MONOGRAPHS ON PHYSICS

EDITED BY

Sir J. J. THOMSON, O.M., F.R.S.
AND
FRANK HORTON, D.Sc., M.A.

OF THE CAVENDISH LABORATORY, CAMBRIDGE

MONOGRAPHS ON PHYSICS

EDITED BY

Sir J. J. THOMSON, O.M., F.R.S.
AND
FRANK HORTON, D.Sc., M.A.
of the Cavendish Laboratory, Cambridge

8vo.

RAYS OF POSITIVE ELECTRICITY AND THEIR AP-
PLICATION TO CHEMICAL ANALYSIS. By Sir J. J.
THomson, O.M., F.R.S. 5s. net. [In the press.

MODERN SEISMOLOGY. By G. W. Watxer, A.R.C.Sc.,
M.A., F.R.S., formerly Fellow of Trinity College, Cambridge.
With Plates and Diagrams. 5s. net.

PHOTO-ELECTRICITY, THE LIBERATION OF ELECT-
RONS BY LIGHT; with Chapters on Fluorescence and
Phosphorescence, and Photo-Chemical Actions and Photo-
graphy. By H. STantey ALLEN, M.A., D.Sc., Senior
Lecturer in Physics at University of London, King’s College.
7s. 6d. net.

THE SPECTROSCOPY OF THE EXTREME ULTRA-
VIOLET. By Professor THEopoRE Lyman, Jefferson Phy-
sical Laboratory, Harvard University, Cambridge, Mass.

[In preparation.

COLLOIDAL SOLUTIONS. By Professor E. F. Burton,
The Physics Department, The University, Toronto.
[In preparation.

ATMOSPHERIC IONIZATION. By Professor J.C. McLENNaN,
i Department of Physics, The University, Toronto.
[In preparation.

THE EMISSION OF ELECTRICITY FROM HOT BODIES.
By O. W. RicHarpson, F.R.S., Wheatstone, Professor of

Physics, King’s College, London. [In preparation.
ELECTRIC WAVES. By Professor G. W. Pierce, Harvard
University, Cambridge, Mass. [In preparation.

LONGMANS, GREEN AN Di Ce:
39 PATERNOSTER ROW, LONDON
NEW YORK, BOMBAY AND CALCUTTA

_
—

-* MODERN SEISMOLOGY

(yd

a yer BY

ge ak yeaa
G. W. WALKER, A.R.C.Sc., M.A., F.RS.

FORMERLY FELLOW OF TRINITY COLLEGE, CAMBRIDGE

WITH PLATES AND DIAGRAMS

GE NI Ins tity ;
iS rales ey
JUL 7 41918
Ah Ve
Sy sival US

meen

BONG VANS, GREE Ni AND CO.
39 PATERNOSTER ROW, LONDON
NEW YORK, BOMBAY AND CALCUTTA

1913

PRE EA ORW NOM Ee

ONLY a week after this book had been handed to the pub-
lisher, the scientific world had to mourn the loss of Dr. John
Milne, who entered into Rest on 31 July, 1913.

It was my melancholy privilege on 5 August to pay a last
tribute to one who had proved a very kind friend.

The assembling of a large congregation in St. Thomas’s
Church, Newport, Isle of Wight, was an eloquent testimony to
the love and esteem with which Milne was regarded by those
among whom his daily life was spent.

No one will deny that Milne was truly the father of
modern Seismology. He founded the subject, he developed
it well-nigh single-handed, and he lived to see the importance
of his life work recognised not only by his fellow-countrymen
but by the whole civilized world.

The credit for several important points in modern Seis-
mology is sometimes assigned to others, and it was only
Milne’s greatness of heart that prevented him from claiming
the priority that was rightly his. ;

But his claim to scientific fame rests not on details, for he
made the whole subject. As Prince Galitzin remarked at
Cambridge only a year ago, “There are not many questions
of modern Seismology that have not been attacked by Milne
long before any other person had thought about them”.

G. W. W.

CONTENTS.

SEISMOMETRY.
CHAP. PAGE
I. General Dynamical Theory of Seismographs . ; 2 . I
II. Methods of attaining Sensitiveness, Damping, Registration . 7

III. Chief Types of Seismographs in Actual Use. Milne, Omori,
Bosch, Wiechert, Galitzin . : é : : : PS a6

IV. Standardization of Seismographs, including Theory of
Mechanical and Electromagnetic Registration . 5 fi e2n

V. Installation of Seismographs and Consideration of Develop-

ment of Instruments ‘ 4 : . 5 5 5 BO
SEISMOGEOPHYSICS.
VI. Theory of a Solid Isotropic Earth . : : : : 37

VII. Interpretation of Seismograms and the Interior of the Earth . 48

VIII. Determination of Epicentre and Focus . : s : O38
IX. Seismic Effects other than those due to Earthquakes see.
X. Statistical . “ : é ‘ , : f ; OS

vi

PLATE
I.

iS)

bo

7A.

7B.

10.

If.

12.

B3:

Sai Ok arly AGS

Milne Horizontal Seismograph for one Hori-
zontal Component . : : : . facing page

. Wiechert Astatic Inverted Pendulum Seismo-

graph for two Horizontal Components

. Galitzin Aperiodic Horizontal Component Seis-

mograph, with Galvanometer and Record-
ing Cylinder for electromagnetic registra-
tion .

. Galitzin Aperiodic Vertical Component Seismo-

graph for electromagnetic registration . i, a
. Time Curves (after Z6ppritz) . t : : ase
. Showing Maximum Depth of Seismic Rays as

Function of Epicentral Distance (after
Zoppritz) . s : ; é : : 55

Portion of Record at Pulkowa, September 18,
Ig10, showing microseismic movement . facing page

Specimen Record (reduced) showing Tilt pro-
duced by Tidal loading . j : ; 5) %

. Earthquake in Yap, West Caroline Islands.

Milne Seismogram, Eskdalemuir, August
16-17, 1911. N-S Component i . following page

. Earthquake in Yap, West Caroline Islands.

Milne Seismogram, Eskdalemuir, August
16-17, 1911. E-W Component ; : % 6

Specimen Record — Wiechert Seismograph
(1200 kg.) with ink registration . : 5 65

Specimen Record—Galitzin Horizontal Com-
ponent Seismograph

Specimen Record—Galitzin Horizontal Com-

ponent Seismograph In the pocket.

Specimen Record—Galitzin Vertical Com-
ponent Seismograph

Vii

59

73

73

83

88

88

INTRODUCTION.

THE present volume owes its existence to Sir J. J. Thomson’s
suggestion that I should write an account of the present
position of seismological investigation.

The book is written from the point of view of Seismology
as a branch of Physics, and particularly as it is determined by
observatory conditions. My qualification to deal with the
subject in this aspect rests on what is probably the unique
experience of having set up at Eskdalemuir and having had
under daily personal observation a Milne twin-boom seismo-
graph, a Wiechert 80 kgm, two horizontal components astatic
inverted pendulum seismograph, a complete Galitzin installa-
tion of seismographs with galvanometric registration for three
components, and an Omori seismograph for one horizontal
component. Simultaneous records of the magnetographs and
autographic meteorological instruments were also available for
comparative study.

This limited treatment of the subject is determined as
much by conditions of available space, as by my ignorance of
the geological side and of the practical application of earth-
quake study to building construction which is of so much
importance to those who live under the daily danger of the
“earthquake”. But the limitation is no disadvantage since
we already have Dr. Milne’s works on “Earthquakes” and
“Seismology” (International Science Series), which deal with
the subject in its wider aspects and with the authority of
Milne’s unrivalled personal experience.

The history of Seismology has been traced back to the

Viil

INTRODUCTION ix

earliest times. It would be interesting to know whether the
ancients possessed any accurate knowledge of the subject, as
they did in the case of Astronomy.

The literature of the subject is very extensive, but until
comparatively recent years it contains much that is speculative,
much that is inaccurate, and much that is false, as is ever the
case with a science until it becomes quantitative. One recalls
Lord Kelvin’s first criterion of knowledge of a phenomenon,
viz., ““ How much of it is there?”

If the newest literature is not entirely free from speculation
and inaccuracy, the study now proceeds on a quantitative
basis which enables the reader to judge for himself as to the
value of the conclusions.

The new Seismology as a quantitative physical science
may be said to have started about thirty years ago, and with
a small band of British scientists in Japan. These pioneers
were Milne, Gray, Ewing, Perry and Knott.

Germany and Italy may also claim pioneers about the
same time.

The horizontal pendulum adopted by Milne appears to
have been independently invented in slightly different forms
by different investigators, and it is difficult to assign priority.
The horizontal pendulum in the forms used by Zollner and
Rebeur-Paschwitz deserve special mention.

The experimental discovery that an earthquake could be
registered by a delicate pendulum at great distances from the
focus marks the first step in the new science. It is undoubtedly
to Milne’s vigorous personality that we owe the application of
this fact to the study of earthquakes. On his return to this
country in 1895 he set up his observatory at Shide in the
Isle of Wight, and by the installation of his instruments at a
number of stations distributed all over the earth, he inaugu-
rated the first Seismological Service. Comparative data were
thus obtained, and rapidly led to an increased knowledge of
the properties of the earth.

x INTRODUCTION

That the seismogram of a distant earthquake represents
elastic waves that have travelled through the earth from focus
to station was early recognised, but the well-known result that
a solid body transmits longitudinal and transversal waves with
different speeds did not at first seem to throw much light on
the meaning of the seismogram, which by its apparent com-
plexity suggested a highly heterogeneous earth.

It was, however, an important thing for seismological
theory when Lord Rayleigh (see “ Collected Papers”) showed in
1885 that there could be propagated along the surface of an
elastic solid a set of waves travelling with speed rather less
than that for transversal waves. Such waves play an important
part in the long wave phase of a seismogram which develops
some time after the first indication of a disturbance. Milne
applied this in 1895 to show that the interval of time between
the apparent “start” and the occurrence of the long waves
on the record provided a means of estimating the distance of
the epicentre. Although the estimate is not very accurate, it
was really the first step in the interpretation of seismograms
and in the location of the epicentre from observations made at
distant points.

About this time we owe to Rudski (‘‘ Physik der Erde’’)
and to von Kévesligethy (‘‘Seismonomia”) theoretical investi-
gations of the path of seismic rays within the earth. The first
application of the well-known theory of longitudinal and trans-
versal waves to Milne seismograms appears to have been made
by Oldham (“ Phil. Trans. R.S.,” 1900). Milne seismograms,
however, partly because of the presence of instrumental vibra-
tion and partly because of the comparatively slow speed of
registration, do not readily lend themselves to an accurate a
priovi estimate of the occurrence of the second or transverse
phase. Thus no great progress to accuracy seems to have
been made until the interpretation of seismograms was taken
up by Wiechert and his pupils at Gottingen.

Wiechert’s investigations began about 1900, when, at the

INTRODUCTION xi

request of the German Government, he made a tour to the
Italian observatories, and as a result of his studies designed
and set up the inverted astatic pendulum now known by his
name. By the introduction of fairly large damping, of in-
creased magnification, and of increased speed of registration,
the accuracy was greatly increased, and the division of a seis-
mogram into three principal phases corresponding to the longi-
tudinal, transversal, and long waves became a comparatively
simple and definite process.

In 1903 Wiechert published a most important memoir on
the “Theory of Automatic Seismographs” (“Abhand. der
K@6nig. Gesell. der Wiss. Gottingen”) showing, among other
things, the quantitative relation of the recorded movement of
the instrument to that impressed on the pier. This memoir
was followed in 1907 by a paper on “ Earthquake Waves” by
Wiechert and Zoppritz (‘‘ Gott. Nachrichten,’ 1907). On the
experimental side greatly improved time curves giving the
time of arrival of the longitudinal and transversal waves as
functions of the epicentral distance are obtained. ‘The results
have been used by Zeissig in the preparation of his interpo-
lated tables giving the epicentral distance for the time interval
between the arrival of the two sets of waves now known as
PandS. These tables (published by the Imper. Academy of
Sciences, St. Petersburg) are now in general use and are the
most accurate we have. On the theoretical side Wiechert adds
greatly to the interpretation of seismograms, and shows how
the time curves lead to a knowledge of the physical properties
of the interior of the earth. Wiechert and his pupils are still
actively engaged in the extension of our knowledge in this
direction.

Galitzin’s investigations began about the same time as
those of Wiechert and have proceeded on somewhat different
lines. It may be said that the problem he set himself was to
make instrumental seismometry a truly quantitative art as

xii INTRODUCTION

measured by the standard of modern experimental physics in
the laboratory.

He was led to adopt electromagnetic damping up to the
limit of aperiodicity and to introduce electromagnetic regis-
tration to get increased magnification. Each point of con-
struction or of theory was submitted to the most rigorous
tests in the physical laboratory until success was attained, and
the observatory of Pulkowa started continuous recording and
publication of observations on I January, 1912.

His separate memoirs have appeared in the “C.R. of the
Imperial Academy of Sciences, St. Petersburg,” and the results
are embodied in his book published last year (“‘ Lectures on
Seismometry”). The whole investigation is a most instruc-
tive and masterly application of physical principles to obser-
vational seismometry.

Perhaps the most striking result attained by Galitzin is a
complete experimental proof that his instruments determine
not only the distance of the epicentre, but also the azimuth
from the observing station, so that it is now possible from
observations at a single station to determine the epicentre
within the limits that must be assigned to the epicentral
region itself.

Seismographs reveal the existence of earth movements
other than those due to earthquakes. Chief of these are the
movements known as microseisms and earth-tides. Seis-
mology is thus brought into intimate connexion with Astron-
omy and Geodesy.

It may truly be claimed that during the close of the
nineteenth and the beginning of the twentieth century seis-
mologists, among whom the names of Milne, Wiechert and
Galitzin stand pre-eminent, have succeeded in dragging the
study of earthquakes from the region of ignorance and super-
stition and in making it a quantitative science proceeding on
the principles of physical philosophy.

P. 3, Equation B, read “ — x#/1”’.

RP. 43, line 17, for ““tané — — cos 2e” vead ““tane = — cot2e
eA nline 235 ufOrvacotee ead: Mey.

P. 47, line 2, read “‘ increasing to 7/2 for = 7”’.

P. 57, line 8, for ‘“‘«” read ‘‘e”’.

P. 65, line 31, for ‘‘ August ”’ read ‘‘ September ’’.

ae)

ERRATA.

?

. 69, line 22, after ‘400 km.” introduce *‘ it”.

?

CHAPTER I.
GENERAL DYNAMICAL THEORY OF SEISMOGRAPHS.

THE most general movement of the ground in the vicinity
of a point on the earth’s surface may be regarded as made up
of the components of a linear displacement resolved along
three mutually perpendicular axes and the components of a
rotation resolved about these three axes, It is convenient to
choose the geographical axes at the point, viz. North, East,
and Vertical.

In practical seismometry the horizontal components have
been mainly the subject of measurement, and it is but recently
that the vertical movement has been carefully studied. The
rotations are not at present recorded, although experiments
directed to that end are now in progress at Pulkowa. The
principal seismic waves recorded are, however, many kilo-
metres in wave length, while the amplitude at some distance
from even a devastating earthquake is but a fraction of a
millimetre, so that the twisting movement is practically small
except in the vicinity of the earthquake, where actual measure-
ments are for obvious reasons of a rough and hazardous nature.

Thus the objective of a seismological station being primarily
the recording of the earth movement experienced there, we have
to consider the instruments by means of which records are ob-
tained.

The instruments are called seismographs, and each seismo-
graph measures, or is supposed to measure, one component.
Thus six instruments are theoretically required to determine
the complete motion, but at present only a few stations are
fitted with three seismographs for the three linear components,
while most stations possess only two instruments for recording

the two horizontal components.
I

2 MODERN SEISMOLOGY

Any stable dynamical system which is set into relative
movement when its supporting platform is moved may be called
a seismometer, because it is purely a matter of dynamics to
determine the quantitative relation between the observed move-
ment of the instrument and the motion of the platform, which
is also that of the ground and is the object of investigation.

The simplest seismometer for horizontal
motion isa simple pendulum supported from a
point ona rigid framework fixed to the ground.
We may imagine that a pencil fixed to the
bob writes on a sheet of paper held horizontally
beneath it, so as to give a record of the move-
ment experienced by the bob when the earth
moves. This simple apparatus would register
on a small scale the horizontal components of
the earth movement ; and since the equation of

M motion is in form precisely that which applies
to any seismograph, we shall do well to ex-

amine it before proceeding to consideration of
BiGs i.

'
'
t
H
1
‘
'
‘
'
i
'
t
'
'
1
'
'
4
1
!

seismographs actually used.
Let 7= length of string,
g = acceleration of gravity,
y = displacement of bob horizontally,
#=prescribed horizontal motion of P the point of
support ;
then for small motions we have
I+(y-+#)gll=0
as the equation of motion. (Rayleigh “Theory of Sound,”
p. 63.)
Now the movement registered is not y but y-- since the
paper must also be supposed to have the motion ~.
Hence if €=y —xthe equation becomes

£4 Ee/l= — Z.
If 6 be the angular displacement of the string we have 6= &//
and hence

0+ 20/0= 8/0) 2) 1)
The distinction between the actual movement of the bob and
the movement recorded is important, as in all seismographs the

GENERAL DYNAMICAL THEORY OF SEISMOGRAPHS 3

registering apparatus, etc., must participate in the earth move-
ment x and thus it is the relative movement that is registered.

The equation may be obtained otherwise by superposing
on the whole system the reversed earth movement and then
taking moments about P now regarded as a fixed point.

We should get an equation of the same form as (A) for a
compound pendulum, Z being now the length of the equivalent
simple pendulum.

The simple pendulum has another feature in common with
all horizontal component seismographs, namely, that it records
not only linear horizontal displacement of the ground but also
rotation about a horizontal axis. Thus if y represents the
angular displacement of platform, etc., about. an axis through
P perpendicular to the plane of the paper measured positive in
the clock-wise direction the equation becomes

6+ Og/l= -#/l+We/lt

where @ is now the apparent angular displacement of the string.
We may incorporate W with 4%, if the latter is now regarded
as the horizontal linear acceleration that would be experienced
by a point coinciding with the nul position of M and rigidly
connected to the earth, while the axis of rotation of yy is moved
to M. As has already been stated the rotation is, in the case
ofan earthquake at some distance, so small that the seismograph
is usually regarded as measuring solely the linear motion.

All vibrating systems are subject to frictional forces and
we must now introduce the necessary modification of the
fundamental equation on this account. The assumption is
usually made that the frictional forces can be represented by a
term proportional to the angular velocity 6. The mathemati-
cal convenience of the assumption is enormous, and in some
cases the assumption is in sufficiently good agreement with
fact.

The equation then takes the form

64+204+220= £/l. i GBy
and this is the fundamental equation in instrumental seis-
mometry.

Wiechert has remarked (‘‘ Theory of Autographic Seismo-
graphs,” “ Abhand, Kon, Gesell. d. Wiss.,” Gottingen, 1903),

1 *

4 MODERN SEISMOLOGY

that all seismographs are fundamentally the same, and if the
frictional term could always be expressed as above no objec-
tion could be taken to the statement. The different behaviour
of instruments in actual practice is, however, mainly due to the
fact that the frictional term is not of this simple form in all
cases.

The equation (B) is of well-known form, and full treatment
may be found in any treatise on differential equations (e.g.
Forsyth).

The free motion is given by

6+2 6+7O=0
and the solution of this is of the form
d=A e-“ sin {(z?-&)3(¢-y)} for n>e
0=A e-@ sinh {(e? — 2”)3(¢- )} for ~<e
@=A e-* G7) for z =e
where A and » are arbitrary constants.

The last case is of special importance in modern seismo-
metry, and the instrument is then described as ‘‘dead-beat ”
or “aperiodic”.

In any case the quantities z and e are instrumental con-
stants which may be determined experimentally by methods
well recognized in ordinary laboratory practice. The quantity
«x is in general a function of time and the recorded movement
@ then consists of two parts: (1) depending on the special form
of x, and (2) depending on the free movement of the instrument
with constants depending on the initial conditions. The
complete solution when + is any prescribed function is given
by Rayleigh (“ Theory of Sound,” p. 74).

We shall consider only the case in which + is a simple
periodic function of the time, say + sin (/¢).

The forced movement 6 corresponding to this is

(= Py si aiaia) ; where tan 7 = 2 5s
Le PP +460 PP
The recorded movement thus differs in phase from the im-
pressed movement. As the actual recorded quantity will be
proportional to 6 say L@

the expression

GENERAL DYNAMICAL THEORY OF SEISMOGRAPAS 5

L 9 9 I\9 949. —i
7 PNG PY + 4ep

represents the “magnification” of the amplitude of the earth
movement.

Terms representing the free motion will also appear on the
record, and these have a diminishing amplitude. Now the
practical problem is to determine the earth movement +
from the recorded movement so that even this simple case
shows us how important it is that the “free” terms should be
made to subside with rapidity. A fortéord it is evident that if
« is undergoing complicated changes, it is difficult to form any
true conception of the earth movement from the seismogram
unless the “ free” terms are made to subside quickly. Thus
the necessity for a large value of ¢, that is very great damping,
becomes apparent.

The expression for the magnification may be written

EN)
where U ={(u? — 1)? + 4u%e?/n?\4 and u=2/f.
Thus U is unity when p= co that is for infinitely rapid vibra-
tions, while U is infinite when /=0. Thus the magnification
is mz¢ for infinitely slow vibrations.

U clearly becomes a minimum for different values of z

when
w=1- 26/2
and we may choose e/z so as to get the minimum for any
prescribed values of z.

If e/z=1/2* we get #=0 as the minimum, and this value
has the advantage of making the magnification for rapid waves
more nearly constant for different periods than would other-
wise be the case.

If the instrument is aperiodic e/z=1 and there is then a
minimum at «=o for U, which now takes the form (w? + 1).
In certain theoretical investigations it is convenient to use
quantities related to ” and e as follows :—

h=eln, p*=1-h* and y=(n? - €)* = un

so that 4=1 or uw? =0 expresses the condition for aperiodicity.

6 MODERN SEISMOLOGY

The simplest seismometer for vertical motion is a small
mass suspended by a light elastic string so that vibration may
take place vertically. .

It is unnecessary to prove here that the equation of motion
takes precisely the form we have already considered, so that
the expressions already obtained are equally applicable to
seismographs for measuring horizontal or vertical motion.

CHAPTER IL.

METHODS OF ATTAINING SENSITIVENESS, DAMPING,
REGISTRATION.

IN general the amplitude of the earth movement to be
measured at a seismological station is small, the convenient
unit being ;,4,,th mm. or micron. Thus a seismograph to be
of practical value must give a large magnification of the earth
movement.

The expression for the magnification

ys

M = TU
where U = {(#? — 1)? + 4e?u?/n?2 and u=n/p

shows that it depends on the indicating length L. It will be

convenient to consider L in connexion with registration and

for the present to treat it as prescribed.

For very rapid vibrations U equals 1 and thus we should
gain by making Z small. I do not think the question of the
best dimensions for a pendulum has received much attention,
but it deserves consideration.

For very slow waves we find that we gain by making 77/
small. Now z?= //and so “7=g and thus the dimensions
are without influence. Thus attention has been directed to
making what we may call the static sensitiveness great, by
reducing the effective control.

It is possible to reduce the influence of gravity in con-
trolling a compound pendulum. If the axis of rotation is
gradually altered from the horizontal to the vertical direction
the effective part of gravity in controlling rotation of the body
about that axis becomes less and less, until when the axis
is vertical the influence is zz/. This is the principle on which
horizontal pendulums are constructed. The name horizontal

7

8 MODERN SEISMOLOGY

pendulum is used because the length of the pendulum lies
nearly along a horizontal axis, not because the pendulum
measures the horizontal component of motion.

It is not possible to make the axis of rotation strictly
vertical as a practical limit is reached at which the instrument
becomes unstable. This depends on the limit of accuracy of
mechanical workmanship.

The arrangement is represented in the diagram (fig. 2) where
BV is the vertical, AB the axis of rotation making an angle
2 with BV, while CM is the rod and mass.

Fic. 2. Fig. 3.

The line AB may be actually a rigid rod pivoted at fixed
points A and B about which the rotation takes place. The
instruments of Hecker and Rebeur-Paschwitz are of this type.
But we can also make C a fixed pivot and support M bya
string or wire toa fixed point A, and AB is then an ideal line
about which the system turns. Types of these are the Milne,
Bosch, and Omori Seismographs. Pivots are liable to become
defective, and in any case introduce frictional effect that cannot
be adequately considered theoretically, so that flat steel springs
replace the pivot C in the pendulums of Mainka and Galitzin
pendulums for second order stations.

Another form of suspension that possesses great advantages

METHODS OF ATTAINING SENSITIVENESS, ETC. 9

is due to Zollner and is shown in the diagram (fig. 3). The
pendulum rod CD is supported by wires AC and BD both
under tension on account of the mass M, and clamped to fixed
points A and B so that AB is the axis of rotation.
This method is used by Galitzin in his aperiodic pendulums.
In all these cases the angle z is practically very small, and
clearly we may regard the system as a compound pendulum
controlled by “reduced gravity” of amount of gsinz or gz.
In this way periods of 20 seconds or more can readily be
attained implying a large increase of magnification.
If Mé? represents the moment of inertia of the system
about AB,
h = distance of the C.G, from AB
and @ in the apparent angular motion, we get the equation
M#6+ Mghi0 = —- Mhi
while if we introduce a frictional term proportional to @ we
may reduce the equation to the form
6 +266 +220 = -£/1
wherein 2*=ghz/k?, and /=k?/k is as before the length of
the equivalent simple pendulum. CM the axis of the pendulum
is very nearly horizontal, and we have to observe that the
pendulum will record not only horizontal motion of the
ground represented by x but also tilting represented by rota-
tion yy about a horizontal axis coinciding with CM and rotation
represented by y about a vertical axis.
It is easy to show that the complete equation is
6+ 2664+ 70= -z/l+gpll+x (2- Al
but in obtaining the equation it is important to remember pre-
cisely what the quantities are, viz. :—
@ is the apparent angular movement.
# is the acceleration that would have been experienced by
a point coinciding with the C.G. of the pendulum
but rigidly attached to the earth.
yy is the angular rotation about the horizontal axis through
the C.G. coinciding with the nul position of CM
and
x is the angular rotation about the vertical axis through
the C.G.

10 MODERN SEISMOLOGY

It is assumed that squares and products of z, 0, x, yy, and x
are neglected.
The signs are such that if OY coincides with the length of
the pendulum, then (fig. 4)
x is+along OX
@ is +round OZ from Y to X
x is— round OZ from Y to X
and > is+round OY from Z to X.

We have already remarked that except in the immediate
vicinity of an earthquake terms arising from Wy and y are so
small as to be negligible in comparison with those arising from
x. In that case we may then simply take + as the horizontal

Z A
0 Xx
Y. P
Fic. 4. Fic. 5.

movement of the ground in the vicinity of the seismograph.
As showing how small z is in actual instruments we may
calculate the value assuming a period of twenty seconds and a
length of equivalent simple pendulum 10 cm.

Amt 47°? 10

T? g 400x981
Of a—Omance

We have sin z=

The diagram (fig. 5) shows the principle adopted by Wiechert
in his seismograph for measuring horizontal movement. It is
known as the inverted pendulum.

The mass M is supported by a rigid rod from a fixed point
P about which it can rotate in the plane of the paper. Flat
Cardan springs are actually used so as to avoid friction. The

METHODS OF ATTAINING SENSITIVENESS, ETC. 11

arrangement would normally be unstable, but it is rendered
stable by means of a spring attached to a fixed point A as
shown acting horizontally through the C.G. of M.

If M&?=the moment of inertia about P
h=the height of the C.G. of M
p= the strength of the spring,

then the apparent angular motion @ of the pendulum is given
by

M226 + (uh? - Mgh)0 = - Mhz
which may be reduced as before to the standard form.

By a suitable choice of w sensitiveness can be obtained. In
practice the larger pendulums with a mass of 1000 kg. give
good results, but with the smaller form in which M is only
80 kg. it is difficult to get a period exceeding eight or nine
seconds, as instability occurs when longer periods are attempted.

We may note that this instrument also registers tilting, but
not rotation about a vertical axis.

Passing now to the measurement of the vertical motion of
the ground, the diagram (fig.
6) represents the principle
on which sensitiveness can
be obtained. The mass M
carried ona rigid rod PM is
capable of moving in a ver-
tical plane about the fixed
point P. Flat steel springs Fic. 6.
are actually used to give an axis of rotation through P. The
pendulum is supported by a spring C D attached to the pendu-
lum at D and to a fixed point at C.

If P D=dand yw is the strength of the spring, then if 2 is
the vertical earth movement and @ the apparent angular move-
ment of P M in the vertical plane we obtain the equation

MAO + u@0= — Me.
By a suitable choice of the quantities wa" can be made small
and sensitiveness thereby attained.

Introducing a frictional term the equation can be reduced
to the standard form.

12 MODERN SEISMOLOGY

We have to note that rotation about an axis perpendicular
to the plane of the paper would also be recorded, and that it is
most important that the line joining the C.G. and P should
be accurately horizontal, otherwise horizontal motion of the
ground in the plane of the paper would also contribute to the
observed motion.

The best known types are those of Wiechert and Galitzin.

Those familiar with the practical difficulties of mechanical
construction will understand that long periods combined with
stability are far more difficult to attain for vertical than for
horizontal motion. The best result obtained by Galitzin was
a period of thirteen seconds. While this is a remarkable
practical achievement, it is only about half what can be
obtained with horizontal seismographs.

We have found it desirable to retain a frictional term in
the equation of motion and we have now to consider this
matter more fully.

The Milne Seismograph is the best known throughout the
world, and in that instrument no artificial damping is intro-
duced. It is, however, subject to such friction as may exist
at the pivot and to the natural damping action of the air. As
the pendulum is comparatively light (only about 1 kg.) we
need hardly expect that with reasonable care the effect of the
pivot should be serious; and my own experience confirms
this. The natural air damping is comparatively small, but
conforms as nearly as one can measure on experimental decay
curves to the law of proportionality to the velocity. It may
be expected to vary somewhat with the temperature and
humidity of the air. The Omori Seismograph is also without
artificial damping, but as the mass is very great, trouble does
arise with the pivot in this case, and the trouble can only be
avoided by the use of Cardan springs.

We have observed that when the damping is very slight,
the record of an earthquake is largely influenced by instru-
mental vibration, making it difficult to determine the period
and absolute magnitude of the seismic waves, especially when
these happen to possess, as they often do, a period nearly that
of the pendulum, viz. about eighteen seconds. Thus for

MEDRHODS OF ATTAINING SENSITIVENESS, LLC, 13

instance the first phase of an earthquake on a Milne seismo-
gram indicates distinct periodicity, whereas on a Galitzin
seismogram the first phase appears extremely irregular. A
most interesting confirmation of this came under my notice at
Eskdalemuir where, owing to the action of some spiders’ threads,
the east component of the Milne Seismograph was rendered
nearly aperiodic while the north component remained periodic.
An earthquake of considerable magnitude occurred, and the
profound difference in the appearance of the component records
was exceedingly instructive.

For these reasons it has appeared desirable to most investi-
gators to introduce large artificial damping in the seismograph
so that the absolute measurement and analysis of seismograms
should be rendered easier. Inasmuch as increased damping
on a given pendulum would reduce its effective magnification,
we must obviously increase the sensitiveness to start with.

The Bosch and Wiechert pendulums are arranged with
artificial air damping. This is done by attaching to the
pendulum a multiplying arrangement with a piston at the
end, so that the piston moves inside a fixed cylinder. The
ends of the cylinder are connected by air passages with each
other and with the external air, so that the amount of resist-
ance offered to the piston can be varied within certain limits.
In this way a frictional term is introduced in the equation of
motion and it is possible to attain aperiodicity if so desired.

The results obtained by the use of artificial air damping
appear to indicate that it is only as an approximation that the
frictional term introduced is proportional to the velocity.

The most important advance in recent years has been made
by Galitzin, who successfully introduced electromagnetic damp-
ing. A horizontal copper plate attached to the Zollner pendu-
lum moves in the field produced bya pair of very strong
permanent horse-shoe magnets fixed above and below it. The
eddy currents induced in the plate when it moves retard the
motion, and here there can be no doubt that the retarding
force is proportional to the velocity.

Aperiodicity can readily be obtained. The magnets have
proved remarkably constant and it is only at intervals of

14 MODERN SEISMOLOGY

several months that they have to be moved a little closer, so
as to increase the field and maintain strict
aperiodicity.

We should expect that light pendulums would
be used in attempting to get aperiodicity, and it
is somewhat curious that the Bosch, Wiechert,
and Galitzin pendulums are actually heavier than
even the Milne pendulum. We shall consider
this in a later chapter.

Ne The values of the angular quantity @ are small

Fic. 7. and we have now to consider the manner in which
a permanent record of the changes of 9 are obtained on a
linear scale of sufficient magnitude. The indicator length L
thus determines the final scale of the record, ie. the sensitive-
ness, but I have preferred to keep this separate from the prob-
lem of the relation of 6 to the earth movement.

We may imagine a very light but rigid rod of length L at-
tached to the pendulum, so that the end of the rod gives the
linear quantity which is to be registered. This is the method
actually used by Milne, an aluminium rod of about I metre
length being attached to the pendulum mass.

We cannot, however, practically proceed to great length of
a straight rod, so that in some instruments that aim at higher
magnification a multiplying arrangement of light levers is in-
troduced. These involve the use of heavier pendulums, and
where pivots are used give serious trouble by introducing solid
friction and often lead to dislocation of the record owing to
loose joints. Galitzin’s arrangement of replacing pivots by
fine wire and spring control gets rid of this objection.

The indicating end may be made to write by means of a
style on smoked paper or by a pen with ink on smooth white
paper. Of the sharpness of the lines so obtained there is no
question and its cheapness is a great recommendation. It is
used in the Wiechert and Omori Seismographs. Unfortun-
ately solid friction is introduced by mechanical registration, the
fundamental equation of motion has to be modified, and recent
investigation has made it doubtful whether the matter can be
dealt with in a satisfactory way.

METHODS OF ATTAINING SENSITIVENESS, ETC. 15

But to Milne we owe the application of photography to the
problem of recording, without the introduction of any friction
or backlash of multiplying levers. The precise method used
by Milne will be described in the next chapter.

The newer method of photographic registration used in the
Bosch and Galitzin instruments depends on the principle that
a pencil of light from a strong source of illumination may be
reflected from a mirror attached to the pendulum and con-
centrated at a point on the surface of the sensitive paper. An
indicating point is thus obtained without introducing the
slightest friction, and so the simple mathematical form of the
fundamental equation is preserved.

The most recent method of multiplying the motion of
a pendulum before applying photographic registration we owe
to Galitzin. It occurred to him that if a coil of fine wire was
attached to the pendulum, so as to cut across the lines of a
strong permanent magnetic field when the pendulum moved,
electrical currents would be set up in the coil, strictly propor-
tional to the angular velocity of the pendulum. These cur-
rents could be carried by wires to a recording galvanometer,
so that the movement of the needle would register photo-
graphically on a large scale the motion of the pendulum and
hence of the earth movement.

The motion of the pendulum being given as before by

6+20+70= -£/l
then the equation for the angular motion ¢ of the galvanometer
needle can be written in the form

P+ 2ept+mih= —kO
where ¢,, 7,, and £ are certain instrumental constants. There is
thus linear relation between ¢ and x.

This is the principle of Galitzin’s electromagnetic registra-
tion method, where in practice both pendulum and galvano-
meter are made to have the same period and be aperiodic
within very narrow limits.

CEVA AE Rag iit:
THE CHIEF TYPES OF SEISMOGRAPHS IN ACTUAL USE.

THE Milne Seismograph is made by Mr. R. W. Munro, London,
and to him and to Dr. Milne I am indebted for permission to
use the photograph shown on Plate 1. The supporting frame
of the pendulum consists of a vertical iron pillar cast in one
piece with a triangular bed-plate supported on three levelling
screws, which rest by hole, slot, and plane on three glass studs
imbedded in the pier. The pendulum boom is a light rod of
aluminium nearly I metre long, and at the inner end it is fitted
with an agate cup which presses against a steel pivot point screwed
into the pillar. The boom is supported at a point a little
beyond the stationary mass by means of a fine steel wire ending
in unspun silk which passes to a pin at the top of the pillar.
The mass (about 1 kg.) itself is not rigidly attached to the
boom, but is balanced on a steel pivot. The object of this
appears to be to reduce the effective moment of inertia of the
pendulum. The adjustments provide for bringing the boom
into the horizontal position along a prescribed line, and so as
to have the desired period of say eighteen seconds. One of
the levelling screws, having a pitch O°5 mm., carries an arm
moving over a graduated arc, and provides the means of giving
a known tilt to the instrument, so that its static sensitiveness
may be determined. The boom is prevented from sagging at
its outer end by a silk cord as shown.

The registration is carried out as follows: The boom
carries at its end a small transverse plate of aluminium with a
narrow slit parallel to the boom. This moves over a fixed
slit at right angles to it in the top of the registration casing.
This arrangement is illuminated from above so that a small

dot of light corresponding to the intersection of the slits is cast
16

jusuodwog jejuozo0Fy 9uO Joy ydesSowsiag [eJUOZOF] sUTIY—I ALVIg

PLATE 2.—Wiechert Astatic Inverted Pendulum Seismograph for two
Horizontal Components

CHI ETAT VPES OR SELSMOGRAPHS IN ACTUAL USE 17

on the surface of the bromide paper wound on the surface of
the recording cylinder. The cylinder driven by a spring clock-
work rotates in about four hours, the paper speed being about
250 mm. per hour or nearly 4 mm. per minute. By means of
a deep helix cut in the spindle the cylinder is made to move
sideways as it revolves by about 6 mm. per revolution, so that
the paper is available to run for one day or two days as the
case may be. Every hour the light is cut off by an electric
shutter operated by a good pendulum clock, so that accurate
time marks are thus put directly on the record and eliminate
any irregularity in the driving spring clock, which cannot be
depended on to give sufficiently accurate time.

In the twin boom instrument the two pendulums are carried
at right angles by the vertical pillar, but the booms are brought
out parallel to each other so that the two horizontal com-
ponents are recorded on one sheet.

The instrument is subject only to such natural damping as
may occur, and this is very small.

The Omori Seismograph resembles the Milne instrument,
but is much larger, the stationary mass being about 100 kg.
A multiplying lever and registration on smoked paper is also
used.

The Bosch Seismograph is also similar to the Milne, but of
a somewhat heavier build. Itis fitted with artificial air damp-
ing and registers photographically by means of a mirror at-
tached to the pendulum.

The Wiechert 1000 kg. astatic pendulum is made by
Herr Bartels, Gottingen. To him and to Prof. Wiechert I am
indebted for permission to use the photograph, Plate 2. We
must refer also to the diagrammatic sketch in Chapter II.

The stationary mass is built up of iron plates and supported
by a strong iron pillar from the pier. The support consists
of a double set of Cardan springs so as to avoid friction and
allow the pillar and mass to rotate about two horizontal axes
at right angles to each other. A rigid framework carries the
registering apparatus, supports for the damping boxes, etc.
Stops are also provided to prevent any large motion of the

mass, which would be fatal to the instrument.
2

18 MODERN SEISMOLOGY

Two light arms engage with points on the top of the mass
so as to give the components of motion in the horizontal direc-
tion. These two arms are connected up to the writing points
by means of a system of similar multiplying levers. We must
refer to the “Phys. Zeit.,” p. 821, 1903, for full details as to
these. It must suffice here to say that connexions are pro-
vided by steel points working in agate cups while axes of rota-
tion of levers are provided by small Cardan springs which also
provide the small controlling force required to make the
pendulum stable. Air-damping boxes are also attached so as
to provide any required degree of damping. The registration
is made on smoked paper in a manner clearly indicated in the
photograph, the writing point being a glass style with a small
ball point.

The whole arrangement is of great delicacy and requires
very careful handling so as to avoid damage to any part.

The speed of registration is about 10 mm. per minute,
and automatic time marks are put on the record every minute
from a good pendulum clock which operates an electrical ar-
rangement for lightly raising the writing points and then
lowering them, so that a small break occurs on the trace.
The dimensions of the enclosing case are 186 cm. high, 138
cm. broad, and 176 cm. long. It is unfortunately true that
in this seismograph the two components are not independent.

The Galitzin Seismographs are made by Mr. Massing,
mechanic in the Imperial Academy of Sciences, St, Petersburg.
I am indebted to His Excellency Prince Galitzin for permis-
sion to use the photographs, Plates 3 and 4.

Plate 3 shows the horizontal component seismograph
with galvanometer and recording drum. The general dimen-
sions of the seismograph may be judged from the fact that the
casing is about I m. high and rather less than 1 m. diameter.

The base is a substantial iron casting supported on three
strong levelling screws. Bolted to this is a rigid iron frame-
work which carries the horse-shoe magnets and clamps for the
supporting wires.

The pendulum which is of Zollner type consists of a strong
brass rod to which is rigidly keyed the stationary mass of 7 kg.

UOHKISISaI MauUSvWIOIIDI]9 IO Japul[Ad Suripiosay puv sojowWoURATTH YA ‘ydeisowsisg yuauodwog jejuozu0fy stpowady ulzyeN—e aLlvig

CHIBI LE YPES OF SEISMOGRAPHS IN ACTUAL USE \.%9

The supporting wires are of steel and platinum-iridium and
pass one from the top of the frame to a point just behind
the mass, and the other from the inner end of the rod to the
foot of the frame.

At the outer end of the pendulum rod are seen the copper
plate and damping magnets, while just behind these are seen
the similar pair of magnets which excite the currents in the
flat coils attached to the pendulum. These coils are connected
to the stout leads of the galvanometer, the connexion being
made by fine bronze strips so as not to interfere with the free
movement of the pendulum. Both pairs of magnets are pro-
vided with screw adjustments so that the damping and magni-
fication may be controlled to the desired extent. Small
mirrors are attached to the pendulum and frame, and these
with a small electric hammer for giving the pendulum a slight
blow are required in the process of standardizing.

The galvanometer is of the Deprez-D’arsonval moving coil
type.

The registration of the movements of the galvanometer
mirror is made photographically. The galvanometer is set at
a convenient distance, say 1 m., from the recording cylinder,
and the mirror is illuminated by means of collimator and slit,
so that the reflected beam falls normally on the cylinder, while
the image is focussed for that distance. The image is concen-
trated to a small luminous point by the intervention of a
cylindrical lens. The cylinder has a circumference of nearly
I m. and revolves in about half an hour, so that the actual
paper speed is 30 mm. per minute. The cylinder also moves
sideways about I cm. per revolution so that the record runs
for twelve hours. In practice we may arrange for both hori-
zontal components, from duplicate pendulums set at right
angles, to be recorded side by side on the same sheet. This is
an economy of expense and possesses certain distinct advan-
tages. But against this must be set the fact that when a large
earthquake occurs the confusion of the record may become
very troublesome. Thus at Pulkowa the practice is to record

each component on a separate sheet, while two scales of regis-

tration are used.
2 *

20 MODERN SEISMOLOGY

Plate 4 shows the Galitzin vertical component seismograph
for electromagnetic registration. The general dimensions of
the casing are rather over I m. in length and less than I m.
in breadth and height. The pendulum rod is now replaced
by a framing to avoid bending. The axis of rotation is very
neatly arranged to avoid friction by using crossed Cardan
springs screwed to the fixed framing and to the pendulum fram-
ing.

The strong supporting spiral spring is fitted with a screw
to get rough adjustment, while final adjustment is made by a
small gravity weight shown to the left of the stationary mass.
Another adjustable mass shown above the axis of rotation is
provided to get the centre of gravity of the pendulum in the
same horizontal line with the axis of support.

The arrangement of horse-shoe magnets is similar to that
in the horizontal seismographs, but they have to be twice as
large, as the attainable period is about half that of the hori-
zontal instrument. Mirrors and electric hammer are provided
for standardization and the registration is made exactly as in
the case of the horizontal components.

Although these seismographs are far more sensitive than
either the Milne or Wiechert Seismograph, they are by no
means difficult to handle. The chief danger to avoid is making
any adjustment of the pendulum while the galvanometer is in
circuit with it.

Further details will be found in Prince Galitzin’s ‘‘ Lectures
on Seismometry,” published (in Russian) by the Imperial
Academy of Sciences, St. Petersburg. These lectures embody
the chief results of separate investigations published (in Ger-
man) in the Comptes Rendus, Imperial Academy of Sciences,
St. Petersburg, and the Permanent Seismological Commission,
St. Petersburg.

uOeIISIS9I M9useUuIONsI[a Joy ydersowsiasg Jusuodui0g [eonI9A dIpowedy uizwyen—b aLvig

CHAP DE RIGIN:

STANDARDIZATION OF SEISMOGRAPHS INCLUDING THEORY OF
MECHANICAL AND ELECTROMAGNETIC REGISTRATION.

IN order that the study of seismograms should contribute
in real measure to a knowledge of geophysics, it is essential
that the results obtained should be expressed in absolute
measure. When we remember that we have to compare
records obtained at different stations with instruments, it may
be, of the same or even of different types, the necessity for stand-
ardization becomes evident. Not only so, but since any indi-
vidual instrument undergoes secular change and _ requires
readjustment from time to time, it must be possible to deter-
mine the constants of the instrument zz sztu at suitable
intervals.

In the first instance it is, however, important that each
instrument should be tested in order to ascertain whether it
conforms to the fundamental equation supposed to represent
its motion. This can only be done on a properly equipped
experimental table by some central recognized authority,
which would then issue with the instrument a certificate giving
any data of importance.

We shall consider how the constants are to be obtained at
the station for the three well-known types, Milne, Wiechert,
and Galitzin.

The latest form of Milne’s twin-boom Seismograph readily
lends itself to the determination of the quantities e and ” on
the photographic sheet itself, a point of great practical con-
venience. If the end of the boom is given a suitable initial
displacement and then left to itself, excellent decay curves are
obtained. The diagram (fig. 8) is an exact reproduction of

an actual curve obtained in this way.
21

22 MODERN SEISMOLOGY

The paper speed is 4 mm. per minute, and for a period of
185 this implies 24 mm. on the paper for twenty vibrations.

Fic. 8.

There is thus no difficulty in measuring the apparent period
T’ to o°1. We may also determine with considerable accuracy
the ratio of successive amplitudes by measuring the ratio for
say ten vibrations. We may then compute z and e as
follows :—

Since
T’ = 27/(x? — e*)

and the ratio of successive amplitudes

eieiebila
we find that
ZANT ;
log, 2 p= F005N/T
where
r= log,, v
and
A Se NCU S 2 aa : 23
N= aR = ar {Lt / (ar log e)*} Tu +0°5372A7H.

From the curve (fig. 8) I found fifteen vibrations in 19 mm.
and the amplitude fell from 5 mm. to I mm, in ten vibrations,
so that we get IT =19'0 and v=1:084 and so T=19°o)
m= 0'331, and e=0'0085.

We have pointed out that Milne has always provided a
screw by which a known static tilt can be given to the instru-

STANDARDIZATION OF SEISMOGRAPHS 23

ment and the observed deflection on the paper noted, and with
his published data of amplitudes in millimetres he gives the
angular tilt required to produce 1 mm. deflection.

Now the fundamental equation

6+ 260+ 7120 = -£¢/l+ g/l

shows that for a steady tilt yw, we get

¢, ==
and since the deflection on the paper say y, is L@, where L is
the length of the boom we may calculate 7 by the formula

pers

wy,
As an actual example we have
i100 n= =0'349

and the experiment gave a deflection of 1 mm. for 043
tilt so that 7= 16:8 and L/7=6.

L and Z are of course constants that may be determined
once for all. Thus while we must admit that in a complicated
record it would be practically impossible on account of the
“free” terms to assign the true magnitude of the horizontal
earth movement in absolute measure, there are certain cases
(notably sharp impulses) in which the earth movement can be
determined from the record. This point has not always been
recognized with regard to the Milne Seismograph.

In the Wiechert Seismograph artificial air damping is
introduced. We shall first suppose that the friction introduced
enables us to write the equation in the form

M#?6 +26 4+ (uk? - Mgh)O= - Mht
or 6+ 266+ 270= — </L

When the expression 1/U which determines the magnifica-
tion is plotted for different values of «and of the damping
ratio v, it appears that the magnification remains more nearly
constant from ~=0 to z=1 when v is about 5 than for other
values of v, and this ratio is aimed at in practice. The corres-
ponding value of e/z is about 0°45. This comparatively large
damping ratio makes it difficult in practice to get a sufficient

24 MODERN SEISMOLOGY

number of vibrations on the paper when an artificial disturb-
ance is given, to determine the apparent period T’ exactly.
Hence the artificial damping is cut out so that ¢ is reduced
considerably to ¢’, and ¢ and z are then determined exactly as
we should do in the case of the Milne instrument. As the
paper speed is one minute= 10 mm, it is generally possible to
determine T’ to o”'1.

The damping is again introduced, and after getting v nearly
5, by trial, its exact value is obtained and z being now known
e is determined. The period T=27/z aimed at is from 10°
to 12° in the 1000 kg. instruments.

Since the instrument also measures tilt we might theoreti-
cally now get /, by giving a known static tilt. But it is not
easy practically to give a known tilt to the 80 kg. pendulum,
and in the case of 17,000 kg. pendulum it is out of the
question. Hence resort is had to another way (cf. ‘“‘ Phys. Zeit.,”
1903, l.c.). Ifa small known mass wz say 10 gm. is placed on
the large mass M at a distance ~ of say 10 cm. horizontally
from the centre of gravity, a small couple is produced which
gives an angular deflexion that may be conveniently measured
on the paper.

We have

(uh! — Mgh)@, = mgp

L6, =;
where y, is the deflexion on the paper and L is known from
the linear dimensions of the multiplying levers. Hence we
obtain (us? —- Mgh) and hence knowing z we get Z. As Z is
proportional to L we really do not require to know L in getting
the magnification. The value of L// is readily made several
hundred units.

Wiechert, however, recognized that the mechanical regis-
tration introduced frictional forces that are not properly
allowed for by a term proportional to 6. He assumes
(Theorie Auto. Seis.) that the solid friction was such that the
equation took the form

| J+ 2ey+n(y tr)=0
where the sign of 7 is such as to oppose the motion, reversing

and

STANDARDIZATION OF SEISMOGRAPHS 25

whenever y reverses. If now y,, #, v3 represent successive
amplitudes while vw is the ratio of successive amplitudes of the
periodic term, it is easy to see that

SS CLG,
Jott Jntr
and hence
pp SE +2 — 27
I2-I3 Vat I3t 27
while

1 = (Yo" — I,Vs) (V1 - Is)
which suffice to determine vy and e/z. In practice it is best
to obtain ~ when the damping is cut out, so that v being
nearly I we get approximately = }(y, — 75).

The writing point may remain at rest anywhere within a
range 27, and discontinuities of this magnitude may occur in
the trace.

It is also clear that the motion can never start unless the
impressed acceleration exceeds a certain amount, and this ex-
plains the fact that so many more earthquakes are recorded
on instruments that use photographic registration even with
smaller magnification.

But a more serious matter arises. Experiments of my own
on an 80 kg. Wiechert showed clearly that 7 was not a con-
stant, but depended on the state of the smoked surface and the
amplitude of the movement. This has been more recently
established by Galitzin (Vorles. ii. Seis.) whose elaborate ex-
periments show that a more complex and non-linear equation
corresponds better with the facts. This, however, rather sug-
gests that if cases arise where the solid friction is so great as to
seriously vitiate the records, we should do well frankly to
abandon mechanical registration. With reasonable care the
magnitude of 7 does not exceed a few tenth millimetres in the
Wiechert instruments.

The motion of the Galitzin horizontal pendulum is given
by the usual equation

6+ 2604+ 20 = -x/0.
In practice T =nearly 24° or = 0'2618, and when in use we
make e as nearly=x as possible. The length / is a definite

26 MODERN SEISMOLOGY

constant for each instrument and is about 120 mm. It may
be determined by observing the static sensitiveness when has
some known value.

Another method used by Galitzin depends on the principle
that 2*=¢2//, so that by changing z by known amounts and
determining the corresponding values of z we get data for
calculating Z.

Thus 7= g(2, - 2,)/(#,? — 19”).

The artificial damping is cut out by removing the magnets,
and observation of the apparent period and damping ratio
(now nearly unity) is made, so that the values of z can be com-
puted. The changes of z are determined by observation of a
mirror attached to the frame, by means of telescope and scale.

The values of e and z do not remain quite constant, so
that it becomes important to determine them at any time zz
salu.

The differences of ¢ and z from their theoretical values are
small, and are determined by observations on the recording
galvanometer, the theory of which we have yet to consider.

The galvanometer is of the moving coil type in which the
suspended system is controlled by the torsion of a fine wire.
When a current I exists in the coil a couple strictly propor-
tional to I arises, due to the strong magnetic field in which
the coil turns. If we neglect the self-induction of the circuit
the current |= E/R where E is the electromotive force in the
circuit and R the resistance. When connected to the coil in
the pendulum E consists of two parts: (1) due to the pendulum
motion and proportional to 6, and (2) due to the motion of the
galvanometer coil and proportional to de

Thus we see that the equation of motion of the galvano-
meter coil is given by

Ue 3g
where e, =¢, +¢/R.

When on open circuit (R = 0 ) the free motion of the coil is
given by

b+ 20h + 2,2 =0
so that ¢, arises from a small damping action to which the
system is naturally subject.

STANDARDIZATION OF SETSMOGRAPHAS 7

The quantities c, and ~, are determined for the galvano-
meter in the usual way. The period is about 24‘ while ¢, turns
out to be a very small quantity of order 0:0005.

When the circuit is closed we see that e, increases as R
diminishes and thus the condition of true aperiodicity «=,
can be secured by suitable choice of R. The value is deter-
mined by experiment thus: Different values of R are intro-
duced and the corresponding values of e, determined. In this
way cis found and is a quantity of order 6 units. We may
then calculate the value of R required to make ¢, = ,, and find
it to be about 25 ohms, Thegalvanometer resistance is about
4 ohms and thus the remainder of the circuit must be made up
to the required value. When this is done the galvanometer
is assumed to be aperiodic (€,=%,) and to remain so as the
quantities involved are not subject to changes that have any
appreciable effect.

Having granted the desirability of great damping, the
passage to the limit of aperiodicity seems obvious as it simpli-
fies the relation between the quantities. Thus the ideal is to
have the pendulum and galvanometer truly aperiodic and to
have the same primary period in the absence of damping, i.e.
€=¢€=%=%n,. Assuming then that ¢,=, we proceed by trial
to make e=”=A,, or yw? =O, and it is easy to get quite near it.
The adjustment would in fact not be considered good if p? rose
to O'I or if the primary periods differed by more than a few
tenths of a second. The pendulum does, however, undergo
small secular changes, and we have now to explain how Galitzin
determines how far x differs from %,, and yp? from 0, and also the
value of & the transference factor of order about forty units.

To simplify matters, suppose the ideal condition secured
and that a small impulse is given to the pendulum. Then

0 = 12,60 ,e~
so that @,, is the maximum value of 6.
The corresponding motion of the galvanometer needle is
given by
p= —knje 0, e~ @ ~ ee)
24258
so that ¢= p=0 when ¢=0,

28 MODERN SETSMOLOGY

We note that ¢ is again 0 when ¢= 4, = 3/m,.
Again @¢ is a maximum when
n,7t? - 6n,t+6=0
or 2,t=3 + ./3.
Thus the first maximum is

RG, pa 3 2
ob, = (2,/3- 3) e~2+ N3 when ¢=(3 - J 3)[%

1
@ then passes through o when ¢= 3/z, and attains a maximum
on the other side

Ons 1S Ez =
ry rg (CVG Se eth (Si 9/3)

and then ¢ gradually diminishes too when ¢= 0.
Hence
k=, cat EES = oun sue NEN :

On) 313) On (2/3 + 3)
Now if e and x differ a little from ,, the motion of ¢ will differ
from the above, but without altering the essential feature that
¢ attains maximum throws on opposite sides. The complete
equations can be written down and observation of @,,, $,, da
and Z, then provide material for calculating f, yu’, and (7 - ,).
The necessary formule and numerical tables have been ob-
tained by Galitzin (l.c. azte). It must suffice here to point
out that for all practical purposes the following are quite
accurate enough, viz. :—

SSeS)
p= 2°94 (2204 z = i)

k= 2817 2,$4/Om = 6°46 m(1 -— 0°34 M?)po/Om
when & and ,»? do not exceed o'1. It is of interest that the
effect of uw”? is much greater in changing @,/0,, than it is in
changing ¢,/6,.

Having determined z,, with the galvanometer on open
circuit, the procedure is to give the pendulum a small impulse
with a small electrically controlled hammer, and then to observe
by aid of telescopes and scales the quantities 6,,, ¢,, ¢:, while
z, is determined by a chronograph. Two observers are required

STANDARDIZATION OF SEISMOGRAPHS 29

and considerable skill is necessary. About ten experiments
are made so as to improve the accuracy.

The standardization of the vertical component seismo-
graph proceeds in precisely the same manner as regards &, py’,
and & In this case the primary period is about 13’ while &
is of order 240 units. The quantity / of order 400 mm. is
determined before the instrument is set up by removing the
controlling spring, turning the instrument so that the pendulum
may hang vertically, and observing the period of vibration.
When in use a small correction may be made on account of
the position of the small adjustable gravity weight.

In the Galitzin Seismographs the indicating length is 2 A
where A is the distance from the galvanometer to the record-
ing drum, usually chosen to be about 100 cm. Thus if the
theoretical adjustment has been .secured the magnification for
periodic waves is

Pele S87

ml (1+)
The expression is a maximum for #=1/3!=0'577 and the
value of w/(1 +z”) is then 0°325. Thus the magnification is
nil for very rapid vibrations, rises to a maximum, and then
falls again to o for very slow vibrations. As an example, if
i245 sees) \— 100 em, 2— 12. cm. £—do the maximum
magnification would be 828 when T= 14 sec.

where «= 2/p=T/T,

CHAPTER: WV:

INSTALLATION OF SEISMOGRAPHS AND CONSIDERATION OF
DEVELOPMENT OF INSTRUMENTS.

THE site of a seismological station is probably determined in
most cases by considerations of policy and finance which do
not concern us here. But we may consider some conditions
that appear desirable from a scientific point of view.

Seismographs are sensitive instruments and thus liable to
be disturbed by artificial causes such as street traffic, so that the
instruments ought to be installed at some distance froma town
or railway line. But such local effects do not penetrate to a
great distance, so that it is only a question of being a kilometre
or so distant from such a source of disturbance. We remember
that most of the European stations from which such important
results have been obtained are at no great distance from busy
centres of industry. The dominant features of a seismogram
of a large tectonic earthquake are not determined by local con-
ditions of the ground, but the smaller details of the seismograms
may be modified greatly by the geological formation of the
rocks in the vicinity of the station. Thus a site where the
formation is known to be of fairly uniform character for a con-
siderable area would be preferable to one where the rocks vary
rapidly. A level plain also recommends itself, while a sharp
ridge or sudden depression are to be avoided. If we remember,
however, that the most frequent wave lengths experienced are
from 30 to 70 km. long, the presence of an isolated obstacle in
the form of a hill is probably not a very serious matter.

There are few recording instruments of any kind that are
not prejudicially affected by change of temperature, and thus
uniformity of temperature in the room containing the seis-
mographs is a highly desirable condition even from a general

30

INSTALLATION OF SEISMOGRAPHS 31

point of view, and becomes of ‘vital importance if the move-
ments of the pendulum zero are to be examined for diurnal
tilting of the ground due to earth tides. An underground
chamber may be used to conduce to such uniformity of tem-
perature, and it has been found at Pulkowa that the disturbing
effects of local wind are considerably reduced in an under-
ground room.

The pendulums have to be carried on substantial piers
which take up the earth movement. Here it is important to
avoid the danger of making a pier which itself becomes a
pendulum and so complicates the recorded motion. The pier
ought therefore to be broad rather than high. Concrete 1
metre square imbedded in clay to a depth of I metre gives
satisfactory results. In the case of a complete Galitzin in-
stallation the pier is enlarged so that all three pendulums may
be carried on it.

The co-ordinates of the station latitude, longitude, and
height above sea-level must be known or determined in some
suitable way, and it then becomes important tc determine the
position of the geographical meridian. It is convenient to
record the N.—S. and E.—W. components directly, and it is im-
portant to make the adjustment correct from time to time or
to determine how far the pendulums have deviated from the
true positions. We also require to know the initial direction
of motion of the recording point, when the earth moves ina
given direction. Thus it is convenient to arrange that a move-
ment up the sheet corresponds to an earth movement to north
or east, and it is useful to remember here that if a sudden
movement of the earth occurs, say to north, the initial motion
of the centre of gravity of the pendulum will appear to be
towards the south. As confusion on this point has occasion-
ally arisen by the notion of forces applied to the pendulum, it
is well to recall that we are concerned not with forces, but with
a prescribed motion given to the point of support in which the
recording part of the apparatus participates, so that it is only
the relative movement of the pendulum that is observed. It
may be that the prescribed initial motion is complicated and
not instantaneous, so that the record is then mixed up with

32 MODERN SETSMOLOGY

instrumental terms. The initial kinematical result remains true,
but whether we succeed in detecting the true apparent initial
movement of the pendulum on the record is another matter.

Accurate time marks must be put on the records auto-
matically, and the station thus requires a good clock and a
knowledge of standard Greenwich Mean Time, so that the
occurrence of events at different stations may be compared
with an accuracy of one second. Absolute time is less im-
portant than the consideration that all stations should have
the same time. The use of the wereless time signal promises
the best solution of this problem.

A word with regard to photographic registration may not
be out of place. If sharp traces are to be obtained only the
highest quality of optical work is permissible. Mirrors must
be optically plane and palladianized on the /ron¢ surface, lenses
must be properly corrected for spherical and chromatic aberra-
tion, and the use of thick plates of glass through which the
light has to pass at a high angle must be avoided.

The number of seismographs that have at various times
been devised is very large, but only a few of these have sur-
vived to practical use at the present time. We cannot attempt
to discuss these obsolete forms, although a study of them will
well repay anyone interested in the improvement of practical
seismometry (for references see Milne, “‘ Earthquakes ”).

We have already remarked that the rotations are not yet
recorded although instruments for measuring tilting have
been proposed. The bifilar pendulum of Darwin and Davison
(“B. A. Reports,” 1881) and the klinograph of Schliiter (“ Gott.
Dissert.,” 1900) have not come into use, as they record other
things besides tilting. Galitzin (Vorlesungen) has recently pro-
posed to record tilting by the combination of two similar
horizontal aperiodic pendulums at different heights working
in opposition on a single galvanometer.

Thus if 6, represents the motion of the lower pendulum we
have

6, + 260, +220, = —(4-gh)/l
while if 6, represents the motion of the precisely similar pen-
dulum at a height s
6, + 266, +220, = -(% -—gh+sp)/d

INSTALLATION OF SEITSMOGRAPHS 33

hence 6, — 0, + 2€(0, — 6,) + 2(6, - 0,)= spl

so that the differential motion is quite independent of x and
depends only on y. Experiments at Pulkowa on an experi-
mental table give very satisfactory results, but it remains to be
seen how this arrangement does for continuous recording.

I am not aware that any experiments have been made with
a view to recording rotation about the vertical, but the use of
electromagnetic registration appears to offer a way of record-
ing this on a large enough scale. Suppose that we have a
heavy rod suspended by a vertical wire which passes through
its centre of gravity so that the rod rests horizontally. If now
the rod carries similar flat coils at its ends moving in strong
magnetic fields, the coils being coupled through a galvano-
meter so as to assist each other when the rod rotates, the equa-
tion will then take the form

6+20+”0= -— x
and the motion will be independent of x and w.

We must remember that, however well we may be able to
adjust apparatus to measure artificial rotations in the labora-
tory, for practical continuous recording of earth tilting we
have to make sure that with slight secular changes of the con-
stants the apparatus does not develop a tendency to record a
part of the comparatively large values of %.

With regard to the recording of the linear displacements
there still seems to be ample scope for the improvement of
existing forms of apparatus. The ideal to aim at is the pre-
cise reproduction of the earth movement on a suitable scale of
magnification. No instrument does this although some come
nearer it than others. Thus consideration of the dimensions
of a seismograph appears to me to merit more attention than
has already been given. The following table shows the diver-
sity of magnitudes in existing forms :—

Stationary Primary Equivalent Damping
Seismograph. Mass. Period. length. Modulus.
M Ti l efit
Milne. c c I kg. 18 sec. 16 cm. 0°026
Wiechert . : 1000 kg. 12 sec. about 100 cm. 0°45
Galitzin : ; 7 kg. 24 Sec. I2 cm. I‘00

34 MODERN SEISMOLOGY

The statement sometimes made that the Omori Seismo-
graph has a natural period of sixty seconds, I take to refer to
the apparent period when friction of the recording system’ is
introduced. Personally I found the Omori became unstable
with a natural period above 16° and I am convinced that there
is no piece of physical apparatus of reasonable dimensions that
could have a natural period of sixty seconds. Instability due
to mechanical imperfection sets in long before this.

The differences in M are very great and Wiechert even used
17,000 kg. in his celebrated instrument at Gottingen. If
we go back to the physical equation of motion of a pendulum,
we see that except for any controlling spring action or solid
friction there is no point in using a large mass. On the con-
trary, if we admit the desirability of high damping we have
everything to gain by using a small mass, as we then make e
larger without altering z and 7 There will be a practical
limit to M depending on very small spring action that might
ordinarily be neglected, but I do not think the practical limit
of small mass has yet been reached.

We have already commented on the advantage of high
damping in removing the effect of instrumental vibration, and
we must now consider more fully the question of aperiodicity.
The mathematical advantage of ideal aperiodicity is to some
extent discounted by the fact that it cannot be precisely main-
tained, and its practical advantage of freeing the record from
instrumental periodicity might be also secured in an over-
damped pendulum. But aperiodicity alone does not solve the
problem of deducing the earth movement. If an aperiodic
pendulum is given a sudden displacement it would still show
I per cent of its initial displacement when z¢= 6°64 or if

277
= a 24 Sec. = )2.5See

Now ¢z can only be reduced by increasing e and we could
then only retain aperiodicity by shortening the primary
period. Further, if galvanometric registration is used, the
record corresponding to a sudden displacement of the ground
presents the appearance of a single wave.

Thus for certain types of movement such as occur in the

INSTALLATION OF SEISMOGRAPHS 35

earlier phases of a seismogram, instrumental terms have a pro-
nounced influence in any case, but the interpretation of the
record is greatly facilitated if we can depend on the rapid
decay of the “free terms ”.

Under the influence of periodic waves the magnification is
given by

L pose pare 2 42h -

F Pat py tga py?
and in the case of an aperiodic pendulum this becomes

7; LBV a bo

We on Dy Vanes
Thus the magnification is dependent on the period of the im-
pressed vibrations, and we can extend the range over which
approximate uniformity is obtained only by an increase of the
primary period T,. But this means that if we use heavy
damping we must be prepared to sacrifice true aperiodicity.

For rapid vibrations the magnification is L/? and for slow
vibrations the magnification is

Py Lp?

We can thus increase the magnification for rapid vibrations
by reducing 7 and that for long waves by reducing z, and this
might be done without any great change in # or 27/T, from
the values at present attainable, say T,=twenty seconds.
Now / may be reduced by reducing the dimensions of the
pendulum, and if z is correspondingly reduced T, would not be
altered. The reduction of dimensions would not greatly alter
e, but the reduction of the mass would increase e considerably.

The point I wish to put is this, that we have much to gain
and little to lose by a substantial reduction in mass and
length of the pendulums as at present used. To be definite it
appears to be practically possible by the use of a fine quartz.
Zollner suspension to make a pendulum in which 7 is of the
order 1 cm., M of the order 1 gram, which is highly damped (e of
order say 1), and which could be placed inside a vessel the
size of an ordinary tumbler. With optical registration at a
distance of 2 metres the magnification for rapid vibrations

3 *

36 MODERN SEISMOLOGY

would be about 240, or forty times that of the present Milne
Seismograph.

Although no seismograph at present gives an exact re-
production of the earth movement in general, we ought not to
regard the attainment of this as impracticable. The relation
expressed by

Yr2ey+My=-rNE
is not the most general that may obtain between the impressed
co-ordinate x and the recorded co-ordinate y. The general
form is
YPr2Zy+wmy= -rA (+ 264 47,74)
and if it should prove possible to get a practical arrangement
in which x=, and e=e, the latter being great, we should
then come very near to a precise reproduction of uniform
magnification.

CHAPTER VI.
THEORY OF A SOLID ISOTROPIC EARTH.

A COMPARISON of seismograms obtained at different stations
suggests at once that we are concerned with mechanical effects
propagated from the region in which the earthquake oc-
curred.

We are thus led to inquire what is the nature of the effects
propagated and to form a working theory as to the physical
properties of the Earth, which will enable us to co-ordinate the
observations.

At the present time the evidence in favour of a solid Earth
is very great, but the alternative view that the interior of the
Earth is fluid retarded for a considerable time the progress of
seismological theory, which requires the Earth to possess the
properties of an elastic solid.

As astronomical theory agrees with seismological in de-
manding a solid earth we accept this as a primary condition.

The simplest assumption we can make is that the physical
properties of the Earth are uniform throughout, and although
we shall find that seismology requires a modification of this
assumption, yet many important features of a seismogram
become intelligible on the basis of this simple hypothesis, and
quantitatively the differences are not so great but that we may
regard a uniform isotropic Earth as giving a good first ap-
proximation to the co-ordination of results. Accordingly it is
instructive to begin by a consideration of the effects to be
expected on this view, as it prepares us to make a first inter-
pretation of a seismogram and to see on what lines the modi-
fication has to proceed.

The fundamental equations of motion of a uniform isotropic
solid are so fully dealt with in treatises on elasticity (e.g. Love’s

37

38 ; MODERN SETSMOLOGY

“Theory of Elasticity ”) that the results are quoted here without
proof.

If the independent variables are x, y, z, the Cartesian co-
ordinates of a point, and ¢ the time, and the dependent
variables are uz, v, w, the components of displacement of a
particle at x, y, z, then the equations are

0? OO
(p UF —- WV); Uv; WwW) = C222 wy x) a

ou ov ow
Py wy - nya
= the density
and ) and yp are Ue defining the elastic properties of the
medium.
If +0 we get

where 6=

(? oa O=(A+ 2m) VE

while if 9=o0 we have

2
(p = uv?) (u, Vv, W)=O

Ou ee Ow

7 yt oe
We thus find that the motion can be analysed into two types:
(1) the longitudinal type @ + Oo in which the velocity of propaga-
tion is V;=( + 2p)/p? and the displacement is in the direc-
tion of propagation, and (2) the transversal type @=o0 in which
the velocity of propagation is V,= ?/p? and the displacement
is at right angles to the direction of propagation.

The components of stress at any point are in the usual

notation

with

CRANEIZ) Sn Agrton (=, =) Gm,
ae
amon (eis
sateen CO)

Although the effects of an earthquake observed at a distant
station may persist even for several hours, we have cumulative

DABLORM OFA SOLID DSODROPIE\ LA RTL, 39

evidence that the primary disturbance at the focus consists of
a concentrated shock or limited series of shocks occurring
within a very short time, a matter of some seconds. In any
case we are certainly not concerned with unlimited trains of
waves proceeding from the focus, so that our discussion must
now proceed in the light of Stokes’ ‘‘ Dynamical Theory of
Diffraction ” (Collected Papers) in which he considers the effect
of an arbitrary initial disturbance produced in the vicinity of
a point.

In an unlimited medium the disturbance spreads in spheri-
cal shells from the origin. If the primary disturbance is of
short duration, the effects observed at a point distant 7 will be
first a short disturbance at the time taken for the longitudinal
waves of velocity V, to reach the point, then a period of
quiescence followed by a second short disturbance when the
transversal waves of velocity V.(<V,) reach the point, after
which the motion at 7 ceases. The relative magnitudes of
these effects depends not only on the distance 7, but also on
how the primary disturbance can be analysed into the two
types, and in particular one or other may vanish. We have
also to note that the effects are not the same at all points at
distance 7, but depend on the axis or axes of the constituents
of the initial disturbance.

When we pass to the case of the Earth, we shall suppose
in accordance with observation that the origin of disturbance
is situated at a point comparatively near the surface of the
earth. We may still expect that the seismogram obtained at
a point on the earth’s surface will, in general, be characterized
by a pronounced movement corresponding to the arrival of the
longitudinal disturbance, and by a pronounced movement when
the transversal disturbance arrives, both of which have travelled
by the brachistochronic path (in this case a straight line) from
the focus to the station. These are accordingly indentified
with the beginning of the first phase P and the second phase
S of a seismogram.

From observations made at comparatively small distances

(<1000 km.) from the focus, Zoppritz and Geiger find that
V,=7'17 km. per second, and V,=4:'0o1 km. per second, and

40 MODERN SEISMOLOGY

these are adopted as the surface values. For greater distances
we have to abandon the supposition that the velocities are con-
stant throughout the earth, but this point we postpone to a
later chapter and meanwhile retain the hypothesis of uni-
formity.

The boundary of the earth introduces many new features
in the seismogram to be observed at a station, over and above
those which we have mentioned and which will be referred to
briefly as P and S.

Following Huygens’ principle, each point of the spherical
disturbances (V, and V,) spreading out from the focus will, as
it reaches the earth’s surface, become a centre from which
spread two spherical disturbances (V,
and V,), so that we have on the seis-
mogram a whole series of diffraction
effects in addition to P and S.

Let E represent the earthquake
focus supposed to be near the surface,
O the station, and C the centre of

(i the earth. Further, let the earth’s
Fi. 9. radius be R and the angular distance
EO be @. Then the arc

EO=4=R@
and chord EO = 2R sin 6/2.

The first effect at O is the beginning of the longitudinal phase
P at a time

Z= 2h sin g

1) WV a
Now consider the disturbance which travels as a longitudinal
disturbance V, by the path EA and then as a diffracted
longitudinal disturbance V, by the path AO. It reaches the
station at a time

t= 7 (sin as sin 2) = - sin ; cos(=")

These disturbances start immediately after P and arrive at
later and later instants for 6, > or < @, until they culminate in
the brachistochronic path of maximum time, which is also that
of regular reflexion, when 6, = 6,.

THEORY OF A SOLID ISOTROPIC EARTH 4i

We may thus expect a pronounced effect at a time
4R

ae sin ie
It isa longitudinal effect and may be identified with Wiechert’s
first reflected effect PR, In practice it is often more pro-
nounced than P in the case of distant earthquakes, and is then
of considerable value in determining the position of the earth-
quake region. The argument may be extended to further
subdivisions of the arc EO.
The second or transverse phase begins with S at a time
z mek sin —
DONT 2
Next consider the longitudinal effect which travels by EA
with velocity V, and is diffracted as a transversal effect along
AO with velocity V,. The time of arrival is
rs 2R sin 6,/2 i 2R sin 0,/2
V; V2
For different positions of A these effects, which begin im-
mediately after P, arrive at later and later instants and culmin-
ate in the brachistochronic path of maximum time which is
that of regular reflexion determined by
Vi
Vz
But here an interesting point arises. Since 6,+0,=0 we
get

cos ,/2 =—cos 6,/2.

Gi (Ni ON:
tan a= ‘a — cos = isin 6/2

and thus we cannot get a real positive value of 9, unless cos 0/2
is << V,/V,. This implies that if @ is less than the value given
by cos 6/2 = V,/V, the diffracted effects continue up to S with-
out any pronounced movement, but if @ exceeds this critical
value the diffracted effects may be expected to culminate in a
maximum transversal effect at a time

_2R sin 0/2
ZUIN A cos0,/2
which is later than the arrival of S. This point is of real

practical importance. With the values of V, and V, as given,

42 MODERN SEISMOLOGY

we get 9=I110° nearly or 4=about 12,000 km. Now it has
been observed that special difficulty attaches to the identifica-
tion of S just when J is about 12,000 km. Thus with an
earthquake in the northern Philippines which are about 11,000
km. from this country S usually comes out very clearly, while
in the case of an earthquake in the Caroline Islands about
12,000 km. from us S is most indistinct and the tendency is to
put it rather late. The result we have obtained throws some
light on the matter.

We may have a disturbance which starts as transversal
with velocity V, along EA and then proceeds as longitudinal
with velocity V, along AO. Here again we cannot expect
any pronounced effect unless @ is greater than the value given
bypcos!0/2\— VAIN.

Lastly we have the disturbances that traverse the whole
path with velocity V,. These start after S and culminate ina
maximum when

t= - sin g
and this we may identify with Wiechert’s SR,.

We must of course add to the cases indicated, the disturb-
ances that travel to the station by the opposite side of the
earth. They may be considered by the method already used,
and we shall point out only the PR, which reaches the station
by the longer path. It arrives when

i= 4 cos 6/4
and will thus be later than S unless
A Vie
sin 0/4 > Th
The critical value is @=about 140° or 4=about 15,500 km.
Thus we have here another critical value tending to indistinct-
ness of the second phase S.
There appears to be no reason why we should not also
have diffracted effects in which @, is negative.
Let us now consider the problem of regular reflexion when
a disturbance of either type is incident at a point on the earth’s

surface.

THEORY OF A SOLID ISOTROPIC EARTH 43

First—Plane longitudinal disturbance incident. The dis-
placement &, 7, ¢, due to the incident disturbance may be
written

Z

X
Q

Fic. Io.

: rcosé+zsine
(&, 1 &)= - A (cos ¢, 0, sin e) flee)
1
This gives rise to a reflected longitudinal disturbance ex-
pressed by
4 cosé- 2 sin ‘)

(&,, 12) f) = > As (cos, (AO) sin e)ifi (1+ V
1

and a reflected transversal disturbance
xcose —2Z sin )

(Ge) =A (sine) @.«cos:e))iif7 (2+ v
9

As the surface must be free from traction we have
Z,=2,=2Z,=0.
This leads to the following relations
A-A,=p A, cos 2é/sin 2e

I ° , ,
A+A,=— Ag sin 2 e/cos 2 €
12

where yw cos & =cos e and p= V/V,

Thus the apparent direction of motion of the ground is given
by ¢/&=tan e= — cos 2¢
Hence cos emai ( 5
This relation is of considerable practical importance.

We have to note that for e=o the resulting motion of the
ground is mz/ whatever A may be. It is thus impossible to
have a longitudinal disturbance in which the direction of dis-
placement is parallel to the surface propagated along the
surface.

44 / MODERN SEISMOLOGY

Second—Incident transversal disturbance, displacement per-
pendicular to the plane of the paper.

In this case it is found that no longitudinal disturbance
arises and that the incident effect is reflected as a transversal
effect without change. The motion of the ground is entirely
horizontal and equals twice that of the incident disturbance.
In this case it is possible to have a tranversal disturbance pro-
pagated parallel to the surface, the displacement being at right
angles to the direction of propagation and parallel to the sur-
face.

Third—\ncident transversal disturbance, displacement in
the plane of the paper.

We assume as the incident disturbance

xcosé+ésin j)

(&,, 7, G,) = A(- sin e, 0, cos e) f (e+ V;

which gives rise to the reflected transversal effect

x cOS €é-2 sin ‘)

(Eo, ta» &) = Aa(sin ¢, 0, cos ¢) f (e422
2
and the reflected longitudinal effect

(Es, 73) G) = Az( - cos é’, 0, sin é) f(t+* cos = Z sin ay
1
Application of the surface condition gives
A-A,= — pA, cos 2 e/sin 2 e

I , ,
A+A,= - ms sin 2 é/cos 2 e

where yp cos e=cos e’ and p= V,/V,
the apparent direction of motion of the ground is given by

ee Al as
¢/E=tan €=-— sin ée’ cos e/cos 2 e.
Mb

This holds as long as cos e+ 1/, but when e is less than the
value given by cos e= I/y, é’ is imaginary, and complex values
have to be assumed for A, and A;. The result is a reflexion
of transversal disturbance with a change of phase while there
exists a type of longitudinal disturbance in which the amplitude
diminishes rapidly away from the surface, but which cannot
in any true sense be regarded as propagated as there is no real
wave front.

It is important to note that at the critical angle the vertical

THEORY OF A SOLID ISOTROPIC EARTH 45

motion of the ground is zero, and only horizontal motion in
the plane of incidence remains. On the other hand when
é= 45° the horizontal motion is zero and only vertical motion
remains. Further when e=o the motion of the ground is zero
whatever A may be, and thus transversal disturbance in which
the displacement is perpendicular to the ground cannot be
propagated parallel to the surface.

Our discussion, which has proceeded on elementary lines,
is particularly useful in showing how difficulty arises in de-
tecting S at considerable distances owing to interference with
other maxima, or it may be the actual vanishing of the hori-
zontal movement. It has indeed sometimes been asserted that
S never reaches beyond a certain distance, and to explain this
an impenetrable core of the earth has been assumed. We see
that no such hypothesis is at all necessary to explain the ob-
servations.

A complete discussion which shall take account of the
magnitude of the diffracted effects as well as of their time of
arrival even for a simple type of initial disturbance would, I
believe, be a valuable contribution to seismological theory, and
in particular I should hope that it would throw some light on
the origin of a class of waves we have still to consider.

We have observed that it is impossible to. propagate along
a plane boundary either longitudinal waves with displacement
parallel to the surface or transversal waves with displacement
perpendicular to the surface. But by combining two types in
which the direction of the wave front is expressed by imaginary
angles, Lord Rayleigh (Collected Papers) has shown that the
surface conditions may be satisfied and that a system of waves,
in which the amplitude diminishes exponentially from the sur-
face, appears to advance parallel to the surface.

If Poisson’s ratio for the material = 1/4 or V,?= 3V,” which
is very nearly the case for the earth’s surface, Lord Rayleigh
finds that we can have a system of waves in which the dis-
placement & parallel to the surface and in the direction of
apparent propagation is given by

E=(€"~— 5773 27 *) sin (7472)

and the vertical motion is

46 MODERN SEISMOLOGY

£=(8475 e~* - 114679 e-*) cos (pt + fx)

where y= 8475f s= 39337
and DIf=V ='9194 Vz,
Thus at the surface

&,= 4227 sin (f¢+ fx)

f= — 6204 cos (pt + fxr)
so that the vertical motion is 1°47 times the horizontal motion
and the apparent velocity of propagation is less than the
velocity V,. Similar waves are possible at any plane boundary
of two media.

It has been sought to identify these waves with the long
waves that make their appearance in a sesimogram after the
second phase S. We shall postpone the discussion to the next
chapter, but meanwhile it is important to observe that we
must not regard Rayleigh waves as propagated in the same
sense as the longitudinal and transversal types in the medium.
We do not know the conditions that determine a surface
separating an undisturbed portion of the medium from a por-
tion influenced by these waves, and since the equations require
the whole medium to be in motion it is difficult to specify the
manner in which they can be originated.

We have so far regarded the focus as being situated at a
‘ point on the earth’s surface. But the focus of a large tectonic
earthquake is probably situated at
some depth of the order of to km.
Indeed it would seem to be the
case that if the focus is very near
the surface the effects are stifled
within a very short distance, and
that it requires a fair depth of the
focus in order that the earth may
be given, so to speak, a good shake
which will be experienced at re-

Ee mote points.
Pesan A most important influence of
a finite depth of focus is the manner in which it modifies the
so-called angle of emergence eat the station. If =the depth
of the focus we get |

BAEORVIOR Ay SOLID ISORKROPIC HART L, 47

R-z%

R sin AFC

COS @é=

so that instead of e starting at o for 9? =o and increasing / to 77/2
for || Or, e begins at 7/2 for @=0, reaches a minimum when
AFC =77/2, and then increases to 7/2 for =7. The minimum
is given by

—h

R i AN}.
cos €=cos 9=5— or sin o/2=(4 4)

This point is important in attempting to determine % from
observations. As an example if Z=10km., R=6370 km., we
set 0= 3°12’, 4=356 km., and the corresponding apparent
angle of emergence @=22°. But for 4=1000 km. the error
in e made by supposing F to coincide with E would only be
about 4°.

CHAPTER VII.

INTERPRETATION OF SEISMOGRAMS AND THE INTERIOR
OF THE EARTH.

Ir may be remarked of most seismograms that, on first
acquaintance, it is difficult to see the wood for the trees.
Only by experience and study is it possible to disentangle
those effects that are characteristic and essential from those
that are accidental. Not only so, but we must keep in view
that any seismogram is influenced by the particular instrument
from which it is obtained. A record from an undamped
instrument is for instance dominated throughout by instru-
mental periodicity. Heavily damped instruments, on the other
hand, agree wonderfully well in presenting the same general
features, and it is chiefly as regards relative magnitude of effect
in different parts of the seismogram that they differ. The
speed of registration also plays an important part, as move-
ments that are resolved with high speeds get crushed together
at lower speeds.

It is unfortunate that general statements with regard to
the character of seismic waves have obtained credence, which
are really dependent on one particular instrument. I am thus
diffident about giving a general description of a seismogram
that may convey a false impression, but as some description
must be given it may be well to state that I have in view
heavily damped seismographs in general, and in particular
Galitzin’s aperiodic pendulums with galvanometric registration
at a high speed. I would add that anyone who desires to
work at any of the theoretical problems awaiting solution will
do well to study actual seismograms for himself and not accept
descriptions made by other people.

We shall suppose that records of the horizontal motion
(X,Y) and of the vertical motion (Z) are available, and in

48

INTERPRETATION OF SEISMOGRAMS 49

practice it is desirable that the fundamental constants of all
three instruments should be precisely the same.

The first phase (undae primae) is initiated by P either as a
sharp impulse (impetus) or rapid succession of impulses, or by
a more gradual development (emersio). This lasts a few
seconds, and is interpreted as the arrival of longitudinal waves.
In many cases P is more pronounced in Z than in (XY). P
is succeeded by a series of smaller movements of a very ir-
regular character, with turning points sharply marked, at inter-
vals of a few seconds. There is in general a marked absence
of periodicity or motion of a sinusoidal nature. We do, however,
sometimes find minute movements with a period of about
I second, and I have seen an instance (earthquake in Yap) where
P started with a few waves of small amplitude of a distinctly
sinusoidal nature. Such cases are, however, rare.

During the first phase we have some outstanding sharp
movements. If these happen to be the P’s of subsequent
shocks they will be confirmed by the later part of the seismo-
gram. They may, however, be the reflected effects PR, etc.,
corresponding to P. I have already mentioned that with
earthquakes in the Philippines PR,, which arrives here about
four minutes after P, is usually much larger than P.

After the first phase, which lasts for a time depending on
the distance from focus to station, the seismogram changes
its type. There is as a rule a large movement denoted by S
which initiates the second phase (undae secundae). Its in-
cidence is less sharply marked than P and it is sometimes very
indistinct. It is clearer in (XY) than in Z. S is interpreted
as the arrival of the transversal waves. Following S the
movements are again very irregular. They are larger than
those occurring between P and S, and occur at longer inter-
vals. The turning-points are rounded, and occasionally give
a suggestion of sinusoidal movement. During this phase we
may have outstanding movements which may be the S’s of
subsequent shocks or the reflexions of P and S. For dis-
tances > 11000 km. it becomes difficult to say precisely when
the second phase starts, and we have explained in the pre-

ceding chapter how this probably arises.
4

50 MODERN SEISMOLOGY

The second phase lasts for a time depending on the dis-
tance, and then the whole appearance of the seismogram changes
and assumes a strongly periodic and sinusoidal character. The
point at which the change takes place is only rarely sharply
marked and is not characterized by a large movement such as
we have with PandS. This phase (undae longae) is initiated
by L. For distances not less than 2000 km. the general
appearance of this phase is marked by first a few waves of
period about 20 seconds, gradually increasing in amplitude and
looking as if they had been drawn with a shaking hand, then
a rapid development of extremely smooth waves of rather
shorter period which reach a maximum amplitude, subside,
pass through a succession of maxima before merging into the
tail of the earthquake or Coda.

For short distances, however, this description does not hold
good. L succeeds S very quickly, shorter periods of about 12
seconds prevail, and the duration of the whole phase becomes
very short.

These remarks apply as a whole to (XY) and Z; but, as a
rule, the development of this phase in Z comes rather later than
in (XY).

Following the maximal or long wave phase we have the
Coda. The amplitudes are now small and the movements are
somewhat irregular and lacking in smoothness. Still the
motion here is on the whole periodic and sinusoidal (about 12
seconds).

If the earthquake is a very large one, we may after about
24 hours observe the arrival of long waves that have
travelled by the opposite side of the earth. In this way
Galitzin has found from the records of the great Messina
earthquake of December, 1908, that the long waves travel
round the earth with a surface velocity of 3°53 km. per second,
which agrees well with the theoretical value for Rayleigh
waves, Viz. O°919 x 4:01 = 369 km. per second.

The view that P and S represent the arrival of the longi-
tudinal and transversal waves that have travelled by brachisto-
chronic paths from the focus to the station may be accepted
without much question. The difficulty that attaches to the

INTERPRETATION OF SEISMOGRAMS 51

interpretation of the first and second phases is that of the
origin of the irregular movements that follow on P and S.
These may in some measure arise from subsidiary shocks either
at the primary focus or at other points, and I have pointed
out that in a uniform earth we have a diffraction effect due to
the surface. This in itself is, however, insufficient, and the
facts obtain an obvious explanation in the multiple diffraction
of the primary disturbance that must go on in the hetero-
geneous mass of rock that constitutes the earth’s crust. There
will thus be not only one principal, but also many subsidiary
brachistochronic paths from the focus to the station.

The suggestion that dispersion analogous to optical dis-
persion may be called in to explain the asserted oscillatory
movement in the first and second phases may be dismissed as
not required, since heavily damped seismographs show that
there is no general oscillation to explain, but only a highly
irregular succession of impulses. The influence of dispersion
is shown in the rounding of: turning-points, so that it is only
a slightly modifying influence and not a determining cause.

This argument is not affected by the minute vibrations of
period about I second that sometimes appear after P on both
Wiechert’s and Galitzin’s instruments. They are only shown
when the earthquake is very great or the station sufficiently
near the focus, and are thus accidental and not essential.
Wiechert’s suggestion (see Wiechert and Zoppritz ‘‘ Ueber Erd-
beben Wellen Gott. Nach.,” 1907) that they represent a natural
vibration of a layer of rock seems to be the only explanation
available.

We have next to consider the long waves. We have
already remarked that they are found by measurement to travel
round the earth’s surface with a general speed agreeing closely
with that of Rayleigh waves. But the long wave phase is a
complex phenomenon, and the fact that the waves are strongly
periodic (mainly 12-second and 20-second periods) presents
considerable difficulty when we remember that the primary
disturbance is an impulse.

With regard to the long wave phase, it has been asserted
that the first portion consists of waves in which the displace-

4*

52 MODERN SEISMOLOGY

ment is entirely horizontal and at right angles to the direction
of propagation, and that there follows the maximum move-
ment in which there is horizontal movement in the direction
of propagation along with vertical motion. This is only very
roughly true. The seismogram reproduced, Plate 11, is a case
in which the first portion of the long wave phase gives horizon-
tal motion in the direction of propagation, while in the follow-
ing maximal phase the horizontal motion is at right angles to
the direction of propagation. What shall we say of cases
where horizontal motion transverse to the direction of propa-
gation is associated with pronounced vertical motion, or
where horizontal motion in the direction of propagation occurs
with little or no vertical motion?

No combination of transverse waves of purely horizontal
displacement (velocity V,) and of Rayleigh waves (velocity
‘92V,) will explain these facts, which, it appears to me, can only
be met by supposing that the long wave phase is complicated by
effects arising from reflexion backwards and forwards between
the Earth’s surface and a layer of discontinuity at some depth.

Wiechert (“ Ueber Erdbebenwellen,” l.c.) introduced the
hypothesis of such a crust resting on a sheet of plastic material
(magma). So far as such a crust provides by its natural
vibration a means of explaining the dominant period of the
long waves (say 20 seconds) we may agree; although the
argument that the thickness of the layer is half the wave
length of the dominant waves, and thus about 35 km., hardly
applies to Rayleigh waves; 40 km., however, as the half wave
length of purely transversal waves travelling across the layer
would give the 20 seconds period, and also about 12 seconds
for longitudinal waves travelling across the layer. But the as-
sumption ofa plastic sheet, which would hardly be accepted
on astromonical grounds, would not serve to contain the long
waves within the layer without at the same time confining the
first and second phase movements, which we have to admit:
penetrate the whole Earth.

At present we know nothing as to whether these long
waves diminish in amplitude as the depth increases, nor does
it appear to me necessary to suppose that they do not pene-

INTERPRETATION OF SEISMOGRAMS 53

trate beneath the crust. What we do know is that there is
a shell of radiation spreading from the focus, within which
there is disturbance and beyond which there is none.

In this connexion it is worth while to remember that the
long waves in a seismogram suggest an importance out of all
proportion to their physical effect. For example in the
Galitzin Seismograph (primary period 24°) we should have to
divide the apparent amplitude of a vibration 20° period by
about 8 in order to compare with the apparent amplitude of a
vibration of 1° period, and if further we remember that to
compare the accelerating effects we should have to divide
again by 400, we find that the long waves dwindle very much
in their physical importance.

This entirely agrees with Wiechert’s remark that the long-
wave phase, interesting as it is, is a residual phenomenon,
Neverthless the elucidation of the Long-wave phase and the
Coda is highly important on account of the information it
promises to afford as to the crust of the Earth, and here it
seems probable that seismic dispersion may play a very im-
portant part.

We shall next suppose that the times of incidence of P, S,
and L have been determined at the station for a well-defined
earthquake, and that similar determinations have been made
at a number of stations distributed over the earth. Further,
we shall suppose that by one or other of the methods to be
described in the next chapter, the position of the focus and
the time of occurrence has been ascertained. We are then in
a position to set out ona diagram the time taken for P, S, and
L to travel from the focus as a function of Mand. The
curve so obtained may be called a time curve (Laufzeit kurve).
For theoretical purposes it is, however, convenient to correct
the curve to what we should have got had % been o,:and we
then obtain a curve expressed by T=/(4). The general
character of the mean results so obtained by Zoppritz and
Geiger from several well-defined earthquakes (Gott. Nach.,
1907) are shown in Plate 5, and the values obtained by inter-
polation are given in the table, p. 54.

54 MODERN SEISMOLOGY

For P.
A P S Y e e
in kilometres. | in seconds. | in seconds. 6/2. peels computed. observe
oO fo) (0) 0750) 0° 22m
500 69 124 PD Tes}! II 23
1,000 136 244 4 30 21 27
1,500 199 356 6 45 30 32
2,000 257 460 C) | © 37 37
2,500 310 555 II 14 44 42 48°
3,000 358 641 | 13° 29 49 47 44
3,500 402 719 15 44 53 52 43
4,000 442 789 17 59 57 54 42
4,500 478 854 20 14 60 58 43
5,000 512 Qg13 22120 63 60 44
5,500 542 971 24 44 65 62 46
6,000 572 1,028 26 59 65 62 48
6,500 601 T,084 29 14 65 63 51
7,000 631 I,140 Bia 2! 65 63 54
7,500 660 1,194 33 43 66 63 58
8,000 688 I,249 35 58 66 64 62
8,500 716 I,301 28013 67 64 65
9,000 743 1,354 40 28 67 65 67
9,500 769 1,404 42 43 68 66 68
10,000 795 1,453 44 58 69 67 70
10,500 820 I,500 AGL 70 67 71
II,000 844 I,545 49 28 70 68 72
II,500 867 1,588 5I 43 71 69 72
12,000 888 1,629 53 58 72 70 73
I2,500 909 1,668 56 12 73 71 73
13,000 929 1,705 58 27 74 72 74

Let EA and EB, fig. 12, represent neighbouring paths, then
BEC ah

COS @=~{5 =

ABT wi

where V is the corresponding velocity of the wave at the sur-
face. This important result, which applies to both P andS

=

FIG. 12.

whatever be the path, is of course meaningless as applied to
L. Since V, and V, are known we may from the time curves

AQ in Megametres.

PLATE 5.—Time Curves (after Zoppritz).

INTERPRETATION OF SEISMOGRAMS 57

determine the corresponding angle e. For the longitudinal
effect P we have

cos ¢= V, ae
and we also have
V, (i -sin é\!
cos @é= V. ( 5 )

where é is the apparent angle of emergence.

Now if the rays travel in a straight line from E to A the
angle of emergence e would be simply 4/2R = 0/2.

The table, page 54, shows at once that as we proceed to in-
creasing distances the value of e obtained from the time curve
is much greater than the corresponding value of 0/2. Thus
the rays dip more deeply into the earth than does the straight
line from focus to station. The rays must on the whole be
concave towards the surface, and we have now to abandon the
hypothesis that the earth is uniform, and instead to assume
that the velocity of propagation depends on the depth. Ac-
cordingly the next step is to suppose that the earth is made
up of concentric uniform spherical shells, but that the velocity
v varies as a function of 7 the radius of the shell. On this
hypothesis the brachistochronic paths are still plane curves in
planes containing the focus, Earth’s centre, and the station, but
are now curved, each curve being characterized by the well-
known equation f/v = c(a constant) where / is the perpendicular
from the centre of the Earth on the tangent to the curve at any
point. From the values at the surface we get
COSA an ale

Tn an Amn aon
Now the path is symmetrical, so that if the greatest depth for
the ray is %,,, the velocity at that depth is given by (R — 4,,)/c.
If we put 7/v=7 we find that 4 and T are expressed as in-
tegrals, viz. :—

b Hil
A et — 1 c| (GEE) Fi log r dan

b 2 EN
2 | @-2) Des ee
where 6=R/v,.

Dig—c—

58 MODERN SEISMOLOGY

If the law of variation of v with v is known we could evaluate
the integrals. We do not, however, know this law, and the
problem before us is whether, from the graphical representation
of T as a function of 4 or @ from observations, we may deter-
mine v as a function of 7.

The analytical solution is expressed by

a y el so
Hel = hy | Co Oe

(cf. Bateman, “ Phil. Mag.,”’ 1910), and

I

2 8 6 [” 2!
Te ee a se Ge) eine

so that if @ and T can be expressed as functions of ¢ or a

we should get 7 as a function of » and hence the velocity at
any depth. Now the observations give T as a function of
A, so that theoretically the problem is solved. But asa
matter of fact time curves are still very inaccurate and do not
justify a very minute analysis at present.

One must proceed by a comparatively rough graphical pro-
cess, and the obvious suggestion would be to take successive
ranges within which @ does not vary much with c.

Wiechert, who first attacked the problem, divided the Earth
into finite layers within each of which the radius of curvature
of the path might be taken as constant, and on this basis
Wiechert, Zoppritz, and Geiger (l.c.) analysed the time curves
for Pand S. The results of the investigation which are set out
in the table, page 61, show that from 4=0 to 4 = 5000 km., 4,,
increases from 0 to about 1500 km., while V, and V, continually
increase as #,, increases. As J increases to 6000 km. &,,
increases very little. Beyond this 4,, again increases until for
4=13,000 km. 4,, attains a value rather over 3000 km. But
from 4,,= 1500 to 3000 km. both V, and V, remain constant.
It is specially interesting that Poisson’s ratio o remains practi-
cally constant.

The variation of velocity with depth may not, however, be
continuous, but we may have surfaces at which the velocity
undergoes a sudden change. Such a surface of discontinuity

———> A in Megametres
VEU VED SAVES RO FO BOE MOS ME Ae NS

hm in
Kilometres
500

1000

1500

3000

3500

PLATE 6.—Showing maximum depth of seismic rays as function of epicentral distance
(after Zoppritz).

Plate 6 shows the maximum-depth (im) attained as a function of the epicentral
distance A, for

I. The first phase P as observed.

II. The second phase S as observed.

Ill. Theoretical straight rays with constant speed.

INTERPRETATION OF SEISMOGRAMS 61

diode lon Vi Vz co
km. km. km./sec. km./sec. Bere
fo) o 7 fae7) 4°01 0°273
1,000 100 7°60 4°24 0272
I,500 200 8°0r 4°47 0°272
1,800 300 8°42 4°70 0274
2,200 400 8°83 4°93 0°272
2,500 500 9°23 5°15 0°274
2,800 600 9°62 5°37 0°273
3,200 700 10°00 5°59 0°272
3,500 800 10°37 5°80 ae
3,700 goo 10°73 6°00 Bons
4,000 1,000 II°07 6°21 0270
4,300 1,100 TI"43 6°41 0269
4,500 1,200 II‘75 6°60 ara
4,800 1,300 T2'08 6°80
5,000 I,400 12°40 6°87
5,300 I,500 et 2.72 6°87
13,000 3,300 12°78 6°87

leads to singularities in the time curve. In particular Wiechert
shows that if there is a sudden increase of velocity, there will
be a corresponding point on the time curve at which the slope
changes suddenly. It would then really consist of two portions
cutting at a definite angle and there would bea certain range
within which the seismograms would show two sharp impulses.
If on the other hand there is a sudden reduction of the velocity
there will be a gap in the time curve corresponding to a range
of distance not reached by the waves.

In this way Wiechert in a recent investigation (Inter. Seis.
Assoc. Manchester, 1911) concludes that there are such surfaces
of discontinuity situated at depths of 1200, 1650, and 2450 km. ;
but I am not aware that any numbers have been published
showing what change this makes in the table of velocities
derived from his former investigation. He further concludes
that for depths greater than 3000 km. the velocities diminish
gradually (see Geiger and Gutenberg, Gott. Nach., 1912).

Interesting as Wiechert’s results are, they must be regarded
as indicating the manner in which Seismology may be expected
to throw light on the nature of the interior of Earth, rather
than as results of great accuracy. Very slight changes in the
slope of the time curve would lead to very considerable changes
in the inferences ; and in this respect it appears to me that we
still require an analytical method which depends on the original

62 MODERN SEISMOLOGY

time curve itself and not on the still less accurate curve ex-
pressing 4 as a function of slope dT /d4. Different investiga-
tors give smoothed time curves which differ sufficiently to lead
to very different conclusions as to the interior of the Earth.
Moreover, we have seen that a smoothed curve may really
involve a quite wrong method of procedure.

The primary curve itself is subject to many sources of
error. Apart from actual errors of the time that do unfort-
unately exist at seismological stations, we have to remember
that the marking of the exact instants at which P and S occur
is a matter of personal judgment, and depends also on the
particular instrument used and the sharpness of the impulses.

The first portion of the curve depends on the elimination
of the effect of finite depth of the focus, and as that is a very
difficult matter, I should doubt if it is often successfully accom-
plished. Again for distances much beyond 10,000 km. S is
often extremely indistinct. There are probable theoretical
reasons for this as we have pointed out, but meanwhile it
introduces uncertainty. Beyond 13,000 km. data are very
meagre, and the determination of the incidence of P becomes
increasingly difficult on account of the smallness of the hori-
zontal movement.

Thus there is room for progress both on the theoretical
and the experimental side, but the growing activity of seis-
mologists is a good augury for the successful improvement of
time curves even to the semicircumference of the Earth.

CHAPTERS VIII

DETERMINATION OF EPICENTRE AND FOCUS.

THE first question that arises when a seismogram indicates
the occurrence of an earthquake is—where did the earthquake
occur?

We have hitherto regarded the earthquake as occurring at
a point called the focus. Strictly the primary shock may
have extended throughout a considerable region, so that in
speaking of the focus we assume some average point from
which the maximum effect appeared to proceed. Again we
have seen that the focus may be at some depth and not at a
point on the surface. For distances over 1000 km., however,
it is quite accurate enough to regard the shock as occurring at
a point on the surface known as the epicentre. Several
definitions of epicentre, based on different physical ideas, may
be given. It may, for example, be defined as the surface point
first affected by the shock, or the surface point where maxi-
mum effect is produced. For our immediate purpose it is
sufficient to define the epicentre as the extremity of the Earth’s
radius that passes through the focus. Until quite recently the
method available for obtaining the epicentre was empirical,
and based on the time curves for P and S as a function of the
epicentral distance 4, obtained from observations of former
earthquakes with well-defined epicentres. The most accurate
of these are the curves obtained by Zoppritz. We shall return
to the manner in which the primary time curves are to be
obtained and meanwhile suppose that the table of values of
S-P in seconds for each 10 km. as interpolated by Zeissig is
available (published by the Imp. Acad. of Sciences, St. Peters-
burg).

If then P and Sare clearly defined on the record the interval

63

64 MODERN SEISMOLOGY

S-P is known, and the corresponding distance 4 of the epi-
centre from the station is determined. The result is free from
any absolute error of time at the station. In many cases,
however, P is so small that its incidence cannot be accurately
assigned, and then one may get an estimate of the distance
from S-PR, or L-S, but these are much less accurate and
ought only to be used as a check.

When JZ is determined thus for three suitably selected
stations the position of the epicentre is determined uniquely
as the common point of intersection of three small circles on
the sphere. Needless to say the circles do not precisely inter-
sect at a point in practice, so that the epicentre is given only
within certain limits. The co-ordinates of latitude and longi-
tude may of course be obtained by computation or graphically
on a stereographic projection.

It was pointed out by Galitzin that if the first impulse
represents the arrival of a longitudinal effect in the plane
containing epicentre, station, and Earth’s centre, the ratio of
the magnitudes of the displacements to north and to east
must give the tangent of the azimuth of this plane, so that the
distance and direction of the epicentre can be determined by
observations at a single station. This principle has been sub-
jected to rigorous examination first at Pulkowa and later at
Eskdalemuir, and the results show quite conclusively that, pro-
vided the first impulse is sufficiently clear and large, the epi-
centre can be determined in this way with great accuracy.
There is a possible ambiguity of 180° in the azimuth deter-
mined in this way from the horizontal seismograms alone, for
the first impulse may be a condensation or a rarefaction. The
vertical component seismograph, however, removes the ambig-
-uity, for if the impulse is a condensation the corresponding
vertical movement is up, while for a rarefaction the vertical
movement is down. There are indications that the first im-
pulse may appear as a rarefaction at one station and as a con-
densation at another. This might be expected on Stokes’
dynamical theory of diffraction, and if it proves correct, it
suggests a means of finding the axis of the primary impulse;
and this would be a valuable addition to seismological know-

DETERMINATION OF EPICENTRE AND FOCUS 65

ledge. When the distance 4 and the azimuth a have been
determined at a station we may calculate the co-ordinates of
the epicentre by means of the formule

sin dg =cos J sin ds + sin A cos $s cos a
and
cos 4-sing sin dr

cos (Ag — As) = cos dg COS bp

where ds, As are the latitude and longitude of the station and
zr, Az are the latitude and longitude of the epicentre.

As an illustration of the accuracy obtained by the use of
Galitzin’s seismographs, compare the independent determina-
tions of the epicentre of the Monastir earthquake of 18 Febru-
ary, I911, made at Pulkowa and Eskdalemuir.

For Pulkowa ¢s=59 46’ N Ns = 30:19 E
and the seismogram gave 4 = 2260 km.=20° 19’
and a=22 53 West of South. Hence for the epicentre
dz = 405° N Az =20'1° E.
For Eskdalemuir ds = 55° 19’ N No sma G
and the seismogram gave 4= 2360 km. =21° 14’
and a=55 50’ East of South. Hence for the epicentre
gn =406 N Ne 2030 E:

As long as the first impulse is really sharp no trouble arises ;
but with a small and gradual start, it is sometimes difficult to
identify the corresponding movements on the horizontal and
vertical seismograms, owing to a phase difference of the maxi-
mum displacement. Thus instruments with the same funda-
mental constants are required to remove this source of error of
judg ment.

It is clear that if the azimuths have been accurately
determined at two stations the epicentre can be determined
from these alone without reference to the determinations
of distance (see Galitzin and {Walker, ‘“ Nature,” August,
GUA\:

The preceding example gives in this way

dz =40'4 N Mo= BOs IE.
for the epicentre, while the deduced distances from Pulkowa
and Eskdalemuir are then 20° 18’ and 21° 26’ respectively.
5

66 MODERN SEISMOLOGY

The three values for the epicentre do not differ by more than
20 km.

The advantages of this method are that it is quite inde-
pendent of (1) the time at the two stations, and (2) the deter-
mination of S, and thus free from any error that attaches to
the empirical time curves. It should thus prove of great value
in improving the empirical time curves, more especially for
short distances where the influence of finite depth of focus is
considerable. For this reason I consider that an instrument
which would give the azimuth directly would be of great
service even if the remaining part of the seismogram had to
be sacrificed.

We have now to consider how the primary time curves are
to be obtained.

We shall suppose that we have available the times of in-
cidence of P and S at a number of stations. Before these can
be arranged we require to know the position of the epicentre
so that the distances 4 can be computed. In some cases (e.g.
the great Messina earthquake, 1908) the epicentre is known
with considerable accuracy from local knowledge. But,in many
cases such information is not available or cannot be relied on,
and then some other method must be used.

We have seen that an extension of Galitzin’s method of
azimuths may give the epicentre directly. So far it has not
been used in the preparation of time curves, but there is little
doubt that it is the most satisfactory method we can have.

When observations of P have been obtained at several
stations known to be not very far from the epicentre, we may
however get a fairly good determination of the position of the
epicentre by a method used by Zoéppritz (Gott. Nach.,
1907, l.c.). If for instance P occurs at precisely the same in-
stant at three stations not too far from the epicentre, the
epicentre would be the unique point which is equidistant from
the three stations. If the times differ we may proceed as
follows: Let A, B, and C be the stations and let X be the
epicentre ; we then have the equations

XA=1(7), XB=2(y+)), XC=0,(7+9)

where # and g are the observed time intervals in seconds

DETERMINATION OF EPICENTRE AND FOCUS 67

between B and A, and C and A, w, the velocity of propagation
of the disturbance, and y the unknown time from epicentre to
A. We may then by trial construct the circles of radii pro-
portional to v7, y+), y +g with centres at A, B, C which inter-
sect in a point, and we then get the position of the epicentre
and also the time y from A to X. The above equations are
approximate and do not take account of the depth of focus.
But as we shall show in a little if the distances are within from
200 to 400 km., the error introduced in the times is less than
half a second even for a focus 40 km. deep, and the observed
times are not accurate to this extent. The time y is then
the time from A to the focus or to the epicentre, to less
than half a second, but we must be careful to observe that the
time from focus to epicentre is not zero. For the formule
become inaccurate beyond the range given.

Having obtained the epicentre we may now set out the
curves giving P and S as a function of the distance J, and if
we accept the time of occurrence at the focus given by deduct-
ing the time y from the time at A, we complete our time curve
giving the interval of time from focus to station as a function
of the arc from epicentre to station. We may not, however,
exterpolate the curve to points quite close to the epicentre,
until we know the depth of the focus.

The curves we have obtained are still time curves depend-
ing on the depth of focus. There is a range of several hundred
kilometres within which the influence of depth is extremely
small, but for shorter distances the influence of depth is con-
siderable and again for greater distances the error may amount
to a few seconds.

The curve cannot be freed from the effect of depth and
so prepared for theoretical investigation unless we know the
depth of focus or have observations sufficiently near the epi-
centre to determine it. Zoppritz (l.c.) proposed the following
method of correcting the time curves when the depth & has been
obtained. Assuming that the path (fig. 13) is symmetrical we
may prolong the path SF backwards to meet the earth’s surface
at O, and the angle EOF = eis equal to the angle of emergence
at the station. Thus OE=EF cot e=4 cot e and the time to

5 *

68 MODERN SEISMOLOGY

traverse OF would be OF /v, = 4 cosec e/v, where % is the depth
of the focus and wv, the velocity of the disturbance at the sur-
face. Thus for great distances we may pass to the corrected

Fic, 13.

curve by applying to the original point (¢, J), the corrections 6t,
and:64 where 6¢=% cosec e/v,, 64=hcote. The corrections
would, of course, differ for the P and S curves and e would be
determined from the corresponding curve.

This procedure is probably accurate enough for dis-
tances >1,000 km., but entirely breaks down as we get
close to the epicentre. In any case no correction can be at-
tempted until Z is known, Thus we may now consider how,
if at all, Z can be obtained by observation.

It seems evident that only observations not far from the
epicentre would be of much use for this purpose, but what I
think one is hardly prepared for is the extreme closeness to
the epicentre required, if we are to depend on the times of
arrival of P for the determination of %.

It is not often that data are available which make any
attempt to determine the depth of focus worth while, but the
occurrence of an earthquake in South Germany on 16 Nov-
ember, I9I11, tempted several investigators to see what could
be made out as to the depth. Galitzin (Nach. d. Seis. Comm.
Petersburg, Bd. v. L3, 1912) went into the problem very
carefully, but it is to be feared that the data finally proved to
be too unsatisfactory to justify an elaborate analysis.

Galitzin first attempts to take account of the influence of
depth on the velocity of propagation of the longitudinal

DETERMINATION OF EPICENTRE AND FOCUS 69

disturbance. He assumes as an approximate law for small
depths
9
PN?
y= (*) = sbaee = 1)
v
where

r=(1-%)
R)?
and v, is the velocity at the surface, v the velocity at depth 4,
and R the earth’s radius. Now Zoppritz’ results give
d= falg cts | Sec.
Oa 7 OO) kms See.
and hence c= 3'529 while R= 6370 km.
Integral expressions for the distance 4 and the time T
from focus to station are then obtained and used to compute
the following among other tables.

Distance Time from focus to station in secs. Differences.
4 km. h=1km. h = 10 km. h = 40 km. Tio — Ti- Ty - Ty.
S. Ss. S. s. S.

£ o°13 1°43 5°52 + 1°30 + 5°39

50 7°35 6°99 8:80 — 0°34 + 1°47
Too 13°33 13°90 14°80 + O13 + 0°97
150 20°80 21°12 21°32 + 0°32 + 0°52
200 27°07 27°57 28°00 — O'1o + 0°33
250 34°80 34°65 34°76 = Ong — 0°04
300 41°74 41°56 41°51 — 018 — 0°23
350 48°71 48°58 48°42 — o'r3 — 0'29
400 55°25 55°44 55°05 + o'19 moo
450 62°44 62°29 61°80 — O15 — 0°64
500 69°43 69°14 68°55 — 0°29 — 0°88

The columns of differences suggest that some error of com-
putation has crept into the numbers.

The table on the following page is obtained on the simple
hypothesis that the velocity is constant for any depth here
considered and equal to 7°17 km. per second.

Several points are suggested by a comparison of these
tables. We notice that the point of inflexion on the time
curve is so ill defined that it is useless for estimating Z. Further,
anywhere between 200 and 400 km. is quite useless to attempt
to discriminate between the two tables or for any value of %

70 MODERN SEISMOLOGY

up to 40 km. by means of observations which are only
given to the nearest second. Only at 500 km. and then only
for 40 km. depth do the two values differ by 1 second, and as
a matter of fact we can hardly suppose the value of v, to be
so accurately known as to give much security.

Distance Time from focus to station in secs. Differences.
A. h =1 km. h =10km, h = 40 km. Ty - Th. Tg) — Ty.
S. S. Ss.
o Only 1°39 5°58 1°25 5°44
50 6:98 rata 8°92 o'13 I°94
Too 13°95 I4°O1 14°99 0°06 I°04
I50 20°91 20°94 21°57 0°03 0°66
200 27°88 27°90 28°35 0°02 0°47
250 34°81 34°81 35°14 0°00 0°33
300 41°83 41°83 42°08 0°00 0°25
350 48°81 48-79 48°98 — 0°02 O17
400 55°76 55°73 55°86 — 0'03 (ode ce)
450 62°78 62°75 62°84 — 0°03 0°06
500 69°75 69°72 69°76 — 0°03 O°O1

We may, however, conclude that on either hypothesis
the observations between 200 and 400 km. should give us the
actual time of occurrence of the shock at the focus to less than
$ second as practically independent of % for 2<40 km. and that
is an important point gained. Next, to get the depth we
must use only the observations for 4<200 km. and even then
it is really only the observations for 4<50 km. that ought to
count heavily. Here also it is impossible to discriminate
practically between the two hypotheses, so that the simpler
one should have the preference.

Turning now to the actual data in the table on the opposite
page, we note that the distances were computed from the
epicentre determined by noting that the times at Ziirich and
Strassburg were the same, as were also the times at Aachen and
Gottingen. The co-ordinates so obtained were

p48) 19 INdand in, Onzsae
Galitzin, from the time at Karlsruhe, Strassburg, and Ziirich,
finds the time at the focus to be 21 hours, 25 minutes, 52°5

seconds, and his conclusion is that the depth was 9°5 km. with
a probable error + 3°38 km. The data, however, show dis-

DETERMINATION OF EPICENTRE AND FOCUS 71

crepancies of as much as 2 seconds. These may be quite real,
for it is not unlikely that the velocity may differ sufficiently in
different directions to account for this.

Station. A. Jey Station. A. 1B

km. leer iy km. Ine, Se.

Biberach . . 39 2I 25 59 | Aachen 36252 2One a2

Karlsruhe . IOI 26 867: | Gottingen 362 42

Zurich I2T to | Bochum . 385 45

Strassburg 123 to | Triest 446 55

Heidelberg 130 12 | Laibach . 463 58

Jugenheim 169 18 | Graz 474 DG Tt
Frankfurt . 222 22 | Wien 510 6 |

Neuchatel 233 23 | Agram 571 8

Krakau 793 42

From the simpler theoretical table we get the following
times to the nearest second.

by To from the data.

hk =1.km.|h=10.km.| kh =40.km. | #=1.km. | 4= 10. km. h = 40. km.
km. s. s s hemes) | ha sms Ss. h. m. s.
39 6 6 8 21 25 53 |- 20 25 53] 25 25) 52

| tor ce 14 15 53 55) 52

| 122 17 17 18 53 53 52

| 130 18 138 19 54 54 53

| 169 24 24 24°5 54 54 53°5

The conclusion is that % was not as great as 4o km. and
that 10 km. is better, but on the data we can hardly say that
h might not have been zero. |

What seems to be clear is that unless the times were
known to o'1 second, only observations at less than 50 km
would be of value to settle the matter. From a human point
of view one hopes that no such case will ever occur, and the
problem of finding the depth of the focus is more likely to be
solved by direct observation of the emergence angle with
horizontal and vertical seismographs combined.

CHAPTER IX.

SEISMIC EFFECTS OTHER THAN THOSE DUE TO EARTHQUAKES.

Dr. MILNE once remarked to me that a seismogram always
has something to show worth knowing even if there is no
earthquake. Those who have had the great privilege of visit-
ing the observatory at Shide and seeing Dr. Milne’s wonderful
album of seismograms will appreciate how true the remark is,
and how thoroughly Milne has devoted himself to anything
that can throw light on the subject which he has so conspicu-
ously adorned.

We must pass over the spurious effects on a seismograph
produced by the presence of the observer, the shutting of doors,
and that bane of the experimentalist, the ubiquitous spider.
They are mentioned here, only to point out that the practical
seismologist must be able to recognize such effects when they
occur.

It was long ago recognized by Milne that a seismograph
frequently shows minute vibrations continuing for many hours.
and that they could not be accounted for by earthquakes or
local traffic.

These effects were called by him ‘‘ Tremors” and although
they occur always with high local winds, they also appear
when it is quite calm.

On the Milne seismograms the tremors present the same
appearance on calm or on gusty days. But with heavily
damped seismographs, using larger magnification and higher
speed of registration, it is found that there are two types of
tremors or microseisms as they are now called. They are
shown to special advantage on Galitzin seismograms.

In the first class, which occurs on windy and calm days
alike, the movements are very smooth and regular, and the

72

ae

= v iS , f
: E eS ‘ i i a’
. f i= ts = -
5 - i] ‘ *
<i h “ ae 0 ; > ' t * % ;
ty ‘ 5 ut { ty
- ; “- Be! a G 5 x
- * 7 '
on A ss = ¢ - A f
7 Z ¥ ~~ r 1
= ¥ = ‘ y > x my
{i 5 u iN ’ ’ ~ A
: ; * : a Z
. - =i, - £
* . r § . = + =
= 5S ~ os
‘ “ = ’ y ~ 4 .
a i ~ r x 5 :
ait ~ J x ; < ey aS a
=) i . : as Oe
é 2 -
. ( fae =
;) a << 4
~ = » “ a

th

< I minute >

PLaTE 7A.—Portion of Record at Pulkowa, September 18, rgr0, showing
microseismic movement

ROYAL

{art Sin ae ae Crock ry wu Vic TORIA
HIGH WATER Ww. Yann
On. 50" ae ee Loe Crup
ee om, ana RYDE, 1.W.
'Oll pos if Yo J. MILNE
APR 2 APR |

settee meine temic encarta ae

PLaTE 78.—Specimen Record (reduced) showing Tilt produced by Tidal loading.

Original scale r mm = o/'"'17 Tilt, and ro ft. Tide gives 5 mm. deflexion.

SEISMIC EFFECTS 73

periods range from about 4 seconds to over 8 seconds.
The periods are not mixed up, for the same period will per-
sist for many hours. In the second class the periods range
from about 12 seconds to about 30 seconds. The movements
are irregular, look like badly drawn sinusoidal curves, and the
periods occur indiscriminately. These occur only on windy
days.

There seems to be little doubt that the second class is due
to the gusts of local wind setting the ground and buildings
into movement, for they start with the wind and cease as soon
as the wind subsides. In my own experience the movements
are not very pronounced until the speed of the wind is about
20 miles per hour, and I should say that the movements
tend to become more regular and of shorter period as the wind
increases in speed. It has been found at Pulkowa that the
amplitude is much reduced in an underground room, and that
it is an advantage to prevent direct access of air to the sides
of the piers.

Microseisms of the first type present an interesting problem
for solution. They are observed at quite inland stations and
at considerable depths as well as at stations near the coast.
A systematic comparison of observations has been undertaken
by the International Association of Seismology, but results are
not yet available. The main features, are, however, fairly
definite. The longer periods are associated generally with larger
amplitude. The longer period movements (8 seconds) come out
strongly in stormy weather, but persist for many hours after
all local wind has ceased, and then the period and amplitude
usually gradually diminish until a normal period of from 4
to 5 seconds prevails it may be for several days. Again
the microseismic movement of this type is more pronounced
in winter than in summer. Indeed there are often occasions
in summer where the movement becomes imperceptible and
this is rarely the case in winter.

Plate 7A is a reproduction of a portion of a specimen
record obtained at Pulkowa. It shows clearly a feature usually
to be observed, that the amplitude rises to a maximum and
then subsides, the maxima being at intervals of about I minute.

74 MODERN SEISMOLOGY

The following table gives the average amplitude and period
observed at Eskdalemuir on the Galitzin horizontal seismo-
grams.

: Amplitude of 2 Amplitude of
IgII Period IQII Period
Month. seconds, CR Month. seconds. ee
Jan. 6°6 2°2 July 4°3 03
Feb. 6:0 2°1 Aug. 4°2 0°3
Mar. 555 Te Sept. : 0'6
B 8)
Apt. 555 o'7 Oct. 4°9 o°5
May 5:2 06 Nov. 5°4 r°8
June 4°5 04 Dec. 5°3 18

Average for year: Period = 5:2 seconds; Amp. = Io.

The vertical movement is quite as pronounced as the hori-
zontal movement, and this suggests that we are dealing with
Rayleigh waves propagated over large continental areas.

The general phenomena and the periods presented by
these microseismic movements correspond so closely with what
one observes of the sea waves on the coast, that one can
hardly doubt that the two things are closely connected. Dr.
Schuster has devised and set up an apparatus near Newcastle
for obtaining a continuous register of the sea waves, but de-
tailed results are not yet available for comparison with the
movements shown by seismographs.

It has been suggested that the land effects are due to the
actual breaking of waves on the coast, but this can hardly be
maintained as an explanation of effects observed so far inland
as central Europe or central Canada. It seems more probable
that, and it is at least worth while investigating theoretically
whether, the motion observed far inland is due to Rayleigh
waves set up at the bottom of the sea by water waves set up
and maintained over large ocean areas by the wind. To take
a simple example: we know that a travelling wind sets upa
train of waves following after it. On deep water, such as mid-
Atlantic, we should get a period of 5 seconds, wave length 40 m.,
with a wind velocity of 8 m. per second, or about 20 miles per
hour; while a period of 10 seconds, wave length 160 m., requires
a wind velocity of 16 m. per second. Such waves advancing

SEISMIC EFFECTS 75

into shallower water would maintain their period but diminish
in wave length and speed, while the amplitude of movement
at a depth equal to the wave length would be 1/500 of the sur-
faceamplitude. This would seem to provide adequate margin
for explaining an observed earth amplitude of I micron=
‘OOI mm. even at a considerable distance from the area of
origin.

The case of waves set up by wind in an ocean of moderate
depth, such as the North Sea (average depth about 100 m.) is
more complex, but is soluble on the lines indicated by Lamb
(“ Hydrodynamics”) and seems to merit investigation with a
view to explaining the microseismic movement observed in
Western Europe.

Dr. Klotz of Ottawa, who has studied the effects observed
there by means of a Bosch Seismograph, is of opinion that the
largest effects are associated with cyclonic areas in the North
Atlantic, and he suggests that the microseismic movement
may appear in West Europe before the cyclone arrives. If -
this should prove to be the case it would be a most valuable
addition to meteorological knowledge.

In Chapter I. we have observed that a pendulum, whether
of simple or of horizontal type, indicates by its relative
motion not only horizontal acceleration applied to the pier,
but also tilting. It also indicates accelerating effect applied
to the mass ina horizontal direction. If these effects are applied
very slowly, the inertia and frictional terms in the equation of
motion have no influence and the pendulum simply shows a
gradual change of its zero position. The equation is now of
the form

0=(@ — gry) or 18 = (% — gy)

wherein @ and ¢ are measured to the right and is measured
in the anti-clockwise direction.

The changes of zero are shown by all mechanical pendu-
lums, but it must be remembered that here the electro-
magnetic method of registration is of no avail, since the zero
position of the galvanometer is not dependent on the zero
position of the pendulum itself.

76 MODERN SEISMOLOGY

It is perhaps needless to remark that the zero of a pendu-
lum is continually changing. Such changes may be merely
instrumental or due to local temperature change. As such they
are of little scientific interest, and are rather a serious nuisance,
and every care should be taken to remove such sources of
change. Careful examination shows, however, that part of
the change of zero is regular and of considerable scientific
importance. The most marked effect in point of magnitude is
that which occurs on the seismograms of the Milne pendulum,
e.g. at Ryde, Isle of Wight, which show in a manner visible
to the eye a regular sinusoidal movement of the zero agreeing
precisely with the rise and fall of the Channel tides. There
seems little doubt that the rock strata bend under the influence
of this periodic alteration of load in the Channel basin. Dr.
Milne has kindly sent me the specimen record, Plate 7B.

Such visible effects are not howevershown at inland stations,
and it is only by careful analysis of results extending over long
intervals that the existence of periodic movement in the pen-
dulum zero can be detected. The effects, although small,
derive importance from their association with earth tides and
the theory of the physical properties of the Earth.

The acceleration of gravity g at any point of the Earth’s
surface is not exactly constant either in magnitude or direction,
but on account of the attraction of the Sun or Moon it under-
goes small changes. The potential of these additional forces
at any point is expressed by a solid spherical harmonic of
order 2 and may be written

te
W, =3mg 3
wherein 7z is the mass of the Sun or Moon,
cis the distance of the Sun (or Moon) from the
Earth’s centre,

(cos?@’ — 1)

g the normal acceleration of gravity,

a the mean radius of the Earth,

vy the geocentric radius to the point, and

@ the geocentric zenith distance of the Sun or Moon.
The solar effect is about half that of the Moon.

If now # is any direction on the Earth’s surface perpendi-

SHISMIG BERFECTS Wal

cular to the original direction of g, the potential W» will give
rise to an accelerating force

which is operative in deflecting the pendulum. But this is
not the whole matter. The Earth yields to the disturbing
potential W. and, in accordance with a well-established prin-
ciple, the surface is deformed by an earth tide of amount
AW,/¢ and the deformation of both earth and sea produces an
extra potential AW2. Thus the additional force operative on
the pendulum mass becomes

(1 +2)

OW,
on
instead of
wW,
On

The tide ZW,/g, however, produces a tilt in the platform
h IW,

so that the recorded displacement of the pendulum zero will
appear to be proportional to
Py)
(2+) Mm

2
or

instead of
W,
Oe
as it would be if the Earth did not yield.

In a similar way it appears that the oceanic tide becomes
(1 -h+£)W3/g in place of Wo/g.

When the potential W2 is expressed in terms of the lati-
tude and longitude and the co-ordinates of Sun or Moon, we
obtain a number of terms representing the component tides.
Chief among these are the approximately semi-diurnal lunar
and solar terms, and for reasons that do not appear quite
adequate, attention has until recently been concentrated on
the corresponding terms in the pendulum zero movement.

The experiments begun by Zollner and Rebeur-Paschwitz
have been repeated by others, and the most recent observations

78 MODERN SEISMOLOGY

are those by Hecker, Orloff, and Schweydar. Hecker’s re-
sults are to be found in ‘‘ Publications of the Royal Prussian
Geodetic Institute,” No. 32, 1907, and No. 49, IgII.

His observations were made at Potsdam with Rebeur-
Paschwitz pendulums at a depth of 25 m. so as to secure
constant temperature.

The azimuth of Pendulum I was 42°E of N, and of Pendulum
II 48° W of N.

The semi-diurnal disturbing potential may be written
ie COS Mae
a m8 (1 — 8 é) cos? zi cos" cos 2 (¢+X- 4)
where ¢ and 2 are the latitude and longitude, e and w the
eccentricity and inclination of the orbit, and ¢ is the lunar or
solar time, as the case may be, referred to some convenient

29 a
M2=?

origin.
It is usual to express the observations not in terms of the
force
oM2
ox

but in terms of the apparent angular change of the direction
of gravity, viz. :—

wee
RuwOre

Hecker’s latest results are as follows :—

Solar Effect. Pendulum I. Pendulum II.
* T OM, , Wa 9
Theoretical = yeh 0’’-00399 cos (2t — 305°5°) 0/:00389 cos (2¢ — 48°77)
5S
Observed. 000353 cos (2¢ — 255°8°) 0'”"00448 cos (2t — 36°6°)
Lunar Effect. Pendulum I. Pendulum II.

rt OM,
Theoretical Sy 0'’-00922 cos (2¢ — 305°5°) 0”’*00900 cos (2¢ — 48°7°)

ox
Observed. 0’’:00560 cos (2f — 293°6°) 000490 cos (2t — 59°7°)

If the lunar effects are computed for the geographical
directions we get
Lunar Effect. N.—S. E.—W.

Theoretical. 0’’*00788 cos (2t — 180°) _0’’"00ggg cos (zt — 270°)
Observed. 0''°00355 cos (2 — 175°) 000665 cos (2¢ — 270°2°)

SEISMIC EFFECTS 19

It is evident at ‘once that these results are not concordant
either in phase or amplitude, as each term gives a different
value for what is presumably the ratio(1-%+). Before pro-
ceeding farther we may remark that the general sensitiveness
of the record was about I mm.=0”-04 and that much larger
zero movements occurred than those expressed by the above
terms. Thus it is open to doubt whether these discrepancies
have any real significance, and whether the apparatus is really
capable of giving more than the general order of magnitude of
the effect.

The ratio for (1 - 4+) given by Hecker’s results are for
the lunar terms.

0°68 for the E—W component.
0°43 for the N—S component.

Orloff (‘‘ Veroff d. Dorpater Sternwarte,” 1911) observing
at Dorpat with Zollner pendulums in the geographical direc-
tions obtained

0°68 for the E—W component.
0°59 for the N—S component.

His apparatus was about four times as sensitive as Hecker’s
and the individual results show better concordance than those
of Hecker.

It may be said that observers on the whole have obtained
something like 2/3 for the value of (1-%+4) from pendulum
observations of this particular lunar term, and this is also the
value obtained by Darwin from his analysis of the fortnightly
oceanic tides in the Indian Ocean. This apparent agreement
seems at first to suggest a simplification of the theory
of the values of Zand &, and that they might be calcu-
lated on an equilibrium theory of the tides and so lead to a
fairly accurate determination of the Earth’s rigidity, But
Schweydar’s recent investigations show that this is not so, and
that theoretically the matter is one of great complexity.

We turn for a little to the theoretical side which we owe
mainly to Lord Kelvin. The matter was one of life-long in-
terest to him, and the investigations (Thomson and Tait,
‘Natural Philosophy’) form the basis of most subsequent

80 MODERN SEISMOLOGY

calculations. The quantities 2 and & are not independent,
but are related and dependent on the physical properties of
the Earth as a whole. The simplest assumption that can be
made is to regard the Earth as a uniform sphere which is in-
compressible, but possesses rigidity mw, and further that the
tides may be computed on an equilibrium theory. We then
find that
k= ph, and = $/(1+ 29H),
2gpa

Thus if we accept the experimental value 2- k= 1/3 we get
h=5/6 and &=1/2 while ~=7°I x10" dynes per sq. em.
This value of ~ which is nearly that of steel, formed the ground
of Kelvin’s estimate of the Earth’s rigidity. Darwin, however,
did not accept this, but regarded the observed reduction of
the fortnightly tides as due to the difference between the
dynamical and the equilibrium theory (cf. Lamb, “ Hydro-
dynamics ”).

The preceding result, however, conflicts with data derived
from the free period of precessional nutation of the Earth as
derived from astronomical observations. Larmor (‘“ Proc.
R. S.,” Vol. 82, p. 89, 1909) shows that

1-(\- PEE)

where T, is the theoretical Eulerian period 306 days,

T the observed Chandler’s period 428 days,

w the angular velocity of rotation of the Earth.
and is the ellipticity of the ocean surface. Thus since
w’a]g=1/289 and e has practically the same value, we get
k= 0°28, and this with 2 - k=0°33 gives =0'61 which does
not satisfy the relation £= 3 and leads to a higher estimate
of the Earth’s rigidity.

Schweydar (“ Veroff. Kon. Preuss. Geod. Instit.,’ No. 54,
1912) investigates the reason for the discrepancies. He takes
account of the oceanic tides, and further introduces Wiechert’s
assumption that the solid part of the Earth consists of a shell
of density 3:2 and thickness 1500 km., and a nucleus of density
8:2, It would perhaps have been an advantage to have

SEISMIC EFFECTS 8I

treated the two separately. His main point is, however, that
while the corporeal tides may be computed at their equilibrium
values, the oceanic tides must be considered dynamically. The
differences of Hecker’s results in the N—S and E—W direc-
tions are attributed to the unequal action of the oceanic tides in
different directions, while 2 and & are substantially changed
from what they would be on the simple theory, by terms de-
pending on the oceanic tides. On certain assumptions with
regard to the depth of the ocean he finds that the general
rigidity of the earth may be from two to three times that of
steel, and that the results obtained from the semi-diurnal lunar
terms may thus be brought into accordance with the astrono-
mical data.

He concludes that the semi-diurnal lunar terms indicated
by seismographs are not of much real value in determining the
value of the earth’s rigidity.

We may remark in passing that somewhat similar numerical
results would follow by taking account of the Earth’s compres-
sibility for one of the most important points obtained by Love
(“ Problems of Geophysics’) is that the compressibility would
substantially increase the estimated value of % without much
affecting &, so that the experimental values when corrected for
compressibility would lead to improved concordance and to
higher values of the rigidity.

Schweydar’s next step is to argue that the nearly diurnal
lunar declination tide due to the potential
AB
ae
is better adapted to give the value of / — #, because ona certain
assumption as regards the depth of the ocean (which is not the
same as that made in the discussion of the semi-diurnal term)
the effect of the oceanic tides may be neglected.

He gives the following results obtained at Freiberg i.S.
with pendulums in azimuth 35° E of N and 55° Eof S.

O.=imeg (1-& é) sin w cos*k w sin 2 ¢ cos (¢+A-4,)

Pendulum I.
Observed. 000412 cos (¢ — 273°). Computed. 0’’-00493 (cos ¢ — 280°).
Pendulum II.
Observed. 0’00318 cos (¢ — 248°). Computed. 0’00363 (cos ¢ — 249).
6

82 MODERN SEISMOLOGY

This leads to (1-2+)=0°35 or h-k=0'15 and along
with =3 h this gives 4=0°38 leading to a general rigidity
about three times that of steel. This, however, neglects the
influence of compressibility.

As a whole the position with regard to earth tides as
indicated by movements of seismograph zero is rather unsatis-
factory. The doubt that may very reasonably be entertained
as to purely instrumental sources of error in the observations,
renders theoretical discussion somewhat futile until we know
exactly what the facts are.

It is gratifying to know that the International Seismo-
logical Association has in view experiments with horizontal
pendulums at different points of the Earth, which ought to
throw much light on the phenomena. It would also be useful
if other means of experimenting could be devised. A solid
surface undergoes, as we have seen, tilting of amount

oWe
Zz oe
This must result in an apparent change of position of any
star. But the effect is so small that it is hardly likely to be
detected by astronomical means. On the other hand a liquid
surface undergoes tilting of amount

a £0) a

The crerential tilting ee a cn surface and a solid
surface beneath it is

= ae -h+khk) ous
The suggestion I a to make is that such an arrange-
ment would show interference fringes parallel to the line of no
resultant tilt, and that the direction would thus change in the
course of the day. It might be practicable in this way to
study the operation of the variable tilt

I OW,
et —h+ ae

CEVA AME Raaxe:
STATISTICAL.

PERHAPS one of the most striking features revealed by the
systematic observations of earthquakes is the large number
detected by seismographs as compared with those earthquakes
which obtain notoriety in the public press. This is owing to
the fact that a large number of earthquakes are of but small
intensity, while of the large earthquakes or megaseisms the
majority fortunately occur at the bottom of the sea or in un-
populated regions without causing loss of human life.

Earthquakes whether large or small are of interest to the
seismologist.

The number recorded at any given station depends on the
position of the station, as well as the sensitiveness of the
instruments. As illustrating the number recorded in a non-
seismic region I give the numbers recorded at Eskdalemuir on
the Galitzin Seismographs in 1911.

Jan. Feb. Mar. Apr. May June Jul. Aug. Sept. Oct. Nov. Dec.
16 Io 8 19 19 20 24 23 28 32 16 20

The total for the year is 235. Most of these were small,
but sixteen at least deserved to be called megaseismic. In
particular the earthquake of 3 January which occurred in
Turkestan (41° N 77° E) wasso violent that the seismographs
at Pulkowa were broken, and even at Eskdalemuir the needle
of one of the galvanometers was thrown out of action.

I ought perhaps to say that none of the above earthquakes
were of local character. I was never able at Eskdalemuir to
detect any indication of earthquakes reported to have taken
place in Perthshire, and even the Glasgow earthquake of

December, 1910, which caused considerable public excitement
83 Onn

84 MODERN SEISMOLOGY

there, produced no perceptible effect on the Eskdalemuir seis-
mographs.

It is now the custom for observatories to exchange bulletins,
and for many years Milne has published (“British Assoc.
Reports ”) annual tables of data from all sources. An annual
table is now also issued under the auspices of the International
Association of Seismology (Strassburg). The importance of
such bulletins and tables can hardly be overrated.

They enable one to confirm or correct inferences and greatly
extend our knowledge of the number of earthquakes which
occur at all points of the earth. Milne estimates that the
annual output from all sources is nearly 60,000 earthquakes.

It has long been noted that the seismograms obtained at a
given station showan extraordinary similarity for separate earth-
quakes that occur in the same region of the Earth. In some
cases the seismograms might almost be superposed. This is
a matter deserving careful investigation as it points not perhaps
so much to a difference of the properties of the interior of Earth
in different directions, as to a characteristic origin of the earth-
quakes occurring in one and the same region.

While there is a general agreement that an earthquake is
caused by a rupture of the rocks within the earth’s crust, we
have no very definite knowledge as to the primary cause of the
rupture. It is not unnatural to look for such a cause in the
tidal stresses of solar and lunar origin. In particular we might
look for a preponderance of the number of earthquakes at the
times of syzygy of Sun, Earth,and Moon. Such investigations
have been made but do not appear to result in clear evidence
of such an association (Milne, “ Earthquakes,” p. 250). Another
way of dealing with the occurrence of earthquakes, and which
is well known in connexion with the analysis of meteorological
and magnetic data, is to express the observations by a Fourier
series in terms of the time, either solar or lunar. Such investi-
gations have been made by Knott (“ Proc. R.S.,” 1897) and by
Davison (“ Phil. Trans. A.,” Vol. 184, 1893). These have
been critically examined by Schuster (‘‘ Proc. R.S.,” 1897).

A question arises as to what should be included in the data
submitted to analysis. It is known that a large earthquake is

STATISTICAL 85

followed by a large number of minor shocks, and the point is
whether these minor shocks should be treated as separate
quakes or regarded as part of the primary shock. Again,
ought there to be a classification according to intensity? I
should doubt if agreement of opinion could be reached a@
priort. It seems to me to rest with the investigator to decide
whether he ‘shall classify and group or not, but it then rests
with him to show that he reaches a conclusion which is a real
contribution to knowledge.

There is a growing doubt whether a Fourier analysis of an
observational quantity is really the best way of expressing
results with a view to physical explanation of the cause, but
however that may be, we must agree with Schuster that there
is a right and a wrong way of making the Fourier analysis,
and that the right way is to take the data as they stand and
not to apply any preliminary smoothing process. It appears to
me that if a smoothing process was permissible it would,
carried to excess, be an argument for never making observa-
tions at all.

It is not sufficient to compute the Fourier co-efficients. We
have to show that any term so obtained is substantially greater
than what might be expected as the result of fortuitous occur-
rence. The criterion given by Schuster is as follows :—

“Tf a number xz of disconnected events occur within an
interval of time T, all times being equally probable for each
event, and if the frequency of occurrence of these events is ex-
pressed in a series of the form

(¢-¢ (¢-¢
a1 +p, COS 27 os FE NN i Ori COS) 2) 7707s —

the probability that any of the quantities p has a value lying

between p and p+ 6p is
ee np?] 4

n
5? 6p e

and the ‘expectancy’ for p is
|r|.”

On this basis Schuster finds that the lunar terms obtained
by Knott must be discarded, but on the other hand the annual

86 MODERN SEITSMOLOGY

periodicity with a maximum in winter and the diurnal period-
icity with a maximum about noon obtained by Davison from
earthquake statistics may be regarded as fairly well established.

Although the small table at the beginning of this chapter
is too limited to justify any general conclusion it will serve to
illustrate the application of Schuster’s method.

I find that the Fourier expression is given with sufficient
accuracy by

N = 20 (1 +04 cos ¢+ 120°+ O'1 cos 2¢+ 120°)
where ¢ is the time reckoned from 1 January at the rate of 30°
per month.

The expentancy is 7/235 or 0°12, and we should thus
argue that the semi-annual term is worthless while the annual
term with its maximum at the end of August is important.

The practical application of Fourier analysis to observational
quantities is really very simple, and since it does not usually
find a place in physical textbooks, a few remarks about it may
not be out of place here.

If the observed quantity fis to be expressed by means of
a Fourier series

n= ow 5
f= tat _ (a, cos 20+ , sin 26)

between the limits o and T where 0 = 27#/T, we have
ip
w= | adi
0
27rut

ta,l -| x9 COS —7 at
0
Beal = io sin “dt
0
If f(o) =f(T) then no difficulty occurs; but if, as generally
happens with observed quantities, (0) +-/(T) then the function
fis not strictly periodic in time T, and this at once sets a
limit. The series represents the function / between the limits
but not at the limits, for the series then gives 4$/f(0)+/(T)} at
the limits.
This difficulty is often dealt with in practice by assuming
that the difference f(o) f(T) is incident linearly, during the
interval T and it is subtracted from / before analysing. This

STATISTICAL 87

merely confuses the issue, and it is less objectionable to take
the function fas observed and to remember that in so far as
J(o) differs from f(T) the representation is incomplete.

The data are, however, usually presented in the form of
hourly values in solar time or lunar time according to the
source we have reason to suspect as contributing to the effect.
If thenf,, fi. . . fa represent the values of / for the various
hours the formule become

3

24a,=4f+fu)t%,, -

m =I

Ven
Mm = 23 °
12an=4$(fotSoa) + > ey Jm COS (mn 15°)
126,= 5)

M = 23

Tm Sit. (em. 15).

m=t1

The numerical process is simple since the terms collect
into groups with the same numerical coefficients.

In these expressions /,, may be the actual value at the
hour 7 or the mean for an hour centering at 7. Neither is
strictly correct for an infinite Fourier series although the
former is correct for a limited series, ending with z= 24.
Here again the representation is incomplete when /, +/,.

Unless the quantity / varies in a very regular manner, one
day’s observations would not be sufficient, and the hourly
values are then averaged for say a month. A similar process
would then be applied to the coefficients so obtained to de-
termine their annual periodicity.

This method, however, fails unless the day or the year are
real periods of the phenomena; and may, as we have seen, give
a false impression of periodicity unless Schuster’s criterion can
be applied.

The only general method of detecting periodicity is due to
Schuster) quote from) his’ paper (Proc, Ra Saale) skowet ly
be a function of ¢, such that its values are regulated by some
law of probability, not necessarily the exponential one, but
acting in such a manner that if a large number of values of
¢ be chosen at random there will always be a definite fraction
of that number depending on 4 only, which lie between ¢, and
t,+ I, where T is any given time interval.

88 MODERN SEISMOLOGY
“Writing
f4ut+T ; ty+T .
A-| y cos kt dt and B=) y sin kt at
ty 1

and forming R = (A? + B?)?

the quantity R will, with increasing values of T, fluctuate
about some mean value, which increases proportionally to T},
provided T is taken sufficiently large.

“ Tf this theorem is taken in conjunction with the two follow-
ing well-known propositions :—

“(1) Ify=cos é, R will, apart from periodical terms increase

proportionally to T.
“(2) If y=cos AZ, A being different from %, the quantity R
will fluctuate about a constant value,
it is seen that we have means at our disposal to separate
any true periodicity of a variable from among its irregular
changes, provided we can extend the time limits sufficiently.”

The method of applying this will be found in “ Camb. Phil.
Trans.,” Vol. 18, 1900.

I have referred to this problem specially, because statistics
about earthquakes are rapidly increasing in number and ac-
curacy, and the search for periodicity will again be taken up.
It is desirable that the search should proceed on the lines in-
dicated by Schuster.

I understand that by application of this method, Prof.
Turner (“ Brit. Assoc.,” 1912) finds evidence of a 452 day
period of earthquake activity. The result is interesting as it
is so near the Chandler period of precessional nutation, and
here we may fitly close the volume with a quotation from
Milne (“ Earthquakes,” 6th edition, 1913, p. 377): “ Speak-
ing generally, so far as I know, neither tidal, barometric,
thermometric, solar, lunar, or other epigene influences beyond
those mentioned, show a relationship to the periodicity or
frequency of megaseismic activity. Their frequency is ap-
parently governed by activities of hypogene origin.”

ABERDEEN: THE UNIVERSITY PRESS

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Prate 13. DARDANELLES EARTHQUAKE. GALITZIN SEISMOGRAM (ESKDALEMUIR) VERTICAL COMPONENT.
1912. SEPTEMBER 13th, * Clock 18. slow.

LN eee tual
A cam Ss
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Puate 11, ZANTE. EARTHQUAKE. GALITZIN SEISMOGRAM (ESKDALEMUIR) HORIZONTAL COMPONENTS.
The N.S. component alternate

8 with the E,W. component. and Is distinguished by

1912. JANUARY 24th. Clock 20s. fast.

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Se
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DARDANELLES EARTHQUAKE, GALITZIN SEISMOGRAM (ESKDALEMUIR) HORIZONTAL COMPONENTS,

The N.S. component alternates with the E.W. component, and Is distinguished by
the automatic time breaks.

PLATE 12,

1912. SEPTEMBER 13th,

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