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Digitized by the Internet Arciiive in 2009 with funding from University of Toronto http://www.archive.org/details/p2proceedingsofs12akad KONINKLIIKE AKADEMIE VAN WETENSCHAPPEN -:- TE A.WSTERDAM -:- PROCEEDINGS OF THE SECTION OF SCIENCES VOLUME XII ( — 2^D pARj _ ) V 1 I JOHANNES MULLER :— : AMSTERDAM : JULY 1910 : : (Translated from: Verslagen van de Gewone Vergaderingen der Wis- en Natuiirkundige Afdeeling van 24 December 1909 tot 29 April 1910. Dl Will.) a 57 -pt.Z /'J. CONTENTS. Page Proceedings of the Meeting of December 2i 1909 503 » » January 29 aad February 26 1910. . . . 547 » March 20 1910 679 » » » » » April 29 » 775 S-& ? KONINKLI.IKK AKADEMIK VAN WETExNyOilArPKN TE AMSTERDAM. PROCEEDINGS OF THE MEETIN(i of Friday December 24, 1909. ('I'raii^hilcd IVoiii : Veislag van de gewone veigaduiiiig dur Wis- en NatuiiikiMidlgo Afdeeling van Vrijdag -2i Ducernbur 190'J, Dl. XVllI). M. J. VAN UvF.K : "'On UiC orbits of a function oHtaini'd by inliiiitcsiiiiul itcraliuii in ils ••uni|il'j.\ I'Ume'. (Communicated by Fiuf. W. K.irTEyNJ, p. 503.. J. W. Gii.TAY and M. UK Haas: "On tlie motion u' tlu' bridyu of the violin". (Ccmniiinicatcd by Prof. II. Kamkklingii Onnks), p. .513. L. Boi.K ; "On the slope of the Foramen n.a;^niiin in I'limatus. 2nd Pajier. On the eum|iarative Craniology of Primates", p. 525. I'll. K011XSTAM.M: "A short reply to Mr. van Laak remarks". (Communicated by Prut. J. 0. VAX UKK Waals), p. 534. II. R. Kulyt: "The e(iuilibrium solid-lillnid-ga^ in binary .system? ot nii.xed crystals". (Com- municated by Prof. P. VAX Koiiui hgii). p. .537. Krrata, p. 545. Mathematics. — "''>// l/ir orJilts of n. function o/jtauici/ hi/ infuti- k'si.iiifil Iti'rntloii lit its co/ii^jle.v plaiw." By M. ,). van Uven. (Communicated Uy Prof. W. K.\ptkv.n). (Comniunieati.'d in the meeting ol' November 27, 190'J). Wlifii ;i I'miftioii // ^ (/ (.)') is iterated, each iteration //„ = </„ {,v) will i^-ive rise to a coiit'orm rt'in'eseiitulioii id' llic coinplex piane.s of X and //„. If we suppose i/ = tf{:) [n Iil' imilt up iiy means of inliiiitesiinal iteration of the functitui Lim n \ =: Liiii (f \ (.(,■), .so that //„ lias also a meaning for broken and immeasurable values of it, then tiie con- form representation of // = <f (.r) will graibialiy appear out of the idenlilv belonging to //„ ^ (/.,(,/;; = ,f. 34 Proceedings Koyal Acad. Amsterdam. Vol. XII. ( 504 ) Wc now reijanl a piano F„ as complex plane of I lie (piaiililv .<■ and we place tlie complex plane I", of the (piantitv // =3 </"(,/■) parallel <o r„ at a (listance h and in sncli a way, that the real axes and the imaginary axes are each other's orthogonal projection. Then to each point .(.' of V,, are conjugated by means of the fnnclion // = y (,*■) one or more points // of Tj. By connecting corresponding points x and y by rays a congruence of rays is formed which can serve as the image of the function y=:<p{.v). For the case y = rp (x) ^^i ic we should obtain in this way the congruence of rays formed by all the noi'nials on the planes 1\, and V^ as representative of the identity. If now we let the function 1/ t= <f {.i-) gradually arise from the identity, then to each stage of the generating process a delinite con- gruence of rays will belong. All these congruences form together a complex of rays. It is clear, that the formation of the function y = (f (.(■) will now be represented by this complex of rays. Let us first examine the complex cones of the ])oints of V„. Each point .(' ^ II -\- ir of this plane is the vertex of a cone counting in any case the normal in ,v on 1^„ among its generatrices; this edge namely intersects the jdane T, in y = 11 -\- ir = .r. The sectio)! of this conn)lex cone with T, will pass through the point ; = ,(; and all points I'cpreseuting the \alues taken by //„ r= «/„(,r) when )i increases from to J. So this section al.so gives us a representation of the genei'ating process of if (r). It goes without saying that we can continue the iteration also past y = <f {.li) and likewise that we can also regai'il negative valnes of n. The whole of the eomiile-N cone end)races in fact all functions //„ =r c/-,, (,c), where II \aries tV(nii - x tn -)- y. Also the section I'egarded as a whole will contain all the \alues of llie fujiction y„ = 7,, (,;■), where .c is constant and 11 xaries t'roin - - o: I'* ~l~ ^- ''^'^''h Nalne of ,/■ possesses its own complex cone and therefore also ils own seciidii. ^\ c> shall indicate this section by the orbit .r -^ //„. We might also luive indicated the increase of </ (,/) by allow iiig the plane T, lo grow gi'adnally out of I', and llial by allowing the distance of the planes to increase regularly iVoni to //, so that <p„ (,t') is represented in the plane ]\ at a height //// abo\'e 1',,. Let us then suppose in each plane l'„ the image y,, ^ (f „ ..r^ belonging to some initial-point .r = // -\- ir to be constiin'ted, then all these point.> will form in llieir regular succession a twisted ciir\e. Lach of the x''' points ,/,■ of I', gi\'es rise to a suchlike liris/n/ c/irri' and the function y =r f/' (.r) with its dillerent stages of develo|ini('nl is thus repi'osented by a roiK/riinirr of tiristi'il ciirrrs. ( 505 ) It is clear tliat the ni'tlio^diial |(npji'i'ii(iii of llic twisted (•iir\-e of .r oil the plane !', cuiiicides with llie orbit .f — » //„. We .shall for liie |)i-eseiit occii|)v ourselves oiilv with ihe siiidy of such an orbil ./■—*//„. To find the orbit .i- —* i/„ we luive hul to solve the funclioiial equation of Auki,. We have namely to find that function /'(.i) of .r increasing with // when for ,/■ is snhstituled yn^<f,/^^r); this function increases for Ihe process of iteration with real contributions, i.e. the quantity s^/(,()= /'-)- 'I' describes in its complex plane the right line V ^ c parallel lo the real a.\is. If once we know the form of the function Z,^f[.v), then we also know the orbit of the quantit}' j;:=/Li(C). The value of I" and the inilial \aliie (?i = Oi of the real part ( of ? represent together two arbilrarv constants, of which we do not dispose until we choose the initial value of x. We shall indicate the current ])oiiit (y„) of the orbit .r —^ i/„ by z, whilst we shall point out ,/: by -„ ; we then have or r 4- /F= U, -f iV„ + n, so that r=z r„ + n , v=z F„. The choice of Ihe initial point ,„ now determines the values U„ and 1"„. When working out some e.xamples we shall not always follow the systematic way sketched above, as it is unnecessarily lengthy in simple cases. In reference to the broken linear fiinction y := we notice that this has been thoroughly investigated already by PoincaioI ') and Klein -I. the latter having also included comple.\ values of «, i3, y, and din the studv. Ki.i'UN loo allows the fuiictioii v ^ to arise gradu- ally out of .'■ and regards the orbit described thereby. For Ihe nou- l)arabolic cases he builds up the function by infinitesimal itei'atiou in the sense indicated by us. For the parabolic case, on the other hand, he takes as parameter of the function in its orbit not the iteration-index //, but a complex nudliple of it. In consequence of J) PoiNCARE. Acta Matliemutica I (1882), p. 1. *| Klein — Fkicke. VovI. li. il. Tlicorie dcr ell. .Modiilt'uiiktiorieti (TEunxEn, 18001, 1G5. 34* ( 506 ) this the (iiiiil of - fouiid by Kt.hin dilfers a little tVoni ours. Alllioiijiii after statiiifi and aiiniilliug this dillereuce we might siillice with a reference (u the results of Klein, we will dwell a little longer on the fnnclion v = , <lie more so as, diifering from Ki.kin, who y.t' + d treats first simple cases and then applies the principle of transformation of the circle correspondence, we shall immediately investigate the most general case. Examples : J. y = w -f- 1-?, //„ — ,(-• + '*i^ 01' - = -0 + "i^- The point z describes the ri(/ht line coimecti)ig the points : = j„ and z ^ Zg -\- li, in such a way that the distance from ; to :„ is pro- portional to 71. II. 1/ =1 <i,i\ t/„ =z a'\c or ; = «";„. Let us put : = Qe>", c„ = 9„t;*", a ^ ae'' , then Qe'" =z (V'e'"'Q„(A>, from which ensues ^ =^ a"f>„, & =z &g -\- uT . . . . , . (1) or ij - % Point c describes a logarithmic sjiinil round the origin. The polar angle ^ increases uniformly with n, i.o.w. the polar angle 6^ increases anthiaflicdllii nni/onnhj : it is clear that the radius vector o increases (jmmetriciillij /iiN/ofinli/. If (( is real, then t = 0. The second e(puition (1) tells us that the polar angle remains constant, so that point : moves along the line connecting (> and :„ and that with a geometrically uniform increase of p. If '//(w/ « =r 1, then <;= 1. The tirst equation (i) then indicates, that the radius \eclor remains constant, so that point - ilcscribes the circle round O as centre, passing through point -„. The polar angle 6 increases arithmetically uniforndy. If T is counncnsurable with rr, i. e. if a is a root (hii of unity, then 1/ =: (t,i: leads back to .r after a whole nundier of iterations. ii V ft — 1 III. II = ax '{ p\ /■(•'■) = log a — 00 ij ... log iz — a) /•(.^•) = for ■<■ = — — <h theretore / (c) = — V ^ lor/ a «— I lo'J t( ( 507 ) If \vc displace tlic origin lo -/ ami it'a(•(•(l|■(iili^l\■ we call c — // ^ q''''''' , \vc lind tni- the (irliit u\' : liie /ni/i/ri/liin/c spiral t)' =z ce'"' round the poiiil '/. If. hiiwevcr. k is real, then ; describes the line from -„ to «■:„ + ,■}, conlaiiiinii' also |)(iint 7 = ■— . Is on the contrary tt — 1 mod a ^ ^, then the (iiliit of : is a circle round 7 as centre. IV. ,/ = — , where (n — rf)'- + V/ ^ *'• Y-r + 6 <^ 1 p.v-i-1 / (.7;) = - loii r, ^^'here / = Mil — =^ log (« + ff)— l/(a^(f)"-+4^7 ■ 4(«(f-|Jy) («— ff)+|/(«— d)' + 4/J7 (a—(f) — y'{a—df--\-4^y We shall take as general case, that ;i, 7, and rf are all complex; then /, /), and ij will also be complex. 1 , pz + 1 1 p 1 Z+p-^ 1 p 1 -o+J^'~' / (;r) = -loc/ = — /()'/ - -I — log z^—loa — I Ion 1- n. From ^w/ — J — log -^ + An (2) ensues that for an infinite \alue of /;. the point c takes either the value — /'"' or the value — q—^. We shall call the points c = — p—^ and - = — q~^ the liinitbuj points and we shall put — p~^ = q' , —<r' =9"- Thus our equation (2) becomes z — (J £•„• — y log = log + (;< -|- iv) u (3) where we ha\'e replaced P. by (ti + iv. Let us choose g' and g" as auxiliary origins and let us call z — g' ^ ^' = Q'e'^' , z — g" =: z" m Q"e'"" , we then find out of (3) log % + ]{e'—8")— loq %^ — ; {a',—a'\) ~ (m -f im, where by separating the real part from the imaginary we find lo(r——lo(i''''^zrz^n . {&'^e") - {6\ — d\)—rn. . . (4) or ( :m ) ^L~'-L±,y„ . H'—H" — 8\~(i\^vu (5) Eliiuiiiutioii of ih loads, wlicii n and r are neitlicr of tlieiu (■(jual to zero, to — 0' — — II" — — V„ — -- 'J"„ By putting we tiiid a t' (/> ■' =C. . . (6) 1 I ., II II .. /ry\ ?='•'" , Q =(■ e' , (7) "'^C." (8) The equations (7) and (8) determine logetlier a so-called /(u/r//7'///////r double sj>/'nil^), with liie points (/' and //" as ])(»les. From the secimd e(piation (5) ensues that the angle f^' — 8" ■=z (f between the two auxiliary radii vectores yc and [f'z increases arith- metically uniformly, whilst tlie first equation (5) shows us that the quotient of the auxiliaiy radii vectores increases geometrically uni- Ibrmly. For the case «, (i, y, and d real, some sinqiliticalions a]ipeai'. We shall distinguish three cases. A. {a — ff)^ + iiiy > 0, «d — /?v < 0. Tlie qtuxntities /> and ij ai'e real, so the points j/ and //" lie on the real axis. Farthermore we have *"<^<', so that r = .t. Hence the orbit of c is a lixjaritlnnic (Imihle siilru/, whose two poles lie on the real axis. A special case is furnished by the coutlitiou (t -\- (i = {), or ;(:=(). Fi'oni the tiist equation (5) now eusues that the quotient of the auxiliary radii vectores is constant, so that the point : describes a circle of Apollonius of the triangle g'<j"~o, whilst the angle t/:</' in- ei'eascs uniformly with ;/. An example of the latter case is furnished I by 1/ =1 ; here g =-\- I, //" = — 1 . a: />'. {« " rf,-^ + 4/Jy > 0, n(f— f?y > 0. The points (/' and </" lie on the real axis, whilst i" ^0, thus r^O. Now the second Ofpiation (5) shows ns, ihal 6' — 0" ^= (f is constant, so that the point z describes the circle passing through ') i''<ii- tlic loL^arillimir (luublc spii-iils Ilic ri'ailrr may cniisiiU : Hdi./.MfLLEit, I'rlier ilio loyanllim. Alibililuiig etc /uilsclir. i. Malli. ii. I'liysik., \'ol. 10. (1S71), p. "281. ( 509 ) All 00^ initial points c„ i'liniisii llms together all circles of the pencil of which g' and /" are the base points. Let us suppose point : ilelermined on its orbit as point of inter- section of this orbil witli an element of the conj ugated Jpencil of circles, intersectiniz; the real axis a. o. in a point s, then evidently (/::(/": =z (/'s : I/" s holds, sO- that the equation (5) expresses that the quotient //.s- ; ,/"s increases geometrically uniformly. (This property enables us to construct easily the jjoints z belonging to given values of ti). Farthermore holds //' = -— x and </" = z^^ . C (li — rif + 4i7< 0. The points if and <i'' lie synunetrically with respect to the real axis, p and q being conjugate complex. As mod. c = 1 we have (I = 0. The ratio q '■ y is now constant, so that the [Kiint describes a circle of Apollonius of Lg' ij" z^, i.e. a circle of the pencil with if and ij" as point circles. We can again regard the point z as if originated by intersection of the orbit with a circle of the conjugated pencil of circles. As the angle ifzif inci-eases uniformly with // we can easily construct with the aid of the conjugated pencil of circles the points c belonging to delinite values of //. It is clear that the orbit of c when n increases ' indefinitely is described innumerable times, so that the function ^„(.i') has a.- a function of « a real period. If Y is commensurable with .t, then this period is a mensurable number. If particularly u -\- 6 ^^() hokis, then r =: .t. This case is a. o. — 1 realized in the function // = — ; here if = i, g" = — /. r. y =r ~ , where (« — d)' + 4,iy =: 0. Here we are in the parabolic case. 2« 2 — a 2« z^ — a a — iTz—ff a — rf 2,— (/ SO that ( MO ) z — a :„—u t'. — (f z. — a Tlic (lili'iM-onco between our iiietliod anil llial nf Ki,kin arises from tlie fact tlial Kuun allows the (|uan1itv — . // to increase viuilltj. 'In If we take '/ and // as anxiliary origins and if we |)ut then the ecjnatioji (9) takes the form of — fj =:-yp« -f- (ft -|- tv) n or (J i' . from which ensnes - cos {d'-8") = -,f COS {e\—S'\) + ft", {>" {J „ ( \ sin {O'—S") = ^-l sin {6\—H\) + rn. \ (10) If we nut ft = <i cos t, v ^ <!•<»// t i. o. w. — = (J*'" we lind out V -'"■ J of (10) when eliminating }i : ^'- sin {&'-e"—r) = ^^ si,> {e\ — 0\-r) — c. . . . (11) i'" i' n It is clear, that the orbit as fonnd by Klein follows from onrs by putting r = 0. The orbit of Klkin can tlms serve as iteration- orbit for real values of the quantitv , thus of -. To investigate the curve determined bv the equation (11) we imagine the circle passing through g and a, and of which the arc (/ii amounts to 2r, so that from each point of the supplementary arc the line (/a is seen under the angle t. (.See tig. p. 511). If we connect g with j„ and :, the connecting lines will meet the circle in ;h„ and m. Now /' gma = /^ gm„a = r Farthermore /^ zam = &' — &" — t, ^' :„<iiii„ = 6\, — 8"„ — t. If we let fall the normals ;„/a„ and :n on uni,, and 'im, then --„//„ = o'„ sill {6\^ — ^"„ — t) and zn = o' sin {ff — 0" — t). The eipialinii (Tl) now demands zn ~aii-„ ztj zn zm ( Ml ) It is therefore evident that we ai'i'ixe from ]ioints //; to ])oiiits c bv diminishing or enhirgmg tlie chords ijin in a definite I'alio. So the orl)it of : is a circle toiicinnii- tlie anxiliary circle [m) in q, wiiose tangent in ij forms in tiiat wav tlie angle t with the Wnega. If the quantities r(, ,?, y, and ff are I'eal, then n and y are real, whilst r ^ 0, therefore also t = 0. The points a and r/ therefore lie on the real axis and the nrliit of : touches the real axis in the point (J. If on the other hand [i = 0, then the centre of the orbit lies on the line (ja. The way in which c changes with n we can read from the equations (10). If we suppose the point ; to be furnished l\y the circle, which passes through ij and : and whose centre lies on (jn, then the first equation (10) tells us that the ri'ci^irncdl value of the radius of that circle increases aritlnneticallv uuiforudv (hat i. o. w. the radii of the circles through (/ whose centres lie on ijn. and which pass through -j, z., etc., form an luinnonic series. If on the othei- hand we suppose that the point - is consti'ucted as point of intersection of its orbit with the circle through : touching the line i/n in y, it then follows easily out of the second eipuition that also the n-eiiiroci/l value of the radius of this circle increases arilhmeticallv uniforndy, that i.o.w. the I'adii of the circles touching ya in ,/ and passing through the points c,, c, etc. form an luinnonic .series. It is clear that for the case (t. ,?, y, and (f real, thus i' ^ and r := (), only the lirsl dcteruuualidii of the cdiirse of c can ser\'e, ( 512 ) whilsl ill llic case ;i ^ i) (iiilv lli" sccdiifl <li'l('i-iiiiii,iliiiii retains its valiililv. VI. ,l = .r^ . >/„ = ■'■'■' , /{■>:) log a loil Umi z log log 2„ log tt lop ft Let us pill /",'' " =^ ," + ""' ^^'^ i\\on lirt\'i' /()(/ /()_(/ z =z log loii c„ -\- (ft -\- ir) II, wo /()(/ c =r loii f,| . I':'" (cox rn -\- i ■'in rii), 1)1' hui o + if) — (Ion Q„ + if)„) !■:'" ('>« rn + ?! mi vii) . . . (12) rniiii wliicli oiisiies (/",'/ c')'^ + ^'^ = !(/",'/ y„r + ^,;^! ''-■'"> I /()(/(> log Q^ cos im — S^sinvn i • • • ■ (^'^) f) liiii pji sill rn -(- ^j cox rn / Out (if these equations follows by eliniiiuitioii of ii the orbit of c. For the case a positive, so r = 0, the .second eipiation passes into loc/ (J log ^d ~^ ~~ ^^^ ^ '*' or Q = C^". The orbit of : is m tiiis case a logarillunic s|Mrai around liie origin, which is uk/cjh'iu/i'dJ of «. If mild ((^1, then f( = 0, so that the lirst etpuition (13) tells us that (/„,/ oY + 6/^ = (/,.„ ^>„y + /y,;-' = T^ or This curve is likewise inde|)endent of the argument of ti. The function y=z.r~^, which we have regarded on one hand under IV A, ft = 0, and which then furnished for the orbit of : a circle, we can also range under the case treated last. If namely we take // = .(;^' as a special case of y = .v'-' {mod. rt = 1, my. a = .t), wc then lind for the orbit of ; (pute a different curve. To this r(Muarkable properly of // = ,(■— ' we hope to refer more explicitly later on. { 513 ) Physics. — "l>a Ihr inoli.m ,•/ lln- hriilij,- nf ll„- rlolin" . I'.v .1. W. Cii.TAY ami rnif. .M. dk llws, Ciiiiiiiiiiiiicaled liv I'lof. H. KAMF.KI.TN(m OnNKS). ((Inmmunicated in the meeting: of Novemjjer -11. 1909). 1. Ill llie rnlldwiu.n- linos ail accdiint is ui\"oii nf an c'.\|ieriiiieiilal researcli tiie object of \\liicli was to make a coiitrilMitioii to oiu- knowledge of the iiiaiiiier in wliicli the \il)rations of iIk^ strings are transmitted to the roof of a violin liv the liridiic. As far as we kno\\- the literature on the |)hysics of bow instru- ments is \ei-v limited aw\ leaves the true nature of tiie motion of tlie bridge undecided. Hkl.mholtz ') savs: "Der eine Fuss des Sieges ruht anf einer relativ festen I'nierlage, n;iinlicli anf dem sogenannten Stimmstocke, einem festen Stabt^hen, welches /.wischen der oberen and nnteren Platte des Kcirpers eingebant ist. Der andere Fuss des Steges allein ist es, welcliei- die elastisclieu Holzplatten mid iniltels deren Hilfe die innere Lufduas.se des Korpers erschiittert." From tills descri[)tion cannot be inferred whether the bridge vibra- tes principally in its own plane i. e. at right angles to the longitu- dinal direction of the strings, or at right angles to its own plane i. e. in the direction of the strings. Vak Schaik ") remarks; "IJy the vibrations of the bowed string a motion of the bi'idge is set up which consists in an oscillation about a line parallel to the length of the violin : in this maimer the movable foot of the bridge communicates \ibration to the roof of the violin and tiius to the air." His opinion therefore is that the bridge vibrates in its own |ilane perpemlicidarly to the direction of the strings. Apian-Bknnewitz ') observes: "dass namlich der rechte Fuss eine viel geringere Bebung als der linke zu machen hat und dass die Thatig- keit des linken Fusses als eine hammernde zu bezeichnen ist." His view is thus the same as van Schaik's, as appears 'further from page 133 of his book. Barton ^) in conjnnciion w itli (iAKKi'.T and afterwards with Pentzer has ') Tonempfindiingen, 3e Ausg. p. 146. -) Dr. J. BosscHA, Leerboek der Natuurkunde, III, beweikt door Dr. W. G. L. VAN SCHAIK, oth Ed., p. 170. 'i Die Geige, der Geigenbau und die Bogenverfertigiiiig. Weimar, Bernhardt Friedrii:h Voigt, 1892, p. 125. ^) Philosophical Magazine, 6th Series, Vol X, XU and Xlll. ( .M4 ) iii\ esliiiiili'il IIh' iiiiliirr of llic \ ilii;iliiiii> (il lln' ^li'iiii;. Iirid^c, and rddl'dl' ;i ist)ii()iii('k>i- as alsii of llic air inside lln.' soiKnncli'r. He cxaniiiR's hotli inotioiis of the liriiliic and fiiids dial t'nr llic >anic |i()iiit of llie liridfje llie (lis|ila('onieid liv llic lidi'izonlal iiiotiim, i.e. in llio dii-cctidii of the sti'in^ti;, is ahoiil 17 limes liic aniplilndc of liic Ncrtical iiiolioii '). As llic l)i'idfie oC Ilio sonmiiek'r is eidiielv dili'erent in shape fi'Oin the bridge of tlio violin aixl tiie soiiomelei- is moreover not fitted with a sound liar, llie rcsidls of the investigation are not immediately applicable lo the motion of the bridge of the violin. Savaut ^) in his very important memoir on string iiisinunents does not refer to the motion of the bridge. 2. ll seemed to iis a jirioi'i soniewliat inqiroluible that as van ScHAiK and othei's suppose a comparatively massive object like the liridge by \ibrating as a whole in its own plane abont one of its corners shonld be able to follow completely the intricate motions of the strings and connnnnicate them to the roof of the vioUn. It seemed to lis moi'e |irobalile Uiat, as I>ai;ton found for the sonometer, both molions slioidd be lakcn into acconiil. In order lo in\esligale Ihis experimentally we jiroceeded as follows. Fig. 1 represents a violin-bridge manufactured by the well known makers Cakkssa & Fkancats of Paris. Fig. 2 shows a small Fi^. 1. metal clamp which can be allached lo the bridge at different points. In order nol lo damage I he bridge llie screw .v does nol press ') Phi!. Mag. Ser. 6, Vol. XIII, p. 451. -) 'Memoirp sur la construction dos iristrument.« I'l cordes et a archet." A lepi'iiil of Ihis paper is to be found in: "Nouveau Manuel complot dii Inthier", by jMauuin and MAKiNK. Paris, librairi(> encyclopcdique de Rorf.t, 3 894, p. 333—398. ( 513 ) directly against I lie I nidge but against a moveable piece of steel p Tiie weight of the clamp is I'atlier more than 7 grammes. If the bridge swings in its own plane about its right foot /, then when we attach the clamp to the bridge at «, the moment of inertia (if tiie bridge ulnjut the axis of rotation at / perpendicular to the plane of the bridge will be much increased. On the other hand when we tix the clamp al A, the eifect on the moniejit of inertia will be much smaller. We found however that there was very little dilference in the soumi of the violin in the two cases. By tixing the clamp at a some dam[)ing influence was noticeable in the // string ; at 1> the e string was somewhat damped. In view of the effect of the clamp being about the same in both cases it is difticult to conclude that the bridge swings principally in its own plane about one" of its feet. Moreover the intluence of the damper was in both cases very small. The following experiment speaks even more clearly. The diistance between the middle of the right foot and the middle of the upper edge of the bridge fc is in our case 38 mms. The distance fa is 37 mms. When the clamp is placed at c a strongly damped sound is obtained : this is the well known mute-effect, but even stronger in our case than with the ordinary mute which weighs only about 4 grammes as against ours which weighs over 7 grammes. At a the effect is as we saw, extremely small. As /f and /a are approximately e<pial, the increase of the moment of inertia of the bridge is about equal in both cases. If the sound were transmitted b,y the bridge chiefly by its vibrations about an axis at /', the damping eifect of our clamp should be about eipuil in both positions. As this appears not to be the case we cannot fmf infer from these expei'imenls that the motion of the bridge in its own [ilane is not of piimary importance for the transmission of the vibrations of the strings to the roof of the violin. We subjoin as an instance some results obtained by two iude|ien- dent observers each playing his own violin. Violin with strong sound, about Old violin by a [lupil of .St.\inkk's, 50 years old, maker rmknow]i, small strotigly aivlied model, model iM.\GoiNi, very large. fine mellow sound, but not strong in tone, (/ string least fine, (I string liy far the best, (' also ver\' good. ( 5.1G ) Metal damper at a. (Fig. 1). Siiiiic (lani|iiii,n eliecl, eisiiecially ij string much less line than with- oii I lie (/ string. Ratiier strong ont damper, nasal sonnd. d harder and interior. a inferior. I' irnproved. none of the strings damped, responil as [)romptl3' as without. iMetal damper at //. Some dainj)ing efi'eet. especially ij string better than usual. on Ihe c' string. <• better than d ,, worse ,, ,, usual. a ,, „ ,, „ <i ,, )> .) >> y, (I and It respond more |)romp1l_v than otherwise. The i' string is slightly damped. i\lelal damper at c. Damping much slrouger than at 'r Mute ell'ect on all strings, l)ut Hlt'ect Ihe same as wilh a mule, much more strongly dam|)ed 0]ily less good than with an than wilh the ordiuaiy mute, ordinary mute. It will i)e seen that the two observers agree entirely as regards the main ellect : Ihe damper at c gives the ordinary mute effect. At '/ and /; the etfect is absent or at least only very small; again both observers find the ell'ect of |)lacing the clamp at a about the same • as at /*. The small diri'erences in the results of the two observers may be due lo indi\ idual dilferences but also to the great dilfereiice between the l\\<i instrumenls. The fnlluwing obscrxations prove also, Ihal the parallel niotitm of the bi'idgc lias little inlbieuce in the ti'ansmission of the string motion Id ihc niiif nt the \ idliii. The observers and \ iolins were ihc same as in the prexious ex- |)('riin('ius and the same damper (if 7 grammes was used. ( 517 ) Metal (laiiii)ei' af <!. (Fig. 1.) Mute etrtH't, strongest oil tlie</ side. '/ string strongly (lani|)0(l. (/ string less, bad in tone. a string .still less, bad. Metal damper at c Daniping, diniinisliing towards r damped, but inni'li less than the '/ side. The ij string has the ij in the il position of the retained its original tone liettei- damper, than the e string in the (/ posi- n less danqjed. tion of the damper. (/ damped, gives tlie ninte-sonnd more than the d string, but is still comparati\ely strong in tone. // less damped than d, \ery ugly. IJoth observers thus found, that in the position </ the damping efteet diminished towards e and viee versa. Thus e. g. in the e position of the damjier the </ string was but little damped, although in this ease assuming the bridge to vibrate ehietly in its own plane, the ij string would act on a bridge with miu-h increased moment of inertia which would invcilve strong damping. We think therefore that we may infer from these experiments that the motion of the bridge does not prin- cipally take place in its own plane about one of its feet, but that it vibrates chiefly transversely, as shown dia- grammatically in Fi;^. 3 where uh represents tbe bridge ill section, (hi this assumption the results of all the above experiments are completely explained : '^' ' \. A damper placed at a has much less damping intluenee than a damper at c, as the moment of inertia about <ih is much less increa.sed in the former case. II. The eifcct is about the same whether liie damper is attached at li or at A. It is clear that the moment of inertia of the bridge, with the (damji attached, about ijli has about the same \alue in the two cases. III. Again the results of the second .set of experiments become ( 518) iiilell:nil»lc when a transverse vilualion ol' flie bridge is ailiiiilted -. we riMind ill ilial ease llial liie damping eli'eet diininislieii inwards llic riulil w lien llie elaiii|i is (ixed at d and vice versa. By weighting- liie liridge at the top corners tiie vibration is no longer symmetrical; tlie |)art wliich is loaded at the top will vibrate less strongly than liu' unloaded pari. 3. An additional (|ueslion with regard to the two motions of the bridge suggested itself in llie investigation. In lig. 4 dc represents the string at rest, he llie bridge: when the string is deflected to the right {da^f), the tension A' of the string lias a component J/ at right angles to llie plane of llie bridge and a com- ponent A' in the plane of the bridge. When llie siring lias its greatest deviation to llie left, llie component M has the same direction as before, the component A' the 0])posile. It follows that llie bridge completes two vibrations in the iliieclioii of the string to one vibration of liie siring itself, whereas the motion parallel to the bridge has the same period as the string. K CX.y ^]V Fig. i. The souiK vibration of of llie \iolin is produced almost e.Kcliisi\ely by the he roof; the siring by itself imparts but a very small aiiKMiiil of energy to the air directly. If we suppose that the sound given liy llie string direclly may be neglected in comparison to the luucli ^1 longer sound which is due to the roof, and that the effect of llic parallel iiiolion of the bridge ma_y also be neglected as against the much greater elfect of ilie transverse motion, all the notes of the violin should be an octave liiglier than the pitch of the siring, assnming Ihal llie strings deviate on both sides of llic position of eipiilibriiiiii. The correclness of this <-oiiclusion however did not ^eenl lo us very |H-obable: jiresumably if real, tins striking fad would lia\e been observed and commnnicaU'il by pre\ ions oiiservcrs. We have therefore investigated llic (jueslion ('xperimenlally by pulling a sleel siring on a violin and making il \ibrale electro- magnet ically. We look a sleel giiilar string and put il in llie posilion of llic d siring. Close lo il a small electromagnet of llie Ro.mkksii mskn type was lixed in a stand about \erli<'ally above the siring, near llie place where il is usuall\ bowed. The coil of the eleclromagnet was in ( 519 ) circuit with three accumulators and a Konig electromagnetic tuning fork {Fa^ = 682 v. s.). The fork was placed in a distant room. The tension of the string was regulated until the violiri when the string was bowed gave a note slightly lower than the fork. The fork was then started and the note of the string raised by pressing it with the linger until no beats were heard. The note given out by the violin was now unmistakably Fa,. Now if there really were a difference of an octave between the note of the violin {Fa,) and the note of the string itself, the string ought under the influence of the electromagnet to have given the note Fa,. This is however impossible: an electromagnet mag- netised by a fork Fa, can produce in a string the notes Fa, Fa,, Fa, etc. but never the note Fa,. The experiment was thus by itself sufficient to show that the note given by the violin has the same pitch as the note of the string itself, even when the excursions of the string on the two sides of its position of equilibrium are about equal. Thinking that the octave might perhaps appear, if the parallel motion of the bridge were damped down, we loaded the left foot of the bridge with our metal clamp, bu^ e\en then flie octave could not be heard. As the question seemed to us of great importance we tried to solve it in a different more direct manner by an experiment in which the sound of the string was heard by itself. On a heavy zinc-block of 80 by 40 cms and 3 y^ cms thick (Fig. 5), two metal bridges are fitted (Fig. 6) at a distance from each other of 3272 cms. An a-string 0,7 to 0,75 mm thick was tied a :2SZ Fig. 5. to a pin s, the other end being attached to a cord going over a pulley and a pan weighted with 6 kilogrammes. When bowed the string sounded a note near Ui,. The friction of the string on the bridges and of the cord on the pulley enabled us to slightly alter o5 Proceedings Royal Acad. Amsterdam. Vol. Xlf. ( 520 ) Fig. 6. tlie pilch by turning the wheel of tlie pullej. In tiiis manner tlie string was accnratelj tuned to Ut^ (1023,9 v.s), so tiiat it produced no beats in a resonator Ut^. Next an a string of the same thickness was put on a violin (lig. 7). The distance of n to h was again 32,5 cms. Tiie violin wa? clamped Fig. 7. on the table with some wooden blocks; in the neck of the violin a hole O was bored, through which the string was made to pass. As the friction of the string on the usual ebony peg would have been too great, a metal peg was substituted which is represented in fig. 8. The string passes over the small metal wheel a. At p a cord was tied to the string which ran over a pulley and had a pan attached to it. The a string was now stretched by placing weights in the pan until the violin on bowing sounded ^ Ut^ accurately. It was found that a weight of 6 kilograms was required to do this, ■^'S- *^- i. e. the same as with the zinc block. That the violin and the string on the zinc block gave the same note, i.e. without a difference of an octave, was confirmed not merely by the ear but also by the aid of resonators : the resonator C/i!, responded to both notes, the resonator Ut^ did not. If the note given by the xidliii iiad been an ocla\c higher than that of the string on the zinc phile (i. c. r'/J the resonator Ut^ would not ha\e responded to the violin note. ( 521 ) We liave also detcnnined tlie note which the string gave at the above tension by calculation. For that purpose the string was cut at a and h (fig. 7) whereby its length shrnnk to 30 ems. The weight of this piece was found to be 0,15 grams. r /P^ By substituting in the formula t^\/ — where t is half the period, /; = 0,15gr, /=32,5cms, (/ = 981,2cms sec-2 , ,s'=r6000gr, 1 it follows t =: sec. 1099 According to this calculation the string would have a frequency of 1099 = 549,5 complete vibrations whereas in reality the frequency was 511,9 (^7J. These numbers agree sufficiently to show with certainty that in both cases the fundamental note of the string was heard. The com- paratively small difference can be explained by assuming that the tension of the string was not exactly 6000 grams in consequence of the friction of the string on the bridges and of the cord on the pulleys. From these experiments it appears that in the mixed sound which the violin produces the fundamental note produced by the parallel motion of the bridge and by the motion imparted to the air directly by the string is still present in sufficient intensity to give the sound the character of the fundamental as far as the pitch is concerned. ') It is indeed well known that the fundamental which determines the pitch of a composite note may be of smaller intensity than the overtones of the mixture, as Helmholtz showed to be the case with the piano. ^) We thus know^ that the sound given by a violin must be ascribed to three distinct causes : a. a vibration imparted to the air by the string. h. a vibration which the roof of the violin acquires from the parallel swing of the bridge. c. a vibration communicated to the roof by the trans\erse \ibration of the bridge. The vibration mentioned under a will be left out of account as being of little importance. 1) Compare Rayleigh, 'Theory of Sound", second ed. Vol. I p. 208 and Barton and Penzek, Phil. Mag (6) XIll p. 452. ") Tonempfindungen, p. 134—135. 35* ( 522 ) If a string is bowed tiie fiiii(iaineiilal of whicli has a period T, tlie note will be acoomiianied by hannoiiics of periods ^/..T, ^//f, y/r etc. respectively. The parallel motion of the bridge will canse a periodical change of pressure of its left foot on the roof of the violin. When the bridge moves to the left the pressure increases and vice versa. The change of pressure may be represented by the following series-. t t t t «; .■*•« 2jt - + «j dn 2jt — -, + «3 sm 2jt -— , + «, sm 2jt —— . . . -^ /i!-' /s-' /t-^ The transverse motion of the bridge will also cause a change in the pressure between the left foot and the roof. When the bridge is pulled forward the front of the left foot will exert a greater pi'essure on the roof; when the bridge moves back the pressure diminishes. This change of pressure may be represented by a series of the form t t t t b, .<!n 271 --;^, + b, shi 2jt — — '-[- b, am 2n -— - -4- b, sm 2.t ^— , . . . /■J-' 14-' 'ts^ /a-' As the foot of the bridge has only a small area com[)ared to the large surface of the violin which is set in motion, we may assume that the pressure changes which are due to the parallel and the transverse motions of the bridge respectively, occur at the same point of the roof. In order to tind the total change of pressure produced by both motions together we must therefore add the two above series. If we assume that the excursion of the roof at the point where the left foot is attached to it is proportional to the change of pressure, the sum of the two series multiplied by a constant will give us the type of motion of the roof at that point. It is well known that in general a sound becomes mellower according as the partial overtones become weaker and that the intensification of the even overtones especially renders the sound shai-per. Manj'^ instances of this are to be found in Helmholtz's work already repeat- edly quoted (p. 129—133 and p. J 51 — 152). As an illustration of the iiilluence of the overtones on a mixed sound we may also mention I he sound of a piano when octaves are played. When an octave is struck on the piano the two notes cannot easily be heard separate, as they can be e.g. with thirds. But only very slight musical training is required to hear in a musical recital that running- octaves are played: the sound is then sharper and rougher. The same holds for miming octaves on the violin. When in the above series we diminish the coefficient* a^, a^, a,' etc. while leaving the />^, b^, b„ unchanged as far as possible, the ( 523 ) fundamental and odd liarmonics are weakened more tlian the even harmonics. In accordance with the above results of Helmholtz the sound will thereby be made sharper. We have proved this in the following manner by experiment : To the bridge of a violin at the lowest possible point a metal clamp, represented half size in Figs. 9a and 9^, was attached. On the left side (i. e. on the side of the ^ string) a copper rod 3 mnis Fig. 9b. thick and 10 cms long was screwed into this clamp. At the end of this rod two ordinary binding screws were fixed, weigliing about 18 grammes each. Tiie moment of inertia of the bridge about the axis through the right foot, perpendicular to the bridge was naturally very much enlarged by these weights. The \iolin now gave a characteristic nasal sound, especially in the g and d strings ; the timbre resembling most the note of a hautboy. Still notwithstanding the great weakening of the fundamental it continued to impart to the sound the character by which the pitch of a note is dislinguished, in other words no change of an octa\e was perceptible. When in addition to the clamp shown in Fig. 9/i the bridge was loaded with two mutes fixed on top of each other and placed on the upper edge of the bridge, the original sound was approximately recovered, as now the transverse as w-ell as the parallel motion of the bridge was damped. Of course the response of the violin at this load was difficult. The two mutes were an ordinary ebony mute with a metal mute, as often used, placed on top. When rtj, rtj etc. and b^, b^ etc. are all diminished in the same proportion the form of the curve of motion will not change, only the amplitude diminishes : the intensity is weakened, but tiie timbre remains the same. If we could diminish the 6's and leave the as unchanged, the sound would become mellower, as in that case only the even upper partials would become weaker, including the first overtone which has the greatest intensity of all. ( 524 ) A mute placed on the bridge damps both motions. But from the fact that it renders the sound mellower we think we may infer that the b's are reduced by it by a higher fraction than the a's. This would mean that the transverse motion of the bridge is damped to a higher degree by putting on the mute than the paral- lel motion. 5. We have also tried to show experimentally that the bridge in its parallel motion turns principally about its right foot. For this purpose we screwed two metal rings into the clamp of fig. 9, which were placed in a iiorizontal position. The violin was fitted with a steel string, as before moved electromagnetically. While the string was moving a small leaden ball was placed alternately in the two rings; the two balls weighed 34 grms each. They were attached to a thin cord ; as nearly as possible at the same moment that one ball was lifted out, the second ball was carefully placed in tlie other ring. We expected that the sound of the violin would be perceptibly weakened as the ball on the right was removed and the left ball simultaneously put in. But we did not succeed in arriving at a trustworthy result in this manner; in the first place a rattling noise was sometimes apparent while the balls were being exchanged and in the second place the tone of the steel string was not always of the same intensity. 6. The conclusion therefore to be derived from our experiments is that the bridge of a violin performs a parallel as well as a trans- verse motion and that the timbre of the tone, given by the violin, is modified greatly when the intensity of one of the motions is altered while leaving the other motion unchanged as nearly as possible. Herewith we have at the same time given the physical explanation of the action of tiie mute and also of the infiuence which the use of too thick or too thin a bridge has on the sound of a violin. The action of the mute is commonly described by calling it "dam- ping" or "deadening" '). But if the mute caused nothing but a general damping or reducing of the bridge motion, the mute would only weaken the sound, and the same eff'ect would be obtained by bowing softly on a violin without as by bowing hard on a violin with a mute. That however is by no means the case as every one knows. Di'I/i, No\ember 1909. 1) Bahton. 'Textbook on sound", \). 419: ''The mute is a small apparatus of wood or metal which fits on the bridge, and thus deadens the sound considerably"' ( 525 ) Anatomy. — "On the slope of the Foramen magnum in Primates". By Prof L. Bolk. (2"d Paper on the Comparative Craniology of Primates). In the tlrst paper on the anatomy of the Primate-skull , the position and shifting of the occipital Foramen in Primates was treated. This paper will be devoted more especially to the consideration of the inclination of this plane. All the writers who have dealt with this subject have pointed out that these two features-, position and inclination, stand in a certain relationship to each other, in so far as the closer the Foramen lies to the occipital pole the more vertical a position does it assume, while as it gradually approaciies the middle of the cranial base the tendency is towards a horizontal position. This variation in the slope, like the shifting, has been connected with the erect gait of the human body. In the typical quadruped, where the skull more or less hangs from the spinal column, the Foramen lies at the occipital pole of the skull, and the plane is vertical ; in human beings, where the longitudinal axis of the body runs vertically, the occipital Foramen lies in the middle of the cranial base, while the plane is almost horizontal. Thus it is seen that this plane is disposed to take up a position perpendicular to the longitudinal axis of the body. Another point of view, first fully developed by Huxley, concerns the conneclion which is said to exist between the slope of the plane of the Foramen magnum and the degree of prognathism '). The more pronounced the prognathism — i. e. the longer the face-skull — the more per- pendicular would the Foi'amen magnum stand. If now a rough comparison be made of an animal's skull with a human skull, the parallelism between these two features is at once noticeable. Huxlky, however, believed he could show it even in the skulls of different races of men. From the superposition of the mediagrams of the highly prognathous skulls of an Australian and a >Jegro on the skull of a Tartar, it was seen that "the plane of the occipital Foramen forms a somewhat smaller angle with the basiscranial axis in those particular prognathous skulls than in the orthognathous". Welcker M holds a somewhat similar opinion, though he does not express it as being a connection between prognathism and the slope of the Foramen magnum, but between prognathism and the position ') T. Huxley, On some fossil remains of man. Collected Essays. VII, p. 198. 2 1 H. Welckeb, Unlersucluingen fiber Wachsthum und Ban des menschlicheii Schadels. Leipzig 1862. ( 526 ) of this opening, wiiicii, however, comes practically to tlie same tiling, if a connection between position and slope be assumed. "Biegt am Vorderschadel", he says (1. c. p. 50), "der Oberkiefer des Menschen niehr nach vorn (Prognatiiismus) so riickt zugleich am Hintersehadel das Foramen medullare melir nach riickwarts". Aeby') does not agree with Huxi.ey : "Huxley glanble die Neigung mit dem Prognathismus in Verbindung bringen zu konnen. Die Steil- heit der Stelhing sollte in gleichem Masse wie die letztere wachsen. In unseren Tabellen findet sich keine Bestatignng dieser Ansicht" (1. c. p. 17). Aeby himself sees a connection between the degree of development of the occiput and the slope of the Foramen magnum : "Die Abfliichung des Hinterhauptes fiihrt eine Erhohung des Foramen magnum im Gefolge." This opinion does not really differ in prin- ciple from Welcker's, for if the occiput be markedly tlattened the Foramen magnum will lie further back, and thus the opinions of Welcker and Aeby coincide after all with the opinion already' expressed liy D.\ubenton, that the For. magn. is the more perpen- dicular in proportion as it is pushed further backwards. The con- nection which Huxley believed he had shown was, howe\er, of another kind, and Aeby is therefore not correct in representing his opinion as being in contradiction to Huxley's. For it is not impos- sible that the slope is proportional on the one hand to the degree of prognathism, and on the other to the position. Is then the relation between position and slope of such a constancy as Topinard made it appear originally when he said ") : "qu'il suflit de raesurer I'un des deux termes par exemple I'inclinaison du trou occipital pour connaitre I'autre, c'est a dire la quantite du deplacement du trou"? This seems a priori improbable, since Topinard's method of deter- mining each of the two phenomena possesses merely a very relative degree of accuracy. Indeed Topinard himself saw this later '), and then expressed himself more cautiously : "Toutefois il n'y a pas un parallelisme I'igoureux entre les deux phenomenes." In general the above writers determined the slope of the Foramen magmim by determining the angle which was formed between the base-line adopted by them, and the line which connects basion and opisthion. The base-line in these researches connected the basion with the nasion or typlilon, and therefore ran through the skull- base. This method has been contested by Broca, and quite justly, for the size of the angle which is formed by these two lines is 1) G. Aeby, Die schadelfonncn der Menschen und Afl'en. Ll-ipzig 1867. ') P. Topinard. L'Aiilliropologie. 4me Edition. *) P. Topinard. Elements d' Anthropologic generate. ( 527 ) not dependent merely on the direction of the Foramen magnum because the direction of the base-line, i. e. one of the legs of the angle, depends ou several factors, e.g. the angles of the basis cranii, the length of the skull, the length of the clivus, the position of the nasion, etc. To avoid this difficulty Broca determined the slope of the For. magn. by an angle made by the plane of this opening with a plane which is entirely independent of the cranial base, viz. that which connects the axes of the two orbitae. He constructed his "angle orbito-occipital" '). Broca here was proceeding from the postu- late that the orbital-plane, in Primates at least, is the natural hori- zontal plane of the skull, as, in the case of normal sight, these animals look straight before Ihem and the orbitae in this circum- stance therefore will have the same direction. The correctness of this opinion will be discussed in a following paper. Rauber ") in a recently published treatise, returns to the old, disused method, takes as base-line again the nasion-basion, and even says that : "eine Beziehung der Neigung des Foramen occipitale auf eine andere Linie als auf die Basallinie fiihrt sehr leicht zu Unver- stiindlichkeiten und entbehrt zugleich der morphologischen Bedeutung". ScHWALBE, also, lately expressed as his opinion regarding the value of Aeby's base-line as follows : "So rationell auch die von Aeby gezogene Grundlinie ist, ist sie doch nicht geeignet fiber die Aus- bildung der verschiedenen Telle des Schiidelraumes Ausknnft zu geben."') There is a certain contradictoriness in this criticism. A rational base-line of a craniometrlcal system must be able to serve as basis for at least a primary division of the skull. I have already briefly stated my objections to base-lines which are drawn through the skull-base, and will come back to this subject in a following paper. Such a line may have a certain value as boundary line between the cerebral- and facial-skull, but as basis of a craniometrlcal system it is absolutely useless. HruER ') finally has determined the slope of the For. magn. in Hylobates with regard to the so-called German horizontal, a method which when the skulls to be examined cannot be halved medially Is preferable to those of other investigators. The probable error will here be less than by tlie use of the basal line and certainly likewise less than by employing the horizontal auxiliary line made use of by ') P. Broca. Sur I'angle orbilo-occipilal. Revue d'Antlaropologie 1897. -) A. Rauber. Der Schadel vou Kegel. Int. Monatsch. f. Anat, und Phys. 1906. 3) G. ScHWALBE. Kritik zu Kohlbrugge's : Morpliologische Abstaminung des Men- schen. Globus 11 Juni 1908. ») L. HuBER, Vergleichung des Hylobates und Menschenschadels. Munchen 1902. ( 528 ) LisSAUER '), running from (he protuberantia occipitalis externa to the point" where the ahi of tlie vomer is joined to the rostrum sphenoi'dalis. By the method 1 have adopted in determining tlie slope of the For. magn. I have proceeded from the base-line which was described in the first paper, and in so-doing have answered the question as to what angle is made by the plane of the occipital foramen with this line. To express this angle in all Primates always as a positive value it is not possible to measure the angle directly. For in the Primates 3 conditions occur : a. the opisthion lies higher than the basion, the For. magn. looks backwards, and the angle is therefore an acute one closed at the back ; b. basion and opisthion lie at equal distances from the base-line, the For. magn. looks downward, is parallel to the base-line, and the angle = ; c. the basion lies higlier than the opistliion, the For. magn. looks forward and the angle is an acute one closed in front. To pre\ent confusion between angles of equal size in cases a and c, a -\- or — sign could be used. I think, howexer, the variations in the inclination might be represented more simply in determining the angle made by the plane of the For. magn. with a perpendicular di-awn from the basion to the base-line. In case n this angle is always acute, in case h it is a right angle, and in case c it is an obtuse angle. Fig. 1. Mecliagram of an Ateles skull, illustrating llio method of determining the slope of llie For. magn. (Vi natural size). In Fig. 1 this method is clearly seen on the mediagram of an Ateles skull. The following table gives the results of the researches on the skulls of full-grown monkeys. 1) LissAUER. Unlersuchungen iiher die sagiltale Kriimmung des Schadels. Arch, f. Anthrop. XV Bud. Suppl. ( 529 ) Lemur. 40. Propitheciis 42. Mycetes. (18), 33. 45. 53. 59. Average 47.5. Pithecia. 54. 56. 60. 64. Av. 58.5. Hapale. 61. 61. 63. 64. 69. 72. A v. 65. Chrjsothiix. 60. 61. 63. 65. 66. 69. 70. 70. 71. 71. Av. 66.6. Cebiis. 63. 64. 64. 65. 67. 67. 68. 72. 73. 75. Av. 67.8. Ateles. 66. 67. 68. 71. 77. 79. 82. Av. 72 7. Cynocephaliis. 63. 64. 66. Av. 64,2. Inuus. 66. 68. 70. 76. 76. Av. 71.2. Macaciis j'. 68. 70. 70. 74. 79. A v. 72.2. Macaciis Q . 67. 73. 75. 78. 84. Av. 75.4. Cercopitheciis. 74. 80. 81. 82. A v. 79.2. Colobus. 64. 72. Av. 68. Semnopitheciis. 60. 6J. 61. 64. 68. Av. 62.8. Siamaiiga. 55. 56. 56. 56. 58. 61. 63. 63. 67. 68. A v. 60.2. Hylobates. 52. 60. 66. 73. 75. Av. 65.1. Chimpanzee. 64. 79. 80. Av. 74.3. Gorilla. 63. 63. 66. 70. 76. 77. 80. 80. Av. 71.8. Orang. 58. 62. 68. 70. 72. 75. 79. 80. Av. 70.3. These figures show in tlie iirst plat-e ihat llie slope of the Foramen magnum varies greatly in individual cases, a fact which is apparent by merely looking at the skulls. This individual variability is espe- cially noticeable in the large skulls such as those of the Anthropoids. And yet the general configuration of the skull is but little influenced by these great variations in the slope of the Foramen. As a .proof Fig. 2. Mycetes. (Vi.) Angle of inclinalion of the For. magn. 18°. of this, 1 have given in Figs. 2 and 3 the mediagrams of two Mycetes skulls, with slope-angles of 18° and 59° respectively. From the figures it can also be seen that a slight shortening of the Clivus is of great iiitluence on the angle of the slope. Now ( 530 ) Fig. 3. Mycetes. ('7^) Angle of inclination of the For. magn. .59°. Mycetes occupies a foremost place in the variability of the inclination as in that of the position of the Foramen magnum for i-easons fully given in the previous paper. For the other skulls, however, the same holds good. Another cause of the individual variations is the striking difference in sagittal measurement of the For. magnum especially in Anthropoids. In the Orang-outang skulls, for instance, which I used, this measurement varied from 25 to 41 mm. Nevertheless, in spite of these individual variations some remarkable features are to be detected between the dillerent primate-genera, especially if the series be compared as a whole with one another. It is noticeable tiiat Clirysothrix does not seem to occupy the place attributed to this family in the literature on this subject. Among the Plathyrliines, Cebus, and more especially Ateles, have greater angles, that is to say, in these genera tiie For. magn. lies more horizontally. In this respect the Chrysothrix is inferior even to most of the families of the Cafarrhines. On an external observation, however, the For. magn. seems in this monkey's skull to lie horizontally in consequence of the enormous development of the occiput, and the large share that the stjuama occipitalis occu|)ies in the formation of the cranial base. (See Fig. 4). Fig. 4. Mediagram of the skull of Ghiysothrix. ('/j) ( 531 ) Among the Catafrliiiie iiiuiikeys, tlie greatest angles, SC and more, oecnr among the Anthro[)oids and the genus Cercopitheeus. This genus thus, also as regards the slope of the F'oranien magnum, takes the high place which we have already awarded to it in the previous paper on account of the position. And similarly the genus Siamanga takes again the lowest place among this group of Primates. In this otherwise so highly developed monkey the Foramen magnum is inclined more vertically than in any other family of monkeys of the Old World, although it is closely followed by the genus Sem- nopithecus. A study of the skull base will afford us the opportunity of pointing out moi-e particularly what a quite distinct place the Siamanga takes in the group of Primates, as regards the general form of the cranium. In the first paper I have already mentioned that it is difficult to believe that original conditions have been here persistent. In the foregoing paper it was also pointed out that during the infantile and juvenile period the For. magn. shifts towards the occiput. It appears now that also tlie slope of the Foramen changes during growth. For in the sliill of a young ape the Foramen magnum lies more horizontnUg tJian in that of a full-grown one. The following may serve as a proof of this. Whereas in a full-grown Siamanga the angle varied between 55" and 68', I found m ajuvenile skull (mixed dentition) an angle of 70°, and in an infantile skull (complete laetal dentition) an angle of 81°. In a Chimpanzee, with a complete set of milk teeth, the For. magn. lay almost horizontally with a angle of 88°. In three infantile Orang-outang skulls 1 found angles of 78', 85°, and 86°, while finally a juvenile Gorilla skull had a angle of 87° and an infantile one even of 95°. In the case of this last skull, thus, the For. magn. looked forwards as in that of man. We shall soon see that as regards human beings also the plane of the For. magn. turns during infantile and juvenile periods in the same manner as with the Anthropoids, though I must here point out that this turning is much more pronounced in Anthropoids and Siamanga than in human beings. Thus both in tlie position and the slope of the For. Magn. the young Anthropoid agrees more with the human conditions than the full-grown one. In respect to the slope of the For. magn., man occupies a distinct place among all Primates, as in him the opening is not turned towards the back but towards the front. This fact, which has already been alluded to by Daibenton, and after him by all the writers on this subject, is illustrated by the figures below. I call to mind that ( 532 ) an angle of 90 ol)taine(l by my nietliod agrees with a position of the For. inagn. parallel to the base-line, i.e. a horizontal position. An(/le of the For. nytgn. in full-grown human skulls. Papuans: 96°, 99, 99, 99,100,101,103,107,107,108, Av. 101,9°. Negroes: 92°, 96, 07, 98, 99,100,100 101,103,106, „ 99,2°. Frisians: 86°, 89, 90, 94, 93, 99,100,103,103,103, „ 96,2°. Zeelandians: 93°, 97, 99, 100, 101, 103, 104, 105, 109, 112, „ 102,3°. Javanese: 92°, 92, 97, 99,100,100,103,103,103,105, „ 99,4°. The averages of three of the groups lie comparatively near each other, and the existence of a ditference between dolichocephalic skulls (the first three groups) and brachycephalic cannot be assumed on the ground of these figures, although the ditference between the long dolichocephalic Frisian skulls and the short strongly brachy- cephalic Zeelandian skulls is very remarkable. It is also peculiar that among the Frisian skulls there were two in which the For. magn. looked slightly backwards (angles of 86° and 89°) and one where it lay exactly horizontal. This was caused by the particularly long clivus in these objects. That the degree of development of this part of the cranial base in human beings has a great influence on the slope of the For. magn. is proved by infant skulls. On an average the For. magn. in young human skulls has without exception a more considerable inclination towards the front than in full-grown ones, as will be seen from the following figures. Angle of the For. magn. in children's sknlls. 0—1 year. 110, 110, 109, 105, 104, 103, 102,101,100, 100,92. 1—2 years. 100, 110, 110, 108, 106. 2 years. 107, 107, 106, 106, 103, 101, 95. 3 years. 110, 110, 108, 107. 4 years. 114, 109, 106, 105, 100. 5—6 years. 114, 113, 109, 107, 105, 103, 96, 96. 7 years. 108, 100, 100, 99, 98. 8—9 years. 104, 103, 101, 97. 10—11 years. 110, 104, 104, 101, 100 92. The average angle of the human full-grown skulls can from the preceding table be set at 100°. And now it is seen that of the 31 ( 533 ) skulls of rliildron iiiidei' 5 years of age oiilv 2 have a smaller angle while of the 23 skulls of children between 5 and 12 years of age this is so in 6 cases. From this it may be inferred that during infancy when, as has been shown in the 1*' paper, a shifting of the position of the occipital foramen takes place in man, also the plane of the For. magn. turns, and in the same direction as with the Anthropoids. Yet, as has been saidj this turning, like the accompanying shifting of position is more marked in Anthropoids than in human beings. We have now seen twice o\er that a shifting of the For. magn. and a change in the angle go hand in hand during the individual development. For in human beings as well as in Anthropoids the shifting backwards diminishes the angle of inclination. To what degree this relation between tliese two features exists in comparative anatomy will be apparent fi'ora the following table. The 2°<^ column gives the average of the angle, while the first column shows the average basal-index as determined in the l"' paper. I may here call to mind that the greater this index is, the further backwards does the For. magn. lie. Index ba.salis. Angle of Inclination For. magn. of the Lemur alhifrions 87 (1) 40" (1) Propithecus diad{ 3ma 80 (2) 42° (2) Mycetes 86 (3) 47.5° (3) Pithecia 74 (6) 58.5° (4) Hapale 71 (8) 65° (8) Cebus 67 (10) 67.8° (11) Ateles 64 (13) 72.7° (16) Chrysothrix 59 (18) 66.6° (10) Inuus 65 (12) 71.2° (14) Cynocephalus 65 (12) 64.2° (7) Macacus 64 (14) 73.8° (17) Cercopithecus 57 (19) 79.2° (19) Semnopithecus 74 (7) 62.8° (6) Colobus 75 (5) 68° (J 2) Siamanga 76 (4) 60.2° (5) Hylohates 71 (9) 65.1° (9) Chimpanzee 64 (15) 74.3° (18) Gorilla 61 (16) 71.8° (15) Orang 61 (17) 70.3° (12) ( 534 ) In brackets after the ligiires of both series is given tlie place number which each of the genera would take in a regular classifi- cation. A comparison of these place numbers shows at a glance in how far the position and the slope of the For. magn. go hand in hand. In general there appears to be a decided parallelism between these features in monkeys, and only in a few cases there is a fairly marked difference between position and slope. This is, for instance, the case in Chrysothrix where the angle is small in comparison to the position, and in Colobus where the reverse is the case. At the beginning of this paper mention was made of the opinion held by Huxley, viz. that the slope of the For. magn. is in proportion to the degree of prognathism. In a following communication, which will deal with the prognathism of the primate skull, tliis view- will be discussed at greater length. Physics. — "A short reply to Mr. van Laar's remarks." By Prof. Ph. Kohnstamm. (Communicated by Prof. J. D. van der Waals). In the proceedings of the preceding meeting of this Academy Mr. VAN Laar made some remarks suggested by a paper by Mr. Timmer- mans and me. Though these remarks do not call in question in anj' point the validity of our results, but exclusively deal with the question wiiether we have done sufficient justice to the share Mr. VAN Laar has had in the construction of the theory, I think that both politeness to Mr. van Laar and deference to the communicator of these remarks forbitl me to leave them unanswered. So I shall try to state as shortly as possible the reasons why I still think I have done full justice to that share. 1. Mr. van Laar writes in point a of his remarks:') "Here I must remark that I have never") represented the special case a^^=^\.^a^a^ as the general case." In wi'iting this Mr. van Laar had cei'tainly forgotten that he wrote in These Proc. Sept. 1906 p. 227 : "In the thh'd paper in These Proceedings (June 24, 1905) the equation: 1 {dT\ , 1 '^l/^fv.- 7,1/1) -1 (3) was derived... for the quite general^) case a, < "i I'-^b^", etc. And on the same page: "Now the restricting supposition ,i =r ') These Proc. XII p. 455. ') Mr. VAN Laar's italics. ( 535 ) was relinquished for the detcnnination of the double point of tiic plaitpoint line, and the quite general case ') a, > a^ b^ < b^ was considered. And on p. 228: "We can, namely, characterize all possible pans ^) of substances by the values of 6 and n, and finally it will only ') depend on these values,^) which of the three main types will appear." And on p. 230: "The calculations get, however, so exceedingly intricate that they proved practically unfeasible for the general case^) (h ^ «i *, ^ *!•" And on p. 231 : "This appears already from the fact that tlie substitution of the quite general assumption ') b-^ < b^ for the simpli- fied assumption b^^b, has made no change in the existence of a double point . . ., and that also the calculations for the limits of type III . . . may be carried out for the quite general case^) b^>b^." And on p. 232 : "The calculation proves that in the quite general case ') 6, < b^" etc. For, everywhere where the general case is spoken of here, it is the case a"ij ^ a, a.^ that is meant, and also the quotation from p. 228 is possible only, l\y an identification of the general case and this special one. 2. In point b of his remarks Mr. van Laar says in connection with our sentence that his investigations: "very onesidedly, lay the stress on the existence of open plaits, a circumstance which by no means can be considered as a resuW), as it immediately follows from the arbitrary, if not erroneous supposition') of the linear dependence of b and x": "Now I have never asserted that d'b — = would always agree with what actually happens ; again I dx- have simply assumed ') this in order to make the calculations ') possible." Yet I read on p. 231 of the cited paper: "We shall once more emphatically point out that the numeric^) results of our investigation will naturally be modified, when b is not assumed to be independent of V and T . . . but that qualitatively'^) everything will remain unchanged." And on p. 233 : "Then further increase of pressure makes the phases 1 and 2 again diverge . . . without the longitudinal plait ever closing again — as mas formerly considered possible^) — [cf. 1) The italics are mine. ) T. and K's italics. ') Mr. VAN Laar's italics. 36 Proceedinors Roval Acad, i^iisterdain. Vol. Xll. ( 536 ) inter alia van der Waals, Coiit. II p. 190 (1900)J. Only a( tempe- ratures higher than 1\ . . . there can be question of homogeneity to the highest pressures." It seems to nie that every unprejudiced reader of these lines must acknowledge that Mr. van Laar thought that he gave a new resxdt here, materially differijig from the result of a closed plait as it was thought possible by van dkr Waals, and that he cannot possibly have I'ealized when writing these lines that this divergent result was oiilv founded on his assumption — ^ 0. d.v'^ 3. As to point c, the sentence mentioned there really refers to a paper by Mr. van Laar earlier than April 1905 (viz. of January 1905). I did not know, however, until the publication of the "Remarks", (and now I only know it from these "Remarks") that Mr. van Laar has abandoned his views of this previous paper. Else we sliould, of course, not have mentioned it. 4. Witii I'egard to point d we must protect Mr. van Laar against himself. We had said : "His results are of importance particularly because they showed that under certain circumstances non-mlscibility can occur for perfectly normal substances, a fact which was generally doubted at the lime." Mr. van Laar remarks in this connection that it was by no means generally doubted up to now whether miscibility could occur for normal substances but only whether some sjjecial "abnormal" forms of non-iniscibility could occur for perfectly normal substances. I must maintain in opposition to this that both Lehfeldt and VAN DER Waals, to whom we referred I.e., had by no means a special case of non-miscibility in view, but \ery decidedly all non-miscibility. So Mr. VAN Laar's merit is decidedly greater than he will own here. On the other hand I must confess that in our endeavours to be perfectly objective to Mr. van Laar, we have really got unjust in the above cited sentence to Mr. van Laars predecessors : van der Waals and Korteweg. The above statement might lead one to think that Mr. van Laar had been the first to demonstrate the possibility of iioii-miscil)ility for noiMual substances. As Mr. v\n Laar justly remarks: lids is incorrect, and it would have been better if our sentence had run like lliis: His results are of importance particularly because he adhered to the jwssibility of non-miscibility for normal substances in a lime in which this was pretty generally doubled, and showed once more that for certain values of (?'s and /;'s, which could not a priori b^ considered as improbable, non-miscibility must really appear". If I wanted to discuss also Mr. van Laar's oilier remarks, I should ( -"^s? ) lia\e lo enter fully into the very heart of tlie matter, as I cannot assume the reader to be fully acquainted with tiie details of these investigations. But tiien I should think I abused the hospitality which this Academy so courteously extends in its publications also to non-members. So I think that the above will suffice. If Mr. van Laar should, however, wish to pursue this discussion elsewhere, I am willin"-, though not ilcsirous, to continue it. Chemistry. — "The cqailifirimn .■io/id-/u/iii(l-(/as in binary .vj.'items lohich present mi.ved cri/stals." By Dr. H. R. Kruyt. (Com- municated by Prof. P. van Romburgh.) First communication. In the Archives Neerlandaises [2] 5 (Jubilee number in honour of Prof. LoRENTz) p. 360 (1900) Prof. Bakhuis Roozeboom published an article "Sur requilibre de cristaux mixtes avec la phase vapeur" in which he described and illustrated the pt.v surface of a binary system when exclusively homogeneous mixed crystals occur as a solid phase. He treats the case of unlimited miscibility in all phases and especially for a system in which the melting point line proceeds without a maximum or a minimum. He has, moreover, limited himself to the case that the three-phase line solid-liquid-gas (i'?LG^) also occurs without a maximum or a minimum. These matters have not been further investigated theoretically ') ; there was in fact no inducement to do so, as there has been an almost entire absence of experimental research. Only two investi- gators, Sper.\nski ■) and Kuster') furnished material as to the equi- librium of mixed crystals with a gas-phase, whereas the researches of HoLLMAN^) belong to a category of more complicated phenomena. I intend to carry out a series of investigations in order to extend our knowledge of the systems showing a miscibility in the solid condition. First of all, I will accept the facts already known and, therefore will discuss at present, theoretically, the various possibilities of the progressive change of the three-phase line indicated by RoozeboOxM (I.e.) and communicate later the results of an invesiic/ation 1) The results obtained by A. Suits (Proc. (1908) XI p. 165, and Zeitschr. f. physikal. Gheni. (1909) 67, 464) do not differ from those of Roozeboom. The only paper 1 know connected with this subject is a communication of Meverhoffer : 'Ueber Keifkurven", Zeitschr. f. pliysikal. Chem. 46, 379 (1903). 2) Zeitschr. f. physikal. Chem. 46, 70 (1903) and 51, 45 (1905). 3) Ibid. 51, 222 (1905). *) Ibid. 37, 193 (1901), ( 538 ) as to the three-phase equilibria in the system /j-dichlorobenzene — /Mlibromobenzene, the same system of which, thanks to Kuster and Speranski (1.c.\ we ah'eady know a series of solid-gas equilibria. Fig. 1. Fig. 1 is a combined FT and T^r-projection : Oa and <>b are the triple points of the components. They are connected by the three- phase line. In the Tx projection this line divides into three branches which indicate, respectively, the composition of the solid (5) liquid (L) and gas {G) phases. Since the influence of the pressure on the equilibrium LS, is very trifling and as triple-point pressures are comparatively low, the branches S and L may, usually, be taken as being equal respectively to the melting-point curve and the freezing-point curve') at 1 atmosphere. In fig. 1 is assumed Po.^Fo^) which case we will call chief type 1. We will now ascertain under what conditions three con- ceivable cases might occur, namely : case a witli a maximum pressure in the three-phase line ,, /> ,, ,, minimum ,, ,, „ „ ,, ,, c without a ma.x. or min. ,, ,, ,, ,, To get an insight as to the change of the pressure with the tem- ^) A (as is customary) is the name of the component with the lovrest melting point and with a vapour pressure greater than tliat of B at the same temperature. -) In wliat follows we shall speak of these curves 'as the branches of the melting diagram." ( 539 ) perature we must first of all proceed in the direction indicated by Prof. VAN DER Waai.s') where lie treats of the three-phase equilibria of a binary compound with liquid and vapour. To the tf)u.«-surface of the liquid and vapour condition another one has to be added which shows the connection between those quantities in the homogeneous solid phase. If we consider the case occurring most frequently that the fusion takes place with an increase in volume this surface will be found between the li(iuul-vapour surfiice and the ifw-plane. As to the form of this new if'w-surface it should be observed that it will practically be a plane with descriptive lines proceeding from the tjM'-plane for ,/■ = 0, to that for x = 1. For the mixing of two solid substances to a homogeneous solid phase takes place either irlikout a change in volume or with a hardly appreciable one '). If we now wish to know which are the coexisting phases we must allow tangent planes to move over these surfaces and thus cause the ajjpearance of the derived surfaces and connodal lines '). Let us commence by considering a surface for a temperature below the triple- point temperatures of the components. The surface for the solid condition will then be situated very low, the tangent plane will rest both on tliis surface and on the vapour part of vapour-liquid sur- face. The lines aj)^ and gji^ in fig. 2 indicate the connodal lines so formed. The derived surface thus obtained will be situated lower than the derived surface which rests on the two jtarts of the vapour-liquid surface and which, there- fore, does not represent stable conditions, but the vapour equilibria of "super- cooled" liquids. The connodal lines {^\d^ and (^,/'i) proceeding therefrom are situated between the connodal lines of the solid- fig- 2. vapour equilibrium. If we pi'oceed to a higher temperature the correlated connodal lines 1) Verslagen Kon. Akad. V, p. 482, (1897). 2) Cf. Retgers, Zeitschr. f. physikal. Chem. 3, 497 (1889) and GossNER, „ „ Kristallographie 44, 417 (1908). 8) In what follows, the question whether a minimum or a maximum pressure is possible for tlie coexistence of two phases Las not been considered. All nodal lines ai'e therefore supposed to proceed in the same sense. ( 540 ) approufli each utliLM-; and also tlie stable ami iiietastable branches on the vapour part especially at the side of the component melting at the lowest temperature '). For if we approach the temperature of the triple i)oint of this component the points (\ and (/, of tig. 2 will have coincided to the point e.,(j^ in fig. 3, which is intended for the temperature of Oa (fig- !)• The two derived suj-faces intersect each other in the tf>«-plane of the component .1 ; that intersecting line is, of course, the tangent to the ij'-line for the gas-liquid condition of A and just the one which is also tangent to the if'-line of solid A (triple point A). By consulting fig. 4 it will be easily seen what happens at a temperature situated between that of the two triple Fig- 3. points. The rolling tangent plane coming from the A side will now rest first on the liquid- and vapour parts; but if a certain nodal line pq is thus reached the tangent plane will rest also on a point /■ of the surface of the solid phase. The angular points of the three-phase triangle IKir give us the composition of the three possible coexisting G, L, and S phases at that temperature. By further motion of the tangent plane a derived surface for GS equilibria is formed, whilst also a similar movement over the liquid part of the fluid surface and over the surface of the solid phase is possible in the direction of the small volumina. Hence a new system of connodal lines for LS equilibria is formed starting from r and q. Fig. 4, however, will be plainly understood without further conniieni and a discussion of the configurations at higher temperatures will also be superfluous. ') Tiie non-related connodal linos ab (solid) and vd (liquid) diverge from each oilier because as a rule the coefTicient of expansion of a substance is smaller in the solid than m the liquid state. ( 541 ) Prof. VAN DEit Waai>s (I. c. p. 490) lias also taught iis how to deduce an expression showing tiie relation between i),l and .<,■. Fi'oni the three equations Vsdp — risdt = dAI^n^ -\- .vsd (il/,fjj — i'/,fXi) Vfj/p — mdt^dM.H^ + A7,c/(i1/,f«., — i/ifXi) Vadp — riGdt=zdM^(i^ -\- xgd^M^n^ — -''^iMi) follows dp dt This gives us a qultn (jeneral expresaloii for the three-phase line in the systems described. It will, however, not be easy to arrive through it to the desired elucidations. If, for instance, we wish to dp know when — -vill be eiiual to the numerator thus becoming nought, dt & b > the question first arising is what do i^l — iiq etc. really represent. KoHNSTAMM M luxs rightly observed that such differences must not be simply called heat of condensation etc. because m and iiq do not relate to the same mixture. And the second question as to the numei'ical value of those quantities in a system to be investigated is still much more diflicult to answer. A'5 ns 1 •■«/> nL 1 I WG na 1 'VsinL—nG) f i«L{nG—ns) + -vGins—ni) xs Vs 1 *-5( Vl- Vg) + xl{ Vg- Vs) +*-g( Vs- Vl) XL Vl 1 XG Vg 1 In order to get a tirst insight into these systems, I have taken another course though of less general applicability. We will see how the pressure changes in regard to the triple-point pressure of ^, when the liquid phase lias the composition xl assuming that .r/. has a very small value, in other words that but a very small quantity of 5 has been added to A. The temperature 'I\ at which that liquid is in equilibrium with a solid phase, the composition of which is xs, is found from Roth- mund's formula ') for very dilute mixtures : RT,- T,= T, + '-{xs 7 *•/.) (1) 1) Proc Kon. Akad. IX p. 647 (1907). 2) Zeitschr. !'. physikal. Ghem. 24, 710 (1897). ( 542 ) in wliicli 7", is tlic temperature of (lie triple point Oa- Tlio vapour pressure P, at the tempeiature T^ is tlie sum of the partial pressures of the components pA and p^j: A = PA + PB for which we may write P^ = {\-xl)Pt, + pb (2) if Pr., represents (lie vapour pressure of liquid 4 at that temperature. If now we call Pr, the vapour pressure of A at its triple point and use van der Waals' well known formula for the saturated vapour pressure we may write Pt. ' T, 13y subtraction we get ; Pl\ ^ 2', lPT,=f'^- + lPT, 1 If now we substitute the value found in (1) for 2\ we obtain thus writing (2) in this form : R2\ f — (■«s— *'X) P.^ = {l-^[)PT,e 9 ^pj, , . , (3) If now case la (maximum pressure) is to occur, the three-phase line must rise from Oa to higher values of P and therefore P, ^ Pq^. The chance of seeing this case realised in a certain system, therefore depends on J'„ having as great as possible a value in regard to Py, and relation (3) shows us when this will be the case. For the first terra — and 5 xs — .'7. will then be characteristic. The value of. ?'^' — ,c/. is indicated by the difference in initial direction of the branches of the melting point lines for solid and liquid and this difference is determined T precisely by — '). When therefore we pay special attention to rs — xi, ') Compare van Laau, Zcitschr. f. phjsikal. Chem. 64, 257 (1908). ( 543 ) Fig. the first term of (3) will he large if ,<;.>■ — x/^ is large, that is to say when the initial directions of the branches of the melting diagram line differ greatly (Fig. 5a). The second term of (3) the partialjpressiire of the component B will as a rule be greater ') when this component gets more volatile; as in the case of this chief type I we have assumed that its tiiple point pressure is smaller than that of A we shall have the most advantageous conditions when they differ as little as possible. For the case la is, therefore, required 1. a type of melting diagram with greatly diverging branches near the ^-axis and 2. about equal triple point pressures. Case 16 (minimum pressure) makes two demands : from Oa an initial fall, but followed by a rise ; if this second demand is not fulfilled we are dealing with \c. This second demand means, of course, a small difference of the triple point pressures ; the first demand, a small 1\ is, therefore, in regard to the value of ijb in (3), opposed to the second and is, in consequence, determined altogether by the first term of (3). In order that this may be as small as possible it is, of course, required that xs — .hl shall approach as closely as possible, a demand which is complied with in a melting diagram v\ith branches almost coinciding in the initial direction. (Fig. 26). We arrive *at an identical result if we start from the triple point of B and examine the vapour pressure P,' of a liquid containing a little of A, when that liquid can also coexist with a solid phase. In this case the relations (1), (2), and (3) become: ;— i-vs — -vi). . . . T' = T\ P'i =PA-\- xlP't., ■tx) + PA (Ibis) (36i») P'. = P'l\ which will be readily understood on considering that the accentuated signs have the same significance for B as the non-accentuated ones had above for A. In the case of lb the three-phase line must descend from B, there- fore P\_<^ P' T\- Now first of all pA should be at a minimum 1) Apart, therefore, from special differences in the critical quantities and of special influences of the components on each other. ( •'^'44 ) vvliicl), oil (lie same supposition as above, again demands about equal triple-point pressures for A and B ; secondly, the exponent of e with a negative sign siiould be as large as possible, which requires widely diverging branches in the melting diagram at the side of the component B. These demands put from two sides are brought into agreement by a conclusion of van Laar (Ioc. cit. p. 265) that closely adjacent branches in the melting diagram at the side of the one component cannot meet a similar contiguration at the side of the other.') If this were possible, the occurrence of a maximum and a minimum in one three-phase line would be quite possible. In the case of lb we therefore, require: 1. Melting diagram with branches nearly coinciding at the side of the J-axis and 2. about equal triple-point pressures. Case Ic finally occurs as an intermediate case between the two previous extreme cases. Of course, the line Oa Ob may be concave or convex in regard to the temperature axis; this depends on whether the conditions for hi or lb have been partially fulfdled. Let us call these cases lc\ and Ic^ respectively. For definite forms of the melting diagram points of inflection may probably occur, but our mode of treatment is inadequate for their investigation. A single remark may be made as to the chance of obserxing a fall of the three-phase line starting from Oa ■ As stated, the follow- ing condition is required : li2\ f ^(■^•.-•^i) If now we imagine the most favourable circumstance, in which p^ may be neglected (because the components differ, for instance, very much in their melting temperature) the factor (1 — ,v^) will cause a decrease and the factor e 1 an increase in the value of the first member in regard to that of the second one. For 1 — x j^ is always <[ 1 ; the other factor is > 1 and only in the case of Xg =1 Xj it is ccjual to I : in that case a fall may be expected, but as soon as x^ and /■/ differ in value the enlarging factor appears and the said difference occurs therein exponential bj. The enlarging influence will, therefore, very soon exceed the other, so that the chance for realising the case Ic will be diminished and that for lb will be reduced to a minimum. 1) At least when we make tlic same suppositions as iu the toolnote on p. 543, ( 545 ) Fig. 6. Let lis now consider a second category- of possibilities, namely Po,<C J^o,. which case we will call ciiief type II. We again distinguish three possiliiiities, viz. II. niaxinnnn jiressnre in tiie three-phase line h. minimum ,, ,, „ „ „ c. no max. or min. ,, „ ,, „ „ It will be snpertluons to repeal the iirevious arguments when we examine the initial directions in the equations (3) and (3A/.s). The conclusions arrived at are that we require for: Case Ila: a melting diagram with closely joined branches at the side of the component B, and but slightly diiiering ti'ipie-point pressures. Case \\b : a melting diagram with closely joined branches at the side of the component A, and but slightly differing triple-point pressures. Case lie will be again the intermediate case between the two previous ones; a concave (IIc'i) and a convex (Ilcj course will again be possible. In a future paper, 1 hope to communicate the results of an expe- rimental investigation of the system /^-dichlorobenzene — p-dibromo- benzene which has been going on already for a considerable time. November 1909. Utrecht, van 't HoFF-laboratory. ERRATA. p. 438 line 16 from the top: for 1000 read 10000. (January 26, 1910). KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN TE AMSTERDAM. PROCEKDINGS OF THE MEETINGS of Saturday January 29 and February 26, 1910. (Translated from: Verslag van de gewone vergadering der Wis- en Natuiirkuodigc Afdeeling van Zaterdag 29 Januari en 26 Februari 1910, Dl. XVUl). II. A. IJuiMWKK: "Pienaarite, a. inelanocratic foyaite from Transvaal". (Communicated by I'rof. (;. A. F. Molengu.4.\fk), p. 547. (With oni; plate;. C. J. C. VAX Hoogeshuyzk: "About the formation of creatine in the ranscles at the tonus and at the development of 'igidity". (Communicated by Prof. C. A. Pekelharing), p. 5.50. E. Rkixdku.s: "Sap-raising forces in living wood". (Communicated by Prof. J. W. Moll), p. 563. K. ZiJLSTRA : "Contributions to the knowledge of the movement of water in plants". (Commu- iiieated by Prof. J. W. Moll), p. 574. P. ZEEM.iS and li. WiSAWER: "The magnetic separation of absorption lines in connexion with snnspot spectra", p. 584. (With 3 plates). H. E. .1. G. DU Bois and Kotaro Honda: "The thennumagnefic properties of elements", p. 596. r. JI. Jaeger: "Studies on Tellurium: I. The mutual behaviour of the elements sulphur and tellurium". (Communicated by Prof. P. vax Rometrgii), p. 602. J. J. van Laar: "Some remarks on Prof. Kohxstamm's reply". (Communicated by Prof. H. A. LORKKTZ), p. 618. H. J. E. Beth : "The oscillations about a position of equilibrium where a simple linear relation exists between the frequencies of the vibrations" (1st part). (Communicated by Prof. D. J Korteweg), p. 619. (With one plate). M. W. Beijerisck.: "Viseosaccharase, an enzyme which produces ilime from cane-sugar", p. 635. (With one plate). M. W. Beijerinck : "Variability in Bacillus prodigiosus", p. 640. Pierre Weiss and H. Kamerlingh Oxxes: "Researches on magnetization at very low teinpc- ratures", p. 649. (With 2 plates). Geology. — "Pienaarite, a inelanocratic foyaite from Tran.waal." By H. A. Brouwer. (Communicated by Prof. G. A. F. MOLENGRAAFF.) (Gomraunicated in the meeting of November 27, 1909). Ainoiit^ llie iieplieliiie syenites on and to the nest of the farm Leeiivvfoiitein to tlie iiorih-east of Pretoria, which show a complete series of varieties in cliemical and niineralogical composition, the "collection Molengraaff" contains a variety very rich in titanite, wliich occurs 7« ™il^ to ^^^ ^'^'sst of tiie Pienaarsriver near the boundary of the farm Zeekoegat. Macroseopically tlie rock shows red felpars to 1 cm. up in length, which Iia\'e a tabular development after (010), and smaller crystals of red nepheline, with which contrast numerous slender prisms of aegirine and bright crystals of titanite, which make up over half of the rock. 37 Proceedings Royal Acad. Amsterdam. Vol. XII. ( 548 ) Under tli'' iniorosoopo the rock is .seen to consist of felspar, nepholine, less socialite, nuich aegirine, (aegirine augite) and titanite and small quantities of apatite, tlnorine, calcite, analcime, and titanic iron ore. Tlie felspars are orthoclase and microperthite in Carlsbad twins. Nearly always nepheline and sodalite are transformed, respectively into psendomorplioses of mica and zeolites. In the crystals of nepheline, which are not entirely transformed into mica, the transformation begins along the tissures, but nearly all the crystals are entirely altered. The sodalite pseudomorphoses consist of zeolites, in which we tind distributed some sn?all flakes of mica. The aegirine is strongly pleocliroic from olive-green to yellowish green, some crystals are homogeneous, other ones contain a centre of aegirine augite, which has for the greater part very low extinction angles; they are very rich in inclusions of small crystals of titanite and apatite, and they are strongly impregnated with fluorspar. The titanite forms the well known twins after (001), in the rhombic sections the long diagonal is the twinning plane ; both individuals are polysynthetically twinned. They are pleocliroic from salmon coloured to colourless. The apatite 'is the first product of crystallization, it is even formed as small idiomorphic inclusions in the titanite, for the greater part the crystallization of the other elements was simultaneous ; the aegirine is idiomorphic in relation to felspars and felspatoids but in general the contactlines are irregular and show simultaneous crystallization. The felspar includes some idiomorphic crystals of nepheline and sodalite, mainly it is the latest product of crystallization. Probably in pneuma- tolytical way, fluorine, calcite, and analcmie crystallized in the remaining cavities. It is evident how much the mineralogical composition of this rock differs from that of the normal types of nepheline syenite by its high content of aegirine and titanite. A. Lacroix ') guxe the name of covite to the mesocratic form of this group and of teralite to the melanocratic form; as the type of covite he considers the rock of Magnet Cove in Arkansas, described by Washington and as type of teralite the alkali felspai'-nepheline rocks from the Crazy Moun- tains in Montana. The chemical composition of the rock here described is shown in I of the following table (analysed by F. Pisani) ; it is compared with the analyses of .some covites and teralites. •) Materiaux poui- la Mineralogie cfe Madagascar. Extr. mmv. Arch, da Museum. 4e serie, Tome 1, pag. 184. H. A. BROUWER. 'Pienaarite, a melanocratic foyaito from Transvaal." Explanation of Figure. (X 30) A part of a large individual of felspar shows polkilitic relation to aegirine, titanite, nepheline (at the top to the left) and sodalite (at the lower edge, in the middle and to the right). The aegirine contains numerous idioniorphic inclusions of titanite, apatite and fluorspar Proceedings Royal Acad. Amsterdam. Vol. XII. ( 549 ) I II III IV V VI SiO. 49/20 l'J.7ii 51 . 10 47.67 44.65 47.85 TiO, 7.1:! 1.33 1.38 - 0.95 — AIsO, 9.i>:! 18.45 21.10 18.22 13.87 13.24 FeaOj 7.73 3.39 90 3.65 6.06 2.74 FeO 3.24 4.32 5.58 3.85 2.94 2.65 MnO — — — 0.28 0.17 — CaO li.Vi 7.01 5.35 8.03 9.57 14.36 MgO 1.35 2.32 2.81 6.35 5.15 5.68 Na.O C.20 5.33 6.35 4.93 5.67 3.72 K2O 4.96 4.95 4.21 3.82 4.49 5.25 PPs 0.06 0.40 - 2.97 2.10 2.74 HjO 2.20 1 34 0.87 1.50 2.42 Som 99.85 99.44 99.65 100.15 99.93 iOO.65 I. Pienaarite. Leeuwfontein (320) Pretoria. Transvaal. II. Covite. Magnet Cove. Arkansas cf. H. S. Washington. Journ. of Geo!. IX. 614. 1901. III. (V)vite. Nosy Koniba. ff. A. Lacuoix Alat. JMineral. Madagascar Extr. Noiiv. Arch, dn Museum 4e Ser. I. 32. IV. Teralite. Crazy Mountains Bull. U. S. Geol. Surv. no. 150. V. Teralite. .. „ „ „ „ „ „ „ „ „ V\. Teralite. (nepheiine pyroxene malignite) of. A. C. Lawson Bull. Dep. of Geo!. lJiii\ . of Califoi'uia I. 337. 189G. We see how the rock, iiere described, ditVers in mineralogical and chemical composition from the melanocratic nepheiine syenites, which are hitherto known, its characteristic features are the lai'ge amount of Fe.,03, TiO,, and CaO (abundance of aegirine, aegirine-augite and titanite) and its low content of lime (diminuation of the felspars and felspatoids). Prof. Moi,.KNGRAAb'F [)roposed to me the name Pienaarite, aftei- the Pienaarsriver because the locality, where he collected this rock, is situated in a region between a tributary of the Pienaarsriver called iMundtspruit, and the above river itself. 37* ( 550 ) Physiology. — Prof. PKKi<;i,iiAi!ii\G offers a (.•(iiiininnicalion, also in the iiaino of Dr. C. J. C. van Hoogknhiyze : "Ahout tlu' Jor- iiittliun of creatine in the nniscles at tlw tvnvs n/id nt tin' development of ryklity." (Commuiiicatfd in the meeting of December 24, 1909). On a preceding- occasion') in this Academy I liave made a com- munication concerning an investigation by J\Ir. van Hoogenhuyzp: and Mr. Verpi.oegh about the excretion of creatinine in man, from which it appeared that the excretion of this substance in a snfficientiy nourished pei-son was not increased by muscular labour. Since that time the investigation, which promised new results, now that a good method for the determination of the creatinine had been furnished by Folin, has been continued by Van Hoogenhuyze and Verploegh and by several others. From these investigations has arisen the opinion that, through the consumption of protein in the tissues of vertebrate animals, creatine is formed, and that this matter is partly decomposed under oxidation, partly, particularly by the liver, changed into the anhy- dride, creatinine, that further the creatinine thus formed is for the greater part removed from the body through the kidneys ""). If this opinion is correct, one cannot but assume that tiie creatinine excreted by the kidneys, originates for the greater part from the creatine of the muscles, not only because the muscles ai-e richer in creatine than other organs, but also because it is especially tiie mus- cles which contain so considerable a part of the proteins which the body contains. This supposition did not seem unacceptable, notwith- standing it has been found that the excretion of creatinine is not increased by muscular labour. Observations were made which pointed to a connection between the excretion of creatinine and another phenomenon, which has to be distinguislied from the muscular con- traction in a narrower sense, the muscular tonus. Van Hoogenhuyze and Veupeoegh found the excretion during the night to be smaller than in the daytime; likewise did they find remarkably lillie ci-eati- nine in the urine of old men and of patients who had a number of muscles paralyzed, whereas in case of fever the urine appc^ared to contain moi-e creatinine than usual. That roalh the tonic shortening of the muscle is brought about in another way than in case of rapid contraction, which has been examined so much more, and that, when a single stinnilus is followed >) Proceedings of the meeting of oO Sept. 1905'. ^ See: Zenliallhl. f. d. ges. I'hysiol. und. r;dliol. d Stoffwechsels. 1900. No. 8. ( 551 ) bj a slow contraction, tlie two ways collaborate, is rendered very probable. Already more liiau 20 years ago Grutzner ') made the supposition that tlie long continued contraction was brought about by anotiier kind of muscular fibres than those which cause a more rapid con- traction. In the first case fibres of the type of the red, in the second fibres of the type of the white muscles, were supposed to be brought into play. Afterwards others, more particularly Bottazzi ^), have defended the theory that the double contraction is caused by two different component parts of the same muscular fibre, the rapid l)y the double-refracting fibrils, the slow ones by the sarcoplasma. Mosso') had objections against this theory and drew attention to the double innervation of the muscular fibres, not long ago once more made clear in this Academy by Boeke *). Meanwhile, whatever the opinion may be, at any rate there is some reason to assume that the contractions of two kinds must be accompanied bj- a chemical action of two kinds. Now that in the usual muscular labour, which is principally based on rapid and tetanic contractions, no increase in the consumption of protein and in the excretion of creatinine was found, it might be asked, whether perhaps in the tonic contraction formation of creatine in the muscles could be proved. That under certain definite circumstances the muscles at their contraction yield more creatine to the blood than otherwise, has already been found by Weber "^j, with respect to the heart treated after Langendorff's method and beating in Ringer's solution. He also found a considerable increase in the excretion of creatine with a dog, after violent cramps had been caused in the animal with cinchonine. Not only in the last case, in which the animal for' an hour "was in violent tonic and clonic cramps', but also with respect to the heart taken from the body, it may be assumed that tonus has played a part. However, in order to draw more certain inferences, we have examined the quantity of creatine in muscles, under circumstances which, as much as possible, allowed to judge about the influence, either of tonic, or of rapid contractions. 1) Pfluger's Archiv. Bd. XLl, S. 280. 2) Journ. of Pliysiol. Vol. XXI, p. 1, Arch. f. Physiol. 1901, S. 377, Arch. Ital. de Biol. T. XLII, p. 1G9. 3) Arch. Ital. de Biol. T. XLI, p. 183. ■*) Proceedings of the Meeting of 23 April 1909. "J Arch. f. exp. Path, und Pharm. Bd. LVIII, S. 93. ( 552 ) The determinations always took )>lat'e in the same way. The minced muscles were for some iiours at a stretch boiled in 0.1 7o tlCl, in consequence of which the tissue breaks altogether and all the creatine passes into the liquid. By eva])oratioii the extract after being freed from protein, was concentrated and then witii then double volnme of normal HCi heated to 115° C. in the autoclave for half an hour, by which the creatine is completely changed into creatuiine. Tlien the determi- nation took place, according to the method of Foi,in, with the colori- meter formerly shown in the meeting of this Academy. First of all we have some observations to mention about decrease of the quantity of creatine in muscles, of which the tonus, in conse- quence of section of the nerve, was eliminated. In the outset we came to very irregular results when examining the muscles of the hinrllegs of rabbits and of a young dog, after a one-sided cutting of the nervLis ischiadicus. Now less creatine was found in the muscles of tiie paralyzed leg, now more creatine in those of the paralyzed side. The cause of this irregularity appeared to be that we did not compare exactly corresponding muscles. Especially in the rabbit the ditference in quantity of creatine in white and red muscles is rather considerable. With 10 rabbits we found in the gastrocnemius on an average 4.463, in rod muscles (soleus, semitendinosus and semimem- branosus examined together) 2.925 mgr. creatinine in 1 gru). of the muscle. When this source of error was avoided, the intluence of section of the ischiadicus became clear. With 3 rabbits we have, taking into consideration what has been said above, repeated the experiment and three days after the section of the nervus ischiadicus, we have examined the gastrocnemius of the paralyzed and of tlie not paralyzed leg. We found : IjOSS after paralyzing. 0.471 mgr. cr. p. gr. of the muscle 0.159 „ „ „ „ „ „ 0.970 „ „ „ „ „ „ Although the differences found lie without any doid)t beyond the limits of the errors of observation, yet we have not contiiuied these exi»eriments, because too little can with certainty be conchuled from them coiiccniing the influence of the tonus. It is true, the muscles of the leg the ischiadicus of which has been cut thi'ough, are distin- guished from those of the other side by the loss of tonus, but there are also other differences, which are perhaps of importance, as Webeh has already observed, who also found the quantity of creatine in Not paralyzed. Paralyzed I 4.703 4.232 11 4.448 4.289 III 4.983 4.013 ( 553 ) the muscles of the paralyzed leg smaller than in those of tiie normal leg in a dog after section of the ischiailicus. That the normal leg continues performing voluntary movements, is no objection, there being no gi'ound to assume that then creatine is formed. But the muscles of the paralyzed leg degenerate. Although we killed the animals already three days after the section of the nerve, yet every time the paralyzed gastrocnemius appeared to be of a smaller weight than the normal one. Nothing is known about the formation and the destruction of creatine in degenerating muscles. Of no less importance seems to be the change in the circulation of the blood after section of the nerve, in consequence of which the removal of creatine from the muscles may be altered in a quite incalculable degree. On account of these objections we thought of entirely giving up the attempt lo inquire into the influence of the muscletonus in warm- blooded animals, and of being obliged to occupy ourselves only with cold-blooded vertebrates, in which without any trouble the blood- circulation can be shut out (in muscles of invertebrates no creatine has been found ; they were not fit for our purpose accordingly) when my colleague Prof. R. Magnus drew our attention to a means of bringing muscles of one half of the body of a cat into strong tonus, whilst the corresponding muscles on ihe other side, without any disturbance in the action of the centrifugal nerves and in the circu- lation of the blood, remain slack. Sherrington ') has found that, when in deeply narcotised dogs, monk-^ys, cats, rabbits or cavias, the action of the cerebral hemi- spheres is excluded by a section in the region of the hindmost corpora quadrigemina, after a short time the so-called "decerebrate rigidity" develops itself, a long continuing tonic contraction of definite muscle-groups, among which especially the extensors of the extre- mities and the retractors of the head and the neck are the chief. This state of things is dependent on impulses which arise in the periphery, and by centripetal nerves are led to the spinal cord. That is why the stiffness does not arise in those parts of which the cori-esponding dorsal roots are severed. Now Prof. Magnus had the kindness to operate upon live cats in such a way that one foreleg was brought in tonus two, three hours at a stretch, whilst the other leg remained slack. The perfectly nar- cotised animal, the narcosis being brought about first by means of ether, then by means of chloroform, after section on the left side of the hindmost roots of the four or fixe lowest cervical nerves and of the two highest thoracic nerves was decerebrated. Soon, the right 1) Journ. of Physiol. Vol. XXII, p. 319. ( 554 ) leg got into tonus, llic left one remaining slack. Somelinies, in order to strengthen t!ie tonus in the foreleg, also the' spinal cord, at about the eleventh breast-vertebra, was cut through. Wlien the tonus had lasted a few honrs, the animal was killed by suifoeation. Directly after the triceps brachii on both sides was prepared, minced and put into hydrochloric acid. In all experiments we found the muscle that had been in tonus, richer in creatine than the one that had remained slack, and that, expressed in nigr. creatinine on 1 grm. muscle, as follows : Tonus Slack Diilerence I 3.690 3.090 0.600 II 4.340 3.848 0.492 III 4.291 3.902 0.317 IV 3.806 3.185 0.621 V 3.198 2.963 0.235 It is remarkable tliat this diiference pretty well keeps pace with the difference that in the experiment of the contraction of the muscles, right and left, was observed. In experiment I and still more in IV, the stiffness on the right was very beautifully developed, in II the tonus was strong, but of a shorter duration, in III the tonus on the right was good, but also the left foreleg occasionally showed some stiffness, which also occurred in V, though in a smaller degree, whilst the stiffness developed here slowly and to not so iiigh a degree as otherwise. We think we are entitle^l to derive from these experiments that by the muscles in tonus more creatine is formed than by those which are slackened. For tiie supposition that the diiference may be attributed to an increased decomposition of creatine in the slackened nmscles, it seems that there is not a single ground to be adduced. Besides this we have made a numi)er of experiments with frogs (Rana esculenta). In the first place the influence of irritation with induction-currents on the quantity of creatine in muscles was examined. About this communications have been made by Mellanby ') and by Gr.\h.\m BiiOWN and Cathcart'-). By direct irritation Meli.anby brought the muscles in tetanus and then he found so sligiit an increase of the quantity of creatine that he came to the result: "lliat tlio pcrfor- 1) Jouin. of Pliysiol. Vol. XXXVI, p. 447. -I Bio-Chemical Journ. Vol. IV, p. 420. ( 555 ) mance of mu.sciilar work leaves creatine unafTected". Brown and C.VTHc.ART sliinulated the niuseles by means of the nerve and found a somewhat more considerable increase, of 7% a l^Vo, in four experiments, at which the circulation of the blood had been excluded. If the circulation was intact, they found a little diminution, not only \vith frogs, but also with rabbits, where it ought, however, to be taken into consideration that, in consequence of the stimulus, the muscle was more amply pi'ovided with blood, so thai an exact comparison with resting muscles is scarcely possible. We have made experiments with frogs, in three diiferent ways, always excluding the current of blood. First, after destroying brain and spinal cord and after section of the heart, the nervus ischiadicus on one side, laid bare high in the thigh, was cut through and stimu- lated with a series of rapidly succeeding induction-strokes. Each stimu- lation lasted Ys minute, after which /^ minute of rest was afforded during about one hour. In the second place the experiment was for the rest . made in the same way, but the nerve was, for about half an hour, with the help of Engelmann's rhythmic polyrheotonie, stimu- lated 24 times per minute, alternately with a closing and an opening induction-stroke. At last two more experiments were made thus : the frog was cut through transversely in the lumbar region, after which the skin of the hind part was taken away. Now this was put astride on the partition of two basins of celluloid, standing against each other and filled wiili Ringer's solution, so that each leg was immersed into the liquid to about half way up the thigh. Then the ischiadicus was on one side, from the pelvis, stimulated for half an hour, 24 times a minute, with single closing and opening induction-strokes. The quantity of creatine, expressed in mgi-. creatinine per 1 grm. of muscle, was foujid as follows : Stimulated Rest Difference Stimulated Rest Difference I 3.490 3.418 + 0.072 1 I 3.366 3.386 — 0.020 II 3.537 3.457 + 0.080 A II 3.616 3.683 — 0.067 III 3.G29 3.550 -f 0.079 f III 3.796 3.856 — 0.060 IV 3.567 3.560 + 0.007 I 3.203 3.230 — 0.027 II 3.585 3.593 — 0.008 c The differences are slight and do not fall, or fall scarcely, beyond the boundaries of the inevitable eri-ors of observation. Moreover the difference is now in favour of ihe stimulated, now of the not-stimulated muscles. Even if one wishes to attach some importance to the greatest ( 556 ) diffei'onces Ibiiiid in lliese experiments, one need not yet derive from them that the muscle during the rapid contraction forms or loses ci'catine. For on the one liand the decomposition of creatine in the muscle is no doubt subject to quite unknown, hut certainly vai'ying influences, whilst on tiie otlier hand long continued irritating can also give rise to some lasting contraction, tonus. That in the frog during the muscle-tonus in contradistinction to the rapid contractions, the (piantitv of creatine in the muscles increases, whilst, in default of tonus, it decreases, appears from the following experiments. In the first place the influence of the elimination of the tonus, by excluding the innervation, was examined, first while the current of blood was stopped, then with undisturbed circulation of the blood. In the former case the ischiadicus was on one side cut through and then an both sides the root of the thigh was so well tied up with an elastic ligature tiiat the blood in the vessels of the webs stood still, in such a way that the ligature ran below the ischiadicus not cut through. In the latter case the ischiadicus was simply cut through. Three days after the section of the nerve the animals were killed and the muscles of the hindlegs examined. intact cut difference I 2.204 2.137 0.067 11 2. 678 2.282 0.396 111 2.990 2.790 0.200 IV 2.987 2.887 O.I 00 V 2.726 2.551 0.175 VI 2.833 2.688 0.145 Intact cut difference I 3.784 3.342 0.442 II 4.000 3.653 0.347 III 4.146 3.688 0.458 IV 3.490 3.192 0.298 V 3.434 3.131 0.303 VI 3.685 3.334 0.351 VII 3.157 2.900 0.257 Without any exception, therefore, there was found in the muscles that had lost the tonus for three days, less creatine than in the unhurt leg. If the cui-rent of blood was stopped, the difference was smaller than when it went on undisturbed. Yet in the tied up legs the quantity of creatine was, also on the side of the unhurt nerve, smaller than it is usually found in the frog. It is therefore probable that on both sides, after the current of blood had been stopped, creatine was decomposed. With respect to the experiments with undisturbed circulation of the blood the same objection may be raised which has been made mention of concerning similar experi- ments with the rabbit, viz. that it is unknown how far, perhaps ( 557 ) by a change of the ciuTeiit of blood, llic removal of creatine from the muscles is altered. It is, however, not to be assumed that the differences observed should be attributed to this. With much more certainty, ho\ve\er, the connection between tonic conti-action and formation of creatine in the frog may, in our opinion, be derived tVom another series of experiments in which, with exclusion of the current of blood, muscle-tonus was caused. We have exposed the muscles to the action of substances of quite different nature, which, however, resemble each other in the fact that they cause tonus, viz. : veratrine, nicotine, calciumchloride rhodan-natrium and coffeine. It is especially Bott^vzzi who has pointed out the tonicizing action of veratrine') If the gastrocnemius of a frog is immersed in Ringer's solution containing 1 : 20000, or even less, veratrine, stimulation of the ischiadicus with a single induction-stroke causes a contraction which lasts much longer than with a muscle immersed in pure Ringer's solution. To the rapid a slow contraction is added. In order to examine the influence on the formation of creatine the hind legs of a frog were brought into the above mentioned cellu- loid basins, of which one was fdled with Ringer's solution, the other with the same solution, in which a definite quantity of veratrine had been dissolved. Now the two ischiadici were, from the pelvis, during half an hour, stimulated 24 times per minute alternately with a closing and an opening induction-stroke. After that the muscles were prepared off, and with the liquid in which they had been immersed, treated in the usual way for the determination of creatine. The result was : nger's solution Veratrine. (1 : 40000) Difference, 1 3.442 3.561 (1 : 20000) 0.119 II 3.189 3.389 (1 : 5000) 0.200 III 3.056 3.430 0.374 IV 3.250 3.670 (1 : 1000) 0.420 V 3.029 3.429 0.400 In III, IV and V the legs immersed in veratrine ceased lo con- tract before the half hour had elapsed. Besides these legs showed in the end some stiffness in these experiments. 1) loc. cit. ( 558 ) The faculty of nicotine to cause tonic contraction of muscles, has been amply studied by Lanui.ky in his experiments on receptive substances'). The forelegs of tlic frog, the flexors of which are so easy to bring in tonus, also by dripping with nicotine, would have been very fit for our purpose, if not the mass of the available muscles was so small that for a single determination of creatine a large number of frogs would be necessary. The experiments of Langley, however, made us surmise that also the hindlcgs would be tit for our purpose, which surmise was corroborated by the result. First an experiment was made as folloAvs : After destroying brain and spinal cord 1 CO of a J "/^-solution of nicotine in Ringer's solution was injected into the abdomen, after which the tonic contraction of the forelegs soon made itself manifest. Half an hour after the injection the current of blood, by section of the heart, was brought to a standstill. Now the left ischiadicus was laid bare in the upper part of the thigh, cut through and for half an honr stimulated 24 times per minute with induction-strokes. Till the end the muscles reacted upon the stimulation of the nerve and at last a slight rigour was to be observed. The stimulated muscles produced 3.491 mgr. of creatinine per grm. of muscle, the non-stimulated 3.090 mgr. Difference 0.401. Then tiie experiments were made in tlie same way as with vei'a- trine, with the following result : Nicotine. Difference. (J : : 100) 3.7tJ6 0.480 3.492 0.402 (1 : : 200) 3.538 0.262 (1 : 100) 3.401 0.364 At the end of the experiment the leg immersed in nicotine did not visibly contract any more and each time this leg was some- what stiff. Through an examination of the action of kalium- and calcium-salts also Guenthek'^) has come to the result that the muscular fibre possesses contractile substances of two kinds, one of which is made more susceptible to stimulation by K, the other by Ca. RiNGEu's sol I 3.286 II 3.090 III 3.276 IV 3.037 1) Journ. of Physiol. Vol. XXXllI, p. 374, Vol. XXXVI, p. 317, Vol. XXXVIl, p. 165, p. 285, Vol. XXXIX, p. 235. Proc. Royal Soc. B. Vol. LXXVllf, p. 170 ^) Amer. Journ. of Physiol. Vol. XIV, p. 73. 559 ) "The first contractile suh^ilance of tlie sartoi-iiis", ho says, "responds qnickly with a contraction wlien subjected to a 1 percent solution of potassium chloride. Calcium chloride in a 1 percent solution produces no contraction of the first contractile element of the sar- torius, gives rise to a slow contraction of the second contractile element, and produces quite a vigorous contraction of heart muscle." We had, therefore, to expect that excitation of muscles immersed in calcinmchioride would make the quantity of creatine increase. Indeed this appeared to be the case. One basin was now filled with Ringer's solution, the other with a solution of CaCl, isotonic with it. For the rest the experiments were made in quite the same way as the preceding one. The I'esults are the following : Ringer's sol. CaCl., 0.72 7„ Difference I 3.177 3.820 0.643 II 3.193 3.703 0.510 III 3.340 3.894 0.554 IV 3.040 3.647 0.607 V 3.156 3.501 0.345 111 the first four experiments the contraction of the muscles immersed in Ca CU left off before the half hour was past and these muscles showed distinct stiffness. In V the contractions of the muscles immersed in CaCl, were at the end of the experiment clearly to be observed and stiffness was not to be perceived. To the examination of the action of rhodane and coffeine we were led by a communication of von Furth and Schwarz '), from which it appeared that these substances, like e.g. veratrine, are able to con- siderably increase the labouring faculty of the muscles. The supposition that also here the tonus, the "innere Unterstiitzung", of which GrDtzner spoke, was playing a part, we found corroborated. The two gastrocnemii of the same frog were hung, one in a vessel with Ringer's solution, the other in a vessel with the same liquid in which some citras cotfeini was dissolved, or of which the sodium chloride had been replaced by rhodan-natrium. After both muscles, under the same tension, had been fastened to registrating levers, they were now and then, by means of the nervi ischiadici laid upon a single couple of electrodes, excited with an induction-stroke. Now wiiile the muscle immersed in Ringer's solution after each contraction returned to its former lengtii, or was even somewhat lengthened, 1) Pfluger's Archiv, GXXIX, S. 525. ( .500 ) the muscle brougiil in coiilacf witli coffeine or witli rliodaiie, whilst it continued reacting well upon the excitalion, became gradually not inconsiderably shorter. The influence upon the quanlit}' of creatine was examined iji the usual way. The following figures were found: Ringers' sol. NaCNS 0.614 V„ Difference I 2.822 3.098 0.276 II 3.106 3.354 0.248 III 3.051 3.537 0.486 IV 3.129 3.459 0.330 V 2.916 3.146 0.230 Towards the eiid of the experiment the muscles did not contract any more. Stiffening was not to be perceived. Ringer's lic(. Citr. Coff. Difference I 3.017 3.432 1 :100 0.415 II 3.055 3.623 I ; ;200 0.568 III 3.090 3.422 1 :400 0.332 IV 3.194 3.551 1 :400 0.357 V 3.316 3.519 1 :800 0.203 The leg brought in contact wilh coffeine was in I quite stiff after 10 minutes, in II after a quarter of an hour, the contractions leaving off. In III, IV and V the contractions remained visible till the end of the experiment. Also in those cases the rigour was clear, though in V not so strongly as in III and IV. In all cases, without any exception, therefore, the quantity of creatine was found to he increased in the muscles that had been in tonus. The difference, except with coffeine, may even be eslimatod somewhat higher llian the figures given, because, leaving the men- tioned exception oul of consideration, the tonus appeared (o be accompanied by a slight increase of the quantify of water. Fhis difference is, how^evcr, so insignificant that it need not be taken info consideration. Increase of the qiuintify of creatine was found only then when the muscles had been brought info tonus by excitation. Immersion of the legs, during iialf an hour, in the solutions, without excitation, had no influence upon the quantify of creatijie. The following expe- riments were made in the usual way, only with this difference, that the nerves were not excited. ( 5fil ) Rixgkr's sol. Veratr. J : 5000 Difference 3.954 3.954 Nicotine 1 : 100 3.544 3.510 0.034 CaCl, 0.727„ 3.399 3.394 0.005 NaNCS 3,340 3.336 0.004 Coffeine 1 : 100 3.295 3.327 0.032 In none of these cases was anything to be perceived of stiffness of the muscles. Therefore our results are perfectly in keeping with the opinion that the muscular fibre, when reacting upon a stimulus with a ra|)id contraction, works in quite another way than when it is brought in tonic contraction. In tlie first case it consumes non-nitrogenous matter, in the second it forms creatine, consequently consumes protein. Against the supposition of Grutzner that each of these actions should belong to a special icind of muscular fibres, tells among others our experience, that, witli the rabbit, it is just the red muscles, which are distinguished for slowness in contraction, that contain less creatine than the white ones. Though the opinion of Bottazzi that muscular fibres show the phenomenon of tonus the more, as they are richer in sarcoplasma, as has already been pointed out by Mosso, is not quite in keeping with the observations, it may, however, especially after Engelmann's important researclies, be assumed that the rapid contraction is performed by the anisotrope elements, accordingly by the muscular fibrils. The seat of the tonus must therefore be sought in the sarcoplasma or perhaps in the parts of the fibrils between which the anisotrope elements find a place. In a further investigation into the two dilferent kinds of contraction of the muscular fibres it will certainly be of importance to keep the attention also directed to the double innervation again demonstrated by Roeke. As to the starting-point of our investigation we think we are entitled to give an affirmative answer to the question whether the formation of creatine, and consequent!}' the consumption of protein in the body, is largely influenced by the tonus of the muscles. Already many years ago it was proved by PelOger ') of how great an importance the muscular tonus is for the production of heat. If our opinion is correct, it also follows from this that limitation in the supply of protein witli the food, which is at the present day aimed 1) Pfluger's Aichiv., Bd. XVtII, S. 247. ( 5fi2 ) at li,y iiiaiiv, lias its dangerous side. Meclianifal lalioiir llio iimscles can perform al tlie cost of food free from nitrogen ; iiowever to be of service to the organism, also in other respects, by means of the tonus, they want protein. The opinion lias often been pronounced that the stiffening of the muscles after death should be considered as a last contraction of the muscles. Especially Hermann has indicated the agreement between the changes the muscle undergoes at coagulation and those which are obsei'ved in the contraction. In the above mentioned paper of VoN FfRTH and Schw.\rz it is proved that it is such substances espe- cially, which are capable of promoting the coagulation of the muscle- plasma, that raise the labouring-faculty of the muscles. It seems that the agreement does not refer to the rapid contraction but to the tonus. We have found an increase of tlie quantity of creatine in frog-muscles which were stiffened by immersion in water of 42° or 45° C. In four experiments the increase amounted on an average to 0.305 mgr. creatinine on 1 grm. of the muscle (min. 0.204, max. 0.460 mgr.). For the rabbit the investigation offered some difficulties, because here the decomposition of creatine, proved by Gottlieb and Stangassinger plays an important part and the so much thicker rabbit-muscle is not so rapidly coagulated as the thin muscles of the frog. When, however, the errors arising from this are avoided as much as possible, also in the majority of cases, both with the red and the white muscles of the rabbit, a distinct increase of the quantity of creatine was observed in the stiffened muscles. Also in the investigation into the spontaneous stiffening of muscles after death, the postmortem disappearance of creatine has to be taken into consideration. When, however, the muscles of one side of the body were, directly after death, put in hydrochloric acid and the corresponding muscles of the other side after three oi- four hours when tiie stilfeiiing liad been well developed, each time there was found more creatine in the coagulated muscles than in those examined in a fresh condition. In the four cases dealt with in this way, we found an uncommonly great difference in one, and in the three others on an average 0.260 mgr. of creatinine more (min. 0.124, max. 0.336 mgr.). The description in details of these and the other observations mentioned we intend to give somewhere else. From our investigation wc think we are entitled to derixr thai in the mu,sclcs of \ertebrate animals, at the heat-coagulation and the postmortem rigour as well as the tonus, a ciiem leal process lakes place which causes the origination of creatine. ( 5r,:} ) Botany. — "Sap-rdisunj fork's in liviiii/ ivood." By E. RkIiNDKKS. (Coniinunicaled by Prof. J. W. Moi-i,.). (_)f the many theories, which have been advanced in expkxiiation of the transpiration-current of trees, most are at present only of historical impor- tance in the literature. The imbibition theory of S.\chs^) ; Bohm's atmos- pheric pressure tiieory -) ; the gas pressure theory of H.artig ') ; the views of Westermaip;r ■*), who regarded the xylem parenchyma as the water conduit and considered the vessels to be reservoirs; Ewart's ') hypo- thesis that the living elements help te overcome the resistance, the cohesion theory of Askexasy "), which neglected to adopt the conti- nuity of water as a conditio sine qua non — all these have been given up. On the other hand opinion is still divided with regard to two hypotheses, the advocates of which combat the views of their respective opponents with remarkable asperity. Godlk\vski ') and his supporters defend the view that the transpiration-curi'ent cannot be explained without postulating the cooperation of the living elements of the wood ; Dixon and Joly '') on the other hand advance the proposition that the living elements have not, and cannot have, anything to do with the process. They explain the phenomenon that water ascends up to the summits of the highest trees by assuming that in these trees the water, enclosed in the narrow water conduits, hangs like a tliread from the surface of the leaf cells, where it is held by capillary or other physical forces. The thread does not bi'eak, because, as is supposed, it is nowhere in contact with air, and in these circumstances water can support a tension of 150 atmospheres. When the water evaporates in the leaves at the summit, this thread is drawn up through the tissues. The keenness with which the two parties oppose each other is best illustrated by a couple of quotations. ScHWENDENER '), an advocatc of the more physiological theory, says : "All der Vorstellung, dass die Lei)enstatigkeit der Zellen irgendwie in die Saftbewegung eingreift ist . . . . unbedingt festzuhalten. Ohne dieses Eingreifen ist die Hebung des Wassers aufHohen van 150-200 Fuss und dariiber einfach unmoglich und alle Bemiihungen, die vor- handenen Schranken mit unklaren physischen Annahmen zu durch- brechen, sind nicht viel mehr als ein Suchen nach dem Stein der Weisen". In the same year 1909 DixGiN '") writes: "The adhesion of writers to the vital hypothesis .... is so 38 Proceedmgs Royal Acad. Aiujlerdaoi. Vol. XII. ( 564 ) remarkable tliat we must devote some space to examine fullj tlie grounds for their contention". When we attempt to trace why opinions diverge so widely, the cause seems to lie principally in a different appreciation of certain experiments and in the somewhat adventurous aspect which the Dixonian explanation presents at first sight. It is necessary to become accustomed to the idea that the life of our trees hangs upon a water- thread, before we can become reconciled to it. Godlewski '') indeed required a much more adventurous hypothesis iu order to reconcile the anatomical structure of the wood with its power of pumping up water. This part of his theory has in consequence received adhesion from no one and so I wnll leave it out of discussion. In what follows below, "GoDLPiwsKi's theory" will therefore mean the view that the living wood must be regarded as the cause of the transpiration current. In order to facilitate a judgment of the state of affairs I will tabulate the most important arguments of the two parties side by side and will then discuss them in pairs. From this table I omit everything relating to the question whether the cohesion of water is sufficiently great to account for the work which DixoN and Joi,y attribute to it. I will assume, if I may put it thus, that there is no technical objection to their theory and I think this assumption may be made with safety. Godlewski c. s. Dixon and Joi.y. 1(7. There is not sufficient con- ih. There is no reason for tinuity in the water columns of doubting the continuity of the the wood to admit cohesion as an water columns"), explanation '^). 2a. The remaining available 2h. Strasrurger's experiments pliysical forces are insuflicient to in which the water ascended in raise the water more than 14 poisoned trees, prove the con- metres '■*). trary "). The cohesion theory has at its disposal forces which would be able to provide a tree of 200 metres and more with water ^''). 8r7. URspRrNo's experiments, with '.ih. In Ursprung's experiments branches wiiicii had been kiikMl for the conduits become blocked and part of their length, after which the the leaves were poisoned because ( 565 ) leaves faded, prove tliut dead wood they got a decoction of wood for cannot transport enongli watei' to llieir drink '"). balance the transpiration '^). 4'/. Tiie struclme of the wood is 4A. "The verv strncture of the in favour of (iouLEwsKi's theory "'"). wood offers the strongest evidence against Godi.ewski's theory" "). Living wood offers the same resistance in either direction to the forcing through of water °°). 5a. Arguments from analogy "'). oh. Arguments from analog}' ''). Ga. The distribution of pressure 6b. The measurements of pres- in living transpiring trunks is op- sure are considered unreliable or posed to the cohesion theory "'). are left out of account. Point 1. The question of the "continuity of the water-threads" in the wood amounts to the following. The cohesion theory requires the assumption that the water in the tree forms one connected mass from the root to the leaves. Every xylem vessel in w^hich there is an air-bnl)ble has according to this theory become useless for the conduction of water, for in such a vessel the water cannot be under negative pressure ; it is at once sucked empty by the adjoining vessels. Every bubble of air therefore puts one vessel out of action. Now if it could be show-n that by far the largest proportion of vessels contain air linl)i)les, only a small percentage would remain available for the conduction of water, and perhaps here and there the required connection of tiie water would be entirely interrupted, so that there could be no question of the cooperation of cohesion. It is of course difticuit to prove the absence of air, for in the iieces.sary manipulations preparatory to the examination there is always the chance that air bubbles in some way or other get into the vessels -*). If air is found in the majority of the \essels this does not prove that it w^as already present in the living plant, for it may have penetrated during manipulation. For the further course of my argument it matters little, however, whether Dixon and Joly or wiiether their opj)onents are right on this point. I will not therefore discuss it any further. Point 2. The proposition, that physical forces alone are insuf- ficient"^) to raise water higher than J 3 — 14 metres is a very weak point in the defence of GoDi.r.wsKrs theory, for Strasburger's intoxi- 38* ( hfifi ) cation experiments have j)roved in the most striking manner, that this proposition is untenable. He fonnd that water still ascended to the highest tops of the poisoned trees, up to a height of 22 metres. The attempts of Godlkwski's supporters to maintain tiieir proposition in spite of this fact give a very unsatisfactory impression. Strasburger is attacked in vague terms ^") ; he is accused of a want of critical insight, he is reproached for not making anj- attempt at explanation : the fact itself remains. The following argument appears to be somewhat more weighty. It is said "') : "with the help of a Jamin chain atmospheric pressure may be imagined to force water up to 13 — 14 metres"; but four- teen is not twenty-two and moreover a Jamin chain can by no way explain anything in this case. It might perhaps be applied to this purpose with some chance of success, if the vessels ran through continuously from the root to the leaf, but certainly not in a system of vesicles like the wood, whei'e the bubbles cannot pass the par- titions, dividing up the conducting tracts, to say nothing of the multitude of other clinching objections. It is further adduced against Strasburger, that continuous liquid threads are formed when the trunk, having been sawn off, is placed in water "), but in the first place it is not clear what objection is really meant by tiiis and in the second place it is difficult to imagine how these threads are supposed to originate. The water which is sucked up cannot remove the air present, for the air is enclosed ; it is moreover saturated with air, and is more likely to give off bubbles than to absorb them, as soon as it is exposed to a lower pressure at a certain height. Sawing off the tree ^vill hardly affect its air-content except to increase it; tlie air which enters does not, however, endanger the cohesion, as it cannot ascend. Point 3. UrsprunCi's experiments'') with branches, which had been killed by steam over part of their length, in consequence of which the leaves faded, do not prove much for Godi.ewski either. The steam not only kills the living elements, but also induces other changes. For some time the vessels must conduct a decoction of wood instead of water and a blocking of the membranes or even of the lumina of the vessels may be the consequence, so that the resistance increases. The cells of the leaves are further more or less poisoned by this liquid, so that it is very doubtful whether the death of the leaves may be attributed to a want of water '"). These experiments are therefore not of much importance in decid- ing the question under consideration. ( 567 ) Point 4. The anatomical strncture of the wood is a better argu- ment for Dixon '') ^'') than for Godi.ewski, for as yet it is (jiiite impossible to imagine in what way the living elements could really exert any successful pumping action. The unidirectional resistance without which such an action can hardly be conceived, has never been observed, in spite of a careful seai-cli for it. This argument is tlierefore no longer always adduced in siippoi-t of GODI.EWSKI. Point 5. In critical cases tlie arguments from analogy are hardly more valuable than illustrations. I will therefore not discuss them here. We see therefore that the arguments which have been advanced so far give little support to Godlkwski's theory. On the other hand the striking and conclusive result of Strasburger's intoxication expe- riments is in favour of Dixon and Jolt. If to this be added the great convincing power which proofs from analogy exert, when well presented (and here Dixon and Jolv are much more fortunate than their opponents), we may I'eadily understand, that the cohesion theory has many supporters. There are, however, two facts which are adduced against this theory with more success. In the first place a second series of experiments by Ursprung '") in which he used ice instead of steam, in order to render part of a branch inactive. This series of experiments does not of course suffer from the objections which deprived the other series of its argumen- tative value. The fact, however, that fading only occurs after several days, makes the result less convincing. Another objection is more important; Point 6. The distribution of pressure in living trees is opposed to the theory "j. In a hanging water-thread the pressure decreases gradually as one ascends and the decrease is at least one atmosphere for an ascent of 10 metres. In living transpiring trees it has been impossible to demonstrate this ; it was found on the contrary tiiat manometers placed at different heights up the trunk, behave quite independently of one another. Sometimes one shows -a lower i)ressure, sometimes the other. It is true that objections can be raised against many of these measurements of pressure, but some of them in Schwendener's opinion proved positively and undeniably that there can be no question of a regular decrease of pressure. For in this case it would be incon- ceivable, "dass ein Baumstamm der nach 2 — 3 Regentagen durch Nachschub von unten etwas wasserreicher geworden, in mittlerer ( oHH ) Holie (wo vorlier Saugen stattfaiid) Liifl in das liier angebraclite Manometer liiiieinpreszt, wahi-end oben in der Krone und insbesondere unten am Stamm weder Saiignng noch Pressung stattiindet"' "). It is remarkable thai Dixon, in liis review of liie slate oi' liie problem in the "Progrcssus", does not at all refer to tlie ice experiments of Ursprung, nor to measurements of pressure, allliough lie there considers at length and refutes much less impoi-tant objections. Thus we have traced the causes of the remarkable phenomenon mentioned in the introduction. The partisans of Godlewski point to the measurements of pressure and maintain that Strasburger's expe- riments are invalid, whereas Dixon points to Strasburger and is not concerned witii pressure measurements. As will be seen the position is somewhat confused. In my opinion no advance can here be made along a theoretical road. Experiments alone can lead us out of the confusion. I think I am able to supply conclusive, experimental proof that the normal living wood is able to pump water actively. In order to give tiiis proof I shirted from the following preliminary conception. If the irregularity of the results of pressure measurements is really caused by a pumping action of the living wood, this irregularity must at once disappear as soon as the experimental trees are killed or paralyzed. This was indeed found to be the case. Moreover, as soon as tlie trunk was dead the differences of pressure, followed the same rule as would be expected to apply to a glass tube. When the conditions became unfavourable to evaporation, as in the evening and when laiu supervened, the indications of the manometers approaclied each other more and more. At midday, in sunshine, on the other hand they dilFered more. This becomes intelligible, when we consider that a more rapid evaporation requires a stronger current ; for a stronger current larger differences of pressure are however necessary', in order to o\ercome the greater resistances. First I will describe the experiments somewhat more in detail. Later I hope to publish the curves of the positions of the manometers, together with tiic I'esult of a more extensive investigation of this subject. Of a ± 2i metres high specimen of /S'or/'H.s' A/Z/AW/Zr/, which divided a little above the ground into two almost equal, strong branches, one branch was left intact as a control ; to the other I tixed above one another some U-shaped opeji mercury manometers, in the following manner. Some lateral branches were cut off from the main branch in such a way that a stump of 5 cm. length remained. A tube was slid over this stunqi, and to it the manometer was afterwards ti.xed ( 5fi9 ) this tube was blown out in tho middle to a small bulb, and was hermetically fixed to the s(iini|i with a piece of India rubber tubing. It was then half filled with water, and momentarily pumped empty that we may inject the cut vessels. I then left it open for half an hour and finally closed it with the perforated rubber stopper, through which the manometer was stuck. Once a day tlie bulb-tubes had to be replenished, for the wood always leaks a little from the inter- cellular spaces. The bark leaks still more and for this reason I always removed it at the place where the rubber tube was to come. As long as the tree was alive, no regularity could be perceived ill the indications of the manometers: they all showed a pressure, smaller than that of the atmosphere, but sometimes one "sucked" more, sometimes another. After a few days I killed the portion of the branch bearing the manometers over its whole length by means of steam. At once the manometers followed the rule indicated above, and did not depart from it. The differences of pressure became very considerable towards midday, showing that the dead portion offered a great resistance to the strong current. The crown and the base of the branch remained intact during this treatment. The leaves showed only after 3 weeks, that they had suffered from the operation; up to that time they remained perfectly fresh. Wiien at last they began to change, Ihey gave the impression that Ihey were diseased, rather than that they suffered from want of water. Two manometers were attached to the small trunk of a Corims and fixed to almost equal stumps of branches, the one 66 cm. above the other. The whole tree was 2 metres high. Befove I cut off the branches, which were to yield the stumps, I killed the trunk at these two places with steam over a length of JO — 12 cm. The manometers were thus attached to dead branch slumps on dead pieces of the trunk, separated by a living portion. I wished to investigate whether the living intermediate portion did pump or not. If so, it would always be occupied in diminishing the difference of pressure between the two dead pieces of the trunk. If it was then suddenly cooled with ice, the manometers would have to diverge suddenly and would once more approach each other if the tree was left to itself. Finally if it was killed, the well-known regularity would be bound to appear. The result was different, however. The intermediate portion evi- dently did not |)ump, for the manometers behaved exactly as in a dead tree. At midday they sometimes differed by 24 cm. of mer- cury. However --- on the fifth day their behaviour changed fairly ( •'^70 ) siiddenlj and on tlic sixlli dav it was as irregular as in living trees! Evidently the intermediate portion liad sntTered too much by this treatment, to function inimedialely, but on the sixth day it had so far recovered, thai it could work again. It lived on until the end of December, as could be seen by the perfectly fresh bark. Now, at the end of January, it is dead. The crown, howex'er, still looks healthy, as also do the buds. Although {hose facts, as far as 1 can see, do not permit of an explanation other than the one given here, a proof may still be adduced that such phenomena cannot be attributed to a change in the resistances. Such, a change would moreover have to be of a very i;emarkable nature to be of any use as an explanation. Four manometers were attached to the trunk of a lilac tree [SyruKia vuhjuris) 2 metres in height, and they were numbered in ascending order 1, 2, 3, and 4. After a short time they all showed an approximately e(iual "suctioji'", which oscillated with diurnal periods between 48 and 28 cm. of mercury. Although the dilferences were small, some times one was the highest, some times another. After J 5 days, whoi 1 knew the course of the pressure curves Huliiiciently, stump 2 was killed, together with the piece of the stem to which it was attached. This was done by passing through it for an hour the discharges of an induction coil capable of giving a spark of 10 cm. long, without sparking in the secondary circuit. Tlie stump ami the portion of the trunk became heated to nearly (50° C: a few pieces of glass cement of that melting point, which 1 had fastened to it, just began to melt. While the induction current was being passed, the suction of stump 2 iirst diminished greatly, as a result of the heating, the other manometers remained constant. Soon the fall of the mercury in no 2 stopped and the suction increased again. After the interruption of the current the mercury rose higher than usual ; this abnormally high suction subsequently persisted ; no 2 afterwards followed the periods of the other manometers, which went on without hindrance, but sucked always strikingly more. How we can deduce fi-om this the proof that tliis phenomenon is not caused by changes in the resistances, will be explained presently. Thus far the description of the experiments. I will now consider what may be deduced from the results. The course of the manometers in Sorhus proves that the water current in a living tree is caused by quite different forces fi-om those of a dead one. The result cannot be attributed to the imperfectness of measurements. Most of these are the same before and after death ( 571 ) and we ounnot suppose tiia( tlie cirouiiislaiifes wliicli are changed in the operation, are altered exactly in such a way as to bring to light the observed i-egularity. Thus the distribution of pressure bet'oi'e death can onl}' be explained on the assumption that there are pressor factors, i.e. pumping actions in the wood. This view receives important support from quite a different side, through the experiments of Zmlstra "). He allowed a solution of Siiureviolett to ascend living and dead branches and then examined them microscopically. In the living ones only the tori of the bordered pits were stained, togetiier \vith a thin layer of the walls of the vessels; in the dead ones, howo\er, the whole of the wood was coloured uniformly. It follows from tliis that the water current takes quite a ditferent course in dead wood from that taken in living wood. That in the lilac only the one manometer was affected, which was attaclied to the portion killed by induction shocks, cannot in my opinion, be explained in any other way than by the aid of God- LKWSKi's theory. If one imagines, with Dixon and Joi.y, that the whole trunk behaves like a dea,d lube, the phenomenon camiot be explained. An increase of resistance cannot be the cause, for then the other manometers would have undergone this influence. If on the other hand, we imagine a tree-trunk to be a s^'Stera of tubes in which everywhere small pumps occur, the phenomenon becomes intelligible. The death of the piece of trunk puts the pumps out of action locally and the suction must there be somewhat greater to get the watei- through the piece of dead wood. This would not necessarily be observed at the following manometers, since the inter- mediate elements bring the pressure back to normal. I regard all the above as proof positive that the li\'ing wood has a hydromotory power. The experiment with Corniis already proves this very clearly : one could almost see the recovering intermediate portion suddenly begin pumping, as it were before one's eyes. After thus having given the positive proof that the living wood assists in the ascent- of the water I will again take up the theoretical considerations with which I started, and see to what extent this proof can modify the condition of affairs. We encounter the difficulty that Stk.^shi'rger's intoxication experi- ments prove that help of living elements is not necessary, whereas the only theory which is not adversely affected by these experiments becomes untenable on account of the pressure measurements. The solution is clear from the preceding. The adherents of GoDU-nvsKi are wrong in asserting that water ( 572 ) cannot ascend more than 14 metres withont the help of life, for Strasburger's experiments show that this is indeed possible. But that is not the question. The position is simply this, that in a living tree the water is p n ni p e d up by 1 i v i n g e 1 e ni e n t s, where- as in a dead one it also ascends, but through o t li e r causes e.g., with the help of cohesion. Let us test this view by the data in the table : Point 1. The question of continuity is only of importance for the cohesion theory. As soon as this has been refuted in another way, the question of the continuity of water, may be left untill it may perhaps arise again in connection with new questions. Point 2. The intoxication experiments of Strasburger have been included in my thesis. Point 3. Although the experiments of Ursprung do not prove anything certain in favour of Godlewski's theory, they certainly prove nothing against it. Point 4. Tlie anatomical structure of the wood can never be adduced as an actiuil objertiou to the view here put forward. As soon as it has been proved (hat tlic living wood pumps, this fact cannot of course be weakened because we cannot at once imagine from its structure how this action may take place. The investigation of this point must simply be left for further research. Point 6. The distribution of pressure is quite in agreement with GoDi.EWSKi's view. When pressor factors are everywhere present in the trunk, the distribution of pressure cannot be predicted as long as these factors themselves are not fully known. We see therefore that the questions discussed iiere do not produce an objection to my view. In this preliminary communication I have of course limited myself to tiie most important matter; afterwards I hope to treat the same subject more completely. I might have omitted the literature entirely, but it seemed desirable briefly to justify my quotations and references. The small figures in the text refer to the bibliography which is appended below. I wish to conclude this preliminary communication by pointing out that the method which is here introduced, may also be of service in the solution of other questions. By its aid we might, for instance, ascertain whether the living elements cooperate, when a branch is made to transport water in the in\erse direction ; the influence of all sorts of stimuli (heat, cold, electricity, stimulant substances) on the activity of these elements can be examined. Should the intoxi- (^ation experiments of Strasburger be repeated with manometers fixed to the experimental trees, they would at once constitute a ( ^73 ) detinite proof in favour of (jodi.ewski. A small tree would, however, be sufficient for this. The method in which a living piec-e of wood is isolated between two dead portions is especially to be recommended. The portions to be killed should not, however, be heated above about 60", in order lo spare the intermediate position (compare the experiment with the induction current). If possible a leafy branch should be left attached to the intermediate portion, for otherwise it must soon die of hunger. GroniiKjeii, January 28''> 1910. B1BLI0GR.\PHY. 1) Sachs, Ein Beltrag zur Kenntniss d. aufsteig. Saftstromes iu transpirirenden Pflanzcn. Arb. des Bot. Inst, in WLirzbui'g, Bd. II. 1882; F. Elfving, Ueber die Wasserleitung im Holz. Bot. Ztg. M'62 p. 706; J. Vesque, Ann. d. Scienc. Nat. Bot. Ser. VI, 19, 1884, p. 188; J. Bohm, Publications before 1884. 2) J. BiiHM, Ueber die Ursache der Wasserbeweguiig, etc. Bot. Ztg. 1881. K. Godlewski, Zur Theorie der Wasseibewegung in den Pflanzen. Pringsh. Jahrb. Bd. XV, Heft 4 1884. ^) R. Hartig, Die Gasdrucktheorie Berlin 1883. E. Godlewski. 1. c. sub. 2), ^) Wes- TERMAiER, Zui' Kenntn. d. osmot. Sang. d. lebenden Parenchyms. B. d. Ueutsch. Bot. Ges. 18S3; idem, Ueber die Wanderung d. W^assers im leb. Parench. Sitzber. d. Berliner Akad. d. Wiss. 1884 ; Dixon, Transpiration and the Ascent of Sap. Progressus Rei Botanicae, Bd. Ill, Heft I. 1900. ■';. A. J. Ewart, The Ascent of Water in Trees, Phil. Trans. Roy. Soc. Ljnden, Ser. B, Vol. 198, 1905 and Ser. B, Vol. 199, 1908. «) AsKENASY, Ueber das Saftsteigen Verb. d. Nat. Med. Ver. zu Heidelberg. N. F. Bd. V. 1895 ; Dixon, 1. c. sub. 4), p. 65. ") Godlewski, 1. c. sub 2). ^) Di.kon and Joly, On the Ascent of Sap. Proc. Roy. Soc. Lond. Vol. 57 (1894) B. p. 3, Dixon 1. c. sub. 4). p. 31 and following. 9) Holtermann, Schwendener's, Vorles. 26 Mecli. Probleme d. Botanik. Leipz. 1909, n. 80. 10) Dixox, 1. c. sub. 4). p. 16. '') Godlewski, 1. c. sub 3), '*) Holtermann, 1. c. sub 9. p. 79. 1-"*) Dixon, 1. c. sub 4), p. 43—46. i*) Holtermann, 1. c. sub 9), p. 80. 1'') Dixon, 1. c. sub. 4) p. 15. i") idem, p. 60 and elsewhere, i') Holtermann, 1. c. sub 9), p. 80. Ursprukg, Die Beteiligung lebender Zellen am Saftsteigen, Pringsh. Jahrb. 1906. Janse, Die Mitwirk. der Markstrahlen bei d. Wasserbew. im Holze, Pringsh. Jahrb. 1887. i'^) Dixon, 1. c. sub 4), p. 19. ") idem, p. 16. a") Janse, 1. c. sub 17); Dixon, 1. c. s. 4) p. 16. -i) idem p. 46, 47, 49, 50, 51, 56, and elsewhere. --) Godlewski, 1. c. sub 3, p. 605 and 598. -^) Holtermann.I. c. sub 9), p. 66 and 67. -*) Dixox, I. c. sub 4), p. 45. -°) Holtermann, 1. c sub 9. p. SO. 2tij Pfeffer, quoted in idem, p. 78. -'') idem, p. 79. ^8) Ursprung, die Beteiligung lebender Zellen am Saftsteigen. Pringsh. Jahrb. 1906. 2i>) Dixos. 1. c. sub 4). 30) ZiJLSTRA, Contributions to the knowledge of the movement of water in plants. Kon. Ak. v. Weteusch. Nat. Afd. Proceedings, this volume p. 574. ( 574 ) Botany. — "Contributions to the hiowledije of the movement of wafer in .plants." By Dr. K. Zijlstra. (Communicated bj Prof. J. W. Moll). (Communicated in the meeting of January 29, 1910). For some liine I have been occupied in the botanical laboratory at Groningeii with the problem of the movement of water in plants and have carried out experiments of a somewhat diverse nature. Various circumstances have prevented nie from continuing my ex- periments in this direction, so that the investigation has not been rounded otf. I did not intend publisliing it, but as I shall presumably have for some time no further opportunity of continuing my studies, I think I may be justified in publishing the data I have collected; possibly they may be of service to other investigators who have chosen for their researches the subject of the movement of water in plants. The experiments referred to may be arranged under three heads, viz.: l'^' . The trunk or stem of intact plants cooled to about 0° C. 2'"'. The ascent of a dye solution in cut branches. 3"'. Interference with the movement of water in a tree-trunk by means of deep incisions. First I propose to discuss the considerations which led to these experiments and the results obtained, and then 1 will give a more detailed account of the execution of the experiments. 1. Trunk or stem of intact plants cooled to about 0° C. As is well known, Godlewski (Zur Theorie der Wasserbewegung in den Ptlanzen. Jahrb. f. Wiss. Bot. Bd. 15) attempts to find the cause of the movement of w^ater in the activity of the living cells of the medullary rays and of the wood parenchyma; these cells would therefore have to act as it were as suction-pressure pumps. GoDLKWSKi did not, however, adduce any direct experimental evidence in support of this theory. His theory is only made plausible with the aid of various data obtained by others, and it is urged that the theory does not conflict with the facts adduced by other investigators. Various botanists (Janse, Sti{asbuiiger, Weber, Ursprung) have after- wards attempted to test the theory experimentally. The most obvious method for such a test would be the following: to cut out the action of the living cells of medullary rays and wood parenchyma, and then to see whether the movement of water had become impossible. ( ^'Ir^ ) This eliiiiiiiatioii of the action of living cells was most easily obtained b^- simply killing these cells by poisons or by a high temperature. This method is, however, open to objection ; snch interference not only attains the elements which it is desired to put out of action ; others also, especially the water-conducting vessels and tracheitis will undoubtedly be atfected, so that it is questionable whether the results of the experiments can only be attributed to the elimination of the activity of the living cells. A method — already used by Urspkung but with a result opposite to mine — which meets this objection, is the cooling of the trunk or stem of the plant to about 0°. By this means it is possible to reduce the activity of the living cells to a minimum, while neither dead nor living elements undergo a permanent change. Moreover the advantage of being able to establish the original conditions after the conclusion of the experiment and therefore bring the plant back to normal conditions, shoidd not be underrated. The experiment and its control can both be carried out on the same intact plant. If by this means the plant could be made to fade, and to assume its original fresh appearance after the cooling had been stopped, GoDLEWSKi's theory would receive considerable support. According to this method I have myself carried out 3 experiments. The trunk of a small apple-tree, 2 stems of Poli/f/onum cuspidatum and 2 stems of Helinnthus tuberosus were cooled to about 0^ C. over a length of 50 cm. The experiments lasted 6, 7, and 8 days, under conditions which were ^■ery favourable for a possible fading. Nevertheless I have in no case been able to observe even incipient fading, although the transpiration from the leaves was strong, as shown by the cobalt test. Cut leafy branches, hung up near the plant, withered very rapidly. We may not, however, conclude from the negative result of these experimeiits that the living cells do not pLay a part in the movement of water. It is quite possible, even probable, that cooling a length of 50 cm. is not enough. This slight obstacle was perhaps easily overcome by differences of pressure present in the trunks. Had the results been positive it would have supported Godlewski's theory. JVIy negative results are, however, not able to oppose this theory. The nature of the results notwithstanding, I think it may be useful to bring them to the notice of others. Description of the experiments. The cooling of the trunk or stem was brought about by melting ice, which was placed in an apparatus indicated by the figure. ( 576 ) Tlie apparatus oonsisted of two equal parts, i.e. of two soinicircular tin-plate resei-voirs with fixed bottom and loose lid. The two half- cylinders had 0)1 the middle- of the flat side a portion which was bent like a half evlinder, so that the two reservoirs when joined to form one cylinder, left in the centre a space for the passage of the trunk, which was to be cooled. The height of the appai'afus was 50 cm., its diameter 30 cm.; the space left free for the trunk had a diameter of 10 cm. Each reservoir was provided at the bottom with a tap, through which superfluous water could run off. The cylindrical surface of each reservoir and also the bottom and the lid, were covered on the outside with a layer of felt, 15 mm. thick and over this there was a covering of asbestos paper, 2 mm. thick. In an experiment the two reservoirs were placed round the stem and screwed together, after a piece of felt had been placed lietween the two flat surfaces in apposition. The reservoirs were filled with ice. The space through which the trunk passed, was closed off above and below round tlie stem by a solid plug of cotton wool through which a thermometer passed. The temperature in the annular air space .snnouiiding the trunk varied between 0° and -\- 3° C. The apparatus was sufKiciently protected by the felt and (he asbestos against the heat of the surrounding atmosphere. Even on hot days it was only necessary to renew the ice twice in 24 hours. ( 577 ) During the experiment the aijparatus rested on some bricks, so that it was abont 20 cm. from the ground. Experiment I. Apple tree. The ice apparatus was fixed round the trunk having a diameter of 37, cm., of a small apple tree, about V|^ metres high, on July 21^' 1904, at noon, tiie weather being hot and sunny. The apparatus was filled wath ice and in the conrse of the afternoon the temperature in the space round the trunk fell to about 1° C. ; not the slightest fading of the leaves could be detected, although such fading would have at once been noticeable by comparison with two other apple trees which stood next to the tree experimented on. Nor could any change be observed on the following days. The temperature of the air space round the stem remained continuously between 0° and 3°. The maximum temperature of the atmosphere on the days of the experiment oscillated between 23° and 29°. On the sixth day, when the temperanire of the atmosphere was 20' and that round the trunk 0", an strong transpiration of the leaves w as demonstrated by means of the cobalt test. On the seventh day the trunk was sawn through immediately above the ice apparatus ; a hole was drilled in the portion of the trunk still inside the apparatus, and a thermometer was placed in it. In this way I was able to show that the temperature inside the trunk was the same as that of the annular air space round the trunk, i.e. in the course of three hours it oscillated between 2° and 3", while the temperature of the atmosphere was 24° to 25°. Experiment II. Polygonum cuspidatum . The ice apparatus was fixed round two immediately adjoining stems, 2 metres in height, on July 6''^ 1905, at noon, and it was tilled with ice. In the course of the afternoon the temperature in the air space round the stems fell to 0°, without withering taking place. The numerous stems surrounding the apparatus served as controls. Nor was any change noticeable during the following days. The temperature round the stems remained continuously between 0'' and 3°. The maximum temperature of the surrounding atmosphere ( .^78 ) on the various dajs during the experiment oscillated l)etweeii 19° and 30°. The experiment was stopped on the seventh day. Experiment 111. Helianthus tuh('rot<us. Tiie ice apparatus was fixed round two immediately adjoining stems of plants, IV, M. in height, on July 14"' 1905 at noon, and was tilled with ice. In the course of the afternoon the temperature of the air space round the stems fell to 0°. No fading could be observed ; several specimens of the same species, standing next to the plants experimented upon, served for comparison. Cut leaves, hung up on the plants were completely withered in a few hours. Nor was withering observable on the experimental plants on fol- lowing days. The temperature in the air space i-ound the stems remained about 0°. The maximum temperature of the atmosphere in the days of the experiment oscillated between 17° and 267,°. The experiment was stopped on the eighth day. 2. Ascent of a dye solution In liviiuj kikI dcud rat brandies. When cut branches, with a fresldy cut surface, are placed with this surface in a dye solution, the liquid will in general ascend into the branches for some distance, and thus may be easily traced by cutting them across at different levels. Various elements of the wood are then found to have been stained. It matters little whether one takes for this experiment living or dead branches, with or without leaves; the fluid always ascends in the branches, even when these are upside down, i. e. are placed in the solution with their cut apex. I generally carried out such experiments with twigs 30 — 40 cm. long ; sometimes with pieces of a branch, which had also been cut at its upper end. After some days the stain shewed itself on the sui-face of the upper section of these latter bi'anches. Although the dye ascends in all branches, the may in which the various elements are stained is not the same in living and dead branches. A sharp difference is observable. In comparing living branches with dead ones, it was of course necessary to use a harmless stain; the experiments of Stracke': investigation of the immunity of the higher plants towards their own poison (Dissertation), led me to choose Smweviplett of Grubler. I used this stain in a '/,(, "/„ aqueous solution. The twigs were placed ( 579 ) separately in a small Ixitlle willi the soluliini ol' llie stain, the iieek of the bottle bein.s closed with a pluu' of eoltoii wool to iire\eiit evaporation. After the experiment the twigs were examined at dilferent levels by microscopic sections. Transverse, radial and tangential sections were examined in oil of ckn-c.s, a medinm in which Siiureviolett is insoluble, so that the stain remained properly localized. The sections wei'e cut without the use of any licpiid and were at once placed in the oil of cloves. The slight water content of these prei>arations did not interfere. After a very short time the oil had thoroughly permeated. This method had moreover the advantage, that after most of the clove-oil hail been wiped away, the preparations could be \ery well enclosed in Canada balsam, without further treatment. A comparison of the beliaviour of the xyleni elements of lixing and dead bi-auches brought out the following dilferences : UviiKj branch dead /jraiich (I. torus of the closing membrane of /i. torus not stained, or oidy very the bordered pits deeply stained. , slightly. f>. adjoining the lumen, a thin h. the walls of the vessels, fibres layer of the wall in the border ; and parenchyujatous cells are of the pits is stained. The walls stained uniformly. of vessels and fibres ai'e only stained in a very thin layer, which is immediately adjacent to the lumen, c. contents and wall of the cells c. contents of the cells are coloured. of medullary rays and wood parenchyma are unstained. The deep staining of the tori in living branches was especially noticeable, also in transverse sections, the more so because the staining of the layer next to the lumen in the walls of vessels and fibres was t)f'ten difficult to see and because the living cells of the medullary rays and parenchyma were ([uite colourless. In the wood of SdLv and of Fiujus, in which the tori cannot otherwise be seen at all, they were made very obvious by this staining of lixing branches. The staining of the tori by eosine in a living branch of (Tiiilyo was already mentioned by Janse in "Die Mitwirkung der ]\Iarkstrahlen bei der Wasserbewegung im Holze" (Jahrb. f. \Vi,ss. Bot. 1887 Bd. XVHIj. In* this case also the stain had ascended the branch : 39 Proceedings Royal Acad. Amsterdam. Vol. Xll. ( 580 ) tlic "iiriiiuirc WaiHlliUiiclIt'" (if lln' nicdullarv rav cells was acfordiiifr to .Iansk all llial had liecii slaiiied. ill iin e\|ie'riiiuMits willi liviiii;- liraiiclios llic staiiniii;' exiemlcd iiol oiilv lo IliL' loi'i of the vessels and fibres, but also to those of the luilf bordered [nts between the medullary rav ceils on the one side and tlie vessels and fibres on tlie other side; the contents of the medullary ray cells however remained colourless, as stated above. The resnlts of other experiments carried out liy me, agree well with these facts. Instead of taking dead branches, 1 caused to ascend in living branches a 7io 7u solution of Saureviolett in strong alcohol, and also a Vm "/» solution in water containing 4% formaldehyde. As controls 1 employed living branches in a Vio °/<, solution of Saureviolett in water. I now found that the living branches in the poisonous solutions were stained practically in the same way as the dead branches in innocuous ones, only not so completely. It was clear that the alcohol and the formaldehyde only gradually exercised their fatal action on the plant. The tori were always unstained; only a few were stained \-ory faintly. The walls iicnerallv showed a uniform staining ; the medullary ray and pareiu-hyma cells with contents were coloured dark blue. Finally 1 may add ihai microscopical transverse sections llu'ough living branches, which seel ions were afterwards placed for 20 hours in an aqueous Saiu'eviolell soliuiou of '/lo "!«, were stained ([uile unifoi'mly dark blue, exactly in the same way as Ihnse sections made after the slain had ascended in ilriiJ brauches; the colour was only somewhat more intense. The Iransvei'se sections through control branches, which had |ire\iously stood in the same solution for 4 days, on the other hand siniwed, as was to be expected, a staiinng quite similar to that which was described above tor living branches. Descriji/iiui t'j llii' i'd'i)fniiti')its. l'An;in\iKNT IV. l-'iiijas silraticii. \ \\\\u'j: leafy Iwiu;, aboiil 4 mm. liiick al ils base, -loud for '.I (la\s ill a sobilidii of Saiirev iiilell. The stain asceii<led lo the top and inlo llie lea\es. The bark, llie cambium and ilie pilli remained (piile nnslaine(l ; llie slaininii- was iimilcd in llii' wood and lier(> liie slain was only in ihe inner la\ei- (a'l.iniiiinL: llie lumen) nf llie walls of vessels and libivs; ihe Idri id' llie luirdered pils w ere slained a very ( o8i ) deep blue-violet, and this was aisn tlie case willi llic liall' lioi-dcrrd })i(s between inedidlary I'liy eells and tibres. Tlie ine(iuliai-_) rays and the xylem parencliyiiia were quite unstained, both as regards wall and contents. ExPEKlMKNT V. Larli; decidua. A living leafy twig, (! iinu. Iliick at its i)ase, stood for 5 days in the solution, after wliicli the slain had penetrated to the apex. Staining completely linnted to the wood, hut no stain in the oldest of the 6 annual rings. The stain only taken up by a very thin layer of the wall, adjoining the lumen of the tracheids and the cavities of the pits. Torus of the pits deep blue-violet, also in the half bordered pits between medullary rays and tracheids. For the rest everything uiistained. Experiment VI. Salix spec. Two li\ing leafless branches, provided at either end with a cut surface, both 30 cm. long and more than '/a t'li- thick, stood for 2 days in the aqueous solution of Siiureviolett ; one of the branches had its lower end in the solution, the other its upper end. The stain ascended readily, and /// llic I tm branches sliiadtaneoudij. The stain only present in a thin layer of the wall adjoining the lumen of the vessels and fibres and the cavities of the pits. Tori deep blue-violet. EXPERHIENT VII. F((l/us siliuitica. Taxus baccatn. Of each of these plants two similar 3 — 5 year old leafless branches were placed with the cut surface in the aqueous Saure\iolett solution for 3 days. Ojie of the branches of each species was alive, the other had been treated as follows. It had stood for Vj., hours in boiling water. Then walei' was suckctl through the boiled branch by means of a filler pump in oj-dcr to remove possible obstruclions, linally a fresh surface was cut. 39* ( 582 ) Afler H (liivs ilio slaiu luiil alniosl rpuclieil llie top in all llii> I'uiir l)iaiiclies. Ill tiic lixiiiij;- i)raiu'lies slaiiiiiig was scarcely \'isil>lc against liie walls of the vessels and traclieids. The tori, iiickuling those of the half bordered pits were deeply stained. Medullary ray- nnd parenchyma cells (|iiite colourless. In (he boiled branch oF F<i^/its liie walls of the liiiriForai libres and of (he vessels were a unifurin pale blue. Against (he walls of the vessels in the s|)ring wood a darker layer. Nowhere however coloured tori. The medullary rays also proved to be colourless. In the boiled branch of 7\i.cus the walls of many tracheids were stained a uniform pale blue; towards the inside against the walls a darker layer. The tori unstained. The medullary rays dark blue. EXI'EKIMKNT VIII. Ttr.cus hdcralit. Two living branches were taken. One was placed with its cut surface in a solution of 0.1 gram of Siiureviolett in JOO c.c. of /ra/t'>'; the other in a solution of 0.1 gram of Saureviolett in JOO c.c. of rt/coAo/. Piotli branches rcuiaiued standing in the solution for 4o hours, after which time sections were made through both at a height of 7 cm. The staining \\as as follows; Branch in n(/i(i'ou-f solution: staining only in the .secondary xylem. A very thin blue layer against the walls of ilie tracheids, and of the cavities of the bordered pits. Tori dark blue, including (hose of (he half bordered |)its. Medullary rays unstained. I>ranch in iilcd/Ki/ir sdJiition : the stain had also |ienetraled into (he cambium and the innermost layers of the corte.v parenchyma, where both walls and contents were dark blue. In the secondary xylem the (racheid walU liuht blue; against the walls also clearly a blue layer, fiirthei- in the cavities of the pits. Tori unstained. Medullary rays dark libie, both as regards walls and contents. 'J'he walls also coloured in the prima'-y xylem. \\ \ P K |{ 1 M 1', N T IX. Til. Ills Jincriita. \ lixiiig bi'aiich was placed with (he v\\\ surface in a solution of 0.1. gram of Saure\ iolett in lOH vv. of a 4" „ fonniihli'liijih' solution (diluted foi nnxliiri. ( 5s:{ ) Aflcr .') (Ia,vs llie liruiicli wuh cxainiiicd ; Iliu .•-hiiii liiul already I'oacliod llic a|K'\. Slaiiiiiiii- (Mily ill tlio secondary xyleni. A^aiiisl tlic \\'alls of the traclieids iherr was a lliiii liliic layer, al>n in ilie caxilies of tiie liordcred pils. Tori coloiiriess. Of llic incdidlary rays hdlli tlic walls and ilie j)rotopiasni dai'U liliie. E.XPKKl.MK.M' X. Salii: spec. A lixiiiii,' twig was placed willi its cut .surface in a solution of O.J jirani of Siiin-eviolell in \(M) c.c. of a V \ fonnaldelnjde solution. After ;i days ili(> slain had |)eiielrate<l to the apex and the twig- was examined. Staining only in tlu^ secondary xylem. The walls of the vessels colonred light Idue with an indication of a soiiiewhal dai'ker layer adjoining the liiiiieii. Toi-i practically colourless. The niednllary ray cells, which adjoined the vessels, are coloured hlne. 3. Intevfcri'ncc. irif/i f/w inon'iin'nt ff irnter in n tr(^e-tfuak by means of deep Incisions. Experiment XI. An experiment with a small willow tree in the Botanic Gardens at Groningen showed, that in a trunk in which the transpiration current had been largely prevented or perhaps completely cut oil' as a result of trans\erse incisions on both sides at various heights, measures were taken in course of time which ultimately led to a complete recovery of this euirent. The experimeni was carried out as follows. The trunk of the tree, li^l.. cm. thick, was sawn into transcerselv to slightly beyond the centre at four places, alternately on either side of the trunk. The incisions were 22 cm. apart, and the lowest was J. 25 Metres from the ground. They were prevented from closing up again by the insertion of tin plates, which in future remained r' in position. At these four places the water current was therefore^ irreparably interrupted. As the trunk had of course been greatly weakened by this" operation the tree was supported by four iron wires, which were attached high up to the trunk and also to four pegs driven into the ground al some distance round the tree. ( •^■^4 ) This cxiiL'L'iiiK'nt was slurk'd mi .liilv J4"' I'.tOS; (lie iufisioiis were read}' and the plales wei'c pushed in al 9.30 a.m. A( 10 a.m. tlie leaves were already droojiiiiji- and diey I'oniaincd so thi'nnuiioiit the day. Ill the course of the live tbllowiiig days, in cool dry weather, tlie leave.s gradually recovered. -On the 7"' i\:\\ of tlie experiment the foliage began to wither from the top downwards ; many yellow leaves also appeared in the crown. In all these days the temperature had not risen above 18° in the neiiihbourhooil of the tree. On the O"'"! day the temperature rose in the afternoon to more than 26°, and probably as a result of this the number of yellow leaxes now increased rapidly. Those leaves which had remained green also began to droop again. The tops of the branches in the upper part of the crown withered completely. The 3 following days were warm and siiniiy with tein[)erature maxima of 27^ and 28°. Most of the lea\es now fell off, while in the upper half of the crown the foliage withered completely. After this time cooler weather supervened and the few remaining green leaves recovered and remained in good condition until the autumn. That the tree had not sutfered greatly however from the incisions, was shown in the following summer, for then the foliage developed as well as before the expei'imeiit, and remained fresh throughout the entire season. Wageninffcn, Dec. 13^'' J 909. Physics. — "Thi' maijni'tic sepitr(iUon of ahsorptioii I'me-'^ la coiwe.L-ion witk Sun-spot spectra:' (I). By Prof. P. Zkkman and Dr. 15. WlNAWKR. 1. As a consequence of the intimate connexion between emission and absorption, there exists closely corresponding to the magnetic separation of emission lines, a magnetic division of absorption lines. The dark lines which ap|iear in a continuous spectrum, if a beam of white light traverses an absorbing tlamc, are divided and pola- rized under the influence of magnetic forces in exactly the .same way as the emission lines. This correspondence between emission and absorption was shown to exist already in some of the lirst experi- ments on the subject by on(> of the jireseiit authors. Our knowledge of emission spectra under magnetic intlueuce has since been extended considerably. The experimental study howevei- of the inverse effect i. e. the luagnelic division of absorption lines has less advanc(Mi. f -,85 ) Al'lcr llic lii->l cxiiciiiiiciil- nt' ihc lii>i iiiiiiicd (if the ;iiiliicpi>> of lliis |);i|iL'i-. tlic clianiii' of ;ili>t)r|iii()ii liiii-- iji u iiiamiclic liclil was slmlicii l)\ KiiNKi ' ami < 'otton \ ; IvM.iii '; ua\e an eiahorale sludy of llic siilijcci, 1(1 wliicli \\(> luue lo rc'liirii later on. Il contains the (iiilv in\x>sti<^-ati()ii ol' the iiiauiic'ic cli'i'd in a ihreelion inciine(l In the lines of" force, ('loselv ennnrclcd with nwv snliject arc linally sonic observations hy JiOixii-: and Daviks ') uii Ihi' inllnence nl' a magnetic field on ilanies, emillnii;- ■■cexersed" lines. The consideration of the inverse effect forms the basis of V'oigt's magneto-optical theories ') ; and it is considered also by Lorfatz ") in his investigation of the magnetic sei)aiation in a direction iiiclineii to the line of force. Theory indicates tliifei-enl |H(int>. w iiicli may l)e tested by e.\(ie- rimeiit. The imerse effect has become of suj)reme interest in solai' pliysics, since Hat.e's ') discovery that the dark lines of the sun-s[)ot spectrnm exhibit the characteristic phenomena of magnetic separation. Tiie e.\|)eriments we intend to describe in the iiresent communi- cation relate to Ihc tlivision of ihe sodium lines l); and D.,. Some of onr results may already be found in the work of the cited authors. In order to present the suiiject in a coiuiected form il seemed necessary not to exclude these. Tlie fads now ascertained in condiiuation with former results appear to be of some value in explaining peculiarities observed in sun-spot spectra. .Some instances will be given later on. 2. Type and relative amount of the magnetic division of the sodium emission lines, Dj and D,, are given in Fig. 1. -^^ Il represents the observations when the line of sight is at right angles to the magnetic field, when it is parallel to ^1 the field. Ill a weak magnetic field D^ exhibits S- ■ (he tiiplet type, at right angles to the n KoNiG. Ann. d. Phys. Bd. 62. 240. 1897. -) Cotton. Eclairage Electrique. 5 et 2(j mars. 1898. ^) RiGHi. Sul fenomeno di Zeeman nel caso generate d'un raggio himinosa comunqiie inclinato sulfa direzione della forza raagnetica. Jlein. di. Bologna, 17 Dicembre 1899. ^) Lodge and Davies. Proc. R. Sec. 61 413. 1897. 5) W. VoiGT. Magneto- und Eielctrooptif;. Gliapter IV and the papers there cited. 6) H. A. LoRENTz. These Proceedings, Vol. XII, p. 321, 1909. ") G. E. Hale. On the probable existence of a magnetic field in sun-spots. Contributions from tlie Mount Wilson Solar Observatory Nr. 30. 1908. ( r>8fi ) lit'ld ; tlic (Ii(iiI)Il'I Iv|i(' if llu' liulil is c.xaiiiiiKMl |iur;illcl lo ilic lines of force. D] seems lo exliihil a (loiiiilel in Uolli principal directions. The FKArMioi''K,K lines in llie specli'a of snn-spots investigated by Hai.ic are either broadened, or ciianijed to doublets (often incom- pletely resolved ipiarlels), or lo triplets. The I'esolntions exhibited by sodium \aponr are therefore the \ery types of special importance to astrophysics ; this and also the facility of producing .sodium vapour in the magnetic field induced us lo commence our experiments with this sni)stance. 3. The explanation of the inverse elfect is easily understood by meairs of the well known law of resonance. If there ai'e in a tlanie nndei' the inflnence of a magnetic Held three |)eriods of free vibrations, then we may expect that tVdui incident while light \il)ra- tions of these very three periods will be taken away. The absorption is a selective one, with tliis pecubarily that the selection refers not only to the |)eriod but also to the direction of \ibration. Consider for example the ceidral componeni of a triplet \\hich iji tiie emission spectrum is due lo \ibralions parallel lo the field. I^'roni incident while light oidy vibrations, corresponding as lo period as well as to direction of \ibration with the middle comiionent, are al)Sorbed. A^ibrations, perpendicular to the field, liiough of the period of die unmodified line, pass unimpeded. ( )n tiie confrai'v white light of periods coinciding with those of the outer components is only deprived of its vertical constituents. It will l)e clear from these vcfv simple considerations what we may ex])ect to observe uith while light nndei- the conditions of the experiment. Tiie arrangement was the following: White light of the incandescent jiositive pole of an arcdamp li-a\erses a sodium flame, placed bet\veen the poles of a or llois-elccUomagnel. This light is analysed by means of a sligmatic s[)ectroscope willi large Uowi.and grating. The observations are made in the first order. If the observation is made at right angles lo llie lines of force, we see in the continuous spectrum 4 daik ct)m|H)nenls in ihe case of />,, I) dark comi)onents in the case of /),, as represenled for both lines under <t in the diagrammatical Figure 1. In order lo observe all ihese components the field unisl be sti'ong and the vapour density adapted to the Held. The groups of lines indicated by /> are seen, if Ihe light is examined axial ly. All these conipcnients, if nai-i-o\\, are seen mdy diffuse and not black. From the considerations above gixcn the reason will be (dear at once; each of the components absorbs only hn/f Ihe incident natui'al light. ( 5«7 ) ^^ itii \erv (liliilcd \;i|K)ur no ali>(>r]ilit)ii ;it all (ir oiilv \eiT weak Inu-es of absorption are seen. 4. The introdiiclioii of a Xicol in the beam before or after llic field entirely cliaiities tlic |ilienoinenoii. Tlie absorption lines can then be seen \cry ]iarro\\ and bhudv. J.et the observation be made at rijilit angles to the horizontal field, then, if the Nicol is placed with its plane of vibration vertical I), exhibits its two, D.^ its fonr onter components. After a rotation of the Xicoi over 90° both D, and D., give onlv the two liorizontally vibrating com[)onents. Let a beam of natnral white light traverse axially the magnetized vaponr [)laced between the perforated poles of an electromagnet. Then by means of a (piarlcr-wa\c plate and a Xicol we may (piench either the rightdianded or the left-handed circnlarly |)olarizcd component. A combination of a (|narlei--\vave |)lale ai]d a Nicol, converting incident light into rightdianded circnlarly polarized light mav be called a right-handed circnlar analyser. The absorption line corre- sponding to a right-handed circularly polarized component is seen with both increased clearness and darkless by examining it with a right-handed circular analy.ser. We introduce here this simjile matter because there has been occasionally some confusion on \\\\> subject. 5. The behaviour of horizoiUal and \erlical vibrations mav be studied simultaneously by using accoi'ding to the suggestion of CoR?ii' and Konio a calcs|)ar rhomb. By means of it we can obtain two oppositely ])olarized images of a horizontal slit of suitable width, placed near the magnetic field. Right-handed and left-handed circnlar \ibrations can be separated on the same plan by the introduction of a Fresnel rhomb between the calcspar and the slit of the s|)ectroscope. It is, however, of considerable interest to examine also the behaviour of the lines in natural light. A separate examination after the removal of the polarizers might be made. The vapour density ought to be the same in both experiments. It seems dilYicult to realise this. The desired end is secured more simply and surely, and Avith only half the labour, by adopting the width of the horizontal slit and the thickness of the calcspar in such a manner that the two images given by the calcspar partially overlap. We now obtain three stripes ; the central one exhibits the phenomena as seen without polarizing apparatus. (See fig. 2). ( r,88 ) Fi! Tlu' ii|i|M.T and low c^l sliipcs >li{iw llic iiilliiciicc' (tf puliuizc'd liiiiil uu I lie plieiioineiioii. The (ibservatioiis ^iveii in tliis coiiinniiiicaiiiiii lune been made by llie descrilied iiielliod. By its use all parlieulars of I lie |ilien(iiiieii(.)ii are siiiinllaueoiish exhibilcd ; we also succeeded in |iiiotoj;ra|)liiiig the essential poiuls. Kxauiples ol' oui- |iliolo,iii'a|ilis ai-e liixeu on (lie plates annexed to our ])a|HT. 6. It' tiie absori>lion Hues are iiol narrow or il' the niagiietic Held is weak, the coniponeiits ol a nuigneticailv divided line will partially overlap. This partial superposition is the cause of some parlicnlanties, especially manifest in the inverse elfect and probal)ly also apparent ill sun-spot spectra. The nature of these i)articnlarities may be illustrated by a fe'v examples. We will consider the case of the magnetic triplet and the magnetic doublet. In Fig. 'A the curves slaiw the distribution of intensity of the ihive com[)onents of a triplet, if the light is examined at right angles to the lines of force. If natural light traverses a source of light placed in a magnetic field, two black bauds are seen, corresponding to the wavelength, for which vertical as well as hoi'izontal vibrations are absorbed. These bla(d< bands are surrounded by less dark parts, which absorb only one of the Fig. 3. principal \ibrations, the other proceeding unimpeded, (cf. ^^ 3 and 4). If iiov\' a Nicol with its |)lane of vibration vertical, is introduced two black bands are again seen. The darkest part of these compo- nents corresponds to the nia.ximnm of the curves relating to vertical vibrations. As a general rule the distance of the com- jionents exceeds that of the lines tirst considered. 7. Parallel to the lines of force a partial, not loo small, overlap|iiug of the components [iroduces a black line limited by two less dark parts. This case is illustrated diagrammatically in I^'ig. 4. The two components may be se|iaraled by '■'S- '*•• a circular analyser. These considerations may lie applied to the magnetic division in ^oYr ( o89 ) sini-s|)(il sj)(M'tr;i; as a geiunal rule wr luav expect llial llic se|)ai'al ion of lines in spot !>pec(ra hecoines more (lislinct and ol' lartioi' aniounl hv the use of analysers. Tlie introduction of a Xicol in liie i)eaui may also reveal lines invisible without analyser. Several peculiarities observed in the distrii)utiou of inleiisily in spot lines, remind one of tiie now specified super])ositi()n pheno- mena'); cf. ^ J9 below. 8. Superposition effects of nearly, though not exactly, the same nature occur if lines with the same direction of vibration are su[)erposed and if the continuous source of light emits unpolarised light. In the more complicated divisions the now specified superposition occurs also. It is Just possible that the superposition of the outer components of the sextet, type D.,, produces only dark, that of the inner and the next outer components, black lines in the continuous spectrum. It is easily seen that al^o in the case of the quartet, type />,, black lines may be produced. The darkest parts may be seen some- what nearer to the middle of the complete figure, than the outer components of the quai-tet. It seems unnecessary to illuhlrale this by figures. Examples of the specified actions will be given presently. 9. Our obser\'ations and spectrograms I'elafe besides to the two principal directions (parallel and at right angles to the lines of force), also to directions inclined to the Meld. In the present, first, communication, observations are discussed, relating to 5 different angles between the field and the direction of propagation of the beam (Voigt's <j:, Lorentz's &). These values are : 90°, (F, 60°, 45°, 36° -39°. The results of the work relating to these angles have been recorded on nearly 100 spectograms. 10. (Jbserv'dioiis periie/ulica/dr fa tlie jichJ. In the upper of the three stri[)es which are present in the field of view (see § 5), the light vibrates vertically : in the lowest one hori- zontally, whei-eas the middle part relates to natural light. Lender the influence of the magnetic field we therefore see the vertically vibrating components as narrow black lines. The quartet of the />! line, the sextet of the I)., line, may be seen very clearly 1) A figure equivalent to tlie one now given concerning the influence of super- position of magnetically divided components was already drawn for emission lines in Zeeman. Doublets and Triplets in tlie Spectrum produced by external magnetic forces. Phil. Mag. July 1897 § 7. ' • ( 590 ) Ii\ Ihi^ iiu'IIkhI. a Miiall ilisiiii-liaiirc is |ii'(i(liiccil li\ llir iianow ivvci-sc'd liiiL's (liic ti) llie cleclric arc lifilil. Tlic iulciisilv of lliese liiiew (lejioiids u|>on soinnwlial variable cii'dimstaiices of ilie arc itself. In some cases these lines are aliuosl iii\ isilile, in olher ones more pfoniineiit. Tlie\ are to he seen on some of our reproductions ; will) our present sMltject lliev liax'e nolliiiig to do. As regards the ceidral stripe wc refer to the remark |)revionsl_v made, liiat llie image of the sepai-atioii must liecome, on ac<'(inni i-f the oiilv partial absor})lion, rather indetinitc ii\u\ weak. (§3). 'I'lie partial sn|)erposition of componciifs gives, at least in (he case of diluted \apour, the most conspicuous lines. (§§ 6 and 7). In the case of llie (piartet. for example, one sometimes sees instead of four. onl\ two components, situated lietween the imier and outer ones. We made experiments with dilfereni \a|)Our densities. The observed phenomena mav be classified inider three jjhases ; 1. The vapour is veri/ dilute. 'I'he com]ioneii(s are clearly- visible m the upmost and lowest stripe. In the central siripe the absorption is either liardly perceptible (Plate 1, Fig. i) or the components of the quartet and the se.xtet are seen as sej»arate, but weak lines. (Plate I, Fig. 2). In this |»hase of the phenomenon the great dilference of detiniteness of the central and outer regions is verv ivmarkable. This contrast is still more marked with eye observation. In order to obtain good pljolograjihs, it was necessary to increase the densitx of the \apour aliove the one reipiired for tlie observation of the very lii'sl trace of al)sorpliou. '2. Va|iour of 'mti'nni'iHiiti' density. The components in the upmost and lowest stripes are now no more seitarateU \isil)le or only in the case o\' the (|uartet. In the ceidral stripe a su|)erposiii(.in of the kind meiititnied in § (j takes place. In |)lace of the qnai'k't an apparent doublet is seen, the components of whicii are situated between the oider and inner components of (he (piurtet. This case is \eiy clearly represented in Plate I, Fig. 3. The phenomena exhibited by the .se.xtet (D^ line) become rather com|ilit'ated. The superposition |dienonienon is often very distinct. The D, line on Plate I, h'ig. 3 shows suHiciently the appearance. :!. With still (li'i/srf \apour. the components become very broail and the magnetic change liardly visible. The pohuimtion of the cdcu's of the broad line may be recognized. This phase is represented in Plate 1, l''ig. A, It corresponds u> the emission effect as it was lir--l discovered: a slight change of broad lines in ti weak field. ( 591 ) Willi still greater ;ilisoi|iti()ii lln' inlliiciHH' tif llie field lioc-oincs imperceptible. All these phases aiipcar with great I'egularity. It' ihe iiileiisitv of the field is known, it seems possible, the resolving power of the spectroscope being given, to deduce the density of the vapour from the Jiature of the observed piicnoniena. The magnetic division phenomena hitherto observed in ^un-spols apj)ear to fall under the second and third phases above mentioned. H.\i,E from measurements of spot lines, compared wilii laboratory experiments, deduces a maximum intensity of the spot tield of 4-5()() Gauss. Hence, one would be inclined to tliink that the density in the layers, which bring about Ihe absor|)tion in the snn-spot spectrum can only be small. Moreover, the non-uniformity of the field of sun-spots produces by itself a widening of the components. Light fi'om a limited portion of the spot would give perhaps very narrow spectral lines. In the light, however, of the critical remarks of Kayskk ") concerning our knowledge of tiie intbieuce of pressure and of tem- perature on spectra ail sucii considerations most be put forward with great diffidence. 11. < Jij.sL'i'ontioiis iKinillid 1o till' lliu's of force. In the present experiments the absorbing vapour sulijected to mag- netic forces is placed between pei-forated poles. After putting on the cnrrenl, one sees in llie conliinious spectrum, 2 dai-k l)ands in the case ()f 1)^, 4 in the case of J)., according to the diagrammatical figure 1. 'J'iie absoi'ption is incomplete also now, because of some wa\e-lengllis only right-handed circularly |iolari/.ed light, bul not lefl-hauded is absorbed and the reverse. In order to obser\e the separation anti the polarization a Fkesnkl rhomb is placed with its lu'incipal [)lane at an aziumth of 45" with the horizon, a horizontal slit being placed in one of Ihe perforated poles. The Fkksnkl rhomb converts circulai-ly polarized into plane polarized lighl. 15v means of a calcspar rhomb also now three stripes are olttained. The first phase {very dilute vajiour) is represented in Plate I, fig. 5. Va|)onr of intermediate density (second phase) exhibits Ihe super- ])Osilion phenomena mentioned in §§ 7 and 8, and diagrammatically ilhi-lraled by Fig. '2. In llie eenli-al strip niie line, at the position of the iinmodilieil one, Min-oiinded by feebly absorbing regions, is ') Kayser. Haiulljuch. Kupitel V. Ud. II, ( 592 ) seen. I'latc I, Fiu'. 6 i^liow.s these lijies for llic iloiihli't ;iii(hlic (|iiarl('t : es|)efiiill\ Willi Dj the effect is very luai-ked. 12. Observations in directions inclined to tlir field. Afcordiiig to Lorentz's elementary tiieoi'v of mayiietic dixisioii one Hciicrally observes in a direction, which is oblique under an anj;le ih with the lines of force, a triplet with clliptically polarized outer components ^). The ellipse, which characterizes the state of polarization of the components with period 1\ -|- r, is the |)rojection on the wave-front of the circle perpendicular to the field, in which the electron with period 7',, -|- r is moving, r is a small (piantit\ . The direction of the niolion of the moving electron also determines the motion in the clli|is('. The ratio of the axes is as I lo r-o.v «>. Foi- the other outer compdnciil \\\\\\ pcrind 7'^, /• holds mnlntis miitdndis \\w ^ama reasoning. The central Hue \\ith the unmodilied period 7'„ always renuxins linearly polarized. The vibrations of the middle compoJient are in the plane determined by the ray and the line of force and the amplitude of the \'iliralious is ])ro])ortii)ual U) sin \h. If we pill il = 0, i.e. in the case of Ihe lojigitudinal effect, only circular motions remain. All this apjilies to very narrow spectral lines in a strong field, the distance of the components being much greater than their width. According lo Voigt and Lorentz we must expect some interesting particularities if this restriclion be discarded. We return to this point hitei' on. As a general rule Ihe deduclions tVoni the elementary theory are verified. Also in the case of the ipiarlet and the se.xtet the outer components become ellij)tically polai'ized, as has been observed already by Ri<iiu '-). In contradiction with the elementary theory, though not strictly applicable to the case, is the very slight diminution of intensity of llu- middle compoiUMils of Ihe (jiiarlet e\en for i)= 45°. i;i ()l,serr,itions at }) = &)'. If the observation is made with a caicspar i-homb, the image 1) cf. lilClll 1. c. -) Rlian's observations I.e. all leler to an angle of nearly ^>'i°. Hie anjjie at wliieh aecordint; to the eleineiilaiy llieoiy llie three einiiponeiils of the Iriiilet are of equal intensity. ( 593 ) rcinaiiis as witli liic transversal ell'ecl. Yel the juvscikh' of elliptic liolurizalioii oiiglil lu iiiaiiifest itself by tlie a[)|)earaiK'e in the lowest stripe of lines, correspond in j^' to the outer components. With very dilute vapour and with that of intermediate density as good as no trace of it is seen. Fig-. 7, Plate II shows the lir.st pluuse with dilute vapour, Kig. 8 the second phase with denser va[)Our. Only traces of absorption, indicative of elliptic polarization can he seen near D.^, Fig. 8. The ellipticity is, however, undoiilitedly proved by means of the Frksnel rhomb, placed with its princi[)al plane at an a/.iniiith of 45° with the horizou. Fig. 9 shows tiie appearance. The outer components of the quartet towards the red or towards the violet, dependent upon the stri|)e and the direction of the Held, are now considerably weakened ; in the case of the sextet they have vanished altogether. All this proves the elliptical polarization of the outer components. For, if the polarisation were linear, as might be inferred from observations with the calcspar alone, then the obser- \ation with calcspar and rhomb c(Hnbincd, ought to siiow no dilference between the u|)must and lowest stripe. The light of all plane polarized components would issue circnlai'ly polarized tVom the rhomb and, the calcspar making no selection between right-handed and left- handed polarizations, the components towards red and towards \iolet would all be alike. Such a condition is disproved by photographs such as Fig. 9. 14. One point must be consitlered somewhat more in detail. What is the leason that the ellipticity is not shown by the calcsjiar rhomb alone, whereas its e.vistence is most clearly demonstrated by means of the Fkksnel rhomb? Let an elliptic vibration with \ertical axis h, horizontal axis a, be incident upon llie rhomb, the principal plane of which is at an azimuth of 45°. It is easily proved that the elliptic vibration issuing from the Frksnkl rhomb has its axes in the same direc-tion as the original It, b — a a motion and a ratio of the axes = -- , the original ratio being . 6, h-^a b If a be small in relation to b (an elongated ellipse), then, the light issues from the Fresnel as a more circular vibration, which is more easily analysed. It depends u|ion the nuxguitude of (^ whether - is ^/'cr/A'/' or /..« than b^a ( 594 ) We (listiiiji-nisli llio Ibllowiiig rases ; b — « a I. (/ \c\-\ .small, llieu ^ -. 6 + f ( ?- •2. a = 0,414 A, then '"" = ". /( — a a 3. ^/ > 0,414 A, then — < . \Vc shall a|)|)lv these results to the interpretation ol'oui' olisei'\atioiis. Two ''ases (le|)eiitleiit ii[)oii the iiiaii'iiitiide oF n are of prineiiuil impoi'tanee. Ill the first case we can observe the efi'ecl of both tlie axes of the ellipse l)_v means of the combination of the Frksnki, rhomb and the eaiespar [litis is ill,- ctisi' (if the qmirti't) {l)^ Fig. !)), whereas without Fkesnel ihomb no eliect of the small axis is visible, lu the secimd ease the eli'ect of the small axis becomes apparent bv the use of the calc- spar, whereas its existence cannot be demonstrated with the Fkesnet,, l>-a ^, . , , , , the \alne ot beiiiff too small. 77us msr is rruri'si'iitiil Ini tin' sr.vtrt, (\), Fig. ;)i. If the ol(ser\'ation is made l»v means of the calc-par rhomb, we indeed see with dense vapour new components in the lowest stripe (see Fig. 8, D„). The theoretical import of this result will be discussed on anothei' occasion. After iutrodudion of the FiiKsNKi. rliond) the comjionent to the left of the cenli'al line (small axis of the ellip.se) remains in\isible. (Fig 9, 1).,, inferioi- stripe). Hence we mav conclude that at the angle now imestigated the (( ellipticitv of the outer components of the se.rtct (the ratio --) exceeds that of the (piai'tel (and is also lai'ger than 0,414). 15. 0/isi'iTii/l<ins lit y- = 45°. The photographs taken with the calcspar alone, show \er_v clearlv the ellij)ticit_\' of the outer components. With va|)0ur of intermediate density the phenomenon is already ver\ marked, especially in the case of D, T'late 11, Fig. JO). Very remarkable is the slight dinnuution of intensity of the innei' com- ponents of the (piartel. According to the elementary theory the inten- sity of the centi-al ''ompoueut of n Ir/plit ought to have dinuiiished alread\ to less than Inil/' (he original \alue. Prof. P. Zeeman and Dr. B. Winawer. The magnetic separation of absorption lines in connexion with sun-spot spectra. Pla D, D, m^ D, D, Ui Dj D, D.. Proceedings Royal Acad. Amsterdam. Vol. XII. Prof. P. Zeeman and Dr. B. Winawer. The magnetic separation of absorption lines in connexion with sun-spot spectra. Pl> (45 ») 11. (45 «) ^ r Dj D„ D, D., (60 ») (39") Proceedings Royal Acad. Amsterdam. Vol. XII. ■^rof. P. Zeeman and Dr. B. Winawer. The magnetic separation of absorption lines in connexion with sun-spot spectra. Di D, S 9 in Types of sun-spot lines. (Mitchell) 5, 6. Widened lines with centres reserved bright. 7. Widened and weakened line. 10. Winged line. 5', 6', 7'. Types of magnetic resolutions in non-uniform fields. 10'. Superposition of magnetic components. Proceedings Royal Acad. Amsterdam. Vol. XII. ( 595 ) It). ll' ;i l''i.'i'',sNKi. rlidiiili ciiiiiliincil wiili a cjilcsiiur rlioiiiliis iiilro- diiced ill llic beam, dih: iif the (■(Hii|i(>iieiits ui' tlie quai'lL't aljso I'litirelv disappears. At an aiiule of tiO' this was only tiio case wiih ilie sextet. (Plalc li. Fi-. 11) 17. (>/>s,'rnitiv/is ,1/ ."> = 39^ Tiie elliptic polarisation tested by means of the calcspar I'liomb is very marked, even with dilute vaponr (Plate II, Fig. 12, Plate III, Fig.13). The inner components of the (piartet are now decidedly less intense than the onler ones. Plate III, Fig-. Hi especially shows the smaller intensity of the components of l)j in the lowest stripe. Indeetl, they are nnmislakahlv thinner than Iliose iji the upmost stripe. 18. According as the angle between the ray and the lines of force is diminished, the intensity of the Held must diminish at the same time. In onler to make it possible for the rays to traverse the field p.nder smaller angles the \ erfex semiangle of the cones iiuist deviate more and more from the theoretical optimum of nearly 55°. The decrease of the magnetic seiiaration is clearly shown in our photographs. We intend :o communicate on another occasion experiments under smaller angles i> and to enter niion some details concerning the case in which the components of the triplet are not neatly separated. Some measurements of the ellipticity of the components will also be given. On the present occasion we only iniended to give a general survey of the inverse effect, illnstrating it by some particnlai' cases. 19. Types of sc pa ration In. spot and /afjoratof//. In one direction we shall now enter upon some more details. The magnetic separation of lines in a iiun-aniform field has been treated on a former occasion. ') The results then obtained and our present observations may be of some interest in connection with certain phenomemi observed by H.\li<:. We intend to return to this subject. Presently it seems interesting to allude to Mitchell's descri|)tions of the various types of spot lines as indicated in the diagram jinblished iJi the Transactions of the International Solar Union"). Our Fig. 14, Plate III has been copied from this source. The types 5, (j, 7, and 10 of the Figure are very characteristic. Tyjie 9 perhaps falls imder the type of lines invisible without Nicol mentioned 1) ZiiEMA.N. These Proceedings, April 19U0, November 1907. -) Transaclions Iiileni. Union Solar Research, p. 199 etc. 1908. 40 Proceedings Royal Acad Amsterdam. Vol. XII. ( 59B ) ^ 7 a!ii)\('. Ill l'"ii;. 15 nrc rc|iros('iiU'(l sonic s('|iaf;vli()us dliscrxcMl in (lie laltoi'atuiT /rit/ioui Nicol uv oilier analyzer, 5', (>', 7' lia\e been lakeii in non-nniform fields. 5' is the quartet of /J, observed across the field ; (i' tlie sextet of D.^ observed axiallv in a non-uniform, in the cejitral part very strong, field; 7' also refers to J), in a weaker field, the observation being made across the lines of force. The type 10' refers to the Z), line, wlieii observed in a direction parallel to the field. The field is unifoi'ui. The separation gives an example of the superposition phenomenon mentioned in § 7. The analogy of the type 10', Fig. 15 and the type of the "winged line" seems very remarkable. Of course observation of the state of polarization would be necessary in order to prove the analogy. EXPLANATION OF l^LATES l-lll. The ligures 1 — 13 are about tliirteenfold enlargements of tlie images given Ijy llie grating of the absorption lines D^ and Do in a magnetic field. The upmost and lowest of the three stripes of these figures relate to (oppositely) polarized light; in the central part the phenomenon is represented as it is seen in natural light. Plate I. 1, 2, 3, 4, otservalions ± lines of force with diflerent vapour density. tj, G, observation // lines of force with ditTercnl vapour density. Plate II. 7, 8, observation at 5 = 60° calcspar rhomb alone. 9, ,5^60°, calcspar combined with Fressel rhomb. 10, 11, & = 45°. 12, S = 39°. Plate lit. 13, d = S9'\ 1 4, Types of sun-spct lines (adopted from Mitchell). 15, 5', C 7', separations in non-uniform laboratory fields. lU' super- position phenomenon g 7. Physics. — "7%' llieDnonvujiwtic properties of e/eiiinnts." By Prof. H. K. .1. (t. nu Rois and Prof. Kotaro Hond.v. (Communication IVoui tli(^ ISossclui-I.aboratory). ((lonuiiuiiiiali'd in llie meeling of .Jiuiiiary 29 I'JIO.) Ill i(Si)5 Cruii'','), llioiigli he had investigated relalively few sulislances, beliexed thai lie could forninlale his icsiills in the following rules: J. l''or paraiiiagiiclic siibsiaiiccs llie specific siisce|itibilily is in- versely proporlioiiaJ to the absoliile lemperalure. 2. h'or diamagnelic substances, mi llie conli-ary. llie siisceplibilily is almost independeul of temperature. 3. l^'or the laller class (d' siibslances, cliaii,i;es of physical state generally liaxc hardly any iiilliience. ')■ P. Guiiii:. .\iiii (Ic Cliliii. ci (Ic I'hys. (7) 5 p. l'S9. ISO,'), - (.»envrcs p. 232 Paris 1908. ( -^97 ) 4. 'I'lic Niiiic iidhU lur viiriiiliuus uf e'liciuicul ^lak' all()lrii|iy;. One oi' lis ill 19(H) iiropused to call tlie first of tliese tlioniioiiuig- iietu- rules ('imik's law ami to iiitrochiee a OovV'.v cvz/.v/'//// such that : Z (^ + 273) = C. It was also expressly stated that very probably this wa> only a kind of "limit-law'" in the sense of the analogons law for ideal gases. In addition it \vas vei'y soon shown that the nsual theory of directed magneeules leads lo such a law, when generalised from a more magnetokinetic poiiil of \ icw ; this was theoretically proved and experimentally coidiriued in ihc Loktatz- and Boss{HA-\olumes of the "Archives" '). Willi all <1ik' regard for CruiK's inijtortant re- searches and for his lirsl rule, the second can and conld have no general signitication, for il at once contradicted the results of other observers, e. g as in the special case of water. With regard to the third and fourth rules even their author pointed out several e.\ceptioiis. As the values of the susceptibilities of the dia- uiagnetic substances tested proved much less than those of the paramag- netic bodies, Curie came to the conclusion that these two oppo.site forms of magnetic induction were due to completely different causes. Starting from these e.vperimental coi. elusions, Langkvin -j in the year 1905 elaborated an electronic theory of magnetism; he also gave a kinetic representation of CVuik"s tirsi law, completely analogons to the one mentioned above, without, hou e\ or. nieiitioiiing it, and which is in addition perfectly independent of the introduction of electrons. It appeared, therefore, desirable to irnestigate the Ihermomagnetic projierties of more substances: in the first place those of elements, in order to Judge whether CntiE's conclusions admit of such an extensive generalisation. Il may be at once remarked that such is not at all the case. Experimental AtTdiKji'iiient. The method, previously used In CiinEand other investigators, of the torsion-balance combined with a non-uniform tield was applied, emjiloying the semicircular electromagnet recently described in these Proceedings. The axes of the two cores formed an angle of 10' to 20": the maximum gradient of the field then lies at a certain distance lo one side of their point of intersection. The tield 1) H. DU Bois, Rapp. Gongr. d. Pliy:^. 2 p. 486, I'aris 1900. — Arcli. Xet-il. (2) 5 p. ^246, 1900, also 6. p. 581, 1901. — Ycih. nat. en gen. Congr. 8 p. 60, Rotterdam 1901. Nutatiom: a. Atomic weight. S, Temperature. C, Cubie's constant. | y. Specific susceptibility. -) I^. Langevin, Ann. tie Ghim. et ile Pliys. (8) 5 p. 70, 1905. Journ. de Phys, (4) 4 p. 678, 1905. 40* ( 5f1S ) ilsc'ir ;il lliis parlicnlai' |i(iii:l ainoimlfd In 25 kilii^iius^es ; it was iiieasiired tVuiii point to point by means ol' a small standardised splierioal test-coil. The sensitiveness of the torsion-balance could be varied; it was determined in the nsiial way by means of applied additional moments of inertia. The fnrjiace consisted of a porcelain lnl>e wonnd with plalinuni wire and insulated with kaolin powder and asbestos. With a con- sumption of 1.2 kilowatts a temperature of 1250° was attained, which was measured by means of a ihermo-elcmcMl, pi-e\iously checked by observations on the melliny-|ioints of tin, tellui-ium, antimony, and gold. Ti'st-.sariiitli's. I'he threat diflit-ulty with all experiments iji this sphere of work is and always will be the prevalence of iron, with its overwhelming ferromagnetic jjroperties, though it hardly ever seems to act (piite freely. In the case of (ifieen elements, their binary alloys w ilh iron were (wamined in I'a.mmann's laboratory, not in the x'ery diluteil slate, howex'er. which generally corresponds to ferrnginoous inipnrilies. ( )!' SI elements, 4o were tested; many of them were supplied as pure as possible by l\Ani.HAi\M ; I'rof. ( 'oiikn and Dr. HorrsivMA of rirechl kindly placed se\eral elements al our disposal; as yel Ihe l<> gaseous elemenls ha\ e noi been tested; Li, Rb, Cs, ('a, Sr, 11a coidd iu)l be obtained sufficiently free of iron; while r>e, Sc. (ia, (ie, ^, IM, and the rare melals could not be procured. Ke, Co, Xi, af course, form a class by themseU'es. Dr. AI. Hanua kindly delei'inined the percentage of iron colorimelrically by the l^erliu blue-reaction. Till' ('.I'lii'fiiin'iildl rfsiills, moreover, furnish certain pli_\sical criteria of their own reliability, for in so far as the susce|)tibility proves indepen- dent of the field there can hardly be (piestion of a ferromagnetic ingre- dient. With about one third of liie samples this was not the case, for the susce[)tibilily dimiiNslieii (in the :i!,ucbrai(' sense) with an increasing field according to a hyperbolic law. l^'i'om tliis Mr. Moitiiis Owen calculated the value w^ wlucii would hold asy mplol ically for an infinite field; and, HI .-iddiliou. the lulbience of the ferrouiagnelic inuredienf, w hich al most amouuled loonl\ one siMJi and generally much less e\'en — of w lull could be imputed to the iron ui the free stale. 'JMie tliermo- nuiguelic projierlies aUo all'oixl a lest of [lurily u|i to a certain point ; a few strongly ferrii,i;iueous substauct's show ;i threat diminution of snsceptilidily between 501)' ami liOO, w lids! alxwc 700^ the inllueiice of iron hardly need lie feai'e(l. In no case is there reason til doubt lli:il ihe \alue of Ihe suscept ibiiily of absolntelv iioii-terru- uineous eleuieiiis woidd icmaiu conshmt. :il leasi within the usual ( .-><.;. ) liclil-runm'. Till.' full (■iiiniiiiiiiicaiiini iif ihr iv-iill- nlilaiin'il wdnld n'i|iiirc iiiaiiv lalili'> ami curves; \\i- llni-i'lini' ilraw allriilinii in llir ]irinci|ial |M)iiil> (inl\ , S^ii'ci/ic sKsci'iitihUilii ') at 18°. Tlie valiiON tumid lie between 1 .111(1 -\- 5 (aiiioi-plioiis ciirbon ami |)alla(liiiiii res()ectivel_v). It camioi he luaiiilaiiieil lliat tlie positive parainagiietie values are on the wiiole iariiiM- liian tlu' neuali\e (iianiaunelic ones. O.wjien alone forms an exceplioii wiili a value of almnl 100: ihc value tor nianjianese \vas ap|)r(ixiuiately 10: tins eiinlaiueil. Inuvever. ' ," „ of iron. ('iKiF. had alreadv p(nule<l oiH ihe intkienee of allotropv in the ca-i- (if pli(isphoi-us and antinionv, and also ihat there is no such inlliKMice w idi sulphur, lh(>u<;h it is so well-known for it.s polymor- phous |)i-opeities. A ditferenee was shown to exist Ijetween diamond ! — 0,49) and am()r|»hous earbon { — 2,02): silicium crystalline (0,12: and aniorpluMis — 0,14): and espiM-ially between eoniuion lelraiional tin ^-j- "'•*'^) <*'"' -'■'^'.^ ''" 0.29,. Ill the ease of tin, the tirst — the tetragonal — was IvAiii.HAt m's very pure eleetrolytie material; it was afterwards inoculated with a small quantity of grey tinpest, kindly sent by Prof. Cohk.n from the stores of the van 't Hoff Laboratory. For weak fields indium seemed to be paramagnetic ; in a field of 7 Kgs. the value of the susceptibility passed through zero and became negative, doubtless in consequence of 0,013" „ iron : this [ihenomenoii is of no conse(pience becau.se it is also discoveced in complicated sub-stances such as certain kinds of porcelain, glass, etc. >>otwithstaiidiiiu many oinissioiis, it was still possible to follow the general course of the curxe >; = fmict. ,«/ ; ihe cur\"e a[ipears to he rather intricate, but still shows a distinct relation to the periodic system. According to the arrangement of Mendklkjeff-Br.wneii's table, the rows (1, 2, 3, 4), (5, 6, 7, 8), and (9, 10, 11, 12) each form a division 1, 11, III in which the shape of the curve repeats itself in a peculiar way. At the junction of I and 11 Cr. Mn, Fe, Co, }vi lie on a positive maximum: between II and III, in the same way, the ■'rare'" metals: within 1, II and III the diainagnelic negative peaks are occupied liy the similar penta\alent elements P, Sb and Hi of the fifth group (3'', 7''"', IP'' row). In more than one respect further magnetic analogies of secondary importance exist, which, however, must be left unmentioned in this communication. Siisce j)f /'/)/' lit 1/ 'it hiijli /i'inji('i-)itin'('s. As a rule the path of the curve X = funct. ((9) for any substance pro\ ed to be the same when the temperature was increased or afterwards decreased ; certain deviations probably depend on a change of condition of the iron present aflei- 1) Everywhere below expressed in millionths 9 ^ I (i(M) ) I' J.. H =r 70 %) i I I > n N Ln ■w 3 O q 2 c 3 =r 3- Q o i 1 3 2; 1 o CT rc £ IS o n 1 1 _ > o CB 1 o '< 3 ^- D. -^ o I m -0 H - H C/5 5* n N N n O CT fD D. "3 r 3 !^ o r o o o o 3 o b i s o o S 1 -o 3- o o ^ tjl ( 'iOl ) liealln.n'. .Mu ami Kn sIil'WimI ihc aliuxi' nn'iiiioncil (liiiiiiiiilKni in a imirUoil iiiainii'r IhMwcci! .MM) ami VAH) . The rt'siills are rollee-ted ill Iho talili- |i. 1)0(1. The (■l('iiiciil> in s(|naic iiracUels have pre- \i()nslv licLMi ('xaniincd l)V (illirr> : llic alnniic wi'iulils iji cacli colunin iiinvasc fidni l(i|i l(i iidlloni; llic I'lcnu'nl'- luidei- coiinnn C show a conslaiil su>cc|itiliilil\ , lunlci- / a niinicric iiicreiise on heating and iind(M- /) a iiinncrir ilcrrease. The fewest number (4) of elements ap[)ears in llic lifih (■olniiin, in the case of wliicii the susceptibility increases on heating, the iiici-ease itcing, liowever, very small in each instance. From a lli(M'nioinagnetic point of view a certain relation also e.xists in connection with the periodic cur\ e / =i fiinct. (a) : the paramagnetic elements under J) ail lie at the principal maxima or at the secondary peaks; on tiie contrary, those under I lie on the ascending branches of the curve. Therefore the sharpness of the bends would be tlattened more and more at higher temperatures; probably at lower tempe- ratures they would become more accentuated, and it may be that only then do they attain their most characteristic shape; of course the temperature of -)- 18^ is (piile arbitrary. Concerning Curik's rules ihe following Statements may be made: J. Only palladium foil from Kahlhaiji, with 0,70"/^ of iron and X = -f- (],1'2, on heating followed, more oi' less. Curie's law, but on cooling it shewed complications. With much purer palladium from Dr. Hekaeus, with 0,03" „ iron and ■/= -|- 5,79, the susceptibility fell less rapidly than would follow from (Jurie's rule; lemperatnre- liysteresis was not observed on cooling'). 2. There are t)nly (i diamagnetic elements which do not vary within the whole tem[)erature-range. Of these P, S and Se had already been experimented upon by Curie. 3. On melting or solidifying, sometimes — not always — a dis- continuity appears, which can be classilied under one or other of the two following divisions: 1, a large or small leap in the curve of /itself, as with P (44°), Ag(961), Sii (233 ), Sb (631°), Te (450"), An (10(54°), Tl(290'), Pb (327°), Bi (268°) ; II, a rather sudden change of dyjde as with Mg (633°), Cu(1065'), Cd (322°), 1(114°); with regard to sulphur, the curve at the melting-point departs slightly from its otherwise absolutely rectilineal character, which variation was probably overseen by Curie. 1) By chance palladium is the only paiamagnetio element examined by Curie ; perhaps it was not pure enough. Tlie important resuits for oxygen, for ferromag- netic metals at very high temperatures anil also for their salts crystallised or in solution, of course continue to hold. ( CAYl ) 4. As I'c^ards ilic i1il'I'iu()iii;i,i;ih'Iic cxaiiiiiialiou of |iul\ iii(ir|ili(His Iranslnriii.'ilidiis, a (lisciiiiliiniiHis (liiiiiniilinii of I •'i" „ nf llic spcrilic siiscr|ilil)ilil\ was I'oiiikI al iIh' li-aiisiliiiii-|Miiiil of (f-llia!liiiiii and ,-?-llialliiiiii al TA-i . liiil llir iiuisi iciiiarkulile pi'opcTlies arc sliewii liy liii: ir (liaina.iiiiclii- ,m-ev liii is shiwlv heated, at 32° tlie specific sii.sce[)til>ilit_v ( — 0,29) cliaiiges almost siiddeniv (liiie tlio density) and at 35° passes through zero. Possihiy this cliange woidd wliolk take place al liie |)oinl of Iransloruialion ( IS") Imi iheii al a inncli slower I'ate. Fiulher healing conlinuoiisly increased ihe sus- ceptibility so thai at about 50' ihe \alne (-|- 0,03) \'uv |iaramag- netic tetragonal tin was reached, which .aflerwards remained practically constant; according to Deukns the poinl of Iranstufmalinii tetragonal ■^ rhombic tin lies at \iM° at which leinperatnre nothing particular was noticed; at the melting |)oint (233") a discontinuity from ■/ ^r -)- 0,03 to /= — 0,04 once more appeared; the diamagnetic liquid metal remained nearly unchanged. An extension of these thermoniagnelic iinestigations Inwards low temperatures is in preparation From the above, especially fnnu ihe conclusions arrived al under I to 4, it seems to follow that Cirik's four corresponding rules are certaiidy devoid of the general meaning, which has rather rashly been ascribed to them. At the same lime ihe expei-imental slarting- points of Lanoevin's theory are e\ideiilly undermined; more solid and broad foundations for future theories can only be laid wiih Ihe aid of more extensive reseairh. Chemistry. — "Sludies nn 'ri'Ihirium -. 1. The niiitunl hi'lmrioiir of tlit' ('lements .'<i(Ijth>ir mul ti'lltirinm" . \\\ Prof V. .M. .lAWiKH. (Connnunicated by Prof V,\n IIomiu udin. ((Jorainunicaled in the meftiiig of .Inmiarv ;^'.t, I'.tjni § I. Whilst we are in the main thoroughly intormed as to the i-elatiou of seleiuum and sulphur, the views as to ihe uiiilual beliax iour nf the elements tellurium and sidphur still dilfer somewhal. Ki.\i'iu>tii ' i has already investigated this sid.jecl. lie slates thai (.n melting together tellurium and sulphur leaden coloured masses aiv formed crvstallising in rays, which, on heating, give off sulphur and yield a porous metallic looking mass, which he takes to be telluriumsul|>hide. 11khzki.Ii s ■'), ihirlv years later again bi'oached the subject; he found thai no compounds were formed nu mellin-. but thoiighl ihal ihe ') KLAMtOTH, GrcUe's Ann. (179S). I'-d. 3) liKHZELius, Gilb.-Pogg. Ann. 8. (182(i|. ii:i. f );o:5 ) (■niii|i()iMi(ls Tl'S_. ;uiiI TcS,, ai'c iircsciil in IIm' lirnw iiisli-lihick |ir('ci- pilatc^, luriniMi wlicn |iassiim II, S liii-dii'jli sdlnlKin-^ nl' Iclliirilc^ and U'lliirah.'s. Il(.' arri\e(i at llial coiiciiisHiM on a.<'C(Miiil nl' llic solnhililv oF these |ireci|)itates in ai|nei)ns pnias^inin or sodium liydrnvidc, wliicli is also tiie case with Te*)., and Te*).,. Bkcker ') was tlie lii-sl to analyze these precipitates and he linallv arrived at the concdiision that liieir c-oniposition aclnally c(n-responds with TeS, and TeS.,. He pro\ed ho\ve\er, that nearly all the snlphni' may be removed from these substances by treatment with carbon disniphide: Te.S., yielded a residue containing 6.14 "/o of sulphur instead of 42.85",,, TeS,, a residue containing 3.69 "/„ instead of 33.4 7„- He concludes that the blacdv precipitates are only mixtures whose composition agrees nearly with those of the supposed com- pounds According to him they are formed primarily as ephemeral compounds, which are at (inro decomposfed by the solvent. Berzkuus^) and Oppknheim ■') obtained double sidphides to which they assigned the formulae SK^S-j-TeS.^, etc. In moi'e recent times, Brauner ■*) and GuTBiER °) again inclined to the opinion that we are dealing here with mixtures of the elements. § 2. Since Dumas placed lelbirinni iji the sulphnrgroup, as the tlrst homologue of selenium, and ihns the well-known difiiculty as to the position of tellurium, in regartl to iodine, in the periodic system introduced later, was created, — the ipiestion as to the relation of tellurium on the one side and snlphui' and selenium on the otiier has again lieconie of actual imjxnlance. For now it is undoubtedly certain that the atomic weight of tellnrinm is I'iT.ti and therefore (/renter than that of iodine. On the other hand the dilferences between tellurium and the other two elements are so strongly pronounced that Retgers on account of the isomorphism between tellurates and osmiates, thought it would be i)etter to include tellurium in the grouj) of the platinum metals. Tellurates to wit, are not isomorphous with sulphates, seleiiates, manganates, ferrates etc. On the contrary, Pellini showed an isodimorphism in the case of (C„H5),SeBr, and (C^Hj^TeBr.; , whilst Nokris and Mommers noticed a direct isomorphism between the selenium- and tellnrinm donble chlorides and bronndes of diuiellni- 1) Becker, Lieb. Ann. d. Ghem. 180. i bST6). '257. -) Berzelius, Traite de Chimie. (1830). •') Oppenheim, Journ. f. prakt. Ghem. 71. (IS.")?). 270 ■*) Brauner, Journ. Ghem. Soc. 67. (1895). 527. '") Gutbier, Berl. Ber. 34. 2114. (1901). ( H(t4 ) ;iiniiii_'. Hill (III llir nllirr li.'iinl inaii\ uhjccl ions lia\i' Itcoii |•ai-^(•^l lo llic |iiisi(ioii ;i>..sijj,iUMl lo Iclliiriiiiii : Ini' iiislance, llie (lillercMil cuiisti- tiilion of telluric acid w liicli, |ii(ilial>i\ . iriiisl he lonkeil upon as H„Te(_)„ and the lolallv diHereiil liydratioii of telliirales in coiiipai-ison widi sid|iliale-i ■iiid ■'^eieiiates. llowevei' lliis may he, il is hiuiiiy desirahle lo ohlaiii more ihitd as to the |iositioii of Iclliiriiiiii aiiioim- the other eiemeiits and I'or liiis re.ison, ihi^ relalioii to sidpiiiir had lo he ascertained in tlie lirst jilace. § 8. The teilnrinni was ohtained from I'/j l<ilo of crude telhirinm I)rol)ahiy derived tVom Amei-icau ore. It appeared to contain tiie following- elements: telluriuni, selenium, sulphur, lead, copper, hisiiiuth, iron, silicon and traces of antimony, zinc and a few other metals. The first puriiicatiou was carried out hy oxidation with aqua rci/ia, evaporation of ilic lillrate to di-yiiess, and re|)eated e.xtraction of the residue with strong hydrochloric acid. The hoiling iiltrate was then precipitated hy snlphnr-dio.xide ; the lirst portions of the precipitate heing rich in seleninni were each time rejected. This operation was repeateil three times. The amorphous tellurium was divided into two parts; one portion was converted, hy liic process given hy Norkis, F.\Y and Edoeiu.kv '), into Itasic tellurium nitrate TeJJ,(( >H)(N(>,) and h\- repeating llie process ti\-e times, wiiich (([leration lasted many weeks, it was linaily ohtaineil (piite pure in the form of the said salt: from this, pure TeO. was then ohtained hy gentle ignition and this, dissolved in pure hydrochloric acid was precipitated hy SOj. The other jiortion was couNerled into telluric acid by means of (JrO,, according to Staidknm wi-.it's process as modilied hy Uutbikr-): this was [inrilied hy precipilaliiig twelve times with idtric acid and then crystallising from water. It is necessary to reduce the adhering C'rOj with alcohol, otherwise the telluric acid ci'ystals retain a yellow colour which is caused hy oi-chuled solid ( 'r( )., ; this mailer I hope to refer lo shortly. The pure tellui-ic acid was then reiiuced completely hy hydrazine hydrate. The crystalline form of (lie liasic intrate has not heen descrihed up t(.) the present. The following data have been ohtained from the substance crystallised from nitric acid. 1) NoKRis, Fay ami Ed(;kui.i:y. Americ, (JIilmu. Jouni. 23. 105. 2) GUTBIER, Z. f. iuiorg. (Jhein. 29. 22. (19U1); 32. 90. (1902). ( oor, ) Fig. 1. Crystalline form of basic tellurium nitrate. ( 'olonrless, xt'vy liisli-oiis needles n|i l(j 5 III. 111. ill leiinlli jiiiil iiMially llalteiied ;il()ii,i; !<*"*!■ I liev e.xliiliil iiiaiiv \iciiial pliUU's |iarliciil;u-lv in the \erlicai /.oiio, and greater angular dilferenfes ofciir also in different individual crystals. Tlie measure- ments must, therefore, lie regarded only as approximations. Rhoi))hir-/jipj/i'(iiii/(/<il. n : f> -.(■ = ().59() : 1 : (».fi()7. Forms; ;//=r|ll()j, h=\{)[{)\ aiHl7/=|J2()J, very lustrous; [)articnlary b, which is also a cleavage plane and possesses a high lustre. On the other hand (/ = \0n\ and s = \()21\ are dull, the form j021j is mostly absent. The crystals exhibit a pronounced inclination to tetragonal svmmetrv. Angular values: Measuied n> -.m = (110) : (iTO) =» Bl- 5' h:q =z (010) : (Oil) =» 58 44 ^ /// : p =(110): (120) /):h =(120): (010): Calculated q:>l II) : ([ III : h 'I = (011): (011) = = (110): (011) = = (110): (010)= .59 = (010) : (021) = (021): (011)= 19 19 25 39 59 62 50 74 54 .59 27i 39 46 19 19° 18i' 40 2i 62 31 74 43 59 27i 39 29 19 16 Completely cleavable towards j010{. The optical axial plane is jOOlj with the r/-axis as first diagonal. Strong rhombic dispersion with o <^ v . the apparent a.\ial angle in cedar oil (1.51) was about 63°. Tt may be observed here that the tellurium [irecipitated from telluric acid by hydrazinehydrate is distinguished from that preci- pitated from a hydrochloric acid solution by sulphurdioxide which is also amorphous, by a perceptible darker colour. It is, as yet, undecided whether this is merely due to another degree of division or to a real allotropism of the amorphous modification. \\ 4. liiilli lii;isscs (if li'lliii-iiiiii niivcil willi T) (i liiiic> llic .■iniuiiiil ol' |Hi\\ ilci cil, IVcsiilv |ii-('|)arc(l |i(itassiiiiii cvaiiidc were I'lisrd Ini' some lioiirs in lariic Kosk criicihlcs in an aliiinsplici-i' (if coal f^as, with iIr' aid of a lariio I'KRKoT-fiiriiarc. In tlie course of a fi'W inonllis, ahoiii 5 kilos of llieso mclls wci'c ohlaiiied. When cand'nilv powdered, tiie dark coloured masses dissolxc in recenllv boiled, lioi walcr hi licaulifnl |inr|ile colonr<Ml sohii idiis, \\hi(di on cold oxida'ion hv |inrili('d air dcposii frinn llie Iv.Te all lln.' lellnrinni in lirilliani needles, ( )n inellin,u llie masses, llie poisoiions iidlneiice (if llie li\ - dro<>eii lelluride, w liicli is formed in small (jiianlilies, was ex|)erienced onlv loo plaiiilv. also llie disu,ureealile coiisc(|nences of lireathiiiii- tlie verv small i|iiaiililies of Te('l,_, formed dnrinu- llie Irealmcnl willi (KUKi ir</i<i. Foi' weeks aflel•war(i^s liic hrealli lias a powerful odour of (CH,).^ Te, wliiidi resembles pliospliiiie and is exceedingly sensitive lo llie olfaclory iierse (if byslanders, ') The crystalline and already vei-y pnre lellnrinni llins obtained is free fi-om seleinnm as proved by llie exceedingly delicate Xoiuns" potassinniiodide-reaclion and by the non-reduction of the TeO,, lyv hydroxylaniine in strong hydnuddoric acid solnlion. All llie seleiiinm lias been removed as KCNSe, whilst llie lellnrinni lias passcil inio K.,Te and llien has again been liberalcd In llic aclion (d' air IVee from II, S. 'Jdie pnritied element was now dislilled in vacuo al about (500--- 700^ in long lubes made of hard niass and conl.uning pings of asbeslos ; a Tkci.i fni-nace was \[<oi\. 'I'iiis operalion was repealed about seven limes, each time abonl 10 grams were used. The pnre tellurium lliiis oblained was siUcry while and coarsely ciy>lalline, much resembling crystallised anlimony. The determinations carried onl have been made with (he prodnci obtained from lelluric acid. The siilphnr was recryslallised Iwice from boiling toluene and healed in a drying oven al !)0 ' for some hours. ') The opinions iis lo llic |iliysiiiliii^i(Ml aclioiis ot lelluriuiu are slill very iiiu(-li divided. Although seleniu'ii is an eleiuenl liaidly le.ss poisonous lliaii arsenic, lellu- rium is considered by Gzapkk and Weill (Chem N (1893), 1098 2) to be com- paratively iiarndess, owing to llie mucli more rapid reduction of the lellurium compounds and the consequent localisalion in the oigaiiisin. The experience gained in my laboratory proves Ibis view lo be incorrccl. Tellurium is nndoubledly poisonous, bul tlic individual sensitiveness to small traces varies widely with dillercnl persons. Tell.,, in particular, is a poison causing severe headache and vomiting: odier lelluriuincoinpounds such as TeCb, for instance are supposed to cause miicli inconvenience only, owing to their conversion into malodorous subslances, hul slill llieie can be no doubt whatever as lo llicir poisonous nature. ( fi07 ) § 5. Tlie coiistnictinii of llie luclliii;^- ;lJ)|nu■alll^ will lie rcadilv seen from fig. 2. Tlie liard giasn tubes always tilled wiili JO grams of the weighed and well mixed comijlex of the two eleuieiils were placed ill iron cylindei's Idled with tine sand. Tube and cylinder were covered with asbestos; the requisite atmosphere of nitrogen was supplied by way of a hard glass gas-inlet-tube. The nitrogen was prepared from NH,C1 and KNO,, freed from oxygen by means of alkaline pyrogallol and sodiumhydrosulphite and dried by sul- phuric acid. The furnace was coiislructed of chauiolte stone furnished F\^ 2. with an asbestos tilling and a central cylinder of unglazed earthen- ware; it was covered with an asbestos board resting on three little chainotte blocks wliicli were either removable or not so, for the regulation of the \elocity of cooling. The icekellle for the cokl solderplace of the plalinum-plaliiinmrliodinin thermoeleinenl {'•> mm.) is double walled and allows of working for some six hours with the single supply of ice; all Ihe eondncling wires were isolated by glass tubes. The galvanometer of .Sikmkns and Halskk was verified by deter- mining the melting[)oiiits of liu, lead, bismuth, cadmium, zinc, anti- mony and silver and by making use t)f the values found by Day and Iloi.iuiKN and by Day and Ci.IvMKNT, which were compared with Ihe gaslhermomeler. The readini;' was taken with Ihe aid of a lens, the counting of the lime by means of a clockwork, which gave a siiiiial e\cr\" 10 seconds. , ( 608 ) § fi. (irciit (lit'liciillies wei'e e\|iericiiro(l in tlic ik'lcniiiiuilidii ; wlicMi we (leall willi luixtiii-es coutaiiiing imicli telluriLuu every pre- caution iiad lu 1)0 taken to prevent the boiling oil' of the .sulphur, and ill the case of complexes containing much snlpiiur trouble arose from the great viscosity of the fusions and the very slow crystallisation of the masses. If the percentage of sulphur exceeds 80, the deter- MELTING POINT DIAGRAM OF SULPHUR-TELLURIUM COMPLEXES. MOI. ";'„ Sulphur o/o by Weight Sulphur Initial solidifying point End Period of solidifying solidifying point in C° seconds (1 (1 4521 — 5 -1.3 440 437 - 10 2.7 435 430 — 1.-. 4.2 431 423 ~ 20 5.9 420 — — 25 7.7 421 - 30 9.7 413 103 30 35 11.9 401 102 09 40 •14.4 394 lOG SO 'i5 •17.1 389 104 - 50 20 387 105 100 55 23.5 385 105 - I'lO 27.4 374 109 — 05 (id. 07 31.7 33.4 308 360 101 105 100° 1'.5 100 70 36.9 3G1 105 - 75 43 348 108 180 SO .50 339 109 180 .■^5 58.7 — 110 210 00 09.3 2SS 108 230 05 82.7 - 109 200 OS 92.4 - 110 - Hill 100 115 - - ( 609 ) nii]i;ili()iis iiftcii Ixx'Oine vcrv iiiicfrluiii ; soiiic uf llioc inixtiiri's milv oxliibited a sharp eiid-solidifyiiigpoint. . Still it was geueially possible to get eoncordant results on repeatijig llie e.xperiiiieiils. The sul»joined table shows the results of the experiments. Fis-. 3. ^ 7. These (Jnfn i-e|ireseiiled graphically in Fig. 3 in the usual manner show, therefore, that the elements sulphur and tellurium when melted together yield, when solidifying, two series of mixed crystals of a diHerent erystalline form. The condition diagram is that which has been noticed frequeully with isodimorphous substances ; there is a very extended hiatus starting fidiu almost pure sulphur to perhaps 27" '„ of sulphur at the side of the trigiuuil mixed crystals. The temperature of the euteclic point A' is 10t)°; the time retjuired for soliditication, as far as could be ascertained, increases continuously with the percentage of sulphur until the pure sulphur is reached. The mixed crystals rich in stdphur have a slight I'uddy colour; as very small amounts of tellurium imjiart to sul]ihur an inlen.sely red colour, their tellurium couleiU must be small indeed. They exhil)it the ihiu Jieedle slia|>ed form of mouoclinic sulphiu-. The transformation at 106^ may be seen beautifully with the eye in the various melts on cooling as well as on warming. The monocliuic mi.ved crystals ( 610 ) rii-li ill siil|iliiir appear Ik cliaii^c iiilo llic rliniiiluc rorin al a 'owrr leiiipi'raliirc. In lliese riiviiiiistances iiolliiiig' ia noticed as to c-oiiipoiuuls between Iclltiriimi and sulpliiir; even at lower teniperatnres no heat effects are observed. The melts of the mixtnres rich in tellnrinin are dark brownish black and in thin layers yellowish brown ; uidike llie iiicils ricii ill siilphnr lliey ai'e ihiii lliiid iqi lo ihcir solidifying points. § 8. Considering all llial is known ii|> to llio prcseni as to the behaviour of the elements sulphur, selenium, and lellurimn (in being melted together, \vc may say thai in this re.specl, tellurium certainly deserves the |)laee assigne<l lo il by Duma-s. Sulphur and selenium form, accdrdiiig lo I{iN(,kk' , a Irimorphons series of mixed crystals, selenium and lelluriiim. accoriling to 1'ki.i.ini and Vio '■') an nninter- riipled series of trigonal mixed crystals; but no compounds are formed, as may Ite expected, looking al \\ii' ex|ierience gained, apart IVnni llie exceptions in sucli Iriads of homologous elements, — at any rale in ihe central groups of llie periodic system. Willi sulphur and selenium llie matter is even somewhal still nmre com|ilicated, as three instead of two heteromoi'phons kinds of mixed crystals occur ill lliis case. If we accept Rktokrs' view according to whom a less stable form, nioslly uiiknowu in llie free state, of each of the compoiienis should correspond to each of these forms, the isotri- morpliisni in ihe case of selenium and sulphur is certainly more diflicull lo explain than the dimorphism of sulphur and tellurium. For of Ihe two monoclinic series in the system; sniphnr-seleninm one, according to Muthmann, is analogous to the form of y-sulphnr, whereas the trigonal series woiikl ab'eady [lossess the form of metallic selenium. l!iil neither of the Iwo known monoclinic modilications of selenium is isoniorph(uis with any iiionocliuic modification of sulphur '), whilst the trigonal so-called f-forin of this element dilfers from the trigonal form of selenium. Looking from RKT(iKKs' standpoint Ixitli these elements should be credited, in addition lo Iheir well known allotropic forms, with at least another two unknown, less stable modilications. In the trigonal series of the system: sulphur-tellurium we are dealing obviously with the same less stable trigonal form of sul|)liur as in Rixokh's investigation, whilst the assumption of an unstable monoclinic form of lelluriiim cannot have anything artificial about it, in \iew of Ihe fact thai this syiiinu>try occurs fre(iuently 1) lilNHliB, Z. f. aiiorg. Gliem 32. 181). (1'.I0l'>. -) Pellini and Vio, Gazz. Chim. It. (1900). 11. 17(i. ^) CiRnTH, Ghuiiiischc Krystallographie, Bel. 1. (I'.JUOj. p. -IS— '6b. ( fill ) both with selenium and sulphur. The research of Pellini and Vio also does not introduce any further complications; the two elements are united there in all proportions to one trigonal series, so that only the sulphur-containing complexes of selenium and tellurium exhibit the Iiiati on mixing in the solid condition. All this admits of the conclusion that the elements sulphur, selenium and tellurium form indeed a natural triad of perfectly homologous elements which are more adjacent to each other than their group- fellow oxygen is to any one of them. There can be only question of true "compounds" when one of the elements combines with oxygen '). § 9. Now there is still the question : what must be thought of the tellurium-sulphur complexes which are formed, at the temperature of the room, by means of HjS from solutions of tellurites and tellurates, and in what sense must the so-called double sulphides obtained by Oppenheim and Berzelius be regarded. First of all, I soon succeeded in showing that the element tellurium and particularly its amorphous modification dissolves, without leaving any residue, when heated with a solution of alkali- or ammonium- sulphide, also that the solubility increases with the concentration of the sulphide ; and further that the solubility also increases when potassium hydroxide is added to the sulphide solution, thus retarding the hydrolysis. Clear yellow solutions are so formed turning some- what ruddy on boiling, and oxidising rapidly in contact with the air with formation of a black precipitate. They are strongly alkaline and give with hydrochloric acid a heavy, black precipitate with evolution of H,S; the precipitate appears to contain tellurium as well as sulphur and is soluble in alkalihydroxyde. The analysis of these black precipitates did not give constant values ; the tellurium content is much dependent on the modus operandi and oscillated between 46.9 7„ and 80.9 7„. Thereupon, the action of H.,S on different tellurium compounds ' was investigated : on the basic nitrate, on the finely divided dioxide suspended in absolute alcohol, on dioxide in hydrochloric acid solution, on telluric acid in water and on the tellurite- and tellurate-solutions obtained from TeO, or telluric acid. A beautiful, somewhat crystalline looking product was obtained from the alcohohc suspension of TeO,; the analysis of the blue-black substance gave 80.1 7o — 80.9 7o of tellurium whereas theory requires 79.97„ for TeS, 66.67„ for TeS, 1) It is, moreover, also known that and S, for instance, never give isomorphous substitutions in organic compounds. S, Se and Te, liowever, behave differently as shown by the research of Pellini, Nokris, Tutton, and others. 41 Proceedings Royal Acad. Amsterdam. Vol. XII. ( 612 ) and 577„ for TeS, so tluit the composition came verj near to tliat of TeS, but with an excess of tellurium. The telluric acid was not reduced in the cold ; the TeO^ dissolved in hydrochloric acid yielded a black precipitate witli 7l.27„ of tellurium and therefore situated between TeS and TeS, . The basic nitrate is rapidly attacked by H,S, but only at the surface; on the other hand it dissolved completely in boiling ammo- niurnsulphide, which solution, after being concentrated in vacuo at 40°, and allowed to crystallise in a vacuum desiccator over CaO, yields hard, pale yellow needles of a compound which may be recrystallised in vacuo without decomposition. The colour of the needles is greenish-yellow: they dissolve in water to a clear, yellow, strongly alkaline solution, which rapidly oxidises when exposed to the air; the crystals also soon turn black on exposure. The analysis gave a varying tellurium-content according to the method of prepa- ration; in one instance were found 20.1% (NH,), 42% Te and 37.9% S, which leads to the formula (NHJ,Te,Sj '). In an analogous way the potassium compounds were prepared from tlie tellurite and tellurate with HjS, by solution of the precipitate in the solution satui-ated with H^S, or by solution in KOH, and by conduction of H^S through it ; the solutions were evaporated in vacuo, and were then left to crystallize over calcium oxide. Apparently the same ;y ellow compounds are formed in all these cases, which crystal- lize in rosette-shaped aggregates of hard, fine needles, which in a humid condition smell strongly of H.S, and yield clear, readily oxidisable solutions. Also the solid salts themselves oxidise rapidly, in which tiiey become greenish yellow, and finally j)erfectly l)lack. On close investigation the colour appears to assume different shades, even if to all appearances the same mode of preparation is used. Attempts to find a constant composition for these salts, have been unsuccessful ; successively it was found to be : 35.47o Te (calculated 1) Tlifi analysis of those complexes Is a very tedious operation. If lellurluni only has to be esllmatcil and no sulphur, the reduction process with SOo or NaHSO;( Is still to be preferred. In our case, the tellurium had to be precipitated from a boiling, strongly ammoniacal solution with hydrazine hydrate, which reduction proceeds very slowly and also incompletely, and had often to be repeated half a dozen times. The last traces of still dissolved tellurium betray themselves on heating by the line steel blue colour of the colloidal tellurium present; this is generally completely precipitated on rendering the liquid acid, and by way of control the siihition may then be heated once more with ammonia and hydrazine hydrate. The tellurium was collected on a weighed filter dried at 100^ and weighed as such. The SO;; was weighed as BaSUi, the K as KCI or KCIO^ the NH^ as NH4GI. For obvious reasons the analysis of the barium salt is a very tedious affair. ( 613 ) for KJe,Sj 35.7%); 33.5°/„ Te, 33.4% S and 33.1% K, which answers to a formula K,„Te,Si,; another time (for a product prepared from KJeOj: 44.7% Te, 31.47% S, and 23.7% K, which would correspond to Ki„S,„Te; ; again another time at somewhat higher temperature: 37.5V„Te,' 34.3°/„S and 28.17„ K, which leads to a formula Ki^Te^Sj,. The behaviour is practically analogous to that found in the poly- sulphides of the alkalies towards sulphur, where, according to KtJsTER's researches '), very complicated equilibria between different polysulphides and their dissociation products occur in the solution ; and to that of selenium towards sulphides where, according to Messinger -), a portion of the sulphur of the complex sulphohydrogen sulphides may be replaced by selenium, forming such compounds as Na, S Se, which, therefore, belong to the type of a /nsulphide. The behaviour of amorphous tellurium towards sulphide solutions as described previously also agrees with the notion that we are dealing here with salts of complex tellurohydrogen sulphides which in such solutions are in dissociation-equilibria with each other and are moreover split hydrolytically. The type of the i'r/sulphides becomes then of particular importance next to that of the rf/sulphides : K,„S,,Te, may be derived from KjS, by isomorphoas substitution of Vs of the S by Te ; K,Te„Sj and (NH,), Te,S. similarly from K^S,, or (NH,), S, ; on the other hand Kj^Te^S,, has again the character of the type KjS^ etc. ^ 10. Although these compounds do not as a rule occur in measur- able forms (the K-salt was obtained a few times as beautiful rhombic parallelopipeds with faint double refraction and without perceptible dichroism) I finally succeeded in obtaining very large, yellow crystals, with many planes, of a barium salt prepared by dissolving the black precipitate formed by the action of H^S on potassium tellurite in BaS-solution. The analysis indeed, did not always yield precisely the same results, but still tlie composition agreed closely witii the fornmla BajS-Te^ ; in one instance the normal composition ; 45,87„ Ba; 25% S and 29,1 7o Te was actually found. These crystals were accurately investigated and proved to be so well constructed that, in their habit, they did not remind us of mixed crystals but, on the contrary, made a vivid impression of belonging to a true chemical compound. The following data were found : Large yellow transparent crystals fairly stable in the air but, after some time, assuming a greyish colour. They are well constructed, 1) KuSTEK, Z. f. anorg. Cbem. 44. 431. 2) Messii»gek, Berl. Ber. 30 805 (1897). 41* ( 614 ) yield constant angular values and have, crystallonouiically, quite the appearance of a well defined compound of constant composition. Fig. 4. The compound has, in moist air, a strong odour of hydrogen- sulphide and is decomposed by water with separation of a black substance which contains tellurium and sulphur. Triclinic-pinacoidal. a -.h = 1.6835 A : 1.5515 .4 = 109° 43' «=:113°7i' 5=122°i0i ^ = 124° 13' C= 90° 32' y= 77° 39' Forms observed: c =; {OOlj, strongly predominating; a =r jlOOj and 6 = {010} equally well developed and lustrous; g = jOllj and ^•^jlOlj quite as much developed as d and />, and yielding sharp reflexes; o = \il2\, lustrous and fairly large; n = {012j .small but lustrous; (r^jlllj, small and subordinate, generally with but one plane ; m := illOj, well developed and lustrous, also without the parallel opposite plane. The habit is flattened towards |001| with slight stretching in the direction of the /!)-axis. A complete or distinct plane of cleavage is not found. The following angular values were measured : ( •;i-'^ ) Measured : Cal ciliated : a h = (100) : (010) =»89° 281 — h C = (010): (001)=:* 70 17 c a = (001) : (100) =*57 49V, — c ,- = (001):(T01)=*53 46V, — c 2 = (001): (011)=* 65 45 — b = (112) : (010) = 67 32 67° 36' r b = (101) : (010) = 68 47 68 47>/, a 9 = (100): (011)= 67 29 67 26 c = (001) : (TT2) = 50 5 49 49 a ;• = (Too) : (TOl) = 68 24 68 24 c n = (OOT) : (012) = 38 17 38 27 n q = (012) : (Oil) = 27 24 27 18 Q b = (Olf) : (010) = 43 56 43 58 r o = (10r):(112)= 52 35 52 47 q = {11% : {Oil) = 32 21 32 19 m A = (110) : (010) = 33 51 33 37 III a = (110) : (100) = 56 4J 56 55 m r = (llO) : (101) = 59 37 59 32V, m q = (110) : (Olf) = 35 10 35 21V, to /> = (lll):(010)= 40 55 40 47 to c = (111) : (OOT) = 85 46V, 85 48V, to r/ = (111) : (100) = 74 28 74 30V, 10 ^ = (111): (Oil) = 38 5 38 3V, The agreement between the observation and the calculation is an excellent one. Etching figures were not obtained. It may be, — looking at the acentric habit and plane-development of some of the forms, — that the symmetry is tvicVmo- peclial. The situation of the optical axial angle could not be determined. That of the optical main directions was such that tiie angle of extinction on j001| with the side (001) : (101), was about 15°, but on {101 1 with the same side it amounted to about 12°, and that with an inclination which on {lOlj proceeds from the left in front to the right at the back, and on {001 1 from the right above to the left below. Here we are, consequently, also dealing with a polysulphide of the type Ba S, in which "/, of the sulphur has been replaced by tellurium. ( (ilfi ) Efforts to obtain this compoiiiid, prepared from Ba S and S, in a measurable form, and thus to obtain an argument in favour of the said view, in tiie event of an isomorphism between the two sub- stances, have been found so far unsuccessful. § 11. In the electrolysis of a dilute solution of the potassium salt, in which the platinum-disii acted as tlie cathode and a disc- shaped platinum-electrode as anode, it looked as if tellurium was precipitated at both electrodes. The liberation of the black substance at the anode is nothing else than an oxidationphenomenon. The tension at the electrodes was 2.6 Volt, the current 0.05 Am- pere; the oxygen formed at the anode oxidises the liquid, so sensitive to oxygen, with separation of tellurium-sulphur complexes which partly stick to the anode and partly collect above the same on the liquid ; if the current passes for some time the precipitate redissolves and the anode again turns bright. I have ascertained that the adhering precipitate contains tellurium as well as sulphur. On the other hand, the precipitation of a black substance at the cathode takes place continuously but ver}^ slowly ; after twelve hours only a small portion of the salt, about one gram and a half in 50 cc of distilled water, had been decomposed by the current. This black precipitate has now proved to be pure tellurium and this observation would, therefore, go against the assumption that tellurium forms a constituent of the anion. This experiment, howevei', cannot be used as evidence against that view, since we know an analogous case in the electrolysis of sodium-sulphantimonate'), where the antimony also proceeds, apparently, to the cathode, although it acts, in the salt, as a constituent of the anion. It has also transpired in these experiments that the metal does not wholly take part in the electric conductivity, but that in the electrolysis of the solution, the sodium sulphide is decomposed, and either the sodium liberated at the cathode, or the hydrogen which it causes to be evolved, precipitates the antimony by a secondary reaction. Only when a very little alkaiisulphide is present, the anion '"SbS^ also moves towards the anode. Obviously, the explanation in our case is the same ; the tellurium is formed secondarily at the cathode, whilst at the anode, as in the experiments cited, fairly complicated and somewhat obscure oxidation phenomena occur. In each case, this apparent contradiction does by no means prove the incorrectness of the view, that the said salts may be considered as derivatives of ') OsT und Klapproth, Zeitschr. f. augew. Chemic (1900). p. 827. C IU7 ) complex tellurohydrogen sulphides. The investigation of this exceedingly complicated subject is being continued in the direction indicated. § 12. Summarizing the results of these investigations, I believe I may say : 1 That the elements tellui-ium and sulpiiur do not form compounds, but mixed crystals. 2. That the elements tellurium, selenium and sulphur behave in quite an analogous manner towards the sulphides of the alkali and alkali-earth metals, and cause the formation of complex sulpho-, seleno- or teilurohydrogen sulphides of a different type, and tiiat it is quite unnecessary to presupjiose the intermediate Ibrmatiou of selenium-sulphur or tellurium-sulphur compounds. 3. That the position, given by Dumas, to tellurium in the sulphur group as the first homologue of selenium is quite justified so far as the mutual behaviour of these elements is concerned, and that sulphur, selenium, and tellurium form a natural triad of elements, whicii are more adjacent to each other than an}' one of them is to oxygen. Gwningen, Inorg. Chem. Lab. of the University. Physics. — ''Some remarks on Prof. Kohnstamm's reply." By Dr. J. J. VAN Laar. Communicated by Prof. Lorentz. In these Proceedings of Jan. 6''' 1910 Prof. Kohnstamm has inserted a reply to my remarks suggested by a paper by Messi's. TimmerxAIANs and KoHNSTAMM. Though I, too, very reluctantly continue the discus- sion, I feel obliged to briefly revert to this matter for tiie last time, in order to prevent further misunderstanding. So I will just point out that Mr. Kohnstamm is quite silent about the cardinal point of my remarks, given in point a second part, point e and point /; viz. that Messrs. T. and K. in consequence of their disregard of the last five of my seven pajiers on the subject in question have wrongly asserted that the "abnormal" type III could not occur for 7iormal substances, and that this would be due to my restricting supposition aj, = V^^ja, . Only to remove this misunder- standing — as I had asserted the very opposite of this — I wrote my preceding jiaper. ( «1« ) Oil (lie other hand a few minor (iiiestioiis are extensively discussed in tlie answer, vi/,. the question n^,^\^a^a, and — ^ 0. I must remark here that when 1 repeatedly' spoke of the "quite general" case fi^ C^ "i , f>. \^ ^^i > this expression "quite general" was obviously meant in contrast to the special ease a, ^ a, , b^^=.b^, treated by me before in the first two Papers, as would be clear to everybody, and that the "generality" meant by me according to the whole tenour of my papers, of course, only holds loithin the area of the once asmmed supposition «i, =: ^^tlrlrt, (Berthei,ot's). For this supposition I explicitly premised in all my papers, and I repeated it more than once (loc. cit.). Now with regard to the question itself of the supposition otj^ = l/aia, (which, however, is not the point at issue), I may be allowed to remind Mr. K. of a paper of his in the Zeitschr. f. physik. Ch. 36 p. 41 (1901), where it, inter alia, says at the end (p. 62): "So weit ich aus dem mir bekannten Material zu schlieszen vermag, scheinen mir jedoch die Tatsachen selir zu Gunsten der (BERTHELOT'sche) Annahme zu sprechen . . . ." [ will add that I, too, consider the supposition a^, = V^a^a^ as very probable, and that seeming deviations from this supposition are attributed by ine to the formation of com- pounds. But 1 hope to treat this more fully on a later occasion. I now demonstrated that even on this simple supposition the ab- normal type HI can occur for perfectly normal substances. And this Mr. K. denied — as my later papers on this subject in wliich this was proved by me, had evidently escaped his notice. (Pb With regard to the supposition -— = 0, Mr. K. refers to my state- ment that "qualitatively everything will remain the same if b is not assumed independent of v and T". This, however, is quite beside d-h the question whether the supposition — ^ = is of intluence on my ^ dx- results or not ; for v and T are not the same thing as x. I fully maintain my contention, and Mr. K. will, no doubt, understand, that this dependence on v and T was only mentioned by me, because VAN DER Waals' later investigations have shown that b still depends on this (juantity. But this is not the point in question. I, however, readily acknowledge that when writing the Hues about the longitudinal plait closing again, quoted by Mr. K., I did ( «19 ) not sufficiently clearly state that the divei'gent result was only founded on the assumption — ; = 0. 1 knew, iiowever, that van der dx'- Waals in his Contijiuitat II p. 24 has already treated this questiou. Yet on theoretical considerations I abide by my opinion that in the neighbourJiood of the limiting volume, so at very high pressures, d^'b — must be ^ 0. dx* And now I think that 1 for my part, have sufficiently elucidated Mr. Kojinstamm's lieply, so tliat further misunderstanding seems almost precluded. Baarn, B^ebr. 21, 1910. Mathematics. — "T/w oscillation,i about a position of eqiiilibriuin where a simple linear relation exists between the frequencies of the principal vibrations." (!>*' part). By Mr. H. J. E. Beth. (Communicated by Prof. Korteweg). Introduction. § 1. In iiis paper'): "On certain vibrations of higher order of abnormal intensity (vibrations of relation) for mechanisms with more degrees of freedom" (Verhandelingen der Koninklijke Akademie van Wetenschappen, Vol. V. N°. 8, 1897 ; Archives Neerlandaises Vol. I, series II, pages 229 — 260) Prof. Korteweg has written down tlie expansions in series for the principal coordinates of an arbitrary mechanism with more degrees of freedom, performing small oscilla- tions about a position of stable equilibrium. From these expansions in series could be deduced that in a certain case it was possible for some vibrations of higher order, having in general a small intensity with respect to the principal vibrations, to obtain an abnormall}' great intensity ; this is the case when between the frequencies n , 71,, etc. of the principal vibrations a relation exists of the form P'Ki- + -?«// + = 9 ; where p, q etc. are positive or negative integers and q is with respect to Ux, ny etc. a small quantity, called residue of relation. Furthermore however il became evident that, when ^S^4(»Sisthe sum of the absolute values of p, q etc.) and at the same time 9 = 0, 1) "Over zekere trillingen van hooger orde van abnormale intensiteit (relatie- Irillingen) bij meclianismen met nieerdere graden van vrijlieid". ( 620 ) the above-mentioned expansions in series lost their validity; we must therefore investigate in a different way what becomes of the movement in the case mentioned. In what follows we shall investigate this for a mechanism with two degrees of freedom. As a base for this investigation a very simple mechanism is selected, namely a material point which moves without friction yet under the influence of gravi- tation on a given surface in the vicinity of its lowest point. Every time one of the cases S^-i is discussed we shall pass to an arbitrary mechanism with two degrees of freedom. Movement on the bottom of a surface. § 2. We shall accordingly first pass on to the treatment of the simple mechanism we have chosen as a base for our investigation. When the surface has positive curvature in the vicinity of its lowest point 0, when plane XY is the tangential plane in 0, and the XZ- and F^-planes are the principal sections of the surface in that point, whilst the Z-axis is supposed positive upwards, then the equation of the surface in the vicinity of takes the form of: z = - {c,a" + c^r + d,,,' + d.x^y + d,.vf + d.f + ...); . (1) [I where c^ and f, are positive. The equations of motion of the material point become : ax dz Availing ourselves of (J) to eliminate : we find ?/:+~(.'/ + '^) = o.j dz d'z . d'z . . d-z . dz ■■ dz ■■ 0.7; da;- O.fOM dy^ d.v dy i (2) .. d^ d^^ . d'z . . d'z . dz .. dz .. ' ^^dy^<^ + d^^''' + 'd;;;dy-'^ + df-'' + d:.-'-^dy'^-'- Let h be the small quantity (small e.g. with respect to the principal radii of curvature R^ and R^ of the surface in ()) which determines the order of greatness of x and y, then the equations (2) become, omitting the terms of order /r and higher: X -j- 2cj A' = 0, 2/4- 2c, 2/ = 0. (3) (4) ( 'iSl ) These equations are in general sufficient to arrive at tlie solution at first approximation. This then becomes: .V ==: Ah COS {nj -\- A) , I 1/ =z Bh cos {n^t -{- (i);) where 7i^ = 1^20,, n, = l-"^ 2cj. Here Ah, Bh, A and ft are constants of integration ; we suppose A and B to be of moderate greatness. At first approximation therefore the horizontal projection of the moving point describes a Lissajous curve, which is closed when wi^ ^ qn^, where p and q are integers. If pn^ =: qn„ -\- q, the curve described is not closed, but it consists of a succession of parts each of which differs but little from a closed curve. These last closed curves have however various shapes which answer to different values of the difference in phase. Thej are all described in the rectangle with 2Ah and 2Bh as sides. § 3. If we wish to take into consideration the terms of a higher order appearing in (2) we generally have but to apply small modi- fications to the first approximation. These modifications are, however, not small in case a relation exists of the form : where *S ^ » -j- o < 4 and — is very small (what is meant here by "very small" will be evident later on). When by applying the metiiod of consecutive approximations, starting from ('4) as first approximation, we try to find expansions in series for x and y, we shall find, when substituting the expres- sions (4^ into the terms of higher order of (2) and developing the products and powers of the cosines, in case — is very small, periodical terms which have about the same period as the principal vibration, to which the equation in which the indicated term appears relates more especially. Such terms in the equations of motion give rise in the expansions in series for x and y to terms with abnor- mally great amplitude. These amplitudes may reach the order k and even seem to be greater still. This proves that in the case supposed our first approximation was not correct. It is evident that in the equations of motion there are terms of higher order, which are of influence even on the first ( 022 ) approximation. So we shall have to find in the equations (2) which terms give rise to the failure of the application of the method of consecutive approximations. These terms we shall have to include in the abridged equations, serving to determine the first approximation. We shall consecutively discuss the cases : 5 = 3 (2h, =n,-\-Q), S = 4 (3ft, = m, + ()), S = 2 {n, = n, + (j). S=^.') Strict relation. § 4. We suppose q ^ 0; therefore In the equations of motion appear for the first time among the terras of order A' terras which, according to what was said in § 3, must be included in the abridged equations. Thej- are : in the first equation 2d„xy, in the second d^.x^. Tiiese are the most important among the terms referred to. Omitting the remaining terms of higher order we therefore have to consider : y -\- 4»j,'' y -\- d^ x" = 0. ' We may also write this system as follows: 0, (5) a; + '»:■ ' .r - dR y + 4ft, ^y - dR ~dy 0; ■ ' ay in which : R~ — d^ A'' )/. To this we apply the method of the variation of the canonical constants. This means, as is known, that the equations, arising dR dR . , . , J . when the terms ^^ and -— are omitted, first are solved; in winch 0* uy solution 4 arbitrary constants appear; we then investigate what functions of the time must be the quantities just now regarded as constants, so that the expressions for x and //, taken in this way, repre- . . dR ^dR ^^ sent the solution of the complete equations containmg r— and :r—. Ine ' dx ay equations in which - and — are lacking, arc solved according to da; dy ') In a following paper we shall discuss the cases S = 2 and S = 4. { fi23 ) tlie method of Hamilton-.Iacobi in order that the C0Il^^tallts we obtain may form a canonical system. If «i, «j, /?j and |?j are the canonical constants then by substitution of the expressions found for x and y in R this R will become a 'function of «;, a^, (?i, ^, and t. The variability of the «'s and iS's with the time is then given by : da, _ dR da., _ dR rf/J, _ dR d^„ _ dR dt ~ dji, ' dt " 6^, ' dt ~ dwj ' d.t ~ d«, ' In case R is a function of the «'s and the (i's alone, and conse- quently does not contain / explicitly, the system has as an integral : R = cotistant (7) § 5. If now we solve the equations dR dR arising from (5) by omission of the terms — and t— , according to da: ay the method Hamiltox-Jacobi we may arrive at : X = COS (?ij< -}- 2wjj}j), y = -—cos(Zn,t-]-iriJ^): where «, , «, , ^^ , jJ^ form a canonical system of constants. We must suppose «i and «j to be of order h'' as the amplitudes of the ,c- and ^/-vibrations nmst be of order h. Substitution of (8) in R^ — d^x'y furnishes 3 terms : ^^—^ cos {2n,t + 4«,i?,), "^ cos [in^t + in, {^, + ,i,)j and o , <^^ 4h, {^, — ^^), on J each term multiplied by — d, . The first two terms contain t explicitly ; setting aside the variability of the «'s and ^'s we can sa}' that those terms are periodical, whilst the period is comparable to that of the principal vibrations. The last term, how^ever, does not contain t explicitly. Onlj- this last term is of importance for the tirst approximation ; the two others we omit (we shall revert to this in § 6). We therefore take : ( «24 ) R= — ~-^a, [/a, cos 4n, (i?,— (?,). Coiiseqiienflj sj'Stein ((i) takes this form : dt dtt^ — = — 2 Nm, a, ai sin w, dt 1 1 2 v' (9) ^ ?rtj ttji COS y, (ft d^2 ^ i ?/!, «, «,~i c'06- O) : d< - 1 1 2 where A^ is written for n^^2n^; further: d. '"' = .Y' ' </) = 2 iV(i3, _;?,). As t does not appear explicitly in R we get according to what has been said at the close of ^ 4 as an integral: «j [/a., cos <p = constant. . . ... (10) Fiirthei'inore it appears at once from (9) that: da, da, dt dt Therefore : «, -[- a„ :=: constant (11) is another integral. The latter gives us reason to introduce a new variable K in such a way that : 1 1 «, =z — E,' N' //' $ , a, = — R^" A" /r (1 — ?) : 4 4 ? is then always situated between and 1, /?„ is of moderate greatness. By this (10) obtains the form : Sl/l^gco.v7 =K; (12) in which A" repi'esents a constant. The first equation of (9) becomes : d^ dR , . -=^^[/Y:r^sin<f.'> (13) By elimination of '/ between (12) and (13) we arrive at: di d,R, = ± -^ h . dt. \^^'{l-?)-K' ^^ ( 625 ) Now put: /(?)-?Mi-S)-i^% then for tlie iiiilial value of S we find /(?)>0. For S^O and S = l we find /(?)<C0. Thus the equation /(5)=rO has two roots between and J . So K^ cannot iia\ e all values ; the possible values of K- lie between two limits; in § 9 we shall revert to this and to the special cases, corresponding to the limiting values of K''. The roots between and 1 which the equation ?'(!-?)- ir= = . . . . , . . (14) has in the general case will be called C^ and 5,, where we suppose ?s^?i- The third root is negative, we call it — ).. The differential relation between ? and t may now be written : di d,R„ ^ = ± -^—^ h.dt (15) l/(?,-?)(?-?J(S+^) ^' So with the aid of elliptic functions $ maj' be expressed in t. It changes periodically between the limits Sj and ?, . Now with the aid of (J 2) we can also calculate <p as function of t. And /Jj and (J, likewise, it being possible to write the last two equations of (9) : d^^ _d,R„K h ~dt~ 2N'' T dii, d,R,K h dt 4iY= 1— S So now X and y are also known as functions of t '). In tig. J the relation (12) between S and tp is represented in polar coordinates, (p is taken as polar angle, \^1 — S as radius vector. The circle drawn has unity as radius. The curves change with the value of K. For A'^O the curves lie to the right of the straight line (p =1 - , for /v<[0 to the left of it: K = furnishes degeneration into the straight line ^f = - and the circle 5 = 0. By the maximal , 2 positive and negative value of A (A ^ =t - l/3) the curve has contracted into an isolated point. The special cases of the motion 2 belonging to A'=0 and to A":= + - |/3 will be discussed in ^ 9. ij These calculations will be found in ray dissertatiou, which will appear before long. ( fi2fi ) § fi. When asti'Oiiomers try to obtain in tlie Tlieorj of tlie distur- bances of the movements of the planets by the appUcation of the method of Lagrange expansions in series for the coordinates of the planets or the elements of their orbits, then terras may appear with abnormally large coefificients in consequence of small divisors, ori- ginating from the integration. This takes place when between the inverse values of the periods of revolution of some planets a linear relation with integer coefficients is almost fulfilled. Besides some other properties the terms are also distinguished according to their class, by which is meant : where (i represents the exponent of n (a small quantity indicating the order of greatness of the disturbing function), m the exponent of /, m' the exponent of the small divisor, as they appear in the coefficient of the term indicated. Now it is the terms of the lowest class which we have to take into consideration if we wish to make the expansions in series to hold for a long space of time. By Delal'nay a method is indicated to determine the terms of the lowest class. It consists principally in omitting all terms of short period (period comparable to the periods of the revolution of the planets) in the disturbing function and retaining the most important of the others. (Comp. e. g. H. Poincare, Lecons de mecanique celeste, vol. I, page 341). The problem under discussion has much resemblance with the one mentioned from the theory of disturbances. In the preceding ^ in omitting some terms in R we have imitated what is done in the theory of disturbances. It is easy to see that the terms omitted have really no influence on the first approximation, when we consider the terms which appear e. g. in «i by introduction of such a term. Osculating curves. § 7. In § 5 we have found that the movement of the horizontal |)rojection of the material point might be represented by : « ^ cos (?«!< + 2«;(?,) , y — -X— ^ COS (2»i< + 4Hiji,); 2«, ( fi27 ) da, r/,f., where «, , «,, 3, and ,?, are slowlv varial)le; tor ana - are ' ' ■ • dt dt '/,:?, d(i^ 01 order //', and - of order //. (Com p. (i))). dt dt ^ 1 V V For evei-y arbitrary moment the «'s and tlie |3's have a definite value. These vahies determine a certain Lissa-joiis cnrve. This curve we siiali call the osculating curve for the moment indicated, which name is in use in the theory of disturbances. (See among others H. PoiNCARE, Lecons de mccani(|ue celeste, \ol. I, page 90). Thus in our problem the osculating cur\'es are the wellknown Lissajoiis figures for 2 octaves. By the change of the origin of time we may write the equations of an osculating curve -. .(' = liJiV^cos n^t, II = ^^UJi V'l — i cos (2ii,t—(f) ; where as in §5 we have introduced $ instead of ((, and a..\ here too <p means 4«j(,-?i — ,i,). We now see that </> is the value of the dili'erence in [thase, to which the osculating curve corresponds when the phase is calculated from the moment of the greatest deviation to the right. The amplitudes of the .c- and //-vibrations being respectively liJi-V'i and hRJi [/I — ?, the vertices of the rectangles, in which the osculating curves are described lie on ihe circumference of an ellipse with its great a.xis along the ,r-a.\is and having a length of 2 Rg/t, and its small axis along the //-axis and having a length of Roll. Now 5 changes its value between ?,, and C^, so the rectangles in which the osculating curves are described also lie between two extremes. Moreover as according to (12) to each value of ^i a value of cuv/' belongs all osculating curves may now lie constructed. It follows from (13) that for the extreme values of ? we tind siiiff^O: so in the extreme rectangles parabolae are described. The distance from (>X of the node of an arbitrai'v osculating curve is "- — , from which it is evident that the nodes and also the vertices of ihe parabolae lie all on the same side of U lying below O for positive values of A' (see tig. 2). Envelope of the osriilafim/ carved. § 8. If we perform the elimination of / and </ from : 42 Proceedings Royal Acad. Amsslerdam. Vol. Xll. ( 628 ) 1 .V — HJi 1/ s ,os n, t, y = -^ RJ, l/l-g cw (2«, «- ,/ ) and S VT^i cos <f = A, wc liiul for (lie Ctiuatioii of the osculating t-urves willi C as [)iii-ainelei' : C^ (A'= + }') + $ (7\')' — X' — A'') + I yv"-' — 2 /v A^ )' + xA = U ; where for the sake of a siniplilieil iiotalion is put : A foi- ''-, )' for - ; RJi RJi Thus the envelope has as e(|uation •1 (A' 4- Y') - K' — 2 KX- Y + A' j — (A' Y — A' - A')^ = o. After reduction and division by X" (.the i'-axis is the locus of the nodes) it may be written : (/v _ 4 r" - y A' )■ -f Yf = (A= + 4 Y' - 1)' (A' -V- )--), or if wc solve A': A'= — (>" ± l/A^4- }'=) + (}' ± 1/ A^ +~n». Tutting r ± vx'^-^Y^ =— , u it passes into ^ _ K K' ^^ — - J^+ -[7,^ [■■^ (1 _ r) — K' = 0. Now this cubic ci|nati(in lias (he same coet'licients as (14i, so it also has the same roots. So the en\elope is degenerated into the 3 parabolae having as equations: U — Z, , l' = Z„_ , ;/= — a: which after reduciion and reintroduclion of ,r and // taUc the form of : 2 --■ . [ = (J s, I'oraliold, RJt A R.-'k^ C, '/ ^., .'('■' 1^ ■' ' —.'■.— 4- -= i«„ iKii-tiliola, II J, A A'/A- C, V i. .'.■' A' 2 •- 4- . =0 /. p'irabol<i. RJ, K lljlr ;. The paiaiiolae are confocal and have (> as focus. When A is pdsilisc the k, andliieC^ parabolae ha\e their o|ieuings turned u|i\vai(ls. ( 629 ) llie ?. |)iir;il)()l;i li;is its upciiin,^- liiriicil ildwiiwunis lliis ivi-l' i^ ivpi'c- seiiled ill tig. 2, where besides some useiihitiii;j,- eiu'\es ilie eiiveiopiiig paraboiae are also given). Spdcial cases. § 9. At liie rlose of § 5 we saw that two speeial cases may occur, viz. wiieii A'=() and wiieii /v'^ i ttI^S. .1. A'^U. We deduce from the rektion b i/r^ CO, (f = K tliree possil)ilities ; 1. > ^ 0. The movement remains eoutined to tlie y/f-planc. 2. si=J. The movement remains confined to the A'.^-phuic. This form of motion iiowe\'er proves to be inipossii)le wiien c z:|= and ^ ^ is substituted in (5). 3. c'cv y = 0, therefore if =^ or '/ = - — invariably. The os- culating curves have iheir nodes at (_). The form of movement approaches asymptotically to a motion in the J'Z-plane. Wliat becomes of the enveloping parabolae has been represented in fig. 3, in which some osculating curves have been drawn too. 2 -1 1 B. A ^ + —y' 3. Then C, = C^ = ,/=-—. Now cos </) =i + 1 invariably, thus </r=() or (f := .t. The same parabola is continiu)usly described, in which also the Si and s^ |>arabc)lae ha\e coincided. (Fig. 4). When K undergoes a slight change, ?i and s,, fall close together. So this form of movement is stable. o ^ 3 , — IS of order ~. § 1(^. The expansions in series written down by Prof. Koi;tf,\\ I'Xi lose for .S ^ 3 their convergency as soon as — - passes into order ft i> h — (.page 18 of his paper) or i. o. w. as soon as - sinks into order — . "i ' "i -/>', We shall now discuss this case. We again take as first approximation : .1' = cos (/(, t + 2hj ,ij) , 2«, 42* ( (i:)() ) and wii iini^l iii\csti<iiile w lial roriii the riiiiclii)ii H now iissiuncs. As we liave supposed (liar the (erins oC tlie order /i in llie ecpialions of luolion would l)eL'oiiic i): -\- Mj' ,1; and y + (2«, — qY .'/• o h Because — is of order — and we lake no lerins o( hi,!.dicr ordoi- " 1 ^A than /<-' in liie etpiations, we may write for the latter: // 4- 4hi' J/ — 4rtj w/. It' we thus take the ai)Ove expression for .r and // as first approxi- mation, then we nuisf adnnt in thi' f'unetion /.' Iiesides the term — (I., ,/■-// also a term '2/i.^ 07". In the expression — <'. •'■'.'/ + -"1 !?.'/' we sidtstitute the alxne expressions for ,/■ and 7 and onnt the terms containin<i' / ex]tiieitly. In this wa;/ we arri\e at : J"^ — ~ v^ "1 ^"-^ "'*■ '-^ + :7Y ""- ' wliere again \ is put for 11.^ -\- c> ^ '2u,. The ecpiatioiis which serve to deternune tlie (t's and |«"s hecome; — = 'JA )/(, a. n, ■■<in it, ''"» ■) Af - ■ — := — IJS in, a, '(., sin <( , dt 1 . . /' d^, ^ 1= III, ((„ cos (C, dt ' ' ' where J J — I — ^ = — o' h \- \ III, a, ((„ cos (( ; dt " ' - 1 ' - d^ , Q We again sec that dt ' dt ila. ili(„ —- I == 0. SO "1 4" "2 ^ coii.-'linil ; for w hicli reason \\e put ; ( <5oi ) 1 1 », = — li' y h' : , (c, r= — A';-' A ' Ir ( I - 0- — ~ «, K it„ COS (f + —^ <f, = condaut. Fiirllier we have according to § 4 as an integral of the system : d, liilroilucing s, it becomes S ^ 1 — S w« y — (/' (1— ?) = A'; wliere /v is a constant and dMJi In the same way as tiiis was done for the case o = we may write down liie differential relation between s and I and find .t" and // in the way indicated there as functions of the time: they get quite the same form as for o = '). In general ? keefJS ciianging periodically between two limits T, and _'.j ; -', and -'^ being the positive roots of Vet there is a considerable ditference between the cases o = () and o of order //. § IJ. We notice this difference most distinctly when we represent the relation eslai)lislied between _' and </ in polar (-(jordinates. If we put ?"' = — ?"' then we liiid : We take (p as polar angle, 1^1 — ^ as radius vector and we inves- tigate the site and shape of tiie curves for positive values of '" and for all possible values of A'. For Ar= q'" there is degeneration into the circle u^O and a straight line normal to the origin of the angles at a distance o" from pole 0^. We have two eases now ; 9'" <[ 1 and {}'" > i «>"'<^1. Let us now investigate the shape of the curves for different values of K. For /v>o"' they lie to the left of the straight line just mentioned, for increasing \alue of A' they contract inoi-e and more until for the maximal value of A', belonging to a certain value of b Virlo Gluiptoi' V of my dissertation. ( ii:^2 ) <y" wo <;('l ;in isoliUcd |)uiiil. Il' ••<C^^''\t' ''"' <"i""\<'^ siirMimid |i()iiil '>, : if A'=() we have ;t curve lliroiiuli <>^, {\>y A' <^ ihev lie In llie Icll (if '■', : Inr llic iiiiiiiiiial valiu^ of A' we auaiii jiel an isohileti point (,liy. •")'. For increasing values of <»'" liie straiglit line sepafating the (ioniains K^ij" anil A <^ o'" in(i\e.s lo the light. The domain A'^<>"' becomes smaller and xaniNlies for o"'^l. For o"' -> J we therefore have curx'es surrounding f\ and curves to the lefi of O^ only. When (>'" increases still more the remaining isolated point a|iproaciies to '>, and the curves farthei iVoni <)^ approach to circles. For f):=0 we had (with the e.xceptiou of the special case K=0) only curves to the I'ight of O^, and curves to ihe left of >>^. Vov o of oi'(ler // we ha\e nK)reo\er curves around '>, , which are even o nioi'e frequent for great values of "" . The curves around O^ point to a form of motion, where <f lakes all values, the nodes of the osculating curves lie then above as well as below the point of fig. 2; the osculating parabolae have their openings tnrnetl to opposite sides. That for increasing \alues of o'" the cni'ves in general begin to resemble circles more and moi'c, indicates Ihal k is abonl constant ; it changes between nai'row limils. This also appears in Ihis way. From (IG) we deduce: A'— o"'(l — L',) =rr ± ;y\—L,, l!y subtraction we llnd : Vov greater values of o" we find u.j — ;-', becoming vei-y small. In this way we a|i|)roach the general case where ihere is no (pieslitiu about relation. § 12. How the transition to this genei-al case takes p,lace is also clearly exideni froiu Ihe limilalion of ihe domain of motion, which limilalion we lind by determining ihe envelope of the osculating curves. In llie same way as this was done for the case «) .-^ 0, we lind thai iIk^ envelope degenerates into three parabolae, of which the e(|ualious are: irahola ( fi33 ) '/ /i'-'>"' ,„ :, .<-•' y.'„// ^ y, ^ ^ K—n" h'/h- y A'— ()'" ,„ z. x'^ 2 — ^ 4- 4 o = ^ — . ^ »., iiarabolii. RJi C, ' K—q'" 11,'Jr - - ^ y k—q'" ,,, )■ *•■ 'J'lie points of iiilL'rsecli(jii of (lie / parabola \\itli the T, and w, parabolae lie again on the ellipse having RJi and '2RJi as axes. The parabolae are confocal ; tiie focus lies on the c/-axis at the height of — ^KJi.q". In lig. 6", 6'\ 6' we find those parabolae (and also the osculating parabolae) corresponding to the cases o"'<[l and In tig. 7 we see how the limitation approaches more and more to a rectangle for increasing o". The r, and _'., paivaiiolae coincide for maximal and minimal A'. ArJiitmr;/ nn'cluiaisni ivlth 2 iLyrees of freedom for irhlcli S := 3. § 13. Let 7, and q., be the principal coordinates of the mechanism ; they remain during the movement of order //, and arc zero in the position of equilibrium. The kinetic energy T and the potential energy i^ may be written : '/' = Y ^^' ^- 2 ''"-"' "' ^'^ ' '^' =" Y ^"'' '^'' + "=' '^-''^ + ^»' where 2\ and U^ are expressions in whose terms h appears at least to the 3''' degree. Let us write down the terms of order A' in 7\-. 1 . ^8 — Y Hi7r + h,<h' + 2 '■q.q.'h + 2 dq.qiq, + cq,q,' +fq,q,') + • • . As far as and inclusive of the terms of order /r the equations of L.\GRANGE now becouie : 1 . ........ Yi + «i''/i — — J "9i' ~ "^i?! - ^"M^ — ''q^q, — <-qiq^ — '^q^.q^ -r ri \- du. 1 .• , dl\ ( '--i ) 111 case llic relation /^, = 2n^ is sirielly salisiied or nearly so, tiie (listnii)ini>; terms are : ill lli(^ lirst eqnation those with 7,7,, <i^<l-. , (]x'l- ^ 'h'l-r „ „ second „ ,, „ 71' , 7i7i , qi"- W at first approximation we try to satisfy the equations by : Yj = A/i cos (;/ji + ;.) , q, = Bh cos (nj. -f fj) where .1, B, ). and n are functions of /, however in such a manner that A, B, ?., {i are of order // or smaller, we may replace in the second member of tlic e([uations: ,/\ i)y n,' {A'/r — 7=,), '/;' hy i>,' {B'/r — 7,=), 7i ''V — "i'2i' V. 'V — ".'7.- If we take this info account for the disturbing terms and if we omii the non-disturbing terms, the e(|uations become: 7i + «i'7i = ('"'1' -r '■"./ + -P) 'h<], — ^"h'J-2 i % + «./7. = (2 ,■«/ - - bn:- + pj q,-^. \ The terms 2/>7i7, in the first equation and j)f/\ in the second originate from a lei'm — /'7%7j, appearing in I ,. To get J-id of the term with 7,7, we use tiie new variable 7' so that : 1 Then : 1 .. 1 '/i = -Zi + 2 '"'' '^' '^' ^ '"^' ''" '^ '^' '^' "^ 1 = 7i + h, 1. - 7/' ("1' + "■') '/. 'h- Therefore : 1 The e(|nations now pass into: 1 I '/■, ^- "7 '/. = C' ".' + " "•/' - 2 ^ "■^' ^ " ^'^ '' ' '''' I 7. + "■/- 7. -= (2 -^ «.' - 2 ''»/+Z')7'.'^- . For we mav replace in the second members 7, by 7,', as liieir dill'ereiice is of order //■. where ( f)35 ) Lei us iiuw su|)|iusc ii.^ to lie = '2ii^ ; tlieii we get : '/. + "i" 'l\ — (-i '• "i' - '' "i' -f - i') q'i '/■!■ So we find : ' 7. + ".^V. + '/,-?'/ = 0; 1 r/._, = — 2 (■?//' -f 2 '"'i' —P The equations (letorniinino- the first apiiroximation have exactly tlie same form as those t'ouiul in § 4. Wiiat was f'(jrnierlj deduced for tlic simple meoiianisiu liolils consc(iuenli_y, if //.. = 2hi, for an arbitrary mechanism witii two degrees of freedom in such a sense that the horizontal ju'ojection of the point moving over the surface may be regarded as the representative point for the arbitrary mechanism. We tlnally observe that any mechanism for which — 2e7i,- + hjm^' --;) = is not sensitive for the relation u, = 2/;i. So this is the condition requisite to make tlie mechanism for n.^ = 2n^ a mechanism of exception in the sense indicated by Prof. Kokteweg (§ 26 of his paper). Mechanisms of exception therefore are among others the symme- trical mechanisms (§ 31 of that paper) ; for here r, h, and /i are all eijual to zero. Microbiology. — " Viscosaccharase, an enzyme which produces slime from /•(iDe-siKjnr' . By Prof. Di'. M. W. Beijerinck. The emnlsi.on reaction. Many spore-producing and a lew non spore-producing bacilli, cause, when growing in presence of cane-sugar or rafHinose on neutral or feebly alkaline agarplates, a very peculiar "colloidreaction", which is also valuable for the diagnosis of these bacteria. This reaction consists in the formation, in and also on the surface of the agar around the colonies or streaks, of a liquid "precipitate", i. e. an emulsion, which can best be recognised in transmitted light, and at the same tinie in a swelling of the agar caused by the increase of \(ilunie produced by the emulsion. The emulsion consists of drops (.see plate) of diffei'ent size, mostly very small, but sometimes growing to 0,2 mm. so that they may ( ti:{(; ) I),' ilisliiiL;iiislici| Willi a iiia,uiiil\ iiiL'' ,uhiss. At Icclilc iiKiiiiiilicaliuii {\\r\ iiii^lil 111' lakfii lor (lr(i|tlets dI' oil siLspeiided in liie agar, l)Ut as sining Mil|iliiwic arid dissolves diesc di'ops iiiiiuedialely, and a feei)i('r acid more slo\vl\ , llicrc can lie no (|iieslion of oil or t'al. ('haraclcrislic lor die rcaclion is dial il can oniv iic dislinclly ol>ser\('il in agar ImiI iiii|ii'i-ri.'cllv in gclalin. In die agar llie process is inipede<l wiieii acid i> produced hv llie nncrobes. Tims lionillon-agar, vea^twaler-aiiar, and wurl-agar wiiii cane-sngar can well he iix'd, Inil the emulsion is more distinctly formed in agar with mixtures (»f siihstanccs that prevent the acidification, to w'iiicli cane-sugar is so \r\-\ apt. l''or that reason nitrates as nitrogend'ooil are esjiccially l'a\ oiiralde, as the wilhdrau'ing of nitrogen then necessarily mnst produce an alkali, while for example ammonium- salts, used as souive of nitrogen, mnst promote the acid reaction. A good experiment to produce the emulsion is the following : A plate is prepared of the composition: tapwater, 2 "/„ of agar, 2% of cane- sugar, 0,(12 7„ KNO, and 0,02 "/„ K,liPO,. Nitrogen food may also be quite left out, so agar-plales with 10 7„ of cane-sugar and bikalinm- phosphate only, are very well fit to demonstrate the emulsion with Azotoliiictt'r and the hereafter mentioned Bucilliis emvl^ioiiis. The (piantitv of canc-sngar can \ary between 0.1 "/,. ^i-"*^' ^^^ ° ,i without much dillerencc in the result. After the solidifying of the agar-plate and the removal of the adhering water, soil-bacilli are dispersed, obtained by shaking some garden-soil with water, and heating it a few minutes at 70" to 80° C. in order to kill (he not s|)ornlaiing microbes. Then the water is poured o\er the ])late and allowed to How oil'. The adhering germs, for so far they livj, ai'c nothing but spores of bacilli, which can gcrnunate a( 30° C. After one or two days the colonics become visible and simidta- neously (he emulsion around some of (hem ; the majority does not produce the emulsion. Cane-sugar may be replaced liy raflinose, which ac(s in the same way; but glucose, levulose, mannose, galactose, lactose, malto.se, trehalose, melibiose, mannite, inulin, dextrin and .xylose, ilo not give the emulsion. The emulsion is distinct round the colonies of /li id /In. < niesenti'iicu.^ vuhjatus (see plate Fig. 1), /). nirifiithcrmm and a not yet described soil-bacillns, commonly also fouml in cane-sugar itself, recognisable by ils small terminal spores, which may be called Bacilhis emul- slonls and whose ti'ansparent colony is likewise given on (he plate (Fig. 2j. The emulsion is wanting in B. suhtUis, B. miicolik's, B. ( '^:i' ) /ii'Ii/iiii/.ni, /J. itilrti.ni.s, II. ^/(//f/c/v/.sy/c/vrv, />'. iii/cKs, liesides in tlic anaerobes (Irniiulolxictci' linhillnini, ilr. .fuccluirohtili/ricniii anil (ri\ pectlnovomiii. The mouklN, tlie \iiriuiis yeast speries, even lliosc wliicli invert eane-sngar, besides all s[)eoies of Strr/itothrid-, and most of the non- spore prodncing bacteria, do not prodncc the emnlsion either. An exception to tln' last iide makes the non-spore prodncing ^4co/o- liartei' chruucoccnia, which on plates of 2 "/„ of agar, 2 to 10 °/„ cane- sngar, and 0,02 7,, I^jUl'*^' ''i water, gives a strong emidsion, which extends to a lai-ge distance ronnd the colonies; later, in their vicinity, perhaps by the influence of a specitic enzyme or an acid it vanishes, while near the colonies of the soil-bacilli the emnlsion is permanent. With the exception of B. chroocuccuiii the other forms of Azotohncter do not produce the emulsion. From cul- tures of Azotobacti'f, [nepared with garden sod and destined for the absorption of free nitrogen, a species related to B. radiobacte.r can be obtained, which produces no spores, but does also give a sti'ong emnlsion. Ane inulsion, from a physical view analogous but quite dirt'erent by the maimer in which it takes rise, was described by me on another occa- sion'). It appears when a 107„ solution of gelatin in water is boiled with a 107o solution of soluble starch, or with a 2" /(, agar-solntion. Even by boiling the two watery solutions do not mix, which, of course, is also the case after solidifying. This I'eposes evidently on the fact that here tw^o colloidal solutions are brought together, which cannot diffuse and whose emulsionated droplets constantly have a positixe surface-tension with regard to each other. The same explanation must hold good for the omidsion formed by the viscosaccharase with regard to the agai-, and as I may add, to culture-liquids wherein Baci/liis I'umlsioni.'' produces the emnlsion also. The enmlsion is prodiiced hi/ mi I'lizi/me. If from the emulsion liekl round a colony a small piece of agar is cut out, without touching the colony, and placed on an other cane- sugar-agar-plate, the emnlsion itself does not diffuse out of it, but into the plate, a substance goes o\er, which produces the emulsion again and with regard to the quantities used rather strongly. This points with certainty to the presence of an enzyme as the cause of the emulsion, an enzyme which must have the property of moving through the agar by diffusion. This agrees perfectly well with the 1) Centralbl. t'. Bacteriologie -J"- Abl. B.i. -1, p. (J27, 1S9G. ( (;;is ) (iriiiiii of llic cnmlsioji i-miiiil the coldiiii's, foi' a siil)slance wliicli is evkleiitly iiisuliible in the agarplutc, can only be found at tiie |)liice wliere it is prodnced. Tliis snbslanee liavin<>' in onr case the nature (if a |iianl slime, liie enzyme may l)e called risrosaccharase. Tlie enzyme is [irepai-ed by lllleriiig- a cidliire of B. nu'siuitcricu.t raljidtiis and pi-eci])ilating the filtrate willi alcohol, whereby, of coui'se, otiiei- enzymes formed by Ihis bacterium such as diastase, and also the slime substance itself, are |>recipilated. Whether to the enzymes, ])resent in this nuxture invertase must be reckoned, which is usually considered as a secretion-product of B. mesentericus, has become doubtful by (he discovery of the viscosaccharase, at whose action, as will be seen below, together with tlie slime, the production of a reducing sugar is slated. Even ill |iiesence of chloroform the emulsion I'eaction takes rise on cane-sugar agar-jilates ihrough the enzyme produced from the iiii'sciift'/'inis culiures, without anything being perceived of the de\elop- ineiil of Ihc germs of B. niescntei'lcus itself, wdiich may be still presoiii after lillei'ing and precipitating. It is not difticult to prepare plates of any size conlaiuing the emulsion everywhere, and fit for e.\|)erimenls lo demonslrale by whal influences it may disap|»ear. 'J'o this end the I'equired cnllure-agar is mi-\ed before solidifying with a not too large number of germs, for example o( B. I'lmi/sioiiis, and then jdaced one or two days in the thermostat ; when the plate becomes cpiile turbid by the emulsion, the sugar is washed out and it is ready for the experiment. A drop of dilute acid thereon rapidly |)rotluces a clear space. .1/ //ir acUoii of lu.scosiicch'trnsc, hesii/cs t/ic sliiin' u reducinii sugar is foiuid. AVlieii small pieces of agar containing the emulsion are introduced iiilo an experimenl-tiihe and cautiously warmed with a little Fkhlixo's copper solution, a strong reduction is seen, which does not lake rise Avilh the same sugar-agar if the eniiilsiou is wanting. The ipicslion arose whellier ihis reaclioii should be ascribed to llu' slime iisclf, oi' if at ihe .--amc lime, ihrough ihe \'iscosaccharase, or in aiiolher way, some other reducing substance is formed. There- fore small pieces of the agar containuig Ihe emulsion were washed out with walei', whereby the slime, which cannot diUnse from the agar iiilo Ihe water, remains behind, bnl ihe reducing power of the agai- is losi, whilst the water used for Ihe washing becomes itself stronglv M. W. BEIJERINCK. Viscosaccharase, an enzyme, which produces slime from cane-sugar. Fig. 1. Bacillus mesentericus "Mfe Fig. 2. Bacillus emulsionis. •r The emulsion-reaction. Proceedings Royal Acad Amsterdam. Vol. XII. C «39 ) rciliiciiiii. Hence it is sure lluit ill llic "eiiiiilsidi: icaclidii", ki,i;-etliei' witii llie non-red iifiiif^' sliine, aji easily diirusinji and reducing suhstanec (|)robabl_v a sngar) is formed. Tlie chemical composition of tliis substance is still unknown, just like that of the slime itself. The possibility exists that the reducing substance is invert-sugar produced Itv iuvei'tase, whicli lattei- enzyme then should always accompany ihe viscosaccharase. L)ecisi\e experiments on this subject in progress. VKriisiicchdiuisi' is a siiiitlii'hciiJhj (icttn<i eiizyiia;. As to the nature of the slime it must be accepted that its molecules are uuicli larger than those of cane-sugar, else it would not be clear why Ihe slime cannot diffuse through the agar, which cane-sugar does veiy easily. Viscosaccharase must therefore be a synthetically acting enzyme. This oircumstance suggests a relation between the slime and "dextran" '). This is, however, a substance forming the cell-wall of the concerned microbes, which substance may spread in water, and even to some exlenl dilfnse into agarplates, liut is not the [)roducl of an exo-enzyme, i.e. of an enzyme able to leave the bacterial body and act outside of it like the \iscosaccharase. In relation to this it is not astonishing that "dextran" can very well originate fi'om gku-ose and some other sugars, which do not produce the emulsion. Very remarkable is the fad that all tiic hitherk) examined bacteria which show the emulsion-phenomenon, are aiile, at (k'linite culture- conditions, for example on cane-sugar gelatin, when no emulsion is produced, to form non-diffusing "dextran", by which their colonies then become \isible on the plates as large transparent drops. This also points to a narrow relation l)etween the two phenomena and leads to the conclusion that the drops of Ihe emulsion must be identic with, or related to dextran. Perha[)s i»y further i-esearch moditicalious of visco.saccharase will prove to exist, which also act on glucose and other sugars and from these may form "dextran", but which cannot leave the body, or rather tlie cell-wall of the microbes, and must be considered as endo-enzymes whose product, which itself does not diffuse, cannot le found beyond the limits of the colony. If in accordance with my expectation, the emulsion is really brought about by "dextran", then light will be thrown on the formation of the wall-substances of iilant cells in general; for there is no doubt ') G. ScHEiBLER, Zeilschr. cl. Vercins fiir Riibenzuckorindustrie, Bd. '24, p. 309, 1S74. L. Maquenne. Les sucres el leurs [n-incipaux durivus. p. 745, 1900. ( «-K> ) tliat "dexlran" is a iiiudilicalioii ol' (•clliilnse, and tlic till iinw iidt explained seeoiulary changes, observed in so many cell-walls, may then freely be ascribed to the action of speciiic enzymes, related to the viscosaccharase. Why the emnlsion is disdnclly obser\ed in agar, and less easily in gelatin-plates, ninst probably be explainetl i»y llie dimension of the molecules of viscosacchai'asc, which are small enough to enter without much troid)le the relatively wide canals of tiie agar, but too large to pass through the much narrower ones of the gelatin. Many of the experiments here related I owe to Mr. D. C. .1. MiNKMAN, assistant in my Laboratory. EXPLANATION OF THE PLATE. Fig. L Colony of Bacillus mtfiejilvricidi ridijatns on: canal wakT, ■2'Vii ol' agar, l",,, of cane-sugar, 0.02"/„ KNO;; and 02'/,, KoHPOj. willi emulsion around colony. Magnilied 8 limes. Fig 2. Colony of Bacillus emulsionis n. sp., on canal water, 2",,, of agar, 0.1",n of cane-sugar, 0.02 7' ClNHi., 0.02'/,, K.HPO,, with emulsion around colony, Maanifled 9 times. Microbiology. — " Vnrldbilihi in Baci/hts /irodi</iosus." l'>y Piof. M. W. Hki.ikrinck. In a former [)aper') I showed how easily new constant variants of Bacillus iirO(li(/i(isiis and olher nucrobes may be obtained. Here follow some further observations, made with the aid of Mr. IL C. J.vcoBSEN, assislaiil in my Ijaboratory. I'hi' kecpiiKj ciiitstunt of ihc culture^: The pi'inciple on which the keeping constant of B. prodii/iosiis seems (o repose is preventing the cultures from becoming alkaline by their own action. Tims, by re-inoculaliug in (puck succession, for instance every 24 hours, into bouillon or on bouillon-agar at 30° C, each form of lidcillns /ini(li(/iosiis. whether the natural or normal form, 01' a variant obtaiiu-d from it, remains unchanged ])robably for an indefinite time. For the transplaidations only \ery little material must be used and ail abundance of food. If some lactic acid is added, for inslance 0,^ to L5 cm'' normal per JOO cm" of bouillon, the cultui-e likewise remains unchanged 1) Royal Acad, of Sciences 21 Nov. PJOO, ( fiil ) afler a [irukiiiyed series of Iraiisporls, if these are al\\avs carried out liefore the acid is neutralised bv the alkali luiiduced tVoiu the bouillon hy the bacteria tlieuiseives '). Addition of 1 to 2 pCt. of glucose acts in the same manner as free acid, B. prodlyio.ms therefrom producing acid which may rise, if sufficient glucose is added, to o to 4 cm" normal per 100 cm' of bouillon. As the litre of alkali, originating in the bouillon alone, can amount to 2.5 cm' N per 100 cm' of bouillon, and as fi'om 1 pCt. of glucose there results Jio more than 1.5 to 2 cm' IS ol acid, addition of J pCt. of glucose is sufficient to [)revent variation, if the re-inoculations take place quickly ; but not if effected with long intervals, for in the latter case more alkali may result from the bouillon than acid from the glucose. If to the bouillon so much ammoniumcarbonate or natriumcarbonate is added that the titre of alkali amounts to about 3 cm' N per 100 cm' of the medium, B. prodigiosut; likewise remains constant after repeated ijioculations at 30° C, whilst the control culture, without carbonate but for the rest under the same conditions, strongly varies. The same I'esult may be obtained with magnesiumhydrophosphale (Mg H P(_)^ . 2 H,(J) to excess; this, however, quickly precipitates, and in order to be active should be used in a bouillon-agarplate or in a thin layer of liquid. In ordinary bonillon-agarplates 1 pCt. of this salt changes entirely into crystals of ammoniummagnesiumphos- phate (MgNH^ PO, . 6 H.j()) the plate becoming quite transparent; a plate with 3 to 4 pCt. on the other hand, remains white and turbid. Although it may be admitted that i)y these various means the formation of secretion products by the bacteria is pi-evented, on whose stimulating action the variability probably reposes, yet it, is not clear how this preventing lakes place. Evidently substances should be thought of here which, once produced, cannot or only with difficulty leave the bacterial body. Of the said means quick transplantation is the simplest for always disposing of constant stocks for the e.vperiments. The oriijln of t/if tufi.ant.i in ijt'iievnl. When cultures, placed under favourable nutritive conditions, but for the rest prepared without special precautions, are growing older between 10° and 30° C, they exhibit a certain variability at which, as formerly described (1. c), variants are thrown oflf, while beside 1) At 4 cm^ of acid per 100 cm"' of ciilluii-' liquid tlic gruwili ol B. jirodifjontis is sliicliened, at *.• cm' it is quite stopped. ( W2 ) these tlif oriuinal I'di-in is rdinid iincliaiigeil. As liv Iransplaiilalions ill rapid siicres8ion (and under constant mid favonrable eunilitioiis) no ciiaiige occni-s during thousands of cell-partitions, this variability cannot repose on some law governed In internal causes only, luit a particular agency is wanted, whicii may have its seat witiiin the cells, hill whicli must yet he enacted on hy external circumstances. Although the variability can reveal itself already in an ordinary same well arranged culture, e.g. in bouillon or in maltworl, allowed to stand for a few weeks, yet this process may considerably be accelerated by repeated transplantations, not atXev a, very short time, but with longer irdervals, for example two days, with cultures kept at '.MY ('.. a not too small quantity of the material for the inoculation being used, c. g. two loo|is of the platinum thread. After three or four repetitions, so after about a week, the variation can then be in full course, the tirst culture, left to itself, not yet showing any perceptible change. This evidently reposes on the following circumstance. The intluence which canses the variability in the culture when it gets older, acts in the chosen conditions already after two days. If now a re-inoculation is performed, the germs alfected l)y that iiilliience can increase as well as those that remained normal, whilst by not re-inoculating, thus in the first culture, the non-affected germs are by far more numerous and remain so as the cell-division slackens after the second clay, because of want of f'ooil. At inoculation after two days there result at each time new modified germs, and those which are modified already, are enabled to augment without losing their modification. In this explanation it must further \nj accejilctl, that a Iransplan- lalion after Iwo days gives no cause foi' atavism: for if this were Ihe case, the re\'ersc ought to take place of what is obserxed : after a week's growth Ihe first ciilliire should be more \ aried than that which has ri'pealedly been Iransplanletl, but this is not so. This shows how carefully Ihe xarialioii exjieriinents must l)e cari'ied o.nt in (irder iiol lo become obscure. rarliciilariv llu' culliires on solid media must very ai'ciiralely be obs(>rved. If these are allowed lo stand for some days or weeks without further precautions, llien in many cases, even with magni- fying glass or microscope no variation ai all can be detected, although it is actually going on, commonly to "rose " or "while". Colony culture then shows that ln're and there varied germs oi' groups of such germs nnisl be presciil. for from ihe seemingly homogeneous matter large iiiiinbers of while and rose \arianls are ( 'i4:j , obtained, vvliicli prove as constant as the iioiinal form itself. However nnchanged colonies, representing tlie pure stock and producing a material as fit for further experiments as the original culture, lie among the variants. Experiences afforded by otlier bacteria seem to prove that the frequent repetition of the thus possible process of selection, produces a form wiucli \aiies less than the original material, iiut it is not here the [)lace to enter u|)on this important fact. All colony cultures of B. piodigio.ms are best made ou bouillon- agar-plates, which after solidifying have been cautiously dried oji a thermostat at circa 40^ C. The water which then condenses ou the glass cover can easily be removed; if this is neglected, _S. />/-f(//;//y.>7w, which is strongly motile, spreads over the surface of the agar and the colonies coalesce. I shall now enter into a short discussion of the most important variants. T/w obtained luiridiits. The variants derived from B. produ/io-sus may be considered as plus- or gain-variants, minus- or loss-variants, and qualitative variants. This is ex|iosed below in the table of descent, which shows the origin of the obtained forms; the (|ualitative variants [an.iuitns and hyallnua) are placed on the sauie line with the noi-mal forui. the plus-variants above it, the minus-variants lieneath. Hence, the arrows not only denote the descent but also whether the variability reposes on gain or loss of characters, or if it is qualitalixe. Dotted arrows indicate that atavism has with certainty been obser\ed. The names indicate the chief qualities characterising the variants. A survey of the variants without I'egard to their descent precedes ; then follows their pedigree, which does not repose on hypothesis, but simply gives the result of the experiments. The obtained variants are : 1. Bncdlus pnid/'/iosiis. Normal form, isolated from nature '). 2. ,, ,, rosi'us 1. ''^- „ „ ,. 2. 4. „ „ a lb us. 5. ,, ,, ,, /ii/alu)iis. 6. ,, ,, vlscosus. 7. ,, ,, „ dibits. 8. „ „ nitralu.'<. 1) About ISDO froiu moukleiiug bonus of a gelatinfactory near Dell't, 43 Proceedings Royal Acad. Amsterdam. Vol. Xll. ( 644 ) 5). i'xicilhis j)j'o(/ii//osiis. (tiiratus rlsrosii.s. 10. JJ. J 2. 13. 14. 15. /ii/ii/i)ms. alhus {= 7 ?) albns {= 4?). V/.SCO.IIIS. ,, a/bus. ulbus (== 5r) The relation and orieiii of these vai'iaiits is li'iven in tiie tbllowiii"- table. aur.viscosus aur.visc.albus hyal.viscosus auratus <- viscosusalbus prodigiosus normal hyal.viscosus a.'bus roseus:/ roseus^ > i^ys aur.albus.- albus albushyalinus hyaLalbus Tlif upwaid arrows denote "gain-variation", tiie liorizontal •■qualitative variation'', the downward arrows "loss-variation". Dotted arrows signify tliat atavism has been observed. The two qualitative colour-variants, (larnttis whirh is orange- coloiireil and Iii/iiI/ihis of a deep \ ine-red, varv in a wav (|uile corres- ponding to the normal form and like this throw otf, under the same circumstances, slime-variants and while vai-ianls. I'.esides, the normal form ma^' return Iw atavism as well IVom iinr.ilns and lujitluLU.-i themselves as fioni the variants derived from them. In lln' petligree table atavism is indicated hy dolled arrows for a few of (he cases where it has been staled wilh cerlainly. Itut there is no doubt that also the oilier \arianls are (lis|iosed lo ala\ism. ll shoidd nioreovei- be noted thai Ihi' (////vir///,v-variant appix)aches, at least in colour, ihe natural \ariely HikuI/hs h'/i'/iensis, but that ihe lall(!r pcssesscs a stronger power of fermentation, and produces much gas (CO, -\- H,) from maltwort with dextrose or cane-sugar, llu' former fernieuliug onlv dextrose. I'or Ihe rt'sl, //. A/VZ/Vz/.v/.s' ilself. which \aries in a \v ;i\ (piUc analogous lo Ihal of Ihe normal form o\' iir(i(/ii//osiis iiere considei-cd, has noi \('l been obtained as a \ariaul fr(un Ihe latter. ( 6-15 ) A new cliamcler vvliicli may rise in addition to (lie already existini; ones, is the prodnction ofa large (luantity of slime substance by exeessi\e growtli of the cell-wall, which slime may suread through the liquids, and makes tiie individuals of the colonies on agarplates cohere into one tough mass. From B. Kielieii.'iis was even a variant obtained whose colonies appear on the agar plates as a very consistent, almost dry zoogloea, but the analogous variant did not till now arise from the common proiligio-sn.'i. The vincosiis (6), dei'ived tVoin the latter, is an ordinary red slime bacterium. This red-coloured, tough-slimy form, which may be called B. pi'O- (Ii</io,ms riscosiis, is no doubt a plus-variant. Its production has been observed under the most different nutritive conditions, between- the temperatures 10° (in a cellar) and 3(P C, but always and exclusively in liquid media, iievei' on a solid one. The latter circumstance is apparently the reason why the numerous experimenters, who have studied B. proilii/iosas, have not seen this variant. It is true that ScHEUERLEN ') obserN'cd that old i)ivdi(^/iosiis-cu\tm-es sometimes turn slimy, but he ascribed it to their becoming alkaline and overlooked that a new constant form was [)roduced. The only distinct condition which seems different in the liquid cultures compared with the solid, is the access of oxygen. In the depth of the liquid this access must, of course, be very deficient for a long lime, or e\en be entirely lacking, as the upper layers of the culture, which are rich in bacteria, take up all the o.xygen. (Conse- quently anaerobiose becomes possible in the depth, which is not the case in cultures lying free on a solid medium, and this jiartial anaerobiose is apparently the stimulus which i)iduces the formation of the sliuie variant. That here a ralher complex influence and not a direct action must he ascribed to the partial withdrawing of the oxygen, follows from the fact that the culture of />'. invtliyiosu^ at complete exclusion of air. as in a clo.'^ed bottle, does not, even with repeated transports, give rise to the slimy variant. \\ temperatures of about 35' C. this variant is no more formed, although the growth of prodiyiosus is then still very strong: at 37' the growth slackens or ceases entirely, according to the food. In the following liquid media the proiluctioii of the slime variant has with certainty been observed, as well after repeated re-inoculations as after prolonged kee|)ing of one and the same culture at 25' to 30" C. : in broth, in bnilli with I pCl of glucose, in malt-w^ort, in tap-water with 5 jiCt of pure gelatin and 0,02 pCt IvJIPO^, and in 1) Archiv. I'iir Hygiene. Bd. :2(j p. 1. 43^' ( (;4(i ) la]i-\valer willi 2 pCt of gluoose, (1.5 j)('l of asparagiiie, 0,02 pCl K„HPO,, always cultivated at 30° C. and with repeated transports after two days or longer. From this we also recognise that there is no question of a dii-ect influence of the food on the production of the variant. The auratus- and hyalimis-vanantii, also, have only taken rise in licjuid cultures, namely in broth and in the glucose-asparagine solution. 31oreover, hynlimis, which is of a deep vine red, is easily obtained from a solution of pure gelatin in tap-water with 0.02 pCt. K„HP( >,, after repeated re-inoculations, at 30° C, whereby also InjaUnn.s viscosus results. The colourless or white variants, which tmly differ from the original form in producing no pigment, should certainly be considered as minus-variants. They are obtaijied with more ease than the slime variants and, at least as to N^ 4, have also been detected by other authors '). Kxcept under the said conditions, iipl l(( keep them constant, all the cultures as well in li(|uid as on solid media, vary sooner or latei' towards white. The original foiin does remain preserved, but a colourless \'ariant is thrown off, which is still more constant than the stock itself. Not always does one and the same variant result in this case: two uncoloured constant forms, N° 4 and 5 can easily be distinguished if they originate at the same lime, and their colonies are on the same agarplate so that they may be compared somewhat magnified, (hie, (ilhiis /ii/ii/ijtiis, then looks more blueish transparent, the other, iilliKs, is more of a cloudy and opake white; under the microscope the former proves to consist of smallei' cells than the latter. The cause of the production of white variants cannot be a more or less abundant access of o.xygen, but must |)i'obably be sought in a stimidus, exerted by seci'etiou producls wliicii lemain enclosed in the interior of the cells. Although the ])reseiu'e of auiuiouiuiucarbouale in the medium (liroth-agar), as also culti\ation at lempci'alui-es higher than 30' ('. e.g. at 33'' ('., pre\'ent pigment produciion, uo liei'i'dilar_\ \arialiou at all is caused by these inlluem-es. If the thus treated colourless cultures are traus|(orte(l al 20' to 25°, no while \ariants are obtained fr(Mu lliem, bul llie normal form i-< found back unchanged, if at least the above uu'iilioued precautious to preserve the constancy of the stock are not neglected. I) 111 F..EHMA.\N and Neum.vnn's Atlas, i'li Ed. 1907, Tablu 30, Kig. .'!, .-liii\v.< a cuJourcd image of a "pure cnlliiii'" of pradyiosits, consisting of red and while Coldllil'S. ( «-t7 ) Wiien llio wliitc variants of llic iiDinial form are ciillivated al 30 ('. ill liDiiillnji or ill inal(-\\iirl, llie cultures will, after a few re-iiiociilatioiis, turn sliniv like those of the red noniial foi-rn itself. (Joloiiy cultni-e on bouillonagar proves that white slinie vai'iaiits are thrown olf, ill tiie same wa\ as the iiornial form tiirows otflhere<l ones. The white slinie >ariaiits (N°. 7 r and 14) corresjioud Ity the nature of their colonies to tlie two wldte forms, (dhiis i4) and kIIhis hi/nJiniis [5), considered aliove. There is still anotiier method to obtain the colourless slime variant from the red one. If this latter is cultivated at 30^ in malt-wort or in liouillon, wo find after one or two transferrings, each time al'ler two days, and when sown on Ixuiilion-agar, many white slime colonies together with the uncliangetl red, moreover a consideralile number of quite normal, not slimy red colonies, N". 1, which is to be considered as atavism, but an atavism reposing on the loss of a character. The white slime variant, thus obtained by minus- variation, and found in the table as N°. 7, seems identic with the one produced by |ilus-variation from the not slimy \vhite variant, which hitter for that reason has not been specially mentioned. Already in my earlier paper I spoke of rose \ariants, which so to say, keep the middle between the normal form and the white variant. They may be produced in various wa^s, for instance, by cultivating the normal form on plates of pure gelatin dissolved in distilled water (H„(), 107o of gelatin) at room temperature, at which rapid growth and vigorous melting occur. By daily streaking off on a bouillon agarplate the same colony obtained on such pure gelatin, and provided the temperature be kept between J4° and 17° C, we tind, on the fifth or sixth day, the first rose variants, either or not with the white, which under these conditions appear later. Two ro.se variants (table N". 2 and 3) are easily distinguished, but it is possible that there are many more whose perception is beyond the reach of our observation. In any case, it is a fact that the character: "the faculty of producing pigment", is divisible in many ways. The here- ditary constancy of at least one of these rose variants proved not to differ from that of the normal form. Another methotl to obtain rose variants is cultivation of the normal form in Itouillon, which by evaporation has been reduced to a threefold concentration. After a. single transf)ort already, a large number of rose variants (3) liad appeared by the side of normal forms; by a much lighter colour they showed a disposition to lose their colour entirely. The variability of the different rose variants is not the .same; the form, obtained by the concentration ( <".4s ) ('xpcriuKMil ;.')) |)i-()iliico.s, niui-e roadilv iIkui llic rose \;iri;inl ("2), as well rod iioniud forms (J) as wliite ones (4). For tlie rest, lliis iiioi'c \ai-ialile \ariant has also |)rovc(l to remain coiislant when i|iiicl<lv lr;ni'^[>lanle(l. Casfs of atavism are Ircquenliv observed in these experiments. Tims, for e.vam|)le, the prodnetion of the normal form from viscosua (6) may easily be seen if the latter grows for a fortnight williout ti'ansi)ort on a bouillonagarplate ; along the margin of the streaks some few normal eolonies (1) will then become perceptible. The (i//jii.t-v<ivhu\ii<, also have a disposition lo throw off a few i-ed normal forms, but lliey do so only after gi'owing for weeks or months on boiiillon-agar; at lirst they are very eonstanl. The to a certain e.\tent completely regular production of the .same variants of BaciUm produ/iosns, suggests the existence of variability in a special and determined direction, of orthogenesis, as EniEit expressed it. As under dilferenl nulritixe conditions the same \ariaiil may appear, the food itself cannot be the stimulus; there nuist be, as said above, another cause in the interior of the cells, which, lor B. jHvdujiosm, seems only active in an alkaline environment. (_)n the other hand, the food, in a wider sense, has certainly a decisive inlluence on the variability, albeit indirectly. So we considered alreadv the inlluence of the alkaline reaction of the medium if this alkali is |)roduced by the microbes themselves. Another example is the following. As well in malt-wort as in bouillon the risvo.'Ots variant is regularly produced; but from mall-wort I lie niinitiis variant, which so readily takes risi' in bouillon, is not obtained at all. Indeed, every culture condition gives a peculiar but con- stantly returning mixture of variants, differing both (piantitatively and (pialitalively from that found under any other conditions. But the real factors here active could not as yet be detected. Ki'om the foregoing the following results may be derived. 1. Jiadlbis iimd'Kjiosns produces as well ipialilative, as gain- and loss-variants, all obtained with certainty by determinetl experiments; the stock-form is always found unchanged in the same culture with the \arianls. All the variants are from Iheir origin as conslaid as their stock. The true factors which govern the variability in these experiments arc still unknown. 2. r>y rapidly repeated re-inoculalions and by other methods, nor- iiial tiinii anil \ariaiils iiiav lie Ui'jil cdiishuil, as it seeius for an iiiiliiiiilcil Iciiuili iif time. 3. All tlu' \'ariaiits varv in a wav analogous to that of the normal fin-ni. lliiis, tlie r/M/r//;<.y-variant produces an (/?/yv///w-slimevai-iant, which must be considered as a gain-variant, and an '//^;/.y-variant, wliicli must he taken for a loss-variant. The natural variety B. KieliensLs, which approaches the nuratus- \ariaut, also varies in an analogous way. The variation thus seems to lie directed or orthogenetic. 4. Gain-ala\ism in loss-variants and loss-atavism in gain-\ ariants, can lie (ilitaii>ed with certainty by determined experiments. Qualita- tive \ariants, too, may gi\c rise to atavism. 5. The experimental variants of B. prodiglosus have not yet been found in nature. From anotiier bacterium, i?rtfi^7/^<5 /ier6/cy/'7, a \ariant, took rise which I had liefore repeatedly isolated from nature aud which I had taken for (piite another species. 6. The variants of prm/ii/iosiis, and this holds good for many (Mher uiici-obes also, dilfer from each other and from their stock forms ill the same way as clo.sely related natural species or varieties do auiong each other. Hut their tlisposition to atavism is much more pronounced. 7. The sub-variants, e. g. the ro.se variants of different colour- intensity, arise in the same way as the chief variants and possess the same degree of constancv. Physics. — "Researches on iii.agnetizdtii'ii atven/Iointeuiperntures." By PiERRK Weiss and H. K.vmerlingh Onnes. Communication N^ 114 from the Physical Laboratory at Leiden. § 1. Object of the research; results. a. Introduction. The extension of Langevin's ^) kinetic theory of magnetism to all ferromagnetic phenomena by means of the hypothesis of the molecular field ^) rendered the testing of deductions from tliis hypothesis by experimental data of great importance. The first results of this comparison were very encouraging ; in some respects a remarkable correspondence was found. For instance the cur\-es 1) Langevin. Ann. China, el Pliys. 8 Ser. I. u, p. 70; 1905. -) P. Weiss. Jouni. de Physique 4e Ser. t. VI, p. GGl ; 1907. ( (;5<» ) calciilalcd I'm- :lic iiiiensiiv iit' llic iiiatiiicliy.alioii a( saliiralioii as a riiiiclidii (if llic Irnijieraliirc t-()rres|)(»ii(l('il xx-rv well with lliosc which liad liocii found cx[)eriiiieiilaliv for uuigiietite at teinperaliu'cs aiuivc llie (>i(niiai-v. Moreover, llie law delerniiniiig the susceptibih'ty aho\e tlie Vv\uv.-/ti>/iit '} develo[)ed fVoiii Ilic iivpothesis of llie inolecnlai' lield was fmiiid in Cuuik's experiments, and in others which will soon be pul)lished, to be accurate over a temperature range of some hundreds of degrees. Finally the sudden changes in the speeitic heat at the ('i!i{iE-|»oin! were in correspondence with the \alues calculated from magnetic data. But other observations do not coi'respojid so well with the theory. Figl, PI. 1 in which the theoretical curve for the change of saluration-magnetizalion with temperature is shown by the full curve a, also shows the experimental results for magnetite, ami the correspoiiding curve, b, for nickel -). The last curve is drawn to such a scale that the best possible correspondence with the theore- tical is obtained at the CuRiK-point. In contrast with what was found for magnetite, nickel shows a deviation from the theoretical gradually increasing o\'er the whole cur\e. Iron and cobalt behaxe practically the same as nickel. When all I his is taken into consideration it is seen that the hypothesis of the molecular lield is of the nature of a working hypothesis; the partial contirmation shows that the hypothesis contains a kernel of truth, and from the experimental deviations one will have to see how it should be moditied or extended while still retaining its essential features. It is not probable that these modifications will attack the property of reacting against the orientation by the magnetic field that has l)een ascribed to the kinetic energy, or that they will come into conllict with the manner in which the MAXWEf.L-BoLTZMANN partition law has been eni|iloyed. Not only are these hypotheses of fundamental import, but they are still further forced upon our consideration by the ease with which they account for the fact that for paiamagnetic substances the huscci)tibility varies inversely as the absolute temperature — an experimental law that is one of the most firndy established for a niuubcr of substances. In their important investigations upon the mag- netization of the elements, of which an account was given at the last 1) 111 this (iomiminication we shall give tlie name CuRiK-point to the tempei-ature a! whicli spontaneous fei-romagnetism ceases. This is l)y no means inconsistent with (Jubie's idea that the transformation temperatnre is a function of the strength of the field, since the temperature at which spnnhnieous ferromagnelism ceases is the temperature obtained by reducing the field to zero. -) Ac.-ording to preliminary measurements. Accuratr I'xiifiiments upon llir IIiiim; iiii'tnls and maiiiii'lilc an' in* progress. { t'.^l ) iiiectiiii;' ',), 11. 1)1 l){ji.s aiul IIdmja liasc, il is Iriie, .shown thai this law of the (li>iteii(leiice of siisceptiliililv u]ioii temperature is not generally valid, and that paraniagiietisni also occurs which i.s inde- pendent of iIr' leniperature or increases v. ilh increasing temperature. But it is by no means the case that the foregoing hypotheses should be discarded on that account; what we learn from experiment in this case is oidy that these suppositions are not sufficient to explain magnetism as a whole. In particular il will be necessary to revise Lanoevin's hypothesis that the magnetic moment of a molecule is constant, or at least (|Urtsi-constant, and also that concerning the nature of the mutual action of the molecules, which until now has been represented by the introduction of the molecular Held. For an at low estimation of the value of both of these hypotheses, ex[)eri- ments temperatures are especially valuable. For it is only at the absolute zero that the magnetization gives the sum (jf I he molecular magnetic moments, as it is only then that hoal-molion can no longer [U'cvent the magnetization from attaining its full value; and at low temperatures, too, is the strongest demon- stration of the mutual action of the molecules to be expected, since they are then at the smallest possible distance from each other. b. Fi'rromiKjnetic suhstmiccs. We ha\e, therefore, aimed at the continuation of the curves connecting nuignetization and temperature in the three ferromagnetic substances and in magnetite down to the neighbourhood of the absolute zero. By utilising the methods and appliances'^) suitable for long-continued accurate measurements at such constant temperatures as are obtainable with liquid hydrogen, we have been able in otu' measurements to reach a temperature of 20°,3 Iv. with hydrogen boiling under atmosphere pressure, and of 14°,() K. with hydrogen near its melting point. The number of degrees on the absolute scale which separate these experimental temperatures from the absolute zero is but such a small fraction of the number between the absolute zero and the CuKiE-poiut (even in the case of nickel this number is still so much as 648 Kelvin degree?) that, considering the nature of the curves, we may regard the saturation-magnetization at the absolute zero as being determined by our experiments. All this, of course, with the proviso that the phenomenon in the region to which extra- polation is carried should give no occasion for adopting another point of view. Since the object of the measurements was a determination of the saturation-magnetization, it seemed suitable to direct the expe- 1) These Proceedings Jan. 1910. -) H. Kamerlingu Onnes, these Proceedings iSopt. '06, Gomm. Leyden N". 94^ ( (;:V2 ) riiiK'iit.s liiwanis (ililaiiiiiiu- ilala loi' iiia.iiiu'li/.aiioii in sh-diiu' licMs, ;uhI from lliese llie deiliiclioii ol' llic> law ;u"i'or<liiig lo wiiieh llie luai;'- iielizalioii aiiproaolieti its liiiiiliiiu' xaliic. I>ul llie metli(Kl rlioseii for the inagnetic measiu'ements, viz.: the delenniiiatioii of the luaxiiimiii value of the couple exerted by a magnetie tield of varying direction ujioii an elli|)soid of the experimental substance, was, as we shall presently sliow. less suitable for this determination of the law of approach than for <'oniparisons of the magnetizations of the substance in the same tield at different temperatures. The data to determine the law of approach were therefore made the subject of a separate investigation ') This gave the following values for the difference between the magnetization ui a tield of iO()()() gauss and that in the limiting case : Iron 0.08 »/„ Niclvel 0.1 „ Cobalt (soft) 1.1 ., Magnetite 0.19 ,, For these substances, the cobalt excepted, the approach of magneti- zaliou as a function of the strength of the tield is hy|)eri)olic, so thai in a Held of 20000 gauss, which we reached in our piesent expcrinieuls, the abo\e differences were reduced to half their values. Observations li\ die ellipsoid method in diiferent fields and at both low and ordinary temperatures have not, indeed, enabled ns to test the law of approach, but they show sufticiently well that there is no essential dilference between the behaviour in this respect at the two temperatures; and that at low temperatures, as could have been supjiosed the magnetic hardness docs not assume an excessive value, the molecules hindering each other in assuming a new direction. Further, bv means of comparative measurements, magnetizations at ordiiiai'y and at low temperatures in tields of great strength were coin|)ared, and it was found that the ratio between the two is pretty well independent of the strength of the tield. Thus, leaving the result uncorrected for the dilatation between the two temperatures (see note 2 on p. 11 ) we found for the ratio of the intensity of niauuetizatioii at 20", 2 K. and at ordinary tem|ierature the following: Nickel (17°.3C.) " 1.054,s Iron (20-. C.) 1.0210 Magnetite (15°.5C.:i I.O.^tJlt The exact value of the ordinary leiiiperaliire is given between brackets. In § 5 it will be e\|)lained why the experiments with ') P. Weiss, Aroh. dos Sc. pliys ct nat. I'l'vi-icr 1'.>U) and .Itmni. do i'hys. 4(> Ser. t IX. uvril I'.IK). ( (i53 ) cobalt luive nut been brought lo a coiichisioii. It is (liflioult to know exactly the degree of accuracy of tliese results. Experinieiilal work in every branch was carried out so tiial an accuracy of J in 1000 or even higiier could be expected, liul when i>ne considei's the disturbing iuiiueuces which made tiieniselves fell iu liie exijerinients upon cobalt, it seems rather incautious — and this is particularly the case with the nuxgnetite measurements — lo ascribe to the results an accuracy greater than 0.5 7„> even though the occurrences which have thrown suspicion u[)on the cobalt measurements were nearly absent in the case of the other substances, and though in all its properties and particularly in its extraordinarily large magnetic hardness cobalt stands evidently alone. Since our experiment's indicate these causes of uncertainty, they show how a higher degree of accuracy may be reached if so desired. The present accuracy is quite sufficient for the treatment of various problems. The experiments with iron and magnetite were carried to 14'' ,0 K. The change of magnetization between 20^ K. and 14^ K. is too small to be expressed in figures. These experiments, therefore, onl}' extend down to 14 K. the temperature region within which the diminution of the kinetic energy and the appioach of the molecules to each other do iiol occasion the appearance of a single new phenomenoji. The portions of the cui'ves for nickel and magnetite which have been newly obtained are given by l)roken lines in fig. 1, Plate I. Magnetite is of particular importance on account of the perfect correspondence between observation ami theory o\er the greatest portion of the region between the CuiUE-point and the absolute zero, and on account of the occurrence of a deviation of observation from theory only at low temperatures. Here, theory gives foi' the ratio between the magnetizations the value 1.139 instead of the value given above, 1.057. The result that theory and experiment clearly differ at these temperatures is cori-oborated by earlier experiments upon four samples of different kinds of magnetite, two obtained from natural crystals, the third from a fused natural crystal, and the fourth from artificial magnetite. These gave the followiiig values for the ratio between the magnetizations at the teuiperature of solid carbon dioxide (— 79°C.) and ordinary temperature: 1.033 ordinary temperature 16° C. 1.042 23^.2 1.043 24° 1.037 21^5 mean 1.039 21 ".2 C. while theory gives 1.053 for Ihe same temperature. ( <!-">4 ) All iiiialon'v lliiis seoiiis to cxisi l)(_'lw(.'('ii llii^ and (•'Mii!ii-c^>i(iii ami cx|)aii.si()ii liv Ileal, tor wliicli \ \n di'.i; W'am.sV theory and law of (_'ori'os|H)iuliii,n- slates are supported better as a rule in tiie iieigiibuiir- liood of the critu'al point than al low retluced teiiipenitiires where the ideal repi'esenlations ol' the molecule and of moleeiilar allruflioii no longer cover the phenomena snlHcienlly well and ihe ddfereucos lietween ihe spccitic pro])erties of ihe real nioiceules ap|)ear. 'I'he liy|iolliesis llial moleciilai' magnets are esseiiliallv imarialiie would lie estahlisiied conciiisi\ely if there e\isle(i simple relations lielweeii the magnetic.' moments as calculated pei' atom, which one mighl he led to sus[)ecl from the increase l\y regular sleps of Ihe satuiaiion-magnetizalion of the three metals. The following table in which the numbers in the fuvsl cobimn are taken from the paper'; referred to above and in which the relative increase for cobalt is estimated from comparison with iron aiui nickel shows that this is not the case. The data are not cm-rected for the dilation (see note 2 on p. TJ ). Specific Increase by i Specific Atomic Moment saturation at reduction to saturation at weight or of temp. ( ). low temp. low temp. '/, mot. wt. gram-atom. Ni MA\ (\~' C.) 1 .o.yi<s .-,7r, ri8 7 :i:«l Co 1(W (17° C.) i.ct llVt.G rs!) 9r.50 Fe ^217 (iO^ C.) 1 e-iOi '221.(1 :>6 12 III.) FeO '/., 90.7.-, (tr.o.S C.) 1 .or)7 y.=. i) 77. :« 7417 111 connection with this we must not lose sight of the fact that although the proof that the abo\e magnitude is of great significanee may have escaped us, still there is nothing wliale\'er lo justify an opposite coiK'Insion. When we look upon our measiiremenis as a whole we remain inclined li^ retain the liypolhesis that in ferromagiielic substances the inagnelic atom does not in ilself change much \\itli temperature. There wore indeed leasons for questioning if this approximate inva- riability, granting that il was proved in other circnnistances, still existed at extremely low lemperatnres. h]lectri''al resistance of metals, pliosphoresc('iice of snlpliiir componnds, absor|)lioii of light by the sails of the rare earths with or wilhont magnetic Held, all, at \er\ low leniiieratures. exhibit charactcM'islics that one may ij 1*. Weiss. Arch. de.s Sc. pliys. ut iial. and .Juuiii. dv I'hys. I'.lUl. ( G55 ) ti'v to e.\|)lain liy u^icribiiig them to Ibrces exerted In poiiderublc matter upon electrons ; these forces in tliat oxphuiation become of primary importance when the temperatnre sinks to that of liquid hydrogen, and it is ascribed to them in particular, that they make the current-carrying electrons in metals suffer an important diminution in number at very low temperatures by their being, as it were, frozen to the atom by the low tem|ierature '). It would also be possible that the motions of the electron> which cause magnetism while renuxining conslaiU or changing not much at other temperatures, begin to show considerable changes at very low temperatures. The negative result that nothing happens even at the lowest tem- peratures, which should throw doubt upoii the relative smallness of the variability of the nuignetic atom itself, is not |)erhaps without importance when regarded as a means of weighing the value of the above assnmptions regarding the phenomena mentioned, or as means of separating the group of electrons which occasion magnetism from groups which form the prime factors of other phenomena. c. \'aiL(i(hinii , rhroimifin, ii/dinjanir:'. The (|ueslion has oflen been asked if a gap \vhich cauuiM In- bridged o\er exists between the ferromagnetic metals of the iron group on the one hand and the paramagnetic metals of the same group on the other, or that the latter metals should also exhibit a \ery low C'rnii:-i)oiut if the temperature wei'e sufficiently lowered. Ch. Ed. GiiiJ.ArMi';") says with refereiu-e to the Heuslek allovs of Mn, Al, Cu and Mn, Sn, Cu which are ferrouuxgnetic : "The reason for this can be found in the fact that aluminium or tin when compounded with manganese, a metal from the magnetic group, raises its trans- formation temperatures, which, following an hypothesis already sug- gested by Faraday, ought to lie \ery low." It can indeed be seen that aluminium and tin raise the meltijig points of various allovs which they form with other metals (the series Al — x\n, Al — Sb, Xa — Sn) and seem to ])0ssess the general property of raising tem- peratures of transformation. We might, therefore, expect that \anadium, chi-omiuni, and man- ganese should at \ ery low temperatures exhibit either the characteristics of ferromagnetism (magnetization not proportional to strength of field. 1) Cf. H. Kamerlingh Onnes, Gomm. fr. the Leyden labor. Siippl. n". 9, p. 27 1904 and P. Len.\rd, H. K.^merlingh O.nnes and W. E. Pauli, These Proceedings .June VM'.\ Comin. IV. llie Leyden Laborat. n". Ill, jj. 3, note "2 1909. -) C.li. Ed. Guillaume. Acles de la Soe. helv. der Sc. nat. Vol. 1 p. 8S. 19U7. f (i5(; ) sjxtiii'iilidii. livsleresis) oi-, in coiitVu'inilv willi {'lkik's law, a sli-onulv ini'i'casc(.l paraniagiiciism. 'J'ht' susfeptiliililv at the teinperalurc nf solid livclrogeii should be alioiit t\veiil_\ limes as great as at ordiiiarv teiiiperaliire '). At lliis tiiue we were not yet aware of the results |)uhlis!ied hxsl mouth bv II. uu Bois aud Honda"), froui which it ap|»ears that the inverse |)i'oportioiialit_v of paramagnetisui to the absolute leuiperature is but one of tiie possible cases. To get an idea of the order of magnitude of the expected phenomena we may suppose that the paramagnetic 7 iron still exists at 14° K. with the same Curie constant (^product of absolute temperature by susceptibility). In that case a value of about 400 is found for the magnetization ot this salistauce in a. field of 20(J00 (ilauss. Some time ago Gkbhardt ") determined the susceptibility of man- ganese at ordinary temperature and found A' := 322.1 0^'' (density 6.4). The above calculation gives a value 134 for the magnetization of this substance in the same circumstances. And as the deflection in our apparatus is proportional to the s(piare of the magnetization, one would obtain a deflection smaller in the proportion of 18 in the case of 7 iron or 160 in the case of manganese than that which was found for iron at the ordinary temperature; as Uiis was 100 cm. the manganese deflection should still be cpiite easily readable. When we now introducetl into oui- apparatus roughly ibrme<l ellipsoids of Moissan vanadium and Goi.dschmidt chromium and manganese in succession, the- aw^aited change did not appear. In every case the deflection at the temperature of solid hydiogen as well as at that of hydrogen boiling under atmospheric pressure remained the same as it was at ordinary Itnnperalure, that is, to a few tenths of a niillim(Mre, and these must be ascrilied to the magnetism of the suspending apparatus. There was therefore no ferroniagnetism and we were obliged to choose between the following two hypotheses foi' these substances. We were either dealing wiiii paramagnetism of a new kind or with diamagnetism, whicii is also 1) A similar supposition formed [he starling point of a rcscarcli jjy H. K.^mek- LiNfiH Onnes and A. Peruier, wiiicli will sliortly be published, and is closely connected with the present research. This investigation has been taken to hand at llie same time with the present subject. Using the method ut the ma.xiniun: couple and the hydrostatic rise the mai^netizalions of liquid oxygen at various tempcraluri'S and ot solid oxygen at the tem)>eratures of boiling and solidifying hydrogen were measured, 'flie inciease of the magnetizalion at low temperatures was found to be very great, though not so much as was expected, and a distinct deviation from Curie's law and a characteristic ciu've were found. -) II. Du Uois and Honda I. cit. '^) Geuhardt. Inaug. Dissert. Marburg I'JO'J. / 057 ) Iniiiid ill t'(i|)[)er wliilc iimsl ol' llie sails of this iiielal are |i;iraiiiauiielic 1)1' Bois and Honda's paper in which tliese tiiree metals are classified under those whose paraniagneiisni is invariable or increases with the temperatnre shows that the first assumption is the correct one. The behavionr of copjter with the present research made us consider the other hypothesis a reasonaiile one. One could always assume that the paramagnetism, which, as a general rule is ascribed to the metallic manganese, results from the presence of its oxides, which are strongly magnetic, or of a small quantity of iron. To put this assumption to the proof we prepared very |)ui'e manganese from Mkrck's pui-e chloride, which had been proved to be free from iron. The preparation was accomplished by electrolysing the salt between a calhode of distilled mercury and an anode of iridium alloyed with 407^ of rhodium which is not attacked by llie cliloridion. The almagam obtained in this way was separated in a stream of pure, dry hydrogen. In this way a grey powder was obtained which when compressed in a glass tube as a mould took the shape of a solid rod. A rod pre|iared in this manner exhibited paramagnetism. A glass tube with the powdered manganese was also pai-amagnetic. The same manganese contained in a magnesia boat was thereupon fused in an electric resistance furnace and in an atmosphere of hydrogen. In this way an ingot was obtained which was co\ered with a light oxidised crust. It was found impossible to grind a\\ay this crust with <|uaitzpowder, since the melal was of the same hardness as (|uart/,. Emery could not be used as it is magnetic. The impure crust was therefore turned otF with a diamond tool, and a small cylinder of pure substance was obtained. This cylinder was found to be f,'iTOiiii((iiii'tic. Fig. 2 PI. I gives the liysleresis curve for this substance. The maxiuium value of the specific magnetization is 100 limes weaker than that of iron, and the coercive field is 670 gauss, that is to say, JO times as strong as the coercive field of steel which is used for the preparation of good permanent magnets. This peculiar substance seems moreover to have striking magneto-crystalline properties. The rod was strongly attracted between the poles of a magnet and placed itself perpendi- cular lo llie field. Manganese of the same degi'cc of purity can therefore occur in two states: iiaramagnetic and ferromagnetic. Gebhakdt's experiments give a su.sceplibilily five limes greater than that observed by w Bois. If Gf.bhaudt's [lowder was not impure or o.\.idised, it is thus [lossible that there are two iiaraniagnetic states. 1) Seckelsox, Wicd. Ami. LXVfi, p. :-J7, 1899. ( (;5^ ) As regards llie reri-oiiuigiielisin ut' iiiaiigaiiese, tliis luul already heeii observeil l>\ Skckelson ') willi eledrolylic manganese which was lihorak'd ai 100 ('. tVoni llie clihiride ii|)()ii a ])hxtiiir.in wire, and \\ilh a i'L'gnlus |ire|iared liy liiNsKN IVoni niangaiiese lluoride. Tiie vei-y indelinile observations concerning llic inagnetizalioii wliicii he pnblished do not contradict our measurements. By a more direct method we have proved tiie absence of strong magnetism in vanadium, chromium and manganese at low temperatures. l''or lliis pui-|)use we inlro«hiced elli|isoids of tlie three substances into a narrow unsilvered vacuum tube whose wails were separated by llie smallest possible distance; this was placed in a second similar tube also as narrow as possible and fdled with liquid air. We then determined the distance from the poles such that the ellipsoids were attracted from the bottom of the tube to the poles of the magnet. This experiment was made first with the inner tube empty, and then with the inner tube filled with liipud hydrogen. The following results were obtained : (Ordinary temperature In liquid hydrogen Vanadium Not attracted Manganese Attracted from distance of ti to <S mm Chromium '/, ., ,, ,, ,, 12 Chromium li ,, ,, ,, ,, tiO J The same as at ordinary \ tenqierature The results for Chromium /i wliicii probably contained a small splinter of irt)U must be lejected. We also found further that a crystal of iron sulphate at ordinary temperature was attracted from a distance of 25 mm. while in litpud hydrogen it was attracted almost from llie base of the maguel. Thus the weak magnetization of the three metals was found lo be ]iraclically invariable, while the iron sulphate exhibited a very great increase in magnetic jiropcrlies. This e.xperiment is well adapted fo,r disjilaying the characteristic difference between the two groups of substances and is a typical e.xanqde of the significance which even the simplest experiments a(U|uire within the fallow region of very low teraperalnres. § 2. Mdhtxls iind i(/i/)aratiis. (t. Discitssiim of llw niethoil nf tin- mii.vl/iiuiii ctniji/f. We measured the iuleusity of magnetization liy measni'ing the couple exerted on a |)rolale ellipsoid of rex'oluiiou of the (\\|)erimcntal substance arraugi'd so that the angle of the field with the major a.xis of the ellipsoid ndaht b(> varied, 'i'lie (\\nrc>siou f(ir the couple is ( fi59 ) M =^ {N^ — iVj) I'^v sin '/. cos (f wliere iV, and K., are the coefiiirients of demagnetization of tiie ellipsoid, /, tlie intensity of magnetization of tlie substance, v tiie volnme, and if the angle l)et\veen / and llio major axis of tlie ellipsoid. Tlic maxinuim value of this couple is 2 for 7 =^ 45°. Hence to measure / it is not necessary to know either the strength or azimuth of the tield which yields the maximum couple. To make use of these methods the ellipsoid is suspended from a torsion-spring whose displacement is determined by a mirror- method, and an electi'omagnet turning round a vertical axis is used. The method has already been described '). Its advantages consist of the small range over which strong fields are necessary and the extreme simplicity of the relative measurements.") We shall now discuss two sources of error which affect it and which, although they may be made as small as one wishes in theory, render it less suitable for the search after the law of approach to saturation, although they do not take away from its value as a method of comparing in the same field two successive and slightly differing states of the same substance. Infiuence of lahomoueneity of the field. The ellipsoid is placed in the centre of a magnetic tield possessing the synnnetry of a body of revolution. The strength of the field at this centre is a maximum for a displacement in the plane of the equator, y, and a minimum for a displacement in the direction of the X axis. It it given by the series which, remembering the equation AF=0 for the magnetic potential V, and converting to polar cooi'dinates /• and &, transforms into 1) P. Weiss, Journ. de Phys. 4 ser. t. VI, p. 655, 1907. -) For comparing the intensities of magnetization I and /' at two temperatures we have to take into account that v = mjd, m being the mass of ellipsoid and d Its density, so j,= -^, . — and -, = — ^, . ^ . The dilatation at tlie low temperatures and therefore the proportion of d and d' being unknown, we have ■omitted the conection for the difference of this proportion and unity, the value of which may be estimated at 0,004. |Added in Translation j. u Proceedings Royal Acad. Amsterdam. Vol. XU. ( 660 ) Now, tlie energy of a volume-element dv of the ellipsoiil vvliicli we consider to be very long and magnetized with equal intensity / in (he direction of the field, is Tl' = — IHdv and tiierefore the moment of the couple exerted by the Held on this element is dW 3 /d-H\ dM' = —- = -- do . I -— 7-= sin 8 cos 6. ^8 2 ■ \d.v'). The conple exerted by the field //„ on the ellipsoid is M r= (iVj — N.^ I'-v sin (p cos <p. In very strong fields the condition is fulfilled that the magnetization is parallel to the external field nothwithstanding tiie demagnetizing forces of the ellipsoid, and therefore 8 = if. Hence the disturbing moment dM' varies with azimuth of the substance in exactly the same manner as the chief couple Af. The maximum value of the couple dM' is dM' — - do I .. ^ wiiich t'lir the whole elli])soid gives M' = — I\ -—- 10 V 3*' where a is the semi-major-axis of the ellipsoid. Tjct us now make the assumption that the field changes coidbr- mallv, and let us call the field 1 cm. from the axis in the direction of the //-axis (1 — e) //„, then and therefore 4 \dx^ J 6 M' — ~ Is IL V . «' and the ratio between the maximum values of the cou|)les is M' 6 H„ = -- e a' M 5 (N,-N,)T Willi consiant magnetization, fhcrefore, the second last equation shows that the disturbing couple increases proi)ortionaIly to the strength of the field. In the JH diagram, a sloping instead of a ( fifil ) horizontal asymptote will be found. This was shown clearly in some of the foregoing cxjieriments. If the field is constant the disturbing couple increases with /. Therefore, if in tlie measurements with the greatest values of J for which the experiments are carried out, made with a certain apparatus the disturbing couple does not make its presence felt, then a fortiori is it negligible for the smaller values of 1. The last equation shows that the relative value of the couple for non-uniformity of the field increases as the intensity diminishes. Hence it is to be feared particularly when one works with small magnetizations, and when, to increase the sensitivity of the apparatus, the torsion spring is replaced by a weaker one. For the purposes of our measurements it is sufHicient to get an idea of the order of magnitude of the error. For this purpose the non-uniformity of the field was measured for three different values of//; it was found to be proportional to ?/'- with e = 0.0087 as factor. '1/' // With a. = 0.15 it follows that — = 0.00023 . For the M {^\-N,)I ellipsoids used JV^i := 1.90 and ^V, r= 5.59, and {JST^ — ^,)/ is almost 6600 gauss for iron and 1800 gauss for nickel. Hence, for iron the correction is scarcely 1 in 1000, while for nickel it increases to some thousandths. Reaction of the ellipsoid on the pole-pieces. When the ends of the ellipsoid come into the immediate neigh- bourhood of the end surfaces of the poles, they exert a noticeable influence upon the distribution of magnetism in the pole-pieces, and the couple becomes increased thereby. This fact was established by previous experiments with a larger electromagnet with flat pole-pieces of 15 cm. diameter. In these experiments was measured the couple exerted upon an ellipsoid with various distances between the poles by a field of the constant value of 9770 gauss regulated each time by passing the required current. In tliis way the following values were obtained for an iron ellipsoid 9 mm. long and 4 mm. thick. 'istance between poles Ml iximum couple 9 mm. 335.6 15 „ 320.45 23 „ 319.32 35 „ 319.18 47 „ 319.08 44* ( fi62 ) Tlie law accoi'diiig (o which tliis magnitude changes shows tliat the cliange is not a coiise(juence of the non-unilbrmity of the field ; ihv jnst when tlie disturbance readies its greatest value, the field is most regular owing to the closer approach of the flat pole-pieces. For distances of 23 mm. and greater the influence is insignificant, and the couple is constant. b. Electromagnet. From what has been said about the influence of the ellipsoid and the pole-surfaces it follows that the distance between the poles should be about three times the length of the ellipsoid. The total thickness of the four walls of the Dewar tubes and of the holder (§ 2c) could not be made smaller than 5 mm. Hence, keeping account of the difficulty on the one hand of obtaining strong fields of wide extension and on the other hand of reducing very small ellipsoids to the correct form, we decided upon an inter- pole distance of 9 mm. and a length of 3 mm. for the ellipsoids. With this distance comparatively strong lields (up to 25000 gauss) may be excited with a magnet whose cores are 9 cm. in diameter. The electromagnet of this power which was used in these experiments has already served for magnetic experiments at high temperatures. It has already been described ') and is represented diagrammatically in tig. 1 PI. II. Comparatively light (132 KG.) and, taking its power into account, easily transported, it was possible to study it in Zurich and to use it in Ley den. Hand-wheels, whose position is read from divided circles, coinniunicate a horizontal micrometric movement to the pole-pieces. The magnet turns upon a vertical axis and for that purpose is mounted upon a ball-bearing support. The azimuth is determined by means of a fixed mark on a cylindrical scale E„ attached to the movable portion of the supporting base. Each of the coils has 1500 turns of 2.5 mm. wire and has a resistance of about 2 ohms. As the coils are arranged for a current of 10 amp. under ordinary circumstances, and as the current can for a short lime be increased to 25 amp. tiip nuniber of ampere-turns at one's disposal may reach as high as 75.000. 'i'he water circulation E,, between the double walls of the coils has this immediate advantage that the duration of an experiment may be doubled, but it is chiefly of importance in protecting the pole-pieces from heat. Such a heating would lead to various diniculties, of which one of the worst would be that the strength of the field would noticeably change, for the expansion of I he com- 1) G. ZiNDEi.. i'.i'vuc electriquu 20 Juiii I'.'OU ami Klcklrul. Zeils^clir. XXX, p. 446, 1909. r fi(^^ ) parativelj long core by heat could distinrtly aller the coni|>ai-alively short distance between the poles. c. Cryogenic apparatus. As it was necessary to shut otf from the air the space in which the ellipsoid was freely suspended since it contained liquid hydrogen and its vapour, a fairly complicated cryo- genic apparatus had to be employed. This is shown diagrammafically in Pi. II fig. 1 and in section in fig. 3. The apparatus consists chiefly of three tube-shaped portions which, naming from outside inwards, we call the cover, the adjusting tube f, and the holder I). The cover consists of a silvered vacuum tube A, a brass tube B, a glass tube C, and a cap D which shuts off the apparatus from the air. Holder. The ellipsoid a (figs. 3 and 5) can turn round a vertical axis with the holder h in which it is fixed. For the greater part of ils leugtii tlie holder is made from a tube l>„ of german silver — a siibslance that is rigid, little magnetic, and a bad heat- conductor. The lower end is joined to a copper rod /;,, which has only a very weak inherent magnetism. The holder is connected to the rod k by the spiral spring g^ (y, was used for iron and cobalt; the weaker spring g^, which was used for nickel and magnetite is shown at the side). To make the equilibrium stable and to prevent the ellipsoid from being attracted to the poles of the magnet the holdei- is held fast underneath by a wire of platinum-iridium of 0.1 mm. diameter, for the torsion of which a correction need hardly be applied (§ 4). The tube />., and the rod A, are carefully adjusted on the lathe, and the ellipsoid a (fig- 4) is fixed carefully in a cylindrical opening, the diameter of which is equal to the minor axis of the ellipsoid. If the ellipsoid is nickel or magnetite it can be fixed in position with a little wax. With iron and cobalt, however, the ellipsoid is subject to such strong forces that it is necessary to clamp it fast by covering it with a thin piece of sheet copper and then driving it forcibly into the opening. The turning of the ellipsoid is transmitted through the rod b^, and the thin-walled german-silver tube ') b„ to the mii'i'or h. From the mirror through the opening /",„ and the window (\ (figs. 1 and 3) the torsion of the spring g^ is read. A 1) A slight twisting of this lube is of do account. Only that portion of the apparatus between the mirror and the cap acts as a spring. Twisting of the portion of the apparatus below the mirror only transmits the couple to that spring, ils sole effect is to slightly, but not noticeably, alter the azimuth of the magnet. ( fifi4 ) glass scale 1.5 meters long, and siil)(li\ idod into half luiUiineters is used; it is placed at a distance of 4.325 m. and is ilhuninated by spherical mirror strips'). The tension of the spring is regulated by the I'od k i^iig. 3), which passes through a stuffing box D, in the cap D. Vertical motion is communicated to k by turning the nut /), and at the same time preventing the motion of JJ,^. The tension is read through the opening y'^, from the pointer / on the scale A^. Before mounting the apparatus, that division of the scale b.,^ is determined which corresponds with the tension that is to be used, by susj)ending known weights from the stretching wire. The apparatus is, like a stretched string, very liable to start vi- brating under the intluence of small impulses. This tendency is counteracted by immersing the vanes of a vane-damper b^ (fig. 6 and fig. 3) in oil contained in a circular vessel divided into different chambers by the partitions b,^. These partitions are attached to a cylinder which turns with slight friction in the adjusting tube and is therefore carried round by the vanes h,,, whenever the holder must experience a somewhat greater torsion (§4)^). The vanes must be wholly immersed in the oil so as to ensure that capillary reactions do not bring forces into play (see § 4), whose torsional effect could not be neglected. In strong fields the torsion oscillations are damped extremely well l)^' the Foucault currents. The whole holdei- and spring hang in the adjusting tube / the upper end of which is screwed to the cap D\ this cap also carries the rod k, and is itself supported by the glass tube C. The adjusting tube, consisting of the portions /2,/s, /,>/&, is three times diminished in cross-section. The lowest portion /\ is narrow and surrounds the rod Aj of the holder as closely as possible. Against the bottom J\ (fig. 5) rests the cone c, which is soldered to the wire (/ and serves to keep it taught. A slit in the bottom allows the conical portion to be placed in position (fig. 5). When the apparatus is put together the adjusting tube sinks into the Dewar vessel A so that the thin tube fr, is centred in the narrow portion of the vacuum tube. The adjusting tube as well as the tube b^ of the holder is made of german-silver. To mount the adjusting tube already containing the holder in the cover, the cap D is screwed to a bronze ring cemented to the glass tube C of the cover; the screws D, are tightened, and the junction ') H.Kamerlingh Onnes, Gomm. fr. the phys. Lab. Leiden, n". 25. (1896). -) It is essential to free the oil beforehand from volatile substances, and also to prevent the accumulation of air bubbles under the oil, since tlie apparatus has to be completely evacuated after it is put together. ( ()G5 ) is made air-tight ijv means of the rubhei- sleeve D^ wliich is smeared with rubber solution and bound with copper wire. Tiie lower end of the glass tube C is cemented to a second bronze ring, wliicii is soldered to the brass tube /> of the cover. To the centre of this brass tube is attached a ring B, carrying the bolts of the supporting rods B^ which hold the vacuum glass in position. The Dewar tube it-elf consists of a narrow lower portion A^ com- pletely silveied and a wiiler upper portion that is silvered up to -Ij (the ujiper ])ortion is left transparent so that we niiglit be sure that we were not allowing loo much liquid hydrogen to enter the glass). It tits into the brass tube Z?, and is protected b^- a wooden ring. The supporting rods B, keep the vacuum tube in position and at such a height that it is just clear of the wooden safety ring. Fig. 7 sho\vs how, by means of the screw B^„, the \acnnm glass protected by a layer of paper is clamped to the thin brass ring 7^^, to which are attached the ends of the supporting rods B^. The lower |iurtion of the vacuum tube has an external diameter of 8 mm. and an internal diameter of 5 mm. The glass walls are 0,5 mm. thick, which leaves only 0,5 mm. as the distance between the two silvered walls. The apparatus is centred by placing it on an auxiliary support by means of the ring B^. Before the vacuum tube it yet in position, the narrow portion f-^ of the adjusting tube is adjusted by a central ring in an adjustable centring-plate. The loose ring is then removed from the plate and a second is fitted such that it just tits the narrow portion of the lower end of the vacuum tube. The nuts B^^ serve to iiring the vacuum glass to its proper position, and, as before, it is made air-tight by a rubber sleeve 7J„, which is smeared with rubber solution and bound with copper wire. By adojiiing this method of attaching the vacuum tube one need not fear alteration of the cover \vhen the apparatus is evacuated, and only small further adjustments are necessary for recentring the apparatus after evacuation. In the tube B is soldered the steel capillary 6, (figs. 1 and 3) of a helium thermometer ') with gerraan silver reservoir 6^ (figs. 3 and 7) and glass stem 8^, which is permanently attached to this portion of the cover. The quantity of helium is so chosen that at the boiling point of oxygen the mercury stands at a mark in the lower portion of the stem, and at the melting point of hydrogen at one in the upper portion. If, as is the case with hydrogen boiling under ordinary atmospheric pre.ssure, the temperature is sufficiently well known ') Compare the apparatus for the liquefaclion of helium. H. Kamerlingh Onnes These Proc May/June 1908, Comm. Leid. N". 108. ( 666 ) widioiil reading tlie tliermometer, (lie tlierinometer is still necessary, however, to indicate the position oi" the npper surface of the liquid gas which is no longer visible beneath A,. As soon as the level siidvs ijclow the npper end of the reserxoir 6', of the thermometer, the mercury in the stem 8, sinks. (/. First the electromagnet is adjusted which operation is independent of the centring of the adjusting tube, the holder and the vacuum tube. The axis round which it turns is made vertical, and then the pole distance is centred round this axis. Next the centre of the truncated spherical socket G^„ (fig. 1, 2 and 3) is made to coincide with the axis round which the magnet turns, It is supported by a plate which is attached by two beams to the freestone pillar @. The cryogenic apparatus is then brought from its auxiliary support and arranged in its proper position by placing the ball-shaped portion of the surface of the ring j5, in the concentric socket G^„ ; the centring of the narrow portion of the vacuum tube on the turning-axis of the magnet is completed by means of wing nuts on the ring B, . This centring must be done with great accuracy, for the magnet must turn freely and the distance between the vacuum tube and either pole is not more than half a millimetre. It can, however, easily be accomplished to 0.25 mm. t'. Li(|uid livilrogen is introduced into the apparatus by a german silver tube B. (cf. Comin. N". 94/'). The gas formed by evaporation escapes through B^ (tigs. 3 and 1) and through the valves 7v", /v, (fig- 1) to a gasometer or to a vacuum pump. By means of the valves the vapour pressure is regulated, and its value is read on a manometer // which at the same time acts as a safety valve. In experiments made in the neighbourhood of the melting point of hydrogen the pressure was kept slightly above that of the triple point. Before introducing liquid hydrogen through the tube B., which is closed by a rubber tube with a glass stopper, the air is pumped out of the apparatus through the valve A',. It is absolutely essential that the apparatus should be air tight, for traces of air would .solidify ill the li(iui(l hydrogen and, owing to magnetic altraclion, would collect in the neighbourhood of the ellipsoid. To prevent the cooling of the upper portion of the apparatus containing the torsion spring by the boiling hydrogen, a number of lai'ge openings are made in the tube /\ (tig. 3) arranged in such a way that no injury is done to its resistance to torsion. In addition to this copper screens surrounding/, and soldered to B^, are arranged so that the tube moves witli sliglit torsion in them. A little cotton- wool placed on the bottom of the vacuumglass and attached to the holder lessens the sudden bubbling') up of the hydrogen'). Further additions of liquid hydrogen are made in the same way as the first. As a rule various series of measurements could be made Avith a single filling with liydrogen. The point of the vacuumglass wiiicli could not bo silvered was protected by a small silvered vacuum beaker L containing liquid air. When the portion of the apparatus above the diaphragms /?,„ is again at ordinary temperature after a filling with liquid hydrogen, one can hardly notice that there is liquid hydrogen in llie iipparatus at all, if it is not above /I3. In the course of time a little mist is precipitated on tiie vacuum tube. By surrounding the tube at A^ with blotting paper, the moisture is prevented from trickling down between tlie pole-pieces. Further- more a stream of air is directed against the tube between the pole- pieces. Hence the pole-pieces are in no way affected by the cryogenic operations. /'. Tlie springs are phosphorbronze. This substance is non-magnetic and acquires very little permanent sei. Springs of the same constant can be made b}' winding a spiral either of a thin short wire or of a much longer thicker one. Of the two, the one which has the greater mass will experience the smaller specific changes, and consequently will be the more perfectly elastic in working. This circumstance has been duly taken into account. The springs are proxided with straight extensions in the direction of their axis and are coiniected with the holder and the rod k (fig. 3) by screws. The turns of the spirals do not touch each other. The temperature of the spring is measured by a mercury thermometer that is clamped against the cap D and with it is insulated with wool. The constants of the two sprijigs used are 261000 and 22300 dyne-centimetres per radian. The corrections for the intluence of the stretching wire and for the temperature change of the spring will be discussed in § 4. The ellipsoids of iron, nickel and cobalt are 3 mm. long and 1,333 nun. thick. They have been made with great accuracy by the Societe Genevoise pour la Construction d'lnstruments de Physique. They were turned under a microscope giving a 30-fold magnification and provided with a camera lucida so that the image of the object and an enlarged drawing could be superposed. Measurements with 1) Should this occur one must ensure that the oil of the damper is not cooled by the drops that arc thrown up. ( fifiS ) tlie dividing-engine ii.ave shown that tlie ellipsoids are very aoeumtely shaped. The iron was obtained by melting pure electrolytic Merck iron contained in a magnesia boat in r-n electrical resistance furnace and in an atmosphere of nitrogen. The nickel and cobalt were prepared in the same way, starting with the purest possible nickel and cobalt powder specially prepared by Mkrck for these experiments. The magnetite was obtained by constructing an approximate ellipsoid from a drop of very pure magnetite obtained by melting very pure Mkrck sesquioxide in an oxy-hydrogen tlame. Since experiment showed that it was only at very high temperatures that the last trace of oxygen was driven out and real magnetite') obtained an iridium cupola was used for this operation. Ellipsoids of approximate shape wcve also constructed from Goi.D- SCHMIDT chromium and manganese and Moiss.vN vanadium. As can easily be seen it is not necessary for comparative experiments that the ellipsoids should be constructed with particidar accuracy. This was, moreover, experimentally denionstrateil for nuignelite, of which various samples roughly worked to various ellipsoidal shapes were used for obtaining curves for the thennal change at high tem])e- ratures, and these curves were in agreement with the theoreiical curve, and consequently with each other. § 3. Expermierital method. As mentioned in the introduction our aim was not to obtain absolute values for magnetization in strong tields at ordinary temperature and at the temperature Of liquid hydrogen, but to compare the \aUies at these temperatures; for we might expect that the change would be only a small fraction of the quantity to be measured. Hence it was an obvious procedure lo make observations at these temperatures alternately in the same Held. The change, however, from the one temperature lo the other neces- sitated operations of such duration as to pr()hil)it the use of this method. Hence we usually began with a series of measurements at ordinary temperature, in which the field was made the required series of strengths. Then an analogous series of measurements was made at a low temperalui'e, and after the ap])aratus had returned to ordinary temperature, some individual measurements were repeated so as to make sure that the apparatus had not in the meantime undergone any change. 1) See also P.Weiss. Arch dos Sc. nliys. ct nal. fevr. 1910 und Jomn. do pliysique, 4e S6r. t. IX mars 1910. ( fiG9 ) Each series of measurements consists in turn of two branches. First by tentative approximation from botii sides for all values of the field those values of the azimuth of the electromagnet are found for which the couple is a maximum. In this way two azimuths are found which are symmetrical with respecl to the major axis of the ellipsoid and which exert couples of opposite sign. This determination can be made accurately to within 0,5° to 1°, which is quite suflicient. Then follows the true measurement in wiiich the magnet without current is placed in one of these positions, the circuit is closed and immediately afterwards the deflection is read. As soon as this is done, the circuit is broken, the magnet is placed in the symmetrical position; once more the current is allowed to flow and the new deflection is obtained. Since these operations occupy only a short time, the after effects in the spring are of no account. The difference between the scale readings gives twice the value of the couple to be measured, independent of the residual magnetism remaining after the current was broken, which however occasioned only an extremely small couple. The field was given as a function of the current indi- cated by the ammeter. For these observations the same ammeter (Siemens and Halske instrument, no temperature coefficient) was used which was employed in the study of the field. This method of evaluating tiie field was quite sufficient for our purpose. The distance between the pole-pieces was read off the divided cylinders of the magnet and was vei-ified by passing between them callipers which had been previously adjusted to the desired distance. The fields given above are corrected for the demagnetizing fields of the ellipsoids. ^ 4. Corrections and controls; auxilinry measurements. The inherent magnetism of the holder is not so weak that the corrections neces- sary for it may be neglected. On that account a series of measurements Avas made with no ellipsoid in the holder at ordinary and liquid hydrogen temperatures. With the weaker spring we found: TABLE I. Correction for the magnetism of the holder. ordinary temperature ^=:r20°.2K. 4000 gauss 0.18 cm. 0.26 cm. 8000 0.29 0.48 12000 0.36 0.61 16000 0.43 0.73 20000 0.50 0.86 24000 0.57 0.98 ( (i70 ) 22300 For tlie stronger spring these corrections are multiplied by nc-,r.no' they are very small. Direct measurements have shown that the values calculated in this way are correct, which indicates that the inherent magnetism of the carrier is not changed by the \arious ojierations of mounting. There is still a correction to be applied to tlie couple-ratio for the .change in elasticity of the steadying wire under the carrier when its temperature changes fiom ordinary to that of liquid hydrogen. To obtain that correction the ratio of the torsion modulus of the platinium iridium wire and that of tlie weaker of the phosphorbronze springs was measured at the two temperatures. This was done in an apparatus similar to the one we have described with the exception that the cap D could turn relatively to the cover. By a mirror method the position of the cap was read on a scale at a distance of 175.9 cm. The cap was turned through an angle of about 360°, and the exact measurement of the angle was obtained from the same scale. This angle is the sum of the torsions of the spring and the wire caused by the same couple. The torsion of the wire was read from the mirror of the holder. In this way the ratio of the modulus of the wire to that of the spring was found to be 0.0125 at ordinary temperature 0.0144 in liquid hydrogen. Tlie fourth decimal is uncertain; hence the correction is two thousanths for the weak spring anil two ten-thousandths for the stronger. The temperature coefficient of the phosphorbronze spring was obtained from determinations of tlie period of oscillation of the same oscillating system while the spring was first at the ordinary temperature and then suirounded with steam. By means of the temperature coefficient thus determined viz. : k = — 0,00053 the observations are reduced to the same temperature. The temperature of the li([uid bath in the vacuum tube was proved to be constant to 0,1 degree, by carrying out temperature measure- ments with a platinum resistance tliermometer placed at different heights in a similar vessel. When placed alongside the thermometer G it indicated temperatures corresponding witli those deduced from the vapour pressures. Caj)illary action in the oil damper. Care was taken to fill tiie oil vessel to such a height that the cylindrical ring carrying the vanes of the (iani|iei' was partly immer.sed ( fi71 ) in the oil so that tlie vanes were completely immersed and should experience no capillary action. But still we wished to know the order of magnitude of the forces brought into play by capillary disturbances; for this purpose we greatly magnified them. A damper as like ours as possible was tilled only to such a height that the vanes and partitions intersected the surface of the liquid. The movable portion was suspended by a platinum-iridium wire 20 cm. long and 0,1 mm. thick ; deflec- tions were read from a mirror on a scale 2 metres away. The oil vessel was placed successively in two different azimuths such that the approach of the vanes towards the partitions would bring into play 'couples of opposite moments. The scale deflection was 5 cm. The moment of the couple is therefore of the order of two thousandths of that of the couple exerted on the nickel ellipsoid. § 5. Details of the observations. Nickel. The first series of measurements was made at 17°.2 C. TAHLE II. //(gauss) f (cm. of the scale) 2230 89.42 6250 89.97 10270 90.12 J 3280 90.34 17760 90.50 20300 90.66 21540 90.79 22760 90.81 The scale reading was always corrected for the ratio of the tangent of the double angle to the double angle of the detlection. The zero as determined by the mean of readings to left and right remained constant to a few tentlis of a millimeter. After this series the apparatus was accidentally damaged; it had therefore to be taken to pieces and remounted. That occasioned a small change in tiie magnitude of the deflections. Since the change of / ^ with H is determined by the foregoing series, only two points were subsequently determined at ordinary temperature before and after determinations in liquid hydrogen. ( 672 ) T At 5LE HI. t = 19°.5 C. 11yd rogeu at atni. pressure (20^.2 K. U (gauss) i 1 ' (cm. of lie scale) H (gauss) /' (cm. of tlie scale) before J 780 93.57 16100 91.74 5410 100.49 20540 92.09 5050 101.54 after 11830 101.84 16100 91.79 16100 102.13 20540 92.20 19050 20540 22020 22840 102.34 102.51 102.48 102.49 The zero determined from tlie mean of readings to right and left changed by about 2 mm. For // = 16100 gauss H == 20540 ha-'.-iK 1.0549 1 .0547 mean 1.0548 not connected for ililation. Cobalt. Tlie measurements with cobalt did not lead to the desired result. It was the extreme difliculty of bringing the magnelizalioji of cobalt to saturation encountered in preliminary experiments Ihat had led to the choice of an apparatus of such small dimensions. For the other substances a weaker tield would have sufficed, and hence a greater distance between the poles would have served. In the observations at ordinary temperature something unexpected already happened. Although the mean of the readings to right and left ought to have given the zero-[)oinl of the apparatus, the |)oint was actually observed to vary with the field. This change was after- wards seen to be about twice as great at low temperatures. The following figures bring this out clearly. (In the cobalt measurements the external field is given uncorrected for the demagnetizing lield of the ellipsoid. When saturation is reached this is 5000 gauss). ( 678 ) TABLE IV. Cobalt I at ordinary temperatiwe. h Cg^aiiss) r (cm. of the scale) calculated zero 4025 17.16 76.73 8050 38.14 77.47 the observed 12075 50.48 78.96 zero was 19560 53.24 78.37 not 23340 53.29 78.18 recorded 25650 53.30 78.63 Cobalt [ at temperatw ■e of sollillfylmi lujdroijen (14°. 3 K.). 4025 13.5 77.62 8050 32.59 78.84 -J 5820 53.23 81.93 observed zero 19560 54.33 81.40 78.26 21800 54.43 81.16 23340 54.45 81.02 24760 54.46 80.08 From this it appears tiiat asymmetric disturbing forces alfect the main phenomenon. It is probable that we are here dealing with phenomena of crystal magnetism arising from the fact that in the small ellipsoid the crystalline elements of the cobalt are not suffi- ciently numerous to realize isotropy by compensation. The magnitude and sign of these subsidiary actions are independent of the main phenomenon, and they can even be of opposite effect for both azimuths of the electromagnet ; they can become of very great importance if the substance possesses a more or less pronounced magnetic plane, and the example of pyrrhotine shows us that their influence becomes greater at lower temperatures. Further, the law of approach to saturation in cobalt which differs from that which holds for the other substances is consistent with the existence of strongly developed magneto-crystalline phenomena '). These experiments were repeated with a second cobalt ellipsoid, and the same asymmetric action, but somewhat weaker, was observed. But in this case a disturbance of another nature was encountered, which shows how concomitant disturbing phenomena may affect the measurement of magnetization: the magnetization at low temperature was now found to be apparently smaller than at ordinary temperature. The following table contains an extract from the results obtained with this ellipsoid. P. Weiss, Arch, des Sc. pliys. et nat. fevrier 1910, Journ. de phys. mars 1910. ( fi'-i ) TABLE V. Cobnh TI at t -- = 18° C. ^c (gauss' /■ (cm. of scale) Calc. zero 402.5 20.33 77.56 12075 54.16 76.49 23340 59.76 76.86 25560 59.94 76.90 Co/' 'lilt If in H^ at atin. presmre (20°. 2 A'.). 15080 53.53 76.61 23340 58.09 77.07 25650 58.46 77.21 ()l)s. zero 77.70 78.90 The same ellipsoid was remo\'ed from the carrier and replaced with Khotinsky cement ; one could easily understand that very strong strain-magnetic phenomena might be occasioned by forcibly driving it into its mount. At the same time it was for the new experiment displaced through a different angle of rotation with respect to its major axis; by this operation the sign of the change of zero point as a function of the lield was reversed. TABLE VI. Cohah n t = i6°.5 a He (gauss) /' (cm. of scalej Calc. zero Obs. zero 8050 40.99 79.12 19560 56.48 78.80 79.45 23340 57.07 78.89 25650 57.34 78.90 Cobnh II in H, at aim. pressure (20°.2 /v.). 8050 34.07 78.88 79.20 19560 53.21 78.24 23340 54.26 78.33 25650 54.63 78.42 The oidy conclusion one seems to be able to draw from these experiments with cobalt seems to be that the increase in magnetizatioji of cobalt between ordinary and liquid hydrogen temperatures is very much smaller than that undergone by magiielilc and Jiickel, for, if this were not the case, (he increase could lud ha\o been oliscured by (he distui'bing influences. PIERRE WEISS and H. KAMERLINGH ONNES. 'Researches on magnetization at very low temperatures." Plate I . ^r~~""--^ 4 ._iVWjl_ M ^ \ \ T— li s 0.« 1 Fig-. 1. is - — u v7 ^ 11 IZ o.i " / > / !.0 - — ' » Fig. 2. Proceedings Pioyal Acad. Amsterdam. Vol. Xll. PIERRE WEISS and H, KAMERLINGH ONNES. "Researches on magnetization at very low temperatures." Plate II 'i£f.P. Proceedings Royal Acad. Amsterdam. Vol. XII. ( 675 ) T A B L E VII. Iron. / (gauss) /- (cm of scale) / = 20' ( ' r=20°.3K. (H„ atm. press.) 7'=14M)K. (Hj solidifving) 1700 95.23 'l01.98 101.95 5675 98.47 •J 02.58 8G80 98.65 103.01 13160 98.91 103.31 15700 99.04 103.31 103.23 16940 99.08 J 03.27 18360 99.06 103.27 19250 99.07 103.25 103.25 for H= 19250 /20'.3K. / 20' C. = 1.0209 183(50 1.0210 16940 1.0209 li 5700 1.0213 mean not corrected fur dilatation 1.0210 111 all the iron e.xiterinients tiie zero as deduced by raking the mean of readings to right and left remained remarkably constant. As a rule its displacement was only a few tenths of a millimetre in any one series, and 6 mm. in proceeding from one series to anotiier. The few measurements at the temperature of solidifying hydrogen are sufficient to show that nothing particular happens between 20' K. and 14° K. Ma<im>tite. We have already mentioned that the jn'eparation of magnetite by heating the sesquioxide needs an extremely high temperature if one wishes to make sure that the last traces of oxygen are removed. A first ellipsoid obtained fn im iron oxide that had been insufficiently heated exhibited only little more that one half of the magnetization that was expected: it showed, too, a very distinct hysteresis, which Wcis about three times as great at liquid hydrogen temperature as at ordinary temperature, while \n all the experiments with the other substances hysteresis phenomena were insignificant. Moreover, the magnetization of this substance was the same at ordinary and liquid hydrogen temperatures, while between them it reached a maximum. These peculiarities were not displayed by a second ellipsoid cut 45 Proceedings Royal .Vcad. Amsterdam. Vol. XII. ( <^''5 ) out of well lieatc'il magnetite, but wiih this set'oud ellipsoid furllier jjlieuomena were observed wliieli liave not yet lieen ex|)lained but which seem to be of secondary importance. The zero point deduced fronr the mean of the two scale readings ditrered noticeably from the observed zero, while in any one series of measurements at the same temperature it remained ])ractically constant. Further — and this is more worthy of notit-e — the deviations dilfer according to tlie direction of the jiekl. It \vould clearly be very rash to attempt to ascribe to magnetite a hemimorphous symmetry like that of tourmaline from this sole observation. It seems more probable that some experimental error has here escaped our notice, and this can the more readily be accepted seeing that magnetite gives results much less regular than those of the metals. The following table contains an extract from the observations; the observations for the positive and negative directions of the field are given separately. TABLE VIll. jM(i<inetiie. / = 15°.8 C. Observed zero — calculated zero -(- 0.9 cm. H (ganss) 8600 18100 21800 23300 24200 8600 18100 21800 23300 24200 18100 21800 24200 From these numbers follow these ratios of the intensities at 20^.3 K. and 15\8 C. + Fiekl — Field /- ^cm. of sc;i de) ["■ (cm. of scale) 71.40 71.72 71.83 72.00 71.75 72.57 71.99 72.57 71.77 72.45 under atm. pressure (20\3 K.) 79.78 79.88 80.69 80.79 80.73 80.96 80.34 81.10 80.08 81.37 Y[„ solidifying (14^ .0 K.) 80.90 81.10 81.12 81.64 SO 96 81.84 ( 'J^^ ) T A i;le IX. Fiel.l +. Field - H gaus8 /20°.3K. -^I5°.8l'. -^15°.8C. 8600 1.0559 1.0553 18100 1.0591 1.0601 21800 1 .0593 1.0567 23300 J .0564 1 .0572 21200 mean 1.0563 -^ 1.0628 1.0574 J .0564 ■^l.V.SC. = i.o; ')69 not corrected for ililatali ion. Simikrly for the ratio of the niaf^netization atll^.OK. to liiat at 15^.8 C. we lind 1.0609 1.0622 hence '- — '=1.0<J1G not corrected for dilatation ^I5°.SC. a ratio which deviates from the foregoing in the expected direction. Collecting the foi'egoing results we find in this iiranch of the research for the ferromagnetic snbstances omitting tiie correction for dilatation (see note 2 pg. 11) Nickel ""-•'''■ = 1.0518 Jl7°.3C. ^90° 3K Iron ^^ =1.0210 /200 c. jTjoO 3 K Magnetite -^-^—=1.0569. (March 24, 1910). KONINKLTJKE AKADEMIR VAN WETENSGHAPPEN TE AMSTERDAM. PROCEEDINGS OF THE MEETING of Saturday March 26, 1910. (Translated from: Verslag van de gewone vergadering der Wis- en Natuiirkundige Afdeeling van 20 Maart 1910, Dl. XVIII). COlsTTElsTOrS. A. K. M. NoTcxs: ■•Communirutiuns about the ek'otroj;ram of the atrium cordis". (Communi- cattd by Prof. U. Zwaakdf.makkr), p. C80. (With one plaie). C VAN WissELiXGii : '•Qn thi; ttsls for tanning in the living plant and on the physiological significance of tannin". (Communicated by Prof. J. "W. Moll), p. 685. H. ZwAARDEMAiLER: "Tlie Camera silenta of the Physiological Laboratory at Utrecht", p. 706. Jak de Vries: "On paias of points which are associated with respect to a plane cubic", p. 711. L. E. J. Brocwer: '"On contiuuous vector distributions on surfaces" ''2ud communication^. (Communicated by Prof. D. ,). Korteweg), p. 71G. H. J. E. Ueth : "The oscillations about a position of equilibrium where a simple linear relation exists between the frequencies of the principal vibrations" (2ud part). (Communicated by Prof. D. J. KoRTEWEc), p. 735. (^Yith one plate). W. VAN DEB WouDE: "The cubic involution of the first rank in the plane". (Commuuicated by Prof. P. H. Schovte), p. 751. J. Bron : "On the surfaces the asymptotic lines of which can be determined by quadratures". (Communicated by Prof Hk. de Vries), p. 759. A. Smits: "A new theory of the phenomenon allutropy". (Communicated by Prof. A. F. Holleman), p. 763. (With one plate;. Erratum, p. 774. 46 Proceedings Royal Acad. Amsterdam. Vol. XII. f GRO ) Physiology. — "Convnunications about the ekcirO(/rnm of the atrium ronl/'s." \W Dr. A. K. M. Noyons. (Communicated by Prof H. Zwaahdkmakkk). (Gomnuniicated in t!io meeting of November 27, 1909). luvoliinhirilv (lie venlririe-image, tiie tops R and 7' of wiucli are |ire\-ailiiit;-. lias, in tiie siudy of tiie electric piienomena of tiie heart, u|) till now heeii the principal subject. The top P h\ its smaller size drew less attention and at the outset was not even observed. At present, however, there are in the literature already some data to be found here and there concerning the top P. Thus EiNTHOVEN 'j has pointed out how with increased action of the heart after great physical exertion P may gain in size, how under certain definite circumstances P may more or less be split up into a dim d(iuble-tO|)[)ed image, and besides how under pathological relations lo|) P may be altered, which is demonstrated by cases of mitral stenosis. In this case of disease P would appear longer and enlarged, which EiNTHOVEN thinks may be attributed to a stronger activity of the atrium for the sake of its compensative function. Divisioji of the lop may also appear. Kraus and NicoL.'Vi ') have corroborated this find, just as Saaio.)loff') and Stkshinsky ^), who have also been able to proxe that ilie phenomenon is not pathognomonic, but de- pends on the relative welfare of the heart in case of mitral stenosis. According to Vaandhagkr ^), tO|) P in absolute measure would be higher with the dog than in man. With N. vagi cut through Vaandragki; fountl in the dog that P gre"w three times its heiglit and conversely could he make F smaller by stimulating the N. vagi. Besides this diminution he got at the same time an alteration in the form of top P. With tuoderate bleeding of a sample-animal P increased in size, whilst after strong bleeding P grew smaller in the dog. Top /' was from the outset attributed by Einthoven to the ven- tricles. 1) EiNTHOVEN : See : Onderzoekingeu van hot Physiol. Lab. te Leyden. Second Series VII and llie liieralui'c pointed out there. -) Kraus F. and NicolaT G. P. Ueber das Eleclrocardiogramm unter normalen und palliologiscben Vci-h;iUnis:cn. Berl. klin. Wochonscbr. 1907 No. 25 and 26. ») Samojloff a. Electrorardiogramme. Jena 1909. Sammlung anal, und piiysiol. Vorltage. ■'j Samojlofi- a. und Steshinskw Ueber die Vorhoferhcbung des Eleklrokardio- gramms boi Milralstenose. Miinch. mediz. Wochenschr. No. 38. 1909. 5) Vaandrager B. Dissertatie Leiden 1908. ( •'■'^1 ) The following grounds may be adduced for this, partially borrowed from my experiments : 1. P always appears with a delinite interval of time before the mechanical change of the atria. 2. P continues existing at the registration of an isolated atrium (Rana, Emys). 3. P is absent when the electrogram is written of the isolated heart-ventricle of Angnilla vulgaris. 4. P continues exisiing when a certain detraction of the ventricle does not show itself. This may be observed both in the patho- logical heartblock and in the heartblock called into existence by experimental causes, among others : a. h\ stimulating the N. ^agus in the dog or the tortoise. b. by administering toxical materials like chloroform. c. by forming, resp. removing a ligature on the boundary of auricles and ventricles in Rana. 5. P may be made to disappear temporarily, when in appropriate sample-objects a heartblock is brought about by "stimulating the N. vagus at the transition cf the sinus to the atria, wiili which, it is true, the sinus-contractions are preserved, but the atrium-contractions with top P in the electrogram disappear. 6. P does not arise from the sinus, witness the fact that a small top may be registrated befoi'e the appeai-ance of top P, which may be attributed to the sinus. 7. Size and form of P depend upon the way in which the atrium is derived. If we consider the electrocardiogram of man and animal super- licially, we get the impression that P has a very simple form. Under quite peculiar circumstances this shape has been seen to alter. On closer investigation, however, it has appeared to me that the electric phenomenon of auricles is practically a whole complex. This becomes clear at the registration of an isolated pulsating atrium. Very fit for this purpose is one of the auricles of the heart of Emys. Thus the adjoined figure 1 renders the electrogram obtained by derivation of apex and basis of an isolated right atrium of Emys with the appertaining myogram registered by simple suspension '). This image, in many respects, makes us think of an electrogram ^) The registration of tlie electrograms was brought about by means of Ei.vthoven's string-galvanometer (Edelman.n's small model) according to the method pointed out before; see: Proceedings of the Kon. Akad. v. Wetensch. 31 Oct. 1908. 46« ( r.s2 ) of the voiiliiciiliis cordis, tis it is to i)e registereil with niiniliers of aiiiiiiais and wiili iiuxii. The lops that are found in this atrioelectrograni are, as it were, analogous in the (ops Q, R and P of the ventricle phenomenon and may respectively be called here Py, P^i, Ps. The tops i-'^ and P^-,, like Q and R, which are analogous to them, fall in the ventricle- image wholly before the commencement of the muscle-contraction. In the electrocardiogram of man and animals derived indirectly, we find, evidently on the ground of the P-elevation, only P^? expressed. Again by derivation of a heart of Rana derobed from ventriculus, accordingly consisting oidy of sinus and atria, we get, a propor- tionately less large, but yet also a complicated electrogram of the ati'ium, as also b}' registration of the isolated ati'inm of the carp, where for example the first half of the atrioelectrogram shows a pronounced diidiasic nature. It is also possible with a sample-object to derive both the atria at the same time The same thing I did also with the cut-out heart of Emys dejjHved of ventriculus, where one electrode invariably found a place on the backside of the sinus, whilst two other electrodes respectively caused the derivation of the atriatops. By means of a swing-apparatus the galvanometer was connected with the sinus and respectively with one of the atria or with both. The electrogram of the one atrium obtained in this way differs at such a derivation somewhat from the image got with the other atrium. The electrogram of the left ati'ium has a strong diphasic character; at derivation from the place of separation between the two atria with the back wall of the sinus, we get a less pronounced diphasic image, whilst the electrogram of the right atrium is only very feebly diphasic. The action-current of the heart is considered as a summary utterance of the electric negativities, which show themselves in the tissue successively in different places and at different times. This negativity, as Hermann formulated it, arises by the circumstance that every point of an irritable tissue at the moment of the stimulation stands in a negative relation to the parts that are in rest. At the derivation of such a tissue in different succeeding points we shall, therefore, every time get a deviating electrogram, but at the same time we shall be able to get an impression of the waj' which the proceeding slimulns has taken through the tissue. For this last purpose I have effected derivations of one atrium in ditfei'ent places, lying in regular order. In this experiment I made use of the heart of Emys de{)rived of its ventricle and then derived with an electrode constantly from the siiuis, whilst the other electrode (store-electrodes with ( r,83 ) movable [)itli) was placed: i. at the apex; 2. '/^ f'"- lower than the apex and 3. 1 cm. lower than the apex of the atrinm. This took place botii for the ri<;lil and the left atrium, and also for the two atria combined. B'ig. 2 {a, h, and c) shows how the amplitnde of top P^a from the atrio-electrogram diminishes in size as we descend with the electrode from the apex along the lateral side of" the right alriuin. The greatest potential dilference, tiierefore, is manifest between basis and point, whilst each point of the atrinm, l.ying lower than the apex, at deri- vation olfers a smaller potential difference with the sinus. This is quite in accordance with the usual representation, at which it is sup|)osed that, taking into consideration the fact that the contraction- stimidns arises from the sinus, the stimulus regularly goes on in the alriuni tissue fi'om the basis to the point of the atrium. In the same tigui'e 2c may also be demonstrated the a|)|)earance of a small elex'ation, with following slight fall in the electrogram, manifesting itself a full second beforu the commencement of the myo- gram of the atria. This elevation may be attributed to the sinus, among others for this reason that this elevation in size increases according as we draw jiearer to the sinus. Already in a former communication I have alleged grounds to prove the indei)endence of the electrical phenomena of the heart with respect to the changes of form. In the alria of Emys this (piality can be demonsi rated \ery clearly. A part of the heart of Emys consisting of only atria and sinus is, isolated, brought into a gas -chamber, and derived in one place from the sinus, in another place b'om one of the I wo atrium-tops or from botii tops at the same time. In this way different combina- tions of derivation may be brought about. The movements of the two alria are, by means of a simple suspension, registered by the silhouellc of the little levers. If 2 cm' of chloroform are administered which are evaporating in the gas-chamber, the mechanic movements are gradually growing smaller, so that at last they stop entirely, 1 J minutes after the administration the chloroform. Also at an examina- tion of the atria no trace of motion is to be observed, whilst however, the electric phenomena continue showing themselves periodically very clearly, through in shape they are a little more complicated than before the poisoning. In fig. 3a and fig. 3i the electrograms have been denoted as they are obtained by derivation of the sinus with the one electrode and derivation of the two atria-tops with a double other electrode. After evaporation of the chloroform by openiiig ( 084 ) the gas-chambei' the ati-ia begin to recover and after 39 minutes the electrical phenomena reach their original size again with (heir meclianical clianges. Snch poisoning-experiments can be repeated a few times without any great harm for the sample-object. It is striking how the right atrium every time recovers first fVom the poisoning, and only later the left atrium begins to show mechanical changes. If the electrogram is examined during and. after the poisoning, it is remarkable that the general form is not really altered here, but that the amplitude of the tops is greater than after the recovery of the poisoning, whilst the mechanical changes are altered in exactly the other way. The form of the atrio-electrogram is evidently also dependent on nervous influences. Einthovkn and Vaandrager have already directed attention to this for the cardio-electrogram of a dog derived indirectly. For the a'ria of Eniys this may be proved very distinctly at direct derivation under the influence of vagus-stimulation. A heart of Emys derobed of ventriculus, so consisting only of the sinus and the two atria, is derived to the string-galvanometer. Derivation of one of the two atria separately or combined makes no particular difference. Therefore the derivation in the experiment takes place from sinus and right atrium-top. The right n. vagus in the neck is prepared free. The electrogram shows a fine double- topped image, accompanied by regular mechanical changes. At stimulation of the rig'ht n. vagus with induction-currents the meclianical utterance undergoes alterations. After a last contraction of the sinus, showing itself in a slight elevation in the myogram of the atria there begins for the object a vagus-standstill, which, as soon as the stimulus is put a stop to, is iiroken otl' and causes a new series of atrium-contractions with a strong sinus-contraction. During the vagus- standstill the object has produced no electrical phenomena that are to be registered. Directly after the vagus-standstill the atrio- electrograms show themselves again, but now altered in form. The double-topped image has been replaced by a phenomenon with a strongly pronounced diphasic character, which however, the stimulation being stopped, passes into the original double-lopped image by a gradual alteration. In this experiment the n. vagus was stimulated by means of tiie sledge-inductorium of or IJois Reymond, without a kernel, at a secondary coil-distance of 5.5 cm. and a LESsiNG-element in the primary chain. When weaker currents are used for stimulation, a state of things may be obtained in which the vagus-standstill does not appear, but in which the ])ecnliar alterations in Ihe form of the electi'ogram of ( im ) the atrium show themselvee, as they have been described above. These alterations in tiie ibrni of the electrogram, apart from Ionic changes, ai'e not accom[)anied by changes in the motoric utterances. Fig. 4. The tonic alteration cannot be considered as the cause of the changes in form of the electrogram, because, in using still weaker currents as stimuli lor the n. vagns, (he same tonic change mav appear, without having any effect on the electric utterances of the atrium. Botany. — "On ike tests for tannin in. the lirin<i /Jaiit awl on tin- pkysiolotjical si(inipc(i are of tannin." l!y Mi-. ('. van Wissi-.i.incii. (Comuiunicateti by Prof. J. W. Moi.i.) (Communicated in the meetins; of t'ebruary 2G, 19 !(.)). In this paper a method will be desciibed for demonstrating the presence of lauiiin in the living plant, a method which enables us moreover to obtain an idea of the amount of this substance in the living cells, and to ascertain vvhelhiM- after a given ])eriod of tiuie the amount has increased or tlimiui.jli'jd : the method does not noticeably affect the living funtious of the plant or damage the latter to an appreciable extent. In addition a few results of experiment?; on the |)hysiological sig- nificance of tannin will be communicated; these results are in my Of)inion a real contribution to our knowledge of this subject. Before proceeding to a discussion of the method which I have worked out, 1 tliiidc it desirable to make some obserxalions on the meaning of the word "tannin" and to give an account of the present state of the physiological fannin-probletn. As regards the meaning attached to the word tannin there is no uniformity. Botanists formerly meant by tannin every thing in the cell wiiicli was coloured blue or green by ferric salts'). This has led to confusion with other substances and to the view that tannin is a generally occurring constituent of plants. Rkimtzek '•') especially has drawn attention to this. As a restdt of his investigations he came to the conclusion that the word tannin is a misnomer, introduced into science from the leather indnstrv. Accordiu"- to him it should •) J. Dekker, De looistotTeu, Bot.-cliem. iiionogLaphie der tamiiden, l'JCi8, V. 1, p. 197 and 210. F. GzAPEK, Biocliemie der Pflanzen, II. Bd. p. 576. -) F. Rei.mtzer, Bemerkungcn zur Piiysiologie de? Gerl)slolI's, Hei-. d d. hot. Uesellsth. Bd. \11, ISS'.), p. 187. ( (!86 ) again disa[)pcar from scientific tei-minoiogv, but lii.s suggestion diil not receive any support. Waage') especially has objected to it. With reference to this question Dekktr^) rightly remarks in his bolanico-chemical monograpli of tlie tannins, that thei'e certainly exist plant substances, wiiich are sharply marked off from other carbon compounds by common characteristic properties, such as the property of transforming animal skins into leather, which depends on tlie property of forming witli protein compounds insoluble in water, the adstringent taste, llic presence of several phenolic hydroxyl- groups in the molecule, the power of precipitating alkaloids from aqueous solution and other properties; these substances must there- fore be collected in a separate group. Until the chemical constitution of these substances is completely known, the group in which they are united, should not be split up. Some authors, e.g. Reinitzek ') and Bkaemer') consider that a group of plant substances cainiot be studied physiologically so long as our chemical knowledge of it is incomplete. Waage"), in my opinion, is qiute light in not agreeing with this. ()f course it will be necessary in the physiological investigation of tannins to ascertain in each case with the means at our disposal, whether the plant under investigation actually contains a sidistance belonging to the tannin class, so that confusion with other bodies may be excluded. The opinion of botanists concerning the physiological significance of tannins has always been much divided. Th. Hartig") supposed that tannins contribute to the building up of the vegetable organism. Schleiden") on the other hand considered that tannin is only a decomposition product of the cell wall. In agreement with Hartig's view tannin is, according to Wigand"), a real factor in the chemical process of plant life and belongs phy- siologically to the group of carbohydi'ates, on the formation and transformation of which the life process of the plant is especially based. In contradistinction to starch, which a|)pears as reserve ma- 1) Th. Waaoe, Die Beziehungen dcs GerbstolTs zur Pflaiizenchemic, Pharm. Genli-alli. f. Deutschl. N". 18, 1891, XII. Jalirg. N. F. p. i247. 2) 1. c. 1908, Vol. 1. p. V; Vol. 11. p. 66; Vol. I. pp. 211 and ^12. ») l.-c. H L. BuAKMEH, Les taimoidus, 1890—91. Ref. Bot. Genlralbl. Jalirg. XII, 1S91, Bd. 47, p. 275. 6) 1. c. ") Th. Hartiu, Entwickehiiigsgoscliiulite des Pflanzenkeims, 1858, p. 103. ') M. J. Schleiden, Grundziige dor wissenschaftlicheu Botanik, 1861, p. 141. *) A. VVuiAND, Einige Siitze iibei- die physiologisclic Budculung des Gerbstoifes Uiid.der Pllanzeiifaibe, Bot. Zeituiig, 20. Jalirg. 1862, N". 16, p. 121 and 129. ( 687 ) terial in the resting periods of vegetation, tannin genei-ally belongs, according to Wigand, to tiie fluid active substances necessary for growth. In some cases it appears, according to the same author, to act as reser\e material. It thus follows that in Wigand's opinion tannin is an extremely important product of vegetable metabolisui. No other investigator has declared this so clearly and so emphatically. Wigand's \iew has been attacked, especially by Sachs, and has not received much support from botanists in general; this is evident, for instance, from the chapter "Die physiologische Bedeutung der Gerbsauren" in Czapkk's Biochemie der Pflanzen '), where Wigand's view and that of Th Hartig -) concerning "Gerbmelil" as carrier of tannin and organized reserve material is reckoned among the "ii'rigen AuffassungHMi fiber die physiologische Rolle der Gerbsiiuren'". Wiga.nd has not published the details of the observations on which his con- clusions are based and this has probably contributed to the ready rejection of his results by other autiiors '). The conception of the role of tannins arrived at by Sachs ^) in his investigations on tiie germinalion of seeds, has received more support than that of Wigand. Sachs considered the tannins formed in germination, to be merely excreloi-y products, by-products or decomposition products. He thought it very improbable that tannins could serve in some way or other as material for tiio building up of cell-walls. The results of some other observers agree with those of S.achs. Thus for instance Kraus '), who was particuiarl\- interested in the physiological significance of tannins, arrived at the conclusion that tannin, once foi'med, in no case takes any further part in metabolism. According to Gerbkr ") the tannins disappear by oxidatioji, without the formation of carbohydrates from them. Af Klercker ') regards ») I.e. p. 588. -) Tu. Hartig, Das Gerbmelil, Bot. Zeiliing 23. Jahrg. N». 7, 1865, p. 53. Weitere Mitteilungen das Gerbmelil betretlend, Bot. Zeitung 23. Jahrg. N*. 30, 1865, p. 235. 3) Compare Emil Kutscher, Ueber die Verweudung der Gerbsaure iin Stoff- wechsel der Pflanze, Flora, 66. Jahrg. N". 3, 4 and 5, 1883, p. 37. *) J. Sachs, Physiologische Uiitersucbiingea iiber die Kcimung der Schmink- bohne iPhaseolus mullUlorus), Silzuiigsber. d. kais. Akad. der Wiss. Wien, 37. Bd., 1859, No. 17, p. 57. Zur Keimungsgeschichte der Dattel, Bot. Zeitung, 20. Jahrg., 1802, No. 31, p. 241 and 24'J. Handbuch der Experimental-Physiologic der Pflanzen, 1865, p. 360. °) G. Kkaus, Grundlinien einer Physiologie des Gerbstofls, 1889, p. 38 and 44=. *>) G. Gerber, Role des tannins dans les plantes et plus particulierement dans les fruits. Gompt. rend. 124, p. 1106. '') J. E. F. Af Klercker, Studien fiber die Gerbstoffvacuolen. Bibang till K.Svenska Vit.Akad. Handlingar, Bd. 13. Afd. Ill, No. 8, 1888. Rcf. Bot. Zeitung, 47. Jahrg. lS8y, p. I'lU. ( «88 ) tannins as excretion prodncts. Waage') calls them by-products of metabolism. Btis&EN ') insists, that the observations which have been made, aft'ord no justification for the assumption that tannin acts as a plastic material. On the utiier hand Schulz ') considers the tainiin of evergreen leaves to play the part of reserve-material. According to a few investigators tannins must, in some cases, be regarded as excretory products or as by-products of metabolism, whereas in other cases they take part in metabolism and serve as plastic material. Such was tlie conclusion of Schei.l '), of Kltschek ^) and of Westermaier "). According to Schroeder ') the tannin of the birch and the maple is not a reserve material, but it is not an excretory product either. In the author's opinion it possibly in these cases constitutes a iinal product of metabolism. He does not, however, attempt to answer the question as to the physiological significance of tannin. Many investigators have adopted the view, that tannins serve to protect plants against harmful external influences. These might be of very different kinds. Stahl ") assumes that on account of its unpleasant taste tannin serves to protect liie plants from the attacks of animals, especially slugs. Kraus ') also considers lainiiu to be a protective agent not exclusively against animals, but ser\ ing in addition to counteract the putrefaction of the plant. In plants with evergreen leaves Warming '") regards the tannin content of the epidermis as protecting the plant from desiccation, while exposed to dangerous dry winds in winter, and as being at 1) 1. c. p. 250. -) M. BiisGEN, Beobaclitungen iibci- das Verhalten des GeibstolTes in den Pflan- zen, Jenaische Zeilscbrift fur Naturwissenschaft, 24. Bd., N. F. f7. Bd., 18'J0, p. 50. Erfautei-ung zu dem Rel'erat iiber Beobaclitungen etc., Bot. Zeitung, 1890, p. 381 . 3j E. Schulz, Ueber ReservestotTe in immergiiinen Bliitteru unter besonderer Berucksichligung des Gerbstoffes, Flora, 1888, p. 256. *) J. ScHELL, Physiologische Rolle der Gerbsiiure, Kazan, 187i (l^ussian), Botan. Jahresber. 111. Jahrg., 1875, p. 870 5) I.c p 73. ") M. Westermaier, Zur pbysiol. Bedeutiing des GerbsloU'es in den Pllanzen. Silzungsbei'. d. konig!. preuss. Akad. der Wissenscb. zu Berlin, Jabrg. 1885, 2. Halbb. p. 1124 and 1125. 7) J. Schroeder, Die Friibjabrperiode dor Birke (Betula alba L.) und der Aborn (Acer platanoides L.), Die landwirtliscb. Versuchs-Slationen, Bd. XIV, 1871, p. 146. **) Ernst Stahl, Pflanzen und Sclinecken, Jenaische Zeit.S('lnift fiir Naturwissen- schaft, XXII. Bd., N. F. XV. Bd., p. 590 and 594. 9) Grundlinien zu einer Physiologic des Gcrbstoffs, 1889, p 21. 1") E. Warming, Beobaclitungen iiber Pflanzen mil iibervvinteriiden Laubblallerii, Bolan. Genlralblalt, Jabrg. IV. Bd. IG, 1883, p. 350 ( «89 ) the same time a means of rapidly restoring lost turgor. Schell *) considers that tannin in seeds is probably a protection against harmful influences from without. Busgen") also supposes that tannin affords protection to the plant. Other authors again have attributed different functions to tannins. According lo Gerbek ') they prevent the transformation and fermen- tation of sugar in fruits and Peepfer *) thinks it very likely, that their role also consists in fixing sugars and other substances in the cell. KuTSCHER ') considers it most plausible tliat tannin serves as a respiratory agent and is oxidized in respiration. Various other functions have further been attributed to tannins in connexion with the metabolism of the plant. Thus Wigand ") supposed that the red colouring matters ai-e formed, from tannins, a view shared by Pick '), Mielice *), and Tschirch ') amongst others. Some authors connect tannins with the formation of resin. Wiesneh^") thinks that starch and cell-wall may be transformed to tannin, and subsequently to resin. Schell ^') and Mielke ^'') also regard tannin as an intermediate stage between starch and resin and between cellulose and resin. Both authors, however, also suppose, that tannin can be converted into starch. Bastin and Trimble '") in their investigation of the resin-passages of conifei's, have also received the impression, that tannin is connected with resin formation. M I.e. p. 877. -) I.e. p. 58. 3) I. C. ') W. Pfeffer, Ubei- Atifnahmf^ voii Anilinrarljen in lel)ende Zellen, Uiiter- suchungcn aus dem botan. Instilut zu Tiihingen, 2. Bil., 18S6 — 188S, p. 310. =) 1. c. p. 73. 6) 1. c. 7) H. Pick, Ueber die Bedeiiliing des rothen Farbsloffes bei den Phanerogamen und die Beziehungen desselben ziir Starkewandei'ung. Botan. Centralblatt, Jahrg. [V, 1883, p. 284. *) G. Mielke, Ueber die Stellung der Getbsaureu im StofTwechsel der Pflanzen, Programm der Realschule vor dem Holstenthore in Hamburg, 1893. Pief. Bolan. Centralblatt, Jahrg. XV, 1894, Bd. 59, p. 281. 8) A. Tschirch. Schweiz. Wocbenschr. f. Pharm. N'. 7. Pharm. Centralbl. N". 10, 1891, p. 141. 1") J. WiESXER, Uber die Entslehung des Harzes im Inneren der Pflanzenzellen, Sitzungsber. d. Wiener Akad., 1865, 52. Bd. II. Abt. p. 126 and 129. Ref. Jahres- ber. fiber die Fortschrilte der Cliemie etc., 1865, p. 627. ") 1. c. 1=) 1. c. 13) E. Bastin and H. Trimble, A coiitrilnition to tlie knowledge of some North Amerikan Conil'erae, Amer. Journ. Phanii. 68, 1896. ( B90 ) According to Buignet ') taniiiu in fruits contributes to tlie formation of sugar and according to Stadler ') it supplies in the nectaries of Oenotliera and Saxifraga the material for the formation of honey. In connexion with the physiological significance of tannins in plant metabolism, I think it desirable to point out, what results botanists have arrived at with regard to the translocation and origin of tannins. Some investigators suppose that tannin can be transported in the plant, namely Kraus, Moeij.kr, and VVestekmaier. According to Kraus 'j tannin travels as such, Moeller ^) thinks it probable, that carbo- hydrates are transported in the form of tannin compounds. Wester- maier *) leaves it an open question whether tannin travels as such, and whether starch travels in the form of a soluble carbohydrate or in that of tannin. The opinions of botanists are dixided as to the origin of tannins in the plant. As was stated above, tannin is according to Schleiden ") a decomposition product of the cell wall. According to Th. Hartog^) it arises from starch during the germination of Quercus pedunculata. Similarly according to Schell ^) tannin is formed from starch in the germination of the seeds of Faba vuhjaii^ and Pimm satloam. MiELKE ") supposes that tannin is formed from carbohydrates, from tannin glucosides, and also from starch and from cellulose. Wester- maier '") regards it as an assimilaiion product but supposes that it is also formed by the decomposition of proteins. According to Schroeder ") it is formed by oxidation from organic material present in the plant. Kraus'') thinks that it is a decomposition product of amido-compounds formed during the synthesis of proteins. The observations of Moeller '") ') H. Buignet, Recherches sur la maliere sucree conteuue dans les fruits acides, son origine, sa nature ct ses transformations, Gonipt. rend. 51, p. 894. -) S. Stadler, Beitrage zur Kenntniss der Nectarien and der Biologie der Bliilhen, Berlin 1880. *) G. Kraus, Grundliuien zu einer Physiologie des GerbstolTs, 1889, ]). 20. ■>; Hkrman Mokller, Analom. Untcrsuchungen fiber das Vorkonimen dcrGerb- siiure, Ber. d. deutschcn bot. Gesellsch. Bd. VI, p. LXXX. • '^) M. Westermaier, Neue Beitrage zur Kennln'ss der pliy.siologischeu Bedeutung des Gerbstoffes in den Pflanzengeweben. Sitzungsber. d. kiinigl. preuss. Aliad. d. Wiss. zu Berlin. Jahrg. 1887. 1. HallDb. p. 134. 6) I.e. p. 141. ■) I.e. p. 102. 8) I.e. "I I.e. lO) Zur physiol. Bedeutung des Gerbstoffes in den Pflanzen, I.e. p. 1124. 1') I.e. p. 146. 1-) Grundlinien, p. 47. '•*) Moeller, Mitt, des naturw. Vereins f. Neu-Vorpoininern und Biigen in Greifs- wald, 1887. ( 691 ) on the leaves of Ampclop.^is licdcraci'a and those of Busgen ^) on germinating seeds of Yicia Fabn and on wounded leaves which were floated on a 10 percent grape sugar solution, have proved that tannin is formed from sugar. Kraus '■') and Westermaiek ') have pointed out tliat in some cases the formation of tannin depends on the influence of light. It is evident from the above, that botanical opinion is much divided on the subject of the physiological significance of tannins. It may be summarized as follows. According to some botanists tannins are of no value to the plant ; they are merely excretory products. Others regard tannins as protective agents against various harmful external influences. A few believe that tannins contribute to the building up of the vegetable organism. A small number think that tannins can fulfil different functions. Various authors e.g. Czapek ") in his Biochemie der Pflanzen and Dekker ^) in his Botanisch-chemische Monographie der Tanniden, have pointed out that the numerous investigations on the physiology of tannins have as yet produced but few results of any importance. Dekker arrives at the conclusion, that if this group of substances is of significance to the plant, whicii he tiiinks probable, it is quite uncertain what function tliey fulfil. Noll") in Strasbirger's Lehr- buch der Botanik expresses himself in tiie same way. The fact that the significance of the tannins is still so obscure, is attributed to various causes. Thus according to Czapek ^) a few observations of microscopical or chemical facts led to generalisations and to the construction of untenable theories. Dekker') further points to the imperfection of the methods of investigation and the one-sided use of these methods, which sometimes causes tannins to be confused with other plant substances. In my opinion the chief cause must be sought in the want of criticism, which often impairs the drawing of conclusions. The physio- logical tannin problem is most certainly a very difficult problem, the answer to which Avill have to take into account a large number of factors. These factors are known to us to a smaller or larger extent, I) I.e. p. 34 and 35. -) Grundlinien, p. 20 and 44. ^) Zur physiolog. Bedeutung des Gerbstoffes in den Pllanzen, I.e. p. 1117. Neue Beitrage etc. I.e. p. 128 and 133. «) 1. c. p. 588. 6) 1. c. V. I, p. 220. «) 1. c. 8. Aufl. 1906, p. 190. 7) 1. c. p. 588. 8) 1, c. V. I, p. 210 and 211. ( 092 ) but unknown factors may also come into play. Hence it is necessary to exercise the greatest caution in drawing conclusions. In advancing an explanation of an observed plienonienon, we must consider care- fully whether it is the only one possible and we must attempt to prove it in various ways by means of comparative experiments. The extent to wiiich these precautions have been observed by no means corresponds to the complexity of the problem. As a result of a few expei'iments many observers have put forward cei-tain expla- nations, when other explanations were equally plausible, or the}' have combated the opinions of other investigators, who perhaps had a more correct insight, although they were unable to adduce sutlicient proof for it. Even serious investigators have made this mistake. I will illustrate this very briefly, by showing the insufficiency of the reasoning which led to the rejection of the possibility that the tannins might serve as plastic material. As was said above Sachs does not believe that tannins can act in any way as plastic material in the formation of the tissues. This opinion he has partly supported by observation and partly by drawing what is in my opinion an erroneous conclusion. Sachs ') found that in the germination of seeds which do not contain tannins in the endosperm or in the embryo, tannins are formed in metabolism and primarily there, where the formation of tissue has just started. He never saw the tannins diminish or disappear during germination. In other cases, namely in that of the acorn and of the chestnut, where the embryo contains tannin, he did not observe a diminution either, but rather an increase. He made similar observations on the development of buds. Sachs concludes from the abo\e-mentioned facts, that tannins remain in plants in the places where they have been formed, and that therefoi'e they do not take part in the formation of tissues, for if this were the case, a diminution would have been observed. I consider this conclusion to be incorrect. Quite a different conclusion might equally well be based on Sachs' observations, namely that the frequent appearance or presence of tannins in tissue-formation shows that these substances have probably a function to perform in this process. Nevertheless I do not at all consider that Sachs has proved, that tannins remain in the places where they are formed and that they do not serve as plastic material in tissue formation. For if in the germination of seeds more tannin is formed than is decomposed, a diminution of the tannin content need not occur, and an increase 1) Physiolog. Untersucliuiigen fiber die Keimiing der Schminkbohne, I.e. p. 111. Zur Keimungsgescbichte der Dattel, I. c. p. 246. Handbuch der Experimental- Physiologie der Pflanzen, 1865, p. 361. ( 693 ) may even take place. Reserve-materials lika starch and fatty oils ma}' not be assnmed to participate directly in tiie building of the cell wall. Tiiey must first be converted into soluble substances. Now suppose that tannins also belong to this category', i. e. to such a plastic material as is present in the plant in a dissolved state, then it is not at all surprising that for the maintenance of growth plentj' of this material should always be present, and that occasionally, when more of it is i>eing produced from tiie reserves than is used up in the growth of the cell-walls, the tannin content increases. Anyhow it has not been proved that, because the tannin does not diminish, it remains unused at the place, where it has been formed, and that it does not serve for the building up of cell walls. Like Sachs, Kraus M also assumes that an increase in the tannin content in germination proves, that this substance is not used up and does not serve as building material. Thus witli regard to the germination of the acorn Krais states, as a result of quantitative tannin determinations, that not only is tannin not used up, but that its quantity even increases, so that it cannot be of service in growth. Whereas Sachs only observed an increase of the tannin content of germinating seeds, Schell ") found in some plants an increase and in others a decrease or disappearance. In the first case Schell supposes, in agreement with Sachs, that the tannins are by-products of metabolism, but in the latter case he regards them as plastic material. With reference to what has already been said, it is a matter of course that I cannot either agree with Schell's conclusions. In my opinion it is not necessary to conclude,, on the ground of an observed increase in the tannin content in some cases and a decrease in others, that tannins behave so differently in different plants. Supposing the tannin to be a plastic material in both cases, then the occurrence of an increase or decrease will depend on the quan- tities produced and used up. I also think it very plausible that in one and the same plant sometimes an increase, and sometimes a decrease takes place, according to circumstances. Several botanists suppose that tannins can undergo translocation in the plant. How this might happen is still a moot point, but there can be no doubt that the possibility of translocation greatly compli- cates the question of the use of tannins as plastic material. The increase or decrease of the tannin content of a particular organ would then not depend wholly on production and consumption, but transport 1) Grundlinien, p. 38. -) I.e. p. 876. ( 694 ) to and from the organ would also have to be reckoned with. The mere increase or decrease of tannin in a seedling or a vegetable organ will not snpply data of any value for the solution of the problem of the significance of tannin as a plastic material. Hitherto botanists have chosen the higher plants for the study of the physiological significance of tannins. For the study of complicated vital processes and of the physiological significance of chemical constituents certain lower plants appear to me to offer advantages above those of the higher ones, the structure of which is so much more complicated. For such an investigation the thicker species of the genus Splrogyvii seem jiarticularly suitable. It is true that the tannin of Spirogym has not yet been examined chemically, but numerous microchemical reactions allow us to conclude with a fair degree of certainty that Spiror/yra contains in its cell sap a considerable quantity of tannin. Dk Vries ^) lias proved this, after abnormal plasmoiysis, with various tannin reagents e. g. ferric salts, potassium bichromate, osmic acid. In addition to these reagents many others also give in the cell sap precipitates which agree completely with those caused in tannin solutions. The advantages which the thicker species of Spirogyra ha\e over the higher plants are the following. Pieces of the fdainents may be examined microscopically without killing them or damaging them, and the changes in the cells can be studied in the living plant. They are particidarly suited to all sorts of experiments. They are not too small to be handled and not too thick for microscopic examination. The various constituents of the cell can readily be observed under the microscope. As in the case of unicellular Algae a transi'ort of foodmaterial from one cell to another is very probably excluded in Spirogyva. This important factor, which must be taken into account wlien dealing with the higher plants, need generally not be considered in the case of Spivogyra. By means of the centrifuge all sorts of abnormalities may be obtained, such as polynuclear cells, cells without nucleus, cells with a large and with a small chromatophores-mass, and even cells without chromatophores. In this way we can eliminate the assimilation process, i. e. the intake by the chromatophores of carbon from atmospheric carbon dioxide under the iiUluence of light. With Spirogyva a number of comparative experiments may be made which are impossible in the case of the higher plants, and because ^) Hugo de Vries, Plasmolylisclie Studion fiber die Waud dor Vakuolen, Pringsh. Jahrb. f. wisscnsch. Botanik, Bd. 16, 1885, Heft i, p. 575. Over looistofreaclien van Spirogyra nitida, Maandblad voor Natuurwetenschap^en, 1885, N». 7, Reprint p. 7. ( 695 ) certain fac-lors are excluded or eliiiiiiuited, ulliers may l)C studied witli a greater cliance of success. Because the investigation of the higher plants has yielded such unsatisfactory results for the knowledge of t!ie physiological signifi- cance of tannins I have attempted to obtain more definite results for the solution of this problem in the case of the lower plants, parti- cularly of Splrogym; to this I was led by the above considerations. The first question to present itself was. which method would be most satisfactory. In the case of the higher plants investigators have followed various methods. Of the many reagents which give precipitates or colour reactions with tannins, ferric salts and potassium bichromate have mostly beeji preferred. Potassinm bichromate especially, which yields with tannins a reddish brown or orange precipitate, has often been used, e. g. by Schroeder '), Scheli. ^), Kutscher '), Rdlf *), ScHULZ '"), IMoeu.er ") and Busgex "). Kutscher made a dish with 8 sections, the colour of which agreed with that of the precipitate, but shaded in such a way that the intensity of the colour in two suc- cessive sections always differed by the same amount. This dish was used for the determination of the strength of the precipitates. Kraus') determined the amount of lannin by means of titration with potassium i)erinanganate or precipitated the tannin with cupric acetate and weighed tlie precipitated copper as copper oxide. The titration with potassium permanganate was also employed bj- Rule'). These titrimetric and gTa\imetric methods cannot, of course, be applied to a small object like Spiwyi/ra; moreover no method satis- fied the tlemand which 1 had imposed upon myself. I desired a method which would enable me to determine the lannin content of one and the same cell at different periods, with sufficient accuracy' to allow me to decide whether an increase or decrease had taken place, and this without killing tiie cell or harming it appreciably. The want of such a method had made itself felt in the investigation of various abnormal cells such as polyniiclear and ;\nuclear ones, 1) I.e. p. i^^. 2) I.e. p. 873. 3) 1. e. p 38 and 39. ■*) P. RuLF, Ueber das Verhalten der Gerbsiiure bei der Keimung der Pflanzen, Zeitschrift fiir Naluiwiss. in Halle, LVIi. Bd. Vierte Folge Bd. Ill, 1884, p. 42. 5) 1. c. 227. '') Hermanx Mueller, Anatomische Unteisuchungen uber das Vorkommen der Gerbsaure, Ber. d. deutsehen botan. Gesellsch., Bd. VI, 1888. p. LXVI. 7) I.e. p. 13. *) Grundlinien, p. 61. 9) 1. c. p. 42. 47 Proceedings Royal Acad. Ajisterdam. Vol. XII. ( ()9(J ) and cells containing nuinv, few or no cliromatophores. While I conld determine the growth of such ceils by measurement andconld deduce from the size of the starch foci whether the starch content had increased or decreased, I was imable to obtain for one and the same cell an idea of the tannin content during the various periods of its existence. The usual reagents only permit of a single examina- tion being made, because during it the cells are killed. 1 had therefore to look for another method. I wondered whether methylene blue might perha])s .satisfy the above re(iuiremen(s. According to Pfeffer') this substance forms a compound with tannin in the livnig cell, and this compound separates as a line blue precipitate. For various physiological investigations Pfeffer strongly recommends aniline dyes particularly methylene blue. Of this he says i.a. the following^): "In alien Fallen werden also Methylenblau und andere Farbstolfe wertvolle Reagentien sein, niit deren Hiilfe, ohne Scliadigung Aufschliisse iiber Vorkommen und Verteilung gewisser Ki'trper in der Zelle zn ei-halten sind. Mitsolcher vielseitig ausnutzbaren Methode lasst sich unter richtiger Erwiigung nach vielen Richtungen hin eine Kontrole des jeweiligen Zuslandes des Zellsaftes und der Veranderungen dieses im Laufe der Ent- wieklung erreichen." When dilute solutions are used, the penetration of methylene blue into the body of the plant and its accumulation in the cellsap continue, according to Pfefker'), without any harm to life and even when complete saturation has taken place, it is still innocuous. Sim-O'nn'n was one of the objects with which Pfeffer experimented. Pfeffer's experiments were repeateti by me a few times with Spirogyra ma.riimt, but with very unsatisfactoi-y results. Even after several days only a slight granular precipitate was obtained in the cells, and at least the greater part of the tannin remained in solution; moreover, even very dilute solutions were found to be harmful. 1 cannot therefore agree with Pfeffer in praising his method of investigation, and after this disappointment a better method was sought. Preliminary cxiiei'iments were carried out on S/)iro</ijr(t iiia.riiiia with various launiii pi'ocipitanls, such as alkaloids, anlipyrinc. am- monium vanadate anil many otlicrs. Of all ihc substances examined, caffeine and antipyriuc were found lo !»> llie least harmful, and therefore the action of Ihesc two substances was investigated more ') 1. c. p. 100. "-) 1. c. p. 191. :') 1. c. p. 195, I'JG and i'.)7. ( 007 ) closely, in order to ascertain their value for tiie study of the phy- siological tannin problem. In doing this, special attention was directed to the following points: whether the substances penetrated ra|)idly into the cell and the cell sap, whether the tannin was completely precipitated, and what concentration of solutions was required for this; the nature of (he precipitate and whether it redissolved on removal of the [)recipitant, whether the strength of the precipitate corresponded to the quantity of tannin in the cells and whether the method was sufticiently innocuous. After a number of experiments with antipyrine and caffeine solutions of various concentrations, which were allowed to act for a longer or shorter time, I came to the following conclusion: The antipyrine and caffeine solutions penetrate rapidly into the cells and in sufficient concentration produce in the cell sap a preci- pitate, consisting of minute grains or globules, which are in constant motion to and fro. In order to precipitate the tannin as completely as possible, it is desirable to have the antipyrine solutions not more dilute than I 7o ''^"d caffeine solutions not weaker than V,, "/„. The greater the tannin content, the hea\'ier the precipitate. Not infrequently the precipitate is so heavy, that the nucleus, which ordinarily can be readily discerned in Spirogyni maxivia, cannot be distinguished at all and sometimes the precipitate is even heavier. If the (SpH'O^yra fdaments are placed in ditch water or in distilled water, the preci- pitate disappears in a short time, say in 10 minutes, and the Spiro- <jijra threads are as before the ex[)eriment. No change whatsoever can be detected. If the Spirogyra fdaments remain in the solution, the precipitate settles down and the small globules or spheres, of which it consists, gradually coalesce to larger globules, which appear perfectly colourless and may sometimes be very large closely resem- bling fat globules. This was generally the appearance of the preci- pitate after a few days. The settling down of the precipitate in the cells and the fusion of the globules to larger, purely spherical masses, proves that it is heavier than water and that it is liquid. From several data 1 deduce that it is not thinly liquid but viscid. The fusion to larger globules proceeds slowly and cannot, for instance, be brought about by a few minutes centrifuging. When the Spirogi/m<-e\h with the globular precipitate are placed in water, the globules dissolve. Solution takes place more slowly, however, than in the case of a recently formed and still finely divided precipitate. If the preparations are placed in ferric chloride solution, instead of in water, the globules are coloured blue, while the cell sap is not coloured. It is rational to use caffeine as precipitant for this experiment, since antipyrine gives 47* ( (i98 ) a reddisli-vioiel coloratiuii witli fiiiik' rliluiide. Since tlie (.•oluiired compound is soluble and easily diffuses tlirough the preparation, the ferric chloride-tannin reaction of the globules may also be detected when antipjrine is used, and the non appearance of the reaction in the cell-sap may be observed, at least when the ferric chloride acts sufficiently rapidly. If the preparations are transferred from tiieanti- pyrine- or catfeine solution to a one percent solution of osmic acid, the globules are first coloured blue and soon afterwards black, whereas the cell sap remains colourless. It is evident from the experiments witii ferric cidoride and with osmic acid, that the tannin is completely or almost completely preci- pitated b^' a one percent aiitipyrine solution and by a 0.1 percent caffeine solution, for otherwise the cell sap should have shown a blue or black coloration. If the antipyrine or catfeine precipitate, whether it be a finely divided recent precipitate or one which has fused to globules, is dissolved by placing the Spiroi/yj'a-i'ilaments in water, and if ferric chloride- or osmic acid solution is then added, the cell-sap is coloured blue or black, just as is the case with cells which have not been treated with antipyrine- or caffeine solutions. Wlien the cells finally die off in antipyrine- or caffeine solution, the globules are stained biown : their solubility in water has then decreased, but they still give with ferric chloride and osmic acid the reactions referred to. By means of comparative experiments with antipyrine- and caffeine solutions, and various other lannin reagents, such as potasssium bichro- mate, osmic acid and ferric salts, with Spirogijra cells containing a varying amount of lannin, I was able to show that the strength of the antipyrine- and catfeine piecipitates agreed with the strength of the precipitates and colorations, given by the above-mentioned reagents. For these experiments 1 used Spirogyra filaments, which had been centrifuged a few weeks before, and in which there were also all sorts of abnormal cells, such as cells without a iiucleiis, without cliromalo- phores, with several nuclei etc. The taimin content of the cells of these filaments vai-ied very much. First the filaments were treated with antipyrine- or caffeine solution and the strength of the precipitate in the various cells was noted ; afterwards the filaments were placed in water, and when ihe prcci|iilalos had dissolved, they were placed in a solution of polassinm bichromate, osmic acid or ferric chloi'ido, and the intensity of the reaction in the various cells was noted. On comparing the various notes it was fouinl thai tiie strength of the antipyrine- and catfeine precipitates agreed with the intensity of reaction obtained with the other reagents, and therefore corresponded to the quantities of tannin present in the various cells, ( 699 ) Tlic strciigtli of the preciiiiLUe- uilli unlipvrine and caffeine was judged in varions ways. Tims it was noted, wiieiher the nucleus, which ill normal circumstances is very clearly visible in Spirogyra nxLvuiiii, coidd still be distingiushed after precipitation of the tannin. Furthermore it was noted whether the suspensory threads, the chroma- tophores and the starch foci above and below in the cell could still be discerned. In order to Judge in which cells the precipitates were strongest, the various cells were not only compared aftei' precipitation, but it was also noted in which cells the precipitate first appeared and remained visible for the longest time after the filament had been transferred to water. I had previously found that the precipitate first appeared in the cells with the largest tannin content and that after the fdaments had been placed in water, it coidd be observed in these ceils for the longest time. In connexion with the use which I wished to make of antipyrine- and caffeine solutions, it was very important to know to what extent these solutions are harmful to life and whether a short stay in these solutions, sufliciently long to obtain an idea of the tannin content, might be regarded as harmless or ])ractically harmless to the Spirogyra filaments. I found that, if a one percent solution of antipyrine, or a 7j„ percent solution of caffeine were used, made up with ditch water or with distilled water (a solution of such concentration therefore that all or nearly all the tannin was precipitated in the cells) and that if the Splrogyni filaments remained in this solution, ■ no further divisions took place and growth was soon arrested or was stopped at once. If, on the other hand, solutions were used which were ten times as dilute, and which did not cause a precipi- tate in the cells, it was found by comparative experiments with Spirogyra filaments in ditch water or in distilled water, that growth was I'etarded by antipyrine and by caffeine, and that fewer nuclear and cell dixisions occurred. 1 made some experiments with a one percent antipyrine solution and with a ' .„ percent and a one percent solution of caffeine, in ordei' to see whether a daily sojourn of 10 minutes in these solutions was harmful to Spirogyra, grown in ditch water. A period of 10 minutes was selected because it is sufficient for an examination of the tannin content. The result of these experiments was, that it could not be ascertained with certainty whether the procedure employed was harmful to the Spirogyra. Sometimes the growth of the controls in ditch water was the stronger, sometimes that of the filaments which had been periodically treated with antipyrine and caffeine solutions. It is not improbable that the differences observed depended ( 700 ) largely on the nature of the cells luidcr investigation. 1 surmise this because the growth of cells in normal and equal nutrient solutions also showed differences. We luay deduce from the results tiiat in general a sliort daily slay in tiie various solutions has at most a slight influence on liie growth and the vital processes of Spirogyra. The above method of investigation of the taiuiin content may there- fore be strongly recommended, especially when it is desired to examine the same cells j'epeatedly at intervals, \vithout harming them. As far as 1 have been able to ascertain, antipyrine- and caffeine solutions have not yet been employed as microchemical tannin reagents. For the sake of completeness 1 point out, however, that such solutions have already been useil by botan'sts in microchemi- cal investigation, namely' by Loew and Bokorny '), to demonstrate the presence of non-organized active jirofein in the living cell. The above-mentioned reagents are supposed to separate this in the shape of small globules, called by these authors proteosomes. Tliis is therefore an explanation of the phenomenon produced by antipyrine or caffeine in the living cell, which is totally different from that given by myself. As a result of my investigations described above, I adhere to my opinion that antipyrine- and caffeine solutions are valuable tamiin reagents, and suppose that Loew and Bokorny have given an inaccurate explanation of the phenomenon which they observed. In the historical survey I pointed out, that, as regards the physio- logical significance of the tannins, there is a great difference of opinion among investigators, and thai in the opinion of various botanists, there is but little, which may be regarded as sufficiently proved, so that we are here face to face with a problem, w liich has in no way been solved. As was stated al)Ove the view thai tannins might serve in the formation of cell walls has received litile support and met with much opposition. With the aid of the method I have worked out, I have now been able to bring to light facts concerning SpirO(ji/ra, which indicate that tannin plays an im()oi-lant part in the formation of cell walls, and that during this process tannin is used up, so that it very probably ser\es as building material. Below I will mention some observations which I'eiate to Ihis. They refer in the first place to the conjugation. Cells which showed a tendency to conjugate, 1 found to be richly ])rovided with tannin. 1 could make out, that the tamiin content diminished during conjugation and in liie adult zygospores which were filled with reserve material, 1 couhl only occasionally observe b 0. Loew and Th. Bukohny, Vorsuchc iibiT akiives lM\v('is.s fiii' Vork'siing uml Praktiiiutii, Biologisclies Ccntiaiblali, iSltl, XI, p. :>. ( 701 ) a feeble tannin reaction wiih fi'nii- cliloriile. It floes not resnlt from this observation wliat is the tate of the tannin, Imt when the conju- gation is followed in greater detail, it is found that there is good ground for supposing, that at least a portion of the tannin serves as |)lastic material for the cell wall. Conjugation is a process which |)roceeds in such a way as to allow us to expect that its study in coiHiexion with the point of investigation referred to will furnish ns with important data, for conjugation does not start simultane- ously in all cells. Some cells are in advance of others; in a smaller or larger number of cells there is evidently a tendency to conjugate, but the conjugation does not succeed, and other cells again do not show a trace of the conjugation process. Whereas the conjugating cells form much reser\e material as starch and fat, those which do not conjugate are apparently very poor in contents and they finally perish. The above mentioned differences seeiB to be determined by accidental circnnislances such at. the coming into touch with cells of other filaments, the proximity of such cells and the position of the cells with regard to each other. They may even be observed with material which before conjugation consists exclusively of healtliy normal cells. The |ioint of interet.t in connexion with the tannin problem is the possibility of comparing, in conjugating ISpirogyra filaments, cells which a short time before were quite equal and afterwards show more or less important differences, induced by accidental and rather su|ierficial circumstances. It is of interest to trace in these various cells what happens to the tannin content. This was investigated with the caffeine- and antipyrine solutions I have recommended, and it was striking to note, how differences in the development of the cell wall corresponded to the quantity of tannin present in the cells. Tims 1 could ascertain, that in cells where the lateral protrusion and mutual fusion had taken place, the tannin content was always appreciably smaller than in cells which only showed the first beginnings of the lateral protrusion. These two kinds of cells onl}' differed as regards cell wall and lannin content; for the I'est they still agreed perfectly. They were distributed promiscuously over the filaments, as is usual in conjugation. These facts seem to me to prove that there is a connexion between formation of the cell wall and the tannin content, and the supposition, that tannin serves as plastic material for the cell wall is very plausible. Furthermore there is a remarkable increase in the tannin content of those cells which have not had an opportunity of conjugating or in which the process was interrupted at an early stage; these cells ( 702 ) degcnei'ate and are gciierallv (k'.^rriUi-il a^ lia\ing a poor coll content. These cells continue to produce tannin for some time and since the tannin in them is not used up in the formation of cell walls or reserve material, the tannin content increases and on the death of these cells a considerable ipiantity of plastic material in the form of tannin is lost. The loss of tannin in nature, e. g. in the fall of leaves in autumn, has repeatedly been used as an argument for the view that tannin cannot be a plastic material and does not take part in metabolism. I cannot share this view and do not think the waste of quantities of a substance, which certain jilants re([nire for their development, to be at all strange, and certainly not a proof that it cannot serve as plastic material in the development of the plant. How often do things in nature fail to attain their end and how many are not wasted without being able to fulfil their purpose! Moreover, it seems to me desirable that the plant should have an excess of plastic material at its disposal, in order that development may never at any lime lie hindered for want of it. The tad that in the autumn the stem is unable to take up all the tannin from the leaves, or all that remains in the leaves from former abundance, hardly proves that tannin cannot serve to build up the tissues. Still less need we wonder at the waste of tannin in Spiro(ji/ra, for evidently it is here not the intention of nature that it should be wasted. Nature ensures a snflicient supply of tannin in Spiroijyrit, because this substance is required in development, as for instance in conjugation and spore- formation. The occasional failure to conjugate, as a resnlt of which then much tannin is lost, does not prove that it is a waste product and not a plastic material. A second series of observations, which show that lauuiii plays a pait in the formation of the cell wall, relate to the formation of transverse walls. On investigating S[)iro<jyra filaments containing cells undergoing division, it at once struck me ihat the tannin content of these cells is somewhat smallei- than that of other cells, not undergoing divisioij. The difference was not large and perhaps, even escapes detection by some of the tannin reagents which have been used hitherto, such as ferric salts and potassium bichnunate, but with antipyrine- and calTeine solutions the existence of a difference in the tannin coident could Ite established with certainty. Not only was it cleai' that the precipitate with antipyrine- or with caffeine solution was somewhat less in the cells undergoing division than in the others, but on treatment of the filaments with these solutions, it was also found, lhat the precipiiale apjieared somewhat later in the cells ( 703 ) ill iirocpss of division ;iii(i llial on I laiisrcrriiiii: them (o distilled water or to ditrii water the precipitate also disappeai'cd somewhat sooner. For the sake of completeness I further mention, that no ditference could be traced between the tannin content of cells in which the nuclear and cell division had just started, and the tannin content of cells not undergoing division, l>ut the tannin content was found to have diminished, when the process of nuclear and cellular division was at its height or could be considered at an end. These results show, that a connexion must i)e looked for between the diminution of tannin content and the process of nuclear and cellular division. This process really consists of two processes, going on simultaneously, and therefore the question arose, which of the two exerted its influence on the tannin content. With reference to this question I carried out some experiments. As has already been stated, the growth of the cells .and the division of cell and nucleus is stopped in a one percent antipyrine solution or in a 0.1 percent caffeine solution. I therefore studied the effect of these solutions on the- formation of transverse walls and on karyokinosis, when the dividing cells and those showing the \ery earliest signs of the process of nucleai and of cell division, were placed in these solutions for some time. Filaments, in which such cells occurred, were left for li hours in the above mentioned solutions, and were then examined next day with regard to the division of cell and nucleus. The transverse walls, in process of formation, had been disturbed in their de\'elopment, and therefore in these cases the cell was incompletely divided. The result in the cells which were on the point of dividing, wheii placed in the antipyrine- or caffeine solution, was more interesting; often in these cells no trace of a transverse wall could be found next day. The pi-ocess of cell division had been completely suppressed. The process of nuclear division was however quite different. In all the cells where it was. going on, or where it was about to begin, it had continued to the end and two normal daughter nuclei always resulted, which were generally situated a little apart in the axis of the cell. It follows from these experiments, that a temporary fixation of the tannin by antipyrine or caffeine prevents the formation of trans- verse walls, but does not directly affect nuclear division. On the strength of this result I feel justified in assuming that there must be a connexion between the diminution of the lanniti content, referred to above, and the formation of transverse walls. Both abolition of transverse wall formafion ihronuh fixation of tannin and the dinii- ( 704 ) imlinn nf llio (aniiiii coiiteiil <liiriim tlic formation of transverse walls, point to till' lanniii being- necessary for, anil iis(>(l nj) in tiie formation of transverse walls. In order to obtain still greater certainty with regard to tins con- clnsion, the influence of antipyrine and cafteine on the formation of transverse walls in C/ii<-lo/>hora was investigated. With ferric chloride, osmic acid, and anti|)\riiie I did not obtain tannin reactions in C/a- do/)h(ij-<i and 1 tiiei'efore was interested in knowing how, for instance, tiie formation of tiansverse walls wonid be atfected by transferring to a one percent antipjrine solntion. I fonnd that transverse walls, which were just beginning to be foi'med, continued to grow until they were completed. This was even the case if the specimens were left in antipyi'ine solution during the whole of the process of cell division. This result still further strengthens my view that in Spiroyyra the inteiTuption or prevention of transverse wall formation is wholly due to the fixation of the tannin. P'or in Chdophora, where no tannin can be used in the formation of transverse walls, a one percent solution of aniipyrinc does not bring about this disturbance. The only ready explanation which, in my opinion, can be given of the results obtained in the conjugation and ti-ansverse wall formation, is this, that the tannin serves as |)lastic material in the building up of the cell wall. I wish to add a few results to those ali'eady mentioned, which point to a connexion between the tannin content and growth of cell wall. In Spirogi/ra lilaments cells are sometimes observed, which, judging from the position of the transverse walls, are distinguished from the others by increased turgor. These cells are generally also distinguished by a larger starch content. On closer examination it is found that the growth of these cells is less than that of the others, or that growth has completely come to a standstill. These symptoms indicate a pathological condition, for generally 1 was able to ascertain that the above-mentioned cells did not divide further and died oil'. I cannot give the reason for this condition, but it is remarkable that the tannin content of these cells as revealed by antipyrine or cafteine solution, is larger, and often much larger, than that of the other cells. Once more it is found, as in the case of cells in which conju- gation tailed, that a cessation of growth is accompanied by an increase in the tannin contejit. As was shown by the investigations of Oek.vssimoi'I' ') and of 1) J. J. Gekassimow. Ueber ilen Einfluss des Kerns auf dns Wachslum der Zelle, Soparat-Abdruck aiis Bull. d. 1. boc. imp. des Nat. de Moskou, 1901, No. 1 en 2, p. 19H. Zur Pliysiologie der Zelle, Separat-Abdruck aus Bull. d. 1. Soc. hup. des Nal. de Moscou, 1904, No. 1, p. 7. ( 7<»5 ) iiivself '), the growtli of cells willioiil nuclei is very slight and gradually stops completely. In aniiclear cells with chromatophores and in those without chromatophores, the two kinds being obtained by centrifuging the cells before or during karyokinesis, the tannin content after a time becomes very considerable, as shown by exami- nation with caffeine- and aiitipyrine solutions. In the absence of a nucleus growth stops, and as a result the consumption of tannin must have fallen olf or has stopped altogether. Its production is however continued for some time; hence the increase of the tannin content in cells without nucleus. In this case also there is cessation of growth and an increase in the tannin content. The results obtained with non-growing nucleated and with uon- nncleated cells, agree with those which I obtained with cells con- jugating and undergoing division, but are of less importance for the e.Kplanation of the physiological significance of tannin, because non- growing nucleated cells must be considered diseased,, and those without nuclei aie very abnormal. The results obtained with con- jugating cells and with cells undergoing division, I consider on the other hand of great importance for the explanation of the physio- logical meaning of tannin, which in my opinion must be regarded in Spirogyra as a substance which serves in the formation of the cell walls. The tannin is here not a reserve-material, however; it belongs to the soluble substances which the plant continually requires for its development. It disappears and gives way to reserve-materials, when the plant forms zygospores and passes into the resting condition. Hence I have arrived at a result, which agrees with the conclusions published by Wigand nearly half a century ago, but which militates against the view of later investigators, such as Sachs, Kkaus and others. For the sake of clearness I must add, that I do not at all claim that tannin is the only substance, which is used in the formation of the cell wall of Spirogyra, nor do I wish to argue that the only physiological significance of tannin is its use as a plastic material. Tills paper is a preliminary one. It is my intention to report at some future time more fully on the physiological significance of tannin in Spirogyra, and to illustrate with tables the conclusions relating to the comparative experiments on the growth of Spirogyra filaments under various conditions, i. e. in antipyrinc- and caffeine solutions, in ditch water etc. At the same time various points of investigation, relating to the tannin problem, and not mentioned in this paper, will be dealt with. 1) G. VAN WissELiNGH. Over wandvonning bij kernlooze celien. Reprint from Bot. .Jaarb. Dodonaea, Vol. 13, 1904, p. 5 and tj. Zin- Pliysiologie der Spiru- gyrazelle, Beihefte zum Bolan. Geutialblalt, Bd. XXIV, Abt. 1, "p. 170. ( 70fi ) Physiology. — '"Tlw (Jinuci-^i silmhi '•'. of th-<' P/ii/.':io/(i(/ic(i/ Liibu- ni/on/ at Utrecht". B_y I'rof, 11. Z\\ aardkmakkk. (Communicated in the meeting of February 26, 1910.) The extension of the means of eoniniunieation calls forth neai'ly every wliere to a higher or lower degree the disadvantages connected with the continnal presence of noise. Therefore we want in many instances a[)artinents free from .sound, and that at first in those cases in which the continuous existence of disturbing sounds forms an insuperable impediment. Such cases present themselves: (I. ill acoustic experiments wlien the observations liave to take place in the proximity of liie minimum perceptibile : I), in public consulting rooms for diseases in the ear where through the coming and going of patients the required silence never reigns, and more frequent visits render every minute investigation well nigh impossible, consequently cause also uncertainty of diagnosis, of advice and of decision in case of examination : c. in moderidy built hospitals, which willi their smooth walls, naked floors, conslruction of stone and iron, etc. show a kind of strong reson- ance, and which, through their many technical 'institutions' can never be quiet; the consequence is the impracticableness of a really efficient percussive and auscultatory examination. Since 1904 a camera silenta (2.28 X 2.28 X 2.20 M.) has been used for the |)urpose mentioned under a in the Physiological Labo- ratory at Utrecht') and also siuc(> that time my advice has repeatedly been asked in the buildiiiu of new laboratories, polyclinics and hospitals in this countiy and elsewhere. In connection with this I venture here to pronounce the conviction that an apartment free from sound, intended for one of the three above mentioned purposes, will have to satisfy three conditions in order to |>reclude disappoint- ment. Tliese conditions are : 1. The inner surface of the aitarlmenl has |o be covered with 1) Silentus, adj. occurring in Gelliu.s, in a I'ragniont from Lakvius used by "loca", is, on account of its shortness, preferable to silentiosus. -) Ned. Tijdsclu-. v. Geneesk. 1905, Part 1, p. r.71. Zeitsclu. 1. Oliienheilk. lid. 54, p. 247. ( 707 ) a iiiateriai tliat does not reverberate sound ; for if this is neglected, not only the involuntary sounds that are made by us, will have a disturbing intiuence, but we shall also be hindered by the small remainder of sound that might still be left on aeeount of incom- pleteness in the construction ; the resonance of the space that is shut off will itself seize definite parts of the small quantity of noise that arises or penetrates into it and make them audible in a higher degree. 2. The isolation must be brought about by a double wall, with interstices of air of such a trifling thickness that resonance of audible lones is quite out of the question and moreover no other contact is left between the two walls than of a few narrow lead-contacts. 3. The isolation of (he outer wall of ihe iiuilding and of its bottom has to be as complete as possible : the first isolation has to take place through a ])urposely constructed secondary apartment. The first condition is fulfilled in our laboratory by means of a covering of horsehair some centimeters thick (trichopiese), as it is used in telephone-cells. Thanks are due to Dr. Biltris of Gent for making me acquainted with this material, which, moreovei", procures an excellent isolation of sound. The second condition is satisfied at Utrecht by making use, in fastening the trichopiese, of a wall of porous stone and by con- structing outside it a second wall, consisting of corkstone of German manufacture. Plates of peatmoss from Klazienaveen in the province of Drente would have answered the purpose even better. The third condition requires the exclusive use of lead-contacts. Espe- cially the bottom has to be well provided for. At Utrecht faults have been made in this respect, which could only partly be made up for by the subsequent addition of an extra-covering. Taking the above-named chief conditions for granted, we shall have to answer the question, whether an apartment free from sound will have to be constructed underground, on a level with the ground or on a higher floor. My answei' is decidedly on a higher floor, for the conduction of the sound coming from the bottom is the obstacle which it is most difficult to overcome. An efficient isolation of the bottom can much more easily be brought about on a higher fioor than on a foundation. In the first case the only thing one has to do is to provide lead-contacts with the stone beams, which in their turn are not directly connected with the bottom, whilst in the second case, under the most favourable circumstances, short columns con- ( 708 ) sistiiig of many sli'ata can be made nse of, uliii-li, however, lia\e a constant direct communication with the ground. As to tlie different tone;, tiie most difticult tiling appears to be to keep away tlie low tones. Inaudible vibrations of very slow perio- dicity are even not at all excluded in our camera silenta, so that a .sensitive microphone, conducted to a gold-thread string-gahanometer does not appear to subside, not e\en when at a complete adaptation of the organ of hearing not a trace of sound is to be observed. (This does not disturb acoustically, luit a somewhal faster periodicity would have been a hindrance). Besides an apartment' free fi'om sound ouglil to iia\e ])orous walls, for if perfectly impermeable walls are chosen, it will appear that in case of long experiments a ventilation is necessary, which in its turn would require the supjily of ventilation-channels, consequently of sound-leaks. For double-door and double-window (the latter in my opinion hygienically indispensai)le) as a matter of course apparatus are Avanted which require much care and a lasting control. When acoustic experiments are made, the supply of sound should come from sound-sources placed outside the apartment, riiilit through a leaden stopper, that the principle that the two walls of the double wall should have none but a lead-contact, is not discounted '). Electric light, telephone, sujiply of air for organ-pipes and sirens through a narrow leaden tube and the necessary conducting-wire to the galvano- meter offer no technical difficulties. An accidental additional advantage of an acoustic apartment with a double wall, double door and double window, duly separated from the outer-walls of the building by means of by-apartments, is this, that it forms a calorimeter. The camera silenta at Utrecht remains without an inhabitant of a constant temperature to within 2 deci- grades. By covering the trichopiese- walls with some meters of extre- mely fine brasswire (0,1 mm.), a bolometer may be made with a Wheatstone bridge and galvanometer placed in a by -apartment, by which bolomelei' the rise of temperature that the space undergoes through an inhabitant, may be measured. The production of heat which this causes is determined empirically (d'Arsonval). x\s a respiration- calorimeter, however, the sound-free apartment is not to be used. This is impossible because the walls are porous, and if this is given u|», it is no longer free from sound for longer expei'iments. A number of investigations may take place in the camera silenta. 1) The leaden slopper.s are 5 em. tliiek and possess a central bore, at its narrowest point being 0.4 em. wide; eorap. Ondcrz. I'hysiol. Lab. Utreclil (5^ VI. p. 13i>, ( 7(i!» ) Those which have been made in tlie last six years, are, it is true, not so miinerons and extensive as I sliould wish, Imt an enumeration with a list o!' the publications ma}' follow here in order to serve as an example of what is to be reached in a sound-free apartment. 1. The sensation of slillness may be ex|)erimenle«l on; unless a perforation of the tympanum exists, a kind of buzzing may be observed, in which at a closer analysis a soft rustling as of the wind in the tops of the trees, accompanied by a high-toned whistling (i*/") may be distinguished ; persons in whom this physiological ear-buzzing is indistinct, perceive a feeling of oppression '). 2. Tlie influence of the ada[)tatioii may be traced; then appears among others a gradual diminution of the physiological tinnitus aui'iiim, ^^■hich after a 3 hours" slay in the sound-free apartment has entirely disappeared (Hoktolotti), whilst at the same time the feeling of oppression, if existing, gradually increases (Minkema) : from this one might be inclined to derive that the physiological ear-buzzing, entirely or partly, possesses the character of an after-image'). .3. The phenomenon of accommodation, discovered by Hensen, may be more closel}' studied, by conveying to a person standing outside the camera silenta through bone-conduction the tone of a tuning-fork, which then from the person's ear is conducted into the apartment through an auditory tube; whenever a metronome placed outside the apartment is ticking, the sample-person accommodates and the observer bears a strengthened sound ((|)uix). 4. From the shortest exposition-time the smallest observable number of sound-vibrations may be derived in the tone of a tuning- fork or that of an organ, conducted to it from the outside; according to Bode this number seems to vary in the scale in a typical manner (de Groot •') and van MexNs). 1) For my ear tlie physiological eai-buzzing can be suppressed : a. by the ticking of a watch; b. by the sound of a tuning-fork of the r'pitcli and a .sound-force of 68.10-^ Erg. per cm.- and per sec. (Erg. d. Physiol. 1905 p. 452). -) According to Bortolotti the buzzing returns directly, after one has left the camera for a moment and then returns. ■5) H. DE Groot, Zl.'^clii'. f. Sinnesphysiol. Bd. 44 p. 18 and Onderz. Physiol. Lab. (5) X p. 1137. ( 'JO ) 5. Tlic miiiiiuiitii |)eroe[)til)ile dining llio iiiiily of time uiiiy l)e fixed iiy tiie se.ale (AIixkema ')). (i. Tiie liiiiil of (listiiictioii iiiav lie trueed and tlie tvpicai variation it undergoes in the scale (Deknik '"')). 7. The sensation of a report, obscr\ ed by Hensen at a sudden intonation or interruption of siren-tones, may be demonstrated in tones of different origin and pitch, with the aid of a sudden opening or closing of a te!e|)hone-conlact or a sudden o|)ening or closing of a particularly constructed lead cock. 8. The spreading of the sound round a tnning-fork with the situation of the well-known interference-planes may be accurately traced, without making the mistakes that must necessarily arise in apartments with echoing walls. 9. The action of the winding mollusc-shells as to their resonance for buzzes may be proved directly. JO. The sound-extinguishing action of ditferent means of isolation may be traced with perfect security ; for reports by dropping steel balls on a steel plate") (fall-phonometer of Zoth), for tones by electri- cally touching purely tuned bells; in both cases the instrument put in a small non-resonant space ; the walls of this space are covered with the materials that are to be examined, and, on the one side the energy with which the bells are touched, and ow the other the distance at which the sound is heard, is defined; the completest isolation with an c(|nal thickness of the walls is got in the case of trichopiese, then, follows the peatmoss-plate from Klazienaveen, then the corkstone; other materials that we examined had a considerably smaller sound-e.\linclion. 1) H. F. MiNKBMA, Oiideiz. Physiol. Lab, (b) VI. p. 134. -) Meeting of this Academy 3 Nov. 1905. ■■^J hi (inler to incvent resunance the stale plate has to b'-' .soldei'eil upon a heavy piece of lead. ( Til ) Mathematics. — "On pairs of points loldcli are associated ivith respect to a plane cubic." By Prof. Jan de Vkies. (Llommunicated in the meeting of February :26, 1910). j . By tlie symbolical equation a^ = a plane cubic c' is represented. If the points A', Y, and Z are connected by the relation each of them lies on the (mixed) polar line of the other two, and every two of those points are harmonically separated by the polar- conic of the third point; thej' form a polar triangle of c'. Let lis look more closely at the case that the three points lie in one line /; then Z is the point of intersection of I with the polar line of X and Y. It is evident that the triplets X, Y, Z lying on / form a cubic involution /o of order two having the points of intersection P,Q,R of c' with I as threefold elements. According to a well known property of the i-) we find that P, Q and R form at the same time a group of the lo. This is indeed directly to be seen ; for, the polar conic of P intersects I in P and in the point H, which is harmonically separated by Q and R from P; the polar line of Q with respect to that conic therefore passes through R. To i'a belongs a neutral pair, U, V forming with each point of / a triplet and therefore having I as polar line. The polar conies of the points lying on / form a pencil ; two of those conies ir and w' touch I in the points V and U. We shall call U and V associated points. Evidently each point U is associated with two points V, viz. with the points which have the polar line and the polar conic of U in common. The associated pairs are thus arranged in an involutory correspondence (2, 2). If / becomes tangent to c\ then in the point of contact L two threefold elements of the /o unite themselves with the two neutral points U, V. For, all polar conies whose poles lie on / pass through L and one of those curves touches / in L. So c' is curve of coin- cidence of the (2, 2) correspondence. 48 Proceedings Royal Acad. Ai£~terdam. Vol. XIl. du = 0, du = 0, dx. dh d^, da (712) 2. We shall see whether there are points L', for which the correspond iiio- points T^ form again an associated pair, so that there is a triplet of points which are two by two associated. If we take the three points as vertices of a triangle of reference, their polar conies will be represented by : ttj.ci -\- i, .fj ,Vj 1= 0, «„ ,v' -(- />, A'j .r, r= 0, a, .r^ -\- h^ ,v^ .r-, = for, each of those points has the connecting line of the other two as tangential chord with respect to its polar conic. If u = is the ecpiation of c', the three polar conies are also represented by d.i/. du dii From this ensues in the first place that the coefficients b^, h,, b, must be equal. Farther on it is directly evident that the equation of c' is : «i •»! + "j •'i'i + «3 *'3 + 36 A\ ,r„ ,«, = 0. The triangle of coordinates is therefore a triangle of inflection, i. e. a triangle of which each side contains three points of inflection of c'. There being four triangles of inflection, the (2,2)-correspondence of the associated points contains four involutory triplets. 3. We shall now determine the locus of the associated pairs, collinear with a given point D. In the first place /) is a node of the locus ; the points D and D" associated with D are the points of intersection of the polar conie (/' of D with the polar line d of D. The locus is tiierefore a nodal bi(|uadratic curve d*. The tangents out of D to e' are at the same time tangents to d\ for in their tangential points two associated points continually coincide. So d'' is the conic of Bertini of d\ For an arbitrary nodal c^ this conic contains besides the six points of contact of the tangents out of the node, the points of intersection of c* with the line connecting the two tangential points of the node, and the tangents in those "funda- mental points" to the conic concur in the node '). The curve (7* is a special curve, because its fundamental points coincide with the tangential points Z)' and D' , so that these are at the same time the points of contact of a double tangent. 1) See my paper 'La quartique nodale" (Archives Teyler, t. IX, p. 263). ( 713 ) 4. It is easy to lincl the equation of d*. The polar conic of Z with respect to al = is represented by a-U:, = 0, the polar line by n.cix — 0. For the tangents out of Z to that conic we have thus 2 3 2 2 , a-a'j-bz ^= o.'zCixbzOj. If these contain the given point Y, then Z is a point of the curve d* belonging to Y. So it has, as equation (in current coordinates ;:): 2 3 2 2 ayttzhz ^n a,,b,ia~b-. From this it is again evident, that the polar line of D is double tangent, and that it touches </' in the tangential points D and D" . For, by combination with a'yn, ^ we find aip'. . b,jb'- = 0. The same is obtained l)y combination with i?i, =r ; by this is confirmed that d'^ is touched in its points of intersection with the polar conic of D by c^ and the polar line d. Out of a'/Cizb: — a ijb yolbl ^ b,fb~a, — b,ja,ib;(C follows that the equation of d* can be transformed into i(aya.bz — - bijb-a-) {a,jb. — a^b,!) zzi 0, SO also into (a,yi- — (izbiiY a-b- -rzz 0. Now Uyhz — u-b,, = (aib-i) (.Vico) + (a-iba) 0/2-3) -f («3^l) 0/3~l)- If thus we represent the coordinates of the line YZ by §yt, the above equation passes into {ab^y a,5- = 0. This equation expresses that the polar conic of Z is touched by the line YZ'). 5. At the same time is evident from this that the line (§) cuts its poloconica in two points. This is more closely confirmed by the observation that the poloconica of (§) is the locus of the points whose polar conies touch (|), from which ensues that it intersects (S,) in two associated jioints. The curve d* can therefore be generated by determining the points of intersection of each of the lines a- through D with the conjugate poloconica a. The poloconica describes there a system with index 2. For, when a passes through any point A' the polar conic of X is touched by 6-. And as two lines .9 satisfy that condition, X lies on two curves a. This generation of f/" with the aid of a system of 1) Clebsch, LeQons sur la geometrie, t. II, p. 278. 48* ( 714 ) conies vvilli index 2 and a pencil projective lo it is characteristic for tlie nodal biquadratic curve 'j . 6. Each nodal biquadratic curve d" of wiiicii the nodal tangents l)ass tliroiigli the points of contact D and D" of a double tangent d is related in the way mentioned above to a c'. The polar curve rf' of D has in D the tangents t' and t" in common with d* and it intersects it in the points of contact R of the six tangents concurring in D. Of the 16 points which d* has in common witli the system of (/' and d six lie in D, four in D' and D' , six in the points R. The tangents i and f contain eight of those points ; so the remaining eight lie in a conic (curve of Bertini). This conic cZ' unites the six points R to the points D and D'. Let us now regard the pencil determined by J'' and the conic d' counted twice; one consisting of the double tangent d and a cubic c' belongs to it. From this ensues that d* is touched by c' in the points of contact R of the tangents drawn out of D to d\ As (/- passes through the points R, it is the polar conic of Z) with respect to c' ; because D' and D" are the fundamental points, so that JJI)' and l)D" are touched by (/- in D and D" , d is the polar line of D with respect to (/■' and of c'. So d* is the locus of the points associated with respect to c' and collinear with D. 12' 3 X 7. If (/' is represented by where ifl =z (fi.fi + CiX-ip') then t:, t", and (/ are indicated by x^ = 0, .y, = 0, and x\ = 0, and (/' by 2x,.v,x, — c3 =3 0. From (■'y''^ + '''i-^'i'^g — "l^s) ~ *'3 (2.ci.f2A'3 — f^) = then follows for d'^ the equation X\X2 — X^ ■=. 0, and from (.ViX-2 — x""-)^ — {x^xl 4- xiX2x^^ — Ara) = we find for c' C^ — S.l'i.fqA's 4" x^ =r 0. For the polar conic if of Y with respect to c' follows from this '■//■J. -- y\''''i-'''3 — y2-VlX3 + ys i'V- — XlXi) = 0, for the polar line of D with respect to tj' -ya'''". — 2/i-^. — y^^i = 0, 1) BoBEK, Denkschriften der Akad. in Wicn, Bd. 53, S. 119. (715 ) thus for the tangents out of D to the polar conic if ^y^ i'^'j'^'l ~ I/.-'^i-'^S — y2XlX;\ + Ji/3 {x-^ — X\,Xi)] = {2l,3.V3 — //i.fo — t/'.iu)-- When one of these tangents passes through ]" we lia\e 4.'/3 (c-;^ - 3.v,//2//3 f yp = (2,y2 - 2.v,//.2)% or ' - -^ II From this is again evident that the curve indicated \)\ tiiis equation is the locns of the points associated with respect to c" and coilinear with D. This special nodal t/^ is characterized by the property according to whicii it is touched by a cubic in tiie six points whose tangents concur in the node. For, when considering the pencil which is determined by c/' with the conic of Bertini counted twice it is immediately evident that the remaining points of fZMying on this conic are points of contact of a double tangent, which must then also lie on the nodal tangents. 8. We shall now see into what a line [I) is transformed by the correspondence of the associated points. To that end we eliminate yk out of the three equations ^y = 0, a,i a- = and a- a^ = 0. Out of the first two we find y\ •!/-2-y3 = {ai S3) a^ : (03 §1) «] : («! Co) a^. Substitution in the third then produces {ai|)(ac5) a^blcl=:Q. A line § is thus transformed into a curve |^ of order five. This could be foreseen, for the two associated points lying on § pass in the transformation into each other, whilst the three points of intersection of 5 and c" correspond to themselves. When the point U describes the line S,, its polar line u envelops the poloconica g% whilst its polar conic ?<' describes a pencil. From this ensues that §* is generated by a pencil of conies and a pencil of rays of index 2 projectively related to it. Consequently $' has nodes in the four basepoints of that pencil and the points associated with U form the pairs of the fundamental involution of pairs ap- pearing on 5* ^). In connection with this s*" is touched by the polo- conica §" in five points (1. c. p. 48j. ij See my paper: "Ueber Guiven funfler Ordnung luit vier Doppelpunkten" (Silz. Akad. Wien, Bd. 104, S. 47). ( 716 > Mathematics. — "On continuous vector distributions on sur/dces". (2'"^ coinnmnieation)'). Hy Dr. L. E. J. Brouwer. (Commu- nicated by Prof. D. J. Koktewkg). (Communicated in the meeting of February 26, 1910). § I. The tangent curves to a finite, uniformly continuous vector distri- bution with a finite ') number of singular points in a singly connected inner domain of a closed curve. Let / be tlie domain under consideration, tlien we can represent it on a sphere, so we can immediately formulate on account of the propert}' deduced in the first communication (see there page 855) : Theorem 1. A tangent curve, ivhich does not indefinitely approach a point zero, is either a simple closed curve, or its pursuing as ivell as its recurring branch shotos one of the following characters : 1^\ stopping at a point of the boundary of y ; 2""^. spirally converging to a simple closed tangent curve; S''^. entering into a simple closed tangent curve. We now shall farther investigate the form (in the sense of analysis situs) of a tangent curve r, of which we assume, that at least one of the two branches (e. g. the pursuing branch) approaches indefinitely one or more points zero, i. e singular points of the vector distribution. We start the tangent curve in a point ^4o (not a point zero) and we pursue that curve in the following way : By (it we understand a distance with the property that in two points lying inside the same geodetic cii'cle described with a radius |?i, and possessing both a distance ^ e from the points zero, the vectors certainly make an aui'le <' - ^T with each other. We farther choose a fundamental series of decreasing quantities «;, f,, f^,,... converging to 0, and of con-e- sponding deci-easing distances 1^ , /?<., ,...., wliich all we suppose, if « is the distance of A^ from the points zero, to be smaller than a— 8,. We then prove in the manner indicated in the first communication p. 852, that, when pursuing r from .1„, a point B„ is reached, possessing a distance ;?,, from ^l^ ; we call the arc .4„Z^„ a /3£,-arc. According to our suppiisition there now exists a finite number n^ in ij For the first communication see these Proceedings Vol. XI 2, p. 850. '^) Tliis restriction we shall drop in a following commumcation. ( 717 ) such a way, that after having completed ?2, |?;j-arcs, but not yet ?i,+l ^j|-arcs, we reach a point ^1,, where for the first time we have approaciied tiic points zero as far as a distance e^. Then again there is a finite number n^ in such a way that, having completed from A^ n,, but not yet n, -)- 1 (^.^-^^'cs, we reach a point A.,, where for the first time we ha\'e approached the points zero as far as a distance f,. From there we pursue r with (?i3-arcs and continue tliis process indefinitely. If we understand by //i(a„) tiie maximum distance from the points zero, which ?• reaches when being pursued after having for the first time approached the points zero as far as a distance s,,, then a first possibility is, that 7n{e„) converges with a,, to zero. In that case the pursuing branch converges to one single point zero and it is an arc of simple curve, stopping at that point zero. We now suppose the second possibility, that m{e,^ surpasses for each 6„ a certain finite quantity e. Tlien we can etfect (by eventually omitting a finite number of terms of the series of 6,/s), that each f ,j <^ - d and each i?. <C g ^• On the pursuing branch then certainly two points P^ and Qi cfii^ be indicated both at a distance e from the points zero, and separated on r by at least one point at a distance fj from the points zero, whilst the distance between P^ and Q^ is <^ i^i.. LetPiASand Q^U be pursuing ji.-^-arcs, and P^R and Q, 2' recurring jl-j-arcs. Fig. 1. Let if; be a point of TU, having from P, the smallest possible distance, then if, cannot coincide with T or U, so that the geodetic arc P,//, is in H^ normal to the vector direction, and the vector directions in all points of that geodetic arc, forming with ( 11^ ) eacli other an angle <^ - jr, are directed to the same side of tlie 8 geodetic arc P,H^. Let Kj be the last point of intersection of tie arc P^H^ of r with the geodetic arc P^H^. Then the arc K^H^ of r and the geodetic arc A'l//, form a simple closed curve, and we prove in the manner indicated on page 853 of our first communication, that either the pursuing branch of /• from H^ lies in the inner domain, and the recurring branch from K^ in the outer domain, or the pursuing branch from i/, in tlie outer domain, and the recurring branch from K^ in the inner domain. Let us first afsume that the pursuing branch lies in the inner domain, then certainly two points P, and Q^ can be chosen on it, both at a distance e from the points zero and separated on r by at least one point at a distance e, from the points zero, whilst the distance between P^ and Q„ is <^ — /?j., . With the aid of those two points we construct in tiie same way as above now a simple closed curve, consisting of an arc K^ H^ of )' and a geodetic arc A', H^, in whose inner domain lies the pursuing branch of r from H^. Going on in this way we construct a fundamental series of closed curves z/i, m,, m,, . . . . lying inside each other. If there is a domain or set of. domains G, common to all the inner domains of these curves (which, as we shall presently show, is really the case) then the boundary of G can only be formed by points belonging to none of the curves u^tU„,u^, . . . but being limit points of fundamental series of points lying on those curves. We assume (jT > 2J, and 5 to be a point of tiq having n distance ^ 3 f ^ and > 3 ft from the points zero. Let C be the first point when recurring from B, and D the first point when' pursuing from B, which reaches a distance — ft from B, then we shall assume for 2 -" a moment that there exists on ?/,y, but not on the arc CD, a point S lying at a distance <[ — ft from B, and we shall show that this assumption leads to an absurdity. 1 ^ ^ 1 Let SV be a recurring — ft- -arc and Ml a pursuiui,^ — ft- -arc 2 i" " 2 /' on u^, tiien the arcs CD and VW can have no point in common, ( 719 ) and the geodetic are Kg Hg, belonging to Ug, has either no point in common with VW, or none with CD. In tiie first case we determine on T^IT a point M, having from B a distance as small as possible. The geodetic arc BM is then in 31 normal to VW, and has a last point of intersection N with CD, so that the geodetic arc XM forms with one of the arcs J^M of Ug, not containing e.g. the point C, a closed curve ; Ug, taken with a certain sense of circuit, would at -1/ enter one of the two domains determined by this closed curve, to leave it no more : further C would lie outside (hat domain ; thus Ug would never be able to reach C, with which the absurdity of our assumption has been proved. In the second case we determine on CD a point M having from S a distance as small as possible, and on the geodetic arc SM the last point of intersection ^V with I'll'. The further reasoning remains analogous to the one just followed : the i)arts of the arcs VW and CD are only intei'changed. Let now B., be the only limit point of a certain fundamental series of points B^, B^, B., . . ., lying respectively on u^,u^,Ug, . . . We assume that B:„ is not a point zero ; it has then for a suitably selected p a distance ^ 4 a^, and ^ 4 i?, from the points zero. Let farther each lUkbe '^ p and let B,,^, B,„„, B,„^, . . . be a fun- damental series contained in the series just mentioned, whose points have all from Bj, a distance <^-- fy, and <^ --- i^, . 8 8 /* If then further on the differentia,,,^ Bm.D,,,, are pursuing, B,,, C,„, recurring — ,?.- -arcs, we prove by the reasoning followed in the first communication p. 854, that there exists a series C,^D„^, C„,D„„, Cr,,D„^,.... converging uniformly to an arc CLZ)^ of a tangent curve u-^ in such a way, that all arcs C„, D„ lie on the same side of CL D... If we describe round Bo, a geodetic circle with radius ■ — /? , 8 'v then it cuts from CL D,^ an arc FI containing Ba ; this arc divides its inner domain into two regions, into one of which, to be called (/, neither the arcs C.D,,, nor any oilier jiarts of the curves u„ can penetrate, as they would get there a distance < — 3s from B,,,. As further the region g cannot lie outside all curves ?/„ , it must lie inside all curves u„ . ( 720 ) So there is certainly a domain or a set of domains G, common to all the inner domains of the curves Uk, and to the boundary- of G belong all points of the limit set ). of the z<t's, which are not points zero, thus also all points of )., which are points zero, as the latter are limit points of the former ones. So the boundary of G is identical to the limit set of the u:'s,, is therefore coherent and identical to its outer circunLfevence, whilst abroad froin the points zero it consists of tangent curves to the vector distribution, which on account of the existence of the domain g can show nowhere in a non-singular point the character mentioned in theorem 1 sub 3. We shall now sliow that a tangent curve r' belonging to the boun- dary of G cannot have the property of r, that its pursuing or recurring branch converges spirally to the boundary of a domain or set of domains (j' . We should then namely be able to form, in the same way as was done above and in the first communication for r, also for r' a closed curve ii'k consisting of a geodetic arc ^ - §s and an arc '/>' of /•', joining the same two points K' and H' . And there would exist arcs of r which would converge uniformly to (f' from the same side, e.g. from the inner side of ii!k- But when pursuing such an arc \^ of r situated in sufficient vicinity of (p' , we should never be able to return between tj' and <p' . As tarthermore in the case considered here, that the pursuing branch of r lies in the inner domain of Mj, it is also excluded, that /•' reaches the boundary of y, only one form remains possible for /•', namely that of an arc of simple curve, starting from a point zero, and stopping at a point zero. (For the rest these two end points can very well be identical). Of such tangent curves there can be in the boundary of G at most two, which possess the same end points, when these end points are different ; but there can be an infinite number, which ai-e closed in the same point zero. Of these however there are only a finite luimber, of which the extent surpasses an arbitrarily assumed finite limit. For, each of these contributes to G a domain with an area, which surpasses a certain finite value. The curves r' whose extent surjtasses a certain finite limit are run along by a Uk of sufficient high index in the same order, as they succeed each other on the outer circumference of G. F'rom this ensues that for all curves r' the pursuing sense belongs to the same sense of circuit of the outer circumference of G. ( 721 ) If the pur3iiin<^ branch of r lies in the outer domain of w,,, the preceding holds with slight modifications. A point of the limit set of the M//s now necessarily bounds a region belonging to 7, and lying outside all Uk^, only then when it is not a point of the boundary of y. The inner circumference, to which r noiv converges spirally on the inner side, consists here again of arcs of simple curve, Avhicli are tangent curves to the vector distribution, but these tangent curves can lie entirely or partially in the boundary of y. However they have all again a pursuing sense belonging to the same sense of circuit of the circuniference. We now agree about the following: When a pursuing branch of a tangent curve reaches a point zero, we continue it, if possible, along a pursuing branch, starting from that point zero, and not meeting the former within a certain finite distance; but if such a continuation is impossible, we stop the branch at that point zero, and so we do likewise when the branch has entered into a closed curve or has approximated spirally a circumference. Then we can resume the preceding reasonings as follows : Theorem 2. A tangent curve is eithr a. simple closed curve, or save its ends it is an arc of simple cui've, of which the imrsuinq as well as the recurring branch shoivs one of the following characters: J*'. stojiping at a point of the boundary of 7; 2"^. stopping at a point zero; 3"^ entering into a simple closed tangent curve ; ^^^\ spirally converging to a circumference, consisting of one or more simple closed tangent curves. From this ensues in particular : .Theobem 3. A tangent curve cannot return into indefinite vicinity of one of its i^oints, after having reached a finite distance from it, unless it be to close itself in that point. That the last theorem is not a matter of course, is evident from the fact that it does not hold for an annular surface. On this it is easy to construct tangent curves of the form pointed out by Lorentz (Enz. der Math. Wiss. V 2, p. 120, 121). We finally notice that the vector distribution considered in this §, does not possess of necessity a singular point (as is the case on the sphere). This is proved directly, by considering in the inner domain of a circle, situated in a Euclidean plane, a vector everywhere constant. § 2. The structure of the field in the vicinity of a non-singular point. To classify the singular points we shall surround each of them ( 722 ) with a domain which we sliall cover entirely with tangent curves not crossing eacli other and we shall investigate the different ways in wliioli tliat covering takes place in different cases. For the sake of more completeness and as an inti'oduction we first do the same for a non-singidar p(5int. Let P be the point under consideration, RS an arc of tangent curve r containing P, UV an arc containing P of an orthogonal curve of the vector distribution. We draw through U and V tangent curves «„ and a^, and through R and S orthogonal curves y and 6, and we let the four points R, S, U, and V converge together to P. Before they have reached P, a moment comes when «o, «i. 7. and 6 form a curvilinear rectangle, inside which lies P, and inside which lies no point zero of the vector distribution, thus inside which on account of the first communication no closed tangent curve can be drawn. We shall cover this curvilinear rectangle with tangent curves not crossing each other. We number «„ with 0, r with -, u^ with 1. Let Q\ be a point 2 4 inside or on the rectangle A^ B^ S R (.fig. 2) having from a, and r 1/ Fig. 2. Non-singular point. a distance as large as possible. We di'aw through Q\^ a tangent curve 4 «i , about which we agree, that, if it meets «„ or r, we shall continue 4 it, by pursuing or recurring «„ or r, until we come upon y or 6. Then «i is a tangent curve joining two points .ll and 7>i of 4' 4 4 y and 6 between «„ and r. In the .'^aine way we construct inside ( '23 ) the rec'langle A^B^SR a tangent curve ^«3^, joining two points 4 A^ and i>3 of y and cf between rand^j. 1\\QveQ.\&i\g\e A^ B„ B^ A^ 4 4 is then divided into four regions. In these we choose in the way described above successively the points Q}^, Q^, Qo, Q±, draw 8 8 8 S through (3l ^ tangent curve «j^ joining two points A\ and B± of y 8 8 8 8 and d, and we deal analogously with the other three points. Going on in this manner we construct for each fraction — <" 1 a tangent curve an joining two points of y and rf; two of these curves chosen arbitrarily can coincide partially, but they cannot cross each other. All these tangent cui'ves must uow cover everywhere densely tlie inner domain of the rectangle A^B^B^ 4,. For, if they left there open a domain G^ then a domain (?'„ bounded by two tangent curves a a -\- \ with indices — and would converge to G. For ?^ sufficiently 2" 2" great however the point %i'+l would then lie inside G, thus in contradiction to the supposition also a tangent curve «2o+i would pass through G. From this ensues, that, if we add the limit elements of the tangent curves «n , which are likewise tangent curves, the inner domain of 2^ the rectangle A„ i>„ B.^ A^ is entirely covered, and further there is for each real number between and J one and not more than one of these tangent curves having that number as its index. § 3. The structure of the field in the vicinity of an isolated singular point. First principal case. We surround the point zero P, supposed isolated, with a^simple closed curve c, inside which lies no further point zero. And we assume as a tirst principal case that c can be chosen in such a way that inside c no simple closed tangent curve exists, inside which P lies. On account of the first communication there can exist inside c neither a simple closed tangent curve, outside which P lies. We now distinguish 2 cases: ( 724 ) n. There exists inside c a simple closed tangent curve q tliroiigh P. We can then choose c smaller, so that it meets p, thus containing in its inner domain a tangent curve p, which (in its pnrsning direction) runs fi'om P to c, and another q^ running from c to P, and we furtlier look foi' such tangent curves inside c which cross neither q^ nor q.^ . Of the possible kinds of tangent curves mentioned at the conclusion of § 1 we shall agree about those, which enter into a closed tangent curve, to continue them along that tangent curve until they reach either P or c, and to stop there. Spirally converging to an inner circumference cannot appear, as the other end of such a tangent curve would be separated from P as well as from c, and so would determine a closed tangent curve, outside which P would be lying, which is impossible. Neither can appear spirally converging to an outer circumference, as P would have to lie in that outer circum- ference and the spiral would necessarily have to cross q^ and p^. h. There exists inside c no simple closed tangent curve through P. Then inside c there exists no simple closed tangent curve at all, so that again spirally converging is excluded. In any case, if we agree not to continue a tangent cur\e, when it reaches P or c, we can distinguish the tangent curves inside c, and not crossing q^ and q^ if the latter exist, into three categories: 1^' . Closed curves, containing P but not reaching c. 2'"'. Arcs of curve, joining two points of c, hut not containing P. Z^'^. Arcs of curve v^hich run from P to a point of c (positive curves of the third kind) or from a point of c to P (negative curves of the third kind). Of this third kind there must certainly exist tangent curves. For otherwise the closed sets determined by the curves of the first, and by those of the second kind would cover the whole inner domain of c, thus would certainly possess a point in common ; through this point however a curve of the third kind would pass. So we can commence by constructing one curve of the third kind and we choose eventually q^ for it. If possible, we then draw a second curve of the third kind not crossing the first and we choose eventually q, for it. Into each of the two sectors, determined in this way inside c, we introduce if possible again a curve of the third kind, not crossing the already existing ones, and chosen in such a way that it reaches a distance as great as possible from the two curves of the third kind, which bound the sector, whilst, if the new curve terminates somewhere on one of the curves bounding the sector, we further follow the latter curve. In each of the sectors, deter- mined after that in the inner domain of c, we repeat if possible this ( '25 ) insertion, and we conliiiue this process as often as possible, even- tually to an indefinite number of insertions. If in this manner we have obtained an infinite number of tangent curves of the third kind, they determine limit elements which each are either again a tangent curve of the third kind, or contain such a curve as a part. And in particular a fundamental series of positive respectively negative curves of the third kind determines in its limit elements again positive respectively negative curves of the third kind. After addition of these limit curves of the third kind we are, however, cpiite sure that no new curves of the tliird kind not crossing the existing ones can be inserted. This is evident from a reasoning analogous to that followed in § 2. The whole of the curves of the third kind, obtained now, we shall call a system of base curves of ■the vkinity of P. An arbitrary positive base curve and an arbitrary negative one enclose inside c a sector, of which the area cannot fall below a certain finite limit. For otherwise we should have a fundamental series of positive base curves, and a fundamental series of negative ones, possessing the same base curve as a limit element, which is impossible, as that limit base curve would have to be positive as well as negative. So the inner domain of c is divided into a finite number of sectors which can be brought under the two following categories : First category. Sectors bounded by a positive and a negative base curve, between which lie no further base curves. The areas of these sectors surpass a certain finite limit. Second category. Sectors bounded by two positive (respectively two negative) base curves and containing only positive (respectively negative) base curves. A sector of this category can reduce itself in special cases to a single base curve. We shall first treat a sector of the first category and to that end we first notice that outside a curve of the second kind Ij'mg in it (i. e. between that curve and c) lie only curves of the second kind, and inside a cur\e of the tirst kind lying in it only cur\es of the first kind. If we draw in the sector a well-ordered series, continued as far as possible, of cur\es of the second kind enclosing each other, then it converges either to a curve of the second kind, or to two curves of the third kind and between them a finite or denumerable set of curves of the first kind, not enclosing each other, and not approaching c indefinitely. If we can construct an infinite number of such series not enclosing ( 726 ) each oilier, then there are among them whicii ent from the sector an area as small as one likes, and at the same time the maximnm distance, whicli such a series reaches from Cj decreases under each finite limit. And analogously, if we draw in the sector a well-ordered series, continued as far as possible, of curves of the first kind enclosing each other, it converges either to a curve of the first kind, or to two curves of the third kind and between them a finite or denumerable set of curves of the second kind, not enclosing each other, and not approaciiing P indetinitelj. If we can construct an infinite number of such series not enclosing each other, then there are among them which enclose an area as small as one likes, and at the same time the maximum distance, which such a series reaches from P, decreases under each tinite limit. From this ensues that for the sectors of the first category we have to distinguish two cases: First case. There are curves of the second kind in indefinite vicinity of P. Then the domain of the curves of the second kind is bounded by the two base curves which bound the sector, and a finite or denumerable number of curves of the first kind, rioi enclosing each other, and not approaching c indefinitely, in whose inner domains, which we call the leaves of the sector, can lie only curves of the first kind. The region outside the leaves can be covered as follows with curves of the second kind not crossing each other: we first construct one which reaches a distance as great as possible from c and the boundary of the leaves; in this way two new regions are determined, in eacli of which we repeat this insertion. This process we continue indefini- tely, and finally we add the limit curves. That then the region outside the leaves is entirely covered, is evident from the reasoning fol- lowed in § 2. And in the same way we fill each of the leaves with curves of the first kind not crossing each other. The whole of the tangent curves filling the sector finally gets the form in- dicated in fig. 3. The sectors being in the discussed first case we shall Fig. 3. Hyperbolic sector. call hyperbolic sectors. Second case. There are no curves of the second kind in indefinite vicinity of P. Then the domains covered by these curves are cut off from { 727 ) the sector hy a finite or denumeiablc number of curves of tlie second kind, not enclosing each otlier, and not approaching P indefinitely. These do- mains we take from the sector (conse- quently modify an arc of c), and there remains a new sector, bounded by the same base cui-ves as the old one, but consisting of one leaf inside which lie only curves of the first kind. This leaf we can fill with carves of the first kind not crossing each other (see fig. 4). These sectors of the second case, which are reduced to a single leaf, we shall call elliptic aectovs. We now pass to the discussion of a sector of the second category, of which, to fix our ideas, we assume, that it is bounded by two positive base curves. Let us consider the set of points lying in the sector or on its boundary, through which cur\'es of the second kind not crossing the base curves can be drawn. This set of points cannot approach P indetinitely, as otherwise it would gi\e rise to a negative curve of the third kind not crossing Ihe base curxes, wiiich is excluded. In the same way as for the elliptic sectors we destroy the regions covered by this set of points, and there remains a sector of the second categoiy bounded by a modified arc of c, inside which no curves of the second kind not crossing the base curves can be drawn. In the modified sector we now consider the set of points, through which curves of the first kind not crossing the base curves can be drawn, and it is clear that this set of points cannot indefinitely approach the just now niftdified cur\'e r. The regions covered by it are therefore bounded by a finite or denumerable nnmber of curves of the first kind, not enclosing each other, not indefinitely a[)proaching c, and each enclosing a domain which forms a leaf, not dift'ering from those appearing in the hyperbolic sectors. By the method applied above already .several times the region outside the leaves can be filled with curves of the third kind (for instance we can choose for them the system of base curves Fig. Parabolic sectu 49 Proceedings Royal Acad. Amsterdam. Vol. XII. ( 728 ) present already in the sector), and finally eacli of the leaves with enrves of the first kind (see fig. 5). The sectors of tlie second category we shall call positice (resp. negative) paraholic sectors. In special cases the whole inner domain of c can reduce itself to a single positive (resp. negative) parabolic sector. A point zero where this occurs we shall call a source point resp. vanishing point. § 4. 7'hc structure of the field in the vicin.itg of an isohiteil singular point. Second principal case. In this case any vicinity of P contains a simple closed tangent curve inside which P lies. We can then construct a fundamental series c, e', c", .... of simple closed tangent curves converging to P, of wliich each following one lies inside each preceding one, and we can fill in the following way the inner domain of c with tangent curves not crossing each other. In each annular domain between two curves c'"'> and c'"+^) we choose a point having from the boundary of that domain a distance as great as possible and we lay through it a tangent curve situated in the annular domain. According to § 1 it is either closed oi- it gives rise to two closed curves, situated in the annular domain with its boundary, into which it terminates or to which it converges spii'ally, and which we draw likewise. (These closed tangent curves can entirely or partially coincide with c(") or c'"+')). So the annular domain is either made singly connected or it is divided into two or three (amiular or singly coimected) new domains. In each of these we again chod^c a ]i((iii( liavin.n' from the lumndai'y a distance as great as |i(issililo and we lay through il a^aiii a taiiueni <Mir\e. A singly connected domain is certaiidy divided by il into two siniily counected domains; on an annular domain it has I he ellecl Just now mentioned. We I'epeat this process inde- l''i", G. I'lolulioii poiui. linilely. l'\)i' each domain il can ( 729 ) happen only once that it undergoes no division ; after that namely it becomes singly connected, so is divided at each new insertion of a tangent curve (see fig. 6). We finally add the limit curves, and we prove in the same \vay as in § 2 that tiien through each point of the inner domain' of c passes a tangent curve. A point zero being in the second principal case we shall call a rotation point. So we can say : Theoreji 4. An isolati'.d sinr/tilar point w eilhti' it rotation point, or a vicinity of it can be divided into a finite number of hyperbolic, elliptic, and parabolic sectors. The tilling of a vicinity of a non-singular point in § 2 furnishes in this terminology two hyperbolic and two parabolic sectors. We must add the observation that in the most general case, where neither in a singular, nor in a non-singular point the tangent curve is determined, sometimes by a modified method of construction, the structure of the lirst principal case can be gi\en to a \icinity of a point zero being in the second principal case. Even the form of the sector division of the lirst principal case is then not necessarily unecpiivocally determined. Out of the reasonings of the following § we can, however, deduce that, if modifications are possible in the form of the sector division, the difference of tlie number of elliptic sectors and the number of hyperbolic sectors always remains the same. ^ 5. I'/ie reduction of an isolated sirn/ular point. For what follows it is desirable to represent the domain y on a Euclidean plane, and farther to substitute for the curve c a simple closed curve d emerging nowhere from c, containing likewise P in its inner domain, and consisting of arcs of tangent curves and of orthogonal curves. In the second principal case this is already attained, and in the first pi-incipal case we have to modifv in a suitable way only those arcs of c which bound the hyperbolic and the parabolic sectors. In a hyperbolic sector we effect this by choosing a point on each of the two bounding base curves, atid by drawing from those points H and K inside into the sector orthogonal arcs not intersecting one another. Then there is certainly an arc of a curve of the second kind 49* ( 7^0 ) joining a point B of one of these orthogonal arcs with a point C of the other, and we bound the modified sector by the orthogonal arcs RB and CK and tiie tangent arc BC. If a parabolic sector is bounded by the base curves k and k' , it is always possible to choose between them a finite number of base curves k.^ ,k\ , . . . . k„ in such a way, that each /■;, and k„j^\ can be connected, inside the sector but outside the leaves lying in it, by an orthogonal arc. By tliose orthogonal arcs and the arcs of base curves joining their eiidpoints we bound the modified sector. The simple closed curve c obtained in this way has a direction of tangents varying everywhere continuously, with the exception of a finite number of rectangular bends. To a definite sense of circuit of c' , which we shall call the positive one, cori-esponds in each point of c a definite tangent vector, and for a full circuit of c' that tangent vector describes a positive angle 2.t. We shall now consider two successive parabolic sectors, .t, and ^Tj , of which (for tiie positive sense of circuit) the first is positive, therefore the second negative, and we suppose them to be separated by a hyperbolic sector f. On the orthogonal arcs belonging to the boundary of .t, the given yector then forms with the tangent vector an angle I 2n J.t (measured in the positive sense), on the ortiiogonal /_ 1 arcs belonging to the boundary ot .t^ an angle 2» -| — The transition takes place along the tangent arc belonging to the boundary of /- , by a negative rotation over an angle -t of the given vector with respect to the tangent vector. The same remains the case if we suppose -t, to be negati\e, -t^ to be positive. But if we suppose f to be an elliptic sector, then the transition under discussion takes place along the tangent arc bounding f, by a positive rotation over an angle jt of the given vector witli respect to the tangent vector. As now the total angle, which the given vector describes for a full circuit of c' , is equal to the total angle which the tan,gent vector describes |)lus the total angle which the given vector describes with respect to the tangent \ector, the former angle is ecpial to ."r (2 -|- », — n.,), where n^ represents the number of elliptic sectors, ii^ the number of hyperbolic ones. Let further / be an arbitrary simple closed curve en\eloping P, bitt enveloping no other singidar point, then we can transform c' into / by coulinnons niodilicalion in siu'h a way, that at evei'v moment ( '-'^l ) J\ lull !Hi odier siii,f!,ulai- |ioiii(, is eiivelopod liy the iiiodilicMl curve. If we I'Oiisider for each of tlie iiiterniediarj curves tlie tiilal angle which the given vector describes bv a positive circuit, then on one hand it can only have continuous modifications and on the other hand it must remain a multiplo of 2.-r. Thus it remains unchanged, and we can formulate: Theorem 5. The total angle mhich, by a circuit of a simfle closed curve enveloping onlj/ one point zero, the vector describes in the sense of that circuit, is equal to n; (2 -|- ?;, — n.^), lohere n^ represents the number of elliptic sectors, n^ the number of hyperbolic ones, which appear irhen a vicinity oj the point zero is covered ivitli tangent curves not crossing each other. In particular for source points, vanishing points and rotation points this angle is equal to -f- 2.t. We now surround P with a simple closed curve x which can be supposed as small as one likes, and we leave the vector distribution outside y. and on y. unchanged, but inside y. we construct a modified distribution in the following way : Let us first suppose that for a positive circuit of y. the vector describes a positive angle 2».t. From an arbitrary point Q inside y. we draw to y. n arcs of simple curve '^^, ^^, ... ^„, not cutting each other and determining in this order a positive sense of circuit. Let us call f,y- the arc of y. lying between /?,, and ^^_j_i , and G/, the domain bounded by i?^,, ^.y. and |i,,+i. Along ^^ we bring an arbitrary continuous vector disti'ibution becoming nowhere zero and passing on x into the original one. Then along [i^ such a one passing on y. and in (I into the already existing vectors, that along the boundary of ^^1 positively described the vector turns a positive angle 2jr. Then along i?, such a one passing on y. and in Q into the existing vectors, that along the boundary of (t.^ positively described the vector turns a positive angle 2jr, etc. As the angle described by the vector in a positive circuit of x is equal to the sum of the angles desci'ibed in positive circuits of the boundaries of the domains G^, G^, .... G,,, it is finally evident, that also for a positive circuit of G„ the vector describes a positive angle 2 jr. In each of the domains G^, with boundary -x^ we choose a simple closed curve r,, not meeting y.^j, of which in a suitable system of coordinates the equation can be written in the form ,v'^ -\- lA = r. Inside and on r^, we introduce a finite continuous vector distribution vanishing only in the point (('.v^- which is directed along the lines ( 732 ) Hr = « iuid from tlie poiiif (",")^,. Tliis veclor descj-ihes along r^, a positive angle 2.t, Jnst as the existing one along y.j,. If then according to ScHOENt'LiKS \vc 1111 the annnlar domain between y.^, and r,y with simi)le closed curves enveloping eacii other and as functions of a cyclic parameter passing continuously into each other, then we can thereby at the same time make the vector distribution along y.^, pass continuously into that along c,,, and in this way give to the annular domiiin between y.^, and c^, a Unite continuous \eclor distribution vanishing nowhere. Inside y.^, we have now obtained a linile con- tinuous vector distribution, having but m/i' jjoint zero, namely the point {o,o)ij, and that a source point of very simple sti'ucture, which we shall call a radiating point- And the inner domain of •■< is covered with a finite continuous vector distribution passing on y. into the original one and possessing inside ■/, instead of the original ])oint zero P, n. radiating points. Let us furthermore suppose that for a positive circuit of ^ the vector describes a negative angle 2n rr. In an analogous way as above we then divide the inner domain of ■/. into ii regions G^, with boundaries :'.,„ and we bring along each of these boundaries such a vector distribution, that for a [)Ositive circuit of k,, the vector describes a negative angle '2.t. The curves c^, are introduced again as abo\e, but inside and on Cf, we introduce a finite continuous vector distribution vanishing oidy in the point (,0,0)^,, which is directed along the lines ,i,'^, >/^, = «. For a positive circuit this vector describes along c), a negati\e angle 2.i, just as the existing vector along y^. So the annular domain between y.„ and c^, can be filled uj) in an analogous way as just now with a finite continuous vector distribu- tion vanishing nowhere, and (he whole distribution inside x^, possesses then only one point zero, namely the point (0,0}^„ having four hyperbolic sectors of very simi)le form (the four separating parabolic sectors are each reduced to a single line), which structure we cha- racterize by the name of reflaxion point. After this the inner domain of /. is covered with a finite continuous vector distribution passing on x into I lie original one and possessing inside y., instead of the oi'iginal jjoiul zero /', n rellexion points. Let us finally suppose that for a circiiil of y. the total angle described by the vei-lor is zero. We can then choose inside y. such a simple ck)sed curve i\ dial in a suitable system of coordinates its equation can be written in the form .v' -\- if- = r''. Inside and on c we introduce a linile continuous vector distribution vanishing nowhere, ( 733 ) wliicli is (lirec'ied along tlie lines // ^ <!. Tlie total angle desfribcd liv this vector along c is zero, Just as the one described bv the existing \ector along y.. The annular domain between y. and c can thns be filled up as in the two preceding cases with such a finite continuous vector distribution, that the whole distribution inside y. is now free of points zero. So we can formulate : Theokkm 6. A finite continuous vector distribution with (i finite mnnhi'r of [joints zero can he transformed, by modifications as small as line likes inside vicinities of the points zero lohich can be chosen (IS sinnll Its one likes, into a nenj finite continuous vector distribution irhirli hits as points zero only a finite number of radiating points, and a finite number of reflexion points. In particular those points zero about which the amjle, described bij the vector for a /lositire c/rcnit. is /josifive, are broken up into radiiitinif fioints : those a/ioat n-hich this angle is negative, are broken II f) into refie.cioii- /)oints ; irhilst those for which it is zero, vanish. In a following communication we shall extend this theorem to distributions with an infinite :dennnierable or continuous) number of points zero. § 6. Remarks on the tangent curves and singular points on a sphere. If we have on a sphere a finite continuous vector distribution with a finite number of singular points, then the reasonings of § 1 lead with small modifications to ; Theorem 7. A tangent curve to a finite continuous vector distribution with a finite number of singular points on a sphere is either a .simple closed curve, or save its ends it is an arc of simple curve, of which the pursuing as well as the recurring branch either stops at a jioint zero, or enters into a simple closed tangent curve, or converges spirally to a circumference con.sisting of one or more simple closed tangent curves. From this ensues that also on a sphere a tangent curve cannot return into indefinite vicinity of one of its points^ after having reached a Unite distance from it, unless it be, to close itself in that point. Out of the reasoning of § I we can deduce farthermore without difficulty that a fundamental series of closed tangent curves with the pr()[)erty that of the two domains determined by one of them, ( '34 ) one contains no points of llic ])ri'C('(linn', the oilier no points of the following- curves, converges either to a single singular point, or to the outer circumference, consisting of simple closed tangent curves, of a domain or set of domains. Tjet now an arbitrary finite continuous vector distriliutionon a sphere be given. On account of § 5 we reduce it by means of in- definitely small modifications to a "reduced distribution", possessing as singular points only radiating points and retlexion points, and we investigate the tangent curves of that reduced distribution. A closed tangent curve can possess no radiating points, but retie.xion points it can possess (its tangent direction shows there a rectangular bend). On the othei- hand a tangent curve can only stop at a radiating point. We now consider an arbitrary tangent curve : according to theorem 7 it is eitlior an arc of simple ciu-ve joining two radiating points, or it gives rise to a simple closed tangent curve j„, which divides the sphere into two domains G and G' . Then on j„ no radiating point can lie, but we shall jirove, that in G as well as in G' there must lie one. If namely there were no i-adiating point in G, we could consider within G a new tangent curve, and as this would not be able to stop in G, it would on account of theorem 2 give rise to a new simple closed tangent curve /i enclosing a domain G,^ being a part of G. Within (tj we could again consider an arbitrary tangent curve, and in this way we should arrive at a simple closed tangent curve j^ enclosing a domain G., being a part of G^ . Contiiuiing this process indefinitely we construct a fundamental series of closed tangent curves ./o'./i',/;'./:i which cannot con- verge to a single singular point, as neither a radiating point nor a reflexion point contains closed tangent curves in an indefinitely small vicinity. On account of the remark made at ihe beginning of this § there must thus be at least one domain (/,„ , bounded by a simple closed tangent curve jo,, and contained in each of the domains G^, G^, G^3, .... Within G',, we could again construct a closed tangent curve j.^^\ bounding a domain Gu-\-\ being a part of Gr„, and we could continue this process to ani/ index of the second class of numbers, which on the other hand is impossible, as the set of domains G — (r\ , G^—G., , . . . G^ — G,,,J^\ , . . . G.^ — Gu^\ , . . . must remain denumerable. So we finally formulate : TnnoRK.M 8. A ri'durcd distrllnijlon on a sjihi'ir possesses nt least two radlallny points. ( 735 ) Mathematics, - "Tin' osi-Uhtiinns ahout a /lositiou of <'</iulifjr/uin ivliere a simple linear relation e.vist'; betineen the fvequevcies of the principal vibrations.'' (Second pari). By H. .1. E. Beth. (Communicated by Prof. D. J. Korteweg.) (Communicated in the meeting of February 2B, 1910). S = 4. •) § 14. In this case tlie ordinary expansions in series hold as long as is great with respect to ( — (see page 7 of the paper by Q Prof. KoKTE\yEG, mentioned aboye). The difficnitj arises as soon as — "i has fallen to the order | - I . The calculations not oetting simpler with the absence of a residue of relation, we sliail immediately assume a residue of relation of order /r. When the relation M, -|- 9 =: SWj exists and we proceed to inyestigate with a view to this wiiicli terms in (2) (page 620 of these Proceedings) become disturbing in the sense indicated in § 3, we easily see that no terms of order Ir appear among the disturbing ones. So when determining the lirst approximation we may omit the terms of order h' in the erpiation of the surface, which terms agree with the just mentioned tei-ms of order /(^ It then becomes 9 for we need not take for the first ap[»roximation in the equations of moyement any terms of higher order than h''. The abridged equations of motion, containing only terms of order /(, still run as follows: X + 2c, .r — 0, j y + 2c,y = oJ Now «i = l/^Ci , n„ = l/2c, are the frequencies of the principal yibrations. -J For the case .5^3 see I'" part, pages (J19 — 635 of these Proceedings. ( 736 ) So We cliangc the al)ridged equations into : but tiien we must admit into tiie function R a term : 3/(j (J I/''. Tiie canonical solution of the abridfted e(|uations is-. y «, .V = (MS (Mj< -(- 27ij|'}j), "l ?/ ^ — cnx {'Sn^t -\- 6«j|?J. 3«, To tind which functions the (t's and ji's are of t, we must in- vestigate which foiMii the function R now assumes. § 15. As the disturbing terms in the e(|uations of motion are of order h' we shall find that a^, a.,, li^, and /-fj can never exceed order h. Of this we ma_y make use to simplify the terms of order /;" containing ,r, y,,*^ and v/\ We may namely replace in those terms: ,v^ by «j — Jij^ ;r^ « ,, — Wj" ,(; and Then the equations become : X -\- H," x + 4ej a:' -\- 5e, ,iry + 2e, .vif -|- c^ ?/' -|- n' 2m/ J 4- — (", + ^f') -^ (•''•' + 81 r) J' = 0. / 9' '9°' ' ( »/■ + 9h.' y — G«j i>y + e^ *■' + 2*', .v^y + 3*-, .r,/'^ + i>', y' + i 9n/ ISn," ] 4 -^ («, + 9«,).v ^ (,«-^ + 81 2/^)// = 0. 9 9 Now the terms of order A' are all disturbing except c.^y' in the first and 3e^ xif in the second equation ; so these may be omitted. The terms Se^.r''// in tiie first and ('.,c' in the second equation owe their disturbing property to the supposed relation. The remaining terms are always disturbing, also when no relation exists. { 737 ) To traiisfonn tlio eiinalioii.s to such u tbriii lljat (he disturbing terms may be regarded as deri\'atives of one and tlie same function resp. to .r and y, let ns consider tlie term with ,(■//' in the first and that witii .),■•'// in liie second equation. If we sulistitute tiie expressions found above as lirst approximation for ,/; and y in these terms, after the development of the products and powers of the cosines among others terms will appear, differing onl}' in coefficient from the expressions indicated for ,*■ and // ; the remaining terms which appear are not disturbing. From this ensues that we nia^y replace : 1 «, in the first enuation : .rjj- by — - — ~ x. in the second equation: .ry l:)y Accordingly the e((uations may be written : / 14o8///\ y + 9n;'y + U., --'~\ ^ + e, ,.» 4- I SWj u Bbi, y = 0. We thus see that they take the form of dR X -j- ?J, d.v y + 9n, dK dy where : 1 / ^ e^ 81w/ \ 6''i C H i «i H 1^ «. .'/' + e., .v' y. "1 9- J § 16. We must now write II as function of the rt's and /}"s by substituting for x and y, in the expressions obtained, the expressions by which they are represented at first approximation, and by retaining only those terms in which t does not appear ex{)licitly. Thus we ai'rive at : 1 1 11 — R^rz ~~- aa^'' 4" ^'«i «a + ~r ^''^-Z + i* ^'" «i + "^1 "] " <^i- '^os tp , ( 7.38 ) a = — -I ^ ~" I8w/ ' '' ~ 108n7 %^ ' ^ Q^ The system of equations giving the tiine-variabilit}- of tiie <f's and ,i's, is now : rf", , ^ - . ^= 2Am, n, - <t.,- sin w, (it 1 1 i >' da^ — — dt 1 1 » A — ~ a«i + bct^ + ~ ?«! «, - ((., - cos <f , — ' =: bet, 4- ea. 4~ o Ii'' -\ m, «, - «„ - cos «; ; where A' is put instead of 3«]. From tins system it ajijiears at once tliat : da, du„ dt dt therefore «, -|- «, = constaiit. So we put : Furthermore according to § 4 : 1 1 It — ((«,' + ?'«j «.^ f — en..'' -\- ()' /(^ «2 + »Mj «i - rr, - fos (f — is an integral of the system. Bv introduction of ? Iliis integral takes liie form of: (17) S J/? (1-g) CO* y=^-?-' +.;?+/• (18) ( -5!» ) where : ^ 1 , 1 ■>> a -1-0 — — c , , 2 + O 1 ' I \ c I ^' ^ '-,,, . '^'-^n^^Bj' '<i V 2 '' n'R'^n, C ' wliere C represents & constant, dependent on the initial state. The tirst equation of (17) becomes by the introduction of fe: j=^^ "h K- -"^'^ ''" ■ i ^^ (^— S) • «■« <P- ■ ■ ■ . (10) By eliminating <f between (18) and (19) we arrive at: I, R," A' h- . di. t/5' (!-?)-(;'?'+??+'•)' '^ Let /{:) = :' {i-L) - (pi^ + qz ■{- ry , then /(-')>0 for the initial value of -', but/(:) < for ^ = and ^ ^ 1 : so f [1) becomes zero for two values T, and -'„ lying between and 1. ' So '^ will generally vary periodically between two limits. It may be expressed in the time with the aid of elliptic functions, after which ji,, ,?.j, ,/■, and y are also known as functions of the time. For the extreme values zero and one of the modulus y. ot the ((J — «) (uj — C,) elliptic functions (y. =\ / ^-, when thee(|uation ;'(^) = has two real roots « and ,? besides uj and 5,) we get special cases. (Jsculatiiui curves. ^ 17. At first approximation we have found : .. = CO. (,<, t + 2«, i?,)> «, y — -5 — t'os (3)ij t A- 6n, (?,), wliere the fi"s and ,i's slowly vary with the time. By introduction of C and <p and by change of the origin of time we iind that we may detei'mine the equation of an osculating curve by eliniiiuiting t between ( 740 ) ,r. ^= K^ h \/L cos «, f and 1/ = "- jRj, /i K 1 — r cos (3«, t — (f). For 'C and y we must siibstitnte the values, wliicli these quantities have at the moment for which we wisii to know the oscuhxting curve. The osculating curves are Lissajous curves answering to the value — for the ratio of the periods of the vibrations. They are described 2 in the rectangles having as sides 2 R„li\''l. and —Egh^ 1 — C. o As u varies between two limits the rectangles in which the curves are described lie between two extremes. The vertices lie on the o circumference of an ellipse having 2 A',, h and -- R^, h as lengtiis of 3 axes. The shape of tiie curve described in a deliinte rectangle is still dependent on the value of y, i. e. on the value of the difference in phase at the moment of the greatest deviation to the light. To an arbitrary value of <f the wellknown Lissajous curve with rr 3;r two nodes of II"-. 8 answers. For <p =z - or — the curve is sym- 2 2 metrical in respect to the axes; tiie nodes lie in the A'-axis on 1 eiilier side of (^ at distances —R^k (tig. 9). For tp ^ or. t we get a curve, which is described in both directions alternately and which passes through (fig. 10). In fig. 11 we find .some of those osculating curves represented for a delinile rase of motion; tmo belonging to (p =: m ■, tiro for 7t f a\ ip = -, and (irw for an arbitrary value ot </ I ^ - . -/: Oul of (19) follows that -" ^=0 for sn/ <f ^ 0. In the extreme rectangles the cur\es are tlescribcd which we ha\e for <p = or .t. Now a uund)er of different cases are possible, of which we gel a clear I'epresenlation by representing eipiafion (18) in polar coordinates. In fig. J 2 .some of the cur\es obtained in this way ai'e I'epresented, where <{ is taken as polar angle, I 1 — ^ as radius vector. The dilferent shapes of the curves cori'es|)ond to the roots of the eij nation : ^(\-^-{p;r + q;^rf = ^ (20) ( 741 ) The cases are : 1. The curve indicated in the ligtire by --- keeps to the riglit or to the left of (J^; ip clianges between two limits; tiie limits are equal and opposite: the positive is smaller than "-. For the extreme values of '1 we find </ either botii times or both times rr. 2. The curve intersects the straight line y ^ - at two points above 0^ and at 2 points below (J^. For the extreme values of l, we again find </ either both times or both times rr. 3. The curve consists of two closed parts (a continuous line in the figure), which surround (>,. >\ow </ assumes all values. For the extreme values of c '/ =r one time y ^ and (/ =: n the other. The transition case between 2 and 3 is represented by . . F'ig. 1 1 relates to the 2'"' case ; for the two extreme values of Z Ave find tf ^ Tt. Special ca.ses. ^ 18. These occur for the extreme values of the modulus x of the elliptic functions; two roots of ecjuation (20) have coincided. 1. y.^l. The elliptic functions pass into hyperbolic ones. The geometrical representation just now discussed of the relation between C and rp and already mentioned as transition case between the second and third cases has a node situated on the axis of the angles. The form of motion approaches asym|itolically to a foi-m of motion belojiging to '/ = or y = :t. 2. y. ::= 0. The elliptic functions pass into goniometrical ones. The CTU've of fig. 12 becomes an isolated point (' (special case belonging to the 1-' case of §17 as limiting case) or it consists of an isolated point and a closed curve (special case belonging to the 3"' case of § 17 as limiting case). If the initial value of ^ coincides with the twofold root of (20) we find that T remains constant ; (p is conti- nually or .T. Thus the same curve is continually desci'ibed. Arbitriiry iiiechanisin irit/i 2 decrees of frecdoin for ichlch ,S^4. § 19. In the case that )i^ = 0!;, -|- !? '''s terms of oitler h.'' can give no disturbing terms in the equations of motion. So we may Avrite : ( 742 ) where if^ represents a liornogciieous I'uii'-tioii of degree 4 in 7, and q^. Furthermore we lind : 7' = i ry/ + i ^,M- k P, qr + I\ q, q, + \ I\ q^\ where : ^1 = «i 1i' -f "2 'h '7a + «3 -72% P^ = ^1 1^"' + '^2 ?! ^2 + ^'» '72% P, -'■, '?/ + ';2'7l '?2 -I- '■\'h'- the a's, //s, and c's being constants. Tlie equations of Lagrange become : 1 dl\ ■ dP^ . . '7. + «i' '?! = — -^1 '/. — P, '/2 ~ Y 9^ '1^' — ^ 9i '72 + 1 /I dP, dPA . ^ df/. ^2 + "2' '/2 = — ^'2 -Zl — A <72 + 1 6P, 6PA . eP, . . 1 dP, . dC/, V2 67, dry, y d,y, 2 d,y, dry. In the same way as was done in § J5 we may rephice (j^, q,, q^" , and Yj" in the terms of order A' by others. Now in the first equation a term — <^2(]x<li(]z a-ppears which we must consider separately (in the second equation also there are terms containing q^ q.,, but these are not disturbing). We introduce for this a new variable q\ in such a way that : 1 '/l = 'h + -J «2 '7.' 92 Then we tind : q\ = '7. + -J "•: 9i"' <h + y "2 '72 ('7i '7i + 'Ji') + '^2 '7i 9i '?=' where ^/, and q.. in the terms of order h' may again be simplitied. Of the terms now appearing in the equations of motion tiie following are disturbing: in the first ecpiation those with h''q^, q^', q-^'q^ and q^q^', in the second those with A-y,^, f/,-', 7/ and q^'^q^. Now just as in § 15 the terms with q^q^'' in the first equation, those with q^-q, in the second equation may still be simplified. If we perform these calculations the result proves that the terms of order li" to be inserted in the ecpialious may be put in this form : PWq^ + ''q^'q-i + '''/i'' in the firsi ecpiaiion. Q.^^'qi + fq\' + "''is' M „ second ( 743 ) Here P and Q are homogeneous quadratic functions of \/a^ and j/«j ; and 1 4 1 ./ = — — ni'«2 + 3i,«i"- — Z. (The terms — 3/ in (' and — / in /' originate from the term Iq^'q, appearing in U^). In the terms of higher order we may substitute 3?;, for n^ in tiie coefficients. We then find : ^=3 if_la, + 3&,Vi'-'L /= (-ya, +3Z>,^«/-;. So we find that e = 3/. We may now write the equations of motion : dR dli ?i +«i'?i = ^.J q, + 7i,-q^ = ^— , Oq, where i? = 1 PPy,' + 1 Q/r5,= + fq^'q^ + 1 c^/ + 1 dq,\ So they get the same form as for tlie simple mechanism so that in case a^S' := 4 also the horizontal projection of the point moving over the surface may be regarded as representative point for an arbitrary mechanism with 2 degrees of freedom. ^ 20. So we suppose that the relation exists : "i = «2 + 9, where — is of order ~ . Howe\er, as we have already seen in the cases <S=:3 and *S'^4 in which way such a residue of relation ma}' be taken into account by inserting in the function R a term with Qci„, we restrict ourselves here to the case that the residue of relation is zero, therefore: Mj =; «j ^ n. 50 Proceedings Royal Acad. Amsterdam. Vol. XII. ( 744 ) For the surface tlie lowest point is an umbilical point. To this lielongs as special case the surface of revolution with the Z-axis as axis of revolution, which case is treated by Prof. F-Corteweg at the close of liis treatise quoted before. Omitting th(> terms of higher order than h\ because in the equations of motion we admit no terms of higher order than /<', and omitting the terms of order A', because in the equations of motion no terms of order h' can be disturbing, we may write the equation of the surface : where we avail our.selves of the fact, that by means of a rotation of tiie .system of coordinates round the Z-axis the coeflicients of ,iv/' and ,<■'// may be rendered equal. The solution at first approximation is: l/''o, *• = cos (7lt -\- 2?ijJ,), 71 y =: ^ COS {lit -f- 2n;i,) ; n where n = I '2e, = V^^c,. § "21. Let us now pass to the simplification of the equations of motion. Corresponding to what was said in ^15 for the case 5=: 4 we may here replace in the terms of order h' of the equations of motion : .r' by «, — n^x', 2/' „ «, — ny, .r ,, — n'x, y u — n^y- The equations become : Here we may omit no terms, for all the terms of order h' are disturbing. The ecjuations may be written as follows: X -\- n',?: — -— z= 0, Ox I dR I ( '45 ) wliere we must take + ^. («. + «.) (■''■' + ir) - h ^-'^ + y')'- § 22. If we siibstilute in tlie i'liiiction Z? for .y and y tlie expressions assumed at first approximation and if we retain only those terms not containing t explicitly, we arrive at 1 1 where 4n* "^ 8g^' b = Se, n- c = 3e, n- 4^^^87' f=- 4n^ 4^" „ 3e, <p = 2n (li, - ^J. The system of ditierential equations indicating the time-variability of the «'s and ^'s becomes : — - = — infa^a^ sin ff cos (p -\- 2n/j («j -{- «,) V a^a^ . sin tp — - = -|- infa^u^ sin <p co^ ip — 2w/, (Oj -\- «,) Va^a . sin (f, dt dt coup. = a«j -f 6«, +/«, sj«' (p + -/j ( 3 V\t^a^ + «, I /^ — j cos tp, ^ = ba, + c«, + ./«, sin^ ip + 1/, r«. [^^ + 3 l/«^^ It appears at once from the system that : da, da, —^A -' — 0, dt dt SO a, -|- «2 = coiistant, 50* (21) ( -40 ) Another integral is according to § 4: 1 1 . -— a«i' -f bct^a^ -f — cffj' + /«i«2 «'»" ''f + /, («i + «,) l^«i«, co« (p=const. ^ 25. Tiie results become very intricate for the general case. This is evidently a consequence of the circumstance, that in the function R appear cosip and .sin"*/) or in other words cos (p and cos2(p. The problem is considerably simplified if we suppose/, =: 0, thus f, =: 0, which means, that we suppose the planes XZ and YZ to be planes of symmetry for the surface. Let ns again introduce C, so that «, = B„Vi'l, «, = R^-P (i — r), tlien the last integral may be written in the form : y^Ci — l) cos (f t:= p C^ -\- q ^ -\- r, SO that we can perform again all integrations in finite form, and x and 1/ may then be found as functions of the time. Osculating curves. § 24. We return to the general case and shall proceed to investigate what becomes of the osculating curves. They are ellipses whose equations are found by eliminating t between j/o, X ^= COS (nt -\- "n^^) and cos {nt -\- 2?jj3,) , By changing the origin of time we see that for a definite osculating curve we can also find the equation by elimination of t between and x nr COS nt y =. cos (lit — rp), so tp represents the difference in phase. When </) has an arbitrary value, the ellipse has an arbiti'ary shape and position. If (/) ^ or jr a sti'aight line is described passing through 0. U <f = — the axes of the ellipse lie along the axes of coordinates. { 747 ) The ellipses are described in rectangles having their sides parallel to the axes and whose vertices, as is evident from «j -j- a, ^ constant, lie on the circumference of a circle. To investigate the change in shape and position we may write down the well-known relations which may serve for the calculation of the axes of the ellipse and the angle of inclination of the long axis with the A'-axis. If Ah and Bli are half the larger and half the smaller axis and if 6 is the angle in view, then these relations become : 1 1 n-R„Vi^ A'li' B''h'' «,«, sin- </) 1 _ n' A'B'h* «i«, sitr if (1) 2 ^/«,«, tg W — -^ .costf (3) «, — «. From (1) and (2) we now deduce at once: The sum of the squares of the axes of the ellipse is constant. ^ 25. From what we have just found we can easily prove that in case the surface is a surface of revolution the osculating ellipse has an invariable shape. Then namely we find : — i? z= i a («j + «,)' -f / «j n^ sin' (f, where : 3«, n* 4n' ^ Sfr As and also we find 2n i(j i a («j-|-ff,)' -|- /«, rt, sin' tp =: constant, «i -|- «, =; constant, «i «, sm- if) = co/ulunt. From (2) it then follows that ABh' z= constan', from which in connection^ with the close of § 24 we may con- clude that ( 748 ) Ah ■=: const. , Bh = const., and so our proposition is proved. If in further consideration of the case of the surface of revolution we wish to see in what way 8 varies, we have to write down the differential eqnations giving the variability of the o's and |3's. They now become : := — 4 nf a^ «j sin ip cos (f , \ dt nr i ttf a^ a^ sin (f cos (p , dt di3, — = a («; + it^) -f / «i sin' <p . rf/J, dl3. We see that in — and — an equal constant term a (a^ -f-aj ^ a iZ/ /t" dt dt appears. This means that the fre(]uency n is nioditied l)y an amount of 2'na R/ h\ When we now ditferentiatc according to t the relation tg 16 = cos (p a, —a, we may ari'ive after some reduction at: dd — = - 2 fn' ABh- , dt from which it is evident, that the ellipse revolves with a constant angular velocity. These results agree quantitatively with those found by Prof. KORTEWEG. § 26. The change in shape and position of the osculating curve does not seem to become simple for the general case jz, = »j. Let us therefore restrict ourselves to ,the case 6^ = 0; then the A'^Z-plane and the F/f-plane are planes of symmetry for the surface. The first equation of i;21) now becomes z=z — A: nj «, Kj sin <p cos (p. dt Or by introduction of C : — =: — 4 nj I\^' Ir Q (1 — L.) sin (f . cos (p. dt ( '49 ) The lelation between Z, and r/ becomes: CO.?' ([: ■= (22) Here ^ again varies periodically between a greater and a smaller d: value. Now however ^ may become eqnal to for sin (/ =2 (It and for cos <f = 0. Thus barring special cases there are 3 general cases : !«' . For the extreme values of 'C cav y> = 0. Tlien in the extreme rectangles ellipses are described with the axes along the A'-axis and J'-axis (fig. 13). 2'"i. For the extreme values of w /iin t = 0. In the extreme rectangles straight lines are described ("fig. 14). 3"'. For one of the extreme values of C sin 'f =0, for the other cos '(■ = U. (tig. 15). Special cases. ^ 27. These we have again for the extreme values of the modulus y. {y. has the same form as in § 16) of the elliptic functions ; which occurs when 2 roots of the equation : /■(;) - (;>;'- + q: + r) \^ a - o - (p:^ + ?: + r)] = o have coincided. Tlie special case corresponding to B of § 9 and the second of § 18 occurs here in two ways. We refer to the cases in which the same straight line is continually described (continuallv .v/^ <^^ ^ ; when the surface is surface of revolution, this form of motion is possible in every meridian) and that coutijiually the sauie ellipse is described [cosfp=zO; this becomes for the surface of revolution the uniform motion in a parallel circle). The special case corresponding to A of § 9 and to the first of ^18 exists iiere too. The form of motion approaches asymptotically the motion in a definite ellipse. Envelope of the osculatinij curves. § 28. Two cases may be indicated, in which the envelope assumes a simple shape. 1. For i^ = — 1, 2 = 1 in (22) (the case of a surface of revolution), the envelope lias degenerated into two concentric circles. 2. Vov /; = and g ^ in (22) the envelope has degenerated into two pairs of pai'allel lines, enclosing a rectangle. Arbitrary mechanism ivith 2 degrees of freedom for which S=z 2. ^ 29. The equations of Lagrange get quite the same form here ( 750 ) as for ;S'^4. In the terms of order //' we may in the same way substitute otlier terms for the terms (/,, </,, g^-,and g„^. dl\ .' . dl\ . . Then we have to reduce tiie terms — - — q^ q^ and — - — 7, q.^ . dq, dq, To this end we introduce ^'i and 7'., in sucii a way, that 1 1 ? 1 = ^i + Y «2 'h' '72 + Y «3 '/: I-,' • 1 1 After these reductions it is evident that the terms of order /;' in the first equation assume the form : 1 «, «, ) / 1 \ «, + 2«. q,' + (^Y «^ + ^') ?>' '^^ - K-^^ + <^i) ?: ?.' - Ty ^-^-S^s)?/- 4 We now substitute — q,' for B^/rq^. This is allowed, because 4 substituting q, = Bh aw (?i/ + ?■) in — (//, we obtain besides a term B'li'q^ terms which ai-e non-disturbing. We wish to investigate whether tlie disturbing terms in the two equations are again derivatives of the same function. For this we need not consider tlie terms with ^i and g/, in the first equation and those witii q^ and g/ in the second. The remaining terms become in the first equation : 1 \ , 1 /", 1 «. + ''>) q.' q. ' («. 26, + c.) q, q,- + - I i, + - .1 ,^^\ In the second : So finally we find that the disturbing terms are derivatives of the same function R\ so the equations become: Mi 7, + «^'7. -.— =^0' where i^ = i'A^ ?,' + Q'*' ?,' + ^4 < when P and Q are homogeneous quadratic functions of Vu^ and Vn.^ and when f/, is a homogeneous function of order four of (/, and g,. The results found for the simple mechanism hold therefore for an arbitrary mechanism with two degrees of freedom. ( 751 ) Mathematics. — "Tlie cubic involution 0/ the first rani in the plane." By Dr. W. van der Woude. (Communicated by Prof. P. H. Sf'HOUTE.) 1. If r is a plane it is in different ways possible to arrange the points of V in groups of three in such a way, that an arbi- trary point forms a part of only one group. If 7\ is a point of V there must exist between the coordinates of Pi and those of the other points of the group, to which P^ belongs, some relations by which those other points are entirely determined. It is however possible that P^ can be chosen is such a Avay that one of these relations is identically satisfied ; in that case /■*, forms part of an infinite number of groups. We now start from the following definition : The points of a plane V form a cubic involution of the first rank, when they are conjugate to each other in groups of three in such a loay that (ivith the exception of some defnite points) each point forms a part of onh/ n e group. A triangle of which tlie vertices belong to a selfsame group we call an involution triangle ; each point which is a verlex of more than one, therefore of an infinite number of involution triangles, we call a singular point of the involution ; each point coinciding with one of its conjugate points is called a double point. If one of the sides of an involution triangle rotates around a fixed point, then the third vertex of this triangle will describe a right line or a curve; loe shall restrict ourselves in this investigation to the case, that one vertex of an involution triangle describes a right line, when the opposite side rotates around a. fixed point. 2. When the points of a plane f form a cubic involution of the first rank which satisfies the just mentioned condition and which we shall furtheron indicate by {i^, we can conjugate projectively to each point of V the connecting line of its conjugate points. Each vertex of an involution triangle and its opposite side are pole and polar line with respect to a same conic, which in future we shall always call y. ; each involution triangle is a polar triangle of y^. It is clear that reversely not every polar triangle of y, is an involu- tion triangle of (/,) ; for each point of V is a vertex of an infinite number of polar triangles of y,, but of only one involution triangle. If however S is a singular point of the involution, then »S must be a vertex of an infinite number of involution triangles, thus each polar triangle of y„ having S as \ertex is at the same time an ( 752 ) involution triangle. If we assume a point G of the conic \\ as a vertex of an involution triangle, then one of the other vertices must coincide with G, so 6^ is a double point of the involution; •/,, the locus of these double points, is the double curve of the involution. Each line / whose pole with respect to y^ is no singular point of the involution is a side of only one involution triangle, namely of that triangle having the pole of / as vertex. On the other hand each line whose pole is a singular point is a side of an infinite number of involution triangles all having that point as vertex. From this ensues that also the lines of I' form a cubic involution {i\) of the tirst rank; the polar lines of the singular points of (i,) are the singular lines of {i\), the tangents of y^ are its double lines and y, is its double curve. Both involutions are with respect to y, polarly related. The involution trianylcs of y^ are all polar triangles of a self same conic y,, which is at the same time the double curve of {i^). TJie lines of V form an involution [i' ,) lohich is ivith respect to y, the polar /igure of (i^). Each polar triangle of y^ having a singular point of the involution as vertex is at the same time an involution trian/le. 3. We make a point describe a line a^ and we ask after the locus of its conjugate points. If we draw through A^ , the pole of a^ with respect to y^ , an arbitrary line /)i , then l\ , the pole of 2^1 , lies on a^, whilst the two points conjugate to P^ lie on p, ; these two points lie also on the locus under discussion. Moreover ylj itself is conjugated to two points of «!, so that J, is a double point of this curve and each line through A^ cuts this curve in a double point and two points more. Hence we tind : If one of the vertices of an involution triangle describes a line a^, then, the two others describe a curve u* of order four with a node in A^, the pole of a, ivith respect to y, . As a^ cuts all singular lines, all singtdar jwints tie on «■*. A few properties of this curve a* may still be given here: 1. Let A^ and A, be the points conjugated to ^1,, then the polar line of A, with respect to y, — that is the line A^A, — must cut «■* in Ai and in the points forming with .4, an involution triangle. These two points are A^ and A^. So will a" be touched in A^ by the lines A^A, and A^A^; A^ and A^ are points of intersection of a, and a". 2. Besides in A, and A^ the curve u^ will be intersected in two points more by a^ ; these points are at the same time the points of intersection of r?, and y. . ( ^53 ) 3. Besides in these last points «' will still be cut by y, in 6 points more, the tangents in these (J points to «" must pass throngh A^ . From this ensues that a* is of the tenth class, by which the PlOckkr numbers of «^ are entirely determined (n = 4, m = 10, d^l). Tliis holds, for it is easy to investigate that a* cannot possess a double point differing from /I,. 4. If a vertex of an involution triangle describes a line, on which lies a singular point, the curve described by the two other vertices degenerates into the polar line of that singular point and a curve which must be of order three. If a vertex of an involution triangle describes a singular line s, then one of the other two vertices will be a fixed point, • namely the pole of 5 and the other point will describe a- itself and as many otiier lines as there are singular points on .S-. As both points together must describe a curve of order four, three singular points will lie on s. In like manner each singular point is point of intersection of three singular lines. If now again a, is an arbitrary line and if a^ has the same signi- fication as above, then the curve a" will cut a line b,^ four times; from this ensues that four times a point of a^ and a point of bi are vertices of a selfsame involution triangle. Tlies3 vertices we call J^i, Qi , iii,>'^i f-iid J\ , Q^,B.,,S.^, whilst the third vertices of these triangles may be represented by J\ , Q,, R,, S^ ; fiirthermore 2\ is the point of intersection of «, and /;, and T.^ and 7\ are the two points forming with 7\ an involution triangle. If now a point describes the line b^, then its conjugate points describe a curve ii' of order four; «" and [i^ have 16 points of inter- section. These are: 1. the two points 2\ and J",; 2. the four points P,,Q^,R^,Sg; 3. ten points moi'e having the property that to each of them two, so an infinite number of pairs of points, are conjugated and which are thus the singular points. Therefore : The involution (ij) has 10 singular points; their polar lines are the 10 singular lines of {i\). These singular elements have such a position that on each of these lines three of these points lie and that in each of the points three of the lines intersect each other; so they form a configuration (103,103). If Sjj is a singular line and S^^ its pole with respect to y^ , then there are besides aS'^ still 6 singular points not lying on s^^ . If »S,, is one of these points and s^^ the polar line of ,§,,, then the point of intersection of s^^ and .s-,, is at the same time the pole of aSu-S'i,. ( 754 ) This point forms an involution triangle with »Si, and with another point of *,2 and an other one with S^, and with a point of .f,, (an "other one", as /S,, and 5j, which do not lie on each other's polar line cannot be vertices of a selfsame involution triangle) ; so the point of intersection of .s\, and *■,, is also a singular point and >S'i,.S'i, a singular line. Each line connecting two singular points not lying on each other's folar line is a singular line ; each iwint lohich is the point of inter- section of two singular lines not passing through each other's pole is a singular point. On 5i3, the polar line of S^„, lie 3 singular points; the remaining 6 are connected with *S,j by 3 singular lines. So each line connecting >Si, with one of the singular points on .s-,. is not a singular line, as only 3 of these lines pass through 5,,. We can indicate the position of the singular points by the following diagram, where the indices have been chosen in such a way that always the points Sik , Sk-i and Su lie on a selfsame line, that the lines Sii; , si^ and sn intersect each other in a selfsame point, and that the point Sik find the line .y,/- are each other's pole and polar line with respect to y, . 5. We make a point describe a conic «, and an other point a line b^ the two points which are conjugated to the former describe a curve «", those which are conjugated to the latter a curve ^*. As ii* and «j intersect each other in 8 points, i, and «" must have 8 points in common, so «" is a curve of order eight ; we shall call it in future «*. As «, intersects all singular lines twice, o" will have in each of the 10 singular points a node. If «5 is described around an involution triangle, then a^ has also Rouble points in the vertices of lliis triangle. As ail involution ( 7.^5 ) .riangles are at the same time polar triangles of a selfsame conic y,, we can describe a conic around each pair of involution triangles ; if a conic i?^ is described around two of these triangles, then tlie curve /i' conjugate to it will have 6 nodes in its circumference. Also the remaining points of intersection of ji„ and /i* are easily indicated ; they are the four points of intersection of ;Jj and y, . We know moreover that a conic described aroimd an invohuion triangle and through two of the vertices of an other involution triangle must also contain the third vertex of the latter. 6. It is also clear, that we can easily construct conies described around three involution triangles ; to that end we make a conic, pass through the vertices of an arbitrary involution triangle and through two singular points not lying on each other's polar line ; for this we choose S^, and S^^. As «, is described around a polar triangle of y^, it is described around an infinite number of these triangles; further each polar triangle of y, having one of the singular points as vertex is at the same time an involution triangle, so that «, is described around three involution triangles. Now the curve a" will have in the circumference of «, nine nodes ; so it must degenerate and a^ must be one of the parts into which it breaks up. If F^ is an arbitrary point of (t^ tiien always one of tlie two points P, and F, forming with F^ an involution triangle will also lie on «,, so also the third vertex lies on «, (5). If now we let Pi describe the conic «,, then P, and P, will describe the same curve; every time however that P, coincides with one of the singular points on «,, F, and P, will be bound to no other condition, than that they must lie on the polar line of that point and must form with F^ a polar triangle of y,. So the parts into which «' degenerates are: 1. the conic «, to be counted double; 2. as many lines as there are singular points lying on «,. From this ensues that besides Si^ and »?!, 2 more singular points lie on «j. This last we can prove still in another way ; we construct a second conic ^,, described around an involution triangle Q^ Q^ Q, and through ,S',j and >Si,; it will cut «, in two points more, which being both the vertices of two, i.e. of an infinite number of involution triangles, are therefore singular points. If we construct anotiier conic d, described around a triangle of involution R^ R, R, and tlirough Si, and *Si,, then this must still cut a^ in two singular points ; these ( 756 ) must he the same as the points of intersection of (i„ and ,1, because on a, no more than four singular points can lie. So all conies passing through S^^ and S^, and farther more described around 07ie, hence around an infinite number of involution triangles will form a pencil; the two other base points of this pencil are also singular points. We determine these first : if we choose as (3, the pair of lines ^S^^ and 535 and as if.^ the pair S,t and S^^, it is evident that Sn and S^^ are the discussed base points. Tiierefore: // the 10 singular points, hence also the double curve y,, of the involution are hioivn, we can generate the involution triamjles in this ivay. We can construct five different pencils of couics of ivhich each conic is described around an infinite number of 2)olartriangles of y^, which are then at the same time the involution triangles in view, the base points of these pencils consist of the sets of points [S^,, Sn., Sn, Sii), (012. 'Sjs, O24, (S'jJ, ((Sij, O33, «Ss4, O35), (514, »S24, O34, O45) and (Ois, 0,5, Ojs, »S 4s). These pencils we shall call in future respectively [B^), {Bj, (B,), (B,) and (B,). If «i and Oj are two conies, the first taken arbitrarily out of(i),), the second arbitrarily out of [B,], these two will have four points of intersection, viz. S^^ and the vertices of an involution triangle. Now it can happen in two diflerent ways that 2 of these points of intersection coincide: 1. S^. can be at the same time a vertex of the involution triangle 2. one of these vertices can lie on the double curve y, I" each of these two cases a^ and a^ will have only three different points in common, but they will touch each other moreover in one of these points. 7. Out of these 5 pencils we choose one — e.g. (i?J — arbitra- rily; an arbitrary conic cfj out of (-Sj is described around an infinite number of involution triangles whose vertices form in its circum- ference an involution of order three. The latter has four double points in the points of intersection of f/, with y„, the double curve of the involution ii^). Inversely the conies of the pencil (i^J determine an involution of order four on y, ; the latter has 6 double points in the points in which y^ is touched by a conic out of {B^). In each of these points three points have thus coincided, forming together a group of (i,). The involution (Z,) has 6 triple points ; in each of the points y, is touched by a conic out of each of the pencils (-Z>'i), (i?,), {B,), {B^), and [B,]. ( 757 ) 8. A point whose conjugate points coincide wc call lx branch poi7it, the locus of these points the hrnncli curve. If we let a point G describe the conic •/„, then the curve of order eight, generated bj the points conjugate lo (r, must degenerate into 2 parts, of which one is y^ itself and the other the branch curve. From this ensues that the latter is of order six and possesses nodes in the 10 singular points ; so it is rational as it should be, as it corresponds point for point to a conic. Also in an other waj we can easily deduce the order of the branch curve; if a point describes a line aj, then the conjugate points describe a curve «■* having with y^ eight points of intei'Section, of which two coincide witii the points of intersection of a-^ and y,, whilst the others point to 6 points of intersection of a, with the branch curve. If (t^^_ is a point of the double curve y„ and (j the tangent in that point to y,, then g will intersect the branch curve in 6 points of which one G^ forms witii the double point G'l, a group of conjugate points; so in the triple points of the involution y^ and the branch curve will have to touch each other. The branch curve is a rational curve of order six, having double points in the singular points and touching the doidile curve in the triple points of the involation. Observation. A rational cui've of order six has 10 double points ; of which howex'er only S can be taken arbitrarily ') ; from (he pre- ceding follows hotvever that 10 points determining a Cf (10,, 10,) can always be double points of a rational curve of order six. In an other form C. F. Geiser (see his paper quoted in the fol- lowing number) makes the same observation. 9. We shall now apply the preceding to some problems out of Threedimensional Geometry. To that end we regard the pencil {B) of twisted cubics which can be brought through 5 fixed points P^, P., P,, P^, and P5. These determine on an arbitrary plane V a cubic involution of rank one; the lines P; Pj cut Fin the sin- gular points Sij, the planes Pk Pi Pm cut V in the singular lines Sij of the involution. Tiirough an arbitrary point of V passes only one curve out of this pencil, through a singular point S,j however pass an infinite number of curves, whidi ha\'e all degenerated into the fixed line I'iPj and a variable conic; these conies form a pencil with Pk, P , Pm and the point of intersection of Pi Pj with the plane Pk Pi Pm as base points. Each double point of the involution in V is now a point, in which a twisted curve out of the pencil (B) 1) Salmon-Fiedler : Hohere ebene Kurven, Zweite Auflage, p. 42. ( 'oS ) touches the plane V; the third point of intersection of this curve with V is a point of the branch curve forming with the point of contact a group of mutually conjugate points of the involution. A triple point of the involution is a point, in which a twisted curve out of [B) is osculated by V. From this ensues : 1. All twisted cuhics passing through 5 given points and touching a given plane V form a surface F^" of order ten, lohich touches V in a conic and cuts V moreover according to a rational curve of order si^r. 2. There are 6 tioisted cubics passing through five given points and having a given plane as osculating plane. As a special case of this last theorem we have still : dirough five given points pass six twisted parabolae. Through the pencil (B) of twisted cubics with P^, P^, i',, P^ and P^ as base points a plane T" is cut according to a cubic involution of the first rank. If « is a curve out of this pencil cutting V in A-i, A^ and ^4j, then « is projected out of A^ by a cone cutting V accoi'ding to the lines A^ A, and A^ A^. If however a curve y out of {B) touches a plane 1' in a point Ctj, and if moreover it cuts F in a point G^, then y is projected out of G^^ by a cone cutting V according to (r,, G^ and the tangent in G^^ to y; y is projected out of Cr's by a cone touching V according to (t, G^^. We have seen that G^^ must lie on the double curve and G^ on the branch curve of the involution, whilst G^ G^, touches the former; if thei-e- fore a quadratic cone is to pass through the base points of the pencil (B) and to touch V moreover, then its vertex must lie on the brancli curve and the tangent with V must touch the double curve. The number of quadratic cones jjas-sw?// through five given points and touching a given plane is singly iyifinite ; the tangents envelope a conic. The vertices of the cones form a rational ciwve of order six.^) The tangential planes of all these cones whose number is oo" envelope a surface of which we wish to determine the class and which for the present we will call *„. If /v, is one of these cones and 6^3 its vertex, then througli a line / drawn in V through (r, one more tangential plane to /v, will pass ; as / has with the branch curve 6 points of intersection, it lies still in 6 tangential planes of '/»„ except in I". Farthermore )' is a trope of */>„ (that is a tangential plane touching (y,) in the points of a conic) to be counted double; the surface </'„ is therefore of class eight. The tangential planes of these cones envelope a surface of class eight ') 1) G. F. Geiser: "Uber Systeme vou Kegeln zweilen Grades". ( '5^ ) We finallv put the (iiiestioii liow many twisted circles can be brouglit tlirougii live points wiiere we understand by a twisted circle a twisted cubic cutting the isotropic circle in two points. All twisted cubics through these five points describe on the plane at infinity an involution ; if now a point describes the isotropic circle, its conjugate points will describe a curve of order eight having with this circle sixteen points in common; four of these points are at the same time double points of the involution, whilst the other lie two by two on a same twisted cii'cle. So tkrough jive yiven points pass ten tivlsted circles, of trhlvh four touch the plane at infinity. Mathematics. — "On the surfaces the asymptotic lines of which can be determined by quadratures" . By J. Bruin. (Com- municated by Prof. Hk. de Vries). In a paper entitled as above A. Buhl {N'oiw. Ann. de Math., 4-= serie, vol.8, page 433, vol.9, page 337, fl«'. A-m. XVII 2, page 62, XYIII 1, page 58j discusses the surfaces given by the parameter representation *• = r cos 6, y ^^ r sin 6, ^{z) = ae^F (r), in which d;y,: refer to a rectangular system of coordinates, so that 2, &, and r are the so-called cylindric coordinates; these are the only ones which are used in the course of the investigation. Buhl now gives the differential equation of the asymptotic lines of (p {:) = a -\- F [r] with (9 and r as independent variables as well as with : and &. It is then evident that this equation embraces many special cases, where the determination of the asymptotic lines comes to quadratures. We can put the question more in general : which are the surfaces of one of the forms z = ff{r,S), or 6=zf{r,z), or r=f{s,&), whose asymptotic lines can be determined by quadratures? Starting from the differential equation of the asymptotic lines D du^ + 2 Z>' du dv + D" dv- = (BiANCHi-LuKAT, "Vorlesungen iiber Differentialgeometrie", page 109), where D, D' and D" have the values, to be found on page 87 of the quoted work, we find for the differential equation in r and 6 of the asymptotic lines of ~ ^ 'f {r, 6) ; 51 Proceedings Royal Acad. Amsterdam. Vol. XII. ( 760 ) dr- V ^^'^^ ^^J V ^^' ">'y TIlis equation gives I'ise to quadratures in the following cases: ,/. : = a (9 +/('•), Bi'HL, 1. c, vol. 8, p. 439, coinp. also Tisserand, "Rec. compl. d'exercices", p. 426. h. l(:)=zad -\-f(:r), Buhl, I.e. p. 440. e. s —Ar dn {0 + a) + ad + F{r), d _z=rJ\{8)-\-fA0), g. z = -&'-^2l(r). 2. The differential equation of the asymptotic lines of dT=f{r,z) in /■ and z we find by eliminating & between fl) and 6 z:^f{r,z). We find: dr'^ dr \drj \ j dzdr ^ ydrjdz^dz\ ^ ' (dr' ^ dr Vd~V i Tills equation gives rise to quadratures in the following cases: a. e = l{r)^f{z), k h. 8 = arc cos \-f(:h Buhl, I.e., vol.9, p. 343, r besides a few others mentioned above. 3. In an analogous way we find the differential equation of the asymptotie lines in z and 6^ of r :=/(;, 6*). It runs: dV i dY 1 df d/i idY 2 rdfX) ^ rfc - -J- 2 — ~ ■'- ^} ihd/9 + \-~ — f I -- da' =0. dc= ' \d:dd f da dz\ ^ \d&' •' f\dOj\ Besides in the above menlioned eases this equation gives rise to quadratures for r ^ /\ (z) f„ {&), surfaces of Jamrt (Ann. de I'ecole norm, sup., 1887, Suppl., page 50 etc.; further: Picard, "Traite d'analyse" I, 2'"^ ed., i)age 433). The classes of sui'faees fouiui above are not strictly separated; some even ai'C to be regarded as subclasses of others. Tiicy can be ranked according to the most general types to be found among them, whilst others fall under these tyi)es, namely as follows: ( 761 ) s =Ar sin {i9 + a) -\- ad -\- F{r) z—ndAr ^ {r) z ■= r^ sin {d\/k -\- c) z =r r—^ e'^l^^+c 5 = - <?' + -^Ar) h 6 = arc cos 1- f{z). r II. Let us discuss oue of the above mentioned classes more closely, viz. ^=r;\(^) +/,{&)') (2) It is the general e(iua(ion of the scrolls with the >axis as directrix. Of these scrolls we can find the striction line in the following way. BiANXHi (I.e. p. 223) deduces that the curvature A'^ — , for which Z)j)"—D" in another place was tound A'=r , is larger in the central point than in all other points of a generatrix. If we make up A' for (2) we lind ^.^ - (//)' Along a generatrix 6 is constant; there only the denominator of the expression for A' changes. If we determine the value of ;■ for which K becomes maximal we find r= -^ (3) So this is the {r,6) projection of the striction-Une. This equation can be found in an other way, too. We have the properly that the tangential plane in the central point of a generatrix is normal to the tangential plane in the point at infinity of that generatrix. We now determine for /=: — ~ -\- >'fi{^) -{- fzi'^) the ()/■ bf df values of ^ ' ^ ' ^ i" ^" arbitrary point p and in the point at infinity on the same generatrix. If then p is to be central point the sum of the products I y I •1^1 ii^ust be equal to 0. This gives again the equation (3). *) In future we shall write /"i and /"o for /i(fij and /".(S). 51* ( 702 ) Let lis now consider in liow far we can find tiie surface (2) for a given striction line. a Let r = 7 [6) be tlie given projection of the striction line. Tins furnishes, regarding (3), the relation between f\ and J\ -. ji 1 + u\f + u:y ./v Thus: The surface ; ^ r/, (^) + | - '^ '^ ^ ^^"^ + ^-^^ ^ ' dd, where /'i is an arbitrary function of 6, has as (r, ^j-prqjectioii of tlie striction line the curve r^^-f'iS). b. Let now be given tliat the striction line must be a plane curve lying in the plane z =z Ax -\- By ^ C ov :^=t'{AccsS-\-J3dn6)-\- C. By substituting this value for : in (2) we must get (3). This furnishes between /\ and /\ the relation ■/; - c ^ yv./V /; — A cos 6 — B sin 6 1 J- (/, )■-■ + (//," or /; ^ 1 + {f.r + (.AT /; — C ./;' (/, — A cos d — B sin 6) r I + ^a;^ +(/■')' JO So the surface 2 = »/, {&) +6'"^ ■^''^■^' " -' "" ' " ^' "" '^ + C, in which /i is an arbitrary function of 6, has a plane striction line lying in 1- = Ax + % + C. c. The most general jiroblem here is : what is the surface (2) for which r = <p{l9), ^=rtp(6') is striction line? To solve this we substitute these values for ?• and z in (2) and (3) and we obtaiu then two equations with /\ and f, as unknown quantities. If we eliminate between these two /\, we retain /^ as only unknown quantity in the equation : r + 'fiAY +f:^P' -'f'AA' = ^^ U /"j is solved out of this we can find /\ out of: n' = 'fj\+A- We can find the solution in explicit form for the special case xp = comt., for which constant we take 0. We then find : from which ensues ( 763 ) whilst further J\ = - 'f.t\ The result is therefore : The surface has the plane striction line /■ = (f, : ^=0. The formulae deduced altove hold for the surfaces (2). As was noticed the general types mentioned at the conclusion of 1 are not strictly separated, however, so that there are still amongst them scrolls with a right directrix, to which then the above formulae are applicable. Examples of this are: Of the type z — Ar sin [6 + n) + nd + F{r) the surface ; = Ar sin [6 -}- «) + a^ + ^»'- Of the type /(.-) = a(9 + /(?•) the surface /(;) = aS + /(/• + p) ; these have the c-axis as directrix. Of the type r=j\{z)J\{d) the scrolls .x-(z + l)r=^"^, or r = {t,jd)\ and m" .c' = (^ - c.k)2'' or r = sec 6 (lqd—cY~'^ c"-' have still the y-axis as directrix. Physics. — ".4 new tlworii of llw phi'norrenon. a//o>ri)j)i/." By Pi'of. A. S.MiTs. (Communicated by Prof. A. F. Hoi,!. i:\i.vn.) Inirod action. In two short communications inserted in the "Chemisch Weokblad" 7, 79 and 155 (J9J0) I have already sketched the main lines of the theory, an extension and experimental coulii'mation of which follow here. Before passing on to this I may, however, be allowed to give the gist of this theory in a few words. Ill the investigation of the phenomenon tautomcrism it has been possible to show by means of the process of solidification that the liquid phases of tautomeric substance; are composed of two kinds of molecules. ( 764 ) Besicies, however, bv de|)Ositi()n of dilierent solid snlistances the ooiuplexity of a li(|iiiiJ pliubo {'an also l>c shown in another way, and investigations in this direction have led to tlie resnlt tliat it may be considered as the rnie liiat tlie liquid phase of a substance is built up of ditfereut kinds of molecules (ions included). Bancroft') and Bakhlis Roozehoom -) have pointed out that when a substance behaves as a unary sul)stance, this is accounted for by the fact that the setting in of inner equilibrium takes place so rapidly in the homogeneous phase that the inner equilibrium if disturbed, is immediately restoretl by the appearance or disappearance of a new phase; the melting-|)oint, boiling-point, critical-point etc. of a substance which beliaves as a unary one, does not i-elate then to a single kind of molecules, but to an equilibrium between ditjercnt kinds of molecules. Bancroft's pupils, viz. Carveth, Soch, and Cameron "i have inves- tigated dilierent tautomeric substances ; it then appeared (hat it may be pretty easily shown in some cases that under certain circumstances the existence of two kinds of molecules in the liquid phase may lead to a binary behaviour, for when the liquid cooled r/ipiiHi/, the inner equilibrium coidd not follow the temperature, and the mass solidified at a ternjierature which tlilTered from (he unary stable melting-point, for then a point wiis realized of one of the melting- point lines of the pseudo-binary system ^4 -|- B, which for the examined substances always showed a eutectic point. As is evident we find the unary stable melting-point where the curve for the inner liquid-equilibrium meets one of the melling-point curves of the pseudo-binary system. Now it is remarkable, as 1 already wrote, that nobody has observed what surprising I'esults are arrived at when it is assumed, what is undoubtedly true, that not only mixeil crystals are always formed in a greater or less degree, but that moreover the I'luwr etpnlibrium, which exists in the liipiid phase, continues to exist in the solid phase. Starting from this supposition we get the relation between hetero- geneous and homogeneous allolro|»y, indicated in Fig. 1, from which it appears that the phenomenon of enantiotropy means uniniving in the solid state, which phenomenon a|)[)ears when the cui'vcs for 1) Jouin. Phys. Chem. 2, 143 (189.<). 2) Zeitschr. f. phys. Chem. 28, 289, (1899). 3) Journ. Phys. Clieui. 2, 159 (181)8). ibkl. 2, 3G4. (1898). ibid. 2, 409 „ the stable and metastable solid equilibria ^^_q and s^n meet the mixed crystal lines ep and din of the pseudo-binary system. In case of nionotropy these meetings between ihe iiiiaiy and the pseudo-binary system do not take place under but (ibar,- I lie unary melling-point temperalure, and lliis is the reason that in tiiis case the second line for the solid inner e(|uililtiia everywhere indicates metastable states. I started from Giubs' principle of equilibrium, which slates that with constant temperature and pressure a number of substances arranges itself in such a way that the thermodynamic potential is a minimum; and then I showed how sharply tiie relation between the pseudo-binary and the unary system can be defined also in this way, when we bear in mind tiiat a state of inner eqnililirinm must always lie in tiie miiunuim of a iiolenlial line. Further the case was considered that the liuee-piiase lenq)crature lies between tiie melting points of the substances A and li. After having discussed tiie phenomena of enantiotropy and inonoli'opy also for lliis case, 1 finally pointed out tiiat when A and B are niiscible ill all |)r()p()rtions in the solid state, Ac^t^/w/'^/'^'t'^.v r///((//v)//// is excluded, and oidy homogeneous allotropy can occur, mdcss iiiiiiii.xiiig orcurs in tlie pseiulo-binary system at lower temperature. Di-icussion of the cun'es of inner equUiln'lam. After this introduction which seemed indispciisaiile to me to make tiie reader ac(|uainled witii tiie main facts, I will consider fig. 1 a little more ciosel}' and discuss the cur\es of inner ecjuilibriiim. U follows from the course of the curve S\S\ that it is assumed here that over the corresponding range of temperatnie the e(|uilil)riui!i shifts towards H with increase of temperature, and so that A:^B — a ml or in words that the transformation from the lefl to the right is endothermic. With ap[ilicati(ni of the equation: dlnK __ Q we know therefore that Q is positive in the assumed case. Neglecting tht; external work we can split up Q into two dilferen- tial h(!ats of mixing, one of which iias the negative sign, because it is a case of unmixing, and further into a heat of transforina- •-ion, SO: ( 766 ) Q = - {Q,„)a + Qr + {Q.„)iJ {Q,n)A = differential heat of mixing of A {Q,„)B= „ „ „ „ „ B Q,. = moi. heat of transformation. It is of importance to point out here that as {Qm) a ^nd {Q,„) a have a different sign, tlie possibility exists tiiat Q has another sign than Q,-; this might e.g. occur when Qr was very small, and tiien we siiould have the special case that e.g. when Q was negative and Q positive, the eqnilibi-inm shifted from B to A with rise of tem- perature, whereas the transformation of A into B is endothermic in itself; this, however, will onlj rarely occni'. If we drop this (piestion for the present, it is noteworthy that in the point S\ unmixing occurs, another solid phase S\ appearing by the side of <S"i. Two cases may be distinguished here. Generally the newly-formed solid phase S' ., will possess another form of crystal than S\ , but it is possible that the two solid phases are isomorphous, for as is known, also isomorphous substances can show partial miscibilily ; if tiiis latter, the simplest case occurs, the heat of transformation will be the sum of a heat of unmixing, a heat of transformation, and a heat of mixing M, another thermal quantity being added to this, viz. that which accompanies the change of crystalline form, when /S''i and S\ are not isomorphous. If we now follow the inner equilibria above the transition-point, it is to be expected that the curves S\ q for the solid-, and l^ k for the liquid inner equilibria will have the same direction as S\ S\ , as is also assumed in fig. 1. SocH, however, has found in liis investigation of ^eHc//6'-c'r//jt)t'a/-/>c)«/c acid that the curve of the inner liquid equilibrium meets the melting- point curve of the modification with the highest melting-point viz. B, and runs to the /l-side for higher temperatures. Further he found that at 65° A passes into B, and combining these two fads, he arrives at the conclusion that the thermal sign of the transformation .4 -^ B must have been reversed between the point of transition and the unary melting-point (137°). When the pseudo-binary T,.(;-iigure for this substance agrees with fig. 1, which is still an open question, we must of course come to the same result also going by this theory, but I will point out here that this conclusion is not yet imperative at this moment, because though it is not probable, the possibility exists that the mixed crystal '■) 1 shall discuss tliis ami the berorc-mcntioncd splitliug up moi'e I'ully later on. ( 767) curve dm of the pseudo-binary system has the same direction as ep; in tliis case tiie three curves of equilibrium aS', 5'i , (S'^ »S', and /,^- might still liave the same direction, and so the sign of Q need not be reversed. If we look upon tiie question of the reversal of the thermal sign from a general point of view, the following may already be remarked. When A and B are isomers, as for benzile-orthocarbonic acid, a reversal of the sign (if Q seems possible, because Qr is probably small in this case'). If, however, we have to do with the pheno- menon polymerism, we may expect with great probability that Qr will ahvaj'S predominate, and that the curves for the inner solid and liquid equilibria will always run in such a way that the equilibrium shifts towards the side of the less complex substance with I'ise of temperature. This leads us at the same time to the question what the 7\.v- figure will be for the case that the substance B is a polymer of A, and that a transition point exists. Fig. 2 shows that when the pseudo-binary system possesses a eutectic point, the curve for the inner liquid equilibria must meet the melting-point curve of the less complex substance, becau.se only in these circumstances all the curves for the innei' equilibria can run to the ^4-side with rise of temperature. Yet this figure will not appear to be quite correct either, in my opinion, as a supposition is implied in it, which is highly improbable. When B is a polymer of A, and the pseudo-binary system pos- sesses a eutectic point, this means that there are liquids (a c) which contain more polymer than the coexisting solid phases {ad), and this is very improbable, so much so that we may disregard this figure altogether, in spite of Hoi.lmaxn's ") assertion that he has found a eutectic point for the system acelaldehyde-pai'aldehyde. Probably this assertion of Hollmann's rests on not quite reliable observations, for my assistant, Mr. de Leeuw, who tested the said assertion at my request, has not found it confirmed. So for the case that _B is a polymer of A and the two substances are not miscible in all proportions in the .solid state, we must conclude to the existence of a I'f-figure as indicated by fig. 3, in which the 1) In consequence of llie considerable displacement of the inner equilibrium at the ti-ansition temperature it is possible, that while Qr predominates below thi:: temperature, abore it the reverse takes place. Qr, too, can reverse its sign, but this seems less probable to me. ■-) Zeitsclir. f. phys, Chem. 43, 129 (_i903j. ( 768 ) three-phase-temperature lies between the mehing-points of the pseudo- components. Now on this assumption, the soUd phase possesses everywhere more of tlie polymer B than the coexisting liquid phase, and if in tiie unary system a transition point oc(;urs, the course of the curves of inner equilibrium must be as indicated by kl^, S,S'., and S^'S',. If the cur\e kl, met the melting-point line of JB, monijtropy alone would be possible, as for enantiotropy reversal of the thermal sign would have to take place in this case, which is very improbable here. Experimental conpnnatiun. It is clear that this theory I'equires that every sui)stances whicli shows a transition point, must consist of two different kinds of mole- cules, which are in equilibrium at every temperature. So if we consider the substance HgJ.,, the red modification oi which passes into the yellom one at 127'', we must assume two dilferent kinds of molecides, the former of which gives rise to the formation of red, and the other to that of yellow HgJ„. Tiie investigation of this substance, which was carried out in collaboration with Mr. S. C. Bokhorst chem. cand. has led to a very ren'iarkable result. That it would appear that working quickly, the substance would betray its binary character, was expected, but that we shoukl find here that case whicli I aUeady mentioned, but considered as an exception, was highly surprising. For the sake of clearness the observed phenomena wUl be dis- cussed here in connection with the schematic lig. 4, in wiiicii « means yellow and (i red HgJj. At 'J 27"' the I'ed [)hase passes into llie yellow one, which new phase remains intensely yellow up to about JSO'; on further heating we observed that this phase assumed a red colour, at first hardly perceptibly, but then more and more pronounced, and that it becomes a dark red liquid at the melting-point temperature 255°,4. This phenomenon, which also appeared with \ery slow rise of the temperature, was studied in different ways with the naked eye and l)y means of the microscope, when it appeared that this change of colour takes place continuously, and is not owing to a second transition-point. This continuous change of colour between comparatively narrow limits of temiierature made it therefore probable that above the point of transition the curve for the solid inner equilibria at lirst ( 7.;y ) runs vertically upwards, after which it bends sharply to the red side, and meets the mixed-crystal curxe of the pseudo-binary system jiear the axis of the red niodificalioji. As therefore, this inner equilibrium curve appears to traverse the 7-.t'-tigure over a larue concenlraiion range, this pointed already to a region of partial-miscibilily in the pseudo-binary figure, which was closed at tiie top, and so also to a continuous mixed crystal curve acb. In order to test this supposition more closely, the following ex- periments were made with HgJ.,, eithei- in thin-walled narrow capil- laries or in so-called alcaloid tubes; it was, namely, quite immaterial which of these were taken, for in either case the experiment yielded the same result. In these tubes the HgJ.j was heated in a melting-apparatus up to a certain tem[)erature nbove llie transilioii-point, and then all at once transferred to an oil-bath of lower tenijierature, but always above the transition point. The considerations which led us lo these experiments, were the following. If it is possible to make the cooling take place so rapidly that the inner equilibrium cannot keep j.ace with the temperature, the pseudo-binary character must appear, and entering the region of partial-miscibility the substance nuist split up into two phases. Suppose that we start from the inner equilibrium p and that we cool this suddenly, in which not the curve of equilibrium, but the curve jiS^ is followed ; then the red phase S^ will ap|)ear by the side of the yellow phase /b'j and will have lo be clearly visible. This three-phase system will be strongly metastable, so that it is not to be expected that it will be very permanent; on the contrary, we may confidently predict tiiat this state will very soon change into the only stable equilibriuui \\liicli must lie on the cur\e SS^. If we now start from the inner equilibrium q, which lies on tlie right of the critical mixing-point A', the mixing-curx'e can be reached in .Sj , and by the side of phase ,V. , the phase ,S', must be found, which has a lighter colour. As appears from the subjoined table (p. 770) not only these phenomena could be observed with great clearness, but moreover it was ascer- tained by these preliminary experiments that the mixing-point K must lie above 147'' '). Though it follows from these experiments that abo\e the transition temperature the ?',.r-figure of the system HgJ., would be as indicated ^) This investigalion is continued iu ditTereut directions. ( 770 ) Temp. HgJj Suddenly cooled down to the temp. Remarks 200° 130° No unmixing as yet. 205° " Unmixing, red phase appears, but has disappeared again after a few seconds and the whole mass is again yellow. 207° „ „ 210° „ „ 212° » ,, 215° „ „ 225^ " Unmixing, but now yellow phase ap- pears and after a few seconds everything is yellow. 230° " The same phenomenon, and still more pronounced. 212° ■140^ Unmixing red phase appears etc. 212° 145° „ 212° 147° No unmixing is to be observed. liere, the question what the rest of the figure, i.e. under the transi- tion point, would look like, remained unanswered. The answer to this question cannot jet be given in this communication, because the equilibrium sets in exceedingly slowlv at temperatures under 100°.^) So the dotted ciu'ves under the transition temperature do not re- present anything i)ut a supposition. For the end in view here, howe\er, the want of certainty helow the transition temperature is of minor importance, as the plienomena observed at higher tempe- ratures furnish a convincing proof for the validity of the theory. Before I leave the substance HgJ., and pi-oceed to another subject, I will only point out, that if the equilibria are considered not at constant pressure, but at the variable va|)Our-pressure, al.so the vapour- curves should be inserted in the 7',.''-figure, which lie on the side 1) If a tube with red HgJo is iramerged in liquid air, the colour becomes indeed mucli lighter viz. orange, but this change of colour has nothing to do with a displacement of the equilibrium. If a mixture of yellow and red Hg.l^ is lal^en, and this is cooled down to — 190'", the yellow colour changes into white, and the red into orange-yellow. When heated to the temperature of the room the heterogeneous mass is found to be entirely unchanged corapured with the initial state. ( 771 ) of yellow ngJ„, because the f/cHoio phase is always deposited from the vapour. If we now consider the question whether the literature mentions results in support of this theory, the answer is affirmative. These are chiefly the results obtained in the investigation of sulphur^) and that of phosphorus-). In the system sulphur we have two ditferent crystalline modifica- tions, and besides them a third modification Sn, which has not yet been obtained in crystalline form. Considered in the light of this theory we must therefore assume three different kinds of molecules, and sulphur being known as a substance which is very slow, we can assume with great probability that sulphur is not pseudo-binary, but pseudo-ternary, i.e. will behave as a ternary system. This, however, be only remarked in passing, as these considera- tions are of no further importance for what follows. If we now direct our attention to the 7',,r-figure of the system Sfi and rhombic sulphur S/- (Fig. 5), it is noteworthy that by extra- polation 110°,6 has been found for the unary melting-point, and 112°, 8 for the melting-point of pure rhombic sulphur. It further appeared, however, that when from rhombic aS was started from, where the equilibrium had set in at 90\ a melting-point was found at 110'. 9, the melting-point amounting to 111°.4 when the inner equilibrium had set in at ± 65°. These are results which support the theory given here, for they point to ihe fact that we have to do here with a curve SS^ for the solid inner equilibrium, which runs to the left with rise of tempe- rature. For this curve shows that as we, working quickly, start from an inner equilibrium established at lotver temperature, this phase will begin to melt at a higher fempei-ature, which was also observed here. The curve for the inner liquid equilibrium, too, runs to the left, so that the two curves of equilibrium have the same direction. Though the sulphur can furnish further proofs, we now proceed to the phosphorus. As Cohen and Olie already mentioned, investigations of Troost and Hautefeuii,i-e, Lemoine, Hittorf, and themselves point to the 1) Kruyt, Z. f. phys. Gliem. 64, 513 (1908). "-) Cohen and Olie, Chem. weekblad 6, 821 (1909). ( 772 ) fact llial for pliospliorus we li;v\e lo ilo will) sulid inner ecinilibi'ia between white and \iulet pliosplioi'us. Tf we consider the folkiwing resnUs of the delei-niinations of the specific gravity : spec. grav. of red 1' obtained al 550' = 2,25 „ „ „ „ „ 450^ = 2,28 „ „ „ „ „ 357° = 2,22 „ ,, » „ „ 255° = 2,20 „ „ „ „ „ 215° = 2,19 we slionld, in view of tiie fact tiiat tlie spec. grav. of white i'^ 1,82, and that of vioiel P may be put at about 2,34, come to tiie con- clu.sion that tiie carve for the inner .solid equilibria runs to the \'ioiet side with rise of teniperatni'e to 450^. As it, however, followed from the e.xperiments of Cohen and Ouk, that when red P was reduced from a higher to a lower temperature, the spec. grav. in general was not lowered, it is clear that they have not investigated states of equilibrium, and that we, therefore, cannot draw conclusions about the course of tlio curve of the inner equilibria from the above results. As to the existence of the inner equilibria, however, this is no longer doubtful. So if we start from this, and if we then think of the phenomenon observed by Chapman ') that red P when melting, gives a colourless li(|uid i.e. a liquid which perfectly resembles melted yellow P, a T, ,i'-figure may be constructed in main lines for the pseudo-binary and the unary system, in which, however, the existence of a eutectic point is still an open question. It has been assumed in fig. 6 that red P{^Pj is a polymer of white P{uP), and therefore no eutectic point is drawn. In this figure the phenomenon observed by Chapman has been iilusti'ated, for heated to the melting-point, the red solid piiase will pass into a liquid 4, which lies entirely on the side of the white P. We see further from tiiis diagram that melted yellow P has about the same composition as melted red P, and that melted yellow P means undercoolcMl Lujuid red /■". AppUcatlons. Besides the phenomena mentioned heiv there, are others which seen in the light t>f thi.'^ theory find a |)lausil)!e explanation. I allude ') Jnuni. Ghem. Soc. 75, 743 (1890). ( 773 ) here to tlie pliciioaiena of retardalioii for so far thcij onhj appear lohen toe worh raj)id/i/^). If we considei- first of all the phenomenon of undercooUng and superheating of the solid, for so far as they are only observed with quiciv change of temperature, fig. 7 gives a satisfactory explanation. Starting from the inner liquid equililirium p, not the curve of equilibrium y^/.^, but another curve e.g. pl^ will be followed with rapid cooUng, and when we get beyond /, the state is not only unarily, but also pseudo-binarily metaatahle. Let us assume for simplicity that in the pseudo-binary system no retardation worth mentioning appears, then the substance will solidify at 4 and the solid substance S^ is deposited. Now this two-phase equilibrium is metastable to a high degree in the pseudo-binary system. In the unary system equilibrium between liquid and solid sub- stance can only exist under constant pressui'e at one temperature, and now it is the rule that a metastable state like that of the system h-\- S^ is at once destroyed. Thus we see e.g. that a supersaturate solution in contact with the substance which this solution must deposit to pass to the stable condition, [generally immediately deposits this substance. So the metastable two-phase equilibrium /, -|- .S', is changed into the stable state ^ -\- S.,, and this being a process which generates heat, the temperature rises to the unary melting-point. Starting from the solid inner equilibrium <j we get just the reverse, because then the substance melts at too high a temperature if quickly heated, as has already been observed for rhombic sulphur. If now the curves of inner equilibrium run as in fig. 8, the liquid can solidify too early if cooled two rapidly, the solid sub- stance can melt too early if heated too rapidly, and then the result is that for a, perfectly pure substance there is a range of temperature over which the solidification and the melting extends, which probablj^ often occurs for organic substances, in which the equilibrium sets in so slowly. With regard to the phenomena of retardation at the transition point I need only refer to fig. 9, which will now be clear without further elucidation. It is further hardly necessary to remark that when a substance is not bi-, but tri-, or polymolecidar, the phenomena discussed here remain essentially the same. 1) The peculiar phenomena, which will .ilso appear for more complicated systems, as e.g. Fe +■ C when we work quickly, will have to be accounted for in the same way. ( 774) 111 conclusion I want lo i)oiiil out tliut Ihis theory gives tlie first plausible ex|tlanation of the metasfah/tU;/ of the metdls. In this it is viz. noteworthy that the cooling of the solidified masses proceeds in such a way that the inner solid eqnilihriiiin can certainly not follow the temperature, and this is one of the reasons why the metals, as we generally Jiave them, are nearly always in metastable state. We must further bear in mind that if we have a metal which is in inner equilibrium, and it is subjected to some mechanical operation, a necessary consequence of tliis will be that the metal becomes metastable, because in stable state a change of pressure is generally attended with a shifting of the inner equilibrium, which, however, in consequence of the inner resistance does not appear at all, or on accouni of the slight velocity of transformation will take place only after a very lung time. The above mentioned circumstances account at the same time for the fact that it hardly ever occurs that two pieces of the same metal are perfectly identical, for this could only occur ^vhen the inner state, stable or metastable, was perfectly the same. Just as so many others the metastable states discussed here can be changed into the stable state by different influences, as increase of temperature, vibration, contact with the stable state etc., in which the transformation whicii lakes place, manifests itself in a recrystallisation.') Amsterdam, March 1910. Anorg. Cliem. Lab. of the University. ERRATA. In the Proceedings of the Meetings of Jan. and Febr. li)l(). p. 652 line 9 and p. 677 line 5 from the bottom, p. 654 line 17 from the top: for 11 read 659. p. 669, 672, 674 for 20.2 read 20.3. p. 670 etc. for carrier read holder. line 9 and 19 from the top: for modulus read constant, line 5 from the bottom : for corresponding read in agree- ment with. p. 672 line 16 from the bottom: for dilation read dilatation. p. 673 for 14.3 read 14.0. 1) It is to be expected that tliis melaslability will not he mot willi only for metals and metal-alloys, btit also for other substances, which have been obtained by rapid cooling and solidification of melted masses. (April 28, 1910). KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN TE AMSTERDAM. PROCEEDINGS OF THE MEETING of Friday April 29, 1910. (Tianslaled from : Veislag van de gewone vergadering der Wis- en Natuurkundii Afdeeling van 29 April 1910, Dl. XVIII). coi>ra?E2sra?s. Jan de Vries : "On polar figures with respect to a plane cubic curve", p. 776. J. G. Sleeswijk: "Contributions to the study of serura-anaphylaxis" (4th Communication). (Communicated by Prof. C. H. H. Speokck), p. 781. L. E. J. Brouwer: "On the structure of perfect sets of points". (Communicated by Prof. D.J. KORTEWEO), p. 785. M. W. Beijerixck: "Emulsion laevulan, the product of the action of viscosaccharase on cane sugar", p. 795. H. Kamerlikgh Onnes and A. Pekrier: "Researches on the magnetization of licfuid and solid oxygen", p. 799. (With one plate). St. Lokia: "The magneto-optic IVERR-Effect in ferro magnetic compounds and alloys". (Com- municated by Prof II. E. J. G. vv Bois), p. 835. (With one plate). Erratum, p. 845. 52 Proceedings Royal Acad. Amsterdam. Vol. XII. ( 776 ) Mathematics. — "Cn jwhtr fujures n-ith i-L'spfct to a plane cubic curve. By Prof. Jan de Vries. (Communicated in the meeting of March 26, 191U). 1. If a plane cubic curve y' is represented symbolicallj l)y r/''=0 then Qx a,, a,c = represents the polar line pxi/ of the points X and Y, i.e. the polar line of A' with respect to the polar conic rr,^ of Y and at the same time the polar line of )" with respect to the poLar conic -Tr of A'. The three polar lines pj.,,, px:, and y>y- will concur in one point TJ' when the three conditions are satisfied axdyC-w = 0, «a«;f'ir = Oi Oi/O-rCw ^= , . . . (1) By elimination of the coordinates wic we (hid out of it (abc) n^(ii/bxb:C,/C, = .... . . (2) So to two given points A', }" belongs a conic y- as locus of the point Z; it passes also through A' and Y, for when Z and A' coincide, we find (abc) axOiib'j-CyCx ^^ {cba] Cj,r,,b2ayar :z: — (abc) (ij,a„blc,jCx ^ 0. As we can substitute {abc) axa,iCxC-b,ib;, =^ for (2), thus also {abc)axa)ib~Cz{bjcC,i — />,^t"x) = 0, we can also represent y- by {abc) axa,/jzC- {bci.) ^= 0, where ^k are the coordinates of the line A' F. Consecjuently (2) can be replaced by (b/i) (bcr,) b._C: = (3) From this ensues that the conic y- is the poloconica Jtt, of the •'7/ '■ lines § and ij. So the poloconica of two lines is the locus of the points i^ which witli relation to the points of intersection A', Y of this conic with one of the given lines are in such a position that the polar lines pi,,~ and p,f- concur on the other one of the given lines, which is then at the same time polar line of A' and Y. 2. If Z and W are the points of intersection of -t^,, with Vj, it follows out of the symmetry of (3) in coiniection with the equations (1), that the four points A', Y, Z, W form a closed group, so that each side of the quadrangle determined by I hem is the polar line of the vertices not lying on it, therefore a po/ar quadrangle (Reye). Out of our considerations ensues that a polar quadrangle is deter- mined by two of its vertices, but also by two of its opposite sides. In the last case the \ertices are determined hj the poloconica of the given lines; in the former case we can use the poloconica be- longing to the polar line of the given points and their connecting line. ( 777 ) Out of axf'yato = and ajXya- = follows Here A and (i can be determined in such a way that Xaio-\-ii(tz="u relates to the point of intersection Z7 of .YF with WZ. As auUx = indicates the polar conic .t„ of U we find that X and Y according to the relation n,,aya„ = lie harmonically with respect to -t,,. In an analogous way ensues from au.az(iy=^0 and ciwazdx =- the relation rt,(,a;rtu := 0, according to which W and Z are also separated harmonically by ^«. But then also the points T=iXZ,YW) and V=(XW,rZ] are conjugated with respect to jt,,, i.e. we have «„«„«; = 0. Now U,V,T are the diagonal points of the complete quadrangle XYZW, so that it is proved that the diagonal triangle of a polar quadrangle is always a polar triangle '). 3. VViien the conic y- degenerates we can take for Z each point on the hue XY. To trace for this the condition, we put ^i^liyt-f-f'^/.; from (2) follows (abc) aj,a,ibif,i {Xh-c + [ih^) ().Cx -f [iCy] = 0, SO X^{abc) axayh'^CxCii -j- Afi {ahc) axdi/i'^c,/ + + ^f« (abc) axa,ihxbiiCxC,i -\- fi- {ahc)axa,jhxhyC,j = 0. By exchanging two of the symbolic factors a, h, c, we see that three of these terms are identically zero; so we have Xn (abc) axa,/bxc'j = 0. For an arbitrary choice of A' and Y this equation furnishes only ?.=:(} and ft = 0, thus the points X and Y. It furnishes each point of XY, as soon as 2 2 (abc) axCijibjc'i = (4) When A', Y, and Z are coUinear, the polar line px of X and the polar lines px,i , p^z concur in one point ; for these three lines are the polar lines of A', Y, Z with respect to the polar conic Jtx- If now (4) is satisfied, then also ly^- passes through that point, hence, the six polar lines px, /),/, /h, Px,/, p<iz, p:., concur in a point W. But when px, p,, and p. are concurrent, the poloconica of i ^ XYZ, degenerates and § is tangent of the Cayleyana. From this ensues that for given Y the equation (4) will represent 1) Mentioned without proof by Gaporali (Transunti d. R. A. dei Lincei 1877 p. 236). 52* ( 778 ) three right lines, namely the three tangents which we can draw out of Y to the Cayleyarid. This can be confirmed as follows. Let Z he a point of the locus of A'^, which is determined by 4) and A' a second point of that locus lying on VZ, so that we have ax^?.a,i-\- na-.. Out of (^4) then follows (ahc) ayc'i {).a,/ -j- fm^) (;.6y + lib.)- = 0. By exchanging a and c we see at once that 2 2 (ahc) a'ic'i {Xhii -\- (ib;)- vanishes identically. Analogously we- lind that [abc) a,jCyazl)~j and (nbc) a,jC'yaJ),,h: vanish identically. As finally the form {ahc) a,ptfizhz is zero because Z lies on the locus indicated by (4) the above relation is satisfied by all points of YZ, so the locus consists of three lines through }". 4. That the line g ^^ X Y is tangent to the Cayleyana as soon as (4) is satisfied, can be confirmed by reducing (4) to the tangential equation of that curve. In the first place we find out of {abc) a^ai/b'ii^J/ = and {acb) axttyCxb^ = the relation (abc) a^a,! {bxCy -\- byCj) {b^Cy — byCj) = 0. The last factor can be replaced by {bc%) where §i indicate the coordinates of XY. After that the equation can be broken up into two terms, which pass into each other when b and c are exchanged. So we can replace it by {abc) axttybiCy {be%) = (5) farthermore it is evident from 2 2 2 2 {abc) axa^jb'xc'j = and {cba) CxCyb^ay = 0, that at the same time is satisfied {abc) b'^UyC,! (ac§) = 0, so also {buc)a\byCy{bci) = Q (6) By combining (5) and (6) we find {abc) a.,c,i {beg) {ab%) = 0. So {abc) Cxtty {beg) {abg) z= 0. Out of the last two relations follows finally ( 779 ) {abc) (ac^ {bci) {al4) = (7) This tangential equation really represents the Cayleyana '). So we have fojind that tlie six polar lines px, Py, pz, Pxp Vy-' P^x concur in one point lohen the points X, Y, Z lie on a tangent of the Cnyleyann. 5. When p^, Py, p- a-i'c concurrent we have {abc) a'^biiC: T=. (8) This equation gives thus the relation between the coordinates of tliree points lying on one and the same polar conic. Vov an arbitrary clioice of X and Y tiiis equation is satisfied except by X and Y by no point of the line XY. If it is to be satisfied by ct = i-^k + l>-yk we must have therefore {abc) ttx by {Xcx + ftc^)- = 0, X(i (abc) ttx bii (> I'y zzz 0. This is satisfied for each value of ;i:(iwhen the relation (4) is satisfied, so when A', Y, Z lie on a tangent of the Cayleyana. Now in general the polar lines pj-j,,j9j::, p^; form a triangle inscribed in the triangle pxp^/P: (see ^ 3). If (4) is satisfied then px,/, Pxz, Pi/-. are concurrent; but then their point of intersection must be at the same time point of intersection of 2)j, p^, p:. If A', Y, Z are three collinear points of the cubic, then p^~,pzx and Px,/ pass successively through A', Y, and Z. For, from ^^ = 0, rt^ = and (P.a^ -\- na,,)' = follows that Ihe point Z is indicated by axa,f (P.a.r + f'"//) = ^*- So we have a^a^a: = U, so Z lies on the polar line px,,- If moreover .V, Y, Z lie on a tangent of the Cayleyana, then p,,:, p-x, Pxy must coiucidc with the tangents px, Pi/, p~ >» A', Y, Z. 6. For i)x,Py, iJ-nd px,/ to be concurrent, there must be a point 11 for which we have a^aic = 0, b,jb,c := 0, and CxC,/C,c = 0. But then iabc) axb'yCxC,, = 0. For arbitrarily chosen Y the locus of X becomes a figure of the third order, passing through Y, because we liave {abc) a'ybyCy^O. But by taking notice- of (4) we see that this figure consists of three tangents of the Cayleyana. Out of a'xatc = 0, aya,(, =: and Oxdyaw :=■ ') S(;e e.g. Glebsch, Le(jons sur la geometric, II, p. 284. ( 780 ) follows indeed {^(ix + (-ta,i)-a,p = ; i.e. if Z lies on A'}" then p^ will puss tiiroiigli tlie point of inter- section ]r of pxipif iind pxij, wliicli bears then at the same time p,i: and pxz- So the three lines p^, p,/, and y^^^ concur only then in one point when X and 1' are united by a tangent of the Cayleyana. Their point of intersection bears then also all the polar lines and mixed polar lines belonging to the points of those lines. The lines y;,, pr,/, and pj-_ will be concurring, when is satisfied, thus also hence also If we put we have the condition {ahc) <i:i'J>,,c,<-z = 2 {ahc) a^CjCi/h^h: ^= 0, 2 (abc) a,/>jf., {lii/C- — h^c,,) ^= 0. {ahc) a%c, (Jcl) = 0. As this can also be written in the forms {abc) 0,ihjcx (at'5) =: and {abc) aj)^c',- {ab^) i= and as out of a, a„ a. a^ I b, b., b, />, I §1 §2 Ss 5^ I follows the relation {abc) §,, = «.. {bci) + b, (c«5) + c, (a/-5) the above condition can be replaced by {abc)^ aAiCxii: = 0- With arbitrary position of X this is satisfied by >;., = 0, i.e. when A'^, )", and Z are coUincar (see § 3). If however {abc)'' aJ>j,:Cj: = 0, SO that A' lies on the Hessian, then A', F, and 2" are quite arbitrary. This was to be foreseen, now namely 7r.r is a pair of lines, so that the lines px, pxp and pxz concur in the Jiode of .t^. ( 781 ) Physiology. — '' ContrUmtions to (he stiidy of serum-nnapUylnxis." (From the "liistitut fur Infektionskranklieiten" at Berlin). By Dr. .]. G. Sleeswijk. (Coiiiiminii'ated hy Prof. C. H. H. Spronck). (4 til communication.) (Communicated in tlie meeting of March 26, 1910). During tiie first months of last year I had an opportunity, in three communications '), to make the results of my investigations about serumanaphylaxis known to the Academy. Since that time the lite- rature about this su[)ject has not inconsiderably increased. It is not my intention at ail here to go in for a discussion of this. Only let it be allowed to me (and I even consider myself obliged to do this before the Acadeni}') to treat here in a few words of these publica- tions, in which my own investigations either directly or indirectly were discussed, and to add the results of a number of further ex- periments. May it be presupposed that in general the facts commu- nicated by me must be acknowledged as correct. Principally we have to pay attention to three points: l""^ . the part of the red corpuscles of the guinea-pig with respect to horse-serum in the phenomenon of Th. Sjuth, 2'"'. the problem of the alexine-fixation and of the haemolysis in the anaphylactic shock, and 3"^ the application of the speeitic hypersensibility for proteins in medicina forensis. Last year I explained why I was of opinion that the sensitizing principle of the first injection and the toxic substance of the second administering of serum must be considered identical and that only quantitative differences are met with here. In the meantime Besredka") has changed his mind and taken' this standpoint. Now the logical consequence of my observation that horse-serum by treatment with the blood of a guinea-pig can be depriveil of its poisonousness for animals made sensitive, was therefore that with this at the same time the sensitizing substance is fixed. Levaditi and Raycpiman, who could also really prove what I mentioned last, do not refrain, therefore, in this connection, from referring to my communication.'') Also Salus corroborated my observation concerning the depoisoning action of the red corpuscles on horse-serum. ^) The problem of the complement fixation has of late attracted attention to a high degree. I remind of my first communication in which I said already, "tiiat a sensitized guinea-pig, which reacts 1) These Proceedings : January, February, and March 1909. 2) Ann. de I'Inst. Pasteur, Oct. 1909. 3) C. R. Soc. de Biol. T. 67, 1909, p. 1078. ■>) Wiener Klin. Woch. 1909, no. 48. ( 782 ) Upon the second seniinadministration with symptoms of intoxication some time after that injection produces a serum that is exceedingly poor in haemolytic alexine." By tlie side of this I have proved that the serum of liypersensitive animals in not a single combination with horse-serum gives a precipitate, nor is it able to fix complement. A remarkable incongruity therefore, and by which the anaphylactic state is distinguished from the phase of immunity, at which prae- cipitines are formed, and at which in vivo as well as in the test- tube alexine is fixed. I point out that here I came in conflict with NicoLLE and Abt'), who had found that the serum of sensitized guinea-pigs does fix complement with horse-serum in vitro. For FriedbehCtEk, who considers anaphylaxis as a peculiar form of the immunity for proteins, at which the praecipitines only for a trilling part have passed into circulation, but principally ha\e remained fixed (sessile) to the cells, the fixation of complement was a welcome phenomenon. He took for granted (evidently without any further control) that the communication of Nicolle and Abt was correct, whilst, on the other hand, he could confirm by his owni investigation my observation aliout the loss of alexine during the anaphylactic shock. ") Yet even here su(;h quantitative ditferences came to light that at first sight my observations seemed to show a shade of in- correctness. 1 had said for example, that the maximum of complement- loss is reached after about half an liour, whereas Friedbkrgfr found this to be* the case already within five minutes. Therefore 1 am compelled to enter into this somewhat more closely. Friedbergeu lias evidently not asked himself where the cause may lie of our diverging results, quantitatix'e as they may only be. He speaks only in passing of "DitFerenzen in der angewandten Anaphylaxie-technik." But here lies the cardo quaestionis, and what I presumed already, appeared to me on closer investigation to be reality. I had namely administered to my animals the toxic serum- injection in not too great a dose in their abdomen ; the reaction is then less violent and has a slower course, so that — through the investigation of blood-samples taken from the animals consecutively at dilFerent times of the anaphylactic shock — I could in general fix the course of the complement-curve. Friedberger, however, injected, his hypersensitive animals intravenously : the reaction then goes so quickly and is so violent, that die guinea-pigs usually die within few minutes. And at the same time also the conqdement-loss has soon reached its maximum. Now in Was.sermann's laboratdi-y I have 1) Ann. dc I'lnst. Pasteur 1908. ") Zcitschr. f. 1mm. forscli. Bd. ill, H. 6. ( 783 ) been able to fix tliese dillerences in a series of exact quantitative conipienient-titrations. From these it appeared among others that a few minutes after intravenous injection of 0.2 cm' of horse-serum the complement-quantity of the testanimal-serum may ha\e decreased nearly as strongly as at intraperitoneal injection of 3 cm' after half an hour. My investigations and those of FRiKDiiERGER, accordingly, do not contradict each other; they complete each other. Therefore I cannot see in Frii:dberger's results anything but an essential corroboration of my observations. The same thing holds good for the haemolysis, which in the anaphylactic shock shows itself in the test-animals. FkiedbI'-.rger corroborates also this fact, just as Poels ') does. My contention, therefore, that Besredka with his exclusivistic opinion that it is only the elements of the central nervous system which are to be brought to hypersensibility, is wrong, finds satisfactory support in this haemolysis which has been proved in more than one hand. Now as to the alexine-fixation of the anaphylactic serum with its antigen in vitro, I think I can inainlain my negative results over- against Nicolle and Aht. In many series with mixtures of falling quantities of iiorse-serum with rising tpiantities of anaphylactic guineapig-serum I could not observe anywhere a specific retardment of the haemolysis. Mot even though 1 stuck accurately' to the quan- titative proportions, as Nicolle and Abt have mentioned. In the meantime, thanks to the necessary controlling expei-iments, I came to the following conclusions. Even normal guinea-pig serum (inactivated) often retard the haemolysis, quite independent of the presence of horse-serum ; nay, we sometimes meet a normal serum which has a stronger fixing power than an anaphylactic serum which has also been examined"). Therefore I abide by my former contention that there is here no question of a specific complement-fixation in the test-tube. This incongruity of the alexine-fixation in vitro and in vivo, to which I drew attention already a year ago, was the other day corroborated with certainty oy Michaixis in the meeting of the "Physiologische Gesellschaft" at Berlin (21 January 1910). Also Friedberger seems after all to share this opinion (Zeitschr. f. Imro. forscli. Bd. IV, H. 5). It now seems (o me that the labile state of physiological equilibrium, in which the hypersensitive organism finds itself, is biologically characterized by the incongruity referred to just now. 1) Handelingen v. li. Nederl. Natuur- en Geneesk. Gongr. te Utrecht, April 1909. -) Ampler details I hope to publish elsewhere. ( 784 ) Not long ago Tsurvn -) tried to reduce the signilication of" llie alexine fixation in vivo in anaphylaxis. Thus already in normal guinea-pigs normal serum from dog or rabbit would cause loss of complement. This certainly does not hold good for horse serum, as had already appeared from my former controlling experiments; indeed, about this Tsurun does not speak. But moreover this investigator has worked with corpuscles sensitized not strongly enough and with insuflicient dilutions of complement, so that his results do not deserve a very great confidence. Leaving lliis out of consideration, he found, just as I did before, that the intoxication-phenomena and the loss of complement need not run parallel, from which I drew the conclusion that these two are not diree'ly dependent upon each other, but have a common cause. I now come to the thiid point that 1 wish to treat of here, viz. the application of anaphylaxis in the practice of medicina forensis. Evidently this application was so clear that about at the same time and independent of my communication, similar results were made known by Thomsen, Uiilenhutii, and H. Pfkiffer. The last lays stress upon the strong fall in the temperature of the body during the anaphylactic shock as a resource for tiie diagnosis. Concerning the technique of this investigation the following may still be mentioned. If a blood spot has to be identified, and if guinea- pigs are treated inlraperitoneally, several cm" of serum are necessary for each animal that is to be examined. If the animals are treated intravenously (in the juguiaris), much smaller quantities of serum are wanted, but then an operation iu the neck has to take place, which, however, after some practice for this purpose offers no objection. It has now appeared to me that also with young rabbits of about 1 K.G. the experiment can be very well made, because here both the sensitizing and the trying injection can be easily made in tiie earvena. A small dried up blood-spot is dissolved in 1 cm", of physiological salt-solution and injected in such an animal ; after a fortnight 1 cm" of the suspected kinds of blood is injected also intravenously. To rabbits, which had thus been previously treated with extracts from human blood-spots, I have administered on conse- cutive days serum from goat, horse, cow, and guinea-pig, without the animals reacting in tiie least. Lastly i cm" of human serum caused them within a few minutes to answer with spasms and paralyses, with respiratory disturbances, incontinentia uriiuie et alvi, etc. The anaphylactic reaction will, in my opinion, in the practice ') Zeitschr. f. linin. forsch Bd. IV, H. 5. ( 785 ) of medicina forensis lienceforth maintain its position hv the side of the precipitation in vitro as a valuable method. Among tiie many questions that show themselves in the study of our subject, there was also the following: what happens to the injected horse-serum during the anaphylactic shock? If there were only a minimal quantity free and unchanged in the circulation of the intoxicated animal, it ought to be possible that with its blood a normal animal could be sensitized. Now it has appeared to me that this is, never crowned with success. From this it may be inferred that all the antigen taken up in the blood-cii'culation is at once fixed by the cells of the hypersensitive organism, resp. deprived of its specific character at the same time. Mathematics. — "On the .■itructuve of perfect sets of points". By Dr. L. E. J. Bkouwer. (Communicated by Prof. Kohteweg). (Communicated in tlie meeting of March 26, f909). § 1. Sets of points and sets of pieces. The sets of points discussed in the following lines are supposed to be lying within a finite domain of a Sp,,- By a piece of a closed set of points ;< we nnderstand a single point or closed coherent set of points, belonging fO'f^, and not con- tained in an other closed coherent set of points belonging to ft. We can regard as elements of ft its pieces as well as its points, in other words we can consider ft on one hand as a set of points, on the other hand as a set of pieces. Let ns choose among the pieces of ft a fundamental series *S\, »§„, (S'j,..., then to ft belong one or more pieces j/S',,, ^S,,„ . . . with the property that ,3,, lies entirely within a for indefinitely increasing n indefinitely decreasing distance e,, from one of the pieces ,,-§„. These parts y.S.„ we shall call the liniitini/ pieces of the fundamental series >j,, o,,, Og, . . . As thus the set ft possesses to each of its fundamental series of pieces at least one limiting piece, a closed set of poi?its is likeirise closed as set of pieces. By an isolated piece of ft we understand a piece having from its rest set in ft a finite distance, in other words a piece, the rest set of which is closed. ( 786 )• Theorem 1. Eddi piece of n is either a limitiny piece, or an isolated piece. Let namely >S be a non-isolated piece, then there exists in {i a fundamental series of points t^, t^, t^, . . . . not belonging to >S', con- verging to a single point t of S. If t^ lies on S^, then >S', has a certain distance e^ from S. There is then certainly a point t\ of the fundamental series possessing a distance <^ f, I'rom S, lying therefore not on /Si but on an other piece S^. Let t^ be tlie distance of S^ from S, then there is certainly a point t', of the fundamental series possessing a distance <C f j from *S', lying thus neither on >Si, nor on S„, but on a third piece S^. Continuing in this manner we determine a fundamental series of pieces S-^, S„ S^, . . . , containing consecutively the points t^,t\,t\,... converging to t. So the pieces *S'i,*Sj, /S'j, . . . converge to a single limiting piece which can be no other than S. By a perfect set of pieces we understand a closed set, of which each piece is a limiting piece. A perfect set of pieces is also perfect as set of points ; but the inverse does not hold. For, a perfect set of points can very well contain isolated pieces. We shall say that two sets of pieces possess the same geometric ti/pe of order, when they can be brought piece by piece into such a one-one correspondence, liiat to a limiting piece of a fundamental series in one set correspondss a limiling piece of the corresponding fundamental series in the other set. So in general a closed set considered as -a set of pieces possesses not the same geometric type of order as when considered as a set of points. A closed set we shall call punctual, when it does not contain a coherent part, in other words when all its pieces are points. § 2. Cantor's fundamental tlieorem and its e.vtensions. The fundamental theorem of tiie theory of sets of points runs as follows : If ive destroy in a closed set an i.'iolated point, in the 7'est set again an isolated point, and so on trans finitely, this process leads after a denumerable number of steps to an end. The discoverers of this tlieorem, Cantor ') and Bendixson -) proved 1) Mathem. Annalen 23, p. 459— 47L -) Acta Matbematica 2, p. 419—427. ( 787 ) it with the aid of ihe noUon of the iiecon( I trnnsjinite cardinal £i, which is ho\ve\er not recognised by all mathematicians. Lixdelof ') gave a proof independent of this notion, where, however, the process of destruction itself remaining non-considered, the result is more or less obtained by surprise. Only for linear sets there have been given proofs of the fundamental theorem, which at the same time follow the process of destruction and are independent of ii "). The rest set which remains after completion of the process of destruction and which we may call the Cantor residue, is after C.\NTOR ') a perfect set of points, however of the most general kind, thus in general not a perfect set of pieces. An extension of the fundamental theorem, enunciated by Schoexflies") and proved by me '"), can be formulated as follows : If we destroy in a closed set an isolated piece, in the rest set acjain an isolated piece, and so on transfnitely, this process leads after a denumerahle number of steps to an end. My proof given formerly for this theorem was a generalisation of LindelOf's method, but at the same time I announced a proof which follows the process of destruction, and which I give now here ; in it is contained a proof of the fundamental theoi'em, which in simplicity surpasses by far the existing ones, is independent of i2, and follows the process of destruction : By means of Spn—\ 's belonging to an orthogonal system of directions we divide the Spn into ?z-dimensional cubes with edge a, each of these cubes into 2'- cubes with edge - a, each of the latter into 2" I cubes with edge - a, etc. All cubes constructed in this way form together a denumerable set of cubes K. Let now [i be the given closed set, then A' possesses as a part a likewise denumerable set K^ consisting of those cubes which contain in their interior or on their boundary points of n. 1) Acta Mathematica 29, p. 183—190. -) ScHOENFLiEs, Beiiclit uber die Mengenlehre I, p. 80, 81 ; Gott. Nachr. 1903, p. 21 — 31; Hardy, Mess, of Matliematics 33, p. 67—69; Young, Proceedings of the London Math. See. (2) 1, p. 230-246. 3) 1. c. p. 465. ^) Mathem. Annalen 59; the proof given there p. 141—145, and Bericht iiber die Mengenlehre II, p. 131 — 135 does not hold. ^) Mathem. Annulen 68, p. 429. ( 788 ) To each destruction of an isolated point or isolated piece in n now answers a destruction of at least one ") cube in K^ ; but of the latter destructions only a denumerable number is possible, thus also of the former, with which Cantor's theorem and Schoenflies's theorem are proved both together. Let us call the rest set, which remains after destruction of all isolable pieces, the Schoenjliea residue, tlien on the ground of theorem 1 we can formulate : Theorem 2. A Sdioenfiies residue is a perfect set of pieces. § 3. The structure of perfect sets of pieces. Let S, and <§, be two pieces of a perfect set of pieces fi. Let it be possible to place a finite number of pieces of (t into a row having aSi as its first element and ^S', as its last element in such a way, that the distance between two consecutive pieces of that row is smaller than a. Then we say, that S^ belongs to the a-group of S^. If *S, and (S'j both belong to the a-group of S^, then S, belongs also to the rt-group of »S.,, so that ft breaks up into a certain number of "(/-groups". This number is finite, because the distance of two different r/-groups cannot be smaller than a. If rtj <;^ a^, and if an fli-group and an (?j-group of ft are given, then these are either entirely separated or the a, -group is contained in the r<, -group. If two pieces »§! and S, of ft are given, then there is a certain maximum value of a, for which ,S\ and S„ lie in different a-groups of fi. That value we shall call the sepai'atinc/ hound of S^ and S^ in ft, and we shall represent it by <?„ {S^ , S^). If fartheron we represent the distance of S^ and S. by a (.S'j , S^), then a„_ (.S, , S^) converges with a {S^ , S.) to zero, but also inversely (I ((Si , S.,) with <J/, (/Si , /S2). For otherwise convergency of a„ (-S'l ,8^) to zero would involve the existence of a coherent part of ft, in which two different pieces of ft were contained, which is impossible. The maximum value of a for which ft breaks up into different a-groups we shall call the width of dispersion oi li, and shall vcpvaent it by (^(ft). This width of dispersion of ft is at the same time the greatest value which <J/;i (5; , <Sj) can reach for two pieces »S'i and »S, of fi. *) Even of an infinite number. ( 789 ) The maximum value of a, for which (i breaks up into at least n diiferent regroups wo shall call the n-partite undlh of dispersion oi n, and shall represent it by rf„ (ft). Clearly tf„ (/i) is ^f*(ft). For fi exists furthermore a series of increasing positive integers /ii(fx'), n^{n), n,{i-i), .... in such a way that fl,,(f«) for n between »i-— 1 (m) and Ilk (f) is e(jual to (J")...////.'')- This quantity '''„,/y)(f«) we call the /:''' ti-idtJi of dispersion of n and as such we represent it by We now assert that it is always possible to break up [i into la.^ perfect sets of pieces f<i, . . . . fi„,, so as to have '^(ft/i) ^ f^MiO-t) and <«/■!> WJE rf«!l('")- Let namely be (f^^ (ft) = d" W(fj) ; we can then obtain the required number m^ by composing each ;x/, of a certain number of dii^i^ji)- groups belonging to a same d'(^'~i)(;i)-group. We are then also sure of having satisfied the condition «0«/,j , fi/,., ) -^ tfm^ffx). Fartheron Ave can place the (f(''')(;t)-groups of a same dC*-— ')(f()-group into such a row that the distance between two consecutive ones is equal to (f('-")(ii). If we take care that each nu consists of a non- interrupted segment of such a row, then the condition d(,'t/,) ^ ('^ihi (f*) is also satisfied. Let ns now break up in the same way each ;»/, into in^ perfect sets of pieces nir. , ■ ■ ■ ■ {i/,m.. bi such a way that (filihij^ffm^il^h) and «(,'t/H'i . f'Ajo) = ff,p..['ih), and let us continue this process indefinitely. If then we represent by r, an arbitrary row of r indices, then we shall always find '^'«./fVv-i)^'^^"'' + "'- + ---- + "'v + '-'(f*) . . . . (A) As n is a perfect set of pieces, the width of dispersion d(ft^ ) can con\ erge to zero only for indefinite increase of v ; out of the formula (A) follows, however, that for indefinite increase of r that convergency to zero always takes place and, indeed, nniforndy for all ri'' elements of decomposition together. At the same time the separating bound of every two pieces lying in one and the same i'''' element of decomposition converges uniformly to zero; so these elements of decomposition converge themselves uniformly each to a single piece. If finally a \ariable pair of pieces of ft is given, then their distance can converge to zero only when the order of the smallest element of decomposition, in which both are contained, increases indefinitely. The simplest mode in which this process of decomposition can be ( 790 ) executed is by (aking all iii/c's equal to 2. If then we represent the two elements of decomposition of the first order by n^ and fij, those of the second order by ;*„„, fto^, fij,,, ft,j, and so on, then in this way the different pieces of fx are brought into a one-one correspondence with the different fundamental series consisting of figures and 2. And two pieces converge to each other then, and only then, when the commencing segment which is common to tlieir fundamental series, increases indefinitely. Let us consider on the other hand, in the linear continuum of real numbers between and 1, the perfect punctual set jr of those numbers which can be represented in the triadic system by an infi- nite number of figures and 2. The geometric type of order of m we shall represent by ?. Two numbers of nr converge to each other then and only then, when the commencing segment which is common to their series of figures, increases indefinitely. So, if we realize such a one-one correspondence between the pieces of [i and the numbers of jr, that for each piece of fx the series of indices is equal to the series of figures of the' corresponding number of -T, then to a limiting piece of a fundamental series of pieces of ft corresponds a limiting number of the corresponding series of numbers in n, so that we can formulate: Theorem 3. Each perfect set of pieces possesses the ijeometric ti/pe of order ?. For the case that the set under discussion is punctual and lies in a phine, this theorem ensues immediately from the following well- known property : Through each plane closed punctual set we can bring an arc of simple curve. Combining Schoenfmes's theorem mentioned in § 2 with theorem 3 we can say : Theouem 4. Each closed set consists of tivo sets of pieces ; one of them possesses, if it does not vanish, the geometric type of order S, and the other is denumerable. M- . The groups ivhich transform the geometric type of order ? in itself. Just as spaces admit of groups of continuous one-one transforma- tions, whose geometric types of order ') are again spaces, namely 1) In this special case formerly called hy inc "Paiameterniannigfaltigkeiten" Gomp. Malhem. Annalen 67, p. 247. ( 791 ) the finite contirmoiis groups of Lie, the geometric type of order ^ admits of groups of continuous one-one transformations, which possess likewise the geometric type of order C. In order to construct such groups we start from a decomposition according to § 3 of the set ii into m^ "parts of the first order" (ij , fij , . . . (x,„,, of each of these parts of the first order into m., "parts of the second order" ft^i, ;^/,2, ftWi • • • /'Am,, etc. The parts of the first order we submit to an arbitrary transitive substitution group of m^ elements, of which we represent the order by !>,, and which we represent itself by g^. After this we submit the parts of the second order to a transitive substitution group g^_ of m^ m., elements which possesses the parts of the first order as systems of impriraitivity and f/^ as substitution group of those systems into each other. We can then represent the order of g^_ by p^ p^. The simplest way to construct such a group y^, is to choose it as the direct product of g.^ and a substitution group y.,, which of the parts of the second order leaves the first index unchanged and transforms the second index according to a single transitive substi- tution group of m., elements. We then submit the parts of the tliird order to a transitive sub- stitution group (/, of ??ii m^_ 7??3 elements which possesses the parts of the second order as systems of imprimitivity and g^ as substitution group of those systems into each other. We can represent the order of gt by p, p, p,. In this way we construct a fundamental series of substitution groups g„g,,g„... Let Tj be an arbitrary substitution of </, ; t„ a substitution of g^ having on the first index of the parts of the second order the same intluence as Tj ; r^ a substitution of 173 having on the first two indices of the parts of the third order the same influence as t, ; and so on. The whole of the substitutions t„ then determines a substitution of the different fundamental series of indices into each other, in other words a transformation r of the pieces of (x into each other. This transformation is in the first place a one-one transformation; for, two different pieces of \i lie in two different parts of a certain, e.g. of the ?''•> order, and these are transformed by x into again two different parts of the ?•"" order. If fartiieron -Sj, S^, S^^, . . . is a fundamental series of pieces, pos- sessing <So, as its only limiting piece, then, if P. (??) is the lowest possible order with the property that ,S'„ and *§-„ lie in different parts of that order, ).{n) must increase indefinitely with n. 53 Proceedings Royal Acad. Amsterdam. Vol. XII. ( 792 ) So bv the ti'aiisformation r tlie fiuidanieiital series passes into a new fundanieiilal series having as its only limiting piece the piece into which aS'.„ piisses by t. As a set of pieces f^ is thus contiauoushi fransfornied by r. Let r\,r\,r\,... be a series of substitutions satisfying the same conditions as tiie series r^,r„,r,, . . . If then Tit'i = t"i; Tjt'3 = t", ; etc., then tlie series t",, r'\, r'\, . . . likewise satisfies the same conditions. If farthermore t' and t" are defined analogously to r, then t t' is equal to r". So the transformations satisfying tJie conditions put for r form a group, inluch lae shall represent by g. To investigate the geometric type of order of this group, we decompose in the way indicated in ^ 3 a perfect set of pieces q into Pi parts of the first order (),,o, , f)^,^ ; each of these into p^ parts of the second order ()/,i, qui , Qhp..\ and so on. The /^i substitutions of r/, we bring into a one-one correspondence to the pai'ts of llie first order of q. Then the p^p^ substitutions of g., into sucii a one-one correspondence to the parts of the second order of o, that, if a substitution of _(/, and a substitution of ^i have the same iniluence on the first index of the parts of the second order of jLt, the part of the second order of q corresponding to the former lies in the part of the first order of q corresponding to the latter. In like manner we bring the pi p^ Ih substitutions of (7, into such a one-one correspondence to the parts of the third order of q, that, if a substitution of //j and a substitution of ^^ have the same influence on the first two indices of the parts of the third order of fi, the part of the third order of q corresponding to the former lies in the part of the second order of corresponding to the latter; and so on. The parts of q corres|)onding to a series Tj, t„. Tj . . . . tiien converge to a single piece of ^>, which we let answer to the transformation T deduced from the series. Then also inversely to each piece of q answers a transformation r, and the correspondenceattained in this manner is a one-one correspondence. Farthermore two transformations t and r' converge to each other then and only then, when tlieir generating series Tj , t^ , t, , . . . . and t', , x\ , t' .,,... . have an indefinitely increasing commencing segment in common, in otiier words when the cori'esponding pieces of q converge to each other. So the correspondence between the trans- formations T and the pieces of (> is continuous. The transformations t, in other words the transformations of the group g, liave thus been brought into a continuous one-one correspon- dence to the pieces of *), so that g posses.ies the geometric type of order ^. ( 793 ) If now we adjoin to each substitution group _(/„ a finite group g'„ of continuous one-one transformations of [i as a set of pieces in itself, tranforming of the pieces of ft the first n indices according to gn , but leaving unchanged all their other indices, then the funda- mental series of tin; groups g\ , g'.,, g\,. . . converges uniformly to the group g. The set whose elements are the groups g of the geometric type of order C constructable in the indicated manner possesses the cardinal number of the contiinium. For, already the set of those series ??)i, m^, »/,,..., wliicli consist of prime numbers, possesses this cardinal number, and any two different series of this set give rise to different groups g. We can sum up the preceding as follows : Theorem 5. The geometric type of order 5 alloios of an infinite number of groups consisting of a geometric type of order y of con- tinuous one-one transformations and being uniformly approximated by a fundamental series of groups consisting each of a finite number of continuous one-one transformations. If in particular we consider those groups g for which each gn is chosen in the way described at the commencement of this § as the direct product of gn—\ and a group y„, we can formulate in par- ticular : Theorem 6. The geometric of order 5 allows of an infinite number of groups consisting of a geometric type of order S of continuous one-one transformations and being uniformly convergent direct pro- ducts each of a fundamental series of finite groups of coiitinuous one-one transformations. ^ 5. The sham-addition in the geometric type of order ?. Let us choose the factor groups indicated in theorem 6 as simply as possible, namely g^ as the group of cyclic displacements corre- sponding to a certain cyclic arrangement of the first indices, and likewise each y„ as the group of cyclic displacements corresponding to a certain cyclic arrangement of the «'•' indices; g is then com- mutative, and transitive in such a way that a transfoi-mation of q is determined uniformly by the position which it gives to one of the elements of jj. Let us further clioose an arbitrary piece of ;i as piece zero. Let us represent this piece by aS„, and the transformation, which trans- fers <S„ into S.y. and is thereby determined, by "+ So,". That the 53* ( 794 ) piece (S,t is transferred hy lliis ti'aiisforination into S^, we sliall express b}' tlie formula Sji -]- Sk = Sy, wliich operation is associative and commutative. Let ns tlnally choose, in order to malce the resemlilance to ordinary ciphering as complete as possible, all ??i„'s equal to 10, let us take for each system of ?i"' indices the digits 0,1,2,3,4,5,6,7,8,9 in this order, and let us give to the piece zero only indices 0. The ditferent pieces of }i we can then represent biuniformly by the different infinite decimal fi'actions lying between and 1, in such a way, however, that finke decimal fractions do not appear and that '30 is not equal to '29, whilst each group •/„ consists of the different ways in which one can add the same number to all ?^"^ decimals, modulo 10. Now according to the above we understand by -5473... I|. -9566... the decimal fraction, into which -5473 ... is transferred by the transformation which transfers -0 into -95(16..., or, what comes to the same, the decimal fraction, into which -9566 ... is transferred by the transformation which transfers -0 into -5473 . . . We shall call the operation furnishing this result, on the ground of its associativity and commutativity, the "sham-addition" of '9566 . . . to -5473 . . . . ; it takes place just as ordinary addition, with this difference that in each decimal position the surplus beyond 10 is neglected, thu.s that dilTerent decimal positions do not influence each other. So we have : •5473 . . . . + 9566 . . . . = -4939 .... Let us understand analogously by 5473 . . . . r^ -9566 .... the decimal fraction, into which -5473 .... is transferred by the trans- formation which transfers -9566 .... into -6, and let us call the operation furnishing tiiis decimal fraction the "sham-subtrnctioti" of •9566.... from •5473....; then this shara-subtraction is performed in the same way as ordinary subtraction with this difference, that "borrowing" does not take place at the cost of the preceding decimal positions, so that here again different decimal positions do not influence each other. So we have : ■5473 . . . . r^ -9566 ....= 6917 .... By operating only with a finite number, great enough, of conse- cutive figures directly behind the decimal sign, sham-addifion and sham-subtraction furnish in the type of order ^ a result agreeing with the e.xacl one up lo any desired degree of accuracy. In this too they behave like ordinary addition and subtraction of real numbers. ( 795 ) Microbiology. — "Emulsion Inevulan, the product of the action of viscosaccharase on cane sugar" . By Prof. M. W. Beijerinck. In tlie proceedings of the Academy of 9 February 1910 an enzyme was described which produces from cane sugar and raffinose a viscous matter incapable of diffusion. My further investigations, made conjointly with Mr. D. C. J. Mimkman, proved that this substance is closely related to the laevulan of Lippmann *) but not identic with it. Our emulsion laevulan originates in watery nutrient solutions in ([uite the same way as in the agarplates, so that these solutions change into a milkwhite emulsion ; the liquid between the suspending laevulan droplets opalises very strongly. In hot water the substance is fairly soluble and the specific rotation of the polarised light, which, on account of the opalisation can only approximately be determined, is about whilst Lippmann gives for his laevulan On account of this considerable difference in its rotating power, a new name, e.g. "sinistran", might seem desirable. But the word laevulan having a collective meaning to wiiich also the more and the less soluble forms of our substance may be brought, we shall here use the general denomination, the more so as it is sure that the laevulan of the literature, like ours, consists of the cell-wall substance of bacteria. Besides by Lippmann the formation of laevulan by bacteria has also been observed by Maassen'), who does not, however, describe the appearance of the emulsion, so that in this case, too, a modifica- tion of our emulsion laevulan seems to be produced. The here concerned microbe is a sporulating fermentation organism, called by Maassen Semiclostridium commune, but not yet found by us. Preparation and properties of emulsion laevulan. We were first of opinion that emulsion laevulan might be best prepared by using Bacillus emulsionis, for we had stated that this species does not decompose the once formed laexulan, whilst B. megatherium and B. mesentericus, which likewise produce emulsion Chemie der Ziickerat-ten S'*" AuH. 1904. Pag. 906, f312. ') Arbeilen aus ileni Kaiserl. Gesundlieitsamte. Biol. Abt. Bd. 5, p. 2, 1905. ( 796 ) laevulaii, atlack tliis substance and use it as food as sooji as the cane sugar fails. We have, however, found tiiat with some precaution it is much easier, especially with B. mesentericus, to produce large quantities of laevulan, than with B. emulsionis ; this reposes on the circumstance that the former species, particularly at high temperatures, about 40', possesses a very strong vegetative power, whilst the latter always grows slowly and has a relatively low tempei'ature optimum, below or near 30" C. Hence we used for the preparation of laevulan the common hay bacterium, which is the form of B. mesentericus obtained by accu- mulation methods, such as the method of potato slices and that of malt solutions. But this form is so common in our surroundings and so well adapted to the life in cane sugar solutions of for the rest different composition, that tliese, after pasteurisation or short boiling and wiien kept warm, of themselves produce laevulan by the devel- opment of the spontaneous spoi-es of the hay bacillus. Such solutions then turn milky and slimy by tlie formation of the microscopic laevulan emulsion. Kor the e.xpei'iments were used large EnLENMEYER-tlasks with 500 cm^ of a medium of the composition: tapwater, 20% canesugar, 0.05% KNO3, and 0.05% K,HPO„ cultivated at ± 27= C. This liquid inoculated with B. mesentericus very soon obtains the said milky appearance. The same emulsion which to the colonies of B. mesentericus and B. emulsionis on cane sugar agarplates gives so peculiar a character, is now in large quantity produced in the culture liquid, saturated besides with laevulan in true solution, which causes the strong and characteristic opalisation, not known to us to such a degree in any other substance. Besides, at the bottom of the llasks a thick transparent slime layer is slowly foi-med, which also proved to consist of laevulan, wherein, however, the bacterial bodies themselves are accumulated, whilst the liquid above it is poor in bacteria but abounds in viscosaccharase and laevulan emulsion. The acid formation in this solution is slight but not absent. The laevulan may be precipitated with alcoliol for which 50 Vg in the solution is sufficient. Only at a much greater alcohol conceidration other substances of the liquid also precipitate. By dissolving in boiling water and again precipitating the further purification is easy. After drying and pulverising a snowwhite nearly tasteless powder results. From a flask as the altove which at first contained 100 G. of cane sugar, 8 G. of pure dry laevulaii \vere obtained after 7 days cultivation, there still being in tiie liquid 20 (J. of invert- and 70 G. of cane sugar; the slime at the l>oltom not being collected. ( 797 ) From another tlask quite alike to the preceding, which also con- tained 100 G. of cane sugar, were gained 15 G. of laevulan after 17 days, 45 G. of cane sugar and 35 G. of invei't sugar still being present. The slime adhering to the bottom, consisting of B. mesentericus with thick cell-walls of laevulan, was used for a new culture for which a solution of 2 7„ of cane sugar, 0.05 »/„ K NO, and 0.05 "/„ Kj HPO^ was used. After 18 days were obtained 2.25 G. from the 10 G. of original cane sugar, accordingly 22,5 % of laevulan was earned. Pure laevulan is somewhat soluble in cold water, much better in boiling; all solutions opalise very strongly. It does not reduce Fehling's coppersolution ; only after prolonged boiling a feeble reduction is observed. It is incapable of alcoholic and lactic acid fermentation, but by butyric acid ferments, in absence of air, it gets into as strong a fermentation as cane sugar, whereby hydrogen, carbonic and volatile acid result. A number of bacteria can feed on it when growing with access of air. Azotobacter chroococcum can use it under fixation of free nitro- gen and formation of some acid. By a treatment with acids, especially when warm, it changes readily into laevulose and so becomes fit for alcoholic and lactic- acid fermentation. After the inversion, bj" heating with resorcine and strong hydrochloric acid, the red colour appears, characteristic of laevulose, whilst with orcine and hydrochloric acid the violet colour, indicating pentose, is completely absent. When distillated and treated with sulphuric acid no perceptible quantity of furfurol can be detected. As said, the specific rotation, which cannot be exactly determined on account of the strong opalisation is and after hydrolysis After prolonged heating with acid in the autoclave al 120^ the rotation lowered even to = — 64°. That of pure laevulose is £) .^=.-92° There is some probability that this diminution is due to destruction of part of the laevulose. ( 798 ) As we had Couiul that tlie slime at tlie bottom of tlie Ihisk is less soluble than that obtained by alcohol from the emulsionated liquid above it, we |)repared laevulan from this slime also by sei)arate experiments, for we supposed that dextran might occur therein, which is much less soluble in water than laevulan. However, it was found tiiat the laevulan obtained in this way gives no other rotation after inversion than the emulsion laevulan, from which it does not ditfer. Hence it is sure that hay bacteria produce no dextran at all, but that their cell-wall consists of various modifications of laevulan of dilTerent solubility. Not only in media of the above composition B. mesentericus pro- duces laevulan, this happens quite as easily in a yeast decoction with 2 to 20 7o of cane sugar, addition of chalk proving favourable. The temperature of cultivation may also vary and even rise to 40^ C, but then care should be taken that the laevulan itself be not attacked bj the pi-oducer. From the preceding it may be concluded that the large lumps of slime so easily formed on cane sugar agar-plates hj B. mesentericus and the other emulsionating species consist as well of laevulan as the emulsion which occurs round the colonies of this species in the agar. Hence, it can neither be doubted that the slime of these colonies, which does not diffuse in the agar, is produced by viscosaccharase from cane sugar, and that this enzyme only partly gets out of the bacterial body proper, the cell-wall included. Evidently in the cell- wall itself the enzyme forms new laevulan by converting the cane sugar, with which both cell-wall and agar-|)late are imbibed. The production of cell-wall substance in consequence of the action of an enzyme, which in my former communication was called pro- bable, must now, as regards laevulan, be considered as proved. Dextran and the dextran bacteria, which we have likewise studied, shall later be treated more thoroughly. For the moment it may be observed that by this substance the |)olarised light is strongly rotated to the right; we found «^ = + 132°, whilst in the literature by various authors is given for dextran «^r= + 199° to 230°. (^uile like laevulan it results exclusively from cane sugar. So lae- vulan as well as dextran are produced by microbes, neither from laevulose, glucose, or any other sugar, but solely from cane sugar and raflinose. The slimy cell-wall substances formed by other microbes from glucose, laevulose and maltose, are of a different Jiature. ( 799 ) Physics. — ''Researches on the mag^ietization oj lUjuid and solid oxygen. " Bj- H. Kamerlingh Onnes and Albert Perrier. Coniraiinioation N°. 116 from tlie Physical Laboratory, Leiden. ^1 1. fiitrod/ictinii. Il is scari'elj necessary to remark tiiat the investigation of tlie magnetic properties of oxygen at low tempera- tures has long occupied a position on the programme of the cryogenic laboratory, or that this has been considered one of the most important items on the programme since the investigation of both liquid and solid oxygen has been made possible by the perfecting of the methods ') of obtaining detailed series of measurements at constant temperatures in the region of liquid hydrogen. Indeed, while the strongly magnetic properties of oxygen of themselves select it from all other substances as especially suitable for the study of para-magnetism, we can in the meantime for no other substance obtain the magnetic equation of state''), which gives a representation of the magnetic properties of a substance in the three states of aggregation at as many successive temperatures and pressures as possible. The investigation of oxygen at very low temperatures and at pressures that can easily be realised was expected to give ar once results of much imjwrtance. Curie") found lor gaseous oxygen between 20° C. and 450''"' C. that the specijic susceptibility (magnetization per gram for H^=l) was inversely proportional to the absolute temperature, and Fleming and Dewar ^) concluded from their latest measurement of the suscepti- bility of liquid oxygen at its boiling [)oint that Curie's law was obeyed down (0 —183° C. Does the specific susceptibilit}- continue to increase so strongly at low temperatures or does it approach a limiting value? Is oxygen in the solid state ferro-magnetic? Does the magnetization finally at extremely low temperatures perhaps begin to decrease and disappear completely at the absolute zero?^) 1) H. Kamerlingh Onnes, Tliese Proc. Sept. 1906, Coaim. from ttie Leyden labor, no. 94/- (1906i. -) H. Kamerlingh Onnes, Commiin, from the Leyden labor. Suppl. no. 9 p. 28. 3) P. Curie. Ann. cbim. phys. (7) 5 (1895) p. 289. *) Fleming and Dewar Proc. Pioyal Soc. London 6o, p, 311, 1898. 5) It has since appeared that the magnetization of ferro-magnetic substances does not yet give any justification when the temperature is lowered to the melting point of hydrogen for the assumption that the electrons whose motion causes magnetization are frozen fast to the atoms and that therefore this disap- pearance at the absolute zero may be expected. (P. Weiss and H Kamerlingh Onnes, These Proc. Jan./Febr. 1910, Gomra. from the Leyden Labor no. 114 p. 9). ( 800 ) These are questions whieli, oonsideriiig the possibility of obtaining important contributions to the knowledge of the influence of density upon susceptibility by high pressures in the region where the gaseous state of aggregation changes continuously into the liquid make it a very attractive work to realise, even from a purely experimental point of view, the representation to which we have just referred. Tiie work was commenced though only when Prof. Weiss ex- tended his magnetical researches to very low temperatures and the measurements on the magnetization of ferro-magnetic and cognate substances at veiy low temperatures, which were communicated to the February Meeting'), were luidertaken. With that investigation which was carried out at the same time, tlie present one is very closely related, and for part of them we made use of tiie same appliances. In our present investigation we have also in various ways made use of Prof. Weiss's method ') of determining the magne- tization by means of the maximum couple exerted by a magnetic field of variable direction upon an ellipsoid of the experimental substance, a method which had been entirely successful in the other research. We must also express the great advantage we derived from the continued support given us liy Prof. Weiss, and we take this opportunity of gratefully acknowledging our indebtedness to iiim. The change with temperature of the specific susceptibility of oxygen, the investigation of which was our first object, is of particular importance seeing that Curie's law follows from Langevin's kinetic theory of magnetism"). It was soon apparent to us that this law was not valid for oxygen, as was thought, down to — 183° C, l)ut that it would have to be replaced by another. According to the impoi'tant paper of DU Bois and Honda communicated to the January Meeting — our experiments had already been completed al that time — various elements were found for which Curie's law did not hold at temperatures above 0° C. This at once increases the impor- (ance of the further investigation of oxygen, for which over a definite region of temperature Curie's law is valid, while over another region it obeys a second law, viz. : that of inverse proportionality to the square root of the absolute temperature. The results concerning this law and also concerning the probability of a sudden change in the value of the specif c susceptibijiti/ on solidii'ication will be discussed in ^ 5. 1) P. Weiss and H. Kamerlingu Onnes. These Proc. Jan. Febr. 1910. Comm. fr. the Leyden labor, no. 114 (1910). ') P. Weiss. Jouin. de phys. 4e seric t. VI, p. 6G1 ; 1907. ') Langevin. Ann. cliim. phys. (8) 5, p. 70; 190ij. ( 801 ) We liave been occupied willi aiiotlier question besides the change of specific susceptibility with temperature, wliich was suggested both by the experimental results obtained by Fleming and Dewar and by the theories of Langevin and Weiss. In the experiments of the first-named there appears sufficient evidence for the conclusion that there is a decided diminution of the susceptibihty as the strength of the field increases (the diminution is of the order of 107o J" ^ field of 2500 gauss). Now, according to the theory of Langevin para-magnetic substances must, it is true, exhibit this phenomenon, but calculation from his foi'mulae limits the magnitude of this change to less than O.l'/o i'l the case of liquid o.xygen at its boiling point. Should a higher value than this be obtained, then one would be led to assume the existence of a Weiss molecular field ^). We arranged our experiments so that the liquid and the solid oxygen could be subjected to a field of 16000 gauss, a field very much stronger (about six times) than that used by Fleming and Deuak, so that we might expect the phenomenon which appeared in the course of their experiments to be exhibited to a much greater degree in ours even at the same temperatures as were used by them. If what was observed h} Fleming and Dewar could really be ascribed to the beginning of saturation then the theory would further lead us to expect that as the temperature sank the change would strongly increase (becoming infinite at 7'=0), and that in our experiments with liquid hydrogen it would become very striking. We have, however, observed only small deviations, which we shall discuss further in § 5. As regards the experimental methods employed by us in our in- vestigation, two completely different schemes were adopted: on the one hand was measured the magnetic attraction exerted upon a column of the liquid, and on the other the maximum couple exerted by a homogeneous field upon an ellipsoid. The second method is more especially suitable for comparative measurements and can also be used for frozen oxygen ; the first can be used only for the liquid phase, but on the other hantl it makes very trustworthy absolute measurements possible ; we have therefore adopted it as the basis of our other measurements. In the carrying-out of each method further precautions are still desirable, so that while we are busy pushing on the investigation, we propose at the same time to repeat it in part in order to increase the accuracy of the results obtained by taking such further precautions as have appeared possible in the course of the work. 1; Weiss, L'hyp. clu champ molec. loc. cit. { 802 ) Liquid oxyjen I. § 2. Method of the maiinetic vise. As mentioned above, we have rendered the method of the magnetic rise employed by Quincke, DU Bois and other observers suitable for nse at low temperatures. One limb of a vertical 0-slia[)ed tube, the upper i)Ortion of which contains the gaseous, and the lower the liquid phase of the experi- mental liquefied gas was placed between the poles of a magnet whose field was horizontal. Let H be the tleld, (//' the tield in the other limb is supposed to be so small that {H'lHf is negligible), // the acceleration due to gravity, c the difference in height of the levels of the liquid under the in- fluence of H, Q and q^ the densities of the liquid and of (he gaseous phases respectively, K and K„ their respective volume susceptibili- ties, then (/v-/t„)//^ = 2^(9-p„)r, (1) or, by introducing the absolute specific susceptibility / (XP— XoPo) It' = ■•^- (?— (>o).'/ If / = x„ tlien the equation becomes simply ' = 77^' <2' which is the formula we have used for our calculations. ^) So there are striking advantages oftered by this method parti- cularly for an absolute measurement, on account of its applicability to the case of a liquid in equilibrium with its own vapour. There are only two magnitudes to Ite determined, the distance :, which can be measured very accurately with a cathetometer, and the field H; nor have we to know the density of the liquid in oi'der to be able to find the specific susceptibility. 3fagnetic rise apparatus. It is a very easy matter to cause an ordinary liquid to ascend under the influence of magnetic attraction, but the experiment is attended by serious dif!iculties when one has to deal with a liquefied gas. Boiling must be completely avoided, and care must be taken that the vaporization is unnoticeable. The first precaution is necessary because the motion of the liipiid or of its surface would render adjustment (piite impossible, and the second ') In § 5 we shall give the reasons why we think that x — Xo< i"J should it be possible that this is not the case there is still the greatest probability that a:o<l-5x; in the most unfavourable case at the boiling-point the correction remains below 0.002 in value, while at lower temperatures it is quite [negligible on account of the small value of o- ( HO^ ) is necessaiy tliat tlie total quantity of iiijuid may not a|)preciably alter dnring the measurement of one rise. Moreover magnetic action itself increases the difTiculties ; it is easy to see that it can occasion the formation of gas-bul)l)les which divide the column of liquid into two parts, so that the one portion remains suspended between the poles, while the other falls back again. In that case measurement of the ascent is out of the question. Starting from the thermodynamic potential it appears that in every case the relation must hold, where H is the field at the surface of the liquid, and H,, the field at a distance y below the surface of the liquid. These conditions shew that there is a limit to the intensity of the fields in which measurements may be made, for they necessitate a range of extended fields (in this case in a vertical direction). Conical pole- pieces are thus as a matter of fact barred. After several preliminary experiments an apparatus was constructed, the most important part of which consisted of two concentric double- walled vacuum tubes, with which we already succeeded in obtaining rather successful measurements. The walls of the double vacuum tube were not silvered, so that we were able to watch how the lit[uid behaved during the experiments. From the experience thus acquired the improved apparatus which we shall now proceed to describe was designed and constructed. It will be seen that the construction of the apparatus lays a very heavy tax upon the art of the glass-blower '). As before, the chief part consisted of two independent U-shaped vacuum tubes, the one fitting inside the other. The double walls of each tube are completely silvered on the vacuum side, except in the case of the inner tube, where the distance which tlie liquid ascends is left free, and in the outer where a sufficient length is left unsilvered to leave a strip of a few millimeters breadth through which the level of the liquid can be read. One of the tubes completely surrounds that portion of the other which contains liquid ; this we call the protecting tube. The narrowest portion J/^ (fig. 2) is placed between the poles of the electro-magnet. The narrow limb of the inner tube must of course be perfectly cylindrical. The ether limb is enlarged and serves as a reservoir. In order to be able to apply equation (2) all care was taken that the temperature of the liquid and vapour up to a height 1) Tlie double vacuum tube was prepared by Mr. Kesselring, Laboratory glass- blower, and the remainder by Mr. Flim, technical assistant at the Laboratory. ( 804 ) somewhat greatei' than tliat readied by the column of liquid was everywhere the same belli in the wide and in the narrow tube; and furtiier care was taken that where the temperature of the vapour above the liquid in the upper parts of the apparatus changes to ordinary temperature it was as far as possible the same at the same height in the two limbs of the 0-shaped space. With this end in view the liquid in the inner tube was, by means of the magnetic tield, repeatedly moved up and down under constant vapour pres- sure, until we might assume that in this tube equilibrium was suffi- ciently well attained. To make this equilibrium possible the inner tube is surrounded with liquid at the same temperature as that which the liquid in it must attain. In the outer or protecting tube the liquid is kept constantly in motion by means of a stirrer consisting of a brass ring ^S'l that can be moved up and down ; it is possible to do this and still keep the space closed by utilising the flexible rubber tube ^S'^. The vaporization in the inner tube is thus very small (between 0.5 and 1 litre of gas measured under normal atmospheric pressure escapes per liour). Notwithstanding all these precautions temperature differences must still be encountered. In the liquid, in which the convection currents maintaining heat-equilibrium can be followed liy the small particles whic^h they carry along with them, these tempe- rature differences must have been very small. In the gas layer in the upper portion of the O-shaped space there must indeed have been considerable dilferences; but on account of the small density of the gas, these have but small influence upon the difference of level in the two limbs, and, moreover, that influence may be almost entirely neglected seeing that the observations are simply comparative measurements with and without the magnetic field. Now, care has been taken that the temperature over the distance that the liquid rises can vary but slightly, while in the upper portions of the tube practically the same state of affairs is maintained during both obser- vations. We have therefore omitted the correction that should still have to be applied for possible temperature differences. Comparing the positions of the liquid in the narrow cylindrical tube with and without the magnetic field also reduces the correction for capillarity to the insignificant differences in form of the menisci, and this correction, too, we ha\'e omitted. The inner and the outer tubes are closed independently of each other by means of the German-silver caps P^; P^, Qi, Qt (fig- 1); the junction is made air-tight by the rubber sleeves M„, iY„, which at the same time unite (he two tubes firmly together. Liquid oxygen ( 805 ) is introduced into tlie protecting tube tlirougli tlie ,sm;dl tube P^, and into the inner tube tlirougli Q^. Tlie two tubes I\ and Q, lead the vaporized oxygen through the valves P^ and Qg (fig. 2) to two gasometers. Two manometers /'„ and Q,, the latter of which is provided with an indicator Q^ so that small vapour pressures may be read off accuratelj", serve at the same time as safety valves. It is not necessary that the oxygen in the protecting tube should be as pure as that in the inner tube; for the latter, with which the obser- vations were made, very pure oxygen was used. A double sliding movement R allowed an easy adjustment of the apparatus each time, so that the meniscus in the measuring tube just reached the desired point in the field between the poles, usually in the axis of the pole pieces. Course of a series of measurements. The field is brought to t!ie desired strength and by means of B the meniscus is made to rise to the desired point, which is read off on a small scale. Then the meniscus is moved up and down several times while care is taken that the field slowly increases. In this way the temperature is made everywhere the same and the walls of the tube are wetted. While the field has the desired value the position of the meniscus is read off; then a reading is made while the field is off; after the meniscus has been three times allowed to rise somewhat higher than the desired position, another reading is made while the field is on ; once more a reading is made with the field off and so on several times. In this manner the error arising from vaporization of the liquid during tlie adjustment of the cathetometer is eliminated '). It is not essential to know the position of the level in the other limb of the tube ; so as to be able to take account of this, we ascertained the ratio of the cross-sections of the two limbs of the tube. We have further made sure that the residual magnetism exerted no appreciable influence upon the position of the meniscus after the current was cut off. For this purpose a feeble current was sent through the coils in the direction opposite to that which had just been iiroken. Had the residual field exerted any appreciable influence we should have seen first a further sinking of the level, and then a rise as the current was slowly increased. This has not been observed. We used the same electro-magnet as was used for the cryogenic investigation of the ferro-magnetic metals"), to which we must refer 1) To control the position of the meniscus without the magnetic field, we measured the quantity of gas vaporized (cf. preceding page). -) P. Weiss and H. KamerlingH Onnes, I. c. ( 8or, ) for details regarding its coiistriictioii. It was only necessary to replace the conical pole-pieces by cylindei-s with flat ends. Their distance apart was micrometrically adjusted to 25 mm. and controlled with an accurate callipers. We may here remark that between the measu- rement of the ascent and that of the field, the pole-pieces remained clamped tight to the cores, so the adjustment of the distance could give rise to no error. Since in the subsequent calculation the strength of the field is involved to the second power, and since we are concerned with an absolute measurement, we endeavoured to make our measurement of the field strength as trustwortliy as possible with our present appliances. With this end in view we measured the strength of an arbitrarily chosen standard field by two different processes, and we compared the strengths of the fields used in our experiments with this standard by successively withdrawing the same coil attached to a ballistic galvanometer from the standard field and from the various fields which we desired to measure. The standard field was set up with the same flat pole-pieces at a distance of 9 mm. apart, and with a current of 5 amp. All pre- cautions were taken to ensure the demagnetization of the magnetic cycle beforehand. This field was first measured by means of Cotton's magnetic balance^). As is well known this method consists of equilibrating weights of a total mass /;/ against the ponde- romotive power of the field H on a straight portion of length / of a conductor through which a current flows of intensity /; then we get m a H=^. 10. For the degree of accuracy, however, which we wish to reach, several corrections must be taken into account. In the first place the various parts of the balance were accurately calibrated. The length / of the current element was determined micrometrically and on (he dividing engine, and so also was the distance between the arcs of the balance which distance ought to be the same throughout seeing that the arcs must be accurately concentric. The very small deviations from this were allowed for by means of a ballistic investigation of the topography of the field. The balance arms of the weights and of the current element were measured with the cathetometer. The topographical study of the field also gave us the 1) For this method of measuring the liuld and for the magnetic balance see : P. Weiss and A. Cotton, Le phenomene dc Zeeman pour Ics trois rales bleues du zinc, Bull. Seances Soc. fran(;. de phys. 1907, p. 140, also J. do phys. 1907. ( 807 ) correction necessary for the force exerted upon the second strai^dit clement of the balance (i. e. that outside tlie [»ole-gap). The sum total of these positive and negative corrections came to some units per thousand. The greatest care had to be de\oted to the absolute \alue of i, which was measui'ed by means of an accui-ate ammeter by Siemens and Halske. This was calibrated in absolute amperes by comparing on tiie potentiometer the potential difference between the terminals of an international ohm lOr for the stronger currents of 0,1 i2) with the electromotive force of a Weston cadmium cell. For the requisite accuracy of the measurements the influence of neighbouring instruments or currents upon the ammeter, or of its position in the earth's field were by no means negligible; we got rid of almost all these irregularities by a suitable adjustment of the distances and of the positions of the rheostats, and we eliminated further possible remaining errors by so connecting all the conductors that the currents //* all except the am- meters could be reversed dt the same time. Finally we always used the ammeters in the same position with respect to the earth's field as that in which they had been calibrat-^d. When all calculations and corrections had been completed it was found that the strength of the standard field was 9857 gauss according to this method. The seconc\ method by which the \alue of the same field was found consisted of the sudden withdrawal from between the poles of the magnet of a coil of wire of which the area encircled by the current was known. The change thus caused in the number of induction lines embraced by the coil was compared by means of a ballistic gahanometer with the number of induction lines embraced by a solenoid the dimensions of which were accurately known. The coil consisted of 19 turns of silk-insulated wire, 0.25 mm. thick, wound round a cylinder of ebonite, 20 mm. in diameter. The dimensions were obtained by various measurements with the micrometer screw and the dividing engine, and were repeatedly controlled. At the same time a control coil was constructed In- winding bare copper wire in a helical groove cut in the curved surface of a cylinder of ebonite ; the area encircled by the current was then measured for this control coil by the same methods and with the same precautions as were adopted in the case of the first. The ratio of the two was in agreement with the ratio of the deflec- tions of the ballistic galvanometer which were obtained by connecting the two coils in sei'ies u itii tiie galvanometer and then withdrawing them successively from an unchanged magnetic field. We may further 54 Proceedings Royal Acad. Amsterdam. Vol. Xll. ( 808 ) say that we liad previously verilied the absence of magnetic pro- perties from I lie ebonite by means of an apparatus after CruiE in wliicli we utilised tlie attraction in a non-uniform field. For the measurement of the field there were placed in circuit with tlie galvanometer the coil on the ebonite cylinder, a manganiii resistance to i-egulate the sensitivity, a secondary coil of 500 turns fitting round the standard solenoid, and iinally, an electromagnetic arrangement which could be used as a danipei' if desired. We also allowed for the very small deviations from the law of pro- portionality between the detlections of the galvanometer and the (piantities of electricity, which had been determined for the galvano- meter (one of the Deprez-d'Arsonv.\l type) by a previous investigation. The solenoid was constructed with the greatest accuracy by winding i)are copper wire on a core of white marble '). The standardisation of the galvanometer was made by reversing the current in the solenoid ; the observations made by withdrawing the coil from the field always took place between two standardisations of the galvanometer; there was, howevei-, no change in the galvano- meter constant to be observed. The corrections and precautions necessary in obtaining the strengths of the current are the same as in llie case of the balance, and have already been deseril»ed. The final result of this ballistic method is 9845 gauss. The relative difference between this and the value given In Cotton's balance is therefore 0.0012; and (his can be neglected especially when one remembers that almost every one of the numerous meas- urements necessitated l)y the one method as much as by the other, beginning with the adjustment of the field by means of the ammeter, is accurate only to 0.0005. It may be useful to comment here upon a particular point that increases the difficulty of obtaining this agreement and therefore enables us to rely more upon the correctness of the numbers which we have obtained. The equation for Cotton's balance involves the strength of the current in the denominator, while this magnitude in calculating according to the iiallislic method occurs in the ninin'rator; a .systematic error there- 1) l''or tlie dimeiisioiis ami tlic description of tlie solenoid and galvanometer see: P. Weiss, Mesurc de I'inlcnsile d'aimantation a saturation en valeur absolue. Arch. Sc. phys. et nat. February 1910, J. de pliys. May 1910. ( H()9 ) fore ill llie nh.wlate iiiiiiihei' of .iiiiperes would, of necessity, occasion a relative difference twice as (jreat between the values of tiie field obtained by the two methods (the same ammeter was used with the balance and with the solenoid). We have given the ballistic method a soiiiewhat greater weight than the other on account of the smaller number of corrections it involved, and thus we have tinally taken as the value of the standard tield 9850 gauss. Once this standard field was definitely fi.\ed all other measurements could be rapidly made by the ballistic method described above. For the conical pole-pieces which are employed in experiments according to the maximnm couple method, and which give much more powerful but much less uniform fields, we need a coil of 7 to 8 mm. diameter accurately centred on the axis of the pole-pieces. In this case direct comparison with the standard field just mentioned was not possible since the flat polepieces had to be screwed off to make room for the conical poles. To meet this case the area of the small coil encircled by the current was determined once and for all by withdrawing it from the standard field before the flat pole-pieces were removed, and comparing the change thns bi'ouglit about in the number of the induction lines with those of the solenoid by means of the ballistic galvanometer. All the measurements that we have given up to the present refer to the field in the centre of the space between the poles. For the fev.' exceptional values of the field, and, consecpiently, of the ascent of the liquid oxygen for which it was necessary to cause it to rise pretty far above the axis of the pole-pieces, the field was determined at those points by simple ballistic comparison with the fields on the axis, and we made use of the cathetometer to adjust the position of the small coil. 54« ( 810 ) Results of observations and calculations. Series of observations with the cipparatiis with unsihered walls. TABLE la. '=- 183'^.OC. ') Position of Obs. rise diff. in height with //in ^••0^ meniscus. s' in cm. reservoir s in cm. gauss. Level of a.xis 1.032 1 .061 2980 1.194 „ 1.046 1.690 3727 1 .210 „ 1.656 1.701 3727 1.224 axis -f- 2Ai cm. 3.024 3.110 5182 1.158 axis 3.198 3.289 5205 1.214 „ 4.16 4.278 5848 1.251 axis + --44 4.90 5.050 6570 1.108 axis 5.124 5.270 0600 1.210 axis + 2.44 7.87 8.094 8075 1.242 axis + 2.44 9.20 9.462 9043 1.158* The difTereiH'e in heiglit ; was obtained from the observed ascent from c = c' (i + 0.0285). The observation "•" was very dillicult and is little reliable. The deviation in the ease of the observation in a field of 5182 gauss is probaiily due to a mistake of 1 in the number of whole millimeters which were read off, but of this we are not certain. Deviations from proportionality with H'' arc considerable but by no means systematic. If we take the mean of all the measurements witii the exception of the last in which special ditHiculties were encountered we reach the value I I' = 1.209 . 10-", and for the specific susceptibility with _(/ = 981.3 lor Leiden X,„MK. = 237.8. 10-e. ') The boiling point of oxygen accoiiling lo 11. Kameulingh Onnes andC. Braak These Proc. Oct '08. Gomm. Ir. th. Lcydcn labor. N". 107a § 6. ( 811 ) TABLE it t=- 201". 75 C. Position of meniscus. Obs. rise s' in cm. diff. in lieighit vvitli reservoir s in cm. H in gauss. m-'" Level of axis l.lOi 1.222 'iUsi 1 1.376 ,j 1.893 1 94 i 3727 1.399 „ 1.881 1.935 3727 1.393 1 3.64.3 3.747 5205 1.383 „ 4.623 4.752 5848 1.389 „ :..9l 6.078 6600 1.395 axis-j-2.5 cm. 7.376 7.586 7421 1.378 axis -(-2.5 cm. 8.715 8.963 8069 1.372 mean -^-^ = 1,386.10- whence it follows that xnT..^ K.= 272,0 . 10 -6 . Finally, at — 209°, 2 C. a single observation was made. Tlie rise was 6.115 t-m. in a field of 6600 gauss, which with the correction for the sinking in the reservoir gives H - = 1.444 . 10-: and yggOpK. - 283.4 . lO-c. We shall now give the series of observations made with the silver-walled apparatus which we have already described. I TABLE II a. f = -183°.0C. Position of meniscus. Obs. rise z in cm. diff. in height with reservoir 3 in cm. H in gauss. «v.'"' • axis 1 llljO 1 . 1 100 2980 1.227 „ 1.009 1.710 3727 1.235 ji 3.169 3.258 5183 1.213 „ 3.220 3.310 5198 1.225 J, 4.035 4. 148 5807 1 230 H 4.093 4.208 5848 1.230 „ 4.101 4.216 5848 1.233 j^ 5.119 5.262 6578 1.216 axis -f 2.5 cm. 7.750 7.967 8075 1.224 „ 8.950 9.201 8659 1 227 axis -|- 3.5 cm. 9 220 9.484 8808 1.222 " 9 266 9.525 8808 1.228 For this apparatus z = z' {1 -\- 0,0280). — is 1,226.10-", X90O.1K. = 240.6.10-6. The naean value of — is 1,226.10-", whence it follows that ( 812 ) TABLE 116. t= = — 20i°.75 Position of meniscus Obs. rise z' in cm. diff. in height with reservoir s in cm. H in gauss S-- axis -1.195 1.228 2980 1.383 „ •1.879 •1.932 3727 1.391 „ 3.625 3.720 5205 1..375 „ 4.5G7 4. 095 5848 l.,373 axis + 3.5 cm. 5 4G1 5.014 0399 1.371 axis + 2.5 cm. 5 8.32 5.995 0507 1.390 axis 5.852 G OIG 6000 1.381 axis + 3.5 cm. G.4G3 0.044 6986 1.305 axis + 2.5 cm. G.899 7.092 7169 1.380 axis + 3.5 cm. 8.207 8.437 7803 1 .305 axis +2.5 cm. 8.G54 8.892 8009 1.300 axis + 3.5 cm. 8.988 9.240 8212 1.370 axis 8.913 9.102 8212 1.358 Mean of all observations is X71°.35K 1.375 vvlience it follow! — 269.9. lO-G. that TABLE \\c. t=- 1 -208°.2 C. Position of meniscus Obs. rise s' in cm. diff. in height with reservoir s in cm. Hin gauss m-^'' axis 1.277 1.313 2980 1.478 „ 1.990 2.052 3727 1.477 „ 3.813 3.920 5205 1 .447 1 „ 4.841 4.977 5848 1.401 axis + 2.5 cm. 0.012 0.180 0507 1.4.33 axis 0.094 0.204 0000 1,438 „ 0.113 0.284 0000 1.4i:i axis + 2.5 cm. 7.116 7.340 7109 1.429 axis + 3.5 cm. 8.579 8.819 7863 1.420 ( 813 ) Mean of all ohserxalioiis 1 .44S, \vli(>iice it follows thai -/6io,,K. = 284-2. 10-G. Finally for finding tlie specitic- suscoptiliiiily llie density of oxygen was found from th,e formula') Q = 1.2489 — 0.00481 (T — 68). From tal)le II we obtain A'9o°.i K.= 275,2. 10-G i!r7i°,3.Mv.= 332,8. 10-G A'61^9 K.= 359,0 .10-6. Table III gives y,\/T for eacli of the temperatui'es and for each of the series. Series with the first apparatus Series with the improved apparatus • T z-IO"^ : yyT.-iO- T [ z.lO« yVTAO' '.111 1 u: '.:.:'. 2.25 71 85 272 2.29 0', 9 283.4 2.26 '.lil.l 240.6 71.35 269 9 64.9 284.2 2.283 2.279 2.289 mean 2.27 ■ 2.284 There is no systematic change to be noticed in the product xj/?'; the greatest deviation fi'om the mean is i"/\ with the first apparatus, and only \/, "/„ with the second ; moreover the deviations in tiie two series at corresponding temperatures are in opposite directions. Hence within the limits of accuracy of the observations the specific suscep- tibility can be represented by the formula 2284 Z = i/r 10 In the comparative measurements which we shall describe in the sequel we shall find the same law, at least as far as its form is regarded. For the discussion of this point we refer to § 5. The differences between the various values of the ratio — - are greater than we should be led to expect from the accuracy obtained ») B.\LY aud DoNXAN-. J. Ghera. Soc. 81 (1902) p. 907. ( 814 ) (0,()5"/„1 ill lliL> ineasiireiiienls wiili tlie callielometer of the dis[)liice- ments of tlie level, and from (lie aeciiracv of the measurements of the fiehl-strengths, of which a discussion is given above. It is certain liiat liie eanse of these deviations must lirise from a source other liian tlie measurement or these two data, fliongh we cannot with certainly indicate what (his may bo. We may in the meantime remark lluit, at least in the case of the first series, the unsteadiness of the apparatus in the vertical direction in the not ([uite homogeneous field, and the slight inconstancy of tlie temperature iiave certainly been contributory causes of these deviations, since the second apparatus which was improved exclusively in these directions gave much more regular results. This remark, howevei-. does not seem to account sulTiciently for certain appreciable changes that occurred without any noticeable corresponding irregularity in the pressure or in the convection current of the liquid, while there was also no iK)liceable change in the shape of the meniscus. Liquid o,ri/_(/ii/ 11. § 3. Measurements hy tlie method of the ■ina.rimum couple e.rerted upon an ellipsoid. Further comparative measurements for liqiu'd oxygen at various temperalm-es were oblained liy means of the method of liie maximum couple exerted by a uniform field upon an cllipsoiii. Tliis method has already been descriliod and discussed in connection with the research on ferro-magnelic substances^); it will be sufficient to discuss the modifications which were found to be necessary owing to the particular circumstances untlcr wliich ihe method had to be applied to the present research. In the first place on account of the small value of the susceptibility it was necessary to make the couple to be measured as large as possible; with this end in view we chose an oblate ellipsoid of revolution, instead of a prolate; its axis of revolution was placed horizontal in a field which could turn round a vertical a.vis. The ratio that is taken between the axes is not a matter of in- dilference; for a given major axis the couple, which is propoitional (,) (jV^ — N..)r, is a maximum for a ratio of Ihe major lo Ihe minor axis that is only slightly smaller than 3; we have therefore taken Ihis value of Ihe ratio for the conslrnclion of the ellipsoids. We used the same electromagnet as served for the measurements made by Weiss and Kamerlingh Onnes (loc. cit.). Two pairs of pole- 1) P. Weiss, .1. de pliy.s. (4) (J (1907) p. Ouj. F. Weiss and II. Kamkhungu Onnes, (Joiuin. N". 114 Tlicso Proc. Jan /Kclu- l'.)10. ( r 815 ) • [ileccK were used; lii'sl llie r_\ liiidriral |)oIe-pieces with {|iiile ll;il end surfaces lliat had been used tor the measurement of the magnetic rise, and then truncated conical pole-pieces the end surfaces of which (slightly concave, see in this connection p. 818) were 4 cm. in dia- meter, and the side surfaces of which were connected by convex surfaces of revolution to the cylinders that formed the cores; these were 9 cm. in diameter. These pole-pieces were constructed to give the strongest possible Held when the distance between the poles was taken to be 20 m.m. l>y this means a Held of about 16000 gauss was obtained. Our observations were made with an ellipsoid that was diamagnetic with respect to the surrounding medium — a solid silver ellipsoid immersed in a bath of liquid oxygen. The ellipsoid was turned by the "Sociele genevoise pour la construction d'instrnments de physique" from a block of very pure Merck silver. A preliminary experiment showed that it was very slightly diamagnetic with respect to air, and that this was quite negligible with respect to the liquid oxygen. The axes were measured microscopically on the dividing engine ; this gave major axis =1.0973 cm. and tixis of revolution ^0.3654 cm. Furthermore, two intermediate ordinates parallel to the axis of rex'olution were measured on the dividing engine, and they were found to 1)6 27„ greater than the corresponding ordinates of a perfect ellipse with the same axes. This deviation from ellipsoidal shape was contirmed by a direct determination of the volume from the weight and the densit}', which gave 0.2329 c.c, while calculation from tlie dimensions of the axes gave 0.2308 c.c. In the calculations we made use of the value 0.2329. The cryogenic apparatus, essentially the same as that used by Weiss and Kamkrlingh Onnes is shown in PI. I tig. 3. Once more we see the cover B, the adjusting tulte /', and the holder //. The cover with its various parts: the cap with the stufiing-box D, glass tube C, window with plane parallel glass plate L\ , the system BG for adjusting the whole apparatus, the tension rods B^ for supporting the Dewar tube, the helium thermometer 8, the little screens to protect the upper portions of the apparatus from cooling, etc. is just the same as before. The Dewar tube is of the same shape, but the lower portion is of greater diameter. The only difference between the adjusting tube /" and that which was used in the other investi- gations is that the lower portion // is of greater diameter. ( 81fi ) 'riic Jiolih'f niid tlie hirsinn .v/y/v//*/ ;ii-c, on the oilier hand, coiiiiilftely iiUored. On acconnl of tlic snialliiess of llie couiilo lo he measured all foreign magnetic actions had to he eliminated as carefully as possible. Preliminary experiments showed us that a metallic holder could not be used, not only on account of the traces of para- or ferro-magnetic impurities that are never absent from workable metals but also on account of the difficulty of keeping the surface sulliciently clean ; this difficulty was encountered repeatedly in the silver ellipsoid that we used in oui- experiments, and it is probable that the constant contact of the hands with iron tools plays a part in causing it. Glass seemed to be by far the most suitable material both on account of the absence of iidierent magnetization and of the fact that the surface on account of its smoothness can be kept quite clean. The holder which we finally adopted was made completely of glass: it consists of a tube // 5 mm. in diameter that at //, is drawn out to a narrow but thickwalled stem, 0.7 mm. in diameter. To this stem the silver ellipsoid was attached ; for this purpose a hole of sufKicieiit width to tit was bored along one of its greater diameters and the ellipsoid was then fixed at the desired height by means of a little wax that completely filled the narrow space between the glass and the metal. The tube was then pumped free from air and sealed oft", so that the liquefaction of the air that it would otherwise contain would be prevented. The flat mii'ror for measui-ing the angle of torsion and the oil-damper were also attached to the holder. The torsion springs. On account of the smallness of the couples to be measured (the constants of the springs were of the order of 1200 c.g.s. while those used for I he investigation of the ferro- magnetic substances were 'some tens of thousands) it was found more suitable to use a straight instead of a helical spring. We took a strip of phosphor bronze about 5.5 cm. long (^/') and 0.2 X"*^^ ''Cl- <^'iii- in cross-section. The upper end was soldered to a sjiii'al spring of three turns made from a much thicker strip than the other: the greatest dimension of this stri[) was horizontal so that in this way it fulfilled its purpose of being elastic to tension while taking no part in torsion ; its presence is essential to prevent the breaking of the thin glass stem or of the platinum-iridium stretching wir'e that is soldered to the stem. This stretching wire is made from a platinum- iridium wire of 0.1 mm. diameter, which was rolled very thin so as to make its torsion constant extremely small without diminishing to any great degree its resistance lo breakage. ' The stretching wire is fused at //, to the lower end of the glass stem, and at its other e.\tremit\' it carries a knob c which is held fast in a ling /' . ( 817 ) Tlie iiKiiuiliiii;' of llie npiiaialii,-^ ludk place willi the same \n-Q- f;iutioii8 i-egardiiii!,' the (•ciilrinu nf ilic whole, the leiisjoii of the springs, etc. and by a nietliod siunlar In ihal which hat; heen dcscril)ed in the research njion the I'erro-niagnctic nielals. T/ii' course of the oh.s-e/vnfiojis is very simple once everything has heen set up in position. First, those azimnths of ihe electromagnet are tentatively determined for which the couple in both directions is a maximum. It was sufficient lo do these experiments two or three times with suital>ly chosen fields, since the azimuth changes bi; t very little with the field, and for other values of the field one can without danger have recourse to interpolation. After tiial the series of obser- vations took place in the following manner : Before making a measure- ment with any particular current this was reversed a certain number of times so as to obtain a well-defined field ; we had not here to deal with a value of the saturation-magnetization, which changes but slowly with the field, but in our case the couple was proportional to the square of the field, so that inaccuraie \alues of the field that might be obtained notwithstanding the fact that the iron of the electromagnet was extremely soft would make their influence very strongly felt in our results. Then the electromagnet was adjusted to one of the determined azimuths, the torsion angle was read off for the two directions of the field, the current broken, the electromagnet tin-iied to the opposite azimuth, and so on several times. At the end of a series a measurement with one of the first fields was repeated as a control. Sources of error, difjiculties, corrections, and controls. I. Inhomogeneity of the magnetic field. As will be seen from the following discussion this source of error is by far the most important in our case and is indeed the only one that need be taken into account. If we assume that the field near the centre' of the pole-gap may be re[)resented by an expression of the form H = H,-V~i-~\{co,-'6-ksin^6) (3) where i/„ is the field in the centre, and r and 6 [lolar coordinates of a point in the pole-gap with respect to the centre as origin ';. Let us now replace the ellipsoid by a vertical disc whose diameter is equal to the major axis of the ellipsoid; by taking the expression for the energy of the magnetized disc in the field and differentiating it with respect to the angle between the disc and the lines of force, we obtain for the couple caused by the inhomogeneily of the field : M Gf. P. Weiss and H. Kamkrlingh Onnf.s I.e. ( 818 ) M ' :=: — _ vr- 1 -— . .v«i (/ cos (p (4) {r = radius of tlie disc). Tiic ratio — of this couple foi' an angle ol 45° to tlie tuudaniental M couple is M' _S ydf If then we suppose that the relati\e change of the field in the space occupied by the ellipsoid is of the order of 1 in 1000, the formula given above shews us that although the disturbing couple is a little smaller than the chief couple, the two are of the same order of mngnihule. Hence we see the great influence that this source of error can have in the investigation of weakly magnetic substances. (With ferro-magnetic bodies it is quite negligible : see the previous paper). We have accordingly devoted the greatest attention to this source of error. The conical pole-pieces were made slightly concave, during which process we every time determined the iiihomogeneity of the field by means of a ballistic galvanometer and a small coil that was slightly displaced. We ascertained that the change in the field in a space of about 1 c.c. was certainly less than 1 in 2000. We have not had time to pursue this investigation further, and, besides, we should have to obtain a much more sensitive ballistic galvanometer. But it will be seen that the homogeneity of the field was sufficient for the comiiarative measurements we proposed to make. We may further remark that all these precautions refer exclusively to the conical pole-pieces; the experiments with the cylindrical pole-pieces were nearly free from these sources of error. We allow for these disturbing couples in the following way : Assuminti- that ( — ) = ^H the expression for the couple due to inhomogeneily given above becomes (<f ^ 45°) : 3 or — vr^ X HI 32 — rr' ).KII\ 32 which we shall represent h\ (ihll- . If 11 is the angle of torsion of ll of the spring, then ic holder and (' the constant ( 819 ) Cct = — {N,~A\)K'ir -\- tmif (5) Thus just as if there were no correction for iniiomogeneity the second side of the equation remains always proportional to the square of the field. Even without knowing that correction, if ji is itself a constant we should be able to deduce from the observations whether K is a function of the field or not. We see, however, that the constancy of /? requires that of ?., i.e. that the field must remain liomothetic no matter how great it should be. Now this is not the 2f) case as can be seen from the quotients — in tables V, VII, and VIII. Table V shows first an increase, then the quotient reaches a maximum and diminishes considerably; tables VII and VIII shew a change in exactly the opposite direction ; this is just what one would expect if ji wei-e variable and K constant, for the tables refer to two practically identical bodies, of which the one is dia- and the other para-magnetic. Now in either case the fundamental couple (uniform field) is in the same direction while the couple due to iniiomogeneity changes sign with the susceptibility ; should, therefore, the correction in the one case first increase and then decrease, it must in the other case first decrease and then increase. We shall return to this point in § 4. Since this determination aims only at relative measurements, we have once and for all taken as the value of the susceptibility of oxygen at — ]83°C. the value that was given by the improved apparatus for measuring the magnetic rise. With the help of this value we have calculated the values of ii for each field from equation 5): (see tables V and VI). These values fall pretty well on a curve of means. Finally the susceptibility at the lower temperatures is calcu- lated by means of the value of |3 as a function of the field given by this curve. We shall take the opportunity of the corresponding series of observations to make some remarks upon the influence of the inhomogeneity for each of the three pole-gaps that were used. 2. llie Inconstancij of the magnetization as a function of the azimuth. The general expression for the couple in a uniform field (iV, — N^ ) /- V sin ifj cog <f oidy reaches its maximum value just at <p = A5' , and consequently sin rp cos 'f =: '/, since 1 remains constant diirimj the torsion. Here again we see a fundamental difference between the application of this method to the investigation of saturation magnetization and to that of a body of constant susceptibility. It is clear that in the first case the condition i=: constant is, as it were, fulfilled by definition. ( 820 ) In (Hir cMse llic deviation fV(.)ni tiiis is hv no means r? y*y/('r/ nenii^ililo; llie two liniilinji' \alues of /"('/i^O and </ ^ DO ) ddler in onr ease by O.o "/„, and sinee /" always elianges l)el\veen these two limits ill the same direction the error caused thereby when sill <i cu.t <f = \ „ is less than 0.1 7o- In contrast with the two foregoing' sonrces of erroi', the reaction t)f the magnetized ellipsoid npon the ilistrihutioii of iiKij/inii.sni over fhr sKrfdci' of f/iii poli'-pk'cc^ can clearly have no ell'ect in the case of a, body of small susceptibility while on the other hand, it had to be taken into account in the case of the ferromagnetic bodies. Indeed, with oxygen we have to deal with a magnetization that in the strongest tields of the electromagnet reaches a value of only a few units (in the case of iron it was J700!). o. I iijliu'iici' of the holder. In this connection we may notice two actions that may go together. In the iirst place there is the iidierent nu\gnetisni of the s'em, and then there is also an action analogous to that which we wish to measure, for if the stem is not a perfect body dl' ri'N'oluliun, it is acted u|)on in the liquid oxygen just as if it were a sii/)p/ei)!entari/ ell/jtsoid. We investigated these two sources of error in a blank experiment in li(|uid oxygen in which the silver ellipsoid was lemoved, and the surface of the glass was carefully freed from all ti'aces of wax. From this we obtained a maximum of only 1 to 2 7oo which need not be taken into account 4. yV/r coiiceiitrettion of the Od-i/</e)i. The oxygen in the bath contained a little nitrogen, the concentration of which constantly decreased during the experiment owing to its faster vaporization. So as to be able to allow for this we analysed the gas at the beginning and at the end of each series of observations. The mean concentration was 1.25°/,, at the beginning and 0.35"/,, at the end (at the moment that the Dewar vessel was almost empty). We allowed for this concentration as far as possible; in this respect there remains an uncertainty of about 0.37o- 5. Calibration of the susjjension springs. The main torsion spring- described above was calibrated outside the apparatus by observing the time of oscillation of a system suspended from it with and without the addition of a known moment of inertia. For the latter we used a bronze ring of rectangular meridian cross section, the diameters and height of which were measured with the cathetometer. Calculation gave the moment of inertia as 582.09 c.g.s. Care was taken that the spring was subjected to the same tension during the calibration as it experienced while in the apparatus (by attaching suitable weights to it by a torsion-less wire). ( «21 ) For the coiislant of the spring' we ibiiiid 1 lcS4,5 c.g.s. The platiiiuiu-iridium stretcliing wire gives a torsion conple as well as the spring; (he correction for this was determined by the same method as was used in liie analogous case by Weiss and Kameklingh Onnes (loc. cit.) and it was found to be 0.0152 times the constant of the spring. The difference between the values of the constant at J8'C. and at — l^O'C. is smaller than the errors of ol).servation. The calculiUions were therefore carried out with the constant 1184.5 (1 + 0.01?2) = 1202.5. 6. O.s-cillations. The silver ellipsoid should be protected sufficiently fioni the influence of oscillations arising from external causes by the occurrence of intensive Foucaclt currents, but tiie occurrence of these currents, which were unusually strong gave rise to great difficulties in the observations. In the first place the holder was extremely slow to reach its position of equilibrium. Further, the smallest ciuxnge in the current flowing through the electromagnet occasioned a sudden kick in the whole moveable apparatus, an immediate result of the oblique pt)sition of the ellipsoid with respect to the lines of force. Hence the regulation of the current had to be done with the greatest care. We retained the oil-damper but removed the fixed partitions, for the capillary action of these gave rise to couples that, although small, were still not negligible. Rt'^-ults of the obsei^ations. TABLE Wa. Cylindrical pole-pieces 21 mm. t = — 183=.0C. apart. H gauss double de- flection 2i cm. of the scale ^•- /(T. 106 'i'J50 0.:',7 0.731 277 4537 1.41 U G8."> 268 8 G676 3.21 0.7200 275.3 8339 5.10 0.7335 278.1 9387 0.44 . 7307 277.5 •10120 7.45 0.7274 277.0 •10685 8. '20 0.7234 276.1 11130 8.99 0.72.58 271.6 •H440 9.38 7107 274.8 11705 9.90 0.7155 274.0 V 1 Mean /iTgoOiK. — 275.6 ( 822 ) TABLE \Vb. Cylindrical pole-pieces 21 mm. r = -2orMC. apart. H gauss double de- flection 2= 2-: cm. of the >/2-*" scale /C.lOe 2250 0..50 1 0.088 324 -4537 2.10 1 .020 328.0 0G7O 4.06 1.041 331.1 8:«9 7.36 1.0.58 334.0 9387 9.17 1.042 331.4 10120 10.63 1 .038 330.8 10685 11 .81 1 034 330.2 •11130 12.70 1 .025 328.6 •11440 •13.41 1.024 328.4 •11765 14.14 1.022 328.0 1 Mean /v-oo.oK. = 330.0. Witlio It being corrected for lack of tiiiiforiiiity in tlie tie!'! tl means yive the fullowin"- valnes : X'JU".1 K. '.0 K. 275.6 1.143 330.0 . 10-6 = 241.1 lU-G 10-6 = 2G8.3. 10 -6. 1.230 Tlic corresjjunding resnlls obtained by tiie luetliod of the magnetic rise were 240.(3 . l(J-6 and 2(iy.3 . 10-6. The dilferences between the results as obtained by the two methods arc scarcely 0.4 "/„. This gives us great confidence in the ellipsoid method even for this j)articnlarly difilicult determination, and it shews that the method is also suitable for absolute measurements if only the necessary care is taken to ensure tiie uniformity of t!ie Held and the correctness of the shape of the ellipsoid. We must remember thai there was a great nundier of absolute ( 823 ) measuremenls whose results liad to be used (axes and volume of the ellipsoid, constants of the springs, magnetic field, density of the liquid oxj-gen) and also that the shape of the ellipsoid was not perfect. On the other hand we must remark that the application of the correction for the non-uniformity of the field might conceivably have diminished the correspondence between the results obtained by the two methods. We liave, however, both theoretical and experimental grounds for the assumption that this correction remains within the limits of accuracy of not more than 0.5°/„ in the case of cylindrical pole-pieces with flat end surfaces 90 mm. in diameter and at a distance of 21 mm. apart, TABLE Va. Conical pole-pieces 20 mm. apart. ^=:— 183°.0C. _g (To determine ,^ we assumed Zg^o ^ = 240.0 . tO ). H gauss. Double de- flection 2o cm. of the scale. '^0 10«/5 3685 1.27 0.935 58.5 4615 1.96 0.920 54.7 6944 4.55 0.9437 60.8 9205 7.96 0.9400 59.7 1-1280 11.90 0.9348 58.5 12835 15.44 0.9374 59.1 14015 18.26 0.9295 57.0 14900 20.19 0.9098 51.9 15585 21.73 0.8945 48.1 16120 22.87 0.8802 44.5 A graph of /? as function of H was made, which was used for the following table. 55 Proceedings Royal Acad. Amsterdam. Vol. XII. (824) TABLE Vb. Conical pole-pieces 20 t = - 208°.2 C mm. apart. H gauss. Double de- flection Vo- cm. of the scale. w.-- /3 . 10" K. 10« 2296 0.79 1.498 56.0 [357] 4015 2.75 i,291 57.8 [328] 0944 6.82 1.414 58.5 344.2 9205 1-2.15 1.435 59.8 340.3 ■1 -1-280 18,25 1,434 59.7 340.2 12835 23.07 1 . i37 59.2 346.9 14015 -27.89 1.420 56.5 346.1 1 4 900 30.84 1.389 52.4 345.1 I55S5 33.24 1.368 48.0 345.0 •10120 35.44 1.303 43.8 347.3 , The mean with the exception of the two values phxced between brackets is 345.9 and it gives ; ! X64O.9K. = 275,0.10 6 while the method of the magnetic rise gave X64°,o iv. = 283,5. 10-6S The dilferenco is IS"/,,; but in this connection we must remember that the correction for non-uniformity is about J67o> and that the temperature of the liquid becomes very uncertain at the pressure of 11 mu). under which the liquid boils at this temperature. Finally, we now give two series of measurements which were made with otiier pole-gaps .so as to obtain other deviations in llic uniformity of the field. They were hastily made and under un- favourable circumstances, since oscillations and disturbances caused by the running of machines in the neighbourhood interfered with the observations. We give them more as examples of how the method of calculalion followed still leads to good results even when the couples due to non-uniformity of the field are extremely large {287o of the chief couple). ( 825 ) TABLE Via. Conical pole-pieces 18.2 mm. apart. t= — i83°.0 C. (To determine (3 we assumed XyQo ,■ =240.6. 10"" '). H gauss double de- flection 2o cm. of the scale S-'°^ ,3.106 ! 5013 3.34 1.328 ir.0.2 7547 7.31 1 .283 147.4 9993 12 64 1.246 137.9 12165 18.33 1.238 136.2 13760 22.39 1.183 121.7 14900 26.26 1 . 182 121.5 15750 28.83 1.162 116.6 17005 35.58 1.230 133.9 ji was again graphed as a function of H, which led to the cor- rection for K in the following table. TABLE V\b. Conical pole-pieces 1S.2 mm. apart / = — 208O.2 C. H gauss double de- flection 2o 2o cm. of the m ' scale ,5.106 AT. 106 1 5013 46.8 1.861 152 341 7547 10.71 1 .880 146.5 348 9893 18.88 1.862 137.5 351 12165 27.92 1.885 130.7 357 13760 35.35 1.868 125.5 359 14900 41.36 1.861 123.0 360 15750 45.67 1.840 122.5 357 17005 51.81 1.791 127.5 347 55* ( 826 ) Tlie ineai) 353 aives y — = 279. 5. 10^'' a value that is not ^ 1.255 iiiiich smaller than 283.5.10-''', which was obtained by the method of the magnetic rise. Solid oxygen. § 4. Ellipsoid of solid oxygeji. In this case observations had to be made directly upon an ellipsoid of oxygen. The oxygen therefore had to be fi-ozen in a mould of approximately the same form and dimensions as the solid silver ellipsoid described above. This new condition nece-!sitated the following experimental arrangement. The cover and the Dkwar tube are the same as for liquid oxygen, willi the exception of the cap 7>. The adjusting tube is also the same, but it is so arranged that it can be moved as a whole up or down, while the whole apparatus remains closed and in its place. With this end in view it is attached to the tube m, which moves through the stuffing-box D'\ ; tliis corresponds to D^ of the liquid oxygen apparatus, but in this case the wide glass tube C'l is lengthened by a rigid bi-ass tube M that serves to give sufficient play to the vertical movement of the whole adjusting tube. The former stem k had to be lengthened by the same amount (L", L",), and is contained in the tube in. The holder is also a glass tube 6"; it is not however closed, but at //', it changes into a very much narrower tube (0.5 mm.) that ends at l)\ in a glass ellipsoid a". To this ellipsoid there is fused a solid stem h\ that connects it with the stretching wire. The oxygen gets to the ellipsoid through the holding tube which it enters at b'\. A rubber tube n ((/ =: 3 mm.) admits the gas from outside; it is attached to the iidet tiil>e »., that passes through the cover and is soldered to it. With this arrangement it is easy to cause the oxygen to solidify inside the ellipsoid. When the apparatus is ready for use the adjusting tube is pulled upwards by the cap A till the glass ellipsoid reaches the unsilvered part of the vacuum glass. The vacuum glass is then tilled with liquid hydrogen. While the ellipsoid is still connected with a reservoir of oxygen, the adjusting tube with the ellipsoid is slowly pushed downwards until it does not quite touch the liquid hydrogen but is in its vapour. The oxygen is then seen to condense slowly, and, if the operation is carefully performed, the whole ellipsoid and supply tube are seen to fill with liquid oxygen. The tube being lowered still further, vapour is reached that is sutlTi- ciently cold to cause the oxygen to solidify. On account of the large ( 627 ) contraction of the oxygen on solidification it is seldom tliut one does not see some empty space in the ellipsoid; the 0[)eration must then be repeated several times, since the oxygen that is still liquid at this temperature has a pretty great viscosity and flows with dii'ficulty from the tube ; we shall return to this point later. When the ellipsoid is completely filled with solid oxygen the adjusting tube may be lowered right down. A mark is made beforehand, so that the ellipsoid may be accurately adjusted to the centre of the gap when the silvered tube is again in its place. Errors, corrections, auxilianj measurements. 1. Couples due to inliomogeneity. As will presently appear, we iiiade measurements not only in liquid hydrogen (solid oxygen), but also keeping everything else the same, at two temperatures in a bath of liquid oxygen (i.e with the same ellipsoid of liquid oxygen). Since the susceptibility of the liquid oxygen was known, we had therefore two measurements of the couples doe to inliomogeneity as a function of the field; they are given in Table VII. As a result of the some- what sniallei- dimensions of the ellipsoid, these corrections are com- paratively much less important. 2. Parity of tlie oxygen. The oxygen was freed fi-om nitrogen by vaporizing a large quantity of impure liquid oxygen under reduced pressure. 3. Density of the solid oxygen. We have alfeady mentioned the difficulty of completely filling the ellipsoid with solid oxygen. ()n account of the opaqueness of the oxygen that has already solidified one cannot with certainty assert that this condition has been fulfilled '). Since the specific susceptibility is determined from a known \ohime this error would have immediate effect upon the result. We ti-ied to eliminate this error as well as possible by deter- mining the density with the same ellipsoid by filling it with solid oxygen under the same circumstances as those obtaining in the experiments and then measuring the quantity of gas formed from it on vaporization. We may assume that the small cavities that may form are pretty much the same in the various cases. Indeed, from two similar measurements the density measured in this way was found to vary by only about I'/o- Ky taking as the mean density that determined by these experiments, the eventual presence of cavities is allowed for. In this way we obtained Q Z= 1.41. The absolute values of the couples due to inhomogeneity of the ') When there is an empty space af a few mm-', however, it can be seen quite well. ( 828 ) Field are not iiiofiified by a cavity formed in the vertical axis, as was usually the case, for it is clearly those portions towards the surface of the ellipsoid tliat are the chief contributors to them. On the other hand, they might obtain a greater relative influence, but as the observations shew, the sum of the corrections arising from this cause is so small that they may be regarded as independent of the susceptibility within the limits of accuracy of the experiments. In I hat case this difllculty completely disappears. 4. Dimensions of the ellipsoid. The internal volume was obtained by lining liie ellipsoid with mercury and weighing it. Tt was 0.1812 c.c. The change of \'olume under atmospheric pressure was found to be of no accouul Ity pumping the space above the mercuiy free from air and observing the position of the mercury in the capillary. The external axes were measured directly. Then the thickness of the glass at ten dilFerent points was determined by focussing a microscope on the image of the outer surface formed on the mercury with which the ellipsoid was filled. It changed but slightly from place to place. The mean was taken and twice that value was sub- tracted from the external measurements. The results were: 1.044 cm. and 0.335 cm. Calculating the volume from these figures we get 0.1925 c.c. which is about 6 % gi'eater than the true volume as directly determined. This is accounted for by the special shape of the meridian section which curves somewhat too strongly at the outer ends. For calcu- lating the coefficients of demagnetization we took a mean ellipsoid with the same major axis and the minor axis small enough to give the real volume'). The data for the calculation were therefore: 1.044 cm. and 0.3173 cm. 5. Opposing couple. The suspension spring and the stretching wire were the same as woie used for the liquid oxygen. We must, however, allow for the rubber supply tube for the oxygen. This (which was chosen as thin as possible) modified both the zero and the constants of the total opposing couple, as soon as the pressure 1) It is clearly not qiiilL' right to do this; llieru are, however, experiinental data to support this method of correcting: V. (Juittker (Diss. Zurich I'.IUS, also Arch, sc. phys. et nat Geneve, Sept.— Nov. 1908) found tliat this method of treatment was sufficiently accurate even for discs, bodies that deviate fai- more from an ellipsoid than those we used. ( 829 ) difference between the inside and tiie outside of tiie tube appreciably altered (on account of the change in sliape of tiie tube). In all oui experiments, therefore, we took care that there was a constant pressure difference of 70 mm. between the pressure inside the cover and that inside the holder (the latter was the smaller of the two). We got a very sensitive indication of the constancy of this difference not only from the manometers but also from the zero position of the holder. Experiments carried out outside the apparatus shewed that the constant of the total couple changed about lO'/o between the complete flattening of the rubber tube by the atmospheric pres- sure and equality between the pressures on both sides. This cor- responds to a deflection on the reading scale of mure than a metre. If we assume rough proportionality we lind that a displacement of 1 cm. would indicate a change in the opposing couple of only 0.1 "/„. The zero was kept constant to a few millimetres. The calibration was made under circumstances exactly the same as in the experiments (pressure difference, etc.). The total constant with the addition of that of the stretching wire was 1503 + 18 = 1521 _egs. Results. TABLE VII. Calculation of the corrections for non-uniformity from observations a bath of liquid nitrogen. Conical pole-pieces 20 mm apart. made in H gauss ^ — 195°.6C. ' = - 2I0°.0C. ,3.10'«(mean). Double weight given — 195".6 20 cm. fk''' P . W 2i cm. §■»' ,5. 104 4G15 1.18 0.554 —0.137 1.59 0.746 —0.199 —0.158 1 6944 2.69 5577 127 3.. 53 0.7322 232 162 9205 4.73 5580 126 6.21 7330 231 160 11280 7.08 5560 132 9.23 7251 250 171 12835 9.17 5564 131 12 10 7341 228 103 I 14015 11.14 5670 100 14.49 7378 219 140 , 14':00 12.90 5812 060 16.94 7613 162 091 15585 14.29"" 5884 039 - - ~ - 16120 15.07 1 6031 O03 20 21 7781 121 OiO ( 830 ) It can be seen that the values olilained for ,i are not the same at the two temperatures. Meanwhile it has to be applied here only as a correction for the susceptibility of solid oxygen which at the most is 3%- A diiference of temperature of J°C. in the bath under reduced pressure gives more than half the dilference between the two values, whence we have given the determination under reduced pressure only half the weight accorded the measurement at ordinary pressure. The uncertainty of the n)ean has less than i7o influence upon the value of the susceptibility of solid oxygen. The curve for ^ as well as its sign correspond with what were found for the silver ellipsoid. TABLE VIII. Susceptibility of solid oxygen. t = — :i5'i°.s (bath of liquid hydrogen boiling under atmospheric pressure). H gauss 2 J cm. of the scale !■- K. W uncorrected corrected according to fab. VII. 2296 0.89 1.69 519 533 4015 3.57 -1.67G 518.3 532.9 GP44 7.92 1.642 512.3 527.3 9205 -14 07 1.CG0 515,0 530 1 1I2S0 21.14 1.661 515.2 530.1 12835 27.92 1.684 518.7 532.8 ■14015 ,32.96 1.678 517.6 529.0 •14900 37.38 1.683 518.4 527.5 -15585 40.77 1.678 .517.6 523.7 •10120 44 05 1.696 520.0 523.0 Mean 529.0 529.0 whence it follows that X2o°.3iv. — 10-c — 375.2 . 10-c . 1.41 ( b31 ) (bath of TABLE IX. Susceptibility of solid oxygen. f= — 258,9° liquid hydrogen under 70 mm. vapour pressure). H '2r: cm. of ! 2.- ,„, A-. no K.U(> gauss the scale //2 •'" uncorrected corrected 2296 1.19 2.257 600.5 614.5 4615 4.80 2.253 000.1 614.6 6944 10.86 2.252 GOO.O 615. 9205 19.24 2.270 002.2 617.1 11280 28.13 2.210 594.3 609.2 12835 37.09 2.250 .599.6 613.7 14015 45.05 2.293 604.7 616.7 14900 51.24 2.306 606.8 615.9 15585 55.73 2.293 605.3 611,1 16120 60.20 2 317 608.5 612.1 614.0 1.41 = 435.6. 10-G. The From llie nieai) 614.0 follows /<„/. u'.sk products into V T are - 252. "8 375.2 . 10-6 \/m7i — ] (390 . iQ-c — 258.°9 435.6 . 10-6 1/14:2 = 1641 . 10-c. Hence we can represent the two observations pretty well by 1690 XW. = -p;^.10-6, which is adjusted to the measurement at the higher temperature. The deviation from this ratio for the lower temperature, however, is somewhat greater than the errors of observation. \ 5. Summary and conclusion. As regards the dependence of specific susceptibility upon temperature our most reliable determination gives X//9.90°.l K. = 240.6 . 10-6. 33700 Curie found / ^ — - — .JO-'' between 20 C. and 450° C. whence it ( 832 ) would ibllow that for 7'= 9()°.l K. /=:374.10-g a number tliat diifers essentially from ours '). There is therefore no possibility of extrapolating Curie's law to the liquid phase of oxygen. This was also the conclusion reached by Fi,EMiNG and Dewar in their iirst treatment of the question, but aftei- more careful experiments they rejected their former result'). The results obtained from the two magnetic rise apparatus at lower temperatures can, within the limits of experimental error, be expressed by a very simple law : tlie specific siusceptibility is inversely propor- tional to the square root of the absolute tempernture. From the observations obtained with the more reliable apparatus we deduce the formula 2284 which holds to within 57„- None of the results obtained bj the method of the maximum couple are in conflict with those deduced from the formula. The results with solid oxygen apjiroximately follow the relation 1690 At the lowest temperatures there is a small deviation indicating a smaller increase at lower temperatures; it is, however, so small that we may still accept the formula given as approximately correct for the solid state of aggregation below the melting point of oxygen and down to 14°. 2 K. Further experiments af more numerous temperatures must show exactly how far these deductions hold for the liquid and solid states. They shew (see fig. 5) that there is a jump in the value of /.at the melting point, since X%^,„ = l)3/,w.r i) R. Hennig's (1893) result should give 27600 •/ = lO-'J and /9uc 1 j^ ^^ 307 . IQ-c. T 2) Fleming and Dew ae's results: 1st paper (1896) '^goo.iK. =200. IQ-''; 2nd paper (1898) 28/ . \0-^, mean 243.5. IQ-'^ pretty much the same as our result. Accord- ing to the mean of the result of Faraday and Becquerel the specific suscep- tibility for oxygen at 0^ is 91 . IQ-''; this gives by extrapolation from Curie's law ZyuiK. = 299. 10-'^. The English savants used this number in their second research for the comparison of tbe susceptibility of liquid oxygen with that of the gas. ( 833 ) We hope to answer the question if tliis jump really exists b}" special experiments arranged for the purpose; we may, in the mean- time, coiisider that it does probably exist. What Curie found in the transformation of y iron to (f iron is analogous to the sudden change which we here assume to exist while the form of the law remains unaltered, and which can occur at the melting point or at a point of transformation to an allotropic moditication. Weiss') has shown that this can be accounted for on the assumption that at this particular point di-atomic iron changes into tri-atoniic. On the other hand we consider it probable that the law according to which the specific susceptibility increases with the temperature, viz : inverse proportionalitj to the square root of the absolute tem- perature at lower temperatures, gradually transforms into that of inverse proportionality (Curie's law) at higher temperatures, and that each of these laws, therefore, may be but approximative to the same function over different ranges of values of the independent variable 7\ The supposition that the change of specific susceptibility with density is of no importance lies at the bottom of the assumption of the gradual transformation of Curie's law into that of T'K If, on the other hand, we assume that this change is of importance, that e. g. when the internal pressure is considerable tlie molecules under its influence undergo not only a compression but also a lessening of their magnetic moments, then a region of great molecular compres- sibility in which the specific susceptibility should change both with the temperature and with the density should exist between the gaseous phase in which the specific susceptibility would be pretty well in- dependent of the pressure, and the liquid phase at lower temperatures, in which the molecules would not be appreciably affected by an additional external pressure on account of their already great internal pressure, and in which, tlierefore, the specific susceptibility would also be pretty well independent of the pressure. As regards the difference between the magnetic moment of the elementary magnets in the condition of saturated liquid and vapour and that at normal or smaller density at the same temperatures, it is to be expected according to that representation, that this difference will change with temperature in consequence of the change of density with temperature. The assumption can also be made that complex molecules are formed in the liquid state, and that these diminish the intensity of the elementary magnets; in that case changes in susceptibility of - 1) P. Weiss, loc. cit. ( ^34 ) mixtures of liquid oxygen with non-magnetic gases should obey the thermodynamic laws that govern the number of such complexes. But all this must be established by further experiments which we hope to complete; in the meantime the most probable assumption is the old one that the spec.itic susceptibility is independent of the pressure. As regards the question as to whether the specific susceptibility at lower temperatures still follows the law of inverse proportionality to tiie root of the absolute temperature, if the ferro-magnetism with a very low-lying Curie point according to Weiss's theoi-y of corre- sponding magnetic states does not exist, then the change to a still slower increase with decreasing temperature and the approximation to a limiting value is, perhaps, more probable. The law of T^^ at once gives rise to the question if instead of the Langkvin elementary magnets whose intensity is independent of the temperature, we should assume tluxt their intensity varies directly as VT; that is, that we should assume the existence of elementary- currents or electrons moving in their paths with speeds proportional to (and, thei-efore, determined by) tlie speeds of molecular heat motions. In other words, while Langevin's theory already supposes that the planes in which the electrons move follow the motions of the molecules, but that the area'; described in those planes are still independent of heat motion, we should now assume that the electrons undergo the influence of heat motion at their motion in their paths, and, if the radius of their path has also become invariable, revolve while remaining in the same position with respect to the atom ; they would be electrons that are frozen fast to the atom, an assumption that has already been made to explain other phenomena. This addition to Langevin's theory, however, does not lead to a specific susceptibility proportional to T' i as one at first sight would be inclined to think, but to a constant specific susceptibility. To substantiate that addition it will probably be necessary to proceed to still lower temperatures tluui those of our experiments. It seems at present that it is not impossible that then the law x proportional to 7'~i changes to %^ const.: our observations on solid o.\vgen seem to indicate a change in this direction. The assumption to which this is equivalent: viz, that the magnetic motions of the electrons cease at the absolute zero, and to which our experiments seem to lead, is much more satisfactory than that the magnetic motions of the elec- trons still persevere even at the absolute zero. The second question to which we devoted attention — the depen- dence of susceptibility upon field strength requires no detailed treat- ( 835 ) nieiit. Tlic method oC the magnetic rise seemed in some instances to give a decrease of the order of 1% in a field of 8000 gauss, while the method of the maximum coupla gave with the cylindrical pole- pieces up to 12000 gauss only a very small systematic deviation and with the conical pole-pieces (16000 gauss) the deviation was scarcely appreciable. The solid oxygen ellipsoid with which a much lower temperature was reached seemed to give a small decrease at 16000 gauss; it is possible, however, tliat a greater deviation is obscured by the cor- rection for the non-uniformity of the field. We consider, however, that, assuming that the experiments were accurate to within I'/othe change of the susceptibility with the field up to 16000 gauss remains within the limits of experimental error. This is in agreement with the theory of Langevin, if this, notwithstanding the deviation from Curie's law, is still applied. Physics. — "The magneto-optic KERR-Efect in ferromagnetic com- jjound.f and alloys". By Stanislaw Loria. (Communication from the Bosscha-Laboratory). It has been shewn by Kaz '), Righi^), Kundt'), Sissingh^), Zeeman°) and also by Kerr') himself that the phenomenon discovered by the last named in 1876 depends not only on the orientation of the reflecting surface with respect to the magnetic vectors, but also (in a somewhat complicated manner) on the angle of incidence and the position of the plane of polarization of the incident beam. In the simplest and by far the most important case of almost normal incidence of light polarized perpendicularly or parallel to the plane of incidence, the reflected light in general is elliptically polarized according to RiGHi'); the rotation of the major axis of the ellipse depends on the magnetisation and the wave-length. According to the measurements made by DU Bois") it is in every case proportional to the former ; as regards the variation with the 1) P. C. Kaz, Diss., Amsterdam 1884. 2) A. RiGHi, Ann. de Ghim. et Phys. (6) 4 p. 433, 1885. 3) A. KuNDT, Wied. Ann. 23 p. 228, 1884; 27 p. 199, 1886. *) R. SissiNGH, Arch. Need. (1) 27 p. 173, 1894. 5) P. Zeeman, Leiden Gomm. no. 15, 1895; no. 29, 1896. Arch. Neerl. 27 p. 252 1894. «) J. Kerr, Phil. Mag. (5) 3 p. 339, 1877. Phil. Mag. (5) 5 p. 161, 1878. ") A. RiGHi, Ann. de Ghim. et Phys. (1) 9 p. 120, 1886. 8j H. DU Bois, Wied. Ann. 39 p. 25, 1890. ( 836 ) latter, tlio rotatoi\y dispersion, according to the same autiior, shews certain regularities. For iron, cobalt, and nickel the rotations visually observed were always neijative ; for iron the dispersion-curve seems to indicate a numerical minimum in the ultraviolet and thence ascends from \iolet towards red; in the case of cobalt the minimum occurs between blue and green, and for nickel in the yellow. These numerical minima of negative rotation may be considered algebraic maxima, their wave-length increasing as the metal's position in the jieriodic system advances. For magnetite the observed rotations were in every case positive, though the curve appeared directed towards negative values beyond the blue ; a distinct maximum occurred in the yellow, corresponding to the above algebraic maxima. More recently Ingersoll ') has contributed important papers relative to this subject; he was able to supplement uu Bois' curves in the infra-red up to about 3 ft. According to this author the complete rotatory dispersion-curves thus obtained shew a marked resemblance to a typical dispersion-curve in the region of an exceedingly broad band of resonance-absorption. The particular cases of nickel and magnetite are notable, for the rotation appears to vanish between 1 and 1,5 fi and then to change in sign. Further progress in this subject was difficult in view of the fact, that as yet the only ferromagnetic substances suitable for a study of the KERR-Elfect were the four above-mentioned bodies. Several attempts to study with reflected and transmitted light the magneto-optic pheno- mena connected with the KKRR-Etfect were made with partially transparent lilms of metals prepared electrolyticaliy, after the manner of KuNDT, or by cathodic discharge. Although the latest investigations ") on the optical properties of these films of magnetic metals brought to light further interesting but confusing results, yet the conditions in the films can obviously depend on their structure and on their mode of preparation in a very complex way. As a reflecting surface such a film is certainly inferior from a physical point of view to a mirror polished on a compact and massive block of metal. An attempt to add to the number of substances which exhibit the KERR-Effect was thus of some interest. I entertained some hopes in this respect, since several chemists of late have synthetically prepared new ferromagnetic substances. In the first place, a number of com- binations of different oxides with iron oxide, so called metaferrites, n L. R. Ingersoll, Phil. Mag. (6) 11 p. 41, 1906 & 18 p. 74, 1909. 2) G. A. Skinner & A. Q. Tool, Phil. Mag. (6) 16 p. 833, 1908. H. Behrens Inaug. Diss. Miinster i. W. 1908. L. R. Ingersoll, loc. cit. ( 837 ) piepared by Hilpekt'), presented an interesting field of researcli. In all these cases, the chemical structure resembles that of ferroferrite (ferroso-ferric oxide), in that the iron sesquioxide plays the acidic part, thus imparting ferromagnetic properties to the compound. Of this class of substances however, only cupriferi'ite and calcitimferrite conld be obtained in a state suitable for my experiments. Secondly, certain alloys of more or less ferromagnetic metals, and in particular those of nickel-iron -), together with the well-known ternary Heusler alloy, and Wedekimd's ') binary manganese-antimony alloy present considerable interest. So far as I am aware, the magneto-optic pro- perties of these alloys have been only partially investigated, the only account of similar experiments, which 1 have come across, being Ingersoll's communication previously referred to and a Russian paper by ToKMATscHEW ^), who described experiments with Heusler's alloy. I ha\e studied the magneto-optic properties of the above mentioned bodies and also those of the w-ell-known magnetic chromic oxide. Below an account of the preliminary results of my research is given. Experimental Arrangement. Solar rays were exclusi- vely used ; they passed through a direct-vision monochromatic illumi- nator^), with divergence 1:4, thus furnishing light of great intensity. The rays passed (Fig. 1) through a lens (L), a total reflecting prism (P), a Lippich's arrangement of two halfshade Nicols (Nj, NJ and falling nearly normal on a mirror between the two poles of an electromagnet, were reflected, tinally passing through an analyser (N,) and a telescope of fourfold magnifying power. The dimensions of the lenses, of the diaphragms, of the width and angle of the conical bores in the cores and poles were all calculated beforehand, particular care being taken to maintain maximum brightness, a uniform field of \'iew, and also the avoidance of all unnecessary reflections "). The observations were carried out with nearly normal incidence. RiGHi ') found, that up to an angle of incidence of 15° there was 1) S. HiLPERT. Ber. doutsch. Chem. Ges. 42 p. 2248, 1909. Verb, deutscli. Phys. Ges. 11 p. 293, 1909. -) Gh. Ed. Guillaume, Les aciers an nickel, Paris 1898. 3) E. Wedekind, Ztschr. f. phys. Chem. 66 p. 6)4, 1909. K. Honda, Ann. d. Phys. 32, 1910. *) S. Tokmatschew, Journ. d. russ. phys.-chem. Ges., 42 (phys. T.) p. 15, 1910. 5) H. DU Bois, Verh. d. D. Phys. Ges. 11 p. 708, 1909. 6) A description of the analyser and polariser mentioned is given by H. du Bois, Wied. Ann. 46, p. 545, 1892." ') A. RiGHi, Ann. de chim. ct de phys. (1) 9 pp. 120, 132, 1886. ( 838 ) scarcely any variation of tlie effect. However, in my experiments the angle between the incident and reflected beams was only 2° or 3°. The incident light was polarized horizontally in the plane of incidence. From RiGHi's observations it is known that even a normal incident beam of linearly polarized light when reflected from a magnetized mirror becomes elliptically polarized, the ellipticity however being only slight; Zeeman ') later measured this ellipticity in the case of iron and cobalt. Up to the present the evaluation of the ellipticity in my experiments has not been attempted ; I considered that the slight reflecting power of some of my mirrors would not warrant such an attempt, and in addition it must be borne in mind, that the i-otations themselves are small. Moreover the ellipticity, if any, must be nearly inappreciable, for by employing the best of my mirrors and by carefully avoiding diffused light, 1 have never been confronted with any difficulties, while the extinction of light in each half of the field of view was satisfactory. Even when the rotations are very small it is possible by means of the half-shade arrangement to observe and to measure them with sufficient accuracy. Hence it was thought unnecessary to use the method of multiple reflections, there- by avoiding new complications and further sources of error. The azimuth of the analyser was determined by means of a vertical scale seen through a combination of mirrors. For the production of the magnetic field a small du Bois semicir- cular electromagnet of resistance 9 Si was employed. To avoid the danger of sparking with reversal of current about 60 i2 were shunted across its terminals. The field was determined by means of a standar- dised thin glass-plate silvered at the back, which could be placed immediately in front of the mirror. The light (;i = 589 fift), being reflected by the mirror as described above, suffers a double magnetic rotation in the glass. The ensuing very slight double rotation of the light in its passage to and fro through the magnetized air could be computed from the data of Siertsema '), but proved quite negligible. Indeed, by using a silver-mirror, it was found that the rotation lies within the limits of experimental error. All the measurements were made with "polar" magnetization and at ordinary temperature. T e s t- S p ec i m e n s. The following substances were experimen- ted upon : Oupriferrite {Ca\ . Fe, Oj), Calcmmferrite (Ca . Fe, 0,), Magnetite (Ferroferrite) (Fe . Fe,0,), llmenite (Ti^ 0, . Fe^O,), ferro- magnetic chromic oxide {Gi\ Oj, "Invar'' (36 Ni, 64 Fe), the Heuslek 1) P. Zeeman, Lelden-Gomm. No 15, 1895. 2) L. H. Siertsema, Versl. Kon. Akad. Wet. Amsterdam 7 p. 289, 1899. ( 839 ) alloii (26 Mn, 18 Al, 61 Cii). The lirst two were kindly prepared by Dr. Hii.PEKT ill the metallurgical Laboratorv of the "Teciinisc.lie Hoelisc'hule" in Charlottenburg; the natural magnetite is from the collection of the Bo&scha-Laboratory, and is the same specimen, possessing a polished octahedral surface, which was formerly examined by Ttv Bois '). A very fine-formed crystal of ilmenite was kindly lent by Prof. Liebiscih. The Heusler alloy was supplied by the de }\\vs chemical factory in Seelze; its interior was full of bubbles, but its surface was capable of polish and supplied a very good mirror. The "invar" contained about 36 7o Nickel and came from France (Societe de Coinmentry-Fourchambault). For the chromic oxide I am indebted to Dr. Koppel. I desire to express my obli- gations to all the above mentioned gentlemen. Throughout this paper 1 shall denote as usual by : ^, the field intensity in kilogausses, j the magnetization, j„, its saturation value, f single rotation of the plane of polarization in minutes, K, Kerr's constant. In the tables, the column under N shows the number of readings in each series of measurements, which depend- ed upon the polish of the mirror and the variable brightness, X denotes the wave-length in mi, L the direct scale-reading in mm. of the double rotation produced by reversal of the current. The average values of the single rotations are given in the fourth column and in the fifth and sixth the average errors in minutes and in percentages. The sense of the rotation is referred as usual to that of the magnetizing current; e.g. in the case of iron the '"polar" Kerr rota,tion is negative. Results. The results obtained with the various substances were as follows : 1. Cupriferrite. Measurements were made on two mirrors of this material with similar results. The relation between the rotation and the wave-length in a field of 10,2 kgs. is shown numerically in Table 1 and graphically in Fig. 2. The dispersion-curve exhibits a type which has not been observed hitherto in the visible spectrum. In the violet the rotation is positive, a maximum occurring in the blue; with increasing wave-length the rotation gradually decreases and ill the neighbourhood of 587 fiji goes through zero, becoming negative for longei- Avave-lengths. Between 640 and 670 wft a rather flat minimum is exhibited, the curve then gradually proceeding upwards. The rotations are small throughout, the maximum value not being 1) H. DD Bois, Wied. Ann. 39, p. 23, 1890. 56 Proceedings Royal Acad. Amsterdam. Vol. XII. { S4() ) TABLE 1. fund {>) Cupriferrite ^ = 10.2.5 Kgs. N y(,.,.) A (mm) s (Minutes) e. li) 436 + 8.3 + i.'M' ± 0.04' = 3.5"/o 48 477 + 11.1 + 1.75' ± 0.05' = 3 „ 43 539 + 8.1 + 1.28' ± 0.04' = 3 „ 40 574 + 2.0 + 0.41' ± 0.02' = 6 „ bi 599 — 2.3 — 0.3G' ± 0.03' = 8 ., 45 G37 - 0.0 — 0.95' + 0.01' = ■! „ 51 688 — 4.9 — 0.78' ± 0.03' = ^ ,, greater tlian -(- 1,75', but they slill admitted of e.xact measurement. The above-mentioned change of sign is analogous to that found bj- Ingeksoll in the infra-red and presents a cliaracteristio and theoretic- ally important phenomenon. The relation between the rotation and the Held was also investigated, and the results are shewn in Table 2 and Fig. 3. For low values of the held the two are proportional to each other, the rotation TABLE 2. e = funct (&) Cupriferrite : 477 /'/' N & (kgs) A (mm) E (Minutes) „%- 40 0.93 + 5.3 + 0.85' + 0.02' = 2"/,, 22 2.25 + 8.4 + 1..34' ± 0.04' = 3„ 59 4.47 + 9.7 + 1.5G' ± 0.03' = 2„ 31 7.19 + 10 2 i- 1.63' + 0.03' = 2„ 20 9.32 + 10.4 -f 1 60' + 0.03' = 2 ^_ 48 10.15 + 11.1 + 1.75' 4- 0.05' = 3„ aflerwards assuming a maximum value, which remained nearly conslant for further increase of the Held. (Jonsidering the form of the curve t = funct (P) and accepting the results previously found ( 841 ) by DU Bois in the case of iron, nickel, and cobalt, we niav assert the proportionality between f and ^"\ with great probability. Bearing in mind this fact we are able to determine from purely magneto- optic measurements the order of magnitude or at least an inferior liiTut of maximum magnetization. As uu Rois ') has shewn in the case of an unlimited homogeneous plane disc, the magnetization of which is uniform and normal, the abscissa of the point of intersection of the straight line f ::= /v j =r yv.p/4:T and of the asymptote E=:co?«.y?. has the value 4-t jm . Accordingly- ji,„^140 c.g.s. in the case of cupriferrile. The small inclination of the uppei part of the curve in Fig. 8 may be explained by the fact that for irregularly formed specimens the real conditions do not correspond to those in the ideal case mentioned above. How- ever this inevitable difference can only produce a decrease in the apparent value of j,„ so that an inferior limiting value is really determined : small fissures, cavities, and impurities in the reflecting surface are particularly capable of exerting such an influence. 2. Mai/netitc. The dispersion of the KEKK-Effect is shown in Table 3 and ¥ig. 4 (continuous line). If we compare this curve with the T A B L E 3 = = funct (/) Magnetite .i;i=rl1.,56 Kgs N > V^y) i_ (mm) 2 (Minutes) 0. 30 436 — 24.0 — 3.81' ± 0.03'= 0.d»/„ 25 442 — 19.9 — 3.15' ± 0.05'= 1.5 „ 15 453 — 9.6 — 1.52' + 0.03'= 2 „ 30 464 — 40 477 + 6.7 + 1.06' + 0.03'= 2 „ 26 510 + 19.4 + 3.07' + 0.02'= 0.6 „ 25 539 -f 24.3 -f 3.84' + 0.02'= 0.5 „ 30 574 + 28 2 + 4.45' ± 0.02'= 0.4 „ 30 599 + 24.9 + 3.94' + 0.02'= 0.5 „ 31 637 + 21.0 -f 3.32' + 0.04'= 1 „ 30 688 + 16.0 + 2.50' + 0.07'= 3 „ 1) H. DU Bois, Wied. Ann. 31 p. 965, 1887; Phil. Mag. (5) 29 p. 301, 1890. (842) previous one given by du Bois (dotted line), which he obtained with the same specimen [a holoedric regular crvstal, possessing a natural octahedral surface) we see, that with the exception of a displacement throughout the whole range of wa\'e-Iengt]is amounting to about 10 to 30 itit — which is explained by the fact that 20 years ago only an imi)erfect method of spectral decomposition was available — - the curves are in agreement in the region between 486 and 671 fiji. The rotation attains a maximum value of 4.45' in the yellow and decreases i-apidly with decreasing wa\e-length. Du Bois '), who was unable to proceed further than the blue on account of insufllcient intensity of light, observed that the rotation probably vanished in the l)lue ; he also considered that a change of sign possibly might occur in the ultraviolet. I have located this zero-point in the visible part of the violet at 464f/;<. For smaller wave-lengths the rotation has rather a large negative value, which seems to approach a mini- mum. Unfortunately it was impossible to carry the investigation beyond 436 ftft since the light at that point becomes loo feeble. At all events, the existing observations establish satisfactorily the fact that the dispersion-curve obtained with natural OTStalline maguelite (FeO . F^Oj) is of the same type as that obtained above with cupriferrile. Without entering into theoretical considerations it may be seen at once that in both cases the curve passes through a maximum, goes through zei'o and probably also through a mini- mum. Experiments are being carried out lo see whether the course of these curves depends on the optical constants of the substances in\ostigated, viz. their ordinary absorption- and dispersion-curves. In the same way as in the case of cupriferrite the relation between I lie rolaliou and the fiekl was also investigated. The results are shown in Table 4 and Fig. 5. They give C<m ^ 358 C. G. S., which agrees with that obtained by du Bois ') (350). The magnetic properties of magnetite crystals have been recently investigated by Quittnkr''), adopting Weiss' methods. From his measurements it follows that the component magnetization parallel to the field, which in this case is alone of interest, reaches a .saturation value of aliout 475 C. G. S. ; this subject and the cause of the discre|iancy ought to be investigated in greater detail. One remark, liowe\er, may be made at once. In many cases the natural magnetite slightly departs from the simple slmctural formi.la (Fe 0, Fe^ 0,) ; 1) H. DU Bois 1. c. p. 38. -) H. DU Bois, Phil. Mag. (o^ 29, p. 301, 1S9U. ■M 1>. Weiss, Journ. de Pliys. (3) 5 p. 435, 1S9G aud (i) 9, p. 373, 1910. V. OuiTTNER, DisserlalloD, Ziiiicli, 1908. ( 843 ) £ = funct (^i) TABLE 4. Magnetite ;. = 574 ,v// N ■»o (Kgs) A (mm) •: (Minutes) 15 2.10 4- 12.9 4 2.07' ± 0.04' = 2 X 15 3.40 + 21.3 + 3.37' ± 0.05' = 1.5 „ 15 5.87 + 28.7 + 4 54' ± 0.07' = 1.5 „ 15 8.87 + 28.0 + 4.43' + 0.05' = ■1 .. 15 10.82 + 28.9 + 4.b7' ± 0.06' = 1 M 30 11 .56 + 28.2 + 4.45' ± 0.02' = 0.4 „ I also Quittner has esfablislied the great diver.sity of samples by measuring tlieir variable densities. It is difficult to foretell the influence of all this on the magneto-optic properties. 3. (Jt/wr fenviiiaynetic coinpotinds. The distinct analogy in the dispersion for substances of similar chemical structure as e.g. cupri- and ferroferrite in contradistinction to iron, nickel, and cobalt suggests whether the properties of other ferromagnetic ferrites and oxides are not similar. The investigation of cnlciiimferrite was in this respect of importance. This substance is very feebly magnetic and brittle. A small piece was suiTOunded by the easily fusible Wood alloy and then thoroughly polished. No KERR-Effect however was observed although the mirror was sufficiently good. The effect, if it exists, must be smaller than 0,35'. A similar result was obtained with ilmenite '). The light was reflected from the base of the crystal as well as from a plane parallel to the principal axis, but in no case could a rotation be detected. «^ 0,3'). It was also impossible to detect any rotation with chromic oxide Ci\ 0,, which without doubt is ferromagnetic. The following alloys were tested : 4. Nickel-iron with 36° /„ nickel, so called -'Invar", known to possess a very small coefficient of e.xpansion, is strongly magnetic and distinctly shows the KERR-Effect. The rotation is exclusively negative in the region of the spectrum investigated, and there is only a slight variation with wave-length. (Table 5, F'ig. 6). The dispersion-curve lies considerably below the zero-line; with increasijig wave-length 1) See B. Bavink, Magn. Influenz in Krystallen, Goltinger Dissertation 1004. ( 844 ) TABLE 5. :funct(/) "Invar" ^1= 13.30 Kgs. N ' {I"A A (mm) ■■(Minutes) .. 15 430 — 74.4 — 11.78' + 0.05'= 0.4% 15 477 — 78.8 — 12.48' ± 0.00'= 5„ 15 539 — 83.5 - 13.22' + 0.00' =r 0.4 „ £0 574 — 80 3 — 13.00' + 0.03'= 0.2 „ 15 599 — 86.8 — 13.74' + 0.05'= 3„ 15 037 — 80.7 — 13.72' + 0.07'= 0.5 „ 15 088 — 80.2 — 13.54' + 0.00'= 0.4 „ il proceeds slowly downwards, passes tliroiigli a (lat miinerical maxiiniiiii in the orange, afler wliicli tiie rotalion tlecreases ver}- slowly. Tlie relation between rotation and niagiielizatioii, as in tlie cases above, exhibits distinct proportionality and we have j„i > 530 (Table ti, Fig. 7). T y\ B L E (>. = funct (ys) "Invar" 1 = 574 /*/* N .&(kgs) A (mm) i (Minutes) <.%- 31 0.54 — 6.5 — 1.02' + 0.02' = 2 n/o 15 1.80 —23.2 — 3.67' ± 0.02' = 0.5 , 15 3.20 —39.0 — 6.17' + 0.03' = 0.5 „ 15 6.32 —69.7 —11.03' + 0.05' ::; 0.4 „ 15 10.37 —84.5 -13.30' + 0.03' — 0.2 „ 15 12.00 —86.6 —13.71' ± 0.02' = 0.1 „ 20 13.30 —86.3 -13.60' + 0.03' = 0.2 „ 15 14.51 —86.2 —13.05' + 0.03' = 0.2 „ It wonld be interesting to study the magneto-optic behaviour of the nearly non-magnetic nickel-iron alloy, which contains 25 percent nickel. ( H45 ) 5. Till' Hi'.isi.KR alluii, sii|)|)u.se(l lo coiilaiii (Jl"/,, <'ii, 2()7„ Mn and J37„ Al is mtliei- stroiigij magnetizable. Dilt'ei-ciil poi-lioiis of two vvell-poiished inirrors were caret'iih' examined in varions parls of the spectrum but proved to be magneto-optiealij inelfective. It is of course possible that tlie KKRR-Effect might be less than 0,3' in this case. Quite recently there appeared a communication by Tok.matschew recording similar experiments on the Heuslkr alloy No. 32 (58,9 Cu, 26,5 Mn, 14,6 Al). From theoretical considerations the author arrives at the conclusion of the probability of an effect capable of measurement occurring in the neighbourhood of 450/»f«. I iiave carried out a series of readings at this wave-length but no rotation could be observed. Ingersoll also failetl to notice any measurable elfect either in the visible spectrum or in the infra-red. The discussion of the theoretical signilication of the above partially positive and partially negative results I resei've for a future occasion; further experiments are in preparation, and the determination of the purely optical properties of the investigated substances is already in progress. ERRATA. In the Proceedings of the Meetings of Jan. and Febr. 19U) p. 672 Table III fui' 5050 read 8050. p. 675 Table VII for 102.58 read 102.85. p. 676 Table VIII for 71.75 read 71.95. (May 26, 1910). CONTENTS. ABDUCENs NUCLEDS (On the motor facialis and) of Lophius piscatorius. 4t. ABEL (Contribution to tiie solution of the functional equation of). 208. ABSORPTION LlNs^s (The magnetic separation of) in connexion with sun-spot spectra. 58-1. ALDEHYDES (On a synthesis of) and indole. 42. ALKALINE EARTHS (The behaviour of the phosphorescent sulfides of the) at various temperatures, and particularly at very low temperatures. 157. ALLOTROPic modifications (The atomic volume of) at very low temperatures. 4o7. ALLOTROPY (A new theory of the phenomenon). 763. AMMONIA and Water (On the compounds of). 183. Anatomy. A. B. Dboogleever Fortuyn: "On the motor facialis and abducens-nucleus of Lophius piscatorius". 44. — C. T. VAN VaLKENBURG: "Surface and structure of the cortex of a microcephalic idiot". 202. — L. BoLK: "On the position and displacement of the Foramen magnum in the primates". 362. — L. BoLK : " On the slope of the Foramen magnum in primates". 525. anturaquinone (The P-T-X- spacial representation of the system Ether-). 231. aporosa campanulata J. J. S. (On Distylinm stellare 0. K. and). 341. ASYMPTOTIC LINES (On the surfaces the) of which can be determined by quadratures. 759. ATOMIC VOLUME (The) of allotropic modifications at very low temperatures. 437. ATRICM CORDIS (Communications about the electrogram of the). 680. BACILLUS PRODiGiosus (Variability in). 640. bacteria (The decomposition of uric acid by). 54. basalt (On micaleucite) from Eastern-Borneo. 148. BASiLicnj" OIL (Javanese) and Methylchavicol. 15. BECQUEREL (HENRI AND JEAN) and H. Kamerlingii Onnes. On phosphor- escence at very low temperatures. 76. BETH (h. j. e.) The oscillations about a position of equilibrium where a simple linear relation exists between the frequencies of the principal vibrations. Is; part. 619. 2nd part. 735. BEYERINCK (M. w.) presents a paper of Mr. F. Liebert: "The decomposition of uric acid by bacteria". 54. 58 Proceedings Royal Acad. Amsterdam. Vol. XII. JI CON T K N T S. B E y E It I N c K (m. \v.) \ iscosaccliarase, an enzyra which produces slime from cane-sugar. 635. — Vnriability in Bacillus prodigiosus. 64(1. — Emulsion laevulnn, the product of the action of viscosuocharase en cane sugar. 7 'J5. BINARY MIXTURES (Isotherms of nioiiatomic gases and their). HI. Data concerning neon and helium. 175. BINARY SYSTEMS (The equilibrium solid-liquid- gas in) which present mixed crystals. 537. BIRDS (A brief contribution to the knowledge of endozoic seed distribution by) in Java, based on a collection made by Mr. Baktiiels on the Pangerango and near Batavia. 108. BLOOD SERUM (On the changes in the) of sharks after bleeding. 377. BOESEKEN (j.). Contribution to the knowledge of catalytic phenomena. 417. BO IS (u. E. J. G. Du) presents a paper of Mr, St. Loria: //The magneto-optic KERR-eli'ect in ferromagnetic compounds and alloys". 835. B o I s (H. E. J. G. uu) and Kot.\ro Honda. The thermomagnetic projierties of ele- ments. 596. B o L K (.L.) presents a paper of Mr. A. B. Droogleeveii Fortuyn; "On the motor facialis- and abducens-nucleus of Lophius piscatorius". 44. — piesents a paper of Dr. C. T. van Valkenburg: "Surface and siructure of the cortex of a microcephalic idiot". 203. — On the position and displacement of the Foramen magnum in the primates. 363. — On the slope of the Foramen magnum in primates. 525. BORNEO (On oceanic deep-sea deposits of Central-). 141. — (On micaleucite basalt from Er.stern-). 148. Botany. Miss C. J. Pekeluaring: "Investigations on the relation between the presentation time and intensity of stimulus in geotropic curvatures". 65. — S. li. KooRDERS: "Some brief remarks relating to the communication of Prof. C. K. A. WiciiMANN: "On fen formations in the Kast-Indian archipelago". 74. S H. KooRDERS : "A brief contribution to the knowledge of endozoic seed dis- tribution by birds in Java, based on a collection made by Mr. Barthels, on the Pangerango and near Batavia" (Contribution to the knowledge of the Flora of Java. V). 108. S. H. KooRDERS: "Some remarks on the nomenclature and synonymy of Xylosnia leprosipes Clos., X fragrans Decne and Fliieggea serrata Miq". (Contribution to the knowledge of the Flora of Java. VI). 116. — Til Wee VERS : 'The physiological signiticance of certain glucosides". 193. J. Kuyper: "The influence of temperature on the respiration of the higher plants". 219. \Y. Burck: "Contribution to the knowledge of water-secretion in phiiits". 306. 400. J. J. Smith: "On Distyliura stellare 0. K. and Aporosa campiinulata J. J. S." 341. 10. A. jr. (;. Went; "The inadmissibility of the statolith theory of geotropism as proved by experiments of Miss. C. J. Pekelharing". 343. — K. Keinders: "Sap raising forces in living wood". 563. C O N T E N T S. Botany. K. Zulstka: '/Contributions to the knowledge of the movement of water in plants". 574. — C. VAN WissELiNOH : ''On the tests for tannin in the living plant and on the physiological significance of tannin". 685. BKAND3EN (P.). On the stable positions of equilibrium of floating parallele[)ipeda. 383. BRIDGE of the violin (On the motion of the). 513. BROUWER (h. a.). On micaleucite basalt from Eastern Borneo. 14S. — Pienaarite, a luelanocratio foyaite from Transvaal. o-i7. B ROUWER (l. E. J.). Continuous one-one transformations of surfaces in themselves. 2nd Communication. !<!86. — On continuous vector distributions on surfaces. 2nd Communication. 716. — On the structure of perfect sets of points. 785. BRUIN (J,) On the surfaces the asymptotic lines of which can be determined by quadratures. 759. B u c H N E B (e. h.). Ou the radioactivity of Kubidium compounds. 154. B u R c K (w.). Contribution to the knowledge of watersecretion in plants. SOfi. 400 BUYTENDYK (f. J. J.). On the consumption of oxygen by cold blooded auimals in connection with their size. 4S. — On the changes in the blood serum of sharks after bleeding. 377. — On the constitution of the urine of sharks with normal and increased diuresis. 3S0. CAMERA siLENTA (The) of the Physiological Laboratory at Utrecht. 706. CANE SUGAR (Viscosaccharase, an enzym which produces slime from). 635. — (Emulsion laevulan, the product of the action of viscosaccharase on). 795. CAKUINAAL (J.). The constructive determination of the velocities of a spaeial system. 12. CATALYTIC PdENOMENA (Contribution to the knowledge of). 417. Chemistry. P. van Romburgh: "Javanese Basilicum oil and Methylohavicol." 15. — P. VAN IloMBURGii: "The essential oil from the fruits of Morinda citrifolia L." 17. — B. A. Weerman: "On a synthesis of aldehydes and indols". 42. — Ern.st Cohen and W. Tombrock: "The electromotice force of zinc amalgams". 9S. — C. J. Enklaar: "On the action of active copper on linalool". 104. — E. li. BucHNER: "On the radioactivity of Rubidium compounds". 154. — A. Smits and S. Postma: "On the compounds of ammonia and water". 186. — J. Boeseken : "Contribution to the knowledge of catalytic phenomena". 417. — A. Smits: "On retrogressive meltingpoint lines". 227. — A. Smits: ♦'TheP-T-X-spacial representation of the system ether-anthraquinone". 23 1 . — A. Smits and J. P. Wuite : "On the system water-natrium sulphate". 244. — F. E. C. Scueffer: "On heterogeneous equilibria of dissociating compounds". 2rj7. — P. VAN Romburgh: "The nitration of diethylaniline". 297. — A. P. N. Fbanchimost: "On sodium alkyl-carbonates". 303. — Otto de Vries: "On the abnormal reduction of an aromatic nitrocompound with tin and hydrochloric acid and an interesting case of dimorphism". 305. — H. DuTiLH: "On partial racemism". 393. — A. P. N. Franchimont and E. Kramer: 'On derivatives of piperazine". 452. 58* IV CON T K N T S. Chemistry. II. li. Kkuyt: "The etiuilibruim soUd-li(|uid-gas in binary systems which present niixeil erystals". 1st Communication. 5'i7. — F. M. Jaegeii: "Studies on Tellurium. 1. The mutual behaviour of the elements: sulphur and Tellurium". 602. CHROMOSPHEHIC LIGHT (On the origin of the). 446. COHEN (e R N s t) and .1. Olte Jr. The atomic volume of allotropic modifications at very low temperatures. 437. COHEN (e u N s t) and W. Tombrock: The electromotive force of zinc amalgams. 98. COMPOUNDS (On the) of ammonia and water. 186. — (On heterogeneous equilibria of dissociating). 257. COPPER (On the action of active) on linalool. 104. CORTEX (Surface and structure of the) of a microcephalic idiot. 202. CREATINE (About the formation of) in the muscles at ihe tonus and the development of rigidity. 550. CRYSTALS (The equilibrium solidliquid-gas in binary systems which present mixed). 537. CUBIC (On pairs of points which are associated with respect to a plane). 711. CUBIC CURVE (On polar figures with respect to a plane). 776. CUBIC INVOLUTION (The) of the first rank in the plane. 751. DEEPSEA DEPOSITS (On oceanic) of Ceutral-Borneo. 141. uiETiiYLANiLiNE (Ou the nitration of). 297. DIMOKPHISM (The abnormal reduction of an aromatic nitrocompound with tin and hydrochloric acid and an interesting case of). 3t)5. DisTYLiuM STELLARE 0. K. (On) and Aporosa campanulata J. J. S. 341. DORP (w. A. van) presents a paper of Dr. E. A. Weerman: "Ou a synthesis of aldehydes and indols". 42. DKOOGLEEVER FORTUYN (a. b.). On the motor facialis- and abducens-nucleus of Lophius piscatorius. 44. DUTILH (h.) On partial racemism. 393. ELECTRIC DISCHARGE in gases (lleniarks on the experiments of Wilson and Mautvn on the velocity of rotation of the) in a radial magnetic field. 428. ELECTROGKAM (Communications about the) of the atrium cordis. 680. ELECTROMAGNET (An improved semicircular). 18^'. ELECTROMOTIVE FORCE (The) of ziuc amalgams. US. ELEMENTS (The thermomagnetic properties of). 596. ELEMENTS Sulphur and tellurium (The mutual behaviour of the). 602. EMULSION LAEVULAN, the product of the action of viscosaccharase on cane sugar. 795. ENKLAAK (c. J.). On the action of active copjjer on linalool. 104. EQUATION of ABEL (Contribution to the solution of the functional). 20?. EQUILIBRIA (On heterogeneous) of dissociating eomjiounds. 257. — (The photo-and electrochemical). 356. EQUILIBRIUM (On the stable positions of) of floating parallelepipeda. 383. — (The) soli(Mi(|uid gas in binary systems which present mixed crystals. (IstConi- niunication). 537. CONTENTS. V FquiLiBiuuM j^Tlie oscilliitioiis iibout !i |iosilion oi) where ;i simi)le liiie;ir relalion exists between the frequencies of the ])riiicipi»l vibrations. 1st ])art. (Jllt. Snd. part. 735. ERKATUM. 88. 179. 545. 774. 845. ETHER-ANTHa.\QUiNONE (The P-T-X-spacial representation of the system). 231. PEN FOEMATiONs (On) in the East-Indian Archipelago. 74. FKNs (The) of the Indian Archipelago. 70. FERROMAGNETIC Compounds and alloys (The magneto-optic KF.RR-eflect in). 835. FLORA of Java (Contribution to the knowledge of the). V. 108. VI. 116. FLCF-GiEA SERBATA MIQ. (Some rsmarics on the nomenclature and synonymy of Xyk'sma leprosipes Clos., X. fragrans Decne and). 116. FORAMEN MAGNUM (On the positiou and displacement of the) in the primates. 862. — (On the slope of the) in primates. 525. FOSSILS (On Jurassic) as rounded pebbles in North Brabant and Limburg. 422. FOY.ilTE (Pienaarite, a melanocratic) from Transvaal. 547. FKANCHIMONT (A. P. N.). On sodium-alkyl carbonates. 303. — presents a paper of Dr. Otto de Vries: "The abnormal reduction ot an aromatic nitrocompound with tin and hydrochloric acid and an interesting case of dimorphism". 305. FRA N CHI MONT (a. P. N.) and E. Kramek. On derivatives of piperazine. 452. FUNCTIONS (Investigation of the) which can be built up by means of intinitesimal iteration. 208. 427. GASES (Isotherms of monatoraic) and their binary mixtures. III. Date concerning neon and helium. 17o. — (Kemarks on the experiments of Wilson and Martyn on the velocity of rotation of the electric discharge in) in a radial magnetic tield. 428. Geology. A. Wiciimann: "The fens of the Indian Archipelago". 70. — G. A. F. MoLENGRAAFF: "On oceanic deep-sea deposits of Central-Borneo". 141. — P. Tescu: "Ou Jurassic fossils as rounded pebbles in North Brabantan(lLimburg".422. — H. A. Brouwer: "Pienaarite, a melanocratic foyaite from Transvaal". 547. GEOMETRY (On pentaspheric). 19. Geophysics. J. P. van der Stok: "On the determination of tidal constants from observations performed with horizontal pendulums". 2. GEOTROPic curvatures (Investigations on the relation between the presentation time and intensity of stimulus in). G5. geotropism (The inadmissibility of the Statolith theory of). 348. gilt ay (j. w.) and M. de Haas. On the motion of the bridge of tiie violin. 513. GLUCOSlDEs (The physiological significance of certain). 193. HAAS (M. 1)E) and J. W. Giltay. On the motion of the bridge of the violin. 513. HELIUM (Data concerning neon and). 175. hermanides (j.) About odour-affinity, based on experiments of (-). 90. H o L L E M A n (a. Fi) presents a paper of Dr. E. H. BCchner: "On the radioactivity of rubidium compounds". 154. — presents a paper of Prof. A. Smits and S. Postma: "On the compounds of ammonia and water". 186. Vf C <) N T E N T S. II o I, I, E M AN (a. r.) presents a ])aper uf Piof. A. Smits aiul Dr. J. P. VYuite: "On the system water-natrium sulphate". 244. — presents a paper of Dr. F. E. C. Scheffrr: "On heterogeneous ecpiilibria of dissociating compounds". 257. — presents a paper of Prof. J. RoesEKEK: "Contribution to tlie knowledge of catalytic phenomena''. 417. — presents a paper of Prof. A. Smits: "A new theory of the phenomenon allotropy". 763. HooG ENHUVZE (c. J. c. VAN). About the formation of creatine in the muscles at the tonus and at the development of rigidity. ri.")0. INDIAN ARCHIPELAGO (The fens of the). 70. — (On fen formations in the East-). 74. INDOLE (On a synthesis of aldehydes and). 42. INTENSITV of slimuius (Investigations on the relation between the presentation time and) in geotropic curvatures. '15. ISOTHERMS of mouatomic gases and their binary mixtures. III. Data concerning neon and helium. 175. ITEIIATION (Investigation of the functions which can be built up by means of infini- tesimal). SOS. 427. — (On the orbits of a function obtained i)y infinitesimal) in its complex plane. 503. J A E G F, B (f. m.). Studies on Tellurium. I. The mutual behaviour of the elements: sulphur and tellurium. G02. JAVA (Contribution to the knowledge of the fiora of). V. 108. V[. 116. JULIUS (w. H.). Regular consequences of irregular refraction in tlie sun. 266. — On the origin of the chromospheric light". 446. KAMERLIXGH ONNES (h.). Isotherms of raonatomic gases and their binary mix- tures. III. Data concerning neon and helium. 175. — presents a paper of Prof. Ernst Cohen and J. Olir Jr.: "The atomic volume of allotropic modifications at very low temperatures". 431. presents a paper of Mr. J. VV. Giltay and Prof. M. de Haas: "On the motion of the bridge of the violin". 513. K A M K 11 L I N G H ONNES (h.) aud Henri and Jean Becquekel. On phosphorescence at very low temperatures. 76. KAMERLlKGll ONNES (u.), P. Lenauu and VV. E. Pauli. The behaviour of the phosphorescent sulfides of the alkaline earths at various low temperatures. 157. kamerlingu ONNES (H ). and Albert Perrier. Researches on the magneti- zation of licjuid and solid oxygen. 799. KAMERLiNcH ONNES (h.) and PlERRE VVeiss. Researches on magnetization nt very low temperatures. 649. K A p T E Y N (w.). presents a paper of Dr. M. J. van Uven: "Investigation of the functions which can be built up by means of infiuiteslraal iteration. Contribution to the solution of the functional equation of Abel". 20S. presents a piq)er of Dr. M. .1. van L'ven: "Investigation of the functions which can be built up by means of infinitesimal iteration". 427. CONTENTS VII KAPTEYN (w.) presents a paper (if Dr. .M. .1. van Uven: "Oil the orbits of a fiiiictinn obtained by infinitesimal iteration in its complex plane". 503. K E R u-EFFECT (The magneto-optic) in ferromagnetic compounds and alloys. 835. KOiiNST.-VMM (ph.). a short reply to Mr. van Laar's remarks. 531-. — (Some remarks on Prof) reply. 617. KOiiNSTAMM (pn.) and J. TiMMERMAXS. On tiie influence of the pressure on the • miscibility of two liquids. 235. — (Some remarks suggested by a paper by). 454. KoORDEBs (s. II.). Some brief remarks relating to the communication of Prof. C. E. A. WicHMANN; "On fen formations in the East-Indian Archipelago". 74. — A brief contribution to the knowledge of endozoic seed distribution by birds in Java, based on a collection made by Mr. Barthels on the Pangerango and near Hatavia". (Contribution to the knowledge of the Flora of Java. V). 108. — Some remarks on the nomenclature and synonymy of Xylosma leprosipes Clos. X fragrans Decne and Flueggea serrata Miq." (Contribution to the knowledge of the flora of Java. VI). 116. K o R T E w E G (d. .1.) presents a paper of Ur. L. E. J. Brouwer : "Continuous one-one transformations of surfaces in themselves" 2ntl Communication. 28ii. — presents a paper of Ur. P. Brandsen : "On the stiible positions of equilibrium of floating parallelepipeda". 383. — presents a paper of Mr. H. J. E. Beth : "'The oscillations about a position of equilibrium where a simple linear relation exists between the frequencies of the principal vibrations" 1st part. fil9. 2nd part. 735. — presents a paper of Dr. L. £. .1. Brouwer : "On continuous vector distributions on surfaces" 2nd Communication. 716. — presents a paper of Dr. L. E. J. Brouwer: '' On the structure of perfect sets of points". 785. KOTARo HONDA and H. E. J. G. DU Bois. The thermomagnetic properties of elements. 596. KRAMER (e) and A. P. N. Franchimoxt. On derivatives of Piperazine. 452. K B u Y T (u. R.). The equilibrium solid-liquid-gas in binary systems which present mixed crystals. Ist Communication. 537. K u Y P e R (J.). The influence of temperature on the respiration of the higher plants. 219. I, aar (j. j. van). On the solid state. II. 26. III. 120. IV. 133. — Some remarks suggested by a paper by Messrs Timmehmans and Koiinstamm. 454. — Some remarks on Prof. Kohnstamm's reply. 617. laar's (van) remarks (A short reply to Mr.). 534. L E N A B D (p.), H. Kameblingh Onnes and VV. E. Pauli. The behaviour of tiie phosphorescent sulfides of the alkaline earths at various temperatures, and parti- cularly at very low temperatures. 157. LIEBEBT (f.). The decomposition of uric acid by bacteria. 54. LiNALOoii (On the action of active copper on). 104. linear RELATION (The oscillations about a position of equilibrium where a simple) exists between the frequencies of the principal vibrations. 1st part. G19. 2"d part. 735 Viri CONTENTS LINES (Tlie de<;Tee of coinpleleiu'ss ol' the circular polarization of maiinetically divide!). 345. LINES OF FORCE (Oil the theory of the ZEEMvx-eti'ect in a direction inclined to the). 321. LiuuiDs (On the influence of the pressure on the raiscibility of two). 235. LOPlilus piscATORius (On the motor facialis and abducens n\icleu9 of). 44. LORENTZ (ii. A.) presents a paper of Mr. J. J. van Laar: " On the solid state". H. 2f). 111. 120. IV. 183. — On the theory of the ZEEMAN-efl'ect in a direction inclined to the lines of force. 321. — presents a paper of Dr. J. A. Voligkafk:" Iteir.arks on the experiments of Wilson and Martvn on the velocity of rotation of the electric discharge in gases in a radial magnetic field". 428. — presents a paper of Mr. J. J. van Laar: "Some remarks suggested hy a paper by Messrs. Timmelimans and Kohnstamm". 454. — presents a paper of Mr. J. J. van Laar: 'Some remarks on Prof. Kohnstamm's reply". 617. L O R I a (st). The magneto optic KERR-ellect in ferromagnetic compounds and alleys. 835. MAGNETIC FIELD (Remarks on the experiments of Wilson and Martvn on the velocity of rotation of the electric discharge in gases in a radial). 428. MAGNETIZATION (Eesearohes on) at very low temjieratures. 649. — • (Resear3hes on the) of liquid and solid oxygen. 7'Jfl. martyn (Remarks on the experiments of Wilson and) on the velocity of rotation of the electric discharge in gases in a radial magnetic field. 428. Mathematics. J. Oardinaal: "The constructive determination of the velocities of a spacial system". 12. — S. L. VAN Oss: "On pentaspheric geometry". 19. — M. J. VAN UvEN : "Investigation of the functions which can be built up by means of infinitesimal iteration. Contribution to the solution of the functional equation of Abel". 2(IS. — L. E. J. Buouwer: "Continuouj one-one transformations of surfaces in themselves." 2nd Communication. 286. — P. Branusen: "On the stable positions of equilibrium of floating ]iarallel- epipeda". 383. — M. J. VAN UvEN: "Investigation of the functions which ran be built u]) by means of infinitesimal iteration." 427. — M. J. VAN UvEN: "On the orbits of a function obtained by infinitesimal iteration in its complex plane". 503. — H. .1. E. Beth: "The oscillations about a position of equilibrium where a simple linear relation exists between the fretpiencies of the principal vibrations". 1st part. 019. 2nd part. 735. — Jan di; Vkies ; "On pairs of points which are associated with resj)ect to a plane cubic." 711. — L. K. .1. HitouwEa: "On continuous vector distributions on surfaces" 2nd Coniuiunication. 710. — \V. Van DEii VVouDE: "The cubic involution of the first rank in the plane." 751. CONTF. NTS IX Matheaiaticr. I. Bruin : "On the ■iurtiK-e? llie asytnptolic lines of wliicli can be determined by quadratures." 75St. — Jan de Vrifs: "On polar figures with respect tu u |)l:me cubic curve." 77(). — L. E. J. Bbouwer: "On the structure of perfect sets of points." 7S5. MELTixG-poiKT LINES (On retrogressive). 237. METHVLCii.wicoL (.Javanese Basilicum oil anil). 15. Microbiology. F. Liebekt: "The decomposition of uric acid liy bacteria." 54. — M. \V. Betjerinck: "Viscosaccharase, an enzym which produces slime from cane-sugar." 635. — M. W. Beyeeinck: "Variability in Bacillus prodigiosus." 610. — M. W. Beijebinck: " Emulsion laevulan, the product of the action of visco- saccharase on cane sugar." 795. microcephalic idiot (Surface and structure of the cortex of a). 202. MisciBiLiTY (On tlie influence of the jiressure on the) of two liquids. 235. M o I, E N G R A A F F (g. A. F.). On oceauic deepsea deposits of Central-Borneo. 141. — presents a paper of Mr. H. A. Brouwer: '-On micaleueite basalt from Eastern- Borneo". 14. — presents a paper of Dr. P. Tesch: "On Jurassic fossils as rounded pebbles in North Brabant and Limburg." 422. — presents a paper of Mr. II. A. Brouwer: "Pienaarite, a melanocratic foyaite from Transvaal." 547. MO ll(j.w.) presents a paper of Mr. E. IIeinders: "Sap raising forces in living wood." 563. — presents a paper of Mr. K. Zulstra: "Contributions to the knowledge of tlie movement of water in plants." 574. — presents a paper of Prof. C. van Wisselincu : "On the tests for tannin in the living plant and on the physiological significance of tannin." 6S5. MORINDA CITRIFOLIA L. (The essential oil from the fruits of). 17. MOTOR facialis (On the) and abducens nucleus of Lophius piscatorius. 44. MUSCLES (.\bout the formation of creatine in the) at the tonus and the development of rigidity. 550. NATRIUM-SULPHATE (On the System water-). 244. NEON and helium (Data concerning). 175. NITRATION (On the) of diethylaniline. 297. NITROCOMPOUND. (The abnormal reduction of an aromatic) with tin and hvdrochloric acid and an interesting case of dimorphism. 305. NOYoxs (a. k. m ). Communications about the electrogram of the atrium cordis. 680. oDOUK-AFFiNiTY (About), based on experiments of Mr. J. IIermanides. 90. OIL (The essential) from the fruits of Morinda citrifolia L. 17. o L I E jr. (j.) and Ernst Couen. The atomic volume of allotropic moditications at very low temperatures. 437. ORBl'ls of a function (On the) obtained by infinitesimal iteration in its complex plane. 503. oscillations CTlie) about a position of equilibrium where a simple linear relation exists between the frequencies of the principal vibrations, 1st part. 019. 2nd. part. 733. X CONTENTS OSS (s. L. VAN). On peiit:isplioi-ic ;;ooinelry. IH. OXYGEN (On tlie ponsimiption of) by cold hloocied animals in conneotinn with their size. 48. — (Researches on the magnetization of liquid and solid). 799. p.viiALLELEPlPEDA (On the stable positions of equilibrium of floating). 38 3. PAULi (w. e), p. Lenakd and H. Kamerlingh Onnes. The behaviour of the phosphorescent sulfides of the alkaline earths at various temperatures and parti- cularly at very low temperatures. 157. PEBBLES (On Jurassic fossils as rounded) in Nortli Brabant and Limburg. ii2. PEKELiiARiNG (c. A.) presents a paper cf Dr. C. J. 0. van Hoogenhuyze: ''About the formation of creatine in tlie muscles at tiie tonus and the develop- ment of rigidity". 55U. PEKELIIARING (Miss 0. J.). Investigations on the relation between the presentation time and intensity of stimulus in geotropic curvatures. 6J. — The inadmissibility of the statolith theory of geotropism, as proved by experi- ments of ( — ). 343. PENDULUMS (On the determination of tidal constants from observations performed with horizontal). 2. PENTASPHERic Geometry (On). 19. PERKIER (albert) and H. K.vmeblingh Onnes. Researches on tlie uuignctizution of liquid and solid oxygen. 799. Petrography. H. A. Brouwer: "On micaleucite basalt from Eastern Horneo." 148. PHOSPHORESCENCE (On) at very low temperatures. 76. Physics. J. J. VAN Laar: "On the solid state." 11 2(i. Ill 120. IV 133. — Henri and Jean BEcauEREL and H. Kamerlingh Onnes: "On phosphorescence at very low temperatures." 76. — P. Lenard, H. Kamerlingh Onnes and VV. K. Pauli : "The behaviour of the phosphorescent sulfides of the alkaline earths at various temperatures, and parti- cularly at very low temperatures.'' 157. — H. Kamerlingh Onnes : "isotherrasofmonatomic gases and their binary mixtures. III. Data concerning neon and helium." 175. — A. Smits and E. C. VVitsenburg: "On the phenomena which occur wiien in a ternary system the plaitpoint surface meet the two sheet three-phr.se surface." 182. — H. E. J. G. DU Bois : "An improved semicircular electromagnet." 189. — J. TiMMERMANs and Ph. Kohnstamm: "On the influence of the pressure on the miscibility of two liquids." 234. — W.H.Julius: "Regular consequences of regular refraction in the sun." 266. — II. A. LoRENTz: "On the theory of the ZEEMA.Nefl'ect in a direction inclined to the lines of force." 321. — P. Zeeman : "The degree of completeness of the circular polarization of magne- tically divided lines." 345. — A. Smits: " The photo- and electro-chemical equilibria". 356. — J. A. VoLLGRAFF: "Remarks on the experiments of Wilson and Martyn on the velocity of rotation of the electric discharge in gases in a radial magnetio field." 428. C O K T K N T S XI Physics. KuNST Cohen and J. Olie jr.-. "'I lie atomic; voiuiiio of allotropic uuidilicatious at very low temperatures." 437. — W. H. Julius: "On the origin of tlie clironiosplieric liglit." 44(i. — J. J. VAN L.AAU: "Some remarks suggested by a ])a])er l)y Messrs. Timmekmans and KoHNsTAMM." 454. — J. W. GiLTAY siud M. T)E Haas: "On the motion of the bridge of the violin.'" 513. — Ph. Kohnstamm: "A short reply to Mr. Van Laar's remarks." 534. — y. Zeemax and B. Winawer: "The magnetic separation of absorption lines in connexion with sun-spot spectra". 584. — H. K. J. G. DU Bois and Kotaro Honda: "The thermomagnelic properties of elements." 596. — J. J. VAN Laar: "Some remarks on Proi'. Koiinstamm's rejjly." 617. — PiEKiiE Weiss and H. Kameklingh Onnes: "Kesparches on magnetization at very low temperatures." 649. — A. Smits: "A new theory of the phenomenon allotropy". 763. — H. Kamerlingh Onnes and Albeut Perrier: "Kesearches on the magneti- zation of liquid and solid oxygen." 799. — St. Loria : "The magneto-optic KERR-eli'ect in ferromagnetic compounds and alloys." 835. Physiology. F. J. J. Buytendijk: "On the consumption of oxygen by cold-blooded animals in connection with their size." 4S — II. ZwAARDEMAKER: "About odour-aflinity, based on experiments of Mr. J. IIermanides. 90. — P. J. J. BcYTENDIJK: "On I he changes in the blood serum of sharks after bleeding." 377. — F. J. J. Buytendijk: "On tie constitution of the urine ol sharks witli normal and increased diuresis." 380. — G. J. G. VAN HooGENHUVZE; "About the formation of creatine in tlie muscles at the tonus and the development of rigidity." 550. — A.K. M. NoYONS: "Communications about the electrogram of the atrium cordis. "680. — H. Zwaardemaker: "The Camera silenta of the Physiological Laboratory at Utrecht." (06. — J. G. SLEESwiJii: "Contributions to the study of serum-anaphyhixis"' 4"i (.'om- munication. 781. PiENAARiTE, a melanocratic foyaite from Transvaal. 547. PiPEUAZiNE (On derivatives of). 452. PLACE (T.) presents a paper of Mr. F. J. J. Buytendijk : "On the consumption of oxygen by cold-blooded animals in connection with their size." 48. PLANE (The cubic involution of the first rank in the). 751. PLANT (On the tests for tannin in the living) and on the physiological significance of tannin. 685. PLANTS (The influence of temperature on the respiration of the higher). 219. — (Contribution to the knowledge of watersecretion in). 306. 400. (^Contribution to the knowledge of the movement of water in). 5i4. Xn CONTENTS POINTS (On p.iirs of) which are ussocliited with respect to a plane cubic. 711. — (On ihe structure of perfect sets of). 78.5. POLAR FiGuiiES (On) with respect to a plane cubic curve. 776. POLARIZATION (The degree of completeness of the circular) of magnetically divided lines. Slo. PC ST MA (s.) and A. Smits. On the compounds of ammonia and water. 1S6. PRESENTATION TIME (luvestiffations on the relation between the) and intensity of stimulus in geotropic curvatures. 65. PRESSURE (On the influence of the) on the miscibility of two liquids. 235. PRIMATES (On the position and displacement of the Foramen magnum in the). 3G2. — On the slope of the Foramen magnum in). 525. QUADRATURES (On the surfaces the asymptotic lines of which can be determined by). 159. RACEMISM (On partial). 393. RADIOACTIVITY (Ou the) of Rubidium compounds. 154. REFRACTION in the sun (Regular consequences of irregular) 266. REINDERS (E.). Sap raising forces in living wood. 563. RESPIRATION (The influence of temperature on the) of the higher plants. 219. RIGIDITY (About the formation of creatine in the muscles at the tonus and the deve- lopment of). 550. ROMBUKGU (p. van). Javanese Basilicum oil and Methylchavicol. 15. — The essential oil from the fruits of Morinda citrifolia L. 17. — presents a paper of Prof. Ernst Cohen and W. Tombrock. "The electromotive force of zinc amalgams". 9S. — presents a paper of* Dr. C. J. Enklaar: "On the action of active copper on Linalool". 104. — On the nitration of diethylaniline. 297. — presents a paper of Dr. H. Dutilii: "On partial racemism". 393. — presents a paper of Dr. H. E. Kruyt: "The equilibrium solid-li(|uid-gas in binary systems which present mixed crystals". 537. — presents a paper of Prof. F. M. Jaeger: "Studies on Tellurium. 1. The mutual behaviour of the elements sulphur and tellurium". e02. ROTATION (Remarks on the experiments of Wilson and Martyn on the velocity of) of the electric discharge in gases in a radial magnetic field. 'I'2S. RUitiDiUM COMPOUNDS (On the radioactivity of). 154. SAP raising forces in living wood. 563. SCHEFFER (f. e. c). Ou heterogeneous equilibria of dissociating compounds. 257. SCHOUTE {v. II.) presents a paper of Dr. S. L. van Oss: "On peutaspheric geome- try". 19. — presents a paper of Dr. VV. van der VVoude: "The cubic involution of the first rank in the plane". 751. SEED DISTRIBUTION (.V brief contribution to the knowledge of endozoic) by birds in Java, based on a collection made by Mr. Baktiiels on the Pangerango and near Batavia. lUS. SERUM-AN.APiiVLA.\is (Contributions to the study of). 4tii Communication. 781. CONTENTS XIII SHARKS (On the cliiinges in the Ijlood serum of) ;ifler Ijleedini;'. u77. — (On the constitution of the urine of) with normal and increased diuresis. 3P". s L E E s w Y K (j. G.). Contributions to the study of Serum-anajihylaxis. ■!"' Commu- nication. 78 1. SMITH (J. J.). On Distylium stellare 0. K. and Aporosa campanuhita J.J. S. 34l. SMiTS (a). On retrogressive melting-point lines. 2-27. — The P-T-X-spacial representation of the system ether-authra(|uii)one"'. 2.'31. — The photo-and electrochemicnl equilibria. 356. — A new theory of the phenomenon allotropy. 763. — and S. PosTM.*. On the compounds of ammonia and water. IS6. — and E. C. Witsenburg. On the phenomena which occur when in a ternarv system the plaitpoint surface meets the two sheet three-phase surface. 182. — and J. P. WuiTE. On the system water-natrium sulphate. 241-. SODIUM ALKYL CARBONATES (On.). 303. SOLID STATE (On the). IT. 26. III. 120. IV. 138. SPACIAL UEPiiESENTATlON (The P-T-X-) of the system ether-anthraquini)ne. 231. SPACIAL SYSTKM (The constructive determination of tlie velocities of a). 12. SPRONCK (c. H. H.) presents a paper of Dr. J. G. Sleeswijk: "Contributions to the study of serumanaphylaxis" 4"' Communication. 781. STABLE POSITIONS (Ou the) of equilibrium of floating parallelepipeda. 383. STATOLITH THEORY (The inadmissibility of the) of geotropism. 34'3. s T o K (J. V. VAN D E r). On the determination of tidal constants from observa- tions performed with horizontal pendulums. 2. SULFIDES (The behaviour of phosphorescent) of the alkaline earths at various tempe- ratures, and particularly at very low temperatures. 157. SULPHUR and Tellurium (The mutual behaviour of the elements). 602. SUN (Regular consequences of irregular refraction in the). 266. SUN-SPOT SPECTRA (The magnetic separation of absorption lines in connexion with). 584. SURFACE (On the phenomena which occur when in a ternary system the plaitpoint surface meets the two-sheet three-phase). 182. SURFACES (Continuous one-one transformations of) in themselves. 2"'^ Communication. 286. — (On continuous vector distributions on). 2"'! Communication. 716. SYSTEM ether-authraquinone (The P-T-Xspacial representation of the). 231. — water-natrium sulphate (On the). 244. TANNIN (On the tests for) in the living plant and on the physiological significance of tannin. 685. TKLLUiilUM (Studies on). I. The mutual behaviour of the elements: sulphur and tellurium. 602. TEMPERATURE (The influence of) on the respiration of the higher plants. 219. TEMPERATURES (On phosphorescence at very low). 76. — (The atomic volume of allotropic modifications at very low). 437. — (Researches on magnetization at very low). 649. — (The behaviour of the phosphorescent sulfides of the alkaline earths at various) and particularly at very low temperatures. 157. CON T E N T S. TEiiNAKY SYSTEM (Oil the ijlieiiomenn wliicli occur wlieii in ;i) tlie plaitpoiut surface meets llie two sheet three-phase surfiice. 182. T E s c U (p.). On Jurassic fossils ns rounded pebbles in Nortii Brabant and Liraburg. 422. •iiiEouY (The) of the ZEEMAN-eftect in a direction inclined to the lines of force. 321. — (A new) of the phenomenon allotropy. 763. THEKMOMAGNETic PROPERTIES (The) of elements. 59(). TIDAL CONSTANTS (On the determination of) from observations jierformcd with horizontal l)endulums. 2. TIMMERMANS (j.) and I'll. KoHNSTAMM. On the influence of the pressure on the miscibilily of two liquids. 3.15. — (Some remarks suggested by a paper by). 454. TOM BROCK (w.) and Ernst Cohen. The electromotive force of zinc amalgams. 'JS. TONUS (About the formation of creatine in the muscles at the) and the development of rigidity. 550. TRANSFOKM.^Tioxs (Continuous oiie-one) ofsurfaces in themselves. 2""^ Communication 286. TRANSVAAL (Picnaarite, a melanocratic foyaite from). 547. uiiic ACID (The decomposition of) by bacteria. 54. uuiNE (On the constitution of the) of sharks with normal and increased diuresis. 380. UVEN (m. .1. van). Investigation of the functions which can be built up by means of intinitesimal iteration. Contribution to the solution of the functional equation of Abel. 208. 427- — On the orbits of u function obtained by inlinitesimal iteration in its complex plane. 503. VALKENUDHG (c. T. VAN) Surface and structure of the cortex of a microcephalic idiot. 202. VARIABILITY in Bacillus prodigiosus. 640. VECTOR DISTRIBUTIONS (On continuous) on surfaces. 2"'' Communication. 716. VELOCITIES (The constructive determination of the) of a spacial system. 12 VIBRATIONS (The oscillations about a position of equilibrium where a sim[)le linear relation exists between the frequencies of the principal), b' part. 619. 2"ii part. 735. VIOLIN (On the motion of the bridge of the). 513. viscosACciiARASE, au snzym which produces slime from cane-sugar. 635. — (Emulsion laevulan, the product of the action of) on cane-sugar. 795. voLLGRAFi' (j. A.). Remarks on the experiments of Wilson and Maktijn on the velocity of rotation of the electric discharge in gases in a radial magnetic iield. 428. VRIES (hk. d e) presents a paper of Mr. .1. BkuiN: '"On the surfaces the asymptotic lines of which can be determined by quadratures." 759. V R 1 E s (J A N u e). On pairs of points which are associated with respect to a plane cubic. 711- — On polar iigures with respect to a plane cubic curve. 776. VKIES (o T T o D e). The abnormal reduction of an aromatic nitrocompound with tin ;ind hvdrochloric acid and an interesting case of dismorphism. 305. C O N TENT S. IV 1V A A L s (j. i).« V AX D E u) presents a pnper of Prof. A . S.MlXi iind E. (;. Witsknuuhg : "On the phenomena which occur when in a ternary system the plaitpoint surface meets the two sheet three-phase surface". 1S2. — presents a paper of Prof. A. Smits: 'On retrogressive melting-points lines." 227. — presents a paper of Prof. A. Smits: '-The P-T-X-spacial representation of tlie system ether-authraquinone." 231. — presents a paper of Dr. J. Timmermans and Prof. Ph. Kohxstamm: "On ihe influence of the pressure on the miscibility of two liquids." 23.5. — presents a paper of Prof. A. Smits: "The photo- and electrochemical equilibria.'' 35S. — presents a paper of Prof. Ph. Kohnstamm: "A short reply to Mr. van Laar's remarks." 534. WATEit (On the compounds of ammonia and). 18G. — (Contribution to the knowledge of the movement of) in plants. 57^. — natrium sulphate (On the system). 244. avatersecretion in plants (Contribution to the knowledge of). SOG. 400. w E E a M A X (r. a.) On a synthesis of aldehydes and indols. 42. w E E V E R s (th.). The physiological significance of certain glucosides. 193. WEISS (p I E R K e) and H. Kamerlingh Onnes. lle-earches on magnetization at very low temperatures. 649. west (f. a. f. c.) presents a paper of Miss C. J. Pekelharixg: "Investigations o;i the relat'on between the presentation time and intensity of stimulus in geotropic curvatures". 65. — presents a paper of IJr. Th. Weevers: "The physiological significance of certain glucosides." 193. — presents a paper of Mr. J. Kuvter: "The influence of temperature on the respi- ra'ion of the higher plants". 219. — presents a paper of Mr. J. J. Smith: "On Distylium stellare 0. K. and .Vporosa campanulata J. J. S." 341. — The inadmissilnlity of the statolith theory of geotropism as proved by experiments of Miss C. J. Pekelharixg." 343. w I c 11 M A X x (a.). The fens of tlie Indian Archipelago. 7U. — (Some brief remarks relating to the communication of Prof). On fen formations in the East-Indian Archipelago." 74. W1L30N and Martyn (Remarks on the experiments of) on the velocity of rotation of the electric discharge in gases in a radial magnetic field. 428. w 1 N .\ w E R (b.) and P. Zekmax. The magnetic separation of absorption lines in connexion with sun-spot spectra. 5S4>. w I s s E L I X G H (c. V A x). On the tests for tannin in the living plant and on the physiological significance of taanin. bS5. w ITS EN BURG (e. c.) and A. Smits. On the phenomena which occur wlieii in a ternary system the plaitpoint surface meets the two sheet three-phase surface. 182 WOOD (Sap raising forces in living). 563. woe BE (w. VAX 1) E k). The cubic involution of the first rank in the plane. 751 CONTENTS wuiTE (j. 1'.) and A. Smito. On the system wnter-iiatriuin sulphate, Hi. XYLOSMA LEPROsiPES CLOS. (Some remarks on the nomenclature and synonymy of), X. frngana Decne and Flueggea serrata Miq. 116. z E E M A N (p.). The degree of completeness of the circular polarization of magneti- cally divided lines. 345. — and E. Winawer. The magnetic separation of absorption lines in connexion with snn-spot spectra. 584. — EFFECT (On the theory of the) in a direction inclined to the lines of force. 331. ZINC AMALGAMS (The electromotive force of). 98. z WA A RD emaker(ii.). About odour- affinity based on experiments of J. llEiiMANiUEs.yO. — presents a paper of Mr. F. J. J. Buytendijk: "On the changes in the blood serum of sharks after bleeding." 377. — presents a paper of iVlr. F. J. J. Buvtekbuk: "On the constitution of the urine of sharks with normal and increased diuresis." 381. — presents a paper of Dr. A. K. M. Noyons: "Communications about the electro- gram of the atrium cordis." 680. — The Camera silenta of the Physiological Laboratory at Utrecht. 70(i. z IJ L s T K A (k.). Contributions to the knowledge of the movement of water in jilantb. UlA. Q Alcadamie van Wetenschappen, 57 Amsterdam. AX'deeling voor A4d de Wis- en Natuurkundige V.12 V/etenschappen pt.2 Proceedings of the Section Physical & of Sciences Applied Sci. Serials PLEASE DO NOT REMOVE CARDS OR SLIPS FROM THIS POCKET UNIVERSITY OF TORONTO LIBRARY STORAGE