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NN Lbbodev. Sur forzulo ,
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31,704.JamiAers aur 1.7r.LITO'Citit p4:4rtiolle,1
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Ncoulq,ue Deo Fluidea
AN Keluogoroff, kicaber of the i tLi lox of
ResintacoeLz C.(..An of TUrN::t Fadw tbrough Snnuth
Tutea , ,
AzitroneEiie
OJbCh1At, Tiember On Law of Planetary
Distances
Physique
IG Hj.Li 3c t t,c;?, de2
acouotiquez dew:7, ife .Tfq17'. iwZ,:viiciusoau et
diceol ethyliqu(o(1. , , , , ?
liergui10? The ved cal T7f:.dper tien
of Antimeuycaoo:Am , ?. , 0 0
if.
Physique Appliqaea
Tchkba4v, 1:mp2bre dt: 1'Aoada6de.
Tranoslsain de 1e17.,? chalew PCY 741 tae ,r,71aJrictue at
une sphert daau 'courant de.
Chimie Physige
VI Blinov. On thc, Lamine Aieh 7:4
SZ liozirsky, CerrezpoudiTAE (Y.Z
AB shokhtEr and 3e1;1Aceova. AiinrrV41;Mr107iL(.;
Study of the Ageir,rx of 1i
CLASSFF f CAPON ,7;.7tit;',11;rc?ir
MAVY X FISPE;
;
. .
',.
; t
AIR
13(
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W;i:3TEICTED '
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Conteno
???14+1?0,....
? Tomahov. Cathodic. rroeez, in totallio Corrosion
Geologic
? Brujewicz, Changoe in tilk Lulo :efiintation in
the Caspian Sea within ilistoricaa Time .
VP Kaziarinov, An Attempt it a (,itlw C1ul3ification
of Fire Clays anr.. ;efraertor7 Clu?
Karabik, Typeu of Nickc). Oopovit of t:Ic Pezhev
Kegion (Uiedle Uraia),,
Hydrof;eolocie
? Keuznetzoi.,, 'J.c! tharEe;!enl, caRpolLtion des
e).ux Eovtevrainn,, Cu kcrel rbol,dore lore
de leur me:ange, 0 0 9 a 0
Genetique
EJ redo.rova, Oytolou of Folp1oid :Sytrid3 traria
grndifIor I F.elatior tnd their Feyti:L4.
Botanique
IV (irus1Nits17. helice of the TiTtzLary Flora of the
Uosue. Region.
Phyaiologie Veuetale
5J Zafnen. On the Theory of GZAtiniAgEi.iic Itish in
Provitamin A... ? ,,cp,,foo.fp C. 0 0
DM Novocruday, On th, Lolstre Gontmtt /oisture
, holdin7 Catacity and flydrophily of Mry Lltter of the
Loaf Series in aoat . ?
Uorphologie Facerimentalo
Vinnikov. Transformation ond I'vollfevation of?
Flemente of the Eye T.,na in TiF$un Culturoa.
YA ?oronzowa and IL 1.4.osn9r. ;,,,1121r:tion Povier of the
Caudal Bud in Rana tsmoraril Embryos,
Zoolocie
YI keashikov. On Geolphir0.. Vprit1;11.1ityn Oorezonne
muksun (Pallas). .o ?. ,
UN Ta1i. Ancestors cd? th Cottod in 'Alpo
Zipiki Lnkes? (Vitimriver uatc;tEm, Zlasin of the Lena)
FA Ohzmov and VR Dubinin, A Ni Er6.mic from the
.,;ounttins of Central Asia, Agama pawlowskii sp nov
(hept11ia2.3aurie) A A OSs s, ? et 9 9. 9. a 0 9
25X1A
00712
n:cr,C1.2.
07
729
741
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COMPTES RENDUS
(DOKLADY)
DE EACADEMIE DES SCIENCES DE tURSS
NOUVELLE SiRIE
1946
VOLUME LII
N2 8
EDITION DE VACADEMIE DES SCIENCES DE VURS S
IvIOSCOU
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25X1A
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AVIS AUX AUTEURS
Les .Crimptes Rendus ,DnIziacly) (le l'Academie des Sciences de l'URSSII pa
raissent tons les dix lours. Neuf numerns crimposent un volume pourvu d'une table
des matieres. If y a ouatre volumes par armee.
Les .Comptes Rendus de l'Acadimie des Sciences de PURSS? en de courtes
communications precisent l'essentiel des travaux en preparation. Ces articles, ne de
passdrit pas en regle generale quatre pages. sont donnes en anglals on en francais.
lis minassent les sciences mathimatiqueA physiques. naturelles et applIquees, mettant
,Itir ies resultats de recherches scientifiques en cours ou qui viennent d'?e
aihcves et representant ainsi les plus re.7entes donnees de l'Investigation scientifique.
..'airteur a droit A 100 exemplaires de son article.
idl?,c7 les manuscritc A la Redaction des *Comptes Rendus*. Moscou, \Vol
iionl.m. 14
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20 Jain
COMPTES RENDUS
(DOKLADY)
DE L'ACADiMIE DES SCIENCES DE L'URSS.
COMITE REDACTION .
D. Benankin, de FAcademie, V. Chopin, de FAcaddmie, A. Prumkin, de FAca
ddmie, A. Kolmogoroff, de PAcademie (vice redacteur), L. Orbeli, de FAca
ddmie ,(vicp rddacteur), A. Richter, de FAcademie, S. Soboleft, de FAcaddmie,"
S. Vavilov, de l'Acaddmie (redacteur en chef)
NOUVELLE SERIE
14me armee
Paraissant tons les dix jours
11) 4 (;
VOLUME LH, Al 8
TABLE DES .MATIE ES
?
Pages
MATIOEMAT1QUES
M. Krein. Concerning the Resolvents of an Hermitian Operator with
? the Deficiencyindex (rn m) 651
N. N. Lebedev. Sur une formule d'inversion 655
G. Pfeiffer, de PAcademie des Sciences de PUkrainu. Sur les equations,
systemes d' equations semiJacobiens, semiJacobiens genkalises
aux derivees partielles. de premier ordre a plusieurs fonctions
inconnues 659
31 ECANIQUE DES FLUES
A. N. Kolmogoroff, Member of the Academy. On the Law of Resistance
in the Case of Turbulent Flow through Smooth Tubes
663
,:ISTRONOMIE
0. J. Schmidt, Member of the Academy. On the Law of Planetary
Distances 667
PHYSIQUE
I. G. lifikhallov et S. B. Gourevitelt. Absorption des ondes ultra
acoustiques dans les m?nges alcool methyliqueeau et alcool
ethyliqueeau
N. D. Morgais. The Optical and Photoelectrical Properties of An limony
caesium Cathodes
673
675
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PHY.Y 1QU E APPLIQUEE
Z. F. Tchnkbanor, menibre correspondant de PAcademie. Transmission
do la chalcur par un tube cylindrique et one sphere dans to cou
rant de gaz G79
CHIMIE PHYSIQUE
V. I. Blinov. On the Burning of Ash Coal Il 683
S. Z. Boginsky, Corresponding Member of the Academy, A. It. Shekhter
:Ind S. V. Sahharova? An Electron Microscopic Study of the Ageing
of Smoke Deposits 6g7
N. D. Tomusbov. Cathodic Processes in Metallic Corrosion 691
GEOLOGIE
S. W. Drujewiez. Changes in the Mode of Sedimentation in the Caspian
Sea within Historical Time
V. P. Kazarinev. An Attempt at a Genetic Classification of Fire Clays
and Refractory Clay ? in West Siberia 699
M. A. lisrasIk. Types of Nickel Deposits of the Rezhev Region (Middle
Urals) 703
695
ii YLIOGEOLOGIE
A. M. KuuzEtetzov. Sur to changement de composition des CJIIX souter
rallies du Perinien et du Carbonifere lors de leur m?nge 707
?
GENi:/./QUE
N. J. Federova, Cytology of Polyplen' Hybrids Fragaria grand/lora
x h'. e atir and their Fertility
711
BOT..4NIQUE
I. V. Grushvitsky. Itches of the Tertiary Flora of the Ussuri Region . 713
P1IYSI0LOGIE VEGE i.t LE
S. J. Zafren. On the Theory of Obtaining Hay Rich in Provitamin A . 717
D. M. Novegrudsky. On the Moisture Content, Moistureholding Capa
city and Hydrophily of Dry Matter of the Leaf Series in Wheal 721
MORPHOLOGIE EX PERI ME _V TA LE
J. A. VinnIkey. Transformation and Proliferation of Elements of the
r:ye Lens in Tissue Cultures 725
M. A. Woronzowa and L. D. Liosner. Regulation Power of the Caudal
Bud in Rana lenapyraria Embryt', 729
?
ZOOLOGIE
M. I. MenshIkov. On Geographical Variability in C.regonas
(Pallas) 7,33
D. N. Talley. Ancestors of no )3ik,1 C,ltuidei in ZipoZipikan Lakes
(Vitimriver system, Basin of the Lena). ......... . . . 737
S. A. Chernoy and V. B. Dubinin. A New Endemic from the Mountains
of Central Asia, Agarrol i?aw),,,vskti sp. nov. (Re pit/ ia, Sauna) 741
Traductions redig6es par D. lt akhman ov et T. Rogali na
tdition de l'Aidemie des sciences de VURSIS
Moscou 194G
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Comptes Rendus (Doklady) de l'Academie tics Sciences de l'URSS
1946. Volume LIT, S
MATHEMATICS
CONCERNING THE RESOLVENTS OF AN HERMITIAN OPERA ?0
WITH THE DEFICIENCYINDEX (7n, in)
By M. KREIN
(Communicated by A. N. kolmogoroff, Member of the Academy, H. H. 1946)
In one of my preceding papers (1.) I have indicated a metLod for
obtainina all generalized resolvents R, of an Hermitian operator A defin
ed in the Hilbert space * with a domain of definition 1) (A) dense in
* and with the deficiencyindex (1,1) *.
In the present note I generalize this result and find the general form
of the resolvent Rz of an Hermitian operator A in the case where its
deficiencyindex is (m, m), m being an arbitrary natural number.
If the operator A is positive (i. e. (Af, f) 0 for f E (A)), then for
it there will be resolvents 11,, the whole spectrum of which is situated
in the interval (0, co); I determine also the general form of all these
resolvents.
1. if the operator A has the deficiencyindex (m, m), then for every
nonreal z the equation A*cp ? zcp = 0 (A* being the operator maximally
adjoined to A) will have exactly m (and not more) linearly independent
solutions c,c), (z), (p, (z),..., cp. (z) which we shall construct in a special
manner as vectorfunctions of z.
Let A? be a certain selfadjoined extension of the operator A, and
= (A? ?zlr (Im z 0), the corresponding resolvent. Then the linearly
independent solutions < p (z) (f=' 1,2, ... , in) of the equation A* cp ? = 0,
where z is an arbitrary nonreal point, or even an arbitrary regular
point of the resolvent R, may be constructed in such a way that for
any two regular points z and 7,
(z) (C) (z ? (j=1, 2, ? ? ? , in) (1)
To this end for some 7,o (Im Co 4, 0) we choose an arbitrary system
of linearly independent solutions p .....p of the equation Ap0p=0
and then putting in (1) = cpi (C0) = cpui CI =1, 2,..., m) determine thence
the cpi (z) (j =1, 2, ... , m) for every regular point z of the operator R.
Consider the matrixfunction of the inth order
Q (z) = qj. (z) = ((z zo) (z) + y0pi cz*0), Pk (zo ))
where z? = x0+ iYo is an arbitrarily chosen regular point of the resolvent R.
By means of (I.) it may be easily shown that the marixfunctions
(z) corresponding to different choices of the point zo can differ from
one another only by an Hermitian, matrix not depending on z.
Another solution of this problem was given by M. Neumark (2).
651
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Denote by 9Z?, the class of all holonnurphic in the upper halfplane
m > 0 matrixfunctions F (z)==li (z) hr possessing the property that for.
mlY complex Za, ? ?
trt
111 I (.!?41;(*). fik (z) ikEi) 0 (Im: >0)
. And
j,
It is easily seen that Q(z)ET?, and, moreover,: that the Hermitian
form un (!i?Q (z) E) corresponding to it, is strictly positive for Im z > 0.
Using the wellknown integral representation of fwictions 1(z) bolo
morphie in and representing thc upper halfplane on a part, of it,
(cf., for instance (s), p. 52), we may obtain a general formula for an
arbitrary matrix F (z) E 9.1?,. Denote by 9tn, the class wn, complemented
by infinite matrixfunctions F (z) of the form
F (z)=.5" CP(19s (2)
0
I,,
N\ here C,, (z) is a, certain finite matrixfunction from 9'4; I,,, the unit
matrix of order y; S. a honsingular numerical matrix of order in;
and S', the matrix, Herniae adjoined to S.
If F (z)E9Zn, and for at least one z (Im z > 0) the Hermitian form
m (Z*F (z) t) is strictly positive. then the same will hold also for arbit
rary z (lm z > 0); in this case the matrix F(z) is nonsingular and
 F1 (z)E
Thus, if F (z)E9i, the matrixfunction (F (z) Q (z))' (1m z > 0)
always exists. If F (z) is of the form (2), then we shall put
(F (z)? Q())1 =, hint (Ft (z)+ Q (z)r (Im z > 0)
where Ft (z) is obtained from F (z) by replacing in (2) the symbol of
infinity by I. This definition has always a sense and it will be readily
found how this limit is calculated.
Recall. now that by the spectral function of an operator A is under
stood (',4) an oneparametric family E (? oo < X < ,r) of bounded
selfaijoined operators, possessing the property that for any /E (E (X)1, 1)
is a nondecreasing function of (i.)1 is a function of continuous
from the left, E (A) 0 for t ? and E (l)f / for t and,
besides, for any / E Z (A)
(Al, ?L' d(E ().)1, It, Af
dE (L) f
To the spectral function E, (it.) correspoads a certain generalized resol
vent IL (Im z > 0) of the operator A
R .1 S
(IL (All (/E)
 
and t he functioa F Inj is completely determined by R, for hit z >0.
Theorem 1. The aggregate of silL ;generalized resolvents R, of the
operator A is given by the formula *
Ji.( ? (2)) /44(2) Pr, (z)
IA I
(1m z > 0) (a)
s By (?,Itisk, whet., +E to.. we ilenote the operator eorrel,iting to .!very vector tE4)
the vector fj,,Up.
1;52
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where
(z) II = (0(z) F (z))'
and ? F (z) is an arbitrary matrixfunction from the class FR?,.
Obsei?ve that formula (3) yields then and only then the resolvent R,
of a certain self adjoined extension X of the operator A.; when
(z) is a constant Hermitian matrix.
2. Consider now the case where the Hermitian operator A is positive.
in this case, according to our preceding investigations (cf. (5), Theo
rem 2), the operator A possesses two positive selfadjoined extensions Afil)
and A(1), the resolvents of which RV and RSkr) possess the property that
for ,any a>0 and tE fr)
(r..,) f) 0) and I E the
points of the complex plane w of the form w (Rd, I), where R, is an
arbitrary generalized resolvent with a nonnegative spectrum of an ope
rator A, fill up a certain convex domain bounded by two circular arcs
intersecting under the angle Tr?arg z (measured irrside the domain).
.T!worern 2 finds some interesting applications in the generalized pro
blem of moments on a semiaxis (`) (of. the Stieltjes type).
Itvolved
14. H. 1944;.
RE FERENC
M. Krei n. C. R. Acad. Sc!. URSS, MAIL No. 8 (19M. 2 M. Ne u mark,
r,1111. Acad. Sri. URSS, sCr. math., 7, 285 (1944). 3 II. A x n t e p )4 M. I p e g
0 lievoropux nonpocax TeOplat mosieuroa. 938. 4 M. Ne u in a r k, Bull. Acad. Sci.
'MSS, ski'. math., 4, No. 3. 277 (i940). M. K re i n, C. R. Acad. Sc!. 'MSS, XLVIII.
No. 5 (19r4). I M. Krei n, ibid.. NUN!, No. c. (194i).
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Comptes Rendus (Doklady) de l'Academie des Sciences de l'URSS
1946. Volume MI, Ne 8
MATIMMATIQUES
SUR UNE FORMULE DINVERSION
Par N. N. LEBEDIENT
(ProsentO par A. F. To/A, de l'AcadOmie, le 4. II. 1046)
Dans la presente Note nous etablissons une formule analogue a, celle
de l'integrale de Fourier
a, CO
K i,(x) ,c sh7C7 CbC S K 1, () I () 4
11:2
?
9
o CI
ot K,,,(x) est la fonction cylindrique de Macdonald, x > 0, 1(x) est une
fonction arbitraire continue ainsi quo sa derivee et verifiant la condi
tion x2/ (x), xf' (x) E L (0 , co)* .
Si l'on pose
(1)
CO
f (x) K (x)dx = F (r)
le theoreme pout etre ecrit sous la forme d'une formule d'inversion
cc,
xf ?,; K i,(x) shircF (T) dT.
2
qui donne pour chaque x > 0 l'expression inverse de /(x) au moyen
de F(r). Les formules d'inversion du type (2), (3) contenant une integration
suivant l'indice des fonctions cylindriques presentent u.n interet parce
,qu'elles sont liees a une classe de problemes de in physique mathema
tique, etudiee par l'auteur on collaboration avec M. Kontorovitch. Dans
un article precedemment publie (') les auteurs donnent la demonstration
d'une formate d'inversion, qui pout 'etre consideree comme inverse par
rapport au tbeoreme de la presente Note, savoir; us demontrent quo
dans cerlaines conditions imposees a la fonction F(T) l'egalite (3) entralne
regalite (2) **. Les applications diverses sont donnees darts les articles (2,3).
(2)
(3)
*Les conditions imposees a f (a) sont suffisantes, cepend.ant le theorem? pout
subsister quand on considere des fonctions de classe plus generale.
**Dans nos notations cos conditions sont les suivantes:
c
F ) est une fonction pair? de la variable complex? p, ? it, hole
, i
znorphe clans le domaine ? i
(9)
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un systerne d'equat ions
0
inAppelons l'equation (8) s( miJaenbiens. Les tolations
? ? , ? (Pk..nr) ?
1 = I, k, r>1
(Pk? functions arbitraires des arguments,
(z?..., .r?..., x?, r?)
11 0'1, (I)2 ?
(10)
(12)
(13)
CIF CA, ? ? ? I CPI Vi)
parametres presentent l'integrale generale du system( jacobien generalise
Xt p? r Zi,
*
XPki X{Pkt + ? ? ? X r Pk.?r
r)
?  P(?), ? ? ? ilk, Tic.'
IIt:1i. ? ? .z,,.1. t> ?  ? Z?k,. X ? ? ? ,
i) 1)k 74, fk..
ft (El . 2k, 11 ..... 1' " ' 1" XJ)
1.03441.. ? ? nr)
X 1)k+n Pi) (14
ZA..?   ? ,
(15)
dont l'integration est equivalent(' A l'integration do systeme d' equations
litteaires ltornogi,tiws
+ .? . +Z+. . + + X (7`).,./ 0 ( Pi)
T
it
Eliminant ks parametres (14) des equations (15) on obtient Si
h=kr1 (17)
unf? equation
f (27., :y, p.:)=0 (18)
et si
It Ar in, in > I (I 9)
un systeme d'equa(ions
?
p?) 0 (20)
t.= 1, 2, . ,in
A ',pylons l'equation (18) et le systeme (20) semiJaeobiens generalises.
L'application de la metliode speciule &integration (') aux equations
(8), (18), aux systitmes d'equat ions (10), (20), ronsiste en deux regles
suivantes.
ot
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La premiere regle 
En cberchant sur ].'equation (8), sur systeme d'equations (1.0), le
systeme Jaeobien (5), on trouve quo ses coefficients dependent de
k ? 1, h, k ? in pararnares. Regardant ces parametres.comnae constan
tes arbitraires et integrant l'equation lineaire homogene (6), on recevra
l'integrale generale (1.) de 1:equation (8), du systeme d'equations (10).
La douxieme reglo
En eliercbant sur 'equation (18), sur le systeme d'equations (20) ,le
systerne Jaeobien generalise (15), on. trouve quo ses coefficients depen
dent de h, kr ? 1, h kr ? m parametres. Ayant determine cos para
metres comnie .fonctiou,S des variables et des h constantes arbitraires essen
tielles de telle maniere que le systeme d'equations lineaires homogenes
(16) soit complet, on reeevra, en integrant le systeme (16), l'integrale
generale (11) de requation (18), du systeme d'equations (20).
Ditanuscrit reca
le 23. II. 1946.
1,ITTL4IATURE CITEE
Pfeifle r, Bull. Acad. Sci. do l'Ukraine, 1, f. I, 41 (1922); C. R., 176, 62
(1923); 10. 11 cl)e li (I) (I) e p, 36ipn. !yam. Inur. MaT. All "SrPCP?M 2, 5 (1939).
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Comptes Rendus (Doklady) de rAcademie des Sciences de l'ITRSS
1946. Volume LH, N S
FLUID DYNAMICS
ON THE LAW OF RESISTANCE IN THE CASE OF TURBULENT FLOW
THROUGH SMOOTH TUBES
By A. N. KOIMOGOROFF, Member of the Academy
In place of the wellknown type of formula for the coefficient of
resistance
lg(ReVr.)+B
(1)
which has come into general use from Karman's classical works, Konakov,
in a note recently published in this periodical ('), suggests formulae of
the type
i/?e=AlgReFB
(2)
From his tuatment of the measurements made by Nikuradze Kona
kov has been led to propose the following values for the coefficients in (2)
1ve 1.8 ig Re ? 1.5
This result invites comparison with the formula
= 2.0 lg (Re VC) ? 0.8
(3)
proposed by Nikuradze for the best approximation of his experimental
data within the range of Reynolds numbers he has investigated (from
3070 to 3 230 000).
For 125 measurements carried out by Nikuradze the Mean quadratic
deviation from theoretically computed values is nearly the same whether
formula (3) or (4) is used. In both cases it will be found to equal 0.07.
Nor shall we detect any significant advantage of one of these formulae
over the other if we were to consider large and small Reynolds numbers sepa
rately and subject the experimental data to a thorough analysis. Therefore
it is rather difficult to understand the reason which made Konakov express
himself in favour of his formula, as regards its applicability to a wide range of
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?
Reynolds numbers. Perhaps he compared it not with formula (fr), but with
;mother formula of Nikuradze
1.9.5 I g (Re):
(5)
whit IL however, had been recomm oiled by its author for large ReyinibIs
numbers only.
Still, it must be admitted that formulae of type (2) are easier to use.
Within the accuracy of the available observations they may indeed from pu
rely empirical poinI of view ,:laim equality with formulae of type (1). On this
,,onsideration the contribution of Konakov may be only wellcorned. Un
fortunately, his PaPer pretends to give a theoretical deduction of formula (2).
From his belief in the theoretical soundness of this formula the author arrives
:it the conclusion that it should be used in extrapolating to higher Reynolds
'lumbers, and with this one can hardly agree. It will be shown presently that
his reasoning, if 4:orrectly completed, would only have led Konakov to the
generally adopted formula (1).
Let us start from Konakoy's formulae
where .V and a are
II', /
V ?2.);
(1 2.'0
(II (2,N + c)13'' N in Wide.
3/
constants, The quantity
,.cry close to unity. So we need not treat it as a variable (its vari
Ability was neglected also by konakov). Accordiugly, Ow expressions (13)
:111,1 (19) may he written as follows
, 4*
(6)
(7)
Ii will be obvious that after 111:, is determined from (6) and substitut
,d in (7) we shall obtain a formula of type (1). Instead of doing ,:41
lionakov assumed
U.
4 (8)
:ind after substituting in (7) the value of W6 derived from this relation
he arrived at (2). However, withik the range of Reynolds numbers consi
dered by him 1/7... varies rather widely, its highest. value being twic.as
large as its lowest (see the drawing adjoined to his note). Anil according
to formula (6) or (13) the variation of the ratio 114fl. is just as wide, too.
Therefore the use of (6) instead of (8) is beyond one's comprehension.
It. may be remarked, in addition, that the assumption
IV;
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together with (13), leads to
c const
or, with good approximation, to the equality
= const
(9)
This is what we have in rough tubes, for which the hypothesis of simila
rity of the flow at different Reynolds numbers (?automodelling?)holds through
out the crosssection of the tube. But in the case of smooth tubes its range
of application is confined to relative velocities at different
points of the ?turbulent core? of the flow.
It will be apparent that Konakov's considerations go only to confirm
the theoretical soundness of formula (1) for large Reynolds numbers. Not
until Reynolds numbers shrink to thousands or tens of thousands, can we
expect to meet with deviations from this formula.
Indeed, the experimental data of Nikuradze seem to indicate that the
real plot of 1/17C as a function of Ig (ReVC) is a straight line only for large Re,
and shows a slight downward bent for small Re. It is to be regretted that the
advantage that may be gained in using for large Re formula (5), which is
built on the experimental indication just mentioned, instead of formula (4),
which satisfactorily interpolates the observation throughout the range of
variation of Re, lies too close to the limits of experimental error. If, however,
the downward bent of the curve should be real, it can certainly receive
no theoretical explanation on the basis of the foliaceous constructions made
by Konakov.
Received
8. V. 1946.
REFERENCES
1 P. K. Ko n a ko v, C. R. Acad. Sei. URSS, LI: No. 7 (19!i6).
2 C.
R. Acad. Sci. URSS, 1948, v. LII, NI 8.
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Comptes Rendus (Doklady) de l'Academie des Sciences de l'URS8
1916. Volume LH, Al 8
ASTRONOMY
ON TILE LAW OF PLANETARY DISTANCES
By 0. 3. SCHMIDT, Member of the Academy
1. The distances from planet to planet increase as we move away from the
Sun. Is this variation of distance subject to mathematical law, and if it is,
just what is the underlying physical reason? The question has long attracted
the attention of astronomers. Wellknown is oBode's law>>, which was made
public in 1772. In terms of the distance between the Earth and the Sun
(put at unity) the distance from the Sun to Mercury is approximately 0.4, and
the distances to the other planets, according to Bode's law, are expressed by
the formula 0.4 + where n is the number of the planet (n =0 for Venus,
n=1 for the Earth, and soon). In the table below the figures obtained accord
ing to Boclq are compared with the actual distances:
a
C.)
r.
A71
,..9 =a
.? .4>
i= I F...
i ' 'I A
El) c
.
A
.,
Saturn
i
Uranus
Neptune
.4>
Bode' slaw . . .
Actual distance .
0.4
0.39
0.7
0.72
1
1
1.8
1.52
2.6
...
5.2
5.20
10.0
9.54
19.6
19.19
38.8
30.07
77.2
39.5
In many cases the coincidence is striking, indeed. But there are also consi
derable departures. We find no planet between Mars and Jupiter, though the
law requires that one should be present there. The asteroids fill the gap badly,
for their total mass is far less than that of any individual planet. Unsatisfac
tory also is the figure for Neptune, and if we refer it to Pluto in order to obtain
a better coincidence, we will find it even more difficult to explain why the
little Pluto should be admitted to full membership in the series, when much
the more massive Neptune is excluded from it.
For close on two centuries Bode's law has continued to be a subject of
discussion. Some scientists considered it a law of nature, unaccountedfor
but none the less real. Others (their number appears to be stronger) looked
upon it as a chance coincidence of two sequencies of numbers. Recently
WeizsAcker (1) has made an attempt to deduce Bode's law in a simplified form
by approximately doubling the distance on transition from one planet to
the next. But the premises on which his conclusion is based seem very arti
ficial.
Nor is it all that can be said against Bode's law. Its most essential draw
back is that the planets are arranged in a single row without taking account
of the fact that they actually fall into two sharply different groups. It is
in fact an important feature of the solar system that Jupiter and the planets
farther away from the Sun are of much larger mass, compared to the nearer
2*
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planets?from Mercury to Mars?not to speak of other differences between
these groups. One may hardly expect that, an adequate law of planetary
distances can be based on the neglect of this distinction.
The author's theory of the origin of planets (2) yields a reasonable inter
pretation at once of the existence of two groups of planets, and of the dis
tance relationships within each group separately.
First, we shall consider the qualitative aspect of the problem, then proceed
to its quantitative treatment, and, lastly, compare the results of the theory
with the known facts.
2. On the author's theory the planets have arisen from a swarm of meteo
rites captured once by the Sun, while it was crossing the central plane of the
Galaxy. Afterwards, as a result of collisions, smaller meteorites settled on
those of larger mass, thus contributing to the eventual formation of several
large bodies, the planets.
Let us examine this process in greater detail. The relatively large nuclei
of the future planets, which had segregated at the early stages of the process,
must, on account of symmetry, have been revolving in the central plane of
the swarm along circular orbits. Collision between such a nucleus and a mete
orite occurs when the meteorite which may move in an elliptical as well as
circular orbit happens to arrive at its node in the central plane just mentioned
at the time that the nucleus is also there (2). Adding its mass to that of the
planetary nucleus, the meteorite also imparts to it its angular momentum of
revolution about the Sun. Thus, the angular momentum of a planet is the
sum (if the angular momenta of the meteorites of which it is composed, and
the position of its orbit (its distance from the Sun) is determined by the value
of this total angular momentum. All the time mutual perturbations make the
meteorites slightly change their orbits, and the neighbour meteorites come
to fill the place of those *scooped onto by the planets.
Let us see now what is likely to happen in the natural course of events to
two neighbouring planet nuclei that are in progress of growth. If close to each
other, they will soon exhaust the store of meteorites that stand a chance to
get between them. With no meteorites to be raptured from these quarters,
the nuclei, provided they do not fuse together, will further increase in mass
and momentum at the expense of meteorites from outside of the exhausted
interval. This means that one of the planets will now aggregate meteorites
revolving nearer to the Sun, and accordingly, having smaller angular momenta
in the mean, as compared to thi meteorites that will add to the other planet.
As a result, the angular momentum per unit mass will gradually decrease
in one planet, and increase in the other, and the difference between their
orbital radii will grow correspondingly. This will continue until the planet
is drawn into the region where it will have to compete with its neighbour from
the other side, which will exert upon it an opposite influence. It, appears there
fore that, the planetary distances have been controlled by the mechanisin
the planet growth from meteorites. A methematical treatment of the results
of this control will be given in ? i.
3. Before we proceed to it we shall dwell on the fate of the planets initiat
ed in the neighbourhood of the Sun. In eapturing meteorites such a planet lied
It) compete not only with its neighbour farther away from the Sun, but with
the Sun itself. Certainly, it was itO match for the latter. The major part of lit"
meteorites was bound to fall to the Sun and not to the planet on account 'if
two factors. First, owing to perturbations, part of the meteorites iwly have
assumed orbits with perihelion distant es shorter than the Sun's radius, and
such meteorites were destined to fall to thl? Sun on the next revolution.Second:
ly, the pressure of Sun light made the particles of matter gradually lose
their orbital momentum (4) with the result that numbers of meteorites approa
ched the Sun in a spiral, and eventually fell upon it. The magnitude of ?this
effect (time of approach) depends on the size of the partiele and its no rat
diatance from the Sun.
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It will be obvious that the influence of the two factors was particularly
strong on meteorites revolving in the vicinity of the Sun. The fall of these
meteorites, as will be shown elsewhere, is responsible for the rotation of
the Sun on its axis. So in the region about it the Sun itself came in for by far
the bigger share of the meteorites present and thus prevented their formation
into bodies of respectable size. From the remains of the meteoric mass in the
proximity to the Sun only small planets could arise, and the first planet whose
mass corresponded to the total mass of the meteorites tevolving in its domain
could only be formed at a distance from the Sun, where the influence of the
abovementioned factors was so weakened as not to affect the result materially.
This is the reason why we have today two groups of planets: the socalled
terrestrial planets, not very different from the Earth in size, and the major
(distant) planets of much larger dimensions.
4. We are going now to derive a law of planetary distances from the theory.
To begin with, let us consider the major planets, to wich we shall assign num
bers in order of distance from the Sun, putting n=0 for Jupiter.
The total angular momentum of the system rests invariable, though indi
vidual meteorites may gain or lose momentum by mutual approach. Of course,
small changes are more probable than great. At this juncture the law of dis
tribution of these probabilities is of no import to us, and it will suffice to
assume that increase and decrease in angular momentum by the same amount
are equally probable.
We shall speak of domains of meteorites belonging to particular planets
and of the boundaries between these domains as of definite notions. For an
individual meteorite the chances are in favour of its being brought in the end
on to that particular planet whose angular momentum per unit mass differs
least from that which the meteorite had upon the formation of the meteoric
swarm about the Sun. Together, all the meteorites whose angular momenta in
this sense are nearest to the angular momentum of an nth planet will be
described as the ?domain of the nth planet)). A meteorite which
stands equal chances to fall to the nth or to the (n 1)th planet
is said to be the boundary between the domains of these planets. As a matter
of fact, all meteorites of a domain do not necessarily fall to the planet con
trolling this domain. Some may land on an alien planet. But as their own
planet is also likely to capture some meteorites from foreign domains, there
will be a tendency to equalize the balance, and we may assume, for the sake
of simplification, that every planet will in time receive all the meteorites
revolving in its domain, and no others. Neither shall we take account of the
( small angle that may possibly exist between different momentum vectors,
so that they might be added arithmetically.
No further simplification is required for the mathematical deduction of the
law.
Let an be the angular momentum per unit mass of the nth planet and my,
the total mass of the meteorites in the respective domain. The angular momen
tum of the meteorite revolving at the boundary between the domains n and
n +1 will be denoted by uci.The mass of an individual meteorite will be expres
sed as a diferential dm, and its angular momentum per unit mass will be
denoted by u. Then, in virtue of the law of conservation of angular
momentum, we can write the following expression for the total angular
momentum in the domain n
U=UL
uninn= u dm (1)
By the definition of the boundary, the angular momentum tin' differs
from u? just as much as from un,i, i. e.
un +1 :141. an' ? Un ?
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or.
?u? u?...
fin =
The paper cited above (') contains a deduction of the relation
P S + e
a =  
2 1?e
which connects the semimajor axis of the orbit with its eccentri
city e for every meteorite captured by the Sun. For the meaning of the
quantity p (the limit distance at which capture occurs) the reader is
referred to that paper. Here it will suffice to mention that p may taken
to be constant in the mean throughout the meteoric swarm. In the
cited paper it was shown also that e in this formula can be used with+
as well as with and that all its values from ?1 to +1 are
equall y probabl e. In virtue of the latter circumstance the mass of
meteorites with e ranging from el to e2 is proportional to the magnitude
of this interval. If the total mass of the swarm be denoted by in, we
shall have
Flt
dm de
(4)
because the interval of variation of e from ?1 to +1 equals two. For
any member of the system the angular momentum per unit mass is
known to be
k (1 ? et) (5)
where M is the mass of the Sun, V is the constant of gravitation. Let
us so select the units as to have k I. For an nth planet moving
in a circular orbit at a distance 11? from the Sun the angular momentum
per unit mass is
For the angular momentum a of a meteorite we have, by (3) and (5),
the expression
a (1 ? ei). 12 + e)
(6)
Denoting eccentricity of the boundary meteorite orbit by e'n we have
respectively
len =? 17.1;. (1 + (7)
Making use of the formulae (4) to (7), we can rewrite the equality (1)
as follows
P 711
linnin ? y( de
or, after integration,
11 n ?
1 4 e:02 ? + e, Os
2
On the other hand, by virtue of (4),
i)
From (8) and (9) follows
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(8)
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= 11/' 2 2
1. e.
u1+ 43
2
(10)
Hence using (2) in order to express Uj and u'n through the angular
momenta of the planets, we get
un =
2
un1 + 1414.1
And as un for the planets, therefore
+ Rn+i
2
This equality can be written also in the form
R?=17 (12)
To put it into words:
The difference between the square roots of their distances from the
Sun is a constant for any pair of successive planets.
This theorem involves the law of planetary distances, as derived from
the author's theory. We can, in fact, denote irk, by' a, and the constant
difference between the successive square roots by b, to obtain
lifF7n=a+bn (13)
which means that the square roots of the distances between the successive planets
and the Sun form an arithmetical progression.
This is precisely the author's law of planetary distances. We have derived
it for the distant planets, i. e. for the region where the direct absorption of
meteorites by the Sun, discussed in ?3, is a factor of minor importance. The
following considerations will show, however, that it holds also for the nearer
planets. In the course of time the Sun had absorbed the main mass of smaller
particles from the region of the nearer planets, so that only those of relati
vely larger size remained, and their orbits were less sensitive to the influence
of light pressure. Moreover, once the meteorites with the longest orbits had
fallen to the Sun, the remaining orbits, being more circular, were also less
liable to be affected by perturbations. Therefore the action of the two factors
mentioned in ? 3 was growing weaker as time went on, and eventually the
conditions were created for planets to form from the remains of meteoric
matter in the regions lying nearer to the Sun. On this consideration we may
expect the above theorem and the law of planetary distances, as expressed by
formula (13), to hold for the nearer planets as well, though, of course, with
modified coefficients a and b.
5. Let us compare our conclusions with the actual data (the values
of R are given in astronomical units).
l/Rn+i?iiRn ? ? ? 0.81 1.29 1.10 0.81
Jupiter
Saturn
TJranus
Neptune
Pluto
5.20
9.54
19.19
30.07
39.52
2.28
4k.09
4.38
5.48
6.29
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From the figures of the last row the square root differences appear to be
not strictly constant. Yet they fluctuate within a rather narrow range about ,
a mean value equal to 1.00. We may look opon this coincidence as satisfactory,
for the law only expresses I he average tendency in the action of millions of
meteorite falls, the process which has not even come to an end as yet.
We shall now compare the law j/ ii,, a + bn with actual data. For a we
take I he actual value of the first planet of the series (Jupiter), and
fur b the mean value of the differew es 1/Rn.1 /R' i. e. 1.00.
Table I
Jupiter
Saturn Uranus
\Neptune
1 Pluto
theoret.
2 28
:3.28
4.28
5.28 6.28
1(N actual
2.28
3.09
4.38
5.48 6.29
R thcoret
5.20
10.7t1
18.32
27.88 39.44
R actual . . ?
5.20
9.54
19.19
30.07 39.52
Depa rt are
0
5%
7%
Here, in contrast to BodWs law, Neptune and Pluto comply with the gene
ral rule.
Let. us turn to the nearer planets. For them the actual differences V R,
Vrin are
0.23 0.15 0.23
the mean value being 0.20. The actinil and theoreti, al figures are brought
together in Table 2.
Table 2
Mercury Venus
Earth
Mars
11R theoret. ?
0.62
0:82
1.02
1.22
fr.R actual
0.62
0.85
1.00
1.23
tbcoret
0.39
0.67
1.04
1.49
R act ua 1 _
0.39
0.72
1.00
1.52
Departure
0
i 4%
 20/0
Insiitute of Theoretical Geophysics. Received
Academy of Sciences of the USSR. 29. IV. 1946.
REFERENCES
C. F. We izsacke r. Z. f. Astinpliysik, 22 alt). 2 O. 3. Schmid t.
C. R. Acad. Sci. URSS. XIX, No. 6 (1944). 3 0. J. Sc h in jilt,ibid., XLVI, No. 9
(19451. II. P. Robe r iso n. Mont. Not. Roy. Astron. Soc., 87. Nu. 6 (1937).
?
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Comptes Rendus (Doklady) de l'Academie des Sciences de PlittSS
1946. Volume LH, Ne 8
PHYSIQUE
ABSORPTION DES ONDES ULTRAACOU'STIQUES DAN ? LES 111.PLAN
GES ALCOOL llaTRYLIQUE?EAU ET ALCOOL killYLIQUE?EAU
Par I. G. MIRHAILOV et S. B. GOUREVITCH
(Presente par A. N. Terenin, de l' Academie, le 18. III. 1946)
On salt quo dans certains m?nges bikaires liquides le coefficient d'absorp
tion des ondes ultraacoustiques depend de la concentration et possedeun
maximum bien exprime. Ce phenomene a ete trouve, par exemple, par
Bazhulin et Merson dans le m?n
ge acetone?eau (1). 1, ;0"
Un des auteurs de la presente
Note a trouve que l'absorption des
ondes ultraacoustiques croit pour
certaines concentrations dans un 80
m?nge alcool ethylique?eau (2).
Dans la presente Note nous don 80
nons les resultats des mesures quart
z0 0 50 80 10
Concentra/en de !Woo& methyl/re
dans l'eaa, %
Fig. 1.
0 20 0 60 80 100
C'oncentratton Iiilcool
dans Pew, %
Fig. 2.
titatives du coefficient d'absorption dans les m?nges alcool methylique?
eau et alcool ethylique?eau*.
* Lorsque ces mesures ont ? rminee s, nous avons appris de P. A. I3azhulin,rqu'
apres la publication de no tre communiqu?2), des me sures analogues oat et?ffectuees:par
J. M. Merson, Lombe sur le champ de bataille. Malheureuse molt, ces resulta Ls n'ont pas
6 te publi6s.
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Lea mesures ont ete effectuies par tine rnethode mecanique. L'on obser
vait les deviations d'une ailette d'aluminium, qu'elle subissait sous l'action
plc la pression sonore. Gellesci etaient inesurees au moyen d'un microscope
avec un oculaire a micrometre. Bien quo retie methode soit simple, son uti
lisation doit etre suivic de ccrtaincs precautions a! in d'obtenir de resultats
satisfaisants. Nous communiquons plus has des resultats qui sont encore pre
liminaires; actuellement nous effectuons l'experience destinee a augmenter
I 'exactitude des mesures.
Les result ats des mesures de afro pour les melanges indiques sont donnes dans
to tableau et sur les fig. 1 et 2. us se rapportent a la frequence 12970 kHz
eta la temperature 18'C. Le coefficient d'absorption a est calcule pour l'am
plitude en cncl. La concentration est volumetrique. On voit que les deux
melanges possedent un maximum bien net qui est. fonction de la concentra
tibn. ?
AlcoA methylique?elu
?..1
Alcool ethylique?eau
P. C. volua
m+Strique
400
P. C. volu
metrique
a
a,
t.
o . ?
0.046
27
8
19
0 0.046
27
8
19
39 . .
0.053
31
10.5
20.5
30 . . . 0.065
38
21
17
59
0 063
37
d2.0
25.0 31 0.085
50
24
26
69
0 073
43
13.5
29.5 0.150
88
26
62
74 .
0.076
45
:4
,
81.0 ' EC . . 0.173
102
27.5
74.5
79
0.078
46
15
81.0 71 . . 0.155
91
27
64
84
89
0 075
0 065
44
38
16
17
28 i
21 t't . . . . . . 0.124
. . . 0.105
.73
62
26
24
47
38
98
0 058
34
19
15
Dans le m?nge alcool triethylique?eau les deux composants possedent
les coefficients d'absorption a peu pres egaux et le m?nge a un maximum
d'absorption aux environs de 80 pour cent. Le coefficient d'absorption pour
l'alcool ethylique est deux fois plus grand quo pour l'eau. Le maximum du
melange a lieu pour la concentration 60 pour cent.
Dans le tableau et sur les figures sont egalement donnees les valeurs calcu
lees par In formule de Stokes en tenant compte de la viscosite ordinaire. Si
l'on tient encore compte de In viscosite volumetrique, l'equation de l'absorp
tion prend la forme
a 2irs ( 4 a' _La'
vs =w 3 7 I .+1
(en negligeant la correction de Kirchhoff stir l'absorption due a la conducti
bilite thermique).
La difference des ordonniTs qv' et eh' per/net de calculer la part de l'ab
sorption ity/0 due a la viscosite volumetrique.
On voit sur les fig. 1 et 2 quo le caractere de variation de l'absorption o'/V1
est determine par la variation de eh en fonction de la concentration. 11
on resulte quo in viscosite volumetrique joue un role principal dans l'absorp
lion des ondes ultraacoustiques dans les m?nges indiques.
Institut de physique de Man crit regu
l'Universite de Leningrad. lo 25. II. 1946.
bITTERATURE CITEE
1 P. A. Bazhulin et J. M. Me rso n, C. R. Acad. Sci. URSS, XXIV, No. 7
1939). 2 I. G. Mikhail ov , ibid., XXV, No. 2 (1940).
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Comptes Rendus (Doklady) de 'Academie des Sciences de FURSS
1946. Volume LII, Jsrs 8
PHYSIQUE
THE OPTICAL AND PHOTOELECTRICAL PROPERTIES
OF ANTIMONYCAESIUM CATHODES
By N. D. MODGULIS
(Communicated by S. I. Vavilov, Member of the Academy, 8. I. 1946)
One of the most interesting sets of questions connected with the problem
,of modern effective photoelectric cathodeselectron emitters which are known
to be of a semiconductor nature?is that of the conditions determining the
absorption of light quanta by these emitters, the excitation of photoelectrons,
and the kinetics of their subsequent motion towards the emitter surface. The
present investigation is devoted to these probleins, the wellknown antimony
ceasium, Sb ? Cs, photocathode being selected as the object of study. The expe
rimental results presented below areas yet of a preliminary character.
Given a wedgeshaped cathode with a continuously varying thickness d,
admitting both direct illumination1 i. e. ordinary illumination from the anode
side of the photocell, and reverse illumination, i. e. from the outer side of
the glass bulb covered with the Sb?Cs layer; let us assume that we have here
a purely volume photoeffect, and that the absorption by the emitters of
both light quanta and excited photoelectrons obeys the exponential law
ng naoe (1)
where 11 and a are the absorption coefficients of the quanta and the photoelec
trons, respectively. Accepting this law for photoelectrons means, assuming
that their absorption results from a single act of adhesion to the crystal
lattice and not from the gradual loss of their energy.
Thus, from. assumption (1) it is easy to show that the intensity of
the photoelectronic current with direct illumination I, depends on the
thickness of the photocathode d and the coefficients p. and a in the
following way
i. e. as d decreases,
a maximum, limiting,
On the other hand,
reverse illumination I,
ing dependence
I, = A a) [1 ? e (11+) d].
(2)
the direct photocurrent I, gradually approaches
value which (for d >>1/(p. + a) is equal to
h1m= A (v. 0) (3)
the intensity of the photoelectronic current with
will, under the same conditions, show the follow
12= A P. . re crd et.t.d]
(P 0) L
(4)
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t. e. the reverse photocurrent /2 will have its maximum /2?1 at a certain
definite thickness of the layer equal to
dm= 41'
?0)
These formulae may be utilized for the solution of our problem,
which is the determination of the values of p and a for various wavelengths
). of the light irradiating the phot1cathode; whence we may 'determine the
values of 1/1.1. and 1/a which characterize the thickness of the zone where the
main absorption of the incident radiation occurs (1)?) and of the zone from
It the excited photoelectrons emerge (1/).
ln our study, for instance, the value of p was determined from (1) by measur
ing the absorption curve of light quanta at various thickness d; the value of
a was then determined from (5) according to the position of the maximum
reverse photoeurrent /2?,.
In order to carry out, those measurements, special photocells were prepa
red. On their walls a layer of antimony in the shape of a long wedge tapering
to a point was deposited by means of evaporation from a sphere made of the
metal concerned. This layer was then completely exposed to caesium vapour
until an antimonycaesium layer was obtained having normal photoelectro
nic sensitivity throughout its surface. Contact with the Sb?Cs photolayer
was obtained by means of two platinized strips placed parallel to the
Sb?Cs wedge on either side. In this manner we obviated the distorting effect
of the longitudinal resistance of the Sb?Cs layer. Since the source of the anti
mony was a small sphere, the subsequent distribution of the thickness of
the antimony layer along our wedge and, accordingly, the distribution
of the thickness of the Sb? Cs layer (making the natural assumption that the
distension of the antimony layer under caesium vapour treatment is uniform)
may be expressed, as can be readily shown, in the following form
d,
(6)
4 ?cep II+ war ris It +(r/a)93/2
The value of the constant d? may be determined for the Sb?Cs layer by
employing some other independent methods. The author, for instance, uti
lized the fact, established for the first time in our laboratory by P. G. Bor
zyak, that the ordinary interference picture may be observed in thin Sb?Cs
layers. In our case, on observing this picture in reflected monochromatic
light, in the direction of the rise in the value of d from its smallest values,
we first see the continuous bright edge of the wedge, passing subsequently into
the ordinary sequence of dark and bright interference bands. Hence, it fol
lows that the index of refraction n of the Sb?Cs layer lies within limits of
n.> n> 1 where no is the index of refraction of glass, for reflection from the
anterior and posterior surfaces of the Sb?Cs layer occurs here with similar
phase; since in our case n0 1.52, we assume n 1.4. Hence the thickness of
the Sb?Cs layer dk in the position of the. kth dark interference band is *
= (2k ?1) 1;171 where k 1, 2, 3,... (7)
Our chief measurements included the determination of the distribution
of the optical transparency D, the direct I, and the reverse I, photocurrents
along our wedgeshaped SbCs cathode with various wavelengths of the
incident monochromatic radiation X 630, 560, 490 and 420 nip obtained
by Means of a Leiss monochromator. The relative intensity of the investigat
?
(5)
* The questions of the precise determination of the value of n and of the effect of
the absorption by the Sb?Cs layer on the position of the interference bands, are deferred
for the present.
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ed radiation in measurements of the transparency of Sb?Cs layer was deter
mined by means of a blockinglayer sulphursilver photocell for X =630
and 560 m, and by means of a vacuum Sb?Cs photocell for ?=490
and 420 mp..
The preliminary results of the measurements of the values D, I, and
obtained in this manner, for our lamp No. 3, for example, are presented in
? JD
ISO ay'
hee 242 88 41.8 50
Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Figs. 1, 2, 3, and 4 for four wavelengths X. The results of computations of
formulae (11) and (5) using these measurements are presented in the table.
1, mil
ll, c1111
lip., cm
dm, .A.
a, cmi.
Va, cm
630
8.4 ?
105
1.2 ?
105
1100
1.0 ?
105
1.0 ?
105
560
1.5 ?
105
6.7 ?
105
740
1.5 ?
105
6.7 ?
105
490
2.7 ?
105
3.7 ?
105
350
3.1 ?
105
3.2 ?
108
420
4.1 ?
105
2.4 ?
105
250
4.2 ?
105
2.4 ?
105
A study of these data leads to the following conclusions.
I. The relative slope of the curves of transparency and, consequently, the
value of the coefficient of absorption v., as well as the mean opaque zone of
the cathode, increase with a decrease in the wavelength X.
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2. In Fig. 1 with X=630 ma. we observe at x =12.5 the appearance of some
local maximum, which vanishes for other wavelengths. This shows that this
phenomenon can hardly be attributed to any optical or structural properties
of our Sb?Cs film at the given place. In the same figure, the line shows the
position of the first dark interference band.
3. The direct photocurrent Ii, contrary to expectation, does not always
yield a monotonous change with a variation in the value of d; with increase
in X and, especially, at X =630 nip. an anomalous character for the changes
in the value of I, is observed, the reasons for which are not yet clear. It is
interesting to note that, fol undiscovered reason, this anomaly of the direct
photocurrent seems to parallel the anomaly of the optical transparency
mentioned above.
4. In complete agreement, with expectation, the reverse photocurrent
behaves perfectly normally at all wavelengths X and passes through a maxi
mum at the values of dm given in the table. These values of dm diminish with a
decrease in the wavelength of radiation.
5. With a decrease in the value of d the values of both photocurrents I,
and I, approach?as follows from (2) and (4)?the same values, which is evi
dence of the fact that here dl/1. and d 1000. Le detachement des tourbillons
de la surface cylindrique a lieu pour He 3050. Il est tout naturel que le
caractere du mouvement influe sur la transmission de la chaleur du cylindre
au courant.
Il est evident quo la transmission de la chaleur dans la partie frontale du
cylindre doit satisfaire dans les conditions normales (jusqu'a Re 100 000)
requation correspondant a une couche frontiere laminaire, comme cello de
Kroujiline (4), par exemple. Al in de l'appliquer pour les buts pratiques ii
I aut savoir la variation de 8 en fonction de x. 1
Pour le cylindre et un courant isothermique, Pohlhausen a obtenu les
donnees (7) pour 8 le long du contour du cylindre d= 9.75 cm, qui a ete etudie
experimentalement par Himentz pour wo =19 cm/sec et Re 185O0. En uti
lisant cos resultats nous pouvons obtenir les courbes theoriques de l'intensite
de l'echange de chaleur de la partie frontale du cylindre. Les courbes calculees
de cette f acon sont presentees sur la fig. 1 pour les valeurs differentes de Re;
de memo on y a donne le changement de Bx suivant le contour du cylindre dans
1' equation
Nu =Bx ? Re?.5 (1)
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On volt que ia coincidence des courbes theoriques avec rexperience sur
la fig. 1 est plus clue satisfaisante. Apres Piritegration on obtient pour la
partie frontale du cylindre requation
? Nu=0.8 ? ile? 5
(2)
Lorsque rangle d'attaque du courant change, Pequation (2) changera
de meme car la forme du contour longe par le courant sera modifiee. On
peut approximativement* tenir compte de ce changement par le changement
Fig. 1. La variation de I 'inte risiti de la transmission de la
chateur sui van t in contour du cyl I mire . Les courbesen I ignes
continues sont trades traprCs I 'equation theorique de
Kroujiline hes oflflt5 e xpkri me ntales: ? K roujillne et
Schwab. = 67 200; 2  uines auteurs, Re = 52 300;
3.? memes aute ors. Re = 32 600; 4 ? me me8 auteurs.
He=21 000: Re = 1 0 000: 6?Joukorsky, KirCev
et Cala witeev, Re= 4 000; 7?mme s a ute urs,* Re= 2000.
respectif de 14 grandeur caracteristique. Sur la fig. 2 nous drinnons la courbe
theorique correspondante, ainsi que les resultats respectifs de Pexperience.
Mais quel est alors In caractere de la transmission de la chaleur de la
opoupe,? du cylindre dims les conditions de formation des tourbillons? Le
domaine du detachernent progressif du courant de la surface du cylindre
jusqu 'an moment de formation ditin ,oiiple de tourbillons depend de la di
nut ion de Nu par rapport a sa valour donnee par (2), par suite de rexchision
(rune partie de la surface du cylindre de In zone de transmission active de la
hale r.
A partir de Re 3050 jusqu 'a Re., 500010000 s'etablit tine relatiton
plus an moms unnst ant e entre la panic, frontale pour cent de In surface
cylindrique totale) et la opoupe,) (......r15 pour cent de la surface). L'equa
t ion (2) donne la definition de Pintensite dii proressus pour la partie frontale.
Le mouvement do courant pre% de In spoupe)) avec des tourbillons
mitres)) a lieu avec la vitesse constante suivant le contour du cylindre. Pour
vela, en vas oil le couple de iourbillons enveloppe toute la surface (55 pour
cent), On petit, probahlement avec line exactitude suffisante, utiliser pour
ceite partie du cylindre Pequation do la transmission de la chaleur d 'une
plaque on regime larninaire. Or, 1 'etude des photographies de mouvement du
* Pour Ia resolution l'xacte du probR!me ii Nut tenir compte dii ehangement du
arartkr14 du courant, ear griice a ce,1 le .nouveau profit. n'est plus eylindrique. Cela a unc
importanvii surtout pour les petits angles d'attaque, lorsquelc courant rencontrant trans
ver.sale inept un scut eyfirulre passe en courant lougeant une oplaque,Y.
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courant mOntre, que l'on a en outro du_ couple principal de tourbillons, des
tourbillons supplementaires, situ& entre ceuxci et le point de leur de
tachement. Ce phenomene no perm.et. pas de determiner la valeur theorique
exacte du coefficient pour Re dans Pequation generale de Pechange de la
chaleur du cylindre avec un courant gazeux. Mais la forme de I 'equation cor
respond a (2)
Nu. = 0.8 ? 0.451105+ b ? 0.55 110.5= a lle?.5
(3)
La valeur de b et aussi de a peut etre determinee, si Pon sait celle de
Nu pour une seule valeur de Re. La courbe de la fig. 3 nous donne pour Re=
DOtle la transmission de la chaleur du cylindre pour Re de
Nap
Nam.
to
0.78
050
025
80 80 70 60 50 0 80 20 70
Fig. 2. La variation de I 'lntensile de la transmission de la
chaleur par un cylindre en function de I 'angle d'attaque:.
I?les exp6riences de Lok.chine et Ornatsky; 2?celles de
Sineinikov et Tchachikhine; 3?celles de Forne m; 4la
courbe the? rique
50 ?40000 (fig. 3) montre quo la coincidence obtenue avec l'experience est
suffisamment bonne.
Pour le domaine de Re10 000100000, presentant le plus grand inte
rot au point de vue pratique, lorsque le courant se meut suivant la ?poupe?
du cylindre avec des tourbillons developpes, l'equation de la transmission
de la chaleur pent etre obtenue de la f non analogue a. cello de (3). Dans cc
but il suffit de determiner Pequation relative a la ?poupe? du cylindre pour
les conditions definies. Tant comme dans le c,as des tourbillons ?laminaires?,
la transmission de la chaleur de la ?poupe? en cas des tourbillons developpes
sera decrite par Pequation relative 'au courant gazeux longeant une plaque
avec une couche frontiere turbulente.
Comme nous avons etabli (8), cette equation* alaforme suivante:
Nu 0 .022114" (4)
Par consequent, Pequation de la transmission de la chaleur d'un cylindre
situ.e dans un courant transversal pour le domaine Re =10 000100 MO
se presentera sous in forme
Nu = 0.8 ? 0.45 Re?.54 Re 0.82
(5)
* Pour une plaque on obtient une approximation plus grande (apres integration)
Nu 026 ftex??82.
3 C. 11. Acpct. Sc!. utAsS, 1946, v. LII, Ari S.
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.determination theurique dee, de menu' quo do h dans Pequation (3),
presente un problem(' independant et suffisaminent taimplexe par lui memo.
Dans notre eas c est Millie par la valour eonnue de Nu (pour Re donne).
Alors l'equation (5) s 'eerit
Nu = 0.3; Re" 0.070 11(0.82
(I3)
Cate equation est. precisement :retie de la transmission de la rhaleur a 'no
4. yli mire dans le doinaine Re ?10 000100 000.
Dans le domaine dit 4slibcritipie*, oil a lieu la turbulence de la couche
frontiere de la partie frontale du tylindre, Pequation no rontiendra qu'un
Zis
???.,
I I
Depres reifeetyen de Pea ezn Nu'll5a744549/letiss
1.L 15
13 (0
?
inpres 1'4i/ellen ore l'azzL'ea
= 0.15Re5 0007Reov
?0 43"
45 0
20 25
lia=0,36Red".1141feas
JO
.75
45 1.0 , .L5
fle
transmission de la chaleur par one parcellesphe
cigoi e et cube: 1?les experiences de Liakhovsky avec des boules fixes *racier
d=2.4314.8'. mm; 2?cellos do Liakhovsky aver des cubes fixes Wader d=6.15 mm;
3celles de Loytiansky et Schwab avec ono boule d=70 et r0 mm; I miles de Vyrou
boffavecuneboule.l.a transmission de la chaleur par un cyIin
dre dans an courant gazeux transversal: 5?les experiences ole
!taped avec des tubes normaux; 6?cellos WEigenson. meme cas.
soul membre avec Re a la puissiince...0.82. L'equation de la transmission
de la rhaleur par uno sphere a nue forme pareille:
Nu = 0.50 + 0.09 Re"! (7)
Sur la fig. 3 est donnee la comparaison des resultats quo nous avons otite
nus au moyen des equations deduites, avec ceux de l'experience.
Manuscrit reca
le 271. III. 1941.
1,1T1IRATtlItE (1TEE
' B. M. A itr y 1,CHH :I. C. F o :1 a 'le ii I u. TCH;1001.3.11.1a H COLMOTH.B.ICIHIC Kon
eerrrintioux uonepXuur100 itarpena. 1938. M. B. I; up to e H. M. A. 141 nxeen
.1. C. 11 react) Tennottoleg.a.la. 19'10_ 3 A. C n e 1, o a Fl A. a H 
XXII, )K11'(1). 11. 91 0 (1941):A. II. 0 p aTctsn. II, Con. noraar4p15ocrpoettue,
94o). ' r. H. 1; p y H n it, ;ETU). VI, 3 (193G). 6 A. A. Fyi a it. tt,nureo
elate ?coma,/ yelmonelpc;mom, 19::4. Modern Development in Fluid Dynamics, Oxford,
1938. H. I'. X all a of ot, llorpaton'tomill caott, 1 936. " Z. h u kita ito v,
C. R. Acad. Sci. URSS, X LV111. No. 2 (194).
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Comptes Rendus (Doklady) de l'Academie des Sciences de l'URSS
1946. Volume LH, .N12 8
PHYSICAL CHEMISTRY
ON THE BURNING OF ASH COAL. II
By V. I. BIANOV
(Communicated by N. P. Chlzhevsky, Member of the Academy, II. II. 190)
1. The rate of burning of ash coal depends essentially on temperature.
It is important therefore to establish what temperature a burning coal will
have under given conditions, and how it will change as the process of combus
tion develops. These problems are discussed in the present paper.
The distribution of temperature is assumed to be steady at every parti
cular instant, and it is also supposed that the coal, when burnt out, leaves a
layer of ash, at the boundary of which the combustion takes place and through
which heat is transferred by conduction. The delivery of heat from the coal
is supposed to obey Newton's law.
There will be deduced the relations determining the temperature in the
zone of burning for a wall, a cylinder and a sphere.
2. A wall of ash coal is taken to be bounded by parallel planes. In its
middle lies the origin of coordinates, the xaxis being directed perpendicular
to the planes bounding the wall. Denote the total thickness of the wall and
the thickness of its unburnt part by 2d and 2C, respectively; the absolute
temperature iii the zone of burning, at the external surface of the wall and
in the surrounding gas medium, by T,, T01 and To; the termal conductivity of
the ash layer by X; the heat conductance by a; the specific rate of combustion
by ks; the thermal effect of the reaction by q.
The temperature distribution in the ash layer should satisfy the
differential equation
PT
dx2
under the boundary conditions
dT
? (df)
x=d= (11 ?T 0)
From equation (1) and condition (2) follows
5:0 (7' ??T 0)
where
1 kco
= 1+ + A (1 ? C.)
ad1:d
NU? ?A ' ? d' k = k .e "IR 1., , A =
Here k :is a constant of the rate of reaction between coal and oxygen;
E is activation energy; R is the universal gas constant; ad is a coeffi
cient characterizing the rate of outward diffusion and analogous to heat
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conductance in Newton's law; 1.) is the coefficient of oxygen diffusion
through the ash layer; and c, is the concentration of oxygen in the gas
medium surrounding the wall. The ratio defining k? is taken from an
earlier work (').
It is from equation (3) that we can determine the temperature at
which combustion will take place in a wall of ash coal at a steady state.
3. Let us now consider an ash coal cylinder of radius r?,. The radius
of the unburnt part of the cylinder will be denoted by r,.
In this case the computations lead to a relationship identical with
formula (3) except that DONV
Iteo
20.= (1 Nu in)

I.',
1. Relation (3) is also true [Or a sphere, but here we have
CL0
?k
;
1  Null
(5)
(6)
the quantities Nu, A and E are determined in the same way as in the
case of a cylinder.
5. Equation (3) may as well be solved by the graphical method
formerly used by the author (2)?
On the iaxis of a rectangular system of coordinates we lay off T..,
and on the yaxis, 7,= qk, ,..=1,(T.,?T.) (Fig. 1). The abscissae of
the intersection points of the curves thus
plotted will give the values of the tempe
rature Ilia becomes established under the gi
ven conditions.
According to the conditions under which
the process goes on, the curves z1 and
intersect at one or at three points. In the
former case one definite state of combustion
is possible; in the latter, three states, cor
responding to three values of the temperature,
0. and 0,.
At 7',?=0, the process is slow and stable
(oxidation), at T..:=0 it is stable and proceeds
rapidly (burning). But. at the process
Fig. I. will not, he stable. If in fact the temperature has
risen above 8,, the balance of heat is positive,
more heat being received than lost, and the temperature in the burning zone
will increase until it reaches the point 0, when the coal bursts into flame.
On the other band, if the temperature remains below 0,, the balance of heat
is negative, and the temperature will decrease down to 0,, at, which point the
proc_ess of combustion dies out. Therefore 0, is the minimum temperature to
which under given conditions the coal must be heated up in order to start
burning. In other words, 0, is the inflammation temperature of the coal.
6. As the zone of reaction shifts, :7, and z, do not remain invariable, and
accordingly the values 0? 6, and 0, vary also. The portion of z, corresponding
to lower temperatures will not be affected by the variation of E. In the case
of a burning wall the upper portion of the curve declines continuously as the
burning zone moves, while in the case of a sphere or cylinder it either goes
Up all the time, or down at first to rise later in aucordance with the ratio of
H to A.
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The curve z, will not change in shape, hut the angie y between z,and the
ac,axis will decrease continuously with in the case of a wail, while it Will
either grow all along or decrease at first and increase later in the case of a
sphere or cylinder. 
If z, and z, intersect at three points, 0.3 will decrease with 7 and approach
Under certain conditions 0, will coincide with 0,, and the coal in the
process of oxidation will, burst into flame spontaneously.
67
Mil
With' increasing y.the value of 02 grows towards 0'3, and, if the conditions
are suitable, will coincide with it. If the coal is burning, it will now cease to
do so.
If at the outset only a slow process is possible?oxidation, and afterwards
as..the..zone of reaction shifts, the curves z1 and z, come to intersect at three
'points,, then no burning will take place unless there is a layer Of ash of ad
equate thickness.
If only stable burning is possible at the outset, and afterwards the curves
zi and ; will intersect at three points, then in the presence of an ash layer
of sufficient thickness a decrease in the temperature of the burning obje(.t
will eventually bring the burning process to an end.
Using this graphical method one can also establish how the process is
influenced by the conditions under which it proceeds.
7. More definite data on the variation of 4,, D2 and 0?3 in a coal sphere,
under certain conditions, as the zone of burning shifts, are given in Figs. 2 and
3, where AT? 4113, 42?T0 and i13 To are plotted. against The plotting
has been made on the assumption that 7, and ad are infinitely great, koco is
put at 5.74.'104 (for the air), E=,35 500, and X and D are taken to be equal
to the respective values for the air.
The curve in Fig. 2 is for a sphere of radius r0 =1 Nil, burning in the air
whose temperature is 400 and 480 ?C. The results obtained for a sphere of unit
radius are shown. in Fig. 3. The temperature of the air in this case is put
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at iSO'C, and the concentration of oxygen in the gas medium at 21, 10.5 and
5.25 per cent.
S. It. has been shown above that on selfinflammation of the coal the
curves z, and z, toueli at the point 8, ?A,. On the basis of this statement one
Fig :t.
may easily show, following the method applied by Semenov (3), that the
selfinflammation temperature should satisfy the approximate relation
qic?coEltri = E
R.evived
ti. It. 19firt.
REFEkENCEs
' V. I. B1ino v. C. R. Acad. Sci. tHISS, Lit, Nu. 1, (1946). B. H. B a it an
Tp. Bopon. roc. yori. XI, 41.?tt. ova., 11. 3 (1939. 3 H. H. Ce H o H, 1teuutk peitit 
mum, 1934, op. 116. 121.
6St;
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Comptes Rendus (Doklady) de l'Academie des Sciences de l'URSS
1946. Volume LII, N2 8
PHYSICAL CHEMISTRY
AN ELECTRON MICROSCOPIC STUDY OF THE AGEING
OF SMOKE DEPOSITS
By S. Z. tOGINSKY, Corresponding Member of the Academy,
A. B. SHEKB TER and S. V. SAKHAROVA
The possibility of directly observing the location and statistical distri
bution of the dimensions of submicroscopic particles in the electron micro
scope has made it possible in principle to observe the variety of changes in
structure that are classed under the conventional terms of (ageing)) and ((re
crystallization)). As an object of investigation we selected the smoke deposits
described elsewhere (").
From the viewpoint of the problem stated above these smoke deposits
possess two essential advantages: 1.) they are incompact to such an extent as
to show predominance of isolated particles, that are in no contact with other
particles but at a few separate points; 2) because of the absence of a supporting
film these preparations can be subjected to considerable heating.
The main observations were carried out on smoke deposits of gold and
silver. In performing the experiments we availed ourselves of the possibility
by repeatedly placing the specimen holder into the apparatus to return to
the same field of vision, properly chosen and containing the characteristic
structural formations.
The following procedure was adopted: first, we took photographs of the
fresh deposit, then the holder with the mounted preparation was taken out
of the apparatus and held at a definite temperature for a certain. period. After
that the holder was again placed inside the apparatus and more photographs
were taken..
Not in a single case did we succeed to detect any change in the deposit upon
keeping it exposed at room temperature. In several cases the preparations were
kept in the mounted state for more than a month. When the temperature was
raised, each preparation was found to possess a region of its own, in which the
.deposit began to undergo an appreciable change.
The changes observed in this case were of a varied nature. In the case of a
smoke deposit of gold (Figs. I and 2), it could be seen that during the
early stages of ageing there was a differentiation of particles according to
dimensions: the number of biggest and smallest particles increased,, and the
distribution according to dimensions expanded, the middle part undergoing
a decrease. The initial spherical shape of the particles was retained. This
phenomenon should be taken as a matter of course, being just another
example of big particles devouring the small because of the difference
in surf ace energy. This takes place at temperatures excluding the possibility
:of either evaporation or. fusion.
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Thp nutt,7 possible rio,hanaccoloilt iig for the redistribution of mat(,tho Wilmer lateral diffusion, the inionsily of which must tie
g. d,posil of !.401d.:..;:H!"?
Smoke deposi giold
al ter ti nig for 211? hours.
I
ityposi
or hotirs.
Smoke deposit. 01 sit vol..
x19
sil ver
Ill inutes.
ote?tderable. triattor h tfte iiir ii hown in Fig. I reprosenis
stretc.bed chain, such a change in disoersity is inevitably acconinarticd by
horwitthiiiia strain. Tho piW?til!st he Hifi,. r ;t set t ling of the whole !lain
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which becomes shorter (as was actually tile case, and is shown in fig. or its
breakup into separate links which are drawn together into more compact
aggregates. Fig. 3 shows a ease of more advanced ageing. in this case the
greater part of the material has collected into large grains.
Heated smoke.deposits of silver (Figs. 4 and 5) exhibit a more pronounced
similarity with the crystal state. It was noted that silver ages easier than
gold.
On the contrary, smoke deposits of zinc oxide and magnesium oxide with
stand much higher temperatures without undergoing any change.
In these cases, apparently, Tammann's rule is observed in general, i. e.
for the solid bodies of the same type the temperature, at which recrystalliza
tion begins, increases with the temperature of fusion, and is much higher
for ionic lattices than for the metals.
Section of Catalysis and Topochemistry. Received
Institute of Physical Chemistry. 18. II. 1946.
Academy of Sciences of the USSR.
REFERENCES
1 A. Schechter, S. Roginskyand S. Sakbarova, Bull. Acad. Sci.,
URSS, ser. china., No. 4 (.1946); A. Schechte r, S. Roginsky and S. Sakh a
r o v a, Acta Phys. Chim.. URSS, No. 8 (1946).
(;89
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Comptes Rendus (Doklady) de l'Academie des Sciences de PULS
DIG. Volume LII, NI 8
PHYSICAL CHEMISTRY
CATHODIC PROCESSES IN METALLIC CORROSION
By N. b. TOMASHOV
(Communicated by A. N. Framhin, Member of the Academy, 2. II. 19,16)
An analysis of practical cases of corrosion leads to the conclusion that in.
most cases the cathodic process is the chief limiting (or controlling) factor
iit'eorrosion. Thus, a change in the rate of corrosion is usually associated
with the kinetics of the cathodic process (except in cases of appreciably pas
sivating corrosive systems). The importance of studying the kinetics of
cathodic corrosion processes has already been emphasized by a number of
authors (1').
The usual cathodic processes in practical cases of corrosion are either the
assimilation of an electron as a result of the ionization of the oxygen dissolved
in the electrolyte with the subsequent formation of OH' ions (oxygen depola
rization), or as a result .of the discharge of the hydrogen ion with the
subsequent evolution of the gaseous hydrogen (hydrogen depolarization).
In so far as the experimental investigation of cathodic processes usually
involves the construction and analysis of polarization curves, it is expedient
to give a graphic interpretation of the regularities observed in cathodic
processes.
The figure shows such theoretical polarization curves, plotted by us on
the basis of analytically established relations between the potential of the
cathode and the change in the density of the polarizing current for different
conditions of operations (6).
Curve ABCshowing the overvoltage of oxygen
ioniz atio n depicts the variation of the cathode potential with the
current density, unless there is concentration polarization.
In that case the overvoltage ? of oxygen ionization (4 i. e. the negative
shift of the potential as compared to the equilibrium oxygen potential in the
same solution, will be connected with the density of the polarizing current
(I) by a logarithmic relation analogous to Tafel's formula for the overvoltage
of hydrogen.
(68) *
0.=a+b lg
(1)
where a is a constant depending upon the nature of the cathode and depolariz
er and b is a constant determined by' the mechanism of the depolarization
process. The coefficient a in our case is taken equal to one volt, this being
close to the experimental values of a which we obtained for a copper cathode.
Coefficient b is given its theoretical value, equal?like the overvoltage of
hydrogen?to 0.117 (for 20?C) (6).
Curve_ ADBN showing. concentration polariza
tion, i. e. the curve depicting the change in the cathode potential depending
* With the exception of very small curre I. densities (when the cathode potential
shifts 3050 mV from equilibrium), in which case we shall have a linear dependence (7, .8)?
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only upon concentration polarization in the total absence of the overvoltage
of oxygen ionization, is determined by formula
RT (
= In . 17; ) (.2)
where 4 is the departure of the potential from concentration polarization
(the decrease of the oxygen concentration at the cathode due to the limited
rate of its transportation). lirre the term RT/ tir is analogous to the similar
term in the Nernst formula for calenlating potentials; n = 4 is the number of
electrons assimilated by imp rtio1ei ide of oxygen; I is the density of the
cathode current when it becomes steady; and Id the limiting current, i. e.
the current density under the maximilin rate of oxygen diffusion possible
in this case.
Like the curve showing the overvolt age of oxygen ionization, the concentr
ation polarization curve takes its origin at the point of the oxygen's equili
Theoretical curves of cathodic polarization.
brium potential, but, has an opposite curvet tire. In distinction to the first
curve, the growth of the current density in this case cannot exceed a certain
limiting value, namely the value of the maximum diffusion current /4. The
value of /d is determined by the conditions of the experiment, and to plot
this curve /d has been taken in accordance with its values obtained in expe
riments ("), viz. 1.75 ruAtetni.
The oxygen polarization curve APFSN was obtained
under such conditions of the Abode's operation, under which the overvolt
age of oxygen ionization was accompanied by concentration polarization
(the rate of oxygen transportation was limited). A.cording to our analysis,
in this case the dependence of the potential 's negative shift f)d upon the density
of the polarizing current / will 1* determined by the following expression:
a+big/?blg(1? I) (3)
/,?
where a auth b are the abovementioned constants, and /4 is the limiting
diffusion current:,
For small polarization currents (appreciably smaller I him the limiting
diffusion current.) this curve will be near to the overvoltage curve for Oxygen
ionization. For polarization currents approaching the value of the limiting
diffusion current, the curve of oxygen polarization will be dose to the curve
of concentration polarization.
An ?increase in the negative potential due to concentration polarization
(hiring cathodic polarization cannot continue indefinitely. As soon as the
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Characteristic Points and Sections of Cathodic
Polarization Curve in Metallic Corrosion
Desig
nation
ofjpoin
or sec
tion
Characteristic features of given
point or section
Location of points on
cathodic polarization
curve
A P
AF
A Q
P Q
Q?G
1. The Maximum rate of the reaction of
cathodic depolarization is equal to the
limiting diffusion rate of the depolarizer
to the cathode
2. The concentration of the depolarizer
on the surface of the cathode is equal to
one half of its concentration in the midst
of the solution
3. The resistance to the cathodic reac
tion is equal to the resistance to the pro
cess of oxygen diffusion, i. e. the cathodic
process is controlled by the rate of the
diffusion of the depolarizer instead of by
the rate of the reaction
The process of hydrogen ion discharge
begins (the beginning of hydrogen de
polarization)
1. The limiting diffusion current, i. e.
the current determined by the maximum
possible rate of diffusion of the depolariz
er under these conditions
2. The concentration of the depolarizer
at the surface of the cathode is equal to
zero
The rate of oxygen depolarization is
equal to the rate of hydrogen depolariza
tion
Section in which cathodic process is
chiefly controlled by the rate of the cathod
ic reaction of oxygen ionization
Section in which cathodic process is
completely controlled by oxygen depolariz
ation
Section in which the cathodic process
is chiefly controlled by oxygen depolariz
ation
Section in which the cathodic process
is chiefly controlled by the diffusion of
oxygen to the cathode
Section in which the cathodic process
is chiefly controlled by the evolution of
hydrogen (overvoltage of hydrogen)
The current density is equal
to one half of the limiting
current of diffusion. The point
potential is equal to the po
tential of the curve of the over
voltage of oxygen ionization
for a current densi ty equal to
the limiting diffusion current
At an equilibrium potential
of the hydrogen electrode in
the given solution
At a current density at which
the polarization curve be
comes vertical, or, approxima
tely, at the point of the se
cond inflection of the cathodic
polarization curve
At the potential at which
the curves of oxygen polariza
tion and hydrogen polariza
tion intersect. At a current
density approximately double
that of the limiting diffusion
current
?
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potential of any new process is reached (in practice this is usually the dis
charge of hydrogen ions), the further dependence of the cathode potential
upon the density of the polarizing current will be chiefly determined by this
um process.
If in accordance with the wellknown logarithmie relation for the hydro
gen overvoltage, VW represent by curve KL. N .11 (see figure) the
variaton of the electrode potential with the current density for the process
of hydrogen evolution (hydrogen polarization curve)*,.
PF,SvC will be the general curve, which for short may be called the curve
of oxygenhydrogen polarization. This ourve may be
plotted by a simple summation of curves A PFSN and KLNJI along the
xaxis.
The numerous experimental curves of 1.,nthodie polarization obtained by
he author for various cathode materials are in good agreement (making
allowance for a few well founded departures observed) with the calculated
curve of cathodic polarization A PFS'QG
Our analysis of the cathodic curve of oxygenhydrogen polarization (')
permitted us to define a number of characteristic points and se
ctions of this curve (see table).
Such an examination of polarization curves is of considerable practical
interest.
Indeed, if we know the potential of the cathode in the process of
corrosion (for small ohmic resistances it is equal to the potential of the
corroding metal), we nifty?on the basis of the curve of cathodic polarization
for the given corrosion process?fully charaeterize the cathodic process:
determine the relation between hydrogen and oxygen depolarization, the
relative value of overvoltage of oxygen ionization and the limiting diffusion
current. This is achieved by simple examination of the location of the point
representing the cathodic potential during corrosion on the cathodic pola
rization curve, according to data given in the table. In the case where the
corrosion process is controlled chiefly by the cathodic process, such a cha
racteristic of the cathodic process is, generally speaking, a characteristic of
the corrosion process as a whole.
The possibilities offered by the use of polarization curves and their
construction from experimental data have been discussed elsewhere in
greater detail (1,9).
laboratory of Corrosion of Alloys. Received
Institute of Physical Chemistry. 12. II. 1o4c,
Academy of Sciences of the USSR.
REF , :Nt:ES
' E., Evans. li Ji.aiiiiister and S. Britton. Proc.. Hoy. SOc. (A). 131,
.:155 (19n). 2 T. P. Uo a r, Trans. Electroch. Soc., 76, 157 (1934 3 T. P. II o a r,
Proc. Hoy. Soc. (A), 142, 628 (1933). F. 13. Anum0 B. Tp. 56 (1938).
'? A. II. (P p y Mit II a. Tp. 2fl notoliep. ho itoppoasit mezaonon aim AU CCCP, 104o, crp.
? H 'I' o m a tile in. ;tom.. Anecepraunn, 1942. MXTII IIM. Ntemeaeena. 7 M. V 0 I 
me r Z phys. Cheni., (A), 166, SO (1933). % I r u in k i n, ibid., 160, 116 (1932);
164, 131 (1933); Acta Phys. Chins. CHB'S, 7. 475 (1937). II. D. To masho
C. Et. Acad. Sd. MISS, XXX, No. 7 (1941); 611, No. 7 (1947;).
Concentration polarization is nid of great importance in the process of hydrogen
evolution, as the hydrogen forma(' on the surfa( e of the cathode in case where it is nut
immediately removed?may be evolved in the form of gas bubbles without any difficulty.
Thus, the curve of polarization occurring at the expe use of the discharge of hydrogen ion's
(the curve of the overvoltage of hydrogen evolution, may in the first approximation be
identified with the curve of hydrogen polarization.
694
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Comptes Rendus (Doklady) de l'Acadinnie des Sciences de FUNS
1946. Volume LII, Jill. 8
GEOLOGY
CliANGES IN THE MODE OF SEDIMENTATION IN THE CASPIAN SEA
WITHIN HISTORICAL TIME
By S. W. BRUJEWICZ
(Communicated by P. P. Shirshov, Member of the Academy, 24. IX. 1945)
While studying the chemical composition of bottom sediments of the
Caspian Sea, the author took and examined in the course of 19351940
coresamples at 25 stations. The variations along the vertical of the bottom
deposits studied give a picture of the general changes in the physical geogra
phical conditions of the sea.
Bringing out a number of regional characters, the vertical run of variation
in the components of the Caspian sediments points at the same time to the
following alterations, which are of rather general nature. At several stations
of the North Caspian, viz. at those of the TiubK aragan gulf l'x 81, l'x 83,.
l'x 85, l'x 92, at the Lbishchenski shoaly slope (St. 19) is recorded a reducti
on in the content of carbonates in the upper horizons, which is especially con
spicuous in the TiubKaragan gulf. This should be taken to mean that the
supply of aeolian material from the land has been intensified here during the
recent time. The rapidly accumulated fluviogenic sediments of the western
part of the Northern, and particularly of the Middle Caspian Sea, are highly
homogeneous along the vertical; they do not point to any essential changes in
the conditions of sediment accumulation which would have occurred
within the recent centuries.
? In the South Caspian, within the area of the sea basin, of the eastern
slope and the eastern shelf, there was observed an increase in the content of
calcium carbonate and a decrease in that of Fe, Mn and P downwards.
In the region of the seabed, in the northern basin of the South Caspian
(station 26) takes .place a wellpronounced transition from the upper layer,
40 cm thick, with a CaCO3 content of 1721 per cent only, to the fiftieth
centimetre and underlying layers, whose CaCO3 attains 45.6 per cent.
In the southern basin of the South Caspian (station 50) the sediments are
poor in calcium carbonate, homogeneous, and no underlying sediments high
in CaC(3), could be found there within a thickness of 1.10 cm of the core.
In the region of the eastern slope (station 28 and l'x 48) the content of
CaCO must likewise show an increase down the vertical, though not so
clearly pronounced. The thickness of the upper layer is 40 cm.
The same phenomenon is observed in the eastern shelf, where it is less
sharply pronounced, however.
Special interest belongs to the sharp increase in the carbonate content in the
under layers in the region of the northern basin of the South Caspian (station
26). The theory of ?landslips from the western slope? should, be flatly dis
carded for the following reason. At the station ?Piksha? 21 (section Kurin
695
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C tP IF tont of Carbonates and of Fe. Mn and P in an 1114 li,xtract
fro HI !?i nil Core.: of the Caspian Sea
. r.Z.T.
,. _
0 ...... .
1,,,"

l'7, 17f
*.oo?.... .....pcve,,,
;::.,'..t.Z5.A !..1t*.6"11.F.
1 I 3
in% FICI extract in per C1111
to absol. dry 'natter
to residue Insoluble in HO
1 Fe Mu
4
IS
Fe Mn
North Caspian. Tinbdiaragan Golf, 1.X1.19391. St. rx 81: 6.5 n. (toiler of the gulf
0
17
25.8
! 53.9 2.07 ! 0.044 0.067 ! 3.86
0.082
:
0.126
17
20
48.6
52.2 1.75 0.042 0.062 3.35
0.081
:
0.119
20
40
28.0
53.8 1.58 0.045 0.063 2.95
0.084
:
0.117
40
50
40.1
48.3 0.85 0.034 0.053 1.76
0.070
!
0.110
50
83
37.7
49.6 1.32 (L414o: 0.054 2.66
0.093
'
0.10,9
St. rx 83: 7 to. north of the st. 1'x SI
o
7 :
:1.1.I
47.5 ' 1.41 0.01:4 0.052 2.95
0.069
0.110
7
15 i
33.8
50.0 1.43 11.em 0.043 I 2.86
0.072
0.086
15
30 1
50.2
41.7 0.30 0.024 0.026 0.72
0.058
0.062
50
60 1
55.4
38.5 0.51 0.02x 0.042 ! 1.32
0.073
0.109
80
92 ;
:10.9
41.S 0.54 0.03n 0.041 1 1.29
0.079
0.098
SI. Cx is:,: 8.5 in
0
5
32.3
:e2.7 1.05 0.039 0.030 .1.00
0.07i
0.057
5
10
10
I
201
34.4
52.4
51.2 1.38 0.041 0.030 2.69
38.8 0.58 0.030 ; 0.031 1.50
0.080
0.077
0.057
0.080
30
67 1
58.4
36.9 0.39 0.022 0.031 1.06
0.060
0.084
North Caspian. open sea. Lbishrhenski slimily slope of the N. Caspian.
St.. 19: 5 in, 26.VI.1940
0
5
31.6
40.7 ! 1.55 1 0.039
0.041 3.40
0.083
! 0.087
5
if)
40.2 0.81 0.031
0.029 2.01
0.077
0.072
10
12
44.0
4n.r. 0.51 : 0.030
0.028 ' 1.18
0.069
0.066
Near Chasoyaya bankJ. SL 4'1: 4.3 01. 2.V11.1940
10.9
68.7 2.62 0.074 0.066 3.83
0.108
0.096
13.0
70.1 2.05 0.078 1 0.060 2.92
0.111
1: 0.086
10
2
11,4
73.0 2.15 0.074 1 0.056 2.94
0.101
0.077
20
It
N.3
79.1 1.34 0.039 0.043 1.70
0.049
0.054
North the Agrakhon Gulf. S. 47: 8.3 in. 4.VII.1940
11.8
67.3 2.65 0.072 0.055 3.93 1
0.107 .
0.082
5
10

10.5
69.9 2.47 0.100 0.055 3.55
0.144
0.079
50
64
!
72.4 :1.29 0.077 0.05:1 ; 3.17
0.106
0.073
Middle Caspian. western part., near Makhaehliala. St. 1: 19 in. 9.14%1940
11S
5 10 ,
50 64
12.6 ! 68.6 1 2.82 1 0.065 0.058 !
12.6 66,2 ! 3.36 1 0.080 I 0.054
ELI 70.4 2.08 0.052 0.051 1
4.10
5.08
2.96
0. 95
' 0.121
; 0.074
!.
!
1
.0.084
0.082
0.072
Noaheast of the kaliaLin spit. St. 130: 258
m. 28.1
V.1140
0 5 :
14.4 : 65.6 : 2.68 1 0.061 0.064 1
4.10
; 0.093
0.097
5 15 ;
11.1 68.4 2.78 1 0.075 0.072 :
4.06
0.124
0.105
45. fa, ;
11.9 70.4 2.77 1 0.067 0.068 ;
3.92
0.095
;
0.096
Eastern shelf, WNW of Gulf Synghyrli. over shell limestone. St. 13; 114m. 1.V.1940
5
15
13
29
!
57.6
%3.$
31.6
!
i
25.6
47.7
57.6
!
0.60
0.79
1.60
:
I
!
0.020
il.025
0.050
0.037
0.031
0.053
I
;
2.34
1.53
2.89
F
1
0.078
0.054
0.087
0.144
0.065
0.092
South Caspian. western part of section Island ZhiloiCone
St. 16; 100 ro,
15.v.194.0
0
,
119.2

I
3.72
1
0.074
1
0.096
1


I

5
10
!
31.2
1
56.3
1.90
!
0.053
1
0.044
!
3.37
0.094
;
0.078
10
20
;
16.5
69.7
:
2.74
1
0.054
1
0.053
!
3.93
0.077
t
0.076
20
38
12.0
1
75.4
!
2.48
1
0.061
e
0.062
i
3.30
0.081
I
0,082
696
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,I 100,;,11(111 extract in per cent
75t,
N0soz,

05
. 515
1535
.Same, in
to aloatt. .1 ry mall er
Fe
3 I, 5 ii
Same, western part. St. 17, 172 m,
21.2
65.6
9,92
0.029
0.109
16.3
71.2
3.14
0.031 ,
0.066
18.1
70.7
2.22
0.035 1
0.066
lo residue insoluble CI
Fe
7
4.V.1940
5.05
4.40 I
3.15
0.044 0.166
0,044 0.092
0.040 0.093,
the troughcutting the submarine Apsheronian ridge. St. 18; 200 in, 14.V.194,0
0,5
13.6
73.1
2.44
0.052
515
13.1
76.7
2.72
0.059.
153Q
11.8
72.8
.1.94
0.050
0.056 3.33 0.085 0.076
0.041 2.55 0,077 0.053
0.038 2.65 0,081 0.052
Northern basin of the South Caspian section Knrinski KamenOgurchinsk . St. 26; ?
0 10
1020
2030
9040
4050
5060
Average
960 m, 31.V.1940
21.2 03.6 289
0.086 0.055
17.5
56.6 9,99
0.108 0.056
20,7
64..2 2.25
0.139 0.052
20 9
60.3 2.53
0.095
0.045
28.6
53.1 2.30
0.118
0.047
45.6
50.3 1.94
0.105
0.049
4.54
4.05
3.50
4.20
4.32
3.85
4.07
. Southern basin of the South Caspian. St. rx 50; 900 m,
015
2545
0080
100110
Average
17.7
21.7
91.4
19.4
64.9
61.5
60.0
63.2
Eustern slope, section
0 5
510
1020
2030
3040
4050
5060
6070
Average
49.8
51.8
54.4
49.8
53.8
67.3
59.1
59.7
37.0
35.0
34.3
35.4
34.1
30.7
27.5
28.7
2.51
2.50
9.41
2.50
0.119 0.064
0.100 0.046
0.091 0.065
0.079 0.066
3.86
4.06
4.07
3.95
3.99
0.136
0.191
0.216
0.157
0.221
0.208
0.188
20.XII. 1936
0.184
0.162
0.150
0.125
0.155
0.086
0.099
4.081
0.074,
0.088
0.097
0.086
0.099, .
0.075
0.107
0.104'
A.096
Kurinski Kamen IslandOgurchinski Island St. 28;
460 m, 1.VI. 940
1.53
1.15
1.14
1.28
0.96
1.04
1.09
0.97
0.086
0.089
0.075
0.051
0.058
0.054
0.052
0.044
0.056
0.057
0.049
0.042
0.036
0.041
0.039
0.023
4.05
3.28
3.32
3.61
9.80
3.38
3.95
3 37
3.47
t3.227
0.254
0.218
0.144
0.170
0.176
0.192
0.453
0,192
0.148
0.162,
0.143
0..119
0.107
0.142.
0.142
0.115
0,135
Same, SW of the Mud volcano shoal. SI. Fx 48 his; 580 in 1.8.XII.1936
020 45.5 40.2
0080 48.9 37.5
8090 55.8 31.5
020
2040
4060
6080
8097
Average
Eastern shelf.
71.8
73.9
76.1
77.4
78.5
1.72
1.48
1.30
0.079 ' 0.046
0.052 1 0.050
0046 1 0.037
4.98
3.95
4.12
0.196
0.138
0:146
SI. rx 47; 33 en, 17.X1 .1936, west of Zelenyi Bu.gor
17.7
17.6
14.6
14:3
14.4
0 68
0.72
0.61
0.62
0.55
0.020
0.020
0.019
0.019
0.018
0.027
0.027
0.022
0.020
0.025
3.85
4.08
4.17
4.34
3.82
4.05
 0.113
0.114
0.130
0.133
0.125
0.123
0.114
0.133
0.117
0.152'
0.153
0.151
0.140
0.174
0.154
Same, section Ku inski Ka nen IslandOgurchinski Island. St. 29; 85 m, 2.V1?10110
05
510
010 I
1040 I
56.6
61.6
26..6
23.6
0.90
0.40
0.031 0.040 I 3.39
0.018 0.021 I
Krasnovodsk Gulf. St. fc7 9.5 in,
43.5 I 35.65 0.97 0.019 0.055
44.9 I 36.96 I 0.87 I 0.019 0.047
C. R. Aced. sci. I1RSS, 1946, v. LIT, N, 8.
0.117 0.150
18.Xl.1935
2.72 I 0.053
I 2.35 I 0.052
0.154
0.128
697
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ski I.Ogurchinski 1. referred to above, east of our station 26), at a depth
of 8(30 in, T. I. Gorshkova observed similar phenomena, viz, a earbonate
content of 19.417.4 per cent, in the upper 40cm layer, and of /13.2 per
vent beginning with the 0011 centimetre (besides these, no measurements
have been made). For the 'landslip* a rise by 100 in is excessive. According
to data by G. G. Sarkissian (core samples up to 2.5 in) within the region
of the basin of the Middle Caspian, at the deepest layers of the sections
Gulf Peschany?Gulf Buynak, and especially Derbent?Sue there occur
deposits showing a much higher content of carbonates and a coarser
mechanical compositinn. The nature of sediments speaks against the
theory of landslips in this case, too. We are thus led to conclude that the
phenomenon of decrease of the carbonate content, in the sediments formed
during the recent epoch is peculiar to the Middle and South Caspian as
a whole, and is not connected with landslips. In so far as under the
conditions of sedimentation that prevail in the Caspian Sea the variation
of carbonate content points to a variation in a given place or the inten
sity of precipitation of fluviogenic and aeolian talassogenic sediments,
the phenomenon here described cannot be accounted for otherwise than
in the following way: During the epoch preceding the recent one the zone
of expansion of solid matter discharged by rivers has been pronouncedly
extended from west to east. This refers both to the sea basin and the lower
part, of the eastern slope. The decrease in the carbonate content in the eastern
shelf took place because of the prevalence or precipitation of aeolian sediments
poor in carbonates and rich in Fe, Mn and P Over the purely talassogeoic
deposits high in carbonates, which was due to the intensification of winds
blowing from the east.
It seems most natural to explain both phenomena by a common cause,
viz, by a general increase in the atmosphere circulation, which is supposed
to have taken place during the last thousand years. Because of the increased
pressure within the area of the Siberian winter anticyclone, the latter phenn
menon is associated with an increase in the Caspian region of winter winds
blowing from the east and carrying aeolian matter from Central Asia. On
the other hand, any increase in the displacement of the upper water layer of
the Caspian westward results in an increase in the eastward compensatory
'movement of the deep waters carrying particles in suspension.
The recent epoch of low carbonate content embraces a time interval of
about 1000 years; the epoch of transition was a very short one, of no more
than 200300 years. The data available are insufficient to determine precisely how remote was the (Torii of high carbonate content.
Received
Xl. 1945.
REFERENCES
S. V.hirujewicz and E. U. Vinogradov a, C. R. Acad. Sci. URS,
1311, No. 9 (194G).
698
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Comptes Rendus (Dellady) de PAcademie dcs Sciences de PURSS.
W46. Volume LH, S
GEOLOGY
AN ATTEMPT AT A GENETIC CLASSIFICATION OF FIRE CLAYS
AND REFRACTORY CLAYS IN WEST SIBERIA
By V. P. KAZARINOV
(Communicated by V. J. Obruchev, Member of the Academy, J. II. 1946)
Fire clays and refractory clays are among the products of MesoCenozoic
weathering crusts widelydeveloped in West Siberia. In the Lower Cretaceous
and Lower Palaeogene time processes of chemical weathering which produced
the weathering crust went on over a vast territory. The chemical weathering
of various stone and loose rocks over vast areas of a peneplained country
would bring about the formation of decomposition