MY ret yt BA Set ROO ON faite hy SEON Ki ite Whey Aa ety 1 1} Nth 1 iii Ls 1 Ie, My Me i it bii i anti SNH wat in) fib Aan ay i ae il Bel ie cig We 7 N i { i a \ De ih ei % oi i AY) Keel ae i) Hij aN edit 4) iP nasi Ii Ri i i i i a ay i i] it hr Wh a Ripa i ce ‘ rey fi Dn Ke Hit sia HER iet Ed sh hee enk td W a un Mh ae aut bi Dn hts a KEN aS ae sins RYO bal \ il Nt ane i i i De TE — Stat STEE TEI, EE sss EE 8: Ens 5 hh ki it Bhat ns at ij ah Pv Nt ih Hien 2 io HED itl EE 1 A) ay Veh aie) iM ap i) ih ICN mi Wait} ne en hl va i / in itt wy Per yise 30) ee AEEA iia Me ay aah eR arias rk Dah Hi WH acht Wik ie ie co Hi iy: is En a ii On eh: Bn kt He Kn a a i AEON rds NAR 7 cS J ey 2 SS ee ee Se See ee Se = Ess sais essere aes Seo et SA Sees et ne ps En in Ren - en ers 3B: afne 5 Eten Ee a Sn i map vatten ne side at ain nh Wi is i i Ai lk H ele rf if tn allt OENE Hi SS ae 3325 == 2, 2s = ASSES eneen See = = =. At ig it yt oe ee on „es: Sins ee ee Sb ote pts EE B D Ee ten, een = SS Sse a = att = SS ps => SSeS en, Ees ent = = ms Eeen ee = 3 zen es Seas En Saas SESS tise Ree SRE SSS ee <= 5 = = Seca See ae EEn: En en se ~ a Se —— ad En En Sar. =e SS = ee ee saa = <= sss -<=-—~ = <= === 2-3 EE En Es i} En ay} ‘ a piety i APS ata Maa iets cara a ae ¢ ae aoe f Nias Bact I ie Ne iy ¥ 4 hy fi : Aa i = = 7S Rs > == Nr 323358 <= “Ste? Pech parte aes Tees ees 05%. Sa == ; ¥ ee pis FOR THE PEOPLE FOR EDVCATION FOR SCIENCE LIBRAR OF THE AMERICAN MUSEUM OF NATURAL HIST Se oe CAN ih Nea : THT hd): Wait , 4 i KONINKLIJKE AKADEMIE _ VAN WETENSCHAPPEN :-- TE AMSTERDAM -:- PREGEEDINGS Ok THE See rion OF SCIENCES VOLUME XXII ap age MULLER :—: AMSTERDAM >: “SEPTEMBER 1930: "ain ay wih. Rhy sisi . es ya i rate ri A a a EG ae Nen { ee ae EE LANE dre Ell Je mi tene Das EA & + ; Be ye Titus nM ore hf wi 7 Jk i ie > fas bl ~ 7 i . “yaorer Jy eel . . fey ji a ne Lew Tak SABES Suse 3 DN Zz ‘KONINKLI JKE AKADEMIE VAN WETENSCHAPPEN -- TE AMSTERDAM -:- PROCEEDINGS OF THE SoeON OF SCIENCES 5.06(49.2) A5 VOLUME XXII — (187 PART) — — (N°. 1—5) — JOHANNES MULLER :—: AMSTERDAM FEBRUARY 1920 14-QUS6S Wammel, IS (Translated from: Verslagen van de Gewone Vergaderingen der Wis- en Natuurkundige Afdeeling Dl. XXVII and XXVIII). Proceedings N°. NY’ NE: Ne N°. CONG EIN TS. edet ND es » „es ft “Ss EL cas se KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN TE AMSTERDAM. Bio oe te DTN io VOLUME XXII N°, 1. President: Prof. H. A. LORENTZ. Secretary: Prof. P. ZEEMAN. (Translated from: “Verslag van de gewone vergaderingen der Wis- en Natuurkundige Afdeeling," Vol. XXVII). CONTENTS. H. B. A. BOCKWINKEL: “On MAC LAURIN’s Theorem in the Functional Calculus” (Communicated by Prof. L. E. J. BROUWER), p. 2. H. B. A. BOCKWINKEL: “Observations on the expansion of a function in a series of factorials”. III. (Communicated by Prof. H. A. LORENTZ), p. 6. H. B. A. BOCKWINKEL: “On the necessary and sufficient conditions for the expansion of a function in a Binomial Series”. (Communicated by Prof. H. A. LORENTZ), p. 18. H. P. BARENDRECHT: “Urease and the radiation-theory of enzyme action”, III. (Communicated by Prof, J. BOESEKEN), p. 29. FRED. SCHUH: “A general definition of limit with application to limit-theorems”. (Communicated by Prof. J. CARDINAAL), p. 46. FRED. SCHUH: “A general definition of uniform convergence with application to the commutativity of limits”. (Communicated by Prof. J. CARDINAAL), p. 48. H. J. PRINS: “On the Condensation of Formaldehyde with some Unsaturated Compounds”. (Commu- nicated by Prof. J. BOESEKEN), p. 51. H. L. VAN DE SANDE BAKHUYZEN: “Photo-growth reaction and disposition to light in Avena sativa”. (Communicated by Prof. F. A. F. C. WENT), p. 57. P. H. VAN CITTERT: “The Structure of the Sun's Radiation”. (Communicated by Prof. W. H. JULIUS), P.tio. A. SMITS, G. L. C. LA BASTIDE, and J. A. VAN DEN ANDEL: “The Phenomenon after Anodic Polari- sation”. I. (Communicated by Prof. P. ZEEMAN), p. 82. H. GROOT: “On the Effective Temperature of the Sun.” Some remarks in connection with an article by DEFANT: “Diffusion und Absorption in der Sonnenatmosphäre”, (Communicated by Prof W. H. JULIUS), p. 89. Proceedings Royal Acad. Amsterdam. Vol. XXII. Mathematics. — “On Mac Laurtn’s Theorem in the Functional ‘'aleulus’. By Dr. H. B. A. BockwinkeL. (Communicated by Prof. L. E. J. Brouwer). (Communicated in the meeting of March 29, 1919). In the third communication of my paper “Some observations on complete transmutation’”’?) | proved a restricted validity of Mac LAURIN's theorem in the Functional Calculus for a normal additive trans- mutation. A normal transmutation was defined by me as follows: 1. There is a functional field /(7'), the functions « of which belong to*) the very same circle (0), and for these functions the transmutation 7’ produces functions belonging to the very same circle (uv), concentric with (6). 2. All rational integral functions are included in the functional field. 3. The transmutation 7’ is continuous in the pair of associated fields F(T) and (a)*). From 2 it can be derived that to any such transmutation 7’ another transmutation P? formally corresponds, which is given by au aa Aus”) age EIR 2/ diene ml! heenescH a where the quantities u”) are the derivatives of the subject of operation u and the quantities a, functions of the numerical variable w, which by means of the formula Am = Sm EK Mm, Em—1 apo eS = ams, 4) +: AET: (2) can be derived from the functions Epa T (a!) [08 NOS) 1) These Proc. Vol. XIX N°. 6 and 8 and Vol. XX N°. 3—7, to be quoted as l.c. 2) A function belongs to a circle, if it is regular within and on it. The symbol (c) means a circle with radius c. 3) See for the definition of continuity l.c. Vol. XIX N°. 8, where also definitions are given of the expressions functional field (F. F.). numerical field of operation (N. F.0.) (here the circle («)) and numerical field of functions (N. F. F.) (here the circle (c)). 4) By mk the binomialcoefficient of order k of the number m is meant. 3 which are the transmutated of the successive positive integral powers of x: the latter functions exist according to 2 and belong to the circle («). The above-mentioned theorem of Mac Lavrin consists in stating the equality of the transmutations 7’ and P, in a certain numerical field (a), which is part of (@) or identical with (©), if the further condition is added, that the latter transmutation is complete in (a’)’). In this statement there is something unsatisfying, if we compare it with Tavror’s theorem for the Zheory of Functions. The latter asserts: “if a function, in a certain circle, has some specified proper- ties (viz. a definite differential coefficient) it can be expanded in TayLor’s series in that circle”. It is therefore not necessary to impose further conditions on that series. Accordingly it would be desirable that also in the Functional Calculus the theorem might be stated in such a way that it were not necessary to impose any further con- dition on the series P corresponding to the given transmutation 7’, but that such conditions were implied in the properties of T. At the time it was our opinion that this was not the case. But now we are in a position to prove the following proposition: The series corresponding to a normal additive transmutation represents a complete transmutation. | For simplicity we consider a circular domain (7) round the origin as a centre and in this the infinite sequence of functions IEEE ta ETE Senate (4) to which, by definition, the infinite sequence of transmuted en L.A corresponds, the latter functions being regular in a circular domain (a). If we denote by e an arbitrarily small positive quantity, then the sequence of functions a x? ym Pir (Gree ool derived from (4) converges uniformly to zero in the domain (a). According to a simple property of continuity (le. IJ, n°. 11) the sequence of the transmuted of the latter functions, which, by the additive property of the transmutation, is represented by (6) 1) A transmutation P, represented by a series of the form (1), is called by me complete in a domain (a), if there is a certain circle (¢), concentric with (4), such that all functions belonging to (o) possess a transmuted, regular in (~). The minimum circle (2), which may be taken for (¢), 1 called the domain corresponding to (a) (lc. Vol. XIX, N°. 6). 1* ed 5, 5, pri ate 6 $0) ’ OSC Roe ON . « . . ( ) o+eé (6 +) (6 + er will converge uniformly to zero in the domain (a); because a normal transmutation is continuous in a pair of conjugate fields. For suf- ficiently large m-values we have therefore in all points of (a) El lo sy ELT ON From the equation (2) it is now easily derived that an analogous inequality holds for the functions a, occurring in the series (1), that is to say, that these functions, too, are less in value than the mt! power of a certain number independent of m. For if (7) is valid for m > m,, we have by (2), |2| being at most equal to a, mo m |am| < Sant Exel + Se mpam—* (6 +. e}k 0 mo +1 m GI “le Ni a 0 (0), (6) being the N.F. F.; and the radius 8 is never greater than o + 2a. Mathematics. — “Observations on the expansion of a function in a series of factorials.” Ul. By Dr. H. B. A. Bockwinxet. (Communicated by Prof. H. A. Lorenz). (Communicated in the meeting of May 3, 1919). 10. We shall now prove that the theorem of NreLsEN is znexact. For this we use the following lemma: co Lf the series Ss: an diverges in such a way that the upper limit 0 of the sum n Sn = Sin am oe Shes kis pce ee (35) 0 for n= ts equivalent to n°, where @ is a certain positive number, then the series oo —~ An eS Mer nnn 0 converges or diverges according as a>>0 or a< 8. Summation by parts gives n n—1 Amn Sn au 1 1 SS SS n Sm ee > 5 ° 37 ~ mt ne zr >» sy (= 0, the limit of the expression s/n is, by the hypothesis, zero for n= oo. Further the upper limit, for m= oo of the general term of the series in the right-hand member is equivalent to m—@1*), if dis a certain positive number, so that the series converges if n is made infinite. So the series in the left-hand member, too, converges fors soon Again, writing n hE Am Sa, n= INGE) . © . . 5 . . (36 ’) me 0 we find on summation by parts n n n—1 =e as a Am ) n= Nr me te = NI San — mM Som [(m +- Le — m*] (38) 0 0 0 From this equality we may infer that, for a< 6, the upper limit 7 for n =o of sn is infinite, and is at least equivalent ton*—*. For, if this were not the case and if s,, were finite for n ==, or equivalent to a lower positive power ‘of n, say n°, then the first term of the right-hand member would be of order n'~*, whereas the terms of the series occurring in that member would be of order mi, and their sum therefore at most of order n’—*. This is in contradiction to the initial hypothesis that the left-hand member of the equality is of order 7’. The lemma has thus been proved. A corollary worthy of notice is that the series (36) for a< @ diverges in such a way that the upper limit of the sum (86’) for mo is exactly equivalent to n'-* (in the second part of our proof we found that it was at least equivalent to this power of n). For, if that limit were equivalent to a higher power of n, say n’—+?, then the limit between convergence and divergence of the series ES 5 == Qe would, by the or Wel just now, be given by B=6@+ Jd, and not by S=@. The same consequence may, for the rest, be deduced by observing that from the equality (87) it follows that for a< 6 the left-hand member is at most equivalent to nf. The same lemma as ae above holds for the series of factorials Gate 39 Basaran Ed passe 8) as may be proved in ae the same way. Both lemmas are moreover a consequence of one another, because the series (36) and (39) converge and diverge for the same values of a, at least if « has not exactly the value 6, which is the limit between convergence and divergence of the series. 11. We now construct a sequence of coefficients ORE Tae mar Ne at Ree Ci) whose upper limit for =o is equivalent ‘i Dn wheres the upper limit of the sum (35) is equivalent to n°, 6 now being a positive number less than 1. It is not difficult to effect this in different manners; but we shall moreover try to secure that the second sum quantity o a n (2) ht en nn m Sm 5 . . . e "3 . . (41) 0 8 becomes of an order which is by less than one unity higher than the first sumquantity (35), i.e. of an order less than n't. We proceed in the following way. For a certain value n =p we take 8) =p, where by the notation p’ the same thing is meant as Z(p’). Now we make the following sums decrease as soon as possible by taking a certain number of coefficients a, for n >> p, all equal to — 1. After p? terms the sum s, has in this way become equal to zero. And after p? terms more s, has attained the value — p’; the value of n is then equal to p + 2p%. It may happen that for this value of n the quantity n° is still equal to p’*). In this case we may say that |s,| has again reached the required upper limit and we assume a certain number of next coefficients a, all equal to +1. If, on the contrary, nf for n=p+2p° is greater than pf, we go on taking a, equal to —1 as long as s, has diminished to such an effect that sa is again equal to 7’. Such an n-value must be reached, if O6<1, for |s,| would become of order n, if we never stopped taking a, ——1. Just as in the former case we assume, after this n-value, the next coefficients a, equal to + 1, until s, has attained the value „? again. And so on. The upper limit for » =o of the sums s, is then equivalent to »? and that of the coefficients a, is equivalent to n°. We call the values of n for which s, is equal to ni critical values and we denote them by nz for k—=1l, 2,.... If we assume arbitrarily a certain value n=p as a critical value, then it follows from the preceding construction that the following critical values are uniquely determined. We might continue the same construction towards the side of the smaller n-values, but this is of no importance, since the asymptotic behaviour of the quantities in question is solely effective. The graphical representation of the coefficient @p, considered as a function of n, consists of a number of horizontal line-segments, which are alternately above and beneath the n-axis, and are at a distance 1, the change taking place in the critical points. The graphical representation of the sums s, consists Jt of joined line-segments including alternately angles of Dn and JT ae with the n-axis; and the alternation again takes place in the 1) We may write (p + 2p 4/9 = pi+26p24—1 (1—e), where lim «=O. From this poe it follows that for 6<1 and large p-values, E (p+ 2p)? is in general equal to E (p'). 9 critical points, where the tops of the lines lie within the parabolic curve y= + nf, at a distance less than unity from this curve. We shall now shew that the upper limit for n= of the second sumquantities (41) is equivalent to n°. The sums Nk net nid m 8m and i Sm 6 we Nk—Nk ne+l are arithmetical progressions consisting of terms of equal signs only; their total value is easily verified to be exactly equal to + (77). Thus we have, but for a certain constant, whose value is of no importance for the asymptotic behaviour of these sums: nk + nk? Sin so (n,°)’ = (n,°)? HE, GOE (nx’)? 0 Since the terms of the sum in the right-hand member increase in absolute value or remain constant over a certain number of n-values, the sum itself is, in absolute value, less than the last and the first term together. So we have (2) 8 nj np where e is a positive quantity which tends to zero for k=o. Further, for an n-value lyiug in the interval between nj — nf = < (ne)? + (2,5)? < (n1°)' (14e) =a + Ba and nz sale the value of s®) lies between the two values of s® for these two limits, because s, remains of the same sign between them, and therefore se) varies monotonely. So we have for all n- values of the interval considered a2] — (nyt? (1-46) Z mx (1-48). For the part of the n-values greater than nz we therefore have a fortiori ec 0 (1 Trajet. 46 ate eer B) For the remaining part we have n> nk — NIS. Now a nye? — (nz — nye? + nj’) = (nk — nj)? (1 + €) where, again, e is arbitrarily small for sufficiently great &. Thus for the latter n-values, too, we have 10 EN ntldr). (2) The inequalities (42) show that the upper limit of s®) for n = ao is, indeed, at most equivalent to n°. On the other hand it is easily seen that 2? is exactly the power of n to which the upper limit of s@) for m= is equivalent, but this is of no importance for our purpose. We now construct a function p(t) by means of the so-determined series of coefficients (40) pO= Sn ae DEENSE ER 0 For this we may also write gy (t) = (l—t) E Sn = (1-1)? Ya se ide (43) 0 The last member of this equality shews of what order gp (t) is at most for ¢= 1-— ò. Since we have seen just now, that the upper limit of s@) for n — oo is at most equivalent to n°, the function eo . n s,(2) te 0 is, for ¢==1, at most of order 1: (1—d)!+?*, Hence p (tf) is for t= 1 equal to zero, if O< 4, and for 4<.0< 1 at most of order 1:(1—2)?*-!. Thus we have == On joes FOS Mor, 6.4: Since 206—1 = 9—(1—0) and 0 <1, we have in both cases 1< 8. From the lemma proved in the preceding paragraph it now fol- lows, since the upper limit of s, for n =o is equivalent to n’, that the series of factorials nl ay POE reen nee oa (6) dwerges for Riv) <0. Hence it is not true here that this series con- verges for R(x) a r'(=0) and Rw) >A, and a fortiori not that a certain integral of the form (1), for all such values of x, might be expanded in such a series. Therefore Nivisun’s theorem is inexact. 11 12. It is convenient to observe on this occasion that by means of the proposition of N°. 10 we may infer from the first part of for- mula (43) that never convergence of the series of factorials (6) takes place for Riv) < 4, a truth already shewn in another way in N°.9 of the precediug communication. For, if the upper limit of s„ for no is equivalent to „, then the function represented by the series oo D n8nt? 0 is, for t=1, at most of order 1: (1—7d)!+* and hence gp (A) is at most of order 1: (1—7)’, so that the number 2 is not greater than 0, whereas the limit between convergence and divergence of the series of factorials according to the lemma mentioned, is afforded by R(x) = 0. 13. At the end of the first communication we said that, if a certain argument mentioned there and used by NrierseN were not erroneous, then another very general case of the possibility of expand- ing the integral 1 2 («) = (po agrar. Er ED ‘0 in a series of factorials would be proved by it, viz. the case when the series of factorials corresponding to that integral converges. We shall now prove that this case, though not established by NIELSEN, is, indeed, exact. In order to do this we shall make use of the following lemmas: l. If the series of factorials converges for a certain value x= c, then it converyes for any value of « whose real part R(x) is greater than that of c. Let mI! am m Pom = Sn, ly . . . 5 . . (44) n Siep we find on denn by parts ml! an net m! am Pe+m+1)_ Brey +1) oem P(c+m-+ 1) POS l-1 Beene (el 1) M(@ +141) as Sn mn T(c 4+- m+ da) br P(e +m + 2) 12 On account of the supposed finiteness of s, 4, the limit of the first term of the last member is for Zr) > R(c) equal to zero, if / increases indefinitely and the series in that member converges abso- lutely. Hence the series in the first member converges also and we have ; bas ml! am A ae Snm T'(e + mt 1) >" Il + m -+ 1) im (ee) Dn IME + m + 2) ij ; (45) n rn by which the lemma is proved. It follows from it that the limit between the domain of convergence and the domain of divergence of a series of factorials is formed by a straight line parallel to the imaginary axis. Moreover we may infer from the equality (45) the following lemma: I]. In any finite part S of the half-plane R@) > Ric) + d, 0 being a fixed, but arbitrarily small real positive quantity, the series of factorials converges uniformly. For nm may be chosen so large, corresponding to any arbitrarily given e, that for all m>n Sim T(e + m + 1) é («+m TT COEN and thus nen Ny en F(c +m pated & | a—e | (46) | wom (re -+ m + 2) R (wone) © Since Mlv—e)>d, the right-hand member of this inequality is, for a sufficiently large n-value, which is independent of the value of xv in the domain S, less than ¢. This was to be proved. From the inequality (46) we may infer the corollary Ha: /f the series of factorials (6) converges for a certain value «= c, then it converges uniformly on the half-line beginning at x=c and having the direction of the positive part of the real axis. For on this line R(w—c) = «—c; hence the right-hand member of (46) passes into e:nRe-e) and this is less than ¢ independently of the place of « on that line. The corollary is an analogon of a well-known proposition of Agen on the uniform convergence of a power-series along the radius of a certain point of its circle of con- vergence, if that power-series converges at that point. From lemma II, in connection with the fact that the terms of the series of factorials (6) are continuous functions of «, it may be inferred that such a series represents a continuous function of 2 in the domain 13 S, even in case S lies wholly or partly in the domain of only conditional convergence; of course it must be supposed that all points of S lie at a distance greater than some fired arbitrarily small quantity } from the straight line parallel to the axis of imaginary quantities which forms the limit between the domains of convergence and divergence of the series. Finally we shall prove that this series represents an. analytic function in S, and for this it will be sutfi- cient to show that the series of differential coefficients, with regard to rz, of the terms of the first series converges in the same domain as this first series. The series of derivatives may be represented by oo oe en n! ay See et ty ell MED) 0 where, in the notation of NieLseN '), w(e) denotes the logarithmic derivative of the Gammafunction: dlog T(z) F(z) es dx i‘ I (z) Summation by parts gives, if the notation (44) is used again i M!dmW(e+m+1)__ sad ml am I (c+-m-+1) w (em 1) De 7 “= wp (2) (48) m I (@ +m + 1) c+m-+ 1) P(e +m +1) ri st V(e+l+1)w(@+l+1) yw SnmI(e+m+ lo . ; == P(e +1+1) phils raat, c)yp(a + m+ 2)—1], where the equation of finite differences 1 satisfied by y(#) is taken into account. Since, for «= the prin- cipal part of We) is equal to loge, the first term of the second member has, for &(«—c)>4d, zero as a limit for /—=o, and the series in that member converges absolutely. Hence the series in the first member converges also and we have ; Has m! an (a + m +1) ay ND Symi (ce + m +1) pie, U (« 4m + 1) nae ae: P(e + m + 2) n n ed) w (w+ m42)—-1]. From this equality it may be easily deduced that, corresponding to any e, there is a number N such that for n > N and for all z in the domain S 1) Handbuch der Gammafunktion, p. 15, Leipzig, Teubner. 14 IN ECI E («+ m + 1) dn? pee 2 slog n n that is <{e, since (logn):n? is zero for n= oo. By this the conver- gence of the first part of the series of derivatives (47) has been proved, together with the wniform convergence in S. The second part need not be examined more closely, since this part forms a series which, save as to the factor We) not depending on n, is equal to the series of factorials itself. On account of its uniform convergence in the domain S and the continuity of its individual terms the series of derivatives in question represents in S the derivative of the function determined by the original series of factorials. The differential coefficient of this function is therefore determined at any point of S, and is independent of the direction in which it is taken: thus the series of factoruuls represents an analytic function in the domain S. 14. We now proceed to settle the question indicated at the beginning of the preceding section, and we first consider the case 4.=0, in which the upper limit of a, for =o is equivalent to ne. We further assume the limit of convergence and divergence of the series (6) to be given by R(x) = 9, 9 being a positive number less than unity. Since, as we showed, the number 4 is in any case not greater than 0, we have lim(l—i)*g()=90 for R(x) >. i=1 From this it follows that the integral (1) converges absolutely for R(«) > 6. At the same time this convergence is uniform in any domain S wholly included in the finite part of the half-plane R(«z) > 6+ 4, where d is a certain fixed, but for the rest arbitrarily small positive number. The same holds for the integral 1 fo (t) (1—t)?—! log (1—+) dt, 0 which may be deduced from the integral (1) by differentiating the latter under the integration-sign with regard to x. Hence this integral represents in S the derivative with regard to 2 of the integral (1), so that the latter is an analytic function of # in the domain S.*) 1) The statements mentioned here briefly have been established by PINCHERLE in his paper “Sur les fonctions déterminantes”’, Ann. de l’Ec. Norm. (3) XXII, 1905, p. 13—17. Their analogy with those we proved in the preceding section is manifest. 15 Now, the two analytic functions represented by the integral (1) and by the factorial series (6) are known to be equal in the domain of absolute convergence of that series, formed by the half-plane R(x) > 1. According to a well-known theorem of the theory of functions they cannot therefore differ in the domain of only con di- tional convergence Bake Thus the indicated proposition has been proved in the definite case 2=0. Further, if 2 >0 and the limit of the domains of conver- gence and divergence of the series (6) be given by R(s)=i! +0, then, again, 20, but in this case the series affords the analytic continuation of the integral in the part of the plane given by MOK Ra) <0. We may also say that a certain remainder of the series (6) is represented by an integral of the form 1 fom (OPOE te se oh ea eee ACS) 0 in the whole domain of convergence of the series. We thus have proved the theorem: A series of factorials, whose domain of convergence is the half-plane on the right of the line R(x) =, is represented in a possibly existing domain of only conditional convergence by the same integral of the form (1) as in the domain of absolute convergence, if §>0. Lf, however, E o—1=4', be expanded in a series of factorials. Further the integral 1 fo. Oet de 0 c.q. an integral containing p,@%(t), can be expanded in such a series for R(«)>w,. Hence the integral (1), being the sum of the two preceding integrals, ¢.q. an integral containing pC (¢), can be expanded _ in a series of factorials for Klep so, of (2) ta, according as w, or 4’ has a greater value. In the first case the series converges conditionally for Bk (nadeel, and in the second for Need In both cases we may state the following proposition : Lf w is the order of p(t) along the whole circle of convergence, and therefore 4’ = aw—1 the characteristic of its coefficients, and if w, is the order of g(t) at the point t=1 of that circle, then the integral (1) (c.g. the integral (8)) can be expanded in a series of factorials for such values of « as satisfy at the same time the ine- qualities R(e) =d and Rl) > o,. 1) „Essai sur l'étude des fonctions”, Journ. de Math., 1892, p. 172. #7 We cannot say, however, that this proposition is a substitute for that of NierseN, because a series of factorials sometimes converges for Re) < w,. Take, for instance, the function of Werersrrass k VE eM Ry eh aria a oe? nef ee ee This function has its cirele of convergence as a natural limit, because all parts of the circumference are equivalent, as appears from the substitution rn ih k (Meee ger eee where k and 4 are arbitrary integral numbers. The order w, in the point t=1 cannot, therefore, differ from the order w along the whole circumference, and this order is equal to 1, the characteristic d’ of the coefficients of the power-series being zero. The quantity à is, however, also zero and the series of factorials converges for Rv) > 0 and is, for all these values of 2, equal to the integral (1) *). 1) The convergence of the series of factorials is, in this case, absolute, in the strip of plane 0 < R(x) <1, owing to the large distance between the coefficients of p(t) which differ from zero. If, therefore, we spoke continually in the preceding investigations of conditional convergence for R(x)< A’ +1, we meant this in general. Proceedings Royal Acad. Amsterdam. Vol. XXII. Mathematics. — “On the necessary and sufficient conditions for the erpansion of a function in a Binomial Series”. By Dr. H. B. A. BockwinkeL. (Communicated by Prof. H. A. Lorentz). (Communicated in the meeting of May 3, 1919) PincHERLE has given a necessary and sufficient condition for the expansion of a function in a binomial series (Binomialkoeffizienten- reihe)*). It runs thus: The necessary and sufficient condition that an analytic function w(a) may be expanded in a series of the form ae at wo (@) = Sn 0 (7) ) Sed oe ae 0 is that w(x) be coefficient-function (fonction coefficiente) of another analytic funtion q(t), which ts regular and zero at infinity and whose singularities lie all within the circle (1,1), with centret=1 and radius r=1, or on the circumference of it, provided that, in the latter case, the order of g(t) on the circumference, taken in the sense defined by Havamarn, be finite or negative infinite *). By a coefficientfunction w(z) of an analytic function g(t) of the kind mentioned PincHERLE means a function which can be deduced from g(t) in a more or less simple manner, according to the order of g(t). The relation between the two functions is, however, always such that conversely q(t), called by PincuerLe the generating function (fonction génératrice) of w(x), follows from w(x) by the equation a + 1 po ye. op nest an ema ae This means: the coefficients of the series of negative integral powers of ¢, in which g(t) may be expanded in a neighbourhood of to, are equal to the values of w(x) for positive integral values of «; the name coefficientfunction for w(x) is due to this circum- stance. The question now arises, how it must be discriminated if a given function may oe expanded in a binomial series. This question is not IS. PINCHERLE, “Sur les fonctions déterminantes’’, Annal. de Ecole Normale, 1905. 2) A circle with centre ~ and radius 7 will be denoted by (a, 7). 19 answered by the theorem of PincuErLE, at least not in a simple manner, as will appear from what follows. In order to investigate the question we should commence to deduce the series (2) from the given function w(x). Next we should examine whether the func- tion g(t) represented by it has the required properties: to be regular without the cirele (1,1), and on the circumference of it of finite order. For this we should try to transform the above series into another according to negative integral powers of t—1 ve) Cn OE The relation between the Rn of the two series is given by the equations cn = w (n+ 1) — & ) w(n) + ...+(—1)"w (1) = A"*[w(1)]') . (4) and ate (Iet +6 er ER deter) By means of (4) we must see if the series (3) converges without the circle (1,1), and further if the characteristic à of the coefficients Cn, defined by n= log n 6) is not positive infinite; the latter being the condition that g(t) shall be -of non-positive infinite order on the circle (1,1). But the relation (4) is rather intricate and so it may be very difficult, if not impossible, to perform the just mentioned research. Suppose this, however, possible, and let 2 differ from + o. Then we have to examine whether the given function w(x) is really the coefficientfunction of g(t). For there are a great many functions Q(z) giving rise to the same generating function g(t), viz. all those contained in the Ee 2 (x) = 0 (2) + F (a) where F(x) is a Mheen that vanishes for positive integral del: of zv. It is therefore necessary to consult the definition of the coef- ficient-function given by PiNcHerLE and to apply it to the obtained y (t) in order to see if the original function w (#) is the result. But this is again not very easy. If the characteristic 2’ defined by (5) is less than —1, then p (4) is finite and continuous along the circum- 1) This symbol denotes the n-th difference of (x) at x=1, the increase Ax of the argument x being equal to unity. 2% 20 ference of the circle (1,1). The coefficient-function is then defined by the integral 1 zei foe LEAS an elven ORN) (1,1) taken round the just mentioned circle, and this integral is easily seen to be equal to the binomial series (1) in the half-plane of « on the right of the imaginary axis, provided definite agreements as to the value of 4! be made. It may, however, be difficult to investigate whether the integral (6) is equal to the given function w (az). And in any case the investigation is intricate if 2’ > — 1, especially when the difference between 2’ and —1 is rather great. For the coefficient-function of p(t) is then brought into relation with that of another generating function, with 2’ < — 1, by means of a polynomial consisting of a very large number of terms. The question therefore naturally suggests itself, if it is possible, to. find simpler tests which are sufficient for a function to be expanded in a binomial series. This is, indeed, the case, and we may, moreover, say that the obtained properties are about necessary *). For we can prove the following theorem: If a function w (w) is regular in the finite part of the half-plane R(#)>y¥?) (y= real), and if, in that domain, tt satisfies the inequality ONES ANC ot) oe ee where M is a positwe and l and 6 are real numbers, the latter such that b+ y>0, further a a complex number on the circumference of the circle (1,1), variable with the argument w of «—y, the argu- ment a of a being equal to —w so that WS DCO TE ed EE eat) then, in the domain (8 = real > v). R(@) Loe.) ae Ae 1) For the sake of comparison we observe that for the expansion of a function in a series of factorials i n! an ETEN Dele in) there is a necessary and sufficient condition, stated by NirrseN and simplified by Pincuerte, which has some similarity with the above condition for binomial series. But it is not possible to find simple tests for the expansion of a function in a series of factorials. The only simple sufficient condition which may be given in this case, is that a function can be expanded in a series of factorials, if it is regular and of zero value at infinity. But this condition is far from necessary. 2?) R(x) means the real part of a. 21 the function w («) can be expanded as a binomial series 7 a BN) w (2) = Si a ( ) ak le ed Dd CLO) at n 0 if L+8—k>y, and otherwise the expansion is possible in the domain R (2)> y. The special value —w of the argument « of a is such that the expression at, w—y being given, has the greatest modulus com- pared to those for other values of a on the circumference of the circle (1,1). Lf the inequality (7) holds for fixed a-value on the circle (1,1), then expansion of w (w) in the series (10) is possible in the domain | BNB ln na en (9% of 1+ 8—15y, and otherwise in the domain R(#)> y. The sufficient condition for the expansion of a function in a binomial-series contained in the above theorem seems, indeed, very simple. If a function w (7) can be represented by the equality D= Et Oye ee) A ee EI) where c is a fixed number within the circle (1,1) and u (z) a function remaining within finite limits in R(z) > y, then it satisfies the ine- quality (7) for a value of / differing arbitrarily little from — oo, and therefore it can be expanded in a binomial-series in the domain R(«)>y. For c=1 formula (11) gives an expression which shows that all functions regular in the finite part of the half-plane R (x) > y and vanishing at infinity may be expanded as a binomial-series in that domain; further all functions becoming infinitely large at infinity of an order lower than a certain finite power of z; so all irrational and logarithmic expressions. The way in which we have arrived at our theorem is substanti- ally the same as that followed in the ordinary theory of functions of a complex variable, in order to obtain the expansion of a function in a power-series; it is founded upon the fundamental theorem of Caucny. According to this we have . ih w (z) dz ot) (2) 271 zE W (12) where the integral is taken round a closed curve W, within and upon which w(z) is regular, and which contains the point z= in its interior. If we wish to deduce from this integral an expansion according to positive integral powers of w—a, a being a number 1) This series is taken instead of (1) for the sake of generality. 22 within W, then we start from the known expansion with known remainder of 1:(z—.2) in such a series. In the same way we may reach our present purpose, if we use the known expansion with known remainder of the just mentioned elementary function in a binomial series, viz. ni I x (ef) ..(@—B—m+41) (w—A)...(@—B—n41) 1 m mara (e—f)...(e—B—m) ee ee ee er es 0 Substituting this expression for 1/z—a in the integral (12) and choosing the path of integration so as to include, besides the point z=, the points z= 8, 8B+1,...,8-+n—1, we find *) n—1 oi" Paro + Ba AE 0 where i 2) (v—6).…… (w—s—n+]1) dz (e—B)...(¢—B—n-+1) y= Tr (13) Zi Formula (13) is the ordinary formula of interpolation of Newton with a remainder added to it and valid for all complex «-values lying within W. If all points z=g8, BH1,...,8 And, are to lie within the integration-curve W, this curve will in general have to be modified with increasing ». It is required to choose W as fit as possible, that is to say: so that the remainder (13) tends to zero with indefinite increase of n, and that vet the aggregate of functions (x) for which this takes place, is as extensive as possible. If, now, the form (7) is taken as majorant-value of these functions, where the number a is, as yet, left undetermined and the number y, in order to have a definite case, is chosen zero (so that 8 > 0), it is found after a rather long but principally not difficult inquirement: 1. that the most favourable integration-curve is a circle with z= 7 as centre and n as radius so that it passes through the origin; 2. that for a a complex number may be taken lying on the circumference of the circle (1,1), with the specifications concerning the domains of validity already mentioned in the Statement of the above theorem. We may further observe that, in case the number ec in formula 1) If a few points @, @+1,..., are excluded from the closed curve W, we obtain an expression the further examination of which leads to the so-called zero- expansions, which are treated in an elementary way by PINCHERLE (Rendic. d. R. Accad. d. Lincei, 1902, 2e Sem.) f 23 (11) is real and not greater than 1, a fired integration-path W for the remaining integral may be chosen, as soon as the number n attains a certain magnitude, and for this we may take the imaginary axis in this case. The proof that im Rk, =O for n= o is then very simple, so that the above mentioned particular cases in which a function can be expanded in a binomial series, may be derived in a short manner from Cavcuy’s integral. As further regards the question, how far the inequality (7) is necessary for the expansion of a function in a binomial series, the way in which the-sufficient condition has been obtained gives us the conviction that the aggregate of functions determined by the latter condition is as large as possible. In order to come to certainty concerning this it is necessary to investigate how a function represented by a binomial series behaves in the domain of convergence of that series. This investigation may be effected by means of the statement, contained in the theorem of PiNcnerre, that a binomial series necessarily represents a coef/icientfunction, at least in the domain of absolute convergence of that series, for to this only the proposition of PINCHERLE applies. For simplicity we assume for the binomial series the original form (1), which is the one considered by Pincnerre. If the characteristic A' of the coefficients c, is less than —1, then, as already mentioned, the binomial series can be represented by the integral (6) in the half-plane R(x) > 0. It can now be proved that this integral satisfies, in the domain mentioned, the condition (7), with y = 0, the exponent l being subject to the inequality Bed Wv! ar be Bll et) where d is an arbitrarily small positive number. This condition can further be specified by evtendiny a certain property of the operation I by means of which, according to the view of PincHErLE, the gene- rating function gp(t) passes into the coefficient-function w(z); we mean the property expressed by the equation 1) (2) I{g™@] = Ea EU" p (O1, This equation is given by PrNcnerue (le. p. 30) for the case r is a positive integer. If 7 is replaced by a and g(t) by g(t): (t-—1), the formula passes into r(: * op 1D oe. |- GE OEE _ (t—1)* (re +e) (t—1)® The last equation appears indeed to be true for arbitrary positive 24 values of a'), if for the derivative of any negative order the defini- tion of Riemann is adopted, which in the present case, a neigh- bourhood of infinity being regarded, can be expressed by the identity en p(t) (w—t)*—! (—1)«D Gie aan nit fj plu)du. 0; <6) Since as domain of ¢ and w the part of the plane outside a certain circle with centre (1) is considered, it will be convenient to assume for path of integration between w=? and w= o the half- line which has the same direction as the vector from u=1 to u==t. The quantities u —t and w—1 then have the same argu- ments and (w— t)*:(w—1)* is real. With these agreements we have the expansion DD Be en EN l +4) (¢—1)»+1 om © (17) so that the derivative of eam order — « of the expression y (t): (¢ — 1)* is, as q(t) itself, regular and zero at infinity. The characteristic of the derivative is, however, a less and this makes it possible, by means of (15), to express the coefficientfunction w («) of a generating function p(t) with characteristic 4’< —1 in terms of another generating function g,(t) whose characteristic is any small amount less than —1. The function g, (¢) is constructed in such a way that the given function p(t) is the derivative of a certain negative order —a of p‚(t):(t —1)* and the number «a is selected from an aggregate of positive values, whose upper limit is equal to the difference between A and — 1. In other words, if ¢ (¢) is given by (3), we take ——~ en 9, (1) = Si Det 0 where the meaning of c’, is given by awk F (n+ 1) Cn en == TAD with gesl Ad 8 ate eee) J, being any small positive number. Then, according to (17) soy ane ee 1) | have communicated the proof of this truth in the Proceedings of the meeting of September 27, 1919. 25 and hence by (15) yp uv “ae ON [ep (t) |= @) | ze | AEN 1) T'(@+a) (t—1)% Not only ¢, (#), but also the function _ top, (l has the property that the operation J applied to it gives a coefficient- function satisfying the condition (7), the inequality (14) for / being left unaltered. We only have in this case y= -—a, instead of y =O, and the domain of validity is determined by A (#) > — a, or, according to (18), by te Ween tects cote Road Te, 0 (20) where Fee ese tic ne at, (Oe eee (A) That is: the domain of validity of (7) is the domain of absolute convergence of the series (1) (for d, is arbitrarily small). For the whole right-hand member of (19), that is for w(«) we therefore have the inequality | w (a) |< M | (@ + by at-C+4)|. 2 2 . … (22) where /, now, satisfies the condition eager eee walt taan geent Nes) If, at last, the characteristic 2' of g(t) is greater than —1 or equal to — 1, then, after Pincarrre, the coefficientfunction can be expressed in terms of that of another generating function p, (0), with a characteristic less than — 1. First, let Ly areal then PINcHERLE considers the additional function p (¢) t) = — D-1| —— }}, p, @) D (25) having a characteristic 4’—1, which, therefore, is less than —4, so that the corresponding coefticient-function w, (#) satisfies, in the domain (re) >a—1-+4d,, the inequality | w, (#) |<< M| (@ + db)! ax—C—-144) | with hee i tod, od The coefficientfunction w (xv) of p(t) is connected with the latter by the formula *) w (cz) = A [(z — 1) w, (# — 1)] 1) PINCHERLE, |. c., p. 64. 26 from which it follows that w («), precisely in the domain (20), satisfies the inequality (22) with, for /, the inequality (23). In this manner we may prove the same inequality for the intervals (0, ®)\- (1; 2) .+- of 2 in succession. If w (x) satisfies the inequality (22) for a certain value of /, then, evidently, for all greater values. Thus there is a lower limit /, for all such values, but this may possibly not be substituted for / in (22). Instead of this we may however write ale) (ebb) at —G), Soe ee with the meaning that (22) holds for any />/,; we may call (24) an equation of equivalence and say that w(x) is equivalent to the right-hand member of this equation. The exponent /, satisfies the condition Tae me ARA Rr /2) since Ò and d, were arbitrarily small. The proposition relating to the necessary condition for a function to be expanded in a binomial- series may thus be expressed in the following manner: A binomial series of the form (A) represents in any half-plane Riv) 54+. 4, differing arbitrarily little from its domain R (x) >4, of absolute convergence, a function w (x), which satisfies the equation of equivalence (24); the exponent 1, satisfies the inequality (25). If, now, this proposition is compared with that relating to the sufficient condition, then, to begin with, we find a complete accord between the majorant valaes (7) and (22). These majorant-values are, therefore, both necessary and sufficient. Further, as regards the domains of validity, the inequality (9) here becomes (a) > +}, since we had 8=1, or we may also write R(w) >1, +4 if 7, is again the lower limit of the /-values which may be taken for the given function. From (25) the same inequality follows with regard to the domain of absolute convergence. Since the domain of possibly conditional convergence extends at most over a strip of unity-breadth on the left of the domain of absolute convergence, the investigation performed by us leaves room for the possibility that a binomial series sometimes represents a function satisfying the con- dition (24) also in a strip determined by L—- tke) +3 or in a certain part of it. In order to come to certainty con- cerning this point, we should have to examine how a function repre- sented by a binomial series behaves in the domain of conditional convergence of that series. To such an investigation we have as yet 27 not arrived; but we may already perceive that the result could not fill up the gap which, as regards the domains of validity, exists as yet between the necessary andthe sufficient condition. First: if a function w(x) satisfies the equation of equivalence (24) for a fixed value of a on the circle (1,1) and for a certain minimum-exponent /,, then, on account of what has been remarked on the expression at, immediately after formula (10), that function satisfies the same inequality, when the number a varies, in the specified mode, together with the argument yw of «. The index /, cannot, however, be dimi- nished, because it must at all events be taken for w— —a, if ais the argument of the original fved number a. The statement belonging to the inequality (9’) informs us, however, that in this case expan- sion of w(e) in a binomial series is possible for R(«) >J/,. The function Eee fg eo (5) 0 for which we have a= 2, /,=0, affords an illustration of this fact, for the expansion is really valid for R (#) > 0, and it is conditionally convergent for R(«)<1. Therefore we can never find R (#) > /,+ 4 as a necessary condition whereas our theorem concerning the sufficient condition only says that expansion is possible in the domain defined by the last inequality. Secondly the last condition only holds in case w(x) has no singu- larities in the finite part of the domain A(x) > l, + 4; for otherwise for the latter domain the one must be substituted where w(2) is regular and that was defined by the inequality A (a) > y. Thus the proposition regarding the necessary condition states that for points in the domain of absolute convergence of the given binomial series we have f(x) >/,-+ 4, but conversely it is not true that in the domain determined by this inequality there is certainly absolute convergence. A simple example is furnished by the function LO aes For this function 7, = — oo, and yet the function can only in the domain of regularity R (7) >>0O be expanded in a binomial series of the form (1). From these remarks it will be clear that in order to fill up the gap existing as yet between the necessary and sufficient conditions we must give more specified propositions for both conditions. In 28 other words we should have to succeed in dividing the functional aggre- gate of all functions satisfying the equation (24) by special charac- terising properties; and so also the aggregate of binomial series, in such a way that between the two kinds of sub-aggregates there existed a complete correspondence, such that functions of some sub- aggregate K could only be expanded in binomial series belonging to the sub-aggregate K and in no others. But the problem to find suchlike characterising properties will perhaps be very difficult, since it is required for it to derive the character of a function from that of the coefficients of the series representing it; a problem which already causes the greatest difficulties when it regards the more known power-series. Chemistry. — ““ Urease and the radiation-theory of enzyme action’, III. By Dr. H. P. BARENDRECHT. (Communicated by Prof. J. BöESEKEN). (Communicated in the meeting of May 3, 1919). 8. The influence of other substances on the urease activity. The question was raised, whether in these concentrated solutions of urea the H-ion concentration was perhaps lowered also through the slightly alkaline substance urea itself. The electrometrie measurement of the py in the 8°/, phosphate solution showed indeed a distinct increase after addition of 8°/, urea. It was, however, soon established, that this increase was not due to a shifting of the true reaction to the alkaline side, but to a diminution of both H-ions and QOH-ions, or, in other words, to a decreased dissociation of H,O in these concentrated solutions of urea. This conclusion was readily arrived at by the following experiments. The py was determined again in 8°/, phosphate solutions. Then a second time after addition of a small amount of ammonia (1 c.c. of 7 N). For each of these estimations 10 c.c. of 9.6°/, phos- phate was brought with water or diluted ammonia to 12 c.c. The results obtained are summarised as follows: PH in: | Diffe- | Phosphate | Diffe- | Phosphate + 1 c.c. Phosphate + 8 %/, urea | rence | alone rence | NH; 51, N 6.79 0.11 6.68 | 0.03 6.71 6.97 0.08 6.89 0.03 6.92 7.59 0.10 7.49 0.07 | 7.56 1.93 0.10 1.83 0.18 | 8.01 8.20 0.10 8.10 0.34 8.44 As will be seen from SÖRENSEN's determinations of pg in standard phosphate mixtures, the buffer-value of these solutions becomes considerably diminished, when py approaches 8. Therefore the same 30 small quantity of alkali, which will leave a py of moderate value practically unchanged, will cause a distinct rise in a py of about 8. This is clearly borne out by the results, obtained by the addition of ammonia. The urea, however, increases the py to an equal degree, whatever its original value. This increase is therefore evidently not due to an alkalinity of urea. A substance, however, which possesses no alkalinity, can only lower the H-ion concentration, if it lowers the OH-ion concentration just as much. This effect may be expected from all neutral substances, when added in such an amount to the solution, that the concentration of H,O is appreciably diminished, provided their ionising power is much less than that of water. In such a case the dissociation product Cu XX Con or, as it is usually called, the dissociation-constant of water &, will show a diminution. Similar changes of the dissociation-constant of water have been studied before, for instance by LÖWENBERz *) in a paper on “The influence of the addition of ethyl-aleohol on the electrolytie disso- ciation of water”. In view of this influence on ky the study of urease action in solutions, mixed with some alcohol, promised interesting results. The alcohol, employed in these experiments had been freed from carbonic acid by boiling in a flask, attached to a reflux condenser. The same 8°/, phosphate buffer-mixtures and Soja extract were used as before in the determination of m in part 4 of this paper. The amount of alcohol, to be mixed beforehand with the phosphate, was calculated so as to obtain a concentration of 5°/, in the final solution. For the sake of comparison the same Soja-meal was used as in the previous experiments. Those beans had been powdered a year ago and the meal had shown already some signs of weakening. Especially in alkaline solution its constancy of activity had been found to be diminished. Therefore in the experiments with high Pu the m of the first stage had to be taken rather than the mean. And this value was multiplied by the ratio of the activities at the same py, found formerly and now. The values, obtained in this way for m in the alcohol solution are plotted in figure 6 /, in which for the sake of comparison the original curve for m of fig. 3 is also reproduced. 1) Zeitschr. physik. Chem. 1896, 20, 283. 31 The third curve II of this tigure has the following meaning: Figure 6. 9 8.5 8 LAT d Ono 6 PH According to our interpretation of the rise of py of 0,10 by 5°/, alcohol, the &,, had increased 0,20, or the poy 0,10. As demonstrated in part 5 of this paper m as a function of py is: 4828 132.6 « 10-8. Alten Le eee (1) kw The &, being diminished by the alcohol from 10-1378 to 10-1398, it was now possible to calculate a new curve II for m by means of this formula. The difference between I and II may be due (at m=18 + 32 least partly), to the experimental errors, which were very large with the old Soja-meal. In an earlier stage of these researches the influence of neutral substances like alcohol on urease-action had been studied with the aid of the ammonium-carbonate —- carbonic acid as a buffer, in experiments like those described in part 6. For instance the experiment of May 25th 1916, in which 0,5 °/, of urea in ammonium carbonate (= 2°/, urea) with and without 5°/, of aleohol had been investigated, produced the results, represent- Figure 7. 5 9/9 alcohol 10 0/, alcohol 40 80 120 160 200 240 40 80 120 160 200 240 minutes. ed in figure 7. The mean of m without alcohol was 0,00381, with 5 °/, alcohol 0,00335. For the calculation of m it was assumed, that the py had been elevated by 5°/, alcohol in this case by the same value as found above in the phosphate solutions. From figure 6 it will be seen, that between py == 7 on the ori- ginal curve and py=‘7%,1 on curve II, theory would predict a decrease in m from 51,3 to 44,5, that is from 1 to 0,87. According to the above results a decrease is found from 38,1 to 33,5, that is from 1 to 0,88. Closer consideration as well as further experimental research showed however, that the problem is more complicated than repre- sented so far. 33 Alcohol not only diminishes the dissociation constant of water by diminishing the concentration Cio in CyCon = kCy,0, but, owing to its ionising power being far less than that of water, it will also diminish the dissociation constant of a dissolved electrolyte. Therefore the dissociation constants of the amphoteric electrolyte urease must be expected to be also diminished by aleohol. The effects of the first influence of alcohol are, as shown in fig. 6, a lowering of the curve and a displacement of the maximum to the left; that of the second one, the diminishing of 4, and k, at the same rate, is evidently an increase of m, the undissociated fraction, without changing the abscissa of the maximum. Figure 8. Proceedings Royal Acad. Amsterdam. Vol. XXII. 34 What we observe is therefore the superposition of a positive and negative effect, the decrease of the dissociation constant of water predominating in this case. Repetition of the experiment of May 25'® 1916, but now with 10 °/, alcohol, gave on May 30 the result, summarised in figure 7. With- out alcohol was found m = 0,00326, with 10 °/, aleohol m= 0,00267. The electrometric estimation of the py in 8 °/, phosphate solutions with 10°/, aleohol had given: with 10°/, alcohol 7,13 io without alcohol 6,89 1,29 difference 0,24 0,24 Thus a constant rise of 0,24, which means a change in A from 10-1378 to 10-1426, Figure 8 shows, together with the original curve of fig. 1 the curve B, calculated for 4828 12364 NOS reed Ol 08 + (Hf (#7) . kw with the value 10-142 for &,,. The results of this calculation are represented by column 3 of the following table. inde TABLE 19. PH 108 Cor m | m (for Ae 20 0/5 | 8.13 14.13 | 41 46.8 8.03 58.88 44 8 51.4 1.80 34.67 50.8 59.— 1.64 23.99 50.3 58.3 1.52 18.20 41.4 54.7 feel 8.90 36.1 40.7 1. 5.495 29.8 32.7 6.67 2.57 let 281 25.— 6.40 1.38 21 == 21.8 8.5 173.8 29.2 ove §.— 1660.— 21.6 22.5 35 This table and curve B of Fig. 8 would predict a decrease in m brom 1<3; forno "BTA, fot pp = 1,24, thatsis-from- 1 to 0.725, whereas according to table 42 experiment produced a decrease from 1 to 0.82. The interpretation of this difference is, that the 4, and ky are also appreciably diminished by 10 °/, alcohol, that therefore the “rest- curve” of urease in this solution has to be placed above the curve B of Fig. 8. The curve C in Fig. 8 and the fourth column in table 19 represent the values which 7 would have, if both ka and k, had . increased by 20°/,, 4, having, as in curve B, the value 10-142, Attention may here be drawn to the fact, that if a neutral body like aleohol, diminishes the dissociation constants of urease as well as of water, the new curve for m intersects the original one, which means, that in such a case a considerable decrease of activity is to be expected at low py (in the present case below py = 7.85) and by the same agency, the addition of alcohol, an increase at high py. The influence of neutral salts may be illustrated by the example atb: KCI. Here, as in all previous cases, 5°/, means 5 gram in 100 c.c. of the reacting solution. In 8°/, phosphate solutions the py was found not to be raised, but to be diminished in a constant degree by 5°/, KCI. Pu without KCl 6.13 6.89 (220 with 5 °/, KCl 6.00 6.79 (ga G2) difference 0.13 0.10 0.10 Nevertheless in the ammonium carbonate solution KCl had pro- duced a considerable decrease of urease activity. Without KCl m = 0,0031, with 5°/, KCl m= 0.0022. The conclusion, to be drawn from these facts, is, that a neutral salt like KCl increases all dissociation constants, that of urease as well as that of water. : We thus have here again and in a marked degree, a change in the nature of urease, a general decrease in its activity, brought about by an increase of its dissociation-constants. The raising of the dissociation-constant of water by salts had already been observed in this study in the determination of the hydroxyl-ion concentration in 8°/, phosphate solutions, the dissocia- tion-constant of water having been found to be 10-1878, instead of 10-1584, as it is in pure water. 3% 36 Moreover, the stimulating action of neutral salts on the catalytic activity of hydrogen-ions has long since been attributed to a raising of the dissociation constants of acids by salts. Another instance of such an effect has recently been supplied by Ko.tHorr '), who showed, that the dissociation-constants of indicators are increased by the addition of neutral salts. Mannitol showed the same behaviour as potassinm chloride. In 8°/, phosphate solutions the following values for py were determined : without mannitol 6.89 (29 with 5 °/, mannitol 6.84 7.24 difference 0.05 0.05 Hence again a constant decrease of py, that is an increase of the dissociation-constant of water. In ammonium carbonate solution 5°/, mannitol produced a small decrease of urease activity (m decreased from 0.0029 to 0.00275), from which it is to be inferred, that the dissociation-constants of urease are increased by this neutral substance also. Figure 9. 5°, Mannitol. 5 0/, Glucose. 40 80 120 160 200 240 40 80 120 160 200 240 minutes. Glucose, added to the amount of 5°/, to ammonium-carbonate 1) Chem. Weekblad 1918, 394. 37 solution, gave only rise to a very slight decrease of urease activity. The cases of mannitol and glucose are illustrated by fig. 9. It will be seen, that the general nature of the curves is not changed by the addition of neutral substances, from which it can be inferred, that only the m in our fundamental formula x — dx = m ——— dt z+ ne is affected. If other substances absorbed the radiation like the hydrogen-ions, the shape of the curves would have been altered, tending with pro- gressive change to the logarithmic curve. The facts recorded above and the theoretical deductions might afford some explanation of at least a part of ONopERa’s observations’), that alcohols can increase as well as decrease the action of urease. The experiments of this author, however, were made without buffer- mixtures and without any estimations of the py, which in the absence of buffers must have varied enormously. Since the long duration of urease action in those undefined, but certainly rather alkaline, conditions, must have had a deteriorating inflnence on the urease, the more so, the higher the py, a neat interpretation of these results is rendered impossible. 9. Reversion of the hydrolytic action of urease on urea. The generally accepted view of the synthetic action of enzymes, shared also by the present author, is, that one and the same enzyme is the active agent in breaking down as well as in building up its specific substrate. The conditions, however, which cause either of the two opposite activities to predominate, have as yet not been made clear. Some observations, made in the course of this study and a general consideration of the enzyme activity in living tissues, indu- ced the writer to venture the following hypothesis: Around an enzyme particle the substrate is broken down as long as the action, radiating from the enzyme, is vigorous enough. Weak- ened by spreading or by other causes, the same enzyme radiation produces the reverse process, synthesis. In the living plant and animal both activities of the enzyme are going on continuously under ordinary conditions as to concentration of substrate, temperature and acidity. As soon, however, as the tissue, 1) Biochem. J. 1915, 563. 38 by extraction and pulverisation, is made into a homogeneous mixture or solution, nearly all synthetic power of the enzyme disappears. As is well known, in order to extract say glucosides from plants, it is necessary first to destroy the enzymes by boiling the seeds or the leaves, before the cells are broken up for extraction. The gluco- side and its enzyme are, as demonstrated already by Guienarp, located in different adjoining cells. The hypothesis that enzymes display their synthetic effect only at a certain distance, affords at once an explanation of these facts. Through the cellwalls the radiation Figure 10. 0.9 0.8 0.7 0.6 9 0.5 ° 0.4 0.3 0.2 0.1 SES ERE EY Sj EE fi hs 8 en ee BPN. 20 40 60 80 100 120 140 minutes. 39 causes synthesis in the neighbouring cells; as soon as enzyme and substrate are brought together, the hydrolytic action prevails almost exclusively. This conception of enzyme activity is borne out by experiment in different ways. The weakening of the radiation may be occasioned, as observed above, not only by spreading, but also by other factors, e.g. by the deteriorating action of temperature and acids or alcalis in a shorter or longer time. As will be seen, in what follows, several times a reversion of the hydrolysis of urea was observed, which could be explained by a decay of the urease. Nov. 18 1915 an experiment was carried out with ammonium- carbonate as a buffer, in which this reversion at the end of the hydrolysis manifested itself clearly. See figure 10, p. 10. i Figure 11. 20 40 60 80 100 120 140 160 minutes. 40 It is to be expected, that the enzyme molecules will not all decay at the same time. At the end of the hydrolysis, when the action on the rest of the urea has become very slow, and on the other hand the concentration of the reaction products has grown correspondingly, it may happen, that a sufficient proportion of the enzyme particles is in the decaying state to cause the synthetic action, having a higher concentration of substrate at its disposal, to predominate over the slow action of the unchanged enzyme on the relatively very diluted urea solution. During such a period the total effect will be reversed, returning to its normal course as soon as the unchanged enzyme particles prevail over the decaying ones, which may be occasioned by a further weakening or total decay. Figure 11 shows the results of an experiment of Jan. 2rd 1916, like that of Nov. 28th, 1915, which shows the same phenomenon in a smaller degree. y In the phosphate solutions of high py also the reversion sometimes became manifest. First a regular experiment of April 6, 1917 may be recorded. TABLE 20. 0.25 gr. Soja + 50 c.c. water + 3.64 gr. Na, HPO,2aq. + 1.16 gr. KH,PO4 10 cc. filtrate mixed with 11.52 gr. Na, HPO, 2aq. + 120 c.c. water. pH = 8.13 0.01 °/, urea 0.0073 4 de 0.01 t (minutes) yp ; oe ly ae wales mn = t 80 O5 0.000029 140 0.305 0.000030 200 0.39 0.000027 260 0.52 0.000028 321 0.625 0.000030 In this case little or no destruction had taken place during this interval; for 10 c.ec. of the extract, left alone for 320 minutes in the same bath at 27° and then mixed with 2 e.e. urea (0.06 °/,) gave in 80 minutes 0.3 c.c. NH, #4 N, which makes m = 0.000026. April 10% 1917. The same arrangement as on April 6 but now with longer time intervals. Part of the same mixture left in the bath at 27° till the next day, when 10 c.c. with 2 c.c. urea solu- tion of 0.06 °/, gave in 260 minutes only y=0.1 (against y about 0.45 originally). Thus evidently a considerable decay of the enzyme had taken place. 41 April 16 and 23¢ 1917. A repetition of the experiment of April 10%. The results of these three experiments are represented Figure 12, minutes. in fig. 12. As will be seen clearly in the curves of April 10% and 27th, there is first a decline, after which the reaction proceeds nor- mally again. As discussed above the inference of this is, that during some time interval the decaying enzyme particles, which cause the synthesis, may prevail, and that after their total decay (or perhaps restoration) the normal enzyme action sets in again. In order to test this conception more directly the following experi- ment was carried out. : In one row of tubes the rate of hydrolysis of an 0.01 °/, urea solution in 8°/, phosphate of p= 8.13 was investigated as before. Another row of tubes, each also with 10 c.c. of the same Soja- extract-phosphate-solution had been placed with all the other tubes at the same time in the same bath of 27°. Immediately after taking out a tube of the first row for analysis, 2 c.c. urea solution of 0.06°/, were introduced into a tube of the second row and the reaction in this one allowed to proceed for 120 minutes. In this way it was possible to get some information about the rate of decay of the enzyme. From Fig. 13 it will be seen, that the nearly horizontal part of 42 the curve coincides with the 7’s of diminishing activity according to table 21. Between 7’—120 and 7’= 230 there seems to have Figure 13. ad 0.6 0.5 0.4 0.3 0.2 0.1 120 230 300 360 420 450 480 540 minutes. been in the beginning another period of some decay, followed by a normal course from 7’= 230 to 7’= 360. This peculiar type of curve, first a retardation and then again pada TABLE 21. T (minutes) | y y (in ee en after 120 0.255 0.175 230 0.385 0.130 300 0.465 0.115 360 0.535 0.08 420 0.53 0.075 450 0.575 0.08 480 0.585 0.075 540 0.625 0.075 43 a normal slope, was observed more than once on repeating the same experiment at py = 8.13. For instance on July 19th., 1917. Fig. 14. Figure 14. Vl 0.6; 0.5 0.4 0.3 0.2 0.1 a SSL 120 230 300 360 420 480 minutes. 10. Influence of the concentration of enzyme. In continuing the investigations of part 4 with smaller enzyme- concentrations it was first observed, that in the same conditions the same quantity of Soja-meal did not always give the same result. In preparing the necessary solutions the distilled water had been freed from carbonic acid by boiling in a tin coated copper flask. After this method of purifying the distilled water had been replaced by distilling the water once more in a Jena flask and glass condenser, rejecting the first part of the distillate, more regularity seemed to be obtained. Still, weighed off in such small amounts, the Soja-meal appeared to be not quite homogeneous. An explanation of this fact will be brought forward further on. Nevertheless important new results could be obtained in this way. For the sake of the necessary constancy of py the urea-concen- tration in these experiments was again 0,01 °/,. Taking the same unit as before the concentration of the urease was now 335. 44 Twice also a series of four experiments with different py was made simultaneously with the same Soja extract to make sure, that in all the same quantity of enzyme had been brought into solution. Evidently the smallness of enzyme concentration involved the necessity of long reaction-time. In order to investigate, if during such long exposures to these - H-ion concentrations at 27° any weakening of the enzyme activity had taken place, the remaining parts of the four phosphate-enzy me- solutions were left for 24 hours in the bath at 27°. On the next day the required experiments were repeated with these ones. In this way it became manifest, that the combined action of H-ion concentration, temperature and time had here, within the limits of experimental errors, no appreciable effect on urease in the acid solutions. If, however, at 27° the H-ion concentration was maintai- ned beyond the neutral point, the activity of the enzyme decreased slowly, but distinctly. In alkaline solutions for the calculation of m, as before, only the measurements of the first intervals had to be used. The acidity, used in these experiments, had itself no hydrolytic action on urea, as was ascertained by boiling samples of 10 c.c. of the liquids with py=6,67 and py == 7,52, incubating them for 24 hours at 27°, adding to each 2 c.c. of 0,06 °/, urea and maintaining these mixtures again for 24 hours at 27°. No ammonia was formed. The results obtained for the specific urease-activity m of low urease concentrations are brought together in fig. 15, in which for the sake of comparison the curve of fig. 3 is also reproduced. The interpretation of these results, based on the theory, put forward in part 9 is the following. An enzyme molecule being the centre of two concentric spheres, an inner one, in which the substrate is broken down, and an outer shell, in which the reverse action is brought about, it is evident, that, if the concentration of the enzyme is large enough to make the inner spheres intersect each other, there is no place in the solu- tion, where the condition for synthesis, that is weakened radiation, is realised ; at least, so long as the enzyme has retained its original activity. A sufficient dilution of the urease will, however, establish a distance between the enzyme-molecules, large enough, to leave the outer shell open for the reverse action. The total power of urease to convert urea into ammonium carbonate, representing the diffe- rence between the hydrolysis and the synthesis it brings about, must therefore be expected to be diminished by decreasing the urease concentration beyond a certain value. The observed facts are clearly in accordance with this theory : 45 Figure 15. 9 8.5 8 1.5 7 6.5 6 pH above a certain value of urease concentration constancy of specific activity, below it a decrease. The peculiarity, mentioned in the beginning of this part, that small concentrations of urease showed less constancy of power than larger ones, becomes now not only intelligible, but even adduces evidence to the theory. For, evidently, in regions, where the specific activity is falling off, a slight change in concentration of the urease will have far more effect than at higher concentrations, where the specific activity is constant. To be continued. Mathematics. — “A general dejinition of limit with application to limit-theorems”*). By Prof. Frep. Scnun. (Communicated by Prof. J. CARDINAAL). (Communicated in the meeting of May 3, 1919). 1. Let us assume an aggregate V of real or complex numbers, in which the same number may occur repeatedly. This can take place, if a mode of arising of the numbers of V has been given and various modes of arising may lead to the same number. Those equal numbers however, are considered as different elements of JV, so that we distinguish between a number having arisen in the former and the same number having arisen in the latter manner. 2. Next we assume a law given by which every positive number JS is made to correspond to a part Vs (consisting of at least one element) of V (covering of the aggregate of the positive numbers by the aggregate of the parts of VV), in this way, that Vw isa part of Vs, if d' a. In the case of complex numbers the theorem is further proved by applying the theorem for real numbers to the real parts and the coefficients of 7 of the complex nnmbers. 5. let V and W be two aggregates of numbers, the elements of which being placed into correspondence. We suppose the coverings of the positive numbers by the aggregates of the parts of V resp. W to be of such a nature, that for each positive number d the parts Vs and JW, in the correspondence between V and W are corre- sponding ones. _We now form an aggregate U by adding the corresponding ele- ments of V and W, at the same time transferring the covering to U. If now with these coverings V shows a limit L, and W a limit Lo, then also U has a limit, viz. L, + Lp, as may easily may be deduced from the definition of limit. Other known limit-theorems also can be stated in this manner in a general way. 6. We now suppose, that the elements of the aggregate V are real numbers. About the existence however of a limit of V, as regards the chosen covering, nothing is assumed to be known. We can then consider the lower boundary of the upper boundaries of the aggre- gates Vs, and call it the upper Limit of the aggregate V as regards the considered covering. The upper limit B is + oo, if all aggregates Vs are unbounded to the right, and — oo, if the aggregate of the upper boundaries of the aggregates Vs is unbounded to the left. We likewise call the wpper boundary O of the lower boundaries of the aggregates Vs the Lower Limit of V as regards the considered covering; this lower limit can also be + o. It is easily proved, that B and O always satisfy the inequality OS B. The aggregate V has a limit Z then only, when O and B are equal and finite; we then have L = O= BL. If O and B are both +o, we speak of an IMPROPER LIMIT + oo, if O and B are both — oo, Of an IMPROPER LIMIT — o. Mathematics. — “A general definition of uniform convergence with application to the commutativity of limits”. By Prof. Frep. Scnun. (Communicated by Prof. CARDINAAL). (Communicated in the meeting of May 3, 1919). 1. Let us assume two aggregates V and W with coverings of the positive numbers by the parts of V resp. W, as described in my paper “A general definition of limit’, p. 46. By these coverings to a positive number J a part V‚ of V and a part W; of W is made to correspond in such a way, that Ve and Wes contain al least one element, Vs being a part of Vs and Wea part of We if I <4. Let VW be the productaggregate, whose elements are formed by combining an element of V with an element of W into a pair (disregarding sequence). The covering by the parts of VW is taken such, that the aggregate Vs Ws corresponds to J. If we replace the coverings belonging to V and W by equivalent ones (c.f. n°. 3 of my previous paper) the covering belonging to VW is likewise replaced by one equivalent to it. 2. We assume VW covered with (real or complex) numbers, so that every element of VW is made to correspond to a number. From this arises an aggregate K of numbers, in which the same number may occur several times. We shall indicate by G an aggre- gate of elements of K, corresponding to a same element of V, com- bined with all elements of W. There likewise arises an aggregate H by considering a constant element of W. The covering by the parts of VW can be transferred to the ageregate K. Likewise the covering by the parts of V or W is to be transferred to every aggregate H resp. G. 3. Suppose that every aggregate G (as regards the covering by the parts of W) possesses a limit Lg; then corresponding to every element of V there exists anumber Lg. We say, that the aggregates G converge UNIFORMLY to their limits Lg, if corresponding to every positive number « there exists such a positive number d, that each element ZE of the part Gs, of G satisfies the inequality |A—Lg| — — = 6—— M. C.S. m m Now the resultant curvature entirely depends on the difference of growth retardation, which corresponds m—1 to an energy difference of 5 M.C.S. If the growth retar- m dation curve were a straight line, there would always correspond 1—1 to an energy difference of b— M..C.S. one and the same differ- m ence of growth retardation, whatever place of the abscissa we may consider; the threshold value would then necessarily be constant. If the growth retardation curve is, however, a curve with decreasing slope, the growth retardation difference corresponding to an abscissa —Ì difference of 6- M.C.S. will become the smaller, the gentler m 1) BREMEKAMP l.c. 63 the slope of the curve becomes, ie. the greater « becomes. We call the energy on the anterior and posterior sides respectively va and x,, the retardations of growth y, and Yq. growth retardation Fig. 1. energy m—1 M.C.S. the difference In a straight with «.—a, constant = b m in growth retardation y.—y, is always = q (fig. 1). 3 : m—1 In a curve with decreasing slope, with «,—«#, constant — 6—— m M.C.S. the difference in growth retardation ya—yp is smaller, according as xq is larger (fig. 2). Now aq will become greater by omnilateral fore-illumination, for —l eo ze + 6 M.C.S. The absolute energy difference 6 MES. oF n m rather the energy of 6 M.C.S. applied unilaterally, will therefore give a smaller difference of growth retardation i.e. a smaller curvature, according as the energy of fore-illamination a is greater. In order nevertheless to obtain a constant difference of growth retardation, m—l giving a just visible curvature, 6 and therefore also 6 will have to be greater. If the energy of fore-illumination becomes greater, the threshold value will rise, as soon as the curve of growth retardation is a curve with decreasing slope. Here we need not yet assert anything about the further relation be- tween the magnitude of the slopeand «. 64 In our case we can however follow the course of the growth retardation curve more closely by attempting to trace the relation- ship between threshold value and fore-illumination. If we calculate growth retardation sd Fig. 2. | energy from Artsz’s tables, mentioned above, the ratio of unilateral after- illumination to omnilateral fore-illumination, we find that, if after- illumination: fore-illumination = 1:11, there is no curvature; with this ratio 1:10, 9.9 or 9.2 a few plants give a feeble positive curvature, with the ratio 1: 7.2 all plants curve positively. This applies, as was already said above, only to a fore-illumination of less than 2000 M.C.S. given within 3 min. with an intensity below 25 M.C. We must, however, recalculate this for the quantities, which the anterior and posterior sides receive. Since we see, that a feeble positive curvature occurs when the ratio after-illumination : fore-illumination (6: a) has reached a definite magnitude 1: 9.7, the m—1 ratio of the energy difference 6—— to the energy of the anterior mm a side — + 6 must also be constant, whatever be the values of m and n. n In order to demonstrate this numerically, | propose to make certain assumptions respecting m and m; in principle it does not matter, what values we ascribe to m and 2. For m I assume 4; the back is then illuminated with 1*/, of the intensity of the front, receives b : therefore in unilateral after-illumination E M.C.S. Since the omni- lateral energy a M.C.S. is distributed uniformly over the whole 65 circumference, each half, viz. the later anterior or posterior sides, will receive, to begin with, half of the energy, '/, a M.C.S. Moreover each point, which is turned away from the source of light, receives during rotation a further *, of the energy of the illuminated side; each half receives therefore altogether £ «x 4a=a M.C.S. Here follows a table, taken from tables 24 and 25 of Arisz; I have added the numbers representing energy difference between front and back La—Ly TOO) Soe [ (= 100 x == | energy of front Lo TABLE Il. Energy of Energy of the unilateral after-illumination in M. C.S. omnilateral eenen 4X5.5 | 8x5.5 | 5x12 | 1012 | 5100 | 10> 100 ee 22 44 6075 120 500 1000 0o= 0| 75 + 15 HH 75 A75 44/75 44/75 ++ 10 5.5= 55| 29.5 + |42.3 HH [47.9 [58.444 [10.2 44 [72.5 HH MOCHT + 27.7 A 33,3 | 46.244 65.24 | 69-8 4+ 865<.12:1= 435) 5.6. (0 | 10.4 -+-71.13.6 23 +. | 48.6 ++ | 58,944 100 5.5=550| 4.5 Ol 85 O 11.1 +219.4 + [44.444 [55.814 Me et 1210 | el Ol OF SP ee) [10.8 ET A fa Explanation: ++ all plants show strong positive curvature. + all plants show definite positive curvature. +? a few plants show slight positive curvature. 0 no plants curved. 105.5 means: during 10 sec. omnilateral fore-illumination with 5.5 M.C. LEE < +6 5 55 + 22 | here assume +? as the threshold value. We see that Ca—e aN gia! itt and: 10/3; and. it 1s for this the quotient 100 X Xa therefore sufficiently constant. Then y.—y, has become so great that a slight positive curvature occurs. If the curvature can be represented by f (@a)—f (a,) and also by o( as can be readily shown by a simple mathematical consideration. Since the region of energy, about which the figures supply information, extends from about 300— 900 M.C.S. we may say that this portion of the growth retardation curve has a logarithmic course. If the \ 5 Va ® : : 2e) then f is necessarily ¢ log x L a Proceedings Royal Acad. Amsterdam. Vol. XXII. 66 2 Tat quotient —£ for the threshold value had not been found constant, va we should only have been allowed to conclude, that the growth retardation curve was a curve with decreasing slope, since the threshold value increases with increasing a. In that case also we might perhaps have obtained some further information about the function. I emphasize here that no fundamental significance should be attached to the logarithmic course of this part of the curve between 300 and 900 M.C.S. Moreover the curve from 0—100 M.C.S. is certainly not logarithmic; in a subsequent paper Ì hope to refer to this point in detail. Arisz regards the omnilateral fore-illamination followed by a uni- lateral after-illumination as a combination of unilateral illuminations, a short one on the posterior side (fore-illumination) and a longer one on the anterior side (fore-illamination + after-illumination). Since the later posterior side has also received energy during the fore-illumination, a tendency to curve in the opposite direction would have to be overcome. “It need cause no surprise, that the excess which must be given on one of the sides, to obtain an ipsilateral curvature, must be greater in proportion as the tendency to curva- ture on the other side is stronger.” Arisz therefore likewise explains the rise in the threshold value without assuming a change in the sensitiveness. This ‘tendency to curve in the opposite direction” however, as has been explained above, also exists in plants which have had a purely unilateral illumination, for the curvature depends on the difference of growth retardation between front and back. As in a combined omnilateral fore-illumination and unilateral after-illu- mination the growth retardation on the posterior side i.e. the ‘“‘ten- dency to curve in the opposite direction”, becomes relatively greater, the difference in growth retardation and accordingly, also the resulting curvature, becomes smaller (Fig. II). Arisz regards the phototropic induction, and hence also the tendency to curve, as a primary reaction, but since according to the theory of Braavw, which I have here worked out further, the growth retardation is primary and the tendency to curve secondary, it is better not to employ the latter expression, but to speak of a greater or smaller growth retardation. The unilateral illumination is a special case of the omnilateral and not inversely. We see therefore, that it is of great importance to ascertain the course of the growth retardation curve. We could not well do this from the figures of table 1; moreover the energy numbers had to be divided by a coefficient in order to make them comparable with those for horizontally incident ligbt. We will now try to ascertain the position 67 of the maximum of the growth retardation curve in another manner. A second phenomenon which shows itself in the combination of an omnilateral fore-illumination with a unilateral after-illumination, is that which Crark *) calls the increase in sensitiveness of the negative reaction. Here also we can find a simple explanation if we suppose the existence of a descending portion in the growth retardation curve. If the front receives in unilateral illumination an amount of light energy just in excess of the maximum, then the back, which receives + of this, will still be on the ascending portion. The ordinate of the front is still greater than that of the back, hence a positive curva- ture occurs. When we now apply more unilateral energy, we shall have to shift the points, representing the back and front energy, along the growth retardation curve, in such a manner that always Ta ta. A negative curvature will then only be possible much further on, because of the great distance between zr, and «,, i.e. we, will have to pass far beyond the maximum before ya becomes smaller than y,; ep need not yet have passed beyond the maximum. When then the negative difference of growth retardation y.—y, is great enough to become visible in a curvature, a negative curvature will occur (about 5000 M.C.S. if administered in 5 mins.). This becomes different, however, if «, and #, both lie on the descending portion and are less remote from each other; in that case, as soon as w, has passed the maximum, the possibility of a negative curvature will arise, the front however requires to receive much less than te, in order still to render a visible negative curvature possible. By means of an artifice we can arrange that vq and 2, come near to each other, although both are on the descending portion. This artifice consists in giving an omnilateral fore-illumination with a quantity of light, which is so great, that front and back both receive a quantity of energy corresponding to the maximum of the growth retardation curve. If we then unilaterally apply an amount of energy, which by itself would have given a strongly positive curvature, there will occur a negative curvature, at least if 7a—a, is so large that y,—Ya can give a visible effect. Here also the magnitude of the difference of growth retardations, and hence the threshold value for the negative curvature, will depend on the slope of the growth retardation curve, as was the case with the positive curvature. Here some data from tables 25 and 26 of Arisz are appended ; I have added the numbers repre- senting the quantities of light in M.C.S. which the front and back receive from omnilateral fore-illumination + unilateral after-illumination. 1) QO. L. Crark. Uber negativen Phototropismus bei Avena sativa. Zeitsch. f. Botanik. 1913. 5% 68 TABLE III. Omnilateral fore-illumination. Energy of - > = unilateral after- 12.1 M.C. during 180 secs. 25 M.C. during 100 secs. illumination ‘3 Eve. inMC.S. | Energy | Energy | | Energy | Energy | to front | to back | Reaction. | to front | to back | Reaction. inM.C.S. in M.C S. in.M. CS; in M: CS: 44 1405 1372 0 1606 1573 0 60 1421 1376 0° 1622 1571 0 120 1481 1391 a 1682 1592 0 500 1861 | = 1486 (3 2062 1687 v4 1000 2562 1812 js | | Explanation: The energy to the front, 1405 M.C.S., is ie ais 5 calculated from rae 6) Er (12.1 x le ns Bs 44) Boos!) b that to the back, 1372 M.C.S, from (+) = (12.1 x 1802 +11) MGs Arisz means by the sign? that “some plants give a feeble positive curvature, but there are always a few which curve negatively”; this has been confirmed by clinostat experiments. He directly connects this phenomenon of ‘increased sensitiveness to the negative reaction” with the fact, that after 300—600 M.C.S. the strength of the maximal curvature diminishes, and considers it possible that “by combining a quantity of light, which gives a curvature in excess of the greatest maximal strength, with a quantity which is maximal or nearly so, a curvature is obtainable towards the weaker illumination’. We may not connect the “decreased sensitiveness to the positive reaction” with the tendency to curve in the opposite direction and as little may we here directly connect the “increased sensitiveness to the negative reaction” with the strength of the maximal curvature, but must explain it from the course of the growth retardation curve. For the maximal curvature will be strongest in that case, where the difference between the ordinates belonging to rq and +2, is a maximum. The decrease in the amount of this difference is primarily connected with the decrease in slope of the growth retardation curve and it is only the rate at which the curvature diminishes at higher amounts of energy or the change to a negative curvature, which is connected with the question whether or no the growth retardation curve presents a maximum. The growth retardation curve will therefore continue to rise, although the intensity of curvature (i.e. the difference between the ordinates belonging to za and }.,) is already declining; the maximum will 69 therefore be situated much higher than 300—600 M.C.S. Now it results from the above table that if the back receives about 1400 M.C.S., negative curvatures may occur. In this neighbourhood there- fore the maximum of the growth retardation curve must lie; there may be considerable individual variation; if the maximum lies some- what higher, a feeble positive reaction will still be possible; if it is at or below 1400 M.C.S., negative curvatures can occur, this depends on the degree of slope of the descending portion. We can also calculate from table IT that a negative curvature never occurs, if the back receives less than 1400 M.C.S. Thus 1756 M.C.S. on the front and 1006 M.C.S. on the back still give a strong positive curvature; here the y of 1756 M.C.S. must be greater than the y of 1006 M.C.S. We may therefore place the maximum of the growth retardation curve at about 1400 M.C.S. With a unilateral illumination the posterior side will not be maximally retarded until the anterior receives m < 1400 M.C.S. This amount of energy must of course lie beyond the threshold value for the negative curvature, for otherwise y, could never become larger than y, and no negative curvature could occur. From this the value of m can be found approximately. I shall indicate yet a third method by which the course of the growth retardation curve can be explored. This can be done by assuming the magnitude of the maximal curvature to be proportional to the difference between the front and back growth retardation. If the maximal strengths of curvature are then plotted against the energy values as abscissae, there results a curve of the differences of growth retardation between the anterior and posterior sides. By a simple mathematical calculation, the growth retardation curve of the front can be calculated from the curve of differences, whereby it is assumed again, that the back receives + of the energy of the front; here the magnitudes of the growth retardation do not of course represent absolute values. The points of the posterior growth retardations are found by subtracting the curve of differences from the anterior curve; these can also be found by plotting the anterior curve with abscissae four times as great (rule of products). The magnitudes of the maximal curvatures I have deduced from tables 1 and 3 of Arisz; the energy was here always applied in 10 sees. Since the front is exposed to the full energy’), we have here again plotted the course of the growth retardation curve. In this way we come to the following result: 1) See footnote p. 61. 70 from O—75 M.C.S. the growth retardation curve is more or less rectilinear, then the slope decreases, so that it becomes approximately logarithmic from 300—700 M.C.S., while a maximum is found at about 1600 M.C.S. This value therefore agrees sufficiently well with that already found. Finally I have also succeeded in finding theoretically the further course of the growth retardation curve, which explains both the phenomena attending more prolonged unilateral illumination and those of unilateral preceded by prolonged ommnilateral illumination. growth retardation I, h Fig. III. Diagrammatic representation of the growth retardation curve for an intensity 1. the growth retardation curve for an intensity 4. magnitude of curvature occurring after unilateral illumination with intensity 4. Here the growth retardation has not been represented as a function of the energy, but as a function of the time, during which there was illumination with the same intensity. Since unilateral illumination, following an omnilateral illumination of more than 5 minutes’ duration, always results in a positive curvature, y, must again be greater than vp. This comes about through the renewed rise in the growth retardation curve, if the plant is illuminated for more than 5 minutes, Since the slope increases here, the threshold value for positive cur- vature must again fall. This latter fact agrees with what CLARK and Arisz found, namely that the threshold value falls if the plants are submitted to a fore-illumination of more than 5 minutes and less EE than 20 minutes. If the fore-illamination is more than 20 minutes, the growth retardation curve has become a straight line, since the threshold value now remains constant; however prolonged the fore- illumination with this intensity is, there is no further change in dispo- sition. Conversely there will be no question of “disposition’’, “change of sensitiveness”, in a process where the effect increases in a rectilinear manner with increasing strength of stimulus. Now since also after unilateral continued illumination (for longer than about 5 mins.) a positive curvature is again obtained *), the growth retardation curve, for an intensity m times as great, will run more steeply, i.e. for the same abscissa (time) there will be a greater ordinate (retardation of growth). If we take, however, the growth retardations of different intensities with equal duration of illumination, and plot these against the intensities, the slope of the resulting carve will of course greatly decrease at higher intensities, as a simple consideration will show. With this two facts agree: firstly that the threshold value after prolonged fore-illumination with high intensities comes to lie higher than after prolonged fore-illumi- nation with low intensities; secondly, that prolonged unilateral illu- mination with a high intensity gives a feebler curvature than illu- mination with a low intensity during the same period. We see therefore that the phototropic curvature is determined by the reactions of the separate longitudinal strips of the front and back respectively. Formerly the curvature was regarded as the direct result of a single condition of stimulation, the phototropic, which was considered to be induced as such. According to the view set out above, the curvature must be regarded as the resultant of the effects arising from the conditions of stimulation, which exist on the side, towards which the ultimate curvature will take place, and on the opposite side. These conditions of stimulation express themselves in photo growth reactions; the difference between the two reactions is expressed by the phototropic curvature. The cause of that which was formerly called “disposition” lies in the peculiarities of the growth retardation curve. These peculia- rities occur to some extent in every process in which the reaction is not directly proportional to the stimulus. A tangent galvanometer also becomes less “sensitive” at greater strengths of current. The “disposition” at a given point of the growth retardation curve, whether we take energy or time as abscissa, depends therefore on the magnitude and the sign (+ or —) of the angle of slope, and of 1) See footnote 2 p. 60. 72 the direction in which the latter is changing. We can ascertain the sign by arranging that «, and #, approximate closely; then we can see from the curvature whether y.—y, is positive (effect: positive curvature) or whether it is negative (effect: negative curvature). In unilateral illumination «, and «, lie too far apart and may be on dissimilar parts of the curve, so that the latter is very difficult to draw. According as #, and «, approximate more closely, the angle of slope can be found with greater accuracy. Since we have found that the growth retardation curve shows an ascending, then a des- cending and subsequently again an ascending portion, the “disposition” must therefore have three phases; finally after 20 mins., it is not possible to bring about a “disposition change” by further illumination with this intensity. These three phases have been repeatedly discus- sed in the literature and explained as processes, involving a change in the perception basis. From the above discussion it is evident, however, that disposition is a concept devoid of specific properties and is simply an expression of peculiarities of the growth retardation curve. Every growth retardation curve has a maximum at about 1400 M.C.S. and a minimum at about 5 minutes. The growth retar- dation curves of two intensities, which are in the ratio 1: m and with which 5000 M.C.S. can be applied within 5 minutes, intersect at two points; the first point is the energy threshold for the negative curvature (+ 5000 M.C.S.), the second is the time threshold for the second positive curvature (5 minutes). It will, however, be easier and more accurate to construct the growth retardation curves, both energy curve and time curve, from the numbers for photo growth reactions than from interpolations according to the above mentioned method, which is only qualitative, but not absolutely quantitative. Phototropism retains its value, however, since the study of photogrowth reactions presents difficulties in so far as we are here certainly concerned with the conduction of stimuli, while we only observe the reaction of the whole plant. Since in phototropic curvature the various zones curve, one after the other, we get this conduction of stimuli here separated into its successive phases. The difference in growth retardation will also be more constant as phototropic curva- ture, than when it is found by examining the photogrowth reactions of two different plants, in one for a quantity of energy J, in the other for a quantity of energy m. For in unilateral illumination both experiments are carried out on the same plant, one on the front, the other on the back. The difference, the resultant of the two reactions, will thus be less influenced by individual variability. Utrecht, April 1919. Astronomy. — “The Structure of the Sun's Radiation’. By P. H. van Citrert. (Communicated by Prof. W. H. Juuivs). (Communicated in the meeting of May 3, 1919). It is known that the intensity of the light on the sun’s disc very apprecially diminishes from the centre towards the limb, and that this takes place in a different degree for the different colours: regions near the limb’) of the sun’s dise are distinctly ruddy as compared with the centre. By the aid of a spectral photometer Voarr*®) determined this distribution of intensity for six regions of wavelength in the visible part of the spectrum. The decrease of intensity towards the limb appeared to be greater for light of shorter wavelength than for light of longer wavelength; the diminution, however, does not take place quite regularly with the wavelength, but presents an anomaly in the neighbourhood of 5000 A. the contrast of the limb to the centre is for this wavelength-region less than the contrasts for the other wavelength-regions would lead us to expect. Voaer’s observations were repeated by Apsor*) in 1906 by the aid of a bolometer. ABBor determined the decrease of intensity for a great number of wavelength-regions in the infra-red and the visible spectrum. He also found a strongly pronounced wavelength- effect: the contrast towards the limb increased very greatly towards the violet. Also his observations presented an anomaly in the neigh- bourhood of 5000 A, though less pronounced than that in Voerr’s observations. In fig. 1 we have plotted as functions of the wavelength, what value the intensity has for places which are at a distance of 0.65, 0.825, and 0.95 of the radius of the sun’s dise from the centre, when in the centre the intensity is put equal to 100 for all the colours. The data are borrowed from Assor’s tables. It is seen that the intensity rapidly decreases towards the limb, and this the more rapidly as the wavelength is smaller, but at the same time the fact 1) In the present paper, the phenomena at the limb itself. and even in regions, more than e.g. 5%5, of the radius of the disc removed from the centre, are left out of consideration. 4) H. C. Voce, Berl. Ber., 1877. 3) CG. G. Aspot. Ann. of the Obs. of the Smiths. Inst., 2, 205, 1908; 3, 153, 1913. 74 . . . © . . . strikes us that in the region 6000—4000 A this decrease of intensify presents an oscillation. It is noteworthy that the maximum of energy of the sun’s radiation lies at the same place in the spectrum. Lr 100 5 90 53 80 82.5 70 93 60 50 40 oO 3000 7000 7000 5000 400A Fig. 1, (ABBor’s observations ranged from 21000 A to 3800 A; the part from 21000—9000 A has, however, been omitted in the figure, because the curves do not present an irregular course there, but gradually approach the line J, = 100 towards 21000 A). To account for this decrease of intensity towards the limb many investigators have considered the sun as a self-luminous uniformly radiating core, surrounded by a strongly absorbing atmosphere. Now the state in the sun’s atmosphere must naturally be stationary on the average: the quantity of energy that the atmosphere absorbs, must be radiated again, even though it be in another form, and half must be radiated towards the outside. Now it has appeared convincingly from the observations of the annular eclipse of April 17» 1912 5) that of the total quantity of energy that the earth receives from the sun, at most one thousandth part can originate from the sun’s atmosphere. It is, therefore, impossible that the absorption should be the chief cause of the diminution of intensity towards the limb. 1) W. H. Junius. These Proc. 15, 1451. 75 Continuing the investigations of Rarreien '), Scnuster ’), Kine *) and SCHWARZSCHILD *) on molecular scattering of light, SPIJKERBOER *) has treated the problem how the distribution of light on the sun’s dise would be for the different colours, if exclusively molecular scattering in a non-absorbing and not self-luminous atmosphere were the cause through which the uniform radiation of a self-luminous solar core was modified. He arrived at a distribution of light which presents close resemblance to that observed by Apsor. The influence of the diffusion (or molecular scattering) is deter- mined by the product H=s.t,in which t= the thickness of the ; } 32 n°(n—l)" 2 dispersive layer and s = aa ee Rarrrien’s coefficient of scat- J tering. When it is now assumed that ¢ has the same value for light of different wavelengths, that the “core” lies, therefore, equally deep for all colours, the wavelength-effect is exclusively determined by the dependence of s on A‘, because, when kinds of light in the neighbourhood of the proper-frequencies are left out of account, (n—1) will vary very little along the spectrum. It appears, however, that the observed dependence of the wavelength is somewhat less great than theory would lead us to expect. This may be due to the fact that besides the diffusion another phenomenon appears, which has a similar influence on the distribution of light as diffusion, but which does not vary so much with the wavelength, e.g. scattering by wrrequiar refraction, and possibly a very slight general absorption. Now it is very probable that, particularly in the deeper layers of the sun’s atmosphere, irregular refraction plays an appreciable part. The existence of a very irregular distribution of density in the solar gases can, indeed, not be doubted, the constant variations in the granulations and floeeuli on the sun’s dise point in any case to the existence of an intricate system of currents in that gas-mass, and these are not conceivable without differences of pressure and irregular density gradients accompanying them. The mean value of these gradients, which is small in the outmost layers of the sun, must at first increase as one gets deeper. At a certain depth the irregular density gradients must then on an average be of the same order of magnitude as e.g. the vertical gradient of our terrestrial atmosphere. A gas-mass of the dimensions of the solar atmosphere, 1) Rayreien, Phil. Mag., (5). 4%, 375, 1899. : *) Scuuster, Astrophys. Journ., 21, 1, 1905. 8) Kine, Phil. Trans. R.S., A (212), 375, 1912. 4) Scuwarzscuitp, Berl. Ber., 47. 1183, 1914. 5) J. SpijkerBoer, Verstrooiing van licht en intensiteitsverdeeling over de zonne- schijf. Proefschrift. Utrecht 1917. Arch. Néerl., (3 A), 5, 1, 1918. 76 quite honeycombed with irregular gradients of such average magnitude, would, as Jurist) has demonstrated, refract, deflect and disperse the rays of light that penetrate there, so strongly to all sides, that the gas would present itself to a distant spectator as a turbid medium; the volume parts in which the density can be considered as constant, and hence the light as rectilinear, would be too small to be observed separately at such a distance. Since the degree of refractional scattering is determined by (m—1), and accordingly varies comparatively slowly with 4, the co-operation of this kind of scattering with the molecular scattering will weaken the mean wavelength effect, peculiar to the latter. Besides, the diminution of the intensity from the centre towards the limb will be greater than would be the case if only diffusion were the cause of this diminution. Irregular refraction, therefore, lessens the difference between the rates of darkening towards the limb shown by tbe different wavelengths, but at the same time strengthens the average contrast between limb and centre. As another possible cause of the fact that the observed wavelength- effect is slighter than the theoretical, SPIJKERBOER suggests that the light of the longer wavelengths, as it gets less greatly weakened by diffusion, might come to us from deeper layers of the sun than the light of shorter wavelength. This supposition, evidently, excludes the older hypothesis that the sun would have to be conceived as a well-defined core surrounded by a sharply defined atmosphere. The idea that the various radiations originate at different levels is more in agreement with the conception of the sun as a glowing gas- mass, of which the density and the temperature gradually diminish towards the outside. Since the light of greater wavelengths is much less weakened than that of shorter wavelengths, it will come to us from deeper layers of the sun. The infra-red light will, accordingly, be relatively more weakened by molecular scattering than would be the case if it only came to us from equally deep layers as the violet light. From what depth the light comes to us is, however, not only determined by the diffusion, but also by the irregular refraction. As this is little dependent on the wavelength, the differences in depth for the different colours will not be so great as in the case that only diffusion played a part. If, as is very probable, irregular refraction plays the principal part, especially in the lower layers, the difference in depth must be comparatively slight. 1!) W. H, Juuius, These Proc. 16, 264, 1913. 77 Let us now consider the question how, seen from this point of view, the radiation emitted by the sun in a direction w (fig. 2) must be composed. The outer layers of the sun will emit very little energy; proceeding towards lower layers the quantity of emitted energy increases, at first slowly, then more rapidly, in consequence of the increase of temperature and density. Let us suppose def to be the layer outside which no appreciable quantity of energy is Fig. 2. emitted. Let us first consider the radiation transmitted by the surface def in the direction w. We shall take the scattering in the atmos- phere outside de f later into account. From the region e, in the centre of the sun’s disc w receives a comparatively slight quantity of radiation of comparatively low temperature, from e, a larger quan- tity of higher temperature, etc. At first the quantity of energy which w receives from the different depths, will increase; the deeper we get, however, the more will the emitted radiation be weakened by absorption, refractional scattering and diffusion. From e, e.g. w will again receive less radiation than from e,, but of higher tempera- ture, from e, very little, etc. Let us suppose that w does not receive an appreciable quantity of energy from layers lying deeper than ). 78 (It is noteworthy that w also receives radiation from e originating from regions e’ and e’’ in the neighbourhood of the radius vector je, which radiation has finally assumed the direction ew through diffusion and irregular refraction). Let us now consider the energy curves of the radiations coming from those different regions, as functions of the wavelength. All of them will probably have the character of the curve of radiation of the absolutely black body: slow increase of the energy from the infra-red to a maximum, and then a comparatively rapid decrease on the violet side of the maximum. The loci of the maxima of the different curves are determined not only by the temperature of the corresponding radiations, but also by the relative importance of diffusion aud refractional scattering. Since the light of shorter wave- lengths is much more greatly weakened by diffusion than that of longer wavelength, the maximum will be displaced by the diffusion towards the side of the longer wavelengths; the irregular refraction, on the other hand, does not displace the maximum, because it is almost independent of the wavelength. For the curves belonging to the radiation originating from deeper layers, the maximum would lie more to the violet than for the curves belonging to the outer layers, if the temperature were the only factor; more to the red, however, if exclusively the diffusion were efficient. The height of the maximum is determined by the temperature of the radiation, the density at the places of emission, and the weakening which the radiation has undergone by diffusion, irregular refraction, and ab- sorption. We may therefore conclude that from the centre of the sun’s dise a quantity of energy is emitted in the direction w of which the energy curve, proceeding from the infra-red towards smaller wave- lengths increases, at first slowly, then more rapidly to a flat maxi- mum, and then runs very steeply down to the violet (fig. 3 1). Not far from the limb of the disc, e.g. at c (fig. 2), we shall receive light from a longer path through the gases (ek > bj), but it will come from less deep layers. Hence at the limb radiation is received from f only from the regions round 47 lying more out- wardly. The quantity of energy that w receives from the limb of the dise is therefore for all colours smaller than that which w receives from the centre. Hence the energy curve of the limb of the sun will as a whole lie lower than that of the centre. // the diminution of energy were the same for all the colours, the composition of the light at the limb would be represented by a curve which had the same shape as that of the centre (fig. 3 II). Now it is directly to be 79 seen that this cannot be the case, for the radiation at the limb will not present such a variety of temperature, irregular refraction, ‘i / Fig. 3. diffusion, ete. as that in the centre, because at the limb only radiation from a smaller number of layers contributes to the total radiation. The radiation at the limb will therefore have an energy curve with a maximum which is not so flat as that for the central radiation, and which, as we only receive radiation of lower temperature, lies somewhat more towards the side of the longer wavelengths (fig. 3 III). When we now compare the curves fig. 31, II, and III, it appears that in the wavelength-region AB the limb radiation will have a relatively stronger, in BC a relatively slighter intensity than would be the case if the distribution of the energy were the same as in the centre. When we now put the central radiation for all the wave-lengths = 100, this is graphically represented by a straight line C parallel to the wavelength-axis (lig. 4). The limb radiation can then be represented by an almost straight line 7, which runs about parallel to the wave-length-axis, but which exhibits an oscillation in the region ABC. These radiations will be weakened by the diffusion and irregular refraction in tbe atmosphere outside def (fig. 2), light of the shortest wavelengths most. When again we put the resulting central 80 Ir Fig. 4. Ee ae B =S Ss: o 5 Knee) Ei (5) 3 Su o ®O Fe DD 2 ® = 2 2 GN P) Sn JZ 3 5 Cen So | he Feats ES a) ee v Ss ste Gh EE —~ D wn o a se} |e ne) S Sh a6 = ÒQ es} 2 re Gees x © 2s Pane = iS Ce @) 2S del rc a a=) a) =) te) 81 ously towards the violet, but presents the same oscillation in the region ABC. Hence the relative distribution of intensity of the light on the sun’s disc reckoned with respect to the centre, must present an anomaly in the neighbourhood of the maximum of energy. This anomaly bears entirely the same character and relates to the same region of the spectrum as the anomaly observed by Assor (c.f. fig. 5 with fig. 1; the part of the curve r outside C in fig. 5 has not been observed by AsBor). Summary. Starting only from the hypothesis that the sun may be conceived as a glowing gaseous body, in which the temperature and the density gradually decrease from the centre outward, and the outer layers of which consist of little luminous, little absorbing, but greatly diffusing and irregularly refracting gas-masses, we have derived an explanation of the anomaly, observed in the decrease of light of different wavelengths from the centre towards the limb of the solar disc. Proceedings Royal Acad. Amsterdam. Vol. XXII. Chemistry. — The Phenomenon after Anodic Polarisation.” 1. By Prof. A. Smits, G. L. C. La Bastipg, and J. A. vAN DEN ANDEL. (Communicated by Prof. P. Zreman). (Communicated in the meeting of March 29, 1919). Introduction. When the experimental electrical potential of iron is measured during the anodic polarisation, we observe that this potential becomes less negative as the density of the current increases. When this experiment is carried out in solutions containing few or no halogen ions, this phenomenon greatly increases above a definite density of current, and the iron then passes from the active into the passive state. Though this transition from the active into the passive state is not observed for all metals, all of them present, when they are anodes, a potential that is shifted more or less in the positive direction. This phenomenon of the anodic polarisation, like that of the cathodic polarisation, has been explained in a simple way by the recent views about the electromotive equilibria’). A new phenomenon, however, presented itself when somewhat more than two years ago Smits and Losry pe Bruyn made use of a rotating commutator according to Le Branc?) during the anodic polarisation of iron. This commutator is constructed so that the potential difference - is not measured during the passage of the current, but immediately after the current has been broken. With this method of procedure the remarkable phenomenon was discovered that when the current density was not so great that the iron became passive, the iron did not show a less negative, but a more strongly negative potential, which means that the iron exhibited a change of the potential after anodic polarisation in a direction opposite to that which was to be observed during the passage of the current. | At first it was supposed that this phenomenon had to be attributed to some mistake in the arrangement of the experiment, but it soon appeared that this was not the case, and that the same phenomenon, but slightly modified, also appeared for nickel, and as Dr. ATEN found afterwards, also for chromium *). 1) Zeitschr. f. phys. Chem. 92, 1 (1916). 2) Zeitschr. f. phys. Chem. 5, 469 (1890). 8) These Proc. XX, 1119. 83 When the experiment was made witbout the said commutator, and the course of the potential of the iron with the time was examined immediately after anodic polarisation, it appeared that after having passed through a minimum, the potential again rapidly rose to the initial value. As a fuller examination of this phenomenon would have led the investigation of Smits and Losry De Bruyn into a bye-way, this study was postponed to a later occasion, and this is the reason why the publication of this phenomenon has been withheld until now. Registration of the Phenomenon. In order to represent the phenomenon graphically we have employed the photographic registration method, making use of Mo11’s galvano- meter. The potential of an iron electrode I was measured with respect to another auxiliary iron electrode II, a platinum tin leaf serving as cathode. The electrolyte was a '/, normal solution of ferro- sulphate. The iron electrode I, which had an area of about 25 m.m’, was made anode for a short time, the density of the current being chosen so that the iron would have become passive with longer passage of the current. Zmmediately after this anodic polarisation the galvanometer was inserted into the circuit iron I-electrolyte-iron II, so that the course of the potential which iron I presented after the anodic polarisation, could be accurately registered. The film now showed: first the potential of iron I with respect to iron II before the experiment, secondly the curve representing FE 6% 84 the course of the phenomenon, thirdly the zero-line, fourthly a line that corresponds to a definite potential difference, and finally time- lines given by a signalling apparatus. Thus in our experiments with iron in ‘/, N. ferro-sulphate solation the following photograplis were obtained. The line DE represents the potential of iron I with respect to iron Il before the anodic polarisation. The other two horizontal lines, which run all over the figure, are the zero-line and the line for —90 milli Volt. The curve A BC represents the phenomenon observed after anodic polarisation of iron | with a density of current of + 0.360 Amp/cm.’. The iron electrode I is at first disturbed in noble direction after this anodic polarisation, but this disturbance ceases with great rapidity and the potential descends to a value which is about 90 milli Volt. more negative than the potential before the anodic polarisation. The iron keeps this negative potential only for a short Kg Ï } Ï i ES, j ; ' et : mn eer Fe Fig:2.- time, and rises, first with increasing, then with decreasing rapidity to the value observed before the anodic polarisation. In fig. 2 the phenomenon has been photographed that was observed when instead of a solution of ferro-sulphate, a solution of '/,, N. ferro-chloride was used. In a few words we may remind here that the increase of concentration in the boundary layer through the short anodie solution has brought about a slight change of the potential in positive direction. In order to find out in how far the found phenomenon is something particular that is caused by the anodic polarisation of iron, also the course of the potential of the iron was photographed after cathodic polarisation. This photo is repro- duced in fig. 3, we see that the iron after cathodic polarisation with a density of current of + 0.360 Amp/em.’. is still pretty greatly disturbed in base direction, the potential of the iron was, namely, in this experiment at first about 80 milli Volt. more negative than before the cathodic polarisation, which potential difference had, 85 however, almost entirely disappeared again after a minute. We will point out here that a small part of the potential in negative direction n . + — — 1 ee SS enn a re ee rn RR FE 4 =e oe he eS Se ee ot SRI Se Bcc ee “+83 mV Fig.3. kb ~ _ _ _ = ee a eee - _ ee a ee | is the consequence of the decrease of concentration in the boundary layer during the cathodic deposition of the metal. As it had appeared before, as has already been mentioned, that nickel, though much more inert, showed the same phenomenon as iron, the same experiments have been made with nickel as with iron, in which the following result was obtained. ~i2.mV. Fig.4. Fig. 4 gives the course of the nickel electrode with an area of 80 mm.” in a solution of '/,, N NiSO, after anodic polarisation with a current density of + 0.075 amp./em., from which appears that here the saine phenomenon is seen as for iron; the potential of the metal becomes more negative after anodic polarisation, and after having reached a minimum value, it slowly returns to the original value of before the anodic polarisation. The phenomenon is the same here as for iron, but it takes place here with much less rapidity. In fig. 5 the result is recorded of the determination of the poten- tial difference for nickel after cathodic polarisation with the same current density. The nickel is at first pretty greatly disturbed in base direction, the potential is about 240 m.V. more negative than before the catho- dic polarisation, but becomes less and less negative with decreasing 86 rapidity, and does not present the potential of before the cathodic polarisation again until after nearly five minutes. As the whole course occupied so much time, we have made the box with the film execute some revolutions, so that the successive pieces have come above each other on the photo. It is of importance to state here that Ranrerr *), who has also carried out experiments on this subject, found a more negative Ni zer 341SEC Fig.5. potential for nickel after anodic polarisation in agreement with us, but after cathodic polarisation a much more positive potential. This is in conflict with what was found by us, and must probably be attributed to a mistake in the arrangement in Raarerr’s investigation. For on earlier occasions the phenomenon photographed in fig. 5 was always found by Smits and Losy pe Bruyn and now by us. As it was of importance to include also a few other metals in our considerations, we have also investigated silver and copper immersed in a silver-nitrate, resp. copper sulphate solution with the following result. The photo fig. 6 is a reproduction of the potential course of a silver electrode with an area of 42 mm.’ in a 0,1 NAg NO*-solution after anodic polarisation witha density of current of 0,333 Amp./cm.’, Fig.6. 1) Zeitschr. f. phys. Chem. 86, 567 (1914). eae denkend 87 from which appears that here the ordinary course is found. Also after anodic polarisation the silver has at first a more positive potential, which becomes smaller with continually diminishing velocity. After cathodic polarisation with the same current density at first a too negative potential is found, which becomes less and less negative, as is shown by the photo fig. 7. 3,41SEC < > Fis.7 a Finally also a copper electrode with an area of 9 mm.’ in a 0,1 NCuSO,-solution has been examined, and then the same thing was found as for silver in 0.1 N. Ag NO,, as appears from the photos fig. 8 and fig. 9. The current density amounted here to 0,666 Amp./em,. ez pn = _ 58mV — CU ne = = < > 31SEC Dee Figs. Cu Ti EN EEE EAN DDE FR ae -69mV SAISEC 88 So far a deviating behaviour has, therefore, only been found for iron, nickel, and chromium, hence for metals that have ions of different valencies, for this reason it was desirable to examine whether the phenomenon would continue to exist when the ratio of concen- tration between the different ions in the electrolyte is everywhere so, that the metal can coexist in state of internal equilibrinm with this electrolyte. This was investigated for iron by boiling a solution of Ferro-chloride with iron-powder for a long time in the vessel in which the definitive experiment was to be made, leading over pure hydrogen at the sametime. The solution thus obtained was 0.6 N. The result obtained with this solution was convincing in a high A ae a ne a ne Fig.10._ degree. As fig. 10 shows after anodic polarisation there was nothing to be seen of a minimum, and at least qualitatively the iron behaved like silver and copper. It is now natural to assume that also for nickel and chromium the phenomenon can be made to disappear in the same way, about which experiments are now being in progress. Laboratory for General and Anorg. Chemistry of the University. Amsterdam, March 27th 1919. Physics. — “On the Effective Temperature of the Sun’. Some remarks in connection with an article by Derant: “ Dijf uston und Absorption in der Sonnenatmosphdre.” By H. Groot. (Communicated by Prof. W. H. Jurius). (Communicated in the meeting of March 29, 1919). In a paper ‘Ueber Diffusion und Absorption in der Sonnenatmos- phare”’ (Sitz. Ber. d. Berl. Akad. 1914) ScuwarzscuiLp treated the problem of the radiation: of a plain layer, which must be imagined as an absolutely black body, and above which there is an absorbing and dispersive atmosphere. When on a layer (see figure 1) bounded by the planes =O and «= H radiation of intensity S, starting from the black body ZZ’ falls from all directions, ScHwaARZsCHILD denotes by b(z,t) the radiation which passes through the plane z in the same sense as the radiation S, and at an angle 7 with the normal to the boundary layers, and then tries to find a formula for 6(0,2). Fig. J. Black body. Accordingly 6(0,2) is the total intensity of light that passes out at an angle 7 at the boundary of the atmosphere, and is built up of direct light and light that is dispersed once, twice ete. SCHWARZSCHILD succeeds in solving this problem for two special cases, and finds: a. Limiting case of exclusive absorption (6 = 0): b cost HO =a + — (1—e—*H sec t) mt VOID 7 (1) 90 b. Limiting case of exclusive dispersion (£ = 0): 6(0,i) car Sa lel ee RE Packie 1+oH 1+oH Here £ = coefficient of absorption, 6 = coeff. of diffusion, H = height of the asmosphere, « and 5 are two numerical constants. In his article: “Diffusion und Absorption in der Sonnenatmosphare”’ (Sitz. Ber. d. K. Akad. zu Wien, Abh. IIa. Bnd. 125 (1914)). A. Devant by the aid of data which he derives from ABsot’s observa- tions on the decrease of the intensity of radiation on the sun’s dise from the centre towards the limb (Annals of the Astr. Observ. of Smithsonian Inst. Vol. III, Washington 1918, p. 158), tries to decide which of the two causes, absorption or dispersion, appears to be most active on the sun. By means of a kind of “trial and error” method he succeeds in deriving a formula: 0.5 Hoos §-+¢—0.0405 sei (0.5 —cos i) 0.3804 -+ 0.3136 cos i 1+0,0405 A+ ‘ which is halfway between (1) and (2) and yields numerically accurate values. This seems to point to this that the diffusion effect by far preponderates, but is yet influenced by a slight absorption. In how far the considerations through which he arrives at for- mula (3), are of value, must be left undecided here. It is certain that the numerical values are pretty accurate, as table I shows convincingly. p07) = TABLE I. A = 0.433 u | A = 0.604 u A=1.031 u cost .. |6(0,é)| Obser- lib Obser- b (On) Obser- BOD | ved | O00) OG vea | 20.0 OUT ved value x value x value 1.0 | 1.2752 | 453 456 0.9486 | 111 111 0.9 | 1.1906 | 423 419 1.0164 | 381 380 | 0.9175 | 107 107 0.8 | 1.0996 | 390 384 0.7 | 1.0006 | 355 348 0.6 | 0.8932 | 317 309 0.5 | 0.7764 | 276 2u 0.4 | 0.6506 | 231 238 0.3 | 0.5180 | 184 192 0.8491 | 99.4 100 0.8137 | 95.2 95.8 0.7765 | 90.9 90.0 0.8476 | 318 313 0.7366 | 86.2 86.2 0.6912 | 80.9 80.9 0.6917 | 259 265 0.5863 | 220 230 —_ ° 5 e a Dp > oO Co © © ie) © © 0.9656 | 361 360 0.8838 | 103 105 91 Explanation of table I: In this table 6(0,/) calculated for the values of cosi is given in the first column for three different values of A by the aid of (3). In column 3 the found values of b(0,i) have been multiplied by a factor in order to render a comparison with ABBOT’s values, recorded in the fourth column, possible. By the aid of (3) and Asspot’s values, which | subjoin, Drerant tries to draw a conclusion on the effective temperature of the sun. Wavelength in « Rn ye cer Ben 0.323 Wn 0.386 338 | 0.433 456 | 0.456 515 0.481 en 0.501 ns 0.534 ae 0.604 a 0.670 Poe 0.699 so 0.866 en 1.031 111 1.225 17.6 1.655 39.5 2.097 14.0 (ABBOT’s values). His reasoning is as follows: For 2 = 0 we obtain (0,0) i.e. formula (3) then gives for every wave- length 2 the intensity of radiation passing out in the centre of the sun’s disc, when that of the area of the photosphere for this 4 is put equal to |. What we measure is, however, not the quantity 6(0,0), but the radiation 2%, actually passing out, which is in relation with 6(0,0) through the formula: U), 4) b(0,0) ( in which /; is the intensity of radiation in the spectrum of the photosphere (considered as absolutely black body) for the wavelength 2. By the aid of (3) and (4) and ABBotr's values the following table can, therefore, be calculated for /; (table II). According to our 92 supposition the photosphere radiates as an absolutely black body, so that PLANCK's formula may be applied, according to which’) : 7.211><10° EN er EN 2.1563 X 2890 nee il en) The quantities J, from the table are expressed in an unknown unity. When we consider this unity and 7’ as unknown quantities, then T can be solved from two values of J, (for 4, and A, e.g.). Lf our basis is correct, we must find the same temperature from all combinations in pairs of L. | Drranr calculates 7’ from the combinations 1=05 1700 } 4,=0.9 I,= 180 | 7” = 8900 and 4,00. hh =300 T — 8700° 7 ae i) and considers the agreement “geniigend’’. (loc. cit. p. 517). TABLE II. Wavelength A | 1 + 0.0405 24 b (0,0) i) | 5 0223 nk A2 0.299 144 481.8 0.386 2.824 0.479 338 105.7 0.433 2.152 0.593 456 769.5 0.456 1.937 0.644 515 799.8 0.481 1.757 0.682 511 748.9 0.501 1.646 0.710 489 688.7 0.534 1.498 - 0.737 463 628.1 0.604 1.304 0.816 399 500.2 0.670 1.201 - 0.853 333 390.5 0.699 1,170 0.864 307 355.2 0.866 1.072 0.903 174 192.8 | 1.031 1.036 0.918 111 120.9 1.225 1.018 0.926 11.6 83.8 1.655 1.005, 0.931 39.5 42.4 2.097 1.002, 0.932 14.0 15.0 1) The constants are those used by DeFANTr. 93 Unfortunately, however, a fatal error has slipped in. For to 2 = 0.6 does not correspond /, = 350, but — (interpolating graphically) — 1, = 506, which yields 7’—= 6600° instead of 8700°, so that agreement is out of the question. A serious objection to the whole method seems perfectly obvious to me, namely this: The assumption that all kinds of light come to us from one photospheric surface, in other words that light of various wavelengths should come from the same depth of the sun, appears more and more untenable in the light of recent researches (see e.g. the thesis for the doctorate of J. SpijkerBoeErR “Verstrooiing van licht en intensi- teitsverdeeling over de zonneschijf” (1917) (Dispersion of light and Distribution of Intensity over the Sun’s Disc)). If, however, in reality light of different wavelengths originates from different parts of the sun, it becomes very questionable whether we shall be allowed to apply PrANCK’s formula, as we saw Derrant do. For this would mean that we supposed every kind of light to have, as it were, a_kind of “photosphere of its own”, which radiates as a black body, the photosphere for the greater wavelengths lying deeper than that for the smaller. It might then be expected that the temperature determined with PANck’s formula, becomes a function of 2, i.e. 7’ would be the greater as 4 increases. In this latter remark we have a means to investigate whether the hypothesis that the photospheres overlap each other like scales ean find a semblance of justification. By graphical interpolation from the values of table II I construed table III: TABLE III. A h A h A h FA h 0.40 709 0.70 342 1.00 134 1.60° (|) 46 0.45 791 0.75 284 1.10 104 1.70 39 0.50 114 0.80 | 239 old Si 1.80 32 0.55 604 0.85 201 1.30 74 1.90 | 25 0.60 506 0.90 174 1.40 64 S00 ok 19 0.65 418 0.95 153 1.50 55 | | As we do not know the unity in which /, is expressed, we require, as was already remarked before, always two values of DP (a, and a,) to find 7. 94 The calculation comes to this: Let A be — 7.210 X 10°, 8 = 2.1563 X 2890, «, and a, the values of /, corresponding to 4, and 2,, f an unknown factor dependent on the unities in which J, has been measured. Then the following equations hold: (6) When we choose the values of 4 so that A, = 24,, and when ge Ors we put 1 = 2, we easily get: Caw ait Nate ics 1 - The root x=1 yields 7 =o, has therefore no physical meaning, so that we find 7’ from: ct, ; ne es ' a, tet? itis Wh sere ene) 7 B n= Alg a | In this way I found: Ar ORS COLO a Oe ea ES | 1.45) "| AOar detd LES ele 2 0 T = 7200 (8800 (8900 (8500 (7900 (7200 6300 5800 5500 5100 (5000 5200 5700 eal ae! ite co hence on an average a decrease of T with increase of à (see diagram 1). 9000 -~ Diagram I. at gi 95 Nor is this manner entirely satisfactory; for now we do not know to what 2 the found 7’ should properly speaking belong, because the two values of 2(2, and 4,), which are required, can lie pretty far apart in this way of calculation. Does for (A, = 0.9, A, = 1.8) T e.g. belong to 2,, to 2,, or to a value lying somewhere between a and AP When we want to avoid this difficulty, we may treat the equa- tions (6) as follows: Let}, 24, De’ ==" Or ek m (9) i= = n we find easily : mB = : 5 nf a a eae 7 a,/\n Woes (10) But: Fs lie EE) ie (11) OAN, then (10) passes into: am Can + (C—1) = 0 (12) When we take care that m is —=n +1, the shape becomes some- what more suitable for numerical approximation, namely: en(e —C) + (C—1)=0 (12a) When z has been sufficiently closely approximated, 7’ follows from : pz el z In this way 4, and A, can be brought close enough together to exclude indefiniteness in the choice of the 4 to which 7’ belongs. Thus we found: EEN REE ENNE KEN KE LN TE EEN NET ENT EERDE EEE EET 0.5 (13) le ROE 0.6 | 0.7 0.8 1.0 1.2 | 1.5 1.8 zene Gee Cet! A KE | ase eer i 2.0 7 | (6400) | 9000 =a 9600 | 8000 | 5500 | 3800 | 5400 = so that on an average: A= 0.5—0.7 T = 9500 0.7—1,2 6000 1,2—1,8 _ 4600 4000 3000 96 hence a similar result as for the first method. (See diagram II), The deviations inter se are now much larger, as was, indeed, to be Di agram II. 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ge 1.2 1.3 14 1.5 expected, as on the small intervals 4,—A, the inevitable errors in /, (an experimental quantity!) make themselves very greatly felt. Thus Sie give an imaginary value for 7, but when for | eed j= 92,0 . J, == 22 is ‘taken instead ‘of Zp 19, then. Ze would become == 18000°. In this manner particularly the smaller values of J, are unfa- vourable, hence the values for 2, == 1,5 and 2, =1.8 are not much to be trusted. The values of 4, for 2< 0,5 are strictly speaking also unreliable, because the graphical interpolation — as indeed every other too — becomes very inaccurate here. : When we leave all these doubtful values of 7’ out of conside- ration we come to the result that particularly in the region of the reliable values of 7’ (the full line in the diagram) there is an un- mistakable tendency of T to decrease on the increase of 4, hence exactly the reverse of what we thought we might expect a priori. In a following paper I propose to discuss the question to what this unexpected result is to be attributed. Utrecht, March 1919. KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN TE AMSTERDAM. EENS VOLUME XXII N92: President: Prof. H. A. LORENTZ. Secretary: Prof. P. ZEEMAN. (Translated from: “Verslag van de gewone vergaderingen der Wis- en Natuurkundige Afdeeling,” Vol. XXVII). CONTENTS, le | „ ROELS and L. MOLL: “On the Index Loquelae”. (Communicated by Prof. H. ZWAARDEMAKER), p. 98. C. A. CROMMELIN, J. PALACIOS MARTINEZ and H. KAMERLINGH ONNES: “Isothermals of monatomic substances and their binary mixtures. XX. Isothermals of neon from + 20° C. to —217° C”, p. 108. A. SMITS: “Metals and Non-Metals”. (Communicated by Prof. P. ZEEMAN), p. 119. je . P. BARENDRECHT: “Urease and the radiation-theory of enzyme-action”, p. 126. P. N. VAN KAMPEN: “On the phylogeny of the hair of mammals”. (Communicated by Prof. WEBER), p. 140. G. NORDSTROM: “Note on the circumstance that an electric charge moving in accordance with quantum-conditions does not radiate”. (Communicated by Prof. H. KAMERLINGH ONNES), p. 145. L. E. J. BROUWER: “Remark on Multiple Integrals”, p. 150. Proceedings Royal Acad. Amsterdam. Vol. X XIL. Physiology. — “On the Inder Loquelae’. By F. Roers and L. Morr. (Communicated by Prof. H. ZWAARDEMAKER). (Communicated in the meeting of December 28, 1918). The idea index vocalis has been introduced by GRADENIGO *) and denotes the quantitative relation between whispers of moderate energy and ordinary speech. Practically this relation is best determined with the aid of the extreme distances at which they are audible. According to GRADENIGO the average index for vowels is */, or’/,*). According to Worr®), ZWAARDEMAKER*), and Reuter’) the average index ranges from */, to */,. For twenty monosyllabic, aequiintensive, isozonal test-words ZWAARDEMAKER—QuIx *) and Reuter’) found the index */,,. The determination of the relative energy of whispers and ordinary speech being rather difficult (as it is difficult to distinguish exactly between what is meant by the one and what by the other), ZWAARDEMAKER has had recourse to RALRIGH's mirror for sound- measurement*). [t then appeared that the normal average index _ reserve-air whisper vocalis ———— = '/,,, the normal average index vocalis ordinary speech stage- whisper : = '/,. These indices enable us to reduce the whisper- ordinary speech values obtained for vowels and words to ordinary speech-values. In experimenting on the intensity of audition separate words are used which consist of isozonal and aequi-intensive sounds, which, for 1) GRADENIGO. Communication au congrés internat. d’otologie. Budapest 1909 GRADENIGO et STEFANINI. Sur l'acoumétrie. Propositions et études. Archives internat. de laryngolie, d’otologie et de rhinologie, 1911. 2) ZWAARDEMAKER. Over den index vocalis bij keuringen en bij de studie der ziektebeelden. (Handelingen van het XIVe. Ned. Nat.- en Gen. Congres te Delft 1913), p. 370. 8) Worr. Sprache und Ohr. 1870. Frankfurt. 4) ZWAARDEMAKER bid. 5) Reuter. Zeitschr. f. Ohrenheilkunde. Bd. 47, blz. 91; Onderzoekingen Physiol. Lab. Utrecht (5) Vol. 5. p. 239. 6) ZWAARDEMAKER. Over geluidmeting. Zittingsverslag dezer Academie van 23 April 1915. Vol. XXIII. p. 1405. 7) Reuter. Onderzoekingen Physiol. Lab. Utrecht (5) Vol. 5. p. 249. 8) ZWAARDEMAKER and Qurx. De studie van het spraakgehoor. Onderzoekingen Physiol. Lab. Utrecht (5) Vol. 5. p. 1. 99 the very reason that every component of the word is heard as distinctly as the others, excludes guessing almost altogether and facilitates the finding of a definite zone of the scale. The question arises, however, though it is not probable, whether the hearing distance extends for whispered sentences as far as for words and vowels. ZWAARDEMAKER believes on the ground of various experiences that the distance at which whispered sentences are heard in a closed space amounts to 1 to 3 m.*) according to the words that are chosen. He calls the relation between this average distance and that at which whispered sounds are heard: inder loquelae. In the present investigation we purposed originally only to determine the index loquelae. However, the combination of the whisper-method with that of the systematic or experimental introspection, procured us a number of data of some importance for the psychology of perception. We availed ourselves of these data only in so far as they influence the determination of the index loquelae, the subject proper of this paper. The material used by us consisted of rather simple sentences. The shortest consisted of 3, the longest of 14 syllables. Most often the sentences were in the indicative mood, seldom in the subjunctive mood or in the imperative mood. Though the present tense was used most frequently, we sometimes also employed the past and the future. Negative sentences were rare. In selecting our material we took an adequate number of symmetrical sentences, i.e. sentences having the same number of syllables before and after the copula, which consisted of one syllable, or an even number of syllables, or we chose such sentences as were divided by a word or by appropiate punctuation into two parts of an equal number of syllables (e.g. waste not, want not; a friend in need is a friend indeed). In most sentences there was a subject, a predicate and an object in the given order; sentences without one of these parts or with a verb split up as the construction required, we term irregular sentences. Proverbs and proverbial sayings were seldom used; sometimes, but not often, a French or Latin sentence was presented to the subject. All in all we made 328 experiments with 4 subjects (R., M., D. and A). Table I gives a survey of the material with respect to the number of syllables, the symmetry and the asymmetry, the regular or the irregular construction of the sentences. It should be noted, that some test-sentences have 1) ZWAARDEMAKER. Ueber die Anwendung von Sig. Exners Akustik von Hörsälen auf die Theorie der medizinischen Hörapparate. Wiener Medizinische Wochenschrift. Nr. 14, 1926. 7% 100 TateyL | | | | | 7 ie =i = |= ore er | got bie lie pe vavo cecal Pe PLT OSE || ve €L0 Lew ee ieonelecic | — Ke RENK ley (too lest || — |— |ezz|| z |wolsez | ¢ |t60 lie || oz leo voe) € \9OT eve || v |160 WEN — |— |G CW € lesols p \zeo lie!) ¢ lego loge || v |oo ee le ort iste || ¢ jsorlsce | v \eri egel — | — |v\ U a le ie al Bes ep eee | cee eee eae a EER ie ee NI OW! CW IAD law) DD AI AW CW OITA Taw] CW AI OW CW Nan W | AD) QW) ‘W CAIAWEN |E ee ie N | oe ie val KN, le dE TRI We Ss sb fei gl ee ‘Ss JOUW pue Ol || ‘Ss 6 | ‘Ss 8 KS | ‘SQ | ‘SG | ‘Sh S¢ Tl SISVL | Ge ere PG 82 | ze | Oe |: 4 ot | 6b | tr | S gece | Tezor | | | ae | = PD | aaa ee = | 3 Eee ke | Z b aces EEE GEG ee en a ee oe ||) AEL ys | | | | | | | — L Ae I € Lois wb. IS oe a elon A eT 98 ‘qs | ZI Ez ZI Il | Zz L Bo bh gies ep PL W Iz | Iz oz | II | Hele venen eee et. En 2 SSI ates | | ee BE = | : saduaqyuas saduaues, ‘Ssaduaques |. : : : ; : É KO Zet | sld Je[n3als] ‘wwuÂs | asowpueg;| ©5 | ©8 | S4 |S dk SS | Da) | set _ equinn fans | 101 contributed values for two of the last three columns; a proverb like “a friend in need is a friend indeed” contributed to the making of the column of proverbs as well as to that of the symmetrical sentences. The material was presented in whispers with reserve air. The subject sat ready to receive the whispers with the right ear, vertical to a line from this ear to the mouth of the experimenter. The maximal distance was 5 m.; if the subject failed to correctly repeat the spoken words at that distance, it was lessened every time by 1 m. until the spoken words were reproduced correctly. When the distance was reduced to 2 m., it was lessened every time by '/, m. Careful introspection, at which the subject recorded minutely what he had heard, followed after each stage of the experiment. At every sitting 12 experiments were made at most, to prevent fatigue. They were all carried out in the over-furnished library of the Utrecht Clinic for Psychiatry and Neurology. Before starting the experiments proper, we determined the audition of our subjects by means of the words given by ZWAARDEMAKER and Quix in “De Studie van het Spraakgehoor” *). It proved to be normal with all of them; the hearing-distance varied for the words employed from 6 to 30 m. In Table II we have calculated the mean distances (M.), for every one of the three snbjects R. M. and D., at which sentences of the given number of syllables were heard (the values procured by A. are left out of account as being too small in number). As precision-indices we also give the mean deviations (M. D.) and the central values (C. V.). lt is obvious that with a larger number of syllables the sentences are heard less distinctly. For R. M. and D. this difference is resp. 1, 3.17 and 1.33 m., in the case of sentences of 3 to 10 or more syllables. Besides to the influence of factors which we cannot discuss further here, these considerable individual differences are to be ascribed chiefly to the various tasks imposed upon the subjects. It stands to reason that proverbs and proverbial sayings are generally heard at greater distances than non-proverbial expressions ; symmetrical sentences farther than asymmetrical sentences; on the other hand irregular sentences are more difficult to hear than those that are constructed regularly. Table III (in which the frequency of the various kinds of sentences is tabulated in percentages of the total number of test-sentences presented) shows clearly that the intensity of audition varied considerably. The difference of the distances at which sentences of 3 and those 1) Onderzoekingen Physiol. Lab. Utrecht (5) vol. 5 pd. 102 of 10 or more syllables are heard, is greatest with the subject M. (3.17 m.). This is owing to a considerable extent, to the high per- TABLE lll ke Symmetrical Irregular Subj sentences. Proverbs. sentences. R 17 13.7 13.7 M. | 16.2 16.2 31.1 D | 14 —— 8.1 centage of irregular sentences (31.3) with their unfavourable influence upon auditory sensation. This influence was much less on R. and D. (resp. 13.7 and 8.1 °/). The comparatively small difference in the case of R. (1 m.) is to be ascribed to the circumstance that symmetrical sentences and proverbs constitute about '/, (30.7 °/,) of the total number of the test-sentences that were used. D., being a FLEMING and conse- quently unacquainted with Dutch proverbs (which were not presented to him) is placed between R. and M., quite in accordance with the difference in the hearing distance with 14°/, favourable and 8.1 °/, unfavourable factors. When we look at Table II more closely, it appears that intra- individual differences are not less conspicuous than the inter-indi- vidual ones. For, though it is true that the hearing distance with the three subjects for trisyllabie sentences is considerably greater than for sentences of 10 or more syllables, the distance decreases far from regularly. In the case of M. the decrease is gradual with only few exceptions. Sentences of an odd number of syllables are generally heard better by R. and D. than those of an even number having one syllable more, they even perceive sentences of 9 syllables resp. 0.23 m. and 0.44 m. farther than those of + syllables. The cause of this phenomenon, so strange at first, is the fact that a symmetrical construction, which largely promotes our hearing faculty, occurs more often with sentences of an odd number of - syllables than with those of an even number. This is because with all symmetrical sentences the number of syllables before and after the copula is the same, while with most of them the copula is made up of one syllable (is, was, had, are etc.) Besides the agreement just alluded to there is also some difference in the shifting of the hearing istance, when the number of syllables increases. In the case of R. 103 the distance generally increases with an increase of the number of syllables from 4 to 9, for sentences of an odd number as well as an even number — with a few exceptions in the latter case. With D., however, a diminution is observed with sentences of an odd number of syllables, an increase with those of an even number. We have seen, then, that while in the case of M the distance decreases regularly (with a few exceptions) from 4 to 9 syllables, with R. it increases for an odd as well as for an even number of syllables; with D. an inerease is observed for the odd and the decrease for the even numbers. What is the cause of the increase of the hearing distance with an increase of the number of syllables in the cases of the subjects R and D, especially the first; and what gives rise to the differences in the subjects? One might expect that with the increase of the number of syllables the retro-active inhibition is more keenly felt, in consequence of which the faculty of perception is weaker. Audition, however, results from the joint action of apperception and assimilation; the latter, just as in the case of reading, plays a prominent part here. Now the .process of assimilation requires a minimum of apperceived elements; it can readily be imagined that in sentences of a small number of syllables such elements are as a rule too small in number to serve as starting-points for the assimi- lation-process. The assertion that with shorter sentences the assimi- lation-process should naturally be less extensive and should therefore require fewer points of contact, does not hold good. We have observed again and again that a certain number of apperceived elements, no matter how short the sentences may be, is required to prevent assimilation from becoming mere guessing. So the number of apperceived elements required for test-sentences of various length is not proportionate to the increase of the number of syllables. It follows that the reproduction-tendencies emanating from the apper- ceived elements are, within certain limits, stronger than the inhibition they exert upon each other. The reverse seems to be the case with all subjeets for sentences of ten or more syllables. It is now easy to explain also the individual differences. With M., for whom we observe a gradual decrease of the hearing distance with an inerease of the number of syllables, the influence of inhibi- tion is stronger than that of assimilation. With R., for whom the hearing distance increases with the length of the sentences of an odd as well as an even number of syllables, the reverse obtains, viz. the unfavourable effect of inhibition is not only arrested, but with the increase of the number of points of contact, assimilation 104 also acts more intensely. Subject D. takes up a place between R. and M. in that for sentences of an even number of syllables assimilation gets the start of inhibition when the sentences get longer, whereas for sentences of an odd number of syllables inhibition proves the stronger. We are unable to account for this strange phenomenon. reserve-airwhisper | ordinary speech — ‘1! By means of the normal average index vocalis which index presumably also holds for whispered sentences, we have reduced in Table IV the whisper-values for sentences of various length to ordinary-speech values. Both sorts of values in Table IV constitute the average of the hearing distances, determined for the subjects R., M., and D., with respect to the length of the various sentences. TABEL IV. | Syllables. | Whispers. Loud speech. 3 | 4.67 | 60.71 peel 3.61 46.93 5 | 3.61 | 46.93 6 | Sell 40.43 d 3.42 44.46 8 | 3.21 41.73 Bist | 3.34 Cils de 10 and more 2.41 | 31.33 Be it observed by the way that owing to the more conspicuous survey of the data in Table IV, our conclusions deduced from Table II are shown here to a greater advantage viz, the great difference between the hearing distances for sentences of 3 and for those of 10 or more syllables (loud speech nearly 30 m.); the com- paratively slight difference for sentences of 4 to 9 syllables (loud speech + 3.5 m); the greater distance for sentences of an odd number of syllables, owing to their most often symmetrical con- struction; ete. We have already pointed out that symmetrical sentences are heard at greater distance than asymmetrical sentences. Table V shows that symmetry is highly favourable to our audition. Here also great individual differences are observed; for R., M. and D. they are resp. OG. Le and 0:24 mt 105 It is easy to understand that proverbs and proverbial sayings are heard at greater distance. The subjects’ familiarity with them, helps TABLE V. In | Symmetrical sentences | Asymmetrical sentences | Subj. | M. | MD. | CV. | M. R (ait \ 019 | 4 3.50 0 ee be EE 2.51 | | 4 | | | | 5 | 3.97 | | ; | Def fat a | the assimilation process consequent on the apperception of elements. The favourable influence exerted by proverbs and proverbial sayings is, however, inferior to that of symmetrical construction, as is borne out by a comparative study of Tables V and VI. The enlargement of the bearing distance for symmetrical sentences is with R. and M. respectively 0.61 and 1.13 m., that of hearing proverbs only 0.74 m. TABLE VI. | | Proverbs. | The other sentences. | Subj. Mean. MD. CV. | M. | Bey) Fl BeBe hist! lant 3.37 | | | | ent i | 3.67 | 0.94 | 4 2.93 | | | From a study of the data of Table VII it also appears that the absence of subject, predicate or object or an inversion of these parts of a sentence may be an impediment to hearing (see page 106). Irregular sentences are heard by R., M. and D. at a much shorter distance than the regular ones. The decrease of distance is resp. 0.73; 0.28; and 1.01 m. The very considerable decrease with D. is pro- bably owing to the circumstance that as a Fleming he is less familiar with the rather uncommon construction of many of our irregular sentences. The relation between the mean distances at which whispered sentences and whispered words are heard we term after ZWAARDE- MAKER index loquelae. In otiatrics the distance at which the latter are heard is estimated at + 18 m. 106 In their experiments ZWAARDEMAKER and Quix ') arrived at the same conclusion. TABLE VII. Een | kak Irreg. sentences. | Reg. sentences. | 5 i I | | | Subj M M.D CN. | M Be. VANS B08 Oena Samael 3.63 OMS ee Nee oie 2.93 DP.” yig2-64 | 0:58 2:50: || 3.65 | The mean distance at which whispered sentences of from 3—14 syllables are heard amounts to + 3.42 m, as shown in Table II 3.42 and IV, so that the index loquelae is enen SUM MARY. tl. Our material consisted of sentences of 3 to 14 syllables. They were presented by means of whispers with reserve-air. 2. We determined the intensity of hearing of our subjects by means of separate words, composed of isozonal and aequi-intensive sounds. With all of them audition appeared to be normal; the distances for different words varied from 6 to 30 m. 3. The difficulty of hearing sentences increases with the number of syllables. The difference in the distances at which sentences of 3 syllables and those of 10 or more syllables are heard, averages 1.83 m. (whispers). 4. Generally the decrease of the distance in proportion to the increase of the number of syllables does not take place gradually. Sentences of an odd number of syllables are as a rule heard better than those of an even number containing one syllable more. Even sentences of 9 syllables are heard on an average 0.33 m. farther than those of 4 syllables. The reason is that symmetrical construc- tion, which aids our hearing faculty, occurred more with an odd than with an even number of syllables. 5. Proverbs and proverbial sayings are heard at larger distances. However, the favourable influence of proverbs and proverbial sayings ') De studie van het spraakgehoor. Onderzoekingen Physiol. Lab. Utrecht (5) dl, 5 blz. 1. 107 is inferior to that of mere symmetrical construction of the sentences. 6. The absence of subject, predicate or object or inversion of the order of words are an impediment to audition; the hearing distance is lessened by it. ; 7. Hearing is the result of the joint action of perception and assimilation. For sentences of 3 to 9 syllables the influence of the reproduction-tendencies, consequent on the apperceived elements, counterbalances that of the inhibition, caused by an increase of the number of syllabies. Only for sentences of 10 and more syllables the latter seems to get the start of the former. 8. The index loquelae i.e. the relation between the mean distances at which whispered sentences and whispered words are heard is *’/,,,. Physics. — “/sothermals of monatomic substances and their binary mixtures. XX. Isothermals of neon from + 20° C. to —217° C.” By C. A. CROMMELIN, J. Paracios Martinez, and H. KAMERLINGH Onnes. Communication N°. 154a from the Physical Laboratory at Leiden. (Communicated in the meeting of June 29, 1918). § 1. /ntroduction. This paper is the continuation of a previous pre- liminary communication '). The reduction of the neon-isothermals has now progressed so far, that what follows may be looked upon as a pretty nearly completed whole. The measurements refer to pressures up to about 90 atmospheres and as regards the temperature go from + 20°C. down to — 217°C.; they therefore embrace the region between the ordinary temperature down to the lowest tem- perature to be reached with liquid oxygen. The region from — 218° C. to —246° C., which can now also be covered by means of the hydrogen vapour cryostat’), is here left out of account; we hope shortly to be able to continue our measurements in this region. As regards the importance of such determinations as will be communicated on this occasion and the apparatus which have been used for the purpose we may refer to previous communications on the isothermals of hydrogen and of argon. *) For the sake of completeness and of a better survey of the whole work we have included in the table the material published in the previous communications on neon, quoted above, viz. the series I, I, STEVEN TME VIL and IX. § 2. The results of the measurements are given in table I, where 6 =the temperature on the international *) Cuxstus-seale, i.e. the temperature on the international Keivin-scale diminished by 273.09 ; p =the pressure in international atmospheres (for Leiden 1 inter- national atmosphere = 75.9488 cms mercury); d4 = the density expressed in the normal density (0° C. and 1 atm.) ; va = the volume expressed in the normal volume (0° C. and 1 atm.). 1) H. KAMERLINGH ONNES and C. A. CROMMELIN, Comm. NO. 147d; these Proc. XVIII (1) p. 515. 2) H. KAMERLINGH ONNES, Comm. N°. 15la@; these Proc. XIX (2) p. 1049. 3) Comp. H. KAMERLINGH Onnes and H. H. FRANCIS HynpMAN, Comm. N°. 69; these Proc. III p. 481; H. KAMERLINGH Onnes and CG. BRAAK, Comm. N’. 97a these Proc. IX p. 754; C. BRAAK, Dissertation, Leiden, 1908; C. A. GRoMMELIN, Dissertation, Leiden, 1910. 4) Comp. H. KAMERLINGH Onnes, Comm. Suppl. NO. 34a. § 5. 109° TABLE I. Series. | NO, | P | dy pv, | —| 6 = + 20°.00 C. | = ae. | if En | VI 1 22.804 | 21.046 1.0835 | VI ge | 25.015 | 23.052 852 | VI 3 26.575 24.464 863 VI 4 29.090 | 26.757 872 VI 5 32.512 | 29,891 897 VIII 34.887 32.002 902 VI Gut: 35.423 32.447 917 VI mk 37.812 34.601 928 VIII gel 39.168 35.843 928 VIII 3 44.762 40.862 955 VIII 5 54.149 49.213 1003 VIII Gn 59.717 54.161 026 VIII a 65.021 58.797 059 VII oy 17.360 69.338 131 VIII 10 | 82.545 | 73.967 160 VIII fetes 63.2392 «Me neee 186 VIII 12 | 93.298 | 83.154 220 6 = 0°00 C. VII 1 | 22.064 21.869 1.0089 VII ad 23.555 23.314 103 VII ge well 25.867 25.558 121 VII Ae «| 28.468 28.089 135 VII 5 | _430.790 30.345 147 IX 1 | 39.753 39.098 168 IX 2 | 44.892 44.030 196 IX 5 59.777 58.234 265 IX 6 66. 104 64.135 307 IX 1 74.059 11.495 359 NE: 79.108 | 76.127 392 BE 9 84.662 | 81.347 408 110 TABLE I (Continuea). Series. NO. Pp | dy | pv, = = 6 = — 103°.01 C. XV 1 35.558 56.40 | 0.6304 Xv” 1 36.697 58.23 6302 XV 2 40.610 64.21 | 6324 XV” 2 42.107 66.53 6329 XV” 4 | 55.136 86.57 6369 XV Adel 58.583 | 91.76 6384 XV 5 78.110 | 120.52 6481 6 = — 1419.22 C. XVI” U 33.840 69.83 | 0.4846 XVI 2 37.707 77.71 | 4852 XVI" 2 38.581 19.50 | 4853 XVI 3 43.319 | 88.97 4869 XVI” 4 49.881 | 102.32 | 4875 XVI 4 51.916 106.42 | 4878 XVI 5 66.471 134.91 | 4927 XVI 6 78.558 158.06 | 4970 | 6 = — 182°.60 C. X 2 32.067 | 99.89 | 0.3210 Xx” 2 32.988 | 102.84 | 3208 X 3 36.438 | 113.69 | 3205 Xx” 3 36.880 115.07 | 3205 X 4 41.371 | 129.44 | 3196 Xx” 4 42.533 | 133.15 | 3194 X 5 49.943 156.61 | 3189 x” 5 50.514 | 158.55 | 3186 Xx” 6 63.320 | 199.21 | 3179 wae TABLE I (Continued). | - Series. | NO, p dy | PU | 6 = — 200°.08 C. | Et | 8 XI 1 26.214 105.10 | 0.2494 eee kl 2/ 28.402 114.38 | 2483 XI 3 HA | 121.24 | 2469 TR 34.268 139.81 2451 XI 4 34.285 139.88 2451 XI 5 39.843 164.30 2425 XI 2 39.891 164.63 2423 BT 3 46.517 194.30 2394 EK 3 46.529 194.51 2392 MEO fog 47.951 200.79 2388 er, ee 61.657 | _ 263.77 2338 HI 2 67.456 | 291.10 2317 II 3 13.850 320.35 2302 HI 4 19.923 348.59 2293 6 = — 208°.10 C. XII 24.071 111.90 0.2151 XII 3 28.844 136.44 2114 XII 4 31.948 153.00 | 2088 XII 5 37.856 185.47 2041 XII 6 41.798 207.95 2010 IV I 58.472 308.32 1897 IV 2 64.451 345.22 1867 IV 3 69.692 377.89 1844 IV 4 74.532 409.18 1822 IV 5 79.228 439.12 1804 | | | Series. | NO, | 112 TABLE | (Continued). | P a, Pv, 6 = — 213°.08 C. | Mit 4 4 23.086 119.92 0.1925 XIII 2 24.810 | 129.82 1911 | Xi 3 26.673 140.90 1893 (is ee 29.365 157.70 1862 ie 5 32.441 177.37 1829 XIII 6 37.418 210.68 1776 II 53.896 334.59 1611 II 2 59.769 382.03 1565 II 3 66.271 | 435.46 1522 Hey ee | 12.858 484.75 1503 I | pies ae urere | 534.62 1491 g= — 21752 & XIV pee kl 21.349 | 123.40 0.1730 XIV Bo | 22.997 134.72 1707 XIV samen 24.686 | 146.67 1683 XIV 4 26.848 | 162.51 1652 XIV 5 30.042 | 186.94 1607 XIV 6 32.795 | 209.68 1564 I 1 49.930 | 358.51 1393 | I 2 53.528 | 395.62 1353 | I 3 59.618 | 458.40 1301 | ie ee 64.975 | 511.85 1269 | I 5 71.649 511.69 1253 | RAE 6 79.417 632.23 1256 A graphical representation of the observations will be found in fig. 1 U in a (Ze: a4) diagram. LT3r % ke) do 456 Bk ore ett eo ee ARE a EE nà LAARNE: Meente plath Tepe Fig: 1: 50 wo os § 3. Virialcoefjicients. By means of the above results some of the coefficients in the empirical equation of state pra = Ag+ Bada + Cada’ + Dada*+ Eadat+ Pada? *) could be computed. These calculations only embrace the coefficients Ba, Ca, Da and in one case Hy; for the densities which were reached are not so high as would be necessary for the deduction of #4 and in most cases of 4 also; these were therefore found from the reduced equation of state VII. A.37) or VIL1.®, in which the coefficients © and & are identical. In some of the calculations, 1) H. KAMERLINGH ONNES, Comm. NO. 71; these Proc. IV p. 125. *) H. KAMERLINGH ONNES and C. A. CromMELIN, Comm. NO. 128; these Proc. XV (1) p. 273. 5) Suppl. N° 19. Proceedings Royal Acad. Amsterdam. Vol. XXII. 114 as will appear presently, Dy, in others D4 and C4 were assumed according to VII. A. 3. For this purpose use was made of the critical constants of neon as published on a former occasion *) Ot Pee oo te Pk — 26.86 int. atm. The calculations were conducted in three different ways: a. for all temperatures only 6,4 was calculated from the obser- vations, the remaining coefficients being assumed, viz. C4 and Dy according to VII. A.3., H4 and F4 according to VII. 1 or to VII. A. 3- (as noticed above, this comes to the same); 6. for the lowest 4 temperatures B4 and C4 were computed from the observations, further as under a; c. for all temperatures B4 and C4 were deduced from the obser- vations, for —200°.08 C., —208°.10 C. and —213°.08 C. also Dy and for —217°.52 C. also Zy, further as under a. The calculations c were made first, with a view to obtaining the best possible accordance with the observations, the coefficients therefore bearing a purely empirical character. When it appeared that the values of C4 could not be connected by a smooth curve, much less those of D4, which proved the observational material to be insufficient for the deduction of Cy and Dy as functions of the temperature, we proceeded to the methods given under a and 8, in which the values of D4 and partly even those of C4 were assumed. Naturally the accordance with the observations is very much inferior with the methods a and 5 than with c. The results of the calculations which were all conducted by the method of least squares are found in tables Il and lil. Table II gives the individual virial-coefficients, as calculated from the obser- vations according to a, 6, and c, table II] the coefficients borrowed from VII. A. 3 as well as the values of 44 computed from the equation A4= Aa, (1 + 0.0036618 6), *) where for A4, the value + 0.99986, as published on a former occasion *), was taken as a basis. 1) H. KAMERLINGH ONNES, C. A. CROMMELIN and P. G. Carr, Comm. NO. 1515, these Proc. XIX (2) p. i058. 2) Comm. NO. 71. 8) Comm. NO, 147d. 1415 TABLE II. B, X 103 B, X 103 | CX 108 | 6 = | According to a. | According to 0. | | | + 20°.00 | + 0.54880 | | 0°.00 471148 | — 103°.01 16653 | — 1419.22 055249 | — 182,60 | — 0.003113 | — 200°.08 15746 = 016119 | 2 0:21531 | — 208°.10 19553 21706 18307 | — 213°.08 22305 24084 18407 — 217.52 24028 | 25880 19649 TABLE Il (Continued). | B, X 103 | C‚ X 108 D, X 10” | E, X 1018 ha | According to c. | | + 209.00 + 0.51578 | + 0.82778 | 0°.00 | 41334 | 1.1538 — 103°.01 069193 1.1515 — 141°.22 — 0.025378 0.71945 |__— 182°.60 13435 33607 — 200°.08 19667 21841 | — 0.24096 — 208°. 10 22926 25304 0.16102 — 213°.08 24625 21123 0.005848 — 2179.52 29313 36427 0.46139 | + 0.57517 116 TABLE III. = = C, xX 106 k Jelle ee E,X108) F,X< 107 | 6 A, = Ek eee at —— According to VII. A. 3. tks | | . | + 20°.00 | 1.0731 | + 0.29747 | | | | | 0.00 | 0.99986 | 25440 — 103°.01 | 62271 | 0.072156 | +0.31445 | —0.1373 | + 40.29 — 141°.22 | 48281 | 39576 | 28409 | 0.03754 | 12.51 — 182°.60 | 33131 | 58524 12718 2409 | — 4.190 — 200°.08 | 26731 | 96581 11124 | 4293 5.666 — 208°.10 23795 | 0.12219 0.081145 | + 0.04550 | 5.367 — 213°.08 | 21971 | 14073 60843 4599 4.836 — 217°.52 20345 | 15882 | 41215 | 4576 4.160 | $ 4. Discussion ana comparison with other observations. The differences between the pv4-values calculated from these equations and the observed values are represented graphically in fig. 2 as functions of the densities d4, the ordinates being the observed minus the calculated pv 4-values, expressed as percentages of the latter. In this manner the character of the deviations is more easily grasped than would be the case, if the numbers were given in the tables. The correspondence between the new and the old series is very satisfactory on the whole; only in the isothermal for — 217°.52C. a marked deviation may be noticed. Whereas for the isothermals of —200°.08 C. the deviations of the observations from tbe most closely corresponding formula (method c) are within 0.1 °/,, differences of almost '/,°/, occur in the isothermals for — 217°.52 C. The differences between the various sets of ABy-values obtained on this occasion from the smoothed B4-values according to VII. A. 3, viz. AB4 = BF (cale.) — By (VII. A. 3) are represented in fig. 3; the corresponding deviations of the B4-values obtained by Carr zat one of us!) from measurements at low pressures are included in the figure. 1!) P. G. CarH and H. KAMERLINGH ONNES, Comm. NO. 152e, presented to the Meeting of the Academy some time ago and shortly to be published in the Proceedings; preliminary values are given by P. G. Carn, Dissertation, Leiden LOL pwc7. i Ld Ty snbicton a a Eo pee a ae nee ae Peo | Fig. 2 118 It may be noticed, that the B4-values according to a agree fairly iel well with those according to VII. A.3. Those obtained by method c deviate much more markedly, as might be expected; especially at the lower temperatures they show much smaller values. Chemistry. — “Metals and Non- Metals’. By Prof. A. Smits. (Com- municated by Prof. P. Zeeman). (Communicated in the meeting of May 3, 1919). Introduction. In a few previous communications I already discussed the un- attackable electrodes and their efficiency as gas-electrodes. Only the hydrogen electrode was, however, discussed in detail. It was pointed out that the nnattackable electrodes are among the most inert metals, being so inert that even in contact with an electrolyte they do not assume internal equilibrium, so that they are almost always in disturbed condition, and we do not even know the potential of the really unary metal. When such a metal is immersed in the aqueous solution of an acid, while hydrogen is passed through, the electron concentration of the metal equilibrium in the electrolyte, which we represent by the equation : — i" . M, 2M, +16, becomes equal to the electron concentration of the hydrogen equilibrium: H, 2H, + 26, in agreement with the pressure of the hydrogen that is passed through, which means that the hydrogen phase and the hydrogen-containing metal-phase possess the same potential, the electromotive force of the circuit metal-electrolyte-hydrogen being given by the equation: Rr On ln A Owe Hence the hydrogen-electrode indicates the potential belonging to the three-phase equilibrium metal phase + hydrogen phase + electrolyte in accordance with the prevailing pressure, temperature, and total ion concentration (/7°)-+ (/”). As, however, as was already stated in the discussion of the #,X-fig., the said electrolyte contains an entirely negligible concentration of ions of the so-called unattackable electrode, we may substitute the words: hydrogen-ion-concentration for “total- ion-concentration”’ k= — (1) 120 When we now consider the case that an unattackable electrode is placed in an electrolyte, chlorine being led through, we get the same thing in so far that the electron concentration of the metal equilibrium in the electrolyte is entirely dominated by the electron concentration of the chlorine equilibrium: CIAO ee el 82) in agreemeut with the pressure of the chlorine that is led tbrough, from which follows that the chlorine-gas phase and the chlorine- containing metal phase will possess the same potential with respect to the electrolyte. Our more recent views about the electromotive equilibria lead to the assumption of an electro-ionisation equilibrium in chlorine gas, though the concentration of ions and electrons in this gas-phase is exceedingly small. That there exists such an electro- ionisation for chlorine, is proved by the exceedingly slight electrical conductivity. Now, however, the above mentioned equilibrium (2) does not suffice, for in electrically neutral chlorine this equilibrium cannot occur alone, because here there is only question of particles charged negatively electrically. Besides the chlorine electrode could not assume a positive charge with respect to the electrolyte. Undoubtedly this difficulty has also been felt in the former view about the electromotive equilibrium. The negative charge which zine assumes on immersion into an electrolyte was explained as follows: The zine tends to go into solution as zinc-ion, and it has evidently always been imagined that this happened through a s¢multaneous splitting up of the zinc-atom into zinc-ion + electrons, in which, however, the zinc-ions only went into solution, and the negatively charged electrons remained on the metal. When we wish to account for the positive charge which the chlorine electrode assumes with respect to an electrolyte in an analogous way, we get into serious difficulties, which have led to the assumption by some physicists that besides the free negatively charged electrons there exist also free positively charged electrons, and that these could render important services for the chlorine electrode and for the other non-metal electrodes. For just as the splitting up Zn Zn +20 was assumed when the zinc went into solution, the process Cl, 2Cl + 2D was supposed to take place at the chlorine electrode when chlorine went into solution. The chlorine ions formed go into solution and the positive electrons would be left behind in the chlorine electrode 197 This solution should, however, be denied all signification, because as yet free positive ions have not been met with, and everything points to the existence of only one kind of electrons, viz. the negative ones. Hence in spite of a desperate attempt, the difficulty remained un- diminished of force. The Chlorine-Electrode. When considering the just-discussed difficulty, I have come to the conclusion that we must necessarily assume that the chlorine atom possesses the power to split off electrons and to absorb them, and that these two processes take place side by side, which we can represent by the following equations: ') XC, 2X Cl ae 2X0. ce ie SF ee and FCI, | + 2YO,22YCl', Ba Vet Sete ( in which X and Y indicate the fractions of the original number Cl, mol. which have undergone a positive resp. negative ionisation. As the electrons, which are absorbed according to (2) proceed from the electron-ionisation (1) it is clear that DS In the limiting case Y = X the chlorine would contain an equal number of positive and negative ions, and no electrons at all. As we have to do here with a non-metal, X and Y will be exceedingly small. It is now the question how the positive charge of the chlorine electrode is to be explained. It is clear that for this we should have to assume that for the non-metal, chlorine, it is the negative ions that go practically exclusively into solution, and possess, there- fore, a much greater solubility than the positive ones. Metals and Non-Metals. These considerations about the non-metal, chlorine, which necessa- rily result from the consistent application of the principles of the theory of allotropy to the electromotive equilibria, lead us to the point of view from which we can survey the metals and the non- metals in a very satisfactory way. ') For the sake of simplicity we assume here that the positive ions are also univalent, En 122 It is, indeed, clear that as the metallic and non-metallic properties in the periodic system of the elements gradually pass into each other, theory will have to make clear that between the metallic and the non-metallic state there exist only quantitative differences, and that there are therefore all kinds of intermediary states possible. This demand can really be satisfied by an in every way plausible extension. For this purpose we must assume that the atoms of all the elements, hence both metals and non-metals, can split off and absorb electrons so that the following reactions take place side by side: Ree Mie Ke sn beeke gh ae me, m, and Nn n vl TEATER Ms me, in which ZE, denotes the molecule of an element and Ex and Hi, the ions. As is known for metals m, is mostly =1, and probably m, will as a rule also be =1. For non-metals m has often been found = 1, but several times also greater than 1. Of the factor m, no doubt the same thing may be expected. With perfect certainty we may only say this that n vi Y—vr, ae MH + calomel = 0,658 Volt <_ > HOH + calomel —= — 0,421 Volt ay + 70H = 1,08 Volt Oxyden electrode anodically polarised: —-> => TH a= Tealomel S= 0,653 Volt _ == ; HOH + Healomel = — 0,736 Volt AH +2%0R = 1,39 Volt The theoretical value for the electromotive force of the oxygen- hydrogen-cell at room temperature is 123. Volt This fact appears to indicate, that the difference between the potential, observed at the oxygen electrode and the theoretical value is due to a polarisation phenomenon, as it can be quantitatively reversed by reversing the state of polarisation of the electrode. 136 13. General remarks. It will be clear from the contents of this paper, that the theore- tical formula ne | 0,434 log es + ay = mt is borne out by experiment in different ways. The hydrogen-ions having been found to absorb the enzyme radiation, the question was considered, if the hydroxyl-ions possess this property also. If such were the case, the equation should be 0,434 1— By repeating the experiments of Part 3, but now with solutions of different pz, in most cases py=7,21, it was tried several times 1 OY aif == ir. y ‚found for two to decide this point. Combination of the factor different py, should give the necessary equations to calculate both n and 7. As will be clear, however, from the considerations in Part 3, the inevitable small experimental errors have an even larger influence at a py below or above the py=7,52 of maximum activity. It proved to be impossible to carry out experiments of sufficient accu- racy for this purpose. Still, the values, obtained for 7, though varying widely, were generally so small, sometimes even negative, that they allowed the conclusion, that the hydroxyl-ions (i.e. negative electricity) do not absorb the urease radiation. Since the writer’s first study on Enzyme-action *) in 1904, in which the probability was first pointed out, that also the catalytic action of hydrogen-ions and of many other catalysts might be due to radia- tion, the conception of catalysis as a radiation phenomenon has been taken up from different sides. In a series of papers Lewis *) has worked out the theory, that hydrogen-ions act catalytically through radiations, a molecnle of the catalysed compound becoming only reactive, if its energy has been increased by absorbed radiation (ultra red heat. radiation) to the “critical” condition. In extensive experimental researches NirRATAN DHAR*) pointed 1) Proc. K. Akad. Wetensch. Amsterdam and Zeitsch. physikal Chem. 49,4. 2) T. Chem. Soc. 1914, 2330, etc. 3) Proc. K. Akad. Wetensch. Amsterdam 1916 and T. Chem. Soc. 1917, 690. 137 out the analogy between chemical catalysts and light. His conclusion was, that: “probably the effects of increase of temperature, of light, and of chemical catalysts in a reaction are intimately connected and are possibly identical in nature”. They all appeared to act by shift- ing the equilibrium between “active” and ‘inactive’ molecules. SUMMARY. 1. The enzyme urease acts by a radiation, which is only absorbed by its substrate, urea, and by H-ions. 2. The mathematical formulation of this conception is et — dt, z+ nc — df =m in which 2 is the concentration of urea at the time ¢, c the con- centration and » the absorption-coéfficient of the H-ions, m a constant, proportional to the concentration of urease, if H-ion concentration as well as temperature are kept constant. Integration gives the formula ne - log —— + ay = mt, ence ae in which a is the initial concentration of urea, and y the fraction of a, still present at the time ¢. 3. By numerous experiments this equation is shown to represent the kinetics of urease action at constant py and temperature. It explains the nearly straight lines of the hydrolysis in alkaline solu- tions equally well as the practically logarithmic curves in acid ones. +. Comparing equal concentrations of urease at varying H-ion concentration, the constant m is found to be dependent on the py; i.e. the activity of a given concentration of urease is a function of the pa of the solution. Plotting m as a function of py, the resulting curve is strikingly similar to the characteristic curves for the undissociated fraction of an amphoteric electrolyte as a function of pz. This connection can be formulated mathematically and leads to the conclusion, that urease is an amphoteric electrolyte, whose activity is greatest when undissociated. The curve obtained represents the excess of activity of undissociated over dissociated urease. 5. This mathematical formulation leads to an approximate deter- mination of the dissociation-constants of urease, which are calculated to be not far from those of carbonic acid and ammonia. 6. The accelerating action on urease, ascribed by previous authors 138 to earbonie acid, is shown to be non existent. Ammonium carbo- nate + carbonic acid form a powerful buffer-mixture, which can maintain constancy of py, the indispensable condition for constant enzyme activity in a urea solution during hydrolysis by urease. Fresh confirmation of the above formula for the rate of hydrolysis is afforded by many experiments with this buffer-mixture. 7. The estimation of initial velocities of bydrolysis, equal con- centration of urease being allowed to act on different concentrations of urea at constant py and 7’, produces results, which appear unexplainable without the radiation-theory. The lower the py, the more these initial velocities increase on increase of the concentration of the urea. With high py, there is first an increase and then a decrease on raising the urea concentration. These facts are shown to be in perfect accordance withthe radiation- theory. 8. The influence of neutral substances is investigated experimentally and theoretically. Both decrease and increase of enzyme action by the same substance are explained by the influence the neutral body has on the dissociation-constants of water or of urease or of both of them. 9. The hypothesis is put forward that urease radiation, weakened by spreading or in any other way, causes synthesis. Experimental evidence of this is afforded by the fact, that at high pa, Where the urease is shown to be decaying, reversion of the hydrolysis is several times observed. 10. A second inference from this conception, that outside the sphere of hydrolytic action around a urease molecule there. must be a region of radiation, weakened by spreading, and therefore of synthesis, explains the fact, established by a series of new experiments, that, diluting the urease concentration beyond a certain value, its specific activity is decreased. For evidently the synthetic action of undecaying urease can only be manifesied, if the spheres of hydrolytic action do not intersect each other sufficiently. 11. A third inference, that in any urease solution, in which the enzyme is decaying through the combined action of alkalinity, tempe- rature and time, a synthesis of urea from ammonium carbonate, proportional to the urease concentration, will be observed, is tested experimentally and confirmed. 12. Description of an apparatus for the simple determination H-ion concentration at constant temperature. The determination of the hydroxyl-ion concentration, which is needed for the calculation of the dissociation-formula of urease, is 139 carried out with the same apparatus, after converting the hydrogen electrode into an oxygen electrode. Laboratory of the Dutch Yeast and Delft. Spirit Manufactory. A complete account of this investigation, with all experimental results, will appear in the Recueil des Travaux Chimiques des Pays-Bas. Zoology. — “On the phylogeny of the hair of mammals’. By Prof. P. N. van Kampen. (Communicated by Prof. Wesen). (Communicated in the meeting of May 31, 1919). In his work on “Die in Deutschland lebenden Arten der Saurier”’ (1872) and later on’), in a paper in which he rejects MAURER'’s well-known theory on the derivation of the hairs of mammals from epidermal sense-organs, Leypie draws the attention to the resemblance between the structure of the hairs and the so-called thigh or femoral organs of the lizards, which he considers to be a transition form between ordinary epidermal proliferations and hairs. Less attention has been paid to this remark than it would have deserved. For the structure of the afore-mentioned organs, whose function is not known (they probably participate in the act of copu- lation) closely resembles in fact the structure of a hair in a sim- plified form *): they are cylindrical rods, composed of horny epidermal cells, and sunken into a follicle of the skin. They differ from hairs principally by the absence of a cutis papilla and by the fact that they do not show a differentiation in medulla, cortex and cuticle. It is true that according to Maurer *) they are composed of two kinds of cells, but the arrangement of these cells is quite different from the one of the elements of the hair. As not one of the hypotheses which try to derive bairs from epidermal organs of lower Vertebrates and among which the afore- mentioned one of Maurer, based on a large body of facts, is best known, has been generally aknowledged (indeed, Borrzar *) in his review on these theories comes to the conclusion that none of them can be maintained and that the hairs in the mammals independently have taken their origin in the skin) it is desirable to examine, whether the idea uttered by Lieypie might contain perhaps a germ of truth. Against a direct derivation of hairs from femoral organs it may be advanced that these organs among the recent reptiles 1) Biol. Centralbl., XIII, 1893. *) LEYDIG compares them, in my opinion wrongly, to a bundle composed of hairs glued together. 5) Die Epidermis und ihre Abkömmlinge. Leipzig, 1895, p. 212 ff. 4) Anat. Anzeiger, XLVII, 1914/15. 141 only occur with the Lacertilians and with these far from generally, often only with tbe male and, besides, always in a very limited number. Elsewhere among Reptiles, organs in some degree compa- rable with femoral organs are only found with crocodiles, where, according to VoerLrzkow’s description *), they lie between the scales of the back. But, though it would be difficult to assume the direct origin of the hairs from femoral organs, the question might still be raised, whether there might not be a connection between them, in so far that they have a common origin. If this were the case, their origin might probably be traced most easily with the last-mentioned, more simply constructed organs, which in this way might throw a light on the origin of the hair. The morphological significance of the thighorgans has been eluci- dated by the research of ScHArnr’). This author not only confirms what has already been recorded by earlier investigators, viz. that with Lacerta the femoral organs of the male are most strongly developed in the breeding time, but he emphatically points out that in that period no keratinisation of the cells takes place. But of more importance is what he found with Sceloporus acantbinus: with this Iguanide no horny cells are formed in the organs, but instead of them a secretion, which is composed ‘‘aus einer völlig zerfallenen, dem Secret von Talgdriisen ähnlich sehenden Masse’’. ScHAFER comes to the conclusion that the thighorgans are glandulae celluliparae, related to those sebaceous glands, which are not connected with hairs. Keratinisation occurs only, when the secretion is slow. In connection with this conclusion the statement of Maurer*) is of importance, that with Lacerta the contents of part of the cells of the thighorgans is of a fatty nature. These facts point to a close connection of: the femoral organs with “holocrinous” cutaneous glands, and the conclusion that they can be derived from such glands is obvious. The difference between them is not great: if the fatty secretion in a sebaceous gland were replaced by keratinisation of the cells, then a horny rod would be formed, which would show great resemblance to the thigh- organs. Now the eleidin, which appears with mammals in the process of keratinisation, according to Maurer has a fatty character *), while on the 1) Abhandl. Senckenberg. Naturf. Ges, XX VI, H. 1, 1899. 2) Archiv f. Naturgesch., LX VIII, Bd. I, 1902. 5). le, ps 220. 4) Goerrr (Arch. f. mikr. Anat., IV, 1868) also describes the occurrence of fat-globules in the young epithelial hair-germ of the sheep. 142 other side it is known, that the cells of the sebaceous glands of the Mammalia contain eleidin-granules and can even partly undergo keratinisation *). One must indeed imagine the sebaceous cells to originate from ordinary epithelium-cells, which had already the capacity of keratinisation and it is not surprising that this capacity reappears now and then. So the femoral organs of Lacerta have arisen from cutaneous glands ; they have preserved the structure of those glands, chemically however they are modified, in connection with the strong kerati- nisation, which is characteristic of the skin of reptiles in general. If the femoral organs can be derived from cutaneous glands, one can imagine the same thing in the case of the hairs of mammals. Only in this case the differentiation has become greater and the structure of the organ is more complicated, in consequence of the more important function the hair has in the life of mammals. The hair papilla is to be considered of secondary origin and to have arisen in connection with the richer nutrition, which had become necessary for the stronger growth. But there is still another phenomenon that can be easily explained by this hypothesis. The origin of the hair as a solid epidermic thickening quite agrees with that of cutaneous glands, but also with that of the femoral organs, according to the descriptions of MAURER and ScHArgr. As to the thighorgans, the first author already directs the attention to this similarity with cutaneous glands of the amphi- bians, but attaches much importance to the difference between them, which lies in the fact that the smooth muscular fibres of the glands of the amphibians are absent in the femoral organs. In this point I cannot agree with him: these muscular fibres, which in the cutaneous glands are necessary for the extrusion of the secretion, are from their very nature superfluous in the entirely horny thighorgans, and so it is perfectly clear, that they have disappeared. And the same is true for the hairs, where they are absent as well. Another point, to which Maurer attaches much importance, is the peculiar arran- gement of the matrix-cells, which appears in the very first origin of the hair and of the dermal sense organ in the same manner. It seems to me however that this arrangement may be explained by the pressure of the surrounding cells upon the growing germ and so in different cases may appear in similar circumstances. Since the researches of Dr MriJere *) an attempt to explain the origin of the hair must take into consideration their arrangement on the skin. ‘) Cf. Scuärer, Text-Book of Microscopic Anatomy, 1912, p. 476. 8) Morph. Jahrb., XXI, 1894. 143 In those places where in mammals scales occur, the hairs are inserted generally in groups behind them. In this point they therefore differ from the thighorgans, which are placed in the middle of scales. It is true, that not too much value ought to be attached to this fact, but yet I want to point out in the first place that the afore-mentioned dermal organs of the crocodiles, described by Vorurzkow, are arranged between the scales, and further, that the similarity in location of hairs and femoral organs becomes greater, if the considerations of Prnkus*) in connection with the “hairdises” described by him, are right. If, as he thinks, a “Haarbezirk’”, that is the whole complex of scale rudiment, hair group and hairdisc, answers to the scale of reptiles, then the hairs are placed in the middle of the region of the scale, just the same as is the case with the thighorgans. Pinkus, who derives the hairdises from tactile spots of reptiles, cannot find an explanation for the origin of the hairs: “Das Säugetierhaar hat kein Homologen in dem Gebiet der Reptilienschuppe; sein Platz ist leer.” By the hypothesis, developed before, this objection against Pinkus’ theory is done away with. Maurer has directed the attention to another arrangement of the hairs; still before the hairgroups are formed, in mammalian embryos the placing of hairs in longitudinal rows may be stated. From this fact Maurer deduces an argument for his before mentioned theory, because epidermal sense organs generally show a similar arrange- ment. This argument however becomes worthless by the observation Fig. 1 Lacerta agilis. Bundle of three femoral organs (1—3). ') Arch. f. mikrosk. Anat., LXV, 1905. 144 of Maurer himself'), that also the first germs of the dermal glands of Triton and Anura occur in rows. The dermal glands of Promammalia have thus Ae in the Mammalia in diverging direction: the hairs, as well as the dermal glands of the mammals have arisen from them. The complex of hair and sebaceous glands is to be derived either from a compound gland, the follicles of which have taken a different direction of development, or what seems more probable to me, from the union of a number of glands into one follicle, in the same way as hairs may be united into a bundle. This last derivation may be strengthened by the fact that a number of femoral organs too, some- times form a bundle with a common follicle. | found this in Lacerta agilis (see fig. p. 143). Dales p. Log. Physics. — “Note on the circumstance that an electric charge moving in accordance with quantum-conditions does not radiate.” By G. Norpsrröm. Supplement N°. 48a to the Communications from the Physical Laboratory at Leiden. (Communicated by H. KAMERLINGH ONNEs). (Communicated in the meeting of May 31, i919). One often hears the remark: Bonr’s atomic theory is at variance with classical electrodynamics in assuming that an electron which is moving according to quantum-conditions does not radiate energy in the form of electromagnetic waves. The assertion formulated in this way does not seem to me to state correctly where the opposi- tion between Bonr’s assumption and classical electrodynamics lies. In the sequel I shall try to substantiate this view. We shall begin by looking at the problem from a general point of view. If an otherwise empty space contains electric charges whose motions are completely fixed, the electro-magnetic field is not singly determined by means of the Maxweri-Lorentz field-equations. In order to obtain a perfectly definite condition certain boundary-condi- tions must be fixed and it is to these that we shall give our atten- tion. Whatever the field may be, it may be represented by the electro-dynamic potentials viz. the vector-potential U and the scalar potential ¢~, which may also be combined in a four-dimensional vector-potential with the following components: Cpe eG pa pepe a Pn) The potentials determine the field-vectors €, 3 completely by means of the equations *) B = rot), 1 0% A IE) € = — grad p — EE (c= velocity of light) On the other hand the potentials U, ~ are not completely determ- ined by the field. For this reason we may submit them to the condition 1 oO v U —_.— =), di U + RE 1) Comp. for instance M. ABRAHAM, Theorie der Elektrizitat Il, 2te Aufl, p. 36. 10 Proceedings Royal Acad. Amsterdam. Vol. XXII. 146 and we then obtain for the various potential-components (using rational units) the following differential equation Pp pa Ppa 1 WY. On? Oy” dz’ * òf (a= 2, y, 2, t). hk 2. oe ee In this equations 6;, 6,, 6, are the components of the electric current, o, is the density of the electricity. The motion of the charges being given, Or, Oy, Oz, 6; are known functions of «, y, z, t. The general solution of the differential equations (3) may then be put in the following form: Let «,, y,, z, be the coordinates of a point for which p„ is to be found. The distance of this point to a point (a, y, z) may be called r, so that r? == (er) + (y—y,)* + (z—z,)* - - - - . (A) The sign of r is not fixed by this relation, but we may leave it undetermined in the mean time. A unit vector with components ee will be represented by r. The direction of vr is evidently dependent upon the choice of the sign of r. Let F be an arbitrary closed sur- face which encloses the point ae Yo. Zo: A surface-element of /, considered as a vector directed outwards, will be denoted by d%. Using these symbols we may write the general integral of the diffe- _ rential equation (3) in the form *): 1 i Ox | r Opa r Pale, = = — fe V —+ | d8| —grad pad —+-9 pe 200) An r r cr Ot or 6 The surface-integral has to be extended over the closed surface F, the space-integral over the enclosed space V containing the 5 point «,, %, Zo. The index t— — is meant to indicate, that at the Cc Nt Ops right-hand side the quantities dz, pa, grad pa, Ps refer to the time Ot a 5 t— —, which varies from point to point. The double sign on the right-hand € I) A proof of equation (6) may be found in Finska Vetenskapssocietetens Forhandl. L. |. 1908—09, Afd. A. n°. 6. If the sign of 7 is not fixed beforehand, it is easily found that the right-hand side has the double sign. 147° side is required as long as the sign of 7 has not been settled, but may be chosen at will; the + sign holds for positive, the — sign for . . . . . if . . negative 7. If the sign is taken positive, the moment ‘— — is earlier C than ¢ and gy, must then be taken as a delayed potential. If the . . . r . . negative sign is chosen, the moment ¢— —is later than ¢ and gs, is C to be considered as an advanced potential. Every function pa (x, y,2, t), therefore, which satisfies the differential equation (3), may be considered either as a delayed or as an advanced potential, if only the contribution to the potential which is due to the boundary surface F (which may also be moved to infinity) is taken into account. It follows that every electro-magnetic field, i.e. every field for which the Maxwewi-Lorentz equations hold, may be calculated for aed ae ; (7) a moment ¢ either from the condition at the time ¢— “+ or from c that at the time 4 + Ee if only the contribution by the boundary surface is taken into account. This contribution is necessarily diffe- rent in the two cases. If the surface F is made to move to infinity and if at the same time the condition is laid down, that at the boundary the surface- integral has the value zero, if the potential is considered as a delayed potential, the ordinary solution is obtained of the problem to calcu- late the field from the charges. But this solution is only one parti- cular one: there are an infinite number of others. The author may be excused for having discussed this question at some length: it seemed to him that it is not always sufficiently kept in view. We shall now prove, that every periodical motion of electricity allows the assumption of a field such that no energy is radiated. The separate points of the electric charges will be identified by means of 3 parameters &, 7,6. Every set of values §, 4, 6, therefore, denotes a definite point in the electricity sharing the motion of the latter. The motion is completely described by the equations (Sr Gale | y =y (31,5, 4), 2 =—12.(5. 1), Gr Alk that is to say, for given values of S,1,¢ the coordinates wz, 7, z are (7) 148 functions of the time. Let us consider the motion represented by equations (7) as being completely given. This motion we shall call motion 1. We therefore assume, that wv, y,z are known functions of ¢7,6,¢ for all values of ¢ from — oo to-+ oo. We then calculate the electro-magnetic field by means of delayed potentials and choose the boundary conditions in such a manner, that the surface-integral in (6) beeomes zero for each potential-component g,, when the surface JF’ moves to infinity. The field is then singly determined by the motion of the electricity. We shall call the field obtained field 1. In this case we obviously have a radiation of energy. We shall further consider «a different motion of the electricity, motion 2, which is obtained from motion 1 by reversing the sign of ¢. 2 == (6, 7, S, —t), y= y (§, 4,5, —4), motion 2. BEM Ge 2) ‘In this system all paths are evidently deseribed in a direction opposite to that of motion 1. For motion 2 we again calculate the electro-magnetic field by means of delayed potentials and with the same boundary-conditions as before. We shall again obtain a field with energy-radiation, which we shall call field 2. If the motion 1 is periodical, this will also be the case for motion 2 and the radiation during “one period is equal for field 1 and field 2.° We now pass from field 2 to a new field 3, by reversing the sign of ¢ and at the same time the components of the magnetic field Bz, B,,B:. It is easily shown, that with this change of sign the Maxwerr-LoreNtz field-equations remain valid. As ¢ changes sign, the motion of the electricity is reversed. The motion of the electricity is therefore precisely the same in field 3 as in field 1. Owing to the reversal of the sign of B, B,, B, (Ex, ©, €.-retaining the same sign) the direction of the energy-stream is reversed, so that in field 3 we have a radiation of energy wwards. For a periodical motion of the electric charges the amount: of energy drawn in during a period is the same as the radiation outwards in fields 1 and 2. It is further easily found that field 3 may be calculated from advanced potentials, with a zero-contribution of infinity. If on the other hand the potentials are taken as delayed, the contribution of infinity is not equal to zero. We now superpose field 1 and 3, which is possible since the field-equations. are linear. In the two fields taken separately the electricity has the same motion, which therefore remains the same in the combined field. The density of the electricity on the other 149 hand is everywhere doubled by the superposition. In order to reduce this to the former value we diminish the strength of the field in the combination to half its value. The density of the electricity is thereby diminished in the same ratio, and in the new field + we thus obtain precisely the same distribution and motion of the elec- tricity as in field 1. The motion being periodical the energy-stream in field 4 evidently gives a total radiation zero during a full period. We have obtained a kind of stationary electro-magnetic waves. Since the motion of the electricity in field 4 is identical with the motion from which we started, it has been proved that every pervo- dical motion of electric charges allows the assuinption of an electro- magnetic field without radiation of energy. Without further enquiry it is clear, that this proposition also holds, if the motion of the charges is not exactly periodical, but consists in, say, a planetary motion with a movement of the perihelion. The question remains, whether Bour’s theory can draw any benefit from the result arrived at, but it seems that such is not the case. If the electrons in an atom were going round in the same orbits for all eternity, there would be nothing to prevent us assuming an electro-magnetic field such as field 4. But the sudden transitions from one allowable orbit to another cause difficulties. A simple calculation shows that in field 4 the energy-density for large distances r is proportional to “4 the energy of the whole field, therefore, becoming infinite. In consequence of this the change of energy associated with the transition of an electron has quite a different value to what Bonr has to assume, and it does not seem to me possible to make the two values agree. The above discussion, there- fore, hardly seems to have a direct bearing on Bour’s theory, but it does seem to me to be of some use for obtaining a broader insight into the question as to where the difficulties of Boar’s theory actually lie. The result to which in my opinion it leads in this case was stated in the beginning of this note and I should like to formulate it in this way: the usual statement that it is inexplicable why an electron moving in accordance with the quantum-conditions should not radiate energy, seems to me to be based on an assump- tion which is not sufficiently general. A more general conception of the problem although unable to solve the difficulty, seems to me to put it in a different light. Leiden, April 18, 1919. Mathematics. — “Remark on Multiple Integrals.” By Prof. L. E.J. BROUWER. (Communicated in the meeting of June 28, 1919). The object of this communication is to make two remarks in conjunction with the first part of my paper: “Polydimensional Vectordistributions’') presented at the meeting of May 26, 1906. Rak I. The proof of the generalisation of Srokes's theorem given l.c. pp. 66—70 provides this generalisation not only in the Euclidean but also in the following ametrie form: Turorem. In the n-dimensional space (a,,...2,) let the (p—1)-tuple integral fF Het, de (es os En) dte: dee +5 STEEN era) be given, where the F’s are continuous and finitely and continuously differentiable; consider also the p-tuple integral IS fue, (rn de de dea, OM ml) where Da OF; Fee, 4 = = ie | Our (indicatrie j, a, aeg. indicatrix a, ... dy). Then, if the two-sided p-dimensional fragment?) G is bounded by the two-sided (p—)-dimensional closed space g, the indicatrices of G and g corresponding and both G and g possessing a continuously varying plane tangent space, the value of (1) over g is equal to the value of (2) over G. 1) See Vol. IX, pp. 66—78; we take the definitions modified in accordance with note') on p. 116 le. I take this opportunity of pointing out that on p. 76 Lc. lines 183 and 14 “finite sourceless current system’ should be read instead of “system of finite closed currents”’. 3) Math. Annalen 71, p. 306. 151 Of this theorem, which was enunciated by Poincaré‘) in 1899 already, without proof however, and in a form expressing the rule of signs in a less simple manner, I shall here give the proof anew, editing it somewhat more precisely than in my quoted paper. I]. In the n-dimensional space (2,,...2,), which we shall denote by S, let the p-tuple integral p j= Pa a, (Ee ERD dite. vann a er eat) a be given, where the p's are continuous. Let Q be a two-sided p-dimensional net fragment’) provided with an indicatrix and situated in S, 6 a base simplex of Q with the vertex indicatrix “A A... AA, A an arbitrary point of o, AP a. the value of Pane, at A, ,v, the value of z, at A,. For every o we determine the value of i =, PEED © Glee 9 P where ey eee | | lt ra sta, “) = ag | x En | ’ | | een et EN U and where, for different terms under the = sign A may be chosen differently; and we sum ‚p over the different base simplexes of Q. The upper and lower limit between which this latter sum varies on account of the free choice of the points A, we call the upper and lower value of (3) over Q. If we now subject Q to a sequence of indefinitely condensing simplicial divisions which give rise to a sequence Q’, Q",... of net fragments covering Q, then, as v increases indefinitely, the upper and lower value of (3) over Q”) converge to the same limit, which we call the value of (3) over Q. Let F be a two-sided p-dimensional fragment provided with an 1) Les méthodes nouvelles de la mécanique céleste III, p. 10. The significance of the rule of signs here formulated, is apparent only after comparing former publications of the same author from the Acta Mathematica and the Journal de PÉcole Polytechnique in which the equivalence of the identically vanishing of (2) and the vanishing of (1) over every g was pronounced, *) Math. Annalen 71, p. 316. indicatrix and situated in S, f a sequence of indefinitely condensing simplicial approximations P’, P",... of HF corresponding to a category w of simplicial divisions. If the values of (3) over P’, P",... converge to a limit which is independent of the choice of f so far as it is left free by w, then we call this limit the value of (3) over F for w. II]. We shall now occupy ourselves with the value of (1) over the boundary 8 of a p-dimensional simplex o provided with an indicatrix and situated in S. In doing so we take it that the indi- catrices of 3 and o correspond, that is to say, the indicatrix of an arbitrary (p—1l)-dimensional side of o is obtained by placing the vertex of o which does not belong to this side last in the indicatrix of 6 and subsequently omitting it. We begin by confining ourselves to the contribution of the single term P = les ee ee Oene: dea, to the value of (1) over 8. By a suitable simplicial division ¢ of the’ Spacen. 2 a, ) we determine a simplicial division of 3, ne ale): pl The family of those (n—p + 1)-dimensional spaces within which are constant, cuts the plane p-dimensional space in whose base simplexes correspond in pairs to those of (wz,,...# Vay 97 2 een which o is contained, in a family of straight lines which connect pairs of corresponding base simplexes of } into p-dimensional trun- cated simplicial prisms. If e, and e, are a pair of corresponding base simplexes of 8, d the concomitant truncated simplicial prism, /a line segment having components 74 ,...7, which leads from, a point Pv of e, to the corresponding point of e,, then the contribution of the term |F. oere , Utyz,.... de, _, to the value of (1) over e, and DE Pe e, becomes n OF. ty = Bee . Pale ° A Oa, { ze é, where A denotes a point of 5 which may be different for the diffe- rent terms under the > sign, and ¢ becomes indefinitely small with nespect tO stats seg j for indefinite condensation of &. p— INOW let BB, s B, be a vertex indicatrix of e, and ,r, the value of «, at B,, then the value of 1%, . eta .-- a) can be expressed as la — pa ta — pha, 0 | ae | (p—1)! | fie, ti i anti Pa, faa ties QO |? oe plz © (Ce bese) sts . Be pez Pa, thus also as ar dina, NA + € so that the contribution of the term [Fa say dn dag, te the value of (1) over e, and e, can be expressed as ; n OF, en e+ a. day... a y=p A Ow, pt » and the value of (1) over 8 is obtained in the form: > P OF; : Dy a PES ha one Of 5 eee ae A Out, : Ps d (indicatria j, a, aeq. indicatriz «‚... Uy); where A represents a point of o which may be different for different terms under the > > sign. Hence it follows immediately, that, if Q is a two-sided p-dimen- sional net fragment situated in S, and A denotes the boundary of Q, while the indicatrices of Q and R correspond, then the value of (1) over Ris equal to the value of (2) over Q. IV. To complete the proof of the theorem formulated in I, we consider a category yp of simplicial divisions of g such that the aggregate of the base sides possesses for y uniformly continuously varying plane tangent spaces, and the ratio of the volume of a base simplex to the (p—1)"" power of its greatest coordinate fluctuation does not fall below a certain minimum for w. Let g',g",... be a sequence of indefinitely condensing simplicial approximations of g corresponding to w. If, ong we construct an approximating simplicial representation g°?” of g®, then, by choosing both u and r above a suitable limit, we can, in virtue of III, see to it that the values of (1) over ge” and g® differ from each other by as little as we please, whilst g®) is covered by ge” with degree one, so that the values of (1) over g'”) and g®”) are equal. Thus there exists a value of (1) over g for ww which, naturally, does not change if, instead of w, some other category of the same kind is chosen. 154 Now let x be a category of simplicial divisions of G analogous to yw. The resulting simplicial divisions of g belong to a category w of the kind just described. Let G',G",... be a sequence of indefini- tely condensing simplicial approximations of (# corresponding to z, then, at the same time, there is hereby determined a sequence g', g’,... of indefinitely condensing simplicial approximations of g corresponding to w. Since, in virtue of III, the value of (1) over g® is equal to the value of (2) over G, there exists, just as there does a value of (1) over g for w, a value of (2) over G for x, both values being equal, and not changing if some other category of the same kind is chosen in the place of either or x. wy In introducing le. p. 70 the notion of a second derivative, we have omitted to give the definition of the underlying concept of normality of an S, provided with an indicatrix and an Seay provided with an indicatrix which are perpendicular to each other in an 5S, pro- vided with an indicatrix. This definition we shall here give. Let 7 be the point of intersection of Sp and S,_,, «,'...a, T the indicatrix of S, and 8,...8,-,7' the indicatrix of S,_,; we call Sp normal to Sep and’ Sj postnormal 10. )5,,.0 ere ip, ann is an indicatrix of S,. Thus, for some values of » the concepts normal and postnormal are equivalent, for other values not equivalent. Furthermore we call a p-dimensional vector system V normal to an (n—p)-dimensional vector system W at the same point, and W postnormal to JV, if, with respect to a rectangular system of coordinates the components of V are respectively normal to and of equal scalar values as the components of JW. In this terminology, the second derivative of the vector distribution a —p pv’ X is the normal distribution of tbe first derivative of the postnor- mal distribution of ’_X. » =e get Je De eerde wrr A om ma en Dt ES ed Janae 2 pe pl BNN Tan Het li aS -< ory EN afd. es B hd EN bim IEA Bs vp En ne eee aan: De ade ed AN ‘ah im Lg Pii a hind atin # e “bi Fast A ae ee nd Mers An - ic ar pe KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN TE AMSTERDAM. PROCEEBINGS VOLUME XXII N°. 3. President: Prof. H. A. LORENTZ. Secretary: Prof. P. ZEEMAN. (Translated from: “Verslag van de gewone vergaderingen der Wis- en Natuurkundige Afdeeling,” Vol. XXVII). CONTENTS. JAN DE VRIES: “Null systems determined by linear systems of plane algebraic curves”, p. 156. H. Ge H. 4 Bs B. A. BOCKWINKEL: “On a certain point concerning the generating functions of LAPLACE,”. (Com- municated by Prof. H. A. LORENTZ), p. 164. K. WOLFF: “On the so-called filtrable virus of influenza described by VON ANGERER”. (Commu- nicated by Prof. C. EYKMAN), p. 172. ZWAARDEMAKER and H. ZEEHUISEN: “On the relation between the electrical phenomenon in cloudlike condensed odorous water vapours and smell-intensity”, p. 175. . GOUDRIAAN: “The zincates of sodium. Equilibriums in the system NayO-ZnO-H,0”.(Communicated by Prof. J. BOESEKEN), p. 179. . VAN LOHUIZEN: “The Anomalous ZEEMAN-Effect”. (Communicated by Prof. H. A. LORENTZ), p. 190. . F. NIESSEN: “A Theory of a Method for the Derivation of the Distribution of Energy in a Narrow Spectrum Region from the Distribution of Energy, Observed in an Interferometer”. (Communicated by Prof. W. H. JULIUS) p. 200. KRUIZINGA: “Some new sedimentary boulders collected at Groningen”. (Communicated by Prof. G. A. F. MOLENGRAAFF), p. 225. W. DE SITTER: “Theory of Jupiter’s Satellites. II. The variations”, p. 236. R. if MAGNUS and A. DE KLEYN: “Tonic reflexes of the labyrinth on the eye-muscles”, p. 242. A. SCHOUTEN: “On expansions in series of covariant and contravariant quantities of higher degree under the linear homogeneous group”. (Communicated by Prof. J. CARDINAAL). p. 251. sl Proceedings Royal Acad. Amsterdam. Vol. XXII. Mathematics. — “Null systems determined by linear systems of plane alyebraic curves’. By Prof. Jan pe Vruxs. (Communicated in the meeting of January 25, 1919). 1. A triply infinite system (complex) S‘* of plane algebraic curves c” contains a twofold infinity of nodal curves; for an arbitrarily chosen point D is node of a nodal curve d” belonging to S®). I shall now consider the null system in which the tangents d, d' of dr are associated as null rays with D as null point. The nodal curves of a net belonging to S‘) have their nodes on the Jacobian, which is a curve of order 3(2—1). It has in common with the Jacobian of a second net the 3(m—1)* nodes, which occur in the pencil common to the two nets. The remaining intersections of the two loci are critical points, i.e. nodes for the curves of a pencil. The null system, therefore, has 6(n—1)’ singular null points. 2. Let a be an arbitrary straight line, P an arbitrary point. The dr, which has its node D on a, intersects the ray PD, moreover, in (n—2) points #. If Eis to get into P, dr must belong to the net that possesses a base-point in P; D lies then on the Jacobian of that net. The locus (£) of the points / passes, therefore, 3 (n—1) times through P, and is consequently a curve of order (4n—5S). Each intersection of (#) with a is node of a 0”, of which one of the tangents d passes through P. There is therefore a curve (D)p of order (42—5) which contains the nodes of the nodal curves 9”, which send one of their tangents d through a given point P. It will be called the null-curve of P. For a singular point S it has in S a triple point. As P evidently is node of (D), there lie on a ray d passing through P (4n—7) points D, for which d is one of the tangents of the corresponding curve dr. From which ensues: an arbitrary straight line d has (4n—7) null-points D. 3. The null-curves (D)p and (Dig have the 6(m—1)* singular points in common; for, a critical point bears oo! pairs d, d’. The two curves pass further through the (42—7) null points of the straight line PQ. Each of the remaining intersections is a point D, for which d passes through P, d’ through Q. In other words, if d revolves round P,d’ will envelop a curve of class 157 (10n?— 32n4-26). To the straight lines d’, which pass through P, belong the tangents of the ò7, which has its node in P. Each of the remaining (10n?—32n+24) straight lines d’ evidently coincides with a ray d, and therefore contains a null-point D, for which the two null-rays have coincided. If such a straight line is called a double null-ray, it ensues from the above that the double null-rays envelop a curve of class 2(n—2) (5n—6) *). 4. The null-rays d, which have a null-ray D on the straight line p, envelop a curve (p) of class (4n—5), which has p as (4n—7)- fold tangent. It, therefore, intersects p in (An—5) (4n—6) — (4n—7) (4n—6) points, which bear each two coinciding null-rays. The locus of the points C, which bear a double null-ray, is, therefore, of order 4(2n—3). The curve (C) is evidently the locus of the cusps of the complex. As the order of (C) may also be determined in another way, it appears at the same time that the curve (p) has no other multiple tangents. 5. The case n=2 deserves a separate treatment. In the first place each line d has now only one null-point; this is the node of the conic, which is indicated by three points of d. The locus (C) is now of the fourth order, and consists of four straight lines cz. For, if the two straight lines of a nodal c’ coincide, Cx is a double line. The complex contains, therefore, four double Lines, and they are at the same time singular null-rays. The vertices S,, of the complete quadrilateral formed by them are the singular points of the null-system. The curves (p), and (q),, cf. $ 4, have, besides the null-rays of the point pg, seven tangents in common, which have each a null- point on p and a null-point on qg, and are consequently singular null-rays. To them belong the four straight lines cy. Each of the remaining three singular null-rays s must belong to oo’ nodal conics. S,, bears as singular point, oo * pairs of lines, which form an involution of rays; so S,, S,, belongs to two, and then to oo *, pairs of lines and consequently must be singular. The diagonals of the quadrilateral, which is formed by the four straight lines c, are consequently the three singular null-rays required. ') In other words, the cuspidal tangents of the cuspidal curves of a complex envelop a curve of class 2(n—2)(5”—6). In my paper on the characteristic numbers of a complex (These Proceedings, Vol. XVII, page 1055, § 13) the influence of the critical points in the determination of the class has been overlooked. ji lg 158 6. If the complex {c*} has a base-point B, it is at the same time singular null-point, for two points on a ray passing through B, determine a nodal d°, with node in 5. The double rays of the involution formed by the curves d? with node B are double lines of {c*}, consequently singular null-rays. Other double null-rays do not exist, for if a straight line d of d? does not pass through B, d’ does. As B is node of the Jacobian of each net belonging to {c’), this point replaces four critical points. Two more singular points, there- fore, lie outside B; they are connected by a singular null-ray. 7. In a fourfold linear system S, each point D is node to a pencil (d”). Two of those curves have a cusp in C= D. I now consider the nz/-system in which to the null-point C are associated the cuspidal tangents c,c’ of the two cuspidal curves yr, which have their cusps in C. The straight line d is touched in each of its points D by a nodal de, which has its node in D. With the straight line PD dr” has moreover (n—2) points / in common. In order to find the locus of the points /, I shall inquire how often / gets into P. In this case J” belongs to the complex that has a base point in P; in it occur (4n—7) d”, which touch at d ($ 2). Consequently (#) is a curve of order (52—9). If # lies on d, PE =d’ touches in that point at a òr, which has its node on J. Every straight line d therefore is nodal tangent of (5n—9) curves 6”, of which the second tangent d’ passes through P. If dis now made to revolve round a point Q, the point D. describes a curve (D) of which every point is node of a 0”, which sends its tangents d and d’ through Q and P. In Q a dis touched by QP, so Q and consequently P is a point of (D), so that this curve is of order (5n—8). If C is one of the (5n—10) points, which (D’) has in common with the straight line PQ, besides P and Q, the tangents d,d’ fall both along PQ, so that C is a cusp of a cuspidal curve y", which has c= PQ as cuspidal tangent. In the above null-system a straight line therefore has 5(n —2)null- points. If e revolves round a point M, the null-points C describe a curve of order (5x—8), with node M (the null curve of MM). 8. The system S@® contains a number of curves with a triple point. If S“ is represented by the equation A+ PB 4 yC + dD+ el =0, 159 the co-ordinates of a triple point have to satisfy the six equations: aAyy + BBei + yCu + Du + el = 9, in which Aj, etc. represent derivatives according to x, and a. The number of points has to be found, for which Ashe Ae Az Ane By BNA ANB Cuero Ce Ge C=O. De) DD Dac We | #, E,, EB. EE, | | According to a well-known rule we find for this (5°—4? + 37—2? + 1?) (n—2)?. There are therefore 15(n—2)? curves c, with a triple point S’). In such a point the nodal curves have the same tangents d,d’. Any straight line passing through S is to be considered as a cuspidal tangent c. The null-system therefore has 15(n—2)? singular points. 9. I now take three points P, Q, R, arbitrarily, and consider (ef. § 7) the curves (D)pg and (D)pr. To begin with they have the point P in common; for there is a dr, which has P as a node, and PQ as tangent and a de, for which one of the tangents lies along PR. Those curves have further in common the (52—9) points D, for which QR is one of the tangents d. Another group of common points consists of the singular points S. Let U be one of the still remaining intersections. There is in that case a dr with tangents U P and UQ, and also a dr with tangents UP and UR. From this it ensues that all é with node U have the straight line U P as tangent, consequently belong to a pencil in which the tangents d,d’ form a parabolic involution. The double rays of this involution have then coincided in U P, and U is cusp for only one cuspidal c. If such a point is called unicuspidal point, it follows from (5n—8)?’—1—(5n—9)—15(n— 2)? that (10n?—25n + 12) unicuspidal curves send their tangent through P. The cuspidal tangents of the unicuspidal points envelop a curve of class (10n?—25n + 12). 10. In any point C of the straight line a I draw the two null- 1) If nm =3, and the system has 5 base-points, the 15 triple points are easy to indicate. One of them e.g. is the intersection of B,B, with Bs By. 160 rays c,c’ (cuspidal tangents), and consider the correspondence between the points Z,L’, which c,c’ determine on the straight line J. If c is made to revolve round Z, the null-points of c describe a curve of order (5n—8), which has a node in ZL (cf. §7). Toa point L therefore belong (5n—8) points C and (5n—8) points LZ’. The point al represents two coincidencies L = L’. The remaining coinci- dencies arise from cuspidal tangents w of unicuspidal points U. The locus of the unicuspidal points is therefore a curve of order 2(5n—9). This may be confirmed in the following way. If C describes the straight line p, the null-rays c,c’ envelop a curve of order (5n—8) which has p as (5n—10)-fold tangent. It therefore has, not counting the points of contact, (52-—8)(52—9)—(5n—10)(5n—-9), consequently 2(5n—9) points in common with p. In each of these points the null-rays c and c’ have coincided. 11. The system S@ produces in a still different way a null- system. Any point F is fleenodal point for five curves g”. In order to find this out we have only to consider the curve that arises if we make every d” that has F’ as node, to intersect its tangents d,d’. This C+? namely, has in F’ a quintuple point ’). I now associate to each point F as null-point the five null-rays f, which are inflectional tangents for the five fleenodal curves gp". Any point D of the straight line a is node for a dr, which touches the ray PD in D. I now determine the order of the locus of the groups of (n—3) points # which each of the curves d* has moreover in common with PD. If £ lies in P, 0” belongs to a complex S®). According to § 2 there are on a (4n-—5) nodes of curves 0” of (3) which send their tangent d through P. So P is (4n—5)-fold point of the curve (EZ) and the latter consequently of order (Sn—8). In each of its intersections F with a a curve ¢ has a fleenodal point, the inflectional tangent of which passes through P. From this it ensues that the locus of the null-points F of the rays f out of a point P (null-curve of P) is a curve of order (5n—8). As it must have a quintuple point in P, an arbitrary straight line f therefore contains (5n—13) null-points. *) 1) In a point S (§ 8) the c” with triple point replaces three of the curves en; for the other two the inflectional tangent lies along one of the two fixed tangents d, d’. For a unicuspidal point (§ 9) one of the curves pr has its inflectional tangent along the fixed tangent d. 2) For n=3 is on —13=2. Each @5 is then the combination of a straight line f and a g°. Each straight line f belongs in S(4) to a figure (f, ¢%); its intersections with p? are the two null-points F. 161 12. In the null-system (Cc) P has a null curve of order (5n—8) with node P (§ 7). Of its intersections with the null curve with respect to the system (/’,/), 10 lie in P. They also have the uni- cuspidal points U in common, for which the tangent w passes through P. In each of the remaining (5n»—8)*—10—(10n?— 25n+-12) inter- sections G, a cuspidal curve has with its tangent g four points in common. From this it ensues that the four-point cuspidal tangents envelop a curve of class (15n*—55n-+42). If n is equal to three, the curves y* with four-point tangents are replaced by conics, each with one of its tangents. The null-system (ff) then has the characteristic numbers 5 and 2; the null-curve (P)’ of P is of class 22, consequently sends 12 tangents f through P, and each of these straight lines forms with the conic touching it a y° with four-point tangent. In conformity with this, the form 15n*—55n-+-42 produces for n = 3 the number 12. 13. In a quintuple infinite system S©) each point D is node for a net of nodal curves. A straight line d passing through D deter- mines in it a pencil, of which all gr touch at d in D. There is consequently one cuspidal y", which has a straight line c passing through D as cuspidal tangent. The curves y”, with cusp D, form a system with index two, for the curves J”, passing through any point P, form a pencil, which contains two curves with cusp in D. If every straight line c passing through D is made to intersect with the cuspidal 7”, which it touches in D, there evidently arises a curve of order (n-+-2), which has a quintuple point in D. From this it ensues that five cuspidal curves have in D a cusp, where the euspidal tangent has a four-point contact. ] shall now consider the null-system (Gg), in which to a point G are associated the jive straight lines g, which are four-point cuspidal tangents for cuspidal curves y" with cusp G. 14. In each point C of the straight line a I consider the cuspidal curve y", which sends its tangent c through P, and determine the locus of the points #, which y” has still in common with PC. If E lies in P, y" belongs to a system S(; in it (5n—8) curves y” occur, which have their cusp on a ($7). So the curve (EZ) passes (5n—8) times through P and is of order (6n—11). In each of its intersections G with a, a 7” has four points in common with PG. The null-eurve of P is therefore of order (6n—11). As it has a quintuple point in P, a straight line g passing through P is nul/- ray for (6n—16) points G. 162 15. The system S)) contains o* curves with a triple point 7’. If S©) is represented by cA HBB 4+7C+6D+ E+ oF =0, the locus of the points 7’ is determined by Axi Bui Crt Dru Eni Fu) = 0. It is therefore a curve (7) of order 6(n—2)’). A w with triple point 7’ determines with a nodal dr which has its node in 7, a pencil of nodal dr with fixed tangents d,d’. The net of the curves d” with node: 7’ therefore consists of o' similar pencils of which the tangents d,d’ form an involution. Each of the two nodal rays c,,c, is common cuspidal tangent for a pencil of cuspidal curves and each of these two pencils contains a y” with four-point tangent. The five null-rays g of 7'are therefore represented by the straight lines c,,c,, and the three tangents ¢,,¢,,¢, of the curve vt”. The points 7’ are consequently not singular. 16. In a sextuple infinite system SS each point 7’ is triple point of at. To 7 as null-point the three tangents /,, ¢,,¢, of t” are now associated as null-rays. In order to find the second characteristic number of this nul/- system, 1 consider the curves 1?, of which the point 7’ lies on the straight line a and | try to find the order of the curve, which contains the groups of (n—3) points 4, in which ¢ is moreover intersected by PT’. If # lies in P, t belongs to an S®), and 7’ is one of the 6(n —32) points which ($ 15) the curve (7’) has in common with a. So F is a (6n—12)-fold point on the curve (£), which consequently has the order (7n—15). The nudl-curve of P is therefore of order (7n—15). As it passes three times through P, a straight line ¢ passing through Pis tangent for (7m—18) curves t+”, which have their triple point 7 on t. A null-ray, therefore, has (7n—18) null points. 17. The curves (7), which belong to two systems S©) comprised in S®, have the 15(2—2)* points 7 of the system S® in common, which forms the “intersection” of the two S©), The remaining intersections are critical points, viz. each of them is triple point for a pencil of curves t”, consequently singular null- point S for (7,4. This null system has consequently 21(n—2)' singular null-points. 1 If, for n—=3, the system S(5) has the base points B, By, Bs, By, the curve (T) consists of the straight lines Be Bi. 163 As the triplets of tangents of the curves rt” of that pencil form an involution, S is triple point with a cuspidal branch for four curves +”. Each singular null-point, therefore, bears four double null rays. 18. The null-curves of P and Q have the singular null-points S and the null-points of PQ in common. Each of the remaining intersections 7’ sends a null-ray through P, a second through Q. From (7n—15)?—21(n—2)’—(7n—18) it therefore ensues that the null-rays ft, will envelop a curve of class (28n’—133n+159), if t, revolves round a point P. The null-rays of P belong each twice to this envelope, each of the remaining tangents, which it sends through P, is evidently double null-ray. The double null-rays, therefore, envelop a curve of the class (28n?—133n-+158). 19. In order to find the locus of the points 7’ for which two of the null-rays coincide, 1 shall consider the curve (p)7z,—15 enve- loped by the null-rays of the points lying on p. It has p as (?7n—18)- fold tangent, is therefore intersected by p in (7n—15)(7n—16)— (7n—18)(7n—17) points. As for each of these points two null-rays coincide, the points 7’ with double null-rays lie on a curve of order (28n—66). It is at the same time the locus of the triple points that have a cuspidal branch. For n= 3 we have a null-system (3,3); the curves t° are three- rays in that case. An arbitrary straight line then forms figures c* with the curves of a net of conics. The Jacobian of that net deter- mines the three null-points of the straight line. If the system S‘) has three base-points, the three null-points of a straight line are produced by the intersection of the sides of a triangle, which has the base-points as vertices. Each base-point is the centrum of a pencil of singular null-rays. Mathematics. — “On a certain point concerning the generating functions of Larrace.” By Dr. H. B. A. BockwinkeL. (Com- municated by Prof. H. A. Lorentz). - (Communicated in the meeting of May 31, 1919). 1. The following remarkable proposition of the integral fer p(r)dr, 0 or of the integral 1 aldi Ld de, oneens vavo BED J derived from the former by the substitution r= —log t‚ has been proved by Lercu’): If the determining function a(x) vanishes for an arithmetical progression of values of « with positive common difference u =d uy, D= © Bad Wy PAAR bel A el gs pte then it vanishes for all values of x, and the generating function f(t) also vanishes. Leren uses for the proof a theorem of Wererstrass, according to which any function which is continuous in a closed interval can be represented by a uniformly converging series of rational integral functions. Since the theorem, which is also mentioned by PINCHERLE *) and by Nirisen’®), has a great many interesting consequences, it seems not unuseful to prove it in a manner which is independent of Werersrrass’s theorem. The reasoning we give in the next pages makes use of the theorem of FourIEr. 2. The following suppositions are sufficient for the purpose: 1. The function f(t) is continuous in the interval of integration, with possible exception as to the value ¢= 0. 1) Acta mathem. 27, 1908. 8) “Sur les fonctions déterminantes’’, Ann. de 'Éc. Norm. 22, 1905. PincHErRte calls f(t) “fonction génératrice’ and x (x) “fonction determinante’, whereas Leren does the reverse. We have followed the nomenclature of PincHERLE in the text. 3) “Handbuch der Gammafunktion"’, p. 118. 165 2. The integral (1) ezists for a certain value «=c of 2. We put (= ff van eN (| 0 Then, by 2, g(é) is continuous in the closed interval (0,1), and zero for t=O. Further, by Ll, g(t) is differentiable at all points of that interval, except, possibly, at t= 0, and we have OOAD mee re ss ly Ng he Hence, if d >0, we may write i 1 1 fr (t) edt = f g (tte dt =[9 () tee], — (a@—c) fg) tret dt é ) ) If, now, z is a complex number with real part & (2) greater than c, the number d in this equation may be made to approach to zero, and thus we find 1 1 fro t? dt = g (1) — (ed) fi ged. —. (5) 0 0 From this it follows: If the integral (1) exists for a certain value «=c of x, it exists in the whole half-plane defined by R(x) > R(c)'). Further it follows from (5) that the integral in the left-hand member represents a continuous function of « in any domain S lying wholly in the finite part of the half-plane R(c) + d, where (f > 0). In the same manner as above it is found that the integral 1 [ioe tog ta NEE EN) 0 exists for R(x) > R(c) and represents the derivative of a (a) at any point of this half-plane, so that @(z) is also an analytic function. These consequences, too, are mentioned by PINCHERLE. The proof Lercu gives of his theorem equally starts from the equation (5). In the following reasoning, however, we shall use an 1) This theorem is fundamental in the theory of generating functions. After PINCHERLE different authors have proved it, though often under less general suppositions. The reasoning in the text is due to Lerou. This reasoning is founded upon the continuity of f(t), which, presumably, is forgotten by LERcH, when, at the end stating his theorem, he says that f(t) may be as well discontinuous. (Of course we do not mean to say that generalization is impossible). 166 equation derived from (5) by repeating once more the process which leads to the latter equation. So we put k= fa de . sje, | ain ee oe ae RA 0 Then, again, A(t) is continuous and differentiable in (0,1) and we have WAG OLY osha ncn «a. etree ae (8) The principal point, however, is that the latter equation is also valid at t=0. Thus the derivative of A(t) is a Limited function in the closed interval (0,1). Further, observing that lam [A (i) >t] =Af' (0) =g (0) = 0, t= we find on integrating by parts, for Rx) > R(c) 1 1 fo (Deel dt = h (1) — (w—c—1) fi Qt dt ao) 0 0 and hence 1 a (w) = g (1) — (w—c) h (1) + (@—e) (w@ —e—l) JA (t) Be? dt. (10) 0 3. The preceding statements are valid independently of any further hypothesis as to the character of f(t. Now, suppose that a(«) becomes zero for the arithmetical progression of values =d U, (OLED DUA DD ae eel Choosing the number c in the preceding equations equal to & we find g(1)—0 and the integral in the right-hand member of (5) vanishes for a=G§+Il+4, (SSO i eis nn te en From this it follows that A(1)=0, and, in connection with the latter result, from (40) 1 frova=o, (u), bende Ades a ES 0 Now we saw that the derivative of h(t) is limited. According to a well-known proposition A(t) can therefore be expanded in a series of Fourier. We have h() = Sn (an cos 2a nt + by, sinQant) . . . . (14) 0 where 1 a, == 4h (i) at 0 Ano Ors == DS. le, 1 1 shan 2 | h (t) cos 2 nt dt, n= 2 fn (4) sin Zar nt dt 0 0 Now the functions cos 2% nt and sin 2a nt are for any value of n expansible in power-series oo oo sae ' zis cos 2 nt = N pap, OF, sit avant — XN u Bu t*, 0 0 converging uniformly in the interval (0,1). Sinee A(t) is limited in that interval we may use the following reduction 1 1 | h(t) cos 2 nt dt = [h(t) Np A, te dt = Se? A, 4 h (t) tr dt Pd Pd 0 0 0 0 0 and in a similar manner we find 1 1 t) sin 20 nt dt = Nz B, A(t) te de [resin nn Di „no C 0 0 0 © co 1 Hence by (13) all coetficients in the expansion of Fourier are zero, and therefore h(t) is identically zero in the interval (0,1). Since, further, g (1) = h'(@, the same thing holds for g (4), and since (jt =g'( (except at t= 0), the generating function f(t) itself is zero in the interval (0,1). This is the second part of Lrrcn’s theorem. Since the first part follows immediately from the second, the theorem has been proved in the particular case that the arithmetical progres- sion of zeros of a(w) has 1 for its common difference. If this difference is equal to the positive number 7 and if, therefore, the zeros are given by formula (2), we substitute ii =o) C= yy. ¢ = He by which the integral passes into 1 5 5 Et é „fre is ds = [roe Mites sting 41-401) yf 0 0 1 Sia Vg ta The function — „(5 Jer “— p(s) has the properties 1 and 2 q of § 2, so that the foregoing arguments may be applied to it. The 168 integral (15) vanishes for the sequence of values (11), hence ¥ (s), and therefore also f(s), identically vanishes in the interval (0,1). The theorem of J.ercn has thus been proved completely. 4. The first part of the theorem, that «(w) becomes identically zero, if this is the case for an arithmetical progression of z-values, may be proved in a direct manner, without first proving the second part; and it is an immediate consequence of the proposition : A function a(x) defined by an integral of the form (1) can, under the suppositions 1 and 2 mentioned at the beginning of § 2, be expanded in a binomial series aj Sal"). re Nagler dae 0 where B is a number lying in the domain of convergence of the integral. Suppose, for a moment, this proposition to be true. If, then, « (x) becomes zero for the sequence of values (11), we take B= £ in the equation (16). Substituting for # the values §&, 5 +1,§+ 2,... in succession, we find that all coefficients c, of the binomial expansion vanish and therefore that a(x) vanishes identically. The first part of Lwercn’s theorem is very easily proved in this manner and it would therefore be desirable that we might derive from it the second part in a short manner. But as yet we are not in a position to do this. The above demonstration is, after all, rather short, but besides, on grounds that, with a view to conciseness, we prefer not to state, we do not think it likely that the zdentica/ vanishing of a(x) is more effective for the purpose than the vanishing for an: arithmetical progression of values of the argument. Nevertheless the first part of Lerca’s theorem has an interest in itself, because remarkable consequences may be inferred from it. Among these Lerch mentions the truth that simple functions such as Ta): (k > 9) cannot be the determining functions of generating functions, in other words that they cannot be represented by integrals of the form (1), neither can products of these functions with others which remain within finite limits in the finite part of a certain halfplane R(x) > c. The proposition concerning the expansion of the integral (1) in a binomial series may be proved in different manners. In the first place integrals of that form belong to the general category of functions of which I showed, in an earlier communication in these sin ka, cos ka, 169 Proceedings (Vol. XXII, N° 1) that they are expansible in series of the form (16). Consider a domain R(x) >c-+ 0, take a positive number Jd, < d and substitute «=c-+ d, + y in the second integral of the right-hand member of (5), then A(y) > d—d, and thus positive, so that we have | fornia g (t) dt where the latter ae! exists, since od is a limited function in the interval (0,1). Hence a(x) is in the whole domain considered of the form 1 z in | t*i—1 g (t) 0) di fer dt, 0 9 (é) a (x) = (@—b) u (2) where w(x) is a function remaining within finite limits and 6 a number lying without the domain. Suchlike functions, however, can always be expanded in series of the form in question. A second, more direct proof, is obtained by substituting t= 1—u in the same integral as considered before, and using the following reduction 5 zg xr—c—1 — (1 —u)Be- (1u Blue! If m (lu) (1 — u) (1—-w) (1—u) Me n(— 1) ke )e where the series for A (w eee converges uniformly in the intervalO R(c) the integral 1 fo (1—wu) (lL—u)#—-¢—1 du 0 converges absolutely (on account of the continuity of g(l—u)), we may, after performing the substitution in question, integrate term by term, and then we find (replacing again 1—uw by ¢ in the partial integrals) 1 1 fo peel dt = Sm (—Ij" cm flames) des (IM 0 0 0 This expansion is, therefore, valid for R(x) > R(8) > Re). Since the product of this series with w--c can be Briana into a series of the same form, the required proposition has been proved again *). 1) In NierseN's book (le. p. 125) we find an analogous proof ofthe proposition in question; this, however, does not part from the integral in the second member of (5), but from the original integral, so that the hypothesis must be made that the latter converges absolutely for limt=0. The reduction (5) makes this hypothesis superfluous. 170 A third proof has the advantage of showing that expansion of (1) according to factorials of «#—c is possible when the integral only exists for «== c, even when the straight line A(#)—= A(c) were the limit between the domains of convergence and divergence in the w-plane, and when the integral did not exist at all points of that line. The proof consists in repeating the process which led to the theorem of Leren an infinite number of times. We write t g(t) =] u T (u) du, (= vo fo u) du, | ; . . (18) 7.0) = 2 {i , (u) du, .- + gn(t) = fan (u) ne 0 0 Then formula (10) may be generalized in the following manner: a (x) = 9 (1) > a (1) (w —c) EE Is (1) re mr Us (1) (ort) ae . 1 = (— Int oid (1) Cz) doe (—1y" lens d'n (t) gend... | Deer The remainder has zero as a limit for R(«) > R(c), for if G i the maximum modulus of the limited function g (é) in the ie) (0,1), we have in succession lg, OIS Ge lg, |< GY, ... ar (OG hence | fo n teen dt afs 1 (é) teen dt 0 ———, voor R(«r—e 4 R(«—e) gi) 4 —Ri = —] Now ( ) is for n =o equivalent to n Eee ‚ and thus the n modulus of the remainder in formula (19) is for all m-values’ less than H (a—e) ! 3 20 R (we) nR (x«—c) ey Al Fete where H is a certain positive number greater than G. For R («) > R(e) the remainder has therefore zero as a limit as 7 increases indefinitely. Moreover the majorant-value (20) shews that on the half-line going from #-=c in the direction of the positive part of the real axis, the binomial series converges uniformly; for R (#—c) = x—c on this line. PiIncnERLE has observed (l.c) that a similar state- ment, which is analogous to a known theorem of ABEI. on power series, holds for the integral (1), and that it follows from the equation (5), which has been found by means of integration by parts. In the same manner the just mentioned proposition may be proved generally by means of summation by parts, both for series of integral factorials (the binomial series treated of in this note) and the series of factorials in the more restrictive sense of the word, which proceed®according to inverse factorials. For the latter series | have shown this in a communication on those series *). The expansion of the integral (1) in such a series is, however, as appears from investigations of NIELSEN *) and PiNcHerLeE®), only possible under restricting conditions for f(é), viz. if it is an analytic function, whose circle of convergence for the point £1 passes through ¢ = 0, and whose order on this circle is different from —+ oo. 1) Proceedings XXII, NO. 1. *) Handbuch, p. 244. 3) Sulla sviluppabilità di una funzione in serie di fattorali, Rendic. d. R. Acc. d. Lincei 1903 (2e Semestre). Proceedings Royal Acad. Amsterdam. Vol. XXII. 12 Bacteriology. — “On the so-called filtrable virus of influenza des- cribed by von ANGeErEr.” By L. K. Worrr. (Communicated by Prof. C. Eykman). (Communicated in the meeting of June 28, 1919). Towards the end of 1918 von ANGERER *) published communica- tions on a virus of influenza, discovered by him. He injected rats with sputa of sufferers from influenza, filtered the blood of these rats germfree, when they were already very ill and put the filtrate into glucose-broth. After incubation at a temperature of 37° C. this broth became turbid, without bacteria distinctly being found in them. Yet von ANGERER describes very small formations, angioplas- mata, which he considers the cause of the influenza. This commu- nication was confirmed and completed by himself and other inves- tigators?). The result of these researches was that the rat was no longer necessary for isolating the virus, but that it was sufficient to add blood of sufferers from influenza to the broth. While investigating, together with Dr. SNAPPER *) the secondary bacteria that are the cause of pneumonia in influenza-patients, we have sometimes observed this turbidness, without being able to find any microbe in the liquid. Yet we were struck by the fact, that a great number of round, gram-negative granules were to be found in such a _ broth, but the unequal size had prevented us from considering these formations as bacteria. After the communications of von ANGERER had been published, I have paid more attention to this turbidness, which is obtained by inoculating the blood of influenza-patients into glucose- broth and by incubating this liquid at 37°C., and I have been able to observe them in three cases of serious influenza-pneumonia. | must add at once however that | found them also in a case of endo- 1) Münchener Med. Woch. 1918, N°. 46 and 47. 2) PRELL: ibidem 1918, N". 52. LescHKE, Berlin. Klin. Woch. 1919, NO. 1. See further OLSEN (Report Aertzl. Verein Hamburg Jan. 7th 1919) and KRONBERGER, Deutsche Med. Woch 1919, N° 9, who consider the results of von ANGERER non-specific. 5) Tijdschr. v. Geneesk. 1919, p. 1483. 173 carditis lenta in a child, where I did not find streptococci in the blood. The epidemic had nearly reached its end, and | should not have been able to continue my researches if not a happy coincidence, a wrong hypothesis, as appeared afterwards, had helped me on. Starting from the fact, observed by myself and also by other investigators, that inoculation of dead bacteria, which complicate the influenza, so pneumo- and streptococci or influenza-bacilli, on persons not only protects them against complications, but also against the influenza itself, I thought that the virus of the influenza would probably be present in the cultures in broth with blood of strepto- and pneumococci, collected by Dr. Snapper and myself, and so | tried to separate the virus by filtration through a Berkefeld-filter and inoculation into broth with blood. It actually succeeded the first times. I obtained liquids in which no ordinary bacteria were present, but which became turbid at 37° C. My results were however varying, at one time the liquid became turbid, at another time it did not. After first having ascribed these varying results to the Berkefeld filters, it became evident afterwards that the presence or the absence of the turbidness was dependent on an addition of a small quantity of hemoglobin and now the riddle was soon solved. If one adds to the broth a liquid containing a small quantity of hemoglobin, this mixture remains clear at room-temperature, but it becomes turbid in the incubator after 24 hours. This turbidness is also for- med in peptone, even in salt solution; the latter must be however very precisely neutral, because otherwise the turbidness is not ob- served. The hemoglobin solution was always made by washing ery- throcytes with salt solution, then dissolvlng them in distilled water and filtering through a Berkefeld-filter. It is easy to give an expla- nation, why this turbidness is obtained in blood from serious influenza patients; in this illness a slight hemolysis of the blood arises intra vitam through the secondary hemolytic streptococci, and the blood we add to the broth will contain not only red blood corpuscles, but also hemoglobin, free in the ‘plasma. And this is broken up in the incubator. To prove this more closely I prepared the carotis of a rabbit free, let a few drops of blood flow into the broth and into a test tube (1). Then I injected distilled water in the earvein and shortly after- wards blood was drawn from the carotis and mixed in the broth and in a test tube (II). The tubes with broth were put in an incu- bator. I let the blood, which was received in the testtubes, coagulate; . 12* 174 the second contained pale red serum and spectroscopic oxyhemo- globin. After 24 hours the broth in the test tube Il was decidedly turbid after slight centrifuging to remove the erythrocytes; the broth in the test-tube I was clear. Both proved to be sterile. The question has still to be answered, what may be the cause of the turbidness. If one adds a little more hemoglobin-solution to the broth and leaves it in the incubator, the next day a turbid liquid and a red precipitate are obtained; the latter does not dissolve, or only with great difficulty in acids, but easily in diluted alkalies. The solution does not show absorptionbands in the spectroscope; in adding a little ammonium sulphide, we get directly a distinct band, characteristic of hemochromogen. If we first add potassium cyanide, and then ammonium sulphide, we get two bands, of which the left one has moved a little towards the red in comparison with the above band. All this points to the fact that we were dealing with hematin. It is obvious that we have an autolysis of the hemoglobin. In most cases the globin will remain dissolved, as the broth is not exactly neutral; in neutral salt solution it may add however to the turbidness. If one wants to obtain the turbidness of hemoglobin in salt solution, one ought to take highly diluted hemoglobin solutions, otherwise it does not appear. This happens, because the reaction of the salt solution changes by adding a great quantity of hemo- globin solution. The fact of getting turbid at 37° C. of tubes of broth and blood that has been drawn from the body a considerable time ago, a well- kown fact to those, who experiment with this cultureliquid, depends of course on the same fact: autolysis and the formation of hematin. Recapitulating the facts, we may say that the turbidness of broth, described by Von ANGERER after adding the filtrate of the blood of serious influenza patients, is not specific, but must appear every where, where in the blood an important destruction of erythrocytes has taken place. The turbidness is not a virus, but hematin (and globin) originating from the hemoglobin present. Laboratory for hygiene of the University. Amsterdam. Physiology. — “On the relation between the electrical phenomenon in cloudlike condensed odorous water vapours and smeli-inten- sity”. Bij H. ZWAARDEMAKER and H. ZeEHUISEN. (Communicated in the meeting of May 31, 1919). In earlier publications’) we set forth that all true odorous sub- stances, a large number of saponins and antipyretica possess the property of imparting an electrical charge. This phenomenon mani- fests itself most distinctly with the first group, less distinctly with the second, and again less with antipyretica. However, the result is largely dependent. on the solubility in water of the individual substances, as it can be of some significance only when, at the spraying, an adequate number of dissolved molecules are present in the water. Suspended particles are of themselves of no value for the phenomenon. Before proceeding we wish to call attention to a fourth group to be discussed later on, viz. the alkaloids. The phenomenon appears with many of them, but their solubility being very slight, it cannot reach a high degree of intensity. To give an idea of the great differences among the four groups, we observe that on comparison, eg. of the charge of a saturated camphor- solution with that of a saturated quinine-solution, the former appears to be at least twice as strong as the latter. It follows then that among the organic substances of physiological activity the electrifying power is always highest in the odorous substances, so that they are most appropriate for the study of the phenomenon. It has been established before, that in homologous series the in- tensities of electrifying power and olfactory capacity rise and fall concomitantly. In order to ascertain the relations of these intensities for odorous substances we selected at random 26 of them, distri- buted over various groups and series and we diluted their aqueous solution to such a degree that on spraying they yielded only in- appreciable electrical phenomena (with an electroscope of moderate sensitiveness 7) a deflection of from 0.1 to 0.2 scale-marks). When 1!) K. Akad. v. Wetensch. Amst. 25 Maart 1916, 27 Mei 1916, 30 Sept. 1916, 23 Febr. 1918, 29 Juni 1918. Arch. neérl. de physiol. T. 1, p. 347; Nederl. Tijdschr. v. Geneesk. 1918 II 980—982. 2) A tension of 220 vlts made the instrument deflect 10 scalemarks. (It had a capacity of + 50 Electrostat.-units). 176 examined in a simple olfactometer these solutions yielded a very weak sensation of smell, which could readily be determined by the length to which the olfactometrical cylinder had to be moved out, in order to procure a minimum perceptibile. Table Ia shows the olfaction-values of the solutions when we try to find the “Reizschwelle”. Table 15 gives of the same substances the lengths to which the cylinder has to be moved out when we search for the “Erkenntnisschwelle”’. TABLE la. TABLE 10. The odorous substances arranged in the | The odorous substances arranged accord- ing to the “Erkenntnisschwelle” in solu- order of the smell-intensity of solutions yielding an approximately equal, extre- mely weak electrical charge tions yielding an approximately equal, (0,1—0,2 scale-marks). extremely weak electrical charge. S : … | “Reizschwelle” »Erkenntnis- ubstances (arranged in a ate Substances arranged \schwelle”incen- ascending order of which the cylin- in ascending order of ae i eae -i i : -j i e n is smell-intensity). der is moved out. smell-intensity. ren nk Caproic acid 5 Caproic acid 9 Artificial moschus 1 Artificial moschus 8 Valerianic acid 1 Valerianic acid 5 Amylalcohol 1 Amylalcohol 5 Cumidin 1 Cumidin 5 Allylalcohol 1 p. Xylenol 5 Iso-amylacetate 0.5 Allylalcohol 3 Terpineol 0.5 m. Xylidin 2.9 Skatol 0.5 o. Toluidin 155 Indol 0.5 p. Toluidin AGS) Pseudocumol 0.3 Iso-amylacetate 153 Xylol 0.3 Terpineol 1 o. Toluidin 0.2 Skatol 1 Anilin 0.2 Pseudocumol i p. Xylenol 0.1 Anilin 1 m. Xylidin 0.1 Thymol 1 m. Toluidin 0.1 Benzol 1 Thymol 0.1 Toluol 1 Benzol 0.1 Xylol 0.6 Toluol On! Indol 0.5 Naphthalin 0.1 Naphthalin 0.5 m. Xylenol 0.1 m. Xylenol 0.4 Guaiacol 0.1 Guaiacol On Nitrobenzol 0.1 Nitrobenzol 0.1 Pyridin 0.1 Pyridin 0.1 Vanillin 0.05 Vanillin 0.1 Average 0.54 c.m. Average 2.16 c.m. In the latter case, it is true, the values are based upon the inten- sity of the sensation, however the psychological consciousness of the quality had been previously established. 4:70 As will be seen, the average olfaction-value of our 26 substances in a dilution with which the electrifying power is next to impercep- tible, is 0.54 cm (determined after the minimum perceptibile without quality). The deviations vary within the tenfold of the mean values. Apparently the liminal perceptibility of smell-capacity, concurs approximately with that of electrifying power. There is, indeed, a certain latitude of variation in the olfaction-values, while the electri- fying power is considered the same for all, though probably it is not quite the same, because it is difficult to distinguish differences of subdivisious of one tenth of a scalemark. One of us surmises that both the intensity of the smell and that of the electrical phenomenon depend in a more or less complicate way on a. the volatility of the substances TABLE Ila. TABLE IIb. Arrangement of 26 odorous substances ; (in the stalagmometer) in a solution 26 odorous substances arranged according yielding a just noticeable electrical to their boiling point. charge. (at 15° C.). Water 49.3 Water 100° Naphthalin 49.3 Benzol 809 Artificial moschus 49.6 Allylalcohol 97° Caproic acid 49.8 Toluol 111° Anilin 50 Iso-amylacetate 116° Indol 50 Amylalcohol 116.3 Nitrobenzol 50 Pyridin 116.7 p. Xylenol 50 Xylol 142° p. Toluidin 50 Caproic acid 155° Guaiacol 50 Pseudocumol 169.8 Thymol 50.1 Anilin 182.5 Allylalcohol 50.2 Valerianic acid 184 Toluol 50.2 o. Toluidin 199.7 Pseudocumol 50.5 p. Toluidin 200.4 Amylalcohol 50.5 Nitrobenzol | 205 m. Xylidin 50.5 Guaiacol 205.1 m. Xylenol 50.6 Terpineol 218 Benzol 50.8 Naphthalin 218.2 Vanillin 51 p. Xylenol 220 Valerianic acid 51 m. Xylenol 225 Terpineol Dl2 m. Xylidin | 226 o. Toluidin 52 Thymol 231.8 Skatol 52.2 Cumidin 235° Cumidin 52.2 Indol 253 Iso-amylacetate 52.5 Skatol 265 Pyridin 54 Vanillin sublimes Xylol 55 Artificial moschus cas Average 50.9 eee 178 b. the lowering of the surface-tension of water, which they bring about. In view of this supposition it avails to know the boiling point of the odorous substances as well as the number of droplets of the diluted aqueous solutions. The subjoined tables (Ila and 115) give us these data. They do not warrant the adoption of an immediate connection, though we may conclude from them that there is a more remote relation *). The sign of the charge may also be of some influence. Whereas for 24 out of 26 substances examined in widely different concen- trations, we invariably found a positive charge, which eventually disappears with progressing dilutions, a negative charge is yielded by caproic acid and valerianie acid in highly diluted solution (positive in somewhat concentrated solutions). In the extremest dilution, in which this negative charge is just noticeable, the smell- intensity of these substances, when compared with the average of our substances, appears to be very slight. Whether the charge increases or not, or whether it decreases, through tbe addition of some common salt to the solution °), does not seem to interfere with the relative arrangement of our 26 substances. With all of them we observe an increase of the electrifying power, with the exception of naphtalin, indol and skatol, whose insignificant charge seems to remain constant, and of artificial moschus, pseudo- cumol, p. xylenol and thymol, whose charges are obviously getting weaker. We conclude, therefore, merely from the facts, without attempting to find an explanation, that also of odorous substances, chosen at random, in approximately similar dilution, the smell-intensity and the electrifying power have reached their limen of perceptibility. 1) Order of the number of droplets of the saturated solution: Naphthalin 49.3, Artificial moschus 49.6, Toluol 50.3, Pseudocumol 50.5, Benzol 50.75, Allylalcohol (1: 500)51, Vanilline 51, Indol 51.5, Skatol 52.2, Xylol 55, Nitrobenzol 55.5, Pyridin (1/9) 57.25, Cumidin 66.75, p. Xylenol 68.5, Anilin 69, p. Toluidin 69, Thymol 73, o. Toluidin 75.5, n. Xylidin 77, m. Xylenol 84, Caproic acid 84.5, Guaiacol 85, Terpineol 90.5, Valerianic acid 106, Isoamylacetate 115, Amylalcohol 131. 4) E. L. Backman, Researches Physiol. Lab. Utrecht (5). Vol. 18, p. 349; 19, p. 210. Chemistry. — “The zincates of sodium. Equilibriums in the system Na,O-£nO-H,O”. By Dr. F. Goupriaan. (Communicated by Prof. J. BÖRSEKEN). (Communicated in the meeting of June 28, 1919). The data we find in the literature about the influence of the strong bases of light metals on the insoluble weak hydroxides of heavy metals, are extremely contradictory and ill-defined. So it is generally assumed that the hydroxides of zinc, aluminium, lead and tin display an amphoteric character, so that in an excess of strong base they ‘dissolve’ under the formation of salts. It is supposed on tbe other hand that the hydroxides of numerous other heavy metals do not show any propensity to form similar salts and by this the stronger electro-positive character of these metals is thought to reveal itself. And yet it is a fact that the hydroxides of the so- called strong electro-positive metals can likewise in some circum- stances display an amphoteric character; nay even cuprum hydroxide can dissolve in a concentrated solution of NaOH or KOH. Hence the differences occurring with the metalhydroxides are probably only of a quantitative nature and it will be worth while to investigate to what extent the different hydroxides show this phenomenon and what compounds arise in this process. This is in the first place of importance for analytic chemistry, where numerous separations are due to the difference in solubility of the hydroxides; next investi- gations about this subject can give us a more definite insight into the mutual affinity of the metaloxides and metal-hydroxides. Up till now it has not yet been ascertained what compounds are formed under the influence of hydroxides on strong bases and under what circumstances they are stable. In a pure state they are not isolated and accurately delineated. We are still quite in the dark about the nature of the hydroxides themselves. For as a rule we obtain these substances as voluminous, gelatinous products, and the question arises whether we are to consider these as solid phases of a constant or variable composition or as liquid ones of a great viscosity. The great difficulties which the procuring of these substances in a pure state, involves, are probably the cause of the very few exact data we have about the subject under discussion. 180 In the following lines we shall now state the results of experi- ments made with a view to get more closely acquainted with the compounds formed in “dissolving” zinchydroxide in NaOH i.e. the zincates of sodium, and accurately to determine their range of existence. Similar investigations concerning the correspondent com- pounds of some other metals are in progress. We shall now briefly summarize the data known up till now concerning the sodium-zincates : By means of solving ZnO in a hot NaOH-solution, followed by the addition of aleohol, Comey and Lorine Jackson ') obtained two products, to which they ascribe respectively the formulae: H, Na, Zn, O,17 H,O and HNa ZnO, 7 H,0. The first compound has a melting-point of 100°, the second proved not to melt even at 300°. Forster and GüNrner ®) found only one compound viz. the formula OH Zn 3 H,O. It formed white silky-glossy needles. O.Na Herz *) precipitated a solution of Zn SO, of known concentration with a KOH-solution and then investigated how much base is required once more to solve the Zn (OH), that is formed. It appeared that to 1 gram equivalent of Zn, 6 gram equivalents of OH were required. Hanrzscu *) denies the existence of sodium-zincates on the ground of conductivity measurements and attributes the solution of Zn (OH), in NaOH to the forming of a colloidal solution. Jorpis °) observes that in the long run, crystals of the composition : Zn (ONa), arise in the cupron-element, in which zine is found as a negative pole in a solution of NaOH. Finally it should be mentioned, that RuBeNBAUER *) and Woop ‘) have determined the proportion of the number of gramatoms Zn and Na in a solution of Zn (OH), in NaOH; these observations can teach but little regarding the existence of definite compounds. We see from these data of the literature how vague our knowledge about such comparatively simple compounds still is. The equilibriums that may arise in an aqueous solution between 1) Amer. Chem. Journ. 11, 145 (1889); Ber. d. deut. Chem. Ges. 21, 1589 (1888) 2) Zeitschr. f. Elektroch. 6, 301 (1899). 5) Zeitschr. f. anorg. Chem. 28, 474 (1901). 4) Ibid. 30, 289 (1902). 5) Zeitschr. f. Elektroch. 7, 469 (1900). 6) Zeitschr. f. anorg. Chem. 30, 332 (1902). 7) Journ. Chem. Soc. 97, 886 (1910). 181 TABLE 1. — System Na,O—ZnO—H,O (Temperature 30,0° C.). NO, Oo OdT ml G OU ok. Oo OD Te ND BD BR BR Rw me eet AD aN. = oo © VO So Um Ne & 25 Composition of the | Composition of the solution. | rest. of, NaO | 0/0 ZnO | °4 H,O |'e/, Na,O | opp ZnO | 0/,H, O 1:8 2.6 | 85.6 LAA $2282 OT 17.4 50) 19.6 1503)" | T1689! | L678 246 1) 1256-2) (62.8 | 20.2 1/2054) “| 507 DARO AAALO | 62s ae 2220 | 23.5.9 |) 545 BSE edu 165.01, lp $9ed |) -Biz2| 49.7, ,,| APES GEO Ar A A OA Er A 28 16-5 | 5.7 = — — 28.0 | 14.9 | 57.1 S551» 2830 43.6 aen 1OSOR 55.6 eee 28- 61 ie 27,9 «1 2435 36.7 9.5 | 53.8 || 31.9 | 28.5 | 39.6 31-8 | iT) 5625 | 31.0. | 20.2 * | 488 sete | 432 ||, soin |) 2000 | 202) alt S22 cA, Qe hy BSG BOLT, Sls 8) las38.5 SRS ONLI So 56.7 202-3204310 36.9 | 10.1 | 53.0 | 34.9 | 17.2 | 47.9 een) 104 540. | 33.7 IP 15.0 || 507 SG LOL 53e. LSA. 0 wilt il: Ty 488 36.8 | 9.9 | 53.3 | 39.1 | 22.8 | 44.1 39.2 9:7 -| 51.1 = = a 39.4 9.0 | 51.6 || 42.6 7.0 | 50.4 39.6 72 53.2 al 410 6.1 | 52.9 40.7 HON STe 4250 1.8 | 56.2 40.5 1.6 | 57.9 || 42.6 11312256: 1 40.9 Wal 580 Ie 42.7 0.4 | 56.9 41.9 Oros — == = PANG ele 2enn 620% 23.6. |'15.2. 1 012 19.9 | 15.2 | 64.9 16. te | ATA | 4562 4.6 1.0 | 96.4 3.1 | 15.4 | 80.9 4.5 0.4 | 95.1 307. |: 20:30 LONG 13.7 1.2 | 19:1 9.3) |-30.4 6073 10.1 4.7 | 85.2 62 lee ST ee | Na,O. Na,O. ZnO: . ZnO BA OF ALO) rang: „ZnO. . ZnO . ZnO Na20 . | Na,O.ZnO.4H,O + | Na,O . 3H,0 .3H,O .3H,O .3H,O .3H,O .3H,O . 3H,O ZnO ZnO ZnO ZnO ZnO ZnO Solid phases. ZnO + Na,O.ZnO. H,0 ZnO ZnO ZnO. ZnO .4H,O „4H,0 4H,O .4H,O 4H,O .4H,O 4H,O 4H,O .4H,0 .4H,O 4H,O Zn (OH), Zn (OH), Zn (OH), Zn (OH), id 182 NaOH, Zn (OH), and their compounds if any, must be considered as those in a ternary system with the components: Na, 0—ZnO0—H, V0. The solubility-diagram of this system could now be completely determined at a constant temperature. As starting-material for these determinations we used NaOH, prepared from sodium; the ZnO being obtained in the following manner: Pure crystallized zine nitrate was precipitated with the required quantity of ammonia, thoroughly washed out with boiling water, the precipitate then being dried at 140°—-150°. The oxide was also obtained by glowing precipitated zinc-carbonate, this oxide being less active, owing to the strong heating, it required a longer time to reach equilibrium. Both prepa- rations showed equal solubility. For the preparing of the hydroxide, vide infra. The water had been distilled and boiled out before use. All the determinations were executed in a thermostat of 30,0° C. In table 1 and the accompanying diagram 1 a survey of the results is given. All the number-values represent weight percentages of the saturated solution. As it proved extremely difficult to accurately isolate the solid phases, we determined the composition in yee all cases according to SCHREINEMAKERS’s rest-method. In determining the curve AB we added ZnO as solid phase *); the solubility of this substance appears rapidly to increase with growing NaOH-concentration. As appears from the second curve BC of the isotherm, the sodium-zincate of the formula: Na,OZnO04H,0 preci- pitates from the solution, while constantly the NaOQH-concentration is increasing. The curve CD, where solutions are saturared by the monohydrate of sodium-hydroxide: NaOH. H,O |Na,O .3H,O] immediately joined to BC. According to the melting-diagram NaOH—H,O determined by PickerinG*) this hydrate is the only stable compound of NaOH with water at 30°. Consequently only one stable zincate arises at 30°, having the formula: Na,O.2nO.4 H,O. All the other zincates described in the literature, must be considered either as metastable or as not existing at all. The opinion of Hanrzscu*) concerning the colloidal nature of the solutions, does not hold true either, as will appear still more decidedly later on. | Properties, preparation, etc. of the sodium-zincate Na,O.Zn0 .4H,O. The diagram referred to already shows that the sodium-zincate 1) Except in the numbers 26 and 29 (vide infra). 3) Journ. Chem. Soc. 63, 890 (1893). 3) Loc. cit. 183 belongs to those salts that form the so-called incongruent solutions. On adding water, these salts do not yield a simple solution, but one of the components, from which we can imagine the salt to be formed, separates on the addition of water. Many instances of this are already known. With the zineate of sodium this phenomenon is specially pronounced. Not only will the addition of water to the solid salt effect a separation of ZnO, hence a decomposition of the compound, but diluted solutions of NaOH will likewise produce the same effect. From the isotherm we may infer, that solutions below a concentration of 33,3 Gr. NaOH to 100 Gr. solution (hence 1 part NaOH to 2 parts of water) will cause a separation of ZnO from the solid salt, i.e. the concentration of the NaOH-solution will have to be raised beyond this boundary in order to obtain pure zincate crystals. The inadequate attention paid to this circumstance is probably the cause of the manifold contradictions in the literature, hardly anywhere do we find the concentration of the solutions indicated, thus obtaining in many cases mixtures of zincate crystals and ZnO, the possibility for this being very great as appears from the situation of the solubility-eurves. In analysing these mixtures investigators attributed the incorrect composition of a compound to them. This occurred among others with the experiments of Comry and Lorie Jackson’), who prepared zineates by solving ZnO in a hot, concentrated NaOH-solution, they then added alcohol to the liquid and in this way obtained two products: one from the layer of water by shaking out so long with alcohol till crystals appeared and one from the alcoholic liquid. I have repeated this method and in doing so I started from 50 Gr. of water, to which 50 Gr. NaOH was added. After introducing 17 Gr. ZnO in small quantities at a time, the zincate crystallized out in an appreciable quantity. According to our diagram we are in the centre of the saturation-region of the zincate. With the necessary precautions the crystal agglomerate was sucked out without any access of air and carbonic acid, and was then dried on porous earthenware. Of course the remaining liquor could not be altogether removed in this way, but washing-out without simultaneous decom- position is impossible. The analysis of the crystals produced: NasOs 208: 2/1 Zn0:537.8°/, HO. 13240 (Theoretically for Na,O . ZnO. 4H,O... Na,O...28,9 °/,, ZnO...37,6 °/,, H,0...33,5 °/,). On microscopic inspection the product appeared to be perfectly homogeneous, the crystals showed as long bars with a 1) Loe. cit. c.f. also GmeLIN-KRAvut?’s Handbuch. 184 Na, 00 450 Ho D MKO Na,0 Zig: blunt extremity. They are faintly double-refractive. Amorphous particles could not be ascertained. The erystals which had formed in the experiments N° 8—18 were likewise microscopically examined ; they appeared entirely to correspond with those obtained in the above-mentioned manner. A specimen of the crystals treated with alcohol of 96 °/, showed a complete change. A superficial examination already teaches us in this case, that under the influence of the alcohol the crystals gradually disappear, while amorphous particles precipitate from the liquid. Microscopically too it appears that in these circumstances the crystals cannot exist, but separate out amorphous particles ZnO (or Zn(OH),): Even alcohol of 96°/, has a hydrolyzing influence on the zincate, and it is not surprising that the above-mentioned inves- tigators obtained products of an improbable composition. Proportional to the quantity of the alcohol and the duration of the operation, a 185- mixture will be obtained containing a greater or smaller quantity of zincate-crystals. The formulae for such products are devoid of sense and should consequently disappear from the literature. Stability, preparation, etc. of zine hydrowide. It is generally known, that as a rule zine hydroxide is obtained as a gelatinous, voluminous mass. Now it was essential for the present investigation to determine whether this phase has a constant or a variable composition and to find out its stability regarding ZnO. There are many indications to be found in the literature that the hydroxides obtained in varying ways do not possess the same composition. HULER') prepared from the nitrate hydroxides which proved to havea varying solubility in ammonia. Herz *) and Hanrzscu *) point out the great difficulties involved in obtaining Zn(OH, in a pure state and specially the tenacity with which it retains sulphate- ions. It seems, that it is easier to obtain a pure hydroxide from the chloride or nitrate than from the sulphate. With the experiments made to confirm this statement, | proceeded from pure ZnO, which was dissolved in hydrochloric acid and nitric acid, upon which the hydroxide was precipitated by means of the addition of the quan- tity of ammonia calculated. If this precipitate is washed out with boiling water it almost immediately becomes more gritty, heavier and less gelatinous. After washing out a few times and following it up by desiccation at 100°, the product appeared to contain only 0,6 °/, water. In conseqence of the washing-out the hydroxide is converted into oxide already at 100°. Hence Zn(VH), at 100° is no longer stable. We then tried to achieve a pure hydroxide by washing out at room-temperature. It appeared however, that even after prolonged and continued extraction the products were still chloridie or nitritic. During the first hours of the extraction we observe a considerable decrease in the concentration of the adsorbed ion, afterwards it falls but very slowly. The following figures will further illustrate this statement: Hence it is practically impossible in this way to obtain a pure hydroxide from the chloride. The products from sulphate and nitrate yielded similar results and though it is stated in the literature, that the nitrate-ion is much less strongly absorbed than the other ions, we determined that even after an extraction continued for days the ) Ber. d. deutsch. Chem. Ges. 36, 3400 (1903). §) Zeitschr. f. anorg. Chem. 30, 280 (1902); 31, 357 (1902). 3) Loc. cit. 186 TABLE 2. Duration of ‘on « | Cl’ degree in 9% after drying extraction in | Boe at 100°. 0 0.59 4 | 0.36 8 0.35 16 | 0.34 48 | 0.30 products were still nitritic. Yet I performed with the gelatinous zinc- hydroxide some measurements concerning the solubility in NaOH, not because the values determined can have an absolute significance, but exclusively with the purpose to investigate whether this hydroxide reveals a higher solubility than ZnO, and consequently must be considered as metastable towards the latter. This proves indeed to be so, as the subjoined numbers demonstrate; the points found le considerably above the curve of solubility (AB) of ZnO. At the same time it appears that the solubility decreases in proportion as the hydroxide is kept for a longer time. This too is in favour of the statement, that these products must be considered as metastable phases inclining to stabilisation to ZnO. TABLE 3. | ition of the solution. Time, given in hours, since the. Composition of the solution reparation of the hydroxide. one 4 % ZnO 0/, Na,O eae | 10.5 11.3 | sy | 9.3 11.4 | 8.2 11.3 Le 7.0 11.3 : | For the solubility of ZnO with a Na,O concentration of 11,3 °/, we find by interpolation 2,3 °/,, hence considerably lower. Crystallized zinc hydroxide. It may be concluded from the above that we must consider the amorphous, gelatinous hydroxides as phases of varying water-percen- tage, they besides being extremely difficult to purify. There now Ho was a possibility that under special circumstances, a erystallized hydroxide of a constant composition could be obtained. There are some intimations in the literature, that Zn(OH), sometimes seems to arise in a crystallized form. Thus Bercqurrer') states that he obtains the crystallized hydroxide by placing a zine bar wound round with a copper-wire in a solution of sicilie acid in caustic lye. In this process isometric octahedrons were formed, to which he ascribed the formula Zn{OH),. Various similar indications are found in the older literature though the observers do not agree as regards the composition of the crystals’). Of late years the zinchydroxides have been newly examined among others by Krein *). He distinguishes three forms of the hydroxide; form A is most strongly soluble in NaOH and arises by adding drop by drop ZnSO,-solution to a NaOH-solu- tion. The analysis of the product dried at a normal temperature yielded: 2Zn0.H,O. In course of time the forms B or C' separate out from the saturated solutions of A; both would have the com- position: ZnO.H,O, but B is sometimes obtained in fine crystals, whereas C is always amorphous. It is clear from the preceding that we need not demonstrate that a constant composition of the amorphous hydroxides is out of the question. The water-percentage of these substances depends on all kinds of factors: preparation, duration of keeping, ete.; hence a definite formula for them is valueless. The case is otherwise with the crystallized hydroxide. I really found it possible, to isolate the zinchydrovide as a crystallized phase of a constant composition. In doing so, [ set about it in the following manner: to 50 ¢.c. of a normal solution of KOH, I added a normal solution of zinc sulphate in drops from a burette. At the outset the hydroxide forming immediately dissolves, but finally a point is reached at which the liquid remains slightly turbid when shaken. On vigorous shaking and especially on scratching the glasswall and allowing to stand for a few minutes a heavy, sandy precipitate arises. Grafting with ervstals already obtained, appeared greatly to accelerate the separation. On mieroscopic examination the product gives an altogether homogeneous impression and it appears to consist of very small, drawn-out bar-shaped crystals. They filtrate very easily and contrary to the amorphous product, the erystals can 1) Lieb. Ann. 94, 358 (1855). 4) Cf. among others BopEKeErR. Lieb. Ann. 94, 358 (1855); ViLLE, Comp. Rend. 101, 375 (1885). 3) Zeitschr. f. anorg. Chem. 74, 157 (1912). See also Woop, Journ. Chem. Soc. 97, 886 (1910). ¥ 13 Proceedings Royal Acad. Amsterdam. Vol. XXII. 188 be washed out very rapidly. On drying at 40°—50° the analysis yielded : 18,06 °/, H,O; 81,91 °/, ZnO (Theoretically for Zn(OH),....18,11°/, H,O 81589 „Za: The concentration of the KOH-solution was varied between the limits of 4,0 and 0,1 normal; the concentration of the zinc sulphate solution likewise; the crystals formed always were of the same shape and composition. Stability of crystallized zinc hydroxide towards zinc oxide. The experiments Nes 26—31 of Table 1 give an insight into the stability-relation of the crystallized Zn(OH), and ZnO. With No. 26 the solid phase was added as crystallized Zn(OH),; the mass was kept for over a fortnight in the thermostat at 30°. It then appeared that the crystalline Zn(OH), had been entirely converted and the solid phase consisted of ZnO. This was confirmed by the ana- lysis of the solution and remainder, the point found falls on the curve AB. So it appears from this, that at 30° the erystallized zine hy- drozide is metastable towards ZnO. With a shorter equilibrium-adjustment 7 proved possible to deter- mine the metastable solubility curve of Zn (OH),. No. 27 was set in with erystallized Zn (OH), and after + 24 hours the solution was analysed; the solid phase appeared to consist even then of crystallized hydroxide. Conformable to this the zine percentage of the solution (c. f. table) was considerably higher than corresponds to the curve AB. Numbers 30 and 31 have been executed in a similar manner, here again crystallized Zn (OH), was added as a solid phase, the solution being analysed after + 24 hours. The determined compositions of the solution again lie considerably above curve 4B. The points representing these solutions form together the metastable solubility curve ZF of the crystallized zine hydroxide. Finally the determinations Nes 28 and 29 have been carried out in the same solution, to which erystals Zn(OH, were added as a solid phase. After about + 24 hours the solution yielded the com- position N°. 28, Zn(OH), being present as a solid phase. Whereas after one day these crystals still appeared to be present, the solution still being of the same composition, we found three weeks later on the composition N°. 29. All the Zn(OH), crystals had disappeared; the point now found lies on AB, while the analysis of the rest indicated too, that ZnO was present as a solid phase. Other circum- stances being equal the solubility of crystallized Zn(OH), is conside: rably higher than that of ZnO. 189 DDM MAR 1. The solubility-isotherm in the system Na,O-ZnO-H,O was com- pletely determined at 30°. 2. We found that the following substances appeared as stable, solid phases: zinc-oxide ZnO, zincate of sodium Na,O-ZnO-4H,0, the monohydrate of sodium hydroxide NaOH-H,O. 3. The sodium zincate forms very strongly incongruent solutions, by solutions below a concentration of 1 part of NaOH to 2 parts of water it is decomposed while separating out ZnO. | 4. The amorphous, gelatinous zinchydroxide must be considered as a phase of a varying water-percentage; it cannot possibly be cleaned of adsorbed ions. It is metastable as regards crystallized zinchydroxide. 5, Under special circumstances zinchydroxide is obtainable as a crystalline phase of the constant composition Zn(OH),. 6. This crystallized hydroxide is metastable at 30° as regards ZnO. The cost of these investigations has been partly defrayed by a subsidy from the Van ’r Horr Fund which was put at my disposal. I here beg to express my cordial thanks to the Board of Admini- stration of this Fund. Anorganic and Physical-chemical Laboratory of the Technical University. Delft, June 1919. 13* Physics. — “The Anomalous Zevman- FL ffect.” By Dr T. van Lonutzun. (Communicated by Prof. H. A. Lorentz). (Communicated in the meeting of May 31, 1919). Different attempts have already been made to explain the ZrrMAN- Effect from the atom model of Bonr'). As yet only the normal Lorentz triplet has been explained, often with superfluous components, which however have disappeared in the theory of RuBiNowicz. The theory of the anomalous Zenman-effect has not yet advanced much, nor is the explanation of the Pascnen-Back-effect much further. It may, however, be tried to bring one of the parts of the problem to a solution, so that possibly the results obtained in this way might be serviceable for the complete solution of the problem. One of the questions that presents itself here, and which I have set myself the task to answer, is this: Does the magnetic splitting up exclusively depend on the initial and the final path in which the electron moves, or does the transition play a part in it? In other words: Is in the presence of a magnetic field the formula: W,—W, B EE Sn) valid, or should it be replaced by a formula as e.g. WW. eM pS = eee PAL EE Bees ee ibe) h ' Az me pe as Bour?) thinks he has to assume. 1 will put the question into still another form, because expressed in these words the solution is easiest to give. It is known that the frequency of the vibrations of every spectrum line may be represented’ as the difference of two functions, so-called “sequences, e.g. : 1) N. Bour, Phil. Mag. 27, p. 506. 1914. K. HerzreLp, Phys. Zs. 15, p. 198. 1914. P. Despre, Gott. Nachr. 1916, p. 142, Phys. Zs. 17, p. 507. 1916. A. SoMMERFELD, Phys. Zs. 17, p. 491. 1916. A. Ruprnowicz, Phys. Zs. 19, p. 441 en 465. 1918. 2) N. Boor, Phil. Mag 27, p. 506. 1914. 19T im IT ER k= 2778 SE an I have extensively set this forth in my treatise “Le Phénomène de Zeeman et les séries spectrales’”’ '), and will henceforth refer to this. There I have demonstrated among others that for every “complex” of spectrum lines, i.e. all the spectrum lines whose frequencies satisfy equation (3) when the functions y and p are given, and 7 and & each pass through the series of the whole positive values, a definite type of anomalous ZerMAN-effect holds, provided the influence of the PascHEN-Back-effect be taken into account. Hence we may briefly say that every type of anomalous Zeeman-effect is determined by the form of the functions w and g. When these functions have once been determined, the difference of these funetions for positive whole values of the argument always yields a spectrum line with a definite type of Zeeman-effect. As has been shown more at length in my cited paper’), these functions may be indicated as: v= wy (i) — ¢ (4) (3) Single p s | d if Double de | Je. S | S' | D | DO F | F' | | | Triple a) a |m| =| =| "| a] a'| a] o lele Accordingly the symbol ME is the brief way of writing: tid ag sree ome es ene See For the Zerman-effects belonging to every complex I refer to (l.c). The above question may, therefore, now also be worded as follows: Is each of the above-mentioned functions (‘‘sequences’’) separately changed by a magnetic field, and is, therefore, the Zruuman-effect that is observed the result of the change of the two sequences together? Or could we ask when speaking of ZF-paths, >-paths ete., by which we therefore express that an electron that jumps from an =-path to a H-path gives rise to a spectrum line belonging to the complex 2’: Is every W-path, -path, etc. in a magnetic field each in itself p=) 2 | 1) T. van Lonuizen, Arch. Musée Teyler (III) 2, p. 165, 1914. 2) Henceforth to be indicated as (I. c.). 192 split up into different paths, each with a somewhat different energy value, so that when jumping from and to these transformed paths the electrons would emit a vibration with a somewhat changed frequency ? If this question should be answered in the affirmative, it would follow from this that given the modes of splitting up of each of the paths (U, = etc), the observed types of ZreMar-effect might be found from this by simple subtraction, in other words, that the anomalous ZreEMAN-effect would follow the so-called ‘“Kombinations- prinzip’’. To answer the questions put I have done exactly the reverse. I have namely tried to determine whether in the material of obser- vation of the anomalous Zreman-effect indications were to be found of the validity or non-validity of the “Kombinationsprinzip”’. In what follows I will communicate some of the results obtained by me. As material has served what I had collected in (l.c.). In order to be able to treat a number of complexes as large as possible with a number of sequences as small as possible I have confined myself for my first investigation to the triple complexes, and added to this some single complexes with strange asymptote. These are collected in the following table: initial path aan ak > A JAN d final path nl Hu Wz HA IEA: Hd m Wish are mA" I'd TT" | Ls TER Eph (ae Te d For the types of Zeeman-effect that these complexes present, see (I.c.). From this material some general conclusions may first of all be made. When the electron jumps from an initial path to a ZF-path, the ZEEMAN-effect is more complicated than when from the same initial path it jumps to a 77'-path, and this in its turns is again more com- plicated than when from the same initial path it jumps to a J7"-path. It is therefore natural to assume that in a magnetic field the 193- I-path will split up into more paths than the 17'-path, and this again into more than the Z"-path. For the double series the same thing applies to the P-path and the P’-path. When only the j polarized components are considered, the same rules hold for them, in this ease “more” should however be replaced by “more or equal”. The number of components polarized // is for WH’ greater than I", and for P equal to P’. Another general conclusion can be drawn. When the electron jumps from & (and also from s and S) paths the components pola- rized 1 and // are always different. Jumps from the other initial paths (4, A’, A", and d) often yield coinciding | and // polarized components. This is e.g. the case for jumps from A paths to 7, HI’, and 7" paths / ” A ” ” u ” 3, Bl ” ” i: and ie ” jp / ” d ” ” Il ” This peculiar behaviour of the © (resp. s and S) paths raises the question whether this may possibly be in connection with SOMMERFELD's *) view: “Infolgedessen drängt sich folgende geometrische Deutung auf: die p- und d-Terme entsprechen ebenen Bahnen in der Symmetrieebene des Atoms, ähnlich den KerPPrer-Ellipsen; der s-Term hat seinen Grund darin, dass die beim Wasserstoff bestehende Punktsymmetrie durch die Atomstruktur von Zi und He anfgehoben ist und dass daher noch andere Bahnen als die in der Symmetrieebene möglich werden”. The = (resp. s and S) paths might therefore be imagined 1 to the equator, and possibly this situation outside the equator might be the reason that in its jump to a path in the equator plane the electron does not give coinciding components, whereas it might be imagined that in jumps between paths lying in the equator plane, the chance to coinciding components is much greater. From a private conversation which I had shortly ago with Professor Bour on this subject it appeared to me that he nad grounds to suppose the X (resp. s and S) paths to be also equatorial. It should further be remarked that the complexes which are the 1) A. SOMMERFELD, Zur Quantentheorie der Spektrallinien, München. Ber. 4 Nov. 1916, p. 153% 194 subject of this investigation, occur exclusively for chemically bivalent elements. For univalent and trivalent elements there occur no Greek complexes. After these more general remarks I will now set forth in what way I have carried out the inquiry as to the possible validity of the “Kombinationsprinzip ’. On a closer examination of the different types of the abnormal ZeRMAN-effect for the complexes mentioned it strikes us that for most of them the distances of the components from the original line are multiples of half the distance of two components from the normal Lorentz-triplet. When we call ¢ the change of energy which a path must undergo for an electron jumping from that path to an unchanged path to emit light corresponding with one of the components polarized L of the normal LoreNtz-triplet, while jumping from the unmodified path it emits light of a frequency of vibration equal to that of the middle component polarized //, then = — h will indicate the difference in frequency between the two before- mentioned components. Accordingly this e must be proportional to the 5. I have now introduced the hypothesis that through the magnetic field each of the initial- and final paths splits up into two or more paths, which present energy-differences with the original path of En. 5 m=. je nk Then I have examined what values of n must be assigned to each of the initial and final paths to enable us to explain the observed components. This yielded the following results: £ ; & de [he 7 path splits up into 7 paths with energy differences 0, + Fc et he = at € II EE) ” IS) 9) a ” ” = —, +6 ” ” ge ” IT" 93 33 ”) > 2 be) 9) ” Se € ” = ” >, be) 9 2 99 be) ” == é be) A >? ” bP) a” 3 3”) ” >’ 0, ze € ” ANS EE) EE) Dj it) a ” ” ” 25 TR) = é 33 d 9) be) 34 bP) 3 be) ” >] 0 zE € Then we get the following types for the Zeeman-effect for the different complexes. es & —E — ZOE — dé g — 0 — —€ -— 2 — 2 2 2 3 - -3¢ —-(-s) e-(-e) —-(-s) 0-(-s) Te &-€ 7e EEEN mat —& —de De) vel (0 MES & 0 —€ 3e ED 2 ” zeke a —E ) e -€ 5 —(-€& &-€ 7 € 5 € —E-E eene MIS 0 — Ze &-€ —E-—E —e-(-3 2 ee ee 3 2 tie eases the 9 € — == — ——— == pee ee Tp ep za 55 ( 0 Se ES 0 —€ — de —-(-s) s—(-s &— ee Ol eh carla ee : oe) LA ele a la gs e-(-6) 0-(-a) 5 - DO 6-0 —*- TC) -e-(-2) ) —-(-8) -s-(-s) —-(-e 9 ( EN é = a zen 5 en BEN 2 ? gee Lani é & é ~& &—(-&) ca tee s-0 —-0 &-& ree —e-0 rae: Sey Pes Cd 9 mA ve DE on 0 —€& — de eé-(-&) e-0 e-3 —e-0 -&-3 196 | TA be 3e ri Fi any ~ 5, me (28) = eh 0 Fi ae En ee 2 2. — 2 = 2 — 2 2 3e 3e (-€ —e\ 3e & de é —& — 3e —-(-s) — e-| — | —- —-& EE —-€& 0e —-& —&-& —_— € 2 2 2 on ele melee 2 2 2 2 € & /—E € & € E -& € se —3e €& m/e EE Dag Cele | PD OS Pad ed ENS PE ls telly Ae a et) dee dlg): 22 2 CNE Nese en OSE ite el (he geisha le ; 2 2 2 2 2 2 —€ — 9 3 pedel) CP mA ? DE En = =e ae 2e — 5 = 0 a -€& — -de 2 = 2 = = 2 -€ & (-& & ES —E €& -€ &—(- e-{|—] —-|—] é«-= — —- —— —-€& —E- ae a) 2 \2 oes aes 5 oe -& & € -& —-~(-&) re Aye (7) —E-— -e (-& ane zee) 1 nr” 3e E a: — de a OTE Oo -—€ Fs —& é = — — = ae) me el —— = a = —€& e- (-&) é 5 g EE & 5 Soe é -2-(-8) — == Id ee ; ——_ be ie 3e ? & -€ ? — 3e En — be DM ET OS oe ee aie de es de 0 A de g 0 -€ — de —-(-s) (—8) —- = == — —- —E —- -&-& —-@ 5 ( Ving é 7 E &-€& 5 € 3 5 eB E 5 g -3¢ ——(-«) 0={-«) —-0 0-0 0- — —e-0 oi te =e — de Gee —¢—(-8) ae Pt €) 3 Ta 36 = g —& a — 3e (2e) 5 e 7 0 a -€ = (— 2e) &—(-6&) ily e—0 Erg EE Api —e-0 mn —E-—E 2 2 2 2 —& ° —€ me 2 -¢—(-8) raat BE | (2e) e 0 ae (- 2e) e-(-e) «0 ee —e-0 —E-E —e-—(-&) Under every component it is indicated from what jump or jumps above it is supposed to have arisen. A — the component expresses under that it is polarized an When both signs occur, the two components coincide or the polarization is incomplete. Between ( ) are placed the so-called superfluous components, which have not been found in the observation. The adjoined notes of interrogation will be dis- cussed further on. First a few words about the superfluous components. By far the greater part are extreme outer components. As the outer components that have been observed, are mostly very faint, it is possible that the theoretically found components are so faint that they could not be observed up to now. These components originate namely by jumps from and to the most greatly deformed paths, and according to SoMMERFELD these are less probable than the less deformed ones, so that the number of jumps of the electron from these greatly deformed paths is relatively much smaller, hence the produced component much fainter. This explanation, is however, not applicable to the question whether the middle components of J’ 2 and IId are superfluous. It would, however, also be possible that the “Kombinationsprinzip’”’ was dependent on a restrictive condition. Prof. Bor was namely of opinion, as appeared to me from a conversation on this question, that in its jumping the electron should also be bound to the condition that the angular momentum in initial and final path may not ’ h ; ir differ more than 1 X BE [ have not succeeded as yet in introducing = 198 this condition as restriction of the ‘“Kombinationsprinzip’. That there must somehow exist a _ restrictive condition, does not seem doubtful to me. That however in some cases the non-appearance of a middle component can yet be ascribed to the observation, may appear from the following example: In my paper (loc. cit.) the ZeumAn-effect for A" is given as an octet without middle component, here as a nonet with middle component. My first statement was among other things based on the obser- vations by Mrrrer *), who has not found a middle component, whereas WeNpr ®) does find the middle component for the same lines: “bei dem zweiten Begleiter ist die mittlere Komponente über- sehen”. (p. 559). From this it may, therefore, appear that it is by no means impos- sible that some of the observed types of ZrrMAN-effect are incomplete, and that possibly some of the “superfluous” components found by me will after all appear to be present on closer observation. The outer components of A" are marked with a note of inter- rogation. These components have been found both by Wenpr (loc. cit.) and Minurr (loc. cit), the distances to the middle component are, however, somewhat smaller than agrees with 2e. The state of polarisation could not be determined by Minter on account of the slight intensity; Wenpr finds 1. The other notes of interrogation are found beside Hd. Wernpt finds for this ten components, whereas Mituer has observed twelve. The latter observer remarks here, however (loc. cit. p. 117): “Die durch Klammern zusammengefassten Linienpaare liefen in eine Linie zusammen, wenn beide Arten von Schwingungen zuge- lassen waren”. When this is taken into account, the notes of interrogation may be omitted, and « and —e may be substituted for them. The O-component has, however, not yet been accounted for. Another factor that remains to be explained, is the state of pola- risation of the components. Though here for the anomalous Zxeman-effects, just as for the normal triplet, the outmost components appear always to be pola- rized 1, 1 must confess that I have not yed succeeded as yet in 1) W. MiLuer, Zeeman-effekt an Mg. Ca. u.s.w. Ann. d. Physik, 24, p. 105, 1907. 4) G. Werpr, Untersuchungen an Quecksilberlinien. Ann. d. Physik, 37, p. 535. 1912. 199 finding a simple rule for the state of polarisation of the components. As regards the intensity, the supposition has already been expressed above that according as the electrons jump from or to strongly eccentric paths, the intensity of the produced components would be slighter. It is to be regretted that the material of observation does not allow us to test this supposition in every detail, because the values found by different observers are often contradictory. Minter (loc. cit. p. 112) gives, indeed, e.g. for the intensities of: TIALS LER Ot A tE Ì MEE ord Vins Ao ck aay 1 1 Pee NAS Sn eh eI and from this appears a rapid decrease of intensity towards the outer components, (with which my “superfluous” components, which are still weaker, are in good agreement). Also the fact that the middle components are stronger, is in good harmony with this that each of the components can be produced by some different jumps. This investigation can, however, not yet-be universally carried through on account of the above-mentioned mutual contradictions. The causes of these differences of intensity have been investigated by Zrrman'), but it can seldom be inferred from the publications of the different observers what circumstances have given rise to their differences in intensity, and which are the reliable intensities. A research as discussed above will not be possible until this has been settled with certainty. From the results of such an investigation important conclusions might be drawn as to the correctness or in- correctnes of the hypotheses given by me in this paper. From what has been found so far I think I am justified in con- cluding that there are indications to be found in the material of observation for the validity of the ‘“Kombinationsprinzip’, also for the anomalous ZwerMaN-effect. It is, however, not excluded that a restrictive condition in the sense as given by Bonr (see above) causes the principle not to be always clearly manifested. At any rate I think I have shown that the “sequences” vary separately in the magnetic field, and that the observed Zurman-effect is the result of the variations of the two sequences together. In connection with this I am also of opinion that Bonr’s equation: WW h keeps its validity in the magnetic field. The Hague, May 7 1919. 1) P. ZEEMAN, Proc. Amsterdam, Oct. 1912 and Researches in Magneto-Optics p. 94 et seq, Macmillan 1913 2 en Physics. — “A Theory of a Method for the Derivation of the Distribution of Energy m a Narrow Spectrum Region from the Distribution of Energy, Observed in an Interferometer’. By K. F. Niessen. (Communicated by Prof. W. H. Junius.) (Communicated in the meeting of May 31, 1919). Micuetson has pointed out the direction in which it should he tried to find such a method'). He first considers two absolutely monochromatic beams, which he causes to interfere with a difference of phase. If this takes place in the interferometer of MicHELson, in which the plane of reference may lie at a distance / before the movable mirror, the difference of phase of the rays that have struck the 2l mirrors at right angles is ni X 20 (or 4rlm, when we work with the number of waves m, by which we understand the number of waves in the unity of length). The ocular is adjusted for infinity, and in the middle of the field of vision (for there the rays striking at right angles interfere) an intensity J/(/) is observed, which can be calculated by the aid of the formula ’): JSA eae ad, cos An ln oenen ee from the equal intensities J, of the interfering beams. An arbitrary beam of light may be thought to be divided into an infinite number of absolutely monochromatic beams of the intensity : Jr WA da Or == in) Gee ee when we use the number of waves m. Ag intensities which are due to different frequencies join scalarly, this light admitted to the interferometer of MrcuersoN, will give rise in the point under consi- deration to the intensity J(/), given by: 1) Phil. Mag. (5) 31, pag. 338, 1891. 4) Drupr, Lehrb. der Optik, p. 123. 201 Ag A ; 4nl | so=2fw (A) da + 2 fu (A) bin ah or rf je Jo=2f4 (m) dm + 2 fx (m) cos 42 lm dm a eee ee. , (9) +a st a Joe] yp (x) de + 2 g(x) cos 4nl (m Hede, = = Here 2, and 4, are the lengths of the smallest and of the largest of the waves present, hence m, and m, are the outmost frequencies on either side, m being the mean of them, so that we may put m=m +2. Further (xs) has been written for y (m + «). The region of frequency 2a of the beam is supposed very small by MicHeLson, so that he examines only a practically monochromatic beam. It should at once be pointed out that we shall not use this restriction in Our reasoning. m being a constant in the integration, the last equation of (3) can be put in the form: +a Jy = afs (w) dw + 2 cos 4x lm x C)—2sn4an lm « S(D). . (4) =e in this C(/) and S(/) are the following functions of 7: = @ +a C (U) =|» (x) cos An le dx, S (l) =| yp (a) sin4aledx, . (5) Our aim is to determine the function y(m), hence g(x). According to Fourtmr’s integral theorem we have generally: 1 co Jo 1 oo + PD = = fe an de fy (8) cos Ea dS + af ux duly (S) sin Sa d5 . (6) 7” n 0 —o 0 —o fo +a In this fv (8) cos Eads may be replaced by fe (xv) cos 4m led, since g(x) for —a > > Ja is zero. +x ie Analogously | p(8) sin Sadi can be replaced by | op (a) sin An lada, — 09 0 202 hence S(/). Evidently 42/ has been taken for «. Now equation (6) passes into: PAD 1 | C (U) cos Aar le dl + 4 js (einAnledl . . (7) 0 “0 Accordingly the function (we) will be known, when we know C and S as functions of £, hence when we can determine C and S experimentally for every value of /. Data to attain this may be derived from “the curve of visibility” Jota —J nin of MrcneLsoN, whose coordinates he defines as VV —=— (in nur: + met which Jar, and Juin. represent the intensities in the successive maxima and minima of the system of interference fringes). It appears from (4) that V is a function of C(l) and S(/); when we assume V(U) to be sufficiently accurately known from photometric observations, we have at our disposal a relation between C(/) and S(/), but without more data we cannot determine these quantities separately. Only in a few simple cases which contained a second condition concerning (U) or SU) has MrenersonN derived the form of g(x) from that of V(!). The aim of our investigation is to find a means through which it is possible to find a second relation between C(/) and S(/) in any given case, and which therefore enables us to solve ¢(2). In equation (4) we can think J(l) experimentally determined +a for a given value of J. Also | ple) de can be measured, this is viz. —a the intensity that one of the beams causes in the middle of the field of vision, when it is not brought to interference with the other. In order to determine this quantity we have, therefore, only to cover one of the mirrors. Hence equation (4) can be considered as an equation with two unknown quantities C and S, to be taken for that value of 7 for which / has been measured. We shall have to find a second equation between C' and S for the same value of / to be able to solve both quantities. This means will fail, however, when in the second equation C and S appear combined in the same way as in equation (4), accordingly +a when they are again derived from fu (x) cos 4 a l (m+ 2) de, en Cl in which a variable parameter is meant by /, which need not have the same physical meaning as just now. : 203: Prof. Ornstein requested me to investigate whether such an equation might not be obtained from the distribution of intensity in the refraction image in the echelon. The result of this investigation is that if the intensity formula no matter of what instrument, contains C or S, it must always contain both of them. And that in the combination in which they already occurred in equation (4), unless one makes use of an artifice to be specified later on. Without this C and S cannot be solved, and ¢ (2) can, therefore, not be determined. We shall first prove that for absolutely monochromatic light the intensity is always an even function of the number of waves, when we premise that the light in the apparatus propagates normally, 1. e. that a change of phase is accompanied with a proportional variation of path, so that e. g. reflection against a denser medium, inetal and total reflection must be excluded. A point Q, where we measure the intensity, receives its light from certain points P of the instrument, which in their turn are again illuminated by the source L. The points P are in different phases, because they receive the light from the source each by another way. As we have everywhere assumed the normal propa- gation of the light, the differences of phase will exclusively depend on differences of paths expressed in numbers of waves, and therefore 1 = ; : en be proportionate to En which the factors of proportionality depend on lengths, angles, indices of refraction ete. We shall leave dispersion out of consideration, because in the instrument that is to be devised later on this will be anyway excluded. Also optically the paths, which the light from the points P must still pass over before it arrives at the point of observation Q, will not be of equal length, so that other differences of phase are added to the already existing ones, which others on account of the normal propa- gation of the light will exclusively rest on differences of path, and will therefore be proportional to m, so that the deviations to which the points P finally give rise in Q, present mutual differences of phase which are proportional to 1. When these deviations are represented as projections of vectors whose angle with the axis of projection is equal to the corresponding phase, then the points P, over which paths of light run from ZL to Q, which are equally long in number of waves, give vectors that overlap. 14 Proceedings Royal Acad. Amsterdam. Vol. XXII. 204 These have been joined to one vector in fig. 1. Thas the action of every instrument where only normally propagated light is used, can be determined in any point Q by the aid ofa vector fan, which revolves round O as a whole in the direction of the A arrow. The intensity of the square will be the result 2 of the vectors. Suppose that for all the points P,, P,, P,,...P; the phases of their disturbances in Q are Aj compared with the phase of the disturbance given by a point P,, the vector of which be AO,, hence a 0 point over which the shortest optical way passes from Fig. 1. the source to the point of observation, then LAGOA is=em, 7 A,OA,=c,m, ete, in which the constants ¢c,,c, ete. depend on various lengths, angles, and indices of refraction. We shall call the direction of the vector OA, the time-direction for the sake of brevity, because it only depends on time; it will appear to play a prominent part in our reasoning, and we shall henceforth draw it always vertical, and omit the axis of projection. We have drawn here a fan whose vertex is small; in reality it will probably contain several times 27. Vectors namely corresponding with paths LQ, which differ a whole number of waves in length, will indeed be superposed in the figure, but will not be drawn as one vector. This happens only when the paths are optically of exactly the same length. Nor need the end points of the vectors form a continuous curve. Let us elucidate this by constructing the fan for an echelon. It consists of a few smaller fans, which are arranged in O like spokes in a wheel, and whose number is equal to the number of plates of the echelon. Only three of them are drawn in fig. 2. All the points P, lying in a vertical line on the front plane of a plate lie optically equally far from the source *), and at the same time equally far from Q. Their vectors are therefore united to a single vector OA, the length of which is accordingly proportionate to the length of the vertical line ®). Any given point of the lefthand side of the first plate can, therefore, be considered as the point P, just mentioned. : The indices at the points P of fig. 3 indicate which vector in fig. 2 originates from their vertical line. When the echelon is viewed at an angle @ with the normal, the rays from P, and P, will dan 1) The light strikes in the direction d in fig. 3. 2) In this way it is also possible to find the influence of diaphragms placed before the plates, which will later on be made use of. 205. +1 Fig. 2. Fig. 3. x, sin Ó interfere under a phase difference Sa 2x. Hence < A,OA, is 2x, sin 0 SE — a, or if in the earlier notation we put this equal to c,, it appears that c, is = 222, sin 6, hence that it is a constant’) that 2x, sin Ô a A ge Ay A depends exclusively on distances; << A,OA, = 26 sin O a — x, when 6 is the width of the glass plates. The other fans are congruent with the first, the last ray of a fan always forming the same angle am with the first of the next; < am is namely the phase difference due to the difference of two paths of light, one going from a point of the righthand edge of a plate to Q, and the other from the lefthand edge of the following plate to the same point Q. When u and d are the index of refraction and the thickness of every plate, then ais = pd—d cos 6, hence again a constant, depending on measures of length and physical constants. This constant distinguishes itself from the first, because it depends on the wave length in consequence of the appearance of u. However u and hence a too, is an even function of m and for this reason we may consider « as a constant in the following discussion. Let us now return to the general case. When the instrument is struck by light of a twice as small wavelength, the fan will be drawn out twice as far, all the angles with the time-direction being proportional to m. It appears parti- cularly clearly with the echelon how in consequence of this the resulting vector, and together with this the intensity in the point of observation Q will change its value. The intensity is, therefore, a function of the number of waves m, and in order to prove that this is an even function of it, we shall reverse the sign of m. This 1) i.e. independent of m. 14* 206 is only a mathematical artifice, to which no physical meaning must be assigned. Points P,, which in their movement were first c,m behind compared with P,, and whose vector lay, therefore, at an angle c‚m on the righthand side of the time direction, will now be as it were c,m in advance of /,, and give a vector which again forms an angle c,m with the time direction, but now lies on the lefthand side of it. The angle between every vector and the time- direction being proportional to m, all the vectors will reflect in this direction, and the form of the fan, and with it the resultant, hence the intensity, will remain unchanged. Hence for any instrument in which light propagates normally, the intensity will be an even function of the number of waves. In this absolutely monochromatic light is supposed, because for compound light the intensity is no function of the number of waves on account of the integration with respect to m between definite limits. 3. Now we shall, however, also admit reflections against a denser medium in the instrument. In this the phase shifts suddenly zr; such an abrupt change of phase we shail call a phase shifting. Let us suppose that some points P’ of the instrument are illuminated by rays which have undergone such a reflection an uneven number of times. Their vectors now no longer make the angle c’m, but the angle c’m + a with the time-direction. Before P, gave a vector by the aid of whose direction the time-direction was defined. When, however, the point P, belongs also to the points ?’, its vector will also turn over an angle x, hence obtain the opposite direction from what it had before. As the other group of points of the apparatus continues to receive normally propagated light, their vectors continue to form the angles cm with P’s former vector. When we now define the time-direction anew, and do so as the direction of the vector originating from that point P, of the apparatus through which the shortest optical way passes from the source of light Z to the point of observation Q, quite apart from all possible phase shiftings, the time-direction will always be the constant direction in the following considerations, with respect to which we determine the position of the vectors. We shall again think it to be always directed vertically upward. A reasoning based on this time-direction will therefore hold both for the case that this point P, receives light that has changed its phase, and for the case that normally propagated light falls on it. We shall let the constants c keep their value, because 207- they only depend on distances; however cm need not be any longer the difference of phase of the interfering rays, this may have become cm + zr. After these extensive considerations, which as will appear later on, go to the core of the method, we proceed to the construction of the fan in the point Q in the case of two groups of points Pand P’. A point P, and a point P,’ give vectors which form respectively the angles c,m and c/‚m 2 with the time-direction. When we now reverse the sign of m, the first group of vectors will again be reflected in the time-direction; a vector from the second will, however, form the angle +2—c’,m instead of the angle a + c,’m with the time- direction, and would, therefore, be evidently reflected in the pro- duction of the time-direction. Ultimately al! the vectors will, therefore, again be reflected in the same line, through which the form of the fan will remain unchanged, hence the value of the intensity will remain the same, which latter is, therefore, again an even function of m. The thesis may, therefore, be extended as follows. In every instrument, in which the light can only undergo phase- shiftings of a whole number of times 2, the intensity is an even function of the number of waves when absolutely monochromatic light is used. The light fulfils these conditions naturally in all apparatus of refraction, also generally in those which are founded on interference. As will appear later on, through the suitable occurrence of phase shiftings among others in reflections against metal mirrors the intensity can also sometimes become a not-even function of the number of waves. Let us now consider the case of a beam continuously composed of some frequencies. Suppose it to be possible to draw up a second equation between 7 and s for a given parameter / by the aid of one of the current apparatus, where we shall suppose a favourable action of the phase shifting to be absent in the metal reflections, which is actually the case with most, if not with all instruments in general use. We can now easily show that the two quantities C and S must occur in the same combination as in equation (4). For +a if the new equation contains e.g. (, ie. f («) cos 4 ar le dx, it would —@ 208 contain J cos4a lr for absolutely monochromatic light, for which can also be written J cos 4 al (m—m). But for single light the intensity must be an even function in m, as we saw; hence cos 4a lm—m) can only occur in that combination which reduces it to J cos a Im, which is even in m, i.e. it can only be met with in forms like: cos Anr lm J cos Aar | (m—m) — sin An lm J sin 40 1 (m—m)_ or cos An Um J cos An la — sin Anlm Jsin4ale. . . . . . .. (8) ba To extend reversely the formula of intensity for single light to that for compound light we must substitute y («)d«# for J in the former, and then integrate with respect to «. The grouping (8) then passes into cos 4x lm C—sin4almS, exactly that of equation (4). The preceding has therefore proved that none of these instruments can yield a second equation between C’ and S which is independent of equation (4). C and S can, therefore, not be separated from ‘their combination, it seems, therefore, that p («) cannot be solved. *) 5. It is now obvious that the first thing to do is to examine whether the function ~(#) can perhaps be determined, if the light in some instrument or other can undergo a phase shifting «, which differs from ze, it being of no consequence whether it is caused by metal reflections or other phenomena. Suppose that part of these points P, now marked by double or triple accents, receives such light. The angles of the corresponding vectors with the time direction would have to be c,"m, c,'m..., ¢,'"m, cm, if the light travelled in the same directions, but everywhere in a normal way; now they are, therefore, c'm + «or c''m tata, according as on the point P" 1) In passing we may remark that C and S do not occur combined in another way in the formula of intensity that determines the distribution of intensity for the echelon over the whole focal plane of the telescope. This depends for the different points on-a parameter, which we might call /’. | have proved by a computation which is left out here that this function of /’ satisfies a differential equation of the second order, in which only combinations of C and S of the kind as in equation (4), to be taken for some values of / in connection with /’, occur as coefficients. Thus the dependence between the intensities in the image of reflection of the echelon and that in the image of interference in MicHELSoN’s interferometer had been proved purely analytically. Though the formula of intensity for the echelon could be greatly modified by the supposition that diaphragms of a particular shape were placed before the glass plates, such a differential equation of the second order remained of force all the same, and so the dependence continued to exist. 209. or P" under consideration light falls that has undergone besides the phase shifting « resp. an even (among which zero) or an uneven number of reflections against denser media. Hence we have to deal here with four groups of points, the classification of which will be clear from the foregoing. When out of each of these groups one point is considered, resp. P, P', P", P"', their vectors make the angles cm, cm + a, c'm + a,c'"'m He 42 with the time-direction. The phase shifting is taken to be positive if it is a shifting back, and negative in the opposite case. When m is now made to change its sign, these vectors are now respectively reflected in the time- direction ¢, the direction # forming an angle 2 with the former, the direction a forming an angle « with the time-direction, and that which forms an angle a+ with the time-direction, being a in fig. 4. Accordingly the vectors can be divided into two groups, one of which is reflected in the line ¢’, the other in a line aa’ forming an angle a. As we have put a=, the fan will now probably also be transformed, and the intensity will no longer remain the same. This is therefore no even function now of the number of waves. In such an instrument C and S will accordingly t certainly not appear in the formula of intensity in a the way of equation (4), which? can again be proved 0 via the intensity holding for single light. Now the separation of C' and S is therefore no longer fun- damentally impossible, hence we are on the right tE track to determine the function g(x). It is finally noteworthy, that a phase shifting in the part LP of the light path LPQ will give the same effect as one directed in the same way and of the same extent in the part - PQ, which remark will appear to be of importance for the inter- ferometer of Micueuson. In what precedes it has been supposed that the phase shifting «a preserved its value and sign in the reversal of sign of m, it must accordingly be a constant or an even function of m. Only an instrument in which it is a constant, i.e. independent of m, will be of practical use to us. Our purpose is namely to examine beams of the most=divergent”’ constitution, and it would be impracticable to have to take a different phase shifting into account for every kind of light. For this reason we shall e.g. not be able to make use of metal reflections. On the other hand it will hardly be possible to do without them, for in almost all interferometers there occur silvered glass-plates or metal mirrors. This“ difficulty is, however, not so serious as it seems. When each of the four mono- chromatic beams that illuminated the groups of points P, P', P", PY Fig. 4. 210 of just now, and which had already undergone the phase shifting discussed then, is besides subjected to a metal reflection, which can take place for all four of them in exactly the same way, this will mean for each of the beams a same phase shifting p. This latter does not deform the fan, as it revolves as a whole over an angle 8, since all the points P have participated in the new change of phase. For some four beams of other frequency the fan will indeed be rotated through another angle 8, but fans drawn up for different frequencies, must yet not be composed vectorially, because two different kinds of light can never interfere. Hence also when com- pound light falls on the apparatus, metal reflections will have no influence on the intensity, if each of the beams undergo them an equal number of times in the identical way; this latter peculiarity is always met with in the current interferometers for reasons of a practical nature. The same thing applies for the phase-shiftings which take place with total reflections. 6. We must, therefore, devise an instrument in which first of all the condition of identical metal reflections and of identical total reflections is fulfilled, but that besides has the property to be illuminated for one half by rays which apart from metal and total reflections, have undergone none or only phase sbiftings of a, and to be illuminated for the other half by rays which apart from metal and total reflections, have been besides subjected to an extra phase shifting «, which differs from 2, and must be independent of the number of waves; the rays of the first and of the second kind must besides be coherent in order to be able to interfere. When we now put the question what constant phase-shiftings we have now at our disposal, we come to the following answer: 1. a phase-shifting of in case of reflections against a denser medium ; 2. a phase-shifting of 2, when a beam of light is narrowed to pass through a focus *); 3. a phase-shifting > forward, when a beam must pass through a focal line *). The last two shiftings take place respectively in the focus and in the focal line; the latter alone differs from zr, and this is the only one we can therefore use for our purpose. We shall, therefore, 1) Gouy, Ann. de Chimie et de Physique (6), 24. 1891. 211 make use of two cylinder lenses, whose focal lines coincide; in fig. 5 they are thought with the descriptive lines normal to the plane of the drawing, and we shall assume this position in all the follow- ing figures. Lines thought above or in the plane of the drawing will be drawn in full, whereas lines that are supposed below it, will be dotted. Rays recurring on the upper side parallel to the plane of drawing (e.g. a), can never be brought to the lower side by the cylinder lenses, because the lenses have no refractive power in the direction of their descriptive lines. Analogously for rays 6 under Fig. 5. and parallel to the plane of the drawing. The cylinder lenses will have to be of such a quality that they do not disturb the phenomenon of interference. It is not possible to place this set of cylinder lenses without any modification into one of the arms of the Micur.son interferometer, because the light then passes twice through the system and the S resulting phase-shifting becomes again st. Compare fig. 6, in which the immovable and the movable mirror are the two halves of the instrument consisting resp. of the points P" and P; the system of lenses now lies as well on the path LP" as on the path P"Q, and we saw that phase shiftings are equivalent in the two parts of Q the light path, when their direction Fig. 6. is the same, e.g. when both are directed forward as here, which causes them to join to 2. For our purpose the inferometer of MicueLson must, therefore, be modified. Fig. 7 indicates how this can be done. The mirrors of Micne.son’s interferometer have been replaced by totally reflecting prisms, through which the advantage is attained that the light goes only once through the lenses. As the two beams undergo precisely the same total reflections, we need not take into account the phase-shiftings appearing in this. When a ray which has originated from a ray originating from the slit S on its reflect- ion on the silvered glass plate through reflection or transmission is resp. provided with the index r or d, the figure shows how the reflected component 17 of the ray 1 leaves the apparatus iinmediately by the side of the transmitted component 2d of another ray 2, 212 when the rays 1 and 2 from the points A and B lie in the same horizontal plane, and SA and SB make equal angles with the radius SC, which passes through the optical centres. The rays 1 and 2 being coherent, the components 17 and 2d e.g. can interfere with each other. The leaving rays can therefore interfere in pairs, and as the rays which form such a pair, run immediately by the side of each other, a sharp interference image will be formed in the focal plane of the objective. Now tbis is not looked at, as usual, with the ocular as magnifying glass, but the ocular must form a real enlarged image of it, which falls outside the telescope, and of which only the central part will be used by us. All the light that has entered the prisms at right angles, will be focussed in one point Q'. In order to calculate the intensity in the point Q’, it is convenient to introduce a reference plane HA, determined by the following equations: ED == CO LPN FE LM Ne which quantities refer to geometrical lengths, and not to optical lengths. Also the two following remarks will greatly facilitate the calcu- lation of the phase-difference between the interfering beams. 1. Let the distance from the front plane to the opposite side be p for the immovable prism, and equal to q for the movable prism; then every ray that strikes the front plane at right angles vertically, covers a path in the prism of the length 2p or 2q, according as this takes place in the immovable or in the movable prism. 2. The system of lenses makes a plane wave again to a plane wave, so that in fig. 5 the optical distance from A to A’ is equal to that from C to C’. Accordingly in fig. 7 the system of lenses will retard the light as much as a glass plate of the thickness 2d, when the thickness of every lens in the middle amounts to d. Besides Jt this retardation the system of lenses causes the phase-shifting — forward, which on the other hand means an acceleration. In order to get from C over the immovable prism to U, the light must pass three times through the glass plate, hence cover a path 3 d in glass, and further the paths of the following lengths in the following media: DE in air, 2p in glass, (FG—2d) in air, and 2d in glass. The light that reaches the point U from C across the movable prism, must also pass three times through a glass plate, hence it must again cover a distance 3d in glass, and further the following 213 distances in the adjoined media: (CO 4 PV) in air, VT inair, 2q in glass, NM in air, (ML + JH) in air. When the first path is compared with this last, use being made of equation (9), the path in the second (the prolongable) arm of the interferometer proves to ni ee AES dea a nen EG i aE: rd : r ? Aa BE Tae ee ee UEA ES 5 N \ an ID KL eve ese Ale ir ee We Fig. 7. be 2u(q—p) + 2 VT + 2d (A—y) longer than that of the first. When : nya 7 we now also take into account the phase-shifting 5 forward when 214 the focal line is passed, (the path is seemingly shortened by this) the phase-difference of the interfering rays at U becomes: Aan VT + An {x(q—p—d) + d} + ; or 4 a Um + 5 when we put: wilg Pd SUS ae en Pe mie EK LO) and : VI el RE) This phase-difference also exists between the rays 1r and 2d, which follows from the figure and from the foregoing remarks. In analogy with equation (3) the intensity for compound light now becomes: my my J) = 2 fix (m) dm + 2 fi (m) cos (43 Um + =| dm or Mo ma my my N= fx (m) dm — 2 fx (m)sin4dalimdm or... . (12) +a TULES 2 fo (e) da — 2 sin Aar Um C(l’)—2cosda lim S(l’) . (13) =a C and S no longer occur in the combination of equation (4). This is owing to the cylinder lenses. J’ will henceforth be the inten- sity measured in Q’, hence with the modified interferometer, J that with the unmodified interferometer. A similar difference will also be made between 7 and 7’: / refers to the original interfero- meter, and is the distance from the shiftable mirror to the reference plane; 7’ is the quantity that characterizes the position of the movable prism in the new interferometer, and controls the intensity. In order to calculate C and S for a given value /, from equations (4) and (13), we should have to place the mirror in one interferometer, the prism in the other in such a way that the quantities / and /’ both acquire the value /,. This would be practically unfeasible, when one worked with the two interferometers separately. We shall, therefore, try to combine them to one instrument, in which the quantities / and l’ both oceur, and are therefore equal. This seems to me practically no more feasible than the adjustment of the two interferometers to one and the same value /, of the parameters. | have, however, 215 succeeded by the application of another slight change in the inter- ferometer in designing a combination of the interferometers, in which the quantities / and /' always differ a same amount, which can be accurately measured. We shall further point out the way to determine this difference, and then demonstrate that the fact that this difference is constant enables us to work just as easily with this instrument and to reach just as much with it as would be the case with the ideal combination of the interferometers. From the modified interferometer we should, namely, immediately get back the original one, when the front planes of the prisms were silvered. Accordingly the two instruments can be joined to one by leaving the upper halves of the front planes of the prisms trans- parent, and by silvering the lower halves. Let the line of division between the two halves lie in the plane of the drawing, then the rays that run above this plane, and have, therefore, been drawn in full, are totally reflected and are joined in a point Q’, when they struck the prisms at right angles. Rays running under the plane of the drawing, and which have therefore been indicated dotted, will be reflected by the silvered lower halves, and collected by another eyepiece K into another point Q, when they too strike the prisms at right angles. Strictly speaking we have here placed the two inter- ferometers on top of each other, which was possible owing to the property of the cylinder lenses of never bringing a ray that runs above the plane of the drawing and parallel to it, under it. Rays that intersect the plane of the drawing could, indeed, pass from one instrument into the other, but these are not joined in the points Q or Q’, and it will appear, that we have to measure the intensity only in these points. Moreover the front plane of every prism can be divided into three regions, the top one transparent, the middle one absorbing, and the bottom one silvered, through which, as it were, a Space arises between the two interferometers, depending on the height of the middle region. The upmost and the downmost regions must remain large, as all the incident light that strikes at right angles is concentrated in the points Q and Q’, and we want to have a strongly pronounced gradient of intensity in these points, when presently the movable prism is shifted. We saw already that the intensity in the point Q’ was determined by equations (12) and (13). The intensity in the point Q is repre- sented by equation (4), in which / was the distance from the movable mirror to the plane of reference, and for which, therefore, V7 of fig. 7 must be taken. We shall, therefore, call the length of VZ'/. Summarising we find in Q’ and Q resp. the intensities - 216 my, my Pl 2 fx (mm) dm —2 fx (m) sin Ax l'm dm ma ma my, my MP 0e (14) J()= 2 fu dm 42 fx (m) cos 47 lm dm ma ms in which ! = lv when v = ulq—p—d) +d A diaphragm with two fine apertures at the place of the points Q and Q’ casts two beams of light of the above mentioned intensity on a photographic plate. When the movable prism is slowly shifted, the intensities of these beams change on account of the change in the para- meters land /’ in (14). The difference /’—/ remains, however, constant, namely equal to v, which quantity is an instrument con- stant, as appears from (10), if u does not appear in it. For that the prisms must be made of different size and so, that g=p+t+d. A small error in their size once for all can be neutralised by turning the plate PL over a small angle. Now » becomes = d, hence an instrument constant. When the photographic plate is shifted over large distances and in its own plane with a slight movement ot the prism through coupling, and when care is taken that the shifting of the plate is always proportional to that of the prism, and further that the shifting of the prism always takes place with uniform velocity *), the two beams will leave behind two bands on the photographic plate, where the blackening is different point for point, and which, when they are afterwards measured photometrically with the photometer and thermopile of Dr. Morr, yield two curves, which will be of great importance for the determination of the desired function (a). We shall call the curves that arise when the bands are measured, Z(el) or Z'(cl’), according as their ordinates represent the course of the blackening in the lower or the upper band, which is, therefore, : ! if at made in Q or in @. When „ denotes the ratio of the shifting of the prism, and the corresponding displacement of the photographic plate, and when in the upper band that point is assumed as zero point N that was placed in Q when the prism passed through the position /=0, then light, the intensity of which corresponds to the position lof the prism, and is therefore indicated by equation (4), will have fallen on that band in a point whose abscissa amounts to c/ measured 1) In order to keep the time of exposure the same for every position of the prism. 217 from the zero point. The blackening is proportional to this; hence when f is a factor of proportionality, the following equation will hold for the lower band: Tif ZEE At ELD) Analogously Ll Zi Velhanern ee for the upper band when on it that point is chosen as zero point N’ that was in Q’ when the prism passed through the position /'=—0, hence when l= — d according to (14). Now equation (14) passes into: my, my fLld).= JE (m) dm + 2 fx (m)cos4almdm. . . (17) ms mna 2 (el)= afs (m) dm —2 fx (m) sin An lmdm . . . (18) The functions fZ and fZ' oscillate round the same constant value: my 2 fund, when / and /' are increased; they will asymptotically ug my, my a] approach this value, for Jam cos 4a lmdm and f x0 sin da mdm De / ma become zero for large values of / and /' on account of the continual reversal of the sign of the cosinus and the sinus, even when we have only to integrate over a small interval with respect to m. According to equation (17) the ordinate Z will reach the greatest maximum for /=0. This can be sharply determined from the course of the Z curve, when we have to do with multichromatic light. The bottom of this ordinate is the zero point MN. This operation is equivalent to the adjustment at the white point in the interferometer of MicueLson, when white light is used, and when we want to make the movable mirror coincide with the plane of reference. The adjust- ment can of course much more accurately be effected graphically than visually. Mi AS Zaman afs (m)dm, it follows from equations (17) and (18) that: Mag mi 2 fx () can dae Im dm =f Z (Cl) — 2 Ze davies, GEO) Me 218 my 2 fx (m) sin Aar Um dm = —fZ' (dl) + kf Zmar. « - (20) ma In this Zijar is the measured ordinate of the highest point of the Z-curve; the bottom point of this ordinate was the zero-point J, from which c/ must therefore be measured along the axis of abscissae of the Z-curve. When a line is drawn parallel this axis and at a distance 4 Znax from it, parts will remain of the ordinates the length of which will again be a function of el, which function we shall call I (cl). According to (19) and (20) we now have: 2 I (d)=27 (a) — § Limar, = — fx (m) cos 4a lmdm . . (21) fe When the same line is drawn through the 7 curve, the remain- ing parts of the ordinates will represent a function II (c/’) so that: ye TI (el) =Z (cl) — A Liman — — = | (m) sin4al’mdm. . (22) i En The zero point N' on the axis of abscissae of the Z' curve may be determined on the following consideration. The function II (cl) is zero for /'=0O, but has several zero points, so that it is not yet known which of them is to be taken as N'; it appears, however, from equation (22) that the function is uneven in /'. When, therefore, of the function I] we determine the zero point with respect to which the other zero points lie symmetrically, we have found the point cl =0, hence the zero point N' on the axis of abscissae. This is the zero point, from which c/' must be measured on the axis of the Z'-curve. Especially for multichromatic light this symmetry will be very apparent. The reasoning involves that observations must have been made also for negative values of /', ie. that we must start with putting the movable prism somewhat nearer than would correspond with /’—0O, hence with /=— d. When we, therefore, begin with placing the prism at a distance greater than d before the plane of reference 2, and moving it from there with uniform velocity, we begin with a negative value both of / and of /', so that the initial points of the bands have negative abscissae. We may cursorily remark that the quantity v= d can be accurately determined from the situation of the zero points NV and WN’. The photographie plate was namely every moment struck by two beams of light, the intensities of which depended on two parameters 219 / and /, the difference of these parameters, if they refer to the same moment, being constant, viz. =d. For points of the Zand Z, curves, which have thus simultaneously arisen, the two abscissae cl and c/' will, therefore, always differ by the same amount, viz. cd. The initial points of the curves are such points that have arisen simultaneously '); we have, therefore, only to measure their abscissae taken to the found zero points on the axes, and to take the diffe- rence between them to find ed. When in the graphically found functions I and II we choose the variables equal, i.e. when we measure equal portions cl and el! on the axes from the points MN and N' (hence independently of the earlier meaning of / and /'), we find from equations (21) and (22): my 2 [i (m)dos 4 ce lmdm= fl (el) 2 2. «1 (28) my 2 fx (m) sin 4x lm dm = — flI(cl) . . . . (24) At first our intention was to solve C'(l) and S(/) from equations (4) and (13), and substitute them in (7). This would come to the same thing as the introduction of C(l) and S(/) into equations (23) and (24), in order to determine them from this, and substitute them in equation (7). The same result, the formula for p(«), can be reached more quickly by the consideration that according to Fourigr’s integral theorem ro + oo oo 1 Ff p(z) = x(m) == fom am aefn (5) cos Sa dE + „en am afs (5) sin Sa da, 0 — 00 0 at oo When we now choose m for §, and put ¢—4a/, and when we besides consider that y(m)=O for m, Can, indeed, be attained without cylinder lenses, viz. by the aid of an oblique-angled paral- lelepiped in which the light is totally A __reflected. We can now work with the unmodified interferometer of Mrcnerson (we have only to put the parallelepiped in one of the arms, and one of right angles —___——}—. - in the other, if we make the paths in glass of the same length; fig. 9), but now meee we must work with light that has been previously polarised either in the plane of drawing or normal to it. For sodium light eg., the direction of vibration of which is normal to the plane of the drawing, we must have < gy = 51°20/21"; then the four phase-shiftings together amount to on oo | that the same parallelepiped could not be used for neighbouring spectrum lines too. After some calculations we find namely cos 2p = zE (n?—1) cos 674°—1 1. However p does not so greatly depend on the wavelength, : ‚in which n is the index of refraction. n Dr. Zernike has pointed out another method, in which the phase- JT = . shifting keeps its value 5 only for one narrow spectrum region, just as this was the case with the totally reflecting parallelepiped: If we turn the compensator in the interferometer of MicnELSON over a small angle and if we place at the same time the movable mirror a little nearer, then in the one arm light has to cover a path in glass, which is d longer, but in air a path, that is d’ shorter than in the other arm. If we choose d and d’ so, that they satisfy ud —d’ = — 42 (A), even when we vary À (however only over a narrow region) the Jt phase difference of the interfering beams will be x + 42 Im, when 1) For * the parallelepiped would assume an impracticable form; index of refraction = 1,5153 for light crownglass. 224 lis the distance over which the movable mirror has been displaced from out its new position. We find d for one definite wavelength du du out of — K d=—tiord x(— À x) == tÀand then d’ out of (1). dà d Perhaps we can replace the unmovable mirror in the interfero- meter of MicneLsoN by a concave one '). In conclusion I wish to express my cordial thanks to Prof. Ornstein and Prof. Juttus for their suggestions and the interest shown in this research. The research which Prof. Ornstein originally asked me to undertake, was the first step on the road by which | have ultimately sueceeded in finding the expedient which was to prove able to solve the problem theoretically in a perfectly satisfactory way. I have followed the logical train of reasoning which has led to this, also in this communication, because through the considerations on the four groups of points P, P’, P", PY it could then be naturally proved as second thesis that the required distribution of energy cannot be determined by any of the current instruments, in whatever way the observations made by them should be combined. The phase- shifting, independent of the wavelength and differing from 2, was accordingly not only a possible, but also a necessary expedient to accomplish the task we had set ourselves. Utrecht, May 1919. Institute for Theoretical Physics. 1) Govy, Ann. de Chimie et de Physique (6) 24, pag. 198, 1891. Geology. — ‘Some new sedimentary boulders collected at Groningen’. By Dr. P. Kruizinea. (Communicated by Prof. G. A. F. MOLENGRAAFF). (Communicated in the meeting of May 31, 1919.) Some ten years ago a favourable opportunity offered for collecting sedimentary boulders at Groningen, where in three different spots at the northern extremity of the Hondsrug and in the neighbourhood of the northern cemetery, which has already become known as a findingplace of erratics, important excavations were performed. First when the foundation was laid for the new tram-shed, and shortly after when new streets were being made, viz. the Tuinbouwstraat and the Koolstraat. When trenches were dug for sewer-pipes, the Dilivium was not reached at the point, where the Tuinbouwstraat joins on to the Nieuwe Ebbingestraat. The presence of sherds of pottery at a depth of more than 1 m., however, made us suspect that the upper soil had been turned over or earthed-up. A little higher up in the Tuin- bouwstraat the Dilivium emerged, and gradually rose to the surface, until about halfway it was seen half a meter above the present level of the street and was covered only by a thin layer of mould. Subsequently it first sank again below the street-level, after which it rose once more to the height just mentioned. Afterwards, on the occasion of the excursion made in 1913 by the Geol. Section of the Geol. Mijnbouwkundig Genootschap (28 p. 83), it was encountered again in the first part of the Verlengde Tuinbouwstraat, also covered with a layer of mould. In the three localities just mentioned boulder-clay was found, which is calcareous but already oxidized. Besides this a number of bands of gravel were to be observed. Among the large number of boulders, found by me during these excavations, there were several interesting specimens. Of the species rarely found near Groningen I mentioned already (30 p. 231) the Upper-Silurian limestone with Pristiograptus frequens Jaek. and the Saltholms-limestone (also the glauconitic variety, the so-called Glau- conitie Terebratula-rock) from the Danian. In the following pages I purpose to discuss three more Silurian 226 boulders, hitherto unknown in our country, the last two not having been met with in any other country as yet. Limestone with Strophomena Jentzschi Gag. Among the erratics found when the new tram-shed was being built, there was i.a. a plate-shaped piece of rock about 2 e.m. in thickness and 1 d,m. in length and in breadth, in which occur a large number of dorsal valves of a typical Strophomena. Of other fossils this boulder appeared to contain only a longitudinal section of a Pleurotomaria and a small pygidium of Asaphus raniceps Dalm. so that from this it is obvious that it belongs to the Lower Silurian. The rock is a fine-grained, rather hard limestone, with scattered small rounded quartz-granules. I have not been able to detect glauconite. The primitive colour is undistinguishable, as through weathering it has changed into a more or less yellowish grey. Also some brownish spots still occur. In looking up the literature I soon became aware that the very same Strophomena has already been described by Gacen of East- Prussia, who termed it Strophomena Jentzschi (15 p. 17 44 pl. V fig. 26). One of the blocks in which this fossil has been found, consists of brownish grey hard limestone with somewhat weathered, yellowish spots. It comes from Spittelhof and contains besides numerous dorsal valves of the above mentioned fossil, also the rests of a large Strophomena and another irrecognizable brachiopod. The other specimen comes from Pr. Holland and is composed of coarse- crystalline limestone in which only one dorsal valve of Strophomena Jentzschi Gag. occurs, beyond corals and rests of crinoids. Their petrographical character induced Gage. to refer both erratic blocks to the Upper-Silurian series. However, it will appear presently that also these boulders, at all events the first-mentioned, have been proved to belong to the Lower-Silurian. On the other hand, according to ANDERSSON, the other may possibly originate from the Upper-Silurian and contain a closely related species. Now in order to make assurance double sure, I begged Prof. ANDREE of Königsbergen to send me one of GaGet’s original samples for comparison. This request was readily complied with. A couple of well-preserved valves of Spittelhof were sent me, for which kindness I still feel greatly indebted. My suspicion came true in every respect. The short description of this fossil by GaceL I quote here for the sake of completeness : Umriss querverbreitert, Schlossrand gleich der grössten Schalen- 227 breite. Schale anfanglich flach, dann allmählig unter einem rechten Winkel nach der Ventralseite zu gekriimmt, so dass die Dorsal- schale convex wird. Oberfläche mit zahlreiehen feinen, aber deutlich runden Rippen bedeckt, deren Zwischenräume durch 2—3 sehr feine Radialstreifen angefüllt sind. Ausserdem befinden sich auf dem flachen Teil der Schale noch eine Anzahl unregelmässiger, flacher, concentrisch angeordneter Runzeln, ahnlich wie bei Strophomena rhomboidalis. In der Mitte der Dorsalschale befindet sich oft noch eine kleine, aber deutliche Kinsenkung”’. The concave ventral valve has not been discovered by Gager either. In consideration of my scanty material I was not enabled to make certain about the petrographic resemblance between the boulder from Spittelhof and the one found by myself; still it seemed to be rather great. The second and latest writer that has described erratics with Strophomena Jentzschi Gag. from Germany is Stotiey (20 p. 136). Without mentioning the finding-place (only the district Schleswig- Hollstein is given), he mentions two more blocks. The one is composed of light-grey limestone, in which here and there vermiform concre- tions of glauconite grains occur. Beyond a number of specimens of Strophomena Jenteschi Gag. this boulder contains only Orthisina plana Pand. The other resembles in a most marked degree the preceding one, but contains only a trace of glauconite and the only fossil accompanying Stroph. Jenteschi Gag. was Orthisina concava var de eBahl: To my knowledge this Strophomena. has not been detected in erratics from Denmark. Only a few years after Gacer, had described the species, J. G. ANDERSSON also recorded a number of erratics with the same fossil from Sweden. One of them originates from L. Brunnby in the parish of Stenasa - in Oeland, one from Källunge Myr in Gotland and four from Gotska Sandon. All these specimens differ largely from the one of Groningen as well as from the German pieces in that they are filled with a number of rolled fragments of brown phosphorite and brown to black phosphoritic sandstone. This makes them true conglo- merates, which induced ANDERssoN to style them Strophomena- Jentzschi-conglomerate. Similar blocks have not been recorded either by SrorLey or by GaceL, who do not make mention either of any phosphorus-content. Neither does my specimen. What typifies ÄNDERSSON's erratics, is that some. phosphorite blocks contain Upper- Cambrian fossils, viz. Peltwra scarabaeoides Wahlb., Sphaerophthal- 228 mus sp. and Agnostus pisiformis Linn., which proves them to come from a region of Cambrian deposits, which was exposed to erosion during the early part of the Lower-Silurian period. ANDERSSON (l.c. p. 79) himself is wrong in inferring from these erratics, that they come from the very locality whence originated also the boulders they contained, and that at the very least in that place the whole Upper-Cambrian must have been eroded away. I think this need not be so at all, and I even believe that it is most likely not the case, but that the region, from which these Cambrian blocks origi- nate, has to be looked for rather in the vicinity of the original locality of the Silurian erratics. First of all we think of the districts near the coast of the mainland of Sweden to the West and to the North of Gotland. The cementing material also which consolidates the phosphorite blocks, varies more or less from the first-mentioned erratics, as, according to J. G. ANDERSSON it sometimes consists of grey to white spotted coarse-crystalline limestone and sometimes of grey, compact limestone, in which occur a larger or smaller number of rounded quartz-grains, as well as occasionally some glauconite. Among the fossils in the last-mentioned erratics are Orthisina sp., Platystrophia biforata Schloth., Strepula sp. Tetradella sp., Asaphus sp., Illaenus nuculus Pomp., Lllaenus sp., some Bryozoa and other non-deseript fossils. Of all these only /d/laenus nuculus Pomp. was known hitherto from a boulder from East-Prussia, as described by Pomprcks (16 p. 69). The author referred it to the Lower-Silurian period. This rock consists of brownish, coarse-grained limestone with many quartz-granules. Finally we refer to one more erratic block with Strophomena Jenteschi Gag. from the North-Baltieum, recorded by Wiwan (23 p. 103), viz. N° 94 of Ekeby. This boulder consists of red Asaphus-lime- stone and does not contain other fossils. The age of all these erratic blocks could be established, because Strophomena Jentzschi Gag. has been found in solid rock first by ANDERSSON (l.c. p. 77) in the northern part of Oeland, afterwards by Lamansky (22 p. 177) on the Wolchow in Russia and finally by Hourepaur (29 p. 46) in South Norway near Vaekkerö and Töien. Lamansky (le. p. 177) suspects that also the brachiopod, which is recorded by Broécerr (5 p. 50 pl. XI, fig. Va) as a Strophomena rhomboidalis Wilek. from the Expansus-shale and the lower part of the Orthoceras-limestone of South-Norway, is identical to Strophomena Jentzschi Gag. The figure alluded to, is not at all like it, as already observed by HoLTEDAHL. 229 This fossil is rarely but regularly found in Oeland, in the lower- most, glauconitic Asaphus-limestone, in Russia in the three divisions of the zone Bij: (Bur, Bur, and Bi) of LAMANsKY. On the basis of his investigations Lamansky parallels the lower half of By, with the Lower Asaphus limestone of Oeland, but the Strophomena-Jentzschi- conglomerate with the Upper Asaphus-limestone and the Gigas-limestone of Oeland, and with the upper part of LAMANsKY’s zÔne Bir, and with his zône Bru, of Russia, so that from this it follows, that Strophomena Jentzschi Gag. is spread over a larger vertical extent than ANDERSSON could have surmised at first. In South-Norway the fossil has been found in the zône 3 ec. If, therefore, we wish to parallel this erratic block with any of the Lower Silurian strata, it is necessary, in view of the varying petrographical character of the divisions, which deserve consideration, and in view of the different character of each of them in different regions, to find out from which region the boulder most probably originates. According to Linpstrém (11 p. 9—12) Asaphus raniceps Dalm. occurs already in the Lower Gray Orthoceras-limestone of Sweden and is still found in the Upper Gray Orthoceras-limestone. According to Scumipt this species is observed in Russia in the zones B 2b—B 36; according to Lamansky (22 p. 169) in the upper strata of the zone By, up to the lower strata of Bur: Brécerr asserts that it is not quite certain whether they are met with in Norway (5 p. 92). Most probably this erratie bloek does not originate from the mainland of Scandinavia, Strophomena Jentzschi Gag being known there only in South Norway. Moreover the rocks from those zônes differ from our boulder. . Likewise the Russian Silurian need not be considered although the latter fossil also occurs in Russia. It bas not been observed yet to the west of Reval. In that region only Bui, of the zone B exists and this division consists of calcareous sandstone. Our boulder, therefore cannot come from the East-Baiticum. As has been said, only one erratic block with Strophomena Jentzschi Gag. from the North-Balticum is known. The petrographical character of it does not agree with this specimen. Boulders of grey limestone have been found there, indeed, which belong to the Asaphus-lime- stone of Wiman and may therefore be of the same age. In Oeland the Lower Asaphus-limestone consists of limestone partly containing glauconite and partly free from that mineral of which 230 the first may agree pretty well with SroLLev’s boulders, but neither of them agree with the Groningen specimen, especially as regards the amount of quartzgrains. The presence of erratic blocks with Strophomena Jentzschi Gag. in Oeland, Gotland and Gotska Sandön leads us to consider also the localities of the Baltic west and north of the last two islands. It is true, the erratics found there, differ largely from the Groningen boulder; still this district is presumably to be considered as their original site. SrouLey and ANDerssoON do the same for their blocks, while the assumption also seems warrantable of the presence of similar erratics in East-Prussia, notably the one described by Pomprcks and the Spittelhof fragment recorded by GAGer. Probably this specimen must be considered to originate from a narrow slip of the Baltie, a little north of Gotska Sandön and at a short distance West of Gotland. From the foregoing it appears therefore, that the place of origin cannot be assigned more accurately, so that we cannot say for sure to which division of the zone Aj,; the boulder belongs. It is therefore, like the Strophomena-Jentzschi conglomerate to be classed provisionally under By). jalcareous Sandstone with Asaphus raniceps Dalm. In the Tuinbouwstraat one boulder was found among the many erraties that, judging from the fossils it contains, must be included among the Lower-Silurian. It is however of a peculiar petrographical character, as it consists of rather hard, fine-grained sandstone with a caleareous cement. The like of it appeared to be quite unknown in the literature of erratics. This erratic block has about the size of a child’s head and its primitive colour was gray to bluish-gray, as may still be observed from the inner part; the outside, however, shows a discolouration to brownish-yellow. For the rest it has suffered little from weathering. The quartz-grains are small, all but colourless and rounded. | did not encounter glauconite, but only some grains of calcite. The rock also contains a few pieces of more or less rounded, coarse-grained limestone, black at the periphery, white in the centre. These frag- ments, which moreover contain a large number of brown, rod-shaped bodies, are presumably little rolled boulders since they differ so much from the surrounding roek. However this is still highly problematical. Beyond one specimen of an Orthis-species this block contains a 231 small but complete pygidium of Asaphus raniceps Dalm. (Length 74 mm, breadth 11 mm) and numerous other indeterminable fragments of Asaphids; i.a. a fragment of an hypostome. In the description of the previous boulder I have already com- municated something about the occurrence of Asaphus raniceps Dalm. in the Lower Silurian deposits in Scandinavia and Russia, so that I now merely refer to it. From the above it appears, therefore that this block is to be classed under the older strata of the Lower-Silurian, specifically under one of the divisions equivalent to the Swedish Orthoceras-limestone. However in Scandinavia or in Bornholm no solid roek is known resembling this rock in any way. Starting from Reval, Bui, of LaMANsKky has developed itself as a calcareous limestone in the Western part of Estland. Fragments of this rock also occur on the beach of Odensholm, so that up to that locality at least this division retains the same petrographical character. There it has sunk already below the sea-level. Having no control-material of this rock I am unable to ascertain its similarity to this boulder. Moreover some boulders have been discovered, which, being composed of limestone, contain a variable amount of rounded quartz- granules and agree in age with yz, as may be gathered from the description of the previous species of erratics. I therefore believe that this piece is to be considered as a quartz- rich variety of the limestone with Strophomena Jentzschi Gag. and of the Strophomena-Jentzschi-conglomerate, especially because in the previous block also occurs a pygidium that belongs to the same Asaphus-species. When examining the fragment more closely with regard to a possible phosphorus-content, both the rock itself and the foreign enclosures distinctly proved to contain at least some phosphorus. The latter, however, did not give off any smell of bitumen when particles were knocked off with the hammer. Furthermore, because they are not fossiliferous, we cannot determine whether these fragments of limestone, as is the case with the erratics of the Strophomena-Jentzschi- conglomerate examined by ANDERSSON, are to be included ander the Cambrian. Most likely the original locality of this erratic block is that slip of the Baltic which covers the prolongation of the calcareous sand- stone in Estland and continues along the North side of Gotska Sandön as far as West of Gotland, thus comprising the region, from which the Strophomena-Jentzschi-conglomerate originates. 232 Limestone with Dinobolus transversus Salt. A piece of fine grained-crystalline limestone having become brownish-yellow through weathering and of about the size of a fist, contains a dorsal valve of Dinobolus transversus Salt. (1 p.59 pl. V fig. 1—6), which in spite of its extreme thinness has been preserved in admirable perfection. This boulder also, which was also found in the Tuinbouwstraat at Groningen, is a completely unknown species of erratics, as the fossil mentioned just now was not met with in any other country. The length of the valve is 3 cm, the largest breadth, across the centre, 4.2 cm. The almost straight hinge margin is 3.3 em. long. The dorsal valve is almost quite flat and reveals on its surface numerous, very faint, concentric lines of growth and an extremely fine radial striation. Whether there are“small spines ou the outer surface, as indicated by Davipson (le. pl. V, fig. 3 and 3a) cannot be made out. Of other fossils this boulder contains besides a number of detached portions of crinoid stems also a valve of Pholidops implcata Sow. (1 p. 80, pl. 8, fig. 13—17) and a valve of Beyrichia Jonesi Boll (13 p.-13, pl. I, fig. 10—12) and a pygidium of Proetus concinnus Dalm. (Pope lS 18rspan 4. pk, LV a fies, AOS ap Ane tig. 5). From all this it appears, therefore, that the block belongs to the upper Silurian, the zône being undetermined yet. Pholidops implicata Sow. contrary to Pholidops antigua Schloth. is probably quite unknown in our upper Silurian erratics as well as in those from Germany and Denmark, which is perhaps due to the fact that various authors have considered the two as synonyms (7 p. 96, 10 p. 173). It appears however, as Mosrre and GRÖNWALL (24 p. 30) have shown, that they were used for fossils which indeed are closely allied to each other but also form a distinct contrast. Only Krusow (6 p. 245) records that Pholidops implicata Sow. (= Crania wmplicata Sow.) is very abundant in West-Prussia in the boulders of the Upper-Silurian Beyrichia Limestone. I think however that he also refers here to Pholidops antiqua Schloth. In solid rock Pholidops implicata Sow. is known only from the island of Gotland (from the zÔnes c—h of Linpsrröm (12 p. 13). Muntue (27 p. 12—13) mentions the fossil from the layers 2—4 distinguished by him and Van Horprn (25 p. 125) from y and es Linpstr6m, also, records the occurrence in Schonen (l.c. p. 26), but MoBerG and GrOnwALL state that the species there differs distinctly 233 from the Gotland species and resembles Pholidops antiqua Schloth. It is not known as yet which of these two fossils occur in Oesel and in Estland. Beyrichia Jonesii Boll. has been reported by Kirsow (13 p. 13 pl. II fig. 10—12) of Gotland from LiNDpsrRÖM’s zÔnes c—h, by VAN HoePeN (l.c. p.132) from his zône s®‚. At Schonen this fossil has not been found, and nothing is known of it in Oesel and Estland. Proetus concinnus Dalm. is mentioned only by Scamrpr (I. c. p. 44) from the Lower Oesel stratum (zône J) of the Russian Baltic provinces, of Gotland by Linpstrém (12 p. 3) from the zônes c—e, and by Van HoePeN from his zÔne y (near Mulde) (1. e. p. 142). Moreover this fossil has been found with Beyrichia Jones Boll in boulders, associated with Leperditia Baltica Kichw. and Beyrichia spinigera Boll. (2 p. au, Lf p.. 902). I, therefore, feel justified in assuming, that this boulder probably agrees as to its age, with the Lower Oesel stratum of the Russian Baltic provinces. In Oesel the Lower Oesel stratum consists almost entirely of blue marl and dolomite. Limestone occurs only in the West of the peninsula Taggamois (+ p. 46). The equivalent layers in Gotland, on the other hand, are composed of marl, marly limestone and limestone, and the equivalent layers on the mainland of Sweden of grapholite-shale, so that this region cannot be considered as the place of origin. Gotland and the part of the Baltic between this island and Oesel and of these probably, first of all, the island of Gotland together with its approximate vicinity is, therefore, in all likelihood to be looked upon as the locality from which our erratic block was derived. Delft, May 1919. REFERENCES. 1. Davinson, Tu. 1864—1871. “British Fossil Brachiopoda. Vol. III: Devonian and Silurian Species.” Londen. 2. Krause, A. 1877. “Die Fauna der sogen. Beyrichien- oder Chonetenkalke des norddeutschen Diluviums.” Zeitschr. d. d. geol. Ges. Bd. XXIX, S. 1—48. 3. AnceLin, N. P. 1878. “Palaeontclogia Scandinavica. Pars I.’ Stockholm. 4. Scumipt, F. 1881. “Revision der ostbaltischen silurischen Trilobiten, nebst geognostischer Uebersicht des ostbaltischen Silurgebietes. Abt. 1” Mém. d. l'Acad. Imp. d. Se. de St. Pétersbourg, 7e Ser., T. XXX, No. 1. 5. Broéeeer, W. C. 1882. “Die silurischen Etagen 2 und 3 im Kristianiagebiet und auf Eker etc.” Kristiania. 6. Kiesow, J. 1884. “Ueber silurische und devonische Geschiebe Westpreussens.”’ Sonderabdr. Schr. d. naturf. Gesellschaft in Danzig, N F., Bd. VI, S. 205 —303. 234 7. Roemer, F. 1885. “Lethaea erratica, oder Aufzählung und Beschreibung der in der norddeutschen Ebene vorkommenden Diluvialgeschiebe nordischer Sedimen- tär-gesteine.” Palaeont. Abhandl. herausgeg. von W. Dames und E. Kayser. Ba. WEL 5: 8. Horm, G. 1885. “Bericht über geologische Reisen in Estland, N. Livland und im St. Petersburger Gouvernement in den Jahren 1883—1884.”’ Sonder-Abdr. Verhandl. d. Kais. Miner. Ges. 9. Linpsrröm, G. 1885. “Forteckning pa Gotlands Siluriska Crustacéer.” Ofversigt af Kongl. Vetensk. Akad. Förh. No. 6. 10. Verworn, M. 1885. “Ueber Patellites antiquus Schloth.” Zeitschr. d. d. geol. Ges. Bd. XXXVII, S. 173—176. 11. Linpsrröm, G. 1888. “List of the Fossil Faunas of Sweden. I. Cambrium and Lower Silurian.’”’ Stockholm. 12. Linpsrröm, G. 1888. “List of the Fossil Faunus of Sweden. IL. Upper Silurian.”’ Stockholm. 13. Kiesow, J. 1888. “Ueber Gotländische Beyrichien.’’ Sonder-Abdr. Zeitschr. d. d. Geol. Ges. Bd. XL. S. 1—16. 14. Moere, J. Cur. 1890. “Om en afdelning inom Olands Dictyonemaskiffer sasom motsvarighet till Ceratopygeskiffern i Norge. Samt Anteckningar om Olands Ortho- cerkalk.”” Sver. Geol. Unders., Ser. C, No. 109. 15. Gacer, C. 1890. “Die Brachiopoden der cambrischen und silurischen Geschiebe im Diluvium der Provinzen Ost- und Westpreussen.” Beitr. z. Naturk. Preussens, herausgeg. von der phys.-dkon. Ges. zu Königsberg, No. 6. 16. Pompecks, J. F. 1890. “Die Trilobitenfauna der Ost- und Westpreussischen Diluvialgeschiebe.”’ Beitr. z. Naturk. Preussens herausgeg. v. d. phys. oekon. Ges. zu Königsberg, No. 7. 17. Krause, A. 1801. “Beitrag zur Kenntniss der Ostracoden-Fauna in silurischen Diluvialgeschieben.” Zeitschr. d. d. geol. Ges., Bd. XLII, S. 488—521. 18. Scumipt, F. 1894. “Revision der ostbaltischen silurischen Trilobiten Abt. IV.” Mém de l'Acad. Imp. des Se. de St. Pétersbourg. Ye Ser., T. XLII, No. 5. 19. ANDERSSON, J. G. 1896. “Ueber Cambrische und silurische phosphorit- führende Gesteine aus Schweden”. Repr. from. Bull. of the Geol. Instit. of Upsala No. 4. Vol. II. Part 2, 1895. 20. Srottey, E. 1898. “Hinige neue Sedimentiirgeschiebe aus Schleswig-Holstein und benachbarten Gebieten”. Schriften der Naturw. Ver. f. Schleswig-Holstein. Bd. XI. 5. 1338 —148. 21. Scumipt, F. 1901. “Revision der ostbaltischen silurischen Trilobiten. Abt. V. Liefr. 2”. Mém de !’Acad, Imp. de Sc. de St. Pétersbourg. 8e Ser., T. XII, No. 8. 92. LAMANSKY, W. 1905. “Die aeltesten silurischen Schichten Russlands. Et. B.’ Mém du Com. Géol., N. Ser., Livr. 20. 23. Wiman, CG. 1907. “Studien über das Nordbaltische Silurgebiet I1’’. Bull. of the Geol. Instit. of the Univ. of Upsala. Vol. VIII, 1906, p. 73—188. 24. Moer, J. Car. and GRÖNWALL, K. A. 1909. “Om Fyledalens Gotlandium”’ Medd. fran Lunds Geol. Faltklubb., Ser. B, No. 8. Lunds Univ. Arsskrift, N. Afd. 2., Bd. 5, No. 1. Kongl Fysiogr. Sällsk. Handl., N. F., Bd. 20, No. 1. 25. Horren, E. C. N. van. 1910. “De bouw van het Siluur van Gotland”’. Dissert. Delft. 235 26. MopereG, J. Gur. 1910. “Historical—Stratigraphical Review of the Silurian of Sweden”. Sver. Geol. Unders. Ser. C. No. 229. Arsbok 4. No. 1. 27. Muntur, H. 1910. “The Sequence of Strata in Southern Gotland’. Geol. För. 1 Stockholm Förh., Bd. 32. H 5. Guide des Excursions du Congr. Geol. en Suéde, No. 19. 28. Jonker, H. G. 1913. “Verslag der tweede excursie der Geol. Sectie naar Gaasterland, Groningen en Noord-Drente onder leiding van de leden J. H. Bon- NEMA en H. G. JonKER’’. Versl. der Geol. Sectie, dl. I, blz. 65—93. 29. Hourepant, O. 1916. “The Strophomenidae of the Kristiania region.” Vidensk. Skrifter. 1, Math. Naturw. Klasse, 1915, N°. 12. 30. Kruizinca, P. 1918. “Bijdrage tot de kennis der sedimentaire zwerfsteenen in Nederland”. Dissert. Groningen, Verhandeling v/h. Geol. Mijnb. k. Genootsch. voor Nederl. en Koloniën. Geol. Serie. dl. IV. p. 1— 271. 16 Proceedings Royal Acad. Amsterdam. Vol XXII. Astronomy. — “Theory of Jupiter's Satellites. IL. The variations.” By Prof. W. pe Sitter. (Communicated in the meeting of June 28, 1919). We still restrict ourselves to the non-periodic part [ 2;| of the perturbative function, as in the determination of the intermediary orbit.') The quantities hj, k;, »;, w;, which were zero in the inter- mediary orbit, are now determined by the equations (23), *) of which the solution is given by (24). For the determination of the five values of 8,7 we have the determinant (28). Then c’;, and c';, are determined from (27) and cj, and c'’;, from the first and last of (25). The expressions for the coefficients a;;, a';;, bis, bis, dis, d'; 5, Eij, C'ij are given in Vol. XII, Part I, of the Annals of the Leiden Observatory, page 31. Then we have Aig = 2) (aitaig + bids), Bij= Zi(eiubij + brveiy), Cig = 21 (Ciraiy + esids;), D;; = 21 (ai bi5 + eters): The details of the computation of the quantities a;;, a’;;, ete. and A;;, Bij, ete. will be published in the Annals of the Observatory at Leiden. Here we shall only give the results. The determinant (28) is: (see formula A, next page). The coefficients are given in units of the eighth decimal place. Denoting the columns by roman, and the rows by arabic numerals, we now perform the following operations: add £.(V)+2.(VI) to (VID pes (Ve Sen URE (AVP See) =d. (1) TG) ” 2. (7) Su (6) ” (5) ee (Oi 2 5 Gi 4) The determinant then becomes: (see formula B, next page). 1) See Outlines of a new theory of Jupiter’s Satellites, These Proceedings, Vol. XX, p. 1289—1308, and Theory of Jupiter’s Satellites. 1. The inter- mediary orbit, These Proceedings, Vol. XXI, p. 1156—1163. 2) “Outlines”? p 1301. The definition of hi and k; is slightly different here, in consequence of the introduction of e; instead of ni. We now have e; cos gi — @: + hi ei sin gj = hy. 237 eet on Cr Oro!) © © 10°0 3 — 89° LIE 86'6LZ1 80% 10°0 Z9°GZ 6£°19E IL 0 000 + 0 SEO se sf] — LELO 0 sd —OLEsE + 0 000 + 2 0 cho tt = 0 Oke 0 O9GIL + = ooo + 00'0 te Lo'ssl — 860 = 4 ZOL PI'819 = 80°8601 — sf — 90'0EG + 100 — 00°0 = SETI + 130 = G6PIL + L8'Z = Kid reen 81811 4 x + +- + Hie ONS oie rd == — 82°L8012 + Mica == Lp zie rd al at 0 0 0 0 cr 0 ar Seon 8008 + LEPOL — ee sf — Leceeiz + + + 3 — 87 Lsol2 + 1G°Gz LL'V + I at EEL — SE6LL + LSLOPI — OL'1 in <9'eg Sj 3 — €6€6be2 + TOEBEH gases Oe Gi EE6r — L810z + Os iste — sf) — 9S LOEIE + IL‘cOl EL 9p + Lo Ltée at I Tr EZLOT — 63 + 99102 + i OGLE 4 so] — Gh'GOLZ@z + Bold ae 0 IE 8 de 602L6 + 99BELE + bE'0 +/4 ETL = LEE = sf — 687682 + 0 =F 8, St 6 1L6 — ELIES + ovo HV AAA ar 6L'OLI + s/ — 6900162 + * de) _ 238 Consider the determinant consisting of the elements of the first five rows and columns. This can be conceived to be the result of the elimination of y; from the linear equations p ZjFip 4; 8 y= rde) where Hi; is the element of the 7 row and the j column. The new unknowns y; are connected with the original unknowns ce’; and c"; of the equations (27) by the relations: Ge Vi EU (HL A) ¢," =Ys + SY + dy, c = Ye + 2 Cn VE To each of the five roots 6*, corresponds a set y;, (2 — Î...5). Then y,, y,; and y, are determined by aj Fijyj B yi= 0. Gn Al Oe We now determine 2; from F'55 = Fos + 2; tj F'5;, G,j—=1...4) Fis + 2; 0; bij — wi F's5 = 9, and we put | ee I el Gi =a) and Yi = 2i + Bi Ys. (@= 1... 4) Then z and #? are determined by the equations (a) Zj F525 — B29, (Cs) eet) and y, is given by Zj Faj aj + (F's5 — 8’) ys = 0. (j= 1...4) This determines four roots 8. The fifth is given by p= Bin and the corresponding values of z; and y, are iti 1 ’ ie |S Ek 4) To solve the equations («) we take z,,= 1. Since the coefficients of the diagonal #';; are much larger than the others, we can take Then the quantities € and n;, are small, and we easily find a set of equations from which they can be conveniently determined. 239 We find F',, = + 1025.18, and the determinant of the four equations (@) is 130250342 <= 263-42 Eede +23296.87— B a 1.72 = 8396 + 0.33 + Ld 214,29 Seriya 289.04 CAMEE yi LOVERS lo B 42651 34.50 +21087.28—B? The five roots 8,?, determined in this way, are (still expressed in units of the-8 decimal place): B? = + 30261.15 8,7 = + 23299.79 =d 2137451 B = + 2108419 B =H 1025.18 We now put, for g=1...4, Wo = Bj Tt Ss ee we ear aes Se Hi See where To VWs ea 0 00 Then =,= gt + Wy. are the longitudes of the “proper” perijoves, aap, Ok ‚4 are the arguments of the inequalities of group II, and w, is the argument of the libration. The mean motions of these arguments are, in degrees per day '): y, = 0°148668 7, = 0 039842 7, = 0 .006949 y, = 0 .001862 B, = 0 16347 El. and Masses. 0° 14407 0 .039593 0 .007046 0 .001864 0 .16252 These values are not yet final, since we have neglected: 1. the effect of the periodic part k;—[R;]| of the perturbative function, 2. the squares and products of the quantities hi, £;,v;, w;, 3. the inclinations of the orbital planes of the satellites on the plane of Jupiter’s equator. Apart from the corrections which must eventually be applied 1) The motions yi and #; are siderial ones, and do not contain precession. Accordingly the precession ‘has been subtracted from the values of yi as given in El. and Masses. 240 later on these three accounts, the values of % and y; as given here are certainly exact and complete to the last decimal place given. The values added for comparison are those of my theory of 1908 °), reduced to the masses adopted here. This theory is SOUILLART's with some errors corrected. The computation of the motion of the argument of the libration, however, was carried one order further than was done by SourLLART. The values of the coefficients cjg, C'iy, ¢'ig and cj, are: 9 1 2 3 4 5 cig + 0.96868 + 0.02754 +4 0.02479 + 0.00230 — 0.00054 fay — 04420 A ogden Re ATB — 01h84 AE 00032 c3g — 00686 + 03813 + .98970 + .08804 + .00004 tg + .00006 + .00012 — .12098 + 1.00000 „00000 cig +0.96038 + 0.05057 + 0.02483 + 0.00229 — 0 00287 da me ONO4A 42 OTT EE Ahab — DTA0D Beenie cso — 00637 + .04010 + .99952 + .08891 +4 .00018 Esp + .00006 + 00012 — .12098 + 1.00000 00000 c'ty + 2.7649 . + 0.9617 — 0.07900 —0.00718 + 0.15668 Slant x COBO. ci 1.8535. ht 57340. 19 05039 OE oly 1400800.) 1 0.7824 .— 11738 I— 01082 AS 02150 oe .0000 0000 + 00001 „00000 „00000 e'ig + 0.01195 + 0.00363 — 0.00028 — 0.00002 + 0.00012 d'or — 02934 + ..01486 + .00405- + 00085 — .00048 er = 200052 == 01118 SR 00149 00013 00007 cae 0 0 0 0 0 If we neglect the squares and products of &, (as well as products ejé,), and if we put Wig = st 4 (Gq Frei), where the lower sign must be taken if either z or q, but not both of them, are 2, then the effect of the variations on the radius- vector and the longitude is: dr; 3 pe aa sat C lig Eq COS Wg — Eg Tig Eq cos (Ai; — Wy) — — } Zy (Cig —C'ig) & cos ( + Wo) Swi Eg eig Eq sin Wy + 2 Ly tig egsin (Mi) + Ly (Cig — ig) eq sin (li + Wy), where we have put On the Masses and Elements of Jupiter’s satellites and the mass of the System, these Proceedings, Vol. X, pp. 653 and 710. 241 Ai = doo + Mio + (ei — 2) 7, ler. The first term for g=1... 4 gives the inequalities of group II, and for q=ö it represents the libration. The second term for q=—1...4 represents the equations of the centre. We have g 1 2 3 4 tty -+ 0.96453 — 0.03905 +0,02481 + 0.00229 T29 + .03186 + .95599 + 16287 + 01492 T3g — .00661 — „08662 + (99461 + .08847 Tag + „00006 — „00012 — .12098 + 1.00000 For ¢=5 this term is better written in the form — 3 (is + 15) &5 cos (l; — U'5) + (¢i5 + €15) €5 sin (l; — Ys). It then represents, like the third term for all values of yg, small periodic inequalities, whose periods differ little from that of. the equations of the centres. It should be pointed out that the theory here given (intermediary orbit and variations) covers the same ground as SourLLart’s, with the exception of the small periodic perturbations and the terms of very long period arising from the action of the sun, Saturn, etc. SovuILLART does not give any term of the perturbative function, nor any inter- action of two terms leading to a term of higher order, which is not taken into account here too; and he omits many terms which are included here. The above theory is certainly complete up to the numerical limit of accuracy which was fixed beforehand. This can certainly not be said of Souirrart’s theory, though it generally gives many more decimais. The new theory has proved eminently suitable for numerical computation. Physiology. — “Tonic reflexes of the labyrinth on the eye-muscles’’. By Prof. R. Magnus and A. pr KreIJN. (Communicated in the meeting of June 28, 1919). It is a well-known fact, that the labyrinths strongly influence the position of the eyes in the orbits. Each position of the head namely, corresponds to a special, definite position of the eyes in the orbits. The inquiries concerning the relation between the position of the laby- rinth and that of the eye is highly impeded by spontaneous movements of the eyes and it is, therefore, easy to understand that more elabo- rate investigations were made almost exclusively with animals, which make but few such spontaneous movements. Among the animals, which are usually experimented on, the rabbit is preferred to all others. | In 1917 an extensive publication on this subject was issued by the pharmacological Institute *), in which the question of the relation between the positions of the labyrinth and of the eye was examined as completely as could be. Starting from a primary position of the head with a horizontal mouth-opening and the lower jaw pressed downward, other positions of the head appeared to involve both constant vertical deviations of the eye and rotatory movements; as for side-movements, in the direction of the eye-opening, no reliable data could be found. The vertical deviations of the eyes always take place in opposite directions for the two eyes, whereas the principal deviations were found with the head in the two side-positions, when the upper eye deviated as much as possible downwards, the lower eye in the same degree upwards. The rotatory movements always take place in the same direction for both eyes, the greatest deviations were found, when the head stood with its muzzle vertically upwards or downwards. During the expe- riments the rabbits were put immovably in various positions with regard to their surroundings, when special care was taken that any shifting | of the position of the head with regard to the trunk was out of 1) J. v. pb. Hoeve und A. DE KLEIN. Tonische Labyrinthreflexe auf die Augen. Pfliigers Archiv. 169. 241. 1917. 243 the question (reflexes of the neck). The various positions of the eyes were determined kinematographically. Minute measuring was made possible by indicating on the photographic plates the shifting of a cross, burned into the cornea anaesthetized by cocain, with regard to a fixed system of coordinates, photographed at the same time. The results were indicated for the vertical deviations of the eye and for the rotatory movements separately, to wit for three per- pendicular rotations of the head round 360° each. In this way we only get to know, of course, the influence of the head, i.e. the labyrinths on the position of the eyes. However, for a minute analysis of the influences of the labyrinths it is desirable to know the influence of the labyrinths on each eye-muscle. Now, as different positions of the head often bring about a com- bination of vertical deviations of the eye and rotation movements, which combination, in its turn, variously modifies the points of insertion of the eye-muscles, it stands to reason that we cannot say beforehand that the greatest deviation of the eye-ball either upwards or downwards, or the full extent of any of its rotations, necessarily implies the maximum lengthening or shortening of the eye-muscles (recti and obliqui). This made it necessary to investigate what position of the head produced the maximum and minimum shortening of the eye-muscles. The above-mentioned inquiry had clearly brought out in what way each position of the head influences the position of the eyes, so that the only thing left fo be done now, was construction of a proper model of an eye, putting the eye-ball of this model in the various positions which had been fonnd, and measuring the length of the six eye-muscles for each position accurately. A short time ago the anatomical relations of the eye-muscles of rabbits were given in detail by Wesserry '). However the accom- panying illustrations do not give us the numbers expressing those relations. For this reason a minute inquiry was made with various rabbits with regard to the length of the eye-muscles, size of the eye-ball, place of insertion for each muscle, etc. and in accordance with this the instrument-maker of the Institute, Mr. F. A. C. Imnor, made a model of an eye-ball with eye muscles to correspond. Starting from the primary position of the eyes and guided by the information obtained before with regard to rotatory movements and vertical deviations of the eye for different positions of the head, now 1) K. Wessety. Ueber den Einflus der Augenbewegungen auf den Augendruck. Arch. f. Augenheilkunde. 81. 111. 1916. 244 the eyes of the model were placed in the corresponding position, so that now the lengths of the 6 eye-muscles for these various positions of the eyes could be measured. In the experiments made before, three rotations had always been performed. Rotation I. Tbe animal originally in ventral-position with horizontal mouth- opening, Rotation of the animal round the bitemporal axis, direction of the rotation head downwards and tail upwards. Rotation II. . The animal originally in ventral position, with horizontal mouth- opening. Rotation of the animal round the occipital-caudal axis. Direction of the rotation: Right eye downwards. Rotation III. The animal originally in side position, left side downward, right eye upward, vertical mouth-opening. Rotation of the animal round the venter-dorsal axis. Direction of the rotation: muzzle downwards. For each rotation the position of the eyes was stated accurately after every 15°. The result of the measurements of the lengths of the eye-muscles of the model for the various positions of the eyes was stated in tabular form. However, the publication of these tabular statements must be put off for the moment, as some correction appeared to be necessary. For when the eye, starting from the normal position, performs rotations, unaccompanied by vertical movements, these rotations of the eye-ball cause the points of insertion of the rectal eye-muscles on the bulb to be removed, by which the length of the rectal eye- muscles is changed passively. However, when a rotatory movement combines with a vertical deviation of the eye, the contraction of the rectal eye-muscles does not take place with the length of those muscles of the normal position of the eye, but with the length they have got by (after) the rotation (contractions of the oblique eye-muscles). So when the eye has performed rotation the lengths of the eye-muscles must be rectified with a value, in accordance with the passive lengthening or shortening, caused by the contraction of the oblique eye-muscles. At the same time, of course, the lengths of the m.m. oblique at different vertical deviations should be rectified with a value, in accordance with the passive lengthening or shorting of those muscles caused by the contraction of the rectal eye-muscles. With the help of the model it was easy enough now, by first putting the eye in the normal position and stimulating either rotatory movements or vertical deviations of the eye-ball exclusively, to 245 state for various lengths of the oblique eye-muscles, the passive lengthening or shortening of the rectal eye-muscles, or conver- sely, for various lengths of the rectal eye-muscles the passive lengthening or shortening of the oblique eye-muscles. The tabular statements, corrected according to this system, have been reproduced as curves on fig. 1—4. So these curves represent the rectified lengths in m.m. of the 4 eye-muscles (both oblique and recti super. and infer.) for the three rotations of the head above-mentioned. For the obliq. superior only the distance from trochlea to the insertion on the bulbus has been reproduced. From all this we learn that curves of the obliq. superior and obliq. inferior form a true reflexion of each other, that is to say that, at the tonic reflex of the labyrinth these muscles act as antagonists, the lengthening of the one brings about the shortening of the other, and conversely. At the same time the curves of the m.m. recti superior and inferior show that these muscles too are absolute antagonists. If we compare the curves of the oblique eye muscles (fig. 1 and 2) with the curves, found at a former period for the rotatory movements of the eye, we see, that they agree with regard to the principal points. Especially the positions of the maxima and minima do not show any essential difference; the rotatory move- ments and the shortening of the oblique eye muscles are greatest when the head with its muzzle points vertically upward or downward. At the same time the curves for the rectal eye-muscles (fig. 3 and 4) agree with the curves, found before for the vertical deviations of the eye. Only the shape of the curve of the eye- muscles at rotation III is a little bit more pointed than the shape of the curve, found for the vertical deviations. However the position of the maxima and minima undergo no essential change. The maximum contraction of the two rectal eye-muscles takes place when the head is almost in side position. Now, comparing the curves of the obliqui and recti, we find the following : At rotation I (—) the obliqui react strongly whereas the recti hardly perform any movement. So, at this rotation we find no vertical deviations, but almost all of them are rotatory-movements. On the other hand, at rotation Il (—.—.) the obliqui hardly react at all, whereas the recti superior and inferior perform strong movements; so for this rotation the vertical movements prevail, whereas rotatory movements do not take place. 246 OBLIQUUS SUPER. i A woe Bd Hennesee 2 3 4 NW TEAN SS NS area 19 ry 20e al 24 25 is) 5 30 45 eo 7 90 Ios 120 135 ISO I8S 180 195 210 225 240 2SS 270 285 300 dis 350 345 367 Fig 1: NE DR. 1 OBLIQUUS INF. (KORRIGIERT) | 2 3 4 5 6 7 8 3 On 2 13 14 IS 16 17 18 19 2! De eo cemnes i) ib 30 45 80 75 90 105 120 135 150 IGS 180 19S 210 225 240 255 270 zes 300 315 330 345 JEF Fig. 2. RECTUS INF. (KORRIGIERT) „We kee EEDE pl 5 12 Sei lit 18 ISO! 24 3s , és 60 rs so B i20 38 So 165 (go 135 210 225 240 255 270 285 300 a8 $0 345 360 Fig. 3. 22 20 o- : mm 28 DR. 1 RECTUS SUPER. … lo (KORRIGIERT) WP See gp « Ast EREN PPP bd 24 22 cf 2 3 4 S 6 ils) 12 20 ° B 90 45 60 75 90 Is Bo 135 iso 165 160 195 210 225 240 255 270 285 300 4s 330 ius zeer Fig. 4. Only for rotation III (—.—.—.) we find a combined reaction of the obliqui and the two recti. From this we may safely conclude that, at rotation 1 the labyrinths influence almost exclusively the obliqui, at rotation II almost exclu- sively the rectus superior and inferior and at rotation III all four eye-muscles. j We cannot yet enter into details about the curves, but will do so afterwards, when determining how far the tonic reflexes of the labyrinth on the eye depend on definite parts of the labyrinth special of the otolithes. Be it sufficient to indicate here that for the oblique eye-muscles the curve of rotation I shows an asymmetric course, whereas for the rectal eye-strings the same thing takes place at rotation II. The exactitude of the former definitions and the measurements now performed, may be derived from the comparison of the corre- sponding points on the different curves. For the three different rotations namely, it occurred several times that the same potition of the head was reached from different directions. The curve shows that, notwithstanding this, the lengths found for the eye-muscles agree wonderfully. Corresponding points are among others: Normal position. Rotation I No. 1 and 25. Rotation II No. 1 and 25. Back position. Rotation I No. 13. Rotation II No. 13. Side position (Left). Rotation If No. 19. Rotation III No. 1. Side position (Right). Rotation II No. 7. Rotation III No. 13. Muzzle upwards. Rotation I No. 19. Rotation IIL No. 19. Muzzle downwards. Rotation I No. 7. Rotation III No. 7. For all these positions the four eye-muscles measured have almost exactly the same length. Conclusions. From this and the inquiry published before we may conclude 248 the following with regard to the tonic reflexes of the labyrinths on the eye-muscles: 1. With the rabbit every position of the head corresponds with a special state of contraction of the eye-muscles and therefore with a special position of the eyes, which lasts throughout the time that the head retains the same position. 2. With the rabbit for the rectus externus and internus no reliable data could be found in the bringing about of these tonic reflexes of the labyrinth. It is especially the rectus superior and inferior which cause the vertical deviations of the eye and the two obliqui which cause the rotation movements. When this happens, the two recti, just like the two obliqui, act as antagonists, on the other hand, changes of the lengths of the recti may combine with those of the obliqui in various degrees. So these two groups of muscles act independently of each other (though of course dependent together on the labyrinths). 3. When the head stands with its muzzle vertically upwards, the two obliqui superiores (right and left) are in a state of greatest contraction, the two obliqui inferiores in a state of greatest relaxation. The upper cornea-poles of both eyes are then rolled forward. When the head stands with its muzzle in a position vertically downwards, the two obliqui superiores are in a position of greatest relaxation, the two oblig. inferiores in a position of greatest con- traction. The upper cornea-poles of both eyes are rolled backward. For all other positions of the head we find contractions, lying between these two extremes, the two eyes always react with rollings in the same direction. 4. When the head is in side-position (left) the right rect. inf. and the left rect. superior are in a state of greatest contraction, the right rectus superior and the left rectus inferior in a state of greatest relaxation. Then the right eye has its maximum deviation downward, the left eye its maximum deviation upward. When the head is in side position (right), the left rect. inf. and the right rect. sup. are in a state of greatest contraction, the left rectus superior and the right rect. inf. in a state of greatest relaxation. For all the other positions of the head we find states of contraction of the rectus superior and inferior lying between these two extremes. Both eyes always react with opposed vertical deviations of the eye. The rectus superior of one side and the rectus inferior of the other side react in the same sense. 5. When we start from the normal head-position and we turn 249 it round the bitemporal axis (360°), it is principally the obliqui which react, whereas the two eyes roll in the same direction. Starting from the normal position and turning the head round the occipital- caudal axis (360°), it is especially the recti superior and inferior which react, and the eyes show opposed vertical deviations. Starting from the side-position and turning the head round its venter-dorsal position (360°), both groups of strings react and the eye-positions are the combined results of opposed vertical devia- tions and rotatory movements equally directed. 6. After extirpation of the labyrinth on one side the vertical deviations of the eye and the rotatory movements continue for both eyes. One labyrinth influences the obliqui of both eyes and the rollings in the same sense; however the recti (sup. and inf.) of the two eyes and the vertical deviations of the eyes are influenced in the opposed sense. For both eyes one labyrinth brings about the greatest vertical deviation of the eye with respect to its normal positions when it is lowest down, whereas the head is in side-position. Then the rectus super. of the same side and the rectus inf. of the crossed side are in a state of greatest contraction. One labyrinth brings about, for both eyes, the greatest rotatory movements by contraction of the oblig. infer., when the head stands with its muzzle vertically downward. On the other hand the greatest rotatory movements of both eyes by contraction of the obliqui superior are brought about by one labyrinth, when the head stands with its muzzle vertically upwards. | | CE end p 4 eo Obl, sup Os e eo oft wf Osse | beer Obl inf IL R eben Keck. sup, epe ing Roect. inf II Boeck. int. O Ben Fig. 5. 250 With one labyrinth the size of these rotatory movements is about half of that of animals with intact labyrinths. 7. For the intact animal it is possible to calculate the changes in the eye-positions by taking the sum of the influences, starting from the right and left labyrinths, on the recti super. and infer. and the obliqui super. and inf. of the two eyes. 8. After extirpation of tbe labyrinths on both sides all tonic reflexes on the eyes, mentioned above, disappear. 9. The minimum number of central courses, necessary for the explanation of the tonic reflexes of the labyrinths of the rabbit on its eyes (so not of the rotations-reactions and caloric reactions), have been drawn in a sketch, accompanying fig. 5. The uninterrupted lines represent the courses of the recti super. and inf., the dotted lines those of the obliqui. Each of the four obliqui is influenced from both labyrinths, each of the two recti (super. and infer,) from only one labyrinth. One labyrinth influences the 4 obliqui, but only the rect. super. of the same side and the rect. inf. of the crossed side. For these tonic reflexes of the labyrinth for the m.m. externus and internus no reliable data could be found. Mathematics. — “On expansions in series of covariant and contra- variant quantities of higher degree under the linear homogeneous group.” By J. A. ScHouren. (Communicated by Prof. J. CARDINAAL). (Communicated in the meeting of March 29, 1919). Notations’). A covariant affinor of degree p may be written as the general product of p ideal fundamental elements’): Pp Marts fe Wer ee aa, BOL ug ei U(L) yaad oe i p P an alternating or symmetrical one as a power of one ideal funda- mental element : 1,...,0 pv = = PU), Oi == WP, pen hed 2 lr ( ) BW ae PEN OLENE Aare Al . P . : . . The p! isomers of u, viz. the p/ products of the ideal factors in pP all possible orders are real rational covariants of u. Each isomer is P formed from u by one of the p/ permutations P, of the ideal factors. N dits : Dat By a penetrating general product o of some affinors u,v,... we pq ; understand any isomer of the general product uv.... An affinor, the different isomers of which are not connected by linear relations, is called a non special one. Classes of isomers’). It is well-known, that the p factors of an isomer can be divided into groups of s,,s,,... factors in one single way, so that in each group the permutation is a cyclic one. The groups are called the permutation regions and the complex of the numbers s,,s,,... in descending order and omitting all numbers 1) See further “Die direkte Analysis der neueren Relativitätstheorie.” Verh. der Kon. Akad. v. Wet. DI. XII NO. 6, p. 7—11. *) Introduced firstly by E. WazLscu under the name of “symbolische Vektoren” in “Ueber mehrfache Vektoren und ihre Produkte, sowie deren Anwendung in der Elastizitätstheorie.” Mon. f. Math. und Ph. 17 (06) 241—280. Er Proceedings Royal Acad. Amsterdam. Vol. XXII, 252 1 the permutation number. For a regions of the same extension we shall write here a.ss. A permutation number is higher than an other one, when its first region is greater or, in the case of equality, when the second region is greater etc. FROBENIUS introduced the name of permutation class for all permutations with the same permutation number. In the same way we shall call the sum of all isomers with the same permutation number divided by p/ the tsomer class with that number. The number of classes is therefore equal to the number & of the whole positive solutions of the equation eo, +2a,+3ea@,4+... nh ae) The classes are arranged in ascending order and written: ey eee Gi p p ,4u——Uu ’ K, u ’ BONG vee ’ Kj. u p! The permutation number may be added as index on the left side eg. for p= 6: Et ’ A ’ sees ’ wah, ’ aie ’ SA u te Ree EN is re A class is called even or odd according as it consists of even or odd permutations. 7 . . " Pp Alternations and mixings. The affinor that is found from u by replacing each of ¢ definite groups of s,,...,s, factors (without displacing them) by the ideal factors of their alternating or symme- . . . . Pp . trical product is called a simple alternation resp. miving of u with the permutation number s,,...., 5, and is written as: (A) p Sirs (A) p A au resp. M u By yeyS t The index (2) at the top on the right indicates the choice of the permutation regions. The affinor is called, locally alternating resp. symmetrical in these regions and in general locally permutable. The sum of all simple alternations resp. mixings with the same permutation number divided by their number is called the general alternation resp. mixing with that number. The general alternations and mixings are arranged in ascending order and written as = p ED eer) Aya Ae den ED MP AE Zep Mum hae Mel ee a eventually, when desirable, with the permutation number as index on the left e.g. for p= 6: 253 When s,,...,s;, is the permutation number of A;, 4; contains de ag aaa simple alternations A;. The same holds for mixings. For a non special affinor all classes and likewise all simple and general alter- nations resp. mixings are linearly independent. When successively , ..,s,4 and "eM are applied to a non special affinor, the result is then and only then zero when more than one factor of a region s belongs also to a region s’. The highest permutation number for which rn P u 515-054 is not always equal to zero is called conjugate to s,,...,s;. From this follows that sne = (t+ p—s, —...—8;) , (8;—1) .t , (Sti —8) . (t—1),...,(8,-8,).2. (5) This relation is a reciprocal one. When the permutation regions of an alternation or mixing are numbered in such a way that a greater region has always a lower number than a smaller one, it is possible, that for all values of « the et? factors of each of the regions are placed in the order of this numbering. In this case the alternation or mixing is called ordered. To an ordered alternation there evidently belongs only one ordered mixing with conjugate permutation number, such that these two operators do not annihilate each other. Then these two are called conjugate. A general alternation and a general mixing with conjugate per- mutation number are called conjugate. Every general alternation or mixing is annihilated by all simple and general mixings resp. alter- nations with a permutation number that is higher than its own conjugate one. For p > 5 the order of the general mixings conjugate to Ag A is not the same as of Mi, M,, for p==0; 6:8. 2Ae ’ 2,243 ’ 3.244 ’ 345 ’ 3,2A6 ’ 23A7 ’ ‚As 14,2Ag ’ 5A10 5 M19 ’ 421 ’ 23M, ’ +Ms ’ 32M ’ 32M, ’ 3Ms ’ 22M ’ 2M: Wiens in … …, ,,Au Dhaene Mu the wnaltered factors are replaced by the ideal factors of their symmetrical resp. alternating product, we obtain a mized alternation resp. an alternated mixing with the same permutation number, written: m p Sips, A p ss 40 resp. Mu. 1 We 254 m a From the operators A and M/ we can form in the same way as m a above general mixed alternations A and general alternated mixings M. As to the order 1,...,%, the independency and the property of being ordered, the same laws hold for these operators as for A, A, M and M. When an A; and a M; with conjugate permutation number do not annihilate each other, the unaltered factors of My; are alternated by A; and those of A; are mixed by M;, so that: m m a Aa A and likewise: A; Mj = Ai Mj = A; Mj = Ai Mj. . oy aH as (7) Expansion of the general alternations and mixings in classes. Theorem I. A general alternation resp. mixing of a non special afinor can be written in one and only one way as the sum of mul- tiples of all classes with the same or a lower permutation number. For a mixing the coefficients of all classes are positive, for an alter- nation those of the even classes positive, the other ones negative. For the same permutation uli their absolute values are Gag A; == =2 J; aj; K; _$+1 for Kj even me sb) da =a 1 gaen odd. (8) Me = S cij K; J The very simple proof of this theorem may be omitted. In order to determine the coefficients «;j we must investigate in how many ways it is possible to choose the ¢ permutation regions s,,..., 5, of an A; or of a M; thus, that each of the f regions s,,.…..,s' of a permutation of a definite A; falls quite within one of those regions. When mij is this number, we have mij p! my p! (a (P—% p—s, — 81-1 Sil eS ile 8,/...81! zn : fy Ba : For p=6 is eg. (see table following page) All A, all M and all A’ being linearly independent, we have inversely : (9) Uij — \ 1,...52 di = By A i ; pias, up! mec apse aera DAE Krt M; ; 255. OE | 098) 1 [pet Hz We * ep 06 | 081 | ZI PE) | > z Wez * spe 06/06 |gilgit\o | 9 oo + | 1 Wee “ vyze 02 | 021 | FP pet) 0 0 EG) . 9 We “ spe 09/09 |91/91I+| F| p | 0 ee allee Wee * ope 0804 (8 8 Xi Pb? | 0 Cece Gee, dake he Wee “ Lpee GI) KCE | ALPEN 140 ea Ae: a 0 We ae SW, “ Spr GP Sib L ec | See er ERE Sales: | Malte F0 IT edel Ole Wer “ _6pzr NCN A ARE Boen Seleen 0 Tet 0 pel We “ ops le | bese cesta PNR EA) nn OI a al: wo "dsor i1p9 EMD us| Vy | ue) wu yu Py | we | Sy | Mu | Ary || Sy | | Oy | | Uy | wz | ty | \os—ts—d7 \ls—d) \d id 256 Expansion of an affinor in alternations or mixings. Thus we have especially for the sum of all classes M/, and for A; : ee iof ae My = — & yi Ai 3 ee ee =» (2) A7 Vki M; i By the application of MM, to the upper equation we learn that Yik = 1, so that when we consider My and A, resp. as o-th general alternation, A, and as o-th general mixing M, we have, / being the identical operator, and because A, = M,= 1: Cee) ea Ol ke ca = = Vii Aji > yu Mi. ee ae © SR (35) 1 t In the same way we can prove: : m 0, igh U == >3 Yk A= = ee MIEREN ee ee ap (14) a For a permutation number s we have in these expansions : vei == (Ii (p—s+)). « - . + (15) and for a permutation number a. 2: ve) Thus we have obtained the theorem: Principal theorem A: Every non special affinor can be expanded in one and only one way into general alternations, into general mixed alternations, into general mixings and into general alternated mixings with the indices 0,2,3....,k. So we deduce e.g. from table (10) for p= 6: (16) 6 a — En == = En == = i= ( Ae eae 4102 Ae Gare (45> ye 6 alee oa, aa “ail ao Sees A, eeN), u (17) 6 = ee on el ume = as = u=(‘M,,—2'M,,—2**M,+3'M,—**M,+6*M,—4°7,+ MM, — En =e b= iG —6*°M,+5°M, HM) u Decomposing the ordinary alternations and mixings into mixed resp. alternated ones we can calculate: 257 m m n= (,4,, — 2,4, + 6 A, + 344, —5 44, + 4inAe 4 Ard m m \ nes as ae 6 +9 sage “oso ade Ae) u 6 Es a En = ah = Ee o—(*M,, —2°M,, + 6**M, + 34M, —5"*M, + 4 MAM, + a a (18) EE ue OE 45 M, 49" M, 45° M, HM u | All expansions remain valid for special affinors but they are then no longer the only possible ones. The numbersystem of the class operators. Fropenius has shown, that the operators K are commutative both mutually and with every permutation /, and that they form a numbersystem with & units, which does not contain any nilpotent numbers viz. numbers of which some power is equal to zero. As k independend units we may choose e.g. K,,..., Kr or A, SA. Aer MM, My, which are thus all commutative both mutually and with each operator P, A or M. According to the theory of the higher complex number systems every system of this kind contains X independent numbers /;,¢=1,...,4, the idempotent principal units (Haupteinheiten) '), which satisfy the equations de foray NN . 9 Fee 0 for iFj 9) The sum of the operators J;, which will be called elementary operators, is the identical operator 7. When these units are expressed in the class operators K: tak ES S wij K; RRT ee ae AEO 4) and when we write K; K; Site IE Di Shy Te” We ce ae so we have Iek Uml = = Umi Umj Lijk « «+ + + + «4 - (22) ty) from which we see that the coefficients u correspond with the group characters x of the symmetrical group as defined by Frosentus’). 1) See e.g. the author's “Zur Klassifizierung der associativen Zahlensysteme'' Math. Ann. 76 (14) 1—66. 2) Berl. Ber. (96) 985—1021, (98) 505—515, (99) 330—339, (00) 516—534 ; see also W. BuRrNsIDE, Lond. M. S. Proc. 33 (01) 146 —162, 34 (02) 41—48, 1 (03) 117—123), 258 The general formula for this correspondence is : Bent eK) LB Thea eS from which follows for un, : Uni = xin? a ee ee es aes (24) From the tables given by Frosenius for the group characters we thus find directly the equations expressing the / in the K eg. for pb: Ki k Kz Ka Koe Rp Rr WENS Ko Ko Ku ane a 1 1 i 1 1 1 ea ed es ees 1 he 1 ui EEE Deh ALT gee Ze ug Ig) MAB Zh OG SE OS. PIS. AB GD eee eg eee ERE ABS AR in BE ia An feos, «RS bag 215 Sa sho whl pa eon leek Beeman Brin Pati NOT alach MRT dace Basie aa Mey er oA Spar lar 9 _217 0 0 0 9 gen Lord 100 20. 20" 30. toe 10 A 0 0 9. eta She Ho. “20 20-20, STO Oe, LO os Dien eo eta Tae Ce 0 eeN O20 OO Oe iCeee *) The indices on the left of the operators / will be explained furtheron. Thus we have found an expansion that is singly determined for every affinor. In fact even for a special affinor no linear relation can exist between the operators / without each coefficient separately being equal to zero, as is seen immediately by application of one of the operators /. This expansion is called the expansion in elementary affinors. Using tables as (10) and (25) we might now m a express the / in A, M, A or M and obtain thusa singly determined expansion with respect to alternations or mixings. The following way however is more simple and more instructive. The elementary operators as the products of two general operators. When M,;, + gh, Mix are conjugate to A: Ei i wea then wale Pee are annihilated by Mj; therefore they cannot contain y, 21 of the principal units 7. In the same way 4,,..., Ar are annihilated by M,, and M;,, so that they cannot contain y,>1 further principal (25) 259 eee | units ete. We thus see, that A, contains only £k—Zy,y21 principal units, from which follows y, = y, =...= y, = 1. Therefore the principal units may be arranged in such way oe 08 En that had EE | Bee Sen Phe woe ee a) L dil The same reasoning holding for the M the principal units may be arranged in such a way Deere eed „er, that Bh he belt JAE Aaen eel) The coefficients of both expansions are equal, but for p >> 5 the operators / do not have both times the same order, e.g. for p= 6, using both indices: A OE EN cpr ENDE TN Furtheron we shall no longer use for the / the indices on the right of p. 257. From (26), (27) and (7) we find the relation: m a sd A, M; 6: A; M; A dele oe) As it is easily proved, that A; M; contains K, just once, the coefficients d’;; are idenfical with the coefficient u; in (20) and the dij therefore equal to the group characters in the first row of Fropenius. We need only know therefore this first row. For the case that 4; —=,A; we have ai dn Re ee Css ZEN p 2a fon) Wea) pms red eS) ie For more general formulae the cited papers of FRopenius and BurnsiDE may be referred to. So we have obtained this theorem: Principal theorem B. Every affinor of the p'* degree can be expanded and for 4;=.2A;: 260 in one and only one way in a series of k elementary affinors. An elementary affinor with the lower index i and the upper j is a quantity that is annihilated by all alternations A', 1 > i, 1 > 1 and by all mixings M,,h>j,h >I. Such a quantity is also annihilated by all A, and M, 5e it can arise from the application of il, A, „As Ms; and M, and it is invariant by application of ‘I, 9;; Apt, p M, and d,, M,. For a definite value of n (number of fundamental elements @) all elementary affinors, for which the permutation number of the A contains permutation regions > n, are zero. For p= 6 e.g. the expansion is: ree. Like fete he ene ee: u—(4,i7,,+254,M,,+814,M,+254,M,+1004,M,+2564,M,+| nk +254,M,+1004,M,+814,M,+25A,,M,+A,,M,) Expansion of an elementary afinor in ordered elementary affinors of the first or of the second kind. Theorem II. Every simple alternation or mixing that is annihilated by all higher ones with the same number of permutation regions of more than one factor, can be written as the sum of multiples of ordered alternations or mixings with the same permutation number that contain in each of their regions only factors from the corresponding regions of the original alternation or mixing. Let us prove this first for an affinor: 0 Pp d as y d mn, m=pgÂn= pleo gv = (U, Up) o (V‚ vg) P When the first factor of m is no u, we may apply a „14, the permutation region of which contains this factor and „u. Then we = may write ,4:4m as the sum of p +1 terms, the first factor of P P only one of which (namely of m itself) is a v. Therefore m may be written as a sum of terms the first factor of which is u. When P now in one of those terms m, the second factor v precedes the second factor u, then we may apply a ‚+14, the permutation region P of which contains the two first v and the p—1 lastu. Tben „+: Am, may be written as a sum of p+1 terms in only one of which P (namely in m, itself) the second and third factors are v and which 261 have only factors of „u in the region p. Therefore i may be written as a sum of terms which are all ordered with respect to the first and second factors. Proceeding in this way we obtain a sum of perfectly ordered terms. In each of these terms the region p contains only factors u and the other region only factors v. P 8 d d d ; Now we consider the affinor m= ,uc,vo,wo.... with Q alter- nating regions of p,q,r etc. factors, p2>q2r2..., which may be annihilated by every alternation with Q alternating regions higher P than „4. When e.g. the first factor of m is w, then we apply a ILA, the permutation region of which contains w and pu. As Pp PR this ,414 annihilates m, m is a sum of terms all beginning with a u. Let now the second factor most to the left be e.g. a v, then we apply a ‚m4, the permutation region of which contains the two first v and the p—1 last u. Proceeding in this way we obtain a sum of terms which are all ordered as to the position of the u with respect to the remaining factors. Now we continue with alter- nations „714, the region p of which always contains the u that are already ordered; thus we obtain ordering of the v ete. until perfect ordering is reached. At eacli passage and therefore also in the final result the factors u of all terms remain in the region p, the v in the region q etc. We shall apply the theorem just proved to the elementary affinor P jfu and we shall prove that the result is singly determined and identical with PD. 1,,...,07 p um Sey A MP)y. ED the summation being extended over all ordered alternations and mixings, the number of which is just dj. For this purpose we use the wellknown property that the numbersystem of the permutations P is an associative system, which may be resolved into & “original” systems with J units. The units of such an original system may be chosen in such a way that Jos for q—r O.. ys Joa Such a system contains at the most dj idempotent principal units, the sum of which is the modulus of the system 7/. Let now for a Ig In = (33) p p definite value of À for any affinor v be AD MO v AO where 4,© 262 P and My) are conjugate, then A‘) Mi v is an elementary affinor and therefore according to theorem IL we have: p p APD MO y= AD MAS A wo. or er BABA) a 9 z 7 3 i ê P : 5 where the w; are elementary affinors, while the summation has to be extended over all ordered alternations A. As Me A= 0 for a B we have: rap \ Ee AOM Dn EMA Dre a Repeated application teaches thus that the operator 4© M® never can be nilpotent. According to a wellknown axiom from the theory of the higher complex numbersystems we conclude from this, that there exists an idempotent number of the form a, (AL? MOD) at a, AG MEO et SE For every ordered alternation there thus exists such an idem- potent number and the products of these numbers being zero, they form a series of idempotent principal units. The number of ordered operators A; and J is therefore d;; or less. It cannot be less, however, as a repeated application of theorem Il teaches that every elementary A Rae affinor may be written in the form > A) u m. If u were 1, then 4 the powers of A‘) M's) would belong to the first class of Prtrcn with respect to one of the idempotent principal units and to the fourth class with respect to the other ones. As in an original number- system such numbers (nilpotent by-units (Nebeneinheiten)) cannot occur, 4 = 1 and the operators d ji =e; Al uy kh eee i are therefore idempotent principal units. In the same way dy ' ) À TS eee sae Re 1G A ig a form a similar system. These operators are called ordered elementary operators of the first resp. second kind and the affinors which may be derived from them, ordered elementary affinors of the first resp. second kind. When y; is the number of permutations in A; or M; (hence for the permutation number s,,...,s; equal to s,/...s,/) and when 8; is the number of operators A; or M; that do not annihilate a definite 263 pt ci a; operator M; resp. A; we may easily calculate that ——-—~— 7 viyj B Bj is the coefficient of A, in Ae MY and MD A™. The coefficient of Kin A; M; being one, this involves that di: Vi; Jj; B Pea 7 ey ely = IN EE ij bj EE a le) p! ay aj Thus the expansion with regard to elementary affinors of the first kind is e.g. for p—=6: 6 bb 0) 1,9 0) Lab yyy BAM AS ZA MH ES AM 42S A) ye 110 (a) (a) acer (a) (a) 1a) (a) (a) de ede KANS ALM 4) Ne. (40) oS Ay a 12S gy 4e gy TN | + 2 = A, M, re B aoe A, M, en = Ay, / 2 a ) u When an expansion of an affinor u is given with regard to ordered alternations or mixings, each of which is alternated by every higher alternation resp. mixing, then this expansion is identical with the indicated one. For on application the operator ¢;; AMM) resp. ej MM) A) all terms are annihilated except those which are derived from A™ resp. M;“. This one term only remains unaltered and is therefore equal to &; A‘) Mon resp. 6 MAO. The indi- cated expansion is therefore singly determined. So we have obtained the following principal theorem. Principal theorem C. Every affinor can be written in one and only one way as a sum of ordered alternations or mixings that are annihilated by every alternation resp. mixing with a higher permutation number. Expansion of an affinor with regard to reduceable covariants of different degree. When 4; and M; are conjugate, then for n >>5 it may be very well possible, that a general mixing lower than M; corresponds to a general alternation lower than A;. When however Arin Ai, then every general alternation lower than A; is of the form (2—8).2, 8.8/4: - The permutation number (@ + p —an), (n—1).e@ is conjugate to «.n and the number (« —8 +¢-+ p—(a—,)n—s,—.. .—S;,), ($y—1), (a—8 + 9, ...,(m—s,).(a—B) to (a—p).n,s,, .. ‚st. The second number 264 is doubtlessly higher than the first, all s,,...... ‚Sn being >a@ we may remark that al’ is a sum of multiples of operators M; Aj, in which the alter- nations always have more than «a permutation regions with m factors. The desired proof will thus be given, when we have shown that p r is annihilated by every operator 2,4, B >a. Thereto we make use of the theorem that a non reduceable quantity possesses no linear covariants of a lower degree '). By p means of an operator 2,4 we may derive from r a penetrating r general product of E? and an affinor w, r= p— Bn. However, r q w would then be a linear covariant of v of degree r 2, because, there being only two sets of variables, only alternations of the form 22A give not identically zero. This is the so-called second expansion in series of GORDAN |). The terms of the expansion with regard to non-reducible covariants may now be further deeomposed in different ways. First each operator m 22A can be decomposed into simple mixed alternations. Then an expan- OF Vive) KP yet a Je P sion of Fm is obtained with regard to locally alternating forms for which in each term the power of a determinant of the variables is the same as the in the same term occurring power of the deter- minant of the ideal factors of m corresponding to those sets of variables. That such an expansion is possible and singly determined has been first proved by A. REISsINGER *). Secondly each elementary affinor may be decomposed in ordered elementary affinors of the first kind. With this decomposition corre- sponds an expansion of the form: iz On et) zo Es 41 mn 7 x — 4d Fm= = D2 97 Per a, oA | Ops, MO) 8 Bo dab Ae af ") g ) P The factors of the form (a, y’) occurring in each term satisfy the condition that they belong to the permutation regions of a definite ordered alternation ,2A acting on xX’ y’? and characteristic of that 1) Srupy, Methoden zur Theorie der ternären Formen § 3 and § 4. The so-called first expansion in series of Gorpan corresponds to an expansion in series of a mixed affinor and is not discussed here. 2) Ausgezeichnete Form der Polaren-Entwicklung eines symbolischen Produktes. Progr. Realsch. Kempten 1906 —07. 3) See Ey p. 263. 278 term. That such an expansion is possible and singly determined has been first proved by W. Gopr’). Thirdly it is possible to decompose each elementary affinor into ordered elementary affinors of the second kind. To this corresponds the singly determined expansion : Le Cs eer Fu Sey BMA mx ys see. we E P EK. Wartscn?) has given another expansion which is also singly P determined and which corresponds to an expansion of m in terms of the form with coefficients that for a definitely chosen order of u?, v’,.... are functions of a,,a,,... It is remarkable that the number of terms of this expansion for a P-linear form is equal to that of the expansion with regard to ordered elementary aftinors of the first kind, eg. 1+5+9+5= 20 for P=6. 7 . Kepansion in series of a form in m sets of n variables. Let P FA MEN en NPN Ee P -P be a form in the m sets of variables x’, y’,... and m= wry’... P the corresponding affinor. We can expand m in non-reducible covariants. Each term is then a sum of ordered alternations each consisting of a penetrating general product of a number « of factors E that is characteristic of that term with a linear homo- geneous non-reducible affinor of degree P—an. To this corresponds ') W. Gopr deduces this expansion in quite another way and this may be the reason that he has not seen the connexion with the group characters of Frozenius and the possibility of an analogous expansion for n > 2. “Ueber die Entwicklung binärer Formen mit mehreren Variabelen”, Arch. f. Math. u. Phys. fe DSj Ald: *) Ueber Reihenentwicklungen mehrfachbinärer Formen. Sitz. Ber. der Wiener Akad. 113 (04) 1209—1217, Warrscm has used for the first time the expansions in series of the theory of binary invariants to decompose directed quantities in parts covariant under the orthogonal group (e.g. the decomposition of the affinor of deformation in scalar, vector and deviator), ‘Ueber höhere Vectorgröszen der Kristallphysik etc.” Wien. Ber. 113 (04) 1107—1119; Extension de l'algèbre vectorielle etc., Gomptes Rendus 143 (06) 204—207. 279 an expansion of the form Fim in a number of terms each of which is a sum of prodnets of a non-reduecible form with a certain number (characteristic of that term) of determinants formed by n ofthe sets of variables. For mn this expansion has been given first by Caprini') and for the general case by J. Deruyts*) and K. Perr®)*). Both Capen and Prrr base their proof upon the property mentioned p. 276, that a form in 7 sets of n variables containing the determinant of the variables as a factor, is not annihilated by (2. The deduction of CaprELii, which is most analogous to the above is based upon the theory of the differential operators Hs). Drruyrs uses his theory of the semi-invariants and -covariants and Perr makes use of differential- operators that can be constructed by means of auxiliary variables. The terms of the expansion in non-reducible covariants may P again be decomposed in different ways. Firstly each term of m can be decomposed into general mixed alternations and these again P into simple ones. To this corresponds an expansion of m in a sum of terms consisting each of a sum of products of a number of determinants with s,,..., s,; rows formed from the characteristic numbers of the sets x’, y’... with one single symmetrical form. All terms are covariants, the sub-terms only then when s,;—=s,=...=n, In each sub-term the power of a determinant of the characteristic numbers of the variables is the same as the power of the determinant of the characteristic numbers of the corresponding ideal factors of yc _m. Under these conditions the expansion is singly determined and an extension of the one given by Rerssincrr for n — 2. P Secondly each term of m may be decomposed into ordered alternations of the form ,,A. Then the determinants occurring in each sub-term must belong to the permutation regions of a definite ordered alter- nation „„4A, characteristic of that sub-term, and acting on x’e vin 1) Fondamenti di una teoria generale delle forme algebriche, Mem. dei Linces (82) § 74; Sur les opérations dans la théorie des formes algébriques, Math Ann. 37 (90) 1—37. *) Essai d'une théorie générale des formes algébriques. Mém de Liège. 2. 17 (92) 4. 1—156; Détermination des fonctions invariantes de formes a plusieurs séries de variables. Mém. couronnés et mém. des sav. étr. de Bruxelles 53 (90— 93) 2. 1—23. 5) Ueber eine Reihenentwicklung für algebraische Formen, Bull. Intern. de Prague 12 (07) 168—191. *) The forms called here non-reduceable are called by Capetu: “formes impro- pres” and by Deruyts: “covariants de formes primaires”. 280 This expansion is singly determined, the corresponding expansion of À p p P m being singly determined *) and an extension of that given by Gopr for n= 2. We have thus found the theorem: Principal theorem B. Every algebraic form, homogeneous and of the degrees 0, 6,... in m sets of n variables x’, y’.... can be expanded in one and only one way in a series of terms, each of which consists of a product of a number « of determinants that are formed from the vart- ables of n of the sets with a non-reducible form, in such a way that the deternunants in each term belong to the permutation regions of a definite ordered alternation „A, characteristic of that term and acting OANA NON x hog tah ALA the number a being characteristic of a definite group of those terms. B Thirdly we can proceed so far with the division, that m becomes a sum of ordered elementary affinors all of the first or all of the P second kind. With this corresponds an expansion of Fm in ordered elementary forms of the first resp. of the second kind, which may be characterized in the following way : Principal theorem GC. Every algebraic form, homogeneous and of the degrees 9, 6,... in m sets of n variables x’, y’,..., can be expanded in one and only one way ina series of ordered forms of the first resp. of the second kind. Examples: The 6-linear form 6 fm = m).. oie Xi. Ke 6 can be decomposed into 76 ordered elementary forms of the first kind corresponding with the affinors. 6 1) 6Ant oM! m N : 6 2,..+; 6) 5A0@Me om, As=1,..., 5 Dee ea Shoe dan eh le me {oel Kan em a Peed REN 26: 2111p BONA eA, ani Aen Bloem AO) snare Ed am tude ailing eae an AT 50) AE Mae mn le a NEET 281 Wabeke sede AMS m, Benen 62,...,70) ssd emd om, 21e A Mn 76) 041 SM Fi for n = 5, 1 becomes zero, forn—4:1,...., G"fOk im = oe oe 25 and for n=2: 1, ...., 56. The expansion in elementary 6 forms corresponds to an expansion of m which is found from the preceding one by taking together the horizontal rows 1; 2,.-..,6; 7...,15; etc. From this can be deduced again ‘the expansion in non-reducible covariants. For n > 6 : 1..., BENTO Ore ee TO Or Woe. ee Oe. TY eee, 76: for a ot AO Oe 76; for AO Pe”: 30; ee Nei FO aS EAA Neat Ole an re TO dn EEN 75; 76. As to the expansion of a form of the sixth degree in a number of sets of variables less than 6 e.g. 6 9 9 2 12 2 9 Fu=nj fiem3 Xix2xs 6 we may remark, that x’,’x’,?x’’, can be obtained by a definite 2 simple mixing *°M,®. Hence, in the expansion Hee all ordered elementary operators of the first kind vanish, when their first factor is an alternation that is annihilated by 3?1/,@. In the first place fuse to... 25:-0f 26—30 remains one term, of 31—46 there remain nine terms, among which three different ones, of 47—56 four equal terms, of 57—60 two equal terms, of 61—69 six terms, among which three different ones, of 70-—75 four terms, among which two different ones, while 76 remains. In total there remain therefore for n<3 twelve terms and for n = 2 seven terms. This last number gives also the number of terms in the expansion of W AELSCH '). Expansion of a form in m sets of n variables of arbitrary degree. P Principal theorem D. Every algebraic form Fm, homogeneous ; a : : Po and of the degrees 0,¢,... in m different sets of variables ®’,y’,... can be expanded in one and only one way in a series of ordered elementary forms of the first resp. of the second kind. ') Comp. p. 278. 282 . . E . The expansion may be obtained by expanding m in ordered elementary affinors. Also the expansion in non-reducible covariant forms (principal theorem B) can be obtained for this most general case by expanding P . . m with respect to non-reducible covariants. Then the determinants of n rows that occur in the expansion of the form generally have only an ideal meaning and therefore the non-reducible forms occurring in each term have only an ideal significance too. The terms themselves however keep their non-ideal significance and are found by taking together definite groups of terms from the expansion in elementary forms of the first kind. Physiology. — “On Cyclopia with conservation of the Rhinen- cephalon”. By Prof. C. Winker. (Communicated in the meeting of May 31, 1919). In These Proceedings of February 1916 I reported the results found after examination of the brains of three cyclopian monstra. I then pointed out, that the characteristic abnormality in all these brains, was the presence of a sack with a thin wall, formed by the roof of the third ventricle, largely extended by fluid. 1 was forced to contradict a sentence, found in ScnwauBe’s “Die Morphologie der Missbildungen” where he says: “das Vorderhirn ist bei den Cyclopen nie in Hemisphären geteilt”. I must assert, that in all cases of cyclopian brain the hemispheres are well differenti- ated at the occipital end. At the frontal pole however they are often united, although in one of the cases then described, there was also a sagittal fissure at the frontal pole of the hemisphere. I therefore deny that the examination of cyclopian brains should give any support to the supposition that the terminating-time of the cyclopia must be placed in a period, preceding that in which the sagittal fissure of the telencephalon appears. Also the view that eyclopia is mevitably accompanied by arhinencephalia, I could only accept under certain reserve. In fact, in all the cases which I examined, the bulbi olfactorii and the lobi olfactorii anteriores were missing. But the lobi olfactorii posteriores were present. They are found along the mesial line of the brain-base. There they were placed next to each other, because the brain-base between them was missing. Ordinarily eyelopian brains are partially arhinencephalic, missing only the frontal part of the rhinencephalon. Since then I have prepared the brains of several cases of cyclopia, largely differing between each other. Now I believe that there is another monstrosity, the so-called synotia, nearly related to the cyclopian one. The former is often combined with the latter. Through the kind cooperation of Professor Kouwer and Dr. Vrr- MEULEN I obtained possession of such-like monstrosities. 19 Proceedings Royal Acad. Amsterdam. Vol. XXII. 284 1. a human fetus with synotia (nose, eyes and maxilla superior are existing). There is an indication of a mouth. No inferior maxilla is found. The ears are united in the mid-line in a single meatus acusticus externus (In this being the brains were unfortunately thrown away). 2. a fetus of a pig with cebocephalia (Proboscis is found above two totally separated eyes in one orbit). 3. a fetus of a lamb with incomplete cyclopia (one oblong eye with a long pupil, two optic nerves; no proboscis is visible by the naked eye, although the X-ray photo shows a little nasal bone at the os frontale). 4. a fetus of a calf with almost complete cyclopia (one oblong eye with hour-glass like pupil, two optie nerves closely situated, no proboscis.) 5. a fetus of a calf with incomplete cyclopia and synotia (two united eyes in the circle of the four eyelids, no proboscis. A mouth- opening, bordered at the upper part by a strong upper-jaw (X-ray photo). No lower jaw (X-ray photo), no tongue. The os hyoideum is completely developed (X-ray photo). The ears are united in the mid-line). 6. a fetus of a lamb with synotia. (A small proboscis, no eyes, “no jaws, no mouth or tongue. There is an os hyoideum. (X-ray photo.) The animal therefore is anophthalmic, agnathic, aprosopic. The ears are united in the mid-line. Larynx and pharynx end towards the top in a blind sack. No thyroid gland.) 7. a fetus of a lamb with cyclopia and synotia, dealt with in this communication. In all the brains of those monstra which were examined, there was found a membranous sack, which shows itself to be the roof of the third ventricle, much extended by fluid. Now and then however peculiarities were seen in this sack at its frontal or its occipital end. In No. 5 and No. 6 e.g. the sack was continued in the like-wise thin wall of the telencephalon and in N°. 7 at its distal end the mesencephalon and the cerebellum were not developed, forming a part of the thin wall of the sack covering also the IV ventricle. The sack is not only existing in the brains of cyclopian monstra, but also in those of the synotic type. In another paper I will describe more in details the differences of the brains in those monstra. Here I intend to demonstrate the monstrosity, mentioned sub N°. 7. It may be considered as a sample of a eyclopian malformation, with 285 synotia possessing a complete rhinencephalon and therefore it already warns us that we should not rashly assert that every cyclopia must be arhinencephalic. Regarding the drawing of this fetus, seen in the face, we notice Fig. 1. Drawing after a photograph of a lambfetus with cebo-cephalia and synotia. The monstrosity possesses a well-shaped and completely developed nose above the two eyes, surrounded by four eye-lids. One can discover the two medial carunculae, The ears are united together. No mouth. that the animal possesses a complete nose with nostrils. Underneath it is found, that what appears to be an orbit, surrounded by four 13" 286 eyelids and in which the two united eyes are placed with a smaller left and a larger right eye. Jaws, mouth and tongue are missing. Directly underneath the eyes the two ears lie by the side of each other in the mid-line and the two external auditory canals possess a combined opening. Pharynx and larynx are well developed and continue in a hollow, ending upwards blind. The X-ray photo confirms that the jaws are totally missing, but it also shows that the os hyoideum is well developed. Moreover the X-ray photo teaches that the ethmoidal bone is fully present. Crista Fig. 2. Drawing of the upper surface of the brains of the monstrosity reproduced in fig. 1. The sack (a) is visible. It adheres to the dura mater (>). At the distal end (c) it continues in the mesencephalon and in the cerebellum, represented by a thin membrane. Through this membrane shines the tela chorioidea of the fourth ventricle. Galli, lamina cribrosa, lamina papyracea and the laryrinth of the ethmoidal bone can be distinguished. Already before removing the brains | therefore knew, that this cyclopian being could not have 287 been formed through the missing of the ethmoidal bone and of the mesial wall of the orbit. On removal of the brains one is directly struck by the presence of the sack (fig. 26) and the epiphysis lying free, seeming to be its point of origin. Behind it one does not find anything of a mesen- cephalon or of a cerebellum. The sack continues in occipital direction and the tela of the 4 ventricle shines faintly through it (fig. 2c). On the other hand both the hemispheres are well developed. There is a deep sagittal fissure, in which the dural septum with its sinus are found, and which has to be cut away from the crista galli in order to make its removal possible. Drawing of the basal surface of the monstrosity reproduced in fig. 1. a = the thin membrane covering the defect in the pes pedunculi cerebri. 6. ol. = bulbus olfactorius. /. ol. = lobus olfactorius. N. // = nervi optici c. m. = corpus mammillare, shining through the membrane. N. VJ = nervus abducens. If we examine the basal surface of the brain (fig. 3), then we are struck by the presence of two well-developed olfactory bulbs with their tract, their lobus olfactorius anterior, posterior and cornu Ammonis, in short of a completely developed rhinencephalon. There are two N. optici (fig. 3, N. II). The corpus mammillare 288 shines through the pia mater. One can surely demonstrate the N. abducens of the cranial nerves. With the nerve lying proximally from this, it is not so. Only after a microscopic examination, it is certain that it is the N. trigeminus. Between these two thick nerves the base of the brain is formed by a thin membrane, which when the brain was removed could only be spared with the utmost care. If the sack is opened at its dorsal part and is folded backward, there appears a local defect in the brain-base, more than 1 cm. wide. There the base of the brain is formed by a membrane, 1 mm. thick, at the utmost, in which some white nerve strings diverge from the mid-line towards a proximally placed mass of nervous tissue. This nervous mass, striatum and thalamencephalon are shining through the pia mater at.the base in fig. 3. There is an interrupted continuation in the brain-base at the level of the pes pedunculi. The bony base of the brain is very remarkable. The crista galli protrudes. On both its sides the lamina cribrosa carries the bulbi olfactorii, which send their fila olfactoria through it. Moreover the optie foramina are normally formed, together with the frontal part of the os sphenoidale. Then however the sella turcica is found missing, also the hypophysis. The base of the crane is not massive, but movable, as there is a great loss of bone distally from the sella turcica. An X-ray photo taken from the upper side makes this obvious. Here a large defect in the bony base of the skull appears. The caudal part of the os sphenoidale is missing in the mid-line and the frontal part of the clivus has fallen out as faras the arch of the Atlas. The os petrosum is intact on both sides. There are no jaws. Through the loss of the facies orbitalis of the upper jaw the eyes have sunk downwards. They are no longer lying in a bony orbit. The lamina papyracea of the ethmoidal bone is placed proximally from the double eye and therefore easily recognisable in the X-ray photo. In this case, it is not because the mesial wall of the orbit has been destroyed that the eyes have met one another in the mid-line, but because the lower wall of the orbit is absent. This ecyclopian monstrosity possesses a complete rhinencephalon, but at the same time it becomes synotic through the loss of a lower jaw. As to the question, what may be the cause of such a monstrosity one must acknowledge that immense difficulties arise in defending that there was an insufficiency of germ material, as far as the brain as well as the bony parts of the skull. is concerned Simpler is the view in taking the sack as point of origin. 289 A pathological process which calls forth the sack (and the latter is found in all the cases of this sort) is able to destroy at the dorsal brainside the mesencephalon and the cerebellum when they hinder its development at the distal end. But at the ventral side the pathological process destroys the pedunculi cerebri and the tissue on which they lie. This is the germ material out of which jaws, mouth and tongue are going to develop. Distally bordered by the second branchial arch (the os byoideum was intact in all the three cases of synotia) all is destroyed that is going to develop out of the first branchial arch (except occasionally of the ossicula acustica) and out of the tissue, which lies proximally from it. In this way the local defect of the pedunculi cerebri and of the tissue forming the middle part of the skull, is easily understood. According to the view which I explained in my previous report, I think the cause of the sack to be a local process of inflammation, which by means of a mechanical influence produces a defect at the base of the brain and of the skull. On the other hand | acknowledge the possibility that the sack and the basal defect, together may be co-effects of another more complicated pathological cause. In debates upon this subject, held at Leiden Dr. Murk JANSEN defended the thesis that the narrowness of the amnion may perhaps produce the sack as well as the defect, by compressing the head of the embryo in a strictly defined plane. The result may be that all the germ material, which is found in this plane, may die. Should such a hypothesis be confirmed, there will arise different possibilities in the formation of these monstra, but I will not yet enter upon these. J only wish to lay stress on the foHowing views. Destroying of tissue at the proximal end of the skull, so that the os ethmoidale disappears and the dorsally placed sack, the roof of the distended III ventricle is formed, gives rise to cyclopian monstra. But they are not the only defect-formations which are found. There is yet another place of predilection, where the tissue that will form parts of the skull, may be destroyed. In such cases the first branchial arch may be destroyed by pathological processes. These lead to synotia. Now and then it occurs isolated. Then we see uncomplicated synotia. Nose, eyes and upper jaw are well deve- loped, as found in the fetus, mentioned sub I. But also the two local destroying processes appear, independent of each other, next to one another. Then cyclopian and synotian deformities are found together. And there may remain between 290 them an intact upper jaw as was the case with the monstrum, mentioned under 5. , Now and then the two local destroying processes are united. Then comes a massive fissure and the monstrosities appear, as described under 4 i.e. synotia with anophthalmia, aprosopia and agnatia. The series of the different eyclopian deformities is joined to that of the synotian deformities. In this series one case is most remark- able, ie. if the local destroying process, commencing at the first branchial arch spreads so far proximalwards that the upper jaw totally falls ont. Then the eyes are going to meet each other in the mid-line, while the maxillar part of the orbit is lost. Then also two eyes are found in a four eye-lid ring, but they are no longer placed in an orbit. In that case there is found a cebocephalic form of cyclopia, with a complete nose above the eyes, and a complete rhinencephalon. To me it seems even possible that the local destroying process may spare the first branchial arch, only destroying the upper jaw. In such a case there results a cyclopia, perhaps always the cebo- cephalic form, but without loss of the ethmoid bone, with a com- plete rhinencephalon and without synotia. Mathematics. — “The Remainder in the Binomial-series.” By Prof. Frep. Scaun. (Communicated by Prof. D. J. Korreweg.) (Communicated in the meeting of June 28, 1919). Dn é ; 7 . mM). fe = (m) 1. We consider the binomial-series XZ u Dad Dl de ee a j gnd sijl =H (m—f). We suppose « real and m not zero and not a = k=0 positive integral number (there otherwise the series is finite). a. “The series is*convergent if vel <1, if e=1,m > —1 and if «= —i1, m > 0, divergent in the other cases. 3. If |w|<1 and if |e)=1, m >—1 we have hm ul” 10); n= @ as appears from N°. 2. 4. According to Mac Laurin’s series we have: dt)" — LS ae j=0 / in which the remainder is given by: En (1—6)"—p n—1 a es m—k)(p 0), TE ae (p > 9) hence: (m) u, Hi rr 5 (1 + Ox)n—m ( ) Me irs =p, m—p (m—1) 5) en 1+ 0e GE A VERN p ee) Kga a 5. The aim of this paper is to show that lim R„=—=0 in all ns cases in which the series perros, that is to demonstrate for those cases the validity of (1 + 2)" = = u exclusively from the remainder in. in the series (which is done incorrectly in a great number of hand- 1) ‘The numbers 6 occurring in these expressions are generally unequal, 6 depend- ing on ” and on p. 292 books). At the same time the advantage of the expression (2) for the remainder in the binomial-series will be obvious. 6. If «20, we find according to (1) if we choose n > m: Bn Sy wm) |, hence lim Rk, = 0 if the series converges. n= 7. If 1e <0, its ensues, from; (2), -if we, take p=1: Al << A|maum |, in which A is the greatest of the numbers 1 and (4 + mt. This leads to lim KR, = 0. ns 8. If v=—l, m > 0, it follows from (3) ‘by taking p= m: (Ott) Al ME a SM ee) In connexion with N°. 3 it follows that lun A, = 0 (the inequality NZL m—1 > —1 being satisfied), a result that cannot be obtained from LAGRANGE's or CavcHy’s form of the remainder if m < 1. It ean further be observed, that (3) leads to the identity _m(m—l) PTO Ne (ei), 1 — mt ee + (— 1)" ———-—— af = „(mln —.2),. … (m — zo) == (== 15) —— Es ES ee =F 1. the identity can also easily be demonstrated by mathematical induction, from which it appears that the identity also holds good for m <0. 9. If we make use of Aper’s theorem about the continuity of power-series, the examination of the remainder in the case | wv | <1 suffices. In order to demonstrate, without distinguishing various cases, that then im FH, == 0; in_(2) we take p= 4 (as in NO. 7): n= Mathematics. — “A problem of combinatorial analysis connected with the determination of the number of different ways in which the greatest common divisor of two products can be found’. By Prof. Freo. ScHen. (Communicated by Prof. D. J. KorTEWEG). (Communicated in the meeting of June 28, 1919). 1. The greatest common divisor (G. C.D.) of the two numbers OS EON AD POD a ep es ade Ap) is obtained as the product of the mn greatest common divisors Gi; Mima ean (ande SL Zire os) LU 1S 10. be understood, that two numbers, whose G. C.D. is determined, are to be divided by that G.C.D. and that for the remaining part of the calculation the quotients are substitued for the original numbers. If such a quotient is combined with another number, we divide again py their G..C..D.-ete.’). So the mn numbers G;; are calculated in a definite order. Gi; ws the G. C. D. of the numbers a; and. b';, a’; being obtained by division of a; by all previously determined greatest common divisors that are connected with a;, while 6'; is derived in the same way from bj. a 2. We now inquire into the nwnber of ways in which the G. C.D. of the numbers (1) can be calculated in the manner indicated m N°. 1. A different order of the mn numbers (;; does not neces- sarily yield a different manner of calculation of the number G sought, for two manners of calculation must of course be considered as identical, if they consist of the same calculations and only differ in the order in which the separate calculations are carried out. Hence it is immaterial which of the two numbers Gj and Grp is calculated first, provided «A and at the same time 7 #/. Conse- quently it is only the order of the numbers G;; which agree either in the first indices or in the second, that is essential. 9: lhe special values- of the numbers @,,...,@mn, 0,,.:-., 6, may 1) The last time that one of the numbers dj, Wg,.-., Ay, 01, Da, +.» bis com- bined with another the division of the first-mentioned number by the G. C. D. found may of course be dispensed with, the quotient being no further involved in the calculation. 294 cause that permutation of two numbers G;;, which agree in the first or in the second indices, does not yield a different manner of calcu- lation of G. If a, and 6, e.g. are prime to each other and a, and 6, likewise, it will be immaterial whether we first determine G,, and then G',, or vice versa, both greatest common divisors being 1 and a division by G,, or G,, consequently not causing any alteration in the numbers. But we shall disregard such special cases, so that we only call two ways of calculation of G identical, if they are the same with any set of values of a,,...,@n,0,,..., On. 4. The number G inquired into in N°. 2 then becomes equal to the number of ways, in which we can arrange the mn symbols CRO CO Er 7) in a sequence, if we consider two sequences identical, if these can be derived from each other by some permutations of two symbols, differing both in the first and second indices. 5. If mm == 2 the number asked for will be the number A, of the ways, in which we can arrange the symbols Gia; Oi Mara: sG tone Re ee Gois Goos raa oars Gone ee eee in one sequence, only paying regard to the order of two of these symbols if in the above scheme they stand in the same row or in the same column. The number A, is the product of n! and the number B, of the arrangements, in which the symbols (2) follow each other in the manner indicated by (2). 6. We can divide the B, arrangements into m groups, thus, that the zt group contains those arrangements, in which Gs; precedes the other symbols (3). That #* group can be subdivided into two — parts, the first part containing the arrangements in which Gs; precedes Gis, the second the arrangements in which Gs; comes after (4;. The first part of the z® group contains as many arrangements as when Gi; and Gs; are cancelled, hence B, 4 arrangements. In the arrangements occurring in the second part of the 2" group the symbols G1, Gie,...., Gi; precede all symbols (3), so that we may cancel the symbols G44, Giz,...., Gi. If we only pay attention to the arrangement of the symbols (721, Goe,...., Gaj mutually and towards the symbols . Ga pi) Gonars sn Gans hy se en ee oD ; : es (considered as identical) we obtain En possible arrangements. n— 1) 295. Each of these arrangements leads to 4, _; possible arrangements, if we also pay attention to the order in which the »—z symbols (4) are placed mutually and respectively towards Gi, Giite,...., Gan. It appears from this, that the second part of the z group contains (n—1)! (nol B: arrangements; this also holds good in the case 7=n, n=} if we interpret B, as 1. . (n—1)/ Hence the 7 group contains altogether B, + rea Re. n—t)! arrangements. This leads to n n—l)! n—2 B == = Bn, + we Bi (ndr (n—1)/ 2 let (5) i=1 (n—1) ! 0 k / 7. If we replace in (5) n by n—1 (for which it is necessary to assume n > 1) we shall find: n—? Bi. B, = (n—1) Bie + (n—2)/ = —, Gey from which follows according to (5): B, == 2n B, — (n—1)? Bs. From this equation in finite differences (which is homogeneous, linear and of the second order) and proceeding from 2B, = 1 and B, = 2 5, we can successively compute B, 5,, B,, ete. Thus we find: ar B IGA, B 1A0922, Be 44a (29. PTAA, Bin 234662231. 8. Owing to A, =n! By, we now find for the number A, of the ways in which we may calculate the G. C. D. of the numbers a, a, "rg Ve Nag sa ee oe —2, A,= 14, A, — 204, A, = 5016, A, — 185520, = 9595440, A, = 659846880, A, = 58130513280. 9. If m and n are both > 2, the determination of the number of ways, in which the G.C.D. of the two products can be found according to the method as indicated in N°. 1, has become considerably more difficult. If m—n—3 we find (by systematic finding out all the various cases) for the number sought 19164. 1) By taking » =1 the formula (5) leads, in connection with By = 1, to B, = 2, n—2 an = then being O (as sum of zero terms). o #: Chemistry. — “On the Phenomenon after Anodic Polarisation.” I. By Prof. A. Smrrs, G. L. C. La Basripe, and Tu. pr CRAUW. (Communicated by Prof. P. Zeeman). (Communicated in the meeting of June 28, 1919). 1. It was shown in a preceding communication that the phenomenon that appears after anodie polarisation of iron in an iron salt solution is owing to this that during the anodic solution the iron gets surrounded by a liquid layer which is very rich in ferro- sens é than outside it. After the current has been broken the ferri-ions will, e É : RE ions. As a rule the ratio Ee will be greater in this liquid layer therefore, diffuse from the surroundings into the boundary layer, (He) Fe”) to become again smaller in the boundary layer. In consequence of this change the potential of the iron, which was at first less negative or positive through the disturbance of the metal, passes-through a minimum value after interruption of the current. That the above explanation actually accounts for the phenomenon was proved by the fact that the phenomenon disappears altogether when the iron salt solution is previously heated in a hydrogen atmosphere with iron powder for some time. We then get a ferro salt solution which is in electromotive equilibrium with unary iron, Pe" whereas the ferro-ions pass outside, which causes the ratio Fes) change when the iron is anodically dissolved. so that in the boundary layer the ratio practically does not 2. In the above mentioned communication it was pointed out that the potential of Nickel after polarisation in a solution of NiSO, likewise passes through a minimum value, so that it was already supposed that this phenomenon would have to be explained in the same way as for iron. To examine this the phenomenon for Nickel was first photographed when the metal was immersed in a solution of NiCl,. In this the mini- mum shows itself very clearly, as the adjoined photo (fig. 1) sets forth. The process is much quicker than when an NiSO, solution is used, in consequence of the positively catalytic action of the chlorine ions. 297 rae | j HH { H i } | | | ; | 1 | | | 4 | | ! H 4 f ' 1 + t manen vereen eter eem rindimn pe } i } } peek Minit Biter atte er ' Sh | wdn Een wt NEREDE itt Gale ee eee ene: 299 Then the NiCl, solution was heated for some time in a hydrogen atmosphere with finely divided Nickel that had been prepared by reduction of NiO at relatively low temperature, after which the experi- ment was repeated. As the following photo shows (fig. 2), the pheno- menon had now entirely disappeared. 3. df was already pointed out in the preceding communication, that as it were, the reflected image of the phenomenon was to be expected, when e.g. iron immersed in a ferro-ferri-salt-solution is made 1. he cathode. The ratio ( will be greatest at the iron surface, because the iron by sending Fe'’-ions into solutions, strives to bring such a change in the boundary layer that in case of unary behaviour, it can be in equilibrium with it. Hence the farther the liquid layer is (Fe) He) iron, immersed in the supposed solution, is made cathode for a moment, the iron will be deposited from the boundary layer, and the ions from the following layer will diffuse into the boundary (Fe) (Fe) iron after cathodic polarisation comes “in contact with a liquid layer When therefore from the iron, the smaller will be the ratio layer, and as the ratio is smaller in this following layer, the in which the ratio a is smaller than before the cathodic polari- sation, which causes a stronger disturbance in noble direction, hence a less negative potential. After interruption of the current the said ratio will now increase again through the solution of the iron, so that the potential of the iron becomes again more negative. As on cathodic polarisation the iron gets covered with a layer of iron, which at first deviates from the unary iron in base direction, a phenomenon after cathodic polarisation was really to be expected also here, which would have to consist in this, that after interruption of the current the potential of the iron, after cathodic polarisation, passes through a maximum value. *) As the photo on fig. 3 shows, this supposition is confirmed. The maximum B lies about 30 m.V. less negative than the initial potential indicated by the line DE. It will now be examined whether the same phenomenon can also be observed for Nickel. Laboratory for General and Anorganic Amsterdam, June 27, 1919. Chemistry of the University. 1) Through diminution of the total ion concentration the potential becomes more negative in a small degree, so_that this circumstance still slightly counteracts the phenomenon under consideration. 20 Proceedings Royal Acad. Amsterdam. Vol XXII. Physics. — ‘Researches on the Spectra of Tin, Lead, Antimony, and Bismuth in the Magnetic Field.” By Dr. P. A. van DER Harst. (Communicated by Prof. P. Zeeman). (Communicated in the meeting of June 28, 1919). To carry out the observations of which the description will follow here, I have made use of the grating apparatus of the Physical Laboratory at Amsterdam, which had been placed at my disposal by Prof. P. Zrrman. The experiments will be described more at length in my thesis for the doctorate, which will be published under the above title. 1. The spectra of tin, lead, antimony, and bismuth have been little, if at all, studied as to, their ZEEMAN effect. The only scientist that made systematic observations on these elements in this respect, is Purvis '). As is known, his results often present, however, great deviations from those of others, which is chiefly owing to an error in the measurement of the intensity of the field. Besides, Purvis measures but few lines of these elements, viz. only those for which no longer time of exposure was required than half an hour to get them distinctly enough on the photographic plate to be measured. It seemed, therefore, desirable to me to subject the magnetic resolution for these elements to a closer examination. 2. The grating apparatus has been described by Mrs. H. B. BILDERBEEK—VAN Mrours*®). The grating is a concave one of Rowland ; radius 3 m., width 8 em, 14488 lines per inch. Slit and grating are rigid with respect to each other, and with respect to an iron circular arch, on which the photographic plates are clasped, so that the exposure can be simultaneous in all orders. The width of the slit generally amounted to about 20 u, ie. 8 1) J. E Purvis. The Influence of a very Strong Magnetic Field on the Spark Spectra of Lead, Tin, Antimony, Bismuth and Gold. Camb. Phil. Soc., 14, 217, 1907; Nature, 76, 166, 1907. *) H. B. BitpeERBEEK—v. Meurs. Magnetische splitsing van het ultraviolette ijzerspectrum (A 2800—A 4500). Diss., Amsterdam, 1909. € 301 times the normal width of the slit *) for the arrangement used. As lens [ used a double lens of quartz-calcium fluorite, transparent to ultraviolet rays, and moreover achromatic. The image of the source of light always fell at the same place of the slit. It was generally necessary to separate the vertical and horizontal components with a calespar rhombohedron, placed between the source of lignt and the lens. One of the two images formed in this way, was projected on the slit. The magnet was a large-size Du Bois magnet, of which the end- planes of the conic pole tops were circular with a section of generally 8 mm. The distance between the poles was never greater than 4 mm. The magnetizing current was measured with a thermic Ampèremeter of Hartmann and Braun, and generally amounted to 15 Amperes. As source of light was used a spark between electrodes of the metal under examination, or alloys of it. The spark was, parallel to a condenser, in the secondary circuit of a transformator (of Kocm and Srerzer at Dresden), of which the primary circuit was fed by the municipal alternating current. Besides in the secondary circuit there was found an auxiliary spark, and an adjustable self- induction, in series with the spark. The particulars of the spark discharge (intensity of light, sharpness of the spectrum lines formed, melting of the electrodes) and the influence on this of self-induction, capacity and auxiliary spark will be fully discussed in my Thesis for the Doctorate. As photographic plates I mostly used ‘Agfa Röntgen” plates, which | made previously sensitive to colour by bathing them in “Pynachrom.” The plates prepared in this way were preferable in my opinion to the commercial colour-sensitive plates. The measurements were performed with a Zeiss comparator. The results were divided into four groups according to their greater or less accuracy, which I gave in the tables the weight: 4, 3, 2, 1. The first group had a _ probable error of 1°/, or less, the second from 1°/, to 5°/,, the third from 5°/, tot 10°/,, the fourth of more than 10°/,. For the precautions taken in the measurements and the determination of the error I refer to my Thesis. Also the preliminary experiments: the focussing of the plate-holder, the determination of the dissolving power, and the scalar value will not be treated further here, but may be found in my Thesis. The resolving power was certainly not smaller than the theoretical. ') SonusreR. The Optics of the Spectroscope. Astroph. J., 21, 197, 1905. 20% 302 3. Observations. The different intensities of the field were always determined by measuring the resolutions which were caused by them for Zn 4680, in which the degree of resolution of this line was compared with that obtained with the well-known absolute measure- ments of Genin, Weiss and Corron'), and Forrar ®). As standard aA value for ane halen I took the mean of the values obtained by them, } viz. 9,376. I mostly worked with fields that lay in the neighbourhood of 30000 Gauss. For tin and lead the electrodes were flexible strips of these metals, which were stretched cross-wise over the pole tops. For tin the spark was still cooled by blowing with a Féhn, because otherwise the electrodes were melted through too quickly, and the discharge passed into a continuous one. This method was discarded for lead, because with the Féhn there would often be a hitch, and an auxiliary spark was inserted in the secondary circuit in series with the illumi- nating spark. By regulating the distance of this auxiliary spark, we have much better control over the action of the illuminating spark. The electrodes of the auxiliary spark must not oxidise, however, because then we get there a more continuous discharge, whereas the very function of the auxiliary spark is to obviate this drawback, which is met with for the illuminating spark, and is not to be avoided there; it has, therefore, to ensure an interrupted discharge. For this purpose the bulbs of brass, of which the auxiliary spark consisted, were coated with platinum hoods. The strips of lead were kept tightly stretched over the pole tops, as otherwise they are apt to bend over towards each other, thus rendering the spark length too small. No flexible bands could be made of antimony on account of the brittleness of this metal; I therefore used small flat rods of this metal as electrodes, which were clasped in a spark stand of brass. An advantage of this metal is that it has a pretty high melting-point, and that therefore the electrodes do not so quickly melt through. I have only used Bismuth as elecirode as alloy with antimony (60 percentages by weight of bismuth) else it combined the drawbacks of tin and lead that it melted soon, and that of antimony that it was brittle. In the alloy the first drawback was eliminated, and it could further be used as antimony. I must state further that in the tables in which ScHiPPers *) records 1) P, ZEEMAN. Researches in Magneto-optics. Mac Millan and Co, London, 1913, p. 67. Deutsche Uebersetzung, Leipzig, J A. BARTH, 1914. 2) R. Forrar. Recherches de magneto-optique. Thèse, Paris, 1914. 8) H. Scurppers, Messungen am Antimonspectrum. Zs. f. Wiss. Phot., 11, 235, 241. 3038 his measurements on the antimony Spectrum, and which are inserted in Karser’s Handbuch der Spectroscopie, 1 came across some lines which I never found on my plates. They are the lines 4370, 4295, 4287, 4091, 4078, 4038, 4024, 4006, 4004, 3979, 3721, 3467, 3460. From some experiments which I made with the purpose of ascertaining whether these lines existed, I think I have to conclude that SCHIPPERS was mistaken, and took lines of the third order for lines of the second order. These lines are in my opinion successively the following lines of the third order: 2913, 2863, 2858, 2727, 2719, 2692, 2683, 2671, 2670, 2653, 2480, 2311, 2306. Besides my own results I have recorded those of Purvis in the tables. I did not, however, use the intensity of the field which he gave, because this value is undoubtedly too high. Instead of this I have ascertained by the aid of my results what field intensity Purvis used by comparing the average of his results with the average of mine; this separately for every element. Thus | found successively for tin, lead, antimony, and bismuth 30400, 31100, 28700, 31100 Gauss. Corron estimates the field intensity used by Purvis at 30800, he himself gives 39980 Gauss. For the better mutual comparison of dd the results those of Purvis for = have been divided in the sub- joined tables by the above mentioned field intensities. The wavelengths are recorded in round values in international ° Angstrom units. 4. Discussion of the Tables. On one of the photos for tin, on which the two kinds of components appeared at the same time, were further seen the quadruplets 2368 and 2762 (are line), which had successively the values 34.6 (3), 48.0 (2), and 49. (2), 56.6 (3) as values for (d4:2°H).10'*. The values between parentheses indicate the weight. 2266 and 2408 are split up. The amount of the splitting up cannot be measured, but amounts successively to less than 76 and 59. 2355 is probably split up. For antimony the air line 3640 given by Scurerrs is also observed as split up. Of this (dé: 2*H) .10'* = 66.2, weight 3. There are some among the Bismuth lines, for which we should be cautious when judging about the splitting up. In what follows I shall indicate by an s everything that refers to vibrations normal to the lines of force, a p marks what refers to vibrations parallel to the lines of force. 3068. The s-figure was on some photos a triplet of asymmetric intensity. Then the corresponding p-figure was a doublet, which 304 looked exactly like a reversed line, so that I supposed the s-triplet to be really a doublet of a reversed line, the two middle parts of which coincided. This appeared to be true when later on, probably through the increase of the self-induction, the s-triplet was trans- formed into an ordinary doublet, the p-doublet into a single line. Also a photo, made with an alloy in which there was less bismuth, presented this latter form, viz. an ordinary triplet. Compare also what Purvis says about this line. 4260. 1 suppose that the same thing applies for this line as for 3068. When the self-induction was carried up in order to get cer- tainty, this line became, however, too faint to allow us to draw conclusions. In favour of the supposition pleads that the splitting up of the p-components is as great as that of the s-components. 4122. This is in my opinion a different case. Exner and HASCHEK give two lines here, 4121.75 and 4122.08, the latter slightly heavier than the former. I too find two lines with a distance 0.24 A.U., and the same ratio of strength. They do not look at all like one reversed line. The s-figure is a triplet which slightly changes in form when the circumstances change (see table). The difference in resolution between corresponding components in the two cases will no doubt be owing to the difficult and therefore unreliable observa- tion. | think, however, that the difference in distance on either side of the middle component actually exists. A triplet is recorded for the p-figure. There is a blurred faint line, which towards red, and also but still fainter towards violet fades over some distance and then ceases more or less abruptly. | think, however, that I see a separation in some places. The measurements are of course worth little. Purvis states 72.3 for the splitting up. This is about equal to what I found for the splitting up of the outer components of the s-triplet. 4723. Wart MomamMmap gives a description of this line as far as its behaviour is concerned for weak magnetic fields, studied by the aid of an echelon.*) I myself find on the s records 2 middle com- ponents, by the side of each of which there is a broad smudge, which stops pretty abruptly. There is hardly any separation to be seen, which renders the measurement difficult. Nevertheless the different measurements of the outer components are in good harmony, those of the inner compo- nents not quite so good. The p-doublet is very close together, so that it is self-evident that a large comparative error arises there. ') GH. Watt MonAmMMaApD. Untersuchungen über magnetische Zerlegung feiner Spektrallinién im Vakuumlichtbogen. Ann. d. Phys. (4), 39, 225, 1912: Diss., Göttingen, 1912. 305 In the table the most violet component is marked by —, the most red by +. In this case the splitting up is not = 2d, as is given in the heading of the column, but = dà; 0 does not mean that the line is at the place of the original unsplit one, but only that there is a component there. 5. General results. When we compare the results in what precedes, it strikes us at once that though the general course of the results may be the same, there yet occur a good many devia- tions, which amount to more than the given probable error. Further it occurs occasionally that a quadruplet is mistaken for a triplet, which is not astonishing with the middle components of these often blurred lines, which are so very close together. That something similar may be the case with other lines which are given as triplet, is not impossible. An agreement as Purvis gives between the quadruplets Pb 3740, Pb 2873, Sb 3723, Sb 3638, and possibly Sb 2668, Bi 2989, does not exist according to my measurements, except perhaps that between Sb 3638 and Bi 2989. When we examine whether there is perhaps some connection between the resolutions of the lines for which Kayser and Runee ') find the known constant differences of frequency, it appears that there is no such connection. VAN LOHUIzEN *) has given series in the spectra of tin and antimony. The correctness of this is, however, doubted by SAunNpers *) and ArNoups *). | have now examined whether the laws of Preston were valid for these series. Of series I, which v. Lonuizen gives for the tin spectrum, I have found the resolutions (i.e. the values for dé: 47H, see table) 29,6 and 56,1 for the lines 3656 and 2785; in this series 2408 has a resolution smaller than 59. In series VII I find for the lines 3801, 2851, 2594, 2483, 2422 successively the splittings up 40,5, 42,5, 44,7 (56 and 46,8), 45,9. In this 2483 gives a quadruplet, the other lines give triplets. Series VII]: 3175 and 2483 give successively 69,7 (56 and 46,8) as splitting up. Antimony. Series XIII: the lines 3268 and 2574 have successively a resolution of 40,9 and 45. Series XIV: 3505 has the splitting up 63,7 and 2719 has 51.5. Series XX: 3233, 2653, 2478 have the resolutions 60,0, 63, 68,7. . Ay ED: KAYSER und ©. Ruree. Ueber die Spectra von Zinn, Blei, Arsen, Antimon, Wismuth. Abhandl. Berl. Akad. 1893; Wied. Ann., 52, 93, 1894. 2) T. van Lonuizen. Bijdrage tot de kennis van lijnenspectra. Diss., Amsterdam, 1912. 5) F. A. SAUNDERS. Astrophys. J., 36, 409. 4) R. ARNOLDS. Das Bogen- und Funkenspectrum von Zinn (von A 7800—4 2069). Zs. f. wiss. Phot., 18, 325. 306 For the other tin and antimony series I have obtained no line that was split up or only one. When we study the above results, we see that of the series for which it is possible to give an opinion in how far they follow Preston’s laws, only series XIII is not in conflict with these rules. In series VII the resolutions gradually increase; the first and the last splitting up lie farther apart than the limit of errors. Besides a quadruplet occurs there in the midst of the triplets. Even if these triplets were at bottom quadruplets, of which only the middle com- ponent was seen unsplit, which is very improbable, even then the amounts of the resolutions of the outer components would not be in agreement with each other. — The foregoing does not plead, in my opinion, in favour of v. Lonuizen’s results, though there remains a possibility that there are series that do not follow the rule of Preston. The efficacy of the Zenman-effects for the discrimination of spectrum series was very apparent when Runew and PascHeN found double lines in the spectra of Mg, Ca, Sr, Ba, which lines were changed in the magnetic field in the same way as double lines in principal and subordinate series in the spectra of the alkali-metals, while similar results were also found for some double lines from the Ra-spectrum. The hopes raised in 1902 and 1904 by the said researches with regard to the finding of series are accordingly not realized as far as the metals examined by me, are concerned. I have further also tried to find regularities myself. For this purpose and also in what precedes I have made use of a graphical representation, which seemed convenient to me. I have arranged the resolutions in every element according to their amount, and then plotted them on the same scale vertically under each other, as is usually done when resolutions are to be compared inter se. Two successive resolutions on the whole differing but little in amount, there arises a curve. A vertical part represents a number of equal splittings up. When the corresponding lines are indicated by the side of the resolution figure, we have at once a survey of all the lines that possibly belong to aseries. Compare the graphical representation. In this the resolution figures are only represented half, which, however, does not give rise to difficulties, as they are symmetrical. The components are not indicated by single lines; I have blackened the whole region, where they can be found according to the probable error. The vertical lines traced in the figure, are at distances = half the normal splitting up. A sloping straight part of the curve means that between two definite amounts of splitting up the resolutions are regularly distributed. When I examined this for the resolutions 307 of the four elements examined by me, | obtained four curves, each consisting for the greater part of a straight sloping part, which Graphical representation of the resolutions. lin, i pod: 4386 | 4145 4020 2802 4168 26137 5609 261% 3672 2873 4038 | 3573 2823 3740 2577 2833 3640 2663 4062 3683 | 2476 | | 2698 | (2718 | "| therefore means that for each of the elements nearly all the resolutions are pretty well regularly distributed over a definite region. For tin most resolutions lie in this way between 40 and 70, for lead between 45 and 70, antimony has a series of resolutions between 60 and 65, and further some in the neighbourhood of 70, of bismuth there exist, indeed, rather too few results, but these lie pretty regularly between 45 and 70. A distinetly vertical part, i.e. several equal resolutions does not 6. Tables. Tin. 2d” (H = 34930) 308 (d A : 22H). 104 vibration TLE Wrist 2 Ae 0.322 0.188 | 0.18 0.24 0.242 0.227 0.291 0.210 0.297 | 0.357 | 0.322 | 0.304 | split. up| 0.270 | 0.385 0.241 0.372 0.21 0.429 0.313 | 0.463 0.368 0.490 0.261 | 0.357 0.30 0.462 0.39 0.276 0.409 0.725 | 0.59 1.316 | split. up 1.02 | 2| EW WW AE EE NPN WwW Ww On wor NP KH WWWWWWHk DY WW WW WW vibration (Aor t: my own observations 2d A .202 | 3 14 | 0| 302 3 | || | | vibrat. S mhoff. 56.1 48.9 68.5 42.5 65.0 35 67.9 48.7 qe! 53.2 69.7 36.1 48.2 40 59.6 49 29.6 40.5 50.6 40 88.5 split. up | 47 split. up. | vibrat. | vibrat. ont | I Oe. 46.8 (=) ee) Sey Ser SS OS Ss SS OS) SS Ss Ss Ss OS 3 oo o0co 0908 © O&O © Purvis’ observ. da sar REMARKS. vibrat. // 1. of Ff. [4g74 0 The splitting up is about as greatas that of 2814. D lep) — oO Lead. 309 : | vibration | een 1 ont Pr | 2a A") fs L | 2476 | 0.294 | 3 2571 | 0.277 | 3 2613.7) 0.23 | i 2614 | 0.24 | 2| 2663 0.308 3 2698 | 0.414 | 3 2713!) | 0.426 | 3 | 2802 | 0.249 3 2823 0.318 3 3883 | 0.34 | 2 2813 | 0.297 | 3 | 3513 0.492 | 3 35162) | | 3640 | 0.57 |2| 3612 | 0.481 | 3 3683 | 0.609 3 3140 0.576 | 4 4020 0.501 3 4058 0.646 | 3 4062 | 0.729 | 4 4168 |0.57 | 2 4245 0.531 3 4386 | 0.543 | 3 5609 1.706 3 !) ee line. ; | 2da(H = 32810) So. Or OO SWO OO Sr ONES DO Or OF ORO SO Oo GG vibration hi LAGE. dlg = alee a)! 21 | 2| „113 | 3 | | si) 2 PAIRS 2208) 03 | ‚468 | 3 | (dA: A2H). 1 014 | Purvis’ observ. | my own observations d A ECE H Sal 1 A” 13.2 63.6 51 53 66.1 Bn 88. 48. 60. 64 ao oOo +s Oo 54.9 58.9 2) Should this perhaps be 3573? glu 0 30 46 0 0 ONE 33.5 | 46.4 0 41.5 36 | 63.0 40.8 | 66.0 0 62.1 0 | 66.0 ODIE 0 69.2 22.6 | 64.4 0 0 | 57.6 o | 68.2 40.8 0 42.2 oO. A 0 0 vibrat. | vibrat. | vibrat. vibrat. | Lloff. | //1. off.) 11. off. | // 1. off. REMARKS. L components: pro- ( bably there are 4lines here, 2 distinct ones and 2 faint ones; the | two middle ones al- | most coincide. Proba- | bly the 2 faint ones | belong to 2613.7, the two distinct ones to 2614. The _ distance | middle faint one to ‚ middle distinct one —0.45 Â.U „the distan- ce according to Klein’s table between 2613.7, fe} | and 2614 — 052 A.U, which corresponds | with what I measure ‚on the photo without field. The agreement | 0.45 and 0.52is bad, but was not to be expect- ed better in view of the components of 2613.7, which are so difficult to measure. // components: The splitting up 130 of 2613.7 rests on only one, very inaccurate measurement ; the | distance from middle to middle amounted ° | to 0.49 A.U. 310 Antimony. | 242 (H = 26290) | (d a: 22H). 1014 _ | vibration | vibration my own | eyes Pel “\ hoff. | //Loff. | observations — 3 : 2.87 Sea ah | 2da |g| 2da lg | Vort left| a Loft oft. | | | | nd 2311. 0,23") 2.0 | Bt 0 2446, 0.152 3 0 | 48.7 0 2478, 0.220 3 | 0 | 68.7 0 2529) 0.20 2/0 | 60 0 55.2 | 0 2574| 0.15 | 2/0 45 0 2598! 0.19 2/0 54 0 55.5 0 2612} 0.217 3 | 0 60.9 0 2617 split.up 0 split. up| 0 2653; 0.23 | 2| 0 63% 3/650, | 2669 | 18.4 34.7) Aretheseperhaps 2671 0.234 3 0.207 |3 628 54.5 | en 2683| 0.229 | 3 | 0 60.9 | 0 | 2719, 0.199 | 3 0 51.5 0 2727021 |,0.| 0 73 0 2770, Med | AiG FO 2851 split.up, 0 split.up 0 2878'0.367a.0 3 0.160 3 |+84.8a.0 36.9 +77.0a.0| 36.4 3 1 and 2//com- 3030| 0.310/4/0 | 64/8. 1 PDP hed. Iep POLES 3233| 0.321 | 4 | 0 60.0 | 0 GORT Hee 3248, 0.309 | 3 | 0 5622 SO 3268 0.233 | 3 0 #,c4059 10. | 3274). 0535-2 | 0.19.0). 251, °. 62 34 3383} 0.571 | 3 | 0 |) al OBA. tae 3505' 0.408 | 3 0 ER er a oe 3638} 0.484/3/0.19 |2| 70.2 | 27 13.2 | 34.4 3683, 0.496 | 3,0 lee HROS ea 3123 05231 405203 3 | 92.3.4) STEE 16-10 4034. 0.28 2 0 33 0 4195 splitup | 0 ‘split. up| 0 4219 0.65 0 0 Log 0 4352) 0.496/3;0 | | 50.0 | 0 | | | | 314 Bismut. Ee SS a ee ee ees 2d A (H = 26290) | (d A: A? H). 1014 | ie vibration | vibration | my own oe RC Si REMARKS kof f, Hak OP Emel observations | a2 > 3.11 RES. | | 7 =) n= mnd — en oo —— i ee — = vibrat. | vibrat. | vibrat. | vibrat. REDE hee And Lon Toll! Hori lL Mork. V1.0 2628} 0.175 |0)| 0 48 0 2898) 0.247 |3| 0 56.0 | 0 (reso, +00 | | | | 2938 0.286 3 0 Gg. 2) }* 0 et) Zenit ag 2989} 0.314 |3/| 0.124 | 2 66.7 26 | 64:0: >| 2654 2993, 0.260 3 0 | 55.3 | 0 Bless 2) 0.0 | 3025] 0.292 | 3 | Split. up? 60.7 | split. up? — 50.2 26.5 | p figure, i.e. | think I find | | | | | the line broadened, on one 3068 0.346 3 0 | 69.8 0 Bend wale is probably Det AAG: Ees Held an too melt 3077) 0.242 | 3 | 0 48.7 0 | for ki line to be seen 3397) 0.241 13) 0 30.8 0 Sho) Deere |= | 3511| 0.461 |3\ 0 ein dites 0 | 3596) 0.321 |3\ 0 | "34782 0 ARGU Wd 3793; 0.285 2 0 38 0 | = KS | | |H = 26290 G ; self 4122|0-99, 0, | 5 —0.32,0, | | — 33,0, + 38 |_35,0,4-34 | andvenonad millitenty: + 0,34 1) “.|4+.0.31 | ’ nog lone —0.28, 0, | | | ‘ ; | H = 30500 Gauss: self 4122! 0.31 2) | 2 too faint I— 31,0, + 35} too faint | | induction 0.8 millihenry. —0.47, 0, | | | bly a triplet of se OA 2490450 49 | "reversed line so that the | : so | Be ed 20.21, | 264 en 30 (8 wfiO.band20.6l 20.9) ts ne en 5145) 0.83 (0. 0 | 60 br | | | | | ') The reddest component is somewhat stronger than the two others, which are equally strong. They are equally sharp. 2) The 2 outer components are about equally strong and sharp, the middle one is fainter and very vague. 3) The middle component is the strongest. The most violet component is fainter than the most red. Only when made more distinct by scratching with a needle this violet component could be measured. 312 occur. Hence in the search for series by the aid of equal resolutions hardly any result can be obtained by means of these curves. I have also tried it for tin and for antimony lines, which had a splitting up between 60 and 65, but likewise with no success. [ further examined whether there existed simple relations between the distances of the components for more than triple resolutions, which were rather accurately measured (weight 3 or 4). For Sb 3740, Sb 2671, and Sb 2878 1 did not find simple relations. For Sn 3331 the distances of the components are in the ratio of 3:2, when the values 59,3 and 39,5, which lie within the limit of errors, are taken for them. For Pb 2802 the ratio is also as 3:2 with the values 49,1:32,7, Pb 2873. gives 54,7:410—4:3; Sb 3723 gives 72,6 : 36,3 = 2:1. There is, however, no mutual connection or a simple relation to the normal resolution. The results may be briefly summarized as follows: The Zreman-effect was measured of 35 tin lines, 23 lead lines, 27 antimony lines, and 16 bismuth lines. In this deviations were found with Purvis’ results which refer only to a few lines, probably in consequence of his less accurate measurements. No relation was found between the resolutions of those lines of the examined metals for which it was proposed to arrange them in series or those which are arranged according to the laws of the 2°¢ kind of Kayser and ~ Runer. The resolutions are pretty regularly distributed between values which amount to about 1 and 1,5 times the normal resolution. As an incidental result [ found that a number of lines given by SCHIPPERS for antimony, are not real. Mathematics. — “On a remarkable functional relation in the theory of coefficientfunctions’. By Dr. H. B. A. Bock winker. (Communicated by Prof. H. A. Lorentz). (Communicated in the meeting of September 27, 1919). 1. Let g(t) be a function having no singular points without the circle (1,1), i.e. the circle with centre t=1, and radius 1. Let g(a) be zero and the order g of p(t) on the circumference of the circle (1,1) be different from —+ oo. Then in the series Ssh, De gp (t) == Dn fl ; 5 e e A @ ° 4 (1) 0 the characteristic k= lim [log g, :logn] of the coefficients g, is n= also different from +o, in virtue of the known relation k—=g—l. If g <0, then the integral vO fy (ie sige olene zakt a (2) a JEL (1,1) taken along the circumference of the circle (1,1) evists for R(x) > 0, because in that case the series (1) converges along that circumference: the value of #1 in it is so defined that the argument of ¢ lies 7 zie with zero limit. The function (x) is called coefficientfunction by PINCHERLE'), Owing to the relation g, = w(n+1). Conversely p(t) is called the generating function of w(x). PixcHertr considers the relation between these functions especially from the point of view of the functional calculus. If we write lope Slower vera artist Lis an additive functional operation, which satisfies a certain number of simple functional relations; these relations may be used in order to define the coefficient-function in those cases in which the integral (2) does not exist. Thus we find easily Iltg()]=o(e¢+ 1), and 1 p'(l) = —(#—1) w (el). . . (4) and by combining these two equations 1[¢'()] = — ad) Lt 0), 1) Sur les Fonctions Déterminantes, Ann. de |’Ke. Norm. (22) 1905, (Ch. IV). of continually between — 5 + d and - J, d being a positive quantity 314 which, by iteration, passes into ll, [*—"g (5] FP (¢—r) It is easy to see that the latter equality is also valid for negative integral values of r. It is, however, remarkable that the same equality holds for not-integral values of r. This property I have made use of in the investigation of a function represented’) by a binomial series 140 (Q)=(-y x—l ze ) the most typical series in which a coefficientfunction n can be expanded. The object of this note is to give a proof of the general validity of the equality in question. 2. We substitute —a for r and replace g(t) by the expression p(t): (—1), in order to have always to deal with functions which are regular for ¢=o. In accordance with Rigmann’s *) definition of the derivative of negative order —a of a function we assume as such the following one ol) te (—1) D-=- — = B (¢—1)+ (a) (uw — 1) t In this we take as path of integration the half-line beginning at u=t, whose prolonged part passes through vw == 1; and we assign the same arguments, lying between add and +2—4d, to u—l and w—t. Then the so-defined derivative is also regular without the circle (1,1) and zero for t=; and by the substitution u—1=(t—1):s we find after a slight reduction the expansion (Ie De pd _ a Du +Vgn (t— 1) met Dn +1 4+ e)(t—1)4! be 0 (6) which, therefore, is related in a simple manner to the expansion for p(t itself. The order of this derivative, as will appear imme- diately from this series, is « less than that of (dé), and therefore negative together with the latter order. On this supposition we may apply the operation / to it in the form (1), so that by this the evistence of the first member of the equation to be proved, ‘viz. Tye ande (7) (le) Mete) [ID] 0 1) These Proceedings Vol. XXII, N°. 1. Nieuw Archief v. Wisk. XIII, 2e stuk (1920). *) See BoreL, Lecons sur les séries a termes positifs, p. 74. The constant a occurring there is taken equal to here, in connection with the regularity of g(t) for t=. 315 has been laid down. Passing to the second member we write l v it a dae ar where the argument of s and of 1 —s is zero. Further t=p(t) “tx +2—ly(t) cE dt, (t—1)+ T Oni (t— 1) (1,1) so that the second member in question is equal to 1 BN Ca city a | acne (t—1)¢ | Cr | i 0 (1,1) If we substitute in the second integral s =u: ¢, this expression passes into t : POUR A iG (t—1)¢ LS '¢— ue! in| ne (1,1) Since the argument of s was zero, the argument of w is equal to that of ¢; thus the variable under the sign of integration goes along the straight line from w=O0 to u=t in the v-plane. But it may go as well from u = —i), on the circumference of the circle (1, 4), along that circumference in the positive direction to the point u == t. On this supposition we consider the system of two integrations, to be performed in succession, as a double integral. Then in the corre- sponding aggregate (w, t) a definite value w, of « has to be associated with all those values of ¢ lying, in the ¢-plane, on the circumference of the circle (1,1) between tw, and the end-point t= + 20 of that circle. Hence the double integral may be replaced by the pair of two successive integrations denoted in the expression 0 dt | du, amin) ki ad where the Pee ah pen to ¢ has to be performed in the positive direction from t—=u to t= + id"). On account of the pro- perties of y(t) the latter integration may be replaced by an integration 1) The here given argument is strong, in so far it is based upon known truths, if the functions under consideration are finite in the whole domain of integration. This is the case for R(x)>1 and R(x) > 1, but, since both endforms are analytic functions of x ‘and of « for R(x) >0 and Ria)>O0, they must also be equal for the latter values. 21 Proceedings Royal Acad. Amsterdam. Vol. XXII. 316 from tu to t=, and one from t=a@ tot=0. The latter gives an amount which is independent of u, and this amount gives zero for the final integration. Therefore, after changing the letters u and 1, we may write for the preceding expression erk (Cm LO zailin af e= (u—1)* dn Ja (1,1) and this, if we take (2) and (5) into account, is just equal to the first member of (7). The latter equation thus has been proved in case g <_ 0. If g>0 and, to begin with, O k=2m—1 plngy + hy) — & (ng = Tile, ‘p)=— ro “i Batti \ p84. In this equation we have t a hp g Y= 3 BE kt NE () (0) == (+) (Di) cot v) Ld mhp 9 ape and the remainder integral A is given by “a R = j— gem vn | 0 Now since h dom ( Es =) — J2m (t) | g (3m) ( q y) dt. q k=+a 1 (2m) ¢ Den DL = =e pen)(t g 1) << 2m ese mes S k=+n 1 1 2m! = TENCO Bak we infer that R\<2\gem (0)| 2m! g2—1 2m ait tl Mei oa Te er Coherence or that RIS me | ga, (0) | gt layer gE (Oe Thus it is shown that |# is less than a finite multiple of the modulus of the last term in the sum that precedes, and we have found the asymptotic expansion ht (iN he An Ti, («,p) = = (-) gl log— frl —](D cot) rip + an k=0 2 EE q = 1 \2e—1 a Kym ( og =| 8 & where the value of is finite and independent of «. Putting h==q—1 h Ag == > Tk (=) (D® cot») _ aps il = —— q 327 we have finally k=2mn—2 k+1 Be L (20?) —qIAet*) = — Hal) SE (5) (loes) ‘| T k=0 | \ ml + kK, gem (tog =) : wv and again A, is finite and independent of z. From (7) we conclude that, if # —1, the function Z (v/v) tends to infinity in the same manner as (7) 1 C — log log— — 2 log g wv ae) - —— +49, q log — . v and that Lim { L («6r) — gL («*)} = —1iq—1)—— = hcot—. (B) 11 2q =d q Thus we have the rather remarkable result that it is only the real part of (v6?) that increases indefinitely in a manner quite independent of p. 2. If we only wish to shew that, when « —1, the function L(v0*) cannot remain finite for all non zero values of &, an element- k=q-—1 ary discussion of the sum > LZ (ef!) suffices to obtain this result. k=0 We have at once k=q—1 n= o > L(#0k) =q = t(ng) ero, k==0 n= and, denoting by D the greatest common measure of n and q, we may substitute n q t = > (dara — ao (3) (3) a Lee Eno! ) Leers. thus DE k=0 Hence, making « tend to 1, we get k=q—1 Lim SL (Gey zi Lle) po ra d where (d) denotes the number of integers less than d and prime to d. Supposing q to be equal to the product p, ! p‚?...pss, we have 328 h=s == 3 09 _ FT (1 se Jer djg d h=1 Ph and therefore, if «—41, for at least one value of & other than zero _L(a@0*) must tend to infinity. 3. It will be readily seen that the method used in obtaining the asymptotic expansion of L(z) can be applied also to the series n= zn NEE bu, if only the coefficient 6, is a simple analytical function of the index n. If we choose, for instance, 1 bn = _- Sa) n we have n= | n n= o M (2) = — J — = = log (l—z") = log M(1— 2") n= N Wen n=1 it and we can put h=q—1 M (« Or) — M (er) = logg+ = Vi (e,p), it where Vile ap) = = {log (L—ara +h Oly) — log (1—2"dGhp) }. n=0 Operating as before we shall find | A=q—1 M (« Or) — M (a?) =logg—— = 7 h h h log 2 sin? | + zina) = | en q jl 1 I 1 k=2m—1 1 k ; I ili k ‘ 1 \2m en uses = Al! ed A oaf ‚gam ni eee (3) ators) + ua (10g ) x where h=q—-1 h Ay aie (~) (D&—-) cote) zap, h=1 q U pie and K, has a finite value independent of «. Now for Eurer’s product ILA”) we have, when 0<2 <1), 1 4n3 — ] 1 : 1 se" lege AES tr rr SORT ADE — + Mle *), 6 log — z 329 hence equation (9) shews completely the behaviour of M («x Op), when v—> 1. In fact, we may write ; | hg …_ hp) | hp Lim} M(«6v)—M(«?)'= logg—— > shlog |2sin — tri (10) x—>1 Q hal | q | g and again we may notice that only the real part of M (x Op) becomes infinite, the imaginary part tending to a finite limit. Taking, for instance, p=1, g = 2, we shall find Lim { M(— «) — M(a*)} = } log 2, 1 or IT (1 + «@?n—1) Bid, ra ra 1 a known result in the theory of tbe 9-functions. 4. Finally, I will state that the discussion of the fundamental equations (1), (2) and (3) furnishes the proof that the function N(z) = Eb, Teil 1—z" cannot be continued beyond the circle |z| =—=1 in each of the following cases : Beb B0. In this case we shall have A om 2s lines q ie 1 log Ne >< ee We bim-b,, A0: n= Now it will be seen that d—= A KiM 2-1 if q log lx … br Il. Lim 2=A 40, s>0. no ns In this case the equation holds A x1 q +s 330 no b IV. Lim }, =0 and either }, > 0, orthe series > —% converges — x n==1 nq for an unlimited number of integers g and the sum of this series is different from zero. In the latter supposition we have n= Ong 3) Lim (1—2) Ne) — = Z, mnd | n==1 1G V. 6,20 and moreover ) l Lim — [both + baga +... + dng pi] = 0 n— Po 7 (SON reeel) for an unlimited number of integers q. 1) By a totally different method FrRANcK deduced this formula in his paper: ~ Sur la théorie des séries. Math. Annalen, Bd. 52, 1899, p. 529. Mathematics. — “Bestimmung der Klassenzahl der Ideale aller Unterkörper des Kreiskörpers der m-ten Einheitswurzeln, wo die Zahl m durch mehr als eine Primzahl teilbar ist”. Aster Teil). By Dr. N. G. W. H. Bercer. (Communicated by Prof. W. Kapreyn). (Communicated in the meetiug of September 27, 1919). In zwei früheren Mitteilungen ') habe ich den Ausdruck hergeleitet für die Klassenzahl der Ideale eines jeden Unterkörpers des Kreis- körpers der /'-ten Einheitswurzeln. Im vorliegenden Aufsatz gebe ich die vollständige Ableitung desselben Ausdrucks für alle Unterkörper aller übrigen Kreiskörper. Der entgültige Ausdruck bat dieselbe Form wie der bekannte Ausdruck für die Klassenzahl des Kreiskörpers selbst. Nur für die zu eyklischen Untergruppen gehörigen Unterkörper des Kreiskörpers ist die Klassenzahl schon vorher bestimmt durch Fucus’). Seine Endformel haben nicht die einfache Form, die ich jetzt für alle Unterkörper bewiesen habe. Seine Untersuchungen könnten sich natürlieh noch nicht griinden auf die jetzige Auffaszung der Theorie. Im Folgenden benutze ich fortwährend den Hi.Berr’schen “Bericht über die Theorie der algebraischen Zahlkérper’*), den ich weiter andeute durch “H”. Weiter benutze ich die Theorie der AprrL'schen Gruppen aus dem “Lehrbuch der Algebra’ von Weber, welches ich andeute durch “W”. Gemeint ist stets Band II. L Der Kreiskörper der meten Einheitswurzeln. $ 1. Die Substitutionen des Körpers. Sie bilden eine Ager’sche Gruppe, die isomorph ist mit der Gruppe der Zahlklassen (mod m)*). Sind also A AE oo aL) die Zahlen < m und prim zu m, und stellt man die Substitution 1) Diese “Proceedings” Vol. XXI p. 455 und 758. % Cr. Journ B. 65. S. 74. 8) Jahresb. d. D. M. V. B. 4. 4) “W.” S. 74 und § 18. 22 Proceedings Royal Acad. Amsterdam. Vol. XXII. 332 ant (Z,Z4i) , Zen vor durch $;, so sind Sie eee 2 eee ewe ee die Substitutionen des Kreiskörpers. Zur Abkiirzung schreibe ich » statt p(m). Die Gruppe der Zahlklassen (mod m) können wir also kurzweg die Gruppe des Körpers nennen. Und wenn wir Unter- gruppen von (2) brauchen, kénnen wir die isomorphen Untergruppen von (1) dafür nehmen.') Die Gruppe (1) bezeichnen wir weiter durch G. Die Zahl m schreiben wir in der Form m = 2e LAr dots . wo 1, J,,.... ungerade Primzahlen bezeichnen. Um die Agrr’sche Gruppe G durch eine Basis darzustellen *) nehmen wir die primitive Wurzeln 7,, 7 resp. von. lela es ge Wir bestimmen die Zahlen A,, Ay, A, .... durch die Congruenzen : A, = —-1 (mod 2'*) = 1 (mod 1,41) =1 (mod lts). A,= 5 5 =1 3 <== os A == i ie fn = - Ao el Bs =1 ie =r, RE Diese Zahlen bilden eine Basis von G. Jede Zahl aus G kann also nur auf eine Weise geschrieben werden in der Form Arto Aue ATH (nod m) wobei die Exponenten durch die Bedingungen OS uy SA OSU Shi OCU pii bestimmt sind. Zur Abkiirzung ist dabei für gv (2%), g(l/).... ge- schrieben worden: x, 9, . Wir machen noch die folgenden Bemerkungen : 1. Wenn kh, =O so fallen A, und Ay weg, 2. Wenn hy = 2 so fällt A, weg. 3. Der Fall Ay = 1 kann für das Folgende ausser Acht gelassen werden °). § 2. Die Zerlegungs- und die Trdgheitsgruppe eines Primadeals. Satz 1. Die Trägheitsgruppe eines Primideals 9 das nicht teilbar ist auf m, besteht nur aus der identischen Substitution. 1) Wir sprechen weiterhin also von der Substitution a, welches bedeutet die Sub- stitution die Z durch Ze ersetzt. 3) ,W” § 18. BW 8 20. 333 Beweis: Wir wissen, dasz am — 1 = (#a—1) (a—Z).... (a). Es ergibt sich hieraus, nach der Division dureh «—1, und nach der Substitution w=l: i A Ee a de rata Jeder Factor des zweiten Gliedes is also nicht teilbar durch }. Es sei nun $=(Z: Z*%) eine Substitution der Trägheitsgruppe; dann musz *): S Z= Z (mod Y) also : Z (Zal — 1) = (mod W) und dies ist nur möglich wenn a= 1]. Satz 2. Die Zerlegungsgruppe eines nicht in m aufgehenden Prim- ideals, das auf die Primzahl p teilbar ist, besteht aus den f Potenzen: Pp, p,...+ pl (mod m) wo f der kleinste Exponent ist, für welchen p/= 1 (mod m). Beweis: Dasselbe welches man findet in “W” S. 742 für den Ball m = 1". Bevor wir nun die Tragheitsgruppe eines Primideals bestimmen können, müszen wir Satz 146 von ,,H.” vervollständigen. Satz 3. Im Kreiskörper finden die folgenden Zerlegungen statt: u ’ Ln he 1. 1—Z == (atic, ode) fir A,’ =0,1,....,h,—1, und n < 24s während n nicht teilbar ist durch 2. m an ma 9. 1—Z " = (fi... bie) -—,.! si . - : Mir i NE LN t_ während n nicht teilbar ist durch &. 3. == (Lo1 meee Éoe,)?* wo @; Primideale sind und /, der kleinste Exponent bedeutet, für welchen 9/* = 1( mod ze) und é& /,—=—¢ ( } Der Grad der Prim- ideale ist /,. | 4. l; == (tie) 3.8 Bie) wo ¥;; verschiedene Primideale sind des Grades /; . f; ist der kleinste Exponent fiir welchen 1) ,H.” SP 251. me Qi 22* 334 fi m F mn l =11 mod ae und ei li = (Gr Li Pee Alle Zahlen, welche die Form 1—Z9 haben, und die nicht sich vorfinden unter 1° und 2°, sind Einheiten. Beweis: Die unter 3° und 4° angegebenen Zerlegungen der Prim- zahlen 2 und /; sind bekannt. 5. Aus (1) $ 2 ersieht man dasz jede Zahl der Form 1—Z9 eine Einheit ist, oder nur teilbar ist durch Primideale die in m aufgehen. Jari Yeh Im Körper e ee die Zerlegung *): m Ahl A es ays “ Ieke 2=1,7 indem {, =(1 ages ) ein Prim-Hauptideal ist. Hieraus ergibt sich die unter 1° angegebene Zerlegung fiir den Fall n= 1. ant TE = Weiter ist im Körper k („7 7) m En —— pt Gt") re NY Se ein Prim-Hauptideal ist. Hieraus ergibt sich die unter 2° angegebene Zerlegung für den Fall n = 1. Um die übrigen Zerlegungen von 1. zu beweisen bemerken wir rE a) Z haha! dasz im Körper t ) die Zerlegungsgruppe des Primideals !, aus allen Substitutionen dieses Körpers besteht. Diese Substitionen m ersetzen die den Körper bestimmende Zahl Ze", durch die Poten- zen dieser Zahl, deren Exponenten alle Zahlen < 2—* sind welche nicht durch 2 teilbar sind. Es folgt hieraus die unter 1. angegebene Zerlegung für alle Falle n > 1. Auf gleiche Art findet man die übrigen, unter 2. angegebenen Zerlegungen. Wir miiszen nun noch den letzten Teil des Satzes 3 beweisen. Das Produkt aller unter 1. und 2. genannten Zahlen der Form 1—Z, ist teilbar durch eine Potenz des Produktes ¥o1.... oe, deren Exponent gleich 1) „H". Satz 125. 2) „H”. Satz 122. 335 @ (Qhe) + Zp (Aal) + 1... 4 iel p (2) = hy Dy ist, und auch teilbar durch eine Potenz des Produktes ¥i1.. Gie,» deren Exponent gleich PED Hip De =p UD ist. Man findet leicht dasz auch m genau durch diese beiden Produkte teilbar ist. Aus (1) ergibt sich also dasz alle Zahlen der Form 1—Z), welche nicht unter 1. und 2. zerlegt sind, Einheiten darstellen. Satz 4. Die Trägheitsgruppe eines in /; aufgehenden Primideals & besteht aus den p; Substitutionen welche die Zahl Zersetzen durch Potenzen von Z deren Exponenten die Zahlen sind, welche nicht teilbar sind durch /; und prim zu m, und die Form haben: m ia Ene ril, eek, nlt os he! t Dasselbe gilt fiir 7; == 2 wenn man überall den Index 2 durch * ersetzt. Beweis: Eine Substitution S ist dann und nur dann eine Substi- tution der Trägheitsgruppe wenn für alle ganzen Zahlen @ des Körpers S 2 = 2 (med ¥) *) ist. Nimmt man nun 2— Z so ergibt sich leicht der Beweis des Satzes. Satz 5. Die Zerlegungsgruppe eines in /; aufgehenden Primideals ¥ entsteht wenn man die Trägheitsgruppe multipliziert mit den #: ersten Potenzen der Zahl Beens i NS Ih wo n so bestimmt ist dasz die angegebene Zahl diejenige von G m Seid / ist, die congruent ist mit /; (mod 1 ist, und nicht jede Zahl 5, durch 7,—1 teilbar ist wenn h‚=l ist; u.s. w. Der Grad der auf diese Art bestimmte Unter- gruppe ist gleich r. Es ist leicht ersichtlich dasz man die Bedingungen für die Zahlen «a, «,,.... in folgender Form schreiben kann: bp (m) ao bon + 2y (a ‚eeb an + YP (ee i, Je bin +.....= 0 (mod p) (3) eee ee . Jedes System a,, dy, a,..-. genügt also allen Congruenzen (3). Die jetzt bestimmte Untergruppe der Zahlen A deuten wir ferner an durch g. Der zu dieser Untergruppe gehörigen Unterkörper durch k. Weil g primär ist, können darin die Zahlen welche m m =] | mod — | oder (moa oe 2 Es sind, nicht vorkommen. Auch nicht die Zahlen die durch Poten- WG 56, 338 zierung (mod m) auf eine der jenigen Zahlen v,y,.. führen, die den Bedingungen m m v= (raa) y=! (moa) ie gentigen.’) Wenn die Zahl m den Factor 2 genau einmal enthält, so gibt es keine primäre Unterkérper’). Wir haben also nur die Falle hy = 0, 2 of > 3 zu betrachten. In einigen künftigen Beweisen nehmen wir nur der mindest einfachen Fall hy D3; es ist dann leicht einzu- sehen welche kleine Abänderungen der Beweis für die anderen Falle erfahren musz. Wir machen nun noch die folgende Bemerkung, die spater von groszem Nutzen sein wird. Die Karaktere einer Untergruppe g von G stellen selbst wieder eine Gruppe dar, die isomorph ist mit der Untergruppe*). Der Karakter eines Elementes A der Untergruppe ist aber bestimmt durch das System der Zahlen Boy Oss Ayn - Diese Systeme bilden also eine Gruppe die isomorph ist mit der Untergruppe g. Die Zahlen B,, welche bestimmt sind durch die Congruenzen BL, = A,’ Ayo Ain, …. (mod m) wo die Zahlen Ay Gg, ere UNG bons Oaneleiiaw rens verbunden sind durch die Congruenzen (3), stellen die reciproke Untergruppe von g dar.*) Die Systeme bons Osas Glny = = bilden wiederum eine Gruppe die isomorph ist mit dieser reciproken Untergruppe. Also können wir sagen: die Systeme OENE NCAA stellen eine Gruppe dar, deren reciproke gebildet wird von den Systemen bons Cans Olne wenn sie verbunden sind durch die Congruenzen (3). Das Produkt der Grade dieser beiden Gruppen ist also stets gleich p*). 1) |W.” § 20 und 21 2) |W.” § 20. BENE ILS. 4) ,W.” S. 56, 8 MWe" Sr 55, 7 339 § 4. Hilfssdtze. 1. In einer primären Untergruppe gy, finden sich, wenn A, > 3 ist, keine Zahlen der Form m 1+n > 5 WO hy Dl > 1 oderh', = 0. nist ungerade. Und wenn hy = 2, so finden sich keine Zahlen vor, derselben Form, wobei h', =1 ist und n ungerade. Auch finden sich in g keine Zahlen der Form 1 + —— wo h > hi > 1, n nicht teilbar durch AG. De Beweis: Die Zahlen der ersten Form sind, für den Fall dasz he =O ist, gerade und kommen daher in G nicht vor; also auch nicht in g. Ist hy=2 und hy’ =1 so kommen die Zahlen in g nicht vor wegen § 1. Wenn nun eine der übrigen Zahlen der ersten Form in der Gruppe g vorkam, so wiirde es ein System ganzer Exponenten a, dy. a,.... geben, die den Congruenzen (3) genügen und für welche er) ny a, aS m JO h a EN (mod m) Es würde sich hieraus eine Congruenz (mod 2!'s/,h...) ergeben und folglich wiirde, nach der Definition der Basiszahlen : ay A, “= 1 (mod I," Hieraus würde folgen: a,=a,—....=0. Man bekäme also: Me he oder he’ en den und hieraus a, — 2/4 a, und a, =—=0, weil A,’ >1. Substituiert man dieses Ergebnis in den Congruenzen (3), so findet man Zhe’ 1 ay! Dyn = 0 (mod Qhx—1) Weil y primar ist, sind nicht alle Zahlen 6,, durch 2 teilbar; darum folgt aus der letzten Congruenz dasz ax teilbar ist durch Maha, Die Zahl ay würde daher teilbar sein durch 2%-? und dies ist unmöglich da ay kleiner ist als diese Zahl. Auf gleiche Art beweist man die ganze Aussage des Hilfssatzes. 2. Es sei /, eine der in m aufgehenden ungeraden Primzahlen ; £ ein Primidealteiler von /,. Der Grad der gemeinschaftlichen Unter- 340 gruppe von y und der Trägheitsgruppe des Primideals £ ist gleich d, wenn diese Zahl den gröszten gemeinsamen Teiler der Zahlen b,, bedeutet. Ist h, > 3 und £ ein in 2 aufgehendes Primideal so ist der Grad dieser gemeinschaftlichen Untergruppe gleich 1 oder 2 jenachdem bon + ban nicht für jeden Wert von n gerade ist oder wohl. Ist hy = 2 so ist der Grad der gemeinschaftlichen Untergruppe = 1. Beweis: Aus Satz 4 und Hilfssatz 1 ersieht man dasz die Zahlen m der gemeinschaftlichen Untergruppe nur die Form 1 +7 ia haben 1 1 können, wo n nicht durch /, teilbar ist. Wenn nun eine solche Zahl zu g gehört, besteht ein System ganzer Exponenten d,, ax, @,..-- für welches As AN en a (mod m) a und das den Congruenzen (3) geniige leistet. Auf gleiche Weise wie beim vorigen Beweise ergibt sich hieraus, dasz alle Exponenten, ausgenommen «,, gleich Null sind. Aus den Congruenzen (3) folgert man dann m p (=) a, bin = 0 (mod y)) ae oder ay bin = 0 (mod p‚) Je Da nicht alle Zahlen b,, durch /, teilbar sind, ergibt sich hieraus (fp dasz a, teilbar ist durch =. Man beweist leicht dasz diese Bedingung En auch genügend ist; denn aus ae AP ROA AY Sx (od) eee ea) ergibt sich: m 1 =+( mod 5) hy L, also 2 == 1 fo Es ist hier m nicht durch /, teilbar denn im vA 1 1 Gegensatz ware „== | (mod l,) (bp und hieraus würde sich ergeben, weil A,=7, (mod /,), dasz Le, 1 341: teilbar sei durch /,—1; also würde a, teilbar sein durch ,; dies ist unmoglich, da a, << ¢, ist. Betrachten wir jetzt den zweiten Fall des Hilfssatzes. Die Zahlen der gemeinschaftlichen Untergruppe können auf Grund des Satzes 4 ap | en m | und des Hilfssatzes 1 nur die Form 1 + n Is haben, wo n unge- rade ist. Wenn diese Zahl zu g gehört, so besteht ein System ganzer Exponenten qd, 4, ... für welches ag Ce ay, ss a Assen ek RAS RY Gy eee eo 3.) Es folgt hieraus wiederum dasz alle Exponenten, ausgenommen die erste Zwei, verschwinden. Die Substitution in (3) ergibt Dhe —3 ao bon + ay. (ee = 0 (mod 2he—?) . : . . : (6) Wir unterscheiden nun zwei Falle: A, >3 und Ah, = 3. Im ersten Fall ist a by. = 0 (mod 2'+—3), Weil nicht alle Zahlen 6,, gerade sind, folgt hieraus a, — 24 —3 Die Congruenz geht dann über in ao bon + aw bn = 0 (mod 2). Nun ist weiter ay < px, also a’, < 2. Wir brauchen daher nur zu. untersuchen welche der folgenden Wertecombinationen den letzten Congruenzen für n = 1, 2,... na genüge leisten: 7 de 07-4," == 0 0 1 1 0 1 1 Die zweite Combination, geniigt nicht, da nicht alle 6,, gerade sind. Auch die dritte nicht. Die erste geniigt immer, und die letzte nur wenn 6,, + bn fiir alle Werte von n gerade ist. Auf gleiche Wiese wie vorher zeigt man dasz diese Bedingungen genügend sind. Ks bleibt noch die Erledigung des Falles h, = 3 übrig. Es geschieht dies auf ganz ähnliche Art. 3. Es sei p eine nicht auf m teilbare Primzahl und jf der kleinste Exponent für welchen p/=1(modm) ist. Es sei d der Grad der gemeinschattlichen Untergruppe von g und der Zerlegungs- gruppe eines in p aufgehenden Primideals. Es sei weiter p = A, “Op Ay A “tp ery S (mod m) und ¢, der gröszte gemeinsame Teiler aller Zahlen 342 m m bp (m) aop bon + 2 p (= Asn ben + Pl — at) | eee EE oh Ifa % Ne a ae ip oh age so Ist: ~==—. dre Beweis: Nach Satz 2 wird die im Satze genannte Untergruppe dargestellt durch die Potenzen von p, deren Exponenten sind Ll okie age d d d f Also ist pd die miedrigste Potenz von p welche zur Untergruppe g gehört. Es ergibt sich daher aus den Congruenzen (3) dasz - die kleinste Zahl ist, welche den Congruenzen je m ie on Lg (m) Wop 5 bon = 2 (sn) Oe p> ben + ...,=0 (mod g) geniige leistet. Daraus folgt die Wahrheit des Satzes. 4. Es sei /, eine in m aufgehende Primzahl, (auch 2) und d,' der Grad der gemeinschaftlichen Untergruppe von g und der cyklischen Gruppe des Grades f aus Satz 5. Nehmen wir weiter an, dasz fiir die in Satz 5 bestimmte Zahl die Congruenz m DE gilt. Es sei ¢, der gröszte gemeinschaftliche Teiler der Zahlen m bp (m) Ao bon + 2 p (=) Use Dagen Se oad l, 4 = bik AeA ..+. (mod m) Qhe Dp n= ld we n Sadie Gat 4 . so ist ae (Man siehe Satz 3 für die Bedeutung der Zahl f,). dy l, : Beweis: Wie voriger. § 5. Zerleguny der Primzahlen in Primideale des Körpers k. Herr BACHMANN ') hat das Zerlegungsgesetz gegeben einer Primzahl in Primideale eines beliebigen Unterkörpers des Garors’'schen Körpers. Seine Betrachtungen können wir hier natürlich benutzen. Sie werden eine leichte Vereinfachung erleiden können weil wir in unserem Fall mit einem Ager’schen Körper zu machen haben: die Substitu- tionen haben also die commutative Higenschaft. 1) Allgemeine Arithmetik der Zahlkörper, S. 495. 343 Satz 6. Es sei p eine nicht in m aufgehende Primzahl und fder kleinste Exponent für welchen pf==1 (mod m). Dann ist p in 4 ed Ie gleich dem Produkte von — verschiedenen Primideale des Grades the O6 f =p. d Beweis: Die von Herrn BACHMANN bestimmte Gruppe welche g und sg,s—! gemeinschaftlich haben, wird hier die Gruppe welche g und g, gemeinsam haben. Diese letzte ist bestimmt in § 2, 8. Im Komplex (42) von B. werden also, für jede Substitution s, immer rf je d einander gleich sein. Es gibt also a verschiedene. Weil G genau . . . .. ed ef Substitutionen enthält, musz man — Komplexe (42) nehmen. yi ed Die Zahl e von B. ist also hier = ai und die Zahlen h; sind hier alle gleich d. Weiter ist auf S. 494 s;9,s;—! hier gleich g,. Alle Zahlen ¢t, von B. sind in unserem Falle = 1. Auch »=1 und daher alle Zahlen d; von B. hier ebenfalls —=1. Aus dem Satze auf S. 495 von B. ergibt sich nun der Satz 6. Satz 7. Es sei /, eine in m aufgehende ungerade Primzahl und f, der kleinste Exponent, fiir welchen [a4 (mod) während 1 1 m : ; } : a ORS dln fs: dann ist die Zahl /, im Körper & die te Potenz ie 1 1 eines Produktes von einander verschiedener Primideale. Die Anzahl e,d,d' ; ~* * und ihr Grad se r d, Jedes Primideal ist im Körper A die d,-te Potenz eines Produktes dieser Ideale ist von einander verschiedener Primideale deren Anzahl ist. teg Ist hy, >3 und bo, + 6,, nicht für alle Werte von n gerade, so ist die Primzahl 2 im Körper & die g,-te Potenz eines Produktes “ae Te von einander verschiedener Primideale des Grades ~- und deren An- * Cull x zahl gleich ist. Jedes Primideal ist im Körper A ein Produkt 5 F eee Uae von einander verschiedener Primideale deren Anzahl ai ist. * 344 Ist hy > 3 und bo, + bn fiir alle Werte n gerade, so ist die Primzahl 2 im Körper 4 die 4,-te Potenz eines Produktes von * einander verschiedener Primideale des Grades a und deren Anzahl * gleich Bal: ist. Jedes Primideal ist im Körper K die 2-te Potenz 7 eines Produktes von einander verschiedener Primideale deren Anzahl ra 2de ist. Ist Ay —=2 so gilt dasselbe welches oben gesagt ist für den Fall dasz bo, + 6,, nicht für alle Werte n gerade ist. f, ist überall der gleich * m kleinste Exponent für welchen 2/+= 1 (mod a) und es ist stets Der Beweis dieses Satzes ist aufgeschlossen in den angegebenen Bacumann’schen Betrachtungen. $ 6. Bestimmung der Discriminante des Unterkörpers k. Satz 8. Für die Discriminante des Körpers & gilt: ( m ) Qi —1) Qhy—1 —2t B ( m ) Ea Uhh 1d; + | Dhue i Ue Ua sss ul; wo die Zahl t=O zu setzen ist wenn 6,, + bn nicht für alle n gerade ist und auch wenn hy = 2 ist; andernfalls ist ¢=1 zu setzen. Beweis: Es sei D die Discriminante des Körpers A; dx die Relativ- discriminante und 2, de Relativ-differente. Es gilten die folgenden Beziehungen : *) D == dr n (Dz); dE == Ni (Dx); Dr. == EH) DS Gri COS VAN NA DRM PE AE A; sind die Zahlen der Untergruppe g (also nicht die Basiszahlen dieser Untergruppe) und A,=1. Weil D nur durch Primzahlen teilbar ist, welche in m aufgehen, so folgt aus obigen Beziehungen dasz das Element Ey) auch nur durch Primideale teilbar ist welche 1) ,H". Satz 38, 39, S. 205. 345 & auf die Zahlen 2 oder /; teilbar sind. Aus obiger Form des Elementes @ folgt weiter dasz alle Zahlen dieses Ideals nur teilbar sind durch das Hauptldeal ZZi das Hauptideal des sa Ist diese letzte Zahl eine Einheit so ist das Element €” daher identisch mit dem Ideale © aller ganzen Zahlen des Körpers K. Es wird dasz Element €” also nur dann nicht identisch mit © sein, wenn A; eine Zahl der Trägheitsgruppe einer der Primzahlen 2 en /; ist. Dateb Zuziehung des $ 2. 2 und des Satzes 3, yy man : De — (Lox ste abe Ees )?t IT (Ea oA nore Bie)” a wo das Produkt sich über alle in m aufgehenden ungeraden Prim- zahlen /; erstreckt. Es folgt oe aus Satz 7: Nz (£0;) )= lo)" und Az B) = lijf Die Zahl D is bekannt). Man erhält nun den Ausdruck der Zahl d wenn man die, im Anfang des Beweises niedergeschriebenen Beziehungen benutzt. HI. Die wichtigsten Hilfssätze fiir die Berechnung der Klassenzahl der Ideale des Körpers k. § 7. Satz 9 ? p n (p)s gi bons ben ° bin, sate. ps Das erste Produkt erstreckt sich über alle Primideale p des Körpers k welche in die Primzahl p aufgehen. Das Symbol im zweiten Produkt hat die bekannte Bedeutung’). Wenn A, = 2 so musz b., im Symbol weggelassen werden und wenn h, = 0 ist, so musz auch bo, weggelassen werden. Beweis: Wir beweisen die beiden folgenden Behauptungen des Gliedes rechter Hand: 1. Das Symbol ist eine Lie Einheitswurzel ; d 2. Eine jede Lie Einheitswurzel tritt im Produkte je “ Mal auf. r Hiermit wird der Beweis erbracht sein denn es ergibt sich daraus dasz das Glied rechter Hand gleich pa d P 1) *H”. Satz 88 und 121. 3) “H’’. § #16. 346 3 ist, und zufolge des Satzes 6 ist dieser Ausdruek dem Gliede linker Hand gleich. 1. Es sei p= + 5!" (mod 2") p =r," (mod phy WO 7, 7,---, die schon vorher benutzte primitive Wurzeln be- deuten. Weiter sei —1 UNC 2 me, fa == 2 : p e, der gröszte gemeinsame Teiler von ra Cy ” ” ” 9 ” Da und 3 Pes Cx Js = 4 Vx é, ” B) ” EE) ” Pr und Lf, ; €, a = pf, Es ergibt sich dann aus der Definition der Symbole: [| = eine /,-te-Hinheitswurzel und keine Niedrigere, oy EN) ” ” ” ” ” ” ” Oe bus P pas , [hy = 00 Jes ” ”” ” ry ” ” ” ” 1 = der gröszte gemeinsch. Teiler von f, und d,, a often de fund d, Se *¥ Il | 9 a” ” 9. zE ” ” ” LE ” bon 5 | = eine LL ea ya ete ist. Io Fal EE Se 2 hz ” Vn ” 9” ” ” Ee eet Jf aaa x” 9, ” >, ” ” Je fs ty Das kleinste gemeinsame Vielfache der Zahlen —, i .... sei an- Jo Jax Jr gedeutet durch v. Dann ist der Wert des Symbols | p | Tm bons b ny bins og Os eine v-te Einheitswurzel. 1) Die Buchstaben @, fo, Em fw €1, fi, u S- W- haben hier eine abweichende Bedeutung von Satz 7. 347 Es bleibt also noch übrig zu zeigen dasz 5 teilbar ist durch v. Aus der Definition der Zahlen p,, p,,.... folgt: p—l ae ee eee =p; P, — aps P, = AUp,----’). Aus der in Hilfssatz 3 gegebenen Definition der Zahl t, ergibt sich dasz es der gröszte gemeinsame Teiler aller Zahlen $ Pra Pibig oo ++ Wp bop nae Pi Ps... dp ben EN Pa Payee e+ dip bin +. A Or er 8 =: ? ist. Alle Glieder dieser Summe sind teilbar durch e, und weil der gröszte gemeinsame Teiler der Zahlen y, und a, auch = e, ist, so ist einleuchtend dasz ¢, teilbar ist durch e, und daher is 4, =e, X gröszter gemeinsamer Teiler aller Zahlen NE Ps... - Mp Don +f; Ps Et EN en Alle Zahlen 6,, sind teilbar durch d,. Daher ist nun ¢,=e,g, gröszter gemeinsamer Teiler aller Zahlen ve b Ti b | EPs — P+: Hop On + — Ps. Ayn On IDR Sa 1 1 1 9, Poe Mar oe et fi bo 5 a Aus dieser letzten Beziehung ergibt sich dasz sich der Faktor e,g, Paso OG. tp Im Nenner bleibt also nur der letztgenannte gröszte gemeinsame Teiler iibrig. Wenn dieser neue Nenner nun noch einen Faktor mit (A) Weiter ist = aus Zähler und Nenner des Bruches heraus heben läszt. der Zahl — gemeinsam hat, so musz dieser Faktor, gemäsz der Definition I, des gröszten gemeinsamen Teilers, auch teilbar sein auf psp, 5 dip . < Vi : Denn die ganze Zahl — ist prim zu p, also auch zu — und nicht 4 91 b n . . alle Zahlen — sind teilbar zu ~,- ut Es hat sich also ergeben dasz jeder Faktor, welchen der neue eA : : : Nenner noch mit — gemeinsam hat, auch sich vorfinden musz im 91 fp - Zahler des Bruches ~~ , Ein solcher etwaiger Faktor kann 1) Man siehe Hilfssatz 3. 23 Proceedings Royal Acad. Amsterdam. Vol. XXII. 348 daher herausgehoben werden, wonach der Nenner prim zu der ganzen . / 5 ade Zahl “+ geworden sein wird. Nun ist weiter 5 eae ganze Zahl und I ( J ‘ Use Gg Decne ile ebenso ~~. Also findet man schliesslich dasz der Bruch —=~—*— „fs 9, ty auch eine ganze Zahl sein musz; m.a.w. die ganze Zahl — ist teilbar Sots ‚ * durch “, wie sich aus (A) ergibt. Dasselbe beweist man für — u.s.w. I; g * ” En! 2 : / Es ist damit bewiesen dasz = teilbar ist durch v. ; 2. Um die Zweite der Anfangs aufgestellten Behauptungen zu beweisen nehmen wir an dasz für zwei von einander verschiedenen Systeme EN Oe ae. und boe. bye, biz, die Gleichheit evn nn eN ern bot, OF bi, Og Ode boz. by25 biz, singels @ al bestehe. Nach Einführung der Werte der Symbole findet man dann leicht die Congruenz p (m) (bo1— boe) ao, + 27 ad (b,1— by2) a,, 4- -. -. = Ô (mod y). Pp / / * * | / gin Wie im Antang des § 3 auseinander gesetzt ist, bilden die Zahl- systeme bon, On,.... eine Gruppe. Es werden also die in obiger Congruenz auftretenden Differenzen wiederum ein System der Zahlen b;; darstellen, das zu der Gruppe gehört. Wir müszen also zeigen ed 4 dasz genau der gegebenen Systeme der Zahlen 6o,, b.,.... der 2 Congruenz Ei m m ; bp (m) bo; aop + Ap ae OF Ce a ei bij Qin t= .«: — O(modyp) (7) 1 A hla NS ed sic Genüge leisten. Denn hieraus ergibt sich dann, dasz genan — Systeme rT A P : bon, ban, ---- dem Symbol | —-~——~ | denselben Wert erteilen ; Ons On ee -_ AN REM eine ee Einheitswurzel kommt daher im Producte rechter Hand der rg We . ed zu beweisenden Gleichheit, genau — Mal vor. , 349 Alle Systeme der Zahlen bon, b‚n,.... welche der Congruenz (7) Geniige leisten, geniigen auch allen Congruenzen die man aus (7) ableiten kann, wenn man der Reihe nach alle Systeme a, otis. 4 welche zur Gruppe g gehören, an der Stelle der Zahlen Bapy Gay Age setzt. Es folgt dies aus der Detinition der Systeme a,, ay,....in§3. Man bekommt dadurch genau 7 Congruenzen. f Die Zahlen p, p’*,....p% gehören, dem Beweise des Hilfssatzes 3 zufolge, nicht zur Untergruppe g. Alle Systeme der Zahlen bon, 6,,,.. welche der Congruenz (7) geniigen, geniigen auch den Congruenzen die sich aus (7) ergeben wenn man anstatt «o,, d...., der Reihe nach die Systeme 2a), Zap, Zap, - jn i ~ Gop, TU rip. ROR Si MT RS in (7) hineinstellt. Man bekommt also d neue Congruenzen. Durch ( Multiplikation der Gruppe der erstgenannten 7 Systeme der Zahlen = IN v6 a) Cor ty,-... mit der Gruppe der zuletztgenannten tee bekommt ( ye man genau 7 Systeme «,, y,.... Setzt man diese, der Reihe nach, q anstatt do, dj, --..- in (7) hinein, so hat man im Ganzen genau rf É J Congruenzen welche genügt werden von den Systemen bon, b.n,.... Pu a welche eben gesucht werden. Der Bemerkung am Ende des § 3 rp ea zufolge, gibt es nun genau y:— —==— solche Systeme der Zahlen d r bon, Oan, ----…, Was zu beweisen war. Im Produkte des Gliedes Rechterhand, der zu beweisenden Gleich- . ; 4 : Je EN : : heit, finden sich — Factoren. Jede re Kinheitswurzel tretet genau Le a ed ed. fk — Mal auf. Da nun — x= so folgert man hieraus dasz auch r Rak ad r te i Tahoe : emme jede ee te Hinheitswurzel im Produkte auftretet. ( Satz 10. Es ist 1 | l, 1 M\ 1 — —— j= 71 | — TT | | n ( DE n bo, ’ Ons bon, ON SMER E s 350 wo /, eine in m aufgehende ungerade Primzahl bedeutet. Das erste Produkt erstreckt sich über alle Primideale | des Körpers 4, die in /, aufgehen. Das zweite Product erstreckt sich über alle Systeme der Zahlen bij in welchen 61, —0 ist. Wenn Ay== 2 so musz das Symbol Dien mervolaseen werden; und wenn i, =O so musz bo, weggelassen werden. | Physiology. — ‘Further Researches in connection with the permea- bility of the glomerular membrane to stereoisomeric sugars”. By Prof. H. J. HAMBURGER. (Communicated in the meeting of September 27, 1919). It has previously ') been shown that when a liquid, whose com- position is efficient, is passed through the bloodvessels of the kidneys, the grape sugar dissolved in the liquid is retained by the glomerular membrane. In this concern the concentration of the Ca-ions and H-ions play the principal part. This retention of glucose is very remarkable, seeing that other crystalloids like common salt, sulphates and phosphates are not retained by the glomerular membrane. The question then arose what the cause could be of this so efficient behaviour of glucose. Had it something to do with the size of the molecule? This supposition had to be dismissed when it appeared that lactose (C,,H,,0,,), which has a molecular weight almost twice as great, passed through completely, which was the case even with raffinose (C,,H,,0,,). Nothing else then remained but to accept that this phenomenon was to be ascribed to the particular configuration of the glucose molecule’). To test this conclusion by experiment other hexoses (C,H,,0,), isomeric with glucose, were examined. It appeared thereby that, in contrast with glucose, the fructose and mannose were allowed to pass through completely. The frog’s kidney isable thus to distinguish glucose from fructose and mannose. The cause thus indeed lay in the difference in chemical structure. Obviously the thought of the wellknown comparison with lock and key occurred: The key glucose thus does not fit in the lock glomerular membrane, while the keys fructose and mannose do. We now asked ourselves the question: will it perhaps be possible to point out a particular group of atoms in the glucose molecule which may be considered responsible for the retention? The attempt to answer this question forms the contents of this article. ') HAMBURGER and BRINKMAN, These Proc. Meeting of January 27, 1917, Vol. XIX, NO. 8. | 2) HAMBURGER and BRINKMAN, These Proc. Meeting of September 28, 1918, Vol. XXI, NO, 4. 302 It is obvious that for this it was necessary to examine a number of isomeric and stereoisomerie sugars with regard to their behaviour towards the kidneys, and then, not only hexoses, but also pentoses. Jhr. ALBERDA VAN EKENSTEIN, Prof. Backer, Prof. BörseKeN and Prof. NeuBeRG were so kind as to assist me in this as much as possible. Hereby | again tender them my cordial thanks. Nevertheless the diversity of material was very limited. The methods pursued did not differ from the previous ones *). They were based upon the reducing powers of the secreted artificial urine, and that of the perfusion liquid. This perfusion liquid, in which the sugars were dissolved, “again had the following composition : NaCl 0.5°/,, KCl 0.02 °/,, CaCl,.6 aq 0.04°/,, NaHCO, 0.285 °/,. The determination of the sugar was made, as in the previous case, according to Bana (1916). Lately however the newest method of this writer was used’). This also is based upon the determination of the reduction. Here the diserepaney which can arise from differences in the intensity of boiling is eliminated, and that because hot steam is passed through the boiling liquid, whereby the cuprous oxide is instantly changed into cupric by means of iodic acid Furthermore the oxidation by the air is avoided. The final titration is carried out with O,OL n thiosulphate and starch solution. If @ ee. thiosulphate are used then the reduction in terms of glucose can be calculated from the formula (1.97— a): 2.8. [f no reducing substance or sugar is present then the answer must be found to be: « = 1.97, which is indeed the case if the substances used are pure’). This new method, it appeared, like the old one also gave very good results. The results are even nice. On the whole the figures obtained for the reduction, by the new method, are higher than those obtained when the old method is used, but, seeing that the purpose here is comparative research it is mostly not of vital importance. In order to have an introduction for our procedure the arbitrary C-OH hypothesis was made that the | group in the glucose molecule H-C-OH caused the retention. 1) Compare also HAMBURGER and BRINKMAN, Biochemische Zeitschrift 88, 97, (1918). *) Bane, Biochemische Zeitschrift, 92, 344, (1918). 5) We wish to call special attention to that because after long searching it has appeared that the KCI “pro Analyse" of Kantyaum contained reducing substances. This was also the case with the KCl which was supplied by the British Drug Houses as chemically pure. After repeated crystallization the KCl was freed from reducing substances and then we obtained like Bane, a value of 1.97 for a. 353 If this hypothesis was true then besides the d-glucose also the l-gulose, |-talose, l-mannose and d-galactose had to be retained. (1) d-glucose (2) l-gulose (3) l-talose (4) I- mannose (5)d-galactose C—OH C—OH COH GE OH | | | | | H—C—OH H—C—OH H—C—OH H—COH H—C—OH | | | | OH—C—H HCO) sG On ~H—COH OH SCH | | | | | H—C—OH OH—CH (= @n;OH—C-=H OH-=0=H | | | | | BSO HO 08. OC OH CH NG): | | | | | CH,OH CH,OH CH,OH CH,OH CH,OH However, of these, only the d-glucose, the l-mannose and the d-galactose were at our disposal. The experiment showed however that the l-mannose passed through completely: In 4 experiments it appeared that in the case of perfusion with a solution of 0.09°/, whose reducing power corresponded to that of a glucose solution of 0.06 °/,, the artificial urine also had a reducing power of 0,06, — there was no retention therefore. Let attention be called to the fact, by the way, that after the preparation of the l-mannose solution the rotation was + 10.5°, and, 16 hours afterwards 13.5°. In the following article we shall return to this. Concerning the d-galactose, it was not retained completely but partly ; to this also we shall return in the next article. Our hypothesis further claimed that the following series should pass through the glomerular membrane: (6) l-glucose (7) d-gulose (8) d-talose (9) d-mannose (10) |-galactose. GOH C—OH C—OH C—OH C—OH OH_CH OH—CH OH_¢_H OH_C_H OH_¢—-H H—C_OH oH_CH OH—¢_H OH_C_H H—¢_OH OH—C—H H—C_OH OH-¢_H H—C_OH H—C_OH OH—C_H oH-d-H H—C_OH H—C_OH OH_—CH | | | | | CH,OH CH,OH CH,OH CH,OH CH,OH We had only l-glucose and d-mannose available. It indeed appeared that the l-glucose in contrast witb the d-glucose is allowed to pass through completely. The 0.1 °/, glucose solution, dissolved in the customary Ringertluid, showed a reduction of 354 0.075 °/,; in the + experiments the urine showed the same reduction. In accordance with the hypothesis also of the d-mannose not a trace was retained. *) We did not have at our disposal the other abovementioned sugars except, however, the d-glukosamine C—OH H—d_NH, OH-—C-H H—C_OH H—C_OH CH,OH The 0.1 °/, solution of glukosamine as well as that of 0.08 °/, passed completely through the glomerular membrane. After this some pentoses were examined. Our hypothesis demanded that the following series should be retained: (1) l-arabinose (2) l-xylose (3) d-ribose C—OH C—OH C—OH H—¢_OH H_—C_OH H_—C_OH oH_¢_H OH_C_H H—¢_OH OH_C_H H—C_OH H_—C_OH CH,OH CH,OH CH,OH Concerning the l-arabinose, a solution of 0:07 °/, was perfused ; it had a reduction of 0.0714°/, glucose, which was also the case with the artificial urine; in contradiction to the bypothesis, there- fore, nothing was retained. In the case of l-xylose there was a retention, but this amounted only to more or less */, part, while not more than */, of the ribose was retained. To this partial retention of l-xylose and d-ribose we shall return in the next article. Lastly there follows here a series of pentoses, which, according to the hypothesis, were expected to pass through completely, 1) Compare HAMBURGER and Brinkman, These Proc. Meeting of 28th Sept. 1918, Vol. XXI, NO. 4, 355 (4) d-arabinose (5) d-xylose (6) d-lykose (7) l-ribose C—OH C—OH C—OH €—OH OH_¢_H . OH—C_H OH_C_H OH-C-H H—C_OH H—¢_OH OH_CH OH—C_H H_—¢_OH oH_¢_H H—C_OH oH_¢_H CH,OH CH,OH CH,OH CH,OH Of these 4 pentoses we were able to investigate only the first two; quite in accordance with what was to be expected there was no question of any retention of d-arabinose. It was different in the case of d-xylose of which again a part was retained. Let us add to this still that neither of the d-l-arabinose nor of the d- and l-arabinose anything was retained. The tetroses could not be examined; if this had been the case it was to be expected that d-erythrose and d-threose C—OH C—OH H—C_OH H—¢_OH H—C_OH OH_C_H CHLOH CH,OH would be retained and l-erythrose and l-threose C—OH C—OH OCH OH_C_H OH_C_H H—C_OH CH:OH CELOH allowed to pass through. To simplify the survey we give here a table in which the results of the experiments are summed up. Let it be remarked here however that with d-galactose and l-xylose a much larger amount of experi- ments have been make. These will find a place in the following article. Summarising the results of the above described experiments we must come to the conclusion that the hypothesis which makes 356 Permeability of the kidneys to stereoisomeric sugars. Date of experim 15 Nov. 12 June 1 July 19 Nov. 11 Aug. 12 Aug. 14 Nov. 20 Aug. 21 Aug. 13 Aug. 13 Nov. 5 Nov. 26 Oct. 26 Oct. 14 Aug. 23 Oct. 30 Oct. 10 July the He ent. | 1918 0.09 1919 0.05 1919025 1918 0.1 1919 0.08 1919.0.07 1918 0.1 19180.1 19180.1 1919.0.07 1918 0.1 1918 0.05 19180.09 1918. 0.09 1919 0.07 19180. 1 1918,0.04 19190.2 C | C OH Perfusion liquid contains: 0/9 L-mannose „ OH d-galactose n d-glucosamine HCl-glucosamine | ” » l-glucose d-mannose l-arabinose l-xylose d-ribose d-arabinose | liquid glucos 0.06 Reduction of| | perfusion ‘expressed in| terms of e percent. ” ‚The urine has a reduction | (11—III). of: » „ Percen- tage of Retention retention calcu- latedfrom I and IV. group of the d-glucose molecule responsible for the retention, an hypothesis, which, as we have already remarked, was of a wholly arbitrary nature, cannot be correct. For if it was, then l-mannose and l-arabinose would be completely retained, but, as a matter of fact, they are allowed to pass through. On the other hand however, in accordance with the hypothesis, d-galactose, l-xylose and d-ribose are retained, but this happens only with a part of each of the sugars which are passed through. As regards the sugars which, from the hypothesis, were expected to pass through completely, they all indeed passed through the glomerular membrane except the d-xylose, which was partially retained. To this partial retention we shall return again, as has already been said, in the next article. If we accept for the time being that even where a total retention had to take place, a partial one satisfies the hypothesis, then in any case there still remain the l-mannose and the l-arabinose of which, in contradiction to the hypothesis, nothing is retained. There will therefore be an inclination to hold a larger group of atoms of the glucose molecule responsible for the retention, for instance C—OH H— C—OH. But also in this case l-arabinose had to be retained | OH—C—H completely. C— OH | H-—C—OH If the group of atoms is taken still larger viz. OH—-C—-H H aide OH then this could be held responsible for the retention of glucose, but then the difficulties of the partial retention still remain. For the present therefore it is risky to explain the retention by means of the image of the loek and key, unless, instead of a part of the molecule, the grouping of the whole is taken. Dr GRAAFF ') came to a more or less like conclusion through his detailed experiments in connection with the behaviour of typhoid and para typhoid bacilli. towards stereoisomeric sugars. As matters stood now there would be the inclination to get a point of contact with the specific physical properties of glucose, and we thought of surface tension, viscosity and adsorption. As regards the surface tension, comparative experiments were made with d-glucose and d-fructose *); of the latter sugar it will be 1) W.G pe Graarr: The biochemical characters of paratyphoid bacilli. Leiden, S. C. van Dogspuren, 1918. 4) We take these especially because, as a rule we have sufficient quantities of them at our disposal. 358 remembered no trace is retained by the kidney. But it appeared through stalagmometric experiments (J. TrAUBE) and through experi- ments where the height of the meniscus through capillary action was made use of, that there existed no difference in the surface tensions of the two sugars. Neither did the viscosities of the two sugars show any difference. We now turned to the adsorption; would the glomerular mem- brane perhaps adsorb glucose, and, on the other hand, not fructose ? As a matter of fact we would not have busied ourselves with these experiments had not O. Conny ') conducted similar ones. For, supposing that the glomerular membrane adsorbed d-glucose, but no laevulose, then it would be extremely difficult to explain the different behaviour of the living kidney toward these two sugars through this. But, in any case, this would at least be an indication in the direction in which the explanation of the facts has to be sought. In addition to this there remains the fact that COHNHEIM could indeed notice differences in the adsorptive power of the kidney substance of warm blooded animals towards glucose and laevulose. However there are several grounds on which this conclusion of ConnHEmM can be doubted. | shall communicate about this in another place and point out through the medium of experiments conducted with frogs’ kidneys, that the methods pursued by this investigator viz., to experiment with mashed kidney substance, to determine the adsorptive power of the kidney for sugars, must lead to results which cannot be relied upon. Summary and Conclusion. To investigate whether a particular group of atoms can be held responsible for the retention of glucose by the kidney, a number of stereoisomeric hexoses and pentoses were examined with regard to their behaviour towards the kidneys. C—OH | The hypothesis that the group eon causes the retention cannot be true seeing that l-arabinose and l-mannose which have this group pass completely through the kidney, while the d-galactose, the l-xylose and the d-ribose in which this group is also present are partly retained, it is true, but still the other part is allowed to pass through (partial retention that is). In keeping with the hypothesis the sugars whose first 1) Connnem: Zeilschr. f. Physiol. Chemie, 84, p. 451. 359 C—OH C—O# d-xylose which had to show the same was partially retained. The experiments have thus yielded two results: 1. Of all the investigated sugars (hexoses and pen- toses) glucose alone is retained completely. 2. This exceptional property of glucose cannot be C—OH ascribed to the group of atoms H—C—OH and just as little to a larger group of atoms in the molecule. Lt seems to be inherent in the whole atomic grouping of the glucose molecule. Still it will appear in the following article, in the light of partial retention, of how much importance the group H—C—OH_ is here. The experiments described above were partly made by Mr. C. L. Arons, Med. Cand. assistant in the laboratory. Groningen, September 1919. Physiological laboratory. Physiology. — “The partial permeability of the glomerular membrane to d-galactose and some other multi-rotatory sugars’. By Prof. H. J. HAMBURGER. (Communicated in the meeting of September 27, 1919). Through earlier experiments it has appeared already that d-galactose is only partially retained *) by the glomerular membrane. Upon closer examination, however, this appears to agree very little with the idea of permeability, because a filter either holds back a substance or allows it to pass through unconditionally. A mean between these two extremes can hardly be conceived when there is question of only a simple substance and: not of a mixture. As a matter of fact at first we were inclined therefore to drop the idea of permeability, but, as appears from the preceding article we endeavoured in vain to clear up the matter with the help of surface tension, viscosity or adsorption. For this reason we then returned again to the idea of permeability. On closer inspection two explanations still seemed possible to us: ist. The concentrations of the galactose-solutions which had been used up to this (0.1°/, and 0.15 °/), might have been too strong that is above the toleration of the kidney for d-galactose. We were mindful here of our experience in connection with glucose; for, did we not find formerly ’), that, when the concentration of glucose in the perfusion liquid exceeded the physiological value only by 0.02 °/,, already a little of the glucose was suffered to pass through, and, that this quantity inereased in proportion as the concentration of sugar in the perfusion liquid became stronger, so that when the concentration of glucose had become 0.2 °/, practically no sugar was retained any longer? There was a possibility therefore that also the concentration of galactose might have been too strong and that it had to be ascribed to this that it was partly retained and partly not. /¢ was therefore desirable to make determinations of the toleration of the kidneys for d-galactose. Should it appear then that the galactose could not, like the glucose, be wholly retained even in weak con- 1) HAuBURGER and Brinkman, These Proc. 28 Sept. 1918. 1) Hampurcer and Brinkman, Biochem. Zeitschr., 88, 97, (1918). 361 centrations, then the phenomenon of partial retention could not be explained by the idea of toleration and then a second explanation would per exclusionem be the correct one. 2nd. This second explanation could be sought in the fact that the d-galactose exists in two modifications — an « and a B variety. In aqueous solutions these two varieties are in a state of equilibrium. Then it had to be accepted only that one of the two modifications was retained by the glomerular membrane, and the other not. 1. Toleration of the kidneys for d-galactose. -To determine the toleration, the perfusion liquid was seasoned with different quantities of d-galactose lying between 0.05 °/, and 0.25 °/,. The following table gives a survey of the results obtained. It will be clear without further explanation. Let it only be remarked that for the determination of the reduction the newest method of Bane (1918) was employed *). Hach time both kidneys of two frogs were perfused at the same time (see table U). What do these experiments teach us? Firstly: That none of the used galactose-solutions, whose con- centrations lie, as has been said, between 0.05 °/, and 0.25 °/, are completely retained by the kidneys. Secondly, that in all cases the retention amounted on an average to a half, independent of the concentration of the galactose-solution that was perfused. In both respects the galactose differs from the glucose; for, was there not found a ¢ota/ retention in the case of glucose when the solution was weaker than 0.05 °/,—0.08 °/, (individual differences)? In the case of galactose, on the contrary, there is no question of total retention. And, as regards the second point, in the case of galactose the toleration remains unchanged in spite of the increase in sugar concentration. Only when the concentration becomes as high as 0.25 °/, does the toleration diminish. Experiments with stronger concentrations were not made since the secretion of artificial urine then became too scanty. We still have at our disposal a number of former experiments in which the reduction was determined by the earlier method of Bane (1916), a part of which experiments have been published already. They are found together in the following table (II). As will be noticed the results are not as uniform as those of table I, but in any case they point in the same direction. 1) Compare for this our previous article in these Proceedings. p 362 « 9 00 OF % IS = OOI X 79600 67700 (AI Pue [] woar pajejnajeo) sÁaupix ay} JO 93eloay ‘osopperes JO uoIjuajor 9SeJUII Ad ‘A £90°0 | 2L0°0 LL90°0 Z80°0 €1S0°0 ¢g0°0 €6€0°0 64700 ¢8Z0'0 cgso'o 6L90°0 GILO'O | 6zF0'0 | 6z¥0'°0 | ero ZLO 0 LL90'0 0s0°0 Z90°0 z90°0 LSE0"0 LGE0°0 €8E0°0 87900 €900 6L90°0 €6£0°0 67700 | Aaupry yay |Leupiy 3USId | | “(II—I1D) vonuezay ‘Al €90°0 1210 £680°0 | LS0°0 L090°0 LS0:0 1z0°0 ¢8z0'0 | 600 | 700 | TLG0'0 | T€e0:0 SES0°0 cec0 0 | ‘AQUPIY Wel €1°0 1210 €680°0 LOT O0 600 50’0 LS€9°0 LSE0°0 ¥0°0 LSE0' 0 L090°0 ILG0°0 IL¢0°0 ces0°0 “Aaupry ysis | 24} JO MO | ay} JO MO | ‘0/) asoon[s Jo 1amod ‘Sulonpes e sey ulin ayy|-ted Jo uononpay ‘Il «< GZ'0 ‘g 5 Áimf Ast €610 a edel, yy ee a . . 40 d 3 oun LS1'0 2560 Vv AES 4“ GIO ‘g r : Aine 4st ZI1'0 “ cI‘0 ‘Vv =" 0 ‘d . a oe : "Ang uw FILO'O 10 V a Tr ry 7 jk a “Ang pig G8L0'0 10 Vv a Ik . . i 8 5 “_aunf Uwi GZ1°0 1°0 Vv : a 2050 a 6161 eunr YIZI #960' 0 lo S0'0 e= | “0/, asoonls ee : : joules : UI passaidxa ; s}uowiadxa pinbty uoisny jo suIejzuod 3014 jo a}ed Tl pinbiy uoisnjsaday y "9S0}IB|E3-P JO} oueIqwaw Iejniswojs 3y} JO UOIJEJaJOF BY TAaTEVvL 368 “ i LOO 600 | €0:0 |. cgc0'0 | ‘g 5 « 5 5 | € | a | « ‘any PACT 67 8000 | SLZ0'O © LOO L0'0 SL60'0 | G10 ‘Vv | | | | Op SL20'0 | 0800 | _SP0:0 L+0"0 a se “ g¢ chz0 0 GLZ0'0 | 8sF0'0 Sh0'0 ‘d “ 0€ 200 g2zo’0 | 82600 | <00 ELO Om Ee ‘V er GI0:0 | _z910'0 |. sLoo‘o | ¢990°0 5) “9% ee | ¢80°0 | __ 8L00 ‘d “ OF gce0'0 | ~—-€€0°0 080°0 | _6#0'0 “64800 0 Nm 4 | ¢zz0'0 ¢2z0°0 | Szeo'o | szeo‘0 “ ¢c0'0 A | | “ ‘Snv uIgi “Ub | ¢zz0'0 ¢7Z0'°0 GZE0 0 ¢zs0 0 “ ¢c0°0 “ae | bo Sar LT “ 6€ | €0°0 Gz0 0 100 | SGr0 0 eaten 00 “2 0 | _ © _‘Snv ugl “ op I= €10%0 z0:0 600 | ¢z0°0 | TEE REN Ee 0 OF Gz0 0 GI0'0 | coo | €00 % HOO | %o 90°0 en 8 il ‘Kouppt ye] Kaup uri, “SPL Bl geel | "0/9 asoon|s (Ij pue | wo. : ae ale | : 3 paje[nojeo) sAaupiy aug | Ul passaldxa sae 3 “juawI.Iadxa Jo ZelaAy ‘asopres | ‘V/o asoonjs Jo Jamod —_— pinbij uorsny ronda ad Lonel eae jo 3eT JO uonuajon aSejuINlad © (III) onware Sulonpal e sey sulin ayy | -Jed jo uononpay pinbyj uolsnjsad ou.L ‘A ‘Al ‘TH all | T | (9161 ONVG JO poyya;) osopere3-p 10 aueiquow JB[NIIWO[S ay} Jo UOIBIIJO} ay IW ATaVL Proceedings Royal Acad. Amsterdam. Vol. XXII. 364 Also from this table it appears, that, in contrast with what was always found for glucose, the galactose was not totally retained in any of the experiments, and further, that the retaimed part forms more or less the half — here a little less than the half. Consequently the first explanation mentioned on page 361 fails here and we are obliged, per exclusionem, to accept the second expla- nation, viz., the one based upon the fact that in aqueous solution the galactose exists in two modifications. 2. The solution of galactose contains two forms of galactose that behave differently towards the kidneys. As is known a large number of sugars exhibits multirotation (DuBRUNFAUT), i.e. some time after the preparation of the aqueous solution they generally possess a slighter specifie rotation than immediately after the solution is made. Several explanations, which we need not discuss here have been given for this. It is agreed upon however that under the influence of the solvent, part of the sugar changes into another form with a slighter rotatory power. The two forms are in equilibrium. This idea is based, in the first place, upon the researches of Tanret’), who separated in solutions, first of glucose and afterwards of galactose and other sugars, three forms with different rotatory power, and several different physical properties; these he termed a, B and y: the « modification of d-galactose with a specific rotatory power of + 135°, a 3 modification with [¢]p>=-+ 81° and a y modification of [a|p = + 58°. More extensive researches, especially of E. Roux’) and further of BourqueLor®) have taught however that the 8 variety of TANRET is no independent sugar, but consists of a mixture of « and y, which are in equilibrium. I find it useful to point out here a misunderstanding in the literature which has given me much trouble personally, and, I am informed, others also. in several scientific treaties and textbooks it is stated or taken for granted that there are two modifications only; these are called then the @ and 3 modifications. It is clear here that the 8 modification is actually the y modification of Tanrer. But, as far / as | know, nobody ealls attention to this fact. To what faulty reports 1) Tanret, Bulletin de la Société Chimique, [3], 13, (1895), p. 728 [3], 16. (1896), 195. 2) E. Roux, Ann. Chim. et de Phys., VII Série, 30, p. 422. 3) Bourquetot, Journal de Pharm. et de Chem. [7], 14, (1916) 225. 365 this interchange of the two can lead appears for instance in the wellknown tables of LANDOLT-BORNSTEIN de Aufl. 1912. There we read in connection with d-galactose: ‘“Anfangsdrehung nach 7 Minuten als « Modifikation + 117°.5; Enddrehung nach 7 Stunden als 8-Modi- fication + 80°.27.” In fact the final rotation which an ordinary d-galactose solution exhibits is 80°.27 or simply 81°. But this is not the rotation of the second modification; that rotation is + 53°. A ~ rotation of 81° results when the two modifications of 117°5 and + 53° are in equilibrium. And this equilibrium comes about when we start out from 8 as well as when we do from a. In both cases a mixture results with a rotatory power of 81°. It is perhaps useful that what has been brought forward for d-galactose should also be applied to d-glucose, which is so much more used. Tanner distinguishes 3 forms of glucose: an « form with [~],}, = 106°, a @ form with [7]; =53° and a y form with [#],)=19°; Both z and 2 forms when dissolved in water finally exhibit a rotation of 53°. The @ form of Tanrer, according to the researches of Roux and others, must be considered as an equilibrium between x and 7, and this is at present generally accepted too. @ is therefore no independent modification, but merely a mixture. But now we read the following: There exist two forms of glucose, — the z and the @ forms and not infrequently there is added: The « form is converted into the @ form. How can those who maintain that we have to deal with a reversible reaction here, speak of an equilibrium between ~ and #? If it were stated that the a form is partially converted into the @ form, it would be clear. We find the case put differently again by Hotteman in his wellknown text book of Organic Chemistry, 5th ed., 1912, p. 300. He also speaks of an z form of 106°, a 2 form of 19° and a y form of 53°. This y form is according to bim a mixture of « and 7. The y of Tanret Hotreman calls thus 7. The question arises why ‘Tanrets’ nomenclature was not stuck to. After Emir FiscHer had shown us how to prepare artificial glucosides !) it appeared to him that stereoisomeric modifications of each glucoside existed. One was acted upon by beer ferment (invertine), and the other by emulsine. *) By way of distinction he called the first the «-glucoside and the second the (-glucoside and the corresponding forms of glucose z and -glucose 4). It is to this nomenclature that later writers seem to have stuck. For this there was an inducement to some extent as if there existed only two forms it was not quite rational to call the second y. Coincidentally that which Em Frscaer calls @ corresponds with what Tanret at the same time gave the name of y. To explain now the partial retention of the galactose solution which is passed through the kidneys, we assume that only one of the !) Emmi Fiscuer, Ber. d. D. Chem. Ges. 26, 2400 (1893). 2?) The same, Ber. 27, 2985, (1894); 28, 1145, (1895). 3) Cf. also E. Fiscuer, Z. f. physiol. Chemie 26, 60, (1898). 366 modifications is retained by the glomerular membrane, and the other is aliowed to pass through. As appears from tables | and II, of not too strong concentrations about the half is retained. It is now, upon closer inspection of what has been handled above, possible to calculate in a simple way the relative quantities of a- and B-galactose. If we call the amount of the « variety (rotation + 135°) in the galactose solution, in which there is equilibrium between « and 8, x, then 1-—x is the amount of the 8 variety (rotation + 53°), and then, because the rotation of the mixture 81° is, the following equation must hold: 135x + 53 (1—x) = 81. x= Ojo: 1—x = 0,66. Therefore the ratio between the quantities of the « and > forms is 34 : 66. These figures cannot boast of great accuracy, because, in the first place, we find with other writers for the specifie rotation of the « variety a value of 117° and not 135°. If this value is the true one then we should get a ratio of 44:56 between the modifications. Further it must be remembered that the concentration and temperature of the galactose solution are not without influence on the equilibrium. In general however it can well be said that the greater half is the 3-form (y-form of Tanrwrt). A similar proportion can also be deduced from the researches of KE. Roux in connection with the rate of conversion of the « into the y-form. , It is now very remarkable also in our perfusion experiments that more or less the half of the galactose is retained. This parallellism ean be considered as supporting our hypothesis. Whether it is the «- or y-form which is retained we cannot venture to say with any certainty at present. That might be the case, if, in the first place, the values of the rotations which we used above in deducing the relative quantities of the «- and g-forms had been taken at the same concentrations as the physiological concentrations (0.05°/,—0.15°/,), which we used in our perfusion experiments. In the second place the values found along chemical lines for the rotation, in concentrations of 6°/,—18°/,, leave much to be desired. And then in the next place it must not be forgotten that, as a matter of course, the degree of accuracy of our determinations of the 367 reduction, by our perfusion experiments is but small. To grasp this it must be remembered that the quantity of artificial urine obtained from our experiments with frogs was only 0.1 c.c. Let us take an arbitrary example to see what influence a small error has on the titration. In table I perfusing with 0.15°/, galactose for obtaining the final reaction. was used: right kidney 1,83 ce. thiosulphate for the urine of frog A | left kidney 1,81 „ | right kidney 1,83 ,, i for the urine of frog B | left kidney” 1,80 ,, From these figures the value calculated for the galactose retention for frog A is 55°/, and 49°/, and for frog B 55°/, and 46°/,. An error of 0.02 °/, thiosulphate, therefore, causes an error of 55—49 — 6 °/, in the retentive power found. To increase the degree of accuracy it is necessary to experiment upon larger animals, which supply more urine, thus with kidneys of warm blooded animals. For this however a room is necessary which can be brought up to body temperature, which, under present circumstances, is impossible. Be it as it may, if the differences between the retained and the not retained had been greater — for, did we not find that approxi- mately half was retained and half not —? and, to correspond with this, the difference between the quantities of « and B modifications in the galactose solution had also been greater, then it would at once have been obvious which form is retained and which is not. We will return to this in connection with xylose. Before we proceed to discuss the behaviour of xylose, we wish to face an objection which may, on superficial inspection, be made against our representation. It could be remarked that when one of the two modifications has been removed by the kidney, from the second modification which remains in circulation, more of the first modification will be formed, and that eventually all the galactose will leave the kidney in that way. Let it be taken into consideration however that in our perfusion experiments the perfusion liquid forms only a very small quantity of urine and therefore only very little of the one kind of galactose is removed. Let it further be femembered that, unlike in the normal body, the same perfusion fluid does not remain in circulation. In our experiments the solution leaving by the renal vein does not return again by the renal artery. But what can be the reason then that in the normal organism 368 the urine does not always contain galactose, i.e. that modification which, according to us, the kidney is permeable to? Is it perhaps just this form which is used in the building up of the cerebrosides? This will have to be determined by further experimentation. Lastly, before leaving the galactose, it may be interesting just to call attention to this: viz., how slight the difference in structure between the «- and g-forms is, which difference is concerned with their retention or non-retention. Thanks especially to the researches of Emin FiscHer (l.c), ALBERDA VAN EKENSTEIN, BOESEKEN*), to which also correspond those of BourquELor’), it must be assumed a little in contravention to the current idea, that the structural formulae of the two forms of galac- tose can be represented as follows: -H—C—OH | _OH—C—H ete oan wy Hel on sie „0 Bee ele 4 OHG ser on ee CZH ne HEC=SOH Ea CH:OH CH:OH ERLE 3-(y-)d-galactose. From these formulae it appears that here mutatis mutandis the retention is exclusively dependent upon the relative places of the OH and H with regard to the first asymmetric C-atom. The partial retention of Laylose. The thought of the explanation given in the previous paragraph for the partial retention of the ‘galactose solution occurred to us when it was noticed that also xyiose-solutions were subject to partial retention. Corresponding to this is the fact that, like in glucose- solutions, also in xylose-solutions two modifications could be sepa- rated by Tanret. Also in the case of xyloses Emit Fiscurr could separate two glucosides, or xylosides rather, an «- and a 8-form, for instance: 1) BöesEKEN, These Proc. 29th June 1912; 25th March 1916. 2) BoURQUELOT, l.c. 369 /H—C—OCHs | CHO—C—H H—C—OH uk Bot OH—C—H ieee oe OH-C-H EC Seita CH20OH CH20H a-methy l-l-xyloside 8-methyl-l-xyloside, the first of which is converted by invertine and the second only by emulsine, and which were distinguished by Fiscner as « and 8. Suppose the CH, substituted by H, then «- and p-xylose is formed again. Similar experiments as were made with galactose were also made with xylose. We give a table which contains the experiments in question (see table III). From this table it appears that the l-xylose, like the d-galactose, is not retained completely either. Always the greater portion passes through the glomerular membrane, a greater portion percent, however, than was the case with d-galactose; between '/, and '/, only of the xylose, used in concentrations which do not affect the production of artificial urine, on an average is retained. We now find, according to von Lippmann’) for the rotation of the l-xyloses: initial rotation + 78°, final rotation + 19°. Therefore, the a-form has a rotation of + 78°; the g-form however has up to this not been isolated, and therefore it cannot be stated, how much of this modification takes part in the final rotation of + 19°. Also for xylose it is therefore not possible to indicate the form, which is retained by the kidneys, and which is allowed to pass through. That the glomerular membrane possesses the property of separating two sugars quantitatively like a sieve, retaining one and letting the other pass through, we have formerly been able to show with mixtures of glucose and fructose, and glucose and lactose’). Let us remark that researehers who are engaged upon distin- guishing of sugars by means of microbes*) have to reckon with 1) Von Lippmann, Chemie der Zuckerarten. *) HAMBURGER and BRINKMAN, These Proc. Sept. 28th 1918. 5) C.f. among others A. J. Kuuyver, Biochemische Suikerbepalingen. Diss. Delft 1914. W. C. pe Graarr, De biochemische eigenschappen van paratyphusbacillen, Leiden, S. C. van DoesBuren, 1919. 370 TABLE III. 2 Toleration of the kidneys for I-xylose. | 1. Il. | II. IV. | V. | tontiguig Reduction of per-| The urine has a reducing Percentage retention of Date of Era Ase Shere are Glee fusion liquid | power of glucose 9p. Retention galactose. Average of experiment. 5: alactaad:: expressed in ({I—III). the kidneys (calculated 8 : glucose 9. = me from II and IV). | Out of the | Out of the | | right kidney. «_ left kidney. . | te A. 0.06 0/0 | 0.059 0/0 0.0357 0.0357 0.0233 39 % Ist July 1919 Bae 0.0393 0.0393 _ | 0.0197 Sc Eee eae eea ie 0.107 , 0.075 0.068 0.035 33, Tth July Be 0.0721 0.075 0.033 Ste ines Oss 0.168 „ 0.125 0.125 0.043 25 Bih July 1919 | 5 0.1321 0.1469 0.028 19 , A. DEAN, a 24l= 0.171!) 0.029 15 , kler SB 1909 Bs ae en 0.185 0.015 se A. | Ope 2 0125 | 0.11 0.10 0.020 16 , 4th Oct. 1918 5 | 0.095 0.1025 0.0987 21 , A. Ot ss 0.105 , 0.075 =. 0.03 28 20th Nove W918 | =p, | 0.075 0.08 0.028 4, became stronger less and less urine was secreted. ”) Both of the last two series marked with * were determined with the help of Bana’s earlier method (1916) ') In this series the urine secreted from each kidney apart amounts to less than 0.1 c.c. the right and left kidneys were added to each other. In general it could be noticed that as the concentration of xylose For this reason the liquids out of 371 the fact that both the modifications in which a large number of sugars occur, need not behave in the same manner towards tliose organisms. Besides that, the circumstances in such experiments are somewhat different from those in our cases; for have we not to do with a disturbance of equilibrium in the ferment experiments, which disturbance is the result of the eventually being used up of one of the modifications, but is adjusted again? (cf. p. 367). The behaviour of the kidneys towards sugars other than galactose and xylose. It has appeared that from the experiments described up to this three cases can be distinguished : 1st. The sugar, if its concentration does not exceed the plhysiolo- ‘gical border for more than to a very slight extent’), is compdetely retamed. This applies exclusively to glucose. ‘ 2nd. A partial retention takes place; This was the case with solutions of d-galactose, of d- and l-xylose, of d-ribose and of maltose. 3°, Nothing is retained. This we found for |-glueose, l- and d-arabinose, l- and d-mannose and lactose. But all sugars mentioned under 1, 2 and 3 exhibit multirotation . and occur therefore in 2 moditications. The question is thus obvious: why do those sugars mentioned under 1 and 2 not behave like galactose, i.o.w. why do they not all exhibit partial retention? I think that the explanation must be sought herein, that, of the glucose both modifications are retained, of the galactose and other sugars mentioned under 2 only one modification, and of arabinose and the others mentioned under 3 neither of the two forms. We will set ourselves the task to test this conjecture by experiment. We are engaged upon this; we have already obtained satisfactory results. Summary and conclusion. The experiments described above are concerned with the question what the cause can be that of a0.1 °/, solution of d-galactose only a part of the sugar is retained and the other not. Two explana- tions were possible. The first was that the original galactose-solution which we used was of too strong a concentration viz. 0.1 °/,. We thought namely of our earlier researches in connection with glucose, where it appeared amongst other things that when there is passed 1) Hampurger and Brinkman, Die Toleranz der Nieren fiir Glukose, Bioch. Zeitschr. 94, 131, 1919. 372 through the blood vessels of the kidneys a glucose solution which exceeds the physiological value (+ 0.07 °/,) only by about 0.03 °/,, already a little sugar is allowed to pass through, and, that this quantity increases according as the concentration of glucose becomes stronger, and that to the extent that with higher glucose concentrations less and less glucose is retained. The kidney cannot endure stronger glucose concentrations, 1.0. w. the glomerular membrane sickens. Towards galactose, however, the kidney behaved quite differently. Indif- ferent to whether stronger or weaker concentrations were used, a portion was always allowed to pass through, and what is remark- able, always about the half (see table | and II). The first explana- tion could therefore not be the correct one. Per exclusionem the second one had to be accepted then, namely this, that, of the two modifications in which glucose is present in aqueous solution, — the « and the 3 modifications, — the one is retained and the other is allowed to pass through. This conception agrees with the fact which we observed previously, ie. that the glomerular membrane is able to separate quantitatively from each other different sugars retaining one and letting pass through the other, which was demonstrated with mixtures of glucose and fructose, and glucose and lactose. The conjecture finds additional strong support in the fact that, according to our calculation, there are present practically equal quantities of the @ and 3 modifications in a solution of d-galactose, the same proportion thus, more or less, in which it is retained and not retained. For this very reason it cannot as yet be said with certainty, which modification is retained, the « or the 3. In the same position we are in the case of xylose, which like the d-galactose exhibits a partial retention. [t has appeared namely from our per- fusion experiments that on an average from */, to */, of the xylose is retained. « From the same point of view the partial retentions which were observed in connection with d-xylose, d-ribose and maltose may be looked at. Also these reducing sugars occur, in agreement with their multirotation, in two modifications; also these sugars exhibit partial retention. However it has appeared that not all sugars that occur in two modifications, show partial retention. In the first place the- d-glucose does not. If present in physiological concentration it is retained completely by the glomerular membrane, and as such it occupies a unique place; and still also the glucose occurs in two modifications. The latter applies also to l-glucose, d-mannose and |- and d-arabinose. Of these sugars nothing is retained. By summarising the multirotatory sugars can be divided into three groups. 1st group, of which we know only one representative viz. the d-glucose of which both modifications are retained. 2nd group (d-galactose, d- and I-xylose, d-ribose) of which only one modification is retained. 3d group (I-glucose, d-mannose, d- and |-arabinose) of which neither of the two modifications are retained. In the cases of the members of the second group viz. d-galactose and l-xylose which have been subjected to more detailed examination, the retention or non- retention is governed wholly by the position which the H and HO linked to the asymmetrie C-atom occupy relative to each other. It is worthy of comment still, in the first place, that the conjecture which we have offered here, and which ought to be controlled by a large number of experiments still, gives a physiological illustration of the existence of modifications, which formerly was found along chemical lines. In the second place it brings to light that if one desires to investigate stereoisomeric sugars with respect to lower organisms, as has already been done by several investigators, the fact has to be reckoned with, henceforth, that the sugar which is investi- gated is not a simple compound but a mixture, the two components of which need not behave similarly towards a microorganism. In these investigations Mr. R. Rorrink has lent his skilled assistance. Physiological Laboratory. Groningen, September 1919. 374 ERRATU M. p. 70 line 14 from the bottom: for: — the growth retardation curve for an intensity 1. read: — the growth retardation curve for an intensity 4. line 13 from the bottom: for: — the growth retardation curve for an intensity 4. read: —- the growth retardation curve for an intensity 1. KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN TE AMSTERDAM. Pon © GEE RINGS VOLUME XXII N25: President: Prof. H. A. LORENTZ. Secretary: Prof. P. ZEEMAN. (Translated from: “Verslag van de gewone vergaderingen der Wis- en Natuurkundige Afdeeling,” Vol. XXVII and XXVIII). CONTENTS. F. SCHUH: “Theorem on the erm by term differentiability of a series”. (Communicated by Prof. D. J. KORTEWEG), p. 376. JAN DE VRIES: “Involutions in a field of circles”, p. 379. H. ZWAARDEMAKER: “On Polonium Radiation and Recovery of Function”, p. 383. W. STORM VAN LEEUWEN and Miss M. Vv. D. MADE: “Researches on scopolamin-morphin narcosis”. (Communicated by Prof. R. MAGNUS), p. 386. A. DE KLEYN and C. R. J. VERSTEEGH: “On the question whether; or no Darkness-nystagmus in dogs originates in the Labyrinth”. (Communicated by Prof. R. MAGNUS), p. 393. N. G. W. H. BEEGER: “Bestimmung der Klassenzahl der Ideale aller Unterkörper des Kreiskörpers der m-ten Einheitswurzeln, wo die Zahl m durch mehr als eine Primzahl teilbar ist” (2ter Teil). (Communicated by Prof. W. KAPTEYN), p. 395. H. J. BACKER and J. V. DUBSKY: “On the preparation of z-sulphopropionic acid”. (Communicated by Prof. F. M. JAEGER), p. 415. J. WOLFF: “Some applications of the quasi-uniform convergence on sequences of real and of holo- morphic functions”. (Communicated by Prof. L. E. J. BROUWER), p. 417. H. ZWAARDEMAKER and F. HOGEWIND: “On Spray-Electricity and Waterfall-Electricity”, p. 429. J. W. LE HEUX: “The Action of Atropin on the Intestine depending on its amount of Cholin”. (Communicated by Prof. R. MAGNUS), p. 438. . F. VAN BEMMELEN: “The interrelations of the species belonging to the genus Saturnia, judged by the colour-pattern of their wings”, p. 447. H. W. BERINSOHN: “The influence of Light on the Cell-increase in the Roots of Allium Cepa”. (Communicated by Prof. F. A. F. C. WENT), p. 457. P. ZEEMAN: “The Propagation of Light in Moving Transparent Solid Substances. I. Apparatus for the Observation of the FIZEAU-Effect in Solid Substances”, p. 462. . E. J. BROUWER: “Ueber die Struktur der perfekten Punktmengen” (dritte Mitteilung), p. 471. B. VON KERéKJARTO: “Ueber Transformationen ebener Bereiche” (Communicated by Prof. L. E. J. BROUWER), p. 475. — is Proceedings Royal Acad. Amsterdam. Vol. XXII. Mathematics. — “Theorem on the term by term differentiability of a series” By Prof. F. Scnun. (Communicated by Prof. D. J. Kortrwne). (Communicated in the meeting of June 28, 1919). ab a. the functions u,(x), u,(a),.... are differentiable in the interval a N, we have | Una (C) + Unpe (Ce) +... + Ur (Ce) | << $d. From the assumption 6 it follows moreover that a number ae can be found, such that for n > N, the inequality | A, (8) | N and for every « within the interval 7. This establishes a; the assumption d can here be dispensed with. 3. Proof of @. If we put u, (@) + u, (#) + .... =p (e) and u, (a) Hw, («) +... =p (e) then the assertion 8 expresses that … @(p + h) — p (p) If we put u, (x) + u, (©) +... + un (e)—= U,(@) and un (we) + + Ure (w) + ....= On (#) then = ee EN DET ol) je ae hae = w(p + Oh) — w(p)— Ri (p + Oh) + © as a EE hence: SE Swee) + |Bu(p+Oh)\ + + [ree oak pe In consequence of the assumptions 6 and d yw (2) is continuous for x =p. Hence, d denoting an arbitrary positive number, the positive number ¢ can be assigned such that | y(p + h) —yVp) | <4 for hi <<; for | h| > N, and any @ in the interval 7 the inequality | Ane) | <4 is satisfied. For n > N, then | B, (p + OA) | < id. Let / be a definite number satisfying | h | <« and h #0, then the numbers MN, and MN, can be so determined that for > MN, and n > N, the inequalities jo, (p + h)| <4 d|h| resp. | en (p) <4 d|h are satisfied. Now, if n be chosen larger than the largest of the numbers N,, 25% 378 | (pp ef, eff) N, and N,, then it follows from (2) that | sin a Wp (p) | N, and > N,, that the inequality |B, (2)| | | Hieraus ergibt i) sich leicht der Beweis. Für die Klassenzahl H gebrauchen wir den bekannten Ausdruck’): 1 1 H = — lim (sl) 1—____ x s—1 pb l1—n (py? wo p alle Primideale des Unterkörpers & durchlaüft. Wenn wir nun die Sätze 9, 10 und 11 benützen und die Factoren, welche sich beziehen auf dem System bo, = bn = bin =... . = 0, von den anderen Factoren abscheiden, so findet sich: WEL g/r 1 | Pp 1 )-! A =— lim (s—1) H W— IT 1 —- 5 x bons b Fil n=? p =p P | 4M) bin ee ps Es ist hierbei angenommen dasz bo; = 6,, = 6,, =6,, =.... = 0 ist. Wir wissen nunmehr dasz: 1 lim (s—1) 7 ————_ = 1. sal P ae Weiter entwickeln wir jeden Faktor des dritten Produktes auf bekannte Weise in einer Diricarer’schen Reihe und multiplizieren all’ diese Reihen. Das Ergebnisz ist: dette Tages 08 n I eo ay OAs bii, - n’ Lene fe ie | fs ns _ T'(e) 0 =| } |#=F@ =! und wenn man noch Gebrauch macht von der Gleichheit ! n n | I= inl wenn 2 =n! (mod m): so findet man Hierin setzt man und 1 1 gfr F (#) Ae x n=2 x (lr) 0 wenn man auszerdem den ersten Hülfssatz dieser $ benutzt. Nach Zerlegung in Partialbriichen kann man die Integration durchführen und findet dann: 1) ,H”. Satz 55 und § 27. * 399 ee kri _—km 1 e/r een 2rke eine en kimi A=— — ~ 28 (e=) : + 4mi— | HK n—2 M kl Nun ist noch m 2hri mm 2 lend m m 2nnki Zr(e j= 3 | n |: m = n [Zen — 0 k=1 = 7 il | | und also bare. Arn 2kri Belen % n=2 mk=1 kri —kri gn kai log 7 a= (9) Um diese Form weiter zu vereinfachen, beweisen wir zuerst vier Hülfssätze. § 9. Hiilfssdtze. *) In allen folgenden Hülfssätzen ist das System 6),=6,,—=6),—= ...==0 ausgeschlossen. Qh ri j 5 tk ie r(e m |= ile =), Der Beweis ist ganz analog mit dem des übereinstimmenden Satzes meiner Abhandlung in “Proceedings” Vol. XXI, S. 758. 2. Es sei 2/* die höchste Potenz von 2 die auf bn teilbar ist. Wenn b,,=—0 ist, so nehme man /',—h,—2. Ist aber auch bon = 0 so nehme man A= hs. Ist h,—=2 so ist h‚==0 zu setzen wenn bon=1 und = 2 zu setzen wenn bon = 0 ist. Es sei weiter /," die höchste Potenz von /, welche auf 44, teilbar ist. Wenn bin =0 ist, so nehme man fh’, =/h,. u. s. w. hs / Es sei nun d = 2 /,4',.. go ist: pal Sa Beweis : Wir faszen zuerst die Symbole k + ik |; ‘le —_____ und | — : 92 92 ins Auge. Ist bon—=0O so sind sie beide —=1, und also einander gleich. Ist bon #0 und n gerade, so ist auch n +> gerade, weil dann 2 teilbar ist durch 4. Die Symbole sind dann beide — 0 und 1) Diese Hülfssälze musz schon Kummer benutzt haben, wiewohl in anderer Form. Die Beweise findet man aber nirgends. 400 wiederum gleich. Ist, zum Sehlusz, bo, #0 und n ungerade, so ist m auch 2 aa ungerade und ( 1%, EEE bon : LN bon Bn ) vo EE 8 zl pe 9 Für die übrigen Symbole kann man die Gleichheit auf analoge Art beweisen. 3. Wenn die Zahl d dieselbe Bedeutung hat wie oben, so ist Qrki "Ae ™ } = 0 wenn der gröszte gemeinschaftliche Teiler von & und m #d ist, und = andi wenn dieser gröszte gemeinschaftliche Teiler wohl = d ist. Beweis : i Qnnki EN 2Qanki ms | n |- mes [eae wales AZU nl 2krt a», 2n(n-+-m/d)ki — ie Chi DE ie a zi e EN nl Es durchlaüft 7 + zugleich mit n alle Zahlen für welche das d Symbol #0 ist, weil zufolge der Definition der Zahl d, die Zahl m/d prim ist zu den Zahlen welche » durchläuft. Also findet man : 2kri Fze d F Ist k nicht durch d teilbar, so ergibt sich hieraus “= 0. Nehmen wir nunmebr an dasz £ durch d teilbar ist; dasz aber _k=dtv, m = dtm’ so dasz dt der gröszte gemeinsame Teiler ist von k und m, t>>1. Man hat nun: 2nki Avi mn Qrvni r(- m J=r(. ©) = D>; | i |e ms n=1 m'_ 2roni de t ome n + sm = er he | und wegen 2: n=l | moni y_4 I ef n + sm —- d De ™ = S| . n—! s=0 Wir werden zeigen dasz der Wert der letzten Summe Null ist. a . rz A mn …. © Es sei w eine Zahl der Form 1 + —y (wo g der gröszte gemein- 3 g 401 same Teiler von n und m’ ist) welche prim zu m ist. Eine solche Zahl « besteht, denn die Form enthält eine unendliche Anzahl Primzahlen. ! m n Nin sist, nw. =n (1 +—y)=n+—ym' und wenn eine Zahl “i 9 _n + sm’ mit. «2 multipliziert wird so gibt es wiederum eine Zahl derselben Form. Wäre weiter x (n+-sm') = wx (ndsm) (moa 5): set so würde vem =azs'm' sein, und a(s —s')m'=0 also s —s — (mod tt) Dies ist unmöglich, da s und s’ m “(a) > 1—.si m fe ef | SE } ; Nun ist L | = 1 weilt durch 8 teilbar ist. 2 by! vei 1 — st [bn moe ; i =sle 7 denn 1 —st == + 58’ (mod 2'e) also 1 == + 5° (mod Qhy—he'—a ) Da die Potenz von 2, welche in den Modul auftritt, >> 2? ist so folgt: s’ = 2Q's—ls-a-2y,. Die Zahl. vy ‘ist ungerade, denn anderenfalls wäre s teilbar durch 2 und dies ist unmöglich weil t oe gröszte gemeinsame Teiler von n+] und m ist, und m gerade . Weiter hat man: 1— st bin dik —— =d h LI — St 5 tee m = eine primitive —-te dt 1 u.s.w. Aus all’ diesem ersieht man dasz J Einheitswurzel ist. Infolge dessen besteht die Gleichheit : Foleli¢h ist MA) Le eA ee wo ¢ alle Teiler von 2/,/,.... durchläuft. Es ist leicht ersichtlich dasz g-Mal die letzte Summe den Wert m hat, womit in unsrem besonderen Falle der Beweis erbracht ist. Um anzugeben wie der Beweis sich gestaltet wenn nicht alle friihergemachten Annahmen erfüllt sind, fassen wir noch den Fall ins Auge wobei h,—/',—a,=0 und a, =1, während alle übrigen Annahmen erfiillt sind. dst : : 3 In der Summe XZ bed nimmt die Zahl s nun nicht mehr alle m Werte an, welche Le jp and prim zu 5 sind ; denn da t nicht durch /, teilbar ist, so kann oe für einige Werte von s, durch /, teilbar sein. Die Zahlen z,z+/,....,2+ ( on 1) l, geniigen der Con- gruenz 1—2t =O (mod /,). Es sind 1) dieser Zahlen prim zu me dt’ In der Summe bleiben also (5) +a) =" +a) Zahlen der Form 1—s¢ iibrig, für welche das Symbol einen von Null verschiedenen Wert erhält. Für diese Werte von s ist ebenso wie früher: nb! y 2nb'ovz an x a ae ee Le He ize hid ga 2he tee : Lat Pin 1—st +, Weiter ist aber | ——— = AG DE Lh Die Zahlen 1—st, welche in der Summe auftreten, können wir m (mod l,, verteilen in sar) Mal eine Gruppe von Zahlen, welche dtl <1, sind. Die letzte Gruppe enthalt /,—2 Glieder, unter welchen die Null und die Zahl 1 nicht vorkommen; denn keine Zahl 1—st ist teilbar durch 7, und ¢ und s sind nicht teilbar durch /,, weil s 405 m ; : Dea prim zu — ist und diese letzte Zahl teilbar ist durch /,. Wir fügen dt 2 st nun die Glieder, in welchen das Symbol En Ln a denselben Wert hat, zusammen. Es wird dieser Wert dann multiplizirt mit der Summe der NE m : : : ; : primitiven ——-ten Einheitswurzeln. Wir sind also zu der folgenden dt l, Gleichheit gelangt: 1 — st m h n bin =| —— | =n x= ; del, n—2 LA Die letzte Summe hat den Wert — 1, wie sich aus dem ersten Hülfssatz dieses Kap. ergibt, denn es ist: Ln A A n bin da, weil 6;, durch /,”—! teilbar ist: n bin n' bin de a | = Fea wenn 2 ==n' (mod l,) Schliesslich: > pia 1 m m chliesslich: ee ve sslich | [ = ul, Der Beweis gestaltet sich weiter wie im vorigen Falle. Wir haben nun hinreichend angegeben wie der Beweis erbracht wird wenn man noch andere der früher gemachten Einschränkungen aufhebt; nur den Fall 6;, = 0 werden wir noch weiter untersuchen. In der Summe # haben in diesem Falle alle Glieder, fiir welche m ix é hae n prim zu -— ist, einen von Null verschiedenen Wert. Und weiter ist Ln n n' ~~ |=| ~ | wenn n=n mod, } Auszerdem kann man alle Zahlen, welche < m und prim zu m sind, (moe 5) verteilen in // Mal die Gruppe der Zahlen welche m DN, 1. oF i und prim zu — sind. Also: 1 Ih Qn (: + a di Qn n i di ha r=z|—|(e> mk m dee) Da d teilbar ist durch //%, ergibt sich hieraus 406 hi Qn n di Qn ndi r= in ines} | | en = L'a F, (- m ) n ooo A m Die Function #, leitet man aus #’ ab, indem man A 1 1 nimmt. Für die Function /’, haben wir schon bewiesen : FR =lh Fl Fi =l%h FF, = anstatt m Et ate he d iy Lai : caer == LJ lb, 1 Damit ist der Beweis in unserem Falle wiederum erbracht. (1) bone Bint ae § 10. Suútze über die Realitüt des Unterkörpers k und Bestimmung der Zahl x. 1. Wenn die Summe bon + din-+...., für alle Werte von n, gerade ist, so ist der Unterkörper /& reell und andernfalls imaginar. Beweis: Die den Körper & bestimmende Zahl ist 40 40 Ae) n= 4 +A wmlhetZ> ©) wo A jede Zahl der Untergruppe g bedeutet. Nun ist Ati) = A," AQ ACH. (modm) wo die Exponenten den Congruenzen (3) geniigen. Aus der Annahme dasz alle Summen bon + bin +... gerade sind, folgt dasz das Wert- system a,—1; a, =—0; a, =}4¢9,;... den Congruenzen (3) genügt. Dann geniigt aber auch das System agi + Ld nt ROTA Nun ist noch Aso! Asi Aci Sera ie Alt) (mod m) Die Zahlen A en — A sind (mod m) verschieden, denn wäre A(t) = — A (mod m) so würde 2 Al) = 0 (mod m) sein, und dies ist unmöglich, weil die A® prim zu m sind. Wir haben also bewiesen dasz die Zahlen A® von g in Paaren verteilt a Sa werden können. Für jedes Paar hat ZAZ caine reellen Wert. Man folgert hieraus leicht dasz 4 reell ist. Nun musz noch der Beweis des zweiten Teils des Satzes erbracht werden. Alle reellen Zahlen des Kreiskörpers A bleiben unverändert für DT) PMs AO: 407 die Substitution s = (7: Z—'). Denn es sei @ eine reelle ganze Zahl, so ist Oad 4.2 ann ei 3 sin 4 a, sin a,——+....=0 m und 2m ie =a, an ees Sei arate es en iia Also: 2—s2— 0. Ebensoleicht beweist man, umgekehrt, dasz eine jede Zahl, welche durch die Substitution s nicht geändert wird, eine reelle Zahl ist, Man ersieht hieraus dasz jeder reelle Unterkörper von A ein Unter- körper ist des zu der Gruppe s,s? gehörenden Unterkörpers und dasz ein Unterkörper, der zu einer Untergruppe gehört, welche die Substitution s nicht enthalt, imaginär ist. Wenn nun nicht alle Sum- men bont-bint.... gerade sind, so genügt das System a, = 1; dy =O; a,=4¢,...-. den Congruenzen (3) nicht; die Untergruppe g kann daher in diesem Falle die Substitution s nicht enthalten. Der Körper 4 ist imaginar. 2. Wenn die Summe bon + bin +... nicht für alle Werte von n einen geraden Wert hat, so ist die Anzahl der Systeme der 5, für ; p welche dieser Wert ungerade ist, gleich = ; * Beweis: Alle Systeme der 6 für welche die Summe gerade ist, bilden eine Gruppe, da 2,97,,9,,.... gerade sind. Um aus dieser Gruppe, die Gruppe aller Systeme der 6 zu bekommen, musz man die erstgenannte Gruppe multiplizieren mit einer Gruppe in welcher jedes System der 6 (ausgenommen die identische Substitution bo 0 b,, =0,....) eine ungerade Summe besitzt. Wenn, in dieser letzten Gruppe, sich zwei Systeme der hb vorfänden mit ungerader Summe, so würde das System, dasz man durch Aufzählung der übereinstim- menden Zahlen dieser beiden Systemen erhielt, eine gerade Summe bon + Oin +.... haben. Ein solehes System kann aber in dieser Gruppe nicht auftreten. In der Gruppe können also nicht zwei Systeme vorkommen (ausser boi = bx, =.-...== 0). Die Gruppe hat daher den Grad 2. (Pp Es gibt also forse Systeme der b, welche eine gerade Summe haben 5 und natürlich gleichviel mit ungerader Summe. Bestimmung der Zahl x*). DH.” S. 229. 27 Proceedings Royal Acad. Amsterdam. Vol. XXII. 408 1°. Wenn der Körper 4 reell ist, so ist w—=2 denn + 1 sind die einzigen reellen Einheitswurzeln. : : p : 3 5 ; Ks ist weiter 7, = 0 und 7, = — weil der Körper ein Gatots’scher ist. 5 af Bho ie p 2. Wenn der Körper # imaginär ist, so ist 7, =0 und 7, = 9. Nun x musz noch die Zahl w bestimmt werden. Ks sei w,—1 wenn alle Zahlen a, = 0 sind; andernfalls w, = 0. Es sei 2”* die höchste Potenz von 2 welche auf alle Zahlen a, teilbar ist und w,—=/,—2 wenn alle Zahlen a,=0. u‚=0 wenn nicht alle Zahlen a, durch 2 teilbar sind und auch = 0 wenn nicht alle a, =O sind; u,—=1 wenn alle a, =O und alle a, durch 2 teilbar sind. Es sei /,1 die höchste Potenz von /, welche auf alle Zahlen a, teilbar ist, und w,=/,—1 wenn alle a, = 0 sind. u, =0 wenn nicht alle Zahlen a, durch 4, —1 teilbar sind und w,=1 wenn dies wohl so ist. U. s. w. Dann ist: g Woh Wy Ua Pl pek ya; wenn hy > 3 ist: w = Wotl1, ww) RE ern abi eho uw) wenn hy == ist wi wenn? = st.) p= 21 Beweis: Der Körper & kann nur diejenigen Einheitswurzeln enthalten, welche Potenzen von Z sind, da der Kreiskérper K nur solche enthält. Nur wenn m ungerade ist, enthält K auch die Poten- zen von Z mit negativem Vorzeichen. Nehmen wir nun an dasz Ze in k liegt, dann bleibt diese Zahl ungeändert für die Substitu- tionen von g. Wenn wir also alle Zahlen von g durch A® darstellen, so ist Ze woraus sich ergibt: (ANIJS mod mee er ee) Wenn, umgekehrt, die Zahl a dieser Congruenz genügt, für alle Zahlen A® von g, so enthält der Körper & die Einheitswurzel Ze. Sind alle Zahlen a, = Oso enthält der Körper die Einheitswurzeln + 7. Beweis: Es gilt für jede Zahl A: AD == 5ax (mod 24) also A 1 (mod 4). Aus (10) ergibt sich daher a=, d.h. die Hinheitswurzel ZA =i liegt im Körper 4. Daher auch —?. 409 Sind alle Zahlen a,—=0O und alle Zahlen a,— 0 so enthält & die Qnt Einheitswurzel ¢2'*. Man beweist dies auf dieselbe Weise. Sind alle Zahlen a, =O und alle Zahlen a, höchstens teilbar durch 272 i ete, Wares ‘af 2 2¥* so enthält & die Einheitswurzel e2”*'?. U.s.w. Alle Beweise werden mit Hülfe der Congruenz (10) erbracht. Man findet nun leicht die im Satze genannte Formel wenn man 2d zit : ee : nl PE beachtet dasz, wenn der Körper die Einheitswurzel Fas enthalt, ; I auch die /+! ersten Potenzen dieser Zahl im Körper liegen. § 11. Ableitung des entgültigen Ausdrucks für die Klassenzahl. Satz: Wenn, für jedes System der 6, die Summe bon + bin + .... gerade ist, wenn also der Körper /: reell ist, so stellt sich die Klas- senanzahl H dieses Körpers, wie folgt, dar: a = | . |? 4 a re | log As yo poe ist Ol bon, bns bin Ged EE R Hierin ist A, =) (1—Z*)(1—Z *) und R der Regulator. Aus- gereden ish angzenommen, 0), biss. 0. Ist h,=2 so musz ban weggelassen werden und ist h, =O so musz auch dbo, weggelassen werden. Wenn die Summe bon + Din +... nicht für jedes System der 6 gerade ist, wenn also der Körper / imaginär ist, so stellt sich die Klassenanzahl dieses Körpers 4, wie folgt, dar: m m 8 2 8 DENN nn ce | 8) RON eee log Ag 1 Gel bons bns Üdrares 4 nsi bon, Diens bin, wee Ji (2m) Par B Hierin ist w die in § 10 bestimmte Zahl. Das erste Produkt ist über alle Werte von n, für welche die Summe bon + bin +... ungerade ist, zu erstrecken; das zweite Produkt über alle Werte von n für welche diese ‘Summe gerade ist. Weiter ist 4, =V(1—/Zs) 4—Z~—) und A! die Determinante der Logarithmen der absoluten Werte eines Systems von Grundeinheiten. Ist Ay = 2 so ist bn wegzulassen, und ist hy =O so ist auch bon wegzulassen. Beweis : Zur Ableitung des ersten Ausdrucks benutzen wir (9) und ersetzen, in der darin auftretenden Summe, die Grösze &£ dureh m—k. Wenn 27* 410 wir dann den ersten Hülfssatz von § 9 benutzen, so finden wir kai aera? 4 EF. —=0. Die Gleichheit (9) läszt sich daher umformen zu: m 1 dpe Wh ae on ki H=— = —— ZF l(e™ jlog A; x n=2 M ki wenn: kr ke ri EEn Ver) kri 2kri é m — € mi — A7, — ~_— 1e m (Eet 1 kere kri Qhri kri ae eg m Va m IEEE m ) Dieses Resultat für H läszt sich noch veränderen mittels Hülfssatz_ 3 und 4 von § 9. Aus Hülfssatz (3) ergibt sich, dasz in der Summe nur diejenigen Glieder übrig bleiben für welche die Zahl & mit m den gröszten gemeinsamen Teiler d hat. Diese Zahl d deuten wir weiter an durch d,. Die Summe wird zu and, 2 k /d =A (Sal | log Ay k Es sei nun == k'd,, so ist, wie man leicht findet: Ap == Ap _A ijd 004 A m K+ bd, Dg n Die Summe wird daher zu: ( andi k/dn —l PF = Dy Ì | loy Aj log: Aven, ANT ME : (lo. + Loy et He ) (11a) n und durch Hiilfssatz 2 $ 9 zu: andi 8 —l (Sal | bog AEN bene dn Sten IL) == denn, weil &' prim zu m/d, ist und in (11a) die Zahl 4 alle Werte annimmt die mit m die Zahl d, zum gröszten gemeinsamen Teiler haben, bekommt man eine Summe, in welcher s alle Zahlen < m durchläuft, welche prim zu m sind oder nur noch teilbar durch diejenigen in m aufgehenden Primzahlen, deren betreffende 5 = 0 [eo ist. sind. s durchläuft also alle Zahlen für welche | In der Formel für A kommt nun das Produkt qe ee Fle ™ ser, om ang eee zum Vorschein. Um dessen Wert zu berechnen benutzen wir Hülfs- 4 satz (4) von $ 9. Da zu jeder Substitution einer Gruppe auch die reciproke Substitution vorkommt, so wird ins Besondere hier zu jedem System bon, On, ... auch das System 2— bon, Ea Osen, P,— Din... auftreten. Wenn diese Systeme von einander verschieden sind, so ist, nach dem ebengenannten Hülfssatz, das Produkt der zugehörigen Funectionen und #" gleich dom. Sind sie aber einander gleich so hat das Produkt den Wert + dm. Das Produkt (12) hat daher den Wert leren ee VAD aeta Gha Care de (13) n=? wenn man das Vorzeichen ausser Acht läszt. Bevor wir nun den Wert dieses Produktes entgültig bestimmen, formen wir die Summe aus (11) um. Zuerst können wir darin den Exponent —1 weglassen, weil, wie schon bemerkt worden ist, zu jedem System der Zahlen 6, auch das System 2—do,,.... vorkommt. Weiter ist m m m s 5 < 2 <2 Ms Zz feu Aa eal ne | log Ams S= s=l s=1 Denn, fiir den Fall dasz m ungerade ist, folgt diese Gleichheit : m aus der bloszen Bemerkung dasz | = 0 ist. Ist, im Gegensatz, m/2 m gerade, so ist offenbar | ~~ |—0O da m durch 4 teilbar ist. Zur weiteren Umformung benutzen wir die leicht zu erhaltende Beziehung : | ri 5 nh men Ë s | und auch die Gleichheit A, ‚== A,, und erhalten so für die Summe: m2 s ed : log As CS) Nach Einführung in die Formel (9) und nach Einsetzung des Wertes von x und der in $ 6 berechneten Diseriminante d, erhält man den im Satze angegebenen Endausdruck fiir die Klassenzabl, nachdem das Produkt aller Zahlen d, welches in (13) auftritt be- rechnet ist. Weil die Berechnung dieses Produktes dieselbe ist beim zweiten Teil des Satzes, so werden wir diese Berechnung bis zum Ende des Beweises aufschieben. Beweis des zweiten Teils des Satzes. Das Produkt ans (9) zerlegen wir in zwei Produkten: das erste 413, ist zu erstrecken über alle Werte von » für welche bon + bin +... ungerade ist, während das zweite zu erstrecken ist über alle Werte von ” für welche diese Summe gerade ist. Das letzte Produkt läszt sich auf dieselbe Weise umformen wie beim Beweise des ersten Teils des Satzes. Beim ersten Produkt erhalten wir, durch Benut- zung des ersten Hülfssatzes § 9, nachdem in (9) m—k statt k ein- geführt ist: m 2 ke lk nde Es bleibt von der Summe aus (9) also nur übrig ni m an ki ee a SI Ay ie ee | M zi Wegen Hülfssatz 3 (§ 9) wird dies zu andi ni wane: kid ee x di kl ed Olt Erle "ze 0 | rz | m a m x ad wo & über alle Zahlen zu erstrecken ist, die mit m die Zahl d, zum gröszten gemeinsamen Teiler haben. Weiter ist, wenn £k=k'd,; d=d,;: KT kid T+ ET KH m/d 1 aa | zee + EW fd} st fe k d k iC k'+- (d—1) me dt 2 (le + (d—1) m/d) } 5 |—— |" ES EEN + =) 0 ist wegen Hülfssatz 1 von $8. Wir erhalten also das Resultat: ni mn \m s 7, ——Ft{ewm SS fie ae 8 mn s=l in welehem der Exponent —1 darf weggelassen werden da, wenn da die Summe bon + bin +..... ungerade ist, auch die Summe der Werte 2— bon, 4 Py — byn,..---- ungerade ist. Es ist ersichtlich dasz in (9) wiederum das Produkt Yd, zum (== Vorschein kommt, wenn wir die Faktoren zu zweien nehmen und jedes Produkt mittels Hülfssatz + (§ 9) umformen. Dabei erscheinen dann Potenzen von 2, und ebensolehe treten auch bei der weiteren Umformung noch auf. Man kann sie jedoch auszer Acht lassen weil die Zahl /7 natürlich eine positive ganze Zahl ist. 415 | p/r $ 12. Berechnung des Wertes des Produktes Hd. n=2 Wir bestimmen die Potenz von 2 und die von /,, welche in dem Produkte vorkommen. Es gibt }@, voneinander verschiedene Zahlen 6,, da die Unter- gruppe g primar ist. Jeder dieser Werte kommt also in allen Systeme ih 3 ; der 6 genau —: 49, Mal vor. Nun gibt es unter den voneinander 1 he verschiedenen Werten der 6,, genau. Ziet Zahlen welche genau durch 2 teilbar sind. Dha—5 Le > ”) ” 2e ” ” 2° ” 9 ” ” Qhe—3 ” ” 1 ” ” ” ” he? be ” Alles zusammen genommen gibt dies die folgende Potenz von 2: . he DL (he) 2 Par wenn man bedenkt dasz das System bon = dyn. . . =O nicht mit- gezählt werden musz. Man musz nun noch Riicksicht nehmen auf den Fallen worein bo, und 6,, beide zugleich den Wert Null annehmen. Für einen jeden solchen Fall musz obenstehende Potenz von 2 noch eae : Pi ie : mit 2? multipliziert werden. Nun gibt es 5, Systeme in welchen ar bon =O ist. Die Systeme in welchen bo,=6b,n—0 ist, stellen eine Untergruppe der Gruppe aller Systeme 6 dar. Die Systeme, in welchen 6, — 0 ist, stellen auch eine Untergruppe dar, welche die erstgenannte Untergruppe enthalt. Diese erstgenannte Untergruppe musz, wie leicht ersichtlich, multipliziert werden mit der Gruppe des Grades 2: Bin BD = OF Oi, eis Gea Og = OS Oi. um die Gruppe der Systeme, in welchen 6,,—0 ist, zu erhalten. Man erkennt also dasz die Hälfte der Anzahl der Systeme in welchen Den == 0 ist, die Anzahl der Systeme ist, in welchen bun = dyn = 0 ist. Dies ist also der Fall wenn nicht fiir jeden Wert von n bon = Dn =O ist. Ist jedoch bon + bun für alle Werte von n gerade, so ist immer bon — On — 0. Es ist nun leicht ersichtlich dasz obenstehende Potenz von 2 noch multipliziert werden musz mit (4) wo ¢ dieselbe Bedeutung hat wie in $ 6 Satz 8. 414 Um die Potenz von /, zu bestimmen fiihren wir das Folgende an: Weil d,*) der gröszte gemeinsame Teiler aller Zahlen bi„ ist, so : p : ; AN: gibt es = voneinander verschiedene Zahlen bi. Kin jeder solchen C 1 ( (fp Wert kommt daher ot Mal vor. Weil g primar ist, ist d, nicht i RO durch /, teilbar, und, wenn hk, =1 nicht durch /,—1. Die durch / teilbaren Zahlen 6;, sind daher 1 ier dle, dl UAE ad a, Von diesen sind WE 8 ET Zahlen genau durch /, teilbar. ¢ 1 iter): lmA 1 9 ” 3 ” [ae —1 Zahlen genau durch /,“—! teilbar. ay Wenn man noch Riicksicht nimmt auf den Fall wo alle 6=0O sind, so erhält man für die Potenz von /, welehe im Produkte aller dn vorkommt : eert pd, d ar hy L, Wenn nun PD die in § 6 bestimmte Discriminante des Körpers & darstellt, so gibt eine leichte Rechnung allgemein die Formel 9), Pan VY D 2 IT dy Si? 2 2 n=2 Im Nenner des Ausdrueks fiir M findet sich nun noch der Regulator R. Nehmen wir ein System Grundeinheiten n;, so gilt: log || = 4 he (mi) *) weil die conjugirten Körper reell sind. Nach Einführung in # kann man die Faktoren 2 aus dem Determinante herausheben, wonach die Richtigheit der im Satz § 11 angegebene zweite Formel bewie- sen ist. 1) Diese Zahl d, hat also dieselbe Bedeutung wie in § 4.2. Eine Verwirrung mit den in § 9.2 bestimmten Zahl d, welche im Anfang des Beweises von § 11 dy genannt ist ist wohl nicht zu befürchten. 2) „H.S. 215. Im Ausdruck auf S. 376 (oben) nennt Herr Hitpert die Zahl R den Regulator. Dies ist nicht in Ubereinstimmung mit seiner Definition auf Seite 221. Chemistry. — “On the preparation of a-sulphopropionic acid.” By H. J. Backer and J. V. Dussky. (Communicated by Prof. F. M. JAEGER.) (Communicated in the meeting of September 27, 1919). The only practical method for the preparation of the «-sulphopro- pionie acid has hitherto been the one found by FRANCHIMONT ') i. e, the action of sulphuric acid with propionic acid anhydride. Besides Francnimont’s general method, an analogical method has been used by Mersens*) for the preparation of the sulphonacetic acid, the first term of the sulphoearbonic acids, namely the action of sulphurie acid anhydride with acetic acid. These two methods show much similarity. In both cases the mixed anhydride, the acetylsulphuric acid, is formed as an interme- diary product, as FRANCHIMONT *) had already supposed and Stinuicn ‘) afterwards established, and as Van Peskr *) proved for the reaction of Mrtsens. | With a view to comparing it with the method of FRANCHIMONT we have applied Mersens’ method also to propionie acid, by treating it with sulphurtrioxide. In both cases the reaction turned out, just as in the ease of the acetic acid, to be indirect, while the mixed anhydride, the pro- pionylsulphuric acid, must be taken to be the intermediary product. GEGE: COH SO, — CH, CHROOSSOSR — = CH3.CH(SO;H).CO,H. (CH. CH. CO),O + H,SO,= ) = CH. CH,.CO,H-+CH,. CH,. CO,SO;H— CH. CH(SO;H). CO,H.$ When the substances are mixed carefully whilst cooling, a colour- less, very viscous liquid is formed, which yields the sulphopropionic acid only at a higher temperature, with development of heat and brown coloration. We were able to follow the process of this reaction by titration, Il. 1) Recueil trav. chim. 7, 27 (1888). 2) Ann. der Chemie 52, 276 (1844). 3) Versl. dezer Akad. 16, 373 (1881). *) Ber. d. dtsch. chem. Ges. 38, 1241 (1905). 5) Versl. dezer Akad. 22, 996 (1914). 416 as the propionylsulphurie acid, when hydrolyzed, neutralizes three equivalents of a basis, and the sulphonic acid only two equivalents. The sulphopropionic acid was separated and weighed in the form of its baryumsalt. From 1 molecule propionic acid with sulphur- trioxide we got an average yield of 0.85 mol. sulphopropionie acid, and from 1 mol. propionic acid anhydride with sulphuric acid (monohydrate) 0.55 mol. Compared with Francuimont’s method, the reaction with sulphur- trioxide is experimentally less simple. The reaction is violent, the product is more coloured, the baryumsalt likewise, and the yield is various. Only for the preparation of large quantities may the reaction be recommended. FRANCHIMONT’S reaction, however, is very easy to carry out; but it is a drawback, that only half the propionic acid anhydride can be transformed into the sulphonic acid, the other half returning to propionic acid. We have therefore tried to combine the advantages of both methods, the better yield of the first and the greater purity of the product obtained by the second. For this purpose, it is possible to mix propionie acid anhydride with sulphuric acid, and to treat the product, containing one molecule free propionic acid, with the equimolecular quantity of sulphurtrioxide. However, it is simpler to let the propionic acid anhydride react directly with pyrosulphurte acid. (CH)2€Q);,0 HSO, = HCH. CH(SO>H). COE, By this method we got an average yield of 0.75 mol. sulphonic acid from 1 mol. propionic acid anhydride. As the pyrosulphurie acid may be added in the crystallized state to the propionic acid, the method is easier than the reaction with sulphurtrioxyde. The reaction proceeds more quietly and the product is purer. We have applied this method to acetic acid anhydride also, but for such an easily accessible substance, the original method of FRANCHIMONT is simpler. For the sulphuration of precious carbonic acid anhydrides, the method just described may, however, be recommended for its higher yield. Groningen, Sept. 8, 1919. Org. Chem Lab. of the Univ. Mathematics. — ‘Some applications of the quasi-uniform convergence on sequences of real and of holomorphic functions”. By Prof. J. Worrr. (Communicated by Prof. L. B. J. Brouwer). (Communicated in the meeting of September 27, 1919). We consider a region G of the complex plane, and a sequence of functions fÁ,f,.... which are all analytical within G. If the sequence converges uniformly in any closed region within G, then the limiting function f is analytical within G. This theorem however, enunciated by Weinrsrrasz, has of late been considerably extended. If it is known that the sequence converges at the points of a set having a point of the interior of G for a limiting point, and besides, that the sequence is uniformly limited in every closed region internal to G, then this is already sufficient to conclude to the uniform convergence of the sequence in any closed region within G, which involves that the limiting function fis analytical *). This same conclusion may be drawn, G being a circle, if only the sequence of functions is supposed to be uniformly limited within oo G and convergent at the points z,,z,,..., such that JI (zs—z,) = 0, i where z, is the centre of G and zj, 4 z, 7); also if it is only supposed that there exist two definite numbers a and 6 such that nowhere in the interior of G one of these two values is assumed by a function of the sequence, and besides that the sequence converges at the points of a set having a point internal to G for a limiting point ®). None of these theorems lead to other sequences of functions than those which are embraced already by Weterstrasz’ theorem. The question now could be raised whether it is possible that a sequence of functions, each of which is analytical within a region G, may converge to a function, analytical within G, without this convergence being uniform in every closed region contained in G. That such sequences actually exist is shown by an example, 1) G. Vrrarr. Annali di Matematica, Serie 32, tomo 10 (1904), p. 65. For a simple proof vide a.o. Verh. der Kon. Ac. v. W., dl. 27 (1918), p. 319. 4) W. BrLASCHKE. Leipz. Berichte, Band 67 (1915), p. 194. 8) C. CARATHÉODORY and E. LANDAU. Sitz. Ber. Ak. v. W. Berlin, Band 32 (1911), p. 587. 418 given by Monten with the purpose of refuting the assertion of Pomprsu that it should be necessary — in order to render the limiting function f of a within G convergent series of analytical functions analytical in G — to suppose that it is permitted to integrate the series termwise along every curve of integration in G''). In fact MonrEL’s series converges to zero throughout the whole plane whilst the termwise integration along the real axis from O to 1 yields not O but 1, so a uniform convergence in every limited region is here altogether out of the question. We state, besides, that it is neither permitted to differentiate Monren’s series termwise for z= 0, as this would yield the result + oo instead of zero. The example of Monte, is embraced by none of the foregoing theorems. The want is now felt of a necessary and sufficient condition that a sequence of analytical functions, convergent in the interior of G, should have an analytical limiting function. Necessary it is that the con- vergence of the sequence be quasi-uniform in every closed region within G, because of the continuity of f and /. If, conversely, this quasi-uniform convergence is given, then it follows that f is conti- nuous within G, but not that f is analytical, as an example of Monte”) shows. In the following a necessary and sufficient con- dition will be deduced among other things. ib We consider a sequence of real functions f,(e), 7,(a),..., each of which is continuous for 0<# 0 and y = 0, or vw =O and y > 0. The limiting function is Dey) = f(x)—f(y) at every point where w > 0 and y > 0 and vanishes at <= 0, y =O, since there every ®, =O. Let it be given that f(w) converges to a finite limit as w — 0. We consider a set of points P, (z,, ,), P, @a» Ya). Where 0 JN. We have (0,0) = 0. In consequence of the continuity of Py 0 a number Jd, can be assigned such that at every point /, where 1) P. MonTEL. Bul. des Se. math., série 2, tome 30, part 1 (vol. 41, 1906, p. 191). 3) Pe MONTEL, li c, p. 190. Thase p. 98. 419 : ; att DE vd, and y, Sd, the inequality Pyle vj) ZS 5 8 satisfied. Since f(x) tends to a finite limit as wv —0 there is a number d, such that for v7, << d, and y,< 4, the inequality Das Y,)) Sn is satisfied. If d< d, and at the same time < d, then ‚Pp (ze, y) — En (es y,) | zal € for ©, <<. d and Y, xa d. There now exist only a finite number of indices for which Ed yv, 2d. At each of the points Br corresponding there to an index N, > N can be found such that en KE At the points of the considered set the sequence of functions DP, therefore converges quasi-uniformly. 2. Conversely, let it be given that at the points of every set Are) Later Ya) a where OX ed, ry Sa, lim v= 0, k= hm y, = 0, this sequence of functions converges quasi-uniformly. k=a Let M denote the maximum of f(w) at 0, m the minimum. It is possible to construct a set of points z,,2,,... where x, >>0 and lim v= 0, such that Zon J (#,) = WM; similarly we can construct a (——— OO == 0) set Yi, Yay---, Where y, > O and my, = 0, such that lim fy) == nt : k=oa k=co If we now consider the set of points ENE BEN ese azen then (2 y,) = M—m, (1) k= unless m=-+ o or MM = — ow. Now it is possible, by hypothesis, to choose at each of the points P, from a finite number of indices 1, 2,...n(s) an index n, such that € |P (a3 yv) PT Py, (es y,) | < Di r F $ 5 5 (2) where « denotes a given arbitrary positive number. In consequence of the continuity of each of the functions ®, (v,y) and since the number of indices here considered is finite, a number Jd exists such that at every point P,, where v, 0 and fim x, =0, the lim f(x) is + oo. k= k= To every «, of such a set a y, >0 can be found such that JW) > Fe) +k and oR — Oee then Ge (ey) =— @, which is incompatible with the formula (4). Similarly it appears that M cannot possibly be — oo. Collecting our results we have the following theorem : If a sequence of functions f (x) be given each of which is conti- nuous for OSe 0 and f (O) for «=O are 46 in consequence of the suppositions continuous atO0, to À = lim f (0) for «=0. If 70) = lim f (0) then it EN aye, RTO ape) liek a x=0 We now consider a set of points «,,#,,..., where lim «= 0. k=o By «,.V we denote two given positive numbers. Then there exists an index NV, > N such that OV EEN ee he We AE (6) . *, a Ie . . Since Sy, = fy) and Sy) is continuous and because of (3), there exists a certain number /, such that for every k > &, ARE) Ae) Se Since at each of the finite number of points where k <%, an index Vz; >> N can kj aa such that AC mis (er) <é, it appears that the ne of functions /" (w) converges quasi- uniformly at the points of the considered set. — If this sequence of functions converges quasi-uniformly at the points of every set of the above-considered kind, then it results from an argument as held in § 2 that 7 (0) = hm En (0). n= We have therefore the following theorem: If a series of functions f («) be given, each of which is continuous 422 for OS Sa and has a finite derivative f (O) at 0, and if f(a) for O "0 L+tn?a2*? 1+n?y* w (x,y) everywhere converges to a continuous function, 2. €. quasi- uniformly in every finite region so that the criterion given in § 3 is satisfied. Accordingly /' (QO) is finite. However, if we choose é<'/, and the number of indices is finite, then for every « which is small enough we have for each of these indices Fe (a) S77 ,, whilst for «40 we have lim f° (v) = 0. The criterion, enunciated n= in this § is therefore not satisfied. Accordingly we have lans HO = KEE 0). „== 00 The foregoing theorems can be immediately extended to the complex plane and are capable of analogous proofs. HI. 5. Let there be given a series of functions f, (2), f,(z),..., each of which is analytical in the interior of the circle |z| <{a, and convergent within this region. We shall now demonstrate the follow- ing theorem: A necessary and sufficient condition that the limiting function f should be analytical within |\z| N exists such that | Wy, (or Yor Zo) — W(X, Yor Zo) | CE From the continuity of wp (B 2): N and w within £' it follows that in 0 &' is contained a region @2 in which everywhere [Wy (1m 2)—P@y2z)|Ce--- 2 2. - (3) If n= Ye then w (,, ¥,%) =9 for every n. Also wp is continuous in a neighbourhood @' of (#,, ¥,2,) and moreover we have lim w (, 4, 2) = 0, so that here too the inequality (3) is satisfied dnek = Zo in a neighbourhood 2 of (w,, y,, 2,): Every point of V therefore lies in a neighbourhood 2 such that throughout $V by one and the same index J, the relation (3) is satisfied. Since V is limited and closed V can be covered by a finite number of these neighbourhoods’), whence it follows that yp. converges quasi-uniformly at the points of V. 7. Conversely, let be given the quasi-uniform convergence of y at the points of every V. Let z, denote an arbitrary point where 1) E. Bore. Legons sur les Fonctions monogénes, p. 11. 28 Proceedings Royal Acad. Amsterdam. Vol. XXII. 424 2, Z, / (2 converges also. If C’ denotes a circle situated within z f (2) =—S,(z) and = f* (0, z) = Sy (0,2), then Sy’ (0, z) 1 n 1 n is continuous for OS |z| 0 we have lim S# (0, 2) Ne) path gd By hypothesis this function mere to a finite limit co as z2—> 0. Now let an arbitrary number ¢, > 0 be given. We construct a set of points z,,2,,... where lim z,— 0. From a limited number =n of indices an index can be chosen at each of these points such that SO, z)—f 0, 2) < Hence an infinite number of points z, exists where one and the same index can be used which we denote by ,. As z, tends to zero Sn, (0, z,) tends to S'n, (0) and f*(O, z,) to f (0). Thence it follows that | S',,(0)—/'O)| Se. Let ¢,,¢,,... BE a KERK sequence of positive numbers, having zero as limit. It is again possible to choose for every z, out of a limited number of indices an index n, >> n, such that | Sn, (0, z,) —f*(O, z | <€,, whence may be concluded, as ny, ( before, to the Ee of an index n, >> n, such that | S',,(0)— f'(O)| Se, Thus pursuing we find that there is a partial series of ee S” (2), S! (2),.-., such that lm S (0) = 7"), which establishes the p=n p theorem *). 12. The theorem of the foregoing § may be reversed as follows: If within \2) NV can be determined such that FOL) N at every point z of the annular region an index n, may be chosen which satisfies ASO 2) — ft (2) | > N such a choice can be made, which establishes the theorem. 13. Some of the results here obtained we shall apply to Montrr’s example, which was cited in the introduction and is reproduced below : The function y (2) =n’ze—" tends to zero as n — op for all real values 1 1 of z. Now consider the three rectangles J, (— nSearSn,— on o throughout the whole plane. According to the theorem established in § 5 the function wy, (a,y,z) must converge quasi-uniformly at the points of every set V: P, (#,,y,,0), P,(#,,y,,0),... where a, F0, yv, #0, lim «x, =O and lin y,=0. That this is really the case may be pro- k=o k= ved as follows: Choose an arbitrary pair of numbers ¢, N. Let 1 N,be > N and at the same time Br From a certain index &, € onwards #, and y, are both internal to /, so that 1Vo 23 2 3 Wy, (es pe Oh Ni (am xy, a Me Me) Ee ; <5 Hence a number 0 exists such that for | «#,|< 5 ef al | <0 we have ws Gy ty DIS This, we remind, is the = from a certain index %, onwards. 4M Besides, a number NV, > N can be found satisfying N, > — me’ 428 where J/ denotes the maximum of «, and y, ,m the minimum for k 1 | k=1,2,...k,—1,) and such that at the same time N eN, 4 1 and a LOO: Rise of temperature, therefore, increases the positive charge of water exceedingly. The influence of addition of a saltsolution is also a fact that cannot be denied. Alcohol, which, without salt, inereases the posi- tive Charge in water considerably, partly loses (his faculty at first in the presence of common salt, and even loses it altogether with a concentrated salt-solution. Then the negative charge of the salt predominates. Salt added to the negatively electrifying camphor heightens this negative charge. Camphor and salt co-operate. It should seem then that in the case of waterfall-electricity a simple summation takes place of the effects of water, salt and the volatile addition *). In endeavouring to account for these phenomena we might look upon spray-electricity as well as upon waterfall-electricity as a form of frictional electricity. In either case the friction, between the liquid and the air in the outflow of the spray, between liquid and zine plate in the waterfall, would set free electrons that are scattered about in the surround- ing air. But then the liquid were invariably to be charged positively, which turns out differently in a majority of cases, as shown by the experi- ments. We presume, therefore, that a more intricate process is at work, in which larger corpuscles arise as carriers of the electric charge. Such formations might perhaps arise from the so-called ions, an equal number ®) of positively- and negatively-charged ions, round ') In the case of spray- electricity the process is a much more complicate one. (See E. L. Backman, Researches Physiol. Lab. Utrecht (5) XIX p. 210. *) H. ZWAARDEMAKER. “Le phénomène de la charge des brouillards de substances odorantes. Arch. Neerl. Physiol. de l'homme et des animaux’ Tome I 1917 p. 347. 436 which the water vapour may condense into droplets and with which salt-droplets may combine afterwards. Lenarp') believes that 7m the superficial layers of every di-electric liquid there is not only an electric double-layer, generated by the molecular forces of the liquid itself, the negative layer being situated on the outside, but also that these layers differ as to material. These differences, which vary ‘with the substances dissolved in the liquid (electrolytes, volatile substances, complex molecules) affect the thickness and the strength of the electric double-layer. It appears then that Lenarp reduces the problem of the origin of waterfall-electricity and of spray-electricity to his hypothesis regard- ing the specifie condition of the surface of every di-electrie liquid. Strictly the origin of the electrification would then be, not an emission of electrons, but a discharge of extremely fine droplets, the so-called “carriers”, which, varying with the surface condition of the liquid, are either very small and charged negatively, because they take their origin entirely from the outer negatively charged layer of the liquid, or they are somewhat larger and may be posi- tively charged, since the majority of them arise from the interior positive layer of the liquid. For further particulars we refer to LiENARD’s article itself. Suffice it to state that most of our results with waterfall-electricity are sufficiently explained by this theory. Not however the intensifying influence of rise of temperature (LENARD’s private opinion, founded on theoretical considerations, was that a lessening influence was to be looked for). Neither does this theory explain why camphor and bornylacetate diminish waterfall-electricity ; no more is the question of the inten- sifying action of the sweet substances and the colloidal substances settled by if. The results obtained before in the Utrecht Physiological Laboratory in experiments on spray-electricity *) are much less easy to explain with the aid of this theory. First of all pure tapwater (Utrecht Water Company) and fresh- distilled water (old-distilled water zs active) give no or an inappreciable charge in spraying. Secondly the intensification of the charge in consequence of addition 1) P. Lenarp. Ueber Wasserfallelektricität. Ann. der Physik. Bd. 47—1915. *) H. ZwaarpemMaker. Het in overmaat geladen zijn vau reukstofhoudende nevels. Verslagen K. A. v. Wetensch. Deel XIX N°. 1. H. ZWAARDEMAKER. Specifieke reukkracht en odoroscopisch ladingsverschijnsel in homologe reeksen. Id. Deel XIX NO. 2. 437 of an active substance, is generally much greater with spraying than with the waterfall. Thirdly, according to Lenarpb’s theory, salt-solutions yield a distinct negative waterfall electricity; on the other hand they did not givea charge with spraying *). Finally Lenarp’s theory proves the bigger positive carriers of a common salt solution to contain sodium; the smaller negative carriers on the contrary consist of pure water molecules. This he demon- strated by bringing the carriers between the plates of a condenser, in which process the bigger positive Na-containing carriers went over to the negatively charged condenser-plate where the presence of sodium could be proved. The smaller negatively charged carriers went over to the positive plate, where no sodium could be found. A similar experiment with sprayed salt-solution *) shows that the big, positive carriers as well as the small negative ones contain sodium. We conclude, therefore, that though there is a correlation between waterfall-electricity and spray-electricity, they are obviously not quite identical. H. ZWAARDEMAKER. Reukstofmengsels en hun laadvermogen door nevelelectriciteit Id. Deel XXV 30 Sept. 1916. H. ZWAARDEMAKER. Le sens de l'adsorption des Subst. Volatiles Acta Oto-lary- agologica. H. ZWAARDEMAKER en H. ZEEHUISEN. Over het teeken v. h. ladingverschijnsel en de bij dit verschijnsel waargenomen invloed op de lyotrope reeksen. Verslagen K. A. v. Wetensch. deel XXVII 1918. E. L. Backman. De olfactologie der methylbenzolreeks. Id. Deel XXV 27 Jan. ’17. E. L. Backman. Ueber die Verstäubungselektricität der Riechstoffe. Arch. f. d. ges. Phys. Bd. 168 S. 351. C. Huyer. De olfactologie v. aniline en homologen. Diss. “Onderz.” (5) Deel RV: 1,589: 1) Afterwards we succeeded in demonstrating a slight negative charge also with spraying a 1°/, common salt solution by lessening the capacity of the receiving disc. This charge, however, is not nearly so great as the one evoked in the waterfall and is far inferior to the spray-electricity generated by additions of volatile substances and of substances that activate the surface. 2) A. Sreranint and G. GRADENIGO. Inhalazione di Nebbié Salina Secche. Lucca 1914, p. 22. Physiology. — “The Action of Atropin on the Intestine depending on ils amount of Cholin”. By Dr. J. W. Le Hevux. (Commu- nicated by Prof. R. Manus). (Communicated in the meeting of September 27, 1919). The action of atropin on the intact, isolated small intestine of mammals has been the subject of a considerable amount of experi- ments, but no explanation, could be found for the various and widely different results achieved by the several authors. *) While Magnus’) established that the isolated intestine of the cat, in’ Rinewr’s solution, is paralysed in large doses of atropin (0,3 °/,), but mostly reacts on moderate quanta of atropin (0,025 —0,075 °/,) with symptoms of stimulation, the pendulum movements, executed by the intestine, getting more regular, especially when the movements of the intestine were previously insignificant, also an inhibiting effect of very small atropin-doses (0,005—0,05 °/,) was demonstrated by Unerr®). Other researchers also sometimes found this (paralysing) effect, sometimes they did not. In the isolated small intestine of rabbits and dogs Kress *) noted stimulation consequent on moderate amounts of atropin, paralysis through large quanta, whereas others again obtained widely varying results. In most cases small quantities of atropin caused inhibition in the small intestine of rabbits, whereas moderate atropin-doses alternately stimulated and paralysed the gut without any regularity. According to P. TRENDELENBURG °) the intact intestine of the rabbit reacts regularly on small quanta of atropin with paralysis, but on moderate quantities now with symptoms of stimulation, now with paralysis. According to the same writer *) an exception. to irregular behaviour 1) An extensive survey of the literature is given by G. LILJESTRAND, Pflüger's Archiv, Bds lva, op. Waly Pore: 2) R. Maanus, Versuche am überlebenden Dünndarm von Säugetieren. I. Mitt. Pflüger's Archiv. Bd. 108, pag. 1, 1905. 3) M. Urerer, Beiträge zur Kenntnis der Wirkungsweise des Atropins und Physo- stigmins auf den Dünndarm von Katzen. Pflüger's Archiv. Bd. 119, pag. 373, 1907. 4) K. Kress, Wirkungsweise einiger Gifte auf den isolierten Dünndarm von Kaninchen und Hunden. Pfltiger’s Archiv. Bd. 109, pag. 608, 1905. 6) P. TRENDELENBURG, Physiol. u. Pharmacol. Untersuchungen über Dünndarm- peristatiek. SCHMIEDEBERG’s Archiv. Bd. 81, pag. 55, 1907. 439 of the gut of various species of animals towards atropin is afforded by the small intestine of the guinea-pig, which is regularly inhibited by atropin, a question to which we will revert in this paper. It is evident from this short survey that the behaviour of the intestine of different mammals towards small and moderate quanta of atropin is varying and inconstant. No doubt the researchers who obtained these various results, have been working under incongruous circumstances. As yet no one has succeeded in accounting for their conflicting results. From experiments by v. Lipra pr JeupE’) it appeared distinctly that the explanation is not to be looked for in the different compo- sition of the salt-solution, in which the isolated intestine was examined. LILJESTRAND ®) showed that the various results could neither be ascribed to the different composition of the atropin-preparations; he is rather inclined to believe that the explanation can be found in the gut itself. The object of the present paper is to verify this conception. Weitanp*) has demonstrated that from the stomach, the small intestine and the large intestine a coctastable substance can be abstracted through extraction with water, which has the property of urging on the movement of the gut, and that this stimulating action can be arrested antagonistically by small quantities of atropin. It afterwards turned out*) that this substance consists for the greater part of cholin and that it can be obtained from the small intestine of the rabbit to such a quantity that it must act a prominent part in evolving the automatic intestinal movements. Now cholin belongs pharmacologically to the group of pilocarpin ; the stimulating effect exercised by cholin, just as by pilocarpin, on the gut is antagonised by slight quantities of atropin. Van Lipta pr Jxupn’) showed by his experiments that the quanta of atropin required to check the pilocarpin-action upon the gut, are very slight; already a concentration of atropin of 1 to 10—50 million will do. Now I found that the atropin-concentrations, necessary 1) A.P. v. LiptH pe Jeupe, Quantitatieve onderzoekingen over het antagonisme van sulfas atropini enz. Thesis. Utrecht 1916. 3) G. LILJESTRAND, Lc. 5) WerLAND, Zur Kenntnis der Entstehung der Darmbewegung. Pfliiger’s Archiv. Bd. 147, pag. 171, 1912. 4) J. W. Le Heux, Choline als Hormon der Darmbewegung. Pfliiger’s Archiv. Bd. 173, pag. 8, 1918. 5) A. P. v. Lipru pe Jeupe, l.c 29 Proceedings Royal Acad. Amsterdam. Vol. XXII. 440 to induce a temporary inhibition of the intestinal movements, fall within the same limits. When combining these facts, the question arises whether perhaps the inhibition of small atropin-doses is to be considered as an antagonism for the cholin present in the intestinal wall. If this is the case, the inhibition of small quantities of atropin, will not appear when the cholin has been previously removed from the intestinal wall, but it will come forth again after the addition of cholin and subsequent administration of atropin. Aside from this inhibition of small quantities of atropin, its actual influence is, according to Magnus’s’) experiments, one that stimulates AurrBacus’s plexus. It is only large quantities that paralyse the nerve centra and muscles of the intestinal wall. ; The action of these latter quantities we leave out of consideration. Experimental evidence in support of the above hypothesis may be obtained in the following way : 1st A gut, inhibited originally by small quantities of atropin, is to be brought into a condition in whieh a small quantity of atropin is without effect, through repeated washings, so that cholin is removed from the intestinal wall. 2nd The atropin-effect is to reappear in this gut after giving cholin. ord. AG cuits which is originally inhibited by moderate quantities of atropin, is to be brought, through repeated washing, into a condition, in which the same quantity of atropin has only a stimu- appeared lating effect. The experiments made to prove this, were performed with the isolated small intestine of rabbits and guinea-pigs, which, as had before, are provided with rich quantities of cholin and — as may be expected, readily give them off to the environing fluid. Experiments with the small intestine of the rabbit. In the experiments with the small intestine of the rabbit a diffi- culty arose in that after some days the spontaneous movements diminished with the washing out of the cholin, which made the results less clear. Turning to account Laqurur’s?) experience that loops of intestine 1) R. Maanus, Versuche am überlebenden Dünndarm von Säugetieren. V. Mitt, Pfliiger’s Archiv. Bd. 108, pag. 1, 1905. 2) E. LAQveuR, Over den levensduur van geisoleerde zoogdier-organen met automatische functie. Verslagen Kon. Akademie v. Wetenschappen te Amsterdam. 24 April 1914, XXII, p. 1818. . 441 kept in horseserum at a low temperature, retain mobility for days, we now proceeded as follows: A certain number of loops were severed from the fresh small intestine, which had been cautiously cleaned with Tyrode solution; their movements were registered by Macnus’s method. The vessels containing the loops were filled with 75 c.c. Tyrode solution of 38°C. When small quantities of atropin (0,002—0,01) were added, the pendulum movements of the loops got invariably smaller. This is illustrated in Fig. 1. Fig. 1: Q Fig. 2. Fig. 3. 15 mg. Atropin 0.01 Atropin 15 mg. Atropin 0.01 Atropin Fig. 1. Pendulum movements of the rabbit’s small intestine suspended directly after killing the animal. By administering 0,01 mgr. of atropin the magnitude of the movement is reduced by half. Fig. 2. The loop whose movements are registered here is the same as in fig. 1. The previous action of the atropin (0.01 mgr) is entirely eliminated through washing three times with Tyrode solution. 15 mgr. of atropin now yields a strong inhibi- tion on the movements while the tonus is only slightly lessened. Fig. 3. A loop of the same gut, standing for 3> 24 hrs. in the refrigerator in horse- serum that was repeatedly refreshed. Subsequently the loop is washed out eight times with warm Tyrode solution and no longer reacts on 0.01 mgr. of atropin. After 15 mgr. of atropin a marked stimulation appears together with a conside- rable increase of tonus. After this the loops of intestine were washed out with fresh Tyrode solution three times every 10 minutes, by which, as we know from our Own experience, the effect brought about by the preceding small atropin-dosis was again completely eliminated. When thereupon a moderate dosis (15 mgr.) of atropin was added, the pendulum movements became considerably smaller in far and 29* 442 away most cases, while at the same time the tonus was more or less lowered. This is exemplified in Fig. 2. In a few cases the initial inhibition, caused by this moderate quantity of atropin, was followed by a rather strong stimulation. This occurred, when the loops had already been cleaned several times, so that the gut had reached a stage, in which the inhibition of moderate atropin doses passes into a stimulating action. The next day some loops were again severed from the gut which had been kept standing during the night in horse-serum at a low temperature ; after baving been carefully washed free from the adherent serum they were suspended as before. Every ten minutes these loops were cleaned with fresh Tyrode solution. One of them (we always took the same) was experimented on to ascertain whether 0.01 mgr. of atropin still evolved inhibition on the movements. If it did not, the other loops of intestine were cleaned again some times and examined with regard to their behaviour towards atropin. In the great majority of cases it appeared again that a small amount of atropin (0,01 mgr.) did not cause the slightest change in movements or tonus, but that after administering 15 mgr. of atropin a stimulation of the intestinal movements together with a large in- crease of tonus was noticeable. See Fig. 3. Over and beyond all this the primitive inhibition of the atropin could be elicited again in this stage of the experiment, if a small amount of cholin (1—2 mgr.) had previously been added to the loops. We did not always succeed in reaching this stage already on the second day, so that it proved necessary to keep the gut in the repeatedly refreshed horse-serum some days longer, in order to arrive at a condition in which small doses of atropin do not affect the gut, which again had been cleaned repeatedly with Tyrode solution. We also succeeded in obtaining this condition by merely cleaning the gut with Tyrode solution, i.e. without the appliance of horse- serum. It is true, though, that, as mentioned before, the movements will become smaller then, and the results less clear. This proves, however, that the results are not influenced by horse-serum. In the foregoing we have thus shown for the rabbit’s small intestine : 1st, that repeated washing, which, as demonstrated before, deprives the intestinal wall of cholin, evolves a condition in which the initial inhibition of small amounts of atropin, is arrested. 2-4. that by administering cholin this inhibition of atropin may be elicited again. 443 3". that the effect of moderate quantities of atropin, which was variable at first and in my experiments was mostly inhibitory, may be altered in a constant, stimulating effect by repeated washing. Experiments with the small intestine of the guinea-pig. We now took the small intestine of the guinea-pig as the object of our investigation. Recently TRENDELENBURG*) has suggested an effective method to register graphically the peristaltic movements of the surviving small intestine of the guinea-pig and to determine to a certain extent in numerical values the action exercised on these movements by various poisons. TRENDELENBURG records that atropin (acting on the small intestine of the guinea-pig) is invariably inhibiting peristalsis. The great thing in our experimentation was to ascertain whether here also, as with the rabbit’s small intestine, its behaviour towards atropin is governed by its condition. We used TRENDELENBURG’s method and proceeded as follows: The guinea-pig was killed by a blow on the neck, the small in- testine was cautiously severed from the mesentery, and cleaned several times, with a warm fluid’) after Locke. Subsequently the intestine was cut into 5 parts, one of which was suspended immediately, the other pieces were put in separate dishes with Lockn’s solution, which was refreshed every now and then. The loop of intestine which was suspended in a vessel of 150 cc. capacity, was first left to itself with an interior pressure of 0 mm. H,O.; then the pressure was gradually heightened and we deter- mined at what pressure peristalsis first appeared (critical pressure). Then the interior pressure was lowered to O and after 3 minutes the critical pressure was again determined. 5 This determination was repeated after 0,1, 1, and 5 mgr. of atropin had been added respectively. In accordance with TRENDELEN- BURG’S report, arrest of peristalsis took place, so that no peristalsis occurred any more even when the pressure was made considerably higher. After being carefully cleaned, a second loop was suspended, which was kept standing for some hours in Lockn’s solution; the same determinations were made prior to and posterior to the administra- tion of atropin. In most cases it appeared already now that the 1) TRENDELENBURG, l.c. *) It is essential that the fluid should be prepared from pure salts and with pure water distilled from glass apparatus. 444 small atropin-dosis yielded a mueh weaker inhibition on the peri- stalis than with the first loop. Subsequently another loop was examined, which had been kept standing in Lockrs’s solution some hours longer again, and had been washed a few times more ete. Table | comprises the results of the complete experiment. From it we see that the gut, which is arrested by quantities of 0,1, 1, and 5mer. of atropin, is brought after a 25 hours’ washing with Lockr’s solution into a condition in which O,1 and 1 mgr of atropin produces a much weaker inhibition, TABEE A: Critical interior pressure in mm H,O. Number Time of : 3 SS of the loop EEE | | ‘ atd 8. \after 0.01 mgr. after 1 mgr. after 5 mgr. we normal. |. of atropin. of atropin. « of atropin. | | 1 0 Tie Inhibition Inhibition Inhibition 2 25 20 25 30 Inhibition | Quick 3 6 20 16 20 ‚20 pendulum movements. 4 8 12—16 13 — | — 5 11 10 7 = = After the process of washing had been prolonged for 6 hours the inhibition of these atropin quantities had entirely stopped, but now peristalsis appears after 0,1 mgr. of atropin already at a lower interior pressure as in the normal period. Also with loops of intes- tine that have been washed 8 and 11 hours atropin causes distinct stimulation, so that peristalsis comes forth already at a lower interior pressure. With loop N°. 3 a very considerable increase of the pendulum movements was also perceptible after 5 mgr. of atropin. It has thus been proved also for the small intestine of the guinea- pig that the inhibition of atropin, already established by TRENDELENBURG, can be arrested by washing and that the atropin-action proper, stimulation of AurrBacn’s plexus, can be elicited by small quantities of atropin. Now it is obvious also why TRENDELENBURG noted only the inhi- bition of atropin. It was because he did not allow the loops to stand in a solution, but always experimented with fresh ones taken from the guinea-pig under urethannarcosis. 445 The results obtained by our experiments put us in a position to view the variable action of atropin on the gut in a new light and to interpret the conflicting results of the various researchers. In the living animal cholin is present in the intestinal wall in such quantities that they stimulate AverBacn’s plexus. On removal, to a certain extent, of the cholin from the surviving gut by a pro- longed washing, the real action of atropin manifests itself distinctly. According to the earlier experiments of Magnus’) it consists in a stimulation of AurrBacu’s plexus by moderate quanta, whereas only very large doses paralyse the centra, the nerve, and the muscle. Originally the intestinal movements are not affected by small quan- tities of atropin. It is, therefore, upon the presence of more or less cholin in the intestinal wall that the atropin-action depends. Cholin has a stimulating effect upon AverBacn’s plexus, which is antagonised by atropin. So long as an adequate quantity of cholin is present in the intestinal wall, a small dosis of atropin will inhibit the cholin action and consequently inhibit the intestinal movements. The result of the action of moderate quantities of atropin will depend on the circumstance whether the immediate stimulating action on the plexus, or the antagonism for cholin preponderates. In case the gut contains little cholin the stimulating action comes to the front, in case it contains much cholin the antagonism (inhibition) predominates. With a moderate cholin-content a stimulation will succeed an initial inhibition. Likewise we are now enabled to account for the results of earlier researches. The fact that with the cat’s small intestine the stimulating effect of moderate quanta of atropin occurs more often than with the rabbit’s or the guinea-pig’s, tallies with our experience that the former contains less cholin than the latter two. On the other hand it stands to reason that with the guinea-pig gut, which was always found to be rich in cholin, the inhibition of atropin appears regularly. It is obvious now why the isolated rabbit’s gut, according to the previous treatment, is now inhibited by atropin, now again is stimulated, while on the other hand the intact gut, which could not be liberated from cholin by washing, is according to TRENDELENBURG *) inhibited regularly. 1) R. Maenus, Le. 2) P. TRENDELENBURG, l.c. 446 Summary. We have demonstrated in an earlier paper that the isolated small intestine yields to salt-solutions quantities of cholin, which are capable of stimulating AurrBacH’s plexus. This loss of cholin results in a changed behaviour of the gut towards atropin. The rabbit's small intestine that is inhibited previous to washing by small doses of atropin, no longer reacts on them; on the other hand it is now stimulated by moderate doses. The normal guinea-pig gut is invariably inhibited by atropin. This effect also here disappears after washing and is substituted by a stimulation through moderate quanta of atropin. This is to be interpreted as follows: the real action of moderate quanta of atropin on the gut is stimulation of AurrBacH’s plexus; if the gut contains much cholin, so that the plexus is readily stimulated, this stimulation is arrested through the antagonism of a small amount of atropin, occasionally weakened, and the result is inhibition. Moderate quanta of atropin are also inhibitory when this antagonism is strong enough, but in the presence of small quanta of cholin in the gut the latter will be stimulated. It is clear, therefore, that here we have to do with a case in which the presence of a well known chemical substance (cbolin) in the tissue determines the manner in which this substance reacts on a poison (antropin). . The Pharmalogical Institute of the Sept. 1919. Utrecht Uniersity. Zoology. — “The interrelations of the species belonging to the genus Saturnia, judged by the, colour-pattern of their wings.” By Prof. J. F. van BEMMELEN. (Communicated in the meeting of May 3, 1919). In his handbook of Palaearctie Macrolepidoptera STANDEFUSS says on p. 106, at the end of his passage on the relative age of the three species Saturnia spini, pavonia and pyri: “If-we exclusively | paid attention to the imagines, much might be said in favour of the opinion, that spini is older than pavonia. For both sexes of spini, aud likewise those of the new species from Kasikoparan (S. Cephalariae Ch.) discovered as late as 1882, possess a remar- kably uniform type. From this type the male of S. pavonia sharply deviates, this species thereby appearing as recently changed in its imaginal dress. But as to the question if we should place the origin of Sat. pyri before the evolution of these minor forms or after it, the imago of this latter species does not seem fit to allow - of a really certain conclusion. In regard to the caterpillar- and pupal-stage, however, things are different. For these it can be clearly proved that S.spini, pavonia and pyri form in many instances three different degrees of protec- tive resemblance against certain hostile factors of the outer world. In this seale everywhere spini occupies the lowest, pyri the highest degree. Keeping in view the excessively near relation and the great similarity of biological conditions between the three species, we are obliged to assume that spini came into existence before pavonia, and pavonia before pyri, or, using the scientific ex- pression for these relations: that phylogenetically spini is the oldest, pavonia a younger, pyri the youngest form. For it evidently would be absurd to assume that in a series of so intimately related forms, the more perfect living being should have originated at an earlier date than the less perfect one”. These considerations of Sranpruss induced me to compare the wing-markings of Saturnine-species, as well among themselves as with those of related genera, to see if this line of investigation did or did not lead to corresponding results as the inspection of the caterpillars, ) 448 In accordance with the general rules for the character of the wing-markings, which | thought myself justified in proclaiming, and which I tested as to their applicability to such families as Hepialids, Cossids, Aretiids and Sphingids, | came to the conclusion that not pyri but pavonia should be considered as the oldest form. For in pavonia the festooned submarginal transverse lines and bands deviate less from the “outer wing-margin and also show a smaller difference between their anterior and posterior extremity, the sub- marginal band therefore having the simplest and least irregular type. In the same way the difference between fore- and hindwing, as well on their upper- as on their underside and therefore also between the superior and inferior surface of each of the wings for itself —, is smaller in pavonia than in the two other species. In comparison to other Bombycids and to the remaining families of Heterocera, the colour-pattern of pavonia shows a greater simi- larity to the general primordial pattern of seven dark transversal bars, which I deduced from the comparison of all these forms, than the two other Saturnids. It might seem that this assertion is contradicted by the fact, that in pavonia the male at first sight looks entirely different from the female by its colour as well as by its inferior size, a difference which has apparently made a deep impression on STANDFUss. But on nearer inspection and consideration the difference is by no means so important as it looks, and need not be regarded as of high importance. For the difference in hues is evidently connected with the frequently occurring feature of discoloration (i.e. partial self-colour) by which on the superior surface the hindwing has partially turned into yellow, while at the underside the same hue has spread over the proximal part of the forewing. On both wings this discoloration is accompanied by a slight and incomplete fading of the pattern. Moreover it may be doubted, on very sound arguments, if the yellow hue — apart from its spreading over the dominion of spots and stripes, which are rendered more or less invisible by it — should be considered as a secondary modification of an older and more original hue, which latter therefore should have persisted on the upperside of the forewing and on the underside of the hind one. For this yellow-brown hue is characteristic for quite a number of Bombycine moths belonging to different genera, and in so far impresses us as a very original colour. It might therefore be assumed, that its occurrence in the male of pavonia should be considered as a reversion to an older condition, instead of being the appearance of a new hue. 449 But we may safely leave these questions unheeded, for the hues in which a pattern is executed, need not be taken into consideration, neither in this case nor in others, when judging the character of the pattern itself. Now, if we carefully compare the markings of the male with those of the female and pay due attention to the half-vanished stripes and spots under the overspreading yellow, it becomes clear, that down to the minutest details, male and female agree in pattern. No more is there any necessity to consider the sexual dimorphism as an important progressive feature, by which pavonia should distin- guish itself from spini and pyri, and be characterised as a recently and strongly modified form, in contrast to the other two, which would have remained more conservative. For secondary and ter- tiary sexual differences oecur in all kinds of Lepidoptera, as well as in other insects. In the Bombycine moths this feature shows itself in a remarkably high number of forms. Now have we really to consider each of these cases as a separate and independent devi- ation from a common original condition, in which male and female were alike in shape, size, hues and pattern? Or did the phenomenon of sexual dimorphism already show itself amongst primitive Bom- bycidae, in the days when the difference between them was less considerable than at present, and they still counted fewer specific and varietal forms; dimorphic species therefore then existing side by side with monomorphic as well as now. In this case the sexual dimorphism of pavonia might repose on the manifestation or the permanency of an old hereditary disposition. But even apart from this question, which can hardly be solved with certainty, it remains doubtful, if the species pyri and spini should really be considered older than pavonia, on account of the similarity of their sexes. For I by no means consider it as proved, that in pavonia the male, which deviates from the general hue of the genus, may be considered as the modified form, while the female, which seems to show so much more similarity to spini and pyri, may be regarded as the unchanged form. When the general rules for the colour-pattern are blindly applied the solution of this question might seem easy enough. In the female of pavonia as well as in both sexes of spini and pyri the fore- and hindwings on their upper- as well as on their underside, show the same clear whitish hue, here and there overspread with a black sprinkling, and subdivided into fragments by nervural and transversal straight or undulating lines. But this 450 general clear tint, reigning over the whole extension of the wings, as well as the similarity in design between fore- and hindwings, upper- and underside, impresses me as a secondarily acquired uniformity, rather than as a really original feature, In my opinion we meet here with a similar case as in the wholly selfeoloured butterflies e. g Gonepteryx rbamni or Aporia crataegi, in which without the least doubt the uniform hue is the consequence of the simplification of the shades, hand in hand with the total or partial regression of the markings. Still more striking is the similarity with the Parnassine butterflies (which, it need hardly be said, is of course wholly superficial and occasional). For in these as well as in Saturninae a set of highly differentiated eye-spots on the forewings, but still more pronounced on the hindwings, form the most conspicuous part of the pattern, though it may undoubtedly be taken for granted that the spotted design of the Parnassines has developed from a far more complete array of simpler and more uniform markings, such as are seen in Thais poly xena and its consorts. As soon as we consider the similarity of fore- and hindwing, upper- and underside of the female of pavonia and of both sexes of pyri and spini as secondarily acquired characteristics, there need no more be any objection against the supposition that the male of the first-named species is more original than the female. From this point of view we may further remark, that the existing contrast between the yellow upperside of the hindwing and the greyish one of the forewing, on which the markings are less hidden under the ground-colour, corresponds in a higher degree to the general type of butterfly-design than does the exact similarity of fore- and hindwing; the overwhelming majority of Lepidoptera showing a similar difference between fore- and hindwing. In the fact that the discoloration (in this case yellowing) occurs on part of the wpperside of the hindwing and of the underside of the forewing, the male of pavonia is in harmony with both sexes of Smerinthus ocellata and of many other Lepidoptera. Up to a certain degree the frequent occurrence and similar extension of this discoloration can be attributed to the influence of the resting attitude of these moths, the discoloured areas being exactly those which during this attitude remain covered by similar parts of the adjoining wing. In his paper: “Sur la position de repos des Lépi- doptères’’, J. T. Oupemans has directed our attention to this cir- cumstance and expressed his opinion that during the development and modification of the colour-pattern in the course of time, the J. F. VAN BEMMELEN: 2 | | | EO Pat. SATURIPAO CREPUSCULARIS | | Fig. 2. OPHIMONDIA PUNCTATA ©. Bis Proceedings Royal Acad. Ams EE oe J. F. VAN BEMMELEN: ‘“Wing-design of the Saturninae”. Fig. 1. SATURNIA BOISDUVALII ¢. Fig. 3. NYCTIPAO CREPUSCULARIS 5. Fig. 2. OPHIDERES DIVIDENS -¢. Fig. 4. EMMONDIA PUNCTATA &. : E. THEYSSEN, phot. Proceedings Royal Acad. Amsterdam. Vol. XXII. 451 hidden and the exposed parts had independently proceeded each along its own course (p. 31—83), on which the one as well as the other could get on at a higher speed and so reach a stage more remote from the original common condition. From a general point of view I feel inclined to join this opinion, but in the case at hand it brings us little light. The difference between the covered and the exposed wing-areas in the attitude of rest, already slight in the male of pavonia, is quite insignificant in the female, as well as in both sexes of pyri and spini. The wing-markings on these areas seem not to be influenced to any notable degree by the habit of passing the forewings over the greater part of the hindwings during day-time. That traces of such an in- fluence are still visible in the male of Sat. pavonia, might be taken as an indication that the original influence of the said habit is now gradually losing its force. The highly conspicuous eye-spots in my opinion must have evolved from simpler discoidal marks, hand in hand with the above men- tioned change in the influence of the resting attitude. This is already proved by their very different degree of differentiation in the several species of the genus Saturnia and of kindred genera. Eye- spots moreover always are special differentiations, secondarily deve- loped on the base of a more primitive and simple colour-pattern, whose elements have occasioned them by modifications in the original shape, colour, size and direction. In my previous paper, on the wing-markings of Sphingides, I tried to prove this assertion for the case of Smerinthus ocellata, in the same way as I formerly did for the Hepialid moth Ze lo- typia stacyi. It must however be possible to prove it as well in other cases, e.g. for Vanessa io and many other Nymphalids, and likewise for Satyrids and Lycaenids. Cynthia for instance shows in what way eye-spots and simple spots alternate in the row of submarginal markings, and also often how an eye-spot on the superior surface is represented by a common one on the underside. The difference between the two seasonal forms in the first place depends on the contrast in the differentiation of the eye-spots: in the dry- monsoon-generation they are scantily developed, in the wet-dito highly so. Now, do the hybridisations of Stanpeuss throw any light on these questions? As far as I can see not much; in general the hybrids are intermediate forms, but as to the width of the submarginal dark seam, they agree more with pavonia than with spini, and assuredly far more than with pvri. 452 But in my opinion in order to judge about the interrelations of the colour-patterns of Saturnia, it is absolutely necessary to compare with each other as many different forms as possible, just as in the case of other groups of Lepidoptera. It certainly cannot stand eriti- cism to draw consequences from the exclusive consideration of three intimately connected species as to the relative age of their colour- patterns. When pursuing this broader way, the well-foundedness of the above-mentioned assertion, viz. that the colour-pattern of Saturnids is a special case of that of Bombycids, and the latter again of that of Heterocera in general, is clearly proved. To begin with: next to the three above mentioned species stands Saturnia (Caligula Jordan) boisduvalif, on whose upper side the submarginal dark seam broadens from before backwards in still higher degree than in pyri, which reduces the somewhat median clearer area under the eye- spot, (broadest in pavonia) to almost nothing by forcing it back in proximal direction. The two dark borders by which this area is limited (and which I suppose to be V and VI) are in one place locally connected by a black transversal link. This part of the pattern of boisduvalii therefore shows the greatest similarity with the corresponding area of the pattern of the male pavonia, but in the latter the connecting link seems to run between IV and V. Generally speaking, it is not easy accurately to make out the exact consecutive number of the bands for each separate form, yet the comparison of the superior surface of the different species leaves the general impression, that the posterior broadening of the dark submarginal area is brought about by the progressive darkening of the colour in a proximal direction, which consecutively incorporates the dominion of a more proximally situated transversal bar. In the female of pavonia this darkening process is restricted to the area of Bar II, in the male it has advanced unto III, in the same way as in both sexes of spini, in pyri it has reached IV, in boisduvalii V. On the inferior surface the broadening of the dark area proceeds more slowly and more equally over the whole extension of the dark submarginal field. Consequently the underside of spini e.g. resem- bles both the upper- and the underside of pavonia to a higher degree than its own superior surface. On the hindwing the back- ward broadening of the dark submarginal border is less pronounced than on the forewing, and this edge is there separated by a light- coloured band over its whole length, from the dark festooned line, that runs along the distal side of the eye-spot. 455 On the underside however (which is never represented even in large illustrated works) we again meet with the same feature, as mentioned just now for pavonia, spini and pyri, viz. that the course of the transversal bars is far more regular and original than on the upperside, this bringing about a much greater similarity between fore- and hindwing on the first mentioned surface. I want to draw your attention to a simple dark transverse stripe on the middle of both wing-pairs, almost devoid of incurvations and rather faint. On the forewings this stripe runs along the distal border of the eye-spot, on the hindwings along the proximal one. Regard- less of this difference, I think we have to deal in both eases with vestiges of bar IV. To this assumption I am especially led by the comparison of the upperside. On that of the forewing the anterior part of this bar is quite apparent up to the eye-spot, which constitutes a marked difference between boisduvalii and the three first-mentioned species. Past the eye-spot however the posterior part of the bar is wanting, but from the postero-interior border of that spot a black stripe runs across to Bar VI, turning sharply at an angle in the middle of its course and then running parallel to Bar VI, perpen- dicularly to the posterior wing-border and as far as this latter. The lastnamed part belongs to Bar V, as is proved by its comparison with the upperside of the hindwing, on which the Bars IV, V and VI may be perceived in their full extension, though faint and half- hidden under the hairy coating. On the underside of boisduvalii therefore the dark submarginal border of both wing-pairs is located between Bars II and III; on the upperside however this is only the case on the hindwing. Bar Ill may even remain separated from the dark seam as an independ- ent isolated stripe, as is shown in the illustrations of Srirz (Vol. II Pl. 31%, in contrast with the specimen at my disposition, where the black internal border was not free from the much broader margi- nal seam. In some respects Saturnia (Neoris Moore) scheneckii corresponds with boisduvalii, e.g. in the presence on its under- side of an extremely faint yet complete transversal stripe, which takes its course along the eye-spots. In this species however no difference exists as to the situation of this stripe on fore- and on hindwing, while the stripe is also present on the upper-side, though in an incomplete and unequal way. Bar V is here the most pronounced and regular, VI and VII are hardly visible. The eye-spots have been removed outward, and in consequence 454 the dark submarginal seam has been reduced, while Bar II and III cannot be distinguished from each other. They are represented by a single dark double-line, which in its anterior part is very much festooned. At the outer side of this double-line a white bar runs along, followed by a particularly broad margin of a light-manilla brown hue. The comparison with the female pavonia makes it probable, that the dark submarginal edge ought to be situated between the white bar and the dark double-line, and that therefore we may assert that it is absent. Yet this is true only to a certain degree: the broaden- ing of the dark submarginal border is, as we remarked before, a consequence of the advancing of the obscuration over the areas between the succeeding transversal bars in a proximal direction, and tbis identical process is also seen to take place in schenckii, viz. on the fore-wing between double-line IL + III and the dark stripe representing IV. At the underside of schenckii the pattern is simple, and moreover pale and reduced. A.o. the eye-spots on the hindwings are much smaller, paler and less complete than at the upper side. Amongst the many genera near-akin, that are arranged around the Saturnids, a great number of additional arguments may be found for the above-mentioned assertion, that their colour-pattern: may be derived from the same scheme of seven dark transversal bars, which proved applicable to Aretiids. We only need point out forms like Rhodinia fugax, probably showing II, III, and V or VI, or Laepadamartis (Seitz II, Pl. 32¢), which on its forewing wears I, IH, III and V or VI, on its hind-wing I, II, HI, IV and V (the last two only in part). Even in such a complicated and special pattern as that of Brahmaeïds it is comparatively easy, with a little attention, to find again the seven primary transversal bars. However it is not only the ground-plan of the wing-design that may be shown with great probability to be common to numerous and various groups of moths, also the moditications of this plan seem to take place after the same rules among the different families of Lepidoptera. In the present case of Saturnine moths it is the broadening of the submarginal dark border in the direction of the hind margin, that constitutes the principal difference in pattern between the various species mentioned and leads to their arrangement in the sequel: pavonia, spini, pyri, boisduvalii, while the pattern of schenckii seems to be due to a secondary regres- sion of the dark band on its forewings. 455 This broadening in a caudal direction makes the impression, that an oblique line of separation runs across the forewing from tip to root, dividing it into an antero-interior lighter field and a postero- exterior darker one. Calling this line the V-diagonal (because the lines of the two wings in expanded attitude form the letter V) we may state, that this V-diagonal-design occurs in numerous Lepi- doptera of various families, though always as a secondary modifi- cation of the original pattern of transversal bars. In the group of Chaerocampine Sphingides, which 1 hope to discuss in a future communication, this feature is particularly striking. We find it however in the same way in the families of Hepia- lidae, Noctuidae and Geometridae, and likewise in numerous Micro- lepidoptera. In the overwhelming majority of these cases the V- diagonal-pattern is restricted to the upper side of the fore-wing; the underside showing the primitive pattern of transversal bars. As specially striking examples may be mentioned the Noctuid genera Ophideres, Nyctipao and Emmondia, the Bombycid genus Eupterote, as well as many Geometrids. The species of Ophideres give rise to the remark that the separation of the forewing-area in an antero-internal and a postero- external part, brought about by the V-diagonal-design, in many species e.g. O. tyrannus, salamia and fullonica (comp. Seitz III, Pl. 66) seems to be connected with the resemblance of the entire forewing-design to a withered leaf, this likeness bringing the said species under the category of the leaf-imitating Lepidoptera. We have to deal here once more with such a feature, as can show itself in many different shapes, and therefore in its real character is evidently independent of the importance it may in some particular cases possess for the establishment of protective resemblance. It further results from the mutual comparison of different species, that the contrast between the anterior and the posterior part of the wing-surface may be very different in quality as well as in quantity. In some species the forepart is light, the hindpart dark, in others they are nearly alike. In O. materna a transversal design of Cossid-markings (traits effilochés Borkn, Rieselung Eimer) spreads over both parts; in the male the V-diagonal is present, in the female absent, while on the contrary an A-diagonal-design is partly developed as a light-coloured-streak. For besides the V-diagonal an A-diagonal can be distinguished, running from before and inside to behind and outside in an oblique direction over the wing-surface, often it seems to possess some connection with the division of the 30 Proceedings Royal Acad. Amsterdam. Vol XXII. 456 wing into a beam- and a fan-part (Spreiten- und Faltenteil as SPULER has termed them). In the species Miniodes discolor, near akin to the genus Ophideres, the dark A-diagonal separates the orange-coloured forewing-area into two parts: both of them well showing the Cossid markings. A remarkable detail in this case is the uniform pink colouring of the hindwing on its upperside, the under one showing just the reverse: self-coloured forewing, Cossid markings on the hind one. On the upper-side of Nyctipao crepuscularis (Suirz Ill, Pl. 58) markings in three different directions coöperate to form the pattern: 1. transversal markings, especially the white line running from the anterior to the posterior margin through the centre of fore- and hindwing, but also the dark bar near to the wing-root, representing Bar VI or VII, 2. the V-diagonal designs of the forewing, 3. a white stripe, forming an obtuse angle with the V-diagonal, and running parallel to the A-diagonal. Inside the angle, made by the V- and the A-diagonals, an eye-spot has been differentiated from elements of the transversal markings. Still sharper these three directions of wing-markings stand out in dark bars against a creamy-white fond on the fore-wings of the arctiid Areas galactina. So numerous are the cases of V- and A-diagonal-design, that I will not even venture to summarize them and compare them to each other. As a general result however of my comparative investigations, I feel justified to assert, that every- where the secondary character of this pattern may be stated with surety, and that in the great majority of cases the underside shows no trace of one of these oblique lines, whereas clear vestiges of the seven original transversal bars nearly always occur, and are also frequently present on the upperside, though generally incomplete. In Rhopalocera the V-diagonal-design, when it occurs, is restricted to the underside, and here serves to establish the leaf-imitating character. Groningen, April 1919. Botany. — “The Influence of Light on the Cell-increase in the Roots of Allium Cepa’. By H. W. Brrinsonn. (Communicated by Prof F. A. F. C. Went). . (Communicated in the meeting of October 25, 1919). Mrs. DROOGLEEVER FortuyN—vaN LeypEN') has found that the cells of young cats increase periodically in such a way, that during the night, the number of karyokineses reaches its maximum, and in the later morning hours and the early afternoon a minimum number is reached. Karsten?) stated likewise that a periodical karyokinesis takes place in the young buds of Zeamais, which reaches its maximum also during the night. However in the roots of Vicia Faba he did not find any periodicity and so he concludes: «Das Wurzelwachstum entbehrt der Periodizitét.” During these experiments the plants remained in the dark. Now he tried to influence the periodicity by exposing the young plants to the light of an electric lamp, in which he succeeded. On the other hand he did not trace the influence of light and dark on the cell-increase in the roots. As the root growth is evidently not a periodical one, the influence of light and dark will be most obvious here. I chose Allium Cepa to experiment upon, because the Alliumeells are easily fixed and stained; because one finds a great number of karyokineses in the roottops and because there are a great many roots, so that it is possible to examine parts of one and the same individual under different circumstances. At 8 a.m. 11 a.m. and 3 p.m. I took a few roottips away from a germinating onion, which was exposed to full daylight. After that I put the same onion in the dark and left it alone until the next day, then I took a few tips off at 6 a.m., at 12 a.m. and at 6.30 p.m., while the onion remained in the dark. During these two days the temperature differed '/,° C. (registered with a maximum and a minimum thermometer). I always took care to take roottips shorter than 25 mm. and of about the same length. The roottips were fixed in sublimate 1) Mrs. DROOGLEEVER FORTUYN— VAN LEYDEN. Proceedings Konink. Akad. Amst. Vol. 19. 1916, p. 38. 2) KARSTEN. Zeitschr. f. Botanik 1915, p. 1. 30* 458 sodium chloride, after passing the different alcohols, they were enclosed in paraffin and then they were cut into series of 10u and the sections were stained according to HerDeNHAIN’s ironhaematoxylin method. I counted the number of nuclei of some of these sections over a length of 1 mm. from the roottop and I fixed the number of mitoses. | took care to count only in the central sections. Table I and II give my results. TABLE I (in daylight). Total Total | Spirema | T Time. | number of | number of ‚and loose Monaster.' Diaster. | ne nuclei. mitoses. chrom. Meh | ru | | | Seam. | 4000 — — — — — lam. | 434 139 53 53 17] deels 3 p.m 2290 41 26 14 5 2 TABLE II (in the dark). Total Total = Spirema | TWo Time. number of | number of ‘and loose} Monaster.| Diaster. Paine nuclei. mitoses. chrom. : 6 a.m. 4702 210 125 66 19 0 12 a.m. 4204 180 147 21 20 4 6.30 p.m. | 4043 124 65 29 10 20 In order to compare these figures, I expressed them in percentages in the following tables. Karsten takes the average of his countings. In my opinion it is more exact to express these facts in percentages, just as Mrs. Droogiever FoRTUYN-VAN LeypEn does, for it is most improbable that Karsten always examined the same number of cells. TABLE Ill (in daylight). Total’ =|... Total Spirema ee aR Time. number of | number of |and loose|Monaster.| Diaster. | Vite; nuclei. mitoses. chrom. | 1 ne ee 8 a.m 4000 0.00 % | 0.00 % | 0.00 J | 0.00 % | 0.00 % il a.m 4345 AOT sed LD lng 1e AFG TIDE ig 0x97 5 3 pm 2290 2505 fy LS 0.61 , 0.21%, 0.08 , - 459 TABLE IV (in the dark). | Total Total Spirema | aoe Time. | number of | number of \and loose Monaster.| Diaster. ducten | nuclei. mitoses. | chrom, — 8 | ys Sin eae 6 a.m. 4702 446-0)... | 3,09 0 I) 1.40%, (10.4 .9/, 1020007 12 a.m. | A004. | 4.98. 4) 3.49 0-40 oee 048 “1 Oa & | | 6.30 p.m. | 4043 BOL zo!) 460 Oh ORDe el 5040 | ‘ | As is evident, the number of karyokineses reaches its maximum between 8 a.m. and 11 a.m. (solar time), which agrees with the well-known fact that good cell-divisions in Hyacinth and Allium are found between 10 a.m. and 11 a.m. From 11 a.m. the number of cell-divisions decreases to 2.05 °/, at 3 p.m. I found the greatest number of karyokineses with the onion in the dark at 6 a.m., 4.46°/,. At 12 a.m. this had slightly decreased to 4.28°/,, and at 6.30 p.m. the decrease was still greater. At 6 a.m. and at 12 a.m. the maximum number of karyokineses in the dark exceeded the maximum number of cell-divisions found in the day- light, while the maximum number of karyokineses in the light sur- passed the minimum number of cell-divisions in the dark only by a slight degree; so that the conclusion seems justified: The number of karyokineses in the rootcells of Allium Cepa increases in the dark, which is stated by Karsten’) for Spirogyra and other plants. When we compare in the tables III and IV the number of spirema, loose chromosome stages with the number of monaster stages, then we see in the first table from 11 till 3 o’clock an in- crease of the number of spirema and loose chromosome stages and a decrease of the monaster stages. This would point to an increase in the number of cell-divisions and nevertheless the total number of mitoses has diminished. We see the same phenomenon on table LV from 6 a.m. to 12 a.m. By considering spirema, loose chromosomes and monaster as one stage (prophase) the number of nuclei in prophase, in table III at 11 a.m. is 2.02°/, and at 3p.m. 1.74°/,, which points to a decrease. The same can be applied to table IV. At 6 a.m. 4.59°/, and at 12 a.m. 3.98°/, and at 6.30 p.m. 2.31 °/, is in prophase, so there is a total decrease. In my opinion this fact is a confirmation of the general conception to consider spirema, loose chromosomes and monaster as one stage. *) EN Karsten. Zeitschr. f. Botanik 1918. 2) PEKELHARING. Weefselleer, p. 67. 460 From similar facts, as are contained in table III and IV, it seems also possible to me, to conclude something about the rapidity from prophase to anaphase and from anaphase to telophase. Let us con- sider table III for that purpose. At 11 a.m. 2.02 °/, were in pro- phase and 0.38 °/, in anaphase. The number of karyokinesis figures in prophase has decreased with 13.8 °/, at 3 p.m. and the number of cell-divisions in anaphase has decreased with 44.7 °/,, so the decrease is intenser, that is to say, the transition from anaphase to telophase is quicker than the transition from prophase to anaphase. The same holds good for the onion in the dark during the whole day, but during the day an inversion takes place in such a way that from 6 a.m. to 12 a.m. the transition from prophase to anaphase is quicker than from anaphase to telophase. Of course these facts are too scanty to draw such far-going con- clusions, but the aim of this calculation was only to show that it is possible to learn the relative rapidity. If one wants to undertake such experiments it is necessary in the first place to fix the time of observation much shorter, i.e. one hour or one hour and a half. It is also possible to derive the duration of one cell-division from such tables. When we consider table I we do not find karyokineses at 8 a.m., and at 11 a.m. we find 16 nuclei in telophase. So the cell-division would take about 3—4 hours with Allium Cepa. JoLiy found with Triton the duration of the kariokynesis 2'/, hours in the erythrocytes at a temperature of 20° C. From the table of Mrs. DROOGLEEVER FortuyN—vaNn LEYDEN | think I may conclude the duration of a cell-division being 12 hours with a cat, because at 2 p.m. + 0.23°/, nuclei were in prophase and no telophases were stated. Only at 2 a.m. 0.20°/, nuclei were seen in telophase for the first time. When we summarize the results, we see that the roottips of the onion show more cell-divisions in the dark than in the light. Evidently light has a retaining influence. Besides it is probable that the transition process from prophase to anaphase is a slower one than the transition process from prophase to telophase. By lack of time I could not control these facts any further. To attain this, it would be necessary to make an investigation into the daily oscillations in the number of karyokineses with the onion, if possible the time of observation ought to be as long as possible (3 to 4 days). At the same time the above-mentioned experiment ought to be repeated. One onion suffices for these two experiments. The bulb is eut into two halves and one half is used for the first series of experiments and the other half voor the second experiment. 461 The two series of experiments are made with parts of one and the game individual. Neither Mrs. DROOGLEEVER FortuyN—van LEYDEN, nor Mr. Karsten have done this, so the results lose reliability. Notwithstanding the incompleteness of my investigation, | thought the facts 1 found, of sufficient importance to be examined further, and for this reason | published this communication. At the same time I make use of the opportunity of thanking Mr. M. W. WOERDEMAN as well for the incitement to this research, as for the kind assistance lent to me. Amsterdam. Laboratory of Histology. Physics. — “The Propagation of Light in Moving Transparent Solid Substances. 1. Apparatus for the Observation of the Fizrav-Bffect in Solid Substances.” By Prof. P. Zeeman. (Communicated in the meeting of May 3, 1919). 1. As a result of an experiment by Araco with a glass prism FrrsneL drew up his bold hypothesis on the convection co- efficient in 1818. When in 1851 Fizeav wanted to put Frrsner’s hypothesis to the test, he experimented, however, with water, and examined whether or not the velocity of light in standing water differs from that in moving water. There are many reasons to be adduced for carrying out an ex- periment, so exceedingly difficult as that of Fiznau, in the first place with water; it is, however, also interesting to examine the motion of light in sold, transparent, rapidly moving substances. In this connection experiments with rapidly moving quartz and glass have been made by Miss Sneruiace and myself. In this communication I will give the description of the apparatus with which these expe- riments have been made. It may be well to call attention to a few points referring to Fiznau’s experiment with water. Let c be the velocity of light in vacuo, u the index of refraction of the water, w the velocity of the water with respect to the tube in which it moves; then the velocity of propagation of the light with respect to the tube is according to Frusne.: Pi] 1 Telt) OEREN oy oro (1) lu u In this the upper or the lower sign is to be taken according as the water and the light move in the same or in opposed directions. In 1895 Lorentz demonstrated that FreESNeEr’s convection coefficient in a dispersive medium must be replaced by: ; i À du EE aa This changes formula (1) into: ieee Wr e(t) at CAREER) u mees ah dA The experiments made by Frizrav in 1851, plead in favour of P. ZEEMAN: “The Propagation of Light in Moving Transparent Solid Substances. I. Apparatus for the Observation of the Fizeau-Effect in Solid Substances”. Proceedings Royal Acad. Amsterdam. Vol. XXII. 463 formula (1). Micnetson and Moruey’s investigation of 1886, performed with MicHeLsoN'’s interferometer in one of the numerous forms into which as a real Proteus this wonderful instrument is capable of being changed, gave with white light a value of the convection coefficient which was in excellent agreement with the coefficient that follows for yellow sodium light from formula (1). Experiments that have been carried out by me with different colours ranging from violet to red, and in which the axial velocity of water in the tube was directly measured have been communi- cated by me in different papers to this Academy *). The validity of the formula (2) with the term of dispersion could be demonstrated with an accuracy exceeding 0.5 °/,. The optical effect that is measured in these experiments, is a displacement of interference bands, which is given in parts of the distance of two bands by the formula: AN (ome mende. 4 =a ow aa uw, ° . 5 ° 5 . (3) in which / represents the length of the whole liquid column which is in motion. 2. The apparatus that has been used for the investigation of the motion of light in solid substances, is shown diagrammatically in Fig. 1. fig. 1 on a scale of '/,,, and might also be used with simple modi- fications for the investigation of costly liquids and compressed gases. The moving, transparent substance is rigidly connected with a piece DE, and can therefore rapidly move to and fro parallel to 1) Zeeman, These Proc. 17, 445, 1914; 18, 398, 1915; 18, 711, 1915; 18, 1240, 1916; 19, 125, 1916. 464 the dotted line, while a beam of light traverses the substance parallel to DZ. The piece DE is moved to and fro, as it is coupled with the rods DC and O'C. Normal to the plane of the drawing, axes have been fixed at O and 0’ in a very strongly constructed frame, on which the bed is fastened, along which DE moves. The axis in O is rotated by a 3 H.P. motor, so that point A describes a circle; B, connected with A by the rod AB, acquires a movement backward and forward, which is transferred to C enlarged. The piece DF of fig. 1 is shown diagrammatically in fig. 2 seen from above on a scale of ~;. In A and B there are bronze shoes which can slide along steel guides. All these have been constructed with great care, so that a rectilinear, horizontal motion of the shoes can be obtained. The rods of the transparent substance, which Fig. 2. rest on a wooden block which is connected with 4 screws to A and B, participate in this motion. The Plate, which is annexed to this communication, gives a general survey of the apparatus. The thickness of the guides amounts to 9 mm., the width to 70 mm., the length to 1.84 m. They rest on heavy rectangularly bent pieces, which constitute the sides of the bed, and which for greater firmness are connected by very solid pieces about half a meter long, which are bent twice rectangularly. These pieces are arranged on the lefthand at the bottom side, on the righthand on the upper side of the bed, as is to be seen in the Plate. The entire length of the upper part of the apparatus is 2.30 m., the length of the stroke is about 1 meter. In order to ensure the regular movement of the apparatus it appeared to be necessary to provide it with two fly-wheels, a large one seen in the foreground, and a smaller one fastened on the other side of the axis of rotation, and just visible on the Plate. The whole apparatus it fastened with solid bolts to a granite slab, cemented to the large pillar of the laboratory. As appears from a consideration of fig. 1, the rate of motion of the shoe is variable, with two, slightly differing maxima of velocity, 465 one in going, and one in returning along the guides. The maximum velocity is practically constant over a distance of about 20 cm. When the fly-wheel performs 184 revolutions a minute, the maximum velocity rises to somewhat more than 10 meters per second. This is the highest value that can be reached. The driving apparatus was constructed by the works Werkspoor at Amsterdam. The execution of the mechanical parts had to be adapted to the optical requirements in the Laboratory. With regard to the optical arrangement of the experiment we may refer to my former communications on the Fizwav-effect. The quickly moving transparent, solid substance takes the place of the running water of the earlier experiments. The length of the moving column of quartz or glass ranged between 100 and 140 cm. in different experiments. After the successive application of numerous improvements it was possible to cause the beams of light to interfere through the moving quartz or glass, and to obtain pure interference lines at the greatest velocity which the apparatus admits. The experiment comes to this, that the interference bands are photo- graphed twice, first with a movement of the column to the right, and then with a movement to the left. These photos should be taken by admitting light during a time of the order of one hundredth of a second and at the moment of the maximum velocity. The optical effect to be expected, when 7 represents the length of the moving transparent substance is: 4l (| 1 A du ie 2 de u' 2th VERDEELT Le Te (4) GNU. ie Se ak It appears from this formula, which will be proved later on (see II), that the optical effect is approximately proportional to u—1. In Fizkau’s experiment the optical effect is proportional to u? —1 according to (3). This difference is connected with the fact that in Frzrau’s experiment in its usual form, the velocity in a definite point of space is always the same, whereas in the experiment now considered the light must overtake the moving bar. As regards the optical effect observed, the method considered now will accordingly be two and a half times less favourable for a w—1 pay —u+i. This is more or less compensated by an advantage with regard to the dispersion term. As follows from formulae (3) and (4), the ratio of the dispersion term to the principal term is in the second case 1,6 times that in the former experiment. value of u = 1,5 than Fizrau’s usual method, because l 466 We shall now discuss a few more particulars of the arrangement and the use of the apparatus. € 3. Determination of velocity. In order to get an insight into the course of the velocity in the movement along the guides the position of DE (see fig. 1) was determined corresponding with 16 different, equidistant positions of the fly-wheel. The graph indicating the con- nection between the positions of the fly-wheel and the deviations, has about the shape of a sinusoid, but the two halves of the curve are not symmetrical, and in particular, the course of the graph in the neighbourhood of the two boundary values is not exactly the same, as already appears from a consideration of fig. 1, when A is imagined to move along the dotted circle. The velocity-time curve can be graphically derived from the path- time curve. At the maxima the velocity is practically constant over a distance of 20 em, of which only 10 cm. are used. As was already stated the maximum velocity amounts to 10005 for 184 revolu- tions per minute, and proportionally the calculated velocity can be derived for another number of revolutions. Whether really the maximum velocity should be taken into account in the calculation, depends further also on the position of the moving column at the moment that the shutter, before the objective of the telescope, transmits light. Im some cases this position did not correspond to that of the maximum velocity, which circumstance was of course taken into account in the interpretation of the photos. In the most accurate experiments the maximum velocity of the column was directly measured (for the method used see one of the following communications), which renders us independent of the supposition that the fly-wheel possesses a constant angular velocity. It appeared in the experiments that the machine ran more uniformly when the fly-wheel rotates clockwise (seen from the side of the larger fly-wheel) than in the opposite direction. Of course this favourable direction was always used. 4. Shutter. Only at the moment that the machine has its greatest velocity may the light be admitted to the photographic plate. The following arrangement was made for this purpose. The axis of the fly-wheel is provided with a toothed wheel, which engages with a second toothed wheel with double the number of teeth. An insulated brass ring with cams is fitted on each side of this second wheel. The cams on the two rings are placed 467 diametrically opposite one another so that they take each other’s places as regards level after every whole revolution of the fly-wheel, and can make contact with a suitably fixed sliding contact. By means of the cam an electric current is closed in a circuit containing also windings of a coil that acts electromagnetically ona shutter or light interrupter. When the second cam makes con- tact, the current passes through a second coil, which closes the shutter. Every time the fly-wheel, and consequently the moving column, have arrived at the same position and move im the same direction, the shutter is opened, and light is allowed to pass for a moment. The intensity of the interference figure not being strong enough to give a satisfactory photo with light that has been admitted once, a photo is taken e.g. thirty times successively on the same plate. As the light is let through three times a second, this takes only ten seconds. To take the second photo, i.e. when the column of quartz moves in the opposite sense, a duplicate arrangement is used, placed sym- metrically with respect to the one sketched. By means of a double- pole throw-switch it becomes possible to make one series of photos succeed the other immediately without loss of time and without stopping the machine. Only the photographic plate must be moved a little. The two large toothed wheels, and between them the small one can be distinguished on the Plate near the bearing at the bottom on the righthand side. As some time passes between the moment that contact is effected by the cam and the opening of the shutter, this time must be taken into account. To do so it is necessary to perform a phase determination. 5. Phase determination, i.e. to ascertain by a separate experi- ment, if really at the moment of the greatest velocity the light passes through the shutter. For this purpose the shutter is put at the place of the greatest velocity by the side of the bed. The wooden beam containing the transparent substance, is provided with a black sereen with an opening. A glowlamp is placed in the line: shutter-place of the greatest velocity, a line which is normal to the longitudinal direction of the apparatus. When the machine is running and when the shutter is in action, the lamp must be observed through it. When it does not work at the right moment, the field of view remains dark. Then the phase can be improved, and can at last be made accurate by gradually shifting the large toothed wheel with respect to the small one. This causes the contact to be formed at another moment. 468 6. Observation and Photography of the Interference-bands. The shutter is placed before the objective of the telescope, which was also used before to record the Fizrav-effect *). In the focal plane of the telescope, which is provided with a negative achromatic system of lenses to increase the effective focal distance, a system of wires has been placed, which is photographed at the same time with the interference bands. The position of the interference fringes with respect to the wires is determined. Immediately behind the wires the photographic plate is in a plateholder mounted quite independently of the telescope with the cross wires. The photographic plate can be put in the required position without the telescope being touched, and be shifted to take the successive photos. The telescope, the plate-holder, the interferometer, and the glass rectangular prism, in which the interfering beams are reversed, are mounted on separate freestone piers, which are cemented to the large pier. This last mentioned prism, which also served in the earlier experiments, is visible on the righthand side of the Plate. 7. Measurement of the tune that the shutter or interrupter is opened. This time, which is of the order of 1 or 2 hundredths of a second, is dependent on the current in the coils of the cireuit of the interrupter and can be regulated by this and by the change of the width of the opening in the moving screen of the interrupter. For the deter- mination of the time the interrupter is placed before the lens of a camera, with which a small lamp is photographed, which revolves on a disk with known velocity. During the time that the interrup- ter is opened, the lamp describes part of a circle, the length of which is measured. 8. Checking and regulating of the apparatus. After the inter- ference fringes have been made as distinct as possible, the beams passing only through the air, a compensator is placed in one beam, consisting of a plane parallel glass plate 7 m.m. thick and with a diameter of 25 m.m., made by Hilger. This plate can be rotated round a vertical and horizontal axis, and enables us to change the slope and the distance of the interference fringes in a simple way. In many cases it was unnecessary to insert this com- pensator, as the desired interference fringes were already obtained with the interferometer alone. Then the column of quartz and glass 1!) Cf. Zeeman. These Proc. 18, 400, 1915. 469 is introduced into the beam of light, precautions being taken which will be mentioned in communication Il. It must then first of all be ascertained whether the apparatus and particularly the guides satisfy the high requirements on which the efficacy of the whole arrangement depends. With a slow movement of the shoes with the quartz column along the guides the interference fringes did not remain stationary, but changed with regard to distance and slope. When the apparatus had first been put together, this movement of the bands was very great. It is clear that it must also occur fora perfectly homogeneous column bounded by parallel planes, when the movement does not take place along a perfectly straight line. For then the whole column of a length of more than 100 em. acts as a compensator of exceed- ingly great thickness. In order to make the circumstances as favourable as possible the steel plates of the guides were laid on the supporting plates about in the correct position. The bolts which were used to keep the apparatus in place, are then screwed down till a sensitive level, which could be placed in longitudinal and in transverse direction, indicated a plane as much as possible horizontal, determined by the upper planes of the steel guides. It was then examined if the inner edges of the guides were as perfectly straight and parallel as possible, and improvements were made in this respect by filing and grinding. At last the free play of the bronze shoes in their movement along the guides was removed as much as possible. Great improvements were successively made to the apparatus in this direction, so that rotation and change of distance of the fringes became comparatively slight. It was, however, impossible to have the interference fringes quite steady when the apparatus was slowly moved. This is, however, not necessary; what is required after all is that the same positions of the interference bands are found again when the shoes have returned to the same point of the guides. The results prove that this is actually the case, and that the occasional deviations fall now in one sense, now in another. The excellent definition of the interference fringes, recorded with the quickly moving apparatus in itself proves already, that every time about the same position of the bands is obtained, as 20 or 30 images are superposed (see above $ 4), which could never produce a definite image, when the single light impressions were not almost identical. Sometimes the system of fringes proved to be rotated, and then the photo had to be rejected. Of course care had also to be taken that the guides were well oiled, and there is one more dynamic particularity that had to be seen to. When the motor has 470 been started for the first time, then the apparatus hardly ever gains the maximum velocity, which corresponds to 184 revolutions of the fly-wheel per minute. It gives the impression that the apparatus is hampered by a resistance, e.g. only 140 revolutions are made. The starting is then repeated a few times, and at the tbird or fourth attempt the machine suddenly runs very smoothly without jerking. Then the feeling of uneasiness of the operator at the exceedingly rapid motion of the large apparatus so close to the delicate optical parts of the interferometer, has abated somewhat and the experiment can begin. *) 1) The experiment is not entirely without danger. When the experiment with glass was to begin, four beautifully finished glass cylinders 20 cm. long and 2.5 cm. thick were placed in the wooden shoe, and optically adjusted. In the very first experiment with this glass column one of the glass cylinders, which evidently had not been properly fastened, got loose, while the apparatus moved at full speed; it smashed all the other pieces and knocked the brass end-pieces off the shoe. The glass cylinders. were entirely smashed, the work of months was destroyed. It was a wonder that the interferometer and the glass rectangular prisms remained undamaged. Mathematics. — “Ueber die Struktur der perfekten Punktmengen”’ (dritte Mitteilung’)). By Prof. L. E. J. Brouwer. (Communicated in the meeting of October 25, 1919). Gel: Kin Fldchensystem bzw. eine Flüche ist im folgenden definiert mittels eines solchen zweidimensionalen Fragmentes *) bzw. zusam- menhängenden zweidimensionalen Fragmentes, in welehem die in einem willkürlieh gewählten Elementeckpunkte mündenden Element- seiten entweder alle oder alle bis auf zwei je zwei Elementen ge- meinsam sind. Im ersteren Falle sprechen wir von einem gewöhn- lichen, im letzteren von einem aussergewöhnlichen Elementeckpunkt, während wir eine Elementseite gewöhnlich oder aussergewöhnlich nennen, je nachdem sie zu zwei oder zu einem einzigen Elemente gehört. Das Flachensystem bzw., die Fläche besteht aus dem Frag- mente mit Ausnahme der aussergewöhnlichen Elementseiten und Elementeckpunkte, welche zusammen die Grenze des Flachensystems bzw. der Flache bilden. Ein Flächensystem bzw. eine Fläche, deren Elemente Grundsim- plexe einer simplizialen Zerlegung des Flächensystems oder der Fläche wm sind, wird ein Teilflüchensystem baw. eine Teilfläche von w genannt werden. Sei a ein auf der Flache w nicht-negativer Charakteristik gelegenes und auf w kein elementares *) Restgebiet bestimmendes, abgeschlos- senes Kontinuum, @ =o, a", c'",.... eine « approximierende Folge von Teilflächen von w, durch welche also « als D(a’, a",....) be- stimmt ist, so dürfen wir für jedes » annehmen, dass «”) auf w kein elementares Restgebiet bestimmt, dass a“ eine Teilfläche von a’) ist und dass die Grenzen von «”) und a®+! keinen Punkt ge- meinsam haben. Sei 4” die Charakteristik von a”, mt”) die maximale Anzahl von einander nicht treffenden, zusammen nicht zerstückelnden einfachen geschlossenen Kurven von a”, so kann für wachsendes v 1) Für die erste und zweite Mitteilung vgl. diese Proceedings XII, S. 785; AIV, 3S. 437. ?) Vel. Math. Annalen 71, S. 306. 5) Ein Teilgebiet von heisst elementar, wenn es nur auf w zusammenziehbare einfache geschlossene Kurven enthiilt. 31 Proceedings Royal Acad. Amsterdam. Vol. XXII. 472 weder 4” noch mm” zunehmen, so dass eine endliche positive Zahl g existiert mit der Eigenschaft, dass 9+) = k® und mate) = m@) für jedes nicht negative u. Hieraus folgt, dass die von ate) in cote) bestimmten Restflächen alle Zylinderflächen sind, dass die topologische Gestalt von a+) für iedes gu gleich derjenigen von a9) ist und dass «9t#+D aus at”) durch Zuriicknahme der Ränder nach Innen entsteht, so dass zwischen den a+”) und a die folgenden Beziehungen existieren: 1. Zu jeder Kombination (e, ¢,, u) gibt es ein solches nur von ¢ und w abhängendes und für festes u mit e gegen O konvergierendes &,, dass jede «-Kette') von av+*) mittels einer endlichen Folge von e,-Abinderungen’) innerhalb «+”) in eine ¢,-Kette von a übergeführt werden kann. 2. Zu jedem « gibt es ein solches mit ¢ gegen O konvergierendes &,, dass jede e-Kette eines a+”) mittels einer endlichen Folge von e,-Abanderungen innerhalb «v+” in eine Kette von a« übergeführt werden kann. 3. Jedes at”) besitzt die gleiche minimale Multiplizität der Basis der Zyklosis®), wie a. 4. Zu jeder Kombination eines hinreichend kleinen ¢', eines €, und eines u gibt es ein solehes nur von e/ und pw abhängendes und für festes u mit € gegen O-konvergierendes ¢,, dass jedes System von (es, &)-Fundamentalketten?) von a+”) mittels einer endlichen Folge von e,‚-Abänderungen innerhalb «&9+/) in ein System von (é,,’é',)- Fundamentalketten von « übergeführt werden kann. 5. Zu jedem & gibt es ein solches mit ¢' gegen O konvergierendes e", dass jedes System von (e, e’)-Fundamentalketten eines a{9+#) mittels einer endlichen Folge von e’-Abänderungen innerbalb a9+# in ein System von (¢, «’)-fFundamentalketten von a übergeführt werden kann. 6. Zu jedem « gibt es ein solches e°, dass jedes System von (e, €')- Fundamentalketten von a gleichzeitig ein System von (e’, é')-Funda- mentalketten eines jeden aYt+?) ist. Wir bringen diese Beziehungen zum Ausdruck, indem wir « als zyklomatisches Katrakt von a bezeichnen. Sei nunmehr-« ein willkürliches auf w gelegenes abgeschlossenes Kontinuum, «,‚ die Vereinigung von « und den elementaren Rest- gebieten von a, a,%) eine a, als zyklomatisches Extrakt enthaltende Teilflache von w. Alsdann gibt es zu jeder Kombination (e, ¢,) ein 1) Vgl. Math. Annalen 72, S. 422. 3) Ibid., S. 424. 473 solches mit ¢ gegen 0 konvergierendes «,, dass jede e-Kette von «,(7) mittels einer endlichen Folge von e‚-Abänderungen innerhalb w in eine e‚-Kette von « übergeführt werden kann. Wir werden auch « ein zyklomatisches Kxtrakt von a, nennen. Insbesondere werden wir ein zyklomatisches. Extrakt eines Ele- mentes als LHlementarkontinuum, ein zyklomatisches Extrakt einer Zylinderfläche als Zylinderkontinuum bezeichnen. rep Sei a eine auf der Flache w nicht-negativer Charakteristik gele- gene, auf w kein elementares Restgebiet bestimmende, abgeschlossene Punktmenge, «' = w, a", a'",.... eine @ approximierende Folge von Teilflächensystemen von w, so dürfen wir für jedes v annehmen, dass a” auf w kein elementares Restgebiet bestimmt, dass «+ ein Teilflächensystem von a”) ist und dass die Grenzen von a”) und acti) keinen Punkt gemeinsam haben. Sei 8 das aus e@@) durch Tilgung derjenigen Stücke, welche die topologische Gestalt eines Elementes oder eines Zylinders besitzen, hervorgehende Flächensystem, S ein willkiirliches Stück von 8”, k (S) die Charakteristilkk von S, m(S) die maximale Anzahl von einander nicht treffenden, zusammen nicht zerstückelnden einfachen geschlossenen Kurven von S, so sind die Zahlen &£(S) und m(S) beide nicht-negativ und nicht beide gleich Null. Mithin existieren zwei ganze nicht-negative Zahlen g und h mit der Higenschaft, dass jedes BYt?) sich aus h Stücken 89+”), B,9T"),.... p+") zusammensetzt, dass die topologische Gestalt von Bate) für jedes u und jedes s gleich derjenigen von 39) ist, dass Bote) aus Bote) durch Zuriicknahme der Rander nach Innen ent- steht und dass das als D (BD, Bio, ...) bestimmte Stüek von « ein zyklomatisches Extrakt von 89) ist. Das hiermit erhaltene Resultat kann in der folgenden Form, welche, wie eine triviale Ueberlegung zeigt, für eine Fläche w negativer Charakteristik ihre Gültigkeit behält, ausgedriickt werden: Zu jeder auf emer Flüche w gelegenen abgeschlossenen Punktmenge « existiert ein Terlfliichensystem yw von w mit der Higenschaft, dass « sich aus erstens von den Stiicken von w je einem zyklomatischen Extrakt, zweitens einer Menge von Elementarkontinuen und Zylinder- kontinuen zusammensetzt. Ist insbesondere w eine Kugel, so sind alle Stücke von @ Ele- mentarkontinua. Ist w eine projektive Ebene, so sind entweder alle Stücke von « 474 Elementarkontinua, oder alle bis auf eines, das ein zy klomatisches Extrakt von w ist. Ist w ein Torus, so sind entweder alle Stiicke von a Elementar- kontinua, oder alle bis auf eines, das ein zyklomatisches Ektrakt von w ist, oder endlich alle bis auf eine abgeschlossene zyklisch geordnete Menge von Zylinderkontinuen; die Teilzylinder von w, von denen diese Zylinderkontinua zyklomatische Extrakte sind, sind alle auf w stetig ineinander überführbar. | Mathematics. — “Ueber Transformationen ebener Bereiche’. By B. von Keréksarto. (Communicated by Prof. L. E.J. Brouwer). (Communicated in the meeting of October 25, 1919). In der vorliegenden Note wird eine Anwendung’) gemacht vom folgenden Brouwerschen Fixpunktsatze : Eine eineindeutige stetige Abbildung der abgeschlossenen Kreisscheibe auf einen Teilbereich derselben lässt wenigstens einen Punkt invariant. Mit Hilfe dieses Satzes beweisen wir nämlich das folgende THEOREM. Eine eineindeutige stetige Abbildung eines von endlich- vielen Jordanschen Kurven begrenzten abgeschlossenen ebenen Bereiches auf einen Teilbereich desselben, bei welcher die Grenzkurven des ursprünglichen und des Bildbereiches paarweise äquivalent sind, jedoch eine und nur eine Grenzkurve in eine dquivalente übergeht, lässt wenigstens einen Punkt invariant. (Hierbei sollen zwei einander nicht kreuzende Kurven üquwalent genannt werden, wenn in ihrem Zwischengebiete keine Grenzkurve liegt). C, sei die äussere Grenzkurve des gegebenen Bereiches, ihr Bild C,’ sei mit ihr äquivalent; die übrigen Grenzkurven seien C,, C,,... Cu, ihre Bilder C,’, C,’,... C',. Man erweitere die gegebene Abbildung durch eine an sie anschliessende Abbildung des Innern von C, auf das Innere von C’, (a = 2,3,...n). Hiermit erhält man eine einein- deutige stetige Abbildung des Innern von C, auf das Innere von C,’, welche nach dem obigen Brouwerschen Satze wenigstens einen Punkt invariant lässt; dieser Fixpunkt kann aber nicht im Innern von C, (a #1) liegen, gehört somit zum urspriinglichen abgeschlossenen Bereich. - 1) Für eine analoge Anwendung nebst daraus gezogenen Konsequenzen vgl. Math. Annalen 80, S. 34. ACA ME Fe dt ras ARS ' + i f - i 7 4 y . e 4 8 . ‘ Simon - KONINKLIJKE AKADEMIE — VAN WETENSCHAPPEN -- TE AMSTERDAM =-:- PROCEEDINGS OF THE SECTION OF SCIENCES VOLUME XXII == {1 PART)- = ed coats ae) PS JOHANNES MULLER :—: AMSTERDAM _ FEBRUARY 1920 pt EF QUNE LA IAR MA) (Translated from: Verslagen van de Gewone Vergaderingen der Wis- en e ein Afdeeling DI. XXVII and ZA S255 9 ‘ * a, vt ed Ge Hh i wl PENN OTE) % yp RU sie 4,1, 1920 24=*" proceedings of ‚ LWIA 10014