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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 ••uniil'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 eumiarative
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 solidlillnidga^ 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 eomiileN 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 initialpoint .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 develoini('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 )ieseiit 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
iterationindex //, 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/ ^ *'•
Yr + 6 <^
1 p.vi1
/ (.7;) =  loii r, ^^'here
/ = Mil — =^ log
(« + ff)— l/(a^(f)"+4^7 ■ 4(«(fJy)
(«— 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 socalled /(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 spiiiils 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 incieases 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'iMonco 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 neiiiroci/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('iiiiiii,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 sMrai 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 \eiv 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.
ApianBknnewitz ') 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 (lisila('onieid liv llic lidi'izonlal iiiotiim, i.e. in llio diicctidii 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 violinbridge 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 muteeffect, 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 iudeien
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 (laniiiii,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 nintesonnd
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 miuh 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 elaiiii 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 Hobable: 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 zincblock 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 astring 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 sec2 , ,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 shaiper. 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 well 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 Primateskull , 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 faceskull — 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
baseline adopted by them, and the line which connects basion and
opisthion. The baseline 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. Llipzig 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 baseline, 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 orbitooccipital" '). Broca here was proceeding from the postu
late that the orbitalplane, 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 baseline again the nasionbasion, 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 baseline 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
baseline 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 baselines which are drawn through the
skullbase, 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 facialskull, 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 socalled 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 orbilooccipilal. 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 baseline which was described
in the first paper, and in sodoing 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 baseline, the For. magn. looks downward, is
parallel to the baseline, 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 diawn from the basion to the
baseline. 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 fullgrown 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 plate 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 slopeangles 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 ieasons
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 Orangoutang 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 primategenera,
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 moie 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 fullgrown one.
The following may serve as a proof of this. Whereas in a fullgrown
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 Orangoutang 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
fullgrown 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 baseline, i.e. a horizontal position.
An(/le of the For. nytgn. in fullgrown 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 fullgrown 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 fullgrown 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 basalindex 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 nonmlscibility
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 noniniscibility 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 nonmiscibility in view, but \ery decidedly all nonmiscibility.
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 iioiimiscil)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 nonmiscibility 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, nonmiscibility 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 nonmembers. 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 threephase line solidliquidgas (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 gasphase, 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 threephase 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 threephase equilibria in the system /jdichlorobenzene —
/Mlibromobenzene, the same system of which, thanks to Kuster
and Speranski (1.c.\ we ah'eady know a series of solidgas equilibria.
Fig. 1.
Fig. 1 is a combined FT and T^rprojection : 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 triplepoint pressures are comparatively low,
the branches S and L may, usually, be taken as being equal
respectively to the meltingpoint curve and the freezingpoint 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 threephase 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 threephase 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(iuulvapour
surfiice and the ifwplane.
As to the form of this new if'wsurface 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 vapourliquid 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
vapourliquid 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 sujfaces 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 gasliquid
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 threephase 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 nonrelated 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 threephase 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 triplepoint 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 threephase 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:
;— ivs — 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 nonaccentuated ones
had above for A.
In the case of lb the threephase 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
triplepoint 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 threephase 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 Jaxis and 2. about equal triplepoint 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 threephase 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 threephase 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'ipiepoint
pressures.
Case \\b : a melting diagram with closely joined branches at the
side of the component A, and but slightly differing triplepoint
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 — pdibromo
benzene which has been going on already for a considerable time.
November 1909. Utrecht, van 't HoFFlaboratory.
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: "Sapraising 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 canesugar", 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 iioriheast 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 olivegreen 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, aegirineaugite
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
peison 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 aie 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 cieati
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 moie 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 doublerefracting 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 onesided 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
coldblooded 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 socalled "decerebrate
rigidity" develops itself, a long continuing tonic contraction of definite
musclegroups, 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
coriesponding 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 breastvertebra, 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 inductioncurrents
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 BioChemical 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 inductionstrokes. 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
inductionstroke. 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 inductionstrokes.
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 eriors of observation. Moreover the
difference is now in favour of ihe stimulated, now of the notstimulated
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 muscletonus 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 cuirent 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
contiaction 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, muscletonus 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
rhodannatrium 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 inductionstroke 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 inductionstroke. 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 inductionstrokes.
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 nonstimulated 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 calciumsalts
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 sartoiiiis", 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 rhodannatrium. 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 inductionstroke. 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 nonnitrogenous 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 startingpoint 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 labouringfaculty 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 frogmuscles 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 rabbitmuscle 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 heatcoagulation and the
postmortem rigour as well as the tonus, a ciiem leal process lakes place
which causes the origination of creatine.
( 5r,:} )
Botany. — "Saprdisunj fork's in liviiii/ ivood." By E. RkIiNDKKS.
(Coniinunicaled by Prof. J. W. Moii,.).
(_)f the many theories, which have been advanced in expkxiiation of the
transpirationcurrent 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 transpirationcuri'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 150200
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 waterthreads"
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 airbnl)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 shown 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 twentytwo 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 aircontent 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 seaicli for it.
This argument is tlierefore no longer always adduced in siippoit
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 waterthread 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, wahiend 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 impoitant 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 Ushaped 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(iinii 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 bulbtubes 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 wellknown
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 fiom 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 fiom 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 iegularity. 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 treetrunk 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 GoDUnvsKi 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 piece 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 treetrunk 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 suctionpressure 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 waterconducting 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 appletree, 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
tinplate reseivoirs 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 iound 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 suiface 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 cknc.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
cloveoil 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 biauches 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 pareiuhyma 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 blueviolet, and this was aisn tlie case willi llic liall' lioidcrrd
})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 blueviolet, 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 blueviolet.
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 ojdcr 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'ouf 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 aK'\.
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 Siiineviolell 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. Toii 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(^etfuak
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.Lion
witk Sunspot 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 apiear 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 );iiL'i. tlic clianiii' of ;ili>t)riii()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
magnetooptical 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 tliifeienl H(int>. w iiicli may l)e tested by e.\(ie
rimeiit. The imerse effect has become of suj)reme interest in solai'
pliysics, since Hat.e's ') discovery that the dark lines of the suns[)ot
spectrnm exhibit the characteristic phenomena of magnetic separation.
Tiie e.\)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
sunspot 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 sunspots.
Contributions from tlie Mount Wilson Solar Observatory Nr. 30. 1908.
( r>8fi )
lit'ld ; tlic (Ii(iiI)Il'I Ivi(' 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 snnspots 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 lia\erses a sodium flame,
placed bet\veen the poles of a or lloiselccUomagnel. 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)mH)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 naiio\\, 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 \;iK)ur no ali>(>r]ilit)ii ;it all (ir oiilv \eiT weak
Inues 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 (piarlcrwa\c plate and a Xicol we may (piench
either the rightdianded or the lefthanded 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 righthanded circnlar analyser. The absorption line corre
sponding to a righthanded circularly polarized component is seen
with both increased clearness and darkless by examining it with a
righthanded 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.
Righthanded and lefthanded 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' iiiM.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'ailis aie liixeu on (lie plates
annexed to our ])aHT.
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 sunspot 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, overlapiiug 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 seiaraled by
'■'S '*•• a circular analyser.
These considerations may lie applied to the magnetic division in
^oYr
( o89 )
sinis)(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 quaitet.
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, wheieas 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 ilisiiiiliaiirc is ii'(i(liiccil li\ llir iianow
ivvcisc'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 sepaiatioii must liecome, on ac<'(inni if
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. Vaiour 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
comilit'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 lirl
discovered: a slight change of broad lines in ti weak field.
( 591 )
Willi still greater ;ilisoiiti()ii lln' inlliiciHH' tif llie field liocoincs
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 ^unspols
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 45()()
Gauss. Hence, one would be inclined to tliink that the density in
the layers, which bring about Ihe absor)tion in the snnspot spectrum
can only be small. Moreover, the nonuniformity of the field of
sunspots 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 peiforated poles.
After putting on the cnrrenl, one sees in llie conliinious spectrum,
2 daik 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\elengllis only righthanded circularly iolari/.ed
light, bul not leflhauded 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 circulaily 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
ilhilraled by Fig. '2. In llie eenlial strip niie line, at the position
of the iinmodilieil one, Minoiinded 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 luaiked.
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 wavefront
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 clliis('. The ratio of the axes is as I lo ro.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 ihomb, 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 righthanded 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 direction 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 uion the nuxguitude of (^ whether  is ^/'cr/A'/' or /..« than
b^a
( 594 )
We (listiiijinisli 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 calcpar 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 cential ''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 sunspot 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 sunspot 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 sunspot spectra.
Di D,
S 9 in
Types of sunspot 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 nonuniform 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 (■(Hiii(>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 iiunaniform 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 rciros('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 nonnniform fields. 5' is the quartet of /J, observed across
the field ; (i' tlie sextet of D.^ observed axiallv in a nonuniform,
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 llll.
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 sunspct lines (adopted from Mitchell).
15, 5', C 7', separations in nonuniform 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(^ ISosscluiI.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 siisceitibilily is in
versely proporlioiiaJ to the absoliile lemperalure.
2. h'or diamagnelic substances, mi llie conliary. 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()lriiiy;.
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 "limitlaw'" 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 torsionbalance combined with a nonuniform
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. Xetil.
(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 testcoil. The sensitiveness of the torsionbalance 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 ihermoelcmcMl, pie\iously checked
by observations on the mellinyioints of tin, telluiium, antimony,
and gold.
Ti'st.sariiitli's. I'he threat diflitulty 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 bluereaction.
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 ui 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 iioiiterru
uineous eleuieiiis woidd icmaiu conshmt. :il leasi within the usual
( .><.;. )
liclilrunm'. Till.' full (■iiiniiiiiiiicaiiini iif ihr iviill nlilaiin'il wdnld
n'iiiirc iiiaiiv lalili'> ami curves; \\i llnii'lini' ilraw allriilinii in llir
]irinciial M)iiil> (inl\ ,
S^ii'ci/ic sKsci'iitihUilii ') at 18°. Tlie valiiON tumid lie between 1
.111(1 \ 5 (aiiioiplioiis 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
cai (if pli(isphoius and antinionv, and also ihat there is no such
inlliKMice w idi sulphur, lh(>u<;h it is so wellknown for it.s polymor
phous )iopeities. 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 espiMially 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
substances 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 MendklkjeffBr.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 rolleeted
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(ii l(i iidlloni; llic I'lcnu'nl' luidei coiinnn
C show a conslaiil su>ccitiliilil\ , 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 iiiciease 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[)eraturerange. 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 meltingpoint 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(irili(His
Iranslnriii.'ilidiis, a (lisciiiiliiniiHis (liiiiiniilinii of I •'i" „ nf llic spcrilic
siiscrilil)ilil\ was I'oiiikI al iIh' liaiisiliiiiiMiiiil of (fllia!liiiiii and
,?llialliiiiii al TAi . liiil llir iiuisi iciiiarkulile pi'opcTlies arc sliewii
liy liii: ir (liaina.iiiiclii ,mev 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 expeiimental 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
ielatiou 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 )
(■niiii()iMi(ls Tl'S_. ;uiiI TcS,, ai'c iircsciil in IIm' lirnw iiislilihick ir('ci
pilatc^, luriniMi wlicn iassiim II, S liiidii'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 ainei)ns pnias^inin or sodium liydrnvidc, wliicli
is also tiie case with Te*)., and Te*).,.
Bkcker ') was tlie liisl to analyze these precipitates and he linallv
arrived at the concdiision that liieir coniposition aclnally c(nresponds
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^SjTeS.^, 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 wellknown 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 coiiipaiison
widi sidilialei ■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 Ameiicau 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 diyiiess, and re)eated e.xtraction of the
residue with strong hydrochloric acid. The hoiling iiltrate was then
precipitated hy snlphnrdio.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 oichuled solid ( 'r( )., ; this mailer I hope
to refer lo shortly.
The pure telluiic 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 liislioiis needles ni l(j
5 III. 111. ill leiinlli jiiiil iiMially llalteiied
;il()ii,i; !<*"*!■ I liev e.xliiliil iiiaiiv \iciiial
pliUU's iarliciil;ulv 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; ;//=rll()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'lliiiiiiiii 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 aliiinsplicii' (if coal f^as,
with iIr' aid of a lariio I'KRKoTfiiriiarc. 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 inrile 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 iiiaiililies 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 veiy pnre lellnrinni llins obtained is
free fiom seleinnm as proved by llie exceedingly delicate Xoiuns"
potassinniiodidereaclion and by the nonreduction 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 fninace 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 gasinlettube. 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 plalinumplaliiinmrliodinin 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 luixtiiies 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 SULPHURTELLURIUM 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 eiidsolidifyiiigpoint. . 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 ieireseiiled 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 )
riili ill siililiiir appear Ik cliaii^c iiilo llic rliniiiluc rorin al a 'owrr
leiiipi'raliirc. In lliese riiviiiiistances iiolliiiig' ia noticed as to coiiipoiuuls
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 Dumas. 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' exierience 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 comilicated,
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; sniphnrseleninm
one, according to Muthmann, is analogous to the form of ysulphnr,
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 socalled fforin 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: sulphurtellurium
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 sulphurcontaining 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 telluriumsulphur 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 socalled 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 telluratesolutions
obtained from TeO, or telluric acid. A beautiful, somewhat crystalline
looking product was obtained from the alcohohc suspension of TeO,;
the analysis of the blueblack 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 greenishyellow: 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 telluriumcontent 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 satuiated 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 rosetteshaped 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 dissociationequilibria 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 Ksalt 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
BaSsolution. The analysis indeed, did not always yield precisely the
same results, but still tlie composition agreed closely witii the fornmla
BajSTe^ ; 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.
Triclinicpinacoidal.
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 planedevelopment 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 platinumdisii acted as tlie cathode and a disc
shaped platinumelectrode 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 telluriumsulphur 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 sodiumsulphantimonate'), 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 telluiium 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
alkaliearth 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
seleniumsulphur or telluriumsulphur 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
dh
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 abovementioned 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 Zaxis 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 . . dz . 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 HamiltoxJacobi 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:
/(?)?MiS)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 eveiy arbitrary moment the «'s and tlie 3's have a definite value.
These vahies determine a certain Lissajoiis 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
liJiV'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 ,ra.\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/lg cw (2«, « ,/ ) and
S VT^i cos <f = A,
wc liiul for (lie Ctiuatioii of the osculating turves willi C as [)iiiainelei' :
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 cinati(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«„ iKiitiliola,
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 oieuings turned ui\vai(ls.
( 629 )
llie ?. )iir;il)()l;i li;is its upciiin,^ liiriicil ildwiiwunis lliis ivil' 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/fplanc.
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'Zplane. 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 inoie 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 aiproaciies 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 veiy small.
In this way we ai)roach the general case where ihere is no
(pieslitiu about relation.
§ 12. How the transition to this geneial 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'lr
„ „ 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 nondisturbing terms, the e(uations become:
7i + «i'7i = ('"'1' r '■"./ + P) 'h<], — ^"h'J2 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 Jid 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 /•(iDesiKjnr' . By Prof. Di'. M. W. Beijerinck.
The emnlsi.on reaction.
Many sporeproducing and a lew non sporeproducing bacilli, cause,
when growing in presence of canesugar 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(itlets dI' oil siLspeiided in liie agar, l)Ut
as sining Mililiiwic 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 iiiiii'iri.'cllv in gclalin. In die agar llie
process is inipede<l wiieii acid i> produced hv llie nncrobes. Tims
lionillonagar, vea^twaleraiiar, and wurlagar wiiii canesngar 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 canesugar 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 agarplales with 10 7„ of canesugar 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 cancsngar can \ary between 0.1 "/,. ^i"*^' ^^^ ° ,i without
much dillerencc in the result.
After the solidifying of the agarplate and the removal of the
adhering water, soilbacilli are dispersed, obtained by shaking some
gardensoil 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.
Canesugar 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
soilbacillns, commonly also fouml in canesugar 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
eanesngar, 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 nonspore 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 laige 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 soilbacilli 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" /(, agarsolntion. 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
surfacetension 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 cultureliquids
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
sugaragarplate, 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 imiiiil 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 [irepaied by lllleriiig a cidliire of B. nu'siuitcricu.t
raljidtiis and pieci])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 secretionproduct 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 canesugar agarjilates 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 cnllureagar 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 experimenltiihe and cautiously warmed with a little Fkhlixo's
copper solution, a strong reduction is seen, which does not lake rise
Avilh the same sugaragar 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 canesugar.
Fig. 1. Bacillus mesentericus
"Mfe
Fig. 2. Bacillus emulsionis.
•r
The emulsionreaction.
Proceedings Royal Acad Amsterdam. Vol. XII.
C «39 )
rciliiciiiii. Hence it is sure lluit ill llic "eiiiiilsidi: icaclidii", ki,i;etliei'
witii llie nonred 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 invertsugar
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 canesugar, else it would not be clear
why Ihe slime cannot diffuse through the agar, which canesugar 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 cellwall 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
exoenzyme, 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 gkuose and
some other sugars, which do not produce the emulsion.
Very remarkable is the fad that all tiic hitherk) examined bacteria
which show the emulsionphenomenon, are aiile, at (k'linite culture
conditions, for example on canesugar gelatin, when no emulsion is
produced, to form nondiffusing "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 iesearch 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 cellwall of the microbes, and must be considered as endoenzymes
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 wallsubstances 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 [nincipaux durivus. p. 745, 1900.
( «K> )
tliat "dexlran" is a iiiudilicalioii ol' (•clliilnse, and tlic till iinw iidt
explained seeoiulary changes, observed in so many cellwalls, 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 gelatinplates, 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 canesugar, 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 canesugar, 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 reinoculaliug in (puck succession, for
instance every 24 hours, into bouillon or on bouillonagar at 30° C,
each form of lidcillns /ini(li(/iosiis. whether the natural or normal
form, 01' a variant obtaiiud 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 cultuie 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 reinoculations 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 bouillonagarplate or
in a thin layer of liquid. In ordinary bonillonagarplates 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 pievented, 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'diin is rdinid iincliaiigeil. As liv Iransplaiilalions
ill rapid siicres8ion (and under constant mid favonrable eunilitioiis)
no ciiaiige occnis during thousands of cellpartitions, 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 loois 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 reinoculation
is performed, the germs alfected l)y that iiilliience can increase as
well as those that remained normal, whilst by not reinoculating,
thus in the first culture, the nonaffected germs are by far more
numerous and remain so as the celldivision 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
agarplates, 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/iosus may be considered as
plus or gainvariants, minus or lossvariants, and qualitative variants.
This is exiosed 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 noimal forui. the
plusvariants above it, the minusvariants 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 "gainvariation", tiie liorizontal •■qualitative
variation'', the downward arrows "lossvariation". Dotted
arrows signify tliat atavism has been observed.
The two qualitative colourvariants, (larnttis whirh is orange
coloiireil and Iii/iiI/ihis of a deep \ inered, varv in a wav (uile corres
ponding to the normal form and like this throw otf, under the
same circumstances, slimevariants and while vaiianls. 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 (lisiosed lo ala\ism.
ll shoidd nioreovei be noted thai Ihi' (////vir///,vvariant 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 canesugar,
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 consideicd,
has noi \('l been obtained as a \ariaul fr(un Ihe latter.
( 615 )
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 cellwall, 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 proiligiosn.'i. The vincosiis (6), dei'ived tVoin the latter, is
an ordinary red slime bacterium.
This redcoloured, toughslimy form, which may be called B. pi'O
(Ii</io,ms riscosiis, is no doubt a plusvariant. 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(^/iosiiscu\tmes 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 reinoculations
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 maltw^ort, in
tapwater 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 diiect influence of the food on the production of the variant.
The auratus and hyalimisvanantii, also, have only taken rise in licjuid
cultures, namely in broth and in the glucoseasparagine solution.
31oreover, hynlimis, which is of a deep vine red, is easily obtained
from a solution of pure gelatin in tapwater with 0.02 pCt. K„HP( >,,
after repeated reinoculations, 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
minusvariants. 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
(lirothagar), as also culti\ation at lempci'aluies higher than 30' ('.
e.g. at 33'' ('., pre\'ent pigment produciion, uo liei'i'dilar_\ \arialiou
at all is caused by these inlluemes. 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
reiiiociilatioiis, turn sliniv like those of the red noniial foirn itself.
(Joloiiy cultnie 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 maltwort
or in liouillon, wo find after one or two transferrings, each time
al'ler two days, and when sown on Ixuiilionagar, 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 ilusvariation 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, niuie roadilv iIkui llic rose \;iri;inl ("2), as
well rod iioniud forms (J) as wliite ones (4). For tlie rest, lliis
iiioi'c \aiialile \ariant has also )rovc(l to remain coiislant when
iiiicl<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.tv<ivhu\ii<, also have a disposition lo throw off a few
ied normal forms, but lliey do so only after gi'owing for weeks or
months on boiiillonagar; 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 maltwort as in bouillon the risvo.'Ots
variant is regularly produced; but from mallwort 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
lossvariants, all obtained with certainty by determinetl experiments;
the stockform 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 reinoculalions 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
finni. lliiis, tlie r/M/r//;<.yvariant produces an (/?/yv///wslimevaiiant,
which must be considered as a gainvariant, and an '//^;/.yvariant, wliicli
must he taken for a lossvariant.
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. Gainala\ism in lossvariants and lossatavism 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 uiiciobes 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 subvariants, 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 xxrv well with lliosc
which liad liocii found cx[)eriiiieiilaliv for uuigiietite at teinperaliu'cs
aiuivc llie (>i(niiaiv. 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 salurationmagnetizalion 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 CuRiKpoint. 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.LBoLTZMANN partition law has been
eniiloyed. 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 CuRiKpoint to the tempeiature
a! whicli spontaneous feiromagnetism 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 (Urtsiconstant, 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
hoalmolion 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 longcontinued 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 CuKiEpoiut (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 saturationmagnetization 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 saturationmagnetization, 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 shdiiu' 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 temierature 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'viicr 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 CuiUEpoint 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 corioborated 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!iic^>i(iii ami
cx)aii.si()ii liv Ileal, tor wliicli \ \n di'.i; W'am.sV theory and law of
(_'ori'osH)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 liyiolliesis 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
satuiaiionmagnetizalion 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 cmrected 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. gramatom.
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 eiOi
'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 currentcarrying 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 temierature ').
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 refereiue 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, chiomiuni, 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 slionulv
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 puie 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 preiared in this manner exhibited
paramagnetism. A glass tube with the powdered manganese was also
paiamagnetic. 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 magnetocrystalline 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 rerioiiuigiielisin 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 ireiared liy liiNsKN IVoni niangaiiese lluoride. Tiie
veiy 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 elliisoids 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(Uuire 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 torsionspring 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 volumeelement 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 /dH\
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 semimajoraxis 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 nonuniformity 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
nonuniformity 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 polepieces.
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 polepieces, and
the couple becomes increased thereby. This fact was established by
previous experiments with a larger electromagnet with flat polepieces
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 nonunilbrmity 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 polepieces. 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 polesurfaces 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. Handwheels, whose position is read from
divided circles, coinniunicate a horizontal micrometric movement to
the polepieces.
The magnet turns upon a vertical axis and for that purpose is
mounted upon a ballbearing 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 ampereturns 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 polepieces 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>aialively
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 tubeshaped 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 platinumiridium 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 thinwalled germansilver 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 vanedamper 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 crosssection. 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
germansilver.
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 airtight 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 itelf 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 centringplate. 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 airtight 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 ballshaped 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 turningaxis 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 polepieces. Further
more a stream of air is directed against the tube between the pole
pieces. Hence the polepieces are in no way affected by the cryogenic
operations.
/'. Tlie springs are phosphorbronze. This substance is nonmagnetic
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 dynecentimetres 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 30fold 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 dividingengine 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 rn 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 oxyhydrogen 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 polepieces was read off the divided cylinders of the
magnet and was veiified 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 coupleratio 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 tenthousandths 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 (ianiiei' 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 platinumiridium 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
magnetocrystalline 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
strainmagnetic 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 notite — 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 venlririeimage, 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(iubletO)[)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 sampleanimal 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 Vcih;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
heartventricle 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
sampleobjects 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 sinuscontractions are preserved, but the
atriumcontractions 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 appeaiance 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
stringgalvanometer (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 musclecontraction.
In the electrocardiogram of man and animals derived indirectly,
we find, evidently on the ground of the Pelevation, 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 sampleobject to derive both the atria
at the same time The same thing I did also with the cutout 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
swingapparatus 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 actioncurrent 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 (storeelectrodes 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
atrioelectrogram 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 atriumtops
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 gaschamber, 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 atriatops with a double
other electrode. After evaporation of the chloroform by openiiig
( 084 )
the gaschambei' the atiia begin to recover and after 39 minutes
the electrical phenomena reach their original size again with (heir
meclianical clianges. Snch poisoningexperiments can be repeated a few
times without any great harm for the sampleobject.
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 atrioelectrogram is evidently also dependent on
nervous influences. Einthovkn and Vaandrager have already directed
attention to this for the cardioelectrogram of a dog derived indirectly.
For the a'ria of Eniys this may be proved very distinctly at direct
derivation under the influence of vagusstimulation.
A heart of Emys derobed of ventriculus, so consisting only of
the sinus and the two atria, is derived to the stringgalvanometer.
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 atriumtop. 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 inductioncurrents 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 vagusstandstill, which, as soon as the
stimulus is put a stop to, is iiroken otl' and causes a new series of
atriumcontractions with a strong sinuscontraction. During the vagus
standstill the object has produced no electrical phenomena that are
to be registered. Directly after the vagusstandstill the atrio
electrograms show themselves again, but now altered in form. The
doubletopped image has been replaced by a phenomenon with a
strongly pronounced diphasic character, which however, the stimulation
being stopped, passes into the original doublelopped image by a
gradual alteration. In this experiment the n. vagus was stimulated by
means of tiie sledgeinductorium of or IJois Reymond, without a
kernel, at a secondary coildistance of 5.5 cm. and a LESsiNGelement
in the primary chain.
When weaker currents are used for stimulation, a state of things
may be obtained in which the vagusstandstill 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 fanninprobletn.
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 teiminoiogv, 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
bolanicochemical 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.
Genlialli. 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 geneially 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 excreloiy products, byproducts or
decomposition products. He thought it very improbable that tannins
could serve in some way or other as material for tiio building up
of cellwalls.
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 ExperimentalPhysiologic
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 byproducts 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 reservematerial.
According to a few investigators tannins must, in some cases, be
regarded as excretory products or as byproducts 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. Erfauteiung 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. VersuchsSlationen, 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 aie 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 cellwall 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 resinpassages 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 amidocompounds
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. NeuVorpoininern 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 Botanischchemische 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 onesided 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 ceitain 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\ementioned 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 tissueformation 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. Reservematerials 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 cellwalls, 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 byproducts
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 chromatophoresmass,
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 faclors 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 eihalten 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. SimO'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 unsatisfactoiy 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 reddislivioiel coloratiuii witli fiiiik' rliluiide. Since tlie (.•oluiired
compound is soluble and easily diffuses tlirough the preparation, the
ferric chloridetannin reaction of the globules may also be detected
when antipjrine is used, and the non appearance of the reaction in
the cellsap 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'ai'ilaments in
water, and if ferric chloride or osmic acid solution is then added,
the cellsap 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 abovementioned 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 vaiied 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 prcciiilalos 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 nonorganized active jirofein in the living cell. The
abovementioned 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()oilant 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
suierficial 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'.^rriUiil 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 piocess 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 tiansverse 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 abovementioned 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,
SoparatAbdruck aiis Bull. d. 1. boc. imp. des Nat. de Moskou, 1901, No. 1 en 2,
p. 19H. Zur Pliysiologie der Zelle, SeparatAbdruck 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 nongrowing 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 reservematerial, however; it
belongs to the soluble substances which the plant continually requires
for its development. It disappears and gives way to reservematerials,
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 leadcontacts.
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 telephonecells. 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 leadcontacts. 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 extracovering.
Taking the abovenamed 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 leadcontacts 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, uliiili, 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 goldthread stringgahanometer
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 ventilationchannels, consequently of
soundleaks. For doubledoor and doublewindow (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 soundsources 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 leadcontact, is not discounted '). Electric
light, telephone, sujiply of air for organpipes and sirens through a
narrow leaden tube and the necessary conductingwire 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 outerwalls of the building by means of byapartments, 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 soundfree 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 soundfree 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 hightoned whistling (i*/")
may be distinguished ; persons in whom this physiological earbuzzing
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 soundfree 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 earbuzzing,
entirely or partly, possesses the character of an afterimage').
.3. The phenomenon of accommodation, discovered by Hensen,
may be more closel}' studied, by conveying to a person standing
outside the camera silenta through boneconduction the tone of a
tuningfork, 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 sampleperson accommodates and
the observer bears a strengthened sound (()uix).
4. From the shortest expositiontime the smallest observable
number of soundvibrations 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 eaibuzzing can be suppressed : a. by the ticking
of a watch; b. by the sound of a tuningfork of the r'pitcli and a .soundforce 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 sirentones, may be demonstrated in
tones of different origin and pitch, with the aid of a sudden opening
or closing of a te!e)honeconlact or a sudden o)ening or closing of
a particularly constructed lead cock.
8. The spreading of the sound round a tnningfork with the
situation of the wellknown interferenceplanes may be accurately
traced, without making the mistakes that must necessarily arise in
apartments with echoing walls.
9. The action of the winding molluscshells as to their resonance
for buzzes may be proved directly.
JO. The soundextinguishing action of ditferent means of isolation
may be traced with perfect security ; for reports by dropping steel
balls on a steel plate") (fallphonometer of Zoth), for tones by electri
cally touching purely tuned bells; in both cases the instrument put
in a small nonresonant 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 peatmossplate from Klazienaveen, then
the corkstone; other materials that we examined had a considerably
smaller sounde.\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
aU:, = 0, the polar line by n.cix — 0. For the tangents out of Z to
that conic we have thus
2 3 2 2 ,
aa'jbz ^= 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 —  bijba) {a,jb. — a^b,!) zzi 0,
SO also into
(a,yi — (izbiiY ab rzz 0.
Now
Uyhz — ub,, = (aibi) (.Vico) + (aiba) 0/23) 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 + CiXip')
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
(.ViX2 — 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 — y2VlX3 + 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\ •!/2y3 = {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 cone
sponding decieasing 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, ?;jarcs, but not yet
?i,+l ^jarcs, 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 (?i3arcs 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 jljarcs.
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 nonsingular 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 aie 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 nonsingular 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 nonsingidar 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. Nonsingular 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 wellordered 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 wellordered 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 cuives 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 nonsingular 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 nonsingular 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 piincipal 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, coriesponds 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 descjihes 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 veilor 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 nhich 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 ciuve 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 iadiating 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' osiUhtiinns 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 teims
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 tiinevariabilit} 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/? (1g) CO* y=^?' +.;?+/• (18)
( 5!» )
where :
^ 1 , 1 ■>>
a 10 — — 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
/{:) = :' {iL)  (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 asymitolically to a foim 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 appears 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 Ay,^, 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^'qi + '''/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 Zaxis as
axis of revolution, which case is treated by Prof. FCorteweg 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 Zaxis 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 timevariability
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 wellknown 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 nR„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 («jff,)'  /«, 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^ faj ^ 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'^Zplane and the F/fplane 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 aie nondisturbing.
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 fiid 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 aSuS'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 , Ski 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) SalmonFiedler : 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
Ai, 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 theie
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 jjassw?// 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«'. Am. 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 socalled 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 =
(BiANCHiLuKAT, "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 strictionUne.
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, ^jprqjectioii 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 caxis 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 yaxis 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, boilingpoint, criticalpoint etc.
of a substance which beliaves as a unary one, does not ielate 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
meltingpoint, for then a point wiis realized of one of the melting
point lines of the pseudobinary system ^4  B, which for the
examined substances always showed a eutectic point.
As is evident we find the unary stable meltingpoint where the
curve for the inner liquidequilibrium meets one of the mellingpoint
curves of the pseudobinary 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 pseudobinary system.
In case of nionotropy these meetings between ihe iiiiaiy and the
pseudobinary system do not take place under but (ibar, I lie unary
mellingpoint 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 pseudobinary 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 liueepiiase 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 pseiulobinary system at lower temperature.
Diicussion 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 eqnilibiinm 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 newlyformed 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 transitionpoint,
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 meltingpoint viz.
B, and runs to the /lside 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 meltingpoint (137°).
When the pseudobinary 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 berorcmcntioncd splitliug up moi'e I'ully later on.
( 767)
curve dm of the pseudobinary 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 benzileorthocarbonic 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 pseudobinary system possesses a
eutectic point, the curve for the inner liquid equilibria must meet
the meltingpoint curve of the less complex substance, becau.se only
in these circumstances all the curves for the innei' equilibria can
run to the ^4side 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 pseudobinary 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 acelaldehydepai'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'ffigure as indicated by fig. 3, in which the
1) In consequence of llie considerable displacement of the inner equilibrium at
the tiansition 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 )
threephasetemperature lies between the mehingpoints 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 meltingpoint 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 meltingpoint 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
transitionpoint.
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 mixedcrystal curxe of the pseudobinary 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 partialmiscibilily in the pseudobinary 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 thinwalled narrow capil
laries or in socalled 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 meltingapparatus up to
a certain tem[)erature nbove llie transilioiipoint, and then all at once
transferred to an oilbath 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
pseudobinary character must appear, and entering the region of
partialmiscibility 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 threephase 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 mixingpoint A', the mixingcurx'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 mixingpoint K
must lie above 147'' ').
Though it follows from these experiments that abo\e the transition
temperature the ?',.rfigure 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 pioceed to another subject,
I will only point out, that if the equilibria are considered not at
constant pressure, but at the variable va)Ourpressure, 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 orangeyellow.
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 pseudobinary, but pseudoternary, 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',,rfigure 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 meltingpoint, and
112°, 8 for the meltingpoint 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 meltingpoint was
found at 110'. 9, the meltingpoint 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 fempeiature, 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,ie, 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 deleiniinations 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 pseudobinary
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 meltingpoint, 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 pseudobinarily metaatahle.
Let us assume for simplicity that in the pseudobinary system no
retardation worth mentioning appears, then the substance will solidify
at 4 and the solid substance S^ is deposited.
Now this twophase equilibrium is metastable to a high degree
in the pseudobinary 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 twophase equilibrium /,  .S', is changed into
the stable state ^ \ S.,, and this being a process which generates
heat, the temperature rises to the unary meltingpoint.
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 extlanation 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 metalalloys, 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 seruraanaphylaxis" (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 magnetooptic IVERREffect 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 nith iL'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
axdyCw = 0, «a«;f'ir = Oi Oi/OrCw ^= , . . . (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'jCyCx ^^ {cba] Cj,r,,b2ayar :z: — (abc) (ij,a„blc,jCx ^ 0.
As we can substitute {abc) axa,iCxCb,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^liytff'^/.;
from (2) follows
(abc) aj,a,ibif,i {Xhc + [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 ai'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 pjj,,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,,:, px, Pxy must coiucidc with the tangents px, Pi/, p~ >» A', Y, Z.
6. For i)x,Py, iJnd 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 serumnnapUylnxis."
(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 guineapig with respect
to horseserum in the phenomenon of Th. Sjuth, 2'"'. the problem
of the alexinefixation 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 horseserum by treatment with
the blood of a guineapig 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 horseserum. ^)
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 guineapig, 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
horseserum 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
guineapigs does fix complement with horseserum 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
Anaphylaxietechnik." 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 bloodsamples taken from the animals consecutively
at dilFerent times of the anaphylactic shock — I could in general
fix the course of the complementcurve. Friedberger, however, injected,
his hypersensitive animals intravenously : the reaction then goes so
quickly and is so violent, that die guineapigs usually die within
few minutes. And at the same time also the conqdementloss has
soon reached its maximum. Now in Was.sermann's laboratdiy 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
conipienienttitrations. From these it appeared among others that a
few minutes after intravenous injection of 0.2 cm' of horseserum
the complementquantity of the testanimalserum 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 testanimals.
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 alexinefixation 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 iiorseserum with rising tpiantities of anaphylactic
guineapigserum 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 expeiiments, I came
to the following conclusions. Even normal guineapig serum (inactivated)
often retard the haemolysis, quite independent of the presence of
horseserum ; 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 complementfixation in the
testtube. This incongruity of the alexinefixation 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
guineapigs 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 intoxicationphenomena 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 bloodspot is dissolved in 1 cm", of
physiological saltsolution 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 bloodspots, I have administered on conse
cutive days serum from goat, horse, cow, and guineapig, 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 horseserum 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 bloodcii'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 nonisolated 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 oneone 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 nonconsidered, 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 ?zdimensional 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. 230246.
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 agroup of S^.
If *S, and (S'j both belong to the agroup of S^, then S, belongs
also to the rtgroup 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 fligroup and an (?jgroup 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 agroups
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
agroups 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 npartite 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,{ii), .... 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 /:''' tiidtJi 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^MiOt) 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
'^'«./fVvi)^'^^"'' + "' +  + "'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 oneone 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 oneone 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 oneone 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 oneone 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 oneone 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 oneone correspondence
to the pai'ts of llie first order of q. Then the p^p^ substitutions of
g., into sucii a oneone 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 oneone 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 oneone 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 oneone 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 oneone 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 oneone transformations and being uniformly approximated
by a fundamental series of groups consisting each of a finite number
of continuous oneone 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
oneone transformations and being uniformly convergent direct pro
ducts each of a fundamental series of finite groups of coiitinuous
oneone transformations.
^ 5.
The shamaddition 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 transfoimation 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 "shamaddition" 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 "shamsubtrnctioti" of
•9566.... from •5473....; then this sharasubtraction 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, shamaddifion and
shamsubtraction 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 cellwall
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 Ziickeratten 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 spoies 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 EnLENMEYERtlasks 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 foimed, 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 cellwalls 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 cellwall 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 pioducer.
From the preceding it may be concluded that the large lumps of slime
so easily formed on cane sugar agarplates 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 cellwall included. Evidently in the cell
wall itself the enzyme forms new laevulan by converting the cane
sugar, with which both cellwall and agar)late are imbibed.
The production of cellwall 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 cellwall 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 paramagnetism, 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 ferromagnetic? 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 ferromagnetic 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 ferromagnetic 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 firstnamed 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 paramagnetic 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 carryingout 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 0slia[)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^(9p„)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<l5x; in the most unfavourable case at the boilingpoint 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 gasbul)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 glassblower '). As before, the chief
part consisted of two independent Ushaped 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
electromagnet. 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 0shaped 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 heatequilibrium 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 Oshaped 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 Germansilver caps P^; P^, Qi, Qt (fig 1);
the junction is made airtight 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 crosssections 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 electromagnet as was used for the cryogenic
investigation of the ferromagnetic 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 polepieces by cylindeis 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 polepieces 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 polepieces 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 [»olegap). 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 accuiate 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 silkinsulated 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 nonuniform 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 iegulate 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 Deprezd'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 polepieces 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
polepieces. 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 polepieces 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 polepieces, 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. 10e.
') 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
axisj2.5 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 tm. 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 . lOc.
We shall now give the series of observations made with the
silverwalled 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. = 240.6.106.
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. lOG.
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. = 2842. 10G.
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. 10G
i!r7i°,3.Mv.= 332,8. 10G
A'61^9 K.= 359,0 .106.
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 zIO"^ : 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 fiehlstrengths, 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 temperalmes 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 ferromagnelic 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 )oIepieces with {iiile ll;il end
surfaces lliat had been used tor the measurement of the magnetic
rise, and then truncated conical polepieces 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 polepieces 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 stufiingbox 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//*/ ;iic, 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 ferromagnetic
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 measuiing the angle of
torsion and the oildamper 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 crosssection. 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 platinumiridium 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 \nQ
f;iutioii8 iegardiiii!,' 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'erroniagnctic nielals.
T/ii' course of the oh.se/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 welldefined field ; we had not here to
deal with a value of the saturationmagnetization, 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
tiniied 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 polegap
may be re[)resented by an expression of the form
H = H,V~i~\{co,'6ksin^6) (3)
where i/„ is the field in the centre, and r and 6 [lolar coordinates
of a point in the polegap 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 ferromagnetic bodies it is quite negligible : see the previous paper).
We have accordingly devoted the greatest attention to this source
of error. The conical polepieces 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
polepieces; the experiments with the cylindrical polepieces 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 weie variable and K constant, for the tables refer to
two practically identical bodies, of which the one is dia and the
other paramagnetic. 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 polegaps 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 Odi/</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 torsionless wire).
( «21 )
For the coiislant of the spring' we ibiiiid 1 lcS4,5 c.g.s.
The platiiiuiuiridium 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.scillations. 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 oildamper 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 polepieces 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 polepieces 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 /voo.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
. 106 = 241.1 lUG
106 = 2G8.3. 10 6.
1.230
Tlic corresjjunding resnlls obtained by tiie luetliod of the magnetic
rise were
240.(3 . l(J6 and 2(iy.3 . 106.
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
oxjgen) 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 nonuniformity 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 polepieces
with flat end surfaces 90 mm. in diameter and at a distance of
21 mm. apart,
TABLE Va.
Conical polepieces 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
11280
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 polepieces 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
12.15
1.435
59.8
340.3
■1 1280
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. 106S
The dilferenco is IS"/,,; but in this connection we must remember
that the correction for nonuniformity 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 polegaps .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 nonuniformity of the field are extremely large {287o
of the chief couple).
( 825 )
TABLE Via.
Conical polepieces 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 polepieces 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 fiozen 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 stuffingbox 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 biass 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 fiom 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 tiied 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 nonuniformity from observations
a bath of liquid nitrogen.
Conical polepieces 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. — 10c — 375.2 . 10c .
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. 10G. The
From llie nieai) 614.0 follows /<„/. u'.sk
products into V T are
 252. "8 375.2 . 106 \/m7i — ] (390 . iQc
— 258.°9 435.6 . 106 1/14:2 = 1641 . 10c.
Hence we can represent the two observations pretty well by
1690
XW. = p;^.106,
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 . 106.
33700
Curie found / ^ —  — .JO'' between 20 C. and 450° C. whence it
( 832 )
would ibllow that for 7'= 9()°.l K. /=:374.10g 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 . IQc.
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 diatomic iron changes into triatoniic.
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 nonmagnetic 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 ferromagnetism with
a very lowlying Curie point according to Weiss's theoiy 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, theiefore, 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 polepieces (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 nonuniformity 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 magnetooptic KERREfect in ferromagnetic com
jjound.f and alloys". By Stanislaw Loria. (Communication
from the BosschaLaboratory).
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 wavelength.
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 dispersioncurve 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 wavelength 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
infrared up to about 3 ft. According to this author the complete
rotatory dispersioncurves thus obtained shew a marked resemblance
to a typical dispersioncurve in the region of an exceedingly broad
band of resonanceabsorption. 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 KERRElfect were the four abovementioned bodies. Several attempts
to study with reflected and transmitted light the magnetooptic pheno
mena connected with the KKRREtfect 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
KERREffect 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
(ferrosoferric 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 nickeliron ), together with the wellknown ternary Heusler
alloy, and Wedekimd's ') binary manganeseantimony alloy present
considerable interest. So far as I am aware, the magnetooptic 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 magnetooptic properties of the above mentioned
bodies and also those of the wellknown 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 directvision 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
iotations 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 halfshade 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 glassplate 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 silvermirror, 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, LeldenGomm. 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&schaLaboratory, and is the same specimen,
possessing a polished octahedral surface, which was formerly examined
by Ttv Bois '). A very fineformed 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 CoinmentryFourchambault). 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 wavelength in mi, L the direct scalereading 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 wavelength in a field of 10,2 kgs. is shown numerically
in Table 1 and graphically in Fig. 2. The dispersioncurve 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 wavelength the rotation gradually decreases
and ill the neighbourhood of 587 fiji goes through zero, becoming
negative for longei Avavelengths. 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 abovementioned change of sign is analogous to that found bj
Ingeksoll in the infrared 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 4t 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 KEKKEffect 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\'eIengt]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 iapidly with decreasing wa\elength. 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 zeropoint in the visible
part of the violet at 464f/;<. For smaller wavelengths 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 dispersioncurve 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 dispersioncurves.
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
discreiancy 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 magnetooptic 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 KERREffect 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. Nickeliron with 36° /„ nickel, so called 'Invar", known to
possess a very small coefficient of e.xpansion, is strongly magnetic and
distinctly shows the KERREffect. The rotation is exclusively negative
in the region of the spectrum investigated, and there is only a slight
variation with wavelength. (Table 5, F'ig. 6). The dispersioncurve
lies considerably below the zeroline; with increasijig wavelength
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 magnetooptic behaviour of
the nearly nonmagnetic nickeliron 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'eiciil poilioiis of two
vvellpoiished inirrors were caret'iih' examined in varions parls of the
spectrum but proved to be magnetooptiealij inelfective. It is of
course possible that tlie KKRREffect 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 wavelength but no rotation could be observed.
Ingersoll also failetl to notice any measurable elfect either in the
visible spectrum or in the infrared.
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 sunspot spectra. 581.
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 abducensnucleus
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 PTX 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 EasternBorneo. 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
canesugar. 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 solidliquid 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 magnetooptic KERReli'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 abducensnucleus 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 deepsea 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 KastIndian 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 watersecretion 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. oi7.
B ROUWER (l. E. J.). Continuous oneone 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: ♦'ThePTXspacial representation of the system etheranthraquinone". 23 1 .
— A. Smits and J. P. Wuite : "On the system waternatrium 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 alkylcarbonates". 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 soUdli(uidgas 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 solidliquidgas 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 CeutralBorneo. 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 abducensnucleus 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 photoand 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.
ETHERANTHa.\QUiNONE (The PTXspacial representation of the system). 231.
PEN FOEMATiONs (On) in the EastIndian Archipelago. 74.
FKNs (The) of the Indian Archipelago. 70.
FERROMAGNETIC Compounds and alloys (The magnetooptic KF.RReflect in). 835.
FLORA of Java (Contribution to the knowledge of the). V. 108. VI. 116.
FLCFGiEA 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 sodiumalkyl 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 deepsea deposits of CentralBorneo". 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 odouraffinity, 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 waternatrium 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 uEFFECT (The magnetooptic) 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 EastIndian 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 oneone
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 solidliquidgas 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 ZEEMvxeti'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 ZEEMANefl'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 KERRellect 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 oneone 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 ■iurtiKe? 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.
MELTixGpoiKT 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
canesugar." 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 CentralBorneo. 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.
NATRIUMSULPHATE (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.
oDOUKAFFiNiTY (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:isplioiic ;;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 threephr.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 electrochemical 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 sunspot 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 magnetooptic KERReli'ect in ferromagnetic compounds and
alloys." 835.
Physiology. F. J. J. Buytendijk: "On the consumption of oxygen by coldblooded
animals in connection with their size." 4S
— II. ZwAARDEMAKER: "About odouraflinity, 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 serumanaphyhixis"' 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 coldblooded 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 solidli(uidgas 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.
SERUMAN.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 Serumanajihylaxis. ■!"' Commu
nication. 78 1.
SMITH (J. J.). On Distylium stellare 0. K. and Aporosa campanuhita J.J. S. 34l.
SMiTS (a). On retrogressive meltingpoint lines. 227.
— The PTXspacial representation of the system etherauthra(uii)one"'. 2.'31.
— The photoand 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 threephase surface. 182.
— and J. P. WuiTE. On the system waternatrium sulphate. 241.
SODIUM ALKYL CARBONATES (On.). 303.
SOLID STATE (On the). IT. 26. III. 120. IV. 138.
SPACIAL UEPiiESENTATlON (The PTX) of the system etheranthraquini)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.
SUNSPOT 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 twosheet threephase). 182.
SURFACES (Continuous oneone transformations of) in themselves. 2"'^ Communication. 286.
— (On continuous vector distributions on). 2"'! Communication. 716.
SYSTEM etherauthraquinone (The PTXspacial representation of the). 231.
— waternatrium 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 threephase surfiice. 182.
T E s c U (p.). On Jurassic fossils ns rounded pebbles in Nortii Brabant and Liraburg. 422.
•iiiEouY (The) of the ZEEMANeftect 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 oiieone) 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 canesugar. 635.
— (Emulsion laevulan, the product of the action of) on canesugar. 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 threephase surface". 1S2.
— presents a paper of Prof. A. Smits: 'On retrogressive meltingpoints lines." 227.
— presents a paper of Prof. A. Smits: 'The PTXspacial representation of tlie
system etherauthraquinone." 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. lleearches 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 EastIndian 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 sunspot 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 threephase 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 wnteriiatriuin 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 snnspot 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.
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