special difficulty of the question whether special arrangements in a plant form an adaptation to a definite animal species is still enhanced in a botanical garden by the circumstance that there nearly all the plants are in a more or less uncommon site or surroundings. Yet here also the mutual behaviour of the animals frequenting the plants may be investigated as well as their behaviour towards the plants themselves, while the results enable us to draw some justified conclusion as to the mutual relations in the natural sites of these plants. I took this point of view when I began my ( 154 ) investigation and among others put myself the following questions to which the here briefly mentioned answers were obtained: 1. On what parts of plants is extrafloral secretion of sugar found ? In the cases examined by me I found secretion of sugar on the branches, leaves, stipules, bracts of different kind, peduncles and pedicels, ovaries and the inner and outer side of calyx and corolla, in each of these organs separately or in a great number of different combinations. The most commonly occurring of these combinations were: a. on leaf-sheaths and calyx together, 5. on the leaf-blade only c. on the leaf-stalks, peduncle and calyx. Of other combinations I only found from one to three examples each. 2. Does the structure or place of the nectaries clearly indicate that they are made for receiving ants? Except in a few cases (as the nectaries occurring in the closely assembled flowers of Gmelina asiatica Scheff. on that side of the calyx, that is turned away from the axis of the inflorescence) this question must be decidedly answered negatively. Although it seems as if the very common cup shape of the nectaries were eminently suitable for storing the secreted honey, yet on the lower side of the leaves these nectaries are for the greater part found with their opening turned downward. I remind the reader of the two large, also downwardly directed cup-shaped nectaries at the base of the side leaves of some species of Hrythrina. The frequent occurrence of nectaries on the calyces, which only in the budding period secrete honey, seems to indicate that these buds require special protection. But inconsistent with this view is the fact that sometimes, according to my observations, only half of the flowers has nectaries in the calyces (e.g. Spathodea campanulata _ BEAUV.). With many species of Smilax only part of the branches attracts ants and these are branches that carry no flowers and so, according to the prevailing conception, would least require protection. It is difficult to make the idea of the protection of the flowers agree with the fact that nectaries occur on the inner and outer side of the upper edge of the tube of the corolla of Nycticalos macrosyphon, Spathodea serrulata and others. Attracting ants to the entrance of the corolla, which is the very place where the animals causing cross-fertilisation have to enter, has certainly to be called unpractical from the biolo- gist’s point of view. Against the conception that these plants should require protection, also the fact pleads that exactly with young plants, where protection would be most necessary, these baits for protective ants are absent. (155 ) A short time ago UE’) has drawn attention to this as a result of his investigation of American plants. 3. Is sugar secreted in al! nectaries ? This is not the case; in some nectaries I could detect no secretion even after they had stayed for a long time under a bell-jar; this was the case e.g. with the leaves of Gimelina asiatica. Consequently they are not frequented by ants, although these insects always occur on the similarly shaped but strongly secreting nectaries of the calyx. The quantity of the secreted substances moreover fluctuates with the same nectaries of the same plant and depends on many external and internal influences. 4. Are all the products secreted by the nectaries always and eagerly consumed by the ants? Evidently this also is not always the case, for whereas the necta- ries of some plants are constantly frequented by ants, with others the nectaries so to say overflow, witbout a single animal visiting them. (So with some species of Passijlora). 5. At what age of the organs do the nectaries secrete sugar? As a rule the nectaries of the inflorescences cease to secrete as soon as the flowers are opened; those of the leaves even only functionate in the youngest stages of development. 6. Are the ants that frequent the plants with nectaries hostile towards other visitors? Although I daily watched the behaviour of the ants with the extrafloral nectaries for hours, I have never observed that they hindered other animals in any way. On the Lufja species one may see the ants at the nectaries peacefully busy by the side of a species of beetles which does great damage to the plant by eating leaves and buds. The results of my investigations of some wild plants in Java in their natural sites agreed entirely with those obtained in the Buiten- zorg Botanical Garden. Exactly those species of ants that occur on the so-called ‘“ant- plants’ of the Indian archipelago, seem to belong to the harmless ones; the dangerous species with powerful mouth-apparatus, e.g. those which are called semut ranggrang in West Java and according to Dr. VorpERMAN are used by the Malay for defending Mango trees against beetles, are carnivorous. So these ants have to be specially allured by hanging animal food (dead leguans) in the trees to be protected. 1) Enawer’s Bot. Jahrbiicher. Heft III, Bd. 37, 1906. ( 156 ) What the real meaning is of the often so highly differentiated organs as many extrafloral nectaries are and of the secretion of sugar which they present in most cases, can only be settled by new investigations which however will have to bear not only on the biology but also on the physiology of the plant. Physics. — “Methods and apparatus used in the cryogenic labora- tory at Leiden. X. How to obtain baths of constant and uniform temperature by means of liquid hydrogen.” By Prof. H. KAMERLINGH Onngs. Communication N°. 94/ from the Physical Laboratory at Leiden. (Communicated in the meeting of 28 May, 1906). § 1. Introduction. Communication N°. 14 of Dec. ’94 treated of the results I had obtained after I had employed regenerators for the cascade method, and especially discussed the way how to obtain a permanent bath of liquid oxygen to be used in measurements at the then observed lowest temperatures. At the end of that paper I expressed the hope to be able to construct a cycle of hydrogen similar to that of oxygen. A mere continuation of the cascade method would not do. By means of liquid oxygen or nitrogen, even when they evaporate in vacuo, we practically cannot reach the critical temperature of hydrogen; for the liquefaction of this gas we had therefore to avail ourselves of cooling by adiabatic expansion. In Comm. N°. 23 of Jan.’96 1 made some remarks on what could be derived from van DER WaaLs’ law of corresponding states for the liquefaction of hydrogen following this method. I had found that an apparatus to liquefy hydrogen beginning with — 210°C. might be constructed almost after the same model as an apparatus that had proved suitable for the liquefaction of oxygen beginning with ordinary temperatures and without any further frigorifie agents. My efforts, however, to obtain an apparatus for isentropic cooling by combining to a regenerator the outlet- and inflow-tubes of a small expansion motor, fed with compressed gas, had failed. Therefore I directed my attention towards the then newly published (1896) application of the Jouur-KeLviN process (Linpr’s apparatus for liquefying air and Drwar’s jet of hydrogen to solidify oxygen). Though the process of LiNpe was the most promising, because he had succeeded with his apparatus to obtain liquid air statically, yet it was evident that only the principle of this method could be followed. ( 157 ) The cooling of an apparatus of dimensions like the first of Linpr (weight 1300 kilogrammes) by means of liquid air (oxygen) evapo- rating in vacuo could not be thought of. And yet, according to what has been said above, this had to be our starting point. It rather lay to hand to magnify the spiral (enclosed in a vacuum glass) such as Dewar had used for his jet of hydrogen to solidify oxygen, and so to get an apparatus with which air could be liquefied, and which could then serve as a pattern for an apparatus to liquefy hydrogen. It was indeed a similar construction with which in 1898 Dewar had statically liquefied hydrogen for the first time. About the installation which apparently afterwards enabled Dewar to collect large quantities of liquid hydrogen nothing further has come to my knowledge. The arrangement of the Leiden hydrogen circulation is based on Dewar’s principle to place the regenerator spiral into a vacuum glass (1896). As to the regenerator spiral itself Hampson’s apparatus for liquefying air (1896) has been followed because it appeared that the proportions of this spiral have been chosen very favourably, and with its small dimensions and small weight it is exceedingly fit, according to the thesis menticned above, to serve as a model for a regenerator. spiral to liquefy hydrogen of about — 205° at expansion from a higher to the ordinary pressure. The other physicists, who after Drwar have occupied themselves with liquid hydrogen, — Travers 1900 and 1904, OrszrwskKr 1902, 1904 and 1905 (the latter rather with a view to obtain small quantities in a short time with simple accessories) — have also built their apparatus after this model. The Leiden hydrogen liquefactor for constant use has enough peculiar features to occupy a position of its own as an independent construction by the side of the apparatus of TRAveERrs and Orszewskr, which do not satisfy the requirements for the Leiden measurements. Moreover I was the first to pronounce the principle according to which this apparatus is built and from which follows that the regenerator spiral fed with hydrogen that has been cooled by liquid oxygen (air) evaporating at a given low pressure, must lead to the goal. The problem of making a circulation in order to maintain a bath of liquid hydrogen —- and of this problem the arrangement of the liquefactor for constant use (which, tested with nitrogen, has really proved efficient) is only a part — has not yet been treated by others. That also at Leiden we had to wait a long time for its solution cannot be wondered at when we consider the high demands which, I held, had to be satisfied by this cycle. For with a view to the intended measurements I thought it necessary to pour a bath of (158) 1.5 liter into the cryostat (described in VIII of the series “Methods and apparatus used in the Cryogenic Laboratory” of these commu- nications) and to keep it to within 0°.01 at a uniform and constant temperature. The requirements were therefore very much higher than they had formerly been for the bath of liquid oxygen. These require- ments could by no means be fulfilled before I had the disposal of a vacuum pump (mentioned as early as Jan. ’96 in Comm. N°. 23), (comp. Comm. N°. 83, March ’03), suitable to evaporate in a short time large quantities of liquid air at a pressure of a few centimeters, and before I possessed compressors for constant working with ex- tremely pure hydrogen. With the former instrument and the com- pressors, described in $ 3, the liquefactor, described in § 2, delivers 3 a 4 liters of liquid hydrogen per hour. Thus I was able to bring to this assembly (28 May ’06) 4 liters of liquid hydrogen prepared at Leiden the day before and to use it in several experiments. Our installation proved quite satisfactory for operations with the afore mentioned eryostat. After we had succeeded in making with it some measurements in liquid hydrogen boiling under ordinary and under reduced pressure the vacuum glass of the cryostat cracked and only by mere accidence the measuring apparatus were spared. Therefore we have constructed another modified cryostat, to be described in XII, which besides insuring the safety of the measuring apparatus has the advantage of using less liquid hydrogen than the cryostat, described in VIII (Comm. N°. 94% June ’05). This new cryostat entirely satisfies the requirements; the temperature is kept constant to within 0°,01. It is noteworthy that while the measure- ments are being made the cryostat shows in no way that we are working with a bath of no less than 1.5 liter of liquid hydrogen. I wish to express thanks to Mr. G. J. Fur, mechanist at the cryogenic laboratory, for his intelligent assistance. Under his super- vision the liquefactor and cryostat, to be described in the following sections, and also other accessories have been built upon my direc- tion in the workshop of the laboratory. § 2. The hydrogen liquefactor for constant use. a. The apparatus does not yet entirely realize the original design’). 1) It might be improved by dividing the regenerator spiral in several successive coils, each opening into the next with its own expansion-cock, where the pressures are regulated according to the temperatures. Compare the theory of cooling with the Joure-Kervin process and the liquefying by means of the Linpe process given by van DER Waars in the meeting of Jan. 1900. ( 159 ) The latter is represented schematically by fig. 1 on Pl. I and hardly requires further explanation. The compressed hydrogen goes successively through the regenerator coils D,, D,, D,, D,, C, B, A. B is immersed partially in a bath of liquid air which, being admitted through P, evaporates at a very low pressure; D,, D,, C and A are surrounded by hydrogen expanding at the cock M/, and D, and D, by the vapours from the airbath in /. As, however, we can dispose of more liquid air than we want fora sufficient cooling of the admitted hydrogen, and the vacuum pump (comp. Comm. N°. 83, March ’03) has a greater capacity than is required to draw off the evaporating air') at reduced pressure, even when we sacrifice the regenerator working of the spirals D,, D,, D, and D,, we have for simplicity not yet added the double forecooling regenerator D, by means of which a large quantity of liquid air will be economized, and hence the apparatus consists only of one forecooling regenerator C, the refrigerator / with cooling spiral B and the principal regenerator A in the vacuum glass £ with a collecting vessel LZ, placed in the case WV, which forms one complete whole with the case U. b. The principal regenerator, Pl. I fig. 2, consists of 4 windings of copper tubing, 2.4 m.m. in internal diameter and 3.8 m.m. in external diameter, wound close to each other and then- pushed together, indicated by A,, A,, A, and A,, (number of layers 81; length of each tube 20 M.). As in the ethylene regenerator (Comm. N°. 14, Dee. °94, and description of Marutas *), fig. 1/’) and in the methyl chloride regenerator (Comm. N°. 87, March ’04, Pl. I) the windings are wound from the centre of the cylinder to the circumference and again from the cireum- ference to the centre round the cock-carrying tube J/,, and are enve- loped together in flannel and fit the vacuum glass Z, (the inner and outer walls are marked with #,, and #,,). Thence the liquid hydrogen flows at #, into the collecting vessel Z,. At M , the four coils are united to one channel which (comp. cock 7’ in fig. 3 of Marnras’ description le.) is shut by the pivot point J/,, moved by the handle M,,. The packing M, hermetically closes the tube M, at the top, where it is not exposed to cooling (comp. Marnras’ description Le). The hydrogen escapes at the side exactly as at the ethylene cock JZ, fig. 2 in Marmras’ description le, through 6 openings M,, and is prevented from rising or circulating by the screens JM/,, and J/,,. c. The new-silver refrigerator case #, is suspended in the new- 1) When using oxygen we might avail ourselves of cooling down to a lower temperature, which then must be carried out in two steps (comp. § 40). 2) Le laboratoire cryogéne de Leyde, Rev. Gen. d. Sc. Avril 1896. ( 160 ) silver case U,, from which it is insulated by flannel U,,. A float F,, indicates the level of the liquid air, of which the inflow is regulated through the cock P,, with pivot P,, and packing P, identical with the cock mentioned above, except that the glass tube with cock is replaced by a new-silver one P,. The evaporated air is drawn off through a stout copper tube F, (comp. § 46). The 2 outlet tubes 6,, and B,, of the spiral B,, and B,, (each 23 windings, internal diameter of tube 3.6 m.m., external diameter 5,8 m.m., length of each 6 M.) are soldered in the bottom. The two inflow tubes B, and £,, are soldered in the new-silver cover, on which the glass tube /’, covering the index F,, of the cork float F,, are fastened with sealing wax (comp. for nitrogen Comm. N°. 83 IV, March ’03, Pl. VII). d. The forecooling regenerator spiral C,, C,, C,, and C, is wound in + windings like A, wrapped in flannel and enclosed in the cylinder of the new-silver case U. The four windings (internal diam. of the tubing 2.4 m.m., external diam. 3.8 m.m., number of layers 81, length of each tube 20M.) branch off at the soldered piece Ci, from the tube C,,, soldered in the cover of U,. They unite to the two tubes C,a and C,b through which the hydrogen is led to the refri- gerator. The axis of this spiral is a thin-walled new-silver tube C, shut at the top. The hydrogen blown off is expelled through the tube U, e. The liquid hydrogen is collected in a new-silver reservoir L,, fitting the vacuum glass Z,, which by means of a little wooden block V, rests on the wood-covered bottom of the insulated case V,, which is coated internally with paper V,, and capoc V,,. Thanks to ZL, the danger of bursting for the vacuum glass is less than when the hydrogen should flow directly from Z, into the glass L,,. This beaker moreover prevents rapid evaporation in case the glass should burst (comp. § 1). The level of the liquid hydrogen is indicated by a float Z,,,, which by means of a silk cord Z,,, slung over the pulleys Z,, and £,, is balanced by an iron weight JZ,,, moving in a glass tube V,,, which can also be pulled up and down with a magnet from outside. The float is a box L,, of very thin new-silver, the hook Z,,, is a bent capillary tube open at both ends and soldered in the cover. The glass V,, fits by means of india rubber on the cylinder VV, which is connected with the case by means of a thin- walled new-silver tube V,,. The hydrogen is drawn off through the new-silver siphon tube N,,, which is continued as the double-walled tube N,, V,,,, leading 501? ( 161 ) towards the delivery cock JN. Here, as at the ethylene cock (description of Maruias le. fig. 2), the packing N, and the screw- thread are in the portion that is not cooled. The pin N,, made of a new- silver tube, passes through the cock-carrying tube V,. Both the outlet tube MN, and the delivery cock NV, are surrounded by a portion of the cold hydrogen vapours, which to this end are forced to escape between the double wall of the tube through N,,, and along Kha (Kd on Pl. II). The outer wall Ns, V;,, of the double-walled tube is insulated from the side tube V,, at the case V,, by means of wool. The glass Z is covered with a felt cover Z,, fitted at the bottom with a sheet of nickel-paper to prevent radiation towards the liquid hydrogen. This cover fits tightly on the lower end ZE, of E and rests on the tube J,,, and the pulley-case J,,. f. We still have to describe the various safety arrangements to prevent the apparatus frem bursting when the cock J/ should sud- denly admit too much gas, as might occur when the opening has been blocked by frozen impurities in the gas, which suddenly let loose or when one of the tubes breaks down owing to the same blocking or an other cause. For this purpose serves in the first place the wide glass tube W,, which ends below mercury. The quantity of gas which of a sudden escapes, and the great force with which the mercury is sometimes flung away rendered it necessary to make a case W,, with several screens W,, all of varnished card-board to collect the mercury and to reconduct it into the glass W, (where a sufficient quantity of it must be present for filling the tube during the exhaustion). If the pressure in the reservoir rises higher than that for which the safety tube is designed, the thin-walled india rubber tube V,,, which is drawn over the perforated brass cylinder wall V,, (separated from it by a thin sheet of tissue-paper), breaks. The safety apparatus is connected with the case V, by a wide new-silver tube V In order to avoid impurities in the hydrogen in the liquefactor through diffusion of air the india rubber cylinder V,,, that is drawn over the rings V,,, and V,,, after being exhausted is filled through the cock V,, with hydrogen under excess of pressure; during the exhaust the india rubber cylinder V,,, is pressed against the india rubber wall V,,. An arrangement of an entirely identical construction protects the case U,, which encloses the principal regenerator, and the case U, which encloses the forecooling regenerator C. As to the protection against pressure which may occur in conse- quence of evaporation of air, it was sufficient to protect the refri- ( 162 ) gerator space F by means of the tube Y opening below mercury. g. In protecting the different parts against heat from the sur- rounding atmosphere, care has been taken that those surfaces of which the temperature might fall below the boiling point of air and which are not sufficiently protected by the conduction from less cooled parts, should not come into contact with air but only with hydrogen. The refrigerator vessel /’, for instance, is surrounded with the hydrogen which fills the cases U and V; hydrogen is also to be found in the space between the vacuum glass Z and the wall of the case V; and lastly a side tube V,, and V,, branches off from the case V in order to surround with hydrogen the double-walled siphon tube N,,, V,,, and the double walled cock N,, N,,,. The new-silver case V, from which the vacuum glass Z is insulated by layers of paper V,, and the refrigerator vessel /” by a layer of flannel, and in the same way the new-silver case U, are further pro- tected from conduction of heat from outside by separate wrappings of capoe V,,, packed within a card-board cover V,, pasted together. To prevent condensation of water vapour, the air in this enclosed space communicates with the atmosphere by means of a drying tube t.dr filled with pieces of sodium hydroxide, as in the ethylene- and methyl chloride regenerators (comp. above sub 6). The air-tight connection between the case U and the case V is effected by the india rubber ring Ua, which fits on the glass and on the strengthened rims U,, and V,, of the new-silver cases. India rubber of somewhat larger dimensions can only be used for tightening purposes when it is not cooled. In this case the conduction along the new-silver wall, which is insulated from the vacuum glass by layers of paper, is so slight that the ring-shaped strengthened rims remain at the ordinary temperature and the closure can be effected by a stout stretched india rubber ring. When the india rubber is only pressed on the glass this closure is not perfectly tight; therefore the whole connection is surrounded with an atmosphere of almost pure hydrogen, which is obtained and maintained by the india rubber ring Ue, which fits tightly on U, and V, and which is filled with hydrogen under excess of pressure through the cock Ud. Thanks to the small conduction of heat of new-silver no cooling is to be feared for the connections of V,, and U,, no more than for the packings of the cocks M, and N,. h. The cases V and U are joined and form one firm whole by the three rods Ub with the serew-fastenings U, and V,,. The vacuum glass Z,, held by the india rubber ring Ua, rests with a wooden 0? ring #, and a new-silver cylinder U;, against the refrigerator vessel /. ( 163 ) The whole construction can stand exhaustion, which is necessary to fill the apparatus with pure hydrogen. After the case U, of which the parts U, and U, are connected together by beams, and the case V are mounted separately, the vacuum glass / is placed in position and the case V is connected with the case U. The entire lique- factor is suspended from the ceiling by means of some rods and is particularly supported by the stout outlet tube /’, for air and the outlet tube U, for hydrogen. Plate II represents the circulation schematically: the pieces of appa- ratus in their true proportions, the connections only schematically. The liquefactor is designated by the letters £iq. The compressed hydro- gen is admitted through Ac, the hydrogen blown off is let out through Khd or Khe. 7. Before the apparatus is set working it is filled with pure hydrogen (the cock Jf being open) by means of exhaustion and admission of pure hydrogen along Kc. In the drying tubes Da and Dh the pure hydrogen is freed from any traces of moisture which it might have absorbed. § 3. The compressors and the gasometers. a. The hydrogen is put under high pressure by means of two compressors in each of which the compression is brought about in two steps. While other physicists use compressors with water injection running at great speed of the same kind as I have formerly arranged for operations with pure gas (comp. Comm. N°. 14 of Dec. ’94, $ 10, and N°. 51, Sept. ’99, § 3), I have used for the hydrogen circulation slowly running compressors (see Pl. IL © at 110 and at 80 revo- lutions per minute) which are lubricated with oil. To enable constant working with hydrogen the highest degree of purity of the gas is required. For if air is mixed with the gas it is deposited in the regenerator spiral and when some quantity of it is collected there it will freeze and melt alternately through the unavoidable variations of temperature in different parts of the spiral, so that even small quanti- ties, taking into consideration that the melted air flows downward, necessarily must cause blocking. And such small quantities of air may easily come in through the large quantity of injection water which is necessary for the above mentioned compressors with water injection or may penetrate into the pieces of apparatus ~lich are required when the same injection water is repeatedly used. Lastly the chance of losing gas is much smaller with the last mentioned compressors ( 164 ) and the manipulation much easier. These compressors are made very carefully by the BurckHARDT company at Basel. In the first compressor (© Pl. II, displacing 20 M*® per hour) the gas is raised in the first cylinder (double-acting with slide) from 1 to 5 and in the second cylinder (plunger and valves) from 5 to 25 atmospheres; in the second compressor $ (plunger and valves) in the first cylinder from 25 to 50 and in the second from 50 to 250 atmospheres. After each compression the gas is led through a cooling spiral. With the two first cooling spirals (those of © Pl. I) an oil- separator is connected. Safety-valves lead from each reservoir back to the delivery; moreover the packings are shut off with oil-holders (Comm. N°. 14 94 and N°. 83, Pl. VIID. The hydrogen that might escape from the packing at is collected. b. The high pressure compressor forces the hydrogen through two steel drying tubes Da and 6 filled with pieces of sodium hydroxide (comp. § 2, 7, and Pl. ID), of which the first also acts like an air- chamber for the regenerator spiral. As in all the operations the gas (comp. c) originally is almost dry and comes only into contact with oil, we need only now and then run off a small quantity of concentrated sodium hydroxide solution. c. For the usual working the compressors suck the gas from gasometers. If these should float on water the separation of the water vapour, which is inevitably taken along by the large quantities of gas displaced, which constantly come into contact with water, would give rise to great difficulties in the compression. Therefore we have used for this purpose two zinced gasometers, Gaz aand Gaz 6, Pl. I, with tinned welds (holding each 1 M.*) floating upon oil *), which formerly (comp. Comm. N°. 14, Dec. °94) have been arranged for collecting ethylene ’). The cock Kpa (Kpb) is immersed in oil; likewise the connection of the glass tube, through which the oil of the gasholder can be visibly sucked up till it is above the cock, with the cover are immersed in oil. The india rubber outlet tube and the connection with the 1) The drawing sufficiently represents the construction which has been followed for economizing oil. The gasometers can be placed outside the laboratory and therefore they are protected by a cover of galvanized iron and curtains of tarred canvas, which can be drawn round them. 2) Formerly it was of the utmost importance that ethylene could be kept pure and dry in the gasometers. But now the purifying of ethylene through freezing in liquid air (comp. Comm. N°. 94e IX § 1) has become a very simple operation and weldless reservoirs for the storage of the compressed gas are obtainable in all dimensions. ( 165 ) copper exhaust tube are surrounded by a second india rubber tube filled with glycerine. From the cock onward the conduction can be exhausted; to prevent the tube from collapsing during the exhaust a steel spiral has been placed in it. A float with valve Kph (Kp?) prevents the oil from being drawn over into the apparatus. Besides these gasometers we dispose of two other gasometers holding 5 M*® each to collect hydrogen of a less degree of purity. They are built following the same system as the zinced gasometers for the economizing of liquid, carefully riveted and caulked and float on a solution of calcium chloride. The oil-gasholders serve only for the storage of very pure hydrogen and this only while the apparatus is working. During the rest of the time the pure hydrogen is kept in the known steel bottles shown-on Pl. IL at Itha. When we wish to liquefy hydrogen, this is blown off into the gasometer through Ky (Khe, Kpe and Kpb for instance to Gaz b), after this gasometer, which has been left standing filled with hydrogen, is washed out on purpose with pure hydrogen. When we stop working the hydrogen by means of & and His repumped along Kpf and Ape through Ka and Kf into the reservoirs Nha. The gasometers may be connected with the pumps or the liquefactor either separately or together. The former is especially required when the cryostat is worked (comp. XII) and for the purification of hydrogen (comp. XIV). § 4. The cooling by means of liquid air. a. The liquid air is sucked into the refrigerator vessel FY (PL. I), which by As (Pl. Il) is coupled to the vacuumpump 8, along the tube Pb connected with the siphon of a vacuum bottle Ya con- taining liquid air. ; This has been filled by catching the jet of liquid air from the apparatus (PL IV, fig. 2) in which it is prepared (comp. XIII), into the open glass (see the annexed fig. 1) and is kept, covered with a loose felt stopper m (fig. 1). To siphon the liquid air into the apparatus, where it is to be used, the stopper is replaced by a cap h (fig. 1) with 3 tubes; one of these d is designed to raise the pressure in the bottle with a small handpump, the other c is connected to a small mercury manometer, and the third 6 reaches down to the bottom, so that the liquid gas can be let out. (When the bottle-is used for other liquid gases, d is used for the outlet of the vapours and ec for the admission of the liquid gas). One of the first two tubes reaches as far as the neck. It may also be used 12 Proceedings Royal Acad. Amsterdam. Vol. IX. ( 166 ) to conduct liquid air from a larger stock into the bottle. With the cap a closed glass tube 6 is connected, in which an index of a cork float dr indicates the height of the liquid. The caps, as shown in fig. 1, were formerly blown of glass and the three tubes were fastened into it by means of india rubber. After- wards the cap /,, as shown in fig. 2, with the three tubes and with a double wall h, of very thin new-silver have been soldered to form one whole, which is fastened on the bottle with an india rubber ring £. The space between the walls is filled with capoc kh, and the whole piece rests on the neck of the bottle by means of a wooden block 7. After it is placed on the bottle the cap is wrapped round with wool. With a view to the transport the vacuum glass is placed in a card-board box with fibre packing. When the siphon is not used it is closed with a piece of india rubber tubing, fitted with a small stopper. When we wish to (54671) siphon over, this stopper is removed and the inflow tube P4 (PI. I) is connected with the siphon-tube 5 (fig. 2) with a piece of india rubber tubing. To prevent breaking of the india rubber, which through the cold has become brittle, the new-silver tubes are arranged so that they fit into each other, hence the india rubber is not strained so much. The admission of liquid air into the refrigerator vessel is further regulated with the cock P, Pl. I. When the float indicates that the reservoir is almost empty, another reservoir is put in its place. The cock Ks is regulated according to the readings on the mercury manometer tube Y. 6. The air is caused to evaporate at a pressure of 15 mm., which is possible because a BurckHarpt-Weiss-pump & Pl. II is used as vacuumpump. The vacuumpump is the same as that used in measurements with the cryostat containing a bath at —217° (comp. Comm. No. 94? June ’05) and has been arranged to this end as described in Comm. No. 83 V. March ’03. The letters at 5 on Pl. Il have the same meaning as on Pl. VIII of Comm. N°. 83. As has been described in Comm. No. 94¢ VIII, June ’05, this vacuumpump 4%, displacing 360 M* per hour, is exhausted by a small vacuumpump, displacing 20 M* per hour’) (indicated by & on Pl. ID. §5. How the liquefactor is set working. a. When the apparatus is filled with pure hydrogen, as described in § 2, and when air evaporating under low pressure is let into the refrigerator, for convenience tbe hydrogen, admitted through © and 5 Pl. II along Ke, is caused to stream through during some time with wide open cock JM, PI. I, for the forecooling of the whole apparatus. Then the cock M/ is regulated so that the pressure in the regenerator spiral rises slowly. It is quite possible for the appa- ratus to deliver liquid hydrogen at 100 atm., it has done so at 70 atm. As a rule, bowever, the pressure is kept between 180 and 200 atm. because then the efficiency is some times larger *). The liquefactor then delivers about 4 liters liquid hydrogen per hour. Part of the hydrogen is allowed to escape along Aha Pl. I fig. 2 (Kd Pl. II) for the forecooling of the siphon N,, Pl. I and the cock N. As soon as liquid hydrogen begins to separate we perceive that the 1) When we use oxygen (comp. § 2 note 2), and a pressure as low as a few mm. should be required, forecooling is required in the second refrigerator like F, where oxygen evaporates under low pressure, for instance towards §. 2) v. p, Waars has shown the way how to compute this (comp. note 1 § 2). 12% ( 168 ) cock M must be tightened a little more in order to keep the pressure within the same limits. When liquid hydrogen collects in Z rime is seen on the tube Ns El Tes 2 meareime cock sy. b. The gaseous hydrogen escapes along Ahd (Pl. ID) to © and to one or to both gasholders. When liquid hydrogen separates, the compressor © receives, besides the hydrogen escaping from the liquefactor, a quantity of hydrogen from the gasholders along Kpa and Kpb. New pure hydrogen is then admitted from Nha, Pl. II, along Kg. c. The float (Z,,, Pl. I) does not begin to indicate until a fairly large quantity of liquid hydrogen is collected. § 6. The siphoning of liquid hydrogen and the demonstration of liquid and solid hydrogen. a. When the float Z,,,, Pl. 1, shows that the glass is filled to the top (this usually happens an hour after the liquefactor is set working) the hydrogen is siphoned into the vacuum glasses Hydr a, Hydr b etc., Pl. II, which are connected behind each other so that the cold hydrogen vapour, which is led through them, cools them successively before they are filled. When one is full the next is moved one place further. They are fitted with caps of the same description as the bottles for siphoning liquid air, figs. 1 and 2 in the text of $4. Pl. III represents on a larger scale 2 bottles coupled behind each other and a third which has been filled, all as on Pl. I, in side- and top-elevation. The evaporated hydrogen escapes along d', and d", and further along K, (see Pl. Il) to the gasholder. The letters of the figures have the same meaning as in fig. 2; for the explanation I refer to the de- scription of that figure in § 4. The conduction of heat in the thin new-silver is so little that the new-silver tubes can be soldered in the caps h, and that they are sufficiently protected by a double wall A, of new-silver with a layer of capoc between, which is again thickly enveloped in wool. It has occurred that the india rubber ring 4’ has burst through the great fall of temperature, but in general the use of india rubber has afforded no difficulties, and hence the somewhat less simple construction, which would lie to hand, and through which we avoid cooling of the india rubber at the place where it must fit, has not yet been made. b. If we desire to see the jet of liquid hydrogen flowing from the cock i Cc. | (469) N, Pl. I, we connect with the tube N, and the india Fig. 3 rubber tube d,, instead of the silvered flasks of Pl. If and Pl. Ill, a transparent vacuum cylinder fig. 3a, closed by an india rubber ring with a new- silver cap with inlet tube. After the cock is opened the india rubber outflow tube d, covers with rime and becomes as hard as glass; soon the first drops in spheroidal state are seen splash- ing on the bottom of the glass and the lively liquid fills the glass. If, as shown by fig. 35, a glass cover is placed on the top, the glass may be left standing in the Open air without the air con- densing into it, which would hasten the evaporation. In the same manner I have sometimes filled non-silvered vacuum flasks holding 1 liter, where the liquid hydrogen boils vividly just as in the glass mentioned before. The evaporation is of course much less and the rising of the bubbles stops when the vacuum glass or the vacuum flask is placed in liquid air. t Fig. 4 To demonstrate the pouring of hydrogen from one open vessel into the other, I use a glass, cap round which a collar of thin india rubber sheet is bound (comp. the accompanying fig. +). The flask from which and the glass into which we want to pour, the latter after being filled with liquid air and quickly turned down and up again (if this is not done quickly a blue deposit of H,O from the air will come in), are placed under the cap, which fills with hydrogen and hence remains transparent, then with the india rubber round the neck of the bottle and round the glass we take hold of the two, each in one hand. Through the cap we can observe the pouring. The escaping hydrogen rises in the air as clouds. In order to keep the half filled glass clear it is covered, under the pouring off cap, with a glass cap, and so it can be taken away from the pouring off cap. c. It is very instructive to see what happens when we proceed to remove this cap and the glass is tilted over a little. Above the level of the liquid hydrogen thiek snowy elouds of solid air are formed, the minute solid particles drop on the bottom through the extremely light hydrogen (specific weight */,,), there they collect to a white pulver which, when the hydrogen is shaken, behaves as heavy sand would behave in water. When the hydrogen is evaporated that sand soon melts down to liquid air *). d. Solid hydrogen may be easily demonstrated when we place the glass, fig. 3a, under a bell as fig. 3c in which a wire can be moved up and down (for instance by fastening it into an india rubber tube) and connect the bell with the airpump. A starch-like white cake is soon formed, which can be moved up and down with the wire. e. To fill a vacuum flask as shown on Pl. LI we first cool it by washing it out with liquid air. The connection at MN, Pl. I fig. 2 and Pl. III, is brought about simply by drawing a piece of india rubber tubing N,, over the new-silver tubes NM, and C, fitting into each other, round which flannel is swaddled. This again is enveloped in loose wool. When some bottles are connected they are filled with pure hydrogen through the tube 6, of Hydr. a after repeated exhaustion and care is also taken that each newly connected bottle is filled with pure hydrogen and that no air can enter the apparatus while the connections are being made. When from the indications of the float Z,,, (PL. I, fig. 2) we conclude that a bottle is full, it is disconnected, but as long as the liquid hydrogen is kept in this glass the evaporating hydrogen is allowed to escape into the gasholder, as is represented by PI. III for Hydr. ce. The disconnection at MN, is simply effected by taking off the flannel band C,, heating the piece of india rubber tubing J,, (unvoleanized) with one’s fingers (or with a pair of pinchers arranged to this end) till it becomes soft again and can be shoved from the tube AN, § 7. Transport to the cryostat, closure of the cycle. a. The vacuum glasses filled with liquid hydrogen (see Hydr. d on PL. II) are transported to the room where the cryostat €7 is mounted 1) All this has been demonstrated by me at the meeting of 28 May. To show the small specific weight of hydrogen I held a very thin-walled glass bulb, which sinks only a little in ether (as a massive glass ball in mercury), suspended by a thin thread in the glass with liquid hydrogen, where it fell like a massive glass ball in water and tapped on the bottom. (4715 into which the hydrogen is siphoned. To this end the tube 6", of PL. II is connected (again by a piece of india rubber tubing, enveloped in flannel and wool) to the inflow tube a, of the cryostat and the tube d, to an inflow tube of pure hydrogen under pressure, which is admitted from RAe, Pl. Il, along Awa. With all these connections and disconnections care must be taken that there should always be an excess of pressure in the tubes that are to be connected, that the disconnected tubes should be immediately closed with stoppers but that first the apparatus after having been exhausted should prelimi- narily be filled with pure hydrogen. The liquid hydrogen is not admitted into the cryostat Cr until the latter has been cooled — coupled in another way (see the dotted line on Pl. Il) — by means of pure hydrogen which has been led from Jthe through a cooling tube immersed in liquid air. This refrigerator is of a similar construc- tion as the nitrogen condenser Pl. VII of Comm. N°. 83 (March ’03). Instead of Nig should be read H, and instead of Ox lig, Aer lig, which is siphoned from the vacuum flask Ye. (comp. § 6). During the siphoning of the liquid hydrogen into €r the rapidity of the influx is regulated after a mercury manometer, which is con- nected with the tube c on the cap A, Pl. III (comp. fig. 2 of § 4). 6. From the eryostat the evaporated hydrogen escapes along Y,, into the compressor ©, Pl. II, which can also serve as vacuumpump and which precautiously through > and Kf at the dotted connection Af stores the gas, which might contain minute impurities, in the separate reservoir hd; or it escapes along Y,, and Kpe or Kpd into the gas- holders Gaz a or Gaz b. [ XI. The purification of hydrogen for the cycle. a. This subject has been treated in Comm. N°. 94d IX. To be able always to obtain pure hydrogen, to make up for inevitable losses, and lastly to be freed from the fear of losing pure hydrogen, which perhaps might deter us from undertaking some experiments, a permanent arrangement for the purification has been made after the principle laid down in IX. The apparatus for the purification is represented on Pl. IV and is also to be found on Pl. II at 3. The impure hydrogen from thd is admitted through An and along a drying tube into a regenerator tube (see Pl. IV) consisting of two tubes enclosing each other concentrically, of which the outer a serves for the inflow, the inner 4 for the outlet. Outside the apparatus a and 6 are separated as a, and b,, within the apparatus from the point ¢ downwards a is continued as a, and subsequently as the spiral (ago a, to terminate at the top of the separating cylinder d, from which the gas escapes through 4,, and the impurities separated from the hydrogen as liquid escape along e and Am (comp. Pl. W). The liquid air, with which the cooling tube and the separating cylinder are cooled, is admitted along / and the cock m (and drawn from the vacuum glass Ub, Pl. II); a float dr indicates the level of the liquid air. The eva- porating air is drawn off by the vacuumpump %& (PL. ID) along Kz. The refrigerator vessel p is protected against heat from outside by a double wall g of new-silver with capoe v packed between, of which the lower end is immersed in a vacuum glass 7, while the whole is surrounded with a layer of capoe enclosed in a varnished cover of card-board pasted together in the same way as for the hydrogen liquefactor. The glass tube Y, opening below mercury, serves among others to read the pressure under which the evaporation takes place. The cock Am is turned so that some more bottles of known capacity are collected of the blown- off gas than, according to the analysis, would be formed by the impurities present in the gas. In this way the purity of the hydrogen is brought to */,, °/,. It is led along Al to the gasholders, and compressed by © and SH in 2thd. 6. A second purification is effected in the following manner. When we have operated with the liquefactor with pure hydrogen we always, after the experiments are finished, admit a portion of this not yet quite pure gas into the apparatus. After some time, usually after 4 liters of liquid hydrogen are formed, the cock is blocked. As soon as it becomes necessary to move this repeatedly to and fro — Travers and OLsznwski say that this is constantly necessary but I consider it as a sign that the apparatus is about to get more and more disordered — the work is suspended and the cock M (Pl. I) closed, after which D, and ®, (PL II) are blown off to the gas- holders along Ke, and K,, and A, is shut. The liquid hydrogen, after being siphoned, is allowed to evaporate and to pass over into the gasholder for pure hydrogen. The impurities are found when, with M and XK, closed, we return to the ordinary temperature and analyze the gas, which in D has come to high pressure. If necessary, the purified hydrogen is once more subjected to this process. When, after the liquefactor with pure hydrogen has been worked, we go on admitting a quantity of preliminarily purified hydrogen of Io /o and take care that the impurities are removed, we gradually obtain and maintain without trouble a sufficient quantity of pure hydrogen. H. KAMERLINGH ONNES. Methods and apparatus used in the cryogenic labora- tory at Leiden. X. How to obtain baths of constant and uniform temperature by means of liquid hydrogen. ENT: . | Roald | | pelo 9000 Ales | Bessa Noy! ae | Bees, Mr } | 9900009.) 1, | 9999 ie be En Us | 9200000}. ol U ze | |o | a Gu | aw Cet Ge Ge | oO GS | o OR it | | oo Geri ' oo kt | Or 8 en H. KAMERLINGH ONNES. Methods and apparatus used in the cryogenic labora- tory at Leiden, X. How to obtain baths of constant and uniform temperature by means of liquid hydrogen. JPL IG Proceedings Royal Acad. Amsterdam. Vol. IX. H. KAMERL tempe Pre it. H. KAMERLINGH ONNES. Methods and apparatus used in the cryogenic laboratory at Leiden. X. How to obtain baths of constant and uniform temperature by means of liquid hydrogen. Proceedings Royal Acad. Amsterdam. Vol. IX. Methods and apparatus used in the cryogenic laboratory H, KAMERLINGH ONNES. X. How to obtain baths of constant and unifcrm temperature by means of liquid hydrogen. eo _ b2 at Leiden. Pl. TEE 4 ( U 5 \ - J - ‘ Al hen sii, Gek) BD Nik SH iS Ko ee i OW SY >. Xe SIDES MS > des Mis ff. es CZ) S - i= rel har (UW 0K tt) *f 9) ) DIC} GS Proceedings Royal Acad. Amsterdam. Vol IX. H. KAMERLINGH ONNES. Methods and apparatus used in the cryogenic laboratory at Leiden. XI. The purification of hydrogen for the cycle. PIAL: ROU af eG MANE Proceedings Royal Acad. Amsterdam. Vol. IX. INGH ONNES. Methods and apparatus used in the cryogenic laboratory at Leiden. Cryostat especially for temperatures from — 252° to — 259°. Pala Pl. - 252° to — 259°. LINGH ONNES. Methods and apparatus used in the cryogenic laboratory at Leiden. Cryostat especially for temperatures from xl : d at Bj Proceedings Royal Acad. Amsterdam. Vol. IX. (173 ) XII. Cryostat especially for temperatures from — 252° to — 259°. § 1. The principle. In X $ 1 I have said that we succeeded in pouring into the cryostat of Comm. N°. 94¢ VII a bath of liquid hydrogen, maintaining it there and making measurements in it, but then the vacuum glass cracked. By mere chance it happened that the measuring apparatus which contained the work of several series of measurements came forth uninjured after removal of the sherds and fragments of the vacuum glass. With the arrangement which I am going to describe now we need not be afraid of an adversity as was imminent then. Now the bath of liquid hydrogen is protected against heat from outside by its own vapour. The new apparatus reminds us in many respects of that which I used to obtain a bath of liquid oxygen when the vacuum glasses were not yet known; the case of the cryostat then used has even been sacrificed in order to construct the apparatus described now. The principal cause of the cracking of vacuum glasses, which I have pointed out in several communications as a danger for placing precious pieces of apparatus into them are the great stresses caused by the great differences in temperature between the inner and the outer wall and which are added to the stresses which exist already in consequence of the vacuum. To the influence of those stresses it was to be ascribed, for instance, that only through the insertion of a metal spring the vacuum tubes (described in Comm. N°. 85, April ’05) could resist the cooling with liquid air. It some- times happens that a vacuum flask used for liquid air cracks without apparent cause and with the same cooling the wide vacuum eylinders are still less trustworthy than the flasks. At the much stronger cooling with liquid hydrogen the danger of cracking increases still. Habit makes us inclined to forget dangers, yet we should rather wonder that a glass as used for the cryostat of Comm. N°. 944 VIII filled with liquid hydrogen does not crack than that it does. In the new cryostat of Pl. V the cause of the cracking of the vacuum glass has been removed as much as possible and in case it should break in spite of this we have prevented that the measuring apparatus in the bath should be injured. The hydrogen is not poured directly into the vacuum glass 5’,, but into a glass beaker Ba, placed in the vacuum glass (comp. Comm. N°. 23, Jan.’96 at the end of § 4) but separated from it by a new-silver case, which forms, as it were, a lining (see X, £ Pl. I). Further the evaporated hydrogen is led along the outer wall of the vacuum glass B’,,. To be able to work (174) also at reduced pressure and to prevent any admixtures of air from entering into the pure hydrogen used, the whole bath has been placed in a stout cylindrical copper case Ub, which can be exhausted. This cryostat is especially fit for hydrogen, yet may profitably replace those described till now, at least when it is not necessary that we should see what takes place inside the bath. A modified pattern, where this has become possible, in the same way as in the cryostat with liquid oxygen of Comm. N°. 14, Dec. ’94, I hope to describe erelong. In the cryostat now to be described, as in the former, the meas- uring apparatus, without our changing anything in the mounting of them, will go through the whole range of temperatures from — 23° to — 90° with methyl chloride, from — 103° to — 160° with ethylene, from — 183° to — 217° with oxygen and from — 252° to — 259° with hydrogen (only for the temperatures between — 160° and — 180° we still require methane). § 2. Description. a. The new cryostat is represented on Pl. V. The letters, in so far as the parts have the same signification, are the same as for the descriptions of the other eryostats; modified parts are designated by new accents and new parts by analogous letters, so that the expla- nations of Comms. N°. 83, N°. 94¢ and N°. 944 on the attainment of uniform and constant temperatures, to which I shall refer for the rest, can serve also bere. Pl. IL shows how the cryostat is inserted into the hydrogen cycle. In chapter X § 7 is described how the liquid hydrogen is led into the cryostat. Especially for the regu- lation of the temperature this plate should be compared with Pl. VI of Comm. N°. 83, March ’03. Instead of Bu Vac on the latter plate, the compressor © serves as vacuumpump here (see Pl. II of the present paper). 6. The measuring apparatus (as on the plate of Comm. N°. 944 VIII I have represented here the comparison of a thermoelement with a resistance thermometer) are placed within the protecting cylinder §, of the stirring apparatus. This is held in its place by 4 glass tubes &,, fitted with caps of copper tubing §,, and §,, at the ends of the rods. The beaker La, containing the bath of liquid hydrogen, is supported by a new-silver cylinder Ba,, in the eylindrical rim Ba, of which the glass fits exactly; the beaker is held in its place by 4 flat, thin, new-silver suspension bands running downwards from Ba, and uniting below the bottom of Ba. The ring Ba, is the cylinder Ba, (175 ) continued, with which it is connected by six strengthened supporting ribs Ba,. At the top it is strengthened by a brass rim ba, with a protruding part, against which presses the upper rim Ua of the case U. On Ba, rests the cover .V',, in which a stopper is placed carrying the measuring apparatus. The india rubber band effects the closure (comp. also Comm. Nos. 83, 94° and 94). ce. In the case U the vacuumglass 5',, of which the inner wall B',, is protected by the thin new-silver cup Db, is suspended by bands Z', and supported by the wooden block L',. The card-board cover 5’, forces the evaporated hydrogen, which escapes between the interstices of the supporting ridges, over the paste-board screen B',,, with notches b',,, along the way indicated by arrows, to escape at 7, The case is lined with felt, covered with nickel paper (comp. Comm. N°.14, Dec. ’94, and Comm. N°. 51, Sept. 799). d. The keeping of liquid hydrogen within an enclosed space, or which the walls have for a great part a much higher temperature than the critical temperature of hydrogen, involves special safety arrangements. That this was no needless precaution appeared when the vacuum glass cracked unexpectedly (comp. X § 1) and of a quantity of more than 1,5 liter of liquid hydrogen nothing was to be seen after a few seconds. Now this disappearance is equivalent with the sudden formation of some hundreds of liters of gas, which would explode the case if no ample opportunity of escape were offered to the gas as soon as the pressure rises a little above the atmospheric. In the new cryostat I have avoided this danger in the same way as at the time when I first poured off a bath of liquid oxygen within a closed apparatus (comp. Comm. N°. 14, Dec. ’94). The bottom of the case U is made a safety valve of very large dimensions; as cover W, of perforated copper with strengthened ridges it fits into the cylindrical case U6, which is strengthened with the rim W. Over the external side of this cover (as in the safety tubes for the hydrogen liquefactor) a thin india rubber sheet W, — separated from the copper by a sheet of paper — is stretched, which at the least excess of pressure swells and bursts, while moreover the entire vacuum glass or pieces of it, if they should be forced out of the case, push the cover W, in front of them without resistance. As the airtight fit of the sheet of india rubber W, on the ring W is not trustworthy and diffusion through contact of the india-rubber with the air must be prevented, it is surrounded with hydrogen; this is done by filling the india rubber cylinder Wa, drawn over the supporting ring Ub, and the auxiliary cover Wb, with hydrogen along We. (176 ) The cords Wd serve to press the auxiliary cover Wb with a certain force against the safety sheet, namely by so much as the excess of pressure amounts to, which for one reason or other we want to admit into the case. To prevent the india rubber from cooling down, for then the arrangement would no longer satisfy the requi- rements, the lower end of the case is lengthened by the cylindrical piece Ub, which between the rim Ud, and the principal body of the case is made of new-silver to prevent the cooling of the lower rim. The entire lower part is stuffed with layers of felt and wool while also a copper flange Ub, by conduction of heat from outside protects the lower wall from cooling. e. The hydrogen is admitted through the new-silver tube a, on which the siphon tube of a vacuumglass (X § 7) is connected with a piece of india rubber tubing a, (which otherwise is closed with a stopper a,, comp. X § 4a). The new-silver tube is put into the new-silver side piece Ud, which is soldered on the case and, being stuffed with capoc held back by a paper tube Ue, carries at the end a piece of cork Uf for support. When the vacuum glass B, with the case U are placed round the beaker Ba, the tube a, is pulled back a little. When subsequently the case is fastened in its position the tube is pushed forward until a ridge on a, is checked by a notch in Ud, so that its end projects into the beaker Ba and the hydrogen can flow’ into it. The india rubber tube a, forms the closure on a, and Ud. § 3. Remarks on the measurements with the cryostat. In chapter X §7 I have communicated how the preliminary cooling is obtained. In one of the experiments, for instance, 3 liters of liquid air were used for it and the temperature was diminished to —110’. Then hydrogen was very carefully siphoned into the eryostat under constant stirring; a quantity of 5 liters was sufficient to obtain a bath of 1.5 liter. About 0.2 liter per hour evaporated after this. During the reduction of the pressure to about 60 m.m. + 0.2 liter evaporated, and then the evaporation remained about the same. The temperature could be kept constant to within 0.01° in the way described in the former papers. The temperature curves obtained were no less regular than those of Pl. II] in Comm. N°. 83 (Febr. and March ’03). If the pressure is reduced down to 54 m.m. the tapping noise of the valves of the stirring apparatus becomes duller. This is a warning that solid hydrogen begins to deposit. E47 9 XIII. The preparation of liquid air by means of the cascade process. §1. Efficiency of the regenerative cascade method. In none of the communications there was as yet occasion to treat more in detail of the preparation of liquid air by the Leiden cascade pro- cess. In the description of the preparation of liquid oxygen (in Comm. N°. 24, Dec. ’94) I have said that especially the ethylene refri- gerator had been constructed very carefully, and that the principle after which various cycles operating in the regenerative cascade can be made was embodied there. When the new methyl chloride circulation (comp. Comm. N°. 87, March '04) was ready and the inadequate methyl chloride refrigerator was replaced by one constructed after the model of the ethylene boiling vessel with application of the experience gained, it was possible to prepare a much larger quantity of liquid oxygen (10 liters per hour easily) with the same ethylene boiling vessel. This quantity will still increase when the regenerator in the ethylene boiling vessel will be enlarged so much as our experience with the new methyl chloride regenerator has again taught to be desirable and when the exhaust tube of the ethylene boiling vessel will have been replaced by one of greater width than could be used originally. The intro- duction of a nitrous oxide and of a methane cycle, which in ’94 stood foremost on our programme, has dropped into the background especially when, also for other reasons (in order to obtain the tem- peratures mentioned at the end of XII § 1), it appeared desirable to procure vacuumpumps of greater displacing capacity (’96) and these, being arranged for operations with pure gases (described in Comm. N°. 83, March ’03) had become fit to be introduced into the ethylene and the methyl chloride cycles (while in general for the cryostats these two cycles were sufficient, cf. the end of XII § 1). Larger quantities of oxygen could be used in consequence, for which (as mentioned in ’94) a BrorHernoop compressor was employed (comp. the description of the installation for operations with pure gas in Comm. N°. 51 $ 3, Sept. ’99). A picture of the cascade method in this stage of development accompanies a description of the cryogenic laboratory by H.H. Francis HynpMan in “Engineering” 4 Mrch ’04. This picture represents how the oxygen cycle is used to maintain the circulation in the nitrogen cycle, described in Comm. N°. 83, March 1903. In the same way as nitrogen we also liquefy air with the oxygen cycle. When it is drawn off the liquid air streams from the tube in a considerable jet; about 9 liters of liquid air are collected per hour, so that in one day we can easily prepare half a hectoliter. (178 ) Liquid air has striking advantages above liquid oxygen when we have to store large quantities or when with the gas liquefied in the cryogenic laboratory we must cool instruments in other rooms. Only where constant temperatures are aimed at pure oxygen or nitrogen will be preferred for refrigerating purposes, and even then the liquid air can be the intermediate agent, for we need only lead the gases mentioned through a cooling tube immersed in liquid air in order to liquefy nearly as much of it as the quantity of air evaporated amounts to. And so the permanent stock of liquid air maintained in the Physical Laboratory has gradually increased, so that for several years liquid air has been immediately sent off on application both at home and abroad. § 2. The airliquefactor. The apparatus for the preparation of liquid air by means of liquid oxygen is in principle identical with that serving for nitrogen, but of larger dimensions (see Pl. VI). Identical letters designate corresponding parts of the apparatus represented (Comm. N°. 83, Pl. VII) for the liquefaction of nitrogen. To liquefy air the ordinary atmospheric air, after being freed by a solution of sodium hydroxide from carbon dioxide, iscompressed to 10 atmospheres in the spiral RgR/, Pl. VI fig. 1. This spiral branches off from the tube Ag, in the soldered piece Ag, and carries four branches Rg,, Ag, Rg, and Ag, Each of these tubes has an internal diameter of 3.5 mm., an external diameter of 5.8 mm., and is 22 M. long. The spiral is wound in 63 layers in the same way as the regenerator spiral of the hydrogen liquefactor (comp. X )and, lined with flannel, it fits the new-silver tube p,, round which it is drawn in the new silver case p. The four windings are united below to one soldered piece to the spiral Rf, 8M. long, which is immersed in a bath of liquid oxygen and whence the liquid air flows through Af; into the collecting apparatus (see fig. 2). This is placed by the side of the principal apparatus (see fig. 2) and contains the collecting vessel 7,, where the liquid air is separated and whence it is drawn through the siphon. The collecting glass is fitted with a float dr. During work we can see it rising regularly at a fairly rapid rate. § 3. Further improvements. The regenerative cascade might still be modified in many points before the principle is fully realized and before one improvement or other, made for one of the cycles, has been introduced also in the others and the efficiency is grown to a maximum ; but this problem is rather of a technical nature, We prefer to spend the time at our disposal on other problems, as enough liquid air is AMERLINGH ONNES. Methods and apparatus used in the cryogenic laboratory at Leiden. XIII. The preparation of liquid air by means of the cascade process. PID VE De a KS: HE CR LS Zi Brigt ceedings Royal Acad. Amsterdam. Vol. IX. ween = woerd vd hoe mi vm ee ae peil 1e) 7: ae u ( 179 ) produced by the regenerative cascade. Enough but not too much, because for operations with liquid hydrogen (comp. X) and also for other experimentations in the realm of eryogenic work it is very important that we should dispose of such a relatively abundant stock of liquid air as is produced by the Leiden cascade. XIV. Preparation of pure hydrogen through distillation of less pure hydrogen. It was obvious that we could obtain pure hydrogen for the replenishment of the thermometers and piezometers ') when we distil liquid hydrogen at reduced pressure *), and then evaporate the very pure liquid thus obtained. Therefore the following apparatus has been constructed (fig. 5). A vacuum glass A is connected with the liquefactor (see PI. 1 and III at N,) or with a storage bottle, exhausted and filled with liquid hydrogen as indicated in X §7. Then C'(exhausted beforehand) in the vacuum glass B is filled several times out of A, and the vacuum glass 6 is connected with 5, to the liquefactor and exhausted like A and also filled with liquid hydrogen and connected with the ordinary airpump at B, so that the hydrogen boils in B at 60 m.m. Then hydrogen is distilled over along c, into the reservoir C, we 1) In Comm. N°. 94e (June ’05) I have mentioned that a purification through compression combined with cooling might be useful in the case of hydrogen even after the latter in the generating apparatus (Comm. N°, 27, May ’96 and N° 60, Sept. 1900) had been led over phosphorous pentoxide. I said so especially with a view to the absorption of water vapour as, with due working, the gas — at least to an appreciable vapour tension — cannot contain anything but HzO and SO,H. How completely the water vapour can be freed in this manner appears from a calculation of Dr. W. H. Kersom, for which he made use of the formula of ScueeL (Verh. D. phys. Ges. 7, p. 391, 1905) and from which follows for the pressure of water vapour (above ice) at —180° CG. 10—!* mm., so that water is entirely held back if the gas remains long enough in the apparatus. This holds for all substances of which the boiling point is higher than that of water (SO; vapours, grease-vapours etc.). The operation is therefore also desirable to keep back these substances. As to a gas which is mixed only with water there will remain, wher it is led in a stream of 3 liters per hour through a tube of 2 cm. in diameter and 8 cm. in length over phosphorous pentoxide, no more than 1 m.gr. impurity per 40000 liters (Mortey, Amer. Journ. of Sc. (3) 34 p. 149, 1887). This quantity of 1 m.gr. is probably only for a small part water (Morey, Journ. de chim. phys. 3, p. 241, 1905). Therefore the operation mentioned would not be absolutely necessary at least with regard to water vapour when a sufficient contact with the phosphorous pentoxide were ensured, But in this way the uncertainty, which remains on this point, is removed. *) This application follows obviously from what has been suggested by Dewar, Proc. Chem. Soc. 15, p. 71, 1899, ( 180 ) Cr Cs E = / LS i= > N e: AN GIEL AN A dé Ex IN | ar NN )3 2 | at As Bi Nt L Cy bend aa Fig. 5. shut c, and disconnect the india rubber tube at a and remove the whole apparatus to the measuring apparatus which is to be filled with pure hydrogen; to this end the apparatus is connected with the mercury pump, intended for this purpose, at c,. To take care that the hydrogen in B should evaporate but slowly and the quantity in C should not be lost before we begin to fill the pieces of appa- ratus, B is placed in a vacuum glass with liquid air. Physics. — “On the measurement of very low temperatures. IX. Comparison of a thermo-element constantin-steel with the hydrogen thermometer.” By Prof. H. KaAMERLINGH Onnes and C. A. CROMMELIN. Communication N° 95% from the Physical Labora- tory at Leiden. (Communicated in the meeting of June 30, 1906). § 1. Introduction. The measurements communicated in this paper form part of a series, which was undertaken long ago with a view to obtain data about the trustworthiness of the determination of low temperatures which are as far as possible independent and intercomparable. Therefore the plan had been made to compare a thermo-element'), a gold- and a platinum-resistance thermometer’) 1) Comp. comms. N°. 27 and 89. (Proc. Roy. Ac. May 1896, June 1896, and Feb. 1904). 2) Comp. comms. N° 77 and 93. (Idem Febr. 1902 and Oct. 1904), ( 181 ) each individually with two gas thermometers and also with each other, while the deviation of the gas thermometer would be determined by means of a differential thermometer’). Nitrogen had originally been chosen by the side of hydrogen, afterwards nitrogen has been replaced by helium. Because all these measurements have often been repeated on account of constant improvements, only those figures have been given which refer to the gold- and the platinum-resistance thermometer *), and these, for which others will be substituted in Comm. N°. 95°, are only of interest in so far as they show thatthe method followed can lead to the desired accuracy. The results obtained with regard to the above-mentioned thermo-element do not yet satisfy our requirements in all respects; yet all the same it appeared desirable to publish them even if it was only because the temperature deter- minations for some measurements, which will erelong be discussed, have been made with this thermo-element. § 2. Comparisons made by other observers. a. Constantin-iron elements have been compared with a hydrogen thermometer only by HorBorN and Wren’) and Lapensure and Krtcet *). The calibration of the two former investigators is based on a comparison at two points viz. in solid carbon dioxide and alcohol (for which —78°.3 is given) and in liquid air (for which they found —189°.1). They hold that the temperature can be represented by the formula t—aE + bE and record that at an observation for testing purpose in boiling oxygen (—183°.2 at 760 m.m. mercury pressure) a good harmony was obtained. LapENnBurG and Kriicen deem HorBoRN and WieN’s formula unsatis- factory and propose =ak | b+ cH’. They compare the thermo-element with the hydrogen thermometer at 3 points, viz. solid carbon dioxide with alcohol, boiling ethylene and liquid air. As a control they have determined the melting point of ether (— 112°) and have found a deviation of 1 deg. With this they rest satisfied. 1) Comp. comm. NO. 94¢. (Idem June 1905). 2) Comp. comm. N°. 93. (Idem Oct. 1904). 3) Sitz.ber. Ac. Berlin. Bd. 30, p. 678, 1896, ‘and Wied. Ann. Bd. 59, p. 213. 1896. 4) Chem. Ber. Bd. 32, p. 1818. 1899. 13 Proceedings Royal Acad. Amsterdam. Vol. IX. ( 182 ) Rotue’) could only arrive at an indirect comparison with the hydrogen thermometer. He compared his thermo-elements constantin-iron at — 79° with the alcohol thermometer which Wisse and BérrcHer ?”) had connected with the gas thermometer and at — 191° with a platinum-resistance thermometer which at about the same tempera- ture had been compared with the hydrogen thermometer in the Phys. Techn. Reichsanstalt by HorBORN and DrrTENBERGER *). The thermostat left much to be desired; temperature deviations from 0°.4 to 0°.7 occurred within ten minutes (comp. for this § 7). As Rotue confined himself to two points, he had to rest content with a quadratic formula and he computed the same formula as HoOLBORN and WIEN. From the values communicated for other temperatures we can only derive that the mutual differences between the deviations of the different thermo-elements constantin-iron and constantin-copper from their quadratic formulae could amount to some tenths of a degree. Nothing is revealed with regard to the agreement with the hydrogen thermometer. This investigation has no further relation to the problem considered here. b. Among the thermo-elements of other composition we mention that of WRoBLEWsKI *), who compared his new-silver-copper element at + 100° (water), —103° (ethylene boiling under atmospheric pressure) and —4131° (ethylene boiling under reduced pressure) with a hydrogen thermometer and derived thence a cubic formula for ¢. He tested it by means of a determination of the boiling points of oxygen and nitrogen and found an agreement with the hydrogen thermometer to within 0°.1. As, however, WROBLEWsKI found for the boiling point of pure oxygen at a pressure of 750 m.m. —181°.5, no value can be attached to the agreement given by him. Dewar’s °) investigation of the element platinum-silver was for the time being only intended to find out whether this element was suited for measurements of temperatures at — 250° and lower (where the sensitiveness of the resistance thermometer greatly diminishes), and has been confined to the proof that this really was the case. c. To our knowledge no investigation has therefore been made as yet, which like that considered in our paper, allows us to judge in 1) Ztschr. fiir Instrumentenk. Bd. 22 p. 14 and 33. 1902. aes 5 Bd. 10 p. 16. 1890. 3) Drude’s Ann. Bd. 6 p. 242. 1901. 4) Sitzungsber. Ac. Wien Vol. 91. p. 667. 1885. 5) Proc. R. S. Vol. 76, p. 317. 1905. ( 183 ) how far thermoelements are suitable for the accurate determination of low temperatures (for instance to within */,,” precise), and also by what formula and with how many points of calibration any temperature in a given range can be determined to within this amount. § 3. Modifications in the thermo-elements and auxiliary apparatus. We shall consider some modifications and improvements which have not been described in § 1 of Comm. N°. 89. The first two (a and 6) have not yet been applied to the element with which the following measurements are made, but they have afterwards been applied to other elements and so they are mentioned for the sake of completeness. a. If we consider that the thermo-element in different measurements is not always used under the same circumstances, e.g. is not immersed in the bath to the same depth ete., and that even if this is the case, the time during which this is done at a constant temperature will not always be so long that in either case the same distribution of the temperature will be brought about in the metallic parts of the element, it will prove of the greatest importance that care should be taken, that the tem- perature of the juncture, given by the electromotive force, differs as little as possible and at any rate very little from that of the surface of the copper protecting block, that is to say that of the bath. The construction of the place of contact shown by fig. 1 is a better warrant for this than that on Pl. I of Comm. N°. 89. The wires a and 5 are soldered on the bottoms of small holes ec, bored in the protecting block and are insulated each by a thin-walled glass tube. If the con- struction of Pl. I Comm. N°. 89 is not carried out as it should be (whether this has succeeded will appear when we saw through trial pieces) and consequently the juncture is a little removed from the upper surface of the block, it may be easily calculated that, owing to conduction of heat along the wires while the thermo-element is immersed in liquid oxygen a difference in temperature of as much as one degree may exist between the place of contact and the block. When the elements are used under Fig. 1. other circumstances, this difference in temperature will have another value and hence an uncertainty will come into the determination of the temperature of the block. Perhaps that also a retardation in the indications of the element will be observed. 13 ( 184 ) Although this construction (fig. 1) (for which a block of greater thickness is required than for that of Comm. N°. 89, Pl. I) has not been applied to the element used, we need not fear uncertainties on this point thanks to the very careful construction of the latter. b. When temperatures below —253° have to be determined we might fill the apparatus with helium instead of hydrogen as men- tioned in $ 1 of Comm. N°. 89. c. The glass tubes of the mercury commutators, described in Comm. N°. 27, are not fixed in corks (see Pl. IV, fig. 4, £) but in paraffin, so as to obtain perfect insulation, which, as experience has taught, is not guaranteed by the glass wall. The tubes are continued beyond the sealing places of the platinum wires c, ¢, c,and ¢,, (as shown Fig. 2 and 3. by figs. 2 and 3) to avoid breaking of the platinum wires as formerly frequently happened. d. The platinum wires of the Weston-elements have been amal- gamized by boiling with mercury (which method has since that time been replaced by the method with the electric current *)). The elements themselves have kept good through all these years. e. In spite of all the precautions which have been described in Comm. N°. 89, thermo-electromotive forces still remain in the wires, which with the great differences of temperature between various points of one wire must doubtlessly amount to a measurable quantity. When, however, care is taken that the circumstances under which the element is used with respect to the temperature along the wires are about the same as for the calibration, a definite value of the electromotive forces will answer to a definite temperature of the copper block. We do not aim at an accurate determination of the electromotive force of the combination of the metals which at the 1) Comp. Jarcer, Die Normalelemente, p. 57. (4185 ) juncture are in contact with each other, but we only require that a definite electromotive force for a definite temperature of the bath in which the element is immersed should be accurately indicated. (for the rest comp. $ 5). In order to lessen the influence of the conduction of heat along the wire at the juncture we shall for the new elements destined for taking the temperature of a liquid bath make a trial with the insertion into the glass tube at 2 c.m. above the copper rim of the copper block of a copper tube, 5 c.m. long, which is soldered on either side of the glass tube and remains over its whole length immersed in the liquid. § 4. Precautions at the measurements of the electromotive forces. a. The apparatus and connections which have been described in §3 of Comm. N°. 89 have been mounted entirely on paraffin, with which also the enveloping portions of the apparatus are insulated. Only the wires running between the different rooms stretched on porcelain insulators, of which the high insulation-resistance has repeatedly been tested, have no paraffin-insulation. The ice-pots are hanging on porcelain insulators. As a matter of course, all parts of the installation have been carefully examined as to their insulation before they are used. 6. The necessity of continually packing together the ice in the ice-pots has been argued before in Comm. N°. 89. c. The plug-commutators are of copper. All contacts between different metals in the connection have been carefully protected from variations of temperature by packing of wool or cotton-wool, from which they are insulated by paraffin in card-board boxes. This was only omitted at the contact places of the copper leads with the brass clips of the resistance boxes. To secure to the Weston-elements an invariable temperature, the latter have also been carefully packed. The accu- mulator is placed in a wooden box. d. With regard to the testing elements, care has been taken that the steam left the boiling apparatus (comp. Comm. N°. 27, § 8) at a given constant rate. e. Before a measurement is started we investigate by short-cir- cuiting in the copper commutators in the conductions, leading from the thermo-elements and the Weston-battery to the connections, whether all electromotive forces in the connections are so small and constant (not more than some microvolts), that elimination through the reversal ( 186 ) of the several commutators may be considered as perfectly certain. § 5. The control of the thermo-elements. It appeared : a. that when the four places of contact were packed in ice, the electró- motive force of the element amounted to less than one microvolt; 6. that the changing of the two places of contact constantin-steel, so that they were alternately placed in the cryostat, indicated only a very small difference in electromotive force. Care is taken, however, that always the same limb is placed into the cryostat ; c. that while the place of contact was moved up and down in the bath no difference could be perceived in the reading (hence the difference of temperature certainly < 0.02). All this proves that the electromotive forces which are raised in the element outside the places of contact, are exceedingly small. ~§ 6. Corrections and calculations of the determinations of the electromotive forces. a. In the following sections Rh, R, R’ have the meaning which has been explained in Comm. N°. 89 $ 3. H,, Ke and H” signify the electromotive forces of the observation-element, the comparison-element and the Weston-battery respectively. If we have obtained ?,, Zi, and ?’ it follows that: R R En w 5 — w 1 Hi R. E or Ly = — EF. As a test we use: a z i TEA Re 6. In order to find A, we read on the stops of the resistance box R'„ (in the branch of small resistance), and A» (in the branch of great resistance) which are switched in parallel to form B a. To none of the resistance boxes temperature corrections had to be applied (nor to those given by A, and Rf’ either). 8. To R'„ we sometimes had to add the connecting resistance of the stops. _y. To R' is added the correction to international ohms according to the calibration table of the Phys. Techn. Reichsanstalt. d. To A, is added the amount required to render the compen- sation complete, which amount is derived from the deflections on (sri) A | 3 (dorewowreyy Bururerredde) 7 "67 =? (err « « ) '8'o8l = # (geog ‘ddry proseuy) UD FP'OL WSloysejowos1eg (96 4 « ) '0'o6r = 72 —= — (sep ‘ON sojomoutIayT) “6° o8b = 17 08 99 0e 99 00°99 03°¢9 GG’ Cg 09 89 GL'79 Og” <9 II IL GL’ 99 67°99 78 <9 GY’ S9 | ¢9° 99 08° L9 ca °L9 OG” 69 I II 4 00° L9 06°99 09299 Stee) OL’ SO GE G9 05 89 G9'’99 II <9 49 OOR ¢8°79 00°S9 II I 96°99 78 99 7°99 €G 99 69° S9 18°79 08° S9 00° L9 I cE 79 Gy 69 09°99 6079 I I 06°99 08°99 0599 Gv'99 gg" a9 ¢9°79 | “ATV ‘ATeS | ‘Uleya qysi | o19z | Jar || WSL} O10Z | JAI pan 2u | osez | ojo |] wyBta | ozoz | oor |IUS | odoz | Jel qystu | o1ez | Jay ij|sIoyepnwwog ‘sUNTZIETJop JJWOURATLD *SUOTPIETJap JA PUOUBA[RY "SUOLPIOTJAP JIJIULOUBA]BH “TOLLE “ON YEISOMUY "689Gb “ON 'H PUB 'S IEISOMUY 'SE6ST “ON 'H PUB “Ss 'F8ISOMUY ; 0088 = “17 0098 = ua 0052 = a OOLL = “ul ‘7008 = 1 ‘(0008 = i N 4 A de b+os = 4 ‘ptos =7a fit+irt+3etos= 4 tHtrtotos = “az “sq ; “DF 8g voneurqwog 5 sjuauoje-UOFs AL zuswore-uostaet mog juouo[e-UOIJEALASYO Typ Gye: OUly, ‘L ‘ON (oanssord poonpor aopun Surrog uosAx0) AX SOLAS ‘GO6T Ane 9 yejsok1o O94} UL O Joezuoo Jo ooerd ‘'g JUOTI9[9-OUL10G} 9} JO UOTVAQITVO T WIAVL ( 188 ) ithe scale of the galvanometer at two values of &", (see tables I and IV). . e. In order to find R,, R', and R"., which with regard to A, have a similar meaning as RF’, and hk", with regard to Ay, are treated like R', and R", concerning the corrections a, 8, y and d. The ‘thence derived result A, holds for the temperature at which the water boils in the boiling apparatus at the barometric height B ‘existing there during the observation. __e. Rl", is corrected to the value which it would have at a pres- ‘sure of 760 m.m. mercury at the sealevel in a northern latitude of 45°. d. To find R' the corrections mentioned sub y and d are applied ‘to the invariable resistance A. e. H', referring to the temperature £ of the Weston-battery, is derived from JAEGER'’s table *). § 7. Survey of a measurement. Table I contains all the readings which serve for a measurement of the electromotive force namely ‘for that at — 217° (comp. § 8). We suppose that during the short time required for the different readings (comp. $ 3 of comm. N°. 89) the electromotive force of the accumulator (comp. $ 4, c) remains constant. We further convince ourselves that the temperature in the boiling apparatus of the comparison-element has remained sufficiently constant and that we have succeeded *) in keeping the temperature of the bath in the cryostat constant to within 0°.01 *) (see table I). In exactly the same way we have obtained on the same day of observation the values for the electromotive forces which are combined in table III. From the preceding survey it appears that the measurements can be made with the desired precision even at — 217°. At — 253° the sensitiveness of the element constantin-steel is considerably less than at — 217°. It seems to us of interest to give also for this very low temperature a complete survey of the readings and adjustments so that the reader may judge of what has been attained there (see Table IV). 1) Jarcer, Die Normalelemente 1902. p. 118. 2) Comp. Comm. NO, 83, § 5 and PI. III. 3) Together with the readings we have also recorded the temperature of the room (tk) and of the galvanometer (tj); these are of interest in case one should later, in connection with the sensitiveness, desire to know the resistance of the galvanometer and the conducting wires during the observation. lor the notation of the combination P3 + Q, of the comparison-elements we refer to Comm. N’. 89 § 2, ( 189 ) From table I directly follows TAB TE, I- Corrections and results. Observation-element. Comparison-element. | Weston-elements. corr. B R',= + 0.001 n | corr. 8 R', =-+ 0.001 n corr. Y R',,= + 0.0080 o corr. y R', =— 0.00015 0 | corr.y Rk’, =—2.40 corr.S R",,=+179 0 | corr.5 R", =+-149 0 corr. 8 A’; =+0.6540 Rr, =50.31630 | arom ght.A5°N.0-=70.21e, | | corr. R'’, =— 0.0373 9 Final results. R = 53.6404 n Re =50.2787 n | F'= 7998.3 0 i’ =18°.8 E'=1.0187 volt. E,)= 6.8312 milliv. ZE, =6.4037 milliv. 4u 3! TABLE III. | 6.8312 6.4037 | 6.8308 6.4039 | | 6.8310 6.4038 | Mean | 6.8310 | 6.4038 $ 8. The temperatures. a. The thermo-element is placed in a cryostat, as represented on the plate of Comm. N°. 944, but there a piezometer takes the place which in our measurements was occupied by a hydrogen thermo- meter. To promote a uniform distribution of the temperature in the ( 190 ) (d9joMOWIeY} Sulureysiodde) —* Zy =? (671 ‘ON ee Gch | (zegg “oN ‘ddry prozeuy) "Md LLL, Tojoworedg (96 "ON Ein, = (BET ‘ON J9JoWOUIEUL) LoS b= # 0667 €9° W 06” 07 GE 07 G9 86 08° 9& 8b 68 OL 68 II II Gh Gr 67 V7 <9 07 LV 07 08°17 LE°GY 0S’ 17 S8'6ENN I II 8L'E7 co" ST 507 Se W G7 OF 60 07 86 GY 1867 Ir OG W 7866 9168 LS 66 II I 99 Ey vs &Y 18° W [66 W 86°07 73 66 B OGAE 06 B || I Cy 86 6886 6807 Sv'6E I | I Sci Gy. OL er €9° WY 80° Wy GE OF | 68°66 RE | "ATV | | | ATe3 | “ware : : (Patera | | | qysta | ofez | Jor |] yqsta | odoz | 3For |Jwwwool ysis | o10z | For || ySt | orez | zor ||3U3ta | o1ez | or [| aySia | oroz | gJoy |Js1oregnwwog | | ‘sUOTJIEYep Ja @UIOUBATYO ‘sUOIJIETJep LJaJsWOULAIEL) ‘suUOTOaTJap JeyaUIOURATeH) “T9ILT “ON “H Pues °s 3eJsooUY ‘68991 “ON “H PUE S I8IS0MUY ‘SGEST ON H PGB SS Fe send = pare) Oe a Voces ‘at | coe ke ie OE “Ue 0058 = a = 0008 = iY Ne s 5 LOrs 0 eh Oe Gat ° zE Pr +r+etos= “wz || rotsotr+e+oo=" "S]U9W19[9-U04S2 AA fae oel “ug waja-UOTJRA LASGO) Cull ug + SWI ‘TE ON ‘(oanssord oroydsouyg opun Surrog ‘uosorpÂH) “XXX setts ‘9O6T AVI G ‘YejsoAxo OY} UI O goezuoo Jo ooerd “q Juowore-ouazeuyg OY} JO UOTVAQITVO Al H'I4VL (191 ) TABLE V. Corrections and results. Observation-element. Comparison-element. Weston-elements. | corr. 2. R',,= +0.00l a | corr. sp. A’, =+0.001 0 corr. 7. R',, =-+ 0.00537 n| corr. 7. R’, =-+ 0.00849 | corr.y Ry =— 2.40 corr.d. R', =-+- 20 a. corr. d. R= — 209 9 corr. ò. R'‚,= 0.8 9 Ri == 00-4133, 0: Bar.hght. 45° N.B. = 76.82 cM. corr. e Rl’, = — 0.1459 0 Final results. R,, = 55.9981 a | R, = 50.2644 9 | R' = 7998.4 a ¢!’ —418°.5 &'=1.0187 volt. Fy =7.1321 milliv. . #,=6.4075 milliv. Qn94! bath a tube is mounted symmetrically with the thermo-element, and has the same shape and dimensions as the latter. Comp. also Comm. N°. 94e § 1. For the attainment of a constant and uniform tem- perature with this cryostat we refer to Comm. N°. 944 and the Comms. quoted there. The temperature was regulated by means of a resistance thermometer. For the two measurements in liquid hydrogen we have made use of the cryostat described in Comm. N°. 94/7 b. With a bath of liquid methyl chloride we have obtained the temperatures — 30°, — 59° and — 88°; with ethylene — 103°, —140° and —159°; with oxygen — 188°, —195°, —205° —213° and — 217°; with hydrogen — 253° and — 259°. c. The temperatures are read on the scale of the hydrogen thermo- meter described in Comms. N°. 27 and N°. 60. On the measurements with this apparatus at low temperatures another communication will erelong be published. ( 192 ) § 9. Results. Column I of the following table VI contains the numbers of the measurements, column II the dates, column III the temperatures measured directly with the hydrogen thermometer, column IV the electromotive forces — /, in millivolts, column V the number of observations, column VI the greatest deviations in the different deter- minations of #,, of which the appertaining /,, is the mean, column VII the same reduced to degrees. TABLE VI. CALIBRATION OF THE THERMO-ELEMENT CONSTANTIN-STEEL. I | Il | Ill | IV | V | VI | VII 20 | 97 Oct. 05 — 58.753 2.3995 3 0.0006 0.016 21 30 Oct. 05 — 88.140 3.4825 3 29 81 17 8 July 05 — 103.833 4.0229 | 3 56 168 16 | 7 July 05 — 139.851 5.1469 3 6 21 18 26 Oet. 05 — 139.873 5.1469 | 4 12 4A 19 | 6 Oct. 05 — 158.831 5.6645 3 15 59 1 27 June 05 — [182.692] 6.2297 3 10 46 28 2 Mrch. 06 — 195.178 6.417 h 28 150 12 | 29 June 05 — [204.535] 6.6382 | 3 31 186 27 2 Mrch. 06 — 904.694 6.6361 4 26 156 14 | 30 June 05 — [212.832] 6.7683 3 8 56 13 6 July 05 — 212.868 6.7668 | 3 15 106 29 3 Mrch. 06 — 7.41 6 8221 3 14 112 45 | 6 July 05 — 17 46 6.8310 | 3 4 32 30 | 5 May 06 .| — 252.93 7.1315 he 17 39 31 5 May 06 — 259.94 7.1585 1 = = The observations 11, 12 and 14 are uncertain because in those cases the hydrogen thermometer had a very narrow capillary tube so that the equilibrium was not sufficiently secured. According to other simultaneous observations (Comm. N°. 95° at this meeting), which have later been repeated, the correction for N°. 11 is probably — 0°.058. The two other ones have been used unaltered. ( 193 ) The mean deviation of /, for the different days from the mean value, and also the mean largest deviation of the values of /. found on one day amounts to 3 microvolts, which amount shows that in the observation of the comparison-element the necessary care has not been bestowed on one or other detail, which has not been explained as yet. We must come to this conclusion because the observation-element yields for this mean only 1,8 microvolt. § 10. Indirect determinations. In order to arrive at the most suitable representation of as a function of ¢, it was desirable not only to make use of the obser- vations communicated in $ 9 but also to avail ourselves of a large number of indirect measurements, obtained through simultaneous observations of the thermo-element and a platinum-resistance thermo- meter, the latter having been directly compared with the hydrogen thermometer (comp. Comm. N°. 95%, this meeting). These numbers have been combined in table VII where the columns contain the same items as in the preceding table, except that here the temperatures are derived from resistance measurements. TABLE VII. INDIRECT CALIBRATION OF THE THERMO-ELEMENT CONSTANTIN-STEEL. I II | TI | IV v VI | VII 29 | 43 Dec. 05 _ ofses | 1.9503 | 3 | 0.0005 0.012 24 | 414 Dec. 05 — 58.748 2.3980 | 4 6 16 93 | 413 Dec. 05 — 88.161 3.4802 | 3 6 17 1 | 93 Jan. 05 — 103.576 4.0100 | 5 9 27 3 | 30 Jan. 05 [— 182.604] | 6.2970 | 4 32 447 5 | 46 Mrch. 05 | (— 182.898] 6.2340 | 3 13 60 4 | 2 Febr. 05 — 195.135 6.4730 | 3 20 107 6 | 47 Mich. 05 | — 195.261 6.4814 | 5 40 53 7 | 30 Mrch.05 | — 204.895 6.6397 | 3 55 330 26 | 26 Jan. 06 — 12.765 6.7637 | 4 33 233 8 | 3 April 05 — 212.940 6.7686 | 4 15 106 95 | 95 Jan. 06 — 217.832 6.8216 | 4 29 232 ( 194 ) § 44. Representation of the observations by a formula. a. It was obvious that the formula of AVENARIUs: t tN lH} 100 -+ b (<3) can give a sufficient agreement for a very limited range only. If, for instance, the parabola is drawn through 0°, —140° and —253°, we find: a= + 4.7448 b= + 0.76117. In this case the deviation at —204° amounts to no less than 7°. If we confine ourselves to a smaller range and draw the parabola through 0°, —88° and —183°, we find: a= +.4,4501 b= + 0.57008, while at —140° the deviation still amounts to 1°.3. Such a representation is therefore entirely unsatisfactory. 6. With a cubic formula of the form R= t b BON A Sn en we can naturally attain a better agreement. If, for instance, we draw this cubic parabola through 0°, —88°, —159° and —253°, we find: a = + 4.2069 6 = + 0.158 e= — 0.1544 and the deviation at —204° is 0°.94. A cubic formula confined to the range from 0° to —183°, gave at —148° a deviation of 0°.34. *) A cubic formula for 7, expressed in / (comp. § 2), gives much larger deviations. ?) e. A formula, proposed by SransrieLp *) for temperatures above 0°, of the form 1) As we are going to press we become acquainted with the observations of Honrer (Journ. of phys. chem. Vol. 10, p. 319, 1906) who supposes that, by means of a quadratic formula determined by the points —79° and —183°, he can determine temperatures at —122° to within 0°.1. How this result can be made to agree with ours remains as yet unexplained. | 2) After the publication of the original Dutch paper we have taken to hand the calculation after the method exposed in § 12 of a formula of the following form: en t b t 3 t 2 t à [ Zan too) + °\ Too tel oro | We hope to give the results at the next meeting. 3) Phil. Mag. Ser. 5, Vol. 46, p. 73, 1898, (195 ) E=aT+blogT +e, where 7 represents the absolute temperature, proved absolutely useless. d. We have tried to obtain a better agreement with the observa- tions by means of a formula of five terms with respect to powers of ¢. To this end we have tried two forms: "100," (oo) + (oo) + (5) ti) make, and t t 3 t 3 5 an el) tik (oo) Hil) ee First the constants of the two equations are determined so that the equations satisfy the temperatures —59°, —140°, —159°, —183° and — 213°. (A) indicated at — 253° a deviation of 113.1 micro- volts, (B) a deviation of 91.8 microvolts. We have preferred the equation (B) and then have sought an equation (BIV) which would represent as well as possible the temperature range from 0° to — 217°, two equations (Bl and BIT) which would moreover show a not too large deviation at — 258°, for one of which (ASIII) a large deviation was allowed at — 217°, while for the other (BI) the deviations are distributed more equally over all temperatures, and lastly an equation (BI) which, besides —253°, would also include —259°. § 12. Calculation of the coefficients in the formula of five terms. The coefficients have first been derived from 5 temperatures distributed as equally as possible over the range of temperatures, and then corrected with respect to all the others without a rigorous application, however, of the method of least squares. In order to facilitate this adjustment we have made use of a method indicated by Dr. E. F. van DE SANDE BAKHUYZEN in which instead of the 5 unknown coefficients 5 other unknown values are introduced which depend linearly on the former*). For these are chosen the exact values of £ for the five observations used originally, or rather the differences between these values and their values found to the first approximation. Five auxiliary calculations reveal to us the influence of small variations of the new unknown value on the representation of the other observations and by means of these an approximate adjustment 1) Also when we rigorously apply the method of least squares this substitution will probably facilitate the calculation. ( 196 ) may be much more easily brought about than by operating directly with the variations of the original coefficients *). After the first preliminary formula was calculated all the 28 observations have subsequently been represented. The values thus found are designated by f#,. The deviations of the observed values from those derived from this first formula are given in column III of table VIII under the heading IV—R,. The deviations from the temperatures in the immediate neighbourhood of each other have been averaged to normal differences and are combined in column IV under the heading (W—-R,). These deviations have served as a basis for an adjustment under- taken according to the principles discussed above. It yielded the following results: leaving — 253° and — 259° out of consideration we find as co- efficients of the equation (4) (comp. § 11): a, = + 4.32044 e, = + 0,011197 6,=— + 0,388466 f, = — des Eet ke be) c, = — 0.024019 If we only leave out of consideration — 259° we find for the coefficients of equation (B) the two following sets (comp. $ 11): a,— + 4.83049 e, = + 0,053261 b, = + 0.436676 f,= +0,003898) . . . (BIZD c, = + 0,048091 and a, = + 4.35603 ¢, = + 0,103459 b, = + 0,581588 f, = + 0,0118632) . . . . . (BD c, = + 0,157678 If we include in the equation all the temperatures, also that of the liquid hydrogen boiling under reduced pressure, we find for the coefficients of the equation (+) a,—= + 4.35905 e, = + 0,111619 | b, = + 0,542848 f, = + 0,0132130' . . . (BIT) ‚== + 0,172014 The deviations from the observations shown by these different equa- tions are found under (W—Rk,) (W—R,) (W—R,) and (W—R,) in columns V, VI, VII and VIII of table VIII. 1) When the polynomial used contains successive powers of the variable beginning with the first power, that influence is determined by the interpolation-coefficients of LAGRANGE. C THERMO-ELEMENT CONSTANTIN-STEEL. (197) TABLE VIII. DEVIATIONS OF THE CALIBRATION-FORMULAE FOR THE AE, BI Ul le VI VII VIII No. t WR, | (W—R,) | (W--R,) | VB.) | (W—R,) | (WR) 22 | — 95 825 | —0.0080 | —0.0080 | —9 0030 | —0 0032 | —0.0013 | —0.0011 OENE 48 | | ~ = Bales odes EE Be SE 90 | — 58.753 0 U | — 88.140} + 44 DE ne ee EE SA 7 93 | — 88.161 | +. 14 | 4 | —103.576 0 he Ran 723 AE A ALE LE A7 | —A03.833 | 4 43 16 | —139.851 | + 5 BEE tl ee Os EA ECN ES La 48° | 439.873 | — 2 | | | | 19 | 458.831 0 GLE Cop O60 EN en 5 3 |[—182.604]) + 4 | ABE AS0e ns 1 “3e jet MAT ed Bt |e Sad 38 Ieke Vsi 5 |[182.898]} + 63 | 4 | A95A35 |H 415 | Be — 495.478 ||. — AE ek rl ss Sle oe WU ae ca A 4 6 | 195.961] 4+ 76 | | 12 |(—204 535]/ + 34 | 27 | —204.694|— 47/1 4 Bun 240 [Lt Big dO Pe 41-90 7 | 204805 | — A | B | 42.765 | 4+ at | | 14 [[—212.839)| + 58 | | ed 404 BH WA 84 22 13 =919-868 |F 38 | 8 | 22.940; + 4 29 | 917.444 |— 36 | 15 | UTM6|H 591} — 7 ape 5 EE ey, oh (Eo 95-| —917.939 | — 36 | 30 | —252.93 0 Oi =F oir DI A ae oan St peor pe A EE LEL 14 Proceedings Royal Acad. Amsterdam, Vol. IX. (198 ) To observation 11 of this table we have applied the correction mentioned at table VI. To the observations 17 and 7 we have accorded half the weight on account of the large deviation from the single determinations mutually (comp. tables VI and VII)". $ 13. Conclusion. For the mean error of the final result for one temperature (when this is taken equal for all temperatures) we find by comparison with the formula found: microv. i RM) VL se R, + 2.6 (2.1 when leaving also out of account — 217°) Rha 18 The mean error of the result of one day, according to the mutual agreement of the partial results, is: + 2.9 microvolts, whence we derive for the mean error of one temperature, supposing that on an average two daily results are averaged to one final result : + 2.0 microvolts. (2 microvolts agree at — 29° with 0°.05, at — 217° with 0°.16). Hence it seems that we may represent the electromotive force of the thermo-element constantin-steel between O° and — 217° by the five-terms formula to within 2 microvolts. For the calibration to — 217° we therefore require measurements at at least 5 temperatures *). The representation including the temperatures of liquid hydrogen is much less satisfactory ; for the mean error would be found according to this representation -+ 3.2 microvolts, agreeing with 0°.075 at — 29° and 0°.74 at — 252° and — 259°. In order to include the hydrogen temperatures into the formula a 6th term will therefore probably be required. But for measurements at the very lowest temperatures the element constantin-steel is hardly suitable (comp. § 7). In conclusion we wish to express hearty thanks to Miss T. C. Jorues and Messrs. C. Braak and J. Cray for their assistance in this investigation. 1) In the calculations for observations 3, 11 and 5 are used temperatures 0°,081 lower than the observed ones. A repetition of the calculation with the true values has not been undertaken, as it would affect only slightly the results, the more because the observations are uncertain. *) If the four term formula (comp. footnote 2 §11) should prove for this inter- val as sufficient as the five term formula, this number would be reduced to four. (199 ) Physics. — On the measurement of very low temperatures. X. Coefficient of expansion of Jena glass and of platinum between + 16° and —182°.” By Prof. H. KAMERLINGH Onnes and J. Cray. Communication N°. 95% from the Physical Laboratory at Leiden. (Communicated in the meeting of June 30, 1906). § 1. Zetroduction. The difference between the coefficients a and 5 in the expansion t iy NG REN a——-+ b 10 100 100 and #, and 4, in the formula for the cubic expansion t GN 8 uma t+ | zip + (oo) 10 | between O° and —182° found by KAMERLINGH Onnes and Hrusr (comp. Comm. N°. 85, June ’03, see Proceedings of April ’05) and those found by Wiese and Borrcuer and Tuirsen and ScueeL for temperatures above 0° made it desirable that the strong increase of b at low temperatures should be rendered indubitabie by more accurate measurements *). In the first place we have made use of more accurate determi- nations of the variation of the resistance of platinum wires with the temperature (comp. Comm. N°. 95¢, this meeting) in order to substitute more accurate temperatures for those given in Comm. N°. 85, which served only for the calculation of a preliminary formula, and then to calculate by means of them new values for a and 5 which better represent the results of the measurements than those given in Comm. N°. 85. By means of the formula W,= W, (1 + 0,00390972 ¢ — 0,0,9861 2), which holds for the kind of platinum wire used in Comm. N°. 85, we have arrived at the following corrections: in table IV read — 87°,14 instead of — 87°,87 and —18]°,42 „ „ —182°,99 im table MV read 86098 7, „ — 87°,71 and —181°,22 „ „ —182°,79 formula for the linear expansion / = J, |: -|- 1) That the coefficient of expansion becomes smaller at lower temperatures is shown by J. ZakrzewsK1 by measurements down to — 103°. This agrees with the fact that the expansion of most substances above O° is represented by a quadratic formula with a positive value of 6. Our investigation refers to the question whether b itself will increase with lower temperatures. 14* ( 200 ) Thence follows Jena glass 16" aoe bi 90 k, = 2343 ki: Thüringer glass Mm’. 50) a= 920 b =120 k, = 2761 k, — 362. i | 1903. Secondly it remained uncertain whether the mean temperatures of the ends were exactly identical with those found after the method laid down in $ 4 of that Comm. The execution of the control- determination as described in Comm. N°. 85 § 4 (comp. § 4 of this paper) proved that in this respect the method left nothing to be desired. Moreover, availing ourselves of the experience acquired at former determinations, we have once more measured the expansion .of the same rod of Jena glass and have reached about the same results which, owing to the greater care bestowed on them, are even more reliable. Lastly it was of importance to decide whether the great increase of 5 at low temperatures also occurred with other solid substances and might therefore be considered as a property of the solid state of several amorphous substances. Therefore and because it was desirable also for other reasons to know the expansion of platinum we have measured the expansion of a platinum rod in the same way as that of the glass rod. Also with platinum we have found the same strong increase of 6, when this is calculated for the same interval at lower temperatures, so that cubic equations for the lengths of both substances must be used when we want to represent the expansion as far as — 182”. After these measurements were finished Scnrer (Zeitschr. f. Instr. April 1906 p. 119) published his result that the expansion of pla- tinum from —190° to O° is smaller than follows from the quadratic formula for the expansion above 100°. For the expansion from ++ 16° to —190° ScnreL finds — 1641 u per meter, while — 1687 u would follow from our measurements. But he thinks that with a small modification in the coefficients of the quadratic formula his observa- tions can be made to harmonize with those above 100°. Our result, however, points evidently at a larger value of 5 below 0°. The necessity of adopting a cubic formula with a negative coeffi- cient of ¢ may be considered as being in harmony with the negative expansion of amorphous quartz found by Screen (le) between — 190° and 16° when we consider the values of a and 5 in a quadratic formula for the expansion of this substance between 0? and + 250°. @ 201") A more detailed investigation of these questions ought to be made of course with more accurate means. It lies at hand to use the method of Fizeau. Many years ago one of us (K.0O.), during a visit at Jena, discussed with Prof. PuLrricn the possibility of placing a dilatometer of ApBr into the Leiden cryostat, but the means of procuring the apparatus are lacking as yet. Meanwhile the investigation following this method has been taken in hand at the Reichsanstalt '). A cryostat like the Leiden one, which allows of keeping a temperature constant to 0,01° for a considerable time, would probably prove a very suitable apparatus for this investigation. Travers, SENTER and JaQqurrop*) give for the coefficient of expan- sion of a not further determined kind of glass between 0° and — 190° the value 0,0000218. From the mean coefficient of expansion from 0° to 100° we conclude that this glass probably is identical with our Thiiringer glass. The mean coeffieient of expansion between O° and — 190° for Thüringer glass found at Leiden in 1903 is 0,00002074. § 2. Measurement of the coefficient of expansion of Jena glass and of platinum between O° and — 182°. The rod of Jena glass used was the same as that of Comm. N°. 85. At the extremities of the platinum tube of 85 ¢.m. length glass ends were soldered of the same kind as the Jena rod. For the determina- tion of the mean temperature of the ends thin platinum wire was wound round these extremities which wire at either end passed over into two platinum conducting wires and was enveloped in layers of paper in order to diminish as much as possible the exterior conduction of heat. The temperature of the middle portion of the Jena rod was also determined by means of a platinum wire wound round it as in Comm. N° 85. The rod was further enveloped in thin paper pasted together with fishglue, and to test the insulation the resistance was measured on purpose before and after the pasting. The tempera- ture of the bath was determined halfway the height of the bath by means of the thermo-element constantin-steel (comp. Comm. N°. 95a, this meeting). This temperature was adopted as the mean temperature of the platinum tube, which was entirely surrounded with the liquid gas and was only at its extremities in contact with the much less 1) Hennine, afterwards Seneer, Zeitschr. f. Instrk. April 1905, p. 104 and April 1906, p- 118. Ranpatt, Phys. Revie ¥ 20, p. 10, 1905 has constructed a similar apparatus, 2) TRAVERS, SENTER and JAgueroD, Phil. Trans. A 200. ( 202 ) conducting pieces of glass, which partly projected out of the bath. The scale (comp. Comm. N°. 85) was wrapped round with a thick layer of wool enclosed in card-board of which the seams had been pasted together as much as possible. The temperature of the room was kept as constant as possible by artificial heating and cooling with melting ice, so that the temperatures of the scale vary only slightly. They were read on three thermometers at the bottom, in the middle and at the top. The scale and the points of the glass rods were illuminated by mirrors reflecting daylight or are-light, which had been reflected by paper and thus rendered diffuse. The vacuum tube (comp. Comm. N°. 85) has been replaced by a new one during the measurements. The evacuation with the latter had succeeded better. So much liquid gas was economized. For the measurement with liquid oxygen we required with the first tube 1'/, liter per hour and */, liter with the second. Of N,O we used with the first only */, liter perst bour: In order to prevent as much as possible irregularities in the mean temperature the bath has been filled as high as possible, while dry air was continually blown against the projecting points. They were just kept free from ice. In two extreme cases which had been chosen on purpose — the bath replenished with oxygen as high as possible and the points covered with ice, and the bath with the float at its lowest point and the point entirely free from ice — the difference of the mean | temperature of the ends was 10 degrees, corresponding | to a difference in length of 4 mierons. The greatest | difference which has occurred in the observations has certainly been smaller and hence the entire uncertainty of the length cannot have surpassed 2 microns. At the lower extremities the difference is still smaller. All this holds with regard to oxygen, in nitrous oxide such variations in the distribution of the tem- perature can be entirely neglected. With some measurements we have observed that the (aes length of the rods, when they had regained their q ordinary temperature after cooling, first exceeded the original length, but after two days it decreased again Fig. 1. to that value. ( 203 ) The cause of those deviations has not been explained. In a case where a particularly large deviation had been stated which did not altogether return to zero, it appeared, when the points were un- wrapped, that a rift had come into the glass. To see whether a thermical hysteresis had come into play a thermometerbulb (see fig. 1) with a fine capillary tube was filled with mercury. First the level of the mercury was compared with an accurate thermometer at the temperature of the room in a water- bath in a vaeuum glass. Then the apparatus was turned upside down so that the mercury passed into the reservoir 5, which is a little greater than A. Subsequently A and also a part of the stem was cooled down during 3 hours in liquid air in a sloping position so that thanks to the capillary being bént near 5 no mercury could flow back TABLE I. — JENA GLASS 16", Date | Ti Temp. L L W. W ate ien. je hek 7 3 d 16 Dec. | 24.35 | 15.7 | 1026.285 | 41026.280 | 40.620 15.9 1904 | 34 50 | 160 „286 | 279 | 40.786 17.0 44.22 | 16.3 999 290 | 40.845 17.4 = 20 Dec | 14.50 | 45.3 | 1095.574 | 1025.559 |s. 3.503 | 5.04 40.6 210} 15.4 560 550 lm.25.029 | 38.28 |- 86.78 24.30 | 15.4 57 561 14, 6.300 | 7.191 4 =224 21 Dec. | 34.15 | 14.6 | 1026.308 | 1026.991 |m.40.523 15.4 34.45 | 14.7 299 284 15.4 Eren Ien 308 „289 Im. 40.583 15.6 22 Dec. 1104 50 | 15.0 | 1025.408 | 1095.091 |s. 2.405 | 5.024 1,=30,8 124.15 | 45.0 112 „095 Im. 9.880 | 38.28 |—181 48 124.50 | 45.0 A15 098 |: 5.005 | 7.494 18.0 93 Dec. 124.30 | 45.8 | 1026.344 | 1026.341 \m.40.606 45.6 3h. 15.6 „339 339 15.2 34.30 | 15.6 335 336 |.40.537 15.2 11 Jan. | 34.40 | 15.4 | 1026.988 | 1096.978 | 40 634 15.9 1905 | 44.30 | 15.5 | „201 „280 | 40.703 16.4 ( 204 ) to A. When A had regained the temperature of the room the s ° mercury was passed again from into A and the apparatus replaced into the same waterbath as before. The deviation of the level of the mercury was of the same order as the reading error of the thermometer, about 0003°. A perceptible thermical hysteresis therefore we do not find. MAB WHat 22>. PLAIN D Temp. 7 7 7 7 ate Time RES t ag? Wy p S 16 Dec. 5h 59 16.5 1027.460 | 1027.464 17.0 1904 16.4 1027.461 | 1027. 459 17.0 17 Dec. | 1445 16 6 1026.620 | 4026630 Oh AD 16 3 1026 .618 622 1045 16.3 613 617 19 Dec. Sh 14 8 1027.459 | 1027.442 15.5 81-39 14.8 457 | 1027.440 15.5 90 sec. a 15.5 1026.627 | 1626.630 |s3.475| 4.993 3h 30 15.5 620 633 | m — 86.32 3h 55 15.4 631 635 147 575 | 8.653 91 Dec. |) 4440 | 44.7 © | 1027.460 | 102744 155 5h 10 14 9 459 444 15 5 6h 14.8 459 449 15.5 99 Dec. | 102 40 15.3 1025.963 | 1025.951 |s2.140| 4.993 AAA 10 15.3 1025 .973 961 | m ~182.6 Ah AD 14.9 1025. 964 947 |i5.649 | 8.653 93 Dec, | 14425 «| 45.7 1027.434 | 1027436 15.0 15 6 440 AAA 15 0 15.7 440 442, 15.2 3 Febr. Qh 15.4 1027.463 | 1027.459 15.2 15.4 459 455 15.2 1) Journ. Chem. Soc. 63. p. 135. 1893. ( 205 ) In table II (p. 204) the temperatures are used which are found with the thermo-element. A control-measurement with the thermo- element placed in the same vacuum tube without rod gave for the temperature in nitrous oxide — 87°,3 instead of — 86°,32. The mean value of the two determinations is used for the calculation. Another reason for the measurement of the temperature of the bath with a thermo-element as a control was the large difference between the mean temperature found by us and the boiling point of nitrous oxide — 89° given by Ramsay and SHIELDs '). As we are going to press we find that Hunter’) has given — 86°.2 for that temperature. § 3. Results. Jena glass 161II a 8385 b 117 } k, 2505 k, 353. | don Platinum a 905,3 b 49,4 k, 2716 k, 148,4. As regards platinum: Benoit finds from 0e to. +905 ~a, SION =.) ADL SCHEEL from.20* to.” 100° — a@° $80.6 ~ 6: 19.5 HorBorN and Day from O° to 1000° « 886,8 5 13,24 As to the differences between the values obtained now and those of Comm. N°. 85 (comp. $ 1), we must remark that these are almost entirely due to the differences in the determinations of temperature. The uncertainties of the latter, however, do not influence in the least the conclusion about 5 and the necessity of a cubic formula. There is every reason to try to combine our determinations on Jena glass above and below O° in such a cubic formula. Taking into account also the previous determination 242.10-® as the mean cubic coefficient from 0° to 100° (Comm. N°. 60, Sept. 1900, § 20) we find in the formula for the linear expansion below O° and in the corresponding one for the cubic expansion t ty & bh kerf SE) ef 104 a} ET ed | Jena glass 16 III a’ 789,4 k', 2368,1 b' 39,5 kl 12012 el 208 k', 86,2 1) With this measurement in N,O we have not obtained a temperature deter- mination with the thermo-element. This determination is not included in the calculation. It is mentioned here on account of the agreement with the determi- nation of 20 Dec., which for the rest has been made under the same circum- stances. *) Journ. Phys. Chem. May 1906, p. 356. ( 206 ) § 4. Control-experiment. The ends of the Jena glass rod were subsequently cut off and sealed together with a short intermediate rod. This short stick was placed in a glass of the same width as the vacuum tube with the same stopper and so short that the points projected in the same TABLE III. — JENA GLASS ENDS. Date | Temp, | L | Ee | y | 7 | ; | Ì scale t “16 t 0 42 April 1905 102 15 15.4 227.684 | 227 683 15.4 11h „686 „685 15.4 117 43 15.4 227.684 | 227.682 15.5 15.4 681 „679 15.5 N,O ; ss 3h 50 15.4 227.533 | 227 536 | ¢ 3.473 5.021 42.3 4h 24 15.4 „543 „541 ; Jm 4h 52 15.4 „590 548 | 4 5.490 TSN 32.3 13 April 17.4 227-677 | 227.681 AeA 14 April 46.2 227.675 | 227.676 15-9 10h 10 0. Sl ke = 2h 50 18.4 227.474 | 227.482 | s 1.941 5.021 35.5 Ai 4h22 18.9 — «482 ‚494 | t 4.683 Thee ks 8.9 15 April 16.6 227.725 |. 227 727 dad 1111 16.6 „724 „726 16.0 4h 20 16.4 227.706 „708 15.8 4h 46 16.4 dea „713 16.0 16 April 14.1 227.706 | 227.702 13.6 17 April 14.2 227.682 | 227.678 14.0 „685 „681 207 } manner as those of the rods in the vacuum glass. Now we have taken only a double glass filled with wool, enveloped in a card-board funnel and tube for letting out the cold vapours. The measurements are given in table III. The 2’s found in the experiment are of the same order of magnitude as those found with the long rods. The calculation with the coefficients a and 5 found in $2 yields: Ly,0 = 227,547 while we have found Ly,o = 227,544 Bo, = 227.487 eh A AE sp Lo, == 227,488. In conclusion we wish to express hearty thanks to Miss T. C. Jorrrs and Miss A. Sinuevis for their assistance in this investigation. Physics. — “On the measurement of very low temperatures. XI. A comparison of the platinum resistance thermometer with the hydrogen thermometer.” By Prof. H. Kameruincu Onnes and J. Cray. Communication N°. 95° from the Physical Laboratory at Leiden. (Communicated in the meeting of June 30, 1906). § 1. Introduction. The following investigation has been started in Comms. N°. 77 and N°. 93 VII of B. Merixk as a part of the more extensive investigation on the thermometry at low temperatures spoken of in Comm. N°. 95%, In those communications the part of the investigation bearing on the electrical measurements was chiefly considered. The hydrogen thermometer was then (comp. Comm. N°. 93 § 10) and has also this time been arranged in the same way as in Comm. N°. 60. Afterwards it appeared, however, that at the time the thermo- meter did not contain pure hydrogen, but that it was contaminated by air. The modifications which are consequently required in tables V and VI of Comm. N°. 93 and which particularly relate to the very lowest temperatures, will be dealt with in a separate communication. Here we shall discuss a new comparison for which also the filling with hydrogen has been performed with better observance of all the precautions mentioned in Comm. N°. 60. We have particularly tried to prove the existence of the point of inflection which may be expected in the curve (comp. § 6) represent- ing the resistance as a function of the temperature, especially with regard to the supposition that the resistance reaches a minimum at very low temperatures, increases again at still lower temperatures and even becomes infinite at the absolute temperature O (comp. ( 208 ) Suppl. N°. 9, Febr. ’04). And this has been done especially because temperature measurements with tlre resistance thermometer are so accurate and so simple. From the point of view of thermometry it is important to know what formula represents with a given accuracy the resistance of a platinum wire for a certain range, and how many points must be chosen for the calibration in this range. In Comm. N°. 95 $10 the conclusion has been drawn that between 0? and —180° a quadratic formula cannot represent the observa- tions more accurately than to 0°.15, and that if for that range a higher degree of accuracy is required, we want a comparison with the hydrogen thermometer at more than two points, and that for temperatures below — 197° a separate investigation is required. In the investigation considered here the temperatures below — 180° are particularly studied; the investigation also embraces the temperatures which can be reached with liquid hydrogen. It is of great importance to know whether the thermometer when it has been used during a longer time at low temperatures would retain the same resistance. We hope to be able later to return to this question. Here we may remark that with a view to this question the wire was annealed before the calibration. Also the differences between the platinum wires, which were furnished at different times by Heraxus, will be considered in a following paper. § 2. Investigations by others. Since the appearance of Comm. N°. 93 there has still been published on this subject the investigation of Travers and Gwyer’). They have determined two points. They had not at their disposal sufficient eryostats such as we had for keeping the temperatures constant. About the question just mentioned : how to obtain a resistance thermometer which to a certain degree of accuracy indicates all temperatures in a given range, their paper contains no data. § 3. Modification in the arrangement of the resistances. The variation of the zero of the gold wire, mentioned in Comm. N°. 93 VIII, made us doubt whether the plates of mica between the metallic parts secured a complete insulation, and also the movability of one of the glass cylinders made us decide upon a modification in the construction _ of the resistances, which proved highly satisfactory and of which we 1) Travers and Gwyer. Z. f. Phys. Chem. LIL, 4, 1905. The wire of which the calibration is given by Oxszewsxt1, 1905, Drude’s Ann. Bd. 17, p. 990, is appa. rently according to himself no platinum wire. (Gomp. also § 6, note 1). ( 209 ) have availed ourselves already in the regulation of the temperatures in the investigation mentioned in Comm. N°. 944, A diffieulty adheres to this arrangement which we cannot pass by unnoticed. Owing to the manner in which this thermometer has been mounted it cannot be immersed in acid. Therefore an apparatus consisting entirely of platinum and glass remains desirable. A similar installation has indeed been realized. A description of it will later be given. The figures given here exclusively refer to the thermometer described in Comm. N°. 944 (p. 210). Care has been taken that the two pairs of conducting wires were identical. Thus the measurement of the resistance is performed in a much shorter time so that both for the regulation of the tem- perature in the cryostat and, under favourable circumstances, for the measurement the very same resistance thermometer can be used. § 4. The temperatures. The temperatures were obtained in the cryostat, described in Comm. N°. 94%, by means of liquid methyl chloride —39°, —59°, — 88°, of liquid ethylene — 103°, — 140°, — 159°, of liquid oxygen — 182°, lo), ——200,,, —- 212°, —— 217°, by means, of. liquid hydrogen — 252° and — 259°. The measurements were made with the hydrogen thermometer as mentioned in § 1. § 5. Results for the platinum wire. These results are laid down in table I (p. 210). The observations marked with { | are uncertain on account of the cause mentioned in Comm. N°. 95% § 10 and are not used in the derivation and the adjustment of the formulae. For the meaning of WR; in the column “remarks” I refer to $ 6. § 6. Representation by a formula. a. We have said in § 1 that the quadratie formula‘) was insuffi- cient even for the range from 0° to —180°. If a quadratic formula is laid through — 103° and — 182°, we find : *) The correction of Carrenpar, used at low temperatures by Travers and Gwrer, Z. f. Phys. Chem. LIL, 4, 1905 comes also to a quadratic formula. Dickson’s quadratic formula, Phil. Mag. June 1898, is of a different nature but did not prove satisfactory either; comp. Dewar Proc. R. Soc. 64, p. 227, 1898. The calibration of a platinum thermometer through two fixed points is still often applied when no hydrogen thermometer is available (for instance BESTELMEYER Drude’s Ann. 13, p. 968, ’04). MA B Ge ( 210 ) if COMPARISON BETWEEN THE PLATINUM RESISTANCE THERMOMETER AND THE HYDROGEN THERMOMETER. | Temperature Resistance Date Wee ake measured Ean 0° | 0° 137 8840 mean cf 5 measurements. 27 Oct 5 he 0 ORS) 421 .587 05 OH ts, 530) == 8-75) 105.640 30 Oct 3h 50 ete 89.277 ’05 8 July 10h. 12 — 103.83 80.448 ’05 26 Oct. 5 h. 20 = 4139587 59.914 05 7 July 4 h. 25 1085 59.920 05 26 Oct Onhe 16 = SES 48.929 05 297 June 1b. 40 [— 182.69] 34.861 W—R4,—— 0.061 05 30 June 11 h. O — 18215 34.858 06 oF June 3h. 50 [-- 195.30] 27.598 W—R 4, =H 0.082 05 2 March 3h. 35 — 195.18 1.595 ’06 29 June 141 h. 6 [— 204.53) 22.016 W_R4}—=— 0.110 ’05 2 March 1 h. 30 — 204.69 29.018 ’06 30 June! 3 he 0 [— 212.83] 11255 W—R 47 = — 0.082 05 5 July bahe55 — DASH 17 290 ’05 5 July 3) li, G40) — 217.41 14.763 05 3 March 10h. O — 217.41 14.770 05 | 5 May Saha) 52805 | 1.963 06 5 Ma aak, — 959.24. 1.444 ¢ 281) Wi W, | 1 + 0,39097 € 5) — 0,009862 (5) f For instance at — 139° it gives WW—R: + 0,084. A straight line may be drawn through — 182°, — 195°, — 204° and — 212° and then — 217° deviates from it by 0°,25 towards the side opposite to —158°. Hence the existence of a point of inflection is certain (comp. sub d). Therefore it is evident that a quadratic formula will not be sufficient for lower temperatures. 6. But also a cubic formula, even when we leave out of account the hydrogen temperatures, appears to be of no use. For the cubic formula through the points —88°,14, —158°,83, —204°,69, we obtain: 3 ‚395008 — —0,0,73677 0,0,58386 It gives for instance at Lares a one of -—0,110, ke es a deviation of + 0,322 *) c. In consequence of difficulties experienced with formulae in ascending powers of ¢, we have used formulae with reciprocal powers of the absolute temperatures (comp. the supposition mentioned in $ 1 that the resistance becomes infinite at the absolute zero). Three of Soak have been investigated : Oe aaa Ga \+0(s55 +5 = fe W, 100 100 100 i i 273,09 ORDE eae) WwW, 100 100 100 i i 273,09 ko 10‘ he mea @ veen: W, 100 100 100 273,09 10° mee ‘(a ep AL, We shall also try a formula with a term Ti instead of = For the first we have sought a preliminary set of constants which was subsequently corrected after the approximate method indicated by Dr. E. F. van pe SANDE BAKHUYZEN (comp. Comm. N°. 954) in two different ways. First we have obtained a set of constants A; with which a satisfactory accurate agreement was reached down to — 217°, a rather large deviation at — 252° and a moderate deviation at — 259°. Column W-—R4r of table II contains the deviations. Secondly we have obtained a set of constants which yielded a fairly 1) These values deviate slightly from those communicated in the original. (255 accurate agreement including — 252°, but a large deviation at — 259°. These are given in table II under the heading W— bar. Lastly we have obtained a preliminary solution B which fairly represents all temperatures including — 252° and — 259° and from which the deviations are given in table IL under W—Rg, and a solution of the form C which agrees only to — 252° and to which W—Reg relates. The constants of the formulae under consideration are : ba? er He 0.399625 |4+- 0.400966 + 0.442793 | +0. 40082 b |— 0.0009575|+ 0.001159 |4 0.013812 | 0.001557 + 0.0049442/+ 0.0062417|+ 0.012683 | +-0.00557 +. 0.019380 |-4+ 0.026458 |H 0.056221 | +-0.01975 | |— 0.0032963] —0.16501 WAL JB Sy Blk COMPARISON BETWEEN THE PLATINUM RESISTANCE THERMOMETER AND THE HYDROGEN THERMOMETER. 8 iy Q Temperature oT obser: eend | Bar ee thermometer. | hydrogen in 0 therm. | ce | 137.884 0 0 0) 0 =) oo gue es 121.587 | 40.025 | + 0.066 | + 0.210} + 0.063 — 58.75 3 105.640 + 0.011 | — 0.011 | + 0.453 | + 0.048 — 88.14 4 89.277 — 0 012 | — 0.050 | — 0.001 | + 0.008 — 103.83 3 80.448 — 0.023 | — 0.061 | — 0.075 | -— 0.015 — 139.87 hl a) gil + 0.004 | — 0.005 | — 0.082 | — 0.005 — 158.83 3 48.929 + 0.023 | + 0.044 0 + 0.008 — 182.75 2 34.858 — 0.029 | + 0.027 | + 0.083 | — 0.035 — 195.18 2 27.505 + 0.009 | + 0.061 | -+ 0.148 | + 0 007 — 204.69 1 22.018 — 0.014 | + 0.012 | + 0.400 | — 0.014 — 212.87 3 17.290 — 0.0 | — 0.065 | — 0.001 | — 0.031 — 27.41 4 14.766 *| + 0.028 | — 0.048 | + 0.270 | + 0.007 — 252.93 2 1.963 + 2.422; + 0.057 | — 0.001 0 — 259.24 1 1.444 + 0.199 | — 4,201 0 ( 213 ) In those cases where the W—R have been derived fiom two deter- minations the values in the 2°¢ column are marked with an **). If we derive from the differences between the observed and the computed values as far as —217° the mean error of an obser- vation by means of A;, this mean error is expressed in resistance + 0,025 2, in temperature + 07,044. The mean error of an observation of the hydrogen thermometer, as to the accidental errors, amounts to 0°,02 corresponding in resist- ance to + 0,010 @, while that of the determination of the resistance may be left out of consideration. We cannot decide as yet in how far the greater value of the differences between the observations and the formula is due to half systematic errors or to the formula. For the point of inflection in the curve representing the resistance as a function of the temperature we find according to B — 180° ®). In conclusion we wish to express hearty thanks to Miss T. C. Jorres and Mr. C. Braak for their assistance in this investigation. Physics. — “On the measurement of very low temperatures. XI. Comparison of the platinum resistance thermometer with the gold resistance thermometer. By Prof. H. KAMERLINGH ONNES and J. Cray. Communication N°. 954 from the Physical labora- tory at Leiden. (Communicated in the meeting of June 30, 1906). § 1. Introduction. From the investigation of Comm. N°. 93, Oct. ’04, VIII it was derived that as a metal for resistance thermometers at low temperatures gold would be preferable to platinum on account of the shape of the curve which indicates the relation between the resistance and the temperature. Pure gold seems also better suited because, owing to the signifi- cation of this metal as a minting material, the utmost care has been bestowed on it for reaching the highest degree of purity and the quantity of admixtures in not perfectly pure gold can be exactly determined. The continuation to low temperatures of the measurements described in Comm. N°. 93 VIII — which had to be repeated because, although MEemink’s investigation just mentioned had proved the usefulness of the method, a different value for the resistance 1) The deviations of the last two lines differ a little from the original Dutch paper. *) Owing to e being negative (B) gives no minimum; a term like that with e does not contradict, however, the supposition wo at 7’=O (§ 1) as the formula holds only as far as —259°. 15 Proceedings Royal Acad. Amsterdam. Vol. IX. ‘ (214) had been found before and after the exposure of the wire to low temperatures — acquired a special value through this peculiarity of gold. As will appear from what follows, the point of inflection of the resistance as a function of the temperature must lie much lower for gold than for platinum. Our favourable opinion about gold as a thermometrie substance was confirmed with regard to temperatures to a little below —217°. With respect to the lower temperatures our opinion is still uncertain. A minimum of resistance seems not to be far off at —259°. § 2. The apparatus and the measurements. About the measure- ments we can only remark that they are performed entirely according to the methods discussed in Comm. N°. 93. The pure gold was furnished through the friendly care of Dr. C. Horrsema. It has been drawn to a wire of 0,1 mm. in diameter by HERAEUS. The gold wire was wound upon 2 cylinders, ie was about 18 m. in length and its resistance at 0° was 51,915 Ohms. The tempera- tures were reached in the cryostat of Comm. N°. 944 as in the investigation in Comm. N°, 95°. The determinations of temperature were made by means of the resistance of the platinum wire of Comm. N°. 95°. The zero determinations before and after the measurements at low tempera- tures agreed to perfection (this agreement had left something to be desired in the measurements dealt with in Comm. N°. 93). The measurements were made partly directly by means of the differential galvanometer, partly indireetly by comparing the gold resis- tance with a platinum resistance, which itself had been compared with the originally calibrated platinum resistance (comp. Comm. 955): § 3. The Results, obtained after the direct and the indirect method are given in column 3 of table III and indicated by d and 4 respectively. For the observations the eryostat was brought to the desired temperature by regulating it so that the resistance of the platinum wire had a value corresponding to this temperature, and by keeping this temperature of the bath constant during the measurements of the resistance of the gold wire. The temperatures given in table III are the temperatures on the hydrogen thermometer according to the observations of Comm. N°. 95¢ belonging to the resistance of the platinum thermometer. | ( 215 ) | TABLE III. CALIBRATION OF THE GOLD RESISTANCE THERMOMETER. Temperature Observed Date. W—R W—R W—R resistance. gold resistance. 4 BI BH 1906 0 51.915 d 0 0 0 1 Febr. 5 b.57| — 28.96 46 137 SOR ee ORO eh 02029 peen, 40e —, 58.58 40 3261 TEE ee #6 Sh 2 — 813 34 640 i zn STEED 2928 | Greens 2 12 June 2h. 20] — 103.82 31.432 d Oe ie Sta Gy) ee ee >» Mh. — 139.86 24.984 d ADS em ON een 0 17 Jan. 3h. 20| — 459.14 20.394 1 Dn a A June A1 h. 50| — 182.75 15.559 d Td RAE eee 12.980 d SO AS tn 0 Den — 204.69 10.966 d BS ese EE OT Dee ey | Di Sh. — 212.87 9.203 d Bfr ORE STE 12 Jan. 11 h. — 216.25 8.460 i EED ee WO ||, age 18 May 4&h.10| — 252.88 2.364 d We inOsake 2 20 RE pe Hen: — 959.18 2.047 d TE Een CG In order to agree with Dewar, we ought to have found for the resistance of the gold wire at the boiling point of hydrogen 1.7082 instead of 2.864 42. Also the further decrease of the resistance found by Dewar’) in hydrogen evaporating under a pressure of 30 mM. is greater than that was found by us. We may remark that this latter decrease of the resistance according to him would belong to a decrease of 4 degrees on the gas thermometer, and that we in accordance with TRAVERS, SENTER and JaQqurrop *) found a difference in temperature of 6,°3 between the boiling point of hydrogen at a pressure of 760 m.m. and of 60 m.m. (preliminary measurements). § 4. Representation of the variation of the gold resistance by a formula. As to this we refer to what has been said in Comm. 1) Dewar, Proc. Roy. Soc. Vol. 68 p. 360. 1901. 2) Travers, SENTER and Jaguerop, Phil. Transact. A. 200. Proc. Roy. Soc. Vol. 68, p. 361, 1901. ( 216 ) N°. 95e, XII. $ 6. The resistance of the gold wire can be represented fairly well as far as — 217° as a function of the temperature by a formula of the form A. WwW; t “2 = 1-0; 39070 —— + 0.017936 (5) + 0,0085684 0.0080999 = ee (A) a a 700... 27309 This formula A is not a to include the hydrogen temperatures. For the deviations W—fy, comp. table III. We have therefore made use of a formula B, and W, t EY = ad 21900102835) —— Wo 5 oe É mo) a 100 ~—-100 18( — | —0,0268911 BI + 0,00352 (5 3) (= ~ an 55) ae (BI) 4 0,0052211 ee eS My) S873 09 is in good harmony down to — 258°, while t 2 EES Ee eS ia) in W, 0,0102889 0,0229106 eae + + OE. aoe 100 TOONT — 0,00094614 | Fees je 273,09 gives a fair harmony also at — 259°"). The deviations are given under the headings W—Rpg 7 and W— kg u in columns 5 and 6 of table III. The mean error of an observation with respect to the comparison with formula B/ is + 0,017 2 in resistance and + 0°,09 in temperature. Formula 6/J gives for the point of inflection of the gold resistance — 220°. (B II) Mathematics. — “Quadratic complexes of revolution.” By Prof. JAN DE VRIES. § 1. When the rays of a complex can be arranged in reguli of hyperboloids of revolution with the same axis, then the complex can bear revolving about that axis. If such a complex of revolution £2 contains also the second regulus of each of the indicated hyperboloids, then it is symmetrie with respect to each plane through its axis 1) The coefficients of the formulae and the values of the deviations, found at a renewed calculation, differ slightly from those given in the original Dutch paper. ( 217 ) and it can be distinguished as a symmetric complex of revolution. This is the case with the complexes of tangents of surfaces of revolution. We determine the general equation of the quadratic complexes of revolution with axis OZ in the coordinates of rays p= ; po yy! 8 PZ, et ee 5 eae Per te A Ps ty — YT. By substitution of p= ap; — Bp, > Ps = Bp, CO ae ane Ee ie he ee ks PP 4 EE 0 (where a° +R? —1) in the ars quadratic equation we easily find that the equation of an 2 can contain terms only with (Py Ps): (Pv IP) Pas Per (P1 Ps—Ps Ps) and (P: Ps + Pa Ps)- As the latter combination can be replaced by — p, p, in consequence of a wellknown identity we tind for @ the equation Api +P 2") + Bps’ +2Cp pet Les + Ep +P s +24 (PPs—PaP J=9- (1) If C=O, equation (1) does not change when z is replaced by — £; so it represents a symmetrical complex. The coordinates of rays gq, = uu 5 g,=v0— : g,==w—w , qd, = vu — we , q; = wu — uw , q, = U —w', where u, v and w represent the coordinates of planes are connected with the coordinates p by the wellknown relations Pi* Us Pz: Ws = Ps? Wo = Pa? A Ps" Ia = Po * Ier So 2 can also be represented by Had) HD +2Cg get Bas FAI +95 +2924 —1198)=0 - « (2) This equation is found out of (1) by exchanging p,; and q, and of A, B, C, D, E, F and ZE, D, C, B, A, —F. § 2. The cone of the complex of the point (2’,7/,z’) has as equation : Ale) + A(y— yy Bae HC (gee (ee) HD 2e!) + HE(Ey-y oP HE(e'e-a2)t AF (o-a)(e'z-e'o) + 2F(y—y'(y'z-2'y)=0. (3) In order to find the equation of the singular surface we regard the cones of the complex whose vertices lie in XOZ and note the condition expressing that the section of such a cone and XO Y breaks up into two right lines. After suppression of the factor 2? which is to be rejected and substitution of z? + 4? =r? for z°, we find the equation ( 218 3 D(AE — F?)r* + (AE 4 BD — C? — F?) 9? (Ez? — 2F2 + 4 ate BEF Fit A =O... NES) As this can be decomposed into two factors of fie form Ir? + M (Bz? —2Fz-+ A), the singular surface = consists of two quadratic surfaces of revolution. These touch each other in the cyclic points 7, and 7, of the plane XOY and in the points B, and B, on OZ determined by Ez? — 2Fz The two surfaces cut each other according to the four isotropic right lines indicated by the equations ey" == 0 And Het 282 A = 0. es. OR If 2 is symmetric (C= 0) the two parts of the singular surface have as equations (AE — F?) (a? Hy) + B(E?—2Fz2+ 4)=0, . . (6) Dig A py Ra Aa 0s 2) a B If we find B=O and D=0O, then = breaks up into the four planes (5) and 2 is a particular tetraedal complex. Out of (3) it is easy to find that the cones of the complex of the points B,, B,, 7, and J, break up into pencils of rays to be counted double. These points shall be called dbisingular. § 3. The rays of the complex resting on a straight line / touch a surface which is the locus of the vertices of the cones of the complex touched by /. This avial surface is in general of order four and of class four and possesses eight nodes. *) We shall determine the axial surface of OZ. The points of inter- section (0, 0, 2’) of an arbitrary cone of the complex with OZ are indicated by the equation lat dy) + Bl2? AF (a? + y*) + Befe + [A (2° + y’) + Be?] = 0. This has two equal roots if (AE — F*) («* + y") + B(Es* —2 Fz + Aa $y) =0 . (8) So the axial surface of OZ consists of the two isotropic planes through the axis and a quadratic surface of revolution which might be called the meridian surface. If 2 is symmetrical, it forms part of the singular surface as is proved out of (6). Also the axial surface of the right line le lying at infinity in XOY breaks up into two planes, and a quadratic surface. Its 1) Srurm, Liniengeometrie Ill, p. 3 and 6. ( 219 ) equation is found most easily by regarding the rays of the complex normaloto XOZ." Krom «== a; a ‘eusues: ‘p, == 0; py 0, Dp, = 2), Pe =O, Pp, = — tp; By substitution in (1) we find (A + De? + Ez? — 2 Fz) p,? = 0, and from this for the indicated surface Die? py Be NRN) For the symmetrical complex this parallel surface is according to (7) the second sheet of the singular surface. The planes of the pencils of rays of the bisingular points 5,, >, form the lacking part of the axial surface of /~. We can show this by determining tbe equation of the axial surface of the right line 2 =0, y’ =, and by putting in it b=. We then find (Eet — 2 Fe + A){D (0? Hy) + Est —2 Fz 4+ A}=0 . (10) The meridian surface, the parallel surface, and the two parts of the singular surface belong to a selfsame pencil, having the skew quadilateral b,/,5,/, as basis. If in the equation of the cone of the complex the sum of the coefficients of &°,y* and z’ is equal to zero, then the edges form 2! triplets of mutually perpendicular rays. The vertices of the friortho- gonal (equilateral) cones of the complex belonging to 2 form the surface of revolution (D+ EB) (et Hy) +2Eet —AFe + (A+ B)=0.. (11) Jt has two circles in common with each of the parts of 2. These contain the vertices of the cones of the complex which break up into two perpendicular planes. § 4. The distance 7, from a right line to OZ is determined by 1 Sen Pe UP teens ah (12) Pi’ + Ps the angle A between a ray and XOY by Ps” N= Siu, Ge”, Wide be, Speier aD : Pi + Pa et So the condition /, tang à = a furnishes the complex DP, =O (Py Psy = he nn on (14) Here we have a simple example of a symmetrical complex of revolution. The equation Ps = apy IEB ee oe ee 1, 5) ( 220 ) determines a complex 2 whose rays form with the axis a constant angle, so they cut a circle lying at infinity. The equation Op ele AN Gah ye ee ke furnishes a complex @, whose rays cut the circle 2?-+y¥’?= a’. For XOY euts each cone of the complex according to this circle. If / represents the distance from a ray O then bi Ps are Per Pi + pa + pa If NOY is displaced along a distance c in its normal direction, p, and p, pass into (p,—cp,) and (p; + cp‚). So for the distance /, from a ray to the point (0, 0, c) we have ie (Pd Pate P Rs = Pals) ae Ct (Pate Ps) Pi HP: + Ps” If in this equation we substitute —c for c we shall find a relation for the distance /, from the ray to-point (0, 0, — €). The equation p (17) (18) elles = 8 furnishes a complex @ with the equation (a, + a) ce — B} (oa = De) — Bias == (a, sE (ty) (pr == Ps == Pe) mi +2(e, —4,)¢(p, Ps — PrP) =9-- « + + + (19) This symmetrical complex is very extensively and elementarily treated by J. NeuBerG (Wiskundige Opgaven, IX, p. 334—341, and Annaes da Academia Polytechnica do Porto, 1, p. 1387—150). The special case a, 1, + a,l, = 0 was treated by F. Corin (Mathesis, IV, p. 177—179, 241—243). For J, =l, we find simply Depo). Dy Ue web otal rents, ze we val This complex contains the rays at equal distances from two fixed points. As c does not occur in the equation the fixed points may be replaced by any couple of points on the axis having O as centre’). § 5. When there is a displacement in the direction of OZ the coordinates of rays p,, P» Ps and p,, do not change whilst we obtain Ps =p, + hp, and Ps =D, = hp, so Pi Pa + Pa Ps = Pi Pa 1 Pa Po The forms (p,?-+ p,”) and (p, Pp; — Ps Ps) are now not invariant. 1) This complex is tetraedral. See Srurm, Liniengeometrie, I, p. 364. ( 2214 ) When in equation (1) of the complex @ the coefficients # and F are zero, the complex @ is displaced in itself by each helicoidal movement with axis OZ. This complex can be called helicoidal. The singular surface has as equation CBI 03) ABE Oe ae a f21) so it consists of a cylinder of revolution and the double laid plane at infinity. § 6. By homographic transformation the complex 2 can be changed into a quadratic complex with four real bisingular points. If we take these as vertices of a tetrahedron of coordinates 0,0,0,0,, it is not difficult to show that the equation of such a complex has the form Ap 2+ Bp rs, + 2 C pia Pas + 2D Pie Pas + 2 E Pig Pas = 9- (22) If we again introduce the condition that the section of the cone of the complex with one of the coordinate planes consists of two right lines we find after some reduction for the singular surface A(D-E) yy, +2{AB-(C-D)(C-E)\9, yaaa B(D-E)y,*y2 =0 - (23) So this consists of two quadratic surfaces, which have the four right lines O,O,, O,O0,, 0,0, and 0,0, in common. For A—0, B=0 the complex proves to be tetraedral. For D= F the equation is reducible to Ap’, si BDk sb 2 (C = D) Ps Ps — 0, and indicates two linear complexes. For the axial surfaces of the edges O,O0, and 0,0, we find @,2,\2A oo, (DB), ==) . .-. = BA) and Sf. 2R oper (DD EB) a, ji Aan For a point (0, y,,0,y,) of the edge 0,0, the cone of the complex is represented by Ayse +2(C—B)y,y,%,%, Bye =0;. . (26) so it consists of two planes through O,0O,. This proves that the edges O,0,, O,O,, O,0,, 0,0, are double rays of the complex *). 1) See Sturm, Liniengeometrie Il, pp. 416 and 417. (293) Physiology. — “A few remarks concerning the method of the true and false cases.’ By Prof. J. K. A. WERTHEIM SALOMONSON. (Communicated by Prof. C. WINKLER.) The method of the true and false cases was indicated by FEcHNER and used in his psychophysical investigations. He applied this method in different ways: first to determine the measure of precision (Präcisionsmasz) when observing difference-thresholds, afterwards to determine these difference-thresholds. Already in the course of his first experiences arose the difficulty that not only correct and incorrect answers were obtained, corre- sponding with the “true” and “false” cases, but that also dubious cases occurred, in which the observer could not make sure as to the kind of difference existing between two stimuli, or whether there did exist any difference at all. Frcunnr himself, and many other investigators after him, have tried in different ways to find a solution to this difficulty. What ought to be done with these dubious cases? Frcuner has indicated several methods, which he subjected to an elaborate: criticism. Finally he concluded that the method to be preferred to all others was that one, in which the dubious cases were distributed equally amongst the false and the true cases. If e.g. he found w true cases, v false cases and ¢ dubious cases, he calculated his measure of precision as if there had been w + 3¢ true cases and }¢-+ wv false cases. Furthermore he showed that a method, employed especially by American experimental physiologists, in which the reagent is urged always to state a result, even if he remains in doubt, practically | means the same thing as an equal distribution of the ¢ cases amongst the true and the false cases. Frconner still worked out another method, by means of which the threshold value was first calculated from the true eases, then from both the true and dubious eases, whilst the final result was obtained with the aid of both threshold values. A most elegant method to calculate the results of the method of the false and true cases has been pointed out by G. E. Mürrer, starting from this view, that as a matter of necessity the three groups of cases must be present, and that they have equal claims to exist; that the number of cases belonging to each of these groups in any case, are equally governed by the well-known law of errors. From the figures for the true false and dubious cases the thresholdvalue may afterwards be calculated. I need not mention some other methods, e.g. that of FovcauLr, ( 223 ) that of Jastrow, because the method of Fovcaurr is certainly in- correct (as has been demonstrated among others by G. B. Mürrur), whilst that of Jasrrow is not quite free of arbitrariness. Against all these different ways of using the method of the false and true cases, I must raise a fundamental objection, which I will try to elucidate here. Whenever two stimuli of different physical intensity are brought to act on one of the organs of the senses, either the reagent will be able to give some information as to the difference between these stimuli, or he will not be able to do so. If he cannot give any information, then we have before us a dubious case, if on the con- trary he is able to give some information, this information may either be correct, — this constituting a true case — or it may be incorrect, when we shall have a false case. If the experiment is repeated a sufficient number of times, we shall have obtained at last a certain number of true cases w, of false cases v and of dubious cases 4. Generally it is admitted that the reagent has indeed perceived correctly w times, that he has been mistaken » times, that he was in doubt ¢ times. If this premiss were correct, FECHNER’s or G. E. Mürrer’s views might be correct too. This however is not the ease. An error has already slipped into the premiss, as will become evident furtheron. No difference of opinion exists as to the dubious cases. To this category belong first those cases, where the reagent got the impression of positive equality, and next those cases, where he did not perceive any difference, and consequently was in doubt. Together they embrace such cases only, in which a greater or lesser or even infinitesimal physical difference was not perceived. Neither need any difference of opinion exist as regards the false cases. In these cases a stimulus has been acting on the organs of the senses, and information was given about the effect, but on account of a series of circumstances, independent of the will of the reagent, his judgment was not in accordance with the physical cause. The physical cause therefore has not been perceived, but accidental cir- cumstances led the reagent to believe that he was able to emit a jadgment, though this judgment, accidentally, was an incorrect one. And now we are approaching the gist of the argument. If it be possible, that amongst a series of experiments a certain number occur, in which the reagent really does not perceive the physical cause, but is yet induced by chance to emit a judgment which proves to be an mecorrect one, then there ought to be also a number of (224 ) cases, in which likewise the physical cause is not perceived, in which however by chance a judgment is emitted, though this time a correct one. These facts being dependent on circumstances beyond our will, the chances are equal that either a wrong or a right judgment may be given. If therefore we had v false cases, we may reasonably admit the existence of » cases, in which practically the physical cause has not been perceived, and where yet a judgment, this time a correct one, has been given. These v cases however have been recorded amongst the true cases, though they cannot be admitted as cases of correct perception: it is only in w_—v cases that we may suppose the physical cause to have been really and correctly perceived; in all other cases, in 2v + 7% cases therefore, there has been no perception of the real difference of the stimuli. In this way we have only to consider two possibilities, constitu- ting the perceived and non-perceived cases, the number of which I will indicate by § and y. The supposition that we may apply the principles of the calculus of probability to them, is justified a priori. This supposition is changed into a certainty, if we apply the mathematical relations, stated by Frcuner to exist between the numbers of true and false cases. As is well known, Frcuner added to the number of true cases, obtained by the experiment, one half of the dubious cases: he used therefore in his calculation a rectified number of true cases ww tt In the same manner he corrected the number of false cases by adding to them likewise one half of the dubious cases : Da Wist KL In calculating the number of my perceived cases, I get § = w—v, whilst the number of non-perceived cases is represented by y= {+ 2v. Evidently I may also express the number of perceived cases by §=—w'—uv'. As Frcuner has given for the relative value of the corrected number of true cases the expression : Dh 1} al Bio, wt En sa ie w+t+tov n Vn 0 and for the corrected relative number of false cases the expression: Dh Me AR wtttyv nn aks Va 0 ( 225 ) we obtain from these immediately for § and 4 the two relations : Dh and We find therefore that the way of dealing with the true, dubious and false cases as proposed by me, allows us to use FECHNER’s well- known tables. I wish to lay some stress here on the fact, that G. E. Mürrer’s formulae give the same result, saving only the well-known dif- ference in the integral-limits: these latter being O and (S,+D) hu. I need scarcely add that my remarks do not touch in the least the question about “thresholdvalue” between FecHNeER and G. E. MüÜrrer. It is evident, that the result of the calculation of a sufficiently extensive series of experiments according to the principles, given in my remarks should give numbers, closely related to those either of Fucuner or of G. B, Mürrer — depending on the limits of inte- gration. Still I wish to draw special attention to the fact that the formulae of G. E. Mürrrer about the true, false and dubious cases are rather the statistical representation of a series of nearly identical psychological processes, whilst the opinion professed by me on the method of the false and true cases, represents a pure physiological view. Finally my remarks show, that Carrer, and FurLERTON’s way of applying the method of the true and false cases is less arbitrary than it seems to be at first sight. They take for the thresholdvalue the difference of stimuli with which the corrected number of true cases attains 75 °/,. Such being the case, § and x are both = 50°/,. They consider therefore the thresholdvalue to be a difference between two stimuli such, that there is an equal chance of this difference being perceived or not. ( 226 ) Chemistry. — “The shape of the spinodal and plaitpoint curves for binary mixtures of normal substances.” (Fourth communi- cation: The longitudinal plait.) By J. J. van Laar. (Com- municated by Prof. H. A. Lorentz.) 1. In order to facilitate the survey of what has been discussed by me up to now, I shall shortly resume what has been communi- eated on this subject in four papers in These Proceedings and in two papers in the Arch. Teyler. a. In the first paper in These Proceedings (22 April 1905) the equation: dT = [aw (1—2) (av—B/Ya)? + alv—b) | - «Ce (1) was derived for the spinodal lines for mixtures of normal substances, on the supposition that a and 6 are independent of v and 7, and that a,, =Wa,4,, while (av—BY/a)* [(1—22) v—3e2 (1—2)B] + + Ve eo] Saro) (arta) + SIE | (2) was found for the v,2-projection of the plaitpoint line, when a=Wa,—Wa, and B=b,—),. b. In the second paper in These Proceedings (27 May 1905) the shape of these lines for different cases was subjected to a closer examina- tion. For the simplification of the calculations =O, i.e. b,=b,, was assumed, so that then the proportion @ of the critical temperatures of the two components is equal to the proportion a of the two critical a b T ‘= ¢, nm (where 7’, is the 0 pressures. If we then put “third” critical temperature, i. e. the plaitpoint temperature for v = 5), the two preceding equations become: t= 4w [a(1—2) + (p +2) (l—o)*] . « . - (le) (p + DP (Lo) (1-30) AL he AT It now appeared that the plaitpoint curve has a double point, when g = 1,43, ie. O—=n—= 2,89. If 6 > 2,89, the (abnormal) case of fig. 1 (loc. cit.) presents itself (construed for g=1, 6=(1 + */,)’= 4); if on the other hand 6 < 2,89, we find the (normal) case of fig. 2 (loc. cit.) (construed for gy = 2, 6 = 2/,). At the same time the possibility was pointed out of the appearance of a third case (tig. 3, loc. cit.), in which the branch of the plaitpoint (120) + Bp + #) (1—@)? + 0. (2a) ( 227 ) line running from C, to C, was twice touched by a spinodal line. Here also the branch C,A is touched by a spinodal line [in the first two cases this took place only once, either (in fig. 1, loc. cit.) on the branch C,A (A is the point «=0, v6), or (in fig. 2 loc. cit.) on the branch C,A (C, is the before-mentioned third critical point) |. So it appeared that a// the abnormal cases found by Kurnen may already appear for mixtures of perfectly normal substances. It is certainly of importance for the theory of the critical phenomena that the existence of two different branches of the plaitpoint curve has been ascertained, because now numerous phenomena, also in connection with different “critical mixing points’ may be easily explained. c. In the third paper in These Proceedings (June 24, 1905)") the equation : eek 1 1 TN o=7(Z)=evslov (1 1va)=i) EN was derived for the molecular increase of the lower critical temperature for the quite general case a, < Das os So, which equation is reduced > to the very simple expression BS OO) eee ts minal ie ete WS) for tHe case n= 1 (p, =p); This formula was confirmed by some observations of CENTNERSZWER and BücHNer. d. The fourth paper appeared in the Archives Teyler of Nov. 1905. Now the restricting supposition 8 =0 (see 6) was relinquished for the determination of the double point of the plaitpoint line, and the quite general case a, < de Us < Db, was considered. This gave rise to very = De intricate calculations, but finally expressions were derived from which 2 1 1 also the values of « and v in the double point can be calculated. Besides the special case 6 = (see b) also the case a =1 was examined, and it was found that then the double point exists for 6 = 9,90. This point lies then on the line v = db. for every value of 6 = — the corresponding value of Desie and ') The three papers mentioned have together been published in the Arch. Néerl, of Noy. 1905. ( 228 ) e. The fifth paper (These Proceedings, Dec. 30, 1905) *) contained the condition for a minimum critical (plaitpoint) temperature, and that for a maximum vapour pressure at higher temperatures (i. e. when at lower temperatures the three-phase-pressure is greater than the vapour pressures of the components). For the first condition was found: Aava Sven for the second: gd 5 EN © which conditions, therefore, do not always include each other ’). After this the connodal relations for the three principal types were discussed in connection with what had already been written before by Korrrwre (Arch. Néerl. 1891) and later by van per Waats (These Proceedings, March 25, 1905). The successive transformations of main and branch plait were now thrown into relief 7 connection with the shape of the plaitpomt line, and its splitting up into two branches as examined by me. J. Im the sith paper (Arch. Teyler of May 1906) the connodal relations mentioned were first treated somewhat more fully, in which also the p,z-diagrams were given. There it was proved, that the points A, A, and £',, where the spinodal lines touch the plaitpoint line, are cusps in the p,7-diagram. Then a graphical representation was plotted of the corresponding values of 6 and a for the double point in the plaitpoint line, in connection with the calculations mentioned under d. Both the graphical representation and the corresponding table are here reproduced. The results are of sufficient importance to justify a short discussion. We can, namely, characterize all possible pairs of substances by the values of @ and a, and finally it will only depend on these values, which of the three main types will appear. To understand this better, it is of importance to examine for what combination (rt, 0) one type passes into another. As to the transition of type I to IL (II), it is exactly those combinations for which the plaitpoint line has a double point. In fig. 1 (see the plate) every point of the ') Inserted in the Arch. Néerl. of May 1906. *) These results were afterwards confirmed by Verscuarretr (These Proceedings March 31, 1906; cf. also the footnote on p. 749 of the English translation). ( 229 ) plane denotes a combination (9, zr), to which every time a certain pair of substances will answer. a = pms, | ; | oy ; Pi | | 100 | 750 en 013 | 0,96 en 0,040 | 257 en 257 119 7A » 0,13 0,94 » 0,036 | 249 » 260 4,74 6,26 » 0,13 0,84 » 0,025 | 226 » 2,68 1,88 5,76 » 0,13 0,78 » 0,021 | 218 » 274 2,04 5,42 » 012 0,72 » 0,018 | 244 » 274 2,92 4,94 » 0,12 0,63 » 0,014 | 2,02 » 2,79 2,89 2,89 » 012 0,24 » 0,003 | 1,73 » 287 9,90 1,00 » 0,1 0,01 » 0,001 | 4,00 » 295 oo — » 0,11 — » 0,000 — » 3,00 In the said figure the line C’APB denotes the corresponding values of 6 and a from 9 —=0 to d= 9,9. For C’ 6=0, x=9, for A d=1, x=7,5; with 6 = 2,22 corresponds w = 4,94. (Case t—G or 4,= 9,9 the double point would lie on the side of the line v=6, where v ET == FT meas (r) sin r sin p — Yn (r) sin p. From this ensues for the force of a plane vortex element with unity-intensity in the origin : An (7) sin p, ( 260 ) directed parallel to the acting vortex element and projecting itself on that plane according to the tangent to a concentric circle; whilst » is the angle of the radiusvector with the Spr—2 perpendicular to the vortex element. V. In the same way as in C § IX we deduce from this the planivector potential V of a vortex element directed everywhere parallel to the vortex element and of which the scalar value is a function of r only. That scalar value U of that vector potential is here determined by the differential equation : 0 Lap | on «dr „cessen cong dp — p Ò an | U sing. sin r dp. ce sin "—3r cos "Sp | dr = r = Yn (r) sin p «sin r dp. dr . ce sin "—8r cos "Ip. dU (n—2) U — Em (n—2) U cos r = Yn (r) sin 7. r dU oe — (n—2) Utg} r= — Yn (r). wT 4 Eene SN =) Stee Dr fo ek onl) ore r a function vanishing in the opposite point, which we put — #, (r). We then find for an arbitrary flux : eae | ay (ea ee ee ce) And taking an arbitrary vector field to be caused by its two deri- vatives (the magnets and the vortex systems) propagating themselves through space as a potential according to a function of the distance vanishing in the opposite point, we find: t= 9 Er od fron G. The Elliptic Spy: Also for the elliptic Sp, the derivative of an arbitrary linevector distribution is an integral of elementary vortex systems Vo, and Vo,, which are respectively the first and the second derivative of ( 261 ) an isolated line vector. For elementary oX we shall thus have to put the field of a divergency double point. dr n—lp The Schering elementary potential if — vn (r) is here a plu- Sin rivalent function (comp. Krein, Vorlesungen über Nicht-Euklidische Geometrie II, p. 208, 209); it must thus be regarded as senseless. IJ. The unilateral elliptic Sp, is enclosed by a plane Sp,-1, regarded twice with opposite normal direction, as a bilateral singly connected Sp,-segment by a bilateral closed Sp. If we apply to the Sp, enclosed in this way the theorem of GREEN for a scalar function g having nowhere divergency, and for one having in two arbitrary points P, and P, equal and opposite divergencies and fartheron nowhere (such a function will prove to exist in the follo- wing), we shall find : Pp AES Pps 0, 1 i.0. w. g is a constant, the points P, and P, being arbitrarily chosen. So no oX is possible having nowhere divergency, so no 5.4 having nowhere rotation and nowhere divergency; and from this ensues: A linevector distribution in an elliptical Spa is uniformly deter- mined by its rotation and its divergency. HI. So we consider: 1. the field £,, with as second derivative two equal and opposite scalar values quite close together. 2. the field #, with as first derivative planivectors regularly distri- buted in the points of a small *~*sphere and perpendicular to that small *—*sphere. At finite distance from their origin the fields #, and Z, are of identical structure. IV. To find the potential of the field Z, we shall represent it uni-bivalently~ on the spherical Sp,; the representation will have as divergency two doublepoints in opposite points, where equal poles correspond as opposite points; it will thus be the field (d), deduced under F’ $II, multiplied by 2: ( 262 ) Ve 7 sin *—|p dr cos p ij sin n—I|p 4 Sat] = Àn (7) cos gp. In the field corresponding to this in the elliptie space, all force lines move from the positive to the negative pole of the double point; a part cuts the pole Sp‚— of the origin: these force lines are unilateral in the meridian plane; the remaining do not cut it; these are bilateral in the meridian plane. The two boundary force lines forming together a double point in the pole Sp,—1, have the equation : Mg 7 sin"—l@ {sin "—'r + (n—1) cot rf sin tig an ie r The Sp‚— of zero potential consists of the pole Sp,—i and the equator Sp,—: of the double point; its line of intersection with the ‘meridian plane has a double point in the force lines doublepoint. All potential curves in the meridian plane are bilateral. V. For the fictitious “field of a single agens point” the potential is f An(r) dr. It is rational to let it become O in the pole Sp,—-1; so we find: Vom af an (rv) dr = F, (r), if and for the arbitrary gradient distribution holds: uo oX 0X = W NO KAAS meee Smee zt We could also have found F, nn out of the differential equation (77) of F § Ill, which it must satisfy on the same grounds as have been asserted there. For the elliptic Sp, also we find: ar. f sin “lr dr —_ =¢ dr sin "lp But here in the pole Sp,-1, lying symmetrically with respect to the centre of the field, the force, thus f sin n—lpdr must be QO; so that we find: ( 263 ) Ig 7 df c Ef sin tr dr. dr sin %—I|p T, VI. In the usual way we deduce the |X, which is planivector potential of the field £,. . dh = ce sin "—2r sin "—2 gp. Force in r-direction: Ig 7 sin %—|p dr 2 2cotr 7 ree EN en tn (r). Dorp |e en en eee) iy = =f» COS P . ln (7) . c& sin "Pr sin "Ip . sin r dg = 0 = Un (r) . ce sin "—!¥ sin ™—lep. = ei aR (7) sin r sin p = x, (7) sin p. From which ensues for the force of a plane vortex element with unity-intensity in the origin: Xn (7) sin p, directed parallel to the acting vortex element and projecting itself on its plane according to the tangent to a concentric circle; @ is here the angle of the radiusvector with the Sp,—2 perpendicular to the vortex element. VII. Here too a planivector potential of a vortex element can be deduced, but we cannot speak of a direction propagated parallel to itself, that direction not being uniformly determined in elliptic space; after a circuit along a straight line it is transferred into the symmetrical position with respect to the normal plane on the straight line. But we can obtain a vector potential determined uniformly, by taking that of two antipodie vortex elements in the spherical Sp, (in their *sphere the two indicatrices are then oppositely directed). The vector potential in a point of the elliptic Sp, then lies in the space through that point and the vortex element; if we regard the plane of the element as equator plane in that space then the plani- vector potential V is normal to the meridian plane: it consists of: ( 264 ) 1. a component U, normal to the radiusvector, according to the formula : wT fee Un-2) Er yn (7) dr + Tr U, 1 cos cos Xn—2) Lp £5 ee fo 2(n—2) Er. Vn (r) dr. sin 2(n—2) Lp Tr 2. a component U, through the radiusvector, according to the formula: U, 1 ae SS >= —_—__ feos Un) L ry, (r) dr — sing cos An—2) Lp Tv 1 fo An—2) Lp .y,(r)dr. US If we regard this planivector potential as function of the vortex element and the coordinates with respect to the vortex element and represent that function by G,, then — 1 / 9X, 7, : = ©) vf eze Xing) die ed holds for an arbitrary flux in the eae Spn- And regarding an arbitrary vector field as caused by the two derivatives (the magnets and the vortex systems) propagating them- selves through the space to a potential, we write: kn VIII. In particular for the elliptic Sp, the results are: Potential of an agens double point: Ek sin *r dr COS + 2cosp ((4 xw—7) — = ——_ . : + cotr}, sin? r LS, EL sin °r or if we put fr—r=y: 2 cos Ja + cot a id sin °r )" Equation of the boundary lines of force: ( 265 ) sn*p(l+yctr) = + 1. Potential of a single agens point: ee COL fe Vector potential of an elementary circular current : re A 1+ ycotr — sin P. ; 8 TT sin 7 So also force of an element of current: ike 1+ y cot r SNP Jt sin Tr Linevector potential of an element of current: . cospl AB Orr according to the radiusvector : = — EL : xz (cos* tr sinr sin? Ar sng | 46° 27r—2 tr? normal to the radiusvector: = nn a (eos ir sin Tr sv tr IX. For the elliptic plane we find: Potential of an agens double point: cos @p cotr. Equation of the boundary lines of force: r sin p = + sinr, or o=| - Potential of a single agens point: | —lsinr. Sealar value of the planivector potential of a double point of rotation: sin p sin r Thus also force of a rotation element: sin p sin Tr Planivector potential of a rotation element : Leot £1. We notice that the duality of both potentials and both derivatives existing for the spherical Sp,, has disappeared again in these results. The reason of this is that for the representation on the sphere a divergency in the elliptic plane becomes two equal divergencies in opposite points with equal signs; a rotation two equal rotations in opposite points with different signs; for the latter we do not find the analogous potential as for the former; the latter can be found here according to the Schering potential formula. With this is connected immediately that in the elliptic plane the field of a single rotation (in contrast to that of a single divergency) has as such possibility of existence, so it can be regarded as unity 18 Proceedings Royal Acad. Amsterdam. Vol. IX, ( 266 ) of field. That field consists of forces touching concentric circles and reat — 8 sun 7 Postscript. In the formula for vector fields in hyperbolic spaces: vz X Ne Pee A: en F. (0) de + f = F,(r) de nothing for the moment results from the deduction but that to \2/ X and \1/ X also must be counted the contributions furnished by infi- nity. From the field property ensues, however, immediately that the effect of these contributions disappears in finite, so that under the integral sign we have but to read \2/ X and \1/ X in finite. For the \i/ at infinity pro surface-unity of the infinitely great sphere is < order e-"; the potential-effect of this in finite becomes