HEH ha ni Hi Tis = == = = = 3 SSS SSS So mn = = == zee en =. we de — SSS = Er = <> ==> i} iia B i th vil i, ki ‘ Ne En En i ee i iv i a | Ii Hi i] WE 1 ee ae ny M anh ies ik Ik iH WM Eee oe oe EE Ki a Hh hat CPR LENT ye Mh Hit MENEREN Winn ih Ni Hh i i ij KK — Sx EEE ee A : (} i EE IE { if A hj ba | a ni ii ae Spore he —— aS = > Sa: SS =~ TES = ~ SPE ed Se ee = TS = : eRe eee ae > 5 = =a = > lh } ies ie an SS a == et lk itt]; Sie ty i tf be | ni : tik : igen es - =. En SS a en EE Eed TEE Se ee = = ————S > = === > ef —— = ee —— es er ee ee A Ee — we. or ———— mn =e = ee 5 eee > 5 ae = < <———— “= —— = is SSS = == a SS ss oA ak Es SS SS Ss =S 2 nn = CN me ee 3 hee jen ae = SS se SSS <= er rn en Si eee diy at Nt ne Het ae Wn i he hikt i hast ini Ha Hath Wy Hi Alt } SS = eo — kt > ee ee - — En = = TS en dE Ei En Tee — Se qe De Een ih ‘ { ti kf itp ae sy i i Ht me { AH 1 iN), a a Ki i B i Hilt ie Ath ay Hails i ty — gen bi | Aen | il en i | maa dr | | ae He ae a Hi dn ie Ee i, Koninklijke Akademie van Wetenschappen te Amsterdam. ee PROCEEDINGS OF THE BCs ON OF SCEE NCD a OME OMG EL VEL a AMSTERDAM, JOHANNES MULLER. July 1905. t q ey VANG PSAE Verslagen van de Gewone Vergaderingen der Wis- en Natuurkunc Afdeeling van 28 Mei 1904 tot 22 April 1905. DI. XIIL) (Translated from Koninklijke Akademie van Wetenschappen te Amsterdam. PROCEEDINGS —————— ————— OF THE mc ba ON OF SCIENCES: ears LE NRE ME VEL. (ist PART) ee PD DE een AMSTERDAM, JOHANNES MULLER. December 1904. t KG, u” pa: cif aM vk ARR 1 ign? 3 (Translated from: Verslagen van de Gewone Vergaderingen der Wis- en Natu 1 end Afdeeling van 28 Mei 1904 tot 26 November 1904. DI, XIII.) Cher Nee VS. : ee Page. Proceedings of the Meeting of May 28 DOQQ A wee ENE kl a Se 8 1 » De 5 » » June 25 » Rieti ont oe = 65 » >» » » » September 24 » EET RE ae eee » » » » » October 29 > Ed RA, em Se ee sE » PEES » » November 26 » tt TEN ; es ae abe 6 OOO OOH E an \ aye as Ane IMG! UN TR UUA BA, 4 YRCTE DK TAR UT KET A An SY) ay EN ae KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN TE AMSTERDAM. PROCEEDINGS OF THE MEETING of Saturday May 26, 1904. DUG (Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige Afdeeling van Zaterdag 26 Mei 1904, Dl. XII). 6 SAGE EM PES: H. P. Barenprecut: “Enzyme-action.” (Communicated by Prof. J. M van BEMMELEN), p. 2 J. P. vay DER STOK: “On a twenty-six-day period in daily means of the barometric height’, p. 18. J. W. LANGELAAN: “On the form of the Trunk-myotome”. (Communicated by Prof.'T. Prace), p. 34. (With one plate). Eve. Dusors: “On the direction and the starting point of the diluvial ice motion over the Netherlands.” (Communicated by Prof. J. M. van BemMe en), p. 40. _ Frep. Scuum: “On an expression for the class of an algebraic plane curve with higher singularities.” (Communicated by Prof. D. J. Korrrwea), p. 42. H. E. pe Bruyn: “Some considerations on the conclusions arrived at in the communication made by Prof. Eve. Dusois in the meeting of June 27, 1903, entitled: Some facts leading to trace out the motion and the origin of the underground water of our seaprovinces”, p. 45. Eve. Dusois: “On the origin of the fresh-water in the subsoil of a few shallow polders.’ (Communicated by Prof. H. W. Bakuvis RoozreBoom), p. 53. C. A. Losry pe Bruynand S. Tymsrra Bz: “The mechanism of the salicylacid synthese”, p. 63. J. J. Branksma: “On the intramolecular oxydation of a SH-group bound to benzol by an orthostanding NO,-group.” (Communicated by Prof. C. A. Lopry bE Bruyn,, p. 63. J. M. M. Dormaar: “The inversion of carvon and eucarvon in carvacrol and its velocity,” (Communicated by Prof. C. A. Lopry pe Bruyn), p. 63, The following papers were read: Proceedings Royal Acad. Amsterdam. Vol. VII. (2) Chemistry. -— “Enzyme Action.” By Dr. H. P. BARENDRECHT. (Communicated by Prof. J. M. van BEMMELEN). (Communicated in the meeting of April 23, 1904.) The following is a preliminary communication of the writer's researches on enzyme actions during the last two years. From the commencement it has been the writers object to ascer- tain in how far a continued research of simple enzyme actions might confirm the hypothesis that the enzymes exert their catalytic action Tee. by radiation. This hypothesis ori- canesugar. ginated in the peculiarity of the action of the enzymes which distinguishes this action so sharply from that of the acids. A graphic representation of the action of the same quantity of acid or enzyme in the same time on sugar solut- Initial concentration of the canesugar. ions of different concentrations Fig. 1. : ee Ei renders this difference very per- ceptible. Inverted canesugar. Fig. 1 gives a scheme of the inversion by acids. The line which remains straight indicates that the quantity of inverted canesugar remains proportionate to the initial concentration. In the case of enzymes the Initial concentration of the canesugar. oeneral course is represented by > Fig. 2. f ig. 2. In the inversion of cane- sugar for instance, the line remains straight up to an initial con- centration of 0.1°/,; it then inflects towards the © axis and runs henceforth parallel to this. This characteristic behaviour of the enzymes is now at once explained by the radiation theory which will be further developed. Let us, for the sake of convenienee, confine ourselves to the action of invertin and let us suppose we have two solutions containing, respectively, 20°/, and 10°/, of canesugar. In the 20°/, solution the radiation from each enzyme particle will be comparatively soon ‘absorbed by the surrounding molecules of canesugar; in the 10°/, solution the sphere to whieh the enzyme action can extend will be larger. So long as the solution is sufficiently concentrated to finally absorb by a sugar molecule all radiation emanating from an (3) enzyme particle, before the distance has become so great that the radiation fails to cause inversion, each enzyme particle is bound to exert the same action. One might compare an enzyme particle in concentrated sugar solutions with a source of light in a fog of varying density; the denser the fog the smaller the region around the source of light which absorbs all the light. If, however, canesugar absorbs the radiation from invertin, we. must expect the same to a greater or smaller extent from the products of inversion. On account of this power of absorbing the active rays these products must retard the inversion. The result of my often-repeated experiments showed that the inversion of canesugar by invertin prepared from carefully dried yeast (we shall see, presently, that the method of preparing the invertin is of the greatest importance) is retarded equally by glucose, laevulose and invert sugar. For instance, the same amount of yeast-extract inverted under the same conditions *) from 10°/, canesugar 49.3°/, 10°/, canesugar + 5°/, glucose 38.5°/. 10°/, canesugar + 5°/, laevulose _ 38.3"/, 10°/, canesugar + 5°/, invert sugar 38.3°/, From the similarity of the last three figures it is already evident that we are not dealing here with a retardation due to a reversed reaction. It was further ascertained that the other hexoses cause exactly twice as much retardation as glucose or laevulose : 8°/, canesugar 43.6°/, inverted 8°/, canesugar + 2°/, galactose 35.5°/, 5, 8°/, canesugar + 2°/, mannose 36.1°/, E8 8°/, canesugar + 4°/, glucose 36.1°/, 3 From these results it is evident that the inversion phenomena behave as if there are emitted by an invertin particle two radiations in equal quantity which we may call, provisionally, glucose and laevulose radiations. Each radiation by itself is capable of inverting a canesugar molecule; the glucose radiation is not absorbed by the glucose but by the laevulose; the laevulose behaves, conversely, in the same way. In accordance with this both radiations are absorbed by any other hexose. We may, therefore, regard invertin 1) All sugar determinations have been made by Kserpaur’'s accurate gravimetric process, 1* KE) as being, probably, a proteid containing glucose and laevulose groups in a peculiar radiating condition. At all events, the radiation hypothesis will be found to lead to further quantitative research, to explain results already obtained and to predict future results. Let us take a sufficiently concentrated solution of « grams of canesugar in 100 ce. of water. A given amount of invertin then yields in the first minute a quantity of invert sugar im, independent of a. As soon, however, as a little invert sugar has been formed, the condition changes. The enzyme rays are now not only absorbed by the canesugar but also by the invert sugar. If we call n the absorption power of the invert sugar in regard to that of canesugar, then after the lapse of a time ¢ when w is the remaining canesugar, the inversion velocity — de will be no longer 7, but Lv eae a ie En w+n(a—e) = ie = oe a— & If we substitute — a if 1—n m l + yet ly n na or, using ordinary logarithms 1 l—n ee log = 0,484 y = — 0,434 t. —y n na =y and integrate, we obtain the equation In this equation two constants occur. The first 7 may be at once determined experimentally from the initial velocity. If we take, m ' for instance the experiments of A. J. Brown’), then — or the fraction a inverted per minute at the commencement is = If in our equation we substitute this value of m, we find during the whole series of Brown’s figures a value for 2 of about 0.5. It is, therefore, evident that the absorption powers of the cane- sugar and glucose, or laevulose molecules are in the proportion of about 2 to 1, that is to say in the proportion of their masses or, perhaps, surfaces. Then we found the retarding influence of glucose and laevulose to be equal to that of invert sugar. Per unit of weight the number of molecules in glucose for instance and in canesugar are in the 1) Journ. Chem. Soc. 1902 pag. 377. (5 ) 342 . proportion of 180 If now the absorption power of a molecule of glucose stands to that of a molecule of canesugar in the proportion of 180: 342 and if we consider that glucose transmits without hindrance 50°/, of the total invertin radiation, the proportion between 0 pro] the absorption power of one part of glucose and that of one part of canesugar becomes: 342 180 1 1 180" 342° 2 7° 2 360 342 sugar. Therefore n, the relative absorption power of the products 360 1 of inversion of one part of canesugar, becomes ie = 0,525. The formula for the inversion velocity thus becomes After inversion, one part of canesugar yields —— parts of invert 1 m log + 0,393 y = 0,827 —t. ly a Brown’s experiments conform ae better to this formula than to l+te Henri’s empirical formula 2%, = = log ; =n The correctness of our ee may further be proved experiment- ally in the following way, If in addition to the « grams of canesugar b grams of glucose, or laevulose are dissolved per 100 ce, the inversion velocity will be represented by : & dt. w+tn(a—e)+4b =<; — Of = a By again substituting =y we obtain, when using ordinary a UL logarithms and calling — 0,484 =k: na 1 egg seo yh ee sE apt b gern Eden 2na 2na / 1 4 og ee A same enzyme quantity acting under the same conditions in a solution containing canesugar only and in one containing canesugar plus glucose or laevulose gave the following figures : (6) 10°/, canesugar. 10°/, eanesugar + 5°/, laevulose. log 40,993 1 “1-y b log +0,393y : 1-2 t y == Le t Tis ome se ‘ t ; t 91 0.412 0.00432 85 0.297 0.00273 191 072 0.00437 181 0.555 0.00276 2514 0.72 0.00296 306 0.765 0.0027 360 0.824 0.00271 According to our formula we have: = Diese k= 0.68 k = 0.00295 2na 10°/, canesugar. 10°/, canesugar + 10°/, glucose. t y k t y k 57 0.191 0.00293 57 0.124 0.00145 118 0.364 0.00287 116 0.287 0.00142 242 0.642 0.00288 237 0.485 0.00142 295 0.522 0.00144 558 0.604 0.00146 calculated 4° = en 5 k = 0.00149, 2na A further control is given by the determination of the initial velocity after previous addition of glucose or laevulose. From a (1—y) : dt ne a(l—y) +nayd En b a dy =m follows that the initial velocity in a solution of a canesugar + 6 glucose, or laevulose is: ? oe dy m whereas without such addition =" = —, AZ These are the experimental results: 10°/, canesugar Ce) 10°/, canesugar + 2.5°/, laevulose inverted 14°/, 12.5°/, ze 5 ERA Rn 10°/, canesugar 10°/, canesugar + 5°/, glucose inverted 12.4°/, vl aa 1E II NE 10°/, canesugar 10°/, canesugar + 10°/, glucose inverted 19.1°/, 12.4°/, ns in TR Er as 8°/, canesugar 8°/, canesugar + 16°/, glucose inverted 16.6°), Gene). = 3 (5 Sie AE It was now to be expected that many other neutral substances would also absorbing the enzyme radiation. These are some of the figures Under the same conditions 10°/, 10°/, 10°/, 10°/, 10°/, canesugar canesugar canesugar canesugar canesugar ++4+++ In another series : 10°/, canesugar 10°/, canesugar 10°/, canesugar -4- a There seems to be some kind carbon atoms and the absorption. obtained: urea mannitol erythrite glucose ~ 5°/, dulcitol 5°/, glucose retard the inversion according to their capacity of the same enzyme-quantities inverted of KO, 9 /o x Se 2 oo: —"/, 28.—°/, 29.8°/, 38. 8. 58.1%, 50.4"/, 7 Q0, «.¢ Ig of relation between the asymmetric In the case of inversion of more diluted solutions of canesugar the above-mentioned simple relations will no longer exist. a dilution will soon be reached diminish the initial concentration, If we where a part of the radiation does not reach a sugar molecule in time, but is either finally absorbed by the water or when arriving has, in any case, become too weakened to cause inversion. The (8) quantity of canesugar inverted by a given amount of invertin will, therefore, go on decreasing. In the end, however, we shall arrive at an initial concentration where, within the sphere of action of an enzyme particle, two canesugar or invert sugar molecules can no longer shade each other. From this point, the inversion caused by the given enzyme-quantity will be just proportionate to the canesugar concentration. Then, during the whole of the process, the reaction velocity merely depends on the average number of canesugar mole- cules present within the active radiation sphere of an enzyme particle. The following are some of the figures obtained which always exhibited the same regularity. Concentration Inverted Inversion canesugar in grms. in grms. per 100 ce. Dg per 100 cc. 0.05 0.022 44,97. 01 0.0448 44.8), 0.125 0.0545 43.8°/, 0.25 0.097 39.—*/, 0.5 0.174 Arijs hes 0.240 ZW, we 0.317 15.9°/, Another series gave Concentration of Inverted canesugar in gr. per 100 ce. in grms. per 100 ce. 5) 0.86 4 0.95 5 0.96 7 0.93 The fact that, in very dilute solutions, the enzyme action really y ” proceeds as a unimolecular reaction according to the formula was further again confirmed by experimenting with a 1 : men log 1 = y solution containing 0,096 °/, of canesugar. Up to the present we have for the sake of convenience disregarded the synthetical action of the enzyme rays. Light, being a catalyzer, can act either as a synthetical or decomposing agent, so we must expect the same from the enzyme rays. That we often do not notice such action is due in the first place to the secondary change of the Ge") decomposition products, at least in the case of invertin. It has already been stated by O’Suniivan and Thompson ') that invertin separates glucose from canesugar in a birotatory condition. Tanrwr?) and Simon *) have afterwards elucidated this birotation question. The birotatory a-glucose is the sugar of the a-glucosides and the semi-rotatory y-glucose that of the @-glucosides; according to the said authors they are the stereoisomeric¢ lactones : ope HOH HOH H HOH ie ate C — C — C — C — CH,OH. Ghent | The form which in solution is stable, the g-glucose conforms to the aldehyde-formula CH,OH (CHOH) COH. This conclusion is opposed by other investigators such as ARMSTRONG‘) and Lowry’), who look upon the stable form not as an aldehyde but as a condition of equilibrium between a- and y-glucose. My investigation goes in favour of the first opinion. Invertin is, generally speaking, the enzyme of the a-glucosides. Canesugar may also be considered as an a-glucoside in accordance with the fact that on inversion, the glucose is always separated in the a-modification. This a-glucose is now, however, gradually con- verted into g-glucose and, therefore, prevents the reconversion into canesugar. Owing to this, all the canesugar is always finally inverted by invertin. A continued research, however, showed that there may be still another reason for the non-appearance of reversal phenomena. The method of preparing the invertin, that is of the yeast extract was found to greatly affect the properties of the enzyme. At first, I used for the preparation of a powerful invertin a yeast cultivated in a solution of canesugar. This yeast after being mixed with “kieselguhr’’, was first dried in vacuum at a low temperature and then for half an hour at 100° in an ordinary oven. The addition of “kieselguhr” facilitates very much the subsequent extraction and filtration. The above experiments have been made each time with a freshly prepared filtrate. Afterwards it was found that ordinary yeast also gives an enzyme with the same properties, provided it has not been dried at too high 1) Journ. Chem. Soc. 1890 pag. 861. *) Zie Dictionnaire de Chimie de Wiirtz. 2e suppl. p. 764. 3) CG. R. 1901 pag. 487. 4) Journ. Chem. Soc. 1903 pag. 1305. 5) Journ. Chem. Soc, 1903 pag. 1314, (10) a temperature. Ordinary laevulose in solution is as a rule less stable than glucose. In invertin a difference in the same direction is also usually revealed. Drying at too high a temperature, heating the “kieselguhr”” mixture above 100°, or precipitation with alcohol and redrying the precipitated enzyme, repeatedly gave invertin, the action of which is retarded considerably more by laevulose than by glucose. Active laevulose therefore generally becomes inert sooner than active glucose. This explains the difference between my results and those of Victor Henrt') who states: “Pour laddition d'une même quantité de suere interverti, le ralen- tissement est d’autant plus faible que la concentration en saccharose est plus grande. Ce ralentissement est produit presque uniquement par le lévulose contenu dans le sucre interverti.” Probably, Hexrr has obtained his invertin from yeast dried at more elevated temperatures or has used commercial invertin, prepared by precipitation with alcohol. That the retardation of a same quantity of invert sugar becomes smaller when the canesugar concentration becomes greater is quite in harmony with the radiation theory. The fact that the laevulose contributed most to that retardation was only a pathological phenomenon of the invertin. We must further bear in mind the possibility that, owing to those harmful actions, the radiation gets so weakened that the reversion can no longer take place, or that the radiating a-glucose has been converted into radiating g-glncose and also that only the latter is capable of inverting. It is certainly to be expected that, if only active glucose or active laevulose is left behind, the power of causing reversion has either decreased or been destroyed. In order to counteract the first cause of the non-appearance of the reversal, namely, the secondary conversion of a-glucose into B-glucose, we may apply much enzyme and so accelerate the con- version. A larger quantity of extract of the above yeast, which had been finally dried for half an hour at 100°, caused indeed a slower inversion of the last remaining percentages of canesugar. he The reversed action was afterwards noticed more distinctly with ordinary yeast, merely dried in vacuum at about 30°. We will first give a mathematical formulation of the phenomena to be expected. Let us imagine an aqueous solution of invert sugar, liable to reversal and consequently containing the glucose in the e-form, in 1) G. R. 1902 Nov. £4. 917. (11) the presence of invertin particles, which render this invert sugar active. The velocity of synthesis will then be proportionate first to the product of the concentration of glucose and laevulose and further to the extent of the active radiation sphere surrounding each enzyme particle. The latter is inversely proportionate to the joint concen- tration of the invert sugar and the other dissolved matters eventually present, each with their own absorption coefficient. The synthetical action of invertin in a solution containing (a—.’) grams of invert sugar and w grams of canesugar in 100 ec. is therefore : : de = mp ——_——- dt. v+n(a—e) The complete formula for the inversion velocity of canesugar, in case the original products of inversion suffered no change, would then be’): £ 1 ZN EN | — Gn) | dt. ) wtn(a—w) 4 : v+n(a—v) The point of equilibrium would then be determined by the equation: 1 : 5 rg ah — v)? = 0. Or returning to relative fractions by substituting a 1 ne hk If now we have introduced into the solution such a quantity of enzyme that this equilibrium point is attained before the birotatory glucose has been converted to any great extent into ordinary glucose, the inversion will not actually come to a standstill, but the line, indicating its progressive course, will exhibit a characteristie pecu- liarity in that place. Then, starting from that point, the formation of S fresh a-glucose by inversion will be dominated by the velocity of the inversion of the total a-glucose present into g-glucose. This velocity is proportionate to the concentration of the @-glucose. If this is continually replenished by fresh formation of «-glucose from canesugar, both the said velocity and the inversion velocity will be 1) The small increase in weight when Cja Hy: Oj), changes into 2 Cy Hy, Og is here neglected; we might also suppose that it is taken into account in the coefficient p. By substituting the variable y this factor in p would in any case disappear again. ( 12 ) constant. From this point the line, representing the inversion as a function of the time, will of course be a straight one until the concentration of the cane- sugar has become too small to make the inversion keep pace with the glu- cose transformation. With 5°/, canesugar (see tig. 3) the Jine commences its straight course at about y= 0,35. This substituted into the equation 50/, canesugar. minutes. 1— y— — pay? =0 4 angie alae ik tically = 4 ig. © gives p — —~—__ or, practically == 4. Fig. de le) / 0,153 ? ] « In the 10°, solution (fig. 4) the equilibrium must then become 10°) canesugar. minutes, pereeptible at the value of y to be calculated from 1—y—10y?=0 therefore, at y = 0,27. Those equilibria phenomena are observed more readily in the inversion of maltose by yeast-extract. The enzyme which converts maltose into glucose is generally called maltase so as to distinguish it from invertin. It seems to me that there is no valid reason for making such a distinction. A veast-extract, which inverts maltose, has always been found to also behave actively (13) towards canesugar '), but not the reverse. This is in harmony with the radiation theory. Maltose, like canesugar, is an a-glucoside. The connecting point of the laevulose in the canesugar molecule with the a-glucose is the C of the carbonyl group of the laevulose. In maltose the «-glucose is attached to the CH, of the otherwise uncombined molecule of glucose. Canesugar is also much more readily inverted by acids than maltose. Both radiating «-glucose and laevulose (probably also radiating 3-glucose *)) are liable to invert canesugar. Maltose is only converted by active a-glucose, but as may be expected from its behaviour towards acids and its constitutional formula it requires a more powerful radiation than canesugar. If, therefore, yeast-extract has been weakened by elevation of temperature or by precipitation, its power of inverting maltose may have been lost or much diminished, although canesugar is still fairly rapidly inverted. The preparation of a yeast-extract with a powerful inverting action on maltose proved to me no more difficult than when inversion of canesugar was intended. The above-described yeast, derived from a canesugar solution, and which had been actually heated at 100° for half an hour, yielded after a year and a half an extract which readily inverted maltose. In this case, | used, of course, by preference a yeast which had been mixed with “kieselguhr” and dried at a low temperature. On extracting the dried mixture, the solution never contains zymase as experiment repeatedly showed. Figures 5, 6, 7, 8 and 9 now clearly show the phenomena of equilibrium. If@-glucose in solution were stable, only 15°/, of the total might be inverted by yeast- 60 1"/, maltose. extract in a 10°/, maltose solution. The fact that maltose is decom- posed by yeast-extract so much minutes, : slower than canesugar is partly 5 10 15 20 25 GR Eie.D due to the circumstance that the point of equilibrium is reached so much earlier. The further decomposition then again merely keeps pace with the transformation of «-glucose into g-glucose. 1) Porrevin, Annales Inst. Pasteur 1903. p. 31. 2) Separate experiments showed that glucose, previously heated and therefore in the B-form, and glucose, dissolved immediately before adding the enzyme and there- fore in the z-form, both retard the canesugar inversion to the same extent. 6-Glucose therefore transmits the glucose rays (then perhaps converted into @-glucose rays) quite as well as the z-glucose. (44 ) 3°/) maltose. minutes. 40 5" /, maltose 10 minutes. 4 70), maltose, 0 minutes. 30 20 10°/, maltose. 10 minutes. (15) The points where the transformation lines commence to run straight, so the points of equilibrium, also conform in this case to an equation similar to that used for canesugar. Glucose is here the only product of inversion; we might, therefore, expect that the undisturbed trans- formation velocity would be here: wv (a) D= Ik Teen ee? vn (ae) ie +- n (a — er) and the equilibrium equation, therefore : lt l1—y—qay =0. For 10 grams of maltose in 100 cc. we experimentally found == O15. 0.85 as for canesugar '). If in the above equation we substitute ¢ = 4; or This gives g= or, practically, 4, therefore the same coefficient l1—y—-—4ay’ =), the calculated points of equilibrium become for a y 10 0.146 7 0.172 5 0.20 3 0.25 4 0.39 This is therefore in accordance with the experiment. The well-known researches of Crorr Hitn*) on the reversal of maltose gave points of equilibrium which were situated at a more advanced transformation; for instance in a 10°/, solution y= 0.945. These equilibria were attained only after days and weeks; the above cited after a few minutes. Afterwards *) Hin himself demonstrated that the resulting biose was not maltose but an isomer, which he called revertose. In Hin1’s numerous experiments, all the glucose was no doubt in the g-form. The synthesis found by Hitt was therefore a combination of two molecules of g-glucose to a new biose, isomeric with maltose *). 1) The diffusion velocities on which depends the velocity of meeting of two molecules, cannot differ much for glucose and laevulose. 2) Journ. Chem. Soc. 1898 p. 634. 3) Journ. Chem. Soc. 1903 p. 578. Emmeruine (Ber. 34, p. 600) had found isomaltose as reversal product. 4) Most of the natural glucosides appear to be compounds of bi- or semi-rotatory hexoses. When endeavouring to prepare lactose from galactose and glucose by ( 16 ) The retarding action of another added hexose is not studied so readily in the case of maltose inversion as in the transformation of canesugar, on account of the immediately occurring reversion. Still it „was found that both laevulose and galactose cause the same retarding action, as might be expected from our theory now we are dealing with glucose-radiation only. For instance, a same amount of yeast-extract gave under the same conditions and in the same time: in: inversion maltose 18.9°/, 20 6°/, 6°/, maltose galactose 15.5°/, ++ l 6°’, maltose 1.5°/, laevulose 15.5°/ Another series gave in: inversion 6°/, maltose 26.8°/, + 1.5°/, laevulose 24.8°/, 6°/, maltose + 1.5°/, galactose 25.-°/, aE 6°/ 6°/, maltose , maltose 1.5°/, glucose 13.5°/, This last figure, verified by other experiments, requires a further explanation. This 1.5°/, glucose was undoubtedly g-glucose. Before mixing it with the maltose, the glucose was dissolved separately and completely converted into the stable form *) by placing the flask for some time in boiling water. The observed order of retardation shows that the 3-glucose also takes part in the process of reversion. Now it is possible that in the maltose molecule the glucose with the still free carbonyl group is present in the g-modification and it is even probable that this free glucose group, when in solution, will be converted into the same stable form as glucose itself. In Hirr’s investigations, veast-extract appeared capable of uniting two molecules of p-glucose; so, probably, also two molecules of a-glucose. The glucose formed in the enzyme-inversion of maltose may, therefore be called homogeneous. Each molecule of that glucose can unite itself under the influence of the enzyme radiation with any other molecule of that glucose to a biose. Therefore, the equation of equilibrium was here means of lactase, Emm Fiscuer and WRANKLAND Armstrong only obtained an isolactose. The synthesis of canesugar has not yet succeeded because we can only add £-glucose and not z-glucose to laevulose. *) Separate experiments showed that unheated glucose causes the same retar- - dation as heated, z-glucose therefore the same as B-glucose. (17 ) 1 —y—4da ys It should also be mentioned that the inverting enzyme of veast appears to be always the same whether canesugar or maltose has been present as a carbohydrate food. In an ordinary cereal extract a little canesugar occurs along with the maltose. A pure yeast- culture, cultivated by myself in a solution of pure maltose (plus the necessary salts and nitrogenous food) gave an enzyme extract which was retarded in its action equally much by glucose and laevulose, and twice as much by galactose. In the enzyme formation, therefore a partial conversion of glucose into laevulose seems to take place. LoBry pu BRUYN and ALBERDA VAN EKENSTEIN') have shown that these two hexoses may be converted into each other in an alkaline solution. The investigations of O'SULLIVAN and THompson?) have rendered it probable that the invertin-molecule (if we may use this expression) contains a carbohydrate group. These investigators have attempted to purify invertin and found that a constant component of the resulting proteid-complex, their so-called -invertan, contained 18 parts of carbohydrate to one part of albuminoid. A further development of the electron theory will probably eluci- date the nature of those enzyme radiations. As Loper *) observed, it is not the occurrence of radiations in matter which need cause astonishment but rather ine fact that not a great many more radia- tion phenomena have already been discovered. Many other catalytic phenomena such as the action of hydrogen-ions and those of Brepic’s anorganic ferments may, after all, be due to radiations. For hydrogen-ions, carriers of loose electrons and dispersed platinum cathodes probably also emit radiations owing to the motion of the electrons in or around the material particle. During the course of a same reaction, Brepia often noticed an increase of the constant 1 == — lor k tee j just as that shown by the invertin action. A retar- nd. dation of the catalysis by indifferent matters has also been frequently noticed, for instance, by KNorvenaceL and Tomacszewski *) in the action of finely divided palladium or platinum on benzoin. If the statements of the French investigators on the physiological 1) Ree. Trav. Chim. 1895 p. 201: 2) Journ. Chem. Soc. 1890 p. 834. 5) “On Electrons” Journ. Electr. Engineers 1905 Vol. 32 p. 45. 4) Ber. 1903. 2829. Proceedings Royal Acad. Amsterdam. Vol. VII. (18 ) n-rays should be promoted to objective truth, our hypothesis would receive a direct experimental support. At all events, the above has demonstrated that the principal mea- surable phenomena, noticed in the enzyme action are in harmony with our hypothesis. Meteorology. — “On «a twenty-siv-day period in daily means of the barometric height.’ By Dr. J. P. vAN DER Stok. 1. A few years ago *) Prof. A. Scuusrer investigated the problem, how to detect the presence of a periodical oscillation, the amplitude of which is small in comparison with large superposed fluctuations which may be considered as fortuitous with respect to the purely periodical motion. Starting from an analogy which may be seen between this question and the problem of disturbances by vibrations in the aether — a problem treated by Lord Rarrrien®) in 1880 — Prof. Scnusrer has endeavoured to apply the theory of probability to the determination of the first couple of coefficients of a Fourter series, and the method he arrives at, and strongly advocates, is applied to records of magnetic declination observed at Greenwich during a period of 25 years. The choice of this material, in Prof. ScHusreR’s opinion not favour- able for the discovery of small effects, is justified by the remark that “the only real pieces of evidence so far (1899) produced in favour of a period approximately coincident with that of solar rotation were derived from magnetic declination and the occurrence of thunderstorms.” In this and in an earlier paper*) the author emphasizes that, in inquiries. of this kind, it is not at all sufficient to come to some result, but that it is necessary to apply a reliable criterion by which a judgment may be formed about the value to be attached to the result arrived at. His mathematical investigation, however, does not, lead to an out- come which in every respect can be regarded as satisfactory, in so far that a method of determining the mean and probable error of the result from the series of observations themselves is not given and, 1) Trans. Cambr. Phil. Soc. Vol. XVIII. 1899. 2) Phil. Mag. Vol. X. Il, 1880. 5) Terrestrial Magnetism Vol. Il, 1898. (19 ) as a surrogate, the author suggests the repeated rearrangement of the records according to different periods, not much differing from the period in question. Thus, in a purely empirical way (“by trial”), a standard may be obtained by which a correct estimate can be formed in how far the outcome arrived at must be considered as a merely accidental one. Now the same problem was treated some 15 years ago *) by the author of this paper after a different method, applied not only to magnetical but as well to meteorological data of different description, and Prof. Scuuster’s important investigation gives a ready occasion for taking this problem in hand again. It is only natural to choose in the first place for this inquiry the series of barometric observations made at Batavia which now extends over a period of 36 years (1866—1901). An investigation into a possible synchronism between the frequency of sunspots and atmospheric temperature, commenced in 1873 *) and recently conducted up to date *), gives some ground to the expectation that, for inquiries of this kind, observations made at tropical stations are of more value than those made in regions where the atmospheric disturbances are such as experienced in higher latitudes. In the second place it seems desirable to look for a shorter way for coming to a reliable criterion than the tedious process of the calculation of ScuustER’s periodograph. 2. An arrangement of quantities according to a given period 7 may be executed by measuring out the successive data from a point 0) taken as origin and along straight lines drawn through this point 2 Try at equal angular distances If we assume the unity of mass attached to the ends of those radii, it is evident that a judgment may be formed about the degree of symmetry in the distribution of the masses with respect to point Y, by simply calculating the average value of all these vectors, or in other words to determine the situation of the centre of parallel forces supposed acting on the masses. If the quantities 7’ show a well marked periodicity as e.g. tidal 1) Observ. Magn. Meteor. Observ. Batavia. X, L888, Append. IL also Natuurk. Tijdschr. XLVIIL. 1889. Verh. Kon. Akad. v. Wet. Amsterdam. XXVIII, 1890. 2) Köppen. Zeitschr. Oesterr. Gesellsch. f. Meteor. VII, 1873. 5) CG. Norpmann. Essai sur le rôle des ondes Hertziennes and: Astron. Phys. et sur diverses questions qui s'y ratlachent, Thése, Paris, 1903. 2* ( 20 ) observations do, it will be possible to draw a line through QO in such a manner, that on the one side all the vectors are greater than the corresponding opposite vectors on the other side of the line. If, therefore, we arrange in this way a great number of quantities which show a slight tendency to asymmetry, the radial momentum will steadily increase, as the mass concentrated in the centre of parallel forces is equal to the total number of observations, whilst the distribution of accidental quantities will tend to a symmetrical distribution. Assuming two rectangular axes going through O, we find for the coordinates, by which the centre of gravity is determined, MV being the number of observations : (ih, a O COS A U — 2 QO sin (ad . . . . (1) The calculation, therefore, comes to the same as the determination of the first couple of Fourier coefficients: 2 =—ZosnO. ia SAN ft al = 0 COS A b and, if the periodical movement is represented by the expression: A cos (nt — C), b 27 A ee tang C= — nn soe a, : This way of representing the arrangement seems preferable to the development in a Fourier series: firstly because the development of a function in a series, as a representation of the function, derives its value from the composition of a great number of terms, so that, in calculating one term only, we are hardly justified in speaking of a Fourierisation of the function. In the second place, because by this way it becomes at once evident that the problem is fully equivalent to that of the determin- ation of a point in a plane by means of a great many inaccurate observations. This problem has been treated by several mathematicians, but certainly in the most complete manner by the late Prof. ScHoLs '), whose original conception of the question leads to the detection of some laws, which are independent of the assumption of any law 1) Over de theorie der fouten in de ruimte en in het platte vlak, Amsterdam, Verh. K. Akad. v. Wet 1eSect. XV, 1875, and: Théorie des erreurs dans le plan et dans l’espace, Delft, Ann. Il, 1886. ( 21 ) of errors and to a remarkable analogy between this problem and that of the moments of inertia in dynamics. If we take NV, the number of observations, equal to unity, the relative frequency of the ends or representative points of the vectors may be represented by the density of these points per unity of surface. This function of probability is called by Scrors “the module” the “specific probability” or the “facilité de Perreur”. We thus obtain a mechanical image of a surface of probability, the density of which will, in general, be a function of the length and direction of the vectors. The determination of what ScHors calls the constant part of the error — the probability of which is NV =1 — is identical with the determination of the situation of the centre of gravity, and the calculation of the mean (not average) error: 5 M= nn with that of the moment of inertia, which leads to the determination of two (in the plane) principal axes of inertia, which, in our case, may be called axes of probability. | Assuming that these errors in the plane are due to the cooperation of a great number of elementary errors, ScHors has proved that the projections of the errors on an arbitrary axis follow the exponential law of errors in a line and that the law of the resulting error can be found by supposing the error to originate in the coincidence of projections of the error upon the axes of probability, these projec- tions being regarded as independent of each other. The application of this theory to our ease can be reduced to very simple calculations. Errors arising from individual or instrumental causes are always distributed in a more or less systematical way, but there is no reason to suppose that the fluctuations e.g. of barometric heights within an arbitrary length of time, and cleared from their constant part, will show any tendency to systematic distribution when arranged around a point in the way described above. ScHoLs’ specific probability of an error in the plane is given by the expression: re Uatake (nihil ato aI aca? 27M,M, (4) in which z and y are the coordinates of the error (polar coord. 9 and 6) ( 22 ) and J, and M, denote the principal axes of probability, so that: M= M,? + M,? = M2 + Mi. The mean error, therefore, can be calculated without any knowledge of the situation of the principal axes, when the mean error of the components relative to arbitrary rectangular axes is known. If F is independent of 6: M M,= M, = — y V2 or, putting : 1 a a M adh 7 Went ek oe en A AE (5) 7u The specifie probability of an error, independent of the direction, is : 2x he | | boet dp= 2 oe. 2. 2 1 ew Jt 0 From this it appears that the probability of an error zero is not, as in the case of linear errors, a maximum, but a minimum, that the curve of the spec. prob. (6) (given in ScHoLs’ paper) shews a maximum for the value of 9: 1 1 ‘lech Tae tise Cg ot me ern toe Mat Me Ss eee and, also, that the computation of the probable error will lead to a coefficient of 7 considerably different from that found for linear errors. We have then to ask for what value 7 of 0: r C 9 4 9 1 21? fo ele do = — i 2 0 p= 0.83956 Od coke ere me This value of the coefficient of the probable error, considerably greater than is found for linear errors, 0.6745, clearly shows that and to what degree results, obtained in investigations of this kind, have to be put to an unusual severe test, and also that there is some reason to adhere to the use of the probable error, which of late years has been somewhat neglected. A reduction of the mean error has no sense if this reduction is (23) always in the same proportion, but it becomes important if this proportion depends on the nature of the problem. If the distribution of errors is not independent of the direction, the coefficient of M/ is determined by the quantity : MS IN k M,? + Ms” The coefficient of the probable error, for which Scrors gives the approximate value : OSSIE OLDER NE cel i US (0) is a maximum for errors independent of the direction and a minimum for linear errors, when N =1. By the assumption, therefore, that all directions are equally probable the most unfavourable case is chosen, which, in doubtful cases, is of course, the safest way of forming a judgement. Whether the operations, which are to be applied to the data, are considered as a determination of the first couple of constants of a Fourier series (the very first, 4,, is left out of consideration), or as a ealeulation of the average or most probable position of the end- points of the vectors, or as a determination of the situation of the centre of gravity — in all cases the result is a quantity determined by two coordinates and the operations we have to perform are: 1stly. to separate the constant part; Indy, if necessary to determine the situation of the axes of probability ; Srdly | 5 to caleulate the mean and probable error, in this case better ealled incertitude. The same method can, of course, be applied to groups of periods, which gives a considerable saving of labour, but also leaves some want of clearness in the result. 3. The investigation of the series of daily means of barometric observations made at Batavia has been conducted in the same manner as it was commenced in 1888. The arrangement has been performed according to a period of 25.8 days, and groups of 30 rows have been taken together so that, out of the 510 periods, 17 groups have been formed. The result of this operation is given in Table LI. If, therefore, an oscillation, periodic in 25.8 days, really exists, its amplitude is not more than : A = 0.055 mm. 30 (22 )) TAB LEM, | Amplitude, Argument. Components. Differences. | | ; RR A 0 a, | b, La | A, enten mn, | mm. mm, HUN. mn. 1 0.69 | 242° —0.32 —0.61 —0.86 0.96 2 || 7.63 | -287° 2 17 7.32 1.63 5.75 3 3.45 | 85° 0 87 —3.34 0.33 AD | 4 3.44 | 68° 41.46 2.91 0.62 4 48 5 1552 | 2158 sl —0) 88 Sl FES 0.69 6 2.08 | 204° —1.90 —0.84 —2 Ah 0.73 7 4.52 | BA5° 438. dh nrd 3.84 | 0.43 8 1.21 | 1040 yan: bY 4.47 —0.84 | 2.74 g 1.78 | Del 0}: 0.01 | —1.78 <= ()), 533) | — 0) | 10 6.31 | 318° 47 —4 2) 4AT —2.63 u 5.00 |) 494 —4 86 —1.A7 —5.40 0.40 12 3.25 | 266° —0 20 —3.% —0. 74 —1.67 Ale} 6.00 | 959° Sl aa —}5.80 —2 08 —4 293 14 948 | 3470 1.59 24010 1.05 | 0.08 15 3.34 195° —3 23 (ERZ | == Ons 16 2 60 23e 0.34 2.57 —0,20 4A4 ZF Rl th (uy Ja4° 7 58 -0.76 7.04 0.84 Mean 1.66 289° 0.54 —1.57 | By subtracting this restant, which has to be regarded as a constant part, from the corresponding values, the differences exhibited in the last columns have been found, which are to be regarded as fortuitous disturbances. 4. The value to be attached to the result may be estimated in different ways. The first and most simple manner is to split up the series into two or more groups. From the data given in Table I we easily find: Number of A C periods. Group 1— 6 1.68 mm. 274° 180 - 7—11 1.62 299° 150 … 12—17 1.76 296° 180 sô 1— 9 1.42 292° 270 … 10—17 1.95 286° 240 ‘ 1—17 1.66 289° 510 From this it appears that there is certainly some indication for the existence of a periodical oscillation, and also that the arrange- ment has been made according to a period which practically leads to a maximum value of the amplitude. The probability that three points, taken suecessively at random, » are situated within an angular space of 30° is (=) and the probability of mere chance would have been even less if we had taken into account that the amplitudes too are in good accordance. 5. A second, equally simple method is afforded by a direct view of the outcome of the arrangement itself, split up into two or more groups. Fig. 1 gives a graphical representation of the differences given in the three last columns of Table II. Fig. 1 shows that the curves of the two series agree satisfactorily and also that a tendency to a double period, with a maximum on the 8—9 day, which in the first group is still well marked, vanishes when the arrangement is continued. If these results are considered as fairly conclusive, so as to justify a more exact determination of the length of the period, this may be easily done by varying the arguments (of Table I successively by ‘/,%, °/,@, °;,@ ete. « denoting the variation of each group- argument which leads to the most constant value of (. In this way 17 equations are obtained from which the most probable values of C and the period 7’ can be calculated. If to each equation the weight is given of the corresponding amplitude, the equations will assume the form: 0.69 (— 118° + */, 2) = 0.69 C 7.63 ( — 73° + */, 2) = 7.63 C ete. | ( 26 ) TAB aes AL: Results of the arrangement. Average values, the general mean value being subtracted. | | | Three subsequent valuestaken together. | ; | i | ae I | I | LH Number | É of 270 | UO | 510 270 | 240) | 510 Groups. nn. mm, mm. mm. mil. min, 4 —0.016 0.034 0.036 | —0.012 0.025 0.014 2 —=() 055 0.047 — 0.008 —0.033 0.031 0.006 3 — 0.026 0.012 —0.009 | —0.025 0.011 —0.010 4 0.006 —0.028 —0.011 —0.015 —0.016 —0.017 5 —0.026 —0.033 | —0.030 | —0.016 —0.029 —0.023 6 —0.028 | —0.025 | —0.028 | —0.023 | —0.033 | —0.029 7 —0.014 | —0.04 220.028. Je 0 014 —0.042 —0.028 8 0.010 — 0.060 —0.029 0.001 —0 066 —0.032 9 0.015 —0.096 —0.038 | —0.019 | —0.086 | —0.052 10 0072) |, 0103 | —0.088 - 0 052 —0.098 | —0.074 u —0.098 | —0.004 | —0097 | —0.076 | —0.076 | 0.077 12 0059: 5! 0002 | —0.047 —0.068 | —0.049 —0.060 13 — 0.047 | —0.021 | 0,036. 1 OOR NED 031. Ie SOUR 14 — 0 004 —0.04 | — 0.022 —0.014 | —(0 025 —0.020 15 0.009 | —0,013 —0.003 —0.011 | —0.010 | —0.011 1G —0.037 | 0.02% | —0.009 —0 010 | 0.024 | 0.005 17 —0.001 | 0.064 | 0.097 0.039 | 0.047 | 0.0% 18 0.054 | 0.058 | 0.055 0.040 | 0.050 | 0.043 19 0.067 | 0.031 0.049 0.064 0.052 0.057 20 0.071 | 0.069 0.069 0.059 0.063 0.060 di 0.039 rer 091 | 0.069 0.046 0.080 {0.061 29 0.029 | 0.079 | 0.051 0.034 0.081 | 0.055 23 || oo | 0.074 | — 0.052 0.036 | 0.064 | 0.048 aa ell soa. 1e 120-088 0.044 0.053 0.080 0.041 25 0.077 | —0.0 | 0.098 0.053 | 0 003 0.028 2% 0.036 | —0,006 0.015 0032 | 0.001 0.026 | | Mean | 0.097 0.103 0.097 | | As might have been expected the result of this calculation shews little or no difference from that of the arrangement. itself. #2 == — 1°,09 C= Ae T= 25,8034, As one day corresponds to *°°°/,,.,, a variation of « degrees for each group is equivalent to: 1.09 258 —__ __—__ = 0.0084. 30 38600 6. By application of the method discussed in $ 2 to the differ- ences A, and A, of Table I we find: SM, =145.05 Si, —212.02 TSM, — 12.44. From the well known formula : tang 2 y = a — i, Me, for the situation of the principal axes of inertia, deduced from the (10) ( 28 ) condition that, when the axes of coordinates coincide with the principal axes, the moment of deviation or centrifugal force M,, will vanish, we find: p= — 18°35) and further Mt = 6.45 M,? = 8.72 N=0.15 and from formula (9) 7 = 0.829. It appears, therefore, that, in this case, all directions of the accidental quantities are equally probable, so that we are fully justified in putting: 7 =— Bea. The mean and the probable error for each group are then : M = 3.89 W = 34 and the final result for each group: 1.76 mM... . probable error 0.810 and for each row : 0.055 mM... . probable error 0.027 so that the probable incertitude of the final outcome amounts to almost exactly half the amplitude. 7. The question may also be put, what will happen if the arguments of Table | are varied in such a manner, that the varia- tions are equivalent to arrangements according to other periods slightly different from 25.8 days. The amount of the variation is limited by the number of rows taken together in one group, which can be shifted only as a whole, and the variation ceases to have any sense as soon as the sums of each group would be sensibly affected by the actual arrangement according to the new period. If quantities, periodical within a length of time 7, are arranged according to a period 7” in m columns, the value at the origin of time being represented by : A. cpa iC, the record to be inscribed in the #% column of the pt" row (fand p counted from nought) will be : , 2 n 5 n A cos —— C+ 2 p— mn n 2 (oy AIF Nn nN T pear ( 29 ) The sum of RA rows is then: Mit Td We N= 0 a == 2 sin Ra HE Od y Pe AR (RL 1) el. Sin a WL m When d is small this expression can be simplified by putting: rp ryy ry m =| [== 2 in the second term under the cosine. The sum of the first, second ete. group of R rows is then: sin heat Bt 2 A —-—— cos B Gee RER ee EE sin ct m sin Ra 27% A —— cos | — T —C + 5 Ra \ ete. sin « m If the oscillation is of a purely periodical description and of equal amplitudes the sum will show a principal maximum, 7 A, for a= 0, and further secondary maxima for all values of @ which satisfy the equation: | KR tang a = tany Ra Le, when AR —510, for values of @ corresponding with periods of: | 25.8724 ( 25.925¢ | 25.728 1-25 but the amplitudes of these maxima will be resp. 5 and 8 times smaller than the principal maximum. The amplitude will vanish whenever tie =, dut, Oe ete. i.e. for periods of ( 25.8502 { 25.9004 | 25.750 | 25.790 The upper curve of fig. 2 gives an image of the fluctuations of these theoretical amplitudes. If we put: se Mr 20,8 Ed the amount of shifting to be given to each group corresponding with 0.01 day, is: dE 20.8 dE # „in the denominator being neglected. The variation has been carried on, as utmost allowable limit, to (30) x— +015, corresponding with a group-variation of about 31°. The group-amplitude is only slightly affected by this variation as: sin Ra -—— = 28.58 sin « instead of 30. When &, the total number of periods, increases, the secondary maxima will become smaller and smaller, and at the same time maxima and minima will approach nearer to the principal maximum. TAN EE HE Results of the arrangement according to different periods by the shifting-process. Period | A | C | Period | A | C AU, nun, | d. d. 25.65 13 4 Jole 95.81 27.4 247° 25.66 6.4 80 82 DEF, 2179 25307 4. 8* 148° 83 15.0 169° 25 68 16.4 76° 84 7.6% 84° 25 69 4.8 34° 85 1236 344° 25.70 30.0 | 345° S86 19.2 ISG Soi 1.9 | ale 87 24.0 | 244° 25.12 A85 2652 s& 23.8 | 907° 2518 14.6 oe 8&9 16.5 174° 95.74 hs Vite 90 Di 1320 US us Cea es 88° Ol 4.9 301° 25.76 10.1 aor 92 (AE OMAN E20 DTI, 14.7 | SIs 93 A510 a 2523 a 25.78 19.8 | 346° 94 150 IDS DEK) 9624) ZO 95 43°37) sie 300° 25.80 28.3 | 989° Table III exhibits the outcome of this shifting of the groups and fig. 2 shews both the theoretical and actual curves. For the sake of comparison the data of Table HI have been multiplied by 2 and the amplitude of the theoretical period has been taken equal to O.1. 25.65 25.70 25.79 25.80 25.85 25.90 25.95 It appears then that, in fact, secondary maxima and minima occur and, at least as regards the first minima, in the right places, but that the secondary maxima, instead of being small as compared with the principal maximum, as might have been expected, are of about equal intensity and, most so on the left side, not at all agreeing with the theoretical lengths of period. This result may be interpreted in three ways: a. We may assume that every one of the three periods 25.80, 25.70 and 25.87 is due to a purely accidental distribution of the quantities under consideration. Db. We may concede that at least for the period 25.80 there is some indication, but that the two adventitious periods are the consequence of the unequal distribution of the group-amplitudes so that they will disappear when the arrangement is continued over a longer series of observations. ( 32 ) c. We can assume that the evidence is equally good for the three periods, and will be enhanced by continued arrangement. Of course only an actual continuation of arrangement for another series of twenty years will enable us to answer these questions. The only test which at present can be applied is to form two or more groups as has been done above for the arrangement according to 25.8 days. Period 25.70¢ 25.874 A Gee A G Group 1—6 1.60 mM. 6 1:40 mM. 252° 7 7—11 woo, 1 2288 Oar 105 ks 12—17 248 ,, AT PE A eeb a 19 Lad 2) Ad A Bos Me 0 10—17 2.98 isd. eGo. 1d +> So far as this criterion allows a conclusion to be drawn, it appears from this result that the evidence for real existence of the periods 25.70 and 25.87 is considerably less than of the period 25.80. In the latter case the arguments for three groups did not differ more than 25°, against differences of resp. 88° and 95° for periods of 25.70 and 25.87 days. The probabilities of mere chance, therefore, are, taking 30° and 90°: 1 1 and — 144 16 i.e. more than 8 times as great. If we take also into account that the amplitudes of the three groups are accordant for 25.80 and widely different for the adventitious periods, we can estimate the probability of chance at 10 times as great. The computation of the probable error (incertitude) of the result for each group also gives an indication for this greater probability, but not in the same degree. Amplitude. — Probable error. 25.80 1.76 mm. 0.810 24.70 KTO 15 0.916 25.87 Lr 0.830 7. If we apply, in so far as possible, the different criteria to the data published by Prof Scuvsrer concerning daily means of magnetic declination for Greenwich, arranged according to 26 and 27 days, we find for the sums of groups, each of which contains resp. 14 and 13.5 rows. 264. 274 A; C AG C Group 1— 5 6.19 267° 8.5: 304° 6—10 4.08 243° 2.19 88° 11—15 3.09 301° 4.75 354° 16—20 1.45 152° 7.05 203° 21—25 2.88 229° 7.56 298 The probability, therefore, of mere chance is: 26 days ma) 0.38 9) C ay 5 epee se ee Vatis er le (a — OC 2464 DaT his voc ( ) — 0,20. j 360 and the final outcome Soedan 0 Mees 0) oe en prob. error 07.348 i eh hth eet Is, ys. 8 se AED If we vary the arguments given in Table VIII of Scnvsrer’s paper for a period of 26 days in such a way that the result is equivalent to an arrangement according to 25.80 days we find: ae C Group 1— 5 10.85 54° 6—10 5.72 104° 11—15 4.88 tle 16—20 4.61 44° 21—25 4.03 89° As these arguments do not differ more than 60 degrees, the probability of chance is in this case: i! Gt The final result, calculated for a group of 14 rows, MEENT prob. error 0/.292. The accurate length of the period and the most probable value of C, calculated after the method discussed sub 5, are then: BSE ANS. a - Ch" Bae G: It appears from these calculations that an arrangement according to 26 and 27 days leads to results the probable incertitude of which has about the same value as the amplitude itself. On arranging according to 25.8 days we find a probable incertitude about four times less than the amplitude. Further, from this investigation, as compared with SCHUSTER’S inquiry, we may draw the conclusion, that elements of terrestrial magnetism, as observed in higher latitudes, allow a more decided judgment to be formed concerning the real existence of periodical aj Proceedings Royal Acad. Amsterdam. Vol. VII. ( 34 ) oscillations of this kind than meteorological observations made at tropical stations. If the outcome arrived at by the arrangement of barometric daily means for Batavia is considered to afford some evidence or indica- tion for an oscillation periodic in 25.80 days, a much greater probability must be attached to the real existence of this fluctuation in the observations of magnetic declination made at Greenwich. Anatomy. — “On the Form of the Trunk-myotome.” (First Com- munication). By Prof. J. W. LANGELAAN. (Communicated by Pro ed Pruacg). The segmented plan of construction of the vertebrate animals, most marked in the muscular system, has led to the conception of the myotome. Two methods are chiefly employed in establishing the form of this myotome. The first method is based on the hypothesis of the primary connection between muscle and nerve; the second, a more direct one, is based on the dissection of the intersegmental tissue. Both methods seem equally restricted in their application, as can be concluded from the researches of BARDEEN *) ; moreover there is reason to believe, that they will not always yield concordant results. The second method is followed in this research. I. Trunk-myotome of Petromyzon fluviatilis. (Fig. 1). The trunk-myotome of the adult animal has in general the form of a crescent, the cornua being directed to- wards the cranial end of the body and slightly inclined to each other. The dorsal cornu (fig. I CD) reaches to the mid-dorsal line, while the ventral cornu (fig. I CV) ends at the mid-ventral line of the body. Both cornua differ in length, the dorsal being about ‘/, longer than the ventral, and while both Bo ae reach to the mid-plane of Fig IL. the body, they are slightly torquated in respect to each other. 1) Anat. Anz. Bd. XXIII, NO. 10/11. (35) The corpus of the myotome shows a kneelike inflection (fig. II A’), which is always situated nearer the mid-ventral line of the body than the mid-dorsal line. In transverse section (along the line FF, fig. 11) Fig. [IL the corpus of the myotome is rhomboidal, this rhombus being more and more flattened towards the cornua; consequently the cornua appear in transverse section as lamellae in juxtaposition (fig. II] “.4.). These lamellae are slightly incurved. One side of the rhombus lies in the body sur- face (black in fig. II). This surface is cylindrical in the middle region of the body, the transverse section being perfectly elliptical. The black-coloured surface of the myotome, must therefore be con- sidered as cut out of this cylindrical surface. The opposite side of the rhombus is turned towards the skeletal-axis and the abdominal cavity. Ns ._ In general it has the same form as the outer side, Transverse section through the trunk of Petromyzon; the inter. C@Vily, Which in this part of the body is cylin- segmental tissue being drical, the transverse section being a perfect circle. black. Both the other sides of the rhombus are con- gruent and bound respectively, a more cranial and a more caudal myotome. being only slightly excavated by the abdominal The position of the myotome as a whole in respect to the sagittal plane, passing through the mid-lines of the body, is such, that the corpus shows an inclination towards the caudal end of the body. Seen in transverse section (along the line FF fig. II) the longest axis of the rhombus makes an acute angle with the sagittal axis of the body, the vertex of the angle being turned towards the head. This caudal inclination of the myotome diminishes towards the cor- nua, so. that the cornua are nearly normal to the surface of the body. In consequence of this caudal inclination the myotomes over- lap to some extent. This muscular overlapping varies between '/, and */, in the neighbourhood of the knee, diminishing towards the cornua on account of the decrease of the caudal inclination of that part of the myotome. The position of the myotome in respect to the dorsoventral axis is variable along the body. If AZ (fig. II) is a dorsoventral axis, at right angle to the sagittal axis, and A a line tangent to the dorsum of the myotome, then we have in the angle A/B a measure for the ante- or retroversion of the myotome in respect to the dorsoventral axis. The first myotomes behind the last branchial cleft, show a little 3% ( 36 ) anteversion, which quickly decreases, so that the 4t and 5% myo- tomes are strictly vertically situated. The following myotomes (as in figure Il) are retroversed, this retroversion reaching a maximum of 10°. Towards the caudal end of the body this retroversion decreases and is again reversed behind the anal aperture, where the myotomes are again anteversed. The description given here of the myotome applies only to the trunk-myotome in the middle region of the body, the branchial apparatus as well as the appearance of the dorsal fins bringing about notable changes in this form. Il. Prunk-myotome of Acanthias vulgaris. (Fig. IV and V). The myotome described in this paper was situated in that region of the body which lies between the thoracic fin and the first dorsal L D Cowl. En Cr Vv; Fig. VL fin. In its most general features the trunk myotome of Acanthias shows a great resemblance with that of Petromyzon, though at first view a considerable difference seems to exist. Looking at that surface of the myotome, which forms part of the surface of the body, we see it interrupted in two places. The lines of interruption are nearly parallel to the sagittal axis of the trunk. The first line (LZ fig. VI) coincides with the linea lateralis, the second (L/L' fig. VI) lies nearer the mid-ventral line J. W. LANG f Acanthias; ventral part in natural size. Trunkmyotom Proceedings Royal J. W. LANGELAAN. “On the Form of the Trunk-myotome,” Fig. L Trunkmyotome of Petromyzon fluviatiles. Enlargement 2 times. Fig. IV. e ma Trunkmyotome of Acanthias; dorsal and lateral part in natural size. Proceedings Royal Acad. Amsterdam. Vol. VII. Trunkmyotome of Acanthias; ventral part in natural size. (37) of the body. At the place of interruption septa of connective tissue descend and seem to divide the myotome into three parts. One part situated between the mid-dorsal line and the line LZ is the dorsal part of the myotome; between the lines L/ and L'L' lies the lateral and between the latter and the mid-ventral line, the ventral part of the myotome is situated. Considered at the line of interruption, the surface of the lateral part of the myotome seems to be cranially displaced in respect to the dorsal part; this displacement amounts to one half of the breadth of the myotome. The same can be observed between the surface of the lateral and the ventral part of the myotome, the lateral part being also displaced cranially in respect to the ventral part: this displacement does not surpass ‘/, of the breadth of the myotome. If we follow the septa of connective tissue at the line of inter- ruption LL, it is easily seen, that the myotome is rolled in towards the axis of the body and then reversed till it reaches again the body surface. In most cases the continuity of the muscular tissue at the bottom of the fold is broken off, but the intersegmental tissue which covers the myotome is always continuous. If we now try to unroll the myotome as much as possible, we find that the dorsal part makes an angle with the lateral part, so that a true knee is formed. The top of the knee is directed towards the head as in Petromyzon. The line along which the myotome is folded in, is parallel to the sagittal axis of the body, and seems to run over the knee, so that the top of the knee lies at the point A’ of figure VI. The same can be observed on following the line L'L’ (fig. VI), the line of folding being also parallel to the sagittal axis of the trunk. The differentiation of the myotome into three parts, ensues there- fore from a process of infolding, the lines of folding being parallel to the sagittal axis of the body. If a model of the myotome of Petromyzon is cut out of paper, and this myotome folded in along the lines FF and fF" (fig. II), we get a precise illustration of the displacements seen in the myotome of Acanthias. The direction in which the outer- surfaces are displaced in respect to each other is a direct conse- quence of the form and the curvature of the myotome at the places of infolding. The difference in the extent of the displacement of the surfaces along the lines L/L and L’L’ (fig. VI) is due to the fact, that the fold along the line FF is longer in the direction from outwards to inwards in correspondence with the dimensions of the myotome. ‘This can be seen in a transverse section through Fig. VIL. the same region of the trunk of Acanthias, where LF (fig. VII) is the intersegmental tissue that divides the dorsal from the lateral part, while L’/” is the septum, that the latter separates from the ventral part of the myotome. The further differentiation of the dorsal part of the myotome takes places by the same process; the lines of folding instead of being parallel to the sagittal axis of the body are in general at right angles to this axis. There are three of these lines, agreeing with the number of ~~ h peaks which the dorsal part of N . XQ the myotome shows. These lines oe considered from outwards to inwards, are originally normal Ne pri to the surface of the body, then Transverse section through the trunk curved with the convex side of Acanthias; the intersegmental tissue turned ventrally and towards black, Natural size. the body surfaces. This curva- ture is most marked in the third line of folding (A fig. VII), reekoned from the mid-dorsal line, the first one being nearly a straight line normal to the body surface and to the sagittal axis. These lines of folding are visible on a transverse section, because septa of inter- segmental tissue stretch out into the fold. The eurved lines (A fig. VII) are the transverse sections of these septa. The lateral part of the myotome shows no further differentiations. The ventral part has only one line along which the folding of the myotome is well marked; consequently this part of the myotome shows only one peak turned to the caudal end of the body. The myotome considered as a whole, as in Petromyzon, has a caudal inclination. This inclination is most marked at the knee. Considering only the dorsal part, we see this inclination diminish towards the mid-dorsal line, so that the most dorsal part of the myotome is about normal to. the surface of the body. This most dorsal part is elongated into a dorsal cornu (CD fig. VI). Im trans- (39) verse section we find therefore these dorsal cornua as lamellae in juxtaposition (fig. VIT CD). When the direction of the myotome is reversed at a line of fol- ding, the same happens with the sense of the inclination, so that these parts show a slight cranial inclination. The folding of the myotome together with the inclination produces the elongated and peakshaped form of the myotome at these lines of folding. The myotomes thus cover each other as hollow, pointed tubes telescoped into each other. In the transverse section (fig. VII) 1 indicates the section of the first peak directed caudally, 2 the peak turned towards the head and 3 the second peak turned to the caudal. end of the body (fig. VI resp. 1, 2, 3). In consequence of the inclination of the dorsal part of the myotome, these lines of folding are not quite at right angles to the sagittal axis, but are also slightly inelined. This is the reason why we find only part of these lines in a transverse section, which is normal to the sagittal axis. From a transverse section we can judge the extent of the muscular overlapse. Concordant with the increasing inclination from the mid- dorsal line to the first lateral line, we see the overlapse increase to about */,. At the knee the inclination rapidly decreases. In the lateral part of the myotome the inclination is insignificant and the overlapse less than '/,. In the ventral part the inclination increases at first and then decreases towards the mid-ventral line; the muscular overlapse does not surpass '/,. The ventral part terminates at the mid-ventral line in a ventral cornu (CV fig. VII, fig. V) turned eranially. This ventral cornu is much shorter than the dorsal cornu. In order to get some idea of the dimensions of the myotome I have measured the length of each of the three parts into which the myotome is divided up. These measurements have been made over the surface of the myotome and this surface was also followed where it is folded in. In this way I have found for the myotome described : Length of the dorsal part 350 mm. ; the lateral part 90 mm.; the ventral part 190 mm. The length of the whole myotome is therefore 630 mm. and of this **/,,, belong to the dorsal region. I have made the same measurements in the myotome of Petro- ‘myzon. If it be conceded that the points A (fig. IL and fig. VI) where the knee is located in both myotomes, are corresponding points, I have found: Length of the dorsal part, from the mid-dorsal line to the knee 33 mm.; the latero-ventral part, from the knee to the mid-ventral line, 26 mm. The whole length of the myotome was therefore 59 mm., and of this also *'/,,, is contributed to the dorsal region. ( 40 ) If we compare the dorsal region in Petromyzon and in Acanthias, it is evident that this region is strongly reduced in the latter ; notwithstanding this, the same part of the whole myotome ‘belongs in both cases to the dorsal region. If this be true in general, it seems to me, that the reduction of the dorsal region is the principal moment which has led to the folding of the dorsal part of the myotome. In figures IV and V the position of the rib, in relation to the myotome, is indicated by a erossed line. The rib is located in the intersegmental tissue that divides the dorsal from the lateral part of the myotome. The junction of the rib with the skeleton, lies a little caudally in respect to the knee of the myotome ; the rib itself is turned to the caudal end of the body concordant with the caudal inclination of the myotome. In a transverse section, three successive ribs are cut through. Geology. — “On the direction and the starting point of the dilwial ice motion over the Netherlands.” By Prof. Eve. Dvusots. (Communicated by Prof. J. M. van BrMMELEN). Referring to the “Beschrijving van eenige nieuwe grondboringen,” V, by Dr. J. Lorié, recently published in the “Verhandelingen der Koninklijke Akademie van Wetenschappen, 2% Sectie, Deel 10, N°. 5”, to which, on p. 20 and 21, the author has added a critique of some conclusions in my communication to the Academie: ‘The geological structure of the Hondsrug in Drenthe and the origin of that ridge” (Proceedings of the meeting of Saturday, June 28, 1902 Vol-We p. 93 sqq.), I beg leave to make the following remarks. The critique of that eminent student of the geology of the Nether- lands is based on such an incorrect and incomplete statement of my conclusions and of the facts, that the reader cannot but regard those conclusions as being of a rash character, which in fact they have not. Indeed, having said that he does well agree with my opinion regarding the structure of the Hondsrug, Dr. Lork continues as follows *): “Another case it is with a particularity mentioned on p. + and 5”. (This refers to 12 lines on p. 96 and 97 of the Proceedings.) “In pit XLI there was found a boulder of quartzite, having a diameter of 0.35 M., cleft into two pieces, in such a manner 1) This quotation has been translated from the Dutch of Dr. Lorm by me. KA) that the upper piece, with respect to the lower one, has been pushed on 1'/, em. in south-easterly direction. So this is a fact! The author however bases thereupon the hypothesis that the ice motion has taken place, as a whole, not from N.E. to S.W., such as is still generally admitted, but from N.W. to S.E., in such a way that the startine point is not to be sought for in Scandinavia but in Scotland. Now it appears to me that here is a strong disproportion between the importance of the observed fact and that of the hypothesis.” Further, at the end, he says: “So to find the explication ot the shifting of the quartzite boulder at Eksloo, over a distance of one centimeter and a half, we have not to admit Scotland as starting point for the iee motion, but can persist in our old opinion.” Thus far Dr. Lori. Now I wish to remind those who take an interest in the matter that it was by no means that one fact, referred to by Dr. Lor, on which I based “the hypothesis” that the ice motion “took place” from N.W. to SE, nor did [ assert at all that the starting point of the ice motion is not to be sought for in Scandinavia but in Scotland. In the quoted Proceedings, to which Dr. Lori refers, I do not speak of a hypothesis, but of a supposition, and this, clearly, is based upon the whole consideration of the structure and the origin of the Hondsrug ridge. Particularly this supposition is related to 2'/, pages of my communication, (the whole text being 10 pages), viz. from p. 99 (in the middle) to p.101 (below). The only sentence bringing in relation the fact of the shifting of the pieces of the quartzite boulder, with respect to each other, to the direction of the ice motion, (p. 100, at the end of the second alinea), occurs half way the explanation of 2'/, pages and runs as follows: “Now with this supposition perfectly agrees the at first sight paradoxical direction of motion as derived from the shifted boulder of quartzite.” And concerning the starting point of the ice motion, on p. 101 of the Proceedings [I most unhesitatingly admit Scandinavia to be the starting point of the ice motion, whereas I only speak of the possibility (“it might be possible, at least”) of a deviation of the Scandinavian ice stream in south-easterly direction, caused by the Scottish ice stream. It will be superfluous to argue that the distance over which the two pieces of the quartzite boulder are separated from one another in the soil ought not to be in any proportion to the large motion of the ice over it. Haarlem, May 26, 1904. (42) Mathematics. — “On an vrpression for the class of an algebraic plane curve with higher singularities.” By Mr. Frep. Scnun. (Communicated by Prof. D. J. KorreweG.) If an algebraic plane curve is given by an equation in Cartesian point-coordinates, its order nm can be immediately read from the equation. The class k of the curve is best defined as the order of the equation in /ine-coordinates. However, in the following it is my intention to restrict myself exclusively to point-coordinates and then the class can be defined as the number of movab/e points of intersection of the curve with the first polar or as the number of proper tangents to be drawn from an arbitrary pot P to the curve. To obtain exclusively different points of contact not situated in manifold points of the curve we must understand by an arbitrary pomt a point that 1st does not lie on the curve, 2e¢ does not lie on one of the tangents in a manifold pomt of the curve, Brd does not lie on a tangent in a unifold point having with the curve a contact of a higher order than the first. A manifold point of the curve is a eurvepoint which cannot be a single point of intersection with a straight line. The lines eonnecting P with the manifold) points must not be counted as proper tangents. From the above mentioned definition of the class another one can be deduced, where no single restriction is made with respect to the situation of the point Z, which thus holds good for any point P. To begin with, we make the restriction that P may not he on the curve. Suppose P to lie on the tangent in a point S of the curve, where S may be a manifold point or a single point with a tangent intersecting in more than two points. If the straight line PS cuts the curve in w coinciding points S, whilst an arbitrary straight line through S cuts the curve in ¢ coinciding points S, then S counts for w— ft of the 4 proper points of contact with tangents from P to the curve, in other words w—# points of contact approach S when P approaches the tangent in S. It is of no importance whether the curve has one or more branches through S, touching SP, neither whether the curve has branches through S not touching SP or not. The above mentioned follows immediately from the following: Tunorem. Let R be a point of an algebraic curve where all branches through R have the same tangent | which intersects the curve in t+-v coinciding points R, whilst every other straight line through R intersects (43) the curve int points R, then R absorbs v proper points of contact with tangents from P when P lies on | outside Rand t+ proper points of contact when P comeides with R.*) Now, if S is such a point, where all branches have the same tangent $P, then wt + r, whilst according to the above theorem S counts for v=r—t points of contact with tangents from P. If besides the branches touching S/ still more branches pass through S, then these latter do not give rise to any new points of contact coinciding with JS; they cause however the same increase of the numbers fand w, so that they leave w_—f unchanged. Then too w—t represents the number of points of contact, which in consequence of the singular situation of P coincide with 5S. | If P is not situated on one of the tangents in S, then w— tf, so 1) For a branch which can be represented by one single Puiseux-development this theorem can be proved i. a. out of the relation existing between the developments in point- and in line-coordinates. By addition follows immediately the same theorem for more branches having the same tangent. In a paper entitled: “An equation of reality for real and imaginary plane curves with higher singularities’ (These Proceedings of April 23rd 1904, p. 764) I made use of the same theorem (p. 765) and veferred for the deduction to Srorz, Zeurnen and STEPHEN Situ. [ omitted however to mention G. Hatpnen, “Mémoire sur les points singuliers des courbes algébriques planes”, Mémoires prés. par divers savants a ? Académie des Sciences (2), t. 26, (1879), n°, 2 (112 p.). This extensive paper was already offered to the Paris Academy in April 1874 (see: Comptes Rendus de Académie des Sciences de Paris, t. 78, p. 1105—1108, where the autor communicates some of his results) so that this paper has the priority, Hateuen formulates the theorem somewhat differently, namely (l. e. Théorème III, p. 42 or Théoréme II, p. 50): Trtorime. La somme des ordres des contacts des branches d'une courbe avec une de ses tangentes est egale a la multiplicité du point correspondant à cette tangente dans la courbe corrélatire. The relation between the developments of a branch in point- and in line- coordinates was first considered by A. Caytey (“On the Higher Singularities of a Plane Curve”. Quart. Journ. of Math., Vol. 7, (1866), p. 212, Collected Math. Pap., Vol. 5, p. 520 or “Note sur les singularités supérieures des courbes planes” Crelle's Journal, Bd. 64, (1865), p. 369, Coll. Math. Pap., Vol. 5, p. 424). If y=Axr+Bar+....(p > is the development in point-coordinates then p 1 Cayrey gives for the development in line-coordinates Z = A! Ar! + B' X pl + …, : ? eA + u Rest, F where the general form of the exponents is = ; here A, w,.... are posi- p— tive integers. Implicilly the Hatrnen-theorem is included in this, Gaytey, however, does not enter farther into the relation between the developments and does not state the theorem. Let me finally notice, that the theorem has also been stated by M. Noruer, “Ueber die singulären Werthsysteme einer algebraischen Function und die singulären Punkte einer algebraischen Curve’, Math. Annalen, Bd. 9, (1876), p. 166 (sp. p. 182), (CART) that then w — ¢ continues to represent the number (uamely zero) of the points of contact coinciding with JS. If S is a point of contact with a tangent out of P where nothing remarkable takes. place, then for that ¢ = 1 and w= 2, so that w ¢=1, whilst S now also counts for one point of contact. By summing up the values wt for all the points S of the curve for which w >t we keep getting for sum / with respect to every point P not lying on the curve, thus CEE NT PERSEN UT EEEN (1) We formulate this in the following way : Trrorem LL. Let P be a point not situated on an algebraic curve and S an arbitrary point of that curve. Let us suppose that the curve cuts the straight line PS in w, an arbitrary straight line through S however in t points coinciding with S; then the class of the curve is equal to w—t summed up for all the points S of the curve for which w >t and for as many other curvepoints as one likes. To continue we suppose that 7? lies on the curve namely in a point of the order ¢t, i.e. £ is the smallest number of coinciding points of intersection of the curve with a straight line through P. For a point S of the curve not coinciding with P? the number of points of contact coinciding with S is still indicated by w—t. Moreover a certain number of points of contact coincides with LP, namely according to the HaLpnen-theorem to the number of £ + + 7, where XE! represents a summation with respect to the different curvetangents intersecting the curve in £ + #!,, f + ¢',,.... coinciding points P. So we get a= tS Suid (eS), a Se REEN where + (w, — ¢,) represents a summation with respect to all the points S of the curve outside P. However we can also include the point P among the points S. The line connecting P and S becomes in that case indefinite. If we take for PS a line which is not a tangent in P, then we get w=t. If however we take for PS a tangent intersecting in ¢ + wv, points coinciding with P, then we get w=? + ,', so w— t'=v,'. So for (2) we can write LEL St A ee if only we extend the summation also to tbe point S lying in P itself, in which case we have but to take for PS those straight lines through P contributing to Y (w,—+t,) (thus the tangents in P) and as many other straight lines through P as one likes. The equation (1) is a special case of (3). If namely P is not on ( 45 ) the curve then each straight line through P has zero points of inter- section ? with the curve; in other words Pis a point of the order zero of the curve, so £==0. So the result of all our considerations is included in equation (3). We formulate this in the following way : Tueorem II. Let P be a point of the order tof an algebraic curve (where t may also be zero) and S an arbitrary point of the order tof that curve. Suppose the straight line PS intersects the curve in w points coinciding with S, then the class of the curve is equal tot’ increased by the sum of w—t over all the points S of the curve. If S is in P we have to regard all straight lines through P as the line connecting P and S. When speaking of all points S or, when S is in P, of all straight lines through P, we mean that we take those points or lines contributing to 2 (w,—t,) and as many other points or lines as one likes. Theorem I is a special case (# =O) of this theorem II. The theorem always holds good for any singularities the curve may have. Sneek, May 1904. Geology. — “Some considerations on the conclusions arrived at im the communication made by Prof. Eva. Dvusois in. the meeting of Sune 27, 1908, entitled: Some facts leading to trace out the motion and the origin of the underground water of our sea-provinces.” By H. E. pr Bruyn. (Communicated in the meeting of September 26, 1903). In the meeting of June 27, 1903 Prof. DuBois made a communication dealing with a problem of great general importance, namely the presence of proper drinking-water in the province of Holland. Although readily acknowledging the many points of merit of this communica- tion and entirely agreeing with many of its conclusions, I differ from the author on a principal point which indeed is essential, namely the origin of the fresh water in our polderland. So a speedy refutation of the author's opinion on this point seemed to me to be desirable. In his communication Prof. Dugois speaks of our sea-provinces; this in my opinion ought to be Holland, since the conditions prevail- ing in Friesland and Zeeland are different, so that considerations which are valid for Holland cannot be applied there. So 1 will only consider the tract of country ehiefly dealt with in the above- mentioned communication, which is bounded by the dykes of the Y ( 46 ) and the Zuiderzee at the north, by the river Vecht at the east, the Rhine from Harmelen to Katwijk at the south and the North-sea at the west. . The geological conditions of this tract inside the dunes are such as are mentioned in the communication: uppermost alluvium, then pretty generally a layer of fen (partly. disappeared) under which a layer known as “old sea-clay”. Under this latter the diluvium, consisting to a great depth of sand, coarser and finer, with here and there banks of clay which are not continuous however. The “old sea-clay” mentioned is called in the paper clay-containing sand and although in my opinion also that layer is permeable to water, yet I think its permeability is smaller than Mr. Dusots assumes and that it is exactly here that the cause of our difference of opinion has to be sought. In some places this layer of old sea-clay is wanting; in special cases this makes an investigation very difficult, for the general condition however, which is here dealt with, this circumstance can be neglected. The communication consists chiefly of two parts, of facts and of conclusions drawn therefrom. The facts I will pass without comment- ing on them, although occasionally objections might be raised against them or rather against the remarks that accompany them. I perfectly agree with a great many of the conclusions, e.g. with the following: that in the diluvium fresh water is present to a certain depth; that ‘in deeper polders the deep groundwater moves vertically upward, in shallow polders downward: that also in the depth a current exists from the dunes to the polders and from the shallower polders to the deeper ones; that no important continuous subterranean current exists from the higher grounds from the east to the west. But I cannot accept the conclusion that the fresh groundwater present in the diluvium also in our polders, owes its origin to rain fallen locally or at a relatively short distance during the wet season. In the following refutation of this opinion | shall speak of fresh and of brackish or salt water. Of course there is no sharp division between these, but in order to avoid cumbrous definitions I shall make this distinction for simplicity’s sake. I base my considerations on quantities, but since only their relative amount concerns us here, I have rounded my figures as much as possible. I intend to show the incorrectness of the conclusion mentioned from the amount of the afflux to in the Haarlemmermeer polder. Now a paper on this amount by the member of this Academy ( 47 ) VAN Diesen is found in the Versl. en Meded. der Kon. Akad. 1885"). IT can by no means accept the amount found there. The chief reason why Mr. van Diesen arrived at an erroneous figure is that he assumes that the groundwater in the Haarlemmermeer polder which is situated at a depth of about a metre below the surface evaporates as much as water at the surface on account of the interstices between the particles of ground. This, I think, is entirely wrong; ground- water at a depth of a metre does not evaporate at all. Now if we assume that the groundwater does not evaporate, the figures given in the paper would lead to a negative afflux, which certainly is wrong too. This is a consequence of another reason why an erroneous figure is found, namely the method of derivation. Mr. van Diesen, namely, calculates the amount of the afflux from two periods of six years, for each of which he derives the equation: we af —b in which & is the amount of the flow, wv the ratio of the evaporation at the surface and the rain fallen; « and / constants, derived from the other data. So he has two equations £ = ax — b and 4, = aur, —b.. Now he determines the ratio of / and 4, from the difference in level of the water in the ““bosom”*) and of the polderwater, a ratio naturally little differing from unity; he further assumes that the . of each period has the same average value, so he puts e= .w,. The two unknown quantities, f and #, can then be found from these two equations. But « and wv, are not exactly equal. Mr. van Dieser himself says: “evidently this value must change according to circumstances.” If x, be equal to «+ 4 we have: k k —b, —b —a, À k, k, = — k k ce a pg — of k Now it will entirely depend on the value of rae and « whether sa 4 has an appreciable influence on the value of wv. Since from the k nature of the case 5 as and @ are great values, differing little between iC 1 each other, however, a small value of 4 has a great influence on wv and hence on 4. 1) Versl. en Meded. van de Kon. Akademie van Wetenschappen, Afd. Natuurk. Be Reeks, DI. l, p. 359—374. ' 2) “Bosom” is called an intermediate discharge canal or basin. ( 48 ) The best estimate of the afflux in the Haarlemmermeer polder is that by Mr. Enink Sterk'). This author bases his calculation on the assumption that the value of rain minus evaporation, averaged over many years, is practically the same for Rijnland and for the Haar- lemmermeer polder. Rain is here tacitly assumed to be rain plus surface condensation, and evaporation, evaporation plus the water withdrawn by plants. From the quantity of water discharged and let in over an average of 14 years Mr. EriNK Srerk then derives with the aid of the assumption mentioned that the afflux in the Haarlemmermeer polder is equal to a quantity of water corresponding to a height of 135 mm. + A (A being the afflux in Rijnland) over the whole surface. Now he puts A=15 mm. Le. */,, of the afflux in the Haarlemmer- meer polder which he calls an ample estimate as I think it is; so he finds for the amount of the afflux in the Haarlemmermeer polder 150 mm. The assumption mentioned that rain minus evaporation is equal for Rijnland and for the Haarlemmermeer polder is not quite correct of course. The rain may be taken equal, but not the evaporation. The rainfall is in my opinion more regular on the average than is indicated by our rain-gauges. Under equal meteorological conditions the rate of evaporation depends principally on water for evaporation being or not being present. In the polders having a high summer- level with regard to the land which mostly consists of meadows, evaporation will be greater than in the Haarlemmermeer polder, since water will always be present at the surface; in the dunes on the other hand it will be less. Considering the character of the grounds in Rijnland we may assume that evaporation there will be slightly greater than in the Haarlemmermeer polder. So if we apply to our figure a correction 4,, making it 150 + L,, 4, will be negative. Mr. ELiNK Srerk has left out of consideration through lack of data: 1. the quantity of water admitted into the Groot Waterschap van Woerden (having the same bosom as Rijnland); 2. The quantity of water let in by locks into Rijnland and the Groot Waterschap van Woerden. Calling these respectively 4, and 4,, the afflux in the Haarlemmermeer polder is k—=150+ 4,+ A, + A, mm. Now the quantities A, and A,, are both small and certainly positive; probably they are together smaller than 4,. So if we omit the three corrections 4,, 4, and 4,, the error can not be large and 1) Verhandelingen van het Kon. Instituut van Ingenieurs. 1897—1898. p. 63—75, (49 ) the figure for k probably becomes too great as it also becomes by putting A= 15 mm. Putting the afflux of water at 150 mm. this gives for a surface of 18000 H. A. 27 million M*. This amount consists of three parts: 1. of what is let in for the higher lands behind the ringdyke through valves and syphons; 2. of the afflux through the ringdyke above the old sea-clay which I shall call the afflux through the alluvium: 3. of the afflux over the whole surface of the polder, moving upward through the old sea-clay on account of the greater pressure, which I shall call the afflux from the diluvium. The first part which is no proper aftluw, is estimated by Mr. ELiNK STERK at 5 to 7 million M* per year. Subtracting this and taking the smallest figure the afflux mentioned sub 2 and 3 becomes 22 million M* per year. How much of this is due to each of the parts sub 2 and 3 cannot be made out, while in those places where the “old sea-clay” is absent no separation takes place. Probably 2 is the greater part, therefore I assume for the part sub 3 an amount of 10 million M* per year; possibly it is much smaller. These 10 million M*. the afflux f/om the diluvium must consequently either flow in as fresh water along the circumference of the polder under the old sea-clay through the upper layers of the diluvium, or rise from below as salt water. Let us for the present asstime that it all flows to in the former manner. In fifty years 500 million M*. of fresh water would in this way have flowed into the diluvium. Now the quantity of fresh water present in the diluvium under the Haarlemmermeer polder is greater; assuming */, to '/, space between the grains of sand this quantity would only correspond to a thickness of 10 metres containing fresh water, whereas this thickness is greater on the average, as is proved e.g. by borings near Sloten. The circumference of the ringdyke being about 60.000 metres, if we assume the afflux to take place over a height of only 20 metres and. the interstices between the grains of sand to be the same as above, this will give a velocity of motion of 30 metres per year and the water flowed to would, even if we neglect the loss of speed further in the polder, have penetrated into the polder only 1500 metres in 50 years and so not have reached the middle. Moreover the assumption that all the water streaming to is fresh, is not probable, if we bear in mind the amount of salt in the Wil- helmina spring which is over 3000 mg. chiorine per litre. Hence it is certain that with a flow of 10 million M* from the diluvium, part of the fresh water nowadays present in the diluvium under the 4 Proceedings Royal Acad. Amsterdam. Vol, VIL. ( 50 ) Haarlemmermeer polder was present there already 50 years ago. Before that time conditions were very different from what they are now. Instead of the deep drainage there was bosomwater. How the fresh water then present, especially in the eastern part, had come under the Haarlemmermeer, is difficult to tell for want of data. Was the water of the Haarlem lake always so rich in chlorine as some old observations show? Certainly the difference in pressure of the deep groundwater was smaller than it is now and accordingly the quantity of water moved was also smaller, while the direction of the current in the deep groundwater e.g. near Sloten must have been exactly the reverse. A thousand years ago when there were no dykes yet to keep out the water of rivers and of the sea, and no mills yet to drain the polders, when the dunes were so much broader at the seaside than they are now, when there were no canals in the dunes yet for sand transport and other purposes, I imagine the state of affairs in the tract of land we are considering, must have been such that fresh water was also present in the diluvium and probably more than nowadays, that in the dunes there existed a high level of ground- water by which water was driven to the diluvium, the pressure at the west side under the “old sea-clay” being greater than that of the groundwater above it. The level of the groundwater in the alluvium of the polderland was then probably much more regular and slightly higher than the average sea-level. How these conditions became pre- valent I must leave to geologists to explain. By making dykes, by enclosing polders, by draining, the level of the groundwater has gradually been lowered, now in one place, then in another. The currents in the layers of fresh water in the diluvium also had their directions changed by this; they certainly were very small, however, before the great drainages were made. In the tract we are dealing with, 3000 H.A. were drained before 1750, 10.000 H.A. between 1750 and 1850 and 26.000 H.A. between 1850 and 1900. The dunes gradually decreased in breadth, while also the flow of water towards the land increased by canals for sand-transport ete. The height of the groundwater in the dunes will consequently also have steadily been decreasing. Bearing in mind the figure for the amount afflux to in the Haarlemmermeer polder, we may safely assume the quantities of water which before the drainages were made, penetrated vertically (51) downward through the old sea-clay, to have been very small com- pared with the amount of fresh water present. Consequently the only source of supply of fresh water to the diluvium has been the afflux from the dunes. At the same time | venture the supposition that part of the fresh water which a thousand years ago was present in the diluvium under the polderland is still present there now. Another question arising here is whether salt water rises upward from below. About former times nothing can be stated with certainty in this respect; for the present time it is rendered probable by the circumstance that the water of the Haarlemmermeer polder contains more chlorine than can be derived from the afflux if no salt water from below is added to it. Therefore I have tried to estimate, though roughly, the quantity of chlorine, discharged by Rijnland and the quantities entering Rijnland in another way than from below, assuming that the quantity of chlorine withdrawn from the ground by plants is equal to the quantity furnished by manuring, which supposition is reasonable. Rijnland discharges annually on the average 476 million M*. of water ; how much chlorine this contains is not known, but from the data for the percentage of chlorine of the water of the bosom?) a figure may be derived which is too small and another which is too large, which figures I take to be 105 and 315 mg. per litre, giving an annual discharge of 50.000 or 150.000 tons of chlorine. The quantities of chlorine arriving into Rijnland are, besides that from the groundwater below 1. the sea-spray ; assuming that this chiefly falls on the dunes we can estimate it; the dunes that discharge water into Rijnland will supply about 20 million M?. of water annually ; this water contains 40 mg. chlorine per litre, making 800 tons of chlorine; adding to this what is blown over the dunes we obtain a total of 1500 tons; 2. the fresh water which is let in. amounting on the average to 125, million M*. per year, containing 40 mg. per litre, which makes 5000 tons; 3. the water through locks; it is difficult to estimate an average percentage of chlorine here, since one lock (Gouda) admits fresh water to the bosom, others (Spaarndam, Overtoom, etc.) water with a high percentage of chlo- rine; I think 2000 mg. per litre a sufficiently high estimate; putting the water let in through locks at 5 million M*. this makes 10.000 tons: 4. what human society discharges into the bosom; this amount is difficult to estimate ; putting it at 5500 tons, the total amount becomes ) Mededeelingen omtrent de Geologie van Nederland, no. 26, by Dr. J. Lon, pp. 8—11. (52) 20.000 tons of chlorine. Assuming that what is supplied to the bosom by all these causes is more than half this rough estimate and less than its twofold we get 10.000 or 40.000 tons of chlorine. In any case there is a deficiency amounting to something between 10.000 and 140.000 tons which has to be ascribed to a supply of salt from a greater depth. When we bear in mind that this will chiefly come from the Haarlemmermeer polder and that this latter discharges on the average about 30.000 tons of chlorine and that the supplies mentioned sub 1—4 occur there in a small degree, the supply of salt from below at the present time is pretty certain. A quantity of 35.000 tons of chlorine corresponds to that contained in two mil- lion M*. of water from the North Sea. The motion of the deep groundwater is generally very slow. If e.g. we consider how long it would take water to travel ina layer of sand between two impervious layers from the sea to the Haarlemmermeer polder, which is a distance of 9000 metres, the difference of pressure being 5 M., if the permeability is the same as that of dune-sand, we find that it would travel in a year (31.557.000 seconds) through a distance of 31.557.000. X 5 X 0.0006 = M. 0000 (0.0006 being the rate of filtration through 1 M. of dune-sand with a pressure of 1 M.'). A distance of 9000 M. would consequently require 900 years. As to the rate with which the salt water can rise from below we find what follows. Assuming that the rise is constant and that under the Haarlemmermerer polder 5 million M*. rises annually, this gives over a surface of 18000 H.A. with a space of '/, to *, between the grains of sand, a rise of about 100 mm. per year; a rise of 50 metres would then require a period of 500 years. The question now naturally arises: since a large quantity of fresh water is present in the diluvium under our polderland and the salt water flows slowly, is it possible to withdraw this fresh water for drinking-water? The part we are considering has, after taking off the littoral margin and the country round Amsterdam which has a different formation, a surface of about 100.000 H.A.; not counting the drainages and other less suitable tracts, half of this territory contains 5000 million M*. of fresh water, if we assume an average 1) Report of the Committee for investigating the supply of water from the dunes to Amsterdam, 1891, supp'ement 16, p. 77. (53) of 10 M* of fresh water per M?, corresponding to '/, to 1/, space between the grains of sand over a thickness of 30 to 40 metres. A consumption of 50 million M* per year being sufficient with the existing dune-water conduit for the need of the population, taking its increase into account, this quantity would be able to supply water for a hundred years; now we may presume that in a hundred years science will have so much advanced that it will be practicable then to convert any water into suitable drinking-water. The answer to our question must in my opinion be affirmative as well as negative. Affirmative with respect to supplying water to single dwellings, to a village, or to a temporary supply in war-time such as the Engineering Corps has made at Sloten ; negative with respect ‘to a lasting demand on a large scale and this because in practice pecuniary considerations would force us to withdraw the water from a limited surface which would be impossible without causing such a diminution in pressure that certainly with a lateral afflux also water from below would flow to, so that after some time brackish water would be obtained. Hence Prof. Dusois’ assertion, that a sufficient quantity of drinking- water is and remains available in the ground under the shallower polders, is in my opinion entirely wrong. Geology. — “On the origin of the fresh-water in the subsoil of a few shallow polders’. By Prof. Eve. Dusois. Communicated by Prof. BaKHvis RoozkBoom. (Communicated in the Meeting of November 28, 1903). In the meeting of the Academy of September 26 ult. Mr. H. E, pe Bruyn, although he agreed with most of the principal conclusions about the origin and the direction of motion of the groundwater in part of our lowland, contained in my communication to the Academy of June 27, gave an elaborate exposition of the grounds on account of which he cannot accept my conclusion concerning the origin of the fresh-water in the subsoil of a few shallow polders. In my opinion this has to be sought in rain, fallen on the spot or at a relatively short distance, which Mr. pe Bruyn thinks impossible on account of considerations about the amount of the afflux in the Haarlemmermeer polder which, in his opinion proves that the layers above the diluvium, especially the “old sea-clay’ than is necessary in my representation. He also supposes that part of the fresh-water which was present under our polder-land a thou- ’ transmit water to a much smaller extent (54) sand years ago, is still there at the present day and that the only source from) which fresh-water has been supplied to the diluvium (the subsoil) has been the dunes. About the velocity with which water can move through our always very impure clay, which Mr. pr Bruyn rightly considers to be a cause of our difference of opinion, I will now state a few facts and at the same time point out the arguments which led me to my conception of a different origin of the deep groundwater mentioned. First however I wish to point out another possible origin which has not yet been suggested and which cannot be at once rejected, and especially a difficulty of a more serious nature even than the one objected to my representation by Mr. pr Bruyn. If we assume the extremely slow motion ascribed to the ground- water by Mr. pr Bruyn, it might namely be that the deep fresh-water under consideration has to be considered as a remainder of what sank away there centuries ago. For not always these polders have been surrounded by brackish water only. According to descriptions from the Roman period, Lake Flevo undoubtedly contained water from the Rhine and no salt water as the Zuiderzee does nowadays. Also the IJ was a freshwater lake communicating with the freshwater lakes Purmer, Wormer and Schermer. Moreover it is well known that the Haarlem Lake (Haarlemmermeer) arose by the union of at least four lakes: the Old Haarlem Lake, the Old Leyden Lake, the Old Lake and the Spiermg Lake which were fed at least partially by one or more branches of the Katwijk Rhine. The map by Bolstra, the able land-surveyor of Rijnland. published in 1745 and incorporated in “Present state of the United Netherlands” '), gives us an idea of the situation of these lakes in 1531 and of their gradual union and the enlargement of the Haarlem Lake, originated in this way, down till 1740. The waves of this large lake could easily erode the steep banks, consisting of fen, as low as the same layer of clay which already formed its bottom, the circumstances for this process becoming more and more favourable, chiefly on account of the “sinking of the lands” in these parts with respect to the sea, described already a century before the draining. This erosion of land occurred at a tremendous rate at the north-east side, where the polders are situated which now have fresh-water in their underground. Le FRANCQ vAN BERKHEY *) 1) Tegenwoordige slaat der Vereenigde Nederlanden. Vol. 6 p. 163. Amsterdam, I. Tirion. 1746. 2) J. Le Francg van Berkuey. Natuurlijke Historie van Holland. Vol. I. p. 227; Amsterdam 1769. about the middle of the 18 century deseribes the water of the Haarlem Lake as “fresh, but in some places, where the erounds become brackish, as near Slooten and towards Amsteldam, the water of the lake is sometimes of a saltish taste. But the abundance of water from the Rhine and the supply from so many small lakes and waters which discharge themselves into it, brine about that the brackish water can by no means get the upper hand, and so the lake has on the whole good fresh-water.’ Meanwhile a quantity of salt amounting to 300 milligrammes per litre is according to the latest investigations not unpalatable. An analysis by G. J.. Munprr’) of water taken from the lake near Sloten in November 1825, shows that it contained 393 mg. chlorine per litre. Now this is the season during which it will probably have been least brackish. Hence it is improbable that the water of the Haarlem Lake was on the whole really fresh. Indeed, the lake had ample opportunity to receive salt from the IJ (which had already become salt towards the middle of the 13" century) through the upper ground of the polder-land which consisted chiefly of fen and which separated the two waters in places (near Halfweg) like a true isthmus. It is also known that at any rate towards the middle of the 18 century those grounds under which fresh-water is found, were brackish. Yet fresh-water of a much earlier period might in places have remained in the underground. Water derived not only from the north and west sides, but also from the east, may have filtered into the polders mentioned at the north-east of the present Haarlem Lake. The Amstel certainly contained for centuries perfectly freshwater, derived from the Rhine. As late as 1530 the Amsterdam canals, fed by this river, had drinkable water, but soon this supply was gradually more and more reduced by natural causes. Is now the motion of the groundwater, not only in the finer alluvium, containing much sand, but even in the coarse and gravelly diluvial sand, which transmits water much more easily, really so slow, as Mr. pr Bruyn believes, that in three or four centuries the influence of the altered circumstances as to level and composition of superficial waters on the deep groundwater will scarcely be perceptible? I believe that numerous facts, of which I will mention a few in this communication, are at variance with this opinion, 1G. J. Murper. Verhandeling over de wateren en lucht der stad Amsterdam. p. 66. Amsterdam 1827. Lori, quoting from second hand, wrongly mentions this same analysis under two different headings and with different amounts of Cl. (Onze brakke, ijzerhoudende en alkalische bodemwateren, Verhandelingen der Kon. Akad. 2e Sectie, DI. 6. N°, 8, 1899, p. 9). ( 56 ) In the first place the actual facts are incompatible with Mr. DE Brvyn’s idea that before the draining of the Haarlem Lake, some 50 years ago, “the direction of the current of the deep groundwater e.g. at Sloten, must have been exactly the reverse” of what it is now (These Proceedings VI, p. 291) and still less I can assume this for an earlier period. For near Sloten the country was not lower, but even a little higher than the level of the Haarlem Lake and not dyked in, so that the incessant washing away of the steep fenny bank of the lake could be enormously great. According fo an accurate investigation, made in 1743, it amounted yearly on the average to as much as 5 to 10 Rijnland rods (about 19 to 38 metres). It is true that the upper side of the layer of fen which now forms the Rieker polder near Sloten, lies at 1.35 metres below A. P. (Amsterdam level) but its lower side is still on a level with the bottom of the Haarlemmermeer polder, as it formerly was with the bottom of the lake, and it rests on the “old sea-clay”. If we now bear in mind that fen, such as that of the Rieker polder consists, When if is completely saturated, for */,, of water, as I have found to be actually the case, and that moreover the “looseness and shiftiness” of these grounds which, as it were, rose and sank with the water, were well known in the time of the lake, it is clear, how, after the draining, in half a century, by losing over */, of their water, they might shrink so far below A. P. and that there can be no question of an earlier current of the deep groundwater under the Haarlem lake towards the country near Sloten. The lake certainly did not allow such a current from the dunes to pass under its bottom. Though the dunes were broader and the level of the water in them higher than nowadays, the hydrostatic pressure imparted there to the deep groundwater must have been exhausted and the horizontal current stopped by the water rising in the alluvial cover, which forms an imperfect screen, long before the opposite side had been reached. For at present the difference in pressure, owing to the water-pressure being now 5 metres less in the polder than it formerly was in the lake, is certainly not less and yet already in the middle of it only a slight upward pressure remains, although pressure is directed from all sides to the middle. Hence in the uncerground we only meet water that soaked the soil from above without any considerable horizontal movement in the underground. I quite agree with Mr. pr Breyn (p. 291) that “part of the fresh- Water now present in the diluvium under the Haarlemmermeer ——— ee ee or (57 ) polder was present in it already fifty years ago”, if I may consider water that contains 200 to 300 and even more milligrammes of chlorine per litre as fresh-water, as I understand he does. Only with this latter not common qualification one is entitled to say that under the Haarlemmermeer polder the diluvium has on an average more than 10 metres fresh-water, for the greater part of that polder has no fresher water in its underground than with these amounts of chlorine. Only where the higher grounds are clearly of recent origin this is different, for the rest the water in the upper diluvial layers of the Haarlemmermeer polder contains about the same amount of chiorine as the water of the former lake. Of the Wilhelmina spring, the amount of chlorine of which is over 3000 milligrammes, the depth -is unknown; undoubtedly however it goes down as far as the salt water which in most places of this polder is to be found below 40.50 metres. The most serious difficulty opposing my view of the origin of the fresh-water in the subsoil of some shallow polders is not men- tioned by Mr. pr Bruyn. It is that the fresh-water in question in all seasons not only is surrounded by, but also rests on and is covered by brackish water. How can the fresh-water under these circumstances owe its origin to the rain fallen on the brackish upper ground ? The explanation of this paradoxical phenomenon I mean to have found in the peculiar hydrological condition of those polders which, like those between Amsterdam and the Haarlemmermeer polder, are themselves at a level only little below A.P. and are situated near deeper drainings. In a similar condition are the shallow polders near Purmerend and Schermerhorn. Like here towards the Haarlemmer- meer polder, so yonder towards the deep polders Purmer, Wijde Wormer, Beemster and Schermer, a considerable flow exists in the coarser diluvium under the more compact alluvial cover and at the same time a vertical downward movement, while in those deeper polders the water tries to rise through the alluvial cover which forms only an imperfect screen. Consequently in boring-tubes the ground- water from the diluvium in these latter rises higher than the field, whereas in the shallow polders it remains far under it and below the level of the groundwater. These circumstances and the geological condition of the soil form in my opinion the solution of the riddle of the presence and the permanence under some shallow polders of fresh-water which on all sides is surrounded by brackish water. I arrived at this conclusion especially by studies in the Rieker polder near Sloten, in which the source for the. military. water-supply for (58 ) the position of Amsterdam is situated. To this 1 was enabled by a few experimental wells which Mr. G, van Roven, 1st lieutenant- engineer, charged with the execution of the works there, was kind enough to have made for the purpose of this investigation. At my proposal seven of these experimental wells were bored to various depths in the middle of a meadow, about 300 metres south of the Sloten road and the farm “Rustvrede”, within a square of three metres side. Under the lower end of the iron tubes which were open below, gravel had been poured to a depth of half a metre. By examining the water that had risen in those tubes as to level and composition one can obtain information about the state of affairs at various depths in relation with the condition of the soil. Letter Depth of the Layer in which the water of the layer of water is found. well. below the field *). A Oe: eo ME fen. A’ A aaa all a ia jh B Dee Jo Upper clay. C 5 eee Peed Layer of sand in the clay. C' Tren nw 7 Lower part of the clay. D 10 cari. | Between deep fen and sand EK Low doo Sand under the denser alluvium. A well situated 30 metres to the west, 44 metres deep, IV, 13, was compared with these experimental wells. Most of these wells were ready at the end of August ult. A’ was finished about the middle of September, C the 19% of October. The level of the water and the amount of chlorine were repeatedly examined by me. After all the wells had been left undisturbed as long as November 20, the following state of affairs was found. Water level in M. + A.P. Chlorine in mg. per L. A 1.465 135 A’ 1.495 202 3 1.495 617 Ë 1.493 780 C 1.677 145 D 2,720 124 E 2 an. 68 Vids 2.747 92 1) The level of the field is 1.33 AP. (59) Of these wells C' may for the present be left out of account since a stationary condition has not yet established itself in it; having its lower end in the clay, the water still rises in it continually Well IV, 18 again has a higher and varying amount of chlorine; determinations at different times gave 114,92,190 me. per litre. I believe that this increase and variation of the amount of chlorine has to be aseribed to the neighbourhood of the deep salt water in relation with fluctuations in atmospheric pressure and also with a proper motion of ebb and flood of the groundwater '). In spite of the very considerable rainfall of the latest months, great variations in the percentage of chlorine did not occur; only in and near the clay, hence especially in C, the amount of chlorine decreased considerably, in C from 850 to 780 mg. per litre. This result does not verify my formerly stated supposition that perhaps during the wet season a continuous freshening of the water might take place. Yet I could not agree with the idea that the fresh-water in the subsoil should have stayed there undisturbed for at least a few centuries. For why then is fresh-water in the diluvium under the Haarlemmermeer polder and the Lutkemeer polder only found at a distance not too far removed from the shallow polders near Sloten and Osdorp? Why does this layer of fresh-water end already before Halfweg, before the Great IJpolder is reached, south of Sloterdijk and also soon eastward of the Amstel? Why does the layer of Purmerend not extend further than a short distance under the Purmer and Beemster polders? Does not this limitation point to an autochthonous origin of the fresh-water in the underground of the shallow polders ? I think to have found the key of the riddle in the stated sudden fall in pressure, amounting to more than 1,20 metre, under the clay and the deep fen which is a consequence of the fact that the level of the groundwater in the Haarlemmermeer polder is almost 3.5 M. lower than in the Rieker polder. So this compressed deep fen, acting as a semi-permeable wall, can transmit to the deeper layers water, but no salt. That fen in a compressed state and under similar conditions of ') On these influences, especially on the proper ebb and flood of the ground- water, see: F. Weype, Die Abhiingigkeit des Grundwasserstandes von dem Luft- drucke, dessen Steigen und Fallen während eines Tages (Ilut und Ebbe), in Meteorologische Zeitschrift of August 1903. The influence of atmospheric pressure was already pointed out in my former communication. These influences become perceptible only in deep wells, because in them the water follows more easily the changing pressure of the atmosphere and gravity than it does in the neighbourhood and so is raised or depressed. (60) pressure as prevail in the Rieker polder, can cause osmosis, | could prove experimentally with apparatus which Mr. A. J. Srorn Jr. at Haarlem was kind enough to make for me in his workshop and with other apparatus kindly put at my disposal by Dr. HerRINGA of Haarlem. The most important of these experiments is the follow- ing one. In an iron tube of 1 M. length and 154 mm. internal diameter newly dug fen from the superficial layer in the Rieker polder was compressed by means of a lever until no further compression was observed. The pressure was gradually raised to 2,8 kilogrammes per square centimetre, a pressure equal to that which is found ina soil of sand or clay at a depth of 14 metres; the layer of fen was three centim. thick. Of water, containing a quantity of sodium chloride corresponding to an amount of chlorine of 1000 mg. per litre, this layer of fen, which on account of its slight thickness, can by no means be so perfect a semi-permeable wall as the deep layer of fen in the Rieker polder which has an average thickness of a metre, water was transmitted which contained temporarily at the utmost 750 me. chlorine per litre; hence at least 250 milligrammes were retained. Now the deep fen in the Rieker polder oceurs as an almost coherent layer, extending from Haarlem, right through the Haar- lemmermeer polder as far as Mijdrecht and from Sloten by Amster- dam as far as Zaandam and Uitdam. This layer is missed in the north-western corner of the Haarlemmermeer polder, i.e. in the place of the former Lake Spiering and farther south. The lower side of this deep layer of fen lies at about 11 to 13 metres below A.P. Still deeper at Sloten in some three borings, parts of a second old layer of fen were found and also repeatedly at Amsterdam and Zaandam. This layer must be distinguished from the former with which it was formerly identified. As a fairly coherent layer this’ deeper fen can be traced above the diluvium, to the north by Purmerend as far as Hoorn and Enkhuizen, to the west by Wormer- veer, Beverwijk, and Velzen to IJmuiden. The upper one of these deep layers of fen can reach a thickness of about 1 M., the lower 1/7, metre thick. one is rarely So we may understand how the underground may have derived in former times and may still derive its fresh-water from the upper ground although this latter always carries brackish water itself. But will the “layer of clay” which is 7 M. thick be permeable enough to render it possible that in the half century, elapsed after the draining of the Haarlem lake, under the polders to the north- ee ( 61) east of this drainage a layer of fresh-water of at least 50 M. thick may have accumulated > This means a yearly increase of at least one metre, or, if we take into account the interstices between the grains of sand, of about 0.30 M. pure water a year. Now this amount is pretty much the same as of the rain that can penetrate into the earth, while also all the other surface-water can furnish fresh-water to the underground, by which also the fresh-water, flown off to the deep polder, can be accounted for. In my former communication I already pointed out that the power of clay to transmit water is commonly underrated. The clay in our alluvial grounds is generally very impure, consists mostly of very fine sand and according as the percentage of this increases, its “permeability” becomes greater. The fattest clay of the Rieker polder at Sloten lies as a thin bank immediately under the superficial layer of fen and contains 30°/, real clay. From the 7 M. “clay” in the Rieker polder, one has to subtract first a couple of metres of sand, the rest is also much richer in sand than the fat upper layer mentioned. Now SPRING has proved that a layer of Hesbay’s loam of 7 M. thickness admitted in 24 hours a movement of water of at least 0.036 to 0.045 M.") which is ten to fifteen times more than the velocity calculated for the Rieker polder. A sample of loam, kindly sent me by that scientist, proved, on analysis by Dr. N. ScHoorr, to contain 21.5°/, clay i.e. about as much as our ordinary, pretty fat alluvial clay contains on an average. Experiments with fatter clay under pressure, as il is in nature, give me a much smaller velocity which however is still sufficient to explain the hydrological condition of the Rieker polder. Of these experiments I intend to give an account on a future occasion. I wish to draw attention to a result of the experiments of SPRING already mentioned in my former communication, according to which, when the thickness of a layer of sand becomes very great with respect to the pressure-column of the water, the rate of filtration may by no means be taken inversely proportional to the thickness of the filter. On the contrary, SPrING found in this case the rate independent of the thickness of the filter. L can confirm this for clay and for this substance the pressure may even be relatively great and the thickness of the laver hundreds of times smaller on account of the so much greater resistance of clay than of sand. A layer of the fattest clay, obtained in the same way as the com- pressed fen, by squeezing out the water, having a thickness of 1) In this time namely a layer of water of 12 to 15 mm. thickness was trans- mitted. ( 62 ) 15 em, under a pressure of 80 cm. transmitted no less water than a layer of the same compressed clay of only 4 em. thickness. Cal- culations like those on page 294 of Mr. pr Bruyy’s communication, in which the rate of filtration through 9000 M. sand from the dunes is simply assumed to be */,.., Of the rate found in an experiment with 1 M. of the same sand lead consequently to erroneous con- clusions. I also want to point out that in the coarse diluvial sand which forms the principal way for the horizontal movements of the water, the velocity of motion is about ten times as great as in sand from the dunes. Under these circumstances I believe to be justified in maintaining my opinion that the water in the underground of some shallow polders is of autochthonous origin. It is also clear now, how in many places in the Haarlemmermeer polder water can spring up which is as fresh as water from the dunes. So in the farm “het Botervat’ on the Y road near the Kruis- wee; in it L found as well in the driest periods as after much rain a quantity of chlorine of 35 to 37 mg. per litre, whereas in the same farm a well has been bored reaching just below the deep fen, in which the water contains an amount of chlorine of 235 mg. per litre. At numerous other spots of the Haarlemmermeer polder the presence of fresh-water in wells (which proved to be no rainwater, as I believed for some time) could be stated; it is also found at about one kilometre east of the just mentioned farm, besides on the Kruisweg between the Sloten road and the Sloter Tocht, on the Sloten road near the Slaperdijk. On the other hand the water in wells in the north-western part of that polder, in which the deep layer of fen is entirely absent, is brackish everywhere. The water flowing under the compact alluvial cover from the higher environs of the Haarlemmermeer polder has there, as I showed in my former communication, a tendency to rise and so the salt- retaining property of the fen can here act in an opposite direction as in the shallower polders in which the vertical component of the water is directed downward. As the old fen forms, as it were, a filter for sodium chloride, so in the shallower polders the “old sea-clay” by its high percentage of iron, keeps the water in the underground relatively free from sulphuric acid. The superficial fen in the Rieker polder contains so much compounds of sulphur that it has a very strong smell of sulphuretted hydrogen, when freshly dug. Water squeezed out from it proved on analysis by Dr. ScHoorL to contain no less than 408 mg. ( 63 ) SO, per litre, while that from a well, 44 M. deep, near the place where the fen had been taken, contained only 17 mg. SO, per litre. Already immediately below the clay, at a depth of ten metres below the meadow, the amount of SO, has become so small. In the iron- containing layer of clay, pyrites is namely formed by the well-known minerogenetie process, with previous reduction of the SO, compounds in the fen, which reduction takes place here with the help of sulphur- bacteria by which the freshly denuded fen is coloured yellowish. Pyrites can indeed be shown to occur in the clay. And so this difference in the amount of SO, between the upper water and the deep groundwater is a proof for the origin of the latter from above instead of against it, as has been supposed. Chemistry. — Prof. C. A. LoBry pr Bruyn also in the name of Dr. S. Tymstra Bz. read a paper: “The mechanism of the salicylacid synthese.” (This paper will not be published in these Proceedings). Chemistry. — Prof. C. A. Losry pr Bruyn presents a paper of Dr. J. J. BrLANKSMA: “On the intramolecular ovydation of a SH-group bound to benzol by an orthostanding NO,-group.” (This paper will not be published in these Proceedings). Chemistry. — Prof. C. A. Losry pr Bruyn presents a paper of J. M. M. Dormaar: “The inversion of carvon and eucarvon in carvacrol and its velocity.” (This paper will not be published in these Proceedings). (June 24, 1904). KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN TE AMSTERDAM. PROCEEDINGS OF THE MEETING of Saturday June 25, 1904. DOCG (Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige Afdeeling van Zaterdag 25 Juni 1904, Dl. XIII). OAT DEN ES: Epmunp LANDAU: “Remarks on the communication of Mr. Krurver: “Series derived from the series Tj (Communicated by Prof. J. C. Kruxver), p. 66. F. M. Jarcer: “On Benzylphtalimide and Benzylphtal-isoimide.” (Communicated by Prof. A. P. N. FRANCHIMONT), p. 77. H. P. Kuyprr: “On the development of the perithecium of Monascus purpureus Went and Monascus Barkeri Dang.” (Communicated by Prof. F. A. F. C. Wenr), p. 83. A. J. P. van DEN BROEK: “On the genital cords of Phalangista vulpina.” (Communicated by Prof. L. Bork), p. 87. P. P. C. Hork: “An interesting case of reversion”, p. 90. (With one plate). B. M. van DarrFsEN: “On the function 5 for multiple mixtures.” (Communicated by Prof. J. D. vaN DER WAALS), p. 94. Frep. Scuun: “On an expression for the genus of an algebraic plane curve with higher singularities.” (Communicated by Prof. D. J. KorreweG), p. 107. Frep. Scuun: “On the curves of a pencil touching a algebraic plane curve with higher singularities.” (Communicated by Prof. D. J. KorreweG), p. 112. C. Easton: “On the apparent distribution of the nebulae.” (Communicated by Prof. H. G VAN DE SANDE BAKHUYZEN), p. 117. C. Easton: “The nebulae considered in relation to the galactic system.” (Communicated by Prof. H. G. van DE SANDE BAKHUYZEN), p. 125. W. H. Juuius: “Dispersion bands in absorption spectra”, p. 134. W. H. Junius: “Spectroheliographic results explained by anomalous dispersion”, p. 140. H. ZWAARDEMAKER: “On artificial and natural nerve-stimulation and the quantity of energy involved”, p. 147. The following papers were read: Proceedings Royal Acad. Amsterdam. Vol. VII. ( 66 ) Mathematics. — Remarks on the paper of Mr. KrurYver on ~~ ri T oe CY ® 5 fe ad ad (m)” page 305 of Vol VI. “Series derwed from the series + —— m by Epmunp Lanpav in Berlin. (Communicated in the meeting of May 28, 1904). In a paper recently published*) Mr. Kruyver treats the infinite series ~ u(mb+h) ul), wb) w(2b6-+h) De moth - Jh b+h 2b+h any a m0 where 6 and h are two positive integers and where / can be regarded as 1, aud in Casey = 2,..., (0), for X (9) > 0. The equation holding good for ® (s) >> 1 and every v(—1,..., ¢(d) ) aad y(n) | 1 nen eh ee TL ps Nl where p passes through all prime numbers, shows that no JZ, (s\ possesses a zero with real part >> 1. The equation (3) gives for p = 1 L.0= |] mal (=) =f} (Sra al p OE plb p/b where p passes through all prime factors of . From (4) it follows that ZL, (s) may be continued across the right line X(s)—=1 and that it possesses in s = 1 a pole, so that 1 lim Ses ois at eta ene Aire, tava ot Meee Loman sl L,(s) Further Dirronrer *) has expressed the quantities Go x %e(%) : Xorb)(n) ies) ae Mt AN Ly) = > Nn fil Ll in finite form by logarithms and trigonometrie funetions and proved moreover — what did not at all ensue from it — that each of the afore mentioned p(b)— 1 quantities is different from zero. So the limits I 1 1 1 lim ——- =——_.,,..., lim —__- > —— . .. .. (6) s=1 £,(s) £,(1) s=1 Lys) (8) Leo) (1) do exist. 1) “Proof of the theorem that every unlimited arithmetical progression of which the first term and difference are integers without common factor, contains an infinite number of prime numbers,” Transactions of the Royal Prussian Academy of Sciences at Berlin, 1837, p. 45—71; Works, Vol. 1, 1889, p. 313—342. („Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthilt”, Abhandlungen der Königlich Preussischen Akademie der Wissenschaften zu Berlin, 1837, S. 45—71; Werke, Bd. 1, 1889, S. 313—342). 5% ( 68 ) Mrs. HapAMARD and pr LA VALLKE Poussin have proved that no i Ly, (s) regular for every » on the right line X(s)=1. In the quoted paper I have proved’) the more general theorem: There is a positive number a so that, when s=o- ti, in the region 1 (3 en — bgt each of the g(b) functions JZ, (s) differs from zero and fulfills the inequality L, (s) possesses on the right line ® (s)=1 a zero, so that is i 5 B < log*t. $ 2. Now I denote by Jp, (s) that analytic function which is determined by the DrricH.eT series nt u(mb Jh) De (mb+-h)s’ m=0 convergent at least for o—®(s)>1 and I shall show that J,;,(s) can be brought in a very simple connexion with the functions ZL, (s). Let the greatest common ‘factor (b,/) of b and A be put equal to d. Without limiting the generality d can be regarded as being with- out quadratic factor; for in the other case u(mb+h)=0, so every member of the infinite series (1) is equal to zero. I. Let then be d=1, so / prime to}. Then there is an integer h,, (determinate modulo 6) for which h, h = 1 (mod. 6). Now ensues from (3) for c= X(s) >1 Lora w(P)\_ % Ou) zo LIG En en ar p ns If we multiply (7) by y,(h,) and sum up with respect to all values of » we get 9(d) 9(0) n—l #(b) ll) EE AOTONEN 1) —_ 8 ] == yr —— (A Pree 8 Ty Ty OD DT nr 2 il sl il Jl Wel Now according to the fundamental property of the characters the 9(0) sum > y,(/) differs from zero only, and then is equal to p (), when y= 1=1 (mod. 5); hence 1) Lc, page 521. Here I put the greater of the two numbers cy, andcs; =a. ( 69 ) #(d) = (6), f hk, n=1, 1. 6. n=h (mod. b) pz % (h, | = 0, ifh, n==1, i.e. n HE h (mod. b) So (8) aoe. into 9(0) ylh,) u(mb+h) _ ae L an AL ) 5) hee (mb + h)s ae p(b) Min (s) ; vl m—0 when with the aid of lh) Hoh) = ll) = wl) = 1 we eliminate h,, we get ¢(b) i. Mss top SECTOR @ II. Let d be >1 and let 5 be put equal to dB, h to dH, so that B and H are prime to each other. Evidently = u(d) u(mB + H ul +1) =n (dmb + H)) cae inki according to bora being prime to d or not. Hence SUE ZEN (10) (mb + h) Oy EN es ae aa m—0 m=0 where the sign =' denotes that m assumes only those values, for which mB-+ H is prime to d. If m,=m, (mod. d), then it is evident that m,B-+ H and m,B-+ H are simultaneously prime to d or not. So those m distribute themselves in certain arithmetical progressions modulo d; i.e. among the d progressions m— 0, 1,. d—1 (mod. d) m has in certain progressions, let the number Hed a to pass through all numbers > 0. This @ is the number of those among the d numbers in mB+ H, (m=0,1,...,d—1), which are (B‚d)p(d) p(B‚d) responding values of m are denoted by m,,...,3,..., mp and m, B + H is put equal to hj, (A=1,...,0), then every / is prime to d and — on account of (B, H)=1 — to B, so to b=db and situated between O (excl.) and 5 (excl.) (as OSHSh=m B+HCd-1l)B+H=)b-B+HLb aud 0 itself is not prime to 6). The corresponding values of mB + + Hare (m+ Md@)B+H=lb+h, prime to d; this number is known to be When the cor- (70 ) where / assumes all integer values >0O. In other words on the rightside of (10), when it is written short u(d) sk) Ot ho” k=1 k assumes all positive numbers belonging to certain @ progressions of the form 70 -- 4), Q@—=—1,2;.2.,@, where OS << Daud (opdr |A Mon POD SEN Ie eben Sa OER | and from the result (9) of the case (I) follows after application to the single members on the right side of (11) ep 9) 1 ud) yo Si 1 g(b) dS dust emt f,(h;)L,(8) A=1/ v=!) Min (s) = (12) Of (12) the equation (9) is a special case, as for d= 1 Boab MEER, ol hts In the following the equation (12) may always be taken as basis. $ 3. The equation (12) proved above for 6 > 1 furnishes in con- nection with the properties of the functions Z,(s) quoted in $ 1 firstly the analytic continuation of J/),(s) across the right line. o6=1. It teaches us that all points of the right line, s = 1 included, are regular places. From the theorem oti at the end of § 1 follows more M, a(S ;) accurately that for t 8 and ; en log is regular and satisfies A the RAE En de Ss 5 ee Se Rl p(b) loge t = @ log? t. Jl) =I | Mi, h Let now a number a >a be chosen in such a way that in the first place for any 43 we have 9 log*t loge t EERE and in the second place pease smaller than the distance of the a right line o—=1 from any singular point of J/,;(s), the imaginary part of which lies between — 84 and 37. If moreover the equation Min (6 + t)| = Min (6 — ti)| is paid attention to, then ensues from the preceding : if (it The function Mij (s) is regular in that part of the plane lying on the right side of the continuous curve (inel. the curve itself) 1 loge = 1 TiN ee for = Saar 5; loge = _-= oa : f t< 3 = TO 23s bg) E and for t> 3,1 aa Ao <2 it is == og® ns — | Mon (o+ t)| = |Mo, (6 — Ey t 23) og tt ne nn 1) Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften in Wien, math. naturw. Klasse, Bd. 112, Abt. 2a, 1903, 5. 537—570. Ri and EF the are of the curve 1 — | — t ; log(—t) g(—t) ie Ee Etn As according to $ 3 in this closed region and on the boundary the function to be integrated is regular, the Cavcny theorem furnishes for the integral appearing in (14) 24-277 Lee a Now I have proved in § 10 of my paper on the arithmetic oS progression for a certain function A‘(s) that, when integrating — K(s) s along the same given path, we have Cc —V log x +f4+ 7+ 74+ ] =0| ze ) AF FE ED DC CB where c denotes a positive constant *). Concerning that function A(s) I have made use |. e. only of the fact ie: it is regular on the path of integration and satisfies for fn 3,1 — ne os 2 the inequality = |K (o + ti)| = |K (o— ti)| log*t. As now according to (13) the function JZ), (s) has exactly these properties, we have for the present case the expression (15) Cc —V log r == (« ) This gives after substitution in (14) x DN u(h) log = oa) ECN PD l==l § 6. Just as in $ 4 of my paper on the function u(t) it could be concluded from il x 2 — log x Su (blog 7 = 0 ( ok ) k=l that 1) May be c= 3a. (73) x 12 —V log «x s wey == O( he ) lll we find here out of (16) « 7 f —PYlogx Aland J ll where y denotes a certain positive constant’). Thus for every n we find & x log” « ; lim ND iPOD ER rae k=l § 7. In § 5 Le. we found out of zx log” « im Ar lim Du) ee that the infinite series @ u (£) log” k pales ll converges for every pair of real values n, ¢; in the same way we find out of (17): the infinite series Blog kg u (mb + hi) logt (mb + h) me fli i (mb + hye — il m=0 converges for every pair of real values 7, ¢. Here is proved in particular for n=0O, t=O the convergence supposed by Mr. Kruvver of the infinite series ~ (mb + h) pi mbh ° m=0 i.e. the existence of the quantity designated by him as 7%,,. As now its existence is proved, it is easy to express its value in finite oo u form. As is known it follows from the convergence of s — that Nn ek 1) May be y=c+1. ( 74 ) Dm ; : : . Un : : approaching from the right side lim he — exists and is equal to gl ns n=1 > =. Making use of (12), (5) and (6) we find therefore n n= pe (mb 4 + h) u (mb — h) h) Sh Ab Memon sl = lim M, bh DE pk > im My» (s) = “mbh (mb + hs + h)s ee j (a) p = 0) a me) ~ yb) d PDE ke ae (1) j=l v=2 § 8. In this paragraph must be proved the lemma: When Q(x) denotes the number of the integers without quadratic factors 1, then 5 being equal to dB and A to dH the numbers mb + h 0...) aah Se OR. LAB LI >) $ 9. We now denote by Ry, (x) resp. by Sy, (x) the number of integers mb -+h< ax containing an even, resp. an odd number of distinct prime factors and we put Pra (7) =H" u), where =' denotes that & passes through every number mb-+-h 0 (B) AS Ov Wied. eee Min < 0. We have just now made use of the last (n-—1) equations of system (V) to calculate the a’s. As we might have chosen (n—1) other equations it is evident that for the existence of a stationary point M,, must always be either >O or <0. The set of inequalities given above is however sufficient to judge the possibility of the mixture. At most one of these n mixtures can be realized. Suppose that two different possible mixtures were to be found GBD ip drent al dn) gal enn Eeen Gta a } A for which ;, were stationary. Now as in consequence of the set of ) equations (V) equal roots 2 lead to equal constitutions, different values 4 belong to different mixtures. If we call the roots 2 belonging to the above mentioned mixtures 2, and 2, we arrive at the following sets of equations 0b Lele ij Ow, wae | ( 0a ‘ db Ee da Em Cok 4 Ee 0 and V. i BP Ams 0 ; Our, fa : Ge ae - fal j ts | da DN, da, (20) _ \ Gals (a) bs sf 5 te is Multiplying the equations V, resp. by (7,),; (@,)s, …- (&), and those of the system V, resp. by (—a,),, (—a,),,.-.(—#n), and summing these up we find in connection with the identity holding good for homogeneous quadratic functions p=n p=n > Ee dj a(S): B 0b (4,—A,) e NE 7 ) = (a as (Ge) oe Se ello) | 223) or, as we supposed 4, == 4, sn \ (ws), [Os (zj) + bso (w,), zet bsn (wi). | = 0. 74 sl All b,’s however being positive and all z’s also for possible mix- tures, the above mentioned equation cannot be satisfied. 4 a So there is at most but one mixture possible for which … becomes ( 97 ) stationary ; the 4 of that mixture satisfies either the inequalities (A) or the inequalities (B). Before passing to the investigation of the nature of the stationary point we shall first prove the theorem pn gn a = > Be es (— by, Ma) — rah vl This theorem can be easily verified for n= 2; it holds still good when 5 „gE = bp, Le. for asymmetric determinants. The Ene proof is supplied by showing that if the theorem is correct for determinants of (#—1) rows and columns (also asym- metric ones), it also holds good for determinants of 7 rows and columns. Now we have An = (a, —2b,) M,, + (a,,.—40,,) My, +…+ (Gin Abin) Min. Let us now make use’ of the following notation : Ang is the determinant derived from A, by omitting the pt row and the g column. ase is the determinant derived from A, by omitting the rows rs with numbers yp and # and the columns with numbers q and We now find BA, — af u : Mab Mi — bin Min + SS SE (41s A En er Further we find Ay 5 = (—1)s—! AGES Ais is an (asymmetric) determinant with (7—1) rows and columns, for which we have supposed the theorem to hold good, so that sn ee rn sak, zap — bi, Mi, + ye (SUL (aj, — 1015) X read p=n q=1,2,..(s).. x a vi boj) IP 2 (17? Ais, ) =d i i ne. where the positive sign must be used for g >s and the negative one for q<-s. Performing the summation according to s first, we find 7 By placing s between brackets we indicate that for the summation all values from 1 to » except s must be assigned to q. 7 Proceedings Royal Acad. Amsterdam. Vol. VII. (98 ) 0A,, / || pn qn Me Dy tr Me + YY boo) Ag Dt (De ji p=2 q=l or ò A, pen eee B= SS tty, p=l q=1 . . a . For a possible mixture for which zi becomes stationary all quan- ) tities JM, have the same sign as is proved. It is now evident that 4 . : e 7 ( 1 5 for such points, whatever the sign of J/,,, may be, en Mn is always 02 negative. With the help of this theorem we can investigate the con- ditions to distinguish an absolute maximum or minimum. Let us now write Ff — a — db. This expression fF regarded as function of 7,,.7,...«, and 2 is zero for every given set of values «,,.7,,...a,, if we assign to 4 the a ; ; ‘ fy : value of ; belonging to that constitution. If inversely we start from a given 4(=4,) then the solutions of the equation f — a—i4,b=0, regarded as an equation in »,,,,...#, furnish all the mixtures for a which „possesses the given value 4,. If moreover that value 2, is ) an absolute maximum or minimum, then only a single set of possible values ,, #,,... £, may satisfy that equation. As fF is a homogeneous quadratic expression in the 7 ws we can write it down as the sum of 7 squares. Let us again call 4, the determinant forming the first member of the equation in 2 of order 1, 4,1 the determinant derived from the former by the omission of the „tk row and the z® column, 4,2 the determinant obtained by the omission of the last two rows and columns, so that finally 4, —— «a, — d,. ‘The transformation into a sum of squares is brought about in such a manner that the first square contains all the terms with .r,, the second all the remaining terms with w,, etc, until finally «, is our last square. In order to evade surds we must every time multiply our function by definite coefficients. It is now evident that by executing the development, if we represent the successive linear homogeneous expressions to be squared by L,, Li, Jin, we have: (99 ) ee Arad SpA DANE = A) Ar) As Ama Alp hy bY 1 Bers en ae aA, Al! LANA, AAE Aig pln AQ k?. + Steer Mere Ap A, Atran rbi. Ly Taal sD Aa GL dh. Here is pn ES (aip — Abip) ep”) p= p=n | a, —Ab) A1p—Abip iN ; ij En aar —Abo, aap — Abs, (a, —Ab, a,,—Ab,,-.-@1,s—i —Ab), s—1 Ap —Abip | gel (Car Ady, Go Ay: a dos ADA sl Aap —Aboy L=\N 5 , Une ps ds -1,1 —)b. 1, IE Baere ds—1,s—1 —Abs—_1, 5—i As—1, p—Abs_1, p as, 1 —46.1 sa Site se ds, s—l — bs, it Asp —Absy | £;, = An Une So we find Le dae Jk ie, + 1 2 =p 2 3 ES -{- -+...——~— 1... —— 5), A, As Ayp4) Ann For a stationary point 4, becomes equal to 0, and so the last . e e LG 7 2 a 3 4 term disappears as it is Berne). n—l 1) For the deduction a continual use has been made of the theorem : LENA AAA pq pars _psqr rs Weger : Lehrbuch der Algebra I, 2ed edition, p. 115. 2) In connection with the following it is easy to find back out of this the system (V). Comp. for a ternary system: van DER Waats, Proceedings Royal Acad. of Sc. 1902, Vol. 5. p. 235. 8) Without looking more closely at the breaking up into squares, as is performed above, Rourn shows in his Rigid Dynamics, Advanced Part, 5th ed., p. 426 with the help of invariants that the coefficient of Lp has the sign of which j—1 agrees to the above. 4) There is a difficulty however when for A= Ay also An—1 becomes 0, so when a stationary point in the mixture (1, 2,....—1) becomes at the same time a statio- nary point for the total mixture. When breaking up into squares we then shall change the order of succession in order to avoid .,? presenting itself as last term. If all first minors belonging to elements out of the principal diagonal An are O then all minors must be 0, if there are to be stationary points. The equations (V) have then a higher degree of dependence and there are an infinite number of b 9 : a : mixtures for which 7 becomes stationary. 7% ( 100 ) Suppose the stationary point to be an absolute maximum or mini- mum. Let 4, be the corresponding value of - = . Then for =A, {here ) is only one constitution; thus for we, #,,... & only one set of values may be found. Now this is only possible when all the coefficients of Li, Lie -- Oya have the same sign for à— 4,. For, if this were not the case then it would be possible for 2 — A, (for which the last coefficient has already disappeared) to satisfy the equation =— 0 without the necessity of L,, /,,.--n—1 being individually zero and then many eer RS sets of adjacent values w,,,...#} might be found for which ) had that absolutely maximal or minimal supposed value; which is absurd. For a stationary point to be an absolute minimum or maximum it is therefore required A, 0 A0 As) A= AND TC Me \4 20 |A, A,<0 or Vn SO ere 7 4 0 JAA Ie pO \A,—9A re Be | VA 40 Benne qa 1 Aris Bet“, “be a ‘roots of Ar U indicating an attainable absolute maximum or minimum, then for 2= 4, the coefficient of the last square (w,”) in the development of # becomes zero. For 4= 4, He dA, , oii the sign of 4r—-1 an determines the sign of that coefficient, whilst 0A, Oh” Now however, as we just before indicated, we find, for a possible stationary point for 4 = 4, —« the sign is determined) py — Bee, dA, 0A, en — Man zo 0. er) en A < So it is evident that for 2=2,-+« the last term is always negative and for 4= A, —e« always positive. From this ensues that in the case of an absolute minimum the inequalities (7) must be satisfied, whilst for an absolute maximum the inequalities (7) must be fulfilled. In the first case the conditions of possibility (A) are still to be added, in the second case the conditions (B). It is clear that by a different numbering of the components other inequalities would have been obtained — evidently however the system (7) or (7) is sufficient to indicate an absolute minimum or maximum. | ( 101 ) Suppose an attainable minimum presents itself in the mixture of some components say 1, 2,...>p. Within the limits of possibility for that mixture a second set of : 04 values of the es for which 4, = 0 cannot be found. As farther ore is negative for that minimum we can draw the following conclusions from the system of inequalities ( 7’): An attainable absolute minimum lies lower than for the com- ) ponents and lower than eventually appearing minima in any mixture to be compounded of the given components. If the original mixture has a maximum and if there is also a maximum in the p-fold mixture (1,2,..p), then for the maximum 0A 8 E 5 p ; in the p-fold mixture 4, is equal to 0 and — A, 1 negative, so / 02 P o 5) ~ 0A a has there the sign of — 4,1. Now (—1p-2A,_, is <0, so US ae i cad) A,—1 has the sign of (—1)P—!. So sal has the sign of (—1). Let 4) represent that maximum, then as 4, becomes but once 0 for possible mixtures 4, is furnished for every value of 4 > 4,, with the sign (—1)r, but for every value of 2< 4, with the sign (—1)p-!. For the maximum in the n-fold mixture (—1)—! A, is <0, and so for A, the sign is indicated by (—1)r. From this ensues that the set of inequalities (7’,) can be expressed as follows: An attainable absolute maximum lies higher than the : for the components and also higher than eventual maxima in mixtures to be formed of the given components. The question now arises whether a maximum or minimum in the n-fold mixture implies anything about maxima or minima of the binary mixtures to be formed of the 7-fold mixture. Suppose the n-fold mixture to show a minimum for 2= 2%, and the constitution of that mixture to be indicated by [uus (@,)m(@n)m]s then we find (B Amb, )(i)m + (a, — Am Dia) (@)m + «+ (ain — An Dn) (@n)m = 0 (a5 An bi) (wm i (a, a An b, )(@)n “he + (an — dm Dan) (@1)m ==) \ (ap, — An Oni) (ij) + (Ana — Am Ong) (&2)m + + (an — am On )(@n)m = 0. Now we know that for a possible absolute minimum a,—dA,, b,>0 whilst of course also (ws)m > 0. ( 102 ) If the above equations are to be satisfied then in every equation at least one of the coefficients must become negative. This is most profitably attained for m= 2 if ra Amb, <0, Ans Amb, <0, and Aar, — Am bot—1, 2 a 0. So then we have e.g. ads — An bs De 0 As, st1 — Am bs, s+1 << 0 Qs] — An bs41 ie 0 sO ds, Ss eS hyp pul ean <> and 4m <5 i ‘ bs, sl DsH1 so that a fortiori as, s+1 wen a s,s+1 Zer Ds, st bs Ds, s+1 bs41 The mixture [s,s + 1] then possesses a minimum. So at least minima are wanted in ¢ binary mixtures if the whole mixture of 2¢ components is to show a minimum. If n=2¢-+1 then there are at least ¢-+1 or $(2-+1) minima wanted in the binary mixtures if the total mixture is to show a minimum. Let us take e.g. G1, — Am ba ll 0 As, — Am by, ZO «oo Qn—2,n—1 — Am On—2, n—1 <0 and dint — Am Oni: For » even the case is taken that each component shows with but one other one a minimum, whilst for 2 odd one of the compo- nents gives a minimum with two other ones. If a component forms with more other ones minima then more conditions are absolutely necessary; if a.o. we assume that » — 1 of the components give mutually no minima, then certainly the last component must give a minimum with each of the (7 — 1) remaining ones, if the total mixture is to show an absolute minimum. Of course the above theorems may not be reversed; so at least $n (resp. }(2 + 1)) minima are wanted, but these do not in the least guaranty the existence of an absolute minimum in the w-fold mixture. In case of an attainable absolute maximum a,— Aub; is <0 and so in our set of equations at least one positive term is required to present itself in each member. From ds — Amb: <0, dye — À bys > 0 and a,.— Ayb,< 0 follows Ars E ds ay => dan whilst ae oe and IMS so a fortiori Ars ds Ars ay —>— and —>-, Drs s Drs b, so that the binary mixture (7, s) then gives a maximum. In the same way as for a minimum we reason here, that to have an absolute maximum in a mixture of 2¢ (resp. 2¢-+ 1) components we must require at least maxima in ¢ (resp. ¢ + 1)-binary mixtures. For a further investigation we shall have to look more closely into the quantities b,, and a. For 5 yg We shall use the formula given by Lorentz Op = 2. ts X (rj sid ro)’ gE where 7, represents the radius of the molecule of the ptt component. This formula holds good for p= q too. The coefficient of 2" in the equation A, =O is Pave Oey aes Ora | (— 1)" |B, 6, --. bo, | =(— 1)" Ay. 2 Gat bna--- by | Now we have Ara = — ti, x? (rr)? [7,6 7,5 +30 737 vr, HBr 7,48 rr] Aims = (U) EE Ori) (rar) (ror) [es ate? + 47, (rohr)t ers (0, ta 7 A= (DD (rr) (rr)! a)? a)? (rar) For five components the eave on the 4’s vanishes identically ; for we find A5 &(5) = Ao) A) ze, 6) = 44 a =(' 2) 9(0, me tai Ohe aken Ty ose x (*/,90)* 9 ne nk (7 ore Parti U mt Ad Ui apes lor Jes =p ass Per)? (o-r) (rr) (rr) 1's) (rar) 3) Par) (37; Ste == So Re as RD is not identical equal to 0. 44 55 For the determinant of the order 6 not only all minors belonging 1) Wied. Ann. 12 p. 134. In reality still another constant factor MN presents itself here. 2) These results have been obtained by remarking that Ay, is always divisible by (rp — 79°; the coefficients of the remaining factor have been found by means of the method of indeterminate coefficients. ( 104 ) to elements of the principal diagonal become O but also all other minors of degree 5 *). Then however it is clear that for mixtures with 6 and more components the determinants on the 6’s disappear identically. Bertuetot and before him Gatritzinn have made about the quantity A), the simple supposition @*,, = 4 4. Although this formula may not be strictly true, as has already been clearly proved by experiments on binary mixtures, it is worth while to consider to what conclu- sions the afore mentioned rule leads us for multiple mixtures. For aj = Vay ag all minors of degree 2 and higher of the deter- minant on the as become 0, from which ensues that the equation An = 0 can be reduced to p=n g=n = x a : Ar N . Bag Ang — A” arj Lamm) po! gl where when developing the determinant 4, the coefficient of 6,, is Bg. So there is an (n—1) fold root 20, which cannot of course indicate an attainable maximum or minimum, not even another stationary point. Let us now consider the different mixtures assuming the rule of GALITZINE-BERTHELOT. n=. 3eside the root 20 a second root appears which can certainly a a a a . . . . 12 1 12 2 not point to a maximum, for from, Bl and ; oe would ensue ) ) ) ee) 1 12 2 * and in connection with 6°,, > hb, 6,, certainly 1) So we find for the asymmetric determinant Dy Dia Dig Ou Os Da Oa Daz Dag Das D = | ba Day Ds Das D35 | On Daz Daz ba Oss | | ber boa Dos bos Des DD =(®/sm)t9(rj — 12) — rra (1 — 74) (13 — rr) 44 55 X(S/5 Or; — 2) — 73) (72 — 73) A — NON (V2 — 153 — 16)('3 — 15)03 — PoF = gm A(T} i) DN hd OA TT 13)(r5 ER ra) (7 Tr VY ER rs)\(V2 = ree) — V=)\(V2 == 1 )(V3 -— vz) (Sla — 72) — 73) (Ng — 73) (0) 2 — V2 — VEN 1'3 — (3 — TS zig ( 105 ) Wer > Gs de. The appearance of a minimum is nota priori excluded; the general conditions now pass into : a, a Bi 2B by and poe, DS i Os So it is not evident, why the appearance of a minimum should coincide with @?,, a, d, & minimum can very well present itself. In fact there are objections to assuming that «*, << a,a,, for this supposition leads when the temperature becomes lower to partial mixability *) and this phenomenon probably does not appear for normal substances *). The following however holds good for a possible minimum : Gis Ais d, 7.,0— and, a, >b,—~, so ee Oss 9) 9 dis a, se a, — 0. For ‘all temperatures excluding partial mixability, we find thus when a minimum appears Waas Saa Z's (a, + 4.) Bd. The equation 4, =0 has a double root 2==0, so we obtain the following series of signs : } | LE | } A3 | kol + Pte! + I a) | or + 0 | Il AN |e vk, a8 a ee 0 0 0 — a | == saris Fh a In case I the simple root 2 belongs to a negative value of a 5» 80 it cannot represent a possible mixture. ) 0A, * ae ; In table I aa is positive for the simple root; thus the minors of w degree 3 must be negative in case the mixture is to be possible, So it is evident, that as soon as we put «,j equal to a, «a, quaternary a mixtures cannot show a minimum in 3: A maximum is excluded, as ) this would lead to maxima in at least 2 binary mixtures to be formed out of the components and these are not possible if a,,* is put equal to a, a. . a . . . For mixtures where 1>4 a mixture for which zi stationary is ( 107 ) certainly excluded as soon as we put a, ? = dp dj, for there is then an (n—1) fold root 4==0 and a root A= Resuming : By assuming the rule of Ganitzinn-BrrTHELOT, we find: OO. n=2. No maximum; a minimum is possible. n=. No maximum or minimum — a stationary point, but no maximum or minimum, is possible. n=4. No maximum or minimum; other stationary points are possible. 1 and higher. All stationary points are excluded. « Or If we assume for 6 a linear function of w, thus b,,=*'/,(b,+ 4,4), then already for „== 3 the determinant on the 6’s becomes identical to O, so that then for ternary and higher mixtures no stationary points are any more possible as soon as we put a pg equal to a, ag. Mathematics. — “On an expression for the genus of an algebraic plane curve with higher singularities.’ By Mr. Prep. Scaun. (Communicated by Prof. D. J. Korrnwne.) Lately I gave the following theorem in these Proceedings’) : Let P be a point of order t/ of an algebraic plane curve (where t’ can also be zero, namely when P does not le on the curve) and S an arbitrary point of order tof that curve. Suppose the straight line PS intersects the curve in w points comciding with S, then U! HE (w,—t) (summed up over all points S for which w is >t) is independent of the situation of point P and equal to the class of the curve. If S les in P we have to regard all straight lines through P as the line connecting P and S or if one likes only those which furnish a contribution to (w,—t,) de. the tangents m P. From this a corresponding and as far as I can see a more important theorem for the genus of an algebraic curve can be deduced, where moreover the straight line connecting Pand S can be replaced by an algebraic curve. Lateron I hope to connect this with problems of contact (numbers of algebraic curves determined by conditions of contact) in particular with respect to the number of normals on a curve with higher singularities (also in connection with the circle points and the line at infinity) let down from a point (which can also have a particular situation with respect to the 1) On an expression for the class of an algebraic plane curve with higher singularities. These Proc. VII, p. 42. ( 108 ) eurve) and the question annex to it after the singularities of the evolute. It seems to me that in no other way known to me these and suchlike questions can be so simply answered. The genus has been introduced in the theory of funetions by RrEMANN and is defined out of the connection of the 7'-leaved RreMANN- surface on which an #'-valued algebraic function is univalent. If s is the number of branchpoints of the function, g the genus, then we have the following relation given by RipmMann hy (Ie; p. 223) s— 2n' = 2(g—1), for which a branchpoint where ¢ leaves of the RimMANnn-surface are connected is to be regarded as #—1 branchpoints. For the theory of the algebraic curves the notion and also the name of genus (“Geschlecht””) has been introduced by CLesscu*), whilst HALPHEN*) has given for the genus of a curve of order » and class & with higher singularities the equation 2 (g—1) = k—2 n+ DE (t,—1), in which (¢,—1) represents a summation over the origins of the separate branches of the curve (which can be represented by one Purseux-development) whose order ¢ differs from 1 and over as many other origins of branches as one likes. If a branch of the curve is represented by the development i+v y—n=a,(e@— 8) Hale) +, 1 according to integral ascending powers of (ge). I call the point (§,7) the origin, the line y—x7—=a, (x—§) the tangent and the numbers ¢ and » the order and the c/ass of the branch, where thus any point of a curve can be regarded as the origin of at least one branch, for which however if the point is a common point of the curve ¢ will be equal to 1. If one and the same point of the curve is the origin of more branches we shall regard this point successively as if belonging to the different branches. The Ha.pnen-relation is an immediate result of the Riemann- relation if only one decomposes the branchpoints into those which 1) B. Riemann. Theorie der Ager’schen Functionen. Crelle’s Journal, vol. 54, (1857), p. 115—155. 2) A. Cregscu. Ueber die Singularitäten algebraischer Curven. Crelle's Journal, vol. 64, (1865), p. 98—100. 3) G. H. Hatpnen. Sur la conservation du genre des courbes algébriques dans les transformations uniformes. Bulletin de la Soc. Math. de France, vol. 4, (1875), p. 29—41. ( 109 ) do and into those which do not depend upon the choice of the system of coordinates. KE a ERC Lose the classes of the separate branches having their origin in P (so that S/, =?) then jerken A. whilst according to the quoted theorem k= 2(¢, +) + = (wt). (equation (2) 1. ce. p. 44) In the two last equations the first S-sien refers to the origins in P, the second -sign to the origins outside P. . From these equations follows 2 (91) = — Ant) = (01) FS (wl) .... (1) Here n—?t'=n' represents the number of movable points of intersection of the curve with straight lines through P. If one draws through Pan arbitrary straight line / which is not a tangent in P, then the points of intersection of that straight line with the curve furnish to © (w,—1) a contribution equal to n— N;, where N; represents the number of origins of branches lying outside Pon the straight line /; '), thus the number of branches over which the 7 movable points of intersection with the straight line /; distribute themselves. If then we draw through P a straight line /; touching MN’; branches through P and if we let NV; be the number of branches over which the ' movable points of intersection with the straight line /; distribute themselves, then NN of these .V; branches have their origin outside P. The points of intersection outside P with these straight lines give a contribution to = (w,—1) equal to (2'— 2 v';) — (Nj—N';) = (nr — Nj) — & (v;'—1), in which Zw; and &(v';—1) are taken only over the branches touching the straight line 7; in P. From this ensues: = (w,—1) = J (n'— NM) +2 (rn —N;) — 2 (v',—1), are the orders, v or = (w,—l1) = DN (xn'— N,) — = (v',—1), in which now & (v',—1) denotes a summation over all the branches with P as origin, © (n'— N,) a summation over all straight lines through P for which #' is > N, and over as many other straight lines as one likes. If we substitute in equation (1) for SY (w,—1) the obtained value and replace n—? by n' we find Gal nn SS (n= Mie ee es (2) We can sum up what has been found in the following theorem: Trroren 1. Lf an algebraic plane curve is intersected by the straight h Counting here also each of the points of intersection as origin of a branch. ( 140 ) lines of a pencil with P as vertex in n° movable points, which distribute themselves for the various straight lines of the pencil over N,, N,....- branches of the curve, then 1—w + 3 = (n'—N,), (where = (n'—N,) is taken over all the straight lines through P), has for every point P the same value equal to the genus of the curve. This theorem however can be considerably extended by making use of the property that the genus of the curve does not change by a one-to-one transformation. If namely we apply to the whole figure a CREMONA-transformation, then y remains the same, but 7’ too. The straight lines of the pencil are turned by the transformation into rational curves having multiple points in the fundamental points of the transformation. Through every point lying outside those fundamental points only one of the rational curves passes, so that we treat a pencil of rational curves; the manifold points here making the curves to rational curves are not present as movable points but as fixed points, which gives rise to linear relations between the coefficients of the curves. The movable points of intersection with a straight line are now transformed into movable points of intersection with a rational curve, and they remain the same in number on account of the one- to-one character of the transformation. By a CREMONA-transformation a branch is furthermore always transformed into one single branch (where we always understand by a branch the whole of the curvepoints whose coordinates satisfy the same Pursevx-development). If thus the # movable points of intersection with a straight line distribute themselves over MN branches then in the transformed figure the # movable points of intersection with the rational curve originated by transformation of the straight line distribute themselves also over N branches. From this it is evident that all quantities of equation (2) are invariant with respect to rational transformation, so that the equation (2) holds good unchanged if the pencil of straight lines is replaced by a pencil of rational curves. This gives rise to the following theorem: Tunorem IL. Zf an algebraic plane curve ts intersected by a pencil of rational curves in n' movable points distributing themselves for the different curves of the pencil over N,, N,,.... branches of the fixed curve, then 1—n’ +42 (n'’ —N,), (where = (n' — N,) is taken over all curves of the pencil), has for every pencil of rational curves the same value equal to the genus of the fixed curve. This theorem can then be extended from a pencil of rational curves to an arbitrary algebraic pencil of curves by means of the following considerations which are however less rigorous than the preceding ones. ‘Sala If we investigate which rational curves of the theorem II contri- bute to En’ — N,) then we find 1st those rational curves which pass through an origin (lying outside the basepoints of the pencil) of a superlinear branch (denomination of Carrey for a branch with order more than one) of the fixed curve, 2°¢ those rational curves touching the fixed curve outside the basepoints, 3'¢ those rational curves where two or more of the movable points of intersection have approached one of the basepoints along the same branch of the fixed curve. In the main theorem Il comes to the determination of the number of rational curves of a pencil touching a given curve and the change which this number undergoes on account of higher singu- larities of the given curve and the particular situation of the basepoints with respect to that curve. Here however it is difficult to imagine how it would cause a difference whether we are working with a pencil of rational curves or with an arbitrary pencil of curves; for in both cases the coefficients of the equation of the movable curve satisfy some linear conditions amounting to one less than is necessary for the definition of the movable curve *). Let us explain the preceding by an example. Suppose the number of cubic curves through eight points touching a given curve were required, suppose further that the given curve has a singular point SS with a singular tangent / and that the C, of the pencil through S also touches /. Point S will then absorb a certain number of points of contact proper with curves of the pencil, and this number will depend on the nature of the singular point S and on the order of contact of C, with the given. curve, but in -no.wise on the fact whether of the basepoints three have coincided somewhere outside 5, either in such a way that in one of the basepoints tangent and curvature are given, or in such a way that the coinciding basepoints form a triangle with finite angles, in which case the condition of the passage through the three basepoints includes the curve having a double point in a given point and being thus rational. Led by the above considerations I think I may state the following theorem : TuroreM Ill. Lf an algebraic plane curve ts untersected by a pencil of curves in n’ movable points distributing themselves for the different curves of the pencil over NN... branches of the fived curve, then 1 —n’ a 4 > (n’ — N,), (= (n’ — N,) taken over all the curves 1) Of course it would be different if the movable curve had to be rational without the singular points reducing the genus to zero being given; if thus e.g. the question were of cubic curves through seven given points, and furnished with a double point not given in position. ( 113°): of the pencil), has for every pencil of curves the same value equal to the genus of the fixed curve. The remarkable thing here of the obtained expression for the genus is that the genus which is invariant with respect to rational transformations is really exclusively expressed in quantities each invariant in itself over against rational transformations. 1 feel the more justified in stating the above theorem having found the theorem confirmed in various simpler cases, e.g. for the case that the given curve admits of double points and cusps only whilst the basepoints can assume any particular position with respect to the given curve, also for the case that the given curve is provided with higher singularities where however only the simplest particular cases with reference to the position of the basepoints have been considered, e.g. the case that one or two basepoints fall in a higher singular point. But all the same a rigorous and simple proof which renders a subtle distinction of the great number of particular cases which can present themselves superfluous, is very desirable. Sneek, July 1904. Mathematics. — “On the curves of a pencil touching an algebraic g Y plane curve with higher singularities’. By Mr. Frep. Scnun. (Communicated by Prof. KoRTEWEG.) In the previous paper I have stated the following theorem : Tf an algebraic plane curve is cut by a pencil of curves in n movable points distributing themselves for the various curves of the pencil over N,, N,,... branches of the fived curve, then 1 —n'+42(n' —N,) (= (n' — N,) taken for all curves of the pencil) has for every pencil of curves the same value which is equal to the genus g of the fixed curve. Expressed in a formula this runs: (gen WE (Sa) ee, ep With the aid of this theorem the following problem of contact can be solved : To determine the number of curves of a pencil touching a plane curve C, of order n, class k and genus g. To this end we first substitute in equation (1) for > (n' —.N,) a summation over the points of C, or better (as we always count a point of C, through which more branches pass as more than one point) over the origins of branches of C,. Let S be ( 113 ) an origin of a branch of ©, whilst the curve of the pencil through S intersects the branch under consideration in w points S, then = (n' — N,) = = (w, — 1), so that equation (1) becomes BG a pa DS Boe Dee! SO Ad eN Here 2 (w,— 1) represents a summation over al/ origins S of branches of (C,, i. e. only over those origins for which w> 1. If one or more basepoints of the pencil lie on C, the summation must also be extended to those origins coinciding with a basepoint B. We must then regard as movable curve through that origin the limiting position of the movable curve through P if we allow P to approach to B along the corresponding branch of C,, in other words that curve of the pencil intersecting the branch at least in ond point B more than any curve of the pencil. For such an origin in B the number of movable points of intersection, approaching B along the branch under consideration when P approaches B along the same branch, is represented by w. The following well known relation Slee ae eb en Ste at a! ES) exists between order, class and genus of C,. Here 2 (f,—1) is a summation over all the origins of branches of GC, whilst ¢ represents the order of the branch, i. e. the number of points of intersection with an arbitrary straight line through that origin coinciding with that origin. From (2) and (3) then follows: Blan he ft (nis net, sas bx ohn eat RC) Turorem |. [f a pencil of curves cuts an algebraic plane curve C, of order n and class k in n’ movable points of which w fall in the origin S of a branch of order t of Ci, we have the relation = (w,—t) =k +2 (n'—n), where XY (w,—t,) must be taken over all the origins of branches of GC, also over those coinciding with basepoints of the pencil. With the equation (4) the given problem of contact for every Cy and every particular situation of the basepoints with respect to C, can, as will be shown, be regarded as solved. We have but to discuss the found equation. If m is the order of the curves of the pencil, then »' = ann for arbitrary situation of the basepoints with respect to the given curve ns SOR . en <=). == Dm (pad Viger ae gt la Peony Od Here we understand by an: arbitrary situation with respect to Cy in the first place that the basepoints do not lie on Cy. Let us 5 Proceedings Royal Acad. Amsterdam. Vol. VIL. ( 114 ) suppose moreover that the basepoints are situated in such a way that not a single curve C,, of the pencil passing through a singular point S of C, touches one of the branches through S and that not a single C,, has with the fixed curve a contact of higher order then the first; then only the curves (C,, showing a common contact with Cls t—=1), furnish to X (ww, —t,) a contribution equal to the number of those touching curves C,. Here however a restriction ought to be made. It may appear namely that a C;, of the pencil has a double or multiple point lying on (, which then furnishes a contribution to S(w,—t,). We can avoid this case by requiring that for an arbitrary situation of the basepoints with respect to C‚ no singular point of C may lie on C,. This is however no longer possible when by mutual coincidence of basepoints the pencil admits of curves showing an infinite number of double or multiple points, in other words when the pencil contains curves, which consist of or contain two or more coinci- ding parts. Though the equation (3) can still be applied to these cases we shall exclude them for simplicity’s sake from our discussion. With these suppositions we find that 2 (w,—t,) is equal to the number of curves of the pencil touching C,. So we find the following theorem: Tueorem Il. For a pencil of curves of order m, none of which contains two or more coinciding parts, the number of curves touching an algebraic plane curve Cy of class k with respect to which the base- points of the pencil have no particular situation is represented by k + 2n(m— 1). If C, is a general curve in point-coordinates then 4 —= n (7 — 1), and the required number is 7 (n + 2m — 3). If we compare this to the number given in the above theorem we find : Turorem UI Every singular point S of C, diminishes the number of curves which belong to the pencil mentioned in theorem IT and which properly touch Cy, by the same number as that with which S diminishes the class of Ch. We now investigate which particularities can present themselves in consequence of a particular situation of the basepoints with respect to C,. To this end we consider in the first place an origin S of branch 7’ of order ¢ of C, supposing „5 not to be situated in one of the basepoints; further we suppose that the curve C,, through S touches the branch 7’ and intersects it in w==t + y coinciding points (so in y points more than when the curve through S of the pencil were not to touch the branch). Then this point S counts (as far as the branch 7’ is concerned) according to (5) for w — f == y¥ points of contact proper. If we restore by a small displacement of the basepoints their arbi- trary position with respect to C,; the curve through S of the pencil (115 ) intersects the branch 7’ in ¢ points S and in y points lying near S. The preceding holds good invariably when the pencil does contain curves containing two or more coinciding parts if only those do not pass through the considered point S. This gives rise to the following theorem : TuroreM IV. Let S be the origin of a branch T of an algebraic plane curve Cy. If now the basepoints of a pencil of curves pass from an arbitrary position to a particular one so that no basepoint approaches SS whilst y points of intersection of C, with the curve through S of that pencil approach S along the branch T, then as many (so y) points of contact of Cy with curves of the pencil approach S along the same branch. Here has been supposed that if the pencil contains curves containing two or more coinciding parts these parts do not pass through S. If the basepoints have the particular position described in this theorem, then S counts for y points of contact proper. So we can formulate the theorem as follows: Tf S ús a point of Cy not comeiding with one of the basepoints of the pencil, whilst the curve through S of the pencil cuts a branch of order t of C, with S as origin in t+ y points S, then S absorbs as far as that branch is concerned y points of contact proper. Theorem IV is an extension of a theorem of HALPHEN and STEPHEN SmitH, which I discussed in a paper in a previous number of these Proceedings *). The indicated theorem can be expressed as follows: Let S be the origin of a branch T of a curve, 1 the tangent of that branch in S. Lf a point P approaches l but not S, then as many points of intersection with PS as points of contact of tangents through P approach the point S along the branch T. This is no other than our theorem IV where the pencil of curves is replaced by a pencil of straight lines. Let us further consider the case of a singular point S of C,, with which coincide one or more of the basepoints. As our only business is to determine the number of points of contact proper coinciding with S we can assume for simplicity’s sake that no basepoints coincide with other points of C. Further we exclude the case that the pencil contains curves admitting of coinciding parts. Let 7#,,7¢,,... be the orders of the different branches flh AOR of C, having S for origin, whilst an arbitrary curve of the pencil 1) On an expression for the class of an algebraic plane curve with higher singularities. These Proceedings VI, p. 42, 8* ( 116 ) cuts those branches successively in z’,,2',,... points S. Then we have n’ =mn — Ez. Further we can break up 2 (w,—d?,) into the share = (w,’ — t,’) of the point S and the share of the other points of C,. The meaning of w', is here, that the curve of the pencil cutting the branch 7", in more then 2, points S does this in 2',+7', points. For equation (4) we can then write = (w, —t,) =k 4+ 2n(m — 1) — VS (w, + 2 2’, —7¢),- - (6) where 2 (w‚—t,) must now be taken only for the curvepoints outside S. If we represent by w', the number vw’, + 2’, of the points S in which the branch 7", is eut by the osculating curve of the pencil, equation (6) becomes : = (w, —t,) =k + 2n (n — 1) — Zw, He, — ¢)). It is evident from this equation that the point S, as far as branch 7”, is concerned, absorbs u’, + 2, —f, points of contact proper. This can be formulated in the following theorem: Trroren V. /f a single or multiple basepoint of a pencil of curves coincides with the origin S of a branch of order t of an algebraic curve Cy, whilst that branch cuts an arbitrary curve of the pencil in z, the osculating curve of the pencil on the contrary in u points S, then the point S absorbs u + 2 — t points of contact of curves of the pencil with C,, in other words for an arbitrary displace- ment of the basepoints coinciding with S the point S furnishes ud z— t points of contact, which are then situated on the considered 1 branch. By allowing the basepoints to undergo not an arbitrary displace- ment but a particular one, another theorem in connection with theorem IV can still be deduced from this. We can namely make the basepoints change their places slightly along the osculating curve of the pencil in such a way, that no more basepoints coincide with S. In that case the point S continues to absorb after the displacement of the base- points a certain number of points of contact proper, and according to theorem IV to the amount of u — t; in fact after that displacement the curve of the pencil through |S intersects the branch in « points S, so that for the point S now w is equal to u. If we compare the number « — t of the absorbed points of contact to the amount given in the theorem V we find: Turorem VI. Jf the curves of a pencil cut the branch T of an algebraic plane curve in 2 fixed points coinciding with its origin 8, then point S gives to that branch z points of contact with curves of the pencil, when the basepoints falling in S move away from S along the osculating curve of the pencil. It is clear that the theorems V and VI invariably hold good when the pencil contains curves degenerated in coinciding parts if only they do not pass through point S. Theorem VI is like theorem IV an extension of the above mentioned theorem of HaALPHEN and Smiru. If namely we substitute for the pencil of curves the pencil of straight lines with S as vertex, theorem VI passes into: Let S be the origin of a branch T of a curve whilst that branch is mtersected in t points S by an arbitrary straight line through S. If now a pomt P moves away from S along the tangent in S, that point S gwes to that branch t points of contact with tangents through P. It is not difficult to see that this is correlative to the formulation given above of the SmirH-HaLpnen-theorem. However when the pencil of curves is not a pencil of straight lines the theorems IV and VI are not correlative and so we have to regard them as entirely different theorems. Sneek, June 1904. Astronomy. — “On the apparent distribution of the nebulae.” By C. Easton. (Communicated by Prof. H. G. van DE SANDE BAKHUYZEN). It being admitted that the results of the visual observations must be kept separated from those obtained by photography, the systematic investigation of the distribution of the nebulae by means of photo- graphy begun by Max Worr should not prevent us from carefully examining the very extensive material regarding nebulae which has been formerly obtained by direct observation and laid down in catalogues; and the less so because it is highly improbable that even in future a visual “Durchmusterung” of this kind, which for the rest is very desirable, will be carried out on account of the different method which now is being followed at Heidelberg. It is noways unimaginable that the distribution of the nebulae as shown by photography will differ greatly from that of the visually observed nebulae; yet it is certain that the latter distribution shows remarkable features which call for an explanation. WitiiAm Herscurn has found that in the main the nebulae are numerous where the stars are sparse. In a certain sense we have here the reverse phenomenon from that of the stars; Newcous (The Stars, p. 187) expresses it thus: (118 ) “The latter (the stars) are vastly more numerous in the regions near the Milky Way, and fewer in number near the poles of that belt. But the reverse is the case with the nebulae proper. They are least numerous in the Milky Way and increase in number as we go from it in either direction.” CLEVELAND ABBE who after the publication of Joux HerscHer’s catalogue in 1864 statistically investigated this peculiarity (Month. Not. R. A. 5. XXVII, p. 257) rightly put the question whether the paucity of nebulae in the Galaxy did not rest upon a mere optical delusion due to the luminous background of the Milky Way. He thought himself justified, however, in answering this question in the negative, because the regions on either side of the Galaxy proper did not show a considerable increase in the number of nebulae; nor was this the case with increasing optical power. With a much more extensive material — 9264 objects — STRATONOFF (Publ. Tachkent N°. 2) arrived at chiefly the same result. With some reservation, however. For in fact if we develop Srratonorr’s data in a somewhat different way we find that, as we go from the galactic plane, the faint nebulae increase more rapidly than the bright ones. This fact, also because it contradicts a preliminary result of Max Worr : that the (photographed) small nebulae are in general distributed more equally over the sky, raises the surmise that in visual observation the light of the smallest and faintest nebulae in the galactic region is to a certain amount extinguished (table I). EA Bl BE Increase of the mean density of bright and faint nebulae in the direction from the Galaxy. Bright nebulae Faint nebulae Beg gph nde ON. peilde (N. faibles) + 35° fee 10.2 + 25° 3.8 7.0 + 15° 2.3 3.4 + 5° 128 diy 5h 2.3 1.8 En 3.7 5.1 —— 25" 6.2 11.0 == oo, 6.5 12.5 In order to investigate this problem more fully I have compared the density of nebulae in the different parts of the northern Galaxy nn ee a ee nn eee (119 ) with the intensity of the galactic light in those same portions. For if really the luminous background of the Milky Way had a highly disturbing influence, a certain parallelism between the distribution of the galactic light over that girdle (which distribution is very irregular) and the distribution of the nebulae in the same region would manifest itself in this sense that the nebulae, especially the faint ones, are least numerous in those patches where the galactic light is strongest. Table II gives the galactic zone proper between —10° and + 10° gal. latitude (northern hemisphere) divided into areas of 30 degs. in galactic longitude; the two upper lines show for each area the mean density of nebulae derived from STRATONOFF’s data; the lower line shows the mean intensity of the galactic light derived from the table on p. 18 of my “Distribution de la lumière galactique’ (Verhand. Kon. Akad. v. Wet. VIII, 3). TA BE BR The density of the nebulae and the galactic light in the Milky Way compared. 180° 90° 0° 0.7 | 4.3 0.9 1.0 | 0.9 Faint nebulae 0.8 Oe 1.8 On7 10 150 1.03 | 0.72 | 0.78 | 4.09 | 1.31 | 4.08 Bright nebulae | 1.2 Galactic light No parallelism is to be detected. Other causes which might influence in the same way as the “extinction” due to the luminous background of the Milky Way, must be disregarded. Up to now the investigation has yielded nothing in contradiction to the view that the peculiar distribution of the nebulae in the sky results, at least in the main, from their particular rea/ distribution in space. CLEVELAND ABBE tried to explain the paucity of nebulae in the galactic region and the (supposed) increase of their number towards the two Poles of the Milky Way by the supposition that the visible universe consists of systems of which our Milky Way, the two Nubeculae and the nebulae are the individuals, which in their turn are composed of stars and (or) nebulae ; that moreover the galactic plane is nearly at right angles to the major axis of “a prolate ellip- soid” in which all visible nebulae are equally scattered. This theory is founded on the supposed symmetrical distribution ( 120 ) of the nebulae with regard to the Galaxy. Also Bauscuinerr (V. J. S. 24, p. 43) and Srratonorr adopted this symmetry. STRATONOFF sup- poses first that the sky from the North Pole to d— 20° has been surveyed uniformly with a view to nebulae (1. c. p. 48); secondly — like his predecessors — that the actually observed decrease in the number of nebulae between about — 50° and the South Pole of the Galaxy must be ascribed to the incompleteness of the obser- vations made in the southern hemisphere. __I shall now proceed to demonstrate that SrravTonorr’s first surmise is certainly erroneous and that the second, considering the present state of our knowledge, is not justifiable and moreover improbable. _ That the nebulae-material from the North Pole to é — 20° would be collected with equal completeness throughout, as would follow from STRATONOFE’S first supposition, cannot, apart from the lack ofa systematic ‘“Durchmusterung”’, be the case on account of the great difficulty to detect faint objects like the nebulae, which arises from the atmospheric absorption in regions far from the zenith. In Lord Rossr’s telescope, for instance, it has never been possible to observe properly the Omega nebula, though it lies only at — 16°. (Dreyer, Trans. Dublin, soc. N.S: Ups): Besides, the circumstance that the summer nights are never totally dark in the relatively high latitudes of the observatories of the northern hemisphere where the nebulae observations have for the greater part been made, must render the number of catalogued nebulae in the southern regions which then rise above our horizon much too small in proportion to the Opposite equatorial region. With regard to this I have investigated the tables of Bauscninarr. I have compared two equally large areas, occupying the same position with respect to the celestial equator and the Milky Way, A: 4 5" to 9h, d + 15° — 30°. B: a 17! to 21", d + 15° 30°, The number of bright and faint nebulae in those areas A and 5 may be seen on table III: TABLE II. Numbers of bright and faint nebulae in opposite equatorial regions. A B bnght nebulae. „0 ey sae. NAG ON law i! TAA tamt mebulae 8 bne) heere HERE EET There appears indeed to exist a difference at the expense of area ( 121 ) B and this difference, as might be expected, is much greater for the faint than for the bright nebulae. It may be remarked that such a difference is not found — rather the contrary — in areas which in the northern galactic zone border upon those mentioned, where therefore the influence meant cannot manifest itself so strongly. It remains possible of course that the number of nebulae in the direction B is indeed smaller than in the direction A; but we cannot consider such a large difference as established where the disturbing influence is undoubtedly at hand. If we consider only the northern galactic hemisphere, then the nebulae seem indeed to increase fairly gradually though not regularly towards the Pole of the Milky Way. For the southern hemisphere such an increase is also visible in the table of Srratonorr as far as about — 60° galactic latitude. SrraroNorr, however, constructed his table omitting the two Nubeculae. Now it seems to me that for such a statistics this omission is not justifiable. The MAGELLANIC Croups must not at all be considered as patches torn off from the Milky Way, which also appears from the fact that the nebulae proper’), which are scarce in the Milky Way are four times more numerous than the star clusters in the MAGELLANIC Croums (JOHN Hersenen, Results Obs. Cape). As to their composition the Nubeculae keep the middle between the Milky Way and the accumulations of nebulae (sometimes intermingled with star clusters and stars) in Coma, Pisces, ete. — and though the latter agglomerations are less dense than the Crouvps it is not allowed, in my opinion, to inelude these agglomerations in our table and to exclude the Croups. Especially also because, as may be seen clearly on STRATONOFF’s own maps (Publ. Tachkent, N°. 2, Atlas, pl. 16 et 18; comp. Srpnry Waters, M.N. XXXII, p. 558) the Nubecuiae are connected with streams of nebulae and not with the Milky Way. If, however, we include the Nubeculae into the statistics then we must substitute my table 5 (table IV) for SrTRATONOFF’s own table A for the southern gal. hemisphere : 1) Where in this paper we speak of nebulae proper or nebulae without more, we mean the relatively regular and well-defined nebulae (‘white nebulae’), while with diffused nebulae are meant the extensive, formless and according to their spectra gaseous masses (“green nebulae’’). ( 122 ) TAB LEBON. Mean density of nebulae in the southern gal. hemisphere, according to STRATONOFF. (A including, B omitting the Nubeculae). A 5 B pal; ABD Ait ow hee AEM ey 0 LO — 75 24 24 — 65 29 29 — 55 36 36 — 45 25 26 — 35 22 31 —- 25 19 19 — 15 10 10 — § eee Agate If we consider what has been said above on the incompleteness of the observations also in the zone between 0° and — 20° gal. latitude, little remains of the systematic increase as far as about — 60° which, according to STRATONOFF’s table A, seems to exist. One more remark. An eminent observer as JOHN HERSCHEL, for the very reason that he naturally avoided the South Pole in his ‘sweeps’, is sure to have accounted for the incompleteness arising from that circumstance and yet he says emphatically (Outlines, Ed. 1851, pg. 596): “In the southern hemisphere a much greater uni- formity of distribution prevails, and with exception of two very remarkable centers of accumulation, called the MAGELLANIC CLOUDS, there is no very decided tendency to their assemblage in any parti- cular region.” We have, however, a means to find indirectly whether the real distribution of the nebulae with regard to the galactic plane is in the main symmetrical, and consequently that the greater incompleteness of the observations in the southern hemisphere is the cause that the tables do not show a similar progression towards the galactic South Pole as they do towards the galactic North Pole. The galactic equator and the celestial equator form a large angle: 60°. A considerable segment of the northern galactic region occurs south of the celestial equator — hence within the less thoroughly investigated portion of the sky —,; on the other hand an equally large segment of the southern galactic hemisphere les within the well-investigated region north of the equator. The strong influence of the incompleteness of the observations (the real distribution of the ( 123 ) nebulae being about the same in the two galactic hemispheres) would then reveal itself in those two segments, so that the southern galactic segment B (lying on the northern hemisphere) is found to be richer in nebulae than the northern galactic segment A (lying on the southern hemisphere), and the difference would be most marked in the small and faint nebulae. RAB IB Nebulae in the segments between the celestial equator and the galactic equator. A, segm. north. gal. hemisphere £, segm. south. gal. hemisphere (southern hemisphere) (northern hemisphere) Pei DELDEN Ae CP ER ey ee Rep TN ECSU ni. tn CE tt eae al TOA In table V the difference meant is indeed very great for the faint nebulae; it is remarkable, however, that for the bright ones there is a large difference in the opposite sense. This raises the surmise that the possible influence mentioned above does not play a prepon- derating part. If we compare the structure of segment A with that of the remaining part of the northern gal. hemisphere it appears that the density of nebulae in the segment is only 0.6 — which is by no means surprising as it borders upon the galactic plane — but it is very remarkable : faint . ) that the proportion ae the segment, 4.96, is exactly the same eht nie 200 in the remaining part of the northern gal. hemisphere, viz. "549 7 4.95. The relation in the segment B, the southern galactic segment, is ee faint on the other hand quite different: en 14.7. This points very 5 markedly to a surplus of faint nebulae in the southern galactic segment situated on the northern hemisphere, and this circumstance together with the presence of the Nubeculae and their relation with the accumulations of nebulae in the southern gal. hemisphere makes it very probable that the structure of the southern galactic sky with regard to the nebulae differs entirely from that of the northern galactic sky. In the very improbable case that by a more accurate survey the northern galactic segment would be enriched so considerably by faint nebulae that the proportion of the numbers of bright and faint ( 124 ) nebulae agrees more or less with that in segments B, i.e. 1:15, we should count there 2500 nebulae against 3200 in the remaining */, of the northern galactic hemisphere, and the increase towards the Pole, at least for the faint nebulae, would then almost disappear in the northern galactic hemisphere. But whether the number of faint nebulae remains 754 or increases to 2500, in either case a symmetrical increase is out of the question and hence there is no reason why we should adopt CLEVELAND ABBR’Ss prolate ellipsoid which moreover was not very probable. Several considerations plead against the view that the nebulae must be considered as distant galactic systems; the most important, which has been expressed half a century ago, is the occurring together, in most cases even in streams, of nebulae and stars, and also the existence of stars and nebulae in the Nubeculae; (it is obvious that in such eases we have to do with objects of the same order of magnitude). Moreover we have the well-established fact that a star may pass into a nebula. (Comp. VALENTINER, Handwörterbuch d. Astron. III, 2, p. 524; also Mourron, Astrophys. Journal XI, 2, SCHAEBERLE, Nature, Vol 69, No 1785, SreniceR, Abh. bayer. Akad. XIX, p. 572). Should on the other hand a “distant galactic system” be visible to us it can only appear to us as a nebula. But the scarcity of nebulae in the galactic region, if that phenomenon is real, points to an un- mistakable organic connection between the great mass of the nebulae and our system of stars. If then we may accept that the basis on which CLEVELAND ABBE built his theory of the ellipsoidal figure does not hold and if moreover we need not consider the nebulae as being situated at enormously vast distances on either side of the Milky Way, but if, on the contrary, it is far more likely that these distances are comparable with those of the stars, it becomes probable that the greater part of the nebulae are contained in a space similar to the oblate spheroid in which SEELIGER places our whole stellar system, in other words: we may begin by adopting that the great mass of the nebulae belongs to our stellar system and that they are asymmetrically scattered on either side of the chief plane. Moreover, if we supposed that the great mass of the nebulae were systems outside our own, the problem would not be capable of further development. We must therefore not start from such an hypothesis. (125%) Astronomy. — “The nebulae considered in relation to the galactic system.” By C. Easton. (Communicated by Prof. H. G. vaN px SANDE BAKHUYZEN. If we consider the nebulae as forming part of our galactic system (comp. my previous paper “On the apparent distribution of nebulae’) their distribution must be considered in connection with that of the other classes of objects in this system. And then, speaking generally, we not only find a contrast between the apparent distribution of the stars (and clusters) and the nebulae, but also between the distribution of the large diffused nebulae (which, as far as we know, occur almost exclusively in the region of the Milky Way) and that of the nebulae proper; on the other hand it is probable that the galactic agglome- rations for the greater part consist of stars of the first spectral type. Staring from the consideration that the nebular and the star-like conditions are phases in the development of matter and no invariable final phases, it is obvious that as the distribution of the star-like matter in some parts of the system differs from that in the others — which is accompanied by a different constitution of most of the centers themselves, as appears from the spectral differences — so the distribution and the constitution of the matter which exists in the nebular condition will not be the same throughout the system. In the galactic region of the system we find a great number of star-like objects probably placed for the greater part at (relatively) small distances from each other, mostly of the first spectral type. In the ‘“extra-galactic” portions of the system we observe a smaller number of suns, separated by vast distances and belonging for a great part to the second spectral type. In the same way we find in and outside the galactic region proper two forms of nebulae. In it are found the “green” nebulae with a spectrum formed of lines, sometimes round and fairly well-defined (planetary nebulae), but mostly extending over immense regions which they cover as with a veil (large diffused nebulae). In the extra- galactic regions we find the “white” nebulae with a continuous or — more probably — a mixed spectrum; isolated and generally widely separated objects, probably as a rule of a spiral form, and certainly more condensed than those of the other kind. As little as we can sharply distinguish between star-like and nebular objects, is the above mentioned distinction intended as a precise classification. But thus taken in connection with the “galacto- phily” of the “green” nebulae, the “galactophoby” of the nebulae proper does not appear as something exceptional; this principal ( 126 ) peculiarity of the distribution of the different kinds of nebulae may then be considered as a natural sequence of the same cause as in the agglomerations of the Galaxy has given rise to a type of stars differing in constitution and distribution from those outside the Milky Way. For the investigation of the problem mentioned at the head of this paper we must take into consideration : a. The place of the sun in the solar system. 4. The great difficulty to detect a nebula as compared with a star at the same distance from us which has the same quantity of matter. Further I adopt as established: 1. That besides a gradual increase in the number of stars towards the galactic plane there exist real agglomerations in the galactic region. 2. That the sun is situated in a region of the Milky Way which is relatively poor in stars, hence as to the type of distribution in a region that must be considered as ““extra-galactic”, or perhaps in a transition layer. (Comp. Kaprnyy, Versl. K. A. v. W. 1892/98, Publ. Groningen n° 11, p. 32; Easton, Astr. Nachr. 3270, Srruerr, Betracht. p. 627; Newcoms, The Stars, Chap. XX). As probable : 3. That the system of stars is contained within a spheroid with the galactic plane as its principal plane (SEELIGER). And from my own investigations (Versl. K. A. v. W. 1897/98; Astrophys. Journal XII, 2; Verhand. K. A. v. W. VIII, 3) it seems to follow: 4. That in the Cygnus-Aquila region of the Milky Way the preceding branch is much nearer to us than the following. 5. That the brightest galactic portion in Cygnus occupies almost a central position in the system of the stars. Now, while discussing the distribution of the nebulae catalogued after visual observation, I shall try to give an explanation founded on the supposition that, the dim light of the nebulae in general taken into consideration, the very distant nebulae escape visual observa- tion (though they for the greater part perhaps may be registered photo- graphically), and hence that, according to what has been said above especially sub 4, the very distant part of the system in which the Aquila branch of the Galaxy is situated, must be disregarded. If the sun lies about in the central plane of the system in a poor region amidst galactic agglomerations, and if that poor region is comparable with the ‘“extra-galactic” regions on either side of the galactic plane, the thus connected ‘‘extra-galactic” regions acquire more or less the shape of dumb-bells (comp. fig. 1, A), the sun lying in its bar n—z. If the nebulae proper are in the main confined to that extra-galactic region, the galactophoby of these nebulae and their accumulation towards the two Poles would be explained by it. If, however, we leave the Aquila branch of the Galaxy’) out of consideration, we must bear in mind that in the remaining and nearer region of the Milky Way the galactic agglomerations on the Cygnus side have a northern galactic latitude; i.e. that the sun now lies south of the plane passing through those agglomerations (C) and the opposite ones in Argo, N; hence it is no longer in the middle but at the bottom of the bar of the dumb-bells, and consequently the apparent crowding of the nebulae towards the North galactic 1) Comp. Fig. 1, B. The section of the Aquila branch is A, that of the Cygnus agglomeration is C, that of the Galaxy in Argo is N. The circle round the sun is the area within which are situated all nebulae of which the luminosity exceeds a certain minimum. The region /7 is considered as a transition region between the (dotted) agglomerations of stars in the galactic plane # and the extra-galactic region contained within the spheroid. The bar of the dumb-bells is represented much thicker. (TB Pole will increase and that towards the South galactic Pole will diminish or disappear. From the circumstance that the nearest galactic region in the direction of XIX h. (towards C) has a northern galactic latitude it moreover follows that the maximum of nebulae of the northern gal. hemisphere lies within more than 90 degs. of the galactic equator in a great circle passing through C and M, whereas in the southern galactic hemisphere the maximum of nebulae is situated on the Cygnus side of the Pole (towards m,). This theoretical consideration has been tested by observation in the following way. On the maps for the distribution of the nebulae by Srratonorr (Publ. Tachkent 2, pl. 18 and 19) we have estimated in a zone 15° wide in the direction of the galactic meridian of 45°, which crosses the Cygnus region the densities in areas from 10 to 10 degrees galactic latitude. The result is given in table VI. (For tables I—V see the paper “On the apparent distribution of nebulae’). The much smaller densities of the southern galactic hemisphere could be expected owing to the greater incompleteness of the obser- vations; for the rest the displacements of the maxima from the Poles are in the same sense as they should be according to theory. In how far the hypotheses made are thus supported cannot be decided because of the incompleteness of the data, at any rate they are not at variance with the results of the observations. If, with the now available data, particulars are to be detected concerning the true distribution of the nebulae, we shall have to look for traces of them in those regions where the galactic and the extra-galactic regions meet. In a mean galactic latitude we shall have to search for the great irregularities in the apparent distribution of the nebulae and compare the places where they occur with the places where the irregularities in the distribution of the stars are found, especially the irregularities in brightness and in width of the diffused light of the Milky Way. To this end the following tables have been constructed (VII, VIII, LX). Over Srratonorr’s maps of the distribution of the bright (n. bril- lantes) and faint (n. faibles) nebulae, a galactic system of coordinates was laid. For each area of 15° in gal. longitude and 10° in gal. latitude we have then estimated as carefully as possible the density of bright and faint nebulae. These values were combined with due regard to the unequal surfaces of the areas in order to obtain the mean densities for zones of 15° gal. longitude between 0° and 50° gal. latitude on either side of the galactic equator (table VII). For the galactic zone proper, between + 10° and — 10° gal. latitude, ( 129 ) VA BLE Vii Density of the nebulae in the direction of the galactic meridian of 45°. Northern galactic hemisphere. Southern galactic hemisphere. x gal. 2 A Neb. z gal. 2 A Neb. 45 Ee 5 2.5 225 — 5 0.4 15 3.5 15 0.2 25 8.0 25 3.7 35 13.8 35 5.7 45 12.5 45 9.3 5D 11.8 55 9.2 65 18.2 65 4,2 75 | 21.0 75 9.5 BER SON et ABO 225 85 8.1 Sree Metre ARP end 45 85 10.3 75 40.3 15 12.0 65 22.2 Nl 65 113 55 17.8 55 16.5 45 13.0 45 15.0 35 11.4 35 11.3 25 12.0 25 5.8 15 6.0 15 3.3 Zn Be ts 8 45. ee 1.9 we have combined in table VIII the regions north and south of the gal. equator, as it was of no use to consider these regions separately. On the construction of the two tables the compensation method used by Srratonorr has probably influence, but in my opinion it is of no consequence that the details of the distribution are more or less obliterated and hence the contrasts are less distinct. For an investigation of the main features this is rather an advantage; the method is only inconvenient with the Nubeculae. In these two tables the data relating to the southern hemisphere have been added, 9 Proceedings Royal Acad. Amsterdam. Vol. VII. ( 130 ) TABLE VII. Density of nebulae in the zones 0° to 50° galactic latitude. A. Bright nebulae. (Values of the upper line north, values of the lower line south of the gal. equator). 3 ro Se S = a = Z 2 R © el oan as) fan) NI nN ON re ton — ler) Ne) ine) Oo |, 2\4. chy ae ‚5|12.1|9 i. cr Sid a 3.7|6.0|4.7/4.5|8.9/11.8]4.7|5.1/4.4/1.9]1.7/2.8 | heete 5.0|28.4/9.9|2.5|5.2|3.8|9.0|2.7/2.4/2.5/7.5| 8.8|41|4.7|4.2|3.1|4.5|3.7| B. Faint nebulae. (Values of the upper line north, values of the lower line south of the gal. equator). 5|4.0!2.8| Kars kak 9] 4.5/8 Ae 2|4.9|4.5| 4.5] 5.2/4.0 6.9|7.1 a; ‘6/4 osleslenlan 5.2 Mee 5|5.8)15.5/11.7/9.0 TABLE VIIL Density of nebulae of the galactic-equatorial areas (— 10 to + 10° gal. latitude). A. Bright nebulae. |1.9|4.1|2.0|1.7|1.9|4.4/3.1/4.1|4.4/2.7/3.4]1.6]4.3]2.0|2.4/1.1|3.1|3.4|2.1|2.5|2.2|2,8/2.4/2.2| B, Faint nebulae. 1|8.0/2.8|1.2/0.9|0.9|1.3]1.40.41.0|1.5|1.8|3.2|2.6|1.9|2.4|8.5|2.4|1.2|2.9|8.0/2.8)3.7|2.2) Do oo oO = ad a So a to 8 5 Si ° S S =) S en) >, j=) TABLE IX. Intensity of the light of the Galaxy in gal. equat. areas, — 14° to + 14° gal. latitude, (Values of the upper line north, values of the lower line south of the gal. equator). | | 1.14 | 1.29 | 1.19 | 0.63 | 0.57 | 0.64 | 0.80 | 1.14 | 1.84 | 1.74 | 0.82 | 1.00 | 0.37 | 0.67 | 0.81 | 1.14 | 1.20 | 1.39 | 1.36 | 1.07 =) 2 ee) le) 1D g KS, Ye) Lie oo Ne) 10 ap} Lan, rd rd rd 0.97 | 1.10 | 1.87 | 1.19 i=) Ie) ° Es we) x ae) re ie, TPO ee 10 ( 131 ) but only for completeness, for especially for the faint nebulae these data have but little value and especially from the curves of table VIII it may be easily seen that the material relating to the southern hemi- sphere cannot be compared with that for the northern hemisphere. Table IX gives for areas of 15° gal. longitude and 28° gal. latitude, 14 degrees on either side of the gal. equator, the brightness of the light of the Milky Way expressed in terms of the mean brightness of the northern hemisphere derived from the table of p. 18—19 of Fig. 2. Crux Monoc. Pers Sagit. QR 8 N a 3 Graphical representation of table VIIL. Graphical representation of table IX. 9g * ( 132) La distribution de la lumiere galactique by the author, (1903); there- for we had only to accept that the diffused light of the Milky Way in the middle areas (—2° to + 2° gal. 8) is equally distributed. A detailed discussion of these tables and of the curves which are constructed by means of them (fig. 2) would lead us too far. We shall only draw the attention to the following points: The general smoothness and the small values of the ordinates of the curve belonging to table VIII B, southern hemisphere, can cer- tainly not lead to the conclusion that the nebulae in the southern galaxy proper are indeed much sparser and more uniformly scattered. It is remarkable that about 110° gal. longitude, even within the galaxy proper the density of faint nebulae nearly equals their average number in the entire northern hemisphere (8.5 and 10.0). If we consider the general shape of the curves for the density of the nebulae we perceive a certain contrast with the curves of table IX for the galactic light — as compared with the minimum of IX the maximum of VII seems somewhat displaced towards 90° — but in the details no complementary shape can be detected. The most remarkable feature of the nebulaecurves is a strong maximum about 100° to 110° in the northern and in the southern hemisphere, which seems to find its counterpart, at least in the northern gal. hemisphere, 180 degrees further on at about 280°. If in the neighbourhood of the galactic zone the space occupied by nebulae extended equally far in all directions from the sun (which, with the suppositions we have made, would mean that the nearest agglomerations of stars in that plane do not lie at greatly varying distances) and if within that space the distribution of those nebulae were almost uniform, there would be no reason why the curves of VII and VIII show considerable maxima and minima. We know, however, that the nebulae show a strong tendency to occur in “streams” and ‘nests’, hence in the details their distribution must be very irregular. Consequently the peculiar positions of the principal maxima of the curves might be explained by accepting one or more nebular streams, running from 100° to 280° gal. longitude somewhat obliquely towards the galactic plane. Obviously this supposition is arbitrary. Another acceptable explanation is that the region of the nebulae, viz. the extra-galactic region, extends in and near the galactic plane farther in the direction 100° towards 280° than in other directions; in other words: that in the direction mentioned the galactic aggrega- tions of stars are lying at greater distances. It deserves attention that the line which in space connects the (133 ) points represented by the greatest maxima of the nebulaecurve (table VID, (excluding the maximum at 280° in the southern hemisphere, due to the Nubeculae) is almost at right angles to the direction of the great agglomeration of stars in Cygnus (comp. table LX, 30° to 45° north). If, as it has been argued in Ap. J. XII, 2 — comp. Versl. K. A. v. W. 1897/98 — the agglomeration of stars in the direction of Cygnus forms the center of the mainly spirally arranged galactic system, of which the unequally dense windings surround the sun, this result (if we also accept the other suppositions mentioned in this paper) was to be expected. For then (comp. fig. 3) the poorer region between the principal Fig. 3 : ‘ > a: ‚A . ‘ac Ee PG te le eee Se Be erg! ee SO ee : e mre . ae ¢ N eN hd k . … 3 . kad a LS ete e Be A P:) a . en = or eee ee ay EEA Pg DK 4 DR otal A uo. 77 a A BE 1 ge ¢-. ery aL a | SE Td EEN Be." ory a’. © ze A : e ,, t eat. "4 ne Kd cS t.'% Wie Si we Ae * S dd REE cm, Q A . ‚9 4 8 TIK : 2 . . ae a B a . 2 e- : . cm a „a a Pd 2 Ke 6 ws = eae x rie a aS Bi oh oats yuh aad UT Ee se ec om = star-windings in the galactic plane — which, as regards the type of condensation of the matter, has been identified with the “‘extra- galatie”” regions (rich in nebulae proper) extending on either side of the galactic plane — will extend farthest out in the direction AB, at right angles to a line passing through the sun and the central condensation of the system. This will especially be the case at 90 degs. from the central condensation or rather a little nearer to it in the direction @ where the great gap between the windings (Perseus ( 134 ) region) is to be found. And indeed we find in table VII fig. 2 the best marked maxima of the nebulaecurves at about 105° gal. longi- tude at less than 90 degs. from the Cygnus region and a less strongly pronounced one in the northern hemisphere at 280°. The available material is certainly not sufficient for us to decide with any probability whether the secondary maxima (at 165° etc.) of the curves of the tables VII and VIII result indeed from the arrangement of the galactic agglomerations or whether they are produced by a merely local accumulation of nebulae. Among such “local deviations” from the uniform distribution we ought then also to reckon the Nubeculae, which apparently have such a great in- fluence on the distribution of the nebulae in the southern galactic hemisphere. It should be borne in mind that the Nubeculae are not connected by streams of nebulae with the southern Milky Way and neither probably by streams of stars. Nor is the influence of the vast nest of nebulae, which constitutes the Nubecula Maior percep- tible in table VIII. Finally, if the nebulae in the very distant regions of the system remain in general invisible and hence are not included in the statis- tical data given here, while they can be more easily photographed, this would in connection with our preceding remarks explain the fact observed by Max Worr (Sitzungsber. München XXX1, II, p. 126) that the mass of the very faint nebulae photographed by him are scattered more uniformly over the sky than those observed visually. Physics. — “Dispersion bands in absorption spectra.” By Prof. W. H. Jurrvs. (Communicated in the meeting of May 28, 1904). The appearance of absorption lines depends on various circum- stances. As to the absorption phenomena in gases and vapours, such conditions as temperature, density, pressure, velocity in the line of sight, intensity and direction of magnetic field, have been fully studied and discussed. In the present paper we purpose to show that anomalous dispersion in the absorbing gas is also, to a great extent, accountable for certain typical features of the dark lines. An originally parallel beam of light, when passing through a mass of matter, the density of which is unequally distributed, will not ( 135 ) remain parallel and, generally speaking, the greatest incurvations will be noticed in those rays, for which the medium has refractive indices differing most from unity, i. e. in those which, in the spectrum, lie closest to the absorption lines on either side. These particular kinds of light, while diverging into space, will spread in many more different directions than the average waves, and, as a rule, a smaller portion of them will fall into the spectroscope, than of waves with refractive indices nearer to unity. Accordingly, there must always be certain places in the absorp- tion spectrum, from which light is absent owing to dispersion in the absorbing vapour, for it may be taken for granted that the latter is never absolutely homogeneous. These darker parts in the spectrum we shall call dispersion bands. It stands to reason that these bands will overlap the regions of real absorption; so they might easily be mistaken for strengthened absorption lines, which no doubt has often been done. We will now look somewhat closer into the characteristics by which dispersion bands may be distinguished from absorption bands. The curvature of a ray of light of a definite wave-length, at any point of a non-homogeneous medium, not only depends on the gradient of optical density at that particular spot, but also on the angle which the beam makes with the levels of equal density. Its diver- gence will be greatest when this angle is zero. Strong ray-curving through anomalous dispersion in vapours may, therefore, be artificially produced in two ways: first, by using masses of absorbing vapour, presenting in a small space considerable dif- ferences in density, such as e.g. occur in the electric arc‘); secondly in larger spaces where the density varies but moderately, by making the light travel over a considerable distance under small angles with the levels of equal density. I have chosen the latter method of investigation, especially on account of the extensive use, which may be made of the phenomena presenting themselves, by applying them to the interpretation of numerous peculiarities of the spectra of celestial bodies *). The absorbing medium was a Bunsen flame, of a peculiar shape, containing sodium vapour and so arranged, that the introduction of the salt could be easily regulated. 1) H. Epert, Wirkung der anomalen Dispersion von Metalldimpfen, BoLTzMANN Festschrift, S. 443. Fig. | represents a section of the burner. A is a cop- per trough, 80 cM. long, 8 cM. wide and 5 eM. deep, thickly coated with varnish and having a broad flange. The planed brass plate B is firmly screwed upon the flange and a leather packing makes the joint air-tight. On this cover, which has a rectangular opening 75 ¢.M. long and 2 eM. wide, are fixed two brass rulers, C and C’, 75 eM. long. They are so adjusted that at O they form a slit, having an exactly uniform width of about 0,1 cM. over the whole length. The prismatic space between C and C" is closed at each end by a small triangular brass plate. The trough is filled to a certain height with a saturated solution of soda, and into the remaining space a mixture of illuminating gas and air is conveyed by means of tubes, entering at both ends. These tubes are fed from a mixing bottle in which the gas and the air are being driven through two separate regulating taps. If now the flame were left to burn without any further precautions, the slit O would soon be closed in consequence of the onesided heating of the rulers. It was therefore found necessary to place the trough in a vessel with running water, reaching up to the burner. In this way a uniform and steady flame was obtained. A few millimeters below the level of the salt solution a platinum wire P is stretched over the whole length of the lamp. Its ends are soldered to insulated copper wires, which pass through the walls of the trough, and are connected to the negative pole of a storage battery of 20 volts. From the positive pole two insulated wires lead to the ends of a long strip of platinum P’, which rests on a glass plate at the bottom of the trough. As soon as the circuit is closed, innumerable minute particles of the fluid rise into the space Mig Al. 1) The abnormal solar spectrum of Hate; the peculiar distribution of light in several of the FRAUNHOFER lines, even in normal conditions; the variations in the average appearance of the spot spectrum accompanying the eleven year period, all these phenomena have been easily explained from the considerations here alluded to (See W. H. Junius, Proc. Roy. Acad. Amst. IV, p. 589—602; 662—666; V, p. 270—302). The present investigation is a continuation of the experiments with the long sodium flame, a short account of which has already been given on those former occasions in support of our theory. ( 137 ) R, and cause the flame to emit a beautiful, clear and constant sodium light, the intensity of which can be controlled and regulated by means of an amperemeter and a variable resistance. At Fig. 2 a. Fig. 2 b. In Fig. 2, a and 6 are shown two dif- ferent ways in which the light travels through this long sodium flame. Z repre- sents the crater of an electric are of 20 amperes. The lens A throws an image of the crater on the slit S,, which, in its turn, is depicted by the lens B on the slit S, of a grating spectroscope. About half of the conical beam of light which leaves A is intercepted by the screen P, and the part, which the slit S, allows to pass, falls almost entirely on the screen Q, which has been shifted so close to the optical axis of both lenses, that only a narrow streak of light can reach the slit S,, through the middle of B. The large gas burner stands on a hori- zontal slide, which is movable up and down and round a vertical axis; thus, by means of screws, it can easily be put in any position required. When the axis of the flame (which we assume to be in its most luminous part, i.e. a little above the blue-green core) coin- cides with the optical axis of the system of lenses, both the D-lines will be seen symmetrically widened in the spectroscope. If not perfect, the symmetry will easily be corrected by slightly shifting the screens Pand ‘Q. Picture 1 Fig. 3 (see Plate) refers to the case when the flame MN is not burning; the narrow absorption lines are due to traces of sodium surrounding the carbon points. When the flame is burning, a very weak current passing through the sodium solution will produce the effect shown in 2. The photographs 3, 4 and 5 were obtained with currents of about 1, 3 and 6 amperes, the flame always being in the symmetrical position. We will now examine the case represented by Fig. 2, a. Here the axis of the flame has been shifted 3 m.M. towards the right. The narrow beam of light which reaches S,, only penetrates that ( 138 ) part of the flame, where the density of the sodium vapour increases from left to right. In a structure of this kind, waves, for which the vapour has a great index of refraction, deviate towards the right, e.g. S,G. They are not intercepted by Q and consequently reach the slit S,. In fact, the presence of the sodium vapour allows similar waves to enter that slit even in larger quantity than they would do without it, for rays of this kind, issuing from the uncovered half of A, which if travelling in a straight line would be intercepted by Q can, when refracted, penetrate the lens B. The case is entirely different for those kinds of rays for which sodium vapour has refractive indices that are smaller than unity. Such rays deviating towards the left (as shown in S, A), are intercepted by Q and consequently will be absent from the spectrum. Nos 6, 8 and 10 are reproductions of photographs taken under these conditions. On the left are seen the smaller, on the right the greater wave-lengths (in fact, in the whole series of photographs the stronger D-line appears on the left side); so it is obvious that really the waves lying on the red-facing side of the D-lines, i.e. those for which the vapour has high refractive indices, are strengthened by anomalous dispersion; and that, on the other hand, the waves on the violet side have been considerably weakened. Alternately with 6, 8 and 10 the photographs 7, 9 and 11 were taken. The position of the flame was now as indicated in Fig. 2, b, i. e. its axis had been shifted 3 mM. to the left, so that the central beam had to traverse that part of the flame where the density of the sodium vapour decreases from left to right. Here we notice that the rays with low refractive indices deviate towards the right and that a larger number of them reach the slit S,, e.g. S,A, whilst the rays with high refractive indices, such as $,G, are intercepted by Q. Nos 6 to 11 show the effect of a gradual increase in the density of the sodium vapour. In No. 12 we again notice the sharply defined sodium lines after the flame has been extinguished at the end of the series of experiments; they are somewhat stronger than those at the beginning of the series, because much sodium vapour had spread through the room during the operations. When carefully examining the original negatives it is possible in most of them to distinguish the rather sharp central absorption lines from the overlaying dispersion bands (especially in the photographs, obtained when the position of the flame was symmetrical; the repro- ductions fail to bring out this peculiarity). Advantage has been taken of this fact in so arranging the twelve photographs here reproduced, that equal wave-lengths occupy corresponding places. Then it is seen 7 W. H. JULIUS: “Dispersion bands in absorption spectra.” Fig. 4. 10 11 12 Fig. 3. Proceedings Royal Acad. Amsterdam. Vol. VIL ( 139 ) that the “centres of gravity” of the two dark bands, as well as the brighter space between them, have been alternately shifted to the left and to the right a phenomenon which needs no further explanation. As a matter of course the interposed flame causes the illumination in the plane of the slit S, to be very irregular, especially with regard to those radiations undergoing anomalous dispersion in the vapour. It is evident that some kinds of rays which are absent from one part of that plane, will be found in excess at another. The distribution of light in this irregular field of radiation might be explored by moving /S,, together with the spectroscope, within it. The same object can be obtained with less trouble by means of a thick piece of plate glass, mounted vertically between B and S, in such a manner, that it may be moved round a vertical axis. When turning it a little we make the whole radiation-field beyond the plate glass shift parallel to itself, thus causing other parts to cover the slit. This influences the aspect of the dispersion bands very materially. In certain positions apparent emission lines of sodium vapour-may happen to be seen, which disappear as soon as the arc-light at S, is intercepted *). In conclusion we wish to draw attention to a peculiarity we repeatedly observed in the dispersion bands. The dark shading in a dispersion band does not become deeper in proportion as we approach nearer to the central absorption line, but seems to reach its maximum obscurity at certain (though not always equal) distances on both sides of the centre; whilst in the space between, the light appears somewhat intensified just as if a wide absorption band had been partly covered by a narrower emission band, the centre of which is again occupied by the fine absorption line. This phenomenon cannot, however, be attributed to radiation emitted by the absorbing sodium flame; for in our arrangement the intensity of the emission from the flame could bear no comparison with that of the are for corresponding waves. In order to make sure we tried to photograph the emission spectrum of the flame, exposing the plate during the same length of time and under the same conditions as had been done for obtaining the absorption spectrum; but not a trace of any impression could be detected on the photographic plate. The light on both sides of the central line therefore originates in the carbon points and this we explain on the principle of ray- curving. The kinds of rays which are most strongly refracted in the flame may, under certain conditions, be curved twice or even more 1) These bright lines originate in the same manner as the light of the chromo- sphere. The chromospheric lines are not emission lines, but “bright dispersion bands”, ( 140 ) times, when passing nearly parallel to the system of the levels of equal density (in the manner described on a former occasion *)) and will therefore have a greater chance of reaching the slit S,, than rays which are less strongly curved. The relative intensity with which the waves, belonging to those central parts of the dispersion bands, appear in the spectrum increases with the distance over which the light has travelled along such a lamellar or tubular structure. Should the true absorption line happen to be exceedingly narrow, the dispersion band may give the impression of a double absorption band, which need not be symmetrical ®). We hold that the dispersion bands play an important part in many of the well known spectral phenomena, such as the widening, shifting, reversal and doubling of lines. In a subsequent communication I purpose to examine from this premise various phenomena observed in the spectra of variable stars and other celestial bodies. Physics. — “Spectroheliographic results explained by anomalous dispersion.” By Prof. W. H. Juus. It is not surprising that the scientific world should be highly interested in the beautiful results, obtained by Hate and ErLLeERMAN with the spectroheliograph *). The brilliant method elaborated and applied by these investigators enables us to see at a glance as well as to study in minute details how the light of any selected wavelength was distributed on the total solar disk at any given moment. W.S. LockYer, in giving an abstract from the paper here alluded to in Nature N°. 1800, rightly entitles it: “A new epoch in solar physics.” Indeed, the spectroheliograph proves capable of providing us with an abundance of new information, which other existing methods could never give and the value of which will remain, whatever may be the ideas on the Sun’s constitution derived from it. But, nevertheless, even the most splendid collection of new facts is useless so long as we have no theoretical ideas connecting them with achieved knowledge. Hate and ELLerman, accordingly, in 1) Proc. Roy. Acad. Amst. IV, p. 596. 2) In Fig. 4 on the plate is given an enlargement of one of the photographs obtained by an almost symmetrical position of the flame. It has been somewhat spoiled in the reproduction. The original is less blotchy and the transition of the dispersion bands to the bright background of the spectrum is there much more gradual. 3) G. E. Hare and F. Errerman, “The Rumford Spectroheliograph of the Yerkes Observatory,” Publications of the Yerkes Observatory, Vol. III. Part. I, (1903). ( 141 ) describing the observed phenomena, lay down quite definite conceptions regarding certain conditions and configurations of matter in the solar atmosphere, by which the observed distribution of the light in the image of the Sun is assumed to be produced. In the cited publication they put forth the working hypothesis that the “caleium-floceuli” or bright regions showing themselves all over the image of the Sun when it is photographed in so-called calcium light, are columns of calcium vapour rising above the columns of condensed vapours of which the photospheric ‘grains’ are the summits (le, p. 15). This hypothesis, though at first proposed mainly as a guide to further research (l.c., p. 13), has been subsequently’) employed by the same authors with much less restriction as the basis on whieh the photographs ought to be interpreted. The great authority of Harre and of such critics as W.S. LOCKYER, J. EversHep and others who, in abstracts from the work of Hare and ELLERMAN, concur in most of the interpretations there given, might eause the value of those ideas to become overestimated and extended beyond the original intention of the authors. It is not superfluous, therefore, to show how we may quite as well account for all the new phenomena thus far revealed by the spectro- heliograph, if we start from the entirely different conceptions of the Sun’s constitution, which the consequences of ray-curving in non homogeneous media and of anomalous dispersion of light in absorbing vapours have suggested to us. Both these circumstances are left absolutely out of consideration by Hate and Errerman. Their conclusions are all founded on the erroneous supposition that the monochromatic light by which their images of the Sun are photographed, has travelled from the source in straight lines, and that they are right, accordingly, in supposing light-emitting masses of calcium vapour to exist in the exact directions, along which calcium-radiations seem to reach us. In making this supposition they fall into the same error as one who would assume the refracting facets of the crystal globe of a burning lamp to be independent sources of light. Our new explanation of the spectroheliographic results will be founded on the hypothesis that the Sun is an unlimited mass of gas in which convection currents, surfaces of discontinuity and vortices are conti- nually forming under the influence of radiation and rotation, so that the various composing elements are mingled as completely as nitrogen 1) G. E. Hare and F. Exrermay, “Calcium and Hydrogen Flocculi,” Astro- physical Journal XIX, p. 41—52, ( 142 ) and oxygen in the Earth’s atmosphere’). This hypothesis too will, of course, want modification in the light of future results; but for the present it seems, so far as the visible phenomena are considered, not to clash with any observation or physical law. The irregular motion of electrons in the deeper layers of the Sun, where the density is very great, gives rise to the radiation with a continuous spectrum. We shall only take fs radiation into account. Peculiar radiations, emitted by the more rarefied outer parts of the gaseous body and giving a bright-line spectrum, may perhaps add a perceptible quantity of light to the bulk, but this selective emission, if present, does not play any part in our explanations. So we behold the brilliant core of the Sun through an extensive envelope, consisting of a transparent but selectively absorbing mixture of gases, into which the core gradually spreads. It stands to reason that the average density of this envelope slowly decreases in the direction from Sun to Earth; but at right angles to that direction the density must be in some places much more variable. For it is a minimum in the axes of vortices; and the average direction of the whirl-cores, lying between the Earth and the central parts of the Sun in the surfaces of discontinuity, differs but little from our line of sight. The rays of the Sun thus reach us after having travelled a great distance along lines, making small angles with the levels of slowest density-variation in a lamellar, partly tubular, structure *). Under these circumstances the solar rays will be sensibly incurvated on their way through the envelope, especially those suffering ano- malous dispersion. As a rule, beams consisting of the latter kinds of rays will show an increased divergence; they will reach the Earth with less intensity than the normally refracted light and so will give rise to dark dispersion bands*) in the solar spectrum. And the degree of divergence will not only be different with waves, which in the spectrum are found at different distances from the absorption lines, but it is also clear that the divergence with which various beams of any definite kind of light arrive at the Earth must differ largely according to the dioptrical properties, exhibited along the 1) A sketch of a solar theory, based on this hypothesis, is to be found in the Revue générale des Sciences, 15, p. 480—495, 30 May 1904. 2) For considerations which have induced us to hold that a similar structure of the Sun is very probable, I refer to former publications: Proc. Roy. Acad Amst. IV, p. 162—171; 589—602; V, p. 270—302. 3) W. H. Jutius, Dispersion bands in absorption spectra, Proc. Roy. Acad. at Amst. Vol. VII, p. 134. ( 143 ) paths of those beams by the system of surfaces of discontinuity. The foregoing inferences really imply the whole of our inter- pretation of the results, thus far obtained with the spectroheliograph. This we shall show by amply discussing some of their main features. The broad dark bands, designated by Hate and ErrerMaN as H, and £,, are not absorption bands, but dispersion bands. Real absorp- tion by the solar calcium vapour we hold to be restricted to the central dark lines H/; and A,. The bright bands H, and K,, predo- minating in the spectrum of the “floceuli” and attributed by Hae and ErrERMAN to strongly radiating calcium vapour, result in our theory from the fact, that with beams of light the wavelength of which is very near to that of the central absorption lines, the divergence may be diminished or even changed to convergence by the tubular structure. Indeed, such rays deviate more strongly than those standing farther from the absorption lines; and as soon as they undergo more than one incurvation, they have a chance of reaching the Earth with increased intensity. This chance improves in proportion as the index of refraction departs from unity, be it in a positive or in a negative sense*). We conclude from it, that the brightness of the calcium flocculi must, as a rule, increase as the monochromatic light in which the Sun is photographed approaches the true absorption line. This consequence of our theory exactly corresponds to one of the chief peculiarities, which immediately struck Hate and ELLErMan on inspecting sets of photographs taken at short intervals of time with the second slit in different positions within the H and K bands. In order to account for the same fact, those investigators are obliged, by their working hypothesis, to suppose that in higher regions of the Sun’s atmosphere the calcium vapour radiates more strongly than in lower levels. This cannot be called a very satisfactory inference ; and less so, as the supposition is added that the incandescent vapour is rising from much deeper layers and, therefore, considerably expanding — a process during which, according to our physical notions, the temperature must fall. Here we meet with a serious difficulty; Hate and ErrpMaN try to get rid of it by means of the rather vague assumption, that some electrical or chemical effect may be responsible for the bright radiation emitted by this calcium layer, which is intermediate between two absorbing layers’). 5) In the experimental investigation on dispersion bands, before mentioned, this brightening in the middle of the dark bands has been distinctly observed. Cf. also; Proc. Roy. Acad. Amst. Vol. IV, p. 596. *) Hare and Everman, Astrophysical Journal XIX, p. 44, ( 144 ) Our theory can dispense with such additional hypotheses. Another characteristic peculiarity, observed in every series of photographs taken at short intervals with the slit set at various points on the broad H and K bands, is the following. When the slit is set, e.g., at a remote point of K,, the structure of the solar image appears relatively fine, sharp and detailed; approaching the central line, we see some of the brilliant spots vanish, others grow more extensive, especially those lying in the vicinity of sun-spots ; at the same time their outlines become less sharp, so that finally the whole image gives us the impression of a coarser and at the same time a more woolly structure’). Harr and ErrerMAN hold that the successive photographs refer to gradually higher levels and conclude that the masses of calcium vapour must have a tree-like shape. W. S. Lockyer, in Natur No. 1800, draws a scheme showing this conception. Against this interpretation we propase the following one. The amount by which the divergence of a beam of light is altered in consequence of the presence of calcium vapour in the streaming and whirling mass depends, of course, on the proportion of calcium in the mixture, and besides on two other circumstances, viz. 18 on the position occupied in the spectrum by the selected kind of light with regard to the absorption lines, and 22d on the steepness of the density gradients in the mixture along directions perpendicular to the path of the beam. Let us suppose the selected light to correspond to the extreme edge of H, or K,, then its index of refraction differs but little from unity. Accordingly, very considerable inequalities of density are required to cause a perceptible change in the divergence of such beams. Similar great inequalities may indeed occur at many separate places, but at each of them they cannot, of course, extend very widely. This accounts for the fine and rather sharply defined reticulation shown by the so called ‘low-level’ photographs. If the second slit were set a little nearer to the centre of the line, the distribution of the light in the solar image would at all events differ considerably from that of the former case; for the indices of refraction being very different for neighbouring waves within a dis- persion band, the divergence of beams, starting from the same point of the Sun, must vary largely with the wave-length. So it is clear 1) Such series of photographs are reproduced in: Publications of the Yerkes Obervatory, Vol. IlI, Part I, Pl, V, VI, X, XI, XII, XIII. ( 145 ) that bright or dark spots, visible on one photograph, may be wanting in the other. Moreover, the general character of the image must change as we approach the central line. For in proportion as the indices of refrac- tion depart from unity, slower variations of density suffice for pro- ducing sensible differences of divergence. And, as a matter of course, in any whirling region slightly inclined density-gradients will take up larger spaces than very steep ones. Besides, when the second slit of the spectroheliograph, having a given width, is set near to the central absorption line, the wave-complex which it allows to pass, covers a greater variety of refractive indices, than when it is set farther from the central line. In the former case the distribution of the light in the solar image must, therefore, be less differentiated. Both circum- stances cooperate in causing the bright-and-dark structure generally to appear coarser and more woolly in proportion as the spectro- heliograph is adjusted for kinds of rays that are more liable to anomalous dispersion. From the same point of view it is not surprising that on photo- graphs, taken in H, or K, light, the calcium flocculi are parti- cularly bright and extensive in spot regions, for in such regions the “tubular” structure of the gaseous mass, by which the strongly curved rays are kept together and conducted, is most developed. Hate and ELLERMANN also mention “dark calcium flocculi’” *), which they describe as special objects, visible in so-called “high-level photographs” and not to be confounded with the general dark back- ground, produced by the absorbing calcium vapour of deeper layers. Dark floceuli often surround the large bright flocculi of spot regions, as is shown e.g. in Fig. 4, Plate V of the cited publication. The explanation given by them is, that we might have here some indications of the cooler K, calcium vapour, which rises to a considerably greater height than the K, vapour of the bright flocculi. In our theory the presence of these darker regions is a direct consequence of the fact, that the particular distribution of the light in the solar image is not produced by local absorption and emission, but by irregular ray-curving. The rays are only caused to change their places; so an excess of light in the bright floeculi must necessarily be counterbalanced by a deficit in the surroundings. H and K are by far the broadest bands of the visible solar spec- trum; even with moderate dispersion the second slit of the spectro- 1) Publications Yerkes Observatory, l.c. p. 19. 10 Proceedings Royal Acad. Amsterdam. Vol. VII. ( 146 ) heliograph could easily be set at different points within these bands. When the dispersion of the instrument was increased by means of a grating, photographs of the Sun could be obtained with light falling entirely within a widened line of hydrogen or of iron. Photographs made with Hz or H, light showed also a flocky structure, differing, however, materially from that obtained with H and K. Harre and ELLERMANN therefore assume dark and bright clouds of hydrogen to exist in the solar atmosphere. Upon the whole, but not in the details, the hydrogen floeculi correspond in form and position to the calcium floeeuli photographed with H, or K, light; the general aspect of the photographs is fainter, they show less contrast, and the detailed structure observed in H, or K, light is wanting. The most striking fact, however, is that the bright calcium floceuli of the H, or Rophotograp hes ane replaced on the Hg photograph by dark struc- tures of similar form. Only in a few places in the vicinity of sunspots small bright hydrogen floeculi occur which coincide with parts of bright calcium flocculi. Hate and ErreRMAN hardly make an attempt to explain these facts which, in the light of their working hypothesis, are really puzzling. We get a much clearer view of the matter as soon as we suppose the widening of the hydrogen lines also to be produced by anomalous dispersion, instead of by absorption only. Indeed, the ray-curving in the solar gases must generally be less with waves belonging to those narrower dispersion bands than with waves lying near the centres of the broad H and K bands. Even in the powerful whirls of spot regions there will only sporadically be found places where the tubular structure is sufficiently marked to keep together rays belonging to the dispersion bands of hydrogen in the same way, as it does gather the strongly curved H, and K, light in the large, bright calcium floeculi. Accordingly, we shall meet with very few places in bright caleium flocculi, where the photographs in Hz or H, light also exhibit brilliant points. All the rest of the bright H, and K, regions correspond to those parts of the gaseous mass where the differences of density — though not so excessive -— are nevertheless very considerable; but whereas in that structure the H, rays are repeatedly curved and may be made to converge, the less strongly incurvated Hz rays will in the same regions diverge and be dissipated in a considerable degree, thus giving rise to dark places in the photographs. Outside the bright calcium flocculi, finally, where the H, and K, photographs are dark in con- sequence of increased divergence of the beams, no strong incurvation (147) is given to the H; or H, light; at those places the image of the Sun, photographed in hydrogen lines, must therefore be less dark. The rather faint character of the hydrogen floeculi, the absence of sharp outlines and of strong contrasts in the structural elements, we ascribe to the dispersion bands of hydrogen being relatively narrow and so allowing rays with a great variety of refractive indices to pass simultaneously through the second slit of the spectrohelio- graph. The hydrogen photographs too would show finer details, like those in K, light, if the dispersion of the apparatus were still greater and the second slit still narrower. We believe that we have shown that every peculiarity, thus far noticed in the photographs obtained with the spectroheliograph, may easily be deduced from the same fundamental hypothesis regarding the constitution of the Sun, which has already proved capable of giving a coherent interpretation of the solar phenomena known before. Not a single new hypothesis was required. Physiology. — “On artificial and natural nerve-stimulation and the quantity of energy involved.” By Prof. H. ZwAARDEMAKER. A living nerve, laid bare, can be stimulated artificially in a number of ways; there are but two kinds of stimuli, however, the effect of which is instantaneous and whose strength can be accurately regulated, namely mechanical and electrical stimuli. Mechanical stimulation has been considerably improved by an artifice of ScHArer'), who used falling drops of mercury instead of electrically driven little hammers. When using droplets the size of which is about equal to the breadth of the nerve, even with a very small height of fall distinct effects we obtain, manifesting themselves by contraction of the muscle which has remained connected with the nerve. Scuirur himself obtained this result with a drop weighing 100 mer. falling through 10 mm. In our laboratory his method gave still better results; an effect was noticed already with a drop of 50 mer. and a height of fall of 5 mm. Such a drop possesses at the end of its fall an energy of 24.5 ergs. Not the energy as such is a measure of the stimulus, however. Apparently the energy has also in this respect to be considered as consisting of an intensity-factor and a capacity-factor *), and this 1) Proc. Physiol. Soc. 26 Jan. 1901. 2?) W. Osrwatp. Ber. d. k. Sächs. Ges. d. Wissenschaften 1892, math. physik. Cl. S. 215. G. Herm. Die Energetik in ihrer geschichtl. Entwicklung 1898 S. 277. > en ( 148 ) after its being transferred to the nerve. That it is not the intensity- factor before the transfer which must be regarded here as the phy- siologically “auslösende” (liberating) force, appears from some curious results obtained on varying the height of fall. A fall of 5 mm. gives an excellent lifting distance when the muscle is loaded with 50 grams, which is reduced to */, with a fall of 15 mm. and to */, with a fall of 30 mm. Consequently a smaller effect is found when the velocity with which the drop comes down is increased, instead of a larger effect. This would be impossible if the intensity-factor of the kinetic energy of the falling drops had been decisive. On diminishing the height of fall again the original lifting distance returns *). The second kind of artificial stimuli can be measured in a very simple manner if at the instance of Cysunsky, Mares, Hoorwre and others, condensers are used. A preliminary trial in the laboratory showed that the best results are obtained with a condenser of a capacity of 0.004 microfarad. This has only to be charged to a potential of 0.012 volts to show already a contraction of the nerve- muscle preparation. The available energy in this case amounts to 0.00029 eres, which is much less than what was found for mecha- nical stimulation *) But it is clear that the one as well as the other is a most unsatis- factory way of stimulating. The mechanical stimulus only reaches part of the axial cylinders constituting the nerve-bundle and it is questionable whether the softness of the mass does not to a great extent obviate the suddenness of the pressure. The electrical discharge, although more instantaneous, spreads in no small measure, besides over the axial cylinders, also over the sheaths and the septa between the separate fibres. A means of diminishing the resistance and so making the time of discharge shorter is hitherto lacking. It is always the full resistance of the nerve, measured across, which one is obliged to put in. These difficulties will never be completely overcome with artificial stimuli; in order to find the real minimum one must have recourse to natural stimuli. It is only with sensory nerves that we have these at our command for the present. There are two organs of sense in which the nerve-cells themselves (or their immediate prolongations) receive the stimulus, viz. the organs of sight and of smell. In the former case it is the rods and 1) An analogous phenomenon has been observed by Wepensky for faradic stimuli (of a frequence of 100 per second). 2) J. Cruzer. Journal de Physiologie et de Pathologie générale 1904 p. 210 gives as smallest values 8, 5 and 7, 2 milli-ergs. ( 149 ) cones, in the latter the fine smelling-hairs which both form part of the terminal neuron. We will consider the stimulation of these senses a little more closely. A. Organ of sight. If one looks through an artificial orifice of 2 mm. at a small Hefner lamp at a distance of 6 metres, a definite amount of the light emitted by the lamp will enter the eye. This energy is con- centrated on a small eireumseript field of the retina, where cones and rods lie ready to receive the light-stimulus. When the room is made dark and the light is made feebler by a system of more or less crossed Nichols, the much less sensitive cones will at last cease to be active and the feeble glimmer that remains, will be perceived by means of the rods only. For this the visual axis will have to deviate a little, as in the fovea centralis itself no rods occur, so that the point-shaped feeble little star seems to be displaced a little upward, at any rate for my eye. It is clear that with such an arrangement of the experiment those rays only will be effective that are absorbed by the purple of the rods. According to A. Könrie *) these are the rays the wave-length of which ranges from 600 to 420 micra. From K. Ancsrrom’s recent determinations *) we know the energy of this part of the spectrum for the Hefner lamp. It is 2.61 X 10~* gram-calories per second and per square centim. at a distance of one metre. At a distance of 6 metres this becomes 0.03 ergs per second and per square centim. In this experiment we supposed the Hefner lamp to be looked at through a system of more or less crossed Nichols, leaving only a feeble glimmer. Besides this we will also insert an instantaneous shutter the time of exposition of which has been adjusted so as to give the most favourable results for feeble visual impressions. From a series of experiments by Messrs. GRIJNs and Noyons, the results of which will be communicated later by these gentlemen themselves, | knew that a time of exposition of the order of a milli-seeond is most suitable. An instrument formerly used by Dr. LAAN and deseribed in his doctor-dissertation *), with slits moving in opposite directions, gave expositions of about 0.6 milli-second, repeated every 0.06 second. By these two means in addition to the narrowness of orifice, hence in all by three circumstances : 1) A. Konia. Stzber. d. Berliner Akademie 1894 p. 585. 2) K. Anastrém, the Physical Review, Vol. XVIIL p. 302, 1903. 3) H. A. Laan, Onderz. Physiol. Lab. Utrechtsche Hoogeschool. (5) IIL p. 182, ( 150 ) 1. crossing of the Nichols 2. shorter time of exposition 8. narrow orifice the stimulus which otherwise would have amounted to 0.03 ergs per second and per square centim. was still considerably enfeebled. The Nichols were completely crossed at 28° 12’ of the scale. Hence full light was obtained at 118° 12. In this position, which is most favourable for the transmission of light, something is lost on and in the Nichols as well as on and in the media of the eye, but we shall neglect this amount, since it is small compared with the uncertainty of the coefficient of absorption of the retinal purple, which we will take into account presently, and of the disturbing influence of adaptation, which cannot be entirely eliminated. When the instantaneous shutter is moving and gives a flash every 0.06 sec. one of the Nichols is slowly turned. The sharp image of the flame disappears and instead of it we see a dot-shaped glimmer, which at last seems to move a little upward. A minimum glimmer I found without previous adaptation to the dark .l coming from the right at 36°. 6 of the scale coming from the left at _ 20° ee ee which means a rotation of the Nichols of 82°.6' and 81°.48' or a mean rotation of 81°.57' reckoned from the position for full light. On account of the crossing of the Nichols we may assume that the intensity in the ratio of cos’. 81°.57' to cos’.0” or as 0.0196 to 1. Besides the time of exposition was only 0.6 milli-seconds every time. Finally the artificial pupil had a surface of no more than 0.0314 em’. By all these circumstances the original quantity of energy of 0.08 eres, contained in the rays that ean be absorbed by the rods, has been reduced to 0.0196 >” 0,0006 0.0314 0.03 ergs = 1.1 10“ ergs. From the measurements of absorption by A. K6niG it follows that '/ of the light of these rays is retained in the retinal purple and since only the really absorbed light must be taken into account for the stimulation, this amount is still further reduced to 0.02 1.1 10=* ergs Rel eres, In a second series of experiments with a time of exposition of 0,00062 sec. repeated every 0.64 sec, and an angle of the Nichols of 84°42' from the position of full light, I found 0,00854 0.00062 0.314 >< 0.03 ergs = 4.9 < 10” ergs. And */,, part of this is again ( 451) OFX ENS 10-8 = 1K 1L0-™ Hess. So the energy capable of causing the impression of an extremely feeble glimmer is, in the case of eccentric vision, of the order of 1.10—' ergs. This quantity would have been found still 100 times less if I had completely adapted myself to darkness. By a corre- sponding method Messrs. GRIJNs and Noyons made such experiments, which will be published later. The transformation of the retinal purple into retinal yellow under the influence of light is a reversible process. Hence it must depend on a displacement of the equilibrium, which towards the end of the adaptation is complete, in a direction opposed to the chemical forces which are the cause of it. The small variation of thermodynamic potential, brought about in the appendix of the retinal nerve-cell by a quantity of energy of the order mentioned, is sufficient to cause a state of stimulation in this nerve-cell. Hence the natural nerve- stimulus can be taken ever so much smaller than the artificial stimulus, even in its most favourable form. B. Organ of smell. If one wishes to make an attempt at caleulating the value of the energy of the natural stimulus in the case of the organ of smell, this can be done in the following manner. In the so-called smelling-box ') (a closed space of 64 litres, having glass walls on all sides), let a smelling substance be diluted to the utmost degree at which it is still perceptible. Let this be done by completely evaporating a few drops of the substance itself or of an aqueous solution of it. Let then a little air from this space be sniffed in. The quantity of air inhaled in a single sniff is estimated by VALENTIN at 50 ee. which I believe to be correct, since in sniffing at my olfacto-meter (which is done unilaterally) 30 ce. are inhaled on the average. If we now suppose that part of this air is directly conveyed to the olfactory fissure, in the most favourable case the air there present will be replaced by fresh air perfumed in the manner indicated. In this case + 0.2 ce. of this air is in contact with the olfactory mucous membrane on both sides of the narrow fissure. We shall now indicate for a few substances the quantity that must be present in 0.2 ce. in order to be exactly pereeived by the smell. This quantity is for 1) Physiologie des Geruchs. Leipzig 1895 p. 34, ( 152.) methyl alcohol 0.000,06 mer. formic acid 0.00012 mer. acetone 0.000.008 mer. camphor 0.000.000.009.6 mer. ionon 0.000.000.000.019 mer. Let us restrict ourselves to these substances for the present. When fully oxidised they are converted into H,Q and CO, Hence their produets of combustion may be considered as entirely indifferent additions to the cell-substance if they are produced gradually and in small quantities. If we regard smell as an intra-molecular property, which is probable for these substances (leaving it an open question whether for substances containing atoms like |S, As, ete. smell may perhaps depend on intra-atomie conditions), the quantity of energy involved in an olfactory stimulus will never exeeed the heat of combustion. For the four first-mentioned substances the heat of combustion is known and amounts to 5.7, 1.5, 7.5 and 9.3 gram-calories per milli- gram respectively; that of ionon is unknown, but may be estimated at 9.6 gram-calory. Then the quantities of energy involved here (for the amount of smelling substance contained in 0.2 ce.) are for: methyl-alcohol 14443 ergs formic acid 1460 5. acetone 25 | is camphor Fr! eee 8 ionon 0.008 ,, Attempts made in order to find out how much of these smelling substances is absorbed in the olfactory mucous membrane were hitherto unsuccessful. Neither in oil, nor in nerve-substance we were able to detect an appreciable quantity of ionon, after they had been left in contact for some time with an atmosphere of ionon. Presumably very little is absorbed. Moreover it follows from the remarkably different degree in which chemically related substances show smelling power’) that only an extremely small part of the intra-molecular energy displays any olfacto-chemical effect. So we are justified in assuming that here also the minimum stimulus in the nerve-terminal will later appear to be of the order of the light- stimulus or even smaller. 1) Onderzoekingen Physiol. Lab. d. Utrechtsche Hoogeschool (5) IV. p. 232, Chloroform, bromoform, iodoform have a specific smelling-power of which the mutual ratio is as 1:69: 155 24. ( 153 ) So of the direct natural nerve-stimuli moments only the light-stimulus has for the present been quantitatively calculated. Bearing in mind that the stimulus we found in what precedes to be of the order 10—-" ergs, is a minimum stimulus on a small field of the retina scarcely measuring 0.002 square millimetres, whereas the total surface containing retinal purple is put by Körie at 700 sq. mm. ; moreover bearing in mind that the light of the sun at Marseilles is estimated by Fapry*) at 100.000 candles and that this enters the eye not under an angle of 23’, under which the Hefner lamp was seen, but from all parts of the field of view, it will be clear that the light- stimuli of ordinary life can by no means be called immeasurably small. Nor do they act for a single milli-second but all day lone. The energy entering the nervous system in this way will not be a fraction of an erg but a number of ergs. It is difficult to make an estimate in this respect as with strong light not the rods but the cones serve as terminal apparatus and in these teleneurons only by analogy a photochemical process is assumed, which for the rest is unknown to us. Therefore we must refrain from such an estimate, but at the same time it seems to us to be undoubtable that the nervous system receives relatively large quantities of energy in a different way from that by metabolism. 1 Compt. Rend. 7 Dec. 1903. (August 26, 1904). KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN TE AMSTERDAM, PROCEEDINGS OF THE MEETING of Saturday September 24, 1904. DEC (Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige Afdeeling van Zaterdag 24 September 1904, Dl. XIII. CENT PEN TS. J. D. van per Waars: “The derivation of the formula which gives the relation between the ecncentration of coexisting phases for binary mixtures”, p. 156. G. C. Gerrits: “On Px-curves of mixtures of acetone and ethylether and of carbontetra- chloride and acetone at 0° C.”. (Communicated by Prof. J. D. van per Waars), p. 162. (With one plate). J. J. van Laar: “On the latent heat of mixing for associating solvents”. (Communicated by Prof. H. W. BAkuvis RoozEBoom), p. 174. A. Smits: “On the phenomena appearing when in a binary system the plaitpointeurve meets the solubilitycurve”. 3rd Communication. (Communicated by Prof. H. W. Bakuurs Roozenoom), p- 177. (With 2 plates). A. F. Horieman: “The preparation of silicon and its chloride”, p. 189. F. M. Jarcer: “On the preservation of the crystallographical symmetry in the substitution of position isomeric derivatives of the benzene series”. (Communicated by Prof. A. P. N. FRANCHIMONT), p. 191. J. C. Krurver: “Evaluation of two definite integrals”, p. 201. C. A. J. A. Oupemans: “On Leptostroma austriacum Oud., a hitherto unknown Leptostro- macea living on the needles of Pinus austriaca, and on Hymenopsis Typhae (Fuck.) Sacc. a hitherto insufficiently described Tuberculariacea, occurring on the withered leafsheaths of Typha latifolia”, p. 206. (With one plate). C. A. J. A. Ouprmars: “On Sclerotiopsis pityophila (Corda) Oud., a Sphaeropsidae occurring on the needles of Pinus silvestris”, p. 211. (With one plate). Eve. Dusois: “On an equivalent of the Cromer Forest-Bed in the Netherlands”. (Communicated by Prof. K. MARTIN), p. 214. H. KaMERLINGH Onnes and C. ZakrzewskKi: “Contributions to the knowledge of van DER Waars’ ¢-surface. IX. The condition of coexistence of binary mixtures of normal substances according to the law of corresponding states”, p. 222. (With 2 plates). H. KAMERLINGH Onnes and C. Zakrzewski: “The determination of the conditions of coexis- tence of vapour and liquid phases of mixtures of gases at low temperatures”, p. 233. (With one plate). J. Weeper: “A new method of interpolation with compensation applied to the reduction of the corrections and the rates of the standardclock of the Observatory at Leyden. Houwü 17, determined by the observations with the transitcircle in 1903”. (Communicated by Prof. H. G. VAN DE SANDE BAKHUYZEN), p. 241. The following papers were read: id Proceedings Royal Acad. Amsterdam. Vol. VII. ( 156 ) Physics. — “The derivation of the formula which gives the relation between the concentration of coexisting phases for binary mixtures.’ By Prof. J. D. vaN DER WAALS. (Communicated in the meeting of June 25, 1904). Already in my molecular theory (Cont. II, p. 10) I derived a formula for the concentration in coexisting phases of binary mixtures. This formula has the following form: | db da | db da a de la ' a de da jer’ sate dn “| =") MRT 1 4 RT 1—wz Jef) Omis 1 — WV N= 8 In the case that the second phase is a rarefied gasphase, the Lv second member is simplified to METS and we find: —W da db a eae da). a Wh Pt ee |= _ MRT a (1) le, ©, 1 a v—b From this 1 have drawn the conclusion that the circumstance that two coexisting phases have the same concentration can only occur ; ; 5 3 sk ; ; Ur for mixtures, for which a minimum value of the quantity 5, occurs, a and so a minimum value for the critical temperature. For the limiting case, with exceedingly low values of 7’, the mixture for Ar LE which b, has a minimum value, would be exactly the mixture, for which the value of « is the same in the two phases; but for in- creasing values of 7’ this concentration shifts to the side of the substance with the lowest value of the size of the molecules. (Cont. Ll, -p: 19 and p. 120): Afterwards I have derived in “Ternary Systems’ the following equation: d for equation (1) f dT 1 dp. 1, ve DE pu de which also holds only approximately for the case that the second phase is a rarefied gas-phase. For the derivation of (2) I have not directly used the equation of state, but I have considered the well- p TT i known formula for the vapour-pressure — aaa | BE a suffi- ciently accurate for liquid volumes which are not much smaller (157 ) than that of the pressure of coincidence (pressure of the saturated vapour for the unsplit mixture). Equation (2) however, can also be found directly from the equa- tion of state. It was to be expected that this was possible, because as I have shown in “The liquid state and equation of condition” !) the formula for the vapour may be derived from this equation. If we want to find also for the factor f the real value of about 7, it is necessary to consider 5 as function of the volume. This not only renders the derivation very complicate, but it places us before the unsolved question: in how far is the decrease of 5 with the volume to be ascribed to real or quasi diminution ? I have therefore confined myself for the moment to examining what follows for the form of (2) from the equation of state, when b is put independent of the volume. We have then to reduce: … db da MRT — = da du is b co TE We write for this successively : db da eee da du a db ] da ——=|[p-++- — = v—b v v/de vde a db b a a \ db tf 1\ da rae et = Se apie Tk Er de da v b? ) de v b de 1o/T 1 1 iy. d Now for a = G — zl = 5 ed we may write: C wv v? b? v b) de a(v—b) (1da v+b1db Ee a(v—b) (1 da 2 db | b\ 1 db bv adc. » bda\. bv a de b da v'}) b da : : a(v—b) . and as according to the equation of state is equal to a(v—b) me ig v—b MRB (MRT — op) — = bv b we find after some reductions: MRT EE i a dl dln Av Av ao ) je DE ) on MRP REET ann v—b v ae de a i de | b (EE p) de a v—b1 db ees eer en RE eee) b bde 1) These Proc. VI. p. 123. 14% ( 158 ) We may also write the second member of (3) as follows: a \ | Ib eee (¢ (|) DI av » rg) aes RS tah Sic ales Pilar cag, oar ee b* de! a a a ii de d log 25 b 2 Ep Bs, a ee ENE RTS ee ee oe Sa da dx v dau In order to examine the general value of the quantity which is to be reduced, we have to distinguish two cases. The first case, that v — hb is small and p(v— 6) may be neglected compared with MRT. In this case (4) may be simplified to: Je / / ab ds denn ei nae da da da The second would hold for high pressures; then the value of p(v— b>) approaches to MRT, when v approaches to 6. In this case (4) is simplified to: a a db b oe eae As we assume coexistence with a rarefied gasphase we have only to deal with the first case. In the second there would not even be question of coexistence with a second phase. We find now for the formula, giving the relation for the concentration of the two phases : a a of = d l a 2 | Aal dee Bend 8 1 b Rs l sten l—w, &, „MRT da dv d ab Ad in which p is neglected, or rather where it is cancelled by an ae almost equal value, which would occur in the second member of the equation given at the beginning of this paper. Let us put: : 8 a MRTT 57 7 and leg Pk = 27 Db: ’ then (5) assumes the form of: ies See ek 27 1 aT, _ dp« la, w Trad SAT ada: prde), 2 ( 159 ) The factor is in perfect concordance with the factor which AE: occurs in the formula of the vapour pressure, when we put the quantity 5 independent of the volume. I have shown before that it must be about doubled, when we assume variability for / — or rather ' cd ee the factor 3 not increased, but the assumption of the smaller value of 5 comes to doubling the factor, when we substitute the a value of 7) for rk Without carrying out the elaborate calculations, which in our case might be the consequence of assumption of the variability of 6, I think to be justified in concluding to the doubling of that factor as a sufficiently approximated value. Then we find back exactly the same value as I had found in “Ternary Systems”, viz. B Dae ndr 1 dp, pas ab aA Wate Seer (OA 1 q a, a be ds Lda pr da in which formula f may put about 7. fig dlp. ee ee Gveeinay put for : 8273p, , we may put for a dlogT;, dlogb da er Hence (6) becomes : PSD i WNT 1 dh 5 l = = = a 1 +- . . . . ( () ft & Ji Ti) de b dx 1 2 db From the form (7) we derive, that only when = =~ 0 änd so Av when the molecules of the mixed substances are of the same size, the concentration of the coexisting phases is the same for the mixture with minimum critical temperature. If the size of the molecules is not the same #, == #, for the mixture for which ve ] dT). 1 db ye a et Ae 1 Ty} da b de . db . . . . . . . If 7, is positive, as is the case for mixtures of acetone and ether ada (ether as second component), then «=, for a mixture for which dT. av is negative. Then the concentration where «, and v, are equal and therefore also the maximum pressure in the p, line has shifted to the side of the component with the smallest molecule. If we multiply both members of (8) by 7, the shifting proves to increase ( 160 ) for increasing value of 7; and so we arrive at a conclusion, to which I came already before, viz. that the concentration of the maximum value of p in the p,# curve is sufficiently the same as that of the mixture with minimum critical temperature only for the very lowest values of 7. It only appears that already at ordinary temperatures the shifting mentioned above may be rather considerable. A consequence of this is, that the shifting between the ordinary temperatures and 7’= 7), may be only slight. This shifting is however the greater as the difference in the size of the molecules is the more considerable, and as the decrease in critical temperature takes place the more slowly. Now that we have found an approximate value for w'‚, we can immediately derive from it an approximate value for w'‚, a quantity whieh must be known, if in the equation: v,, dp = (*,—#,) (5 *) de, + a dr re) pT 1 the factor of dz, is to be considered as known. We have viz. (53) URT | L B | Ik ET sn On? wT (lar) nt We find then: a a d— el— , 1 h Lb Be SRT Gok Soave or ieee ie KA dT Fal Mede da? So for small vapour pressure the equation : heek lpr, 1 (dp 1 Pp de, T Cet ole) mn da, 54 da? holds approximately. 277 In general the quantity will be positive, and this is certainly Avs” so when there is a minimum value for 7; the value of the other term may change this of course. But as a rule uw’, will be found negative for normal substances. T (dp In the value of the quantity (5) only one of the two parts ) en ded £ Er BAe dlp. Oly, OCEUFS WIZ. Zn and not the other part AL Av ey af part depending on 7'is kept. In different ways the value of this quantity So only the ( 161 ) may be found. It is easily found from the equation, occurring in Cont. Il p. 146, slightly reduced, viz: — — = (Ll — e,)+ ®, el u hike Wik el From this form we derive, keeping «, constant: py! du, CD He dp Le,” : ar d (Ux, — %, Ux, —1) pdT {A ea NEL dT for which we may write: dp i Wy de We) pd gl es dT dT or dp ih dur > Ute, Wa) ae ee pdl | di dl du’, y, du, Bog —— eres dim = and. for” we ge ralue qr i pide OT ag We get the value ih 1 oY : T? k 4 Hence: Tdp ia dT). En ee by &,—@ i aos i | Rare ae it) dx, Multiplying the second member by MRT, we find w,,. For wy, we find then 2 terms, the first J/2 f 7), representing the heat of evaporation, when the mixture «, evaporated as an unsplit substance, and therefore the vapour phase would have the same concentration mn dT}, (U) denotes the mo- f de, dification, which is the consequence of the circumstance, that the vapour phase has another concentration than the liquid phase. This as the liquid phase. The second part modification can be very considerable in certain cases, viz. when Lt, is very large. If 7, should depend linearly on 2, then my \ dT}, ryy . ryy . ° : ly (ty — @) a 7’., and for 7), we might write in that case, at < id 1 denoting the components of the mixture by « and 6: 7),= Ti, . = C Ee Ge Lie T dp (1—@,) -- Ti, z,. (Cont. II, p. 155) or ai = (1—a,) —- — + ) vo fi ee Pa dd gh dp, wv Then w,, = Me ra A —a,) + Myro 2,, and the process 7 ny 7M of mixine in the liquid state will take place without heat of mixing, o le) 21 ( 162 ) If the graphical representation of 7% as function of x, is a curve (Cont. IL, p. 45), lying everywhere above the tangent, which is the a, a, ad A dT}, case when IE + RN is positive, then 7, + (7, —4#,) de. smaller than 7. If we draw a tangent to the curve in the point «,, this tangent cuts the ordinate of 2, in a point which lies lower than the curve, and the distance from that point of intersection to the curve is a measure for the quantity of heat required for mixing the condensed vapour with the liquid phase considered. As u”, consists of two terms, the latter of which is only negative, when the mixing in the liquid state is attended by absorption of heat, we are not justified in expecting that this latent heat of mixing alone determines the sign of wu"). is Physics. — “On Pr-curves of mixtures of acetone and ethyl-ether and of carbon tetrachloride and acetone at O° C.’ By G. C. Gerrits. (Communicated by Prof. J. D. vaN DER Waats). (Communicated in the meeting of June 25, 1904). The imperfect concordance found by Cunaxus *) between the rela- tion deduced by var per Waars ®) in his theory between the vapour tension over a mixture of two liquids, the molecular concentration of the vapour and that of the liquid, induced us to take up the investigation once more according to the same method as had been used by Cunanus and with the same substances, acetone and ethyl-ether. Afterwards also mixtures of carbon tetrachloride and acetone were examined. It had viz. appeared, that improvements might be applied to the method of investigation. By means of the determination of the refractivity of the vapour, both of the simple substances and of the mixtures, the molecular concentration of the vapour was determined by means of the law of Bror and Arago. This determination of the refractivity was made according to the method of Lord Rarreien *) also followed by Curarvs *). 1) Cunaeus, Proefschrift, Amsterdam, 1900, blz. 47—51. 2) van DER Waats, Arch. Néerl. 24, blz. 44; Continuität des gasf. und flüss. Zustandes II, blz. 137. 3) Rayreien, Proce. Roy. Institution, Vol. XV, Part. 1, pag. 1; Proc. Roy. Soc. Vol. 59, blz. 201. 4) Cunaeus, Proefschrift, blz. 4—6. Proc. ( 163 ) Rarreren always compares the gas, the refractivity of which he wishes to determine, with dry air free from carbonic acid. He takes care that the gas and the air are both in such circumstances that their refractive indices are the same in these circumstances, which may be tested by means of a Fraunhofers’ diffraction phenomenon. This equality of refraction indices is obtained by regulating the pressure of the two gases. With the aid of a cathetometer and the barometer the pressure is determined, and from the two pressures the refractivity of the gas. This is applied to mixtures of vapours coexisting with mixtures. The pressure of the air, the refractivity of which is compared with that of the vapour, is regulated in such a way that the vapour and the air are in such circumstances that their refractive indices are the same. From the relation between the pressures the molecular refractivity of the vapour (or of the vapour mixture) may be determined, the law of Bror and Arago gives then the concentration of the vapour mixture «,. The vapour pressure is of course, also known. The concentration of the liquid was obtained by weighing the original quantities of liquid brought into the apparatus, and diminishing each with the weight of the quantity of that substance in the vapour. The arrangement agreed on the whole with that of Cunagus. It was a modification of one given by FRAUNHOFER based on the interference of light, which fell through two vertical parallel slits, after having passed through two tubes of equal length, shut off by two plates of plate-glass. Then it was converged by the object-glass of a telescope and the diffraction phenomenon brought about in this way, was observed by means of an eye-glass consisting of two cylindric lenses. In one of the tubes mentioned the vapour is found, in the other air or dry carbonic acid, the refractivity of which was known, and the pressure of which was now regulated in such a way that the diffraction phenomenon occupied the place which it would also have taken, when the two tubes had been filled with the same gas under the same circumstances. The two tubes were connected with open manometers and an apparatus to increase or decrease the pressure of the air. The improvements which were applied, were chiefly the following : dst. The two substances which were used for the determination were sealed in two separate small pieces of glass tube, so that the air could be removed from the liquid as much as possible by boiling the liquid under decreased pressure. ( 164 ) KonnstaAmM ') has proved that the way in which these little tubes are filled, may give rise to errors in the determination of the vapour pressure, even when they are filled with a simple substance, and not as with CuNArus with a mixture. ”) 2nd, By adjusting a glass spiral between the globe in which the liquid was brought and the experimental tube the globe could be violently shaken, which prevented insufficient mixing of the layers of the liquid. 3°, The difference in appearance between successive fringes was very slight with Cunanus, which rendered the adjustment of the movable interference phenomena compared with the firm, pretty un- certain, By taking a more favourable relation of the distance between two corresponding edges of the slits to the width of each of those slits, spectra were distinctly visible on the sides of the bands, all turning their violet rim to the middle fringe. So there was a means of ascertaining the latter. Moreover it was investigated whether possible absorption of vapour on the plates of plate glass would have influence on the course of the pa-curves. This appeared not to be the case, however, so that this was no longer done in the examination of the second mixture. The temperature at which the substances were examined, was 0°C. The substances were obtained from KAHLBAUM at Berlin and were purified before use by distilling them a few times by means of a Youre and Tuomas’ *) dephlegmator with twelve constrictions. The mixture acetone-ethylether. The results of the experiment are stated in Table I. (p. 165). For the refractivity that of dry air free from carbonie acid has been taken as unity, the pressure is given in mm. of mercury at O°C. Ethylether must be considered as admixture. . The pe, ev, diagram obtained in this way is graphically repre- sented in fig. 1. From the course of the curves appears: There is a maximum pressure, just on the side of the ether. The point of inflection found by Cunrarvs is also found here, and af about 2, = 0:66. 1) Konystamm, Proefschrift, Amsterdam, 1901, blz. 170—180. 2) How difficult it is to get the substance pure and free from air has been lately observed once more by Teicuner, Ann. de Phys. 13 p. 603. 3) Sypney Youre and Tomas, A dephlegmator for fractional distillation, Chem, News, 71, blz. 177. Pressure. atta Ta ry 69.08 3.7658 0. 0. 80.27 | 4.040 | 0.173 0.024 98 .03 4.3097 | 0.342 0.083 107.03 4 4353 0.421 0.124 127 Ol 4.6458 | 0.993 0.232 134.40 4 7036 0.590 0.295 164.72 4.9548 0.748 0.561 180 15 5.1112 0.846 0.796 185.60 5.3562 1. ie The p= / (@,) curve is convex from «=O to = 0,66. The curve p—=f(«,) has a very simple course: it turns every- where its concave side to the z-axis. From the course of the curves follows finally, that the vapour is always richer in admixture than the liquid, the greatest difference between 7, and 2, amounting to about 0,33. These curves were now compared with the differential equations derived by vaN DER WAALS: dp Pt 2 1 Pp \ de, Er En Geer), which the curve p= /(r,), must satisfy throughout its course, when the volume of the liquid may be neglected by the side of the volume of the vapour and when the vapour phase may be considered as being rarefied, and: 2 1 At dp ot p ; da, bi ©, (1—2,) which must hold chiefly for the borders of the curve p= (@,) under the same conditions. 4 dp ; : From the figure p and ee have been found for the values ate of x indicated in Table II, then «,—, is derived from the equation and from this #, and these values are compared with the observed values : ABUL ee, dp apes | “ 5 dat sae E DEE. caf. | observed. 0.1 74 71 0.086 | 0.084 0 014 0.016 0.2 82.5 88.5 0.172 0.170 0.028 0.030 0.3 93 109 0.246 0.238 0.05% 0.062 0.4 104 133 0.307 0 292 0.093 0.108 05 119 158 0.332 0.315 0 168 0.185 0.6 136 188 0.332 0.314 0.268 0.286 0.7 15% 190 0.258 0.250 0.442 0 450 0.8 174 140 0.129 0.122 0.671 0.678 0.9 183 73 0.004 0.004 | 0.896 0.896 The differences between the observed and the calculated values of v, are evidently slight, so that the observations may be considered as satisfying the equation in question. When testing the observations by means of the second equation, we got the following result: TAB Lel ank dp ba) %g 2 de, cale. | observed ) DRIE | observed 0 05 90 385 0 203 0.22% 0.253 0.274 0.1 102 252 0 222 0.280 0 322 0.380 0.2 122 198 0.260 0 322 0.460 0.522 0.3 138 156 0.239 0.314 0.539 0.614 0.4 149 5 119 0.191 0.276 0.591 0 676 0.5 159 102 0.160 0.224 0.660 0.724 u.6 168 83 0.119 0 168 0.719 | 0.768 0.7 176 79 0 094 0.120 0.794 0.820 0.8 180.5 16 0.044 0.052 0.811 0.852 0.9 183 5 12 0.004 0.006 0.904 0.906 ( 167 ) The closest concordance between the observed and the calculated values of v, seems to be found at the edges of the curves. (ow | Van DER WAALS derived from the equation of (sr) for both phases: d v.T' - p 2 U. é ee — 1 tk it ois in which: ; ae mA | a Al a Ti a7 Kn ( ) and (ty, = MRT joe — MRT log (v — ba) — — | ; v With the aid of the observed wv, and w, we may derive w',, as function of #, from this equation. The values found in this way are given in Table IV. For x, smaller than 1 and larger than 0,9 the values of ws, are too inaccurate, so that they are not given for the borders of the curves : dE rs gel ed Gy a, fs Pr AEN 0.4 0.380 5.516 | 1.74 0.2 0.522 4.368 | 1.47 0.3 0.614 So ol wr 0.4 0.676 3.130 | A44 0.5 0.724 2.623 | 0.96 0.6 0.768 2.207 | 0.79 0.7 0 820 1952 | 0.67 0.8 0.852 1.439 | 0.36 0.9 0.906 1.071 | 0.07 The function w'‚ as function of wv, has been graphically represented in fig. 2. Evidently the points give a pretty continuous curve. It appears from the table that gw’ approaches to O for higher values of w. This is in accordance with the fact that the p2-line has a maximum on the ether side, so for vw, = 1. Van per Waats has given *) an explicit expression for p as function 1) van per Waats, Versl. K. A. v. W. Amsterdam, Januari 1891, bl. 409; Continuität II, p. 146. ( 168 ) * dp of z,, from which ze may be derived : dx, rf IJ dp (e Te) pst SS > . ’ du, „ J tb eee 71 Bo, fora de dp ce FES NR ET = If p is a maximum for 7, = 1 then ty, = 0. Van per Waars has also given an approximate value for ws, in “Ternary Systems” *) : F der d log per EA de, dz, where 7 and p,, represent the critical temperature and pressure of the unsplit mixture and / the known constant, about 7, which inter alia also occurs in the formula, given much earlier by VAN DER ' : Pp T,—T Waars for a pure substance: Nep. log — = — f —— Pk 1 This formula for ws, has been derived by van per WAALS also directly from the equation of state *). If it only contained the first term, then a minimum critical tem- perature would be attended by a maximum pressure, the minimum would therefore lie near the pure ether. Now however this minimum will not be found there, as appears from the occurrence of the ! Eo == second term. If we now assume that 7, depends linearly on « and that this is T dp, T is about — 48 and about — 15, C vy ( vy in consequence of which we find for w,, about 0,8, a value which also the case for p,,, then really lies between the values found. Also u’, the differential quotient of w’,, with respect to 7, may be determined. The accurate relation derived by vAN DER WAALS for the dependence of the vapour pressure on the molecular concentra- tion of the liquid mixture, where it is only assumed that the liquid volume may be neglected by the side of that of the vapour, and that the vapour phase may be considered as a rarefied one, is 1 mn on re 1 dp Pp de 1) van DER WaAts, These Proc. V, p. 9. 2) van DER Waats, These Proc. VII p. 156. ,-= - —=- " ( 169 ) From this u’, may be found. These values, calculated, are given mm Fables U: EA BE BV. ©, p a Ta — 2 zi) ten | Bey 04 102 252 0.280 8.8 aya 0.2 122 198 0.322 5.0 EE We) 0.3 138 156 0.314 3.6 x #2 0.4 149.5 | 119 0.276 2.9 NE 0.5 159 102 | 0.24 2.9 ie 0.6 168 83 0.168 2.9 ane 0.7 176 79 0.120 3.1 =e 1d 0.8 180.5 46 0.052 4.9 By ee 0.9 183.5 12 0.006 410.9 dig: The values given for #,=0,7, 0,8 and 0,9, however, are not very reliable, on account of the small value which «,—., then has, by which the first member is to be divided. For the same reason w'‚ cannot be given for z,, smaller than O,1. Evidently all the values of w'‚ are negative and as to their absolute value, they are smaller than 4. This in accordance with the 2 0°5 fact, that for stable phases, (==) must be larger than 0. wv p “1 From the course of the curve ws, =/(#,) appears that the diffe- rential quotient of about «0.2 to «= 0.6 must have a constant value. And this is really found for u. With regard to u’, we may still remark, that its numerical value may be found also for the x, of the inflection point and the corre- sponding «,. VAN per Waars has namely derived : *) DM ul 2 ST % 5 dp dp de,[ 2(e “—1) La" Ame | Ks I) — hd 1 pi oie 7 Tt RET — e dx*, dat, dit, 8 Ti # be ry 7 Js ry —1 | 1- w,+a,e law, He e i d*p : ; : ue } ; ‚dp de, —— being 0 for the point of inflection, the factor of — .—— dar,” de, de, must be 0. 1) These Proc. Ill p. 172. (170 ) / By ©, lg With the aid of the relation e Dee. we fint from this: 9 (v,— 2)" a x, (le) 2 (ea) He, (le) For the inflection point we have about z, == 0,66 and wv, = 0,40. For u, we find then about — 1,5. Taking into consideration that the place of the point of inflection cannot be determined accurately, and that with change of the z, of this point at the same time also its x, is changed, the agreement may be called satisfactory. ” ae Un = The mixture carbontetrachlorid-acetone. These two substances, carbontetrachloride and acetone were chosen with a view to the critical pressure, which is about the same for them. Now Van per Waars ®) has derived, that for pw’, = constant the relation between p and a, is represented by a straight line, that between p and w, by a hyperbola. On the supposition that at low temperature for gw’ ‚, may be written : 1 bs GALITZINE —BERTHELOT a@,, = Wa, a, holds good, the condition wt’, = constant involves, that the critical pressures of the components are the same. Now the above mentioned improved approximation for w‚, has already been given by Van per Waars himself instead of the one ‚ for b,: 4 (6, +h,) and that also the relation of mentioned here; moreover the relation a,,—= Wa, a, has already been discussed and rejected by KonnsraMM.*) It is therefore not surprising, that a strong deviation from the straight line and the hyperbola was found for this mixture *). The px,v, diagram is given in Table VI. Acetone is to be consi- dered as admixture : 1) These Proc. [IL p. 168 — 169. 2) Kounstamm, Proefschrift, blz. 99. 3) From the survey of the investigated mixtures given by Konystamm, Zeitschr. f. Phys. Chem. XXXVI. 1, it appears also, that equality of critical pressure of the components does not necessarily involve the existence of a straight line for the p=f (x) line. Pressure. | ah re fs vy 34.20 | 6.1407 0. 0. 41.46 5.7302 0.172 0.044 16.43 5.4772 0.281 0.122 52.27 5.1253 0.430 0.23% 59.47 4. 9261 0.515 | 0.333 60.43 4.6135 0.647 0.477 64.49 4.3219 0.771 0.696 69.32 daalde | fl. 15 In fig. 3 it is graphically represented. Both the curves have a very simple shape. They turn their con- cave side to the v-axis. The vapour is always richer in the admixture than the liquid. The greatest difference between , and z, is about 0,190. Both the curves are now compared with the differential equations: 1 dp Dd, P de, U, (la) | dp dd, p dx, uv, (l—«,) The results are given in table VII and table VIII: FABLE VI. | dp | Td, zy is | - dit, | Sica’ | been | eaten later abe 04. | 39 39 0.090 | 0.090 0.010 0.010 0.2 | 43 37 5 0.139 | 0 t4t 0.061 | 0.089 0.3 | LE HE | 0.178 | 0.135 | 0 192 04 | st 36 0.169 0.190 | 0.931 | 0.910 05 | 55 35.5 | 0.161 | 0.18% | 0.339 0.316 0.6 | 39 35 40.482 | 50.165 “)) 0,458 0.435 0.7 | 62.5 | 29 | 0.097 0.122 0 603 0.578 0.8 | 65 24.5 | 0.060 | 0.080 | 0.740 | 0.790 0.9 | 67.5 | 93 | 0.031 0.033 0.869 0.865 12 Proceedings Royal Acad. Amsterdam, Vol. VIL eves TABLE VII. dp Ts => Tj | Tg dl p tia aa | calculated | observed | calculated observed 0.05 | 41.5 «402 0.117 0.120 0.167 0.170 0.4 45.5 72 0.143 0.160 0.243 0.260 0.2 50.5 48 0.152 0.180 | 0.352 | 0.380 0.3 54,5 | 39 0 150 0.182 0.450 0.482 0.4 | 38 33 0.137 | 0.172 | 0.537 | 0.872 0.5 61 24 0.098 0.154 0.598 0.654 0.6 63 20 0.076 | 0.120 0.676 0.720 0.7 64.5 15.5 0.050 0.078 0.750 0.778 0.8 66 15 0.036 0.040 0.836 0.840 0.9 B | Us 0.019 0.020 0.919 0.920 Evidently the curves satisfy the first differential equation much better than the second. For the second the closest agreement is obtained, here too, on the borders of the curves. Also for this mixture mw), as function of 7, was determined by means of the relation : & 1 1 — ~ fn a= lr, In 2 The values, found in this way, are given in Table IX: TA aa Ae des ee ot | 0.20 | 3462 | 1.45 0.2 | 0.380 | 2.452 | 0.90 0.3 | 0.482 | 2.471 | 0.78 0.4 | 0.872 | 2008 | 0.70 0.5 | 0.688 a) 4.890 "| “0:6: 0.6 | 0.720 | 4.714 | 0.54 0.7 | 0.778 | 4.502 | 0.41 0.8 | 0.84 | 41.313 | 0.97 0.9 | 0.920 | 1.278 | 0.25 / G. Che at O° C.” mM. 200 180 160 740 20 100 40 ad 06 QF 48 a9 Proc C. GERRITS On Px-curves of mixtures of acetone and ethyl-ether and of carbon tetrachloride and acetone at 0° C.” G. C. Eas, mM. 7M. Of 02 03 0 05 06 or 08 09 ar 02 03 a4 5 06 A7 C8 09 Fig. 2. Proceedings Royal Acad. Amsterdam. Vol. VII. mM. aM. 70} 70 de 65 les LA Me Gok 55, je 50 150 45 45 40 40 35 35 = 3 30} 4 a AE 7 3 D5 30 de 01 02 0.3 04 05 06 0. ¢ Fig. 3. (173 ) Here too the values of uw, are not accurate on the borders of the curve. In fig. 4 the results have been represented. Here too the points give a continuous curve. The critical pressures of the components being the same for this mixture, the second term in the equation: je Qa oe d loy Per ! Ul nk an ro ; i Saas da, will be small compared to the first. If we take for w'‚ only the Pi dT cy ge . rn term — Td and if we assume linear dependence of 7, on 2,, Ae 1 nd : : then is here about — 40, so ws, about 1. In this neighbourhood av 1 also the values of u',, are found. Also in this case u, is determined from the equation: l d 1 == nee Ei — dt PE Jen — ge . Wo t 1 Ka p de, uv, (bw) The values found from this are given in Table X. Web Elin X. Ot | 455 72 0.160 9.9 eta : Ie le BOBs 48 0.180 5.3 1.0 03 | 545 | 39 0.182 3.9 —0 9 0.4 | 58 33 0.172 3.3 —0.9 05 | 6t 4 | 0 184 2.6 id 0161. “63 20 0.120 26 —45 OF Gh || 15.5 0 078 3.1 a iy 08 | 66 15 0.040 5.7 —0.6 0.9 | 68 14.5 0.020 10.7 | 0.4 Here too the value of u’ for 7, =0,7, 0,8 and 0,9 is not very reliable on account of the small value of wr, The quantity «",, proves again to be negative and in absolute vaiue smaller than 4. From the course of the curve wy, = f(7,) appears that it seems to have two points of inflection, one at about v, = 0,3, the other at about v, = 0,7. It is remarkable that this also follows from the change of the u“, in Table X; just as we had to expect from the course of the curve w,, = /(7,) u’ gets a maximum at about 2, = 0,3, and a minimum at a, = 0,7. 12* (174) Chemistry. — “On the latent heat of mixing for associating solvents.” By J. J. van Laar. (Communicated by Prof. H. W. Bakauts RoozeBoom). (Communicated in the meeting of June 25, 1904). 1. When some substance is solved in an associating liquid, as e.g. water, and we try to find an expression for the latent heat of mixing of these two substances, we shall in the first place have to take into account, besides the change of the potential energy, the heat of tonisation of the solved substance, if this substance is an electrolyte. The fact, however, that the state of association is changed by the solving, is nearly always overlooked. We are inclined to reason, that in much diluted solutions the influence of the addition of a few molecules of the solved substance must necessarily be exceedingly slight, with regard to the degree of association of the solvent; but in doing so it is overlooked that the umber of molecules of the solvent which each undergo a very slight change in their state of association, is very great. For infinitely diluted solutions therefore, a value is obtained approaching to OX oe, and I shall demonstrate in what follows, that the absorbed eat in consequence of the change in the state of association, approaches to a definite value, which is finite and even comparatively high. 2. In diluted solutions — which we solely have in view in the following pages — the state of equilibrium of the associating mole- cules of the solvent may be expressed as follows: (Ce 2 29 — K, ‘/, (Ae) (1—8) N eae 1—p N 56 For, given 1 — rt mol. H,O, normally reckoned, « mol. salt (calling the solved substance sa/t for convenience’ sake), then there are */,(1—w) mol. H,O, if all are double. Therefore if the degree of dissociation of these double-molecules is 3, then there are: /, (1—#) (1 — B) double mol. ; /, (1—«) 28 = (1 — 2)@ single mol. The total number of particles is .V. If the degree of dissociation of the saltmolecules is a@, then there are (in binary electrolytes): «(1 — a) neutral mol. ; 2va Tons. We have therefore: N= */, (le) (1 + 9) + 2(1 + a), or with */,(1+8)=y , 1+ @=7, where therefore 7 has the usual meaning, and y is the reverse of the so-called association-coefficient: N=y (1 — 2) + ie = y (1 — 2) |: + — ae y le Our equation (1) becomes therefore: or y being = /, (Ll + 8): B 5 tw a Ue KEY Wt Se ska oon a ee ( a oo @) ve! . ae NG From this follows, putting GE Y iN git tA ye (la ON te Posh, ie va ltd On ERE (DE GEEK i SES ie n= ae zend 1+/, K 1. e. the value of 9, if « or d— 0, so that we have got the pure Now evidently Bo 8 : solvent, for which the equation EE —='/, K holds. Therefore we — Bo obtain : ltd 1— 8? i ee B, VA wes Sap € + ers —— s) 9 Po = if, JS being very small and approaching to O, we content ourselves with a first approximation. Substituting for dits value, and taking into account that y= '/, (1+8), we obtain : 3 2 1 1—8,” i # =o IER 1/, (148) Ln . or with 1+ g=1l+4g8,: B=B, (fd — Dis) eee 7 (176 ) So this is the sought-for expression for the change in 8,, caused by the addition of « gr.mol. salt. 1 —wv t v 3. Now on 1 saltmol. there are or.mol. H,O (normally 1—a 1 reckoned), among which there are evidently iS Ve emo B Lv single mol. In consequence of the fact, that the state of dissociation of the watermolecules is changed by the solved substance, this number according to (3) will amount to 1l—@z | — B= v Lv a lr ne Be ie Kante X B, (LB) {ae Sat that is to say an mcrease of d wv Tnt De (1 -——B,) u pz £ lr And now it is clear that, as was already observed above, the lr one factor of this product, viz. ——-, approaches to oo, while the other L v factor, viz. B, (l— B)? IES approaches to 0. The product however is — WV evidently finite, viz.: Sy B een Now if Q is the heat, absorbed when 1 gr.mol. (18 Gr.) H,O changes from the state of double molecules to that of single molecules, then the heat, absorbed in consequence of the state of association being changed by 1 mol. of the solved substance, is: AMW ee EN And this heat it is, whieh we have to take into account for associating solvents. For H,O at 18° g= 0,21 *), so that the factor 8, (1—8,) becomes =—= 0,17. Further Q (as J calculated some time ago ’)) = + 1920 gramecalories, so for water (at 18°) will be: WSO Peen se If the solved substance is no electrolyte, then {— 1, so for much diluted solutions about 325 gr.cal are absorbed with every concen- tration, if 1 gr.mol. is solved in the water, only in consequence of the change in the degree of association of the water; for salts, acids 1) Zeitschr. fiir Phys. Ch., 81, p. 4 (1899); Lehrbuch der math. Chemie, p. 36 (1901). 2) Z. f. Ph. Ch., 81, p. 5 (1899); Lehrb. der math. Chem., p.87 (1901). ——_—- nnn ee and bases, where 7== nearly 2, this number becomes 650 gr. cal. So e. g. for KCl, of which the heat of ionisation of 1 gr.mol. = — 720 er. cal.'), the total heat of mixing with much H,O, (excluded the change in potential energy) will therefore be not —720 er. cal., but only — 720 + 650 = — 70 gr. cal. So it is seen, that the order of magnitude of the heat to be expected, can be totally modified, and that in general a great mistake would be committed, when we neglected the above calculated 3267 gr. cal. in the calculation of the heat of mixing. Therefore, with di/uted solutions of non-electrolytes in associating sol- vents, 325 er.cal. on each gr. mol. of the solved substance must always be subtracted from the absorbed heat determined by experiment, in order to calculate the pure (absorbed) heat of mixing, that is to say that heat, which is caused solely by the change in potential energy. Physics. — Prof. Baknuvis RoozrBoom, in the name of Dr. A. Sits, presents a paper, entitled: “On the phenomena appearing when in a binary system the plaitpointeurve meets the solubility curve.” (Third communication). *) (Communicated in the Meeting of June 25, 1904). The previous qualitative examination of the binary system ether- anthraquinone showed that a good survey of the whole could only be obtained by continuing the examination in quantitative direction with the aid of the pump of CAILLETET. | Some difficulties were to be foreseen; the investigation would have to be extended over a range of temperature from + 170° to + 300°, in which the pressure might be expected to reach a pretty conside- rable amount — and the combination of high temperature and high pressure being exactly the thing against which glass is but seldom proof, it seemed at first that we should meet with great experimental difficulties in the quantitative examination. The experiment however showed that the pressures were not exceedingly high; it appeared a maximum pressure of 100 atm. would suffice, and this pressure Jena-glass could withstand up to more than 300° *). 1) Z. f. Ph. Ch., 24, p. 611 (1897); Lehrb. der math. Chem., p. 53 (1901). 2) This paper is a continuation of the two preceding ones on the system ether- anthraquinone. The title shosen first seemed to me undesirable and was therefore modified. 3) With pleasure I avail myself of this opportunity to thank professor KaMeRLINGH Onnes for his kindness towards me in procuring the necessary information and in lending me some instruments wanted. ( 178 ) The object of the experiment was to determine the p-r-sections of the p-r-t-surface at different temperatures, and if possible also the p-r-sections of the v-r-f-surface. At the same time I should get to the knowledge of some projections already spoken of in the previous paper, viz. the projections of the p-a-f-surface on the p-¢ and the t-v-plane. I shall briefly state the result here. In order to have the same succession as was chosen in the prece- ding communications, the p-f-projection will be treated first. 179* 180 dop Lo 210 L350 Luo 450 Ih Vig. 1. 1, ea represents the vapour pressure curve of pure ether go dbo ayo 260 299 300 In fig. with the critical point in « (198° and 36 atm.). ep and qd represent the portions of the three phase curve which can be realized. Up to 193° the three phase curve practically coincides with the vapour pressure curve ea of pure ether, in consequence of the very small solubility of anthraquinone in ether. On the curve ap lie the plait- points of the unsaturated solutions of anthraquinone in ether, and p denotes the first plaitpoint of a saturated solution (203° and 48 atm.). The second plaitpoint of a saturated solution of another concen- tration lies in g (247° and 64 atm.) and on the curve gh lie the plaitpoints of the second series of unsaturated solutions. Probably this curve, which runs on to the critical point of anthraquinone, has a maximum. The hne fd, which partly coincides with the 7-axis, is the vapour pressure curve of solid anthraquinone, and dy that of liquid anthra- quinone. dh is the meltingpoint-curve, which (as vaN per Waars *) has proved) marks the direction of the three phase curve near the meltingpoint d. These last three curves are drawn here schema- tically. The main result represented by this p-tfigure is this that by the meeting of plaitpointcurve and three phase curve a part of the latter has vanished or rather has become imaginary, and in the examined system that part that contains the maximum. The plaitpointeurve is metastable between p and g and therefore still to be realized, but this is not the ease with the three phase curve. However it appeared to me that at temperatures between p and g, with concentrations greater than those of point g, three phases could temporarily appear together, if they had originated at a tempe- rature above 247° and if afterwards the system in equilibrium had quickly cooled down to less than 247°. The three phases however were not in equilibrium now, for at a constant temperature a slight change in volume proves to cause a great change in pressure. The liquid therefore, thongh in contact with solid anthraquinone, was supersaturated ; it was very viscous and passed very slowly, at times not until after an hour, to the stable condition of solid fluid, under secretion of solid anthraquinone. Fig. 2 gives a number of p-z-sections for different temperatures, the pressure being given in atmospheres and the concentration in 1 mol. total of the mixture. We may immediately point out here that all the lines in this figure joining points of equal value, as plaitpoints (4), liquids coexis- ting with vapour and solid anthraquinone (¢c), vapour coexisting with liquid and solid anthraquinone (©), are all projections on the p-a- plane of curves, which occur in the p-r-t-surface *). The branches ¢, p and e, p, which pass into each other conti- nuously, represent the series of liquids and vapours which if we come from a lower temperature, coexist with solid anthraquinone. In p, the point of confluence of the two branches, we have the first point, where a saturated solution reaches its critical condition. This takes place at a concentration 0,015, temperature 203° and pressure 435 atm. If we pass on to higher temperatures a stable solution is impossible over the range of temperature 203°—247°, and instead we 1) These Proc. VI p. 230. *) If the plaitpointcurve has a maximum, it must possess a maximum also in fig, 2, In the ¢-v-projection on the contrary no maximum occurs. ( 180 ) get fluid phases coexisting with solid anthraquinone. Above 247° liquids can again exist and the continuous curve dc, c, C, Cy Cs q Cs €, 5 € &; consisting of two branches then represents the series of liquids and vapours which coexist with solid anthraquinone above 247°. The point of confluence here is g, in which therefore for the second time a saturated solution reaches its critical condition. This occurs with a concentration 0,15, temperature 247° and pressure 64 atm. The liquid branch c¢, p of the first loop and the liquid branch q ¢, d of the second loop are what we are accustomed to call two parts of the solubility curve. As however the two liquid branches pass conti- nuously into their vapourbranches, there is no objection to calling the two continuous loops solubility curve. Branch c¢,p of the first solubiliy curve and branch dc, q of the second show here a particularity. The circumstance that these branches pass continuously into the branches e, yp and dc, q and that the point of confluence coincides with the highest pressure involves the pheno- menon of retrograde solubility. Cop points to retrograde solubility in the liquid branch (ef. also fig. 4) and de,q to retrograde solubility in the vapour branch. The extent of these phenomena however surpassed all expectations. It was known that the liquid and the vapour branch of the curve de, qe, d from q to a higher temperature have to separate first in order to come together again afterwards, but it was not to be foreseen that the distance would be so large as to make the vapour branch extend to the concentration 0,01. From this particular situation results the very interesting phenomenon that, after we have reached point p, with a concentration 0,015 or in other words after the saturated solution has reached its critical condition, at a higher temperature there may again occur three phases. The vapour branch ge, d extends namely as already mentioned, to the concentration 0,01, and the concen- tration of point p is 0,015; therefore we get from point p at a higher temperature into the region on the right of the vapour branch de,g, in which three phases may occur. This phenomenon was observed at a temperature nearly 60° above the plaitpoint- temperature of the concentration 0,015 (p), that is at 260°. After the formation of the three phases, first the solid and then the liquid might be pressed away by raising the pressure, so that finally only a fluid phase was left. Fig. 2 shows further the p-a-sections at temperatures above that of point q, beginning at 250°. The p-v-section corresponding with this temperature is separately drawn in figure 2a. The continuous curve c‚ £, e, which represents the coexisting unsaturated liquids and (ASR) vapours, has a peculiar sbape and shows that retograde condensation is wanting. a Fy, +Anthraquinone (solid) F, + Anthra- quinone (solid), Ether. yi Anthraquinone. Fig. 2a. The curve g,e, on the contrary indicates a tolerably strong retro- grade solidification. The curve ¢, s, shows that here the solubility of anthraquinone in ether decreases with increased pressure. The curves J, @, and c,s, are two portions of a continuous curve, of which the partly not realizable intermediate part is schematically represented by a dotted curve. I propose to call this continuous curve henceforth isotherm of solubility. Passing on to a higher temperature, we see that the p-r-section at 255° is still of the same type as that at 250°. At 260° (fig. 25) however, the situation is already considerably changed; not only the p-«-loop c,k,e,; has become much larger, because the points e, and c, have become more widely separated and /, has moved to higher pressure, but also it is clearly visible that the retrograde condensation, which is still wanting here, will have appeared at a slightly higher temperature. In the part g,e¢,, though in a smaller degree than at 250°, the ( 182 ) Qs 0 01 G2 43 OY OS Go 7 0 a 7d Ether. xX Anthraquinone. isotherm of solubility shows still clearly the phenomenon of retrograde solidification. At a still higher temperature the region of retrograde condensation becomes greater and greater, so that at 270’ we get a p-r-section like that drawn in fig. 2e. . The retrograde condensation is here very strong and undoubtedly ranges over more than 40 atm. A retrograde condensation of such streneth, however, could not be observed because the volume of the compression tube was too small; the strongest retrograde condensation observed by me covered a range of pressure from 55 to 39, of 16 atmospheres therefore. The small volume of the tube prevented us from observing whether any retrograde solidification still existed at 270°. As, however, it is not very probable that we still should have retro- grade solidification here, it is not represented in the figure. Above the melting-point of anthraquinone (283°) the retrograde condensation is enormous, so that I could observe it at 290° over a pressure-range of 83 to 40 atmospheres. Further L mention that most of the p-a-sections are crossed in ( 183 ) O Of (LO 07 05 IT oF os Og Ether: X Anthraquinone. Fig. 2. different ways. In fig. 2a the regions passed are marked with arrows. 1 indicates the transition from the region for /’, + solid anthraquinone into the region for /’7, + solid antraquinone, the three phases appearing intermedially. /”, denotes here a fluid phase which in ordinary cir- cumstances, that is to say below the critical temperature of ether, would be called gas-phase; and 4, denotes a fluid phase which in ordinary circumstances would be styled liquid phase. It is evident that the difference between /, and #/, exists solely in their foregoing history. 2 marks the transition from the region /, + anthraquinone into the region /’;, the three phases appearing intermedially. 3 indicates essentially the same as 2, but yet the phenomenon is somewhat different, because now we do not in the end pass the liquid branch, as in 2, but the vapour-branch; this is marked by the sign /, over the branch e, /,. 4 is a very remarkable transition, as here we pass directly from the region for /’, + solid anthraquinone into the region for /%. As to the lack of retrograde condensation at temperatures between 247° and + 260° and its appearance at higher temperatures, | want ( 184 ) to say a few words about it in connection with the appearance of retrograde solidification. If in fig. 3 the p-v-loop dckeR represents the liquids and vapours which may coexist at a given tempera- ture, but of which a series of liquids and vapours are not to be realized in a stable state because of the appear- ance of the threephase pres- sure curve’), then several cases are possible. If the threephase pressure curve, as drawn in fig. 4 lies above the critical point of contact R, then no retrograde con- densation will occur, not- A X B ENG MEE “si 5 withstanding its possibility is ‘ig. 3. strongly pronounced in the character of the p-v-loop, because the part giving rise to the retrograde condensation lies in the metastable region. Now this occurs in the system ether-anthraquinone from 247° to 260°. The dotted vapour and liquid curves below the threephase-pressure curve ec f are metastable; the stable state here is solid 5 by the side of a fluid phase, and now the question was raised: “how is this part of the isotherm of solubility situated?’ Evidently this stable curve must lie left of the metastable curve d Re or in other words towards smaller B-concentrations. This conclusion is of great importance for us, for from it follows that, if the threephase-pressure- curve lies above the critical point of contact of the vapour curve coexisting with liquid and for that reason the retrograde condensation falls in the metastable region, retrograde solidification must occur instead ef retrograde condensation, and this retrograde solidification must be stronger than the retrograde condensation would have been. If the threephase-pressurecurve passes exactly through the critical point of contact, retrograde solidification is no longer necessary. tesuming, we conclude that, given the case that the plaitpoint- 1) | propose to give this name to the curve that in a p-v-section denotes the pressure at which the three phases coexist. This curve refers therefore to one temperature, whilst the ¢hreephasecurve embraces a series of temperatures. ( 185 ) curve meets the solubility curve, it is possible to prove in a very simple way the necessity of the appearance of retrograde solidification in p and g. Here however we must at once point out that, as will be discussed presently, retrograde solidification also occurs between pg. The fact that theory requires this, can only be proved mathematically *). Returning to fig. 2, we must still state that the curve q 6 uniting the plaitpoints of the different p-z-loops, is very steep and, as far as it has been observed, parallel to the first part of the plaitpointcurve ap. This course however will probably change towards a higher temperature, for if the plaitpointeurve possesses a maximum, which is probably the case, then the projection of the plaitpointeurve on the p-z-plane must also show a maximum. The p-x-sections below the temperature 203° are not drawn in fig. 2, as the scale is too small to render the par- ticulars conspicuous. There- fore this part of fig. 2 is separately reproduced on a larger scale in fig. 4. In accordance with the pre- ceding we see that, though at 200° no retrograde con- densation occurs, instead of it there appears retrograde solidification. Soon however the situation changes here, for already at 196° retro- hd Y grade condensation could be Fig. 4a. observed, What was observed when going from point q to a higher tempe- rature, is naturally also found in point p, but here towards a lower temperature. This is illustrated by figures 4a and 44; tig. 4a applies to temperatures above point g and fig. 44 applies to temperatures below point p. In both figures three p-v-sections are represented schematically ; the sections 1 and 2 differ but slightly in temperature, and 3 applies to a temperature considerably different from that with which 2 corresponds. In fig. da section 1 applies to the lowest and 3 to the highest of 1) van DER WAALS, |. c. ( 186 ) the observed temperatures; in fig. 45 the reverse is seen, but all the same it is seen that in the two figures the same things are to be met with in the succession of sections 1, 2 and 3. The three Fig. 5. x Fig. 4D, pressure curve lies highest in 1 and lowest in 3. In 1 and 2 we do not find any retrograde condensation, but retrograde solidification, and in 38 we find retrograde condensation only. In fig. 4a however the plaitpointpressure increases in the order 1, 2, 3, and decreases in fig. 46, but this is due to the fact that in the first case the order 1, 2, 3 means towards higher temperatures, and in the second to- wards lower. Concerning the course of the iso- therms of solubility above the three phase pressure curve, VAN DER WAALS has shown the probability of a course as given in fig. 5, from which results that the branch cs also shows retro- grade solidification. This case, in which c 2 ea —— (187 ) the whole isotherm of solubility points to a double retrograde solidi- fication, has not been ascertained as yet. What has been found, is that below 240° the upper part of the isotherm of solubility runs toward the right, which points to an increase of solubility of anthra- quinone in the fluid phase with increase of pressure *), whereas above 250° a reversed course was found. Between 240° and 250° a change of direction seems to have taken place, and in this range it might be possible to ascertain the course foretold by vaN per WAAIs. As however the small range of temperature 240°—250° corresponds with a great difference in concentration, the point when the change of direction takes place is not easily ascertained. The results obtained at temperatures between 203° and 247° are represented in fig. 6. Here the isotherms of solubility for the fluid phases at 210°, 220°, 230° and 240° are drawn. All these isotherms show, as predicted by van DER Waars®), the phenomenon of retro- grade solidification, and the nearer we get to point g, in other words the nearer to 247°, the larger the region of this retrograde solidification. Ether. X Anthraquinone. Fig. 7. 1) This influence of pressure has been examined up to more than ‘0) atmospheres. 2) loc. cit. 13 5) Proceedings Royal Acad. Amsterdam. Vol. VIL. ( 188 ) This isotherm of 210° has the steepest course; with increase of temperature the course becomes at first less steep, but at 240° a steeper course seems to reappear, which is probably connected with the change of direction which appears above 240°. The projection of the solubility curve and the plaitpoint curve on the #-z-plane is represented in fig. 7, where the dotted curves represent the vapourbranches. The projections of the two parts ap and qb of the plaitpointeurve are almost straight lines. If we examine the course of the line g, in order to see at what temperature this line will meet the line for pure anthraquinone, we shall find + 800°. Lastly we find in fig. 8 the course of the molecular volumina of LUO Kther. X Anthraquinone. the saturated solutions. Here too we have two continuous branches, each of them consisting of a liquid and a vapour branch. dq and cp are the liquid branches and ge, and pe are the vapour branches. p and q denote the molecular volumina of the two critical saturated solutions. The dotted vapoureurve ge runs on to the concentration 0.015, so that from this figure also directly follows that at higher temperatures and larger volumina three phases may again be obtained with the con- centration with which point p may be realized. Here too the curves cp and ge indicate clearly the phenomenon of retrograde solubility. So the investigation described here has furnished proof positive of the general points of view which were prominent in the qualitative Plate i A. SMITS. “On the phenomena appearing when in a binary system the plaitpointcurve meets the solubility curve.” Plate I. Fig. 2. 10u it Lo ooo Mp Kg. 280 Mig ie 8 yy. KY L60 250" Jp 7 Proceedings Royal Acad. Amsterdam. Vol. VIL. Yoo es ned Plate IL A. SMITS “On the phenomena appearing when in a binary system the plaitpointcurve meets the solubility curve.” Jo Fig. 4. go do go 30 10 Fig. 6. Plate IL GOs Proceedings Royal Acad. Amsterdam. Vol. VII. =. EEN ( 189 ) investigation, and of which the theory of van per Waars could give a closer description. The peculiarity of the examined system, which lies in the fact that the vapour pressure of the one substance (ether) far exceeds that of the other (anthraquinone), caused some wholly unexpected phenomena, and made it on the other hand possible to realize retro- grade solidification on a much larger scale than had been thought possible till now. Laboratory for Anorganic Chemistry of the University. Amsterdam, June 1904. Chemistry. — “The preparation of silicon and its chloride.” By Prof. A. F. HoLLEMAN. (Communicated in the meeting of June 24, 1904). The numerous proposals which have been made for the prepara- tion of the element silicon in both the amorphous and erystallised form prove that a simple method has not as yet been found. W. HrurpeL and yon Haasy') have published in 1899 an additional process consisting in the decomposition of silicon fluoride with sodium. They melt this metal in small portions at a time in an iron apparatus and then pass over the mass a current of silicon fluoride, which is then very readily decomposed. The brown porous mass, which has been brought to a faint red heat is allowed to cool for two or three hours in the current of silicon fluoride. An attempt to convert it into silicon chloride by heating the mass without previous purification in a current of chlorine was unsuccessful. It was impossible to remove the Na Fl and Na, Sif l, by boiling with water ; so in order to obtain pure silicon it was necessary to fuse the mass with sodium and aluminium. The latter dissolves the silicon which is then left insoluble on treating the regulus with dilute hydro- chloric acid. Mr. H. J. SrisperR who has repeated these experiments in my laboratory showed (1) that by a small modification of the process the crude product may be purified to such an extent by boiling with water that it may be used for preparing silicon chloride ; (2) the reason why the crude product on being treated with chlorine does not yield silicon chloride. 1. It is known that sodinm fluoride readily absorbs Si Fl, and 1) Zeitschr. f. anorg. Ch. 23, 32. 13" (190 ) passes into Na, SiFl,. By allowing their apparatus to cool for 2 to 3 hours whilst transmitting this gas Hremprn and von Haasy practically converted the sodium fluoride, which had been formed according to the equation: 4 Na + Si Fl, = 4 Na Fl + Si, into sodium fluosilicate, which is soluble in water with great difficulty. If, however the action of Si Fl, is stopped as soon as all the sodium has been introduced into the apparatus, it is easy to almost completely avoid the formation of Na, SiFl,. 100 grams of sodium yielded to Mr. SrijperR 219 grams of crude material (4 Na Fl + Si) instead of 213.6 the quantity calculated; 55 grams of the Na gave 119 grams, theory 117.2, and in some further experiments the theoretical quantity was but little exceeded. By washing and boiling with water and with dilute hydrochloric acid the 119 grams were reduced to 20 grams whilst the product may contain 16.7 grams of silicon. The product so obtained is not, however, pure amorphous silicon, only about 40 per cent is volatilised in a current of chlorine and may be condensed as silicon chloride, and a residue is obtained, which is only to .a slight extent soluble in water and principally consists of silicon dioxide. This must have been formed during the washing; for if the crude product is heated in a current of chlorine there remains besides sodium chloride only a very small quantity of insoluble residue. As the crude product when immersed in water causes a visible evolution of gas with the odour of SiH, it is probable that the 510, has been formed by decomposition of SiH, which may have been pro- duced by the action of water on some sodium silicide. Motssan has recently shown that on treating silicon with boiling water the dioxide of that element is formed. 2. In accordance with Hremprn and von Haasy, Mr. Super found that on heating the crude product in a current of chlorine not a trace of silicon chloride is obtained. As the said product consists mainly of 4 Na Fl + Si, it was surmised that this must be attributed to the fact that the primary formed silicon chloride reacts with sodium fluoride according to the equation Si Cl, + 4 Na Fl = Si Fl, + 4 Na Cl It appeared indeed that on heating sodium fluoride or sodium fluosilicate in the vapour of silicon chloride the said decomposition takes place. If, therefore, chlorine is passed over a mixture of 5i and NaFl as is present in the crude product the reaction must proceed in this manner : Si + 4Cl+4NaFl = Si Fl, +4 NaCl. | | (191 ) That such is practically the case was shown by the fact that the gas evolved consisted of Si Fl, and that the substance left behind -in the boat was found to be almost pure sodium chloride. A better method of preparing amorphous silicon seemed to be the decomposition of silicon chloride by sodium. When boiled in benzene-solution with sodium or potassium no action took place. A reaction, however, took place on heating sodium in the vapour of silicon chloride, but it became very violent; the brown powder obtained could certainly be readily freed from sodium chloride by means of water, but on heating in a current of chlorine a large amount of SiO, (about 30°/,) was left behind showing that even this process does not lead to pure amorphous silicon. Much more simple is the preparation of crystallised silicon accor- ding to the method recently published by R. A. Künre (Chem. Centr. 1904, I. 64) if we introduce a slight modification. A mixture of 200 grams of aluminium shavings or powder, 250 grams of sulphur and 180 grams of fine sand is put into a Hessian crucible placed in a bucket with sand. Upon the mixture is sprinkled a thin layer of magnesium powder and this is ignited by means of a GorpscHaipr cartridge. The mass burns with a beautiful light and the contents of the crucible become white hot. When cold, the mass is treated with dilute hydrochloric acid, which dissolves the aluminium sulphide and leaves the silicon in a beautifully crystallised state. The yield amounts to about 30 grams. On heating in a current of chlorine SiCl, is very readily formed, only 3°/, remaining as non- volatile products. It is a material eminently suited for the prepa- ration of SiCl,, but Mr. Striper did not succeed in converting it into silicon sulphide by heating with sulphur. Groningen, Lab. Univers. March 1904. Crystallography. — “On the preservation of the crystallographical symmetry in the substitution of position isomeric derivatives of the benzene series’. By Dr. F. M. Jancer. (Communicated by Prof. A. P. N. Francuiont). (Communicated in the meeting of June 24, 1904). Some time ago when engaged in a research as to the connection between molecular and erystallographical symmetry of position-isomeric benzene derivatives'), | demonstrated, that of the six possible fribromo- 1) Jazcer, Crystallographic and Molecular Symmetry of position-isomeric Benzene- derivatives. Dissertation, Leiden 1903. (Dutch). ( 192 ) toluenes, the 1-2-4-6 and the 1-2-3-5 derivatives exhibit an isomorphy bordering on identity. I then endeavoured to explain the similar molecular structure of these two substances by referring to the analogy of the (CH,)-group and Br-atom in the positions 1 and 2, particularly in a spacial respect. The small differences in crystalline form, melting point etc., are then caused by the difference which of course exists between CH, and Zr. We may now inquire how matters will be in both molecules as regards the substitution of the two remaining H-atoms of the core for instance. It is interesting to notice that as regards the substitution by NO.) the two H-atoms in each of the two isomers are quite equi- valent and — what is still more striking — that this substitution does not perceptibly alter the molecular symmetry of the two molecules, so that the crystallographical relation of the initial products is pre- served in the substitution derwatives. If we nitrate the 1-2-4-6-tribromotoluene with nitric acid of 1,52 sp. gr. a dinitro-product is obtained, as shown by Nevine and WINTHER *). Wrosiewsky *) had previously obtained a mono-nitro-derivative which differs in its melting point but little from the dinitro-product. But notwithstanding many efforts, I have never succeeded in obtaining a mononitro-compound not even as a bye-product, when using fuming nitric acid. The mere formula of 1-2-3-5-tribromotoluene does not at once raise a suspicion that a dinitro-product will be formed in this case. If, however, the analogy of (CH,) in the position 1 and Zr in the position 2 is really so great that the difference amounts almost to nothing, we may surmise that the 1-2-3-5-tribromotoluene will behave on nitration also nearly quite analogously to the other isomer. The experiment shows that in this case also not a trace of any mononitro- derivative is obtained; we obtain exclusively a dinitro-product, which is in all respects quite analogous with the above named dinitro- derivative. After nitration by red fuming nitrie acid (sp. gr. 1,516 at 16°) in the cold, separation by adding an excess of water, agitation with benzene and ether and recrystallisation from benzene, in which both isomers are very soluble, the two substances are at once obtained pure in large colorless or pale sherry-colored crystals, whose bromine- amount corresponds with that of the dinitro-derivative. 1) Nevire and Wintuer, Journ. Chem. Soc. Vol. 37. 438; Berl. Ber. 13. 974. 2) Wrostewsky, Ann. d. Chemie 168. 147. ( 193 ) The 1-2-4-6-tribromo-3-5-dinitro-toluene melts at 220°; the 1-2-3-5- tribromo-4-6-dinitro-toluene at 210°. Like the two tribromotolwenes themselves, these substances are again quite %omorphous and owing to peculiar twin-formation, they so resemble each other, that at first sight we cannot distinguish the two kinds of erystals from each other. a. 1-2-4-6-tribromo-3-5-dinitro-toluene. Br. Br. Br. (NO). (NO). (CH,); melting point. 220°. 6) @ (2) (5) (3) (1) From benzene this substance crystallises in large apparently qua- dratic, colorless crystals which are nearly all twins, — which may be recognised at once by a very fine diagonal on two of the broadest planes. We also may obtain needle shaped or very elongated thick pillar shaped crystals. The planes are generally angular and give plural ( 194 ) reflexes. The crystals are also frequently bordered by curved planes and by vicinal forms in the prism-zone. These circumstances render an accurate investigation very difficult; occasionally, however, I obtained better formed crystals, which gave very sharp reflexes and served for the following accurate measurements. They are monoclino-prismatic with the axial relation : abio sAlT st 0 7805 B= 85°12". Forms observed are: m == {110} and p = {120}, broad and lustrous; a = {100} and n—{130}, very narrow; a is generally hazy ; b= {010}, a little broader, but is generally absent; c—={001}, large and very lustrous; 7 = {101}, well developed and lustrous; t= {104!, narrower and is often absent; 0 = {112}, generally small and dull, occasionally a little broader and better reflecting; s = {132}, large and lustrous, but generally there are only two parallel planes present. Combinations of a// the forms rarely occur. Generally such with mep. ers INE D0, CF, tse and 059, Pr h and ce, ete. The more typical crystals are shown in figures 1-3. Fig. 2. Fig. 3. In the properly formed erystals, the angular values are very constant; the reflexes are as sharp as possible. The substance has a decided tendency to twin-formation; in this, one form {102} is a nh na Sa Cn a bbe (195 ) twin-plane with a twin-axis normally standing on it. On the plane of p may be often observed a delicate line parallel Dies in this vertical zone the most important geometrical anomalies are found. The following are the calculated and observed angular values. Observed: aleulated: *m : m = (110) : (110) = 54° 561/,' — m:a = (110): (100) = 27 281, 27° 281/, my pe (FLO): (101) == 45° 20 43°13 EG B= OA: (101) ===) ey) = r:a = (101) : (100) = 35 20 35 33 c:a = (001): (100) = 85 15 85 12 #¢:m = (001) : (110) = 85 441), ee e:p = (001) : (120) = 86 40 86 36 e:n = (001) : (130) = 87 30 87 25 c:b = (001) : (010) = 89 58 90 0 pin = (120): (130) = 11 12 4:43 nep (ELO (620) == 18.35 18 39 p:r = (120): (101) = 55 25 55 29 GOOL (104) == 24-12 21 -2 GELE — Al BIEN 41 52 ers = (001) : (132) — 55 49 55 54 m:s == (410) : (132) = 47 361, 47 354/, A distinct cleavability was not observed. On c, 7, and a the position of the optical elasticity-axis is orien- tated perpendicularly to the direction of the orthodiagonal; the symmetrical angle of extinction on m amounts to 23° with regard to the vertical axis. An axial image could not be observed. The sp. gr. of the erystals is 2,456 at 15°; the equivalent volume is, therefore : 170,6. The topical axes are: rp: = 3,9087 : 7,4921 : 5,8461. b. 1-2-3-5-tribromo-4-6-dinitrotoluene. C,. Br. Br.Br.(NO,).(NO,) .(CH,); melting point: 210° C. 5) (38) (2) (6) (4) (1) ( 196 ) This compound crystallises from benzene in very large, colorless, iso- metrical-developed crystals, which are always twins and of exactly the same form as that of the previous compound with which this substance is isomor- phous. The geometrical anomalies caused by the curvature or angulation of the planes are more considerable with this derivative, than with the previous one; the development of the crystals is less perfect and they also exhibit a smaller number of combining forms. From ether and acetone we obtain besides Fig. 4. twin-crystals also single needles which can be measured more accurately. The symmetry is monoclino-prismatic; the axial relation is: a:b:¢= 0,892 :1 : 0,7574. [= 86°28". Forms observed are: m= {110} and p = {120}, broad and lustrous; c= {001}, very lustrous and well developed; 7 = {101}, smaller but properly measurable; 6 = {010}, narrow and often absent. Angular values : Observed : Calculated : #

da the primitive of which, involving two arbitrary constants A and 5, may be written in the form y= AL (a, m) + Ba?m-l M (2, m), where Aer: (5 ie Aen) = Lt ear | h PT(—m+3y4h) h=0 2 \2h BG) M (a, m) == a rr ar ema ’ h=0 and the constants 4d and B must now be determined so that y . Ld a represents the function f(x, m). To find A, we suppose m > 3 and put z equal to zero. In that case we have dt ra) P(m—} A f (9, m) === (3) ['(m—}) — AE (Om ) 2 Pun) I'(— m + 3) and hence id I'(4) 2 cos am T'(m) A=— For the deduction of the constant B it is convenient to consider first the function /(v,m) in another form, Let the real part of m still be positive, then we have ( 202 ) F (m) En (142) Kn ae (pay 7 0 and hence val (oa) ao x? 3 . —u ml — ut? FAT NE AT I'(m) f(am)= fe u dufe cos wt dt = 3 arte u du. 0 0 0 From the latter integral a simple functional relation is derivable. de Changing the variable w into 4, We may write 5 ee) f 1 wv \2n—1 —y— OM E'(m) 7 (wm) = — Va (; =) |. vy 2 dv—= 0 av \2m—1 = (=) (1 — m) f (x, 1 — m) and so it follows that the function P(m) f(a, m) x 1 a \—m eS —_—_ — EE Gl ik 7, tz Ie (5 ik 2 cos” m 2 2 bn 2 -_ a \m—1 + 22m—1 B I'(m) (5) M (wx, m) 2 remains unaltered, if 7 is replaced by 1 — m. Now obviously the series ZL, and J/ are connected by the relation L (a, 1 — m) = M (a, m), hence we must have n 1 22m—1 B UM (m) = dr (5) 2 cos 1 m zt T'(&) 1 2n—1 B= —— , = — |, 2cosxm T(m) (5) and therefore Ss U 5, IME: 2m—1 Tike; == eee = ee. G) — Le, m)+(5 M(a,m)}. (Lt +f? a? ~ 2cos am T(m) Now it will be observed, that the series L (w,m) and M (a, m) converge for all values of « and m, and so we must conclude, that the function f(r,7m) exists over the whole z-plane, that its only singularities are «=O and «=o, and that therefore the integral, we started with, represents the function ina very incomplete manner. ( 208 ) Numerical evaluation of the integral for not too large values of x offers no difficulties, as the series 4 (vm) and MZ («‚m) converge rapidly. Because of the equation le} ° 1 Tt “== n— P (m) f (w‚ m) = Ei Vs e du y od 0 the result will always be a positive number and the integral will not vanish for any real value of z. _ A few further remarks may be made. Firstly we may state, that J (v,m) is intimately connected with Brssur’s function J, (7). In fact by means of the usual expansion of ./, (+) we may verify the relation 1 ; - . wu PG ) al —F(=-3) Fac PM) = — ge 2 LN nee a pe 148) 2cos zen _P'(m) (5) —m+3 (« ) ml ] me Holme), 219 —e ? mites hE er ad: NO 2 From this we infer, that for positive integer values of m the origin v=— 0 ceases to be a singular point, and that f(z, 11) can be expressed in finite terms. We shall find by actual substitution of the finite expressions for J__, sl: (« tol a SEY, ste] 5 oe El | nH EN 2 an | ro| 3 ey. hm 7 4 et wv \n—i So (m—1-+-h)! 1% B, m) = a es eA |B J a "(m— hy, dant ji; (mik)! Ze ==Û) However this result may be obtained in a simpler way as follows It can be shewn, that /(r,7) obeys the relation zin —_ 1) F'(m-h), +5 ag) A Cel Wa m| = = ee f (we, mm + h), and since we have / De “vos at Va / WwW ra ra (wu Va == ave = ( 5 Den 2 we get for all positive values of Ww (Er Ss —, D ll, 2° (ml! s=ll Va a result that can be identified with that obtained before The singularity in the origin 2m is an odd integer 24 + 1. v= 0 becomes logarithmic, when The expression of /(@, m1) is in this 14 Proceedings Royal Acad. Amsterdam. Vol. VIL. ( 204 ) case somewhat intricate. By repeated differentiations it is derived 1 from /(« =k for we have ft Pe - EDE 1 DE J (- Va, =) at TG) 7 (« k + 5 ) : 1 1 To evaluate / é = we put 7 == oy + d in the general expres- sion for f(w, 1m), and make d tend to zero. In this way we get Lim ——_—— ~ p—0 2sin ik = h! T(h+1—9) »\ 2h eB h=o a 1 os at te) 1 1 1 1 av a eee =S viens = Of == a est oat 5 2 yYite NE Heth 2° 2 h 2 0 Ah=v LT 2h aL Yh [es =S She G An f ( 5 J= Ld 57 a Gp i, eee a ’ 9 Ens. / We shall now pass on to consider the second integral sin «et p(w, m = {arn a dt and it will appear that its character is quite similar to that of the first. Again we transform the integral by the aid of the identity = Go - m 0 and obtain a) an I'(m) p (er, 1m) =| e—u ym—l iw | eu son at dt. 0 0 A further transformation gives ei : Py Hi x re Tw 1 —ul? , ed oo 73) v EE == e siete = ASB di = — fe 44 (l—w) ? dw. ) 2u 4u 0 0 0 and therefore je l 0 aw sb aay hie Tae ee FP (m) p (wz, m) == w (1—w) 2 dw é du u du. 0 0 Comparing this equation with the equality obtained before * 1 > = fj Ea) 7 (a, m) = va | e du yy 2 du, bo it follows, that we may write wv 5 EENS | T(m) p (#, m) = oe fan 2 dw F'(m — 3) (ero iki ) a a nr ik 2 0 : 1 We now expand I(m— 3) f(a Vw, m — 5 ad ) and make the sub- stitution V'(m— 3)f Vw,1 : 2 T(} fi : »— 1} £ M= j= ———— , 1 a ay Dd err : EE zel 2 2sin am (3) Adie 2 i vw \2m—2 1 a a) M (« Yw,m — =) Then integrating with respect to a, we find the desired expansion of gp (ze, 7m) in the form sin at Ft w (4) | N 2m—1 p (w, m) fare RT xam) (w,m) 4+- (; ) Mx, mj where N (a, 7m) represents the new series au \2h+l h=a AY 9 N (té, m) = DE f P(h Te m +- 2 + DN h=0 The same remarks as were made concerning the first integral J (x, m), can here be made again. The integral has only a meaning for real values of « and for positive values of m, but from the expansion is inferred, that the integral incompletely represents a function of « which exists over the whole z-plane, quite indepen- dently of the values assigned to the parameter im. Again the origin v=0O and ew =o are the only singularities of the function. The proses are logarithmic, when im is an integer and the origin becomes a regular point, when 2m is equal to an odd integer 1) It is possible to invert this relation. It may be shewn that we have also 1 FP (m) f (we, m) — 7% Va C(m — })= J Mad oss ran ate paca aor == WE if (1—vw) du FP (mn pl aw, m gk 0 Pd 14% ( 206 ) 24+41, but in no case is a finite expression by means of elementary funetions obtainable. The function g (v, m7) as well as / (x, 1) satisfies the relation h mes Pi ) ASA EF (m-h) Den E "9 Cs Aes 7) ey F'(m) p (ew, m Jh) 1 and by means of this rule expansions for g (w,/) and ¢ (« ka zi may be deduced from the equations h=a sun wt peh+l 1 I 1 1 ZE > = — eee en GL a p(el) friet sb Mr (; NE As ma we) h=0 1 — : Li (et?) — et Li (e-*) | ; 2 and al ( s) sin vt Fr a 2 4 ‘sin « en WEA VIE eN l=0 Py + ] 2 Botany. — On “Leptostroma austriacum Ovp., a hitherto unknown Leptostromacea living on the needles of Pinus austriaca ; and on Hymenopsis Typhae (Fuck.) Sacc., a hitherto msufpiciently described Tuberculariacea, occurring on the wt thered leafsheaths of Typha latifolia.” By Prof. C. A. J. A. OUDEMANS. ‚1. LEPTOSTROMA AUSTRIACUM Ovp. (Plate I.) On the 13 of June 1904 I received from Dr. J. Ritzema Bos, Professor at Amsterdam, a number of specimens of transplanted seedlings of Pinus austriaca, originating from Schoorl, all dead and of which the accompanying letter informed me that the roots showed here and there cushionlike prominences, the surface of which was covered with shuttle-shaped conidia, divided into cells, and the microscopic properties of which resembled most those of conidia of the genus Fusarium. Besides I found, without my attention having been directed to it, that most needles of the dead plantlets were spotted on both ( 207 ) sides with small black specks and streaks, the external appearance of which showed most resemblance with the perithecia peculiar to Leptostroma or Leptothyrium. The plants sent to me, provided with a here and there ramified tap-root of about 1 decimetre length and 1— 8 millimetres thickness, proved on closer inspection to have much suffered, since in various places the bark was loose from the wooden kernel, if it was not entirely lacking. These circumstances justified the supposition that the young pine-trees had succumbed under the attack of the Fusarium-plantlets and that the Leptostroma- ov Leptothyrium- individuals had chosen the sickly, lingering or dying needles as the seat of their fatal activity. The /usarium-cushions that had remained were little numerous, 1—3 mm. in diameter and had a light rosy tint. Lacking suitable objects for investigation, I had to restrict my answer to the com- munication that here very likely Fusarium roseum had been active, and I left the further elucidation of the devastation caused by that fungus to the care of Prof. Rrrzema Bos. A closer examination of the very numerous specks and streaks found on the needles of Pinus austriaca, induced me, on account of their generally elongated, sometimes more, sometimes less hysterium- like shape, their little tendency to loosen at the circumference and to fall off, the fact that nowhere a parenchymatic structure of the perithecium-wall could be distinguished and that the basidia had not developed, to think rather of the genus Leptostroma than of Leptothyrium, and besides to mark the fungus as non-deseribed and to give the name Leptostroma austriacum to it in order to distinguish it from other fungi. One of the characteristics of Leptostroma austriacum is that the perithecia are never united to continuous series, but rather form greater or smaller groups of streaks or small shields, which differ greatly among each other in size, and are rather dull than glossy. Their length varies from '/, to 1 mm. and their breadth from '/, to , mm. Their perithecium-wall is ‘halved‘, as the term is, does not reach further than the epidermis of the leaf, and consequently has the shape of a cupola. This wall has no foundation or basis. Moreover it is black, carbonaceous and structureless, so that there can be no doubt that we have here a cuticle (Fig. 2 and 3), from which follows that the space, occupied by spores, rests on the epidermis, as is clearly shown by Figs. 2 and 3. By reasoning more even than by observation, one is lead to the conclusion that the spores are produced by a very thin layer of threads extending over the epidermis, ( 208 ) Above this layer the spores form two layers or storeys. A third layer does not exist, as the space, required for it, is occupied by the spores which have loosened themselves and have become entangled. The spores have an elongated (cylindrical?) shape and are colour- less and undivided. Their foot is rounded and encloses (Fig. 4 and 5) a circular or oval, glossy vacuole; their top is more pointed and empty. They measure 7—8 wu in length and 1°/,g in breadth. The difference between Leptostroma austriacum and other Lepto- stromata, peculiar to pine-needles, like L. Pinorum, L. Pinastri and others, is: that in the latter the perithecia form mostly narrow parallel series; that the spores are not broader than 0.5 u, and finally, that no vacuoles are found. The Latin diagnosis of the new species is as follows: “Peritheciis cuticulam inter et epidermidem occultatis, amphigenis, irregulariter distributis, majoribus et minoribus, item longioribus et brevioribus intermixtis, dimidiatis, nigris, opacis, diu clausis, tandem irregulariter ruptis, persistentibus neque decedentibus nee circumeiree a substrato solutis. Sporulis sessilibus, eylindraceis, hyalinis, continuis, vuleo 7.51 u, basi rotundatis guttulaque sphaerica vel ovali, mieante, bd praeditis, apice acutiusculis, vacuis.’ EXPLANATION OF THE FIGURES OF PLATE I. Fig 1. A piece of a needle of Pinus austriaca with small heaps of perithecia (p.) on them. (°/;). „ 2. Vertical section of a not yet fully mature perithecium. a. Cuticle. b. Epidermis. c. The two layers of rod-shaped colourless spores. (5°°/;). „ 3. Vertical section of a ripe perithecium which has burst open. a. Cuticle. b. Epidermis. c. Spores, partly undamaged, partly in a displaced position. (500/,). „ 4. Spores, with a rounded foot and a sharper top. At the foot a vacuole. (1000/,), The same (?209/)), 4 or 2. HYMENOPSIS TYPHAE (Fuck.) Sacc. (Plate II). This fungus, found for the first time at Nunspeet in July 1904 on the withered leaves of Typha latifolia, was sent to me among many others by Mr. C. A. G. Berns, fo a ( 209 ) Unlike the Sphaerellae and Leptothyria it has not the appearance of small specks but of raised black spots (Figs. 1, 2 and 3) which are spread in the grooves between the nerves and have a length of 1—4 and a breadth of '/, mm. Fucken described the fungus first under the name of Myrothecium Typhae (Symb. 364), in the following words: “Peridiis hemisphaericis, '/, lineam longis, aterrimis ; conidiis oblongo-ovatis, utrimque oblongis, obtusis, simplicibus, biguttulatis, 18 6 u, pallide fuscis,’ and gave a not quite satisfactory picture of a conidium in Fig. 21 of Plate 1. He was succeeded by Saccarpo (Syllabus IV, 745), who agreed with his predecessor that the fungus belongs to the Tuberculariaceae, but nevertheless removed it to the genus Mymenopsis, on account of the spore-bed (sporodochium) of Myrothectum being surrounded by a circle of fringes, which is not the case with Hymenopsis. In a very successful drawing by Mr. C. J. Konina of a vertical section of Hymenopsis Typhae, (Plate 2), the structure of the fungus is excellently seen, much better than in other pictures, also of other species of the same genus. Where the black disks or specks rise above the surface of the leaf-sheaths (Figs. 1, 2 and 3), one does not find, as FuckEL writes, a “perithecium” (i.e. a more or less completely occluded fruit-body), but a globular assemblage of reproductive cells or conidia (Fig. 4 s.s.), covered by the cuticle and produced by a layer of peculiarly shaped sporophores (Fig. 4 wr), collectively called stroma or fruit-bed. Under this stroma the epidermis is found (Fig. 4 0.): a layer easily recog- nisable by the width of its cells. It deserves to be mentioned that the black colour of the prominent little disks (Figs. 1, 2 and 3) must not be ascribed to the colourless cuticle (c), nor to the colourless epidermis (0), but only to the conidia (Fig. 5 y) which have been left uncoloured, however, in Fig. 5, in order not to make the picture too full. One of the most important Figures of Plate 2 is Fig. 5. At x it shows the favoured club-shaped threads or basidia, whose task is the production of the conidia; these latter, let free by their bearers, being seen in their neighbourhood in a free condition (4). The conidia have an elongated, cylindrical shape, are more or less asymmetrical or curved, rounded at both ends, somewhat more transparent at the base and of the grey colour of mice. (Sacc. Chromotaxia, pl. I, Fig. 3). They contain 2—4 consecutive vacuoles each and have a length of about 10 and a breadth of about 4 u. Comparing the Figures of Plates I and II, one might get the impression that in the Figures 2 and 3 of Plate I a perithecium is ( 210 ) lacking as well as in those of Plate II, although this term is usual in descriptions of the Leptostromaceae. Therefore we remark that this latter family of the Sphaeropsideae forms a transition between the perithecium-bearing and the peritheciumless forms and that in judging these two cases weight has been attached to the black colour of the upper half of the shields, which sometimes consists of the cuticle only, sometimes of a combination of the cuticle with the epidermis. In addition to this the Leptostromaceae do not produce well- developed basidia and have remarkably small spores. The Latin diagnosis of Hymenopsis Typhae is as follows: “Sporodochiis amphigenis, hemisphaericis, inaequaliter in vaginarum suleis distributis, majoribus et minoribus, item longioribus et orbieu- laribus intermixtis, primo cuticulam inter et epidermidem caelatis, 1—18 w in diam., aterrimis; denique expositis, calvis, thalamo basidio- phoro basilari praeditis ; basidiis dense fasciculatis, elongato-clavatis, hyalinis, continuis; conidiis oblongis, rectis vel paullo curvatis et utplurimum inaequilateralibus, utrimque obtusis, basi vulgo elarioribus, 104 u, murinis (Sacc. Chromotaxia Tab. I, f. 3), 2—4-guttulatis, guttulis hyalinis, nune binis sibi oppositis, tune iterum ternis (aut quaternis) in seriem dispositis.” EXPLANATION OF THE FIGURES OF PLATE II. Fig. 1. Piece of a leafsheath of Typha latifolia, studded in the grooves between the nerves with sporodochia (spora — spore; docheion — receptacle), (natural size); p.p. perithecia. „ 2. Piece of a leafsheath with two sporodochia, of which one is opened, the other closed (#9/;). „ 3. Piece of a leafsheath with two sporodochia, of which one has a groove on the dorsal side (#/,). „ 4. Vertical section of a ripe sporodochium. — c.c. cuticle; 0.0. epidermis; s.s. conidia; vb. vb. vascular bundles ; z. club-shaped basidia. „ 5. A bundle of club-shaped basidia (x...) with some conidia (y. y.), in which two or three vacuoles. The end of the conidia resting on the basidia or turned towards tiem is always somewhat more transparent than the other. =~ > Var ie : ra Î at i> . a EN C.J. Koning, del. PROCEEDINGS ROY. ACADH J. Buret Lith, P. J. MuLper Impr. Leiden. (aat) Botany. — “On Sclerotiopsis pityophila (Corpa) Oep, a Sphaeropsidea occurring on the needles of Pinus silvestris. By Prof. C. A. J. A. OuprMans. In the “Nederlandsch Kruidkundig Archief”, 3¢ series, vol. II, pag. 247, I mentioned a fungus found in 1901 by Mr. C. A. G. Burns at Nunspeet on the needles of Pinus silvestris, which fungus, dis- covered in 1840 on the same host near Prague by the botanist A. J. C. Corpa, was described in vol. IV of his “Tcones Fungorum” on page 40, under the name of Sphaeronema pythiophilum *) The same fungus received a place in Saccarpo’s “Sylloge Fun- gorum’, vol. III (A° 1884), p. 101, this time under the name of Phoma pityophila, whereas on account of a new investigation of fresh specimens I thought it necessary myself, in the article quoted above, to change the name Phoma again and to replace it by that of Selerotiopsis. Besides Saccarpo, also ALLEScHER, in the 6!" vol. of WiNrer's Kryptogamen-Flora (1901), page 199, uses the name Phoma pityophila for this fungus, which name is changed into Sclerotiopsis, by way of improvement, in vol. VII, p. 847 of the same work. Having been enabled through the kindness of Mr. Berns in January 1904, to examine again some fresh specimens of Sc/erotiopsis pityophila, I availed myself of this opportunity of testing once more my former experience by facts and had the advantage of having at my disposal the drawings by Mr. C. J. Koning, chemist at Bussum, which accompany this article. | have to thank Mr. Korixe for the kindness which he has repeatedly shown in assisting me on former occasions as well as on this. Some particulars supplementing former communications may be mentioned here. The reason that induced Saccarpo in 1884 to change the name Sphaeronema into Phoma was that some very characteristic properties of the former genus had been passed over silently by Corba, viz. that in his paper no mention is made either of a beak- or brush- shaped prolongation of the peritheciumwall or of spores which, conglomerated to a ball, should have been found at the surface of the perithecia. The generic name chosen by Corba could not be retained and so no other name seemed more appropriate to the Italian mycologist to replace it than that of Phoma, which judgment has not been doubted by any subsequent writer. 1) The Greek for pine being xitug, in what follows Corpa’s wrong orthography has been corrected, ( 212) Meanwhile it was evident as well from the very brief deseription of Phoma pityophila in Saccarpo’s Sylloge as from his silence on the microscopic properties of the fungus, that this author had not been able to examine freshly collected specimens, so that mycologists working after him under more favourable conditions might possibly find something to improve. Having had this opportunity myself it may not be superfluous to return once more to my Sc/lerotiopsis pityophila and to consider more fully the difference between Sclerotiopsis and Phoma. First of all it must be mentioned that the perithecia of Phoma, when produced by leaves, although they lie concealed below the epidermis, yet are by no means buried deep in the tissue as is the case with Sclerotiopsis (Fig. 3—5) and probably on account of this are much more irregularly shaped, sometimes coalesce and come forth with a stronger and less rounded appearance. Secondly any one who has examined many specimens of Phoma must have noticed that with Sc/erotiopsis stronger and denser peri- thecia are found which are carbonaceous at the surface, whereas those of Phoma belong to the forms that offer little resistance, and are tender and light-coloured ; finally that the perithecia of Sc/eroti- opsis have no orifice but decay or burst, whereas with Phoma the rule is that a small round ostiolum is found through which the spores are discharged. In addition to this we remark that the spores of Sclerotiopsis do not lie loosely together like those of Phoma, but remain long con- nected by means of a sticky substance (fig. 3 and 4), the consequence of which is that a few drops of water are sufficient to cause Phoma- spores to diverge in all directions whereas with Sc/erotiopsis a slight pressure or friction is required to make them fit for a closer exa- mination. This latter peculiarity was exactly the reason why Corba imagined to have found a Sphaeronema, overlooking that the beak- or brush- shaped prolongation of the peritheciummouth was absent and that consequently no cluster of spores could be formed at the top of such a prolongation. The question whether the spores of Sc/erotiopsis are produced on the top of sporophores is difficult to answer, although analogy pleads for it, since there is no distinct division between the wall of the perithecium and the gleba (the cluster of spores) but a gradual transition of one into the other. Yet not far from the surface of the perithecia ((Fig. 6) a segmentation seems to take place and the formed spores seem to be slowly pushed to the centre. C. A.J. A. OUDEMANS: „On Sclerotiopsis pilyophila (Corda) Oud”. C. J. Koning, del. PROCEEDING ROY. ACADEMY AMSTERDAM, VOL, VII. J. Bijtel lith., P. J. Mulder impr. Leiden. ( 213 Sclerotiopsis pityophila (Corda) Ovp., a saprophyte, appears as black, fleshy grains (Fig. 1), '/,—-2 mm. broad, which are expelled from the tissue of the needles. They consist of polygonal parenchym- cells which at the circumference are larger, harder and darker but in the interior become smaller, softer and colourless and seem to border on a small cavity, which is soon filled with spores. These latter are oval or egg-shaped, straight or slightly eurved (Fig. 7), unicellular and undivided and have rounded tops. They vary from 7—8X38—4u, have no polar drops and no appendices. Germi- nating spores were not found. The first Sclerotiopsis was found by Spreazzixi in the Argentine republic on rotting leaves of Eucalyptus Globulus and was called S. australasiaca. A second and third species (Sel. Cheiri Ovp. and Sel. Potentillae Ovp.) were found by myself and Mr. Berns, the former on the stems of Cheiranthus Cheiri in the Botanie Garden at Amsterdam, the latter on the leaves of Potentilla procumbens at Nun- speet. Finally Corpa first mentions Sel. pityophila (Corda) Oep. which was collected in 1840 on pine-needles at Prague and 60 years later at Nunspeet. EXPLANATION OF THE FIGURES. Fig. 1. A few needles of Pinus silvestris studded with perithecia of Sclerotiopsis pityophila (Gorda) Oup. — Natural size. „ 2. A needle of Scl. pityophila loaded with some perithecia. Cross-Section. y Magnification 100. » 93, 4 and 5. Vertical sections of Sel. pityophila, magnified 400 times. The carbonaceous wall of the perithecium is clearly visible here everywhere. In 3 and 5 the perithecia have broken through the epidermis, in 4 not yet; in the former two also the conglomerated spores ere discerned. » 6. A piece of a peripheral part of the wall of the perithecium with some stalked spores. Magn. 1000. » 7. Single spores, 1000 times enlarged. (214 ) Geology. — “On an equivalent of the Cromer Forest-Bed, im the Netherlands.” By Prof. Eve. Dusois. (Communicated by Prof. K. Martin). On the eastern frontier of the Netherlands, along the middle third part of the province of Limburg, there is the steep west border of a plateau, made up of gravels and sands, which, for the greater part, is enclosed between the valleys of the Meuse, the Niers and the Roer and rises to an average height of about thirty meters above the adjoining low land. That border is falling within the Dutch frontier opposite Venlo, Tegelen and Belfeld, further, east of Swalmen and of Herken- bosch. The plateau is a piece of a formerly coherent, much larger plateau, extending to Nimeguen and Cleves, of which, according to Dr. Lork, the Veluwe is also a part. This author is inclined to suppose, that the large mass was still entire at the time of the principal extension of the Scandinavian Ice-sheet and that only after the retreat of that ice-sheet, by melting, the great eroding process began, which divided it into a number of pieces and also assailed each of these ; so that under consideration was partly divided by the Swalm and the Nette with their affluents. Dr. Lorm showed that the northern and eastern parts of the plateau do not merely consist of ‘Rhine- diluvium’’, as SrArING supposed for the whole till Nimeguen, but that these northern and eastern parts expose traces of having been reached by the Seandinavian Ice-sheet of the great Glacial Epoch, in consequence of which they consist, at least at the surface, of “Mixed Diluvium”. This is not the case with the western piece of the plateau of gravel, which we consider now more particularly. In this only stones are found, which have been transported by the Rhine and its large tributary, the Meuse; further, the horizontal stratification has not been disturbed by an ice-sheet having moved over the plateau. Nevertheless here too the characteristics of the co-operation of ice in the transport of the sand and the gravel, out which the mass has been made up, are not wanting, but these occur to great depth in it, till 30 metres and more below the surface, and, as already stated above, the horizontal structure has not been disturbed by an ice-sheet having moved over the surface. This stratification, with fluviatile current-bedding, can be observed in a number of places where gravel is dug. At the same time there are repeatedly found, among sand and finer gravel, big angular stones. So I observed in a gravel-pit in the Jammerdaalsche Heide, opposite Tegelen, the following boulders, which were found on an area of ( 245 ) about one hectare, and 2 or 8 M. above the basis of the sand and gravel deposit: A large boulder of Zhonschiefer, of 1.35 Xx 0.75 Xx 0.35 M., and three smaller ones, of about 0.5 M. greatest dimension, a boulder of veined gray quartzite, of 0.80 x 0.75 x 0.50 M., another gray quartzite of 0.67 Xx 0.36 x 0.20 M., a flint nodule of 0.60 < 0.35 Xx O.15 M. Other large stones were knocked into pieces. East of Belfeld boulders are not so frequent in the gravel. Amongst others I observed there a basalt of about 0.40 M. largest dimension. From these observations we are led to suppose a transport on a large scale by floating ice, and we can imagine that iee having had its origin, in the upper-course of the Rhine and the Meuse, from bottom-ice. The basal part of the deposit, 2 M. thick east of Tegelen, 5 M. thick east of Belfeld, is, however, entirely devoid of pebbles, it consists of rather fine sand. All this induces us to consider this ‘“Rhbine-diluvium” as a glacio- fluvial formation of the first Pleistocene Glacial epoch, chronologi- cally the equivalent of the tluvioglacial Deckenschotter of the Dilu- vium of the Middle-Rhine. This interpretation is now affirmed by the character of the bed underlying the gravels and sands in the plateau in consideration. Save gravel and sand there is dug clay, which furnishes the material for the many tileries and stone-factories, in a great number of places, of the Netherland province of Limburg and of the adjoining region of the Rhine-Province of Prussia, chiefly on the borders and along the transverse valleys, of the Swalm, ete. That clay is lying confor- mably and with not eroded, rather well horizontal separating plane under the “Rhine-diluvium’’, the equivalent of the Dechkenschotter. Her own planes of stratification are also generally horizontal. In the clay-pit of the well-known stone-factory of the firm CANOY-HERFKENS, on the western border of the Jammerdaalsche Heide, her upper surface is at 27 M. + A.P. Kast of Belfeld, near Maalbeek, 4.5 K.M. 5.S.W., I found that surface at 35 M. + A.P. East of Reuver and 8.5 K.M. 5.S.W. of the pit opposite Tegelen, it is at 48 M. + A.P. East of Swalmen, near the Dutch Custom-house on the frontier, 14 K.M. south-west of the pit in the Jammerdaalsche Heide, it is at 50.5 + A.P. *) The same clay is also dug roundabout Briiggen, on the Swalm, in the Rhine-Province, 5 to 8 K.M. east of the pit near the Custom- house. It is probably also the same clay, which is met with, at the surface, east of the Zwartwater, (north of Venlo), and west of the plateau, in the communes of Tegelen, Belfeld Reuver. Evidently this clay constitutes a continuous bed underlying the this english version they are given from exact determinations by levelling. ( 216 ) “Rhine-diluvium”’, which has a regular gentle upward slope to the south and probably also to the east, and of which rests appear in the low country bounding the plateau to the west, where the ‘Rhine- diluvium” has been removed. There is not much known about the total thickness of the bed, by reason of the under layer of it having not yet been attained in any pit. In the Jammerdaalsche Heide the clay is dug out 6 M. deep, and it has further been ascertained by means of bore holes, that even 2 M. deeper, so at about 19 M. + A.P., the clay makes place for sand. It is however probable that another layer of clay is underlying that sand. In bore holes put down on several places in and about Venlo, some of which approached the last mentioned pit to 2*/, K.M., they met with similar clay, in a layer of 8 M. thickness, resting, at 4 M.—A.P., on coarse white sand and gravel with much mica, and covered at 4 M. + A.P. by 3 M. sand and about 12 M. of gravel’). Along the right bank of the Meuse, south of Venlo, the edge of this very ferrugineous and somewhat consolidated gravel appears, covered by loam, at about 14 M. + A.P. Even at very low watermark, ofa _few decimeter above 8 M. + A.P., generally the underlying of the gravel cannot be seen in this outcrop. On a few spots however, about 1 K.M. south of the Meuse-bridge and 2 K.M. north-west of the mentioned clay-pit, I observed similar clay as that of Tegelen in the original situation, over 7 M. in horizontal connection, under the gravel. It reaches there upward to 11 M. + A.P. Evidently this clay in the bank of the Meuse belongs to the same bed as that which was met with in the bore holes at Venlo, the bed having been unevenly eroded a long time before the development of the present river channel. In such a way a difference of 7 M. could arise in the upper sur- face of the clay. In that clay on the right bank of the Mense I found a tibia of Rhinoceros, which is only assimilable with that bone of R. etruscus and R. Merchi. The bone was still a little fastened in the clay, for the greater part enveloped with the consolidated gravel. This clay is thereby characterised as interglacial or preglacial (pre- pleistocene). If belonging to the same bed as the Clay of Tegelen, the whole thickness of the latter, including sandlayers, may be estimated at about 30 M. In this computation it has been supposed that its under surface, from Venlo to Tegelen, is horizontal, which seems 1) According to a communication of Mr. pe Waar Marerur the top of the clay was 5 M. lower in a bore hole put down on the right Meuse-bank at Venlo. Pro- bably the clay which has been met with in bore holes, 24 K.M. south of Venlo, on the east side of the Meuse in the neighbourhood of Roermond, at about 3 M.-+A.P., under as much gravel, is geologically identical with that under Venlo« (-247) allowable over such a small distance, in comparison with the great extension of the bed, and with regard to the horizontal structure of the clay. Then we have to regard it as prepleistocene, an inter- pretation entirely confirmed by the following palaeontological facts. Much more improbable I hold it that the clay under Venlo and on the Meuse near that own, was deposited after the first Pleisto- cene Glacial epoch, that of the “Rhine-diluvium’. In that case we should be obliged to suppose two periods of erosion. In the first one the “Rhine-diluvium”, with the underlying Clay of Tegelen in the valley of the Meuse, should have been eroded, afterwards (during an interglacial period) clay should again have been deposited in it, which was attacked in a second period of erosion, on which than in the Second or Great Pleistocene Glacial epoch a deposit of gravel accumulated. For the chronology of the different beds of the Dutch Pleistocene formations now it is of great importance to ascertain, by means of enclosed fossils, the age of the Clay of Tegelen, which was deposited in the time preceding the accumulation of the ‘Rhine-diluvium’’. I am much obliged to Mr. L. Stuns, at the time medical student, now physician at Roermond, for having shown me, already in 1897, fossil remains of Mammals (especially Progontherium and Deer) and of Molluses, together with such of plants, which he had found in the clay-pit of Messrs. Canoy, HERFKENS and SMULDERS, a number of which he has yielded to me for a closer examination. I have further to acknowledge the benevolence of the last named gentlemen for the opportunity of collecting some fine and characteristic fossil remains of Mammals, especially of Cervus, Rhinoceros, Kquus, Hippopotamus and Trogontherium (now in Teyler Museum at Haarlem), by the aid of which the fixing of the geological horizon has been arrived at. The shells and plant remains (especially seeds and wood) and many bones are found at about 5 M. below the upper surface of the clay bed, where this is rather sandy, another, more abundant, ossiferous niveau is at nearly 3M. below that upper surface, in stiff clay '). Opposite Belfeld bones 1) To an average of 2.70 M. below the upper surface, from below which a very stiff clay begins, the clay in ihis pit has a yellow colour, caused by the action of the atmospherilia on the ferrugineous compounds in the clay, which action is lower down shut off by that stiff clay. The latter itself is of a bluish colour and at the bottom of the pit it is nearly black. Excepted near the upper surface, the yellow clay is on the whole sandy, only at a few places in the pit it is rather stiff. In those places the blue colour continues up to a higher level and the limiting line is not at ail right and horizontal, on the contrary the yellow clay, there, is sinking down, in that blue clay, which continues to a relatively higher level. Agatiform wrinkling brown parallel lines, in those yellow insinkings, imitate then contortions, ( 218 ) are mostly found at a depth of 4 M. in the clay; at 1.25 M. below its upper surface it there encloses a layer of sand, 0.30 M. thick. The outside of the bones is always absolutely uningured, they do not look rolled worn at all. The following enumeration of fossil forms will suffice for the determination of the geological horizon. I hope to be able to work out and to complete the list on a later occasion. As regarding the Molluses, it is in the first place noteworthy, that these for the most part belong to forms proper to fresh water, and especially to stagnant or very slowly running water; a few land- snails belong to species which may have lived upon the vegetation on the shore. Till now I have recognised : Fig. 1. — Cervus teguliensis, sp. n. Left antler, lateral aspect. (1/g). The figured specimen belongs to the collection of Mr. Sruns. Several other specimen do not possess the strong, curvature of the beam at the origin of the tres-tine, in such a manner that the beam is on the whole straighter. Paludina, 2 sp., Planorbis sp., Helix hispida L., Helix arbustorum L., Helix sp., Limnaeus sp., Pisidium, 2 sp, Unio sp. Of the Mammals the following species are well determinable : Trogontherium Cuvieri Owen, Cervus Sedgwickü Fale. (= Cervus dicranius Nesti), Cervus teguliensis, sp. n., Cervus (Aris) rhenanus, sp. n…, Cervus (Avis) sp., Hippopotamus amphibius L., Equus Stenonis Cocchi, Rhinoceros etruscus Fale. such as they have been produced elsewhere by the motion of the Pleistocene ice-sheet, but here we have indeed only before us a result of the process of the blue clay to yellow. Elsewhere, as opposite Belfeld, where still the origmal thick gravel bed covers the clay, and consequently the underground water is at a higher level, has preserved the greyish blue colour up to its upper surface. ( 219 ) There are found too remains of: Cistudo lutaria Marsili, of a Frog and of Fishes. Of great importance are the remains of plants, from which especially seeds were carefully collected by Mr. Stuns. They enable us to form a conclusion concerning the climate and thereby concerning the time in which the plants lived. And that is so much the more desirable as remains of Elephants have not yet been encoun- tered, among the mammalian remains. The species already determined are : Viburnum sp., Prunus sp., Trapa natans L., Cornus mas L., Vitis vinifera L., Sta- phylea pinnata L., Juglans tephrodes Ung., Nuphar luteum L., Stratiotes Websteri Pot., Abies pectinata DC., Chara sp. That is an assemblage of animals and plants which can only be preglacial in the sense of prepleistocene. The group of Mammals is distinguished from that of the Sands of Moosbach, which are now gene- rally regarded as a deposit of the inter- Fig. 2. — Cervus (Axis) rhena- nus, sp. n. Right antler, ; lateral aspect. (1/3). glacial period before the great or second Pleistocene Glacial epoch, by the possession of guus Stenonis *), of two species of Deer of the Arts type and an other species belonging to a type not represented by any living Deer. They give to the whole a Tertiary character and make, for themselves, the equivalence of the Clay of Tegelen with the Cromer Forest-Bed probable. From the last mentioned deposit we know one species Deer of the Avis-group (C. elueriarum, probably nearly related to the new species from Tegelen), a second species has been described from the somewhat older Norwich Crag; from the Pliocene of central France there are described as many as six species. Cervus teguliensis closely resembles C. tetraceros, Boyd Dawkins, from the youngest beds of that Pliocene, characterised by Mlephas meridionalis, and from the Cromer Forest-Bed, but the antler of the large Deer of Tegelen never obtained more than three tines. The other Mammals of the Clay of Tegelen are all known from the Cromer Bed. The presence of Equus Stenonis and Rhinoceros etruscus together with Trogontherium Cuviert and Hippopotamus amphibius major leaves no doubt on that equivalence. 1) Of this species it is the variety distinguished by Mr. M. Bovre as the one with great dimensions, which he believes to be the immediate ancestor of #. caballus. 15 Proceedings Royal Acad. Amsterdam. Vol. VII. ( 220 ) What is known from the species of plants in the Clay of Tegelen, points even to a somewhat warmer climate than that indicated by the flora of the Cromer Bed, and so, seemingly, to a somewhat older age. The Prunus-species is certainly different from P. spinosa L., which belongs to the Cromer fossil flora, and which is now also indigenous in northern Europe. The seed is only assimilable with that of species, which now appear to be spontaneous in Turkey, south of the Caucasus, in Armenia and northern Persia. Amongst the plants of the Cromer Bed Vitis vinifera is also wanting, which grows now spontaneously in temperate West-Asia, especially in Armenia and south of the Caucasus and the Caspian Sea, also in southern Europe, Algeria and Marocco. Remains are also found in Pleistocene travertines of Tuscany and southern France, where the species, with Ficus carica L., is considered to be a remnant of the Tertiary flora, further in Italian lake-dwellings. Amongst the numerous species of plants of the Cromer Bed is wanting too Staphylea pinnata. In wild state this species does not grow now more northerly than southern Germany, it is especially indignous in the Pontine countries and also in Asia Minor. No species of Juglans has been found in the Cromer Bed, Juglans tephrodes, the nut of which, like that of some nearly related forms, is hardly discernable from the present American Juglans cinerea L., is a Tertiary species of Italia and the middle of Germany. The seed of the Stratiotes is very different from that of S. aloides, on the contrary strikingly similar to that of S. Websteri from the Upper Miocene of the Wetterau. In the Cromer Bed as yet no Viburnum was found. The seed of the species from the Clay of Tegelen closely resembles that of V. Opulus L., it is only larger and a little less flat. The circumstance, that the genus Viburnum played an important rôle among the Tertiary flora seems to me not to be without bearing, in connexion with the above mentioned facts. A similar consideration applies to the genus Cornus, of which another species, C. sanguinea L. is found in fossil state on the coast of Norfolk. C. mas appears to grow, besides in Asia, only in southern and central Europe. C. sanguinea, on the other hand is also indige- nous in northern Europe. It wants no demonstration that the flora from the Travertines of Taubach, to which, on conclusive grounds, the same age as that of the Sands of Moosbach is now attributed, is a much younger one than that of the Clay of Tegelen. The former contains arctic and alpine forms, which are wanting here, on the other hand the fossil flora of Taubach lacks the mentioned Tertiary forms and_ those pointing to a warmer climate. From the fact that the flora of Tegelen ~ on » > EN co en — >» ( 221 ) apparently lived in a somewhat warmer climate than that of Cromer we are, however, not obliged to conclude that the former is older than the latter. For we have to consider, that the situation of Tegelen is about 2° of latitude more south than Cromer, but especially, as Prestwich and Cremerr Rem have shown, that local circumstances must have made the climate of Cromer relatively a less genial one. Of the Mammals, by which the older Pliocene deposits of Norfolk are distinguished from the Cromer Bed none are found in the Clay of Tegelen. Taking all these facts into serious consideration, there seems to me hardly to remain any room for doubting the equivalence of the latter with the Cromer Forest-Bed. Like this celebrated fluviatile and estuarine deposit and like the undermost gravel beds of Saint-Prest near Chartres, the alluvia characterised by Hlephas meridionalis in central-France and the lignite beds of Leffe near Gaudino, not far from Bergamo, they must be placed at the top of the Pliocene. On good reasons it is generally accepted, that at the end of the Pliocene period the continual subsidence during that period, the unmistakable proofs of which have been found as well in the Netherlands and Belgium as in England, has been interrupted by an uprising of the region, properly a flattening of that great concave zone or geosynclinal, in which the marine Pliocene sediments were deposited. In consequence the southern half of the North Sea was converted into land and England united with the continent. The great river of that sedimentation basin, the Rhine, as has been shown by Ciement Rem and by Harmer, then poured its waters over the east of England into the North Sea, and in Norfolk the Cromer Forest-Bed is a deposit of that river. Also Harmer rightly remarked, already in 1896, that this river, before it reached England must have passed somewhere over the Netherlands; so we should perhaps one day find the equivalent of the ossiferous beds of Cromer in our Country. In the Clay of Tegelen we now have really met with such a bed, which evidently accumulated in a shallow fresh-water lake, flow through by the Rhine. On good reasons it is also accepted that with the beginning of the Pleistocene period the geosynclinal became steeper, in conse- quence of which, over the greater part of the present Netherlands, sand, gravel and clay could accumulate, attaining a thickness, in Holland, up to. more than 150 M. But, at the same time, on the border of the steeper basin, in consequence of its larger angle of slope and the increased transporting capacity of the running waters there, first deposition of coarser material, the ‘Rhine-diluvium’’, took place 15* ( 222 ) and, as the slope increased still more, an important erosion. By this erosion the Meuse could excavate a broad valley through the Rhine-diluvium and deep in the Upper Pliocene clay, in which, the slope having somewhat decreased, probably already in the second or great Pleistocene Glacial Epoch, there accumulated a mighty deposit of gravel. Physics. — Contributions to the knowledge of VAN DER WAALS’ w- surface. LX. The conditions of coexistence of binary mixtures of normal substances according to the law of corresponding states. By Dr. H. KAMeERLINGH Onnes and Dr. C. ZAKRZEWSKI Supplement N°. 8 to the communications from the Physical Laboratory at Leiden. (Communicated in the meeting of February 27, 1904). 1. The graphical treatment of the conditions of coexistence. In this paper where the theory of mixtures of vaN DER Waars is illustrated, as in the former contributions we have placed in the foreground the law of corresponding states. The data required calculating vAN DER WAALS’ w-surfaces for all temperatures may be defined in the following way from the point of view of this law: 1°. An equation of state agreeing with reality, must be given for one normal substance over the whole range of temperatures and pressures to be obtained, (comp. $ 2). 2°. For the different mixtures of the two substances considered, as well as for these substances themselves, the deviations from the law of corresponding states must be known (comp. $ 8). 3e We must know the critical temperature 7, and pressure Pek Of each mixture taken as homogeneous *) with the molecular proportion 2 of one of the components, derived from the law of cor- responding states, as functions of those of the simple substances and of « (comp. $ 4). With these data at our disposal van DER Waats’ theory will teach us all possible cases of coexisting phases of those substances if we roll tangent planes over the w-surfaces of each pair of substances for different temperatures. In the treatment of the problems concerning conditions of coexistence 1) Whenever we speak of critical temperature, a maximum vapour tension ete. of a mixture without more, we always mean “taken as homogeneous”, ( 223 ) which have been solved in general by van per Waars, the following simplifications have been made: 1°. The first form with two constants a and 5, in which vaN DER Waats has written the equation of state, is used instead of the real equation of state; 2°. The supposition has been made that also the equation of state of each mixture with the proportion « has the same form with two constants ax and 6,, whereby the law of corresponding states is rigorously satisfied ; 3°. it has been supposed that the critical quantities determined by ad, and 6, are related to those of the simple substances determined by a,, and 6,,, a,, and 6,, by means of the relations dy = Ai 2° + 2a,,2(1 — 2) Ha, (1 — 2)’ by =b et? + 26,,2(1 — 2) Hb, (1 — a)’, so that the entire behaviour of the mixtures of two known substances is determined by two additional constants a,, and 4; and 4°. it has often been assumed that the vapour phase satisfies the laws of ideal gases. In this way vAN DER Waars has obtained important approximation formulae. Though they do not always represent numerically accurately the behaviour of the mixtures, most of the particularities of the con- ditions of coexistence are sufficiently explained by these in general valid formulae *). If in the treatment of these problems we want to use equations of state which over the whole range of temperatures and pressures agree accurately with the observations, if therefore we do not require the simplifications ad 1°, 2° and 3°, and if finally we want to consider other than rarefied vapour phases, so that the neglection of the deviations from the law of Boyie-Gay Lussac-AvoGapRo mentioned ad 4° is not allowed, an analytical treatment of the conditions of coexistence in general becomes impossible. Such problems we come across, for instance, when we derive the conditions of coexistence for mixtures of oxygen and nitrogen (critical state of air and relation between composition and pressure in the boiling off liquid air) from equations of state which also at the ordinary temperature accurately represent the compressibility of these substances and of their mixtures. An instance of an entirely different kind is given by the following group of problems : determine at ordinary temperature the absorption of hydrogen in ether and the 1) van per Waats, Die Continuität etc. IL p. 52, ( 224 ) deviations from the law of Henry for this pair of substances, investigate the variations of this absorption with a small variation of temperature and finally find for the same temperature the pressure to which Kunpt ought to have gone in his experiments *) on the removal of the capillary ascension of liquid by pressing gas on it in order to see the meniscus of ether disappear under the pressure of hydrogen, which, as has been remarked in van Erpimk’s thesis for the doctorate p. 7, comes to a determination of the plaitpoint pressure of the mixture of ether and hydrogen of which the plaitpoint lies at this temperature *). In such cases we can only obtain solutions by means of the graphical treatment described in Comm. N°. 59a (Sept. 1900). It is true that the graphical method lacks the general character of the approximate solutions just mentioned, yet by means of it a better numerical agreement may be obtained for each special case. *) By a proper choice of special cases some data may also be derived for the qualitative characters of phenomena in mixtures ‘). If we only aim at such qualitative results we may simplify the graphical treatment as well as the analytical by introducing different approximations according to the nature of the problem, while it lies at hand to derive wanting experimental data or results of calcula- tions from empirical formulae on a larger scale than in the analytical treatment. For instance, everything that may be neglected in the analytical treatment may also be neglected in the graphical method. Occasion for this exists only, however, with problems which neither qualitatively are solved by means of the analytical method. If we do not want to neglect to such an extent as in the analytical treatment, we might for instance retain the negleetion of the deviations mentioned sub 2° of different normal substances and mixtures of normal substances from the law of corresponding states, which first occurs in Comm. N°. 59a and is kept up in this whole series of contributions: 1) Repeated by van Expix (thesis for the doctorate; Leiden 1898) for hydrogen- ether and ethylene-methylchloride. 2) Comp. also van per Waats, Die Continuität etc. IL, p. 136. 3) For instance, one of us and Retneanum have derived (Comm. N°. 595 Sept. 1900) a numerical fairly approximate representation of the retrograde condensation observed by Kurnen with mixtures of methylchloride and carbon dioxide from the isothermals observed by him at other temperatures. +) So for instance the character of the retrograde condensation (comp. note 2), and also the peculiarity in the conditions of coexistence in mixtures of which the critical temperature varies almost linearly with the composition, far below the critical temperature. Comp. ‚these contributions Comm. NO. 59a § 8 at the end and III, Gomm. N°. 64. Harrman, Livre jubil. Lorentz, p. 640. The simplifications relating to the case that one of the two coexisting phases as compared with the other has a very small density, will be considered in the sections 7 and 8 of this Commu- nication. § 2. Empirical reduced equation of state. As to the supposition mentioned in $ 1, we are now under much more favourable circum- stances than at the time when Comm. N°. 59a was written. The beauty of VAN DER Waats’ theory lies above all in the fact that it brings under one point of view phenomena in mixtures which are distributed over a large range of temperatures and densities. Hence for a satisfactory illustration of this theory we require first of all an equation of state which holds true over a large range of tempe- ratures and densities. Now most of the equations of state — this has been made clear especially by D. Bertartot — hold only for a limited range. For considerations as are meant here probably only those equations of state can serve which are developed in series and made to agree with the observations over a very large range. Such equa- tions of state which are very suitable for the calculation have been obtained in Comms. N°. 71 (June ’01) and N°. 74 (Livre jubil. BosscHa p. 874) by combining as well as possible the known pieces of reduced equations of state for substances with different critical temperatures. As now we neglect the deviations from the law of corresponding states in the different substances and in their mixtures, we may without more base our considerations on a similar empirical reduced equation of state. We have used a form which does not differ much from the more preliminary one given in Comm. N°. 74, which was indicated by V2. We obtained it by making it agree with hydrogen 0° C. *), oxygen and nitrogen 0°C. (all of AmacaT) and ether 0°C., 100° C., 195°C. (Amaat, Ramsay and Youne). This polynomial, which contains for instance all the reduced temperatures which occur on the y-surtace for ether and hydrogen at 0° C., will be designated by V/1. As in Comms. N°. 71 and 74 we have for a substance with the critical temperature 7, and pressure pz, if v is expressed in the theoretical normal volume, Be sE VE a BO Ee Da We Tire i fea sae ie ap EEE dd where at an absolute temperature of t° above freezing-point A= 1+ 0.0036625 t 1) For hydrogen the critical quantities of Orszewskr are still used in the calculation, 2) Comp. Comm. No. 71 form. (10). Ti Te Ti Te Te B ED, | fae ayo D=—>9, k= a R= — 8°) Pk k Pk Pk ed Vi and the reduced virial coefficients B, €, D, €, % with it are k determined by Reduced virial coefficients VI. 1. 3 2 1 é 1 108 B 18 OSS "AOR Er 2121 —184410 5 2 5 1 1 1 108 € 58.508 ¢ + 23:55 — 14.451 +159.936-5 ei 21.692 & Oy i 1 : 1 1 10%8D| 482.544¢ —379.527 — 562.94-— 1203.384 5 — 1582155 bs 1 1 10% € | —1910.43t +6797.37 — 5322 +1143.47— | i 1 10% § | 2052.16¢ —7742.41 +7204.66—- —1843.03 5 + 192.5575 The calculations which show the systematical deviations of different normal substances from a similar equation of state are progressing. § 3. Validity of the law of corresponding states for mixtures. We can judge of this much better now than when Comm. No. 59a was written. The different applications given in this series of Contributions to the knowledge of van DER WAALS’ y-surface *) seem to indicate that the deviations in mixtures of normal substances are not much larger than those which occur in normal substances inter se. This is very striking, especially if we look for the basis of the law of corresponding states in the mechanical similarity of the move- ments, as a mixture even geometrically is not at all similar to a simple substance. It would seem to follow from this, that the linear quan- tity which determines the geometrical scale of similarity is a mean value of very different linear quantities, which play parts in different collisions. In that case we should have to attach to the “volume of a molecule” not so much a physical meaning and rather the geome- trical meaning of a sphere drawn with this linear quantity as a radius. To establish a systematical relation in the deviations of the mix- tures of normal substances from the law of corresponding states is not likely to be feasible until this has been done for the devia- 1) Comm. No. 595 KamertincH Onnes and Retneanum. Comm. No. 64, HARTMAN. Comms. Nos. 65, 81, Suppls. Nos. 5, 6, 7 Verscuarrett. Comms. No. 75 and No. 79 Keesom. ( 227 ) tions of the normal substances themselves. Still a beginning has been made with calculations which aim at a representation of those deviations. § 4. Determination of the critical quantities of the mixtures taken as homogeneous. As we neglect the deviations from the law of cor- responding states, we may derive these quantities (occurring in the unstable region and hence not to be determined directly) from any observed range of the equation of state of any mixture. The most obvious means is the shifting of logarithmical or partly invariant diagrams of isothermals in the area near the critical state. pv oi log v, in Comm. N°. 65 to log p-isothermals with regard to log v, in In Comm. N°. 595 it was applied to isothermals with regard to Comm. N°. 88 to log 2 isothermals with regard to log p and to v, ; ele log = isothermals with regard to logv. We may also imagine, however, that we have at our disposal a sufficient number of obser- vations of another range. Thus, to give a simple instance, we know the critical temperature of a mixture if the temperature is found, at which under relatively small pressures it does not deviate from the law of Boyre. And it is also possible that we may derive the data from observed conditions of coexistence. If the critical quantities for some mixtures are found, it will in graphical solutions be preferable to derive the 77 and p,, as a function of x also graphically. For the experiments of KugeNEN have made us doubt whether the suppo- sition made ad 3° is in general possible, and this doubt is strengthened by Kerrsom’s experiments. If on the other hand we confine ourselves to qualitative investi- gations of mixtures of substances about which all the data which belong to the mixing are lacking, the supposition first lies at hand that dia =Va KET 4) and bis = 3 Oe Dak § 5. The reduced w-curves. In Comm. N°. 59a is briefly set forth how the different y,-curves can be derived from those that have been calculated once for all for a simple substance. If we write a little more extensively, and if v, indicates such a large volume that with this the mixtures are in the ideal gaseous state, v Wey = — | pdv + RT fa log « + (le) log (1—2)} omitting a temperature function linear in z. This with 1) GauirzinE and D, BERTHELOT. ( 228 ) Pk Uk 1 Pp v =P rn Tend here C, Pk Vie may be easily transformed into: » Wy ih JE = = C, fra — log x + «log « + (1—2) log (le) . (1) n if we put m a certain large number and neglect other temperature functions. p For convenience we may call foes as function of » the curve of n reduced free energy for t. In the construction of each given Y-surface occurs the group of curves of reduced y that lie between the extreme values of reduced temperature which occur on this surface. On the planes ws —=0 and z=1 of the y-model we can draw the y-curves of higher and lower temperature. In passing over to a w-surface of higher temperature the w-curves (at least in the most common case) are moved on the surface from the side of the highest reduced temperature to that of the lowest, while the linear dimensions in the two directions yw and v undergo a certain variation. $ 6. The w-surface for mixtures of methylchloride and carbon dioxide at — 25°C. As an example of the application of the graphical method and the empirical reduced equation of state we have now chosen the prediction of the composition of the coexisting phases and the coexistence pressure for mixtures of methylchloride and carbon dioxide at — 25°C. We were led to this choice by the following considerations : 1°. we can derive the critical quantities from the experiments of Kurnen'), they lie tolerably far from each other, are given in Comm. N°. 594, while for + 9.°5 C. a model has been constructed by HARTMAN; 9°. — 25° C. corresponds to the lowest reduced temperature for methylchloride for which the empirical reduced equation of state has been calculated ; 1) As critical quantities are used, comp. Comm. NO. 595, 7 Tek Pak carbon dioxide 0 303 129 1/4 336 3 1/, 363 71.8 3/4 391 68.9 methylchloride 1 416 64.8 i . rh - , ier fe Pig . H. KAMERLINGH ONNES and C. ZAKRZEWSKI. „Contributions to the knowledge of VAN DER WAALS’ J-surface. IX The conditions of coexistence of binary mixtures of normal substances according to the law of corresponding states.” Fig ile Proceedings Royal Acad. Amsterdam. Vol. VII ( 229 ) 3°. with no other equation of state suitable for a calculation we can reach such a low temperature ; 4°. van DER Waars, for similar circumstances as are found on this surface, has also derived analytically the particularities in the coexistence-phenomena (Contin. il, p. 146, sqq.), and the agreement between these results and the observations of carbon dioxide and methylchloride, found partly already by HARTMAN at + 9.55 C., will probably appear even more clearly at — 25° C., 5°. finally that a paper on an experimental determination of these conditions of coexistence, which also van per WaarLs thinks very desirable (Contin. I, p. 154) will, as we hope, soon be published. The numerical agreement becomes less accurate because methy|-chlo- ride is not similar to ether, with which substance the empirical reduced equation of state for the reduced temperature of the methylchloride on this w-surface is made to agree whereas this is the case with carbon dioxide at the reduced temperature, which it has on this w-surface. An empirical reduced equation of state in good harmony for the reduced temperature of 0.6 with methylchloride of — 25° C., for the reduced temperature of 0.8 with carbon dioxide of — 25° C., would have been more favourable for the obtainment of a numerical agreement. The plaster model obtained is represented on the annexed plate fig. 1, it is 0,7 m. long (v-axis), 0.4 m. high (y-axis), 0,8 m. broad (x-axis). The large dimension in length was made necessary by the great difference in density of the vapour phase between carbon dioxide and methylchloride. The binodal curve and the tangents which connect two coexisting phases, the nodal lines, are found by rolling a piece of plate-glass. Fig. 2 shows the binodal curve with nodal lines, and also sections v = const. projected on the ep-plane, fig. 3 the same on the zv-plane; fig. 4 shows the values of the pressure as function of the composi- tion of the coexisting phases. It is obvious that: 1°. the liquid ridge, i.e. that part of the surface which lies on the side of the small volumes, when the dimensions of the surface are not taken extraordinarily large, becomes very thin and the con- struction is practically possible only if we take for it a plate of uniform thickness (for instance a sheet of tin); 2°. while the tangent plane is rolled over the ridge of the liquid part and over the convex vapour surface, as the ridge near the pure methylehloride rapidly changes its direction, the point of contact moves a long way on the vapour branch of the binodal line, while the node on the liquid branch moves only a little. Hence in ( 230 ) the directions of the tangents the double fan-shape is very prominent. 3°. the vapour branch of the binodal line in the projection is almost a straight line, and in agreement with this, according to VAN DER Waars, the law of Henry holds over the whole area of composition variation, while the vapour branch on the pe-diagram is again almost a hyperbola. § 7. Simplification of the determination of conditions of coexistence when the liquid phase is far below its *) critical temperature. In order to determine the binodal line of the transverse plait we need only know two zones on either side of the plait. Let the border curve of the homogeneous mixtures be that curve which on the w-surface con- nects the vapour and the liquid phases in which the mixture, with the composition # taken as homogeneous, would be in equilibrium, then the binodal line wanted lies beyond *) this border curve, which it meets for the compositions O and «. Therefore it is more or less indicated, which zone on the vapour side we have to calculate. If, as in the case of methylchloride and carbon dioxide at — 25° C. (or with ether and hydrogen at the ordinary temperature for the part of the ether side) the zone on the liquid side is shrunk to a plate, we need only calculate a single curve on the liquid side. For then the point of contact is so near the rim curve — that for which w is a minimum —- that this curve may be substituted for the w-surface. The v of this curve may be easily derived for each « from the d *) = — p=0. With the value Vp=o OF equation of state because ( 3 t,—o we then find by integration „—o, while in the way as described in Comm. n° 66 $ 5,°) the tangent through the point w,—o, v,=9 to the curve yw, on the vapour surface is drawn graphically and the vapour tension of the homogeneous mixture is obtained. If we only wish to determine the pressure and the composition of the coexisting phases we have also a neglection of little importance in the supposition that vige = Vig, & + Vig, (l—e), in other words that the rim curve lies in a plane, while in many cases we may suppose that this plane coincides with the yr-plane. In order to find the conditions of coexistence we roll a plane (piece of plate-glass) over the rim of a thin plate cut after the calculated rim curve (vette still the rim curve for ze) and over a model 1) Comp. § 1 footnote. 2) Van per Waats Gontinuität II p. 100. 3) Arch. Néerl. Livre jubil. Lorentz, p. 665. ( 231 ) (made for instance of plaster) of the vapour surface (which if we ~ Pea ed ty i ef é il wi a A ie Fa Bs ak f Pr ¥ sv JRL 8 > ( of; « 7? ‘- ê me * 7 a J re 7 dik wis? 2 ee eee Li H. KAMERLINGH ONNES and C. ZAKRZEWSKI. ,,Contributions to the knowledge of VAN DER WAALS’ v-surface. IX. The conditions of coexistence of binary mixtures of normal substances according to the law of corresponding states.” 1.6 0 3 2 fig. <2: Pax 18 X 0.94 4 16 fs.5. Proceedings Royal Acad. Amsterdam. Vol. VU. ( 233 ) The considerations, constructions and figures given seem therefore suited to illustrate this part of the theory of vaN DER WAALS. Er is in the case of carbon dioxide and methylchloride a curve slightly bent downwards. Fig. 5 shows how the liquid branch of the binodal ) Tieke line (the rim curve of the liquid part) (comp. also the other d u figures) follows from the curve Wn Yh and the curve pz. As to the calculation of gat it should be remarked that Paz as function of t is known through the vapour tension law, and hence also from the empirical equation of state »,,,, and that n n f pdo = 3e pdv + Pmax (Ovap — Mig). Especially if vig is small and vliq Crap hence Wap large, (so that at the utmost — comes into consideration v for the deviation from the ideal gas laws) this, when at the same time neglecting %;,,, leads to an important shortening of the calculations. Neglecting entirely the deviation of the vapour phases from the ideal ! / 1—t gaseous state and accepting for a simple substance log pmax = — f —— t we return to the developments given by vaN DER Wats in his theory of the ternary mixtures*) in which theory many problems about the binary mixtures are developed more in detail. Physics. — “The determination of the conditions of coexistence of vapour and liquid phases of mixtures of gases at low tempe- ratures.’ By Dr. H. KAMERLINGH ONNEs and Dr. C. ZAKRZEWSKI. Communication N°. 92 from the Physical Laboratory of Leiden by Prof. H. KAMERLINGH ONNES. (Communicated in the meeting of June 25, 1904). § 1. Introduction. The two methods for the determination of the molecular coexistence compositions x; and 2, of the liquid and vapour phases of substances which are gaseous in normal conditions, it is known, can be described as follows. Following the first method we separate small quantities from the two phases at a series of coexistence 1) Proceedings, May 31, 1902, p. 1, sqq. ( 234 ) pressures p and determine each time the composition of those two quantities either by chemical or by physical means. Fig. 1. Fig. 2. On the wv z-projection of the binodal curve of the transverse plait on VAN DER Waars’ wav-surface for a given temperature 7, fig. 1, and also on the pa diagram of that binodal line, fig. 2, two such phases are indicated by a and 5, for instance. The determination of several pairs of values ab, a’ b' ete. gives then the whole course of p, v« and v over the transverse plait for 7. If we follow the second method we observe in a series of mix- tures with a known composition x, the beginning- and endcondensation phase and determine for them p and v, hence pur and p‚rr, and vier, and voer. This investigation comprises each time the phases represented in fig. 1 and 2 by 4 and c. By combining the results be, b’c’ ete. we can derive again the binodal line and hence the values for Llp; LypT, VipT and v‚pr. The application of this last method to low temperatures especially under moderate pressures, forms the subject of this paper. It is possible to follow also the first method in the case of low temperatures as it has been applied by HARTMAN in Comm. N°. 43 (June ’98) for ordinary temperatures. Yet as a rule the analysis of a gaseous mixture is much more difficult than the preparation of a mixture of a definite composition (among others, by means of the mixing apparatus of Comm. N°. 84, Dec. ’02) and it is difficult to obtain certainty whether the quantities of vapour and liquid run off have the same composition as» the phases which are brought to equilibrium by stirring. Therefore it is important to solve the difficulty which accompanies ( 235 ) the second method for temperatures below — 48° (melting-point of mercury). When we made a first trial in this direction, a high degree of accuracy was not aimed at. In order to answer several questions the accuracy we have reached is sufficient and for the caleulation of corrections for more accurate determinations it is quite sufficient. In our measurements we have used the apparatus which is represented schematically in fig. 3. It is in principle a twice bent tube of CarLLePET, of which one end is immersed in the cryostat of the temperature 7’ and filled with a known quantity of a mixture of known composition «, which by forcing up mercury at the ordinary (or higher) temperature 7” is brought to condensation at 7. The mixture of which the quantity and the composition . are known, is then only partly contained in the vessel 7’ at the tem- perature of investigation. Another part is necessarily in a tube at the ordinary or higher temperature 7“ which is connected with the vessel by means of a capillary tube. This involves complications, not with regard to the measurement of the begin condensation pressure p‚ for the composition x; for until the condensation of the first traces of liquid, with respect to which the vapour is a phase of equilibrium, the composition of the vapour phase in the space at low temperature remains as it was originally, and hence the com- position of the vapour phase of equilibrium is perfectly known; but with regard to the determination of the end condensation pressure Pic of the mixture with the composition v. For we cannot condense the whole quantity of the mixture at low temperature. Hence the composition of the liquid phase at 7, above which there is a vapour phase of a composition differing in the main from this or the original composition, is no longer indicated by the latter and therefore unknown. We can, however, find this composition by applying a correction to the original composition wv which, as long as the vapour phase occupies only a small volume and remains under a moderate pressure, is not very large (comp. $ 5). § 2. General arrangement of the measurements. A schematical representation of it will be found on Pl. I, fig. 1. The letters are the same as those used on the plates to which we shall refer. The volumenometer / (with manometer JZ, comp. Comm. N°. 84, 16 Proceedings Royal Acad. Amsterdam. Voi. Vii. ( 236 ) March °08, Pl. II, figs. 1 and 2) contains the gaseous mixture which has been prepared in it by means of the apparatus connected at r, and r, (comp. Comm. n°. 84). Through the cock 7,, the steel capillary g,' and the steel three-way stopcock / (see Comm. N°. 84, Pl. I, fig. 2) it is led to the pressure tube 4 with calibrated stem (see Comm. N°. 69, April’O1, Pl. Il) placed in the pressure cylinder 4, (Comm. N°. 84, Pl. I) in order to be compressed by means of mereury from the pressure reservoir (,’ (comp. Comm. N°. 84). Thence through the three-way stopstock / and the steel capillary g," it can be brought under pressure into one of the two apparatus Dor YP. For the present, according as we wanted to determine either the begin or end condensation pressure p‚‚r Or Pier, We have connected either the capillary of 9,2), or that of ® W, to the capillary g," by means of sealing wax. Different from Comm. N°. 84 Pl. | fig. 1 the mercury of the pressure cylinder (see our plate fig. 1) is also connected by means of » with that of a manometer to determine the pressure in D or Wp. By means of the three-way stopcock # the apparatus D or } may also be connected directly with the volumenometer and then the pressure is measured by means of JM. A detailed representation of the apparatus 2 and is given in figs. 2 and 3 of Pl. I and a description in §§ 4 and 5. In both cases the glass tube to which the mixture which is to be investigated is conducted by means of the capillary g,", p in ® and « in D, is immersed in a cryostat where the desired temperature is kept in the same way as described in Comm. N°. 83 (Feb. and March ’03). The regulation is brought about for through the exhaust-tube 7,, and for D through the tubes 7, and m,. In our measurements we brought liquid methylehloride into the cryostat and the tempe- rature was regulated according to the indications of an alcohol ther- mometer. By means of the cock 7, the volumenometer £ is connected not only with the gasreservoirs but also with the mercury airpump so that also the pressure-tube and the test-tube in D and $ can be exhausted. We shall not expatiate on the process of filling the apparatus with a definite quantity of the desired mixture and of measuring any quantity of gas which we allow to escape from them. For the rest we refer to Comm. N°. 84 for the volumenometer and the mixing apparatus, to Comms. N°. 69 and N°. 84 for the pressure cylinder and its appurtenances, to §§4 and 5 and Comm. N°. 83 for the cryostats. § 3. Determination of the molecular volume of the coexisting phases. As we did not aim at a very high degree of accuracy a few remarks will be sufficient. As to the liquid phase: with the limitations and conditions to be treated in $ 5, part of the gaseous mixture in the apparatus J, called for shortness the piezometer, may by conden- sation attain a liquid phase of a composition which, as said in § 1, is derived from mz of the original gaseous mixture by a correction. Measurements with the volumenometer yield the volume that the liquid phase would occupy in its gaseous state. And by means of the divisions at p,; and p, Pl. I fig. 3 we read the volume of the liquid phase itself. The molecular volume of the gaseous phase may best be derived from the coexistence pressure and from isothermals which with not too small pressures can be determined after the method of Comm. N°. 78, April ’02, if necessary with the dew-point appa- ratus D itself. With not too large pressure it will in most cases be possible to apply the correction for the deviation from the law of Boye-Gay-Lussac-AvoGapro by means of the empirical reduced equa- tion of state (Comm. N°. 71, June ’O1) according to the law of cor- responding states. | § 4. Beginning of the condensation. In order to determine this we have applied the principle of Rrerauur’s hygrometer '). To this end we observe the first condensation which is formed in a part of the apparatus of which the temperature is a little below that of the surrounding gaseous mixture when the gaseous mixture is brought to the begin condensation pressure. In order to be able to observe a very small condensation we have taken for the place of the lowest temperature a shining mirror, and next to it is placed another mirror which is not cooled. The apparatus is blown of one piece in the same way as an ice ‘alorimeter of Bunsen. The outer space « has a capacity of about 20 ee. and is provided with a capillary to which the steel capillary g',, through which the gaseous mixture is led, may be sealed on. One of the mirrors d is sealed on to the bottom. The inner tube ec, carries at c, the second reflecting surface and serves at the same time as a cryostat to keep the temperature of this surface constant, a little below the temperature of the gaseous mixture. To this end we have devised in the same way as for the outer cryostat the cover m,, fitting hermetically on the tube c,, the small stirrer 4, of which the rod h, projects through an india-rubber tube 1) Barren, (Ann. de Chim. et de Phys. ser. 6, vol. 25, p. 59, 1892), has found that for pure substances the deposition of liquid on a not cooled mirror placed in the vapour is a suitable means for the determination of the condensation point. 16* as in Comm. N°. 83, and the thermometer 7 sealed on to m, while the vapour of the liquid gas in c, is exhausted through m,. On the tube c at c, the glass cap f, is sealed on, through which with an india-rubber stopper the capillary connected to c passes at f This cap forms the cover of the larger cryostat. Through this the apparatus has become very firm and very easy to handle. The capillary 6, is protected against the stirrer by a metal frame n. Through the cover at /, pass the wires by which the stirrer is suspended. At 7, the vapour of the bath of liquefied gas is exhausted. The surfaces of d and c, are made reflectors by platinizing them at a redhot temperature with platinum chloride in camomileoil. Though the platinum surface is not so shining as that of a silver mirror, yet the advantages of platinum for this purpose are evident. The regulation of the temperature in the two cryostats of the appa- ratus is performed by means of the same exhaust-pump that keeps the pressure constant by means of a pressure regulator (according to the principle of Comm. N° 87, $ 3, March O4) in a main tube which branches off into two exhaust-tubes, each shut with a cock. By adjusting these cocks we can take care that the temperature in the inner cryostat is a little below that of the outer cryostat. What difference of temperature can be kept constant in the two depends on the temperature itself and on the liquid gas. In measurements to be described in the continuation of this paper the temperature of the large cryostat was — 25°.0 C., that of the small one — 25°.1 C. The required decrease of pressure in the main exhaust-tube could be easily kept up (boiling-point methylchloride — 23° ©.) with a water airpump. The pressure regulator was adjusted at about the pressure belonging to — 25° C. By means of the regulating cocks an assistant took care, as signs were given by another assistant according to the thermometer readings, that both temperatures, hence also the diffe- rence between them, remained constant. This could be attained to within 0°.05 C. The accuracy which we can reach in the determination of the dew-point with our apparatus depends in the first place on the difference of temperature in the two cryostats. As a matter of course it is smaller as the temperature coefficient of the begin condensation pressure is larger. At a given difference of temperature it increases according to the difference of the pressures at which the conden- sation appears or disappears. The amount of this difference is also determined by the illumination and this is much impaired by the two walls of the cryostat. In our experiments, observations made with the ——— a a nnn ( 239 ) naked eye with side-illamination of the mirror proved to be the best. The difference ranged between limits which amounted to 2°/, of the pressure. The optical part of the method may certainly be much improved. The accuracy attained will, however, be sufficient in many cases. Adiabatic pressure-variation must naturally be avoided. Yet all such difficulties arise also with measurements at ordinary tempera- ture. With a view to the large dimensions of the vapour space a we have not made use of a stirrer and have tried as much as possible to surmount the difficulties by operating slowly. § 5. Determination of the end condensation pressure. For this determination the mixture, after the begin condensation pressure is measured, is led back to the volumenometer, the dew-point apparatus is disconnected from the steel capillary and in its place the piezometer p in 9 PL I figs. 1 and 3 is connected with the steel capillary g",. The piezometer consists of a wide glass tube p, Pl. 1 fig. 3 fastened to a capillary, both graduated and calibrated. The dimensions are chosen with regard to the quantities of gas that may be intended for the measurements. If these are decided upon, the exact quantity of gas, necessary for filling the piezometer at a suitable position of the mercury in the pressure tube 6 with liquid to near the end of the capillary, must be determined before each determination by means of a preliminary experiment. The equilibrium of the phases in p, is reached by means of a magnetic stirrer g moved by the coil S. The immediate effect of this coil is not sufficient to move the stirrer forcibly through the liquid meniscus. Therefore a soft iron tube z, with a groove z, which enables us to read on p,, is moved up and down at the same time with the coil. This movement ought to be independent from that of the stirrer in the cryostat. But as we did not require a very high degree of accuracy in our experiments we have for simplicity devised the iron tube < as a connection between the upper and the lower part %, and x,, of the ringshaped valved stirrer (comp. Comm. N°. 83). This is moved up and down mechanically and with the hand in turns, one time to stir the liquid bath, the other time to establish equilibrium between the liquid and the vapour while we simultane- ously move the magnetizing coil S. The essential difference between a determination of the end con- densation pressure in our apparatus and that in a Camerer tube does not lie so much in the circumstance that we do not liquefy the whole quantity of gas, as in the fact that, as remarked in ( 240 ) § 1, the temperature is different in the different portions of the gaseous mixture. The portion (see fig. 3 $ 1) at the temperature of investigation 7’ is separated from that at ordinary (or higher) tem- perature 7” by a series of layers (in the capillary) of which the temperatures range from the highest 7” to the lowest 7. One of these temperatures we shall call 7”. This circumstance involves ” several restrictions and conditions in the application of the method. In order to make measurements possible for all compositions w the temperature 7” must be taken or raised so high above 7’, that on the pr-diagram (fig. 4) the vapour branch of the binodal curve of 7” does not intersect the liquid binodal curve 7. Then we need not fear that condensation begins at 7” while condensation takes place during IJ) a fe] . . . . the transfer of the mixture from Mig. 4. the compression tube (or perhaps the volumenometer) to the piezometer. At 7” we then always have a gaseous mixture of the original composition. When the above- mentioned curves intersect, as is represented in fig. 4 for the case of 7", we can make measurements only for the compositions represented by points outside the region included approximately between d and e. Even if, without condensation taking place in the compression apparatus (or perhaps volumenometer) the gaseous mixture can be transferred to the piezometer, the part of the capillary where the temperature falls from 7” to 7’ still offers another difficulty of the same kind. Here we necessarily find temperatures 7” at which the vapour branch of the binodal curve of 7” intersects the liquid branch of 7". If drops are formed at 7", the composition .,,7 of the liquid belonging to the observed coexistence pressure can no longer be indicated. By flowing down and by distillation (the effect of capilla- rity exceeds that of gravitation) the drops gradually pass over into the liquid phase at 7, if care is taken by means of the cock 4 (see Pl. I fig. 1) and by adjustment of the pressure in 5 (see PI. I fig. 1)) that gas streams only into and not out of the piezometer, until finally When we stir with open cock & it appears that equilibrium is reached and the capillary contains vapour only. In order to further the distillation and the disappearance of the H. KAMERLINGH ONNES of mixtures of gases RLINGH ONNES and C. ZAKRZEWSKI. „The determination of the conditions of coexistence of vapour and liquid phases ME : ” Ë or mixtures of gases at low temperatures. Fig. | Vig. 3. Proceedings Royal Acad. Amsterdam. Vol. VII. (edad) condensation, it is desirable that the capillary at the place where the temperature is between 7’ and 7” should not be too narrow. Moreover the capillary is surrounded by an air jacket p,, made of a glass tube tightly closed with india-rubber rings p, and fish-glue. To avoid diffusion the other portion of the capillary g", must be narrow. If by previous determinations with the dew-point apparatus we have determined wr (as a first approximation it will in some cases be possible to use a preliminary y-surface as constructed in Supple- ment N'. 8 see p. 222), it is easy to apply the correction necessary to derive the composition of the investigated liquidphase at 7’ from «,, the original composition. On the piezometer divisions the volume of the vapour is read. Let WV, be this volume reduced to normal cir- cumstances and corrected for the first virial coefficient ZB (comp. for instance the continuation of this paper), let V be the entire volume of vapour and liquid, measured and corrected in the same way, then V;—=V—VJ/, is the volume of gas, measured and corrected in the same way, which would form the liquid phase. Hence : x 1 Vi— Kop TV, Vey : V, KT ee VY (X= Xe) Se: If we operate under moderate pressures, the correction will be always small and even if «,,7 is not very accurately known, it can be applied in a satisfactory way. Astronomy. — “A new method of interpolation with compensation applied to the reduction of the corrections and the rates of the standard clock of the Observatory at Leyden, Houwi 17, determined by the observations with the transit circle in 1903.” By J. Weeper. Communicated by Dr. H. G. vAN DE SANDE BAKHUYZEN. § 4. In the Proceedings of Nov. 29, 1902 occurs a paper “On interpolation based on a supposed condition of minimum,” of which the present paper is to be considered as a continuation; this explains the numbering of the sections. In order to interpolate between the ordinates S belonging to the abscissae t— a, b...y,7, 1 have there determined the interpolating curve for which the total sinuosity as? 5 fg (=) dt has the least value. I found that between two sue- a ( 242°) cessive points which are well defined by observations this curve satisfies an equation of the third degree S,= S,;-+ gg T+ ¢, T° +e, T°, where 7—=t—q and q<{t day. This difference of temperature, apart from variations in connection with mete- orological conditions, clearly shows a yearly periodicity; in winter it is small, in summer on some days it increases to above 0°.50 CG. It can hardly be doubted that, at least to begin with December 1898, the yearly periodicity must be ascribed to the temperature gradient; for it is since that date that the standard clock has been placed in the vestibule in a niche cut out from the pier of the 10-inches refractor. Since about March 1899 the place on the Eastern pier in the transit room where the standard clock hung during the former periods 1862—’74 and 1877—"98, treated by KE, F. van pe Sanne BAKHUYzEN, has been occupied by the clock Hohwii 46 with a Rieffler-pendulum. In order to obtain accurate data about the temperature gradient also for this place, two thermometers, which had been compared with each other and were graduated to tenth parts of a centigrade were suspended in the case of Hohwii 46 at a vertical distance of 65 ems. on February 27, 1903. It appeared that here the differences in temperature were in general greater than in the pier-niche, and in July 1903 it was derived from the observations that a difference of temperature between the thermometers of 1 centigrade corresponded with a variation of OS.40 in the daily rate of this clock. A yearly periodicity in the temperature gradient appeared also distinctly in Hohwii 46, as will be seen from the fo'lowing monthly means, which are given by the side of those of Hohwii 17. Hohwü 17. Hohwü 46. 1903 March +-0°.14 +025 „ April + 0.09 + 0.16 „ May + 0.31 + 0.42 » June Si + 0.37 » July + 0.29 =O » August + 0.18 + 0.22 „ September -+ 0.16 . + 0.26 » October + 0.06 = Oe » November + 0.92 + 0.10 „ December +. 0.01 -+ 0.03 1904 January — 0:02 + 0.08 „ February + 0.05 + 0.12 Moreover in the temperature gradient in the clockcase in the transit room a daily inequality was observed, which was hardly perceptible in the niche. In the case of Hohwii 46 the mean values of the temperature gradient from Ob to 12" mean time, are regularly greater than the mean values from 125 to 24» mean time. From the differences between these mean values in connection with the differences in the relative rates of the two clocks for the corresponding half days | could derive the influence of a variation in the temperature gradient on the rate of Hohwü 46 for a shorter period. The investigation is not yet finished, but in connection with the theoretical ( 245 ) A 4th inequality in the rates was due to the difference between the personal equations of the observers. From direct determinations we have obtained the following differences in the clock corrections : B—P 1 April 1901 22) OPO 18 March 1903 SWS 15 Mareh 1904 = (994 For the time being I have adopted for this difference during the period Jan. 14, 1903—Jan. 14, 1904 : — 05.250 and have reduced the clock corrections determined by P. to b.’s system. Besides the corrections for atmospheric pressure, temperature and difference in temperature I introduced a correction for the assumed personal equa- tion; these 4 unknown quantities are introduced into the equations and will be derived from the observations. I represented them by wv, y, 2 and wv and chose the following units so that these quantities should have about the same values: for atmospheric pressure (/) the unit ='/, mm. mercury of 0° C., for temperature (9) the unit = '/,, degree Celsius, for difference in temperature (V) the unit = ing degrees Celsius, for clock rates per day the, unit. ‘haa second of time; w, y and z represent the influence of each of the three units on the daily rate, expressed in thousandth parts of a second of time, while w is the 10 part of the correction for the assumed difference in the personal equations of the observers during the same unit of time, viz. B — P = — 250 — 10 u. § 6. The observed clock corrections were reduced beforehand to midnight of the data of observation. This was done with clock rates according to a preliminary formula with due regard to atmospheric pressure, temperature and temperature difference in the period from the instant for which the clock correction was determined to mid- night. This formula is derived from observations during the first half of 1903, it has been regularly tested by each new determination of the clock error, and has proved very satisfactory for the purpose get here: Constans = daily rate =O? (bar. — 760) + 0.032 (temp. — 10°C.) 20:31 (temp. difference). development of B. Wanacu of Potsdam on the influence of the temperature-gradient on clocks (A. N. Nrs. 3967—68) I think it worth mentioning that in this way | found that per 1 centigrade difference in temperature between the thermometers, there was a variation in rate of about O%.80, twice the amount derived from the before mentioned observations. ( 246 ) From readings of the barometer and the thermometers, daily means of atmospheric pressure, temperature and difference of temperature from midnight to midnight have been derived. Let the clock correc- tions at midnight be S, and the mean daily rates in the intervals between the successive determinations of the clock corrections be Q, in accordance with the letters previously used by me. From the said daily means, the mean values of the atmospheric pressure, of the temperature and of the difference in temperature were derived for the same intervals. In the interpolation these quantities correspond with (QQ, hence I call them Q”, Q5, Q'. For the barometer readings I used the deviations from 76 cms. The temperatures 9 are those of the upper thermometer. In this paper I shall also use the letters S”, S*, SV to indicate quantities that can be computed for each observation by taking the sum of the daily means of B, 9 and J’, to begin with a. certain date, say Jan. 14, 1903 till midnight of the day for which the clock correction is determined. SS’? denotes a value which is + 10 for each observation of PANNEKOEK, and zero for each observation of 3AKHUYZEN, and the series Q’” relates to the series SP in the same way as each series Q relates to the series S according to the formula S, — Sy Y ' 8 haf Shae . . M= Ris S, and S, are two successive quantities of the series, and n is the interval between them expressed in days. L=S =a Sues ys” — 2 od mea is —f must then be considered as a formula for a reduced and compen- sated clock correction. Each of the letters 4, S, f represents a series of discrete values, one for each observed clock correction, but by means of the inter- polation along the least sinuous line they can also be taken as con- tinuous variable quantities of which the derivatives of the first and second order also vary continuously, but of which the derivatives of the third order vary abruptly. To determine such a variable, say SB, for the instant t=q-+7' between the epochs g and r of the determinations of the clock correction, we use according to § 4 the formuls B B B B B ST = S, ii9) T + Cy Melle, where g is the epoch of the observation which immediately precedes ¢. The coefficients g? c? el can be derived from the series Q? if we use the formulae C in § 2 of the previous paper on this method of interpolation. Taking the quantities L, S, f in the sense as explained above ( 247 ) we can develop the total sinuosity of the reduced and compensated correction as follows: PL Eafe PSE FS ASV, PSE = dt = pine Se AL jn Ya [ dt. dt? dt? dt? de” de de In zele to Zn the greatest probability to the series we must choose w, y, z and w so, that the partial derivatives of /;, with regard to each of these are zero; i.e. that they satisfy the following relation : — dl! mer | F —ez nn 44, ——— —— — ; dt? dt? dt? dt? dt? a and 8 others, ae we obtain by substituting for the last factor Sb d°S3 d?SV | d°SP —_—, successively —— anc dt?’ dit di? dt? EN : SB d? SS : Definite integrals, such as FR as dt, which here occur as at” C a coefficients, can be computed in the following way: "PSB | aS [ASB dS) pasa se | EE ETR de eal Aer den | ae a at a Pp a, 8 > aa . —~ B a = 2 le? g ‘| — 62 5 en (5 — Sy)* =2 Kik gl 6 21 Bi Qu: If in the last term we also substitute for B B B ne, , (Cr — ey) , we have ade ‚B ) 1/ a i a we dt? dt = Le 3 9 | ae > Qs; (cy — c‚) a and after interchanging the indices B and 9 in the second member also 2 B =) [ey] +E Qu (ey — ee). I have computed these coefficients according to both formulae, and thus obtained a rigorous test. At the beginning and the end of the interpolation the quantities c are always zero, so that products such as ec? g* would be zero at either end. With a view, however, to the continuation of this com- putation for next year, I have not closed the interpolation-compu- ( 248 ) tation for the atmospheric pressure, the temperature, ete. on January 14, 1904. Hence on Jan. 14, 1904 c®, c%, c’, cP differ from zero and in the formula I had to retain the term for the end. | "d: SB dt (S—f In integrals such as = ep dt? dt? a to zero and computed the known terms of the four equations accord- dt I first put all errors equal ing to formulae of the form : rd? SB aS eeu B ie ge Teng > lee en AD. /2 dt? dt? K | ae (cy ( ) Q By solving these equations, I have obtained preliminary values for x, y, z and w, and in the way to be explained in § 7, also for the most probable errors /; from the mean daily rates computed with these preliminary values, using the formula B ea V Ap Tr — Sa Ore 08 — 9 2 2.0 a | have derived as a second approximation the corrections , y, 2 and w to these preliminary values. The influence of those corrections on the most probable errors was of little importance. | EN Integrals such as | ( =) dt have been computed twice, first de a CSB Ss apa PSN dt ea mihi according to the formula for */, a “(eS a, ( It? ) dt = Le 7] + = Q:, (ec, rn ey) a and secondly according to the formula, deduced in § 2 BN 4 r dl dt = — En(eg + eg er + Cr) dt” B) The following are the 4 equations expressed in numbers : 54601 —252y— 142+ 59u—-+ 21664 — 145 A Ay FA Sar = dl or ee 2 V4 ae A Sj deb M= PASS ai + 5de 8y— 22z410lu=— 101 21 | ( 249 ) The second members are written in 2 parts. The first part is derived from the observed rates Q, the second from the prelimi- narily reduced compensated rates (27. The solution of the two sets vielded : v= + 3.90 — 0.04 = + 3.86 y = — 2.30 — 0.34 = — 2.64 = + 2.72 + 0.46 = + 3.18 u= — 2.86 — 0.09 = — 2.95 || &Q According to these results the influence on the daily clock rates of the atmospheric pressure per 1 mm. mercury of 0° C. is + 080154, of the temperature per 1 centigrade (upper thermometer) — 0.0264, and of the difference in temperature per 1 centigrade (upper—lower) + 0,318, while for the personal difference BAKHUYZEN— PANNEKOEK in the clock corrections is found — 05.220. Table I contains the values required for the said computations ; for each interval between two consecutive determinations of the clock correction : the number of days 7, tbe observed clock rate Q, the values Q®, Q, QV and Q?, and also the values (c, — c‚)B, (Cots, (Co-—Gr) "5 (¢y—c,)”. To this must be added in order to render the computation possible the values for the last epoch Jan. 14, 1904, : g? = — 14.0 g? = + 57.7 g¥ =— 3.1 gE =4+24 (ea OL ey 2.01 oy SS EG ine) eet tes gy =— 146, adopted in case we use the series of the observed clock rates QQ. Ga 194 A a E u „ preliminarily reduced Q,. § 7. Here follows the reduction of the relations between the most probable errors and the determination of the latter according to these relations. Sub § 5 I have derived for each error f, the relation : Oly, 5 OL 7, Kn À — tet dû or ES = zag == 0 òf, ur òf, Ju,” I have thought myself justified in adopting the same value for the mean errors in the determinations of the clock correction, although for 1 of those corrections only 2 stars, for 18 corrections 4 stars, and for all the others 3 stars were observed, since the difference in accuracy resulting from the different numbers of stars is relatively ( 250 ) OD a Sy Be | Obeerscd Be fi Atmospheric Im, en | Temperature Personal a ee pressure (£). pee id difference (/). | equation (£). > a « De | = 5 B | Si 3 | V sn B De Je Ug om Q Q (¢,-—¢,) Q (ej—e,) Q (ej—e,) Q (ej—e,) fy | — 067) ale MO 2.99). 80.471’ 1.0.84 | g.9 06s Aes 8. | — 235) —31.0 | —14.05] 88.3 | 10.54 | 4410.6 | +0.18| 0.0 | —0.51 3 000| +23.2 | 444.99} 87.7 | 1.48 | 449.3 | —0 45 | 43.3 | +2.81 9.) | = 063) 440.7 | =. 0.90)" 80.7 | 1.73.) 480 [A95 3.35) IN , | — Aly] = 9.9) — 9.97) 81.8 | —0.82) A75) AD OO ATD 6 | 04 17.4) HAA 57) 89.5 | 1.36 | 414.81’ —1.00. | "0407008 5 | — .201] —23 1 | —42.98| 105.0 | 1.62 | 499.8 | 41.99 | +20 | 44.46 7 | — 439 — 2.7 | +. 6.05] 408.3 | +1.49 | +19.1 | +0.38 | —1.4 | —0.98 12 | — 192] — 2.7 | + 0.67] 196.6 | 41.63 | +95.3 | —0.24 | +0.8 | 41.22 3 | — 404 499.4 | — 5.19] 195.7 | 3.82 | HUT | 2.84 | —3.3 | 2.94 3 | — .o5e] +40 1 | 444.67] 138.3 | 44.92 | 192.3 | 44.44 | 43.3 | +3 29 B | — .490| 4-84-9490) 483,819. en | term ALO |) 1 00, | ete 5 | — 995] 45.0 | — 4.61] 474.0 | 47.85 | 153.2 | 46.98 | 0 0| 40.99 4 | — 439| 430.8 | 412.40) 157.3 | —5.25 | 430.2 | —5.55 | 0.0 | +0.30 10 | — 314) —14.8 | —40.00| 156.7 | +2.97 | 497.0 | 42.44 | 1.0 | —0.90 RE gop |g 00 144.2 | 9.91 | 416.2 | —2-47 | 49.5 | 44.95 4 | — 453) HAA + 4.89) 147.8 | —9.51 | 490.8] 4.94, 0.0 | 0.8 4 | — 458] +26.4 | + 0.40] 167.0 | +0.57 | +36.2 | 41.84] 0.0 | +0.09 4 | — 209) 420.3 | + 0.39) 180.8 | 44.27 | 440.5 | 42.88 0.0 | +0.42 gg Penn”) Amoah (rengersare | BEE |) ES / : EEE Eee He) al la? Men) | hao)" a ER 6 | —0.243| + 9.4! 4 0.49] 174.0 | 1.93 | +97.7 | —9.55 | 1.7 | —0.65 7 | — 34 — 4.9) — 2.98] 168.6 | —0.63 | 427.4 | +0.05| 0.0 | 40.36 8 | — 344 —5.4 | + 2.45 175.0 | +0.61 | +29.6 | 4+4.07| 0.0 | —0.09 41 | — 370 -12.0| — 3.95] 175 9 | +0.29 | +24.7 | —0.88 | 0.0 | +0.02 8 | — 338 + 4.8] + 7.33| 173.3 | —0.56 | +20.8 | 410.42] 0.0 | +0.02 6 | — .436] —30.0 | — 8.031 173.5 | +146 | 449.8 +0.80| 0.0 | 0.13 12 | — 365) — 2.4 | + 3.54] 167.4 | 4.73 | 414.6 | 4.81 | -£0.8 | 10.93 3 | — .988| 440.5 | — 2.83] 174.0 | —0.84 | +22.7] 40.02! 0.0 | —0.41 4 | — 271] 43.9 | + 8.68] 190.5 | +3.45 | 497.0] +240 | 0.0 | —0.16 8 | — .363| —12.2 | —13.53] 163.4 | +0.99 | 447.4 | —0.13| 0.0.| +0.6t & | — .196| +28.2 | 4 9.44) 143.8 | —2.04 | + 6.8 | 4.40 | —9.5 | 1.05 5 | — .909| +26.9 | — 1.18] 146.2 | —0.55 | 47.6 | —0.56| 0.0] 40.84 4 | — .937| +46.4 | + 2.75) 452.8 | —0.70 | 4+13.0 | —0.04| 0.0 | —0.49 7 | — 373 13.5 | — 4.14] 160.7 | +9.46 | 447.4 | 44.53 | 44.4 | 40.49 8 | — .429| —35.0 | — 5.03] 149.5 | —0.06 | +40.0 | —0.34 | 0.0 | +0.09 44 | — .333] 46.4 | + 6.48] 131.4] —0.72 | + 2.2 | —0.45 | —0.9 | —0.91 3 | — .447| 37.3 | — 2.68) 124.7 | 1.02 | + 0.3 | 40.94 | +33 | 44.87 2 | — .395| —35.4 | —11.48| 426.0 | 44.47| 0.0|—0.90| 0.0 | 4.31 6 | — 493| 449.6 | 442.45] 191.2 | —0.37 | + 2.2] 41.95} 0.0] —0.04 12 | — 147] +92.9| — 0.56) 441.9] +0.15 | + 0.8] —1.68] 0.0 | 40.53 3 | — 435 + 8.4 | + 4.55] 104.0 | —0.34 | 47.7 | +282 | —3.3 | —0.70 40 | — .243| —18.0 | — 0.58] 95.2 | 44.54] +0.2| —3.65| 0:0 | —0.05 3 | — .274| 40.5 | — 1.52] 77.0 | +0.02 | + 9.0 | +3.96 | +3.3 | 40.96 7 | — .965| —43.7| — 0.09] 66.7 | 331 | —4.4|—259| 0.0 | —0:85 3 | — .309} 4.5 | — 3.32 73.7] +0.29| — 2.7 | +0. | 0.0 | +0.99 4 | — .935| —20.4 | — 4.54 80.5 | 44.48 | —2.5|—0.86| 0.0| —0.07 5 |— 067 412.2 | +5.15] 78.2) 44.19] 4+0.8] 40.58 | 0.0] 40.03 7 | — 008 418.4 | — 0.81] 61.3 | +0.89 | + 2.6 | —0.26 | 0.0] —0.09 5 | + .060| +45.9| + 3.28] 39.6 | —2.43/4 3.2| 44.31] 0.0| 40.34 49 MON ALS = 5.04) 49.9 | 9/46 VO ITD (— OET cost a iy Proceedings Royal Acad. Amsterdam. Vol. VII, ( 252 ) small as compared with that depending on other causes which are difficult to account for. But the computation would not become much more difficult if we should assign different weights to the determinations. I now use the following thesis for the interpolation with regard to the smallest sinuosity that is proved in § 3 of my previous paper viz.: the partial derivative of the total sinuosity /s of a series of clock corrections with regard to one of the corrections Sj, is equal to twice the abrupt variation which occurs in the derivative of the third order of the interpolation curve near the abscissa g and the ordinate S,. Hence according to § 4 ds ÒS, I apply this relation to the interpolating curve after compen- sation determined by the corrections /, and I obtain oly, OL, in case that each part of that curve is represented by an equation : Ly = Ly + Gat + Cot + Ent. In accordance with the previous paper I have used capitals for the interpolation coefficients belonging to the corrections which are freed from errors. Also the meaning of = for this interpolation corre- sponds entirely with that of 0 used for the interpolation without compensation. = 12 (e,—em). = 12 (£,—E,) = 12 2, , ol "J A a . . For —~ we may substitute — ia , because Zij + fy is invari- OLig òf, able, hence 0L, = — 0f¢- OL or 3 After the substitution of pare —=— 12 =), each relation 4 pee un -- OF ; df, Hg i takes the form : = = Tae!” The first member of this relation depends on 2, y, 2, U and the errors f, and all these quantities occur in it in a linear form: B 5: V B Ei ero rn WA tE If we use the approximate values obtained for z, y, 2 and « in the supposition /—= 0, we can compute the expression : B 3 4 thy = Og — £6, —YO, —26, — UG, - I have made this computation by determining the differences between ( 253 ) the successive coefficients of ¢ for the reduced clock rates QE — y — zQ’—uQp, according to the new method of interpolation. ie : : : 6, may be developed in the following way: i OR ED en wk ET by bat Oy == allee . Reet J- Kad» - Kg Jg + Ky Jr + Ky Js + ane : RvO GY We have also en 15 ae where Jy = (2) dt. Hence: 0 t 2 ba Ps 05 eo Ly 0 I ols dos -4 annen | a a |S a q Of, 12: 0f,0f. of, \12 of Oy therefore the coefficient of f, in the expression for of is equal to the coefficient of f, in the expression for of. Let us consider the case that /, = 1 and all the other values of f=0 rp Meto =H or MG or Ke , 6; = Kor = K, , ete. The series of the quantities of for that ease gives us directly all the coefficients K which are required for the development of of . In this manner an interpolation was made between the numbers..., 0, 0, 100,0, 0... as many times as there were observations, each time moving the number 100 further over one interval between two successive determinations of the clock correction. If this computation is properly arranged, it can be made in a very short time and the accuracy can be tested by the results themselves as the coefficients are derived twice independently of each other. [ shall now describe in detail how I have arranged the compu- tation of those coefficients for a determination of the clock correction. According to $ 2 of my paper “On interpolation ete.” of Dec. 10, 1902 we have the relations ; _ Ja + Ir — 2 Vn Ip + Jg — 2 Qn Og = En — @ ey = TT — q n m n n° a m? which allow us, once the series of the g being found; to deduce the series of the o from the series of the g and Q. The series of the g is determined by the equations (C’) of § 2 viz. Nn Oa + m Gh n (Qn = 9) m (Qn — 9) RETE: m+n mda) | 2m)! 17% ( 254 ) which for the first and the last epochs a and z become: Qu — gp Q, — 4, 9a = Qn + 2 = A+ — where u and v refer to the intervals at both ends. In the example given here, also one of the latter relations occurs, i.e. that for the observations of January 20, 1903. The successive intervals with the values of Q belonging to them are here: Go o 3 8 3 6 LOS a = 4G ay 2,0) Oe a Tan 0e Dre PIO) The line separates what precedes this observation and what follows. As a first approximation for the quantities g I have adopted n Qm+m Qn only three of these values differ from zero, namely : min ’ l 0 Ip = EG Qn 5 gg On A 3 += a Qn In my example: Qn = 30; M= — 12; LS m= 8, ns o=8; hence g=-+ 17; 9,= + 21; 9 = — 3. With these three values begins the annexed scheme, in which the successive corrections of the g are computed. (see p. 255 sq.) Annexed to this paper is a table (II) (p. 256 sq.) containing the coefficients 100 A for this computation; under the heading ‘Sum of the squares....” have been given the sums of the squares of the coefficients for each date, and to the first and second approximations the values of 100 w which belong to the observations. Each equation of errors is reduced to the form: 20+ Ko fo + Ke fo Hg + 2) fo + Kg fp + Ky fs t+-+--=% 1 : where Ss pe is constant for all equations, and must be deter- pu mined in order to enable a solution. In the relation: Eg det Bih + KG ty + Ka tq + Kaf + Ky fe Hg which is independent of the hypothesis of the probability of a group of clock corrections, the quantities f represent real errors of observa- tion, of which the mean value is u. The error of interpolation gq in the interpolation with regard to the least sinuosity, i.e. the difference between the real clock correction L, and the interpolated correction for the same instant is ae From the expression of pro- q ( 255 ) 5 — Gat IZ + om - + + ff Jo sanjeA any Ca T+ SAUL -- 0 : ‘ 8 uoyeurxoidde ysty oUI O1 ei c + g — + = “S° saomoptroo-qery [LS en t+ Z — "+ “7 dogosuod puodes G GG GG Goo Qe Grae ae Cal ge se “yBnouo [[VUs ore sUOIDOL —109 Jsvy ayy [yun ‘suoyvorydyjuu ayy Ort VG GE dae TG. = ; yara Suruutsoq suoyetado osoyy qpaden gj =<. = ) 8) 8 + Glan OT + : sooejd omg Aq FUSE OY} 0} POAOW OT MOTEL * + 7 =e + e + nen 3 SIVAIOIUT “LL + yy 9 © Q e e : oo udis oduviyo pus MOleq nn £ ; 2 Ten od iens qo] ayy 0} Stoquinu ayy Aq 8 APTATP “Ol 0 0 ie id zen q = 6 RS Buorgoe noo JS S[VATIJUT DAISSAIONS OM AOL JO WINS 9YF OY EI"G ‘ : : . 7 5 ; : ST GG GG Gib ones epee RT Gb TOL DE ie a Mojeg Yo] ay} op SIoquinU X SLAM ZL CS Oni 66+ Cri 0 a Z oaoqe JYStt vy} 0} stoquinu X S[BAHJUI Q © aso i) 0 ie + ge = s (qy8t1 043 0} Aponbigo 1 Woay 5 younqns) *G ~~ a Se: Se eed Os 0 4 5 (IED re er en gD 8 8 6 6 ee (ajar oy} 03 Kppubijgo yowyqus) eg 7 : ‘ - ; ° HR) Me 0 ie 4- alae À r+ (uoyrunxosdde GEIT OT EE A RE he) Co IZ + LT 0 Coefficients for the compensation. 34 sle 8 = SEE | 23 os 100 | 100 | 400 | 100 | 100 | | dS: Ki | Kf | Ki | Ky’ | Ke | | he =e S - Be +) - | Flite |+ 1.0[2.4|4.2|0.2|0.A | 6.8|— 30|— 20 2.4/5.1) 3.5] 0.9/0.4] 0.0 | 43.1) 4+ 27 Ee 1.2/3.5 |2.914.3]0.8/04]0.0|2%2/+4 30/4 43 0A|0.9|1.3|2.5|2.4|0.6 | 0.2 [O4 | 14.6 | — 80|— 79 oalo4losl2a4les|1.0|o.4|o.9 |o.0 |45.7 | + 50 |+ 57 0.0 | 0.1 [0.6 | 4.0 | 0.7 | 0.6 | 0.4 | 0.0/0.0; 24/— 4 | Ln 0.0 |0.2|0.4|0.6 |4.9|1.9 | 0.5 | 02:04") 80 == S94 cae 0.4 0.2/0.4) 41.9] 2.3] 0.9] 0.4 | 0.3] 0.4 | 40.3} — 15|— B 0.0/0.0/0.5/0.9/]0.7/0.8}0.6/041]0.0| 26|+ |H 57 0.0 | 0.2 | 0.4 | 0.8 | 2.8 | 3.8] 4.8 | 0.3 | 0.4 | 26.9 | — 301 | — 320 0.1/0.3] 0.6) 3.8] 7.2] 5.1] 4.5] 0.3 | 0.4 | 95.7 | + 548 | + 580 0.1} 0.2/4.8|5.1] 5.6/2.7] 0.8| 0.2 | 0.0 | 68.3 | — 380 | — 405 0.0| 0.3/4.5} 2.7} 2.3] 4.9] 0.5 10.4 | 0.0 | 46.8 | + 191 | + 492 0.0/0.2 | 0.8 | 1.2 | 15/4.0/0.3]01/0.0| 5.5|/— a8|— 40 0.4 | 0.2 | 0.5 | 4.0 | 1.2 1o.6|0.2!0.010.0| 3.2/4 UH 414 0.0/0.4) 0.3/0.6] 0.7]}0.4)01/0.0]0.0] 1.2)4 14)4 14 0.0} 0.1) 0.2/0.4] 0.4] 0.3 | 0.4 10.0] 0.0| 0.5 | — 44|— 42 0.0|0.0)0.1]0.3|0.4/0.3]01/0.0}00] 04/4 9/+4+ 8 0.0}0.0/0.4]0.4)0.4|02/02/041/00] o4|— a|— 47 0.0 | 0.0 0.1/0.2/02/0.3)0.3)01}0.0] 03|H |H 44 0.0-1"0:0. |. 04} 0:39) 24 13-2 1 102 OA ATA SNE 0.0/0.4 |0.313.2|/6.6/4.5/41.2/0.3)0.4| 7.2/4 8|— 6 0.0}0.4/4.6|4.5| 44/24/08] 0.2] 0.0; 47.6] — 42|— 27 0.0 | 0.2 | 1.2] 2.4) 9.1/1.6) 0.6/0.1) 0.0 / 43.2) 4 70|-+ 69 0.1/0.3/0.8/1.6|2.3/4.5/0.4/0.2/04/41.9|— 4 |— 6 0.4/0.2/0.6]1.5/1.3/0.6/0.4/01/0.0} 48/— 51+ 7 0.0/0.4/0.4]0.6}4.3]4.7/0.9/0.2/0.0] 63|H 51|H 4 0.0/0.2] 0.4] 4.7] 3.3] 2.7/4.4] 0.3} 0.0] 92.4] — 82|— 80 0.41104] 0.9| 2.7/3.7] 2.7/4.0] 0.2} 0.0] 30.0) + M |H 47 0.0] 0.2} 4.1 | 2.7 | 3.512.2/0.6/0.4|0.0|26.0)+ 55| 4 48 0.0 | 0.3] 4.0 | 2.2 | 2.4 |4.0] 0.3 | 0.4 | 0.0 | 44.4 | — 101 | — 102 ( 257 ) Coefficients for the compensation. EN ae EE BE 100 400 | 400 | 400 | Ee 25 oe | | sis Ek | ZE Ky | KP | Kq | Ky’ | ae | EE Top MAS 4908-04) Jer ot ope A fe ee pe], — | July 40| 0.0 /0.2/0.6|4.0|0.9|0.5|0.2|0.0/00| 2.5|+ 63|H 67 » 47| 0.0 | 0.4 | 0.3 0.50.6 0.3 O02) 0071010: 08 EL NE » %/ 0.0 | 0.4 | 0.2 | 0.3 | 0.3) 0.2/0.4] 0.0/0.0] 0.3|— 28] — 30 Aug. 5| 0.0 0.0 | 0.1 0.2 | 0.3 |o0.4|o.2 |oo |0.0| 03 4 SN » 43/ 0.0 | 0.0 | 0.4 | 0.3 | 0.6 | 0.5 | 0.2 0.2/0.0] 0.8|— 9 | — 80 » 1910.0] 0.0 |0.2|0.5 [0.5 /0.4|0.4|0.0|0.0| 0.8| + 33| 4 80 » 31) 0.00.0 0.2 0.4 1.9 | 2.6 1.0/0.2|04/]44.6|— 64 | — 58 Sept. 31 0.0 | 0.2 | 0.3 26|42|2.2|0.6|0.3|0A|D6|H |J 96 » 710.0 0.0 1.0}2.2/4.7}0.9/05/04/00] 98|/— 6 0 » 45/00/0.2/0.6|0.9/ 4.6/1.6] 0.7/0.2] 0.0] 7.0/4 27 4. 95 » 49/0.1/ 0.3/0.5) 4.6/2.5 /1.8]0.7]04/0.0]43.0|— 2|— 2 » 4 0.1 0.4 0.7 |4.8| 2.5 |4.7 [0.4 | 0.4 | 0.0 | 43.4 | — 58 | — 56 » W|0.0|0.2|0.7|4.7|4.7|0.7|0.2|OA|OA| 69|H 63| 4 61 Oct. 5| 0.0 | 0.4 | O4 | 0.7 0.6 | 0.3 | 0.2 22. \ Oe 0 | ys) SS Oe ne », 43, 0.0/0.1 10.2] 0.4}0.4/0.5]0.4/0.4] 0.0! 0.7 | — 16|— 47 » 410.0101 10.910.519.51 4.5 2.5/0.4 | 0.0 | 32.8 | + 249 | 4 954 > aal 01 | 0.2 | 0.4 | 4.5 11.6 | 8.3 1.0 | 0.2 | 0.2 [295.7 | — 619 | — 629 » 99/ 0.0) 0.41 9.5} 8.3} 7.1 | 4.4 | 0.4| 0.2 | 0.0 [198.8 | + 448 4. 455 Nov anya 1.1/4.4] 0.6/0.5 | 0.3] 0.0/0.0} 3.8|— 60] — 64 » 16 0.0/0.2 0.4/0.5 | 15 | 4.5 | 0.5 |0.3|0.0l 5.2] — 8 | 4 40 » 19 0.1 02 | 0.3 £25 kas LoLo: 0.6)! 0:41 -0.0 1 64 PE OE » 9910.0] 0.0/0.5 | 0.9|2.0/2.0/0.8] 0.4/0.4] 9.9 |H 70| + 97 Dec. 2/ 0.0 | 0.3 | 0.6 | 2.0 | 2.5 | 4.6 | 0.9 | 0.2 | 0.0 | 14.4 | — 123 | — 142 » 90.0} 0.4 | 0.8 | 1.6 | 3.4 |3.4|1.2| 0.2 | 0.0 | 25.7 | + 146 | + 150 » 412/0.0 |0.4|1.0|3.4|49|2.8|0.8|0.2 | 0.0] 44.9 | — 54 | — 65 » 160.1 | 0.2 | 4.2/2.8/2.7) 4.40.4] 0.1/0.0}19.0}— 95} — 90 » | 0.0 [0.2 | 0.8] 4.4] 1.3 | 0.8 | 0.3 | 0.0 4.9 | +101 | + 401 » 928| 0.0 | 0.2 | 0.4 | 0.8 | 1.0 | 0.6 | 0.4 93 0d Jan. 2/ 0.0/0.4 | 0.3 [0.6 | 0.5'| 0.4 | 0.8 |H 64| 4 55 » 44} 0.0 | 0.0 | 0.4 | 0.4 | 0.0 | 0:6) | “gpa ide ( 258 ) bability e "L we may deduce the mean value of g,. Let us suppose that all the real clock corrections remain unaltered with the exception of that for the instant g. Then /; will be a minimum for Ly — $4; and for any other clock correction adopted for the same instant, which differs from this interpolated value by «, /7 is equal to its least value augmented by 6 Kj €”. Hence the probability of the group of clock corrections contains the factor e—’K,*, whence follows that the mean value of ¢, or also bi that of pg is equal to . Then the mean value of gy WK, is V/ 122K, constant and equal to this constant is called v. V122" Hence the 2°¢ power of the mean value of the first member of the above relation is: HG nt [kt Pie AR EEG rae ok The 2"4 member w, computed with approximative values of z, y, 2 and w is known. I had no direct data for the determination of «’. It consists of one part which is independent of the number of stars observed for the clock correction, and another part which is in inverse proportion to this number. For one star observed by Bakuvyzen, the latter part amounts to about 900 and for PANNEKOEK it is a little less. I wished to avoid a too large value for u° for fear of exag- gerating the regularity of the clock at the cost of the accuracy of the observations, and therefore I have put for each of those parts of u? 300, together u? = 600. According to the value given above for § we have v*: u’ =$ and tbe mean value of the expression : we VK, KREKELS HK, + KP HES + will be equal to pu. With different suppositions for §, I could derive from this relation the corresponding value gu’. My result was that for £ —*/,, the value of u? is 592 and therefore I have retained this value of & in the further computation. > The hypothetical expression of the probability was tested in two different ways. In the first place I investigated whether indeed y* might be regarded as equal for the different intervals of the period treated. Therefore I have arranged the observations according to their coefficients A, and have derived from the half with least A, separately the value of u? belonging to § = '/,,. ‘The result was u? — 591. ( 259 ) In the second place I have investigated whether also for intervals of longer duration, the constancy of »? remains the same. In that case I could derive »*? from the total sinuosity Zr of the real clock corrections. If we imagine that from the series of those clock corrections one is dropped, which deviates from the interpolated value by Po; then Zr is diminished by 6 A, g,?. 59 times we can drop succes- sively one observation from the 61 observations at our disposal, and thus each time diminish the total sinuosity of the remaining clock corrections by 6 Ag? At the end /, is zero and hence /, may be considered as consisting of 59 parts, each of a mean value of 6 v*. Hence the mean value of /y is 354 v?. From the reduced rates Q—z Q£ —y Q?—z QV—uQ? I deduced the total sinuosity of the clock corrections L+/ and obtained Erp 8136. In my previous paper on this interpolation it is demonstrated that if the errors f and the clock corrections expressed by I are independent of each other /74,=/7-+/y. Instead of the formula derived there: IL;==68,(f,—/,) I now write [>= S6f(en—en)—= =6f,0,f and substitute for ol its value expressed in terms of the errors f. In this way we get: eee Kof Erf Ke fei In the 2ed member occur under + many products of real errors. The mean value of these terms is zero. | omit them, substitute u? for the square of each error and find as mean value for Jy 6u’ + Ko. The computation of > K, yielded 1.39. In this way I have found the following relation between v and u: 8756 = 354 pv? + 8,34 u? whence, if u? is equal to 600, »? = 11, which result is in good harmony with the value first found for §. The set of equations by which the most probable errors are connected is readily solved, if we use as a first approximation fy = 0.48 ESS After I had found by the substitution of these Kg aw values that the 2nd member required still another correction A, w, I used Aw 0.80204 as a correction for the first approximation. Ki, +8 When computed according to the empirical formula I¢= = : Kg + 8 ( 260 ) the errors appeared to satisfy fairly well the set of equations. Where ia Asti it was necessary I took away the last differences by adding ——— K, +8 to the errors. Table III contains the results of the compensation which are necessary to compute the clock correction at any instant between the observations. Let this instant be q + 7, the epoch of the im- mediately preceding clock correction being q; if moreover during the interval from q to q+ 7: the mean atmospheric pressure, expressed in mms. of mercury of 0° C. is 760 + Br, the mean temperature in centigrades 10°C + #7, the mean difference in temperature in centigrades Vr, then we can compute the clock correction S,47 according to the following formula: ‘yy 1 En es ease, ST Sg hat roooGet 1 The values of S, f, G, C, E, occurring in this formula can be derived from table III. The 5™ column shows the mean rates for each interval between two successive determinations of the clock correction after the reduction and the compensation. From this we may judge of the constancy of the rate of the clock. It must be remarked that the small yearly inequality occurring in these values is very probably due to a little maccuracy in the coefficient of temperature obtained as described above. The last column of table HI shows the quantities >, the diffe- rences between the successive values of £. They give us a simple test for the computation of the compensation, because they must be equal to the errors / multiplied by 20, or f, = +, : 20. The adopted series of errors satisfies tolerably well this relation, if we admit small differences, which in thousandth parts of a second of time do 26,497 4318 Vr 4+C,THET*). not exceed the intervals m or # expressed in days, and hence give rise to a difference less than O*,001 in the mean daily rates. ( 261 ) PCBT: LL | Error (Gompen-| & Clock corrections) accord- ind Date | & | reduced toB |imstothe) gay | G c E = D | compen- : 1 ra = S sation | Fätes 2 | f QL 1903 is | Jan. 44] P | — 3. 28222 |—0.008| . 474 0.0 — 0.44 0178 — 0.44 ve A a 27.561 |+ .004 | 176 |— 4.3 +. 0.12 — .180 — 0.02 5 201 97.453 |+- „010 | 184 |— 4.5 + 0.44 15488 + 0.42 SdB 26 864 |— .014 184 |L 4.5 — 0.24 hi de — 0.09 = 3th B 26.744 |H .006 VT? 0.7 + 0.04 OLS — 0.05 Febr. 6 | P 96.921 |H .004 175 |— 0.2 + 0.07 =. AT + 0.02 , 1618 26.740 |— .004 MA | 0.5 — 0.08 — 170 — 0 06 Ì „49 | P 96.189 |— .001 170 0.0 + 0.09 Ait — 0.04 21960 oR 26.690 |H .005 AT 0.8 + 0.42 a + 0.08 Mrch. 6 | B 98.568 |— .029 173 |4+- 4.0 — 0.50 — ATA — 0.42 ERE 98.539 |J .038 10 98 + 0.80 DE E:77 + 0.38 a AT) B 98.757 |— .021 186 |+ 0.6 — 0.36 183 + 0.02 Andor Bs: 29.225 000 480 |4+ 0.9 — 0.02 Es Afk 0.00 ENE 99.475 |— .004 168 |4+ 0.9 — 0.09 = 166 — 0.09 ue | ae. 30.449 |+ .002 166 |— 0.5 + 0.05 AT O08 Apr. 3 | B 31.453 |J .007 480 |— 1.4 + 0.12 220186 + 0.08 a a 32.506 |— .004 189 |+ 0.2 — 0.07 = 486 + 0.01 » 20] B 33.565 |+ .004 183 [4 0.4 + 0.02 EN Vi: + 0.03 kP 35.575 |— .007 472 |H 4.4 — 0.42 — .169 — 0.09 May 7| B 38.051 [4 .007 175 |— 1.6 + 0.43 = 189 4- 0.04 „ 19] P 40.326 |+ .002 497 |— 0.3 — 0.01 EEE + 0 03 epee Ne 10.667 |— .004 198 0.0 — 0.10 — .199 — 0.07 » %| P 40.794 |— .002 200 |— 0.6 + 0.02 — .205 — 0.05 » 30] P 44.746 |+ O14 MA | 4.. + 0.24 — 03 +. 0.48 June 4| P 42.872 |— .010 240 |+ 1.4 — 0.24 — .205 — 0.02 2 Sa BR 43.427 |— .002 200 |H 1.2 — 0.04 | — .194 — 0.06 ek Ee 46.570 |+ .008 194 |— 0.5 + 0.47 — 194 + 0.44 » |P 41.654 |— .012 193 |+ 0.8 — 0,25 | — 192 - 0.14 45 IP 48.265 |H .009 194 |— 0.9 + 0.20 — .196 + 0.06 mess B, 48.896 |+ .006 198 |— + 0.09 — .496 | + 0.45 ( 262) | Z Clock corrections vee Compen-| | be ‘ling to the sated Date 5 EE to B compen- ae G C E | = S es Q, | 1903-04 = July 4| P| —3- 49.732 |—0.020 een 192 + 16 ie „ 10| B 51.220 |H .016 196 |— 2.3 REVAN: 3.488 |4 002 hig ik ap „ |B 56.086. | 20M | - ds aal 10 |T pe Ams 5 Bali, =e ABS) A is ie 10 |— 1.8 a ihe Mee eile: 3.415 |— .026 |) 908 27 | „ 49 | B 5.734 |4 04 | ese ane tae Pi 4e BEP 10.082 5 012 a A 15 1 2.3 i iN Sept. 3'| P 10.94 0m] MLA eae a 12.029 |+ 005] | MB OA | 451 P 14.936 |+ .008 12 |— 41.7 — 216 + 0.48 „ 49 | B 15.752 003) 17 |H 0.4 i „ |B 16.795 |— .010 |_ a 10 | 4.4 dor „ | B 47.744 |+ 013 | hee ee oe 1. Oct. 5] P 20.328 |— .009 | rd 204 |4+ 4.5 BB i „ 13 P 93.762 |H .001 Gi Ai 192 |4 0.4 me oa B 27.460 | 020] | A 1.9 oie „|P ET Je MNT en Ee „ 20 | P 29.561 | .021 eee ce ae : Nov. 4| P 30.720 |— .004 | es eas bea AGP Ae cae READ bo fess he See „ 19| B 32.918 |4 002 | te 190 4+ 0.3] uae „ 20 | B 35.348 |4 OM | ie: ig pele ee Dee Bs) Pe baad = abe foes | aR a a os ae Sa OP 37.998 |4+ .096 | eed Mies Sele Gia , 12|P 38 926 |— .005 | re 201 1 0.4 za „16 | P 39.866 |— 04 | id 190 14 2.8 |T isis RR: 40.201 |4+ .o19 | Ender bx peas ieee 40.254 |— .047 2 ts 179 4 4.3 za fan kol P 39.953 |H 045 ae oe Se „ 141 B meo Soos met Saat Mo a (October 20, 1904). 4 KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN TE AMSTERDAM. PROCEEDINGS OF THE MEETING of Saturday October 29, 1904. SIC (Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige Afdeeling van Zaterdag 29 October 1904, DI. XIII. SO Ne PEN TE St JAN DE Vrins: “The congruence of the conics situated on the cubic surfaces of a pencil’, p. 264. A. F. Hotreman: “The nitration of disubstituted benzenes”, p. 266. A. P. N. FRANCHIMONT and H. Frrepmann: “On zz’-tetramethylpiperidine’, p. 27C. P. J. Monracye: “On intramolecular atomic rearrangements in benzpinacones”. (Communicated by Prof. A. P. N. FRANCHIMONT), p. 271. A. P. N. Francuimonr presents the dissertation of Dr. J. Morr van Cuarante: “Sulpho- isobutyric acid and some of its derivatives’, p. 275. W. A. Verstuys: “The relation between the radius of curvature of a twisted curve in a point P of the curve and the radius of curvature in P of the section of its developable with its osculating plane in point P”. (Communicated by Prof. P. H. Scmovurp), p. 277. L. J. J. Muskens: “Degenerations in the central pervous system after removal of the floc- culus cerebelli”. (Communicated by Prof. C. Winker), p. 282. H. KAMB&LINGH ONNES and C. ZAKRZEWSKT: “The validity of the law of corresponding states for mixtures of methyl chloride and carbon dioxide” (continued), p. 285. B. Meminx: “On the measurement of very low temperatures. VII. Comparison of the platinum thermometer with the hydrogen thermometer”, p. 290. — VIII. “Comparison of the resistance of gold wire with that of platinum wire”. (Communicated by Prof. H. KAMERLINGH ONNES), p. 300. (With cne plate). Jan DE Vries: “A congruence of order two and class two formed by conies’, p. 311. W. EixrmoverN: “On a new method of damping oscillatory deflections of a galvanometer”, p. 315, (With one plate). W.H. Junius: “Dispersion bands in the spectra of © Orionis and Nova Persei”, p. 323. J. Orie Jr: “The transformation of the phenylpotassium sulphate into p-phenolsulphonate of potassium”. (Communicated by Prof. C. A. Losry pr Bruyn), p. 328. J. F. Suyver: “The intramolecular transformation in the stereoisomeric z- and 3-trithioacet and z- and -trithiobenzaldehydes” (N°. 11 and 12 on intramolecular rearrangements). (Communicated by Prof. C. A. Losry pe Bruyn), p. 329. J. W. Diro: “The viscosity of the system hydrazine and water”. (Communicated by Prof. C. A. Losry pe Bruyn), p. 329. J. M. van BEMMELEN: “On the composition of the silicates in the soil which have been formed from the disintegration of the minerals in the rocks”, p. 329. Erratum, p. 329. The following papers were read: Proceedings Royal Acad. Amsterdam. Vol. VII. ( 264 ) Mathematics. — “The congruence of the conics situated on the cubic surfaces of a pencil.” By Prof. JAN DE Vrins. (Communicated in the meeting of September 24, 1904). 1. On each cubic surface S* of a pencil lie 27 systems of conies, of which each system has one of the 27 right lines for common chord. Through any point P of space passes one .S* of the pencil; so P bears 27 conics C? of the congruence formed by the C? of all 35": The pairs of lines of the congruence evidently form the skew sur- face of the trisecants of the basecurve 7’; for the points of inter- section of a right line of one of the S* with a second S* belong to all S* of the pencil. Quite independent of this consideration we can find the degree of the above mentioned skew surface in the following way. Let vj be a homogeneous function of degree / in w,y, 2. If we take a point of Zi’ to be the origin of the system of coordinates, the pencil has for equation u, Ju, Hu, =O, where the coefficients contain a parameter 4 to the first degree. If a right line through QO must lie on 5%, for all values of m the equation mu, + mu, + mu, = 0 must be satisfied, so we have simultaneously us (| Resor A ee U. =): By eliminating 2, y, z out of these three equations a relation is found containing the coefficients, so 2 too, in degree Ne DO Wa Se The basecurve A’ is thus an elevenfold curve on the skew surface of the right lines lying on the surfaces S* of the pencil. Each surface S* bearing 27 right lines, its section with the indicated skew surface is of degree 27 + 9X 11 = 126; the skew surface is thus of degree 42 *). 2. Any right line / is a chord of 42 conics, whose planes are determined by the 42 right lines of the skew surface g** resting on /. Let us now consider the surface (P) formed by the C*, the planes 1) See a.o. CreBscu, Legons sur la géometrie, Il, 13, 2) See a.o. Kivyver, Kenmerkende getallen der algebraïsche ruimtekromme (Characteristic numbers of the algebraic twisted curve.) Versl. K. A. v. W., 3rd series vol. VII, page 152, 1889. ( 265 ) of which pass through the point P. An arbitrary ray through P is intersected outside P in 84 points. As P bears 27 C* the degree of (P) is 84+ 27 = 111. According to the notation of Scuupert we have thus pe? == 42 and up ae BEI: From the well known relations *) dy — 24 + d+ 4u and 30 = 4 + 2d + 2u, where in our case 4 is equal to 0 we deduce by symbolic multi- plication the following system of relations : dSuv=—dut+4u, SLO 20-27; op —=drvt4ur, avo =2dr+2ur, dsvo—de+ 4uo, 30 =20d0+2uUQ. We have here six equations for nine characteristic numbers of which we have already determined two. But the number dr we ean find directly. For on the arbitrary right line / rest 42 right lines of g'*; each of these right lines is intersected bij 10 right lines of the surface S* to which it belongs ; so it furnishes 10 pairs of lines resting on /. Consequently dv — 420. We now find successively p= 288% Ou = 165, vo = 304, Botha, dok, on = 402°. 3. Out of »v? = 288 follows that the surface A formed by the conics cutting the right line l is of degree 288. Evidently / is a 27 fold right line of 4 and a chord of 42 conics lying on A. It is evident that on A lie 462 right lines, which are situated three by three in 210 planes. If lis a trisecant of R’, thus a right line of a surface &,°, then A**“ breaks up into the surface S,* counted double and the loci of the conics passing through each of the three points of intersection of the trisecant. The conics having a point of the basecurve R* in common, form thus a surface of degree 94. The surfaces 7'** belonging to the points of intersection 7’, and 7’, of the trisecant have evidently the 10 conics in common which are determined by the 10 triseecants through the third point of inter- section 7, 1) Scuusert, Kalkiil der abzdhlenden Geometrie, p. 92. {8* ( 266 ) If / is a chord of &*® then A degenerates into two surfaces 7'°* and a surface of degree 100. If / is a line cutting A’ once then A consists of two parts, which will be successively of degree 94 and 194. 4. The numbers do = 510 and du—165 furnish well known results. *) The first tells us that the skew surface o** of the trisecants of f° possesses a double curve of degree 255 ; for each plane through a double point of a pair of lines is to be regarded two times as tangent plane. The second number furnishes the property that the threefold tangent planes of the surfaces S* of a pencil envelop a surface of class 55. The surface (P) contains thus 165 right lines lying three by three in 55 planes. The numbers ve and we furnish with reference to the plane at infinity the following properties : The parabolae of the congruence form a surface of degree 354, their planes envelop a surface of class 138. Each S? contains 108 parabolae. *). As a definite S* can cut the parabolae on the other S* only in points of the basecurve F° the locus of the parabolae passes (3 x 354 — 2 108) : 9 = 94 times through A”. ; So through each point of A” pass 94 parabolae. Chemistry. — “The nitration of disubstituted benzenes.’ By Prof. A. F. HorLLEMAN. (Communicated in the meeting of September 24, 1904). If we introduce into a benzene derivative C,H,X a second atom or group this takes up in respect to X a position either chiefly meta or para-ortho depending chiefly on the nature of X. The cause of this is as yet obscure. The efforts for elucidating this phenomenon are totally inadequate, first of all because they are too vague, secondly because they do not take into account the relative quantities which are formed from the isomers; in fact they could not do so, as these were still unknown at the time that these ‘explanations’ were given. A better insight into this problem can only be rendered possible by the quantitative study of the substitution process, which has 1) Kruyver, page 152. 2) J. pe Vries, La configuration formée par les vingt-sept droites d'une surface cubique, Arch. Néerl., sér. 2, t. VL, p. 148. ( 267 ) already been made in a number of cases of nitration of the sub- stances C,H.X. For the present we must content ourselves with accepting the results of those quantitative studies as facts. Doing this, we may put the following question: Given a benzene derivative C,H,XY, in which a third group Z is introduced. If now we know the relative quantities of the isomers of both C,H,XZ and C,H,YZ which are formed by the introduction of Z into C,H,X and into C,H,Y; can we then deduce from this the structure and relative quantities of the isomers C,H,XYZ which are formed by the introduction of Ze into; Cf, XY P Suppose (by way of an example) that we have determined how much para- and ortho-compound is formed in the nitration of chloro- benzene and how much meta- and ortho-benzoic acid in the nitration of benzoic acid; can we then determine beforehand which and how much of the possible nitrochlorobenzoic acids will be formed in the nitration of chlorobenzoic acid? Qualitatively this problem has been studied rather fully, but as a rule not very systematically.. In a great many cases it has been determined which of the possible isomers C,H,XYZ are formed by the introduction of Z into C,H,XY and one has tried to draw con- clusions therefrom which render it possible to predict what may be expected in unknown cases. Brinstrm has summarised these as follows: “In the introduction of a group Z into a substance C,H,XY both X and Y exert an influence but that of one of these groups is predominant and directs Z.” Undoubtedly, this rule is correct in a great many cases, but not in a good many others. For instance it cannot be applied to the nitration of m-nitroanisol, which I have investigated. In any case if shows that the groups X and Y do not exert their directing influence independently of each other but that this is modified by their simultaneous presence. This has been fully confirmed by a quanti- tative investigation in the case of a number of nitrations of the compounds C,H,XY. If the groups X and Y exerted a directing influence on a third substituent independently of each other we ought to have the following: If we call the proportion in which the three isomers are formed when Z is introduced into C,lH,X Portho * (meta + Ypara and that of the three isomers when Z is introduced into C,H,Y ' PE mal Portho + {meta * © para ( 268 ) the quantity of the isomers, when introducing Z into C,H,XY, would be expressed by products as pq’ etc. as shown by the subjoined scheme: X p/ xX A Pr aa a OE 2 ' | | pal q vl r Mea! qr de q rd In this it has been supposed that on introducing a second group into a monosubstituted benzene derivative all the three possible isomers are formed, which in practice will most likely be the case even if the quantity of one of these should be so small as to be generally overlooked. In fact, in a number of cases where at first only two substituents had been found, such as in the nitration of nitrobenzene, a careful investigation also revealed the presence of a third one. The quantitative investigation as to the relative quantities of the isomers which are formed in the nitration of substances C,H,XY now showed that those quantities generally differ very considerably from the products pq’ ete. so that a serious diversion of the directing influence on the third substituent must be admitted. This diversion was found to depend not only on the nature of the substituents but also on their place in the molecule as proved by the following example. In the nitration of chlorobenzene at O° para-, ortho- and meta- nitrochlorobenzene are formed in the following quantities: COH Cl In that of benzoic acid ee HME | | | | | the nitrobenzoic acids in | | ‚0.3 3 : 5 80.2 ee, the following proportion : HE 1.3 69.9 If CO,H and Cl did not modify each other’s directing influence, nitro- derivatives obtained in the nitration of ortho- and metachlorobenzoie acid would be formed in the quantities indicated in the subjoined schema : CO‚H COH va Cl nal 69.9 < 18.5 AN 29.8 X 18.5 69.9 {80.2 29,8 < 80.2 Cl Na eA of the other possible isomers only very insignificant quantities. These were in fact so trifling that they were not found. Of the two isomers to be expected in both cases, the relative quantity ought to be just as large as that which is formed in the nitration of chlorobenzene itself. Instead of this was found: ( 269 ) COT COSI (Ne 91.3 (Ne en Pan Ned In the subjoined table a number of such observations have been collected. By “diversion” is meant the quotient of the quantity of the byeproduct actually found and that of the quantity calculated. Amount of byeproduct OE on 100 parts of main prod. at —30° at 0° C,H,Cl 36.4 42.0 C,H,Br 527 60.5 OCOAH 16.9 23.1 Diversion of the directing influence of the halogen | of carboxyl. | at —30° at 0° | at —30°| at 0° | o-C,H,Cl.CO,H 16.3 19.4 048 | 0.455 in Wan o-©,H4Br.CO,H | 20.6 UA 0.304 | 0.403 on HO MCO Of 9.5 0.250 | 0.2 | 0.539 | 0.444 m-C,H,Br.CO,H 13.4(2)| 12.9 GORA) O88, |) > eh oredr oC HON 5.5 7.8 0.451 Fis: > m-C,H,Cl, 9.7 Bel aie Or 0.098 = = In the nitration of o-halogenbenzoie acid and of o-dichlorobenzene the NO,-group in the byeproduets places itself adjacent to the halogen; in that of the m-dichlorobenzene between the carboxyl and the halogen. On comparing the diversion figures of the directing influence of the halogen in these acids and dihalogen-compounds, those of the meta-compounds amount to about half of the ortho-compounds. This is one of the many cases which show that the introduction of a substituent between two others meets with a particularly great resistance. | Groningen, Lab. Univers. Aug. 1904. Chemistry. — “On aa’-tetramethylpiperidine.” By Prof. A. PLN. FRANCHIMONT and Dr. H. FriepMann. j (Communicated in the meeting of September 24, 1904). This substance, which was obtained in 1885 by Canzoneri and Spica but in an impure condition, was prepared by us in another manner, namely, by reduction of y-bromotetramethylpiperidine with a copper-zine couple (Gladstone-Tribe’s method) in absolute alcoholic solution. It is a liquid boiling at 155,°5—156°5 at 760 m.m. pressure having a sp. gr. of 0.8367. With water it yields a erystalline compound which melts at 28° and loses its water totally or partially in a dry atmosphere. The compounds with hydrogen chloride, hydrogen bro- mide and sulphurie acid form very beautiful crystals; those with the two first-named acids sublime on heating without previous fusion, those with sulphuric acid melt: the acid one at 174°, the neutral one at 270°. Compared with piperidine, this amine reacts remarkably slowly on acid chlorides such as benzoyl chloride, chloro-formic esters, pierylehloride ete. In aqueous solutions the reaction takes place hardly at all, in ethereal solutions extremely slowly. However, there were obtained : methylurethane as a liquid with a strong mint-like odour boiling at 227°? at a pressure of 760 m.m.; sp. gr. 0.9848 and the benzoyl derivative as crystals melting at 41°— 42; the pieryl derivative melts at 225°. An effort to prepare a urea from the hydrogen chloride compound and potassium isocyanate has resulted as yet in failure. This reminds us of experiments of Dr. K.H. VAN DER ZANDE in 1889 with di-isopropylamine, where urea could only be obtained with difficulty and in very small amount, whereas dinormalpropylamine presented no difficulties *). If we compare the formulae of di-isopro- pylamine and a a@’-tetramethylpiperidine we notice that they only CH, \ ae HO see Pe (CH), © € (CH), (CH,),C CCH), Ed N N H H 1) Some years before, I had already noticed an analogous phenomenon when treating propyl- and isopropylmalonic acid with nitric acid; the first compound is much more readily attacked than the second. ( 274 ) differ in this way that the two hydrogen atoms of the first compound (indicated by asterisks) have been replaced in the second one by the bivalent group CH,—CH,—CH, ; piperidine and tetramethylpiperidine differ because the first one contains hydrogen atoms where the other possesses methyl groups, namely at the @ C atoms in regard to the nitrogen. As piperidine reacts strongly with the above substances and tetra- methylpiperidine does not do so and as there exists an analogous difference between dinormalpropylamine and di-isopropylamine it is natural to look for the cause of this in the methyl groups. As, however, their nature does not explain this difference we are bound to consider their mass and their position in space in regard to the nitrogen. This is then a case of so-called sterical obstacle which is to a certain extent comparable with a number of other cases whieh have been chiefly observed in the aromatic compounds; a case which may, perhaps, affect the views held as to the nitrogen atom. It must be finally observed that tetramethylpiperidine yields like di-isopropylamine a crystalline compound with nitrous acid, which is fairly stable and is only decomposed at a higher temperature into water and the nitroso-compound. Chemistry. — “Qn intramolecular atomic rearrangements in benz pinacones.” By P. J. Montacne. (Communicated by Prof. A. P. N. FRANCHIMONT). (Communicated in the meeting of September 24, 1904). The following research originated in an effort by Ner') to explain the intramolecular atomic rearrangement in the conversion of benz- pinacone into benzpinacoline by assuming the presence of an inter- mediate product. His explanation when put into formulae is as follows: 3enzpinacone is dissociated into water and unsaturated hydrocarbon: OH C.H. OH CH, PR rd Z EN de | Nc | Ne a ee | Oe C,H, . H i OH CH, ve GE: | NC, then addition to: C,H 5 1) Ann. 318 p. 38. OH edly then dissociation: | C,H,—C C—C,H, | NOCH, we | ó C,H, then addition to: C,H;—C C—C,H, | No, O ra then formation of C,H,—C——C(C,H,),. If, however, we represent the matter by structural formulae we get: OH | | C.H, OHT OH, 5 : ‘ai nl On Dn ee On | | | | Gree “\ | bet ae i CH: fn C.H, OET He en ee 4 eke 4 Cat ae | bes By An OH: 0 aw = | ea . 675 5 00362 23 pe ee ae Ces This shows now in the plainest manner that the core I is attached to different C-atoms before and after the migration. If we, therefore apply this representation to derivatives of benzpinacone, ihe group or the atom in the core L of benzpinacone then occupies a different .position from that in the benzpinacoline obtained from the same. 1) Which H-atom of core I migrates to the group C,H, is not stated by Ner; the ortho-placed H-atom was taken arbitrarely. Some time ago') I pointed out the possibility of the existence of a similar intermediate product in the transformation of hydrobenzoin into diphenylacetaldehyde OH H OH H Pal ae H—G oe a ee ea | Nor net A OH A nord re but at the same time I have shown experimentaily the incorrectness of that supposition. In an appended note I have already observed that in view of my results obtained, the theory of Nur was not acceptable. It appeared to me therefore, to be of importance to extend my researches also to a derivative of benzpinacone and thus to form a definite opinion as to the correctness or incorrectness of Nur’s theo- retical explanation. For this research I took 44’4"4'" tetrachlorobenzpinacone obtained by reduction of 44’ dichlorobenzophenone. On being heated with acety | chloride it passes into tetrachlorobenzpinacoline. If this is boiled with aleohotie potassium hydroxide it is resolved into trichlorotri- phenylmethane and p. chlorobenzoic acid according to this scheme: (4). Cle Hy) sCOr COM (CHE Clie == (4). Cl. CHN as ECO ZOEN (fis 2 CI GHZ O (4). Cl. CH), / 5S OH + C— OH, . Cl (4). ECE. CH, OH This trichlorotriphenylmethane now appeared to be identical with the 44'4" trichlorotriphenylmethane obtained from 444" triaminotriphenyl- methane (p. leucaniline). This explains the para-position of all chlorine atoms in the first-named trichlorotriphenylmethane. This further shows that the phenyl group is attached to the same C-atom before and after the migration and that therefore the intermediate product as suggested by Ner is an impossibility. The explanation given by Nur for this intramolecular atomic rearrangement is, therefore, incorrect. The views held as to the transformation of e-glveols into aldehydes are two in number: 1) Rec. 21. p. 30. 1. Splitting off of H,O, in such a manner that the group OH departs with the H of a C-atom, for example: CE zon fou CH, OH ll, > c—C—cn, : Se NCH, Ris ove. Ner’s view of benzpinacone is in accordance with this. The object of this representation is to abandon the idea of an intramolecular atomic rearrangement and to substitute so-called nor- mally-proceeding reactions. 2. Splitting off of H,O in such a manner that the group OH departs with an H of the second OH group, for example: CH, CH, CH CH \CcOH—COHS ae ON 6) ie Sareea CHA CH, Cn, 7e OE Ner'): “Es ist jetzt vollkommen klar, dass diese Reaction (Umwand- lung der 1.2 Glveolen in Ketonen) bei weleher eine scheinbare Verschiebung der Hydroxyle eintritt, auf eine intermediaire Bildung von Alkylenoxyd zurückzuführen ist.” H H C,H,—C—OH C,H,—C vit CH 07 CH OOH Nu In this representation we still admit an intramolecular atomic rearrangement; not, however, with the 1.2 glycols but with the oxides. In the transformation of hydrobenzoin into diphenylacetaldehyde, and now again in that of benzpinacone into benzpinacoline, | have shown that the first theory is untenable. In view of this I consider the existence of a trimethylene-ring also in the transformation of pinacone into pinacoline too as being less probable. Of course a direct proof, as in the case of the aromatic «-glveols, cannot be produced, but, provisionally, this theory seems to me to lack all foundation. It looks to me as if Ner himself is abandoning this theory, because, whilst formerly *) he considered the trimethylene-ring as very pro- 1) Couturier. Ann. chim. phys. [6] 26 p. 434, Ertenmeyer. Ber. 14 p. 322, Note. Zeuinsky and Zeukow. Ber. 34 p. 3251. 2) ERLENMEYER. Ann. 316 p. 84. 3) Ann. 335 p. 243. 4) Ann. 318 p. 38, bable, he now, to judge from the above quotation, definitely adopts the oxide-ring *). The results of my researches, | may draw up in the following rule: In the transformation of the 1.2 glveols into aldehydes, a real intramolecular atomic rearrangement takes place, which cannot be explained by any normally-proceeding intermediate reaction; it has not, however, been decided as yet whether this atomic migration takes place with the 1.2 glycols themselves or whether the oxides are formed first and then undergo an intramolecular rearrangement. IT am now making experiments in that direction with Dr. MeerBerG. Chemistry. — Prof. A. P. N. FRANCHIMONT presents to the Library of the Academy a dissertation from Dr. J. MOLL van CHARANTE, entitled: ““Sulpho-tsobutyric acid and some of its derivatives” and offers the following explanation. (Communicated in the meeting of September 24, 1904). Dr. Morr van CHARANTE has commenced at my instigation to tho- roughly investigate sulpho-isobutyric acid. He prepared it according to the process which I had published many years ago for the preparation of those aliphatic sulphocarboxylic acids in which the sulphonic acid group is attached to the same carbon atom as the carboxyl group (namely from the acid anhydrides with sulphuric acid). These acids are not only important from the fact that they are bibasic acids, of which our knowledge leaves generally much to be desired, but also because the two acid functions are of themselves, and not merely on account of their position, of different strength, and are situated together more closely than in the case of the aromatic acids, and can therefore, exert a greater influence on each other. The difficulties experienced in the case of sulphoacetic acid, sulphopropionic acid ete. caused by the mobility of the hydrogen atoms which are placed at the same carbon atom could not present themselves here, because the atom to which the two acid functions are linked, does not carry hydrogen. The said method of preparing, which had never been fully eluci- dated, in which two mols. of acid anhydride react with one mol. of sulphuric acid to yield one mol. of sulphonic acid is thus explained by Dr. Morr van CHARANTE: a diacylsulphurie acid is formed which 1) At least if the quotation is meant for all the 1,2 glycols. ( 276 ) on being warmed is converted into monacylsulphonic acid, which in contact with water yields sulphonic acid and carboxylic acid: Ca Hani CO.0.502.0.CO.Cy Hani passing into Chn Har CO.O. SOs.Cn Han CO.OH and then by HO into C‚ Ho CO.OH and HO.SOs. C‚ Ho,.CO.OH. Specially undertaken experiments led him to this conclusion and also taught him that when the acid chloride was used instead of the acid anhydride also two mols. of the latter are required to one mol. of sulphurie acid. The action of chlorosulphonic acid on carboxylic acids, which is also given as a method of preparing sulphonic acids, is understood by him to first yield the acid chloride and sulphuric acid, which then react on each other with formation of the sulphonic acid. Sulpho-isobutyrie acid itself is a very hygroscopic substance con- taining two mols. of water of crystallisation. The barium salt contains three mols. of water, the sodium salt half a mol. The neutral silver salt is anhydrous like the acid salt, which latter can only be obtained in the presence of a large excess of the acid. When acting on the sodium salt with phosphorus pentachloride Dr. Morr vaN CHARANTE obtained, according to circumstances, either the dichloride or a chloro-anhydride, which is the chloride of the carboxylic- and the anhydride of the sulphonic acid function. The dichloride is a colourless liquid, which distils at about 55° under a pressure of 9 20 1—'/)m.m. mercury, with a sp. gr. d Pan 1.4696 and a refractive = power np = 1.4887; it solidifies at — 10°. The sulpho-anhydride- carboxy-chloride is solid, erystallises from ligroin and melts at 61°. With a little water the dichloride yields sulpho-chloride-isobutyric acid, which is crystalline and melts at 134°. With more water, sul- pho-isobutyrie acid is formed. With methyl alcohoi the ester of the carboxylic function is generated whilst the sulpho-chloride function remains. This ester sulpho-chloride is a liquid, which passed over at a pressure of 1'/, m.m. at about 60° and solidified at 21°.5; the 20 sp. gr. was d =1.3436, the refractive power np = 1.46658. Treatment with sodium methoxide dissolved in methyl alcohol yielded not the dimethyl ester but the ester sodium salt of the sulphonic acid. The dimethyl ester prepared from the neutral silver salt with methyl iodide was a liquid which passed over at a pressure of 1—*/, m.m. between 82°—78", solidified on cooling and then melted at 4°; the 20 : sp. gr. was oa = 1.2584, the index of refraction np = 1.44481. (277 ) The neutral ester is saponified by methyl alcohol and then yields an acid one like all sulphonic esters. With ammonia it yields an ammonium salt of the sulphonic ester function, which is also an ester of the carboxylic acid. | The acid ester, namely the carboxylic ester of the sulphonie acid, was also obtained from the sodium salt of sulpho-isobutyric acid by means of hydrogen chloride and methyl alcohol and is hygroscopic. Its isomer, the carboxylic acid of the sulphonic ester, which was prepared from the acid silver salt with methyl iodide, is not hygroscopic, it crystallises from benzene and melts at 90°. Dr. Morr, vaN CHaran'rr’s experiences with the esters of sulpho-isobutyrie acid agree fairly well with those of Werescuriper with metasulphobenzoic acid. The melting points of the compounds obtained behave as might be expected ; those of the sulphonic acid chlorides are more elevated than those of the sulphonic esters; those of the carboxylic chlorides are lower than those of the carboxylic esters. The melting points of the esters as well as those of the chlorides of the carboxylic acids are lower than those of the carboxylic acids themselves. Mathematics. — “The relation between the radius of curvature of a twisted curve in a point P of the curve and the radius of curvature in P of the section of its developable with its osculating plane in point P”’ By W. A. Vurstvys. (Communicated by Prof. P. H. Scnovuts). (Communicated in the meeting of September 24, 1904.) § 1. Turorem. Kor each twisted cubic C* the ratio is constant of the radius of curvature in any point P to the radius of curvature of the section of the osculating plane in the point P with the developable O, belonging to C°. Proor. If we take P to be origin of coordinates and the tangent, principal normal and binormal of the curve C* in the point P to be the axes of coordinates, then C* is the cuspidal curve of the surface O, enveloped by the plane At—3BH+4+3Ct—D=0, where D= C2264, B=b,«#+b,y+ ), 2, A=a,e+a,yt+a,z+a,. ( 278 ) The coordinates of the points of the curve C° satisfy the conditions : AG RB 23 AS sj whence a, b, Cy hk a, b, €, e a, b, ¢,—a, ¢, t+(a, b,—a, b,)?—a, b, C, es N ; ; CNN a, t (e‚—b, t—b, c, t°) y= ——, go IN N Now the radius of curvature F, of the twisted curve C? in the point P is the same as the radius of curvature of its orthogonal projection on its osculating plane in P, the curve with its projection in P having three consecutive points in common. The parameter expressions for the coordinates of this projection are d, be a PR) t (c, —b, t—b, c, t°) N N ng OY From the value of y we find me for t=O; so for the general ad == formula a Sk + dy)" 3 de dy — dy @u giving the radius of curvature of a plane curve, can be substituted the simpler expression : da? = 5 i 2° d a 2a, 6, 2 b, The equation of the surface O, enveloped by the plane A®—3Be+3Ct—D=0 AAD—6ABCDL4AC LAB D—-838BC’=—0. The curve of intersection with the osculating plane D = z= Ois: C?(4AC—3 B)=0. So the equation of the conic d, lying in the osculating plane is: A(a,xtayta)cy —3(b,¢+ 6, y) =0. The equation of the parabola osculating this conic d, in the origin 1s: 4a,c,y—d3b?x’?=0. This parabola has in the origin the same radius of curvature 7, as the conic d,. The radius of curvature in the vertex of the parabola ( 279 ) is the parameter. So the radius of curvature 7, of the conie d, in tl set . a d, Cy je driein® is —— = re a, € 2 a, C 4 “9 as Rk, == ‘and ry, = = 2 AT > b? now follows: MR end, QO: Hel). $ 2. The theorem can be easily expanded to a general twisted curve C. Let P be an ordinary point of C, the tangent and the osculating plane in P showing no particularities. Through point P and five consecutive points of C' a twisted cubic C° can always be laid. The radius of curvature /, in the point P is the same for the curves C and C*, having six consecutive points in common. The osculating planes of the curves C and C* in the point P will coincide too. This common osculating plane ( intersects the developables belonging to C and C° according to the tangent in P counting double and moreover according to two plane curves d and d,. If the curves C and C* had but a three-point contact in P, the curves d and d, would have a common tangent in the common point P, so that the curves d and d, would have in P at least two consecutive points in common. If the curves (and C* were to have a five-point contact, a common generatrix of the two develop- ables not lying in the common osculating plane © would meet the osculating plane Q in a third common point of the curves d and d,. Now that the curves C and C” have a six-point contact in P the curves d and d, will have at least four consecutive points in common. These two sections d and d, have thus in P the same radius of curvature 7,. Consequently in the ordinary point P of the twisted curve C we have: § 3. When two arbitrary twisted curves have in a point P a three-point contact, they have in that point the same radius of cur- vature Zè. If now the common osculating plane © in P of the two curves cuts the two developables belonging to the curves in the plane curves d and d@’ then the radii of curvature in P of these sections 4 d and d are both 5 R and therefore equal. The curves d and d' have thus in P also a three-point contact. From this follows the theorem : 19 Proceedings Royal Acad. Amsterdam. Vol. VIL. ( 280 ) If two twisted curves have in P three consecutive points in common this will be also the case with the plane curves forming part of the sections of the common osculating plane with the developables belonging to the twisted curves. The radius of curvature of the section d in the point P being four thirds of the radius of curvature of the cuspidal curve C in this same point, the curves d and C have in P but two points in common. From the theorem proved here, follows once again the theorem communicated by me before, concerning the situation of the three points which a twisted curve has in common with its osculating plane. (see These Proc., Febr. 27, 1904). § 4. By expansion of the coordinates of an arbitrary algebraic or transcendent twisted curve in the proximity of an ordinary point P into convergent power series of a parameter f, the theorem ot § 1 can be proved also directly for such a twisted curve without using the twisted cubic. Let P be an ordinary point of the curve C'; if the tangent, the principal normal and the binormal in / are taken respectively as X-axis, Y-axis and Z-axis, then the coordinates of the twisted curve C become : C=O, bt) 2G, P= ws ee be Su Bees En 4 zt ee oe a oe et e The point P corresponds to the value zero of the parameter t. If P is an ordinary point the coefficients @,, 6, and c,‚ cannot be zero. Let R, be the radius of curvature of Cin point P, thus the value obtained by the radius of curvature A for ¢=0. The radius of curvature in P of the projection of C on the osculating plane z= 0 is also A, this projection having in P three consecutive points in common with C. The coordinates of the points of this projection are: = 08 a Vn ny gb, Cit Ore aes 5 LS di As = is equal to 0 for t=O the general formula for the radius C of curvature R (da” + dy?)*l2 de d?y—dy Bx’ ( 281 ) transforms itself into the simpler one da? k= ne dy i—0 It is easy to find a, R, == 25, The coordinates §, # and § of an arbitrary point Q on the developable belonging to C can be expressed in the parameters ¢ and 7 where r represents the distance from point Q (§, 7, $) to point (x, y, 2) of the cuspidal curve measured along the tangent of C passing through Q. The coordinates of Q are: dt dx ba an dt dy n=y + OTT ; dt dz PT For the points Q situated in the osculating plane § = 0 the relation dt dz es ce must exist between the parameters 7 and ¢. By eliminating 7 out of this relation and the equations for § and 4 we find expressed in functions of ¢ the coordinates of the points @ situated in the plane $=0. These coordinates of the points of the curve of intersection d are bs dau dz S=, Z dt dt dy dz NZ=Y zie, : dt dt IE 8 6. tere...) (a. kl A en 2 Ee ‘ 3c, t?-+ 4e,0°+... 3 Ct --¢,t°-++..:).(26, 4-30.07 -L... 1 7 = b, t? oa b, t3 + ae ( cen En 4 : ( 2 3 at: Ee ) — = b, t? + ek E Best de t° +... Ii dh ! 3 : As here too 7 is equal to O for f=0 we find as above that the t ( radius of curvature # in point P of the curve dl is: 0 19% ( 282 ) This formula gives for 7, the value: 0 4 9 oa LA 9 2a," ee — 2 3b, aie 3 2 > ; a,” Zn From the obtained values Rk, = and r, == we get 2 b, 3b, SS ye a ee Delft, Sept. 1904. Physiology. — “Deyenerations in the central nervous system after removal of the jflocculus cerebelh”. By Dr. L. J. J. Muskens. (Communicated by Prof. C. WINKLER). (Communicated in the meeling of September 24, 1904). In 6 rabbits the flocculus of the right side was extirpated. This organ lies, as is well known, in these animals in a separate bony hole, so that we here have the possibility to remove a part of the cerebellum without disturbing the nervous structures of the neigh- bourhood in their conditions of nutrition as well as of pressure. The animals were killed after 8 days to 5 weeks and complete series stained after Marchi, were prepared. The degenerations of fibres after this lesion in 4 of the 6 cases were found exclusively directed upward i.e. to the superior crus- cerebelli and to the pons. In one case there was a fine degeneration all over the restiform body; in this case however it could not be made out with certainty whether we had to deal with really descending degeneration, because firstly all through the cord fine, black spots were found, and secondly the black spots were of so little dimensions, that there is much doubt about the genuineness of such a fine degeneration. In this animal the staining was insufficient, irregular and not limited to degenerated nerve-fibres, for an unknown reason, so that we do not think much value can be attached to this single case, in which downward degeneration was found. In another wellstained case in the restiform body a number of degenerate fibres on the operated side was found; also in the longi- tudinal posterior fascicle and in the field of the tecto-spinal bundle, ~ equally on the operated side. In the superior cervical region also a small field with the base lying towards the margin, the point towards the restiform body was found full of degenerate fibres. Lower down than the upper cervical segments, these degenerate fibres do not reach. In this case not only the flocculus and the floeeular peduncle, but also the vestibulary nucleus was severed, so that also this experiment cannot be recognized as a clear experiment. Although allowance must be made for an eventual different result after extirpation of other parts of the cerebellum of the rabbit, so we think, that these experiments show clearly the absence of descending degeneration after a sharply localised lesion of the floc- cular cortex. The discussion in the literature between Marcu, Frr- RIER and Turner, Tomas, Brepi and Raster Resserr regards the question, how much of the found degenerations must be aseribed to lesion of the neighbourhood, because, as THOMAS justly remarks, exactly in this region of the cerebro-spinal axis it is characteristic, that also without direct lesion by the severing instrument yet by haemorrhage or an alteration of pressure, extensive degenerations can be caused. As in these experiments certainly no such lesion of the neighbourhood can have arrived and in the completely successful cases the cord was found free of degeneration, we may be sure, that from the ganglioncells of this part axis-eylinders with centrifugal course to the medulla are not found, so that for this part of the cerebellum at least, the original data of Marcni are not confirmed. Thus these observations as also those of ProBsr can be regarded to agree with the English observers, after whom only after lesion of the nucleus-Derrers des- cending degeneration of the anterior and lateral tracts is found. In judging this result it is important to observe, that also in another point than by its own bony capsule the rabbit must be regarded as an abnormal form. The floceulus of the rabbit contains viz. except its part of the cerebellar cortical gray matter and its af- and ef-ferent fibres also a nucleus of large multipolar ganglioncells, such as are found in the nucleus dentatus. The study of the development of kindred animals (squirrel) leave not the least doubt, that indeed a part of the dentate nucleus is dislocated in the flocculus. It appears that it is not always in connection with the principal nucleus. Now I do not think that for the elucidation of the question, whether there exist descending cerebellar tracts, this circumstance must be regarded an indesirable complication, but rather we may reckon this a useful detail, in so far as it allows to exclude at the same time, ( 284 ) that such efferent fibres descending in the cord, should spring from (this part at least of) the dentate nucleus. Regarding the ascending degeneration in the different operated animals the most complete accordance is found. Two bundles are found in all successful cases, very clearly and in exactly the same place of the cross-sections and both find in the same region of the cerebrum their end, viz. in the regio subthalamica. In the first place the supe- rior cerebellar peduncle being the most voluminous bundle, where we find fibres of heavy caliber. This degeneration shows especially gross fibres, compared with the fine degenerations, found elsewhere in the rabbit. The degeneration is found especially in the middle third part of the superior cerebellar peduncle, whereas the medial and lateral thirds are nearly entirely free from degenerate fibres. Arrived about at the posterior quadrigeminal body, the degenerate fibres curve downward in a nearly rightangle, as this is repre- sented by the authors, building the wellknown peduncular deeussa- tion. Only a few sections separate the commencement and the finish of the decussation in the sections. In the substantia reticularis the direction is again purely longitudinal to the long axis of the cerebral stem, where as in the region of the red nucleus it becomes clear, that especially the ventral part of the red nucleus comes in contact with the crossed peduncle. This crossed connection is, as far as the floeeulus is concerned complete. Here it may be recalled, that Probst has shown, that after extirpation of more dorsally situated cerebellar parts of the cat also non-crossed fibres run to the subtha- lamie region. Besides this most important upward degenerating bundle, there is another tract up to now only described as far as I am aware by Prosst, which is constituted of finer fibres than the first bundle, takes its course by the substantia reticularis, of the contra-lateral side, and joins the first tract about its arrival in the red nucleus. Both together run frontalwards, and end in the ventral part of the nucleus ventralis thalami. The sections leave no doubt, that no fibres from the floceulus arrive in the thalamic region unerossed, but all decussate either in the decussation of the superior peduncle or as far as the second bundle is concerned in the pontine region, right near its emergence from the flocculus. Also THomas has designed this degeneration, but he thinks, that here we have to deal with descending collaterals of the frontal cerebellar peduncle, which leave the principal bundle after the decussation of this peduncle. Progst on the other hand thinks, that these fibres arise from the dentate nucleus, pass directly through the region of the vestibular nuclei, to the substantia ( 285 ) reticularis of the crossed side and ascending frontalwards are found in the same region up to «their junction with the superior crus cerebelli. My own sections suggest very strongly indeed, that these centri- fugal (from the cerebellum, or rather from the nucleus dentatus) fibres, take their course by the superficial layers of the middle cere- bellar peduncle and then can be followed right through the pyrami- dal bundles or partly winding around them to the reticular substance. In different series it becomes clear that proceeding in the series of sections from below upwards there, where are found the first dege- nerate fibres in the reticular substance, also the first degenerate fibres appear in the middle peduncle. While by THomas no sound reasons are given for his conception about the significance of this bundle, it pleads against the opinion of Progsr that in the region of the vesubulary nuclei, no degenerate fibres are found. Finely the sections show, compared with the sections gained by other experiments, that the ventral thalamic bundle originates for the greater part from the ventral portions of the cerebellum, especi- ally of the floeculus. Sections of cats-brain after similar operations leave no doubt, that after lesion of more dorsal cerebellar portions, there exists a very marked contrast between the very pronounced degeneration of the crus cerebelli ad corpora quadrigemina and the very slight degeneration of the ventral thalamic bundle, whereas as well in the cat as in the rabbit after exclusive lesion of the flocculus, both bundles are affected about equally. Physics. — “The validity of the law of corresponding states for mietures of methyl chloride and carbon dioaide,’ by Prof. H. KAMERLINGH OnNES and Dr. C. Zakrzewskt. Communication N°. 92 from the Physical Laboratory at Leiden by Prof. Dr. H. KaAMERLINGH Onnes (continued). (Communicated in the meeting of June 25, 1904). § 1. Introduction. In n°. IX of the “Contributions to the knowledge of vaN DER WAALS’ y-surface’” we have expressed the hope of giving an experimental contribution to the investigation of the co-existing mixtures of methyl chloride and carbon dioxide at low temperatures in connection with the test of the law of corresponding states for mixtures, which for many years has formed a subject of experimen- tation at Leiden. Of the extensive territory of reduced states, which the mixtures of carbon dioxide and methyl chloride afford for mea- surements on either side of the critical state, (reason why in about 1890 it was chosen for the first investigations of the w-surface) a considerable portion round the critical state has been immediately investigated by Kurnen (Comm. N°. 4, April ’92). HARTMAN in Comm. N°’. 48, June ’98 has added to this area that of the coexisting phases at 9.5 C. We have extended the area investigated in two directions, albeit only by a few preliminary researches. The results of some of those measurements, though a few are only preliminary and served chiefly as a means for us to decide upon the method of investigation, seem important enough now that still so little is known about the different degrees of approximation to which the law of corresponding states holds for mixtures in diffe- ; ; 7 rent fields of reduced state (: nn jk DJ. 1). Our measurements refer in the first place to gaseous mixtures under almost normal conditions, in the second place to coexisting phases at low temperatures. For the normal gaseous phase we found the law of corresponding states to be confirmed to a high degree of approximation. The virial coefficient B, which determines the deviation of mixtures of methyl chloride and carbon dioxide from BorLe's law at small densities, can be sufficiently derived by means of the law of corresponding states. Greater deviations were found when we investigated the coexisting phases at low temperature. Here we have determined by means of the dew-point apparatus, described in the first part of this communication, the begin condensation pressure of the mixture 7 = */, at — 25° C.: the temperature for which we have constructed the y-surface in Suppl. N°. 8, Sept. ‘04. The deviations found are rather great, they point to an increase of the deviations from the law of corresponding states in the mixtures at low temperatures in the liquid state. The deter- mination of the end condensation pressure for the same mixture 7 = !/, at — 25°C. with the piezometer of the first part of this communication would involve complications (comp. ibid. § 5). In order to obtain an idea of the deviations of the liquid branch of the binodal curve at «= 1/, from that according to the law of corresponding states, we have Asif investigated the condensation pressure for 2 == '/, at a lower tem- perature, viz. —38°.5 ©. This corroborated the result of the investi- gation of the vapour phase at — 25° C. I. The compressibility in the neighbourhood of the normal state. § 2. Determination of the seeond virial coefficient. The mixtures ( 287 ) were , prepared and the compressibility determined in the mixing apparatus and volumenometer deseribed in Comm. N°. 84, March 03. The method of observation and calculation has been treated in detail by Krrsom, Comm. N°. 88, Jan. ’04. The gases were prepared by distillation first in ice, subsequently in solid carbon dioxide. From previous communications it will appear that in this way pure carbon dioxide is obtained. Of methyl chloride the same will be proved in the continuation of this paper ($ 8). The values found at the temperature ¢ of the pressure p, volume V and molecular composition of methyl chloride w are given in table I. TABLE I. Compressibility of mixtures of carbon dioxide and methyl chloride. x= 1 (CH, Cl) N°. pinmm Pmt: | t L MST 537 67 | 20.05 LO ON 90.07 UT. 479:23 | 199633 | 90:07 I. 4900.32 | 537.49 | 20.09 Il. 604.45 | 1043.50 20 10 IIL. 503.49 | 4297.01 20.08 : Fr | PE vr — 0.5030 I. | 1173.08 | 537.73 19 97 Il. 608.87 | 4043.66 | 419.87 IIT. 490.88 1296.30 | 19.87 The values for =O may be borrowed from Comm. N°. 88. For the calculation of these observations we shall use the empirical reduced equation of state of Comm. N°. 71, June ’01, which is particularly suited for the investigation of the degree of validity ( 288 ) of the law of corresponding states, in the form as laid down in § 4, which deviates little from that of Suppl. N°. 8, Sept. ’04. In the first place the observed pressures must be reduced to the same temperature 20°C. For this purpose we have calculated the real coefficients of pressure-variation of carbon dioxide (0.003460) and methyl chloride (0.008586) with the equation of state mentioned and the coefficient of pressure-variation given below for ideal gases, and we have taken linearly interpolated values for the mixtures. Owing to the small differences in temperature the errors ensuing from this remain below those of the observation. Let v be the volume expressed in terms of the theoretical normal volume (introduced in Comm. N°. 47 Febr. ’99), then we have approximately B py A+ ae where A=1-+a,t the coefficient of pressure-variation of an ideal gas. One of the advantages of the empirical reduced equation of state is, that it teaches us the degree of approximation to which the higher terms in 1 L and a, may be omitted. Then we have for the calculation of the second virial coefficient as a first approximation, if B? also is neglected (for further approximations see $ 5): > P12; jeje a ae Ri) nn fa NR Pers A and with @,, = 0,00366195 (instead of 0,0036625 of Comm. N°. 71) we derive from table I: TABLE IL. Second virial coefficient for mixtures of carbon dioxide (er =O) and methyl chloride («= 1) to the first approximation. composi- | B WEE (petals Aa pe from T and hl 2 from Land nn „2 mean B ee Alge, An) eer Et. | il — 0.01797 — 0.01800 | — 0.01798 —0.02071 0.6945 — 0.01302 200139 ea a0 ELD —0. 01509 |. 050304) = O70da4 — 0.01005 | —0.0i019 —0.01175 0 KEEsoM, Comm. NO, 88. —0.0065% ( 289 ) § 3. The virial coefficient Bas a quadratic function of the molecular composition «. According to VAN DER WaAAts’ equation of state B= RT bpp — drow, if appr and bypy represent VAN DER WaAas’ constants with regard to the theoretical normal volume. Hence, to the first approximation, we must have for the mixture with the composition « (Cl Me. COs) (Clie) (Cl Me. CO) (C'O3) B+ Bar +2 apBer(l —a)+ Bl) By means of least squares we found’): (Cl Me) Bago = — 0.020772 (Cl Me. COs) (12) aoe = 0.010067 (0s) The agreement appears from the following table: i B observed £8 computed Obs.—Comp. 1 EN DOT 22002077 4. 0.00006 0.6945 — 0.01509 — 0.01490 — 0.00019 0.50380 — 0.01175 —= (0.01190 + 0.00016 0 — 0.00654 — 0.00652 — 0.00002 The deviations are less than 2°/,, hence also less than the devia- B tions of the single values of WT inter se. Thus the agreement with the quadratie form was sufficiently proved, so that for the time being measurements with other mixtures, not exceeding this accuracy, could be left off. § 4. Validity of the law of corresponding states for the virial coefficient B. According to the law of corresponding states the virial coefficients are derived from the coefficients of the reduced equation of state through multiplication by functions of 7, and pp, (comp. Comm. N°. 71 and also Suppl. N°. 8, Sept. ’04, the first four sections). As the eritical data of mixtures of carbon dioxide and methyl chloride have been derived in Comm. N°.595 from KurNrEN’s experiments, we may determine B for a given temperature, for instance Bay by Boo — 7 Pk 1) The coefficients given here have been derived from values for B which do not differ essentially from those given in table If. ( 290 ) where Be is the value of the function ® of the reduced temperature 293.04 =a For B we have used a function of a form differing slightly from the form V1.4, given in Suppl. N°. 8, which did not only agree with belonging to t= hydrogen, oxygen and nitrogen but also with ether, viz. a form VI. 2, which instead of agreeing with ether in the same way as VL 1, agrees with the average of ether and isopentane : KA 1 10°. 8 — -+ 179,883 t — 374,487 — 181,324— — 110,267 — t (3 The agreement appears from the following table, where we find in the first column the values calculated according to the last formula, and in the second column those of the quadratic formula of $ 3; according to according to difference corresponding states quadratic formula I — 0.021920 — 0.020772 — 0.001148 ne — 0.016502 . — 0.015866 — 0.000636 af — 0.012179 — 0.011855 — 0.000324 0 — 0.006485 — 0.006515 + 0.000030 The deviations on the side of the methyl chloride are larger than those of the errors of observation and those of the quadratic formula. Methyl chloride, therefore, does not agree so well with ether and isopentane as carbon dioxide. This same result is also arrived at in another way. It appears, however, that the mixtures do not deviate more than the methyl chloride itself. (To be continued). Physics. — “On the measurement of very low temperatures. VI. Comparison of the platinum thermometer with the hydrogen thermometer’. (Continuation of Comm. N°. 77. Febr. 1902). By B. Meimixx. Communication N°. 93 from the Physical Labo- ratory at Leiden by Prof. H. KamEriincu ONNEs. (Communicated in the mecting of June 25, 1904). § 5. The measurements at low temperatures. The thermometers were mounted as described by KAMERLINGH ONNES in Comm. N°. 83 Febr. 1908 § 5. During the first preliminary measurements, the hydrogen thermometer and the resistance thermometer (ef. Comm. N°. 77 § 2) were piaced in the eryostat (described in Comm. N°. 51, Sept. 1899), ( 291 ) which was modified as described in § 2 Comm. N°. 83. In this cryostat a vacuum vessel was placed inside of B, (Pl. I, Comm. N°. 51), which vacuum vessel by means of cork was pressed against the walls of B, (ef. end of $ 2 and also Pls. I and II of Comm. N°. 83). The inner wall took the place of &, in PI. Il, Comm. N°. 83 (the same parts of Comms. N°. 83 and N°. 77 are marked with the same letters). These preliminary experiments had shown, that, after repeated mea- surements at the lowest temperatures, the original value was again found for the resistance at 0° C., hence that the platinum wire, though its expansion differed from that of glass, was not lengthened and that it also remained properly in the notches. Further it had become clear that an accurate comparison of the two thermometers was only pos- sible when the temperature of the bath was kept constant with the utmost care, and there we met with the difficulties treated in § 2 of Comm. N°. 83. It was attained by arranging the cryostat as described in $ 5 of Comm. N°. 83. It may moreover be remarked that the liquid gas was always kept higher than §,’ (Comm. N°. 83, Pl. IL); else, notwithstanding the level in and outside the protecting cylinder would go up and down through the motion of the stirrer, no circulation would be produced. The course of a measurement was as follows. As soon as the cir- cumstances under which we desired to make a measurement were established, the resistances of the leads were determined, then the resistance of the platinum wire was adjusted and, by giving signs to the assistant charged with the regulation of the pressure, care was taken that this resistance, and hence the same temperature, were maintained. After about ten minutes we began, while constantly reading the galvanometer, the measurements with the hydrogen ther- mometer and continued them until the liquid was evaporated or until we deemed that sufficient data were obtained. At the end the measurement of the resistance of the leads was repeated. The observer at the galvanometer had, therefore, only to look after the continual closing and breaking of the currents and the noting down of the values of the galvanometer readings and of the time belonging to them. Afterwards the deflections were derived from this (see Pl. II] Comm. N°. 83) and the mean deflection during the time of observation was found by means of a planimeter. § 6. Zero after the measurements. By a too rapid decantation of liquid oxygen, numerous bursts had unfortunately come in the cylinder of the resistance. To repeat with it the above described operations for the determination of the zero seemed rather dangerous, especially ( 292 ) as the refastening in the cryostat would have involved many difficult operations. Therefore in order to bring the resistance thermometer to a constant temperature near 0° C., the case U of the eryostat (PL. I, Comm. N°. 83) was screwed off from the cover N, (Pl. I, Comm. N°. 51), while the other parts of the cryostat remained fastened to the cover, “and it was replaced by a zine cylindrical vessel, which could be managed more easily. This vessel was provided with a rim fitting on to N, and was placed in another larger zine vessel, so that a jacketing space of 5 em. remained which was entirely filled with ice. Then isopentane was distilled into £,, (PL HU, Comm. N°. 83) and the apparatus was left to itself during one night. The next day the temperature (near O°) had become constant and we determined it (while stirring) by means of a thermoelement (@ PI. II, Comm. N°. 83). § 7. Corrections. A survey of the mounting of the WHRATSTONE's bridge (ef. § 3 Comm. N°. 77) is given in fig. 5. Mè indicates the tE: ml C. | | Py A 48 | 5 HR | Rl] IA A) 3 z , Ss EH 1/ 3 = t, € Fig. 5. resistance to be measured, f, and fF, the two coils of manganin wire, B, and F," the resistance boxes of Hartmann and BRAUN and ee RR," which with 7,, forms the fourth arm of the bridge; C, and C, are the commutators with mercury contact (Comm. N°. 27, May and June oF 1896), C, is the copper commutator treated in § 3 Comm. N°. 77 and of Sinmens and Harskr giving together the resistance #,= lt, — represented there in fig. 2. Putting for the factor for the inequality of the branches of the R bridge — =1 — a, we found a = 0.00216 (as mean value of twenty 2 ( 293 ) values ranging from 0.00219 to 0.00214). As according to § 3 Comm. N°. 77 the resistance of the platinum wire F is equal to the difference between two measured resistances, one ¢ (the resi- stance of the leads), another /? + ¢, and as the branches of the bridge are so nearly equal, 7,, is eliminated, and hence we need not know the -value: of °7,;: ; To the resistances read on the box /,', the corrections found by calibration must be applied. We may easily convince ourselves that the arrangement of box f," parallel to box /?,’ has no perceptible influence on the value of the corrections at /,'. The corrections of the errors in the nominal box values could be neglected for all coils below 1 Ohm. For the measurements at low temperature a correction had to be applied, because during the regulation of the temperature the mean deflection was not zero. In order to express that deflection in terms of the resistance, the platinum thermometer in the bridge, after the measurements were made, was replaced by an equally large box resistance and for a known modification of this resistance the deflection was observed. The regulation of the temperature was in most cases so successful that it was hardly necessary to take the correction into consideration. The resistance, measured near 0°, was reduced to 0° C. with the approximate formula W = 110.041 (1 + 0.0038644 ¢— 0.000001051 £), derived from preliminary observations. § 8. Survey of a measurement. The course of a measurement is described in $ 5. The quantities which are derived directly from observation are given in table I (p. 294). Under the head “Connection” I have recorded between which blocks of- the commutator C, a conducting connection existed. Therefore commutator C, was not used while the measurement lasted. This had become possible because the platinum wire was wound free from induction, so that no induced current was observed when the principal current was closed. From these data we now derive for each connection the value of R,, i.e. the resistance of the branch of the bridge in which the resistance boxes are placed (apart from 7,,). If the value of FR, which is found when we measure the platinum wire with the leads, is diminished by the value of 2, which is found with the leads alone, we obtain the resistance of the platinum wire, in the supposition 1°. that the arms of the bridge are equal, 2°. that during the measurement the mean deflection was zero and 3°. that the resistance box requires no correction. For each of these suppositions =: TABLE I. Calibration Platinum Thermometer i in n Oxygen Boiling under Reduced Pressure (May | 29, 1902). ~ Resistance Measurements. > | Position Position of Den ; Time. Connection. | R’, | R', | commutator equilibrium of | | C; galvanometer. galvanometer. : os a 1-52-6 | 0.4 | 1.5 58.5 i Leads. Left. Right. 41.7 1.6 Left. 43.4 58.2 1-3 2-4 | 0.4 | 1.6 Left. 43.8 58.0 capes. 1—5 2-4 (2042 3200 platinum | wire. ED 0.3 | é 45.2 3h_49! Left. 44.0 | 45.0 46.0 46.7 47.0 45.0 50’ 44.2 45.4 de to Right. | = 4h95! | and so on ey for all _— minutes to EL | | | 4h 25’. ix is | ESO | | | Se 4h27' | Leads. En | 1359-4 | 04 | 45 Left. 46.0 — a ch oi. | : : 2 | 15 Beb WOM de Left. 3. ae | 46.0 | | en ( 295 ) we must apply the correction mentioned in $ 7 in order to find the true resistance. The mean deflection during the measurement is found, according to Comm. N°’. 83 $ 5, by means of a planimeter. (See for the graphical representation Pl. III, Comm. N°. 83, which does not, however, bear upon our case). In the following table the corrections are combined. TABLE II. Calibration Platinum Thermometer in Oxygen Boiling under Reduced Pressure (May 22, 1902). Resistance measurement, Corrections. Ratio arms. . . . . . . . +0,00216 Correction to R for box values 20 and 2 + 0.0005 Mean deflection . , . . . . +01 cM. After what has been said above about the method of calculation the further calculation will be sufficiently clear from the following table. | TABLE III. Calibration Platinum Thermometer in Oxygen | Boiling under Reduced Pressure. Resistance at — 197°.08 C. ; : ‘ , ; En Resistance Time | Connection | R’, RY | R", mean R, Drika | 1—5 26 0.4 | 1.54 15:55 0.3182 1—3 24 0.4 | 1.57 3h. 49— | 1—5 2-4 (2042 3200 22.1457 4h 25 +0.3 | | 1—3 24 OEI Ae 5a | 1.53 0.3171 | 15 2-6 | 0.4 | 4.53 21 8981 Correction arms of the bridge. . . . .. . , + 0.0485 Correction resistance box. . . . . . . . . . +0,0005 Correction to mean deflection O0 . . . . . . , + 0.0002 | | | Hastabance 4 [igs delhi. eres | abe) ort Ue ey OT RS aN nl Pe AN teva 20 Proceedings Royal Acad. Amsterdam. Vol. VII. ( 296 ) Determinations made at other temperatures did not yield anything particular. Only for the zero determination the corrections are some- what different, as that for the reduction to the mean deflection O is no longer necessary. A new one, however, is added because the determi- nation has not been made exactly at O° C. but at a little higher temperature. After what has been remarked about this in § 7, it seems superfluous to illustrate this small variation by an instance. § 9. Determinations of the resistance at 0° C. They are made in three series. For the first we still used leads of 0.5 m.m. (Comm. No. 77 § 2 and fig. 3), the insulating liquid was petroleum ether or amylene ; for the second the leads were 5 m.m. thick (Le. fig. 4), the insulating liquid was isopentane; and the third (insulating liquid isopentane) was that treated in § 6. TABLE IV. Calibration of Platinum Thermometer. Zero. | Number of 4 â5 : Resistance. determina- Rr pale Pes Mean value. Honig: Smallest Largest | Series 1. June 701 4. 110.031 140.048 110.040 2, Nov.Dec.Ol 7 033 57 43 3. Nov. 702. 3 043 51 48 Mean resistance at 0° C. | 110.045 | = = : et ee a =S § 10. Determinations at low temperatures. The measurements were made at fairly gradually decreasing temperatures; at the lowest temperatures the intervals are smaller. The measurements with the hydrogen thermometer (see Comm. N°. 77 $ 2) are made by Dr. W. Hevusk to whom my best thanks are due for the trouble he has taken. The determinations are made up of two series. Thé first series was made between May 18 and July 10, the second series between Dec. 10 and Dec. 22, 1902. It seems desirable to consider the two series separately. The first series has yielded results that may be derived from the following table. In order to judge how the values given here agree inter se, I have first calculated the formula of the form ww, (1 + at + bt’), ( 297 ) TABLE V. Calibration of Platinum Thermometer. First Series. | Temperature | Resistances | | Bath in which the Date. determined with the | | l ‚hydrogen an measured, ‚measurements were made. | | Oet: 110.045 comp. § 9. May 24, 'O2 — 51°.43C. 87.760 methyl chloride boiling | under reduced pressure. May 13, ’02 — 104° .66 64.256 ethylene. 3 — 104°.38 64.374 id. ag el a ethylene boiling under ; __ 198° 8 52 379 reduced pressure. May 16, ’02 — 1619.15 38.676 methane, Ps — 1619.45 38.672 id. | A — 161°. 47 38.515 id. May 22, 02 — 182°.63 28.692 oxygen. | | July 10, ’02 (— 195°.75) 1) (22.600) nitrogen. | h, ? My mo ©. Lr haw | May 22, "02 197° .08 21.877 oxygen boiling under | ” — 197°.58 21.673 ERA PESSE | July 10, ’02 — 209° .93 16.025 nitrogen boiling under | reduced pressure. which agrees with the observations at 0°, at —104°.66 C. and at —182°.63 C., the temperatures which best correspond with those which as a rule are also used by other observers. The formula becomes w = 110.045 (1 + 0.0038788 ¢ — 0.000 000 9257 7°). The deviations of the observed resistances from the formula are given in the column Obs.—Comp.; of table VI, and are quite appreciable. In the case of methyl chloride the deviation amounts to 65 on 87760 or a difference in temperature of 0°.15 C. For methane these deviations are 63 on 38674 or a difference in temperature of also 0°.15 C. In oxygen, boiling under reduced pressure, the deviation is 90 on 21637 or about 0°.2 C. 1) This observation is less reliable because an uncertain correction to the hydrogen thermometer attained a rather high value. 20* : ( 298 ) To find out whether these deviations are perhaps due to irregular errors in the measurement, it will be useful to investigate whether, by addition of another term, the differences between these obser- vations and the calculation might be reduced to within the limits of the errors of observation. It succeeded indeed fairly well as may be seen in column Obs.—Comp.77 of table VI. The calculated values are derived by means of the formula w = 110.045 (1 + 0.0039167 ¢— 0.000 000 3432 t? + + 0,000 000 002069 7°). TABLE VI. Test of a Parabolical and of a Third Degree Formula. First series. Temperatures deter- | Measured | | mined with the hydro- | Obs.—Comp. ,| Obs.-Comp. 7, | gen thermometer. | resistances | 0e | 140.045 | 0 | 0 | — 51°.43 | 87.760 | = 0,065. 4 4- 0.012 — 104°. 38 64.371 | — 0.011 | — 0.017 — 1049.66 64.256 | 0 | — 0.005 — 197°.74 | 53.910 | + 0.039 i — 1289.88 | 53.372 | + 0.030 | — 0.014 — 1619.45 | 38.674 | +0,068 |. + 0.019 | — 161°.47 |_ 38.515 | + 0.054 | + 0.008 | 484063 | 98.692 0 a 105008 (— 195°.75) | (22.600) (0.014) | (4+ 0.078) — 1979.08 HRT Dee | "0 ER (MLE AEON — 197°.58 | «a 637 }o Jono lotor — 209° 98 | 46.098 | + 0.07% | + 0.299 The deviations from formula I, with the exception of the last, although they are not entirely within the limits of the errors of observation which were expected, are only little in excess. In the ease of methane, where the deviation is 19 on 38674, an error in the temperature of 0°.04 C. is sufficient to explain this amount, ( 299 ) In the case of nitrogen boiling under reduced pressure, however, the deviation has become very large, so large even that it cannot be explained by errors of observation. Hence the circumstance that the formula is not fit to represent the resistance so near to the absolute zero must account for this deviation. All the same it is remarkable that this turn appears so suddenly. At —197° C. the formula still holds, at —210° C. there is a deviation of 229 on 16025, i.e. a deviation of O°.49 in temperature. But if we take into consideration that, according to the formula, the resistance at —-248° C. would become zero and that we are only about thirty degrees from this point, we need not wonder at this result. In order to gain certainty that there was indeed a fairly rapidly increasing variation in the shape of the curve that represents the resistance as a function of the temperature, | resolved to repeat especially these measurements at very low temperature in nitrogen. These constitute what | have called at the beginning the 2ed series. Unfortunately the result was unsatisfactory. Though the observations indeed point in the same sense, yet one error or another seems to have crept into them and it could no more be detected at the time when the calculations revealed it. We shall omit them here. Therefore the results as to the amount of the deviations remain more or less uncertain; yet it is very probable that even in nitrogen boiling under reduced pressure, a beginning may be observed of the variation in the course of the temperature function which, as follows from Dewar’s experiments, appears so strongly at the temperature of liquid hydrogen. The conclusions to which the measurements lead may be summarized as follows, A representation of the resistance by a quadratic formula, according to the temperature, even if we do not go below — 180° C., is only permitted when no higher degree of accuracy than 0°.2 C. is aimed at. When a greater precision is desired we require for the calibration of a platinum thermometer a greater number of points of comparison. Tie APE ats For a comparison to within 50 °C. a number of at least 6 tempera- tures of Comparison is considered very desirable. Below —197°C. the deviations of the platinum thermometer become so large that before using it for this range an investigation must be made of the course of the resistance as a function of the temperature. ( 300 ) Physics. — “On the measurement of very low temperatures. VI. Comparison of the resistance of gold wire with that of platinum wire.’ By B. Memik. (Communication N°. 93 (continued) from the Physical Laboratory at Leiden by Prof. H. KAMERLINGH ONNEs). (Communicated in the meeting of June 25, 1904.) $ 1. The investigation deseribed in this paper forms part of the subject mentioned sub 2 in $ 1 of Comm. N*. 77, Febr. 1902 and had for its chief object the establishment of the method of observation. The gold wire was made of the material kindly given us by Dr. C. Horrskma, inspector and assay-master general of the Mint, according to whom no impurities could be detected in it by means of chemical processes, which, with regard to the accuracy of the gold The piece was drawn out to a wire of '/,, mm. in diameter., The great length of 0 1/ 0/ DOD analysis, excludes an impurity of more than the wire used, however, had rendered a soldered joint in the middle necessary. § 2. Arrangement of the wires. The same advantages which made us prefer a naked platinum wire to one enclosed in a glass tube (cf. Comm. N°. 77) exist also when the wire is made of an other metal, though the difficulties, especially with regard to the action on the metal, are greater. The difficulties of the arrangement increased, however, considerably with metals of such high conductivity as gold, because then the wire must be so much longer in order to produce a sufficient resis- tance. With the first forms that were tried, the metal wire lay in a screw-thread etched on a glass cylinder. But with the longer wire the latitude for the expansion became so great that it could slip too easily from the screw grooves and thus cause short-eircuiting. In order to obtain deeper grooves the glass cylinder was coated with a paste of oil and carborundum and slowly spirally moved with a speed of */, or */, mm. The cylinder grinds against an iron or copper disk, which is kept in rapid rotation and thus by means of the carborundum a groove is ground in the glass. This groove proved to be much deeper than that formed by etching’); the wires of 0.1 mm. in diameter were entirely enclosed in it. On a cylinder of 37 mm. diameter and 55 mm. height we could wind more than 12 meters of wire. 1) Later we have again succeeded in making still deeper grooves by etching. ( 301 ) The investigated wire covered two of these cylinders ¢, and ¢, (see fig. 1, Pl. I), the one fitting into the other and leaving a jacket of about 2 mm. for the circulation of the liquids. A third eylinder c, round these two served to protect the wire. The cylinders rest on a copper star with 3 teeth d in which concentrical grooves are made to hold the glass cylinders. At the other end each of the cylinders, by means of copper ridges ¢,, e, and ¢,, lying on the glass rim, is pressed against the star by means of a tightening rod f and nuts g,, g, and y,, thus forming one tightly connected whole. As the ridges ¢, and v, and also the lower star d served at the same time as connective places for the wire, they were insulated from the tightening rod and the nuts by glass cylinders 4, and /, and plates of mica 2,, 7, and 4, The winding and the mounting was done in the following way. We began by fixing the inner cylinder ec, between the star and the ridge ¢,, the wire was soldered on to the ridge ¢, and led downwards along the groove of the serew-thread on the cylinder. At the bottom it was soldered on to the star. Then the second cylinder was placed round it, the wire was turned upwards along it and fastened to é,. If the two serews in the glass are wound in the same sense, the wire is almost free from induction, (Cf. the platinum thermometer of the previous paper where this was attained in a different manner). As in the case of the platinum wire which was treated in the previous paper, there were 4 leads. For the method with the differential galvanometer (cf. § 4) it does not matter that they have little resistance, hence wires of 1 mm. were taken, flattened over the last 5 ems. The entire apparatus was suspended by a copper tube 4, which was screwed on to the tightening rod /, and which, in order to prevent too much conduction of heat along it, had a piece of ebonite inserted in it (not shown in the drawing). § 3. Determination of the zero. The zero was determined in the same way as described in Comm. N°. 77, when thin leads were used. Besides with the gold wire, determinations were also made with a copper and a silver wire. A single determination never offered any difficulties. With copper, however, the values determined at different times did not agree. They showed a regular increase of the resistance at 0° C. This must probably be ascribed to a chemical process. Copper oxydises so easily that the greatest precautions must be taken to avoid moisture during the storage. If in distilling the insulating liquid into the zero-vessel the vapour was passed over phosphorous pentoxide and if care was taken that while the wire was kept, the air could only enter over phosporous pentoxide, we ( 302 ) succeeded in stopping that process. Yet at any rate this experience obliged us to perform measurements at low temperatures, like those treated here for the gold wire, in a very short time in the case of copper wire. With gold and silver this phenomenon did not appear. § 4. Comparison of the resistances. In order to investigate the variation of the resistance of the wire, wound as described in § 2, the ratio of this resistance to that of platinum had to be deter- mined in different baths. To this end previous investigators have always measured the resistance of the two wires alternately and hence derived the mean ratio. The first experiments made by me were also arranged in that way. In order to attain a higher degree of accuracy I have followed the advice of Prof. KaMmprLINGH ONNes and arranged the measurements so that at a definite moment the ratio itself can be read. If we use the Wuerarsronn’s bridge, it seems that this may be attained by arranging the wire to be compared in parallel to the box of Hartmann and Braun (/’,) instead of arranging it in parallel to the platinum resistance. An insuperable obstacle for this simultaneous determination is, that at any rate a connection is required between the two wires, which connection may be made by two of the leads from the bath, in which we measure, to one of the angles of the bridge. Elimination of all the resistances of the connections except that of the stops used in the measurement, as it was obtained in the previous paper, is impossible. Errors may then creep in of which the amount may only be estimated. Resistances of the connections may occur to a considerable amount and be brought about by minor causes. If we cannot constantly test their amount the results remain uncertain. Moreover in the Wueatstonr’s bridge the unavoidable resistances of the connections are so large that we cannot reach the accuracy proposed in the outset with wires of so little specifie resistance as gold. For in that case, even though we succeeded in winding a wire of more than 20 meters in the necessarily small compass of the bath, a resistance of no more than 35 Ohms at 0? C. was obtained, which. in liquid oxygen or nitrogen fell to below about 6 Ohms. All these reasons led us to choose the method of the differential galvanometer for the comparison of the metals inter se. It enables us to determine at once the ratio of two resistances, while resi- stances of the connections have no-appreciable influence. It is true that a measurement cannot be made in such a short time as with the Whmrarstonr’s bridge but this inconvenience is sufficiently balanced by the advantages mentioned. ( 303 ) § 5. The mounting with the differential galvanometer. The mounting as it finally was made is represented in PI. I, fig. 8. The commutator C, serves for the element. The commutator (, enables the observer to compare either the platinum wire with the resistance Mi, (complete lines), or the gold wire with the platinum wire (dotted lines). It is convenient that by a single commutation we should be able to interchange these two mountings, because for the comparison of gold with platinum wires it will be always desirable to know the approximate value of the platinum resistance for the determination of temperature (cf. § 8). C, is a commutator to interchange the two conducts of the differential galvanometer. This is necessary for the determination of the ratio between the currents in the two not perfectly equivalent conducts when the deflection is zero and moreover it seemed desirable to me to test continually whether this ratio remained unchanged. The galvanometer was first a thin wire THoMson with two pairs of coils each with a single wire. To attain the required symmetry, the four coils were replaced by new ones, in each of which two wires were wound together. This was done in the workshops of the laboratory. The sensibility was a deflection of 1 mm. on the scale with a difference in intensity of current of 10~'° Ampere, period of oscillation 20". Then the galvanometer was aperiodic. Like the galvanometer of the Wueatstonr’s bridge in the previous paper, this was protected against disturbances arising from terrestrial mag- netism by a soft iron ring. The resistance boxes &',, R, and &,, are all of manganin wire, having therefore an almost negligible temperature coefficient; 2, is the carefully investigated resistance box of Hartmann and BRAUN, also used in the measurements with the Wueratstonn’s bridge; &, is the box wound in our own laboratory, which produced two branches in the Wueatstonr’s bridge (cf. previous paper), of which I also determined the absolute values. Af, is a resistance box of Sremens and Harskr tested by the Reichsanstalt. I have determined a few times the ratio between the units Zi’, and /&, and found 1.00255 (11 determinations, greatest... 264, smallest... 242). § 6. The measurements at low temperatures. After the zero of the wire that was to be compared, had been determined a few times, and sufficiently harmonizing results were obtained, the wire was placed into the eryostat inside the platinum thermometer at the place occupied by the hydrogen thermometer when the former was calibrated (cf. Comm. N°. 77, $ 4 and N°. 83, PL. II). When enough liquefied gas was poured off, the ratio between the ( 304 ) resistances of the two wires was determined while we stirred. Before and after this determination, we measured the ratio between the platinum resistance and one of the resistances in A&,. As a rule that resistance in PR, was taken of which the value corresponded best with that of the platinum wire. Thence the temperature could be derived. In later measurements a thermoelement was sometimes. placed inside the cylinder C, of the wire, and after the temperature had been determined once by the measurement of the ratio to /,, an assistant at the thermoelement took care that the temperature remained constant in the way described in Comm. N°. 83. Consequently the measure- ments could be made in a still shorter time. § 7. Calculation and corrections. The mounting is drawn schema- tically in Pl. I, fig. 2, where ry and 7; are the resistances to be compared, IH’, and IW’, the resistances of the galvanometer conducts provided with the resistance boxes #", and f,. Suppose that with certain values in the boxes there is equilibrium and that we can then represent the ratio between the two currents in the two galvanometer conducts by 1+ 3, where 8 may be considered as a small number; then we have the relation | et. ge W, B Pz Vien © WTR W, and W,, however, are as regards the gaivanometer coils copper resistances, and if a rather high degree of accuracy is required, IW, and JW, must be determined before each determination of the ratio between 7, and 7/. If the galvanometer could be placed in a space of constant temperature the greatest difficulty of this would be removed and one adjustment would show us the ratio. HW, and Wy, being unknown, we can proceed as follows. We add to Wy, and W,, a, and «, units of resistance so that again equili- brium is attained, then we also have: i 1 kes W, fh a, ms B alde Lat Ber > Wyte (1+eW,+ 4, H 1) 9, a Sed ea, Re sg x a, or — = (1 I 8) — Tot a, How large we shall choose @ depends on circumstances. The variations in JV. and WW, with regard to the leads during the course of one experiment, as appeared in the measurements with the Wuwar- sToNE’s bridge in the previous paper, certainly never exceeded 0.01 ( 305 ) Ohm. If the galvanometer is carefully packed in eotton wool it appeared that during the time required for one measurement the temperature was sufficiently constant and hence the total variation in J, and W, did not exceed 0.1 Ohm. If then we take for the smallest of the «, and «, 1000 Ohms, the inaccuracy due to this uncertainty need not therefore be larger than */,,,,,. It is likely, moreover, that the first adjustment after the measurement where a, and a, were added to W, and JI’, will be repeated in most cases so that we then can more or less judge of those variations in W, and W r a : 4 3 8 In the relation —- = (1 + B)—, 2 is still to be determined. To this Pot a, end by means of commutator (, (PI. I, fig. 3), the conducts of the galvanometer are interchanged, so that the turns which first were parallel to 7, are now arranged in parallel to rj and reversely. ! « . . Tr 4 a, . . = A new determination yields — = (1 — #) — (3 is considered small). Pyt a I 2 . . 2 Vy a, a Rivas LeU Men Oy Te Wier TMG a = Ph ee N Ppt a, Qa, Besides we can determine (, and find for it elt NEN a a 2 2 == ——— a, a, Te Oem tis If a rapid determination of the ratio is desirable, as it was naturally always the case with the measurements at low tempera- tures, it is better to determine 2 beforehand. We must then have convinced ourselves by preliminary experiments that $8 remains sufficiently constant. This was the case for observations made at not too long intervals. Thus 11 determinations from June 14 to July 12 yielded as smallest value 0.000764, as mean value 0.000780 and as largest value 0.00083 (°). Thus supposing the value of 8 to be known we can derive the ratio = from a single determination of @, and a@,; @, and a, were pt read on the resistance boxes Ff’, and &,. $8. Lhe determination of the temperature. The determination of the temperature, now that we do not measure the resistances separa- tely but determine the ratio at once, need not be made with the accuracy which was required for the test of the platinum thermo- ve meter. It was obtained by measuring the resistance of the platinum ( 306 ) wire after each complete determination of the ratio. If in the arrangement of PL I, fig. 3, the commutator C, is placed differently, we can, instead of comparing the gold with the platinum resistance, compare the latter at once with the resistance in the box #,. In determining the ratio between the resistances of the platinum wire and the values in box AR, we have not, however, repeated the whole measurement as described above for the determination of the ratio between the platinum and the gold resistances, but one measure- ment sufficed. For with A, instead of r‚ as in $ 7 we find: Din eee ae ae LR ri. Mi +1 W, where JI’, and WW, are the resistances of the circuits where the galvanometer turns and the resistances occur. RR; : If we know the resistance of the turns, — may be derived from PS pt one determination of a resistance in the boxes. The resistances of the turns can be derived from the two deter- minations of the ratio between the platinum and the gold resistances. Let A and 5 be the resistances of the turns round the galvano- meter, and 7, and 7, the resistances in the boxes, then they give with whence A and B may be derived and consequently W, and W, may be found. Imaccuracies occur, that is to say we neglect the resistance of the connecting wires between the commutator holes; but as the value of A and B was about 940 Ohms, an inaccuracy of 1 Ohm was allowed and this resistance certainly remained below this amount. Se LS lr For the calculation it is important to remark that — — — is equal WEEDE a, (I a a, ey Ve a to — and ——— is equal to —, while — and — may be derived at, 1— Bro; a: a, a, 8 . . : : t : . directly from observation. Besides — — is small, so that in the last 1+8 term of the equations an approximate value for A and B is sufficient. § 9. Survey of an observation. The way in which the observations were made will be seen best from an instance which at the same B. MEILINK. “On the measurement of very low temperatures. VIII. Com- inum wire. tance of gold wire with that of plat 1S f the res parison o RRM AA ache ree ree PATTI IIIT TET IOI I IITI IIE Fig. 3. S&S al ‚oh in Rx Ro Proceedings Royal Acad. Amsterdam. Vol. VII. (307 ) time will be developed as an instance for the calculation. It refers to a measurement with the gold wire in oxygen boiling under reduced pressure. The values of the following table (p. 308) were read direetly. In this measurement the temperature was kept constant by means of the thermoelement so that the determination of temperature need not be repeated. For the derivation of the results from the observations see table Il, which does not require further explanation. § 10. Results. The determinations of the zero yielded the following results. | Date! Resistance Method of the measurement gold wire. of resistance. April 23, 1902 31.5506 Wueatstonr’s bridge. May. 43, …, 31.556 5 i had ee y 31.565 Differential galvanometer. May 26, „ MO, WHEATSTONE’S bridge. Our chief object of the determination with the differential galvano- meter was to ascertain that the two methods gave the same results, so that the latter could also be employed for a determination of temperature. During this measurement we did not stir and so it is possible that the temperature has increased a little. In connection with the following results the agreement is sufficient. After the measurements at low temperatures, which are made between June 17 and July 12, the zero was redetermined in October and then a deviating value was found, viz. 51.045. I have searched in vain for the reason of this deviation. It does not probably lie in the measurement of resistance. A known resistance determined in the same way gave the true value. While | searched for possible causes the gold wire broke so that I have not attained any certainty ou this subject. [t may be that during the interval between the two determinations a short circuit has been formed between the two ends of the gold wire, in consequence of which the resistance is apparently so much diminished. A change in the gold wire itself would probably always have produced an increase of resistance. The values for the further determinations of the ratio between the platinum resistance and the gold resistance will be sufficiently clear from the following table (p. 310). The measurements show that this method to determine the ratio between the resistances is certainly a good one and preferable to that of measuring the resistance of each wire separately. It will ( 308 ) TABLE I. Comparison between Gold and Platinum Resistances in Oxygen boiling under Reduced Pressure. Observations. = | Resistance in = a3 jk 5 ES ol EE | 33 Ce ve 29 Sot Tinie: SS (|Sremens &| Hartmann | S8 | SS =e 2a 2e 5 2 oe ea ol & Braun. |‘2 | age | OS 2 ALSKE, é RAUN. 2 5 5 af al Temperature determination. 2h. 30! 500 140 27.00 | 33.00 = 136 23.00 Determination of the ratio. 27.50 0 1000-+-400-+- | || 31.50 30020410 | = 23.90 == || 31.50 +2 28.05 u 27.00 |2h. 45! | 28.30 1000 31 .1000| | 26.50 443. 100) = 27.70 | 26.60 | 25.40 +. 3 == 29.20 La 4) ned 1000 3 +1 . 1000 30.30 442. 100; = 28.30 | 30.30 | — = | | | 29.12 | oaker en CE, | 29.62 | Ort stooge nto 21.00 | 443. 100| || 38.20 (244. 10 Neem ee ee) | 50) 41. 3 = 21.15 Ket | 34.85 ate en i es ! AA Del TGOD | | == [44-3. 400; = | 26.70 241. 10} || | 31.65 | — 26.80 1000 | 33 “bh 1 . 1000 == IN | 29.40 4+2. 400 || | 97.50 = 29,25 ( 309 ) 6E IG = 1% ‘SMYO ZO'RG=“Y UOonepmojen aAnyvsodUloz 10f GULZSISE| | i SRB Sc ROMA eh 06 4627~ | eae | 006007 |] ago, ce6780 || £96VE'0 | i oe Jl grot |-ee-gt | 0°0— }108°9986 | 000F |__| Hog enn lln | | OF FELT | 087 — |} anr doon 0 a : = zr 2 Ae RER | | | | OFT | gog +007 +onor || | | 90” ILT Lon Hog Hoog || | PE GLRE 7 OG LV — | + oov-+ oor || 9 | zeere' 0 || YOGYE "0 He ii Jil grot | cae | e0:0— |] 99° F488 | 000T Spe ee ay | ee | 0¢ 0 + Ov + 007 4 007 x nite + 0001 +000 | | 9 L6S7 teen anr +- anr pr ae 005 + 007 | Ee tS Scene! 00° | +aror+oo0e || OOF | | EE re | | ED iar pet 006 007 =| por | veore-ollgoere:0 | 0% L988 :7 || eyo | eee | c0-0— ||a6:908e | 0007 | +0001 + 0008 _||_ | YooVs GHG 06-5001» 5 sk 6 5 ) DSG ie fae a a1 ze | | zt | | et Mas aoSe) Se Sea elas ete |e oel is + 00% + 0001 ie pone ee 5 | ga ‘Ud NH MOE Ne ee | Er one %8 SE ek ‘HSS || soyoueiq OMA || S 'KIUVH tk Dace NAVU XcWauvy || CSN 0) ae: ia 5 Les ay} UI JEE Send N SNANAIS || 103} | E aIULJSsISse IETS fs Gd soude mpoy SANA ne woroemed PSISOY dUL}SISAY IJ-nu mos) SUOT}JIAIION peice | ‘one oyr Jo UOIPL[NITKD ‘amnssorg poonpey 1opun Surplog UesAX() Ul SoOULISISoYy WAUW puv POL) WooArjoq uostreduog "TT ATA ( 310 ) Comparison between Gold and Platinum | TABLE III. | Resistances. B Ratio Ratio between fResistanae ee Reo the resistances of a TR Et gold gold wire and a Dat P old wire, |_Zesistance_| platinum wire a“ un 5 “| platinum {which at 0° C. have : resistance |the same resistance. 110.045 Simone 0.28672 April 23, 1902. (at 0° C.)| 31.556 0.28675 1 May 23 ot. DOS 0.28684 id. 34 1555 (0), 28674 May 26. 0). 295677 0.295736 June 17, 64.78 sree 0.29554 in ethylene. 0.29596 0.28674 ~ 9. 295564 : 0.295531 = 1.0307 64.85 0.295562 52.34 0.301954 0.301950 June 17, 0.301952 0.98674 in ethylene 0.301949 ow boiling under ee 0.301946 - = 1.0530 reduced pressure. 52.68 28.71 0.330287 June 23, 0.330314 0.330295 in oxygen. 0. 330285 Sosa = 0). 330204 0.28674 0.330256 s 0.330289 Se 28.74 2970 0.34490 0.34497 Wate = ple 0.34490 benen te fae 0. 34498 0.28674 sik 22 74 Bik | 0.34464? == 192029 0 34495 22.79 | 21.39 8 0.34933 June 23, in | 0 34934 a oxygen boilin | YR < ys 8 | | 0.34932 0.28674 under reduced 21 47 Bales — 4.2173 Dees ESL) be necessary, however, at least for low témperatures, to take great care that the temperature is kept constant during the time required for a measurement. In these measurements the determination of the temperature. was less accurate than seemed desirable with a. view to the accuracy of the determination of the ratio during the measure- ments on one day. In the results there is a striking difference between the gold and the platinum. Though the values found do not help us to fix the temperature function for gold for want of certainty about the zero to which they belong, yet they show that the curvature of the line whicb for gold represents that temperature function is much smaller than in the case of platinum and that the curve is bent more towards the absolute zero. Hence a gold wire would be more suited for extrapolation than a platinum wire, because here the deviations which we cannot but expect, are much smaller. Mathematics. — “A congruence of order two and class tivo formed by conics’. By Prof. J. pe Vrins. For a twofold infinite system of conics (congruence) order is called the number of conics through an arbitrary point, class the number of conics with an arbitrary right line for bisecant. The congruences of order one and class one arise from the pro- jective coordination of a net of planes to a net of quadries*). Under investigation were furthermore the congruences of order one and class two and those, the conics of which cut a fixed conic twice *). In this communication the characteristic numbers are deduced of the congruence determined by the tangent planes of a quadric Q? on the planes of a net {Q*| of quadrics to which they are project- ively conjugate. 2. To obtain this conjugation we project the points P of Q? out of a fixed point P, of Q? on a plane ®. A projectivity between the points P' of ® and the surfaces of [@Q*| furnishes then imme- diately a projectivity between [Q*] and the system [ar], of the tangent planes a of Q?. To a pencil (Q?) in [Q?] corresponds a range of points (/) in 1) D. Montesano, Su di un sistema lineare di coniche nello spazio, Atti di Torino, 1891—1892, t. XXVII, p. 660. *) M. Piert, Sopra aleune congruenze di coniche, Atti di Torino, 1892—1893, t. XXVIII, p. 135. 21 Proceedings Royal Acad. Amsterdam. Vol. VII. ( 312 ) ¢, thus a conic on (/, thus the system of the tangent planes 2 passing through a fixed point 7 Through T and a point X of the base-curve of (Q*) two planes a pass; wherefore X bears two conics of the congruence, which is thus of order two (P= 2). 3. To the tangent planes a through an arbitrary point T corre- spond the points ? of a conic not passing through P,, having thus for image a conic in ®. So to this system (2),, of index two, is conjugate a system ((*), possessing likewise index two, having two surfaces in common with each pencil (Q?). When considering the ranges of points determined by the projective systems (2), and (($°), on an arbitrary right line we find that they generate a surface 7° of degree six, which is the locus of the conics of the congruence the planes of which pass through a fixed point A. Hence we get WP. 4. Through two arbitrary points pass two tangent planes 2, hence the planes of two conics; so an arbitrary right line is bisecant of two conics, and the congruence is of class two (u*? = 2). The numbers P=2, wy =—6 and w?=2 satisfy the well known formula P= pv — 2 w’. Through a right line of (/ pass an infinite number of planes zr; the conics they bear form a cubic surface. As each ray through T meets two conics, T° has in T a double point. If A? is one of the conics on which 7 is situated, T° is touched in T by each bisecant of A? out of T. So T is a biplanar point. If T is one of the eight base points of the net { Q? | then T° has in Ta fourfold point; for on every ray through T hie but two points besides 7. 5. Let us take for Q? the paraboloid a 7 = z, then the substitution “«=a0, ¥ = 8.0, £ Ye furnishes first 0 —y-@P and then N fB ERE ke So the tangent planes a are represented by Bye + ayy—ape—y7 =. The above-indicated conjugation is arrived at by putting aA+BpB+yC=0, where A, B, C are quadratic functions of x, y, 2. We represent their coefficients by api, bri, ce and we write brietly dij = a a + Bb + Y “ik: ( 313 ) If a Q of the net is to be touched by the conjugate plane 2, then A A By | d.. da. aes de. ay ieee esd a, der. —ap esi St ee eer ee ee By er nt () must be satisfied. We find here a relation EP ef, 10, which is homogeneous and of degree 7 in a, 8, y. If we regard these parameters as homogeneous coordinates, this relation represents a curve of degree 7 possessing nodes in the points’ d (8=0, y=0) andr (@—— Oy == 0); 6. For the conics passing through point T(7,, y,,2,) we have the relation M, (a, 8, y) ==, By + y, ay — 2,08 —y? = 0. It is represented by a conic passing through A and 5. Besides 4 and B the auxiliar curves D* and J/ have ten points in common. So through T pass the planes of ten conics each degene- rated into two right lines (du = 10). That the points A and B must not be taken into consideration is shown as follows: For a= 0, y=0 we find B—=0 and y= 0:0, thus the pencil of planes around (LY; of these tangent planes of course only one is conjugate to B==0 and the conic determined by it does not form a pair of lines generally. . Out of the relation *) uv — 2yu + du + 4u’? Glistes, as ur 1b, du — 10 and u° — 2, nu 0. This could be foreseen, for the cones of [Q?] form a system ao!; the number of those cones touched by the homologous planes a is thus finite and all twofold symbols in which aj appears have there- fore the value zero. 7. The right line e=0, y=O is cut by the conics for which we have ~ a ‘ ~2 ») a is ape + y?=0 and d,, 27 + 2d,,z+d,,—9, 1) Compare my communication in these Proceedings, p. 264. 21* ( 314 ) thus N, (a, B, y) = 4,, y* — 2d,, aBy’? + d,, a? B — 0. The curve .V* representing this relation has evidently nodes in A and B. By connecting N* with M* and DD? we find anew ur =6 and farther dp = at. The pairs of lines of the congruence form a skew surface of degree 27. 8. To find the characteristic numbers containing the symbol @ we consider the pairs of points which the conics of the congruence have in common with the plane z= 1. They are indicated by Bye + ayy = a8 + 7°, d,,«?+2d,,ey+d,,y°+2(d,,+¢d,,)¢+2(d,,+¢,,y+(d,,+2d,,+d,,) = 0. So for the conics touching z= 1 i, thy dis + dy, By Ld. de, ee oe Ree | ds + d,, dj, Hd, Berle Od eds ay ae | ==", | By ay — ap— 7’ 0 This is a relation hoe, Be ri U which is represented by a curve #* having A and ZB for nodes. By combining F* and D*, JJ? and NM? we find successively do = 34, Ho = 8; vp Az. From this ensues that the skew surface of the pairs of lines has a double curve of degree 17 and that the comes touching a given plane (in particular thus the parabolae of the congruence) form a surface of degree 22. Out of the relations Sr: = dr + 4uv and 307 = 2de + 2u0 we finally find for the missing characteristic numbers Pp? = it: and Ae =e. So the conics cutting a fixed right line form a surface of order 17. ( 315 ) Physiology. — “On a new method of damping oscillatory deflections of a galvanometer”. By Prof. W. EINTHOvEN. (Communicated in the meeting of September 24, 1904). In a number of investigations, requiring the use of a galvano- meter or electrometer, it is desirable to damp the oscillatory deflect- ions shown by most of these instruments under many circumstances. Either mechanical damping is applied or electromagnetic damping or both are combined in order to obtain a stronger effect. In some instruments, e.g. the Deprez-p’ ARSONVAL galvanometer, in which the coil is movable in a stationary magnetic field, the electro- magnetic damping may without any special arrangement be so great that the deflections have lost their oscillatory character and have become quite dead-beat. The movements are thereby retarded. This retardation may be very considerable and so become troublesome, even to such an extent that the instrument becomes impracticable. Means of diminishing the damping are then applied, e.g. by increasing the resistance in the galvanometer. In order to apply electromagnetic damping in a needle-galvano- meter the rotating magnetic system is to a greater or less extent enveloped by a mass of pure copper in which during the motion of the needles damping vortex currents are raised. Mechanical damping is applied as liquid or air damping, thin plates of aluminium or mica or insect wings being often used. The method of damping to be described in this paper is entirely different from the methods just mentioned. It consists in inserting a condenser between the ends of the galvanometer wire as is indic- ated in fig. 1. In the figure FE represents a source of current by means of which an arbitrary potential difference can be esta- blished between P and P,. G is the galvanometer and C the condenser. The action of the condenser Fig. 1. is most easily understood by assuming the mass of the moving parts of the galvanometer to be zero and the eventual causes by which the motion is damped to tend to zero. If under these conditions the capacity of the condenser is zero, when a potential difference between / and /, is suddenly established, the galvanometer will also at once assume the corre- ( 316 ) sponding position of equilibrium. If on the other hand there is a certain capacity, the deflection will require some time. The way in which the image of the mirror or in the string- galvanometer the quartz thread then moves, is entirely determined by the way in which a condenser is charged or discharged. Calling a the deflection of the galvanometer at the time ¢ after the potential difference is established and A the permanent deflection, we have t oe A if é ie where e is the base of natural logarithms, ¢ the capacity of the condenser and 7’ a resistance of which it is easy to give a nearer definition. In the closed circuit containing the source and the galvanometer the external resistance be IW, the resistance of the galvanometer be IV;, then, if we neglect the resistance of the wires joining the condenser and the galvanometer, we have W, W. we en me e . Py . a e e e (1) Wi + W, The value v/e is the time constant of the deflection wie en 7 Expressing 7’ in Ohms and c in Farads, 7’ is given in seconds. When the deflection of the galvanometer is recorded on a uni- formly moving plane, a curve will be obtained which is the ex- pression of an exponential function and which agrees entirely with the wellknown normal or standardising curves of the capillary electrometer *). The constants of the curve, besides being determined by the rate of motion of the recording plane and the amplitude of the deflection, will depend only on the value of 7. By changing w’ and c we can regulate the value of 7 at will. This means that we are able to retard or damp the deflection of the galvanometer to any extent. The reasoning given is confirmed by the observations. As an example we reproduce three curves, figs. 1—38 of the plate, recorded by the string-galvanometer *). The connections are. schematically 1) See e. g. W. Enytuoven. Priiiger’s Arch. f. d. gesammte Physiol. Bd. 56, p. 528. 1894. And ,Onderzoekingen” Physiol. laborat. Leyden, 2nd series I. 2) See W. Ermtuoven, Ann. der Phys. 12. p. 1059. 1903 and 14. p. 182. 1904. Also in Kon. Akad. v. Wetensch. te Amsterdam, Report of the meeting of June 27, 1903 and March 30, 1904, bhi in fig. 2. Here # a battery, S a key, G the or. tee! galvanometer and C the condenser, A, B and R repre- senting resistances. The sensi- tiveness of the galvanometer has ETT been kept about equal in the three cases so that a deflection rie ok of 1 mm. corresponds to a current of 2107 Amp., the electromotive force / of the battery and the resistances A, B and A being so chosen that when the current is passed a permanent deflection of 20 mm. is obtained. The rate of motion of the recording plane is 500 mm. per second. Hence in the net of square millimetres on the plates') 1 mm. of absciss = 0.002 sec. and 1 mm. ordinate = 210-7 Amp. The circuit was automatically made and broken at S by an arrangement attached to the recording plane. For Fk a carbon resistance was taken with large resistance and B was small compared with R. WW, could be put equal to 7? without an appreciable error. In figs. 1 and 2 of the plate WW, was 1.11 megohm, whereas JI’, in fig. 3 amounted to 117000 ohms. The resistance of the galvanometer J’; was 8600 ohms. In fig. 1 of the plate the capacity of the condenser is 0. The string is seen to make oscillatory movements with a period of about dem OI __ These movements are damped by inserting a certain capacity in the condenser. In fig. 2 of the plate that capacity is 0.94 microfarad, in fig. 3 of the plate 0.2 microfarad. Caleulating the value of w' from JW; and IV, by formula (1) and then the time constant Z’= w'c, the time constant of fig. 2 is found to be 8.05, and that of fig. 3 1.66 and it is clear that the amount of retardation or of damping is determined by the value of the time constant. For clearness’ sake we started in the above reasoning from the simplest case and assumed that the mass m of the string and the forces which independently of the condenser damp its motion and which we will collectively indicate by 7, may be neglected. This hypothetical case will the more closely agree with reality, the larger, other conditions being equal, 7’ is taken. Hence in this respect fig. 2 1) On the way of recording and the net of square millimetres see Annalen der Phys. hee 2) 1 ¢ = 0.001 sec. ( 318 ) of the plate answers better the conditions required than fig. 3, but the great practical importance of the method is exactly that it is possible to. damp the oscillations and at the same time to retard the deflection as little as possible. When measurements are made one will always try to ehoose 7’ such that exactly the limit between oscillatory and aperiodical motion is attained. In this case 7’ is rela- tively small and m and 7+ may no longer be neglected. The question now arises -how for known values of m and r the value of 7 must be calculated in order to obtain the limiting case mentioned. In passing it be remembered that with the capillary electrometer the damping of the motion of the mercury meniscus is also composed of mechanical friction and of retardation by capacity.) And from the combination of these two results a motion which can be expressed by a simple exponential function either quite accurately or with only small deviations. The resistance of air or liquid damping as well as electromagnetic damping influence the motion of a body having mass, in exactly the same way as conductive resistance in- fluences the motion of electricity when a condenser is charged or discharged. A simple reasoning will show, however, that adding a condenser to the galvanometer has not always an influence on the movements of the string of the same nature as an increase of the damping forces which we called +. For the addition of the condenser has the effect of a temporary change of the active force. And the way in which the force is increased or decreased from moment to moment is not determined by the motion of the string, as the mechanical and electromagnetic damping, but by the product of the conductive resistance and the capacity 0’ e=. When applying the condenser method, the character of the motion of the string near the limiting case of aperiodicity can only be represented by a more or less complicated formula. I have therefore for this limiting case preferred direct experimental determination of the value of 7’ to calculation. Some curves have been reproduced which exemplify the motion ') Some investigators have been of opinion that the motion in the capillary electrometer is dependent on the charge of the mercury meniscus only. But in reality damping by mechanical friction is much more active here. See Prrücer’s Arch. f. d. ges. Physiol. Bd. 79, p. L. 1900; and ,Onderzoekiugen”’ Physiol. laborat. Leyden, 2nd series, 4. of the string in the limiting case in question’). Figs 4, 5 and 6 of the plate were taken with the same string as the former figures. For the connections we refer to figure 2 in the text. The deviation is now 30 mm. Again 1 mm. of absciss is 0.002 see. and 1 mm. of ordinate = 2X 10-7 Amp. R = 1300 Ohms. Br we W; == 8600 » , from which we calculate W.= 1327 and w' = 1148 Ohms. In fig, 4 the capacity of the condenser == 0, hence 7 = 0. Biase Oe yy en rs = O6p/,, 5, Foie 2) LE) 6 LE) LE) 9 LE) es — OT PE jl 0,80 0. One sees that the oscillatory motion, the period of which is about 2,7 6, is damped by the application of the condenser method and that the time constants 7’ of 0.69 and 0,80 6, obtained by means of capacities of 0.6 and 0.7 microfarad, are required in order to reach the desired limit of aperiodicity. In fig. 5, where a capacity of 0,6 uf. is used, the limit has not yet fully been reached, in fig. 6 the limiting value has already been passed with a capacity of 0.7 uf. The two last-mentioned plates show that the motion of the string in the neighbourhood of this limit is not very simple. In the small oscil- lation which has remained in fig. 5 the string, after having deflected through 30 mm., passes the new position of equilibrium by 0.5 mm. and then returns to a point which is still 0.8 mm. lower than the position of equilibrium, The ratio of the values of these deflections does not agree with the laws which damped motions generally obey. Moreover the first turning-point is reached after 2 6, the second after 1 6, whereas with damped vibrations, such as generally occur, these times are equal. In fig. 6 the string comes to rest after about 0.002 sec. at a distance of 0.3 mm. from the new position of equilibrium and reaches its equilibrium after a small movement in the opposite direction. If in the measurement of a current one is contented with an accuracy of 2°/, the result is known in about 1,5 o. Another example is found in figs. 7 and 8 of the plate. These photograms were taken in the same way as those immediately 1) The process by which the photograms of the plate have been reproduced does not reveal the minor details of the curves. [ shall be pleased to send direct photographic copies of the original negatives to those who are interested in them, ( 320 ) preceding, but the string is lighter here, has a greater conductive resistance and is slightly more stretched. 1.mm. absciss = 0,002 sec., 1 mm. ordinate = 3 X 10-* Amp. W; = 17800, W, = 20000, hence aw’ = 9420 Ohms. In fig. 7 the capacity is 0; in fig. 8 it is 0,05 uf, hence 7’ = 0,47 o. In this latter photogram the string shows a turning-point after about 11 5 exactly on the new position of equilibrium. It moves back through 0.9 mm. and then reaches its equilibrium again and finally. If in the measurement of a current one is contented with an accuracy of 3°/,, the result is obtained in 0.86, If an accuracy of 0.3°/, is wanted, the result is only obtained in 2.2 6. These examples may suffice to see what can be expected of the method. It is obvious that when seeking the exact value of 7’ for reaching the limit, we were led by theoretical considerations although we could not use a rigorous formula. One of these considerations was that for a given string and constant resistances, the capacity required for the limit must be the smaller the more strongly the string is stretched. For with greater tension of the string the period t of its oscillations becomes smaller and we may expect the wanted value of the time constant 7’ to change in the same sense as the period ¢. This consideration leads to some paradoxically sounding predictions. So stretched string that has been made dead-beat by applying the con- , for example, it is to be expected that the motion of a strongly denser method, will become oscillatory again as soon as the tension is diminished and thereby the motion is retarded. Such an expectation seems at variance with the experience gained with other galvano- meters, we might say, gained without exception with all instruments in which vibratory motions are observed. The result, which was expected with some anxiety, completely confirmed the prediction. A quartz-thread of such a tension that a permanent deflection of 1 mm. corresponded to a current of 2 X 107 Amp., showed, when a current was suddenly passed or interrupted, (see figure 2 in the text) a number of oscillations. By inserting a capacity ¢ = 0.135 uf the motion was damped to such an extent that the limit of aperiodicity was reached. Next the tension of the string was exactly 4 times relaxed so that a deflection of 1 mm. was caused by 5 & 10-8 Amp. The oscillations then re-appeared, and could not be checked again until the capacity was increased to 0.40 uf. With a 4 times smaller tension, i.e. with a 4 times greater sensitiveness the capacity and at the same time the value of 7’ had ( 321 ) to be increased 2.96 times in order to reach the limit of aperiodicity. Observations with other quartz-threads, the tension of which was varied, always gave corresponding results: with strong tension a small value, with a more feebly stretched string a larger value of we is required in order to check oscillations. If w' is kept unchanged, one has an easy means of accurately regulating the desired degree of damping in a commercial condenser, in which capacities are shunted in by means of plugs in the same way as the resistances of a resistance-box. And it is remarkable that less of the means of damping is sufficient as the oscillations pass the zero-point farther and last longer and consequently the need of damping is greater. The phenomenon that, leaving the other circum- stances unchanged, diminution of the tension only, i.e. — with the same deflection —, diminution of the moving force, changes an aperiodic motion into an oscillatory one, stands quite isolated and has, as far as is known to me, no mechanical or electrical analogon, no more in scientific instruments than in industry. We shall now give some results of measurements which although they cannot compensate the lack of a simple formula, may yet be helpful to form an idea of the method in practical work. 1. When the damping influences already existing are increased, for example when the eiectromagnetic damping is strengthened by diminishing the resistance in the galvanometer circuit, a smaller value of 7 will suffice in order to reach the limit of aperiodicity when the quartz-thread is the same and the tension is not changed. 2. If the change in the electromagnetic damping which is caused by varying the value of IW, is taken into account, it makes no difference how the single factors iw’ and c are chosen. If only their product wie = 7’ retains the same value, also the damping influence will remain the same. This latter is only determined by the product 7’ 3. If the motion of the quartz-thread is oscillatory it will be observed when the condenser method is applied, beginning with small values of 7’ and gradually rising until the limit of aperiodicity is reached, that increasing 7’ does not always cause a regular increase of the damping. Especially with feeble tension of the quartz-thread, when only a few small oscillations are normally produced, one sees an irregularity appear. The addition of a very small capacity can then even slightly enlarge the existing vibrations. When such a value of 7’ has once been taken that the limit of aperiodicity is reached, 7’ has only little to be raised in order to obtain a regularly shaped curve. With further raising of 7’ the motion ( 322: ) is more and more retarded, the regular form of the curve being retained. 4. That we may form some opinion about the value of the time constant 7’ that is required in various circumstances in order to reach the limit of aperiodicity, we give the following table, containing the results of measurements, part of which have already been men- tioned above. | | | | 3 | , A C | Twe t k dns Me ig | a ‘in thousandths|in thousandths . in micro- AED af | Damping- in Ohms. in Ohms. | in Ohms. | farads. | ee ay | LEET eatin: 8600 | 4117000 | 8000 | 0.40 3.2 | 7.7 7.6 8600 | 147000 | 8000 | 0.435 | 1.08 2.7 34 | | | | | | 8600 | 4144405 | 850 | 042 | 1.02 | 2.64 3.1 8600 | 41397 | 4448 | 0.65 | 0.75 | 2.7 4.5 17800 20000 940 | 0.05 | 0.47 | 1.41 3.16 The first five columns of this table need no nearer explanation; they give the conductive resistances, the capacities and the values of the time constant 7. For the values of 7’ mentioned the limit of aperiodicity was just reached. The two last columns indicate how the string vibrates when the capacity of the condenser and together with it 7’ is zero. In the last column but one we find the period ¢ expressed in thousandths of a second, while the last column gives the damping ratio 4. The observations have been arranged according to the values of 7’. Finally some remarks may find a place here on the circumstances under which the condenser method will be useful in practice. For the present the applications will presumably be restricted to such measuring instruments as possess a great internal resistance and a short period of oscillation. A galvanometer for thermo-electric currents with a small internal resistance and a great period of oscillation would for damping by the condenser method require an enormous capacity. The mica or paper condensers, which admit of easy regulat- ion, would be out of the question here, since even the largest sizes of the trade would turn out to be still a hundred thousand times too small. So one would have to have recourse to another kind of condensers, e.g. electrolytic ones, and it would require a separate investigation how far these can indeed be rendered practicable for the purpose in view. EEn einen uae SSSR USS ees gape deld ' 8 BERENT 4 any 8 = Pd al Acad Absc. 1 m.M. gs Roy : eedin mie W. EINTHOVEN: “On a new method of damping the oscillatory deflections of a galvanometer.” Absc, 1 m.M. = 26, Ordin. 1 mm. = 210 Amp. Absc. 1 m.M. = 26, Ordin. 1 mm. = 2X 10 Amp. T=0.696 7 Absc. 1 mm. = 26, Ordin. 1 mm. = 3><10 Amp. 8 T=0.476 Proceedings Royal Acad. Amsterdam. Vol. VII. (323 ) The conditions of a short period of oscillation combined with a relatively high internal resistance are fulfilled by only one instrument besides the string galvanometer, as far as is known to me, namely by the oscillograph. Here the damping is effected by means of oil which is heated *). | The temperature of the oil determines its viscosity and the regulation of the degree of damping is consequently obtained in the oscillograph by regulating the temperature of the oil. It is doubtful whether the instrument would greatly gain in practical usefulness if the oil with the heating arrangement were done away with and replaced by a condenser. In the string galvanometer the condenser method will be success- fully applied in cases where it is desired to measure variations of current of very short duration. Taking a very short and strongly stretched quartz-thread, it will be possible to obtain deflections whose quickness leaves little to be desired. Without a condenser these would be useless for many purposes on account of the oscillations, whereas now they may become useful for a number of physical and electro- technical investigations by a judicious damping. In these cases the string galvanometer will for equal quickness of deflection appear to be a much more sensitive apparatus than the oscillograph. Also in a number of electrophysiological investigations we can avail ourselves of the condenser method, while the study of sounds will be particularly facilitated by it. IT hope to make a nearer com- munication on this subject in a following paper. Physics. — “ Dispersion bands in the spectra of JS Orionis and Nova Persev’. By Prof. W. H. Junius. When light, giving a continuous spectrum, passes through a selectively absorbing, non-homogeneous mass of gas, the spectrum of the transmitted light contains places which, according to circumstances, may contrast as bright or as dark regions with their surroundings *). Though resembling emission and absorption lines, these bands have a wholly different origin. They ave due to anomalous dispersion and, therefore, the name dispersion bands has been suggested for them *). 1) Also a mixture of two liquids is used, of which one has a great, the other a small viscosity. The mixture is so chosen that the desired viscosity is just obtained. 2) Proc. Roy. Acad. Amst. IH, p. 580 (1900). 3) Proce. Roy. Acad, Amst. VII, p. 134—140 (1904). ( 324 ) Dispersion bands always appear in the proximity of absorption lines, covering them more or less symmetrically; they show great variety in width and strength, and the distribution of the light in them may be irregular, so as to give the impression that one is witnessing cases of shifting or doubling or complicate reversal phe- nomena of widened absorption lines. All these cases can be produced almost at pleasure in the absorption spectrum of sodium vapour by merely varying the structure of the non-homogeneous medium through which the light is made to travel. In the spectrum of the various parts of the solar image dispersion bands play an important part’). We can scarcely doubt that they are also present in stellar spectra; for the light coming from the stars must, as a rule, have travelled through immense gaseous envelopes and suffered ray-curving and anomalous dispersion, just as well as the light from the Sun. Taking for granted that most of the visible stars are rotating gaseous bodies, with or without a solid core, we must suppose them to have a structure, deseribable by surfaces of discontinuity with waves and vortices, and resembling the peculiar structure of the Sun, by which it has proved possible to explain solar phenomena’). Consequently, the stars too give existence to “irregular fields of radiation” rotating with them. Our line of sight continually cuts other parts of the refracting mass; it may pass closely along sur- faces of discontinuity, now on the one, now on the other side of them; so the light reaching us must vary in strength and in composition. The variability of many stars is very likely to result from this cause; and from the same principle it necessarily follows that their spectral lines should be liable to every kind of change in place and in appearance. In many cases where the application of Doppier’s principle leads to very unsatisfactory conclusions, the dispersion bands afford a plain solution. Let us consider, for instance, the spectrum of d Orionis. In this spectrum rapid changes in the position of the lines had been observed by Desnanpres (1900), who concluded from them that J Orionis was a spectroscopic binary having a revolving period of 1.92 days. Some observations made by. J. HARTMANN®) did not agree with this period. Professor Hartmann, therefore, submitted the 1) Proc. Roy. Acad. VII, p. 140—147 (1904). 2) Proc. Roy. Acad. Amst. V, p. 162—171; 589—602; VI, p. 270—302 (1903). 3) J. Hartmann. Untersuchungen über das Spectrum und die Bahn von 3 Orionis. Sitzungsber. der Kön. Preuss. Akad. d. Wissenschaften, XIV, S. 527—542, Marz 1904. star to an extensive spectographie investigation in the winter months of 1901—2 and 1902—38, and, from the 42 plates obtained, drew the following conclusions. The spectrum contains chiefly the lines of hydrogen and helium; besides a few belonging to silicium, magnesium, calcium. The calcium line at 2 3934 (corresponding to A of the solar spectrum) is extraordinarily weak, but almost perfectly sharp; all the other lines (nineteen in number) are very diffuse and dim, often appear crooked and unsymmetrical, sometimes indeed double. While every prepossession of the observer was most strictly avoided during the measurements, it was found, that the centres of the diffuse lines really oscillate, the period being 5,7333 days; but, owing to the unsymmetrical appearance of many of the lines, no evidence could be obtained that the values of the displacements were in mutual agreement for all the lines on one and the same plate. From the average displacements HARTMANN calculated the ‘variable velocity in the line of sight”, and finally the elements of the orbit. An utterly surprising result, yielded by the measurements, was that the calcium line at 2 3934, does not share in the periodic displacements of the other lines, but shows a constant shift corresponding to a velocity in the line of sight of + 16 km. (reduced to the Sun). HARTMANN rejects the idea that this line should have originated in the Earth's atmosphere; also the assumption that it belongs to the second component of the binary system. He is thus led to the hy- pothesis that at some point in space in the line of sight between the Sun and d Orionis there is a cloud of calcium vapour which recedes with a velocity of 16 k.m. By examining the spectra of neighbouring stars no further information as to the existence of such a cloud was obtained. A quite similar phenomenon, however, had been exhibited by the spectrum of Nova Persei in 1901: the lines of hydrogen and other elements were enormously broadened and displaced and continually changing their appearance, but during all the time the two calcium lines at 23984 and 23969, as well as the D-lines, were observed as perfectly sharp absorption lines, yielding the constant velocity of + 7 km. Harrmann therefore assumes that also in the line of sight between the Sun and Nova Persei there exists a nebulous mass consisting, in this case, of calcium and sodium vapour, and moving from the Sun at the rate of 7 km. a second. It must be admitted that these hypothetical clouds do not form a satisfactory solution to the problem. ( 326 ) A much simpler explanation of the phenomena may be derived from our conception of the irregular fields of radiation caused by the stars. We need only suppose that the outer parts of d Orionis and of Nova Persei, like those of so many other stars, contain much hydrogen and helium, little caleium and sodium. The currents and vortices in the gaseous mass, which produce the irregularities of the field of the star’s radiation, bring about very broad dispersion bands in the vicinity of the lines of hydrogen, helium, ete. The darkest parts of these bands will be displaced when, by the star’s rotation, masses in which the density is variously distributed, pass our line of sight. The dispersion bands of calcium and sodium, on the other hand, are so narrow, that the varying position of their darkest parts cannot be distinguished from the fixed position of the corresponding absorption lines. The constant displacement of the latter indicates that d Orionis recedes from the Sun with a velocity of 16 km., Nova Persei of 7 km. a second. According to our opinion d Orionis, therefore, is no spectroscopic binary. In the spectra of a great many stars oscillations and duplications have only been observed with diffuse lines. In those cases too the displacements are, as usual, expressed in so many kilometers a second, because no other interpretation than motion in the line of sight is thought of. From the above considerations it follows, however, that the observed oscillations are very likely to be executed by dispersion bands and not by the absorption lines; then no sufficient ground remains for classing such stars among spectroscopic binaries and for calculating orbital elements. Several difficulties to which the conclusions derived from DorPLumr’s principle lead us, will then disappear at the same time. How, for instance, are we to realize the physical conditions of the orbital motion in such so-called binaries as « Orionis, 57 Cygni, @ Orionis and many others, all of which are involved in nebulous matter, but Whose motion in the line of sight is nevertheless — according to Frosr and Abams — subject to periodical variations of 70, 90, 140 km. a second, in spite of our physical notions concerning resistant media? When, on the other hand, the observed displacements of spectral lines, as well as the oscillations of the brightness of similar stars, are supposed not to result from motion in orbits, but from irregularities in their fields of radiation, there remains nothing astonishing in the fact that such variations often occur with stars involved in nebulosity. € 327°) In order to explain certain peculiarities in the spectra of Novae the principle of anomalous dispersion has already been applied by H. Esrrt'). A characteristic of those spectra, viz. the presence of double lines consisting of a bright and a dark component, the bright one being displaced towards the red, the dark one towards the violet, is very suggestively explained by this author in connection with the theory of Swrnicur. According to this theory the appearance of a Nova results from a dark or faintly luminous celestial body entering at a great velocity into a cosmic nebula. During this process the front part of the star’s surface will become excessively heated and luminous, and a dense gaseous atmosphere will be formed, in which, as EBerr shows, the incurvation of the rays must necessarily be such as to cause the dispersion bands appearing in the spectrum to be bright on the red-facing and dark on the violet-facing side of the absorption lines. Eprrt expresses the opinion that displacements and duplications of lines in the spectra of many variables of short period might be explained in a similar way, i.e. by admitting that the radiating power of such bodies is very unequal in different parts of their surface, and that they are surrounded by dense atmospheres. Their rotation will then cause us to see, as it were, the phenomena of the Novae periodically repeated. In certain cases this interpretation may undoubtedly account for the peculiarities observed in the spectra of variables; nevertheless we cannot generalize the idea without meeting with some serious difficulties. First, it is not easy to form a clear conception of the physical conditions prevailing in a star, the incandescent surface of which is supposed to contain permanently large regions radiating very much less than the rest. The Sun with its spots may certainly not be adduced as an analogous case. Moreover, there are plenty of instances that in the spectrum of a variable, bright bands appear at the violet side, dark bands at the red side of the absorption lines, i.e. just the reverse of the phenomenon presented by the Novae; and it happens that with one and the same star bright and dark dispersion bands change places in course of time with respect to the average position of the absorption lines. This occurs e.g. in the spectrum of Mira Ceti, as will appear when comparing the obser- vations made by Voerr and Wiisine in 1896 (Sitzungsber. der Berl. Akad. XVII) with those made by Campsenr in 1898 (Astroph. Journ. IX, p. 31) and by Srespins in 1903 (Astroph. Journ. XVIII, p. 341); 1) H. Expert, Ueber die Spektren der neuen Sterne. Astron. Nachr. Nr. 3917. Bd. 164, p. 65, 1903. 22 Proceediugs Royal Acad. Amsterdam. Voi. Vili. ( 328 ) also in the spectrum of # Orionis observed by Hueerns in 1894 and 1897 (An Atlas of representative Stellar Spectra, p. 140), ete. In those cases the explanation suggested by Esert would require the addition of special hypotheses. Our fundamental hypothesis that the structure of most stars is similar to that of the Sun (it being admitted, of course, that the stars may greatly differ as to the extent of their respective gaseous envelopes, the average steepness of the density gradients in them, their chemical composition, temperature, ete.) seems to admit of the interpretation of a greater variety of facts. It makes displacements of the dispersion bands towards the long and the short waves almost equally probable — if we leave the asymmetry in the form of the dispersion curves out of question and provisionally assume the directions of the axes of the stars to be distributed at random through space. The direction in which we see a star may be regarded as a steady line in space, allowance being made for aberration and parallax. If, now, the distribution of the matter constituting that celestial body remains nearly unchanged for a long time, then after each rotation of the star our line of sight will again pass through the same points of the “optical system”, and we shall observe an accurately perio- dical course in the stars brightness and in the appearance of its spectral lines. In most cases, however, currents and vortices will cause more or less considerable alterations to arise in the distribution of the density of the gaseous mass, and, consequently, in the com- position of the beam of light reaching the Earth at a given phase of the star’s rotary motion. Thus the strictly periodical succession of phenomena is open to any degree of disturbance. The very irregular and sometimes rapid changes in the brightness of objects like o Ceti, SS Cveni, u Cephei, ete. are much more intelligible from this point of view, than from interpretations based on the assumption of violent eruptions, large spots, or eclipses caused by dark companions. And it is so difficult to make a sharp distinction between variables of long period and Novae, that we should not resent the idea of com- paring even the appearance of a new star to the sudden gleam of a revolving coast-light when the optical system, giving to the beam — a considerable decrease in divergence, passes our line of sight. Chemistry. — Prof. C. A. Lopry pe Bruyn presents a paper of J. Orm Jr: “The transformation of the phenylpotassium sulphate into p-phenolsulphonate of potassium’. (Communicated in the meeting of June 25, 1904). (This paper will not be published in these Proceedings). ( 329 ) Chemistry. — Prof. C. A. Losry pr Bruyn presents a paper of J. F. Suyver: “The intramolecular transformation in the stereoisomeric a- and 8-trithioacet and a- and B-trithiobenzal- dehydes”. (N°. 11 and 12 on intramolecular rearrangements). (Communicated in the meeiing of June 25, 1904). (This paper will not be published in these Proceedings). Chemistry. — Prof. C. A. LoBry pr Bruyn presents a paper of J.W. Dito: “The viscosity of the system hydrazine and water”. (Communicated in the meeting of June 25, 1044). (This paper will not be published in these Proceedings). Chemistry. — Prof. J. M. van BEMMELEN read a paper: “On the composition of the silicates in the soil which have been formed from the disintegration of the minerals in the rocks.” (This paper will not be published in these Proceedings). Bik AFT, DM: p. 238, 1. 5 from the bottom, for “increases” read “deereases”. (November 23, 1904). Lai ven KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN TE AMSTERDAM, PROCEEDINGS OF THE MEETING of Saturday November 26, 1904. nne) (NC es (Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige Afdeeling van Zaterdag 26 November 1904, Dl. XIII). EE NME SS. Tu. ZieneN: “On the development of the brain in Tarsius spectrum”. (Communicated by Prof. A. A. W. Husrecur), p. 331. P. H. Scnourn: “On the equation determining the angles of two polydimensional spaces”, p. 840. J. CARprNaAr: “The locus of the principal axes of a pencil of quadratic surfaces”, p. 341. A. SOMMERFELD: “Simplified deduction of the field and the forces of an electron, moving in any given way”. (Communicated by Prof. H. A. Lorenrz), p. 346. C. Easron: “Oscillations of the solar activity and the climate”. (Communicated by Dr. C. H. Winxp), p. 368. (With cne plate). W. Karrryn: “The values of some definite integrals connected with Bresser functions”, p. 375. I. KAMERLING Onnes and C. ZAKRZEWSKI: “The validity of the law of corresponding states for mixtures of methyl chloride and carbon dioxide” (Continued), p. 377. Corrigenda et addenda. p. 382. The following papers were read: Zoology. — “On the development of the bram in Tarsius spectrum.” By Prof. Tu. Zinuen of Berlin (Communicated by Prof. A. A. W. Hvsrecut). (Communicated in the Meeting of September 24, 1904). Owing to the kindness of Prof. Husrecut seven series of embryos of Tarsius spectrum were at my disposal, among them a sagittal series. With regard to the central nervous system of the adult animal I refer to two short papers published by myself, Anat. Anz. Bd. 22, N*. 24, p. 505 seq. and Mon. Sehr. f. Psychiatrie u. Neurol. Bd. 14, p. O4 seq. Proceedings Royal Acad. Amsterdam. Vol. VII. ( 332 ) The first stages of development are only known to me from Husrecut’s paper. In the youngest embryo the segregation of the two hemispheres has only just commenced. The next youngest embryo shows the hemispheres already developed, namely at the lower posterior periphery of the anterior vesicle. They are separated from this latter by a sulcus hemisphaericus which anteriorly forms a pretty deep and sharp groove but occipitally terminates in a shallow groove. The segmentation of the posterior brain (5 segments) is clearly shown especially by the youngest embryo. The following stages are very similar to those of other mammalian orders. In frontal sections through the hind-brain the uncommon depth of the sulcus limitans in the distal parts is especially striking. Frontally it soon becomes smooth. A sulcus intermedius (Groenberg) is indicated, The inner and outer labial grooves (“Lippenfurchen”) are present. The cerebellum consists of two symmetrical lamellae, one to the right and one to the left, joined by a thin and narrow medial part. On the outer surface of each lamella a broad medial longitudinal groove in the immediate proximity of the median part and a narrower but relatively deeper lateral groove are to be seen. Corresponding to these two grooves we find on the ventricular surface of each lamella two longitudinal ridges and three grooves (an unpaired sulcus medianus dorsalis, a sulcus medialis dorsalis and a sulcus lateralis dorsalis on each side). The roof of the mesencephalon is rather pointed and edged like a keel. The pharyngeal part of the hypophysis shows an almost compact appendix, extending backwards and downwards. Also two lateral continuations in a backward and downward direction of the ventricle of the mid-brain deserve notice. The chorioid fissure is already developed and shows some bulgings. The suleus hemisphaericus has also become much more marked occipitally. The sickle-fold (“Sichelfalte”) forms a sharp but shallow groove and is enclosed by the bifurcating ventricle of the fore-brain. The sulcus hemisphaericus lies at the right and left in close proximity of it. The Ammon fold (Hippocampal furrow) is still entirely absent. At a stage which for the rest has only little advanced, the shape of the fore-brain has already materially progressed in development. The sickle-fold is a deep groove. In its wall the Ammon fold is noticed to which corre- sponds on the surface of the ventricle a distinet Ammon ridge. In an occipito-parietal direction the siekle-fold reaches as far as the anterior limit of the mid-brain, basally it finally terminates smoothly in the lamina terminalis. The suleus Monroi is very sharply marked. On one side it terminates smoothly in the neighbourhood of the stalk-fold of the optic vesicle and on the other side in the neigh- ( 333 ) bourhood of the floor fold of the primary foramen Monroi (not in the foramen itself). The above-mentioned appendices of the lumen of the midbrain have already become rudimentary. The characteristic enclosure of the cerebral part of the hypophysis by the pharyngeal part is met with here in a similar way as in other orders of mammals. The lamella of the cerebellum has grown thicker. Of the longitudinal grooves, it is only the suleus dorsalis medialis and the suleus medianus dorsalis that are well marked on the inner surface. On the outer surface the ridges and grooves have been almost entirely flattened out. The posterior longitudinal bundle, the spinal root of the trigeminus and the lower olive form already distinct prominen- ees. The chorioid plexus of the fourth ventricle has already invagi- nated itself considerably. The differences in level of the fossa rhomboidea have already a little more flattened out. The sulcus intermedius is lacking, the sulcus limitans is distinet. The labial erooves have become flatter. The next-following changements may be briefly summarised as follows : a. The hemispheres show a deep groove corresponding to the thalam- encephalon, vallis diencephalica. The medial wall of each hemisphere shows on horizontal section three ridges projecting towards the lumen of the ventricle, which we will denote by R, S and 7 in their order from before backwards. Between S and 7’ we find, following up the series in a basal direction, a great diminution in the thickness of the ventricular wall (part d). In this thinner part and much nearer to S than to 7 the formation of the chorioid fissure occurs and the invagination of the plexus chorioideus ventriculi lateralis. S and 2 coalesce more and more. Meanwhile from the lower posterior part of the wall of the hemispherical ventricle the broad ridge of the caudate nucleus arises. The lateral ventricular wall shows only a very slight thickening, resp. elevation .V in its posterior part, which at higher (i.e. more parietally situated) levels, together with the caudate nucleus marks a narrow slit and coalesces with the caudate nucleus at lower levels. Between the caudate nucleus and the ridge 7’ there is a fold, which may be denoted by r. A very shallow prominence P? is also shown by the lateral wall in its most anterior part. The further the series is followed in a basal direction the more conspicuously a short anterior portion is) marked off on the medial hemispherical surface of the vallis diencephaliea, Whieh portion is not contiguous with the thalamencephalon, but is separated from the corresponding part of the other hemisphere by 23% ( 334 ) the primitive sickle fold only. Within reach of this portion we find now also a slight ridge projecting into the ventricle, which we shall here designate by @ for briefness’ sake. To all these just mentioned prominences of the interior wall-surface correspond only very slight grooves of the exterior surface or none at all. Only the ridge S corresponds in future pretty accurately to a shallow groove S', which must be interpreted as the fissura hippocampi. This groove belongs to that part of S which is nearest the chorioid fissure and finally has an almost hook-shaped bend near the neighbouring lip of the chorioid fissure. The more in the following sections |S is arched, the more also the fissura hippocampi is deepened, while at the same time all other ridges are levelled. Only the caudate nucleus remains entirely unchanged. On its surface a very slight groove is tempo- rarily seen. The grooves rp and t become gradually obliterated for the greater part, so that the caudate nucleus coalesces with P and T. The thinner part d of the medial wall does not coalesce with the caudate nucleus. So the bottom of the groove t finally corresponds exactly to the border of d and 7. When we progress still further ina basal direction, the first connection between the thalamencephalon and the hemispheres appears immediately below the bottom of the groove t, ie. in the former region 7’ and rapidly increases first in an occipito-basal direction. At the same time the lamella d now seems to originate in the bend between the caudate nucleus and the optic thalamus. In the following sections we find «/ more and more connected with the lateral wall of the thalamencephalon, i.e. with the optic thalamus itself. The insertion of the lamella @ seems to shift more and more towards the roof of the thalamencephalon and the lamella itself seems to become shorter. It is one of the most difficult questions of cerebral development whether this shifting and shortening of the lamella / and also the coalescence of the caudate nucleus with 7 and the lateral wall of the thalamencephalon must be interpreted as a secondary coalescence of originally separated parts. My histological investigations of Tarsius do not allow me to give a definite answer to this question. Still more basal sections show the disappearance of the chorioid fissure. Since d has in the meantime also disappeared, S passes immediately into the epithelial root of the thalamencephalon or primary fore-brain. The groove-shaped longitudinal depression on the surface of the corpus striatum becomes a little more distinet. The fissura hippocampi resp. the fissura prima is pointed so that in the cross-sections the well-known picture appears, ( 335 ) resembling a lance point. Also the roof of the median-mantle-slit which was at first formed by the folded roof of the thalamen- cephalon, now becomes sharply pointed. The fissura hippocampi or rather its lower lip marks pretty sharply, even after the separation of the hemispheres from each other, the point reached by the for- mation of the pallium. It is also very remarkable that S and &? no longer exactly correspond to each other topographically, but that S’ becomes situated somewhat ventrally of S. Below S the medial wall of the frontal brain shows a second feeble prominence U. U and the caudate nucleus terminate in the medial respectively lateral wall of the ventriculus lobi olfactorii and in doing so grow smoother. After the disappearance of the fissura hippocampi the medial wall first remains quite undivided in many sections, but then (almost exactly in the corresponding place) the medial terminal part of the rhinal lateral fissure appears as a shallow groove. 8. The eavity of the thalamencephalon shows in its posterior parts two grooves, an upper one #, which is continued in the lateral groove of the midbrain ventricle (aqueduct) and frontally very soon smoothes out, and a lower one 2, which can be traced as far as the region of the foramen Monroi. The latter one is accompanied during the anterior part of its course by a parallel groove u of a slightly more basal situation. Further folds of the surface are temporarily observed within reach of the optie stalks and of the corpora mammillaria. I regret not to be able to decide whether 2 or u has to be interpreted as the sulcus Monroi; to me it seems more likely that w deserves this designation. The picture is materially completed by studying a sagittal series belonging to about the same stage of development (length of the embryo 11 mm). In a section situated somewhat laterally of the medial plane, we find as follows: The fossa mesodiencephalica is sharply marked. Before it and for a smaller part also in it, lies the long-stretched cross-section of the posterior commissure. A very shallow groove which T designate by I) corresponds approximately to the frontal plane of the epiphysis. There is no objection to designating with v. Kuprrer (ef. v. KUPFFER in Hertwie’s Vergl. Entw. geschichte p. 95) as synencephalon the roof part between the fossa mesodiencephalica and /. The fossa praediencephalica is not sharply marked. It might be sought at the place where the invagination of the plexus chorioideus ventriculi tertii occurs. It must be emphasised that it remains doubtful whether this spot corresponds to the velum transversum of lower vertebrates. The region above the invagination of the plexus is the posterior roof portion of the primary fore brain. Below the invagination of the plexus lies the lamina reuniens, the lower part of which cor- responds to the lamina terminalis of the adult animal. Before the praeoptie recess we find on the basal inner surface a shallow trans- verse prominence, which is laterally prolonged into the central mass of the striated body. This transverse ridge of which even in the median plane we can recognise indications, corresponds to the crus metarhinicum corporis striati of the human embryo, described by His *). Consequently it passes medially into the lamina terminalis. Only in somewhat more laterally situated sagittal planes we meet before the crus metarhinicum with a second transverse prominence, which sinks away into the olfactory lobe. I designate it as crus rhinicum corporis striati. It corresponds to the crus mesorhinicum of the human embryo. A crus epirhinicum is scarcely indicated. In my opinion it is only counter- feited by the fold of the lateral rhinal fissure. On the outer contour we find corresponding to the border of the crus metarhinicum and rhinicum a shallow groove, corresponding approximately to the posterior edge of the cappa olfactoria and also approximately to the anterior edge of the olfactory tubercle. It by no means corresponds exclusively, as His seems to assume, during its whole course to the erus rhinicum or mesorhinicum. I believe that its formation is essen- tially independent of the morphological condition of the striated corpus and has rather to be explained by thickening of the wall of the olfactory lobe by superposition of the olfactory ganglion (cappa olfactoria) on one hand and of the olfactory tubercle *) on the other. To this is added a sharp bend in the brain tube in a basal direction when it passes from the hemisphere to the olfactory lobe. Moreover we must bear in mind that the lumen of the ventricle, when passing from the hemisphere to the olfactory lobe, at first tapers very rapidly, but then again very slowly *). Especially at the base this behaviour is very conspicuous. Obviously this must result, quite independently of a thickening of the wall by the striated corpus, in a basal transverse groove. The designation “fissura meso- rhiniea” which His has given to this latter, does not seem to me appropriate under these conditions. I propose to speak of a vallis mesorhinica. A second transverse groove is found caudally of the 1) His, Die Extwickelung des menschlichen Gehirns während der ersten Monate. Leipzig, S. Hirzel, 1904. S. 61 (cf. also fig. 34, p. 56). 2) The two borders only coincide by chance and not accurately. 3) Die Entwickelung des menschl. Gehirns etc. Leipzig 1904, p. 54 and p. 60, vallis mesorhinica behind the origin of the olfactory tubercle and before the region of the praeoptic (— optic) recess. I propose to denote if as vallis praeoptica, whereas His seems to look upon it as a continuation of his stalk-fold (my suleus hemisphaericus) *). It also is brought about independently of the striated corpus by the bulging of the olfactory tubercle on one hand and of the region of the chiasma on the other. The niche in the ventricle corresponding to the olfactory tubercle is designated by His (at least in man) as the posterior olfactory brain in opposition to the olfactory lobe s. str, which he calls the anterior olfactory brain. To the posterior olfactory brain he reckons particularly also the substantia perforata anterior. Against the designation “posterior olfactory brain” I only object that it favours confusion with the lobus piriformis, i. e. with the posterior part of the rhineneephalon *). 1) Cf. on this point also His, die Formentwickelung der menschl. Vorderhirns, etc. Abh. d. math. phys. Cl. d. Kgl. Sächs. Ges. d. Wiss. Bd. 15, fig. 32, p. 725. 2) Concerning the nomenclature in this region I would make the following remarks. All cerebral parts thatlie basally of the fissura rhinalis Gectorhinalis=rhinalis lateralis of many authors) I denote as rhinencephalon. As lobus olfactorius @lobus olfactorius anterior of many authors) I designate in a purely topographical sense the anterior part of the rhinencephalon as far as it is quite separated from the lower surface ofthe pallium. For the posterior part of the rhinencephalon the designation ‘Tobujs pinileormis “might be reserved, but it is more advise able to give up this name entirely. Part of the lobus ol- factorius is covered with a microscopically sharply cha racterised formation, the so-called formatio bulbaris. biecrcaver i designate as cappa “olfactoria From. tee developmental point of view it corresponds to the gang- kon folfdctorium of Hrs. The. separation of the) lobus olfactorius and the pallium is exclusively brought about bythe fissura rhinalis (lateralis). Hence we must also tecard as a part of the fissura rhinalis. the separating groove of olfactory lobe and frontal lobe which is visible in the median plane, i.e. l assume that the fissura rhinalis incises at the front as far as the median plane. The cappa olfactoria is. marked off from the free sur- face, & 6:-firom the surface. of the olfactory. lobe that is moteovered by the formatio bulbaris, by a. limiting groove, the margo cappae olfactoriae.: Laterally it #s much more clearly marked than medially. In many ani- mals (hedgehog, Echidna) the cappa olfactoria covers almost the whole olfactory lobe. The olfactory tubercle is also marked off against its surroundings by a shallow ( 338 ) Finally we mention as self-evident that in exactly medial sections all the just mentioned borders, grooves and prominences are almost entirely lacking. Sagittal sections further prove concerning the mid-brain that at this stage it covers the isthmus to a relatively small extent, less e. g. than in a human embryo of 2 to 3 months. The cerebellum shows in sagittal sections still a horn-shaped form. Of the occipitally directed ridge, which is so characteristic for man, nothing is to be found vet. The oldest of the embryos at my disposal had been dissected into a somewhat slanting frontal series. Here also nothing can be detected of a crus epirhinicum corporis striati. Notable is the considerable thickening of the wall in the basal portion of the medial ventricular wall. It only becomes distinct after the lateral ventricle and the ventricle of the olfactory Jobe have already coalesced for some time’). The thickened lower portion and the not-thickened upper portion of the wall are separated on the inner surface of the ventricle by a very distinct groove which can be followed almost as far as the frontal plane of the terminal lamina. It has nothing to do with the marking off of the olfactory lobe, as it appears considerably later and also lies somewhat higher than the fissura rhinalis lateralis. In the same way it is entirely independent of the ammon fold, since it lies considerably lower than this latter. Microscopically it forms a pretty sharp basal limit for the pallium formation. Therefore | designate it as margo pallii medialis internus (v. also below). In the section where the third ventricle is visible for the first time, it appears as a paired structure ; between the two terminal points of the ventricle a median groove incises into the ventricular roof (cf. above). Special notice deserves the floor of the third ventricle. Its median groove forms a very sharp incision. The lateral parts of the floor rise, so to say, in three gradations. The most lateral prominence groove, the, margo tuberculi olfactorm: This groove alse is generally not so distinctly perceptible at the medial edge, on the other hand at the lateral and anterior edges it is well developed and hence has here often been desig- nated as fissura rhinalts medialis, s:-entorbimalis: Aw thee posterior edge, towards the substantia perforata anterior it is generally rather shallow. When the cappa olf. reaches far- backward, the’ anterior “mareo, tub, ol. Co ine es entirely or partly, with the margo cappae olf. ') In what follows the series is supposed to be examined from before backward. which at the same time is the broadest, proceeds from the erus rhinieum of the corpus striatum and forms in future the principal mass of this ganglion. The middle elevation corresponds to the crus metarhinieum of the corpus striatum. Ht disappears with the separation of the hemispheres. In the lamina terminalis it meets the homologous opposite prominence. It is interrupted by the anterior commissure. The most medial and smallest elevation only becomes visible before the lamina terminalis and is at first very flat; then it rises pretty steeply frontally but remains narrow. From the homologous opposite elevation it remains separated by the shallow median floor groove. Following the series farther in a frontal direction, the two hemis- pheres split finally within reach of the median groove of the floor and the most medial elevation coalesces before the foramen Monroi with the upper portion of the medial wall, corresponding pretty accurately to the margo pallit medialis internus. From this deseription we must conclude that also this most medial elevation can by no means be interpreted as the crus epirhinicum in Hrs’ sense. The optic thalamus projects in the following sections between the middle and lateral ridge just mentioned. More sharply developed than in preceding stages a longitudinal groove on the outer surface of the thalamencephalon (sulcus fastigialis thalami) is now visible dividing the pointed crest of the optic thalamus from the broad basal mass of this ganglion. It is situated somewhat higher than the above mentioned groove x, Which for the rest is now much less distinet. 2 and u are no longer clearly divided. Instead of them we find a broader groove, which doubtless must be designated as suleus Monroi. The hind brain presents no peculiarities. Reviewing the whole of the peculiarities in the development of the brain of Tarsius that have been noted in the preceding pages, a far-reaching agreement with the development of the brain of the primates is obvious. The essential differences are sufficiently explained by the rather pronounced macrosmatic character of the brain of Tarsius. It is much more difficult to determine the relations of the Tarsius brain in the descending direction. Carnivores and Ungulates are out of the question. The development of the brain of Chiroptera is unfortunately too little known as yet but certain analogies are certain. Very great is also the agreement ‘with the development of the brain of Rodents, only one must not consider the brain of the rabbit as the typical representative of the brain of Rodents, as is often done. As the rodent brain in its turn is not far. distant in its ( 340 ) development from the brain of the Insectivores, it is clear that the Tarsius brain shows unmistakable genetic relations with this latter also. A more detailed account with figures will be published in the Handbuch d. Entwicklungsgeschichte edited by Hrrtwic ~ Mathematics. — “On the equation determining the angles of two polydimensional spaces”. By Prof. P. H. Scrnourr. The problem which we wish to solve is the following : “In a space S, with 7” dimensions a rectangular system of coordi- nates O (X,X, ... Nn) has been taken and with respect to this system a space S, passing through O has been given by the equations Up biz Ui + Arita oee HL Api Op 3 Ce ee A eco MIP) supposing this space „5, to have with the space of coordinates OKK or Dut AAE point Y in common, the p angles a,,. . .@, are to be determined between these two p-dimensional C / SE spaces.” 3y means of geometry we should set to work as follows. Suppose in the given space JS, a spherical space having ( as centre and unity as radius and thus forming in S, the locus of the points at P distance unity from 0; if this spherical space projects itself on the space of coordinate 0 (XY, X, .. . Xp) as a quadratic space with the half axes a, a,,. . - Ap, we get A, == COS, A, =S COB Ay, - + - » A — COSA). In an almost equally simple way the tangents of the demanded angles are connected analytically with the central radii-vectores of an other quadratic space. If P is an arbitrary point of S, and Q its projection on the space of coordinate (AT Krt vp) the angle PO Q=a is also determined by the relation n—p : NATL ONRI MRS Api ©)? OP: — 00 = ela ee T Spi dp) () p SR ite ll It we consider in S, the points / the coordinates of which are bound to the condition n—p se = (aise + 2, #2 +... +} Gp Pps Re (Le il ( 341 ) containing only a, v,,...«, and thus expressing that the projection Q of P on OLX, X,...X,) remains in this latter space on the quadratic space represented by (1), then the relation holds good 1 CI ==: ——. OR If 4,,6,,...6, are the half axes of this new quadratie space, we shall find 1 td Ce 7° tq LA k EES tat), = D A 2 I Now (1) passes into the symbolic form (A, Wi = A; Ws = ees = A, # y)) es] by the substitutions np n—p Be a Nan pee a, A ee 5 AN df es AS il | so the well know secular equation dier Ged PA | Abs Ale —— ji Ay» | == i", At» Ao» App — À | furnishes by its roots A,, 2... 4p the coefficients of the equation of that quadratie space reduced on the axes. From the relations ensues immediately that the demanded equation is arrived at by replacing in the above mentioned determinant 2 by ty? a. Mathematics. — “The locus of the principal axes of a pencil of quadratic surfaces”. By Prot. J. CARDINAAT. 1. The envelope of the axes of a pencil of conics was investigated among others by M. Tresirscnmr'). He found that the axes of the above mentioned conics envelop a curve of class three touching the right line at infinity of the plane in two points conjugate to the directions of the axes of the two parabolae of the pencil with respect to the 1) Ueber Beziehungen zwischen Kegelschnittbüscheln und rationalen Curven dritter Classe, Sitzungsber. der Kaiserl. Akademie der Wissenschaften, Bnd, LXXXI, p, 1080, isotropic points 7 and J. So the curve is of order four, Le. rational. This result is mentioned, in the “Eneyklopädie der Mathematischen Wissenschaften” III, p. 101. However, if we consult in the same work the theory of the quadratic surfaces we find no evidence of an attempt to determine the locus of the principal axes of the sur- faces of a pencil. The present writer makes it his aim in the following to publish some investigations on this locus. 2. We presuppose a simpler special case of the pencil and we take a pencil of concentric quadratic cones, of which the locus of the principal axes is a cone the order of which can be determined. Let us suppose to this end the section of one of the cones with the plane at infinity; the conic formed in this way determines with the isotropic circle a common autopolar triangle and the vertices of that triangle determine the directions of the principal axes of the cone. From this follows: The principal axes of all the cones of the pencil cut the plane at infinity in the vertices of the common autopolar triangles of the conics situated in this plane and the isotropic circle. These triplets of points form the Jacobian curve of the net of conics determined by two of the conics and the isotropic circle. So the cone of the axes is a cone of order three cutting the plane at infinity in the just mentioned Jacobian curve. To realize the position of the principal axes of this cubie cone we choose a generatrix a,. If we assume a plane through the vertex normal to «, this will cut the cone according to three rays a, a, 0,: a, and a, are normal to each other, 4, belongs to an other trieder of axes, obtained by assuming through the vertex a plane normal to 4,; this plane passes through a, and cuts the cone moreover in the two principal axes 6, and 6, normal to each other. As a rule this cone will not have a nodal generatrix, so it will not be rational. 3. Suppose a pencil of quadratic surfaces be given. Out of a point O in space as vertex we construct the parallel cones of the asymptotic cones of the various surfaces; in this manner a pencil of cones is formed, with respect to which we can construct the cone of the axes. The trieders of axes of this cone are parallel to the trieders of axes of the surfaces of the pencil. Let further a skew cubic g¢, be constructed, which is the locus of the centres of the surfaces of the pencil: if then out of each centre a trieder is constructed parallel to the corresponding trieder of axes of the cone, the surface is formed which is the locus of the principal axes, From this ensues: ( 343 ) The locus of the principal axes of quadratic surfaces belonging to a pencil is a skew surface of which one of the directrix curves is a skew cubic gy, possessing a director cone; each point of the skew cubic is homologous to a trieder of rays of the cone. 4. The order of the surface can be determined by investigating by how many principal axes an arbitrary right line / is cut, or how many planes possessing a principal axis can be made to pass through /, whieh comes to the same thing. Let A be a point of g,, to which three points A’,, A’, A’, cor- respond on the Jacobian curve C, in the plane at infinity P, Let moreover P be a plane through /; then this cuts p‚ in three points A, B, C, to. which correspond again in P, the points A’, 4',...C’,, so to the plane / correspond through / nine planes P, PP If reversely we assume a point A’ on C, only one point A on g, corresponds to it. If we now make a plane / pass through /, it cuts C, in three points A’, B, C", to which correspond three points A, B, C; so to a plane P’ correspond three planes P. From this ensues : The two coaxial pencils of planes ? and /” have a (3,9)-corre- spondence. So the number of elements of coincidence amounts to 12. From this reasoning, however, we may not conclude that the order of the skew surface is to be 12; this number must be dimi- nished by the number of points common to p‚ and (,. The three points of intersection of g, and P, are namely situated on C,: if we call one of these points S, then S, coincides with S quite in- dependently of the position of the assumed right line /. So of the 12 planes of coincidence 3 pass through the points of intersection of gy, and C,; so 9 remains for the order of the skew surface. 5. A full investigation of this surface U, is a very extensive one; however, we can consider some properties and trace some particularities. From the plan of the problem ensues that from each point of g, three generatrices can be drawn meeting P, in the three corresponding points; so g, is a threefold curve of (,. The section of P, and QO, possesses some particularities which we shall look into. In the very first place lie on it the three centres re the present. Out of each of those points two principal axes can be S,, S, of the paraboloids of the pencil supposed to be real for drawn having therefore twelve common points of intersection. More- over each of these axes cuts C, in two more points, which can thus be regarded as double points. One of these points belongs however to a triplet of points corresponding to a point of intersection of gy, and C,; so it can be regarded as a point of contact of the plane ( 344 ) P, and O0, If we combine these results, we arrive at the following theorem: The section of O, and /, is a degenerated curve of order nine broken up into a plane cubic and six right lines. On this section are situated twelve nodes, points of intersection of the principal axes two by two; moreover six nodes are situated on it, formed each time by one of the points of intersection of a principal axis with (,, and six points of contact, which are the remaining points of intersection. So P?, is a sixfold tangent plane of Q,. So we come to the conclusion that ©, possesses besides the three- fold curve g, still a nodal curve of which for the present we cannot make out how it is composed, but of which the total order is 18. The number of points of intersection of this curve with one of the generatrices of O, can be determined. Let « be one of the right lines connecting a point A of g, with one of the corresponding points A’, on C. An arbitrary plane Q through « cuts g, in two more points B and C to which correspond on C, two triplets of points 3', B',, B', and C’,, C’,, C',. In like manner a plane Q’ through a cuts the curve C, in two more points to which correspond two points on g,; so there exists between the pencils of planes Q and Q a (6,2)-correspondence and the number of planes of coin- cidence amounts to 8. So all together @ is cut by 8 principal axes. As in the general case this number must be diminished by 3, for now too the three points of intersection of g, and Cy must be taken into account; so « is cut by five principal axes. Each gene- ratrix of QO, has thus five points in common with the nodal curve. From the preceding is apparent that the general section of the surface possesses 18 nodes and 3 triple points; if we have in mind that the latter are equivalent to 9 nodes we see that the general section is not rational, as a curve of order nine can have 28 nodes and the curve under investigation possesses only 27 nodes. 6. We shall consider a single case, where the surface U, is simplified. We have already noticed that the cone of axes is of order three without nodal generatrix; there would be one if the net of conics possessed in /, a point, common to all conics. As however to this pencil belongs the isotropic circle this case is excluded; it may however happen that the cone of axes breaks up into a qua- dratie cone and a plane, or into three planes. 7. We choose an example of the first case. When the cone of axes breaks up into a quadratic cone and a plane, then the Jacobian curve in 2, must degenerate into a right line (/ and a conic C,. This happens : ( 345 ) a. When the conies of the net pass through two fixed points. 4. When the net possesses a double right line. We restrict ourselves in this communication to the first of these cases; then the base curve of the pencil of surfaces is circular. It is in the first place necessary now that the cone is degenerated into two parts to consider the distribution of the axes on cone and plane. If the base curve of the pencil of surfaces is circular, there is a system of parallel planes so that each plane is cut according to a pencil of circles. Of each surface of the pencil one principal axis runs parallel to these planes. From this ensues: When in consequence of the existence of a circular base curve the cone of axes degenerates into a quadratic cone and a plane, then of the three points A’,, A’,, A',, homologous to a point A on p‚ one lies on the right line C, in P. and two on the conic C,. So the skew surface QO, degenerates into two other skew surfaces intersecting each other in their common directrix g,. For one skew surface p‚ is a nodal curve, for the other it is single. This already suggests that the former of the two skew surfaces is of order six, the latter of order three. This can be reasoned more minutely in the following manner: Let / be once more a right line; a plane P through / has three points A, B, Cin common with p‚ to which six points A’,, A’,...C',, 'C', on C, correspond; so six planes P?’ correspond to P; if reversely we make a plane /” to lie through /, it cuts C, in two points to which on g, two points correspond, so that between the planes Pand P” a (2, 6)-correspondence exists. However gy, has a point in common with C, as C) contains the point of contact of a hyperbolic para- boloid of the pencil with 7. ; so there remain for gy, two points of contact with C and the order 8, which would arise on account of the (2, 6)-correspondence, must be diminished by 2; so we get a skew surface (, of order six. The second skew surface is of order three. In the general case the section of PP, and Q, consisted, besides of C,, of three pairs of right lines, to be called a,a,, 6,0,, c,c,. If O, degenerates in the manner described above these right lines will also be distributed themselves on QO, and Q,. Let A’ again be the point where g, cuts the right line C,, thus the point of contact of a hyperbolic paraboloid of the pencil; through A’ pass the two prin- cipal axes ‚a, and these belong to O,, whilst the principal axis not lying in P, through A’ belongs to O,. To QO, belongs thus one principal axis of each of the pairs 6,6, and c,c,, so P, is a double tangent plane of QO, and the section of OU, and 7, consists of the conic C,, the pair of axes a,a, and the principal axes 5, and ¢,. Of a and «, the point of intersection «‚«, is the node in the curve 1 2 ( 346 ) of intersection of P. and @,, one of each of the points of inter- section of a, and a, with C, is point of contact; so on a, as well as on «a, another node is situated. Of each of the points of inter- section of 6, and c‚ with C » one is point of contact, the other is also point of intersection of g, and (. So the points of intersection h,, besides the point of intersection counted already a,a, belonging to g,. So the entire number of the nodes of the section of O, and P. not Cs Aj, a4, mutually are left as nodes; these are five in number lying on g, amounts to 7. From this ensues that , has besides g,, another double curve of order seven. If we make a plane to pass through a generatrix Y, and if we investigate how many right lines are situated in it, we shall find the number to be 3 corresponding to former results. The nodal curve of order seven is thus intersected three times by the right lines of O,. 8. The closer investigation of the surface 0, as well as that of Y, and the other possible forms appearing by variously assuming the pencil of surfaces, gives rise to very extensive considerations, which are not to be included in this communication, as for the present its aim was but to show the general properties of the discussed locus. Physics. = “Sunplified Deduction of the Field and the Forces of an Llectron, moving in any given way.” By Prof. A. SOMMERFELD. (Communicated by Prof. H. A. Lorenrz). § 1. Summary. In the “Göttinger Nachrichten” *) | communicated a general repre- sentation of the field of an electron, moving in any given way, which seems to be simpler than the formulae, hitherto known, which are based on the work of H. A. Lorentz. This is the difference: My formulae express the potentials by a simple integral, extending over the past time and containing only the varying distances of the point in question from the centre of the electron, supposed to be spherical, whereas the formulae hitherto known are double or triple integrals, extending over the space, charged with electricity, and containing the distance of the point in question from the position of the charge at a certain former time. It may be remarked, that 1) Nachrichten d. K. Gesellschaft d. Wissenschaften 1904 Heft 2; in the follo- wing to be cited as “first paper”, ( 347 ) P. Herrz') has published a method, though only for special cases, equivalent to my general representation of the field, for which he very happily uses the figure of a sphere contracting itself with the velocity of light. In the “Göttinger Nachrichten” | start from rather a toilsome Fourier’s integral, whereas [ shall now choose a very simple way, using only the theorem of Green. In this way I represent the potential in the first place by a quadruple integral, ($ 2), one integration extending over the time, the others over the space occupied by the charge. Here the road divides: Either you can calculate the integration over the time; this leads to Lorertz’s representation; then the inte- gration over the charge gets rather a complicated form, relating no more to simultaneous positions of the elements of the charge, but to positions occupied by each element at a certain former time, or, as you may say, relating no more to the real shape of the electron, but to a deformed one. Or you can calculate the integration over the charge; this leads to my formulae; it is then no longer the integration over the time, in general cases of motion, that you have to evaluate (§ 3). § 4 applies our formulae to problems, essentially known, viz. to the determination of the field in a great distance from the electron, and to the case of stationary motion, especially with a velocity exceeding that of light, in order to complete the statements of my first note and to study in detail the behaviour of the field in the neighbourhood of the “shadow of motion”. In the last § I pass on to the representation of force, exerted by the eleetron’s own field. This force is computed exactly for any motion, excluding rotations, according to the principles of H. A. Lorentz. At first sight the general formulae I am using here, seem to be more complicated, than the more explicit formulae, I have published in the “Göttinger Nachrichten” *) but in reality they are very easy of application to the case of stationary motion. For you may derive immediately from them the known result, that the stationary motion with a velocity less than that of light is in every case a possible free motion of an electron. Moreover you deduct easily the value of the force, necessary to maintain a motion of a bodily charge with a velocity exceeding that of light. This force is distinctly finite, even 1) Untersuchungen über unstetige Bewegungen eines Elektrons. Dissertation Göt- tingen 1904, § 3. 2) Nachrichten d. K. Gesellschaft d. Wissenschaften 1904, Heft 5, in the follo- wing to be cited as “second paper”. 24 Proceedings Royal Acad. Amsterdam. Vol. VIL. ( 348 ) in case of infinite velocity ; its value has been calculated, for the first time, as far as I know, in my second paper. Further you derive from the same formulae the surprising result, stated in my second paper not only for the case of stationary, rectilineal motion, but for» any motion you like: The motion of a surface-charge with a velocity exceeding that of light, is actually impossible, requiring continually an infinite supply of force. In order to make this more evident, let me point out: the more the charge is concentrated, the more the force will increase; in case the charge is concentrated at one point, the force is infinite even in the case of a velocity less than that of light. . It may seem unsatisfactory, to be restricted to the special shape of a sphere. Only a few of the following results are independent of this restriction, that is those, which do not contain the radius of the electron, e.g. the approximate formulae for the field of a charge, stationarily moved, in the case of a velocity less than that of light, and those in case of a velocity exceeding that of light, in the regions I and III ($ 4), whereas the formulae relating to the limit of the shadow of motion, that is to the region Il, depend on the special spherical shape. Yet it is evident, that on more general suppositions, you could probably not proceed so far. It is kown, that H. A. Lorentz’) has lately supported the hypo- thesis, that the shape of the electron is variable, conforming itself to a “Hravisipe-ellipsoid”’, according to its momentary velocity. As for velocity exceeding that of light this hypothesis fails, because in this case you can hardly speak of a “Heravisipe-hy perboloid”. So I have not been able, to use this hypothesis. § 2. Green’s Theorem. All natural philosophy proves the wonderful power of GREEN’s theorem. We shall use it here very much like Krrenmorr*) in his enunciation of Hureexs’ principle. Let p be the scalar potential, satisfying the differential equation : De na ye eae ° . ae gts is = hts (1) where ¢ means the velocity of light, and g the density of charge of the electron; as regards the choice of units see H. A. Lorentz, Ency- klopädie der Mathem, Wissenschaften Bd. V. Art. 13, N°. 7. 1) K. Akademie van Wetenschappen te Amsterdam Mei 1904. Proceedings p. 809. *) Vorlesungen tiber Mathematische Optik, 2te Vorlesung, § 1. Leipzig 1891. ast ( 349 ) Let » be an auxiliary function (2) 2 r the distance of a certain point in question A from any point P, ¢ the moment, for which the value of p at the point A is required, f a variable moment of time. Our auxiliary function v then satisfies the differential equation : Like Krrcunorr we shall suppose, that the function /’(v) is represented by a narrow prong, enclosing the area 1, viz. that /(w) vanishes for all abscissae different from =O, but in the point v= 0 increases so strongly and so suddenly, that notwithstanding He AEN B a eee NEVEN e —@ If we apply GREEN’s theorem in its most common form to the functions p and v, we have: Ov dp : fear ramus f(e Senge |e Jo rt me (ON The surface integral on the right hand side is to be extended over the border of the space S and over an infinitely small surface, enclosing the point A, in as much as this point is contained in S. This holds good, because we shall let S finally expand into infinity. The part of the surface integral, relating to the surface enclosing A, is known to give: 4nxngp,t F(—e(t —t)). If we use on the left side of (5) the differential equations (1) and (8), substituting in g as variable time # and noticing, that dv dv ae ae it follows: io Ov p dw aul (eren) at foanf(ox an an) We A (Set €) ?. NEN The second integral on the left is extended over the charge of the electron, the first on the right no more than over the surface of S. Multiply the last equation with cdt and integrate with respect to { from —o to +o. Thereby the first term on the left vanishes on account of the nature of the function /#. In fact this member relates, after the integration is performed, only to the moments 24% ( 350 ) {== +o, and certainly v — 0 stands for t' = + oo according to the definition of £. Further we suppose, that the first integral on the right vanishes also, on account of the nature of the potential g. In order to understand this, we may perform the integration with respect to ¢’, as follows: 0 0 i 1d p Po fo ed = ed == = one. On IE ; r On where g, means the value of p for that #, for which the argument 7 of F vanishes viz. £ — t— —. Similarly we show: C +o Ov Nee pl lor 1 0g, fue Te ne hin Fhe If the electron was at rest originally, for instance until the moment t,, we can in any case expand the bordering surface 6 so far, that the value ¢ just now defined gets less than ¢,. In this case p‚ becomes fo ot integral in question is then reduced to the following expression of 1D i @ ae we a which we know by the potential theory to vanish, if it is calculated for a surface sufficiently distant. So you keep in equation (6) only the second member on the left and on the right, and you have: +o - +a fe at {ov dS te fy jn F(—e(t—t))e dt’. Perform the integration on the right in the way used repeatedly and denote for short with g the value at the point A at the time 4. We get conclusively : +a Aap =|: af? F (rc (—?)) dS. A SER = The scalar potential is represented here by a quadruple integral, viz. a time-integral and a space-integral. the electrostatical potential of the electron and = 0. The surface ( 351 ) § 83. Transition to Loruntz’s potential-formulae on the one hand, and to those given by myself on the other. It is tempting, to perform in (7) the integration with respect to . Ly t. As F is different from O only for the moment # =t— — C Sot tngaf{ Bas ONE Pate) where fo} means the density, contained in the element dS at the hl b we get immediately time t =i — =. As for the proof of formula (8), H. A. Lormntz') refers to the expression (8) satisfying the differential equation (1). Wiecuert*) and others do not start from GRrEN’s theorem, but from BerurraMr’s, which naturally is only a transformation of GREEN’s theorem and, it strikes me, not a very transparent one. Instead of performing the integration in (7) with respect to time, it is better, to evaluate that in respect to space. Now for this pur- pose it is necessary to add a certain supposition as to the shape of the electron. In the first instance we suppose the electron to be an injinitely thin spherical shell of the radius a, on which ‘|. the charge e is uniformly distributed. So we take : € ———d 6 instees l RY ee instead of 9 dS J and get from (7) me ne é ! 1 1 | EE c dt eee = sam Fig. 1. +. Let O be the centre of the spherical shell. Round OA we count the azimuth 4 and from OA the angle 9, so that 4, 9 mean the geographical longitude and the complement of the latitude on the surface of the electron. Let R be the distance OA from the centre to the point in question, it follows : r= kh? - a? — 2 Racos3, rdr= Rasin 9 dd, 1) La théorie électromagnétique de Maxwerr, Leiden, 1892, pag. 119. ) Elektrodynamische Elementargesetze. LoRENTz— JUBELBAND, pag. 560, Haag 1900, ( 352 ) it Nr Ye is =o" | Hie ot Gs, ox LT] SU papa | ak SE sin 9 d= 0 0 ho 0 Ra 3 rdr 2 na sne R Ral SO op Ra € Sat basen li: oe et Ln Hed (9). 2a a Ee The lower limit |R—a| equals R—a, if R >a, but equals a—h, if Recrand |R—a|a, and a certain value rt. Equations (11), in which 7’ is to be substituted for a, show, that A—1, if cr lies between A — 1" and / + 7’, or, what comes to the same thing, if 7’ > | R—er|. Now two cases are possible: | R— ct | may either be smaller than a or larger. In the first case a triangle with the sides (a, 2, er) is possible, not in the second ease. In the first case we have: a a ' 1 IE rdr' =| dr == d (a? — (R— cr)’), u | R—cz| in the second : a IE zede = 0 0 If we define a quantity * by 3 Ket? <= ete (| ro by wo Pare et.) 2 a according as the triangle (a, F, cr) is possible or not, we can write for an exterior point instead of (12) sv de tag=mof seri ait fh Sea) oe In the case of an interior point, for which & cr— R and < cr-+ FR. Now three cases are possible: a). er— Ra, er+h>a, triangle (a, PR, cr) possible. b). er + Ra, consequently cr + R >a, triangle (a, R, cr) im- possible, « not the largest of the three sides a, /, er. In these three cases we evidently have: k 1 fe r dr =| r dr = ae — (er — R)’). A 0 ct— ht a crt farar=frar=te ree ° (b) “0 lez ER Now if we define the quantity x by (18) in the cases a) and c), but by er NE GN 9 a- in the case 4), the potential is given by (14) for interior points as well, according to equation (20) of my former paper. Notwithstanding the simplicity of our quantities 2 and xz, it is easier for further purposes, to replace them by an analytical expres- sion, holding good for all values of r. As for an expression of 2, we know, that 5 5 ds JT ST - sin st — == + aos == We We es (15) % [ according as « is positive or negative. Now: dh ye | sin sa sin sR sin set = 7 je s (a + BR — er) + sins(a — Her) — — sins (a + RJ er) — sin s (a — R—et) . ... .. . « (18) As for the four quantities at R—cr,a—R+ecr, —a—R—cr,—atRoter three are positive and one negative if the triangle (a, FP, cr) is possible, nd een end ( 355 ) two positive and two negative, if the triangle is impossible. After having ee ( ds ' : ¢ multiplied equation (15') by — and having integrated with respect to \) : 5 5 % s from s=0 to s =o, you get in the first case yin the second case 0; i.e. you have in both cases: io 2) ‚ (16) sin sa sin sR sin set - Sn „| & | | | That is the required expression for 2; substituting it in equation (10), we have in the case of surface-charge: EC “dr ds p= | — |] smsasem rhein: supr Oak ML U Za R 8 0 0 Replacing further a by 7” in equation (16), we get simultaneously for the integral, contained in (12): ! | 4 ds jk 5, ! ! ! e ‘ a7 de = — | — | sinsr' 1! dr' sin SR sin set = JT Ss 0 0 0 0 . 4 sin SA — SA COS sa , . ds — sin sR sin set — TT 3? Ss 0 Therefore we can write instead of (12) in the case of bodily-charge : an oc DEC dr (‘sin sa—sacossa , k ds M= sin sR sin scr —. . . (18) Anta) R (sa)? s 0 0 It may be remarked, that in my first paper the foregoing equations (17) and (18) appear as primary and the equations (10) and (14) are deducted from them by performing the integration with respect to s. Moreover it is probable, that the quantities 2 and * may be replaced in several other ways by a uniform analytical expression. Almost the same formulae stand for the vectorpotential, if the motion is free from rotations. Our deduction proves immediately, that it is only necessary, to multiply the integrand by , ¥y_- meaning c the value of the velocity v at the time ¢—r. If on the contrary the motion is accompanied by rotations, you must add to the part due to translation another part due to rotation, where the quantities 2, x are to be replaced by some quantities 7’, x rather more com- ( 356 ) plicated. The expressions for this are derived in my first paper and may be derived more easily by the present method. § 4. The field of stationary motion especially with a velocity exceeding that of light. In my first paper I have applied the foregoing representation of the field in order to derive the well known approximate formulae of Liinarp and Winscnert for the field at a great distance of an electron moved anyhow. It strikes us in these formulae, that the cases of velocities smaller or greater than that of light seem to differ from each other only by the sign, whereas in reality a funda- mental physical difference must exist between the two cases: If the velocity is less than that of light, the whole surroundings of the electron is seized by the effects of the moving electron, if the velocity exceeds that of light, only those points are seized which lie in the “shadow of motion” of the electron so to speak. This incongruency is cleared, if the roots of equation (11) are discussed, what was not sufficiently pointed out in my first paper. In general we note this (details depend on the special character of the motion). If veiocity is less than that of light, each of the equations (11) always has a positive root; if the velocity exceeds that of light, imaginary and negative roots are possible as well; they appear in all those points which are situated so to speak in the front of the electron; positive roots exist only for those points that lie in the shadow of motion; and here for each of the equations (11) even a pair of positive roots exist. Only for a narrow region bordering on the shadow of motion and about équal to the diameter of the electron we have not two but only one pair of positive roots. It follows: The approximate formulae mentioned before, which I have derived formerly supposing two roots rt, 7, to exist, hold good absolutely if the velocity is less than that of light; in the opposite case they are to be replaced by O out of the shadow of motion, and they are to be completed by a member similarly formed within the shadow of motion. Fig. 2 explains, what shadow of motion means. Here the momentary position O of the electron and its preceding path OP is marked. Round every point P of the path the sphere may be constructed with the radius ct, where t denotes the time, in which the electron gets from that Fig. 2. point to OU. The envelope of these spheres —— ee 7 ( 357 ) defines the shadow of motion. Evidently it is the smaller, the more the velocity of the electron exceeds that of light. The region bordering on the shadow of motion, which was mentioned before, is also sketched in the figure as a narrow strip. The foregoing general remarks are corroborated by closer discus- sion of stationary motion with constant velocity v. The field of stationary motion can be found exactly by a singular process of reciprocation ') if » c, has been explained by prs Couprrs following the steps of Hpavispr. Compared to DES CouprEs’ treatment the following is hardly new. It may merely be pointed out, that the infiniteness of the Hravisipr-prs Couprus *) solution near the borders of the shadow of motion is not real, the formula no longer holding good in this region. The infiniteness mentioned just now results from prs CouprEs treating the case of a charge concentrated in one point, which is passing to the limit of vanishing dimensions of the electron. We shall adopt in general this simplification and thereby dispense with a rigorous solution, but at the same time we shall point out, that this simplification is not legitimate near the border of the shadow. We suppose bodily charge, as it will be shown later, that in case of surface charge any motion’ with v > e is actually impossible. Let the stationary motion be directed towards the positive axis of z. The system of coordinates has its origin O at the position of the centre of the electron at the time ¢. Let the coordinates of the point in question be «, y, z, let its distance from O be r = Va? + ye + 23, so that 7 now has a different meaning from that in § 2 and § 3. At the time ¢—vt the centre of the electron was in the point —vr of the axis of w; the distance of the point in question from this point is ier. be (aoe a att AAP Te (19) The conditions, under which the triangle (2, a, cr) is just possible, are given by the equations (11) R, —a cert, ‘ Ve POE ae in the case of an exterior point (more correctly R > a); we can combine (11) into («+ vr)? Hy? + 2? =(er + a)? or 1) V. the summary of H. A. Lorentz in “Encyklopädie der mathematischen Wissenschaften”. Bd. V. Art. 14. Nr. 11. *) Zur Theorie des Kraftfeldes electrischer Ladungen. Lorentz-Jubelband, p. 652. Haag 1900, ( 358 ) fer 07) c= Alor =e ace) ee FF SOE ATD the upper sign relating to the roots t,, the lower one to the roots t,. The product of the two roots t, or the two roots t, is: As r >a in a point of the exterior, this product is negative, if CS, ib 1s positive, if ev. We conclude: If the velocity is less than that of light, the two roots of our quadratic are real and have opposite signs. Each of the two equa- tions therefore has one available positive root. If the velocity exceeds that of light, the two roots may be con- jugate-imaginaries; if they are real, they have equal signs, and therefore they may both be either positive or negative. Hach of the two equations has therefore either no roots or two available positive roots. We distinguish between real and imaginary roots by consulting the discriminant of our equation (20). The roots are imaginary if (ve = ac)? < (7? — a’) (v? — c?), for which we may also put (Cares av) ey? = Sa i wade la no ta eee We introduce the abbreviations : Bs = Pea ey? + 2 so that @ means the distance of the point in question from the direction of motion, § the distance of the same point, measured in the direction of motion, from two points P, P, (see fig. 3) of which the coordinates are x= + a. Replacing in (21) the sign < by =, we get EOD = lice eee This defines a cone of revolution about the direction of motion, of which the apex lies in the point P, or P, according to the meaning of § and of which the generatrices are inclined towards the direction of motion in the angle arctg (5° ma i) For points in the interior of these cones, i.e. between the conical surface and the conical axis, the roots of (20) are real, for points in the exterior they are imaginary. In case of the reality of roots the distinction between positive and negative values results from the sign of the coefficient of r in equation (20). ( 359 ) Both the roots are eos if O2 = aC ; $ a c ti B they are negative, if OL = .@.0 a en RCN Dern a » ° r Evidently the planes «= + 3 are polar planes of the points P, P, with respect to the surface of the electron. We distinguish a back and a front of these planes judging from the direction of motion. Fig. 3 gives the result of our discussion. Here the points P,, P, and their polar planes are constructed. From P,, P, the cones K,, K, diverge, which touch the surface of the electron at its inter- section with #7, /,; they appear in the figure as two pairs of straight lines. We call such points region J, for which both pairs of roots are either ünaginary or negative. Region IT consists of such points, for which only one pair of roots (t‚) is posttwe. Finally region II is that, in which doth pairs of roots are positive. The regions I, II and III are distinguished in the figure by different shading. We need not concern ourselves with the interior, where the field acts differently. We now proceed to the computation of the scalar potential. If vc, we have to distinguish, whether the point in question lies in region I, IL or III. III). In this case there are two positive roots t, and two positive roots t,. We distinguish them as 1,', t,", vr, 7," and easily see that they arrange themselves according to their magnitude as follows: ! ! " " Ghee HAT T Re Sr Indeed if we imagine a diagram in which for the abscissa cr the curves y= h—a, y—=H-+a@ and the straight line y= cr are drawn, the latter intersects the former curves in four points, viz. firstly, beginning from O, the curve y = Rk — ad, then the curve y= k-+a, then the curve y= R+ a for a second time, lastly the curve y= ll — a for a second time. These four points belong to the values Tt, 7,, t,", t,", before mentioned. Moreover the diagram shows immediately, that the triangle (/, a, cr) is possible only for Weser values” oft, for which either T‚ SET, OET CET On this account we obtain from (13) and (14): 1 OL, de TN Ne OER R—er\?\dr ND ees ee eee i eae Nik DEL a 3 oss my ( ( a Vie | oul ( a Ve zee We introduce by (24) the variable u. It is to be noticed in expression (26), giving t by wu, that the denominator 1—* is negative as well as the term au + gein the numerator, the latter being so because we are in region Ill. From this follows, that the negative sign of the root in (26) belongs to the interval of the larger values of r(r‚ , the second a Tv equals — 5 or rr 5 according as v<¢ or v >c (see equation (15) in $ 3). We therefore have: ie a) Ò sinvst , zr sin est dt = CSE EE >) Open — 0 The result of the equations Maca consequently is: Jn case of stationary motion with a velocity less than that of light we have 6 =0; this motion is a free possible movement of the electron. Further the equations (42) and (43) give, in the case of bodily charge if » > c: oo Ana’ _ vr—c? sin ds—as cos as\° ds oo mad) nn Er Tree 8 ge? v? a's? s ( 366 ) The value of the integral, still to be calculated, is a mere number, 1 ‘ namely BeOne Seen introducing the new variable p= as and transforming as follows (note, that the expressions taken in [ | disappear) : GO 0 dp sin p—p cos p)* d (sin p—p cos p)? en Ed me We sin p (sin p—p COS p) — pete pe 4 p* : 2 Pp 0 ° 0 0 [sin p (sin p—p cos dp — A Aad Lah p) +— * (conp (sin p—p cosp) + p sin’ p) — 2 = Se Prete p* 0 0 1 (/1 sin2p pcos2p 1 (d sin2p 1 / sin 2p i — {| — ———— dp = — — | — dp = — ——oe 4 Dap Pp 8, oP Pp 8 p 4 0 p= Thus the force exerted by its own field on an electron bodily charged and moving with a velocity exceeding that of light becomes: fa 9 co \¢? tape (1-55 1 Se This force acts contrary to the movement. The opposite force is to be exerted in order to maintain the motion and to balance the less of energy caused by radiation. The force is absolutely finite and and remains so for v—o. For v=c we have § = 0, a value which is connected continually to the case of velocity less than that of light; for v= © we get i oe OS ees De this equals the attraction of two point charges Ti in the distance a, according to CouLoms’s Law. Although the stationary motion with velocity exceeding that of light is no free possible movement of the electron, yet this motion is not impossible from a physical point of view as requiring (even if the velocity is infinite) im every moment only a finite expense of force and also for every finite path only a finite expense of work. We finish by studying the motion of a surface charge with a velocity exceeding that of light, returning to equations (41) and (43). These give us with v >c: oo Ara’ _ vic? ds — ii Lim |-sinrssinas—. . ... (40) 2 S vu ra Q ( 367 ) In order to evaluate this integral we divide it into one part from 0 to a quantity ¢ to be conveniently chosen and another from e to oo. In the second part we express the product of the sines by the difference of the cosines: ee ls 2) oo ; 4 ds ds haf ds 1 Is sin rs Sin ds — — | sin rs sinas — + — | cos (r —a) s— — —|cos(r r +a) B: ej " s 9 Ss Ss s ‘ + oe 0 0 In the second and third integral we introduce the new variable of integration p= (r—a)s and p= (r+-a) s respectively. Then the difference of these two integrals becomes: (ra ‘ra io a) dp 1 dp 1 dp L (dp — cate cos pS EP SS SE 2 p 2 p 2 p zr za (ra) (7 - a) (ra) (ra) (ra) cos p—1 1 ra p dp je : dp = — 5: 129 sin? £ zl r—a 2 p ae s(r—a) or, if we sum up: o : ra) ds f ds 1 rda dp sin rs sin as — == | sin rs sin as — + — log = ik sin® — —. 8 s 2 “r—a 2p 0 0 3(r— a) Now if we choose ¢ sufficiently small, the first and second integral of the right-hand side may be made as small as we like. Namely in both cases we have to integrate an entirely finite function within two limits indefinitely close to each other. Therefore for any given rand a (7 >a) there results rigorously : 3 EN [sin resin as = lag nlite a> | pane 8 0 Making 7 converge to a, our integral becomes positively logarithmic infinite. It follows, that the force necessary to act on the electron in order to maintain its uniform motion also becomes infinite. The stationary motion of an electron, charged uniformly over its surface, with a velocity exceeding that of light, is actually im- possible; it would require an infinitely great expense of force and energy. ( 368 ) Meteorology. — Oscillations of the solar actiwity and the climate by Dr. C. Easron. (Communicated by Dr. C. H. Winn). The parallelism between the oscillations in the “solar activity” and the variations of the magnetic elements of the earth is certain, and a similar parallelism is suspected for some other terrestrial phenomena. The meteorological elements, however, have always seemed to be subject to so many different perturbations, as to obscure the corresponding parallelism, which most probably does exist in this case also. Brickner has considered this point in his well-known investigations on oscillations of climate '), but he only reached a negative result. It is true that KöPPeN and NorpMann *), restricting themselves to tropical countries, have established a parallelism with the 11-year period of the solar spots for the period from 1840 to 1900, and KörPeN’s curve also shows this parallelism tolerably well for the southern temperate zone, while for the northern temperate zone LANGLEY’s bolometric observations’) give us a right to expect much from his method for the future. For the non-tropical zones on the whole, however, (and therefore also for the earth as a whole) the disturbing influence of terrestrial causes would seem to be such that the oscillations produced by a cosmical cause are entirely obscured. The reason of this is apparent. The direct influence of the solar radiation can only be visible in the general temperature-curve for regions where the difference between the seasons is neither very large, nor their change very irregular. This reason is already sufficient to explain why Briickner’s so-called “temperature-curve for the whole earth” {on which the observations in the northern temperate zone have a preponderating influence} differs so widely from the curve representing Rupotr Wo tr’s “Relativzahlen” for the sunspots. It appeared to me therefore that this question must be considered from a different point of view. Dr. W. J. S. Lockyrr has recently published an investigation *) in which he reaches the result that BRÜCKNER’s period of 35 years in the climate is also found in the irregularities of the 11-year period of solar activity. He tries to show this by comparing the variable quantity 1/—m (which represents the interval of time between a minimum of sunspots and the following maximum), with Brtcknrr’s 1) E. Brückner. Klimaschwankungen seit 1700., Geogr. Abh. IV, 2 (1890). 2) W. Köppen. Zeits. Oesterr. Ges. f. Met. VIII, XV, XVI. —Cu. NorDMANN. Comptes rendus T. 136, p. 1047 (1903). 3) S. P. LanareY. Astroph, Journal XIX, p. 305 (1904). ) | 4) W. J. S. Lockyer. Proc. Roy. Society, LXVIII (1901). ( 369 ) eurve for the rainfall. This meteorological element appeared to me to be a very unsuitable one for comparison on account of its ex- ceedingly large local variations, it might however be of some interest to compare Brickner’s curve for severe winters. Further the very few oscillations recorded by Lockyrr proved very little, in my opinion, unless the previous oscillations of the solar activity, though less accurately recorded, agreed at least approximately with the result. An investigation in this direction led me to a negative result with reference to the confirmation of Briécknur’s climate-period, which was suspected by Lockynr. Another very surprising result appeared however, viz. a parallelism (though imperfect) between the J/—m curve of solar activity mentioned above and the curve of the frequency of severe winters. I do not give these curves here, since they are of no direct further use. This parallelism suggested however two important conclusions, viz. 18 that the A/—m curve [or preferably the deviations of the maxima and minima of solar activity from their normal positions as determined by Nrwcomsp')| could be of great value along with the frequency-curve of sunspots (Relativzahlen), while it appeared at the same time that these deviations are real, at least for the greater part, and 2°¢ that in the records about severe winters we possess a rough but important material from which we can derive a judgment concerning the general course of the weather in the past. The parallelism which I found is in this sense that the more severe cold corresponds with the larger number of sunspots (de. with the greater solar activity, to retain this term). This does not agree with Sir Norman Lockygr’s views. It is in accordance however with the view, which is now generally accepted, that the spots do not represent excessively heated areas. It is also well in keeping with the result of an experiment by SavÉriër *), and with the con- clusions arrived at by Prof. Juris in his solar theory *). That the inequalities in the eleven year solar period cannot be attributed in the main to errors of observation had already been indicated by the investigations of Fritz and Loomis on the aurora *) and of Harm on corrections to the inclination of the ecliptic, on the variations of latitude, ete. *), which show corresponding inequalities. 1) S. Newcoms. Astrophys. Journal, XIII, 2 (1901). 2) SavÉuir. Comptes rendus T. 118, (1894). 5) W. H. Junius. Archives néerlandaises, Série Il, Ts. VII, VIII, IX. 4) H. Frrrz. Das Polarlicht. Leipzig 1881; — E. Loomis, quoted by Harm, A.N. 3649. 5) J. Harm. Astron. Nachrichten 3619, 3649; Nature Vol. 62, 1610. ( 370 ) The records concerning severe winters naturally are a very rough material, though the remark has already been made that tbey are far more reliable than e.g. records about hot summers, the excep- tional formation of ice being an unmistakable criterion. The data must however be carefully and critically arranged and compared. This has already been done by Prof. W. Köppex in his well known investigations on the periodicity of severe winters. The data as given by Körper have therefore been used instead of those of Brickner, which are simply taken from PrLGraM. On comparing KörppeN'’s list with BRÜCKNER's curve I was struck by an indication of periodicity in very long periods, different however from what K6ppeN sought (vz. the regular occurrence of severe winters in determined, equidistant years), and also not consisting of regular oscillations like those suspected by Brickner, but of a recurrence of the general character of the weather in periods of about 180 years. The distribution of the winters within each of these periods is the same, w/z. very many severe winters in the first 60 or 70 years of the period (e.g. the 60 years following 1220, 1400 or 1580), very few in the next 20 years, many in the following 20, few and irregularly distributed winters in the remaining part of the period. In accordance with what was said above this phenomenon, if it is real, must be attributed to a secular oscillation in the solar activity, presenting itself to us in the form of systematic variations of the eleven-year period. For this reason I took as the basis of my investigation a period of 16 x 11.18 = about 178 years, 11.13 years being the normal period according to Newcoms. The available material covers a period of more than a thousand years, viz. from the middle of the ninth century to our own time, including the additional data procured by Körper himself. The reality of this periodicity was made very probable by a statistical investigation in which the year 848 was taken as the first of a period of 178 years. We denote the influence of a “normally severe” winter on the climate by a “eold-coeffieient”” unity. To an exceptionally severe winter (“winter of first class” of KöprpeN) the coefficient 3 is assigned, and 2 to winters of intermediate severity. It then appears — taking all the periods since 848 together — that the four sub-periods of 67, 22, 22 and 67 years have total coefficients of 114, 15, 39 and 62 respec- tively, de. 1.70, 0.68, 1.77 and 0.93 respectively for one year. These oscillations are of such amplitude that the proportional number of severe winters in these cold periods of 67 years (e.g. from 1561 to 1628) is nearly twice that of the succeeding relatively mild periods of 22 years, the ratio in the case of exceptionally severe winters (371! ) being seven to one. The mild 22-year periods have, up to the present time, contained altogether 12 severe winters, only one of which was exceptionally severe. It is now important to ascertain the character of this oscillation. On the one hand there seemed to be some indication of a period consisting of two consecutive 178-year periods (i.e. of 356 years), but even our material does not cover a sufficiently long interval of time to allow any reliable result in this direction to be derived. On the other hand the rise and fall of the curve in the middle of the 178 year period is the most characteristic feature, and points to the possibility of dividing the period into two. In fact the period can be divided into two halves of 89 years, which show a remarkable curve, somewhat different on the two sides (see diagram fig. II). Perhaps this period of 89 years may be further divided into two periods of 44.5 years. The depression in the middle of the 89-years period is but indifferently indicated for all the severe winters together, but becomes more marked, if we take into account only the excep- tionally severe winters. In the following list (Table I) the twelve 89-year periods which are available since 848, have been entered, each divided into intervals of 11 years, corresponding to the computed normal maxima of solar activity (according to NewcomB). The last interval of each period contains 12 years, which however has only A eo ae Ey. Ty Distribution of severe winters in periods of 89 years (848—1916) (divided into intervals of 11.1 years), I 848 Sh May id ae a 1 Hr Pr te i 37 | - = = Sie | Ay ee | 1G. 9086. De Ten 1 MS 7. aah ee AEP RD Oe Ne AE he EG So irl aa V_ 1204 1 EN | 1 A | VI 1293 ADE) Fe ARNE i Sak! PEN en VIT 41382 | RE a A ER NE KME | 1 EET De AZ ME Dit 1 End IX 1561 A EN RORE 1 X 1650 Ee ION nae ANN aa XI 1739 teler de. 7 adh | Tre 1 XII 1828 si Se SAN SAMEN, rk eet Ane ol 41° 38 26 39 27 19 9? an appreciable effect on the total of this interval. The totals of the exceptionally severe winters alone are: 7.11, 365 2410) VOR ON In order to keep the division according to whole years I have dropped one year — the year 1560 — near a minimum of the period. The last interval of the last period being of course unknown, I have taken for this interval the mean of the last column, viz: 1, but even a much higher coefficient would not appreciably alter the general results. The division of the period can certainly not be continued beyond a period of 4 >< 11.13 years. In our material there is no indication of a regular alternation (at least not in the majority of the cases) of cold and mild periods of 22 years. There is even less evidence of a regularity in the succession of the 11-year-periods. Taking a period of 44'/, years as a basis, we can express our results as follows: There exist oscillations of climate with a perrod- veity of: 441), ‘years -and: (multiples thereof, Chien thus, that. one period of 11.13 years contains less cold than the three preceding and the three following ones; that at intervals of 89 years there is one period with very little cold; that in two consecutive inter: vals of 4.78 years theslast 5 or -6 periods of One them are colder than the corresponding periods of the other interval. This oscillation of climate comme sponds to an oscillation in the “solar activity’, jam a higher order’ than the well known eleven-year period. The very doubtful difference between two consecutive 178-year periods, which was indicated above, would be in this sense that the greatest amplitude of the oscillations in one period occurs in the beginning of the period, and is displaced in the next towards the middle of the first 89-year sub-period. Nothing however can be ascertained on this point, and still less on the existence of still longer periods. It seemed interesting to investigate whether the 11-year variation of the solar activity itself is shown by this material. For this purpose the distribution of the ‘‘cold-coefficients” over five phases of the eleven-year sunspot period was investigated, vz.: ‚mm = 2 years on both sides of the observed minimum, J/—= 2 years on both sides of the maximum, ap — ascending phase, dp, and dp, = two halves of the decreasing phase. The observed maxima and minima are taken in accordance with Nrwcomp (the deviations from R. Worr’s last list’) are of no importance). The unequal duration of the phases has, of course, been taken into account. The periods have been arranged in four groups, A, B, C, D in order of decreasing cold. We then find the following values of the frequency for one year. The values in parentheses are derived from those periods only for which the weights assigned by NrewcomB (le. p. 7) to the determ- ination of the maximum and minimum are together at least equal to 8. Hees We IE Distribution of cold winters over the phases of the 11-year sunspot period. (Groups arranged in order of coldness). d B C D m | 1.50 (4.67) 0.75 (0.75) 0.49 (0.42) 047 (0.17) ap 4.13 (1.44) 4.13 (1.13) 0.60 (0.50) 0.32 (0.39) M 1.08 (1.50) 1.25 (1.25) 0.33 (0.25) 0.2 (0.25) dp, 0.98 (1.09) O21 (0.21) 0.76 (0.65) 0.57 (0.57) dp, 1.06 (1.03) 0.48 (0.48) 0.36 (0.22) 024 (0.24) For all the groups together, and also for the two coldest groups, we find a curve corresponding with the sunspot curve; for the two mild groups the cold-maximum seems to be displaced towards the descending phase. This phenomenon however does not necessarily depend on a really different distribution of the cold winters within the eleven-year periods, no more than the high vaiue of the minimum _ in the coldest group. The reason probably is that the variations of temperature precede or follow those of solar activity by a certain interval of time. This is apparent from a comparison of the temperature variations with the curve of the Relativzahlen for the sunspots. In the diagram fig. I, A is the latter curve according to R. Worr, B is the second half of the temperature-curve the period being taken at 356 years, C is the temperature curve, if a period of 178 years is adopted, D is the last 178-year period alone *). In presenting these curves it is not my intention to contend that they agree with each other in details, — as is the case with the curves of the variations of magnetic elements and the sunspots. Such a parallelism could not be expected 1) R. Worr. Astron. Mittheilungen LXXXII (1893). *) In drawing the curves B, C and D no other process of smoothing was ara applied than that given by the formula OAT Ter The cold-coefficients have been taken for each year, only in the case of D for every two years on account of the small number. The scale is: coefficient unity = 40 Relativzahlen. ( 374 ) a priort having regard to the material on which the curves are based. I only wish to show that the general character of the temperature curve is the same as that of the curve of solar activity. It is important to remark that the mean curve B is more similar to A than D. The deviations in position of the variations of temperature relatively to those of solar activity seem to be at least partly system- atie. On comparing the temperature curve and the solar curve in the separate eleven-year periods we must however not only consider the greater or smaller number of spots, but also another phenomenon (which, it is true, is certainly connected with the abundance of spots) viz. the deviations of the minimum and maximum from the normal. This phenomenon appears already vaguely in the correspondence of the values J/—m with the number of cold winters, to which I referred in the beginning of this paper. It is shown more clearly however in fig. TIL on the plate. For the last 17 periods of 11 years the dotted line represents the acceleration (—) or retardation (++) (expressed in years) of the observed solar minima and maxima, compared with the normal period of 11.13 years. The corresponding cold-coefficients are represented by the continuous line having four different ordinates corresponding to the four degrees (A, B, C, D) of cold. On the whole we find that to an acceleration of the solar period corresponds a more intense cold, and to a retardation a greater heat. This rule also holds for the individual periods, provided the deviations are large. Combining this result with the distribution of cold-coefficients over the phases of one and the same eleven-year period which was mentioned just now, and with the larger oscillations found above, we find the following rule, which is presented as a hypothesis : On the whole the oscillation of terrestrial tem. peratureis accelerated relatively to the eleven- year variation of solar activity in the colder part ofthe larger period, and retarded inthe hotter. part. In cold 11-year periods the centre of gravity of cold is near the minimum of sunspots, and often there are very cold years preceding the minimum; in warm periods it is situated near the maximum or thereafter; in periods of intermediate character it falls between the minimum and the maximum. For the individual cases we find that, in cases of considerable retardation or acceleration of the solar minimum or maximum, the centre of gravity of cold tends to be exceptionally retarded or accelerated. If the minimum of a sunspot period occurs very early and the subsequent maximum is retarded, the cold period also is largely extended. i] a nk C. EASTON, “Oscillations of the solar activity and the climate.” ze) N D EN = \ 7 Ac if \ Token / , 5 ary > ‘ N ‘ Ne SNS ms. oS EN DAS ad as A JOU ’ \ 1 Ee aie ‘ 2 iF wl ’ v N 7 ‘ Pa be aN ZEN Pere ee N 1 N AES Ei ‘ v \ AFS an Z Ee Pe ae bet y f hi A id on 1 sof id IS 7 ie Me Ü ins N Ai ‘ ae ~ 4 Fi a / x 5 ‘ “ae Td Re Dl id See A a Ah N , N 0 eier Ne nS MS ee Nov SE AC IA fe iy an ie I os ~ jens a SON \ 1 PEN 1 \ -. ae eN Af ea } Ù ‘ ' lies Ô J ets We SARS IR \ Woy fi . " ry a =) = IEN 4 Wo pte ; AAR gi a : : eae Nite a a8 ca see TN / D fj x FS fi CN, 5 . y Uy - 1 Sor t \ 1 ye Sree hy Aa OU F \ ~ ra L N N a dites 1 AB ees Nees N A 5 CAN . = Gi Mas 0 fs 9 soe i ji 0 Mt or) - . \ … re . - aa ae zr 0 M s u sn p a 5 EG 7 Be 4 a , \; ENS Bs Al y f Ks ws ~ yf < De oe ERO 2, 1 is ty a Á 5 nos rant 1 t 7) B 1 is / \ Ev 4 rs i BS i Sen ‘ in GG pS iM ' 25 ‘ ‘ 7 1 1 ! \ t EN 4 14 ! 1 RA ’ \ U 1 Lay: he LAS 1 1 \ yy ty ry 1 1 1 i a ech 1 1 i H 1 Hf Den i A i q \ s\ UT SN ry 1 \ ‘ 1 (PN 1 ea 1 BNO ie Lt \ Oy i i ‘ GY PAN Le ie Loar (RN t \ IN 1 1 1 LAS N VA SN 1 1 a \ \ \ 1 ‘ N 1 Nal eN \ ZN 1 } oY i wy) Verse 14 i ’ oS a i Sg \ 1 1 Hf No Yo ‘ ° 1 Def) \ ’ N i] A ‘ * , o a ee A, v ae EE TE „ SA een ed Ad, W) we! Noe Seeds Kn mn Ne 109-4 A N ZN a \ 50-/ LN | \ / \ / \ ¥ i? \ ij ee 7 BN DR J Ny ae i T T = T — T T = i= T FAG T a aa T r 1750 1800 1850 1909, oF oe - EN 4 AAN 7 \ / \ 3 4 \ \ A N 4 Se 2 B = Om À as \ 1 7 kK Dp ke \ zl \ > IX ar’ Re ? —— Wie 1800 bl) I. Sunspots and col vi s i i 4 ere I a f II. Period of climate of 89 years. A, Re numbers (Relativzahlen) of sunspots according to R. Worr A wae i since 1745 5 Eto Ty OLE, z, Mean of all 12 periods since 848, all severe winters. B. Mean of Nrs. II, IV and VI of the six periods of 178 years of climate f. The same, for the exceptionally severe winters only. (cold winters) since 854. , MM C. Mean of all six 178-year periods of climate. D. The last 178-year period of climate alone, Proceedings Royal Acad. Amsterdam. Vol. VII. Cold winters and inequalities of solar activity. 4 Cold winters: mean of 178-year periods since 1712, in four groups. 8. Deviations of phase of 41-year sunspot-period from normal. Mathematics. — The values of some definite integrals connected with Bessel functions’. By Dr. W. Kaprnyn. The integrals referred to are oz i cos (w sin 0) — cos (w sin p) dO, cos J + cos p 0 27 Q * sin (a sin B) sin O — sin (w sin p) sin p 16 Ts a) ess O + cos p 0) 2 cos (a cos A) — css (x cos p) p— f ( ) (a cos p 0 2 LG cos G +- cos p pee zi g cf sin (x cos A) cos 9 — sin (w cos p) cos p on = U 10. cos + cos p U If in these integrals we insert the wellknown equations cos (w sin B) = J, + 21, cos290+21,cs490+... sin (a sin 0) sin @ = I, (1 — cos 20) + I, (cos 20 — cos 4 0) +... cos (a cos 0) = I, —2 I, cos2A0+2I,cos4O—... sin (w cos B) cos 0 = I, (1 + cos 2 6) — I, (cos 20 4 cos40) 4... where /, stands for the Bessel function /, (7) of order p, and if we write 2 cos 2 nd — cos 2ngp Aon —- = d p cos 4 + cos p 0 DE he ae es Ae aie it is easy to find Gd A = an eer: Er i oie IN DT a jes ni A, ak i (A, ae A,) 5 De (A, ie A) DN In order to determine As, I notice that cos 2n4 — cos 2ng (a) sng - = — sin 2np +- 2 sin (2n — 1) p cos 6 — cos O +- cos p 2 sin (2u = 2) gp cos 20 -|- odes + 2 sin p cos (2n — 1) 6 This formula can be proved as follows : ( 376 ) If we multiply the 2"2 member of this equation by cos 6 + cos p we find Di) 2 | X top, for “indignous’ read “indigenope’’. (December 21, 1904). De N #7 cr a 4 « + # . a4 . $F + Pd ae oe ‘ . » = . eed . 2d , Mer, vert oF = 4e . - oo d ha ‘ . vik); “iy a, ¢ ma +4 ‘ 5 vi eal | § ite é st Vats ar og Zes VT AN rte Ka U k 4 - “ * _ er 4 hade ie Tae 1 PNL ; ki py Sen ak (en | . * , © Ld ‘ mI 4 . st ¢€ ’ oe ak, 1e- wr ‘ pe dr f F . oA 7 ++ Sy it . * ee * a « é 5 ee Par} ‘4 40 Go a ® TS « 3 . 5 8 7 EN 5 Ps 8 4 te hl i) o- Re Le Li my: id E! L a7 “a y ' 2 Ry HAR was, . -e > Rae " Sh EN Et Ne te EE TT 7 zt ; my. ee be tba . N a 3 : re ae | oe 7 . tt: 5 : Eph - . 3 « ~ hy Ed . 2 sj ke pias . ei +" 5 ES . 7 = - =. x Set & rey A | EL 5a ig glial sé : : : hes he ae N is a Ee | A, : “3 Koninklijke Akademie van Wetenschappen te Amsterdam. PROCEEDINGS OF THE BBC db ON O0 FS Cil.EN: Caban dd WV, Get ME B VLT: (ist PART) rr 1 AMSTERDAM, JOHANNES MULLER. December 1904. anstated from: esi van de Gewone Vee cdeniigen a Wis- en » Natuurke 4 i RRS 2, u : Mat er ae’ 25 ee inl ar gy! PRINTED BY DE ROEVER KROBER & BAKELS AMSTERDAM. Pe al) eh : ne ag s Ke O5~« 100139139