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THE

JOUENAL

OF THE

COLLEGE ,0E S( lENCE,

IMPERIAL UNIVERSITY OF TOKYO.
Vol XXXVII.

:^ :5ï ^ s ;^ ^ pp ff

PUBLISHED BY THE UNIVEKSITY.
TOKYO, JAPAN.

1914-1917.
TAISHO 3-6.

J a.a

Publishing Committee.

Prof. J. Sakurai, l^'f^. f-, /.'/'/f//./('("A-'fv/a', Director of the College (e.r njficin).

Prof. I. Ijima, I'li. !>., IH'jah-nhx/.-nshi.

Prof. F. Ömori, llhjahih'ikx^hi.

Prof. S. Watasé, P/'. /', lliiinkuhalntsJii.

/ % Î 7 Ô

CONTENTS.

Art. 1.— Researches on the electric discharge of the isolated electric
organ of /Isfrapei Japanese electric ray) by means of oscillograph.

(With 30 plates). By K. Fiji — Piibl. I'ec. 11th, 1914.

Art. 2.— Recherches sir les spectres d'absorption des ammine-complexes

métalliques. I. By Y. Suhim -.— Pnl)l. Sept. 30th, lülö.
Art. 3.— Considerations on the problem of latitude variation. Py K.

SoToMK. — Pull]. Nov. 30th, 101."..

Art. 4.— On the distribution of cyclonic precipitation in Japan. P-y
T. Ti.KADA, T. YoKoiA and S. OiiKi.— Pnbl. Jan. 27tli, 1910.

Art. 5.— On the relatively Abelian corpora with respect to the corpus
defined by a primitive cube root of unity. Py T. Takixotciu. —
Puhl. Mav. SUtli, 1910.

Art. 6. — Numerical calculation of the Jacobian ellipsoids. By P. Kaibara.

— Publ. July 20tli, 1910.

Art. 7.— On the elastic equilibrium of a semi-infinite solid under given
boundary conditions, with soîiie; applications. By K. ïep.azaw.a.

— Puhl. Pec. 7tl), 1910.

Art. 8.— Eeoherches sur les spectres d'absorption des ammine-complexes
métalhques. II. Py Y. Shuata — Publ. Dec. '29th, 1910.

Art. 9.— On rapid periodic variations of terrestrial magnetism. By
T. ÏKi-.Ai.A.— Publ. May 25tlK 1917.

Art. 10.— On the photographic action of «, ß and r rays emitted from
radioactive substances. (With o plates). By S. Kino^hua and H.

I a-T-Ti.— PuM. Nov. 20th, 1917.

PRINTED BY THE TOKYO PRINTING CO., LTD.

.TOURNAI. OF THE COLLEGE OF SCIENCF, IMPEEIAL UNIVERSITY, TÔKTO.

VOL. XXXVII., ARTICLE 1.

Researches on the Electric Discharge of the Isolated

Electric Organ of Astrape (Japanese Electric Ray)

hy Means of Oscillograph.

K. Fuji, BigcdtisJii.

With SO Plates.

Contents.

Pare

T. Tntrodnctioii 1

If. description of the fisli, and preparatioii of the organ 3

IIT. Plan of the experiments G

lY. Fonnnla expressing the discharge curve ] 4

V. l^elation hetween the form of stimuhis and the discharge caused by it. ... t2G

YI. Discharge by two successive stimuh 4'.i

YIT. Fatigue phenomenon 65

Y ITT. Speed of propagation of excitation through the nerve 70

IX. Miscellaneous problems 72

Summary ... 77

Ai^peiulix. Tables of experimental data and calculated numbers. ... 80

I. Introduction.

It is now sonic twenty years since the discharge cni'ves of
electric fishes began to be investigated by such physiologists as
Gotch, Schcnlein, Garten, Koike, Cremer and others. The
method of Gotcli consisted in tracing the curve, point by point, by
means of a ballistic galvanometer, a single point being determined
by each discharge. Scliënlein apphed Bernstein's rheotome;
Garten and Koike used tlie capillary electrometer and tlie string

2 Vol. XXXVII.. Art. 1.— K. Fuji :

galvanometer, whereas Cremer employed his string electrometer.
Although the metliod of experiments used by Gotch was certainly
the most ingenious at that time, the curves obtained are not
sufficiently accurate to be aiial,ysed quantitatively. The method of
Schönlein is inferior to tliat of Gotch, for the form of the discharge
curves obtained l»y liis metliod cannot fail to be mucli affected by
the fatigue of the tissues. As the capillary electrometer, the string
galvanometer and the string electrometer cannot faithfully follow
so quick a change in the electromotive force as happens in the case
of the discharge plienomena in question, tlie curves olttained by
til em are only qualitative.

In April 190G, in conjunction with ]Mr. S. Ginuma, now
I(jahuhahusJù, the autlior began his experiments on the same
subject by means of an oscillograpli in the Physiological Institute
of the Imperial University of Tokyo. In the course of this investi-
gation, his collaborator went to Europe, and the author was then
obliged to continue the experiments alone.

The electric hsli used Ijy us was one of the electric rays called
Astrapa japonica, which was brought from Misaki, a fishing town
in Sagami peninsula. Perhaps on account of the imperfectness
of our aquarium, the fish could not be kept in a healthy state ex-
cept for a few months in Spring and Autumn; and in these most
favourable periods, even a single fish sometimes could not be ob-
tained. ()n account of these and other hindrances, and although
lengthy and exhaustive experiments were continued up to 1910,
the problems first proposed could not l)e thoroughly investigated.
The results so far obtained l)y these experiments arc now presented
in this paper.

With regard to the form of the discharge curve which will be
discussed in § IV., it is very interesting to note that some physio-
logical phenomena can be explained from the standpoint of the
theory of probabihty. It is a known fact that many quantities
characterising a class of animals or plants are subject to individual
variations; and if the observations are taken on a sufiiciently large
number of individuals of the same class, the values representing any
of these characteristics distribute themselves, among the number of

Ecsoarclies on the Discharge of the Electric Organ. ^

individuuls taken, in sucli a numner that the variation appears to
occur in accordance witli the law of errors. Since now the tissues
of an organism consist of a large numher of structural elements, it
is natui-al to suppose that a characteristic helonging to these ele-
ments varies in different individuals according to a similar law.
Then, it may be remarked, that the treatment of the physical
phenomenon in a tissue must l)e based on the principle of prob-
ability, and especially in the treatment from the standpoint of the
''all or none "* theory, the application of this principle must be
effective and indispensable, for a physical phenomenon exhibited
by a tissue must be the integral effect of the phenomena occurring
in each of the component elements.

II. Description of the Fish, and Preparation
of the Organ.

The structure of the electric organ in Astrajx' jajwiiica was
recently described in detail by K. Ishimori +, and therefore need
not be entered into here. However, a few points having direct
bearing on the present investigations may here be recorded.

The external asj^ect of the fish and its electric organs are re-
presented in Plate I. The length of the mature fish is al)Out 25 cm.,

* Lucas inferred from his investigation on the contraction of a skeletal muscle, M.
cutaneus dorsi, of the frog (Journal of Physiology, Vol. 38, 1909.), that, when a stimulus
greater than a certain threshold value is given to a single nerve-fibre, the excitation evoked
in it appears to have a definite value independent of the intensity of the stimulus. Then,
it seems very probable that, with regard not only to a nerve but to tissues in general, tlie
excitation evoked at a point in the excititory eleuients follows the so-called "all or none" law.
In other words, there exist only two alternatives, i.e., whether the excitation does occur in a
definite intensity or not at all,— no intermediate value being i>ossible. In this view, the
liropagation of the excitation should take placj by the successive action of the excited
point towards its neighbour, the projiagation in reverse direction being impossible in
virtue of the existence of the refractory period. The energy of the excitation of each, portion
should be supplied by some chemical change in it, and therefore its intensity should
depend on the physiological state of that portion only. A case may occur, that an excita-
tion is enfeebled on its way to pro^^agation, by passing through a portion in an abnormal
state, to such a degree as is incapable of evoking the excitation of the neighbouring portion.
Then the farther propagation is impossible and stops there. For proper understanding of
the discussions throughout this pajjer, it is necessary t? keep these considerations in mind.

t Beitrage zur Pliysiologie. Festschrift zu Ehren der 25 Jihrigen Lehrtätigkeit von
Kenji Osawa.

4 V..1. XXXVII., Art. 1.— K. Fuji :

measured from liead-eiid to tip of tail. Tlic electric organ is present
in a pair in the fni-in of hnge flat bodies, situated one on each side
iuid lateral mainly to the head and the branchial regions. The
upper and lower surfaces of the organ are in direct contact Avith the
integument of the flsh. P]ach organ is an assemblage of vertical
hexagonal prisms al)out 200 in number. The height of the prisms
measures 1-1 -5 cm. Kach prism consists of a pile of disc-like body
called the electric jjlalc numberiug about 400. The plates consist
each of a clear jelly-like mass inclosing a number of large nuclei
and are surrounded as well as separated from one another by a
connective-tissue layer in which the nerves and the blood vessels find
their course. From the Looks electricus of the brain, there arise
on each side five special nerves, the electric nerves, which supply
the organ in question. Distally the nerves undergo successive
branching, Anally to terminate in fine network on the ventral
side of each electric ]:»late. In the discharge of electricity the side
of the plate just mentioned is always negative, while the dorsal side
Avithout the nerve-endings is positive.

Before explaining the ])lan of the experiments, we shall
describe the method of preparing the organ for the purpose. Except
Avhen the spontaneous discharge was to be studied, the organ Avas
separated from the l)ody and the Avhole or a part of it Avas used
according to eircumstances. When the flsh was brought from
the aquarium to the lalioratory, in order to avoid tlie setting in of
fatigue, the brain Avas extracted l)y applying a cork-borer to the
head and tlien sti'iking its upper end Avith a hannner. The spinal
coi'd Avas next destroyed by pushing a Avire into the spinal
canal. Having thus avoided tlie reflex action of the nervous centre,
the organ Avas separated from the fish.

The electric dischai-ge of the organ is evidentl_y a very complex
phenomenon, for it must be the integral efïect of unit discharges
of the electric plates. When a stimulus is given at a point in
the nerve-trunk, the excitation is distally transmitted through
nerve-fibres of different lengths, and as the speed of transmission
through them is finite, it should arrive at the different electric
plates not simultaneously l)ut at difïerent periods. Hence it is

Rosearches on the Discharge of the Electric Organ. 5

advantageous, for the investigation of the discharge curve, to
stimulate the organ not through the nerve^trunk but directly,
so that each plate may receive the stimulus at the same instant.
For the direct stimulation of the muscle, it is customary to use
curara in order to l)enuml> the nerve-endings. In the case of
the electric organs, however, there is known no such drug that
can be used witli the same effect. Hence the term direct stimula-
tion used in tins paper in relation to the electric organ has a some-
what different meaning from that used in the case of a muscle.
Tiiough the stimulating electric current is sent directly through
the electric organ, tlie stimulus is probably in the main imparted
to it through the nerve-fibi-es distributed in it. Nevertheless, in
that way the condition of simultaneous stimulation can be better
fulfilled tlinn when tiie stimulation is given through the nerve-
trunk. ^L)reover. when the nerve-trunk is left attached to the
organ, the discharge cui'rent flowing through the nerve-trunk iè
very lial;)le to act as a second stimulus and to cause a secondary
discharge of considerable magnitude, by which a tertiary discharge
is called forth, and so on. This makes the phenomenon very
complex. But. in tlie case of direct stimulation of the organ, if
the nerve-trunk be carefully taken away from the organ, and if its
temperature l)e kei)t sufficiently low, there Avill result only a very
Aveak secondary discharge (Plate 1., Fig. 2). In accordance with
these considerations, we empk\ved direct stimulus in general, ex-
cept when the [)i-operty of the nerve was to be investigated. In
the former case the ti\e trunks of the nerves .were cut off as near
as possible to the (jrgan. and the whole or a part of it was utilised.
In the latter case, when the stimulus was to be sent through thé
nerve-trunk, al)Out a quarter of the organ with one nerve-trunk
was employed.

Vol. XXXVII., Art. l.-K. Fuji.

III. Plan of Experiment.

The diagram of the plan of the experiinent is represented in
Fig. 1.

The oscillograph used by us was that designed by Duddeli
and made b}^ The Cambridge Scientific Instrument Company,
and that which is known as the high h'equency type. The period
of strips in an undamped state measures about ]/K)i)00 sec. By
means of this instrument, an alternating current having a period 50
times that of the strips may be photographed without an appreci-
able error.* As the duration of a discharge of the organ measui-es
about 1/100 sec, the curve obtained by this instrument may be
relied on for quantitative analysis. As the instrument is designed
for an alternating current, it has a pair of strips, one for the
current and the other for the apphed E. ]M. F. In our
experiments, when the organ was stimulated directly, only one (jf
them was employed, the stimulating current and the resulting
discharge current being made to flow through the same strip; and
when the stimulus was sent through the nerve-trunk, one of the
strips was used for the stimulus and the other for the discharge.
The fact that both the stimulus and the discharge may be
photographed upon one and the same film is very convenient for
accurate work. The sensitivity of the oscillograph is about 2G mm.
for Ol ampere, the film being at a distance of 50 cm. from it.
The resistance of each strip is about 8 ohms.

The fight source Q for illuminating the oscillograph consists
of a hand-regulating arc lamp of 15 amperes, the image of which
is formed on a slit s l>y a lens L.

The registering drum D is supported within a camera and has
a circumference of 50 cm. (Plate III., Fig. 1). This is revolved by a
small motor M of 1/30 H.P. The number of revolutions used
was 5-20 per. sec, accoi'ding to circumstances. A photographic
film was wound around the drum and kept in position by means
of two india-rubber rings which keep each side of the film pressed

* Zeitschrift für Instrumenten Kunde, S. 240, Bd. 21, 1901.

Kesearches on the Discharg-e of the Electric «Jroau.

Fig 1.

O Oscillograph.

Q Are lamp.

l-i ('onverg-jng lens.

••; Slit.

^' shutter.

i.-j Cylindrical lens.

D Registering clruin.

M Motor.

ii'i llegnlating resistance for motor.

J3j, 11, Break-circuit contrivances.

i' Organ-preparation.

T Thermostat.

« Electrodes for indirect stimulation.

The Plan of Experiment.

Z], Z. Zinc electrodes for electric organ.

P Eelay for thermostat.

E Stimulation-apparatus.

/ Induction coil.

A Ammeter.

/.'._. Regulating resistance for primary

current of induction coil.
/,■ Morse key.

K Chronograph.
(.' Chronometer.
Il Resistance box.
/r To Kohlrausch bridge.

^ Vol. XXXVIL. Art. 1. K. Fuji.

firmly against the dram and also jjy a thin j)i('cu ut hanihoo which
presses the two overlapping ends of tlie film. When the stand
supporting the drum is pulled, tlie axis of the <h-uni is disconnected
from its driving shaft by means of a special contrivance (Plate I\^,
Fig. 1, a), and the drum with its stand may he taken out of the
camera, the operation l)eing accomplished in tlie dark williout
any trouble.

On the shaft of this registering a})paratas. there are two
break-circuit contrivances like that of a chronometer (Plate IV.,
Fig. 1, ^1 b>] Fig. 2), one for the chronograph by whicli tiie number
of revolutions of the drum is measured, and tlie other for the
shutter described below.

A kymograph K made by Zimmermann, witli accom|)anying
time-markers, was used as the chronograph, the record being
oljtained on smoked paper. When the speed of tlie rotation of
this instrument is increased to its maximum, the number of
revolutions of the registering drum may be accurately determined
to three figures.

The organ-preparation was supported between two zinc plates
that served as electrodes. To secure a good contact, and also to
avoid the effect of polarisation, if such existed, the surfaces of the
zinc plates were covered with kaolin paste, soaked with a saturated
solution of zinc sulphate, and then with a layer of the same
material kneaded with a jihysiological solution of natrium
chloride.

Since it is found that the form of the discharge curve is much
affected by the cliange of temperature, the oigan is ])ut into an
electrically regulated thermostat. As the favourable range of
temperature for our investigation was found to be <S C. — 15°C. . and
because in the warm season this range of temperature is lower than
that of the room, the thermostat must be so designed as to ensure
that a constant temperature lowa-r than the surrounding temperature
be maintained. For this purpose the author constructed a s])ecial
thermostat shown in Fig. 1. The moist chamber, in which the
organ supported by the zinc electrodes is put, is surrounded by a
jacket of transformer oil. in which a toluene I'egulator and • an

Researches on the Discharge of the Electric Org-an. 9:

electricall,y-li Gated resistance are immersed. Outside of this comes
a layer of ice, and the whole is protected from external heat by
an air jacket. The thermostat works well for G — 7 hours without
any fresh supply of ice.

Next we sliall describe the construction of a shutter designed
hy the author (Plato IL). The function of this shutter is as
follows. (a) When a certain place on tlie film just comes to
receive light from the oscillograph, the shutter opens, (b) then the
stimulation-apparatus is set in work, whereby the organ is excited
and the oscillograpl i prints tlie motion due to the ^stimulation
und the discliarge current, and (c) when tlie drum has made just
one revolution after tlie shutter first opened, it is made to close
so as to protect the film from unnecessary exposure to light.
The operations a and c are accomplislied by sending to tlie
electro-magnets mi the sluitter instantaneous breaks of current,
wliicli occur once in eacli revolution of tlie registering drum. Fig.
4 in Plate IV. repi'esents the construction of the shutter. It consists
essentially of twe) almost identical aluminium sectors A and B,
placed one lieliind the other witli a circular hole in eacli of them,
and two electro-magnets A' and />'. one fcir each sector. When
the sectors, by means of cords, are pulled to the left against
the action of the springs, and there held in position liy the
detents attached to the armatures of the electro-magnets, tlie shutter:
is ready for action. When the current is broken for an instant,,
at the break-circuit contrivance attaclied to the shaft above
mentioned, only the electro-magnet A' ])elonging to the front sector
is put into action and the sector A is released and turns to the
right, and then a free passage of light tlirough the shutter is.
admitted, (operation a). I>y a simple mechanism attached to
the front sector, the electric connection is changed automatically,
without interrupting the current in the operation, so that at the,
next break of the current, which takes place after one revolution
of the drum, the back sector is released and the passage of light is
intercepted (operation c). The contrivances for these operations
are illustrated diagrammatically in Fig. 2. The mechanism of
the operation b will be described under the paragraph of the.

10

Vol. XXXVII., Art. ].-K. Fuji;

/;;.','VW^-,

-WAVWIT — '

Fig 2. Connection.«
of shutter.

stimnlatiuii-appavatus. Tlie sliuttL']' docr^ not
work udIi'ss a ^lorse key k is pressed wliich
serves as a sliunt to tlie l)reak-circuit con-
trivance. By adjusting tlie ])hase of the break
witli reference to the drum, the shutter may
be opened when any desired position on the
fihïi comes to receive hglit from tlie oscillo-
graph. For the sake of economy of time and
of film, a series of discliarges are pliotograph-
ed m order on one hlin. liv tlius shifting the
phase of break with reference to the drum.
The shutter may l)e used as a time-shutter if
desh"ed, wlien it is connected to tlie ordinary
Morse key as sliowii in Fig. .*>.
T3efore describing the stimulation-apparatus, we sliall ex])]ain
the form of stimulus used in our experiments. The stimulation ordi-
narily used was an induced current in the
secondary circuit of a Du Bois Reymond's
induction coil usually employed by physi-
ologists. When the primary current is
suddenly opened , the secondary current
increases from zero to a maximum value in a very short pei-iod,
and then decreases exponentially to zero. By shunting the second-
ary current at a desired instant in its decreasing stage, it can be
made to decrease to zero suddenly. Such a current was ordinarily
used as our stimulus in direct stimulation. In many experiments
the strength and the duration of such a secondary stimulating
current were changed and their influence upon the discharge curve
was investigated. The oscillogram in .Plate I.( Fig. 2) shows a
form of such stimulus followed by the discharge current of the
organ in response to it. Here we may remark that we can see in
this oscillogram how well the shutter works. In the indirect
stimulation, however, the induced current caused by an instan-
taneous contact in the primary circuit of the induction coil is
ordinarily used. This forms a stimulus of very short duration as
shown in Fig. 4. For brevity we shall hereafter call such

Fig. 3. Connection of Morse
key for time-shutter.

Eesoarchcs on the Discharge of the Electric Organ. XI

a stimulus tlio momentary dimii-
lus. Sometimes it was required
to experiment with two such
momentary stimuh separated by-

Fig. 4. Momentary stimulus. .^ g|^,^,j-^ interval of time.

In the direct stimulation (^f the organ, the secondary of the
induction coil, the organ and the oscillograph must ho connected
in a series. But wlien the discharge is to be photographed, it is
necessar}^ to exclude the secondary coil h'om the circuit and to
introduce a proper resistance if required. For, if the secondary
coil is in the circuit, its enormous self-induction influences and
modifies the form of the discharge curve, so that, after a stimulus
is given, the exclusion of the secondary coil is necessary.

To fulfil the requirements above mentioned, a stimulation-
apparatus represented in Plate III. and Plate V. was designed.
A brief description of the apparatus will l)e given below.

A small car ^1 liaving three wheels B on each side lies on a
pair of cylindrical rails C. The rails have grooves in their upper
and lower sides to guide the motion of the wheels. A helical spring
is wound round each rail and pushes the car to the left end of the
rails. An armature made of laminated iron is fixed to the right
end of the car. An electro-magnet fixed on tlie right end of the
apparatus, whose core consists of a. l)undle of soft iron wires,
attracts the armature and keeps the car at the right end of the
rails against the action of the lielical spiings. The current of the
electro-magnet flows through tlie I)reaking contrivance of the
shutter. When the sliutter is opened (operation a Ix'foi-e men-
tioned), the electro-magnet releases the car (operation b) and the
latter is jnished away along the rails l)y tlie action of the helical
springs. The car has a knocking contrivance on its lower side
called a knocher, whose function will ))e explained later on. In
order to protect Xho. car from damage caused by a strong collision
against the sto]:> at the end of its leftwar<l motion, a special
device is attached to the stop. This consists of a pair of jaw's
closed by a strong spring. When the car comes against the stop,
a plate forming part of the car plunges into the jaws and dissipates

12 Vol. XXXVir., Art. l.~K. Fuji :

its energy by friction. Behind the electro-magnet, there is a
circuit-breaker for oj^iening tlie current of the electro-magnet. To
its level", an armature made of laminated iron is fixed, so that
when the armature is pushed against the electro-magnet, the
contact on the upper end of the lever makes the current by means
of which this armature as well as the armature fixed to the car is
kept in position at the same time, and the magnetic circuit is thus
closed through these armatures and the cores of the electro-magnet.
The service of this contrivance is twofold. When the current
through the electro-magnet is l;)roken at the shutter, this armature,
together with that of the car, is released, and the magnetic circuit
is broken at once, so that, on account of the increase of the
demagnetising force, the electro-magnet loses its magnetism very
soon. This is essential for the quick and unfailing action of the
stimulation-apparatus. The second service of this contrivance is
evident, for wlien once the circuit is opened at tliis place, it does
not close again.

On the base-board of the stimulation-apparatus and under the
rails l)efore mentioned, lie three grooves each furnished with a
millimetre-scale along its length and each parallel to the rails.
Into these grooves fits any one or a combination of small mechan-
isms, which are put into action by the knocker attached to the car.
The position of tlie.-^e mechanisms can be read to one-tenth of a
millimetre l)y means of verniers attached to them. :

One of these mechanisms is called a circuit-breaker and is used
for breaking the primaiy circuit of the induction coil. Plate V.,
Fig. 2 shows the construction of the circuit-breaker. The circuit
is to be broken at tlie contact between a knock-down lever / and the
platinum point of an adjustable screw .9. The lever / is kept vertic-
aUy and ]iressed firmly against the screw by a holder h by means
of a spring. When the car moves leftwards, the knocker kicks down
the lever I and the contact is broken. The peculiar form of the
end of the lever and of the holder protects the former from making
a second contact with the screw.

Plate v.. Fig. 3 is another mechanism which we shall call the
connection-changer. The rôle of this is to short circuit the secondary

Researches on the Diseharo-o of the Electric Organ. 23

coil at tlio pi'02)er instcint after the primary cuiTent is broken and
then to exckidc the secondary coil from tlie oscillograph circuit.
Since the latent period of tlie discli arge measures about 80 x 10"*
sec, this change of connection may be made Avithout disturbing
the discharge current. Referring to the figure, A and B are two
springs attached to an arm of a T-shaped lever. .Vt first, the spring
A is in contact with A.' In this case the organ, the oscillograph
and the secondary of the induction coil are in a series. When the
knock-down lever is kicked, the spring B comes into contact with
7>' and then ^1 parts from A.' At the moment when B touches
B', the secondary coil is short circuited, and when A parts from
A', it is excluded from tlio oscillogram)] i circuit. 'J'hen tlie organ,
the oscillograph and an appropriate inductionless resistance are in
a series, ready to receive the response of the organ. (^ is a catcher
of the knock-down lever to prevent it from rebounding. The
mechanism here mentioned is the final form used by the author.
In the earlier experiments mercury contacts were employed, but
the trembling of the surfaces of the mercury which generally formed
a very complex stimulus could not ])e avoided. Next we attempt-
ed simply to cut the secondary current instead of short circuiting
the secondary coil, but the spark at the instant of breaking the
circuit, prevented the instantaneous decrease of the current. After
such experiences, the author was convinced that it is very dangerous
to presume on the form of the stimulus l)y theoretical considera-
tions only; actual experimental evidence is indispensable. Plate
v.. Fig. 4 is a contrivance for giving two successive momentary
stimuli with any desired interval of time. Two electric contacts
are made between the ends of the levers and the knocker of the
car. Usually such stimuli are given to the nerve and consequently
in this case the connection-changer is not necessary. The distance
of the two points of contacts may be adjusted from zero to 25 mm.
and may be read to one-tenth of a millimetre by means of a
vernier.

These mechanisms are fitted into the grooves fixed on the
base-board of the apparatus. The grooves being provided Avith
millimetre-scales, their positions may be read to one-tenth of a

14 Vol. XXXVII.. Art. 1.— K. Fuji :

millimetre by means of verniers. AMieii in good ^\■orkiIlg-coIKli-^
tion, the length of one millimetre on this a[)[)aratus corresponds
to 4-4x10"* sec.

IV. Formula Expressing the Discharge Curve.

From the oscillogram in Plate I., we see tliat the discharge
curve, caused by a direct stimulus, resembles the probability curve
of Gauss, except that it is not symmetrical with respect to the
maximum ordinate. After some assumptions, a formula is obtained
for expressing the curve that agrees very closely witli tlie experi-
mental curve.

In the first })lacc it is assumed that tlie discharge of each
electric plate, caused by a single stimulus, is of \ery short duration.
It is a known fact that only the first small time-interval of a closing
current influences the height of the discharge curves, and this will
be discussed more fully afterwards. When this fact is considered,
it will not be unnatural to suppose, that a stimulus of sucli short du-
ration causes a discharge of an instantaneous nature. In the second
place it is assumed that the interval between the stimulus and the
discharge of a single plate in response to it, which interxal may be
called the latent period of a single plate, may have various values, and
among them there is a certain value, which predominates in
number, so that other values deviate more or less from it according
to the law of errors, though in a somewhat modified form. This
predominating latent period may be called the modal latent period,
which, as will be explained afterwards, represents the interval be-
tween the stimulus and the instant corresponding to the maximum
point of the discharge curve. In biological phenomena there exist
many instances which are governed by a law that involves the idea
of probability. The phenomenon of contingency and of correlation
treated by Pearson and others are such examples. Here it is
simply assumed that a similar relation exists in the quantity — the
latent period of a single plate.

In Gauss's probability curve, the freedom of deviation from
its most probable value is symmetrical with respect to it. In the

Researches on the Discharge of the Electric Organ. l5

discussion liorc. llic discliarge of any single plate cannot of course
occur before a stimulus is given, so that the freedom of deviation
from the modal value can never be symmetrical. There must exist
a certain instant before which the chscharge can never occur. We
shall caU this instant the origin of the discharge. This origin may
be at the instant of stimulation or it may be later.

T(j adapt the present case to that of Gauss's it is assumed that
the equivalent elementary interval with respect to the deviation is
proportional to the interval between the origin and that instant.

Denoting tlie latent [)eriod <>f a single plate l)y .r, the al>ove
assumption gives

i. e. Jlog.r = A-,

where /;■ is a proi)ortional constant. Tins nieaiis that, when the
logarithm of x is taken as the measure of the abscissa, the case
becomes identical with that of Gauss's.

The well known curve of probal»ility is expressed by

7/ = ^f -&-'T-, (1)

w'here tlie origin of x is at the maximum of y. Transforming the
origin to — .t,i. then

is ol)tained. Substituting the logarithm of x and of x^ instead of
them, the above reduces to

y = Ae " .Co \~')

Since the total electromotive force at any instant must be pro-
portional to the number of plates simultaneously discharged at that
instant, then, wlien y is taken as the electromotive force at a point
on the discharge curve and x as the time-interval measured from
the origin of the discharge to that point, the curve expressed by
the above formula ought to represent the rlischarge curve obtained
by the oscillogragh.

16

Vol. XXXVII., Art. 1.— K. Fuji :

Before entering into an examination of expeiinicntal curves,
the method for determining the constants .1, //' and Xo must he
treated.

Fio-.

To determine Xj, take two points P and Q with the abscissae
'Xi and Xi on the curve in question on its ascending and descending
branches Avhere the vaUies of i/ are equal. Then it is obvious that

X.,

or

X} X- 1 — ■ X(\ •

m

Xq X]

Let M l.>e the intersection of FQ with the maximum ordinate <)f
the curve. Measure PJ/ and QM, whicli may be called i^, l^o, then

or

(^.)

Thus we can determine the vakie of x^, so that the position of
the origin is made known. Tlien we can calculate b" by the follow-
ing formula.

Ir = |Qgu.-^ -^Q^io^/., X 04343, (5)

(log,oa;-logioi'o)-

where x and y are the co-ordinates of F or Q.

Researches on the Dischar<^e of the Electric Organ. J'^

Siiieo the (kuioiuinator of tlie fraction of .t'o in (4) is very >snjall,
eiTors incurred in ineasui-ing ç, and ç.. influence very much the
value of À'o. Evidently the error «luo to ?-— <ri is made smaller, the
lower the points 7^ and Q are taken; Init near the foot of the curve
on its descending lu'anch, tliei'e exists prohahly secondary dis-
charge, and therefore we cannot take F and Q too low. It may
he shown tliat in the calculation of //', it is least affected hy the
error in //, caused l>y the error on the position of the .I'-axis, when
y is equal to Ac \ !^o also, for the determination of :i-j, we prefer
the points F and Q whose heights are equal to A(r\ Since it is
very difficult to accurately determine the position of the maximum
ordinate, and since the error in its determination influences the
difference ^.- ^i douhly, we must make full allowance for the exist-
ence of an eri'or of not insignificant importance in .To due to this
cause. To eliminate this error as far as possible, Ave take another
pair of points F' and Q' of equal heights and calculate x^ as above.
If the values of .i\, in the two cases do not coincide, let us assume
tliat the discre[)ancy is due solely to the error of the position of the
maximum ordinate, say o. Correcting for this amount and equa-
ting the \alues of Xj in the two cases, Ave haA'e

■'= WW^M ' ^'^

Avhere the terms of the second order are neglected.

1^'irst the agreement l^etAveen the theoretical and the experi-
mental curve Avas examined on Oscillogram No. 54 (Plate XIIL).
This Avas a series of experiments for flnding the relation betAveen the
magnitude of the stimulation cui'i'ent and the discharge in response.
The stinmli Avere direct and their durations Avere taken pretty long
in order to avoid the effect of small clianges in them.

The folloAving are the data of the experiment: —

Oscillogram No. 54.

Object of experiment: Eelation between tlie intensity of

the stimulation current and the
magnitude of tlie discharge.

18

Art. 1. K. Fuji:

Date of experiment :
Organ :

Nov. G, 1909.

Left organ (wliole) of a fish of mid-
dle size.
Temperature of the organ: 8-5° — 7-2°C.
Resistance of the organ: A])Oiit 240 olims.
Reading on tlie first rail of the stininlation-apparatus (circuit-
breaker): 64-0 mm.

Reading on the second rail of tlie stimulation-apparatus
(connection-changer): 60- 0 mm.

No. of stimulus
in order of time.

1

2

3

4

5

(j

7

8

Primary current of
induction coil in amp.

1-5

2-0

2-5

3-0

3-5

4-0

4-5

4-95

Number of revolutions
of registering- drum.

()-.51

6-38

6-50

C)-.53

6-40

6-18

6-00

6-36

The ordinates of tlie points on the curves were measured for
e\'ery one mm. of abscissa on the oscillogram, and they were
plotted on a section paper, the value of the abscissa being converted
to the unit of 10"'' sec. (Plate VI.). The following table shows
the constants obtained by applying the method a])ove described
on the reproduced curves: —

T.Vi-.LK I.

L. P. means latent period (ordinary usage).

M. L. P. means modal latent period.

Full details of calculations are given in the Appendi x.

No.

L. P.

in 10-1 sec.

M. L. P.

obs.
in 10-* sec.

0
in 10-* sec.

M. L. p.

corr.
in 10-* sec.

Ä

in mm.

//-'.

in 10-* sec.

1

76-8

107-5

Too lo^

?v to be

1-6

2

79-2

1064

measured.

50

3

73-9

107-8

+ 0-68

1085

101

4-60

592

4

71-9

1071

+ 1 12

108-2

17-3

4-69

59-8

5

73-6

1096

-0 10

109 5

26-7

563

64-2

6

69-7

108-5

+ 140

109-9

35-7

5 37

60-8

7

68-3

106 6

-M06

107-7

44 8

449

54-8

8

67-5

109 9

-1- 1-46

111-4

55-6

5-95

618

Researches on the Discharge of the Electric Organ. 19

Referring to tlie tiiblo we see tliat the value of tlie modal latent
period is toleral)l\' constant in spite of the wide changes of the
magnitude of the stimulus, while the latent period decreases with
the height of the discharge curve. The small increase of the modal
latent period may be due to the progressive decrease of tempera-
ture, as is given in the data of the experiment. In this case it is
a remarkable fact, that o is always positive except in Curve No. 5,
and the constancy of the modal latent periods corrected becomes
rather worse than that of the observed modal latent periods. The
values of .^•o and of <^" vary with the increase of the height of the
stimulus and the ^'alues of b" are equal to 5-0 in order of magnitude.
Here we ma^^ remark ; that it is very difhcult to determine accurate-
ly the value of .iVj, but b" may compensate for the error of Xq in
the final result. So we cannot put too nmch weight on the
changes of the value of Xq and of ^^ At anyrate we calculated the
values of 2/ for every one-thousandth of a secondi from, the values
of constants in the table, and plotted these calculated values on the
reproduced curves. These are shown in Plate VI. The agreement
between the calculated and the observed curves is veiy satisfactoiy
except for a small part on both feet of the curves, wehere the calcu-
lated points are in general lower than the observed. The discrep-
ancy on the descending foot ma}^ be due to the superposed
secondary discharge.

Next Oscillogram No. 37 (Plate XIII.) was examined. This
was a series of experiments for in\'estigating the relation between
the duration of the stimulating current and the magnitude of the
discharge. The durations of the stimuli were very small compared
with those of Oscillogram No. 54. The cur\'es on the oscillogram
were measured and reproduced in the same way as in No. 54
(Plate VIL).

In this series of experiments, the previous method for the
determination of a^o failed to give consistent values to x^ on account
of the smallness of ^-^—^v or, in other words, the values of Xo were
greater than those of No. 54. After many laborious trials, we at
last found that, when we took the origin of the discharge at the
beginning of the stimulus, the square root of the product of Xi and

20

Art. 1.— K. Fuji:

X.2 for any value of y, agreed very closely witli the observed modal
latent period. So assuming that the origin of discharge was at tlie
beginning of the stimulus, we calculated as in No. 54.

The following table shows the constants of tlie curves: —

Tai'.li: II.

Oscillogram No. o7. Left organ (whole) of a hsh, the same

preparation as No. o5 and No. oG.
Temperature of the organ: iTC.

M. L. P.

]Meaii

Xo.

obs.

of

/j-.

A

in 10-* sec.

VxiX.j.

in mm.

1

1230

1220

23 7

31-5

2

1242

1246

25 8

30-8

3

125-7

126 6

27 1

28 3

4

126-7

1274

27-4

27-5

5

125-4

1266

261

250

6

128-2

128-2

270

23-3

7

129-5

129-2

26 8

19.0

8

1275

1282

27-8

12-5

The data and the details of the ol>served ;ui(l the computed num-
bers given in the Appendix.

Here the modal latent pei-iod is tolerably constant as in No.
54, having a tendency to increase a little with the progress of the
experiment. The constant /r increases with the decrease of the
breadth of the stimulus ^"'or the height of the discharge.'"'

Calculating with these constants the values of y for every one-
thousandth of a second, and plotting the values obtained on the
reproduced curves, the agreement of the points is very satisfactory
as we may see in the figures (Plate VIL).

* For the sake of brevity, let us call the duration of the stimulating current the Ireadth
of stimulus, the maximum intensity of it the heùM of stimulus, and the maximum electromo-
tive force of the discharge the heinid of discharge.

Kesearches on the Discharfice of the Electric Ortjan.

21

Oscillogram No. 40 (Plate XVIL), whicli is the same kind of
experiment as No. Ill and in which the breadths of the stimuli are
very small, was examined and tlie constants were calculated as
in No. :]7 (Plate VIIL).

Tlie following is the table of the constants obtained: —

Taiîli: III.

()scillo2;rani No, 4<).

Left organ, the same prepara-
tion as No. :38.

Temperature of the organ: 11-5° C.

M. L. 1'.

M.Mll

Xu.

oJjs.

OÏ

b\

A

in 10-* sec.

Vxi'T.,.

in mm.

1

P21

121-G

200

44-5

'2

119

119-Ü

208

410

3

120

1204

20G

40 0

4

120

120-2

20 4

380

5

122

122 2

2b8

390

6

122

1228

23-2

33-5

7

122

1220

22 2

35-6

8

122

128 4

22-0

39.5

Tlie data and the details ol the observed and the com[)uted num-
bers are given in the Append! .\.

Here the modal latent period is constant again, and // seems
to increase with the decrease of the heiglit of the discharge. The
agreement between the calculated and the observed curves is
shown in Plate VIII. The values of // in No. .'>? and No. 40 are
equal to :^0 in order of magnitude.

Tlius having proved the close agreement of tlie theoretical
formula with the experimental curves, we shall next seek the source
of the discrepancy between the constants of No. 54 and those of No.
37 and No. 40. In the foregoing considerations, we did not take
into account the effect of the stimulus due to the break of the current

22

Art. 1.— K. Fuji

at all. The stimulus used in these experiments consisted of tlie
closing and the opemng-stimulus.'^ The stimulus on the break of
the current must be due to the recovery from some sort of
polarisation, though not limited to electrical polarisation, caused by
the current. It is not therefore unreasonable to suppose that,
the closer the opening-stimulus is to the closing-stimulus, the small-
er is the effect of the former. ]Moreover, when we consider the
existence of the refractory period, we may suppo'^e the effect of the
opening-stimulus neglected when the breadth of the stimulus is
sufficiently small. Comparing ( )scillogi'ams No. o7 and No. 40
with No. 54, we find that the breadths of the stimuli in No. 54
are gi'eater than those in No. 37 and in No. 40. Examining many
other oscillograms, we see that, where the breadth of the stimulus
is gi'eat, the unsymmetry of the curve is manifest. Hence we
may conclude that the anomaly in No. 54 must ])e due to tlie
supei*position of the effect of the opening-stimulus.

In No. 54 we see that, when we take the magnitude of the
latent period into consideration, the actual commencement of the
discharge by the opening-stimulus must be a little later than the
culmination of the curve. Hence assuming that the discharges
caused by the closing-stimuli had their origins at tlic begiiming of
the stimuli and the ascending branches of the curves were wholly
due to them, we calculated Ir h-om tlieir respective co-ordinates x-^
and 2/j. These values of h^ corresponding to each curve in No. 54
are represented in the following table. Let us denote sucli Ir \\\
hi

Table IV.

Xo.

in 10-* sec.

hi

Xo.

in 10-* sec.

hi

1

107-5

—

5

1096

188

2

106-4

—

(J

108-5

19-8

3

107-8

t208

i 7

106 0

24-3

4

107 1

'ill

109 9

'26-5

* In this paper, the stimulus on the growth of the stimulating current is called
the closing-stimulus, and the stimulus on the decay of the current is called the openhig-sthnuhis,
•whether the current is derived directly from the battery or is induced by an induction coil.

Eesearcîies on the Discharo^e of the Electric Organ. 23

Here we sec tliat tlie values of h^ tlius obtained agree in order
oi magnitude witii those of No. 37 and No. 40. To prove that our
consideration was correct, Ave took Curve No. 7 in No. 54, and
using tlie new constants we calculated the values of // for every
1/1000 sec. Plotting the calculated values on the experimental
curve, we see that they fall so very closely on the ascending
l)ranch. that even tlie disagi'cement previously observed at the
begiiming of the curve disappears; while the descending branches
separate from eacli other widely. The experimental and the
calculated curves are shown in Plate IX. Calculating the differ-
ences of tlie ordinates of points on the descending l)ranches of the two
curves foi- eveiy 4/10000 sec, we have drawn a residual curve
whicli would no doubt be caused by the opening-stimulus only.
The I'esidual curve thus found is plotted on the same Plate.
Having assumed the origin of the residual curve to be at the
instant of bieaking of the stimulating current and having found the
value corivsponding to .r„ by the formula Xq=^'j:iX.,, Ave calculated
the value of //' co]-i-es]ionding to the new curve, and we found that
it was e(jual in order of jnagnitude to that (jf the main curve. The
constants thus found for the i-esidual curve were .l=]0-0, />'"=:l7-2
and .(-,— 100-8. Calculating tlie values of // for eveiy 1/1000 sec.
by usiug these constants, and })lotting them on tlie residual curve,
we see that the agreement is rather Avonderful. The value of x^ of
this curve is a little smaller than that of the main curve, but if Ave
consider that tlie effective instant of the closing-stinnilus is a little
later tliaii the beginning of tlie stimulus, we have reason to suppose
that tlie modal latent ])eriods are equal.

Tables \. and VI. shoAV the constants oljtained on
Oscillograms No. <)2 and No. bo. Avhere broad stimuli are used.
Oscillograms No. (*)2. No. 03 and No. 04 (Plate XVI.) form a set of
experiments for finding tlie relation betAveen the intensity of the
stimulus and the magnitude of the discharge. The breadth of the
stimulus is greatest in No. 02, medium in No. 03 and smallest in
No. 04. The temperature of the organ Avas ]3-5°C.

24

Art. 1.— K. Fuji:

Tai'.li: \

No. (^2.

So.

■''o
in 10-* sec.

M. L. P.
in. 10- i sec.

i^

b,;.

A

in mm.

1

46-5

82-0

li-69

12.8

33-7

2

45-0

83-5

3-76

U'.i

2t') G

3

55-3

83-5

5-10

11-0

19-7

4

90-5

90-5

5 -96

ll:i

13-7

5

80-5

88-0

9-48

10-7

8-7

6

64-5

84-1

5-0.5

1-24

46

7

—

—

* —

2-3

1

Taule VI.

No. 03.

No.

in 10-1 sec.

M. L. r.

in 10-* sec.

b-.

M

A

in mm.

1

45-8

84-1

3-28

14-5

25-3

2

441

74-2

3-35

14-0

19-3

o
o

54-5

84-0

4-96

14-0

14-3

4

44-1

83-5

2-84

1.3-8

10-4

5

53-8

86 0

4-47

13-8

6 4

6

53-0

83-7

4-93

14-7

3-4 .

7
8
9

89-7

8!r5

12-4

14-7

1.5. .

^)8"8

830

2 -OS

14 1

24 4

lu tlK'so tublc.-; the nuinbLTs in tlic columns dcuotcd hy/v are
the values of Ir calculated ou the assuuiptiou that the origin of tlie
discharge is at the l)eginniug of the stimulus. Here again the values
of Z»" seem to vary witli the decrease of the height of the
stimulus. I)Ut tlie variation of //■ in No. 54. No. ()2 and No. Go
should l)e ap[)aivnt. and it shows that the efïect of the opening-

Researches on tlie Discharge of the Electric Organ. 25

stimulus is small when the closing-stimulus is very great or when
the opening-stimulus is very small. The increase of Ir in No. 37
and No. 40 with the decrease of the breadth of the stimulus Avould
show that the narrower the stimulus the smaller the effect of the
opening-stinuilus.

The last cui-ve in Plate VIII. shows that the formula inay he
applied to the discharge caused by an indirect stimulus, if a small
part 'of the organ is taken. The figure represents a curve re-
produced from Oscillogram No. C. 3. which is not shown in this
paper. The constants obtained by the ordinary way are : —

No. C. 3.

A = 55-4 mm.,
.f^ = 57G X 10~* sec,
Ir = Ö-83,
M. L. P. = 102-2x10-' sec.

The temperature of the organ was 10 5° C. The stimulus was
indirect and descending. From the order of magnitude of the
constants, Ave may suppose that the discharge is not simple. It
.seems to the author that the opening-stinuilus liad some effect on
the response curve.

From the altove discussion we lind that: —

i.) When the discharge is simple, the time-curve of the
electrcnnotive force of it may be represented by the formula

7/ = Ae~^'^^S'7,. where the origin of the discharge is at the stimulus.

ii.) The departure h'om the above law, as in No. 54, may be
considered to be apparent, the superposition of the discharge by
the opening-stimulus changing the values of the constants.

iii.) The modal latent period of a simple discharge remains
constant in spite of the change of the magnitude of the stimulus.

iv.) 44ie value of //' seems to vary a little with the height
of the discharge, but the relation is not clear on account of the
overlapping of the errors of the same order of magnitude.

V.) It is very doubtful whether the so-called latent period
has a definite meaning. It should be a function of the sensitivity
of the instrument l)v which it is determine»!.

26 Art. l.-K. Fuji :

V. Relation between the Form of Stimulus
and the Discharge Caused by It.

Tlic relation belwuen the form of the electric stimulus and the
excitation in a nerve caused hy it, has been investigated by many
physiologists. Some give a formula representing the relation
which contains a function of an unknown form. König' s formula
is such an example and is expressed by

i'=/F(/) :;;<;/,

0

where E denotes the excitation evoked, and / the instantaneous
value of the stimulating curi'ent at a time t which is measured from
the commencement of the stimulating current. .Vccording to him
F {t) is a function of an unknown form that acts as a decrement
factor and that has a finite value only when t is very small.

In 1892 Hoorweg,* in his experiments on the stimulation of a
condenser discharge, found a relation connecting the potential
difference, by which the condenser was charged, the resistance of
the conductor in the circuit, and the capacity of the condenser that
caused minimal shock in a nerve. He deduced from it a general
expression representing the relation betAveen the electric stinndus
and tlie excitation caused by it. His formula is expressed l)y

E

ajie-^'dt, \1)

where « and ß are constants. According to his opiniim the element-
ary excitation evoked in a nerve at an instant is proportional to
the strength of the current l at that instant, in opposition to the
prevailing opinion in whicli it depended on the rate of change of the

currents, e. ^; and as a decrement factor he introduced e~^'. Al-
though many authors opposed his formula, it is the only one that is
expressed by a definite function. He afterwards tried to deduce

* Plumer Archiv, Bd. 52, S. 87, 1892,

Kosfîarclu's on the Dischar"-c of the Electric Orjxan.

'J.i

his foi-mula irom tlic liypotlicsis of Neriist, but liis solution of tlie
(liffercntial equation does not satisfy the initial condition of uniform
concentration. The main «lefects of In's formula are: (1)
the definition, explaining how the excitation in a nerve is
measured, is very obscure; (2) since his formula is deduced from
the exj)erinîental data at a point of minimal excitation, it is not
possilde to extend it into the region of finite excitation.

In 1010 Hill,''^ following Nernst's hyj)othcsis and introdu-
cing the assumption of Lapicque, found excitation formulae in
several cases of electric stimuli. Since his consideration is based
on the theory of " all or none " first pi-oposcd l)v (Jotch, the
formulae have a somewhat different meaning from that of
Hoorweg. 'i'hey i-epresent the progression of a local change which
on attaining a definite value causes an actual excitation. At
anj'rate, in the case of a constant current, his formula contains the
exponential function as the term that varies with the duration of
stimulating current. According to the *' all or none '' theory the
magnitude of the response depends on the number of elementary
portions that receive a stimulus greater than the threshold
value to evoke the response. The number may depend on the
distribution of tlie current in a tissue or on the variety of the
elementary portions whose minimal stimuli are difïei'ent h-om one
another. In the former case, the relation between the electric
stimulus and its response reduces to a mere ])hysical probk-m.
and as in the latter case to that of some kind of probability.

Since it was considered that the discharge of an electric organ is
the best means for the investigation of such j)i'oblems, many experi-
mentsin these subjects were made. Let us here explain the superiori-
ty of the discharge as a means for the investigation of the genei'al
properties of excitation in tissues. The means ordinarily used are
muscular contraction and negative variation in a nerve or in a
muscle. In the former case very trouljlesome factoi-s of elasticity and
viscosity (if the latter term may be allowed) in a muscle complicate
the phenomena. On the contrary, the negative variation is an
ideal means for the purpose, but to obtain its record w^e are obliged

* The Journal of Phy.siolo<jy, V^ol. 40, p. 101, 1910.

28 Art. 1.— K. Fuji :

;it present to use siicli nii iiistruiiient ms tlie eapillary electrometer
or tlie string galvanometer. As we remarked at tlie beginning of
tins paper, snch an instrument does not give a true picture of the
electromotive force as in this case, unless the change of the
electromotive force is very slow as in a cardiogram. Some people
tr}^ to correct the curve obtained (»n these instruments by finding
the inclination of the tangent to the curve from point to point.
But since a small error in the measurement of the inclination may
cause a considera])ly great error in the correction, especially when

tlie inclination approaches -y, it is dangerous to rely on the

result obtained by analysing the curve thus corrected, even if it be
allowed that the ])rinciple of tbe correction is right. On the
contrary, tlie electromotive force of the discharge of an electric
organ is very lai'ge and therefore in this case we may employ such
an ingenious instrument as the oscillograph, in which, if used with
proper care, correction is unnecessary. ]\Ioreover the stimulus must
be very sti-ong for exciting the organ so that even a subminimal
stimulus may be recorded by the same insti'ument. It must,
however, be admitted that the record due to the abrupt change of
current is not so correct as that of the discharge. The only
diflficulty met with was our local trouble in obtaining lisli.

lieturning to the problem: we made many experiments
regarding tlie magnitude of the electric stimulus and the same of
the corresponding discliarge. The experiments were made in the
main with direct stimulation, and its form was that described in
§ III. As it Avas morj convenient to make the picture of the
stimulus on the oscillogram on the same side of the zero line as
that of the discharge, the direction of the stimulating current was
always homodrome. By changing the strength of the current of
the primary of the induction coil, or by shifting the position of the
circuit-breaker with regard to that of the connection-changer on the
stimulation-apparatus, stimuli having different strengths and equal
durations, or stimuli having different durations and equal strengths,
were given to the organ and the resulting discharges were in-
vestigated.

Eesearches on the Discharge of the Electric Organ. -29

Assuming that the opening-stimulus, when very near to the
closing-one, lias not an appreciable effect on the discharge curve, its
influence Avas not considered in the first plan of the experiment.
But in examining the discharge cur\-e \Qvy minutelj^ as in the
preceding section, we found that it may be influenced more or less
by the opening-stimulus. If that be the case, in the experiment
for the relation between the duration of the current and the magni-
tude of the corresponding discharge, the maximum electromotive
force of a discharge may, for two reasons, diminish with the
decrease of the duration on account of the opening-stimulus: (1)
since the opening-stimulation would be due to the i-ecovery from
some kind of polarisation, the magnitude of its effect may l)e due
to the duration of the current that flowed before; and (2) when a
stimulus occurs imnn-diately after another it has a smaller effect,
being influenced by the preceding one — a phenomenon to ))e
discussed latei". Though I Ijclieve, after the result of the
analysis of No. o7 and of No. 40 in the preceding section, that
the decrease of the maximum electromotive force is due mainly
to the decrease of the duration of the current of the closing-
stimulus, it is very difficult to know how much the opening-
stimulus affects the discharge-height. Therefore without making
any assumption, we shall now regard the set of our stimuli to be
a single stimulus as a whole and deal with its relation to the
coiTesponding response. In the preceding section we see that a
discharge curve has three parameters to characterise its form, L c.
A, b^ and ;ro. Since, as we see, the changes of bo and .Vo with
respect to the stimulus are not conspicuous, we shall use ^1 as the
measure of the excitation. The area of the curve, which, if we
assume the theory of '"all or none," would correspond to the
number of the electric plates discharged, may be found by
integrating the formula of the discharge curve i. e.

Area = I Ac~ ""^ -'ulx ^"^Ax,ß^^' (8)

0

Indeed it was tried to use the area as the measure of the excitation
in the reduction of some of our experiments, but it did not give a
result much differing from that of A in general. Therefore leaving

30

Art. 1.— K. Fuji:

this problem to more precise investigations, we shall now confine
ourselves to tlie discussion of the relation of A to the stimulus.

Tables in i-egard to many experiments for the stimuli with
equal l)rea(hhs and different heights are given next, and graj^hs re-
presenting the relation between the height of the stimulus and the
height of the discliarge are drawn in Plate X. Fig. 1, 2, 3.
Here we may remark thai a small error must ]je taken into account
for the height of the stimulus measured on the oscillogram, because
even our oscillograph does not follow such an al)rupt increase of
current as tliat observed in our stimulating cui'rent.

Table VII.

Oscillograms No. 35 and No. 3G (Plate XIV.).
Temperature of the organ: 1 ] -0° C.
Resistance of the organ: 130 ohms.

No.

Height of Stiui.
in mui.

Height of discharge
in mm.

No. 35.

No. 3Ü.

Mean.

No. 35.

Xo. 36.

Mean.

1

34-7

35-4

35T

17-0

17-3

17-2

2

41-2

41-7

41-5

25-0

24-0

24-5

3

47-0

47-2

47-1

31-0

30T

30-G

4

52-3

53-3

52-8

37*5

37-0

37-3

5

58-0

58-7

58-4

40-8

40-3

40-6

G

62-0

G3-3

62-7

43-3

44-5

43-9

7

68-5

69-0

G8-8

48-0

48-0

480

Table VIIL

Oscillogram No. 38 (Plate XV.).
Temperature of the organ: 11-5° C.

Xo.

Height of Stim.
in mm.

Height of dis-
charge in mm.

Xo.

Height of Stim.
in mm.

Height of dis-
charge in mm.

1

2
3

2G-8
33-2
39-2

11-0
20-8
32-8

4
5
G

45-7
51-5
G4-2

41-5
47-0

58-5

Kesearches on the Dischartrc of the Electric Orçfan.

Ol

Table IX.

Oscillograiiis No. 42 and No. 4o.
Temperature of the organ: 11 •5'' C.

No.

Height of stiiimlus.
in mui.

Height of dischargt
in mm.

No. 42.

Xu. 43.

Mean.

Xo. 42.

Xo. 43.

Mean.

1

12-5

13-0

13-0

2-5

2-5

2-5

2

19-5

19-5

19-5

3-5

3-5

3-5

o

»J

2Ö-0

2G-5

26-3

10-0

8-5

9-3

4

82-0

33-0

32-5

20-5

17-5

19-0

5

39-0

38-5

38-8

29-5

24-5

27-0

G

45-0

46-0

45-5

38-0

33-5

35-8

7

51-0

52-0

5l'5

43-7

40-5

421

8

58-0

58-5

58-3

48-0

48-8

48-4

9

G3-5

03-.5

()3-5

51-0

55-0

.53-0

Table X.

Oscillogram No. 54 (Plate XIII.).
Tempera tm-e of the organ: 8-5° — 7-2° C.

lîesistance of the organ;

240 ohms.

Xo.

Height of Stim.
in mm.

Height of discharge
in mm.

1

19-0

1-5

2

25-5

50

3

31-5

10-0

4

38-0

17-3

5

44-3

2G-5

6

50-2

35-8

7

5G-8

45-5

8

G4-5

5G-0

m-

Art. 1 . — K. Fuj i :

Table XL

Oscillograms No. 55 and No. 56 (Plate XV.).
Temperature of the organ: 104° C.
Kesistancc of the organ: j?40 ohms.

Heig-ht of Stiu

.

Height of discharge

No.

in mm.

m mm.

No. 55.

Xo. 5(3.

Moan.

No. 55.

No. 55.

Mean.

1

70-0

07-0

(')8-5

48-5

46-0

47-3

2

(33-0

(53-0

()3-0

39-0

40-0

39-5

:5

58-0

56-8

57-4

32-.5

32-5

32-5

4

51-5

50-5

51-0

22-5

23-7

23-1

5

45-0

44-5

44-8

15-5

J7-3

lG-4

6

88-5

37-5

38-0

9-5

10-5

10-0

7

—

32-0

—

—

G-2

—

8

25-0

25-5

25-3

0-8

2-5

1-8

9

68-0

(>G-0

(;7-o

45-5

4.5-7

45-()

Table XII.

Oscillograms No. 02, No. Go and No. 04 (Plate X\'I.).
Temperature of the organ: ].'*)-5" C
Resistance of tlie origan: P)0 ohms.

No

G2.

No

G3.

No

6i.

No.

Stim.

Discharge

Stim.

Discharge

Stim.

Discharge

m mui.

m mm.

m mm.

m mm.

m mm.

m mm.

1

63-7

33-7

62-3

2.5-3

63-3

24-0

2

57-7

26-G

56 "5

19-3

57-4

17-4

3

50-8

19-7

.50-3

14-3

50-6

12-5

4

44-3

13-7

44-4

10-4

45-C

9-2

5

38-7

8-7

38-5

6-4

38-2

5-5

6

31-7

4-6

31-4

3-4

3P8

3-2

7

25-6

2-3

25-4

P5

25-5

1-4

8

—

—

62-(3

24-4

63-5

22-5

Researches on the Discharge of the Electric Organ. 33

Here in general we used very broad stimuli to avoid the effect
due to the difference of their breadths. In No. 35 and No. 36 the
preparation of the organ was the same, and in the former case the
stimuli were given in the increasing order of height and in the latter
in the decreasing order so as to eliminate the progressive change in
the organ. Tlic two curves (Plate X., Fig. 1) representing them run
very closely to each other, the latter being somewhat lower than
the former. This shows that the progressive change, which might
be due to the decay of the organ or slow cooling of the same, was
very small during the experiments. The mean curve of No. 35 and
No. 36 is shown in Plato X., Fig. 2. This may be considered to be
the true course of the curve when there exists no progressive
change. Oscillograms No. 42 and No. 43 are of a set of experiments
of the same kind as the above. The preparation was the same as
that of No. 40 and No. 41, which were a set of experiments for
the investigation of the relation Ijetween the breadth of a stimulus
and the corresponding height of the discharge, and which will be
discussed later. The stimuli are in the increasing order of height
in No. 42, and in the reverse order in No. 43. Here w^e see that
the curves (Plate X.) rejoresenting them do not coincide; showing
the existence of a progressive change, so that the discharges
became relatively smaller with the course of the experiments.
Assuming that the progressive change Avas uniform, the mean curve
may represent the true course of the curve. Oscillograms No. 55
and No. 5(j (Plate X. ) are of experiments of the same kind. Here
we have a check on each film, which sIioavs that the effect of the
progressive change Avas very small. In the figure, the check points
are marked Avith ©. Oscillograms No. 38 and No. 54 have no check
for the progressive change (Plate X., Fig. 3). Oscillograms No. 62,
No. 63 and No. 64 (Plate X. , Fig. 3) relate to a series of experi-
ments in Avhich stimuli given Avere in the decreasing order Avith
respect to their heights, having a check on each film. The
breadths of the stimuli are different in t]ie three films, broadest in
No. 62, and narroAvest in No. 64.

Tracing the general course of these several curves Ave see that
they converge to the origin of the co-ordinates. The height of the

o^ Art. 1.— K. Fuji :

discharge increases very slowly from zero. Then comes a steep
increase of the height of the discharge, where the curve has an in-
flexion point and then becomes concave with respect to the axis of
the abscissa. By direct stimulations we could not arrive at the
upper part of the curve on account of the ^'cry large stimulation
current required. Nevertheless, it is very probable that the curve
approaches an asymptotic value of the ordinate wliich may easily be
attained in the case of indirect stimulation. Thus the curve
forms an S-shape analogous to that, found by Waller, relating to
the negative variation in, the nerve. Here we may remark that it
is very doubtful whether the words minimal und subminimal stimidus
as customarily used have any definite meaning. B,y the Avord
(minimal stimulus or liminal current, is certainly meant the stimulus
by which the response becomes just sensitive to a certain in-
strument. Of course, if Ave acknoAvledge the truth of the ' ' all or
none " theory, there certainly exist the minimal and the maximal
stimulus in the literal meaning. But that Avhich is ordinarily
obtained by experiment Avould not be the true minimal stimulus,
i. e. not for the response of a unit element. Noav, if Ave take the
height of the discharge as a measure of the excitation, the
proportionality between the current and the response in Hoor-
Aveg's formula does not hold good for the finite range of the
discharge. It is not clear whether the inclination of the tangent
to the curve at the origin is equal to zero or not. If the latter is
the case, the proportionality in the.ver}^ small portion at the
beginning of the curve, in agreement Avith the formula of Hoorweg,
is nothing but the general property of any curve, that a small
portion of it may be regarded as a straight line. ■Many trials Avere
made to formulate the relation extending to the finite region of
the stimulus, but the results Avere not satisfactory. As Ave re-
marked before, if aa^c may assume that the magnitude of a discharge
is due to the number of elements evoked by the stimulus, and that
the number depends on the A'^ariety of the elementary portions
Avhose minimal stimuli are different from one another, then the
curA^e representing the relation betAveen the strength of the stimulus
and the area of the discharge should be the integral curve of the

Eesearches on the Discharge of the Electric Organ. 35

category of a probability curve. Since I cannot ascertain the
i^roper method of anal3^sing the curve from tlie standpoint of
this hypothesis, I am not able to give the rigorous proof for it. But
it is obvious that, qualitatively, the course of the ex})erimental curve
agrees with this view.* I am nov/ working on this principle;
I only refer to it here, and sliall leave the problem for a later report.
Next we shall treat the relation between the Ijreadth of the
stimulus and the height of the discharge. With reference to the
problem, the fdm first examined was that of No. 37, an anah^sis of
whicli was made in the preceding section. Tlie organ-prep-
aration was the same as that of No. 35 and No. 3G, in which we
knew that the effect of the progressive change was very small.
Assuming Hoorweg's decrement factor to be true and not taking
into consideration the efïect of the opening-stimulus separately,
wliich Hoorweg, in his condenser investigation, also ignored, we
examined whether his factor held good in our case. According
to his formula the excitation E caused by our stimulus is
equal to

./

t

ai

0

where i represents the maximum current in the secondary of the
induction coil, A is a constant determined by the self-induction
and the resistance of the secondary circuit, and wliere e~^' means
Hoorweg's decrement factor. Of course, in our stimulus there is a
steep rising portion of the current before its maximum, which is
not considered here. But in treating of tlie variation only, the
part common to all stimuli has no influence on the result, if we
take the origin of the time at a proper instant. Hence measuring
tlie time from a suitable origin the formula becomes

which, for tlie sake of brevity, we write

* A trial examination was made on the curves of Oscillogram Xo. 5t. The curve represent-
ing the relation between the height of the stimulus and the area of the resulting discharge
curve was graphically diä'erentiated by driiwing the tangents to the curve, point by point,
and by finding the corresponding rates of increase of the ordinates with respect to the
abscissa. Then it was found that the differential curve might be expressed by a formula
analogous to that of the discharge curve given in the previous section.

36

Art. 1. — K. Fuji ;

E = B[\ -e-':{t-fo)] (9)

After laburious trials B, r and ^o were found from the plotted
curve of No. 37 (Plate XI., Fig. 1). As the breadth of the
stimulus, we took the interval Ijetween its beginning and the
instant of abrupt decrease of the current. Though our oscillo-
graph would not give a true picture for such a quick change of
current as our stimulus, it does not influence the value of the
stimulation-breadth taken as above. The result of the experiment
and the corresponding calculated values are given in the next
table, and represented graphically in Plate XL, Fig. 1.

Taijle XIII.
(Jscillogram No. .']7.
Temperature of the organ: llO'C.

Resistance of the organ: 130 ohms.

Formula used in calculation: // = 34 0 x (I -10-'-'''-^'- ■'■'>)

Xo.

M. L. r.

iu 10-'4 sec.

Breadth of

Stim.
in 10--* sec.

Heif:^ht

of discharge

obs.

in mm.

Height

of discharge

calc.

in mm.

Diff.

1

1242

18-2

31-5

;U-5

0 0

2

1227

161

30-8

302

+ 0-6

3

125-4

142

28-3

284

-01

4

126 7

130

27-5

26-9

+ 0-6

5

1271

11-8

250

25 0

00

6

1265

113

233

241

- 0-8

7

127-9

9 1

190

188

+ 0 2

<S

127-7

7-7

125

13 9

- 14

As we see in the table and in the figure, the agreement of the
observed with the calculated values is very satisfactory.

Oscillogram No. 58 (Plate XVII.) is an example in which
the breadths of the stimuli Avere too large, so that the heights of
tlie discharges showed a nearly constant value. In this oscillo-
gram, it may be seen that the height of the discharge never
becomes smaller with the increasing stimulation-breadth, even

Kosearclies on. the D'.seliarge of the Electric Org-an. 3y

when the beginning of the opening-discharge appear.-^ later
than the summit of the discharge curve. Tliis show» that the
variation of tlie discharge-height with respect to tlie stimulation-
breadth is not tlie effect of the opening-stimulus, but is due to the
variation of the duration of the closing-stimulation current.

Next we examined Oscillograms No. 40 and No. 41 (Plate
XVII.). These are a series of experiments on the same preparation
as No. 42 and No. 4o. In No. 40 the stimuli were given in
the decreasing order of Ijreadtli and in No. 41 in the reverse order.
For the breadth of the stimulus in this case, we took the breadth
on the bottom of the grapli, for the measurement as in No. :)7 was
very difficult to ascertain accurately. In this case, it was found
that the heights of the stimnli differed a little from one another.
Since the small differences in the heiglits of the stimuli exceeding-
l.y influence the heights of the discharges, the latter were reduced
into those corresponding to their respective stimuli having an
equal height 44d mm. The proportional constants in these
calculations were derived from the mean curve of No. 42 and No.
43. Here the stimulation-apparatus did not work well on account
of the contraction of the wooden part, by which the two rails were
bent a little. So the l)readth of the stimuli did not decrease
regularly. IMoreover, on plotting the points on a section paper, it
becauie clear that some progressive change had occurred during
the experiments, so that the points on the figure arranged them-
selves in a very complex manner. Therefore we assumed a fatigue
factor e~"% where n represents the number of the order of a dis-
charge, the first discharge being regarded as zero. The numbers
observed and calculated are given in the next table and the graph
is represented in Plate XL, Fig. 2.

Tahli: XIV.

Oscillograms No. 40 and No. 41.

Temperature of the organ : 11-5 °C.

Formula used in calculation:

No. 40, 7/ = 48 0xl0-3""«""'[l-10-»-3^-^^'-^î],
No. 41, y = 4G 9xlO-°-^°«°="[l-10-"-o^-^^'-°'].

38

Art. 1. — K. Fuji:

No. 40.

No.

M. L. P.
in 10-* sec.

Breadth of

Stirn,
in 10-4 sec.

Height

of discharge

obs.

in mm.

Height

of discharge

calc.

in mm.

Diff.

0

121

28 0

445

45 0

-0-5

1

119

184

39 9

39-7

+ 0-2

2

120

19 8

391

401

-10

3

120

174

37 6

37-8

-02

4

122

192

382

38 6

-04

5

122

15 0

33 5

34-6

-11

6

122

14 3

35 0

33-3

+ 1-7

7

122

11()

29 7

29 7

00

No. 41

No.

M. L. P.
in 10-1 sec.

Breadth of

Stim.
in 10" 4 sec.

Height

of discharge

obs.

in mm.

Height

of discharge

calc.

in mm.

Diff.

0

118

13 9

35 9

35 0

+ 09

1

119

131

33-7

336

+ 01

2

120

15-7

33 0

35 9

+ 01

3

119

170

36-7

36-6

+ 01

4

120

18-8

364

37-4

-10

5

125

17-5

386

360

+ 2-6

6

118

230

386

38-7

-01

7

120

22 4

38 4

37-9

+ 0-5

The details uf the data are given in the Appendix.

Next, Oscillograms No. 59, No. 60 and No. 61 are of a series of
experiments of the same kind made on the same organ, in which
tlie heights of the stimuli are different in the three films. In each
.'jeries, the experiments were in decreasing order witli respect to
the breadths of the stimuli. On each film, a check was taken at
the end of tlie experiments, and it was found that there Avere

Eesearchos on the Discharffe of the Electric Ortran.

39

progressive clianges. So, using e """ as a fatigue factor, in whicli
a' s were calculated from the checks, and finding the other
constants by laborious trials, the corresponding values of A were
calculated. These results an? given in the next tables (Plate XI.,
Fig. :y:-

Tai'.lk XV.

Oscillograms No. -19, No. 60 and No. 01. (Plate XVIII.).

Temperature of the oi'gan: 13-5 °C. g

Referring to the accompanyiug figure, AF'
in No. 50, AC \\\ No. GO and AF' in No. (H
were taken for the stinuilati()n-l)readths.

Fornuila usc<] in calculation:

in No. oi), 7/ = '21 3x 10-<=-033'""[l — lo-3-o^c"('--)],
in No. 00, 7/ = 3-d xlö-°°^-i"[I — 10-°-™s^<'--;],
in No. (')], y = 38 s x 10-'-°"i^"[l -lO-"-"«-^^'--^].

No. ol).

Fio-. 6.

No.

M. L P.

in 10 ~* sec.

Breadth of

Stim.
in 10-J sec.

Heio;ht

of discharge

obs.

in mm.

Height

of discharge

calc.

in mm.

Diff.

0

90 0

27-3

205

205

00

1

91 7

194

195

18 9

+ 06

'2

89 5

198

193

18-8

+ 0-5

3

91-2

19 3

181

185

-04

4

88 6

16-7

174

175

-01

5

881

149

170

165

+ 0 5

G

83 0

81

110

110

00

7

86 9

27 4

19 3

193

00

40

Art. 1.— K. Fuji

No. GO.

Xo.

M. L. P.
in 10-1 sec.

Broadth of

Stirn,
in lü-i sec.

Height

of discharge

obs.

in mm.

Height

of discharge

calc.

in mm.

Diff.

0

84-5

26-4

320

32 0

00

1

840

22 0

30-8

310

-02

2

82 9

190

297

30 0

-03

3
4
.5

809

173

29 0

29 0

00

87-4

171

27-5

27-4

+ 0-1

6

780

148

260

26-2

-0-2

7

835

271

26 3

26 4

-01

No. Gl.

No.

M. L. P.

in 10-1 sec.

Breadth of

Stim.
in 10-^ sec.

Height

of discharge

obs.

in mm.

Height

of discharge

calc.

in mm.

Diff.

0

831

29-4

38 0

38 0

0 0

1

795

21 2

36-9

361

+ 08

2

840

19 9

36 0

35-4

+ 06

3
4
5

81-5

181

34 G

34 3

+ 0 3

830

164

32 0

32 8

-0-8

G

85-0

13 8

30 4

30-4

00

7

792

25 8

36 0

35G

+ 04

111 the calculation, wc assumed that tlie values of Z^, are equal in the
three cases. But as we .^ee in the tables, the ineasurements of the
breadths Avere not ascertained in the same way. Hence the values
of /o must differ in these three cases. Indeed we attempted to find
the more correct values of B, r, i) and a by applying the method
of least squares to the residuals of the approximate values obtained
by the trial method, and to com])are the values of these constants

Researches on the Discharge of the Electric Oro-an.

41

for various licights of the stimuli. But tiio coefïicieuts of tlie normal
equations became very different in order of magnitude, and perliaps
for this reason, no significant values could be obtained. Hence
we abandoned these calculations, and we shall leave the problem
without any presumption upon tlie variations of the constants
witli regard to tlie height of tlie stimulus, until we shall have liad
an opportunity of obtaining more trustworthy results free from
i:)rogressive changes.

In short, except No. 37, the results of tlie experiments were
not perfect; but taking into consideration the difficulty of the
measurement of the small duration of the stimulus, and the influence
of the troublesome factor of the progressive change, these results
rather prove the correctness of the decrement factor of Hoorweg.

Next we shall give an experiment of indirect stimulation, in
which momentary stimuli of different heights were given. The
relation between the height of tlie stimulus and the height of the
corresponding discharge is given in the next tal»le and in Plate
X., Fig. 3.

Table XVI.

Oscihogram No. 40 (Plate XXVI IL).

Temperature of the organ: 8-2X'.

Additional resistance inserted in series: 200 ohms.

The stimuli given: Descending.

(The stimulus of Curve No. 1 was ascending by accident.)

Xu.

E. I'.

M. L. P.

Heio-ht
of Stiui.

Height
of discharge

in lo-* sec.

in 10~* sec.

m mm.

in mm.

1

142

173

4-0

9-0

2

> Too low for m

easurement.

G-0
7-0

0-5
1-0

4

IG'2

185

9-0

2-3

5

162

188

11-0

8-3

6

159

185

12-5

15-0

7

15Ü

189

13-0

17-0

8

156

189

14-0

18-3

42 Art. 1.- K. Fuji:

The course of the curve representing tlie relation between tlie
height of the stimnkis and the height of the discharge resembles
that of the direct stimulus. Here we may remark that the modal
latent periods have very consistent values except in Curves No. 1,
No. 2 and No. o, in which, since the curves are very low, the
ineasurements were very inaccurate.

In closing tliis section, tlie results of our experiments are
summarised as follows: —

i.) The height of the discharge A slowly increases with the
increasing intensity of the stimulation current from zero.

ii.) Then it increases very steeply, where it passes an inflexion
point; and then approaches an asymptotic value.

iii.) With respect to the duration of the stinuilating current,
Hoorweg's decrement factor seems to l)e correct, but it is not clear
whether the constant ß in it does really depend on the strength of
the stimulating current or not.

VI. Discharges by Two Successive Stimuli.

Ah'eady in one of the preliminary experiments, we discovered
that M'hen two successive stimuli not too far apart in time from
each other are given to tlie nerve, the discharge l)y the second
stimulus shows greater delay compared witli the normal. A pair
of momentary stimuli which were caused by a contrivance on the
shaft of the registering drum were sent periodically, i.e. once in
each revolution, to the nerve. At first only the preceding one of
the pair, then the succeeding, and finally the pair together were
sent througli the nerve. These were photographed on the same
film. The second stimulus was so given that, if the two discharges
appeared at the normal delay, the second discharge should appear in
a position superposed on the descending branch of the first. But
from the film we saw, on the contrary, that the second dischai'ge
occurred after the first had been over.

For the precise investigation of the phenomenon, we constructed
a special contrivance in our stimulation-apparatus, by means of
Avhich two consecutive momentary stimuli separated by a desired

Rosearchts on the Discharge of the Electric Organ. 4:3

interval of time may bo given. The details of the mechanism
were explained in § III. and illustrated in Plate V., Fig. 4. As for
the strength of the stimuli, care was always taken to obtain the
maximal discharges. Tlie sense of tlie stimulating current was
mainly descending, tlie distance of tlie two electrodes being about
o mm. This conies from the consideration that, in a descending
stimulus, the excitation in the nerve evoked by the closing-stimulus,
occurs at the cathode which is nearer to the organ than the anode
and therefore the excitation is evoked in an undisturbed part in the
nerve and propagates to the organ without any obstacle, while
the excitation caused by the opening-stimulus occurs at the anode
and it is arrested l)y the cathodic block from propagating to the
organ. (3n the contrary, if we use an ascending stimulus, not only
it has frequently been observed that the excitation due to
tlie closing stimulus propagates to the organ, though greatly en-
feebled in passing the anode; but the excitation by the opening-
stimulus must 1)6 affected by the stimulating current which flowed
before that instant. In short the result in the case of the descend-
ing current should be more simple.

Returning to the problem, the influence of the flrst
stimulus to the second consists of two parts: namely (1) the effect
on the modal latent period of the second discharge, (2) the effect on
the height of the second discharge. The former effect almost dis-
appears Avhen the second stimulus departs from its predecessor
about one-hundredth of a second, while the latter remains a little
later. These may be the effects of some kind of fatigue that re-
covers itself in a siuall fraction of a second. We shall call the
phenomena the temporary fatigue.

Oscillograms No. 74 and No. 75 (Plate XIX.) are a series of
experiments for such phenomena. Two stimuli, separated by
various intervals of time, wx^re given at a point on a nerve, and
the influences of the flrst stimulus on the sec(jnd discharge were
measured accurately. The following table shows the numliers
deduced from these oscillograms: —

44

Art. 1. — K. Fuji :

Tajîlk XVII.

Oscillügrams No. 74 and Nu. 75.
Temperature of llie organ: 14-5 °C.

No. 74.

No.

Interval

between

two Stim.

in 10-1 sec.

Ratio of
2n(l M. L. P.

to 1st.

Height of
1st discharge

in mm.

Height of
2nd discharge

in mm.

Ratio of

2nd height

to 1st;

1

120-5

1-000

53-0

39-0

0-73G

2

110-4

1-015

51-5

41-5

0-806

8

99-5

1-030

48-5

45-5

0 989

4

90-2

1-023

48-0

49-0

1-020

5

79-9

1-063

47-0

48-5

1-030

6

72-5

1-100

4G-0

4G-7

1-015

7

63-2

1-120

4G-0

45-5

0-990

8

.55-8

1-175

45-0

42-0

0-935

N

o. i o.

Xo.

Interval

l>etweeu

two Stim.

in 10-i sec.

Ratio of
2nd M. L. P.

to 1st.

Height of
1st discharge

in mm.

Height of
2nd discharge

in mm.

Ratio of

2nd height

to 1st.

9

43-9

1-301

45-0

24-7

0-550

10

33-6

1-411

43-3

4-5

0-104

11

34-7

1-411

41-5

4-5

0-109

12

27-4

—

42-5

9

—

13

14-9

—

41-8

9

—

14

9-1

—

41-2

—

— ■

15

3-0

—

42-5

—

—

IG

0-0

—

41-5

—

—

N. B. -.—In this paper, for the sake of brevity, the word latent -period or modal latent iwriod
is used to represent the interval between the stimulus and the commencement (observed) of
the discharge or between the stimulus and the instant of the maximum electromotive force
respectively, whether the stimulus is direct or indirect, /. e. the values of them are inclusive
of the time for the propagation of the nerve-excitation.

Eesearches on the Discharge of the Electric Organ. ^^

Full details of the data and tlie computed number^ are given in tlie
Appendix.

The numbers in the table are plotted in Plate XIL, Fig. 1.
Keferring to Curve 1, the abscissa represents the interval be-
tween two momentary stimuli in K) "* sec, and the ordinate
represents the ratio of the two jnodal latent periods subtracted by
one, L e. the excess of the delay of the second discharge with
respect to the first. Here we see that the ordinate decreases
exponentially with the nicrease of the abscissa. AVe assume the
curve to l)e expressed by exponential function ij = Mc''-\ and
determine the value of J/ and of ?., in which we may find M = 1-74,
X = 409. Calculating the values of y from the formula, and plot-
ting these numbers on the experimental curve, we see that they
agree very well within tlie limits of the experimental errors.

The other curve in the figure is that of the heights of the
second discharges. To eliminate the individual errors of each
experiment, the ratio of the height of the second discharge to that
of the first is taken as ordinate. Referring to the curve Ave see
that, even in 1/100 sec. from the first stimulus, the height of
the second discharge is smaller than the normal. Proceeding to-
wards the origin from tliis point it increases even greater than the
normal, and then decreases quickly to an almost imperceptible
height where the second stimulus enters the so-called refractoiy
period. The refractory period in this experiment is measured to be
32 Ox 10-* sec. In tliis experiment it is not certain whether in the
refractoiy peri(jd the response does not occur absolutely, or whether
some small responses continue to exist up to the origin of the C(j-
ordinates. Indeed there exist many small discharges that cannot
be distinguished from the discliarge of the higher order of the
preceding experiment. To clear this point further investigations
will be necessary.

The increase of the height of the second discharge towards the
origin seems to l)e paradoxical at first sight. But on examining
the film, we can find that when the second discharge is great, the
second stimulus is at an instant corresponding to a point on the
ascending branch of the first discharge, so that the second discharge

46

Art. 1.— K. Fuji:

overlaps the secondary discharge of the first stimukis. Hence Ave
see that the apparent increase of tlie height may be due to the
summation effect of the secondary discharge. To make this fact
clearer, Ave may refer to the following table: —

Table XVIIL

Oscillogram No. 74.

From the first set of curves the interval l)etAveen the first

stimulus and the summit of the secondary discharge may ]>e

found to be 193 2x10"* sec.

luterval between first stimulus

Xo.

Eatio of two heig-lits.

and summit of second
discharge, in 10"* sec.

1

0-73G

22G

2

080G

214

8

0 989

208

4

1020

201

5

1030

193

G

1015

191

7

0 990

183

S

0 935

185

In this table we see that tlie maximum ratio between the
height of the second discharge and that of the first, occurs Avhen
the summit of it just coincides in position witli that of the second-
ary discharge.

Oscillogram No. G5 (Plate XX.) is another series of experi-
ments of the same kind. The corresponding numbers are tabulated
in the next table and plotted in Plate XII., Fig. 2.

Eesearclies on the Discliarçre of the Electric Ororau.

47

Tai-.le XIX.

Oscillogram No. 05.

Temperature of the organ: lo-5 °C.

No.

Interval
between two

Stiui.
in 10-4 sec.

Ratio of
2nd M. L. P.

to 1st.

Height of
1st discharge.

in mm.

Height of
2nd discharge.

in mm.

Katio of

2nd height

to 1st.

1

133

1-790

24-5

9 0

0-367

2

—

—

22 6

17-8

0-788

3

110

0-970

21-5

19 0

0-885

4

97

1025

21-0

19-0

0-925

5

88

1-130

20-3

18-0

0-900

C)

—

—

200

15-0

0-7.50

7

69

1-230

19-5

16-0

0-820

8

60

1-300

19-9

11-0

0-553

9

49

1-480

18-8

3 9

0-207

The general course of the curve is similar to the preceding.
On the right liand side, the arrangement of the points is not so
regular as in No. 74 — No.?'), especially the position of the last
point in the figure deviates exceedingly from the others. But
on examining the oscillogram, we see that the second discharge
occurs after the secondary discharge of the first. Thus the second
stimulus was influenced l)y the first discharge which acted as a
preceding stimulus. Here we may find that M = 3 00, ;. = 376 and
the refractory period=4G-0x 10"* sec. The values of y calculated
from these constants are plotted on the figure. The interval of
time, in which the value of y l)ecomes one half, may be called the
2)eriod of recovery of the temporary fatigue. In the preceding two
experiments the period of recovery is equal to 17-0 x 10"* sec. in
the former and 18-5 x 10"* sec. in the latter, tlie temperature being
14-5 °C. and 13-5 V. respectively.

With respect to the height of the second discharge, we may
remark that it decreases with the decreasing interval l)etween the
two stimuli. But, if we take tlie ratio of the height of the second

48

Art. l.-K. Fuji:

discharge to that of tlio first, we can see that it iias a maximum
value as in No. 74 — No. 75. In this oscillogram the interval be-
tween the first stimulus and the summit of the secondary discharge
may be measured from the first set of the discharges, and as tliis
value we obtain '22o x 10"^ sec. Here we see again that tiie
maximum of the ratio of the height of the second discharge to
that of the first occurs Avhen its summit just coincides with that of
the secondary discharge. The relation is shown in the following
table : —

Table XX.

Oscillogram No. 05.

Interval between the first stimulus and the sunimit of tlie
secondary discharge: 223 x 10"* sec.

No.

Eatio of two heights.

Interval between first stimulus
and summit of second
discharge, in 10" * sec.

1

0-367

348

2

0-788

22G

3

0-885

229

4

0-925

222

5

0 900

225

(3

0-750

222

7

0-820

217

8

0-553

215

9

0207

220

Now since it was not plain to me whether tlie seat of tlie
phenomena is in tlie nerve or in the organ, an experiment Avas
made to determine the question. By giving a pair of stimuli with
a fixed interval between them, and by inserting proper resistance
in the discharge circuit, we carried out a series of experiments with
equal stimuli and difïerent heights of discharges. Changing the
magnitude of discharges by varying the resistance, the delay of the
second discharge with respect to the first was measured. The fol-
lowing table shows tlie results.

Researches on the Discharge af the Electric Orj^'an.

40

Table XXI.

0>cillügrani No. S4 (Plate XX.).

No.

Eesistanco inserted
in ohm.

Interval between
two Stim.
in 10-1 sec.

Ratio of two
M. L. P.

1

0

30-0

1-14

I.

2

200

30-6

1-16

3

300

30-2

I-I7

f 1

0

21-8

1-27

II.

2

200

24-6

1-24

3

300

24-9

1-25

I

0

15-4

1-46

III.

2

200

14-6

1-67

3

300

23-2

I-3G

In eacli set of tlie experiments, the ratios of the two modal
latent periods were found to be nearly equal. The disagi'eement
in tlie third set must have oecurred in virtue of the differences of the
interval between the stimuli as shown in t]ie table. So we liave
to conclude tliat the temporary fatigue, so far as it is defined by the
prolongation of the modal latent period, is not the result of the
discliarge.

In December 1901), Lucas ^"reported tlie same phenomenon in
a somewhat different way. According to his paper, experiments
were made on the negative variation in the sartorious muscle of the
frog. The instrument used by him was a capillaiy electrometer
and the curve ol)tained was corrected l)y the method of Burch. As
the time of reference he alwa_ys took the latent period /'. e. tlie
interval between the stimulus and the commencement of its re-
sponse. Even Avhen he measured the interval between the stimulus
and the summit of the response curve in his later paper, he sub-
tracted a constant value in order to obtain tlie latent period. For
my part, to measure on a curve the commencement of the response

The Journal of Physiology, Vol. 39, p. 331.

50 Art. 1.— K. Fuji:

is not only very inaccumto, but it is doubtful wlietlierthc so-called
latent period measured on a graph has any definite meaning. For
example the theoretical curve discussed in §IV. , which has its
commencement at the stimulus, may give some finite value for the
so-called latent period, if it is measured as is usually done on the
experimental curve. The response in a muscle or in a nerve may
differ in its property from the discharge of an electric organ, though
I am inclined to think that the form of the negative \'ariation de-
pends on a similar cause to ours. As a matter of fact, the modal
latent period so-called by us, gives more consistent values among
themselves than latent periods either in our case, as always observ-
ed before, or in the case of the negative variation in a muscle,
as was shown in the paper by Lucas. Whatever it may be, he
said in his j^aper: " It is found that the second electric response
begins at a constant time after the beginning of the first, whether
the stimulus by which it is pro\'oked occurs immediately after the
end of the refractory period or considerably later. If the second
stimulus occurs immediately after the end of the refractory period
the latency of the resulting electric response may be many times
the normal. As the second stimulus is made later the latency of
the resulting response becomes proportionately less. Only when
the second stimulus occurs so late that it would otherwise have a
latency less than the normal does the second response depart from
its fixed time of occurrence." He also shows that the nearer the
second stimulus approaches the end of the refractory period, the
smaller the resulting response becomes.

After this paper appeared, Gotch^ reported his analysis of a
similar phenemenon on the negative variation in the sciatic nerve of
the frog, and concluded that: "I. The electrical response of the
excised sciatic nerve of the frog to a second stimulus may show a
great increase in delay as compared Avith the response to the pre-
ceding stimulus. IL This increase of delay is augmented in pro-
portion as the second stimulus approaches the end of the period of
complete inexcitability (refractory period) which is developed

* The Journal of Physiology, Vol. 40, p. 250, 1910.

Reseatolies ou the Discliarwe of the Electric OriJ-an.

51

during the first response, but becomes imperceptible if the second
stimulus occm^s at a sufficient interval after the first." He report-
ed the efïect of temperature u\K)n the phenomenon and also gave
the discussion al)out the locality of this peculiarity. More recently
Lucas '"'reported his researches on the locahty of the same phenom-
enon by his ingenious method, and concluded that it is an after
effect of the disturbance propagating through a nerve or a nniscle.
In this paper he plotted many curves which represent the relation
between the interval of the two stimuli and the same from the first
stimulus to the connnencement of the second response, and not
only did he accept the behaviour of the delay of the second response
au;reeing with that of Gotch, so that he modified his first statement,
but he found that the interval between the first stimulus and the
second response becomes greater in some cases, when the second
stimulus is l)rought very near to the refractory period. Quoting
his words: "In the more complex cases with which I have
dealt, namely the excitation of nerve and the recording of the
consequent electric response in the innerviated muscle, it appears
that a new phenomenon inust be recognised; the electric response
becomes still later when the second stimulus is brought very near
to the first. This suggests an important difference associated with
the passage of the propagated disturbance through successive tissues
having unlike time relations." Now we shall reproduce here two
figures of such kind given in his paper.

.s

t1.

stz/

0.0}

-

/

-

Resp.2
/

0 005

:

/ /

(

/

' /

'^ \ 1 I

1

1111

1 1 1 !

o oos
E.rj}. 11.

The Journal of Physiology, Vol. 41, p. 63, li)10.

52

Art. 1— K. Fuji:

Here the ordinate represents tlie interval between the two stimuli,
and the abscissa the same between the first stimulus and the
commencement of the second response. The length of the abscissa
measured from the straight Hue inchned at 45° to the axis, repre-
sents the interval between the second stimulus and the commence-
ment of the second resj^onse. The curves have a minimum with
respect to the abscissa.

Now from the numbers given in the tables in Lucus's paper,
which are reproduced below, I constructed two curves representing
the relation between the interval from the second stimulus to tlie com-
mencement of the second response and tlie interval between the two
stimuli (Plate XII.). Then it was found that the newly reduced
carves were simply exponential in agreement with our results. Of
course, in this case, the ordinate shows the absolute value of the
so-called latent period instead of the ratio of the two modal latent
periods. But since the values were obtained by subtracting a con-
stant value which may be considered to be the interval Ijetween the
beginning of the response and its summit, the nature of the curve
must be the same as if it were drawn in our way.

The tables found in the paper of Lucas are given below.

Table XXII.

Exp. o and Exp. 1 1 l>y Lucas. The Journal of Physiology, Vol.

41, p. 337 and p. 391).

Exp. 3. 8ciatic-gastrocnemius preparation. Frog. Temp. ITöW

Obs.

Time first stiumhis
to second stimulus.

Time first stimulus
to second response.

Time second stimulus
to second resi^onse.

A.

•0010 sec.

•0081 sec.

•0071 sec.

B.

•0017

•0074

•0057

C.

•0025

•0064

•0039

D.

■0038

•0066

•0028

E.

•0051

•0081

•0030

F.

•0072

0092

■0020

Researches on the Discharo'e of the Electric Ori^au.

53

The times of tlio bogiiniing of the second responses are estimated
on the assumption that tlie response begins always at a fixed time
(0 0031) before it attains its maximum IM).

Ex. 11. (irastrocnimius-seiatie prc})aration. Frog. Temp. 17-5 °C.

B. Time of commencement of second response calculated from
time of its maximum P. ]). (as the interval between the connneuce-
ment of the curve and its sunnnit, ■{)i)'22 sec. is taken)

T. With second stimulus at same point as first.

Obs.

Time first stimulus
to second stimulus.

Time first stimulus
to beginning- of
second response.

Time second stimulus
to beginning of
second response.

A.
B.
C.

1).

•0021 sec.

■0030
■0042
•0067

■0069 sec.

•0070
•0074
■0093

•0048 sec.

•0040
•0032

•or>26

n. A\'itli second stimulus 11 mm. nearer to muscle than first.

Obs.

Time first stimulus
to second stimulus.

Time first stimulus
to beginning of
second response.

Time second stimulus
to beginning of
second response.

E.
F.
G.

H.

•0015 sec.

•0029
■0042
•0064

•0069 sec.

•0067
•0074
•0084

•0054 sec.

■0040
■0032
•0020

Correcting bv time of conduction •(•(>()(').

Oils.

Time first stimulus
to second stimulus.

Time first stimulus
to beginning of
second response.

Time second stimulus
to beginning of
second response.

E.
F.

G.
11.

•0009 sec.

•0023
•0036
•0058

•0069 sec.
•0067

•0074
■0084

(0060) sec.

(0045)
(■00^38)

(0026)

54 Art. 1.— K. Fuji :

The numbers enclosed in the brackets are calculated l)y the present
author.

From these data we calculated M and X. The vahies of the
normal latent period could not be discovered in his paper and we
therefore found them as follows: —

Taking thive ordinates //,, y-i and i/i on the figure (Plate XII.)
Avith equal intervals in succession, we put

//i = / + Ö,
?/._, = / + mh,

1/3= ^ + "i' ^.

where I is the normal latent period and m a constant to be de-
termined by the period of recovery. So in Exp. 3 we may find
7=0 0019^ and m=OàOr,, the interval between the successive ordinates
being 0 002^5 sec. From these values we obtain M =00098, A = 646,
The numbers calculated from the data are plotted on the curve by
the mark . (Plate XII., Fig. o and Fig. 4). In experiment 11 we
got / = 0 0016s and m = 0-48, the interval between the successive
ordinates being 0 0024 sec. and using these constants, we obtain
M = 0 0058, X = 300. Taking the value of 0 0031 in exp. 3 and of
0 0022 in Exp. 11 as the interval between the summit and the com-
mencement of the response, we may calculate the values of J/
and of / in the same units as in our case. These constants are
given below: —

Exp. 3. M = 2 25,

/ = 646,

Exp. 11. M = 2 66,

/ = 300.

In Plate XII. we sei; that the agreement ])etween the experi-
mental and the calculated curves is very good. From these results,
we may say that our exp(jnential law for the recovery of tho
temporary fatigue holds good in the results obtained by Eucas on
the gastrocnemius-sciatic preparation of the frog.

Now denoting the inter\'al l)etween the stimuli by (, and the
normal latent period l)y /, the interval between the first stimulus

Researches on the Discharge; of the Electric Organ. 55

and the commencement of the second response is given hy

t + lMe-'-' + l.
This has a minimum with respect to t at

t = -]]ogJ3Il

The vahies of t,„ at this minimum are evah-ated as follows:

Experiment hy Lucas:

No. 3. 0-0029 sec.
No. n. 00020

Experiment l)y tlie author:

No. 74-No. 75. 000495
No. 65. 0-00695

Thus we see tliat the existence of the minimum is a consequence
of the exponential property of the recovery, and therefore little
weight should he placed on this minimum. Since, as we see, the
minimum occurs very near to tlie end of the refractory period, it
would sometimes not be discoveix^d in an experimental result. It
is very proha])le then that the S(j-called irresponsible period report-
ed by Eucas in his first paper was an apparent phenomenon ap-
pearing for the reason that tlie observations were restricted in tlie
vicinity of this minimum, where the variation becomes zero.

The corresponding phenomenon U) the supernormal increase
of the height of the second discharge was o1)served by Samojloff*
on tiie electric response of a muscle indirectly stimulated. Kecently
Adrian and Lucast interpreted the phenomenon by the summation
of two successive disturbances propagating through a nerve-
ending, 'iliey considered that a propagated <listurbance whose
propagation is stopped at the nerve-ending l)y its decrement, if
alone, is successfully transmitted through the block, when the like
disturbance passed before. In our case, it seems rather probable
that thé supernormal magnitude of the second discharge is not the

* Arch. f. (Anat. u.) Physiol., Snppl-, 1908.

t The Journal Physiology, Vol. 44, p. 68, 1912.

56 ^t- 1'-^- ^^Jiî

direct effect of the distiir])ance evoked by the fir.-^t stimukis, l)ut it
is the effect of the first discliarge acting as a stimulus. In this
view we must assume tliat the maximal disturl)ance in a nerve-
trunk cannot cause the maximal discharge of the organ. The
possibility of the fact may ho considered in several Avays. For
instance, some elementary portions of an end-organ, say electric
plates or muscle fibres, cannot l)e excited by the disturbance
through a nerve, on account of the enormous decrement of the
nerve-endings of those individuals, or else the disturl)ances in some
nerve-fibres are too small to l)e transmitted to the end-organ for tlie
same cause. In our exjieriment, when we take a i>art of an organ
with a nerve-trunk attached, there certainly exist electric
plates which do not receive the supply of the nerve-tibres from the
trunk through whicli stimuli are gi\'en. Such plates cannot be
excited by stimulating the nerve-trunk, but can be excited by
the discharges of the electric plates belonging to that ner\'e-trnnk.
Then, when the discharge of such secondary nature is super}:>osed
on the discharge evoked l>y the second stimulus, the result
must be the increase of the magnitude of the second discliarge.
If we may allow that the electric response of muscle may act
as a stimulus, then the same interpretation may be appliedi n
the case of nuiscle.

Our experiments hitherto described were made in such cases
as the discharge evoked by the first stimulus had an asymptotic
values in practice. Without presuming the law of " all or none "
it is A'ery important to examine the case when the discharge is so-
called submaximal. We have three series of such experiments
that were obtained accidentally. These are Oscillograms No. 50,
No. 47 and No. 51.

In ( )scillograni No. 50 (Plate XXL), when two stimuli separated
by an interval 77 5 x 10"* sec. , the response l)y the second cannot
be observed. When they approach CG 3x Kj^Vec. , we see a low
discliarge scarcely distinguishable from the secondary. Tlien they
approach nearer /. e. to 55 0xl0~* sec. and the second discharge ap-
pears on the descending branch of the first discliarge. When they
approach each other still nearer /. e. 29 9 x 10"* sec, the second

Researches on the Discharge of the Electric Organ.

57

C'0ivi08 on the ascending branch of the first. At last wlien the
interval between the two beconx'S 164x10"^ sec, the discharges
coalesce into a single one. Here since the position of the second
summit would be displaced on account of its lying on the inclined
branch of the hrst, we cannot measure tJie modal latent period
accurately and therefore we also measured the interval l)etween
the stimulus and the commencement of the curve. The next table
shows the numbers obtained on the oscillogram.

Tatjle XXIII.

Oscillogram No. ">(,).
Temperature of the organ:

8-2 v.

Xo.

Interval

between Stim.

in 10-' sec.

L. P. of

1st discharge

in umi.

L. P. of

2nd discharge
in mm.

Ratio of
two L. P.

1

77-5

69 0

Xo response.

2

G6 3

70 0

715

102

3

55 0

710

70 5

099

4

442

710

65 0

0-92

5

29-9

70 0

650

0 93

G

7

16 4

5-S

65 0

Coalesce into f

I single curve.

Xo.
1

2
3
4

5
(•)

7

M. L. P. of

Isb discharge
in mm.

810
820
82 0
79 0

M. L. P. of

2nd discharge
in mm.

78 0

75-5

Ratio of
two M. L. P.

0 95

0-96

58 Art. l.-K. Fuji :

As a mere experimental result, it seems that l)efui'e the so-
called refractory period there exists a period in wliieli the response
may occur. The modal latent period or the latent period of the
second discharge seems to decrease even to a smaller value than
that of the first for some value of the interval between the two
stimuli. Indeed the shortening of the latent period from the
normal is equal to (3 nnn. in the maximum length on the oscillo-
gram. But in other experiments we know that, while the modal
latent period gives very consistent values, the latent periods differ
widely according to the height of the curve. It will not therefore
be useless to calculate roughly the value of the displacement of the
summit of the second discharge curve on account of its lying on
the descending branch of the first. Suppose a discharge curve

y = Ae~^''^^^''Tj superposed on an inclined straight line

ij = (p — x)ta,n(f.

Then the resultant curve is expressed by

y = (p-x)tan<f + Ae-^''^''^''X'

Differentiating the expression with respect to x, and equating the
result to zero, we have

t ■'■0 J^ ,

A value of .i- satisfying the equation must be that for tlie maximum
value of //.

Let tho displacement of the summit l)e represented l)y o,
which, in the first approximation, may be considered very small
when compared witli .^o. l^utting x=^Xa—d in the above expression

and neglecting the higher orders of — than the first, we have

*- Xq

^n^tan^

2.46'-' ^ ^

Taking the third experiment in the oscillogram No. öO, we may
put very roughly

Researches on the Discharge of the Electric Oro-an.

59

and

tan

Xq = '29 ram.,

6-' = 10 7,

A = 4 mm.
15

10

at most,

wliere tlic value of .I'oaiid oi' /rare detenuined from the first expcri-
meiit. From those data we may get 14 mm. as the value of o.
Referring to the table, the difference of tlie two apparent modal
latent periods in that ex])eriment is equal to 4() mm. So giving
full allowance to the effect of o, tlie (hfference needs not be smaller
than 2 6 mm. The value cannot be considered to be an error of
measurement.

Oscillogram No. 47 (Plate XXI.) is another example of the
same kind, and tlie numbers obtained on this film are tabulated
in the next tal)le.

Tadle XXIV.

Oscillogram No. 47.

Temperature of the organ: 8-5 °C,

No.

Interval of
two Stim.
in 10-' sec.

M. L. P. of

1st discharge

in 10-* sec.

M. L. P. of

2nd discharge

in 10-4 sec.

Height of

1st discharge

in mm.

1

79 1

174

233

310

2

G9 4

171

235

31-2

3

58-1

172

248

30-8

4

450

175

—

29 0

5

87 0

174

—

29 5

G

28-9

173

188?

27 0

7

8-3

176

—

33-2

8

00

—

—

332

Here the modal latent period of the first discharge is toler-
ably constant. In the first three, the modal latent period of the
second discharge increases a little witli the decrease of the interval

60 Art. 1 -K. Fuji.

l)etwL'on tliu two stimuli. When tlio interval l)ocoine.s 45 0x10-*
sec. the second discharge seems to disappear. When it is di-
minished to 37GxlO-* sec. the second discharge becomes percep-
tible on the descending Ijranch of the first discharge, but the
modal latent period cannot l)e measured, although it is certain that
it is made smaller. AVhen the interval Ijecomes 28-9x10"* sec. the
modal latent period of the superposed second discharge ma_y rough-
ly be determined to be 190x10"* sec. Finally, the interval becom-
ing 8 3x10"* sec, the two discharges coalesce into a single one.
From this onwards, the height of the first discharge becomes greater
than those of the preceding, thus showing the summation of the
second discharge.

Oscillogram N(.». 51 (Plate XXI.) is a series of the experinients
of the same type. Here we may find as the ratios of the second
modal latent period to the first the values of lOO, 0-98, 0 95 and
0-93 for 4th, 5th, Gtli and 7th curve respectivel}^

Let us now discuss the results with reference to the theory
of "all or none.'' As the stinuilus is sul)maximal, a })art of the nerve
fibres is excited by the first stimulus and the other part is not excit-
ed, but only receiving the sul)minimal stinuilus. Let us call the
former the first part and the latter the second part. ^Vhen the
second stimulus, whicli is also subminimal to the second part, is in
the refractory j^eriod and on the outside of the summation Interval"
so-called l)y Lucas, which must be assumed to be smaller than the
refractory period, the second response cannot occur. But when
the second stimulus enters into the summation interval it excites
the second part in concordance with the first stimulus, while it
does not excite the first part. Hence the experimental results can
l)e explained. For the correctness of this interpretation, it must
be allowed that the refractory period in this case is greater than
775 X 10"* sec. and the summation interval is greater than 66-3 x 10"*
sec. The latter numl )er is \-ery great compared with the same measur-
ed I )y Lucas on the motor nerve to the gastrocnemius muscle of
the frog which was equal to 5 x ]0-*sec. AMiether the modal latent

Tho .Journal of Physiology, Vol. 39. p. 461.

Kosearchcs on the Dischartre of tho Electric Orran.

61

period, nieasfured iroin the second stiuiulus, beluDgiiig to two
subminimal stimuli may take a value smaller than the normal or
not is a question which appeared on Oscillograms No. 50 and 51,
and wortli wliile investigating hereafter.

Next we sliall give an example of summation of two dis-
charges in response to a closing and an opening-stimulus (Plate
XXIL, Oscillogram No. 70). Stimuli of the same type as those
used in tlie direct stimulation were given to a nerve-trunk with a
proper shunt, by whicli tlie stimulation current through the nerve
remains submaximal and yet the current flowing through the
oscillograph was sufficiently large to ti-ace its form. The experi-
ment was made for another purpose /. e. for the comparison of
the intensity of the stimulating current and the magnitude of the
resulting discharge. At flrst the stimulating current was increased
step by step for five successive experiments. Then the duration
of the stimulation current was changed, and commencing with the
smaller current, it was increased step by step in the five succeeding
experiments in an exactly similar way. In the latter half, the
forms of stimuli were not good, and we have not therefore used
them in the analysis. The following tal:)le shows the numbers
measured on the oscillourram.

Table XXV.

Oscillogram No. 70.

Xo.

Height of

Stiin.

in mm.

L. V.

of 1st

discharge

in 10-^sec.

I.. P.

of 2na
discharg-e
in lO-^sec.

Ratio.

M. L. P.

of 1st
discharge
in 10~^sec.

M. L. P.

of 2nd
discharge
in 10-^ sec.

Ratio.

1

110

96-5

101-8

1055

113-9

1190

1045

2

23 3

880

101-5

1150

1120

1152

1030

3

35-5

81-3

1030

1270

109-8

111-5

1015

4

470

79 0

100 2

1-270

1085

108-5

1000

5

Out of film.

79 5

102 5

1-290

1090

1090

1-000

Since the duration of the stimulating current did not exceed
21x10""^ sec, it must l)e allowed that the opening-stimulus
had occurred before tlie end of the refractory period due to the

Ü2

Art. 1. — K. Fuji

closing-stimulus, and therefore the response to the Ojoening-
stimulus must be due to the nerve-fibres that were not excited by
the closing-stimulus. Referring to Oscillogram No. 70, we see
that when the intensity of the stimulating current is very weak,
the discharge by the opening stimulus is higher than that Ijy
closing. When the intensity of the stimulating current is made
larger, the discharge due to the closing-stimulus becomes larger while
the same due to the opening-stimulus becomes first larger and then
smaller. Putting the base for tlie explanation of this phenomenon
npDu the theory of "all or none," when the stimulating
current is sufficiently strong so that all the plates are discharged
by the set of two stimuli, the area of the superposed resultant
curve, which corresponds to the number of the plates discharged,'^'"
must be constant. In this view we measured the values propor-
tional to the areas of the curves on the oscillogram by weighing
the pictures cut out from bromide paper, and then multiplying
each of them by its respecti\'e inter^•al corresponding to one-
millimetre of the abscissa, we got the values proportional to the
number of the plates discharged. These values are given in the
next table.

Table XXVI.

No

Weight of

picture cut out,

in mgr.

I unu. of aljscissa

corresponds to,

in lu '* sec.

Product.

1

324

345

112

2

41-3

3 39

140

3

44 1

3-38

149

4

454

329

149

5

4.5-8

331

151

* When we take a small part of the organ as in our experiments, the resistances of all in-
dividual columns may be considered to be equal to one another. Let the resistance of one column

be denoted by r, the current flowing through each of the columns at an instant by ii, fj, i- i,«

respectively, the resistance of the external conductor by B, and the current flowing through it
by /. Let the electromotive force of an elementary discharge of a plate be e, and the number of

Researclies on the Discharge of the Electric Orgau. Q3

The above table sliows that the area of the discharge curve be-
comes constant when the stimulus is made strong enough. While
the experimental result gives support for the theory of "all or
none,'' it seems to reveal two new facts : (1) the opening-stimulus
can evoke its response, which could not be caused on closing the
same stimulating current, or, otherwise, the opening-stimulus may
be summed up into something that occurred at the anode on
closing tlie current : ('2) the delay of the second discharge is longer
than the normal. The cause of the second phenomenon is not
clear. It may be seen that the modal latent period measured on
the oscillogram seems rather to have normal value i.e. not prolonged.
But since the second summit lies upon the inclined branch of the
first discharge curve, the displacement of the apparent summit
towards the stimulus is not small. On calculating the displacement

bv the formula <J= t .... . it mav l)e found that the disiilacement
is not smaller than 2 nun. on the film. Therefore the prolongation
of the modal latent period of the discharge by opening-stimulus
seems certainly to exist.

Before closing this section we ha\-e tw(» more oscillograms of
allied problem to be explained. Since I thought that the rhythmic
response of a ner\'e or of a muscle to high frequency stimuli,
might be the effect of the refractory period, a few experi-
ments were made regarding that subject. (!)scillograms No. ()7 and
No. 85 (Plate XXIII.) represent the results obtained. In No. 67
the make and break of the high frequency of the primary circuit of
an induction coil was made by a contrivance like the commutator

the plates dischargiag per second at an instant in each column be ?ii, ?*_•, «■,, n,,» respectively.

Then by Kirchhofes law, we have

cnie = Rl+rii,

c)u>e = RI+ri.2,

£nie = IiI+rto_,
cn„ie = El + ri„„

where l = ii + ù+i:+ + im, and c is a proportional constant.

By adding each side of the equations, we have

ce'^n,u = {Ji + r) I,
1

and therefore / iI/(„/7f = — ^ / LU.

i. e. the total number of the plates dischar-^ed is proportion il to the area of the curve.

64 Art. 1.— K. Fuji:

of a dynamo, and the nerve-trunk attached to the organ was
stimulated by the induced current. For this and the other ends,
our shutter was so constructed that, wlien we made proper connec-
tions, the stimulating current could flow only when the shuttei'
opened (P^ate IV., c). The number of the momentary stimuli
given amounts to 885 per sec. In this case the stimuli varied a
little .in their strength, and hence another experiment No. 85
was made in which the induction coil was not used. Here,
as stimulus, a current from eight accumulators was interrupted
by the same contrivance as No. 07, no induction coil being
used. The stimuli were regular, and tlieir number amounted to
1412 per sec. As may be seen in the oscillograms, tlie dis-
charges were not so regular as to account for one determinate
l)eriod, being a series of high and Ioav superposed discharges. This
nn'ght be the case because the stimuli were not maximal, though
in the beginning part it seems to be so. From many other experi-
ments it seems to me, that tlie liminal value of the stimulus for the
maximal response in tlie nerve increases with the temporary
fatigue, and by this assumption the irregularity of the discharges
may be easil}^ interpreted. Hence assuming that the stimuli were
sufficiently strong at the beginning, we may take as the period of
response, the inter\^al between the first and the second discharge,
and we get as its value ]17x]0~*sec. in No. S6, the temperature
of the organ being 15°C. For the more advanced discussion the
more precise analysis should l)e necessary, and hence we here
allude to this as a mere experimental result.

The summary of this section is : —

i.) A\'hen two successive stimuli separated l»y an interval a
little greater than the refractory period are given at a point in a
nerve of the electric organ, the modal latent period of the second
discharge is prolonged with regard to the normal.

ii. ) The recovery of this prolongation follows an exponential
law with respect to the intei'val between the two stimuli.

iii.) The prolongation of the modal latent period of the
second discharge is accompanied by the variation of the maximum
electromotive force of the second discharge.

Kesoarches on the Discharge of the Electric Organ. ß5

iv.) The electromotive force increases probably from zero
when the second stimulus separates more and more from the end
of the refractory period and when at a certain interv^al between the
two stimuli it Ijecomes even greater tlian tlie normal and then
decreases again.

V.) The abnormal increase of the electromotive force may be
interpreted as the summation effect of the second discharge and
the secondary discharge of the first stimulus.

vi.) The second discharge caused by two successive submaxi-
mal stimuli given at a point in the nerve may be superposed on
the first discharge, and the modal latent period of the second
discharge seems to be even smaller than the normal at a certain
interval between the two stimuli.

vii.) The discharge l)y an oi)ening-stimulus given in the
nerve may l)e superposed on that by closing-stimulus.

viii.) Phenomena vi. and vii. may give support for the
theory of "all or none," but for this we must assume new
subordinate properties ef the nerve which is worth while investi-
gating hereafter.

VII. Fatigue phenomenon.

Fatigue phenomenon was investigated in two different ways.

(1) The registering drum Avas rotated very slowly by connect-
ing the shaft of it with the clockwork of a kymograph, and succes-
sive momentary stimuli about 25 in a second were given to the
nerve and the deviation oi the strip of the oscillograph was photo-
graphed on a film. The periphery speed of the drum being about
1 cm. per sec. , a discharge curve reduces to a straight line and the
locus of its summit forms the so-called fatigue curve.

(2) The drum was rotated rapidly and two stimuli in each
revolution were given to the nerve so that tliey might be photo-
graphed at two fixed places on the film. The stimulus consists of
the induced current of an induction coil, which was caused by
the make and break of the primary current flowing for a very
short interval during which a peg on the shaft makes instantaneous
contact with a stationary conductor (Plate IV. , Fig. 1). The direc-

(3(5 Art. 1.— K. Fuji :

tion ()1 the 8tiinulatiiig current was in general descending on
closing tlie primary circuit. But in one case the ascending and
the descending stimuli were alternately gi\'en to the nerve by a
proper contrivance on the shaft of the registering ch-u m (Plate IV.,
Fig 3). In another case the stimuli were given directly to tlie
organ. Here, as remarked at the outset of this paper, the in(hic-
tion coil must be excluded from the circuit during the dis-
charge of the organ. This was accomplished by a proper contri-
vance lielonging to the shaft of the registering drum (Plate IV.,

Fig- '^) . . .

Beginning with the former case, the fatigue curve resembles

very closely that of muscle. Oscillograms No. 1, No. 2 (Plate
XXIV.), No. 3, No. 4 and No. 5 (Plate XXV.) show such curves.
In No. 1, which shows the typical form of the fatigue curve, the
electromotive force increases a little during a few discharges at the
outset of the fatigue, and passing over a maximum, it decreases
almost exponentially. The rate of decrease increases after pass-
ing the maximum, then decreases gradually, again increases
a little an<l then decreases again. Thus repeating the same
type of variation, in such a way that the locus of the top of the
discharge curve forms a mild wave form, the nerve or the organ is
tired out. Two low loci on the Ijottom of the oscillogram are
those of the secondary and of the tertiary discharge. The marks
of regular intervals on the bottom ai'e those of a second. Here
we may remark that the tops of the discharge curves form a
smooth curve at the beginning of the fatigue, but when in deep
fatigue their lieights become very irregular (Oscillogram No. 2).

Oscillograms No. 3, No. 4 and No. 5 (Plate XXV.) form a
series of experiments. Oscillogram No. 4 was taken at an interval
of one hour after No. 3, and Oscillogram No. 5 at 40 min. after No. 4.
The number of stimuli is about 25 per sec. Here we see, that the
fatigue may be partially recovered by a repose of the stimulation
so that there must be two kinds of fatigue, namely that which can
be recovered and that which cannot be recovered. I am of
opinion that, that part of the fatigue which can be recovered is
the same as the temporary fatigue of tlie nerve discussed in the

Eesearches on the Discharge of the Electric Organ. G7

previous section, and tliat whicli cannot l»e recovered is due to
the alteration in the organ or in the nerve. Of course, the interval
hetween two consecutive stimuh is ^'ery large compared with the
period of recovery discussed in the former section, hut there is still
a fatigue, for not only small residuals of tlie fatigue may accumulate
in such cases, hut also a part of the nerve trunk may still he in
deep fatigue on account of the stimulation hy tlie discliarge of the
higher oi'der, when a succeeding stimulus is given. As we re-
marked in the preceding section, when the fatigue advances, the
hminal vahie for the maximal stinudus increases in nerve, so that
the stimulus of the given magnitude, which is sufficiently maximal
at the outset of the fatigue, hecomes submaximal, and a little
N'ariation in its magnitude may cause the discharge of a difïerent
magnitude. May not this l)e the cause of the irregularity of the
discharge in the tired state ?

In regard to the increase of the contraction at the outset of
fatigue in nuiscle, Fröhlich interprets the phenomenon as the ap-
parent efïect of the prolongation of the wa\'e length of the contraction
propagating through a muscle. He thinks that, the fact that the
same phenomenon could not l)e observed in negative A'ariation,
gives support to his theory. lUit since a similar interpretation
fails to apply in our case, it is i)robal»ly incorrect. In view of our
experimental results, it seems to me that the phenomenon is
analogous to the abnormal increase of the electromotive force of
the second discharge discussed in the preceding section. In that
section, we explained the phenomenon as the efïect of the super-
position of the second discharge on the secondary discharge. Here,
of course, the principal discharge evoked by the succeeding
stimulus is very far from the secondar}^ discharge, l)ut it may be
superposed with a discharge of one of the liigher orders. To
express my idea clearly, suppose a discharge followed by a series
of higher orders, to be CN'oked by one of the stimuli separated
by a fixed interval of time as in the experiment. When the
fatigue advances, the modal latent period is prolonged gradually, as
may be observed in tlie fatigue investigations of the other form,
and the secondary, the tertiary discharge et seq. delay in propor-

(33 Art. l.-K. Fuji:

tioii tu tlic iutervcils butwcen tliose and tho stimulas, so tluit tlie
discharges of tlie higlier orders chase the succeeding stimulus. At
•the instant at which one of the discharges of the higher orders just
comes at the j^rincipal discharge due to the succeeding stinuüus,
the height of the discharge must be increased. By this consider-
ation not only tlie tirst maximum of the fatigue curve may be
explained, Ijut its wave form may ho interpreted also. In the
case of muscle, ability for response of the higher order may l)e
observed in the so-called tetanic state which, in my opinion, is
caused by a series of responses successively evoked by the preced-
ing electric response acting as a stimulus on the tissue in a highly
excitable state. This view may be supported Ijy the fact that the
decrease of the proper period Avith the rise of temperatures follows
the same law as that of the increase of the speed of propagation
through the nerve. ''^ Then may not the same interpretation be
applied to the case of muscle ? Concerning this point further
investigation should be necessary. Here we describe this as an
interpretation wliich seems to he true in so far as our experiment
goes.

Next we shall explain another form of fatigue oscillogram.
Oscillograms No. (*), No. 78 and No. 78' (Plate XXVI.) show such
experiments. From these oscillograms we see that, when the
fatigue advances, tlie modal latent period is prolonged and the
lieight of the discharge curve decreases. Oscillogram No. 7 (Plate
XXVII.) is an example in which the discharges appeared in a
. very complex manner. Perhaps the discharge was affected l;)y
the tertiary discharge belonging to the preceding set, but no
satisfactory explanation has l)een obtained. The following
, explanation looks somewhat prol)al)le. The discharges due to the
opening-stimuli at tirst appeared, then those due to the closing-
stimuli were superposed on their ascending branches, and finally in
the advanced stage of the fatigue, only those by the closing-stimuli
remained. For tlie sake of correctness of this consideration, it
must l)e assumed that the modal latent period belonging to the open-

jDittler and Tichomirow. Pflüger Archiv, Bd. 125, S. 117.
""'1 Dittler and Oinuma. Pflüger Archiv, Bd. 139, S. 293.

Researches on the Discharj^-o of the Electric Organ. 69

ing-stiiiiulus was greater than the iiorinaL In tlie.-e experiments
tlie direction of the stimnlating current was generally descending
on closing the primary circuit, ])ut in the above case the stimulat-
ing current was ascending. If tlie above phenomenon may be
explained by Pflüger' s contraction law in tlie muscle-nerve prepara-
tion, it must l)e recognized that tlie value of the maximal stimulus
with respect" to the nerve increases with the progression of the
fatigue and hence the same stimulating current will be strong at
first, then it will become medial and finally weak with respect to
the nerve. If the above explanations l)e correct, here again we
liave evidence that the value of the maximal stimulus with respect
to the nerve increases with the progression of the fatigue.

Oscillogram No. <S (Plate XXVI.) shows the case in which
the stimulating currents are ascending and descending alternately.
This oscillogram shows that the fatigue phenomenon is not tlie
consequence of the polarisation at the electrodes. Oscillogram?;
No. 78 and No. 78' (Plate XXVI.) show the fatigue by the
descending and by the ascending stimuli respecti\'ely.

Oscillogram No. 1) (Plate XXVII.) is an example of the fatigue
phenomenon l)y the direct stimulations. In this case, it is a,
remarkable fact that, although the modal latent period is prolonged
with the progression of the fatigue, the descending branches of tlie
discharge curves coincide with one another into a line. Since the
discharges in the two sets of curves have the very same aspect,
notwithstanding the stimuli are different in their breadths, the
responses must be due to the closing-stimuli only, — the case wlien
the closing-stimuli are strong enough to evoke the maximal dis-
charges. Therefore the oscillogram shows the fatigue phenome-
non in the"case when the stimuli are direct and simple.

The principal point of this section is that the increase of the
magnitude of tlie response at the outset of the fatigue exists in the
case of the electric organ as well as in muscle. Therefore the
explanation by Fröhlich in tlie case of muscle cannot l)e considered
correct.

70

Art. 1.— K. Fuji

Fig. 8. Nerve-organ preparation with six
pairs of the electrodes.

VIII. Speed of Propagation of Excitation
through the Nerve.

vSince the time relation can be measured very accm'ately l)y
our method, some experiments were made regarding the speed (jf
the propagation through tlie nerve-trunk.

Oscillogram No. C. 5
(Plate XXVIII.) shows an
experimental result regarding
the uniformity of the speed
along the nerve-trunk. On
an ebonite plate about 1 cm.
in breadth, six pairs of electrodes were fixed as shown in the
accompanying hgure. The distance between the electrodes in a
pair was 1 mm. and the distances l^etween the corresponding
points of such consecutive pairs were 5 mm. .V nerve-trunk with
a part of an organ Avas laid on this plate along its length and the
momentary stimuli were seut to the points on the nerve in the
order of 1, 2, 3, 4, 5, 0 and 2, the last one being for the check. The
folloAviug numbers were ol)tained on this oscillogram: —

Taiîle XXVII.
Oscillogram No. C. 5.
No additional resistance.
Tem])erature of the organ : 1 1C» °C.

Xo. of

«lectrodes.

Time stimulus

to beginning

of discharge

in mm.

Number of
revolutions of
registering-
drum.

1 mm.
corresp. to,
in 10-* sec.

Time stimulus

to beginning

of discharge

in 10-^sec.

5 and (J of
2nd column

inter-
changed*

1

814

1323

1515

123 0

'2

(36 0

13 04

1-533

101-3

3

60 U

]279

1565

95-5

4

58 3

1317

1520

88-6

5

50 3

13 27

1-508

76 0

830

G

48 7

1213

1-650

804

73-5

2

60 0

11-70

1-710

102 3

* In 5 and 6 the electrodes should have been interchanged by accident, therefore for
their values we used the numbers in the last column.

Kesearclies on the Discharge of the Electric Organ. 7]^

The modal latent periods could not be measured, for the-
summits of the curves were on tlie outside of the film. Therefore
the time between the stimulus ai^d the beginning of the discharge
was measured.

In Plate X. a grai)h representing the time between the stimulus;
and the l)eginning ()f tlie discharge with regard to the position of the
electrodes is drawn. In this figure we see tliat the clieck point
lies very closely on the curve. Tliis shows that the experi-
ment was not affected by any progressive change. Tlie points 2,3, ■
4, 5 and 0 arrange themselves on a straight line, while the point 1
falls on tlie upward side of the line. This sliows that Ijetween the
points 2 and (3 in the nerve, tlie speed of propagation was uniform
Avhetlier tlie point concerned was near to or far from the stimulated
point. As the value of the speed, we (obtained Toi metres per
sec, the temperature of the nerv(3 l)eiiig 11 (> °C. The discrepancy
of the point Xo. 1 must have its cause in the alteration or decay of
the nerve at that portion which occurred near the excised end of
the nerve. This becomes very olnious when we examine the
oscillogram, for tliere we see that the discharge curve coriespond-
ing to Xo. 1 is very low compared with the other discharges, while
the stimulus is ii()t smaller than the others. The prolongation of
the latent period of the discharge evoked by a stimulus given at a
point near the excised end of the nerve, may be explained by one
of tlie following as-uinptions. (1) The sjieed of pro})agatioii in the
portion near the excised end of the nerve altered l)y its injury
is smaller tli.an the normal. (2) The local latent ])eriod at the
nerve-ending or in the organ, I.e. the interval between the instant,
at which the disturbance arrived at the nerve-ending, and that of
the beginning of the discharge is greater when the disturbance in
the nerve is smaller. (.'>) The speed of ]^roi)agation of the nerve
excitations differing in magnitude has different values, the stronger
the larger. (4) The local latent period at the point of stimulation
is greater when that portion of the nerve is altered. The cases
(2) and (8) cannot be considered to be true from the standpoint of

72

Art. 1.— K. Fuji

the "all or none" theoiy. Moreover Oscillogram No. 40 denies
these assumptions, for the modal latent periods of the discharges
are equal, though the magnitude of the discharges varies very
widely. Then the cause of the prolongation must l)e due either to
(1) or to (4). The causes (1) and (4) cannot he separated in our
former experiments. .1 am, however, inclined to helieve that the
cause would he that of (1).

A'. />.— Oscillogram Xo. 49 is an experiment intended to find the relation between the
strength of the stiunilus and the height of the discharge as shown in § V. The latent
periods and the modal latent periods are given again in the following table : —

Table XXVIII.

No.

Height of discharge
in mm.

Latent period | Modal latent period
in 10" ''sec. in 10-''sec.

1

90

142 173

2

05

Too low to be measured accurately.

3

10

Too low to be measured accurately.

4

23

162

185

5

83

162

188

6

150

159

185

7

170

156

189

8

18-3

156

18'.»

(The stimulus of Xo. 1 was ascending by accident.)
Here we see that the latent period decreases with the increase of the heiglit of the dis-
charge and therefore, when our eyes are restricted to L. P., it seems that the speed of the
propagation of the excitation increases with the increase of the height of the stimulus.
But when our eyes are turned to the modal latent periods, we see that they give toleraV)ly
constant value, and show no regular variations with the increasing .strengths of the stimuli.
This shows that the speed of the propagation through the nerve in the normal state does
not depend on the strength of the stimulus or on the magnitude of the excitation. Also
this experiment shows that the so-called latent period is not suitable to be used as the time
of reference for exact Avork.

IX. Miscellaneous problems.

Oscillograms No. 53 and No. lOü (Plate XXIX.) represent
photographs of spontaneous discharges of a li^•ing hsh. At first we
tried to open our shutter by tlie fiist discharge of the fish and to
photograph the motion of tbe strips of the oscillograph caused by

Researches on the Discharge of the Electric Organ. 7S

the succeeding discharges. Tlie shutter could be opened as desired,
but tlie chscharge curve could not be obtained. So laying a living
fish on a wooden board having a hole of the shape of the organ for
an electrode, and applying a pair of electrodes to the dorsal and the
ventral side of the organ b}^ my hands, the registering drum was
rotated. Judging the proper instant, which could be readily
known by the behaviour of the fish before its discharge, the shut-
ter, used as time-shutter, was opened. When the discharges
occurred, which were percei\'ed l)y the shocks felt, the shutter was
closed. In the two oscillograms taken under difïerent conditions,
we see that a shock always consists of two discharges followed ]jy
a very low one, the second discharge being a little smaller than the
first. The proper period of the discharges of a living malaptervnis
and its variation by the change of the temperature were
investigated by Koike ''" with tlie string galvanometer. In his
results a shock consists of many periodic discharges, which Garten
and Koike thought to be theefïectof rhythmic central excitations.
In our experiment, tlie kind of Hsh was difïerent from his, and in
his experiment the fish was placed in water whereas ours was
placed in air. Therefore it is not strange that the results do not
agree in the two experiments. But in the face of our experimental
results, perhaps the second and the succeeding discharges would
l)e of a secondary nature, for a ^-imilar phenomenon 7. e. two large
discharges followed by a very low one was frequently observed in
many experiments of the indirect stimulation given to the prepara-
tion of the isolated nerve-organ.

We shall next show two experiments regarding the discharge
l»y the stimulating current of long duration in the a^icendiiig or
in the descending direction. Oscillograms No. 7(j and No. 77 (Plate
XXX.) are two such experiments. These are the results with
difïerent preparations. The original object of tliese experiments
was to test Hoorweg's considerati(tn that the opening stimulus
was the efïect of the polarisation current flowing tlirough the shunt
ordinarily used for the regulation of the stimulation current.
The result was of course negative. But on examining the

* Zeitschrift für Biologie, Bd. 51, 1910.

74

Art. l.-K. Fuji:

oscillograms precisely, we found tliat the stimulating current of
long duration sent through the nerve influences the strength and the
modal latent period of the succeeding discharge in a peculiar
way. The experiments were performed l)y closing the stimulation
circuit and by shunting it at the proper instant. These operations
can be made by our stimulation apparatus by using two connection
changers, one of Avliich is a mercury key of earlier design. The
current was directly derived from eight storage cells. TJie
numbers obtained from these oscillogi'ams are tal)ulated i]i the
following table: —

Table XXIX.

Oscillograms X\). 7(> and No. 77.
Stimulus : Indirect.

Temperature of the organ : 145°

No. 70.

Xo.

Direction

of

stimulating

current.

Duration of

current
in 10-^sec.

Modal latent

period

in 10"^sec.

Height 01

discharge

in mm.

1

2
3
4

Ascending.
Ascending.
Descending.
Descending.

1699
3040
3000
172 0

1107
1100
1135
1204

20 0

18-5

8-0

2 0

Eesearohcs on the Discharge of the Electric Orsan.

No. 77.

Xo.

Direction

of

stimulating

current.

Duration of

current
in 10-^sec.

]\Iodal latent

period
in 10-^sec.

Heio-ht of

«lischarge

in mm.

1

2

3
4
5
6

Descending.

Descending.

Ascending.

Ascending.

Ascending.

Descending.

187
124-8
16G0
43-7
230
212

129-3
133-3
213-9
1482
138-5
1305

Out of

film.
47-5
46-5
53-5
590

The modal latent periods of 1 and 2 in No. 77 were found by the formula .''o= V'.ii.c..
Since the stimulus was indirect, it must be allowed that the values of the modal latent period
thus found may be somewhat larger than the correct values.

In the doseending stiuuilus, tlio discbarge occurs by the
closiiig-stimiüus and in tlie ascending by tlie opening-stimnkis.
The positive direction of tlie ordinate on the oscillograms shows
the descending stimnlus.

In Xo. 77, the modal latent period of the third experiment
shows extraordinary prolongation. The discharge was caused by
an ascending stimulation current of tolerably long duration and the
preceding discharge was that due to the descending current. The
succeeding discharges were evoked by the stimulus of the same
direction and the result was that, the shorter tlie duration the
shorter the modal latent periods, tliougli it is longer than the
normal. The prolongation may be interpreted either as the
effect of some local cliange caused l»y the ascending stimulating
current of long duration, of wliich the longer the duration,
the greater is the ])rolongation ; or as the influence of tlie
preceding stimulus which is recovered Ijy repeating the opposite
current. The former is more probal)le if the observation is
limited to this film only, l)ut in tlie discharge evoked by the
stimulus of the same direction in Oscillogram No. 70, it cannot
be observed that the modal latent period is prolonged. The
number of the series of experiments of the above kind is
onlv two, and therefore we cannot give a definite conclusion.

76

Art. 1.— K. Fuji ;

In Oscillügram No. 71 (Plato XXII.), similar proloDgations of the
modal latent i:)eriods may be observed in the experiments of the
ascending-stimnlns which were made after a series of experiments
of the descending-stimulus. But there is no proof that the
prolongation was tlie influence of the previous descending stimulus.

Tai'.le XXX.

Oseillo

gram No. 71.

No.

Heio-lit of
stimiilns
in mm.

Heij>-lit of

of discharge

iu mm.

Modal latent

period

in 10-^sec.

1

120

260

96 5

2

230

245

92 7

3

34 0

24 0

97-4

4

460

24 0

94 0

5

—

230

91-5

6

—

180

106 3

7

45 0

18 0

110 0

8

340

160

107-5

9

100

20 5

990

Here the stimuli of No. 1. — No. o and tliat of No. U were
descending, and those of No. (J, No. 7 and No. 8 were ascending.

In the oscillograms we may observe tliat wlien tlie modal
latent period exhibits its prolongation, it is alwa^^s accompanied
by the decrease of the magnitude of the corresponding discharge,
although the converse is not true always. We may remark here
that this law applies to all cases whetiier tiie cause of prolongation is
the temporary fatigue or any other alteration in the nerve, or that
of an unknown cause like tliat above described. On tlie contrary,
Avlien the magnitude of the discliarge is small on account of the
smallness of the stimulus, the prolongation does not appear. This
shows that, when the })rolongati(.)n appears, the nerve-fibres are
changed in its state and in consequence the discharge becomes small,
i. e. evei'v " all " of an excitation itself in the nerve fibres becomes

Eesearclies on the Discharge of the Electric Organ. 77

small, jind perhaps tlio iinmlxT of disturbaneos in the nerves
Avhich can pass through the decrement of nerve-endings or of some
altered part in the nerve becomes less ; and when the prolongation
does not appear, the smallness of the discharge is caused on account
of a small part of nerve fibres in a trunk being excited. Tims
tlie phenomenon gives support to the " all or none " theory.

Summary.
Since several new phenomena liave been 1)rought to light
after the analyses of the oscillograms, and since at present we have
not an opportunity to confirm them by further experiments, we
are obhged to leave many problems as not positively decided.
We shall conclude this paper, by summarising the results.

1) A formula expressing the time relation of a simple dis-
charge may l)e got from tlie tlieory of prol)al)ility and is expressed
by

.r
— li!o(i-

ij =. A e ' -'0 ,
where y denotes the electromotive force at time x measured h-om a
certain fixed moment, ô^ and Xq being constants.

2) In the case of the direct stimulus of short duration, the
origin of time in tliis formula is in agreement with the instant of
stimulation.

3) When the direct stimulation is of a longer duration, the
dischaige in response to it may be analysed into two simple dis-
charges corresjionding to the closing and to the opening-stimulus,
and eacli of them may be represented l)y the al)Ove formula, having
its origin at the instant of the corresponding stimulus.

4) Taking the value of A as the measure of the excitation,
the relation Ijetween the intensity of a stimulating current and the
excitation in response forms an S-shaped curve which rises from the
zero-stimulus very slowly, then c[uickly, and after passing an
inflexion point on its way, finally ap])roaches asymptotically a
constant maximal value.

5) With regard to the relation between the duration of a
stimulating current and the excitation in response, Hoorweg's
decrement factor seems to hold good.

Yg Art. 1.— K. Fuji:

6) When a stiniulatioQ is given to a nervo, there remains a
fatigue which recovers in a very short interval and wliich is called
the tem'povary fatigue by the author.

7) Tlie fatigue is characterised by

a) The prolongation of the modal latent period of the
discharge in response to a stimulus given in the
inter^'al influenced by that fatigue.

1)) The decrease in the intensity of the discharge denoted
by A in the discharge formula.

8) The time relation of the recovery of this fatigue measured
by the excess of the modal latent period of the second discharge
compared with the same of the first may be expressed by an
exponential function Me~''\ and tliis law of the recovery holds
good also in the case of similar results obtained by Lucas in nerve-
muscle preparation.

0) The abnormal increase in the intensity of the second of
the discharges, evoked by two successive stimuli, may l)e inter-
preted by the superposition of the secondary discharge.

10) The behaviour of the discharges in response to the two
submaximal indirect stimuli, separated by an interval shorter
than the refractory period, gives support to the "all or none"
theory.

11) In this case, a new phenomenon is probably involved, i.e.
the shortening of the modal latent period of the discharge, caused
by the summation of the two subminimal stimuli.

12) The discharges in response to submaximal closing and
opening-stimulus indirectly given may superpose, and the modal
latent period corresponding to the opening-stimulus shows the
prolongation in the modal latent period. In this case also, the
phenomenon ma_y be explained by the "all or none" theory,
together with the assumption that the subminimal stimulus may
cause some local change, which is made apparent from the prolonga-
tion of the modal latent i)eriod of the succceeding discharge and
which is not yet otherwise confirmed.

: 13) The fatigue curve in the case of the electric organ very

closelv resemldes that of tlie contraction of muscle. In both-

Etsearclies on the Discliarge of the Electric Ori^-iu. 7Q

cases, thoro exist the so-called staircase phenomenon and the other
details in a similar way, and this fact indicates the failure of
Fröhlich' s explanation in the case of muscle.

14) The speed of propagation of the excitation is uniform
throughout the nerve-trunk Avliether the point concerned is near
to or far from the stimulated point, if the nerve is in tlie normal
state.

15) In the altered part of a nerve, near its periphery end,
the speed of propagation hecomes smaller and is accompanied by
the decrease of the intensity of the corresponding discliarge.

10) A spontaneous discharge curve of the living fisli Astrajje
consists always of two peaks, sometimes followed by a very low
irregular one. The second and the following discharge may l)e
considered to l)e the secondary discharge, etc. evoked l^y the first.

17) The prolongation (jf the modal latent period from
its normal value, of whatever cause it may l)e, is necessarily
accompanied by the enfeeblement of the discharge, while the feel:)le
discharge evoked by the weak stimulus does not indicate the same
prolongation.

In closing this paper, I wish to express my best thanks to
Prof. K. Osawa, Director of the Physiological Institute, whose
liberality enabled me, for such a long period, to carry out in the
Institute, these costly experiments. Also cordial thanks are due
to Dr. 8. Oinuma who was my zealous collaborator at the begin-
ning of these researches, who kindly collected and placed at my
disposal the literature relating to the subjects discussed in this
paper, and who gave me many valual)le advices.

80

Art. 1.— K. Fuji:

APPENDIX.

Tables of Experimental Data and Calculated Numbers.

(Table I. — V. are those for the form of tlie
discharge curve.)

Table I.

Oscillogram No. 54.
Date of experiment :
Object of experiment:

TreTiaration:

Temperature of the organ:

Resistance of the organ:

Reading on the stimulation-apparatus:

Circuit-breaker: 640 mm

Connection-changer: (30 0 mm

Nov. G, 1909.

Relation between the intensity

of a stimulating current and the

magnitude of the corresponding

discharge.

Left organ (whole) of a fisli oî

middle size.

8-5°-7-2° C.

240 ohms.

1 Xo.

Current through

primary circuit

of induction

coil

in amp.

]S" umber of
revolutions of

registering
drum per sec.

1 mm. of

oscillogram

corresponds to,

in 1U-* sec.

Height, of
stiumlus
in mm.

Height of

discharge

in mm.

1

1-50

6-51

307

19 5

1-6

2

203

6-38

313

257

50

3

2-50

6-50

308

31-8

. 101

4

3 00

G -53

306

380

17-3

5

3-50

G-40

313

447

26-7

6

400

618

324

500

35-7

7

450

GOO

333

57 0

44-8

8

4 95

G-3G

314

65-5

55-6

Researches on the Discharg-e of the Electric Oro-an.

81

No.

Latent period

Modal latent period

in mm.

in 10-4 sec.

in mm.

in 10—* sec.

1

2
3
4

5
6

7
8

25-0
25-3
240
23-5
23 5
21-5
20 5
21-5

76-8
79-2
739
71-9
73-6
69-7
68-3
67-5

350
340
350
35-0
350
33-5
32-0
35 0

107-5
106-4
107-8
107-1
109-6
108-5
106-6
109-9

Co-ordinates of the points on the discharge curve for every 2 mm.

of X on the oscillogram.

Curve No. 3.

Curve No. 4,

X

in mm.

X

in 10-4 sec.

y

in mm.

in mm.

X

in 10-4 sec.

in mm.

00
24 0

00
739

Beginnino-
of stimulus.

00

00
235

00
71-9

Beginning
of stimulus.

00

5-0

77-0

05

250

76-5

12

7-0

832

21

70

82-6

3-8

90

89-3

49

9 0

88-7

8-6

31-0

95-5

7-7

31-0

949

13-5

3 0

101-6

9-5

30

101-0

16-5

50

107-8

10-1

50

107-1

17-3

70

1140

9-7

70

1132

16-6

90

1201

8-8

90

1193

15-0

410

126-3

74

410

1255

12-8

30

1324

61

30

131-6

104

50

1386

4-8

50

137-7

8-1

70

1448

3 4

70

1438

62

90

150 9

2-7

90

1499

46

510

157-1

1-8

51-0

1561

34

30

163-2

1-3

30

162 2

25

5-0

169-4

11

5-0

1683

1-7

7-0

175-6

0-7

7-0

1744

13

90

Sm

350

181-7

limit of the cu

107-8

0-3

rve.

1 101

90

Sui
35 0

1805

limit of the cu

1 107-1

10

rve.

17-3

82

Art. 1.— K. Fuji :

Curve Xo. 5.

Curve Xo. G.

X

iu mm.

.r
in 10—1 sec.

y

in mm.

in mm.

.r
in 10-i sec.

V
in mm.

00

00

Beginning
of stimulus.

0-0

00

Beo-inning
of stimulus.

235

736

0 0

21-5

697

00

250

78 3

2-4

230

745

16

7-0

84-5

7-0

5-0

81-0

45

9-0

90 8

14 2

7-0

87-5

115

31-0

97-0

21-1

9 0

94 0

22 8

30

103 3

25 3

310

100 4

315

5-0

109-6

26-7

30

106 9

35-5

7-0

115-8

25-3

50

1134

35 1

9-0

1221

22-2

7-0

119 9

314

410

128 3

18-5

90

1264

26 3

30

134-6

145

41-0

132 8

20 6

5-0

140-9

110

30

139-3

157

7-0

1471

8 3

50

145 8

115

9-0

1534

6 0

70

152 3

85

510

1596

45

9-0

1588

61

30

165-9

33

510

165-2

46

50

172 2

24

30

1717

35

70

1784

20

5-0

178-2

2-8

90

184-7

16

70

184 7

24

35 0

mmit of the eu

j 1096

Yve.

26-7

9-0

Su

33 5

191-2

mmit of the cv

108 5

2-2

rve.

35-7

Researches on the Dischartje of the Electric Orofan.

83

Curve Xo. 7.

.r
in umi.

.1-
in 10~* sec.

y

in mm.

00

00

Beofinnino-
of stimulus.

20-5

683

00

3-0

76-0

3 2

5-0

83-3

. 9-3

7-0

89-9

22-7

90

96-6

35-7

310

1032

436

30

109-9

44-8

50

1166

407

70

1232

339

9-0

1299

26-5

41-0

136-5

200

30

1432

146

5-0

1499

102

7-0

1565

7-5

90

1632

5 4

510

1698

41

30

176 5

33

50

183 2

23

7-0

189-8

21

Sui

limit of the cu

rve.

32 0

106-6

45 0

Curve Xo. 8.

.r
in mm.

in 10-1 sec.

in mm.

00

00

Beginning-
of stimulus.

21-5

67-5

00

23 0

722

11

50

78-5

3-8

7-0

84-8

92

9 0

911

220

31-0

97-3

37-5

30

103 6

52 2

5-0

109-9

55 5

7-0

1162

534

90

122 5

46-4

410

128-7

381

30

1350

294

5-0

141-3

21-8

7-0

147-6

16-3

90

1539

11-9

51-0

160 1

8-5

3 0

1664

6-3

5 0

172 7

4-7

70

179 0

3-8

9 0

185 3

3 0

Sui

limit of the cu

rve.

35 0

109-9

55 5

84

Art. 1.— K. Fuji :

From the values given in the above tables, the curves from
No. 3 to No. 8 were reproduced on section papers whose one
division is equal to two millimetres. These curves are shown in
Plate VI. For the ordinates, one division of the section paper
represents one millimetre of the original; and for the abscissa,
one division represents 2xl(r* sec. On these curves, '^i. ^2 and
^1, ^-j were measured; and from these values, ^> ^0 and ^' were
calculated. The following table shows these values. As the unit
of y> ^i, ^-2, Vi, ^1 and ^Z, one division of the section paper is taken.

No.

y-

?!•

1

2

3

3 5

10 u

4

02

10 0

5

98

11-3

(3

130

100

7

105

9-8

8

'20-5

9-7

&l'.

Too low to be used.

185

1-4,

134

18 4

31

129

172

43,

141

17-2

0 5

12 5

171

82

121

104

10 1

124

201
251
248
240
24 0
234

in 10-i

sec.

+ 00,

+11,

-Olo

+ l-4o
+ 10,
+ l-4o

M.L.P.

correct.

in 10-1

sec.

108-5

108 2

109 5
109-9
107-7
111-4

in 10-1

sec.

59-2
59-8
04-2
00-8
54 8
01-8

400
409
5 03
5 37
449
5-95

By using these constants, the ordinates of each curve for
every 1/1000 sec. were calculated. These are shown in the next
table.

Eesearches on the Discharije of the Electric Orjj-au.

85

X

ill 10-4 sec.

11 iu mm. (calculated).

Xo. 3.

No. 4.

Xo. 5.

Xo. G.

No. 7.

No. S.

00

Ori^

1

^in of discharge.

1

10-0

OOo

00,

00,

00,

00,

0 0,

20-0

00,

02

OOi

00,

0-5

0 0,

30-0

11

10

0-9

24

8-8

2-5

400

50

8-2

7-3

13-9

28-8

181

50-0

8-9

148

18-4

29-1

43-3

42 0

GO-0

101

17-3

25-9

35-7

43 4

55 3

70-0

88

154

25-7

32 1

314

50-7

80-0

G 5

115

20-0

23-8

237

374

90-0

44

7-8

144

150

14-9

240

100-0

2-7

4-8

90

9-4

8-9

141

1100

10

2-9

5 4

5-4

51

7-7

1200

0-9

17

3 1

3 0

2-9

41

1300

0 5

10

1-7

10

10

2 1

1400

0-3

0 5

0-9

0-9

0-9

10

150 0

0-2

0-3

0 5

0-4

0 5

01

These values are plotted in Plate VI. hy the marks ^.

86

Art. 1.— K. Fuji

Or-cillograni No. 37.
Date of experiment:
Object of experiment:

Temperature of the organ:
Resistance of the organ:

Table II.

Oct. 23, 1909.

Relation between the duration of a

stimulating current and the magnitude

of the resulting discharge.

110° C.

130 ohms.

Xo.

Keadings on stimulation-apparatus.

XuDilior of

revolutions

of registering

drum per sec.

1 mm. of

oscillogram

corresponds to,

in 10-* sec.

Circuit-breaker.

Connection-
chano-er.

1

2

3
4
5
6

7
8

(3j, 0 "i"i-
63-8

636
634

63-2
630

62-8
62 6

.54 8 li^m-

6-60
6-59
6-79
6-94

7-28
7-27
7-66

7-77

3-03
303
2-95
3-88
2-75
2-75
261
2-58

Stmulatio

n l.ireadth

Modal latent period

\o

\

in mm.

in 10-1 sec.

in mm.

in 10- * sec.

]

60

18-2

40-6

123 0

31-5

2

5-3

161

410

124-3

30-8

3

4-8

142

42 6

125-7

28-3

4

45

130

440

126-7

27-5

5

4 3

11-8

45-6

125-4

25-0

6

41

11-3

46-7

128-2

23-3

7

3 5

91

49-5

129-5

]9-0

8

3 0

7-7

49 5

127-5

125

Researches on the Discharge of the Electric Ortjan.

87

Co-ordinates of tlie points on the discharge curves for every
2 mm. of X on the oscillo2;ram.

Curve No. 1.

Curve Xi). 2.

.1"
in mm.

in 10-' sec.

in mm.

in mm.

X

in 10 -^ sec.

11

in mm.

00

00

Beginnino-
of stimulus.

00

00

Beginn ino-
of stimulus.

-.20 0

78-8

00

27 0

81-8

00

80

84-8

1-G

8 0

84-8

1-4

300

90 9

4-4

300

90-9

2-6

2 0

97 0

9 0

2 0

97-0

6-4

40

103 0

15 0

4 0

103 0

11-Ç

(50

1091

22 4

GO

109 1

184

80

1151

290

8 0

115-1

25 6

400

1212

31-5

40 0

1212

30 0

'2-0

127-3

300

2 0

127-3

30 0

40

133 3

25-8

4-0

133-3

26 8

GO

139 4

20-G

G-0

139-4

22-0

8-0

145 4

15-0

80

1454

16-8

500

151-5

10-G

50 0

1515

120

2 0

l57-(')

7-0

2 0

157-G

8 0

40

1G3-G

4-G

40

1G3-G

5 4

(')0

1G9-7

2-8

GO

169-7

3-4

80

175-7

20

8 0

1757

2 2

GOO

GOO

181-8

1-8

Su

mm it of the ci

irve.

Su

mm it of the ci

rve.

40G

123-0

31-5

410

124-3

30-8

88

Art. 1.— K. Fuji:

Curve Xo. 3.

in 10-* sec.

0 0 0 0 Beginnino-

^^' ^ ^^ of stimulus.

28 0 82 6 0 0

300 88-5 10

2 0 94 4 3 G

40 1003 70

6 0 loa 2 12 0

8 0 112 1 18 8

40 0 118 0 250

2 0 123 9 280

4-0 129-8 27C)

60 1357 24-4

8 0 1416 20 0

500 147-5 150

2 0 153 4 10 4

4 0 159-3 6 8

60 165-2 4-4

8-0 171-1 28

600 1770 18

Summit of the curve.

426 125-7 283

y

in mm.

Curve No. -4.

.r
in mm.

X

in 10-* sec.

in mm.

0-0

00

Beginning
of stiuiulus.

29-0

83 5

0-0

32 0

92 2

24

40

97-9

54

6 0

1037

9 0

8-0

109-4

142

40-0

115-2

20-2

20

122 0

25-4

4-0

12G7

27-5

CO

132 5

26 0

8-0

138 2

22 6

500

1440

18 2

2-0

149 8

136

40

1555

94

6 0

161-3

66

80

1670

44

600

192-8

32

2-0

178-6

20

Sui

limit of the cu

rve.

44-0

126 7

27 5

Researches on the Discharçfe of the Electric Or^au.

89

Curve No. 5.

x
in. mm.

in 10 ~i sec.

in mm.

0-0

0-0

Beginning
of stimulus.

30-0

82-3

00

20

880

1-0

4-0

93-5

2-8

60

990

5-2

8-0

104-5

9-4

40-0

110 0

14-0

2-0

115-5

19-6

4-0

121-0

24-0

6-0

12G5

24-8

8-0

132 0

23-8

50-0

137-5

20-6

2-0

143 0

16-4

4-0

148-5

12-6

60

1540

9-0

8-0

159 5

60

60 0

1650

4-0

20

1705

2 4

40

176 0

14

Su]

Jim it of the cii

rve.

456

1254

250

Curve No. 6.

.r
in umi.

.T

in 10-4 sec.

in mm.

0-0

0-0

Beginning
of stimulus.

30-0

82-3

0 0

2-0

88-0

10

4-0

935

22

6-0

98-0

46

8-0

1045

7-8

40-0

110 0

116

2-0

1155

168

40

1210

216

60

126-5

232

8-0

132 0

22-8

50-0

137 5

20-4

2 0

143 0

16 6

40

1485

13 2

6-0

1540

9-8

80

159-5

6 6

600

165-0

46

2-0

170-5

3 2

4-0

176-0

2-0

Sm

nmit of the cu

rve.

4G-7

1282

233

90

Art. 1.— K, Fuji

Curve Xo. 7.

.r
in uiin.

.1-

in 10" 4 sec.

V
in mm.

0 0

00

Beginnincj

of stimulus.

330

86-1

0-0

40

88-7

06

GO

940

2 2

80

99 2

3 8

40 0

104-4

5-8

20

1096

8-8

4 0

114-8

13 0

GO

1201

164

80

1253

18-8

50 0

130-5

190

20

135-7

17-6

40

140 9

152

60

146-2

12-4

80

151-4

9-6

600

156-6

7-0

20

161-8

5-2

40

1670

3-4

60

1723

20

Sui

limit of the cnrve.

49-5

129 5

190

Curve Xo. 8.

in 10-4 se<

on no Beo;innins

^^ ^ *^ ^ of stimulus.

35-0 90-3 0 0

60 929 1-2

8-0 98 0 2-4

400 103 2 3 8

2 0 108-4 5-6

4 0 113-5 8 0

60 118-7 106

8 0 123 0 11-8

50 0 129 0 12 4

20 134-2 11-6

4-0 139 3 10 0

60 144-5 8-4

8-0 149-6 6-4

60-0 154-8 4-8

2-0 160 0 3 2

4 0 1651 2 0

Summit of the curve.

495 127-5 125

in mm.

Ecsearohos on the Discharere of the Electric Orjran.

91

From the values given in the above tables, the curves from
No. 1 to No. 8 were reproduced on section papers in the exactly
same way as in No. 54. These curves are shown in Plate VII.
On these curves ^i and a% for several values of y were measured.
Then the mean of V^ for every curve was found and compared
with the value of the modal latent period observed. The units of
'^v ^-2 and y are one division of the section paper. Using tlie
mean of Vxix. as the value of Xo we calculated h".

yo.

y-

•î'i-

Xu

V-ri-^-j-

.'/'•

.Vi'.

X-J,

v/a-i'.Tj'.

y"-

.fi".

x^ .

V-'l'-'-L'".

1

116

50 0

74-y

611

60

46-6

79-8

610

210

53-7

69-4

610

2

11-4

51-4

75-8

62 4

60

48 0

80-7

62 2

200

55-2

70-4

624

3

104

52 4

76-7

634

50

48-5

81-7

630

190

56-3

71-3

63-5

4

101

52 3

771

63 6

50

488

82-7

63 5

19 0

570

71-5

639

5

9-2

52-2

768

63-4

50

490

810

63 0

170

56 3

71-3

63-5

6

86

52-7

77-7

640

40

490

836

640

150

56-8

72 8

64 3

7

70

53-4

78-5

64-8

40

500

82 9

64-4

13 0

57-5

72-7

646

S

46

52^8

77-5

640

20

48-2

82-5

(634)

80

566

72-6

641

the

The number in the
mean.

bracket Avas not taken in the calculation of

Xo.

Mean of

thus found,
in 10-1 sec.

//'.

1

610

122 0

23-7

1 2

62 3

1246

25-8

3

63-3

1266

27 1

4

63-7

1274

27-4

5

63-3

126 6

261

6

641

128 2

270

7

64-6

129-2

263

8

641

128 2

27-8

92

Art. 1.— K. Fuji :

By using thu constants given in tlie above tables, the
ordinates of tlie curves for every I/IOOO sec. were calculated.
These values are shown in the next table.

.1"
in 10-4

y

in mm. (c

alculated)

sec.

No. 1.

Xo. 2.

Xo. 3.

Xo. 4.

Xo. 5.

Xo. G.

Xo. 7.

No. 8.

0-0

00

00

00

00

00

00

00

00

700

oa

00

00

00

00

00

00

0 0

80 0

0 5

0 2

0 2

01

01

01

01

0 0,

900

35

20

12

10

11

0-8

0-7

04

1000

125

9-2

6-4

5-6

5-8

45

3-5

23

1100

245

20 6

166

15-3

148

124

96

6-5

120 0

31-3

29-7

26 2

24.3

23-2

20-7

16-4

111

1300

28-G

29-4

27-8

27-2

245

232

18-8

124

1400

201

21-7

21-5

21-5

191

18-9

16 0

101

1500

115

12-9

13 0

13-2

11-6

120

106

63

1600

5-5

61

64

6-8

5-8

62

5-7

32

1700

2-3

2-6

2-7

2-8

2-5

2-9

26

14

180-0

0-8

0-9

10

10

0-7

10

11

05

1900

02

0-3

0-3

03

0-3

0 0,

04

0-2

200 0

00

01

01

0-2

01

0-0

01

01

Eoscarches on the Dischar<re of the Electric Orcran.

Ü3

Table III.

Osciilograni No. 40.
Date of experiment :
Object of experiment

Temperatm'e of the organ:
Resistance of the organ:
Current of the primary of the induction
coil:

Nov. 24, 1009.
The relation between the
duration of a stimulating
current and the magni-
tude of the discharge.
11-5° C

0-5 amperes.

No.

Eeadings on stimulation-apparatus.

>; umber of

revolutions

of registering

drum jjer see.

1 mm. on

oscillogram

corresponds to,

in 10-4 sec.

Circuit-breaker.

Connection-
changer.

1

2

3
4
5

6

7
8

(35-0 mm.
64-2

63-8
636
63 4
63-2
630
62-8

.54-8 m"i-

700
6-^3
6-77
6-99
6-67
6-67
6-99
7-05

2-86
2-88
2.96
2-86
3 00
300
2-86
2-84

stimulation-breadth

Modal latent period

No.

A.

in mm.

in 10^^ sec.

in mm.

in 10-^ sec.

1

9-8

28-0

425

121

44-5

2

6-4

18-4

41-4

119

41-0

3

6-7

19-8

40-7

120

40-0

4

6-1

17-4

422

120

39 0

5

6-4

192

40-7

122

390

(3

5-0

150

40-8

122

33-5

7

5-0

14-3

42 5

122

35-6

8

41

11-6

43 0

122

29-5

94

Art. 1.— K. Fuji

Co-ordinates of the points on the discharge cur\'es for every

2 mm. of X on the oscillogram.

Curve Xo. 1.

X

in mm.

.T

in 10-* sec.

in mm.

00

18-7

00
53-5

Beginning-
of stimulus. '

0-0

197

563

0-0

21-7

G2 1

0-3

3-7

67-8

10

5-7

73-5

14

7-7

792

23

9-7

84-9

44

31-7

90-7

8-5

3-7

96-4

165

5-7

1021

25-1

7-7

107-8

341

9-7

1135

410

41-7

119-3

443

3-7

1250

436

5-7

1307

49-8

7-7

136-4

34-0

9-7

142-1

27-6

51-7

147-9

21-0

3-7

1536

156

5-7

159-3

113

7-7

1650

81

9-7

170-7

5-9

61-7

1765

44

3-7

182 2

36

5-7

187-9

34

7-7

193-6

3-6

9-7

199-3

42

71-7

205-1

4-7

Sui

nmit of the curve.

42-5

1212 445

Curve No. 2.

00

22 0

260

80

30 0

20

40

60

80

400

2 0

40

60

80

500

20

40

60

80

60 0

20

414

in 10-* sec.

00

634

74-9

806

86-4

92 2

97-9

103-7

1094

115-2

1210

126 7

1325

138-2

1440

1498

155 5

161-3

167-0

1728

1786

Summit of the

119-0

V
in mm.

Beginning
of stimulvis.

00

09

3-1

60
109
185
26-5
35-0
39-6
40-9
380
32-9
26-3
20-1
146
10-6

7-5

5-5

40

3-2

ve.

410

EesearcL.es on the Discharore of the Electric Oro-an.

95

Curve Xo. 3.

x
iu mm.

in 10-1 sec.

y

in mm.

00

00

Beginnino-
of stimulus.

23-5

690

00

5-5

75-5

19

7-5

81-4

2-7

9 5

87-3

GO

31-5

93-2

11-1

8.5

991

18-3

5-5

1051

27-1

7-5

1110

35 5

9-5

llG-9

39-2

41-5

1228

39-8

3-5

128.8

36-6

5-5

134-7

31-3

7-5

140G

24-3

9-5

146-5

17-9

51-5

1524

127

3-5

1584

8-9

5-5

1643

G -2

7-5

170-2

4-5

9-5

176-1

3-5

Sui

umit of the curve.

40-7

120-5 40-0

Curve Xo. 4. i

.r
in mm.

.1-

in 10-1 gee.

in mm.

00

00

Beg-inning
of stimulus.

23-9

68-4

00

5-9

741

12

7-9

798

2 4

9-9

85 5

4-3

31-9

91-3

8-2

3-9

970

140

5-9

102-7

21-8

7-9

1084

29 8

99

1141

360

41-9

1198

39 0

39

125 4

37-4

5 9

131-3

330

7-9

1370

27-4

9-9

143 0

214

51-9

1484

15-4

3-9

154 2

HI

5-9

159-9

7-2

7-9

165-6

50

9-9

171-3

36

61-9

177 0

2-6

Sui

nmit of the curve.

42 2

120-7

390

96-

Art. 1.— K. Fuji:

Curve Xo. 5.

.(.■
in mm.

X

in 10-4 sec.

in mm.

00

00

Beginning
of stimulus.

211

63-3

0-0

41

72-3

0-8

0-1

78-3

1-4

8-1

84-3

2-8

30-1

90-3

6-0

21

96-3

11-8

4-1

102-3

18-8

61

108-3

27-8

8-1

114-3

34-7

40-1

120-3

38-9

21

1263

38-5

4-1

132-3

34-S

6-1

138-3

28-2

8-1

144-3

21-5

50-1

1503

15-5

2 1

156-3

10-7

41

162-3

6-3

6-1

168-3

4-8

1 - 8-1

174-3

3-3

GO-1

1803

3-8

Sui

limit of the curve.

40-8

122-4

39 0

Curve No. 6.

X

in mm.

.1-

In 10-4 sec.

in mm.

0-0

0-0

Beginning
of stimulus.

25 0

75-0

00

7-0

81-0

1-7

9-0

87-0

3-9

31-0

930

6-3

30

990

11-5

5-0

1050

17-9

7-0

1110

25-8

9-0

1170

31-7

41-0

1230

33-5

3-0

1290

31-9

5-0

135 0

27-1

7-0

1410

21-9

9-0

1470

15-2

51-0

153-0

n-3

3-0

1590

7-9

5-0

1650

5-3

7-0

1710

3-7

9-0

1770

2-7

61-0

1830

2-7

Sui

nmit of the cu

rve.

40-8

1224

33-5

Researches on the Discharge of the Electric Orfi-an.

97

Curve ]S'o. 7.

X

in mm.

in 10—* sec.

in mm.

00

00

Beo-innino-
of stimulus.

241

68-9

0-0

6-1

74-6

0-0

8-1

80-4

1-4

30-1

86-1

4-4

2-1

91-8

6-1

4-1

97-5

11-3

6-1

103-2

18-2

8-1

1090

26-5

40-1

114-7

32-3

2-1

120-4

35-3

4-1

1261

34-5

6-1

131-8

31-0

8-1

137-6

25-5

501

143-3

19-8

21

1490

14-8

4-1

154-7

100

6-1

160-4

6-5

8-1

166-2

4-3

00-1

171-9

3-5

Su]

Jim it of the cu

rve.

42-5

122-0

35-6

Curve Xo. 8.

.r
in mm.

.r
in 10-^ sec.

in umi.

0-0

00

Beginning-
of stiuiulus.

26-5

75-3

0-0

8-5

80-9

1-6

30-5

86-6

2-2

2-5

92-3

4-7

4-5

98-0

8-8

6-5

103-7

14-2

8-5

109-3

20-6

40-5

115-0

26-2

2-5

120-7

29-5

4-5

126-4

280

6-5

1321

25-0

8-5

137-7

20-5

50-5

143-4

160

2-5

1491

12-0

4-5

154-8

8-5

6-5

160-5

6-0

8-5

166-1

4-0

60-5

171-8

2-6

2-5

177-5

1-8

Sui

oimit of the cu

rve.

43-0

122-2

29-5

*J8

Art. 1. — K. Fuji :

From the values given in the above tables, the eurves No. 1
— No. 8 were reproduced on section papers in the same way as
in No. 37. These curves are shown in Plate VIII. In these
curves, ^u ^-i for several values of y were measured. Then the
îxiean of Vï^ for every curve was found and compared witli the
observed value of the modal latent period. Tlie unit of x and of
y are one division of the section paper. Using tlie mean of Vx-^r.,
as the value of a-o, we calculated K.

No.

y-

Xy.

X,.

y'j'lXj.

.'/'•

x{.

xj.

^.q'.r.'.

Mean

of

thus

found

in

10-1 sec.

w.

1

16-3

48-3

761

GOG

8-2

451

82-5

G 1-0

608

121-6

200

2

150

47-9

74-5

59-5

7-5

44-4

80-7

59-5

59-5

119-6

20-8

3

14-6

48-3

751

GO-2

7-3

44-4

81-0

60-2

60-2

1204

20-6

4

14-2

48-5

750

GO-3

71

44-3

80-8

59-8

601

1202

20-4

5

14-3

49-5

75-7

61-2

7-2

45-9

80-8

60-9

611

122-2

21-8

G

123

500

75 G

Gl-5

6-2

460

81-7

61-3

61-4

122-8

23-2

7

130

49-5

75-5

611

G-5

46 0

80-4

60-8

610

1220

22-2

8

107

49-6

754

Gl-2

5-4

45-8

81-0

610

611

123-4

22-9

Researches on the Discharire of the Electric Ortran.

99

From the constants given in the above tables, the ordinates
of the curves for ever}^ I/IOOO sec. were calculated. These values
are sliown in the next tal)lo.

.r

in

10-^ sec.

h

in mm.

(calculated).

yo. 1.

Xo. 2.

Xo. 3.

Xo. 4.

Xo. 3. :

Xo. 6.

Xo. 7.

Xo. 8.

00

BeginE

ling of sti

tnviliis.

GOO

00

00

00

00

0-0

0-0

00

00

700

0-1

01

01

01

00,

00,

0-0,

00,

800

14

1-4

1-3

1-3

0-8

0-5

0-7

0-5

900

7-3

7-6

70

70

51

3-G

4G

3-G

100-0

210

21-3

19-9

19-6

164

12-8

150

11-9

1100

36-4

35-5

33-8

32-9

30-7

25-3

28 1

22-9

h200

444

410

40-0

38-G

38-7

331

354

29-3

1300

40-7

35-5

35-4

33-7

35 9

310

31-8

270

1400

29-9

24-5

25 0

240

2G-0

22-5

234

19-3

1500

18.5

141

14-8

14-2

15-5

13-2

13-8

11-2

1600

9-9

70

7-5

7-3

80

G-G

7 0

5 G

1700

5-3

31

3-4

3-3

3G

2-7

31

2-4

1800

21

1-3

1-4

14

1-5

11

1-2

0-9

1900

0-8

0-5

0-5

0-5

0-6

0-5

0-5

0-3

200 0

0-3

0-2

0-2

0-2

0-3

0-1

0-2

0-1

100

Art. 1- K.Fuji:

Taele I\'.

Oscillogram No. 54, curve No. 7 and its residual cuvxe.

y

= 44.8 e-*-i« '»y- ^TsT

7/^=44-8 e--^-^ '"^"-^ "i(lc.

.(■

,'/

.'/(I

y—'!h)

in 10-1 sec.

in mm.

in mm.

in mm.

1060

44-6

44-8

0-2

1120

43-6

42-2

1-4

60

410

37-6

3-4

1200

37-3

31-9 ■

5-4

40

330

25-8

7-2

8 0

28-6

19-9

8-7

1320

24-4

148

9-6

60

20-6

107

9-9

1400

171

7-4

9-7

40

140

5 0

90

8-0

11-5

33

8-2

1520

9-3

21

7-2

60

7-5

1-3

6-2

1600

60

0-8

5-2

40

4-8

0-5

4-3

80

3-8

03

3-5

1720

30

0-2

2-8

60

2-4

01

2-3

1800

1-8

00c

1-7

40

1-5

00,

1-5

80

1-2

OOi

12

1920

0-9

OOo

0-9

7/=10-0e-^'--^'-'-

in 10-* sec.

in mm.

00

00

700

0-9

800

40

90-9

81

100 0

100

1100

8-8

1200

5-9

1300

3-3

1400

16

1500

0-7

1600

0-3

Table A^

Oscillogram No. C. 3.
Date of experiment:
Prex^aration:

Temperature of the organ:
Stimulus :

One millimetre on the film
corresponds to:

May 13, 11)10.

About a quarter of the organ with

one nerve-trunk.

10-5° C.

Indirect momentary stimulus.

1-64x10-' sec.

Eesearches ou the Discharge of the Electric Orgau.

101

Coordinates of the points on
the discharge curve for every
'2 mm. of X on the (oculogram.

.('

,;

'J

in mm.

in 10—1 sec.

in mm.

Beginn ing

00

00

of stimuhis.

78-0

1280

00

S20

1.34-5

2-4

40

137-9

5-4

60

141-1

10-4

8-0

144-4

19-4

'.)0 0

147-G

29 0

20

1510

39 0

40

154-2

47-0

00

157-5

52-4

80

1G08

55-0

lOOO

1G4 0

550

20

1G7-3

53-G

40

1 70 G

508

(30

173 8

470

80

177-1

42-8

10 0

180-4

38-4

20

183-7

33-8

40

187-0

29 0

GO

190-2

24G

80

193-5

20 4

200

19G-9

lG-8

20

2000

1 13-4

40

203-4

110

60

20GG

^■^

8-0

2100

G-8

:;oo

213-2

5-G

20

2 IG -4

4-4

40

219-8

3-4

6-0

223 0

2-4

80

22G-4

2 0

400

229 G

1-4

1480

242-8

00

Sill

nmit of the oi

irve.

990

1G2-3

55-4

The values of the ordinates

for every yrvTSn ^'^^•' <^'^lcu-
lated by the formula

?y = 55-4 ^-•'^•ss '"3' sT-exio-i

in 10-1 sec.

0-0

0-0

30-0

4-3

400

25-5

50-0

49-8

GOO

54-9

70-0

44-4

80-0

29-5

90-0

17-3

100-0

9-8

110-0

4-8

120-0

2-4

130 0

1-2

1400

0-6

J=+0-84.

102

Art. 1. — K. Fuji :

Tables A'L— VIII. arc tho^e for tlie
relation between the l^readth of tlie stimulus
and the height of the resulting diseharge.
AC et seq. written on the heads of the columns
of the breadth of the stimuhis are referred to
the accompanying figure.

Tai-.le \'L

Oscillogram No. o7.
Date of experiment:
Preparation:

Temperature of the organ :
The formula employed in
calculation:

Oct. 'l:\, VMV.).

A wliok' organ (the same prepara-
tion as that of No. 35 and No. MO.).

7/==34-0[l-10-°''^«-f-^-i)j^

Eeadino- of

Eeading of

Breadth of

^Numljcr

1 mm.

No.

position of

position of

stiinulns

of

on filui

C.B.

C. Ch.

(AC)

revolutions

corresi^onds to,

m mm.

m mm.

m mm.

per sec.

in 10- 1 sec.

I

640

548

60

6-60

303

'1

63-8

5-3

6-59

303

3

63G

4.8

6-79

2-95

4

63-4

4-5

6-94

2-88

5

63-2

4-3

7-28

2-75

6

630

41

7-27

2-75

7

62-8

3-5

7-66

2-61

8

620

30

7-74

2-58

Height

Breadth

Heio-ht

Hei<4-ht of

Xo.

of
stimulus

of
stimulus

of
dischar^-e

discharge
calculated by

Difference.

m mm.

in 10-* sec.

m mm.

formula.

1

45-5

18-2

31-5

31-5

00

2

45-5

161

30-8

30-2

+ 0 6

3

46 0

142

28-3

28-4

-01

4

46-5

130

27-5

26-9

+ 0 6

5

46-3

11-8

25 0

25 0

00

6

460

11-3

23-3

241

-0-8

7

46-3

91.

190

18-8

+ 0 2

8

45-6

7-7,

12.-)

13-9

-14

Researches on the Discharcre of the Electric Organ.

10[^

Taele VII.

Oscillograms No. 40 and No. 41.

Date of experiment:
I 'reparation:

(Jet. [24, 1909.
Left organ (whole). (The same
organ as Xo. -)8 and No. 39.)
11-5° C.

Temperature of the organ:

Primary eurrent: 35 amperes.

The formula employed: No. 40 7/==48 0xI0-"X"-°«»'[i-10-^--"'^ -°>]

No. 41).

No.

L'eadiny- df

i)Osition of

C. B.

L'eadino- of

position of

C. Ch.

Breadth of
stimiilns

(AD)

Ximiber of
revolutions

1 mm.

on film

corresponds

to,

in mm.

in mm.

in mm.

per sec.

in 10-* sec.

1

65 0

54-8

9-8

7-00

2-8G

2

64-2

6-4

G 93

2-88 !

3

03-8

G-7

G-77

2 -96

4

(33C)

GI

G-99

2-86

5

G34

G-4

(')G7

300

G

G3'2

5 0

G-G7

300

7

G3 0

50

G-99

2-86

8

G2-8

41

70.->

284

No.

Height of
stimnlns

in mm.

Breadth

of
stimulus

in 10-4 sec.

Height

of

discharge

in mm.

Height of

discharge

reduced

i to Stm. 44-5,

in mm.

Height of
discharge
calculated,

in mm.

Difference.

1

44-5

28 0

44-5

44-5

450

-0-5

2

45 0

18-4

40-5

' 399

39-7

+ 0-2

3

44-8

19-8

39-5

391

401

-10

4

450

17-4

38-2

i 37-G

37-8

-0-2

5

45 1

19-2

390

38-2

38- G

-0-4

6

44-5

150

33-5

33-5

34G

-11

7

450

143

35-G

] 350

33-3

+ 1-7

8

440

11 G

29 ]

29-7

29-7

+ 00

(C. B. Circuit-breaker.

C. Ch. Connection-changer.)

104

Art. 1.— K. Fnji

Xo. 41.

Xo.

Reading of

position

of

C.B.

Reading of

position

of

C. Ch.

Breadth of

stimulus

(AD)

Number of
revolutions

1 mm.

on film

corresijonds

to,

m mm.

m mm.

m mm.

per sec.

in 10—* sec.

1

62-8

.548

4-2

605

3-31

2

630

5 J

40

612

3-27

3

(53-2

>>

4-8

612

327

4

(i34

5>

5-5

647

308

5

63-6

)>

60

6-38

314

6

63-8

,,

5-3

608

3-29

7

64-2

>'

72

625

3-20

S

6.50

"

71

6-35

316

No.

Height of
stimulus

Breadth

of
stiuiulus

Height of
discharge

Height of

discharge

reduced

Hight of
discharge
calculated.

Difference.

lu mm.

in 10-* sec.

in mm.

in mm.

in mm.

1

45 0

139

36 5

359

35 0

+ 0-9

2

44-5

131

33-7

33-7

336

4-0-1

3

45 0

15-7

36-6

360

35-9

+ 01

4

45-5

170

380

36-7

36-6

+ 01

450

18-8

37 0

36 4

37-4

-10

6

450

17-5

39-2

38-6

36 0

+ 2-6

7

44-8

230

390

38-6

38-7

-01

8

450

22 4

390

38-4

37-9

+ 0-5

Roseàrches on the Discharjxe of the Electric Organ.

lO:

Taijle Vlll.

Oscillograms No. öU, Xo. ()0 and Xo. 01.

Date of experiment:
PreparatioD:

Temperature of the organ
R.esistance of the organ:
Current in primary:

No. 51)

No. (')()

No. (U
Formula employed :

No. .7.)

No. (;o
No. r.i

Noy. 10, VM).
Left organ (whole).

Ahout 200 ohms.

Ô amperes
4 ,,

!j='2l3 X 10-"^-''«'>[ 1- 10-°-^'^«^^'-->},
// = 3-2 1. X \0-"'"'-' [1- lo-°-^3'5('-'-^)],
7/ = :3.S 8 X 10-»>^""'i'^[ 1 _ 1 o-^-*2X«-t;)j _

N

o.

No.

Heading of
position
of C. B.
in mm.

Heading of
position

of
C. Ch.
in mm.

Breadth

of
stimulus

lAP')
in mm.

Ximiber of

revolutions

per sec.

1 mm.

on film

corresponds

to,
in 10^1 sec.

Height

of

stimulus

in mm.

1

()-2 0

62 0

8-2

()00

3.33

:-380

o

(51-0

))

61

6-28

3-18

37-5

a

(iO'S

, J

54

5-46

3-66

:^7-2

4

006

,,

55

5-70

3-51

37-9

5

604

,,

4-9

5 87

3-41

381

C)

60-2

,,

43

5-77

3-46

37-5

7

(500

)î

2-3

5-67

353

32-8

S

620

■ 9

8-2

5 08

3-34

:37.5

Xo.

Breadth

of

stimulus

in 10-* sec.

Height of

discharge

in mm.

Height of

discharge

reduced

to Stm.37'5,

in mm.

Height of

discharge

calculated.

in mm.

Differt'nee.

Height of

discharge

calculated by

neglecting

fatigue

factor.

1

27-3

210

20-5

20-5

00

20-5

•2

194

19-5

19-5

18-9

+ 0-6

190

3

19-8

190

19-3

18-8

+ 0-5

191

4

19-3

18-5

181

18-5

-04

190

5

10-7

180

174

17-5

-01

181

6

14-9

170

170

16-5

+ 0-5

17-3

7

8-1

6-3

110

no

00

11-6

8

27-4

19-3

19-3

193

00

20-5

106

Art. 1.--K. Fuji :

Nu. GO.

No.

Eeading of

position

of

B.C.

in mm.

Heading of

position

of

C. Ch.

in mm.

Breadth of

stimulus

(AC)

in mm.

Number of

revolutions

per sec.

1 mm.

on film

corresponds

to,
in 10-4 sec.

Height of
stimulus
in mm.

1

620

62 0

11-9

900

2-22

500

'2

010

y -7

883

2-27

500

3

(50-8

8-5

891

224

50-5

4

GO-G

81

940

213

500

5

G04

—

889

2 25

G

GO -2

80

9-33

214

50-8

7

GOO

64

—

—

500

8

G2 0

11-7

8G1

232

500

Xo.

Broadtli

of

stimulus

in 10-* sec.

Height of

discharge

in mm.

Height of

discharge

reduced

to Stm. 50-0,

in mm.

Height of

discharge

calculuted

in mm.

Difiereuce.

Height of

discharge

calculated by

neglecting

fatigue

factor

1

264

320

320

320

00

320

o

22 0

30-8

30-8

310

-0-2

31-9

o

190

30-2

29-7

300

-0-3

31-7

4

17-3

290

290

290

00

31-5

5

—

—

—

—

—

—

6

171

28-3

27-5

274

+ 01

374

7

14-8

26 0

26 0

26-2

-0-2

310

8

274

26-3

26-3

264

-01

320

Researches on the Discharw of the Electric Ort^an.

10<

No. Gl.

No.

Reading of

position

of

C.B.

in mm.

Eeadino- of

position

of

C. Ch.

in mm.

Breadth of

stimulus

(AF')

in mm.

Xumbor of

revolutions

per. sec.

1 mm.

on film
corresponds

to,
in 10-1 sec.

H(üght of
stimulus
in mm.

1

620

02 0

120

8-59

2-33

a

2

010

9-5

8-90

2-23

p

3

GO-8

8-7

8-74

2-29

o

4

60-6

8-8

9-73

2 00

2

5

00-4

—

9-35

2-14

—.

Ü

GO-2

7-0

907

207

7

000

71

10-25

1-95

8

()'2 0

12-7

9-84

2-03

No.

Broadtli

of

stimulus

in 10-* sec.

Height of

discharge

in mm.

Height of

discharge

calculated.

in mm.

Difference.

Height uf
discharge

calculated by
neglecting

fatigue factor.

1

29-4

380

38-0

00

38-0

2

21-2

30-9

30-1

+ 0-8

36-4

3

19-9

30-0

35-4

+ 0-0

35-9

4

18-1

340

34-3

+ 0-3

35-0

5

—

—

—

—

6

10-4

32 0

32-8

-0-8

33-9

7

13-8

30-4

30-4

00

32-1

8

25-8

30 0

35 0

+ 0-4

37-4

108

Art. 1.— K. Fuji

^J\VBLE IX.

Oscillogmms No. 74 — TT).
Date of experiment:
Preparation:

Teniperature of tlie organ:

Nov. 17, loot).

A quater of an organ witli one

nerve-trunk.

14-5°.

No.

Distance betw.
two cmtiicts

on

sfmalatioii-

appavatius

111 mm.

Xuuiber

of

revolutions

per sec.

i mm.

on film

corre-
sjjonds to,
in 10—* sec.

Interval
between

two
stimuli
in mm.

Interval
between

two

stimuli.

in 10—1 sec.

M. L. P.

of

first Disch.

in mm.

M. L. P.

of
second
Disch.
in mm.

1

2

8

4

5

6

7

8

9

10

11

12

13

14

15

16

260

240

22 0

200

180

160

140

12 0

100

80

6 0

40

2 0

10

0-5

02

5-98
606
6 18
6-22
5-84
605
6-17
611
6-34
6-57
6-62
6-58
6-71
()-60
6-67
6 02

3-35
3-30
3-24
3-22
3-43
3-31
3-24
3-28
3-16
305
302
304
2-98
3 04
300
200

36 0

33-5

30-7

280

23-3

21-9

19-5

170

13-9

110

11-5
90
5 0
30
10

Coincide.

1205

110-4

99-5

90 2

79-9

725

()3-2

55-8

43-9

33-6

34-7

27-4

14-9

91

30

00

31-5
320
32-5
33-5
31-0
32-5
330
33-5
330
340
340
34 0
340
34-5
350
35-5

31-5
32-5
33-5
34-3
330
35-7
370
39-4
430
480
480

Xo.

Ratio

of

two

M. L. r.

I \lerval

Ijetween

first Stim.

ana secoiiJ

rli sell arse.

in mm.

Ditto

in

10-4 sec.

Heio-ht

of

first Disch.

in mm.

Heio-ht

of
second
Disch.
in mm.

Ratio

of

two

heights.

1

1-000

67-5

226

530

39-0

0-736

'_)

1015

6()-5

214

51vj

41-5

0-806

^)

1-030

64-2

208

48-5

45-5

0-989

4

1-023

62-3

201

480

49 0

1-020

5

1063

56-3

193

47 0

48-5

1-030

()

1-100

57-()

191

4()0

46-7

1015

7

1-120

56-5

183

4()-0

45-5

0-990

8

1175

56-4

185

45 0

420

0-935

9

1-301

56-9

180

45 0

24-7

0-550

10

1-411

590

180

43 3

4-5

0 104

11

1 411

59-5

180

415

45

0109

12

—

—

—

42-5

?

?

13

—

—

—

41-8

?

?

14

—

—

—

41-2

—

—

15

—

■ —

. — .

42-5

— -

—

16

—

—

—

41-5

—

—

IJesoarches on the Disoharse of the Electric Ort^an.

109

Taj]

lj:

X

Oscillogram No. G5.
Date of experiment:
Preparation:

Temperature
Stimulus:

)ftl

le oryan ■

Nov. 10, 1900.
l*art of an organ
trunk.
13 5° C.

Ascending;.

with one nerve-

Xo.

Disiance letw.
two contiicls

on
stimulation-
apparatus
in mm.

Number

of

revolutions

IDer sec.

1 mm.

on film

corre-

sjionds to,

in 10—* sec.

Interval

between

two

stimuli

in mm.

Interval
between

two

stimuli

in 10-* sec.

M. L. 1'.

of

first Disch.

in mm.

M. L. P.

of
second
Discli
in mm.

1

260

6-34

316

420

138

380

68-0

2

240

6-57

304

—

—

39-6

—

3

22-0

6-55

305

36-0

110

400

890

4

200

6-20

3-22

300

97

380

390

5

180

6-27

319

27-6

88

380

430

6

160

6-36

314

—

—

37-5

—

7

140

6-47

309

22-2

69

390

48-0

8

120

6-55

306

19-6

60

390

50-8

9

100

6-54

306

160

49

390

560

Xo.

Eatio

of

two

M. L. P.

Interval
(between
1st Stim.

ana

2ucl Discli.

in mm.

Ditto

in

10-4 sec.

Height of

first Disch.

in mm.

Height of

second Disch.

in mm.

Eatio

of
two heights.

1

1-790

110 0

348

24-5

9-0

0-367

2

—

74-4

226

22-6

17-8

0-788

3

0-975

75-0

236

21-5

190

0-885

4

1025

690

222

21-0

190

0-925

5

1130

70-6

225

20-3

180

0-900

6

—

71-0

222

200

150

0-750

7

1-230

70-2

217

19-5

16-0

0-820

8

1-300

70-4

215

19-9

110

0-553

9

1-480

72-0

220

18-8

3-9

0-207

Published December 11th, 1914.

K . FIJI.

RESEARCHES ON THE ELECTRIC DISCHARGE OF THE ISOLATED ELECTRIC ORGAN
OF AST RAPE BY MEANS OF OSCILLOGRAPH.

PLATES.

I. III., Photographs.

IV.- v., Drawings of instruments.

VI. ^XIL, Curves.

XIII. XXX. , Oscillograms.

Abbreviations used in Plates.

Stim. or Stra.: Stimulus.

Eesp.: Eesponse.

C. Stim.: Closiug-stimulus.

O. Stim.: Opening-stimulus.

E. T.: Equivalent time i.e. time interval corresponding 1 mm.

of abscissa on oscillogram.

^m> iSni) i^m- All tlicsB abbreviations represent " Stimulus."' Figures

in suffix show the order of experiment with respect to
time. Where two successive stimuli were used, ]S,„
represents the predecessor in those of the »(th experi-
ment, and nSm the si\ccessor in the same set.

I'm) lEm, ^Em- AH tlicsc abbreviations represent " Piesponsc "". The

numbers in prefix and suffix show the correspondence
of the response to the stimulus having the same
respective affixes.
2ry D.: Secondary discharge.

Sry D.: Tertiarv discharge.

Elecù-i

X /^obus elff^/ctts.

ETectrt'e nerves-,
ffirec iriink.9 nrnon^ /h-e.

Ctce/réC' <*r-ffa7t'.

ri^.

A direct

IJa/tini Une.

■ lies on the '■
(Japanese Elf

iations used

Stimuliu

X.

iu sufltix sliow-

>K>ucliug 1 mm.

ruent, and

oecond;i.v;,- dl.-
Tcvtuii'v dii^cî

■«\1^.S«M>\v>4\

Jour. Sei. Coll., Vol. XXXVII., Art. 1, Plate I.

Fie:. 1.

Astrape japonica.

Anterior part of the skin has been removed to show the electric or<>aus and the electric nerve«.

Fig. 2.

A direct stimulus and the resulting discharge.

K. Fuji: Researches on the Electric Discharge of the Isolated Electric Organ of Astrape
(Japanese Electric Ray) by Means of Oscillograph.

Flg. 1.

Jour. Sei. Coll., Vol. XXXVII., Acrt. I, Plate II.

Shutter.
Front view.

Fig. ±

]îack view.

K. Fuji : Eesearches on the Electric Discharge of the Isolated Electric Organ of Astrape
(Japanes*^ Electric Ray) by Means of Oscillograph.

Fm. 1.

Jour. Sei. Coll., Vol. XXXVII., Art. 1, Plate. III.
Eegisteriiig apparatus (cover removed).

Fi"- 2

Stimiilation-apparatus clamped to a table.

K. Fuji : Researches on the Electric Discharge of the Isolated Electric Ofgan of Astrapc
(Japanese Electric Eay) by Means of Oscillograph.

Jour. Sei. Coll., \^ol. XXXVII., Art.l Plate IV.

Drùinff Shftff of UtB^ Reffistering Driiriv nnd Its Accessories.
C^/h Actual, Size.. )

c

i3°0

Circuit -breaJters .

o

F ig. 2.

Fug. 1.

OJC

Flg. 3.

ÈÀ

n

:^

\Ç)\ Heyisterijig DruTn .

Coupling .

Fig. I . Drivithg Shaft- ojifL It^ Coupling .
Fig. 2 . Circuit- breaJee^r.
Fig. 3. Connection -chfmge.r.

Shuti^r. ( ^/s AcIu-clL ■St'zey.)

CentraZ Se^fyion/.

Ba.rk Yi^eiv.

Fi

,<7 • 4- .

K. Fuji : Researches on the Electric Discharge of the Isolated Electric Organ of Astra'pe
(Japanese Electric Ray) l)y Means of Oscillograph.

1^

Oscï/Zogram No. 54-.

Jour. Sei. ColL 1^0 1. XXXVII., Art. 1, Plate VI.

( Red-tLced to U^^ qJActnalùf v.fail,Sc(iJf.)

:

'Z2.

Li.

. i

Full Uji-e on tJt^i mldttLe, parC g/*

Or iff in- ^ caordJ7Ui£t!X is tttJten. tU.
Out heffinninff of sOnuiLua .

1
<('^'

3.

—

/-■

^

^

^

y

1

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--^

w..

4-.

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X

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X

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^

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J

Curve

Ö.

f^

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30

/

/

\

\

/

\

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/

\

/

V

N,

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' 1

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S.

/

/'

\

/

\

/

à

\

/

\

f

\

1

/

1

è

^

^^

--^

e

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7.

i

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1

\

\

/

\

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\

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1

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€uyt/e

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î

:

.£

y-

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K. Fuji: Researches on the Electric Discharge of the Isolated Electric Organ of Astrape (Japanese Electric Ray) by Means of Oscillograph.

Oscillogram No. 37.

Jour. Sei. ColL,i^ol. XXXVII., Ârt.l Plate Vll.

Curve

J.

/

Ts

B

3U^<t

repr
ht ma.
rid, bi

X, -

1 Uu^

xi.

t/t£-

f

] S

\

>

f

\

\

1

1

\

^

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y

f

N

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Curve

2.

-'-

/

\

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N

t

\

1

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s

/

\

s

'^

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^

i-

Curve.

3.

>

n

\

/

i

\

/

\

/

]

\

\

/

/

i

\

^

1

y

1

N

^,

Curve

*.

^

\

/

\

/

\

/

i

-^

\

f

/

i

\

"^

Ï — ,

^

/

i

--,

^

0

*

0

a

o

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»r-^

n

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,

to

o

Cu^,

s.

.o

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k

/

1

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,o

)

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/

\

o

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j

N

-^

"urve

6.

7

"

.o

r^\

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/

\

\,

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\

^,

^

/

V

iv^

7.

20

f

^

N

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1

Curve

8.

>^

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i

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V,

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,

2

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4-

0

e

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so

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li

o

1*0

leo

lao lO'^sec

K. Fuji : Eesearohes on the Electric Discharge of the Isolated Electric Oryan of AstTn]K (Japanese Electric Eay) by Means of Oscillosjraph.

Jour. Sei. ColLVol.XXXVII..Art.l. Plate VM.

Oscillogram .Vo. 4-0.

Turw

,.

j

/

\\

^

1

\

.

/

\

1

j

\

\

,

'

\

\

!

/

1

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i

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*

i

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r--

—"

^.™,

2.

—

—

f\

\

,,»

_

1

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/

\

,

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\,

/

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—

/

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N

„

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Clow

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1

7

H

~

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/

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V

i _

1

1

1

r

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A

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\,

~~

-^

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i

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_

■~~',

u

t.

/\

1

\,

1

\

1

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\

/

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^

J

\

^

^

y

^^

rurfe

6.

/

r\

/

\

1

\

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i

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/

\ I .

/

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V

_,

y

?lirw:

g

/

^

/

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/

^

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f

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/

^

1—

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,

_.

^

__

■^

'_ ,

^

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,.

1

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s,

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1

1

\

L

/

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X

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^

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■

lum

a.

1

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(

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;

/

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y

-H^

«

Oscj'Ilogra

Ac

No. C. J.

i

^

/

\

/

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1

1

i

' 1

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.X

\

/\

1

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^

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-<.

I

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VA

_

,

K.FujL: KtiBü

1 tliö Electric Dischiirgc of the leolated Electnu Organ of A,ir<i^t (Japanoso Elootno Rny) hy McauB of Osüilk>gwph.

H

Jour. Sei. Coll.,Vol.XXXVII., Art.1, Plate X.

OseiZlagrams ^o. 35 ofLti No. 36.
.Va. i-2 anti No. 4-3.
.Vo. 55 and, No. 56.

OsdUograna No. 38

' 1
and. No. 54-.
No. 63 <^ N,

\

-^v

1

' , 1

No. 62
No. i-9

.64-.

/

/

' '

1

^ /

Î

4°'

/

/

'

\

/

A

/

V

/

;

'^ //

*

A

—

—

-

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j/

f

/'

J/

—

//

/

^2!L

/

^' 1

1

/ /

/yy

v^

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' k

^

X-.

:^^

f^^^

^^^

1

1
1

'

»o :iu

l_

I .

MÏa^ou

rue \r No

.4-2

and. No. 4-3.

I, so

m.

"

No

.55

and. No. ..

6.

y\

/

\irj,

/

/

"^A

/m

/

■/

/

r

ao

/

/

/

1/

/

A

/

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'

1

(^

/

/

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/

y

J

/

/

/

/

/

y

V

i

.^-.-

.é:::.

"

I
1

1

'III!

OsetlLoffT'extru \ No. C. 5.

1

!

\l20

1

1

\

i

1

^210

\

■""" T'

V

-

2

Y

!..

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.

1

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1 1

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4 _ _

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>x

5

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lO 20 30 4-0

SO 70 mm.

S 10

IS 20 25 SO mm

f.. ^ Post/ion ar elmJJ-adei i

K. Fuji ; Eeaearches on the Electric Discharge of the Isolated Electric Organ of Attrape (Japaneso Electric Bay) by Means of Oscillograph.

Jour. Sei. Coll.,Vol.X/XVII.,Art.1, Plate /I.

Observed^-

i ,

Cetloalaled^.

,

yÀ^

—

/

ï^-r^

I'-

"■'

/^'

/

'

/

/

OfciUn

1

grams No. 4-0 afttt 4--

/

Red

poults'! Xo. 4-^

/

j

1

!

/

' T' '

!

/

o

'

"

'

"

"

'

\^''

■■ W

'r

7--f

'/

Jfo.\ i-0.

No.

4-1.

V -

BLtt^/o pointa: observe.dy.
itetl- poùtt^ : c<tif'ii/ftf^Tfl. .

-nXOOeOi -00t29X(t--0)

JVo. 4-0. y- 4-SO XIO r (1 - lO ).

~nmoeo4^ -o oé2sxii-o)
yo.4-1. y-4-6 9 xlO X (/ - 10 ).

1

-4-

1

4-

1

1

1

.^

/I

\

'/

\

/

OscUlogram. No. 37,
• observetL.

/

i

-0 oaesxU
y~34-0X(t-10)

1

-S-1)

/

1

1
1

/ 1

Î

\

Pia. I. ßreeuHA of st£/nuJi/s .

i 1 1 Mill

\OaciILograjnx \ No.S9, No.SO cauiNo.ej\

!

1

i

;

-*-„

-m.ei.

A^

4-

"

%'

•

-<r

i

"'

,g\ —

M^

1

k"

1

//

',

.f

'1

/

■^

'

-^m.

a».

i

^

Ä

i

J/.

1

1

k

/

^ Bla£/e paints : observed. ■

m.ai y-3a a .lô'Ti"-'' 10°'.°°" '"'"'

if

f^î

/

^ 1

1

1

i

1

BretuUA qf stimulwv.

Fig. 3.

Fig. 2.

K. Fuji ; KesearoheB on the Eleotrio Disoharge of the Isolated Electric Organ of Astroje (Japanese Electric Ray) by Means of Oscillograph.

Jour. Sei. Coll.. l/ol. XXXVII., Art.t, Plate XII.

^

1

1

i

^.^

___!

!
1

\

-1

i y

! .

\

h-

«

\

\

\

V

»

\

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!

\

î

1

i

k.

1
1

Oscilioffram.

No. 7* -No

75.

\

i

1

o|

observsdy.
caloüUUed'

1

V

^<

1

1

•n

/

1

\

^^^

. i_ 7 i

.'\bsci»sa..'. tn
Ortiinujte. :

ij-t .

Un . to Zfici fle^p.

Se

y/.

I

^

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1

\/

JUsp.2.

- \

\/

lteîp.2.

/\

/

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/

\/

/

\/

f-

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/

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y

1

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/

\

/

/

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i

A

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1

\

\

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1

1

/

1

\

K

/

1

/^

T

(

/

\

'

20 t-0 60 80

Ejcp. 3. \Luca^}

O 20 4-0 60 SO

14-0 10'

\

OmOU.

from.

2V

o. 65

i

L^

" '

obaeri

fed..

^

-v

^

-

-*/ i

i

\

V

(

V

1
1

\

/

i

V

/\

X

1

\

i

i

1

N

1

\

^

1 ' 1 :' 1

Aàactssa: limeß^m let Stm-.to

1
Jntl Stfih.

«o

observed^.

Ordinate-: timejront' 2 net

Shn-tD 2nd

Reap. 1

calcuÙLted:

2nd. iîesp

\

\

2nd ReA-p.

\

___

\

\

\

V

!

\

!

X

^

\

■"

-^

he- ß«9p.

/st /iéiap.

20 4-0

O 2 1) 4-0 60 SO 10

Fig. 4-. i'J'p. II.

K. Fuji ; Eesearohes on the Eluctric Discharge of the Isolated Electric Organ of .Ulraf (Japanese Electric Eay) by Means of Oscillograph.

Jour, Sel. Coll.. Vol. nXill.. Art. 1. PhU XIII.

fia- 1- Oscillogram No. 54,

K. Fuji ; Resuttrüliee on tUo Eleotrio Dieclia

e Isolated Electric Ors

m/)« (Jupanos-; Elcotrie Riiy) by Moans of Oaoillograpli.

Jour. Sei. Coll., Vol. XXXVII.. Art. 1. Plaie XIV.

T.-mpoi-ntm-c r,f tbo oivni. : 1 1 0» C,

■2 HO 200 a-92 %<.H J 80 3 24 3 00 m 10-'

A\ ^

Fig. 2. nsciUoerain No. 30. I'roparation some as No. .35. Oidor of stimuli reveispil.

anO 2-61 200 270 20ïi 258 2 7fi in 10"'

■ i:Uctric DiBülmrt,'*' of tti.- laoIuU'd Elocfriu Orgiin c.f .l.irape (JivpanosL- Electric Ray) ti.v M.snns ..f Oficillu

Jour. Sei. Col, Voll. XXXVII., Art. I, Plate XV.

Fig. 1. Oscillogram No. 38.

TL-miierature of the organ : 11-5'' C.

Jour. Sel. Coll.. Vol. XXXVIl., Art. I. Phle XVI.

Kig, J. Oscillogram No. 02

Tcmperatore of the organ : 13 5° C

l'ig. 2. OâiWognim î^o. 03,

ii'ig. U. Oscilloftrain Ko. til.

K- Fuji ■ ^tKTihtm on tl.e Elrdric D>a<^»r.f of tVo Iwlat»] El»ctrie Orj>o of .Utnju (Jipta«- Ktwtris \l\y. Uy M '\w of 0«:illa?

Jour. Sel. Coll.. Vol. XXXVII.. Arl. I. Plaie XVII.

FiR. 3.

Oßillogi-aiii No,

, r,fi.

Temperature of tlie organ :

13-8° C.

l;.-si^la

"'■■■'■' "'^^'-

i: 200 .y,

i;'i'

:;M : :-, .;■.. ■.:: :: Is^ :: ,.

;.,; , |s i„ lO-soe.

i

1

v_A

1

\ '

■

\

^*^

^

■^

K

lî

li,

r

—^

i;, i;.

Il Kliîotria Diaohtirs« ni tho laolatod Elootric Organ of .iHraj-r (.Upnnoso Electric Tiny) l.y Moana nf OBciIIogTft].1i

Jour. Sel. Coli., Vol. XXXVII., A't. I. Piate. XVIII.

Fig. 1. Oscillogram No. 50.

Tempentare of the

Fig. 2. OBcillogram No. 60.

Preparation same as No. 59,

^Eg

Fig. 3. Osillogram Xo. G I

s No. 59, No. (10 and No. Ijl are of a set of experiments carried nut witli the same praparation.
K. Pujii lîosowoliCB on t!io Kleotrlo Diaolmrgo of tlio leolatod Elootrio Organ of Attripe (J-ipaneso Electric Ray) by Meana of OBOillograph.

Jour. Sei. Coll., Vol. XXXVII., Art. I, Plaie XIX.

Pig. 1. Oscillogram No. 74.

K. Fuji : Roflearcliee on the Electric Discharge of the laqlutud Electric Organ of Aurape (Jupaaest- Eluctric Ray) by Means of Oecillograpli.

Jour. Sel. Coll., Vol. XXXVII., Art. I. Plate XX.

Fig. 1. Oscillogram No. G.5

Tmiperature of the ovgm : 13-5° C.

.105 .3 22 31'J 3 1.1 3 0!) .3.00 .TOG

=ij. ■ .iJ.

,5; ;b

.ft, .H..

'\ /" ^"nii. i"'a

"55

II ,v. t). l'a, -B,

Fig 2. Oscillogr.im No. 84,

K. Fuji : lîesearoIioB ou tlio Electric Diaclinrge of the Ieolat«d Electric Organ of Aitrij:r (.liipuncRc Electric Ray) li.v Honns of Oscillogrnpti.

Jour. Sei. Coll.. Vol XXXVII., Art. I. Plslr XXI.

Fig. 1. Osrillogram No. :".0. Tl-mpr

\n. nf,.N|, : 1 2 :1 4 .'",

i^tllllcn: •jnoii, i;'!' 'J Ml ä.w 250 2i;o 2uo

2 71

71 in 10-

Pi,, 2 0«ül..._MinH X... .17

FiR. :l. Osiillnsram No. :,\.

/t

TRT ..s,

,1î., .,1;.. ,li, .n, ,r,, ,R,

'IVmprraliiiv of llir oi-Knn- .K.'.T TiisovtBil rosintaiire : WO S! .

r

V.

A.

r

"v>

V

^

\,

~.

,>,

.s.

i ,

"^T"

1^. "N.

,1

,i,\ ,1;,

^t;.,

,1:.

..l;,,i;,

..F., ,1

^

u, ,,,i:.,

:r"Ä AT"

_i; ,u, .IÎ. ,1;, ..1;,

^uji ; Reuoiirclii-s on tt.e Eloctric Disohargo of the laolatod Electric Organ of .l^frnpe (JiipanoBO Electric Ray) by Means of Oscillograpli,

Jour. Sel. Coll., Vûl. XXXVII.. Art. I. Plate XXII

Fig. l- Oscillopifim Xo, 70,

Fig. ':. Oscillot^raui ^a 71.

il . ?i : Descending.
6 R : Ascencliiig.
Descenclbiy,

No. of exp. : 1 2

E.T.: 3-ôl 3:S1

340 :i-51 355 355 3 53 3-41

K. Fuji ; RcsoarohPB on the Eloctrto DiacharRo of tho laolatcd Electric Organ of Aftritje (JaptinoBO Electric liay) by Means of Oaoillo^iipli.

Jour. Sei. Coll., Vol. XXXVII., Art. I, Plaie XXIII.

Fig. 1. OaciUogram No. 07.

Nnmber of stimtili : 885 per s

E.T. : 2 80x10-

A

,v,v.*.^v.y.wdv

Fig. 2.

OBcilbßraiii No. 8.'J.

Nnmljer of stimuli :

1412 per sec.

Temperature of the organ : 15 5^0.

E.T.:

ôOSxlO-'sec.

1

m

■■H|

■

Éj

ËÊÊ

1

■■

1

^^^H^^^^^^

^^^^

■

fP

■

'

9

!

mnzz:- i^^^^H

■

K. Fuji : RwoarcliOB oa the Eloctrio Discharge of the Isolated El.-clric Organ of J'lrajie iJapanoBo Electric Itay) by Mwana of OflCillograph.

Jour. sa. Coll.. Vol. XXXVII., Art. f. Phte XXIV.

Fig. 1. OscillofTi-ara. No. 1.

Typical form uf fatigue

Xumbf-r oE stimuli : About "27 per.

K. Fuji : Reeentcbca on tlie Electric DiBcliargû of tbo Isolated Electric Organ of AAr,yj>e (JaiKmeao Eluctvic lî.iy) by Means of Osuillograpli.

Jour. Sei. Cell . Vol. XXXVII.. Art. I. Phile XXV.

Fin. I. Oscillngium No. :l.

Niimlicr of stimuli : Alioiit 2i per aeo.

Tîilîen at (in intevval ol" one lionr aftev No. il.

Osdllof-rnins No, ;i. No. 4 and No.:.5 aip of a spt of cxpoiinipnts canietl out with tlic saliio in-opavatioii.

K. Fuji 1 Roioiiroli,.. on llir El.'ctric Di«oliar|;o of 111.) I«.l»b>,> EU'ol.rio Or^.v« nl .Ulmpe (JupamiBe Elcotri« E»y) by Moan, of OmilloBnipli.

Jour. sa. Coll., Vol. XXXVII.. Art. 1. Plaie XXVI.

Fig. 1. Oscillogram No.

Fatigae by descending stimuUitiDg current,

Fig. 2. OBcillogrom No. 8.

Fatigue by descending and ascending stimulating cuiTents altci-nately given.

Ascendinp-stimuli,

Fig. :i. OsciUnpram No. 7h.

Descondiug-stimnli.

Fig. 4. Oscilln^miiii Nn. 7h'.

ABccnding-fltimuIi.

*'Poji lUwMrcbM OB tb<- Glwtric iHaclikr)^ of tbv laolftt«-! Elc«thoOrpui cf Attr^ft (iipui«» E1f«trie Eaj) by Mmiia of O«:illofnikp))-

4oiir. Sei. Cell., Vol. XKXVII.. Ait. 1. Plate XXVII.

I'tt,'. 1. Uscillogmm Ho.

K, Fuji : Ito»earuhta on tlic Ek-ctrtc Diaclinrgo of tlio IboIiiU-iI Eloctrio Organ of .Utrnj'f (Jiipancee Electric Roy) Ky Mohiib of OBCilli.yrui.li.

Jour. Sel. Coll., 1/0/. XnVII.. /(/■(. 1, Phie XXVIII.

Fig. 1. Oscillogram No. C

Ik. Fuji : ItuoAroliEd on the Ek-otric Dischargo o£ the Isolated Electric Org.ia of Atlr.tpc (Jupiinose Electric Uiiy) by Mciins oï Oscilloürni.li

Jour, Sei. Coll-. Val. XXXVII.. Art. 1. Plate XXIX.

Via. 1. OsciUoBram N\.. 1011.

n~ ilisclmv".'. IiKnrt.'il irsisbmcc : 100 S . Tmiiwratniv ■ ■2:î'0<'C.

Pig. 2. OBcillogmm No, 53.

K. Fuji : ItosBatcUos oa tho Electric Diachart.'O of t!w Isoliited Elüotrio Orgaa of .Iilr.ijte (Japanoao Elootrio KayJ l)y M«ftDS of Oacillograpli.

. Oall., Vol xnVII.. Art. 1, Plate XXX.

Yii-. I. üscilloRmm Nu

'IVmperalnre of tbo organ ; 1 1 ■'' C.

— i(

Fig. 2. Oscillogram No. 77.

Tempci-îitnre of the or^'an ; 1 1 .5^ C

i^-.:-^0è^

^feltPääS^

;;«^

K. Fuji : Keaearahea on tHo Eleotrio Disoharge oî tbe laolated Elactrio Organ of Ästrape (Japanoae Electric Eiy) by MeaiiB of Oeoillograph.

JOURNAL OF THE COLLECrE OF SCIENCE, TOKYO IMPERIAL TNIVERSITY.

vol. XXXVII., ART. 2.

Recherches sur les spectres d'absorption des
ammine complexes métaHiques.

I

Les spectres d'absorption cfes solutions aqueuses

des ammine-complexes cobaltiques et (eurs

constitutions chimiques.'^

Par

Yuji SHIBATA, LUijakvuhi.
Laboratoire de chimie minérale de l'Université impériale de Tôkiô.

Avec 17 /îyiires.

Quoi que ce soit un fait bien connu et très intéressant que les
sels complexes cobaltiques ont toujours les couleurs très diverses et
vives, et cle plus, que le changement de ces couleurs, d'après les
substitutions de quelques restes dans les ions complexes, est très
brusque, une étude systématique sur ce sujet, au point de vue
de l'optique, n'a pourtant été entreprise que par peu de chimistes.
Depuis que j'ai entrepris la présente étude, une notice intitulée
"Über die Beziehungen zwischen den Absorptionsspektren und
der Konstitution der Komplexenkobaltamminsalze" a été publiée
par R. Luter et A. Nikolopulos. "'^ Mais ce dernier travail bien
intéressant, a été exécuté avec un spectrophotomètre et, en
conséquence, l'étude n'est pas sortie de l'échelle spectrale visible,
tandis que la mienne, parce que j'ai employé un spectrographe de
quartz, s'est étendue vers l'extrême ultraviolet. Les matières
choisies dans les deux études ne sont pas non plus les mêmes. A

1) Une note brieve d'une partie de ce travail a été rapportée, sous les noms de
Gr. Urbain et Y. Shibata, dans " Comptes rendus des Séances de l'Académie des
Sciences " (Paris), 157, XV, 593. [Oct. 13, 1913]

2) Zeitschr. physik. Chem. 1913, 82, 361 378.

Art. 2.— Y. Shibata:

part cela, M. A. Werner a eonsacn' un chapitre de son omrage
aux stéréoisoméries des sels complexes de cobalt. ^^ Mais il n'a
traité que de la relation entre les couleurs vues à Tœil nu et des
constitutions des sels complexes. Je me suis donc mis à cette
étude en employant la méthode d'absorption spectrale des solutions
aqueuses des sels complexes cobaltiques les plus divers. Une
série de spectrogram m es du visible et de T ultraviolet que jo décris
précisément ci-dessous m'a permis de mettre en évidence des
relations assez intéressantes entre la constitution de ces complexes
et leurs absorptions.

La méthode et les matières.

Dans ce travail, j'ai employé le spectrographe'' de quartz,
construit par M. Adam-Hilger à Londres. Comme la source
de rayons, j'ai préféré l'arc de fer à cause de la facilité, avec laquelle
on peut connaître la longueur d'onde à l'extrémité d'absorption.

Les solutions des sels complexes ont été prises toujours à la

même concentration de j^^- et ^q— ; seulement pour quelques sels,

N
celles ont été étendues jusqu'à j^^q^^-- Les mesures ont ete trcKkutes

par des courbes en portant, suivant deux axes rectangulaires, les
logarithmes des épaisseurs des solutions et les fréquences corres-
pondant aux limites de l'absorption, d'après la façon de Baly-

Hartely.

Les sels complexes cobaltiques que j'ai pris comme objets de
cette étude comprennent les 26 espèces suivantes :
Cobaltihéxammines

CoiNH,%Gk Coen^Ck [Coen-lNH-^-^jCl,

Cobaltipentammines

[CoiNH^XCqCl, [Co{NH.;),ILO]CI, [CoiHN.;),OH]CL

lCoiNH,)NO.;]CL Igo{NH.:;)ONO]GI, [cof^.j[NH,)Br]Br,

[Coß^^{NH,)Br] {Ikomocampf er sulfonate), {d, â et l, l) [Co{'NH,)lNCS)\Cl.

1) A. Werner : Ann. Chem. 1911, 386, l-27:i

* J'ai commencé ce travail au Laboratoire de cbimie minérale de M. le prof. G. tlrban.
\\ la Sorbonne de Paris et je l'ai fini à l'Institut de chimie de l'Université impériale de
Tokio. Comme les deux laboratoires possèdent le memo appareil du même fabricant
londonien, j'ai pu heureusement ichever co travail, sans interruption ni emp vehement,
dans les deux laboratoii-es.

Eecherches sir los sooetres d'absorption des amuiine-complexes mélalli lues. 3

Cobaltitétrammines

[Co(NH.XC,0,']CI [Co:NH,;),CO,]CI [Co fJi,CO,]Cl

[Co{NH,\{H,0)CqGI, Ico':NH,\(H.JJ).;]CI, [CoiNH,%NO,(OH}]Cl
lcoiNH,X{NO.j).i\CI(Crocêo et Fiavo) [CoInH,;)NO./JI]CI

[Cn eii,CI.j]Gl(Prasco)

Cobaltitrianimine

Co(^^ff,),(^"o,);.

Cobaltidiammines

[Co[HN.;)JNO,),]NH, [Co:NH,\(C,0,){NO.^;}NH,
Colaltihéxaiiitrite
[Co[NO.;),]Na,

Les sels nommés ci-dessus ont été préparés par 1" auteur au
laboratoire de M. A. AVerner à l'Université de Zurich et au
laboratoire de chimie minérale de ]M. G. urbain à la Sorbonne de
Paris. Quant à leur pureté, elle a été assurée par l'analyse et par
les formes cristallines.

Partie théorique

Les diagrammes faits ainsi présentent, dans l'étendue étudiée
du spectre, deux ou trois minima des courbes (maxima d'absorption)
très nets, dont les fréquences se trouvent toujours respectivement
voisines de 2000, 3000 et 4000. Cette troisième absorption n'existe
qu'en quelques complexes contenant le groupe de nitro, dont on
verra la discussion dans la partie expérimentale. L'absorption qui
a lieu à la fréquence de 3000 montre phis de déviation que la
première à la fréquence de 2000, selon la constitution des ions
complexes.

C'est seulement la première bande d'absorption qui ne manque
pas à toutes les sohitions des sels cobaltiques, soit les sels d'ammine-
complexes, soit les sels ordinaires. ^^ Par conséquent cette première
absorption à la fréquence de 2000, nous semble e re due à l'atome
de cobalt, qui se place au centre des ions complexes,* tandis que les
autres doivent appartenir aux autres atomes métalloïdes qui sont en
connexion immédiate avec l'atome de cobalt.

* D'après l'opinion de M.A. Werner les sels cobaltiques, comme chlorure, nitrate, sulfate

etc., en solution arjueuses, font aussi des ions complexes avec l'eau [Cn{H-jO)f^]."
1) Comparer A. Hantzscli u. Yuji Shibata: Zeitschr. anorg. cliem., 1911, 73, 3)9-324.

4 Art. 2.- Y. Shibata:

D'après les recherches des physiciens modernes, les causes de
la production des spectres liniaires et de ceux de bandes sont très
probablement dues à la vibration respective des électrons positifs et
lies électrons négatifs.'^ A la suite de cette théorie fondamentale,
J. ^tark'^ a présenté une hypothèse, qui est expérimentalement
constatée à certains degrés. Son raisonnement peut être brièyement
exposé comme il suit : les valence des atomes chimiques ne sont
autre chose que les électrons négatifs qui sont liés avec les lignes de
force sur la surface d'un atome, petite particule élémentaire chargée
électriquement au signe positif. Ce savant a classifié ces électrons
de valence en trois catégories, selon la relation qui existe entre eux
et les atomes. L'électron de valence saturé est telle modification
qui a lieu entre deux atomes et, en conséquence, leurs lignes de force
se terminent à la surface de ces atomes. La deuxième modification,
nommée l'électron de valence insaturé, s'attache à un seul atome
et, en conséquence, toutes ses lignes de force n'atteignent qu'à la
surface de ce même atome, tandis que la troisième, appelée l'électron
de valence relâché (gelockert), coexiste nécessairement avec la
modification saturée, qui lie deux atomes, et il est caractérisé par
ses lignes de force qui ne se terminent qu'à la surface d'un atome.
Cette dernière sorte d'électron de valence, se rencontre, par
exemple, dans le cas de la double liaison des carbones de combi-
naisons organiques insaturées.

Or, d'après J. Stark, ce sont ces électorons de valence, qui
jouent le rôle du résonateur, dont les oscillations, excitées par une
énergie quelconque, soit celle de la chaleur, soit celle des rayons
lumineux, produisent des spectres de bandes. Le calcul de
J. Stark rend compte du fait que c'est principalement les électrons
de valence relâchés, qui donnent les absorptions de bandes dans
l'échelle spectrale visible, et ultraviolette ; c'est-à-dire l'intérieur de
l'enceinte de la longueur d'onde ca 7000-1500 A. C'est donc bien
la raison pour laquelle les combinaisons organiques insaturées,

1) Voir J. Stark : Die Principieu dtr Atouidynamik, II, Die elementare Strahlung. [HW.
S. Hirzel, Leipzig], Paul Kuggli : Die Valenz-Hypothese von J. Stark vom chemischen
Standpunkt [191:i, Ferdinand Enke, Stuttgart] et Ci. Urbain : Introducion a l'Elude
de la Spectrochimie [1911, A. Htrmunn et Fils, Paris]

2) loc. cit.

Recherclies sur les spectres d'absorption des amuiiae-complexes métalliques. 5

possédant évidoinmeut les électrons de valence relâchés aux points
des liaisons doubles ou triples des atomes du carbone, sont
vivement colorées, ou donnent des absorptions de l)andes bien
nettes à la partie ultraviolette.

Comme je l'ai déjà indiqué, les solution-; acjueu-es des sels
cobaltiques montrent deux ou trois Ijandes d' al »sorption très nettes
dans l'échelle spectrale visible et ultraviolette . L'une d'elles qui
est la moins réfranejible, se trouve, sans exception, voisine de la
longueur d'onde 5000 A. et elle est peu influencée par une
substitution quelconque dans les ions complexes. En conséquence,
cette bande la moins refrangible est l)ien probal)lement causée par
l'oscillation des électi-ons de valence relâchés qui s'attachent à
l'atome central de col)alt, tandis qui' les autres, qui sont plu? ou
moins sensibles à la substitution, sont caractéristiques des atomes
métalloïdes qui se tiennent en directe connexion avec l'atome de
col )alt.

Si donc on considère que l'électron de valence joue le rôle
d'un petit résonateur qui oscille par l'agitation de l'énergie de
rayons avec une certaine longueur d'onde, on peut réprésenter la
relation entre cette énergie et le nombre d'oscillations par la
formule fondamentale de M. Planck: £=h'^, où 8 réprésente un
quantum de l'énergie d'un résonateur oscillant, et '■" le nombre
d'oscillations, tandis tjue // est appelé la constante universelle de
Planck, ayant la valeur de G*548x 10"''.

Si l'on replace maintenant le nombre d'oscillations v par les
termes de la vitesse des rayons lumineux (3x 10^*^ cm/sec) et de sa
longueur d'onde A, la formule de Planck peut ê:re transformée ainsi,

g ^ 6- 548 xlO--^x3xlO'-° ^ 19-64x10-^'

Puisque l'atome de co1>alt possède les électrons de valence
relâchés cpii donnent une l>ande d'absorption toujours voisine delà

c

longueur d'onde 5000 A, on pourra alors calculer l'énergie moyenne
de ces résonateurs, d'après la formule donnée ci-dessou-.

S = ll^iiil21'=4xlO--W.G.S.)
5x10-' •

6 Art. 2.— Y. Shibata :

A présent, il n'y a naturellement aucun moyen direct pour la
détermination de la valeur de ce genre-là. Seulement, on pourra
examiner cependant, si de pareils sels complexes colorés, par
exemple, de chrom, de nickel, de platin etc, donneraient les
absorptions caractéristicpies aux atomes' métalliciues. Dans le cas
ou cela sera possible, la comparaison de la grandeur des éuergies
d'oscillation des électron de valence de plusieurs métaux doit être
un problème bien intéressant. Je vais continuer davantage mon
travail dans ce sens.

Partie expérimentale
I. Cobaltihéxammines (Série lutéocobal tique/"'

Les sels complexes de cette série sont toujours colorés très
jaune. Ils se cristallisent à aiguilles tiues et sont solubles assez
facilement dans Teau.

Comme on le voit dans la figure I, ils donnet deux absoi'ptions,
dont les minima des courbes (maxima d'absorption) se trouvent à
2100 et 3000 de fréquence. Il est bien intéressant de constater que
les absorptions de ces trois corps, chlorure de cobaltihéxammine,
I>romure de cobaltidiammine-diéthylènediamine et chlorure de
cobaltitriéthylènediamine, sont pratique naent égales l'une à l'autre,
à propos des positions et des formes des bandes d'absorption.
C'est seulement leur absorption à l'extrémité qui n'est pas la même.
Ces parties des courbes sont déplacées de plus en plus vers le rouge
par la substitution de la molécule d'éthylènediamine à celle
d'ammoniac ; c'est-à-dire que la courbe de l'absorption cVextrériiité
du chlorure de cobaltitriéthylènediamine est poussée le plus
sensiblement vers le rouge et celle du sel héxammine se trouve à la
partie plus refrangible, ttmdis Cj[ue la courbe de l'absorption
d'extrémité du sel de diammine-diéthylènediamine se place entre
les deux précédentes.

En ce qui concerne la nomenclature des sels complexes de cobalt, j"ai adopté celle
donnée dans im ouvrage de M. G. UrVjain " Introduction à la chimie des complexes,"
[1913 ; A. Hermann tt fils, Paris.]

Eecherches sur les spectres iVabsorptiuu d<-s auiuiine-comi^k-xos métalliques.

1500

3.5

3.0

^ 2.5

ä 2.0

1.5

0.5

2300

Fig. I

Frénueiice
2500 3000 350Q

4000

4500

• V.

) ■'*'

^

■ r

J 1

•.V

/ '" *

\

\|

1'

\

//\ \

V

,À

\

\ \

> >
\

\

'

\

\

M

\

1000

100

[Co NH.;),]CI, —
[Co[NH.;),eu.;]Br:
[Co(^,:]Cl,

Comme conséquence du fait important que les formes et les
positions des deux bandes d'absorption eont presque les mêmes
pour ces trois corps, on peut tirer la conclusion suivante : la cause
de Tabsorption des rayons des sels complexes de cobalt ne réside
qu'aux points de connexion de l'atome central du cubait et des
atomes métalloïdes (dans le cas actuel, les atomes d'azote dans les
molécules d'ammoniac et d'étbylénediamine) qui sont encliâinés
directement avec le premier. Quant à la constitution des molécules
ou des groupes, qui forment les ions complexes autour de l'atome
du cobalt, elle importe peu à l'égard de rabsor})tion des rayons.

Donc, on doit donner les constitutions ^uivantes aux ions
complexes des trois corps en question :

Art. 2. -Y. Shibata:

HHH HHH

H\ \l/ /H \\/

.Co. rr I X^'^x I

CH.,-NH./ 'r ^NH..-CH.

H\ /T\ /H

H -N N N-H ^^ - ^'^ -■ N

H/ ZW \h /W

HHH HHH

liéxauimiiie cliamiiiiiie dit'-tln^lrnediaunne

CH.,~CH.,

\ ' \ '
NH, NH,

CH.-NH.,^^^ yNH.,-CH.,

j ■ -\Co/ - I '

ch-nh/ ^nh-ch,

triéthylônediamine

Il nous semble que T influence des anions sur l'absorption des
l'ayons est tout-à-fait insignifiante, parce que le chlorure et le
bromure montrent peu de différence dans les courbes d'absorption.
Des cas pareils se rencontreront encore souvent dans la suite
de ce travail.

II. Cobaltipentammines (Séries purpurdcobaltique
et roséocobaltique)

Les sels purpuréocobaltiques sont les produits, faits par la

substitution d'un atome de chlore à une molécule d'ammoniac des

sels lutéocobaltiques. Dans cette étude, j'ai choisi deux sels de ce

groupe: le chlorure [Co(A7f,lC/]C7, et le sulfate acidique [Co(A^ff,>,

cq.iHSO,)..
SO,

Les sels roséocobaltiques sont également des corps substitués
clés séries lutéocobaltiques, dont une molécule d'ammoniac est
remplacée par une molécule d'eau.

La figure II nous indique que les formes et les positions des
bandes d'al)Sorption de ces trois corps ne différent prcsciue pas
l'une de l'autre. -

Recherches sur les spectres d'absorption des aDimine-complexes métalliques.

Fig. II

Fréquence

1500

2000

2500

3000

3500

4000

3.5

3.0

2.5

2.0

1.5

1.0

1000

[CoiNH,XCl]Gl,

[CoiNH,%H,0]CI,

La preniièro bande qui est moins refrangible montre son
maximum d'absorption a 1950 de fréquence, tandis que celle plus
refrangible se trouve à 2800.

En comparant ces figures avec celles des sels lutéocobaltiques.
on remarque facilement que les deux bandes, spécialement la
deuxième, plus refrangible, sont sensiblement déplacées vers le
rouge; c'est-à-dire que la substitution de l'eau et du cblore
à la place d'une molécule d'ammoniac a produit une influence
bathochromatique.''' Dans la seconde bande plus refrangible on
voit l'influence hypochromatique en même temps.

* Comme il est évident ciue la sensibilité relative des bandes influe autant sur
la coloration que la position des bandes dans Téchelle spectrale, nous dirons
des groupes auxochromes, qu'ils fonctionnent comme hyperchromes lorsqu'ils
augmentent cette sensibilité, et comme hypochromts, lorsqu'ils la diminuent, de
m-me qu'on dit qu'ils fonctionnent comme batho- ou hypsochromes, suivant qu ils
provoquent un déphxcement de bandes vers le rouge ou VL-rs l'ultraviolet.

10 Art. 2.-Y. Shibata:

Comme je Tai déjà iiidi(|ué plus haut, il me semble que les
secondes l)andes plus réfraugibles sout très i)rol)ablemeut causées
par les oscillations des électrons de valence qui s'attachent aux
atomes métalloïdes joints directement à l'atome du cobalt.

Dans le cas actuel, il faut se rappeler que l'atome d'ox3'gène
dans la molécule d'eau (dans le sel roséccobaltique) et l'atome de
chlore (dans le sel purjjuréocoboltique) ])rcduisent la n.enic
intluence opticjue; cela est pourtant extraordinaire. Considérons,
par consé(iuent, que les deux sels complexes, pui'puréo et roséo,
sont en équiii])re dans la solution aqueuse:

[Co{NH.;)ßJ]CI, + H,OZ[Co{NH.;),H,0]Gl,

Mais le sel roséocol)altique est, en général, moins stable que le
sel purpuréocobaltique. En effet, le premier est préparé, en faisant
précii:>iter d'une solution ammoniacale du sel purpuréocobaltique
par l'addition soigneuse de l'acide chlorhydrique à O^. Il est alors
bien vraisemblable que la réaction montrée par Tecpiation marche
plutôt de droite à gauche dans la solution aqueuse très étenthie, et
qu'il n'y existe que le sel purpuréocobaltique. Or l'influence
optique parue sur la seconde bande, doit être causée probablement
par l'atome du cJilore.

On voit ici l'inHuence insignifiante de Tanion sur rabsorj)lion,
parce (jue les formes des courbes du chloruie et du sulfate acidi(|ue
de chlo]])entammine ne diffèrent que peu l'une de l'autre.

III. Cobaltidihydrotétrammiue et Cobaltichlorohydrotétrammiiie
(Séries de roséotétrammine et de hydropurpuréotétrammine)

Ces deux sels sont les ])r()(hiits obtenus par la suljstilution de
deux molécules d'ammoniac du sel col)altihéxan:imine et ont
respectivement les formules suivantes:

[Co(NH,UH,0).;]Cl, [Co(NH.;),{H,0)Cl']CI,

chlorure de roséotétrammine chlorure de hydropurpuréotétrammine
(rouge fade) (violet rougiâtrej

La ligure III nous montre (jue les courbes d'absor])tion de ces
deux coi'ps coïncident à peu près, ayant les maxima d'absoi-ption à
1000 et 2S00 de fréquence. Par consétjuent, il doit y avoir eu,

Eoelierchos sur les spectres cValisorption des auimiue-coniplexes métalH'Xues

11

comnio dans le cas prccecdent, le cluiDgement suivant elans la
solution aqueuse de ces deux sels tétrannnines

[CoiNH,UH,0).;]Cl,-^[Co{NH,;),H,0-CI]CL + H,0,

parce que le sel roséotétmmniine est assez labile n:ême à l'état
solide et incline à se transformer spontanément en sel hydropur-
puréotétrammine.

Fi^. III

1500

2000

i'réquence
2500 300D

35B0

^ 2.0

[Co(NH,:),(H,0).;]Ch

lco(NH..),H,0-a]GI,

4000

— 1000

^

1/iiilluence iiy[)erchromatique de disubstitution pour héxam-
mine est l)ien remarquable dans la première bande, (^uant à
l'influence sur la seconde bande, dans T ultraviolet, elle ne se
discerne prescpie pas de celle qui se produit dans le cas de pentam-
mines.

Ce dernier fait, que les courbes d'absorption du sel purpuréo-
pentammine et du sel liydropuréotétrannnine possèdent presque les
mêmes minima à 2800 de fréquence, est bien comprébensible, si
l'on prend la courbe d'absorption (hi monobydroxypentammine
\_Co(NH.^^-OH]Cl2 en considération. Connue on le verra plus tard,

12

Art. 2. -Y. Shil.ati :

il iiiaiKjue la deuxièiiu' haïuk' dans 1" ultraviolet on ce sel complexe,
qui contient un groupe d'hydroxyl, dont Fatonie d'oxygène se lie
directement avec l'atome de cobalt. Vn atome d'oxygène étant
ainsi indiffèrent pour la deuxième bande, elle doit donc ê:re due
seulement à l'atome de chlore du sel i^urpuréopentammine et
hydropurpuréotétrammine, bien que ce dernier contienne encore
en plus une molécule d'eau.

IV. Carbonatotétrammine et Oxalatotétrammine

Les sels de cette série se cristallisent bien en aiguilles,
quelquefois a^sez grosses. Le carbonatotétrammine [Co(A^iT,,),CO,dC/
et le carbonatodiéthylènediamine [é'o ^?A,rO,]r7 out la couleur du
carmin foncé, tandis que l' oxalatotétrammine lCo{NH:?itC.O,]Cl est

rose.

Fio-. lY

Fréquence
250Q 3030

[Cu^/,C0,dC7 -

Cö{nh:),c,o,]ci-

co{Nh;),co,]ci

Kechorchts sur los s^poctros d"ab£orption des ammine-coruplexLS mctallicßios. 13

r.a figui'u I\^ rend compte du fait, que los trois sels absorbent
les rayons également. C'est alors encore une preuve que l'absorp-
tion n'agit qu'aux points de connexion entre l'atome cobaltique et
les atomes métalloïdes, qui sont en coordination avec le premier,
et que l'inégalité de structure des gi-oupes de carbonato etd'oxalato
importe peu. On peut donc donner les constructions suivantes
aux cations complexes de carbonato et oxalatotétrammine

Os ,0-C=0

(NH,\Co( ^C=0 {NH.^,Co(^ \

Carlonatotétrammine Oxalatotétrammine

Les maxima d'absorption de ces deux séries de complexes se
trouvent respectivement à 1000 et 2700 de fréquence ; par consé-
quent les deux groupes en question causent une influence
botlîocbromatique sur le sel héxammine. En comparant les
absorptions de ces sels tétrammines avec celles du complexe
purpuréocobaltique, on ne remarque qu'un peu de difïérence à
l'égard de la position de la deuxième bande plus refrangible,
c'est-à-dire une influence un peu bathochromatique. Pourtant
l'intensité d'absorption se distingue assez sensiblement l'une de
l'autre, comme cela est montré dans la petite table ci-dessous, dont
les cliifferes rendent compte de l'épaisseur, ou les maxima d'absorp-
tion de chaque sel commencent de paraître.

Purpuréo- Purpuréo- Carbonato et

pentamminc tétrammine Oxalatotétrammine

1«^° bande (1900) 250-300^^ ISi.^^ 100-UG""»

•i''"'«-^ ,, (2800--2700) 800°^"^ 800-900'"'" 500-560''^"'

Or on y trouve que les groupes de carbonato et d'oxalato sont bien
h y pe rch ro m atiq ue s.

14

Art, 2. — Y. Shiluita

V. Combinaisons qui contiennent le groupe des nitros
dans les ions complexes.

Les aramine-complexes qui contiennent respectivement un.
deux, trois et quatre nitros dans les ions complexes absorbent
bien semblaljlement et mettent en évidence quelques faits fort
intéressants.

Il s'agit des dix sels complexes suivants

[Cü{NH.;).^NO.;]CI,
cliloriira de xautliop3ntamminî

[Co(NH,%ONO]Ch
clilorn 3 d " isoxantli op 3ntamm i n e
ou de nitritopentaniniine

['^^^^âi:!]^'

chlorure de flavotétrammine

Co(iVJÎ:0a(A^O,),,

cobaltitrinitrotriammine

[Co[NO.;)c:\Na,
cobaltihéxmitrite de natrium

[CoiNH,;),(No.,;)Ci]ci

clilorure de inonocliloromononitro-
tétrammine

chlorure de crocéotétrammine

[CoiNH,U^O.;),]NH,

cobaltidianimonionitrite
d'ammonium

[CoiNH,;),{NO.XOH)]CI-H./)

chlorure de mononitromono-
hydroxyltétrammine

[CoiNH,%{C,OX^^O.^;]NH,

col)altidiammoniooxalonitrite
d'ammonium

Tous les sels de cette série sont colorés jaune ou jaune rougeâtre,
spécialement les deux derniers ont la couleur de T orange
rou'jfeâtre.

■Recherches sur les spectres iV.ibs^rption d ^s ani nitio-camplL-xes ;n''tilli mes. \!^

Les épaisseurs (mm) «les solutions correspondant à jy,\,- -V

»

(

\

__

■

J

~^^

i

:

^

snoTi^nps sip s.ia'^ssnîd.i »."ïp sr>raq;tai;.oO|; sa^j;

Les épaisseurs (mm) des solutions correspondant -i — ' - A*

I 1

9

d'
'<

S"

suoT^ups sip siaasstedv» ssp s3tnt[-4t.n3Sax si^j;

16

Art. 2. -Y. Shibata. :

Ties rpaissciu-s (umi) des solutions corrtspcndant à — ! .V

^ 10 0 0

/ci, ,r

O

snoT;pilos s.ip s.inossit;do sop Sc»inqc)T.iiiSo[ so^j^

Les épaisseurs (uim) des solutions correspendant à — '- A'

y

■^

n

(

N

o

1

CO

^^____^

y

c

e
a

>

I

snor;n|os s.>p s.IuossI^'dfl sop sainn;iai;.ooi^ so-^j

Recherches sur les spectres d'absorption des ammine-complexes métalliques. ^7

Les éjjaisseurs (mm) des solutions correspondant à ' N

X 2

c

^^^ /

) y

J^-

^--<*^

^<^'

^y^'

^,.<>-

b

'J

• ^-

-— — îî*'^

(>■

■v>.

^

^yf^

-<S>

r;^ o >

snoi:ju[os sap sJuassi'Bdo sop somi[c>Ta'Bj>oi sai

Les épaisseurs (nmi) des solutions correspondant à ^^^ N

<

?
^

suoi:^nios sap saaasstBdi? sop soinii'^iai3.oO{ sofj

18

Art. 2.— Y. Shibata ;

Les épaisseurs (mm) des solutions correspondant à ^ — N

suoT^^nps sep sriiessT'eda sep s^ml(î}Ta^'.oO]; se^x

Les épaisseurs (mm) des solutions correspondant à y^tû ^"

I — I

d"

i

o

X

t>X) ^"3^ "

PR ^

O

o

i

o

snoT-^njos sep sjuessT-eda sep seinq';Ta'BSo^ seq

Eecherches sur les spectres d'absorption des ammine-complexes métalliques.

19

1500

2000

Fig. XIII

Fréquence
2530 3000

3500

4000

1000

[Co(ONO%]Na,

Ils se cristallisent en aiguilles fines et sont solubles plus où
moins facilement dans l'eau. Les solutions sont bien stables, sauf
celle de l'isoxanthopentannnine et de l'héxanitrite ; il me semble
que ce dernier se dissocie en ses composants, c'est-à-dire le nitrite
de cobalt et le nitrite de natrium dans la solution par la simple
dilution, parce que ses absorptions aux concentrations de tôt ^Vet
de rv^ ^ ne s'accordent pas avec la règle de Beer. Quant au sel
isoxanthopentammine, il est assez labile même à l'état solide et se
transforme en xanthopentammine en quelques jours, tandis que la
solution change sa couleur rouge au jaune de son isomère bien
rapidement, bien que la solution fraîche satisfasse complètement la
règle de Beer.

Comme on voit dans les figures V— XIII, on peut diviser cette

20 Art. 2.— Y. Shibata :

série en deux classes: Tune qui montre deux bandes d'absorption
dans l'échelle spectrale et l'autre qui en a trois. Si l'on remarque
que seulement trois corps parmi dix donnés ci-dessus, c'est-à-dire le
crocéotétrammine, le trinitrotriammine et le diammoniotétranitrite
appartiennent à la deuxième classe avec la troisième l)ande, on
comprendra l)ien qu(,^ deux nitros à la position de trans (ou 1,6)
causent cette troisième absorption à Textiême ultraviolet (ca 4000
de fréquence), parce que le sel crocéotétrammine a évidemment,
d'après A. Werner'^ ses deux nitros à la position de trans, et que
ceux qui contiennent quatre nitros doivent en avoir deux nécessaire-
ment à cette même position. Quant au cobaltitrinitrotriammine,
il est théoriquement possible, qu'il apparaisse en deux isomerics
stéréochimiques, l'une d'elles ayant deux nitros à la position de
trans, et l'autre ses trois nitros à la juxtaposition. A en juger par
l'existence de la troisième absorption, il est bien vraisemblable que
ce complexe est composé de telle manière que deux de ses trois
nitros sont à la position de trans. Alors ces trois sels complexes
doivent être réprésentés par les formules stéréochimiques suivantes

NO, NO^ NO-. NOo

NHo I NH^ NHaJ NO; NH.J NO. NOoJ NH-^

/ / / / / / "'' z /

NH-x I iV//.. iV//.. I NU., NH^ I NO: NH.^ \ NO.,

NO., NO, NO; NO;

crocéotétrammine trinitrotriammine diammoniotetranitaite

En dehors de cette différence concernant la troisième absorption,
toutes les courbes d'absorption des ammine-complexes contenant
des nitrose ressemblent fortement l'une l'autre.

En général, les maxima d'absorption de ce groupe se trouvent
à 2100 et à 3000 de fréquence ; la troisième bande, si elle existe, a
son maximum d'absorption à 4000.

En comparant ces absorptions avec celles des héxammine, on
aperçoit tout de suite que l'introduction du groupe du nitro dans
les ions ammine-complexes cause une influence hyperchromatique

1) loc. cit

Recherches sur les spectres d'absorption des ammine-complexes métalliques. 21

qui est très remarquable, spécialement dans les deuxièmes bandes.
Seulement, le cobaltihéxanitrite de natrium [Co(NO.;)e]Na, et le
chlorure de nitritepentammine (isoxantho) [Co{NH,%ONO]Cl, se
montrent un peu exceptionnels, c'est-à-dire que leur première bande
est assez hypochromatique, comparée avec celle d'héxammine.

Les chiffres suivants rendent compte de ces relations. P, dans
les tables, signifie l'épaisseur des solutions correspondant à i^ N
où se trouvent les minima des courl)es.

Les premières

bandes.

Les sels

P-

liUteohéxam mines

'280-800""^'.

Nitroam mines

100-L50 „

Héxanitrite

520 „

Nitritopentammine (isoxantho)

340 „

imm

Les secondes bandes.

Lutéohéxammines 470"^

Héxanitrite 5G „

Nitritopentammine (isoxantlio) 45 „
Xanthopentannnine 4 ,,

(Jrocéo et Flavotétraiiiinines 8 „

Triniti-otriammiiie 4 „

Téti-anitrodiammine 5 „

Monoh^^droxymononitrotérammine 9 ,,

Monochloromonnitrotétrammine 18 „
Oxalodinitrodiammino 7 „

La troisième l^ande à 4000 de fréquence, n'existe que dans les
solutions très étendues (,-^hrö ^V) des sels qui contiennent plus de
deux nitros, parmi lesquels deux sont à la position de trans, comme
je l'ai indiqué plus haul.

(^uant aux épaisseurs des minima (^9) où commence ral)sorp-
tion pour la troisième bande, elles ne diffèrent pas beaucoup les
unes des autres entre ces trois corps.

22 Art. 2. - Y. Shibata :

Les sels p

(correspondant à la concentration de ^^l^„ ^)

Crucéotétrammine 17"".

Trinitrotriammine ^^'^ "

Tétranitrodiammine 1^1 »

En résumé, on peut tirer de là quelques lois données ci-dessous,
à regard de l'absorption des sels ammine-cornplexes qui contien-
nent des nitros ou de nitrito dans leurs ions complexes.

1°- Les nombres des groupes du nitro dans l'ion complexe
influent peu sur la quantité et la qualité d'absorptioji, parce que le
mono-, di-, tri- et tétranitroammines montrent de très semblables
courbes d'absorption, si l'on met la troisième bande à part.

2°- L'al)Sorption ne dépend ni du signe, ni des valeurs des
ions complexes, parce qu'ils absorbent très semblablement les uns et
les autres, sauf la troisième bande, bien que le xanthopentammine,
le crocèo- et flavotétrammines et le mononitrohydroxytétrammine
soient les cations respectivement de di- et monovalence, et que
le tétranitrodiammine et le dinitrooxalodiammine soient les anions
de monovalence, tandis que le trinitrotriammine est une molécule
sans charge d'électricité.

3°- Les nitros à la position de trans donnent la troisième
bande dans les solutions très étendues de tttfitü ^, A part ce
point, les sels stéréoisomeriques, comme crocéo-et flavotétrammine
absorbent tous également.

4°- Le coljaltihéxanitrite et le nitritopentammine (isoxantlio)
absoi'bent bien semblal)lement Tun et l'autre. Sans doute, la même
construction du groupe du nitrite ONO dans les ions complexes
a causé cette ressemblance de l'absorption.

VI. Isosulfocyanopentammine

La courljc d'al)sorption du chlorure de ce complexe [CoÇNH;^^-
(NGS)']Ch a deux bandes très nettes à 1950 et à 3350 de fréquence
La première est un peu bathochrome et la deuxième sensiblement
hypsochrome, comparée avec les bandes du lutéohéxammine.

Recherches sur los spectres fl'aljsorptiou des ammine-comploxcs métalliques.

23

La première bande

fréquence p.

Lutéohéxamniine '-^100 300„,„,.

Isosulfocvanopentaii inline 1950 100 ,,

La (lenxième bande

Lutéohéxammine 3000 470„,„,.

Tsosnlfoe3;anopentanimine 3350 18 „

1500

Fig. XIV

Fréquence
2000 2500 3000 3500

4000

4500

1000

lCo{'NH,\{fiCN)\CL

Il est bien remarquai )le que la'foi'me et la position des bandes
de ce sel complexe sont presque les mêmes que celles de la solution
alcoolique du sulfocyanate de cobalt" C'o(,S' Cyy\ bien que les couleurs
des deux solutions soient très différentes: la solution aqueuse
d'isosulfocyanopentammine est brune rougeâtre, tandis que la

1) A. Hantzsch und Y. Shibata : Zeitsch. anorg. Chem., 1911 73, 309.

24 Art. 2.— Y. Shibata :

solution alcoolique du sulfocyanate de cobalt a la couleur bleue vive.

Pourtant, si l'on examine d'un peu plus près les deux courbes,
on trouve bien facilement que la raison de cette contradiction
superficielle réside dans la différence des positions des minima
d'absorption (des maxima des courbes). En l'isosulfocyanato-
pentammine (Fig. XIV), ce point se trouve à TOO,,,,,, de l'épaisseur
de la solution correspondant à nroT N, tandis que dans le cas du
sulfocyanate de cobalt, le minimum (ral)sorption se place à une
épaisseur tellement grande qu'on n'en a pu trouver trace dans la
concentration en question.

Or la solution alcoolique du sulfocyanate de cobalt laisse passer
les rayons entre le bleu et le violet, tandis que la solution aqueuse
assez concentrée (ou assez épaisse) de l'isosulfocyanatopentammine
absorbe tous les rayons plus courts que 6000 A; c'est-à-dire
qu'elle est transparente seulement pour le rouge et le jaune.

VII. Praséotétrammine

La substitution de deux atomes d'balogène à deux molécules
d'ammoniac du lutéohéxammine produit deux stéréoisomères : le
praséotétrammine (trans) et le violéotétrammine (eis). Les deux
sels qui sont respectivement colorés en vert et en violet, sont assez
stables à l'état solide, cependant leurs solutions aqueuses cbangent
rapidement leurs couleurs et prennent à la fin la même couleur
carmine. Seulement le chlorure de chloropraséodiéthylènediamine
\Goen2Ql'^l^ IcZ étant un peu plus stable que ses antres dériA'és, j'ai

préféré ce corps pour l'objet de cette recherche d'absorption.

Comme on le voit dans la figure XV, la forme de la courbe
d'absorption montre une anomalie : c'est-à-dire qu'elle ne renferme
qu'une seule petite l)ande à 2100 de fréquence dans l'échelle
spectrale mesurable. Cependant, la branche descendante de la
courbe au rouge indique qu'il y aurait très probalilement une large
bande à l' ultrarouge.

La courbe qui est indiquée par les lignes Ijriseés est celle de la
solution du praséotétrammine qui a été laissée pendant 24 heures
pour faire complètement changer la couleur. On j aperçoit
facilement la ressemblance entre les formes de cette derinière et

Eecherclies sur les spectres d'absorption des ammine-complexos métalliques. 95

celles du chloropurpuréotétrammine et du roséotétrammine. Alors,
le changement de la couleur de la solution du praséotétrammine du
vert au rouge s'est passé vraisemblablement dans le sens suivant :

[Go en,Ci;]Cl + H,0^[Co e)h/^H,Oyci]CI,
vert camiin

Fig. XV

Fréquence
1500 2000 2500 3000 3500 4000 _

1000

3.b
3.0

\u

2.5

J\

i
i

vS^ /' '\

\

2.0

\

y

1.5

\

\

1.0

\

i

100

10

La solution verte -

carmine —

VIII. Bromopurpuréopentammine et Mononhydroxypentam-
mine (série du rhodocobalticomplexe.)

Pour la première sorte des complexes, les trois comlnnaisons
suivantes sont choisies :

Co 6112 -D ^..,. \Br-2 (lö corps racémique)
â^Co e>^2^f^'|!J yiiC,,H,,0 Br SO,),
/[Co en^^f'^^ ]KG>oH,,0 Br SO^,

26

Art. 2.— Y. Shibata :

Les épaisseurs (mm) des solutions correspondant à Yoâg ^

S

o

a

0)

t£

VI)

^

Ph

a;
|3

'^

o-i

r-«<r

g

^-^

vO)

o

fH

^

CO

9

'M

'«k„

o

Pq^

^

te

~ïl

o

^

(^

w:i

o

/ V

'^ J

tjr

^l

;^ßq I^Pq

71

5ÏÏ 1

"^ 1

^ 1

^ 1

O

c

O

o

__/

saot^nios sap sanasstBda sap sdxnqc^iJijSo^ sd^

Les épaisseurs (mm) des solutions correspondant à ytoïï ^

ce

te

suoic^nps sap sanessi'Bda sap saraqcjia'cSoi; sa^j;

Recherches sur les spectres d'alisorijtion des amuiine-complexes métalliques. 27

Tous les sels de cette série ont la couleur violette fade, tandis
que leurs solutions sont colorées en carmin. Bien que le bromure
et le bromocampliersulfonate fassent des anions d'une grandeur très
difïérente, et de même que deux bromocamphersulfonates soient
les antipodes optiques l'un de l'autre, leurs courbes d'absorption
(Fig. XVI) coïncident très bien l'une a l'autre. Voilà encore un
témoignage pour la conclusion, à laquelle je suis arrivé plus haut
en telle sorte que la cause de l'absorption existe seulment aux
points de connexion entre l'atome du cobalt et les atomes
métalloïdes dans un ion complexe, et que, par conséquent, la
grandeur moléculaire des anions importe peu à cet égard, dans le
cas oïl ces anions eux-mêmes ne contiennent aucun chromophore.
Pour cette même raison, les antipodes optiques doivent aussi
également absorber.

La forme de ces courbes et celle du monohydroxypentam-
mine, qui est un corps rouge et donne une solution de couleur
carmine, montrent une exception curieuse, comme on, le voit dans
les figures XVI et XVII; c'est-à-dire que chacune des courbes ne
renferme qu'une bande à 2000 de fréquence. Cette ressemblance
des courbes dans deux séries des complexes avec les atomes de très
différents caractères dans les ions complexes, comme [(NH3)iCo-OHj'

et [ß^Goß^, H est encore inexplicable. Si cela se rencontre seule-
ment par hasard, ou s'il y a quelque raison à cela, je laisse la
question à futures recherches.

Résumé

1°- Lorsque l'atome de cobalt fait un ion complexe en se
coordonnant avec les atomes métalloïdes ou avec quelques groupes
de ces atomes, la cause de l'absorption des rayons se trouve au
point de connexion de l'atome cobaltique avec les atomes métal-
loïdes. L'absorption caractéristique à l'atome du cobalt, qui est
causée probablement par l'électron relâché de ce dernier élément
métallique, est très peu influencée par des substitutions quelconques
dans l'ion complexe.

2°- La cause de l'absorption existe également dans le- a ornes
métalloïdes qui sont liés directement avec l'atome du cobalt.

28 Art. 2.— Y. Shibata :

Mais, dans ce cas, la bande d'absorption sera plus ou moins
sensiblement déplacée par les substitutions dans l'ion complexe.

3°- Si les constitutions cbimiques des ions complexes cobalti-
ques sont semblables entre elles, leur absorption est aussi semblable,
et de même, si les atomes métalloïdes dans un groupe de pareil
caractère, sont égaux, la complexité et la grandeur moléculaire de
ce groupe import peu à l'égard de ral)Sorption. Par exemple,
l'éthylènediamine-complexe et Tammi ne-complexe correspondant
absorbent bien semblablement, tandis qu'il n'en est pas de même
dans le cas de nitroammines et du nitritoammiue.

4°- Les différences des signes et des valeurs des ions e(.)mplexes
eobaltiques n' influent presque pas sur l'absorption, si les ions sont
pareillement construits. Les antipodes optiques et le corps
racémique ne montrent aucune différence d'absorption non ]:>lus.

5°- En général, les isomères stéréocbimiques absorbent
différemment. Dans le cas du praséotétrammino (trans) et du
violéotétrammine (eis), par exemple, leurs couleurs à l'oeil nu sont
déjà très différentes, tandis que le flavotétrammine (eis) et le
crocéotétrammine (trans) absorbent également jasqu'ala concentra-
tion de r^ JV ; mais dans une dilution plus grande (xTriru" -^X)» 1^
crocéotétrammine montre encore une bande à 4000 de fréquence.

Je me fais un devoir d'adresser ici à j\L'. le Professeur
G.Urbain à la Sorbonne, qui m'a aidé de conseils bienveillants et a
eu l'amabilité de me procurer les appareils necéssairs à ces
expériences, lorsque je me trouvais à Paris, mes remerciements les
plus sincères et les phis empressés.

Je suis aussi ])ien reconnaissant à mon préparateur privé
Mr. T. Kato qui m'a donné ses aides vigilantes pour ce travail
au Laboratoire de Cliimie de l'Université impériale de Tôkiô.

Yuji Shibata

Juillet 1915.

Published Sej>t. JOtli, Win.

JOURNAL OF THE COLLEGE <iF SCIENCE, TOKYO IMTEIUAL UNIVERSITY.

VOL. XXXVII., ARTICLE 3.

Considerations on the
Problem of Latitude Variation.

By

Kiyofusa SOTOME, Pii'jahnshi,
Assistant Professor of Astronomy, Tokyo Imperial University.

Introduction.

Tins paper covers some studies on the zénith telescope as an
astronomical instrument extensively used in observing the varia-
tion of latitude. Herein the writer intends to call attention to tlie
fact that there are, according to his experience, some remarkable
phenomena in the behaviour of the zenith telescope, and further,
to show the possibility of finding a method of explaining the z-term
and closing sum (schlussfehle]") in the variation of latitude in con-
nection with tliese phenomena. The development is made in the
foll()wi]\g tliree sections.

I. Singular behaviour of the zenith telescope.

During the years l'.JU()-lUn, the writer was in charge of the
observations of latitude variation in the Tokyo Astronomical Obser-
vatory^, under the auspices of the Geodetic Committee. This was
in continuation of the series of observations made successively by
Professors 11. Kimura and K. llirayama. The instrument used
was a zenith telescope of the usual form, made hy Wanschaff in
Berlin, No. 8U0, aperture 8r'""- and focal length lOO'""- .

In order to determine the value of one turn of the micrometer

Art. 3.— IC. Sotome:

screw, I made observations of polar stars at their greatest
elongations every summer and winter. The total numl)er
of determinations during the period amounted to 58. When
observing the polar stars in their eastern and western elongations,
I found a remarkable fact, viz. that the ]>ubbles of the two levels
attached to the zenith telescope moved gradually towards the south,
both in the eastern and the western positions of the telescope.
Moreover, the magnitude of displacement was larger in winter than
in summer, and it varied generally Avith the length of the ol)serva-
tion. This can be seen from the following table, in which the
first column is the date of observation, the second and third
columns the bubble displacements of the first and second levels in
units of division, the fourth the observed stars, the fifth the interval
■of observation, and the sixth the temperature of the observing
room.

Table I.
Telescope West.

( + increase, — decrease)

Date

Level I.

Level IL

Star

Interval

Temperature

1906 August

4

d
+ Ü.20

d
+ 0.05

Polaris

m
33

o
22.0

„ Sept.

5

0.00

+ 0.25

»

27

21.8

1907 August

17

+ 0.20

+ 0.10

51 H. Cepliei

22

23.8

,.

18

+ 0.15

+ 0.10

..

22

23.4

„

19

0.00

- 0.05

»

23

24.7

190S August

21

+ 0.15

+ 0.05

,.

22

23.0

,.

22

+ 0.05

+ 0.25

..

23

25.5

,•

26

+ 0.10

+ 0.05

.,

25

22.6

»

27

+ o.rs

+ 0.50

..

23

22.6

1909 August

8

+ 0.15

+ 0.25

.,

23

22.8

„

9

+ 0.35

+ 0.55

,.

23

2 > <>

1911 Sept.

2

+ 0.20

+ 0 30

:.

23

19.8

,•

3

+ 0.15

+ 0.45

..

23

21.8

Mean.

1+ 0.13)

- I
(+ 0.18)

Considerations on the Problem of Latitude Variation.

Date

Level I.

Level 11.

Star

Interval

Temperatr.n'

1909 January

23

d
+ 0.15

d
+ 0.20

I H. Draconis

m
8

O

1.0

,,

29

+ 0.45

+ 0.25

»

8

99.8

,.

30

+ 0.35

+ 0.25

..

8

99.9

„

31

+ 0.25

+ 0.50

„

8

2.0

1910 February

2

+ 0.50

+ 0.30

„

8

0.4

,.

7

+ 0.05

+ 0.05

,.

8

5.0

„ March

5

+ 0.70

+ 0.85

0 Ursae Min.

19

98.1

„

7

+ 0.75

+ 0.70

„

19

O.S

»

10

+ 0.85

+ 0.75

„

19

09.9

„

13

+ 0.40

+ 0.55

,,

19

O.j

1911 January

25

+ 0.85

+ 0.4;)

1 H. Draconis

8

2.2

Mean

(+ 0.42)

(+ 0.44)

Telescope East.

Date

Level I.

Level II.

Star

Interval

Temperature

1909 July

21

d
+ O.lJ

d
0.00

0 Ursae Min.

m
19

o
24.0

..

25

0.00

+ 0.10

„

20

24.6

,,

28

- 0.20

- 0.25

„

20

24.1

„ August

2

0.00

O.OJ

,,

L'O

23.0

1910 October

26

- 0.25

- 0.25

76 Draconis

9

11.9

Mean

(- 0.07)

(- 0.08)

1907 February

2

- 0.25

- 0.25

Gr. 7.50

14

96.8

..

5

- 1.00

- 0.75

>,

14

99.7

„

9

- 0.20

- 0.35

.,

14

99.2

,.

14

- 0.25

- 0.25

„

14

99.8

1908 January

12

- 0.53

- 0.65

Polaris

40

2.7

»

16

- O.W

- 0.(55

„

50

2.6

„

17

- 0.45

- 0.30

„

45

0.0

„ February

22

- 0.10

- 0.15

Gr. 750

14

7 8

Art. 3 — K. Sotome:

Date

Level I.

Level IL

Star

Interval

Temperature

l£i09 January

16

d
- 0.10

d
- 0.20

Polaris

m
53

o
99.2

,,

17

- 0.55

- 0.4 5

»

56

• 98-6

1910 i'ebruarj'

3

- 0.45

-- 0.50

Gr. 750

14

08

..

7

- 0.10

- 0.10

„

14

2.8

,.

9

- 0 35

- 0.45

,,

14

2.0

,.

12

- 0.50

- 0-Ö5

,,

14

98.5

„

]5

- 0.45

- 0.45

.,

14

0.7

, March

5

- 0.70

- 0.35

51 H. Cephci

23

98.0

„

7

- 0.20

- 0.35

„

23

0.2

..

10

- 0.60

- 0.60

„

23

99.4

'> »

13

- 0.45

- 0.50

„

23

0.6

1911 Febinary

3

- 0.30

- 0.[0

Gr. 7C0

14

0.6

»

4

- o.io

0.00

43 H. Ceph. i

8

2 2

„

7

- 0.60

- 0.60

Gr. 75J

14

0.0

,.

9

- 0.30

- 0.45

,,

14

99.7

»

10

- 0.50

- 0.35

43 H. Cephei

16

3.3

..

11

- 0.15

- 0.45

„

16

6.6

>j >)

12

- C.50

- 0.45

:,

16

2.5

»

•ZI

- 0.30

- 0.45

„

7

3.8

„

22

- 0.45

- 0.30

„

16

5.4

March

2

- 0.55

- 0.50

„

16

7.8

Mean

(-^ 0.41)

(- 0.41)

l'i of 1=1. "15 l'i of II = 1."15

From this table we see the fact that the displacement in
the winter is about four times as large as that in the summer, when
reduced to the same interval.

Under such circumstances, the question suggested itself, in
making the reduction of observations, " Does tlie bubble displace-
ment correspond exactly to the change of inclination of the
telescope ? ' '

Thereupon I made tentative reductions from two extreme
standpoints:

Considerations on the Problem of Latitude Variation.

I. Tlic displacement of bul:)bles is entirely due to some cause
in the level itself, and is not due to the variation uf inclin-
ation of the telescope. The telescope did not move
during the observation.

II. The displacement corresponds exactl}^ to tlie change of in-
clinition of the telescope. This is the assumption usually
adopted. In the reduction, thechange between tworeadings
\ya< considered to be proportional to tlie time interval.

AVith the first assumption, western elongation gave a greater
value of tlie micrometer than eastern elongation; witli the second
assumption, on the contrary, eastern elongation ga\'e a greater
value than western elongation.' This fact is easily seen from the
series of observ^ations during February-March lOliJ, in the follow-
ing table, in wliicli the first column is the date, the second the
polar stars, the third the elongations, the fourtli the micrometer
value with the first assumption, and the fifth the same with the
second assumption.

Table II.

JJate

Star

Elongation

Micrometer Values with,
the Assumption
I. IL

190 ; Au--ust

4

Polaris

E

51."öS-2

51." 701

September

5

„

E

579

583

1907 February

o

Gr. 750

W

(319

595

".

5

„

"W

712

6 51

>■ V

9

W

G98

682

„

U

>,

W

()33

• •.22

„ August

17

51 H. Cephei

E

628

6U

»

18

„

E

645

653

..

19

„

E

654

656

1908 January

12

Polaris

^V

706

est

..

16

„

W

723

687

„

17

,.

W

732

608

„ February

22

Gr. 750

W

638

631

1 Western elongation was observed in the eastern position of the telescope and eastern
elongation in the western position.

Art. 3 — K. Sotome

Date

Star

Elongation

Micrometer Vaincs with
the Assumption
I. IL

U08

Au<,'ust

21

51 H. Cephei

E

5l"597

51"603

..

»

22

»

E

658

665

.>

>,

26

„

E

625

630

>.

..

27

„

E

651

654

19G9

January

16

Polaris

W

687

678

..

.,

17

..

W

681

661

»

„

23

1 H. Draconis

E

668

678

>.

..

29

»

E

656

675

..

»

30

<•

E

692

708

■ »

..

31

„

E

669

690

..

July

21

0 Ursae Min.

W

662

665

»

»

25

»

W

659

662

..

..

28

„

^V

645

630

..

Aiigust

2

:,

w

643

643

..

"

8

51 H. Cephei

E

667

679

..

..

9

„

E

619

644

1910

February

2

1 H. Draconis

E

637

660

..

»

3

Gr. 750

W

708

C83

..

.'

7

1 H. Draconis

E

675

678

.'

..

7

Gr. 750

W

689

684

"

..

9

..

W

747

724

"

..

12

..

w

667

651

„

»

15

,.

w

687

662

.-

Mnrch

5

ô Ursae Min.

E

664

708

"

.'

5

51 H. Ceplni

W

639

629

..

.

7

& Ursae Min.

E

662

705

•'

■.

7

51 H. Cephei

W

658

642

..

..

10

ô Ursae Min.

E

692

741

»>

i>

10

51 H. Cephei

W

721

(93

..

..

13

S Ursae Min.

E

630

655

"

"

13

51 H. Cephei

W

689

664

Considerations on the Prolilom of Latitvde Variation.

Date

Star

Elongation

Micrometer Values with
the Assumption
I. II.

1910 October

26

76 Draconis

W

51."62)

51. "606

1911 January

25

1 H. Draconis

E

661

698

„ February

3

Gr. 753

W

652

634

„ X

4

43 H. Cephei

w

707

704

.

7

«ir. 750

w

734

700

:!

9

,,

w

663

641

^•

10

43 H. Cephei

w

717

693

,.

11

,

w

646

629

.

12

w

716

687

,.

21

„

w

674

652

„ ' ..

22

\y

705

683

March

2

„

w

629

598

„ September

2

51 H. Cephei

E

595

609

:',

E

5S^

619

Although systcinatically different values are obtained from
tlie different elongations on the different assumptions, the mean
value from hoth elongations on the first assumption is nearly equal
to that from both elongations on the second assumption. ï:^o tliese
mean values may be looked upon as giving approximately the real
value of the micrometer.

An assumption, to hold good, must l)e of sut-h a nature as to
yield theoretically the same value from both elongations. From
this point of view both of the above assum])tions are to be rejected,
and Ave must find an intermediate one which will bring Ijoth elon-
gations into conformity.

It is manifest that such a reduction, provided its validity be
detinitely granted, Avill need a diminishing factor less than unit}^
In order to determine the diminishing factor I made a complete
reduction of my series of elongation observations, forming the
equations of condition in the following form, —

A + Bt + CO = a + Db

g Art. 3 — K. Sotouio:

where A unknown constant

B yearly change of tlie micrometer vahie

C temperature coefficient

]) (hminishing factor

t time

d temperature

a vakie of micrometer corresponding to the first

assumption.

1) excess of the micrometer vakie on the second

assumption over that on the first assumption.
For the sake of convenience, I took the year 1910.0 and 7^0 for
the origin of the time and temperature respectivel3\

Assigning equal weights to all the equations of condition, I
treated them l^y the method of least squares. The solution of the
normal equations gave, among others, as the most probahle value
of the diminishing factor D,

]) = 0.54 ± 0.17 mean error.

This result shows that the regular displacement of the level
bubbles is due to two different causes :

I A southward displacement of the bubbles due to some

cause in the level, independent of the motion of
the telescope. This accounts for al)out half of the
total motion.

II A regular northward depression of the telescope,

corresponding to nearly half of the total displace-
ment of the bubbles.
Owing to these phenomena we should obtain for the micro-
meter too large a value from western elongation and too small a
value from eastern elongation on the hrst assumption; conversely,
too large a value from eastern elongation and too small a value
from western elongation on the second assumption.

The first phenomenon may possibly be a disturbance due to the
heat of the observer and of the reading lamp. When observing lati-
tude by the Talcott-Horrebow method, the proximity of the observ-
er to the instrument is of short duration; so that the effect would

Consideratious on the Problem of Latitude Variation. 9

not appear, owing to a reason to hu discussed later. And, even
if the effect of this were considérai )lc, it would be ehminatcd from
the final result, as the relative position of the observer and the
levels does not change in the two positions of the telescope. There-
fore I did not try to make any further investigation on this point.

As to the second phenomenon, we can take into consideration
the following three causes. —

i. Flexure of the telescope,
ii. Differential change of the telescope stand.

iii. Gradual tilting of the pier and the ground.

A gradual change of the flexure of the telescope tube may be
probal)le. according to K. Hirayama^\ although it does not seem to
be very effective, considering the structure of our zenith telescope.
And this cannot l>e the sole cause ; for if it were, the displacement
of the bubble must be wholly attributed to a cause in the level itself,
as the flexure of the telescope would by no means api)ear on the
level. As this consequence is of course unnatural and also im-
probable, we are led to seek some other causes. At any rate, as
the effect of flexure on latitude from the observation of a star pair,
when we 1)egin Avith the southerly star, has the tendency to cancel
that from a star pair, when we begin with the noi'therl}^ star, the
final effect will tend to vanish, if these distinct pairs are impartial-
ly contained in a group. Moreover, even if there be a residual, it
would be eliminated I)y the chain method reduction, as it can l>e
looked upon as persistent with the star pairs. So I shall not at-
tempt any further discussion of this subject.

As to tlie disturbance of the telescope mounting, I can first
of all take into consideration the thermal effect of the observer's
body and the unsymmetry of the meteorological conditions with
respect to the instrument. In winter the wind blows mostly from
the north, producing a draught of cold air ; in summer, the south
wind prevails, forming a warm air current. These disturbances,
combined together, may cause a certain unsymmetrical distribution
of heat in the telescope mounting in some way, and therefore a
certain change in the inclination of the telescope. As the heating

li Astronomische Nachrichten, Xr. 4332.

10

Art. 3 - K. Sotomu:

effoct varies witli the difference of temperature between the body
and the instrument, tlie (hsturbance would vary inversely with the
temperature, i.e., it would be greater in winter than in summer.
This sequence agrees well with the observed plienomenon, and wo
may take this thermal effect as one cause of the behaviour of the
zenitli telescope.

Witli respect to the third cause, I may notice the fact that tlie
horizontal pendulum observers in several parts of the world have
revealed a remarkable unsteacHness of the ground. Tlie most con-
spicuous of the regular movements has a period of one solar day,
being due to the effect of the solar radiation on the earth's surface.

E, von Rebeur-Paschwitz did pioneer work with the horizon-
tal pendulum of his original form at Karlsruhe (1887), Strassburg
(1892-94), and Nicolaiew (1892), to the effect that the pendulum
swinging in the prime vertical showed a regular movement in the
period of one solar day, and took the soutliernmost position in the
evening and the northernmost position in the morning, as is here
shown, — ^^

Table III.

Component in the meridian (pendulum swinging in tlie prime
vertical) North +, Soutli — .

Mean Time

K;irlsruhe

Strassbv.rg

Nicolaiew

0-'

+ 0."C8tD

+ 0.'017

+ 0.040

2

-0. 0J5

-0. 016

+ 0. 024

4

-0. 151

-0. or.7

+ 0. 033

('>

-0. 200

-0. 079

-0. 016

8

-0. 184

- 0. Oôt

-0. 030

10

-0. 192

-0. 034

-0. 037

12

-0. lOS

--('. Oil

-0. 037

14

-0. 016

+ 0. 015

-0. 029

16

+ 0. 135

+ 0. O^iO

-0. CIO

1) .\strunoinische Nachrichten, Nr. 3148.

Considerations on the Problem of Latitude Variation.

11

Lu cal
Mean Time

Karlsruhe

Strassburg

Xicolaiew

h

18
20

0'7

+ 0.225
+ 0. 237

+ 0. 204.

+ 0."073
+ 0. 0(:;5
+ 0. 041

+ 0."013
+ 0. 035
+ 0. Ot5

His result at Strassburg (1892-94) was exi^'ossed in the
following résumé : ^^

The epoch of daily miniiuuui or southern elongation nï tlie

pendulum lies generally l)etween 5'' and 6''. The daily

maximum or northern elongation comes always in the

morning between lS''-20'', or slightly later in winter. The

amplitude varies l)et\veen 0. "1 and 0. "2.

After the early death of von Rel)eur, Ehlert took chai'ge of

the observations at Strassburg (1895-9(')). His result was quite

similar to that of his predecessor, as may be seen from the table

l)elow."^

Tap.le IV.

Date

Southern t'longation

Xorthern elongation

Amplitude

Jan. 15

3,'o

20. 5

0."049

Feb. 15

3. 5

20. 0

0. Of, 7

]\Lirch 15

5. 5

20. 8

0. 138

May 1

6. 0

18. 0

0. 201

May 23

5. 9

18. 0

0. 138

June 15

5. 7

17. 9

r. 124

July 15

5, 3

17. 9

0. 135

Aug. 15

6. 0

19. 2

0. 187

Sept, 15

5. 5

20. 2

0. 208

Oct. 15

4. 5

20. 5

0. 053

Xovember

and December, indete: mi

nato

Mean

5. 1

18. 9

1) (3! erlands Beiträge zur Geophysik, Bd. IL
2i Ditto Bd. IV.

12

Art. 3 -K. Sotome :

He concludes :

Die ,, tägliche Periode" besteht in einer Schwankung des
Erdbodens, welche im Mittel 0. "112 beträgt ; in der Tiefe
von 5ni ßndet das ^hiximum der Nordableidcung im Mittel
um 18''.9, derjenigen nach l^üd um ö/'l statt. Die Ursache
liegt in der Ausdehnung der von der Sonne erwärmten Erd-
oberfläche, welche sich auch nach der Tiefe hin unter A))-

schwächung und Verzögerung geltend macht Klarer

Himmel und grosse Temperaturoscillation verstärken das
Plienomen

According to Hecker' s long series of olrservations at Potsdam
in an underground room at the depth of 25 metres (1902-1909)'\
the pendulum occupied the extreme southern position in the even-
ing and northern in the morning, ju-;t as in the preceding cases.

Southern elongation

Northern elongation

Marcli, April, May

&

17'

June, July, August

(>''

18'

September, October,
November

G''

15''

December, January,
February

G''

15''

'J'he liorizontal pendulum observations at Kyoto (1910-1911)
by Shida"'^ gave a similar iv-^ult, showing a solar daily variation
mainly in a SW and NE direction; the pendulum took the extreme
south-west position in the evening and the extreme north-east posi-
tion in the morning. The elongations in the meridian run as
follows: —

Ij Veröffentlichung dt-s KCinigl. Preuszisehen Geodätischen Institutes, Neue Folge Nr.
32, 4.9.

2j Memoirs of tie College of Science and Engineering, Kyoto Imperial University.
Yol. IV.

Considerations on the Problem of Latitude Variation.

13

Southern elongation

Northern elongation

April - Juno

8''

21''

July - September

10'''

21^^

Octobe r - December

6''

■20^

January - March

6*

20''

Ol)<ei'V;itions at Freiburg i. S. made by Hecker at a depth of
rJO metixs below the surface of the ground (1910-1911) scarcely
showed the solar efïect. It was almost insensible or negligible, the
amplitude not exceeding 0. "001. From these facts we can infer
that the solar thermal effect is the minore pronounced the shallower
the gi\iund work, as might also be guessed from mere conjecture.

While most horizontal pendulum rooms are deep and are kept
in as invariable a temperature as possil)le, our astronomical telescope
piers are, on the cont]'ar3% rather shallowly founded and are sul)-
jected to the diurnal change of temperature in its full range, from
their exposure to the free air. Moreover, astronomical observations
are naturally only practicable in clear weather, when the change
of temperature of the environment of the pier is great and rapid.
Under these circumstances we are led to conclude that the ground
and pier, on which the telescope is set up, are undergoing a con-
tinuous tilting whilst it is being used in the night, the meridian
component of which is in the direction, north downward or south
upward, and the magnitude of which is far greater than that experi-
enced in the horizontal pendulum observations. There is, however,
a contradiction between the European and Japanese observations.
In Europe the amplitude is at its maximum in summer and its
minimum in winter ; in Japan^\ on the contrary, it is at its maxi-
mum in winter and its minimum in summer. This difference may
be due to some meteorological causes, depending on the latitude.
For low latitudes the latter variation may hold good. It is also

1) Shida, loc. oit.

14 Art. 3-K. Sotome:

to be noticed tliat the rate of the tilting motion increases from
evening until miclniglit.

In this way we are led to the conclusion that the observed
phenomenon is to be understood as principally due to the second
and third of the above causes, whose efïects can be looked upon as
generally varying inversely with the temperature during the time
of our observations.

The above cited singular Ijehaviour of the zenith telescope was
deduced from the regular shift of the level bubbles, as experi-
enced by myself. In order to ascertain whether the same pheno-
menon is noticed in other ol)servatories, I made enquiry of several
latitude observers in \'arious i)arts of the world respecting this sub-
ject. Fortunately, almost all the specialists were kind enougli to
favour me v/ith detailed answers, for which the writer tenders his
most sincere thanks.

Prof. H. Kimura and Dr. M. Hashimoto of the Mizusawa
Latitude (Jbservatory recognized the phenomenon in the zenith
telescope there used. We can also see the tendency from the re-
sults of their elongation observations, as given in " liesultate des
Internationalen lîreitcndienstes," Bd. I., pp. 19-20. In the deter-
mination of the value of tlie micrometer during the period 1899
Dec. 27-1900 March 11, eastern elongations constantly give larger
values than western. This corresponds to the case Avhen the re-
duction was made on assumption II. (siqira).

I could hear nothing from Tschardui observers. But the series
of determinations given in "Resultate," Bd. IV., page 74, makes
me suppose that there is also the same tendency there. In the
observations during the period IDOG January 19 — February 28, the
mean value from western elongations (60. "2G8) is decidedly larger
than that from eastern elongations (60. "148). This may be caused
l)y tlie method of reduction on assumption I. If such is really the
the case at Tschardui, we can fairly well account for the fact that the
temperature coefficient of the Tschardui micrometer comes out
notably lai^ger than those of the other stations. This is because
winter observations are principall}" of western elongation while
summer observations are mostly of eastern elongation; so that the

Considerations on the Problem of Latitude Variation. 15

winter value coines out too large, l)ut the summer value too small,
giving finally too large a tempei'ature coefficient.

Dr. G. Bemporad of the International Latitude Station at
Carloforte reported that his instrument showed variations during
elongation observations, but that the sense was not systematic.
However, Dr. V. Fontana, the director of the station, wrote me
afterwards that he had l)egun some researches on the problem of
the systematic sliift of the level bubbles, and desired to know ni}^
results.

Dr. F. E. Ross of the Gaithersburg Latitude Observatory in-
formed me that he experienced a similar phenomenon, and con-
sidered it to be principally due to a temperature effect in the
ground, which usually shows a progressive change one way or the
other.

Dr. E. L Yowell of Cincinnati Observatory favoured me with
a letter, stating that his telescope shows a similar tendency. He
ascribes that effect to the heat radiation from the observer and
the reading lamp.

Dr. ^V. F. Meyer of the Latitude Observatory at Ukiah kindly
reported that his own experience was limited, but that his predecessor,
Dr. Schlesinger, had observed the same tendency. The following
was extracted from the record book, which gives Dr. Schlesinger' s
opinion in the matter: '' there is reason to believe, from an inspec-
tion of the results, that these changes in the levels are not wholly
due to a real change of inclination of the telescope. No doubt the
presence of the observer for so long a period at one side of the
levels has an injurious effect upon the levels."

At Union Oljservatory, Johannesburg (Astronomer, Prof. R.
I. Innes), the method of polar star elongations was not employed.

At Kasan Observatory (Dr. i\L A. Gratschew) the instrument
and method were different.

The following astronomers have kindly let me know that they
experienced no systematic movement of the level bubbles.

Prof. E. Doolittle of Flower Astronomical Observatory,

University of Pennsylvania. (Warner ct Swasey's zenith

telescope).

Iß Art. 3— K. Sotome :

Prof. J. Bonsdorff of Pulkovo Observatory.

Dr. E. JSchoenberg of the Universit^^- Observatory at Jurjew

(Dorpat): (transit instrument of broken type made by Repsokl).

After all, it may be concluded that the majority of the most

careful latitude observers (especially the international observers)

experience a systematic shift of the level bubbles, in the same

manner as I have already described.

Thus I have reached the position of being able to make the
following resume, in so far as I have ol)tained results from the
al)0ve investigations: —

In most cases the level bubbles of the zenith telescope make a
southward shift s^^stematically, when the observer has been
near the telescope for a tolerably long time, the magni-
tude of which varies inversely with the temperature.
This movement may be considered as due, partly to the
observer's direct effect on the level, and partly to the tilting
of the telescope mounting, <nving to the observer's heat disturb-
ance, in conjunction with some meteorological conditions.
The regular change in the ground from the solar radiation may
also contribute to the latter, altliough to a comparatively
slight degree.

It may not be out of place to remark that we may expect the
counter effect to the above phenomenon; that is, when the observ-
er recedes from the instrument after a tolerably long stay near it,
the bubble would move back in a northerly direction towards its
original position, being freed from the thermal disturbance. This
was often experienced by some observers. It seems also to be very
desirable from my standpoint that hereafter the elongation observ-
ations of polar stars should be more frequently and regularly made
and the reduction performed more exhaustively, so as to throw
further light upon tlie subject.

2. A theory of the motion of the level Bubbles.

In using a screw micrometer we are accustomed to elinnnate

Considerations on the Problem of Latitude Variation. X7

the so-called ''dead motion" of the screw by turning it always in
the same direction. The same principle is applied in determining
the value of a division of the spirit level, always making the bubble
come to rest from the same direction. This precaution implicitly
recognizes the existence of a failure in the function of the level.
It seems to me rather curious that in the practical work of star
observation the level is taken as a perfect instrument and the pre-
caution needed in the above case is utterly neglected. Here I
intend iu the following lines to show that there actually exists such
a failure in the function of the spirit level. I will first proceed
from the theoretical standpoint and then give an experimental
proof.

The motion of a level buljble can be looked upon as analogous
to a damped oscillation of a simple pendulum ; the equation of
motion accordingly takes the following form, —

d'd ^ dd „^ _

4- 2p — - + n-d = 0

df "■ dt

where/) and n are constants and 6 is the deviation of the bubble
from the position of equilibrium. The applicability of this equation
was experimentally tested by Bonsdorff'^ and further discussed by
Orloff.')

The integral of this differential equation comes out in the fol-
lowing three forms, according to the cases, n'>'j), n-'p, yi<p respec-
tively, —

( i ) nz>p, d = e~"' [cii cos (it + a.sln fd]

(ii) n=p, 6=6-"' [a,t + a,]

(iii) n'<p, 6 = b, f;-V + 6,e-V ip' = iv—p-

whereJrta = pVi-Jl-'j^. |

Here the quantity n depends on the radius of curvature of the level,
and the quantity p is a function of the bublde length and of the
viscosity of the liquid.

1) Mitteilungen der Nikolai-Hauptsternwarte zu Pulkowo, Bd. II, p. 43.

2) Ditto Bd. II, p. 137.

13 Art. 3-K. Sotoine:

When the radius of curvature is small, and the bubble length
fairly long, the case (i) occurs, exhibiting a periodic motion. But
in the sensitive level, the radius of curvature is necessarily long,
but the bubble length cannot be proportionately long. This class
of instrument corresponds to the cases (ii) or (iii), which represent
an aperiodic motion. In these two cases the bubble comes to
rest asymptotically, that is, after the lapse of an infinite time. In
other words, the bubble of our sensitive level does not come to its
destination theoretically in a finite time. In practice, however, to
wait even a pretty long time is useless, as some other disturbing
cause may interfere with it in the meanwhile.

Let tx be the time interval, during which we wait for the rest-
ing of the bubble (practically, a minute or two), and d^ the corres-
ponding value of 6. Thus, —

Then we are referring to d^ instead of 0 as the resting point of the
bubble. As this discrepancy can be looked upon as caused by
internal friction or viscosity of the liquid, the sense of d^ will be
naturally opposed, when the direction of motion is opposite, giving
-6^. Therefore,, when the bubble has moved from opposite direc-
tions the resting points would difïer by the quantity 26^. This
property would afïord a method of determining the magnitude of
^i, which we may call the resistance of the level.

The occasional lack of parallelism in the two level bubbles of
the zenith telescope, commonly experienced by latitude observers,
can partly be explained by the above phenomenon, as the parallelism
would change by the sum of the resistances of the two levels, when
the two bubbles come to rest from opposite directions.

Having thus examined the existence of a defect of the level
from the theoretical standpoint, I proceed now to ascertain the
order of magnitude of the resistance d^ from the experimental side,
depending on the principle cited above.

The instrument I made use of in my experiments was a level
trier made by Hilldebrand in Freiburg, and the levels subjected to
the examination were the following nine pieces : —

Considerations on tho Problem of Latitude Variation. 1 Q

No. 1. The hanging level of the transit instrument,

Bamberg No. 7958.
Nos. 2, 3. The latitude levels of the above.
No. 4. The hanging level of the transit instrument,

Bamberg No. 7959.
Nos. 5, 6. The latitude levels of the above.
No. 7. The hanging level of the transit instrument,

Bamberg No. 11508.
Nos. 8, 9. The latitude levels of the above.

First the level was fitted to the trier in such a way that the
bubble moved in increasing sense of division in accordance with
the increase of the micrometer reading of the trier. Then the
micrometer was turned in increasing sense and set, say, to a read-
ing a. The l)ubble would move increasingly and come to rest at
/9-^„ where Q, is the resistance. Next, the micrometer was further
turned slowly up to, say, division « + 10. Hereupon it is turned
back to «-1; then again it is set at the reading «, theoretically
bringing the level Ijack to the original position. To this operation
the bubble conforms with considerable lag of time, owing to its
inertness, and would come to rest in decreasing sense at the point
/9+(?.. Let the reading of the bubble centre for the first position
be /S'i and for the second position be ^-i, then we have

This process was applied to two such points alternately ; for the
first point, the motion was in tlie order increasing-decreasing, for
the second point in tlie order decreasing-increasing. This will
eliminate the effect of the gradual change of the pier, not to speak
of determining the value of one division of the level. The result
of my experiments is given in the following table V, in which
the resistance d, is expressed in terms of the unit of division of the
levels.

20

Art. 3 -- K. Sotoine :

Table V.

Level

Date

Bubble Length

Kesistance 9^

No. of Observations

No. 1.

1915 Feb.

17
18

d
24

24

d
0.034

0.042

58

30

No. 2.

1913 Dec.

31

1914 Jan.

8

»

10

,,

12

„

13

»

15

„

31

Feb.

7

„

12

28

21
24
24
24
24
23
20
20
21
20

d
0.022

0.026

0.035

0.042

0.047

0.029

0.049

0.031

0.029

0.029

1 division = l."32

52
42
44
24
32
28
50
26
54
24

1 division = 1."32

No. 3.

1913 Dec.

31

1914 Jan.

8

»

10

..

12

..

13

,.

15

„

31

Feb.

7

„

12

28

26
26
26
27
26
26
20
21
21
20

d
0.020

0.012

0.023

0.016

0.024

0.050

0.030

0.031

0.060

0.027

52

44
24
32
28
50
26
54
24

1 division = l."33

Considerations on the Problem of Latitude Variati

21

Level

Date

Bubble Length

Resistance Oi

No. 4

1915 Jan.

6

25

d
0.C69

"

10

27

0.025

•'

18

26

0.057

'•

19

26

0.062

'•

21

28

0.043

Fel:>.

13

37

0.072

No. 5

No. 6

No.

No. 8

1914 Oct.

2

..

4

..

9

1915 Feb.

14

..

15

10

26
26

26
27

d
0.587

0.464

0.523

0.479

0.462

0.441

(This level is not fit for use.

1914 Oct. 2

26

4,

26

9

25

1915 Feb. 14

26

15

27

16

27

1915 Feb. 20
22

1915

Jan.

20
21
21

24
24

d
20

20
32

d
0.053

0.202

0.080

0.055

0.047

0.084

d
0.045

0.012

d
0.050

0.073

0.030

No. of Observations

24
18
30
34
14
21

1 division=l."22

32

28
32
52
16
28

1 division = 1."01

32

28
32
52

16

28

1 division = 1."06

42
08

1 division = 1."00

24
20
10

22

Art. 3— K. Sotome ;

Level

Date

Bubble Length.

Resistance öi

No. of observations

No. 8

1915 Jan. 23

d
33

d
0.006

42

27

26

0.040

20

28

26

0.033

48

29

26

0.029

52

No. 9

1915

Jan. 20
21
21
26
27
28
29

19
19
33
34
25
25
25

d
0.079

0.060

0.020

0.010

0.040

0.022

0.021

1 division = 1. "26

24
20
10
42
20
48
52

1 division = l."28

These results show tliat the resistance ^^ of the level is not of
a negligiljle magnitude, so far as the degree of accuracy required in
the zenith telescoj^e observations is concerned, and we can take it
as established that this defect is a common property of the spirit
level, as we have confirmed its existence both from the theoretical
and the experimental standpoints.

Hitherto we have been considering the motion of the l)ubl)le in
case the level itself is at rest. But as the motion is purely of a re-
lative nature, we can convert the above obtained result to the case
when the bubble is at rest and the level moves.

Thus, when the level is put into slight movement from rest,
the bubble will accompany it, and will fail to show the real move-
ment of the level. If the movement of the level is less than or
equal to the resistance ^i, as above obtained, the bubble would
show no displacement relative to the level.

Moreover, when the level continues to move further, the
bubble will follow it to a certain degree. The equation of motion

Considerations on the Problem of Latitude Variation. 23

takes then the following form, taking the origin at the highest
point of the level, —

where the quantity A depends on the quality and movement of the
level. Under the same conditions as before, in the case of our
sensitive level, we have

^ n

This equation shows that another perturbing term interferes in
this case. Thus it is to be concluded that the indication of the
level bubble has an error greater than the resistance 6^, when the
level itself is in motion.

Now I showed in the foregoing part of this essay that there
are reasons to believe in an unsteadiness of the telescope and pier,
not to speak of the seismic movement of the ground. From the
above investigations, we see that the level cannot indicate instan-
taneously the varying position of the telescope in presence of these
disturbances. So it is manifest that the spirit level which we now
use is not suited to fulfilling our requirements with the degree of
accuracy demanded in modern astronomical measurements. We
are faced with the necessity of using some other means in order to
realize the present expectations of practical astronomy.

3. Application to the Talcott-Horrebow observations and
deduction of effect on the variation of latitude.

In the first section of this essay I discussed certain systematic
motions of the levels and zenith telescope, which can be looked
upon as due, firstly, to the disturbance from the observer, and
secondly, to a terrestrial cause. In the second section I investiga-
ted the failure of the function of the level which can l)e taken as a
common defect of the level. Conversely, I can now conjecture
how the telescope and level behave under such circumstances.
When the observer approaches the telescope, the stand would first
suffer a thermal disturbance, and cause the said efïect, owing to the
unsymmetry of meteorological conditions. To this the level bubble

24 Art. 3-K. Sotome:

will not respond immediately, because of the defect discussed above.
When the disturbance has exceeded a certain limit, the level begins
to indicate it to a certain degree. The thermal effect on the level
would appear later, as the levels are more distant and better pro-
tected. After a fairly long time, the resulting effect would be the
observed systematic shift of the level l)ubbles, in Avhich the regular
tilting of the ground partakes to some extent.

Now the observation of latitude variation is based on the
cyclical system of star groups, consisting of pairs of stars selected
for the Talcott observations. This method is the so-called chain
method. As the result of this procedure, we ol)tained the polar
motion and Kimura's z term. The closing sum is also a product of
the chain method. The z term and closing sum form the principal
enigmas of present-day practical astronomy.

Now for the first subject numerous causes have been proposed,
among which we may mention the following, —

1. Yearly atmospheric refraction.

2. Yearly cosmic refraction.

3. Improper value of the parallax and proper motion of the

oliserved stars.

4. Ditto of nutation and aberration.

5. Actual change in the earth's centre of gravity.

6. Eesult of computation.

7. Latitude variation of short period.

All these hypotheses may be in a greater or less degree probable,
and at the same time no one of them has yet such a firm I)asis of
proof, as to secure our universal assent.

The same may be said of the closing sum, for which the fol-
lowing hypotheses may be mentioned, —

1. Erroneous value of aberration constant.

2. Diurnal atmospheric refraction.

3. Latitude variation of short period.

The first explanation has been universally accepted. But the
usually adopted value 20. "47 is quite consistent with the recently
determined value of the solar parallax, and does not permit so

Considerations on the Problem of Latitude Variation. 25

much increase as to account for the whole closing sum. The second
and third are nothing more than conjectures.

Under these circumstances, it does not seem utterly super-
fluous to make new suggestions, to be further discussed and inves-
tigated by a wider circle. With tliis intention, I venture to declare
that the above puzzling subjects may possibly l)e explained l)y the
phenomenon above discussed, in connection with the defect of the
spirit level.

Now, provided that the considerations in the earlier part of
this essay on the disturbance of the zenith telescope are applicable
to tlie case when the Talcott-Horrel)OW observation is made, in
which the proximity of the observer is of short duration, the first
part of the disturbing efïect only may come into play, so tliat the
change of inclination of the telescope, due to the thermal disturb-
ance, would not appear in the position of the level bubbles.
Therefore, the level bubbles Avill be situated too far north of their
due position. The result of this is that the corrections depending
on the level reading are positively too large or negatively too
small, giving finally too large a value of latitude. The principal
part of this error varies inversely with the temperature. So the
correction to be applied to the latitude is of negative sign and of
varying magnitude, depending on the seasons and the hour of the
day.

Let the star groups selected for the chain method be from I to
XII, as is adopted in the International Latitude Service. And
further distinguish these groups by suffixes according to coml)ina-
tions ; sufiix 1 when coml)ined with the ])receding group and
suffix 2 when combined with the following group. The period of
observation and the corresponding group are as follows, —

Date.

Corresj^ondin

<^- Group.

Nov.

2 — Dec.

6

I. &

Ill

Dec.

7 — Jan.

4

IL, &

nil

Jan.

5 — Jan.

30

III, &

IVi

Jan.

31 — Febr.

24

IV. &

Vi

26

Art. 3— K. Sotome ;

Date.

Corresponding Group.

Febr.

25 — March

21

y-2

&

VIi

March

22 — April

15

VI,,

&

VI Fl

April

16 — May

11

VIL,

&

Villi

May

12 — June

8

VIIL,

&

IXi

June

9 — July

9

IX,,

&

Xi

July

10 — Aug.

13

X,,

&

XIi

Aug.

14 — Sept.

22

XIj

&

XIIi

Sept.

23 — Nov.

1

XII,,

&

II

As the temperature continues to fall during the latitude ob-
servation in one night, we may tentatively but reasonably assign
the following mean corrections to the latitude corresponding to
each of the groups, —

II

-0.05

h

-ao5

III

-0.07

IIa

-0.07

nil

-0.08

Illy

-0.07

IVi

-0.08

IV2

-0.06

Vi

-0.08

V2

-0.05

VIi

-0.07

VI2

-0.04

VIIi

-0.0(5

VII2

-0.03

villi

-0.04

VIIT2

-0.02

IXi

-0.03

1X2

-0.01

Xi

-0.02

X2

-0.00

XIi

-0.01

XI2

-0.01

XIIi

-0.03

XII,,

-0.03

Now in the reduction of the chain method it is assumed that
the latitude is constant during the observation of two groups in one
night, and under this supposition the star places are reduced to the
mean of the whole system. But when the apparent latitude ob-
tained from observation requires the said correction, the reduction

Considerations on the Problem of Latitude Variation. 27

to the mean would consequently need a corresponding revision.
Thus in order to find liow the above corrections enter into tlie re-
duction by chain method, i. e. , how to reduce them to a true
homogeneous system, we must treat these corrections in a manner
exactly identical with the chain method reductions.
I form, tlierefore, —

Correction to III-IV = Difference of corrections to Ills & IVi = +o!bl

IV-V = „ IV2 & Vi = +0.02

V-VI = „ v. & VI, = +0.02

VI-VII = „ VI2 & VIIi = +0.02

VII- VIII = „ VII2 & VIII, = +0.01

VIII-IX = „ VIIIo & IX, = +0.01

IX-X = „ 1X2 & X, = +0.01

X-XI = „ X2 & XI, = +0.01

XI-XII = „ XT2 & XII, = +0.02

XII-I = „ XII2 & Il = +0.02

I-II = „ I2 & III = +0.02

II-III = „ II2 & III, = +0.01

Hence,

the correction to the closing sum, in the usual sense, = +0.18

Further, in order to find the corrections to the reductions to
the mean system, I form, —

for the gi'oup III ; correction to

IV-III

= -0.01

V-III

= -0.03

VI-III

= -0.05

VII-III

= -0.07

VIII-III

= -0.08

IX-III

= -0.09

X-III

= -0.10

XI-III

= -0.11

XII-III

= -0.13

I-III

= -0.15

II-lII

- -0.17

Sum

= -0.99

I apply similar treatment to the other groups and divide the
sums by 12.

28

Art. 3— K. Sotome :

The correction to tlie closing sum is to be distributed equally
among the groups, so that the quantity, —

0.18
12

xll-^2 = 0.083

is to be added to the above. The resulting quantities constitute
the reductions of the star places to a true homogeneous system, —

I

II

III

IV

V

VI

VII

VIII

IX

X

XI

XII

o!boo

+ 0.Ü05

o.'ooo

-0.Ü05 o'.OOO

+ 0.005

+o!oio

+ 0.005

O.UOO

-o!ü05

-o!6io

- 0.005

Combining these with the said corrections to the latitude ob-
servation, we apply the usual process of reduction and form differ-
ences with the mean of all of them. Arranging them with respect
to seasons, w^e have, —

Mean Date

Corrections

Mean

Eesultin;^ Corrections

Jan. 18

-d!070, -0.085

-0.078

- d.'033

Febr. 12

-0.0Ü5, -0.080

-0.073

-0.028

Mar. 10

-0.050, -0.065

-0.05S

-0.013

Apr. 4

-0.035, -0.050

-0.043

+ 0.002

Apr. 29

-0.020, -0.035

-0.028

+ 0.017

May 26

-0.015, -0.030

-0.023

+ 0.022

June 25

-0.010. -0.025

-0.018

+ 0.027

July 27

-0.005, -0.020

-0.013

+ 0.033

Sept. 3

-0.020, -0.035

- 0.028

+ 0,017

Oct. 13

-0.035, -0.050

-0.043

+ 0.002

Nov. 19

-0.050, -0.065

-0.0.58

-0.013

Dec. 21

-0.065, -0.080

-0.073
Mean -0.045

-0.028

These are the corrections to the latitude variation which may
be looked upon as common to all stations. Therefore naturally

Considerations on the Problem of Latitude Variation. 29

it is the correction to that term of latitude variation which is in-
dependent of polar motion, or z term.

Now, according to Ross^\ the mean value of z term during
the years 1900- 1005, can be put in the following analytical form, —

+ 0."()27 sin (® + 170°) (® = sun's longitude)

when tlie effects of stellar parallax, Oppolzer's term in latitude
variation and nutation are taken into account and excluded. Hence
it follows that the result obtained through my argument is practi-
cally sufficient to account for the closing sum and z term.

Now, as the phenomenon discussed by me is considered to
arise from some unsymmetry of the meteorological conditions and
also from the solar radiation on the ground, its sense should be
inverted for the southern hemisphere and is to be considered as
an odd function of latitude, vanishing at the equator. So it can be
looked upon as varying with sm f. Under such a conception, the
effect on the southern observations should be opposite to tliat
on the northern, the correction to the latitude being of positive
sign and the correction to the closing sum negative.

The correction to be applied to the latitude variation indepen-
dent of longitude would be

positively small for January, and

positively large for July.
Therefore, the correction to Ije applied to the yearly term in the
variation of latitude, or z term, is

negative for January, and

positive for July.
The sign is the same as that of the northern observations, and is
sufficient to interpret the result obtained from the southern observ-
ations.

As to the amplitude of the z term, it depends jointly on the
said phenomenon and the seasonal variation, Ijoth of which can be
looked upon as varying with sin <f> ; so we can consider the ampli-
tude of the correction of z term as dependent on sl/i^ <p. This
signifies that the amplitude of the z term increases from the value

1) Astronomische Nachrichten, Xr. 4593.

30 Art. 3— K. Sotome

zero at the equator, towards north and south latitudes symmetri-
cally. To ascertain Avhether this is actuall}^ the case or not,
observations at the equatorial zone are very desirable.

The numerical results so far obtained I do not intend to regard
as definitive. But they may serve as a clue for the yet unsolved
problems, or an indication of the direction in which we must pro-
ceed in order to account for them.

Now, as my argument has been advanced from two causes, viz.

1. some regular movements of the zenith telescope;

2. a defect of the spirit level,

I may also state with diffidence the results to which my considera-
tions have led me, as follows: —

If we assume a regular change of inclination of the zenith
telescope, varying with the seasons and the hour of the day,
the three phenomena, that is, z term, closing sum, and the
regular shift of the spirit level, can all become explicable to-
gether. Therein the defect of the level makes the error enter
into the latitude value and gives rise to the z term and closing
sum.
My idea, therefore, leads to the conclusion that these enigmatic
subjects in the latitude problems would disappear, if the level were
really a perfect instrument, or if we were to use another and an
ideal means instead of this insufficient instrument.

The photographic observations made with the Cookson floating
zenith telescope at the Greenwich Observatory by Mr. Jones' are
very -important in regard to the present problem. The latitude
variation obtained from this series was compared with the results of
the International Latitude Service. Herein the agreements was
particularly improved, when the z term was subtracted from the
latter series. As to the aberration constant, the value

20. "467 ± 0."UOG

was obtained. In other words, this novel series of observations was
practically free from the z term and closing sum. Arguing from

1) Monthly Notices of the E.A.S., Vol. LXXV, No. 7.

Considerations on the Problem of Latitude Variation. 3!

my standpoint, this is to be looked upon as a natural and necessary
consequence.

The notion that tlie level is not suited to the most delicate
measurements is not novel. Already as earty as in the year 1893, E.
von Rebeur-Paschwitz^^ insisted on the inadequacy of the spirit level
to meet the disturbance in precise astronomical measurements from
pulsatory or microseismic movements of the ground. Also G. H.
Darwin is said to have declared to the same effect, — ''I venture
to predict that at some future time practical astronomers will no
longer Ije content to eliminate variations of level merely by taking
means of results, but Avill regard corrections derived from a special
instrument as necessary to each astronomical observation."

Altliough the motive of my idea is not strictly the same as
that of these authorities, our resulting conceptions are convergent,
and I am pleased to conclude my paper with the expectation that
Darwin's prediction will be promptly actualized in order to meet
the degree of accuracy required in up-to-date astronomical ob-
servation.

Lastly, the writer desires to express his sincere thanks to the
several astronomers and observers referred to above, by whose
efforts and information he has been enabled to accomplish this
work.

The Tokyo Astronomical Observatory, September 1915.

Published Nov. 30th, 1915.

1) 4stronomische Nachrichten, Nr. 3177.

JOUKKAL OF THE COLLECrE OF SCIENCE, TOKYO IJirERTAL UKIVEKSITV.

VOL. XXXVII., ARTICLE 4.

On the Distribution of Cyclonic Precipitation in Japan.

(Contribution I. from the Geophysical Seminary in the
Physical Institute, College of Science).

By

Torahiko TERADA,

UifldlnilinhiisJii,

Takezo YoKOTA, -, Sy6hu OTUKI.

,,. , , . unci ^

The distribution of cyclonic precipitations has been investi-
gated by many meteorologists for different localities. H. Hilde-
brandsson^"^ found that in Upsala, the precipitation is most abundant
on the W side of a depression. Akerbiom"^ found the heaviest rain-
fall in Thorshavn on the front right quadrant of a centre, whereas
in Vienna it was on the opposite side. Since these classical investiga-
tions, it seems to have been a favourite problem to inquire on which
side of a depression the precipitation is most frequent or al)undant.
Krankenhagen^'^ found greatest precipitation in Swinemünde on
the rear side of a cyclone and attributed the fact to the influence
of seawind prevailing on that side. V. Drapczyanski'*'^ came to the
conclusion that the precipitation in 8t. Louis, U.S.A., is most
frequent on the rear side, l)ut the average amount of rain on rain-
days most abundant on the S side. H. R. ^lilP^ investigated the
trace or " smear " swept by the rain area accompanying a number
of remarkable depressions passing over Great Britain, and came to
the result tliat " the belt of cyclonic rains is much wider on the left

1.) H. Hildebrandsson, Sur la distribution des éléments met. autour des minima ft des
maxima barométriques, Upsala 1S83 ; v. Bebber, Met. ZS., 18S4.

2.) Ph. Alcorblom, Sur la distribution à Vienne et à Thorshavn des éléments etc.

3.) Krankenhagen, Met. ZS. 1885. See also Polis, Met, ZS., 1904.

4.) Drapczyanski, Met. ZS. 1903.

5.) H. E. Mill, Sjmon's Met. May. 39, 1904 ; Quart. Jouru. E. Met. Soc. 36, 1908.

2 T. Terada:

of the patli tlian on tlie right, and the heaviest falls occur in ad-
vance of the centre" regardless of the direction of progression of
the depression. He also remarks that the distribution of cyclonic
rains seems to have no apparent relation to the physical feature of
the country, though data were wanting to decide the point. J. A.
Udden^"^ who attacked the proldem for Davonport, Rock Island,
111., U.S.A. and also for a number of other districts, found that
the position of tlie area of the greatest rainfall-frequency relative to
the depression, varies largely for difïerent localities ; he suggested
that the relation may vary w^ith the hours, the seasons and the
-courses taken ]\y the depression. W. G. Reed^-\ following the
method adopted by Mill, made an extensive investigation of the
" smear " for a number of cyclones passing over the U.S.A. His
results are not so simple as that of Mill. He notes rightly the con-
siderable influence of large water bodies, such as the Atlantic Ocean
.and the Great I^akes and also the fact that the rain area forms a
series of patches, the heavier ones being connected by lighter ones
■or rainless areas. The latter fact is also attril)uted to the influence
•of important water bodies, though he concludes : " every relation
■expected occurs, l)ut there seems to be as yet no classification
which will reduce the relation to a system." Hann^'^ discusses
these results in his Lehrbuch and emphasizes the influence of the
•districts over which the wind comes.

One of the authors^'^ has discussed the influence of the distri-
bution of land and water in modifying the meteorological feature of
a region and proposed a simple theory which may explain many
regional irregularities of barometric distribution. As an illustration
of this theory, he drew attention to the difference of the Pacific and
Japan Sea Coasts, ^vith regard to the relative position of the rain area
and the centre of Ijarometric depression. The present paper may
be regarded on one hand as the continuation of the above cited one,
in so far as the results of the statistical investigation are reviewed
under the light of the theory, but on the other hand as a contribu-
tion to the literature on the general precipitation problem.

1.) J. A. Udden, Symon's Met. Mag. 41, 1906.

2.) W. G, Reed, Monthly W^eather Review, Oct. 1911.

3.) Hann, Lehrbuch d. Met. 3te Auf. S. 524., 552.

4.) Terada, Proc. Tùkyù Math.-Phys. Soc, 7, 1914..

On the Distribution of Cyclonic Precipitation in Japan.

Method of Investigation.

The " smear " method adopted Ijy Mill and Reed, though very
useful and convhicing for its proper purpose, is not convenient for
revealing the influence of the physical nature of land, since this
effect essentially lies in the peculiar distribution of precipitation
determined l)y the momentary position of the given configuration
of land and water, or of flat land and mountain range, relative to
the centre of depression, and may he quite obliterated if only the
smear is compared with the track of the centre. To make the
effect apparent, it is necessary to find the probability or the
amount of precipitation at different districts for different positions
of the depression. If these statistical data be at hand, we may
construct the isohyets for the different positions of the depression,
or draw for each district a diagram showing the distribution ()f the
probability or the amount for the different positions of the centre
relative to the district in question.

As material for the present statistical investigations the daily
weather charts and " Kisyôyôran " (a brief monthly weather re-
view) of the Central Meteorological Observatory, from January
1905 to December 1915 were used. The different districts of which
the precipitations were to be investigated were at first classified
into three groups : Pacific, Japan Sea and middle regions. Each
group was again divided into six subgroups including the meteoro-
logical stations mentioned Ijelow :

Pi Kagosima, Satazaki, Toizaki, Miyazaki.

P2 Asizuri, Koti, Hinomisaki, Siomisaki.

P3 Namikiri, Tu, Nagoya, Hamamatu.

P4 Numadu, Xagaturo, Yokosuka, Yokohama, Tokyo, jNIera,
Tyôsi, :\Iito, Tukuba.

P5 Kanayama, Kinkwazan, Isinomaki, Miyako.

l\ Tokati, Kusiro, Nemuro, Abasiri.

^Ii (Jita, Matuyama, Hirosima, Kure.

]M2 Okaj^ama, Tadotu, Tokusima, Wakayama, Kolje, Osaka,
Yagi.

^ T. Terada :

Ms Gihu, licla, Kôhu, Takayama, ^Matiunoto, Nagano.

M4 Kumagaya, Maebasi, Asio, Utunomiya.

M5 Hukusima, Yamagata, INIidnsawa.

Mg Aomori, Tappi, Hakodate.

J Kumamoto, Nagasaki, Saseho, Saga, Hukuoka, Simono-
seki.

J Hamada, Sakai.

J Kyôgasaki, Maiduru, Kyoto, Hikone, Hukni.

J Kaiiazawa, Husiki, ]\Iinatuki, Niigata.

J Kamo, Akita.

J Suttn, Sapporo, Kamikawa, Soya.
The above order of grouping may appear someAvhat arbitrary and
be liable to the oljjection that it takes no account of tlie peculiar
local cliniatic or physical features. This may, however, be justi-
fied if we consider tliat the principal aim of the present investiga-
tion is to sort out the general influence of the configuration of laiid
and water with respect to the centre of depression. Korea, Sagha-
line and Formosa were excluded, since these regions may obviously
stand under the influence of depressions existing bej'ond the limit
of the weather chart.

The position of the centre of depression was determined not
only l:>y the daily chart, but also by consulting the chart of the
track of depressions given in " Kisyôyôran." The chart area was
divided into 2.5° square meshes and the position of the centre was
fixed Avithin one of these meshes.

Our first procedure was to examine the daily charts one b}' one
and for each position of tlie centre to note doAvn the districts Avith
or Avithout precipitation. Here Ave met of course Avitli a difficulty.
In man}^ cases more than one centre appeared in the chart, to sa}'
nothing of the supposed shalloAV depressions appearing side by side,
especially in days Avitli generally small barometric gradient all over
the chart area. In .such a case, Ave are liable to an arbitrary pre-
judice in discriminating Avhether tlie precipitation in a given dis-
trict Avas due to the one or the other of the coexisting depressions.
Again, AA^hen an extensive depression is bifurcated, as is often the
case Avhen it is crossing over the main island, Ave cannot fix the

On the Distribution of Cyclonic Precipitation in Japan. 5

position of the ''effective" centre, M'ithout making more or less
uncertain assumptions. Since it Avas our immediate purpose to
investigate the distribution for an isolated simple depression and
discuss the results from the theoretical standpoint, all these ambi-
guous cases were excluded, confining our attention only to the
simplest cases where only one conspicuous depression is shown in
the chart. ^ It must be admitted that in adopting this selection,
we are taking only those precipitations in consideration which
correspond to a quite limited weather type, and hence that the
whole subsequent discussion has no reference to the cases of preci-
pitations of noncyclonic type. Again, since each district includes
a numljer of stations, it occurs as a rule that for each position of the
centre, many districts have only partial rain or snow. Those cases
with precipitation in only one station, Ijut with none in the others,
were counted as " no precipitation,"' otherwise as " precipitation "
for that district. It must be remarked that the hours to which the
weather charts refer are limited to 6a, :2p, lOp, while the distribu-
tion of precipitation in other hours may often A'ary widel}^ lUit
f()r the present investigation, the three observations in a day seem
to be more than sufficient. Finally it must be i-emarked that the
cases in which there exists a centre of depression in the chart. l)Ut
with no precipitation in either station, were excluded. Such a c;ise,
wliich is only met with when the centre is near the margin of the
chart, is rather rare and the corresponding position of the centre
more or less doubtful. At any rate, the weight of these extreme
positions for the result is small and ]nust be taken into considera-
tion, if at all, with precaution.

The statistical part of the investigations was chiefly carried
out by Yokota and Otuki. The number of times n in which the
''precipitation" occurred in a given district, say Pi, corresponding
to a given position of the depression, say A, divided 1)}' the num-
Ijer of times N in which the centre was found in the area A, ex-
pressed in percent, was called the " expectation " of precipitation
for tlie pair (Pi, A). Tlie result of the statistical part is shown in
Table I. The first column gives the positions of the centre of

* In such cases, the ijosition of the centre could he determined with a fair degree of
certainty.

T. Terada ;

depression ; all those positions of which the total number JV was less
than 5 were excluded. The second column gives the number JV.
The remaining columns give the percentage expectation 7ilN%,
for each district named at the head.

Table I.

Position of Centre.

N Pi

1

40

Ps,
0

30

P6

0

0

Ml

60

0

Ms
20

M4
20

0

Me

Ji Ji'

Ja
10

0

J.^ ^ Jg

i

22^5- 25°
122.5-125

E

10 70

0

40 20

0 0

125 -127.5

E

7 67

71

30

33

0

0

33

40

14

14

0

0

50 33

0

0

0 0

127.5-130

E

9 33

44

33

22

0

0

33

29

0

0

0

0

33 11

0

0

0 0

25 - 27.5
125 -127.5

X

E

18 0

39

0

28

0

0

39

28

22

11

17

0

56 33

28

11

6 0

127.5-130

X

E

18

56

44

28

1

45

0

0

17

7

28

22

11

0

44 17

1

11

22

6 0

130 -132.5

X

E

9

56

78

1
28

33

0

0

44

17

45

22

0

11

22 55

55

33

11 0

27.5- 30
125 -127.5

X

E

13

0

23

15

15

0

0

38

15

8

8

4

«

46 23

15

15

0 0

127.5-130

N

E

28

62

43

14

11

0

0

43

36

14

4

11

0

57 36

14

11

0 0

130 -132.5

E

19^ 64

47

47

47

5

0

63

52

37

42

8

0

53 47

41

26

0 0

132.5-135

X
E

12 58

50

25

42

8

0

42

33

17

25

0

0

33 33

i

8

14

0 0

30 - 32.5
122.5-125

X

E

5

0

0

0

0

0

0

0

0

0

0

0

0

80 0

0

0

0 0

125 -127.5

E

19

60

32

5

^

0

0

42

26

0

0

0

0

74 26

10

5

0 0

127.5-130

X

E

24' 37

75

21

17

4

0

67

50

17

8

0

0

71 54

21

16

7 4

130 -132.5

X

E

30 71

76

54

27

3

0

73

70

32

19

0

0

78 65

1

38

24

3 0

132.5-135

X

E

37 76

87

73

43

16

5

62

81

68

32

14

3

49 70

1

73

41

14 0

135 -137.5

E

37 22

48

87

84

24

8

24

49

65

62

27

11

16 22

49

49

11 3

137.5-140

X

E

22 11

18

59

77

37

0

13

23

55

59

32

14

23 23

j

32

32

9 0

140 -142.5

X

E

12 5

0

25

84

25

0

0

18

42

50

33

17

8 42

42

42

25 8

1

32.5- 35
122.5-125

X

E

: * ''^

50

38

0

0

0

38

0

0

0

0

0

75 25

0

0

0' 0

125 -127.5

X

E

12j 66

58

25

8

0

0

67

58

9

0

0

0

92 75

17

8

0 0

127.5-130

X

E

24

i

1
50

67

35

8

0

0

62

54

38

13

4

0

83 71

50

21

0 0

On the Distribution of Cyclonic Precipitation in Japan.

Position of Centre

A'

P] Pi

P:

P4

P.5

Pf

M,

_

M,

Mo

M,

M,

M,

Ji

J.

Js

J.i

J,5 Jg

32.5'-35'
130 -132.5

N

E

21

62 76

48

43

5

0

41

59

57

38

14

0

67

67

62

43

0

0

132.5-135

X

E

24

13 72

63

58

38

0

54

79

71

58

29

13

58

75

75

54

21

4

135 -137.5

N
E

24

8 50

79

79

46

8

25

58

83

67

33

12

25

59

75

67

42

la

137.5-l'40

X

E

28

4 25

47

90

39

7

71

25

50

68

25

14

14

14

57

39

11 36^

140 -142.5

X
E

27

4 0

15

74

56

11

0

7

37

44 29

7

7

19

33

51

45 7

35 - 37.5

125 -127 5

X

E

16

31 50

13

6

6

0

50

50

19

12 6

0

62

31

25

19

13 O'

127.5-130

X
E

7

35 57

43

43

14

0

72

43

43

19

14

0 86

86

43

57

0 0

130 -132.5

X

E

20

35 75

50

40

15

5

75

75

50

35

10

5

66

55

60

65

15

132 5-135

X

E

10

10 40

40

80

50

0

40

60

90

80

30

10

50

80

70

90

30

0

135 -137.5

X

E

19

0 0

32

58

52

21

5

10

53

63

53

47

21

26

48

47

26

16

137.5-140

X

E

12

0 0

25

67

58

8

0

9! 42

67

58

25 0

8

58

75

42

O'

140 -142.5

X

E

20

0 0

5

45

80

35

5

0

20

40

55

40

0

10

26

55

50

20

142.5-l'45

X

E

14

0 7

4

14

50

22

0

0

21

29

29

36

14

14

36

36

29 14

37.5- 40
130 -132.5

X

E

9

11 11

0

0

11

0

22

0

0

0

0

11

44

22

11

11

0 0'

132.5-135

X

E

12

42 67

50

42

42

8

50

92

58

33

50

33

33

33

33

25

33 33

135 -137.5

I

22

14 41

27

39

45

9

32

41

50

32

22

36

23

36

69

68

50

14

137.5-140

X

E

20

10 5

5

30

75

40

1

10

2

40

45

55

10

15

25

45

50

25

140 -142.5

X

E

9

0 0

11

11

33

44

0

0

11

22

44

78

0

11

22

33

55

56

142 5-145

X
E

17

6 0

6

0

41

70

6

0

6

0

29

59

0

12

29

29

41

47

40 - 42.5
130 -132.5

X

E

6

0 60

33

0

17

0

16

17

0

0

0

0

17

17

50

17

0

33

132.5-135

X

E

14

7 2

29

29

36

25

14

14

36

21

29

29

14

33

29

50

29

43

135 -137.5

X

E

12

17 16

50

33

8

25

8

33

67

42

8

42

33

20

42

50

42 25

137.5-140

X

E

8

0 13

13

25

38

63

0

0

13

0

0

25

12

0

0

25

50

50

140 -142.5

X

E

12

0 8

8

8

17

50

0

0

27

8

8

8

0

17

25

50

42 67

142.5-145

X

E

1(>

0 6

6

12

12

56

J

0

6

19

12

6

38

0

13

19

31

50

63

T. Terada;

Position of Centre

.V

Pi

P-

Ph

P4

P5

Pfi

Ml

M.j

M,

M,

M5

Me

Ji

J...

J.

J4

J5 Je

40° - 42.5° N
145 -147.5 E

10

10

0

0

0

10

60

10

10

20

0

10

50

0

30

30

10

40 50

42.5- 45 N
135 -137.5 E

7

0

14

0

17

0

lu

33

0

K)

0

0

0

33

0

33

17

50 50

137.5-140 E

12

0

0

0

8

0

82

0

0

0

0

8

33

0

0

0

17

25 Ü7

N
140 -142.5 E

15

7

7

7

7

7

40

0

7

7

0

7

47

0

0

7

20

0 73

N ■
li2.5-145 E

9

0

0

0

0

0

5ij

11

0

0

11

11

22

0

0

22

33

0 55

1

N

145 -147.5 E

7

0

0

0

0

0

42

0

0

14

0

14

0

0

0

0

28

14 43

45 - 47.5 X
145 -147.5 E

7

0

0

0

0

29

14

0

0

14

0

29

57

0

0

57

43

29100

147.5-150 E

7

0

0

0

0

0

42

0

0

14

0

14

100

0

14

43

28

29 8G

47.5- 50 N
145 -147.5 E

5

0

0

0

0

GO

20

0

0

40

0

20

40

20

0

40

CO

20 GO

N

147.5-150 E

10

0

0

0

0

20

0

0

0

0

10

50

70

10

10

30

40

GO 40

From this table, we may easily trace the cun'es of equal ex-
pectation corresponding to each given position of tlie centre, or the
locus of the positions of the centre bringing ecpial expectation for
each given district. For the sake of simplicity call tlie former
curves " the isohyets " for the given position of the centre, though
here, instead of the amount of the precipitation, the expectation is
meant ; the latter sets of curves may be called " the centre loci "
for the given district. To smootli down local irregularities, the
average was taken uf the expectations of four adjoining 2. 5° meshes
and the mean value was attril^uted to the centre of the four meshes.
Those sets of four meshes of which any one had the numl.)er of oc-
currences of depression, i.e. iYless than 5, were excluded. In this
way, the uncertainty due to the defect of data in tlie marginal re-
gions was avoided. From these averaged expectations we con-
structed the two sets of diagi'ams mentioned above.

Figs. 1 to 20 show the isohyets for different positions of the
depression wliicli is marked with ¥: in each diagram. Figs. 21 to
08 gives the centre loci for each district, the middle point of which
is marked with •.

Oa the Distribution of Cyclonic Precipitation in Japan.

Fig. 1.

Fig. 2.

125 KIO 135 140 U5

Fig. 3.

Fig. 4.

10

T. Terada :

125"

I30"

135"

140'

145'

■>' — /-— -J— _JI__

_IIU\_1 — \ —

/nr^

ilwA-

lirî^2

J

A

-tS-^

~~^ L— L^

^■

A

\M^l

\ \

^mXl

AA \

Ivj

-H

^

ML.\S^

Mj'

^nf\

W

IX

j [ e^W~

L- J — V=

j ^^ ■

14

^~Y^

■■

\ . 1

1

Fig. 5.

Fig. G.

r25° 130' 1:15 1+0" 145

130 135

Fig. 7.

Fig- S.

On the Distribution of Cyclonic Precipitation in Japan.

11

Fig.

Fig. 10.

Fig. 11.

Fig. 12.

12

T. Terai.la :

Fig. 13.

Fig. 14.

Fig. 15.

Fig. IG.

On the Distribution of Cyclonic Precipitation in Japan.

13

Flrj. 17.

Fig. 18.

Fig. 19.

Fig. 20.

14

T. Terada :

Fig. Qi, Pj.

Fig. 22. P.

Fig. 2S. p.,.

Fig. 2é. P,.

On the Distribution of Cyclonic Precipitation in Japan.

15

125* lOO' 135' 140* 145'

Fig. 25. P,.

Fig. 26. P,

125' 130' 135' 140] 145'

—4 L j 1

._1UU4— H=

ijinz

Jul^Vv-

hXt^

/

H-K^

~~~4- LL^

^

/

^^^^^^^

^

\

kîÎX^

N

3MA-V

1 f ri

' — ^

^01^44^

i^^

^

ïQ-4--'r

^IXiW^

— so- 1

LhK

"v

1 '^'

k

No

1 1

f 1 ^ — r

lao 1.15

Fig. 27. M,.

Fig. 28. ]\L,

T, Qerada :

Fig. 20. Mo.

Fig. SO. il/,,

Fig. SI. ill,.

Fig. 32. il/,.

On the Distribution of Cyclonic Precipitation in Japan.

17

Fig. 33. J y

Fig. 34. J.,.

Fig. 35. Jy

18

T. Teraclfi :

Fig. 36. j;.

Fig. 37. J,.

Fig. 3S. J,

On the Distribution of Cyclonic Precipitation in Japan. 19

Discussion of the Results,
1.) Isohyels.

A passing glance at the isohyets will reveal tlie undeniable in-
fluence of the land and water. Referring to Figs. ], 2, 5, 14, 17,
19 and 20, we see that in front of a distant cyclone, the Pacific
coast has decidedly greater expectation than the Japan Sea coast,
whereas on the rear side the reverse is the case. This fact has al-
ready been noticed and explained by one of us in the above cited
paper. This relation is not so conspicuous as in the above exam-
ple when the cyclone approaches tlie main island, though the same
tendency is still suggested by the shape of some isohyets, as in Figs.
3, 4, 6, 7, 8, 1], 13, 16 and 18, especially in the regions remote
from the centre of depression. In the immediate neighbourhood
of the centre, the difference between the Pacific and the Japan Sea
coast is not conspicuous, as may be seen from Figs. 3, 4, 6, 7 etc.
It is quite evident that the above cited theory requires a modifica-
tion in the inner region of a cyclone where the curvature of the
isobars and the variation of the gradient could no more be neg-
lected, and the ascending air current proper to this region is more
conspicuous, so that the effect of the topography does not appear
so pronounced as in the external region.

Comparing Figs. 5, 12, and 15, it may be seen that the dis-
trict Mo and probaljly also J3 shows some local irregularities, which
may probably be attributed to the peculiar configuration of land
and water in this region. Next, referring to Figs. 15, 16 and 18,
a tendency is suggested that wlien the centre lies in tlie Japan Sea,
M districts often show smaller expectations than the neighbouring
P or J districts. This peculiarity is most pronounced in the iso-
hyets for the position of the centre, 132.5° E, 42.5° N which were
omitted in the above diagrams, since the number of occurence JV
in two of the four meshes surrounding this point were less than 5.
If we nevertheless calculate the corresponding expectations, we
obtain

Ml M, M3 M, M, Mo Pi P. Pa P4 P5 P6 Ji J2 Ja 'h J5 Jg

8 8 9 5 7 7 2 16 16 7 13 6 8 13 32 29 20 32

20 T. Terada :

Thus the coiTesponcling isohyets show a ' 'valley' ' along the central
axis of the main island. If this tendency is any thing real, the ex-
planation must be sought in the draining influence of the Pacific
mountain range lying on the wind side of this district, enhancing
the ascending air current and condensing abundant moisture on the
Pacific side. Moreover, it is interesting to notice that this parti-
cular position of tlie centre lies nearly at the centre of curvature of
the circular axial line of Honsiu, and hence the geometrical relation
of the different parts of the land with respect to the centre are
similar to each other. This may at least explain why the isohyets
in this case run nearly parallel to the land. On the contrary, when
the centre of depression lies on the Pacific as in Fig. 8, not only
the distance of, but the angle made by the coast line, with the
radius vector drawn from the centre varies widely for different re-
gions. This explains why the expectation in this case varies rapidly
along the coast line. Nevertheless some peculiarities of the M
districts similar to the above are suggested by the isohyets corres-
ponding to some more remote positions of the centre than those
shown in the above figures. These cases were, however, omitted
on account of their small weight, and may better be postponed for
a future research.

Again, comparing the Pacific and the Japan Sea coasts, for
examples Figs. 7, 8 with 10, 13, or 4 with IG, the expectation
seems to be generally greater for the Pacific districts than for the
Japan Sea side, when the centre lies on the sea not far from the
district in question. This may probably be attributed to the differ-
ence of temperature of the extended water bodies over which the
wind comes.

Thus far we have considered only the average distribution of
precipitation for different positions of the centre of depression. For
the actual cases, the influences of an accidental nature, i.e. of the
trivial local irregularities in topography, meteorogical conditions,
etc., may give rise to various discrepancies compared with the
average relations. Still it is not difficult to find a number of tj^pical
examples among the daily charts, which may well illustrate the
above general inference. A few of these examples are shown in
Figs 39 to 42.

On the Distribution of Cyclonic Precipitation in Japan.

21

'ig. 39. W 2^.vi., July 30, 1913 ;
compare with, Figs. 11 and 14.

Fig. 40. G" a.m., April 23, 1913 ;
compare with Fig. 15.

I2.V I -'10

130 13.

Fig. 41. r? a.m., Octoher 19, 1913 ; Fig. 42. 6" a.m., Novemver 23, 1913 ;
compare jvith Figs. 13, 16 and 18. compare loith Figs. 17 and 19.

22 ^^^ ^^^*' Distribution of Cyclonic Precipitation in Japan.

2.) Centre Loci.

As already explained, Figs. 21 to 38 sh(3w for eacli of the 18
districts, tlie percentage expectation brought about Ijy all possible
different positions of the barometric depression. For example, in
Fig. 21 the curve marked with 50 shows the trace of the positions
of the centre which may bring precipitation t(j the district Pj in 50
cases out of 100 on an average.

From these figures, it will be at once seen that the area with
the greatest expectation lies mostly on W, SW or S side of the dis-
trict in question. This implies that the expectation is generally
greatest on E, NE or N side of a cyclone. On a closer examina-
tion, however, we may easily discern the characteristic difference
between the Pacific and the Japan Sea coasts. For the P districts,
the bO% curves generally pass along the immediate neighbourhoods
of the middle point of the districts concerned and the areas with
the greater expectations lie entirely on the W to S sides of the dis-
tricts. For the J group, however, the districts in question lie
decidedly apart from the 50% lines and nearer to the point with
the maximum expectation ; moreover, the extent of the belt with
10 to 50% expectations in front of the district, is decidedly greater
than that in the case of the P districts. Besides, J-., J4 and J5 show
a belt with the lesser expectation projecting far in advance of the
district, while J3, J4, J,, also suggest another maximum near Sagha-
lien. For J« the district in question apparently lies on the rear side
of the area with the maximum expectation. For the M districts,
the relations are intermediate between P and J as may be
expected.

One interesting relation revealed b}^ this way of graphical re-
presentation may be worth a special description. Referring to
Figs. 22 to 26, suppose a line drawn along the longer axis of the
elongated area with expectations above 30 or 50 %, for each of P,,
P2, P3 Pe- The lines seem to turn round clockwise as we pro-
ceed successively from Pj to P«. The same tendency is more ap-
parent for Ms to Me. The interpretation of this peculiar relation may
probably be sought in the influence of the Japan Sea depressions.

T. Terada 23

We have already shown that when a centre hes far in the Japan
Sea, the entire Pacific coast stands under nearly equal conditions,
as far as the effect of the position of the coast line relative to the
centre is concerned. Hence the influence of these depressions re-
mains persistent when we proceed along the different districts.
For the Pacific depressions, the case is quite different ; the isohyets
cross the land more or less transversally, the expectation var^dng
rapidly along the coast line. In other words, the part of the centre
loci on the Pacific side moves with the centre, while on the Japan
Sea side, it remains comparatively stationary.

General Theoretical Considerations.

Though we are afraid that we may he drawing our inference
rather too far on the basis of too scanty materials, it will not be
quite out of place to attempt here a discussion on the general theo-
retical aspect of tlie problem at hand.

Among the numerous factors determining the unsymmetrical
distribution of precipitation due to a cyclone, we may conveniently
distinguish the following three as the most essential :

(1) The first maj^ be called '' thermal and planetary " for the
sake of simplicity. It consists in the difference of temperature witli
the latitude and may be considered always present regardless of the
distribution of land and water. This influence would predominate
if the earth were completely covered with ocean or land only, and
would bring, according to the usual simple theory, more abundant
rain on the eastern side of a depression.

('2) The second may convenient!}^ be called "thermal and
geographical" and consists in tiie difference of tliermal condi-
tions governed by the distribution of land and water, especially of
continents and oceans. If this influence predominates, we may
expect in the northern hemisphere the following : In summer
when the land is generally warmer than tlie water, the area with
the heaviest precipitation will lie in that direction whicli, viewed
from tlie centre, has the land on tlie right side, provided of course
that the land is sufficiently humid and the air kept nearly in
saturation. But if the land be very arid, the reverse may occur, if

24

On the Distribution of Cyclonic Precipitation in Japan.

Wa\er

we can expect any precipitation at all. In winter, the relation will
be different ; we may generally expect heavier rain or snow on that
side of the depression which viewed from tlie centre lias the ocean
on the right side.

(3) While the above two factor are essentially thermal or
thermodynamical, there remains the third one to be considered,
wdiich may conveniently be called ' ' hydrodynamical and topo-
graphical." This consists in the effect of the forced ascending air
current brought al)Out by the discontinuity of the horizontal flow of
air across the coast, due to the difference of " friction "'^ over land
and w^ater, or flat land and mountains. This effect has been dis-
cussed in the previously cited paper and may be summarized as

follows : In the annexed figure, AB is
the coast line bordering the land on its
left side. PQ makes with AB a certain
angle depending on the coefficient of
friction on land and water. The ascend-
ing air current is induced w^hen the
gradient of the l)arometric pressure is
directed toward the B side of PQ, while
the descending current occurs wdien the
gradient points to the A side. The ab-
solute intensity of the current is maxi-
mum when the gradient is perpendicular
to PQ, while it is zero Avhen the gradient
coincides with OP or OQ. This influence will appear most con-
spicuous wdiere the land is in the shape of a narrow strip having
a large extent of water bodies on both sides, provided the temper-
ature and humidity on the two sides are not very different. Along
the coast of a continent, how^ever, the thermal and geographical
influence mentioned under the previous paragraph will generally
combine with the hydrodynamical influence, so that the relation
may vary widely according to season or the physical conditions
of the continent.

Fio-. 43.

In the sense of Guldberg and Mohn's theory.

T. Terada:

25

In actual cases, these three influences are generally super-
posed, the resultant effect varying largely according to the degree
of relative importance of each factor. For example, in the case of
a deep depression of small extent, the first factor plays no greater
part than slightly shifting the area with tlie heaviest precipitation
toward E, while the second and the third factors may be important
not only for determining the precipitation, l)ut also the subsequent
course of progression. It will Ije especiall}^ interesting to in-
vestigate the relation with respect to the cases of tliunderstorms
of the cyclonic type, i.e. those with circular depression in the
centres.

The mathematical discussion of these different influences will
not l^e easy, till we have at hand a more or less complete theory of
cyclones in general. In the following, however, an attempt is made
to illustrate the essential influences of the above mentioned factors
for very simple ideal cases, and to found the starting point for analy-
sing the actual complicated phenomena into their essential ele-
ments. It must be emphasized that tlie whole is nothing more
than qualitati^'e considerations expressed in mathematical forms.

(1) Planetary thermal influences : Referring to the annexed
figure consider a centre of cyclone at 0 and draw two concentric

isobars with the radii r and r + dr, in the
inner region, i.e. the region where the
ascending current proper to the cyclone
is taking place. Suppose now all the
isotherms are artificially maintained
parallel to the .r-axis. Assume tlie angle
of deflection of the wind </', i.e. the angle
between the barometric gradient and the
wind, everywhere constant — an assump-
tion apparently contradicting the idea of
varying latitude, but since here we are
only considering the effect of the tlierinal gradient, the small
difference of </' is evidently not essential. Then we may put

Fig. 44.

2Q On the Distribution of Cyclonic Precipitation in Japan.

where ^,„ is the mean kititude and ;/ the coefficient of friction in
Guldberg and Mohn' s sense. The flux across the circle with the
radius ;■ is ^7:rücos,</> where v is the resultant velocity of wind at P,

given by

_ Gcos</>

in which G is tlie gradient aecelaration :

dr

where B is tlie barometric pressure and K a constant,''^ The excess

of the llux across the circle with the radius r-^-dr is

o / d(rv) 7 o cos-ç^/^ , dG \ ,

'2-cosc^— ^ — ^ dr=-^~ ^(G-^r-^ — )dr.

dr n dr

Hence the total amount of the ascending current per unit length of
the circular belt with the breadth fZ/-is

cos'YVr' I dG^n
}•>( dr'

If the air is kept artihcially always in saturation corresponding to
the momentary temperature as assigned hj the given temperature
distribution, the condensation of water vapor due to the ascending
current will be proportional to the mean absolute humidity or
nearly to the mean maximum vapor tension E between P and Q.
The latter is well known as the function of temperature

E=ÂT) say.
Hence the condensation per unit liorizontal area at P will be pro-
portional to

B = I^±l{G + r^'-^) cosV- (1)

;//■ dr

;//• dr
It lor example, G = co)ist. = G^, A = ^^^^— — -^ , (2;

}cr

or it G = br, B = ^^^-^ -> (3)

in which f{T) may always be regarded as the function of the space
coordinates of P, since T is given as such. Tliis holds indeed

* Though of course, in actual cases, K involves the temperature of the air column,
the above assumption may be allowed for the present purpose.

T. Torada : 27

for any (listribiiti(jii of tciuperaturo, provided it is maintained
stationary by any cause. In tiie present particular case, the curve
of equal B will be parallel to the isotherms and be straight, if
G=br, and R will increase toward the direction of the increasing
T. When G=const. and for example T=T,—Gy, the curves will
be given by f[T„ — Cij)/\/x- + y'-=const. in Cartesian coordinates,
which are generally concave toward the direction of increasing T.
This will liold within the limit of the inner region and the effect
will in any case tend to shift the centre of precipitation area
toward the direction of the increasing temperature.

The quantity B is, however, not the only one in determining
the precipitation. When the air proceeds from Q to F, the
temperature must vary, due to the assumjittion that the isotherms
are stationary. It will be easily seen from the figure that the
temperature decrease is given by

COS^

when T=l\ — Cy. Hence the unit volume of air, in proceeding
unit distance along /•, condenses out an amount of water given by

clE clT _ df(T) c_sin(t^=E' - u)

(IT dr dT cosç-'' ■ ^^

Tbis will 1)0 zero for d = <J', positive for d = - + </' to 2- + ^^-, and
negative (i.e. evaporation instead of condensation) for ^ = ^ to

Tbe total ])i'ecipitation will then be proportional to B + Ii' in
which tlie effect of B' is in ;my case to shift the centre of the

heaviest })recipitation toward the (hi'ection 6 — — + </'■ In the

above, it has been assumed tliat Ji at any place is merely due to
the condensation caused by the vertical current at tliat very spot.
In the actual case, however, the ascending air is transported
leewards by the horizontal current to an extent depending on the
ratio of the horizontal vel()City to the vertical an<l also t(_) tlie
height of the cloud layer. 'J'bis effect will result in deforming and
twisting the isohyets as a whole m counterclockwise sense.

28

On the Distrilmtioa of Cyclonic Precipitation in Japan.

Moreover, even if the isotherms were originally parallel to the
latitude, they will be distorted counterclockwise as is well known,
since the artificial assumption does not hold that the earth's
surface as a sensitive heat reservoir keeps the air always in the
temperature prescribed for the latitude. These efïects taken
altogether, will tend to drive the area of the maximum precipita-
tion toward the eastern side of the depression.

(2) Geographical thermal influences. When the coast line is
straight and the air is everywhere saturated, the reasonings of the
preceding paragraph exactly apply to this case, only that the
temperature gradient is here everywhere small, except in vicinity
of tlie coast line. The part of the precipitation due to the term
R of the preceding paragraph will be abundant on the warmer
half-plane. The influence of the term R' will be only sensible in
a narrow zone along the coast line if the variation of temperature
across the coast be abrupt or discontinuous. If in the land area
the air is not saturated, the case may differ, according as the
absolute humidity is greater or less than on the sea. In the
special case when in summer the absolute humidity on a very arid
land is small compared with that on tlie colder sea, the result will
1)0 reverse to that in tlie above case, so far as we take no account

of the horizontal transport. The as-
cending current must, however, in
either case be carried leeward as al-
ready remarked. Hence we can expect
precipitation on the coast of a dry land
where the wind comes from the sea.
In the normal case, when the air is
everywhere saturated, the isohyets will
simply be twisted clockwise, so as to
increase the precipitation on the coast
lying on that side of the centre, which
viewed from the centre have the warm-
er half plane on the right side.

(3) Hydrodynamical topographical influence. Suppose AB
is a portion of the coast line bordering the sea on its right

'""<?

T. Terada : 29

side. OG shows the direction of the barometric gradient which
makes an angle 6' wâth OA. Here the essential difference of
land and sea is assumed to consist in the difference of the
coefficients of friction, n and n' respectively; the other thermal
behaviours are supposed to be equal on both sides. It has been
shown in the above cited paper that the intensity of the ascend-
ing current due to the discontinuity of the horizontal flux across
AB is given by

o = D cos(^' + £)

where D cos £=-|(cos5^-cos5^0. tg^=7'

D sill £ = |(sinÀ?^ — sin^^^''), tgç^' = ~^-

If il'=Q,V, ^' = 42', then D= -0.3255, ^--Vè\

Since D is usually negative, the ascending current at 0 will be

maximum when 6'=7T-e, and zero when ^'=— ( ^ + ^) ^^ 2"""^'
For ö'= -('.^ + e) to -^)-^» the vertical current will be directed

downwards; hence the expectation of precipitation at 0 will be
none for these directions if there is no general ascending current
proper to the depression. In the case wlien O lies within the
inner region of a cyclone, the ascending current at this point
proper to the cyclone will be enhanced when the centre lies on the
B side of PQ, but weakened wdien it hes on tlie opposite side. If
there exists no such hydrodynamical influence, the trace of the
cyclonic centre bringing equal expectation to O, will evidently be
a circle with O as the centre. In the special case, when the
gradient is proportional to the distance from the centre, or G=br,
R will be independent of r, as may be seen from (3) since here the
temperature is assumed constant. In general cases, however
R^R' will be a function of r and 6' as may be seen from (2).
Denote this by

Taking the hydrodynamical influence in account, the centre loci
will be given by

F{rd') + Dco^id' + 0 = const

30 On the Distribution of Cyclonic Precipitation in Japan.

where r and 6' are the polar coordinates, with O as tlie origin and
OA as the prime vector. It wih be seen, that whatever may be
the form of F^rd'), the effect of the second term will be to shift the
centre of area of the loci loci toward the direction d = - - e.

From the above, it will be seen that the first and the second
factors are essentially not very different, and there would have been
no special need at all to distinguish them, if Ave are everywhere
concerned with humid land only. In the latter case, the essential
factor is only tlie distribution of the isotherms, and the results of
the al)Ove considerations in the paragrai)hs under (1) and (2), may
be summarized as follows : The maximum precipitation will be
expected on the right front quadrant when we look out from the
centre of depression toward a direction having the warmer half plane
on the right side, provided the isotherms are maintained nearly
stationary regardless of the cyclone. The latter condition will be
nearly fulfilled when the general temperature distril)ution is de-
termined by an extensive " centre of action."

When we consider the three factors as separate, it will be seen
that the second and third generally conspire in winter, but tend to
cancel each other in summer, provided the land and water are of
sufficient extent. For a narrow strip of land such as Japan, the
second factor will play no important part when a cyclone of con-
siderable dimensions is concerned, since in such a case the assump-
tion does not apply that the land behaves as a reservoir of heat de-
termining the stationary isotherms, regardless of the disturbance
due to the cyclone itself. Meanwhile the third influence remains,
since it will be independent of the latter disturbance.

Next the first and the third influences conspire when a coast
line running more or less in west-easterly direction, face to the
water on its southern side, but tend to cancel each other when the
water is on the northern side. The resultant effect is well illu-
strated by comparing the centre loci diagrams for P and J districts.
That the first influence predominates in these cases, is shown by
the fact that the district in question lies generally on the eastern
side of the point of maximum expectation, though yet the eccentri-
city is more decided in the case of the Pacific coast than in the

T. Terada : 31

Japan Sea districts. The latter fact still shows the unerring effect
of the third factor.

Hann remarks that on the northern side of the Alps, the rain
fall is most ahundant on the rear side of a harometric minimum.
This fact may prohahly be explained l>y the predominating influ-
ence of the hydrodynamical factor, since the effect of a great moun-
tain range is equivalent to the increase of the coefficient of friction.
That northern Germany, including Swinemiinde and Breslau, has
maximum rain fall on the rear side of a depression, may be under-
stood by the combined effect of the second and the third factors.
The case of Great Britain investigated l)y Mill, seems at first sight
iiTeconcilable with the above considerations, since the heaviest
rainfall area occurs on the left side of the track of the centre, con-
tradicting in most cases the influence of the third factor. But it
must be remembered that Mill's results refer to the "smear" in
which the rainfall on the front and rear sides of the centre are
supesposed. He states, indeed, that the heaviest rainfall occurs in
advance of the centre. The influence of the third factor combined
with that of the first one may in some cases shift the centre of the
smear to the left side of the track ; the front rain falling on the
eastern side of laud or mountain is abundant on the northern side
of the track, due to the third influence, wliilc the rear rain is
abundant on the southern side of the track, on the western side of
the land, or any orographic irregularity ; hence if due to the first
factor, the rear rain is less abundant than the front rain, the centre
of the smear will lie on the northern side of the track. This seems
to explain most cases given by ^till where the depressions proceed
toward E. In order to explain different cases in which the relation
is apparently not so simple as considered above, exact knowledge is
of course necessary of the thermal and topographical conditions of
the district concerned for each particular case.

Finall}", it must be remarked, to avoid misunderstanding, that
the present discussions involve no essential novelty as a theory of
cyclonic rainfall, except emphasizing the importance of the hydro-
dynamical influence due to the difference of the coefficient of fric-
tion. The alcove may only be regarded as suggesting a way toward

32 On the Distribution of Cyclonic Precipitation in Japan.

the better uiiclerstanding of the coniphcated rain problem. The
discussions refer to different ideal cases, underlying several artificial
simplifying conditions. Above all, the assumption is made that
the thermal conditions are prescribed independent of the cyclone,
and the secondary disturbances both thermal and topographical due
to the cyclone itself, are entirely put out of account. The latter
disturbances may in some cases modify the resultant effect of the
other primary factors in no small degree. Still, we are inclined to
believe that the above way of analysing the complicated influences
in these principal factors may in many cases facilitate a better
understanding of the phenomena of cyclonic rainfall, and if pro-
perly understood and applied may be utilized for the purpose of
Aveather prediction.

In conclusion, we wish to express our best tlianks to Prof. T.
Okada of the Central Meteorological (observatory for many valu-
able informations.

PubUshed January 27th, 1916.

JOURNAL OF THE COLLECrE OF SCIEN'CE, TOXYO IMl'EF;[A.r, UX^VERSU'Y.

VOL. XXXVir. ARTICLE 5.

On the Relatively Abelian Corpora w ith respect to the
Corpus defined by a Primitive Cube Root of Unity.

By

Tanzo Takenouchi, Ui'jah-n.'i/ii,
Professor of Matliematics in the Eighth High School.

It wa^^ conjectured by Kronecker that all the relatively
Abelian corpora {KOiyer) with respect to an imaginary quadratic
corpus are probably exhausted by those which arise from the
ecjuations of transformation of elliptic functions with singular
moduli. Prof. T. Takagi^-* investigated this problem in the
remarkably interesting special case, in which the fundamental
corpus is defined by the imaginary unit i, and proved that the
relatively Abelian corpora with respect to k{_i) are completely
exhausted b}^ tlie division-corpora {TeilungshJriJer) of the function
sn with the singular modulus j<=l. Following his example, I am
going to treat of another interesting special case in whicli the
fundamental corpus is h{l'), (' denoting a primitive cube root of
unity.

The present paper consists of two parts. In the first part it
will be proved that the relatively Abelian corpora with respect to
h{l') are completely exhausted by the division-corpora of the func-
tion sn Avitli the singular modulus ;£ = /,".

Now, if CO be a quadratic number whose imaginary part is
positive, and m a natural number, then the invariant j(mco) is
called a class-invariant. By adjoining the class-invariant jimoi)
and a primitive ??^th root of unity to the quadratic corpus Ic(oj), we
obtain a corpus, to whicli I shall (§. 10) give the name strahl-
coiyus. It is known that the strahl-QOY\\\\ii is relatively Abelian

1) Takagi : Journal of the College of Science, Tokyo Imperial University, Vol. XIX, Art.
5 : Proceedings of the Tckyo Mathematico-Physical Society, 2ncl Ser., Vol. VII, Xo. 21.

Art. 5.— T. Takenouchi :

with respect to k{oj). But the relatively Ahelian corpora witli
respect to 1c{(o) are not complete^ exhausted bj^ s^ra/J-corpora.

The chief object of the second part of the present paper is to
make these points clear in tlie special case in which the fundament-
al quadratic corpus is /"(/>).

PART I.

§. 1-

Consider tlie function ^(li), Avhose periods lo, to' are in the ratio

a)

p, p =

l + V-3

For sucli ^-function, we have

rj, = 0,
and consequently

Tills îf-function admits of complex multiplication. Namel}^,
if we denote by /^ anj^ integer in the quadratic corpus /•(/'). then

Hf>')=p- (1)

where P and R are rational integral functions of ^^(«) of degrees
//I— I and m respectively, m being the norm of /^. Let ns suppose,
once for all, that tlie coefficient of the highest power of ^-(?/) in F is
equal to /^'. Then, putting u=0 in (1), we find that the coefficient
of the highest term in R must be equal to unity.
Let us now introduce a function c^v, !^nch that

(/',, = n, when vi = 0 or 1,

where the product is to be taken for all such incongruent residues
> with respect to the modulus /^-, that are not divisible by «. Then
we get

P = 6"-

Ou the Eelatively Abelian Corpora.

and we may put'^

if )!i, he odd,
it" ))i he even,

Avhere ÂV is a rational integral function of \^-^[/f) whose higliest term
is

fA-'itt)

or

— ctWÖ - y

according as m is odd or even. The sign of (/'a is thus completely
defined. And it can be seen that, if we expand ^v ii^ a,n ascending
power series of u, the first term is always equal to

ÎÙ

m ■ 1

Now, 1)y (1),

Since the expression on the left-hand side vanishes only when

fxu = +11, (mod. oj, (1)'),

it follows that the numerator on the riglit-hand side can differ from
(/'.u^i v^e-i only by a constant factor. Putting u=0, we find that this
constant factor is equal to —1. TJierefore

If we assume that

n = a + ho,

a and /> ]>eing rational integers, we get

s-.™)-i'» = -|^.

when (a, h) = (0, 0)

^ _n^6y-.SWi>Vi^ when (aJj) = i\,0)

Sß.

when (a- b) = (0, 1), (1, 1)

{mod, t2).

1) Cf. Weber: Lehrbuch der Alo-ebra, Vol. Ill, §. 58 and ?, ]52.

Art, 5. -T. Takenouclii :

i 2.

Let u and fi bo two integers in k(f'). It can be shewn exactly
as in the case of ordinary (not complex) ninltiplication'*, that

Eeplacing (y, (j) in this relation l:)y (//-+1. //—I). (/^ + 1. /^)> (/^- + .^'' /-«)'
(/^ + l + ,o, /7.) respectively, we get

If we express these formnlae in terms of S, we obtain the
following recursion-formnlae for S : (n=a + ho)

(i) when (a, h) = (0, 0) (mod. 2),

(> S.2rAri-> = ^^'00' '^'/'-M+z' '^'/'-l-i-/' ^l ~ '^A'+l 'S^/'-l S'/'+/) »

(ii) when (rt, /^) = (1,0) (mod. 2),

_ S,, = S,^, Sp S,U - >^. '^.-2 s,?^, ,

() S.,aj^p = 'S/.^-i+z) Sf-i-i-p Sn — 6"'(?^ )" /SV+1 S,,^i S/r^p ,

(iii) when (a, h) = (0, 1) (mod. 2),
e o OS C3C'

O.jit^l — Oft^-'jOy, — 0,14-1 »J /«_! >
/■> S'2/. J-/, = S,,j^.i^p >S';._i+/) *S',7 0^\?/)' 'S',,.;.! 0,._l O/.V^ ,

— jO-S'o,.-; 1^-/) ^ ^'(n)' Sy.A..2J.pSp+pS,,. — S,,+i o,,_io,r+i+/, ;

1) Weber : Algebra III, §. 58.

On the Relatively Abeliau Corpora. k

(iv) when (a, b) = (1, 1) (mod. 2),

The expressions of ^S; for small A-alues of a and h can easily be
found by direct calculation. They are

s, = 1, s,^, = 1 +;■>, s,.„ = a-o)p,

S, = 32;'-%^), 5n+, = (3 +;i)jr-og„ S,., = (3-/V
,S,= -^jf + lOrj.p' + rj,-,

where p stands for ^(u'). All the otlier .S' can be found by the re-
cursion-formulae.

The general form of A',., is as follows'':

'^>' = — oi^ - +c^g.p -' +

n. - G

when m = 0

(7j being even)

^y- — P-P - +c,g.p '-^ + +[

W-l »7-7

S,.= HP - +c,g.p -' +

when m = 1

(7> being odd)
— (— 1) - if-Q'A^ p. when ;;i = 3

) (mod. G),

'^." = - -tj-^J - + <^i .^3P - + + [ J j/> ^, « , when m = 4

where Ci, c,, are integers in /<;'), and ('^ J ') means Legendre-

Jacobi's symbol for quadratic character.

1) T. Takenouolii : Tolioku Mathematical Jonrual, Vol. VII, Xos. I, 2.

G Art, 5. -T. Tiikenoiiclù :

§. 3.

Let us now introduce a new function:

./ -, H«) .

■ '-"^ = W '

the ambiguity arising from the cube root of g^ Avil be remo^'ed
afterwards. If we suppose that <« is relatively prime to 3, then

where ii^(:i') and F,,(.c) denote rational integral functions of the
forms

i?/,(aO = x"' + a^x""^+ +a;X""^'-'+ + cim-ix,

P,(rr) = f_{^x—' + h,x"'-'+ +b,x'-"'-'+ + {-iy"-\o-^'+'\

with integral coefficients in Z(/0-

Hence, if we regard t(){) as an unknown quantity-, the equa-
tion

Iir--(fm)P, = 0

is of the ??zth degree, and its absolute term is associated with -(/m).
All the roots of this equation are given by

where ^ is to take all the values of the coinplete system of residues
with respect to tlie modulus /^. Hence we obtain the relation

the sym1)ol =^ meaning "is associated with."

Xow, this last relation exactly corresponds to that of tlie same
form, which presents itself in the comiilex nuiltiplication of the
function sn." Hence, starting from this relation, we can deduce the
the folloning results: ''

1) Weber: Algebra III, §. 157 i22i.

2) Weber: Algebra III, §. 157.

On the Relatively Abelian. Corpora. 7

If « be an integer in h{(') and relativel}' prime to 3, and if /^ l)e
I'olatively prime to «, then the nnmbers

\ a

are all associated with one another.

If /^ be a power of a prime, then the reciprocals of the
numbers

_/ vco

\ fi
are divisors of [J-.

If /^ consists of more than one distinct prime factor, then the
numbers

^\ //

are algebraic unities.

Next let us consider the case where i*- is not relatively prune
to .']. In this case, y- is necessarily divisible by l + 2o, for

3==-(l + 2.v-
Since

-Kl+2/'(a") — -P'— 1'
-t i_j_2^) \X) —— OX ;

we get

x'' + 37jx- — l = 0, (2)

where x = t(ï{), y = t[{\ +2f>)u].

This equation shews that x is an algebraic unity, provided that ?/ is
an algebraic integer. '

Now it can easilv be seen that

1+2/^
Hence, by the repeated application of (2), we conclude that

Vco

(i+2.y

are all algebraic unities.

], A->I,

Art. 5. — -T. Takenouclii :

By the same reasoning, since

is an algebraic unity, when /^ contains more tlian one distinct
prime, so must also Ije

.(1 + %>)>/'
Lastly, to determine the nature of

K(l+X>)v)' ^' ~ ^'

when li- is a power of a single prime, observe that the three roots
of the equation (2) are

which are so related tliat

X1X.2+ x._,x^+ x^Xi = 0.
Therefore we get

1 + 2.7/ \ l+2;>/ 7(lt)

, (0 \ . [ (O \ 1

Putting for u. in these relations a fraction in Iclf), whose denomi-
nator is a power of a single prime relatively prime to l + 2^o, we see
that

U(0

^(l + 2.>;«.
is a divisor of /-«, and its square is associated with the reciprocal of

Thus, the number

(l + 2;>)«

On the Relatively Abelian Cor^Dora. 9

being an algebraic integer, it follows as before, tbat

are algebraic unities.

We sball now sum up the results obtained in the present
section.

Let -, -', denote distinct primes in kQ'), not associated with

l + 2o, then

iT^)=''

vco

'\Jü^^^~^' ^■^■^'

vco \ _ J_

x

- = 0 (mod. x),

y-^x,

— 1, k > 1,

(1 + 2;.)V'

where v is any integer in kQ') relatively 2Jrime to the denominator in
each case.

In particular, when ii is a prime in Z'(;'), not associated with
l + 2;7, all the roots of the equation

are divisors of u., and are associated with one anotlier. Since the
coefficients in this equation, except the last one, are associated
with the elementary symmetric functions of the roots, and also
since these coefficients all belong to the corpus 1c{f>), we infer that
they are all divisible by ,«.

When a = l + 2o, we have

10

Art. 5.- T. Takonondii :

Hence, when « is any prime in h{(>), the coefficients of P,.(.t), ex-
cept the constant term, are all divisible hy /^-.

The same property holds also for /iSV; whence follows, by
Eisenstein's theorem, that ^^,. is irredücil)le in /{/').

§. 4.

The only thing we have hitherto assumed concerning the
function ^^(^/) is the relation

which gives

r/, = 0.

Here, as well as hereafter, we shall completely specialise the
function b}' the additional conditions :

C: =

f.,= ^^

1'

,-1

,^1 0) + 0)'

-1'

(3)

and consequently

(J., = àe^c^f. =

Then, let us put

.(„) = «")

V r/.i

-^4

The ambiguit}' in the former definition of t(i/) is thus removed.

Now, let 0, A, 7>, C'be the vertices of the parallelogram in
the Gaussian plane, corresponding respectively to the values

qi := 0, (O, oj-\-(o\ CO ;

and let OE, OF be ch'awn perpendicular to AB. BC respectively,
and P(r, Pi/ perpendicular te CO, 0^4 respectively. Since

1) Takcnouchi.: loc. cit.

On the Eelatiyely Abelian Corpora. W

-f-

we get

u 1 /•" d-

This shews that, in tlie parallelogram OABC, the locus of the
point u, for whicli z(;ii) is real, consists of two straight lines, on

whicli

i( ill.

or —

is always real. Taking into account tlie values (."')) and the relation

f>-(fn() =z f'-'ifu) = r(u), (4)

we arrive at the following conclusion :

z(^u) is real along the straight lines OB, AC,
pz^7() „ „ „ OC, AB. OE, BG,

frT(7() „ ., „ OA, BC, OF, BII.

The direction of OB, along whicli t{v) is real and positive, makes
witli the real axis of the (Jaussian plane an angle, which is equal
to the phase of the numl)er V/' — k i.e. the angle of 75^.

In our specialised ^-function, since ^.j = 0. we liave j[co) = 0.
Hence

which gives

;,' = i: //> or -^^ io'-.

Between our ^-function and the function sn with the singular
modulus 7c = io, til ere exists tlie relation

Hn) =^.^— r^- (5)

Bll- It 1 —o

and 2K = (0, '2iK'= oj + o/.

If we comhine (4) with (5). the followiug formulae can easily
he deduced :

12

Art. 5. — T. Takenouchi

SU pit =

sn?i
an IL '

sn<

o sna

ru = o-

cnu

en, ou =

1
dn2^ '

CD,

dn /^
en u

dnou =

cnu

dn.

1

ru = .

' dnu ' cnu'

Also, tlifferentiatiiig (5) witli respect to n, we get

^'('0 =

'2 en ;i dn //,
snhi

2

sn/t sn.v^^sn /ryt

(6)

(7)

Assume that

//. = a + bp, I

a=l (mod. 4), h = 0 (mod. 2), J

« and Z» being rational integers. Then we know that'^

su fJ.U

X = sn/t,

(8)

where £.-!„ (.r) = P/^ + J;,,!-- + + A,„_2x""^ + ^ir-'^x"-\

Da (x) = 1 + D,„..x- + -f D,x""^+ Bay/ot"'-^x"'-\

111 = ^iv/^-)' '' — ^'^ £'=1,

the coethcients yl's and JJs l)eing integers in IQ', i)'\ such tliat
^^^'"--'+1 = A, i = 3,4, •••, ;;i-2.

To find the exact value of £, o1)sei've that
1 1

sn-fiu sn-u

1) Wober: Algebra III, §. 157.

2) It follows from (9)ou the next pag-e, that the cooffieionts of .4ft{.i), i.e.,

A:

i = 3, t, , m-2.

are integers in 7.(p).

On tlie Relatively Abelian Corpora. 13

consequently

o snhi S,?

Hence, if we regard /S',„ as a function of .T = sn?^ and express it by
S'(i-l-i-,;). ^veget

-l^.,A,.{x) = sJX- /^ ). (9)

In the limit when a-=oc>, we get

lv.- = .sv(-^,!,).

The value of the right-liand side can lie calculated successively by
means of the recursion-formulae of /iV given in §, 2. (The supposi-
tion that h is even is essential in this calculation.) Tlie general
form of it is

1 V nArl-\ h

Therefore

£ = C-iyy-V^'""' = ^"-'

and the product of all tlie roots of the equation
is equal to

(-i)V'-v.

Now, it is Avell-known that, if ^wu ha any one of the roots of
this equation, tlien l)Oth cn?^ and dn?/ can l)e rationally expressed
by sn?^ in the corpus /r(/'). Therefore, it follows from (G) that the
corpns ol;)tained b}' adjoini]:ig sn?^ to Z-(/') must remain unaltered, if
we change p- into another integer associated with i'-. ( )n the other
hand, it is certain that, if there be given any integer with the odd
norm in ^{z'), then among the integers associated with it Ave can
always find one, and only one, possessing the form (8). Hence,
in treating the nature of the corpus /•(/', sn?/), wo may entirely
confine ourselves to the case Avhere /'• is of the form (8), provided
that the norm m lie odd.

]^^ Art. 5. —'J'. Takenouchi :

Hereafter we shall call prime integers of the forni (8) simply
primär J/ integers,

§• S.

Let jJ- Ije a primär}^ integer relatively prime to 3, and m its
norm, then

m = 1 (mod. G) ;

and the equation of the (w- — l)th degree

is irreduci])le in /'(/O- ^^he roots of this ecjuation are
^, = sn(r^-^], ^'^^O,!,^, ,m-2,

y «lenoting a primitive root of f^.

Now, to find the discriminant of this equation, we make use
» of the equality:

(sn a — sn ü)(sn u + sn v) [siVjt + v) — sn(?/, — r)]

= 2 sn V CD u tin u su(?^ + v) sn(« — y)

_ "Isn-it sn?? sni'2^+y)sn(?/, — v)

tinflu sn/r»,

(10)

i I

which can immediately he verified 1)y the addition-theorem of the
function sn and the relation (ß). If we put

H = y — , V =

ft

/., /:= 0, 1, -2, ■•-, >ii--2, / * ;/ (mod. '-^^ï

then, corresponding to eacii of the (m—l){m — S) different comhina-
tions of values of P^ and ?^\ we ol)tain an equation of the form (10).
Multiplying all these equations side hy side, we get

_ 0(".-l)(«.-3) ,/(«-S)_

Hence, |)utting

we get

On tlie EL'latively Abelian Coprora. ^5

Avhere s i> a certain ca])e root of unit\\ If we denote tlie required
discrinmant 1)y I), tlien

(,„_l)(„,_s) «,-i ;.

= J-2"'-V-.
On tlie other liand, let D^ 1»e the discriminant of tlie equation

>svG/) = o,

which is of degree ^ , and whose roots are

Then, mi account uf the relation

srf \ \r.f \ 1 1 (snu — sn?Ä)(snz; + sn?Ä)

sn-u nn-v sn-7tsn-u

we ol)tain

;:(«.-l)(„.-3)

It was shewn in §. 2, that Ä(y) i^5 an integral function of //.
Hence, if we regard // as an unknown quantit}', the al)Ove equa-
tion is of degree i- ? '^^^'^ its discriminant y/> must satisfy the
relation

1 ;.

2

-1 ^ma+b)^-Y-

Therefore

/r

Comparing this result with (11), and remembering tliat e must he
a culje root of unity, we find that

J(î Art. 5.— T. Takenouchi;

Thu;^ we obtain

9 G

/^ .-.
Lastly, let D^ be the discriminant of the equation /iSV(//) = 0
regarded as an equation for

yl '

Then wo can shew without any difficulty that

— n ^

Avhence follows

B.= D.,—, ^,

(,„-l)(„,-7) w-

Hitherto Ave have supposed that /'- is a primary integer. Uut,
the same method of calculating the discriminants can also be
applied to the case where ,« is a composite integer, provided that it
is odd. The result is exactly the same as in the above case. But;
it must be observed that the equation A„{x)=0 in this case is not
irreducible. The corpus of /^-division is determined Ity a root of
the equation of the v"(/^)th degree

A',(x) = 0,

where ^-J'.(^') is an irreducible factor of A,{}-). The abo\e method
cannot give the discriminant of this latter equation. For, if /^ be
not a prime, then in the form

71 ±V = -^ — ,

ii

the numerator may have a common di\'isor with /'-, without being
divisible by <« ; and consequently sn(?/±?') is not necessarily a root of
A'.ix) = 0.

On the Relatively Abelian Corpora. 27

Since it is evident, liowever, that the discriminant of the
latter equation must be a divisor of that of the equation Af.{x) = 0,
the following conclusion is valid:

The discrlmmanf of the equation of f^-diviston of jxrlods of sn
cannot contain an odd 2'>ri7nc, which is relatively j^i'imc to /-«, /^ Ijeing
an odd integer in h{i>).
Hence it follows, exactl3^ as in the case of primary /^, that

tJie discriminant of the corres'ponding division- equation for the
function ^Tj cannot contain apriine, which is relativeh/ privî<^ to 3 and. /^.

§. 6.

Again, let /^ be a primary integer relatively prime to 3, and ?/?
its norm. By adjoining a root x of the equation

Ä^(x) = 0

to k(f'), a corpus is determined, which we shall denote by A"(a-),
instead of k(f', x), for the sake of simplicity. This corpus is of the
(w— l)th relative degree and relatively cyclic, when we take k(f>) as
the fundamental corpus. In the following, when the fundamental
corpus is not explicitly mentioned, it should always be understood
to Ije k(p).

Since the discriminant D of the above equation contains only
two distinct primes, viz. 2 and /^, the relative discriminant of the
corpus Jy^(x) contains no other prime than these two.

It is well-known that all the roots

sn(;-^ — ), ; = 0,1,2, •••,m-'2,

are associated with one another, and

Hence (sn— ) is a prime principal ideal {Ifauptideal) in K{;x)^ and [^

is associated with its (m— l)th power. Therefore the relative dis-
criminant of K{x) contains i« to the (wi— 2)th power.
As for the prime 2, we proceed as follows :

18 ' Art. 5.— T. Takenouchi :

Sicco A^,(^x) contains only even powers of x, tlie corpus K(x)

must be relatively quadratic with respect to K{x'). But, this latter

corpus is identical w^ith the corpus K{y) defined by adjoining to
k{f>) tlie number

which is a root of tlie equation

SV(7/)^0. ' (12)

This equation can also bo regarded as an equation for if of degree
, — . Hence the corpus K{;y) is relatively cubic witli respect to
K{if). Consequently, 7v(a-) is relatively sextic witli respect to

A'(r)-

The corpus K{y") can be defined l)y the number

as well as by y\ The former is an algebraic integer as shewn in
§. 3, while it is not the case with the latter. But, as we have
found in the preceding section, tlie discriminant D^ of the equation

(12), considered as an equation for -^, does not contain the prime

factor 2. Hence the relative discriminant of K{;if) cannot contain
the factor 2.

Next, let us consider the relative discriminant of K{;y). i.e. of
Jv(x'), with respect to /v(;y"0-

Let

JJi = ^(^0' Z/i = Hv)

])e two roots of (12), sucli that yi=hy[, then

y,= ^{it + ü), y{,= ^(u-v)

are also two roots of (12), •dudijo=kfjL From the relation (7), we

t^et

^'(,,) ^ }^'(v),

and then, from the identity

Ave see that

(ur-u'ÙXy-yd^n^^f-

Oa the Relatively Aboliau Corpora. i n

Changing Y^, vmtou + r, u-r re^pecüvely, we get

where y,- ^(-2,,), y'^^ ^^2yj.

Let us repeat the same process successively, until we arrive at tha
relation

where /4, //,; are derived from y,_^. ijl_^ hy changing u, r into
u + v, u-v respectivel3\ From these h relations, it can l)e con-
cluded that

iv -Uif^ ^"'Ot)'. (13)

From the multiplication-formula

(t— /Orw'
we get

If r</0 be a root of (12), so must also be ^[(l-,o)»] and hX^'ic), and
they are evidently unequal. Hence, putting

in (lo), we get

^■'{jty^a-oyKiuf.

Therefore, l»y (7),

(i-i'yHuy^^

If we put

then a, ;i, y are all algebraic integers in K(.r), such that

Hence, in the corpus Atr), the principal ideal (•_>) mu^t l^e equal
to the cube of some ideal, say

(^) = [v.v, y.

20 Art. 5.— T. Takenonchi :

Now, the numljer a breaks up into two factors in Jy{->'')^ as
follows :

a = (1— /y)^(?^)sn-M

= (1 — 10) — sn-?/

= (en 2( + ,o-)(cii « — .(/).

Since the difference between these two factots is 2^o-, both of them

iiiust be divisible by the same power of PiPz Hence we find

that

Consequent^ the relative différente of A"(a:'), i.e. of ^(^), with
respect to the corpus K{;if), must contain the same power of Pi p^ ...
as the relative différente of the number cnw+//^ with respect to the
corpus K{if) does.^-*

Under our present supposition that p- is primary, the following
relation holds for a variable ii :

cn/i« = cn?/jR(sn-w),

where it(snV) denotes a certain rational function of sn^^"' From
this relation, putting

we obtain

Hence, if we put

u -

o) _ 'Mv

11 [X

en u

1

~ B{&nhi) '

en pu

1

~ B{ün\on)

en f ru

1

B(s,n'f)-u)

f-

cnu+/r,

r =

— cnpu + p-,

$'' =

en fPu + fr,

1) Hilbert: Theorie der algebraischen. Zahlkorper (Jahreshericht dor Deutschen
Mathematiker-Vereinigung, IV), §. 126.

2) Weber: Elliptische Funktionen und algebraische Zahlen (1st edition), §. 117.

On the Relatively Abelian Corpora. 2T

then these three numbers are conjugate to one another witli respect
to Jy(y"); and

consequenth'

(^') = {n = {n = \\v.

Now, the ralative différente of ? with respect to Jy(i/^) is

{^ — ■?')(l^ — ^") = (en u + en oiL)(cn u — cii o-u)

_ (en u dn u + l)(on"^i — dn u)
cnuànu

Hence, observing that both en?/, and i\nit are algebraic unities, and

also that

cmiàniL + l = ç"cn-u—frç- + 2.o~,

ciihi — dn /<; = o|' d nit + ^-'— '20- en ?^ — 2 dn ii,

we see that tlie différente in question contains ViV-z to the

second power.

Next, to investigate the relative différente of K{pc) with respect

to K{/). put

_ l + en?i + sn;;.
.r + CTiu

and let ^' Ije conjugate to f] witli respect to K(^x^)j then

, l+en?i — sn?^
;7- + en a
and

-^ , ,/_ '2(1 + en, a)
.)- + en ?t

. . , _ 'icnujl + en «,)

which shew that ^ and ■'^' are integers in AX'O- Also, since ■
1 + en ?i = 2 + ;v + {o- + en ?/),

the number 1 + cn?* is relatively prime to 2. We conclude there-
fore that each of the ideals p,, p,., must break up into two equal

prime ideals in K{:c). Let us put

\\\h = (^:^. r,

then the integers ^ and '^/ are l)oth divisible by -^^ ^,, , Ijut by

22 Art. 5.— T. Takenonchi :

iiü higher power of it. Hence tlie relative différente of tlie corpus
K{;x) and of the number '^j, with respect to K(x'), nuist contain
^1 ^.j to the same power. But, since

f>- + en 71

we conckitle that the poAver in question inust l)e the fourtli.

Therefore the relative différente of Kix) with respect to KQf),

as well as to k(f'), contains ^i ^2 to the eighth power. Thus,

we find tliat the relative discriminant of A"(-*') with respect to Z'(.^')
is equal to

4(m-])

2~ 3 ^/'.--.

The corpus /v(.r) is a relatively cyclic corpus of relative degree
m— 1. Hence, if d, 1)0 a divisor of ?>? — !, tliere exists certainly a
divisor (Unterkörper) of JC(.r), which is of relative degree d, and of
course also relatively cyclic. Wo shall denote it l>y d.

Let

w-1 = 2"-i3"'+Vï.^^ ,

Avhere j9i, jh, î^i'c distinct natural primes, different from 2 and

0. Then the corpus Jv(.c) can be looked upon as the result of
composition of the relatively cyclic corpora C2'>+^, Csn'+i, C^,^;-., C^i^^.-,
, taken all together.

Lot us now determine the relative discriminant of eacli of
these corpora.

In 62''+!, the number y- l)reaks up into 2''+^ equal prime ideals.
Therefore the relative discriminant of G'^+i contains /^- to the power
2''+i_l. The same reasoning applies also to otlier corpora Gv+i,

^i'l''- ? •

On the other liand, it can be shewn generally that the relative

discriminant of a relatively cyclic corpus C of relative degree /•)". p

being a natural prime, cannot contain. the factor 2, unless |9=:2 or

3. For, if ^9=^2 and 2 enter into the relative discriminant of C,

then, since 2 is relatively prime to the relative degree p'', the

corpus of ramification {Verzicelriungslcöriier) of 2 in C must l)e tlie

On the Eelatively Abelian Corpora. 23

corpus C itself, while the coi'pus of inertia {Tnigheitshöiyer), whose
relative «liscri minant cannot contain 2, must necessarily l)e a proper
divisor of C. Hence the relative degree of the corjms of ramification
with respect to the corpus of inertia must he //', h'^h^l. Then,
if / he the degree of a prime ideal contained in 2 in the", corpus 6'
we get'*

4^-1 = 0 (mod./).

But, since/nuist ])e a power of ^?, it follows from this congruence
that

4-^-1 = 4-1 = 0 (mod. p),

wliich is imp()ssil»le, unless ]j='d.

Therefore, among the corpora Ave are considering, only the
two, viz. t'^h+i and CV'+i, rnay contain '1 in their relative discrimi-
nants. The relative discriminants of other corpora CpK , Cph,,

are equal to //J'i'''-i, /^?'v"-i, respectively.

Tlie corpus Jv(ff), i.e. Ck-J, has the relative discriminant
relatively prime to 2. Hence the corpora C2'' and Csv, l)oth heing
divisors of dtri, must also have the relative discriminants relativelv
jmme to 2. Their relative discriminants are therefore /^^''~^ and
^ß'>'-i respectively.

As for the excei)tional corpora CV'+i and Cs'-'+i, we have to
consider them a little further. If we put

m' = 9]^ 'pi'ipl^ ^

the relative discriminant of the corpus C,,, is relatively prime to 2.
Composing C,„. with (3''' or Cs'i'+i, we ohtain K{y^) or 7v"(//) re-
spectively. Hence, the relative cl'ifferente of CV+i with respect to
63''' must contain as many ideal factors of 2 as the relative différ-
ente of K{y) with respect to K^jf) does. Therefore the relative
discriminant of 63" +1 is S^-^"' //'''+'-i.

Next, to investigate the relative discriminant of Ck'^+i, let us
consider the corpus -Ä"(~), where

-^7)-

1) Wtbtr: Algebra II, §. 181.
Hillxrt: loc. cit.. Ç. 11.

24 Art. 5.— T. Takenouchi :

Since

the corpus K{z) must either be relatively quadratic with respect to
J^df) or coincide with it. But, if Jv(z) coincide with K{;if), the
corpus AX'/), i.e. 7v(.r), must contain l)Oth :; and sm^^. ïlien. by
the relation

z — — ^ ,

K{lf) must also contain snw ; which is of course impossible. Hence
K{z) must be relatively cjuaclraric with respect to J^(jf), and con-
sequently K(x) is relatively cubic with resj^ect to IC(zy

The principal ideal (2) is equal to the sixth power of an ideal
in KÇx). Hence, in AXt;), (2) is equal to the square of an ideal, say

(2) = (Q.q, f.

If we regard q^ q,, as an ideal in Ä"(a-), it follows that

QiQ. =i%'^-2 f-

Now, in general, if a prime ideal p in a certain corpus k is
equal to the cth power of an ideal $ in another corpus JC, which
contains k as a divisor, and if e be divisiljle Ijy p, then the relative
différente of Jv with respect to k contain >p at least to the eth
power. ^^ Hence, in the present case, there lative different e of K{z)

with respect to At//) must contain at least {c\i(\» )'. On the

other hand, if the différente in question contain (qi t]2 )^ the

relative différente of K{x) witli respect to K{;}f) must be divisible

at least by ('l^i ^o )', Avhich is contradictory to the result

obtained in the preceding section. Therefore the relative différente
of K{z) contains Qi q.^ exactly to the second power.

Composing Cs^'+i or €& with Cm" , where

we obtain K{z) or K{}/') respectively. Tlius, here tlie corpus K{z)
playing the part of K{if), \\Q can shew just as in tlie case of Gv+i,
that the relative discriminant of Ck'^+i is equal to 22''"*"^ /^^'^^^-i.

1) Hilbert: loc. cit , §. 12.
AVeber: Algebra II, §. 174.

On the Relatively Abelian Corpora.

25

We close the present section with tlie following list of the
divisors of Jv(.v^.

Divisors

Eelative Discriminants

Cpi. ,

>^-:=l,2,-

■,Jh

/.P^-i

-'s

L=],2,.

■Jl,
■Jl'

/^2'/— 1

Cs;-',

?^'=l,%-

lßl'-\

C's'i'+i

22.3''' ^«3'''+l - 1

Co?. ,

/ = 1,2,.

■Jl

;.2^-l

6'2''+l

23''+! ,^3''+l-l

§. 8.

Before proceeding to the discnssion of the division of periods
of sn l)y powers of /^, let us insert here a digression on the classes
of congruent integers in an algebraic corpus.

Let p" he a power of a prime ideal in an algebraic corpus.
The integers in the corpus can be classified into classes of congru-
ent integers w^ith respect to the modulus p\ These classes can be
composed with one another by multiplication, They form an
Abelian proup of order ^^-'^"-^^ (/>'' — l), wdiere ^ is a natural prime
divisible l)y p, and/ the degree of p.

The problem of determining the rank and the invariants of
this group has already been solved to some extent by G. Wolfï'^
and by the present author."' The result obtained by the latter is
as follows.

The Abelian group in question can be decomposed into two
component subgroups 31 and 35 of orders ^:)-^("-i) and 75-^—1 respec-
tively.

1) Wolff: Ueber Gruppen der Reste eines beliebigen Moduls im algeraischen Zahl-
kürper (Dissertation ; Giesson, 1905).

2) Takenoiichi: Journal of the College of Science, Tokyo Imperial University, Vol.
XXXVI, Art. 1.

26 Art. 5.-T. Takenouclii:

^ is a cyclic proui^. If

P--^ =^Hi^j^. ^

Avliero 2'^^^ P-^ ^^'^ distinct primes, the invariants of 23 are

7>"S 7?."-,

3( is an Abelian group, wliose invariants consist of powers of
21. If ^'' be the highest power of p contained in 77, then tlie I'ank r
of %i is

r = fd + l or fd, if n^d + l; /.• = T-^],

according as the congruence
2) + oc'~^= 0 (mod. p''+i)
lias a solution or not,
r =fd, if n — d + J,-,

r-/(7^-[|-]), if d + l->n>l,

r = 1, if ;i = 1 ;

where the sj-mbol [.c] denotes the natural number, such that

rr + 1 r> l-c] ^ .r.

In the case where d is not divisible by 75 — 1, we can find not
only the rank, but also a system of bases as follows.

Let 7: be such an integer in the given corpus, that is divisible

by p, l)Ut not by P", and ç;(/ = l,2, ,/) such integers in the

same corpus that

c^ç^ + c..ç.2+ +Cj Çy ^ 0 (mod. p),

for all rational integral values of c, (/ = 1,2, ,/), except when

Cj = r._. = = cv = 0 (med. 2^).

Then, the following yH numbers represent a system of l»ases:

«, = 1,2,3, • , f7 + /<• — 1 , excluding" multiples of »,

^ -1,2,3, ,/.

Even if d be divisible by ^7 — 1, we obtain similar results, pro-
vided that n ^ d + k. But, if 77^>d-\-kj we cannot determine a

On the Relatively Abelian Corpora. 27

system of bases in general. Supposing that h is not divisible b^^p,
the onlj^ thing we can conclude is as follows:
Determine an integer (^ from

1) = -''(> (mod, p'^+^).

Then we distinguish two cases according as the congruence

(>-{-x^-^ = 0 (mod. p)
has a solution or not.

If it has no solution, the /(/ numbers given above represent a
system of bases.

If it has a solution, say x = .r,„ then / rational integers a,
{1=1,-2, ,/), such that

a\ , = a^ç^-\-a.,ç.,+ + «^ ç. (mod . p)

can 1)6 uniquely determined witli respect to mod. j/. Hereljy,
Avithout losing generality, we may suppose «i^ 0. (mod. ^?). Then
determine an integer c,,, such that

for all rational integral values of c^, except when

r,, = C.J = c~= ■ = Cf= 0 (mod. 2J)-

Now, it can be shewn that all the elements of 3( can be re-
presented ))y means of they</ + l numbers:

l + ^i~\ a = 1,2, , fZ + A— 1, excludiDg m.iiltiples ofp,

i = l,2. ,f,

and l + ç„-'+^

But, these /(/+1 numbers do not in general represent a system of
bases. In fact, tliey represent a system of bases only when o.i=p,
where ^a-i denotes the exponent to which the number 1 + ^^-'-' belongs
with respect to mod. p".

Let us now confine ourselves to the corpus /{/'). f denoting a
primitive cube root of unity:

^ _ -l + v^H

28 -^rt. 5.— T. Takenouchi :

(i) When p^l (mod. G), ;? can be decomposed into two
different prime factors of the first degree in /■(,'>). Hence

and consequently r=l. There exist primitive roots of p".

(ii) When p=— 1 (mod. G), j^ is itself a prime number of tlie
second degree in k(/>), and

d = l, f='2, /. ■ = 1.

If n = l, then r=l. There exist primitive roots of p.
, If n>l, then r=2. The invariants of 31 are ^9"-^ p"-^
Putting

Ç being a primitive root of p, we obtain a system of leases of 51^^;
viz,,

l+p, l + ^p.

As for the root of 33, we may take such a power of ç as pre-
scribed in §. 1 of my paper just cited.

(iii) The natural prime 3 is associated with the square of a
prime in /{;>), viz.

Hence

and thus we meet with a case where d is divisible by p— 1.

If 71=1 or 2, then r=l. Consequently there exist jnimitive
roots of l + 2;7 and (i+2//)-, e.g. —1 and —o respectively.

If n = 3, then r = 2; the invariants of 3t Ijeing o, 3. Putting

--.-1, e,= l,

we obtain a system of bases uf 3C :

p, l — 3p.

If ;^>3, then ;-=3. If we put

^ = —(1 + 2..:/), ç^ = fr, ç^,= //-,

1) Strictly speaking, we obtain a system of numbers representing a system of bases of
?(; for, the elements of 31 are not numbers, but the classes of numbers. But, for the sake of
simplicity, we shall herafter use this abridged term.

On the Relatively Abelian Corpora. 29

Ave obtain the three numbers

1 + ç^- = o, ij^^-"-=4. + 3,0, 1 + ,-,- • = 4 - 3/s

wliich belong (mod. (1 + 2/0") to the exponents

respectively. Thus €^^ = ?>, whence we conclude that the above
three numbers form a system of l)ases of SI. The exponents to
Avhich they belong are the invariants of 3( in this case.

As for 33, the invariant being '1, we may take —1 as its base.

(iv) The natural prime 2 is itself a prime number in /•(/>), and

d = h /=2, Ä = l.

Thus we have another case where d is divisible l^y 7>— 1.
If n — \. then there exist primitive roots of 2, e.g. p.
If ??=2, then ?-=:2; the invariants of 3( being 2, 2. Putting
- = -'2, ^^=\, l^=p,

we obtain a system of bases of 5f :

-1, 1-2^.

If 7?>2, then r = 3. Putting

- = -2, ?! = 1 , I: = [', c, = -fr,
we obtain the three numl)ers

l+^^- = -l, 1 + Ij:= 1-2o, 1 + ç~=1-V,

That these numbers form a system of bases of 3( can be shewn as
in (iii). The invariants of 31 in this case are

2, 2"-\ 2"--.

Since the last one (1— 4/''') of the above bases can 1)e decomposed
into 1—2/' and 1 + 2;', we can represent all the elements of the
group also l)y means of

-1, l-2^o, l + 2,o.

But these three do not form a s^^stem of bases.

As for ^. the invariant being 3, we may take p as its base.

3Q Art. 5.— T. Takeuouchi :

S.

Let lis again iissume tliat /^ is a piimary iiitogcr relatively
prime to 3, and m its norm. It was shewn in §. 4 that

sn ait = — '—^ , X = sni(,

7.1

where </\ = A,,(x), /, = D,,{x).

Ô

Substituting x-^-^ in place of .r, we get

x^

where '/". is a rational integral function of x of degree wr— 1, wliicli
must he divisible 1)}^ ^j. If we put

then (I'.^ is necessarily of the form

<J>., = Sx" + a.x'--+ + rto, A'"--" + + n,

where £ is an algebraic unity and «o, , «o;., are all integers

in /•(/') divisible by y-.

It can 1)e shewn ])y matliematical induction, tliat in general

where <^„ is of degree m''~'^(/n—l), and its coefficients possess the
same property as those of ç\,.

Hence, if we put

this equation is irreducible, and gives as its roots the values of sn^^
corresponding to the /^-''-division of jwriods. Tliat tlie discriminant
uf this equasion can contain only two distinct prime factors 2 and
/•Î has already been remarked at the end of n^. o.

To find the Galois' group of tliis equation, we have to dis-
tinguish the following two cases.

On the Eelatively Abelian Corpora. Q1

(i) A\'lion r- is not real, put n=-, m^ji. Then
p = ^- (mod. 6),

hence there exist primitive roots for -''. The roots of tlie above
equation are

s"(/'^-^)> ^--0,1,2, ,y-iQ;_i)_i,

Ï being a primitive root of -'*.

The corpus obtained by adjoining these roots to h{p) is rela-
tively cyclic, the relative degree being p^-iQ^-l). There are two
divisors of this corpus, of relative degrees p— 1 and ^V'-^ The first,
of relative degree ^?-l, is nothing else but the corpus discussed in
§. G and §. 7. The second, of relative degree 79''-^ is also relatively
cyclic, and its relative discriminant is a power of -.

(ii) When ,« is real, put iJ.=q. Then
q = —1 (mod. Ö)

and m=(f. In this case we have to use two integers, say r and r\
which belong to the exponents q^'-\rf—l) and q^'-'^ respectively, in
order to obtain all the roots in tlie form

^ '// //= 0,1,2, ,^"-^-1.

The corpus obtained by adjoining these roots to /r(/>) is rela-
tively Abelian, the relative degree being q-^''~'^\q- — \^. The divisor
of this corpus of relative degree y'-l is nothing else Init the corpus
discussed in §. 0 and §. 7. The other divisor of relative degree
q-^^"^^ is relatively Abelian, and its relative discriminant contains
no other prime than q. Since the Galois' group of this latter
divisor is of the form

sH', ./.,,9 = 0,1,2, ,q"-'~\,

it contains q"\q + l) difïerent cyclic subgroups of relative degree (J"'.
Namely, if we use the notation (.t) to denote a cyclic group con-
sisting of all the powers of a substitution x, then these subgroups are

(4 M, {.sei , {sti'-'-^i

(0, (.s't), (/'{), , (.s^V'-^-i)/).

32 Art. 5.- T. Takenouchi :

Corresponding to these subgroups, there are as many relatively
cyclic corpora of relative degree <7''"\ whose relative discriminants
contain no other prime than q,

§. 10.

The primary integer /-« has hitherto been supposed to be rela-
tively i^rime to 3. In the present section, let us consider the
case where /^ is a primary divisor of 3:

/^ = l + 2,v.
If we put

iO

then, on account of the relations (5) and (7), it follows that the
corpus K{x,^ can be composed of tw^o corpora Ä"(^^(i';,)) and K{^'{v,^).
From the multiplication-formula

S'{(l + 2,,)«}=M^|, (14)

we find that

îp(r,) = 0 and ^-(tg'=r/3.

Hence, if we put

Vk

-^Oo

where ^^ = _^^

then, since the corpus K(X\Vi)), Ji>\, must necessarily contain the
number ^(^'s), we get

The relation (14) gives

y^ + ^vn-m -1 = 0. (15)

Differentiating (14) with respect to u, we obtain

(i+2o)îp'{(i+2o).] = ^^l^n^o^

which gives

-3(l + 2,«)!>-'{(l + '2/.W=-|||^!f-'(«). , (IC)

On the Eelatively Abelian Corpora. 33

Hence, putting

we get '

2il + 3(\+'2o),,_,z-^^9z,-3(\+2o)^n-i=0. (17)

Since, by (10), the number ^-'(t\) = ±V—g-^ can always be rationally
expressed l)y ^•''(r,,), we see that the corpus Ä"(lr' '(?'/,)) is identical
Avith 7v"(^/„ V— (/s), i.e. K(z/„ Vl— />). Thus we conclude that

Now, if we put as before

the relation (14) becomes

where /^ =r(«/' — 1, r/j = — 3r(?/)-.

Therefore, by successive iteration, Ave get

f/;i-i

Avhereyj,_i and ^,,_i are integral functions of r(?^), of degrees 3'"^ and
3''"^— 1 respectiveh'. Tlie recursion-formulae for/,"_i and //,,_i are

All the values of i//, are the roots of the equation

Since each factor in this equation is of the 3''"'th. degree, the rela-
tive degree of the corpus K{iji) cannot be greater than o''"l

But, on the other hand, since l + 2/> is a primary integer, it
can be shewn exactly as in the preceding section, that the corpus
Ä'Xsn^v.), i.e. Ä"(K^v,)), is of relative degree o''~\ Therefore, since
the equation of degree 3''"^ for determining ^-(r/,) can be looked
upon as an equation for ^'('v,)"' of degree 3''''. the corpus K{)^-(^r,)^),

Q^ Art. 5. — T. Takenouclii :

i.e. Ki^ifi), is of relative degree 3''"'. Consequently, K{y,) must Ije
at least of the 3''"^th relative degree.

If follows therefore that the relative degree of Ki^ih?) must be
exactly equal to 3''""; and consequently the equation (15) is irredu-
cible in ]v(y,,_i). At the same time, it follows tliat

Jl(yJ = K(yfX (18)

Now, since

7/i=0, z,= ±l,

we find that

K(y,) = K{.z,) and K{jj,) = it(,3-,).

More generally, it can he sliewn l)y mathematical induction, that

for all values of h. For, if we suppose that

K{y n-i) = K(^k-i)>
then, by (17), Jv(z/,) nmst be at most relatively cubic witli respect
to Jv(j/n-i). On the other hand, however, since

-1= l-%fe>

the same corpus Jv[z/i) contains Jy(f/l). Hence, by (18), it contains
Ä"(3//,) ; consequently it must be at least relatively cubic with respect
to K(;f/,,.i). We conclude, therefore, that Jv(z/) must coincide witli

Thus we find that the corpus Jv(x/,) can be decomposed into
the following three divisors :

§. 11.

Let us now investigate the nature of lv{yi). But. since this

corpus for a certain value of h is contained in another corpus of the

same type having a greater value of h, we may confine ourselves

to the case

7i = 2;i + 2>3.

On tlie Eelatively Abelian Corpora. 35

It follows from ^. S (iü), that the roots of the equation

^„^1 = iu.-ü.xu-p<h:)(u-:ro.:) = 0 (19)

are the values of r(?^), where

(; = 0, 1, 2,
/(4 + 3o)''(4-3/>>" n-0^ o .... 3'._l

^ ' h =0,1,2, ,3'-l;

•jr, in other words,

o^(; + y.,j) ,- = 1,4, ,3(3"-l) + l,

(1 + 2/;)-+- /^ =0,3, ,3(3"-l).

It is evident that, of all the roots of equation (19), those
which arise from one and the same factor on the right-hand sido
must correspond to the same value of c in the above expressions
for II. But, the consideration of the parallelogram of periods of
r(?/), as shewn in §. 4, gives

Avhence we infer that the roots of the equations

correspond respectively to tlie values

c = 2,0,1.

Since these three equations define one and the same relative
corpus with respect to /{."), we may confine ourselves to the one

This equation can be reduced to the following series of cubic
equations :

//: + 37/,.,^?-l = 0, (20)

/.■ = 3,4, ,2y?, + 2, y,= 1.

If we consider (20) as an equation for -, its discriminant is

36 Art. 5. -T. Takenouclii :

But it follows from (17) tliat z,._-^ cannot contain any ideal factoi
relatively prime to 3. Hence it must also be the case with D, and
consequentl}^ also witli the relative discriminant of the corpus
K{:lli^. Therefore tlie relative discriminants of the corpus of de-
composition (^ZeTlegvmjslcorper) and of the corpus of inertia of the
number l + 2/> in K{yi) must be an algebraic unity. Hence both of
these corpora coincide with /{/'). It follows therefore that l + 2,o is
equal to the 3^"th. power of a prime ideal of the first degree in AX///.).
That this prime ideal is a principal ideal can be shewn as
follows.
Put . _ l + 2,> z. -3 4 o,, , o

and let its conjugate numbers with respect to K{i/i.^ be denoted by
Zl and cl. Then, observing that, if ij,„ y',, y",, denote the three
roots of (20),

Ave ol)tain

2(l + '2o).

./.-/. I ~ /.■ - /.■ T^ -ï /.■ ~ :■

2//.-1
1 .

- /. - 1

' ■ ' Ul

Since //;,_i is an algebraic unity, C/, must be an integer, provided
that r;_i is an integer. But, this is really the case with

for

C:w^+:^r^+:^r:; = 2(i+2/.n

Hence C/. is certainly an integer. Its relative norm taken with
respect to AX.V/,_i) is, as shewn above, associated with ^/._i. Hence
we infer that the relative norm of C/, taken with respect to k(f>)
must be associated with that of C3. Thus we get

(i+2o) = (r,„,,)^"

On the Relatively Abelian Corpora. 37

ïlie rc4ative différente of w- with respect to K^jj^,.^ is

ylÀyj:-i-yi:-^{y,-yk-i)
= (i+2.>)r,_,[(i+2;.)^,_,r;-i] ^f^.

Therefore tlie relative différente of AX'//.) with respect to K{ii;,.^
must be equal to {l + ^f>)^,:-\- Consequently, that of K(i/-2n+-2) witli
respect to k(p) is

whence we conclude that the relative discriminant of K{y-zn+'i) is
equal to

(1 + '2.y)-"- 3'' + 3 + 3-+ + 3-"

(4» +3)3-'' -3

As in case (ii) of ^. 0, it can be shewn that the corpus K()/-2„u.i)
contains as its divisors 4. o"'^ different relatively cyclic corpora of
relative degree 3", whose relative discriminants are powers of 1-1-2^6».

Next, let us consider the corpus

which is relatively cubic with respect to Z{;'). Since the discrimi-
nant of the equation

x^-^ = 0

is —2'. 3", the relative discriminant of C cannot contain any other
prime than 2 and l + 2;>. If we put

l+2/>

^ = v2, ■'/ = "rfrr'

then

i+.2,„ = -^yL.

1 — X

Therefore, since 1—x is an algebraic unity, we get

2 = x^,
1 + 2;,^ y ', .//--^1 + ar.

38 Art. 5. —T. Taken ouch i :

The relative différente of y with respect to h{{>) is

(l + 2.o)-V-rJ— _i — V^Jt , ^ ., ) = a-(l-rr)(l+a-)--a;y.

^ '-^\l+2; l + /'a- /\ 1 + rr l + ^ra- / v a / j

Hence the relative diffcrente of C with respect to h{p) contains y
to the fourth power. The relative discriminant of C is therefore
equal to 2'3^

In the relatively quadratic corpus

Q = KVT^>) = HVY^,},
all tlie integers can he represented in the form

«+/5a/T+27/

2 '

wdrere « and ß are integers in k(f'), such that

«- — (I + 2p)ß- = 0 (mod. 4).
From this congruence, it follows that

op—ß-=(a + ß)(a—ß) = 0 (mod. 2). '

Hence or—ß- nmst the divisible by 4. Therefore, from the original
congruence, we see that ß, and consequently also «, must Ije divisi-
ble l.y 2.

Thus the numbers 1 and \/l-l-2/^ from a system of leases
{Mui'tmcdôasis) of Q, so that the relative discriminant of Q is
2^(1 4-2/0-

§. 12.

In this section we consider the division of periods of sn b}' a
power of 2.
Put

then the duplication-formula for j9(«^) is

On the Eelatively Abelian Corpora. QQ

Differentiating (21) with respect to u, and making use of the
relation

we obtain

Hence, if we put

then

q(:ii) =

_ n^i]

9

Let ?//, and Z/, denote the vahies of p(?/) and q(;^f) respectively,
when

(0

Then, from (21) and (22), we find tliat

yi' = 1 aiîd -, = 0 ;
therefore

K(y,) = ZC",) - lif>).

To find 1/.2 and -^2, we have to solve the equations

y 2 - 4^/1 y^ + 8 //, + 4?/i = 0

and .ro* + '2(l + 2/>),r| + l = 0

respectively. Both of these equations are reducil)le. In fact, they
break up into factors as follows :

(ij^-^y^y-^y^r = 0,
(,-,- + 2/) .r.-l)(?,--2^o^,-]) = 0.
Therefore

7/,= (l±V3)yi,

2.2= ±f>i^i:f>i) .
Thus we see that

!%,) = AX.,) = Hr, i).

Now, from (21), we can derive by iteration the formula

40 Art. 5. -T. Takenouchi :

where/, and (ji, are defined l>y the recursion-formulae
with tlie initial values

By similar reasoning as in the preceding section, it can 1)6
shewn that the roots of the equation of the 2'''"''th degree

fu-x-iffju-i =0, c = 0, 1, 2, h > 2, (23)

are the values of />(?/), where

./-Xl + 2or(l-2o)'' « = 0,1,2, ,2"-i-l,

2 b =0,1,2, ,2"-^-l;

or, in other words,

,,_ />^-'(^+^/>),, ^ = 1.5, ,4(2"--r)+i,

^ --^ = 0,2, ,2(2"-i-l).

As will he shewn hereafter (§. 17), the corpus Ki^iji) is the
class-corpus {K lasse nkor per) corresponding to the group of numhers
a in /•(,'/), such that

a = 1 (mod. 2'').

Therefore the relative degree of J^(>/1) cannot he less than^^

V'C^"^ i.e., 2-'.

(3

Hence, of course, the relative degree of A"(//a) cannot be less than
2'''"'. Thus we see that the equation (23) must he irreducil.de in
/{,"). .Vnd it follows, at the same time, that

K(yf,) = K(y,) ;

whence it can l)e proved, as in the preceding section, that

for all values of //.

1) Weber: Algebra III.. §. 167.

On the Relatively Abelian Corpora. ^[

In virtue of tlie relations :

2 sn?^, cnitdnu

üir2u

l + .i-^W^i
1

q(u) =

sn^u

it can l)e concIiKled tliat ^ii'2// can 1)0 rationally expressed by />(?/)
and q(ff) in the corpus kQ'). Therefore, if we put

then the corpus JC(x,,.i) must a divisor of IvQ//,, z,), i.e. of K{_y/).
But, on the other hand, this latter corpus K{i/i) is evidently a
divisor of K{xi). Thus we find that the two kinds of corpora

K{x^ and AXyJ, ^ = 1, '2, ,

can be arj'anged in such a series

-^%i)' -^^X^i)' l'^^UÙ^ ^A-A'X i'^iUi^y K(x„), ,

that each corpus is a divisor of the next one. It Avill also be
proved presently that none of these corpora coincide witli the
neighboring ones, except

K(!j,) = K{x,) = ]c{o).

Hence, in the following, we shall consider AX///,) instead of K(x,X
where i/,^ is a root of the equation of the 2'''"^th degree:

fi,-i-!Jh-i = 0.

By (l^l), tliis equation can l)e reduced to a series of the follow-
ing l)iquadratic equations:

ul-^!J,-i!jl + ^!J, + ^!Ji,-i = 0, (24)

^ = 2,3, ,h, ,j,= l.

The irreduciljility of (24), excepting the case k=2, follows at once
from that of (23). It is, however, impriniitive ; for, any one root
of it can be rationally expressed in /t;(/') in terms of any other root

42 ^rt. 5.— T. Tak-enouchi:

of the same equation. In fact, if

be a root of (24), then evidently the other three roots are

/ , ^/\ o7/,. + 2,0-

Putting

Ave get

Vic-

5^. + :^^ = 47/,.,,

r,Y;: = -4(7/,., + l),

y.+2 - 7/,7/r.

Thus the solution of the biquadratic equation (24) can be reduced
to those of the following two quadratic equations :

Y|-47/,_ir^-4(7/,_i+l)=-0,

7/|-r,7/,+(r,+2) = o.

The discriminants of these equations are I'ospectively

4= 17-4(7, + 2).
Dk can be transformed as follows.

This shews that the corpus J^iYi.) can l)e derived from K{i/,,_-^ l)}-
adjoining a:;/,_i to it; in other words, K{Y,) is identical with K{xj,_-i).
It follows therefore that K{xk-^ cannot coincide with /vXva-i), if
k>2.

On the Eelatively Abelian Corpora. 43

Next, since

dj: can be transformed as follows :

f7, = 4(y,_i-lxT/. + l)-
Now, from the relation

we conclude that l)oth y,-{> and ji,-o are divisible by />-l. Con-
sequently both

r>.- + 1 ^ VirzJ^ ^ Ilk' - .'^'

/y— 1 // — i /' — I

, and zi+L = i^+j/;-."-

must be algebraic integers. But, since

it folloAvs that

Tlie corpus K{Y,) can be defined l)y an integer

But, the relative discriminant of this integer with respect to

B,

^=16,rtxa:^.i-16,

since it can easily be inferred from (22) that z,., is an algel^raic
unity. Hence Ave see that the relative discriminant of K(Y,) with
respect to Ä%_i) can contain no ideal factor relatively prime to 2.
The corpus Ki^y,) can l)e defined l)y an integer

whose relative discriminant with respect to K{Y,) is

(Z'-l)"^'

44 -^'^t. 5. — T. Takcnonehi :

Hence the relative discriminant of /vX///.) nmst consist of i<leals,
which are all divisors of 2.

It follows therefore that tlio relative discriniinant of K{}/i)
with respect to /■(/') cannot contain any other prime than 2.
Hence the principal ideal (2) must l)e equal to the 2''''Hli power of
of a prime ideal of the first degree in Jv(j//,).

Tliat this is a principal ideal can l)e shewn as follows. Put

then, as shewn in §. 3, '/, is a divisor of 2. The norm of ^/, taken
with respect to A"(//,,_i) is, 1)y (24),

44

(4z/;.-i)

n"- ) = ^^ = —2 •

(-2?/i-,)"

whence we conclude that

(2) = U,,y , Q. E.D.

Let us next consider the relative différente of C/, with respect
to Jy(i/u-i), supposing that A>2. If we indicate the conjugates of
any num1)er l>y placing accents upon it, the dlfflrentc of C/, is

4-

1 1 \/ 1 1 \/ I

u'h u'i: J^ Vk uT J\ yt y'l

j^-Jyl+'^f+^yL^y::

yû-iyk

Therefore the différente is divisil)le hy 2'' CS, hut l)y no ihgher
power of C/,.

In the corpus AX'/O? ^■'^' A^/), the numl)er (2) is decomposed
as follows :

2 = (V3+ixvä-i) ^ {V3+iy.

Hence the relative différente of A"(/yO is equal to 2.

On the Eelatively Abelian Corpora. 45

The relative différente of At///,) is therefore
Accordingly the relative discriminant of K\ii,) is

0(3/(— '*)- - .

Now, the corpu. At.V„). /'>2. is relatively Abelian, and ils
Galois' group is ot the tonn

s'C', « = U, 1,.;, ■••■,-

;, = o,i,2, ,v-'-\.

In this Galois' group, there are 2'-= cyclic suhgroups of order 2'-,

^'^''" (.,), (rf). Uf'^. ■ (st"'-=-'l-

Besides them, there are also 2'- cyclic subgroups of order 2'-=, viz.

if), im, (s't), (s"-p'-'-«0-

Therefore the corpus A'(/a) contains as its divisors 2'-= relatively
yclic corpora (we shall call them A) of relatno clegree_2 . and
also ■->"- relatively cyclic coi-pora (B) of relative degree 2

Similarly, the corpus A%,.«) ^vill contain 2'- relatively cychc

aivrso (C) of ;elative degree 2'-, an.l 2^ U» of relative degree 2 .

'n^e relation between C an.l A is such that each one of C contains

^ attain one of J, and convesely, each one of A is contained m

certain two of C. Similar relation exists between 1) an.l A.

Hence we see that, by the division of perio.ls by powers ot -,

^™°'*''" 2..-. + o'.-. = 3.2'-

different relatively cyclic corpora in all, whose relative degrees are

all 2»-', and whose relative .liscriminants are powers ot -.

§. U!.

We have considered all the division-corpora arising from the
division ot periods ot sn with the singular modulus x=lohy powers
of primes in iO»); and analysed them into relatively- cyclic compo-
nents. Let us here tabulate these component corpora.

4G

Art. 5.— T. Takenouclii

In the following table, /^ <lenotes an odd prime in /•(/>) rela-
tively prime to 3, and m the norm of !'■. Tlie decomposition of
in— I into distinct natural primes is supposed to he

m-1 = 2''+i 3''+! p'--

Relative Degree

Distinct Primes

in Relative

Discriminant

Xumber of Corpora of the
same Relative Degree

I

P\ ^-=1,2, Jc

F-

II

3^ -^. = 1,2, ,h'

!^

III

3^.'+i

jj. and 2

IV

-y-, '^^ = 1,2, ,],

P-

V

jx and 2

VI

3

l + 2/>and2

VII

2

l + 2;>and2

VIII

m'\ 11 being any natural
number

I'-

{{ill + 1);m""^ when a is real
\l, when fi is not real

IX

3", n being any natural
number

l+2;7

4- 3"-^

X

2", n heiwg any natural
number

2

3- 2"-i

Hereafter we shall call tliese corpora >-imply elementary coiyora.

If we analyse tlie division-corpus for a composite divisor ,«, no
other corpus will be found than the elementary corpora. For, if ,«
contains more than one distinct prime factor, we can decompose ,«
into two factors relatively prime to each other, say i.i=aß. TJien
two integers ? and '-y can 1)e found, such that

1 = 1+^

and, i)utting

10 =

u. =

a

Tea

r

we get

sn?6" =

_ sn u en V dn v + sn r en u dn ?i

l + o"-sn-/6 sn-'ü

On the Relatively Abelian Corpora. 47

But, since cn?Mln?i and cni'dnv can be rationally expressed by
riuu and snv respectively, even in the case where ic or v h ix power
of 2 (§. 12), we conclude that

K (sn w) = K (sn u, sn v), Q. E. D.

§. 14.

We are now in a position to prove the following important

theorem.

Then: can he no other relatlvehj Abelian corpora ivith respect to
k{r) than those contained in the dimsion-corpora of the elliptic function
sn 'with the singidar modulus 7c=i;K
The same thing may also be put in another way as follows :

(jiven any rclativehj Abelian corpus with respect to h{p), ice can
aUraijs find such a corpus which is composed of elementary corpora
only and contains the given corpus as a divisor of it.

It is well known that every Abelian corpus can be decom-
posed into cyclic ones, whose degrees are powers of primes. This
is true not only for absolutely Abelian corpora, but also for rela-
tively Abehan corpora with respect to 1c{p). Hence we shall prove
our theorem only for such relatively cyclic corpora.

Let Ch be a relatively cyclic corpus of relative degree p>\ p
being a natural prime, and C\ its divisor of relative degree p. If
the relative discriminant of G contain a prime factor in l{f>) rela-
tively prime to p, let it be denoted by «. Now, consider the
corpus of inertia of « in Ci,. Since the relative discriminant of it
must lie relatively prime to !'■, this corpus of inertia cannot contain
C'l in it; consequently it must coincide with ^-C/»). On the other
hand, the corpus of ramification of i^ is (J,, itself, since ,« is relatively
prime to the relative degree / of C,,. Therefore the number /^ is
equal to the j/'th power of a prime ideal of the first degree in C,..
If the norm of /^ be denoted by m, we get'^

m~l = 0 (m.oà.p'').

1) Weber ; Algebra II, §. 181.
Hilbert : loc. cit., §. 41.

48 Art. 5.-T. Takcnonehi:

Hence, iî ]-> ]>e not equal to 2 or 3, we can always find an ele-
mentary corpus I, whose relative degree is p' and whose relative
discriminant is a power of /^. Let it be called U. Composing J^
with C/n we obtain a corpus of relative degree ^7", where h^n£2h.
In this corpus JSQ,, the principal ideal (/^) will be equal to the p"'th
power of an ideal, where h£ii'^2//. Then, since JlV/, is itself the
corpus of ramification of /^ in EC/,, it follows that tliis corpus must be
at least of relative degree p'', and relatively cyclic, with respect to
the corpus of inertia C of f^-. But, since it is evident that the
Galois' group of EC/, cannot contain a cyclic subgroup whose
degree is higher than the ^/'th, the relative degree of EC with
respect to C must be exactly equal to p''; whence follows that the
relative degree of C is jf''.

The two corpora E and C have no other common divisor
than /•(/'), for their relative discriminants are relatively prime to
each other. Hence we get

CE = CE. (25)

If 21=2 or 3, the same reasoning can be applied, provided that
p' be not the highest power of 2 or 3 contained in m — l.

If p' be the highest power of 2 or 3 contained in w— 1, we
must take an elementary corpus V, YII or III, VI in place of E,
according as p= 2 or 3. Then the relative discriminant of ^con-
tains the factor 2. It may happen, therefore, that tlie relative
discriminant of C is not relatively prime to that of E. Neverthe-
less, it i> still valid that C and E cannot have a common divisor
other than Z'{/'). For, if there be a common divi^oi'. it must be a
relatively cyclic corpus, whose relative discriminant is a power of
2. But. it is evident that none of the corpora V, VII, III, VI
contains such a divisor. Thus again Ave arrive at (25).

The relative discriminant of C contains all or a part of the
prime factors in the relative dircriminant of 6/„ with the exception
of /^.

If the divisor 6'/, of relative degree ]?, of C still contain a
prime f'-' relatively prime to ^7, then apply the above process once
more to C to get rid of this factor y'. In this way, repeating

On the Relatively Abelian Corpora. 49

this process a number of times, we sliall arrive at a corpus C^, of
relative degree ^/ (ji^h), the relative discriminant of whose divisor
Öl, of relative degree p, docs no longer contain a prime relatively
prime to p.

§. 15.

It was proved by Hubert" that every cyclic corpus, whose
degree is an odd prime ?«, and whose discriminant does not contain
any other prime than u, must necessarily be a kreishörpcr. Fol-
lowing his example, we can now prove tliat the corpus 6\ of rela-
tive degree p is nothing else but an elementary corpus VIII, IX or
X, according as/7>-3, p=3 or^) = 2.

In the case p>'), the proof can be effected without any
difficulty by Hilbert's method"'. As for the method, the reader is
referred to his original work, as it will occupy too much space to
reproduce it here.

Next we have to piove that the relatively cyclic corpora of
relative degree 3, whose relative discriminants are powers of 1 + 2/*,
are exhausted l)y the four relativel}^ culiic ones of the elementary
corpus IX.

From what has been shewn in n^. lU and §. 11, it follows that
the relatively cyclic corpus obtained by adjoining

(1 + 2/0^

to /'{/') is of relative degree 0, and that this corpus contains four
different relatively cubic divisors, Avhose relative discriminants are
powers of 1 + 2/'.

But, on the other hand, it can be shewn that there are no
more relatively cubic corpora, whose relative discriminants are
powers of l + 2/>, tlian the above four corpora. For, since the
fundamental corpus lie) contains the primitive cuIjo roots of unit3^
every relatively cyclic corpus of the third degree can be obtained
from k{l>) by adjoining to it a cube root of a certain integer, say «,

1) Hilbert: loc. cit., §. 103.

2) Also cf. Takagi : loc. cit., §, 13 and §. 14.

gQ Art. 5.— T. Taken ouchi:

properly chosen in k{py'. Without losing generality, we may
suppose that this integer a does not contain such factors that are
perfect cubes in k(t^). Then, any prime factor in a, since it be-
comes a cube in Äl^^v^«), must necessarily enter into the relative
discriminant of this corpus. Hence there can be no other relative-
ly cubic corpora, whose relative discriminants are powers of l + ^o,
than the following ones:

E(^f>"(l + 2py), a,, h = 0,1/2 (mod. 3).

Here the combination « = ^» = 0 should of course be aA'oided ; also it
is to be observed that the two pairs of values (a, b) and (— «, —b)
give rise to one and the same corpus. It follows therefore that
there can be at most four different corpora of this kind, o. E. D.

Lastly, let us consider the case p—l. From §.12 we see that
tlio relative corpus defined by ?^("^) i^ oi relative degree 8, and
contains three different relatively quadratic divisors, whose relative
discriminants are powers of 2. On the other hand, it is evident
that there can be only three relatively quadratic corpora, whose
relative discriminants are powers of 2 ; viz. Al^y*), Jv(V'2), Jv{\^i).
Our proposition is thus proved.

The case h=l having been finished, we can now prove, by
mathematical induction, that C,;, h>l, is a divisor of a corpus
composed of elementary corpora only.

The theorem is true in the case h=l. Suppose that it is true
for all Gz, Jkzu, '?^ being a certain natural number ; and consequent-
ly that it is also true for all relatively cyclic corpora of relative
degree jV', not necessarily of the kind C.

Now, let Ml, denote an elementary corpus VIIL IX or X of
relative degree p*^, according as p>3, ^9=3 or p=2. Since the
divisor C^ of C„ must be identical with a certain J^i contained in JSn,
we can find such a relatively cyclic corpus I) that ,

L/n-tiln =^ U 111 „ ,

1) Hubert : loc. cit., §. 101.

On the Relatively Abelian Corpora. Kl

ami sucli tliiit its relative degree is less than ]f.'^ Therefore D, and
consequently also C„, must be contained in a corpus composed of
elementary corpora only.

PART II.

§. 10.

Let ji^nip) be a class invariant, m being a natural number. By
adjoining J(wo) to lc{r\ we obtain a relative corpus called order-
corpus (Onlnuiigs-korper). Order-corpora are often called class-corjiora
(Jvlassenkörjm^). But, to avoid confusion, we shall never use the
^vord class-corpus in this sense. As for the définition of class-corpus
used in the following, the reader is referred to Weber's Lchrhuch
der Algebra, Vol. Ill, §. 104.

Tlie order-corpus K{j{mp)) is relatively Abelian witJi respect
to /•(;'), and its relative degree is equal to the number of classes of
the order [w] {Ordnung mit dem Fährer m). Hence, if the decom^
position of m into distinct natural primes be

))l — nh\ ph. f)h

•'- 1 -'- '2 -'- i

the relative degree is

h ^ Lii pin~\(p -^i ll

y,

3 ^^ V' \3
It is known tliat the irreducible equation of the Ath degree,
wliich gives J{m;>) as a root, becomes reducible, when we adjoin

( — 1) '-^ pi, i =\,1, , supposmg that p^^'I,

a/— 1' if m = 0 (mod. 4),

V—l , V'2 , if m = 0 (mod. 8),

to tile fundamental corpus k{p)r' But, no further reduction of the
equation can take place, whatever roots of unit}^ may be adjoined

1) Takagi : loc. cit., §. 9.
Hilbert : loc. cit., §. 103.

2) Fueter : Der Klasseukörper dor quadratischen Körper etc. (Dissertation • Güttiu'i-en

1903). ' '' '

Weber : Algebra 111., §. 13S.

52 -'^'"t« ^—'i-'- Takenouclii :

to the fundamental corpus'^ Hence, if a primitive 7nih root of
miity, in terms of which all the square roots above given can l)e
rationally expressed, be adjoined to the order-corpus JC(jivif>)),
then the resultant corpus must be of relative degree

Avhere r denotes the number of distinct odd primes in w, and

.9 = 0, if VI ^0 (mod. 4),

5 = 1, if 7n = 4: (mod. 8),

s = 2, if 7)1=0 (mod. 8).

This corpus of the ITth relative degree is relatively Abelian with
respect to Ii{p). We shall denote it by the symbol Ä"[wi].

It is known that tlie corpus Ä^[7?i] is the class-corpus corre-
sponding to the group of numbers «, which belong to tlie order
[??^] . and satisfy the condition

Ilia) = 1 (mod. 77i),

all the prime factors of ni l^eing taken in the excludent.-' Such a
group of numbers is called a straJd by E. Fueter^*. Hence the
corpus Ä"[w?] may well be called a strahl-corpus.

Weber concluded tliat his so-called division-corpus, i.e. the
corpus obtained by adjoining to k(p) the numbers m and >S|-^I,
where

ß being an integer in k{('). can always l;)e looked upon as a divisor
of a certain s^7'a/?/-corpus**.

It seems to me, however, that there are two defects in his
proof. The one is the misconjecture that his division-corpus for

1) Weber: Algebra III, §. liJ8.

2) Weber : Algebra III, §. 16S.

3) Fneter : Crelle's Journal, Vols. 130, 132 ; Math. Ann., Vol. 75.

4) Weber : Algebra III, §. 169.

Og the Relatively Abelian Corpora, §3

any composite value of ,« can ahva^ys ])e composed of those for
Avhicli /^"s are powers of single primes' \ The other is the confusion
of terminology. His proof that the division-corpus is the class-
corpus corresponding to the group of iiuml)ers «, such that

« = 1 (mod. n),

is correct only wlien the word d/vision-corpus means

^<Hf))

Avhile he used the word to mean liis so-called division-corpus
mentioned above'\

Recently Fueter has given a simple example^\ which denies
Weber's conclusion. However, as for the relation between strahl-
corpora and division-corpora, there still remains a great deal of
obscurity.

Let us now begin our own investigation upon this point.

§. 17.

Let - be an odd prime of the first degree in 1c{p), which is
relatively prime to o, and such tliat

- = a + ho, h = 0 (mod. 2),

a and h being rational integers. Tlien as we liave seen in §. 4,

„ _ _ ^lc"-'^x'' + Äp..2.x''---\- -^-A^x^ + B-x

^<inuii -^.^j^p-ij.p-i_^j^^^p-6^ +o^_,^^'+l '

where x = snw, /> = n(-),

a+'>-l _
;r = I.,, £ = (-1) -• ^n^-'^.

The coefficients ^4' s and X>"s are all integers in k{(\ l), and divisible
by ::. Squaring l)oth sides of this formula, we obtain

'^ " ;rV-<^-i) + (7^_,a;--'("-^> + '+C,x'+V

i) Weber : Algebra III, §. 158.

2) Weber: Algebra III, §. 167. In his original paper (Math. Ann., Vol. 00, p. 22), ^ve
find a necessary precaution against this point.
•è) Fueter : Math. Ann., Vol. 75.

54 Art. 5.— T. Takenouchi :

where Bi< and (7s are integers in k(p), and divi8i)>le Ijy r. Intro-
ducing the relation

r, r- =r>

and also remembering tliat

(p-l)"-' = l (mod.-).

we find, after a few steps of transformation, that

(,-n^(.„) = -^^^;^f+'?. (20)

where (^ and it are rational integral functions of ^-(i(), the coefficients
being all integers in k(f>) and divisilde 1>y -.
On the other hand, if we put

■we get, from §, 1 and §. 2, the formula of the form

the coefficients a' s and the a' s being integers in lip).

We have alread}^ shewn (§. 3) that the <^'s are all divisible
by -. XoAV, comparing (27) with (26), we see that the «'s must
also be divisible by r. It follows therefore that

zijziif = zi'ufi' (mod. -). (28)

Hitherto we have supposed that ô is even. But, if it be odd, we

may use [>- or [>-- instead of - ; for

f>- = -h-{-{a-b)o,

f>-- = —a + b— ap,

and either a— h or a must certainly be even. Therefore the relation
(28) holds good, even if b be odd.

This premised, we are going to shew that, if p be an integer
in Ä(,o), and

O)

U =

On the Eelatively Abelian Corpora. 55

then the relative corpus determined by r(z/)-\ i.e. by ^-(w)', i^ the
class-corpus corresponding to the' group A of numbers «, such that

a = ±1, ±f>, ±f>- (mod. //),

Avhere tlie excludcnt consists of all the prime factors of i'-.

Tlie relative degree of K{r{uf) is not higher than '^^, where
m is the norm of /A Hence, in proving the above statement, no
generality will be lost, if we suppose that m>l. Under this sup-
position it can easily be verified that the group A does not contain
a divisor of 3,

As shewn in §.3, if

/i=^7r'" or (1 + 2/0"'",

Tz' being a prime in Tiip), then the reciprocal of z{n) is an algebraic
integer. In all other cases z{u) is itself an algebraic integer. At
any rate, it follows from (27) or (28) that, if t be a primitive integer
of the corpus 7vXr(?/?), and p any prime ideal factor of - in the
same corpus, we get

t = P (mod. v), ('29)

supposing that ~ belongs to the group A. If - be a prime, which
is relatively prime to the cxcludent, but does not belong to A,
then the congruence (20) does not hold. For, since the discrimi-
nant of the equation for r{uf is not divisible by p (§. 5), we get

z{t(?rP = ziziif ^ Ti^if (mod. p).

Hence we see that tlie congruence (20) holds good, when and
only when p is a prime ideal factor of -, where - is a prime integer
of the first degree in ^-(,0) belonging to the group A. But, this is
the very condition, necessary and sufticient, in order that the
corpus K(j{u>f') may be the class-corpus corresponding to the group
A. Our statement is thus proved.

Now, in general, if G be any group of numl)ers in an imagi-
nary quardratic corpus, and E the group of algel)raic unities in the
same corpus, then the class-corpus corresponding. to G is identical
with that Avhich corresponds to EG'\ Hence, for the sake of

1) Weber : Algebra III, §. 167.

56 Art. 5. —T. Takenouclii :

simplicity, we shall hereafter characterise the group Ä, to which
the class-corpus K{z{iif) corresponds, by the congruence

« = 1 (mod. ij).

It is also known^', tliat the relative degree of the class-corpus
corresponding to the group G is not lower than

~ {EG, G) '

where 0 is the group of all the numbers in Jc{p) (of course, taking
the excludent into consideration), and (O, G) and {EG, G) denote
the indices of G with respect to 0 and EG respectively. If we put
G:=A, then

supposing that A« is associated neither with 2 nor with l+2/>. This
value of d coincides with the degree of the equation for ~{nf.
Therefore the relative degree of K(j{iif) must l^e exactly equal to

^(fi'j). AVhen y- is associated with 2 or 1-1- 2/^, the relative degree

reduces itself to unitv-

Not only /v(~(»)0' ^^^^ '^^^^ other division-corpora, e.g. K{^{ii)),
K(finu), etc., can be looked upon as class-corpora corresj^onding to
some groups of numbers. But, as it is not at all necessary for our
subsequent investigations, and moreover, as it will occupy too
much space, we shall not here enter into a detailed discussion of
them. However, it will not be out of place to give here the results
I liave arrived at.

1) Weber : Algebra III, §. 167.

On the Eelatively Abelian Corpora.

57

Class-corpus

Group of Nurabers

Ex-
cludent

Relative Degree

i-j)

{a = a + bp)

K(T(u.f)=K{^iuf)

a = 1 (mod. //.)

r-

ifW

K(T(l0))

« = 1 (mod. fi), b = 0 (mod. 3)

3//

fv^(/^)» when //^O (mod. 3)
[-^^(/i), when fi = 0 (mod. 3)

K{^(u)) = K(sirii)

a = 1 (mod. u), 5 = 0 (mod. 2)

2//

^^ ^(/-/), when //.^ 0 (mod. 2)
[-^ (p(/jt), when //. = 0 (mod. 2)

((f(,u), when /< ^ 0 (mod. 2)

K{>{, sn-«)

a = 1 (mod. //), /j = 0 (mod. 4)

2//,

1 4"^C")' when fi = 2 (mod. 4)

(Weber's

division-corpus)

Ä'(snw)

Z^ = 0 (mod. 2),
« = 1 (mod. /i),

^ ' a + Z^ = l(mod.4)

2/.

(^4f (/>«), when tJL = 0 (mod. 4)

i 18.

Hereafter, for the sake of simplicity, we shall denote the
corpora A'(sn^}), k{^{^^)), A'(lf(^/) by SO'), A». PXt') re-
spectively.

As shewn in the preceding section, F^m) is the class-corpus
corresponding to the group ^1 of numbers a, such that

« = 1 (mod. m).

On the other hand, l^lm] is the class-corpus corresponding to the
group B of numbers ß, such that all these numbers ß belong to the
order [î^i^] and satisfy the condition

7i(ß) = 1 (mod. 7n).

Now, if we suppose that

a = a + bp = 1 (mod. m),

58 Art. 5.— T. Takenouchi :

a and h being rational integers, then taking the conjugate number,
Ave get

«'= a + hrr = 1 (mod. m).

Therefore

7?(«) = aa' = 1 1

'mod. m).

I

a — o!— hiji — o-) = 0

From tlie latter congruence, it follows that h must Ije divisible by
m. Hence we see that all the numbers of Ä arc included in i> ;
syml^olically we may write

B ^ A,

the sign > standing for "includes, as a proper divisor." There-
fore it follows tliat

Let //' be the relative degree of P\m) with respect to Txif). In
general, H' is equal to '^^(f{m)^\ m being regarded as an integer in
}c{f>), not as a natural number. Hence, if

m = ;)''! 9)''-' p^'i ,

1 ;.' ■'- i

wdicre ])i, p-2, , Ph are distinct natural primes, then

the meanings of r and s are as in §. 10. The only exceptional case
is when w«=2, in which case we have

Hence the equality

P\m) = K\_m']

holds only in the following cases:

(i) î' = l, s = 0; i.e. m =//', 2) being an odd prime,
(ii) r = 0, s = l\ i.e. m=^,
(iii) 7» = 2.
NoAV, since the class-in\'ariant ji^co) can always be rationally

1) Weber ; Algebra III, §. 154.

On the Eelativelj Abelian Corpora. 59

expressed by j^nw), n 1 )eing any natural number, we get

P\m) ^ Jv[7;il ^ A'[/>Ji] K\_pl^ Kip]"] (30)

Avhere . -^^^C^jfO = P\pl'')^ ^i general,

the only exception being

P''(2") > 7v[-2"], 7/>2.

§. 19.

Before proceeding further, let us insert here a few preliminary
considerations.

I. Let Ci, Co, , Q, be absolutely or relatively cyclic

corpora of degrees |/'i, p^-, , ^/"* respectively, p being a prime.

We suppose that these corpora are independent of one another,
i.e., we suppose that none of these corpora has common di^'isors
with the corpus composed of all the others.

If we compose these cyclic corpora all together, we get an

Abelian corpus of degree j??''i+"- + +"''-, which we" shall call A.

The Galois' group of ^1 must be of the form

s? s;' s:' 5^ c, = o, 1,2, ,pai-^i.

{i - 1,2, ,//)

Here >Si denotes a substitution, by which all the numbers in Ci are
changed into their conjugate numbers, while those of other C's
remain unchanged. We may express this fact by saying that the
cor})us

CiC.2 C',_i C(.Li Cj,

IkIoikjs to the cjx-lic group

S'^ , r,= 0,l,2, ,pn-K

Now, let C- be the divisor of G of degi'ee p'^'"^, then the
Abelian cordons

c[a c; d

belongs to the groujD of the form

>sr^f^ ^f ^':\ si=sr" ^-^1^0^...^,^

c,^0,\,2,.-,p-\,

60

Art. 5.— T. Takenouclii ;

All the cyclic subgroups of this Abclian group are of the form

(Är''S'f-' ST' s',f"r, c=o,i,2, ,p-i,

Avhere a^,(u, ,«i, ,«•,, are some fixed integers for each sub-
group. Each of these a's can take p values: 0,1/2, ,2^ — ^.

Hence there are (j^ — 1)''"^ special cyclic subgroups, in which all of
these a' s are different from zero. If Ave denote by ^ a divisor
of A belonging to one of these special cyclic subgroups, then B
must possess the following properties :

(i) A>B^ C[C', C',,

(ii) A is of thepth relative degree with respect to B,

(iii) none of the original cychc corpora C'l, C., , C,, is

contained in B completely.

It can. easily be seen that these three properties are characteristic
of B.

Let us call B a derived corpvs of C\, C., , C,,. There are

in all (i)-l)''"^ different derived corpora. In particular, when 79=2,
the derived corpus is uniquely determinate.

II. Let there be systems of cyclic corpora

C„C,, , A

A, A.

E,,E,,

<ph, qd'2,

)of degrees^ '•'^' ^■'''-'

r'l, v%

J \

respectively, where p, cp , r, 11, v, are all primes. Suppose

that these corpora are independent of one another, and that

21, q, , r are all different from one another, but not necessarily

all different from u, v, Composing these corpora all together,

Ave obtain an Abelian corpus, which Ave shall call A.
Now, let

C-i, 0-2, , ^-h, A, , Jl^i, Jli-j,

On the Eelatively Abelian Corpora. Q\

be tlie divisors of

of degrees

^;ci-i,_??c.'-ï, ,g''i-i, gf?j-i, , ,r^i-],r"--'-i,

respective]}- ; and let B be a divisor of A, snch that

(i) A^B^ac;, A'-D^. EiE!, S,S, 1\T, ,

( ii ) the relative degree of A with i-espect to B is pq ?%

(iii) B does not contain any one of Cj, C.,, , D^.D., ,

,E^,E.,, completely.

Then it can easily- be seen that this corpus B must be composed of
the following corpora:

a derived corpus of C^,C.., ,

a derived corpus of Di,D.,, ,

a derived corpus of E^, E.,

S„ S,, ,

T„ r.., • ,

§. 20.

Let 7)1 be a natural number, such that

VI = r:;'i -«^

where -j,-.,, are distinct primes in the corpus l(f>). Tlien

Ä(m)-,5'(-^)5'(;7^-->)

The relative degree of 'S'(-^'), i = 1,2, , is

ç'(-^''), when 77^ is odd,

y(^')' when - = 2.

Therefore the relative degree of >S''(w?.) is

f (?»), when 7u is odd,

^(f(i)i), when 771 is even.

62

Art. 5. — T, Takenouchi

Now, it is evident that

S(m) ^ P(m) ^ P\m),

and fS(^vi) is at most relatively quadratic with respect to iXm), and
JP(m) at most relatively cubic with respect to P'^{m). l]ut the rela-
tive degree of P\m) is in general \(f{m), the only exceptional case
being when î«— '2. Hence, Avhen m is odd, the relative degree of
F{m) must be ],(f{m). AMien m is even, the relative degree of P(?«)
must be either \(p{m) or \(p{m). ]jut, since l\m) cannot be relative-
ly c[uadratic with respect to FXm). the value y^{w) is inadmissible.
Hence F^n) is of relative degree \(f{m), and consequently coincides
with F''{m), provided that m be even.

Between the three kinds of division-corpora ^S', F, P", theres
exist the relations:

S{m) ^ P{m) ^ P(-;'>)P(<''-') , i (31)

If Ave decompose >y(-fO, i = 1, '2, , into elementary corpora, we

get the con:!ponents of the folloAvin'g types:

(i) I„ II„ III,, IV„ V„ when -,=#=l + iV. -,=^2, a^^l,

VIII,,
(ii) VII,
VI,
.IX,
(iii) X,

when -, — A+ 2//

when -,=2,

«i>l,

(:m>2

(32)

i'he sufiix I is attached to the components in (i) to make clear
their dependency upon - .

The corpus F{-'-''), -/. being an odd prime, can be decomposed
into two components (cf. §. 9.). Tlio one is of relative degree
^f('-), and is no other tlian F^-i); the other is of relative degree
y^(~i)'''~\ <^ii^^ i^^ relative discriminant is a power of - . The former
is a divisor of /S'C-,); and the latter is a corpus composed of VIIL or
IX or X. Hence, if avc decompose P«0 into elementary corpora,
Ave get :

On the Relatively Abelian Corpora.

63

when -,=^l + 2/>, -=^-2, (^^^1, ^
when -=1 + 2;^, (ti^^l,

when -=2, rti>*l, >

> (33)

(i) I,, II,, III,, IV,,

VIIL,
(ii) VI,

IX,
(iii) X.

These promised, let us now consider t]ie relation between
Pi^iii) and the elementary corpora. In virtue of (31), we see that
the elementary corpora given in (33) are certainly contained in
P(m). But, as for the corpora Vi and VII, which are contained in
(32), l)ut not in (33), we have to make a further investigation.

To begin with, suppose that m is odd. In this case, both V«
and VII can never be contained in l\m). For, if Vi be contained

in P{_)n), tJie number sn(^-J, and consequently also ü^Y _,, V must

be contained in P{;m). Then, from the formula

^i'w + r)— ^-(ii — v

it follows that F^jii) must contain the number ^'i^.i:, ), and

consequently also sn( _;(., j. But, in terms of this last number,

all the numbers snf^^V i = 2, 8 , can be rationally ex^n'essed.

It follows, therefore, that Pi^m) contains all the corpora V»,

i = l,2, , and VII. Then P{:m) w^ould coincide with S{m)\

wdiich is impossible. Similar I'casoning applies also to Vo. ^ ;•„

and VII.

Hence, making use of the lemma given in the last section,
we see that the corpus P{m) must be composed of the following
components:

(i) I., II„ III,, IV.,

VIII,,
(ii) VI,
IX,

i — 1,2, , excepting the value, ''

for which -^=1+2/?,
when m^O (mod. -j),
when 1)1 = 0 (mod. 3),
when m = 0 (mod. 3-),
(iii) the derived corpus of V, (/ = 1 ,2, ■ • • ; V,= VII, if -t = 1 + 2o).^

)m

64 Art. 5. —T. Takenoucbi :

If m be even, but not divisible by 4, tbe relative degree of 7^^^
is equal to that of F{ '^^ \. Hence

p(,„) = p(^) .

all the components of which are given in (34).

Next, let m be divisible by 4. In this case, one or more X's
ma}^ be contained in P{in). Composing these X's all together, Ave
obtain a relatively Abelian corpus, which we shall call X^,. Simi-
larh', composing all the X's contained in Â'(m), we obtain an Abelian
corpus X.v. It is evident that X^, is a divisor of X,. But, X^ can
never coincide with X,. For, if X^,=X,. then P{m) must contain

sn(-^ä), where 2' is the highest power of; 2 contained in m. That

this is impossible can be shewn by exactly the same reasoning as
we have done for Vi and VII in tlie above.

It must be remarked, however, that the same proof does no
longer hold for V, and VII in the present case, since the assumption

that -o, -o, are all odd is essential in that proof. In fact, both

Vf and VII are completely contained in P(m). For, since X^,<:X,,
the relative degi'ee of X_,, is at most half that of X,. But, since
S{m) is relatively quadratic with respect to P{iii), it follows that
the relative degree of X^, must be exactly half that of X„ and all
the other elementary corpora in S{'ni) must be completely contained
in P{in). Thus Ave see that P{)n) is composed of

(i) I^, Hi, III^, IVi, Vi, ^=:l,2, , excepting the values,

for which ^^ = 1+2,0 or 2,

VIII;, when vi = 0 (mod. t:?),

(li) VI, VII, when m = 0 (mod. 3),

IX, when m^O (mod. 3-),

(iii) X.

Now, from (o4) and (ou), the following conclusions can be
deduced.

All the elementary corpora arc included in the division-corpora
P{m), if m be made to assume all the positive integral values j or at

W35)

On the E-elatively Abelian Corpora. 65

least all tJie integers of the form 4];)^ where j) is a natural j)t'hne (JJk
case i')=:2 he'imj included).

Therefore all the relativehj Abelian covpora tvith resjiect to h{j>)
are exhausted by the division-corpora PyVi), and consefjnenth/ also In/
Weber'' s division-corj^ora.

When ni is odd, tlio relative discriminant of P'\m) does not
contain the factor 2 (§. -")). Therefore PX/n) cannot contain tlie
elementary corpora III. V, VI, VII. Hence we see that P\rn).
m being odd, is composed of

(i) I,, IIj, IV, / = 1, 2, , excepting the vahie,

for which -^ = 1+2//,
VIII,, ^Yhen m = 0 (mod.-;),

(ii) IX, when m=0 (mod. 3'-'),

(iii) a derived corpus of III,- (/ = 1,2, ; III,=YI, if - = 1 +2o),

(iv) the derived corpus of Y,- ( / = 1, 2, ; Y, = VII, if -,= 1 + 1f>).

Bnt, as we l]a^'e remarked before, the corpns P\m). when
m is even, coincides with P{iii). whose constitution is given in (34)
and (35).

Therefore we conclude that

the relativehj Abelian corpora with respect to k(p) are exhausted
also by the diivision-corpora P\i)i).

§. 21.

Let us now proceed a step further and consider the constitu-
tion of the s^ra/^ /-corpus K\i){\.
As we have shewn in ^. 1<S,

when ^) is an odd prime, or wrien^9 = 2, h = \, 2. In otlier cases, in
which j9=2, /i>2, the corpus PX'2!') is relatively quadratic with
respect to 7\'[2''].

Hence, if we remember the relation (30) and the value of
the relative degree 7/, the decomposition of K\_ni^ into elementary
corpora can be effected by §. 19 without any difficulty.

66 -^rt. 5. -T. Takenonchi :

AVheii III is odd, the components are

( i ) II, II;, IVi, 1 = 1, -2, , excepting the value,

for which -^ = I + In,
VIII;, when ;;i = 0 (mod. -f),
(n) IX, when m = 0 (mod. 3-),
(iii) a derived corpus of III; (i = 1 , 2, ; III, = YI, if .- = 1 + 2/>).

AVlien m is even, Init not divisible ])y 4, we get
( i ) (ii) as above,

(iii) III,, i = 1 , 2, , excepting the values,

for which r:i=\L or 1+2/v,
YI, when m = Q (mod. 8).

AVlien III divisible by 4, we get
( i ) (ii) (iii) as above,
(iv) X.

Observe that, in these lists, we miss two kinds of elementary
corpora, viz. V and VII.

Bnt, is it not possible that the corpus V or VII is contained
in the whole corpus K\_iii] ? We are now going to shew that it is
impossible. AVe shall, however, confine ourselves to the proof that
Vi is not contained in K{m). I'he same reasoning also applies to
other V's and VII.

Decompose K[;m'\ into elementary corpora as al)Ove. Then,
reject IVi from the components thus obtained, and recompose all
the rest. We oI)tai]i a relatively Abelian corpus, which we shall
call K'. Now, after the rejection of IVi from the components of
K\jn\, there may be still some elementary corpora, whose relative

degrees are powers of 2; viz. IV,-, 'i=2,3, , or some X's, if m be

even. Composing these elementary corpora all together, we obtain
a divisor of K' . We sliall call it K" . Then, every divisor of K'
having a power of 2 as its relative degree must necessarily l)e a
divisor of A'"; consequently its relative discriminant cannot con-
tain the factor -,. It follows therefore that K' is relatively prime
to Vi. Hence, if we compose Vi with K\ Ave obtain a relatively

On the Relatively Abelian Corpora. 67

Abelian corpus, whose relative degree is evidently twicy that of
Jvlm]. Therefore

YJ^[m] := \JC > K[i)i],

which shews tliat Vi is not a divisor of K[ui\.

Thus we are led to the following important conclusions :

Relatlvchj Ahcllan covpora wlfli, ra^pcct to h{i>) are not exhausted
Inj strald-coiyora.

If l^ he a)i odd 'prime In lc{p)^ m its norm, and if 2'' he the highest
power of 2 contained in m—1, then tJte relatively cyclic corpus of rela-
tive degree 2'' contained in the division- corpus SÇf-'-) can never he con-
tained in a strahl-corjius.

Now, in §. 14 and §. 15, we have given a method of finding
a corpus composed of elementary corpora only, such that any given
relatively Al:)elian corpus may he a divisor of it. In that method,
V and VII are used only in the case when the relative degree of
the given corpus is even. Hence we see that

relatively Ahelian corjyora of odd relative degree with, respect to
kÇ") arc exhausted hy strahl-corpora.

I '1-2.

According to Fueter, " all the prime factors of tlie relative
discriminant of K\_m\ must be contained in m; and conversely, all
the |)rime factors of m must, in general, be contained in the rela-
tive discriminant of A"[m]. The only exceptional case is when m
is divisible by 3, Init not by 3"; in this case, tlie relative discrimi-
nant of 7l'[7;^] does not contain the factor 3.

But, I think, this is not quite correct. In the first place, the
most obvious exceptional case m = -l is not noticed in the above
statement. Secondly, his so-called exceptional case is not neces-
sarily exceptional. For example, take the value m=6, tlien

A^t^) =^A-3 + 3V_3) =j{^~ri).
But, we know that

where x'—?>x- — '^x — l — Op i.e. x =1 + ^2^ + ^4-

1) Pneter : Math. Ann., Vol. 75.

2) Weber : Alg-ebra III, p. 722.

53 Art. 5.— T. Takenonchi:

Hence Ä"[6] is nothing else l)ut the elementary corpns \I, whose
relative discriminant is divisihle by 3.

Let us now determine the nature of the relative discriminant
of Ä"[??2] , as a sequel to our preceding investigations.

If Ave assume, as in §. 18, that

VI =^y'i2)^'-i jj^'i

then we liave

K[iJi:]^K[p^qK[p!;q K[p^!q ,

where Z[p''i] = P-'Q^'O, if j;,=^'2 or ^^, = '2, //,r=],'2,

and PX2'*0 > AT2'''] > P%2'''''), if //, > 2.

But, the relative discriminant of F'Xp^-0 is a power of />„ except
when p^^' = 2 or 3. Hence the relative discriminant of 7v"[?«] must
contain all the prime factors of m, except in the following cases :

(i) m=0 (mod. 2), but #;0 (mod. 2-),
(ii) w = 0 (mod. 3), but ^0 (mod. 3-).

On the other hand, since

"\ve see tliat the relative discriminant of J'^lm] docs not contain
prime factors relatively prime to m. Thus, if we exclude the
above tAVO cases (i) and (ii), we may conclude that the relative
discriminant of Ä"[?>?] contains all the prime factors of ??z, but no
other.

In case (i), since w is divisible by 2, the corpus K[7n] con-
tains the elementary corpus HI or VI (§. 21), except when ;«=2.
Therefore its relative discriminant must contain the factor 2. Thus
we see that this case (i) need not be excluded from tlie aboA'c state-
ment, provided that w=f-2.

Next, consider the case (ii), and suppose that ??i=^3. If ^//i be
even, then K[m] contains VI, and consequently its relative dis-
criminant is divisible ])y 3.

If, on the contrary, m be odd, then 7v [?«] does not contain

VI itself, but a derived corpus of III,- (? = 1,2, ,_pi=/:3)and VI.

Hence we have to inquire, whether the relative discriminant of

On the ■Relatively Abolian Corpora. G9

this derived corpus is divisible by 3 or not.

Without losing generahty, we may suppose pi=#=3. Let C be
a relatively cubic divisor of IIIi. There are two different derived
corpora of C and VI. At least one of these derived corpora is
certainly a divisor of Ä"[m] . Let us call it D. If the relative dis-
criminant of K[m\ were relatively prime to 3, so would also be that
of D. Moreover, since m is odd, this relative discriminant cannot
contain the factor 2. Hence D would be a relatively cyclic, cubic
corpus different from C, with the relative discriminant, a power
of pi, provided that the relative discriminant of K[m\ is relatively
prime to 3.

But, this is evidently impossible, if G be a proper divisor of
nil. For, if C does not coincide with IHi, then the relative dis-
criminant of C must be a power of ^i, without containing the
factQr 2, and consequently it can be proved that D cannot be
different from C. '^

If (7=111,, then the relative discriminant of C is divisible by
2. Compose C with D, and consider the corpus of inertia of a
prime ideal contained in p^. This corpus of inertia must be rela-
tively cyclic, and its relative discriminant must be a power of 2.
Also, since D is not identical with C, this corpus of inertia cannot
reduce itself to h{p) (cf. §. M). Therefore it must be relatively
cubic. But, we can shew, as follows, that this is again impossible.

We know that every relatively cyclic, cubic corpus, with
respect to A(/'), can be o])tained by adjoining to k{(>) a cube root of a
certain integer in h{p). Hence, if there be a relatively cyclic corpus
of the third relative degree, whose relative discriminant does not
contain any other prime than 2, it must be one of the forms

In the corpus («), however, the following decomposition holds :

1 + 25 =(l + ^'2£X/^+ '28)(r + ->^2£)

1) Takagi : loc. cit., §. 10.

70 Art. 5.— T, Takonoiichi :

Hence the relative discriminant of this corpus must contain the
factor l + 2e. The relative discriminant of the corpus (b) does not
contain the factor 2, since the discriminant of the equation

is — 27£-, Thus we see that there can be no relatively cyclic, cubic
corpus, whose relative discriminant is a power of 2, Q. E. D.

The relative discriminant of K[m\ must therefore be divisible
by 3, when m is divisible by 3, provided that m^d>.

Thus we arrive at the following conclusion.

All the prime factors of the relative disc7'iminant of Kirn] are
contained in m; and conversely, all the prime factors of m enter into
the relative discriminant of K[m\, except when m—2 or 3.

In the two exceptional cases, Ä"[w] coincides with the funda-
mental corpus Ti{p).

Nagoya, November 1915.

Published March SOth, 1916.

JOUENAL OF THE COLLEGE OF SCIENCE, IMPEKIAL UNIVERSITY, TOKYO.

VOL. XXXVII., ARTICLE 6.

Numerical Calculation of the Jacobian Ellipsoids.

By

Ryosuke KAIBARA, Hi<jaktishi.

Professor of Physics, Imperial Peers' College, Tohtjo.

1. The Jacobian ellipsoid, or the ellipsoidal figure of equili-
brium of a rotating mass of homogeneous fluid, was first calculated
numerically by Darwin" for the whole range of the ratios of axes
and angular velocity. In a previous paper, -^ I solved the same
problem by a different method of treatment, and then interpolated
values which I intended to compare with those given in Darwin's
table. In his Scientific Fajier^^^ Darwin has recomputed a part of
his table, and pointed out the discrepancy between his results and
mine; he has also given an evidence for the incorrectness of some
of the interpolated values. The discrepancy is, in general, very
small, but in a few of the results, it amounts to several units of the
third decimal place. These considerable discrepancies occur in those
values, for which, by the nature of the method I adopted, the
calculation was the most tedious, and an inaccuracy in the inter-
polations of the order mentioned was therefore unavoidable. It
may be remarked, however, that a majority of my results agree
better with Darwin's corrected values than with his original ones.

More accurate results on this subject being desirable, I have
greatly modified the method so as to remove its imperfection in
the previous paper as far as possible, and have repeated the whole
calculation.^^ In the following paragraphs, this improved method
will be explained and tables of numerical results added.

1) O. H. Darwin, Proc. Roy. Soc. London, 41 (1887), p. 319.

2) R. Kaibara, Proc. Tokyo Math.-Phys. Soc, 4 (1907), p. 98.

3) G. H. Darwin, Scientific Papers, 3 (1910), p. 130.

4) Some of my notations in the former paper are also altered to conform to those usually
adopted.

Art. 6. — R. Kaibara

2. Let «, h, c (a^-b^c) be the semi-axes of the ellipsoid, <t,
its density, and w, the angular velocity (about the axis c); then,
if we write

/•" ds

P = abc —. ., , N ,-'

where

(«- + ,
/•° ds

p ds

R = abc ^T", ^^, '

Û ^

J = V'(«"'' + s) {^' + s)(c~ + s),

the condition for the equilibrium of the ellipsoid is expressed by
the well-known equations:

where

d\P-iè) = b'-(Q-O) = c^B,
'A-a

(1)

From these equations, Ave get

(2)

this represents the relation to be fulfilled by the axes a, h, c.

For the sake of brevity, we shall denote the ratios of axes and
eccentricities of the principal sections thus: —

_ c _^ cr—c'

^ '^ ~ a "^ ~~

a

'■^= b—

V ä- — b'-

(3)

' " a " a

It can be easily proved that P, Q, R satisfy the identities

P+Q + i? = 2, 1

a-P-\-J)'Q-\-rR = S,j
ds

(4)

where S = abc J -^.

' 0

By combining (-1) with (1), various forms of expression can be

Xumerical Calculation of the Jacobian Ellipsoids. 3

obtained for the angular velocit}^, of which the following two will
be utilised hereafter: ^^

1 + f'i + (h- + Pii'i - 4/>i' ' ^'^^

Let li, e, i be the quantities which measure the moment of
momentum il/, the kinetic energy E, and the intrinsic
energy / respectively of the ellipsoid, and which are defined
according to Darwin thus: —

Writing m for the whole mass of the ellipsoid, we have

this we put equal to

5

3. 1

m"- (abc) ^. (X,

SO that /^ = ^(^>.ft) '^(l+piWii. (7)

Again, E = —^^oM = m-(ahc) . e,

where e = ^L^^^^^ij^ (8)

Lastly, if V be the potential of the ellipsoid at an internal
point, and d/\ an element of its volume, then

2.

2./

= --^-o' Ysßv-Flx-dv-QJy-dv-BjzHv\,

or, evaluating the integrals and introducing (4), we find

5

2 _i

1= ——^7:amS = m\ahc) "*. (i — 1),

1) In the previous imper, SL was calculated from an equation deduced from

-C = 1 —{ 1 + p;- + 9r)li.

But I have found that the two forms in the text, though less simple, are more suitable for
obtaininof accurate numerical results.

Art. 6. — R. Xaibara :

where

= 1

10

{piPÙ ''Sa

(9)

The problem now is to reduce equation (2) into a form which
is numerically soluble, and with its sohitions, to determine systems
of values for />' s, t s, ii, //, e, and i.

3. First, the integrals P, Ç, R, iS must be evaluated. Put

a- + S ■= -^'"'(^/r— 6;.),

C- + S = /-(^/ü — P]),
and let s = 0 correspond to w = u, so tliat

then we have

n dw }^'u

P= -^'iiJ^ -

(10)

(ei-e.Ofe-es)

\ +e,u\.

This result may be expressed in terms of t^-f unctions^* with
argument v=îi/2oj, and modulus z—w'jco, thus:

Tj ?5i(2y)

In like manner, we obtain

iy/^;'(iO H ^y. ^,(yj J

and

(11)

^^^ - "^^ .>,(^0^,(^) ^''^•

(12)

By means of (1<))» ^^^^ ratios of axes and eccentricities may be
written

ri =

= /y/C;

au _ d,d,{v)

(13)

1) The notation here adopted for the functions is in accordance with that given in
Weierstrass-Schwarz, Formeln u. Lehrsätze zum Gebrauche d. ellip. Funktionen. Formulae
in that book are used throughout this jjaper.

Pi =

Numerical Calculation of the Jacobian Ellipsoids.

(13)

4. In bringing (2) into a form suitable for our purpose, the
same method as in the previous paper is adopted, which may be
briefly sketched as follows: —

Substituting from (11) and (13), équation (2) becomes, after
reduction

[^Ä"^o(2iO + &:d:ß,{9,v)^ 2r = a;äJ(^v) + âX(2v)-7râ.^â,(2v). (14)

On replacing the ^-functions Ijy their expressions in A-series, it is
immediately seen that each member of the above equation is
divisible by /r; hence a solution of this equation is A=0, from
which it follows that 62=^3 or a=b. This solution determines the
well-known series of ]Maclaurin's spheroids, the consideration of
which, however, is not an aim of this paper.

Dividing out the factor /r on both sides of (14), arranging
the quotients in powers of h, and transposing, we obtain

A, + AJr + ÄJi'+ +A,h'-+ =0, (15)

where ^4' s are known functions of -2' alone.

If, in the last equation, we put ^ = 0, then

A, = (4-cos2^)2,j-16sin^ + 5sin2." = 0, (16)

where ^ = 2-i\

The value of z, sav 2^0, which satisfies (16) corresponds to the
starting point of the Jacobian from the revolutional series, and can
be found, as will be given below, by the method of successive
approximation.

Now, Zf) being known, the general value of 0, which is defined
by (15) as a function of h, can be expanded in the neighbourhood
of A = 0, thus: —

z = Z(, + aji- + aJi' + + aji'- + (17)

The numerical coefficients 2^0, «1, «2, , «5 were recomputed and

«6 was newly found for the present paper, the result being

6 Art. 6.— E. Kaibara :

;^o= +1-8974390607

a, = -0-l673'2092

a., = -0-4747820

a. = +0-380062

a,= +0-40908

a-^ = —0-3475

«e = -0-894.

By means of the equation just obtained, the vahie of z can be
determined for an assigned vahie of h, provided the latter is not
near unity.

5. Formula? for jo's and e's given in (13) and thence those
for i? and i given in (5), (6) and (9) can be adapted to numerical
calculation, by quadratic transformation of the ;?-functions and l^y
introducing auxiliary angles.

With a pair of corresponding values of h and z, let the follow-
ing four functions be computed:

d, = 2^Ti{l + ¥ + h''+ ),

/?3 = l + 2,V + 27t«+ ,

d^^v) = 2v/î(cos^ + 7i-'cos3^ + 7t'2cos5,?+ ■■■')
6,(2v) = 1 + 27r cos 2^ + 27i' cos 4,s + ,

where the symbol 6 signifies that the modulus of the function is 2r,
i. e.,

6, = â,iO ! 2r), dX^v) = mv 1 2r), {i = 2, 3).

All these series converge very rapidly; in most cases, the first two
terms of each series will give a sufficiently accurate value of the
function. Next, let angles «, ß l)e determined by the equations

6.

(18)

tail a =

(19)

^3

tan ^y = — — =^^ — — ;
^3(2^)

then, we have the formulée:

ê.? = 16. ß., = ;;^sin2«,

7%^ = di+di = X\ \ (20)

Xumerical Calculation of the Jacobian Ellipsoids

â-'(v) = dßl1v)-d.ß.i:lv) = ;.//cos(«-,9)J

(20)

where

;. = ^

cos a

Hence (13) becomes

., tan (a — ,5")

,«o-

taii'ia

sin(a— y5)
sin 2« cos(« + y9)

cos2acos;^a + ,9)

^/2r)

COS^Î

£2 =, sin(V/ + ^9)

sin 2a cos (a— ^5)

^ o _ tan (a + ß )

tan 2« '

o_ sin2a sin(« + ji5')

cos(«— /9)

) (21)

cos(a — /:^)

From the definition of -v's and e's, it foUows that

/V + V = l, (i = l,2,3);

hence s^ can be found from />,., or vice versa. But, actually, these
six quantities were all calculated independently from (21), using
the above identities as a means of verification.

6. Substituting in (5) the value of S given in (12), and using
(20), (21), we get

îi =

( 1 + 2 f ''- ^^"^ ^"- + ^^^ W 6 f ^'' ^^^ ^'^ + ^^ ^
\ cos '2a J

If, herein, we put

tan^ = — =-

V2o2cos(a + /î)
^ \ cos2.i / '

sm;^ =

•Si

Idß.zp.

■cos^,

then

(> = 3

sin^

£^tan^ tan;/

(22)

(23)

The second form of expression for i?, viz. (0), can be treated
in the same manner: putting

Ai"t. 6. — ïi. Kaibara :

4r//

and defining an angle w by

SlllW =

2^., cos a cos ;9

t^^î'z I ^ ^s(2î')tan'2acos(a— ^5j
it can be j^roved that (6) becomes

J2 =

^it^zp.

tan (l^ cos oi

tan ^'

£.,tanc

(24) ■

(25)

(26)

^•i^3 V^ ^P?, f^S ^-2 f ^ ^ ^i Sill <p

The values of .Q were always found from both (23) and (26).
Since these two equations lead to identical results under the
condition that (17) is satisfied, the comparison of the two values
for JQ thus obtainable, enables us to verify the calculation and
confirm the accuracy of the values of z found from (17).

Lastly, the quantities //, e, and i are determined by

f^ = ^-(ruOÙ'\^+riW^^^ ^

e = ^âLa-^'lJ,

i = l

5 £.

(27)

By the method hitherto explained, the calculation was
carried out with values of h taken in the range from 0 to 0'24 at
equal intervals of 0"02. The result is shown in the first 13 rows of
Table I at the end of this paper. Following the scheme of
Darwin's table, the quantities

a 1 ^

(!\<h)^

{ahcf

h
{ahcf \ /'•-'

^ - -i

(28)

03 = VTTf = ^(''''-"^
{ahcy

are tabulated instead of ,^'1, p-i, ft.

Xumerical Calculation of tlie Jacobian Ellipsoids.

7. As li tends to unity, the foregoing equations become
gradually inconvenient. In tliis case, the ^-functions ^vith
modulus r may be transformed into those with modulus r^ = — r-^,
according to the formuke

(29)

where a- denotes that the modulus of the function is z,. By this
transformation, h goes over to /^j, wliere

lognit// ■ lognat//i = tz'.

"(30)

It must be borne in mind that the values of A,, with which we
shall be concerned hereafter, are exceedingly small, being less than
0-0009920 which corresponds to the extreme value 0-24: of h
referred to al:)ove.

By the modular transformation, (14) becomes

(31)
In this case, an equation analogous to (17) cannot be obtained, for
the argument z^v increases without limit as //•, approaches zero.
We may, however, proceed thus: —

1

Writing

= —'IrziT.v = ^^Icgnat-

h:

m

and substituting for the functions in (31) their respective
/ii-series, this equation may be put into the form

(ÄU+BV)z, = CX+Y,

where U = cosh z, + //,-cosh 3^^ + h.'^cosh 5z, + ,

V =l + 2Ji, cosh 2z, + 2h,' cosh éz, + ,

X = h, sinb 2z, + 1h ,' sinh 4.r, + ,

Y = D ahûxz, + Di/i, sinh 3,?, + DJ/, ^ sinh 5^ +

(33)

(34)

10

Art. 6. — E. Kaibara :

A =-

-ä,^a.^'l(8:zVi,)_

B=-

-âM^'KUzVi;)

C =

m^K)

D =

{â,^-â,^)/8h,

A =

(3^/ + ^o')/8

D,=

(5^:/-^o'.V8

V35)

= 1 + 8h, + Mh,- + 367i,^ + 56/;,' + ,

= 2;i + 13/ir + 42V+ ),

= 8(1 + 5V+10V+ \

= i(6-ll/?, + 90/v-121//,' + 238V----),

= 1(1 + 4Ä, + 517^-10/^^^ + 211 V- ),

= i(l + 16/«, + 29/i;-+170/;,^'-31/ii'+ ).}

Here, tlie values of ^-1, B, Q D, can be determined for a given

value of /h either by calculating â-n, «?3, i^, à-^", ^3", or preferably
by direct evaluation from the series in (35). Then the correspond-
ing value of ^1 is found from (33) by the method of successive
approximation.

For convenience of subsequent interpolations, values were
first assigned to h at the intervals previously chosen, and then the
corresponding values of A, were accurately determined from (30).
A further advantage of this procedure is that equation (17) is still
available, for, from the value of z furnished by this equation, we can
calculate that of z, by (32), which serves as a first approximation.
If h be not ver}^ near to unity, this approximate value of z, is
fairly close to the true one, so that by applying (33) once or twice,
a final approximation of sufficient accuracy is arrived at. When k
is close to unity, the extreme smallness of h, facilitates the work
with (33), and thus the trouble arising from the roughness of the
first approximation is compensated.

8. For the calculation of o's, e's, /i, etc., the process is the
sam.e as before. Let

^, = 2V^,(1 + V+ )'

d^(2T^v) = 2v'//i(cosh^i + /;,^cosh 3,.-, + \

"dj^lr.v) = 1 + 2// ,- cosh 1z, + 2/? ,^ cosh 4.:, + ■ • ■

then the angles a and ß introduced in § 5 are now given by

(36)

tan (45° -r/) = -L£-,

taii(45° + y9) =

(37)

Ximierical Calculation of the Jacobian Ellipsoids. 11

Formula^ similar to (20) can be easily written down for ^'s; then
b.y means of them and (29), it may l)e verified that exactly the
same expressions as (21) hold good for «,, joa, i>z, £i, £2, £3-

Again, the angles (p and (p are determined by the same equa-
tions as in (22), but the equation giving ■/ now transforms into

siny = ( 3-5— jcosc. (38)

Then, expression (23) for /2 also apphes in this case.
As to the second expression for il, put

f7'= _^?^,- = l + 9/^-->25/^/'+ ,

' ^ ' '' ^nih^ 2 ' 2 2

and let a new angle co^ be defined by

(39)

sm CO,

/ 2t,'iT,v) \W 2^,cos(45°-a)cos(45°-/9) U

then we have

Lastly, the expressions for /^ and e given in (27) remain
unchanged, while that for l becomes

3 / IK, \^_dA^

^ = 1—5 177/ ~V"

(42)

The method of §§ 7, 8 w^as applied for A=0"24— 0"6 at
intervals of 0'04; the last 11 row^s of Table I show the result.

9. As an example, let A=0. Here, we may conveniently
proceed from the original equations (5), (6) and (13), which
simplify in this case thus:

,->, = .:>., = COS-^, ft = l.

£, = £., = sin ^, £:; = 0,

12 Art. (5. — R. Kaibara :

and -" z — 7^ :, — '

(l-cos,,,)tan-|^ 5-2COS-.,

where, as was already given,

z,, = 1-89743906 = 108° 42' 54"- 900.
Hence we find

^o^ = IK, = 0-582724'2. ,o, = 1 ,

£, = s, = 0-8126700, £, = 0,
// = 0-1871148.

This set of values determines the ellipsoid of transition referred to
in § 4. Of course, the same result can be obtained from (21), (23),
(2G) by the use of the auxiliary angles a, .3, <p, etc.
Next, take A=0'24. From (J7) we get

z = 1-88630305 = 108' 4' 37"-934 ;
lien ce, from (18),

log d. = T-9925741, log d.i'iv) = T-4791308,
log d, = 0-0473614, log d,(2v) = T-9576028,

and therefore by (19),

a = 41= 23' 44"-00, ^9=18'' 2-2' 53"-59.

Then from (21) we obtain

^;- = 0-0537271, ;r>/ = 0-7828731, /v = 0-0686280,
£^2 == 0-9462730, e^ = 0-2171269, £/ = 0-9313719.

Also, we find

<p = 22° 34' 59"-76, </> = 2° 50' 35"-75,

y = 55" 12' 41"-96, w = 29^ 59' 24"-59.

Hence from (23), -^^ = 0-09485087,

and from (26), -'^ = 0-09485084.

Corresponding to //=0'24, we get from (30),

h, = 0-0009920-201 ;
whence by (32), z^ = 4-1524265.

Xnmerical Calculation of the Jacobian Elliiisoids. ^3

By appljàng (33), it can bo proved that this value of z^ is correct to
the seventh decimal figure, so that the approximation is sufficient,
if the accuracy required for //s, s's, /^, etc. be that attainable by
the use of seven-figure logarithms. Now (37) gives the same
values of « and ß as shown above, and (41) gives

U = 0-09485085.
As the last example, take 7^=0 '(3, to which corresponds

h, = 0-000000004064856.
The first approximation for z^ as given l)y (17) is

z, = 11-047.
B}^ means of (33), we find that the second approximation is

z, = 11-0505,

and that the third is

z, = 11-0505035,

which can be proved to be correct to the sixth decimal.
From (23) and (41), we find respectively

.Q = 0-0005114177,
and /2 = 0-0005114175.

10. To compare the result with that obtained by Darwin' \
Table II was calculated thus: —

In Darwin's method, the auxiliary quantity to which arbitrary
values are attributed is an angle r which is defined in our notation

by

COS/' = p^,

or cosr = -^A^'A = A^oiMl. C43)

From the computed values of ,oi=cosr, the value of Ji corresponding
to an assigned value of r was first interpolated, a correction was
added, when necessary, to this value of h so as to satisfy (43)
accurately, and then, Ijy tlie usual method, the corresponding
values of Oj, J^, Oi and -| i? were interpolated from Table I. The

1) G. H. Darwin, /. c.

14

Art. 6.— E. Kaibara

results are shown in Table II with Darwin's values side by side.
From this table, it will be noticed that the results are in good
agreement, excepting the values of '\ and o.^ for ^-=60'' and

TABLE li.

Auxiliary Quantities

Axes

Anof. Vel.

0

0 02

0-04

006

008

01

012

014

016

0-18

0-2

0-22

0-24

0-28

0-32

0-36

0-4

0-44

0-48

0-52

0-56

0-Ö

1

0-99202015.10-'^
0-4293505110- a
0-17306452. 10- i
0-63756224.10-4
0-20994321.10- i
0-60121544.10-s
014457357.10-5
0-27878215.10-s
0-40503148.10-7
0-40848561.10-3
0

: = 1-89743906
1-89737206
1-89717013
1-89683057
1-89634886
1-89571876
l-8949323i
1-89398010
1-89285104
1-89153282
1-89001190
1-88827376
1-88630305

1 = 4-6436612
5-1717957
5-7462979
6-3781799
7-08098i5
7-8720639
8-7743890
9-8192530
11-0535035
00

{abc)*

1-19723
1-26294
1-33395
1-41091
1-49456
1-58578
1-68554
1-79500
1-91551
2-0i864
2-19626
2-36057
2-54420
2-98276
3 -51.635
4-28736
5-28644
6-67793
8-69008
11-73766
16-62132
25-06353
CO

[abcy

1-197234
1-136253
1-079494
1-026513
0-976915
0-930348
0-886497
0-845082
0-805349
0-768571
0-733042
0-699075
0-666502
0-604938
0-547279
0-492629
0-440321
0-389744
0-340513
0-292410
0-245465
0 199798
0

{abcy

ß =

0-697657
0-696854
0-694449
0-690458
0-684903
0-677818
0-669244
0-659229
0-647831
0-635111
0-621137
0-605982
0-589722
0-554205
0-515239
0-473467
0-429603
0-334220
0-337942
0-291353
0-245105
0-199695
0

2t,(t

0-187115

0-186278

0-183786

0-179694

0-174096

0-167115

0-158905

0-149645

0-139533

0-128780

0-117604

0-106223

0.0948509

0-0729128

0-0531468

00364823

0-0233988

0-0133538

0-0074594

0-0035774

0-0014360

0-0005114

0

Xnmerical Calculation of the Jacobian Ellipsoids.

15

those of IIJ for ;'=80' where the discrepancies are still sensible.
In writing this paper, I owe very ninch to Prof. H. Nagaoka
for his kind advice.

TABLE I2.

h

Eccentricities of Sections

Mom. of
Mom.

Energy

£■ =

a

£._; =

£a =

(J.

Kinetic
e

Intrinsic

i

Total
E

b

a

0

0-812Ö70

0 812070

0

0-30375

0-080461

0-41495

0-49541

0-02

0-833990

0-789857

0-436535

0-30512

0-080042

0-41549

0-49613

004

0-853803

0-765606

0-587468

0-30923

0-081180

0-41712

0-49830

0-06

0-872075

0-739981

0-680048

0-31611

0-082059

0-41983

0-50189

0-08

0-888817

0-713075

0-756800

0-32584

0-083255

0-42364

0-50690

0-1

0-904046

0-681978

0809817

0-33818

0-081733

0-42856

0-51329

0-12

0-917797

0-055805

0-850520

0-35415

0-080450

0-43459

0-52104

014

0-930119

0-625Ö83

0-882241

0-37298

0-088355

0-44174

0-53010

016

0-941073

0-594750

0-907201

0-39514

0-090388

0-45003

0-54042

0-18

0-950732

0-533153

0-920960

0-42084

0-092483

0-45946

0-55194

0-2

0-959174

0-531048

0-942055

0-45032

0-094570

0-47004

0-56461

0-22

0-966488

0-498597

0-955142

0-48387

0-090573

0-48177

0-57834

0-24

0-972766

0-465969

0-905076

0-52182

0-098415

0-49466

0-59307

0-28

0-982587

0-400867

0-979218

0-01260

0-101307

0-52386

0-62516

0-32

0-989390

0-337135

0-988021

0-72711

0-102049

0-55750

0-66015

0-36

0-993883

0-276195

0-993377

0-87135

0-101918

0-595-34

0-69725

0-4

0-990692

0-219295

0-990525

1-05439

0-098707

0-03081

0-73558

0-44

0-998343

0-167767

0-998295

1-29009

0-092987

0-68136

0-77434

0-48

0-999244

0-122049

0-999232

1-00008

0-084028

0-72803

0-81266

0-52

0-999092

0-084752

0-999690

2-01974

0-073977

0-775Ö7

0-84965

0-56

0-999891

0-054504

0-999891

2-00922

0-001594

0-82275

0-88435

0-Ü

0-999908

0-031990

0-999968

3-47997

0-048192

0-80773

0-91592

1

1

0

1

GO

0

1

1

16

Art. 6. — E. Ivaibara :

TABLE II.

Cl

c

-

S3

eß

T

h

Darwin

Kaibara

Darwin

Kaibara

Darwin

Kaibara

Darwin

Kailiara

55°

0-0059319

1-216

1-2162

1-179

1-1787

0-698

' 0-6976

0-0935

0-09352

57°

0-0245838

1-279

1-2787

1-123

1-1229

0-696

0-6964

0-093

0-09293

60°

0-0531895

1-3831

1-3840

1-0454

1-0442

0-6916

0-6920

0-09060

0-09063

65°

0-103149

1-6007

1-6009

0-9235

0-9233

0-6765

0-6766

0-08295

0-08295

70°

0-157340

1-899

18988

0-8111

0-8110

0-6494

0-6494

0-07047

007046

75°

0-218351

2-3i6

2-3i63

0-7019

0-7018

0-6072

0-6073

0-0536

0-05358

80°

0-291436

3-1294

3-1293

0-5881

0-5881

0-5434

0-5434

0-03307

0-03350

85°

0391275

5 0406

5-0407

0-4516

0-4516

0-4393

0-4393

0-01293

0-01297

Published July, 20th, 1916

JOURNAL OF THE COLLEGE OF SCIE^'CE, IMPERIAL UNIVERSITY, TOKYO.

VOL. XXXVII., ARTICLE 7.

On the Elastic Equilibrium of a Semi=Infinite Solid

under given Boundary Conditions, with

some Applications.

By
Kwan-ichi TERAZAWA, JHnakushi,

I. Introduction.

§1. The statical problem concerning an infinite elastic solid
bounded by a plane subjected to a given distributiou of traction or
deformation has attracted the attention of numerous eminent
elasticians. The first solution for the case of a purely normal load
was given by Lame axd Clapeyron^^ by means of Fourier's
theorem, through which an assigned function of two variables is
expressed as a quadruple integral. The credit of first improvement
on this subject may well be claimed by J. Boussinesq,'^ who
introduced several kinds of potentials — direct, inverse and logari-
thmic with three variables — into the theory of elasticity, and
opened a new field of treatment in it. Almost all conceivable
cases have been solved by him, especially in relation to what takes
place at the boundary surface. Besides Boussinesq, many other
authors have touched on this problem, employing the method of
integration by Green's functions. Not long ago, Prof. H. Lamp.^^
solved a special case of this problem, viz. that in which the
boundary condition is a normal pressure symmetrically distributed
about a point on the surface, by making use of the integral theorem
of Fourier's type concerning Bessel function of the zeroth order;
and thus. Lamé and Clapeyron's method, which was considered to

1) Crelle's Journal, vol. 7 (1831) p.p. 400-40i.

2) Application des Potentiels, Paris, (1885;.

3) Loncl. Math. Soc. Proc. vol. 34 (1902) p. 276.

2 Art. 7. — K. Terazawa :

be extremely unsuited for obtaining physical results, seems now to
have gained practical importance.

§2. The present paper deals with the problem in the case in
which the bomidary is subjected to any given normal pressure, by
generalizing the method adopted by Lamb.^^ In the first two
sections the general solution of the problem is obtained in the type
of the Bessel-Fourier expansion of a function. The third section
discusses several examples in the case of symmetry about an axis
normal to the boundary, and forms the main part of this communi-
cation. Most of these special examples have been investigated by
the autliors above cited: the behaviour at the surface especially;
and yet it ]jiay be worth while to discuss them again more closely,
referring especially to the beha^■iour inside the boundary.

The last section is added as an appendix, supplying the
general solutions corresponding to several boundary conditions,
excepting that of normal pressure, in the case of symmetry about
a normal to the boundary.

§3. The results of tliese special examples applied to find the
limit of rupture of a foundation over which a heavy load is
distributed. Strictly speaking, ]jy applying the mathematical
theory of elasticity, we can treat of rupture only, for some kmds of
brittle solids like cast iron, in which tlie linear relation of stress
and strain holds and, moreover, the strains are so small that their
squares are negligible up to the point where rupture takes place.
For a ductile material, such as mild steel, and for an imperfectly
elastic material, like cen:ient or sandstone, we must bear in mind
that the theoretical results indicate only roughly the state of stress
when, in the first case, it begins to take permanent set, and, in
the second case, when it breaks.

§4. Another application will be found in a problem of
geophysics. In his elaborate observations on the lunar deflection
of gravity, Dr. 0. Hecker has pointed out that the force acting on
the pendulum at Potsdam is a large fraction of the moon's force
when it acts towards the east or west than when it acts towards the

1) Though the writer had not read his paper until the work was almost finished.

Elastic Equilibrium of Semi-Infinite Solid. 3^

north or south. ^^ Various explanations of this anomaly have been
proposed, among them one, suggested by Prof. A. E.H. Love,^^
is that a possible cause may perhaps be found in the efïect of the
tide wave in the North Atlantic. Recently Prof. A.A. Michelson^^
has found a similar result in his arduous task of measuring the
lunar perturbation of a very long water-level at Chicago. Prof.
Sir J. Larmor kindly suggested to me a query whether the excess-
pressure of the tide in the North Atlantic would affect much the
measurement of the water-level at Chicago, owing to the depression
of the solid earth that it would produce. A calculation is under-
taken in order to ascertain to wdiat extent the consideration of the
tilting of the ground is important for the explanation of this
geodynamical discrepancy; we may in a first estimation neglect
the curvature of the earth. ^^

II. Solution of Equation of Equilibrium,

§5. The equation of equilibrium of an isotropic elastic body
free from the action of a body force is

curl curl «. = ^ grad A, {\)

where u denotes the displacement, / and !^ Lame's constants which
specify the elastic nature of the body concerned,' and A is the
amount of dilatation defined by the equation

J = èiwn. (2)

Our first object is to find the solution of the equation of
equilibrium, which is appropriate for the discussion of the problem
concerning a semi-infinite elastic body.

Since div. curl of any vector quantity vanishes, if we perform
the operation div on both sides of equation (1) we have simply

div grad J = 0, (3)

1) A similar result has been found by A. Orloff at Dorpat and T. Shida at Kyoto.

2) Some Problems of Geodynamics, Cambridge, (1911) p. 8S.

3) The Astrophysical Journal, vol. 39 (1914) p. 105-.

4) The geodynamical application will appear shortly in the Trans. Roy. Soc. London.

4 Art. 7. — K. Terazawa :

which determines the dilatation J. If J is found from this
equation, the displacement u can Ije determined by solving the
equation

X-\-[JL

grad div u — curl curl u = — grad A. (4)

A*

The elastic body which we deal with is supposed to be
bounded by an infinite (say) horizontal plane in its natural state
and to extend without limit both horizontally and downwards. Take
the cylindrical coordinates (r, d, z) such that the axis of z coincides
with an inward normal to the boundary and the origin lies on the
boundar}^ surface of the body in its natural state, then we have,
for z^>o, the equations

i^+J-. -M,+J-ü^+ü^ = o (5)

to determine J, and

+ „.^^ + Ji^__±^ ,. -j:-^ +^

a<? /^ a?-

?r'^ /■ ■ Ir Iz-^ r ' 16- i^ ' ^z'

(6)

where ur,v^,uz are the components of the A^ector u. The equation
(5) follows from (3), and (6) from (4).

§6. To solve these equations, assume

^ ^ smj

where h is a positive constant so that there may be no dilatation at
;2=oo, and m is an integer, positive or negative, or zero so that the
solution may be unique round the origin, then the equation (5) gives

dr- r dr \ r

of which the solution is

1) The second solution is rejected for the reason of its singularity at the origin which
we exclude from our present investigation.

Elastic Equilibrium of Semi-Infinite Solid. 5

where Jm (r) is the Bessel coefficient of order m, and Cm is a
constant of integration. Thus we obtain

§7. To find n corresponding to

assume that

11, = Urdosmd,

Ug = Ü g sin VI d,

U,, C/q, U„ being functions of r and z. Then equations (6) trans-
form into

^ilL + l. M^ + ^:il^-J!l+lü,-^ü, = -'^^ kC,,e-r„»r), (8)
or" 1 àr dz r r !^

^!^ + l l^ + ^^._i!^f7,=A+i^A-C..-^-^/„,(ÄT) (10)

in which Jm (.*•) means dJm {jx)\dx as usual.

The last equation suggests that TJ^ has the form VJmQcr),
where F is a function of z only and satisfies the equation

d'V

.],^V=^^l-C„,e-'--'

The solution of this equation is

Dm being an arbitrary constant. Thus

w, = -(^-^C,„^-D,„)^-^V,„(At)cosw^. (11)

To find TJr, and C/q, write

Art. 7.— K. Terazawa :

U-U^= Y,
then combining the equations (8) and (li) we have

(5?- T Tir Tiz^ r" 1^

Tlie-e equations suggest tliat X and Y vaiy as Jm+\ {hr) and
Jm-~i(kr) respectively with regard to r, and they can l:)e solved
in a manner similar to the equation (10), The result is

^ A II. I

Em and Fm being arbitrary constants. Writing

■E,-F,„ = 1B„„
and simplifying, we have

"Ifj. kr \- kr -J

Thus we have a system of solutions: —
J = CJ„ikr)e-'''co^md,

{ ['^^^'^^'"''"^"G '^'"(A-r) + .-l.^J".(Äv) Y'-co^md
II. = - { ^'^ C,„z -D„ \j,„{kry cos m 6.

e~^ smmu,

) (12)

Elastic Equilibrium of Semi-Infinite Solid.

The constants Bm, Cm and Dm are not independent of one
another, but are connected bv the relation

2/^

(13)

which follows from the definition of J.

§8. Next, the solutions corresponding to ^ which has sin md
in its factor can l)e found in a similar way. They are distinguished
from the above by placing a bar on every quantity.

J= C,J,XJ^r)e-''ämmO,

y L '2//. J Av J

uh= I r ^''^'" C,„z-B,l JILj^x^r) + A,J-,Xkr)\e-''co?imd,

(1^)

with

k{B„-Dj

_J_+_3/^(7

(15)

§U. The stress can be calculated Ity using the formulae

ZZ = U + 2n--"-'-,

'bz

zd

p;^ III, \
I Iz Ir )'

;•/• = /J + 2«

(16)

' I Ir r r W J

Thus corresponding to the first solution (12), we obtain

Art. 7. — K. Terazawa :

e "^coämu,

+ !-iTiA,jr„l kr) \e"' sin m d,

( (17)

etc. ;

zz

ZT

and corresponding to the second solution (14),
= U>^ + fJ.]hC,,z~[xC,,-'Zi,.l{T)\ J,Xh-)e-''smmd,

r = -{ ^{X + /x)kC,„z-^^ C-fJtkÇBi-D,,,) ] J\„ßr)
+ /jiJi-Ä,J',„,{lr) I e'''co6iu 0,

(18)

etc.

III. Lamé and Clapeyron's Problem.

§10. We will apply the solution obtained in the last article
to discuss the effect caused by a given normal pressure applied
locally on the boundary. Suppose as a preliniinary that a system
of stress of the form

zz = \z eus m 0 + Z^h\md\ J,„{kr^,
7r =0, ^ =0

(19)

is given at the surface z=0. Then we must have the following
relations between the arbitrarv constants: —

Elastic Equilibrium of Semi-Infinite Solid.

^ + /^

^ + /^

-fiC„-2fukD,„ = Z,

G,„+fxMB„, + ï),„) = 0,

An = 0.

From these equations, combining (13) and (l-")), we have

B,„ =

Z
Z

c, =

'^ + ,n

^ ^ _ U + 2;.)^

A,„ = 0,

B... =

Z

J) = _ (A + '2iJ.)Z

'2//Ät;.+//.)

A,„ = 0,

Putting these vahies in the formuhe (12) and (14) and adding them
together, we have an exact solution of the form

= (^ ^ 1 {zco^md + Zün\md\j'Jh-)e-'\

We

= U \ )(

= -{

2// 2(>Ï + //)A;
z _^ X + 2/J.

Zcosmd —Zäinmd

]{

Zcosmd + Zsin^nd

kr

(20)

2/^ 2iJi[X + [i)h

corresponding to tlie l»omidary conditions (19). Tlie formuho for
stress are

Tz = (1 + kz){z coii md + Z sin /// o\j,„iJir)<'^'',

zr = —kz I Z coamd + Z sin md\j',„(kr)e~''^

7d = -kz Izcoümt^ -Zs'mitid] ^"'-- J„ß.r)e"',
I ) kr

etc.

(21)

10

Art. 7.— K. Terazawa

§11. To generalize the above so that the sokition may be
suitable to discuss the effect of any given normal pressure at the
boundary surface, suppose that Z and Z are functions of h of the
form Zm{k)dk and ZmQcJdk respectively, and superpose the
corresponding solutions for all positive values of h, then add them
together for all integral values of m. Thus, corresponding to the
boundary conditions

zz= 2 /* ^ {^-(^'■'>-) COS m 6 + Z,„ (k) sill m 6 \j.,Xh-) dl:.
7r=0, 7ti = 0

at z=0, we have

CO

■i'r = ^ ~J lz„,[l-)cosmß + Z,,,(k)mi)nd\e-''J' ,,,{l:r)dl:

CO

an

«Ö = y -^f{z,XJc)co^md-ZJl-)smmd]e-'^T,,lkr)

(22)

dk
k

dk^
k

y. =

->\,;7 , /"{z„^(Ä;)cos;;.^-Z„,(^-)siii;;.4«''^^"'(^-'')4^r '
^-t^3 2(/ + /ij/-,/ I J ^-

= - ^—-1 [ZrX'k) COS ;^i ^ + X,„(Ä;) sin m e\e-''J,Xky)dk

7)1= 0 f^ Q

^ CO • _

«;eo2/<;. +/.)./ I ' -^ ^ • i "^ ^ k

(23)

and

^^ = |] /(l + Ä.-^4^.,(Ä:)cosmÖ + Z„XÄ-)si]i;;;^l'-*^/„,|A,7>ZA-,
^ = -J ^/Ä;{z,„(/>;)cos;;i^ + Z„,,(A;)sinwöje-V'„,(A7-)f?Ä;,

CO

7d = -^JI^/ iz,Xk)coHmd-Z,Xk)mim.d\e-''J,,Xkr]dk,
etc.

(24)

Elastic Equilibrium of Semi-Infinite Solid. W

More generally, if the boundary conditions are given in the
form of an arbitraiy function, e.g.

7z=f(r,d), ]

zr = 0, zd r=o]

at z = 0, the general solution can ])e obtained by making use of
the integral theorem

Fir) = I J„ikr) MJ:fF(a)J„, (Ä;a) ada, (26)

'o 'o

provided the function /(r,Ö) caii be expanded into a trigonometrical
series of the form

/(r, ^) = i {Mr)cosmd+fJr)dnmd] , (27)

where />»(r) and /»«(r) are supposed to satisfy tlie above integral
theorem.

Comparing the expansion (27) with (zz^ which follows from
the first formula of (24) by putting z={), we see tliat the functions
Zvi{k) and ZmQc) which correspond to the boundary conditions (25)
or (27) are the solutions of the integral equations

Mr) = I Z.,„{}^)J„i'kr)d-k,

0

f,u{r)=jz.^{h)J„{hr)dh.

Looking at the integral theorem (20), the solutions of these integral
equations appear easily to be

/.CO

Z,X'k) = A,7/,„(aV„,(Av/)«r/«,

) (28)

Z.„ik) = Ic/fja)J.,Xka)ada.

Thus, substituting tliese values of ZmQc) and Zm(k) in the
formuke (23) and (24) we get the solution answering to the

12

Art. 7. — K. Terazawa :

boundary condition (25). In his book of differential equations,
H. Weber^^ solves the same problem by using the Cartesian
coordinates. But it seems that his mode of using Fourier's
theorem was anticipated by Lamé and Clapeyron.

IV. Examples in the Case of Symmetry.

§12. The solution for the case of symmetry round the origin,
which is discussed by Boussinesq with numerous examples, has
been afterwards obtained l)y Prof. Lamb in the same way as
adopted here. This case is implied, of course, in our solution.
Suppose

(29)

zr = 0, zd = O!

are given at the surface ;:;=0. The corresponding solution will
then be obtained from (23) and (24), l)y taking only the first term
(m=0) in the summation. Thus

' 0

2(; + //.)./ ^ W.-

2',, = 0,

11^ =

lz{k)e-'-%ßT)(U-

(30)

and

2/' ■'.

n- = -^|l(Ä:)e-'"J"o(Ä-r)MÄ'+^-|z^A-)e-''/,(Ä-r),

0 0

J ('^> + /-<)^" •' 1^'

)cR-

+

1) Part. Diff. Gleiclmngen, vol. 2 (1912) §76-.

Elastic Equilibrium of Semi-Infinite Solid.

13

Öd

À + fxJ

(31)

U:

+

'o *0

^? = 0,

T'y = z'fz{l-)e-''J,{kr)Mk,

0

7d = 0;

where the auxiliary function Z is connected with the prescribed
condition by

Z(k) = k ff[a)J,(ka)ada. (82)

0

We shall now apply the general theorem to a few special
examples.

Example I.

§13. The first example, which is discussed by Prof. Lamb, is
to find the effect of a given normal pressure concentrated at a
jDoint on the boundary, on the supposition that/(r) is zero for all
values of r except those in the immediate neighbourhood of the
origin, where it becomes infinitely great in such a manner that

/f(r).27rr.dr= -II,

(83)

in which // is the total amount of the pressure applied and is
finite. The function Z{k) now is

Z(k) =

n

Putting this in (30) and (31), and then integrating we get

14

Art. 7. — K. Terazawa

Ur =

n

and

u.

66 =

zz

^JL\

: +

/. + 2/^

^^\'

77

77

n f 3.rr- _ /^ r 1 _ ^ "] I

// /,-(2r + ^^) ._ 1 I

77

3^^

77 3zh'

76 = 0, 6z = 0.

At the surface (^=0) tbey I'educe simply to

n 1

("Jo =

47r(i^ + //) ?•

77(yl + 2//) _1_
47r/i(>^ + fj.) r

,.-^ 77// 1

( ^n, =

77//

'2-(;. + ^«) r-

(34)

(35)

(36)

(37)

{zz\= 0.

§14. It is interesting to show by graj^li the state of
deformation at different depths from the surface. For the sake of
simplicity we assume that the material satisfies the Poisson' s
relation ^=[J; and wu take only one component of the displace-
ment vz for reference, which is now written in the simple form

47r/. l(r^ + /-/.3 2 (i^ + rf- i
The attached diagram is di'awn on the supposition that Il=4.-p

1) The integrations have been carried out by Lamb, and so it is unnecessary to recapitu-
late them here.

Elastic Equilibrium of Semi-Infinite Solid.

15

s-

Fis. 1

As seen from the diagram the state of affairs in the neighl)Our-
hood of the origin and even at a finite distance from it is an
impossible one, and the mathematical theory of elasticity does not
apply to such a case. The above argument must, if possible, be
amended l)y a suitable process of analysis. The general solution
found above is restricted to a special class of functions f^r) which
satisfy the integral theorem (26). The hypothesis of point con-
centration of given pressure does not, in a strictly mathematical
sense, satisfy this important condition, and the solution deduced
from it may not be looked upon as a legitimate one, at least in the
vicinity of the origin.'^ What follows from the assumption of
point-concentration of given pressure may, however, be considered,
except locally, as the limiting case of the effect of a pressure which

1) A quite similar failure of the solution will occur in the problem of the deep-sea
water-wave and allied problems which can be solved by the aid of Fourier's Integral
Theorem.

16

Art. 7. — K. Terazawa :

acts on a slightl}^ extended area of the boundary, whicli will be
discussed in the next example.

Example II.

§15. Suppose the normal pressure at ^ = 0 is given in the

form

fir) =

n

a>0,

271 {d' + j-f
n being its total amount. The function Zijc) is

(38)

03

^71 ./ fcr + a-r-

77

2;r

he'

(39)

Thus the solution corresponding to the boundary condition (38)
Avill be given by

u, = - — j c-^'^"^''-J,{'kr)'kd'k

Jh

II

^-Mi')

-fe-^'^'^yj,ik7-)(U;

Attu. J

+ ■

4;r/</l + //) /

and

7r = -^ l7'^^">%lhWdl-~^fe-''^''^VXh-)k
27r ./ ^ 2;rr •/

dk

II

'M

2<T(/l + /.);

66

= -^ fe'^^^'^^'J.ikrkdk- /'f" , fh^^-'-J^ik

•)dk

in

27r(; + ^) J^

fe-^'+'-'^-J,<kr)kdk,

(40)

(^1)

Elastic Equiftliriimi of Seuii-Infinite Solid.

17

"' 'o ""■ 0

zr = -^ /e-<^+^'^V,(A7-)/rV/A-,

0

To = 0, d): = 0.

These integrals can he expressed in terms of zonal liarmonics
and the associated functions. If Pn (x) denotes the zonal harmonic
of ?üh order and

X X

then

fK-JjMä^ = -J-+»)

dx"

.p:

n).

By making use of these formula?, we have

,, _ // / z{z + a) , ;> + 2/. 1 \.

4-Ü I (r- + iz + a)'-^)^''- / + // * (r- + (^ + a)- )i'" / '

(42)

(43)

and

-__/// a{r- + {z + of) + 3^r- _ jj. F 1 _ ;g + a ~||

' 2;r I (r- + (^ + a) J"" T+ /^ " L >•-' t%7^ + (;? + a)^^ J / '

fa_ ft [ ^ ,« r^__ z + a ~\

~~ YA (rH(;J + «)-)^'- T+TT L r- r'V + C? + a)7^J

_ ^ z+a \

7+y 7^+(^ + a)T- )'

2- ■ (r^ + (,, + ,,/)-v2 '

(44)

1) See K. Terazawa, Proc. Eoy. Soc. London, A. vol. 92 (1915) p. 57—.

18

Art. 7. — K. Terazawa

zr

_ _ /7 ?>zr{z-\-a)

2r (,.H(^ + rt)T'

rd = 0, fz = 0.

If we proceed to the limit a -> 0, we have the same result as

that due to the pressure of point concentration. But this limiting

process is, in general, not permissible. A little examination of

the value of -y-' shows tliat tlie quantity a has a lower limit,

such that

27^, _ /7(2; + 3«)^

a-

(45)

A^ixX + iif '

in order to avoid the impossible state of affairs near the 2;-axis.
At a distance from the origin great compared with «, these solutions
reduce, in a first approximation, to those in Ex. I., so that the
solution which follows from the assumption of point concentration
>of given pressure may be valid at a great distance from tlie origin,
though only approximately.

^16. It is desirable here also to see how the displacement
varies with the depth. On the same hypothesis as before, that
A = //, the variation of Uz is shown in the attached diagram, in
which the upper curve represents the distribution of applied pressure
(38) and the lower ones represent uz on a proper scale, a is put
equal to unity for the sake of simplicity.

^17. At the surface, the expressions for displacement and
stress become

I ^ r{r- + a-)' f
1

47r(>^ + /^)

(46)

,-~ n J a fz r I _ a -]\

mR\ - -AU /^ r ^ ^ 1 + ^^ «^

^ ^' " 27: \x + fx L r' rXr' + aT' J >^ + /^ ' {r' + aT'

n

)(^T)

^''^' ~ 2- • (r'^ + a'O"'-^ '

(zrX = 0,

Elastic Equilibrium of Semi-Infinite Solid.

19

Fis. 2.

By integrating (t(z)o over the surface, it may be easily seen
that the total depression of the surface appears to be infinitely
great, though it is caused l;)y a finite normal pressure of total
amount /7. This seems again to be paradoxical, but that is not the
case; if we calculate the work done by the given pressure, instead

of total depression, it will appear to be finite, equal to -0 — • \)2^ '
inversely as a.

§18. Let us now proceed to appl}' the results of this example
to the theory of rupture of a foundation over which a heavy load is
spread. There have been proposed several hypotheses concerning
the conditions under which an elastic body is ruptured or nearly
so. Among tliose hypotheses usually adopted there are two in
which a limitation on stress is taken as the measure of tendency to
rupture: the one which was introduced by Lam/, is that the
greatest tension should be less than a certain limit which is

20 -A-rt. 7.-K. Terazawa:

different for different materials; the otlier which was recommended
by G. H. Darwin asserts that, as mere h^'drostatic pressure can
hardly affect the case, the maximum difference of the greatest and
the least principal stresses should be less than a certain limit. ^^
These two hypotheses lead, in general, to different results. Eitlier
of them will give w^arning that danger is l)eing approached, and in
any case a certain factor of safety must, in practice, be adopted.
Here we shall calculate the limits following from these two
h^^potheses and compare corresponding results.

For tins purpose we liave, in tlie first place, to find the
distril)Ution of principal stresses throughout the body concerned.
Let ^Y;, jVo, Ns denote the values of principal stresses at the point
(/', ^, z). Owing to the hypothesis of symmetry round the axis of
z, the component 66 is one of the principal stresses, say J\\, as is
to ])e seen from the formulae (31); and any plane passing through
the ;i-axis is one of the princi])al planes of stress. The other two
principal stresses will be found by

^1 = -J- {>r + 2z) + -^J{rr-zzy + ^rz\

N:. = ^{^■r + zz)-^J(f?-^f + 4fP.

(48)

2 ^ ^2

At the surface, since rz = (), the stress components Pr,66,zz them-
selves are the principal stresses.

§19. Now, to apply these formulae to this example, let 'us
assume"'^ that the pressure modulus >^ is so great compared with the
rigidity /^ tliat the material may be considered to l>e incompressible.
Thus, substituting tiie values of the stress-components found in
(4-4) in the formulae (48) and making ?. = x, we have simply

// Sz + a

N,=

Qrr

N, = N. =

J J ^^ a

~2^' -(r'+iz + afy-

1 There is another view often adopted, in which a limitation on strain is taken as the
measure.

2) This supposition is not at all necessary, but it makes the calculation extraordinarily
simple.

Elastic Equilibrium of Seuii-Infinito Solid.

21

iVi being greater tlian N.2 and N:- which are equal. Thus the
stress quadric at any point in the interior is a spheroid, of whicli
the axis of rotation lies in a plane passing through the 2;-axis.

If we denote by D the difference of the greatest and the least
principal stresses, tlien

77 Sz

D = N,-N...= —

2- (r' + (z + afr^'
D is a maximum at the point (r=0, ^ = -|-J and its value is

(50)

B,.o.. =

2/7

9-r

(51)

wliile iV'^i, the greatest princi})al stress, is a maximum at the point
(r = 0, ^ = 0) and

n

N =

"lûïCl?

(52)

Thus the latter maximum is greater than the former, and, more-
over, the position where the rupture might occur is quite different
for the two hypotheses: it is at a certain distance ])elow tlie surface
in the former, while it is at the surface in tlie latter.

In Fig. 3 it is shown liow tlie greatest principal stress and the
greatest difference of principal stresses along tlie .r-axis vary with
the depth from the surface.

sAÇ

D

;x

Eig. 3.

22

Art. 7. — K. Terazawa :

§20. To fix matters, suppose an isolated mountain or a high
tower of uniform density with a circular base standing on the
ground; the surface of the mountain or the other being given by
the other by the equation

mVi^

(wVr + (w^-l)rf2

(53)

This equation is so adjusted that the height at the centre is h
and that at the point r=mh is hjn. We shall make the rough
assumption that each point on the surface of the ground is pressed
normally downwards with a pressure given by the product of the
specific gravity and the height of the mountain at that point. Tlie
quantity «, used in the above, is now

a =

mh

V;i^"-1

In the annexed diagram, the upper curves are supposed to
represent the profiles of mountains and the inner ones those of
columnar buildings such as chimneys or monuments, the height
being taken as unity, and m and n chosen properly.

Curve

71

m

a/2

Ci

1-837

1

0-7071

Cj

1-837

1/2

0-3536

c?.

2-828

1/3

0-1667

Ci

5-196

1/5

0-0707

Cr,

11-180

1/10

0-0250

If we denote the specific gravity of the mountains or other
bodies by w, then the total amount of pressure will be

n

The maximum of the greatest principal stress becomes

N.

= —wh,

(54)

Elastic E>iuil)brium of Çeuii-lnfinite Solid.

23

/

Fig. 4.

acting at the origin; and the maximum uf the difference between
the greatest and the least principal stress is

D,no^= "-9" '^'^''

(55)

acting at the point {r=0, ^=-|-) • The values of a/2 corresponding
to the curves are calculated in the last column of the above
table, and the positions of the critical points corresponding

to them are shown also in the figure by the points F,, F2,

It will be interesting to find the limiting height of mountains,
which stand on bases of several kinds of materials without crushing
the latter. In the foUowing table a few examples are given: the
third column contains the greatest heights of a mountain given by

24

Art, 7. — K. Terazawa :

the formula (54), and in the fourth cokimn are given those
calculated by means of the formula (55); the value of w being
taken as 3 grammes weight per eul)ic centimetre.

Material in the base

Strength to resist
crushing in
kg. per cm.-'

Max. height
in km.
by (54)

Max. height
in km.
by (55)

Cement

320

1-07

2-40

.

Strong sand stone

800

2.67

6-00

Strong granite

1600

533

12-00

Strong glass

2100

8-00

IS 00

Wrought iron

3200

10-67

24.00

Example III.

§21. Next we will consider the case in wliich a uniform
normal pressure acts on the surface within a circular area of radius
a, outside of which tlie surface is free from traction.

Suppose

/(?•) = — , , for a :> r,

(56)

= 0

,, a<zr

n being again the total amount of the given pressure. The
function Z(Jc) will be

Z{T,)

II

Tta

- J ~a

(57)

and the components of displacement can l)e computed from the
form u he

tr = ^^ fc-''J,a-r\J.{la)âl-

II

2ïï-a(; + ^)

fe-''M,(lT^J,a-a)^

(58)

u^ =

Elastic Equilibrium of Semi-Infinite Solid.

Hz

25

'^Ttafx '

fe-''J,(h-V,{l-a)dk

+

\7tati(XA- a) >' k

and those of stress are given l;)y

7r = ^- fl-''llhr)J,{ka)kdk- '^^ fe-"J,(ky)JA[ka)dk

0 0

^ re-'-^Jlkr)J,{ka\lk + ^— P-'-J-{kr)J,ika)^ ,

Tta J ' 7:{X-\- fx)ar •■' ' k

dd

Trar J ~(X + ,n)ar •'

dk

Ilk

— fe-''-'J,(kry,(ka)dk,

zz

(59)

= - ^^- fe-"J,ar}J,ika)kdk--^-^ / e-"J,{kr\I,ika)dk,

0 0

7r-= - ^'' fe-'--J,(kr)Jn-a)kdk,

rd = 0, dz = 0.

§22. The integrals required liere cannot be evahiated in a
very simple way. Some of them are closely connected to the
magnetic potential due to a circular current, or to the velocity-
potential and stream function of a circular vortex and have been
discussed by various authors. In his paper on the inductance of
circuhu- coils, '^ Prof. H. Nagaoka has devised a comparatively
simple method which may he applied to evaluate all the integrals
needed liere. Let us follow his method and descril)e it briefly.

Pat

il' = y/ d' — -lar cos 6 + ?•-), (60)

then we have

1 /*"' \

J^ki-y^ß-a) = -^ / J^,ikB) cosd da,

0

-r/7 NT'7 ^ 1 /" " a — rcosä

J,(ky)J,{ka) = -_-/ J^ikB)- - ^

-dd,

{VA)

1) Phil. Mag. YI. t; 1903 , p. 19-.

26

Art. 7. — K. Terazawa :

sin-ö

J,{kr]J,{Tia) = / J^[IB)—^

71 ' M

cW

(61)

which follow from Neumann's addition theorem for the Bessel
function. Making use of these formulae and on referring to the
formulas (42), we obtain

fe-"J,{j£r)J^il.a]dk = ^/

1 /•'^ cos<^

fe"'Jç,{kr)J,{ka)dk = -^ f

cW,

a — rcosd iß_ z /"^ a—rcosd ^a

B'

/ '^ ' '^ ^ I r: J B- ^ J Bl

siii-

J'e-'^Joih-y.ika)-^' = -^ / " V-^' + ^' (a - ?■ cos d)dd

_ ^ f~ a—rcoaO

71 J

dd,

B'

dd

(62)

To find these integrals, put

so that

2 \i r. a'+r' + z-

P =
ar

bar

_ 2/9 _ a- + 7-- + z~
« 'à ara

_ 1— /5 _ _ a."' + r'' + 2'' — 6ftr

« &arf/-

_ _ 1+/5 _ _ a?-^ r-\-z"- + ^'>ar

()ara

m)

^1 + 62 + 63 = 0,
and change the integration variable from 6 to .5 by putting

cos, 6 = as + ß,

then we have

Elastic Equilibrium of Semi-Infinite Solid. 27

P^os^_^^^ = or a

similarly for the others.
Put again

s = ^(u)

and denote b}^ (o^ and oj, the real and imaginary half-period
respectively and w.^—w^ + oj^, tlien, since s or }^{u) is real and lies
between eg and e^, s=e3 and s = 62 correspond to u=(o.^ and w=w2
respectively, if we take the sign of }^''(iù) to be positive.'^ Thus

r " cos/? ,n o f<^-2( 1 . -J

= «'(-^ei^'^.--'^i)- (64)

For the evaluation of the otlier integral, write

2ara 6ara

then we have

/'^ a-rcoQd ^jß = J^ ,0, - r-ar r^^^ du
{ B'^B' + z' 2a ' 4ä^V ^ )^^^)-)^{,u) '

etc.

Now

(66)

in which the term mTzi enters l:)ecause of the many-valued property
of a logarithm. The actual value of ??^ and ^'(v) will l)e determined
by the following consideration.

1 If we assume 1?'(î() to be negative, then s = 6', and s = e-j correspond to »=(»_, and
ii = 2coi+Gûi respectively. But the saoie result will, as a matter of course, be obtained after
integration.

that

28 Art. 7. — K. Terazawa :

From the definition of }S'iv) and 61,^2,^3, it follows immediately

^aro.

ia-7f

2ara

(a + r)-

2aYa

accordingly

62 <: !f (î;) <r e^

The last inequality shows that the value of v must be one of the

following:

(i ) V = ()in + l)w, + (2?/ + d)(o.^ , \

(ii) V = {2n+l)(o, + i2N' + -2-d)(o, J ^''

where 7i and 71' denote any integers, positive or negative, or zero
and d a positive number less than unity.

To determine the value of m in the formula (66) for the value
of V given in (i) of (67), observe that the integral on the left hand
side of (66) and the function v/i^ — oj^^(v) change their values
continuously^ as 6 varies from (J to 1, wliile m I'emains unchanged
during this variation. In the limit as ^->0, - the value of the
integi'al is nil and

and therefore we have

(i) m = —n'.

Similarh^ for the value of v given in (ii) of (67), proceeding to the
limit ^->0, we find

(ii) VI = -{n' + l).
The value of ^'[v] will l)e ol>tained from

Elastic Eiiuilibrium <ii Semi-li. finite Soli.l.

29

In the present case, we get ,

^ [v) = + I. — ^ ^ .

~ 2a7-

It may easily be shown that the vahie of }S''{v) is a pure imaginary
quantity for tlie vahies of v given in (G7), with positive sign for (i)
and witli negative sign for (ii). Therefore we liave to take

r{v) = + I -^- '-

'2ar

for
and

for

in which

7}i = —)l ,

r(y) = -t, —

j 2ar

m = — (m' + 1),
0-<^<l.

Hereafter we shall, for the sake of simplicity, take into
account only the value of v which will be ol)tained by putting
71 = 0, n =0 in (i) of (07) \va.

c = (t), -f- d(o.^.

By this convention tiie term mni disappears and the vahie of ^'C^)
is to l)e taken as

(68)

iP'(v) = +i

"Aar

Thu^

P^i-rco^ ^ = -^<o, _ JIz:«^ . _A_{rr,-aj,:(v)]. (69)
/ BV^M^ 2a 2aV riv) I '' "^-7 ^ '

Similarly we have
sin-^

if VÜ) y

30

Art. 7. — K. Terazawa:

r- — a'

a-ya-

+-~-r^-— ^^77^U'^i-^<(«) • (71)

a-ra^ (? (v) l i

a'ra' Jr'(

The other integrals in (62) are found without the knowledge
of elliptic functions.

/'^ a—rcosd -la 71 ,

/ do = lor r <; a,

•/ R- a

0

= 0 „ ?• > a ;

./-

-dd = -^ „ r <: a,
2a'

(72)

(73)

?r-'

'A)

„ r > a,

§23. Substituting these values of integrals in (öS) we obtain
the following expressions for the displacement:

2a~'/2 I '2 J

//

2a7T(X + fjL) ] a

2a

(r < a)

2r

-(r > a)

orz

+ ^r>[-^i-^'^i(W+e.)]

2;:

+

^ [^^ (V-.J . [^^ (.)-..J ^^-^-f >Ö^} , (74)

?/. =

Hz

2a7T[x

— (r <; a)
a

0 {r>a)

azcoi , z(r' — a-) v/]^ — co^^[v)
2a>T 2(2-V>T

2a7t[j.{k-\- [I.)

~{r < a)
0 (r>a)_

S='(^)

, 2 ,■ , >, ?•- — a^
««TT za'va-

+ ^(f^ [W-e,] ^^^^If^l • (75)

7ra ra'' i^ iv) )

Elastic Efjuilibrium of Semi-Infinite Solid. 31

§24. For purposes of calculation, it will be very convenient
to have the forniuhe expressed in terms of Jacobi's ç-series. q is
defined by

CO,

After Weierstrass, if we put

then q can be computed from

of which the ftrst two or three terms are usually sufficient. The
ç'-series of the functions needed here are as follows: —

2 \ 2 ' "^ ^

■&l'{o)

and

'' V2co, ' â,{o) '

â,{o) = 2q^(i + q' + (f+q''+ ),

ä["(o) = -27rY(l-3Y- + 5V-7V-+ ),

â/(o) = 2;rg^(]-3gH5g«-7gi^+ ),

ê:{o) = -2>T'g'(l + 3V + 5V + 7V--^+ ).

§25. The calculation of the term vy], — io,^(v) requires a little
explanation. By the fermula

e • (t{v) = Ico,-

we have

32 -'^rt. 7. — K. Terazawa :

„ N 1 \ '2io,

Making use of the expansion formula ()f the right member, we get
vr.-conv) = --""^ jcot-''-^ +4V ?"''"- - . sin "^^ \ . (76)

The quantity v may l)e calculated by utilizing the Weiersteass'
formula to any degree of accuracy,^-* and therefore the value of the
function v/^i—w^^(v). The approximate value of this function can,
however, be found in tlie following wa}^:

Put

and

1 (ei-ev)HS'(iO-e,)--(gi-e3)'(^(^)-g-)- = ^

1 (gj — e.y a.(v) — (e^ — e^)^ (7;(v)

2 (e, - ej' (Tjv) + (e, - e.,)* a^v)

g COS h (/COS +

l + 2(7^cos "^+2g^«cos ^^" +

then, since e^ < lf'(tO < «: . we shall liave

As ç-^ is usually a very small quantity, we may take

COS — ^ ^ = = s say

lo, q

with a close approximation. Since v—co, = dco, purel}' imaginary,
Ave ma}^ put

—^ ^— = IX,

1) This method has been adopted hy Prof. Xagaoka. I.e.

2) Halphen, Traité des Fonctions Elliptiques, I. p. 274.

Elastic Equilibrium of Semi-Infinite Solid. 33

X being a real quantity, then

COshjL' = .9,

and we have

cot -^"^ — = — itanh-'' = —ii

'ho, 2 Vs+1

sin ^^ = — /sinh.r = — /Cs- — 1)',

CO,

sui ~" = /siiilr2a' = •2is{s'- — iy,

Substituting these values in the equation (7G), we have

This approximation formula is recommended for the case in which

For the case ^ > I-, if we put

(e, - ej' (e, - ej - (e, - ^v))' f

then we shall have

Q^v — (o.j) = t',

and making use of this formula, we shall obtain an approximation
formula which is convenient for the case d z>h.

§26. The expressions obtained in §23 are rather complicated,
and the state of deformation can hardly be grasped at a glance.
The formula for the displacement at the surface is, however,
comparatively simple and can Ije calculated witli any accuracy.

Putting z = 0 in tlie general formulae, we obtain

— (r :> a)\
and

3^ Art. 7. — K. Terazawa :

OO. = -^ . -^"y- ■ -'-{^ fe + -».)-^». } (78)
Tzar 'A(À+u)u -vi« ara' )

2(/l + /.)/.

where

2 \J

a =

ar

cr + r- «- + ?•■ — 6ar (X" + r- + 6a>-

3<xra " 6ar« Gar«

At the surface the vahie of / Ijecomes simply

(>• + «)*+ lr—a|*

so that the use of the ^'-series is very advantageous for the calculat-
ion, especially at the points near the origin or at a distance from
it. For example, at the point r = 3«, we have

— = 0-085786, q = 0-085796 ;

thus, even for the value ;- = 3a,, we may neglect the terms after /^
and rf in the series. For larger values of r the ^-series converge of
course very rapidly."

§27. The formula for {ih\ can he transformed into the form
which is convenient for the nse of Legendre's table of elhptic
integrals. If we remember that

-fj^ + eiw^ = ^'e^ — e-^E,
K

Ä'and jË* being the tirst and the second complete elliptic integrals
with the moduli k and k' , and

the expression for {;uz\ beccwmes

1) For the point near the edge of the circle, the calculation may be undertaken by
using similar series, specified by qi and /j. See below.

Elastic Equilibrium of Semi-Infinite Solid. 35

At the centre of the loaded circle, since
Jc = 0, K = E = ^,

o

we get

(^,X = ^ . J;:;^^\ /) (80)

7^a 2{X + ii)n

and at the periphery of the circle (>' = «)> since

A- = 1 , E' = 1, Ji -^ log — , r,— a) K -> 0,

we have

The values of (il), at the centre and at the periphery of the circle
bear the constant ratio n-/2, and this is independent of the elastic
constants and radius.

At the centre, as will be seen from tlxe formula (SO), the
vertical displacement varies inversely as the radius, when the same
amount of total pressure is applied to different circular areas, while
it varies directly as the radius when the pressure of the same
intensity is applied to different areas. The same relation liolds at
the edge of the circle, with regard to both the components, radial
and vertical, of the displacement.

By the aid of the Legexdre table of K and Jß, we can trace
by a graph the general march of (i/,:)o. The next diagram is drawn
in this way, where the radius a is taken as unity and the pressure
Z7 is taken equal to 27vafJt(?< + fj.)/{^^ + '2>/j.)

§28. To find the formuke for stress we need two more
integrals which can be also carried out by the same method as
before. AVe have, from (01), tliat

1) This result may be obtained from (58) directly hj usjng the foruiula ./ Ji[ka) -7— =1-
The i-esult (80) and (81) agree with those givtn by Bodssinesq. p. 140, I.e.

36

Art. 7. — K. TerazaAva ;

Fie. 5.

j:-''Ul.-)ma)Mk = -l^f-^^^ dd,

fe-'^ih-Wla^ldk = -j^j

1 r'^ a—r cos, 6

(B'+zT'

dd

(82)

B being detined by (.GU). Of course we may find their values by
differentiation of tlie integrals already found with respect to z, but,
owing to the complexity of the elliptic functions, it will l:)e seen
that the direct method of integration is much easier. By the
same transformation of variables as before, we liave

—^ du — — üj,,
4 J e,-lf(«.) 4

and by the aid of the formula

|S=(?0 — 6) (61—^^(61 — 63)

the integral can be found to Ije

J{E'+z-f'- 2a>-l 2(^1- e,.) (6,-63) '

Elastic Equilibrium of Semi-Infinite Solid.

37

Siniilarlj' the second integral in (82) is

Substituting tliese values and those found ])efore in the
formuke (59), we have for tlie stress

rr

27r'-VrV I 2ry.(ej— e2)(ei — e^)

aa

+

+

2 [îf(io-^j • [s-(^)-.d- '^'w":f' -] }

//

TT«

-^(r<a)
0 {r>a)_

{)- + !i]r

2«

— (?• < (X)

(85)

-"■-(r>a)
_^ 2r ^ .1

r<^f,

r^— cr vrj^—ci}^^{v)

ao.

X4-a L'

^ '^ *' ' ^ ^ r .

I ^— (r <: a)
0 (r>«.)

]

Î5='(î;)

//

7t(X + //)«;•

2«

•)..

(r >«.)

(86)

J I

-_ //.? f (2a — 3are,)(-/^i + e,(o,) y^ — a^ ^ Vfj

zz =

TY \

77 (—('•<«) 1

!î= (t\'

TT«

('

27rVr l2(e,-e,,)(e2-e.O '

r^ = 0, ^ = 0.

(87)
(88)

38

Art. 7. — K. Terazawa

§'29. At the surface these expressions for the stress reduce to
simpler ones.

{rr)o =

2(^ + /.)

=

TTV

m =

-

2/1 + //

2(/ + /z)

. ^' (r < a)
Tza-

/^

2(^ + ;.)

(F^)o =

—

11

-ar

{r^a).

=

0

(r > «, 1 ;

{^r% -

0,

(89)

All the tractions acting on the boundary vanish, as they
ought to, under the conditions of our problem, except a uniform
pressure on the circle of radius a. The state of stress just below
the surface is made up of a simple and beautiful scheme of the

pressure system with a radial tension equal to — ,-, o , ^ • ., inside
the circle, and -ötV^^ — ^- — ^ outside it; and a transverse tension

lU-{- [X) Ttf '

2;>+/i

//

2(;.+/i) Ticv

2" inside the circle, and

/^

77

2(/î + /^) ' nr''

equal to

outside it.

§30. Along the edge of the loaded circle there occurs
a singularity of stress. We have seen ah-eady that as a rule the
component stress zi^ vanishes at the Ijoundary surface. But this is
not always the case. If we put r=a in (88) and then proceed to
the limit z -^ o, we shall have

{zr\ = — -

77

Tta-

~ , (r = a).

(90)

Thus the tangential traction (£?')o does not vanish at the periphery
of the loaded area, which is contradictory to our assumed boundary

Elastic Equilibrium of Semi-Infinite Solid. 39

condition. It appears that along the circumference of the loaded
circle a radial shearing stress of magnitude equal to the given
normal pressure, divided b}' ~, should be applied. This was
also pointed out Ijv BoussixESQ.^-* Bat the area on the boundary
over which this shearing stress applies is infinitely small, so that it
is practically of no account at all. This singularity possibh' means
that the region in the interior of the body in which the stress
component i)* exists has a cuspidal edge, which touches the
boundary surface at the periphery of the circle. To avoid the
above difficulty, Bous<;ixesq supposed tliat at the edge of the
loaded area the pressure decreases more or less rapidly to zero,
instead of vanishing abruptty."' If, in the actual problem, there
were no singularités tliis consideration might lead to legitimate
results.

^ol. In this example, it is not eas_y to calculate the
maximum of the greatest principal stress or that of the difference
between the greatest and the least principal stresses, even when
the material is incompressible, consequently we shall abandon the
general discussion concerning the conditions of rupture. But if
we confine our attention only to the condition which determines
liow much load the body can sustain without breaking at the
surface, the problem l)ecomes tractable.

The equation (89) gives

_ _ E-fx n

n

for >■-<:«, in which the elastic constants X and />« are replaced by
Young's modulus E and rigidity «. Since 3/^ > ^ in ordinary
materials tlie component (B)„ is the greatest. The difference
between the greatest and the least principal stresses at the surface

is

1) BOUSSINESQ p. 148. I.e.

Ä

2) For example, we might take f{r) = — —- or a similar relation. But the analysis

might be very complicated.

40 Art. 7.— K. Terazawa :

D = - -V--^ //

The values of (zz)o and Du might give the condition of rupture of
the surface.

§32. Now we shall apply this solution to the geophysical
phenomena mentioned in the introduction. Dr. C. Chree"^
followed by Prof. Nagaoka^* finds a formula. l;>y using the solution
obtained by Boussinesq, to calculate the deviation of the direction
of gravity due to the attraction of a material loading on tlie
surface of the earth. The same result will be attained of course
from our solution. The expression of the vertical displacement
at a point on the surface

ivX = ./^^ fliJcrjdJ^ l'piy)J,hr')r'dr'

where pO'') is the pressure produced by the material load, can be
transformed into

(y\ = —- . ' / — — r Circle

I ^ ' i - 0 0

bv making use of Nei'maxn's addition theorem for the Bessel
function, where R' stands fur

B' = ^y--2yy' CO:, ç+r'-)

On the other hand, if we denote the attraction constant by r, and
gravity, prior to the application of the load, l)y (/, then the
gravitation-potential at a point on the surface due to the load can
be expsessed by

a .' J R'

q •' •' R

provided the height of the material load is negligibly small
compared with the distance of the point under consideration from

1) Phil. Majj. (V) vol. iS (1897) p. 177.

2) Tokyo, Sug. But. Kizi (VI) (1912) p. 20S.

Elastic Equilibrium of Semi-Infinite Solid. 4.1

any point in the loaded area. Comparing the above two ex-
pressions we have

VzX

Thus tlie direction of gravity becomes, in consequence of the
attraction of the loa(b inchned to the vertical at the angle Ç-'' which
will be determined bv

ta„^. = ^L.i^^(4^) (91)

while its tilting effect is expressed by

tanc = ( ^^''^) (92)

§33. In the present exam])le, in which a uniform material
loading is confined in the circle of radius «, we have, from the
formula (58),

// /Î + 2«

fe-''J,(lry,(l-a)df]

Referring to tlie fcirmula^ (^>1)- (^^~) 'Ti^<1 (^J-t)i ^^'C' obtain

taiic = - — . ^ . aa-[ e,co,-r,, ) (9d)

tan/' = — — . — '— . aa?\—e^co^ — -f]A (94)

-a'- g'' ^ 2 /«=o.

The function h'^^co^ — ri^ has been discussed already and the
expression which is suitable for the calculation of its value at a
point not near to the edge of the circle has been established in
terms of q. Using the ^-series in §24 we shall have

aa?l^^e,oj,-rj\ = 2-yjL . q'%l-\-'èq'-'iq'-9q' + 2^,f+ ). (95)

It is equally interesting and important to find the value of 9
at the point near the edge of the loaded area. This will be

42

Art. 7. — K. Tera-îawa;

accomplished Ijy using tlie quantity qi, instead of y, wiiich is
defined by

g, = e-"^i,

Now, by the aid of the relation

ê[{o) ä, ß, &,

the expression of èe/«;,— --y, found in §24 may be transformed into

Making use of the transformation formulae of Theta-functions it
will he easily shown that

^o'(o/t) _^. ,.., ß:/(o/T,)

'>o(o/r)

^,(o/rJ '■

'UolT) - ' Hoir,) '

consequently we liave

-- L I J, loir,) &loiT,) J J

The </rseries for tlie functions needed here are as follows:

^., =z 3gi(l-|-g,- + gi' + g;'-'+ ),

.>;' = - \i7i\q, + 2--g,^ + '^'q,' + ),

d,= l + 2g, + 2g,'' + 2g:«+ ,

/A, = l_2g, + 2g,^-2g,"+

The quantity q^ will be found from

Elastic Equilibrium of Semi-Infinite Solid.

43

z-

Fie. 6.

Î1 =

Thus the déviation of tlic directiou of gravity at any point can
be calcnlate«! with any aceni-acy.

In the next diagram the aj^p-roxiinate eour^^e of aa-{y.co^—rj^
is exhibited as a function of the distance of the point of observation
from the centre of the loaded circle, the radius of which is taken as
unity.

§34. If we liken tlie North Atlantic to a circular basin of a
large radius and determine the I'elative ]iosition of Potsdam or
Chicago referring to the centre of it. the attraction effect of the
periodic fiUing and emptying of tide, which might assist in
producing the extra east-west force in observations of the lunar

^zj. Art. 7. — K. Terazawa :

disturbance of gravity, may be computed l)y our formula. If we
suppose the place of the observation not to be very near to the
circular basin, the effect, as we see from the above diagram, is of
course small, but it increases repidly as the edge is approached.

For the water-level measurement, the efïect of a material
loading will appear in the form ^ + ^, instead of ^ only, where 'P is
due to the attraction exerted by the material loading and ^ to the
deformation caused by its weight.

For example, suppose the radius of the North Atlantic basin
to be 2000 km, the position of Chicago to be 3000 km from the
centre, and the level of the water in this area to be raised one
metre, then

-Z^ = 1-5, q, =0-00255.

a

aa^^^-'^y-r^, \ = 0-8639.

Further assume that the density of sea water is unity and in c.g.s.
;- = 0-65 X 10-^ 0 = 980,

^dt^^ = jL, /. = 6 X 10",

then we shall have

y'' = 1-17x10-« = 0"-0024,
Ç ^ .^siTxlO-« == 0"-00(39,

accordingly the total effect amounts to

<pj^cr = 4-54x10-« = 0"-009.

It will be noticed that the effect of tilting is about three times
as great as that of tlie attraction, so far as the material constants
are assumed as abo\'e. According to Lord Kelvin,'^ who
initiated these investigations, tlie direct lunar effect on the
deviation of a plumblinc is a maximum when the moon is at the

1) Xatural Philosophy, Part II. p. 383.

Elastic Equilibrium of Semi-Infinite Solid. 45

altitude 45° and aiiionnts to 0. "017 neaiiy. The total effect of a
tide of amplitude one metre (which is possibly two or three times
the actual amount) found here is not small enougli to be neglected
compared with the direct effect of the moon. As the tilting effect
and the attraction effect of the tide wave are directly proportional
to the height of the tide, the total effect oscillates in time in
accordance with the law which the tide obeys. There is, in
general, a difference in phase between the lunar effect and tidal
effect, which is worthy of closer investigation. But we must bear
in mind that the calculation adopted here is nothing but a rough
estimation of order of magnitude, since the north Atlantic is far
from circular, the tidal loading in it is never uniform. Neverthe-
less the above analysis shows that tlie tidal effect on the w^ater-
level measurement, even at a point as far from the coast as
Chicago, plays an imY)ortant role and cannot be regarded as a
small correction.

Example IV.

§35. Let us take another example by assuming the normal
pressure of the form

/(r) = — '^- / a- — }-- for >• <r « ^ ^

•^^ ^Tzar V I (97)

= 0 „ r > a J

to be given at z = 0, II l)eing its total amount. In this case the
function Z{}c) becomes

0

_ _ 3// f aml-g — kg cos Jca \ .qo.

Therefore the components of displacement are given by

46

Art. 7. — K. Terazawa

4-a/i./ l A-V- J

) (99)

Ta/j.

The integrals contained in the above can be obtained by
expanding the trigonometric functions into power series of k and
making use of the formuke (42). In this way we have

u, =

_M_ ^_ y( i)n-i nj-ln-^)] ( a \^---' \

2-a/. • VrH/„4/ ' [%i + \)\ \ Vj^+¥'I "■^^-

2</i + ^) ' V,---' + r^t\ (2« + 1) ! ^ Vr^+^/ '""'^ ^'

^ = A^L ^ ^/ nn-i ^K2»-l)!

2.Ta/i Vr' + ^' n = 1 (2m + 1) ! ^ V*-- + ^'

•2n-\

P.n-l(>)

):ioo)

_^ 3//(>t + 2/z)
Avliere

^+^U ■ (2m+1)! lvr^+7^/ ""^^j

*> I 5

Vr^ +

These series converge for Vr^+;?'>«, and are apphctible in this
region.

At the boundary, we have to put z = 0 and v = 0. Since

Pô\o) = h PL.Xo) = o,

2.4 (2n — 2)

1) For the first term (h = 1) of tlie second series we have to take — ;- P-l ('y").

Elastic Eqiiilibi-ium of Semi-Infinite Solid.

47

we have

/

0

J

J^{h-)cU- = -— ,

k a or

siuka — kacoska r/i \n ^ tt-/ 1 1

^-^; jAknak = F[ , —

Jrci? ^ ^ 3>- \ 2 2

5 (T

{ (101)

2a I ^. 2 a- / r ^2a ^ r /

for a ^ r. Consequently

{ii,%

II

1

47Ï.X+IJL) r

>(102)

for a ^ r.

§30. To ßncl the expressions for the displacement within the
loaded circle, we proceed as follows:

Making use of the power series of the Bessel function, we
have

0

a ,^„ ;,!(,,+ !)! \2aJ " ^"^
y ;_,.| ,inka-kacoska y^,j^.,.^^jj^.

_ 1 ^-. (-1/ / r V-^j
a „4i («!)' \ 2a/ ""\ a

(103)

where i2 stands for

0

By the aid of the formuke

48 -^rt. 7.— K. Terazawa :

^•;_;„. sin/l ,;_ , ^^_i 1

,/'

f

~^-''coü?uU =

l + x'

the evaluation of the function Il,„ (:r) can l»e undertaken. A httle
calculation will give us

IJ^^(x) = - — •|a: + tan"^a: — a;-taii"^ > ,

4 2 1 X )

ßi(x) = 1 — j:tan~^ ,

X

(L{x) = tan-'--- —*-- ,
X 1+a?"

Î2,{x) ^

(l+x')-
and in general

i2,„U) = (-1)"-' -^{-^1

{l + xj

Thus the integrals on the left liand side of (103) can be expanded
in accending power series of rja which probably converge for
limited values of r if the value of z is fixed. These series and
those found in (lOO) ha\'e a common region in wliich the}^ are
both convergent and tlierefore they must be congruent to each
other in that region. On the proof of tliis proposition we shall
not enter; but we shall find the region of convergency of these
latter series at the boundry. Let us take the first series of (103).
Expand the function //„+i(-^) for n^l into a power series of zja,
supposing zja to be sufficiently small, then tlie first term of it wiU
be {-l%i{%i-'2) !. Thus if we put ,? = 0 in the first series of (103),
its general term will tlien be

%i{-2n-'2)\ / r ^-"+^
n\ (n+1)! 2-"+^ \ a

Elastic Equilibrium of Semi-Infinite Solid.

4&

The series which lias this expression as its general term converges
obviously for the value of r smaller than a. Similarly for the
second series.

Since, for z = 0,

■i

IJJo) = — ,

-'^=(o) = l,
<J.Jo) = 2,

we have, after suimnation.
'" * sin Jia — }ca cos ka

/■ "^ sin ka — kacoa ka

(104)

ff»r r £ a.

Consequently we have

/y

16(/ + ;/> ' a\ -2 ■ a^/'

(105)

Fig. 7.

50

Ar*. 7. — K. Terazawa :

for ré a. lu the annexed diagram the approximate course of tlie
vertical displacement is shown in proper scale.

§37. By a similar process the distribution of stress can l>e
found. Here we shall calculate the stress at the boundary. It
ma}^ be shown that

/

"^ sin ka — ka cos ka

.2^2

k'a

J^[kr)dk = 0

(lOGj

and, therefore, as tlie expressions for the stress-components at the
surface we liave

H/x

27r.7 + /^) r^
27r(/ + //) ■ r

(107)

(.a)o = 0, (H-)(i = 0
for r ^ a, and

2
3//

)(108)

27r«-

1-4^*

a'

{^r\ = 0,

for r £ a.

The result of this example may be looked upon as a special
case of what has been discussed by H. Hertz in his papers'^
concerning the contact of two elastic bodies. He assumed the area
on which pressure acts to be an ellipse instead of a circle. If we
put b = a in his results, we get exactly the same formulae for the
pressure and for the vertical displacement. And therefore this

1) Oesammelte Werke, I, p. 154 — and p. 175.

Elastic Equilibrium of Semi-Infinite Solid.

51

example may be applied to the discussion of contact of an elastic
body upon another with plane-surface.

§38. Another application will be considered here. Suppose
we have a material loading of a semi-spheroidal form whose
equation is

-^ + ^ = 1. (^<0)

and of uniform density p. This load may be likened to the tidal
inequality in the North Atlantic ocean which afïects the gravity

measurement. In this case 77 will be replaced b}^ T^a gp j^^

before, the deviation of the direction of gravity produced by tlie
attraction of this load is given b}^

tan (Jj

^nd' g^ '/ k^a

Ji{kr)dk,

and the level-change due to tlie deformation of the ground arisen
from the load 1)y

tan^ = —

3^ À + 2/1

-./-

sin ka — ka cos ka
k'a

J.ykr)dk.

The evaluation of this integral can be undertaken in a manner
similar to those of (99) and will appear to be

./

■" sin ka — ka cos ka

kra

J,{kr)dk

r
2^

nr
4a

\ ■ -^ a a /. or I ^

sm — /I — ÏÎL- } , a é r

\ r r V r' /

r é a

(109)

Tlius we have

tan ^ = - J^ . ^^1 . Ji- (sm -^^-JL /i _ Zl , ,1 10)
^ 27ra" g' a \ r r V /•' / '

52

Art. 7.— K. Terazawa :

tan^ = —

_ _ 3/7 (/ + 2;^)

Ind' 4//(>^ + ju)

L|sm--^-^/l_4},(lll)
i \ r r y r^ )

for the point r :> a.

In the next diagram, the general march of the function

assin"^ — — \— is exhibited, where x is the ratio of the distance

of the point under consideration from the centre of the loaded
circle to its radius. The course of the curve is very similar to that
of Fig. 6, except at the point very near to the edge of the loaded
area, where, in this case, it remains finite.

For example, with the same assumption regarding the
material constants of the earth as in the former example, and
supposing that the total amount of the load is the same as before,
i.e. the mean lieight of the tide is one metre, anda=2xl0^, >•=
X 10^ cm, we have

Elastic Equilibrium of Semi-Infinite Solid.

58

and

iP = 112x10-« = 0"-0023,
<p = 3-21 X 10-« = 0"-0066,

,/, + <f = 4-33 X 10-« = 0"-()09

nearly the same as the results in the former example.

If we suppose the place of the observation to ])e very near to
the edge of the loaded area, tlien

,/> = 5-0x10-« = 0' -01.
0 = 14-4x10-« = 0"-03,

and

greater than the maximum of the direct effect of the moon.

Example V. •

§39. Lastly, we shall take another example in whicli tlie
normal pressure of the form

A.r) =

II

1

27r« ^a?—r-

(112)

is applied to the boundary within a circle of radius a. which is
otherwise left free from ti'action. This problem has 1)een «iiscussed
also by Boussinesq and others, and the expression for the vertical
displacement at the l)oundarv has been found. In this case the
function Z{k) becomes

^(«) = - ,. I' ^-ir — ; =-^ — ^mka. (113)

Consequently we have

àrra/i

II

4:7ta{)^ + fl)

..CD

dl

(114)

54

Art. 7. — K. Terazawa ;

Uz

■^nafx ''

/■ a
c~''^ sin ]i:aJo{Ji:r)dk

+

47Ta{X + fx]i.

sin^-aJ",/^'^')

äli

(114)

The integration can be carried out by expanding öin/;« into a
]»ower series and making use of the formulae (42). Thus we
olitain

77

-2-

(-Î)"

- /

4to(;.+/^) i^^ 27?(2w + l) \ v^- + ^-

> (115)

where

VrH/

These series apply for the region ^r^+z^ > a.

As in the last example, the expressions which may be applied
for small values of r and z can be found by using the power
series of the Bessel function and the formula

./

Ä;"«'^* sill Art . dli =

n !

{z' + a?y

n+l

sm

r(n+l)tau-i-^1

Thu^

Ij For 71 = 0 we have to take P,-^(v).

Elastic Eqiiilibrium of Semi-Infinite Solid.

55

i, = — . > — ^^ ^ — - — - sm •2;i + -2)ti'

_ A X- (-i)"(2^0

■^-a{). + [x) t,n\ («+!)! 2"-^"+i \ ^z' + d'

n

^ (-lÄ2n)!/ r V^",

^2 Zj ^^|\29rn \ /,2_i_^2y

4-a// V-^'+a^^o (w!)'2-» \ V'^H

sin(2» + l)^A,

sill (2??,+ \)ip

+

where

-'^^^±^i:^=^^:^(-..^) -'w.-

4-ii(;. + ^);/„^o (w!f2--^" \ Vr + ^^

cA = tail"'

_i a

V(116)

vi40. At the surface {z = o) they reduce simply to'^

for /'é <i'. and

{uX = -

fj a — y/à' — r-

1

1

47T{À + /i) ' r

A- A-\- a)na

(117)

(118)

for r ^ a.

As seen from the formula (117) the vertical displacement is
constant over tlie loaded area, and therefore this solution is
applicable when an absolutely rigid body of circular base is pressed
normally against an infinite elastic body; the problem has been
attacked by various writers from this point of view.

§41. Similarly the expressions for the stresses at the surface
will be found to be

1) For n = o we have to take -L instead of zero.

2) If we wish to know the expressions for displacement only at the surface, these can
be olitained with less calculation. See Lamb I.e.

56

Art. 7. — K. Terazawa :

(rr)o

n

m

l^

a—'s/cC^—r-

+ //

-}•]

_ //

{zz\ =

dl 9)

27ra, 's/f^—T-

for r <: a; and

{fr\ =

/7^

1

^ '" 2<yl + />'.) 7--^

{Jz\ = 0,

(1-20)

for 7' :>• a.

The stress at the peripher}' of the loaded area is iiifinitly
great, so that the elastic body would 1)e ruptured at the edge. The
present problem may, therefore, tlirow considerable light upon the
explanation of the phenomena of punching.

§42. The expressions for the stress-components in the
interior of the body which, so far as I am aware, have not been
treated by any one can be found without using any special
functions. We shall take here a simple case, for example, in
whicli the material is incompressible.

Since the integral

y'>-^Vo^ç/)r7/ =

1

I

is valid for all values of C and ç, real or complex, provided the
real part of ~ is not smaller than the al^soluto value of the
imaginary part of ç, putting Ç—: — ia, ç = r in tliis integral and
equating the imaginary parts in both meml)ers we have .

fe-'-'^ sin l-a. J,:krâl- = ^^llli£±f-Z^ [Y2\\

Elastic EquiliVn-ium of Semi-Infinite Solid.

57

where *S" stands for

S' = ^[/' + i--a?f + 4.d'z'. (122)

For z = ^). til is formula is still applicable, if we take

8' = r — d' for r >- a, |
T= d' — y- for y <; a. J

Siiuilarly. putting

r = ^S'-~{r + y'-d'), I

(123)

we have

/"-*' ■ 7 TH \in aU^ - r- + d')Q + z{z' + r^ + a.^)P ..o.x

P

"'shila . jn-yMl-

aQ-zP

(125)
(120)

Til us. for the case of incorapressibility, we have

rr =

// jP z(aQ-zP) z[dz'-r' + d)Q + z(z--\-r' + a')P]\ ]

r-^S'-

S"*

7'

dß

^_ // (_P_ziciQ-zI>)\

- ^ _ // { P ^^a{r-i' + d)Q+j(f+r^^-)P'] \

> (127)

zz = — ., .^ ■»-- +

Fr = —

/y^ f r(-r + r--a-)P + 2a.^g |
2V2WI Ä" i'

for the stress coinponent^

V. Boussinesq's Problem,

§43. The problem of Lamé and Clapeykon is a special case
of those known as Boussinesq's, which can be stated as follows:

58

Art. 7. — K. Terazawa

A limited portion of the surface of a large mass of elastic
material is subjected to local stress or to local deformation, it is
required to find the strain and stress in the bod}^ due to these local
disturbances.

In the case of symmetry about an axis perpendicular to the
surface of the body, this problem may be discussed, in a general
way, by applying our method of analy.-is. We shall sketch the
results here as an addendum.

The t3q3ical solution of the equilibrium, in this special case, is

•128)

and

? (129)

etc. ;
with the relation

in which the components ?/e and zd follow from the supposition
that J is nil. Since these do not give very interesting i-esults, we
shall not consider them here.

§44. Case in which all the surface tractions are given.

We suppose first of all that

zz = ZJdhr),
B = B.J,{l-r)

(130)

Elastic Equilibrium of Semi-Infinite Solid.

59

are given at z — o. then we sliall liave tlie following values for the
arbitrarv constants:

B =

/ + -2//

i? +

1

-In'/.+ ii'l -2 /.+ a)l-

z,

D=--^±^/^Z+ 1

1

-R,

C= ^ (Z-B).

Putting these in (128) and (129) we have the solution corre-
sponding to the boundary conditions (130).

If the traction over the surface is given in the form

zz = p{r),
T>- = T(r),

(131)

p and r being any prescriljed functions of r, the corresponding
solution can be obtained by making use of the integral theorem
(26), on the supposition that the functions p(r) and r(r) do not
violate that theorem.
Thus

Î,,. = - /"^(^LrziA-i -i? A- 1+ ^• + '-^^-' . B(k)

- -— ^ . Z\ A) \e-''J.(kr'dk,

2(/ + /i)A /

u,= - f ( -[ZA -i?(A] + -i;t?L.Z(A)
./ \2a ^ 2uU+n^k

2(/. + /.)A f

(132)

and

rz = f[kz[Z(k)-B k)] + Z[]i)}e-''%(kr)dk,

0

I? = ßkz[Z[k)-Bk)] + Bxk)]e'''J,{kr)dk,
etc. ;

(133)

60

Art. 7. — K. Terazawa :

where tlie fu^ctiün^^ Z and B are determined by

0

(134)

If we put ry7-) = 0 in tliis solution, we get as a matter ol course
tlie solution (30) and (31).

§45. Case in which the normal traction and radial displace-
ment at the surface are given.

If

II,. = ii(r),
Tz = pir)

(135)

are given at z — '<), tlie corivsjionding solution is:

0 / V ' i

./ 1 2/.:;> + 2/7.) ^ ^ - ^ -J 2//;7 + 2//)7.-

\ (136)

and

zr

= /""( ''t^+-'"^'' [Z(/,-) + 2«Är(//)] + Z/.-)l'-'-J-,,(ÄT)rfA-,
./ l / + 2// /• + 2«

) (137)

>^. + 2/^ )

+ 2/.

etc. ;

in which T^and Z are iïiv

en i>\'

Elastic Eqiülibriuui of Semi-Infinite Solid.

61

TJ ' k) = ^ / it\o.)J^lka)ada,

0

Z{z) = hi p{oM,lU>.a(lo..

(138)

46. Ccise ill wliicli the tangential traction and normal
displacement at the sm'face are given.
If

F? = r(r),
Uy = w(r)

are given at .z = o, then we obtain

(139)

''-vi-^él^rt^^^^

+'i/^

WiJc)

+ \V(k)\e-''J,ßr)dl- ;

> (140)

and

Si = - /^l j(l+/^^_Ä^ [i^(A-)2/.A-IF(A-)]+ '^4^i^lF(Ä-)
^ I / + 2/^ /. + '2/i

- --^ E(A0l-'^^/„(A7-)f7Ä-,
rr = - f[-A^±^ [B{k) + '2!.kW(k)]
+ i? (A) }.-■-' J',(/,r)'7/.-,

) (141)

etc. ;

wher(

62

Art. 7. — K. Terazawa

B{k) = l- 1 z{a)J,{ka)a.da,

0

W{h) = l- j w{a)Jlko)ada.

(142)

§47. Case in which both the displacement components at
the boundary are given.
If

ti^. =. uir), \

(143)
u, = io(r) j

are given at z=o, we have

,,^ ^ / I ^(^■ + /^^^'[T7(A-)- TJ{k)'\ + W{k)\e-^.likr\lk :

./ I l + dfX J

and

+ 3^

2^(;. + 2^)A- w{k)\e-^^Ukr)dk,

etc. ;

Ü'(A;) = k I ii{o)J ^(ka)ada,

' 0

/.CO

TF(Ä;) = k / w{a)JJ^ka)ada,

in which

144.

(145)

(146)

Cambridge, April 2-', 1910^.

Elastic Equilibrium of Semi-Infinite Solid. ßg

CONTENTS,

I. Introduction

§§ 1— 4.

II. Solution of Equation of Equilibrium.

§ 5. Differential equations concerning the equilibrium of a semi-infiuite

elastic body.
§§ 6 — S. Solution for displacement.
§ 9. Solution for stress.

III. Lamé and Clapeyrons Problem.

§10. Statement of the problem and typical solution.

§11. General solution for any given normal pressure at the boundary.

IV. Examples in the Case of Symmetry.

§12. General solution for the case of .symmetry.

Example I.

§13. Normal pressure concentrated at a point.

§14. Failure of the solution.

Example II.

§§15 — 17. Diffused distribution of normal pressure according to the law Al{d-+i-)l.
§§18—20. Application to the problem of rupture of foundation. Stability of a
mountain.

Example III.

§21. Normal pressure distributed uniformly over a circular area.

§22. Evaluation of certain integrals.

§§28 — -27. Expressions for displacement.

§§28 — 29. Formulae for stress

§30. Singularity at the edge of the loaded area.

§31-. Application to the rupture of the surface.

§§32 — 34. Eff'ect of a material loading on the water-level measurement.

^jA Art. 7. — K. Terazawa :

Example IV.

§§85 — 37. Normal piessnie distributed over u circular area according to the law

§38. Application to the water-level measurement.

Example V.

§§39 — 42. Normal pressure of the form ^/{a" -y')^ distributed over a circular area.

V. Boussinesq's Problem

§43. Statement of the problem.

§§44 — 47. Solutions when at the surface all the surface tractions are given ; the
normal traction and radial displacement ; the tangential traction and
normal displacement ; the radial and normal displacements.

Published December 7th, 1916.

JOURNAL OF THE COLLEGE OF SCIENCE, TOKYO IMPERIAL UNIVERSITY.

VOL. XXXVII., ART. 8.

Recherches sur les spectres d'absorption des
ammine=complexes métalliques.

Par

Yuji SHI BATA, Fuf/akii.shi.
Laboratoire de Chimie minérale de l'Université Impériale de Tokio.

II.

A) Spectres d'absorption et conductibilités éleclrolytiques des

solutions aqueuses des nitro-ammine-complexes cobaitiques,

qui sont des coordination-polymères.'*

Fait en commun avec K. Matsuno, iiigakiuhi.

Arec 5 /h/ lires.

Si un métal Me forme deux ions complexes s' enchaînant
tantôt avec un groupe atomique A, tantôt avec un autre groupe
atomique A' et si, de même, un autre métal Me' se comporte
comme le premier, on appelle coordination-isomères les deux sels
complexes suivants ainsi formés :

[MeA] • [Me'A'] ' et [MeA'] * [Me 'A] '

cette catégorie de 1' isomeric est aussi possible, dans le cas oii les
atomes métalliques centraux dans les ions négatifs et positifs ne
sont pas différents. Il y a de nombreux exemples de tels isomères.
En voici quelques uns :

[Pt(NH3),] [PtCl,] et [Pt[^^jj3) J [^^cf ']

[Co(NH3)e] [Co(NO,).,] et [Co[^g^] [Co^^^^^^;]
[CrenJ[Cr(CA).s] et [Cr ^-^^ J [Cr^^f ^^^'] etc.

1) Comparer „Neuere Anschauungen auf dem Gebiete der anorganischen Chemie" par M.
A. Werner, P. 260. [1£09, Friedrig Vieweg u. Sohn, Braunschweig].

2 Art. 8.- Y. Shibata :

De plus, dans le cas où les noyaux des ions négatifs et positifs
consistent en un même métal, la polymérie est aussi possible et les
complexes, qui sont en une telle relation les uns avec les autres,
sont appelés coordination-polymères.

Les six complexes suivants, dont j'étudie ici les spectres d'ab-
sorption de leurs solutions aqueuses, appartiennent à cette dernière
catégorie de 1' isomeric, et peuvent être représentés par la formule
générale n[Co(NH,KNOo)3]''"'' :

[Co(NH3)gN0,J [Co(NH,,),(NO.)4]o (n = 3)

[Co(NH3),^'^;[^]] [Co(NH,,),(NO,)J (n=2)

[Co(NH:0,^I^^<^]] [Co(NH,),(NO,)J (n = 2)
[Co(NH3)5NO,], [Co;ONO)J, (n = 5)

[Co(NH3X^^^[^]] ^[Co(ONO),] (n=4)

[Co(NH3),^g^[^]] ^[Co(ONO)J (n=4)

Dans le travail de l'un de nous^\ l'auteur a démontré, que les
anions et les cations de ces sels complexes sont chromopliores
remarquables et, que leurs solutions aqueuses donnent deux ou trois
bandes d'absorption très nettes dans l'échelle spectrale entière,
quand ils forment des sels complexes ordinaires, en se liant avec
les anions ou les cations simples, comme Cl', SO4", Na' ou NH^ etc. ,
selon leurs signes électrolytiques.

Le but du présent travail est donc d' examiner, quelles pro-
priétés optiques ont les six sels complexes donnés plus haut, dont
les anions et les cations sont, à la fois, complexes et chromopliores,
sous le rapport de l'absorption des rayons, et si les ions complexes
et chromatiques exercent quelque influence les uns sur les autres,
quand leurs solutions absorbent des rayons.

1) s. M. Jörgensen : Zeitschr. f. anorg. Chem. 5, 175 (1894).

* Parmi 8 polymères préparés par Jörgensen, les deux qui contiennent le leutéo-com-
plexe [Co(NH3)6]"' n'ont pas été étudiés dans ce tra.vail, à cause de leur insolubilité dans l'eau.

2) Tuji Shibata : Journ. of tlie College of Science, Imp. Univ., Tokyo. Vol. XXXVII, Art.
2 (1915).

Eecherches sur les spectres d'absorption des ammine-coniplexes cobaltiques, II. 3

Afin do comparer plus facilement, les absorptions de ces sels
à celles de leur corps mère, trinitro-trianmiine cobaltique, nous
avons pris 1/n molécule de chaque sel et préj^aré des solution de

1Ö0 i^^squ' à -jj^ööö d'équivalent. Les solutions ainsi préparées sont
toujours tout à fait stables, et, en conséquence, la loi de Beer sur
ral)sorption des rayons est complètement satisfaite par elles.

En examinant les courbes d'absorption de ces six polymères
complexes, on peut remarquer facilement, qu'ils peuvent être classés
en deux catégories: les polymères qui possèdent 1' anion complexe,
diammine-tetranitro cobaltique [Co(NH3)2(N02)4] ', dans leurs
molécules, et les autres, qui contiennent l'héxanitrite cobaltique
[00(0X0)5] '", comme leur anion.

Ceux qui appartiennent à la première catégorie montrent une
bande d'absorption très caractéristique à 2200 de fréquences. Elle
ne devrait peut-être pas être appelée ,,la bande" dans le sens strict,
parce qu'elle est tout à fait plane, et ne montre aucun point étroit
du maximum d' absolution, en présentant l'aspect du point d'in-
flexion de la courbe mathématique. Les spectres d'absorption des
solutions des sels de cette catégorie montrent encore une seconde
et une troisième bande respectivement à 3000 et à 4000 de fréquen-
ces; la seconde de ces bandes est commune à tous les nitro-ammine-
complexes cobaltiques, tandis que la troisième est caractéristique de
r anion tétranitro-diammine cobaltique [Co(N02)4(NH3)2] '.

Les polymères complexes de la deuxième catégorie, qui con-
tiennent r anion complexe, héxanitrite cobaltique, montrent deux
ou trois bandes d'absorption très nettes et normales, dans l'état de
solution aqueuse.

En résumé, la propriété optique concernant l'absorption des
rayons des six polymères complexes, qui sont formés cl' anions et de
cations à la fois complexes et chromatiques, est généralement addi-
tive, sauf seulement le cas de la première bande d'absorption a-
normale des trois sels déjà nommés.

Pour rechercher d'où provenait cette dernière anomalie, nous
avons alors entrepris la mesure de la conductibilité électrolytique
des. solutions aqueuses des polymères complexes, parce que si l'ano-

4 Art. 8.— Y. Shibata:

malie est causée par un changement quelconque do la constitution
chimique de ces sels complexes à l'état de solution, leurs conducti-
bilités électroly tiques, qui rendent compte des nombres des ions,
doivent montrer aussi quelques anomalies. Pourtant les résultats
de ces mesures, comme on verra dans la partie expérimentale, se
sont trouvés être tout à fait normaux. Alors on ne peut plus
attribuer la cause de l'anomalie des premières bandes d'absorption
des poljmières de la première catégorie à la transformation de leur
constitution chimique. Nous avons donc essayé d'expliquer cette
anomalie par le fait que les oscillations des électrons de valence
relâchés,^-* s'attachant aux atomes cobaltiques dans les deux ions,
cation et anion, des polymères complexes, sont fortement limitées
par leurs connexions mutuelles. Nous avons renvoyé la discussion
précise sur ce sujet dans la conclution à la fin.

Partie expérimentale.

1.) Spectres d'absorption des solutions aqueuses des polymères
complexes de la première catégorie.'

Les trois sels, — [Co(NH3)5NOo]" [CoCNHsKNOo),],',
[CoCNHsXno^D * tCoCNHsKNO,^^^ et [Co(NH3).no;(6)] *
[Co(NH3)2(N02)4] ' — qui contiennent 1' anion monovalent, diam-
mine-tétranitro cobaltique, ont été groupés dans la première
catégorie des polymères complexes présentement étudiés. Ils ont
été préparés par la méthode de la double décomposition, en mélan-
geant la solution aqueuse saturée du cobalt-diammine-tétranitrite
de potassium, [Co(NH3)«(N02)4]K respectivement avec celles des
chlorures pentammine-mononitro-cobaltique (xantho), cis-dinitro-
tétrammine-cobaltique (flavo), et trans-dinitro-tétrammine-cobal-
tique (crocéo).

Le xantho- et le flavo-diammine-tétranitro-cobalt ont été
étudiés dans les concentrations respectivement de 0.0025-0.000025

1.) Comparer le dernier travail de l'un des auteurs, Yuji Shibata ; loc. cit.
* Pour ce qui concerne la méthode de l'étude de l'absorption des rayons et des reiîrésen-
tations graphiques, l'un des auteurs, Yuji Shibata, l'a décrit dans son dernier travail, loc. cit.

Recherches sur les spectres d'aljsorption des amminc-complexes cobaltiques, II. 5

équivalent et de 0.01-0.0001 équivalent, tandis que le crocéo-
diammine-télranitro-cobalt n'a puêtre étudié que dans la solution
de la concentration de 0.0001 équivalent, à cause de sa faible
solubilité dans Teau.

Fig. I

Frônuenco

03 O

r!

[Co(NH,),N(3,] [Co(NH,),(NO,),],

La figure I représente la courbe d'absorption du xantho-
diammine-tétranitro-cobalt. Les lignes courtes, tirées horizontale-
ment dans la figure, significient les épaisseurs recalculées corres-
pondant à O.OnUl écpii valent; les chiffres placés aux deux côtés
des lignes expriment naturellement les épaisseurs (à droite) et leurs
logarithmes (à gauche). Comme nous l'avons déjà indiqué, la
bande d'absorption à 2200 de fréquences de ce sel est très caractéris-
tique, ayant presque la forme du point d'inflexion de la courbe
mathématique.

6 Art. 8.— Y. Shil:>ata :

Les deux autres bandes sont, au contaire, tout à fait normales
dans les nitro-ammine-complexes, dont au moins les deux groupes
de nitro se placent aux positions de trans l'un et l'autre. Dans ce
cas, en effet, 1' anion remplit cette condition.

Fig. II

Fréquence

avjo 3000

\

\

\

V
v
\\

NX

"^

V

X^'

looo© i^s

[Co(NH,),(NO,X,][Co(NH,,,),(NO,)i]

Flavo

Crocéo — —

Dans la figure II, on aperçoit les courbes d'absorption des deux
sels qui contiennent les cations complexes respectivement du crocéo
cobaltique et du flavo cobaltique. La forme de leurs courbes
coïncide parfaitement avec celle du complexe précédent.

Eecherches sur les spectres d'absorption des ammine-complexcs cobaltiques, IL 7

2) Spectres d'absorption des solutions aqueuses des polymères
complexes de la deuxième catégorie.

Les trois sels appartenant à cette catégorie — xantho-, ßavo- et
crocéo-héxanitrite cobaltiqne ont été préparés de la même manière
que les sels précédents. Cependant les solubilités des produits
ultimes étant considérables dans ce cas, il a fallu qu'on fasse refroi-
dir les solutions mélangées avec le mélange réfrigérant, pour les
faire se séparer par cristallisation.

La figure III représente la courbe d'absorption du xantlio-
héxanitrite cobaltique. Elle a deux bandes d'absorption à 2100 et

Fig. Ill

Fréquence

I500 2000 250O 3000 3SûO <r

OO *5C

«)

\

\/

\

Tß

\

30

\

.^

2.5

\

\

20

\

1.0

^

0

M

V

'.OOOO

p

ai

a>

Ö

i-S

Oi

B

a

pj

a>

m

m

lOCO

[Co(NH3)5NOJ:, [Co(ONO)«].,

à 3000 de fréquences; les deux bandes paraissent aux épaisseurs de
solution sensiblement plus grandes que celles des autres nitro-
ammine-complexes.

Art. 8.— Y. Shibata:

Fig. IV

Fréquence

,5 qp

\

■

1

u

1 \

\ \
\ \
\ \
\ \

\ \

//\\

•■

'\ / \

\

H

►ö

[Co(NH;0,(NO,),].[Co(ONO),]

Flavo

Crocéo

Les courbes d'absorption de flavo- et de crocéo-héxanitrite
cobaltique, qui sont tracées dans la figure IV, sont presque les
mêmes, pour leurs formes, que celles des chlorures de flavo et de
crocéo; c'est-à-dire que les flavo- complexes ne montrent que deux
bandes d'absorption, tandis que les crocéo-complexes en ont trois,
dont les deux premières coïncident dans les deux cas pour la
position et pour l'épaisseur, où elles paraissent.

En résumé les polymères complexes appartenant à cette caté-
gorie ne montrent aucune anomalie à l'égard de l'alisorption des
rayons.

Recherches sur les spectres d'absorption des ammine-camplexes cobalticiues, II.

3) Conductibilités électrolytiques des solutions aqueuses des
coordination-polymères cobaltiques.

M. M. Werner, INIiolati et leurs élèves ont mesuré les conducti-
bilités électrotytiques des solutions aqueuses des plusieurs ammine-
complexes métalliques, dans le but de connaître le nombre de leurs
ions dans l'eau, en comparant les valeurs de leurs conductibilités
moléculaires.

Nous avons suivi ces exemples de mesure de conductibilités
électrolytiques des pol^nnères complexes cobaltiques, pour connaître
aussi les nombres de leurs ions dans l'état de solution aqueuse, pour
la raison déjà donnée dans l'introduction.

Comme les chiffres, que nous allons donner ci-dessous, l'ex-
pliquent bien, les résultats des mesures des conductibilités électro-
lytiques ont été tout à fait normaux, c'est-à-dire qu'il n'y a eu
aucun changement des constitutions chimiques des polymères
complexes au moment de leur préparation.

Quant à la méthode de mesure des conductibilités électroly-
tiques, cette mesure a été exécutée d'après le système d'Ostwald,
et les dilutions des solutions observées ont été prises, selon les
solubilités des polymères complexes, entre 25G et 1G384, les tem-
pératures d'observation étant toujours 25°.

Les polymères complexes cobaltiques étant des electrolytes
assez faibles, leurs conductibilités moléculaires n'atteignent presque
pas à la valeur constante, quand même les dilutions des solutions
sont suffisamment grandes. De même, la tendance minima de
l'hydrolyse des solutions fait monter la valeur de la conductibilité
moléculaire de plus en plus, quand on les mesure, pour un sel
d'une certaine dilution, à certains intervalles de temps. Par cette
raison, il nous a fallu faire la comparaison des valeurs de
nos mesures avec celles des auteurs déjà cités, dans la dilution
fixée ; pour cette dilution de comparaison nous avons choisi la
dilution 1024.

10 ^rt. 8.— Y. Shibata :

Table I. Table II.

[Co(NH3). NO J [Co(NH3), (NO,),], ^ ,y^ F ^^ (NH.)^

-^7/-n-i f \ ^(Conductibilité (Conduct. •- ^ ''^^ ^ ^'- ^-^-' L (NO.j^J

V(Dilution) k^ spécifique) /^moléculaire) y j,

10'24 22-00x10-5 225-2 256 36-48 xlO-^ 93-35

2048 1216 // 249 0 512 20 21 // 103-5

4096 6-798 // 278-4 1024 10-69 // 109-4

8192 3-740 // 306-3 2048 5-982 // 122-6

4096 3-399 // 139-2

Table III. Table IV.

[Co(NHa(NO,),['>][Co[^^^] [Co(NH3),NO,]3[Co(ONO)J,

V k /^ V ki k^ ixy H.2

2048 87-98x10-6 180-2 256 33-99x10-* 870-2

4096 49-85 // 204-2 512 18-24 // 934-0

8192 29-33 // 240 3 1024 9-84 // 93-48 x 10-M008 0 957-2

16384 19-68 // 322-4 2048 5-157 // 49-85 // 1056-0 1021-0

4096 2-876 // 26-71 » 1178 0 1093-0

8192 1-558 // 14-3S // 12760 1250 0

Table V. Table VI.

[Co(NH3),(NO.,),g|]3 [Co(ONO)J [Co(NH3),(NO,),[^l]3 [Co(ONO)e]

V k ^

256 12-26 X 10-* 313-8

512 6-50 // 333-0

1024 3-32 // 340-3

2048 1-824// 373-6

4096 9-906 x 10-^ 405-5

8192 5-540// 453-8

Pour permettre la comparaison, nous citons ici les résultats des
mesures des conductibilités moléculaires, obtenus par M. ]M.
W^erner, Miolati et leurs élèves* pour les sels complexes de platine
et de cobalt dans la dilution 1000.

*A. Werner et A. Miolati : Zeitschr. f. phys. Chem. 12, 35 (1894).
A. Werner et Ch. Herty : „ „ „ „ 38,331(1901).

V. Kohlschütter: Ber. d. deutsch. Chem. Gesell. 36, 1151 (1903).
A. Miolati : Zeitschr. f. anorg. Chem. 22, 445 (1900).
A. Miolati et Pizzighelli : Journ. f. Prakt. Chem. 77 417 (1908).

V

1024

k
37-40x10-5

391-8

2048

20-48 //

419-6

4096

10-69 //

437-5

8192

6-503 //

532-7

Rechcrclics sur les spectres d'absorption dos auiuiinc-complexes cobaltiques, II. H

a) Complexes qui consistent en deux ions

[Pt(NH.O;, Cl] Cl /^r= 115-8

[Pt(NH,)Cl5] K 108-5

[Co(NH,)4CO,]Br 106-0

[Co(NH3), (NO,),[i]]Cl 100-7

• Tco^^^^^^ "1 Tco^^^^'-'^n 61-19
L (NO.XI'lJ L (NH3)J ^^^'^

(Temperature d'observation 25°).

Notre valeur pour /^ (dil. = 1024) du flavo diamniine-tétranitrite
cobaltique, qui est donnée dans la table II, coïncide alors très bien
avec celles des complexes consistant en deux ions de platine et de
cobalt, tandis que la valeur donnée par les auteurs cités ci-dessus et
celle donnée par nous pour le même complexe du flavo cliammine-
tétranitrite cobaltique diffèrent plutôt d'une façon considérable.
Mais nous croyons que notre valeur est meilleure que celle de
M. M. Werner et Miolati, parce que notre conductibilité moléculaire
de ce complexe montre une excellante coïncidence avec la majorité
des complexes consistant en deux ions de platine et de cobalt.

Alors il est bien sûr que ce sel complexe possède la formule

normale de [Co(NH,), |^;|!J] [co(NH,),(NO,),] et qu'il n'y a

eu aucun changement de sa constitution au moment de la prépara-
tion par la double décomposition entre le chlorure de flavo
cobaltique et le cobalti-diammine-tétranitrite de potassium.

Quant à la conductibilité électrolytique du crocéo-diammine-
tétranitrite cobaltique, isomère stéreocliimique du sel précédent, elle
n'a été mesurée qu'à partir de la dilution 2048 à cause de sa faible
solubilité. Nous n'avons pas, par conséquent, sa valeur de /^ à la
dilution 1024 pour faire la comparaison. Au surplus, on obtient
toujours les sensiblement hautes valeurs de /^ pour ce sel, parce
qu'on a besoin d'un peu plus de temps pour sa dissolution complète
dans l'eau.

b) Complexes qui consistent en trois ions

[Pt(NH,),Cl,]CL /i = 228 9

[Pt(NH4(N0,,),] (N0.,)o 234-4

[Co(NH;,), (NO,)] (NO,). 231-4

12 Art. 8.— Y. Shibata :

[Co(NH3).5 Br] Br, //=257G

[Cr(NH,)5 Cl] CL 2Ö0-2

(clil.= 1000; temp. =25°)

Notre valour /^ (dil. = 1024; temp. =25°) pour le xantho-di-
ammine-tétraiiitrite cobaltiquo, qui est donnée dans la table I, est
225-2 et coïncide encore très bien avec les ^\a]eurs pour les com-
plexes consistant en trois ions de platine, cobalt et de chrome,
dont les chiffres donnés par M. "SI. Werner et Miolati sont cités
ci-dessus.

c) Complexes qui consistent en quatre ions

[Co(NH3)5 H,0] Bi-, // = 412-9

[Co(NH3)3 (H,0).J CI3 ' 383-8

[Co(NH3), (H2O),] Biv 399-5

[Co(Nn3),] (N03)3 421-9

(dil. = 1000; temp. =25°)

Dans les tables V et VL on voit nos valeurs des conductibilités
moléculaires pour le flavo- et le crocéo-héxanitrite cobaltique.
Ces valeurs /^io2i=391-8 et /^]024^340-3 montrent encore une coïnci-
dence satisfaisante avec celles des complexes cobaltiques consistant
en quatre ions cités ci-dessus.

d) Complexes qui consistent en cinq ions

Comme nous l'avons montré dans la table IV, notre valeur /^m4
pour le xantho-liéxanitrite cobaltique [Co(NH3)5N02] 3 [Co(ONO)g]2
est à peu près 1000, tandis que celle mesurée par M. M. Werner et
Miolati'^ pour le même sel est de /^2oo:,=572-2 et pour [Pt(NH3)6]Cl4,
elle est de «1000= 522 9. Par conséquent notre valeur est sûrement
trop haute pour le sel de cinq ions, quoique nous ayons exécuté
deux séries de mesures aussi attentivement que possible.

Cette haute valeur extraordinaire est causée très probablement
par l'hydrolyse partielle de 1' anion, [Co(ONO)6]"' dans la solution,
parce que cet ion est assez instable dans l'eau, comme l'un des au-
teurs l'a indiqué dans le cas de l'étude spectroscopique de la solution
aqueuse du cobalthéxanite de natrium dans son dernier travaiP^

1) loc. cit.

2) Yuji Shibata: loc. cit.

Recherches sur les si^cctres d'absorption des amininc-complexes cobalticmes, II. J 3

En effet, nous avons ()l)tenu encore les trop liantes valeurs de
p- pour le cobalthéxanitrite de natrium; nous allons en montrer
les chiffres dans la table suivante :

Table VII.

[Co(ONÜ)J Na,

V

k

II.

256

16-62x10-^

425-4

512

8-798 //

450-5

1024

4-532 //

464-0

2048

2 837 //

478-6

4096

1 -268 //

5190

La valeur de l'-wn est alors 464U, tandis que celle d'un com-
plexe de quatre ions compte env. 390, comme nous l'avons dit plus
haut à propos des observations de M. M. Werner et Miolati.

Maintenant ce que nous venons d'apprendre des résultats de
nos mesures de conductibilités électrolytiques des solutions aqueuses
des coordination-polymères, est que les doubles décompositions,
dans le cas des préparations des polymères, se font tout à fait
régulièrement, et peuvent être représentées par la formule générale
suivante:

m[EjX,+n[EjMe,„=[Rj,,[Rj,-f-mnXMe.

Conclusion.

Comme nous l'avons indiqué ra et là plus haut, il ne faut pas
attribuer la cause de l'anomalie de la première bande d'absorption
des trois polymères complexes, qui contiennent 1' anion [Co(NH3)2-
(N02)4], au changement de leurs constitutions chimiques, mais
cette cause doit être attribuer absolument à la restriction de
vibrations des électrons de valence relâchés, qui s'attachent aux
atomes de cobalt dans les ions complexes.

Avant de rentrer dans la discussion sur cett supposition relative
à la restriction des vibrations des électrons, nous voulons exprimer
une opinion hypothétique, qui est tenue par l'un de nous dépuis
long temps, sur l'état des sels dissous dans les solvents, spéciale-
ment dans l'eau. L'un de nous pense que les anions et les cations

X4 Art. 8.— Y. Shibata :

d'un certain sel dissocié dans un solvent ne se trouvent pas
séparé sans ordre, mais qu'ils sont liés encore par couple; l'affinité
entre eux est naturellement très affaiblie peut être par l'hydratation
des ions respectifs. Maintenant si l'on fait passer un courant
d'électricité dans la solution, cette liaison légère entre les ions
est coupée, et leurs mouvements vers les électrodes sont ensuite
observés.

Cette supposition est aussi appuyée par le fait que les
pouvoirs rotatoires des solutions des sels optiquement actifs sont
fortement influencés par les anions (ou cations) qui s'accouplent
avec ces cations (ou anions) optiquement actifs. Piir exem|)le,
dans le cas des ammine-complexes cobaltiques optiquement actifs,
dont la plupart ont les cations complexes asymétriquement
construits, les pouvoirs rotatoires de leurs solutions aqueuses
changent régulièrement d'après la grandeur des anions'^'', bien que
l'activité optique ne soit causée absolument que par la structure
asymétrique des cations, et que les anions n'aient aucune relation
avec elle. Si on admettait l'idée que les ions négatifs et positifs
se trouvent séparément dans la solution, ce dernier fait serait
évidemment improbable. Quant à l'espèce d'électrons de valence,
qui sert à lier légèrement les anions et les cations dans la solution,
elle ne peut être naturellement déterminée en général. Mais
revenant à notre sujet, nous osons dire que, au moins dans notre
cas, ce sont les électrons de valence relâchés attachant aux atomes
cobaltiques dans les ions complexes, qui les lient dans l'eau.

Prenons d'al)ord en considération les nombres des valences
auxiliaires des atomes cobaltiques dans les anions et les cations, qui
forment la molécule de chaque coordination-polymère :

*Comparer A. Werner : Ber. d. deutsch, ehem. Gesell. 44 1887 (1911)

„ 3272 „

A. Werner et McCutcheon
„ „ Y. Shibata :

„ „ Tschernoff :

„ 3279 „

45 121 (1912)

„ 3281 „

„ 3287 „

„ 329-1 „ etc.

Recherches sur les spectres d'absorption, des ammine-coinplexes cobaltiques, II. ^5

Table YIII.

Polymères de la première catégorie.

Nombres des
valences auxiliaires.

[Co(nh3).noj.[Co;nh3Kno,),], I^^:;^ }f,^?^;r l

[Flavoji
[Crocéoji

[

Jdan« le cation 4

Polymères de la deuxième catégorie.

[Co(NH3)5 NOJ, [Co(ONO)J,

[Flavo]:, [ // l

[Crocéojj [ // ]i

-^^ \ // r anion 3

{dans le cation 15

// l'anion 6

i // le cation l'2

\ V l'anion 3

D'après l'hypothèse de J. Stark, c'est très probablement
l'électron de valence relâché, qui donne la bande d'absorption dans
l'échelle spectrale visible ou ultraviolette^. C'est donc notre avis
que dans les composés minéraux, cette espèce d'électrons de valence
se trouve toujours aux points de connexion produite par la valence
auxiliaire, parce que presque tous les sels minéraux colorés con-
tiennent des ions complexes dans leurs molécules, sauf quelques
iodures et sulfures, où les derniers atomes sont les chromophores
remarquables. De plus, l'un de nous a constaté autrefois que le
changement de nombre de coordination de certains sels complexes
cobaltiques provoque la transformation de leurs couleurs.^'

Dans la figure V nous avons représenté la manières de liaison
hypothétique des ions d'un polymère complexe, par exemple du
xanthodiammine-tétranitrite cobaltique, selon notre opinion que les
électrons de valence relâchés se trouvent aux points de connexion
produite par la valence auxiliaire et que des électrons de valence
de cette espèce servent à lier les ions dans l'eau.

Comme on le voit dans la table et la figure, la différence de
nombre des électrons de valence relâchés dans les cations et les
anions étant égale à 1, dans les trois polymères de la première
catégorie, il n'y en a qu'un qui puisse osciller librement, tandis que

* Voir le dernier travail de Yuji Shibata : loc. cit.

1) A, Hantzsch et Yuji Shibata : Zeitschr. f. anorg. Chem. 73 309 (1911)

16

Art. 8.— T. Shibala

Fig. V.

NQ,

NH,

NQ,

NOL

[Co(NH3),(NO,) J' [Co(NH,)5NO,] •• [Co(NH,,X,(NO,) J'

( valence auxiliaire, valence principale,

o électron de valence relâché)

les autres sont enchaînés les uns aux autres dans les anions et les
cations, et leurs vibrations sont, par conséquent, forternente
restreintes. Si cette dernière supposition est admise, il sera bien
naturel que l'aspect de la première bande d'absorption devienne très
plane, ayant un maximum et un minimum d'absorption bien
insignifiants, parce que, comme l'un de nous l'a fait remarquer dans
son dernier travair\ la première bande à env. 20U0 de fréquences
des sels cobaltiques est provoquée probablement par les vibrations
des électrons de valence relâchés s' attachant aux atomes de cobalt.

Le fait que tous les trois polymères de la seconde classe donnent
la première bande d'absorption très nette et normale, est facilement
compris, si l'on observe, comme l'indique notre dernière table, que
les différences des nombres des valences auxiliaires dans les anions
et les cations sont bien considérables c'est-à-dire 9; en d'autres
termes, il y a neuf électrons de valence relâchés, s' attachant aux
atomes de cobalt, qui peuvent osciller librement.

Parmi les trois polymères de la deuxième catégorie, le xantho-
héxanitrite cobaltique [Co(NH3)5NOj3 [Co(ONO)j2 donne les
bandes d'absorption sensiblement hypochromatiques, et la forme
de sa courbe ressemble beaucoup à celle du cobaltihéxanitrite de
natrium ^^ [Co(ONO)6] Nas, tandis que les deux autres donnent les

1) Voir le dernier travail de Ynji Shibata, loc. cit.

2) , ■.
loe. cjt.

lliichc relies sur les spectres d'absorption <les ainmi ne-complexes cobalticpies, II. 17

courbes d'absorption caractéristiques dans leurs cations, fiavo et
crocéo, et ne sont presque pas influencés par 1' anion [Co(ONO)c]"\
qui est un hypochrome remarquable. Cette différence des formes
des courbes d'absorption de ces trois sels : [Co(NH3).3N02]3 [Co(ONO)cl.,

[co(NH,X^g^;;;]^ [CoCONO)«] ^ [CoCNH.,), ^g;>;]jCo(ONO).]

doit être causée par l'inégalité des nombres des groupes de nitro et
de nitrito dans chaque molécule des polymères ; en effet, le xantho-
■ cobalthéxanitrite possède 12 nitrites pour 3 nitros dans sa molécule,
tandis que la flavo-et le crocéo-cobaltbéxanitrito n'ont qu'un
nitrito pour 0 nitros.

Recherches sur les spectres d'absorption d<'s ammine-complexes cobaltiques, If. \^

B.) Spectres d'absorption des solutions aqueuses des ammine-
complexes cobaltiques des poly-noyaux^\

Avec 6 figures.

On connaît de nombreux ammine-complexes métalliques, dont
les cations complexes contiennent plus d'un atome métallique.
Nous devons la découverte de la plupart de ces complexes à M. M.
Jörgensen, Werner et à leurs élèves. Spécialement M. Werner a
fait des travaux très intéressants sur la détermination de leurs con-
stitutions chimiques, au point de vue de sa théorie de coordination.

J'ai étudié les absorptions des solutions aqueuses de quelques
sels complexes appartenant à cette catégorie, corps purs dont les
préparations sont comparativement faciles et, dont la constitution
est bien claire.

Ce sont les cinq sels suivants que j'ai choisis comme objets de
la présente étude :

j 2) S)

Co [^g Co(NH,),] ^ Cl« [(NH3),Co^Hco(NH3),] Cl,

[(NK0,Cogg^'Co(NH3),] Cl„ 4 H,0^' [(NH3),Co^^^'Co(NH,x] (N0,%'

EH,0 "H

(NH3),CoNH,Co(NH3),

Les concentrations des solutions mesurées ont été, comme
toujours, d'un centième normal jusqu'à un dix-millième normal ;
seulement dans le cas du chlorure de dodécammine-héxol-tétra-
cobalt, la moitié seulement de cette concentration a été prise, parce-
qu'il contient des atomes de cobalt deux fois plus que les autres,
tandis que l'azotate d'octammine-/-«-amino-sulfato-dicobalt a été

1) Comparer " Xeuere Auschanungen auf dem Geliete der anorg. Chemie " par M. A.
Werner, Page 185. [1909, Vieweg u. sohn, Braunschweig.]

2) A. Werner : Ber. d. deutsch, ehem. Gesell. 40 4426 (1907)

3) „ „ „ „ ,. « ,, 4434 „

4) „ „ „ .. „ „ „ 4605 „

5) A. Werner et Baselli: Zeitschr. f. anorg. chem. 16 111 (1898)

6) A. Werner : Ber. d. deutsch, chem. Gesell. 40 4605 (1907)

n
CL.

20 Art. 8. -Y. Shibata :

étudié dans les solutions des concentrations de -^ et :^ à cause
de sa faible solubilité.

A mesure que je les ai étudié, j'ai remarqué que ces sels com-
plexes ne sont pas du tout stables à l'état de solution aqueuse;
dans tous les cas, elle change rapidement sa couleur originale qui
devient d'abord très brune, et puis cette couleur s'assombrit peu à
peu jusqu' à ce qu'elle paraisse presque opaque. Par conséquent la
loi de Beer sur l'absorption des rayons n'est point satisfaite par ces
solutions. De même, on remarque que l'inclination à l'abscisse des
courbes d'absorption de ces solutions instables est toujours sensible-
ment plus faible que dans le cas de la solution acidique des
•complexes, qui est tout à fait stable.

Cette faible inclination des courbes d'absorption des solutions
aqueuses fait penser qu'il s'agit, sans doute, de l'absorption de la
solution colloïdale de l'hydrate de cobalt, qui est formée par la
décomposition hydrolytique. Cette considération a été bien con-
statée expérimentalement, comme on le verra dans la partie
expérimentale.

Comme je l'ai indiqué plus haut, j'ai ajouté ensuite aux
solutions des sels complexes un peu de l'acide correspondant à ses
anions, pour empêcher la décomposition hydrolytique. Dans ce
€as, les solutions étaient bien stables et la loi de Beer a été
■complètement satisfaite et les résultats des observations de l'ab-
sorption m'a permis de mettre en évidence quelques faits bien
intéressants que je vais décrire et discuter dans la partie expéri-
mentale et dans la conclusion.

Partie expérimentale.

1) Chlorure dodécammine-héxol-lélracobaltlque.

Ce sel complexe bien intéressant a été récemment coupé en
des composants optiquement actifs par M. Werner'^ La solution
aqueuse de ce complexe est assez instable et, si on l'abandonne

1) A. Werner : Sur l'activité optique de composés chimiques sans carbone ; Comptes
rendus dos séances de TAcademie des Science (Paris), 159, XVII é2ô ^1914).

Kecherchcs sur les spectres d'absorption des ammino-complcxes cobaltiquts, II. 21

quelque temps, ou remarque que la couleur de la solutiou est
complètement perdue et que la coagulation totale de l'hydrate de
cobalt, formé par la décomposition hydroly tique, l'accompagne en
mGme temps. En faisant évaporer l'eau mère, on n'obtient que le
chlorure d'ammoniaque.

Les courbes d'absorption de la solution aqueuse de ce com-
plexe sont tracées aux lignes ponctuées dans la figure I. Cette
solution ne satisfait pas la loi de Beer, et chaque branche des
courbes corres])ondant à plusieurs concentrations ne continue pas;

Fig. I.

Fréquence

150O

200O

300O

3600

loo \

•A

» \

V

1

1

'^

\
*

X

\

looo \

4500

1Û00C

w.

_Co(^gCo(NH3X)3]cie

solution aqueuse

„ acidique

22 Art. 8.-Y. Shibata:

celle de la solution la plus concentrée (^) montre encore une
bande d'absorption, parce que l'hydrolyse de la solution n'était
pas encore très sensible au moment de l'observation de cette
partie, tandis que l'autre partie de la courbe de la solution dix fois
étendue met déjà en évidence le développement de l'hydrolyse par
sa faible inclination.

La solution acidique, qui a été préparée en dissolvant le
complexe dans ^^ l'acide chlorohydrique, est cependant très stable
et satisfait la loi de Beer, donnant une cour])e d'absorption l)ien
continuelle dans toutes les concentrations des solutions. Cette
courbe, tracée dans la figure I, par une ligne noire, montre les
deux bandes d'absorption très nettes, dont la première se place à la
fréquence 2000, comme on voit la même bande d'absorption dans
tous les sels cobaltiques déjà étudiés, tandis que la seconde se
trouve à la fréquence 3400. C'est cette seconde bande nouvelle
qu'on n'avait jamais observée dans les sels cobaltiques, soit les
simples, soit les complexes. Elle paraît d'une épaisseur assez
mince, en effet elle est d'env. 60 mm cori'espondant à la concentra-
tion de -^3555- normal. Il me semble que cette bande d'absorption
hyperchromatique est bien caractéristique des sels cobaltiques, dont
les ions complexes instables sont construits d'une manière sem-
blable, comme ceux présentement étudiés ; c'est-à-dire que chaque
atome cobaltique (du noyau) est lié avec quatre molécules d'am-
moniaque et deux groupes basiques ou acidiques.

Ces dernières connexions aux groupes basiques ou acidiques
doivent être bien probablement si faibles que leurs affinités ne
peuvent plus être actives dans l'eau, et la décomposition hydro-
lytique commence, sans doute, aux points de ces connexions
faibles.

Je reviendrai encore une fois sui- ee sujet plus tard dans la
<';onclusion.

2) Chlorure oclammine-diol-dicobalique.

Ce sel violet rougeâ're est bien soluble dans l'eau et la
solution est aussi très instable ; la précipitation totale de l'hydrate
de cobalt et, par conséquent, la décoloration complète de la solution

Kocliorclies sur los spectres d'absorption dos ainmine-cimploxes cobaltiques, II. '23

sont observées en quelques lieures. Son ejui mère ne donne que
le chlorure d'ammoniaque en s' évaporant.

Fig. IL

Prcqucnce

1500 2000 2500 3000 3500 4000 4500

JL

/\

^ lOO

/ \

^

/

■"\-

-/,

V

/ \

s '

s \

X

'^^ \

X

♦^ \

\ ^^

^— ^

'"'^ ^v

^ \

\

\

\

N '

•^ ^

V

looo

X

[(NIL),Co^gco(NH3),] CI,

_. 1000

solntioii aqueuse

: ,, acidiquc

Les courbes tracées en lignes ponctuées dans la figure II
représentent les absorptions des solutions aqueuses de ce complexe.
Dans le cas de ce seL la courbe de la preniiùi'c concentration de
1ÖÖ ^^^'i"^ ^^^ montre aucune bande d" absorption (^t son inclination
à l'abscisse est sensiblement plus faible que celle de la solution
rucidique qui est tracée avec la ligne noire dans la môme figure.

Cette dernière courl)c crabsorption do la solution acidique

M

Art. 8.— Y. Shil.ata

niontro encore deux l)andes très nettes, cunnne le se] con'iplexé
précédent. De même, les positions des bandes et la forme totale
de leurs courbes sont presque les mêmes l'une et l'autre. On verra
toujours cette coïncidence curieuse et remarquable des courbes
d'absorption entre les différents sels complexes étudiés dans ce
travail.

3) Chlorure oclammine-/'-amino-ol-dicoballique.

Fig. III.

Fréquence

1500 2000 2600 3000 3500 "î'^O'^ '*^0'3

,5 3.0

155 loooy

.r

.

~--\

i

\

s....

^

"\

.

\ \
\ \

a:

[(NH3),Co™^Co(NH,),] Cl„ 4H,0

solution aqueuse

„ acidique

La décomposition liydrolytique de la solution aqueuse et la
production de l'hydrosol de l'iiydrate de cobalt est peut être la plus

Eoc'Iterchcs snr L s sjieetrcs d a I sorption dts ainmint-coniplcxcs colaltiqnts, TT. '25-

facile en ce sel complexe, parce (jue, coinine on le voit dans la
figure III. F inclination des courbes de cette solution ne change
plus par les dilutions, et elles continuent assez bien dans chaque
concentiatioîi, en satisfaisant la loi de Beer.

L'absorption de la solution acidiquc doiuie deux bandes, dont
la position et la forme sont égales à celles des autres, dont nous
avons parlé, bien que ce sel complexe contienne un nouveau
groupe d' amino.

4) Azotate octammine-//-amino-suirato-dicoballique.

Fig. IV.

Fréquontv^

1500 2000 250O 3000 3500 4000 4500

4 0

\

v/\

1

3.5
3.0
2.5

2.0 ■

-*-.

',

\

o

'^ --.

* ^"^ \

""■--

V:::-.

..,

*«» **..

til

â

ai

A

»

\

<

2

\

ci

15
1.0
0.5

\

o
1-^

[(NH3),Co^^^Co(NH3),] (NO3).

solution aqueuse

,, acidique

r26 Art. 8.-Y. Shibat-.i:

Ce sel complexe, étant nioin^ soluble, lutus avons mesuré les
absorpti(.)ns de ses solutions dans les concentrations de ^^^q et de
:;^. Sa solution acjueuse est nussi ti'ès instable et les courbes
d'absorptiou donnent une inclination bien caractéristique à la
solution colloïdale, tandis que la solution .-icidique est normale et
•donne une courl)e tout à fait sc^mldable aux .-mires. I.e nouveau
groupe de suliato. non i)lus, n'exerce aucune intluence sur la forme
•de la courbe d absoi'ption.

Les deux lignes borizontales dans l;i ligure IV, qui ])()rtent les
■chiffres lOOO et 100 à leur extrémité droite, indiquent les épais-
seurs recalculées cori-espondant :i la concentration de O.OOOIN.

5) Chlorure octammine-monochloro-monoaquo-/i-amino-

dicoballique.

\ai solution aqueuse de ce sel est un peu plus stable ijue celle
■de<, autres, parce que sa courbe d'absorption <le la solution avec la

concentration ^Ô^ donne encore une bande à la fréquence 2000.

Dix l'ois étendue, elle ne satisfait plus la loi de lîeei' et la
•coui'be de cette partie présente l'inclination faible caractéristique à

l'bydi'osol de l'hydrate de cobalt.

La solution acidique de ce sel aussi donne une courbe d'al)sorp-
tion complètement pareille à celle des autres, quoi qu'il contienne
encore deux nouveaux substituants, c'est-à-dire, un atome de
•chlore et une molécule d'eau, dans son cation complexe.

Recherches sur )(« spectres diibsarption des ammine-eomplexos cobaltiques, IL -27

1500 2000

Fig. V.

Fréquence

3000 3.'>0&

4000 4500

lOO \ \

./A

looo

1 \
't \
» \

s

\^

%

r\

V
\

\

*^^^^^^^'CoNH.,Co

LH,

Cl

o

Cl,

(NH3)J
solution aqueuse
,, acidique

6) Hydrosol de l'hydrale de cobalt.

La solution eolloidalc de l'hydrate de cobalt a été préparée
•<l'a})r(js la méthode donnée par M. Theo Svedberg dans son livre,
intitulé ''Herstellung colloidaler Lösungen." Selon cette méthode,
la préparation de T hydrosol de h hydrate de cobalt a été exécutée
-de telle sorte qu'on fait précipiter d'abord l'hyrate de cobalt de la
solution aqueuse de son azotate, en y ajoutant l'alkalie., Le
précipité est donc filtré et parfaitement lavé en grande hâte avec

28

Art. 8. -Y. Shibatii

de l'eau; pure. Quand on bouillit ensuite cet hydrate de cobalt
filtré avec une solution étendue (gQ N) d'acide chlorohydrique, on
obtient, une solution colloïdale de l'hydrate de cobalt d'un brun
foncé. La teneur de cobalt dans l'hydrosol ainsi préparé a été
dosé et la solution a été donc étendue jusqu'à la concentration de
i^, pour mesurer son absorption des rayons.

Fig. VI.

Fréquence
2500 3000 3500

^.

4.0

-^

3.5

^^^>.^^

œ

^**v,

a

^^^^

o '

^^^^^

-i-J

^^w

3

3.0

V

O

>v

K

X

a?

X

v

X

•^

\

.•/.

>

i^

^

2.5

«.

O!

"S

ft.

■1>

œ

■-J

a

2.0

"^

5

^

;->

^

ëc

1.5

h^

1.0

0.5

Sohitioii coUoidale de l'hydrate de cobalt

La courbe d'absorption de cette solution, comme je l'avais
pensé d'advance, montre une inclination faible et donne mie forme
très semblable aux courbes d'absorption des solutions hydrolysées
des sels complexes avec poly-noyaux.

Tfechiircln's sur los spectres d'absorption des ammino-complexcs eobaltiquos, IT. 29

Conclusion.

Dans les résultats expérimentaux que je viens de décrire avec
précision, on peut apercevoir deux faits bien remarquables — i)re-
mièrement l'hydrolyse parfaite des solutions aqueuses des com-
plexes étudiés, et deuxièmement la même propriété de leurs
solutions acidiques, relativement à l'absorption des rayons.

La cau5e de ces deux phénomènes chimique et physique se
trouve sûrement dans le fait que les connexions de deux noyaux
(les atomes cobaltiques) des ions complexes, qui sont liés indirecte-
ment à l'aide des groupes d' amino, d'hydroxyl, de sulfato ou de
■deux d'entre eux, sont extraordinairement faibles. La décom-
position hydrolytique de leurs solutions aqueuses, il me semble, se
produit aux points de ces faibles connexions, et enfin la molécule
•complexe se décompose complètement, donnant l'hydrate de
cobalt et le chlorure d'ammoniaque, l'azotate d'ammoniaque ou
le sulfate d'ammoniaque.

Les modes de ces décompositions hydroly tiques peuvent être
^bien comprises, si on représente les sels complexes par les formules
•constitutionelles :

OHv •

--Co(NH3X]ci,+ 2H,0
a) 2H,0+ Cl, [(NH3),Cor^/^CoJ^H >4Co(OH)3+ lîNH.Cl+GîïH.

OH— Co(N Hj n Cl,+ 2H,0

Chlorure dodecammine-liexol-tétracobal tique.

h) 2H,0 + Cl, r(NHACotOO^°(^^»M Cl2+2HîO— ^2Co(OH)»+4NH.Cl+4NH,
Chlorure octammine-diol-dicobaltique.

,30 Art. 8. -Y. Shibata:

l;) 2H,0+C1, [(NHJjCo^/^CoCNH,),! CI2+2H2O — >2Co(OH),+4NH,Cl + 5NH.

+ ILO

Chlurure octau)uune-[j.-:iriiin>>-ol-dicobaltiqu<>.

a) 2H5O + (NO,), r(NH,)«Co ' ><fjbo(NH,)4l N0,+ H,0— >2Co(OH)3 + 3N0,NH,
^ ^0?*\ " -^ ' +(NHJ,S0.^4NH,

+ 2H,0

Azotate octau)uiine-[x-auuno-siiIfatG-(lico1)a]tiqiiö.

c) 2H,Ü + Cl,[(NH3XCo':

. Ji,ü Cl.

X

>Co(NH3)n Cl, + 3H,0— >2Cc(0H),+ 5C1NH, + 4NH,
-' +H.,0

Chlorure octammine-monocliloro-ir.onoqno-iJ.-amino-dicobaltiqne.

Selon la loi d'action de masse, une petite quantité des acides
correspondant aux anions, ajoutée à leurs solutions aqueuses, sert
à arrêter parfaitement la marche do la décomposition hydrolytique.
Dans ce cas mOme, les affinités entre les atomes cobaltiques et les
groupes atomiques, qui lient les noyaux, atomes cobaltiques,
doivent être encore beaucoup moins fortes que les autres affinités
de coordination, parce que les différences chimiques de ces groupes
n'exercent aucune influence sur l'absorption des rayons, comme on
l'a vu dans leurs courbes d' absorption, qui sont presque les mêmes
les unes et les autres. A la fin, j'ajouterai encore quelques lignes
sur la (îause de cette égahté remarquable des courbes d'absorption
des solutions acidiques.

Considérons encore une fois les formules constitutionelles
des complexes relativement aux distributions des valences prin-
cipales et auxiliaires autour (Tini noyau, atome cobaltique, de
chaque sel :

Eecherchcs sur les spectres d'absorption des amiiiine-coiui^lexcs cobaltiquos, II. ^Jj

. NHi-::,.
^-' NH3-.:''
NH,-'

,ÜH-,

Co.

NH3-

NH,
NH,'

'^'' NH,-:-''"°'

NH-T.

.^ NHî-V.V../

'OH
^NH,

"'OH''
^iNH,-

y

Co'

Co'r-

NHî

NH:

,/

co'-:._

H2O Cl'
( A'ulence principale,

NH,''

NH,-.. /NH,.

^) NHr:-' \ ,

NH-' so/

Coc:

Valence auxiliaire)

Un coup d'œil suffit pour nous faire comprendre qu'ils sont
construits très semblablement les uns et les autres, et que, par
conséquent, ils absorbent des rayons de la même manière.

Published December 29th, 1916.

JOURNAL OF THE COLLECrE OF SOIENOE TOKY'J IMPERIAL UNIVERSITY.

VOL. XXXVII., ARTICLE 9.

TO Professor Aikitu Tanakadate

ON TUE OCCASION 0_iMMEMli];ATING HIS TWENTY-FIVE YEARS' SERVICE

DEDICATED BY

HIS DEVOTED PUPIL, THE AUTHOR.

On Rapid Periodic Variations of
Terrestrial Magnetism.

By

Torahiko TERADA, Uiiiakitliakmhi.

PART I.
Introduction.

1. Immediately after the organization of the Imperial Earth-
quake Investigation Committee, regular magnetographic obserA'a-
tions were begun, on the proposal of Prof. Tanakadate, at four
stations, Kumamoto, Kyoto, Sendai and Nemuro, with the special
purpose in view of detecting any magnetic disturbances which
might reveal themselves associated with destructive earthquakes.
The instruments used were of the ordinary Mascart type. (3n the
other hand, a general magnetic survey of the empire was under-
taken, 1893-96, the result of which has been published in the
Journal of the College of Science, Vol. XIV. In discussing there-
in the local disturbing field due to Mt. Huzi, Prof. Tanakadate
came to the conclusion that even if the whole mass of the moun-
tain be suddenly removed, the disturl^ance in the vicinity would

2 Art. 9. -T. Terada:

scarcely amount to 1 j-. Hence it was considered futile to continue
the observations with instruments of such low sensibility, and the
regular observations were suspended in 1900. At the same time,
he devised and constructed, with the able assistance of I)r. H.
Kadooka, now expert to the Military Telegraphic Department, a
set of extraordinarily sensitive magnetographs, and laid before the
Committee the plan of a provisory magnetic ol)servatory equipped
with these instruments. The proposal was approved, and the
necessary arrangements were promptly made under his supervision.
An underground room was excavated for the purpose, in the
vicinity of the Marine Biological Laboratory of the Science College,
at Misaki. The regular observations were commenced early in
1910. In the summer of 1911, Dr. Kadooka was appointed to his
present position, and the author took charge of the Observatory,
until April 1914, when the observations were suspended for an
indefinite term.

The observers who were successively resident at Misaki and
took charge of the instruments were :

Mr. Hideo Momose, formerly Hitotuvanagi, now teacher in
the Nagoya Sôdô-syû Third Middle School,

Mr. Takeo Tatiiri, now Assistant in the Meidi Technical
School, Tobata, Hukuoka Prefecture,

The late Mr. Kisaburô Matui, at the time of his death teacher
in the Sibusi Middle School, Kagosima Prefecture,

Mr. Murato Nakata, now teacher in the Hakodate Com-
mercial School,
to whom the author's sincerest thanks are due for their untiring
alertness to their duty, which claimed their utmost patience and
attention, to say nothing of the inconveniences of living which
had to ho endured on account of the lonely situation of the
Observatory.

The entire task of examining and studying the magnetographic
records in detail and drawing up the report thereon, was entrusted
to the author. Though the investigations are as yet by no means
completed, it seems now to be the proper time to summarize here
the principal results hitherto obtained.

On Rapid Periodic Variations of Terrestrial Magnetism. 3

2. Observatory. — The observatory is situated at the foot of
the northern slope (139°37'*5 E, 35^9'*4 N.) bordering the Bay of
Aburatubo, ^lisaki. Province of Sagami, a few minutes walk from
the Marine Biological Laboratory belonging to the College of
Science, Imperial University of Tokyo. The station was chosen

K

S >

*« 2

►0 " ^

o p

"C -i-

K

■^ 30.

P rr^

1 rD

2. CfQ

-■ ^

^ "^

M-

s*

H.^

?5

c

N ^

C P

■"

1 o"

^

i %,

^

li

rr ■••

H-

a X

^ ^

^T3

m ::-

^ o

M

3 ri-

o

i J-.

3

o ;:;

w

•C

tn S-'

Oj

J5 a

rn

--■>.

-CT

3

4 Art. 9.— T. Terada :

on account of its remoteness from any powerful electrical plant^^ and
also for the conveniences which its proximity to the Marine Labo-
ratory afforded. The edge of the terrace overlooking the bay was
partly cut ofï to form a vertical cliff. An underground room was
then excavated with a narrow entrance (Fig. 1, Ei) opening at the
foot of the clifï. Since the rock was entirely of a soft tertiary forma-
tion, the excavation was comparatively^ easy. The approximate size
and the arrangements of the room may be seen from Fig. 1. A is
the antechamber where the photographic recording apparatus R and
the acetylene lamp L used as the source of light, were installed on
the pedestal Pj or Pa. In the interior chamber B, communicating
with A by the narrow entrance Eo and also by the windows Wi
and Wo, two long pedestals P3 and P4 are laid for mounting the
magnetic instruments. Ci, Co and Co are small stone pillars on
which were fixed the stands for fitting the magnetic bar used for
determining the sensibility or constant of the magnetographs.

In the beginning of the observation, two similar sets of
instruments were arranged on Pi P3 and Po P4 respectively and run
simultaneously. The results were almost identical, as was to be
expected, and the later part of tlie observations was almost
exclusiveh" made on pedestals Pj and P3.

The temperature of the interior chamber was kept tolerably
constant, the daily variation amounting to only a fraction of a
degree and the annual range scarcely reaching 3°C.

A serious difficulty met with was, however, the extreme
dampness of the chamber during the summer months. Not only
does the moisture of the external air, saturated at very high
temperature, gradually condense in the cooler interior, but
luimidity is constantly supplied by the percolation of underground
water, tlirough numerous fissures in the wall and ceiling, fed by
the abundant rain during " Baiu,'^ the rainy season on our entire
Pacific coast in early summer. This caused so much trouble that
during a certain period observations were almost rendered im-
possible. Though it was not impossible to overcome this difiiculty,
the funds at our disposal were not sufficient to carry out the

1) The nearest tramway line is at Kamakura, nearly 20 km. N. from the station.

On Eapicl Periodic Variations of Terrestrial Magnetism. 5

neces>^aiy reconstruction of the room. In any future work of the
kind, it will he ahsolutely neccessary to provide first of all for the
removal of this obnoxious humidity.

o. Instruments. — Since it was the immediate ^purpose of
the present investigation to detect the most minute disturbances
possible, the sensilulity of the usual instruments was far from
being sufficient. To meet this need. Prof. Tanakadate and Dr.
Kadooka devised a specially sensitive set of instruments which
will l>e separately described in the following paragraphs:

a) Wed- East- or Y- Component Instrument (West taken as
positive). Maxwell, in liis discussion of the theory of bifilar
suspension^^ suggested the utility for the measurement of the WE-
component, of a suspended magnetic needle twisted nearly 180
from its natural direction, which can be made highly sensitive by
properly adjusting the breadth of the bifilar suspension. This
principle was adopted in the following manner. The magnetic
needle (Fig. 2, N) with a length of 20 mm. and a diameter of 1*5

Fis.

VI w

Hr^-^ft

i__t

0 t

2 3^ scr:

mm., was held by a hook AB, made of aluminium wire of O'T
mm. diameter, bent in the form as shown in the figure. A kind of
light stirrup for supporting the hook with the magnetic needle
was made with pieces of thin fused quartz rods, welded together

1) Maxwell, Electricity and Magnetism, 3rd. Edition, Vol. II, p. 118.

6 Art. 9.— T. Terada :

into shape as shown in Fig. 2, T. This was made to hang on the
looped end of the suspension wire ivw, for which fine Wollaston
wire of 5-10 /^ diameter was used. Tlie upper arc of the alumi-
nium hook is made to ride on the central V-shaped recess cc of
the stirrup. JM is a liglit plane mirror of 5 mm. diameter, attached
to the hook. The whole system was hung in the metallic case of
Mascart's magnetograph made by Carpentier, Paris, from which
the usual attachment in the interior was removed. The regulating
screw attached at the top of the case for adjusting the Itreadth of
the supension was utilized as such. For damping the natural
vibration of the suspension, an electromagnetic damper ]) made
of a copper block was introduced. The damper could Ije rotated
on a cylindrical brass block P, which in its turn could be rotated
about the screw S, fixed to the bed plate of the case, passing
through the groove cut along the projecting arm F at the foot of
L. The damping was very effective, making tlie vibration com-
pletely aperiodic.

To set the suspension in working condition, we proceeded as
follows. At first, the stirrup only was hung on the wire, making
the breadth of the suspension sufficiently large. After hanging
the magnetic needle carefully in its natural direction, twist the
torsion head slowly till the needle is turned about 180°, and the
luminous image of the slit formed by the reflexion of the mirror
M appears within its proper range. Then, after bringing the
damper in position gradually narrow tlie breadth of the bifilar
suspension, at the same time adjusting the torsion head so as to
keep the luminous spot always wdthin the range assigned to it.
Proceeding very carefully in this way, the sensibility can l)e made
extremely great, till at last the suspension attains its unstable
position. It must l)e remarked that near this extreme position
any minute chan'ge in the elastic property of the wire, the breadth
of the bifilar suspension or the weight of the suspended system,
may sensibly affect the deflection. Nor is the system independent
of the slight inclination and the vertical acceleration of the
instrument. In one instance, an inclination of about 4' produced
a dis]3lacement of about '1 cm. on the photographic record. Since

On Rapid Periodic Variations of Terrestrial Magnetism. 7

the latter effect seems to be due chiefly to the rigidity of the
suspension wire, it is advisable to use as fine a wire as possible.
In any future work, a thorough annealing of the wire after
hanging the suspended system is very desirable.

The tripod support of the instrument rested on a thick glass
l^late, which was provided with a hole and a V-groove for
receiving the feet of the levelling screw, and was rigidly fixed to
the face of the stone pedestal.

The lens in front of the instrument case was replaced with
another with a focal length of 5 m.

The deflecting magnetic rod for determining the sensibility,
was fixed to a special holder'^ such as is usually attached to
Mascart's magnetograph for a similar purpose, and placed on the
support pasted on the pillar C3 in the same meridian as the
instrument. After starting the photographic apparatus, the
deflecting magnet was brought in position, i.e. horizontally in WE
direction, and reversed every 3 or 5 minutes. The magnetic
moment of the rod mostly used was M = 77 e.g. s. at the room
temperature. The distance of the deflecting rod from the needle of
the instrument was 123 cm., so that with the rod " side on," the
change of the WE-component corresponding to the reversal of the
deflector was 8'3 "/'. The sensibilit}'' was so adjusted as to make
the corresponding deflection nearly 50 mm.; i.e. 1 mm. on the
record corresponded to 0"17 ;-. The sensibility, though fairly
constant, showed occasionally a tendency to decrease slightly after
running for 24 hours, though not at all so serious as to affect the
general aspect of the results obtained. It was therefore considered
preferable to test the sensibility at least once every day and
redetermine the constant. This gradual decrease of sensibility is
probably due to the influence of a slight elastic time-effect of the
suspension wire, made apparent on account of the extreme position
of the bifilar sj^stem. The presence of a sensible time-effect in
the wire used, is also made evident on the photographic record,
wdien an abrupt deviation is effected by means of the deflecting
magnet. After an immediate deflection, a slow creeping up of the

1) Mascart, Magnétisme Terrestre, p. 195, Fig. 46 c.

Art. 9.— T. Terada :

luminous image is always to be traced, whicli in most cases
practically attains the final position after a few minutes (see PL
I, marked with^). This effect will of course affect the accuracy of
the record, especially in relatively magnifying the disturbances of
the longer period compared with the shorter ones. The error
may, indeed, amount to several percent in unfavorable cases,
if the relative amplitudes of the disturbances with decidedly
different periods are to l)e compared with accurac3^ However, the
general inferences which will be given in the following communi-
cation will not be seriously affected, since here the comparison of
amplitudes is either made of the X- and Y-components for waves
of the same period, or of the X- and Z-components for waves with
the periods usually longer than 1 minute. In the former case,
both instruments show after-effects very similar to each other.
Even in the latter case, where the Z-instrument is comparatively
free from such effect, any serious error Avill occur only in the case
of very short waves. This very disagrea])le time-effect could
prol)ably l)e avoided by the use of quartz fil)re suspension, directly
welded to the quartz stirrup, though in this case we must devise
a necessary modification of the suspension head.

h) North-South- or X-Component Instrument (North taken as
positive). The NS- insti'ument is essentially the same as that of
Mascart's magnetograph. The suspension system is, however,
replaced by one quite similar to that of the WE- instrument above
described, except that in the present case the plane of the mirror
is perpendicular to the axis of the magnetic needle.

Keeping the breadth of the l)i filar suspension at first suffici-
ently wide, the torsion head is slowly twisted, till the luminous
spot appears in the assigned range. Then cautiously turn the
torsion head, at the same time regulating the breadth of the bifilar
suspension, so as to keep the luminous image always within
proper range. The unstable position is attained in a certain
azimuth of the torsion head, which may be noted down for
subsequent readjustment. The determination of the constant is
carried out in a way similar to that in the case of the Y-instrument,
with the only difference that in this case the defiecting magnet is

On Eapid Periodic Variations of Terrestrial Magnetism.

9

applied "end on," so that for the same distance of the deflector,
the deflection must be made twice as large as in the case of the Y-
instrument to insure the same sensibility. The distance of the
deflecting magnet from the needle of the instrument was 127 cm.,
so that 1 mm. on the record corresponds to 0*15 T.

The gradual creeping of the luminous spot is also observed in
this case. Hence the same remark applies as to the accuracy of
tlie records.

Fig. 3, A.

1—4 — h-f-

5 loom

H 1— ^H 1 1 1

n

CM

7=î

Ü

ma

(2

CE

lï

m

10

Art. 9. -T. Terada :

Flg. 3, B.

c) Vertical or Z-Compone7it Instrument (downward taken as
positive). Instead of the ordinary Lloyd's balance, a new in-
strument was designed and constructed by Prof. Tanakadate and
Dr. Kadooka, whicii proved very satisfactory. The magnetic
needle (Fig. 3, NS), 4 cm. long and 2 jnm. tliick, was fixed to
the lower face of a rectangular plane mirror M (Gx 17x 1 mm.),

made of fused quartz pkite, platinized on its

upper suface by means of cathode discliarge.

1" ' \-s^ The mirror has a i)air of projecting arms aa

N on l)oth sides, to the pointed ends of wliich

are Avek.led quartz filtres ^^ of 0'05-0-08 mm.

diameter. Tlie welding may l)e easily made,

after some practice, l)y pointing the sharp

point of a fine oxyhydrogen ßame to the

spot where the fibre is attached to the arm

by sliglitlv wetting the surface. The fibres

yj, i"^' ff '^i"e stretched in EW direction by means

-^'5r^^ ^"-^ of helical springs SS^-* made of fused quartz

rod, of which the one end is welded to the

fibre and the other is rigidly fixed to the brass liolder hh by

means of solder applied in the cavity for receiving the c^uartz rod.

By means of the nuts nn, the tension could be adjusted without

twisting the fibre. By turning the scrcAv t, a slight torsion could

be given to the fibre, in order to adjust the zero position of the

mirror. Damping is effected by means of the copper block Dj^,

which serves also for clamping the suspended system when lifted

up l;)y means of the screw Ar at the lowest part of the instrument.

Pr is the reflecting prism originally used in MascarFs vertical

component instrument for a similar purpose. The inclination of

the prism could be adjusted by means of the screw i. Dc is the

desiccator containing calcium chloride and TA a thermometer.

The points of junction of tlie mirror arm aa with the fibre are
originally so adjusted that tlie centre of gravity of the suspended

1) The spring may be dispensed with, provided that the temperature is kept sufficiently
constant, without sensibly impairing the reliability of the instrument, though there is the
danger of breaking the fibre by accidental shock given to the instrument during mani-
pulation.

On Rapid Periodic Variations of Terrestrial Magnetism. jl

system is very near tlie line connecting tliem. The sensilnlity may
be then adjusted by means of a small weight w^ playing on tlie fine
screw projecting on the upper face of the mirror, while the inclina-
tion of the mirror can be regulated by means of anotlier pai]' of small
nuts iv». running on the horizontal screw as shown in Fig. o, B.

For determining the sensi):)ility, the same deflecting magnetic
rod as used for the preceding two instruments, was placed on the
pedestal Ci in the same meridian as the instrument at a distance
of 128*5 cm. In this case the deflector was of course applied and
reversed in tlie vertical position. The sensibility was so adjusted
tJiat the reversal of the deflector produces the displacement of
al)out 5 cm. on the record, or 1 mm. corresponds to 0'15 /'.

The instrument worked very satisfactorily. Once carefully
adjusted, it remained so constant in every respect that it miglit
have run many years witJiout any furtlier trouble, except for the
excessive humidity of the room, which caused gradual rusting of
the instrument, especially of the magnetic needle, also a gradual
deterioration of the platinized mirror, and necessitated the com-
plete rearrangement of the instrument. The great advantage of
quartz fibre is very clearly shown in this instrument compared with
the others, since here no sensible creeping up of the luminous image
after abrupt deviation is observed, as may be seen from PI. L An
instrument of a similar construction may be replaced with great ad-
vantage for Lloyd's balance, for the usual work of lower sensibility.

d) Illuminating and Photographic Apparatus. The source of
light used for photographic purposes was an ordinary acetylene
burner with two orifices facing each other, which was fed by a
capacious tank placed outside the room. The l)urner was fixed in
the interior of the metallic case of the lamp originally attached to
Mascart's magnetograph, in front of which a suitable vertical slit
and a cylindrical lens was inserted. The lamp was hung over the
pedestal Pi of the antechamber, close by the window Wi, and
immediately above the photographic apparatus placed on Pj.
The drum carrying the photographic paj^er was 24 cm. in diameter
and 30 cm. in length, and revolved once every 3'76 hours on an
average, by means of a suitable clockwork, so that 1 mm. corres-

12 Art. 9. -T. Teracla:

ponds nearly to 17*8 sec. Closely in front of the drum a fine
horizontal slit and a cylindrical lens are introduced, hoth of the
same length as the drum. During the wet season, a desiccator
dish containing calcuim chloride was placed under the drum.

For giving time-marks on the record, a clockwork was placed
in front of the lamp to interru})t the light for a few mimutes at
the end of each liour.

The three magnetic instruments are arranged in succession as
shown in Fig. 1. For the Z-instrument an auxiliary lens with the
focal length of 2 m. was placed in the path of the incident beam
of light, by means of a special holder fixed to the pedestal.

To save the considerable breadth of the photographic paper
required, owing to the remarkable sensibility of the instruments,
the following device was adopted Avith success. Two long strips
of thick plane glass plate (breadth 10 cm., length 3 m.) were
placed along both sides of the long pedestal Pi (Fig. 1, T). These
were held firmly by means of special holders, with their refiecting
surfaces vertical and parallel to each other, so that when the
luminous beam reflected from the magnetic instrument is deviated
just beyond the limit of the horizontal slit of the photographic
apparatus, it is caught by one of the glass plates, or ' ' optical
traps" as we have called them, and reflected back to the slit.
For a still greater deviation the opposite glass caught the l:)eam
reflected from the first one, and so on. Except in the extremely
damp season, even the third reflection produced a luminous spot
intense enough to affect the photographic paper; By properly
adjusting the position of the three instruments and also the mean
direction of the reflected beams produced l:)y them, an uninter-
rupted record of the three components could be ol)tained, even in
the case of remarkable magnetic storms.

The mutual magnetic influences of tlie three instruments
were tested by mechanically disturbing each instrument, while
the photographic record was being taken. No sensible effect was
noticed.

Since the drum carrying the photographic paper revolved
once per o'76 hours, paper of a length of about 4*85 m. was

On Rapid Periodic Variations of Terrestrial Magnetism. 23

required per day and night. To save the excessive use of paper,
it was replaced only twice during 24 hours, i.e. usually at G o'clock
in the morning and evening. Hence tlie record was run over
three times hy each component, which caused considerahle
confusion of the record, on quiet days by tlie superposition of the
trace of the same component, and on disturbed days, by the
mutual intersections of the different components either direct or
reflected b}" the traps. The confusion is, however, merely ap-
parent in most cases. The time marks, the difference in the
intensity of traces and also slight differences in the optical error of
the mirrors of different instruments, causing different shading of
the photographic traces, served together as convenient signs for
disentangling the intricate record. Ambiguous cases were ex-
tremely rare, even if two branches of traces were nearly super-
posed. After we became familiar in the course of the investiga-
tion, with the characteristic behaviours of short period disturbances
for different components, the distinction of the different com-
ponents became still easier.

During the summer months, the pedestal P3 was covered
with large cases of zinc, which consisted of seven pieces and when
put together in series, formed a continuous channel extending along
the entire length of the pedestal. Five or six dishes with CaCl2
were placed inside the channel. These arrangements were,
however, far from being efficient for the j^revention of the ex-
traordinary dampness of the chamber during a certain period of
the warmest season.

The clock for giving the time mark was occasionally compared
with that of the Post Office in Misaki and also checked by means
of the sundial placed in the yard of the Marine Laboratory, so that
the accuracy of the time was only of the order of a minute at the
most. A convenient control could be made by the very frequent
occurrence of slight local earthquakes which left distinct marks on
the photographic records and could be easily identified with the
seismographic record obtained in Tokyo. ^^

1) Ttie distance between the Observatory and the Tokyo Central Meteorological Ob-
servatory is about 60 km., so that the time taken by the principal phase of seismic waves
will never exceed 18 seconds.

14 Art. 9. -T. Terada :

PART II.
General Results of the Investigation.

4. The general character of the photographic records ob-
tained by our system of instruments may be seen in PLI. and II.
The time taken as abscissa increases from left to right. Upward
corresponds to the positive direction of each component for the
non-reflected trace. PI. I. is the representative record for a quiet
day. The trace of the sensibility determination is to be seen at
the beginning of the carves (marked witli-^). PI. III. is one of
the typical records for disturbed days, the parts of the curves
reflected from the " traps " are marked witli the letters Ri, Ro, the
suffix giving the number of reflexions.

Even on the most quiet days, the records shows as a rule
numerous trains of more or less regular waves or pulsations, the
periods of which range from about 20 sec. to several minutes.
Allied phenomena seem to have been studied first by Balfour
Stewart^^ who found periods of 30 seconds. Kohlrausch "^ found
by direct eye observations a wave of 12 sec. period. Arendt^^
investigated waves with periods of several minutes, frequent in
night hours, in connection witii his researches on the magnetic
disturbances associated with the phenomena of thunderstorms.
Eschenhagen'^-* found prevalent waves of 30 sec. which appeared
most frequently during daytime. Birkeland^-' studied the pheno-
mena for Haldde as w^ell as for Potsdam, and obtained most
frequent groups of waves with periods of 10 and a little longer
than 30 sec. More recently, van Bemmeln in Batavia"^ studied
similar waves with 1-4 minutes periods which he called " pulsa-

1) Balfour Stewart, Phil. Trans. 18Ü1, p. 425.

2) F. Kohlrausch, Wied. Ann. 60, p. 336.

3) Th. Arendt, Das Wetter, 1986, p. 2-il and 265.

4) Max Eschenhagen, Sitz. Ber. d. preuss. Akad. d. Wiss., Berlin, 32, 1897, p. 678 ; Terr.
Mag. and Atm. Elec, 2, 1897, p. 105.

5) Kr. Birkeland, Expédition Norvégienne de 1899 — 1900 pur l'étude des aurores boréales.
Résultats des recherches magnétiques. Christiania, 1901.

6) W. van Bemmeln, Verslag. Konink. Akad. v. Wet. t. Amsterdam, Proc. of the Sec. of
Sc, 2, 1899-1900, p. 202.

On Rapid Periodic Variations of Terrestrial Magnetism. ]^5

tiens" to (listingnisli them from peculiar disturbances called
" spasms," and discovered a remarkable daily period of frequency
with a maximum near midnight. Afterwards'-" he compared the
materials from Zi-Ka-Wei and Kew, and found for the former
station a similar nightly maximum as in the case of Batavia,
while for the latter station, the maximum frequency was found in
the day time. Recent authorities seem to agree in the opinion
that the magnetic waves in question are chiefly due to some
fluctuations of the electric current existing in the upper at-
mosphere, though the actual modes of fluctuation still remain
obscure.

Though the original purpose of our investigation was to
detect any abnormal disturbances associated with earthquakes,
it was in any case necessary to study tlie characteristic pulsatory
waves in some detail, even if these waves should have no direct
connection with earthquakes, in order to be able to distinguish
which were the normal and which the aljnorinal disturbances.
The present paper is chiefly confined to the study of these
characteristic pulsatory disturbances, since unfortunately no
positive results have yet been obtained with regard to earth-
quakes.

In the following, we shall enumerate the most interesting
results obtained by the detailed study of the magnetograpiiic
records comprising observations extending over four years.

5. «) Generally speaking, the magnetic waves in question
are decidedly more regular in the night than in day time when
waves of different periods seem to be very irregularly superposed.
In the great majority of cases remarkably lon^ continuous trains of
moderately regular waves with period of 30-60 sec. appear at
5^-7'' in the morning and continue up to 9^-11^, with occasional
interruptions (PI. IL, III., IV. and V). The number of hours, in
the course of which such trains occurred, was noted for successive
days, quite regardless of the length or the number of the trains.
The number shows occasionally an apparent periodicity of 25-30
days, though generally not so regular as to allow us to deduce any

2) W. van Bemmeln, Natuurk. Tijdschr. v. Xederlandsch-Indie, 62, 1902, p. 71.

16 Art. 9.— T. Terada:

definite period from the scanty material at hand. The general
aspect will be seen from the annexed figure (Fig. 4).

Fig. 4

Since it was suspected that the number in question may
have some relation with the solar activity, it was compared with
the "provisory sun-spot number," published in Meteorologische
Zeitschrift, but no convincing relation could be found. ^-^ It will
however, be interesting to compare the present number with the
occurrence of sun-spots on a definite central area of the solar
disc.'-'

b) These short period waves appearing simultaneously in the
X- and Y-components run remarkably parallel to each other
during the morning hours, viz. 6^ — 8^ say, every detail in one
component being repeated by the other with marvellous similarity,
no noteworthy phase difference being observed between the two
components (PI. V). ^Moreover, the amplitudes of the two
components are generally of nearly the same order of magnitude.
It seems as if these waves were due to the fluctuation of an electric
current running from NE to SW, making an angle nearly 45° to
the meridian. For these short waves, Z-component is com-
paratively insignificant and may be clearly discerned only when
the photographic trace is very fine.

In the later afternoon hours, the parallelism between X- and Y-
components becomes imperfect. Some waves are present either in

1) W. van Bemmeln compared the frequency of the " pulsations " with the sun-spot
ntimber and arrived at a negative result, loc. cit.

2 1 E. W. Maundernoticed a 27 days period of magnetic disturbances and came to the
conclusion that the cause of the disturbance is to be sought on a limited portion of the
sun's surface, Monthly Notice of E. Astr. Soc, 65, 1904-05, p. 2, 538. The paper was
criticized by A. Schuster, ibid, p. 186. E. Marchand found a connection between the
magnetic disturbances and the sun-spots passing the central meridian of the sun's disc,
Congr. intern, de metéorolog., Paris, 1900, p. 148.

On Kapid Poriodie Variations of Terrestrial Magnetism.

17

one or the other component only. Even if a train of waves may
be traced in both components, the variation of the disturbing field
is generally of irregular rotatory character.

c) Generally, no corresponding regular trains of short waves
with such remarkable duration can be seen during the evening
hours, nor is the remarkal)le parallelism between X- and Y-
components so conspicuous as in the morning.

cV) During the later hours of the evening, the short waves
with periods of less than one minute generally disappear, while
very regular waves with 2-4 minutes periods appear instead,
frequently forming beautiful trains of nearly simple harmonic
waves. Another remarkable characteristic of the evening waves
is that the Y-component waves are generally inverted with respect
to the X-waves, i.e. the fluctuation of the horizontal magnetic
field is such as could be caused by the periodic variation of a
current running from NW to SE. Generally speaking, liowever,
the horizontal components show more or less rotatory character,
X- and Y-components showing frequently decided phase dif-
ference. This latter point wnll be fully investigated later.

For these longer waves, the Z-component is more con-
spicuous than for the shorter waves frequent in day time, and
runs remarkabl}^ similar to the X-component, though lagging
behind it by a considerable fraction of the period (see Plates
I- IV).

Fig. 5, A (reduced to \ original size).

Note the rotatory character of the horizontal component in the upper curves.

Ton 24 1913

Tun. 2 6 .

18

Art. 9. -T. Terada:

e) The beginning of these waves is sometimes gradual, but
frequently abrupt, starting quite suddenly after a period of
dormant calmness (Fig. 5). The trains with the abrupt beginning

Fig. 5, B (J original size).

Remark the regularity with which similar disturbances are repeated.

td> Z9. 1913

Feb 2g J9I3

are frequently accompanied by an abrupt increase of the average
value of the X-component whicli gradually attains a maximum
and then falls slowly (see Figs. 5 and 6). The sudden increase of
the X-component is usually accompanied by the simultaneous
sudden change of the average value of tlie Y-component, which
is at the outset generally toward E before midnight and toward W
after midnight. It seems as if an electric current were suddenly
started Avith rapidly increasing, and fluctuating intensity, with
its direction NW to SE, or SW to NE according to the hour of
occurrence. Disturbances of this kind seem to have been studied
by Birkeland who succeeded in tracing a system of whirling
currents (tourbillons de courants) extending over the N-hemi-
sphere, fixed relative to the sun. The slow change of the
horizontal components gradually attains a rotatory character, the
sense of which seems to generally confirm the result obtained by
R.B. Sangster,^-* i.e. mostly counterclockwise (N-W-S) before, and

1) R. B. Sangster, Proc. Roy. Soc, 84 A. 1911.

On Eapid Periodic Variations of Terrestrial Magnetism.

19

Fig. 6, B {ditto).

clockwise (N-E-S) after a certain hour near midnight. This seems
just as if the magnetic disturbing vector, initially inclined to the
meridian, rotates in the sense to l^ecome parallel to the meridian.

Characteristic disturbances of the above type are met with
most frequently just at midnight. In other hours, especially near

IBS' cases occur not
Fig. 6, A (^ original size). ^^rely where an abrupt

Xote the parallelism between X and Z. increase of the llOri-

zontal component is
not accompanied with
any conspicuous train
of waves.

It is a very re-
markable fact that
even for these abrupt
nonperiodic disturb-
ances of long duration,
the Z-component

follows the various
fluctuations of the
X-component with
utmost faithfulness,
except for a reduced
amplitude and a
definite retardation.
This characteristic

behaviour served as a
very convenient means
for disentangling the
cliaos of photographic
traces in the records
of disturbed days.

At the beginning of
the al)ruptly starting
train of waves, the
amplitude is some-

T'emark the a^jparent " beat " of the waves in tlie
lower curves.

Fig. Ü, C {ditto).

20

Art. 9. -T. Terada:

times very large, in one instance amounting to 5'4 ï with a period
of 1-2 minute (see Fig. 6, A). The amplitude is generally
damped gradually, with a sensibly constant period. On closer
examination, cases were also found where the period of the waves
changed gradually in an apparently coherent train. Sometimes,

Fig. 7 (f original size).

'^^nt 13. 19)3. M^-^r

1 Dec. 7. AT = +5"'

On Riipid Periodic Variations of Terrestrial Magnetism.

21

a very cliaracteristic train is repeated two or three times with
nearly equal intervals of 15™ 40™ etc., as if the same train of
waves were recrn'ring to the point of observation with a definite
period (Fig. 5, B).

/) It is frequently observed tliat while a fairly regular train
of waves is traced in both the X- and Y-components at the same
time, the periods of waves are quite different for the two com-
ponents (for examples, see Fig. 7). Among (:)S conspicous cases
chosen, 42 were those in which the periods were longer in X-
than in Y-component. The distribution of such cases in different
hours of the day may be seen in the following table:

Table I.

Hour.

1^

>

Ty

; Tx

<_

Tï 1

Hour.

Tx

>Tï

Tx <Tr

0- 2

0

2

1

12-14

8

0

2- 4

4

2

14-1Ü

3

1

4— 6

7

3

16-18

0

1

6- 8

2

2

18-20

2

6

8-10

6

3

20—22

3

0

10-12

4

2

1

22 - 24

3

4

A tendency is suspected, though not very apparent, that the
cases where the periods of X-waves are greater than those of Y,
are most frequent near midnight and noon, while they are com-
paratively rare in the morning and evening hours.

g) Though the most conspicuous regular trains are chiefly
observed witli periods ranging from 30^ to 300' still longer waves
are not at all rare, if we take those trains together in account,
with only a small number of maxima and minima. The longest
wave traced was of a period nearly amounting to one hour, which
may of course be detected in the ordinary magnetographic record
of low sensibility, when the amplitude is sufficiently large. The
intermediate periods are rather evenly represented in our list of

22 Art. 9. -T. Terada:

the waves, though there seems to be a maximum of frequency
near 15"

h) Slight local earthquakes, which are very frequent, affect
the instruments, leaving a trace similar to that of the Milne
seismograph (see for example PL V). This is in all probability
mainly due to the mechanical shock, to Avhich our instruments,
especially X- and Y-instruments, are rather sensitive. These
traces, however, served as very convenient and trustworthy time-
signals, as already mentioned. Besides, remarkable earthquake
waves with long periods, originating in distant regions, are
beautifully recorded in our magnetographs (PI. IV). The most
conspicuous periods are of the order 10'-20', i.e. decidedly shorter
than the usual magnetic waves. These peculiar waves appear in
the vertical component as conspicuous as in the horizontal com-
ponents, ^-^ while in the case of the usual magnetic waves of such
a short period the vertical component is almost nil in comparison
with the horizontal ones. Since our instruments are neither
quite free from the influence of inclination nor of acceleration,
even in the case of the Z-instrument, we can not be sure whether
these waves are not largely due merely to the mechanical effect,
though it seems not altogether improbable that an actual magnetic
wave is produced by the earthquake. To decide this point, it
will be desirable to carry out simultaneous observations with
quite different systems of instruments with the same magnetic
sensibility, but with considerably different mechanical sensibilities,
though it will be difficult to obtain instruments perfectly free from
mechanical disturbances. '-*

i) All slight or conspicuous magnetic disturbances which
may serve as premonitory signs preceding earthquakes, were
carefully sought, throughout the records in hands. At first, a
possible connection was suspected between the earthquakes and
the characteristic abrupt occurrence of the regular wave trains,

1) The vertical-force magnetograph as now constructed may be regarded as a sort of
seismograph, inasmuch as the line of suspension does not pass through the centre of the
mass of the suspend system.

2) The trace of the same earthquake on the X-component record of the Eschenhagen
magnetograph at Kakioka Magnetic Observatory, was about 6 mm. in maximum ampitude,
while the magnetic sensibility was about 5 y per mm.

On Rapid Periodic Variations of Terrestrial Magnetism. 23

described in paragraph e). But the trains in question which are
most frequent during the night hours, are distributed quite
liapliazard with respect to the earthcjuakes, if we either take
shght and moderate earthquakes together, or conßne our attention
to the moderate ones only. In any case, a further study in tliis
direction seems scarcely promising, in so far as the characteristic
magnetic waves are of no local plienomena confined within a
very limited area of the earth's surface.

On some occasions, slight irregular disturbances of very
abnormal type were observed in both components, but usually in
one component only, no corresponding simultaneous anomaly
being marked in the other. These irregularities are most probaljl}^
due to some slight accidental shock acting as an impulse to
settling down a slight unstable position of tlie suspended system;
for example, a slipping of microscopic magnitude, or the in-
troduction of a minute dust particle at the point where the
suspension wire touches the adjusting screw, may cause a quite
sensible displacement in such an ultrasensitive state of suspension
as adopted in our instruments. To detect premonitory, if any,
magnetic disturbances, it will therefore be necessary to protect
the instrument more thoroughly from any mechanical shock or
other accidental disturbance.

However, during the entire course of the observations, no
remarkable destructive earthquake occurred in the neighbouring
districts, so that we are not yet in a position to draw any definite
conclusion as to tlie value of magnetic observations as a means
for predicting strong earthquakes — which was the original purpose
of our present installation.

PART III.
Special Statistical Investigations in Detail.

6. In the following, some of the results of the detailed
statistical investigations with respect to some of the general results
above described, will be given, w^hich are on one hand a con-
firmation of .the qualitative remarks stated in the preceding

24 -^rt- 9.-T. Terada.

paragraphs, and on the otlier hand constitute the Ijasis of the
theoretical considerations to be propounded later on.

It is the pleasant duty of the author to acknowledge here his
indebtedness to Messrs. T. Tatiiri, the late H. Suzuki, M. Nakata
and S. Hana who successively served as his assistants and com-
putors in examining and making toilsome measurements of the
records, and carrying out many tedious calculations, with in-
defatigable zeal and true scientific candour.

Some parts of the statistical work wliich necessitated the
utmost care in measurements, foi' example the determination of
the phase differences of difïerent components etc. w^ere made by
the author himself.

7. At first, the photographic record were examined sheet
by sheet, the conspicuous weaves constituting more or less regular
trains were selected and the approximate mean periods determined
by dividing the time interval by the number of waves contained
in the train. ^^ It is indeed difficult to decide wliich trains to take
and which to omit. It was therefore decided to take as many
trains as possible with different periods, provided that the trains
comprise at least three very regular simple waves with sensible
amplitudes. When during the course of an hour a very long
continuous regular train with nearly constant period occurred,
the period was determined only for one or two portions of the
train chosen arl)itrarily from the most conspicuous parts.

The case is more difficult when the fundamental waves are
superposed with different " overtones." Very complicated waves,
liable to ambiguities, were omitted, even if it were possible to
analyse them into simple components, and the apparently regular
trains only were taken into account. The results of the ex-
amination wei'e systematically tabulated, separately for each of
the three components, according to the successive hours of day.
These tables formed the Imsis of some of the following in-
vestigations.

1) Strictly speaking, the periods are sometimes variable even in an apparently reg-ular
train of simi^le waves. The inconstancy is too considerable to he explained Ijy the ir-
regularities of motion of the drum carrying the j^liotographio paper.

On Rapid Periodic Variations of Terrestrial Mafjnetism. 25

8. Hourhj distribution of ivaves of different j)eriods. The
different periods were classified into the following arbitrary
groups: 30•0^-49•9^ 50•0^-69•9^ 70•0^-89•9^ 90•0^-129•9^
]30^-20(/. The occurrence of waves Ijelonging to the different
groups, in different hours of successive days, was plotted down in
diagrams similar to those drawn by Bidlingmeier'^ in his " Ue-
bersicht über die Tätigkeit des Erdmagnetismus."

From these diagrams, it could at once be seen that waves
with periods shorter than about 50' are most frequently met with
during the day time, while the longer waves over 90^ periods are
generali}' most frequent during the night. This is naturally true
for each of the three components, though in Z-component the
shorter waves are generally of almost insignificant amplitudes.
To make this remarkable fact more apparent, the fohowing
procedure was adopted. Different days of a month, or of a
season, were taken together, and the number of those days in the
said interval, in which a certain given hour was disturbed by the
waves of a certain given period, was counted and then divided by
the total numljer or days in that interval, in which the ol)-
servation was actually made at the very hour in question; the
latter reduction was necessary, since especially in summer, the
records were often imperfect. The number tlius reduced, multi-
plied with 100, was called the "frequency" of that given group
of waves in that particular hour, for the month or season in
question. The following tables give the hourly frequency of the
different wave groups in X- and .Y-component, for different
seasons, all tlie records available being taken into account. Z-
component is omitted, since it is always a reduced facsimile of
X-component, as already mentioned.

1) Bidlingmeier, Veröffentlichungen d. kais. Observatoriums in Wilhtlmshaven

26

Art. 9. -T. Terada

Taiîle II.

The nnmbers give the frequency of the waves of different i^eriods in %.

X. 80^-50'^ Y. 30^— 50'^

Month.

ir

V

VIII

X[

II

V

VIII

XI

III

\^I

IX

XII

Year.

II [

VI

IX

XII

Year.

Hours.

V[

VII

X

I

IV

VII

X

I

0- 1

3-2

2-8

6-6

39

4-1

2-0

3 0

3-9

23

2-7

1- 2

5-7

5-2

8-7

6-4

6-4

3-3

3 0

7-8

2-3

3-7

2- 3

4-5

6-7

7-8

7-1

6-4

5 3

6-0

3-9

2-8

4-4

3- 4

7-5

8-3

8-4.

101

8-9

8-2

10-1

7-8

2 0

6-7

4- 5

12-2

9-7

10 3

12-5

11-3

10-3

10-2

11-2

3-6

8-5

5- 6

18-0

14-7

14-1

171

16-2

11-7

13-4

6-6

4-4

8-9

6- 7

18-4

14-9

lö'O

19-7

17-5

15-4

13-4

16-8

10-9

13-9

7- 8

2)-0

13-7

21-4

27-3

20-1

19-8

15-6

25-5

24-2

21-2

8- 9

18-7

14-7

154

31-5

20-8

17-8

. 14-9

16-7

23-5

18-6

9—10

18-6

151

li-9

25-9

191

15-0

14-5

12-9

19-1

15-8

10—11

13-8

19-6

19-2

211

18-4

12-2

13-3

14-5

16-2

14-1

11-12

15-8

20-8

20-Ù

22 3

19-9

12-5

lG-4

7-8

16-9

14-0

12-13

20-7

19-1

20-4

19-5

19-9

12 2

7-2

16-4

16-9

13-5

13-14

19-2

21-8

2>8

16-1

19-3

13-3

6-7

15-9

16-5

13-5

14-15

14-4

21-1

22'2

12-7

17-2

11-7

6-2

U-3

.12-6

11-4

15-16

11-6

15-5

10-8

9-9

11-8

4-4

4-7

7-8

81

63

10-17

10-8

14-2

9-0

7-1

10-1

4-3

26

2-8

7-1

4-7

17-18

6-8

10-4

9-2

10-3

92

2-9

0-6

4-1

7-0

40

18-19

4-2

13-6

6-9

10-8

8-9

3-9

51

8-4

5-4

5-4

19-20

5-3

12-8

(3-8

6-6

7-8

4-2

6-1

7-0

2-9

4-8

20-21

4-2

6-5

3-4

5-5

5-0

2-7

6-0

1-3

2 2

31

21—22

5-5

5-3

3-4

5-6

5-0

2 3

37

1-9

3 0

2-7

22 23

1-G

31

6-3

3 0

3-4

1-5

2-3

3-8

19

2-2

23-2t

3-2

4-0

tV3

38

4-3

1-f,

33

5-1

0-7

2-4

X.

50^—70-^

-70^

Month.

II

V

VIII !

XL

II

V

vin

XI

III

VI

IX

XII

Year.

III

VI

IX

XII

Year.

Hours.

IV

VII

X

I

IV

VII

X

I

0- 1

8-6

7-9

12-2 '

7-4

8-9

3-6

3-5

10-2

3-8

4-8

1— 2 i

11-1

5-6

5-6 1

9-4

8-2

7-8

5-0

4-5

4-2

5-5

2- 3

7-5

8-6

104

15-2

10-6

9 3

5-5

6-5

5-5

6-4

3- 4

12-8

9-7

90

17-6

12-7

6-5

106

6-6

9-5

8-4

4- 5

11-4

12-6

7-6

15-2

120

11-2

11-7

7-4

8-3

9-8

5- 6

14-6

15-7

13-6

19'9

16-2

9-6

10-7

130

10-6

10-7

6— 7

19-2

20-3

16-6

14-7

17-6

11-2

17-2

18-2

6-7

12-5

7- 8

17-5

21-7

22-0

21-6

207

18-2

17-1

25-8

16-9

18-9

8- 9

19-6

16-6

16-1

25-5

19-8

16-9

11-8

19-9

23-5

18-3

9-10

18-6

17-i

14-8

18-2

17-3

14-2

7-5

14-0

19-3

14-2

10-11

16-2

18-2

16-1

17-1

17-0

9-4

13-3

11-6

17-6

13-2

11-12

16-0

15-0

16-7

19-8

17-0

6-9

8-5

ll-O

20-2

12-4

12 -13

15-8

16-4

14-4

19-8

16-4

103

5-2

13-2

17-8

12-4

13-14

17-1

12 3

18-3

16-1

16-0

9-5

5-3

11-1

17-1

11-5

14-15

13-5

12-6

15-6

13-9

14-3

9-3

3-4

6-8

14-4

9-4

15-16

10-7

14-1

12-4

11-6

12-1

6-1

2-0

2-7

6-9

4-9

16-17

7-8

9-5

8-0

8-9

8-5

1-7

2 6

3-3

7-0

4-0 '

17-18

7-2

7-0

11-8

7-5

8-2

1-6

0-6

6-6

6-2

3-8

18-19

i 6-6

11-4

10-S

9-0

9 3

3-9

6-6

7-0

4-6

5-3

19-20

6-8

8-9

13-6 1

6-2

8-6

5-0

13-2

10-4

5-1

7-9

20-21

5-0

7-0

12-6

5-9

7-3

2-7

10-6

11-0

5-1

6-8.

21—22

4-7

8-9

10-1

7-1

7"7

4-8

6-4

8-5

2'2

4-8

22-23

4-3

6-7

7-8

7-2

6-4

31

51

6-8

5 9

51

23 - 24.

72

S'5

9-3

120

8-8

2-3

4-2

9 '9

22

41

On Rapid Periodic Variations of Terrestrial Magnetism.

27

Table II. {continued).

X.

70^—90'^

Y.

70^—90^

Month.

II

V

VIII

Xi

II

V

VIII

XI

III

VI

IX

XII

Year.

III

VI

IX

XII

Year.

Hours.

IV

vri

X

1

IV

VII

X

I

0— 1

10-2 '

9-3

ll-I

12-0

10-7

7-2

3-0

6-4

5 0

5-4

1— 2

11-5

6-4

10-8

I3-0

10-8

7-3

3 0

7-1

4-3

5-4

2- 3

11-2

9-9

9-4

18-8

12-7

8-9

4-5

7-1

6-7

6-9

3— 4

9-5

8-5

7-9

130

9-9

5-7

6-5

3-8

5-6

5-5

4— 5

12-7

11-5

8-7

12-9

11-6

5-0

61

5-3

6-4

5-7

5- 6

: 13-4

12-8

8-1

14-0

123

6-2

7-5

40

5-2

5-8

6- 7

12-2

9-2

8-3

10-8

10-3

6-2

4-8

6-0

3-6

51

7— 8

14-2

12-1 '

7-9

130

120

7-4

7-3

7-3

7-4

7-4

8— 9

10-6

8-8

9-6

13-3

10-8

10-1

6-7

6-7

12-1

9-3

9-10

10-5

8-1

5-4

11-1

9-1

6-5

4-5

2 0

8-2

5-8

1Ô-11

8-1

9-6

6-8

9-5

8-6

4-5

5-5

2-2

8-3

5-5

11—12

! 9-4

10-1

6-4

7-7

8-4

4-7

1-2

1-6

7-7

4-5

12-13

120

10 4

10-4

7-6

100

5-1

2-6

0-8

10-2

5-6

13-14

8-8

4-7

7-6

9-6

7-8

5-2

27

4-8

8-2

5-7

14-15

93

9 -.7

10-9

8-9

9-6

5-6

2-0

30

7-5

5-1

15-16

9-0

8-2

8-8

3-8

7-3

3-5

3-3

3-6

4-6

3-9

16-17

3-5

4-9

4-0

2-6

3-7

0-9

1-3

1-4

3-7

20

17-18

40

4-9

7-7

7-4

6-0

2-4

0-0

' 4-1

4-0

2-8

18-19

4-6

8-0

9-4

5-4

6-6

51

4-1

7-1

4-3

50

19-20

' 7-6

8-9

10-2

100

9-1

9-5

8-5

7-0

5-1

7-4

20-21

5-0

6-4

9-3

7-0

Ü-8

3-8

7-4

11-3

4-8

6-3

21—22

1 71

9-3

9-1

10-4

9 0

4-2

5-5

5-6

5-5

5-2

22-23

1 7-9

9-8

9-7

9-8

9-3

5-4

4-2

4-5

4-1

4-6

23-2i

10-8

r.-i

8-8

11-5

0-5

5-9

28

3-8

4-1

4-3

X.

90^—130^

Y.

90^—130^

Mcmth.

Hours.

II
III
IV

V
VI
VII

VIII
IX
X

XI
XII

I

Year.

0— 1

1— 2

2— 3

3— 4

4— 5

5— 6

6— 7

7— 8

8— 9
9-10

10-11
11—12
12-13
13-14
14-15
15-16
16—17
17—18
18-19
19-20
20-21
21 — 22
22-23
23-24

19-6

20-1

18-2

15-4

13-9

176

10-4

5-4

9-0

9-7

5-7

8-2

9-1

7-5

11-4

7-7

5-2

40

5-4

7-6

8-1

11-0

123

16-4

9-6
9-6
9-3
9-5

10-6
9-2
7-0
7-7
6-8
6-1
8-1
6-7
6-8

10-2
7-5
50
8-5
3-5
7 0
5-0
8-1

10-0
6-4
6 0

10-6

9-2

8-8

13-2

8-0

7-1

5 0

4-2

7-6

5-9

7-8

6 '4

6-5

6-1

9-4

8 '8

4-0

4-6

11-4

14-5

1-2 -2

11-6

10-2

131

24-8

277

21-8

19-6

16-4

14-8

12-8

8-4

10-0

S-5

120

8-8

6-4

10-0

6-3

5 0

56

7-1

8-9

14-4

12-1

IM

19-6

22t'.

16-8

17-5

15-2

14-8

12-6

12-6

91

6-6

S-5

7-7

8-5

7-6

7-2

8-5

8-5

6-5

5-8

50

8-1

10-4

101

10-9

12-5

150

II
III
IV

V
VI
VII

VIII
IX
X

10-6
8-9
8-6
7-4
60
3-8
6-2
8-1
6-1
4-5
2-6
1-9
4-3
2-8
2-6
0-9
2-8
7-0
91
8 0
9-5

112

lis

70
30
2-5
7-5
6-6
5-9
4-3
31
31
1-5
0-6
1-2
1-3
2-7
1-4
2-0
2 0
0-6
6-6
6-1
101
10-1
8-8
5-1

8-3
77
7-1
8-3
8-6
3-9
47
3 3
67
27
1-5
2-3
00
1-6
0-8
3-6
07
1-4
4-5
8-3
107
130
77
51

XI

XII

I

Year.

11-5
8-2

11-1
7-2
6-3
4-8
53
5-8

12-5
8-2
3-9
4-2
51
47
4-8
37
4-1
3-3
5-4
6-6
7-8
7-4
9-3

10-1

9-1
7-6
77
7-9
7-3
5-2
4-5
4-9
8-0
51
2-9
2-8
2-5
3-6
2-8
31
2-1
23
6-0
7-5
8-9
97
9-5
8-5

28

Art. 9.-T. Terada:

Table II. {continued).
X. 130^—200' Y.

130^—200'

Month.

II

V

VIII

XI

1 II

V

VIII

XI

^^

1 III

VI

IX

XII

Year. ' III

VI

IX

XII

Year.

Hours.

IV

9-6

VII

X

I

IV

VII

X J

I

0- 1

6-5

1-5

17-1

9-2

6-8

5-0

1-3 1

7-3

5-5

1- 2

7-6

8-5

7-2

16-7

10-3 4-9

5-0

3-2

5-8

4-9

2- 3

61

8-1

6-2

12-1

8-3 1 4-0

2-5

1-9

50

3-6

3- 4

6-5

10-2

4-7

9-8

7-9

1-2

3-5

3-2 i

3-6

2-3

4- 5

7-4

8-3

3-2

9-8

7-4

3-7

2-0

3-3

3-6

3-2

5- 6

3-3

5-4

4-4

8-2

5-4

4-6

4-3

3-9

4-8

4-5

6- 7

3-7

4-0

0-6

39

3-2

1-7

1-6

1-3

1-6

1-6

7- 8

1-2

3-3

2-6

2-7

2-4

0-8

05

0-7

2-3

1-2

vS- 9

1-6

3-3

1-5

3-7

2-6

2-8

31

0-7

3-8

2-8

9-10

0-8

1-4

4-4

6-3

3-3

2-0

0-5

2-0

4-1

23

10-11

1-2

2-7

3-4

6-6

3-6 ! ! 1-6

0-6

0-7

4-2

20

11-12

2-4

1-8

4-4

6-6

3-9 0-4

0-6

00

1-9

0-9

12-13

4-1

4-1

25

10-2

5-5 ' 0-5

00

0-0 '

20

0-8

13-14

4-1

3-2

3-6

5-0

4-0 1-9

20

08

16

1-6

14-15

3-7

3-7

31

5-4

4-1 : ! 0 5

00

0-8

0-4

0-4

15-16

2 5

3-8

2-1

2-7

2-8 !

0-0

0-7

0-7

1-5

0-3

16-17

1-7

2-8

0-5

30

2-1 1

0-9

0 0

0-7

2-6

1-3

17-18

0-4

1-4

0-5

6-4

2-4 ' i 0-8

00

0-0

3'7

1-4

18-19

2-7

1-8

3-5

6-5

3-7 ; 4-3

15

1-3 1

7-2

4-1

19-20

4-1

1-8

5-3

6-9

4-6 4-2

4-2

6*4

80

57

20-21

5-7

4-8

7-3

9-6

6-9 ! Ö-3

5-5

7-6

8-5

67

21 — 22

6-9

4-9

6-2

14-6

8-4 1 ' 7-6

10-6

3-7

5-9

7-1

22-23

10-5

6-7

6-8

19-6

11-3 5-4

8-4

2-6

9-3

6-8

23-24

11-0

58

4-4

15-3

9-5 6-6

2-8

5-1

10-1

6-5

X.

3œ

—200^

Y.

30

^—200^

Month.

II

V

VIII

XI

II

V

VIII

XI

III

VI

IX

XII

Year.

III

VI

IX

XII

Year,

Hours.

IV

VII

X

I

IV

VII

X

I

0— 1

51-2

36-1

420

65-2

49-7

28-4

■

21-5

30-1

29-9

27-5

1- 2

56-0

353

41-5

73-8

53-2

33-9

19-0

30-3

24-8

27-1

2— 3

47-5

42-6

42-6

75-0

53-2

j 36-4

21-0

26-5

31-1

29-0

3- 4

51-7

46-2

43-2

70-1

54-2

30-2

38-2

29-7

27-9

31-3

4— 5

57-6

52-7

37-8

G6-8

54-9

37-6

36-6

35-8

28-7

34-5

5- 6

66-9

57-8

47-3

74-0

62-7

38-1

41-8

31-4

29-8

351

6- 7

63*9

55-4

46-5

61-9

57-7

38-3

41-3

47-0

28-]

37-6

7- 8

58-3

58-5

58-1

730

61-8 52-4

43-6

62-6

56-6

53-6

8- 9

59-5

50-2

502

810

62-5 55-7

39-6

50-7

75-4

57-0

9-10

58-2

48-1

45-4

70-0

56-5 43-8

28-5

33-6

58-9

43-2

10-11

45-0

58-2

53 3

66-3

561 322

33-3

30-5

50-2

377

11 — 12

51-8

54-i

54-5

65-2

56-8

271

27-9

22-7

50-9

34-6

12-13

61-7

56-8

54-2

63-5

59-0

30-0

163

30-4

52-0

34-8

13-14

56-7

52-2

56-4

56-8

55-6

34-2

19-4

34-2

48-1

35-9

14-15

523

54-6

61-2

47-2

53-7

29 9

130

25-7

397

29-1

15-16

i 41-5

46-6

42-9

33 0

405 16-6

127

18-4

24-8

19-0

16—17

29-0

39-9

21-5

27-2

30-2

8-7

8-5

8-9

24-5

141

17—18

22-4

37-2

33-8

38-7

30-8

10-5

1-8

16-2

2i'2

14-3

18-19

; 23-5

418

42-0

40-6

36-6 24-2

23-9

35-4

26-9

25-8

19-20

31-4

37-4

50-4

44-1

40-5 32-0

38-1

39-1

277

33-3

20-21

28-0

32-8

44-8

40-1

361 225

39-6

41-9

23-4

31-8

21-22

35-2

38-4

40-4

48-8

41-0 28-4

36-3

32-7

24-0

29-5

22—23

36-6

32-7

40-8

59-2

42-9 26-6

28-8

25-4

30-5

28-2

23— 2i

48 -G

30- i

41-9

65-2

47-1 28-2

18-2

29-0

27-2

25-8

On Eapid Periodic Variations of Terrtstrial Magnetism,

29

The results are also plotted in Fig. <S. As will be clearly
seen, tlie waves with periods less than 70' show decided maximum
frequency in the da}^ time, more or less near noon, while those

X

'h

30-

50

i

îflïH

H\

1 '^

/

\

'^

jyvyt

/

\

/

^

\

\

/

r

4

'Si'BES

\i

(

/

\

^

""

\f

/\

"nmfn

\

G \Z \S Z^

YEAR

\

\

"■

\

I ^

0 6 li ;i z^ 0 6

A

\

r

\

T~

^U

i\

^

/

^w

1

;h

\'^/

y7

^^

A,

^^

\,

\ ,.

1

V

0 6 1

. IS zt.

/^

/

\

J

Vv/

Fig. 8, X.

70-90 90-130

(30-200

30-200

^ -J

-j^x,zi

\j

7 /vV Mj /

^ ^^x'

-J^K^

r ^^

v^ ^-a57^ 5 .

^ A ^

'WV

-J/^A, f\ za.

'^^ N^^

L- 4—

C t /

K

A - V- -/i

^ ^u ^ t t

\

K ^\. \ i. J

6 1Î 1

J 2^ C 0 IZ 18 1+ 0 Ô rl 15 24

::^5^iz

7tt\^

^%'

An

k'

V"

IS 1+ 0 6 li IÎ

"\,

s

./

[v^

'X

1

1

S

/'

L^

^'

5 (

1

I 1

B 34

1

1

1

_y|'>^

N

V.

^

0

J 1

( i-v

longer than 90^ are more frequent during the night. For 30'-50^
waves, a tendency is suggested in the X-component that the hour
of the maximum frequency is earlier in winter than in summer.
Besides, for X- as well as for Y-components a secondary minimum
near noon is èuspected in spring and autumn. For 50^-70^
waves, the hours of the day time maxima seem to fall somewhat
earlier in the morning than for the shorter waves, and a secondary
maximum in the early evening is suggested in some of the
Y-diagrams. 70'-90' weaves form apparently a transition stage to
the still longer waves. For 90^-130^ and 130^-20^ waves, the

30

Art. 9.— T. Terada:

maxi man frequencies fall decidedly in the night hours, mostly a
little before midnight, while at the same time, a secondary
maximum in the day time is suspected in some of the graphs.

Y

IDQ

Fig. 8, Y.

30'-50' 50'-7(3' 70'-90' qo'-l30' i30-200'

HVVI

mffln

3Z1SI

A

^^\

^1 I

/ 1 ^v_

^"^

t V=^

r- A^^

(i

t^

A'

*r \j\

' viV/

fV

S

-^-V

i "t — J;— H

0 6 ,2 \3 i^ c

1

1

h

1

f \

1

rl ^

M.

U-"^

1

!

1

A^

K

/ ^

"^ ^

1
1

1

\

-A

A U

ir^

\/^

\J

A

r

\

/

%

T^A.

K-j

A

i^

v

~^^.,

i\

-' ] \

-

1

1

1

-\

1

A

A

K

Ki 1

1

,

.-i^^'V

^

Sa

ö-

AU

V^

ft

TÎ n~J4 e

^'

fv

-/-~v\^

r

/\

Va

J^

^/ ^

>

r-j^

L. '^

1

i

1

S

v^

^

l£V

v^l

ij C4 0 s 12 li 24

YEAR

1 1

1

'\i

^ u

7

1 À

y

1 "i^

^

/

V.

s,

/

\.

^

1

1

-1^^7

f^

I 6 II IS i1 0 e 12 I« ^^ 0 o 11 IS :+

! 1 i 1

j

1

1

Ua--.^

fC^

im;^

^

II I! Z4 0 6 II '» -24

30'- .200'

•y«

I

'a

i

^. LJ

0

1

i 1

i 1

1

/iS h^

J ! ^^ / -

1

100

i

K ■ '

.Ar^

-^r

1

V

1

1

1

1

1

1 1

lAi i

J KU

0

n

s 6 à 15 2I

Ö u I«

The frequency of all waves, 30'-200', shows a maximum in
the morning and a minimum in the evening.

It is to be remarked that the diurnal variation of the number
of ' ' spasms ' ' as well as ' ' pulsations ' ' studied by W. van
Bemmeln, and also that of the frequency of the disturbed hours,
as could be deduced from the diagrams in Bidlingmeier's paper
above cited, show a character quite similar to the variation of the
frequency of the "longer waves" here studied, up to the
appearance of the secondary daytime maxima.

On Eapid Periodic Variaticns of Terrestrial Magnetism.

31

On closer examination of the diagrams mentioned in p. 25
above, it is frequently found that wave trains with a certain
period appear in almost the same isolated hours of two or three
successive days, as if to repeat the particular program of the
preceding daj^s. This is interesting as one of the facts strongly
suggesting that the agent producing these waves is intimately
connected with the position of the earth relative to the sun. In
some cases, it also happened that the hours of tlie successive days
in which a particular train appears seemed to shift gradually in
either direction.

9. The ^' spectra^' of the magnetic waves. To find the most
frequent period, the o1;>served periods were at first classified second
by second and the total number of trains (not that of days or
hours in which they occurred) belonging to these groups were
plotted in a diagram with the periods as abscissa. The diagram
obtained showed a great number of maxima and minima with

intervals of several
seconds. Most of
these maxima and
minima were, how-
ever, found to have
no real physical
meaning, being

either due to
chance,^-' or due to
some involuntary
tendency of par-
ticular persons to
some par-
ticular fractions of the scale division when measuring the wave
trains on the records with a millimeter scale. ^^ Hence, finally the
following groups were taken: 20'-30', 30'-40' etc. up to 190'-
200'. The results are given in Table III. and also plotted in
Fig. 9. As will be seen, the maximum frequency is generally

II T. Terad», Proc Tokyo Matb.-Phys. Soc, 8, 1916, p. 492.

2) Note that a tenth of a millimeter corresponds to about I'S sec.

Fig.

9.

7o

12

^

!

n^-

.

i Î i ;

10

A

1

à

6
4
2

t

i

V ^

N

/

1

«*

0 20 40 60 50

100

120 140 IfcC

T'' prefer

32

Art. 9. -T. Terada :

at the interval of periods 5(/-60! The curve reminds us ratlier
of the energy distribution of the kiininous spectra than the usual
probability curves. A secondary maximum is seen near the
period 90! The results for other years are of quite similar
character.

Table III.

For each season, the first column gives the number of trains and the
second the percentage of the total number.

1912.

X.

Season.
Period.

I 11

ni

IV ^

^ VI

VII VIII IX

X XI XII

Year.

10s— 20s

3

0-3

2

0-2

0

0-0

2

0-1

7

0-1

20— 30

20

1-8

20

2 3

10

0-6

36

2-0

86

16

30— 40

120

10-9

75

8-8

126

7.8

143

7-8

46-1.

8-6

40— 50

145

13-2

114

13-3

207

12-8

250

13-6

716

13-2

50— 60

134

12-2

106

12-4

231

14-3

224

121

695

12-8

60— 70

116

10-5

100

11-7

2S9

17-9

227

12-3

732

13-5

70— 80

85

7-7

63

7-4

168

10-4

145

7-9

461

8-5

80— 90

77

7-0

56

6-5

136

8-4

141

7-6

410

7-6

90 — 100

107

9-7

66

7-7

139

8-6

154

8-4

466

8-6

100 — 110

70

6-4

51

6-0

92

5-7

127

6-9

340

6-3

110 — 120

59

5-4

36

4-2

56

3-5

114

6-2

265

4-9

120 — 130

45

4-1

35

41

38

2-4

74

4-0

192

3-5

130 — 140

33

3 0

36

4-2

34

2-0

62

3-4

165

3-0

140 — 150

24

2-2

23

27

16

1-0

37

2 0

100

1-9

150 — 160

21

1-9

20

2-3

29

1-8

35

1-9

105

1-9

160 — 170

24

2-2

21

2-5

19

1-2

37

20

101

1-9

170 — 180

8

0-7

9

1-1

17

1-1

17

0-9

51

0-9

180 — 190

5

0-5

5

0-6

4

0-3

10

0-5

24

0-4

190 — 200

5

0-5

18

2-1

7

0-4

10

0-5

40

0-7

1101

856

1618

1845

5420

The result must not be hastily interpreted as showing that
the absolute maximum of frequency is at the above mentioned
interval. If we could multiply the sensibility of the instruments
and turn at the same time the recording drum correspondingly
faster, the waves most conspicuous in our case might possibly be
replaced by others with a decidedly shorter periods. Besides,

On Rapid Periodic Variations of Terrestrial Magnetism.

38

it must be taken into account tliat tlie shorter the period, the
greater will be the chance of revealing isolated trains in a given
interval of time, even if the mean intensity and number of
maxima of the trains be independent of the periods. This
partially explains the falling off of tlie frequenc}^ curve tOAvard
the longer period in a form of h3"perbola. Hence the real
meaning of the above result can be simply interpreted as follows:
As far as the sensibilities of the present instruments reveal, every
period is rather evenly represented as in a continuous spectrum,
showing no very sharply defined maximum, except two rather

Fig. 10.

XI XI. I II ill ,v

1^12 j;i ,x

M XII

I I

II III

(ßJ

ce;

0^ To 20 ctai^S

100

A

/A

\

/

fv

v'

5" 10 cUv5

flat relative maxima near 60^ and 90? The above holds good
when we take the different hours of day and night altogether
into account. AVhen the different hours arc taken separately,
the results are somewliat different as already remarked in §8.
This latter point will be referred to again later on, with respect
to difïerentty chosen materials. Here it may be remarked only

34 Art. 9. -T. Terada :

that the secondaiy maximum of frequency at about 90' corresponds
to the most frequent waves observed during night hours.

No conspicuous group of waves with about 10' period,
observed by Birkeland, could be confirmed in our records, though
such waves might have been detected if the amplitudes were
somewhat larger than 0'2 y.

10. Fluctuation of frequency of regular waves in successive days.
The number of hours (not that of trains) disturbed by regular
waves was counted for each of the successive days and plotted in a
diagram with the days as abscissa. The results showed apparently
very irregular fluctuations, either if we take all waves with the
periods 30^-200^ or only 30^-70' waves. Though we must be
very cautious in deducing any periodicity from such material, the
results show nevertheless a remarkable tendency to suggest the
existence of a period, or rather of an "interval" of about 25-30
days. The most conspicuous examples are illustrated by the
interval from October, 1912, to April, 1913, especially when the
waves with the periods 30^-70^ alone are taken into account
(Fig. 10, A). Fig. 10, B and C were obtained by superposing the
successive series of 14 or 28 days^^ respectively. These graphs
seem to suggest the existence more of a 14 days period than a 28
days one, the amplitude amounting to the same order of magnitude
as the mean vahie. The last procedure will however, have a
definite meaning only when the phenomena are simply and purely
periodic, but not when the frequency maxima are repeated with
conspicuous amplitudes only a small number of times in a definite
period, and then replaced by anotlier of a similar kind, but setting
in more or less abruptly with quite accidental time relation to the
former. The general aspect of the frequency diagram here in
question strongly suggests that the nature of the phenomena
concerned is of the type above mentioned. In the above
example, we ma}^ probably assume the accidental superposition
of two independent serijs witli the longer period, say 28 days.
The lunar period is to be excluded, since the above fluctuation

1) 28 days instead of 27 days was taken simply because it is an even number, no weight
being laid here on the exactitude of the period.

On Rapid Periodic Variations of Terrestrial Magnetism. 35

does not keep pace with moon's phases. It will be more plausible
to suppose that the period in question is related to the sun's
rotation. ^^

It is to be regretted that our observations were not very
adequate for investigations of the kind mentioned above, since
in the warmer half of the year the records were occasionally
defective. For the same reason, we must unfortunate^ refrain
from carrying out the statistical investigations with respect to the
annual or seasonal variation of the daily frequency.

11. Probable relation of the frequency of waves with meteor-
ological elements. Since the phenomena of the magnetic pulsations
were suspected of being closely related to some atmospheric
phenomena as will be seen later on, it seemed possible that they
might have a sensible correlation with some of the meteorological
elements. Among the latter elements, cloudiness was considered
as a most promising one for investigation, since its correlation
with the solar activity is already acknowledged by some au-
thorities,^-' and moreover, considering the layer of cloud as a
conducting sheet covering a considerable part of the earth's
surface it may in some way or other pla}^ a sensible rôle where
electromagnetic waves of not indefinitely long periods are in
question. To carry out the comparison effectively, it will
however, be necessary to take the cloudiness over a sufficiently
wide area of the earth's surface in order to eliminate local
irregularities of secondary nature which are very conspicuous
in the case of this element. As the only available data in hand,
the Monthly Reports of the Central Meteorological Observatory
were used. Different groups of the local stations were taken
and the average amounts of cloudiness were calculated. The
results of comparison were not conclusive, though some correlation
was suspected between the frequency of the magnetic pulsations
and the average cloudiness for Tokyo, Tyosi, Mito and Maebasi,

1) Maunder and Marchand, loc. cit. Curiously enough, Schuster, on examining
Maunder's data, also noticed a pronounced 14 days period. If the 14 days period was to be
invariably found in allied phenomena, then the seat of the direct or indirect cause of
magnetic disturbances may be sought in two antipodal portions of the sun's surface.

2) Hann, Lehrbuch der Meteorologie, 2nd. Ed., p. 47ö.

36 Art. 9. -T. Terada:

which is shoAvn in Fig. 10,1). Tliis point decidedly deserves
further investigation.

12. To carry out a more detailed quantitative investigation
regarding the nature of the magnetic pulsations under con-
sideration, the records of 1913 were specially chosen, since in
this year the determination of tlie sensihilities of the instruments
was most regularly made and lience most adequate for the
quantitative comparison of the different components. It is,
however, to be regretted that during the summer months the
records were too frequently defective, chiefly due to damage to
the photographic paper on account of the extreme dampness
notwithstanding tlie use of the desiccator, and also due to the
condensation of the atmospheric humidity in minute drops or
mist which was especially dense in tlie daytime and caused a
remarkable absorption of light. It must therefore be remarked
that in some statistical studies to be described later, tlie winter
season has a rather overweighing influence on the general results
though the statistics extend over a full year. As far as the
present investigations are concerned, no serious modification of
the general results will, however, l)e required on that account.

13. Ratio of amplitudes of X- and Z-components. As already
mentioned, the waves appearing in the Z-component arc generally
the reduced facsimiles of those in the X-component, except that
the former always lags behind the latter in definite amounts
depending on the periods, but usually less than a quarter of a
period. In other words, the end of the magnetic vector repre-
senting the periodic disturbing field revolves on more or less
elliptic orbits, with their major axes mostly dipping towards N.
To find first a quantitative relation between the amplitudes of
the two components, the following reductions were made.
Specially regular portions of wave trains were carefully chosen
out from among all the records of the year, and for each train
the ratio of the mean amplitude of corresponding waves for the
two components w^as calculated, the daily value of the sensibilities
being duly taken into account. The results were tabulated
together with the corresponding periods and tlie hours of

On Rapid Periodic Variations of Terrestrial Magnetism.

37

occurrence. Plotting the ratios as ordinates with the periods as
abscissa in a diagram, the points representing different trains are

1-2
10

0-8
0-6
04

02

0^-

-3

*

..

•,

.

. .

•-•♦.

-.;

'■'.

*

•

■• •

s ^"

■f^

0

1

0

2

J

3

0

-?

■0

5

0

7

Fig. 11.

12
1-0
OS
06
0-4
0-2
0

z'-

-\b

X

>

1

i

•

\

.

.

•

. .

''.'

•

i '

■

10 20 30 40 50 T

1-2
i-0
0-8
0-6
0-4-

f-

- h

to

5--

17

K

10
C-8
0-6

1

•

^

.

'

•

.

,

1

.

..-

•

••^

•

'•1

.%

;•

'h'

•

02
0

s^"

rr

1 i

...

10 20 3Û vy 50 T

10 20 30 40 50 T

)-2
10
OS
0-6
0-4

0-2
0

6--

-«?■■

^

•

'

♦

•

7^'"^
^•^•'

.. .

IÛ 20 30 40 50 T

1-2
10
08
0-6

1

s-

-21

K

•

■-

^,

.

•••

;•*'

,

•

02
0

*
■x^-

',

12
10
0^8
06

0-4

0-2

9'--

■12'

1

•

•

*

'.i.i

;,i--

C-»"i

,. .

•

1

10 20 30 40 50 T

12
10
0-8
06

04-

0-2
0

<

if-

-2^

i'

1

'

•

•

■

•;':

:••*

'/

.

•

'i

c^

T

L_

(0 20 30 4-0 50

38

Art. 9.— T. Terada

dispersed rather irregularly as shown in Fig. 11 in which
the results are grouped with respect to the hours of occurrence.
Next, the results were grouped according to the periods and the
mean ratios were calculated for each group of the periods. The
results are shown in the following Tal:)le and also plotted in
Fig. 12 where tlie mean periods^^ are taken as abscissa.

Fig. 12.

AX

0%

0-6

04

O

. .

^^

9

-"^

\

^-^C'

J

"^-■^ 1

/

^

i

■---__

1
i

10

20

30

40

50

60

T

O-l

0-2

0-3

0-t

0'3

OÇ>

'/tl

Table IV.

Eange of T.

m m
0-2-5

m m
2-6-5-0

m m
5-1-7-5

m m
7-6-10-0

m m
10-1-15-0

m m
15-1-20-0

m m
20-1-30-0

m m
30-1-40-0

m m
40-1-50-0

m m
51-0-600

Mean T.

m
172

m
3-49

m
6-23

m
8-90

m
12-86

m

17-41

m
23-76

m
33-90

m
47-50

57-50

AZ

AX

0-235

0-291

0-421

0-459

0-536

0-580

0-574

0-672

0-621

0-745

tg-i AZ _
AX

IS-"]

16-° l

22-°6

24-"'5

28-°0

29-''9

30-°4

33-''4

31-''7

36 -°5

As will be seen, the ratio increases at first rapidly with the
periods, then gradually tends to an asymptotic value probably
nearly equal to unity. The fact is rather remarkable and must
serve as an important basis for explaining the phenomena in

1) The mean of the actual periods corresponding to the trains taken was calculated, not
the simple mean of the interval of the period concerned.

On Eapid Periodic Variations of Terrestrial Magnetism. 39

question. For the purpose of a subsequent reference, the vakie
of the ratios are also plotted with ^|T, i.e. the reciprocal of the
periods as abscissa (the dotted line in Fig. 12).

The irregular dispersion of the points in Fig. 11 is too
remarkable to be considered as due to the inaccuracy of the
measurements of the record or to the inconstancy of the
instruments, but must be regarded as actually inherent to the
nature of the phenomena. Neither is the irregularity^ at all
eliminated, even if we take the ratio of the amplitude of Z to
that of the resultant horizontal component, ^àX- + AY^ instead,
which was actually calculated for the earlier period of the year.

Though we have not calculated the ratio AZj^ AX^+AY^
throughout the year, it will surel}^ be an overestimation of the
Y-component if we take AZj^\AX for it, since, as will be seen
later in §15, the azimuth of the horizontal component of the
disturbing periodic field is generally less than 45° and about 23*°7
on an average (see Fig. 16).

14. Phase retardation of Z-component with respect to X-
component. Next, the phase relation of X- and Z-components
were to be investigated. Since the corresponding waves in the
two components appear as a rule widely apart on the photographic
records where no special zero-line was recorded, a device was
necessary to accurately mark off the corresponding time in each
of the components at any part of the record. For this purpose,
a kind of sliding T-square, originally constructed by Dr. Kadooka,
was found very convenient. For the waves of shorter periods,
utmost caution was still needed to avoid serious mistakes, very
liable to be committed b}^ the slight inclination of the sliding
lineal which was to be kept parallel to the time-mark lines.

The mean retardation of the maxima and minima of AZ
relative to those of AIL were calculated in fractions of the periods.
The corresponding values for maxima and minima respectively
were often sensibl}^ different, especially when the waves are not
of simple form, in which case the mean of the two values was
simply taken. Plotting tlie values for different trains with the
periods as abscissa, the annexed figure was obtained (Fig. 13, A).

40

Art. 9.— T. Terada :

Classifying the period in the same groups as in the case of Fig. 12,
we obtain the following Table and also Fig. 13, B.

^ 05

23-
Z

0.1
0 )

TT 0-5"

0-3
01
0-1

Z

o'^-i,'^

It

«■■:

'" •

Fig. 13, A.

■K 0-5
04
0-3
0 1
01

fO 20 30 T

6'- 1 2»^

•

\

\

..'-.

. .

• *

•

-

0

■n Ofr

of

0-3

02

10 20 30 I

is'- 24''

•

■':'

••*

• «V» •'•

*

iO 20

30

10

10 30 T

Table V

Eange of T.

m m
0- 2-5

m m
2-6- 5-0

m mm m
51- 7-5 7-6-100

m m
10-1-150

m m
15-1-20-0

m m
20-1-30-0

m m
30-1-10-0

Mean T.
Phase.

m
1-59

0-215

m
3-34

0-172

m m
5-95 î 9-11

0-134 0-112

m
12-80

0-100

m
17-22

0-079

m

24-45

0-057

m
35-15

0-019

On Rapid Periodic Variations of Terrestrial Magnetism.

41

As will be seen from Fig. 13, the retardation is remarkable
for shorter waves, sometimes amonntin«; to decidecUv more than

Fig. 13, B.

0-2

0-!

\

1 1

\

\

^

\

1

K

•\

1

^

1 --^

-■•-^_

r

!

10^

ZO"

ZÖ'

40^

a quarter period in the case of 3(/ waves and gradually decreasing
in a hyperbolic cur\-e apparently tending to zero for longer waves
over one hour. Combining the result with that obtained in the
preceding article, we may trace the meridional projection of the
elliptic orbit described by the end of the periodic disturl)ing
vector of different periods.

15. Äzimutli, of periodic disturbing fields, or the relation of
amplitudes of X- and Y-comj>onents. While the amplitude ratio
between JX and JZ depends very much on the periods, but not
sensibly on the hours of day, the ratio l)etween the amplitudes of
X- and Y-waves varies remarkably with the hours of occurrence,
but not sensibly with the periods. To investigate the case more
closely, the following procedures were adopted.

At first, all available records were exami^ied and the numljer
of regular trains were counted in which the two horizontal
components are nearly parallel to each other, i.e. of phase
difference zero, and also those in which the two are nearly
inverted, i.e. of phase difference ^. The results are given in the
Table VI. and also in Fig. 14, regardless of the periods of the
waves.

42

Art. 9.-T. Terada

Takle VI.

The numbers are given in % of all cases for each season,
denotes the cases where J-V and J Y are nearly parallel to each other,
denotes the cases where JX and Ji' are nearly inverted to each other.

III IV V

VI VII VIII

IX X XI

XII

I 11

Year.

Hours.

+

-

+

-

+

-

+

-

+

-

0- 1

2-5

1-1

3-5

0-4

1-8

1-0

1-7

1-6

2-2

1-2

1— 2

2 "2

1-1

0-7

0-0

2-4

0-0

1-0

1-0

1-6

0-7

2- 3

2-5

1-4

2-1

0-4

2-5

0-4

1-6

1-2

2-1

1-0

3- 4

1-9

1-4

4-6

1-8

1-2

0-0

1-2

01)

1-8

0-9

4- 5

2-2

1-9

5-6

00

2-9

0-2

1-6

0-5

2-5

0-7

5- 6

2-5

1-4

6-7

0-0

1-2

0-4

0-8

0-5

2-1

0-6

6- 7

8-6

0-3

11-6

0-4

4-1

0-2

2-0

00

5-5

0-2

7- 8

7-2

1-4

13-4

0-0

14-7

0-0

6-9

0-7

9-5

0-6

8 9

8-9

1-4

5-6

0-0

157

0-6

13-0

1-1

11-5

0-9

9-10

4-7

0-9

5-6

0-4

11-5

1-0

8-5

1-1

7-7

0-9

10—11

2-3

0-3

1-8

0-0

5-7

0-2

6-1

0-6

4-4

0-3

11—12

2-3

0-0

1-8

0-7

6-1

0-0

6-4

0-5

4-7

0-3

12-13

1-4

00

1-4

0-4

3-5

0-0

5-4

0-7

3-4

0-3

13-14

1-9

0-2

0-7

0-4

31

0-8

4-6

0-6

31

0-5

14-15

1-1

0-2

0-7

0-0

1-6

0-2

22

0-3

1-6

0-2

15-16

0-6

0-5

1-1

0-0

0-2

0-2

0-7

0-5

0-6

0-3

16-17

0-2

0-3

0-0

0-0

0-2

0-4

0-7

0-9

0-3

0-5

17-18

0-6

0-6

0-4

0-0

0-4

0-6

1-4

1-8

0-8

1-0

18-19

1-6

4-1

1-8

2-1

10

2-2

1-1

2-7

1-3

2-9

19-20

1-9

4-7

2-5

4-6

0-4

2-5

0-7

3-3

1-1

3-7

20-21

0-6

5-3

3-2

1-1

0-4

1-8

0-7

2-6

0-9

30

21—22

2-2

4-2

35

2-8

0-4

2-5

0-7

2-4

1-4

3-0

22-23

1-7

31

M

3-9

0-2

2-2

0-3

2-3

0-8

2-7

23—24

1-1

1-7

1-1

0-7

0-8

1-0

1-2

2-5

11

1-7

It will be seen that the number of cases with the phase
difference zero attains a maximum in the early hours of morning,
while that with the phase difference tz shows a maximum in the
evening (Fig. 14).

Secondly, those cases w^ere chosen where either the one,
X- or Y-component, is alone conspicuous, while the other is
quite insignificant. The distribution of these special cases in
different hours is given in Table VII. and plotted in Fig. 15.

On Eapid Periodic Variations ef Terrestrial Magnetism.

43

As may be seen, the cases where X-waves are alone conspicuous,

or in other words, the disturbing field is directed nearly in the

meridian, are most frequent near midnight and noon, while the

opposite cases, i.e. those in which the disturbance occurs nearly in

the WE direction, are most frequent in the morning and

evening.

Fig. 14.

Parallel.

Inverted.

iOO

80

<oO

40

20

i

!

1

i

/

I

1

\

/

1

\

\
1

I

u

.

1

1

A

\

-->-

^x'

\

\

//

y\^

0 ^ 8 IZ 16 20 Zf

Fig. 15.

X alone.

Y alone.

1 i

f

i

r^

\

/^.

/

U V

x

^"•\

\
\

/

\

/

1
r —

N,.; ' yy

/\

-A

- -f^

,x.>

\

C Z 4- G

10 11 14- 16 IS 20 22 2.4

44

Art. 9.— T. Terada :

Table YII.

Hour.

X

only.

Y

only.

Hour.

X

only.

Y

only.

Hour.

X

only.

Y"

only.

Hour.

X

only.

Y
only.

0—1

37

5

6- 7

9

16

12-13

73

3

18-19

20

12

1 -2

39

4

7— 8

6

30

13-14

53

0

19-20

9

20

2—3

30

4

8- 9

6

35

14-15

63

0

20-21

22

9

3-4

21

5

9-10

18

23

15—16

51

2

21—22

24

4

4-5

11

7

10-11

37

4

16-17

46

4

22-23

30

3

5-G

12

14

11-12

68

2

17-18

30

4

23-24

44

2

Thirdly, taking the records of 1913 only, those cases were
chosen in which the phase difference of the two horizontal
components was eitlier o or nearly t:, i.e. in which the periodic
distnrbancc is "polarized'' in a certain azimuth, and the ratio
of the amplitudes was carefully determined which corresponds
to the tangent of the azimuthal angle «. The latter angle was
counted from N, positive value taken toward W. The results,
if plotted in a diagram, witli the hours as abscissa, regardless of

Fig. 16.

Co'

20'

-10

-40'

-60'

/\

'A,

/
/

\/

v^

■^^

y

\

/

\

y

u

~' —

^

0 :

)

f i

s i.

Î 1

C (

2 1

4- <

é 1

s z

Û Z.

2 2'

f

On Kapid Periodic Variations of Terrestrial Magnetism.

45

the periods, sliow very irregular scattering of the points
representing different trains. Nor is the irregularity lessened I»}'
choosing the waves of a definite period only.

The most frequent value of the azinuith for eacli hour was
tlien determined, not by taking the simple mean value, l)Ut hy
plotting graphically the frequency of different azimuthal angles
for each hour and taking the maximum })oint of tlie frequency
curve thus obtained. The results are given in Table VIII. and
plotted in Fig. 10, Avhicli generally confirms the result to l)e
inferred from Fig. 15. Neither is the present result in con-
tradiction with tliat shown in Fig. 19, since the maximum and
minimum value of JY/^X must correspond to cases where the
direction of the disturbing force makes the largest angle with
the meridian, and the zero value of this ratio must correspond
to the case where the field is nearly in the meridian. These
results show that the azimuthal angle, on an average, undergoes
a continous and fairly regular diurnal variation. Tliough the
present data are too defective for drawing conclusive inferences
with respect to the seasonal difference of the above relation, they
suggest that in summer, the azimuth has a secondary minimum
near noon. Besides, it is suspected that the mean azimuthal
angle in day time is somewhat less for longer periods than for
shorter. These points must, however, be postponed for a future

investigation.

Taf.le VIII.

Hour.

a"

Hour.

a°

Hour.

a°

Hour.

a°

0—1

— 13

G-7

52

12-13

25

18-19

—37

1—2

0

7- 8

43

13-14

35

19-20

—25

2-3

5

8- 9

37

14-15

13

20-21

—23

3—4

7

9-10

45

15-16

10

21 22

-25

4-5

13

10-11

30

16-17

— 10

22—23

-27

5-6

25 i

11 — 12

35

17-18

— 13

23—24

-20

46

Art. 9. -T. Terada:

16. Since in the above investigations, the trains chosen
from among the records of 1913 were those especially regular
and moreover, the periods were determined with special care,
the material seems appropriate for studjdng the frequencies of
different periods in different hours of the day, in spite of the
relatively small number of trains taken. The results of the ex-
amination are tabulated below, which show remarkable predomi-
nance of the longer waves in the night hours, as already remarked
in §9.

Table IX. Number of trains.

Period in
m.
Hour.

0-0-5

0-5-1-0

1-0- 1-5

1-5 -2-0

2-0-2-5

2-5-30

30 -3-5

3-5-40

40-4-5

0- 4

0

19

26

29

29

7

20

6

0

4— 8

4

66

39

21

22

10

7

3

1

8-12

4

93

49

20

15

7

5

5

3

12—16

1

45

40

10

■

S

7

2

1

1

16-20

0

14

12

19

16

6

3

5

6

20—24

0

6

25

41

31

24

13

6

8

Sum

9

243

191

140

121

61

50

26

19

17. Rotatory character of the horizontal components. As
already mentioned, the horizontal components of the periodic
disturbing field show as a rule more or less rotatoiy character.

Fig. 17.

clockwise
cou-nterclocRwisc

To investigate the relation
regular waves

in detail,

with decidedl}^ rotatory
character were chosen from
the records of 1913, at the
same time with the investi-
gation of the pieceding
article. The data obtained
were classified into two
groups, i.e. those which
showed regular rotation in
the clockwise and in the counterclockwise sense respectively.

30

20

I 0

/\

A

\
\

/-

-A /

-^ \

/■

\->

/

I z

16^

20"

24-^

Ou Rapid Periodic. Variations of Terrestrial Magnetism.

47

The frequencies of the two groups in different hours were
calculated, as shown in Table X. and also in Fig. 17.

Table X.

Hour.

Counter-
clockwise.

Clockwise.

Hour.

Counter-
clockwise.

Clockwise.

0- 2

24

G

12-14

7

1

2- 4

16

4

14-16

5

1

4— 6

24

6

16-18

1

9

6- 8

2

13

18-20

2

15

8-10

3

15

20-22

6

16

10-12

3

13

22-24

10

4

For each group, the frequency shows apparentl}^ a seinidiurnal
period. Comparing the result w4th Fig. 16, it may be noticed
that the clockwise rotation predominates in those hours where
the direction of the disturbing field is at the maximum deviation
from the meridian, while the counterclockwise rotation falls
most frequently in tlie intermediate hours. It must be also
remarked that the sense of rotation during night hours is
generally opposite to that determined by Sangster^^ for distur-
bances of decidedly longer durations, and therefore also opposite
to the sense in which the abrupt disturbing field described in §5
e) tends. In Sangster's case the sense of rotation showed a
diurnal period, being of the same sense throughout twelve hours,
while in the case of the short waves here in question, it shows
a semidiurnal period as given above.

PART IV.
Discussion of the Results.

IS. The results of the present investigations so far described
seem to tlirow some light, however faint, on the actual origin

1) Sangster, loc. cit.

48 Art. 9.-T. Terada:

of the magnetic pulsations in question, tliough it is at the present
stage rather difficult to draw anything like a conclusive inference,
for even if the observations had been carried out for a much
longer period with more reliable instruments, the data were in any
case confined only to a single station. In the following, we wdll
try, instead of hastening to any premature conclusion, to consider,
merely by way of tentative suggestion, different possibilities
regarding the probable cause of the phenomena in question.
The considerations will inevitably be of a speculative cliaracter,
but may serve at least as useful hints for projecting future
investigations of allied phenomena, especially of the as yet very
obscure nature of the electrical as well as meclianical behaviour
of tlie upper atmosphere.

19. A fact strongly impressing us in the first place, is
that the occurrence of the magnetic pulsations in question is
subjected in more than one respect to a remarkable diurnal
variation. This evidence alone is sufficient to infer the important
rôle played by the sun, wdiether its influence l;e direct or indirect.
The position of the sun, not only determines the length of the
periods of the most frequent waves in different hours, but also
affects the direction in wliich the periodic magnetic field
fluctuates. It seems quite plausible to assume from the outset
that the seat of the primary cause of the phenomena is chiefly
to be sought in our atmosphere subjected to solar radiation of
different kinds. The periodic heating of the superficial layer
of the earth crust, though it may possibl}^ cause a slow variation
of the terrestrial magnetism, may scarcely account for the periodic
nature of the disturbances in question. The direct magnetic
influence of the sun itself^^ seems also improbable, since if such
be the primary cause, the more or less complete screening off of
waves shorter than 50^ during night hours must be explained,
while 100^ waves are so conspicuous in these hours. Moreover,
the unsymmetrical distribution of the cliaracteristic waves at noon
is rather difficult to explain on this view. On the other hand,
the existence of remarkable electric currents in the upper region

1) Bosler, Journal de Physique, [6] 2, 1912, p. 877.

On Eapicl Periodic Variations of Terrestrial Magnetism.

49

of our atmosphere may be regarded as almost an established fact
since the classical investigations of Schuster/^ Birkeland and
Störmer.'^ Now, according to Schuster and Bezold, there exists
in the upper atmosphere a definite system of currents whose
position is nearty fixed relative to the sun, wliich produces the
remarkable diurnal variation of the terrestrial magnetic field.
The first suggestion which naturally arises as to the cause of the
magnetic pulsations, is the fluctuation of this permanent system
of currents.^-' This seems the more plausible, if we remember the
very regular diurnal reiteration of the characteristic phenomena.
The material nearest in hand for testing this conjecture is the
daily variation of the azimuth of the liorizontal components of
the periodic disturbing field. Examination of Fig. 16 will show
that a similar daily variation could be joroduced, if a nearly

circular zonal system*-* of
electric currents, fixed with
respect to the sun and having
its pole situated at a con-
siderable distance from the
earth's axis, undergoes some
periodic fluctuations in its
different parts. Referring to
Fig. 18, let JV be the earth's
astronomical pole, while P
is the pole of the zonal
current. Denoting the latitude
of the point of observation A by <p, and its longitude counted from
the meridian containing P, by X, the azimutlial angle « of the

1) A. Schuster, Philosophical Transactions, 180 A, 1889, p. 467; v. Bezold, Gesammelte
Abh., p. 404.

2) Birkeland, loc. cit.-, C. R., 147, p. 539. C. Störmer, Archiv for Math, og Naturvidenskab.,
31, 1911, Nr. 11.; Archiv des Sc, phys. et nat., 24, 1907, pp. 5, 113, 221, 317 ; numerous papers
in C. R.

3) The idea is not at all new, being expressed already by Eschenhagen, Birkeland, van
Bemmeln etc., though more or less vaguely.

4) We do not necessarily mean a continuous system of current flowing in a given
direction at the same timr, but only that the directions of the different portions of the
current causing pulsations, when put together, form the 25oî'<îo7îs of a circular zone with a
given pole.

50

Art. 9.— T. Terada

magnetic field due to the current passing near the point A,
counted positive toward W from N of A, is given by

sin 6 sin ?.

sni a =

\/l— (cos 6 sin ^ + sin 6 cos ^ cos /Î)-
Putting for example, <P—od° and (?=30°, 40° or 50°, the variation
of a as the function of ?. is shown in the annexed figure (Fig. 19).

Fig. 19.

360 X

To compare with Fig. 16, the present figure must be properly
shifted along the axis of ?<, which is equal to the hours, up to an
additive constant. It seems that the meridian corresponding to
the pole P, (i.e. ?-=0) of the assumed circular current may be
placed somewhat in the early hours of the morning. The shapes
of the curves, however, do not agree so well as to enable us to
determine the most probable value of d. Indeed, it will be too
much to infer at once the existence of such a simple circular
current in view of the above consideration, based on the
observation at a single station. Still the orientation of the
different portions of the atmospheric current, of wliich the
fluctuations may produce the magnetic pulsations, must resemble
in some measure that of the corresponding portions of the ideal
simple system above considered, and must in any case be directed

On I7apiil Periodic Variations of Terrestrial Magnetism. 5|^

nearly SAV-NE or rever.-e during tlio day time and N\V-SE or
reverse during niglit. The supposed system of currents can not
be directly identified with that given by Schuster or Bezold, since
in the latter case, « passes the value ^ twice during '24: hours, and

the corresponding curve of « witli the hour as abscissa shows no
ai:>parent resemblance either with Fig. IG or with Fig. 10. Nor
is the system of currents deduced l)y Birkeland'^ from the
disturbances of longer durations observed during his memorable
auroral expedition similar to the supposed one. Remembering,
however, that the daily variation of the terrestrial magnetic
field can be represented in the first rough approximation l.^y a
system of currents having its axis considerably inclined to the
earth's axis, we may regard provisionally the total current system
given by Schuster and Bezold consisting of tw^o parts, and that
the one part which is the princij^al and represents a system of
zonal currents, shows a more conspicuous regular fluctuation than
the remaining secondary part more or less converging toward the
pole (see § 28). As a matter of fact, Fig. 19 represents only
the averafje distribution of the most fi-equent azimuth in different
hours. In actual cases, the points corresponding to different
trains of waves are so remarkably scattered that the adoption
of the mean value evaluated in the usual manner seems scarcely
justified for seeking the most frequent value. It is very probable
that among these widely scattered points, there are inany which
actually correspond to tlie fluctuations of the part of the current
belonging to the higher harmonics. At any rate, our conjecture
seems to be justified in the first approximation, though the
position of the pole of the circular current can not be determined
with certitude.

20. As to the actual modes of fluctuation of the atmospheric
current causing the magnetic waves concerned several possibilities
are suggested at the same time. Firstly, we may consider the
total intensity as well as the distribution of the current as
constant, but oscillating as a whole about its mean position,

1) Birkeland, loc. cit, PI. X.

52 Art. 9.-T. Terada:

either vertically, horizontally or in a rotatory manner. Secondly,
it is also possible that tlie cnrrent itself undergoes a periodic
fluctuation, either in the total intensity or in the distribution
in its different parts. IMore probable is the combination of the
above two modes. Tliirdly, we may suppose a system of parallel
currents arranged at nearly equal intervals propagated per-
pendicular to itself, over the point of observation with a finite
velocity, in which case the waves must show a time difference in
difference stations. If the results of observations confirm the
exact simultaneity of waves in widely distant stations beyond
all doubt, the last hypothesis will naturally fall out. Birkeland,
indeed, observed the approximate simultaneity of some waves
with short periods in two stations so widely apart as Potsdam
and Bossekop. The two examples reproduced in his report,
however, refer only to nearl}^ the same midnight hour, where
the direction of the atmospheric current might well have been
approximately parallel to the line joining the two stations.
Again, according to the result of the simultaneous observations
made at Kyoto, Misaki and Sendai. in April, 1909, by Dr.
Kadooka and Prof. Tanakadate, a similar simultaneity is observed
within the limit of experimental error; but in this case the
distances were not very great. Though these two observations
strongly speak against the progressive nature of the periodic
disturbances, a further accumulation of evidence Avill not be
considered superfluous for deciding the point beyond all doubt.
On the other hand, some disturbances of a longer duration
investigated by Ad. Schmidt^^ were actually progressive.
Birkeland also attributed a velocity of translation at the rate of
10Ü km. per minute to some class of perturbations. A possibility
of occasional occurrence of progressive waves, if not of regular
phenomena, seems to be not yet dispproved. On this view, it
will be of some theoretical interest to include the case of
progressive waves among tlie possible cases and see how far the
hypothesis is favourable or unfavourable for explaining the
different peculiarities of the observed phenomena.

li Ad. Schmidt, Met. Zs., 16 1899, p. 385.

On Rapi«! Periodic Variations of Terrestrial Magnetism. 53

In the following, we will recapitulate some of the most
remarkable results of tlie statistical investigations and try to
revie-sv them in the light of the different hypothetical atmospheric
currents.

21, First, take the relation between X- and Z-components
which show such an intimate connection as regards their
amplitudes and phases, as desci'ibed fully in previous |)aragraphs.
We will proceed to consider the influences of different possible
ideal systems of atmospheric currents separately, and compare
the results with the observed facts by way of seeking a most
plausiljle explanation.

Since the disturbances ^vith tlie short periods here in
question most probably extend to a limited portion of the
earth's surface, as may l)e judged from their remarkable
dependency on the hours of the day, it will be allowed for a
first approximation to consider both the surface t^f the earth and
the atmospheric layer carrying the current as a plane.

'22. a) Consider an infinite linear current / running per-
pendicular to the meridian at a height A from the earth's surface
considered plane, and at a horizontal distance x from the point
of observation A. Then the X- und Z-components will ]je given l)y

jX=_plL__ ^ jz = -^^ (1)

x' + lr xr + lv

If the intensity of the current fluctuates in any manner, while
its position remains unchanged with respect to the earth, the
two components w^ill follow the fluctuation simultaneously,
provided that the variation is slow enough for neglecting the
effect of the induced current. The ratio of the amplitudes being

JX h '

it may assume any value when x varies from — :o to +00.
Besides, AZ will be of opposite phase on both sides of the current.
h) If the above current moves perpendicular to itself from
x= — -jD to +x, X-component at A evidently attains a maximum
at x=0, while JZ is zero at ^-=0 and has a maximum and

54

Art. 9.— T. Terada :

minimum at x=±h respectively. The end point of tlie vector
describes a circle with the radius ijli.

c) Instead of a linear current, we consider next a current
of uniform density 7, with a rectangular section having a breadth
of 2/ and a height of 2h, the middle point lying at a height h
and the horizontal distance :i- from A. 'Hie two components
are respectively given l)y

i ^ {x^rf+{h-hr ' {x-if+iji+hy
"" {x-^if^{h-hf ' (x-iy+(ji-hr

(h + b)- + x- — I- {h — by + x- — t

JX=2i'to-^

h- + x^—P

"^ {x-iy + Jr
Again, wlicii the current slieet is vertical with a breadtli '2/),

x- + {h — by

x- + Jr — b'

(2)

,/.io^- C^-+0'-'+(/^+6)'-' (x-ty+jJi-by^

"" [x+iy+ih-by {x-iy+{h+by

.. (x+iy+(h+by {x+iy+ih+by

^ {x-iy+ih-by ' {x-iy+(h+by

AVhen the current degenerates into a current sliect of the
horizontal breadth 21 and tlie linear intensity i\

'2Ih

•(3)

•(4)

In the former case (3), tlie maximum vahie of JZ. J/,, say,
which is attained at .f=±V7'+7^', wih increase indefinitely, while
the maximum value of JA^, or JX,,, tends to 2-1', when the
ratio bjh increases indefinitely. For ^=0. tlie ratio is 1/2. In
the latter case (4), JZ,„ varies hom (.) to 4j when 20 increases from

On Eapid Periodic Variations of Terrestrial Magnetism.

55

0 to GO , 7i being kept constant, while JX,„ increases indefinitely
with b. The ratio AZjAX,,, tends to 1/2 when b decreases
indefinitely, as is already evident from the former results.

d) Let a number of such currents be arranged with the
directions alternately positive and negative, and parallel to each
other at definite intervals, eacli moving perpendicular to itself
witli a uniform velocity. Neglecting the effect of the induced
current, the magnetic field at A will undergo a train of periodic
variation as illustrated in the annexed figure (Fig. 20) for a special

h-b- I

Fig. 20.

Viilnnct, . - X = 6

Positiue Ciiyy^nt oX
Negoliue Currenï at

case. As will Ije seen, the phase retardation of AZ is somewhat
similar to that in the actual case, though here at the same time
with the short waves, a rather remarkable general swelling of
the curves appears. The latter may probably be avoided ])y
properly reducing the intensity of the currents toward both ends
of the train. In this case, the pulsations must necessarily be
progressive.

23. Next, we will consider the effect of the current induced

56 Art. 9.-T. Terada:

in the earth, especially for a simple case convenient for mathe-
matical treatment, i.e. the case when the current is arranged
in an infinite train of ivaves, either stationary or progressive ^

Take the surface of the earth, considered plane as usual,
as :f?/-plane and the positive direction of z downward. For
the positive value of z, the space is considered to he filled with
a conducting medium with the uniform specific conductivity h
and the magnetic permeability ix, while the negative side of z is
regarded as a vacuum. Assuming the electric and magnetic force
independent of y and denoting their components respectively by
^x> ©2. âa:. -Ös» the usual fundamental equations for the slow
variations reduce to

-f^

It Iz

"ht 2>X ' '^X 'èz

.l^J.Q _ ^â. _

3.Ö.

?,J

Ix

Sâ.r _j. ^Qz

=0.

(1)

since ©^=©^=0, §^=0. Next, assume that the electromagnetic
field varies periodically with the frequency nl'Irr and the distri-
bution of the fields represents a two-dimensional wave with the
wave length

a

in the case of stationary waves, we may assume

e^=:e'«^-(? + '-^)~'sin ax, (2)

where ß and T are considered real. From

M Tax- S^'

we obtain

ATZfikin = — «- + [ß + iy)'
or

ß- — 'r=a-, '2ß}'=:4z/jik}i, (4)

1) The mathematical solution of the case Avas kindly carried out by Prof. S. Sano, to
-whom the best thanks of the author are due.

On Knpid Periodic Variations of Terrestrial Magnetism.

57

wlience

From tlie first set of the fundamental equations results

^.

i-r + iß) Jnt-(^ + h)=- ,,,

sm ax,

an

[Ill

Taking tlie real parts and putting 5;=0, we obtain at last
(©z/)a=o==cos nt sill ax.

.(6)

/Ä \ iry « sin nt

an

•(7)

Similarly in tlie case of progressive waves, we ma}^ proceed by
assuming

(^^^ = J.'^'±ccv)i-i?±fi)= ^Q^

and obtain

JX= — 1 ycos{nt ± ax) + ßsiii(nt±ax) \ ,

fjin \ )

AZ=^ cos(;i^±aa;),

where the values of ß and /' are the same as above (5).
Putting

ß=.Aco^ (p, ^=^siii if,

or

.(9)

(7) becomes

AX:

dZ=

[xn
a

siu(;iiî + ^)siii ax,\
sin nt cos ax.

.(10)

.(70

un

58 Art. 9.-T. Ter.'ula:

The same substitution makes (0), for progressive waves toward N,

JX= co^7it — ax-{- — + (p),

an ^ '2 /

.(90

fxn

JZ= — '— cos())t — ax),
[Jtn

and for retrograde waAes toward S,

JX=— üm(nt-{-ax + <f),

^^" _ } (9")

JZ=—-^ äm()}t + ax-h^]
im \ 2 /

Hence for the case of stationary waves (7'), tlie ratio of tlie
amplitudes of Z and A^ is given by

ÂZ

JX,„ A

cotg ax (11)

which can assume any value whatever between — cc and +x,
as we proceed normal to tlie wave ridges. Moreover, âZ is
retarded after âX by <p given by (10).

Again, in the case of progressive and retrograde waves,
the ratio of the amplitudes is

^^=^=— ^= (12)

JA„, A V/?- + r

which is always less than, and tends to, unity as the frequency
decreases, since

^^=^9- + r=V'a^+l6;ry^W (13)

Besides, âZ lags behind JA" by f + ^ or f— if according as the

Avaves are progressive or retrograde.

Now, the angle determining the pliase is given by

to- c = J— — l'^"-' "^^ ^■^'i^-'khv — é

which becomes zero for small value of kn and tends to unity for
large values. Hence, in the case of stationary waves, the
retardation of the vertical component will increase from 0 to
-^ when kn increases from 0 to c/: . In the case of progressive

On Rapid Periodic Variations of Toircstrial Magnetism. 59

waves, when kn increases from 0 to x , tlie retardât ion will vary

from 4r to -7r+--r <>i' ""^^ while in the ease of retrui;Tade waves,
'2 2 4 4 _ _

it may l^e considered as negative and \'aries from — ^ to ~ r-

Now, retnrning to the actual case, Ave will tiy to examine
if either of these h^'pothetical waves could be reconciled with
the observed facts regarding the relations between the horizontal
and vertical components of magnetic pulsations.

If the stationary waves alone were concerned, we would
have to assume that the distance of the observing station from
the node of the electric current was in any case greater than
-5- but less than -f^, in order to explain the fact that the ratio

o 4

AZjàX,,, was always less than unity, at least in the case of the
slow oscillations where ajA in (11) is nearly equal to unity and
cotg ax may become less than unity only when o.x or — ,— is greater
than -7-. This will be a rather awkward, though not impossible

assumption, if the wave length were small in comparison with
the earth's quadrant, since Ave must then also assume that the
location of the current is always limited to a rather narroAV range
favorable to the above relation. TJie only plausible hypothesis
reconcilable with this assumption is that the Avave length of
the current producing the sIoav waves is of the order of magnitude
of the earth's meridian, and also that the station lies ahvays
not far apart from the loop of the current. In such a case,
hoAvever, either of our assumption as to the planeness of tlie
earth's surface and the existence of the infinite train of Avaves,
Avill fail to apply; but the case Avill ratiier approach that Avliich
Avas discussed in the preceding paragraph. The present result
may be interpreted as merel}' indicating that the effect of
the induced current has the tendency to retard the vertical
component Avith respect to the horizontal one. For such a case,
the mathematical calculation giA'en by Lamb^^ for spherical
conductors Avill directly apply, which also shoAVS the retardation
tending to -^ for the large A'alue of kn. At auA^ rate, the observed

(1) H. Lamb, Phil. Trans., 1883, p. 52Ü ; ihid. ISO, 1880, p. 513.

60 Art. 9— T. Teracla :

waves witli periods longer than 101" which show the retardation
generally less than -j-, may easily l)e explained Ijy a current

of rather diffused character suhject to fluctuations. Indeed,
these longer waves are generally not very regular and rarely
form a train with the numher of successive maxima greater than
three or four, — a decided contrast with the shorter waves of a
few minutes periods whicli form as a rule remarkably regular
trains and show occasionally phase retardation of JZ greater than
^. If the above conjecture be justified in some measure, the

current, of which the local fluctuation produces these longer
Avaves, may plausibly be identified with the circular current
discussed in § 18 and compared with the princijDal part of the
current causing the diurnal variation of the terrestrial field, since
in this way, the variation of the azimuth of the disturbing field
may also be explained satisfactorily, at least to the first ap-
proximation.

The maximum current intensity may be expected probaljly
near the equator if the phase relation between dX and JZ is
never inverted throughout the N-liemisphere. It will l)e in-
teresting to see if the relation is actually opposite on the ^5-
hemisphere, v. Bemmeln found ~t^ in Batavia invarial^ly
insignificant.

On the other hand, the most regular waves with periods
less than about 4™ show as a rule remarkable phase retardation
of the vertical component after the horizontal ones, generally
greater than — . Tliis seems apparently difficult to explain on

the assumption of stationary waves, if we solel}" rely on the above
calculation.

Next, turning for a while to the case of progressive waves,
we remark that one fact seems at first siglit to be in favour of
this assumption. As described repeatedly above, the actual ratio
of the amplitudes AZjAX,,, never exceeds unity and decreases
gradually as the frequency of the wave increases. From (12)
and (lo), we have indeed

On Eapid Periodic Variations of Terrestrial Magnetism.
àZ^ a

which may be written = 4 , ,

_ 4:7r/j.kn _ ^/ik?-^

if we put

27r

61

denoting the period by T^ i.e. putting ''^—-rp

The variation of the calculated ratio AZjaX,,, witli x as altscissa,

may be seen from the annexed figure (Fig. 21). Referring to

Fis. 21.

'Sx,

OS

0'Q>

o-'t

\

\

\

01

^^

~~ —

1

1

fO

zo

30

40

X-^'

Figs. 11 and 12, and comparing the present theoretical curve
with the curve of the observed ratio with IjT as abscissa, we
may observe a striking resemblance, thougli tlie ratio of the
scales for the two diagrams is not yet hnoAvn. If we take, for
example, the 3'" wave for the purpose of comparison, the
observed value of dZ.JdX,,, is nearly 0*28. jMaking a discount
of 10 % on account of tlie reason discussed at the end of § 13,

(52 Art. 9.— T. Terada :

we obtain 0'25. The corresponding value of x in Fig. 21 is
found to be nearly 17. Hence, we must liave, assuming for
simplicity's sake /^=1,

2H-^17x 180 = 3000;

, -. 1530 2-10 • . 1 1 AT • +1

lience ^''= , or — - — \\\ round number. JNow assunnng the

k Ic

conductivity of tlie earth to l)e 10"'^ as is suggested by the result
of Schuster's investigation, we ol)tain

;. = 1-2 X 10^ cm. or 1200 km., say.
Nearly the same value may l)e oljtained, if we take 10'" waves
instead. We have no theoretical ground at liand for assuming
tliat the wave length varies with the periods; still, judging from
the remarkable resemblance of the two diagrams above compared,
it seems plausible to assume provisionally that the wave length
is at least of the same order of magnitude, and the different
periods are determined not so much by the difference of the
wave length as of the velocity of propagation. In the above
example, where i'.= r2x]0' km. for T=180' the velocity of
propagation v will be about 7 ^^^ The above estimation was

solely based on the assumption that the equations (11), (12) etc.
hold rigorously. Judging from the analogy of the case in-
vestigated hy Schuster, it is probable that the actual reduction,
of the vertical component will be decidedly more remarkable
than the calculated value. If such be the case, the value of x
and accordingly the values of X and v will become less than those
above estimated.

A closer examination of Fig. 12 sliows that the asymptotic
value of AZjàX,n is not 1, but somewhat near 0*8. If we compare
Fig. 12 with a diagram obtained b}^ multiplying the ordinales
of Fig. 21 by 0*8, tlie acceptable value of x become about 10
instead of 17. But this does not alter the order of magnitude
of the results.

1) According to Birkeland, the lateral velocity of the current producino- some dis-
turbances was of the order of 100 km. per minvite, while that of some auroral bands observed
in the polar region was about 300 m. per sec, which is very small in comparison with the
above calculated velocity of the hypothetical current.

On Rapid Periodic Variations of Terrestrial Magnetism. (33

As for the phase retardation of AZ after AX, the serious
difficulty against the hypothesis of the progressive waves is that,
in the theoretical result, the phase angle must be included within
the range 90° to 135° or —45° to —00°, wliile in the observed
results, it crowds most densely within tlie limit 45° to 90°.
At present we are at a loss to judge whether the difficulty may
be evaded by assuming the presence of more conducting layer
below the earth's crust, as argued by Schuster to explain a
similar discrepancy in the case of the diurnal wave.

On the other hand, if we once admit in the case of the
stationary wave that cotg ox is somehow of the order of unity,
the ratio of amplitude JZJjX,^ is just the same as in the case of
the progressive waves, and the aljove calculations generally
apply also to this case, except that part concerning the velocity
of propagation which is zero in this case. The general mode of
dependency of the ratio on the periods is equally favourable for
the stationary waves, and we see no strong reason for preferring
the theory of progressive waves, as far as the amplitude ratio is
concerned. If we take cotgaa;=l or 0'8 and k=10~'^^ the wave
length estimated seems, however, too small.

At any rate too much weight must not be laid on the
numerical results of the above calculations, since they have no
claim to accuracy of the quantitative results, if we remember
the utmost simplicity of tlie assumptions on which they are
based, even if they may be legitimate in essential features.

24. Next, consider the ideal case where an atmospheric
current is subject to a rotatory oscillation remaining parallel to
itself. Taking the simplest case of a constant linear current
running parallel to the 7/-axis and oscillating about the mean
position, given by its elevation h^ above the surface of the earth
and the horizontal distance ^o from the point of observation taken
as the origin. Let the position of the current I be given by

x=Xq+x^co^ nt,

Ä = 7^0 + /jjCOS {7lt — (f).

We may obtain, neglecting the efïect of induction

I '■'

64

Art. 9.-T. Terada

X _ h

- + A^cos(nt— </',.),

Il xl+lil

2 9,7') ^ ^ T ~ '

where

h'ö+xö y xô+ni \

. . . /'of ^h 2/i,|/^i 1 •

lil-\-xl\ ho M^ + xl }

/«S + a^ü '- a?o ^û + a^o
^5 sin (l',=

2hji^

iIq -jr x^

)cos,|'

M + xl

'- COS <p

\

^JiJh

- sm (p.
lil-\-xl Kl-]rxl

For the special case ç=0, /.<?. when the oscillation is linear,

A.
A,

h

JA

hl + x

'in + a^r! t a:n

hönrXo y Xçi

/^5 + a^5 J I

/? Q + X^j

or

1+e^

ii i+r- \

1 + r

2(1 + «^

if we put
Hence,

. + e \ 1 + Ç-

7ip=l, a'n=ç^ and A,/a:i = «.
A, _ AZ,„ _ (l + g>.-2(e+«)

/I.

JX,

•(2)

.(3)

•(4)

.(5)

(6)

i+6^-2(e+«)

This is greater or less than unity according as a>or<l. Since
</f^=(/'^=0, there is no difference of phase, neglecting the induction.
If Ave take the induction into account, Z-component will probably
lag behind A'', though we are at present at a loss to carry out the
calculation.

Again in the special case ^"—^, 'i-^- when the current
oscillates ehiptically, with the axes of the ellipse horizontal and
vertical respectively, we have

On Rapid Periodic Variations of Terrestrial Magnetism.

A.

9P

tg ér =

21

V

'+1,

.^ +«%

tg Ç-^'^= ^f_2 « .

■(>)

(^)

It may Ije seen that

A,^ JZ,„

wbicli varies from 1/« to «, and tlien fr(.)ni « to ]/«, when ? in-
creases from 0 to 1 and 1 to go . The phase difference is given by

tgfs^'.x-vg=

(i+F)^

(9)

If c<l, then ^^-^
If |<1, then c'' -''-

or <::90" according as «>- or <:1.
ç'^ <i or >90'* according as «>» or «c:!.

Thus we see in this case that the phase difference as well
as the ratio of the amplitudes may assume different values according
to the values of « and ?. If the effect of the induced current be
taken into account, these values wdll of course be altered
considerably. Fig. 22 illustrates the variation of the amplitude
ratio and phase difference as functions of ç for a special case.

In the above discussion, /- was considered constant. If it
varies at the same time, the results will be more complicated, in
some cases giving rise to a higher harmonic oscillation of the
magnetic field. In the case treated in § 22, the external source
of the disturbance was such as could be represented by the wave-
like distril)ution of current in a ßxed horizontal layer. If such
current layer is disturbed into a wave motion simutaneous with
the fluctuation of the intensity, each elementary current will be
subjected to motion similar to that expressed by (1) of this
paragraph. This motion will greatly modify the relations between
the X- and Z-component waves discussed in the preceding paragraph
as is suggested by the al)ove simple example. It seems plausible

<■)()

0-9
OS

0-7
06
0-5

0^
0-3
Ö-2
öl

oo

/60
120
SO

40

0

40

Art. O.-T. Terada:

Fig. 2-2.
OLIOS'

\
1

\

1

1
1

!

Az

A

1

1

f

\

'

\

r

\

f

JZ^

1

\

ix^.

\

\

2Q
V5
1-6
1-4
1-2

\

\

\

\a

■

i

\

w

^ - ''

-• -^

\

^ ■■

'Az

â:

\

\

^

^

Ax

'ai

i*,

10

\

\

\

,/'

OS
0 6

\

\

A

■^■

y'

^\

s,

0-4
ö'2

^

<:^

-_

-^

^^

» 2 3 + 567

— ■
\

\

- A-

/*■

\

/

\

V

\

\"\,

fe-

%.

1

—

--

\

^

:^

—

—

- ■

\

^

■"-^

^

V-

\

""

jt.

--

--

■--

--

---

^

^^

-----

,^

%

-ÜJ

—

— _

On Eapid Periodic Variations of Terrestrial Magnetism. Q(

to consider tliat the difficulties met with in the preceding
paragraph witli regard to tlie phase relation may partly be
accounted lor hy a similar combination of the motion and the
variation in the intensity of the current. Such a combination
is quite i^roliable in view of tlie reciprocal action of the magnetic
field on the conducting layer carrying the current. Though these
points seem to suggest many interesting problems worth in-
vestigation, it will at present be rather too far fetched to introduce
any further liypothesis.'-'

25. The fact described in n^ 5, (/) that two trains of waves
with different periods sometimes appear simultaneouslj' in the X-
and Y-components respectively, may probably be explained
by the combination of fluctuations in the direction as well as
in the intensit^^ of the current. Take the simplest case, when
a linear current at a distance d from the point of observation A.
in its mean position, is fluctuating in its intensity as given by

/z=/„ cos pt,

while it is osciUating statioiiarily about the position of equilibrium,
in sucli a manner that it were always tangential to a string
vibrating in a horizontal plane with its node at the foot of the
perpendicular from A to its mean position. If the angle made
by the current to its mean position at the time t be «. then tg «
will vary as Ccos(qf-\-(f), where C is the tangent of the maximum
angle of inclination. If the mean position of the current be
perpendicular to the meridian, tlie periodic variation of X and
Y will be given by

JX=— - cos pt,

i C
Jr=— ^* — cos 2)t cos(qt + <f)-
d

Thus a "combination tone " will arise in the component parallel
to the mean direction of the current. The effect of the induced
current will probably enhance the "summation tone" in
comparison with the "difference tone." If the oscillation is the

1) These points will be touched upon again later in § 28.

ß8 Art. 9. — T. Terada :

neccessary result or cause of the fluctuation of the intensity, p
must be equal to q, in which case the period of dY will be
octave to that of JX. This latter case was frequently observed,
though not always.

The above consideration seems to explain the fact that
the cases where the period of the X-waves is longer than that
of Y are decidedly more frequent than the opposite cases, if we
remember that the hypothetical current is generally inclined
to the circle of latitude by an angle decidedly less than 45.°
It is however strange to observe that the hourly distribution
given in § 5 is rather conformable to the supposition of the
"difference tone" being more conspicuous. This latter point
requires further investigation.

Different facts remain still to be considered in the light
of the above hypotheses, i.e. the dependency of the prevalent
periods on the hours of occurrence, the rotatory character of the
horizontal components, etc. We shall consider these points
later, after having tried to ascertain the nature of the supposed
atmospheric electric currents.

26. Thus far, we have tried to find some plausible ex-
planations of the observed phenomena on tlie assumption of a
system of fluctuating currents existing in the upper atmosphere,
neither alluding to the possibility of such, nor giving hint as to
the cause of the supposed fluctuations. It will be by far the
more interesting physical problem to inquire into the origin of
these remarkable periodic variations. Tliough ^ve are not yet
in a position to answer the question in any definite sense, it will
not be quite out of place to say a few words on the subject,
merely by way of suggesting many interesting problems regarding
the electrical behaviour of the upper atmosphere.

The existense of a permanent s.ystem of atmospheric currents
pru<kicing the remarkable diurnal variation of the terrestrial
magnetism, first effectively elucidated by Schuster, may in these
days be regarded as a universally accepted hypothesis and requires
scarcel}^ any further comment. It is the fluctuation of this
current that we are here chiefly concerned with, and the

On Eapid Periodic Variations of Terrestrial Mag-netism. (59

question is, win' and in what manner these \ariations are
produced. Schuster attributed the origin of the current to the
induction caused by the daily oscillation of the upper atmosphere,
which may be regarded as a conducting layer enveloping the
entire earth. The first suggestion strongly appealing to us is
that this diurnal wave may occasionally^ be accompanied by
numerous trains of secondary waves with decidedly shorter
periods compared witli tlie daily or semidiurnal periods. An
analogy is afforded in the case of the secondar}' undulations of
oceanic tides. '^ Whatever may l)e the cause, any mechanical
wave motion produced in the upper atmosphere will produce
corresponding waves of the induced current whicli will very
much resemble the case discussed in § 23. SujDposing such to be
actually the case, let us see how far tlie assumption may Ije
justified.

Taking first the case of the progressive waves, a serious
difficulty confronting the assumption is to explain the enormous
velocity of propagation as obtained in the preceding calculation
wdiich is far greater than the sound velocity, even if the upper
atmosphere be entirely of hydrogen at the temperature of the
stratosphere. The case Avould become more favorable only, if
by any modification in the method of estimation, we could
l)riug down the calculated vehjcit}- to a plausible range.

Another alternative hypothesis of the stationary wave,
where the wave length is uf the order of magnitude of the earth's
merichan, must then be considered. If the atmospheric oscillation
be purely mechanical and he one of the modes of the natural
tangential vil)ration of the ivhole atmosphere, the period corre-
sponding to such harmonics of low order can scarcely be of the
order of a few minutes, as is most frequently observed, in so far
as we can judge from the results of investigation^^ on the
phenomena of the daily variation of atmospheric pressure.

These considerations alone seem to lead us rather to

1) K. Honda, T. Terada, Y. Yoshida and D. Isitani, Secondary undulations of Oceanic
Tides, Journal of the Coll. of Sc, Imp. Univ., Tokyo. 24, 1908.

2) Rayleigh, Phil. Mafj., [5] 29, 1890, p. 173: M. Margules, Sitz. bor. d. kön. Akad. d. Wiss.
z. Wien, 99, 1890, p. 204, <-tr.; H. Lamb, Proc. Eoy. Soc. A 84, 1911, p. 551.

70 Art. 9.-T. Terada:

conviction, tliat if the tiuctuations of the current be either of tlie
kinds here considered, it is not merely due to the mechanical
wave motion of the upper atmosphere. This leads the question
towards the weakest side of our knowledge.

The natural period of the usual electrical oscillations cannot
in any case be as long as several minutes, even if we take into
account the effect of the upper atmospheric shell; nor can it be
of such indefinite periods as actually observed.

Though it seems .50 far difficult to say anything definite
aVjout the physical cause Avhich may produce the fluctuations in
question, the existence of the fluctuations above considered
seems to be suggested, on the other liand, l)y the phenomena
of tlie aurora polaris. Firstly, the frequently observed auroral
arc vividly reminds us of our hypothesis of the difïused
zonal current considered in § 23. According to the numerous
descriptions of the phenomena, the luminosity seems to undergo
different modes of variation, thougli we could find no record
definitely stating a periodicity of a few minutes. Such a slow
variation, even if present, would prul)al)ly have eluded discovery
by ordinary simple eye-ol)servations, not equipped Avitli special
means for that purpose. A special investigation in this direction
seems in any case desirable; for example a minute photometrical
study of cinematographical I'ecords as obtained b}^ Stürmer.'^
Again, among the numerous records of the auroral displays,"^
w^e frequently meet with descriptions of peculiar bands of
luminosity, running nearly perpendicular to the meridian and
showing wave-like propagation mostly toward S. It is very
probable that similar phenomena are of quite frequent, or rather
of almost daily occurrence, although usually the intensity may
be so weak that it rarely shows a luminousity strong enough to
attract the attention of the naked eye. Indeed, it is well known
that the characteristic "auroral lines" of spectra are almost
invariably visible in different latitudes. If such bands of

1) C. Stormer, Videnskapsselskabets tkrifter, Math.-Naturv. Klasse, I'Jll, No. 17;
Astrophys. Jonr., 38, 1913, p. .311 ; Bull. Soc. Astr. France, 1913, Xov.

2) E. E. Barnard, Ob.servations of Aurora made at the Yerkes Obs., 1897-1902, Astrophys.
Jour. 14. 190J. p. 135. M. Brondel, Uober das Xordliclit von 30, Juni, Met. Zs., 1908, p. 552.

On Rapid Periodic Variations of Terrestrial Magnetism. 7]^

luminosity correspond to the traces of tlie horizontal eleclric
current, as Birkeland supposed,'- they remind us strongly of
the liypothetical parallel cm'rents considered ahove. It must
be noticed that according to the calculation cited, the progressive
motion must ])e S to N instead of N to S, in order to explain
the observed phase relation of X- and Z-components. It miglit.
however, be surmised that the motion of the current in tlie
upper atmosphere is in many respects of opposite directions in
the lower and liigher latitudes, as is suggested by Schuster-
Bezold's system of currents. A similar observation in a con-
sidérai >ly high latitude will be in any case very desirable,
especially witli a reliable instrument for tlie vertical component.

27. As to tlie remarkable dependency of the periods of the
most frequent waves on the hours of day, we may suggest first
of all that it probably has some relation with the difference in
the temperature as well as in the state of ionization of the upper
atmosphere. According to Prof. Nagaoka,'^ the tliickness of
tlie conducting layer in the atmosphere must also be decidedly
greater during the day than in the night.

If the cause of the periodic fluctuatie)ii of the atmospheric
current is to be sought in the purely horizontal oscillation of the
atmosphere, the dependenc}^ of its |)eriods on the hours of day
is rather difficult to explain, even if we suppose the atmosphere
consisting of different layers with remarkably different composition
and temperatures.

Another, and probabl}' the last possibility is that in the
upper atmosphere there may occur a vertical longitudinal wave
(.)f limited extent, the period of which may sensibly depend on
the effects of the solar radiations. That the vertical vibration
of the atmosphere is possible, and may have a definite natural
period depending on the sound velocity, has been fully

1) According to the recent investigations of Stornier, the auroral band is produceil by
the vertical inflow of positive corpuscles. If this be actually the c isr, t!ie idea must be
abandoned.

2) H. Nagaoka, Proc. Tokyo Math.-Phy.«. Soc, 7, I'Jll, p. lû;î : Eevue générale des
Sciences pures et appliquées, 26, p. 570.

72 Art. 9.-T. Terada:

investigated by H. Lamb and also by Prof. S. Sano.'^ Both
authorities agree in the result that the period must be roughly
of the order of ö minutes, if we assume a plane earth and take
the ordinary value of the velocity of sound, which is assumed
to be uniform throughout the atmosphere. According to S.
Sano, the period is given by

y_ 47rc
Ï0

where c is the velocity of sound and ?'=1'41. Since c=VrBd, i.e.
the period increases witb the temperature, we must assume a
lower temperature on the side of the upper atmosphere facing
the sun than on the opposite side. This seems at first sight
strange, but is in accordance with the fact that the diurnal
variation of temperature in the upper layer shows a tendency to
become opposite in phase compared with the lower layer';
though the amplitude is of course small, a tendency is suspected,
that it increases with height. The diflficulty is to explain the
actual amount of the difference of the characteristic periods, the
niglit-wave being nearly twice as long as the day-waves. The
absolute temperature must then be assumed nearly 4 times
higher on the night side, if the difference of the periods is to be
exclusively attriJnited to the difference of IK If the corpuscular
radiation from the sun chiefly frequents the night side of our
atmosphere, as is supported by different phenomena pertaining
to aurora and magnetic storms, the effective temperature of that
side of the atmosphere may be raised considerably by the presence
of the additional kinetic energy of the free electrons. As to the
increase of c due to the ionizations, no conclusive experimental
evidence is yet afforded,^' especially for the highly rarefied state

1) H. Lamb, Proc. London Math. Soc, '2] 7, 1907, p. 122. S. Sano, Bull, of the Central
Meteor. Obs., Japan, 2, No. 2, 1913.

2) T. E,eo-er, Arbeiten d. kön. preuss. aeronautischen Observatoriums bei Lindenberg,
S, p. 229.

8) W. Küpper found a sensible increase of c for gases ionized by radiations of different
kinds, Dissertation, Marburg, 1912 ; Ann. d. Phys. [4] 43, 1914, p. 905. W. H. Westphal
obtained, however, a negative result. Verb. d. deutsch, phys. Ges , 1914', p. (313. These results
refer to gases at ordinary pressure.

Ou Eapid Periodic Variations of Terrestrial Magnetism. 73

of gases. 111 tins respect, many interesting physical problems
may be suggested which seem worth special consideration.

On the other hand, if a more or less limited portion of the
atmosphere be subjected to the oscillation of vertical type, the
period will surely depend on the extent of the area disturl)ed.
According to a personal communication by Prof. Sano, such a
mode is actually possible and the period will decrease with the
area disturbed. At least, the periods shorter than five minutes
may partly be explained in this way. The period uf vertical
vibration may, however, also depend on the modes of laminar
structure of the atmosphere in the distribution uf the temperatures
and the wind velocities of the different layers. The inquiry in
these directions involves many intricated problems for mathe-
matical physicists, and may better be left for the specialists in
these lines.

If the vertical vibration of the atmosphere be the actual
cause of the magnetic pulsation as appears most "probable, the
investigations of the latter phenomena will in any case afford
very valuable material for studying the actual physical conditions
of the upper atmos})here, and may offer probably an unexpectedly
wide field for new researches in different directions.

According to Prof. Sano, mechanical disturbances of any
kind must gradually suljside into a regular vertical natural
vibration, which is comparatively slowly dissipated. This tlieoreti-
cal result is in harmony with the observed phenomena, viz.,
the subsidence of an abrupt change of the terrestrial field into
a train of regular damped waves, and also the peculiar persistent
character of some trains. These points will be touched on once
more in ^ 28.

28. Finally, let us consider the last crucial test of the
different h3^pothese^. the peculiar ])e!iaviour of the rotational
oscillation of the horizontal components. If we assume the
observed sense of rotation for different hours, as generally
applicable for the entire northern hemisphere, the distribution
of the different senses of rotation in different hours will be roughly
represented l)y the annexed figure (Fig. 23). where the semicircles

74

Art. 9. -ï. Teratla

represent tlie hemispheres for day and night respectively. The
distril^ution, expressed in other words, is such that when tlie
periodic magnetic field is directed toward due south, the WP]-
component, just passing zero, is increasing toward the direction

Fi.a. '2d.

fiiçWiSiè€

^a^^5<.êi

6>.:n

M,N

Qi'am.

fieen.

Qym.

of the meridian at 6^ a.m., or 6^ p.m., lor all points of a.m.,
or p.m. region respectively. A similar variation might have
been produced, if there existed two sets of horizontal atmospheric
oscillations corresponding to the zonal harmonics of the second
order, the one having its axis at the earth's axis and the other
in a direction passing through the equator at the meridians
corresponding to 6^' p.m. and G^ p.m.; the two component
vibrations having a proper phase difference. Purely mechanical
vibration of such a type must, however, have a more or less
definite, and certainly much longer period, and is scarcely apt to
explain the observed phenomena in the case of the most frequent
short waves.

Another suggestion is that along the atmospheric current
of more or less linear cliaracter, running near the place of
observation, a sinuous motion is propagated, in which case
the periodic disturbance may become rotatory and the sense
of rotation will be opposite on both sides of the current. It
seems to suffice, therefore, to assume a belt of current making
no considerable angle with the meridian at (f' or 12Î' on which
a sinuous motion is propagating with a proper velocity- The
assumption is so far not utterly contradictory to the assumption
of the circular current mentioned earlier, since the latter has

On Rapid Periodic Variations of Terrestrial Magnetism.

75

the pole in the morning lionr, decidedly deviating from the
earth's pole. But in this case the phase relation of JZ an JX
must be opposite on Ijoth sides of the current which is never
actuall}^ the case. If the above be the case, the magnetic
waves will naturally be of progressive character.

Agairi, to explain the phenomena in question on the
assumption of a progressive wave of the sort considered in § 2o,
it may probably suffice theoretically to consider two intersecting
systems of waves propagated with proper velocities, though as a
matter of fact, we have neither an evidence nor a physical
ground for assuming the existence of such a complicated system
of currents. The wave length of sucli waves must, however,
be of the order of the earth circumference, in order to explain
Fig. 23; hence the case is essentially similar to that considered
at the outset of this paragraph.

Fje. -24.

Finally, we may su})})(ise that tlie atmosi)heric electric
current is not purehj horizontal as was generally assumed in tlio
altove discussions, and also that tlie vertical component current
is subjected to a periodic fluctuation Avitli a proper phase relation

76 Art. 9.-T. Terada:

to tliat of tlie horizontal one: or less p^obal3h^ to a rotatory
oscillation of the kind considered in § 24, but in a horizontal
elliptical orbit instead of the vertical one. These will probably
remain the only admissible hypotheses, if the progressive nature
of the magnetic waves be ultimately denied Ijy the observations.

The simplest ideal case conceivable of this kind is that
in which a linear current is subjected to a see-saw motion in the
vertical ptane, about its horizontal mean position. Referring
to the annexed figure," where .T^z-plane corresponds to the
earth's surface, let HH' be the mean position of tlie current j,
at the height A, and let it be vibrating about its mean position
in the vertical yz-^h\\\Q, as if it were a tangent to a string
subjected to a stationary vibration, at its node at H. Denote

the frequency of vibration b}^ -fr . A is the point of observation

at a distance x from the origin, and AD the perpendicular from
it to the current HT at time t. If AP represents the magnetic
field at A due to the current at t, it is perpendicular to AD
in the plane perpendicular to HT and of the magnitude

2^ _ %

if we denote by tl the inclination of HT to HH'. Let Q and
N be the feet uf the perpendiculars from P to 0:r and :r//-plane
respectively. Then, denoting the angle ^OAD by </',

AQ= X =ÄPsiu ip.

QN= - Y=PN sill tf,

PQ=-Z=PN cos ä,
since the angle ^NPQ = ^. Put PN=ÄP cos é and tg (/>= '— •
Hence, we have for the fields of tlie thi'ce components:

'liJlCOH 6
;T- + //-COS-/^

2 /j;' bill ä
x' + h'-cos-d

2?iccos ti
x^ + h'-cos-ä

1) The coordinates axes are chosen lefthandid to correspond to our initial convention.

^ 2///COS â 1 ^~ X

X = --^ — - — - - lor rs -comx^onent,

,. 2/j;'bili ä 4- T^ ,

— 1= —^^ _^ tor h-component,

Z = ^ _^ ;^ lor L pwara-coiiiponeiit.

Ou Rapid Periodic Variations of Terrestrial Magnetisui. 77

According to our assuiiiptioii, ^ i^^ given ]»y

If C is small and we neglect the small quantities of the second
order against unity, we may put cos ^=^1 and sin d^C cos{pt+<p).
Hence

= ; Y=— ^ ^C coä[pt-\-e) ; Z—— - -— .

x^ + Jr xr + Jir ' ' x'^ + h'

Thus, the Y-component onh^ will be sensibly affected while the
others remain constant, if i be considered constant. If l be
variable and given by

i = i^-\-i^ cos qt ,

where y'o and ii are constants and /i is small compared with /o,
the periodic parts of the three components will be given by

JX= — pA_cos qt,
x^ + h"

Jr=---^M^cos (pt + f),
x- + 7r

dZ=-^^^'^ cos qt,
X +h-

neglecting the small quantities of the second order in JF. If
p=q, as may be the case when the see-saw motion is the direct
cause or result of the fluctuation of the current and moreover if
<^=^0, we will have a rotatory motion of the kind desired.
Besides, we notice that the sense of rotation is opposite on both
sides of the ^5;-plane. The relation of JJC and JY is just the
same as in § 22 a), as is evident. JZ must be of the opposite
sign on both sides of the current. The results of observations
show, however, that the phase relation between JX and JZ is
generally the same all through, JZ never being ahead of JJT.
Hence the maximum intensity of the current in question can
not be running largely in the meridional direction, but 23robably
more or less near the equatorial zone, since otherwise we should
expect that in some hours JZ may run in advance of JJC. To
reconcile the present case with the result shown in Fig. 23, it
will be plausible to assume that in passing along the four

78 Art. 9.- T. Terada :

quadrants 0'^-6!^ 6^-12;^ 12"-18f 18"-24'; the phase angle differs
successively by -, if /o be of the same sign throughout; or û
clianges the sign alternatively, if ç remains constant. It we
are to identify /o with the equatorial portion of the circular
current assumed in § 19, then it will suffice to suppose that a
portion of the circular belt or ring carrying the current is subjected
to a radial vibration, as if it were a portion of the ring carrying
out a dilatator]/ vibration in such a way that the amplitude is
given by cos-V, say, where ^' is the longitude counted from noon.
The fluctuation of the intensity must be considered as keeping
pace with the vibration and the period must be different for day
and night.

Though the above solution is by no means unique, yet it
may at least serve as a provisional explanation as far as the
phenomena of rotatory pulsation are concerned. It is, moreover,
not contradictory to the principal facts of observation already
enumerated: viz., not irreconcilable with the mode of the daily
variation of the azimuth of tbe horizontal components, since
the circular current assumed may l)e considered to have its
maximum intensity alwaj^s near the equator; and again not
inconsistent with the observed amplitude ratio and phase relation
of JZ and JX, since the see-saw motion chiefly affects JY,
and tlie discussions on these relations given earlier hold good
liere also without any essential modification.

According to van Bemmeln, the ratio /iZjilX at Batavia
seems to be invariabl}^ of insensible magnitudes. This fact is
in harmony with the above assumption that the maximum current
is near the equator. On a theoretical ground, this seems also
admissible if we may attribute the origin of the disturbing
current to the vertical motion of tJie atmosphere, since general
planetary circulation of the whole atmosphere must be in any
case vertical in these regions, where the horizontal intensity
of the magnetic field is also greatest.

29. Tlie above taken as granted, it seems plausible to
assume next that th(? impulse causing the vertical vibration, if
not solely mechanical, is to be attributed to the reciprocal action

On liapitl Poriodic Variations of Terrestrial Magnetism. 79

of the magnetic field on tlie cluuige of the current,^^ assun)iiig
the cm-rent already caused l)y other agents. In § 11) we have
suggested that the zonal part of the atmospheric current seems
to be more liable to the fluctuations than tlie part running in
the meridional direction. This is in some measure favourable
to the hypothesis.

If the mechanical vibration of the atmosphere ])e once
started, the periodic fluctuation of current naturally results on
account of the induction due to the general terrestrial field.
The intensity of the induced current will be greatest at tlie
equator for a given amplitude of the oscillation.

The cause of the initial impulsive increase of the current
may on one hand be considered as the result of the motion of the
atmosphere brought al:>out in a purely mechanical way, for
example, an abrupt advent of the vertical flow of the upper
atmosphere caused by the release of some instability of equi-
librium; or we may conceive a limited portion of the atmosphere
excited» to a vigorous vertical current, as in tlie centre of a
cyclone, caused as a secondary disturbance accompanying the
daily exchange of air between the day and night hemispheres.

On the other hand, the abrupt increase of current may
also be attributed to an external agent, for example, a sudden
increase of the conductivity of air caused by the inflow of the
corpuscular radiation from the sun.'^

Among the above two possibilities, the latter is more
plausible, being in accordance with the commonl}^ accepted view
as regards the origin of magnetic disturbances, which is strongly
supported by the intimate relation existing between the solar
activity and the disturbances in general. Fig. 10 is also
favourable to tliis hypothesis. The former is rather doubtful,

1) In view of this consideration, an interesting physical problem presents itself :
Consider u current passing through highly rarefied gases, acted upon by an external
magnetic field ; will the gases really flow «.s- a whole as Fleming's rule specifies for a solid
conductor? As far as we are aware, the answer is not yet given either experimentally or
theoretically, especially for such a case as is analogous to the atmosperic currents.

2) A. Schuster concluded on the basis of energy consideration that the E.M.F. of the
current causing magnetic disturbances is to be attributed to the motion of tlie atmosphere,
the sun's radiation serving only as an agent increasing the conductivity.

yO Art 9.-T. IVrada:

since it is an open question whether in tlie liiglier part of our
atmosphere there may occur any remarkable vertical convection,
as is the case in the troposphere. At any rate, the fact that an
abrupt increase of the general fiekl is most frequently accom-
panied by a train of remarkable pulsations, is in harmony with
the above idea that the increase of the current is associated with
the increase of the vertical impulsive motion of the atmosphere
which subsides into a vertical natural vibration. The charac-
teristic disturl^ance of this type occurs as a rule near midnight.
On the other hand, examples are by no means rare, especially
in the evening, where an abrupt increase of the horizontal
component occurs, but not at all accompanied l)y a train of waves.
How is the fact to be explained ? It may at first be suggested
that in this case the abrupt increase of tlie current takes place
in the direction of the general terrestrial field; but as a matter
of fact, the increment of the Y-component in such a case is not
generally of a different order of magnitude compared with that
of the X- component. Another possible altervative is that in the
hours which are near the boundary between the day and night
hemispheres, the regular vertical vibration of the atmosphere
is not so easy as at midnight. The latter assumption seems
in some measure plausible, if Ave consider very probable
heterogeneity of the atmosphere in that region forming the
transition stage of the illuminated and non-illuminated halves.
The minimum of the frequency of pulsations of all periods falls
actually in the evening hours (Fig. 8). The morning hours
are not necessarily in the same condition as the evening hours.
In the former, the atmosphere is being rapidly heated up
and ionized, but on the night side of tlie boundary a dormant
homogeneity of the night-atmosphere may probably prevail;
whereas in the latter, the gases are cooling down and the ions
recombining, and gradually passing into the nightly condition.
In the morning, the transition may therefore be abrupt and the
area of the heterogeneous atmosphere comparatively small,
W'hile in the evening the heterogeneity may extend to a consid-
erable area of the earth's surface.

On Rapid Periodic Variations of Terrestrial Magnetism. y]^

iVgain, cases are quite common where remarkable trains
of waves occur without any general swelling of the horizontal
component. Among such trains, we may distinguish two types,
viz., those with the al)rupt beginning and those gradually
increasing in amplitudes. The former, which is most frequent
during night, may prol)ably be explained b}^ a transient increase
of the atmospheric current with a duration comparable with, or
shorter than the natural period of the atmospheric oscillation.
Indeed, the first wave of the train of this type is often sensibly
shorter than the following regular waves. The latter class,
whicli is generally the case with the shorter day- waves and also
sometime with the longer night-waves, may be caused eitlier
by the mechanical disturbance of the atmosphere propagated
from a remote region, or by some different agents. Among
other conceivable causes of the vertical vibration of the upper
atmosphere, we may cite the instability of the discontinuous
horizontal motion of the higher layers. That the upper layer
has a considerable angular velocity with respect to the earth's
surface is well known from the observation of the "illuminated
night cloud." It is then probable that a favorable vertical
distribution of density may cause a reinarkaljle wave motion of
different types. The gravitational wave, as occurs in the case
of an incompressible fluid, will not answer our purpose. P)Ut
it is more than probal)le that a radial or vertical expansional
wave whicli is the most persistent type, is excited in this way — a
case somewhat analogous to the excitation of the organ pipe
by the stream of air running along the loop of the vibrating air
column. If the upper atmosphere be arranged in different
layers with different temperatures and different general wind
velocities, as is the case in the troposphere, different periods
may occur at the same time. Tlie irregularities of the pulsations
during day time might well have been produced in this way,
if we consider that the laminar structure may I)e enhanced in
some way or other, by the influence of solar radiation. Beside
the thermal efïect, light pressure may also play some sensible

82 Art 9.— T. Terada :

rôle in this respect, since it acts in differentiating the gases with
different absorbing powers and molecnlar sizes. '^

Again, if there be present any sensible fluctnation of the
solar radiation^^ with a short duration, the thermal effect, and
also the radiation pressure in some measure, may contribute
to the initiation, if not the direct excitation of the vertical
motion of the upper atmosphere. The remarkable irregularities
of the day-waves may partly be accounted for in this way.^^

As to the origin of the magnetic waves with the periods
longer than 10'" we may probably suggest the slow atmospheric
waves possible in the case Avhen two layers with different
temperatures are superposed. ^^

The fact described in § 17, that the hourly distribution of
the senses of rotation of the disturbing magnetic vectors are quite
different for the short waves and the long undulations, requires
an explanation. It may only be suggested for the present that
the difference is in any case due to the difference in the modes
of atmospheric free vibration. Any further discussion must be
postponed till the nature of the vibrations of the atmosphere
has been more fully investigated from the theoretical side,
especially for a spherical atmosphere.

1) Adopting Debye's approximate formula for an ahaorhinfj sphere (Ann. d. Pliys., [4] 30,
1909, p. 117), tlie pressure of tlie light wave with the wave length ), on a sphere of radius
(( may be put

„ 27Trt ,S'

1-83 . w;2 ,

> c

where S is the energy of the radiation per sec. per cm.- Assuming « = 10-^, ). = 5xlO-5 and

3
Ä = — ; — X4"10'', we obtain a pressui-e 0'48XlO-22 dynes. Taking the mass cf the sphere as

l"GXlO~'-*gr. for a hydrogen atom, we obtain an acceleration of 30 cm/sec. The mean
free i^ath at the pressure of 0007 mm. being about l-7tî cm. the mean velocity may be
estimated to be of the order of 5'1 cm/sec The path traversed is therefore only 184 m. per
hour. This is probably too small to answer our purpose. But if we take a " Dipole "
instead of the absorbing sphere, the value of the pressure may in some cases be estimated
to be decidedly greater. Near 200 km. from the earth's surface, the pressure will vary
rapidly with the height ; in the higher layer, the path traversed by molecules or atoms
may be enoi-mously greater than the above estimation.

2) Together with the back-radiation from the earth.

3) The corjmsculnr radiation being deviated by the terrestrial field, will not be confined
to the day hemisphere, or rather prefers the night side. The fluctuation of this radiation
may be considered to be of the type considered in p. 79.

4) H. Lamb, Proc. R.S., A 84, 1911, p. 551.

On Kapid Periodic Variations of Terrestrial Magnotisui. 33

30. From all we have discussed at length in the preceding
paragraphs, we can as yet draw no convincing conclusion
regarding the actual mechanism producing the magnetic pulsations
in question. Many things, indeed, turn on the crucial observation
regarding the universal simultaneity of the periodic phenomena.
If it be ^rovecZ ultimately beyond doubt as seems proljable, there
remain only those hypotheses at disposal, which lead to the
simultaneous occurrence of the phenomena for an area of
considerable extent. It that case, the magnetic waves may
probabl}' be explained by the combinat'w7i of 2^(^r iodic fluctuations^
in intensity as ivell as in position and inclination of the atmospheric
current, which is then to be considered of rather diffused, but
not of linear character, having the maximum intensity near the
equator. The origin of these fluctuations might then very
probably be sought in the vertical stationary oscillation of limited
portions of the upper atmosphere accompanying the diurnal
oscillation of the entire atmosphere.^-*

At any rate, it must be admitted that the present results
of observation refer to a single station, and the observations are
far from being complete. A further study of the allied pheno-
mena, es2:>ecially the simultaneous observations in at least three
stations, sufficiently apart from each other, will be desirable.
The results will not fail to advance our knowledge on the nature
of the atmospheric current of which we have at present only a
ver^^ vague idea. The interest attached to the problem at hand
is by no means confined to the limited subject of terrestrial
magnetism. The phenomena have a mucli wider bearing than
at first sight appears, on various interesting problems in different
branches of physics, for examples those regarding solar physics,
meteorology and also especially those regarding the electrical
and mechanical behaviours of highl}' rarefied gases under the
action of different kinds of radiation.

In conclusion, the author wishes to express his sincerest

1) In this case, however, the phenomena of the propagating auroral bands accompanied
with no corresponding magnetic maves, become rather incomprehensible and throw some
doubt on Birkeland's original conception pertaining to the nature of the luminous band.

84 Art 9.-T. Terada:

thanks to Prof. A. Tanakadate, under whose supervision the
entire work was carried out, for his kind guidance throughout
the course of the investigations. Most cordial thanks are also
due to Prof. H. Nagaoka and to Prof. S. Sano, for the interest
shown in the investigations and for many valuable suggestions
and instructions given. Last, but not least, we feel very much
indebted to Prof. I. Ijima, the Director of the Marine Biological
Laboratory, for his kindness and generosity in affording lodging
and many other conveniences to the observers resident in
Misaki, for which the best thanks of all the participators in the
present work are due.

Summary of the Results of Investgations.

1. The period of tlie magnetic pulsations has no sharply
defined value, varying from about 20 seconds to nearly 1 hour.
Nor is it exactly constant even in a coherent train.

2. During the day time, waves of 0*5-1 minute periods
predominate, whereas during the night hours longer periods
l'D-2'5 minutes are most frequent.

3. A periodicity of 25-30 days is suspected in the daily
frequency of the pulsations.

4. The vertical component of the waves is a reduced
reproduction of the NS-component except the phase retardation.
The shorter the period, the more remarkable is the reduction
of the amplitude as well as the phase retardation.

5. The azimuth of the linearly pulsating magnetic field
undergoes a remarkable diurnal variation, showing maximum
deviations from the meridian a few hours before noon and
midnight.

6. The disturbing field is generally more or less rotatory.
The sense of rotation shows a semidiurnal variation. The clock-
wise rotation is most frequent duiing the hours between sunrise
and noon, as well as between sunset and midnight, wdiile the
opposite rotation falls most frequently in the remaining hours.

On Rapid Periodic Variations of Terrestrial Magnetism. g5

7. The observed results may probably be explained by the
fluctuations of the horizontal electric current existing in the upper
atmosphere, and causing tlie diurnal variation of the terrestrial
magnetism.

8. Two diverging lines of theoretical considerations intended
for the explanation of the phenomena in question, are given;
the one based on the assumption of the simultaneity of the
phenomena in a wide area, and the other on the assumption of
a progressive nature of the pulsations, though some evidences
at hand seems to speak rather strongly against the latter
assumption. The results of the discussions turn out rather
favourable for the hypothesis of simultaneous disturbances than
for that of progressive waves. If the simultaneity be universally
establislied, the phenomena may probably be accounted for by
the fluctuation of the atmospheric current, in its intensity as well
as in its location. The fluctuation must then very probably be
attributed to the more or less vertical oscillation of limited portions
of the upper atmosphere. If sucli be actually the case, we
have in the phenomena of the magnetic pulsations a very
valuable clue for studying the physical conditions of the upper
atmosphere unattainable by the usual means, and then, it may
be hoped, for following the hourly or daily changes occurring
in the remotest part of our atmosphere.

Published May 25th, 1917.

Jour. sa. Coll.. Vcl, XXXVII.. Art. 0. PI. I.

iy^t 3 - V .!'<

Night rcTOi-a on a craiiparatively i-aliii da,y, witli diavactimsticlmif; waves in midiiiglit aiul slirat wavfB in Ihc njnvniiif;. April 3-4, lftI3. (Keiluccd to | original size).
T. Tera<la. On Rapid Periodic Variations of T( rrestrial Magnetism.

Jour. Sei. Coll., Vol. XXXVII., Arl. 9, PI. II.

Chanicteiistic day record. Note the short waves in the moiiiiiig. In this example simik

March 17, 1913. (Eeduced to J original size).

^"N

T. Terada. On Rapitl Periudic Variatious of Terrestrial Maguetiaii

Jour. Sei, Coll., Vol. XXXVII.. »ri. 9, PI. III.

I >i-,Liiili.'(l tliiy. Iv'K'il'^"' '""fî vviivrs cluiracteristic to midnight not cniispii

spite nt the iviiiarkuhlc clistin'lmiico of the RcDcTal fu-lcl. Note tlii> inorniiis; trains of slir
T. Terada. On Kapid Poriotlie Variations of Terrestrial Magnetism.

,ilnr trains not occurring in the evening. .Tan. .3-4, Mll.l. (Kwliiceil to ' original si:'i-

Jour. Sel. Coll., Vol. XXXVII., Arl. 9. PI. IV.

., oiiginated at a sontlierii imi-l of North Pacific: Mai'cli 1 4-1 5, 1918. (Eeduced to J origmal size)-
T. Terada. On Rapid Periodic Variations of Terrestrial Magnetism.

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JOUENAL OF THE COLLEGE OF SCIENCF, TOKYO IMPERIAL UNIVERSTTY.

Vol. XXXVII., Article 10.

On the Photographic Action of a, ß and r

Rays emitted from Radioactive

Substances.

By

Suekichi KiNOSHITA, Ui<j<ikuhah-us]ii.

and

Hadime IKEUTI, lH'jakushi.

Wit It S Plates.

1. Introduction. In 1909 one of the writers^^ showed that
whenever an « particle strikes a grain of silvei* hahcle in the
sensitive fihii of photographic phites, that grain becomes suh-
sequently caj^able of development. It was also shown that this
is the case throughout the whole range of « rays, no matter what
may be the quality of the plate. It is remarkable that the
behaviour of the photographic plate regarding this phenomenon
should obey such a simple law, when we take into consideration
its complex behaviour with regard to light.

From the atomistic point of view, on the other hand, the above
result is also highly interesting, as it affords a third independent
method of recording the emission of a single a particle, in addition
to the electrical method devised by Kutherford and Geiger^\ and
the scintillation method first systematically studied by Regener^\

In the light of the above investigation, it was to be
anticipated that if a particles were projected tangentially on a
photographic plate, the halide grains encountered by each «
particle would on development appear in a train. This effect

1) S. Kinoshita, Proc. Eoy. Soc. A, 83 (1910), p. 432.

2) E. Eutherford and H. Geiger, Proc. Eoy. Soc. A, 81 (1908), p. 141.

3) E. Eegener, Verh. d. D. Pliys. Ges. 10 (1908), pp. 78 and 351.

2 Art. 10.— S. Kinoshita and H. Ikenti :

was examined first l)y Keinganum^^ On a photograph ol)tained
in this way, silver grains were found arranged on distinct hnes,
showing themselves as the a ray tracks. He also found that
some of the tracks showed the efïect of scattering. Experiments
on this subject were later made in detail by Michl"' and by
Mayer^\ Some microphotographs of the a ray tracks showing
deflexions were ol)tained by Walmsley and Makower*\

For the last three years, we have been engaged in devising
some simple methods of obtaining radial a ray tracks and making
some investigations upon them. The present paper contains
the main results in this line of investigation, most of which
appeared separately in other publications*^

2. Methods of obtaining radial a ray tracks. In investigating
the photographic traces of « rays it was thought most ad-
vantageous to work with the smallest possible source of the rays.
For, if a point source be established and placed on a photographic
plate, the expelled « particles will leave on it a set of radial traces,
which can Ije followed with greater ease and certainty^\

The source which we first utilized was a fine needle point
coated with a trace of active deposit of radium'''. This was
prepared by lightly rubbing the point on a small ball of iron
which had previously been exposed to a few millicuries of radium
emanation. By 1 »ringing the needle in contact with a photo-
graphic plate, a part of the active deposit was detached from the
needle and left on the plate at the point of contact. Leaving the
plate for a certain time to allow a sufficient number of a particles

1) M. ßeing-anum, Phys. Zeits. 12 (1911), p. 1076 ; Verb. d. D. Phys. Ges. 13 (1911), p. 848.

2) W. Michl, Akad. Wiss. Wien. Ber. 121, 2a, (1912), p. 1431.

3) F. Mayer, Ann. d. Phys. 41 (1913), p. 931.

4) H. P. Walmsley and W. Makower, Proc. Phys. Soc. 26 (1914), p. 261.

5) S. Kinoshita and H. Ikeuti, Proc. Tokyo Math.-Phys. Soc. 7 (1914 , p. 360; also Phil.
Mag. 29 (1915), p. 420; H. Ikeuti, Phil. Mag. 32 (1916), p. 129; Proc. Tokyo Math.-Phys.
Soc. 8 (1916), p. 465.

6j In the first publication, we have referred to the paper of Michl of which we learnt
through the Beiblätter z. d. Ann. d. Phys. and the Science Abstracts, as the number of the
Wien. Ber. containing the original paper could at that time not be found in our Library.
In these reviews nothing was mentioned of the method by which a ray tracks had been
obtained. When we obtained the original paper afterwards, we found that the method
employed by him had, in some ways, been similar to that of ours. However, his lines of
investigation were different from ours.

7) For the case of polonium and of active deposits of thorium, E. R. Sahni's paper in
Phil. Mag. 29 (1915), p. 831 may be referred to.

On the Photographic Action of a, ß and y Eays emitted from Radioactive Substances. ^

to emit, it was developed wlien a fine spot became visible to tlie
naked eye. On examining the plate under a microscope, the said
spot was found to consist of a multitude of separate trails of silver
grains running radially from a common centre.

In the photographs taken in this way, however, there is
around the centre a dark area, which is no doubt the portion
where the active needle touched the plate during the above said
process. In this case, the radial tracks do not emerge at the
centre but at the rim of the dark area. Since the tip of the needle
had the sliape of a truncated cone terminating in a flat section of
about 10 //in diameter, it was impossil)le to obtain by this method
a photograph showing no central dark area. Still, this method
was found useful when, as reference will be made later, deflexions
of a ray tracks had to be investigated.

Several trials were made to obtain photographs with nuclei
as small as possible. When an iron ball coated with the active
deposit was held above a photographic plate and knocked with
a hammer or the like, some fine dust particles adhering to the
iron ball seemed to I)e set free by the shock and to settle down on
the plate, becoming thus the sources of the « radiations. On
developing the plate several spots appeared and some of them
were found to possess nuclei whose linear dimensions are very
small compared with the length of the « ray track. The spots
themselves were so small that they were hardly detected by the
naked eye.

To obtain a large number of these spots, it was found
effective to strike the photographic plate directly with the active
iron ball on the film side. To illustrate the general feature of
the photographs obtained in tliis way, one of the plates is
reproduced in fig. 1, enlarged 120 diameters. The irregular dark
areas seen in this figure are the spots where the plate was struck
by the iron ball. Around these areas, there can be seen several
circular spots, each of which consists of a set of a ]'ay tracks
]'unning radially from a common centre.

It may be remarked that, in a photograph in which a large
number of « ray trncks radiate from a common centre, each trail

4 Art. 10. — S. Kinoshita and H. Ikenti :

does not appear distinct from the otliers along the whole length,
but, in the vicinit}^ of the centre, comes very close to or overlaps
the neighbouring ones, even when the radiant nucleus is
practically a point. This effect also gives rise to a dark area at
the centre, which has no definite shape. The dimensions of the
dark area vary of course with the number of the trails.

3. Two different types of radial a ray tracks. As microscopic
examinations at once show, the trails of grains may, in general, be
divided into two difïerent types.

In the first type, the trails emerge directly at the centre or, in
case wdien the dark nucleus is present, at the rim of it. Conse-
quently, when a set of homogeneous a rays have been utilized, all
the tracks terminate very nearly at the circumference of a circle
drawn around the centre, and present themselves as a halo (figs.
2-8), resembling the pleochroic halo seen in minerals such as
biotite, and investigated in detail by Joly^\

Without question, the radial tracks do not all lie on planes
parallel to the surface of the emulsion film, and as can Ije traced
by focussing the microscope, some of them are running oblique to
the surface. Since the film on ordinary photographic plates has a
thickness equivalent to only about two centimetres of air in stop-
ping a rays, the photographic halo obtained on these plates can
never be that of a complete sphere.

The trails belonging to the second type spread out around the
centre over a wide region of no definite boundary, as can be seen
in fig. 9, which is reproduced in the same magnification as the
haloes in figs. 2-8. In this case, most trails are found to have
their seat within the uppermost layer of the film, showing that
they are the tracks of the a particles projected tangentially to the
surface of the film from the part of the source just above the sur-
face. At first sight, some of the trails seem to be much longer
than those constituting the haloes. Closer inspection, however,
shows that this is only apparent and that each of them consists of
two or more elementary trails following one another in succession.

1) J. Joly, Phil. Mag. 13 (1907), p. 381 ; 19 (1910), p. 327; J. Joly and A. L. Fletcher, Phil,
Mag. 19 (1910), p. 630.

On the Photographie Action of a, ß and y Eays emitted from Radioactive Substances. 5

4. Haloes due to radium C. Tliese can be obtained Ijy uti-
lizing as the source of « rays the active deposit of ra(hum, in which
radium A has ah'eady disappeared. Figs. 2-4 are the micro-
photographs of these haloes, enlarged 50U diameters. It will be of
interest to see the different stages of formation. In the halo in
hg. 2 about 4'") tracks are to ]>e seen, this being of course the num-
l)er of the tracks on a plane focussed in reproducing the micro-
photograph and forming therefore only a small fraction of the
total. The haloes in figs. 3 and 4 are seen to contain about 70 and
1 20 tracks respectivel3\

The ends of the tracks constituting a halo do not lie strictly
on a circle owing to the difference in the struggling of the rays
through the medium. \A'e have tlierefore taken as the radius of the
halo the radius of the circle drawn so as to pass through the ends of
most of the far reaching tracks. The radii of the haloes determined
in this way vary slightly, but their smallest limit is found to be
o2 [JL. Since, on the other hand, isolated tracks of this length
are found in such pliotugraphs as fig. 1), this may be taken as the
range of til e « rays from radium C in the substance, so that the
radiant nuclei of these haloes must l:)e ver}- small.

5. Haloes due to radium A and radium C. To obtain haloes
of this kind, we have exposed an iron ball to radium emanation for
a few minutes and ])erformed the above stated process as quickly
as possible. These haloes are reproduced in figs. 5-8, in the same
magnification as before. They indicate that the tracks of a set of
homogeneous a rays from radium A give rise to another concentric
circle inside that due to radium C, as in the case of the pleochroic
halo.

It will l)e seen that in fig. 5 the outer circle is more conspicu-
ous than the inner one, wliile with fig. (i the reverse is the case.
Which of the circles comes out more conspicuously depends upon
in what proportion radium A and radium C have been mixed in
the source utilized.

Tlie inner circle is in each case smaller than the outer by K) //
in radius. Thus tlie ratio of the ranges in the sul)stance of the
two a rays becomes(52-l()) : 52 or 1)9, which is the same as its value

6 Art. 10. — S. Kinoshita and H. Ikeuti :

in air. According to Marsden and Ricliardson'', the above ratio
for silver is much higher, amounting to *S6. In the present case,
the absorption of the a rays is mainly due to gelatine, which is the
main composition of the sensitive fihn, and this substance seems
to be of a character similar to air concerning absorption.

6. Number of silver grains along an a rag track. Silver grains
along an « ray track are naturally not equidistant. To find the
distribution, Ikeuti has measured, by means of the ocular micro-
meter of a micro.scope, the distance d between successive grains on
the track for a large number of pairs of the grains. When the
number of pairs of grains was plotted against the corresponding
values of d, and a mean curve was drawn, this showed a maximum
at d=l'Sö fj. in the case of Ilford Process Plates, falling quickly to
zero on the side of the origin and somewhat slowly on the other side.
From tliese measurements the mean value of d was calculated to be
!2'85 /ji. This corresponds to 350 grains per millimetre of the track.
Since the longest tracks of the «rays from radium A and radium C
are 36 and 52 // respectively, they will consist on the average of 12'6
and 18'2 grains respectively. It must, how^ever, be remembered
that the actual numbers are subjected to faii'ly large fluctuations.

7. The nature of the 'photographic action of a particles. Alpha
ray tracks obtained in a sensitive film do not exhibit even the
slightest difference along their wliole range; neither in the com-
pactness of the grains nor in their size. This is a very important
fact and confirms conclusively the previous experiment of Kino-
shita"\ in which the constancy of the ])hotographic action of an
a particle along the whole range was photometrically established.
Now, if it be considered that there are all possibilities of regarding
ionisations in a solid and in a gas to be of a similar type, the
question may arise, why the photographic action, which is
nothing but the result of ionisation, should not be represented by
the characteristic curve known as Bragg' s ionisation curve. As
a matter of fact, darkening in the pleochroic halo is particularly
pronounced near the boundary, in spite of that the number of a

1) E. Marsden and H. Eichardson, Phil. Mag. 25 (1918), p. 191.

2) I. c.

On the Photographic Action of a. ß and y Eays emitted from Radioactive Substances. 7

particles traversed, taken per unit volume, is least in that region,
and this fact can be explained as the result of an intense ionisation
near the extreme end of the range of the « particles^^

A theory put forward by Kinoshita to explain the singularity
ol the photographic action was that, if some of the halide
molecules witliin a grain are initially ionised by one or more a
particles, the whole grain l)ecomes subsequently capable of
development, l)ut the reduction cannot extend to other grains
which have not been initially ionised. When the film is
completely developed, all the grains struck by the « particles are
reduced to a constant limit, which depends on the size of hahde
grains in the emulsion film but not on the degree of the primary
ionisation in them, and we thus obtain silver grains as a secondary
consequence of the ionisation, so that the photographic action is
constant throughout the whole range of the a particles in the
substance. A further consideration will be presented later^\

In variance with the above theory, MichP^ states that only a
part of the halide grains encountered l)y an a particle become
subsequently developable. This conclusion is based on a micro-
scopic comparison between the compactness of silver grains on a
photographic plate which was obtained by exposing it to light,
and the difïuse arrangement of the grains on an « ray track
obtained on the same plate.

In order to show that the above reasoning is by no means
adequate, the original paper of Kinoshita must be again referred
to. It was shown that the photographic density D of a photo-
graphic plate produced by normally falling a particles varies with
their numljer n per unit area of the plate as

D=D,(l-e-), (1)

where D^ is the maximum value of D attainable by an indefinitely
large number of the a particles, and c- is a constant depending on
the quality and the thickness of the emulsion film on the plnte.

As is generally known, the photometric density of a plate is

1) J. Jo]y, /. c.\ E. Rutherford, 'Radioactive Substances and their Radiations'(1913),p.310.

2) cf. p. 15.

3) I. c.

8 Art. 10. - S. Kinoshita and H. Ikeuti :

proportional to the mass of silver contained in a unit area of the
plate. Therefore, if it he assumed, as a first approximation, that
all the grains are of equal mass, the above equation may he
written in terms of .the number s of the grains per unit area in the
plate.

.9 = .<?„(l-e— ), (2)

where Sq is the value of s for the plate for which 7)=7)„ and would
be the total numl)er of the halide grains initially pre-ent in tlie
emulsion film per unit area.

Consequently, the number of the falling « particles required
to increase one more developable grain will be

^ = 1 . (3)

ds c (s^—s)

If we take the corresponding values of s and n in a plate of
very small density, .s is negligibly small compared with s,y In this
case, the above equation reduces to

dn _ 1
ds csq

Giving values experimentally found:
c= 1-18. 10"' and So=l-]6. 10' for a Wratten Instantaneous Plate,
and c='93.1U-'' and s„=-97.10' for a Wratten Ordinary Plate,

——= '73 for the Instantaneous Plate, and
ds

= I'l for the Ordinary Plate.

On examining microscopically, it was found that the Instan-
taneous Plate of very high density was entirely covered with silver
grains, while the Ordinary Plate of maximum density was almost
covered with grains, but about one-tenth of the area was estimated
as allowing « particles to pass through without contact with the
grains, and that in an Instantaneous Plate of small density al)Out
one-third of the grains were overlapping others, while in an
Ordinary Plate the overlapping was practically negligible, so that
one a particle did not strike more than one halide grain.

Taking these facts into consideration, it can be concluded
with certainty that the number of « particles required to change
one halide grain into the developable state is one, whenever it

Oa the Photographic Action of a, ß and y Eays emitted from Radioactive Substances. 9

strikes the grain. For, if, in the case of tlie Instantaneous Plate,
the numl)er of the overlapping he taken as 37 per cent, of the
total, eacli « particle would in the average change To? grains on
its passage through the film; in other words, the numher of a

particles required to change one halide grain would l)e ^r^^ or

•73, which is the value of ~^ calculated from the values of c and

.?o for that plate. Similarly, in the case of the C)rdinar3' Plate, if
the area uncovered with the grains be taken as 10 per cent, of the
totals, PI a particles on average would be required to change one
halide grain. We are thus lead to the conclusion that one a
particle is sufficient to change a halide grain into the developable
state, whenever it encounters the grain and whatever may l)e the
quality of the plate.

In dealing with an « ray track, it must be rememlK^red that
we are observing those grains only, the centres of which lie, as
w^ill be directly shown, within a very nari'ow cylinder of a cross
section equal to that of the halide grains, l)ut not those having
their centres outside the cylinder. Since grains at various depths,
the difïerence of which is much greater than their linear dimen-
sions, can be seen simultaneously in a microscopic vision even of
fairly high magnification, mere comparison of the compactness
of visi1)le grains on the two difïerent plates, one acted on by fight
and the other by individual a particles, would never lead to the
conclusion already stated.

8. The size of (/rains. The size of the silver grains has been
determined, as far as we know, only by microscopic methods,
visual, projecting or photographing. These methods, bowever, do
not seem to give very reliable results, unless care is taken with re-
gard to the diffraction phenomenon, because the grains to be mea-
sured are so small that their linear dimensions are of the same order
of magnitude as the wave-lengths of the visible light. In the fol-
lowing, we shall descril)e two methods of deducing the size of halide
grains from the data obtained in the investigation on the « i-ay
photographs. Although the methods are rather indirect, tlR\v may
still be of use in practical application.

IQ Art. 10. — S. Kinoshita and H. Ikeiiti :

(a). Having proved that each halicle grain is rendered capa-
ble ()f development whenever it is encountered by an a particle,
equation (3) which can be written in the form :

cls = c (sq—s) dn, (3')

may be interpreted as: the rate at which the number of acted or
developable halide grains increases with the falling a particles is
proportional to the number of halide grains not yet acted upon.
And the proportional factor c will be the probability of one halide
grain being struck by an a particle, and consequently, the ratio of
the area of the projection of each grain to a plane perpendicular to
the direction of motion of the « particles to a unit area.

Therefore, if Vq is the average radius of the halide grains, sup-
posing these to l)e spherical,
Tiro

c =

1

from which

H^y

(4)

Thus, from the values of c already given, we obtain:

ro='Gl /^ for the Instantaneous Plate, and
ro="5-t fi for the Ordinär}^ Plate.

If each grain of silver bromide of radius ^o be completely re-
duced to metallic silver, its mass m and radius r would be

4 , 108 4 / c \l 108 ,..

8 188 3 \ 7T I " 188 ^

where />„ and •> are the densities of silver bromide and metallic sil-
ver respectively.

As a verification of the result, the mass M of silver contained
in a unit area of a plate deduced from the mass m calculated by
equation (5) and the number s of the silver grains per unit area,
viz.

.r 4 / c \§ , 108 .„x

On the Photographic Action of a, ß and y Rays emittod from Radioactive Substances. W

may i)e cunipared with- that deduced from its photometric density
D, viz.

M=kD, (8)

where k is the photometric constant, being for green light equal to
rOo.lO* gr. per sq. cm. of the plate.

Giving tlio following values experimentally found:
.s = l-21). lOMor an Instantaneous Plate, for which i) = '412, and
s= 1 -06. l(y for an Ordinary Plate, for which 7>=-210, we get
iH=4-52. 10"' gr. per sq. cm. by (7) and

M=4'24A()'^ gr. per sq. cm. by (8) for the Instantaneous Plate,
and

i¥=2"55. ]()"■' gr. per sq. cm. by (7) and
-^i" = 2'16. 10'* gr. per sq. cm. by (8) for the Ordinary Plate.
Bearing in mind the fact that the conversion of silver
bromide to metallic silver is not carried out to completion, the
agreement betw^een the values obtained l)y the two methods is
seen to be quite satisfactory.

The average radius of silver grains calculated by equation
(6) corresponds, for the reason just stated, to the superior limit.
It is

r=*43 j" for the Instantaneous Plate, and
r='38 fJ- for the Ordinary Plate.
The size of the silver grains was actually measured under a
microscope, by means of an ocular micrometer for a number of
grains. It was found that the above calculated values of r are
within the range over which the size of the observed grains varies.
(1)). Ikeuti showed that the average radius /'o of halide
grains can als(j be deduced from the average number Si of silver
grains per unit length along an a ray track, when the thickness
of the emulsion film and the mass of silver halide contained per
unit area of it are known from other determinations.

Rememljering that every halide grain becomes subsequently
developable whenever struck by an a particle, it can easily be
>^eeYi that silver grains presenting themselves as an « ray track
on a developed plate must have their centres within a circular

X2 Art. 10. — S. Kincshita and H. Ikeuti :

C3^1inder drawn round the path of the a particle with radius equal
to that of halide grains. Consequently, .Si, which is the reciprocal
of the average distance d between successive grains along the
track, will be the number of the halide grains having their centres

Q

within the cylinder of unit length; thus — V will l)e the total

number of the halide grains initiall}^ present in the emulsion film
23er unit volume.

If t is the thickness of the film, the total numl)ar ,So of the
halide grains in a unit area of the plate will be
sd
Tin
Another relation l)etween ,^o and )\ can be obtained from a
consideration on tlie amount J/o of silver bromide contained in a
unit area of the plate,

■M"o=— -- ro ,Oo Su»
o

where /r>o is, as before, the density of silver bromide.

From the above two equations, we obtain

3 lU

From a set of measurements made on an Ilford Process Plate,
e.g.

Sj=3,500 per cm.,

-^0=9*6. lO^^gr. per sq. cm., which was determined by chemical
analysis, and

^=15 J«, which was measured by Zeiss' s Dickenmesser,
the average radius of the halide grain is found to be

By using the relation in equation (0), tlie average radius of
the silver grains becomes

V 188 /> / '

A suitable photometer ])eing at present not at our disposal,
we are unable to compare this result witli that deduced from the
photometric density. It may only be noted that the above

On the Photographic Action of a, ß and j Kays emitted from Radioactive Substances. 13

calculated value of r is also within tlie range over wliich tlie size
of the grains actually measured by means of the microscope varies.

9. Deflexions of a 'particles, on the passage tJirough the eynulsion
film. Evidences have already been given by the previously cited
investigators that some of the a particles suffer sudden deflexions
on the passage through the emulsion fihn.

In dealing with the deflexions of a rays, the importance of
utilizing a single source may be emphasized. If a set of radial a
ray tracks are obtained with an active needle in the way already
described, the possibility is excluded that two tracks, running in
different directions, happen to fall at a common point and present
themselves as if they were a single track suffering a sudden
deflexion.

In the microphotograph in flg. 10, which is enlarged 1,500
diameters, the tracks of four « particles, running from left to right,
are visible. The source of the particles lies outside the figure to
the left about 15 centimetres on this scale from the left end. We
can see that while the second « particle from the top passed
straight on, the other three suffered sudden deflexions of 10° to
15° downwards after traversing some distances nearly parallel to
one another.

We have examined hundreds of sets of the radial a ray tracks,
but so far we have not been able to find any which can be said
with certainty to have suffered the deflexion of an angle so large as
90°. The smallness of the proportion of the largely deflected tracks
to the total will not be inconsistent with the experimental results
arrived at by Geiger^-" and by Geiger and Marsden''^^ and also with
the theory worked out by Rutherford'^ It must be borne in
mind that we are, in the present case, observing only such
deflexions, which have taken place within a very thin layer
bounded by planes parallel to the surface of the fllm, whereas the
deflexions occur, in general, equally in all planes which contain
the initial line of motion.

1) H. Geiger, Proc. Roy. Soc. A, 81 (1908), p. 174-, and 83 (1910), p. 492.

2) H. Geiger and E. Marsden, Proc. Roy. Soc. A, 82 (1909), p. 495.

3) E. Rutheiford, Phil. Mag. 21 (1911), p. ÜÜ9.

14 Art. 10. -S. Kinoshita and H. Ikeuti :

When a photographie plate was plaeed in contact with a flat
piece of glass coated witli the active deposit of radium and thus
exposed to the a rays coming out of the source of large area, a con-
siderable proportion of then* tracks seemed to have sufïered large
deflexions. This is not in conformity with tlie result just stated.
It appears likely that inost, if not all, of the tracks which look as
if they were deflected are only apparently so. This view is sup-
ported by the fact that there were as many tracks showing large
deflexions as small ones.

Experiments made with tlie object of flnding if the magnetic
field has any influence upon the a ray tracks gave negative results.
In a field of ten thousand gauss, up to whicli the experiment was
extended, an « particle with a velocit}^ of 10" cm. per sec. (one-half
of the initial velocity of the a particles from I'adium C) will describe
a path for which the radius of curvatui-e is as great as 20 centi-
metres. It would not be possible to recognize such a slight curva-
ture, as the track under examination is only 52 // at most in
length.

10. The 'photographic action of ß particles. It has long been
known that ß rays possess the property of acting on a photographic
plate; but owing to difficulties involved in the experiments, very
little is known about the efïect of an individual ß particle.

When the a and ß rays from the active deposit of radium were
allowed to fall separately upon two areas on a thinly coated plate,
the photometric density produced by the ß rays was, for an equal
number of the particles, found to be one-sixth to one-eighth of that
produced by the a ra^^s. Therefore in going a unit distance through
the emulsion film, a ß particle brings out at most the above frac-
tion only of the silver grains which an « particle would, the actual
path of the ß particle being, on account of deflexions, greater than
the tliickness of the film. It is thus to be anticipated that the
silver grains acted on by a ß particle follow one another with too
wide intervals to present themselves as the track of the particle,
much more so when the liability of the particle in suffering deflex-
ions through matter is considered. The difference shown by the
tracks in air of a and ß particles in the photograph obtained by

On the Photographic Action of a, ß and y Rays emitted from Eadioactive Substances. ]^5

C. T. R. Wilson in liis well-known condensation experiment may,
though not quite , similar, serve as an analogy. While an a
ray track would still 1;)e detectable, if the number of water drops
per unit length were reduced to one-thousandth, say, in a ß ray
track this would l)e far from being the case. It must be remem-
Ijered that /9 particles will suffer greater deflexions in gelatine films
than in air. A direct evidence for the above conclusion is found
in the fact that it is possible to obtain a halo due to the a particles
from radium C, outside of which no perceptible track of any sort
is present, although the source emits simultaneously ß particles
as well and the most of them travel far beyond the boundary of
the halo.

The photographic action of [i particles should l)e explained by
an hypothesis of the kind we have made in the case of « particles;
for, in the latter the l)asis of the hypothesis is the ionisation of
halide molecules, which is the efïect also common to /? particles.
We shall consider that a halide grain becomes capal)le of develop-
ment when it is encountered by a (i particle, but only when the,
encounter takes place under certain favourable conditions. This
seems plausible, because a ß particle, while it would encounter on
a path per unit length as many halide grains as an « particle does,
renders developable silver grains of only a small part compared
with those similarl}^ affected by an « particle. Since it is very
likely that, whenever a halide grain is encountered by a /? particle,
some halide molecules in it, however few in number, are ionised,
we may assume that the process of development in the grain cannot
start unless the number of initially ionised halide molecules exceeds
a certain value. An analogous phenomenon is met with in the case
of an electric discharge between two electrodes. The discharge is
facilitated and starts at a lower but definite potential when the inter-
posed gas is initially ionised beyond a certain degree. The above
assumption apparently contradicts what has been said in the case
of a particles; but it seems quite possible to an « particle, which in
gases shows an ionising action several thousand times stronger than
that of a ß particle, to ionise a number of halide molecules over the
threshold value, whenever it strikes a halide grain.

Iß Art. 10. — S. Kinoshita and H. Ikenti :

11. Considerations on the ^photographie action of y rays. The
general feature of tlie photographic action of y rays may Ije inferred
from the facts liitherto accumulated. As is well known, X rays
do not directly ionise the molecules of a gas which the rays
traverse, but liberate from them corpuscular radiation which is
responsible for übe ionisation'^ The velocity of the emitted cor-
puscles is independent of the nature of the emitter and depends
only upon the wave-length of the exciting X rays, the smaller the
wave-length the greater the velocity. The excitation of the
"corpuscular radiation by X rays is not limited to gases 1)ut is also
true of solids and liquids.

There is every reason to believe that this is the case with y
rays, which, as we know, differ from X rays only in wave-length.
It has already been shown that the corpuscular radiation set up by
X rays from a radium salt is nearly as swift as the primary ß rays
emitted by it^-*. Thus, the effect of exposing a photographic plate
to X or /' rays will be that corpuscular radiation is set up through-
out the exposed portion of the plate. Since individual ß particles
leave no detectable tracks of silver grains on a photographic plate,
it Avill be impossible to obtain a track of X or y rays on a photo-
graphic plate similar to that of X rays illustrated by the photo-
graphs obtained by C. T. E. Wilson in the condensation method.
It will be evident that the halide grains rendered capable of
development by a Hash of X or y rays are scattered very diffusely
throughout the volume of the emulsion film, but with no definite
arrangement.

In an experiment we have exposed a plate to a flash of X rays
through an extremely narrow slit between two thick brass plates,
to see if any particular effects were produced in- and outside the
region upon which the X rays fell. Microscopic examination of
the plate gave negative results, as was naturally to be expected.

1) C. G. Barkla and L. Simons, Phil. Mag. 23 (1912), p. 317; C. T. K. Wilson, Proc. Eoy.
Soc. A, 87 (1912), p. 277 ; C. G. Barkla and A. J. Philpot, Phil. Mag. 25 (1913), p. 832.

2) A. S. Eve, Phil. Mag. 8 (1904), p. 669 ; S. J. Allen, Phys. Eev. 23 11906), p. 65.

On the Photographic Action of a, ß and 7 Rays emitted from Eadioae-tive Substanoes. \~i

In conclusion, we wish to thank Professor Nagaoka for pla-
cing the resources of the Laboratory at our disposal and for his
continual interest ^luring the progress of this investigation.

Pli_ysical Laborator}^, University of Tokyo.

Explanation of llie Piales.

Fig. 1. The general feature of a plate coutaiuiug several spots, cacb of wbic-li consists

(^f a nnnil)er of radial a ray tracks. The isolated spot at the top and the

large one at the bottom of the figure are reproduced in figs. 5 and G

respectively in a higher maguiiication.
Figs. 2-4. Haloes due to radium C, at different stages of formation,
Figs. 5-8. Haloes (hie to radium A and C. In eacli halo two concentric circles are

to be seen. Tiie inner circle is due to radium A and the outei' to radium C.

In fig. 5 the inner circle is more conspicuous than the outer one, while m fig.

(> the reverse is the case.
Fig. 9. The tracks of a particles emitted from a nucleus of radium C, reaching to

varioiis distances.
Fig. 10. a ray tracks showing sudden bents. On tliis figure are visible the tracks of

four a particles which passed from left to right. Three of tliem suffered

sudden deHxions.

Published November 20Lh, 1917.

Jour. Sei. Coll., Vol. XXXVII., Art. 10, Plate I.

>-^.-.T~ - >'ZäfS^ '^

-»"^ • 'Jf' •■ ^",••0. • - .

Fig. 1.

X120.

Fig. 2.

X500.

^*?ä^2^fc^>«-^:^--

^>^ ■-• ■ V '•/■:• ^>:vv. -c'y- ••-. . ^\\« >l::W^^-\-:^:> •

::T^^m

Pig. 3.

X500.

Pig. 4.

X500.

S. Kinoshita and H. Ikeuti : The Photographic Action of the a, ß and y Rays
ejnitted from Eadioactive Substances.

Jour. Sa. Coll., Voi. XXXi/l/., Art. JO, Plate II.

' ' • >-^* -^^

f^vK:'-

Fig. 6.

X500.

'■:■< > v-^

BBS;* '<*■ • I • ' ., . • *.-' ■ . - 1. ■' ' 0'"\>'-^ ' -.'

• .^ -./•'*<

.T,'^

tt'

?V^i

. • . . .V...:. './.r K*» /*« .'i v

Fig. «.

X600.

'• ' ■• '' •.^i»'''-"'K'W .••.. •

Fie.

X500.

Fig. 8.

X 500.

S. Kinoshita and H. Ikeuti: The Photographie Action of tho a. ß and y I^ays
emitted from Radioactive Substances.

Jour. sa. Coll., Vol. XXKVII., Art. 10, Plate III.

' ^- . ■ < ::. . • - * î* . ■'.'■if .'/"■ :*▼*.,' .< .

-.■■^'

Fig. 9.

X500.

». ••

Fig. 10.

X 1,5U0.

S ,. JKiCosV.lï aiid H. Ikeuti : The Photographie Action of the %, ß and y Eajü
emitted from Radioactive Substances.

Il

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