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Proceedings of the

Royal Society of Edinburgh

/^t,i

T ^ • >

HARVARD UNIVERSITY.

LIBRARY

OF THE

MUSEUM OP COMPARATIVE ZOOLOGY.

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PROCEEDINGS

OP THB

ROYAL SOCIETY OF EDINBURGH.

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PROCEEDINGS

THE ROYAL SOCIETY

EDINBURGH.

VOL. XXV.

(IN TWO PARTS.)

PART I.

(Coiitaiuing pages 1-592.)

NOVEMBER 1903 to MARCH 1905.

""EDINBURGH;

PBINTED BY NEILL AND COMPANY, LIMITED.

HDCOCOVI.

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CONTENTS.

PA6B

Election of Office-Bearers, Session 1903-4, .... 1

The People of the Faroes. By Nelson Annandale, B.A. (Oxon.).
Communicated by Professor D. J. Cunningham, F.R.S. Issued
separately November 30, 1903, ..... 2

Seiches observed in Loch Ness. By R Maclagan-Wedderburn.
C<mimunicated by Professor Chrystal. Issued separately Janu-
ary 15, 1904, . . . .26

The Bull Trout of the Tav and of Tweed. By W. L. Calderwood.
(With a Plate.) Issued separately January 30, 1904, . . 27

The Relative Efficiency of certain Methods of performing
Artificial Respiration in Man. By E. A. Schater, F.R.S.
(With a Plate.) Issued separately January 29, 1904, . . 39

Physico-Chemical Investigations in the Amide Group. By
Charles E. Fawsitt, Ph.D., B.Sc. (Edin. and Lond.). (7am-
municated by Professor Crum Brown. Issued separately Feb-
ruary 6, 1904, ....... 51

The Theory of Cfeneral Determinants in the Historical Order of
Development up to 1846. By Thomas Muir, LL.D. Issued
separately February 12, 1904, ..... 61

Man as Artist and Sportsman in the Pakeolithic Period. By
Robert Munro, M.A., M.D., LL.D. (With Eleven Plates.)
Issued separately February 13, 1904, . .92

The Theory of Continuants in the Historical Order of its Develop-
ment up to 1870. By Thomas Muir, LL.D. Issued separately
February 26, 1904, ...... 129

On the Origin of the Epiphysis Cerebri as a Bilateral Structure
in the Chick. By John Cameron, M.B. (Edin.X M.R.C.S.
^Eng.), Carnegie Fellow, Demonstrator of Anatomy, United
Collie, University of St Andrews. Communicated by Dr
W. G. Aitchison Robertson. Issued separately March 17, 1904, 160

Theorem regarding the Orthogonal Transformation of a Quadric.
By Thomas Muir, LL.D. Issued separately March 17, 1904, . 168

Ocean Teniperatures and Solar Radiation. By Professor C. G.
Knott Issued separately April 4, 1904, . .173

On Deep-water Two-dimensional Waves produced by any given
TTiitiAting Disturbance. By Lord Kelvin. Issued separately
April 4, 1904, ^186

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vi Contents.

PAGE

Some Field Evidence relating to the Modes of Occurrence of
Intrusive Rocks, with some Remarks upon the Origin of
Eruptive Rocks in general. By J. Q. Goodchild, of the
Geological Survey, F.G.S., F.Z.S., Curator of the Collection
of Scottish Mineralogy in the Edinburgh Museum of Science
and Art. Communicated by R. H. Traquair, LL.D., M.D.,
F.R.S. leeued separately May 20, 1904, . .197

Note on the Standard of Relative Viscosity, and on " Negative
Viscosity." By W. W. Taylor, M.A., D.Sc. Communicated by ;

Professor Crum Brown. Issued separately June 16, 1904, 227

The Viscosity of Aqueous Solutions of Chlorides, Bromides, and
Iodides. By W. W. Taylor, M.A., D.Sc, and Clerk Ranken,
B.Sc. Communicaitd by Professor Crum Brown. Issued
separately June 16, 1904, ..... 231 j

On the Date of the Upheaval which caused the 25-feet Raised j

Beaches in Central Scotland. By Robert Munro, M.A., M.D.,
LL.D. Issued separately June 18, 1904, ... 242

The Complete Solution of the Differential Equation of Jin].
By the Rev. F. H. Jackson, H.M.S. "Irresistible." Com-
municated by Dr Wm. Peddie. Issued separately August 16, ;
1904, 273

A Differentiating Machine. By J. Erskine Murray, D.Sc.
Issued separately August 15, 1904, .... 277

On the Thermal Expansion of Dilute Solutions of certain |

Hydroxides. By George A. Carse, M.A., B.Sc. Communicated |

by Professor MacGregor. Issued separately August 16, 1904, . 281

Effect of Transverse Magnetization on the Resistance of Nickel at
High Temperatures. By Professor C. G. Knott. Issued
separately July 30, 1904, ..... 292

Observations on some A^ed Specimens of Sagartia trt^lodytes, and
on the Duration of Life in Ccelenteratea By J. H. Ashworth,
D.Sc, Lecturer in Invertebrate Zoology in the University of
Edinburgh, and Nelson Annandale, B.A., Deputy-Super-
intendent of the Indian Museum, Calcutta. Communicatea by
Professor J. C. Ewart, M.D., F.R.S. Issued separately July 21,
1904, 296

Note on the Molecular Condition of Nickel (and Iron) de-
magnetised by decreasing Reversals. By .fames Russell.
Issued separately August 22, 1904, .... 309

On the Front and Rear of a Free Procession of Waves in Deep
Water. (Cmtinued from Proc R.S.E., Feb. 1st, 1904.) By
Lord Kelvin. Issued separately August 22, 1904, . .311

Some Results in the Mathematical Theory of Seiches. By Pro-
fessor Chrystal. Issued separately October 6, 1904, . 328

A New Form of Spectrophotometer. By J. R. Milne, B.Sc,
Carnegie Scholar in Natuial Philosophy, Edinburgh University. i

Issued separately November 5, 1904, .... 338

A New Form of Juxtapositor to bring into Accurate Contact
the Edges of the two Beams of Light used in Spectro-
nhotometrv, with an aj)t)lication to Polarimetry. By J. R.
Milne, B.oc., Carnegie Scholar in Natural Philosophy. Issued
separately January 17, 1905, ..... 365

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Contents. vii

PAGE

The Three-line Detenninants of a Six-by-Three Array. By
Thomas Muir, LL.D. Issued separately January 20, 1905, . 364

The Sum of the Signed Primary Minors of a Determinant. By
Thomas Muir, LL.D. Issued separately January 20, 1905, . 372

Ciystallographical Notes. By Hugh Marshall, D.Sc, F.R.S.

Issued separately February 1, 1905, . . . .383

The Effect of Simultaneous Removal of Thymus and Spleen in
young Quinea-pigs. By D. Noel Pal on and Alexander Goodall.
(From the Laboratory of the Royal College of PhysidanSy Edin-
burgh,) Issued separately February 1, 1905, . . . 389

Networks of the Plane in Absolute Geometry. By Dimcan M.
Y. Sommerville, M.A., B.Sc., University of St Andrews.
(Abstract,) Communicated by Professor P. R. Scott Lang.
Issued separately February 1, 1905, . . .392

A Specimen of the Salmon in transition from the Smolt to the
Grilse Stage. By W. L. Calderwood. (With Two Plates.)
Issued separately February 1, 1905, .... 396

A Comparative Study of the Lakes of Scotland and Denmark.
By Dr C. Wesenberg-Lund, of the Danish Fresh- water Biol(^cal
Station, Frederikwal, near K. Lyngby, Denmark. Vom-
municated by Sir John Murray, K.CTB., F.K.S. (From tJie
Danish Fresh-water Biological Laboratory, Frederiksdal,) (With
Two Plates.) Issued separately March 3, 1905, . .401

Variations in the Crystallisation of Potassium Hydrogen Succinate
due to the presence of other metallic compounds in the Solution.
(Preliminary Notice.) By Alexander T. Cameron, M.A.
Communicated by Dr Hugh Marshall, F.R.S. Issued separately
February 4, 1905, ...... 449

A Laboratory Apparatus for Measuring the Lateral Strains in
Tension and (Jompression Members, with some Applications
to the Measurement of the Elastic Constants of Metals. By
R G. Coker, M.A. (Cantab.), D.Sc (Edin.), F.R.S.E.. Professor
of Mechanical Engineering and Applied Mathematics, City and
Guilds Technical CoUege, Finsbury, London. (With a Plate.)
Issued separately March 3, 1905, . .452

On Astronomical Seeing. By Dr J. Halm, Lecturer in Astronomy
in the University of Edinburgh. Issued separately March 3,
1906, 458

On the Graptolite-bearing Rocks of the South Orkneys. By
J. H. Harvey Pirie, B.Sc, M.B., Ch.B. Communicated 6y
Dr Home, F.R.S. With a Note by Dr Peach on Specimens
from the South Orkneys. Issued separately March 30, 1905, . 463

A Possible Explanation of the Formation of the Moon. By
George Romanes, C.E. Issued separately March 30, 1906, \ 471

On Pennella: a Crustacean parasitic on the Finner Whale
(Bakenopiera musculus). (Abstract.) By Sir William Turner,
K.C.B.,LL.D. Issued separately March 30, 1906, . 480

The Diameters of Twisted Threads, with an Account of the
History of the Mathematical Setting of Cloths. By Thomas
Oliver, B.Sc (Lond. & Edin,). Communicated by Dr C. G.
Knott Issued separately April 8, 1906, .481

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viii Contents.

PAGE

A Study of Three Vegetarian Diets. By D. Noel Paton and
J. C. Dunlop. {From the Research Laboratory of the Royal
College of Physicians^ Edinburgh.) Issued separately April 8,
1905, ........ 498

Continuants whose Main Diagonal is Univarial. By Thomas Muir,
LL.D. Issued separately April 8, 1905, . . . 507

On Professor Seeliger's Theory of Temporary Stars. By J. Halm,
Ph.D., Lecturer on Astronomy in the University of Edinburgh,
and Assistant Astronomer at the Royal Observatory. Issuea
separately April 15, 1905, ..... 613

Some Suggestions on the Nebular Hypothesis. By J. Halm,

Ph.D. Issued separately April 15, 1905, . . . 663

Deep Water Ship- Waves. {Continued from Proc. R S.E., June
20th, 1904.) fey Lord Kelvin. Issued separately April 18,
1905, 662

On Two Liquid States of Sulphur Sa and S^ and their Transition
Point. By Alexander Smith. {Abstract.) Issued separately
April 18, 1905, ....... 588

The Nature of Amorphous Sulphur, and Contributions to the
Study of the Influence of Foreign Bodies on the Phenomena
of Supercooling observed when Melted Sulphur is suddenly
Chilled. By Alexander Smith. {Abstract.) Issued separately
April 18, 1905, . . . . . .690

For Index to Part I. see end of Part II.

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PROCEEDINGS

OP THE

ROYAL SOCIETY OF EDINBURGH.

SESSION 1903-4.

No. I.] VOL. XXV. [Pp.i-ia8.

CONTENTS.

PAOS

The People of the Faroes. By Nblson Annandalb, B.A.
(Oxoil). Communicated by Professor D. J. Cunning-
ham, F.R.S., ...... 2

{Issued separately November 80, 1903.)

Seiches obeerved in Loch Ness. By E. Maclagan-Wedder-

BURN. Communicated by Professor Chrystal, 25

{Issued separately January 15, 1904.)

The Bull Trout of the Tay and of Tweed. By W. L.

Calderwood. (With a Plate), . .27

(IssiLed separately January 30, 1904.)

The Relative EflSciency of certain Methods of performing
Artificial Respiration in Man. By E. A. Schafur,
F.R.S. (With a Plate), .... 59

{Issued separately January 29, 1904.)

[Continued on page iv of Cover,

^EDINBURGH:
PuBUSHXD BY ROBERT GRANT & SON, 107 Pbinces Street, and
WILLIAMS k NORGATE, 14 Henrietta Street, Covent Garden, London.

Price Seven Shillings.

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fiEGULATIONS REGARDING THE PUBLICATION OF
PAPERS IN THE PROCEEDINGS AND TRANS-
ACTIONS OF THE SOCIETY.

Thb Council beg to direct the attention of authors of communications to
the Society to the following Regulations, which have been drawn up in
order to accelerate the publication of the Proceedings and Transactions,
and to utilise as widely and as fairly as possible the funds which the
Society devotes to the publication of Scientific and Literary Researches.

1. Manuscript of Papebs. — ^As soon as any paper has been passed
for publication, either in its original or in any altered form, and has been
made ready for publication by the author, it is sent to the printer,
whether it has been read or not.

The * copy ' should be written on large sheets of paper, on one side
only, and the pages should be clearly numbered. The MS. must be
easily legible, preferably typewritten, and must be absolutely in its final
form for printing ; so that corrections in proof shall be as few as possible,
and shall not cause overrunning in the lines or pages of the proof. All
tables of contents, references to plates or illustrations in the text, etc.,
must be in their proper places, with the page numbers left blank ; and
spaces must be indicated for the insertion of illustrations that are to
appear in the text.

2. Illustrations. — All illustrations must be drawn in a form im-
mediately suitable for reproduction; and such illustrations as can be
reproduced by photographic processes should, so far as possible, be
preferred. Drawings to be reproduced as line blocks should be made
with Indian ink (deadened with yellow if of bluish tone), preferably on
fine white bristol board, free from folds or creases ; smooth, clean lines
or sharp dots, but no washes or colours should be used. If the drawings
are done on a large scale, to be afterwards reduced by photography, any
lettering or other legend must be on a corresponding scale.

If an author finds it inconvenient to furnish such drawings, the Society
will have the figures re-drawn at his expense ; but this will cause delay.

When the illustrations are to form plates, a scheme for the arrange-
ment of the figures (in quarto plates for the Transactions, in octavo for
the Proceedings) must be given, and numbering and lettering indicated.

3. Peoops. — In general, a first proof and a revise of each paper will
be sent to the author, whose address should be indicated on the MS.
If further proofs are required, owing to corrections or alterations for
which the printer is not responsible, the expense of such proofs and
corrections will be charged against the author.

All proofs must, if possible, be returned within one week, addressed to
The Secretary, RoycU Society, Mound, Edinburgh, and not to the printer,

{Continued on page iii ofCov^r,

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^R U

PROCEEDINGS

OF THE

HOYAL SOCIETY OF EDINBURGH.

VOL. XXV. 1903-4.

Thb 1218T Session.
GENERAL STATUTORY MEETING.
Monday, 26th October 1903.
The following Council were elected : —

President.
The Right Hon. Lord KELVIN, G.C.V.O., F.R.S.

Fice- Presidents,
The Rev. Professor Duns, D.D.
Prof. James Geikie, LL.D., F.R.S.
The Hod. Lord M*Laren, LL.D.
TheRey. Professor Flint, D.D.

General Secretary^Frofeaaor George Chrystal, LL.D.

Robert Munro, M.A., M.D., LL.D.
Sir John Murray, K.C.B., LL.D.,
•F.R.S.

Secretaries to Ordinary Meetings,

Professor Crum Brown, F.R.S.

Ramsay H. Traquair, M.D., LL.D., F.R.S.

Treasttrer—TniLiF R D. Maolaoan, F.F.A.

Curator of LUnrary and Museum — Alexander Hugh an, M.A.,
LL.D., F.RS.

Ordinary Members of Council,

R. Tbaill Omond, Esq.
DrGso. A. Gibson, F.R.C.P.E.
Sir Abthur Mitchell, K.C.B.
Professor J. G. MacGbegor, LL.D.,

F.RS.
John Horne, LL.D., F.RS.
C. G. Knott, D.Sc.
Abthub T. Mastsrman, M.A.,

D.So.

Professor Ralph Stockman, M.D.,

F.RCP.E.
Professor James Walksb, D.Sc.,

Ph.D., F.RS.
Professor Andrew Gray, MA.,

LL.D., F.RS.
Robert Kid8Ton,-F.R.S., F.Q.S.
Professor D. J. Cunningham, M.D.

LL.D., F.RS.

PBGC. ROY. SOC. EDIN. — VOL. XXV.

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Proceeditigs of Boyal Society of Edinburgh. [suss.

The People of the Faroes. By Nelson Annandale, B.A.
^Oxon.). Communicated by Professor D. J. Cunningham, F.RS.

(MS. received Oct. 7, 1908. Eead Nov. 2, 1908.)
Part I. — Anthropometbical.

The physical anthropology of the Faroes has recently been
described in a very elaborate manner, as far as the island of
Suderoe is concerned, by Dr F. J0rgensen (1), who was resident
there as a medical man for some years. While pointing out, how-
ever, that the people of Suderoe differ considerably from those of the
* northern islands,' he only gives a comparatively small series of
data regarding the latter, nor does he state to which of the northern
islands the men he examined belonged, or even whether they
came from one island or from several. Apart from Suderoe, there
are sixteen inhabited islands (fig. 1) in the group, and between some
of them very little communication exists even at the present day.
In historical accoimts of the Faroes the six following islands are
usually called the * northern isles,* — viz., Kalsoe, Kunoe, Boroe,
Wideroe, Fugloe, and Svinoe, — but I take it that Dr J0rgensen
would include at least Osteroe, Stromoe, and Waagoe also. His
elaborate, laborious, and presumably accurate tables serve so
well to point the moral that until a uniform method, a imiform
standard, and a uniform set of anthropometrical instruments are
adopted by anthropometrists of all nationalities fiiicd work in this
branch of science will be impossible, that I have thought it well
to put on record a small series of measurements taken by myself
in the Faroes recently, and at the same time to point out wherein
some of the data pretty generally adopted fail in accuracy, differing
with the observer as well as the observed.

My measurements were taken in Thorshavn, the chief town in
the islands, in August 1903, upon twenty adult males. The only
value that can be claimed for so small a series is that it was
obtained at a definite period and within a very limited area, for
the men examined were all resident in the town. The length and
breadth of the head, the length and breadth of the nose, the

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1903-4.] Mr N. Annandale on the People of the Faroes, 3

Wtf«CMAf^CC>. Wx

THE FAROES

^ I * I t A

, THe MONK

Fig. 1.

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4 Proceedings of Royal Society of EdirOmrgK . [bbs*.

length of the face, the bizygomatic and bigonial breadths, were
taken with callipers of a simple type, while the height of the head,
the auriculo-nasal and the auriculo-alveolar lengths were taken
by means of Professor D. J. Cunningham's craniometer ; all these
measurements, therefore, were obtained directly, not by projec-
tions or estimations. The statures given can only be approximate,
as all my subjects were measured with shoes or boots on their feet,
and I was obliged to extract a varying number of millimetres in
accordance with the kind of footgear worn.

The individuals measured are too few to make a rigid mathe*
matical examination of the data regarding them legitimate, and
they can give at best but an approximation to the race characters
of the people of Thorshavn. With so small a series perhaps the
rough and ready method of examination by the aid of means
and extremes is the best, as having the least appearance of
finality.

The length of the head, as may be seen by the table, varies in
the twenty adult men from 176 to 157 millimetres, while the
mean of the series is 166, only '5 less than the mean of the two
extremes. Though the extremes in the breadth of the head are
less divergent from one another than those of the length, their
mean is more divergent from that of the series, the former ex-
ceeding the latter by '9, and the variation is also greater. The
mean index derived from these two measurements varies from
86*8 to 76*3; twelve of the men are brachycephalic, though five
of these have an index between 80 and 81, while the remaining
eight are mesaticephalic, only three being between 78 and 80 ; the
mean, 80*6, is brachycephalic. If the skuUs of these twenty men
had been examined instead of their heads, it is probable that not
more than four would have been brachycephalic, and that two
would have been dolichocephalic ; the mean index would certainly
have been mesaticephalic. The mean cephalic index of Dr
j0rgensen's series of thirty-three men above the age of twenty
from the northern islands is 80*4, and the extremes are 75'4 and
85*3 ; and the variation, as might be expected in a larger series,
in slightly greater than in mine, while the differenBe between the
mean of the series and that of the extremes is less. Taking the
two series together, the mean is 80*7, and the mean of the extremes

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1903-4.] Mr N. Annandale on the People of the Fa/roes.

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Proceedings of Boyal Society of Edvnbiirgh, [i

9
e3

§
^
H

Onathie Index,

105-6

•
S

Nasal Index,

?
«

BigoniaZ Index,

FacUd Index,

FeHical Index.
CephcUie Index,

1-4

S5
&

Auriculo-alveoUr
Length.

Auriculo-Daaal
LeDgth.

Breadth of Nose.

Length of Nose.
Bigonial Breadth.

Bizygomatic
Breadth.

Length of Face.

Height of Head.
Breadth of Head.

El:

Length of Head.

cm. ' niin.
176 J202

Stature.

8*
1

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1908-4.] Mr N. Annandale on the People of the Faroes. 7

Table of other Particulars.

Serial
Number.

Name.

Age.

Colour
of Eyes.

OiUmr
of Hair.

Andreas Diurhuas,
Jacob Jacobeen, .

brown

Andreas Jacobsen,

blue

brown

Jacob MikkelsoD, .

blue

fair

Christian Christiansen,

blue

fair

Ole Hansen,

blue

brown

Rasmns Andreassen, .

blue

fair

Paul Nichodemussen, .

blue

fair

Joen Gjoueraa,

blue

fair

Paul Hansen,

blue

dark

Tomas Yule Nichalsen,

blue

red

William Paulsen, .

bine

brown

Daniel Samuelsen,

blue

fair

Peter Hana SiSrensen, .

dark brown

fair

Andreas Olsen, .

dark

Peter Haraldsen, .

bine

brown

Hans Mikkelsen, .

dark grey

black

NilsJoensen,

blue

fair

Djone Isaksen,

blue

brown

Augnst Mouriksen,

...

blue

brown

is 81*1. If we consider 75 as the upper limit of dolicbocephaly
and 80 of mesaticephaly, eighteen of Dr J0rgensen'8 are brachy-
cephalic and fifteen mesaticephalic. We may say, therefore, that
were a large series of skulls of the people of the Faroes, leaving
the island of Suderoe out of account, to be examined, it is probable
that the great nm'ority of them would be found to be mesati-
cephalic, while a comparatively small number would be dolicho-
cephalic, and a less small number brachycephalic. Dr J0rgen8en's
data show that the proportion of individuals with dolichocephalic
or low mesaticephalic heads would be greater in Suderoe than
elsewhere in the Faroes, as is noted below.

The vertical height of the head, measured between the vertex
and a line joining one external auditory meatus to the other, is,
in every individual in my series, less than the greatest parieto-
squamosal breadth, and in every case but two, very considerably so^
Professor Cunningham's craniometer permits this measurement to
be taken on the living person with considerable accuracy, but the
question how far it corresponds to the basi-bregmatic height of the
skull is a difficult one. The centre of the external auditory meatus
is certainly, in most cases, several millimetres higher than the

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8 Proceedings of Royal Society of Edinburgh. [siss.

basion, but the limits within which this difference in level varies
will be discussed later. At any rate, the thickness of the soft
tissues of the scalp and the hair must quite compensate for it, if
they do not cause the vertical height, taken as described, to be
slightly greater, as is possible, than the true basi-bregmatic height.
It is very unlikely, however, that in more than two cases at most
the basi-bregmatic height of the individuals under discussion
would equal their parieto^uamosal breadth in the skull, and it
is improbable that this would be found to be the case, could the
skulls be measured, in a single instance. In the living men the
mean breadth-height index of the head is 87*9, and the extremes
are 98*6 and 77*9 ; the mean height is 136*4, and the extremes
are 151 and 126 mm.

The length of the face, measured directly with the callipers
between the bridge of the nose and the tip of chin, varies from
134 to 106 mm., with a mean of 122*3 mm., while the interzygo-
matic (or bizygomatic) breadth varies between 156 and 152 mm. ;
in two cases out of twenty the length of the face is greater than
the bizygomatic breadth, and in one the two measurements are
equaL The complete facial index, calculated from these two
measurements, varies from 101*8 to 77*9, and the man with the
shortest face, which is considerably shorter than any other in the
series, has the lowest index, though the man with the longest face,
which is not so much longer than any other, has only the third
index, the breadth being equal to the length. The measurements
for the cephalic and vertical indices are easy to take with a fair
degree of accuracy, and do not depend upon the play of the sub-
ject's features ; but it is far otherwise with those for the facial
index — an unfortunate fact, seeing that, provided all the measure-
ments are taken by the same person, no index is of greater import-
ance as a racial character. It makes all the difference in the
world whether the length of the face is taken directly, or by pro-
jection from the vertex to the nasion and to the chin and by sub-
sequent calculation, and it makes just as much difference whether
the features of the subject are perfectly at rest or in any way
distorted. I am not aware in what manner exactly Dr J0rgensen
obtained what he calls the "longitudo naso-menthalis," or what
degree of pressure he exerted in measuring his *' latitudo bizygoma-

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1903-4.] Mr N. Annandale on the People of the Faroes. 9

ticusy" but the fact remains that the facial index he calculates
from these measurements differs considerably from that which I
obtain from the nasio-mental length and bizygomatic breadth. Of
course we measured different individuals, possibly from different
islands — though at present I am only considering the thirty-three
men from his series to whom I have referred — and I have known
the facial index to be very different in two villages no further
apart than, say, Thorshavn and Klagsvig, but this was in the
Malay Peninsula, in a district where there was far more reason to
«uspect admixture of foreign blood in different degrees in neigh-
bouring localities ; and the difference in the figures between the
two series from the Faroes, without including Suderoe, is so great
that I cannot help thinking that either my own measurements,
Dr J0rgensen's, or both, must have been taken in a manner not
altogether satisfactory. The mean index of his thirty-three sub-
jects, calculated from the figures he gives, is about 11 per cent,
lower than that of my series ; and while he makes a very large
proportion of his subjects mesoprosopic,^ and a considerable pro-
portion actually chamaeoprosopic,^ eleven out of my twenty are
leptoprosopic,^ five mesoprosopic, and only four chamseoprosopic.
In his series no man has a face of which the length even approaches
closely to the breadth, and the mean of his series is chamaeopro-
«opic, while that of mine is leptoprosopic. This is a very consider-
able difference ; and although the facial index taken on the skull
is probably, at any rate in normal individuals, considerably higher
than if taken on the living individual, as the combined thickness
of the soft tissues on both sides of the face is probably greater
than that of the soft tissues at the tip of the chin, yet I am
inclined to think that the Faroe men have narrower faces than Dr
J0rgensen*s figures would suggest, though it is quite possible that
my own data may err in the opposite direction. What strikes one
in a visual examination in the faces of a group of Faroemen, as
distinguishing them at a glance from those of the Icelanders, and,
to a less extent, from that of one type of Dane, is the narrow-
ness of the zygomata, and the oval outline longitudinally.
It should be noted, however, that in Icelanders the cheek bones

* My usage of these terms is that adopted by Sir William Turner in his
recent papers {Trans. Boy. Soe, Edinburgh, vol. xL, 1908, part iii. pp. 605, 606).

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10 Proceedings of Royal Society of Edinburgh. [j

are often very prominent, and the face is frequently so flat, the-
eyes are so narrow, and the mouth is so big, that one is inclined
to speculate as to the possibility of environment having induced
some latent Mongoloid strain, inherited from prehistoric times, ere
Iceland was colonised, to develop, or even whether environment
alone could possibly have produced a similarity to the Esquimaux,
not inherited at all. However, the time has not come to settle, or
even to seriously discuss, such questions, and, in any case, they are
beyond the point in dealing with the Faroemen, in whom there is
little, if any, trace of any such phenomenon. All that can be said
with reference to the point at issue is, that two observers who have
examined the faces of the Faroemen get very different results with
regard to the facial index, and that there is reason to believe
that were a large number of Icelanders examined, they would be
foimd to have considerably broader and flatter faces than the
Faroemen.

The bigonial breadth is another measurement that depends very
largely upon the individual observer, and probably has a very dif-
ferent relationship to the same measurement on the skull in different
subjects. In taking it on the living person it is by no means easy
to regulate the pressure exerted by the points of the callipers upon
the soft tissues, and the degree or absence of such pressure makes
a very great difference in the results obtained, while the extent to
which the muscles which work the jaw are developed also influences-
them considerably. Personally, I now make it a practice to draw
the skin as tight as possible in taking this measurement, and to
press in the points of the callipers as far as they will go without
injuring the subject, believing that in this way it is possible to get
a more uniform standard of comparison, both as regards different
individuals and as regards the difference between the skull and the.
living head. It is probable, however, that many anthropometrists
take care to exert as little pressure as possible, though it is obvious
that if this be done, the measurement must vary even more with
the muscular development and the amount of adipose tissue than
^vith the true breadth of the skeletal support. The mean bigonial'
breadth in my series, taken as described, is lll'S mm. — 21*6 mm.
less than the mean bizygomatic breadth — and the extremes are 128^
and 100. The bigonial index, that is, the index obtained by the^^

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1908-4.] Mr N. Annandale on the People of the Faroes. 11

, bigonial breadth X 100 . .,. ,....,

fonnma -^. rr-r — ttv— , vanes witnm narrower limits than

bizygomatic breadth '

the facial index, or than either of the separate measurements from

which it is calculated, showing that the longitudinal shape of

the face is fairly constant ; the mean is 83*87, and the extremes

are 92*6 and 76*8. This is by no means a high index, and it

probably shows that the faces of the Faroemen, as might be

expected from a visual examination, narrow considerably from

above downwards, though they are by no means broad across the

cheek bones ; but it must be borne in mind that my method of

taking the bigonial breadth is very possibly not the general one,

and 1 have been able to find very little information as to how

it is obtained by other anthropometrists.

The measurements of the nose, again, seem to vary considerably
with the individual observer ; and, as the figures which express
them are comparatively small, the variation in the index is
magnified proportionately by an error or difference of method.
In European peoples there is rarely any difficulty in finding the
points of measurement with fair approximation, but this is
always provided that the subject's face is in a state of perfect
repose, and that no imdue pressure is exerted on the callipers,
especially in taking the breadth. In my opinion, it is quite
impossible for the ordinary observer to take these measurements to
within half a millimetre, as it has been suggested by Professor
Haddon (2) that he should do. These things being so, 1 am surprised
at the extent of agreement, rather than disagreement, with regard
to the nasal index, as estimated on the living person by different
observers. The mean nasal index of my series of Faroemen is
65*66, and the extremes are 78*8 and 55*0, so that they appear
to be a very distinctly leptorhine people. The mean of Dr
J0rgensen's series from the northern islands is 67*5, and the
extremes are 81*1 and 58'6. In shape the nose is generally
straight and prominent, the rather flat, coarse type common in
Iceland occurring but seldom, and the Roman or aquiline being
rarely if ever seen, in the Faroes.

As already stated, the auriculo-nasal and the auriculo-alveolar
lengths were taken by means of Professor Cunningham's
craniometer between the external auditory meatus (or rather

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12 • Proceedings of Boyal Society of Edinburgh, [sbss.

the line joining the centre of this opening on one side of
the head to the same point on the other) and the hridge of the
nose, and the central point of the upper jaw between the two
central incisor teeth, respectively, the upper lip being lifted out
of the way in the latter measurement. The index calculated from
these two measurements appeared to make the people far more prog-
nathic than I would have expected, if the centre of the auricular
orifice, as has often been assumed, corresponded, as far as the measure-
ments from which the gnathic and vertical indices are calculated are

Fig. 2. — Diagram illustrating the relation of measurements taken from the
basiou to those taken from the auricular point. A = basion. B = auricular
point C = nasion. D = alveolar point

concerned, in some degree with the basion; and, at Professor
Cunningham's suggestion, I commenced a series of measurement on
ekulls in the Anatomical Museum of the University of Edinburgh,
in order to see how far this assumption was legitimate. Before I
had gone far in this investigation — indeed, on the same morning on
which it was commenced — the solution of the problem became
obvious, with the result that I find that the two points do not
correspond with one another, for the following reasons, which are
made clear by the diagram (fig. 2). In every skull examined I
discovered that while the centre of the auricular orifice was several
millimetres higher than the basion, it was also several millimetres
posterior to it, so that while the auriculo-nasal length and the basi-

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1903-4.] Mr N. Annandale on the People of the Faroes. 13

nasal length were approximately equal, the former being very slightly
the longer of the two, the aoriculo-alveolar length was considerably
longer than the basi-alveolar. In five Irish skulls the difference
between the vertical index when the height was taken from the
basion and when it was taken from the auricular point, that is to
say, from the centre of the external auditory meatus, varied from
2*9 to 6*3, so that it is very evident that the two measurements
have little relationship to one another, except that the auricular
height is, probably in all cases, the less of the two. In the same
skulls the two gnathic indices obtained in a similar way diflfered by
from 5*4 to 14*7, but in this case the auricular index was the greater
of the two. It must be remembered, however, that measurements
taken on the living head diflTer considerably from those taken on
the skull ; while the thickness of the soft tissues of the scalp and
of the hair must go far in bringing the auriculo-bregmatic height
up to the same figure as that of the basi-bregmatic, if they do not,
in some cases, cause the former to surpass the latter, yet the
comparatively greater thickness of the soft tissues and of the hair
on the occiput and of the forehead must again reduce the vertical
index, in whatever way it is obtained, to a result of which the
degree cannot ever be arrived at with exactitude. In the
gnathic index, on the other hand, the soft tissues that cover the
nasion must make the index on the skull considerably higher than
one obtained from the same measurements taken on the living
head, and it is obvious that thickness of the fleshy coating on the
nasion differs considerably in different persons; so that persons
with thin faces will have, ceteris paribus^ a gnathic index higher
than that of persons with fleshy faces. It is therefore worth noting
that the Faroeman in my table with the highest gnathic index was
a very thin and unhealthy man, who suffered greatly from asthma,
I do not see that there is any possibility of reducing measurements
taken on the living head, as far as the vertical and gnathic indices
are concerned, to a common denominator with those of the skull,
no matter what the points may be from which the lengths are
measured, and it would be dif&cult to persuade craniologists to
give up measuring from the basion, even though the auricular
point is one which can be found with equal ease in both cases.
The statures given in my table can only be regai-ded as approxi-

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14 Proceedings of Boyal Society of Edinburgh. [i

mate, for all of them were taken, as mentioned above, on men who
were not barefooted, and allowance had to be made for different
kinds of footgear in different individuals; for these reasons I
have only given the results in centimetres, though the measure-
ments were originally taken in millimetres, and I believe that when
recorded thus they are fairly accurate. The statures seem to fall
into two very distinct series, those of 170 cm. and above and
those below that figure ; it is noteworthy that the last four men
-examined fall within the former category, showing how necessary
a large series of measurements must always be in estimating the
mean stature of a race. Dr J0rgensen's series of thirty-three men
from the northern islands gives a mean of 169 cm., with extremes
of 155 and 178 mm. Again, a very serious discrepancy exists
between my measurements and his, for my mean is 166 cm., and
my extremes are 157 and 176 cm., but I have not been able to
discover whether his measurements were taken on barefooted
subjects, or, if not, whether allowance was made for footgear. In
any case, a visual inspection of the Faroemen makes it obvious that
they are a very short race, perhaps as a result of in-breeding,
though they are robust and well-built, and not, so far as I have
been able to discover, degenerate in any other way. It is
difficult, however, to discover to what extent insanity prevails
among them, as all bad cases of madness are removed to Denmark;
but on the little island of Naalsoe, where several families, con-
sidering themselves to be descendants of the kings of Scotland,
Tefused to marry the inhabitants of other islands, imbecility
and total hereditary deafness are said to have been unusually
common (3).

I have not thought it worth while to record my observations on
the skin colour in detail, as I believe that this is due far more to
the degree of exposure to which the individual has been subjected,
to climate, and even to altitude, than to race, at any rate within
reasonable limits ; for no amoimt of protection from the elements,
no cold, and no altitude would make a Negro white, or even give
;an Italian the complexion of a Dane. All that can be said on
this point as regards the Faroemen is, that those men who
have dark hair have also a dark skin, which in some cases is as
-dark as that of an Italian, and that such persons have frequently

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1908-4.] Mr N. Annandale on the People of the Faroes, 15

^features ^ more marked, and especially a more pronounced promi-
nence, often combined with a tendency to be hooked, of the nose,
than the m^ority of their fellow islanders.

It is probable that the twenty persons examined give a very fair
approximation, at any rate as far as the island of Stromoe is con-
cerned, to the general colour of the hair and eyes of the Faroemen,
l)ut the series of observations is not sufficiently extensive to permit
the calculation of a percentage index of nigrescence on Beddoe's
system (4). They show, however, that while the great proportion of
the people have light eyes and light or neutral hair, there is a
distinct dark element among them, which, as Dr J0rgensen has
shown, and as Landt (5) had anticipated, is more pronounced in
Suderoe than in the northern islands of the group. The danger of
•drawing conclusions, however, regarding this point is well illus-
trated by a fact in the history of a family living near Thorshavn.
Several members of this family are very dark indeed, and have
:almo6t an Oriental appearance, which I was inclined, before I
knew their history, to put down as due to an extreme development
•among them of the dark type that occurs sporadically in all
Scandinavian countries, and is far from xmcommon in the Faroes
and Iceland. Quite incidentally, however, I learot that the grand-
mother of the present head of the family came from somewhere in
Eastern Europe, and that her grandchildren took after her. It
would seem, on prirnd facie evidence, that hardly any place in the
world was more unlikely to harbour an Oriental European than the
Faroes, but facts are liable to run counter to evidence of the kind,
-and it is, moreover, certain that this unlikely importation, who was
met by her husband when both were being educated, I believe in
Switzerland, has proved, in zoological language, prepotent, and
may conceivably have an ultimate effect on the population of the
Faroes, though, the present head of the family having married an
Icelander, also met in the course of education, the problem becomes
«ven more complicated. I may also say that this family is one
which prides itself on keeping up the old customs of the Faroes,
though some people in Thorshavn have told me that the conspicuous

1 Some ezoellent photographs of Faroemen are reproduced in a paper just
pabliflbed by Dr Bormeister Norburg {Oldbus, vol. Ixxxiv., 1908, No. 14,
pp. 219-222). Oct. 29.

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Proceedings of Royal Society of Ediriburgh, [sess.

'old-fashioned* costume, which the men of the family delight to wear
on festive occasions, is partly the result of their own imagination.

Having now dealt with the measurements and observations in
my tables severally, I propose to inquire whether there are
any obvious correlations between them, such as can be shown in
even so small a number of individuals as twenty. If we take the
mean stature of the five tallest men, the mean stature of the five
who come nearest to them, of the next five, and finally of the five
shortest, and if we take the mean of all the head indices of the
same five individuals in each of the four batches, we get the
following results : —

Stature.

Fire tallest.
Next five, .
Next five, .
Five shortest,

173-4
167-4
163-4
159-8

Cephalio
Index.

Vertical
Index.

Facial
Index.

94-5

Nasal
Index.

Gnathic
Index.

93-9

82-0

70-4

63-0

81-1

72-0

88-1

68-5

98-4

79-8

70-9

90-1

66-4

96-9

79-3

70-6

94-4

64-8

99-8

As one figure is apt to throw out the mean in batches so small
as five, we may further consider the head indices in the same way
from the point of view of the cephalic index, as the five tallest
men are not those who have the five shortest indices : —

Cephalic
Index.

Vertical
Index.

Facial
Index.

Nasal
Index.

Gnathic
Index.

Five shortest heads,

84-4

70-6

90-7

70-6

94-0

Next five,

81-1

71-1

93-0

68-3

98-6

Next five.

79-3

70-8

90-2

61-4

96-8

Five longest heads, .

77-2

69-8

91-2

67-9

100-3

From these tables it would seem that there is a certain relation-
ship between the stature and the shape of the head, and also,
possibly, between the cephalic index and the gnathic index.
J0rgensen*8 data for Suderoe appear to indicate no connection

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1903-4.] Mr N. Annandale on the People of the Faroes. 17

between the stature and the cephalic index in that island, but it
is clear that a longer head and a shorter stature differentiate the
population of Suderoe as a whole from that of the northern islands,
for all observers agree that the former are distinguished from the
latter by being smaller and darker, while the following details
exhibit tbe diflference in the cephalic index in a sufficiently striking
manner. Dr J0rgensen, who adopts the number 77*5 as the lower
limit of mesaticephaly, states that of the adult males of Suderoe
44 per cent, are brachycephalic, 27 per cent, mesaticephalic,
and 29 per cent, dolichocephalic. If my twenty observations
from Thorshavn are combined with his thirty -three from the
northern islands, and if the same standard of brachycephaly is
adopted for the sake of comparison, we get as a result that in the
two series together, decimals omitted, 56 per cent, are brachy-
cephalic, 32 per cent, mesaticephalic, and 12 per cent, dolicho-
cephalic.

Part II.— Historical.

Before discussing the history of the Faroes and the traditions
current among the people as regards their origin, it may not be
superfluous to consider for a moment the personal names given in
my table. With two exceptions the second or third name of each
man is a patronymic, but one adapted to modern Danish orthog-
raphy, and become a regular surname, which, at any rate in
Thorshavn, is not changed either from generation to generation or
according to the sex of the person who bears it. Mr Henry
Balfour has called my attention to the fact that the initials carved
on objects from the Faroes, even if these be women's belongings,
are the first letters of Christian names and surnames, not, as would
be the case on Icelandic objects, those of a Christian name, another
Christian name and an 8 (for 8on) or a (i (for ddttir), according to
the sex of the owner, and that there is no special indication of the
name of the woman's husband, as would be the case on objects
from the country districts of Norway. In a list of names of people ^
living in the Faroes between the years 1600 and 1709 there appear
to be but a few real surnames, but married women adopt their
husbands' patronymics without change ; single women are known

* N. Andersen, FcBr^eme, 1600-1709. Copenhagen, 1895.
PROC. ROY. SOC. EDIN. — VOL. XXV. 2

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18 • Proceedings of Royal Society of Edinburgh, [«ess.

by their personal names, followed by those of their fathers with
datiir added ; men are for the most part referred to in the same
manner, but with sen instead of dattir, while occasionally they
adopt the name of their place of abode or birth instead of a
patronymic. In the present list one man has a surname which has
probably been introduced from southern Denmark or from the
Schleswig-Holstein provinces, namely Djurhuus, while another has
simply taken the name of his birthplace, Gjoueraa, a small village
on the island of Stromoe, surnames being by no means a fixed
institution in the country districts of the Faroes even at the
present day, though they have gone far further in this direction
than in Iceland. It is also worthy of note that a very large pro-
portion of the names in my list are Biblical, and only a very small
proportion Norse; while in a similar number of names from
Iceland the majority would probably be found to be such as Gisli,
Herjolfur, Arni, or the popular Magnus —a name introduced into
Scandinavian countries through a misunderstanding of the latinized
name of Charlemagne, a very popular hero in the ballads of the
Faroes as in other Norse folk-lore.

The Faroes, we know, were colonised by vikings of Norse ex-
traction, many of whom were also descended from the Iberian chief-
tains of the Hebrides and Ireland. There is no reason whatever to
think that the islands had other human denizens when the vikings
came, except perhaps occasional anchorites seeking to outdo the
records of their fellows in the way of finding * solitudes.' There
is good reason, however, to believe that Faroe, or, as it is properly
spelt, Fseroe, means * sheep island,' though Landt (5) gives other
derivations, and that the group got its present name because the
vikings found a breed of sheep already established there; and if
this assumption l»e correct, the fact is difficult of explanation
without supposing either that the island had already been
colonised by some race which had disappeared, or else that the
sheep had originally been accidentally introduced by a wreck, as
was the case with the "great" or brown rat (5) in 1768. The bree<l
appears to have been similar to that of Soa in St Kilda, but is
now quite extinct, having been purposely extenninated by the
islanders ; it could hardly have come spontaneously into being on
small islands separated by a very deep channel from any consider-

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1903-4.] Mr K Annandale on the People of the Faroes, 19

Able mass of land, but its origin must, for the present, remain a
mystery, and its existence in no way militates against the view
that the Faroes were devoid of human inhabitants when they were
first visited by the wanderers of more or less mixed race who are
known in British history as the * Danes,' although comparatively
few of them had any connection with Denmark. Professor York
Powell (6), in the introduction to his translation of the Fareyinga
SagOj shows that during the early history of the Faroes their
I^orse families were closely related to several of the Icelandic
chiefs both by blood and marriage, and it is probable that the
Faroes were colonised in the first half of the tenth century, a little
later than Iceland, which commenced to be peopled in 874 a.d.

In Icelandic history the people known to the vikings as ^ men of
the West,' that is to say, Irishmen and inhabitants of the outer
Hebrides, occasionally make their appearance, chiefly as captives of
war ; it is to them that the Westmann Isles, ofif the south coast of
Iceland, owe their name, a party of mutinous slaves having
occupied them after slaying their master on the mainland, whence
his avengers soon came to exterminate the murderers. In the
Faroes, Westmannhavn, a fine natural harbour near the north-west
comer of Stromoe, is said to have at one time been a favourite
resort of the Western ships, while Saxen, a place with a similar
but smaller harbour a few miles to the north, is believed to have
attracted Scotch and Dutch smugglers until comparatively recent
times, when the land-locked bay became silted up in the course of
a single storm. The people of Suderoe claim themselves to be of
Western descent, and a curious story (3), told me some years ago in
Stromoe to account for their physical and dialectic peculiarities,
makes them to be descended from an Irish captain's wife who was
kidnapped from her husband's vessel by a native chief. The story
has evidently been embellished by an ignorant person in order to
account for the name of a village in Suderoe, but, for all that, may
contain a germ of truth.

A far more circumstantial tradition links the island of Naalsoe
with Scotland. Certain families on this island, which has a popu-
lation at the present day of about two hundred souls, believe im-
plicitly that they are the direct descendants of * Jacobus the Second
of Scotland,' whose daughter eloped with a page of her father's

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20 Proceedings of Royal Society of Edinburgh, [sbss.

court named Eric and came with a great following to the Faroes.
Naalsoe had been utterly depopulated by the Black Death, which
raged in the islands at that date, and so the princess and her
followers settled there. There 'she bore a son to Eric. Years later
her father followed her, and when he came to Naalsoe he saw his
grandson, whom he recognised because he was very like her,
playing ou the shore. Struck by the boy's beauty and manly
appearance, he offered to forgive his daughter and her lover if
they would return to Scotland with him. This they refused to do>
remaining in the Faroes and having many other children there.
The first-bom sou fell on a knife with which he was playing and
killed himself, then the king of Denmark confiscated half the
island from the princess because she was a Boman Catholic, but
she and her other children, her followers and their descendants,
peopled the island, and some of her descendants still refuse to
marry outside the families who claim her as their ancestress. The
present amptmand of the Faroes, the first native to be appointed
to this position by the Danish Government, is of her kin. The
whole story is, from the point of history, ridiculous, but I am in-
clined to agree with Robert Chambers (7), who heard the outlines of
the tradition on a visit to the Faroes in the middle of last century^
that in the main it may be true, any foreign lady of birth and
wealth being easily transformed into a 'king's daughter' in a
region so remote as the Faroes.

All these floating traditions, in any case, probably set forth a
real fact, viz., that there was, subsequent to their original colonisa-
tion, a considerable influx of blood other than Norse into the
Faroes ; but whether the immigrants came as single ^individuals or
in parties we cannot say with any more accuracy than we can give
their advent an exact date. Throughout the later Middle Ages,
and as late as 1874, the crown trading monopoly, instituted by the
kings of Denmark, shut off the Faroes from commerce with Iceland
on the one hand, and with the rest of Europe on the other ; and
though extensive smuggling doubtless occurred, smuggling is not a
form of trade likely to lead to intermarriage. The fishermen of
the Faroes met with fishing-smacks from Shetland on the high
seas, and frequently hired themselves out to Shetland shipowners,
learning to speak English from their mates, but they came home

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190B-4.] Mr N. Annandale on the People of the Faroes, 21

with a scorn of Shetlanders as intense as the Icelanders' scorn of
faroemen, and it is worthy of note that the old dialect of Shetland,
recently extinct, took a totally different line of development from
that of the Faroes (8), though both sprang in the early Middle Ages
from the old Norse, a language practically identical with the
Icelandic of to-day. Young Faroe men and women who are
anxious to make a little money still visit the west coast of Iceland
during the fishing season, to help on the boats and with the pre-
paration of salted fish, but the men rarely, if ever, bring home an
Icelandic wife, and if a girl marries an Icelander she stays in
Iceland.

As I have frequently heard it hinted that the dark strain in the
population of the Faroes, especially of Suderoe, is due either to
casual intercourse with Breton fishermen or to the raids of the
Barbary corsairs, it may be well to consider whether there can be
any truth in either or both of these insinuations. With regard to
the Bretons* visits to the Faroes I have no information, but I have
never heard it said that any of them settled in the islands ; and
the Faroe women are extremely modest, viewing the custom, so
common in Iceland, of postponing marriage until a child is bom or
expected, with abhorrence. In Iceland, however, it is just possible
that temporary connections formed between these foreign seamen
and native women may have made dark complexions commoner in
^Reykjavik, as they certainly appear on casual inspection to be,
than in the country districts, although, of course, a dark strain ex-
isted among the vikings themselves, and still exists in parts of
Norway where Bretons and Algerians alike have been unknown,
whether as a remnant of the aboriginal population, as is very
possible, or as a result of intermarriage in the ninth century or
earlier between the Norse raiders and their Irish captives, is very
hard to say ; probably its origin is mixed, perhaps even more
mixed than has been suggested.

As regards the Barbary corsairs, I am doubtful whether they
ever raided the Faroes. There is a tradition, it is true, on Naalsoe
to the effect that once, while all the men of that island were away
at the fishing, the * Turks ' visited their homes and seized their
women, but the women leapt into the sea from the ships to which
they were hurried, and the * Turks ' cut off their breasts in the

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22 Froceedings of Royal Society of Edinburgh, [sww.

water, so that they sank and were drowued. Mr Stanley Lane-
Poole (9), moreover, in his Barhary Corsairs^ states that Murfid, a
German renegade, "took three Algerine ships as far north as
Denmark and Iceland, whence he carried oflF four hundred, some
say eight hundred, captives . . . ," and I have heard it stated in
the Faroes that this expedition also visited these islands. Some
years ago, while staying in the Westmann Isles, I took the trouble
to translate the contemporary Icelandic accounts of Murad's raid,
and of another, led by three Moorish captains, which also took
place on the coast of Iceland in the same summer, that of 1627.
These records (10) were collected and printed in Reykjavik about
half a century ago. They contain no mention of a visit to the
Faroes, and show that it is exceedingly improbable that any admix-
ture of Algerian blood now exists even in Iceland. Between three
and four hundred persons were taken prisoners by the two expedi-
tions, and not more than forty, some of whom were women, got back
to Iceland, the great majority being from the Westmann Isles, to
which those who were ransomed by their friends or by the sub-
scription raised for the purpose in Denmark returned. It is just
possible that the women may have brought home with them
children by Algerian masters, but it is exceedingly improbable
that this would have been permitted ; and even if they did, those
who returned to the Westmann Isles, at any rate, have almost
certainly left no descendants behind them, for all children, almost
without exception, who were born there died within a fortnight
after birth of tetantis neonatorum'^ until quite recently, and the
islands were constantly being repeopled from the north of Iceland,
a region which the corsairs did not visit (11, 12).

Conclusions.

?^y object, as regards the first part of this paper, has been
critical rather than constructive, for I do not believe that
measurements on the living person, even in series of considerable
magnitude, can give more than a rough sketch of the physical

^ The islanders ascribe the recent extinction of this disease to the fact that
while new-born children were formerly laid on a ma^s of uncovered feathers,
they are now placed on a covered mattress.

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1903-4 ] Mr N. Annandale on the People of the Faroes. 23

characters of a race, and we do not yet know at all what is the
physical result of crosses in the human species. The fact noted
r^arding the Faroe family whose ancestress came from Eastern
Europe is of interest in this connection, although I am not
able to give statistical details, for it shows how necessary
ifc is that anthropologists should pay attention to that mysterious
quality inherent in certain races and certain individuals
— prepotency. Personally, I must express the great debt I
owe to Professor D. J. Cunningham for calling my attention to
this factor in ethnology, though it does not make the ethnologist's
task the easier. With regard to the measurements themselves,
it must be remarked how great an allowance must always be
made for the idiosyncrasy of the observer in anthropometry on
the living person. Some men naturally measure too short, some
too long, and a couple of millimetres' divergency from ideal ac-
curacy will often make a very much greater proportionate difference
in an index where the numbers combined are small. If the observer
would have even his own measurements of equal value on different
occasions, he must take care to reproduce the conditions exactly,
not only as regards his subjects, but also as regards himself ; and
above all, he must not attempt to measure more than a very few
individuals at a sitting, for no other kind of purely mechanical
investigation is more fatiguing to the mind and body, and a tired
man is not in a condition to measure accurately.

By the combination of anthropometry with history and tradition
it is possible to arrive at legitimate conclusions regarding the
ethnology of the Faroes. The people, descended in the main from
ancestors whose blood was somewhat mixed, but chiefly Norse,
have remained more or less isolated for about a thousand years,
except for casual immigration of persons and parties, who were
probably * Celtic' or Iberian, and who, it is safe to say, came
either from Scotland, from Ireland, or from the intermediate isles.
This casual admixture has taken place more frequently or in
greater proportion, or the immigrants may have been more pre-
potent, in the most southerly island of the group. In-breeding may
possibly have dwarfed the stature of the race, but details regarding
imbecility and deafness are so indefinite that they may be well
ignored, and after many weeks spent on diflferent occasions in

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24 Proceedings of Royod Society of JSdinburgh, [sbss.

different Faroe villages, I see no reason to believe that the race is
physically or mentally degenerate. A point which needs investi-
gation even more urgently than the ethnology of the Faroes is the
development of the Icelandic race, which has been more strictly
isolated than the Faroemen, and in which some interesting peculi-
arities, I believe myself, might be discovered, even with so rough a
method of examination as a large series of measurements of living
individuals.

It only remains for me to express my thanks to Sir William
Turner for his encoxiragement in the study of physical anthro-
pology, and to Professor D. J. Cunningham, at whose suggestion
the investigations embodied above were undertaken.

BIBLIOGRAPHY.

(1) F. J0RGEN8EN, Anthvopologiske Unders^gelser fra Fseroeme
{Anthropologia Fxroica) : A/handling for DoMorgraden i Medecin
ved Kj^penliavns Uniuersitet Copenhagen, 1902.

(2) A. C. Haddon, Ths Study of Man, London, 1898.

(3) N. Annandale, Blacktcood^s Magazine^ No. dccccxciv., 1898,
pp. 244-260.

(4; John Beddoe, The Races of Britain. Bristol, 1885.

(5) G. Landt, a Description of tlie Feroe Islands, London^ 1810.

(6) F. York Powell, Tlie Tale of Thrond of Gate. London, 1896.

(7) Robert Chambers, Tracings of Iceland and the Faroe
Islands, Edinburgh, 1856.

(8) Jacob Jacobsbn, Fser^sk Anthologi (U. V. Hammershaimb's).
Copenhagen, 1891.

(9) Stanley Lane-Poole, TheBarhary Corsairs, London, 1890.

(10) Bj6rn J6N880N OP ScardsA, TyrJgarans Saga; (1643).
Reylgavik. Hallvab^^ur Hjengsson and HRiBRSKUR Hrolfsson,
Litil Saga umm herlaup Tyrhjans a tslandi arid^ 1627, Reyk-
javik, 1852.

(11) George Steuart Mackenzie, Travels in tJie Island of Ice-
land during the Summer of the year MDGCCX, Edinburgh, 1811.

(12) N. Annandale, Man, 1903, art. No. 79, pp. 137, 138.

{Issued separate fy Xoveinber 30, 1908.)

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1908-4.] Mr E. Maclagan-Wedderbum on Seiches in Loch Ness. 25

Seiches observed in Looh Ness. By Mr E. Maolagan-
Wedderbum, Communicated by Professor Chrtstal.

(Read November 16, 1908. MS. received December 22, 1908.)

{Absirad.)

The first observations on seiches in Scotland were made last
summer by members of the Lake Survey, the differences in level
having been measured by a foot-rule. A Sarasin limnograph was
procured by the Survey and was set up at Fort Augustus on Loch
Ness in June of this year, and has been recording since then, with
only a few stoppages. The biggest seiche so far recorded had an
amplitude of about 9 cm. The boat-house of St Benedict's Abbey,
kindly put at Sir John Murray's disposal by the Lord Abbot, gave
shelter to the instrument both from wind and waves.

Three types of seiches are common on Loch Ness, with periods
of approximately 31*5, 15*3, and 8*8 minutes. The first of these
is probably the uninodal seiche. It seldom occurs pure, or of any
considerable magnitude. This may be due to the influence of Loch
Dochfour, which is a continuation of Loch Ness at the north-east
end. The two lochs are connected by a narrow channel about 20 ft.
deep, through which a strong current sometimes flows, and for this
reason, in calculating the theoretical period of the seiche, it was
thought proper to omit Ix)ch Dochfour.

The period was calculated in two ways. First, by the formula

t^2ldl ^bjag^ where b is the breadth and a the area of a cross

section at any particular point. This is the formula obtained by
assuming the hypothesis of parallel sections. The value obtained
was 42 minutes, which is considerably in the excess of the observed

value. The period was then calculated by the formula t — 2\ dl/^Jgh,

and the value obtained for t was 30*9 minutes, which agrees very
closely with the observed value. This method assumes that the
period of the seiche would be the same if the shores of the loch
rose perpendicularly instead of obliquely.

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26 Pi'oceedings of Royal Society of Edinburgh, [i

The binodal seiche, whose period is about 15*3 minutes, is-
usually very well marked. It is the commonest type, and lasts
longer than the uninodal seiche. The node is probably some-
where in the neighbourhood of Inverfarigaig, but has not yet been
accurately determined. It is also interesting because its period is^
less than half the period of the uninodal seiche, although, accord-
ing to Du Boys, it ought always to be greater than half ; and in
most lochs it is so, the most notable exception being Lake Geneva.
The basin of Loch Ness is so regular that it is difficult to explain
it, as waa attempted in the case of Lake Geneva, by assuming an
oscillation of part of the loch.

The polynodal seiche, whose period is 8*8 minutes, is always of
small amplitude, but sometimes very regular. There are also*
oscillations of shorter period, but they do not occur regularly
enough to allow of their measurement with any degree of accuracy.
On one or two occasions there were embroideries on the curve,,
which may have been due to transverse seiches. Owing to the
narrowness of the loch, the period of such a seiche would only b©
about 1 minute. These embroideries may be due to a variety of
causes, such as the wash of steamers, the opening of the lock gates
in the canal, etc. It will only be possible to determine whether
they are vibrations or transverse seiches by simultaneous observa-
tions at the two sides of the loch.

The range of atmospheric conditions at Fort Augustus included
thunderstorms and earthquakes, but these had no very marked
influence on the loch. It seems probable that the cause of seiches
is sudden local variations of atmospheric pressure ; and this view-
is supported by the records of a barograph at Fort Augustus. The
polynodal seiches, and perhaps the uninodal and binodal seiches
also, may be started by sudden gusts of wind. The wind blows
down the various glens in strong, almost vertical gusts, and this
may be sufficient to start the oscillation.

All the speculations, however, regarding the causes of seiches
can only be satisfactorily tested by quantitative measurements of
the forces at work, and it is hoped that something will be done in
this direction in the summer of 1904.

{Issued separately January 16, 190^.)

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1908-4.] Mr Calderwood on Btdl Trout of Tay and Tweed, 27

The Bull Trout of the Tay and of Tweed.
By W. L. Calderwood. (With a Plate.)

The particular bull trout with which I desire to deal in this paper
are the important migratory fishes which are commonly referred to
by this name in Scotland. I make no mention of more or lesa
monstrous examples of the common brown trout, or even of those
examples of S, fario which have assumed a semi-migratory habit,,
and have become much modified by reason of their life in the
estuaries of our larger rivers.

Amongst the true migratory salmonidce are two fishes which I
hope to show are distinct from one another, but concerning which
considerable confusion seems at present to exist, because they are
both called bull trout This somewhat ambiguous term * bull trout ^
is a familiar one throughout Scotland, but the two forms to which
I here refer are well represented, the one in the Tay and the other
in the Tweed, and it is convenient, therefore, to mention these twa
rivers specially, since they are, as it were, the homes of the separate
forms. Pamell, in his essay on the Fishes of the Firth of Forth,
describes and figures eight bull trout, to some of which he gives-
the name of 'salmon bull trout.' These fishes are placed aa
varieties of the species S, eriox^ and are, curiously enough, included
in part by Giinther under his species S, trvMa (Brii, Mue. Gat.y
vol. vi. p. 26).

During last summer I had the opportunity of examining many
Tay bull trout, and I am satisfied that this fish is the same as the
•salmon bull trout' of Pamell; and further, that it cannot be
referred either to S. eriox or to S, trutta.

The bull trout of the Tay grows to a size beyond that ever
attained by any variety of sea trout. Examples occur from 5 lbs.
to 60 lbs. I have not myself seen any example approaching 60 lbs.,
and such are naturally extremely rare, but records in the possession
of the Secretary of the Tay Salmon Fisheries Co. are suflicient to
show that the fish attains as great weights as the salmon. During
the past season two or three occurred well over 40 lbs., the heaviest

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28 Proceedings of Boyal Society of Edinburgh, [suss.

«almon being 51 lbs. On 6th July of this year (1903) seven bull
trout were weighed together, and turned the scale at 214 lbs.,
showing the high average of 30 lbs. A small run of fish between
5 lbs. and 8 lbs. appeared with the grilse in July ; and I may remark
in passing that the Tay grilse are heavy as compared with the
grilse of other rivers.

In general outline this so-called bull trout is in no way different
from the shapely Tay salmon, and the appearance of the head,
the outline of the gill cover, and shape of the preoperculum are
identical This is seen in PI. fig. 1. The caudal fin also and the
<»audal peduncle are alike in like sizes of fish. The opportunity
given me of viewing salmon interspersed with bull trout laid out
in rows upon the sloping cement floor of the Tay Fisheries Co.
Fish House at Perth enabled one not only to compare bull trout
and salmon, but to note the variations which occur in both ; and
those variations I found to be in no way dissimilar.

The distinguishing feature of the bull trout is primarily one of
mirface marking. The dorsum is more or less thickly speckled
with small black spots, and these are also to a varying extent
displayed on the side, and more especially on the * shoulder* of
the fish below the lateral line. A well-marked bull trout has
the spots below the lateral line continued backwards as far as the
level of the dorsal fin. But when one examines a large number
of fish, examples are readily found with few spots ; and one notices
that a diminishing gradation blends ultimately into an appearance
which in no way differs from that seen in fish which are unquestion-
ably pure salmon.

A peculiar characteristic of these fish, however, is the presence
of 'maggots' (Irente(>po^a salmoneay Linn.) on the gills, the parasite
which commonly infests the gills of salmon kelts in fresh water.
These bull trout coming from the sea into the river, and with tide
lice (Lemeopthirus) upon them to prove their comparative clean-
ness, are nevertheless usually infested by gill maggots.

I know of no other special features other than the two just
mentioned whereby this so-called bull trout may be distinguished
from salmon, and in my opinion no real structural difference
•exists.

A detailed examination reveals nothing in the dentition, fin-ray

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1903-4.] Mr Calderwood on Bull Tr&id ofTay and Tweed, 2&

formulae, number of scales, from adipose fin to lateral line, or in the
relative proportions of the head, which can be regarded as of any
specific importance.

By fishermen these bull trout are judged by their spotted or
speckled appearance and by the presence of maggots on the gills.
In cases where the spots are so few as to render decision doubtful,
the gills are examined, when, if maggots are present, the fish i&
regarded as a bull trout. For the table, the fish is considered aa
of inferior quality to the salmon, and it does not realise quite as
high a price in the market.

I subjoin particulars of eleven of these fish examined at Perth
on 15 th August last, two of the examples being from Loch Kess^
the others from the Tay. Length measurements are in each case
made on the fiat, without taking into account the round surface
of the fish. Scales are counted from posterior margin of adipose
fin obliquely forwards and downwards to lateral line.

No. 1. Female.
Length 32'x7J'' (81-7xl8-4 cm.);

weight 14i lbs.
Length of head 15 cm.
Post, margin of gill cover to back of

eye 8*5 cm.
Teeth only on head of vomer.
Tail straight; caadal peduncle 5*8

cm.
Spots below lat line.
Scales 12.

Fin rays, D 13, P 18.
Maggots on gills.

No. 2.
Length 394" x 84" (101x217 cm.);

weight 26^ lbs.
Length of head 20 cm.
Eye to post margin of gill cover

11 cm.
Teeth absent fh)m vomer.
Tail straight; caadal peduncle 7*2

cm.
Spots, none below lat line or on

head.
Scales 12.

Fin rays, D 18, P 13.
Maggots on gills.

No. 8. Female.
Length 424* x 9" (107*8x23 cm.);

weight 33 lbs.
Length of head 21 *3 cm.
Eye to post of gill cover 12-3 cm.
Teeth absent from vomer.
Tail straight ; caudal ped. 8 cm.
Spots numerous below lat line and

on head.
Scales 12.

Fin rays, D 14, P 12, A 12, V 8.
Maggots on gills.

A well-marked example.

No. 4. Female (Tay).
Length SOj^xei" (78-6 x 16-9 cm.) ;

weight Hi lbs.
Length of head 14 cm.
Eye to post, of gill cover 8 cm.
Teeth on head and one on shaft of

vomer.
Tail concave ; caudal ped. 5*2 cm.
Spots, only two on shoulder, below

lat line.
Scales 11.

Fin rays, D 14, P 12, V 9.
Maggots numerous on gills.

A shapely, salmon-like example.

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Proceedings of Boyal Society of Edinhurgh. [sess.

No. 6. Female (Tay).
Length 81" x 7" (79-2 x 17*8 cm.);

weight 18 lbs.
Length of head 14*3.
Eye to post of gill cover 8*2 cm.
Teeth absent from vomer.
Tail straight ; caudal ped. 5*6 cm.
•Spots below lat. line to level of

dorsal fin.
.Scales 12.

Fin rays, D 12, V 9.
^laggots, very few.

No. 6. Female (Tay).
Length 28J''x6i'' (73-8x16 cm.);

weight 10| lbs.
Length of head 13*8 cm.
Eye to post, of gill cover 7*8 cm.
Teeth on head of vomer.
Tail concave ; caudal ped. 5*2 cm.
^pots on shoulder below lat. line.
iScales 12.

Fin rays, D. 13, V 9, P 12.
Maggots numerous.

This example had a marked
salmon appearance.

No. 7. Female (Tay).
Length 82* x 7" (81*7x17*8 cm.);

weight 12i lbs.
Length of head 15 cm.
Eye to post, of gill cover 8*8 cm.
One tooth on head of vomer.
Tail straight ; caudal ped. 5*8 cm.
■Spots below lat line to level of post.

margin of dorsal fin.
i^cales 12.

Fin rays, D 18, P 12, V 9, A 10.
Maggots, only two present.

No. 8. Female (Tay).
Length 861" x 74" (93*2 x 19-2 cm.);

weight 18i lbs.
Length of head 17 7 cm.
Eye to post of gill cover 10 cm.
One tooth on head of vomer.

Spots, a small patch on shoulder

only.
Scales 11.

Fin rays, D 13, P 12, V 9.
Maggots not numerous.

No. 9. Female (from Loch Ness).
Length 341* x 8" (88x20*8 cm.);

weight 194 lbs.
Length of head 17 cm.
Eye to post of gill cover 10 cm.
Teeth absent from vomer.
Tail straight
Scales 12.

Fin rays, D 14, V 9.
Spots all along dorsum and also below

lat line to level of front of dorsal

fin.
Maggot, only one present.

No. 10. Female (from Loch Ness).
Length 31" x 7" (79*2x17*8 cm.);

weight 124 lbs.
Length of head 15 cm.
Eye to post, of gill cover 8*5 cm.
Tail straight ; caudal peduncle 5*8.
Scales, 11 on right side, 10 on left,

distinct
Fin rays, D 18, P 11, V 9.
Spots below lat. line to level of dorsal

fin.
Maggots absent

No. 11 Female (Tay).

Length 844" x 78" (88x19*7 cm.);

weight 17J lbs.
Length of head 17 cm.
Eye to post, of gill cover 9*3 cm.
Teeth absent from vomer.
Scales 12.

Fin rays D 14, V 9.
Spots, very few below lat. line (}).
Maggots numerous.

Had appearance of ill-
conditioned salmon.

In this series some fish were selected as having specially notice-
•able bull trout markings, others were less distinctly marked,
while No. 6, when selected from amongst the other fish, gave
rise to much discussion amongst the men present as to whether it

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3903-4.] Mr Calderwood 07i Bull Trout of Tay arid Tweed. 31

'yras a bull trout or salmon. It is the smallest fish of the series,
being only 281" long and lOf lbs. in weight, but it is interesting to
■compare it with the distinct bull trout nearest it in size, viz., No.
10, which is 31" long and 12^ lbs. in weight — a Loch Ness fish.

Head.

TaUfin.

Scales.

Fin fonnulee.

offish.

No. 6

13-3

concave

D13P12V9.

5§ times

15-0

straight

11/10

D18P11V9.

5 times

No. 6 had a few spots on the shoulder below the lateral line
^md numerous maggots in the gills.

No. 10 had spots along the side to a level of the posterior
margin of dorsal fin, but had no maggots.

The total absence of maggots is, I believe, rare.

That the bull trout of the Ness is quite similar to the Tay
bull trout is well seen by comparing Nos. 9 and 1 1.

Length.

Depth.

Weight.

Head.

Scales. Fins.

u 9

34^

17 cm.

12 D14V9.

344"

17 cm.

12 D14V9.

The measurements of the head in each case show that, in the
series, the length of the head is contained in the length of the
fiah from 5 times to 5^^ times, all measurements being of females.
In the same way, the vertical measurement of the caudal peduncle
is contained in the length of the fish from 13 J to fully 15 times.

The belief that these Tay bull trout are in reality salmon
receives what I think may almost be considered practical confir-
mation from certain recaptures of marked salmon which have
recently been reported to me. Six Tay fish have been recaptured
.as bull trout which, when marked, were not noticed to show any
trace of bull trout characteristics, but to be ordinary salmon.

i marked 14 lbs. : 86": kelt: 9 : 17th Jan. 1902: at Battleby.
recaptured 33 lbs. : 48" : clean : 27th July 1903 : ** Skin the Goat "
station, near Newburgh.

(This fish may have ascended, spawned, and descended in
the interval)

N 8311 > ^^^^' • 32": kelt: 9 : 23rd Jan. 1902: EastHaugh, r. Tummel.
J 17 lbs. : 38" : clean : 16th Apr. 1903 : Flookie station, in tidal water.

I 6 lbs. : 24" : kelt : 6 : 10th Feb. 1903 : East Haugh, r. Tummel.
:No. 8348 ] 14} lbs. : 33i" : clean : 20th Aug. 1903 : Pyeroad station, in tidal
( water.

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32 Proceedings of Royal Society of Edinburgh, [i

( 24i lbs. : 37i'' : clean : 6 : 14th Nov. 1902 : Weetshot, Linn of
No. 8882 < Campaie.

r 22i lbs. : 40" : clean : 13th Feb. 190S : Flookie, in tidal water.

!7i lbs. : 27" : unspawned grilse : 9 : 22nd Nov. 1902 : Almond-
mouth.
12 lbs. : 31i" : dean : 13th Aug. 1903 : Needle station.

N 9402 / * 1^ • 26" : grilse kelt : 9 : 6th Feb. 1903 : Logierait, Upper Tay.
\ lOi lbs. : 30 : clean : 3l8t July 1908 : Flookie station.

The intervals of time are, in order, 556 days, 447 days, 191
days, 91 days, 295, and 176 days. In other words, we have one
recapture after 18 months, and, at the other extreme, a recapture
after only 3 months, but this latter is peculiar, since the fish was
clean run when marked. It is just possible that this fish, No.
8882, may have been descending (without having spawned) when
recaptured. The loss of weight is significant.

I have already noticed that the gill maggots are commonly
found on kelts. Lemeopoda aalmonea is usually believed to be
exclusively a fresh-water parasite. My attention was first called
to the fact that this may not be the case in the results obtained
by the marking of salmon which has been conducted by the
Fishery Board for Scotland during recent years. A grilse kelt^
marked in the Deveron on 11th March 1901 by a silver label
numbered 6508, was recaptured on 11th July of the same year,
at Cove, just south of Aberdeen. To have travelled in four
months round the coast, passing, as it had done, the mouths of the
rivers Ugie, Ythan, Don, and Dee, is sufficient to show that the
fish must have been some time in salt water, and between marking
and recapture it had gained 2f lbs. in weight, yet quite a number
of maggots were still attached to the gills when I received the
fish. This induced a more careful examination of the gills of fish
ascending rivers from the sea, and during the continuance of
salmon marking, Mr H. "W. Johnston, who kindly associates himself
with me in all the Tay markings, has noted, as I also have noted,
many autumn fish with a few maggots in their gills — indeed, late-
running fish are very commonly found with maggots. In salmon
and grilse proper the maggots are never so numerous as in * bull
trout,' or fish with certain bull trout markings, but I regard it
as most significant that fish fresh from the tide- way in the lower
Tay should be so found. Our marking experiments have shown

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1903-4.] Mr Calderwood on Bull Trout of Tay and Tweed, 33

that in our large rivers kelts frequently remain for surprisingly
long periods after spawning. During a prolonged stay in fresh
water the maggots remain fixed to the gills, and in some cases
the fish do not regain their silvery appearance before entering the
sea. The suggestion which I would venture upon is, that if such
fish remain only a comparatively short time in the sea, or, it may
be, remain a considerable time in the vicinity of the mouth of a
large river like the Tay, the maggots will still be found attached
to the gills on their return. Further, T think it very probable
that the peculiar spotted appearance may arise under similar con-
ditions ; that the fish having, as it were, failed to visit good feeding
grounds, and being, it may be, less fully nourished than the
average salmon, exhibits to a varying degree this peculiar speckled
appearance.

Since examining these fish, I find that in an addendum to
Giinther's Brit Mus. Catalogue, vol vi., reference is made to his
seeing other specimens of bull trout taken from the Beauly. He
states that in Lord Lovat's opinion some of those Beauly fish arc
hybrids between the salmon and the sea trout, " yet,** he adds, " the
relative size of the scales on the tail is in all these bull trout the same
as in the salmon. Captain H. Fraser believes that other specimens
of ' bull trout ' are true salmon, which, having gone down to the sea
as kelts, return to fresh water before having attained to the con-
dition of well-mended fish. Thus, as regards the river Beauly at
least, fishes named ^bull trout' do not constitute a distinct
species." This was written in 1866, and I gather from it that
Dr Gtinther would afterwards have probably altered the position
which he assigns to * the salmon bull trout of Pamell ' taken from
the Forth.

Captain H. Fraser's surmise is, I think, a correct one, applied not
merely to Beauly fish but also to the so-called bull trout fomid
in the Forth, Tay, Spey, Ness, and other rivers.

Tweed Fish.

Turning now to the bull trout of the Tweed district, we find
at once a very different fish, and in this case a trout in reality.
We have seen that Pamell classed his salmon bull trout under
S, erioxy and I have ventured to assert that S, solar would have

PROC. ROY. SOC. EDIN. — VOL. XXV. 3

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34 Proceedings of JRoycU Society of JEdiriburgh. [sbss.

been a more appropriate title. This Tweed bull trout, otherwise
known as the grey trout or round tail, is the 8, erioz, as
described by Yarrell, who, better I think than any other writer,
seems to have recognised the rather distinct character of the fish.
Giinther refers to Yarrell's S. eriox under S, cambricus, the sewen,
or English and Irish equivalent of our Scottish sea trout; and
Day places the fish in the same category, with this difference, that
he does not consider cambrieus as specifically distinct from
trutta.

Without entering at length into the wide question of the
genealogy of migratory and non-migratory trout, it is advisable to
recollect both the apparently great differences which exist between
what I prefer to call local races of trout, and the infinite gradations
which certainly exist to join such local races with one another and
with the typical sea trout or the typical brown trout. The result
of transporting brown trout eggs to New Zealand has shown how
rapidly change of environment will produce a fish which our
British Museum authorities diagnose as typical sea trout
{S. trutta).

The late Sir James Maitland showed by different methods of
feeding how Loch Leven trout could be made to resemble either
S, fario or S, trutta ; the beautifully silvery trout (fario) of some
of our West Highland lochs inaccessible to ascending fish; the
characteristics of estuary trout, of the Orkney trout, or, let us say,
of the creature usually described as Salmo ferox, are enough to
show that either we must have a great many species, in accordance
with the view adopted by Giinther, or, laying stress on the inter-
mediate gradations, we must regard all trout as belonging to one
species, and that a plastic, and therefore perhaps a comparatively
recent species. The name S, eriox is as old as the thirteentli
century. In 1824 Sir Humphrey Davy classed all our varieties
under the name S. eriox ; but it being maintained in 1878 that the
fish Inferred to by Linnseus was in reality the young of 8. eaJLar,
the term eriox^ as applied to trout, was discarded, and by a process
of gradual disentanglement from amongst the many specifically
named creatures which in the interval had been described by
naturalists, our present name of 8, trutta has been brought into
common use.

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1908-4.] Mr Calderwood on BiUl Trout of Tay and Tweed. 35

If we examine the Tweed bull trout, locally termed simply sea
trout, as it comes from the sea at Berwick, its appearance is very
different from that of the typical S, trutta. It is not a very
silvery fish, and the sides are profusely spotted. This condition is
constant in Tweed trout of all sizes. In examining a large
number of these trout at Berwick last August, I was fortunate
enough to find at the same time a single small specimen of the
typical trutta, a fish of 2| lbs. The brilliant sheen of this fish was
very distinct from the rather faded grey appearance of the Tweed
trout of the same size. The head had the conical appearance so char-
acteristic of S, trutta — small in proportion to the length of the fish,
with the maxillary bones well sunk into the surface, so as to give
that smoothness and compact appearance which always seems to
me a noticeable feature in typical examples of the species. The
operculum and suboperculum united also in a rounded angle only
slightly below the level of the eye. In the grey trout the head
is flatter on the sides and the bones of the mouth more prominent,*
thus giving a coarser appearance to the head. The giU cover is
more angular, and the angle is at a lower level, being in a line with,
and sometimes even rather below, the level of the posterior
extremity of the maxilla. On this account the lower margins of
the suboperculum and interoperculum are straighter and more
horizontal than in trutta or solar, A rather marked peculiarity of
the preoperculum struck me, which does not appear to have been
referred to by any of the authors whose works I have consulted.
Instead of the posterior margin being gently curved or slightly
sinuous, I found that the great majority of these fish have a
crescent-shaped notch in the posterior margin of this bone. In
a few cases two less distinct notches occurred, while in one or two
examples three less deep notches were present, giving to the
outline of this bone a rippling or undulatory appearance. In
only one case out of the twelve or thirteen dozen fishes examined
did I fijid no trace of indentations on the preopercular bones,
while in one other case I found the bone of one side of the head
with the usual deep single notch, while the bone of the other side
of the head was unindented.

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36 Proceedirifjfs of Boyal Society of JBdinburgh, [sbss.

The typical gill cover I would represent thus : —

Tweed Trout. 8. trutta.

The general appearance of the head will be seen in the photographs
of the male and female clean run fish (figs. 2 and 3). Belativelj
to the total length of the fish, I find that the head is contained
from 4^ to 5} times. The males examined in August varied from
4| to 4f times. The females in each case had the head measure-
ment 5f times in the length of the fish (measured on the flat).

The caudal fin is also a well-marked feature. At a comparatively
early age this tail fin becomes truncate or rounded at its outer
margin. In so/or and in irtUta proper this never happens, so far
as I am aware, except in distinctly large fish. In the Tweed trout,
however, fish between 6 and 7 pounds, or about 25 inches long,
show this rounded tail — whence the name ' round tail/

The female specimen photographed is 7^ lbs. and 26 inches in
length. The rounded tail is well seen. An example weighing
2^ lbs., and which was 18| inches long, was found to have the
caudal fin slightly forked when fully extended. From this
slightly forked condition in young fish, the tail fin becomes first
* straight,' then, with increased size and age, the rounded outer
border appears. Finally, in fish of 10 lbs. and upwards, a stunted
aspect is frequently noticeable, the tail being not only rounded, but
apparently so much thickened and grown-over by the caudal
peduncle as to have the free portions of the caudal fin rays notice-
ably short. All large specimens have not this appearance, but it is

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1908-4.] Mr Calderwood on Bull Trout of Tay and Tweed. 37

common amongst laige examples; the tail is thick, short, and
•clumsy. The male of 12^ lbs. represented in the photograph has
not this stunted tail. The Tweed trout is not often found of
greater weight than 15 or 16 lbs. The heaviest fish of which I
-can find any record is one of 22 lbs., said to have been caught at
'ComhiU boat-house in either 1841 or 1842 (William Rochester,
Tweed Salmon Reports, 1866, p. 102).

The caudal peduncle is, trout-like, comparatively broad, varying, 1
:find, in the proportion of 12 to 13^ times the total length of the fish.
In the finer-tailed salmon this measurement gives 13| to 15 times.

The fish appears to retain its teeth on the shaft as well as on
the head of the vomer to a more advanced age than is the case in
the ordinary sea trout. No gill maggots were present in the fishes
•examined.

The following are particulars of a few selected specimens : —

No. 1.
Length Sl^xej" (79*2 x 17-5 cm.);

weight 13 lbs.
Head 16-7 cm.
Eye to post, margin of gill cover

9*6 cm.
Tomer teeth absent
Scales 14/14.

Tail tmncate ; caudal pedmicle 6*6 cm.
Fins, D 12, P 18, A 10.

No. 2.
Length SlJ^xS}" (80*4 x 17 cm.);

male ; weight IS lbs. 3 oz.
Head 17*0 cm.
Eye to gill cover 9*6 cm.
Soalm 12/18.
Fms, D 11, P 18, A 10.
Tail truncate; caudal peduncle 6*5

cm.

No. 8.

Length 80^x6! (76*5x17 cm.);

male ; weight 12^ lbs.
Head 16*0 cm.
Eye to gill cover 9*0 cm.
Teeth, two on head of vomer.
Scales 13/18.
Tail markedly truncate (2*2 cm.) ;

caudal pedunde 6*2 cm.
Rna, D 12, P 18, A 11.

The specimen photographed.

No. 4.
Length 26''x6i'' (66*8 x 14-6 cm.)

female ; weight 7i lbs.
Head 11*7 cm.

Eye to gill cover margin 7 '1 cm.
Teeth, two on shaft and two on head

of vomer.
Scales 14/14.

Tail truncate; caudal peduncle 5*0 cm.
Fins, D 12 (very distinct), P 12,

A 10, V 9.

Specimen photographed.

No. 6.
Length IS^xSi" (46 x 10 cm.) ;

weight 2^ lbs.
Head 8*5 cm.
Eye to gill cover 5*0 cm.
Teeth, 8 on shaft and also on head of

vomer.
Scales, R 18, L 11.
Fins, D 11, P 12, A. 10.

No. 6.
Length IS'xSI (46 x 9*2 cm.) ;

weight 2 lbs. 10 oz.
Head 8*6 cm.
Eye to gill cover 4*9.
Teeth all along shaft of vomer and on

head.
Tail very slightly forked.

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38 Proceedings of Royal Society of Edvtiburgh, [sbss*

I am indebted to Sir Bichard Waldie Griffith, Bart., Chairman
of the Tweed Commissioners, for specimens in spawning condition
taken later in the year.

Though the Tweed trout cannot, in my opinion, be regarded a»
a species distinct from trtdta^ it is perhaps the best-defined variety
of migratory trout in the British islands, and on this account-
might well, I think, retain the distinguishing name of erioXy in
contradistinction to the variety cambricue, I am not familiar with
the trout of the Coquet, but there seems no reason to doubt that
the Tweed trout and the Coquet trout are of the same local race,,
and that Berwickshire and Northumberland form, as it were, the
headquarters of the variety. Moreover, the history of the local
fisheries seems to show that this variety haa almost entirely super-
seded the sea trout proper. A point upon which more information
is required is the relative distribution of this fish at the mouths of
many of our Highland rivers, as referred to recently by Mr Harvie-
Brown {Fishing Gazette, Oct. 10, 1903). In the Tweed, clean bull
trout have been taken in January during netting for experimental
purposes ; and although the greatest runs are in early summer, and
especially in late autumn, a certain number of fish are entering fresh
water all the year round. They affect certain tributaries more than
others, but push up to high spawning grounds.

In particulars of Estimated Annual Produce of the Fisheries of
the River Tweed from 1808 to 1894, it appears that, whereas at
the beginning of that period trout were less numerous than either
salmon or grilse, in process of time trout became more numerous,
first than salmon, and afterwards than grilse.

In 1808 the figures are 37,333 sahnon, 25,324 grilse, and
21,033 trout. In 1844, the year of the maximun trout crop, there
were 21,830 salmon, 88,003 grilse, and 99,256 trout In 1894 we
have a marked shrinkage, viz. — 3271 salmon, 7877 grilse, and
18,535 trout.

The surprising manner in which this trout has asserted itself
leads us more clearly to understand the well-defined character
which the variety ei'iox now exhibits.

{Issued separately January 30, 1904.)

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Pror, Roy, Sory. of Ed in.] [^y^jj ^^-^y

Mk W. L. Caldkrwood.

Fig. 1.

Fig. 2.

Fio. 3.

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1903-4.] Prof. Schafer on Artificial Bespiration in Man, 39

The Relative Effloienoy of certain Methods of per-
forming Artificial Beepiration in Man. By B. A.
Sch&fer, F.R.S. (With a Plate.)

(Read December 21, 1903.)

Preliminary observations upon this subject, which were made
by the author on behalf of a committee of the Royal Medical and
Chirurgical Society of London, are published in a report presented
by the committee and read on May 26th of this year before that
Society.

The methods which were then investigated comprised traction
by the arms with alternate relaxation, with and without chest
compression ; and pressure upon the chest walls alternating with
relaxation from removal of the pressure; the subjects of the
experiment being for each method placed successively in the
supine, the prone and the lateral positions (in the last-named case
one arm only being used for traction). In addition, the method of
MarshaU Hall was similarly tested. In this, the subject is alter-
nately rolled over from the lateral to the prone position, expiration
being assisted by pressure upon the back whenever the subject is
brought to the prone position.

It was evident from those experiments that it is possible by
nearly all the methods investigated to obtain an exchange of air
per respiration as great as that of the tidal air, the sole exception
being the methods in which traction alone, without alternating
pressure upon the lower part of the chest, was employed.

The number of experiments which we were able to make at
the time was, however, too limited to enable us to draw any
positive conclusion regarding the relative value of the several
methods of performing artificial respiration in man which have at
various times been recommended, although the experiments clearly
show the very important part which alternating pressure upon the

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40 Proceedings of Eoyal Society of Edinburgh. [i

lower part of the chest plays in effecting the emptying and (by
resiliency) the consequent filling of the lungs. It has seemed
desirable, therefore, to supplement them by further experiments,
having for their object the exact determination of the amount of
air exchanged, not only per respiratory movement, but also per
unit of time, a factor which was left out of account in
the earlier experiments, but one, nevertheless, of considerable
importance.

The apparatus which was used in the experiments referred to
in the report consisted of a counterpoised bell-jar, filled with air
and inverted over water ; to or from this the air of respiration
was conducted from the mouthpiece (or mask) by a curved tube
which passed through the water and opened into the bell-jar.
When, therefore, air was drawn by the movement of inspiration
from the bell-jar this sank in the water, and when air was forced
into it by the movement of expiration it rose. These movements
of the bell-jar were recorded upon a slowly moving blackened
cylinder, and the diameter and corresponding cubic contents of
the bell-jar being known, the amount of air exchange was found by
measuring the ordinates of the curves described on the cylinder.
The readings, however, must be looked upon as only approximate,
because, firstly, the bell-jar which was used was only approximately
cylindrical ; and secondly, because the counterpoised bell-jar
acquired, with the somewhat rapid movements imparted to it, a
swing of its own which must have affected the record.

In order to obtain more accurate measure of the amount of air
exchanged in respiration, the apparatus which was employed in
these earlier experiments has been discarded, and we have used a
carefully constructed graduated gasometer (spirometer), counter-
poised on the principle devised by the late Dr W. Marcet to avoid
the error which arises from the fact that the more a gasometer is
raised out of the water in which it is inverted, the greater is the
pressure exerted upon its contents. The air which is pumped
out of the chest is alone measured, but it is clear that an equal
amount must afterwards pass in to take its place. The air is
respired through either a mask or mouthpiece. In practice the
latter is found to be the more convenient, as less liable to
accidental leakage. When it is used, the nostrils must be occluded

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1908-4.] Prof. Schiifer on Artijicial Bespinttion in Man, 41

by pinching the nose either by the fingers or by a spring clip.
The tube which leads from the mouthpiece is forked, and each
ftirk passes to a water valve, one for admitting air to the mouth-
piece, and the other to enable the air Avhich is driven out of the
chest to pass through on its way to the gasometer. The air which
is pumped into the gasometer can either be read ofE at once on a
scale attached to the instrument, which is graduated in litres and
tenths of a litre, or it can be graphically recorded by attaching a
pen to the moving (ascending) gasometer, allowing this both to

Fig. 8. — Silvester method.

register the extent of each movement and also the number of
respiratory movements per minute upon a blackened drum revolving
slowly by means of clockwork, and upon which a time tracing is
also recorded. The tracings so obtained can be afterwards studied
at leisure.

Fig. 1 is a photograph showing the arrangement of the apparatus.

Fig. 2 shows the manner in which any respiratory method is
investigated by it. The method shown in the photograph is that
of intermittent pressure upon the lower ribs, with the subject in
the prone position.

Figs. .^, 4 and 5 are samples of tracings obtained by this
method. The 'steps' upon eiicli curve mark the successive

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42 Proceedings of Royod Society of Edinburgh, [siss.

respiratory movements ; each * rise ' gives the amount of air
expired ; inspiration occurs during the * tread ' of each step ; the
intervals between the horizontal lines represent 500 c.c. ; the time
tracing shows a mark eveiy ten seconds.

The tracings reproduced in figs. 3, 4 and 5 were all taken
at the same time and from the same individual. The experiment
begins in each case at the bottom, and is continued until the pen
has nearly reached the top of the paper. The drum was then
stopped and the cylinder (and pen) lowered (continuous vertical

Fio. 4. — Supine pressnie method.

line), and after a brief interval of natural respiration another
record of the particular mode of artificial respiration Avhich Avas
1)eing investigated was taken. Fig. 3 illustrates the amounts of
air exchanged in the employment of the Silvester method*
(forcible raising and subsequently lowering the arms, followed by
lateral pressure upon the chest); fig. 4, the amount exchanged when
the Howard method t was used ; and fig, 5, the amount exchanged
by intermittent pressure over the lower ribs, witli the subject

* H. R. Silvester, The Discovery of the Physiological Method of inducing
Respiration in Cases of apparent Death froni Drowning^ Chloroform^ Still-birth,
Noxious OaseSt etc. etc., 3rd edition, London, 1863.

t B. Howard, Plain Rules for tlie Restoration of Persons apjyarently Dead
from Drcvming, New York, 1869.

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1903-4.] Prof. Schafer 07i Artificial Respiration in Man, 43

in the prone position. The amount of pressure used in the last
two methods was approximately the same, having been produced
by throwing the whole weight of the fore part of the body of
the operator upon his hands, which were placed over the lowest
part of the thorax of the subject, the only diflference being that
in the one case (Howard) the subject was supine, in the other
prone. The pressure was in every case applied and removed
gradually; a pressure of about 60 lbs. was thereby exerted.

Fio, 5. — Prone pressure method.

Fig. 6 shows two tracings obtained by permitting the subject
to breathe, under approximately natural conditions, into the
spirometer, and the steps on these tracings give, therefore, an
idea of the amount of tidal air. The rate of respiration on this
occasion was about 16 per minute, and the average amount of
air exchanged at each respiration {i.e, the amount of tidal air)
was 385 c.c, or 6160 c.c. per minute. Before and after these
two tracings, others were made with employment of the prone-
pressure method ; and these, which are also shown in the figure,
illustrate well the efficiency of that nietliod in providing a due
exchange of air.

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44 Proceedings of Royal Society of Ediriburgh, [sbss.

The following tables will serve to show the results yielded by
the four principal methods which have been recommended for
Artificial respiration in man. In each case the respirations were
performed during five minutes, but as the spirometer was only
graduated to ten litres, it was necessary to take the amount of
«ir yielded by each minute separately. In the intervals the
subject was allowed to breathe naturally. There are also two
tables (I. and II.) giving the amount of air breathed naturally
into the spirometer, the circumstances being otherwise similar.

Fig. (5.— Two mi<UlIe traciiig^i, uatural respiration ; Iwo lateral
tracinj(8, artificial respiration 1»y prone pressure method.

In the one series of these the subject was supine, in the other
prone. Since, from the result recorded in these two tables, it
appeared that the normal rate of respiration was about 13 per
minute in the subject under the conditions of the experiment,
this was the rate aimed at in performing artificial respiration.
The same operator and the same subject took part in all the
experiments. The amount of pressure produced by the weight
of the upper part of the body of the operator when thrown
forward on to his hands in performing the artificial respirations,
shown in Tables FV. and VI., was determined to be about 60 lbs.
The statistics of the subject of experiment are as follows: —

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1903-4.] Prof. Schafer on Artificial Respiration in Man, 45

Male ; age, 23 ; occupation, laboratory attendant ; height, 5 feet
7\ inches (1*71 m.); chest measurement (at mammary line and
in full inspiration), 38^ inches (0-978 m.) ; weight, 10 stone 1 J
lbs. (64 kilog.) ; vital capacity, 4450 c.c.

Table I. — Tidal Air of Natural Re$piration — »upine poBiiian.

Number of
Bespirations.

Amount of Air
in Cubic Cent

Ist minute

2nd „

3rd „

4th ,

5th „

14
13
14
13
12

6,700
6,200
6,500
6,600
6,800

In 5 minutes,

66
respirations.

82,800 O.C.
air respired.

Remarks, — The average number of respirations per minute was-
13. The average amount of air exchanged per respiration was-
489 C.C., and per minute 6460 c.c.

Table II. — Tidal Air of Natural Retpiration — prone position.

Number of
Respirations.

12
12
12
13
18

Amount of Air
in Cubic Cent.

1st minute,

2nd „

8rd „

4th „

5th „

5,800
6,000
5,000
4,200
5,700

In 5 minutes

62
respirations.

26,200 C.C.
air respired.

Remarks. — This gives about 12 J respirations per minute, with
an air exchange per respiration of 422 c.c, and per minute of
5240 c.c.

Combining the results given in Tables I. and II., the tidal air of
the individual under experiment averages 456 c.c.

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Froeeedifigs of Royal Society of EdivJtmrgh.

Table III. — Silvester Method. (Forcible traction upon the anns,
followed by bringing of the arms back to the side of the chest
and pressure upon the chest.)

Number of
Respirations.

Amount of Air
in Cubic Cent.

Ist minate,

2nd „

8rd „

4th

6th „

13
12
13
13
13

3,700*

2,100

1,600

1,700

2,300

In 5 minutes,

64
respirationsL

11,400 cc.
air exchanged.

Remarki, — The average number of respirations per minute was
12*8, and the amount of air exchanged per respiration averaged
178 C.C., and per minute 2280 cc.

The amount of physical exertion required to effect even this
■amount of air exchange was very great, and it would have been
impossible to continue it for any length of time. Moreover, the sub-
ject could scarcely sustain the effort not to breathe, for the amount of
air he was receiving was quite inadequate, his natural tidal air being
about 450 cc. per respiration, and 5850 cc per minute (see Tables
I. and IL). The subject was on the ground, with a folded coat under
the shoulders ; the operator at his head, in a semi-kneeling posture.

Table IV. — Supine Preisure (Hofoard^e) Method. (Intermittent pres-
sure over the lower ribs, with the subject in the supine position.

Number of
Respirations.

Amount of Air
in Cubic Cent.

1st minute,

2nd „

8rd „

4th „

6th

14
14
14
13
13

4,000
4,100
3,900
3,500
4,600

In 5 minutes,

64
respirations.

20,100 CO.
air exchanged.

* The relatively large amount recorded here was probably due to the lungs
having been unusually well filled by the subject just before the experiment
•commenced.

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1903-4.] Prof. Schafer on Artificial Respiration in Man, 47

Bemarks. — The average niunber of respirations was 13*6 per
minute, and the amount of air exchanged works out at 295 c.c.
per respiration, and 4020 c.a per minute. Very little physical
exertion is required with this method, especially with the patient
on the floor, since it merely consists in throwing the weight of the
operator's body forward upon his hands and alternately swinging
back to relieve the pressure. The amount exchanged in this
experiment, although far more than by the Silvester method, was
not up to the tidal air standard, but the deficit was not sufficient
to cause any feeling of distress to the subject of the experiment
during the minute that each bout of respirations lasted.

Table V. — Marshall Hall Method* (The patient is laid prone
and rolled over to one side and back again, and so alternately.
When in the prone position, pressure was during three of
the five-minute intervals exercised upon the back of the
chest.)

Number of
Kespirations.

let minute (with pressure),

2nd „ (with pressure), . . |

8rd „ (withoutpressure; rolling

only), '

4th minute (without pressure ; rolling

only),

5th minute (with pressure),

13
14

12
12

Amount of Air
in Cubic Cent.

3,100
8,600

2,400

2,200
8,300

In 5 minutes.

63
I respirations.

14,500 cc.
air exchanged.

Remarks, — The average number of respirations was 12*6 per
minute, and the amount of air exchanged per respiration comes to
230 c.c. If the three minutes during which pressure was alter-
nated with the rolling over are alone taken into consideration,
the exchange with each respiration works out at 254 c.c. The
rolling without pressure gave 192 c.c. per respiration. Since the
method as recommended by Marshall Hall embraces alternating

* Marshall Hall, Prone and Postural Respiration in Drovming^ etc. ,
London, 1857.

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48 Proceedings of Royal Society of Edinburgh, [sess.

pressure upon the back, the highest of these three numbers may
be adopted, viz., 254 c.c. per respiration (3300 c.c. per minute).
This amount, as compared with the tidal air of 450 cc per
respiration, and 5850 c.c. per minute, is obviously inadequate ; and,
conformably with this, the subject experienced distinct distress
towards the end of each minute, even when pressure was used.
In the experiments without pressure, the minutes had to be cut up
on this account into two periods of half a minute each.

Although not a great deal of physical exertion is required to
roll a body half over in this way some 12 or 13 times a minute
and alternately to press upon the back, yet the labour is much
greater than that required by the simple pressure method. Such
efficiency as the method may have depends largely upon the
alternating pT*e8sure, for without this the rolling is quite ineffective.
The reason why this pressure produces less effect than in the
method next to be considered appears due to the fact that the
time taken up by the rolling enables less time to be given to the
pressure, so that this is almost necessarily inadequately performed
if the normal rate of respiration is kept up.

Tablb VI. — Prone Preeeure Method.* — (This is similar to the
Howard method (intermittent pressure on the lower ribs),,
but the subject is in the prone position.)

Number of Amount of Air
Respirations. in Cubic Cent

1st minute,

2nd „

8rd „

4th ,

6th „

12 6,100

13 6,800

14 6,760
12 7,000

14 7,200

6 minutes,

66 88,860
respirations.

Remarks, — The rate of respiration was on the average 13, and
the amount of air exchanged averaged 520 c.c. per respiration,

* This method is described in a paper communicated by the author to the
Royal Medical and Chirurgical Society, which was read on December 8th, 1903,
and will be published in the Med, Chir, Trans,

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1903-4.] Prof. Schafer on Artificial Respiration in Man, 49

and 6760 cc. per minute. It is the only method which, in this
series of experiments, gave an amount equal to the normal tidal
air of the individual — which was, in fact, somewhat exceeded.
Not that it is impossible by other methods (especially those of
Howard and Marshall Hall) to obtain larger figures for the ex-
change air than those given in the tables here shown — figures
equal to or even larger than the tidal air — ^but merely because it
is more difficult to do so at the rate of artificial respiration at
which these experiments were carried on. The most important fact
which the tables show is that at this rate (which is the normal
rate of this particular individual, and not by any means a fast
rate), it is easily possible to pump far more air into and out of the
chest by the prone-pressure method than by any of the methods
generally employed. The actual pressure exerted upon the prone
subject was not greater, probably rather less, than upon the supine
subject, in which the fvll weight of the fore part of the operator's
body was certainly thrown upon the lower ribs, whereas in the
similar experiments upon the prone subject the outflow of air on
making pressure on these ribs was so abundant and easy that there
was a tendency for the operator not to throw the whole weight on
the hands; even more air, therefore, could have been exchanged
if desired.

Table VII. — The follotcing Table gives ilie main results of all
the foregoing Tables in a summarised form.

Mode of Resi^ration.

Number
per Minute.

Amount of Air

exchanged per

Respiration.

Amount of Air
exchanged
per Minute.

Natural (supine),
Natural (prone),
Prone pressure, .
Supine pressure,
Boiling (with pressure), .
Kolling (without pressure).
Traction (with pressure), .

12-6

13-6

12-8

489 cc.
422 „
520 „
295 „
264 „
192 „
178 „

6,460 cc.
6,240 „
6,760 „
4,020 ..
3,300 „
2,300 „
2,280 „

Results similar in character to the above have been yielded by
many experiments, both upon the same andupondifferentindividuals.
These experiments all show that by far the most efficient method

PROC. ROY. SOC. EDIN. — VOL. XXV. 4

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50 Proceedings of Roycd Society of Edinburgh, [sess.

of performing artificial respiration is that of intermittent preuure
upon (he lower ribs with the subject in the prone position. It is
also the easiest to perform, requiring practically no exertion, as
the weight of the operator's body produces the effect, and the
swinging forwards and backwards some thirteen times a minute,
which is alone required, is by no means fatiguing.* This statement
also applies to the supine-pressure method when eflfected slowly and
without undue violence. But not only is this method less efficient
than the prone-pressure method, but there are undoubted dangers
attending it, especially in those cases where the asphyxial condition
is due to drowning. For in drowned individuals the liver is
enormously swollen and congested, and ruptures easily, as Dr
Herring and I found when endeavouring to resuscitate drowned
dogs by this method of artificial respiration.! And further,
the supine position is contra-indicated both in drowning and in
asphyxia generally, since it involves the risk of obstruction of
the pharynx by the falling back of the tongue, and also fails to
facilitate the escape of water, mucus, and vomited matter from
the mouth and nostrils.

The Silvester method, as compared with the others, has nothing
in its favour. It has all the disadvantages of the supine position,
is most laborious, and is relatively inefficient. As regards the
Marshall Hall method, the most effectual part of that method is
the exertion of pressure in the prone position ; the rolling over is
quite unnecessary, and attended by manifest disadvantages. The
addition to this method which is advocated by Bowles, % consisting
in raising the one arm over the head after the body is placed in
the lateral position, has been found, in measurements we have made,
to introduce no serious augmentation in the amount of air ex-
changed, but merely serves to render it still more difficult to per-
form the respiratory movements efficiently at the necessary rate.

• I have on one occasion continued it for nearly an hour without experi-
encing the least fatigue, and without the subject having any desire to breathe
naturally or feeling at all inconvenienced.

t Report of CJommittee of Royal Medical and Chirurgical Society, op. eU.

t R. L. Bowles, A Method for the Treatment of th^ apparcrUly Drowned,
Loudon, 1903.

[Isstied separately January 2P, 1904.)

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Ptoe. Ruy, Socy. of Eiiin.]

[Vol. XXV.

Prof. E. A. Sch.\fer.

Fig. '1.

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i»08-4.] Physico-Ghemical Investigations in Amide Oroup, 51

Physico-Chemical Investigations in the Amide Group.
By Charles R Pawsitt, Ph.D., B.Sc. (Edin. and Lond.).
Communicated by Professor Crum Brown.

(MS. received December 14. Read December 21, 1903.)

Some time ago, while studying the chemical dynamics of the
changes which occur in solutions of urea or carbamide,* I came
upon some rather unexpected results which led me to hope that
investigations conducted on somewhat the same lines with other
substances of the amide group might prove to yield results of some
interest. The amides referred to are those derived from carboxylic
acids. While proceeding to this investigation I noticed some
measurements, t obtained in connection with the viscosity of
aqueous solutions of carbamide, which appeared of sufficient
interest to demand an inquiry into the nature of solutions of
this class of substances before proceeding further with the subject
of inquiry in the manner at first intended.

The Viscosity of the Amides in Aqueous Solution,

The viscosity of solutions is a problem on which a considerable
amount of work has been carried out, and the way in which the
viscosity of a solution changes with the concentration of the sub-
stance dissolved has been found to be generally in agreement with

the formula

7. = A«(i),

where -rj, is the viscosity of a solution of concentration «, the

viscosity of water being taken as unity and where A is a constant

Some observers have shown that results occasionally follow the

formula

i7,= l+aa;(ii),

where *a' is a constant. It will be noticed, however, that if

• ZeU. fUr phygikal. ChtmU, 41, 601 (1902).

t Rudorf, ZeU. filr physikaL Chemie, 43, 267 (1903).

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Proceedings of Royal Society of Edinburgh. [«

' a ' is small and also z, equation (ii) ib really a particular caae
of equation (i) ; for we may put (i) in the form

i7*»l+a;IogeA +
or, putting log^ A « a

a:*log^2A gfilogiJi

i7,= l+aar+-2y + -3y +

(iii).

Considering aqueous solutions, we may (roughly) divide the
dissolved substances into electrolytes and non-electrolytes. In
the former class substances are known, e,g, potassium chloride,
which do not follow the above formula (iii), but possess what may
be called a * negative ' viscosity. Thus the viscosity of } normal
potassium chloride is less than that of water. Up to the present
no non-electrolyte has been found to show this ' negative *
viscosity. In the paper mentioned above, Rudorf drew attention
to the fact that carbamide in dilute aqueous solution shows a
'negative' viscosity. I have repeated these measurements, and
have also made determinations of the viscosity of acetamide in
solution. These substances show a normal behaviour in their
depression of the freezing-point.*

Carhamidft (Urea).

Concei

itration.

mol.

1-005 ..

. 1-005 ..

-0

»> •

1012 ..

. 1011 ..

. - -001

1-024 ..

. 1-022 .

. --002

mol.

1-0 16 ..

. 1-046 .

»»

1-089 ..

. 1092 .

'. +-*()03

Acetamide,

Concentration.

1 mol.

1-013 ..

. 1-014 ..

. +-001

■

f$ *

1-028 ..

. 1-028 ..

[

»i •

T067 ..

. 1-057 ..

mol.

1-117 ..

. 1-118 ..

*. +-001

f * •

1-260 ..

. 1-250 ..

T7i is the viscosity determined experimentally ; 172 that calculated
from equation (iii). A is difference of the calculated value from
that observed ; (a) in the case of carbamide being taken as -044,
♦ ZeilschnfifUr phytikal Che'mie, 2, 491 (1889).

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i9(»-4.] PhysicO'Chemical Investigations in Amide Oroup. 53

and in the case of acetamid as '111. The calculated and observed
values agree well with one another. There is no indication of
any negative viscosity in the case of carbamide. As the substance
employed was very pure, I have some difficulty in explaining the
different result obtained by Budorf. In case the solution used by
him had undeigone any decomposition (into ammonium cyanate),
I heated some \ moL solution of urea for an hour at 100* C. to
see whether the production of ammonium cyanate would affect the
result: the solution had a viscosity almost identical with the
result previously obtained for pure urea, an increase of *002 being
found.

Thb Chemical Naturb of ths AmoBa

The amides are above described as non-electrolytes, but I
thought it might be of interest to inquire as to how far this was
the case, and to what the amides owe such conductivity as they do
possess. In the following measurements I have used urea as the
amide.

The amides are known to form compounds with acids. Thus
urea and hydrochloric acid give the compound CO(NH3)2,HCL

These compounds are split up very largely into amide and acid
again by dissolving in water.

Walker showed * that the concentration of free acid in a solu-
tion is gradually decreased by the addition of urea, and the
relations here may be. represented by the formula

CcO(NHa), X ChCI _ ^
CoO(NH,)B,Ha

where C, is the concentration of the substance x and K is a
constant

He found that if the concentration of H* ions in normal
hydrochloric acid be represented by the number 315 (25* C), then
the concentration after addition of urea was as follows : —
Norm. HCl 315

1. XJ.Vyl . . . . .

-i-imoLCO(NH2)2

. 237

1 + ^ » >»

. 184

f "r -* l» »»

. 114

>9 +3 „ „

. 82

» + 4 >» »

. 60

* ZeiL fUr physikaL. ChemU, 4, 319

1889).

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54 Proceedivgs of Royal Society of Edinburgh, [j

I have represented these results in fig, 1.

The diminution inVoncentration|offthe''H'^ionsmay befolsened.

J«o

*•«

Fig. 1.

Fig. 2.

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1902—4.] Physico-Chemicdl hivestu/atiom in Amide Group, 55

by making measurements of the electrical conductivity. On tlie
addition of urea we have the ion CO(NH2)2H' forming at the
expense of tlie H* ion, but the mobility of this new ion, as indeed
of all other kations, is considerably less than that of H*. Below
are results I have obtained for urea and hydrochloric acid at
34*2* C. (Tlie relations are practically unaltered at other tempera-
tures between 25' and 100* C.)

Urea ami HydrocfUortc Acid,

i nomi

HCl

Concentration.

+ iniol CO(NHj)a
+ i ..

+ 1 M

+ J'6 >,
+ 3-2 „

Moloc. Conductivity.

406*8

353

812

250

206

147-6

These results are reproduced in figure 2, giving a curve very
similar to that in figure 1. It will be noticed in these curves that
the effect produced by the urea falls off greatly in the higher
concentrations.*

To show the effect of urea on the electrical conductivity of a
neutral salt in solution, I have measured the conductivity of a
solution of potassium chloride with varying additions of urea.

Urea and Potamum Chloride; 25" C.

Concentration.

Molec. Conductivity.

i norm. KCl

116-4

+ J mol. COCNHa), . . .

115-3

+ 4

114-6

»i « • II |} ...

111-9

+ 1-6 „ „ . . .

1087

+3 2 „ „ . . .

100-1

It w^ill be seen that the percentage decrease here is very much
less than in the last case. The results are given in the curve
(fig. 3). The form of the curve is also different from the last
case, being almost a straight line, but slightly concave towards the
abscissa axis.

In the present case we may assume that there is no measurable
salt formation in solution. The decrease of conductivity may be

• Compare also t/itmm. Chem. Soc,, 79, 707(1901).

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Proceedings of Roycd Society of Edinburgh,

[sBsa.

looked on as due to increased viscosity of the solution, as will be
shown further on.

An amide is usually represented by the formula R - C - NH^

where R stands for some radical. The formula R - C - OH has

I
NH

also been suggested, although recent work * favours the adoption

of the former. In investigating the constitution of such sub-

//4

/08

stances, it is generally agreed that physical methods give the most
reliable results to draw conclusions from. Now, if R - C - OH

ii
NH

represented the formula of an amide, we should expect a substance

of this kind to show at least feebly acid properties. I have

♦ BtrU Bcrlchte, 84, 3142, 3161, 3658.

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1908-4.] PkysicO'Ghemical Investigations in Amide Oroup, 57

investigated this by measuring the electrical conductivity of
sodium hydrate solution with varying additions of urea.*

Sodium Hydrate and. Urea; 25* C.

CoDcentimtion.
i norm. NaOH,

+ ^ urea,

+ M »

+ 2M „

Molec. Condactivity.
194-2
191-8

188-7

183-0
1720

By adding ^ mol. urea to hydrochloric acid, potassium chloride

/rr

ft 2

Fio. 4.

and sodium hydroxide, we obtain depressions of the conductivity
by 23*6%, 1*6%, and 2*8% respectively. As, among anions, - OH'
wanders faster than any other ion, we would have expected a much

* Winkelblech {Zeit, fUr physiJcal, Chemic, 86, 676 {l90l}) has experi-
mented with dilute solntions ^ to yf^ molec. ; at these dilations signs of
salt formation could hardly be expected.

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58 Proceedings of Rayed Society of JSdiriburgh. [sbss.

larger decrease in the last case than that actually found if there
had been any acidic character at all about urea. Further, the
form of the curve obtained here (figure 4) resembles very closely
that obtained for the case of urea and potassium chloride. We
conclude, then, that there is no measurable acid function in the
amides. As the basic character is itself only a slight one,
we should expect that aqueous urea solutions would conduct
the electric current feebly. The ions here in the case of
urea are CO(NH2)2H" and - OH', and the dissociation constant

K=«?5|^^)£l^j55^ha6 been calculated from the amount
CO(NH2)2H20

of salt formation between urea and hydrochloric acid* to be
1*5 X 1 0"*" (25* C. ). The value of the dissociation constant for water
is '8 X 10"". Such water has a specific conductivity of '05 x 10"*,
but it is impossible, under ordinary conditions, to prepare water any-
thing like this. With water purified by ordinary methods we should
be able to prepare a solution of urea having almost identically the
same conductivity as the water used. Using water of spec.

conductivity 1*5 x 10"*, I have prepared urea solutions (— ) having

a conductivity indistinguishable from that of the water. The purest
specimen of urea obtained by recrystallisation from alcohol gave
a molecular solution (60 grams per litre) of spec conductivity
2*8 X 10"^ There is little doubt that this small amount of con-
ductivity, in excess of that of the pure water, is due to impurity in
the urea, but the determination is of interest in so far as it shows
how pure such substances may be obtained by the ordinary process
of recrystallisation. In preparing other amides in a pure state I
have found the determination of electrical conductivity a very
useful means of following the purification.

TTie Viscosity of some of the above-mentioned Solutions.

I next give some measurements of the viscosity of solutions
containing (a) potassium chloride and urea, (b) hydrochloric acid
and urea; and in making these determinations I have had the
valuable assistance of Mr Clerk Ranken, B.Sc, to whom I wish

• Wood, Joum. Chtm. Soc., 88, 484 (1903).

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1908-4.] Phy8ic(hChemiccd Investigations in Amide Chroup, 59

to express my thanks. With these solutions I have calculated
values of the viscosity from the formula

a^^x^ . a^ji^

where R is the viscosity of the KCl or HCl and the other letters are
as before.

Potamum Chlorirfe and Urea (25* C).

Concentration.

Viscosity

ObservtKl.

Calculated.

1*()06
1-017
1*026
1-040
1-068
1-068

i norm. ECl

„ +J mol. urea, .

+ i », . .
+ •7 „ . .
+ 1 „ . .
+ 1-4 „
+ 1-6 „

•996
1 007
1-017
1-080
1-048
1-066
1-076

-•001

-'•004
-•008
-008
-•007

' a ' is here taken equal to '044, as also in the next series.
ffydrnchloric Acid and Urea (25° C).

Viscosity

Conoen trat ion

Observed. Calculated.

i norm. HCl

1088 '

„ +i mol. urea .

1046 1-046

...

+ 4 M . .

1064 1066

+ •002

+ •7 „

1068 1-066

+ •002

+ 1 M . •

1-081 1 080

-001

+ 1-4 „

1-102 1099

-•003

+ 1-6 „

1-114 ! 1-109

-006

The observed and calculated values for the case of KCl and
CO(NH2)2 agree very well up to 1 mol. urea. For the case HCl
and CO(NH2)2 the agreement is also fairly good.

The viscosity of KCl and CO(NH2)2 is represented in fig. 5 :
it appears as almost an exact reverse * of the conductivity curve

(fig. 3).

* Compare also Phil. Mag., 6, iii. 487 (1902).

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Proceedings of Royal Society of Edinburgh, [i

Summary, — (1) The amides show no acid character, and ac-
cording to this view they are better represented by the formula
R-C-NHathanbyR-C-OH.

II II

O NH

(2) The non-conductivity of the amides in aqueous solution is
a good criterion for their purity.

(3) The viscosity of pure aqueous solutions of acetamide and

/•*3 •

Fio. 5,

carbamide follows the formula rj, = A', where rj^ is the viscositj-
of a solution of concentration x and A is a constant.

(4) A comparison of the viscosities and conductivities of a
solution of potassium chloride, to which varying amounts of an
amide were added, shows that the two are very closely related.

{Issued separately February 6, 1904.)

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1908-4.] Dr Muir an General Determinant. 61

The Theory of General Determinants in the Historiccd
Order of Development up to 1846. By Thomas
Muir, LL.D.

(MS. received August 10, 1908. Read November 2, 1908.)

Since the year 1889, when the last of a series of six papers
with a title similar to the above appeared, further research has
led to the discovery of a number of writings belonging to the
period then dealt with, viz., 1693-1844. Of those an account
is now given before proceeding to the papers of later date than
1844.

Fontaine (1748).

[M^moires donn& h, PAcad^mie Hoyale des Sciences, non im
prim^ dans leurs temps. Par M. Fontaine* de cette
Acad^mie. 588 pp. Paris, 1764.]

These memoirs of Fontaine's, sixteen in number, cover a con-
siderable variety of mathematical subjects : it is the seventh of
the series which indirectly concerns determinants. There is not,
however, even the most distant connection between it and the
work of Leibnitz. The heading is " Le calcul integral. Seconde
m^thode," the sixth memoir having given the first method. The
date is indicated in the margin.

The matter which concerns us appears as a lemma near the
beginning of the memoir (p. 94). The passage is as follows : —

" Soient quatre nombres quelconques

al , a2 , a3 , a\ ,

* The full name is Alexis Fontaine des Bertins, The very same collection
was issued in 1770 under the less appropriate title Traits de calcul diff^entiel
el inUffral, Vandermonde is said to have been a pupil of Fontaine's {v, Nouv,
AwnaUs de Math, , v. p. 155).

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62 Proceedings of Royal Society of Edinburgh, [ssas.

et quatre autres nombres aussi quelconques

al , a2 , a3 , a4 ;
faites

al a2 - al a2 = oM ,

a2 aZ - a2 a3 = a^2 ,

a3 a4 - a3 a4 = a^3 ,

al a3 - al a3 = a^l ,

a2 a4 - a2 a4 = a22 ,

al a4 - al a4 = a*^l ,
vous aurez

an ai2 - an an + aU a^Z = 0."

^ranifestlv this is the identity which in later times came to be
written

\<hhV\HW - l«AI-l«2M + l«i&4l-l^2^8l = 0.

and which, so far as we know, appeared first in its proper connec-
tion in the writings of Bezout.

It is curious to note that Fontaine was not satisfied with the
lemma in this form, but proceeded to take " autant de nombres

quelconques que Ton voudra al, a2, . . . . , alO, " and wrote

the identity one hundred and twenty-six times before he appended
^* et cetera," the 126th being

aSGa^T - a26 a^? + a^G a^8 = 0.

Cauchy (1829).

[Sur Tequation k Taide de laquelle on determine les in^galit^s
seculaires des mouvements des plan^tes. Exerdces de
Math,, iv. ; or (Euvres (2), ix. pp. 172-195.]

As the title would lead one to expect, the determinants which
occur in this important memoir belong to the class afterwards
distinguished by the name " axisymmetric," and thus fall to be
considered along with others of that class. Since, however, the
proof employed for one of the theorems therein enunciated is
equally applicable to all kinds of determinants, it would be
scarcely fair to omit here all mention of the said theorem.

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1903-4.] Dr Muir on General DeterminajUs, 63

la modern phraseology its formal enunciation might stand as
follows : —

S being any axisymmetric determinant, R the determinant got by
deleting the first row and first column of S,Y the determinant got
by deleting the first row and second column of S, and Q t]ie
determinant got from R as 'R from S, then, if R = 0

SQ = - Y2;
and the theorem in general determinants whose validity is
warranted by the proof given is in later notation —

I AjBg I - I a^b^c^d^ \'\c^d^\.

Jacobi (1829).

[Exercitatio algebraica circa discerptionem singularem frac-
tionum, quae plures variabiles involvunt. Grelle^s Joum.,
V. pp. 344-364.]

In the ordinary expansion of (ax + by + cz-t)-^ there are
evidently only negative powers of x and positive powers of y and
z; in the like expansion of {b'y + cz + ax-t')-^ there are only
n^ative powers of y and positive powers of z and x; and
similarly for (c^z + a^x + b^y - f)"^. It follows from this that the
ordinary expansion of (ax + by + cz-t)-^ . (b'y + cz + ax - 1')-^,
{c"z + a''x + b''y-f)-\ looked at from the point of view of the
powers of x, y, z, contains a considerable variety of terms; for
example, terms in which negative powers of x occur along with
positive powers of y and 2, terms in which x does not occur at all,
and so forth. There is thus suggested the curious problem of
partitioning the fraction

1 ^^____

(ax-hby + cz-t) (b'y + cz + ax-t') (c^z + ax + b^y -t")

into a number of fractions each of which is the equivalent of the
series of terms of one of those types. This is the problem with
which Jacobi is here concerned.

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64 Proceedings of Roycd Society of Edinburgh. [i

In the case of two variables he counts three types of terms,
viz., that in which the indices of both x and y are negative, that
in which the index of x only is negative, and that in which the
index of y only is negative. In the case of three variables he
counts seven types, viz., that in which the indices of x, y, z are all
negative, the three in which the index of only one variable is
negative, and the three in which the index of only one variable is
not negative. These two cases are gone fully into, with the result
that the expressions for the three aggregates in the former are all
found to contain the factor (ab')~\ and the expressions for the
seven aggregates in the latter the factor (a b'c")-^. The reciprocal
of each of those factors is recognised as the common denominator of
the values of the unknowns in a set of linear equations, a
denominator "quam quibusdam determinantem nuncupamus et
designemus per A." Its persistent appearance in the problem
under discussion, — a persistency, in fact, sufficient to suggest the
change of the numerator of the given fraction from 1 to (a b') in
the case of two variables and from 1 to (a b'c") in the case of three,
— is remarked upon: — "Quam determinantem in hac quaestione
magnas partes agere videbimus, videlicet ofunes illaa series infiniias^
quae ut coefficientes producti propositi evoltUi invenimus^ ex
eooltUione dignitatum negatiiHirum determinantis provenire.*' Then
fixing the attention on a unique term of the expansion Jacobi
ventures on the generalisation that the coefficient of

(XX^X^ Xn-^)'^

in the expansion of

(uu^u^ Wn-l)-^

that is to say, of

{ax + by-k-cz+ . ,.y^ (b'y + cz + .. , )-i(c"z + .... )-i

is the determinant

(a6V' )-i.

No proof, however, is given, save for the cases where n = 2 and
n = 3. The proposition is most noteworthy in that it supplies the
generating function of the reciprocal of a determinant.

To obtain a generalisation in a different direction, viz., from
(air + /v/)-i(^j// + aiar)-i to {ax-\rl/y)''^ (b^y-^ayc)-'^, Jacobi pro

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1.. + ...

P-a -^

a-fi

1903-4.] Dr Muir on General DeUrminaTUs. 66

ceeds in a veiy curious and interesting way. Denoting

by*

since it is the sum of the infinite series for ()8 - a) " ^ and (a - jS) " ^
he proyes after a fashion that its product by ^ - a or a - )3 is 0,
and that therefore its product by

1 1

7 + mp-a) ^' y + w(a-/i)

IB sunply its product by . Turning then from this lemma

to the product
/ 1 1 \ / 1 1 \

where u^ = a^x + h^ , t^ = b^y + a^x , he substitutes for the
first factor of it

h -. h..

\%h\^ - I Vol + ^0 K-^) I Vol - l«oM^ - h K-^i)

bis justification being the fact that

^K - ^o) = ISM« - IVol + M^i'h) y
^ti on account of the said lemma, he leaves the term 5q (wj - t^
out of both denominators. For the second factor there is thereupon
substituted

l«oM

h{ I %h \'y - \ Vi i } + «i { I «o^ I « - I Vo I }
. !.?o?!il ^

^{|«o<il - l«o^U} + «i{l Vol - \%h\^}
* Jaoobi writes it — ^ + — — - with the caution that the two parts are not

to be taken as cancelling one another. Of course, also, lower down he does
not write lo^ | but o^j - a^b^ or later {ajb^),

PROC. ROY. SOC. BDIN. — ^VOL. XXV. 5

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66 Proceedings of Royal Society of Edinburgh. [i

on the ground that we have the identity

«oM-K - 'i) = h{WA\y - I Vil} + <h {l«oM« - I Vol}.

the term a^ { | ajb^ | « - I ^i^o I } ^^^% subsequently left out of
both denominators for the same reason as before. The result thus
reached is consequently

./ i^oM . Ky \

\ I «o*i I y - I Vi i i «o'i I - 1 «o*i I y A

or, if we write f , ij for the values of «, y which make «,-<,« 0,
«i - <i - 0,

Since the general terms of the four doubly-infinite series here are
we deduce

^ _ C^"

2_IVoJM^oM:_

where m, n on the one side and fi, v on the other are to have all.
integral values from - oo to +00. Since the coefficients of
tfj^t^^jx^y on the two sides must be equal, we obtain the theorem : —

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i90»-4.] Dr Muir on Oenerai Determinants. 67

The coefficient of in the expojisum of-

M the same as the coefficient of to"ti" in the expansion of
(Vo - VlV^'Ha^t^ - H^^Y'^ it i^ng remembered that m and

n are of the same sign as fi and v respectively and tluU m + n «
ft + V - 2.

In similar fashion the , author deals with the case of three
functions «o » **i > ^2 ^^ three variables a; , y , 2 , proving labori-
ously and not very neatly the neat result

-(iH^,l.) (,4-,^,) ishf^ «

thence deriving

I «0 ^'l «2 ! ^M^'^+l ttj-+l ^2^+1 ^ a^+l y^l ;^P+l

and ending with the theorem : —

The coefficient of in the expansion of

(a^ + b^y + c^y-^\b^y + Ci^ + a^xf-^\c^ + a^-^ b^yY+^

is the same as the coefficient of tQ^H^^tg' in the expansion of

it being understood that m, n, r are of tJie sarm sign as fi^ v, p
respectively and tliat m + n + r = /A + v + p-3.

The corresponding result^ for n functions of n variables are
evident. They had already been enunciated in the introductory
section of the paper, and Jacobi now merely adds " Omnino
similia theoremata de numero quolibet variabilium, quae § 1
proposuimus, eruuntur." It has to be noted, however, that belief

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68 Proceedings of Royal Society of Edinburgh. [sns.

in the general fundamental theorem, viz., that which includes
(a) and (fi) above, is more strongly induced by the elegance of the
form of the theorem than by the mode of prool In § 1 it stands
approximately thus—

/ 1... ^„1 \ / 1 + \\ / 1 + \ \

\%'K h'-^J ^-^ ^i-V \t*,-,-d Ci-tt-i/

.l(..l.-. 1 )( 1 -H \ .)....(. 1 + \ )

and then follows the passage containing the two deductions, viz.,
"quam aequationem etiam hunc in modum repraesentare licet:

designantibus Oq , a^ , etc. /8o > i^i > ^^' i^^^iii^®i^>s omnes et
positivos et negatives a -oo ad +'». E quo theoremate
videmus, coefi&cientem termini

xfo+' xfii+' — «f!!r'^'

in expressione

M,,*o+l t*/i+^ .... t**2-l+^

aequalem fore coefficienti termini </i ^/i ... t^-i
in expressione

^ »t-i

The use here of ^Sq + 1 , )8i + 1 , . . . . rather than the change made
in the two special cases to the less natural )Sq , )Si , . . . is worth
noting.

The theorems of the remaining four pages of the paper have a
less direct bearing on our subject.

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190S-4.] Dr Muir on General Determinants. 69

Jacobi (1833).

[De binis quibuslibet fimctionibus homogeneis secundi ordinis
per substitntiones lineares in alias binas transfonnandis,

quae solis quadratis variabilium constant : una cum

OreUe^s Joum., xii. pp. 1-69.]

Jacobi's mode of -proving his theorem regarding a minor of the
adjugate occupies § 6 (pp. 9-11). Temporarily denoting by X,^ the
left-hand member of the m^ given equation

Oi^^Xi + fla w^« + +«i?*'i*?n = y»*,

and by Y« the left-hand member of the m^ derived equation

and explaining that by

[u]

«1«2 • • • i*^n

he means the coefficient of Xi'^x^"^ • • • a;„"* in a certain specified
expansion of U, he recalls his paper of the year 1829 on the
" discerptio singularis/' and affirms that he had there proved

" fore

LxA..x,J L

sive etiam, quod idem est,

LY,y,...yJ L

^1^2-

ac generalius

J^ri+rt+ ' ' +rn+^

r Y/'Y/'.>>Yn^1

y^t-'^Vn

designantibus n , r.^ , . . . , r„ ac «i , «2 > • • • i «n numeros
quoslibet integros sive positivos sive negativos/*

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70 Proceedings of Boycd Society of Hdinburgh. [ana.

A glance, however, suffices to convince one that the concluding
general theorem here given differs considerably from the theorem
which he had previously enunciated and possibly proved. As
originally stated the theorem was —

L^o*""*"^*'"*^^ . . . «JL7^"^U

which being altered into the notation of his present paper by the
substitutions

becomes

«0i

»«i,

= UJi,

«,,....

t*0:

»«*i,

- X,

, X3 , . . . .

fPi*

Y,
" A

Y,
'A '••••

>'h,

- n,

r, , . . . .

^0 » A >

= «i,

«j

= A,

•X,

rn+ij

■ • *»•»+'

A'»+^

1
1+ • ■ •

+#»+i

[Yi'Y,^ .

v..]

yi'

%'^---i

Using on both sides of this the fact that if an expanded function
be multiplied by the product of certain powers of the variables,
any particular coefficient in the original expansion has now for
facient its original facient multiplied by the said product, we
obtain

OJjXj • • • ^«

• • • +'"+^Lyr'+ v/«+^ • • • y/»+ J L

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1908-4.] Dr Muir on Genet'ol Determinants. 71

--a statement differing from Jacobi's in having r's and s^s on the
right-hand side where he has ^s and r's respectiyely. The over-
sight was probably not noticed by reason of the fact that in the
special instances considered by him the values of any r and the
corresponding 8 are the same.
In the first of these instances he puts

ri = r, « . . . - r, = - 1
and obtains

^"^ LyxY,...yJ_j_ B '

thus arriving at Cauchy's theorem regarding the aci^ugate, viz.,

B - A"-^
In the second instance, he puts
r, = ra = . . . = r„ = - 1 , r^+j - r«+5 « . . . - r, - 0 ,

and obtains

r 1 1

yi^a • • • ym

He then recalls the fact that by the conditions attaching to the
expansion of the expressions enclosed in rectangular brackets the
powers of a^i , Xj , . . . a*^ contained in the one and the powers of
y^ 9 Vm+i f'iVn contained in the other are all positive ; and
argues that as we are concerned only with terms that do not
involve these variables, it is quite allowable to put them all equal
to 0. This being done it is seen that

^m+l^+i '

2±

<«■,"<«■,"• ••aJT

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72 Proceedings of Royal Society of Edinburgh, [sbbb.

and

yiVi'-Vm

80 that there is obtained

as was expected.

Jacobi (1834).

[Dato systemate n aequationum linearium inter n incognitas,
valores incognitarum per integralia deHnita (n- 1) tupHcia
exhibentur. OrelWs Journ,, xiv. pp. 51-55]

This short paper is, as it were, a by-product of the investigation
which resulted in Jacobi's long memoir of the preceding year.
Its only interest for us at present lies in the fact that values
which are ordinarily expressed by means of determinants are here
given in the form of definite multiple integrals. Indeed, instead
of viewing the result obtained as being the solution of a set of
simultaneous linear equations, it might be equally appropriate to
consider the investigation as belonging to the subject of definite
integration. It will suffice, therefore, merely to give a statement
of the theorem arrived at. In Jacobi's own words, it is, —

" Sit propositum inter n incognitas «i i % , . . . , 2„ sjrstema n
aequationum linearium

6i,z, + />,o^, + + bi^„ - m, ,

Kl^ + ^nS^a + + Kn^n = ^n ',

statuamus

X = [b,^x^ + b^x^ + • • • + bnix^]

+ [byJXi + b^2 + • • • + ^nyCnJ

+ [hnifh. + b^x^ + • • • + b^>^ ,

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1908-4.] Dr Muir an OenercU DeterminarUs. 73

porro

M = THi^i + Wi^ + • • • + m^n
ubi

radical! positive accepto ; porro ponamus

V = ± 2 ± ^ii^a • • • ^iw >

signo ancipiti, ante ipsum 2 posito, ita determinato, ut valor
ipsius V positivus prodeat Quibus omnibus positis, erit

n ^1 __ /"

n_,^^ r^'^^ihv^ + 6»»2 + • • • '^h^^)hxMt-' &Pn-i^

int^;ralibus (n-1) tuplicibus extensis ad omnes valores
reales ipsorum a^ , aij , . . . , ir„_i et positivos et negativos, pro
quibus etiam x^ realis sit sive pro quibus

aJi' + i»-^^ + • • • + 4-1 < 1 ;
et designante S aut

2.4. ... (n-2)W *"* 1.3. 5 ... (n-2)w'
prout w aut par aut impar."

M0UN8 (1839).

[D^onstration de la formule g^n^rale qui donne les valeurs
des inconnues dans les ^juations du premier degr^. Joum.
delAouviUe^ iv. pp. 509-515.]

The real object of Molins was simply to give a rigorous demon-
stration of Cramer's rules. His literary progenitors, so far as
determinants were concerned, were apparently Cramer, Bezout,
Laplace, and Gergonne, the last of whom, it may be remembered,
^rote a paper which might well have borne the same title as the

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74 Proceedings of R&yal Society of Edinbv/rgK [j

above. The writer, however, whose work that of Molina most
closely resembled was Scherk, and very probably the two
were unknown to each other. Both had the same purpose in
view, and both used the method of so-called " mathematical induc-
tion." The difference between them may most easily be explained
by using a special example and Inodem notation.
To make the solution of the set of three equations

ck^x -I- a^y •{- a^ ^ a^

c^x A- c^y + CgZ = c^

dependent upon the already obtained solution of two, Scherk put
the first pair of equations in the form

= ^ - V J,

b^x + b^

solved for x and y , and substituted the values in the third equation.
Molins, on the other hand, having used the multipliers m^ , m, ,
1 , with the equations of the given set, performed addition, solved
the pair of equations

m^a^ + m^b^ -J- c^ = 0

^^a^ + rn^b^ -J- c^ = 0 |
itts + mj6g + Cg = 0 J

for m^ and mg , and substituted the obtained values in the result
His exposition is laboured and uninviting.

Boole, G. (1843).

[On the transformation of multiple integrals. Cambridge Math,
Joum., iv. pp. 20-28.]

Boole had to use in his paper the resultant of a system of n
linear homogeneous equations, and he therefore thought proper, by
way of introduction, to state a mode of forming the resultant, and

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1908-4.] Dr Muir on Oeneral Determinants, 75

to prove that the result was correct. As the mode is that in which
the rule of signs is dependent on the number of interchanges,* or,
as Boole calls them, ''binary permutations,'' any interest attaching
to the little exposition is connected with the '* proof." The first
essential paragraph is : —

"The result of the elimination of the variables from the
equations

Oia?! + agar J + • • • + a^n = 0 ,
Ml + ^a^ + + *«^ ^ ^ y

is an equation of which the second member is 0, and of
which the first member is formed from the coefficient of
x^x^- • • a;^ in the product of the given equations, by assum-
ing a particular term, as a^^' ' *^n > positive, and applying to
every other term a change of sign for every binary permutation
which it may exhibit, when compared with the proposed
term ai&2' ' '^n •

The curious point worth noting here is that we are directed first

to form the terms of the expression afterwards denoted by

+ +

I Oj ftg • • • r^ I and called a " permanent," and then to alter the

signs of certain terms of it. Boole then proceeds : —

" The truth of the above theorem is shown by the following
considerations. The elimination of o^ from the first and
second equation of the system introduces terms of the form
ai62-«2^i> ^^s-^^u etc., in which the law of binary
permutation is apparent, and as we may begin the process of
elimination with any variable and with any pair of equations,
the law is universal. From the same instance it is evident
that no proposed suffix can occur twice in a given term,
which condition is also characteristic of the coefficient of
x^x^ - - >Xn in the product of the equations of the system,
whence the theorem is manifest."

* See Rothe's paper of the year 1800.

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Proceedings of Boyal Society of Edinburgh.

L»

It will be observed that neither the word " detenninant " nor
the word "resultant'* occiirs: indeed, throughout the paper,
instead of resultant he uses ** final derivative," a term which prob-
ably may be traced to Sylvester.*

Catlby (1843).

[Chapters in the analytical geometry of n dimensions. Cam-
bridge Math, Joum.^ iv. pp. 119-127 ; or Collected Math,
Papers^ i. pp. 55-62,]

Of the four short chapters which compose this paper, the only
one which concerns us is the first, although in the others deter-
minants are constantly made use of. At the outset an important
notation is introduced which afterwards came to be generally
adopted. The passage in regard to it is : —

" Consider the series of terms —

Kj Kg

Kn,

the number of quantities A , . , . , K being equal to

q {q<n). Suppose g + 1 vertical rows selected, and the

quantities contained in them formed into a determinant,

, , . » (w-1) • • • (7 + 2) ,.^
this may be done m , ^y — - — . _ .. v dmerent ways.

The system of determinants so obtainexi will be represented
by the notation

1 .r.

K^ Kg

K„

* See Sylvester's paper of 1840.

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1903-4.]

Dr Muir on General Detei^minants.

and the system of equations, obtained by equating each of
these determinants to zero, by the notation

(3)

K»

A.'!

.0."

K»il

A theorem is next enunciated in regard to the expression of
any one of the determinants in terms of n - g of them.

"The } a — r~z 1^ equations represented by this

formula reduce themselves to n — q independent equations.
Imagine these expressed by

(1) = 0, (2) = 0, ..... in-q)^0,

any one of the determinants is reducible to the form

®i(l) + ®s(2) + • • • + ®n-,(n-!?)

where 0^ , 0^ , . . . , 0n-« ^^ coefficients independent
ofa^,X2, . . . , a;„."

No proof is given.

The introduction of the notation is fully justified by two
theorems which follow. The first is virtually to the effect that
we may multiply both sides of (3) by the determinant

(5)

just as if (3) were a single equation instead of C„,g+i equations,
and as if the left-hand side were a determinant ; and the result,
written in the form

(6)

;iiKi + . . . + A,K,, ^iKi + . . .+/A,K^

T1A1 + .

• + ^«A,.

•-Ht„K,

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78 Proceedings of Boyal Society of EdiTiburgh, [i

will be true; that is to say, we shall have a new set of
^n.ff+i equations, which follows logically from the original set.
Further, and conversely, if the set (6) hold, we can deduce the
set (3) provided that the determinant (5) be not zero. The other
theorem is quite similar, being to the effect that the equations
(3) may be replaced by the set

and that conversely from the set (8) the set (3) is deducible
provided the determinant

; Xj ftj • . . CD

Xj /ij • • • CD

be not zero.

As the " derivation of coexistence " came prominently before us
in examining Sylvester's early work, it may be noted here in
passing that Cayley's second chapter, extending to about a page,
consists of the enunciation of a theorem on this subject.

Caylky (1843).

[On the theory of determinants. Trans. Cambridge PhUosph.
SoCj viii. pp. 1-16; or Collected Math, Papers, i. pp.
63-79.]

Up to this point Cayley had dealt with determinants, only, as it
were, incidentally. Now, however, he devotes a memoir of sixteen
quarto pages to the study of them.

The introductory page shows a pretty wide acquaintance with
previous writings on the subject, the authors mentioned being
Cramer, Bezout (1764), Laplace, Vandermonde, Lj^range,* Bezout

* As the memoir of Lagrange which Cayley refers to is not one of those
brought into notice in the early part of our history, but is one bearing the
title " Sur le probUme de la determination des orbites des cometes d^apr^ trois

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1908-4.]

Dr Muir on General Determmants.

(1779), Gauss, Binet, Cauchy (1812), Lebesgue, Jacobi (1841), and
Cauchy (1841).

The first aectiou of the paper is said to deal with 'Hhe pro-
perties of determinants considered as derivational ftmctiona" As
a matter of fact, however, a close examination shows that the
fimctions whose properties are investigated are not strictly deter-
minants, but belong to a class afterwards named bipartites by
Cayley himself. It is true that it is the determinant notation
which is employed in specifying the functions, but this is due to
the fact that the bipartite under discussion is of a very special
type, and so happens to be expressible as a determinant.

The function U from which he considers his three determinants
to be ** derived ** is

a:(a^ + )8i7 + . . . . )

+ x\ai + jS'iy + )

there being n lines and n terms in each line. This at a somewhat
later date (1855) he would have denoted by

(

o fi ....
d fi ....

[i,v

,...$«

and called a bipartite. A still later notation is

i V ••..

a fi ....
a fi ....

•

from which each term of the final expansion is very readily

observations," it may be well to mention that the substance of the only
sentence in it which concerns us had already appeared in the memoir of 1773.
The sentence is

" De 1& il s'ensnit aussi qu'on anra

{tTu' - fuy = (a^z' - Tfsry' + (/«' - yV)^ + iafy' - xY)\

— Ncvx, JfAn. dA VAcad, Roy (Berlin)^ ann. 1778, p. 160.

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Proceedings of Boyal Society of Sdinhirgh.

obtained by multiplying an element^ jS' say, of the square array by
the two elements (i;, x') which lie in the same row and column
with it but outside the array. The three determinants which are
viewed as " derivational functions " of this function U are

a p ....
a' ^ ....

R^ + Si; + . .
R'f + S'l; + . .

Aa; + A V + . . .
a
a

Bx+Vx' +

and

1 A'f + B'v +

Rx + RV + . . . S« + Sy +

These are denoted by KU, FU, lU; and the closing sentence of
the introduction is, "The symbols K, F, 1 possess properties
which it is the object of this section to investigate."

KU, it will be observed, is what afterwards came to be called
the discriminant of U; and FU, lU are the results of making
certain linear substitutions for the elements of the first row and of
the first column of the determinant

1
a

It
a

It is this determinant, therefore, which is under investigation and
under comparison with U. That it is a bipartite function of
Xf y, z, , , . and f , 1;, £, . . . is manifest when we think of expanding
it according to binary products of the elements of the first row and
of the first column, the expression for it in the notation of

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i»03-4.] Dy Muir on General Determinants.

bipartites being thus seen to be

X y z

-1^7' • •

1 lay---

1 -la'r-.-l ----

1^/...

1 -lay"...

1 |«)8"...| ....

-\H---

1 ky...

1 -|a;8'...| ....

z ....

«1

Og ....

\ ....

<=!

c, ....

IXow the properties of this which are investigated by Cayley are
properties possessed by the more general bipartite

which is not expressible in the form of a determinant. So far,
therefore, as this section of the memoir is concerned, it is evident
that the title is somewhat misleading, and it is unnecessary to enter
into detail regarding the properties in question.

In the course of the section, however, having occasion to use
Jacobi's theorem regarding a minor of the adjugate, Cayley gives
at the outset a formal proof which it is most important to note, as
it is the natural generalisation of Cauchy's proof for the ultimate
case, and consequently has since become the standard proof given
in text-books. The passage is

" Let A ,

....,A',

A= ff

y • • .

y • • •

A' = ±

i8"

y" • • •

be given by the equations

B =

y 8' ...
y 8" . . .

y 8" . . .
y 8"' . . .

the upper or lower signs being taken according as n is odd or
even.
PROC. ROY. SOC. EDIN. — VOL. XXV. 6

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82 Proceedings of Royal Society of Edinburgh, [s

These quantities satisfy the double series of equations

Aa + BjS +.... = K
Aa + Bj3' +.... = 0

A'a + B'j3 +.... = 0
AV+ B'^+ . . . . = *c

Aa + A'a +.... = «
A^ + k'ji' +....= 0

Ba + W'a +.... = 0
B/^ + B'fi +.... = *c

(6)

the second side of each equation being 0, except for the /*"*
equation of the r*^ set of equations in the systems.

Let X , fi , . . represent the r^ ^{r + 1)^ , . . . terms of the
series a , )3 , . . . ; L , M , . . . . the corresponding terms of
the series A , B , . . . , where r is any number less than n ,
and consider the determinant

A , , L

A''--^^

L(r-l)

which may be expressed as a determinant of the n^ order, in
the form

A , . .

. . , L ,0

0, . . .

A"-\ . .

. . , L'-", 0

,0, . . .

0 , . .

... 0 ,1

,0, . . .

0 ,..

.., (. ,0

1 , . . .

Multiplying this by the two sides of the equation

« = 1 a , /i , . . . I
I a , /i' , ...

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1903-4.]

Dr Muir on General Detei*minants.

and reducing the result by the equation (©) and the equations
(6) , the second side becomes

K 0 . .

Ok..

* •

0 0 . . .

fl^*"' v<*"> . . .

ich is equivalent to

0
0

l/*-* ....

we have the equation
A L

^C-l) JJ''-'^^

K*--*

»(r+l) ylr+1)

which in the particular case of r = n becomes

A .... B
A' . . . . B'

The Second Section is said to concern " the notation and pro-
perties of certain functions resolvable into a series of determinants,"
and it is at once seen that the functions in question are obtainable
from the use of m sets of ?/ indices in the way in which a deter-
minant is obtainable from only two sets. Sylvester spoke of them
later (1851) as commutants.*

Caylby (1845).

[On the theory of linear transformations. Camh. Math.
Joum., iv. pp. 193-209; or Collected Math. Papers, i.
pp. 80-94.]

* See Postscript to Cay ley's paper " On the Theory of Permutauts," Cainh.
and Dub. Math. Joum.^ vii pp. 40-61 ; or Collected Math. Papers^ ii.
pp. 16-2(?.

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84 Proceedings of Eoyal Society of Edinhurgh. [siss.

[M^moire sur les hyperd^terminants. Crelle^s Joum^ xxx.

pp. 1-37.]*
[On linear transformations. Gamb, and Dub. Math, Joum,,

i. pp. 104-122; or Collected Math. Papers^ i. pp. 95-

112.]

These memoirs, afterwards so famous in the history of what is
now known as the algebra of quantics, contain exceedingly little
on determinants. It is important, however, to direct attention to
them, because the basis of them is a generalisation of determinants.
Using language which came into vogue two or three years later,
we may say that just as the idea and notation of determinants
provided the means of expressing one of the invariants (viz., the
discriminant) of a function, the idea and notation of hyper-
determinants were brought forward for the purpose of expressing
all the invariants.! The generalisation is of great width, hyper-
determinants including as a very special case the generalisation
previously made, viz., comrmUants.

The first memoir gives incidentally a more general mode of
using what we may call the notation of multiple determinants than
that specified in his paper of 1843. The first usage, it will be
remembered, is exemplified by

\ h^ b^

which is meant to signify that

«2

«i

«8

«1

«4

«2

= 1

«S

«2

«4

«»

«4

''.

'^2

b»

t>.

= 0.

A corresponding example of the new usage is

a^ a.f Og a^
^1 ^2 ^8 h

Xj X.2 iCg x^

Vi Va Vz Va

* This is stated to be a translation of the preceding paper, with certain
additions by the author ; and as such it is not reprinted in Collected McUh,
Papers. It also contains the substance of the paper which follows, the latter
having been delayed in publication.

t And indeed the covariants also.

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190S-4.]

Dr Muir on General DetemiinarUs.

where six equations are again intended to be specified, viz.,

a, Oj

as, arj

*1 *2 I I ^1 Vi 1.

each determinant of the one group of six being meant to be equal
to the corresponding determinant of the other group.

The example actually employed by Cayley is a result of the
multiplication-theorem, and fully justifies the usage. It is

Xa+\'a'+...,A^ + A'j8' + -
' lJia + iia' + -'-,ixfi + ii.'P^ + -

X>'

where, of course, the number of columns in the multiplier must
be greater than the number in the determinant which is its
cofactor.

It may be worth adding that the MSmoire mr les hyper-
dMerminards affords the first instance of the occurrence of Cayley's
vertical-line notation in GreU^s Journal.*

Db F^rubsac (1846).

[Sur la r^lution d'un syst^me g6n^ral de m Equations du
premier degr^ entre m inconnues. Nouv, Annales de
Math., iv. pp. 28-32.]

This is a belated contribution, having no connection with any
of those immediately preceding it. The author in all probability
knew nothing of the subject, with the exception of Cramer's rule,
which by this time was almost a century old.

The theorem which he seeks to establish is : —

*'Connaissant les valeurs des inconnues d'un syst^me de n
equations k n inconnues, pour avoir le d^nominateur commun
des valeurs d'un syst^me de n + 1 Equations kn-\-\ inconnues,
on multiplie le d^nominateur du valeur du premier syst^me,
par le coefficient de la nouvelle inconnue dans la nouvelle
^nation. Puis on en retranche les produits respectifs des

* In JAouvUle^B JoumcU brackets, [ ] or { }, were used in Cayley's own
papers of the year 1845. See vol. z.

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86 Proceedings of Royal Society of Edinburgh. [sess.

numerateurs des n inconnues du premier syst^me par leurs
coefficients, dans la demi^re du nouveau systeme. Quant au
num^rateur il se forme toujours du d^nominateur en rempla^ant
le coefficient de Finconnue que Ton consid^re par le terme
tout connu."

The method of proof is that known as " mathematical induction.'*
The details of it need not be given, as they correspond closely
with what are to be found in Scherk's paper of the year 1825, the
main differences being that F^russac uses no special determinant
notation, and, while clear and simple, is not nearly so lengthy nor
so laboriously logical.

Tbrqukm (1846).

[Notice sur T^limination. Nouv, Anncdes de Afafh,^ v. pp.
153-162.]

This is a continuation of Terquem's paper of the year 1842.
Just as the previous portion dealt with Cramer and Bezout, this
deals with Fontaine (des Berlins), Vandermonde, and Laplace,
explaining concisely and clearly their main contributions to the
subject.

The only portion of it calling for notice is that in which
attention is drawn to the curious fact that Laplace makes no
reference to Vandermonde's paper read to the Academy in the
preceding year. In regard to this Terquem's remark is —

"II est extr^mement probable que Laplace n'a pas pris
connaissance du m^moire de son confrere : on sait, d'ailleurs,
que les analystes fran9ais lisent peu les ouvrages les uns des
autres. Ceci nous explique ^galement comment la r^lution
de r^quation du onzi6me degr^ k deux termes, la plus impor-
tante d^couverte de Vandermonde, soit rest^e ignorde jusqu'ii
ce qu'elle ait attir^ Tattention de Lagrange, apr^ la d^couverte
similaire de M. Gauss."

Not only, however, does this explanation not carry us far, but
the question arises whether the point sought to be explained is
really the point which stands most in need of explanation.

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1903-4.] Dr Muir on General Deteiininants. 87

Vandennonde's paper was read at the very beginning of 1771 and
Laplace's in 1772: yet in the History of the Academy for the
latter year Laplace's occupies pp. 267-376 and Vandermonde's
pp. 516-532, and neither refers to the other's work.

It may be noted here that, notwithstanding Terquem's knowledge
of the early history of determinants and his manifest desire to
induce his readers to take up the subject, he does not himself hold
the new weapon with a very firm grasp. For example, in giving
in this volume an account of a paper of Grunert's in Crelle'e
Journal, viii. pp. 153-159, in which the author says—

" Entwickeln ^

wir nemlich x'.

!/',z', (lurch Elimination

aus

den Gleichungen

ar = Ax'+ By
y = AV+BV

+ Cz',

' + cv,

2 = AV+ liV+CV,

so erhalten wir :

(B'C--I

\"C')x + (B-C -

B(r)y +

(BC-

B'C)2

•C ^

y' =

«' =

wenn wir

L = AB'C- A'BCr+ A'BC- AB''C'+ A'B"C - A'B'C

setzen " —

he paraphrases the passage as follows : —
'* Les Equations donnent

, x[B^(r| + y[l^G\ + e[BC']

oil les crochets repr^entent des bindmes altemSs ;

[B'Cr] = B'C"- B'C,
et ainsi des autres: L est la resultante, d^nominateur

commun.

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88 Proceedings of Ruycd Society of Edinburgh. [siss.

The •imultaueous use of bindme alteme and rSmdiante is far
from happy.*

Catalan (1846).

[Recherches sur les de^terminants. Bvlh de VAcad. ray, ... de

Belgique^ xiii pp. 534-565.]
As is known, Catalan had already dealt with determinants in
the year 1839 in a memoir regarding the change of variables in a
multiple integral In the paper which we have now come to be
leads up to examples of the same kind of transformation ; but the
greater part of it — seventeen out of the total twenty-two pages —
is occupied with determinants pure and simple. Half of this
amount consists of an elementary exposition of known properties,
and calls for no remark save that what Cauchy called ''produit
principal " or " terme indicatif ^* is here called '* terme carac-
teristique/' and that he makes constant use of the symbolism

d6t.(A, B, C, . . . )
to stand for the determinant whose first row consists of a\ second
row of h\ and so on : for example,

d^t.(B, A, C, . . . ) = - det.(A, B, C, . . . ) ,

d^t.(A, A, C, . . . ) = 0,

d^t.(A + M, B) = d^t.(A, B) + d^t.(M, B) ,

When we come to § 13, however, we find fresh ground struck.
The exact words are : —

" Supposons maintenant qu^^tant donn6 le syst^me —
A,

A,,

(A)

♦ Two years later we find him, in referring to a paper of Cayley's where the
determinant ' L T S ^

T M R i;

S R N f

I U f

occurs, calling it a '* fonction cramerienne," and writing it

r L T S I A

I T M R 7, I

I S R N C (

^ ^ n C ^'

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1908-4.] Dr Muir on General Determinants, 89

dont le determinant est A , on ait combine par voie d'addition
et de soustraction lea ^nations dont les premiers membres
sont repr^sent^ par A^ , A^ ,...., An ; et, par exemple,
qu'on ait d^duit du syst^me (A) le systeme suivant

Ai + Aa + . . . + Ky\
Aj — Ag ,
A, - A, . } (B)

An_i — An

dont la consideration nous sera utile plus loin. Soit A' le
determinant de ce nouveau systeme: d'apr^ les n** (3) et
(4), nous aurons

A'= det. (Aj , - Ag , - A3 , . . . , - A„)

+ det. (Aj , Ag , — A3 , - A4 ,..., — An)
+ det. (Aj , Ag , A3 - A4 , . . . , - An)

+ det. (An , Aj , A 2 , . . . , An_i) .

On sait que si Ton change les signes des termes d'une colonne
horizontale, le determinant change de signe ; done

A' = (-1)-^ det. (Ai , A2 , . . . , An) + (-l)«-« det. (A2 , Aj , A3 , . . . , An)

+ (-l)"-»det. (A8,Ai,A2,A4,...,A„) +

+ (-det.(A„ i,Ai,A2,...,A„_2,An) + det.(An,Ai,A2,...,A„-i).

Dans la premiere parenth^se, il n'y a pas d'in version ; dans la
seconde, il y a une inversion, etc. ; done

A' = (-l)«-^w A."

The theorem thus reached may be enunciated as follows : — 1/
from a determinant A of the n** order, we form another A' such that
the first row of ^' is the sum of all the rotes of A and every other
row of ^' is got by suhtracting the corresponding row of A from the
row preceding it in A, then

A' = (-l)«-in A.

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90 Proceediivjs of Royal Society of Edinburgh. [s

In Catalan's notation it is

d^t. (Ai + A2 + . . . + A, , Aj - A.^ , A2 - A3 , . . . , A,_i - A,)
= (-l)"-^n.d(5t. (A,,A2,...,A.),

although, strange to say, it is never so formulated by him.
A generalisation of it is next given by saying : —

"Si la premiere ligne du syst^me (B) avait renferme
seulement p des quantites A^ , A^ , . . . , A^ , nous aurions
trouv^, pour la determinant de ce syst^me,

A' = (-!)"> A,"

and then there follow a number of applications to the evaluatioD
of certain special determinants.

Thus, to take the simplest example, having

A = 1 . . . = 1
. 1 . .

the theorem gives

-1

= ( - 1)M A = - 4 .

The other illustrations all concern determinants of the special
form afterwards known as "circulants " ; for example, C ( - 1 , 1 ,
l,...,l),C(-l,-l,l,l,...,l),etc.,C(l,l,...,l,0),
C (I , 1 , . . . , 1,0,0), etc. They therefore fall to be dealt
with in a different place.

Sarbus, p. F. (1846).

[Finck, P. J. E. Elements d^Alg^bre. Seconde ^tioa.
iv + 544 pages. Strasbourg.]

In the course of his discussion of the solution of a set of linear
equations with three unknowns, the author interjects the following
paragraph (No. 52, p. 95) : —

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190:i-4.]

Dr Muir oil General DetermiiuDUs,

"Pour calculer, dans un exemple donn($, les valeurs de x, //
et 2, M. Sarrus a imaging la m^thode pratique suivante, qui
est fort ing^nieuse. D'abord on pent calculer le d^nomina-
teur, et k cet efifet on ($crit les coefficients des inconnues ainsi

a h c
a' b' c

n -lit n

a h c

On repute les trois premiers a h c

et les trois suivants a h' c

Actuellement partant de a, on prend diagonalement du haut
en has, en descendant h, la fois d'un rang, et reculant d'autant
k droite, a h'c : on part de a' de m^me, et on a a h'c ; de
a , et on trouve ah c ; on a ainsi les trois termes positifs
(c*est-k-dire k prendre avec leur signes) du denominateur. On
commence ensuite par c et descendant de m^rae vers la
gauche on a c h'a" , clfa , ch a , ou les trois termes n^gatifs
(ou plutdt les termes qu'il faut changer de signe)."

This **methode pratique" or mnemonic is the original form of
the so-called " r^le de Sarrus " which came later to have un-
necessary prominence given to it by writers on determinants when
ilealing with those of the third order*

* The date 1883 has been assigned to this '* rule *' in a recent German text-
book on detenninants (Weichold's) : if 1833 be the correct date the '* rule '*
probably will be fonnd in a publication by Sarrus entitled Nouvelle mdhode
pour la r^lutian dcs iquatums^ which appeared at Strasbourg in that year.

LIST OF AUTHORS
whose writings are herein dealt with.

PAGE

PAGB

1748. Fontaine

. 61

1843. Cayley .

. 76

1829. Cauchy .

. 62

1843. Cayley .

. 78

1829. Jacobi

. 63

1846. Cayley .

. 83

1833. Jacobi .

. 69

1846. De FfiRUssAC .

. 86

1834. Jacobi .

. 72

1846. Terqukm

. 86

1839. MoLiNs .

. 73

1846. Catalan

. 88

1843. Boole .

.74

1846. Sarrus .

. 90

(Issued separately February 12, 1904.)

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92 Proceedings of Royal Society of Edinburgh. [sbss.

Man as Artist and Sportsman in the Palaeolithic
Period. By Robert Munro, M.A., M.D., LL.D. (With
Eleven Plates.)

(Ad Address deliyered at tlie request of the Coonci], Not. 28, 1903.)

I. Introduction.

So long as Homo sapiens was believed to occupy a higher
platform in the organic world than other animals by virtue of
his special endowments, no one, apparently, thought of looking
for evidence of his origin and history in the obscure vista of
prehistoric times. The long cherished traditions and myths
which had gathered around the inquiry left little room for any
other hypothesis than that his apparition on the field of life was
the last and crowning achievement of a long series of creative
fiats which brought the present world-drama into existence. In
the cosmogony thus conjured up, the multitudinous phenomena
of the material world — animals and plants, the distribution of
land and water, the recurrence of seasons, etc. — were regarded as
having been specially designed and arranged to administer to the
life-functions of this new being.

Nurtured in an environment so full of legendary romance, we
need not be surprised that the philosophic schools of Britain, as
well as of other countries, continued to teach some such theory
of man's origin up to about half a century ago, when the doctrine
of organic evolution captured the scientific mind of the day. But,
notwithstanding the far-reaching significance of the evolution
theory, the evolutionary stages of man's career on the globe
remained almost as great a mystery as before ; for, at the outset,
the new doctrine appeared to go no further than to point to the
direction in which the trail of humanity was to be looked for.
The erect attitude, bipedal locomotion, true hands, and a unique
handicraft skill, amply difierentiated him from all other animals.
But for a long time no rational explanation of how he acquired
these distinguishing characteristics was forthcoming; and, even

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1903-4.] Dr Munro on Man in the Palceolithic Period, 93

now, their origin and development are among the most obscure
problems within the whole range of anthropology.

In the address which I had the honour of delivering in 1893,
as president of the Anthropological Section of the British Associa-
tion for that year, I advocated the hypothesis that the origin of
the higher mental manifestations of man was primarily due to
the attainment of the erect attitude, which, by entirely relieving
the fore-limbs of their primary function as locomotive organs,
afforded him the opportunity of entering on a new phase of
existence, in which intelligence and mechanical skill became the
governing factors. With the completion of the morphological
changes involved in the attainment of this attitude, the evolution
of the present human form, with the exception of some remark-
able modifications in the skull and facial bones, which will be
subsequently referred to, was practically completed. As soon as
bipedal locomotion became habitual and firmly secured on an
anatomical basis, it does not appear that the osseous characters of
the lower limbs would be sensibly affected by any subsequent
increase in the quantity or quality of brain-matter. For example,
the function of the femurs being henceforth to support a certain
load, i,e. the entire weight of the body, it would not influence
them in the least whether that load contained the brains of a
fool or of a philosopher. The important and novel element which
the permanent assumption of the erect posture was the means of
introducing on the field of human life, was the use to which the
eliminated fore-limbs were put. By substituting, for nature's
means of defence and self-preservation, a variety of implements,
weapons and tools made with their own hands, the subsequent
well-being of these novel bipeds became dependent on their
ability to interpret and utilise the laws and forces of nature.
As time went on they began to recognise the value of the faculty
of reasoning as the true source of inventive skill ; and hence a
premium was put on this commodity. In this way, stimulants to
the production of new ideas and new inventions were constantly
coming within the scope of their daily avocations, the result of
which was a steady increase of human intelligence, and conse-
quently of brain substance. Now, according to the well-
established doctrine of the localisation of brain function, the

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94 Proceedings of Boyal Society of Edinburgh. [sess.

additional brain molecules and cells thus acquired had their seat
of growth for the most part somewhere in the cerebral hemi-
spheres which lie well within the anterior portion of the brain-
casing. The mere mechanical effect of this increment to the
physical organ of thought would be to increase the weight of the
anterior half of the head, and so to upset its finely equipoised
position on the top of the spinal column. But as any interfer-
ence with the free and easy rotatory movements of the liead
would manifestly be disadvantageous to the individual in the
struggle of life, it became necessary to counteract the influence
of this disturbing element by some other concurrent morpho-
logical process, which would not be prejudicial to the general
well-being of the human economy. This object was partly
attained by a retrocession or contraction of the facial bones,
especially the jaw bones, towards the central axis of the spinal
column, and partly by a backward shifting of the cerebrum over
the cerebellum. As the gradual filling up of the cranial cavity pro-
gressed necessarily pari passu with these cerebral modifications,
we have, in the facial angle of Camper, a rough mechanical means
of estimating the progress of mental development during the
period of man's existence as a human being, i.e. since he
attained the erect attitude.

One of the results of this retrocession of the facial bones was
the gradual contraction of the alveolar borders of the jaws, thereby
diminishing the space allotted to the teeth, — a fact which plausibly
accounts for some of the peculiarities which differentiate the older
fossil jaws from modern specimens. Thus, in the dentition of the
former, the third or last molar is the largest, whereas in the latter
it is the smallest. Not only so, but among Neolithic and some
European races of to-day these four molar teeth (wisdom) make
their appearance at a later date in the individual's life than for-
merly, so that they seem to be on the highway to become vestigial
organs. It is interesting to note that the shortening of the dental
portion of the human jaw attracted the attention of Mr Darwin,
who, however, attributed it to " civilised men habitually feeding
on soft, cooked food, and thus using their jaws less."

Another peculiarity of civilised races is the greater prominence
of the chin, a peculiarity which may also be due to the contraction

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1903-4.] Dr Monro on Man in the Pcdceolithic Period, 95

of the alveolar ridges and the consequent more upright setting of
the teeth in their sockets. But whatever the precise cause may
have heen, there can he no douht that the gradual formation of
the chin has a striking parallelism with the progressive stages
of man's intellectual development, ever since he diverged from the
common stem line from which he and the anthropoid apes have
descended (see fig. 18).

From these general remarks it will he seen that there are two
distinct lines on which investigations into the past history of man-
kind may be profitably conducted, both of which start from the
attainment of the erect attitude. The evidential materials to be
gathered from these different sources consist, in the one case, of
some fragments of a few skeletons of former races, which, by some
fortuitous circumstances, have to this day resisted the disintegrating
forces of nature ; and, in the other, of a number of specimens of
man's handicraft works, which, being largely made of such en-
durable substance as flint, are more abundantly met with. The
successive modifications which these respective materials have
undergone during a long series of ages, though different in kind,
are found to bear a decided ratio to the progress of human intelli-
gence. Thus, taking the human skull at the starting-point of
humanity as comparable to that of one of the higher apes, we
know, as a matter of fact, that during the onward march of time
it has undergone some striking changes, both in form and capacity,
hcfore reaching the normal type of modem civilised races — changes
which can be largely classified in chronological sequence (see pp.
99-108). Similarly, the artificial products of man's hands show
a steady improvement in type, technique, and efficiency, commen-
surate with his progressive knowledge of the laws of nature and his
ability in applying them to mechanical and utilitarian purposes.
Indeed, the trail of humanity along its entire course is strewn with
the discarded weapons and tools which, from time to time, had to
give way to others of greater efficiency. Such obsolete objects are
now only collected as curiosities to be preserved in archaeological
museums (see pp. 109-117).

The main object of these preliminary remarks is to emphasise
the nature and true significance of the methods by which anthro-
pologists have been enabled to prosecute their researches far

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96 Proceedings of Royal Society of JEdinhirgh. [sess.

beyond the limits of the historic period. Without a clear notion
of the logic and grounds on which their deductions are founded,
it would be impossible to enlist the attention of a general audience
to an address involving data so different from those of ordinary
scientific worL

The special subject on which I have to discourse consists of
some exceptionally interesting human relics, chiefly belonging to
the Later Palaeolithic period in Europe. These remains have been
most abundantly found among the culinary d&nis of a race of
hunters who inhabited caves and rock-shelters in France, Switzer-
land, South of England, and other parts of Europe. Among the
more remarkable objects collected in these localities are representa-
tions of various animals carved, and sometimes sculptured, on
pieces of ivory, horn, bone and stone. As illustrations of most
of these artistic productions have been published, I am enabled
to exhibit some of the more characteristic specimens on the screen.
But before doing so, there is one question which I had better
dispose of at once, viz., that of their supposed age, because the
answer is itself a typical object-lesson of the resourceful means
by which anthropological investigations are being conducted.

Whatever views may be held as to the anthropological value of
the famous skull of Pithecanthropus erectus (figs. 4 and 5), dis-
covered some ten years ago by M. Dubois in the Upper Pliocene
deposits of Java, the femur (fig. 6) found in the same stratum
with it conclusively proves that there had been then in existence a
being of the genus Homo which had assumed the erect attitude as
its normal mode of locomotion — i.e. at a time prior to the advent
of that great landmark in the physical history of the northern
hemisphere known as the glacial period. Now it was only towards
the end of that period, just when the ice sheet and its great
feeding glaciers were creeping back to their primary centres of
dispersion in the mountainous regions of Britain, Central Europe,
and Scandinavia, that the European troglodytes, whose antiquity
is now suhjudice^ flourished. Hence, they and their works must
be assigned to an intermediate period between the present time
and the starting-point of humanity. As the first part of this
chronological range may be equated with nearly the whole duration
of the glacial period, the task of converting it into so many cen-

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1903-4.] Dr Munro on Man in the PcUceolithic Period. 97

turies or millenniums may be left in the bands of astronomers and
geologists, wbo, in more recent times, bave appropriated among
them tbe solution of this part of tbe problem. It is witb the
second part of tbe range, viz., tbe time wbicb bas elapsed since
the Palseolitbic artists and hunters lived, that we are now chiefly
concerned. It embraces tbe entire duration of the Historic, Iron,
Bronze and Neolithic Ages, together with an interval of unknown
length between tbe Neolithic and Palseolithic civilisations. It has
long been supposed that during this obscure interval there had
been a hiatus in the continuity of human existence in Western
Europe — an idea which, however, is now justly discredited in face
of more recent discoveries, throughout the same geographical area,
of transition deposits containing human relics. Of these later
discoveries the rock-shelter of Schweizersbild, near Schaflhausen,
is one of the best examples known to me, as its d^hria indicates
that tbe site was a constant rendezvous for bands of roving hunters
from the Palaeohthic period down to the Bronze Age. Dr Niiesch,
its explorer, has expressed the opinion, founded on the relative
thickness of the deposits and the character of the fauna represented
in them, that the antiquity of its earliest human relics cannot be less
than 20,000 years. Now, since the art-remains found in tbis station
and in tbe adjacent cave of Kesslerlocb are precisely similar to
those of the analogous stations in France, we can accept the above
estimate as equally applicable to the latter. The nature of the evi-
dence on which Dr Niiesch founded his opinion is briefly as follows : —

According to Professor Nehring, who bas made a special study
of the animals now inhabiting tbe arctic and sub-arctic regions,
those characteristic of the former are — Band-lemming, Obi-
lemming, arctic fox, mountain bare, reindeer and musk - ox.
With these are frequently associated a number of animals of
migratory habits, such as northern vole, water - rat, glutton,
ermine, little weasel, wolf, fox and bear. Now, the extraordinary
fact was brought out that of these fourteen species only tbe Obi-
lemming and the musk-ox were imrepresented in the lowest
relic-bed of tbe Schweizersbild. The latter was, however, found
in the dShris of the Kesslerlocb cave in the vicinity. It appears
that the Band-lemming (Myodes torquatus) and the arctic fox
are the most persistent animals of the arctic fauna, so that the

PBGC. BOY. SOC. EDIN. — VOL. XXV. 7

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98 Proceedings of Royal Society of Edinhtrgh. [sbss.

presence of the bones of these two animals in the debris of
this rock - shelter was alone suflficient to prove that the climate
of the period was of an arctic character. In the upper portion
of this deposit relics of new animals, indicating a change to a
sub - arctic climate, began to appear, and had their greatest
development in the next succeeding layer.

The result of careful analysis of the contents of the other
deposits showed that this arctic fauna became ultimately dis-
placed by the true forest fauna of the Neolithic period. Among
the newcomers were the badger, wild cat, hare, UmSj Bos longi-
frons^ goat and sheep; while of those represented in the Palae-
olithic deposit a large number was absent. Thus both the
arctic and sub-arctic fauna had given way to a forest fauna,
and, synchronous with these changes, the Palaeolithic hunters
and reindeer vanished from the district.

Among the few art specimens found at the Schweizersbild is
a stone tablet, having rude outlines of a wild ass and of a
reindeer incised upon it. The whole collection, among which
were 14,000 worked flints, 180 fragments of bone needles, 41
whistles, 42 pierced ornaments made of shells and of the teeth
of the arctic fox, glutton, etc., is typical of the latest phase of
Palaeolithic civilisation of the Dordogne caves.

The chronological deductions founded on the investigations at
the Schweizersbild are, from their very nature, more or less
hypothetical. But, after all allowances for possible errors are
made, I can see no objection to Dr NUesch's lowest estimate of
the date of man's first appearance into Northern Switzerland,
viz., 20,000 years ago.*

I now proceed to exhibit some illustrations selected from the
evidential materials on which the opinions and conclusions ad-
vocated in this address are founded. The slides are arranged
in two series, corresponding to the two lines of research on
which, as mentioned in the preliminary remarks, anthropological
investigations are most usually conducted. Afterwards I will
add some further comments on the phase of human civilisation
thus so singularly resurrected from the lumber-room of oblivion.

•See Neice DenkschrifUn der allgemHncn schweizerischen Oesellschaft fiir
die gesainmUn Natunrisscnschaflen^ vol. xxxv.

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1903-4.] Dr Munro on Man in the Palceolithic Fe7*iod. 99

II. Illustrations,

The following illustrations are not in all cases reproductions of those
exhibited on the screen when the address was deliyered, as it was im-
practicable to conyert some of them into printing blocks. They are, how-
ever, with few exceptions, substantially the same, only grouped differently,
and are specially selected to elucidate the various points touched upon in
the text. The remains of fossil man are, as yet, too meagre to afford much
choice of illustrative materials ; but of the handiworks of the artists and
hunters of the Paleolithic period there is no lack, as, indeed, most of the
principal musetuns of the world contain more or fewer specimens in addition
to casts of the most remarkable pieces. Even in the Scottish metropolis,
anyone desirous of becomii^ conversant vrith their characteristic features
has only to visit the ethnological department of either the Museum of Science
and Art or of the National Museum of Antiquities. The literature of the
subject is also voluminous and much of it readily accessible, among which
I would particularly mention the recently issued Guide to the Antiquities
of the Stone Age in the British Museum. Owing to the roundness of the
beam of an anUer, on which these engravings are generally executed, the
whole of the incised outlines of an animal cannot always be seen from one
point of view, and hence a drawing is sometimes more effective than a
photograph. The illustrations here supplied are the result of a combination
of all available sources— original specimens, casts, photographs and drawings
of objects not at hand being requisitioned into the work.

A. — Evidence of Progressive Changes in tJie Human SkvlL

Among the bodily features which distinguish man from other
animals the following are particularly worthy of note, viz., the
upright attitude, with the head balanced on the top of the spinal
column; the double curvature of the spine; the great difference
between the hands and feet; the power of firmly opposing the
thumb to each of the other four fingers ; the prominence of the
frontal bone; and the almost vertical profile of the face. It
may, however, be observed that, as regards the prominence of
the forehead and degree of prognathism of the facial bones,
some strikmg variations occur among the different existing races.
To show the extent of these differences I reproduce, from Owen's
Comparative Anatomy (vol. ii. pp. 558, 560), figures of two skulls,
one (figs. 1 and 2) labelled "Craniimi of a native Australian,"
and the other (fig. 3) "Skull of a well-formed European," from
which it will be at once seen that the former has, relatively, a
retreating forehead and a highly prognathic profile, while the
latter has a well-filled forehead and an orthognathic face.

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100 Proceediyujs of Royal SocUty of Edinbiirgh. [siss*

The next step in the argument is to show that some fossil
skulls possess, to a more or less degree, the features of the
Australian skull — the degree of divergence from the normal
European type being in direct proportion to their antiquity.
As bearing on this important generalisation, let me, in the first

Figs. 1 and 2. - Front and side views of the skull of a native
Australian. (After Owen.)

place, refer to the famous calvaria of Pithecanthropus erecttis
(figs. 4 and 5), discovered (1891-2) by Dr Dubois, in the
detritus of a Pliocene river in Java, which shows a remarkably
low and retreating forehead. In the absence of the facial bones

Fio. 3. — Skull of a well-foiined European. (After Owen.)

we can only surmise that the individual which originally owned
this skull presented a highly prognathic appearance, approaching
even to that of Hijlohates, to which Dr Dubois compares it.
(See Pith, eredus, Plate I., 1894.)

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1903—4.] Ik MuTiTo on Man in the PalceolUhw Period, 101

The femur (fig. 6) discovered by Dr Dubois in the same place
has been pronounced by most of the anatomists who had criti-
cally examined it to be human; but, as it lay at a distance of
15 metres from the calvaria, there is no absolute certainty that
the two bones belonged to the same individual. There can,
liowever, be no doubt that this femur was that of an animal

Fio. 4. — Side view.

Fio. 6. — Top view.
The skull of PUheeanthropus erectus, Java (i). (After Dr Dubois )

which, at that early period, had attained the erect attitude —
an animal which therefore must have belonged to the genus
Homo, The logical deduction from these data is thus necessarily
limited to probability ; but if the hypothesis of organic evolution
be correct, the Java skull is precisely in that stage of cranio-
logical development which would be expected at that early time
in the history of humanity.

The skull of the human skeleton discovered in 1856 in the

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102 Proceedings of Boyal Society of Edinburgh. [(

cave of Feldhoven, situated at the entrance to the Neanderthal
ravine, on the right bank of the Diissel, and since known as
the 'Neanderthal skull/ presented such remarkable peculiarities
that, when first exhibited at a scientific meeting at Bonn,

Fig. 6. — Femur of Pithecanthropus erectus^ found in Java (J).
(After Dr Dubois.)

doubts were raised by several naturalists as to whether the
bones were really human. Figs, 7 and 8 represent two views of
this relic, outlined from figures published by Professor Huxley
{Collected Essays, vol vii. p. 180), from which its characteristics,
especially the low retreating forehead, may be seen at a glance.
Writing in 1863, Professor Huxley made the following remarks

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1908-4.] Dr Munro on Man in the Palceolithic Period, 103

on the Neanderthal skull : — " There can be no doubt that, as
Professor Schaaffhausen and Mr Busk have stated, this skull
is the most brutal of all known human skulls, resembling those
of the apes not only in the prodigious development of the
superciliary prominences and the forward extension of the orbits,
but still more in the depressed form of the brain-case, in the

Fi«. 7.— Side view.

Fig. 8.— Top view.
The Neanderthal skull (^). (After Huxley.)

straightness of the squamosal suture, and in the complete retreat
of the occiput forward and upward, from the superior occipital
ridges." — (LyelFs Antiquity of Man, p. 84.)

The skull (cephalic index 70) of one of the Spy skeletons
(figs. 9, 10 and 11) also shows a low retreating forehead, marked
prognathism, a sloping chin, and large third molar teeth. These
skeletons were discovered in 1886, buried 12^ feet in fallen
debris at the entrance of a grotto in the province of Namur,

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104 Proceedings of Bayal Society of EdivhiLrgh.

Belgium. The worked flints found in the cave were of the
type known as Mousterien, and among the fauna represented
were Rhinoceros ticlwrhinus^ cave -bear, mammoth, hya?na, etc
No works of art were among the relics, so that the Spy troglodytes

Fio. 9. —Side view.

Fio. 10.— Top view.
Skull from the Orotte de Spy (i). (After Fraipont)

arc justly regarded as Ijelonging to an earlier period tlian that
in which the reindeer hunters and artists flourished.

The larger portion of a lower human jaw (figs. 12 and 13) was
disinterred in 1 865 from the debris in the Trou de la Nanlette, at
a depth of 4-50 metres beneath the last floor of the cave. Above
it was a succession of five stalagmitic layers, intercalated with

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1908-4.] Dr Munro on Man in the Palasolithic Period, 105
fluvial deposits from the river Lease. The fauua represented in

Fio. 11. — ^Tracing showing size of teeth in the lower jaw of Spy skull (§).
(From photograph. )

Fig. 12.— Naulette jaw— side view (|). (After M. Dupont.)

Fio. 13.— Naulette jaw— view from above (|). (After M. Dupont.)

the same stratum included the mammoth, rhinoceros, horse, and a
number of animals common to Neolithic times. The special

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106 Proceedings of Royal Society of Edinhargh, [sess.

features of this jaw are its small height in proportion to its thick-

FiG. 14.— Side view.

Fio. 15. — Front view.
Skull of the • Old Man of Cro-Magnon ' (j^).

ness, the backward slope of the chin, and the large size of the
socket of the third molar.

Figs. 14 and 15 show front and profile views of the skull of

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1903-4.] Dr Munro on Man in the PalceolUhic Fei'iod, 107

the 'old man of Cro-Magnon/ which discloses a decided approach
to the normal type of civilised man. Its cephalic index is 73 6
and its capacity 1590 c.c. The height of this individual was 1*82
metres (5 feet 11^ inches). The lower jaw has a large ascending
ramus, behind which, on both sides, the third molar is partly
hidden. These two teeth have also the peculiarity of being
smaller than the other molars, being in this respect more allied to
the dentition of Neolithic and modem races. For these reasons,
as well as the fact that it was found on the surface of the Palseo-
lithic debris, some anthropologists maintain that the * old man of

Figs. 16 and 17. — Two skulls from the Grotte des Enlauts, Meutone.
(After M. Verneau.)

Cro-Magnon' belonged to the early Neolithic period — a point
elsewhere referred to in this address.

Figs. 16 and 17 are reproductions of illustrations by Dr
Vemeau of two skulls found in the Grotte des En/ants, near
Mentone. That on the left belonged to a young man, and that on
the right to an aged female. They are part of two skeletons
which lay close together on a hearth-layer at a depth of 7 '75
metres. The cephalic index of the former is 69 72 and of the
latter 68*58. These skeletons were those of small individuals,
their respective heights being 1*54 mfetres (5 feet OJ inch) and
1*58 metres (5 feet 2 inches). About 27^ inches higher up in the
debris another skeleton, measuring no less than 1*92 metres in
height (6 feet 3 J inches), was found, which presented all the

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108 Proceedings of Royal Society of Edinburgh, [i

characteristics of the Cro-Magnon type (cephalic index 76'26).
The debris in which these skeletons were discovered contained
relics comparable to those of the latest phase of the Palaeolithic
civilisation (VAnthropologie, vol. xiii. pp. 661-583).

Fig. 18 represents a series of lower jaws illustrating, accord

Cliimpanzee — Troglodyte*
Avbryi.

2. The Kaulette jaw, from the
valley of the Leise, Belginm.

3. Melanesian, from the New .
Hebrides. 3

4. The Arcy jaw, from the Z
GrotU deg Fit* (Yonne). ^

6. From Uie dolmen of Cha-
mans (Oise).

6. Modem Parisian.

Fio. 18. — Profile of various lower jaws. (After Broca.)

ing to the late Paul Broca, the gradual evolution of the human
chin. M. Broca exhibited the drawing in support of his views at
the International Congress of Anthropology and Prehistoric
ArchfiDology held in Paris in 1867 {Goraptes ReTidtts, p. 399). The
Spy jaw, which of course was then unknown, would take its place
in the series between Kos. 2 and 3

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1903-4.] Dr Munro on Man in the Palceolithic Period, 109

B. — Evidence of progremve skill in tlie handicraft works of Man,

Plate I. gives a full-sized view of a flint implement found, along
with an elephant's tooth, at Gray's Inn Lane, London, about the
end of the 17th century, being the first recorded discovery of the

Fio. 19.— Palaeolithic flint implements from the Terracogravel
at Galley Hill (i).

kind in Britain. It is a typical specimen of what French archae-
ologists call the * coup de poing,' probably the first definite type of
hand-implement which came to be widely imitated among the earlier
races of man. Implements of this kind vary considerably in form
and size, the degree of variability being, however, strictly compat-
ible with its function as a hand-tool. Fig. 19 shows a variety

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110 Proceedinffs of Royal Society of Ediriburgh, [i

of such implements from the terrace-gravels of Galley Hill, Kent*
Of course it is not denied that stone implements were used by man
long before he invented the * coup de poing/ but I am unable to
classify those earlier forms into any chronological sequence. Nor
would I hazard a guess, in the present state of our knowledge, as
to whether it is by centuries or millenniums we are to reckon the
duration of that earlier stage of man's career.

Worked flintis of the * coup de poing ' type are largely collected
from the river-drift gravels of England and France, as well as
elsewhere, and nearly all have the peculiarity of being made by
chipping a nodule so as to convert it into a useful hand-tool — the
flakes struck off being apparently of no use. When, however, it
was discovered that some of the larger flakes could be utilised as
sharp cutting tools, attention began to be directed to the art of
producing them for teleological purposes. After some experience a
skilled workman could produce a flake of any required size and
shape. By subjecting these flakes to secondary chipping, imple-
ments of great variety and efiiciency were ultimately obtained.
This was indeed an important step in flint industry, evidence of
which is to be found in the fact that henceforth flakes were the
useful products, wliile the residuary cores were rejected as waste.
The worked flints found in the earlier inhabited caves of France
and Belgium, such as Moustier and Spy, show that the flaking
stage was already in full progress — thus proving that their habita-
tion was later than the formation of the river-drift gravels.
Towards the middle of the PalsBolithic civilisation (Soltttreen) the
flint industry had attained a state of great perfection, scarcely sur-
passed in any subsequent period.

That these cavemen did not confine their awakening intelligence
to the working of flint objects is amply shown by the array of
broken or lost harpoons, lance- and spear-heads, pins, needles,
and nondescript articles made of bone or deer-horn which now
appear in the debris of their inhabited sites. Some idea of their
skill in this new industry may be gathered from an inspec-

♦ These flint figures are from the Quarterly Journal of Vie Oeologieal
Society (vol. 11.). The block was kindly lent to me by the Council for uae in
Prehistoric Problems^ and it ia here reprinted from the clich6 then made for

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i90»-4.] Dr Munro on Man in the Palceolithic PejHod. Ill

tion of Plate III. Indeed it would appear as if bone and horn
had almost superseded flint in the manufacture of weapons of the
chase. This partly accounts for the large number of small flint
tools, such as knives, saws, scrapers, borers, etc., found on

Figs. 20 and 21. — Bovidte incised on stone, from the rock-shelter of
Bmniquel (3). (After British Museum Catalogue.)

Magdalenien sites (Plate 11. ). It was, no doubt, by means of these
finer flint instruments that the artists were able to bore the eye
of a fine needle, to carve hunting scenes, and to sculpture their
dagger-handles and hdtons de commandenient into the conventional
forms of familiar animals.

The artistic skill displayed by these primitive hunters has been

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112 Proceedings of Boyal Society of Jidinburgh. [sbss.

one of the most astounding revelations of prehistoric archaeology.
Typical specimens of their skill in carving and sculpture on bone,
deer-horn, and ivory may be studied on Plates HI. to X. Figs.
20 and 21 represent two stones from the rock-shelter of
Montastruc, Bruniquel, with outlines of bovidae incised on them,
the forms of which might have been intended for the Bosprimi-
genius. The originals are now in the British Museum.

C. — The Carving and Painting of Animals on the Walls of
PaUBolithic Caves.

Within later years interest in the art remains of these

Fio. 22.— Incised figure of horse on the wall of the QroUc de la Moathr.
(After E. Riviere.)

Palseolithic hunters has been greatly stimulated by the dis-
covery of large engravings, and even coloured paintings, of
various animals on the walls of some newly-explored caves in
the South of France, more especially those of Combarelles and
Font-de-Gaume, both situated in the Commune of Tayac (Dor-
dogne), and within a short distance of the well-known station
of Les Eyzies. Obscure indications of this kind of art had been
observed as early as 1875 in the cave of Altamira, near San-
tander, in the north-east of Spain. Subsequently, and at various
intervals, more pronounced examples were notified in the caves
of Chabot (Giird), La Mouthe (Dordogne), and Pair-non-Pair

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1903-4.] Dr Munro on Man in the Palaeolithic Period. 113

(Gironde), in all of which figures of animals regarded as
characteristic of the Palaeolithic period occurred.

Of the earlier discoveries I reproduce (after M. Riviere) illus-
trations of two horse figures engraved on the walls of the cave
of La Mouthe {Bvll. de la Socieie d^ Anthropologies October 19th).
These designs were incised on a panel 128 metres from the
entrance. The first (fig. 22) represents an animal with a small
head, slender neck, and well-formed fore-quarters; but the
posterior part is heavy and altogether out of proportion. The
other (fig. 23) has a stout neck, a long head, with a front
directed almost vertically, and a heavy chin. Whatever may
have been the defects of the artists, the originals of these two

Fig. 23.— Head of horse, QrotU de la Mouthe. (Riviere.)

drawings must have been very different animals, if not differ-
ent species. Among the other animals figured in this cave
were bison, bovidae, reindeer, goat and mammoth.

On the 16th September 1901 MM. Capitan and Breuil sub-
mitted a joint note to the Paris Academy of Sciences on "A
New Cave with Wall Engravings of the Palceolithic Epoch."
This was followed a week later (23rd September) by a second
note, by the same explorers, on "A New Cave with Painted
Wall Figures of the Palaeolithic Epoch." A noteworthy dis-
tinction in the art illustrations of these two caves is that one
(Combarelles) has its walls adorned almost exclusively with
engravings, cut more or less deeply, and the other (Font-de-
Graume) with paintings in ochre and black, or sometimes only

PROC. ROY. SOC. EDIN. — VOL. XXV. 8

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114 ProceediTigs of Royal Society of Edinburgh. [}

in one colour, forming real silhouettes of the animals thus de-
picted.

Some of the engravings in the cave of Combarelles have been
carefully copied and published by the explorers, from which
the following figures are reproduced {Revise de I'JScole d^Anthro-
pologie, January 1902).

Fio. 24. — A group of animals on the wall of the cave of Combarelles.

Fig. 24 shows a group of animals on a portion of the wall.
Fig. 25 represents a pony with a large head, shagg}' mane, and
a bushy tail. It has been suggested by MM. Capitan and
Breuil that the animal was domesticated, bridled, and draped
with some kind of ornamental covering. Reindeer, wild goat.

Fig. 25.— Outline of horse supposed to be domesticated. (Combarelles.)

and mammoth will be readily recognised under figs. 26, 27,
and 28. It will be of interest to compare with the latter
figure that of the skeleton of the mammoth (fig. 29) whose
carcass was discovered in 1799 embedded in frozen tundra at
the mouth of the Lena, Siberia. Seven years later it was
purchased by Mr Adams for the museum of St Petersburg, but
in the interval dogs and wild animals had eaten the flesh, and
only the bones and fragments of the skin with its long hair
could be recovered. The carcass of another mammoth was

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1903-4.] Dr Munro on Man in the PcUceolithic Period, 115

observed in 1901 near the town of Stredne-Kolymsk, and an
expedition under Dr O. Hertz has recently transported the

Fig. 26. — Reindeer incised on wall of Combarelles.

entire animal in sections to Moscow, with the view of mount-
ing it with its skin.

Fig. 27. — Figure of wild geat from the cave ol Combarellea.

The total number of engravings in the cave of Combarelles, so
far as they could be distinctly made out, is 109 : — animals
entire but not identified, 19; equidsB, 23; bovidse, 3; bison, 2;

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116 Proceedings of Boyal Society of EdiTibv/rgh. [sbss.

reindeer, 3; mammoth, 14; heads of goats, 3; heads of ante-
lopes, 4 ; heads of various animals, chiefly horses, 36 ; human
face, 1 (?); cup»mark, 1. These engravings, in the opinion of
the explorers, betray so much artistic skill, precision of details.

ilf^"^

Fio, 28. — Incised figure of mammoth in cave of Combarellos,
Figs. 24 to 28 are reduced from the drawings of MM. Capitan and Breuil.

and knowledge of animal life, that they must be regarded fas
vahiable documents in Palaeontology,

Fig*. 29.— Skeleton of the mammoth found in Siberia in 1799,
now in St Petersburg.

^fore recently, M^f. Capitan and Breuil published ilhistrations
of some of the painted figures on the walls of the Grotte de Font-
de-Gaume on two plates, one of which is here reproduced (PI. XT.)
on a smaller scale — (Revue de VEcole d^ Anthropologies July 1902).

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1903-4.] Dr Munro on Man in the Fakeolithic Period. 117

This plate represents an excellent picture of a bison (fig. 1) and
a still more striking one of two reindeer (fig. 2). The original
drawing of the former is painted in ochre, and measures
1 m. 50 in length and 1 m. 25 in height ; that of the latter
is 2 m. 10 in length and 1 nu 50 in height^ and presents the
peculiarity of having portion of the figure on the left executed
in incised lines.

The total number of painted figures in this cave is 77:—
aurochs, 49; indeterminate animals, 11; reindeer, 4; stag, 1;
equidse, 2 ; antelopes, 3 ; mammoth, 2 ; geometrical ornaments, 3 ;
scalariform signs, 2. The authors suggest that these paintings
belong to a later period than the engravings on the walls of
Combarelles, founding their opinion on the frequency of the
figures of the bison, and the rarity of those of the reindeer
and mammoth. Time will not allow me to enlarge on the
details of these remarkable rock carvings and paintings, more
than to say that MM. Capitan and Breuil have, by their ex-
plorations and published reports, greatly added to our know-
ledge of Palaeolithic civilisation.

III. Human Culturb and Civilisation in thb Pal^outhic

Period.

These illustrations, though only covering a small portion of the
available materials, are sufficient to give a general idea of the
salient features of the stage of culture to which the inhabitants
of Europe had attained towards the close of the PalfiBolithic period.
We have seen that all their works were characterised by a gradual
development from simple to more complex forms* Implements,
tools and weapons were slowly but surely being made more
efficient, thus evincing on the part of their manufacturers a pro-
gressive knowledge of mechanical principles. Hence, French
anthropologists have arranged these cave-remains in chronological
sequence, using the names of the most typical stations to define
various stages of culture, as MoustSrien^ SolutrSen^ and Mag-
daUnien. The earliest troglodytic station, according to the
classification of M. G. de MortiUet, was le Moudier^ situated on
the left bank of the Vezere (Dordogne). During its habitation by
man the climate was cold and damp, and among the contemporary

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118 Proceedings of Royal Society of Edivhurgh. [sbs.

fauna were the mammoth, woolly rhinoceros, cave-bear and musk-
ox. The special features of the industrial remains of this period
were the scarcity of the coup de p<nng^ which is so character-
istic of the older river-drift deposits, and the splitting up of
flints into smaller 'implements, such as scrapers, trimmed flakes,
etc. The next station in ascending order was the open-air encamp-
ment of Solutrd (Saone-et-Loire). The stage of civilisation here
disclosed was characterised by great perfection in the art of
manufacturing flint implements, especially spear and lance-heads,
in the form of a laurel leaf (Plate II. No. 12), and by the abundance
of horses and reindeer, which were used by the inhabitants as
food. The climate was mild and dry, the great glaciers were on
the wane, and the rhinoceros seems to have disappeared from the
scene. The third and last of the typical stations was the well-
known rock-shelter of La Madelaine (Dordogne), characterised by
the abundance of objects made of bone and horn, the development
of a remarkable artistic talent, the predominance of a northern
climate and fauna, and the extinction of the mammoth towards the
close of the period.

With regard to the ethnological characteristics of these people
little information is to be gained from their artistic productions, as
the few engravings and sculptures of the human form hitherto
discovered are too rude or fragmentary to be of much value in this
respect. That these artist-hunters should have displayed less
aptitude in the delineation of their own form and features than of
those of the animals hunted, shows how restricted was their con-
ception of human life and of the dignity of man. Evidently the
cult of humanity was still in the womb of futurity, and the
struggle of life alone was uppermost in their minds. It may be
stated, however, that, so far as this line of research leads us, these
anthropoid figures -represent both sexes as nude and covered with
hair, some of them also being, from our point of view, indecent^
On the other hand, there can be no doubt, judging from the
number of bone needles and pins collected on their inhabited sites,
that they wore clothing probably made of skins. Indeed, it
would be impossible for human beings who had their origin in a
warmer climate to endure with impunity the inclemency of the
sub-arctic climate which then obtained in Central Europe without

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1903-4.] Dr Munro on Man in tike PcUceolithic Period. 119

personal protection of some kind. Our knowledge of their
physique and general appearance is, as already mentioned,
mainly derived from a comparison of a few of their fossil
skeletons with those of modem civilised races. On this phase
of the subject we have a considerable amount of evidence to
show that since man parted company with the lower animals,
there has been a gradual expansion of the cranium, corresponding
to an enlargement of certain portions of the organ of thought.
All such materials have, however, to be carefully sifted and
scrutinised before being admitted as valid assets in a scientific*
inquiry ; and even then, this kind of evidence seldom amounts to
more than probability without being corroborated by other dis-
coveries. The subject has grown so much of late that it was
impossible in the limits at my disposal to do more than giv« a
few pertinent examples. The race represented by the skulls of
Neanderthal and Spy was long anterior to the time of the Palseo-
lithic hunters of the reindeer period, who so greatly distinguished
themselves as artists ; and as to the Java skull and femur, they
are probably the oldest osseous relics of man yet known. The
human remains found in the rock-shelter of Cro-Magnon have been
for a long time regarded as belonging to, and typical of, the latest
Palaeolithic people ; but as they were merely lying over the culture-
debris, they are regarded by some archseologists as burials of a
more recent date. The fact that the last molars were smaller
than the others gives additional support to this view. It does
not, however, appear to me that this point is of much conse-
quence, as the amount of superincumbent talus under which the
skeletons lay shows that they could not be later than the transition
period. Moreover, there are other human remains with regard to
which no such doubts have been raised, as, for example, the well-
known skulls of Chancelade and Laugerie Basse, both found in the
Dordogne district, which show equally advanced cranial characters.
The recent discovery of two skeletons, which Dr Verneau, of
Paris, describes as belonging to a new race intermediate between
the Neanderthaloid and Cro-Magnon races, marks an important
addition to fossil craniology. From the preliminary facts already-
published, and from what Dr Verneau has told me, anthropologists
may look forward with high expectation to the full report of these

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1 20 Proceedings of Royal Society of Edinburgh. [i

and other discoveries in the Mentone caves, which is now being
prepared under the direction of the Prince of Monaco. We have
already seen that in the same cave, and only 0*70 metre (27^
inches) above the site of the two skeletons just referred to, another
skeleton of the Cro-Magnon type has been discovered, thus bring-
ing two different races almost on the same chronological horizon.
But this by no means discredits Dr Yemeau's theory, as it is not
at all unlikely that, while a higher race was being developed,
some individuals of lower but vanishing races still survived in
» Europe. Indeed, the point is no longer a matter of conjecture, as
recently two skulls of a distinct negroid type have been found
among Neolithic remains in Brittany.* The skull of the 'old
man of Cro-Magnon' is large and well-proportioned, both pos-
teriorly and anteriorly, thus indicating a great stride in the
development of mental capacity, but perhaps not more than might
be ex(>ected of a people who displayed such artistic feeling
and mechanical skill as the authors of the art gallery of the rein-
deer period. But how radically their aims, hopes, aspirations, and
manner of life differed from those of their Neolithic successors we
shall immediately be in a position to realise.

It would appear from these combined sources of investigation
that the earliest Palaeolithic people of Europe entered the country
from Africa, at a time when there was easy communication between
these continents by several land bridges across the present basin of
the Mediterranean. At that time man's mental predominance over
other animals was not so conspicuous as it now is, as shown by the
fact that his mechanical ingenuity was only adequate to the pro-
duction of one typical implement — the coup de poing. Implements
of this kind are chiefly found in the stranded gravels of former
rivers, and, from their wide distribution in the Old World, they
must have been then regarded as the ne plus ultra of human
craftsmanship. Their original owners are supposed to have in-
habited the wooded banks of these rivers, wandering about in
isolated family groups till the advent of the glacial period roused
their dormant energies. It is difficult to realise how much the
severe climatal conditions which then prevailed in Europe con-
tributed to the perfection of human attributes, and consequently
* Bull, de la SocUU d* AMhropologie de Paris, series y., vol. iv. ]». 482.

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1903-4.] Dr Munro on Man in the PalceolUhic Period. 121

to the progress of civilisation. The beneficial effect of this uncon-
genial environment on these early pioneers of humanity was to
stimulate their natural capabilities of improvement — for the adage
that necessity is the mother of invention was as applicable then
as now. Entering Europe as naked, houseless nomads, living on
wild fruits and the smaller fauna of a sub-tropical climate, they
were ultimately forced by the severity of the climate to take
refuge in caves and rock-shelters and to cover their bodies with
skins. The natural food productions of a warm climate gradually
disappeared, until finally there was little left but fierce animals,
such as the mammoth, reindeer, chamois, horse, bison, etc., which
came from northern regions into Central Europe. To procure the
necessary food and clothing in these circumstances greatly taxed
the skill and resources of the inhabitants. But this difficulty they
ultimately solved by the manufacture of special weapons of the
chase, with which they successfully attacked the larger wild
animals which then occupied the country. The coup de poing,
which for a long time served all the purposes of primitive life^
gradually gave place to spear- and lance-heads fixed on long
handles, together with a great variety of minor weapons and tools
made of stone, bone, horn and wood. When the Palceolithic
people finally emerged from this singular contest with the forces
of nature, they were physically and mentally better than ever
equipped for the exigencies of life. A greater power of physical
endurance, improved reasoning faculties, an assortment of tools
adapted for all kinds of mechanical work, and some experience
of the advantage of housing and clothing, may be mentioned
among the trophies which they carried away from that long and
uphill struggle.

The civilisation thus developed represents the outcome of a
system of human economy founded on the free play of natural
laws, and little affected by the principles of religion or ethics
— subjects which were as yet in their embryonic stage. The
mysteries of the supernatural had not then been formulated
into the concrete ideas of gods or demons. The notions of
good and evil, right and wrong, were still dominated by the
cosmic law that might is rightw Neither gloomy forebodings
nor qualms of conscience had much influence on the actions

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122 • Proceedings of Boy cd Society of Edirtburgh. [skss.

of these people. Their philosophical and sentimental speculations,
if they had any, centred exclusively on th6 habits of the animals
they hunted, and on the strategic means by which they could
be waylaid and captured. During this time they made great
progress in the development of mechanical appliances, as shown
by the number of flint implements — saws, borers, scrapers, etc.
— with which they manufactured needles, pins, ornaments,
weapons and other objects, including the so - called bdtons de
commandement Upon the whole, it would appear as if their
minds were engrossed with the chase and its exciting scenes and
incidents, for their domestic economy indicated little more than
the art of broiling the flesh of the captured animals and con-
verting their skins into garments. Possibly some round pebbles
abundantly found in the debris might have been used as 'pot-
boilers,' but a few stone mortars (PI. II. No. 14), which
occasionally turned up, would seem to have been used only for
mixing colouring matter to paint their bodies, as some modem
savages do. Of agriculture, the rearing of domestic animals,
the arts of spinning and weaving, and the manufacture of pottery,
they appear to have been absolutely ignorant. But yet, in an
environment of such primitive resources and limited culture
associations, these M'ild hunters developed a genuine taste for
art, and cultivated its principles so effectually that they have
bequeathed to us an art gallery of over 400 pieces of sculpture and
engraving so true to their models that many of them bear a
favourable comparison with analogous works of the present day.
They adorned their persons with perforated teeth, shells, coloured
pebbles, and pendants of various kinds. They depicted the
animals with which they were familiar, especially those they
hunted for food, in all their various moods and attitudes, often
with startling fidelity. Harpoons, spears and daggers • of horn
and bone were skilfully engraved, and sometimes the handles
of the last were sculptured into the conventional form of one
or other of their favourite animals. (See Pis. III. to X.)

They also in some instances adorned the walls of the caverns
they frequented with incised outlines of the neigh Ix^uring fauna
(figs. 22-28), and made actual •colour paintings of them in black
and ochre, or in one of these colours (PI. XI.). The discovery

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1903-4.] Dr Munro on Man in the PcUceolithic FeiHod, 123

of so many art specimens is of considerable importance among
the more notable facts disclosed by these anthropological re-
searches, as it proves that the origin of the artistic faculty was
independent of, and prior to, the evolution of religion, ethics,
politics, commerce, and other elements of which our modem
civilisation is built up.

The other characteristic feature in the lives of these people
was, that they lived exclusively on the produce of the chase,
for, without agricultural and pastoral avocations, what else could
they do but organise daily hunting or fishing expeditions! To
capture the big game of the district was a formidable task,
requiring not only great strength and agility of person and
limb, but also strong and well-made weapons. During the
later stages of the Palaeolithic civilisation their principal prey
consisted of reindeer and horses, both of which animals then
roamed in large herds throughout Western Europe, thus rendering
themselves more liable to be ambushed, trapped or speared by
their wily enemies. It is not likely that they would take the
initiative in attacking the hyaena, lion, or cave-bear, except in
self-defence. That, however, these formidable creatures were
occasionally captured by them is suggested by the fact tihat their
canine teeth were highly prized as personal ornaments, or as a
memento of their prowess in the chase. The weapons used by
these hunters were harpoons, generally made of reindeer-horn,
spear- and lance-heads of flint, and short daggers of bone or
horn. Before these weapons were invented it is difficult to
imagine that any member of the genus Homo would have the
courage to attack such a formidable animal as the mammoth
armed only with a coup de jpoing, but yet there are facts which
suggest that such was the case.

When the physical conditions which called these accomplish-
ments into existence passed away, and the peculiar fauna of the
glacial period disappeared from the lowlands of Central Europe
— some by extinction, and others by emigration to more northern
regions or to the elevated mountains in the neighbourhood — we
find the inhabitants of these old hunting grounds in possession
of new and altogether different sources of food. Finding the
former supplies becoming so limited and precarious that it was

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124 Proceedings of Royal Society of Edinburgh. [sess.

no longer possible to live a roaming life, now gathering fruits
and seeds, and now hunting wild animals, they fell somehow
into the way of cultivating special plants and cereals, and rearing
certain animals in a state of domestication. Whether this new
departure was a product of the intelligence of the descendants
of the PalsBolithic people of Europe, or derived from new
immigrants into the country, is a debatable question. At any
rate, the expedient was eminently successful. It was in reality
the starting-point of Neolithic civilisation, and henceforth there
was a rapid increase in the population. They cultivated a variety
of fruits, wheat, barley and other cereals ; they reared oxen, sheep,
goats, pigs, horses and dogs ; they became skilled in the ceramic
art, and in the manufacture of cloth by spinning and weaving wool
and fibrous textures ; they ground stone implements so as to give
them a sharp cutting edge; in hunting the forest fauna of the
period they used, in addition to spears, lances and daggers, the
bow and arrow; they built houses, both for the living and the
dead — thus showing that religiosity had become an active and
governing principle among them. But of the artistic taste and
skill of their predecessors they had scarcely a vestige, and what-
ever they did by way of ornament consisted mainly of a few
scratches, arranged in some simple geometrical pattern. The
fundamental principles of the two civilisations are really so
divergent that the Neolithic can hardly be regarded as a local
development of the latest phase of that of the Palaeolithic period
in Europe. The probability is that, while the isolated colonies
of reindeer hunters were still in existence, people of the same
stock were elsewhere passing through the evolutionary stages
which connected the two civilisations together.

The far-reaching consequence of securing food supplies by means
of agriculture and the domestication of animals led to more
sedentary and social habits. The existence of large communities
concurrent with the development of various trades and professions
was but a matter of time, the outcome of which is now a vast
system of international commerce. Already the greater portion of
the earth capable of being cultivated is converted into gardens and
fields, whose choice productions are readily conveyed to all the large
cities of the globe. Flesh diet is abundant, but it is no longer

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1903-4.] Dr Munro on Man in the Pateeolithic Period, 125

necessary to hunt the animals in primeval forests. Skin-coats,
dug-outs and stone weapons are now lineally represented by
woven fabrics, Atlantic liners and Long Toms.

Were it possible for one of our Palaeolithic ancestors to sit in
judgment on the comparative merits of the two civilisations, I
fancy his verdict would be something like the following : " You
have utilised the forces of nature to a marvellous extent, and
thereby greatly increased the means of subsistence to your fellow-
creatures j but, at the same time, you have facilitated the physical
degeneracy of your race by multiplying the sources of human
disease and misery. The invention of money has facilitated the
accumulation and transmission of riches to a few; but it has
impoverished the many, and supplied incentives to fraud, theft, and
aU manner of crime. Patriarchal establishments have given place
to social organisations, governed by laws founded on moral senti-
ments and ethics ; but their by-products are extreme luxury and
extreme poverty. Hence, to support the weak and the unfortunate
is no longer a matter of charity, but a legal and moral obligation.
Notwithstanding the size of your asylums, hospitals and alms-
houses, they are always full and always on the increase. Your
legislators are selected by the voice of the majority : what if that
majority be steeped in superstition, prejudice and ignorance?
You have formulated various systems of religion, but whether
founded on the principles of fetichism, polytheism or monotheism,
they are still more or less permeated with contradictory or contro-
verted creeds and dogmas. Natural sport, as practised with
weapons of modem precision, can only be characterised as legalised
killing of helpless creatures. To shoot pigeons suddenly liberated
from a box at a measured distance, or overfed pheasants, even
after they have managed to take wing, or semi-domesticated deer,
especially when driven to the muzzle of a rifle — all, of course,
within sight of a luncheon basket — is a poor substitute for the
excitement and field incidents of the chase in Palaeolithic times.
With no better weapons than a spear, or lance tipped with a
pointed flint, and a small dagger of bone or horn, we had, not
infrequently, to encounter in mortal combat the mammoth,
rhinoceros, cave-bear, or some other fierce and hungry animal, which,
like ourselves, was prowling in quest of a morning meal. Such

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126 Proceedings of Royal Society of Edinburgh, [;

scenes had many of the elements of true sport, and being essential
to our existence, were of daily occurrence. Moreover, from the
standpoint of modem ethics, our method put the combatants
on something like a footing of equality, or at leadt gave our prey a
fair chance of escape. "We cultivated physical and manly qualities
by the natural exercise of the senses, and personal prowess was the
distinguishing prerogative of our heroes. Thus we acquired the
experience, skill, strength, agility and courage of practised athletes
— qualities which left no room for cowardice. With us * brain
power ' passed almost directly from the generator to the muscles
of the administrator; with you it has to pass through a complicated
system of accumulators and distributors, liable to various degrees of
leakage, and it is this leakage which often sucks dry the life-blood
of your civilisation. Finally, the permanence of your civilisation
remains to be tested by the touchstone of time. For civilisations,
like the genera and species of the organic world, have their life-
histories determined by laws as fixed and definite as those that
govern the resultant of the parallelogram of forces. To cosmic
evolution, under which our race and civilisation to a large extent
flourished, you have superadded altruism, which means the sur-
vival of the weak as well as of the strong. But altruism will
continue to be a living force among civilised communities only so
long as present and prospective food suppUes hold out. For,
after all, the essential problem of your social existence is to
procure food for an ever-increasing population. Whenever these
necessaries of life become inadequate to meet the d'emands of the
inhabitants of this globe, then your boasted civilisation comes to
the end of its tether, and the only solution of the crisis will be to
reduce your numbers by a recurrence — sauve qui pent — to the
cosmic law of * the survival of the fittest."

DESCRIPTION OF PLATES.

I. A flint implement in the British Museum found, with a skeleton of an
elephant, near Gray's Inn Lane, London, about the close of the seventeenth
century. Reproduced from plate i. of Guide to the Antiquities of the Stone
Age in the British Museum.

II. Specimens of flint tools illustrating the progressive skill of the Paleo-
lithic cavemen of France, chiefly from the Lartet and Christy Collection, now

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190 :-4.] Dr Munro on Man in the Palctolithic Period, 127

in London and Paris. Nos. 1-7, 9-11, 18 and 19 represent saws, borers,
sorapers, etc. from the later stations. Noe. 12 and 16 are illustrations of the
laurel-leaf-shaped lance-heads commonly described as belonging to the Solu-
tr4en period. The former was found at Laugerie Basse (Col. Mass^nat-Girod),
and the latter (made of agate) in the Grotte de I'^glise (Dordogne). Nos. 8,
15y 17 and 21 are specimens of the earlier implements from Le Moustier, and
are all trimmed flakes, with the exception of 17, which is a small c(mp de
poing. No. 13 represents a core from Les Eyzies, showing on the left a small
portion of the original surface of the flint, and No. 20 a well-made flake from
La Madelaine. A small mortar made out of a waterwom pebble from Les
Eyzies is shown under figure 14 ; others like it have been recorded from
La Madelaine, Laugerie Basse, Bruniquel, and probably elsewhere.

III. Weapons and ornaments made of bone, teeth, deer-horn, ivory and
shells. Nos. 1-14, 16, 17-19 (ivory), 20, 25 (ox), 26 (fox), 27 and 28 are
from La Madelaine (Col. L. and C). Nos. 6-14 are from Laugerie Basse
(CoL Mass^nat-Girod). Nos. 24 and 29, representing a supposed whistle and
a sculptured dagger, are from Laugerie Basse (Col. L. and C). No. 16 is
a thin plaque carved of bone, probably an ornamental pendant, found at
Bruniquel (British Museimi). Nos. 21-23 are from Kent's Cavern. The
precise use of the pointed objects figured under Nos. 12-14, 28 and 30 is
not known, but it is probable that they were the tips of small lances pro-
pelled by means of such an implement as is figured under No. 8, Plate IV.
The small harpoon (No. 27) might have been used as an arrow-point, but
we have no evidence that bows and arrows were then in use.

IV. On this Plate there is a collection of objects from various stations
illustrating the art of the Palaeolithic people. No. 1 shows a portion of
reindeer-horn with a rude representation of a prone man, ap}>arently in the
act of throwing a spear at a male auroch. The hands are imperfectly repre-
sented, the body is covered with hair, and a cord, possibly attached to the
head of a harpoon, falls behind the legs. This specimen was found at
Laugerie Basse (Col. Mass^nat-Girod). Nos. 2 and 14 represent portions of
darts with badly-executed human hands, showing only four fingers. Nos.
3, 4 and 6 are from La Madelaine (CoL L. and C). One (8) represents a
piece of stag's horn (hdUm de ccmmandeineTU), having a stag iivith complex
antlers incised on it. Another (4) is a plate of the canon bone of a reindeer
with incised figures of bovine animals. The third represents a truncated dart
ornamented with flowers, and what looks like the outstretched skin of a
fox. No. 6 is from Les Eyzies, and shows a ruminant having a spear
entering its breast {ibid. ). A portion of a bevelled dart-head from Laugerie
Bfisse, with a sequence of half-fledged birds, is shown by No. 7 {ihid. ). No. 8
represents a dart-propeller from Laugerie Basse, ornamented with a horse's
head and an elongated forepart of a deer (iMd,). Nos. 9, 10 and 16 are
also from Laugerie Basse (Col. Massenat-Girod), and represent the well-
extended antlers of a reindeer (9), an otter eating a salmon (10), and a
hare (16), sculptured in ivory. No. 11, unmistakably sho\iing the hind
portion of a pig, is from the Kesslerloch, Switzerland (after Conrad Merk).
On the canine of a bear (No. 12) from Duruthy Cave a seal is engraved
{Beliquice Acquitanicce, p. 223). The palm of the brow antler of a reindeer
is incised with the figure of some kind of horned animal (No. 13), probably
intended for an ibex.

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128 Proceedings of Royal Society of Edinburgh. [sbss.

V. This Plate shows a famous relic in the form of a piece of ivory ^m
the outside layer of the tusk, having incised on it the outline of a hairy
elephant (Col. L. and C). The lofty skull and hollow forehead of the animal
here represented are characteristic of the Siberian manmioth, as shown by its
skeleton (fig. 29). On comparing it also with the figure of the mammofch
incised on the wall of Combarelles (Ug. 28), one cannot fail to be struck
with the striking resemblance between them.

VI. Portion of a reindeer-horn (bdUm de eommandemtiU), having salmon
engraved on one side and eels on the other.

VII. Two bdUms de commandemerU from La Madelaine, one showing a
human figure with an upraised club, as if going to strike a horse in front
of him, while a serpent (?) seems to be in the act of biting his heel; the
other shows four large-headed ponies in sequence (0>1. L. and C.)*

VIII. Figures of a reindeer, horse, and three ornaments from the Eesaler-
loch C)ave, near Schaffhausen. The two former are among the chef-d^CBUvres of
Paleolithic art. Of the hanging ornaments two are made of shale. All the
figures are after Conrad Merk.

IX. Two carved handles of daggers like the complete specimen from
Laugerie Basse figured on Plate III. No. 29. The reindeer is carved in
ivory and the mammoth in reindeer-horn. These interesting relics, as well
as a third handle of the same kind, are from the rock-shelter of Bruniquel,
and are now among the antiquarian treasures of the British Museum. The
highly conventional manner in which the artist has adapted horns, tusks and
trunk to serve his purpose, shows power of imagination and a fSncility of
execution which even now could only be acquired by long experience.
Figure 3 represents some fantastic animal with large mouth and no teeth.
It comes from Laugerie Basse (Col. Mass^nat-Girod).

X. One of the sculptured horse-heads here represented is most remarkable,
as the original seems to have been partially skinned. M. Piette, writing in
1889 {Congris IntemaHonal^ «te., Paris, p. 159), makes the following state-
ment : — ** L'homme a toigours en Tamour du beau. . . . Pour se perfectionner
dans Part de repr^senter le vivant, les artistes du Mas d'Azil sculptaient
Tecorche et le squelette." Also M. Cartailhac {La France Frehidorique, p.
70) thus notices the above piece of sculpture : — " Le relief de la t^te en partie
decham^e est tout k fait ^tonnant line t^te isoUe, de la meme grotte, est
^galment figur^ sans le peau. De tels ouvrages donnent k Part de l*%e du
renne un aspect inattendu. Les d^ouvertes rentes nous ont appris que cet
art connut la fantaisie."

XI. Bison and two reindeer painted in ochre on the walls of the Orotte de
FoTU-de-OaumCf reduced from illustrations by MM. Capitan and Breuil (Bevue
de r£cole d^Anthrapologie, July 1902, pi. ii.).

{Issued separately February 13, 1904.)

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Proc, Rify, Sitcij. of Edin.]

[Vol. XXV.

Plate I.

Flint iiiipleiiient — 'coup «lc poing' -from riveiMliifr gravels ({).
Dk Munro.

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: Proc. Roy, Socy. of Edin.] [Vol. XXV.

:• Pl.ATK II.

Objects illuHtr.itiiig flint indiistry anion*,' tin- Civcnicn of Fmnco (\).
Vr Munko,

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Proc, Boy. Soey. of Edin.] [Vol. XXV.

Plate III.

Weapons and ornaments made of bom*, teeth, deer-hoiu, ivoiy and sliells {h).
Vr Munko.

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/Voc Roy, Socy. of Edin,] [Vol. XXV.

0
^

I ^

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Ptoc. Roy. Socy. of Edin.]

[Vol. XXV.

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Proc. Roy. Sory. of Edin,]

[Vol. XXV.

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Proc, Ritij. Sijry. of Edin.]

[Vo\. XXV.

5 1

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Froc. Roy. Socy. of Ed in.'] [Vol. XXV.

Plate VIII.

Fio. 1. — Reindeer on a portion of reindeer-horn (|)

Fig. 2. -Drawing of h horse on portion of reindm'r-horn (|).

Fic;s. 3, 4, 5. — A perforated shell and hanging ornaments made of coal {\).

Engraved figures of animals and ornaments from the K^Hslerloch Cave, ncNir
SchafThausen. (After Conrad Merk. )

Pr Munko,

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1*ror. Jioif. Socy. of Etlin.]

[Vol. XXV.

Platk IX.

Fig. 1. — Handle of a dagj<er sculptured into the form of a reiudeer.
Rock-shelter of Bruuiquel (a).

Fig. 2. — Maiumoth sculptured in reindeer-horn. Rock-shelter of Bruuiquel ('^).

Fkj. 3. — Uuknowu animal sculptured in reindeer-horn. Laugeiic Basse {{).
Animals sculptured in ivory and horn.

Dr MrsRO.

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Vn^. Hoy. Svcy. of Klin.] t^'«l. >^'XV.

Plate X

Portion of reindeor-liorn from Mas d'Azil, scnl|(tiirf(i into two liorse-heads
(Col. Piette). After E. Cartailhac — L(( Fram-e Preldstoriqur.

Dli MUNKO.

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Proc. Roy. Socy. of Eilin.] [Vol. XXV.

Plate XI.

Fio. 1. — BUon i>aiijted in oclire.

Fig. 2. — Reindeer paitly painted and partly incised.

Specimens of painted animals from the Cave of Foiit-de Gaume, after MM.
Capitan and Breuil.

Vll Ml7NR0.

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4. Additions to a Paper after it has been finally handed in for
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5. Brief Abstracts op Transactions Papers will be published in
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The author receives fifty of these reprints free, and may have any
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To prevent disappointment, especially if the paper contains plates,
the author should, immediately after, receiving his first proof, notify
io the printer the number of additional copies required.

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Tiinsactions.

MODEL INDEX.

Sehafer, E. A. — On the Existence within the Liver Cells of Channels which can
be directly injected from the Blood- vessels. Proc. Roy. Soc. Edin., vol ,
1902, pp.
Cdb, Liver, — Intra-oellnlar Canaliculi in.

R A. Schafer. Proc Roy. Soc. Edin., vol. , 1902, pp.
liver, — ^Injection within Cells of.

E. A. Schafer. Proc. Roy. Soc. Edin., vol. 1902, pp.

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IV CONTENTS.

PAGE

Physico-Chemical Investigations in the Amide Group. By
Charles K^Fawsitt, Ph.D., B.Sc. (Ediu. and Lond.).
Communicated by Professor Crum Brown, . 51

(Issued separately February 6, 1904.)

The Theory of General Determinants in the Hisyrical
Order of Development up to 1846. By Thomas
MuiR, LLD., ...... 61

(Issued separately Fehnuiry \2i \^0i,) ^ ^

Man as Artist and Sportsman in the Palaeolithic Period.
By Robert Monro, M. A., M.D., LL.D. (With Eleven
Plates), ...... 92

(Issued separately Febrtuiry 18, 1904.)

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PROCEEDINGS

OF THE

ROYAL SOCIETY OF EDINBURGH.

SESSION 1903-4.

No. n.] VOL. X XV. [PP 129-192.

CONTENTS.

PAQB

The Theory of Continuants in the Historical Order of its

Development up to 1870. By Thomas Muir, LLD., 129
{Issued separately February 26, 1904.)

On the Origin of the Epiphysis Cerehri as a Bilateral
Structure in the Chick. By John Cameron, M.B.
(Ediu.), M.K.C.S. (Eng.), Carnegie Fellow, Demon-
strator of Anatomy, United College, University of St
Andrews. Communicated by Dr W. G. Aitchison
Robertson, . . . . . .160

{Issued separately March 17, 1904.)

Theorem regarding the Orthogonal Transformation of a

Quadric. By Thomas MriR, LL.D., . . .168

{Issued separately March 17, 1904.)

[CarUinued on page iv of Cover,

^ EDINBURGH:

Published by ROBERT GRANT & SON, 107 Princes Sireet, and
WILLIAMS k NORGATE, 14 Henrietta Street, Covent Garden, London.

Price Three Shillings.

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REGULATIONS REGARDING THE PUBLICATION OF
PAPERS IN THE PROCEEDINGS AND TRANS-
ACTIONS OF THE SOCIETY.

The Council beg to direct the attention of authors of communications to
the Society to the following Regulations, which have been drawn up in
order to accelerate the publication of the Proceedings and Transactions,
and to utilise as widely and as fairly as possible the funds which the
Society devotes to the publication of Scientific and Literary Researches.

1. Manuscript of Papers. — As soon as any paper has been passed
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made ready for publication by the author, it is sent to the printer,
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The * copy ' should be written on large sheets of paper, on one side
only, and the pages should be clearly numbered. The MS. must be
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2. Illustrations. — All illustrations must be drawn in a form im-
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If an author finds it inconvenient to furnish such drawings, the Society
will have the figures re-drawn at his expense ; but this will cause delay.

3. Proofs. — In general, a first proof and a revise of each paper will
l)e sent to the author, whose address should be indicated on the MS.
If further proofs are required, owing to corrections or alterations for
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corrections will be charged against the author.

All proofs must, if possible, be returned within one week, addressed to
The Secret art/ y Royal Society y MoumJ, Edinburgh, and not to the printer.

[Continued on jxige iii ofCorer,

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1908-4.] Dr Muir on the Theory of CofUinuants. 129

The Theory of Ck>ntiniiaiit0 in the Historical Order of its
Development up to 1870. By Thomas Muir, LL.D.

(MS. recdred October S, 1908. Read Norember 2, 1908.)

The more or less disguised use of continued fractions has been
traced back to the publication of Bombelli's Algebra in 1572,
eighty-four years, that is to say, before the pubUcation of Wallis'
ArWimetiea Infinitorum^ in which Brouncker's discovery was
announced and the fractions explicitly expressed.* The study of
the numerators and denominators of the convergents viewed as
functions of the partial denominators was first seriously under-
taken by Euler in his Specimen Algorithmi Singtdarie of the year
1764, in which denoting by

the convergents to

a + -- ,

c +

he established a long series of identities, such as

(a, 6, c, d, . . . )-a(&, c,d,... ) + (c, rf, . . .)
(a, &, c, . . . Z) = (Z, ...,(;, ^ a),
(a,6)(6,c)-(5)(a,6,c) = l,
(a, b, e,){d, e,f)-{a, b, c, d, «,/)= -(a, b)(e,f).

The study was pursued by Hindenburg and his followers during
the last twenty years of the eighteenth century, but not with any
great profit; and, although in the first half of the nineteenth
century considerable attention was given to the theory of con-
tinued fractions as a whole, little advance was made in elucidating

* For the early history see Favaro's Notueie stcriche auUe frazioni continue
cUU aeeolo decimoUreo al dedmoaeUimo published in vol. viL of Boncom-
pagni's Bollettino : and as regards Bombelli see a paper by O. Wertheim in
the AbJutndl. zur Oeseh, d. Math,, viii. pp. 147-160.

PBGC. ROY. SOC. BDIN. — VOL. XXV. 9

ljr_

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130 Proceedings of Royal Society of Edinbwrgh, [i

the properties of the functions referred to.* Their connection
with determinants, after the awakening of interest in the latter
about 1841, was sure sooner or later to be detected : there is no
evidence, however, of the discovery having been made before the
year 1853.

Sylvkstbb, J. J. (1853, May 13).

[On a remarkable modification of Sturm's theorem. PhUos,
Mag. (4), v. pp. 446-457.]

The mention of Sturm's theorem in the title of a paper renders
not improbable the occurrence therein of matter connected with
continued fractions. Especially likely is this in the case of a
writer like Sylvester when in a characteristic mood ; and, assuredly,
the present communication is in structure, style, and originality
redolent of its author. It must have been written in the white
heat of discovery. The main part of it consists of six pages:
this is followed by a " Remark '' a page and a quarter long ; then
comes a " Postscript '' of three and a half pages ; and finally a
small-page footnote as long as the '' Remark."

It is the postscript which particularly concerns us. It begins
thus : —

"Suppose that we have any series of terms, i/^, «2> ^s»

. . . , t/„, where

<^i = Ai, u^ = A^A^-\, W3 = AiA2A3-Ai-A3, . . .

and in general

«< = A<tt<_i-w<_,,

then 1*1 , «*2 , Wg , . . . , w„ will be the successive principal
coaxal determinants of a symmetrical matrix. Thus suppose
n = 5 ; if we write down the matrix

* The state of the theory in 1833 can best be gathered from Stern's
monograph published in vol. x. of CrelWs Journal,

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1903-4.] Dr Muir on the Theory of Continuants^

131

(the mode of fonnation of which is self-apparent), these
successive coaxal determinants will he

1, Ai,

A, 1

A, 1 0

Ai 1 0 0

1 A,

1 Aj 1

1 A, 1 0

0 1 A,

0 1 A, 1

0 0 1 A,

etc.,

t.e.

1 , Aj , AjA2 - 1 , AjAjAj — Aj — A3 ,

•^1 AjAjA^Aj — A^A2Ag — AjA^Ag — AgA^A^ — A^A2A3

+ A5 + A8 + A1.

It is proper to introduce the unit hecause it is, in fact, the
value of a determinant of zero places, as I have observed
elsewhere."

After using this as an aid to prove his proposition regarding
Sturm's theorem, he returns to his new determinant in the
following words: —

"I may conclude with noticing that the determinative
[determinantall] form of exhibiting the successive con-
vergents to an improper continued fraction affords an
instantaneous demonstration of the equation which connects
any two consecutive such convergents as

2^' and 2-'

D...

viz. N,.D,.,-N<_,D<=1.

For if we construct the matrix which for greater simplicity
I limit to five lines and columns,

and represent umbrally as

*2 ""8 **4 '*6

^1 h h h h>

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1 32 Proceedings of R&ycd Society of Edivbu/rgh. [smb.

and if, by way of example, we take the fourth and fifth con-
vergents, these will be in the umbral notation represented by

t* ^ 5;» ?* ^

a, Oj

\ *2

<h <»4

Oj a^ ci^ a^ (tr,

h ^2 ^8 ^4 ''&

respectively.

Hence

N,D,-N,Dj

^4 h

6, ftj 6^

h »8 ^

?! - ?« ^« ^* X ?2 ?»

6i 6j 6j 64 b^ 6g

04 "e "1 .

6, ^2 6j b^ b^ 62 ^3

»4 ftl

''2 *$ *4

ft'

ftj ft, ft^

ft*

1 B
0 1

1 0

C 1 ""

1 0 0
B 1 0

0
0

0 0

1 D

1 C 1

0 0

0 1

0 1 D

1 X

1 = 1,

as was to be

proved. And the demonstration is

evidently

general in its nature."

regard

to this there has to be noted, first, the use of

«s

a$ «4

6s h

when it would have been equally effectiye to use

2 3 4

2 3 4;
and, second, the use of a theorem for expressing the product of a
five-line determinant and one of its secondary minors as an
aggregate of products of pairs of four-line determinants.

Following on this comes the assertion that

**"We may treat a proper continued fraction [i.e. vnth positive
imit numerators] in precisely the same manner, substituting
throughout ^ - 1 in place of 1 in the generating matrix.

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1908-4.] Dr Muir on the Theory of Continuants. 133

and we shall thus, by the same process as has been applied
to improper continued fractions, obtain

N^,D, - NA+i = ( ^^^7x ( ^3T)*
= (-l)V'

This would seem to imply that as yet Sylvester had not observed
that an alternative mode of representation was obtainable by
merely changing the sign of the units on one side of the diagonal.
The footnote contains two additional observations, the first
being to the effect that the new mode of representation

" gives an immediate and visible proof of the simple and elegant
rule for forming any such numerators or denominators by
means of the principal terms [term f| in each ; the rule, I mean,
according to which the i^^ denominator may be formed from

(?i> ?2» • • • > S'* being the successive quotients) and the i^
numerator from

^8^4 • • • ?<

by leaving out from the above products respectively any
pair or any number of pairs of consecutive quotients as ^p^p+i.
For instance, from q^q^^q^q^ by leaving out q^q^ , q^q^ , q^q^
and q^f^ we obtain

and by leaving out q^q^-q^^^ MsMs » ^^z^db ^« obtain

^6 + S's + 3i ;
so that the total denominator becomes

and in like manner the numerator of the same convergent is
r, ^ 1 ^ 1 ^ 1 ^ 1 )

M«?4?6 ^ 1 + + + + f

i*e.

qa^i^i + Mft + ^^5 + Ms + 1 •"

The " rule " here spoken of is that enunciated for the more
general case of

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134 Proceedings of Boydl Society of £dinbu/rgh. [i

in Stem's Theorie der Kettenbruche, the fourth section of which is
given up to the consideration of such rules {OreUe^s Joum,, z« pp.
4-7).

The other observation is to the effect that

" every progression of terms constructed in conformity with

the equation

may be represented as an ascending series of principal coaxal
determinants to a common matrix. Thus if each term in
such progression is to be made a linear function of the three
preceding terms, it will be representable by means of the
matrix

B'"

C""

B""

A""

indefinitely continued, which gives the terms

1, A, AA'-B, AA'A"-BA"-AB" + C" *•

This exhausts the paper so far as determinants are concerned:
the results announced in it, one can readily own, were such as
fairly to entitle the enthusiastic author to express his belief that
*' the introduction of the method of determinants into the algorithm
of continued fractions cannot fail to have an important bearing
upon the future treatment and development of the theory of
numbers."

Spottiswoodb, W. (1853, August).*

[Elementary theorems relating to determinants. Second edition,
rewritten and much enlarged by the author. Orett^s
Joum., U. (1856) pp. 209-271, 328-381.]

Save the utilisation of the fact that the denominator of any
convergent of the continued fraction

* This is the author's date at the end of the paper (p. 381). The first two
parts of the volume, however, are dated 1855, and the remaining two 1856.

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1903-4.] Dr Muir on the Theory of Continuants.

135

is the differential-quotient of the numerator, Spottiswoode did
nothing but report the fundamental result reached by Sylvester.
The full passage (p. 374) is as follows : —
•* The improper continued fraction

where

A-

i-i

-k^^

7 -

A 1

...00

1 B

...00

0 1

...00

0 0

...Ml

0 0

...IN

in which any number of rows may be taken at pleasure, and
the formula will give the corresponding convergent fraction.
The same holds good for the continued fraction

^+¥ + -

if we write

1
B

Sylvbstbr, J. J. (1853, Sept.).

[On a fundamental rule in the algorithm of continued fractions.
Philos. Mag, (4), vi. pp. 297-299.]

Without any reference to his previous paper on the subject
Sylvester here announces that if

(Oj, Og, . . . , a<)
be the denominator of the f^ convergent to

1 1 ,

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136 Proceedings of Roycd Society of Edinbv/rgh. [i

then

+ («1 > • • • > «m-l)(«m+2 » • • -I «fii+*)»

— a possibly new result which he considers *'the fundamental
theorem in the theory of continued fractions." This, he says, is
an immediate consequence of the fact that (o^ , . . . , 0^+^) can

be expressed as a determinant, all that is farther necessary being
the application of the " well-known simple rule for the decomposi-
tion of determinants." Thus, e.g., the determinant

a 1

-1 b 1

-I c 1

-1 d 1

-1 e

-1

is obviously decomposable into

a 1
-1 6 1
-1 c

X d 1 +
-1 e 1
-1 /

a 1 X el
-1 b -1 /,

or into

alxel +ax<21

-1 h

-1 d I

-1 e 1

or into

-1 e 1
-1 /

-1 /.

a X

6 1 +

c 1

-1 e 1

-1 d 1

-1 e 1

-lei

-1 /.

-1 /

Following this is what is called " Corollary 1. " viz.,

- ( - r(am-M«m+i-i ... to t - 1 terms),

its connection with the expression for the difference of two con-
vergents being illustrated by the instances t ^^ 1, 2, 3, 4, . . .

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1908-4.] Dr Muir on the Theory of Continuants.
The next "coroUaiy," viz.,

137

*P+/J

')(«!,.

*P > ^P+1 »

(«!>•.•> «p> «p+i I • • • , «p+a)(«i » • . • > «pi «p+i >
' ( - )^{(«P+l> • • • »«P+/)(«|M-1> • • • .«»»+») - («p+i. • •

••»«P+*) '
• • » «P+0
>^p-w)(^P+i> '

>«P+*)}

is clearly incorrect, • it being impossible for the value of the left-
hand side to be independent of the elements Oj, Og, . . . , Op.
Further, as the author gives no accompanying word of comment,
the difficulty of suggesting the true theorem is increased. A
" sulHsoroUary " is appended dealing with the case where all the
0*8 are equal, and leading up, nob without some misprints or
inaccuracies, to a theorem of Euler^s quoted from the NouveUes
Annates de Math., v. (Sept. 1851) pp. 357-358, to the effect that
if T^+i—aT«-&Tn_i be the generating equation of a recurrent
series, then

^w+i

aT.T^^.-i-CT,

is a constant with respect to n. Of course the more natural form
of this expression is

Tn-n ~ ^n^n-k^^

the numerator of which being

'•"+1

*•*»+!

is successively transformable by means of the recursion-formula
into

■•»H-1

■■•-l

J2I T..1 T„

6«

Tn-2 T„_j
Tn-j T„«2

SO that the constant in question is

T T I
-T T

This, however, Sylvester does not show.*

* An interesting extension of this is given by Brioschi in the NouveUes
AnmaUi d$ Matk., lav. (Jan. 1854) p. 20. i

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138 Proceedings of Royal Society of Edvrimrgh, [i

Finally, and to more purpose, it is noted that if we pass from
(<ii > % 9 • • • 9 a<) to the readily-suggested extension

the corresponding fundamental theorem is

^ »»i »»H^ / \ Wj n<_, / \ n<+i fii+j /

/ ^1 ^<-s \ / ^<+i h-^i \

- Z<ni wij, m,, . . . , ?»<^x j( ^i+ii ^<+si • • • > »»<-h/+i )

Sylvbstbb, J. J. (1853, Oct, Nov.).

[On a theory of the syzygetic relations of two rational integral
functions, comprising an application to the theory of
Sturm's functions, and that of the greatest algebraical
common measure. PhU. Trans. Boy, Soc London^ cxliii.
pp. 407-548.]

Although this lengthy memoir in its original form bears date
" 16th June 1853," it is the equally lengthy "supplements" added
later while passing through the press that claim attention in the
present connection. In the first of these (§ L, p. 474) the de-
nominator of the fraction

1 1

-1

is denoted by [^^ , gg > • • • > Qn\i and termed a " cumulant," and
throughout the later portion of the paper this name constantly
recurs. It is not^ however, until we come to the second

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1903-4.] Dr Muir on the Theory of CorUintiants, 139

" sapplement " that anything apparently new in substance is met

with. There in § a (p. 497) the following lemma occurs : —

" The roots of the cumulant bi , ft , . • • > yj in which
each element is a linear function of os, and wherein the
coefficient of x for each element has the like sign, are all
real: and between every two of such roots is contained a
root of the cumulant bi i ft , . . . , ^i-i] and ex eonverso a
root of the cumulant [ft , ft » . . . , ^<] : and (as an evident
corollary) for all values of f and f intermediate between 1
and 1 the greatest root of [^^ , ft , . . . , <^J will be greater,
and the least root of the same will be less than the greatest
and least roots respectively of [ft , ft+j , . . . , ft.i , ft*]."

Even this, however, may be placed under the well-known theorem

regarding the roots of the equation

= 0

which had been enunciated by Cauchy in 1829.*

The next noteworthy result occupies § i. (p. 602). As a
preparation for it the theorem

[Oj, ag, . . . , a«, 6i, 2>2, . . . , *n] = [a^ ag, . . . , aj[6i, ij, . . . , K]

- [a^, Oj, . . . , aw_i][ft2, 65, ... , 6J

may be recalled, the group of elements on the left being now
viewed as consisting of two sub-groups. This theorem Sylvester
writes in the form

[o,oj = [o,][oj - [0',]['0J

and he succeeds in including in it a general theorem, not explicitly
formulated, in which the number of groups is », the next two
cases being

[0,0,0,] = [OilMM

- [o'i]['oJ["3] - ["J[o'J['Os] + [o'lro'JC'Osl

* V. The theory of orthogonants ... in Proc, Roy, Soc Sdinhu/rgh,
xziT. p. 261.

a,i-ar

«1S

«18 • •

«12

ajs-x

«2S • •

«1S

<hs

Ojs-a; . .

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X4(> Proceedings of Royal Society of Edvnbmgh. [«

and

[n,o,n,oj-[oj[o,][oj[oj

- [0'ir«2][0s][0j - [Oi][0'J['OJ[OJ - [0j[0J[0'a]['O4]

+ [o'i]['o' J['«,][«J + [«'i]['« Jo'slL'oJ + [Oi][n'2]['o'8]['o J
-[o'J['o'J'n'J['oJ-

The general theorem is described as giving an expression for
[OjOj . . . O J in terms of

ro*-i],['oo

that is to say, in terms of all the unaltered O's, all the curtailed
O's except the last, all the beheaded O's except the first, and all
the "doubly-apocopated" O's except the first and the last; and it
is pointed out that the number of products (or terms) in the ex-
pansion is 2*"^ " separable into i alternately positive and negative
groups containing respectively 1, (i-1), K*-^)(*~2)» • • • >
f - 1, 1 products." Further, it is noted that " in every one of the
above groups forming a product the accents enter in pairs and
between* contiguous factors, it being a condition that if any O
have an accent on the right the next O must have one on the left^
and if it have one on the left the preceding O must have an
accent on the right, and the number of pairs of accents goes on
increasing in each group from 0 to t - 1."

In a footnote the case where each O has only one element, and
where, therefore, each singly-accented O becomes 1, and each
doubly-acceuted O vanishes, is stated to be identical with the
"rule"

[oj , Oj, . . . , a<] = OyO^ . . . a< - ^—— . a^a^ . . . Oi

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1908-4.] Dr Muir on the jTheory of Continuants, 14 1

Sylve^tbb, J. J. (1854, August).

[Th^or^e sur les determinants de M. Sylvester.
Annalea de Math,^ xiii. p. 305.]

This communication in its entirety is as follows : —
" Soient les determinants

Nouv,

X. 1

1 X.

A.,

0
0

2
0

. <

. •

, ,

la loi de formation est ^vidente ; effectuant, on trouve

X, X2-1, X(X«-2«), (X«-12)(X2-32)^ X(X«-22)(X2-42),

(\2 - 12)(X« - 3«)(X2 - 52) , X(X2 - 22)(X« - 42)(\2 - 62) ,
et ainsi de suite."
That Sylvester was the author of the implied theorem may be
considered proved by an entry in the index of the volume (v. p.
478), and by a statement of Cayley's in the Quarterly Journal oj
MatheniaticSy ii. p. 163. Probably the title of the communication
v^as prefixed by the editors, who, knowing of Sylvester's papers in
the Philosophical Magazine^ felt themselves justified in applying
the name ** Sylvester's determinanta"

ScHLAPLi, L. (Nov. 1855).

[Reduction d'une int^grale multiple qui comprend Tare de
cercle et Taire du triangle sph^rique comme cas particuliers.
Joum, de Liouville, xx. pp. 359-394.]

Here there appears the equation

A(/i,

A,ri)

1 -

cos^a

1 -

COS^jg

1 -

__C082f
1 - C082i7

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142 Proceedings of Bayal Society of Edinfywrgh. [i

where, in view of the contents of a subsequent paper (see under
year 1858), it would seem that A(a,jS, • • • i ty'^) was used for

1 cosa

- cosa 1 cos)3
— cosjS 1

- cosf 1 cosi;
— cosi; 1

No properties, however, of this determinant are given.

Ramus (1856, March).

[Determinantemes Anvendelse til at bestemme hoven for de
convergerende Broker. Oversigt . . . danake Vidensk, Selsk.
Forhandlinger . . . (Kj0benhavn), pp. 106-119.]

Ramus' introduction consists in recalling the result of the appli-
cation of determinants to the solution of a set of linear equations,
his mode of stating the result being that given by Jacobi in the
De formatione ... of the year 1841, — that is to say, he takes for
his set of equations

V^o + <y\ + «a% + •

and puts the solution in the form
where

(cu)

* It is in this mode of writing Aj, viz., with the negative sign, that Jaoobi's
peculiarity consists. Not content with removing from Rn the row and
column in which a^ occurs and prefixing to the minor thus obtained the sign-

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«o

exists the set of

equations

- "hyo

+ yi

- Vo

-0^1 +

2/s

- ^tUx -

«»!'« +

1908-4.] Dr Muir on the Theory of Gontimuints. 143

He then recalls the further fact that if ^o > ^1 » ^2 > • * • » Vn ^ the
numerators of the convergents of the continued fraction

-\
= 0
= 0

and he thereupon draws the natural conclusion that the previous
result can be applied to the determination oiy^y yi^y^, . . . , ^„ .
Making the necessary substitution for the u's and for R^ he of
course obtains

y^ = a^An^ + b^A^\

A^®, A^^ being now determinants which for want of Cayley's
notation he cannot accurately specify, but which he persists in
writing in the form

- Z ±«oW •••<:!. - Z±«oSV • • • C!-

From this result he calculates in succession the values of y^, y^,
y^j y^; but it will readily be understood that the process is neither
elegant nor short.

In the remainder of the paper (^ 4-9) no further use of the
properties of determinants is made, the contents of the last ten
pages being such as might appear in any ordinary exposition of
continued fractions. First there is established the old "rule" for
writing out the value of y„, above referred to as being given by
Stem. This is followed by the results

factor ( - 1)*+*, he takes the further step of moving the row with the index k
over jc- 1 + 1 rows, thus arriving at

or course there is at the second step the option of moving the column with
the index i over te-i+1 columns, and this Ramus does.

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144 Proceedings of Royai Society of Edinhwrgh. [i

+ i+...(n6'8)" ^a2 + 46\V 2 J "V 2 / /'

which by putting a^l^h give the number of terms in Y„, — a
number also obtained in the form

Cn+J.1 + Cn+j,8*5 + C„

Anything else is of small moment.

5«+ ... I

Caylby, a. (1857, April).

[On the determination of the value of a certain determinant
Quart. Joum, of Math., ii. pp. 163-166 ; or Collected MatK
Papersy iii. pp. 120-123.]

The determinant in question is rather more general than Syl-
vester's of the year 1854 {Nouv. Annalee de Math.^ xiil p. 305),

being

^ 1 . . .

X 0 2 .

X'l e 3

. a;-2 6

e ti-1
x-n+2 $

while the other is obtained from this by putting ic = n- 1. De-
noting his own form by Un, Cayley, with Sylvester's results before
him, found

U,=(6«-l) - (x-1),

Ug = 6(^-4) - 3(3!- 2)«,

U^ = (^-l)(tf«-9) - 6(a:-3)(tf'-l) + i(x-S)(x-l);

so that, if he put H, for the value of IT, in Sylvester's case (viz.,
when « = «- 1), he could write

Uj=H2-(a!-l)H,

U3 = H,-3(x-2)H,

U, = H« - G(x - 3)H2 + 3{x - 3)(x - 1)R^ ,

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1908-4.] Dr Muir on the Theory of Continuants, 145

and thence, doubtless, divined the generalisation

U„ = H^ - B^i.(a;-n + l).H,_, + B„,2.(a;-n + l)(a--n+3VH„.4 - . ...

where

H, = (^ + n-l)(^ + n-3)(^ + »-5) to n factors

and

n(w-l)(w-2) {n-28+\

^n,.- 2*. 12. 3 8

The establishment of the truth of this is all that the paper is occu-
pied with, the procedure being to expand U„ in terms of the elements
of its last row and their complementary minors, thus obtaining

U, = ^U„., - (n-l)(x-n + 2)U..,
and thence

Un+{(n-l)(a'-7* + 2) + (n-2)(a?-n + 3)-^jU„.,
+ (n-2)(n-3)(a;-n + 3)(ar-n + 4)U„_4 - 0,

and showing that the above conjectural expression for U,, satisfies
the latter equation. The process of verification is troublesome, and
was not viewed with satisfaction by Cayley himself.

As a preliminary the coefficients of the H*s in the value of U„
are for shortness* sake denoted by A„,oi - A«,i, . . . , and for
the same and an additional reason the coefficient of U„., in the
ditference-equation is denoted by

M„,- |^-(n-2«-l)2l,

which is equivalent to putting

M,^,= (n-l)(x-n + 2) + (w- 2)(.r-n + 3) - (w-2«-l)l
The operation to be performed being thus the substitution of
A^oH„-A,.iH„,2+ +{- VA„.,H„.3.+

for U« in the expression

XIn + [M,M- {^-(n-2«-l)2|]u,., + (n-2)(7i-3)(aj-7i + 3)(a:-« + 4)U.
PROC. ROY. SOC. BDIN. — VOL. XXV. 10

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146 Proceedings of Royal Society of Edinburgh. [siss.

it is readily seen that the result will be an aggregate of expressions
like

+ (n - 2)(n - 3)(x - n + 3)(ar - n + 4)A„ .,.,H,_,_2, .
Now if we bear in mind that by definition

the second of the three terms of this

" '"ln,rA^-2^11n-2-2» ~ ^n-%^n-Xty

or, if we write « - 1 for « in one case,

■• "~ ^n,«-l'A^-2,»-lH«-2« ~ A^-2,«ll»i-25

and the third, by writing s - 2 for «,

= (n - 2)(n - 3)(^ - n + ^){x - n + 4)A„_,,.,H,.^ .
Consequently the sum of the three will vanish if
An,, - (M„.,.,A„.2.,_i + A„.2.,) + (w-2)(n-3)(ic-n + 3)(a:-/i + 4)A,_;,_, = 0,
and therefore if

B,.Xa--n+l) - H„_^,(a--w + 2^+l)
- B„_2.,_iM„.,_i + B„.,.,_,(n-2)(n-3)(x-n + 4) - 0,
that is, if

{x - n)[B,., - B„_2., - (2n - 3)B„.,^.i + (n - 2)(n - 3)B,..,^_,J

But this is the case ; for, as Cayley shows, both the cofactor of a; - n
and the other similar expression following it vanish identically.
The verification aimed at is thus attained.

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1903-4.] Dr Muir on the Theory of Continuants.

147

Painvin (1858, February).

[SuT un certain syst^me d'^uations lin^aires. Joum. de
lAoumUe (2), iii. pp. 41-46.]

The system of equations referred to in the title of Painvin's
paper had presented themselves to Liouville in the course of the
research which led to his '* M^moire sur les transcendantes eUip-
tiques ..." {Joum. de Liouville (1), v. pp. 441-464). Painvin's
reason for taking up the subject was his belief that one of
Liouville's results could be more simply arrived at by the use of
determinants ; and in a few lines of introduction he succeeds in
showing that the result in question can be viewed as merely the
resolution of the determinant

r a . .....

n(a-l) r-1 2a

(n-l)(a-l) r-2 3a

(w-2)(a-l) r-3 .,.

... r -n+l na
a-1 r-n

into factors.

In explanation of the process followed the case of the fourth
order

r a . .

3(a-l) r-1 2a

9(a-l) r-2 3a

a-1 r-3

will suffice. Increasing each element of the first row by the corre-
sponding elements of the other rows, — an operation which we may
for the nonce symbolise by

rowj + rowg + rowj + ,

— he removes the factor r + 3a- 3 and finds left the cof actor

1111
3(a-l) r-1 2a

2(a-l) r-2 3a

a-1 r-3

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1 48 Proceedings of Royal Society of Edinburgh. [j

•On this are performed the operations

colj - C0I2 , C0I2 - C0I3 , C0I3 - C0I4 ,

the result being a determinant of the next lower order

3a-r-2 r-2a-l 2a

2-2a 2a-r r-3a-2

1-a a-r+2

Finally, after changing the signs of all the elements here, [the

operations

rowj + rowg + rowj + . . . , roWj + roWjH- . . . , row3+ . . . , ...

are performed, the result

?• - a a

2(a-l) r-a-l 2a

a- 1 r-a-2

being a determinant exactly similar in form to the original but
with r-a instead of r. This, therefore, in turn may be trans-
formed into

(r + a-2)

a
r-2a-l

r-2a

a-1
and 80 on.

The value thus obtained for the above-written determinant of
the (n+1)'*' order is

(r4-wa-»)(r + na-n-2a+l)(r + na-n-4a + 2) . , , (r-na)
each factor being less than the preceding by 2a- 1, and the whole
a function of a{a - 1).

The special case is noted where a — |, and where therefore all the
n + 1 resulting factors are alike. This Painvin writes in the form

r J

n
"2

r-1

w-1
2

2 ^

n-2
2

r-3

-n + 1

n
2

-2 ^-^

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1903-4.] Dr Muir on the Theory of Continuants.

149.

but a preferable is, evidently,
P 1 . .

-» p-2 2
. -«+l /a-4 3
-n + 2 /a-6

p-2n'\-2 n
-1 p-2n

= (p-n)-

Hbinb, E. (1868, Sept.).

[Auszug eines Schreibens iiber die Lameschen Functionen an
den Herausgeber. Einige Eigenschaften der LamSacYieD.
Functionen. Ordle^s Joum., Ivi. pp. 79-86, 87-99.]

In tbe case of Heine the functions afterwards known as '* con-
tinuants" made their appearance under totally different circum-
stances, viz^ while he was engaged in transforming a special
homogeneous function of the second degree by means of an
orthogonal transformation. It will be remembered that if the

quadric

a^^x^^ + 2a^^x^ + 2a^^x^ 4- . . .

be transformed hy an orthogonal transformation into

Ajiti + -^22%2 + -^38^8 + • • •

the coefficients of the latter expression are the roots of the
equation

Oil -A 0^2 Ojg ...

^12
«18

a22-A

^28

*28

Ojs-A

Now Heine's peculiar quadric was

+ (C32 + C>22.

+ (ci<r-l+4rK'

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150 Proceedings of Royod Society of Edinburgh. [si

where in every case the coefficient of the product of two a^'s
vanishes if their suffixes differ by more than 1, and where

«'o»-K«)(«+i).

Ci2 = i(n-l)(n + 2),

c,«-J(n-r)(» + r+l), (r>0)

C*n-l = }«i

He was thus naturally led to the equation in z

1 2-Co^ f^o^ • • • • •

I f^A^ Z ~~ C^ "" ^2 f ^^ .... .

KC^C^ Z-C^-C^ ....

KCia-i C2«r-l

= 0,

where either c^a^ is 4_i, or 4,_i is ci_i and, if the latter, 4r = 0.
From a knowledge of Painvin's paper he recognised the left-hand
side of the equation as being the numerator of the continued
fraction

^0 ,-r^2~r-,2^

but he ventured nothing in elucidation of it. Even the special
case where 5 = 0 and where therefore k^I appears to have proved
at the time too troublesome, although he knew otherwise that in
' this case the continued fraction

2(2-22)(2-4g) . . . (g-n2)

and

if n be even

(z'V)(z-S^){z^5^) .... (2- n^) .
(2-22)(2-42) .... (2-n^2) '

-^2\ if » l>« odd ;

for his words are — " Einen directen Beweis ftir diese Summirung
des Kettenbruchs habe ich noch nicht aufgefunden.*'

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1908-4.] Dr Muir on the Theory of CorUimuiTUs,

151

SCHLAFLI, L. (1868).

[On the multiple integral / dxdy , , , dz whose limits are

p^ = a^x + b^y + . . . + /*i2: > 0, 2?2 > 0, . . . , i?n > 0,
and a;^ + y2 + . . . +2^ > 1. Quart, Joum, of Math., ii.
pp. 269-301, iii, pp. 64-68, 97-108.]

The determinant which makes its appearance in the course of
Schlafli's research is

1 - cosa

-coea 1 -cos^....

- cos^ 1 . . . .

1 - cos 17

- cos 17 1 - cos ^

- cos^ 1

which for shortness' sake he denotes by

A(a,Ar, • • . ^V>0)
and whose connection with continued fractions he therefore
specifies by the equation

A(a,Ay»

yVy^)

A(j8,r, . . . ,v.O)

= 1 -

cos^a

n" -

cos^

C08*17

1 - cos^^
The first property noticed is, naturally,

A(a,/3,y, ...,<?)- A(^,y, ...,<?)- cos^a.A(y, . . . ,0).

Later there is given what may be viewed as an extension of this,
viz.,
A(a, . . ,MUO. . . .A) = A(a, . . .M'HvA • • • A)

-C0S«f.A(a,...,8).A(«, ...,X),

the proof being said to present no difficulty. The third is a little
more complicated, and is logically led up to by taking four instances
of the first property, viz.,
A(a,Ay, . . .,0 = A09,y, . . . ,f) - cos^a • A(y,8, . . . ,{),
A(/)\y,8 . . . Zv) = A(yA . . .,17) - cos2^- A(8, . . .Zv)^
A(y,8 . . . ,U0) = A(y.8, . , . ,rj) - C08«<? • A(y,8, . . . ,{),
M«, ' • .»U«,a) = A(8, . ..,U0) - cos^a. A(8, . . , Zv).

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152 Proceedings of Royal Society of Edinburgh. [b

using in connection with these the multipliers

A(8, . . .,^,17), -A(8, . . .,0, A(8, . . .,0, -A(y,8, . . .,0,

respectively, performing addition, and then showing that the right-
hand sum vanishes, the result thus being

A(a,/3,y,8, . . . ,0 • A(8, . . . ,^.17) - A(8, . . . ,U«,a) • A(y,8, ... ,0
= {A()8,y,8, . . .M - A(y,8. . . . ,£,i7,«)} • A(8, . . . ,0 .

The fourth property concerns the determinant

A(Ay, . . . ,17,(9) A(a,/3,y, . . . ,17,^)
A(i8,y, . . .,1;) A(a,Ay,. 17)

which by reason of the first property can be shown equal to

A()8,y, . . . ,i7,<?) - A(y, . . . ,17,^?)

A(Ay, ... ,17) - A(y, ... ,17)

COS^O,

A(y, . . ,,ri,e) A(i8,y, . . .,,7,^)
A(y, ... ,,7) A(^,y, ... ,17)

COS^O,

and ultimately, " by repeating this sort of transformation," equal
to

cos^a cos^jS cos^ .... cos^^ .

If we use for a moment the present-day notation for continuants,
viz., where

a, + ^1 K

^2 + TT ^

yg^ og 03 . . . y

Vh «8 • • • /

Schlafli's results are seen to be

■'(I^^'^..)->^(lY•..)-ft'^(.v^.).,

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1908-4.] Dr Muir on the Theory of Continuants,

■■^(l''•....^^w.^...''-l)^(-K.^...^)l

«(A..^) >^(l^■l...^)

■•{

153

1 1...

.),

^U\..'-\) K\J-\)

the only change heing the writing of jS^ , jSg ,

(-l)-/3^^,...)8.,
. for - cos^a ,

WoRPiTZKY (1865, April).

[Untersuchongen tiber die Entwickelung der monodromen
und monogenen Functionen durch Kettenbriiche. (Sch.
Progr.) 39 pp., Berlin.]

Of the six sections into which the paper giving the results of
Worpitzky's painstaking investigation is divided it is only the
first headed '* Fundamentalrelationen '' which concerns us, these
relations being nothing else than what we should now call ** pro-
perties of continuants."

He takes his continued fraction in the same form as Schlafli,
viz..

* * 1 +

showing of course that it equals

^ r

where

2"»

^..n

I 1

-o. 1 1

- a,., 1

- On-l 11

-a, 1

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154 Proceedings of Royal Society of Ediiibv/rgh. [sks.

The first matter of interest is the expansion of N. „ as a sum of
products of a« , a,e-i , . . . , a„ , e.g.,

^1.8 = 1 + («i + ^2 + «8) + «A-
This is written in the form

1 + <n + a,i. = . . . .

where, he says, "a^y^ die Summe aller moglichen (als Producte
aufgefassten) Combinationscomplezionen ohne Wiederholung be-
deutet, welche sich aus o^ , ok^^ , . . . , o^ so zu je r Elementen
bilden lassen, dass nicht zwei neben einander stehende Elemente
a«, a«+i dieser Reihe in den einzelnen Producten zugleich vor-
kommen." By way of proof it is pointed out (1) that the term
independent of all the a's is

1
1

i.e. + 1 ;

(2) that the cofactor* of (-a^)( -a,)(— a,)
the a's are consecutive is

when two of

1 1

0 1 1

0 1 1
1 0 0
1 0
0

I 1

0 1
0

1
1

i.e. 0;

* To obtain the cofactor of the product of a number of a set of elements in
a determinant Worpitzky puts a 1 in the determinant in place of each element
occurring in the said product, O's in all the other places of the rows to which
these elements belong, and O's for all the other elements of the set.

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1903-4.] Dr Muir on the Theory of Continuants,

and (3) that the cofactor of ( - a^)( - a,)( - a,) . . .
two of the a's are consecutive and their number i« j?, is

1 1
0 1 1

0 1 1
1 0 0
0 1 1

155
when no

0 1 1
0 1

t.e.

1 1
1 0

t.e. (- \y,

and that, therefore, the cofactor of a^/i^ ... in this case is + 1.
In exactly similar fashion by partitioning N. „ into terms which
contain - a, and terms which do not, he finds

N,.n = Do - a,D,,

where

I>o =

1 1
- a^i 1 1

-a«.i 1 1
-a„ 1

1 1

-a,., 1 1

1 1

a„-l 1 1

-fln 1

= N,,.,.N,^,„,

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156 Froceedings of Boyal Society of Edinburgh. [i

and

1 1
-a» 1 1

I>.=

a,_ 1 1

-a,-i 0 1

1 0 0

- a*+i 1 1

1 1

0 1 I

1 0

'On 1
1 1

-a*-8 11 I -a^.i 1 1

-«*-2 1 -a« 1

and thus reaches the result

already obtained in a different way by Schlafli.

Lastly, taking a determinant of the same form as Njt,„, but
having

- a„ - a,_i, . . . , - Oj+i, - a^, - aj, - Oi^.,, . . . , - o^i, - a^

for its minor diagonal of a\ he obtains for it by isolating the first
Qj^ the expression

and by isolating the second a^
and thus deduces

It is then noted that the bracketed expression on the right differs

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1903-4.] Dr Muir on the Theory of Continmants. 157

from the expression on the left merely in having A; + 1 in place of
k ; so that there results

This also, it will be seen, is connected with a result of Schlafli's ;
for putting «=n - 1 we have*

which becomes identical with Schlafli's last proposition on trans-
posing the two rows of the determinant and (what is equally im-
material) putting A;= 1.

Thiblk, T. N. (1869, 1870).

[BemflBrkninger om KJ8Bdebr0ker. Tidaskrift for Math, (2),

V. pp. 144-146.
Den endelige Kj»debr0ksfunktions Theori. Tidsskri forft
Math. (2), vi. pp. 145-170.]

The first of the two notes comprising Thiele's first paper con-
tains only one result, viz.,

«i+;i^ + ^

ag-h

(Oj , tto , . . . , a„)
(ttg, . . . , a„) '

where (a^y ag, ...,««) is used to stand for

«! ^

- 1 a. 6«

-1 a„

♦ In giving to N,+i.,, Na+2^, N«+8^ the values 1,1,0 which are necessi-
tated by assuming the generality of the recursion-formula

Worpitzky forgets to note that in these cases the proposition N».n=Nn.ft , used
by him in the demonstration, does not hold.

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158 Proceedings of Royal Society of Edinhwrgk.

[8B8S.

There is nothing to indicate that this is not viewed as a fresh
discovery, notwithstanding the fact that Ramus' paper of 1856 con-
taining virtually the same identity was published in the same city.

The other paper may be described as a careful study of finite
continued fractions with the help of determinants. Instead of
6j, 6j, . . . are used ai2> ^2S> • • ''y ^^^

Wl ^iH-ljH-l

«g-l Clq-\,q

is denoted by

Further, this determinant is spoken of as a " Kjaedebr0ksdeter-
minant," or, shortly, a " K-Determinant " ; and a section (§ 3, pp.
149-152) is devoted to a statement of its properties.

There is no need to rehearse all of these, the last portion (D) of
the section being alone that which contains fresh matter. Opening
with the double use of a previous property, viz.,

K(M*) = K(/^A;-l).K(A^w) - a*_,^(M- 2).K(A;+l,m),
K(;i,/i) - K{h,k-\yK{h,n) - a;fe_,^K(A,A;-2).K(A; + l,n),

where h, A:, m, n are in ascending order of magnitude, the author
eliminates K(/i,A;- 1) and obtains

K(V») TL{k,m)\ _ „JK(&,m) K(&+l,m)|

K(A,») K(&,n) -«*-M--K'('^*- 2)- 1 K(;fc^„) K(;t+ l,n) | . ^">

Then by taking the particular case of this where k appears in
place of h and A; + 1 in place of k there results

K(A:,m) K(A;-Hl,w)
K(A;,n) K(A:+l,n)

I K(A;+l,m) K(/c + 2,m)
^*'*"''| K(A;+l,n) K(A;+2,n)

which when applied to one of the determinants occurring in itself
gives

lL{k,m) K(A;+l,m)
K(A;,n) K(A;+1,»)

— ^ft.*+l^*+l,*+2'

K(A: + 2,m) K(A; + 3,m)
K(A;+2,n) K(A:+3,70

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1903-4.] Dr Muir on the Theory of Contimmnts,
and finally

159

K{m+l,7n) K(m + 2,7w) I
K(m+l,w) K(m + 2,n) |,

= a*.*+i«*+M+J • • • • «m,«+i • K(w + 2,n) .
Further, by using this to make a substitution in the previous result
(a) there is obtained

which on putting k — h-^l and m = n - 1 becomes
K(M-l) K(;i+l,n-l)

K(M) K(h+l,n)

= *A,»+l^A+l.A+i •

<»n-l.

— a result which may be compared with one of Schlafli's and
Worpitzky's, but which is more general in that the main diagonal
of each " K-Determinant " does not consist of units.

LIST OF AUTHORS
whose writings are herein dealt with.

PAGE

PAGB

1853. Sylybsteb .

. 130

1857. Cayley.

. 144

1853. SpomswooDE

. 134

1858. Painvin

. 147

1853. Sylvbstek .

. 135

1858. Heine .

. 149

1853. Stlvester .

. 138

1858. SCHLAFLI

. 151

1854. Sylvester .

. 141

1865. WORPITZKY .

. 153

1855. SOflLAFLI

. 141

1869. Thiblb.

. 157

1856. Ramus .

. 142

1870. Thielb.

. 157

{Issued aepartUely, February 26, 1 904. )

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160 Proceedings of Royal Society of Edinbu/rgh, [i

On the Origin of the Epiphysis Cerebri as a Bilateral
Structure in the Chick. By John Cameron, M.B.
(Edin.), M.R.C.S. (Eng.), Carnegie Fellow, Demonstrator of
Anatomy, United College, University of St Andrews. Gont-
municated by Dr W. G. Aitchison Robertson.

(MS. received January 4, 1904. Read same date.)

CONTENTS.

(1) Results of thb pbesrnt Research .

(2) compaeison of results

(3) summaet and conclusions .

(4) Literature ....

(5) Explanation of Illustrations

PAGE

160
163
164
165
167

(1) Results op the present Research.

Till within recent years the epiphysis cerebri has been generally
regarded as a mesial outgrowth from the roof of the thalamenceph-
alon in Vertebrates. The researches of B^raneck (6), Dendy (11),
Hill (17), and Locy (19), however, tend to demonstrate the fact that
this structure arises in the form of two bilateral outgrowths ; while
Gaskell (12) has drawn attention to its bilateral nature in
Ammocoetes. Some observations which the author made on the
development of the epiphysis in Amphibia (8 and 9) were found to
agree in the main with those of the above-mentioned workers. The
present research was therefore undertaken with the view of cor-
roborating the results which had been obtained in the Amphibia, and
it was found that these received support in the case of the chick.

A number of early chick-embryos (chiefly between the 60th
and 60th hours of incubation) were examined ; and although it was
difficult in every instance to obtain distinct evidence of the bilateral
nature of the epiphysis, still in the majority of cases this con-
dition was distinctly marked. The reason for the difficulty of
demonstrating in all cases the presence of the bilateral epiphysial
condition will be explained later.

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1903-4.] Origin of the Epiphysis Cerebri in the Chick, 161

Fig. 1 is drawn from a chick-embryo at the 50th hour of incuba-
tion, and represents a transverse section of the thalamencephalon
in the pineal region. The larger of the two evaginations lies
distinctly to the left of the mesial plane (which is represented by
the dotted line in the figure), while on the right side a much

I
Fio. 1.

smaller evagination exists. The latter was found to bej^evident in
the whole series of sections of the pineal region in this embryo,
but it was in every instance much smaller than the left
evagination.

Fig. 2 is from a chick-embryo at the 60th hour of incubation,

Fio. 2.

and represents a transverse section of the roof of the thalam-
encephalon in the pineal region as in the previous instance. The
resemblance between this fig. and the fig. No. 5 which illustrates
Dendy's paper (11) is most striking, as will be at once recognised on
comparing the two. Fig. 2 shows with marked clearness the simul-
PEOC. ROY. 800. EDIN.— VOL. XXV. 11

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162 Proceedings of Royal Society of JSdinburyh. [stss.

taneous presence of both the right and left primary epiphysial
outgrowths. Here, again, the left is by far the more marked of the
two ; but the right outgrowth is also well developed — more so than
in the previous instance (fig. 1). This section seems to the author
to afford distinct proof of the fact that the epiphysis in the chick
arises in the form of two distinct evaginations. Many other figs,
of this early stage could have been represented ; but those already
given amply demonstrate the presence of the right and left
epiphysial outgrowths. In all the many sections showing paired
outgrowths the left was better developed than the right.

A study of the later stages of development of the epiphysis in
the chick shows that the duration of the bilateral condition is very
brief — the right and left primary outgrowths blending with one
another to form the single unpaired epiphysial evagination. This
is found to take place towards the end of the 3rd day — after
the 60th hour of incubation. The bilateral condition is thus best
observed between the 50th and 60th hours of incubation, so that
it has a very transient existence (just as in Amphibia) ; and this
probably explains why the bilateral origin has not been previously
recognised. But it should also be noted that in some instances the
right or smaller evagination was present, but only faintly distin-
guishable, so that it was quite possible either to overlook its presence
altogether (more especially if a single embryo was being examined
instead of a series), or to consider it was as a small fold of the cere-
bral wall due to faulty fixation, or, lastly, to look upon it as an
anomalous condition. All the eggs which were examined in this
research were incubated under a * broody ' hen, so that the occur*
rence of those anomalies which ensue from the use of an artificial
incubator was avoided. All the embryos were carefully fixed in
Bles' fluid, which is an excellent fixative for embryonic tissues, and
all risks of shrinkage were thus entirely obviated.

As has been already stated, the bilateral condition of the
epiphysis ceases to exist about the end of the 3rd day of incuba-
tion; but one cannot draw a hard-and-fast line of demarcation
regarding the duration of the bilateral condition, as it is a well-
recognised fact that chick-embryos vary considerably in their rate
of growth. In some cases, therefore, the presence of the bilateral
condition was observed previous to the 50th hour of incubation,

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1903-4.] Origin of the Epiphysis Cerebri in the Chick, 163

while in the other cases this condition was distinctly evident after
the 60th hour of incubation.

Fig. 3 is, like the others, a transverse section of the thalam-
encephalon in the pineal region, and is from a chick-embryo at
the end of the 3rd day of incubation. This figure represents what
might be termed the unpaired condition of the epiphysis. On
close examination, however, the presence of two small angular
recesses within the evagination will be noted, and it may be sug-
gested that these are probably lingering evidences of the previously
existing bilateral outgrowths — the process of coalescence having
apparently just taken place.

Fig. 3.
It therefore appears that what in its earlier stages of development
used to be looked upon as a mesially placed epiphysial evagination
is really situated to the left of the mesial plane, tohile a more
feebly formed evagination exists on the right side. This bilateral
condition is, however, very transitory, and soon gives 7-ise to the
impuired condition of the epiphysis by a coalescence of tJie
primary elements.

(2) Comparison of Rksults.

The results of this research are of interest in so far as they
support the observations previously made by the author in the
Amphibia (8 and 9). They also agree in the main with the
results obtained by various observers in reference to other classes
of the Vertebrata. In Amphibia the author has described the
presence in the early stages of right and left recesses from the roof
of the thalamencephalon, of which the left is the better developed
of the two ; and has shown that these very soon coalesce to form a
single epiphysial structure. It will be at once observed that these
conclusions are corroborated in the case of the chick.

It is also interesting to compare the results of the present re-
search with those of Dendy (11) on Hatteria. This observer has

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164 Proceedings of Royal Society of Edinburgh, [siss.

demonstrated in embryos of this reptile the presence of right and
left epiphysial outgrowths, which remain distinct and separate from
each other. Of these, the left is the more important, and gives
rise to the pineal eye, while the right never becomes transformed
into anything resembling a pineal eye, but retains its attachment
to the roof of the thalamencephalon, and constitutes the epiphysial
stalk. So also in the chick the left evagination is the more important
of the two. It is, however, uuable to remain separate from the
right evagination, and thus fails to retain its individuality.*

Hill (17) has described right and left epiphysial evaginations in
Teleosteans and in Amia ; but in the specimens examined by him
the right outgrowth was somewhat more vigorous than the left,
while they showed no tendency to blend with one another.

Locy (19) has been another worker in this field of research.
He describes the epiphysis of Elasmobranchs as developing from a
pair of united accessory optic vesicles. In this group of Fishes,
therefore, the paired elements tend to blend with one another as
in the case of the chick and the Amphibia.

This research was conducted in the Anatomy Department of the
United College, University of St Andrews, under the terms of
my appointment both as a Carnegie Fellow and as a Research
Fellow of St Andrews University. I wish here to express my
best thanks to Professor Musgrove for many valuable facilities
which were afforded to me diiring the progress of the work. I
intend to study the early stages of development of the epiphysis in
Mammalia in order to ascertain whether any evidence of the bi-
lateral condition of the epiphysis can be found in this class of
Vertebrates.

(3) Summary and Conclusions.

(1) The epiphysis cerebri in the chick-embryo first appears in

the form of right and left outgrowths or evaginations. Of these,

the left is the better marked of the two.

* My attention has been directed to a statement in Bateson's MateriaU for
the Study of VaricUion^ to the effect that the functional eyes of Vertebrates,
like other structures near the mesial plane, tend in certain rare instances to
coalesce. This cyclopian condition has been described in the chick (see page
458 of the above work), while on page 461 there is illustrated a specimen of
the worker-bee (Aj^ mellifica) with the two compound eyes fused together in
the mesial plane.

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1903-4.] Origin of the Epiphysis Cerebri in the Chick, 165

(2) The right primary evagination blends with the left at an
early stage of development to form a unified structure.

(3) These observations correspond for the most part with those
already made by the author in the case of the Amphibia. They
also agree in many ways with those of B^raneck, Dendy, Gaskell,
Hill and Locy in other classes of the Vertebrata. As a result of
this, it is evident that in the four lower Vertebrate classes the
epiphysis cerebri arises as a bilateral, and not as a mesial structure.

(4) It is probable that the ancestors of Vertebrates possessed a
pair of parietal eyes, and not a single unpaired structure.

(4) LiTBRATURB.

Literature consulted in connection with the present research : —

(1) Balfour, F. M., Comparative Enibryology^ vol. ii., 1881.

(2) Beard, J., "The Parietal Eye in Fishes," Nature, vol.
xxxvi., 1887, pp. 246 and 340.

(3) Bbard, J., " The Parietal Eye of the Cyclostome Fishes,"
Quart. Jour. Micr. Sci^ vol. xxix., 1888, p. 55.

(4) B^RANBCK, E., " Sur le nerf parietal et la morphologie
du troisi^me ceil des Vert^br^s," Anat. Am,, Bd. vii., 1892,
8. 674.

(5) B^RANBGK, E., " Llndividualit^ de Toeil parietal," Anat.
Am., Bd. viii., 1893, s. 669.

(6) BuRCKHARDT, R, " Die Homologien des Zwischenhirndaches,
und ihre Bedeutung fur die Morphologie des Hirns bei niederen
Vertebraten," Anat. Am., Bd. ix., 1894, s. 152.

(7) BuROKHARDT, R., "Die Homologien des Zwischenhirndaches
bei Reptilien und Vogeln," Anat. Am., Bd. ix., 1894, s. 320.

(8) Cameron, J., "On the Origin of the Pineal Body as an
Amesial Structure, deduced from the Study of its Development in
Amphibia," Anat. Am., Bd. xxiii., 1903, s. 394. Also in Proc.
R(yy. Soc. of Edin., vol. xxiv., 1903, p. 572 ; and in Proc. Scot.
Micr. Soc., vol iii, 1903.

(9) Cameron, J., " On the Bilateral Origin of the Epiphysis in
Amphibia," Proc. of Brit. Ansae, 1903, Section D.

(10) Dendy, A., " Summary of the Principal Results obtained in
a Study of the Development of the Tuatara {SpheTwdon punctatus),"
Proc. Roy. Soc, vol. Ixiii., 1898, p. 440.

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166 Proceedings of Royal Society of Ediriburgh. [sess.

(11) Dendy, a., "On the Development of the Parietal Eye and
Adjacent Organs in Sphenodon {Ifatteria)" Quart, Jour. After. Set,,
vol. xlii., 1899, p. 111.

(12) Gaskbll, W. H., "On the Origin of Vertebrates from a
Crustacean-like Ancestor," Quart. Jour. Micr. Sei.^ vol. xxxi.,
1890, p. 379.

(13) Graap, H. W. db, "Zur Anatomic und Entwickelungs-
geschichte der Epiphyse bei Amphibien und Reptilien," Zool. Am,^
Bd. ix., 1886, s. 191.

(14) Hill, C, "Development of the Epiphysis in Coregonus
cUbus" Jour, of Morph,, vol. v., 1891, p. 503.

(15) Hill, C, "The Epiphysis of Teleosts and Amia'' Jour, of
Morph., vol. ix., 1894, p. 237.

(16) Lbydig, F., "Das Parietal Organ der Wirbelthiere," Zool.
Am., Bd. X., 1887, s. 534.

(17) Loot, W. A., "The Derivation of the PiQcal Eye," Anat.
Am., Bd. ix., 1894, s. 169, s. 231.

(18) LocY, W. A., "The Mid-brain and the Accessory Optic
Vesicles," Anat. Am., Bd. ix., 1894, s. 486.

(19) LocY, W. A., "Accessory Optic Vesicles in Chick-
embryo," Abstract in Jour, of Roy. Ulicr. Soc, 1898.

(20) Marshall, A. M., " Vertebrate Embryology," 1893.

(21) Prbnant, a., "Sur Toeil parietal accessoire," ^«a^. Am.^
Bd. ix., 1894, s. 103.

(22) Rabl-RCckhardt, H., " Zur Deutung der Zirbeldriise
(Epiphysis)," Zool. Am., Bd. ix., 1886, s. 405.

(23) RiTTER, W. E., " On the Presence of a Parapineal Organ
in Phrynosoma," Anat. Anz., Bd. ix., 1894, s. 766.

(24) Spencbr, W. B., " The Parietal Eye of Hatteria," Natvre,
vol. xxxi v., 1886, p. 559.

(25) Spencer, W. B., "Preliminary Communication on the
Structure and Presence in Sphenodon and other Lizards of the
Median Eye described by de Graaf in Anguis fragilis" Proc Ray.
Soc, 1886, p. 559.

(26) Spencer, W. B., " On the Presence and Structure of the
Pineal Eye in Lacertilia," Quart. Jour. Micr. Set., vol. xxvii.,
1886, p. 165.

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1908-4.] Origin of the Epiphysis Cerebri in the Chick. 167

(5) Explanation of Fiourks.

[The figures were drawn with the aid of Zeiss's oamera lacida apjMiratus.
ZdsB^s objective A and ocular No. 3 were employed.]

f./., subcutaneous connective tissue; ep., epiphysis; epib.^
epiblast; /. ep, ev.y left epiphysial evagination ; r. ep. ev.,
right epiphysial evagination; thcU,, cavity of thalamen-
cephalon.

Fig. 1. Transverse section of the roof of the thalamencephalon
in the pineal region. Embryo-chick at the 50th hour of incuba-
tion. The right and left primary epiphysial evaginations are seen.
Two germinal nuclei in a condition of karyokinesis are observable.
The dotted line represents the mesial plane.

Fig. 2. Transverse section of the roof of the thalamencephalon
in the pineal region. Embryo-chick at the 60th hour of incuba-
tion. The right and left primary epiphysial evaginations are
especiaUy well marked. Several germinal nuclei are seen. The
mesial plane is represented by the dotted line.

Fig. 3. Transverse section of the roof of the thalamencephalon
in the pineal region. Embryo-chick at the end of the 3rd day of
incubation. The unpaired condition of the epiphysis is shown.
The presence of two small angular recesses, however, within tlie
epiphysial evagination may denote traces of the previously existing
bilateral condition.

Issued separately March 17, ld04,)

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168 Proceedings of Royal Society of Edinbrnrgh, [j

Theorem regarding the OrthogoncJ Transformatioii
of a Quadric. By Thomas Muir, LL.D.

(MS. received July 27, 1903. Read November 2, 1903.)

(1) The theorem in question arises out of a consideration of
several passages in Jacohi's important memoir of 1833* on
orthogonal transformation. Having determined the suhetitution
which simultaneously changes

and

2«it\««^A into Giy,2 + G2i/22+ . . . +G^«2,

Jacohi proceeds to show (p. 12) that, by the same substitution,
Gi V + ^'2V + . . . + G,'y„«,

where p is any positive integer, can be expressed in terms of
OJj , ajg I • • • i^n{^* expressionen per ipsas x^^x^^ . . . , ar„ exhibere
licet "). The actual result, however, is not sought for. Later on
(p. 14) he reaches a theorem which would enable him to remove
the restriction on ^ so as to admit negative integral values as well,
but the opportimity is not used. The reason for the seeming
neglect probably is that he has in view a second return to the
subject when prepared to deal more effectively with it. However
this may be, certain it is that he does return to it, and gives a
hypothetical form of the desired expression in x^^x^^ . , . ,x^.
His words (p. 20) are : —

"Statuamus G^y^^ + G^V + • • • +G///„2 „ ^p^^^x^^ ubi

and where, we may add, the a*s are the coefficients of the substitu-
tion. Regarding the validity of this nothing is said, but proof is

* Jacobi, C. 6. J., De binis qaibuslibet fonctionibus homogeneis secuDdi

ordinis per tubstitutiones linearen in alias binas transformandis

CreUc'BJoum,, xii, pp. 1-69. (Aug. 1888.)

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1903-4.] Dr Miiir on Orthogonal Transformation of a Qtuidric, 169

adduced to show that whether j> be a positive or negative int^er
the coefficient of x^)^ is a rational function of the coefficients of
the original quadric.

With this general statement of the case before us, let us take
up the individual results in order, and see what is obtainable
therefrom in the light of later work.

(2) The primary result is the transformation implied in the
equation

This, for our purpose, it is essential to write in a form which
brings into evidence the matrix M of the discriminant of the
quadric, viz., in the form

a'*

«11

«1J

«l.

«M

«!2

<hi

«B1

«8«

- Giy,2 + G^y^^ + G32/32,

where, merely for shortness' sake, only three variables are taken.
Now, as Jacobi himself showed, any equation which holds between
the ar's and y's will still hold if we put

( «ii ^'12 «i3 )(-^ I ^s I ^s) f <>r a?! , jTg , arj
«ai ^ «28

«81 «82 «88

and

Gi2/i , ^2^2 > C^s^'s ^^^ yiyy^yVv

Thisjysubetitution, however, in the bipartite function on the left
results simply in the matrix of the discriminant being twice
multiplied by itself* so that we have

M«

(^iW + G2V + GsV-

* Trans. R. S, Edinh., xxxii. p. 480.

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170 Proceedings of Boyal Society of Edinbv/rgh.

[«

The continuation of the process, and the same treatment applied
to the equation

= ^1^ + 2/2^ + ^8^

. . 1

thus lead us to the result that, for any positive integer p, we
have —

^^ ^ ^ = Gi^'y^^ + GiV + ^iV-

Not only therefore do we know that ^Gx^'y^* can be expressed

in terms of the jt's, but the actual form of the expression — and a
beautifuUy simple form — is obtained.

(3) If this result is to hold for n^ative values of p, some^con-
vention must be established as to negative powers of a matrix^
Now according to Cayley the first negative power, M"^, is
defined by the equation

(

«11 «12 «13 V = ^ -^

«21 ^2 «28
«3l «32 «M

^12

•"21

^28

Asi )
A

^88

where A = | a^ Oj^ ^ss I and A^^ , A^g , ... are the cofactors'^of
^u > ^12 > • • • in A : consequently the p^ negative power, M~^,
may be viewed either as

( «ii «i2 «i8 y\

«21 «22 «28
«81 «82 «88

i-1

( 4ii ^ ^1 y

AAA

_12 ^« -2^
AAA

-^18 ^ ^88

AAA

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1903-4.] Dr Muir on Orthogonal Transformation ofaQuadric. 171
With this before up let us return to the primary result

=^1

«11

"12

"18

«il

«S2

"28

«S1

"82

«88

and make use of the theorem^ that any equation which holds
between the x^s and y's will still hold if we put

( ^1 -^21 ^81 ) V*^! » ^2 > ^s) ^^^ ^1 » ^2 > ^8

AAA

Ai2 ^22 Ajj

^^13 '^as —^
AAA

and

g.g.^" for ,.,y,,3.

xi VX2 Gj

The performance of the substitution on the left-hand side
changes the matrix M into M"^ M M"*, that is, M"\ and we have

or. or. cr. = |l + Vj, + tl ^

M-

G«

The repetition of the substitution upon^this equation, and the
application of the same process to the equation

* Jacobi's enunciation of this is ' ' In relationibos omnibuSi quae inter
Tariabilee arj , xij , . • . , «n et variabiles ^n ^s » • • • i ^** locum habent,

simnl loco ym poni posse ^, atqne loco ar^

GjOa . . . G«

SiOiiOa,

Onn

where the Va correspond to the modem A's, and the tign of equality is used
for *or/

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172 Proceedings of Royal Society of HdinJmrgh.

lead to the result

th"

Vh"

(4) Combining this with the result of § 2, we have the general
theorem : —

Tfie orthogonal substitution which changes

(xi , X2 , X3)(M)(xi , X2 , X,) into GiJi^ + Ggy./ + Gsj,*
will change

(xj , X,, X3)(M'')(xi , X,, X3) into G/y,2 + G/y/ + G3 V
where p is any integer, positive or negative,

(5) Since Gj , Gg, Gg, are the roots of the equation

«18

a22 - a: a,

*82

28
«83-^

it is at once suggested from § 4 that the equation whose roots are
the p^ powers of the roots of this equation is got by substituting
for din a^2f • • ' 9 ^^® corresponding elements of the matrix
which is the p^ power of

( «11 «12 ^8 )

' ^21 ^22 ^28 i
I ^1 ^82 ^88 >

a theorem first formulated by Sylvester in 1852 (v. Nouv.
Annales de Mat?t., xi. p. 438).

{Issued separately March 17, 1904.)

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1908-4.] Prof. C. G. Knott on Ocean Temperatures, etc. 173

Ocean Temperatures and Solar Radiation.
By Professor 0. G. Knott.

(Read February 15, 1904.)

Two years ago I communicated a short paper on Solar Radia-
tion and Earth Temperatures {Proc, vol. xxiii., pp. 296-311).
This paper had its origin in a critical discussion of certain results
deduced by Dr Buchan from observations of Mediterranean tem-
peratures which had been made by the staff of the Austrian war-
ship Polo. The mathematical method by which I discussed the
relation between the solar energy incident on the surface of earth
or sea, and the comesponding fluctuations of temperature in the
rock of the Calton Hill and the surface waters of the Mediter-
ranean, has attracted some attention in America ; and correspondence
with Professor Cleveland Abbe has drawn my attentidT again to
the subject. In this paper I propose to consider more carefully the
significance of the observations made and published by the Austrians.
These are contained in four quarto volumes, which Dr Buchan has
kindly placed in my hands for the purposes of a thorough investi-
gation from the point of view of solar radiation. Dr Buchan
clearly saw that something might be made out of these ; and the
results he gave two and a half years ago before the Society indi-
cated a penetration of solar heat every day to a depth of more
than 100 feet. The results were based upon means of tempera-
ture at different depths grouped according to the time of day at
which they were taken. As I showed in my former paper, the
results so deduced indicated a daily penetration into the waters of
the Mediterranean of an amount of heat greater than the sun
could supply.

From the point of view of the present inquiry, the method
adopted by the Austrian observers is not altogether satisfactory.
Their immediate object seemed to have been to accumulate a
sufficient number of temperature and salinity observations at
various depths and at various stations, so as to enable them to
draw isotherms and lines of equal salinity at different depths in

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174 Proceedings of Royal Society of Edinburgh. [i

the eastern half of the Mediterranean Sea. This they have
accomplished, and no doubt their results in this respect are &irlj
acciirate. With this object in view they took complete sets of
observations at as many different stations as possible, and at
stations in as many different situations as possible. After finish-
ing a set of observations at one station at early morning, they

Tabls a. — List of Selected Stations^ with Latitude, Longitude
and Time of Observations,

Station.

Long. E.

LatN.

Date.

Ti me of Observation.

188

80**

14'-1

32**

5' -8

Sept 5

H.15 to 7 a.ni.

68-2

4.40 „ 5.80 p.ni.

210

14 '9

41 -4

5.30 „ 6.15 p.ni.

212

19 -9

89 -5

6.10 „ 7.80 a.m.

218

7 7

45-8

5.35 „ 6.80 p.m.

219

28 -9

20-9

6.80 „ 7.10 a.m.

220

38 *9

15 -8

8.10 „ 4.15 p.m.

222

64-1

14 -5

6.10 „ 7.15 a.m.

228

19-6

6 „ 6.45 p.m.

226

7-8

47 -3

6.15 „ 7.30 a.m.

226

62-6

47-6

6 ,, 6.46 p.m.

228

21 -6

6.10 „ 7.80 a.m.

229

28 -6

6-7

3.15 „ 4.20 p.m.

281

57-7

10-5

6. 5 „ 6.50 a.m.

282

46 -1

85-7

1. 6 „ 2 p.m.

285

8 -5

5.55 „ 6.15 a.m.

248

29 -6

6.45 „ 7.20 a.m.

260

2-6

2. 6 „ 2.80 p.m.

262

60-2

57 -2

7.15 „ 9.45 a.m.

268

7-4

4. 2 „ 6. 5 p.m.

257

29 -1

82 -1

2.10 „ 6 p.m.

269

6-5

27 1

6.10 „ 6.55 a.m.

260

21 -7

8-9

2.10 „ 6 p.m.

262

40 '9

10-4

6.30 „ 7. 5 a.m.

264

19-8

5 -2

1.17 „ 2.15 p.m.

would, for example, steam off to another station twenty or thirty
miles distant, and make similar observations at the new station at a
later hour the same day. They never made two sets of observa-
tions in the morning and afternoon of the same day at the same
place. For our present purpose a few days* steady observations
at the same station would have jjiven more useful results than can
be derived from the observations as made. Still, by comparing
the temperatures at different depths at contiguous stations, for
which the times of observation did not differ by more than ten
or twelve hours, we may hope to get some data available for our

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1903-4.] Prof. C. G. Knott on Ocean Temperatures, etc. 175

purpose. It should be said that the Austrian observers deserve
great credit for the manner in which they carried out the work.

A little consideration showed that only a selection of the
numerous stations were available for the present inquiry. Dr
Buchan had already pointed out the necessity for confining the

Tablb B.-

—Temperatures at Various Depths,

Depth.

*-■ _

100

Stfttlon.

24-0

188

26-0

26-1

25-9

25*2

24-3

20-0

191

27-0

26-8

267

26-6

25-3

24-8

21-8'

212

27-8

27-6

27-5'

27-4

27-3

26-4

22-4

19-9'

17-5

213

28-3

27-9

27-8'

27-8

27-3

26-5

22 7

20-2'

180

219

277

27-5

27-4'

27-6

27 0

25-6

20-4

18-4

17-4

220

281

27-8

277'

27-8

27-2

25-8

20-6

18-5

17-6

222

27-4

27-2

27-0'

27-0

267

25-5

20-5

18-8

17-6

223

28-3

27-9

27-5'

27-1

26-9

25-6

20-5

18-7'

17-4

225

27-8

27-5

27-4'

27-3

26-8

25-8

21-1

18-6

17-3

226

28-1

27-6

27-5'

27-5

26-9

26-8

21-5'

19-2'

17-9

228

277

27-8

277'

27-5

27*0

26-5

21-3

19-2'

17-9

229

27-9

27-8

277'

277

27-2

24-8'

19-6

18-0

17-3

231

267

26-9'

27-0'

26-8

24-4

19-2

18-1'

17-4

232

277

27-8

27-8'

27-6

25-6

22-0'

19 0

17-9'

16-8

235

27-0

26-9

26-8'

27-0

257

21-3

19-0

17-8'

167'

238

27-4

27-0'

26-2

23-4

20-4

18-3

17-3

16-8

248

267

26-6

26-4'

26-2

26-4'

22-4

19-8'

18-2'

16-8'

250

26-9

267

26-6'

26-4

26-2

23-8

19-5

17-6'

16-4'

252

27-0

26-9

26-9'

27-0

26-1

24-4

20-3

18-4'

16-9

253

27-1

26-9

267'

26-6

25-4

22-4

19-1

17-7'

16-5

257

26-3

26-2

26-0'

25-8

25 1

22-8

18-1

17-0'

16-3

259

26-1

257

25-5'

25-3

24-4

20-8

17-8

16-6

16-1

260

27-0

26-6

26-4'

26-4

25-8

21-3

18-5

17-4'

16-5

262

27 0

26-9

267'

26-6

25-5

22-2

19-5

18-3'

167

264

27-4

27-3
27-12

27-0'
26-98

267
26-85

26-5'
2608

23-2'
23-96

20-0
20-02

18-1

16-6
17-05

Means

27-3

18-26

stations chosen to those of deep water. Thus all the stations
near land, however important their temperatures and salinities
from the point of view of a general survey, must obviously be
discounted when the question was one of the direct penetration of
solar radiation. Dr Buchan accordingly picked out the stations
characterised by great depths of water. I think, however, that
his method of selection is not altogether sound. He seems to
have aimed at getting as many stations as possible without paying

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176 Proceedings of Royal Society of Edinburgh, [suss.

sufficient heed to the necessity for having them in contiguous
pairs, so as to have for every morning set of observations a corre-
sponding afternoon set not more than twelve hours apart Guided
by this and other considerations, I found myself compelled to take
a very limited selection of stations, all situated in the Levant.
These selected stations are given in Table A, along with their

Table C. — Temperature Differences at Various Depths,

D ths.

— ^>__

100

Station.

•7

•8

1-8

191-188

1-4

•8

218-212

•5

•8

•4

•1

•3

"•8

•5

220-219

•4

•8

•2

•1

•2

220-222

•7

•6

•7

•8

•6

•8

•1

- 3

-•2

228-222

•9

•7

•6

•1

•2

•1

- -1

-•4

228-226

•5

•4

•1

— "2

•1

- -2

- -6

•1

226-225

•3

•1

•2

•1

•6

•4

•6

226-228

•4

-•2

- -1

- -2

•2

229-228

•2

-1-7

-1^7

-1-2

-•6

229-281

1-2

•8

•4

- 1

-•1

232-281

1-1

•9

•6

-1-2

-2-4

- '2

- 2

-•6

288-286

•4

•5

•2

-•8

-2-8

- -9

- '7

- -6

-•4

260-248

•2

•1

•2

•8

- 8

- -6

-•4

250-262

-1

-•2

-•3

-•6

•1

- -6

- -8

-•6

258-252

•1

-•2

-•5

- -7

-2^0

-1-2

- -7

-•4

257-259

•2

•5

•7

2-0

•8

•4

•2

260-269

•9

•5

•7

•8

•4

260-262

-•8

— '2

•8

- •o

-1

- -9

-•2

264-262

•4

•3

•1

1-0

•5

- -2

-1

Meana

0^-48

0''87

0**-29

0^-22

O'^OO

-0"-08

-0'-09

-0'-18

-O'^ll

Probable (
Error S

±057

±065

±•062

±•088

±12

...

latitudes and longitudes, and the date and hour at which the
observations were made. The number of the station is the number
in the Pola reports. All the observations here discussed were
made in September of 1892.

Table B contains the corrected observations of temperature for
all these stations at the depths 0, 2, 5, 10, 20, 30, 50, 70, 100
metres. Most of the observations at the depth 5 are interpolated,
and are so entered in the Beport. The interpolation can be
effected with considerable accuracy since the law of diminution of

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1908-4.] Prof. C. G. Knott on Ocean Temperatures, etc. 177

temperature with increase of depth is very steadily maintained
throughout the whole series of oheervations, and is hest given by
the means of all (see Table B, and the figure on page 181). -

Table C contains the differences of temperatures at correspond-
ing depths at pairs of stations, at which the times of observations
differed by approximately half a day. The precise difference in
time in any case can be found from Table A. In all it will be
seen that there are just nineteen pairs of stations available for the
inquiry. If the waters to a depth of 100 metres were heated up
during the day by direct solar radiation, and cooled off again during
the night, these differences should all be positive. A glance shows
that out of the nineteen there is one negative value at the surface,
three at a depth of 2 metres, four at 5 metres depth, five at 10,
four at 20, eight at 30, eight at 50, eleven at 70, and eleven at
100. At depths greater than 20 metres there is no evidence of
penetration of solar radiation. Even at 20 metres it is doubtful
if we can find any evidence of direct daily heating. We may,
however, take the means of the differences at each depth, and then
test the sufficiency of the observations by calculating the probable
errors in the usual way. The result is as follows : —

Depth iB

Mean DaUy Oifferenoe

Probable

of Teinpentnre (C).

Error.

0-48

±0-067

0-87

±0065

0-29

±0-062

0-22

±0-088

0-09

±012

-0-08

-0-09

...

-0-18

...

100

-0-11

...

The thermometers read to tenths of degrees, so that little value
can be attached to the second decimal place.

It would obviously be wasted labour to calculate the probable
errors for the last four depths. At depth 20 metres the probable
error is numerically greater than the mean ; so that we can say
nothing definite as to the effect of solar radiation at this depth.

The errors are so great that we may, without running any risk
of introducing greater errors, combine these numbers by a linear
formula, assuming that the difference of temperature t between
morning and afternoon in the waters of the Mediterranean during

PROC. ROY. BOC. BDIN. — VOL. XXV. 12

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178 Proceedings of Royal Society of Edinburgh, [i

the month of September is connected with the depth by the

formula

< = a + fee.

Combining the first four temperature diflferences down to a depth
of 10 metres by the method of least squares we find

t » 0-44 - 0-025X.

If we include the difference for the 20-metre depth we find

t = 0-42 - 0018a;.

Another result obtained by using twenty-seven selected pairs of
stations instead of nineteen is

/ = 0-47 - 0"02a;.
For this last case the mean differences at the four smaller depths
were 049, 042, 0-33, 028.

If we compare the values of the mean differences of temperature
here calculated with the values given in the former paper, we see
that the present values derived from a carefully-selected number
of stations are distinctly smaller, and that no confidence can be
placed upon the means for depths greater than 10 metres.

We may now complete the investigation by calculating how
much heat accumulation and loss of heat day by day this fluctua-
tion of temperature in the Mediterranean means. This is at once
done by integrating the expression tdx from x = 0 Xo x equal to
the value for which t vanishes. These values are for the three
formulae given above — 17*6, 23-3, and 23*5 respectively. Integrat-
ing for these cases and using the corresponding superior limit

for X we find

0-44^ - 00125a;2 = 3*9

0-42x - 0-009 x^ = 4-9

Oilx - 0-01 ir2 = 5-5

Changing the unit from the metre to the centimetre we find 390,
490, 550 calories as estimated values for the amount of solar
radiation which heats the Mediterranean waters daily during the
month of September. The probable errors in each of these
determinations are large, so that only the first significant figure is
of any real value. Let us consider 450 ± 50 as a fair average, and
compare this with the amount of solar energy available as cal-

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1908-4.] Prof. C. G. Knott on Ocean TemperaJtv/res, etc. 179

colated in the previous paper. On page 299 in that paper a table
is given from which we may estimate the amount of solar energy
available in one day in the middle of September for localities in
the latitude of 33*" N. Taking the average declination of the sun
during September at about 3*, we find for the solar energy supplied
in one day the value 6x117 = 700. According to the present
calculation we conclude that about two-thirds of the solar energy
incident on the surface of the Mediterranean Sea heats the surface
waters through a depth of nearly 20 metres. This, perhaps, is
not an unreasonable result, and is an important correction upon
the earlier result, as showing that the Austrian observations are
from this point of view in sufficient accordance with Langle/s
valuable investigations into the value of the solar constant.

Dr Buchan has drawn attention to the importance of the obser-
vations in relation to the manner in which the ocean waters (first)
gain their heat in the day, and (second) lose it again at night. But
here again their value would have been greatly increased if the
observers had had this particular problem present to their mind when
the observations were being made. Had the Polaj on one particu-
larly quiet sunny day, in the centre of the Levant, far from land,
made throughout a complete day of twenty-four hours a succession
of complete sets of temperature readings at the various depths,
at intervals, say, of two or three hours, a great deal of valuable
information bearing on this question would have been obtained.
The conditions of the survey undertaken quite precluded this.
Fortunately, however, observations of the temperature of the
surface waters at midnight were frequently, though not regularly,
taken. By comparing these with the preceding afternoon tem-
peratures and the succeeding morning temperatures, and taking
into consideration the air temperatures at the same times, we gain
distinct evidence of convection in the surface layers. The data
are given in Table D, sixteen diflferent cases in all. In only two
cases was the early morning temperature lower than the immedi-
ately preceding midnight temperature; in two cases it was the
same ; in all other cases it was higher, sometimes markedly so.
In thirteen out of the sixteen cases the air temperature was lower
than that of the water at early morning ; and in eleven of these it
was lower even than the contiguous midnight temperature. We

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180 Proceedings of Royal Society of Edinbv/rgh. [i

may therefore safely conclude that the warming of the water
between midnight and early morning was not due to atmospheric
influence. The simple reason is, in fact^ not far to seek« By
whatever processes the daily heating of the waters is produced, it

Tablb D.-

-Table Showing Convection During Cooling.

Station.

Hour.

Surface
Temp.

Air
Temp.

SUtioii.

Hour.

Surface
Temp.

Air
Temp.

198
194
195

210
211
212

7.40p.
I1.45p.
6a.

5.80p.

12.20a.

6.10a.

5.85p.

12.30a.

6.8a.

26-9
26-5
27-1

27-6
26-0

238
234
235

12.80p.
la.
5.55a.

26-9
26-6
27 0

30-2
26'-5

26-6
26-9
27-8

28-0
25'-6

238
289
240

243
244
245

6.5p.
12.1a.
6.45a.

27-4
27-2
27-6

28-1
26*6

218
214
215

28-8
27-5
28-1

26-9
26*7

121p.
la.
6.14a.

27-8
27-2
27-2

27-6
2*6*-5

27-8
24-6

217
218
219

220

221
222

2.10p.

12.15a.

6.80a.

28*8
27-6
27-7

28-5
26-0

253
254
255

257
258
259

260
261
262

268
269
270

4.2p.

12.20a.

6.10a.

2-lOp.

12.80a.

6.10a.

2.10p.

12.80a.

6.80a.

1.45p.

12.10a.

6.10a.

2.45p.

12.10a.

6.20.1.

27-1
25-9
25-9

8.10p.

12.30a.

6.10a.

28-1

27-2.

27-4

27-5
27-8

26-3
25-8
26 1

26-5
2*4-2

27-6
2*7*-3

24-5
23*-2

228
224
225

6p.

12.80a.
6.15fl.

6p.

12.30a.
6.10a.

3.15p.

12.20a.

6.5a.

28-8
26-9

27-8

281
27-3
27-7

27-9
26-8
267

27-9
27'l

28-3
28'-5

80-0
27-5

27-0
26-5
27-0

226
227
228

24 1
28-4
237

229
23n
231

272
273
274

26-3
25-6
24-6

27-6
2*8*-9

is evident that as the sun sinks the surface layer of the water will
begin to cool by radiation. Suppose it to cool by half a degree
centigrade : it will then become denser than the slightly warmer

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1903-4.] Prof. C. G. Knott on Ocean Temperatures, etc. 181

water beneath ; and if it could sink without loss of heat it would
find its position of equilibrium at a depth of about 5 metres.
This, of course, is a very crude description of what really occurs ;
but it is sufficient to indicate the general nature of the convective
process. The steady cooling by radiation of the surface waters
must be accompanied by a steady vertical convection determined
by the average temperature gradient and the viscosity of the
liquid. This will go on steadily until an approximate equilibrium
is reached, probably towards early morning ; and it is evident that
by this process a considerable depth of surface waters will be
cooled.*

Of no small importance with respect to the question of the
penetration of solar heat through the surface waters of an ocean or
lake is the manner in which the temperature falls off as the depth
increases. The curve shown in the figure represents the means
given in Table B, and may be taken as typical of all cases in
which the body of water is above the temperature of maximum
density.

It will be seen at a glance that the
vertical distribution of temperature
follows a somewhat complex law. As
the depth increases the temperature
falls off, first fairly rapidly, then more
slowly until a depth of 20 metres is
reached. Thereafter a rapid rate of
diminution sets in, which attains its
maximum at a depth of about 30 metres. The rate of decrease of
temperature with increase of depth then begins to diminish, and con-
tinues falling off till the greatest depths are reached. It is evident
that this fairly permanent vertical distribution of temperature can-
not be explained by conduction alone. Probably for depths greater
than 40 metres the main factor is conduction of heat from the upper
warmer layers to the cooler lower layers. But it is quite clear
that some other factor powerfully affects the distribution of tem-

* For an interesting discnssion of similar phenomena in the fresh-water
lakes of the Austrian Alps, see ''Seestudien," by Professor E. Richter (^^o-
grvbphische Abhandhmgm, edited by Professor Penck, Vienna, Band VI.,
Heft 2, 1897)— an important memoir.

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182 Proceedings of Boyal Society of Edinburgh. [

perature in the surface layer above 20 metres depth. This factor
can only be convection, or, let us say, division of liquid. As
already shown, this convection will set in as the sun sinks and the
day cools towards night, and will continue till early morning. No
doubt also surface waves and ripples due to wind will aid this con-
vection ; nor can we leave out of account the vertical migration of
fish and other denizens of the deep. Gonvective movements may
also occur during the day in bodies of salt water, the surface layer
of which, in virtue of evaporation and consequent increase of
salinity, may become denser than the slightly cooler water immedi-
ately below it. This last-named factor we should not expect to
find in the case of fresh-water lakes. That the main causes are,
however, the same in fresh-water lakes as in salt-water seas is
proved by the general resemblance in the law of variation of
temperature with depth in the two types of cases. From the data
furnished by Professor Bichter in the memoir already referred to,
and from similar data supplied by W. F. Ganong, who studied the
vertical distribution of temperature in certain American lakes, we
notice, however, one striking diflference between the fresh-water
lakes and the Mediterranean Sea. In the Mediterranean Sea the
most rapid vertical variation of temperature occurs at a depth of
30 metres ; in the fresh-water lakes, on the other hand, the corre-
sponding maximum gradient occurs at much less depths — namely,
from 6 to 12 metres. The reason for this diflTerence may probably
be found in the following considerations. In the first place, the
somewhat higher temperature of the Mediterranean Sea will no
doubt mean a greater depth of the layer of quickest variation ; but
this can hardly explain the magnitude of the difference. It must
be remembered, however, that in the case of the fresh-water lakes
the vertical distribution of temperature experiences a complete
change during the winter months when the mass of water is at or
below the temperature of maximum density. Hence the summer
distribution of temperature, which resembles in type the distribu-
tion throughout the whole year in the waters of the Mediterranean,
has just time to establish itself before the autumn and winter
conditions set in again, and finally overturn the whole type of
distribution. On the other hand, in the Mediterranean the waters
are never cooled sufficiently so as to come within sight of the

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1908-4.] Prof. C. G. Knott on Ocean Temperatwes, etc. 183

temperature of the maximum density even of fresh water, and
consequently the same type of vertical temperature distribution
remains permanent throughout the year. In the Mediterranean
we are therefore dealing with a permanent average distribution of
temperature which is the steady resultant eflfect of ages of solar
radiation, convective cooling, and heat conduction, down from the
warmer surface waters and up from the slightly warmer earth
below the cold bottom waters.

Superposed upon this steady average distribution we have the
daily see-saw of temperature due to direct solar radiation and to
the complex indirect effects which accompany it. As the sun rises
the surface waters become heated, and to some extent evaporate.
This may cause increased salinity in the surface waters, and give
rise to gravitation convection currents. Ripples, waves, migration
of fish aid the mixing of the waters, so that down to a depth
of perhaps 5 or 10 metres the temperature distribution is largely
affected by these causes, the pure conduction effect being compara-
tively unimportant. The direct heating effect of solar radiation at
depths greater than 15 metres may be regarded as negligible,
because of the great absorption of solar energy in the water near
the surface. From the Pola records we know that luminosity
can penetrate to considerable depths, for white objects at depths of
50 metres were frequently visible. But these rays must be robbed
of by far the greater part of their original energy, which, indeed,
has gone to heat the surface waters. As evening comes on
evaporation will largely cease, the surface waters will cool off by
radiation, and convection will be set up which will last well
through the night, warmer water continually welling up to replace
the cooler heavier water which sinks. By this means the tempera-
ture throughout the upper layers becomes steadily reduced, and
the heat gained in the day is lost at night. During the day the
process of heating is mainly due to the radiant energy of the sun
being absorbed by the water near the surface, aided by mechanical
mixing of the layers of water. At night the process of convection
tends to bring to the surface all the water comprised within a
layer whose depth will depend upon the temperature reached
during the day, the rate of cooling of the surface during the
night, and the viscosity of the water. The depth to which

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1 84 Proceedings of Roycd Society of Edinburgh. [^

solar radiation penetrates in the waters of the Mediterranean
does not exceed 20 metres, and the accumulation of heat within
this layer during the sunshine of a September day may be
estimated at 450 calories per square centimetre of surface, or
about two-thirds of the available radiant energy incident on the
surface.

These, broadly speaking, are the conclusions to which a study of
the Pola observations seems to lead. But it is obvious that a
much more valuable set of data would be obtained by the use of
several platinum thermometers permanently fixed in mid-ocean at
convenient depths, and read at fairly frequent intervals through-
out the day and night, under different atmospheric conditions as
regards cloudiness and wind.

{Isiiied separately April 4, 1904.)

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1908-4.] Lord Kelvin on Two^imemional Waves, 185

On Deep-water Two-dimensional Waves produced by
any given Initiating Disturbance. By Lord Kelvin.

(Bead February 1, 1904. MS. receiTed February 18, 1904.)

§ 1. Consider frictionless water in a straight canal, infinitely long
and infinitely deep, with vertical sides. Let it be disturbed from
rest by any change of pressure on the surface, uniform in every
line perpendicular to the plane sides, and left to itself under
constant air pressure. It is required to find the displacement and
velocity of every particle of the water at any future time. Our
initial condition will be fully specified by a given normal com-
ponent velocity, and a normal component displacement, at every
part of the surface.

§ 2. Taking O, any point at a distance h above the undisturbed
water level, draw O X parallel to the length of the canal, and O Z
vertically downwards. Let ^, ^ be the displacement -components
of any particle of the water whose undisturbed position is (a, z).
We suppose the disturbance infinitesimal ; by which we mean
that the change of distance between any two particles of water is
infinitely small in comparison with their undisturbed distance ;
and the line joining them experiences changes of direction which
are infinitely small in comparison with the radian. Water being
assumed frictionless, its motion, started primarily from rest by
pressure applied to the free surface, is essentially irrotational.
Hence we have

^=^*(«.M); {^^(-.M); ^=^^^; ^4/ • (D;

where tf>(x, z, t), or <^ as we may write it for brevity when con-
venient, is a function of the variables which may be called the
displacement-potential ; and ^{x, 2, t) is what is commonly called
the velocity-potential. Thus a knowledge of the function <f>,
for all values of x, z, t, completely defines the displacement
and the velocity of the fluid. And, by the fundamentals of
hydrokinetics, a knowledge of <f> for every point of the free

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186 Proceedings of Royal Society of Edinburgh. [sns.

surface suffices to determine its value throughout the water ; in
virtue of the equation

dx^ dz^

The motion being infinitesimal, and the density being taken as
unity, another application of the fundamental hydrokinetics shows
that, as found by Cauchy and Poisson,

^.n = ,(.-/. + £)-|t = ,(.-.)./|-^ . (3);

where g denotes gravity ; n the uniform atmospheric pressure on
the free surface ; and p the pressure at the point (a:, z + {) within
the fluid.

§ S. To apply (3) to the wave-surface, put in it, « = ^ ; it gives

«(SL-(^).-. <')^

and therefore if we coidd find a solution of this equation for all
values of 2, with (2) satisfied, we shoidd have a solution of our
present problem. Now we can find such a solution ; by a curi-
ously altered application of Fourier's celebrated solution

r-" dv d^v 1

« + .)-..-- for ;^ = *^.J

his equation for the linear conduction of heat. Change t + CyXj kj
into z + tXyt, g~^ respectively : — we have (4), and we see that a
solution of it is

7(huf^' <^)'

which also satisfies (2) because any function of z + la; satisfies (2)
if I denotes J -l. Hence if {RS} denotes a realisation* by
taking half sum of what is written after it with ± t, we have, as
a real solution of (4) for our problem

^^x,z,t)={RH} j^-^^e*'-^^ .... (6).

* A very easy way of effeotiiig the realisations in (6) and (9) is by aid of
De Moivre's theorem with, for one angle concerned in it, x^tan-^x/a ; and
another angle = ^^/4(i? + a^.

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1903-4.] Lord Kelvin on Tivo-diviensional Waves. 187

where p^ = z^ + x^

('),

(8).

where 0 = tan"

The sign of »J{p - z) changes when x passes through zero.

Going back now to (5), and denoting by {RD} the difference of
ite values for ± i divided by 2t, we have another solution of our
problem essentially different from (6), as follows

,i>(x.z,t)=m) J^^^e^> . . . (9).
= ^[70> + .)^^^-V0>-)cosf^].^ (10).

^^Un(^^o-^y^ (^»>-

§ 4. The annexed diagram, fig. 1, represents for ^ = 0 the solu-
tions 2^ and i<f> as functions of x, with z = 1 for convenience in
the drawing. The formulas which we find by taking ^ = 0
in (7) X J2 and (10) x J2 are

Before passing to the practical interpretation of our solutions,
remark first that (12) contain full specifications of two distinct
initiating disturbances; in each of which <f> may be taken as a
displacement-potential, or as a velocity-potential, or as a horizontal
displacement-component or velocity, or as a vertical displacement-
component or velocity. Thus we have really preparation for six dif-
ferent cases of motion, of which we shall choose one, - {= ^2 x (7),
for detailed examination.

§ 5. Taking a = /i = 1, for the water surface, let the two curves of
figure 1 represent initial displacements^ (12), of the water surface,
left to itself with the water everywhere at rest. The displacements
at any subsequent time t are expressed in real symbols by (7) (10)
without the divisor ^2, and by (8) (11) with a factor J2 intro-
duced ; either of which may be chosen according to convenience
in calculation. One set has thus been calculated from (8), with

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188 Proceediiiys of Royal Society of Edinburgh. [•■

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1908-4.] Lord Kelvin on Two-dimensional Waves. 189

^ - 4, and «» 1, for six values of t ; '6, 1*5, 2, 2*5, and 6 ; and for
a sufficiently large number of values of a; to represent the results
by the curves shown in figs. 2 and 3. Except for the time < = 5,
each curve shows sufficiently all the most interesting characteristics
of the figure of the water at the corresponding time. The curve
for t = 5 does not perceptibly leave the zero line at distances
x<\'S ; but if we could see it, it would show us two and a half
wavelets possessing very interesting characteristics; shown in
the table of numbers, § 7 below, by which we see that several
different curves with scales of ordinates magnified from one to
one thousand, and to one million, and to ten thousand million,
would be needed to exhibit them graphically.

§ 6. Looking to the curves for < = 0 and < = ^ ; we see that at
first the water rises at all distances from the middle of the
disturbance greater than x « 1*9, and falls at less distances. And
we see that the middle (x = 0) remains a crest (or positive maximum)
till a very short time before < = J, when it begins to be a hollow.
A crest then comes into existence beside it and begins to travel
outwards. On the third curve, <= 1, we see this crest, travelled
to a distance a:=l'7, from the middle where it came into being;
and on the fourth, fifth, sixth, seventh curves (figs. 1, 2) we
see it got to distances 2*9, 4*8, 6*5, 22, at the times 1^, 2, 2^,
5. This crest travelling rightwards on our diagrams has its
anterior slope very gradual down to the undisturbed level at
X = 00 . Its posterior slope is much steeper ; and ends at the bottom
of the hollow in the middle of the disturbance, at times from ^ = |
to ^=1^. At some time, which must be very soon after ^=1 J,
this hollow begins to travel rightwards from the middle, followed
by a fresh crest shed off from the middle. At t-2, the hollow
has got as far as « = '9 ; at ^ = 2 J, and 5, respectively, it has reached
z = 1*75. and x = 6'7. Looking in imagination to the extension of
our curves leftwards from the middle of the diagram, we find an
exact counterpart of what we have been examining on the right.
Thus we see an initial elevation, symmetrical on the two sides
of a convex crest, of height 1*41 above the undisturbed level,
sinking in the middle and rising on the two fianks. The crest
becomes less and less convex till it gets down to height Tl, when
it becomes concave; and two equal and similar wave -crests

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190 Proceedings of Royal Sooieiy of Edinburgh. [ssba.

are shed off on the two sides, travelling away from it rightwaids
and leftwards with accelerated velocities, each remaining for ever
convex. Thus we see the beginnings of two endless processions of
waves travelling outwards in the two directions; originating as
infinitesimal wavelets shed off on the two sides of the middle line.
Each crest and each hollow travels with increasing velocity. Each
wave-length, from crest to crest, or from hollow to hollow, becomes
longer and longer as it advances outwards ; all this according to
law fully expressed in (8) of § 3 above.

§ 7. Here is now the table of numbers promised in § 5 above ; it
practically defines the forms and magnitudes of the two and a half
wavelets, between a* = 0 and a; = 2, which the space-curve f or / = 5
(figs. 2 and 3) fails to show.

p2 = aj2 + /ia; h^X, g = i', < = 6; ~£= V- sin (^Vd)c"^.

Col.l.

Col. 2.

Col. 8. ! Col. 4.

Col. 6.

Col. 7.

^/^

1
II

0||8

1^0000

•sis ***

14142

+ 10-10-1963

1-4142

10000

10-10-1357

•06 ' 1-4140

•9997

1-4140

•3434

„ -1478

„ » 0717

•064

•10 ; 1-410

•9987

1-409

- -7641

n -1778

-lO-w-1891

•16

1-407

•9972

1-403

- -8997

„ -3066

,. M -8882

•20

1-401

9962

1-393

- 0032

„ •3682

M ,. 0016

•202

•30

1-884

•9894

1^370

•8997

.. 1-094

+ 10-W1-862

•868

...

•40

1^862

•9820

VZU

- -6461

„ "4-866

- 10-W3-243

•60

1-309

•9688

1-262

- -2341

„ 108-9

„ .. 8r84

•682

...

..•

•80

r249

•9437

1179

•7598

10-5 -02896

+ 10 « -0227

1-00

1190

•i^239

1-099

•8962

., -2958

., „ -8162

1*26

rii8

•9015 1-007

■6831

„ 5-798

„ M 4-424

1^50

1-063

•8817 -9287

•4923

., 46-63

,. „ 23^67

1^517

1-76

•9961

•8661

•8616

- •6832

„ 212-5

-10-5 144^6

2-00

•94.56

■8606

•8043

- -9997

„ 848-2

„ „ 801-9

2-60

•8612

•8243

•7142

- 1688

•08180

., n 447-3

2-64

8 00

•7952

•81 13

•6450

•8296

•08210

•0642

8-50

•7411

•7980

•5917

•9473

•1616 ^1064

CorUimied on p, 198.

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1908-4.] Lord Kelvin on TwO'dimensional Waves. 1 91

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192 Proceedings of ItoycU Society of Edinburgh. [i

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MODEL INDEX.

Schafer, E. A.— On the Existence within the Liver Cells of Channels which can
be directly injected from the Blood-vessels. Proc. Koy. Soc. Ediu., vol. ,
1902, pp.
Cells, Liver, — Intra-cellular Canaliculi in.

E. A. Schafer. Proc. Roy. Soc. Edui., vol. , 1902, pp.
Liver, — Injection within Cells of.

E. A. Schafer. Proc. Roy. Soc. Edin., vol. 1902, pp.

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CONTENTS.

PACK '

Ocean Temperatures and Solar Eadiation. By Professor

C. G. Knott, ...... 173

{Issued separately April 4, 1904.)

On Deep-water Two-dimensional Waves produced by any

given Initiating Disturbance. By Lobd Kslvik, . 185
{Issued separaUly April 4, 1904, )

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PROCEEDINGS

OF THB

ROYAL SOCIETY OF EDINBURGH.

SESSION 1903-4.

No.ra.] VOL. XXV. [Pp. 103-272.

contp:nts.

Some Field Evidence relating to the Modes of Occurrence
of Intrusive Rocks, with some Remarks upon the Origin
of Eruptive Rocks in general. By J. G. Qoodchild,
of the Geological Survey, F.G.S., F.Z.S., Curator of
the Collection of Scottish Mineralogy in the P^din-
burgh Museum of Science and Art. Communicated
by R. H. Traquair, LL.D., M.D., F.R.S.,
{Issued separately May 20, 1904. )

Note on the Standard of Relative Viscosity, and on ** Nega-
tive Viscosity." By W. W. Tayu)R, >r.A., D.Sc.
Communicated by Professor Crum Brown,
{Iss tied separately June 16, 1904.)

Tlie Viscosity of Aqueous Solutions of ('hloridei*, Bromides,
and Iodides. By \V. W. Taylor, M.A., D.Sc, and
Clerk Rankbn, B.Sc. Communicated by Professor
Crum Brown, ......

{Issued separately June 16, 1904.)

PAGE

197

227

231

[Continued on page iv of Cover,

\A\p

'EDINBURGH:

PuBUSHBD BY ROBERT GRANT k SON, 107 Princes Stbeet, and
WILLIAMS & NORGATE, 14 Henrietta Street, Covent Garden, London.

Price Three Shillin/js.

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All proofs must, if possible, be returned within one week, addressed to
TJie Secretary^ Royal Society^ Mound, JSdinbargh, and Tiot to the printer.

[Continued on page iii of Cover,

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I908-4.] Lord Kelvin on Two-dimensional Waves.
fc-1; j,-4;< = 6; -{= ^? sin (^ + «)«?.

193

Col.L

Col. 2.

CoLS.

1 Col. 4.

•5490

1 Col. 5.

Col. 6.

1 Col. 7.

€P^"-

i|ir

4-0

•6965

•7882

•4866

•2298

•07771

4-41

...

4-6

•6588

•7798

•6189

- ^0944

•3088

-•01917

6 0

•6262

•7788

•4848

- 6684

•8823

- 1386

5-6

, •5981

•76-78

•4592

- -8457

'4498

-•2273

1 -5783

•7629

•4875

- -9781

•6122

- -2872

6-5

i -5518

•7587

•4185

- •9956

•5641

- ^3096

7-0

•5318

-7556

•4018

- -9374

-6066

- ^3024

7-5

•5150

•7522

•3868

-•8383

•6462

- 2778

8 0

•4980

-7494

•8734

- •7053

•6808

- -2892

9-0

•4699

•7461

•8601

- ^4289

•7872

- -1486

1 -4461

•7416

•3308

- -1679

•6846

- -04768

10-62

, -4255

•7885

•3U2

•05698

•81*47

•01975

1 -4076

•7369

•2999

•2428

•8416

•08876

•8916

•7889

-2874

•8940

•8644

•1834

•8775

•7318

•2762

•5176

•8808

•1721

•3648

•7302

•2663

•6168

•8954

•2018

•3683

•7286

•2574

•6953

•9082

•2281

•3331

•7266

•2419

•8098

•9260

•2498

•8160

•7256

•2290

•8881

•9396

•2622

•3014

•7230

•2179

•9818

•9497

•2666

•2885

•7216

•2082

•9627

-9679

•2661

•2772

•7206

•1997

•9816

•9638

•2622

•2672

•7193

•1923

•9916

•9685

•2566

•2681

•7187

•1856

•9979

•9727

•2505

•2500

•7181

•1795

•9999

•9759

•2489

•2425

7173

•1740

•9993

•9786

•2371

•2294

•7163

•1648

•9938

•9828

•2240

•2182

•7155

•1661

•9840

•9847

•2188

•2084

•7147

•1490

•9734

•9883

•2005

•2000

•7141

•1429

•9623

•9902

•1905

•1906

•7136

•1360

•9486 ,

•9917

•1794

•1826

•7129

•1802

•9361

•9931

•1697

•1690

•7120

•1208

•9125

•9949

•1535

•1581

•7114

•1125

•8931

•9961

•1407

100

•1415

•7108

•1005

•8626

•9977

•1217

•7071

roooo

§ 8. Look at the values shown in the previous table for the
three factors which constitute £; — we see that the first factor
(col. 2) decreases slowly from a?«0 to a-^oo ; the second factor
(col. 5) alternates between + 1 and - 1 with increasing distances
(semi- wave-lengths) from zero to zero as x increases. The third

PBGC. ROY. SOC. EDIN. — VOL. XXV. 13

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1 94 Proceedings of Royal Society of Edinburgh, [i

factor (col. 6) increases gradually from c "**/** at a; = 0, to 1 at
a; = 00 . At a; = 507i, the third factor is '99, which is so nearly
unity that the diminution of amplitude is, for all greater values of
ar, practically given by the first factor alone, Avhich diminishes
from '2 at or = 50/i, to 0 at ar = oo .

§ 9. The diagrams hitherto given, figs. 1, 2, 3, may be called
space-curves, as on each of them abscissas represent distance from
the centre of the disturbance. Fig. 4 is a time-curve (abscissas
representing time) for x = 27i. It represents a very gradual rise,
from < = 0 to ^= *G, followed by a fall to a minimum at < = 2 "8, and
a succession of alternations, with smaller and smaller maximum
elevations and depressions, and shorter and shorter times from
zero to zero, on to / = oo . The same words with altered figures
describe the changes of water level at any fixed position farther
from the centre of disturbance than a; = 2. The following table
shows, for the case a:=100/i, all the times of zero less than 717/,
and the elevations and depressions at the intermediate times when
the second factor (col. 5 of § 7) has its maximum and minimum
values (±1). These elevations and depressions are very approxi-
mately the greatest in the intervals between the zeros, because the
third factor (col. 6, § 7) varies but slowly, as shown in the first
column of the present table.

7i=l; a;=100; p= 100-005.7i; ^ = <a»-^^^^J = 45' 18'.

Times of Zero

-f2

and of

Approximate
Maximum

-ta

and of

Approximate
Maximum

€>

Approximate
Maximum

€P«

Approximate
Maximum

Elevations and

ElevaUous and

Elevation and

Depressions.

Elevation and

Depressions.

•9922

Depression.

•7718

Depression.
50^90

+ -1091

8^383

+ -1403

...

15-33

62-42

•9616

19-80

- -1360

-7478

68-90

- -1058 1

...

23-43

55-34

•9817

26-67

+ -1317

-7247

6674

+ 1025 1

29-38

58-10

•9031

81-94

- -1277

•7023

59-45

- 0998

...

34-31

...

60-75

•8750

86-54

+ •1237

•6806

62-03

+ 0962

...

S8-62

63-29

•8480

40-61

-•1199

-6696

64-51

- -0988

...

42-60

65-72

•8219

44-31

+ •1162

•6392

66-90

+ -0904

...

46 04

68-07

•7964

47-72

-•1167

•6195

69-21

- -0876

...

49-34

70-34

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1908-4.J Lord Kelvin on Two-dimeneional Waves.

195

CO >Q

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1 96 Proceedings of Royal Society of JSdinbtirgh. [sicss.

§ 10. Our assumption A « 1 for the free surface involves no
restriction of our solution to a particular case of the general
formula (7). Our assumption g*^i merely means that our unit
of abscissas is half the space fallen through in our unit of time.
The fundamental formulas of § 3 may be geometrically explained
by, as in § 2, taking 0, our origin of co-ordinates, at a height k
above the water level, and defining p as the distance of any
particle of the fluid from it. When, as in §§ 6-9, we are only
concerned with particles in the free surface (that is to say when
z = h)f we see that if ar is a large multiple of 2, /»%«. See for
example the heading of the table of § 9. And if we are concerned
with particles below the surface, we still have p=x, if 2 is a
large multiple of z. Thus we have the following approximation
for (7) of § 3 :—

Suppose now d<l>/dt to represent £ (instead of <^, as in g 6-9) ;
we have

which is easily found from (13) without farther restrictive
suppositions. But if we suppose that z is negligibly small in com-
parison with z ; and farther that

S-^ 05).

we find by (14)

This, except the -sign - instead of -H, is Cauchy's solution;* of
which he says that when the time has advanced so much as to
violate a condition equivalent to (16), "le mouvement change
" avec la m^thode d'approximation." The remainder of his Note
XVI. (about 100 pages) is chiefly devoted to very elaborate efforts
to obtain definite results for the larger values of t. This object
is thoroughly attained by the exponential factor in (8) of §3
above, without the crippling restriction z/x'-O which vitiates (16)
for small values of a:.

• CEuvreSf vol. i. note xv'i. p. 193.

{Isstied separately April 4, 1904.)

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1908-4.] Mr J. G. (xoodchild on Intrusive Bocks. 197

Some Field Bvidenoe Belating to the Modes of Oocur-
rence of Intrusive Books, with some Bemarks upon
the Origin of Eruptive Books in General By J. Q.
Qoodohild, of the Geological Survey, F.G.S., F.Z.S.,
Curator of the Collection of Scottish Mineralogy in the
Edinburgh Museum of Science and Art. Communicated by
R. H. Traquair, LL.D., M.D., F.R.S.

(Read Dec. 6, 1P03 ; MS. reooived Jan. 6, 1904.)

SYNOPSIS.

1. Introduction, pp. 197-199. History of preyious opinion, pp. 199-202.
Eridenoe bearing upon the question whether intrusive rooks displace or
replace the rocks they invade, pp. 202-218. Basic sills in sandstones,
pp. 202-204 ; in shales, pp. 205-207 ; in limestones, p. 208; in coal seams,
pp. 208-210. Basic dykes in the same connection, pp. 211-212. Acid
intrusions, pp. 212-218. Anomalies in the mode of occurrence of dykes
discussed, pp. 218-217. Relation between dykes and sills, p. 217.
Evidence cited from other sources, pp. 217-218. Summary of the
author's conclusions, pp. 218-226.

It is commonly believed by geologists, as well as by coal miners,
that the inner faces of the rocks which enclose intrusive masses
were at one time in contact, and that each of these surfaces is the
counterpart in form to the other, from which it has been severed
by the forces to which the injection of the intrusive mass was due.
In the case of a sill, for example, this belief implies that the rock
floor below the sill and the roof above it were in imbroken
contact at some time before the sill was intruded, and that the
floor and the roof have been forced apart to a distance equal to
the thickness of the intrusive mass. In like manner, so it is
believed, the waUs right and left of a dyke are supposed to have
been thrust apart from their original position. In other words, it
is evidently the common belief that these intrusive rocks,
whatever their volume may be, have added that volume to the
rocks they invade. To put this statement into yet another form,
it is evidently believed that two seams of coal, or beds of black-
band, or of oil shale, which occur under normal conditions at ten

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198 Proceedhigs ofRoyai Society of Edinburgh, [t

fathoms apart, are thrust to twenty fathoms apart if there happens
to he ten fathoms of intrusive rock between them. A reference
to ahnost any treatise on geology in which this relationship
between intrusive masses and the "country rock" is discussed
will at once prove that the view referred to has evidently been
the one that the author had in mind.

Amongst colliery people, who have to deal with these questions
in a practical way, there has long been some difference of opinion
upon this point; some believing that trap rocks cut out the
measures. But as they are "only practical men," their opinion
upon a geological matter is apt to be ignored. Furthermore, as
will be evident from the sequel, many field geologists are now of
opinion that intrusive masses usually replace the rocks they
invade.

It is obviously a matter of considerable commercial import-
ance to test by field evidence whether the current view referred
to above is or is not the correct one. This is especially the case in
connection with the Scottish coal-iields, which are in many cases
"much troubled with whin," as the increasing demand for coal
is leading to the prospecting of parts of coal-fields which have
hitherto been left untouched, because the areas referred to have
been known to be afi'ected by intrusive masses. A little con-
sideration will suffice to show that the question is one of at least
equal interest to geologists, as it is one of wide- reaching import-
ance, and as, moreover, it raises many questions in both chemistry
and physics which are much more easily asked than answered.
One may indeed go farther than even that, for if it can be shown
that the current view is not in accordance with the facts, it is
obvious that our views on the origin of eruptive rocks in general
will have to be reconsidered, and we may even have to modify our
opinions on some matters relating to the succession of events
which took place in the earlier geological, or later astronomical,
periods of the Earth's history.

Fully realising, therefore, the importance of the issues about
to be raised, I shall endeavour, in the first part of this paper, to
keep rigidly to a statement of the facts which bear upon this
question, and then, after summarising the evidence, I shall go on
to point out the conclusions to which the study of these facts

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1908-4.] Mr J. G. Goodchild on Intrusive Bocks. 199

appears to lead. Id the latter part of the paper, while passing
additional facts in review, I shall venture to suhmit for the
consideration of field geologists * an hypothesis which appears to
me to be in full harmony with the facts.

The question whether intrusive rocks displace or replace the
rocks they invade has often been raised before. A brief notice of
two or three of the more important papers dealing with the
subject cannot be out of place, and accordingly they are given
here.

In 1852 or 1853 the late Prof. J. Beete Jukes wrote in the
Cfeologiccd Survey Memoir^ "On the Greology of the South
Staffordshire Coal-field," pp. 246-7, as follows : " I was assured
also by almost every one engaged in the works of this neighbour-
hood that, notwithstanding the variation in thickness of 'The
Green Bock' [a basic sill], there was no change in the total
thickness of the measures; that, for instance, the thickness
between the Xew Mine Coal and the Blue Flats Ironstone
remained the same, whatever might be the variation in the
thickness of *The Green Rock.' In other words, it was afl&rmed
almost universally that *The Green Rock* not only intruded
between the measures, but obliterated [the italics are the author's]
a mass of beds equal to its own thickness." Jukes then goes on
to express a doubt about the miner's conclusions ; nevertheless, on
the next page (247) he adds: "At Union Colliery, north of
[Walsall], the Bottom Coal is cut out entirely by * green rock.' "
I do not give the evidence cited by Jukes in support of his own
view, as the fact that he was informed of evidence of the trap
cutting out the coal is all that need be referred to here now.

There may have been other evidence published before that, or
since, of which I have at present no information. But, in 1867,
Mr Hughes (now the Woodwardian Professor of Geology at
Cambridge, wrote as follows in a review of Nicholson's "Essay
on the Geology of Cumberland and Westmorland," Geol. Mag,,
dec. L, vol. v., pp. 466-7 (1868) :—

" One point seems often to come out from a careful examination
of a granite mass. The granite seems to replace a certain portion

* The questions raised are of a petrographical as distingtiished from lith(h
logical character.

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200 Proceedings of Boyal Society of Edinburgh, [i

of the sedimentary strata, and not to displace them, leaving them
pushed out on all sides. If we suppose the intruded rock to eat
its way into the sedimentary strata, assimilating portions of it,
we allow a good deal of what is asked hy those who hold the
metamorphic origin of granite rocks, i.e., the possihility of changing
a sedimentary into a granitoid rock. The advocates of that theory
may take their stand upon the assimilated portion, and ask is it
the heat of the intruded mass, or the new conditions under which
the minerals have heen hrought into contact with the sedimentary
rocks, which has produced the change, and then point out that
both the one and the other may be obtained by a sufficient
depression of the sedimentary rocks" [the above italics are the
author's].

In a later reference, made in the Geological Survey Memoir
on 98 S.E., pp. 41-42, the same author repeats the statement
chiefly with reference, on this occasion, to the dykes of minette,
porphyrite, and quartz felsite which occur in the region described.
He adds the remark : " It may be worth consideration whether in
some cases it might not be possible that the action of gases or
of hot water holding minerals in solution, communicating along
lines of fissure with the joints, might produce the phenomena
observed."

As I happened to be working with the author at the time when
both of these remarks were penned, and had abundant opportunities,
then and on later occasions, of observing the facts upon which
his conclusions were based, I can confirm them in every particular.
Attention may be directed to the fact that no mention was made
of any lithological passage from that of the dyke to the country
rock. Nevertheless, in the discussions which followed the
publication of the above passages, only side issues were raised,
mainly on the ground that no evidence of a lithological passage
could be made out; and the statements of fact, thus apparently
discredited, were allowed to drop out of sight.

In 1879 Mr Clough of the Geological Survey took up the
matter again, in connection with the Whin Sill of Teesdale, and
read a paper before the Geological Society of London, in which
similar views were advanced, and supported by an excellent array
of facts and arguments. Again a side issue was raised, and the

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1903-4.] Mr J. G. Groodchild on Inti-usive Bocks. • 201

paper was not allowed to appear in the Quarterly Journal. But
in the Geological Magazine^ decade ii., vol. xii., pp. 434-447
(October 1880), the substance of that communication appeared
under the title of " The Whin SiU of Teesdale as an Assimilator
of the Surrounding Beds.'' Besides the materials collected in the
field by himself, Mr Clough was able to get corroborative evidence
in support of his views from Dr James Geikie, Dr Peach, myself,
and other of his then colleagues. Mr Clough was quite as fully
aware of the fact as any of his predecessors in the field that
though the dolerite in question replaces beds of very diverse
chemical composition, its own mineral constitution remained
uniform, and he was equally well aware that there is no trace of
any lithological passage from the country rock to the intruder, or
vice versa. To meet this very formidable chemical difficulty,
which still looms very large indeed in the eyes of cabinet geologists,
he wrote (p. 442), referring to objections likely to be raised on
these grounds: "But any force which this objection possesses
depends upon the assumption, that if sedimentary beds were taken
up by the Whin, they would remain in it close at hand in their
original situation, whereas there may have been a very general
circulation, both on a large scale and molecule by molecule,
reducing all the parts of the mixture to a general uniformity of
composition. The very possibility of forming alloys and of
modifying the properties of metals by adding to them small
portions of other substances depends upon this principle of
circulation or diffusion, so that it cannot be said that we are
without warrant for it."

I may add that the paper has always appeared to me to be
a very valuable one, and that I can adduce abundant corroborative
evidence in support of the author's statements of fact, partly from
a knowledge of the areas adjacent to Teesdale, where similar
phenomena are seen, and partly from an examination of the part
of Teesdale referred to, after the Geological Survey map of the
district was published.

Again, in Mr Clough's case, were the facts ignored or explained
away, apparently on no other ground than that it appeared very
unlikely that an extensive sheet of dolerite could, by any means,
eat up large volumes of sandstone without showing a higher silica

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202' ProceediTigs of Boyal Society of Fdiriburgh. [i

percentage than usual, or that it could assimilate thick beds of
limestone without the development of any additional lime silicates,
or that it could eat up shales without any perceptible increase in
alumina-silicates being evident in any part of the invading rock.

It must occur to any reasoning person, however, that the
FACTS, at least, either do exist as stated, or they do not. If they
do, then it is very illogical to close our eyes to them. It would be
much better to face those facts at once, and either to accept them
as such without attempting to explain how they came about, or
else to re-examine the evidence and endeavour to frame some
hypothesis which would harmonise what is known about them ;
or, at least, to think out some explanation which would serve for
the time being as a working hypothesis until a better on^ could
be suggested.

Bearing these considerations in mind, I have collected much
additional evidence which bears upon this controverted question.
Most of the facts have been obtained in the Lowlands of Scotland,
and I have aimed, as much as possible, at citing instances which
are either to be seen without difficulty in such easily-visited
localities as the Queen's Park, or else at other places withia a
short distance of Edinburgh. The behaviour of basic intrusive
rocks will be considered first, taking sills in the first place and
dykes next.

In view of the fact that many geologists think that mechanical
disturbance always accompanies the intrusion of eruptive masses,.
I have thought it well to give first an outline drawing (fig. 1)
taken from a photograph by Mr A. G. Stenhouse, F.G.S.^
of the well-known example in the quarry at the south end of
the foot of Salisbury Crags, which is the example illustrated in
Hay Cunningham's Fig. 3, Plate III., Mem. Wem. Soc,, vol, vii.
In this case a wedge of dolerite has been, so to speak, arrested
while in the act of forcing off a fragment of one of the beds of
Cornstone there. The section to the left of the wedge follows
the method of attack usual in such cases. Fig. 2, traced from a
photograph taken at Hound Point, Dalmeny, by the weU-known
vulcanologist Dr Tempest Anderson, shows a similar wedging off
of the country rock by the intrusive mass, which in this case is
also a dolerite. It may be remarked that within six feet of this

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190 '-4*.] Mr J. G. Goodchild on Intinmve Bocks, 203

wedge the dolerite is seen, as in the last case cited, to have eaten
its way into the rock, across joints and faults as well, without any
signs of disruption.

Fig. 3 shows the top of the dolerite in the old quarry at the
north end of Salisbury Crags. The dolerite in this case has made
its way upwards into the Cornstones there in a very irregular
manner, and has consequently left a downward extension or tongue
of sandstone (now altered into a quartzite) with the intrusive rocks
on either side of it. The figure, traced from a photograph by Mr
Fingland, of the Glasgow University, shows irregular tongues of
the dolerite in the sandstone, which have evidently made their
way there without causing the least mechanical disturbance. Two
or three cases are seen in this example in which the dolerite has
tunnelled into the sandstone, and has left an unbroken ring of the
sandstone around. At the bottom right-hand side are included
fragments * of the country rock still remaining undissolved within
the dolerite. The section at the foot of Salisbury Crags described
by Hay Cunningham {op. ciL\ and figured on Plate TV. of his
Greology of the LothianSy is one of very considerable interest in the
present connection. One aspect of it is represented on fig. 4,
traced from a photograph by Mr Stenhouse. It shows several
tongues of dolerite ending oflf against unbroken country rock
(Cornstones). With these finger-like processes there are several
protrusions of dolerite completely surrounded by the unbroken
sandstone. One example of this has been detached, and is now
exhibited in the Gallery of Scottish Greology and Mineralogy in
the Edinburgh Museum of Science and Art, along with other
examples to be referred to in detail presently. On the south side
of the Queen's Park alone nineteen cases of dolerite, either ending
off against unbroken rock, or else completely surrounded by it,
have already been noted, and there are probably many others
there, as well as more in other parts of the Park. In Hay
Cunningham's treatise (pp. cit, PI. III. fig. 1) is an example of
the same kind occurring at the base of the intrusive basalt of St
Leonard's Hill. Again, there are masses of sandstone caught up
in the curious dyke-like mass of dolerite which rises into the rock of
Salisbury Crags from below the Radical Road, near its western-
* Why should these be called Xenoliths ?

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204 Proceedings of Royal Society of Edinburgh. [s

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1903-4.] Mr J. G. Goodchild on Intrusive Bocks. 205

most extremity, and which has so often been likened to the stem
of the mushroom of which the Crag forms the cap. In this
sandstone there are several examples of the same nature. In
connection with the dolerite sills which give rise to the beautiful
scenery around Hawk Crag, Aberdour, there are many remarkable
and most instructive examples of the same kind. Some are to be
seen at the foot of the crag N.N.E. of the outer end of the stone
pier; but the best occur just above high-water mark on either
side of the base of the pier. The sedimentary rocks consist of
carbonaceous shales and sandstones belonging to some part
of the Oil Shale subdivision of the Lower Carboniferous Bocks.
The rocks on the north side of the pier base are chiefly sandstones.
The dolerite has tunnelled its way into these rocks in several
places, so that it now occurs in apparently isolated masses entirely
enclosed within sandstone. These arc shown in fig. 5, which
is from a photograph taken by Mr Steiihouse. One of these was
got out, and is now exhibited in the Collection above referred to.
At the roadside facing the south edge of the pier occurs a bank
of shale which is traversed by at least nine small sheets and
wedges of dolerite. In a generalised way this also was figured by
Hay Cunningham (op, cit, PL XIV., and here, drawn from a
photograph, in fig. 7). It is an excellent example of the manner
in which bands of dolerite interdigitate amongst the strata near
where rapid variations in the thickness of the intruder are taking
place, or near where it is dying out. Amongst these tongues or fingers
are several which end off abruptly against unbroken shale. One
of these, which is in the Edinburgh Museum, is shown in fig. 6 ;
wliile the irregular junction of the larger mass in the east side of
the harbour with the sandstone beneath, taken from one of Mr
Stenhouse's photographs, is shown in fig. 8. Fine examples of
this lateral passage by interdigitation of an intrusive mass into
the country rock may be observed also at the west face of The
Dasses, in the Queen's Park, about midway between The Washing
Green and The Piper's Boad. It is quite a common occurrence
for sheets of dolerite (and also sills of other kinds) to end ofi* by
interdigitation in this way. A good example is that presented by
the dolerite sill which forms Fair Head, on the coast of Antrim.
In Geikie's Aricient Volcanoes^ vol. ii., p. 304, fig. 317, is given a

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206 Proceedings of Royal Society of Ediriburgh. [i

section at Farragandoo Cliff, at the west end of Fair Head, which
shows this indigitation of dolerite with the country rock in a
manner which is thoroughly typical of the behaviour of sills in
that respect. It will be observed that there is little evidence, if
any, of mechanical disturbance. On the contrary the whole mass
of field evidence seems to point to the intrusive rock having taken
the place of the shale, without causing any uplift of the rock
surfaces which are supposed by some writers to have been thus
laccolitised. The relationship of the one rock to the other is
certainly not of the kiml that might be illustrated by thrusting
one's fingers between the leaves of an otherwise closed book lying
upon its side. The separation of the leaves forming the upper
half of the book from these forming the other would in such a
■case bear an exact proportion to the size of the fingei^s thrust in ;
and there must in all such cases be a certain amount of curvature
of the upper part, which occasions some lateral movement of the
■ends of the separated parts relative to their position before the
" intrusion." This shortening, supposing the uplift to take place on
one side only, would be proportionate at either end of the uplifted
part to half the difference between the length of the arc formed by
the lifted portion and half the length of the portion undisturbed.
In a rock thus acted upon the adjustment to the changed lateral
dimensions must occasion some mechanical disturbance. Traces of
such I have never met with. In actual examples the case is rather of
that kind which might happen if part of the leaves, corresponding
in shape and in volume to those of the fiugers thrust in, had been
<jut out. In the former case, supposing we are dealing with a
<jlosed book lying on its side, the outer cover would be lifted ; if
the case were of the latter kind, the cover might remain quite
undisturbed while the fingers were pushed in. I shall adduce some
further evidence in support of the view that the case last
illustrated is the usual one; though there may well be some
occasional exceptions to it. Fig. 9, on page 207, shows a well-
known case of intrusion at Dodhead Quarry, near Burntisland Golf
Course. Fig. 10, from the same quarry, is traced from a photograph
taken by Professor Reynolds of Bristol, in which yet another
example occurs of an intrusive rock eating its way into shales, which
remain undisturbed above and below. Its position is shown by a B

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im-i,] Mr J. G. Goodchild an Iivtrimve S^ycks, 207

on fig. 9. In this, as in the other cases cited, it is perfectly evident
that the intrusive mass has not added its volume to that of the

Fio. 9.— Eastern face of Dodhead Quarry, within the Golf Links,
Burntisland, Fife.

The sedimentary rocks here are mostly sandstones and shales, more
or less carbonaceous in character Thoy belong to some part of
either the Oil Shale Series or to the subdivision of the Lower
Ciirboniferous Rocks yet below that. In the lower part of the quarry
occurs a thin band of a more calcareous type, which might be regarded
as a finely-laminated shaly limestone. It is shown in the section by
vertical ruling. Two or three sills of basic rock have been intruded
into the sedimentary rocks hereabouts, and one of these, altered by
the carbonaceous matter into '* White Trap,*' traverses the quarry
from the present section northwards, maintaining throughout nearly
the same thickness, and keeping to nearly one horizon. In Dodhead
Qaarry the ''trap*' begins to thicken, thin, die out, and reappear, in
a very irregular manner, as shown by the figure, which has been
carefully drawn in the quarry from a series of photographs taken with
the express object of showing the phenomena in every possible aspect,
and checked on the ground by actual measurement.

It will be noticed that the distance between the limestone and the base
of the sandstone remains the same at either end of the section, alike
where the trap is present and where it is wanting. The evidence of
replacement of the country rock by the *' trap'* is quite clear. There is
also quite clear evidence of local displacement below the trap. This
phenomenon sometimes occurs in the cases in which there are two sills
present (as there are in the present case, the second occurring a little below).

It is presumed that the forcible injection of the magma forming the
sill displaced part of the magma forming the dyke-like extension of the
mass and rui>tured the sediments in the manner shown. The patch
marked.!) is separately represented in fig. 10.

The section embodies examples of nearly all of the phenomena which
usually accompany the intrusion of eruptive masses, and hence it has
been selected as a typical section.

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208 Proceedings of Royal Society of EdviH»irgh, [sios.

country rock around it The volume remains just the same,
whether the intruder is present or not ; just as the Staffordshire
coal miners told Jukes was the case in their district. Evidently
the older rock has been gradually removed by some means, and
the newer one just as gradually introduced into its place.

Mr Clough cited some cases in which limestone had been eaten
out when the Whin Sill was being intruded. I can corroborate
his statements from my own observations along the Cross Fell
Escarpment, which I mapped in connection with the Geological
Survey of that district. Quite recently the Berwickshire Natural-
ists' Club paid a visit to Dunstanburgh Castle, on the coast of
Northumberland, where the Whin Sill occurs in the upper third
of the Yoredale Rocks. In Queen Margaret's Cove, at that place,
a mass of sandstone, capped by limestone, has been caught up in
the lower part of the dolerite, and in the caught-up portion
several protrusions of the Whin Sill into the limestone are clearly
shown, some of which are surrounded by limestone in an un-
broken condition, just as occurs in the sandstones and shales
already mentioned.

Turning for the occasion to the evidence afforded by an
intrusive mass of dolerite from a foreign locality, it may be men-
tioned that Mr Walcot Gibson of the Geological Survey of Great
Britain has a photograph which shows the very uneven upper
surface of a bed of dolerite which has been intruded into sand-
stones. This photograph has been traced, and is reproduced in
outline in fig. 12. It will be observed that in this instance again
there is absolutely no evidence of the beds above the dolerite
being lifted, or " laccolitised," so that their dip conforms to the
surface of the sandstone. On the contrary it is quite evident
that one of two things has happened in this case : either the sand-
stone has been deposited after the dolerite, or eke the latter has
eaten its way into the sandstone. As there is abundant evidence
of contact metamorphism in the rock in the marginal zone next
the dolerite, the alternative explanation may be at once dismissed
from further consideration.

Passing now to notice cases in which the basic intrusive
mass comes into contact with coal seams, beds of oil shale, of
blackband ironstone, or other carbonaceous rocks, it may be men-

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1908-4.] Mr J. G, Goodchild on Inttnmve Hacks.

209

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PBGC. ROY. SOC. EDIN. — VOL XXV. 14

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210 Proceedings of Royal Society of Ediiiburgh, [sess

tioned that, when this paper was read, Mr Cadell cited a case in his
own collieries at Bo'ness. A bed of dolerite one foot in thickness
had been intruded into a three-foot coal seam, and it left one
foot of coal above and another foot below : one foot of coal had
disappeared and one foot of dolerite had taken its place ; the
upper surface of the seam remainmg three feet above the lower,
just as if no dolerite were present. Mr John Smith of Kilwin-
ning, amongst other practical men, has furnished me \idth a
similar instance which occurs in a quarry 350 yards N.K of
Dykeneuk farmhouse. Fig. 1 3 is an outline taken from Mr Smith's
sketch sent to me. It may be added that my colleagues Mr Grant
Wilson, Dr Peach, and others have assured me that these are
typical cases. Mr Dron, the author of an important work on the
Scottish Coal-fields, has mentioned other cases. I would specially
mention the cases illustrated by figs. 24 and 25 in the Survey
Memoir on the Geology of Central and Western Fife.

I-Astly, a reference may be made to two of many cases that might
be cited in which a dolerite sill invades schistose rocks. Fig. 14
is traced from a photograph by Dr Bernard Stracey, F.G.8., and
is from near Beinn ladain, Morven. It shows well the abrupt
termination of the sill against quite unbroken schist. The other,
fig. 15, is from Torr na Sealga, Ross of Mull, from a photograph
by Mr David Russell of Markinch, and a drawing made on the
spot by myself.

We may now consider a few cases in which the relationship
of DYKES to the country rock can be made out. The current
belief in regard to these certainly is clearly enough expressed in
nearly all treatises on the subject. The relationship implied in
these statements may be well illustrated by taking a row of books,
placed on edge and side by side, to represent the country rock,
and then by intercalating other books here and there between
them. This illustration makes it clear that there must be a
lateral shift corresponding in amount to the aggregate width of
the volumes intercalated. If a small book happens to be thrust
between the leaves of a large one in the row the pages are sepa-
rated from each other to an extent determined by the size of the
smaller book in question, just as was illustrated by the " intrusion "

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1903-4.] Mr J. G. Goodchild on Intrimve Rocks. 211

of one's fiugers into a book, referred to above in connection with
sills. References to the letterpress of almost any text-books on
Oeology will suffice to show that this relationship is what the
authors had in mind when they wrote. Strangely enough the
Jigures of dykes in these books are usually drawn in accordance
with the facts, just as figures are which relate to sills or to other
forms of intrusive rocks.

Out of a large number of cases a few will suffice to show that
■dykes generally replace their own volume of the rocks they invade.
This is the case, just as it is with sills, quite irrespective of either
the lithological character or the structure of either the intruder
or the country rock. Fig. 18 is traced from a photograph show-
ing the upward termination of a Tertiary basalt dyke in New Red
Sandstone, near the Borough Cemetery at Belfast, and figs. 16 and
17 other dykes traversing Chalk at Whitewell Quarry, Belfast.
These show an entire want of correspondence between the opposite
walls of the country rock, such as could not have occurred had
the dykes filled simple rents. For both of these I am indebted to
Miss Andrews. Fig. 1 1 is taken from a photograph by Mr Voge,
showing the upward termination of a similar dyke in Chalk
at the White Rocks, near Portrush. The rounded patch seen
above the end of the dyke is probably the continuation of the
same dyke, which has bent in its upward course, so that it passes
behind the face of the cliff for a short distance. Fig. 19 shows
a tertiary basalt dyke, which ends oflf abruptly in a remarkable
melange of (Devonian) granite and Highland Schist at Torr na
Sealga, in the Ross of Mull, already referred to. This locality will
be referred to presently in another connection. Again, in the cliffs
formed by the basalt lavas of Skye and Mull, many fine examples
of the same kind are clearly laid open to view. This is especially
the case in the grand range of precipices forming the cliff below
Beinn an Aonidh, on the* south shore of Mull, west of Carsaig.
There may be seen dykes and sills of basic rocks which zigzag
their way up the face of the cliff through the various beds of lava
without producing the least disturbance of these volcanic rocks,
and without adding their own thickness to that of the pile in
which they occur. Fig. 20 shows some intrusions at Carsaig
Arches, sketched from the sea.

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212 Proceedings of Boyal Society of Edinburgh, [sess.

Fig. 21 is traced from a photograph by Miss M. K. Andrews of
Belfast at a quarry in the Upper New Red Sandstone of Scrabo-
Hill, County Down, in which some dolerite sills of Tertiary age
traverse the sandstone without the intrusion being accompanied
by the slightest evidence of any mechanical disturbance, or of any
" laccolitisation " of the overlying strata. The sills are^ traversed
by a later dyke, as shown.

Basic dykes and sills have been considered first in relation
to the country rock because they are of more common occurrence.
But it can easily be shown that precisely the same inter-relation
exists also in the cases in which rocks of a more acid type are
concerned. There is only one acid intrusion of any size near
Edinburgh, which is that of the microgranite of Black Hill in
the Pentland area. This, geologically, is an intrusive mass of
Devonian age, which appears to represent a subterranean mass of
the more acid type of rock whose lavas form the trachytes of the
Caledonian Old Red Volcanic Series of the Pentlands. It ha&
evidently been formed at a late period in the history of the
Pentland volcanoes, and has been intruded into, amongst other
rocks, the conglomerate which lies at the base of the volcanic
series. Close to Logan Lee Waterfall its relation to the con-
glomerate can be easily examined. At several places its upper
surface has welded itself to the old gravel which forms the con-
glomerate referred to, and the union has been so firm that many
patches of the conglomerate may be observed still adhering to the
face of the granitic rock. At the foot of Logan Lee Waterfall
the conglomerate is much hardened, and veins and protrusions of
the microgranite traverse it in exactly the same manner as in the
cases of the basic intrusions already described. The veins are
not easily photographed, though they are readily seen on the
ground. But the relationship between the one rock and the
other may be seen to be of exactly the same kind as that so
well illustrated by Mr Griffith Williams' beautiful photograph in
the Brit. Assoc. Series (G. J. W. 603), of the case which occurs
at Tan y Grisiau, in North Wales. Mr Williams kindly
outlined the granite protrusions upon a print of the photograph
and sent it to me, and a tracing made over these lines is given
here on fig. 23. Field geologists must be fully aware that the

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1908-4.] Mr J. 6. Goodchild on IrUnrnve Bocks. 213

aase cited is a perfectly typical one so far as the relation of veins
-of granite to the country rock is concerned. There is not the
slightest evidence of any disruption of the rock invaded by the
granite ; but, on the contrary, it is perfectly clear that there has
been, in these cases also, a concurrent removal of the country
rock going on while the introduction of the material that after-
wards consolidated as granite was in progress. But before passing
on to consider in more detail the mode- of attack followed by these
acid intrusive rocks, I may perhaps be permitted to repeat the
statement that the acid and subacid dykes (of Devonian age) which
traverse the Ordovician and Silurian Rocks of the Kendal and
Sedbergh districts, referred to -at the commencement of this paper,
behave in precisely the same manner as the granite veins just cited.
The lamprophyre occurring at Swindale Beck, Knock, near
Appleby, which was figured in Teall's British Petrography as a
typical minette, certainly eats its way into the country rock
in the manner already described in so many other cases. I
have figured it in plan in the Geological Survey Memoir on
Sheet 102 S.W., to which the reader may be referred.

Lastly, so far as the mode of occurrence of dykes is concerned,
the well-known pitchstoue of Corriegills Shore, on the east coast
of Arran, sends finger-like ramifications into the enclosing rock,
some of which are clearly seen to terminate against the Bunter
Sandstone around it in the manner already described in connection
with the dykes of basalt. One specimen showing this mode of
•occurrence of the pitchstone is exhibited in the Scottish Collection
■already referred to.

Leaving this part of the subject for the present, it may be
remarked here that there are some singular features about basic
•dykes in general which may be noticed in the present connection.
These are (1) the very small proportion which their width bears to
their length (and usually to their depth) ; (2) their wonderful uni-
formity of composition as a whole, which they maintain throughout
the whole of their extent ; (3) the remarkable parallelism of their
-enclosing walls as a rule ; (4) the fact that the dykes most extensive
in their range are those in which lime-soda felspars predominate.
Furthermore, the mode of occurrence of a basic dyke suggests
that) the attacking surface formed by its magma was limited to its

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214 Proceedings of Eoyal Society of Edinburgh. [i

U»itt-ft»aa. G<<UiXi>aM<Su>ag. ^40

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1903-4.] Mr J. G, Groodchild on Intrusive Bocks.

215

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Fig. 26.

Fig. 26 bos been drawn up so as to afford a conspectus of the proportions in
which the Essential Minerals of the Eruptive Rocks occur in any one of
the sections into which the whole lithological series can be divided. For
example, taking the second band, the proportions in which the plagiodase
felspars occur relatively to the ferro-magneeian silicates in any one of either
the sub-basic or the basic eruptive rocks, can be estimated by comparing
the distance above the thick curved line traversing the middle with that
below, measured at any point along a line perpendicular to the base of
the diagram. The same method can be employed in the case of any
other of the subdivisions of the series.

The principle of arrangement followed is based, primarily, upon the
percentage of silica present — the rocks containing highest percentage
being represented at the top lert-hand, and those with the lowest at
the bottom right ; and, secondarily, with reference to the nature of the
dominant alkali, or alkaline earth, which characterises each of the
compounds.

The classes of rocks formed of these components may be grouped under
three primary categories, to each of which one subdivision of the diagram
is devoted. At the top are represented the Mineral Combinations arising
from the action of a Potash Magma upon other rocks in which the
dominant alkali is Soda. The middle of the diagram includes those
which are here regarded as due to the action of a Soda- Lime Magma
upon sedimentary rocks. The lowest subdivision is intended to represent
the products of consolidation of a Ferro-magnesian Magma. Further
subdivisions, which are sometimes convenient for use, are made in
accordance with the dominant substance, and are as follows: rocks
characterised by minerals containing Potash, Potash-Soda, Soda, Soda-
Lime, Lime-Soda, Lime, Lime-Magnesia, Magnesia.

The graphical method here employed can be used also to illustrate the
proportions of each of the mineral constituents present in the Aplites (or
more acid segregations of each group), as well as those of the Pegmatites
and Gneisses whose comi)ositiou allies them to that of their massive
prototypes.

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216 ProceediTigs of Roycd Society of Edinburgh, [i

extremities, >.e., to the ends and the upper side of the intrusive
mass. Wedge-shaped intrusions are much less common in the
case of the dykes composed of hasic, or of sub-basic, materials
than in those which contain potash felspars. Why this is the
case is not clear.

Occasionally basic dykes are clearly seen to terminate down-
wards. Sir Archibald Geikie has lately figured some examples
from Fife which are seen to do this. But all those which do so
l)elong, I think, to a diflferent category from the one which is here
specially under consideration, and they will be considered in that
connection in another paper.

It seems to be generally assumed that d^es often coincide
with lines of fault. In the course of an extensive field experience
I have but rarely met with cases in which it was quite clear that
this was so : but as geologists of good repute say that such cases
are of common occurrence, I will not press my own convictions too
far. It seems to me that in many cases where a dyke has risen in
contiguity to a fault of older date that the dyke is not in the least
influenced by the old plane of weakness. Quite commonly, how-
ever, older dykes may deflect the course of a newer one which has
cut obliquely across them, in a manner analogous to that which
happens where a newer fault is "trailed" by an older one — a
phenomenon quite diflferent in its nature from the " heave " pro-
duced when an older fault and its enclosing rock are bodily shifted
by a later thrust. This is only referred to here because there
seems to have been some misunderstanding regarding the relative
ages of two dykes of which one has gone oflT on one side of another
dyke in a different plane from that at which the two met on the
other. I have previously discussed this matter at some length in
a paper on "Faults" in the Trans, Edin, Geol. Sac. for 1889,
pp. 71-74.

There is a fine example of the influence of an older sill upon thi^
upward course of a dyke on the west shore of Carsaig Bay in Mull.
The dyke rises through Lias Shales, and on coming near to the
base of the sill the dyke suddenly spreads out laterally, so as to
pass on both sides into a sill, which it does, however, without
coalescing with the older one, or even quite reaching it. On
either side the lateral extension of the dyke thins out within a

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1903-4.] Mr J. G. Goodchild an Intnisive Rocks. 217

short distance. It may well have been the case that a difference
•of relative temperature of the country rocks and the magma at the
ipart where the dyke passes into the form of a sill may have had
something to do with the change of direction. (See fig. 22.)

The fact just referred to suggests the question, why should
the same magma eat its way in a horizontal plane at one pa^t and
4it another within the same type of country rock make its way
upwards in a nearly vertical plane 1 I^ossibly the answer to the
question may be that the magma was injected from below obliquely
upward and outward from the focus, and that its course, as a
lyhole, has really followed the oblique direction; but as it tra-
Tersed strata of very varying degrees of resistance to the thrust,
the magma eats its way upwards in a zigzag manner, forming a
sill on one platform, then going off as a dyke, again as a sill, and
^o on (see fig. 27, p. 226). The phenomenon may be illustrated by
attempting to scarp a fluted surface by drawing the end of a walk-
ing-stick in an oblique direction across the flutinga. The stick will
run along one of the flutings, make a jump to the next, along that
•again in a line nearly parallel to the first one, and so on. This is
what is above referred to as " trailing," which is a phenomenon of
-common occurrence wherever a newer set of faults crosses an older
set in an oblique direction.

On the view just set forth, the abundant Tertiary dykes of
North Britain may be represented by sills at no great depth below
the surface, and need not be supposed to extend downwards to any-
thing like the depth with which they are credited.

A few additional examples, out of a great many that might be
selected from amongst Scottish writers on Geology, will now be
referred to, in which those writers have figured the relationship
which actually exists between an intrusive rock and the rocks it
invades. For this purpose I give a list selected from Sir
Archibald Geikie's Ancient Volcanoes^ and his two recently-issued
memoirs on the Geology of Fife ; the references preceded by an
asterisk are particularly noteworthy :

Ancient Volcanoes of Great Britain^ vol. ii.. Figs. 238, 241-
245, 248, 249, 251, *255, 304, ^322, 323, 329, 349, 351, 353-5,
561, 371, 380, *381. "Geology of Central and Western Fife"

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218 Proceedings of Royal Society of Ediiiburg]i, [sbss.

{Menis, Geol Survey), Figs. 18, *20, *23-25. "Geology of Eastern
Fife " {Mems, Geol. Survey), Figs. 32, 60, 62. To these reference
may be made to Mr David Bums' diagrams relating to the Whin
•Sill which illustrates his paper in the Proceedings N. of England
Institute of Mining and MechaniccU Engineers, vol. xxvii., Plate Y.
The illustrations cited relate to a considerable variety of petro-
graphical types, of both the intruding masses and rocks invaded.
They include several figures of sections in which eruptive roc1(s
are clearly seen to cut out coal seams — not merely by altering
their quality, so that they have been rendered unfit for ordinary
uses, but by actually replacing the coal seams, in the same manner
as many intrusive rocks occupy the place of other materials which
have been removed, concurrently with the act of intrusion. As
before remarked, this feature is one of considerable importance both
from a commercial point of view and on account of its bearing
upon the questions here under consideration.

I commend the facts above stated to the careful considera-
tion of all unprejudiced geologists. It must be quite evident to
such workers, after a study of the foregoing considerations, that the
views commonly held with regard to intrusive rocks will have to be
modified to a very considerable extent. That must be done, what-
ever view one may entertain with regard to how these facts have
been brought about. It may be well to remark here that I do not
wish the readers to understand that any other signs of mechanical
rupture than those specially referred to do not exist ; but I
certainly do intend to convey the idea that such evidence is of
very much less common occurrence than most people seem to believe^
Furthermore, I state emphatically that even in the cases where
there undoubtedly is evidence of a certain amount of displacement^
the extent of that displacement is, as' a rule, by no means com-
mensurate with the volume of the rock intruded. It appears likely
that the degree of viscosity of the magma on the one hand, and
the resistance presented to the intrusive force on the other, are the
chief factors which determine the mode of occurrence of intrusive
masses. Where a viscous, or a half-consolidated, mass is being forced
between imperfectly consolidated materials, and under relatively
small superincumbent pressure, it is most likely that the overlying.

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i»03-4.] Mr J. G. Groodchild on Intrusive Bocks, 211>

rocks would actually lift and thus conform to the upper boundary
of the intrusion. But where the magma is more fluid, and th&
pressure to be overcome surpasses some, as yet undetermined,
amount, solution ensues, and the process becomes a physico-
chemical one instead of a purely mechanical act.

At any rate, and by whatever means the process may have been
carried out, I can confidently assure my fellow-workers that the
replacive mode of occurrence of intrusive masses is the rule and
not the exception. The belief founded upon these facts is by
no means what it has lately been described — a superstitious belief
entertained by ignorant miners, but is one that geologists in
general will have sooner or later to accept, whether that belief i»
in accordauce with preconceived ideas or not.

Taking it for granted that the evidence of replacement ia
admitted, there next arises the question as to how the missing rock
has been removed. Evidence bearing upon this, and helping to
furnish some kind of answer to that question, is certainly not
entirely wanting. It will be found in many cases that Nature
has not always finished the work of removing the rock so neatly
that no trace of the mode of attack can be found. Yarioua
stages may be seen when a large number of junctions come to bo
examined, and by patient investigation it is quite possible to-
arrive at a tolerably good idea regarding the method that has been
followed. A brief description of a few cases observed by myself
may be given first, and to these may be added some observations,
made by other geologists, selected from the writings of those whose
claim to be regarded as careful observers probably no one will
question. Choice will be made of the phenomena at first on a
large scale, and I shall choose the mode of attack followed by
granite as being the most suitable for the purpose in view. One of
the best examples is that presented by the marginal zone of the
Ross of Mull granite. That granite rises through some ancient
rocks of sedimentary origin, which pertain, I think, to the lower
part of the Highland Schists. They are chiefly greywackes and
flaggy quartzites which had been much affected by dynamic
metamorphism long prior to the intrusion of the granite. The
marginal zone is one of considerable width, and is by no means a
mere line, as one is apt to suppose is usually the case. For quite

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220 Proceedifigs of BoycU Society of Edinburgh, [sess.

a quarter of a mile, iu some parts, it is difficult to say whether
the rocks should be described as schists traversed by veins of
granite, or granite enveloping blocks of schist, I do not, however,
mean to convey the idea by this that there is any lithological
passage of the one type into the other ; for that there certainly is
not. On the contrary, the line between the granite and the schist
is clearly seen in hand specimens to be quite sharp and well-defined,
and, under the microscope, the presence of crystalline felspar on
one side of the boundary line and its absence on the other can
also readily be made out. The field relations of these rocks, as
aeen.at Torr na Sealga,- is shown in fig. 15 already referred to.*

It may be remarked, in passing, that having regard to the
fact that a zone consisting of closely interwoven, or spliced, granite
and schist extends for a considerable distance around the granite
proper, one is led to speculate what the result would be were the
whole area subjected to extensive dynamic metamorphism. The
granite would deform into muscovite-biotite gneiss, the plexus of
granite veins and fragments of hornfelsed greywacke, quartzite,
and mica schist, would form a gneissoid complex of a second kind,
while the schists themselves would form a third group, the only
feature common to the whole being a general parallelism of the
planes of schistosity. There cannot be much doubt that many
older complex areas of this kind occurring in the Highlands and
elsewhere have been affected in this manner, and it may well be
the case that some of the anomalous groups of gneisses and
gneissoid rocks of the Central Highlands of Scotland owe much
of their present character to the fact that the parent rocks were
of the type seen in the marginal zones of the Ross of Mull granite.

But, to return to the consideration of the mode of attack
followed by the granite in this area ; what has really happened can
easily be made out. The granite sends forward into the schist thin
wedges of its own material, which thicken as they advance along
the joints or other divisional planes, and do so at the expense of
the schist. The impression one gathers from a study of numerous
examples of this nature is that the whole periphery of the granitic
magma exercised a corrosive effect wherever it came into con-

♦ See a paper by the present author, ** On a Granite Junction in the Isle of
Mull," Oeol. Mag., dec. til, vol. ix. pp. 447-451 (1898).

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1903-4.] Mr J. G. GoodchDd an Intrusive Rocks, 221

tact with the rock invaded. Hence the magma was enabled to
advance along the joints and other divisional planes of the country
rock. Every stage of the process can be traced, from the first
insinuation of a thread, or a knife edge, of granite, through the
later stages of development, where the advancing mass has widened
out, and has begun to form a thick wedge, up to the point where
it has eaten its way so far into the adjoining rock that the portion
attacked has become surrounded by the fluid magma, and thus
ready to float away as an isolated mass into what one may term
the trunk stream. (Here, perhaps, it may be as well to repeat
the remark that I do not entertain the belief that the fluid granite
is simply so much quartzite or greywacke in a different state from
what it was at first. Granite cannot be made simply out of
greywacke, much less out of quartzites, for there are several im-
portant constituents present in the eruptive rock which are absent
from the other.) But the advance of the veins of granite into
the schists, the enlargement, ramification, and coalescence of
contiguous veins, carried on until the two are closely spliced into
one, can be seen in every stage of progress. Whatever may have
been the particular solvent, its mode of operation is sufficiently
evident from a study of the various intermediate stages in the
process of, what may be termed, the mastication and assimilation
of which records have been left. The process has clearly been
of a physico-chemical nature, and one in which the continual sub-
division of the rock undergoing attack has been effected by the
erosive action of the peripheral parts of the magma. Each stage
in the process of comminution has led to an increase of the area
being exposed to attack, and has led, finally, to the complete solu-
tion of the fragments. I have long regarded the basic inclusions so
often found in plutonic masses as incompletely assimilated portions
of the country rock. This view, I am glad to notice, is now
being adopted by many of the rising generation of field geologists.
Some reference has already been made to the different mode
of attack followed by the more basic as compared with the more
acid magmas which, by the way, I should like to refer to hence-
forth under the respective terms soda magma and potaj^h magma.
The evidence appears to suggest that the soda magmas in general
acted with more corrosive effects at the extremities of their masses.

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1222 Proceedings of Royal Society of Edinburgh, [sim.

Avhile the potash magmas often appear to have possessed equal
-corrosive power over the whole of their surface in contact with the
rock undergoing attack. A thin dyke, or a thin sill, of a basic lock,
has made its way underground iis a nearly parallel sheet, in some
■cases over an area which may be hundreds of square miles in
extent, and, what is still more remarkable, it has done so notwith-
standing the fact that the rock invaded was at a lower temperature
than the soda magma. Had the corrosive effect been equal over
the entire surface in contact with the country rock, it must be obvious
that the part first invaded, that is to say, the part nearest the conduit
which gave emission to the fluid magma from below — would be the
parts where the intruded rock would be very much thicker than at
the points near the extremities. But many intrusive sheets appear
to retain nearly the same thickness for a distance of many miles.
The Whin Sill, for example, varies but little from the mean thick-
ness throughout the greater part of the extensive area it occupies.
The potash magmas, on the other hand, usually give rise to short and
thick lenticular masses, and it is very rarely indeed that they appear
^is sheets with parallel boundaries. One is, of course, reminded by
these facts of the similar behaviour of basic lavas, which may flow
with comparatively little variation in thickness for thirty, forty, or
«ven fifty, miles, while a lava stream of acid composition but rarely
extends more than a very few miles from its point of emission, and
in many cases does not get more than a few hundred yards away
from that point before it comes to a standstill. Of course the
temperature of the country rock must be an important factijr in
this connection in the case of all intrusive masses, even in those of
trappean, as distinguished from plutonic, origin. Still, the fact
remains, that potash magmas erode over their entire surface, so
that they tend to eat tlieir way outward in the form of gradually-
enlarging wedges. It follows tliat the rock surfaces on either side
of one of these wedges may retain much similarity of form, and
that the shapes of the opposite sides of a wedge may nearly or
quite match, even though a considerable quantity of the interven-
ing rock may have been removed.

For the information of those who may wish to examine the
evidence, it may be mentioned here that the best sections where
tlie relations of the Ross of Mull granite to the country rock can

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1908-4.] Mr J. G. Goodchild on Intrusive Bocks. 223

be studied are all within easy distances of Bimessan, where the
Dunara Castle calls twice weekly from Glasgow. There are
large quarries at Camas Tuadh, Ardalanish, and other ]>laces near,
^and there are exceptionally fine coast sections at Carraig Mhor and
Torr na Sealga, which can easily be examined from Bunessan.
Even in passing by steamer from lona to Oban the broader features
-can easily be made out with the aid of a good field-glass.

On referring to the older literature of the subject I find that
some of the statements here put forth regarding the granite margins
had, to some extent, been anticipated by previous writers. Thus
M'Culloch gives a most interesting account of the relationship
between the granite of Cruachan and the schists around, which
tallies in almost every respect with what I observed in the Ross of
Mull (see Trans. Oeol. Sac. Lond., vol. iv., pp. 126 et seq.),
Jameson noticed the same features in connection with the granite
-of Braemar (Annals of Philosophy, vol. iv., p. 419). Mr Came has
recorded similar facts around the granite of Cornwall ( GeoL Trans,
•of Cornwall, vol. i., p. 22). So did Dr Davy; also Dr Boase,
De la Beche, and others. But as these observers were not well
acquainted with modern petrographical methods, it may be ets well
to add to their testimony the evidence lately put forth by one of
our ablest workers in that department of science, which is accord-
ingly subjoined.

Since my paper " On a Granite Junction in the Ross of Mull "
was published, my colleague, Mr Kynaston, has mapped the area
around the granite mass of Ben Cruachan, which is probably of
the same age as the granite of the Ross of Mull, and, like that mass,
it rises through the Highland Metamorphic Series. In the
JSummary of Progress of the Geological Survey of the United
Kingdom for 1900 an outline is given of Mr Kynaston's conclu-
sions. These are so pertinent to the subject at present under
consideration that no apology is needed for quoting them nearly
in full. The quotation, pp. 73-74, is as follows :

" Great difficulty was experienced in mapping out the boundary
line between the granite and the schists owing to the complicated
nature of the marginal area. Indeed, in some places the granite
and the schistose rocks are so intennixed that no sharply-marked
boundary-line can be drawn between them. . . . The contact zone

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224 Proceedings of Royal Society of Edinburgh. \\

consists of a network of sills, veins, bands, and tongue-like pro>
trusions of granite, covering a belt of mountainous ground sometimes
more than a mile broad. The vein-like offshoots do not, as a rule,
anastomose with one another, but tend to run in a roughly-parallel
direction, coinciding with the original planes of foliation of the
schists, although irregular intrusions of granite, having no apparent
relation to any planes of weakness, are not uncommon. The com-
plication is such that a line can only with difficulty be drawn
between schists crowded with granite veins and sill-like bands,
and granite crowded with strips and inclusions of schist of every
size up to a mile or more in length. ... As we approach the main
mass of the granite the schists are frequently seen to be sa
impregnated with granitic material that it is impossible in a
hand-specimen to distinguish the igneous portion from the material
of sedimentary origin. ... In many places the schists have been
broken up under the process of injection and a breccia has been
formed .... consisting of a confused mingling of altered schistose
fragments in a granitic matrix .... [Some of the] fragments are
usually crowded with flakes of secondary biotite in more or less
parallel layers, and are somewhat suggestive of the origin of certain
ill-defined patches rich in biotite, occasionally seen in the granite.'*
[My own remarks about these inclusions, which form a most
'conspicuous feature in the granites of Ballachulish, were written^
but not published, before I knew that Mr Kynaston had published
the note. J. G. G.]

As contact or thermo-metamorphism of the country rock
must play an important part in the subsequent processes of
conversion, especially in the cases where the preliminary changes,
have taken place under plutonic conditions, a few remarks here
upon that subject may well be given. In the case of certain
schists, and of some of the older grey wackes, both of which may
have contained mineral matter of eruptive origin before they were
affected by thermo-metamorphism, there is usually some advance
towards the conversion of the rock into homfels, knotted schist,,
andalusite rock and the like. Kadiolarian cherts have been altered
into granular quartz, almost into quartzite, around the Galloways
granites, and graptolitic mudstones into graphitic schist. In
Mull, in Glenco, and in the Lake District, the Green Earths^

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1903-4.] Mr J. 6. Goodchild on Intrusive Rocks, 225

which formerly occupied the vapour-cavities of the lavas have
been converted by subsequent thermo-metamorphism into various
forms of Epidote and the associated zeolites into Albite or other
felspars. These are common eifects in the areas that have been
affected by thermo-metamorphic action.

But some of the most striking cases of the development of
minerals by the causes which have given rise in adjacent areas
to eruptive masses of deep-seated origin are to be found in the
case of the metamorphic marbles which occur in various parts
of the Highlands of Scotland and elsewhere. Referring
to the specimens in the Scottish Mineral Collection, I find
the foUowing species occurring within the substance of these
altered limestones: — Quartz, Andesine, Anorthite, Tremolite,
Diopside, Forsterite, Biotite, Phlogopite, Sphene and Apatite,
besides Graphite, Idocrase, Garnet, Zoisite, Wollastonite, and a
variety of other minerals with which at present we are not
concerned. The feature of special interest in these cases is the
development within the limestone by the same causes to which
the formation of eruptive rocks is due (whatever that may be), of
an assemblage of rock-forming minerals which are either identical
with those which characterise rocks of eruptive origin, or else are
allied to them. Amongst these are Quartz, two felspars (or more
than two); Tremolite, as a representative of the Amphiboles;
Diopside and Wollastonite as representatives of the group to which
Pyroxene belongs ; Forsterite, which is closely allied to Olivine ;
two micas (perhaps three), and other rock-forming minerals. Yet
no one seems to doubt that these minerals have been developed by
metamorphic changes out of impurities which occurred within the
marble. But it does not matter in the present connection whether
the limestone was impure to begin with, and contained in those
impurities the substances required for making the silicates referred
to, or whether part of these requisites may have been introduced
into the rock through the agency of the thermal waters which
have been concerned in bringing about the final result. Any way,
the fact is one of great importance in the present connection, and
must on no account be allowed to drop t)ut of sight.

PROC. KOY. SOC. KDIN. — VOL. XXV. 15

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226 Proceedings of Royal Society of Edinburgh, [sbss.

Fio. 27.

(Note added April 22, 1904.)

The remainder of the paper dealt with theoretical considerations, which
may be summarised as follows : —

In explanation of the facts, it is suggested that four chief factors are con-
cerned, which are as follows : — (1) Earth movements, which generate the heat
required for volcanic action, and also furnish the motive power by which the
magma is forced outwards from the focus. (2) The presence, at the focus of a
volcano, of saline waters, whose dissolved salts become concentrated by pro-
longed boiling, and the consequent escape of steam at the surface. These
saline solutions, operating at high pressures and temperatures, dissolve the
rock in various directions around the volcanic focus, ana add their own alkalis
to the magma so formed. (3) An excess of alkalis (es{)ecially of soda) in the
magma, whereby it is enabled to gradually extend its ramifications into the
rock around its focus. (4) Circulatory movements from the extremities of the
system to the volcanic focus and back, analogous to the movements of the hot
water in the pipes of a heating apparatus. This circulation behaved in a
manner analogous to that of the circulatory system in a tree, in which the
leaves generate one set of products, and the roots carry in another, in the
shape of water and alkalis. These commingle, and then travel outwards from
below, to bo finally left in the solid form, and thus contribute to the extension
of the whole.

An ordinan' sedimentary aggregate, to which the dissolved constituents of
sea-water had been added, oj^erating under high temperatures and pressures,
might furnish the materials of the basic and sub- basic eruptive rocks ; while
the granitic materials constituting the floor of the Eiarth's crust could supply
the additional potash and silica required for the formation of acid and sub-acid
series of rocks.

It was further suggested that many basalts, and most gabbros, were of
secondary origin, ana that their present structure is due to changes which have
originated within the core of a volcano. Some basaltic tuffs had thus been
softened and reconsolidated as pseudo-massive rocks ; while many basalt lavas,
dykes and sills, occurring within the same zone of reconstruction, api>ear, in
like manner, to have been softened and then recrystallised into gabbro. Most
granophyric granites associated with gabbros may represent such changes
carried further still, and may be due to the solvent action of a granite magma
upon an older set of basic rocks (see fig. 27 above).

The bearing of these considerations upon various other metamorphic pro-
cesses connected with the origin of gneisses and rocks allied thereto, was
discussed in some detail.

(Issued separately^ Miuj 20, 1904.)

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1903-4.] Note on the Standard of Relative Viscosity, etc. 227

Note on the Standard of Relative Viscosity, and on
" Negative Viscosity." By W. W. Taylor, M.A., D.Sc.
Communicated by Professor Crum Brown.

(Read March 21, 1904.)

The Unit of Rblativb Viscosity.
The absolute viscosity calculated from the formula

wprH

(where jp = the pressure, t the time, r the radius, I the length of
capillary, and v the volume of liquid), which connects the viscosity
of a liquid with the rate of flow through a long capillary tube, is not
often made use of, mainly on account of the difficulty of accurately
determining some of the constants (r in particular). Further, a
correction has to be made if the velocity of outflow is not sufficiently
©low.* For most purposes the viscosity is referred to that of a
given liquid as standard, and is calculated from the formula

where t/q, Sq, t^, are the viscosity, density, and time of flow through
a tube of a given volume of the standard liquid, and -q, s, t are the
corresponding data for the other liquid. Of t/^, Ostwald-Luther
{Phys, Chem, Mesmngen, p. 260) say, " the viscosity of water at 0° C.
(or at the temperature of experiment) is put= 1."

It is the general practice to take the viscosity of the solvent
(whether water or other liquid) at the temperature of experiment
as ly^j = 1 . In place of this, it would be an advantage if the
viscosity of w^ater at 0* C. were taken as standard, and the relative
viscosity of liquids and solutions referred to this alone.

For certain purposes, e.g. demonstration of the additive
character of the viscosity of salt solutions, the relation between
viscosity and atomic weight, or between viscosity and concentra-
* Cf. Ostwald-Luther, Fhys. Chem, Messungen, p. 369.

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228 Froceedings of Royal Society of Edinburgh, [sbs«.

tion, where all the experiments are made at one temperature, the
general practice is not inconvenient, but it has several disadvan-
tages:—

1. It is possible, and maybe desirable, to determine the viscosity
of a solution at temperatures below the freezing-point or above the
boiling-point of the solvent ; in this case *q, t^ cannot be deter-
mined.

2. It affords no good way of graphically representing the
relation between viscosity and temperature.

3. It may lead to misunderstanding. Most of the experiments
on solutions have been made at 17* or 25** C, and a comparison
of the relative viscosity of, €,g,, 1 n KCl is as follows : —

Temp. Temp.

15* 25**

Water 1 1

1 n KCl 0-972 1-001

Water 0-640 0 501 I . .^^ . ^. , v.

In KCl 0-622 ^.^^^ } (^^^eratO =1),

from which it appears that the relative viscosity of the solution ii>-
creases with increase of temperature. In this connection it may be
remarked that Euler,* referring to the influence of temperature,,
says, — "whilst the specific viscosity of all solutions of non-
electrolytes decreases with rise of temperature, the solutions of
strongly-dissociated electrolytes are affected in the opposite
direction." Without a definition of "specific" viscosity this
statement might be misunderstood.

If the viscosities are referred to water at 0" as unit, it is seea
that they do not increase with rise of temperature, but that they
do not diminish so rapidly as the solvent; in other words, the
temperature coefficient of the solution is smaller than that of the
solvent, but is of the same sign. Of course, there may still be
a fundamental difference between the two classes of solutions.

As to the unit, no maximum of viscosity for water is known
(as there is of density at + 4° C), and there is not much to choose
between water at 0' and -I- 4" ; in either case, Sq can be put = 1
without appreciable error in 77, which is ordinarily not more
accurate than one in 500 or 600.

• ZeU.f, Phys, CJicm., 25, p. 536 (1898>.

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1903-4.] Note on the Standard of Relative Viscosity, etc, 229

There is no need to determine t^ directly ; the simplest way is
to determine t for the solvent at the temperature of experiment,
and to calculate t^ from it by means of the table of viscosity of
water at various temperatures.

" Nbgatutb Viscosity."

The bearing of this on " negative viscosity " (a term frequently
used to denote that the viscosity of the solution is less than that
of the solvent at the same temperature) is indicated below.

In general, the temperature coefficient of the solution will be
(a) less or {h) greater than that of the solvent.

(a) If at a given temperature the viscosity of the solution is
greater than that of the solvent, and its temperature coefficient is
smaller than that of the solvent, at higher temperatures the
viscosity-temperature curves will diverge, but at lower tempera-
tures they will approach, and finally intersect at some temperature,
below which "transition temperature" the solution will exhibit
"negative viscosity."

(b) If, on the other hand, the temperature coefficient of the
solution be greater than that of the solvent, the curves will
diverge on lowering the temperature, whilst they will approach
and intersect on raising the temperature. In this case the
solution will exhibit " negative viscosity " at higher temperatures.

The particular case where the solution and solvent have the
same temperature coefficient needs no discussion.

Aqueous solutions of electrolytes appear to belong to group (a),
and in some cases, at any rate, a solution has " positive viscosity "
at one temperature and " negative viscosity " at lower temperatures,
e,g, KCl, KNOg,* etc.

Until quite recently no solutions other than aqueous solutions
of electrolytes were known to exhibit "negative viscosity," and
on this Eulert based his explanation, — "the electric charge of the
ion causes a compression (electro- stricti on) of the water, on
account of which the viscosity is diminished." But Miihlenbein, |

• Sprang, Pogg, Ann., 169, p. 20 (1876). t Loc. cit,, p. 541.

t Diissertation, Leipzig, 1901. Also Wagner, Zeit, /. Phys, Chem., 46,
p. 872 (1908).

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230 Proceedings of Royal Society of Edinburgh. [siss.

a pupil of Wagner, has found that some organic substances in
organic solvents do also exhibit it, e.g. cyanobenzol in ethyl alcohoL
In the known cases of group (a), increase of concentration
raises the transition temperature : there is very little to show in
what way concentration affects the transition temperature of
solutions in class (b), whether decrease of concentration will
lower it or not, but measurements by Rudorf * on aqueous solutions
of carbamide indicate that at 25° C. the relative viscosity decreases
with dilution, and even becomes " negative," e.g, —

CoDcentntioD.

q( Water at 25*=]).

0-937n

I^OIO

•469

roo2

•234

0996

•117

•993

•058

•995

— but the viscosity is so nearly the same as that of water that it is
not safe to base any conclusions on these data.

Increase of molecular weight, in the known cases of class (a),
raises the transition temperature, and this affords another means
of bringing it within the range of experiment.

The general case, where the viscosity curves of solution and
solvent intersect twice, is of some interest. According as the one
curve or the other represents the solution, there will be a transition
from "positive" to "negative" viscosity, or vice versct, at both
high and at low temperatures. It may not be possible to realise
this case, except perhaps with a very soluble substance, and a
solvent which permits of a wide range of temperature, but there
should not be much difficulty in realising the particular case of it
where at one extreme of temperature and concentration the one
part of the curve is obtained, and the other part at the other
extreme.

I hope to commence experiments, in the near future, with a
view to verifying these conclusions.

* Zeit. /. Phys. Chem., 43, p. 257 (1908).

{Issued separately June 16, 1904.)

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1903—4.] The Viscosity of Aqueous Solutions oj Chloi-ides, etc. 231

The Viscosity of Aqueous Solutions of Chlorides,
Bromides, and Iodides. By W. W. Taylor, M.A., D.Sc,
and Clerk Banken, B.Sc. CommuTiicated by Professor
Crum Brown.

(Read March 21, 1904.)

In a recent investigation on the aluminium anode, by one of us,
in conjunction with Inglis,* a striking difference was found
between chloride and bromide during some preliminary experiments
on the rate of solution of aluminium in sulphuric acid : —
addition of a small quantity of potassium chloride to the sulphuric
acid greatly increased the rate of evolution of hydrogen, but
addition of an equivalent quantity of potassium bromide^ under
the same conditions, appeared to have no effect at all. Subsequent
investigation, not yet completed, has shown that, under similar
conditions and with solutions of pure hydrochloric acid and
hydrobromic acid which are isohydric (have the same concentration
of H*), the rate of evolution of hydrogen from hydrochloric acid is
about thirty times as great as from hydrobromic acid. No experi-
ments have yet been made with hydriodic acid.

Such marked differences between chloride and bromide are by
no means common ; so far as we are aware, the only one previously
recorded is by Ostwald,t that chloride, bromide, and iodide have
very different effect on the periodic dissolution of chromium in
acids. Another interesting instance has since been found by Elbs
and NUbling J — that with a lead anode aijd hydrochloric acid as
electrolyte, a compound of quadrivalent lead is formed ; but that
when hydrobromic acid or hydriodic acid is the electrolyte, no
similar compound is formed. It is a curious circumstance that in
each of these cases the reaction is one which takes place at the

♦ Phil Mag, (6), 6, p. 312 (1903).

\ZeU.fiir Phys. Chem., 35, pp. 33, 204 (1900) ; 38, p. 441 (1901).

t ZaU, fur Elektrochemie, ix. p. 776 (1903).

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232 Proceedings of Royal Society of Edinburgh. [=

surface of a metal in contact with a solution. In the paper on
the Aluminium Anode {loc. o^.) it is suggested that the permeability
of the surface film of aluminium hydroxide by CI' and imperme-
ability by SO^ " is the cause of the differences observed between
hydrochloric acid and sulphuric acid ; and if this be so, differ-
ences of permeability by CI', Br', and I' are to be expected.

As it seemed probable that similar differences might manifest
themselves in other physical properties, we decided to determine
the relative viscosity of solutions of chloride, bromide, and iodide
under various conditions of temperature and concentration. The
viscosity of solutions of potassium chloride has been determined
many times at one temperature (17** or 25' C.) and one concen-
tration (usually 1 n). Sprung* determined the viscosity of
potassium chloride, bromide, and iodide over a considerable range
of temperature (5** C. to 50* C), but at only two concentrations of
chloride, and the other solutions were not at comparable concen-
trations. Wagner t also made determinations of viscosity of
hydrochloric acid at various concentrations and temperatures.
Their results are referred to later on.

£XPBRIMBNTAL.

• The potassium chloride and bromide were purified by repeated
precipitation from hot aqueous solution by addition of ethyl
alcohol ; the iodide was twice recrystallised from water. The
hydrobromic acid was made by the direct union of hydrogen and
bromine in contact with hot platinised tile, the gas absorbed in
water, and the solution redistiUed ; no rubber or cork joints were
used in the apparatus, so that the bromine and acid never came in
contact with organic matter. The most concentrated solutions of
the salts were made up by weight, and the others were prepared
from them by dilution ; the concentration of each solution was
further checked by titration with silver nitrate. The concen-
trations of the acid solutions were ascertained by titration with
barium hydroxide solution.

The densities were determined by means of an Ostwald-Sprengel
pyknometer. The viscosity apparatus used is the form figured

• Pogg, Ann., 159, p. 1 (1876). t ^"ied. Ann,, 18, p. 259 (1883).

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1908-4.] The Viscosity ofAqueom Solutions of Chlorides, etc. 233

and described in Ostwald-Luther {Phys. Ghent. Messungen^
p. 260). In every experiment the time of flow was observed
six or seven times and the mean of all the readings taken ; also,
in many cases duplicate determinations were made, but no
difference in the mean result was obtained except at 0** C, where
a difference of 0' 1-0*2 sec. in 150 sec. were obtained; the times
were measured by means of a stop-watch, giving 0-2 sec.

In every case the viscosity of the solution is referred to the
viscosity of water at 0* C. as unit = 1 ; for convenience of com-
parison, the viscosity of water at the temperature of experiment is
added. The temperature at 15° and 25* did not vary 0-1*, but
the low temperature varied between 0*1* and 0-15*, and the data
are corrected to 0** C. We made determinations of the relative
viscosity of water with each of the three tubes used in the other
experiments, and the results given below are the means of all the
five values obtained at each temperature : —

0* C. 1000

15° 0-6395

25° 0-601

1-000

0-638

0-501 (Thorpe and
Rodger).*

1000

0-637

0-500 {Ho»Ung)A

Table I. — Potassium Chloride,

Temp.

Mol. per
litre.

1
Density. Viscosity.

Viscosity of
Water.

1
2
3

1-0480
1-0935
1-1371

0-931
-886
-880

1-000

15"

1
2
8

1-0455
1-0901
1-1333

•622
•615
•625

•502
•507
-517

0-640

26-

1
2
3

1^0488
V0877
M295

0-501

• Phil. Tram., 185, p. 397 (1894).
t Phil Mag. (5), 49, p. 274 (1900).

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•234 Proceedhigs of Royal Society of Edinburgh, [a

Table II. — Potassium Bromide.

Temp.

Mol. per
litre.

Density.

1-0858
11692
1-2521

Viscosity.

Viscosity of
Water.

O**

0-911
-837
•815

1-000

16»

1 10831

2 11662

3 1-2453

•601
•685
•582

0-640

25''

1
2
3

1-0S04
1-1623
1-2413

•483
•477
•486

0-501

Table III. — Potassium Iodide.

Temp.

Mol. \^eT
litre.

15''

1
2
3

25*

1 s

I 1^1212

r2415

' 1^8621

1^1188
1^2365
1^3552

1-1159
12823
r3499

Viscosity.

0-854
•778
•748

•583
•552
•544

•467
•458
•459

Viscosity of
Water.

1000

0*640

0-601

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1908-4.] The Viscosity of Aquemis Solutions of Chlorides, etc. 235
Table IV. — Hydrochloric Acid.

m,^^ MoL per

Density.

Viscosity.

1-020
1-041
1-069

Viscosity of
Water.

1
2
3

1-0160
1-0327
1-0489

1-000

1
2
3

1-0144
10808
1-0454

0-667
•695
-725

•529
•557
-585

0-640

26-

1
2
3

1-0123
1-0278
1-0426

0-501

Table V. — Hydrobromic Acid,

T*«»« I Mol. per
^'"P- litre.

Density.

Viscosity.

Viscosity of
Water.

0 987

1-000

•970

•962

-650

0-640

-657

-671

-514

0-501

-529

•544

Kesults.

In the first place, it may be pointed out that the value we have
obtained for 1 n KCl solution at 25' is slightly greater than the
viscosity of water at that temperature, whereas it is generally
stated to be less than water; the value for 17-6* C. (interpolated
between 15** and 25*) agrees extremely well with that given by
Arrhenius. A certain amount of confusion has arisen regarding

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236 Proceedings of Royal Society of Edinburgh. [sbss.

various determinations of these data : e,g, Rudorf * gives a table
comparing the data for various salt solutions by Abegg,t
Arrhenius, J and Reyher, § stated to be for 25' ; whereas Reyher's
alone are for that temperature, those of Arrhenius were for
17*6' C, and those of Abegg apparently for 15' or 16'. It is not
surprising that the data do not show good agreement.

H-

W-

0» 5» 10* 15* 2Xf 25*

Fio. 1.— Concentration of solutions 1 mol. per litre.

The results contained in the above tables show that there is a
considerable difference between chloride, bromide, and iodide, not
only at any one temperature and concentration, but especially
in the effect of variation of temperature and concentration on
the viscosity. The experiments have, unfortunately, not been
extended over a sufficient range of temperature and concentration

♦ ZeU,f, Phys. Chcm., 43, p. 257 (1903). + Ibid., 11, p. 248 (1893).

X Ibid., 1, p. 296 (1887). § Ibid., 2, p. 744 (1888).

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1903-4.] The Viscosity of Aqueous Solutions of Chlorides, etc, 237

to warrant general concluBions, but some points worthy of notice
may be referred to.

The Effect of Temperature, — In every case the viscosity decreases
with increase of temperature, but at diflferent rates for the three
salts, the rate for chloride being greatest and iodide the smallest.

H-

H)-

•9

•5-

-L

(f S" 10* 15* 20" 25*

Fio. 2.— Concentration of solutions 2 mols. per litre.

It will be noticed, too, that a solution can at one temperature
exhibit "negative viscosity/'* and "positive" viscosity at
another ; e.g. potassium chloride at each of the three concentrations
is "positive "at 25' C. and "negative" at 15° C, while all the

* The term " negative viscosity '* has been frequently employed to express
the fact that the viscosity of the eolation is less than that of the pnre solvent
at the same temperature.

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238 Proceedings of Royal Society of Edinburgh.

[SBSS.

and

solutions of hydrobromic acid are "positive" at 15' C.
" negative " at 0" C. {cf. figs. 1, 2, 3).

Another effect of temperature is well seen in fig. 4, in which,
for the purpose of comparison, the viscosity of water at each
temperature is shown by a thick black line. At 0° hydrochloric
acid alone has viscosity greater than that of water at all con-
centrations, at 15° the viscositj' of hydrochloric acid and hydro-

II -

1-0

•8-

•4

0** 5' l(r 15* 2(r 25*

Fio. 3.— Concentration of solutions 3 mols. per litre.

bromic acid is greater than that of water, while at 25" potassium
bromide and iodide still have viscosity smaller than that of
water, but the one normal solution of potassium chloride has
practically the same viscosity as water, though at all three con-
centrations it is greater than that of water. Sprung (Zoc cit.)
has shown that at higher temperatures the viscosity of the
concentrated solutions becomes greater than that of water.

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1903-4.] The Viscosity of Aqueom SoltUiona of Chlo7^e$, etc. 239

IJffect of Concentration. — The eflfect of concentration on the
viscosity depends very much on the temperature, as is seen in fig 4.
The viscosity of hydrochloric acid increases with increase of con-
centration at all three temperatures ; this is in accord with
Wagner's results {loc. cit.).

Increase of concentration increases the viscosity of hydrobromic
acid at 25° and 15*, but decreases it at 0° C. In the case of the

•8

15*

25**

-L

Concttntration Im. 2 m. 3m.

Fig. 4. —Eflfect of concentration at different temperatures.

salts the viscosity decreases at 0** with increase of concentration,
at 15"* bromide and iodide still decrease, while chloride passes
through a minimum ; and at 25° chloride increases, while bromide
and iodide pass through a minimum. This is in agreement with
Sprung's * conclusions, qualitatively at least, as will be seen by com-

* Loc. dt.

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240 ProcetdiTiys of Royai Society of Edinburgh, [ssss.

parison of his curves ; * it is plain, however, that experiments over
a much wider range of concentration are required before any
satisfactory conclusions can be reached.

Electric Conductivity at 0* C.

The equivalent conductivities of dilute solutions of chloride,
bromide, iodide of a metal are practically the same at 18' C, though
concentrated solutions do show small differences. In order to see
if greater differences exist at a lower temperature, we have deter-
mined the conductivity of all the solutions employed in the
viscosity experiments at 0° C. The method was the usual
Kohlrausch alternating current method, with bridge and telephone.
The results are corrected for the slight variations in temperature,
and the cell constant was determined by means of the value at 0** C.
of 1 n KCl, as given in Kohlrausch {Leitverjndgen, p. 204).

Whethamt has recently determined the conductivity of a
number of solutions at 0' C, potassium chloride being one of them :
for 1 n KCl (r2 n was the most concentrated solution employed)
he found A = 69*0. There is also in Kohlrausch {Leitvermdgen^ p.
199) a table of temperature coefficients of conductivity for dUute
solutions of HCl, KCl, KI, as determined by Deguisne.

Table VI.

Mol. per litre.

Eqairalent conductivity.

KCl

65-4

631

62-4

KBr

68-3

67-6

65-8

700

69-5

68-0

HCl

187-0

165-7

143-5

HBr

203-0

175-0

148-3

* These are viscosity-percentage concenti-ation curves, and are not compar-
able, as are viscosity -molecular concentration curves,
t Proc. Roy, Soc, 71, p. 332 (1903).

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1 90 ;-4.] The Viscosity of Aqueoiis Solutions of Chlorides, etc. 241

The differences between the conductivity at 0' C. of equivalent
solutions of KCl, KBr, and KI are very similar to the differences
at 18" C. (cf. Kohlrausch, — Leitvermogen),

An explanation of the equal mobility of CI', Br', and I' has
been ^suggested — that molecules of the solvent may be associated
^ith an ion ; that the number of molecules associated depends
on the electro-affinity of the ions ; and in this case the difference
in number of molecules associated with CI', Br', and I' causes the
mobilities to be the same. The formation of complexes is also
referred to the electro-affinity of the element.* Some influence of
this might be expected in the viscosities of the solutions ; but
whether the differences observed, especially with variation of
temperature, are to be connected with this, it is not possible to say ;
we have worked with concentrated solutions only, and possibly
dilute solutions would be better for this purpose.

Summary.

1. We have determined the relative viscosity of aqueous solu-
tions of KCl, KBr, KI, HCl, HBr at 0', 15', and 25' C. ; and at
concentrations of 1 mol., 2 mol., and 3 mol. per litre. Also the
equivalent conductivity of the same solutions at 0" C.

2. The change of viscosity with change of temperature dimin-
ishes from Cl-Br-I.

3. The effect of concentration on the viscosity depends on the
temperature : it may affect the viscosity in opposite directions at
different temperatures.

4. There are considerable differences in viscosity of chloride,
bromide, and iodide, and especially in the effect of changes in con-
centration and temperature.

• Cf. Abegg and Bodlander, 2eil,/. anvrg. chem.y 20, p. 468 (1899), and
Baur, AhrerCi Sammlung chem. u. chem, teeh.^ Vortrage viii. No 12 (1903).

{Issued separately June 16, 1904.)

PROC. ROY. SOC. EDIN. — VOL. XXV. 16

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242 Proceedings of Royal Society of Edinburgh, [

On the Date of the Upheaval which caused the 25.feet
Raised Beaches in Central Scotland. By Robert
Munro, M.A., M.D., LL.D.

(MS. received March 28, 1904. Read May 2, 1904.)

About forty-two years ago Mr Archibald Greikie (now Sir
Archibald), then an energetic member on the staff of the Geolo-
gical Survey, propounded and advocated the doctrine that the
change in the relative level of sea and land, indicated by the 25-
feet raised beaches which have been long known to geologists as
fringing the winding shores of the firths of Central Scotland, took
place subsequent to the occupation of the district by the Bomans.
Further researches, together with a more careful examination of
the archaeological phenomena on which Sir Archibald mainly relied
as evidence, convinced later observers that the facts did not justify
this conclusion. Hence for some years I have been under the
impression that the post-Roman theory was abandoned, not only
by the general body of geologists and archaeologists, but, as I
understood, by the author himself. The following statement of
opinion on the subject,* recently urged in the interests of the
Trustees of the British Museum by a distinguished Professor of
Greology, and one who has had exceptional opportunities of making
himself conversant with all the factors of the problem, will, how-
ever, show how wide of the truth that impression must have been.
Professor Edward Hull, F.R.S., said " that he was formerly Director
of the Geological Survey of Ireland. The spot where the articles
were found was part of what was known to geologists as a raised
beach. The raised beach extended all along the north coast of
Ireland, and down the east coast as far as Wicklow. In the north
it was about 15 feet high, but towards the south its height was
only about 4 ft. Its general character was, that it was a nearly

* Evidence given in the recent case of the Attorney-General v. The Trustees
of the British Museum with regard to the remarkable hoard of gold ornaments
found near Lough Foyle, Ireland. {Times Law Reports^ June 13, 1903.)

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1903-4.] Date of Upheaved of Raised Beaches in Scotland, 243

flat terrace, of varying width, with the old coast-line on the inland
side, and a slope down to the sea on the other side. A similar
formation was found in Scotland, but there the height was
generally greater — about 25 feet. The Carse of Gowrie was an
example. In the raised beach in the north of Ireland were found
not only shells of the present period, but flint arrow heads and
other articles made of flint. In Scotland there was stronger
evidence of the date of formation. There had been found skeletons
of whales, and canoes, some hollowed out of single trunks, but
others clinker-built of sawn planks, with holes for riveting. Iron
anchors and boat-hooks had also been found in the raised beach
in Scotland. The raised beaches in Ireland and Scotland were a
simultaneous formation, in his opinion. The iron implements were
important in fixing the date. He should say that the beaches
began to be formed about the fourth century a-d. His opinion

was founded upon all the sources of information available

It was a disputed question when the sea retired from these beaches.
The flint implements dated from the Celtic era, which might be
from the second century b.c. to the second century a.d."

Differing from Professor Hull with regard to some of the items
in the above statement, more especially that the finding of cetaceous
remains, canoes, iron anchors, etc., entitles him to fix the date of
the upheaval to so recent an epoch as the fourth century a.d., I
propose in this paper to reopen the former discussion on the sub-
ject, though to many it may seem to be slaying a dead animal.
For this purpose it is necessary to go back to the early sixties of
last century, when the post- Roman theory was first promulgated by
Sir Archibald Geikie, whose researches were evidently the fans et
xrrifjo malt of the Professor's statements.

In his first published essay on the subject {Edinburtjh New Phil.
Journal^ vol. xiv., 1861) Sir Archibald restricted the field of his
researches to the Firth of Forth. The principal evidence then
adduced was the discovery by himself and Dr Young of small
pieces of two kinds of pottery " in a regularly stratified deposit "
in the lower reaches of the Water of Leith, which they considered
to be of Roman origin. In support of the validity of this argu-
ment he writes : " Since the examination of the sand-pit at
X.eith I have visited all the localities along the shore where

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244 Proceedings of Boycd Society of Edinburgh, [i

Roman remains are known to have existed, and I have found no
authentic evidence that in any way militates against the recent
elevation of the land, but, on the contrary, several facts that tend
to confirm it." {Ibid., p. 107.) At Inveresk and Cramond all the
Roman remains were, so far as he could discover, 60 or 70 feet
above present high-water mark. He ridicules the tradition that
some old carving to be seen on the Eagle Rock, near Cramond,
and situated only a little above present high-water mark, was
Roman workmanship.

^* Antiquaries,'' he writes, '' have grown eloquent at the eight of this
relic of the creative genius of the old legionaries, but the carving has
really about as much claim to be considered Roman as the famous pne-
torium of Jonathan Oldbuck. In a niche of the soft sandstone crag stands
a rude figure, as like that of a hmnan being as of an eagle, with a very
short stump by way of legs, surmounted by a long and not very sym-
metrical body, on one side of which an appendage that may be an arm
hangs stifily down, while the corresponding one shoots away up at an
uncomfortable angle on the other side. Like other carvings on the shores
of the Forth (as the figure near Dysart and Queen Margaret's footstep at
South Queensferry), it must take i-ank among the handiworks of idle
peasants or truant schoolboys." (/Wd, p. 110.)

By way of strengthening his theory, he further observed that
the Roman wall commenced at the Hill of Carriden ; that, accord-
ing to the author of Caledonia Roniana, the remains of the
Roman Portm ad Vallum existed (near Camelon) down to the
last century, and that an iron anchor was dug up in the same
locality. These statements will be dealt with later on.

In restricting his observations to the valley of the Forth, the
author did not then think it necessary to the truth of the con-
clusions of his paper **that the west coast of Scotland — as, for
instance, at the termination of the Wall of Antonine — should be
proved to have experienced any elevatory movements at all.'*
However, in the following year he recurred to the subject in a
more comprehensive communication to the Geological Society
of London (Journal, March 19, 1862), entitled, "On the Date
of the Last Elevation in Central Scotland," from which it will be
seen that he no longer confined himself to the east of Scotland^
as he included in his purview the Firth of Clyde, and, indeed^
*• the greater part of the British Isles."

Before proceeding to discuss the scientific value of the evidence

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advanced in support of these views, it is desirable to start with a

clear idea of what is meant by a * raised beach.' In reality, the

elevated portion includes not only the former sea-margin, or

beach proper, but also wide patches of sea-bottom which, in course

of the terrestrial process of upheaval, came to the surface, and

have remained dry land since. As an authoritative description of

the composition and general appearance of these beaches, I know

nothing better than that which Sir Archibald has himself put on

record— for in geological matters he is to be implicitly trusted.

It is only when weighing archseological facts in the balance of

probability that he becomes vulnerable. In the following extract

he brings both p>arties in perfect agreement to the very core of

the controversy, and admirably places before us the materials

on which our keenest deductive faculties are henceforth to be

exercised ; —

" The Firths of Clyde, Forth, and Tay are each bordered with a strip
of flat land, varying in breadth from a few yards to several miles, and
having a pretty uniform height of 20 or 25 feet above high-^^'ater mark.
This level terrace is the latest and, on the whole, tlie most marked of the
raised beaches. It must have been formed when the land was from 20 to
30 feet lower than at present, and evinces an upheaval which was nearly
uniform over the whole of the central valley of Scotland. What, then,
was the date of this upheaval ? The discovery of human remains in the
sands and clays of the raised beach affords the only ground for an
answer to this question. From these strata canoes, stone hatchets, boat-
hooks, anchors, pottery, and other works of art have been exhumed on
both sides of the island.''

Sir Archibald first deals with the Clyde Canoes, and, at the
outset, makes some judicious observations on the nature of the
evidence to be derived from their study. " It must be borne in
mind," he writes, "that the occurrence of these canoes in the
same upraised silt by no means proves them to be synchronous,
nor even to have belonged to the same geological period." After
discussing the various degrees of technical skill displayed in their
construction, he concludes that "the only evidence that remains
is that which may be afforded by the character of the antiquities."
But yet, in face of this well-selected and, indeed, unassailable
position, he deliberately pens the following remarks as his final
opinion on the evidential value of the Clyde canoes on the
upheaval problem : —

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246 Proceedivgs of Royal Society of Edinburgh. [siss.

** It is plain that the inlanders who buUt this primitive fleet were not
only acquainted with the iiae of metal, but that before they could have
cut out the more highly-finished canoes they must have been long
familiar with its use. They must have had serviceable metal tools
wherewith they could saw an oak through cleanly and sharply at its
thicker part, make thin oaken boards and planks, and plane down a
large tree into a smoothly cut and polished canoe. They had advanced,
too, to a high d^ree of mechanical ingenuity." . . . "Two of the
canoes were built, not out of a single oak stem, but of planks. That of
Bankton, already described, had its deals fastened to strong ribs like a
modem boat ; its prow was turned up * like the beak of an antique
galley,' and its whole build suggests that the islander who constructed it
may have taken his model, not from the vessels of his countrymen, but
from some real galley that had come from a foreign country to his
secluded shores. Nor is this the sole ground for inferring that, at least
at the time indicated by some of these canoe«, the natives of the west of
Scotland had some communication with a more southern and civilised
race How otherwise are we to account for the plug of cork ? * It could
only have come from the latitudes of Spain, Southern France, or Italy.
By whom, then, was it brought ? Shall I venture to suggest that the old
Briton who used it was not so ignorant of Roman customs as antiquaries
have represented him, and that the prototype of the galley-like war-
boat may have come from the Tiber to the Clyde? But whether
such a suggestion be accepted or not, it is abundantly evident that the
elevation of the bed of the estuary, by which the canoes have attained
an altitude of sometimes 22 feet above high-water mark, cannot be
assigned to the rude ages of the Stone period, but must have taken place
long after the islanders had become expert in the use of metal tools."
(Journal^ p. 224.)

The above sweeping deduction, with which he brings the Clyde
canoe-controversy to an end in conformity with his own views, is
the weakest link in the whole chain of his arguments, as there is
really no logical connection between the premises and the con-
clusion. Nor does it require much critical acumen to expose
where the fallacy comes in. Some of these Clyde canoes have
been found above, at, and below present high-water mark. In
discussing the chronological problems suggested by their respec-
tive positions, it must be borne in mind that, as boats may be
submerged in any depth and afterwards become silted up, their
final positions afiford no reliable criterion for determining the

* One of the Springfield group had n hole in its bottom said to contain a
cork plug. The Clyde canoes were found at au average depth of 19 feet
beneath the surface of the ground, and about 100 yards back from the
original edge of the Clyde, chiefly in a thick bed of finely-laminated sand.
(Smith's JV<fu?cr Pliocene Geology, p. 163.)

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relative level of sea and land at that time. It is only when they
are found in marine stratified beds above high-water mark that
their final positions can have any bearing on this point. Mr
Robert Chambers (Ancient Sea Martjins^ p. 206) describes the
situation of the boats found under the Tontine and Trades' Lands
as twenty-one or twenty-two feet above high- water in the river,
but this is the only instance in which such a height has been
recorded. The canoe containing a stone celt, found under St
Enoch's Church, lay at a depth of 25 feet from the surface, but
of course that does not indicate the height of tlie site above high-
water level. Since the publication of Mr John Buchanan's paper
describing the discovery of eighteen canoes in the bed of the
Clyde, and from which Sir Archiljald derived his data, seven
additional canoes have been recorded from the same place, five of
them being prior to the 2nd February 1869.

On that date Mr Buchanan, in an address to the Glasgow
Archaeological Society, made the following statement: — "The
last of the five canoes was found also last summer, a h'ttle
below Milton Island, near Douglas. It is 22 feet in length
and about 2 feet 10 inches in breadth. The interior is well
scooped out Some interesting relics were got inside. These
consist of six stone celts, an oaken war-club, and a considerable
piece of deer's horn." To what age would Sir Archibald assign
this canoe? Judged by the character of the antiquities, which,
according to his own dictum, is the only chronological criterion
admissible, the Stone Age is undoubtedly here indicated.

It must not, however, be forgotten that canoes do not neces-
sarily carry us back to prehistoric times, as they are frequently,
if not invariably, associated with crannogs and other mediaeval
structures. It is therefore extremely probable that some of the
Clyde fleet may have been comparatively modern. A few years
ago a fine specimen of the dugout was discovered close to the
site of the so-called crannog of Dumbuck, in a kind of dock
of artificial construction, and just barely covered with mud. At
low-water its site was exposed for several hours, but at high
tide it was submerged to a depth of 8 to 12 feet. Again, some
years ago four canoes were discovered in the Loch of Kilbirnie,
one of which contained a lion- shaped ewer and a three-legged

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248 Proceedings of Royal Society of Edinburgh. [skss.

pot, both made of brass or bronze — relics which, of course, relegate
the cauoe to late mediseval times (Ancient Scottish Lake Dwelliv-gs,
p. 66). The canoe exposed during the excavation of the Buston
crannog had been mended by boards fastened to its sides by
wooden pins. A gold coin of the sixth or seventh century found
in the debris gives some clue to the date of this crannog. (Ibid.^
p. 206.)

As to the difficulty about the cork boat-plug, if the material
really was cork, there is no valid reason why it would not
have been brought to the Clyde by trading vessels in Roman
or post-Roman times. Had the clinker-built boat been deposited
in stratified marine sands anywhere within the substance of
the 25-feet raised beach above present high-water mark. Sir
Archibald's deduction would have some foundation in fact. But
the record is silent on this crucial point, and only states that
the boat lay keel uppermost, as if swamped in finely- laminated
sands, about 250 feet back from the ancient river-margin. Its
position relative to sea -level may, however, be approximately
inferred from the fact that it was found near Mr Thomson's
new shipbuilding yard. Allowing its depth below the surface
to have been 19 feet (see footnote, p. 246), it is manifest, from
the lowness' of the locality, that its site could not have been
much above, but possibly greatly below, the level of present
high-water mark.

It is therefore quite evident that canoes were used on the
Clyde, without any break of continuity, from the Stone Age
down to mediaeval times. But no specimen, to my knowledge,
showing evidence of having been made in the Iron Age, or in
post-Roman times, has been recovered in circumstances which
would suggest that it was abandoned while the level of the
Clyde estuary stood 25 feet higher than at present. While,
therefore, the opinion that some of the Clyde canoes foundered
in the Stone Age prior to the formation of the raised beach,
has some foundation in fact, the inference that this change
had taken place "long after the islanders had become expert
in the use of metal tools" can only be regarded as a mere
gratuitous assertion, unsupported by any kind of evidence.

Sir Archibald Geikie next deals with the archaeological phe-

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1903-4.] Date of Upheaval of Raised Beaches in Scotland, 249

nomena of the Forth valley. He begins by giving an excellent
account of the composition of the Carse lands, with a description
of the whale skeletons, and the deer -horn implements found
along with them. It may be mentioned that since then another
deer-horn implement associated with a whale skeleton has been
foimd, and, having fortunately come into the possession of Sir
William Turner, is now carefully preserved in the Anatomical
Museum of the University of Edinburgh (fig. 1). It is the
only one of its kind now available for study, all those previously
recorded having been lost. By Sir William's kind permission
I have had the privilege of publishing an illustration of this
unique object (Prehistoric Scotland, p. 58), from which it will
be seen that it is not a harpoon, but a veritable hammer-axe,
made of a portion of the beam of a stag's antler, and perforated

Fig. 1. — Hammer-axe head of stag's horn, found with a whale'6 skeleton
at Meikle wood,' near Stirling^. (J.)

for a handle. Judging from their de^criptive records, the other
horn implements (some two or three in number), which were
found associated with cetaceous remains, were evidently of the
same kind, and had been . used by the natives to cut the blubber
from the stranded whales. **The circumstances under which
these remains were found," writes Sir .Archibald (p. 226), "leave
no possibility of doubt that the land here has been upraised
at least 24 feet, and that this upheaval has been witnessed by
man. The horn weapons do not, indeed, indicate an advanced
state of civilisation; yet they unquestionably prove the presence
of a human population, perhaps contemporary with that wliich
built the ruder canoes of the primitive fleet of Glasgow."

While cordially agreeing with the inferential statements in
the above extract, let us note the admission that some of the
Clyde canoes might have been contemporary with the whale

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250 Proceedings of Boyal Society of Edinhurgh. [

catastrophe in the Forth, i.e. when the Carse lands were still
submerged — for it is not admissible to suppose that the date
of elevation was different in the two localities. The fact of
the matter is, that neither the whale skeletons nor horn implements
have any bearing on the date of the raised beach, beyond proving
that primitive races inhabited the Forth valley when the school
of whales were stranded in the shallow sea which then occupied
its upper reaches. Had the horn axe-head been made of iron
or had worked objects of undoubted Roman origin been found
along with any of the cetaceous remains, the date of upheaval
would unquestionably have been brought down to post-Roman
times.

The evidential materials of the Forth valley, by which the
upheaval is brought within the domain of positive chronology,
are thus set forth: —

"In the elevated alluvial plains of the Forth, canoes similar to
some of those of the Clyde have also been found. One was dug up
on the Carse, not far from Falkirk, from a depth of 30 feet Early
in the last century, too, a flood in the river Carron, which flows through
the Carse, undermined a part of the alluvial plain, and laid bare what
was pronounced at the time to be an antediluvian boat. It lay 15 feet
below the surface, and was covered over with layers of clay, moss,
shells, sand and gravel. Its dimensions were greater than those of any
other canoe yet found in Scotland, for it reached a length of 36 feet
with a breadth of 4 feet. ' It was described by a contemporary news-
paper as finely polished and perfectly smooth, both inside and outside,
formed from a single oak-tree, with the usual pointed stem and square
stem.'

" These features," he goes on to say, " seem to harmoniBe well with
those of the more perfect of the Clyde canoes, and to justify the inference
that they were produced by the employment, not of stone, but of metal
tools.

" But on the Carse of the Forth an implement of metal has actually
been found, and one formed not of bronze, but of iron. It was an iron
anchor, dug up a little to the south-east of the place from whence the
Dimmore whale was obtained. The exact depth at which it lay is not
given ; it was probably about 20 feet above high- water. . . . Pieces
of broken anchors have also been found below Larbert Bridge, near
Camelon.

" Putting together, therefore, the archieological evidence to be gathered
from the contents of the elevated silt of the Forth, the inference, I think,
can hardly be avoided that not only was the upheaval effected subsequent
to the first human immigration, but that it did not take place until the
natives along the banks of the Forth had learned to work in metals, and

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until vessels sailing over that broad estuary had come to be moored with
anchors of iron." (/few?., p. 216.)

The non sequitur of the latter half of the above conclusion is too
transparent to mislead any cautious reader, but yet, so as to leave
no loophole for escape, we will consider seriatim the various items
on which it is founded.

(1) In the absence of precise details of the relative positions of
the Carron and Falkirk canoes to present high-water level, and of
the general circumstances in which they were found, it would be
sheer folly to draw any inference as to whether they were swamped
or abandoned before or after the upheaval. If depth or thickness
of the superincumbent materials be a valid criterion of age, then
both these canoes must have been far older than the whale
skeletons, which lay only a few feet beneath the surface of the clay.
Then again, the well-known shiftings of river and estuary detritus
during floods are the effects of powerful natural agencies, which at
one time unearths the works of antiquity, and at another buries
those of modernity under fathoms of gravel and mud.

(2) The story of the iron anchor said to have been discovered
near the site of the Dunmore whale skeleton is thus recorded by
Mr Keddoch in a letter to Professor Jameson (Edin. Phil Jour.,
vol. xi. p. 416): —

^ Many years ago an iron anchor was dug up arlittle to the south-east of
it (the whale skeleton). The fleuks (sic) were much decayed, but the
beam, which was of a rude square form with an iron ring, was tolerably
perfect. It hung many years in the old tower near Dunmore, but was at
length stolen. Dunmore Moss extends a great way to the south-west, and
in it, at about 300 yards from the skirts of the wood, are found the roots
of large oaks."

From this record we have no certainty that the writer had ever
seen this anchor, or examined the conditions under which it had
been found, so that he is merely repeating hearsay evidence. We
are informed that the skeleton of the Dunmore whale was 200
yards from the then bed of the Forth, so that " a little to the south-
east of it " would be in the direction of the river ; but it would be
useless to speculate on the precise distance. From the constant
shifting of the windings of the Forth, there is nothing very
improbable in the discovery of a small anchor belonging to a
comparatively modern boat in this raised beach. Such anchors are

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252 Proceedings of Roycd Society of Edinhurgh [sBaii.

not usually thrown in deep water, like those of large vessels, but
on the shore, and one might have been easily lost and buried in
the mud during a storm. At any rate it would be a violation of
the rules of scientific archsBology to admit such vague statements
as evidence that the raised beach was formed after iron anchors
came to be used in the Forth, or that this particular one had any
chronological relationship with the " Dunmore whale."

(3) The chronological value of the pieces of anchors found
below Larbert Bridge may be estimated by the perusal of the
following extract from Nimmo's History of Stirlingshire^ one of
the authorities quoted for the statement : —

^* After the river hath left the village and bridge of Larbert, it soon
comes up to another small valley, through the midst of which it hath
now worn to itaelf a straight channel, whereas, in former ages, it had
taken a considerable compass southwards, as appears by the track of the
old bed, which is still visible. The high and circling banks upon the
south side give to this valley the appearance of a spacious bay ; and, as
tradition goes, there was once an harbour here. Nor docs the tradition
appear altc^ther groundless ; pieces of broken anchors have been found
here in the memory of people yet alive, and the stream-tides would still
flow near the place, if they were not kept back by the great damhead
built across the river at Stonehouse. There is reason, too, to believe that
the forth flowed considerably higher in former ages than it does at
present ; so that there is no improbability in supposing that at least
small craft might have advanced thus far. In the near neighbourhood
of this valley stands the ruins of ancient Camelon, which, though we
have no ground to believe that it ever had possessed that d^ree of
extent and splendour which some credulous authors mention, yet might
he inhabited by the natives of the country for several ages after it was
abandoned by the Romans." (Page 73, 2nd ed.)

Of all the explanations that might have been offered as to how
small anchors came to be dropt in a locality to which even now
the tides reach, the hypothesis that the level of the sea was then
25 feet higher than at present is surely the least satisfactory.
Would it not be more rational to suppose that in earlier times the
embouchure of the river Carron was more inland, and that
consequently the tides flowed further up? * Is there no allowance
to be made for the accumulation of the detritus brought down by

* On referring to the Ordnance maj*, I find the highest point to which the
ordinary spring tides now flow is at a sluice in the Carron Ironworks, from
which Camelon is less than a mile distant.

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its floods during so many centuries ? Besides, the flowing of the
tides 25 feet higher would by no means help to explain the
position of the anchors, as it is more likely that they would be lost
on the shallow margin of a tidal river than in a depth of 25 feet
of water.

As a preliminary to the discussion of the more important
archeeological phenomena of the Firth of Tay, Sir Archibald
points out, in the words of Mr Robert Chambers, that " along the
Carse of Growrie many of the hillocks and eminences which rise
above the general level of the plain bear names in which the Celtic
word inch (island) occurs ; such as Inchyra, Megginch, Inchmichael,
Inchmartin, Inchsture — as if a primitive people had originally
recognised these as islets in the midst of the shallow firth."
{Ancient Sea Margins, p. 18.) To this is added the evidence of
tradition to the effect that the Flaw Craig and the rock on
which Castle Hiintly stands bore iron rings, to which ships were
fastened when the sea covered the surrounding carse lands.
Finally, we have the following statement of the discovery of
specific objects of iron, to which the author seems to attach great
importance : — " Between 60 and 70 years ago a small anchor was
dug up, not many feet beneath the surface, on a piece of low
ground near Megginch (N, St Ad., "Perthshire," p. 378). Mr
Chambers refers to another anchor as having been met with in
casting a drain below the Flaw Craig (Ancient Sea Margins, p. 19).
But the most important and the most carefully investigated relic yet
discovered in the district was an iron boat-hook (fig. 2), found
in 1837 by some workmen on the farm of Inchmichael." (Ibid,,
p. 19; and N, Phil, Journal, 1850, p. 233.)

It is not surprising that the discovery of such an array of
relics associated with early navigation, especially whjBn brought
before us by so skilled a writer, should carry some weight with
general readers. It is therefore all the more necessary to inquire
what their chronological value may be.

With regard to the philological argument that the Gaelic word
inis (an island) appears in the composition of several place-names
in the Carse of Gowrie, it will be sufficient to observe that its
English equivalent, inch, has often been applied to low-lying
meadows near water, such as the North and South Inches in the

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254 Proceedings of Royal Society of Edinburgh. [srss.

town of Perth, which never were islands. The story of the
existence of iron rings in the adjacent rocks for the mooring of
boats wants the essential link of an eye-witness to make it
admissible as an argument in this inquiry. There i-emain, there-
fore, to be seriously considered the circumstances under which the
two anchors and boat-hook were discovered.

The Megginch anchor is thus referred to by the author of the
article on " Perthshire " in the N. St. Act. of Scotland (p. 378) :—

"The writer has conversed with a man who told him that he
recollects distinctly of hearing his father state that, at a period of

FiO. 2. —Boat-hook of iron, found in Caree of Gowrie. (^. )

about forty years ago, the latter was engaged in digging in a piece
of very low ground on the estate of Megginch, not many feet
beneath the surface, when he and his fellow labourer found a small
anchor, the figure of which was tolerably preserved, but which
mouldered down or went to pieces when lifted."

The discovery of the other anchor and the boat-hook is recorded
by Mr Robert Chambers (Ancient Sea Margins, p. 20) : —

" In the same district, which is fully a mile from the margin
of the firth, a boat-hook was discovered 8 feet below the surface,

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1903-4.] Date of Upheaval of liaised Beaches in Scotland. 255

sticking among the gravel, as if left by the tide on the sea-shore.
This relic has been preserved by the farmer who found it.*

^' I am also assured that what was considered as the remains of
an anchor were found some years ago in casting a drain below
Flaw Craig, a clitf which overlooks the Carse, between Kiimaird
and Fingask."

Mr Chambers takes the precaution to state that for these
remark)!, and others which followed, he quotes from " a letter from
a lady, the daughter of one of the chief proprietors of the Carse.''
Subsequently, however, owing to the importance of the subject,
he recurs to it (Edin. Phil. Journal, vol. 49, p. 233, 1850), and
informs us that he ''took some trouble to ascertain the precise
local and geological circumstances of the relic, as observed at the
time of the discovery.

It is unnecessary to epitomise the result of this inquiry, the
upshot of which was that the spot where the boat-hook lay was
8 feet below the surface, 20 feet above the level of present
high tides, and about a mile distant from the estuary of the Tay.
It is advisable, however, to quote tlie following incidental remarks,
which seem to contain the germ of a more natural explanation of
its presence in the locality than that of Sir Archibald Qeikie.

"One important feature of the Carse in this district is now to
be adverted to, namely, a trench or ditch in which a little rill
crosses the plain obliquely to join the estuary in one of those
creeks locally called paws. The distance of this rill is not
more than 150 yards from the spot where the boat-hook was
discovered. It is, in these days of high cultivation, a narrow
ditch of well-defined sides, but no one can doubt that in other
times it would comprehend a wider space. Now, the bottom of
the ditch at this place is so little above the level of the sea that
an abnormal tide might reach it."

After describing several instances of great floods Mr Cliambers
writes:— ""With such events as those on record, within the period

*Thi8 object (fig. 2) is now in the National Museum of Antiquities, Edin*
bnigh, and consists of a socketed spike, 11 inches in length, from tlie middle
of which the hook curves backwardn. The socket is formed by the backward
folding of the irou, the edges only partially meeting, and in it the handle
was fixed by a rivet. From its appearance, it mi^ht belong to compara-
tively recent times.

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256 Proceedings of Royal Society of Edinburgh, [ssst.

during which iron implements have been in use, it does not appear
very difficult to account for the loss and embedding of the Inch-
michael boat-hook, without calling any greater geological forces
into operation in the case."

Mr Chambers' idea, that a flood might account iov the stranding
of the boat-hook, was opposed by Sir Archibald Geikie, on the
ground that the effects of a storm would not adequately explain
the geological phenomena. "We can hardly conceive," he writes,
" the sea rising upwards of 28 feet above high- water mark, and
flowing for more than a mile inland ; still less can we believe that,
if it did so rise, it could deposit 8 feet of sediment over the
surface of the Carse." But, waiving the intervention of a flood,
is there anything very improbable in the supposition that the pow^
described by Mr Chambers as little above present sea-level, wias
formerly sufficiently deep, either by natural or artificial means, to
admit of a boat being rowed to the spot? Before the days of
railways, harbours, and piers, trading vessels were beached on
convenient places for the purpose of loading or unloading their
cargoes. But surely it is unnecessary to discuss the possible ways
in which such a portable object as a small boat-hook might have
got strayed. The suggestion that it was lost by a sporting sailor
in a wild-boar hunt is as feasible an explanation as that it was
dropt from a sailing-vessel while the Carse lands were still sub-
merged. But whatever the true explanation may be, there can be
no doubt that this boat-hook is a relic of post-Roman times, and
probably much nearer the present day than the Roman period.

Sir Archibald's next and final argument in support of his thesis
is the relative positions of the ends of the Wall of Antoninus to the
high-water marks in the adjacent estuaries. It is thus presented
to us : —

'^ Mr Smith of Jordan Hill was the first to assert that since the Antonine
Wall was built (about a.d. 140) there could have been no change in the
relative position of sea and land, inasmuch as the ends of the wall were
evidently constructed with reference to the existing level {Mem, Wtm,
Soc.y viii. p. 68, and Edin. New Phil, Journal, vol. xxv., for 1838, p. 386).
This statement has been the foundation of all the subsequent geological
arguments as to the long period at which the British Isles have been
stationary. If it be true, then we must allow that the upheaval, of which
the evidence has been adduced in the present conmiunication, is referable

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to a period certainly previous to the Roman invasion. If the statement
be erroneous, the other alternative remains, that the upward movement
may have been wholly or in part effected after the Roman invasion,

" After carefully examining both extremities of the wall, and reading the
narratives of the vaiious antiquaries who have treated of the Roman
remains in Scotland, I have no hesitation in affirming that not only is
there no evidence that the wall was constructed with a regard to the
present level of the land, but there is every ground for believing that it
was built when the land was at least 20 feet lower than it is at present.
To begin with the east end : from the Avon, west of Borrowstounness,
eastward to Carriden, the ground rises from the old coast line as a steep
bank, the summit of which is from 50 to 100 feet above the sea ; between
the bottom of this abrupt declivity and the present margin of the Firth
there is a narrow strip of flat ground, about 200 yards broad, on which
Borrowstoimness is built, and which nowhere rises more than 20 feet
above high-water. It is a mere prolongation of the Falkirk carse,
already described, and beyond doubt formed the beach where the sea
broke against the base of the steep bank. Now the Roman Wall was
carried, not along this low land bordering the sea, but along the high
ground that rose above it. The extremity at Carriden, therefore, instead
of having any reference to the present limit of the tides, actually stood on
the summit of a steep bank overhanging the sea, above which it was
elevated fully 100 feet. If the land here were depressed 25 feet, no part of
the wall would be submerged. The only change on the coast-line would
be in the advance of the sea across the narrow flat terrace of Borrowstoun-
ness and Grange, as far as the bottom of the abrupt declivity.

"The western termination of the Antonine Wall stood on the little
eminence called Chapel Hill, near West Kilpatrick, on the north bank of
the Clyde. Between this rising ground and the margin of the river lies
the Forth and Clyde Canal, the surface of which is 20 feet above high-
water mark, and the base of the hill at least 5 or 6 feet higher. Hence
the wall terminated upon a hill, the base of which is not less than 25 feet
above the present level of the sea. In making the canal, a number of
Roman antiquities were found at various depths in the alluvium : these
seem to have been part of the ruins from the fort above. If we admit
that the wall was constructed previous to the last elevation of the land,
we see a peculiar fitness in the site of its western termination. The
Chapel Hill must, in that case, have been a promontory jutting out into
the stream, and at high-water the river must have washed the base of the
Kilpatrick Hills— a range of heights that rise steeply from lower grounds,
and sweep away to the north-east. Hence, apart altogether from con-
siderations dependent upon the strategic position of the hills, which were
infested by the barbarians, we obtain an obvious reason why Lollius
Urbicus ended his vallum at Old Kilpatrick."— (/Wd, p. 228.)

For the purpose of homologating these views, he quotes passages
from the writings of various antiquaries, the most pertinent of
which are the following : —

PBGC. ROY. see. EDIN. — VOL. XXV. 17

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258 Proceedings of Royal Society of Edinburgh [sjsss.

" If tlie Falkirk carses were not entirely overflown in the time
of the Romans, it is probable at least that they were then salt-
marshes, subject in some degree to temporary inundations in high
spring tides." (Roy, Military Antiquities, book iv. c. iii. sect. 2.)

Mr Stuart, author of Caledonia Roiiuma (p. 177), declares his
belief that " the whole of this lower district (towards the mouth of
the Carron) had in all likelihood been covered by the sea when
the Roman forces occupied the Wall of Antonine. It is likewise
probable that the entire plain between Inneravon and Grahams-
town (that is, the whole of the Falkirk cai-se) was at the same
period subject to the influx of the tide, which may even have
penetrated the deeper hollows of the Carron as far up as
Dunipace."

In a footnote at the end of his long communication, Sir Archibald
writes as follows : —

" I have not deemed it necessary to increase the length of this com-
munication by controverting the alleged Roman origin of certain road-
ways and other traces of art. found along the present coast-line at a
height of less than 20 feet al)ove high-water mark. The causeway of
logs, for instance, which crossed a part of the Kincardine Moss, in the
Carse of Stirling, is commonly spoken of as Roman, but this is mere
conjecture. The bronze vessel found in the same moss, and cited by
some writers as a Roman camp-kettle, is most certainly of ancient British
workmanship."

The final conclusions drawn from these elaborate investigations
are thus stated : —

" Putting together all the evidence which the antiquities yet dis-
covered along the Scottish coast-line afibrd as to the date of the last
upheaval of the country, we are led to infer that this upheaval must have
taken place long after the first human population settled in the island —
long after metal implements had come into use, after even the introduc-
tion of iron ; and reviewing the position and nature of the relics of the
Roman occupation, we see no ground why the movement may not have
been effected since the first century of our era ; nay, there appear to be
several cogent arguments to make that date the limit of its antiquity "
(p. 232).

The publication of Sir Archibald's essay naturally attracted
attention. His theory as to the date of the 25-feet raised beach
was accepted by some of the leading geologists and archaeologists
of the day, among whom were Sir Charles Lyell {Antiquity of

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i90i-4.] Date of Upheaval of Raised Beaches in Scotland. 259

Man, 3rd ecL, p. 50 et seq.)* Sir Daniel Wilson {Prehistoric
AnnalSy vol. i. p. 38), and Professor Ramsay (Geology and
Geography of Great Britain, p. 251). On the other hand,
several local geologists raised objections on various grounds to
the validity of some of his arguments. Mr Alexander Bryson,
F.R.S.E., contended that the so-called Roman pottery from the
Leith sand-pit were merely fragments of dishes made, within the
memory of living persons, at a Portobello manufactory, and of
glazed flower-pots which skippers were in the habit of bringing
from Holland to adorn their parlour windows (Proc. Roy, Phys.
JSoc, vol. iii. p. 284). In 1873 David Milne Home, Esq.,
fluccessfully controverted his deductions from the height of the
ends cf the Antonine Wall above present sea-level (Trans. Roy.
aSoc. Edin., vol. xxvii.) — a result mainly due to the discovery in
1868 of a Roman sculptured tablet which definitely fixed the
eastern termination of the wall to be at Bridgeness, and not at
Carriden, as was generally supposed when Sir Archibald wrote his
paper.

Mr Home's chief argument was that the position of the tablet
sX Bridgeness proved that the Antonine Wall terminated so close
to the sea as to preclude the idea that, when that wall and tablet
were inserted, the land could have been 25 feet lower than now.
The spot where the tablet was found was exactly 19 feet above
ordinary spring tides, and at the place where it lay there was a
quantity of squared stones in a confused heap, some of which bore
the marks of masons* tools, evidently forming part of the wall in
which the tablet had been fixed. At the point, and only one or
two feet above present high-water mark, a portion of a building
was discovered, a few yards in length, consisting chiefly of large
whinstone boulders. The line of this building pointed towards
the place where the tablet was found, " so that if the building had
continued on the same line, it would have passed through or near
the site of the tablet." The effect of these discoveries on the
post-Roman theory of the upheaval is thus stated : —

* It Hppears that Sir Charles Lyell, in consequence of the articles of
Mr Milne Home, abandoned the post-Roman theory, and accordingly his
remarks on the subject were deleted from the fourth edition of his Antiquity
of Man. Trans, of the Roy. Soc. Edin., vol. xxvii. pp. 39-41.

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260 Proceedings of Royal Society of Edinburgh, [sess.

"If the land was then twenty-five feet lower than now, then the
tablet, and the wall in which it was fixed, must have been six feet under
the sea at every tide, and must also have been so exposed to the beating
of the waves that neither tablet nor wall could have stood many weeks.
It is impoesible to suppose that the tablet, with elaborate sculpturing,
and bearing a dedication to the emperor, could have Ijeen set up in such
a position. Moreover, the neck of land which joins the ness or knoll to
the mainland being only twenty-three feet above high-water, must have
been submerged and exposed, so that any wall or rampart on that neck
would soon also have succumbed to the waves. Then there is the old
building at the point of the ness, which, if Roman (as it appears to be),
must have been aJt all times under water, even at the lowest tide, were
Professor Geikie's theory correct." (Trans. Roy, Soc. Ed.y vol xxvii.
p. 45.)

In criticising Sir Archibald Geikie's speculative deductions,
founded on the geological and archaeological phenomena connected
with the western termination of the Antonine Wall on the top of
Chapel Hill, Mr Home thus expresses himself : —

" If the Roman antiquities here mentioned (see page 257) be the same
as those described in the Statistical Account^ their position is not
correctly stated by Professor Geikie. They can in no sense be re-
presented as having fallen from the fort above. The relics were found,,
not (as he says) at various depths in the alluvium, but in a subterranean
recess — i,e, in a cavity which contained them. As there were vases as well
as coins, the probability is that it was a grave. Now, as this recess,
when formed, must have been several feet below the surface of the
ground, and as the surface of the ground is admitted to have been only
twenty feet above the present high-water mark, the * recess ' must have
been at least seven or eight feet imder the sea if, during the Roman
occupation, the land was twenty-five feet lower tlian now." (Ibid,, p. 48.)

Hitherto my chief r61e in this controversy has been to meet the
statements and logic of the advocates of the post-Roman theory
with a non sequitur on all the points raised — of course utilising for
this purpose the arguments advanced against it by previous ^vrite^s
on the subject. Henceforth, however, I become a direct supporter
of a theory about these beaches which I have elsewhere formulated,,
and which for distinction may be called the pre-Roman theory,
viz., that the upheaval took place " subsequent to the appearance
of man in the district, but prior to its occupation by the Romans."
This was the conclusion come to in an address which, as president
of the Antiquarian Section of the Archaeological Institute, I gave
at Lancaster in 1898 (Journal , vol. 55, pp. 259-285).

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1908-4.] Date of Upheaval of Riisei Beaches in Scotlaind, 261

In looking about for positive evidence in support of the pre-
Roman theory, we shall first of all deal with the wooden roadway
and the so-called Roman camp-kettle, which Sir Archibald Geikie
did not think of sufficient archaeological value to be discussed
among the evidential materials from the Forth valley.

Nothing can be more certain than that the chronological
sequence in the physical phenomena of the Forth valley was sea,
forest, peat, and modem cultivation — the last stage being due to
the removal of the peat by the hand of man. Now, objects of
human workmanship which happened to be lost or abandoned in
these woods became ultimately covered over with peat, and so were
less liable to the ordinary processes of decay. Hence such
relics, when recovered in these circumstances, are often in an
excellent state of preservation. Of the condition of the peat mosses
of Kincardine and Flanders towards the end of the eighteenth
century, we have a good account by the Rev. Christopher Tait,
minister of the parish of Kincardine {Tram, Roy, iSoc. Edin.y
vol. iii.), from which the following is an interesting extract : —

" The trees are oak, birch, hazel, alder, willow, and in one place there
are a few firs. Among these the oak aboimds most, especially on the
west side of the moss, where forty large trees of this species were lately
found lying by their roots, and as close to one another as they can be
supposed to have grown. One of these oaks measures 50 feet in length
and more than 3 feet in diameter, and 314 circles or years' gro\vths were
counted in one of the roots." (/6m/., p. 272.)

He further observes that the trees were not blown down, but cut
about 2 feet from the ground. " The marks of an axe, not ex-
ceeding 2| inches in breadth, are sometimes discernible on the
lower ends of these trees."

The Roman roadway is thus described : —

"That a people more civilised than the ancient Caledonians must
have been in this country before the moss of Kincardine existed is
completely established by the discovery of a road on the surface of the
clay at the bottom of that moss, after the peat, to the dej^th of 8 feet,
had been removed. The part of this road already discovered is about
70 yards long ; the breadth of it is 4 yards, and it is constructed of trees
measuring from 9 to 12 inches in diameter, laid in the direction of the
road. Across these have been laid other trees about half their size, and
the whole has been covered with bmshwood. The depths of the materials
varies in conformity to the nature of the soil ; the trees, which arc laid

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262 Proceedings of Royal Society of £dinburgh. [i

lengthwise, being generally on the surface of the clay, but in the lowest
and wettest parts they are sunk about 2 feet under the surface.

^'This road lies across a piece of ground lower than the adjacent
grounds, and its direction is from the Forth across the moss, where it is
narrowest, towards a road, supposed to be Roman, that passes between
the mo58 and the river Teith. The vestiges of this last road have been
traced, from about four miles north-west of the Bridge of Drip, where
formerly there was a ford across the river, south-east of Torwood and
Larbert, to Camelon on the wall." (Ibid , 276.)

The signiiicance and bearing of this road on the upheaval
question is concisely stated by Mr Milne Home as follows : —

"The tide now comes up to Craigforth, which is about half a mile
below Drip, and with a fall of only 4 feet between the two points. If,
therefore, the land wa^ during the time of the Romans 25 feet lower
than now, neither the Drip Ford nor any river could then have existed,
for the whole country west of Stirling must have been covered by the
sea, even at the lowest spring tides." (Ibid.y voL xxvii p. 49.)

The finding of portions of similar roadways in Flanders Moss is
noticed by several writers of the period. One such structure,
described as having logs lying across each other like a raft, with
a general direction from south-east to north-west, is supposed to
have been a branch of the Roman way from Camelon.

The general evidence, over and above tradition, which associates
these roads with the incursion of the Romans into the valley, has,
in my opinion, considerable weight, certainly more than can be
expressed by the words '* mere conjecture.'* Historians are almost
unanimously of the opinion that the march of the soldiers of
Agricola to the estuary of the Tay was from Camelon, via Stirling,
Dunblane, Ardoch, and Stratheme ; in which case the most con-
venient place to cross the river Forth would be a few miles
to the west of Stirling (as shown on the map in Gordon's Itiner-
avium Sepientrionale), and just in line with the wooden causeway
in the Kincardine Moss. In support of this view the following
fact is worth mentioning. It will be recollected that the Rev.
Mr Tait, in noticing the cutting marks on the felled trees found
in the Kincardine Moss, describes the axe cuts as not exceeding
2^ inches in breadth. Now it is very significant that the only
iron axe-head found in the Ardoch camp, during its recent ex-
ploration by the Society of Antiquaries, measured 5| inches in

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1903-4.] Date of Upheaval of Raised Beaches in Scotland, 263

length by 2 J inches across its cutting face (ProCj vol. xxziii.,
fig. 14, p. 463).

If it be true, then, that when the Romans invaded Scotland,
* towards the close of the first century a.d , the areas subsequently
covered by peat within the 25-feet raised beach were then occu-
pied by great forests, it is but natural to suppose that objects lost
in these forests would be recovered, in modern times, in course of
the operation of removing the peat, so as to convert the rich clays
underneath into arable land. On this point Mr Milne Home
writes : — ** Stone hatchets and other stone implements of a very
primitive people have been found also on Blair-Drummond estate,
lying on the surface of the carse clay, after the peat moss lying
above it was removed. These implements were, as I understand,
in localities below or within the line of the old sea-cliff, and not
very far from where the Blair-Drummond whale was found. I
have seen three of these implements : one was in the Macfarlane
Museum, Stirling ; the other two in the possession of the late Mr
Home Drummond, who showed them to me at Blair-Drummond
in September 1863." (The Estuary of the Forth, p. 116.) This
would seem to show that the elevation made some progress in the
Stone Age.

Among other relics thus brought to light, there is one which
has a special chronological value, viz., a large bronze caldron
(fig. 3), now preserved in the National Museum of Antiquities,
Edinbui^h. It is recorded as having been found in 1768,
"upon the surface of the clay, buried under the moss.'* It is
made of thin plates of beaten bronze riveted together, the
rounded bottom portion being fashioned out of one piece, and
measures 25 inches in diameter and 16 inches in depth. The
everted rim is formed of a couple of bands of sheet bronze
fastened to the upper edge of the vessel, and bears marks of
the rivets by means of which a pair of ring-handles had been
attached. Sir Daniel Wilson informs us that two rings (pre-
simiably its detached handles), each measuring 4J inclies in
diameter, were found along with it. "No question," writes
Sir Daniel, "can exist of its native workmanship. The rings
and staples are neatly designed, but rudely and imperfectly
cast and finished, and are decorated exactly as those of the

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264 Proceedings of Royal Society of Ediiiburgh, [skss.

Farney caldron. The circles embossed on the side of the vessel
are, in like manner, such as have been frequently noted on
objects of the Bronze period, both in Britain and on the Continent.
Nevertheless, in accordance with the classical system of desig-
nation, which is even yet only partially exploded, this remarkable
native relic figures in the printed list of donations in the
Archoeologia Scotica as a Roman camp-kettle." {IbvL, p. 409.)

The acceptance of Sir Daniel's opinion as final carries with
it strong presumptive evidence to show that the surface of the

Fig. 3. — Bronze Caldron found in the Moss of Kincardine (25 inclies
diameter).

clay beneath the peat was already dry land in the latter part
of the Bronze Age — an admission which would at once give the
coup ffe grace to the post-Roman theory of the raised beaches.
But as this opinion may be controverted on the ground that the
caldron might be regarded as a survival from a former to a later
age, it is desirable to determine as accurately as possible the
chronological range of the class of objects to which it belongs.
Spheroidal bronze caldrons, similar in type and make to t)ie

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1903-4.] Date of Upheaval of Raised Beaches in Scotland. 265

Kincardine caldron, have been discovered elsewhere in Scotland,
as well as in various localities in England and Ireland. Of the
Scottish finds, some consist of merely ring-handles or other frag-
ments, such as were among the bronze hoards found in Budding-
stone Loch and at Kilkerran (Prehistoric Annals, vol. i. p. 349).
Entire specimens were, however, among the Bronze Age relics at
Dowris, King's Co., Ireland, and at Heathery Burn Cave, Dur-
ham (Ancient Bronze Implements, pp. 361 and 412; Proc, Soc.
Antiq., 2nd series, vol. ii. p. 132). On the other hand, analogous
caldrons, but perhaps not so artistically finished, have been
discovered at Cockbumspath, Berwickshire, and in Carlingwark
Loch, Kirkcudbrightshire, associated with iron tools and other
objects undoubtedly of post-Roman date. The former of these
Iron Age finds are thus described: —

"They included two lai^e vessels of extremely thin sheet bronze,
apparently with traces of gilding externally, and measuring, the one
about 21 inches in diameter and 10 inches in depth, and the other
13 inches in diameter and 7^ inches in depth. When found these vessels
were entire, and the one appeared to have been inverted on the other,
with the articles within them. The large one has obviously l^een much
exposed to the fire, and repeatedly repaired ; the smaller one has had
handles fastened to it on opposite sides by three rivets, the holes for
which remain, and it lias probably also been strengthened by a rim
of iron, without which it would collapse, from the extreme thinness
of the metal, if lifted full of water. It is probable that the whole were
contained in a large wooden pail, as there were two large rings with
staples and nails, the latter of which are bent in, indicating the thickness
of the staves to have been about | of an inch. The rings measure 4j
inches in diameter. There are also a number of iron hoops, broken and
crushed together, but which there can be little doubt encircled the
wooden paiL

" The objects enclosed included a bronze Roman patella of the usual
form, 6} inches in diameter, and with the bottom composed of concentric
rings in lx)ld relief, but wanting the handle ; the large iron chain figured
above, measuring 27 inches in length ; a circular bronze ornament,
apparently the shield to which the handle of some object has been
attached, measuring nearly 3 inches in diameter ; an iron lamp-stand,
similar to examples frequently foimd on Roman sites ; two iron knives,
one of them with a wooden handle ; an iron gouge ; two iron hammers ;
an iron tankard or jug, crushed flat ; two ornamental ends of pipes, like
the mouth-piece of a trumpet, of bright yellow bronze, and a mass of the
same metal weighing nearly 1 J lb." (Proc. S.A,, Scot,, vol. i. pp. 43, 44.)

The Carlingwark caldron, though of the spheroidal type, is

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266 Proceedings of Royal Society of Edinburgh. [sess.

slightly different in shape. It measures 26 inches in diameter
across the mouth, the sides being straight, but bulging out to
the extent of 1 inch above the rounded and somewhat flattened
bottom. When dredged up it contained a number of iron tools
and other objects — axes, hammers, staples, rings, a file, a saw,
a bridle-bit, a tripod, portions of chain mail, a bronze vessel,
green glass, etc. (Ibid., vol. vii. pp. 7, 10.) One or two other
spheroidal caldrons have been found in Scotland, but not being
associated with objects which furnish any chronological data
bearing on the problem at issue, they need not be discussed
here.

We now come to another series of caldrons which, though
made of plates of thin beaten bronze and riveted together in the
same way as that found in the Kincardine Moss, differ from
it in having a bucket-like shape and a flat bottom. A caldron
of this description (fig. 4) was discovered, some two generations
ago, in the north-west comer of Flanders Moss, on the Cardross
estate, "in what had always been considered to be a Roman
camp." This vessel, hitherto unique among Scottish antiquities,
was exhibited at a meeting of the Society of Antiquaries of
Scotland on 9th January 1888 by H. D. Erskine, Esq. of Cardross,
and a full description of it by Dr Joseph Anderson is inscribed
in their Proceedings for that year. It measures 19 inches in
height, 10 inches in diameter at the base, and 14 inches at
the mouth, widening to 16 inches at the shoulder. Two large
rings for suspension, passing through ornamental loops, are attached
to the inside of the lip. Although this is the only specimen
known to have been found north of the Tweed, several have
been met with in different parts of the British Isles, especially
in Ireland. The conjunction of both types of caldrons — the
spheroidal and bucket-shaped — in the Dowris and Heathery Bum
Cave bronze hoards shows that they were contemporary in Britain
at the close of the Bronze Age.

The foreign models, from which both these types of British
and Irish caldrons are derivatives, became first recognised among
the grave goods of an early Iron Age cemetery at Hallstatt
(Austria), which dates from about the eighth to the second century
BjC. These Hallstatt relics showed that the people of the dis-

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1903—4.] Date of Upheaval of Raised Beaches in Scotland. 267

trict had acquired the art of making thin plates of beaten
bronze, as vessels of that material analogous to the British cal-
drons just described were among them. They differed, however,
from tlie British types, inasmuch as the spheroidal forms on the

Fio. 4.— Bronze Caldron (19 inches in height) found at Cardross.

Continent had no suspension rings, but only handles riveted to
their sides, while the buckets had generally bow handles like
those of our common water-pails. As this Hallstatt civilisation
spread westwards in Europe, it gathered so many new ideas in
France and Switzerland that it became necessary to distinguish

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268 Proceedings of Royal Society of Edinbicrgh. [hess.

its art and industrial products in these countries under the
designation of La T^ne civilisation — a name derived from the
shallow outlet of Lake Neuchatel, where stood the Helvetian
oppidum which yielded its most characteristic relics. That both,
these culture streams had reached our shores is proved by the
discovery in Britain and Ireland of a number of objects whose
origin can be clearly traced to prototypes in Hallstatt and La
Tfene. But our insular artists, in the process of imitation, so
handled their materials as to give their works a sufiSciently
distinctive character to differentiate them from their original
models, and hence originated the style of art known as *Late
Celtic' Wlien the Romans took possession of Britain in the
first century a.d., this native art was in a highly flourishing
condition, but its further development in the southern portion
of the island was cut short by the introduction of the civilisation
of the conquerors. How long it was in existence previous to
this event it is difficult to say, but it is safe to assume that
some of its foreign prototypes reached the British Isles some
three or four centuries before the Christian era — a period which,
however, may be equated with the early Iron Age of Central
Europe. The presence of both the spheroidal and conical caldrons
in Britain and Ireland during the late Bronze Age shows that
their importation into or development in these countries was
altogether independent of Roman influence. I am unable to
agree with the general opinion that all these caldrons are of
native origin, although undoubtedly such vessels were made
at home. We are told in the Tripartite Life of St Partick
that the saint, when a boy in slavery in Ireland, was sold to
some mariners at the mouth of the Boyne for two caldrons of
bronze; also that Daire gave him an aerieum mirabilem trans-
marinum, i.e. " a wonderful brazen caldron from over the sea "
(Joice, Social History of Irelandj vol. ii. p. 124). At any rate
the most artistic specimens — in which category that found in the
Kincardine Moss must be reckoned — were not only prior to the
Roman occupation, but probably earlier than the most flourishing
period of Late Celtic art.

In corroboration of these views it may be observed that among
the antiquities found in Oppidum La Tene were about a dozen

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1903-4.] DcUe of Upheaval of Raised Beaches in Scotland. 269

caldrons, including both the spheroidal and conical types. The
former were always constructed on a uniform plan, the special
feature of which was a lower rounded portion made of thin
bronze, and an upper band of iron to which the lower was
riveted, and to which also were fastened two large suspension
rings. (See Gross, Oppidum Helvhte^ p. 45 and pi. xiii.)

It will be remembered that one of the Cockburnspath caldrons

was supposed to have had its mouth strengthened by an iron band.

Similar caldrons made of iron have been found in Ireland, two

being among the collection of relics from the Lisnacroghera cran-

nog, which also contained a number of Late Celtic objects {Lake

Ihcellinys of Europe^ p. 386). It would thus appear that there

was an evolutionary sequence in the manufacture of these caldrons

in the British Isles : first, those made of bronze ; second, those

made of bronze and iron; and third, those made exclusively of

iron. On the Coutinent, caldrons were generally found associated

with sepulchral remains, except those from Oppidum La T^ne,

but in the British Isles they were undoubtedly used for culinary

purposes. In protohistoric times in Ireland they were so highly

prized that they are often referred to as heirlooms in families, and

as forming part of the special property of kings. Tradition tells

us that among the treasures brought to that country by the Tuatha

De l)anoan was the Coire an DaghdhOy or Magic Caldron. On

these grounds I see no reason why the Kincardine caldron, though

belonging to an earlier date, should not have been used as a Roman

camp-kettle; and the association of the Cardross bucket with a

military camp, traditionally believed to be Roman, lends additional

support to this view. The general argument on this phase of the

subject may be thus briefly stated : — The finding of bronze caldrons

of pre-Roman types, and of a wooden roadway, presumably of

Roman construction, in association with the debris of great forest

trees, some of which showed over 300 ring-growths, all buried

beneath a bed of peat from 8 to 14 feet thick, affords something

more than presumptive evidence that the site of this forest had

become dry land at least some centuries before the Christian era.

But before attempting to assign a more precise date to this
upheaval, it is desirable to know something of the terrestrial move-
ment which caused it, especially as to the rate of its action. Was

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270 ProceediTtgs of Royal Society of Edinburgh, [sess.

the elevation effected suddenly, or in a few years, or in a few or many
centuries? From what I can gather of the history of land oscilla-
tions in other parts of the world, the probability is that it was a
very slow process, so much so that its progressive littoral changes
were too small to be appreciated during the ordinary lifetime of an
observer. If that be the case, it follows that there is a correspond-
ing difference in the dates when the shallower and deeper portions
of the sea-bottom reached the surface. We have already seen that
the upheaval must have l^een practically completed in the vicinity
of Drip Bridge before the wooden roadway was laid down, the
carse lands there being only a few feet above present high-water
mark. Hence the chronological value of antiquarian relics found
within the zone of the 2 5 -feet raised beaches depends to some
extent on their position above sea-level. There are several recent
discoveries which help to elucidate this point, one of the most
instructive being a Bronze Age cemetery near Joppa, the situation
of which is thus described by Mr W. Lowson, F.S.A.Scot. : —

*• In the beginning of December last (1881) workmen b^an to excavate
a piece of ground, little more than an acre in extent, lying between
Magdalen Chemical Works and Eastfield Cottages, Joppa, on the north
side of the road from Edinburgh to Musselburgh. The level of the
ground is about 12 to 14 feet alx)ve high-water mark. On the top was
ordinary soil, and beneath that a layer of sea-sand from 4 to 8 feet thick,
and beneath tliat gravel. On the 21bt January last I learned from the
person who had feued the ground that in the course of removing the sand
the workmen had discovered a large cinerary urn, filled with calcined
human bones." * Subsequently, six other urns, varying in size, and all
contained in stone cists, were recovered from the same locality. Besides,
there were two or three cists without urns, and one with a skeleton. All
these intenuents were from 4 to 6 feet below the surface of the ground,
and about 3 feel down on the bed of sand. " The piece of ground," writes
Mr Lowson, " in which these remains were found lies along the sea-shore,
and is now faced with heavy stones towards the sea ; but I saw an old
man in Fisherrow who remembers that he used to dig out sandmartins'
nests in tliat bank Ijefore the stones were put there. He had seen similar
urns taken out in his boyhood."

These facts conclusively prove that the sea had retreated to
close upon its present limits before these interments had taken
place. For if the surface of this sandy beach is 12 to 14 feet
above high-water mark, and the graves from 4 to 6 feet in depth,

* Fi'oc. S.A. Scot., vol. xvi. p. 419.

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1903-4.] Date of Uplieaval of Raised Beaches in Scotland, 271

it is evident that the sea-level could not have been much more
than 8 feet higher when the interment took place, without
occasionally submerging and damaging the cemetery. Unless the
high-tide limits were several feet lower, it is not likely that
people who paid such respect to their dead would select an
exposed beach as the final resting-place of their friends.

The hypothesis that the formation of the 25-feet raised beach
on the west of Scotland was not completed till about the beginning
of the Bronze Age was first suggested to me some years ago by the
discovery of five bronze axes of the flat type (fig. 5), while work-

FiG. 5. — Five Bronze Celts found together at the '* Maidens,"
Ayrshire. (^).

men were excavating the foundations of buildings on the sea-shore
near Culzean Castle. These axes — which were bound together by a
strong bronze wire, and had the remarkable peculiarity of being
graduated in size — evidently formed the *kit of tools* of a
Bronze Age workman. They were lying in a crevice beneath a
ledge of rock, against which were heai)ed up a few feet of gravel.
The spot was about 100 yards from the sea-shore, and 25 feet
above present high- water mark. In recording the discovery {Pror,
S.A, Scot, vol. xvii. p. 436), I suggested, as an explanation of

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272 Proceedings of Royal Society of Edinburgh, [i

the phenomena, that the rocky ledge under which the axes had
been deposited, apparently for temporary concealment, was at that
time open towards the shore, and that subsequently, during a
storm, the crevice had been covered over with coarse sea-gravel.
It does not appear that the owner, when finally parting with his
kit of tools, suspected any danger frum the proximity of the sea ;
and hence there is some ground for supposing that the ordinary
high tides were not wont to reach the spot. Now, had the
relative level of sea and land been the same then as now, a
storm could hardly account for their being covered over with
sea-gravel. It is not, therefore, unreasonable to suppose that the
upheaval had already, i.e, at the beginning of the Bronze Age,
made considerable progress, for these axes are among the earliest
objects of that period known in Scotland.

In conclusion, I have only to express the opinion that the facts
and arguments here advanced warrant us in assigning the upheaval
which caused the 25-feet raised beaches of Central Scotland to a
more restricted chronological range than that expressed in my
former theory on the subject, viz., " that it was subsequent to the
appearance of man in the district, but prior to its occupation by
the Romans." The additional evidence points to the well-founded
inference that the process of elevation had been virtually com-
pleted about the beginning of the Bronze Age. When it com-
menced there is little evidence to show, beyond the fact that it
was a considerable time posterior to the stranding of the school of
whales on the tidal shore of the shallow sea which then covered
the carse lands to the west of Stirling.

{Issued separately June 18, 1904.)

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Index Slips, to be issued with each part of the Proceedings and
Transactions.

MODEL INDEX.

Schafer, E. A.— On the Existence within the Liver Cells of Channels which can
be directly injected from the Blood-vessels. Proc. Roy. Soc. Edin., vol. ^
1902, pp.
Cells, Liver, — Intra-cellular Canaliculi in.

E. A Schafer. Proc. Roy. Soc. Edin., vol. , 1902, pp.
Liver, — Injection within Cells of.

E. A. Schafer. Proc. Roy. Soc. Edin., vol. , 1902, pp.

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iv CONTENTS.

PAGK

On the Date of the Upheaval which caused the 25-feet
Raised Beaches in Central Scotland. £7 Kobrrt
MuNRO, M.A., M.D., LL.D., .... 242
{Issued separately June 18, 1904.)

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PROCEEDINGS

OF THE

ROYAL SOCIETY OF EDINBURGH.

SESSION 1903-4.

No. IV.] VOL. XXV. [Pp. 273-336.

CONTENTS.

PAGE

The Complete Solution of the Diflferential Equation of J^y
By the Rev. F. H. Jackson, H.M.S. "Irresistible."
Communicated by Dr Wm. Peddib, . . . 273

{Issued separately Att^ist 16, 1904.)

A Differentiating Machine. By J. Erskine Murray,

D.Sc., 277

(Issi^ separately Augtcst 16, 1904.)

On the Thermal Expansion of Dilute Solutions of certain
Hydroxides. By George A. Carse, M.A., B.Sc.
Communicated by Professor MacGregor, . . 281

(Issued separately Auifust 15, 1904.)

[Continusd on page iv of Cover.

^EDINBURGH:
Pttblishbd by ROBERT GRANT & SON, 107 Princes Street, and
WILLIAMS & NORGATE, 14 Hbnbietta. Street, Covent Garden, London.

MDCCCCIV.

Price Four Shillings.

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All proofs must, if possible, be returned within one week, addressed to
The Secretary, Royal Society^ Mound, Edinburgh, and not to the printer.

Continued on page iii qf Cover,

,Coo^V

.If

1803-4.] Solution of the Differential Equation of J^^j . 273

Tbe Ck>mplete Solution of the Differenticd Bquation of
Jf„j. By the Rev. P. H. Jackson, H.M.S. "Irresistible.*'
Communicaied by Dr Wm. Peddie.

(MvS. received April 28, 1904. Read July 4, 1904.)

In connection with the function J[nj, it may be of interest
to give briefly the complete solution of the differential equation
satisfied by the function Jfnj . The method of Frobenius will be
employed. Consider the differential equation

^■y' {i-H-[-.ii."/"V)

in which

r 1 P'-^

If j; = 1, the equation reduces to

xf +x/' -^-i^-v?)/ = 0,

which is Bessel's equation for functions of order n .
Substituting an expression

in the equation (a) , we have an indicial equation

[a + n][a-»] = 0,
PBGC. BOY. SOC. BDIN. — VOL. XXV. 18

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274 Froceedings of Royal Society of Edinbwrgh, [i

and an indicial function

/r a:[a+2i

''"[a-n + 2][a + n+2][a-n + 4][a+n + 4]-. J
The principal roots of the indicial equation are

a= +«, a=» — n.

If n be not an integer, the corresponding integrals are J,^, and
Jem (^) = [2]-r^([n'+ i]) 1 "" [2][2n + 2] ^ * * I

If n = 0 these integrals are identical, while if n be an integer,
one or other becomes ineffective according as » is positive or
negative. In these cases, then, it is necessary to form a second
distinct and effective integral corresponding to Hankel's solution
of Bessers equation.

When a is integral, we write *

/W = C { [a + 2»]-[«] } { [a + 2«]-[-«] I

[

, ,^ a^a+21

^*""[a-n + 2][a + 7i + 2]+ ...

..-(-!)"., "•*■"-"

'rj({[a+2r]-[»]}{[a+2r]-[-«]}J
+ ELt.+i!«]_ |[a + 2»+2] -fn]l I [o + 2« + 2]-[-n] I

]

= 0)1 + 0)2. -■

* Gf. Forsyth, Th$<yry of Differential Equatiom^ vol. iii. pp. 101, 102.

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1908-4.] Solution of the Diff&remiiaX Equation of J^^^ . 275

From

we obtain

(«>,).= -„ = 0 (1)

(»,).= .„ = [2]"^.([n+l])EJ,„,(*) (2)

n-1
Iog_p ^^_i^r [2n]

r=l

>A=-n ^p-l 2l} ^^^^[2] [4] . . [2r] • [2 - 2n] . . . [2r - 27i]

a;I«r-nj(3)

pn+2r^n+2r)

(2)»+r

- E^l^< - ^> i [2] + [4] + • • • • •*• [2r] ■" [2« + 2] + ■

r=l

■■■■^[2» + 2r]/[r]![w + r]!(2),(2)n+r ' " ^'

IfwegiveCthevalue-<^).Lz^1UthatE = (2^^
or what is equivalent

[2]»-ii;([„+i])

we obtain an integral from (3) and (4) which may be termed
Wj + Wj . If ^ = 1 , this integral reduces to that given on p. 102,
vol iii., Forsyth's Theory of Differential Equations,
In the case when n = 0 , the integral is

/= <J,o,(P, «) log ar - c^ { ?^ + ||J + . . .
2p2« \ d^]

These functions satisfy the same recurrence equations as the

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276 Proceedings of Royal Society of Edinburgh, [i

function Jj^j given in the Transactions of the Society, vol. xli.,
part i., Nos. (1) and (6). The expression for IJ([ar]) given on
p. 105, No. 6, vol. xli., should be

The class of differential equations integrable by Bessel's functions,
and discussed by Lommel in vol. xiv., Mathematische Annaleny
may without difficulty be formally extended in the same way that
EesseFs equation and its solutions have been extended in the
above work.

(Isstud separately August 16, 1904.)

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1903-4.] J. Erskine Murray oii a Differentiating Machine, 277

A Differentiating Machine. By J. Erskine Murray, D.Sc.

(Read March 21, 1904. MS. received May 28, 1904.)

It was pointed out to me a few months ago, by my friend
Professor W. H. Heaton, that our knowledge of the laws of physical
variations might be greatly increased if their study were facili-
tated by the invention of a machine which would automatically
deduce the rate of change of a function from the curve represent-
ing that function. In cases where the physical law is already
known, and is expressible in terms of known mathematical
quantities, such a machine is not essential, though it provides an
excellent illustration of mathematical laws ; there is, however, a
vast and ever-increasing mass of numerical results awaiting
discussion and co-ordination, and it is in reducing these to law
and order that the differentiator should prove a useful tool. As
instances of a few cases in which rates of change are of the first
importance, I may mention the following : —

(1) Meteorological observations of Temperature, Pressure,
Humidity and Rainfall.

(2) Terrestrial Magnetic records.

(3) Experimental results in Physics and Chemistry which
involve changes, whether in time or space. The determination of
thermal conductivity by Forbes' method is an example.

(4) Statistics of Population, Mortality, and Migration.

(5) Statistics of Wages, Prices, and Commerce.

(6) Medical records.

(7) Engineering calculations, such as the deduction of Tractive
Force from a Time and Space or Time and Velocity diagram.

Up to the present all determinations of rates of change of
quantities like those above mentioned have had to be made by
laborious arithmetical or graphical methods, involving so great an
expenditure of time for their completion that but little has been
done. The differentiator reduces enormously the necessary labour,

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278 Proceedings of Royal Society of Edinburgh. [asss.

and even the roughly constructed instrument shown will give
results sufficiently accurate for most purposes.

The construction of the diflferentiator depends on the well-known
fact that if the values of a variable quantity be represented on a
diagram by the ordinates of a curve, its rate of change, at any
point of the curve, is measured by the slope of the tangent at
that point.

The machine, then, is guided by hand so that one line on it
remains tangent to the curve, while a tracing point describes on a
second sheet of paper a curve whose ordinates are proportional to
the slope of the tangent. Thus if y=f(x) be the equation to the
original curve, the derived curve will have for ordinates the
corresponding values of d(f{x))/dx. The abscissae are the same
on both curves.

In order that a line may be tangent to a curve it is necessary
that two consecutive points on each should coincide. In practice,
two black dots on a piece of transparent celluloid are used, the
distance between them being about 2 mm.

The plan of the machine is shown in fig. 1. It consists of
three parts. Firstly, the large drawing-board A B C D, on which
the original curve is placed. Fixed to each long side of this
board is a metal rail, one, CE, having a plain surface, and the
other, D F, a longitudinal groove of V-shaped section. The second
part is a smaller board, CHI, having three spherical feet, two
of which run in the groove and the third on the plane rail.
This arrangement permits free motion of the smaller board in the
direction of the length of the larger one, i,e. parallel to the Y
coordinate. The small board carries the paper on which the
derived curve is traced by the machine. Attached to its edge
are guides, JKLM, which hold the principal part of the
mechanism, allowing it free motion in a right and left line.

This part, shown in fig. 2, consists of a frame A B C D, at
one corner of which is a pin. A, which serves as the vertical axis
about which the rod P Q revolves in a horizontal plane. P Q has
a slot in it, through which passes the pin R fixed to the rod S T.
S T is controlled by guides E and F, so that it can only move in a
direction parallel to 0 Y.

Below the arm P Q, and fixed rigidly to it below A, is a small

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1903-4.] J. Erskine Murray on a Differentiating Machine, 279

plate of celluloid, not shown in the diagram, on the under side
of which are two dots by which the machine is guided along the
curve. The line through the dots is parallel to PQ. The
celluloid rests on the paper ou which the original curve is drawn,
thus supporting the outer end of the frame A B C D.

Since the distance AV between the pin and the centre line

Fio. 1.

of ST is constant, and since RY/AY = di/ 1 dz, it is clear that
the distance II V which R is displaced above or below the zero
line AV measures the tangent of the angle of slope of the
curve, i,e. dyjdx, A pen at the end T of ST records the
movements of R, and therefore traces a curve of which the
ordinates are proportional to the rate of change of the ordinate
of the original cxirve. It should be noticed that the purpose of

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280 Proceedings of Royal Society of Edinhargh, [i

the second board is to eliminate the Y coordinate of the original
curve. In using the machine the anu PQ is moved so that
it remains tangent to the original curve, while the frame A B G D
is moved from left to right, and it and the smaller board to and
fro as may be necessary in following the curve.

The machine shown has been constructed to deal with curves in
which the tangent of the angle of slope does not exceed 5 ; this is
sufficient for almost all experimental or observational results, since

Fio. 2.

it is always possible to flatten out the curve by making the
horizontal scale large in proportion to the vertical.

It is, of course, easy to obtain the higher derivatives of the original
curve by a simple repetition of the process on the successive curves.

In a future communication I hope to lay before the Society the
results of the study of a number of meteorological and other curves
by aid of the differentiator.

{Issued separatdy August 16, 1904.)

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1908-4.] Thermal Expansion of Solutions of Hydroxides, 281

On the Thermal Bzpansion of Dilute Solutions of certain
Hydroxides. By George A. Carse, M.A., B.Sc.
Communicated by Professor MagGrboob.

(Read March 21, 1904.)

In a paper communicated to the Nova Scotian Institute of
Natural Science,* Professor MacGregor has shown that in the case
of weak aqueous solutions of certain hydroxides, the volume of a
solution is less than the volume of water used in its preparation.
At his suggestion I have investigated the hydroxides of sodium,
barium, and strontium, to ascertain whether they exhibit this pro-
perty, and how the excess of the volume of solution over the
volume of constituent water varies with the temperature. From the
observations made, I have also determined the thermal expansion
coefficients, and found how they vary with temperature and with
concentration.

Freparation and Determination of Composition of Solutions,

The substances were purchased as chemically pure from E.
Merck, Darmstadt, and were found to be of sufficient purity, the
sodium hydrate being tested for carbonate, chloride, and sulphate,
and the barium and strontium hydrates for strontium and calcium,
barium and calcium, respectively.

The original solutions were prepared by dissolving the substances
in twice-distilled water, and they were analysed volumetrically by
titration with acid, phenolphthalein or methyl orange being used
as an indicator. The concentration of the acid had been
determined by means of sodium carbonate made by heating sodium
bicarbonate. The value of the chemical composition of any
solution thus analysed was got by taking the mean of several
determinations. The values of the atomic weights used were those
given by the International Atomic Weight Table of 1904, and the

♦ Tram. Nov, Scot. Insl, Nat. 8c. , 7, S68, 1889-90.

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282 Proceedings of Boycd Society of Edivhurgh. [sms.

densities of water at the various temperatures those given by
Landolt and Bornstein.*

Other solutions were made from those prepared directly by
mixing measured volumes of the solutions and distilled water at
15' C. The percentage concentration was then got from the

formula, p = _-— — -^.^^r-, where G is the number of grams of salt
VJD+ W A

per c.c. of original solution at 15"* C, V the volume of the solution,
D the density of the solution at 15° C, W the volume ef water,
and A the density of the water at 15° C. The volumes were
measured out by pipettes and burettes which had been certified
correct by the Physikalisch-technische Reichsanstalt, Berlin.

The accuracy aimed at in the estimation of the chemical com-
position of the solutions was the greatest attainable, and in the
estimation of the solutions of barium, and, to a lesser degree, of
strontium, the errors were greater than in the case of the solutions
of sodium. The so-called " probable errors " in the estimation of
concentrations were found in no case to exceed '00003 per gram of
solution.

Detennination of Density.

The density determinations were made primarily to measure
expansion on solution, and I found that the error introduced into
the measurement of expansion by the error in the concentration
set a limit to the density accuracy necessary. It was found
unnecessary to measure densities to any greater degree of accuracy
than 5 in the fifth decimal place. Accordingly, the pyknometer
method of determining density was adopted.

My attention was drawn to a method devised by Mr Manley t
of eliminating the error in a density determination by the
pyknometer, due to a difference in the amount of moisture con-
densed on the glass of the pyknometer in different weighings.
The method consists in using as a counterpoise a similar, sealed,
pyknometer, which is treated as regards heating, handling, etc in
exactly the same way as the pyknometer containing the liquid
whose density is to be measured. Mr Manley finds that " when

* Pkysikaliach'Chemische Tabellen, 1894.
t Proe, R,S.E., 24, 857, 1902-8.

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1903-4.] Thermal Expansion of Solutions of Hydroxides, 283

it is desired to obtain a value for the relative density of a water,
which shall be as nearly correct as possible to the fifth decinial
place, the use of a counterpoise for automatically eliminating
certain incidental errors is absolutely essential."

From calculations I made, based on a paper by Dr G. J. Parks *
" On the Thickness of the Liquid Film formed by Condensation at
the Surface of a Solid," it was found that had the pyknometer
had maximum deposition of moisture in the one case and none at
all in the other, the difference between two weighings of the
pyknometer empty could not exceed -004 per cent. Parks found
that the thickness of the film of moisture deposited on the surface
of the glass after 16 days' exposure, when the maximum was
attained, amounted to 13*4 x 10"* cm. This moisture if all
present would increase the weight of my pyknometer by '0008
gms., which is equivalent to -004 per cent, of the weight of the
pyknometer empty.

The difference I am dealing with is not the absolute amount of
moisture deposited, but the change in the amount of moisture that
may occur from experiment to experiment, and therefore is much
less than that calculated above.

To find whether it was necessary to use a counterpoise or not,
when I wished an estimation of density which should have no
greater error than 5 in the fifth decimal place, I determined the
specific gravity of a solution at various temperatures, both with
and without the counterpoise.

I took two pyknometers of the Sprengel-Ostwald type, of the
same kind of glass and of nearly the same external volume. 1
weighed each one, reducing the weight to weight in vacuo. One
of the pyknometers was then sealed by closing the end of the
tube of large bore and melting the end of the tube of small bore
till it was almost closed. The whole pyknometer, except about a
quarter of an inch of the capillary tube, was immersed in a beaker
of water, and the beaker covered with layers of paper to prevent
the heat of the sealing flame reaching the water. The pyknometer
was left in the water till the air inside had reached the temperature
of the water, and the capillary end was sealed with a fine small
flame. Knowing the temperature of the water, the height of the
♦ Proc, Phys. Soe, Lond., 18, 410, 1908.

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284 ProceediTUfs of Royal Society of Edinburgh. [

barometer and the internal volume of the counterpoise, we can
calculate the weight of the air enclosed.

Let w be the observed weight of pyknometer with liquid in it
using the counterpoise, w^, w^ the true weights of pyknometer
and counterpoise respectively (lo^ including the weight of air
inside counterpoise), I the true weight of liquid in pyknometer, v^,
V2 the volumes of air displaced by pyknometer and counterpoise
respectively, A. the density of the air at the particular temperature
and pressure at which the observation is made, and p the density
of the weights ; then

l = w-\-W2-w^- X( — H- v^ - Vj).

The volumes v^ and Vg were determined by finding the weight
of water in the pyknometer at a given temperature, and thence
calculating the volume occupied by the water, and by finding the
weight of the pyknometer empty, and the density of the glass, and
thence getting the volume of the glass.

All the terms on the right-hand side being known, we can find /.
If the pyknometers have nearly the same surface, then the weights
of moisture on their surfaces balance.

I now give my own experiments with and without the
counterpoise, showing that the use of the counterpoise was needless
in my work. The observations are as follows : —

Temperature
degrees Centigrade.

i Specific Gravity using
1 Counterpoise.

Specific Gravity not
using Counterpoise.

1-18566

1-18416

1-18418

1-18269

1 18266

1-18174

1-18176

The pyknometers used in the two series of observations given
above were different, and each weight of liquid was the mean of
two weighings. The pyknometers were not left standing exposed
to the air for more than 20 minutes (the time occupied in a
weighing). As the diflferences (the maximum being -00003) in
the specific gravity vary indiscriminately on either side there is no

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1903-4.] Thermal Expansion of SoltUions of Hydroxides. 285

indication that the one method is any hetter than the other from
my point of view. I therefore did not use the counterpoise.

In the determinations of the densities of the solutions, the
pyknometers weighed about 20 grams, and had a capacity of about
20 c.c. The pyknometers, after being filled, were placed in a
thermostat, the temperature of which was kept at 15** C, 20* C,
26* C, 30* C, as was required ; the bath did not vary more than
•04* C. from the required temperature during any experiment. The
stirrer was driven by an electric motor, or latterly by a Heinrici
hot-air engine. The thermometer which gave the temperature of
the bath was graduated to fiftieths of a degree centigrade, and had
a table of corrections from the National Physical Laboratory, Kew
Observatory. After the pyknometer had been for some time in
the bath (the period varying from 2 hours to 20 hours, as the
apparatus was kept going day and night), the meniscus was made
to coincide with the mark on the stem. A short time after, if the
meniscus still coincided with the mark, the pyknometer was taken
out, dried with a cloth and weighed. All weighings were corrected
for the buoyancy of the air by adding on to the observed weight
of the pyknometer the weight of air displaced by the excess of the
volume of the pyknometer and liquid over that of the weights.

To get an accuracy of '001 per cent, in a weighing the
thermometer in the balance-case should be read to '14* C. and
the barometer to '35 mm The thermometer in the balance-case
read to •!* C. and was correct to •02*' C, and the air in the case
was kept dry by means of sulphuric acid. The barometer, which
had been corrected at the National Physical Laboratory, read to
•1 mm. In the correction for buoyancy the density of the air
was taken from Landolt and Bornstein.* The error introduced
by taking the air in the balance-case as perfectly dry was
calculated and found to be negligible.

All weighings were the means of at least two observations, and
the deviation of any weighing from the mean of two weighings
was found not to exceed '002 per cent, for 94 weighings examined,
thus giving a rough estimate of the accuracy in weighing.

The so-called " probable error " in the estimations of density
was found not to exceed 00002.

*Loc. cit.

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286

Proceedings of Royal Society of Edinhwrgh. [i

Expansion on Solution,

The volume of unit mass of the various solutions examined
was calculated, and also the volume which the solvent water
contained in unit mass would occupy if its temperature were the
same as that of the solution. The amount hy which the volume
of unit mass of the solution is greater or less than that of the
solvent water employed in its preparation is the difference of these
quantities. Knowing the density, p (gma. per c.c), of a solution

at t° C, we can find the volume, — , of I gm. of the solution at

P
that temperature ; and knowing the concentration of a solution (c),
and the density of water at t**, A, we can find the volume that
the water in 1 gm. of solution would occupy if it were free, viz.,

— ^— ; hence the excess of the one volume over the other

is —.J- -. This may be called the expansion on solution.

P A

The " probable error " in the determination of the expansion was

found to be '00004, the values of the expansion varying from

•00957 to -00001.

The following tables give the results found. The headings are

self-explanatory.

Sodium Hydroxide.

Grams of

Volume of

Volume at t*

substance in

Temp.

Density

1 gram of

C. of water in

Expansion

100 grams

t'C.

grams per c.c.

Solution at

1 gram Solu-

V-Vcc

Solution.

t** C. (V C.C.).

tion (V C.C.).

16-3829

1-18468

•84415

•83740

-h -00675

1-18208

•84596

•83815

-h -00781

1-17891

•84823

•83935

-f -00888

t y

1 17676

-84988

•84031

+ -00967

6-0785

1-06884

•93559

•94003

- -00444

))

1-06699

-93721

•94087

- 00366

1-06452

•93938

•94222

- -00284

1-06294

•94078

•94829

-•00261

3-1805

1-03532

-96589

•96954

- -00866

1 03373

•96737

•97041

- -00804

1-03180

■96918

-97179

- -00261

1 -03074

•97045

•97290

- -00245

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1908-4.] Thermal Expansion of Solutions of Hydroxides. 287

It thus appears that for solutions of this hydrate below a
certain dilution the expansion is negative, and that this negative
expcmsion becomes less numerically with rise of temperature, i,e, it
increases algebraically with the temperature, just as is the case
when the expansion is positive (see fig. 2).

The following are curves for sodium hydroxide showing
expansion on solution plotted against concentration for the various
temperatures.

The solution exhibiting the maximum contraction at 15* C. is

Sodium Hydroxide.

o s ./o /6"

Fig. 1.

one containing 6*07 per cent, of the hydroxide, while the corre-
sponding value deduced by Professor MacGregor is 6 per cent.
The maximum contraction, as deduced from the above graph, is
•0044 c.c, while that given by Professor MacGregor is '0045. The
crosses on the diagram indicate values taken from Professor
MacGregor's table. It is also to be noted that contraction decreases
with rise of temperature, and that the maximum contraction-

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288 Proceedings of Royal Society of Hdinburgh, [

point slowly shifts towards the concentration origin with rise of
temperature.

Barium Hydroxide,

Grams of
substance in Temp.
100 grams I t" C.
Solution. I

Density
grams per c.c.

Volume of Volume at t* i

1 gram of C. of water in ' Expansion
Solution at ' 1 gram Solu- ' V-V cc.
t°C.(Vc.c).!tion(V'cc.).'

•89387

•08212

•04303

I-

15
20
26
30
15
20
26
30
16
20
26
30

1-01079

1 00998

1 00847

1-00721

1-00000

•99913

■99766

•99656

-99957

-99870

•99728

•99611

•98933

•99010

•99160

'99285

1-00000

1^00087

1-00234

1-00345

1 00043

1-00130

1-00273

1-00390

•99192

•99281

•99423

•99537

1 00005

1^00095

1-00238

1 00352

1 00044

1-00134

1-00276

1-00392

- -00259

- -00271

- -00263

- -00252

- -00005
-•00008

- -00004

- -00007

- -00001

- -00004

- -00008

- -00002

It thus appears that all the solutions of barium hydrate examined
have a negative expansion. This hydrate is thus so far analogous
to sodium hydrate. The effect of temperature on the expansion is
not very marked, and for the last two concentrations the numerical
values of the expansions are subject to considerable variations in
the fifth decimal place, although they all agree in giving negative
expansion (see fig. 2).

Strontium Hydroxide,

Grams of

Volume of

Volume at t"

substance in

Temp.

t^a

Density

1 gram of

C. of water in

Expansion

100 grams

grams per cc.

Solution at

1 gram. Solu-

V-V'cc.

Solution.

1-00363

t"C.(Vc.c.).
-99639

tion (V ca>

-32744

-99759

- -00120

))

1-00263

-99738

99848

- -00110

1-00114

•99886

•99992

- 00106

))

•99996

1 -00004

1-00105

- 00101

12162

1-00072

•99923

-99965

- -00042

))

•99971

1 00029

1-00055

- -00026

•99831

1-00169

1-00197

- -00028

»»

•99708

1-00293

1-00313

- -00020

•02354

-99946

1-00054

100063

- -00009

•99849

1-00151

1-00153

- -00002

))

•99700

1 -00806

1-00296

+ -00010

»»

•99600

1 00432

100411

-f -00021

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1903-4.] Thermal Expansion of Solutions of Hydroxides. 289

Here also solutions of strontium hydrate exhibit this negative
expansion, and this negative expansion becomes less numerically
with rise of temperature, and in the case of the last solution
examined it changes from being a negative to a positive expansion
with rise of temperature. Strontium hydrate is thus analogous to
sodium hydrate (see fig. 2).

The following are curves exhibiting expansion on solution plotted

Fig. 2.

against temperature for the hydroxides of sodium, barium, and
strontium.

TJiermal Expansion.

Adopting the formula V< = Vi5[l + a{t - 15) + b{t - 15)2 + c(<- 15)8]
where V, is the specific volume at t** C, and a, b and c are con-
stants, I have determined by a modified method of least squares
the constants a, h, c ; the formula gives the volume at any
temperature between 15** C. and 20" C. correct to within 5 in the
fifth decimal place. By the aid of the above formula the expan-
sion coefficients, a< = — , ', where a, is the expansion coefficient
Vt df

at t** C, were calculated.

PROC. ROY. SOC. EDIN. — VOL. XXV. 1 9

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290 Proceedings of Royal Society of Edinburgh. [i

The following are the tables : —

Constants and Goeficienis.

Concen-
tration.

axlO«

6xlO«

cxlO»

««

xlO»

clO»

oas

xlO*

0,0 >^ 10=*

Sodium Hydrate.

16-8829
6 0785
3-1805

+ 480
+ 810
+ 300

+ 000
+ 890
+ 124

+ 116 42

- 327 31

- 21 80

44
37
81

47
88
82

Barium Hydrate.

'8989
•0821
•0430

+ 97

+ 1200

-180

+ 130

+ 980

-210

+ 148

+ 630

- 28

51
86
88

84
28
31

Strontium Hydrate.

8274

+ 180

-^340

+ 50

1216

+ 220

-410

+ 380

•0235

+ 165

+ 600

- 12

81
85
33

The expansion coefficients, since they involve small differences
of volume, are subject to large errors in the fifth decimal place,
and can only be considered as approximate.

The following curves show expansion coefficient plotted against

Sodium Hydroxide.

O lo

CDNwC4>l^Jl3vaJ&y4nv .

Fio. 3.

2.0

concentration, the first set being for sodium hydroxide alone, while
the second set are for the three hydroxides. In the second set

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1908-4.] Thermal Expansion of SoltUions of Hydroxides, 291

the concentrations and expansion coefficients are plotted on scales
20 and 2 times those of the first set respectively.

In the case of sodium hydroxide the expansion coefficient
increases with concentration, and does so at a less rapid rate as the
temperature rises.

The strontium and barium curves seem to indicate that the rate
of variation of expansion coefficient with concentration reaches

«
1

*«oeto

o -5

Fig. 4.

stationary values in the range considered, hut no great stress can
be laid on this conclusion, because of the uncertainty caused by the
large errors in the expansion coefficient.

The above experiments were carried out in the Natural Philos-
ophy Laboratory, University of P^dinburgh. I have to tender my
best thanks to Professor MacGregor for the assistance he has
afforded me in this work, both by way of suggestions and advice.

{Issued separately August 15, 1904.)

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292 Proceedings of Royal Society of Edinburgh. [si

Effect of Transverse Magi^etization on the Resistance of
Nickel at High Temperatures. By Professor O. Q.
Knott.

(ReadJune 20th, 1904.)

Abstract.

In a previous communication * it was pointed out that the effect
of transverse magnetization on the resistance of nickel wire was
inappreciable in fields below 500 C.G.S. units, thereby differing
from the case of longitudinal magnetization, in which the effect was
easily measurable in fields below 20. t The reason of this is no
doubt to be referred to the thinness of the wire in the direction of
the magetizing force. To measure the effect of transverse magnet-
ization it was necessary to form a flat coil and insert it between
the poles of a powerful electro-magnet. Considerable difficulty
was experienced in winding this coil with interwound asbestos in-
sulation, for great care had to be taken that no part of the wire
cut the lines of force obliquely, otherwise there would be a resolved
component of longitudinal effect, which in certain cases might
altogether mask the effect looked for. The coil used in the final
experiments was suitable in all respects. It was coiled between
glass plates, the successive coils being separated by threads of
asbestos. Round the coil another coil (of Beacon wire) was wound
anti-inductively, so that any current passing through it would have
no magnetic action upon the nickel wire inside. By varying the
current in this external coil I was able to heat the nickel to any
desired temperature up to 400" C. In any one expeiiment the
final temperature came to a steady state, and not till this state was
reached was it possible to begin the observations on the resistance
change. This was measured in the manner already described in
my paper on the effect of longitudinal magnetization, and it will
suffice meanwhile to call attention to a remarkable result obtained

♦ Proc,, vol. xxiv. p. 601 (1908).
t Trans., vol. xli. pp. 39-52 (1904).

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1903-4.] Prof. Knott on Effect of Transverse Magnetization. 293

when the temperature approached that at which nickel ceases to
be strongly ma^etic.

The nature of the phenomenon is indicated in the following table,
which gives the change of resistance of 100,000 ohms of nickel
wire at the temperatures shown when the wire is subjected to
a transverse magnetic field of about 3800 units.

Temperature.

Resistance change
in Field 3800.

Temperature.

Resistance change
in Field 3800

750

320' C.

320

640

330

270

390

335

170

250

340

100

190

345

201

350

250 !

10** c.
100
200
250
290
300
310

The peculiarity consists in the marked minimum at temperature
290* and the still more abrupt maximum at temperature 320*.
The very rapid fall oflF to zero as the temperature rises from 330
to 350 is also worthy of note. So limited is the range of
temperature within which these changes take place, that the
phenomenon might easily have escaped notice. It was fortunate
that in one of the earlier series a temperature very near the
minimum point was hit upon. The peculiarity was at first
ascribed to the inherently greater difficulties of making the
experiments at the higher temperatures : but time after time, by
means of small successive changes of temperature between the
critical limits, exactly the same results were obtained. There can,
therefore, be no doubt as to the existence of a peculiar molecular
change as the nickel wire is raised in temperature from about
290* to 350". In my paper on the eflFect of longitudinal
magnetization (see especially the curves at the highest temperatures,
p. 46, l.c,\ a similar peculiarity was indicated. It was, however,
so slight — being merely a slight upward bulging of the isodynamic
curves— that it was not at the time regarded as of any moment,
but, in the light of the present result, it can no longer be looked
upon as due to small errors of measurement.

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294 Proceedings of Boyal Society of Edvaburgh. [sbss.

In this connection I would draw attention to a paper published
in the Philosophical Magazine for June 1904, bearing on a cognate
line of research. In that paper Dr E. P. Harrison shows that
pure nickel undergoes curious changes of length as the temperature
approaches the temperature at which its magnetic properties are
lost. This is strictly analogous to the behaviour of iron at
red heat, as discovered long ago by Grore. Tait found that
the thermo-electric properties of iron had peculiarities which
occurred at this same temperature ; and that similar thermo-
electric peculiarities were possessed by nickel. He tried, but un-
successfully, to find a Gore effect in nickel at a temperature of
400". This has now been very satisfactorily accomplished by Dr
Harrison. It is possible, however, that the result obtained by
Dr Harrison may be partly due to variation in the magnetic strain
caused by the circular magnetization accompanying the strong
current used for keeping the nickel wire at the required high
temperature.

As to the cause of the curious effects described in this note,
more than one hypothesis might be advanced, but it would be
premature to attempt any complete discussion until further facts
are made out. These I hope to communicate in due course.

{Issued separcOely July 80, 1904.)

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1908-4.] On Aged J^edmms of Sagartia troglodytes, etc. 295

Observations on some Aged Specimens of Sagartia
troglodytes, and on the Duration of Life in
Coelenterates. By J. H. Ashworth, D.Sc, J-ecturer in
Invertebrate Zoology in the University of Edinburgh, and
Nelson Annandale, B.A., Deputy-wSuperintendent of the
Indian Museum, Calcutta. Communicated by Professor J. C.
EwABT, M.D., F.R.S.

(MS. i-eceived June 10, 1904. Read June 20, 1904.)

We have, during the last two years, made a series of observa-
tions upon specimens of Sagartia troglodytes which are at least
fifty years old, and have thought it worth while to give a some-
what detailed account of these, as, so far as we can ascertain, there
is only one other recorded case of longevity in Coelenterates (see
p. 302), and very few in the whole of the Invertebrata.*

These specimens of Sagartia troglodytes were collected by Miss
Anne Nelson (Mrs George Brown) on the coast of Arran, some few
years previous to 1862 (the exact date has not been recorded), and
were placed in bell-jars containing sea- water. In 1862 they were
transferred to the care of Miss Jessie Nelson, in whose possession
they still remain, and to whom we are indebted for the opportuni-
ties of observing these interesting anemones. Sixteen of the
original specimens are still living, so that they have lived in
captivity for about fifty years. They are kept in a bell-jar about
13 inches in diameter and 9 in depth. The original specimens
are all together on a piece of stone, which bears a number of deep
depressions in which the anemones have ensconced themselves.
These conditions closely resemble those in which S. troglo-
dytes is usually found, the specific name of this anemone
being derived from its favourite habit of dwelling in holes and
crevices of the rock. These specimens have been under constant
observation since 1862, and there can be no doubt that they are
the original ones.

* See the appendix to Weismann's Essay on the Dnration of Life, 1891,
p. 80.

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296 Proceedings of Royal Society of Edinburgh. [sbsk.

As the conditions under which these anemones have lived for so
long may be of interest, the following particulars are given. The
bottom of the bell-jar is covered with small rough stones on which
several species of green algae are growing. On these rests the
large stone containing the cavities in which the anemones are fixed.
The sea-water in the jar (about four gallons) is changed every six
or eight weeks, and is usually aerated every morning. From time
to time a little fresh water is atided to keep the density of the
whole constant The anemones are fed about once a month on
small pieces of raw lean beef. They usually reject fish or mutton,*
but appear to digest the beef very thoroughly, a small mass of
white flocculent matter being ejected from the mouth a day or two
after feeding. In addition, the anemones catch and feed upon the
small isopods which abound among the algae One of us lately
observed a specimen seize and engulf an Actinia mesembryanthemum
which had freed itself from a neighbouring stone and come inVo
contact with the tentacles of the Sagartia. Two days later the
victim, almost intact, but quite dead, was ejected. Those tentacles
of the captor {Saijartia) which had first touched the Actinia
remained for some days dimuiished in size and opaque in colour,
but finally recovered their usual appearance. Sagartia trofflodytes
is evidently not immune to the poison of Actinia viesembryanihemum^
but, so far as could be ascertained, only the tentacles of the former
suffered from the effects of the poison of the latter. Probably the
nematocysts of the latter became inoperative soon after its capture,
either owing to the death of the Actinia or to some other cause, so
that the internal structures of the Sagartia remained practically
uninjured. Grosvenor {Proc. B.S.L., vol. 72, 1903, pp. 478-479)
ascribes the discharge of nematocysts to osmtitic action. His
experiments show that the contents of the capsule are able tx) take

• Owing to the value of these aged specimens, we have not been able to
make sufficient experiments upon them to determine whether they have a
sense of taste, but the above observations seem to suggest tbat snch a sense is
present, though feebly developed. For an account of such experiments see
O. H. Parker, **Tho Reactions of Metridium to Food and other Substances,"
Bull. Mus, Comp, Zod. Harvard, vol. xxix., 1896, pp. 107-119. Parker
concludes that the tentacles of this anemone when stimulated with meat juice
move so as to point to the mouth ; similar stimulation to the lips gives rise to
peristaltic movements in the stomodseum, reversal of the ciliary action of the
lips, and contraction of the sphincter muscle of the oral disc

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1903-4.] On Aged Specimens of Sagartia troglodytes, etc. 297

up liquid from sea-water until, on the pressure reaching a certain
amount, the thread is shot out. Such discharge would probably
take place only in sea-water, or in some fluid which differs but
slightly in density from sea-water.* The Actinia , on entering the
ccelenteron of its captor and becoming surrounded by the denser
mucous secretion poured out upon it, would probably be rendered
innocuous, its nematocysts becoming inoperative. Even if the
mucous secretion merely served to delay the discharge of the
nematocysts (as is almost certain, for it would prevent or retard the
access of sea-water), it is probable that the density of the fluid in
the ccelenteron (after closure of the stomodsBum) would, from other
causes, soon increase to such an amount as to then render the
discharge of nematocysts impossible. That such a change in the
contents of the ccelenteron does occur soon after closure of the
stomodflBum is evident from the behaviour of the young anemones
described below. Then, again, the mucous secretion which the
captor forms over its prey would also act as a shield against any
nematocysts of the latter which might be discharged. We may
account in one or other of these ways for the apparently uninjured
condition of the internal structures of the captor.

Miss Nelson's specimens of SoAjartia troglndytes and also of
Artinia mesembrtjanthenium have been very prolific, though only a
small proportion of the young produced has survived. As a rule,
most of them disappear within a week or two after birth, some
being devoured by the adults of their own or other species, and
the rest disappear in other ways not ascertained. Both species
breed in early spring : Actinia commences to bring forth young as
early as the beginning of February, and Sagartia about a month

* The fact that the uematooysts of Hydroids are able to pass undischarged
through a portion of the alimentary canal and into the dorsal processes of
jEoliSf but may be discharged on being extruded into sea-water, 8Upi>orts this
view. Again, some fish appear to feed with impunity on anemonen and other
Coelenterates, e.g. Peachia hastaia is found in the stomach of the cod (M 'Intosh,
The Marine Invertebrates and Fishes of St Andrews^ p. 37), swarms of an
Edwardsia in the stomach of the flounder (p. 38), while VirgvZaria mirabilis
is also occasionally seen in the cod's stomach (p. 39). Anemones are some-
times used on parts of the Scottish coast as bait for cod, and are found to
answer this purpose well (see, for example, M'Intosh, The Resources qf
the Sea^ p. 129). Off the south coast of Iceland one of us has seen the
stomach of a cod fhll of specimens of Pennatula,

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298 Proceedings of Royal Society of Ediriburgh, [

later. As a rule, only a few young are extruded at one time, and
generally early in the morning, and one individual may repeat this
operation every morning for several weeks. The young, the
majority of which when extruded already possess the first two
cycles of tentacles (t.e. twelve tentacles), are not expelled with
violence, but gently, and usually lie for a time, with their tentacles
retracted, on the disc of the parent. They are dispersed in a
manner which is no doubt very useful and effective in a tidal pool
on the sea-shore. At or soon after extrusion the basal portion
of each young anemone is much swollen, owing to the presence of
a considerable amount of fluid in the coelenteron, so that the pedal
disc becomes strongly convex. This is probably due to the fact
that the tentacles being retracted and the mouth closed, the
products of metabolism are unable to escape. In addition to their
mere accumulation, the soluble products exert some osmotic action
which causes sea- water to diffuse through the thin body-wall into
the coelenteron, thus strongly inflating the basal portion of the
young animal. Owing to this Imsal inflation and the retraction
of the oral end the young anemone has an almost globular shape,
so that the slightest current in the water causes it to roll ofi" the
oral disc of its parent, and often carries it some distance before it
sinks to the bottom, as its specific gravity is not much greater than
that of sea- water. As soon as the young anemone finds the
bottom of the vessel it becomes orientated in the proper direction
and fixed by the pedal disc, apparently possessing already that well-
marked polarity which is characteristic even of pieces of adult
anemones which include a portion of the pedal disc (see A. P.
Hazen, Arch,f. Enttvickelungsmechanik d, Org,^ Bd. 14, 1902, pp.
592-599, and Bd. 16, 1903, pp. 365-376, Sagartia lucim). We
have occasionally seen adult specimens of Actinia mesemhryan-
themum assume this globular and buoyant form, the pedal disc
becoming free from its attachment, the basal part of the animal
swollen and the oral disc retracted. Both S, troglodytes and A.
mesembryarUhemum are frequently found in this condition at birth,
but adult specimens of the former rarely adopt it, though a case is
mentioned by Gosse (1860, p. 95). S. troglodytes seems to rarely
change its station when once settled in a cavity which is to its
liking.

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1903-4,] On Aged Specimens of Sagartia troglodytes, etc. 299

S, troglodytes is, or may be, viviparous. As stated above
(p. 298), all the young which we have seen extruded were already
provided with six, or more usually twelve tentacles. Our
experience agrees with that of Mr Sydney Chaflfers, Registrar
of the Owens College, Manchester (see also p. 301), whose
specimens have invariably reproduced in a similar manner. He
informs us that he has seen many batches of young born, and
has succeeded in feeding some of them within a few minutes
after extrusion. Neither Mr Chaflfers nor ourselves have seen
any ova or ciliated larvae issue from the mouth. Oskar Carlgren,
however, states (" Die Brutpflege der Actiniarien," BioL Gentrcdhl,,
Bd. 21, 1901, p. 469) that in S, troglodytes, S, viduata, and
S. undata, fertilisation of the ova takes place in the sea- water
outside the parent. It appears, therefore, that S. troglodytes may
be either oviparous, as in Carlgren 's specimens, or viviparous, as
in Mr Chaflfers' and ours.

The mode of reproduction in anemones is evidently subject to
some variation. For example, Bunodactts (Bunodes) gemmaeea is
usually viviparous, "living and well-formed young" with twelve
tentacles being brought forth (Gosse, 1860, p. 193, and Carlgren,
Biol Centrcdbl, Bd. 21, 1901, p. 469). Mr Chaflfers, who has
also observed the reproduction of anemones of this species, states
that he has found them to be in all cases but one viviparous.
He observed on one occasion the extrusion of four or five ciliated
larv®, which swam vigorously for some minutes.

We have carefully observed the old specimens of Sagartia
troglodytes during the last two years, with the view of noting
any points of interest in their appearance and physiology. It
was not possible to obtain one for dissection or histological
examination. On comparing these old ones with younger specimens,
there is seen to be little diflference in their external characters.
Certain younger individuals, the progeny of the old ones, and
now about fourteen years old, are living in another aquarium, to
which they were removed soon after birth. They have been under
very favourable circumstances as regards volume of water, feeding,
etc., and are now larger than their parents. The latter are rather
more variegated in their coloration than is the case with their
oflfspring, but these differences are not important. The coloration

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300 Proceedings of Royal Society of Edinburgh. [sEsfs.

of this species is, as Gosse has pointed out (1860, pp. 89-92),
extremely variable. Specimens of this species collected by one
of us in the Faeroes are both smaller and more intensely pigmented
than others from the Scottish coast. Specimens kept in captivity
show little tendency to increase in size, but become decidedly
paler in colour. These old captives are lighter in colour than
individuals which have been more recently taken from a rock pool.
All the individuals of this species which we have observed are
sensitive to changes of light and of temperature, becoming and
remaining semi-contracted during cold weather and at night, but
expanding to their fullest hi warm, bright weather. The old ones
are much more strongly affected by unfavourable conditions than
those which are more than thirty years younger, and also are
longer in recovering when conditions become again favourable.
When the aquaria are examined in early morning or in fine warm
weather succeeding a period of cold, it is found that the old
specimens remain contracted for some time after their children
and grandchildren are fully expanded.

The most notable difference between the old (fifty years) and
the younger (fourteen years) individuals of Sayaiiia troglodytes is,
as would be expected, in point of fertility. In 1903 the sixteen
old ones did not produce altogether more than half a dozen young ;
indeed, it is doubtful whether they bred at all, as the few young
found beside them may not have been tlieir progeny. During
the same period their children and grandchildren reproduced in
large numbers (hundreds), though, as mentioned above (p. 297),
only a few of these survived.

Sagartia troglodytes^ in these aquaria at any rate, ap{)arently
takes tliree years to reach maturity.

In the early part of 1904 the aquaria were somewhat
neglected, the water was aerated less frequently and not changed
for over three months, and the animals remained unfed for a longer
period than usual. Probably as a result of these less favourable
conditions only a few young, much fewer than usual, were produced,
even by the younger specimens of Sayartia ; these younger ones
were abnormally thin and transparent, and were not extruded until
early in April. The sixteen original specimens produced no
offspring whatever in the spring of this year (1904).

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1908-4.] On Aged Spedmens of Sagartia troglodytes, etc. 301

Specimens of Actinia meaemhryanthemum living under identical
conditions and in the same aquaria as the Sagartia were more
fruitful, two in particular being very prolific, though their breeding
season was somewhat retarded. It would therefore appear that
S, troglodytes is more sensitive than A, Tnesemhryanthemum to
changes in the environment, and that these changes exert a
considerable effect on the reproduction, though it is obvious that
there is some individual variation in this respect.

In August 1903 two specimens of S. troglodytes were brought
from Thorshavn in the Faeroes and placed in the aquaria. In the
following October each produced several young, and in April 1904
one of them gave birth to a single young anemone. All the other
specimens of S. troglodytes which were under the same conditions
breed only in the spring, and it is improbable that October is the
normal breeding-time of specimens under natural conditions in
the Faeroes, as by this late season of the year the sea is already
running high, and there would be a great risk of the delicate
young anemones being unable to fix themselves, and being destroyed.
It is probable that the change of environment (perhaps temperature
was largely responsible) had induced these anemones to breed out
of their usual season (see also p. 303).

We are indebted to Mr Sydney Chaffers for sending to us
some particulars regarding anemones which he has kept in captivity
for a number of years (see also pp. 299, 303). These specimens
have in most cases been returned to the sea. He lias kept for
a period of eight years, without any difficulty, specimens of
Actinia mesenibryanthemum, Sagartia troglodytett, and Bunodactis
(Bunofles) gemmacea in aquaria containing about seven gallons of
sea-water. These anemones were fed regularly twice a week on a
portion of the mantle of Mytilus, and the water was aerated every
other day by means of a glass syringe. Mr Chaffers states that
during these eight years there was no appreciable alteration in the
size and appearance of these anemones. This supports the view
that under favourable conditions they may live to a great age.

Miss Nelson informs us that Actinia mesnnhryanthemum is the
only other anemone which she has been successful in keeping for
any length of time, and that no specimens of this species have lived
in her collection for more than about eight years.

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302 Proceedings of Royal Society of Edivhurgh. [

A specimen of this species collected by Sir J. Graham Dalyell
(1848, p. 203) at North Berwick in August 1828 reached the age
of about sixty-six years. So far as we can ascertain, this is the
only recorded example of longevity in anemones, and is quot^ by
Gosse (1860, p. 182), M*Bain (1878, p. 280), Weismann (1891,
pp. 6, 55), and others.

Dalyell computed, after comparison of the size of this specimen
with that of others which had been bred in his aquaria, that it
must have been at least seven years old at the time of its capture.
After DalyelFs death in 1851 this anemone was placed successively
under the care of several naturalists, and died in August 1887,
being then about sixty-six years old. Unfortunately, nothing is
known with certainty as to the cause of its death. The obituary
notice which appeared in The Scotsman states that the anemone
" appeared to be in excellent health up to a few weeks ago, when
it was attacked by a parasitic disease, which finally proved fatal."
Mr R. Lindsay, who had charge of this anemone during the last
five years of its life, informs us that this report is unfounded, and
that "the death of the anemone was not due to any parasitic
disease," but was apparently "natural." There is also a footnote*
to this effect on p. 55 of Weismann's Essays (1891, vol. i.). It
was kept in a comparatively small volume of water (the vessel in
which it lived is described as a large tumbler), was fed on half
a mussel once a fortnight, and the sea-water was chemged soon
afterwards.

During the first twenty years of its life it produced 334 young
(Dalyell, 1848, p. 213), and then remained unproductive for some
years, but during the spring of 1857 it gave birth to 230 young
during the course of a single night (M*Bain, 1878, p. 286). For
the next fifteen years it was unproductive, but in August 1872 it
produced a brood of 30, and in December of the same year one
of 9. It continued to reproduce each year, the number of its
young being from 5 to 20 at a birth. During the seven
years beginning August 1872, over 150 living young were bom.
Two of these were isolated and regularly fed, and at the age of

* *' It died, by a natural death, on Aagust 4th, 1887, after having appeared
to become gradually weaker for some months previous to this date." -Foot-
note by Professor Poulton, from information obtained by Mr J. S. Haldane.

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i9a3-4.] On Aged Specimens of Sagartia troglodytes, etc, 303

four years produced over 20 young ones, so that the offepring
produced by DalyelPs Actinia when it had reached the age of fifty
years were quite normal and vigorous.

A few of the statements regarding the breeding of these
anemones in captivity may be brought together here. As noted
above (p. 297), Miss Nelson's specimens of Sagartia troglodytes,
which are usually fed once a month, breed in the spring. During
the spring of this year (1904), however, when they were somewhat
neglected, and feeding, aeration and change of water occurred
at longer intervals, they were much less productive. A Faerish
specimen of this species placed in the same aquarium bred in
autumn 1903 and in the spring of 1904, the latter being probably
its normal, and the former an unusual breeding season, induced by
change in the environment, rise of temperature being probably an
important factor (though better feeding may have contributed to
the result).*

Mr ChalTers states that his specimens of S. troglodytes and
A. mesembryantkemumy which are fed twice a week, bring forth
young at all times of the year except during the cold weather.

Dalyell (p. 214) states that ** feeding certainly promotes fertility ''
in Actinia mesembryanthemum.

From these facts it appears that temperature and feeding
exercise a very considerable influence upon the production of
young in these forms of life.

Sagartia troglodytes and Actinia mesembryanthemum are
viviparous; the former may also be oviparous (see p. 299).
Bunodactis {Bunodes) gemmacea is usually viviparous, but Mr
Chaffers has observed, on one occasion, the extrusion of ciliated
larvae.

Little is known concerning the rate of growth and the duration
of life in Coelenterates, but it may be useful to collect here some
of the scattered references to these subjects.

Hydrozoa. — Evidence shows that Hydroids grow rapidly, for, as
Hincks (1868, p. xliii) remarks, "timber immersed in the sea is

* This specimen was taken from a pool near high- water mark, where food
was probably not abundant. We have noticed, on the west coast of Scotland,
that the largest specimens are almost invariably found in the pools near low-
water mark, those living in pools higher up the beach being distinctly
smaller.

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304 Proceedings of Boycd Society of Edinburgh. [i

soon found to be covered with a luxuriant growth of zoophyte ....
a Eudendrium has been observed to cover the bottom of a boat in
fifteen days.

One of us has observed off the coast of the Malay Peninsula
hydroid colonies {Obelia, sp.) several inches in length attached to
the cast skins of sea snakes {Enhydrina valakadien and others).
These therefore had grown upon the skins before the latter had
had time to disiptegrate, for such colonies were not present on any
of the hundreds of living sea snakes examined.

Hincks states (p. xliv) that some species of hydroids, especially
such as grow on fronds and stems of seaweed, are annuals. The
larger arborescent masses of the stouter kinds of Sertidaria^
Helecinm, Eudendrium^ etc., are, however, probably the growth of
several seasons.*

Some of the Siphonophora are probably annual. A species of
PorpUa t is common in calm warm weather (February to April) in
the Indian seas, but completely disappears in the stormy season
(about July). This animal has no power of sinking, and its com-
plete disappearance seems to indicate that it has perished, and
those which appear in the next warm season probably belong to
the following generation.

* Thero is a complete alisence of hydranths in some forms dnring the winter,
but the coenosarc persists, and new polyps develop by budding in the following
spring. Weismann {Die EnbsUhung der SexuaZzellen hei den BydromeduaaCf
p. 102, Jena, 1883) states that in Eitdendrium racemosum the hydranths are
wanting during the winter in those colonies which are situated in exposed
stormy places, but they may persist in those which live in more protected
situations. The hydranths of Tubularia indivisa (Allmaii, Oyntnoblastie
Hydroids, p. 403, Ray Soc, 1871) are in greatest perfection during spring
and summer, and when the racemes of gonophores have attained their greatest
.size the hydranths are *' perpetually cast otf and renewed." Towards the end
of summer the renewal of tlie hydranths ceases, and the upper parts of the
perisarcal tubes are empty, and probably remain so duriuj;; the winter, new
hydranths being formed in the spring. Van Beneden ( " Recherchcs sur la
Fanne Littorale de Belgique (Polypes),'* M^m. VAcad, Boy, de Belgiqwt^
t. 36, 1867, p. 101) records specimens of Tubularia and Campanularia which
have lived in his aquaria for several years without any diminution of their
powers of growth.

t It may be of interest to refer here to what we believe is the first reference
in English to Porpita, It occurs in a letter written from 6oa by Thomas
Steevens in 1579. In describing his voyage to India he says — '*The first
sign of laud was certain fowls which they know to be of India. The second
was boughs of palms and sedges. The third, snakes swimming on the water,

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1903-4.] On Aged Specimens of Sagartia troglodytes, etc. 305

Antliozoa,

(1) Actiniaria, — The instances given earlier in this paper show
that the age at which an anemone hecomes mature varies with the
species and conditions. For example, Dalyell (p. 2 1 7) records a
specimen of Actinia niesembryanfhemum, one of the progeny of his
famous specimen, which was mature fifteen months after its birth ;
while M*Bain (p. 287) states that another of the progeny of this
same parent, although carefully tended and fed at least once a week,
was four years in reaching maturity. Sagartia troglodytes seems
to be at least three years in reaching maturity, at any rate in
captivity. These anemones may continue productive, either
regularly or at intervals (this being apparently largely determined
by the external conditions and regularity of feeding), for over
fifty years. The only information available respecting the actual
duration of life in anemones is that derived from the statement
that DalyelPs Actinia apparently died *'a natural death " at the
age of sixty-six (see p. 302). Miss Nelson's specimens of Sagartia,
which are now about fifty years old, show little sign of loss of
vegetative vigour, but, as noted above (p. 300), breed either
sparingly or not at all.

(2) Madreporaiia. — The only reference known to us upon the
duration of life in corals is contained in a paper by Mr Stanley
Gardiner (1902). He describes (pp. 465-468) the life history of
Flabellum rubrum, and states that by the time the corallum
measures 15-17 mm. along the long axis of its calicle, the
mesenteries bear testes, and spermatozoa are being discharged
from those on the larger mesenteries. Coralla of this size bear
"five lines of growth, which correspond probably to annual
periods." Later, the male organs gradually disappear and ova
are found on the mesenteries. In specimens in which the axis
of the corallum is over 25 mm. in length, ripe ova are present.
As the two or three large ova on each mesentery are extruded,
a similar number of smaller ones take their place, and this

and a substance which they call by th^ name of a coin of inoney as broad and as
round as a groat^ vxmderfully printed and stamped of nature like unto some
coin.*^ — Voyages and Travels^ mainly during the Sixteenth and Seventeenth
Centuriss, C. R. Beazley, 1908, p. 168.

PROC. BOY. SOC. EDIN. — VOL. XXV. 20

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306 Proceedings of Royal Society of Edinburgh, [j

process is continued for a considerable time, there being no dearth
or vacuity in the ovary. Mr Grardiner finds, however, that in
a specimen 40 mm. long some of the mesenteries bear no ova,
but on most of them isolated ova are present. On none of the
mesenteries are there any small ova to take the place of those
which had escaped or were about to escape. ** It seemed obvious
that a critical period had been reached, after which ova ceased
to develop. . . . There is no direct proof — indeed it is only a
presumption — that the polyp now dies." This seems, however,
very probable, for the largest specimen among over 600
from the Gape of Good Hope measured 42 mm. in length, and
Mr Gardiner dredged eight dead ones in the Maldives which
average about 38 mm. His largest living specimen, the one
described above, measured 40 mm.

Mr Gardiner has been good enough to re-examine his material,
and to give us some valuable information respecting the number
of growth-lines on these old specimens. These growth-lines are
difficult to count in specimens in which the calicle is longer than
20 mm. He found that the maximum number of lines, allowing
for the cut-oflf base, is about 24 in the largest specimens. We
may assume, therefore, that these specimens of Fldbellum^
which were obviously nearing the end of their reproductive
powers, and probably also near the end of life, were about
twenty-four years old.

Mr Gardiner states (1902, p. 469) that he examined, on the
reefs of Rotuma, a large area covered by Madrepora ptdcra, Brook,
var. cUaeolata, Brook, and found that most of the polyps were dead.
The living polyps were all female, and the reproductive organs
were in the condition described above for the 40 mm. Flabellum,
that is, the ova were either few and isolated, or had been
already discharged. In this and in other similar cases mentioned
there were no external conditions, such as silting up, which might
account for death. Each colony has presumably originated from

single ovum, and the limitations in the size of the colonies point
to some reason innate in the organisms themselves. '* There can
be no rejuvenescence, and the operative cause is probably the
same as that which ultimately produces the death of our forest
trees," but Mr Gardiner does not consider that he is able to offer an

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1903-4.] On Aged Specimens of Sagartia troglodytes, etc, 307

explanation whioh is complete or quite satbfactory (but see his
paper, 1902, pp. 470, 471).

We are grateful to Mr Gardiner for permission to bring forward
here some of his observations, not yet published, on the probable
age of certain large colonies of Maldivan corals * which seemed to
be dying. His method of estimating the age of these colonies is
as follows : — The number of polyps on colonies, the age of which
is approximately known, t is first determined. Each of these
colonies presumably originated from a single primary polyp, and
the numerous polyps have been produced by successive budding.
The number of polyps so produced would increase in approximately
geometrical progression. Knowing the period required for the
production of the known number of polyps on the colony of known
age, it is possible to make an estimate of the age of the old colonies
of the same species from the number of polyps of which they are
composed. Mr Gardiner finds that the results of his examination
of several colonies are strikingly uniform, giving a maximum age
of twenty-two to twenty-eight years.

It is therefore probable that the duration of life in solitary
corals like Flabellum is about twenty-four years, and in colonial
corals such as Goniaairasa, Pnonastrasa, OrbiceUa, and PociUoporOj
from twenty-two to twenty-eight years.

LITERATUKE.

1848. Daltbll, Sir John Graham, Rare and Remarkable
Animals of Scotland^ vol. ii. ch. 10, London, 1848.

1860. GoasB, P. H., A History of the British Sea Anemones and
Corals, London, 1860.

1868. HiNCKS, T., A History of the British Hydroid Zoophytes,
vol. i., London, 1868.

* CfonioMtrwa reti/ormis, Prionastroea fuseoviridis, Orbieella laxct, and
yarioiis species or fades of PocUlop(yfa.

t These colonies must have grown up (from ova) within a period ** certainly
less than three years, and probably not more than two years and ten months."
They were obtained from a canal cut through the reef of Hulule, which is
regularly cleaned out once every three years. See J. S. Gardiner, The
Fauna and Geography of the Maldive and Laecadive Archipelagoeif voL L
pp. 329, 330, Cambridge, 1908.

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308 Proceedings of Royal Society of Edivhurgh, [sbw.

1878. M*Bain, J., ** Notes on Actinia mesembryanthemumy'
Proc, Roy, Phys. Soc. Edin,, vol. iv., pp. 280-88, Edinburgh, 1878.

1891. Weismann, a., Essay 8 upon Hereflity and Kindred
Biological Problems^ edited by E. B. Poulton, S. Schonland, and
A. E. Shipley, vol. i.,— Essay on "The Duration of Life," Oxford,
1891.

1902. Gardiner, J. S., "Some Notes on Variation and
Protandry in Flabellum rtibrum, and Senescence in the same and
other Corab," Proc. Camb. Phil, Soc, vol. xi., pp. 463-471,
Cambridge, 1902.

{Issued separately July 21, 1904.)

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1903-4.] On the Molecular Condition of Demcigiietised Nickel. 309

Note on the Moleciilar Oondition of Nickel (and Iron)
demagnetised by decreasing Reversals. By James
Bussell

(Read July 18, 1904.)

In a former communication* it has heen shown that iron
demagnetised hy decreasing reversals of a directional force ab,
develops an induction component at right angles to the sub-
sequent magnetising force H, when the angle $ between these two
forces is other than 0^ and 90^. This component after reaching a
maximum tends to disappear as saturation values are reached.

It has now been found that these transverse induction effects
also exist in nickel.

The curves for nickel resemble those for iron in the following
respects: —

(First) They change sign either if the direction of H be re-
versed, or if a6 be rotated through an angle of 90' ;

{Second) Their maxima are sharpest when d = 45' ; and

(TJiird) They vanish in the horizontal axis when tf = 0' and 90*.

The curves for nickel differ from those for iron in the following
respects : —

(First) The smallness of the transverse induction is extreme.
When ^ = 45', the nickel curves reach a maximum of about
13 C.G.S. units only. In iron, under the same conditions, the
maximum attained is equal to fully 230 C.G.S. units. In order
therefore to compare by superposition the curves obtained for
nickel and iron, the nickel ordinates require to be increased
eighteen times.

(Second) If o^ be rotated so that d is gradually reduced from
45' to 0', and the values of H be not too small, the curves are
relatively increased in value to an extent greater than the corre-
sponding curves for iron. Further, if ^ be not too small, the 45'
maximum is even exceeded.

* ''The Molecular Condition of Iron demagnetised by variouB Methods,"
Proceedings Roy, Soc Edin. , vol. xxiv. p. 544.

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310 Proceedings of Royal Society of Edinburgh, [sess.

The above results may be equally well illustrated if transverse
induction be plotted not against H for various values of ^, but
against d increasing from 0' to 90' for various values of H. If the
values of H be low, the curves for both metals appear to reach
their maxima when $ is approximately equal to 45'. If, however,
H be taken higher, maximum values are rapidly displaced to the
left, the curves rising very abruptly between 0' and 15'. In iron,
on the other hand, this displacement occurs slowly, and is (within
present experimental limits) much less in amount.

The above experiments were made with hollow cylinders, so
constructed that the shell of each cylinder was itself hollow. Or,
they may be described as hollow anchor rings flattened so that the
difference between the internal and external radii was less than 1
in 10. The width of each hollow ring was made nearly equal to
IT times its average radius. The smallness of the transverse effect
in nickel necessitated the elimination of the demagnetising effect
of the ends of the hollow cylinders previously used.

I take this opportunity of acknowledging my indebtedness to
the Royal Society of London for placing at my disposal a Govern-
ment grant for the purposes of this research.

{Issued separately August 22, 1904.)

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1903-4.] Lord Kelviu on a Free Procession of Waves. 311

On the Front and Bear of a Free Procession of Waves
in Deep Water. {Continued from Proc. R.S.E., Feb. Ist,
1904.) By Lord Kelvin.

(Read June 20, 1904.)

§ 11. The present communication is substituted for another
bearing the same title, which was read before the Royal Society
of Edinburgh on Januarj' 7th, 1887, because the result of that
paper was rendered imperfect and unsatisfactory by omission of
the exponential factor referred to in § 10 of my paper of February
1st, 1904. I shall refer henceforth to the last-mentioned paper as
§§ 1 . . . . 10 above.

§ 12. I begin by considering processions produced by super-
position of static initiating disturbances, of the type expressed in
(12) of § 4 above; graphically represented by fig. 1 ; and leading
to motion investigated in §§ 1-3, 5-10. The particular type of
that solution which I now choose, is that chosen at the end of § 4,
which we, with a slight but useful modification,* may now write
as follows : —

where p= v/(22 + x2), and X=tan-i(a;/2)

Here - f denotes the upward vertical component of the displace-
ment of the fluid at time t from its undisturbed position at point
(x, z), which may be either in the free surface or anywhere below
it. Taking ^ = 0 in (17), we have, for the initial height of the
free surface above the undisturbed level,

§ 13. We shall first take, as initiating disturbance, a row
extending from - oo to -|- oo of superpositions of (18); alternately

* The substitution of JX, for ^w- tan" *^/? — ^- , saves considerable labour

V p-z

and use of logarithms ; especially when, as in our calculations, 2=1.

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312 Proceedings of IU>y<d Soddy of Edinburgh, [

SI8S.

positive and negative ; and placed at equal successive distances \\ :
so that we now have

-i, = P(j;,0)= 2"(-l)'«^(^+»'-^.o). . . (19).

or, as we may write it,

where

D(^,0) = <^(;.,0)-<^(x+^,0) (20).

In (19), P denotes a space-periodic function, with X for its period
This formula, with t substituted for 0, represents - {„ being the
elevation of the surface above undisturbed level at time t, in
virtue of initial disturbance represented by (19).

§ 14. Remark now that whatever function be represented by ^,
the formula for P in (19) implies that

P(.r + \,0) = P(a-,0) (21),

which means that P is a space-periodic function with X for period.
And ( 1 9) also implies that

P(x-hiX,0)=-P(^,0) (22);

which includes (21). And with the actual function, (18), which
we have chosen for <^(a;, 0), the fact that 4>(x^ 0) = ^( - a;, 0) makes

P(.c,0) = P(-x,0) (23).

Thus (19) has a graph of the character fig. 5, symmetrical on each

Fig. 5.

side of each maximum and minimum ordinate. The Fourier
harmonic analysis of P(jr, 0), when subject to (22) and (23), gives

P(ar, 0) = A, co8--^ + A3 cos 3?^ + A^ cos 5?^ + • • • (24).

AAA

§ 15. Digression on periodic functions generated by addition of

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1908-4.] Lord Kelvin an a Free Procession of Waves. 31 3

values of any function for equidifferent arguments. Let f{x) \ye
any function whatever, periodic or non-periodic ; and let

P('')=^/(x + i\) (25);

which makes

P(a!) = P(a: + X) (26).

Let the Fourier harmonic expansion of ¥{x) be expressed as
follows : —

P(x) = Aq + Ai C08a + A2C0s2a + A3C083a+ . • • • i 2w«

-I-Bi8ina + B28in2a + B38in3a+ • • • • / A.

. . . (27).
Denoting hyj any integer, we have by Fourier's analysis

iX^; = />P(.)-y?^ (28);

which gives

2irx

JXA^^T rdxf (x + i\) COB j^^=r''dxf{x)coaj

JXB, = y" rdxf(x + ik) sin/""' = l^dx/(x) Binj-"^
iT:.,J 0 A J -» A ^

§ 16. Take now in (29), as by (19'), (20),

. . (29).

/(a!) = <^(a-,0)-<^(a: + ^,0)

(30).

This reduces all the B's to zero ; reduces the A's to zero for even
values otj ; and for odd values of^ gives, in virtue of (22),

^.27ra;

/+• 2

dxfl>{x, 0) cos^*

.... (31).

Go back now to §§ 3, 4, (6), (12), above ; and, according to the
last lines of § 4, take

^(a.,0)=fRsi-/2-, = ^^i^±^) . . . (32).
Hence, for the harmonic expansion (24) of P^a;, 0), we have

2Trsr
T

. . . (33).
The imi^inary form of the last member of this equation facilitates

:the evaluation of the integral Instead of cos^ in the last

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314 Proceedings of Royai Society of Edinburgh. \\

factor, substitute

cos^ — - +tsm - — , or fvx (34).

A A

The alternative makes no diflTerence in the summation

ion I rfr.

because the sine term disappears for the same reason that the
sine terms in (29) disappear because of (30). Thus (33) becomes

put now J{z + ur) = or ; whence = 2^<r, and iaj= - <t* - 2.

V(2 + ur)

. . . (36).

Using these in (35) we may omit the instruction {RS} because

nothing imaginary remains in the formula : thus we find

A,^^^\'j<r.--y. .-^=C-¥. ?f2 • J^ ■ ^/. . (37).

Swtr 8
= c--r.-= (38).

The transition in (37) is made in virtue of Laplace's celebrated

discovery / dae'^^^s/-

§ 1 7. Equation (38) allows us readily to see how near to a curve
of sines is the graph of P(a:, 0), for any particular value of k/z .
It shows that

d 2wZ 4wt 4ir2

A,= 46-r; A^'A,= ^^c--; A^A,= Vf.€-T; . (38).

Suppose for example X = 42 ; we have

c-X = €-'= 043214; Ag/A^ = 02495 ; A5/A8= "03347 . (39).

Thus we see that A3 is about 1/40 of A^ ; and A5 , about ^j^ of A^ .
This is a fair approach to sinusoidality ; hut not quite near enough
for our present purpose. Try next X= 2* ; we have

Ai = — . -043214; €-«'= -001867 ; Aj/A^ = '001078 . . (40).
vX
Thus Ag is about a thousandth of A^ ; and A5 about 1 J x 10"* of
A J . This is a quite good enough approximation for our present
purpose : Ag is imperceptible in any of our calculations : Aj is
negligible, though perceptible if included in our calculations (which

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1908-4.] Lord Kelvin on a Free Procession of Waves, 315

are carried out to four significant figures) : but it would be utterly
imperceptible in our diagrams. Henceforth we shall occupy our-
selves chiefly with the free surface, and take z = h^ the height of 0,
the origin of coordinates above the undisturbed level of the water.
§ 18. To find the water-surface at any time t after being left free
and at rest, displaced according to any periodic function P(a;)
expressed Fourier- wise as in (27) ; take first, for the initial
motionless surface displacement, a simple sinusoidal form,

- Jq = A cos(77ia; - c) (41).

Going back to (2), (3), and (4) above, let w {z^Xyt) be the down-
wards vertical component of displacement. We thus have, as the
differential equations of the motion,

dw d^w .^

d^o dhjo ^ ,.«v

^-^ + 5?=-° (*')•

These are satisfied by

w = C€""" cos(ma; - c) cos ^\/(7m . . . . (44),

which expresses the well-known law of two-dimen«ional periodic
waves in infinitely deep water. And formula (44) with Cc""'* = A
and ^ = 0, agrees with (41). Hence the addition of solutions (44),
with jm for m ; with A successively put equal to A^ , Ag . . . ,
Bj , Bg . . . ; and, with c = 0 for the A's, and = ^w for the B's, gives us,
for time ty the vertical component-displacement at depth z-li below
the surface, if at time ^ = 0 the water was at rest with its surface dis-
placed according to (27). Thus, with (38), and (24), we have P(a;, t),
§ 19. Looking to (44) and (27), and putting m = 27r/\y we see
that the component motion due to any one of the A's or B's in the
initial displacement is an en«lless infinite row of standing waves,
having wave-lengths equal to \/J and time-periods expressed by

Jim ^ jg
The whole motion is not periodic because the periods of the
constituent motions, being inversely as Jj, are not commensurable.
But by taking X = 2^ as proposed in § 17, which, according to (40),
makes A3, for the free surface, only a little more than 1/1000 of
A I, we have so near an approach to sinusoidality that in our illus-

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316 Proceedings of Royal Society of Edinburgh, [ssah.

trations we may regard the motion as being periodic, with period
( 45) f or y = 1 . This makes t = ^tr when, as in § 5, we, without loes
of generality (§ 10), simplify our numerical statements by taking
.7 = 4; and A = 1, which makes the wave-length = 2.

§ 20. Toward our problem of " front and rear,'* remark now
that the infinite number of parallel straight standing sinusoidal
waves which we have started everywhere over an infinite plane of
originally undisturbed water, may be ideally resolved into two
processions of sinusoidal waves of half their height travelling in
contrary directions with equal velocities 2/Vir.

Instead now of covering the whole water with standing waves,
cover it only on the negative side of the line (not shown in
our diagrams) YOY', that is the left side of 0 the origin of
coordinates ; and leave the water plane and motionless on the right
side to begin. At all great distances on the left side of 0, there
will be in the beginning, standing waves equivalent to two trains
of progressive waves, of wave-length 2, travelling rightwards and
leftwards with velocity llJir, The smooth water on the right
of O is obviously invaded by the rightward procession.

§ 21. Our investigation proves that the extreme perceptible rear
of the leftward procession (marked R in fig. 10 below) does not,
through the space 0 R on the left side of 0, broadening with time,
nor anywhere on the right of 0, perceptibly disturb the rightward
procession.

§ 22. Our investigation also proves that the surface at O has
simple harmonic motion through all time. It farther shows that
the rightward procession is very approximately sinusoidal, with
simple harmonic motion, through a space O F (fig. 9) to the right
of 0, broadening with time ; and that, at any particular distance
rightwards from 0, this approximation becomes more and more
nearly perfect as time advances. What I call the front of the
rightward procession, is the wave disturbance beyond the point F,
at a not strictly defined distance rightwards from 0, where the
approximation* to sinusoidality of shape, and simple harmonic
quality of motion, is only just perceptibly at fault. We shall find
that beyond F the waves are, as shown in fig. 9, less and less high,
and longer and longer, at greater and greater distances from O,
at one and the same time; but that the wave-height does not at

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1908-4.] Lord Kelvin on a Free Procession of Waves. 31 7

any time or place come abruptly to nothing. The propagational
velocity of the beginning of the disturbance is in reality infinite,
because we regard the water as infinitely incompressible.

§ 23. Thus we see that the front of the rightward procession,
%vith sinusoidal waves following it from 0, is simply given by the
calculation, for positive values of x, of the motion due to an initial
motionless configuration of sinusoidal furrows and ridges on the
left side of 0. Fig. 8 represents a static initial configuration,
which we denote by Q {x, 0), approximately realising the con-
dition stated in § 20. Fig. 9 represents on the same scale of
ordinates the surface disphcement at the time 25r in the sub-
sequent motion due to that initial configuration ; which, for any
time tj we denote by Q (ar, t) defined as follows : —

Q(^, 0 = i*(«» t)-<l>{x + l,t)'^tt>(x + 2,t)- ... ad. inf, (46),

where tf> is the function defined by (17), with z=l and g = i.

§ 24. The wave-height, at all distances so far leftward from O
that the influence of the rear of the leftward procession has not
yet reached them at any particular time, t, after the beginning, is
simply the 'P{x,t) of § 13 calculated according to §§ 18, 17;
and the motion there is still merely standing waves, ideally
resolvable into rightward and leftward processions. Let I,
beyond the leftward range of fig. 10, be the point of the ideally
extended diagram, not precisely defined, where the leftward
procession at any particular time, f, becomes sensibly in-
fluenced by its own rear. Between I and K the whole motion is
transitional in character, from the regular sinusoidal motion P(a:, t)
of the water on the left side of I, to regular sinusoidal motion of
half wave-height iP(a;, <), from R to 0 ; and on to F of fig. 9, the
b^inning of the front of the disturbance in the rightward proces-
sion. Hence to separate ideally the leftward procession from the
whole disturbance due to the initial configuration, we have only
to subtract ^F(x, t) from Q(a;, t) calculated for negative values
of X. Thus the expression for the whole of the leftward pro-
cession is

Q'a;, t) - iP(.J^, t) for negative values of a; . . . (47).

Fig. 10 represents the free surface thus found for the leftward
procession alone at time t = 25t.

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318 Proceedings of Jioyal Society of Edirtinirgh. [i

% 25. The function D(a:, t\ which appears in § 13 as an item in
one of the modes of summing shown for P(a;, 0) in (19'), and
indicated for P(a;, t) at the end of § 13, and which has been used
in some of our summations for Q(a:, t) ; is represented in figs. 6 and
7, for t « 0, and t = 25t respectively.

§ 26. Except for a few of the points of fig. 6, representing
D(a:, 0), the calculation has been performed solely for integral
values of ar. It seemed at first scarcely to be expected that a fair
graphic representation could be drawn from so few calculated points;
but the curves have actually been drawn by Mr Witherington with
no other knowledge than these points, except information as to all
zeros (curve cutting the luie of abscissas), through the whole
range of each curve. The calculated points are marked on each
curve : and it seems certain that, with the knowledge of the zeros,
the true curve must lie very close in each case to that drawn by
Mr Witherington.

§ 27. The calculation of Q(j:, t), for positive integral values of Xy
is greatly eased by the following arrangements for avoiding the
necessity for direct summation of a sluggishly convergent infinite
series shown in (46), by use of our knowledge of P(fl^ t). We
have, by (46) and (19),

Q(0, t) = i</>(0, t) - <^(1, t) + 1^(2, 0 - ad, inf. (48),

P(0,0= 2"(-l)W,0 .... (49).
<— »

Hence, in virtue of 4>{ - 1, t) = <^(i, t)^

P(0,0 = 2Q(0,0 (50).

Again going back to (46), we have

Q(«,0 = i<^(^>0-<^(^+i,0 + *(^ + 2,o-«(« + 3,0+

Q(x+1,0= i<^(j^+l,0-<^(a;4-2,0 + <^(» + 3,0-

By adding these we find

Q(a; + 1 , 0 + Q(.^, t) = ^[4>{x, t^<l>(x+ 1, t)] = ^I){x, t) (51 ).

By successive applications of this equation, we find

2Q(x + t,0 = ( - l)'2Q(uj.0-(- iyD(^,0± • . +I>(^- + *'- l»0(-'i2).

Hence by putting a;=0. and using (50), we find finally

2Q(t, 0 = ( - 1)'P(0. 0 - ( - 1)'D(0, 0 ± . . + D(* - 1, 0 (53).

This is thoroughly convenient to calculate Q(l, t\ Q(2, t) , , . .
successively ; for plotting the points shown in fig. 9.

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1903-4.] Lord Kelvin 07i a Free Procession of Waves, 319

♦,!f^

H'^^^ ^ >

^ (N lO !*• 5

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320

Proceedinf/8 of Royal Society of Edinburgh. [a

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1903-4.] Lord Kelvin on a Free Procession of Waves. 321

FROC. HOY. SOC. KDIN. — VOL. XXV.

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322 Proceedings of Roycd Society of Edinfmrgh, [sess.

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1903-4.] Lord Kelvin on a Free Procession of Waves. 323

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[(54).

324 Proceedings of Royal Society of Edinburgh. [s»m.

§ 28. For fig. 10, instead of assuming as in (47) the calculation
of Q{x, t) for negative values of a*, a very troublesome affair, we
may now evaluate it thus. We have by (46)

Q(^xJ)^i<f>{-x,t)-<f>{-x+ht) + <f>(-x+'I,t)-

Hence

-<^(-r+l,0 + <^(-a: + 2,0- .
Now by the property of 4>, used in the first term of (54), that its
value is the same for positive and negative values of x, we have
<^( - jj + 1, i) = <t>(x - i, t). Hence (54) may be written

Q(^, 0 + Q( - ', 0 = 'x ( - 1 )'*(^ + »'')= Pe^. «) • (-"^s)-

Hence Q( - ^S /) = H(^, <) " Q(-»^. ') (56).

Using this in (47) we find

iV(jr,t)-qix,t) (57),

for the elevation of the water due to the leftward procession
alone at any point at distance x from 0 on tlie left side, x
being any positive number, integral or fractional. Having pre-
viously calculated Q(x,t) for positive integral values of x, we
have found by (57) the calculated points of ^^. 10 for the leftwanl
procession.

§ 29. The principles and working i>laus described in §§ 1 1 - 28
above, affortl a ready means for understanding and working out in
detail the motion, from ^ = 0 to< = oo, of a given finite i)rocession
of waves, started with such displacement of the surface, and such
motion of the water below the surface, as to produce, at f = 0, a
procession of a thousand or more waves advancing into still water
in front, and leaving still water in the rear. To show the desired
result graphically, extend fig. 10 leftwards to as many wave-lengths
as you please beyond the i)oint, I, described in § 24. Invert the
diagram thus drawn relatively to right and left, and fit it on to the
diagram, fig. 9, extended rightwards so far as to show no perceptible
motion ; say to a; = 200, or 300, of our scale. The diagram thus
compounded represents the water surface at time 25t after a com-
mencoraent correspondin^jly compounded from fig. 8, and another

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1903-4.] Lord Kelvin on a Free Frocessian of Waves, 325

similar figure drawn to represent the rear of the finite (two-ended)
l)rocession which we are now considering.

§ 30. Direct attack on the problem thus indirectly solved, gives,
for the case of 1000 wave-crests in the beginning, the following
explicit solution,

i»'JO00

-i= ^(-im*-*.o (58),

where i/r is a function found according to the principles indicated in
§ 4 above, to express the same surface-displacement as our function
<;^ of § 12, and the proper velocities below the surface to give, in the
sum, a right ward procession of waves. Our present solution shows
how rapidly the initial sinusoidality of the head and front of a
one-ended infinite procession, travelling rightwards, is disturbed in
virtue of the hydrokinetic circumstances of a procession invading
still water. Our solution, and the item towards it represented in
figs. 6 and 7, and in fig. 2 of § 6 above, show how rapidly fresh
crests are formed. The whole investigation shows how very far
from finding any definite " group-velocity " we are, in any initially
given group of two, three, four, or any number, however great, of
waves. I hope in some future communication to the Royal
Society of Ekiinburgh to return to this subject in connection with
the energy principle set forth by Osborne Reynolds,* and the inter-
ferential theory of Stokes t and Rayleigh { giving an absolutely
definite group- velocity in their case of an infinite number of
mutually supporting groups. But my first hydrokinetic duty,
the performance of which I hope may not be long deferred, is
to fulfil my promises regarding ship-waves, and circular waves
travelling in all directions from a place of disturbance in water.

§ 31. The following tables show some of the most important
numbers which have been calculated, and which may be useful
in farther prosecution of the subject of the present paper.

* Nature^ vol. xvi, 1877, pp. 343-4.

t Smith's Prize Paper, Camh. Univ, Calendar, 1876.

t Sound, ed. 1, vol. i., 1877, pp. 246-7.

[Table I.

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326 Proceedings of Royal Society of Edinburgh. [sess.

Table I.
P

2
8

10
11

I 13
14
15
16
17
18
19
20
21
22
23
24
25
2()
27 I
2S

29 I

30 I

31 ,
32
33 -

1-4142
1-0987
-8045
•6452
-5490
-4843
•4375
•4018
•3784
•3502
•3308
•8142
•2999
•2874
•2763
•2663
•2574
•2498
•2420
•2352
•2290
-2232
"2179
•2129
•2082
•2039
•1998
•1959
•1923
•1888
•1855
•1824
•1795
•1767

0)=-D(0,OJ

<»(«,0)

D(-l,0)=-D(o,()

Xa:,0)

D(x,u)

•8155

•1740

•0026

•2942

•1714

•0025

•1698

•1689

•0023

•0962

•1666

•0028

•0647

-1643

•0022

•0468

39 1

•1621

•0021

•0357

40 ,

•1600

-0020

•0284

41 1

•1580

•0019

•0232

•1561

•0019

•0194

48 '

•1642

•0018

•0166

-1624

•0017

•0143

•1507

•0017

•0125

•1490

•0016

•0111

■1474

•0016

•0100

-1468

•0016

•0089

•1443

-0016

-0081

•1428

•0014

•0073

•1414

-0014

•0068

•1400

•0014

•0062

•1886

•0013

•0058

•1373

•0018

•0053

•1860

•0012

•0050

•1348

-0012

•0047

-1336

•0912

•0043

-1324

•0011

•0041

•1318

•0011

•0039

■1302

•0011

-0036

•1291

•0011

•0035

-1280

-0010

-0033

-1270

-0010

•0031

•1260

•0010

-0029

•1250

•0010

-0028

•1240

-0009

•0027

•1231

0009

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1903-4.] Lord Kelvin 07i a Free Procession of Waves, 327

Table II.

t = 25t ; T = ^TT ; x = ^^

16
17
18
19
•JO
21
'2*2
23
24

2r»

26
27
28
29
30
31
32
33
34
35
36
37
i 38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68

... (!*•)_ .

41 IT + 133" 43'

39»+ 30' 41'

:5(3ir + 16r 31'

34ir + 157° 22'

33t H ir 3'

31ir+ 77' 25'

29jr hl7r 23'

28ir-hl09^ 24'

27ir+ 68' -20'

26ir+ 45* 33'

I5ir+ 39' 6'

24,r+ 46** 38'

23ir+ 67" 0'

>2ir+ 98" 45'

21» + 140-' 41'

1t+ ir 47'

;0t+ ir 11'

9ir + 138" 6'

9ir-h C:' 50'

8r + lir 49'

.8ir+ 17" 29'

7»- + 108" 23'

7ir+ 24* 6'

6ir + 124* 14'

6ir+ 48* 27'

5ir + 156* 27'

L5ir+ 87' 58'

5»f 22* 44'

4ir+140* 32'

4t+ 81° 8'

4ir+ 24* 24'

i3ir + 150* 6'

3ir-f 98* 9'

3ir+ 48* 20'

3ir+ 0* 82'

L2ir + 134* 40'

2ir+ 90° 36'

•2t H 48° 10'

2ir+ T 2.'/

Iir-fl48* 9'

lir + 110' IS'

lir+ 73' 48'

1t-H 38" 35'

lir+ 4' 34'

[0ir + 15r 43'

.Or + 119* 67'

Oir+ 89* 14'

0»+ 69* 30'

0ir+ 80* 43'

0ir+ 2* 50'

9ir + 155* 48'

9ir+129* 36'

9ir + 104* 9'

9ir+ 79* 28'

T'^^n/I^K?)'"^^''^"^'

•0002
•0005
•0011
•0024
•0044
•0075
•0118
•0174
■0246
•0333
•0434
•0550
•0679
•0820
•0917
•1131
•1299
•1472
•1651
•1832
•2016
•2'201
■2385
•2569
■2752
•2934
•3112
•3287
•3459
•3629
•3794
■3956
•1112
•4267
■4416
•4560
•4702
•4840
•4973
•5101
•5226
•5348
•5464
•5580
•5690
•5797
•5900
•6001
•6098
•6193
•6284
•6872
•6469
•6540

^x, 25r)

D(a-, 25t)

•0000

+ •OOOI

- •oooi

- ^0002

+ -0001

- ^0002

+ •ooos

+ ^0006

- ^0003

+ ^0020

- •00-23

- -0018

- -0005

- ^0055

+ -0050

+ •0117

- -0067

- 0136

+ -0069

+ -0146

- -0077

- •oiss

+ 0111

+ -0281

- -01 70

- ^0386

+ -0216

+ -0377

- -0161

- •OlOl

- -0060

- ^0372

+ •0312

+ -0558

-•0246

- 0032

- -0214

- 0626

+ ^041 2

+ ^0267

+ •0145

+ 0637

- ^0492

- 0266

- -0226

- ^0713

+ •0487

+ -0021

+ ^0466

+ •0728

- 0262

+ •0425

- ^0687

- -0410

- ^0277

- -0761

+ ^0474

- ^0290

+ -0764

+ ^0434

+ •0330

+ -0741

-•0411

+ ^0429

- -0840

- •oigo

- ^0650

- ^0642

- ^0008

- ^0657

+ ^0649

" ^0282

+ 0931

+ ^0224

+ ^0707

+ -0582

+ •01 ^25

+ -0643

-•0518

+ •0417

- -0935

+ -0035

- '0970

- •0332

- 0638

- ^0556

- •0082

- ^0578

+ ^0496

- -0421

+ ^0917

- ^0162

+ ^1069

+ •0141

+ -0928

+ -0373

+ -0555

+ -0501

+ ^0054

+ ^0506

- ^0452

+ -0403

- -0856

+ -0226

- •lOSl

+ ^0022

- -1103

{Issiud snparatchf August 22, 1904.)

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328 Proceedings of Royal Society of Edinburgh, [sess.

Some Results in the Mathematical Theory of Seiches.
By Professor Chrystal.

(Read JuTy 18, 1904. MS. received July 29, 1904.)

{Abstract.)

I propose in this preliminary communication to lay before the
Society some results of investigations in the theory of Seiches m
a lake whose line of maximum depth is approximately straight,
and whose depth, cross section, and surface breadth do not vary
rapidly from point to point.

As the seiche disturbance is small compared with the length of
the lake, I shall make the assumptions usual in the theory of long
waves : — viz., that the squares of the displacements and of tlieir
derivatives are negligible.

The a:-axis, O X, is taken in the undisturbed level of the lake,
in the average direction of the line of maximum depth ; the c-axis,
0 Z, is taken vertically upwards. The horizontal and vertical dis-
placements of a water particle originally in the undisturbed surface,
at a distance x from the origin, are denoted by f and ^. A(x) and
h{jc) are used to denote the area and the surface breadth of tlic
cross section at a distance x from O.

AVe suppose that the vertical disturbance at every point in the
surface line of any cross section of the lake is the same ; in other
words, we neglect the dynamical effect of any flow perpendicular
to 0 X due to the gradual increase or diminution of the area of
the cross section of the lake. As in the theory of long waves, the
vertical acceleration is also neglected ; and we also neglect the
(usually small) effect due to the shelving of the shore.

With these assumptions, the equations which determine i and ^
are found to be

= ^^W-^. (1)

C^2

f - - s <^)

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1903-4.] ProL Chrystal o)i Mathematical Tkeoinj of Seiches. 329

where 2< = A(ar) ^^v^ldx b{x) , cr(r) = A{z) h{x) ; and (j and t have

the usual meanings.

A natural* seiche of frequency n is therefore deterraine<l by the
equations

A(a;)f=<i = P8inw/ + Qcosw/, .... (3);
where P and Q are solutions of

Since ^{v) is a slowly varying function of r, wo might take it to be
either a linear or a quadratic integi-al function of v. On the former
assumption the solution of (4) is found to depend on BesseFs Func-
tions. It is found, however, that the assumption a(v) = A(l - v-/a^)
is more convenient for obtaining approximate representations of
the cases that occur in nature. The solution in tliis case is found
to depend on certain functions which wc may call the Seiche
Functions, defined, for - l<w< + 1, by the following convergent
series : —

r ^ r{r-\.2) . .•(f-1.2) 0' - 3.4) ,

c ^ r(r--i.3) , c(c-2.3)(6'-4.5) .
S(c,f.)=«;- 2:3,^3 + 23 ^4 gt.- 2.3 X 4.5 X 6.7 "^•■*-- ' • '

,,, , , '• o c(c+1.2) ^ r{c -h 1.2) {c + 3 A) ^
(£(c,t.)^l-j;^t.2+j-2~3^tr^--^2^3^~5, ,.«+...,

^, , <• 3 c(c4-2.3) , c(c + 2.3)(c + 4.5) .

c(c,t.) = r.-g^^.3 + __--^^^,,., 2.3x4.5x6.7 «''+••••

The functions C and S are synectic integrals of tlie differential
equation

* As opposed to ^forced seiche, whose period depends jwirtly on the period of
the disturbing agency. Some of the seiches on Lake Erie arc. I believe, of this
nature.

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330 Proceedings of Royal Society of Edinburgh, [sRis.

and are connected by the relation

C(c,t^)S'(c,tt7)-CXott7)S(c,t^?)=l . . . (6),*

where the dashes denote differentiation with respect to w. On
account of the fact that C and S have certain of tlie properties of
cos w and sin «;, and in a certain limiting case reduce to these
functions, we may call them the seiche-cosine and the sticlie-sine
respectively. From another point of view they are limiting case*
of the hypergeometric function ; but from this fact no practical
advantage has been found hitherto.

In like manner S(c, w) and (2(c, w), which we may call the
hyperbolic seiche-cosine and hyperbolic seiche-siiie, are integrals of

(i+»^^;S+cP=o, (7)

and

{i{c,w)Z\c,w)-{^\c,w)^{c,w)^\ ... (8)

For the particular values w = 1 and w — i (where % is the imaginary
unit) we have

C(c.I) = (l-i-2)(l-3^)(l-5y J

6(<-..-) = 0^iy0^o)0^5?6) I

€(<:,.■) =<l + 2?3) 0^0)0^6-7) '

(0)

(IU>

It follows from Sturm's Oscillation Theorem regarding the solu-
tions of a linear differential equation, such as (5), that, for any
given real value of t; -^ 1 , there are an infinite number of positive
real values of c which satisfy the equations

C(e,i-) = 0, S(c,v) = Oj
(i(c,«') = 0, S(c,v) = 0;

and that the roots of either of the equations of one of these pairs
separate the roots of the other.

* The analogue of the relation co8'j; + sin^ = l for the circular functions.

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1903-4.] Prof. Chrystal on Mathematical Theory of Seiches, 331

It appears at once from (9) that the real positive roots of
C(c, l) = Oare

r= 1.2, 3.4, 5.6, Le, 2, 12, 30, . . (11)

andof S(c, 1) = 0

c=2.3, 4.5, 6.7, i.e. 6, 20, 42, . . (12)

The roots of @(c, 1) = 0 and @(c, 1) = 0 are neither commensurable
nor so easily found. A somewhat laborious arithmetical calcula-
tion, in which I have been kindly assisted by Dr Burgess and Mr
E. M. Horsburgh, has given for the smallest positive root of (S(c,l)
= 0 c = 2-77 .... , and for the corresponding root of (S(c, 1) = 0
c = 12.34

It should also be observed that, when c has one of the values
(11), C(c, v) reduces to an integral function of v; wid the same
happens to S(c, v) when c has one of the values (12).

If we assume <r(v) = A(l + v^la^\ the equation for P is

which, if we put w = vja , and take

reduces to either (5) or (6). Hence A{x)i can be expressed in
terms of the seiche functions ; and f is given by

a dw

In the case where the breadth of the lake is constant and the
cross section rectangular, but the depth variable, say h{x) =
h(l -x^ja-), we can replace the variable v by x. The constants h
and a are then linear magnitudes (whose meanings are obvious)
instead of a volume and an area as in the general case. It will be
observed, therefore, that all the general features of the phenomena
of seiches are to be foimd in this more special case, regarding which
we now give some particulars.

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332

Proceed uiys of Royal Society of Edinhcrgh. [seb^.

Lake with Symmetric Longitudinal Section of Parabolic
Concave Form h{x) = A x (1 - jc^ ^2j

If c-=v{v-{- 1) , and T^ be the period of the v-iiodal seiche, then
T^ = 'Iirhi = 27ralJ(Cyg/i) = irlij{ v{v + 1 )yh) } . . (13)
where /( = 2a) is the whole length of tlie lake.

a O a A

Fig. 1.
Fji* SL'iches Avith odd and even numbers of nodes we have

and

A C(r2,_,,»r)

l-^6-'-*

sin nt , X.^ 'a ^^^'^'-^ » *^^ ^^" "' '

S a

(14)

(15)

respectively.

Un I NODAL Seiche.
., = 1.2; Ti = 7r//V(2^//) (IH)

Node

A 2Aa;

^= -J- sin n^ , (; = -^ sin id ,

(17)

If 1,^ denote the maximum horizontal and vertical displace-
ments of a particle on the surface at the end of the lake, and ^ the
maximum horizontal velocity of displacement, then

l^lllih, l^irmh\ (18)

It should be observed that here, and in the cases that follow
under the present head, the boundary condition at A and A' is not
that i =• 0 , but that the motion be tangential to the shore.

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1903-4.] Prof. Chrystal on Mathematical Theory of Seiches. 333

BiNODAL Seiche.

C5 = 2.3; T2 = 7rZ/7(67/i), .... (19)

5 = - sm w^ , C = -r. ^^ «in w^ , . . (20)

Nodes .T= ±a/V3- ± -HT . . . a (21)

We have

T,7T. = 72/^6 = -574 (22)

Hence the period of the binodal seiche in a concave lake of
symmetric paralx)lic section is greater than half the period of
the uninodal seiche.

Also the nodes are more than half way towards the ends ; i.e.
they are displaced towards the shallows.

If ^ , ^, and \ have the same meanings as before, we have

|=Z^/4/i, i^irltl2hT^_ (23)

at the ends of the lake. At a node the values of f and ^ are
reduced in the ratio '57 . . . : 1. At the centre ^=0 at all
times ; and I has half its value at the end of the lake.

Trixodal Seiche.

^3 = 3.4; T3 = 7r7/V(l-2r//0.
^= A^(a2 - 5j:2) sin nt, ^= ^{V2(i^x - 20u.-3) sin n/, . (24)

Xodes x = 0, x= ±aj3/Jb=- ±-7746 a, . . . (2.^))

'yTi = N/2/x/12 = .4082 (26)

QUADRINODAL SbICHE.

r,= 4.5; T, = 7r//V(20r///), (27)

f=Jl^(3a^ -7.^2) sin n/, t^^^T'J -3a* + 30a^u^^3r)x^)m\vt (28)

^Vles a;- ±.3400. . .rt, ±-8621 a, . . (29)

T/l\ = .3162 (30)

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334 Proceedings of Royal Society of Edinhiryh. [sbss.

QUINQUINODAL SbIOHS.

^ fia*
^= -^(30a4^- 140aV+ 126r5) sin nt,

(31)

(32)

Nodes

a: = 0, ±.5384 a, ±.9058... a, . . (33)

T,/Ti = .2582 (34)

Lake with Symmetric Longitudinal Section of Parabolic
Convex Form h(x) = hx{l+sc^/a^).

A 0

Fio. 2.

If q, Cj, Cj iv .... be the real positive roots

taken in order of magnitude of the equations @(c, 1) = 0 and
3(<J, 1) = 0, so that Ci is the smallest positive root of Ci(<^, 1) = 0,
Cg the smallest positive root of S(c, 1) = 0, and so on, then, for
seiches with an odd number of nodes,

^=X^-W~«'"'"' r=-|®'(f-..«')8m«/,. . (35)

for seiches with an even number of nodes

^ B S(C2,_,,tr) , , ^ A^,,

^ = X ~TVw^ ^'"^ "*' ' ^" - "^^ ^^^-^' '^^ ^'^ nt, . . (36)

Uninodal Seiche.
Ci = 2.77..., Z^^ttII J{%11 ...gh), . . (37)

Hence Xi<Ti; that is to say, for the same central depth and
the same length, the uninodal period is less when the lake is
convex than when it is concave.

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1903-4.] Prof. Chrystal on Mathematical Theoi^ of Seiches, 335

BiNODAL Seiche.
C2=12.34, 3:2 = 7rW(12.34...^;i), . . . (38

Hence %^<T^,

Also 2;.,/2:i= V{2.77 . . . /12.34. . . } = .474. . . (39)
In other words, in a convex lake of symmetric parabolic section
the period of the binodal seiche is less than half the period of the
uninodal seiche.

It follows, of course, from the fact that the seiche functions
degenerate into the circular functions when the curvature of the
bottom is infinitely small, that when the lake bottom is flat
T., Tj = ^, etc., as in the case of vibrating rods, or strings.

Case of Concave Lake with Unsymmetrio Biparabolic
Section.

The depth from 0 to A is given by /i(ic) = 7i(l - i^^^a^) ; from

If w = xia, w' = x:a ; c = n^a'^lgh, c' = n^a^lfjh, then for the two
portions 0 A and 0 A' we have respectively

^h(l - w^) = ^^ ;^ ^--{ S{c , l)C{c , w) - C(c , 1)S(. , w)}mi nt ,

C= -^^f^i){S(r,l)C'(c,f^)-C(c,l)SX<^,fr)}sinn^; . (40)
and
^h{\ - ,.'=)= --^ {^c', \)C{c\ w) + C{c, 1)S(<-', rc')}sin nt,

r - - ,7S^){«('''' 1) ^y^ «'') + C(c', \)^'{c, tr')}sin nt . (41)

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336 Proceedings of Royal Society of Edirtburgh, [

The values of C and C which determine the periods are given
by dc = a^la^ together with the period-equation

aC(c,l)S(c,l) + aC(c',l)S(c,l) = 0 . . (42)

If we put a"^c = c^c = n^a^a'^lgh — z, the period equation may be
written

"(•-ii.)(>-5:b) ■•■(•-w.X'-ii-.)

*»■('- ,.i-0(' - 5:^-.) ■ • (' - s.)(' - 4..y ■••-»•■(«>

Unsymmetric Lake with onb Shallow and two Maximum
Depths.

/ w

h l^ Ti' ri* O d J> I, 3 ^

Fig. 4.

A good approximation to the form of lake section in many cases
that occur in nature can be obtained by piecing together six
parabolas, as in figure (4), so as to form one continuous curve. If
B be the minimum^ and h and li the two maximum depths, D and
D' the points of inflexion ; A B = a^, A' B' = a j, B D = ^, B' D' = h\
O D = 6?, 0 D' = iVy then we may represent the portions A B, B D,
I) 0, 0 D; D' B', B' A' by the six parabolas \—1i(x) = /i(l - x^la^) ;
li{x)^li{y'X^la^)\ h{x) = s(l+xVa^^); h(x)=^8(l +zVa^^); ?i\x)
= h\l - xVa\^) ; h{z) = h\l - xVa\^),

The conditions of continuity lead to

a.^ = hh{d + h)j(h - 8\ a^ - ^d(d + h)l(li - «) ;

(44)

All the magnitudes marked in the figure may be arbitrarily
determined ; but after tliis has been done the depths at the points
of inflexion are not at our disposal.

The formulae for i and i and the period-equation have been
worked out for this case. I'hey involve all the four seiche-

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MODEL INDEX.

Schafer, E. A. — On the Existence within the Liver Cells of Channels which can
be directly injected from the Blood-vessels. Proc. Roy. Soc. Edin., vol ,
1902, pp.
Cells, Liver, — Intra-cellular Canaliculi in.

K A Schafer. Proc. Roy. Soc. Edin., vol. , 1902, pp.
Liver, — Injection within CeUs of.

E. A. Schafer. Proc. Roy. Soc Edin., vol. , 1902, pp.

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iv CONTENTS.

PAGE

Effect of Transverse Magnetization on the Resistance of
Nickel at High Temperatures. By Professor C. G.
Knott, ...... 292

(Issued separately July 30, 1904.)

Observations on some Aged Specimens of Sagartia troglo-
dytes, and on the Duration of life in Coelenterates.
By J. H. Ash WORTH, D.Sc, Lecturer in Invertebrate
Zoology in the University of Edinburgh, and Nblson
Annandale, B.A., Deputy - Superintendent of the
Indian Museum, Calcutta. Communicated by Pro-
fessor J. C. EwART, M.D., F.R.S., . . .295
[Issued ftejKtratch/ July 21, 1904.)

Note on the Molecular Condition of Nickel (and Iron)
demagnetised by xlecreasing Reversals. By Jambs
Russell, . / . . . 809

{Issued separately August 22, 1904.)

On the Front ami Rear of a Free Procession of Waves in
Deep Wate/ {Continued from Proc. R.S.E., Feb. Ist,
1904.) B^ Lord Kelvin, . . .311

/ {Issued separately August 22, 1904.)

Some Results in the Mathematical Theory of Seiches. By

Professor Chrystal, ..... 328
{Issued separalehj October 6, 1904.)

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rt. -

•Jf!

PROCEEDINGS

OF THE

ROYAL SOCIETY OF EDINBURGH.

SESSION 1904-5.

No.V.l VOL. XXV. [Pp. 337-400.

CONTENTS.

PAGE

A New Form of Spectrophotometer. By J. R. Milne,
B.Sc, Carnegie Scholar in Natural Philosophy,
Edinhnrgh University, . . . .338

{Issued separately November 5, 1904.)

A New Form of Juxtapositor to hring into Accurate
Contact the Edges of the two Beams of light
used in Spectrophotometry, with an application to
Polarimetry. By J. R. Milne, B.Sc , Carnegie
Scholar in Natural Philosophy, . . , 355

(Issued separately January 17, 1905.)

The Three-line Determinants of a Six-by-Three Array.

By Thomas Muir, LL.D., . . . .364

{Issued separately January 20, 1905. )

[Continued on page iv of Cover,

^EDINBURGH :

PuBLisHKD BY ROBERT GRANT & SON, 107 Princes Street, and
WILLIAMS & NORGATE, U Henrietta Street, Covent Garden, London.

MDCCCCV.
Price Four Shillings.

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1. MANUSCiiiPT OP Papers. — As soon as any paper has been passeil
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[CojUinucd on page iii of Cover.

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Air..

1908-4.] Prof. Chrystal on MathenuUical Theoi^y of Seiches, 337

f onctions ; and are naturally somewhat complicated. We therefore
omit them from this preliminary communication.

In a more detailed paper which I propose to submit hereafter to
the Society I shall give particulars regarding the establishment of
the above results, further developments of their application, a
discussion of the agreement of the results in particular cases with
observation, and a comparison of the above theory with that given
by Du Boys in his " Essai Theorique sur les Seiches " {Arch, d, Sc.
Phys, et Nat, d. Geneve, P^r. iii. t. xxv., 1891).

In the meantime I cherish a hope that the above summary may
help to encourage and to guide the ardent observers who are now
engaged in procuring for us accurate data regarding the interesting
natural phenomena with which they deal*

(Issued separately October 6, 1904.)

PBGC. ROT. SOC. KDIN.— VOL. XXV. 22

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338 Proceedings of Royal Society of EdvrJburgh. [i

A New Form of Spectrophotometer. By J. R Milne,
B.Sc., Carnegie Scholar in Natural Philoeophy, Edinburgh
University.

(Read July 4, 1904. MS. received Aogast 1, 1904.)

The present paper is the continuation of a note sent to the
Society in July of last year,* and is for the purpose of describing
the developed form of the spectrophotometer whose principle was
indicated in that communication.

The former paper described the employment of a divided
spherical lens to bring together the two slightly separated spectra
seen in any ordinary form of spectrophotometer. This divided
lens is placed at about twice its focal length behind the two spectra

Fig. 3.
produced by the objective of the telescope, and, when suitably
adjusted, gives rise to two spectra in contact with each other, as
shown in fig. 1 of the former paper. It has been found, however,
to be better to modify the action of the divided lens, and to use
it as indicated in fig. 3. The defect of the former arrangement
can be seen from fig. l,t where the point b is beneath O, per-
mitting light from h to pass straight along beneath the lens-half
L, to prevent which an opaque stop is required to fill up the space
00', the stop being so contrived that freedom of relative motion is
still preserved to the two halves of the lens L and L'. In the
present arrangement, which is depicted in fig. 3, no such device is

• Proc. Roy. Soc. Ediv., vol. xxiv. p. 496, 1908.
t See t«)riner note.

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1903-4.] Mr Milne on a New Form of Spectrophotometer. 339

needed, provided that the light rays from any the same point of
ab and c^ are all in a horizontal plane through that point, and this
condition, as will be seen later on, is actually fulfilled.

The edg^ a' and d' of the two spectra formed by the objective of
the telescope are necessarily somewhat hazy and ill-defined, whether
the gap between the spectra has been produced by the menisci of
a liquid or by the edge of a solid. To remedy this, a strip of
metal ad is placed so as to cut off the extreme edges of the two
spectra, and by this means the edges which are afterwards brought
into contact in the plane SS' are beforehand made perfectly
straight, and are sharply delimited. This strip of metal or
" trimmer " ad (fig. 3) really consists of two similar pieces, which
by means of a slow motion screw can be arranged in such a way
that the compound strip is slightly wider at one end than at the
other. This arrangement the author has found to be necessary,
as in his model instrument, for reasons of economy, the divided
lens is a simple one, and so the neighbouring edges of the images
of the two spectra formed by the flivided lens are not parallel
to each other. This difficulty is perfectly overcome, however, by
making the trimmer slightly wider at one end or the other as
may be required. The point is mentioned because even with a
more perfect lens the device might be necessary to obtain the
most exact results.

It inevitably happens that the two beams of light • falling on
the trimmer adinfig. 3 suffer marked diffraction, and if (say) the
lower beam be stopped off, obvious diffraction bands at the lower
edge of the remaining spectrum may in general be seen on looking
through' an ordinary telescope eyepiece, placed behind the divided
lens at a distance of about twice the focal length of the latter. If,
however, the eyepiece, originally somewhat too far off to focus
objects in the plane SS', be slowly pushed nearer that plane, the
diffraction bands, which in this case are dark and are situated
upon the bright strip, are observed to begin closing in towards the
edge of the image, and when the eyepiece is exactly focussing the
plane SS' no bands are to be seen at all. On continuing to move
the eyepiece towards the plane SS' the bands reappear, being now
bright lines situated outside the bright strip, and they continue to
move out from its edge with the motion of the eyepiece. These

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Proceedings of Royal Society of Edinburgh. [i

340

results show then, that when the eyepiece is correctly focnssed no
trouble will be experience* 1 from diffraction effects.

In the former note all that was contemplated was a device for
attachment to an ordinary spectrophotometer to briqg the two
spectra exactly together, that the judging of their relative inten-
sities might be made more accurate. The author, however, had
in view the object of measuring the light intensities of various
liquids, which were to be contained in tubes about a metre long,
and it was found that for this purpose, in addition to the above
device, a further modification in the form of spectrophotometer
was desirable. This new design of instrument also presents
advantages for general spectrophotometrical work.

Fig. 4 is intended to give a diagrammatic view of such an

Fio. 4.
[That some parts may be more easily seen, this diagram is not drawn to scale.]

apparatus. The collimator A is so far distant from the prism R,
that there is room to insert between the two the long tube B
containing the liquid. The ends of the tube are made of plane
parallel glass, so as not to interfere with the parallelism of tiie rays
of light passing through it. Before the customary vertical slit of
the collimator, there is placed a thin piece of opaque metal pierced
with another slit whose opening is horizontal, so that the effective
aperture of the two is a very small rectangular hole. This
arrangement results in the production of a beam of light from
the collimator lens, which is sensibly parallel, and, the tube B
being only half filled with liquid, all the upper half of the beam
of light passes entirely clear of the latter, while all the imder half
of the beam passes through the full length of the liquid.

Without this arrangement, and using the light as it comes

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1903-4.] Mr Milne on a New Form of Spectrophotometer'^ 341

naturally from the light source with its different rays inclined in
various directions, there would be the following difficulty. Some
of those rays which are inclined downwards would enter the tube
above the liquid, but before leaving the tube, they would pass
into the liquid, and so the emergent lower beam would consist
only partially of rays that have passed through the whole length
of the liquid.

At first sight it might be supposed that this error is compen-
sated by a similar addition from the lower to the upper beam, but
this is not the case, for, as will be seen on reflection, each beam
would thus gain eqiLoL quantities of light, whereas, did complete
compensation occur, the gains of the lower and of the upper beams
respectively would bear a ratio to one another which is equal to
the fraction of the total light, incident upon it, which is trans-
mitted by the absorbing liquid.

In reality too the number of rays passing from the lower to the
upper beam within the absorption vessel is not equal to the
number passing from the upper to the lower, because a large pro-
portion of the former rays will be totally reflected down again at
the surface of the liquid; and consideration will show that this
fact will make the error spoken of above still greater.

There is also the further point that with non-parallel light and
a long absorption tube the number of rays that pass out through
the sides of the tube will be different for the upper and for the
lower part of the tube, owing to the presence of the liquid in the
latter.

With a non-parallel beam not only do the two above noted
difficulties arise, but there comes in the additional error that the
source of light is in effect brought some distance nearer in the
case of the beam that passes through the liquid, and hence the
light intensity of that beam is increased, that of the other beam
being left unchanged.*

Even were the beam of light employed to be the cone of rays
proceeding from a very small hole in an opaque screen placed
immediately in front of the light source and at the level of the

*In this connection see a paper entitled "On the Absorption Spectra of
some Ck)ppeT Salts in Aqneous Solution," by Thomas Ewan, 6. So., Ph.D.,
Pm. Mag, (5), No. 208, p. 881, April 1892.

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342 Proceedings of Royal Society of Edinburgh. [sess.

liquid in the tube, of these three errors just noted only the first
would be done away with, besides which such a plan would give
less light intensity than the arrangement of the collimator described
above.

In a spectrophotometer as ordinarily made there is no room
between the collimator and the prism for an absorption vessel,
and to comply with the above parallel light condition it becomes
necessary to take off the collimator and to mount it by itself in
front of the spectrophotometer at such a distance as permits of
inserting the absorption vessel between the two.

In the ordinary type of spectrophotometer there are two difficulties
that would arise were parallel light to be used. It will be
seen that as all the rays of both the beams of light which emerge
from the absorption vessel are parallel to the general optic axis of
the instrument, these two beams of light, after duly passing
through the prism and the object glass of the telescope, will give
rise to one and the same spectrum; and that the width of this
spectrum will be very small.

Taking the latter difficulty first, the width of the spectrum
produced by any spectroscope must be equal to the length of the
collimator slit midtiplied by the focal length of the telescope
objective and divided by the focal length of the collimator lens.
Now in the above arrangement the ** length" of the small hole
which acts as a collimator slit may be about j^^th of an inch, so
that the spectrum formed by the two beams of light will have a
quite insufficient width for our purpose. Besides, we require
each beam to give rise to a separate spectrum, and we must not
have the two spectra formed in the same position one upon the
other. Both difficulties, however, are readily solved by using as
the telescope objective a cylindrical lens (C, fig. 4) whose axis of
figure is placed vertically : the focal length of the lens being
identical with that of the spherical lens whose place it has taken.
In this way, while using a strictly parallel beam of light to paas
into the absorbing vessel, we obtain two separate spectra placed
one above the other, and formed respectively by the " comparison "
and by the *' absorbed '' beams of light ; and the widths of these
spectra are amply sufficient, for they are respectively equal to
the heights of the cross sections of each beam of light. See

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1903-4.] Mr Milne on a New Form of Spectrophotovieter. 343

p and g, fig. 4, which represent the intersections by the
plane of the paper of the two spectra formed by the cylindrical
lens C.

In the case of a spectrophotometer where the light intensities
are regulated and measured by means of a Vieroidt double slit on
the collimator, the latter cannot be removed and placed in front of
the absorption vessel without the loss of this means of controlling
the light intensity. Of course the collimator might be left on the
spectrophotometer and another collimator might be arranged in
front of the absorption vessel, the two beams of light from the
latter being directed upon the two Vieroidt slits respectively.
With such an arrangement, however, the intensity would be
reduced by the narrow openings of the Vieroidt slits, as well as
by the small rectangular opening of the first collimator. The
writer tried a modification of the above plan designed to obviate
this loss, in which the second collimator being provided with
Vieroidt slits, the latter were made to open at the maximum to a
width equal to that of the two beams of light, while the lens of
this collimator was discarded. Those changes are legitimate
because the light rays have already been made parallel by the first
collimator, and all that we wish to retain of the Vieroidt double
sUt collimator is its power to regulate the intensities of the two
beams. The difficulty with this plan is that the beam of light
produced by the first collimator is apt not to have the same
intensity at every point across a normal section, and if, for example,
the jaws of one of the slits be closed together till only the half of
that beam is permitted to pass through, we shall not in general
have reduced the total light intensity of that beam by one half.
The uniformity of the distribution of intensity in the cross
section of the beam of light after leaving the first coUimator
depends to a lai^e extent on what source of light is employed ;
lime light, owing to the small area of its light source, being
markedly inferior for such a purpose to a flat acetylene flame.
Even with the latter, however, a doubt may exist as to the perfect
equality of the intensity throughout the cross section of the beam,
and 80 this modification of the Vieroidt double slit was abandoned
and another device for intensity regulation was substituted which
will be discussed later.

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344 Proceedings of Royal Society of Edivhmgh, [obs.

The plane P and Q (fig. 4) in which the two spectra p and q
are formed, is occupied by a screen whose function is to limit
the field of view of the eyepiece to two narrow strips taken one
from each spectrum for the purpose of having their intensities
compared. This screen is shown diagrammatically in fig. 7. By
means of the sliding piece A, the colour of the strip taken from
the upper si»ectrum can be altered at pleasure, while by means of
the second sliding piece a, mounted on the first, the width of the
strip can be altered. The sliding pieces B and h perform similar
offices for the lower spectrum. Through the opening of the slides
the trimmer T may be seen. The latter is fixed at the side of the
screen adjacent to the divided lens, and fulfils a function that has
already been explained.

After the screen there follows at a distance of about twice * its

Fio. 5.

[That some of the parts may be more easily seen, this diagram is not drawn to

scale, nor does C show the true cross section of the lens at that place. ]

focal length the divided spherical lens D (fig. 4), and, as shown
above, the resulting images in the plane FF (which is conjugate to
the screen in the plane FQ) of the strips of the two spectra p and
q can be arranged by adjusting the lens-halves so that their edges
are in complete contact.

It may be mentioned here that there is an alternative arrange-
ment of the parts just described which has the merit of shortening
the telescope tube. The latter point is important, because if an
ordinary spectroscope prism be employed, all the parts of the

* It will be recollected that the mmimnm distance between an object and
the image of it formed by a convergent lens is equal to four times the focal
length of the lens ; the divided lens has been placed at a distance of twice its
fooal length from the two spectra p and q (fig. 4), so that the telescope tube
may be as short as possible.

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1903-4.] Mr Milne on a New Form of JSpectrophotoifteter, 345

optical train that follow mnst be capable of rotation about a
vertical axis through the prism to an extent sufficient to cause
all the various colours of the spectrum in their turn to fall upon
the eyepiece and to be seen by the eye of the observer. In the
usual form of spectrophotometer this is achieved by supporting
the telescope only at its objective end, which is pivoted so as to
have the required rotatory motion. Now in this instrument the
telescope tube must have a length equal to the focal length of the
cylindrical telescope objective, plus a further length, equal to four
times the focal length of the divided lens. Such a length of tube
makes it difficult to secure the necessary rigidity without resorting
to a cumbrous form of mounting.

The alternative form of apparatus just mentioned, which is
shown diagrammatically in fig. 5, is provided with a single lens,
C and D, which takes the place of the two lenses C and D
of fig. 4, with the result that the telescope tube is shortened
by a length equal to the distance between the planes PQ and FF'.
In order to find the specification of the lens required in this case
two points must be borne in mind. In the first place the lens,
when placed behind the prism R (fig. 4), must give rise to two
pure spectra formed from the two beams of light respectively.
Now a cylindrical lens with its axis of figure upright will fulfil
the above condition. Its focal length may equal the distance
from C to the line PQ, so that the spectra will be formed in a plane
normal to the paper through the latter line. In the second place,
as already explained, to avoid diffraction eflfects the trimmer must
be situated in a plane conjugate to that in which the spectra are
formed. To fulfil this condition along with the other the lens,
having its front face ground to the cylindrical curvature deter-
mined above, must have its back face ground as a cylindrical
lens whose axis of figure is horizontal. The exact focal length
of the curvature on the back face of the lens we shall discuss
later. At present it will merely be specified that it is to be less
than the focal length of the curvature formed on the front face.
This lens will bring the beam of parallel light ABCD (fig. 6a) to
a line focus EF, where £F is situated as before at a distance from
the lens equal to the distance from C to the line PQ (fig. 4).
Before reaching £F, however, the beam is first brought to

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346 Proceedings of Hoyal Society of Edinburgh. [i

another line focus OH, the distance from OH to the lens being
equal to the focal length of the curvature on the back face of the
lens. Now this lens, if placed behind the prism R in fig. 4, will
form two pure spectra in the plane normal to the paper through
the line PQ. Further, if in that figure the strip of metal called
the trimmer be placed immediately behind the lens of the
collimator, we can arrauge, by properly choosing the radius of
curvature of the back face of the lens, that the plane in which
the trimmer is placed shall be conjugate to the plane in which
the spectra are produced ; and this fulfils our second condition.

The two spectra so formed from the two beams respectively
will exhibit a dark gap between them, and therefore, as before,

Fio. 6a.

the lens is to be cut through the centre in a horizontal plane,
and then on separating the lens-halves to the required extent the
two spectra can be moved towards each other till their edges come
into perfect contact. In fig. 6jS, there are shown two beams of
homogeneous light, and the resulting lines EM, MF (which are
two elements of the two spectra that would be formed in the
general case) are drawn as they would be if brought with their
ends just to touch each other by an appropriate separation of the
lens-halves L and L'.

With a simple lens, on bringing the edges of the two spectra
near each other, it can be seen that they are not parallel This
is due to a mixture of the errors of distortion and of chromatic
aberration of the lens. To remedy this it would be of no
avail to make the trimmer wider at one end, as explained on
page 339 ; for reflection will show that that would merely reduce
the intensity of the light which forms the adjacent edges of the

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1903-4.] Mr Milne on a New Forvi of Spectrophotometer. 347

two spectra, but would not alter the positions of those edges.
Elach spectrum, however, may be tilted to the required amount
by slightly rotating the corresponding lens-half about the general
optical axis of the instrument. In order to preserve symmetry
the respective rotations in opposite directions of the lens-halves
should be to equal amounts. Even with so-called achromatic
lenses this device will probably be found necessary.

The limitation of the field of view seen by the eye to a similar
narrow strip from each spectrum is obtained in this form of the
instrument by an appropriate screen in the eyepiece.

Fio. 6/8.

Finally, it may be noted that it is desirable, in the interests of
good definition, to use spherical lenses in preference to cylindrical,
and to avoid curvatures of too small radius. Accordingly, instead
of the theoretical lens discussed above, it is better to substitute
one having the curvature on one of its faces spherical, and having
the other face a convergent cylindrical lens whose axis of figure
is horizontal. The proof that such a lens can be equivalent to
the former is part of the general theory of optics, and neither this
proof nor any details as to the necessary focal lengths of the
curvatures, etc., need be entered upon here.

The author's experiments with this form of the instrument have
not been numerous, because he found that the cheap divided lens
used by him in the model gave less satisfactory definition than the
spherical divided lens employed in the model of the instrument
first described. He believes, however, that with a well-made lens
this second arrangement of instrument might perhaps be better
than the other.

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348 Proceedings of Royal Society of Edinburgh. [i

It may, however, be noted that the chief objection to a some-
what long telescope tube can be done away with by the use in the
spectrophotometer of a "constant deviation prism,"* a construc-
tion of prism which permits the telescope of the instrument to be
permanently fixed, while the prism alone rotates to bring the
different parts of the spectrum to the observer's eye. In the case
of a spectrophotometer furnished with such a prism the rigid
mounting of even an unusually long telescope tube of course
presents no difficulty.

Either of the above described arrangements of spectrophotometer
having been adopted, it might be supposed that the similar strips
of the two adjacent spectra could be satisfactorily observed on
looking at them through any ordinary eyepiece. What is thus

Fio. 7.

seen, however, is unsatisfactory. The two luminous strips are not

like natural objects, which give out rays of light in all directions

from every point, but on the contrary the edge of each of the

strips brought into contact gives out rays of light only in a single

plane, as indicated in fig. 8. From any point a of the upper edge

of the lower image rays proceed only in the plane normal to the

paper which passes through the line aB, and similarly from any

point b of the lower edge of the upper image the rays proceed only

in the plane normal to the paper which passes through the line bA.

Accordingly, the coincident edges of the two images are seen by

means of two sets of rays which respectively fall on the optical

system of the eye at places some distance apart. Now through

the effects of the eye's spherical aberration, and probably also

because of general irregularities in the refractive parts of the eye,

the two sets of rays from the coincident edges of the strips will

not be brought to the same line on the retina. Any slight move-

* As employed, for example, by Messrs Hilger, Ltd., on oertun of their
speotrometers.

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1908-4.] Mr Milne on a New Form of Spectrophotometer, 349

ment of the head will alter the paths of the two sets of rays
through the optical system of the eye, and the effect of such a
moYement will be to cause an apparent relative motion, as seen by
the observer of the really coincident edges of the two spectra.
As a matter of fact the edges of the two spectra are seen by an
observer to be slightly overlapping each other at one moment,
while a moment later a slight gap will have made its appearance
between them. This, no doubt, is due to movements of the head
or eye.

The author at first sought to remedy this defect by giving to the
divided lens a focal length of about half a metre, which caused a
reduction of the angle AaB of fig. 8, and a consequent reduction
in the distance between the two sets of rays aB, bA when

Fig. 8.

entering the eye. A specially short eyepiece also was used, so
that the eye of the observer might come as near the diverging
point a as possible. These alterations, while undoubtedly effecting
much improvement, were after all only palliative in their effect,
and the comparatively great focal length of the divided lens
necessitated a somewhat unwieldy length of telescope tube, a point
that has already been dwelt upon.

After various other methods had been considered without
success the following means of overcoming the difficulty was
finally discovered. Advantage was taken of the well-known fact
that if a ray of light fall normally upon one of the faces of a
Wollaston double image prism there proceeds from the other face
two divergent rays which are polarised in planes at right angles to
each other. If now — reversely — there fall on one of the faces of
the Wollaston prism two converging rays of light inclined at the
proper angle, these two rays will emerge from the opposite face of
the prism in one and the same straight line normally to the face.
It is true, of course, that unless the entering rays be each

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350 Proceedings of Royal Society of Edinburgh. [siss.

polarised, and polarised in the proper planes respectively, then in
addition to the two coincident exit rays there will be two other
non-coincident exit rays, making four exit rays in all, but the
divergent rays have no connection with our present purpose, and
may be disregarded, as will be shown later. Suppose now that
a suitable Wollaston prism be placed in the plane FF', fig. 4,
then all the rays which go to form the edges of the two spectra in
the plane FF* (two of which rays are indicated by C6 and Do,
fig. 8) proceed, after passing through the Wollaston prism, in one
and the same horizontal plane through the eyepiece E. In this
way all the rays from any point conmion to the coincident edges
of the two spectra fall on the cornea of the observer's eye in one

Fig. 9.

and the same straight line, so that the optical defects of the eye
spoken of before do not cause any difficulties.

As mentioned above, each ray incident on the Wollaston prism
gives rise to two emergent rays. Considering then, for example,
the point b (fig. 8), we see that the ray proceeding from it in the
plane of the paper will, after passing through the prism, give rise
to two emergent rays SH and KL (fig. 9). The ray KL will not
be seen at all by the observer unless the angle COD be small and
the power of the eyepiece low. In the model that the author has
had constructed the distance CO is about 6*5 inches and CD is
equal to -6 inch, while the eyepiece is one of moderate power, and
such rays as KL can only be seen by moving the eyepiece either
up or down until the junction of the two bright strips has passed
out of the field of view, so that only one of the two bright strips

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1903-4.] Mr Milne on a New Form of Spectrophotometer, 351

can be seen by the observer. It will then be noticed that this
strip has superimposed upon it another bright strip (G or D, fig. 10),
at the end which has just been brought, into view by this move-
ment of the eyepiece. Hence it appears that really there are in
all four bright strips, disposed as shown in fig. 10, the two
with which we are concerned being the middle pair with their
edges in contact along the line AB. Now the two additional
bright strips G and D are formed by KL and the other rays
whose refraction is analogous, and it will be seen that the state-
ment made above — that the rays so refracted may for our purpose
be ignored — is justified.

But there is a further advantage to be gained by such a use of
a Wollaston prism. It will be remembered (see p. 343) that the
use of a Yieroidt double slit in connection with this instrument

Fio. 10. — Under normal oonditions only the middle portion of the above can
be seen through the eyepiece, C and D lying outside the field of view.

to regidate and measure the light intensities of the two beams was
found to be unsatisfactory owing to the great loss of light which it
entailed, while a modified arrangement of the same kind had
also to be discarded. Now, with the arrangement of apparatus
described above, by the mere addition of a Nicol prism to the
eyepiece there is provided the necessary appliance for regulating
the intensities of the two strips of light seen by ^^he observer until
a perfect match is attained. The rays r and » (fig. 4), after trans-
mission through the Wollaston prism, pass out along the same
straight line /, but remain distinct in this, that they are polarised
in planes at right angles to each other. Accordingly, because of
the Nicol in the eyepiece, rotation of the latter about its axis
causes every possible variation from zero to infinity of the ratio of
the intensities of the two strips of light seen by the observer. By
means of a circular vernier or other device the position of the

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352 Proceedings of Royal Society of Edinburgh. [sias.

eyepiece as regards its rotation is ascertained after each setting to
equal intensity of the two bright strips. The ratio of the bright-
ness of the two strips is equal to the square of the tangent of
the angle of displacement of the eyepiece, if the zero of the
latter be that position in which the light of the under strip
is completely extinguished. As the under strip is that due to
the comparison beam, that is, to the beam that does not pass
through the absorbing liquid, the tangent of the displacement
angle is equal to the fraction of the incident light transmitted
by the absorbing substance.

It is to be noted that, in common with other polarising spectro-
photometers, this instrument suffers from the defect that the light
in passing through the main prism is partially polarised in a
vertical plane, for which reason, when there is no absorbing sub-
stance in the path of either beam, and when accordingly the
analysing Nicol ought to give equality of illumination when set
at an angle of 45**, it is found that the Xicol has to be turned
round slightly from that position before the intensities of the two
beams will exactly balance. The amount of this error, which
depends on the refractive index of the glass of the prism, etc.,
can be calculated by Fresners formulsB, and in a case computed
by the author it is about 4** 40'. As, however, it is hoped later on
to publish some experiments on this subject, the mathematical
discussion need not be entered into here. It only remains to be
said, that the observations made with such instruments are to be
reduced by assuming that a certain (constant) absorbing body has
been permanently placed in the path of one of the beams.

It should be noted that the two beams of light are in some
ways asymmetric as they pass through the Wollaston prism, and
hence it is possible that different fractions of the light may be
transmitted in each case. As regards absorption the existence of
such a crystal as tourmaline shows that this may be very different
in the case of the ordinary and of the extraordinary rays. With
the crystal mentioned the ordinary ray is practically non-existent
after transmission through one or two millimetres of the substance,
while the extraordinary ray in the same circumstances is only
slightly absorbed. In the case of this instrument, however, the
Wollaston prism is of quartz, which is a substance where no such

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1903-4.] Mr Milne on a New Form of Spectrophotometer, 353

marked disparity in the absorption coefficients for the ordinary
and the extraordinary rays exists ; and, further, any difference in
the intensity of the transmitted beams after passing through the
WoUaston prism could, in any case, only be a small one, because
the rays of each beam are transmitted for the length of approxi-
mately half their path through the crystal as ordinary (extra-
ordinary) rays, and for the remaining half as extraordinary
(ordinary) rays.

Another possible source of asymmetric error lies in the fact
that the rays from any, the same point of the image p (fig. 4) may,
after passing through the Wollaston prism, diverge to a different
extent from the rays from the corresponding point in the other
image q, after they have passed through the Wollaston prism.
Were this the case, and were the difference sufficiently marked,
the eye would see the strip due to the less divergent beam to
sensibly greater advantage as regards intensity than the strip due
to the other beam. And indeed the two images themselves,
because they are formed inside the Wollaston prism, may not
correspond in brightness to the original beams, for the rays of the
two beams respectively may be converged to a different extent on
entering the prism.

Any such errors, however, did they exist could be at least very
approximately got rid of as follows. The light absorption of any
liquid for any particular wave length would be twice measured,
once with the Wollaston prism emitting the upper beam as the
ordinary ray, and then with the Wollaston prism turned upside
down and emitting the same beam as the extraordinary ray. The
mean of these two measurements would give the true absorption
very nearly.

The model instrument which has been made, while it shows
the general soundness of the principles involved, is not capable of
measurements of the accuracy required to definitely settle this
question. All that can be said in the circumstances is that no
such discrepancy can be seen with the present apparatus.

In the note of last year the use of the instrument for Murphy's
method of mapping the visual intensity of a spectrum was pointed
out, and it only needs to be said that the necessary adjustments
of the apparatus are those described in that communication in the

PROC. ROY. SOC. BDIN. — VOL. XXV. 23

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354 Proceedings of Roycd Society of Edinburgh, [sm.

case of the form of this instrument which is depicted in fig. 5.
In the case of the form which is depicted in fig. 4 the sliding
pieces A and B of the screen (fig. 7) are first set respectively to
the two neighhouring strips of the spectrum whose intensity it is
desired to compare, and then the lens-halves L and L' (fig. 3) are
moved sideways normal to the plane of the paper to bring the two
images of these strips one above the other in the plane SS'. The
perfect contact of the edges is secured by moving the lens-halves
vertically either nearer together or further apart, as has already
been explained.

{Isstud seiarately November 5, 1904.)

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i90d-4.] Mr J. B. Milne on a New Farm of Juxtapositor. 356

A New Form of Juxtapositor to bring into Accurate
Contact the Bdges of the two Beams of Light used
in Spectrophotometry, with an application to
Polarimetry. By J. R. Milne, B.Sc, Carnegie Scholar
in Natural Philosophy.

(Read June 20, 1904. MS. received June 23, 1904.)

In the ordinary spectrophotometer and in Laurent's "half-
shade" polarimeter, two neighhouring patches of light of the
tame colour but of different intensities are presented to the eye
of the observer,, who by an appropriate means reduces the
intensity of the brighter until in his judgment it is brought down
to the same intensity as the other. The accuracy of such a
measurement must depend on two factors. The first factor is the
accuracy with which the observer's eye can judge of the equality
of the two patches of light, and the second factor is the accuracy
with which the instrumental reading indicates the intensity of
the comparison beam, i.e., of the beam whose brightness is reduced
till it becomes equal to that of the other. Now it is found
that in ordinary cases the error of the eye's judgment in such
measurements amounts to about 4% or 5%, while the measurement
of the instrumental regulation of the light can be made much
more accurately. Accordingly, the error in the measurements
made with a spectrophotometer cannot be much less than 4% or
5% unless some special means be employed for improving the eye's
power of judgment in such a case, and the mere provision of a
finer instrumental graduation will not meet the difficulty. Con-
siderable assistance would be rendered to the eye were the two
patches of light, whose equality the eye is to judge, brought with
their edges accurately to touch each other so that no hiatus existed
between them. As a rule however such a hiatus does exist, for
should the two lights be' from different sources, the edge of the
mirror or other appliance which directs the comparison beam into

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356 Proceedings of Roycd Society of Edinbv/rgh. [sbss.

the instrument invariably shows as a more or less badly defined
dark space between the two spectra ; while in those cases where
only one light source is employed, one part of the beam being
absorbed by any given substance and the other part used for
comparison, the edge of the substance, if the latter be a solid, or
the meniscus, if it be a liquid, brings about the same result. The
object of the present paper is to describe an appliance by which
this difl&culty may be overcome.

The instrument (see fig. 1) is constructed of two separate
pieces of glass which are cut from the same block to ensure

Fiol.
The two glass blocks cemented The two glass blocks shown

together. apart as they are before

being oemented.

similarity of optical properties. These pieces having been worked
truly plane on the faces which transmit the light, are silvered
over the portions shaded in the figure, and are then cemented
together along their common interface PQRS. The effect of the
cement, whose refractive index is practically the same as that of
the glass, is to make the joint nearly optically homogeneous with
the glass blocks on each side. As will be seen from the diagrams,
in every case the various faces of the blocks are either perpen-
dicular, or are inclined at an angle of 45* to each other.

The glass block thus built up is encased in a metal shell, with

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1903-4.] Mr J. K. Milne on a New Form of JuxiaposUor. 357

appropriate openings for the entrance and exit of the light. The
best place for the attachment of the apparatus to the spectro-
photometer will, of course, vary to some extent with the pattern
of the instrtiment — in the author's case it is mounted immediately
in front of the collimator slit. When the juxtapositor is so
situated with regard to the spectrophotometer, the upper or " com-
parison " beam of light enters face AB (fig. 2a) and meets the
interface CD at an angle of 45**, and the part of it falling on the
area OC is reflected upwards by the silvering. The other part,
which falls on the unsilvered surface OD, passes straight on
and out through the face CF, and is not used. In the same

/• ,/

Fio. 2

way the lower or " absorbed " beam enters face DE, and is
reflected upwards by the silvering on the face EF, and the
part of it incident on the lower half OD of the interface
DC continues on its vertical course upwards. The other part,
which falls on the silvered surface OC, is reflected out through
the face CF, and is not used. The two beams which are re-
spectively reflected and transmitted by OC and OD pass upwards
in a common vertical direction, and have their edges in complete
contact along a plane through OL normal to the paper. The
beams thus brought into contact are reflected once more at the
silvering on the face GH, and pass out through the face HC
parallel to their original direction.

In the above discussion the action of the juxtapositor has been
explained in a particular case — namely, when attached to a spectro-
photometer immediately in front of the collimator slit — and we

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368 Proceedings of Royal Society of Edinbv/rgh, [i

have spoken of the *' comparison " beam as entering the face AB^
while the absorbed beam was supposed to enter the face D&
But obviously these are merely the special circumstances of a
particular mode of application of the apparatus, and the latter
might be attached to a spectrophotometer in any other way, and
would work equally well, provided that one of the beams — it doea
not matter which — is made to fall normally on the face AB, and
the other to fall normally on the face D£ ; and provided also that
the point O (fig. 2) be in a plane optically conjugate to the retina
of the observer's eye. The latter condition is necessary to avoid
the appearance of diffraction effects caused by the cutting off of
the edges of the two beams at the edge of the silvering on the
interface ; and also because the juxtapositor cannot be so exactly
made that the two beams emerge from it quite parallel to each
other ; but as can easily be seen, their edges in such a case will
once more be brought in contact in any plane where a real image
of the point 0 is produced by the parts of the optical train of the
spectrophotometer.

An important, and indeed one may almost say essential, principle
of such an apparatus has been successfully observed, namely, that
each of the two beams of light should pass through exactly the
same length of glass. When this condition is not fulfilled the
light from one beam will be absorbed to a greater extent than the
light from the other, and an error will thus be introduced. Of
course, in theory at least, an appliance faulty in this respect might
be used correctly were its differential absorption found accurately
beforehand ; but the correction would have to be ascertained for
a great number of different wave-lengths throughout the visible
spectrum, and every observation made with the spectrophotometer
when the appliance was in use would have to be individually
corrected. That the passage through even a short length of glass
causes marked absorption in a beam of light, particularly at the
blue end of the spectrum, has been shown by various workers,
among others by Nichols and Snow;* and the knowledge of thia
fact caused the author to reject an earlier design which, though

* ** Note on the Selective Absorption of Light by Optical Glass and Calc-
spar." By Edward L. Nichols and Benjamin W. Snow. PhU Mag. (6),
No. 208, pp. 379-882, April 1892.

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1903-4.] Mr J. K. Milne an a New Foimi of Juxtapositor, 359

otherwise satisfactory, could not claim to be entirely symmetrical
as regards absorption with respect to the two beams of light. Of
course, as regards symmetry, absorption is not the only thing to
be taken into account : the reflections and refractions of the two
beams must be the same ; but an examination of the figures will
show that each of the two beams of light in this apparatus sufiers
two reflections and four refractions {i,e, into the glass, into and
out of the cement, and finally out of the glass). This form of juxta-
positor, as the author originally designed it and had it constructed,
was arranged in what at first sight appears to be a symmetrical
manner, and the fallacy involved was not observed till later on.

Fig. 3.— The letter 0 cannot be shown in the above diagram, but its
position is the same as in fig. 2 (a) and (/3).

In this older form the upper of the two blocks of glass which
compose the apparatus was cut through at an angle of 45*", as
shown by the line GB (fig. 2j8). The triangular comer so detached
was cemented on again, the cemented junction GB in the path of
the upper beam being for the purpose of balancing the cemented
junction DO in the path of the lower beam. The reasoning as
to the symmetry of this form with regard to the two beams of
light is as follows: — Each beam is twice reflected at a silvered
surface. Each beam passes once from air to glass and once from
glass to air. Each beam passes through the same total amount of
glass. Each beam passes through one cemented junction. Hence
the juxtapositor is symmetrical with respect to the two beams.
In this reasoning, however, we are assuming the eflfect of a junction
to be the absorption of the light owing to its cement layer, while

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360 Proceedings of Moi/al Society of Edinburgh. [ssss.

in reality this effect is inappreciable, the cement layer being -so
extremely thin; and we are leaving out of account the effect of
the reflection by such a junction, which is not inappreciable, as
will be shown later. Now, were it the case that all the light of
the upper beam fell on the silvered part OC of the interface DC,
and none of it on the unsilvered part OD, then each of the two
beams would lose the same fraction of its light as it passed
through the cemented joint in its path, i.e., as the upper beam
passed through the junction GB and the lower beam passed
through the junction DO. It is necessary, however, that the
lower edge of the upper beam should fall at least some distance
below the point O in the figure, because only in this way can
the full intensity of light be ensured right up to the edge of the
silvered part OC of the interface CD. Assuming then that we
have the lower part of the upper beam of light falling on the
unsilvered part of the interface DO, there must exist the following
state of affairs : — A certain fraction of the light of the upper beam
is reflected by the junction GB and passes out through the face
AG, leaving the beam that passes on towards the interface CD so
much the less intense. The light lost in a similar manner by
the lower beam, however, by being reflected at the junction DO
and sent out through the face CF is more or less made up for by
the light of the lower part of the upper beam which is reflected
vertically upwards from the same junction DO.

If, however, the junction BG (fig. 2)8) were to be omitted, and
the upper beam of light arranged to cover the whole face AB (fig.
3), then the gain and the loss to the light of the lower beam,
caused by the interface at OD, would exactly balance each other.
Provided always that is, that the juxtapositor is placed in the
optical train after that piece of apparatus, whatever its particular
form, whose function it is to equalise the intensity of the two
beams of light, for then we have two beams of equal intensity
falling on the same surface (OD) at the same angle, and
accordingly the reflections will be of exactly the same magnitude.

In those cases where the juxtapositor is not so placed we have
the loss or gain of intensity of the lower beam given by a quantity
which is the reflection at the cement of the difference of the in-
tensities of the two beams, and even here the error introduced

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1908-4.] Mr J. R Milne on a New Form of Juoctapositor. 361

will, iu general, be less than it would be were such a junction as
BG (fig. 2)8) arranged in the path of the upper beam. Moreover,
in such cases to make the junction CD optically more homo-
geneous, such a liquid as the well-known a-monobromonapthaline
might be used between the faces of the glass blocks instead of
the ordinary cement.

The following experiment was undertaken with the view of
approximately ascertaining the amount of light reflected by such
a cemented jimction OD as occurs in this juxtapositor. The face
D£ of the latter was blocked up by an opaque screen, so that no
light could pass through. The apparatus was then brought near
a window, and the image of the latter produced by reflections
at the silvered part CO of the interface, and at the silvering on
the face HG was observed by looking into the face HC. No
image whatever could be observed caused by a reflection from the
unsilvered part OD of the interface, and not even an increased
darkness could be seen corresponding to the places where the
images of the window bars would fall. As a still more stringent
test, the juxtapositor, with the lower face DE blocked up as
before, was brought quite close to an incandescent electric lamp.
In this case an image caused by reflection from the unsilvered part
OD of the interface could be seen, but the image did not show
the glass or brass fittings of the lamp, but only the glowing
filament itself. Accordingly, it is clear that while there must be
some difference between the refractive indices of the glass and of
the cement used in the juxtapositor, which gives rise to reflection
of light at the cemented surface, the fraction of the total light so
reflected is very small indeed. It was noted also that the colour
of that part of the glowing filament which was reflected by the
unsilvered part OD of the interface appeared to be unchanged,
which indicates that the small difference in the refractive indices
of the glass and of the cement must be at least approximately
constant for different wave-lengths.

The edge of the silvered part of the face PQRS (fig. 1) of the
upper block of glass is cut off very trim and sharp by means of an
ivory chisel and nitric acid. This is a most important point in the
construction, because it is at this place that the two beams of
light unite, and on the abruptness of the termination of the

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362 Proceedings of Royal Society of Edinhwrgh, [i

reflecting surface depends the perfectness of the joining of th&
beams.

It is to be noted that the use of polarisation for r^pilating and
measuring the light intensities is not prohibited by the adoption,
of this appliance, even when the light is polarised before being
passed through the latter. No change of polarisation or produc-
tion of polarisation can be caused by the entrance to or exit from
the glass, for that only takes place normally to the various feces.
If the two beams are plane polarised vertically and horizontally
before entrance, with a view to the adjustment of their relative
intensities later on by means of a Nicol prism, then because the
plane of polarisation in each case is either in or normal to the
plane of incidence on the silver surfaces no change of polarisation
can occur. On the other hand, if the two beams have their
respective planes of polarisation inclined to the vertical and to the-
horizontal, these beams, because they are each twice reflected at
parallel silver surfaces, will emerge plane polarised still, though
the plane of polarisation of each has been rotated to some extent.
Hence in both cases the analysing Nicol can be used as before
for the purpose of measuring the light intensity, although the zero
will have been permanently displaced through a definite angle.

A suggested application of the juxtapositor described above will
be readily understood by anyone conversant with the construction
of Laurent's '* half-shade " polarimeter. In that instrument two-
parallel beams of light polarised in planes at an angle to one
another are passed through a substance whose rotative power it is^
desired to measure, and are then analysed by means of a Nicol
prism. By properly adjusting the position of the latter, the two-
half-circles of light seen in the eyepiece of the instrument, due Uh
the two beams, can be made equally bright. It is found that in
this way a much more accurate setting of the rotating Nicol can be
obtained than when, as in the ordinary case, only one beam of
light is employed and the Nicol is set to extinction. But the^
accuracy of the measurement in lAurent's improved form of instru-
ment turns on the degree of precision with which the eye is able Uy
determine when the two halves of the circle seen in the eyepiece
are equally bright. Now, these two halves are separated by a
dark line, and accordingly, as explained above in the case of tbe^

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1908-4.] Mr J. R Milne on a New Form of Juxtapoaitor. 863

spectrophotometer, increased accuracy of measurement would result
from getting the two bright semi-circles into perfect contact along
their common diameter. By the use of this juxtapositor it is
hoped this may be accomplished, and the accuracy of polari-
metrical measurements correspondingly improved.

The experiments which led up to the designing of this form
of juxtapositor were made in the Physical Laboratory of the
University of Edinburgh. The apparatus employed was in part
supplied by a grant fropi the Moray Endowment Fund, to the
trustees of which the author's best thanks are due.

{Isiued separately January 17, 1906.)

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364 Proceedings of Royal Society of Edinburgh. [a

The Three-line Determinants of a Six-by-Three Array.
By Thomas Mnir, LL.D.

(Seoond copy of MS. received September 12, 1904.*
Read November 7, 1904.)

(1) If the array in question be

<h h

9\
9^
9%

*8 »

its score of three-line determinants | a^p^f^ \ , | o^^^/s I > • * • -
may be viewed as consisting of two complementary sets of ten,
each of the first set containing at least two columns taken from
\ 0^62^3 I , and each of the second set at least two columns taken
from |/i<72^|. Further, either set of ten may be viewed as
consisting of one unique member and three sub-sets of three
members each, the members of a sub-set being derivable from one
Another by performing the cyclical substitutions

In this way a convenient notation for the twenty determinants
will be found to be

1 «lV8 1

1 "i Vs 1 . 1 \<^^% 1 . 1 <'i«s/s 1
1 ai6,//j 1 , 1 \cj^ 1 , 1 e^a^g^ |

-■

0
1,2,3
4. 5. 6
7, 8, 9

1 /i!7A 1 ]
1 <\9^t 1 . 1 «i Vs 1 . 1 \f^t 1
Cl^2/« 1 . 1 ai/2^8 1 » 1 *1?S*S 1

. = ■

0'
1'. 2', 3'
4'. 5', 6'
r, 8', 9'

* The original MS. was despatched by the aathor from Cape Town on
20th!March 1904, but was lost in transit through the post.^Sec R.S.E.]

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1904-5.] Three4me Determinants of a Six-by-Three Array. 365

(2) T?ie prodtk:t of any two complementary determinants of a
six-by-three array is expressible in six different toays as an
aggregate of three similar products.

Taking as an example the product | a^b^c^ \'\f\9^z I ^'^' ^^\ ^^
have from a well-known theorem by interchanging /, g^ h in
succession with a

«iVf H /i^a^ 1 = \fi^^\'\<h9Jhi\ + \lh^Wfi<^\ + \fhf>^\^f^9^\*

%.e. 00' = 88' + 22' + 55'.

By interchanging f g, h in succession with b and f g^ h in
succession with c two similar identities are obtained, viz.

00' = 99' + 33' + 66',
00' = 77' + 11' + 44',

which, however, it is simpler to view as derivatives of the first by
cyclical substitution. On altering the order of the factors in the
given product the same procedure leads us to

O'O = 8'8 + 6'6 + I'l,
O'O - 9'9 + 4'4 + 2'2,
O'O = 7'7 + 5'5 + 3'3.

It is clear (1) that what is here done with 00' can be done with
any similar product; (2) that each product on the right, by
reason of the mode of obtaining it from the product on the left^
will consist of factors that are complementary, (3) that the
theorem used will not give more than six expressions, because
the interchanging of two letters with two, — which is the remaining
possibility, — is the same in effect as interchanging one with one.

(3) The nine products in the first triad of expressions for 00',
.... are the same as the nine in the second triads and further
can be so arranged that a row-and-column interchange wUl produce
the latter triad, any five of the expressions thus giving the sixth.

Thus in the case of 00' such an arrangement is

88' + 22' + 55' 88' + 66' + 11'

66' + 99' + 33' and 22' + 99' + 44'
11' + 44' + 77' 55' + 33' + 77'

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366 Proeeedingi of Boyal Socidy of Edinburgh, [sm

That this must in all cases hold is evident from consideiiiig
that the interchanges, which when made on 00' produce the nine
products of the first triad, are

fi), CO. C).
I). D, Q

and that these when taken in columns are exactly the interchanges
which need to be performed on 00' to produce the products of the
second triad.

(4) The ten groups of such sets of six expressions may thus be
compactly exhibited as follows : —

00'

11'

00-

-44'

65'

-77'

-66'

22'

-88'

33'
44'

99'

00'

-22'
33'
66'

77'

-99'
56'

88'

-11'

-77'

00'

-11'

-44'
88'

-33'

22'

-55'

99'

66'

11'

44'

77'

88-

22'

66'

99'

33'

22'

00'
-44-

-65'
66'

-88''
~33'"

-99'

11'

77'

56'

00'

-22'
-88'

00'
-11'

-6G'

-33'

-77'

11'

66'

44'

88'

99'

-22'
33'

-55'

99'

77'

44'

33'

00' 1 - 66'

-99'

- 55' 44'

- 77' i 22'

11'

88'

66'

00'

-11'

-88'

-33'

22'

44'

-99'

56'

77'

99'

00'
-22'
-44'

-33'
11'

-66'|

77' 1

88'

55'

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1904-5.] Th/ree4ine Determinants of a Six^by* Three Array. 367

Of course in the sixty equations here implied every distinct
•equation is repeated four times; for example, the equation
00' -11' -44' -77' = ^ occurs under each of the headings 00',
ir, 44', 77'. The number of distinct equations is thus 15.

(5) These fifteen equations are not all independent, the fact
being that any one of the ten sets of six gives rise to all the
remaining nine equations. Thus, taking the first set of six, viz.

00'-ll'-44'-7r = ^,
00'-88'-22'-56' = 0,
00'-66'-99'-33' = 6?,
00'-ll'-88'-66' = 6>,
00'-44'-22'-99' = 0,
00'-77'-55'-33'-(?,

we can eliminate from pairs of them the nine binomials

00' -11', 00' -44', 00' -77',
00' -22', 00' -55', 00' -88',
00' -33', 00' -66', 00' -99',

thus obtaining nine other equations of the same form, which are
the nine in question. It is thus seen that the connecting
equations will be better viewed as statements of the equality of
binomials; and the theorem which this view leads to is that
either the sum or the difference of any two of the products
00', 1 1', .... is expressible in two ways as the suni or diflference
of other two. The forty-five possible binomials may be arranged
as follows to show these equalities : —

00'-ll' = 77' + 44' = 88' + 66']
00' -22' = 88' + 55' =99' + 44',
00' -33' = 99' + 66' = 77' + 55'^

00'-44' = ll' + 77' = 22' + 99'|
00' - 55' = 22' + 88' = 33' + 77' /
00' -66' = 33' + 99' =11' + 88')

00'-77' = ll' + 44' = 33' + 55')
00' - 88' = 22' + 55' = 1 1' + 66' '

uu - 00 = zz + 00 = 1 1 + bb /
00' - 99' = 33' + 66' = 22' + 44' )

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368 Proceedings of Royal Society of Edinburgh, [sbs.

11' - 22' = 65' -66' = 99'- 77' \
22' - 33' = 66' - 44' = 77' - 88' \,
33'-ll' = 44'-55' = 88'-99')

11' -55' =22' -66' = 33' -44',

11' -99' = 22' -77' = 33' -88',

44' -88' = 55' -99' = 66' -77'.

It will be seen that the second line is derivable from the first,
and the third from the second, by the cyclical substitution : and
that the number of such triads is four. The last three lines are
not so related : the cyclical substitution if performed on any one
of these would simply reproduce that one.

(6) It is interesting to note that to each of the foregoing fifteen
sets of three equivalents a fourth equivalent of a different form
may be added. Thus for the seventh line we have the additional
equivalent

\<hh\ IVsl l^sM

I/1I72I \f^z\ l/s^il

\<hh\ l^sl 1^1

for this can be shown equal to

l^'^V' l^'fll i.e. 44'+ll',

and as the interchanges

OX')

alter only the sign of the three-line determinant,* the latter must
also be equal to

\figA\ \fi9^\

and

t.(9. 3'3 + 5'5

* The other similar interohange \ j\ gives nothiDg nei

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1904-5.] Three-line Determinants of a Six-hy-Three Airay. 369

(7) Turning now from the products whose factors are comple-
mentary to those whose factors are not, we see that the taking
of 0 along with any other of its own set {e.g, 01, 02, . . .) would
be nugatory, because the two factors of any such product would
have two columns in common. But 01, 02, . . . , 09 being on
this account unfruitful, it follows that the same cannot be said of
or, 02', . . . , 09'. As for the products which begin with 1, they
must be nine in number, because if they cannot be taken along
with any particular one that follows it in its own set, this very
fact ensures fruitfulness if taken along with the corresponding one
of the other set : as a matter of fact the useful cases are

12, 13, 14', 15, 16', 17; 18', 19.

Similarly the useful products beginning with 2 are

23. 24', 25', 26, 27, 28', 29' ;

those beginning with 3,

34, 35', 36', 37', 38, 39' :

and so on. It is thus seen that if we confine ourselves to the
products whose first factor at least is taken from the first set of
ten and is represented by a smaller integer than the second factor,
the number of fruitful products is

9 + 8 + 7+ . . . +3 + 2 + 1 .

From every one of these products, however, another fruitful
product is obtainable by changing each factor into its complemen-
tary. The total number is thus 90.

(8) Taking the first of the ninety, viz. 01', we have on inter-
changing c, ^, ^ in succession with a

I «lV3 M '''\9J^Z i = I 1/1*2^3 hi '•l«2^3 I + I KhH H ^l5'2«8 I »

t.c. or = 23 - 59.

Now, no new result is got by interchanging c, g, h in succession
with 6, nor by interchanging c, ^, h in succession with c.
Further, by reversing the order of the factors in 01' and
applying our theorem, we merely repeat the same result. We
thus learn that each of the ninety products of pairs of non-cow-
plementary three-line minors formed from a dx-hy-three array can
PROC. ROY. SOC. EDIN.—VOL. XXV. 24

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370 Proceedings of Royal Society of Edinburgh, [sess.

he expressed in one and only one way as a sum or difference of tvoo
other such products.

(9) We are thus prepared to learn that if we take either of
the two products whose sum or difference has been obtained in
this way as an equivalent for a given product of the same kind,
and apply our theorem as before, we shall merely get another
repetition of the previous result. Thus

1 .'/i Vs 1

•1 ^i^A 1 = 1 «i Vs M ^i^A

+ 1 /^iVs II <^l«2^3 1

i.e.

23 = or

+ 59,

and

1 'h Vs

^i(f/^s\ = 'l/iVaHqVsl

+ 1 a, Vs M <^i^A 1 »

i.e.

- 59 = - 23

+ or.

It follows therefore that since there are ninety products and
each can only occur once in an identity along with two others,
the number of such identities is thirty. Probably the best
arrangement of the thirty is that which brings into juxtaposition
those that form a triad, and places opposite to each other those
that are complementary. The result of this is : —

or - 23 + 59 = (? = O'l - 2'3' + 5'9'
02' - 31 + 67 = (? = 0'2 - 3'r + 67'
03' - 12 + 48 - (? = 0'3 - r2' + 4'8'

04' - 66 + 38 = (? = 0'4 - 5'6' + 8'8'
05' - 64 + 19 = (? = 0'5 - 6'4' + r9'
06' - 45 + 27 = (? = 0'6 - 4'5' + 27'

07' - 89 + 26 = ^ = 07 - 8'9' + 2'6'
08' - 97 + 34 = (? = 0'8 - 97' + 3'4'
09' - 78 + 15 = ^ = 0'9 - 7'8' + 1'5'

14' + 82' + 69' = ^ = r4 + 8'2 + 6'9
25' + 93' + 47' = ^ = 2'5 + 9'3 + 4'7
36' + 71' + 58' = (? = 3'6 + 7'1 + 5'8

16' -r 49' + 73' = ^ = r6 + 4'9 + 7'3
24' + 57' + 81' = (? - 2'4 + 57 + 8'1
35' + 68' + 92' = (? = 3'5 + 6'8 + 9'2.

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1904-5.] Three-line Determinants of a Six-hy-Th/ree Array. 371

(10) The relations between products of three factors it is less
necessary to study, the foundation of them being laid in what
precedes. For example, there are numerous results like

l(or+ 59) = 2(02'+ 67) = 3(03'+ 48)

which is clearly obtainable from the first triad of § 8.
easily verifiable from the foregoing are the pair

Less

1 86
4 29
753

the process being —

- 000',

r8'6'
4' 2' 9'
7' 5' 3

O'O'O,

1 86
429
763

= 1(23-59) + 4(56-38) + 7(89-26)

= 1 or + 4 04' +7 07',

= 0(11'+ 44'+ 77')
= 000'.

{Issued separately Ja/imary 20, 1905.)

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372 Proceedings of Royal Society of Edinburgh. [si

The Sum of the Signed Primckry Minors of a
Determinant. By Thomas Mnir, LL..D.

(MS. roceived July 25, 1904. Read November 7, 1904.)

(1) The fundamental propositions in regard to the sum of the
signed primary minors of a determinant are —

(A) An expression for the negative sum of the signed primary
minors of any determinant is got by taking a determinant of the
next higher order whose first dement is zero with the given deter-
minant for complementary minor y and whose remaining elements
are units all positive or all negative,

(B) The sum of the signed primary minors of any determinant is ex-
pressible as a determinant of the next lower order, any element (r , s)
of the latter being the sum of the signed elements of a tuKhline minor of
the former, viz,, the sum (r, s) - (r, s + 1) - (r + 1 , s) + (r + 1 , s + 1 ) .

(C) If the elements of a determinant he all increas&l by the same
quantity ta, the determinant is thereby increased by <o times the
sum of its signed pnmxiry minors,*

(2) By the application of the first of these the following results
are readily obtained —

77ie sum of the signed primary minors of the alternant
\ a^'^cP .... I M equal to the alternant itself, (I)

The sum of tliA signed primary minors of a circulant of the n**
order is equal to n times the quotient of the circulant by the sum of
its variables. (II)

Thus, the sum of the signed primary minors of C(a , b, c)

111

= - 1

1 1 1

^(a-^b-^e),

I a b c

a-^-h-Ve

a b c

I c a h

a + b-^c

cab

1 h c a

a + ^ + r

h c a

-3 11 1

-^(a + i> + c),

1 . a b c

, c a h

. h r a

3C(a, b, c) ^ {a + b-{-c).

* Proceedings R

oy. Soe, Edinburgh^

zxiv. pp. 387

-892.

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1904-5.] Signed Primary Minors of a Determinant. 373

TTie sum of the signed primary minors of a zero-axial sketo deter-
minant is equal to a similar determinant of the next higlier order,
and therefore is zero if the order of the original determinant he even,
and is the square of a P/affian if the order he odd. (Ill)

(3) By the application of the second fundamental result (B)
the case of a centro-synunetric determinant can be equally easily
dealt with, the result being —

The sum of the signed primary miiurrs of a centrosymmetric
determinant is equal to a similar determinant of the next lower
order, and therefore is resolvable into two factors. (IV)

Thus, the sum of the signed primary minors of

a b c
d e d
c h a

a-h-d+e b-c-e+d
d-e-c+h e-d-h+a

= (a-c)(a-26-c-2(; + 2c).

= (a-6-(/ + e)2-(ft-(;-e + d)2,

(4) The case of a continuant requires and is worthy of a little
more consideration. Restricting ourselves, merely for shortness'
sake, to the six-line continuant

/ *i h b, h h \
I «i a, Og a^ Oj a, »

\ ^'l ^2 ^8 ^4 ^6 / »

and denoting the sum of its signed primary minors by prefixing to
it an M, we know to begin with that this sum equals

1111

^8 «4

Fixing the attention on the last column and last row, the non-

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374 Proceedings of Boyal Society of Edinburgh, [sbs.

zero elements of which are (1, 7) , (6, 7) , (7, 7) , (7, 6) , (7, 1), we
obtain the equivalent expression

(7,7)cof + (6,7)(7,6)cof + (l,7)(7,l)cof + (lJ)(7,6)cof + (7,l)(6,7)cof

/ h,,,, \ / ^y " \ / ^1 ••• \

1 «! ^ . .

1 Cj O^ 63

I . Cg Oj 63 I

1 . . (•« a^
1 . .

- K

^8 '*4

11111
Oi &! . . .

. • Cj a, b^

Of the two determinants here written at full length the first is
seen to be

= ( «1 • • • «4 ) + «4

I ai 61 .

1 C^ Og ^2

1 . c^ a, ^

1 . . c '

■\rC^Cfyr

and the second

. ( Oi . . . a, » - 6J a, . . . a, I + Vsl «i<'2 ) - ''AM«i) + hh^A

It thus follows tliat
M

+ ( «! • • • «5 j - (Cs + M «i ' • • «4 j

- (V4'

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1904-6.] Signed Primary Minors of a DeteimtinanL 375

and consequently that the sum of the signed primary minors of a
continuant of the n^ order can be got when the corresponding
sums for the cases of the (n - 1 )*** and (n - 2)"* order are known.

(5) By repeated application of the preceding resiilt we obtain
ultimately an expression involving only the continuants

I Oi ag . . . a^ I , i Oi , 02. . . . , a^ I , . . . and

their co-

efficients. The following is the general theorem thus reached : —
If the cof actors of a^ , anan_j , anan_^an_2 , . . .in the continuant

I *i> %> • • • > ^ i
\ Ci . . . /

denoted by K„_j, K^.j, K^.j, . . . , and

the eof actors of a^, a^aj, a^ajag, . . , be denoted by H„_i , H^.j,
Hn-8 y ' ' ' 1 tlie sum of the signed primary minors of the con-
tinuant K„ is

K^i + K,_,(l , b,_^ + c,. J Hj , - 1 ) (VI)

+ K^-8(l . K-2 + ^n^2 > ^n-i^*i-2 + <'*«-l^n-2 $ ^^ , - Hj , 1)
+ K„_4(l , 6h_3 + C„_8 , 0„_20n_3 + C^^^C^^^ , 0„_jO„_2"n-8

+ CH_iCn_2C«-8 5 Hg , - H2 , H, - 1 )
+

+ (1, ^ + Ci> Vi + Vi> • • • .$H„_^,-H«_2, . . . ).
For example, the sum of the signed primary minors of the con-

tinuant

/ W h \

Kg, i.e, { a^ a.^ Og j is

^2 + <^2K» -1)

+ (1 , 61 + c, , b^b^ + C2C1.5 a^a^ - /^jCg , - Og , 1) ,
i.e.

«i«2-^^ + ^(«8-^2-^2)

+ «2«8 - V2 - ^si^l + ^) + (^^ + ^2^) »

t.c.

aiOj + agOg + OgOi - 01(62 + ^2) - 03(61 + c^)
— 6|Ci — 62C2 + 61^2 + C^C2 .

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376 Proceedings of Hoyal Society of Edinburgh. [sess.

(6) For the case of a * simple' continuant^ ue, when each of
the 6*8 is 1 and each of the c's is - 1, the expression (VI
hecomes

(ojOa . . . a„)
+ (0102 . . . o^_i) -o^
+ (Oi02 . . . o,_2). {(a._iO,) + 2}
+ (ai02 . . . a«_3) • {(a^-2'^-ia„) + 2o«}
'\r{a^a^ . . . o,_4). {(o„.80^_j0^.iO.) + 2(o^_iOh) + 2}
+

and therefore, like the continuant itself, has all its terms
positive. (VII)

For example, the sum of the signed primary minors of the
continuant (Oj , Og , Oj , oj is

+ {(«2«8«4) + 2aJ

ue,

(h!h<H + «i + «8 + («i^ + 1)«4 + «i(^8«4 + 3)

+ «2«8«4 + ^ + 3«4»

i,e,

+ 401 + 02 + 03 + 404.

(7) If the expression (VI) in § 5 be arranged in the order of the
U's and their cofactors, it becomes

H,_, + H,_j(l,6, + CiJKi,-l) (VIII)

+ H,_,(1, 6, + Cj, 4i6, + CiCj$K,,-Ki, 1)
+

which accoiding to (YI) is the sum of the signed primary minors

/ 6»-, .... b, \

I a« a»-i . . . Oj Oi I
\ e,_, . . . . c, / ,

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1904-5.] Signed Primary Minors of a Determinant.

377

— a result to he expected, since generally
, . 1 1 1 . .

«2

1 1

. . ..

1 ...

<^i

1 ...

«i

and therefore

MllehVs l} = M{| cjb^a^ | }. (IX)

(8) When each of the a's is equal to a, each of the h's to 6, and
each of the c's to c, the H's and K's are no longer distinguishable,
and the expression (VI) becomes

K,_i + K«_2.(l,6 + c$Ki,-l)

+ K«_3.(l,6 + c,62 + c25K2,-Ki, 1)

This, however, is best arranged in portions containing 1 , ft + c ,
ft^ + c^ , . . . . , and their respective cofactors, the result then being

(K«_i, K,t_2, K„_8, . . . , Kj, 1 jj 1 , Kj, . . . , K^_3, Kn-2} ^-i)
+ (ft + c) ( K„.2, K„_g,. . . , K„ I § 1, Kj, . . . , K._3,K,_2)

+ {b^ + c2)( K„.3, . . . , K,. 1 § 1, K„ . . . , K,_3)

+ (X)

or, say,

Now since

and the known ultimate form of K„ is an expression consisting of
terms descending by second powers of a and ascending by first
powers of 5c, viz.

it follows that there must be for Xn ^^ expression of similar

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378 Proceedings of Bayed Society of Edinburgh.

character. Towards finding this we first note the following
property of continuants, viz.

The cof actors of the elements in the places

(w, n), (7A-1, w), (n-2,?0, • • •
of the continitant

( ^ ^2 ) or

(l,w)

are

'^-1 > '^«-2»

, #Cj, I.

Changing K„ into the fonn ( ^^ 22

and putting in the said places of it

1 > '^i > ^ > • • • • > ^-1

we thus learn that the resulting determinant is equal to

(K^-i 1 K^. 2 > • • • > M ^ > ^1 > • • • » ^-1)
and therefore is equal to Xn-r ^^ other Avords we have

(XI)

X»»-i~

a
-1

-be

a -be
-1 a

^-2

a
-1

(XII)

This determinant, however, may be developed in another way, viz.,
in terms of the elements of the first row and their respective co-
factors ; and doing this we obtain

— a recurrence-formula which readily gives
Xn-i = na-i - {n-\)Cn^^„a^-^bc + (« - 2)C,.3, ^ a^Wc^ - . . . (XIV)
In illustration let us take the case where n « 4. We then have

the sum of the signed primary minors

/ b b b \

of I a a a a\

\ c c c /

= X8-(^ + ^)X2 + (^' + ^')Xl-(^' + ^),

= 4a8 - 6abc - (fe + (j)(3a«-26c) + (b^ + c^)2a - (6» + c»),
= 4a8 - 3o2(6 + c) + 2a(b^-Sbc + c^) - (6 + c)(6« - 36c + c«) .

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1904-5.] Signed Primary Minors of a Determinant, 379

(9) If it be desired to have the general result arranged
according to descending powers of ti, we have only got to sub-
stitute in Xn-i - (^ + c)x»-2 + (^ + ^^)Xn-s ~ . ... the expressions
for Xn-i 9 X«-2 » • • • obtained from (XIV), and then coUect the
coefficients of like powers of ((. The theorem thus arrived at is —

/* * \ .

TTie sum of the signed primary minors o/l a a a . . . a Uts

\c e /

-(n-l)a-^b + e}

+ (n-2)a"-»{(i« + c«)-C,._i,,i<-}

- (« - 3)a»-*{(6» + c») - C,_j , i(6 + c)bc]

+ (n- i)a'-^(b* + r*)- C,_, , ,(62 + c')bc + C,_a , ,6V}

- (» - 5)a-«{ (6» + «») - C._, , i(6» + c»)6<; + C„.a , # + c)6V}
+ (XV)

The cof actors here of na^'^ , - (w - 1 )a^^^ , ... are related to
one another in a curious way, which is worth noting if only for
use as a check in computation. Denoting them by X , X^ , Xg , . . .
we have

X»„+i = (6 + c)X„^

X^ =(6 + c)X^_,-(«+l)J-C, -hn^-^r <^^*)

-™.,„-i- ft"*"- j"

the demonstration of both resting on the facts

(6'- + c'X6 + c) = (6'+' + c'+') + 6c(6'-' + c'-'),
C,,,, = Cp_i., + Cp_i,,_i .

(10) It is thus suggested to examine the result of multiplying
the whole expression by a + (b + c). Taking it in its original
form

X,-i - {b + e)xn-i + (6* + c*)x»-8 - . • • •

we readily see, to begin with, that the product is

«A&.-i - a(6 + c)x»-2 + a(^ + c*)x»-» - • • • •

■^{b + c)xn-i- (f^ + <^))^ +(6» + c»)>

- 6c(6» + c») i "+ 6c(6 + c)/^""' ■ ■ ■ ■

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380

Proceedings of Royal Society of Edinburgh, [ass.

This, however, if arranged in parts containing W + <fi, h^ + c^t
bf^ + c^ , . . . . and their respective cof actors, is

{axn-, - 2/>cx„-2} + (^ + c) {x*-i - axn-2 + f^Xn-z}

- {b^ + C^){xn^^ - axn-s + bcXn^^}

Now it can be shown that

«X— i-26cxn-2=»»K„,
and, as we have already seen,

Xn-i - «X«-2 + ^<^X»-8= ^-1 J
we thus reach the following interesting result —

The 8um of the signed primary minors of

is the quotient of

nK, + (^^ + c)K,., - (62 + c2)K,.2 + (63 + c^)K^.^
bya + b-hc,

(11) It has recently been proved* that

a + d b + d d d
c + d a + d b + d d
j d c + da + db + d
d d c + d a-k-d

I a a a ... a I

\ c c /,

(XVU)

'•+i/»**i

j-y

s(a^lS)

11 1 « + (n+l>f
1 a~+l l+a+... + a"
1 i8"+l l+j3 + ...+j8"

where ij = a + 6 + c and a, fi are the roots of the equation <»* + ox +
6 = 0. Now, in the first place, the determinant on the left here is

/bb X

by the third theorem (C) of § 1, the continuant (a a a ... a I

\ c c /"

increased by d times the sum of its signed primary minors : that

is to say, is equal to

K^ + d.U{JQ.

In the second place, the determinant on the right is equal to

1 1 n+1
+ d 1 a"+i l+a+ . . . +a*
1 p^^ l+i8+. . .+)8*

1 )3^i

By Dr F. S. Macaulay in Math. GazeUe, iiL pp. 44-46.

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1904-6.] Signed Primary Minors of a Determinant. 381

It follows, therefore, because of the known result

1 a"+i

K,=(-rv

^n+l _ ^n+1

that

1 Pn+l

M(K„):

( - )-+ic»

s{a-IS)

1 1 n+1 I

1 a"+i l + a+ . . . +a'» (XVIII)

, 1 )8"+^ l+j8+ . . . +^ |.

This curious result ought to agree with (XVII) : in other words,
we ought to be able to show that

.-/i

(^^~- 11 «4-l

1 a"+» l+a+.
1 i3«+» l+j8 +

Towards doing so it has first to be noted that the determinant on
the left

1 1 n

+ a« ' -(62 + c2)K,_^
+ )8" . +

a + a* +

+ a"

1 a"+^
1 p^+'

1 )^+^ /i + iS2+ . . . +i8-

+ 11

1 a"+* a + a^+ . . . +a*
1 j3"+^ i« + ie2+ . . . +i8"

= //

1 a"+»

= »

1 a"
1 p"-

a'^^^ a + a^ + .-.+a"

+ a"+^ a

-1 a"

I 1 P^

1 a + a2 + . . .4-a"
1 /i-^l3^-h...+fi^\

pn+i p2

1 p^''

Multiplying this now by ( - )"'*"^c'*/(a - /8) and substituting K^ for
( - c)*(a'^^ - jS"-^')/(a - j8) we obtain

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382 Proceedings of Roi/al Society of Edinburgh, [sbss.

«K, + (fe + c)K,.,-(62 + c2)K,., + . . .

as was desired.

Since « = c(l - a)(l - P) the result (XVIII) may also be written
in the more symmetrical form

M(K.) = , ,

(-C)-

1 a"+' o + a2+ .

. . +a"

1 /3-+' )8 + /S2+ . .

.+/3»

1 y"+' y + y*+ •

.+/

(y-a)(y-P){fi-a)

where a, )8, y are the roots of the equation

rx^-{c- a)2^ + (6 - a)a; - 5 = 0 ;

and, noting that the coefficients of this equation are the non-unit
elements of the determinant

I h-a ^b

1 a c b - a -b

1 c a-r h-a -b

1 . c a-c b-a .
1 . . c a -c .

.... 1

. - 1 ft

which is another form of M(K«) , we have at once suggested the

problem of evaluating the determinant

led

I b c d

lab c d . , . .

I . a be ....

1 . . a 6 .-. . .

in terms of the roots of the equation

aa^ + bx^-\-cx + d = 0.

After doing this, however, we should only have reached a simple
case of a known theorem of wide generality.*

♦ Fide my Text-Book of Determinants, p. 173, § 127.

{Issued separately January 20, 1905.)

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1904-6.J Dr Hugh Marshall: Crystallographical Notes, 383

OrystaJlographiccd Notes.
By Hugh Marshall, D.Sc., F.R.S.

(MS. received November 21, 1904. Read December 6, 1904.)

I. Axes of Compound Symmetry of the Second Order.

In recent years, since the more or less general adoption of the
systematic classification of crystals under the thirty-two possible
types of symmetry, it has become usual to dispense with the
* centre of symmetry * as one of the elements of crystal symmetry,
and to adopt in its place the ' axis of compound symmetry of the
second order.' The derivation of symmetrical directions from any
given one by means of a compound axis of order n involves not

Fig. 1.

merely rotation through the angle 2irJ7i about that axis, but also
reflection in the plane at right angles to the axis. If the axis A
(fig. 1) is of second order, then B, by rotation about A through tt,
would give B', but this by reflection in the normal plane P gives
B", and the latter (not B') is therefore symmetrical to B with
reference to the compound axis A. But B' is evidently parallel
to B, and oppositely directed, so that it follows that in crystals
possessing an axis of compound symmetry of the second order (or

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384 Proceedings of Royal Society of Edwhurgh, \\

of order n such that n is divisihle by 2 but not by 4), opposite
directions are equivalent to one another. So far as exteroal shape
is concerned, this involves, essentiaUy, the occurrence of parallel
faces on every form. These are the characters of centre-
symmetrical bodies, however, so that at first sight it appears as if
the symmetry might be referred indifferently either to the com-
pound axis or to the centre. There is, unfortunately, one grave
objection to the former method which seems to be generally over-
looked. In all other cases an axis of symmetry is some perfectly
definite direction in the crystal, and the number of axes is never
large — not exceeding six of any one order, even in the most
symmetrical classes. An axis of compouwi symmetry of the
second * order, however, is not a definite direction in the crystal,
and every centro-symmetrical crystal possesses not one such axis,
but an infinitude of them, because any direction whatsoever may
be chosen as the axis without affecting the final result. It is
therefore much better to avoid this lack of definitiveness in the
expression * axis of symmetry ' by giving up the use of the * axis
of compound symmetry of the second order,* and restoring the
* centre of symmetry ' to its former position.

II. The Classification of Trigonal and Hexagonal Crystals.

For teaching and ordinary crystallographical purposes, the
classification of crystals is largely a matter of practical convenience ;
questions of structure or arrangement of crystal molecules may be
entirely overlooked in this connection. Bearing this in mind, it
is a matter of some importance that the crystal systems which
resemble one another in possessing one principal axis of symmetry
(the trigonal, tetragonal, and hexagonal systems) should be so
arranged as to accentuate their similarities ; by doing so it becomes,
for students beginning the subject, a much easier matter to
appreciate and remember the various classes (nineteen out of
the total of thirty-two) included in these three systems.

The tetragonal system is defined quite sharply, and the seven
classes belonging to it present no characters which would lead to

* This does not apply to compound axes of liigher order, because an axis cf
compound symmetry of order n is necessarily an axis of ordinary symmetry of
order n/2.

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1904-5.] Dr Hugh Marshall: Crystallographical Notes, 385

the inclusion of them in any other group. We may therefore take
the tetragonal system as a standard, and compare the trigonal and
the hexagonal with it. The seven tetragonal classes and their
characteristic symmetry are as follows : —

1. Bi-sphenoidcd doss, — One axis of compound tetragonal
symmetry. (Representatives of this class are not actually known,
however.)

2. Pyramidal class. — One axis of tetragonal symmetry.

3. Trapezohedrdl doss. — One axis of tetragonal symmetry ; two
pairs of lateral axes of digonal symmetry.

4. Scalenohedrdl doss. — One axis of compound tetragonal
symmetry ; one pair of lateral axes of digonal symmetry ; one pair
of planes of symmetry intersecting each other, normally, along the
principal axis.

5. Di-tetragonal pyramidal doss. — One axis of tetragonal
symmetry; two pairs of planes of symmetry intersecting, all at
equal angles, along the axis of symmetry.

6. TdragoTwU U-pyramiddl dass. — One axis of tetragonal
symmetry ; one plane of symmetry normal to the axis.

7. LHrtetragonal bupyramiddl dass. — One axis of tetragonal
symmetry ; two pairs of lateral axes of digonal symmetry, all
equally inclined to one another ; one principal plane of symmetry
and two pairs of planes of symmetry, each plane normal to an axis
of symmetry.

At first sight it might be expected that, corresponding to these,
there would be possible seven classes in each of the other two
systems, the only differences being those due to the lower or higher
order of the principal axis of symmetry. So far as regards the
classes in which the axis is not one of compound symmetry, this
is the case ; but not so when the symmetry is compound. Axes
of compound symmetry of even order are possible, but axes of com-
pound symmetry of odd order are not possible merely as such,
therefore two classes must be lacking in the trigonal system. This
is easily seen by a reference to the usual symmetry diagrams re-
presenting projections on a plane at right angles to the principal
axis of symmetry.

Fig. 2 represents the case in which the only symmetry assumed
is that of a trigonal axis of compound symmetry. An upper face,

PROC. ROY. SOC. BDIN. — VOL. XXV. 25

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386 Proceedings of Roycd Society of Edivhurgh. [sbss.

1, on rotation through -— and reflection in the normal plane,
o

would give a lower face, 2 ; and by repeating these operations an

upper face, 3, would result. Repeating the rotation once more

would bring the face back to its original position, but the ensuing

reflection would give a new face, immediately below the first. It

is therefore evident that in order to return to the original position,

by repeating the operations characteristic of the symmetry, two

complete revolutions are necessary, and this produces six faces, as

shown in fig. 3 — three above and three below. The diagram now

/ N
/ N

' X3 ^ / \

i A-

01 \

^<s.._

Fig. 2. Fio. 8.

exhibits the higher symmetry of an ordinary trigonal axis combined
Avith a plane of symmetry at right angles to it ; but this is the
symmetry of the trigonal bi-pyramidal class which corresponds to
the tetragonal bi-pyramidal class. There can, therefore, be no tri-
gonal class corresponding to the tetragonal bi-sphenoidal class.

Similarly, there can be no trigonal class corresponding to the
di-tetragonal scalenohedral class, as a trigonal axis of compoimd
symmetry combined with vertical planes of symmetry leads neces-
sarily to the symmetry of the di-trigonal bi-pyramidal class. Each
of these two classes — the trigonal bi-pyramidal and the di-trigonal
bi-pyramidal — therefore represents, in a sense, two classes of the
tetragonal system. It is noteworthy that not a single substance is
known to crystallise in either of them ; they are only * theoretically .
possible.'

As hexagonal axes of compound symmetry are possible, there are
the full number of seven classes possible in the hexagonal system.
The classes corresponding to the tetragonal bi-sphenoidal and
scalenohedral are the rhombohedral class and the hexcujonal scaleno-
hedral. Representatives of both are known, especially of the

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1904-5.] Dr Hugh Marshall: Crj/stallographical Notes. 387

latter; they are the classes of dioptase and calcite respectively.
Instead of being classed in the hexagonal system, however, they
are generally placed in the trigonal system, the scalenohedral one
being known as the di-trigonal scalenohedral class. The principal
axis of symmetry is, of course, a simple trigonal axis, as well as
one of hexagonal compound symmetry, but that is no sufficient
reason for departing from the strictly systematic method of treat*
ment. The result of doing so is to complicate matters for the
student quite unnecessarily.

For the purpose of introducing the student to the various crystal
classes, it would therefore appear to be best, after treating of the
triclinic, the monoclinic, and the rhombic systems, to take up the
tetragonal system, and, after this has been gone over, to proceed to
the hexagonal and, lastly, the trigonal systems : the close analogies,
allowing for the exceptions in the trigonal system as referred to
above, render the study of the latter systems quite simple.

The various classes might then be tabulated as follows, the sym-
metry of the different systems being expressed in general terms : —

Systems and Classes.

Symmetry.

n=4
Tetragonal.

7t-gonal axis of com- \
pound symmetry /

n-gonal axis

n-gonal axis ; n lateral \
axes (digonal) )

n-gonal axis of com- '
pound symmetry ; n/2
planes intersecting in
axis ; n/2 lateral axes
(digonal)

n-gonal axis; n planes \
intersecting in axis /

n-gonal axis ; one plane \
normal to axis /

n-gonal axis ; n lateral \
axes (digonal) ; plane j-
normal to each axis j

Bi-sphenoidal

Pyramidal

Trapezohedral

Scalenohedral

Di-tetragonal
pyramidal

Bi-pyramidal

i>i- tetragonal
bi-pyramidal

?t=6
Hexagonal.

Rhombohedral

Pyramidal

Trapezohedral

Scalenohedral

2>t-hexa^onal
pyramidal

Bi-pyramidal

2>i-hexagonal
bi-pyramidal

n = 3
Trigonal.

Pyramidal
Trapezohedral

Dt-trigonal
pyramidal

Bi-pyramidal

Z>i-trigonal
bi-pyramidal

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388 Proceedings of Hoycd Society of Edinburgh, [i

When considered in this way the trigonal and hexagonal
systems are referred to the Bravais axes, using the appropriate
symbols. It is important, however, that students should be made
acquainted with the mode of referring certain crystals to rhombo-
hedral axes, with Miller's original symbols ; therefore those classes
for which such axes can be adopted should subsequently be
brought together into a rhombohedral system by themselves. The
classes to which this is applicable are those, belonging to the tri-
gonal and hexagonal systems, which do not possess elements of
symmetry higher than those pertaining to a (geometrical) rhombo-
hedron. Consequently, all classes possessing a simple hexagonal
axis, and also those which possess a principal plane of symmetry,
are excluded from the rhombohedral system, which therefore
includes —

The trigonal pyramidal class
„ „ trapezohedral class
„ di-trigonal pyramidal class
„ hexagonal rhombohedral class
^ „ scalenohedral class

The above list contains all the represented classes which are
usually included in the trigonal system, and doubtless this is the
principal reason why two classes which are, strictly speaking, hexa-
gonal, are generally placed in the trigonal system. It appears to
me, however, that considerable advantage is obtained by first
deducing the trigonal and hexagonal classes in a strictly systematic
manner, and, after the student has become acquainted with them,
introducing the use of rhombohedral axes as an alternative method
of dealing with a certain group, represented in both of the preced-
ing systems, before passing on to the cubic system.

{Issued separately February 1, 1905.)

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1904-5.] SimuUane(ms Removal of Thymvs and Spleen. 389

The Effect of Simultaneoiis Bemovcil of Thymus and
Spleen in young Guinea-pigs. By D. Noel Paton and
Alexander GoodalL {From tlie Laboratory of the Royal
College of Physicians^ Edinburgh.)

(Read December 5, 1904.)

We have already shown that removal of the spleen (1) or of the
thjrmus (2) has very little effect on the animal economy. Since
the spleen and thymus together comprise the largest amount of
lymphoid tissue in the body of young animals, it would appear not
improbable that although removal of either of these organs causes
no marked disturbance, their simultaneous extirpation might be
expected to give rise to some more manifest change. Friedleben
(3) states that, while in his series of experiments no dog died of
removal of the thymus, and that the removal of the spleen in
young dogs does not influence the course of life, the simultaneous
removal of the thymus and spleen causes a marked deterioration
of blood formation, and leads to death.

Since his experiments were made without aseptic precautions,
and since his results may therefore have been due to sepsis, it
appeared desirable to repeat these observations on young guinea-
pigs, in which animals removal of the thymus and of the spleen
separately has been found by us to cause no disturbance of
importance.

In the following series of observations D. Noel Paton is
responsible for the operations, which were performed under full
anaesthesia. The animals invariably recovered rapidly. There
was never suppuration, or any evident discomfort to the animal.

The observations on the blood were made by A. Goodall.

Experiment I. — On 9th April two female guinea-pigs were
brought under observation. A. weighed 200 grms and B. 160 grms.

A. had thymus and spleen removed : — thymus '3 grm., spleen
•18 grm.

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390 Proceedings of Royal Society of Edivburgh. [i

On 25th April A. weighed 330 grms. and B. 230 grms. A. had
11,200 and B. 7800 leucocytes per c.cm. Both were killed on
5th October. A. weighed 870 grms. The thymus was completely
gone. A small piece of splenic tissue weighing '67 grm, was
found. B. weighed 550 grms. Thymus *55 grm., spleen '93.

Experiment IL — A guinea-pig weighing 260 grms. had thymus
(•32 grm.) and spleen ('18 grm.) removed on 25th ApriL On 2nd
May it weighed 290 grms. and had 6800 leucocytes. On 12th
May it had 6600 leucocytes. On 2nd June it weighed 410
grms. and had 13,000 leucocytes. It became pregnant, and aborted
on 26th July, giving birth to three young, weighing in all 123 grms.
It was killed the same day. The thymus was completely removed,
while a small scrap of spleen was found.

Experiment III, — Tavo female guinea-pigs had thymus and
spleen removed on 2nd May.

A. weighed 220 grms. . Thymus -275 Spleen '345

B. „ 280 „ . „ -280 „ -405
On 12th May A. = 280 with 1200 leucocytes.

B. = 310 „ 8800
26th A. = 355 „ 5000

„ B. = 385 „ 7200 „

2nd June A. = 370
„ B.-400
Both were killed on 6th June. Removal of thymus and spleen
was complete.

The number of leucocytes compared with that of normal animals
of the same age showed the same slight diminution that we have
noticed after removal of the thymus alone, but, as in the case of
removal of thymus only, this leucopenia does not persist after the
animal has attained the age of three months.

DifTerential counts of the leucocytes showed no departure from
physiological limits.

We conclude that simultaneous removal of the thymus and spleen
in the young guinea-pig in no way interferes with nutrition, blood
formation, growth and development of the animal.

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. l»04-5.] SimvManeous Removed of Thy mm and Spleen, 391

Refebbnces.

(1) Noel Paton and GtOODall, Jour, ofPhys,, xxix., 1903, p. 41 1.

(2) „ „ „ xxxi., 1904, p. 49.

(3) Fribdlbbbn, Die Physiologic der Thymusdriise^ 1858.

{Issued separately February 1, 1906.)

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392 Proceedings of Royal Society of Edinburgh. [&

Networks of the Plane in Absolute Qeometry. By
Dunoan M. Y. Sommerville, M.A., B.Sc, University of
St Andrews. Communicated by Professor P. R. Scott Lang.

(Read December 19, 1904.)

(Ahgtrad.)

The problem to divide the plane, without overlapping, into a
network of regular polygons with the same length of side, has been
completely worked out for the three geometries for the case in
which the polygons are all of the same kind. The resulting net-
works are called regular.

On the Elliptic plane there are five regular networks. These
correspond to the five regular polyhedra in ordinary space. On
the Euclidean plane there are three, consisting respectively of
triangles, squares, and hexagons. On the Hyperbolic plane there
exist an infinite number.

To investigate the extension of this problem to the case where
the polygons are of different kinds, i.e. to find the semi-regular
networks, I consider first how the space about a point can be
exactly filled with regular polygons. I take the three geometries
separately.

I. The Euclidean Plane. — The angle of a regular polygon is
definite. If there are p^ n^-gons, p^ n^-gons, etc. at a point, the
condition that the sum of the angles at the point is 360* leads to
an indeterminate equation which may be denoted by A = 0, A being
an integral function of the n*s and p's. The solutions of this
equation in integers give the possible combinations of polygons. Of
these there are 17. I call them the "kinds of angles." They are
divided into three Classes according to the number of kinds of
polygons involved. The development of some of the kinds of
angles leads to impossible combinations of polygons. Rejecting
these, there are left 11, involving triangles (T), squares (S),
hexagons (H), octagons (O), and dodecagons (D). They may be
denoted as follows : —

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1904-5.] Networks of the Plane in Absolute Geometry. 393

Class A. 1. Te. 2. S^. 3. H3.

Class B. 4. TgSg. 5. T^Hj. 6. T,H. 7. TDg. 8. SOg. 9*

Class C. 10. TSjH. 11. TgSD. 12. SHD.

Out of these all the semi-regtdar networks must be built up. I
distinguish types of networks according to the kinds of angles of
which they are composed. If there is only one kind of angle the
type is called simple^ otherwise it is composite. The types are
classified into Groups according to the kinds of polygons which are
inyolTcd, and the groups into Classes according to the number of
kinds of polygons. There are four classes. Class A. consists of
the regular networks.

The simple types are first considered. There are four unique
types, T4H, TDj, SO^, and SHD. T3S2 admits of an infinite
number of varieties of the simple type. In TjH^ two distinct
varieties can be recognised, an infinite number of varieties being
obtained as mixtures of the two. With TSgH there are three
distinct varieties with an infinite number of mixtures. TgSD
does not admit of a simple type, nor, of course, does Class D.

The composite types in general admit of infinite variation. In
any group a composite type corresponds to a possible combination
of the kinds of angles contained in the group. Thus in the group
of triangles and squares there are the three angles 1, 2, 4, and
the composite types 1, 4; 2, 4; 1, 2, 4; the combination 1, 2
being impossible. The method of investigating these is chiefly
experimental, and consists in testing the various combinations.
It is easily seen, however, that certain combinations are impos-
sible. For example, H3 must be accompanied by T^H^ in order
that the gap of 120"* may be filled up. The following are the
numbers of composite types in the various groups :
B. I. (T, S) 3 ; IL (T, H) 8. C. I. (T, S, H) 47 ; II. (T, S, D) 10.
D. (T,S,H,D)169 + .t

II. The Elliptic Plank. — Here we get a relation of the form
A>0, and by giving positive integral values to A an infinite
nimiber of kinds of angles are found. Only a few of these,
however, can be developed. For example, if there are at a point

* No. 9 is 2 pentagons and 1 decagon, bat this is not a developable angle,
t I have not exhausted all the composite types in this class. There
cannot be more than 222.

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394 Proceedings of Rayed Society of Hdinbicrgh. [i

an nj-gon, an n^'gon, and an »3-gon, nj, n^ and n^ must all be even,
for the nj-gon must be surrounded alternately with rig-gons and
nj-gons. With the angles which remain there are thirteen simple
t3rpe8, two with two varieties each and one with five, and two
infinite series of simple types, one corresponding to right prisms
on a regular polygonal base, the other with triangles instead of
quadrilaterals.

Of composite types it is probable that none exist, if we make
the condition that the angle of a regular polygon must be less
than 180**. When a polygon occurs in a particular combination
its angle is thereby determined, and if it occurs in another com-
bination its angle must be the same, which is not in general the
case.

III. Thb Htperbolic Plane. — The number of simple types
here is infinite. For example, one n-gon and two 277»-gon8 at a
point determine a simple hyperbolic network for all values of n
and m for which the network is neither Euclidean nor Elliptic.

As regards the composite types, the same considerations hold
here as in the case of the spherical networks.

{Issued separately February 1, 1905.)

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1904-5.] Salmon in transUum fr(ym Srnx)lt to Gril^. 395

A Specimen of the Salmon in transition from the Smolt
to the Ghrilse Stage. By W. L. Calderwood. (With
Two Plates.)

(Read December 19, 1904.)

In October of this year (1904) there came into my hands a very
interesting specimen of a young salmon. In round terms, the fish
is 1 pound in weight and nearly 14 inches long.

Up to the present time very little is known of the life history
of the salmon during the transition from the stage of the smolt
leaving the river, a fish of about 3 ounces, and that of the grilse
returning to the river for the first time, a fish of 3, 6, or 9
pounds in weight.

A great deal of speculation has arisen as to the length of time
occupied in this change, and most of the earlier writers have
upheld the view that three or four months is sufficient, or, in other
words, that the smolt of May or June is the grilse which appears
in the summer of the same year. This view was mainly based, I
beh'eve, upon results which it was held had been obtained by mark-
ing the fish by the mutilation or removal of the adipose fin. But
since the adipose fin grows again to a greater or less extent, a con-
siderable amount of uncertainty in recognising the recaptures was
inevitable; and I may add that recent observations made in
Devonshire by the instructions of the Duke of Bedford, in which
the marking was carried on in precisely the same manner, have
been held to show that the grilse do not come back the same
season, or within four months or so of the seaward smolt migration.
All the recaptured grilse obtained in the Tavy were caught in
the succeeding season. If any still remained in the sea and
ascended during the second season succeeding, they would probably
be unrecognisable. Further, the few s»olts which have been
recaptured after being marked by the attachment of a foreign
body of some sort — I refer to those of the Early Tweed Experi-
ments— have been got as grilse in the summer of the year after
that in which they were marked.

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396 Proceedings of Boyal Society of Edinburgh. [ssss.

The specimen (PL I.) now exhibited throws some light upon the
question of rate of growth at the period between smolt and grilse.
It is an Irish fish, and was taken on a small fly on 25th August
of this year (1904). It was caught by Mr W. N. Milne when
angling a quarter of a mile above tide reach iu the river Cralway.
It was therefore taken at the season when grilse proper are
commonly found to be several pounds in weight, and when, if the
old observers were correct, the fish could not have weighed, as it
does, only \b\ ounces. It is more than a smolt, is evidently a
quite young fish, and cannot fairly be called a grilse. It has
attained, I believe, about a third of the growth of the grilse, as
this stage is commonly recognised, and requires another year of
sea feeding to accomplish the transition. I am not aware of any
similar specimen existing in this country, if we except a few that
have been artificially reared, and, as smolts, transferred to salt
water aquaria or sea ponds. In this way Dahl in Norway has
reared examples up to 31*5 cm. ; and recently in Scotland a
sea pond at the mouth of the Spey, belonging to the Duke of
Richmond and Gordon, has produced rather larger examples. I
am able to show one of these, which is 33 cm., or almost the
size of the Galway fish (PI. II.).

I have heard of two occasions on which fish approaching the
stage of the Galway fish have, in the wild state, been caught in
Scotland. In other cases which have been brought to my notice
the identification is uncertain. A specimen weighing \ pound was,
Mr S. Gurney Buxton informs me, caught by him when spinning
%vith natural sand eel in the Kyle of Tongue in 1886 ; and two
fish, each weighing \ pound, were reported to me by the late Mr
Anderson, salmon tacksman in the Forth district, as having been
taken by his father in 1863, he himself being present, in the Dundas
net which used to be fished between Hopetoun and Queensferry.
The fish were not preserved, or, so far as I can find, identified
scientifically, but the reports are, I consider, worthy of record, my
informants being in e^h case men with long experience in salmon
fishing.

Dahl, in his valuable report of inquiries into the early stages of
the sea trout and salmon,^ refers to three yoxmg salmon which
* CErret og Unglaks^ Christiania, 1902.

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1904-6.] Salmon in transition f row Smolt to Grilse. 397

were sent to him by mackerel fishers near Oks^. Those measured
43, 36, and 17*5 cm. Two other specimens he found in the
Zoological Collection of Bergen University, which, though unde-
scribed, are believed by Professor Collett to have been found
amongst young mackerel in the Christiania fish market. They
measure 23*5 and 28 cm.

Dahl's special netting in Norwegian fjords and some special
netting which I have carried on in Scotland have as yet produced
only negative results. Sea trout can easily be obtained in all
stages at and near the mouths of rivers, but it is clear that on
entering salt water the salmon smolt separates himself from the
sea trout, and has a habitat in the sea which has not yet been
discovered.

The particulars of measurement, etc. respecting the Qalway
fish are given below. They are those most approved by the
British Museum authorities for the purpose of identifying the
species of salmonidsB. I may add that I have already submitted
the specimen to Mr Boulenger in London, and that he and his
colleague Mr C. T. Regan, who made a separate identification,
agree that the fish is a salmon.

The measurements are given in millimetres.

1. Sex, 6

2. Length to centre of caudal fin, .... 850

3. Weight 16i ounces

4. Length of head from end of snout to posterior

border of gill cover, 77

5. Length of head to anterior border of eye, . . 21

6. Diameter of eye, 11

7. Length of month from end of snout to posterior

border of maxillary bone, . . . .85

8. Length of caudal peduncle, measured in a

straight line from base of last ray of anal fin to
base of lowermost ray of caudal fin, . .47

9. Least depth of caudal peduncle, . . .29

10. Length of longest ray of anal fin, . . .89

11. Shape of posterior border of tail, . . , Fully notched

12. Number of scales, cousting from poflterior

extremity of base of adipose fin downwards and
forwards to lateral line, 12

13. Number of gill-iakers, .... 7 + 6 (gills damaged)

14. Presence or absence of black spots below the

lateral line in the region of the ' shoulder ' . Present

In general appearance (PI. I.) the fish has to my eye certain

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398 Proceedings of Boyal Society of Edinburgh, \i

sea trout characteristics. Owing to the more or less familiar
appearance of artificially reared specimens, a series of which I
show in PI. II., one is prepared for the presence of spots
below the lateral line (although in the largest specimen which I
show, and which has been twelve months in a sea water pond,
spots are less conspicuous), but the rather noticeable breadth of
the caudal peduncle in the Galway specimen is certainly not
in keeping with the shapely specimens which can be reared
artificially, as it is opposed to the characteristics of young salmon,
as insisted upon and figured by Dahl in Norway.

The measurement of the caudal peduncle is contained in the
length of the fish only llf times.

In a Fochabers smolt retained in fresh water till three years old
similar measurements give 13*6 times. In the Fochabers smolt
placed for a year in a sea pond the measurements give 15 '2
times.

In a small Beauly grilse of 1 lb. 15^ oz. similar measurements
give 15 times.

In a small Tay grilse of 2 lbs. ^ oz. the measurements give
15*9 times.

These two grilse are exceptionally small, and have been pre-
served by me on this account. Without any doubt the caudal
peduncle is broad, but on inquiry I am informed by Mr Milne, w^ho
has had a wide professional experience as a salmon fisher both in
Scotland and Ireland, that the fish of the Galway river "are
thicker above the tail than the East of Scotland grilse. They are
rougher altogether, fins and tail larger in proportion." In spite,
however, of this unusual depth of caudal peduncle, the number of
scales, counted forwards and downwards from the posterior margin
of the adipose fin to the lateral line, is on each side 12. This, in
my view, is by far the most reliable test by which to distinguish
between salmon and sea trout, the former having almost invariably
11 or 12, the latter having almost invariably 14 or 15 scales in
the line indicated. The present specimen, therefore, in spite of
its sea-trout-like caudal peduncle, has the salmon number in the
matter of scales. Mr Milne reports that on capture the scales
came off very freely. This accounts for the rather patchy appear-
ance of the side in the photograph of the fish which accompanies

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1904-5.] Salrnon in transition fro77i SnioU to Grilse. 399

this paper. The smolt, as is well known, has this characteristic,
as also has the newly run grilse and the spring salmon. In
other words, when the salmon is found in a very silvery condition,
at a time remote from the season of its spawning, the scales are
very deciduous. Grilse and salmon, in a more or lees gravid
condition, after a stay in fresh water, do not show this pecuUarity,
the scales heing apparently enclosed firmly in the skin pockets.

A number of scales from the fish have been examined by my
friend Mr H. W. Johnston, Strathtay, who has recently made a
special study of salmon scales, and is more able than I am to deal
with the question of age and growth as shown by scales.

From notes he has kindly sent me, it appears that in his opinion
this Uttle salmon has attained the age of rather less than two and
a half years, and that fully two years have been spent in fresh
water. Mr Johnston writes — " The area of the fresh-water scale
growth is larger than is usual in Tay fish, and corresponds more
to that of hand-fed smolts from a hatchery." I am informed
that no hatchery exists in the Gal way district. It is possible,
however, that the conditions of feeding may vary greatly in
diflferent localities. "In the early part of the third year," that
is, when the fish is two years old and has reached the migratory
stage, " there is slightly improved growth, owing perhaps to (a)
tidal feeding or (b) increased temperature, followed immediately
by probably continuous sea feeding, and corresponding growth of
comparatively brief duration, resembling from half to three
quarters of that generally shown by a grilse in its first summer
in the sea. There is no trace of river feeding after the sea
growth."

I am therefore inclined to the view that the presence of the fish
in the Galway river, a quarter of a mile above tide reach, is not
indicative of any habit which the salmon at this stage develops.
The presence of the fish in fresh water at this stage I am inclined
to regard as exceptional, or at least unusual. The specimen is a
male, with genitaha quite undeveloped. The stomach is empty
and contracted, but the pyloric appendages are fairly well sur-
rounded with fat. The vomer bone has the usual complement of
teeth on the head, while on the shaft of the bone two pairs of
teeth are still present. The dorsal and caudal fins are blackish

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400 . Proceedings of Royal Society of Edinburgh,

[«

as in the grilse, and the caudal fin still carries several spots. The
adipose fin is, like the dorsum of the fish, a dark steel colour. The
fork in the tail fin is well marked. Measurements taken with the
caudal fin imextended, as in the photograph, show that the lower
lobe of the fin extends 2*5 cm. beyond the central part of the fin.

The length of the head is contained 4^ in the total length.
The maxillary bone shows a condition midway between that notice-

able in the parr or smolt and that of the grilse or salmon. In the
parr the posterior margin of the bone reaches to a point vertically
below the centre of the eye. In the adult fish the maxillary bone
is prolonged backwards to a point vertically below the posterior
margin of the eyeball, or beyond the eye altogether. In the
Gal way fish the point at which the maxillary bone ends is
vertically below the posterior margin of the pupil of the eye.
This and the arrangement of the opercular bones will be seen from
the accompanying outline drawing of the head.

(Issued separately February 1, 1905.)

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Proc. Roy. Socy. of Ed in ^ [Vol. XXV.

Plate I. — Tlie Gal way R. specinien.

Mr W. L. Calderwooi).

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Proc Rmj. Son/, o/ Blin.] [Vol. XXV.

Plate II. — Artificially reared salmon, one, two, and three years old. The largest fish
is, like the fish shown immediately above it, three years old, but has been one year
in a sea-water pond. It is 33*0 cm. long.

Mr W. L. Calderwood.

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To prevent delay, authors residing abroad should appoint some one
residing in this country tp correct their proofs.

5. Brief Abstracts op Transactions Papers will be published in
the Proceedings, provided they are sent along with the original paper,

The author receives fifty of these reprints free, and may have any
reasonable number of additional copies at a fixed scale of prices which
will be furnished by the printer, who will charge him with the cost.
To prevent disappointment, especially if the paper contains plates,
the author should, immediately after receiving his first proof, notify
to tlie printer the number of additional copies required.

MODEL INDEX.

Schafer, E. A. — On the Existence within the Liver CeUs of Channels which can
be directly injected from the Blood- vesaels. Proc. Roy. Soc. Edin., vol. ,
1902, pp.
Cells, Liver, — Intra-cellular Canaliculi in.

E. A. Schafer. Proc. Roy. Soc. Edin., vol. , 1902, pp.
liver, — Injection within Cells of.

E. A. Schafer. Proc. Roy. Soc. Edin., vol. ,1902, pp.

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iv CONTENTS.

PAGE

The Sum of the Signed Primary Minors of a Detenninant

By Thomas Muir, LL.D., . . . .372

{IssvM separately January 20, 1905.)

Crystallographical Notes. By Hugh Marshall, D.Sc.,

F.R.S., 383

{Issued separately February 1, 1905.)

The Effect of Simultaneous Removal of Thymus and
Spleen in young Guinea-pigs. By D. Noel Paton
and Alexander Goodall. {Frwn tlie LahmaJory of
tJie Royal College of Physicians, Edinburgh), . . 389

{Issued separately February 1, 1905.)

Networks of the Plane in Absolute Geometry. By
Duncan M. Y. Sommerville, M.A., B.Sc., University
of St Andrews. {Abstract). {Communicated by Pro-
fessor P. R. Scott Lang), .... 392
{Issued separately February 1, 1905.)

A Specimen of the Salmon in transition from the Smolt to
the Grilse Stage. By W. L. Calderwood. (With
Two Plates), . . . . . .395

{Issued separately February 1, 1905.)

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PROCEEDINGS

OF THE

ROYAL SOCIETY OF EDINBURGH.

SESSION 1904-5.

No. VI] VOL. XXV. [Pp. 401-464.

CONTENTS.

PAGE

A Comparative Study of tlie Lakes of Scotland ana
Denmark. By Dr C. Wesenbkrg-Lund, of the
Danish Fresh-water Biological Station, Frederiksdal,
near K. Lyngby, Denmark. (Coimnunicafed by Sir
John Murray, K.C.B., F.R.S.) {From the DanisJr
FresJi'Water Biolofjiral Laboratory, Fredeinksdal.)
OVith Two Plates), . . . . .401

{Issued separately March 3, 1905.)

Variations in the Crystallisation of Potassium Hydrogen
Succinate due to the presence of other metallic com-
pounds in the Solution. {Preliminary Notice.) By
Alexander T. Cameron, M.A. {Comimmicated by
Dr Hugh Marshall, F.R.S.), . . . 44D

{Tssi'cd separately February 4, 1905.)

[Continued on page iv of Cover.

JL - -

^EDINBURGH :

Published by ROBERT GRANT & SON, 107 Princes Stkbkt, and
"WILLIAMS k NORGATE, 14 Henrietta Strekt, Covent Garden, London.

MDCCCCV.

Price Three Shillings ami Sixpence.

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EEGULATIONS REGARDING THE PUBLICATION OF
PAPERS IN THE PROCEEDINGS AND TRANS-
ACTIONS OF THE SOCIETY.

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the Society to the following Regulations, which have been drawn up in
order to accelerate the publication of the Proceedings and TransaetioDs,
and to utilise as widely and as fairly as possible the funds which the
Society devotes to the publication of Scientific and Literary Researches.

The * copy * should be written on large sheets of paper, on one side
only, and the pages should be clearly numbered. The MS. must be
easily legible, preferably typewritten, and must be absolutely in its final
form for printing ; so that corrections in proof shall be as few as possible,
and shall not cause overrunning in the lines or pages of the proof. All
tables of contents, references to plates or illustrations in the text, etc.,
must be in their proper places, with the page numbers left blank ; and
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appear in the text.

If an author finds it inconvenient to furnish such drawings, the Society
will have the figures re-drawn at his expense ; but this will cause delay.

3. Proofs. — In general, a first proof and a revise of each paper will
be sent to the author, whose address should be indicated on the MS.
if further proofs are required, owing to corrections or alterations for
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All proofs must, if possible, be returned within one week, addressed to
77ie Secretary, Royal Society, Mouncl, Edinburgh, and not to the printer,

[Continued mi page m of Cover,

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a904-5.] Study of the Lakes of Scotland and Denmark. 401

A Compcurative Study of the Lakes of Scotland and
Denmark. By Dr C. Wesenberg-Lund, of the Danish
Fresh-water Biological Station, Frederiksdal, near K. Lyngby,
Denmark. Communicated by Sir John Murray, K.C.B.^
F.R.S. {From the Danish Fresh-waier Biological Laboratory ^
Frederiksdai.) (With Two Plates.)

(MS. received January 13, 1905. Read January 23, 1905.)

Introduction.

In June 1904 I received an invitation from Sir John Murray to
visit Scotland and spend three or four weeks in exploring the
Scottish lakes, in order to make a comparison between them and
the Danish lakes : he was of opinion that such a comparison of
the lakes of a highland and a lowland country, which had hitherto
not been attempted, would lead to some interesting results. The
admirable bathymetrical and physical explorations carried on by
Sir John Murray in Scotland, and more especially in Loch Ness,
being far advanced, the question as to the scope of the biological
observations called for consideration ; so he desired me to indicate,
from the impressions derived during my visit, my views as to the
most useful lines of investigation that might be taken up with
reference to the biology of the Scottish lakes. I was much
interested in the task imposed upon me, and at the same time
gratified at the prospect of assisting in the design of the biological
explorations in the lakes of a foreign country ; and as it was of the
greatest significance to me to learn the nature of alpine lakes, I
immediately accepted the invitation. I spent three weeks in
Scotland, — the first two at Fort Augustus, on Loch Ness, and the
third in Edinburgh. From Fort Augustus I explored the lakes of
the Caledonian Canal, and thus became acquainted with alpine
lakes ; from Edinburgh I explored a few lowland lakes, especially
Loch Leven. The steamer Mermaid, belonging to the Marine
Biological Station at Millport, fully equipped for deep-sea work,

PROC. ROY. SOC. BDIN. — VOL. XXV. 26

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402 . Pro(xediiigB of Royal Society of Edinburgh. {i

under the direction of Dr Gremmill, was sent into the Caledonian
Canal, and many hauls were taken with the dredge and trawl, as
well as with different kinds of tow-nets, in Lochs Lochy, Oich,
and Ness, down to the greatest depths (500 and 750 feet).

Before entering on the suhject of this paper, I beg to tender to
Sir John Murray my most cordial thanks for his invitation and
for his kindness to me during my stay in Scotland. As regards
limnological explorations, Scotland was a few years ago a complete
terra incognita, but when the work of the Lake Survey is com-
pleted there will undoubtedly be no other country in which the
lakes have been better studied than in Scotland. On Loch Ness
I learnt the methods employed in taking the temperature and
other physical observations ; and when the numerous observations
and enormous mass of material have been worked out, I think that
Loch Ness, as regards the bathymetrical and other physical con-
ditions, will be one of the best explored lakes in the world —
perhaps only equalled by the I^ke of Geneva.

It has hitherto been difficult to give equal prominence to the
physico-chemical investigations, on the one hand, and the biological
investigations, on the other, in the study of the lakes in different
countries, owing mainly to the lack of scientists versed in the
different branches of limnology, and interested alike in these two
great departments. The admirable explorations carried on by
Professor F. A. Forel and his pupils show what excellent results
may be obtained when the investigations are planned on a uniform
basis. I trust that Sir John Murray and Mr Laurence PuUar will
agree with me in expressing the hope that, on the completion of
the bathymetrical and physical survey so admirably commenced
by Sir John Murray, and continued at the joint expense of both
gentlemen, the work may be still further carried on in such a
manner as to utilise the results yielded as to the biological
study of lakes. I am quite well aware, as will be seen from
the following pages, that the study of organisms, and especially
of the influence of organic life upon the general conditions of a
lake and its environs, presents greater difficulties in alpine
countries than in lowland countries. The problems presented
by the local conditions of lakes can perhaps be better studied
in Scotland than in any other country ; and I sincerely hope that

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1904-5.] Study of the Lakes of Scotland and Denmark. 40Ji

the investigations relating to the extremely interesting plankton,
the bottom-fauna, the Diatom flora of the shores, and the influence
of water rich in humic acid upon fresh-water organisms, may be
studied in accordance with the knowledge which has been gained
of the life-conditions common to all organic life. It would be
most unfortunate for the study of fresh-water and its organisms if,
in a country where the knowledge of the life-conditions is so
prominent, this knowledge should not be fully utilised.

During the last fifteen years I have spent most of my time in
the study of our own lakes and their organic life, and I hope
that my statements in the following condensed and brief account
of the Danish lakes may prove reliable ; time will show whether
I have carried my generalisations too far. What I learnt re-
garding the Scottish lakes brought to light many differences
between them and our own lakes ; and I had occasion to make
some observations which, if carried further, would have served
as starting-points upon which to base my working theories. My
knowledge of the Scottish lakes is, of course, very limited, but I
hold it to be the duty of a scientist not only to make known the
actual facts observed by him, but also his ideas as to the bearing
of these facts. Strictly speaking, new ideas should be regarded
not so much from the standpoint as to whether they may be
right or wrong, but rather as to their value in the promotion of
scientific knowledge ; and I hope the following pages may contain
ideas useful in some measure in future investigations.

Gbnbral Rbmarks on the Natural Conditions op thb
Danish and Scottish Lakes.

A. The Danish Lakes.

My explorations have shown the most remarkable differences
between the Danish and the larger Scottish lakes in nearly all par-
ticulars, which was to be expected, considering the wide divergence
in the geological structure of the two countries. I would here
merely point out that Denmark is a lowland country, the highest
eminences not exceeding 500 to 550 feet above sea-level, and.

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404 Froceedings of Boyal Society of Edinburgh, [i

geologically speaking, it is of recent origin, being built up of very
light and friable soil— mostly the moraines of those enormous
^aciers which covered Denmark and the surrounding seas during
the Ice Age. It is probable that lime strata of different geological
ages occur nearly everywhere beneath the soil, rising in certain
places to the surface, and in other places not far below the surface.
The soil itself is commonly very rich in lime, which is washed out
by the rivers and carried into the lakes. The rainfall is not great^
only about 614 mm. (24 inches) per annum; and this, in con-
junction with the lowness of the country and the friable soil,
accounts for the fact that the rivers are all small— rarely more
than about 50 feet in breadth, with level courses (falls being
quite unknown), and transporting only a small quantity of water.
The outflow of water from the rivers is greatest in spring after
sudden thaws, and least in summer (especially in dry seasons) and
in autumn, increasing considerably in November and December,
with their abundant rainfall. As an example we may take the
liver Skem in Jutland, which in summer discharges at its outlet
only about 500 cubic feet per second, while in spring it may dis-
charge about 7500 cubic feet per second.

Denmark is now rather deficient in lakes, though at an earlier
period they must have been more numerous. They are all very
small, the largest covering an area of only about 40 square kilo-
metres (about 14 J square miles), while the great majority are much
smaller. Their depth is inconsiderable, as was to be expected in
a< low and flat country; exceptionally, depths of about 120 feet
have been recorded, but the majority are only 40 to 60 feet in
depth, while some of the largest lakes are in fact merely great
pools, with a maximum depth of only 10 ta 12 feet. Denmark is,
on the whole, a flat country, with no-deep depressions, and most of
the lakes are roimdish in outline, long and narrow lakes being rare ;
formerly the lakes were much more irregular, but owing to the
silting up of the bays and shallower parts the shore-lines show very
few sinuosities, though some of the larger lakes are very irregular.

The renewal of the water in the lakes goes on very slowly. As
the amount of water carried into the lakes by rivers is always
greatest in spring and slowly diminishes in summer, it will be
understood that the level of the lakes is highest in spring and

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1904-5.] Study of the Lakes of Scotland and Denmark. 405

lowest in August and September, the diiference amounting to 2 or
3 feet in the two seasons. Hence it follows that in our shallow
lakes the breadth of the beach increases in summer and autumn
to the extent of several hundred feet, and in winter and spni^
the ice or the waves cover places over which one might walk dry-
shod in summer.

The sides of the lakes are gently sloping ; and the same remark
applies to what the Germans term "uferbank," and the French
and Swiss term " beine." The deeper parts of the lakes are floored
by more or less level plains, the greatest depth being often found
near the centre. Islands are not common, though both islands
and banks occur. Owing to the small amount of detritus carried
down by the rivers, deltas are usually inconspicuous, and well-
marked banks at the embouchures of the rivers are rare.

Erosion by waves upon the shores is seldom conspicuous, as the
force of the waves is broken in rolling over the shallow plains,
often covered and bound together by vegetation. The wind-
blown sides of the lakes (especially the east-south-east shores) are
frequently sandy, or covered with stones and pebbles, while the
west and north-west shores are often peaty. On the other hand,
certain parts of the lake-shores show remarkable indications of
erosion, and these are most conspicuous where the shores are
covered with wood ; here one may see trees with scars and rifts
2 to 3 feet from the ground, and often showing remarkably
irregular forms. Further, one may find many overthrown trees
and dead shrubs standing high upon their washed-out white roots.
In the few cases where the shores rise precipitously from the
water's edge marks of erosion are often found, and abundance of
stones and pebbles washed down from the slopes above. This
erosion, however, is to be ascribed rather to the action of ice than
to that of waves. In spring, when the ice breaks up, it is often
piled into heaps 2 to 4 feet high, in front of which one always
finds a very conspicuous "end-moraine," consisting of gravel,
stones, broken PhragmiteSy shells of mussels and Limncea, and
various drift-materials. It may be pushed 20 to 25 feet from the
shore, and even — to the amazement of a naturalist— remain there
from one year to another. The ground over which the ice has
travelled will show, after the disappearance of the ice, a very

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406 Proceedings of Royal Society of Edinburgh, [\

conspicuous ''bottom-moraine,^ consisting of the shells of
AftodorUa, etc., which may be scattered over the ground iu
thousands ; stones are polished, and the ice, striking against the
trees, causes the rifts and wounds referred to above. Many treea^
on the prominent points bordering the lakes, are killed by these
heaps of ice, which are piled up year after year on the shores of
our lakes (see fig. 1). We may also mention that the ice-slabs in
spring often break great apertures in the closed stocks of Phragmites
and Scirpus, detaching large patches of rhizomes a square metre
(over a square yard) in extent and throwing them on shore ; the
ice may in the course of a few hours cover over a peaty shore
with sand, or cover a sandy beach with peat-forming material.

With reference to the temperature of the Danish lakes, it is to
be regretted that the observations are rather deficient. Still, it
may be stated generally that the temperature of the water follows
very closely the changes in the temperature of the air. Having
exemplified this statement in my Plankton paper, I shall here
only remark that the surface waters of our lakes are generally
very warm in summer, often attaining a temperature as high as
23** C. (73* F.), and in hot summers the water may maintain a
temperature of 20* to 23* C. (68* to 73' F.) for more than a
month : it is very rarely that the surface temperature in summer
falls below 16* C. (61* F.). Almost every winter most of the
lakes are frozen over, though the length of the period during
which they are ice-bound varies greatly in different years, but
never exceeds more than about four months. The observations I
have made show that the lakes are usually frozen for one or two
months, generally from about 15th January to 15th March, but
exceptionally they may not be frozen at all. As we have often a
short spell of frost in November and December, followed by thaw,
usually followed again by the customary long period of frost in
January to March, the smaller lakes may have two ice-bound
periods — a short one in December and a longer one in January to
March, but in the larger and deeper lakes only the latter period
prevails. As most of our lakes resemble each other as regards
height above the sea, latitude, depth, and form of basin, it will be
understood that they vary little in temperature. It may generally
bo said that the deeper and narrower the lake and the steeper the

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1904-6.] SPudy of the Lakes of Scotland and Denmark. 407

sides, the more will the temperature of the water differ from that
of the air; it will take a longer time to freeze over, but will
remain ice-bound much longer than a shallow lake, and the
temperature of the water will rise more slowly, never attaining the
high temperature of the shallower lake. Only in oiie Danish lake
(Haldso) does the temperature of the water appear to differ
essentially from that of the other lakes. Thus the mean tempera-
ture of the air in July 1901 was extremely high, — 19*9* C.
(67-8' F.), and the surface temperature of all our lakes except
Haldso was 21" to 23* C. (70' to 73" F.), while in Haldso the
temperature never exceeded 18' C. (64* F.); in the winter
of 1901-2 the other lakes were ice-bound from 39 to 65 days,
whereas Haldso was only ice-bound for 35 days. It may be
added that Haldso is one of our deepest lakes (about 120 feet),
and has more precipitous shores than any of the others.

The transparency of the water in our lakes is small, and varies
regularly with the season of the year, being always greatest in
spring, diminishing during the last days of April, and least in
August. During the ice-bound period the water becomes much
clearer, all the detritus and huge masses of phytoplankton being
precipitated to the bottom.

The colour of the water in the Danish lakes in April, after the
ice has broken up, is nearly always a bright blue, but this colour
only continues till the beginning of May, when most of the lakes
become of a yellowish-green colour, which continues to be the
predominant colour till the frost sets in. In hot summers the
surface is generally covered by a coating of ** wasserbliithe,'' and
then the colour changes to blue-green or green ; in cold summers
no ** wasserbliithe " appears on the deeper and colder lakes.

As regards the chemical composition of the water,, very few
observations have as yet been made, but I hope this will soon
be remedied.

B. 77w Scottish Lakes,

Comparing the natural conditions of the Danish lakes, as
indicated in the foregoing pages, with those of the Scottish lakes,
we shall find the greatest differences in nearly every detail. It
must be borne in mind that geologically Scotland is a very old

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408 Proceedings of Roycd Society of Edinburgh, [ssas.

country, for the most part built up of hard rocks. Nearly all
the lakes belong to the Highlands, the highest mountain peaks
attaining an elevation of more than 4000 feet above sea-leveL It
is unnecessary in this short paper to enter into the chemical
composition of the rocks, but I think I am right in stating that, as
compared with Denmark, lime generally plays a subordinate r61e
in the chemical composition of the Scottish Highlands, and I am
of opinion that the amount of lime washed out by rivers and
carried into the lakes is nearly everywhere inconsiderable. The
Scottish rivers, with their rapid currents, their sources high up in
the mountains, their great eroding powers and waterfalls, are quite
different from our little brooks. As far as I could gather from the
members of the Lake Survey staff, there are no special seasons in
which the rivers carry exceptional quantities of water into the
lakes or into the sea. At different times of the year, though
probably mostly in spring, the rivers after heavy rains become
swollen, and after periods of drought they become low, but this
rise and fall are not, to the same extent as in Denmark, restricted
to certain seasons, and the suddenness with which the Scottish
rivers come down in flood has no parallel with us.

These differences are closely connected with the wide divergence
in the geological structure and climatological conditions of the
two countries — the one a low country, with moderate rainfall ; the
other mountainous, with a heavy rtdnfall,* the hilltops shrouded in
mists, and the hills themselves clothed with peat or peat-mosses,
which suck up the water . like a sponge and feed the rivers.
While Denmark has few lakes, Scotland has very many ; and
though generally of moderate size, many of them are much larger
than the Danish lakes. The main difference is the great depth of
the Scottish lakes, often exceeding 500 feet, and in one case (Loch
Morar) exceeding 1000 feet, and they are nearly all long and
narrow, none of the larger ones being circular, as is the case with
many of the Danish lakes. Their narrow form facilitates the
renewal of the water, and the sudden flooding of the rivers at
nearly all seasons of the year causes rapid changes in the level of
the lakes. With these phenomena we have hardly anything to

* In the western Highlands the raiufall is five to seven times greater than
in Denmark.

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-1904-6.] Stvdy of the Lakes of ScotlaTid and Denmark 409

compare in Denmark ; and the regular, but slow and comparatively
slight, rise in the level of our lakes in spring, and the fall in
summer, have, generally speaking, as far as my information goes,
no, or only a slight, counterpart in Scotland.

I consider the steep and precipitous shores of the Scottish lakes
to be one of their most prominent characters (see figs. 3 and 4).
From what I know (unfortunately only from the literature) of the
alpine lakes of S\vitzerland, the Scottish lakes generally surpass
them in this respect; in Scotland the mountains often descend
almost vertically into the lakes, and depths of 500 feet may be
found only a few yards from shore. Consequently there may be
no beach, or only a very narrow one, and I suppose the same
may be said of the " beine."

The Scottish lakes resemble the Danish ones in that the
greatest depth is generally found near the centre of the lake, and
that banks and well-marked deep holes are rare. Owing to the
large amount of detritus carried down by the rivers, banks are
common opposite the mouths of the rivers, and well-defined delta
formations seem to be a frequent feature. Where beaches occur,
they very often consist of pebbles and cobblestones, which during
storms are agitated by the waves ; the erosion of the waves upon
the rocks is often very conspicuous.

With reference to the temperature of the water, the excellent
observations of the Lake Survey show great differences between
the Scottish and Danish lakes. The larger Highland lakes are
never ice-bound, the surface temperature in winter being generally
from 5' to 7* C. (41* to 45" F.). On the other hand, the maximum
temperature in the same lakes in summer will never (I am
informed) exceed 18° C. (64° F.). It will thus be seen that,
while the surface temperature of the Danish lakes varies from a
little below zero to 23* or 25" C. (73" or 77" F.), the amplitude of
the variation in the surface temperature of the larger Highland lakes
is only from about 5" - 7" C. (41" - 45" F.) to 18" C. (64" F.).

The tranaparency of the water in the Scottish lakes is, strange
to say, not much greater than in the Danish lakes. ForeFs disc in
Loch Ness disappears at 24 or 25 feet, and I am told that in other
Scottish lakes the transparency is even less. This fact is very
remarkable, and, so far as I know, at variance with what one might

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410 Proceedings of JRoyal Society of Edinburgh, [am.

expect from the observations in other alpine lakes. As regards
the transparency, there is still this great difference between the
lakes of the two countries — that in the Danish lakes the trans-
parency is always and everywhere greatest in spring, and slowly
diminishes with increasing temperature, whereas in the Scottish
lakes, according to my informants, the transparency is nearly
constant all the year round, but may at any season, especially
after heavy rains, be suddenly greatly reduced.

As to the colour of the water, another great difference between
Danish and Scottish lakes is to be noted ; for while the colour of
our lakes undergoes a regular alternation, strictly dependent on the
different seasons, the colour of the Scottish lakes varies very
little at all seasons. The larger Scottish lakes never show that
turbid yellowish-green colour so characteristic of nearly all our
lakes from May to November, nor the deep blue colour displayed
by our lakes in April, neither are they covered with "wasserbliithe"
caused by blue-green Algse. The water in all the Scottish lakes
seems to be very clear, but has a yellowish-brown colour, quite dif-
ferent from the blue colour of most of the alpine lakes in Switzer-
land, which are also characterised by the great transparency of the
water: in both the Swiss and Danish lakes the transparency is
much greater in winter and spring than in summer and autumn.

As will be noted in a later chapter, the colouring of the Danish
waters is due to the plankton ; the colouring of the Scottish lakes
has quite a different origin. It must be remembered that the
Scottish rivers nearly always drain through peaty bogs and the
moss-covered sloping sides of the mountains, and only very
rarely, and for a short period of the year, do the rivers obtain
their water directly from the snow. 1 am told that the layer of
peat on the mountains may attain a thickness of 1 to 2 feet,
and it will therefore be easily understood that the water of the
Scottish lakes must necessarily be peaty and very rich in humic
acid, and this fact accounts for their yellow-brown colour and very
slight transparency. In my opinion we have here the most strik-
ing and the most interesting difference between the alpine lakes of
Switzerland, with their clear blue water, their rivers fed directly
from the vast eternal glaciers, and the alpine lakes of Scotland, with
their yellowish-brown water, their rivers rising in bogs and travcrs-

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1904-5.] Stttdy of the Lakes of Scotland and Denmark. 411

ing the moss-covered precipitous mountain sides. I have been told
that Loch Morar, the deepest of all Scottish lakes, has the clearest
water, Forel's disc being visible at a depth of 44 feet ; and in this
connection it is of great interest to note the fact that the rocks
along the shores of Loch Morar and all over the drainage area are
not covered with peat and mosses, but are for the most part quite
bare. As far as I know, we have no particularly peaty water in
any of our larger lakes, though it is, of course, a very common
feature in the smaller lakes surrounded by peat, and whose floors
are covered by peaty mud, many of which are quite artificial,
being due to the digging of peat.

The foregoing remarks refer only to the character of the Danish
and Scottish lakes, but I feel convinced that many of the facts
stated are common to lakes belonging respectively to the great
Central European plain and to alpine countries. As traits common
to all the first-mentioned lakes, I would specially point to their
shallowness, their gently sloping shores, their roundish outline, the
high temperature of the surface water in summer and the freezing
over in winter, the ice-erosion on the shores, the small trans-
parency, and the yellow or yellow-green colour of the water in
summer, due to the huge plankton-masses. Differences may be
looked for with regard to the chemical composition of the water
and bottom-mud, owing to the varying chemical composition of
the soil in different countries ; I anticipate that further investiga-
tions will prove that the large amount of lime carried by streams
into our lakes is one of the most characteristic peculiarities of the
Danish lakes. On the other hand, 1 am of opinion that the
features mentioned in connection with the Scottish lakes are
common to alpine lakes in general. Especially would I call
attention to their great depth and long and narrow form, their
precipitous shores, the sudden flooding of the rivers and the rapid
changes in the level of the lakes, and the slight amplitude in the
annual variation of the surface temperature. Peculiar to the
Scottish lakes are the small transparency and yellowish-brown
colour of the water, to which may undoubtedly be added the
large amount of humic acid. These peculiarities may be traced to,
and are closely connected with, the strongly-marked climatological
and geological conditions common to the whole country.

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41 2 Proceedings of Royal Society of Edinbwrgh, [sob

II.
The Organisms, and their Relations to the different

LlFB-CONDITIONS, IN THE DaNISH AND SCOTTISH LaKES.

It will be easily understood that the life -conditions offered to
fresh-water organisms differ widely in the Danish and Scottish
lakes respectively, and that there are great differences between
the vegetable and animal life in each case. Generally speaking,
it may be said that the low temperature and freezing over of the
Danish lakes in winter have not hindered the immigration of
most of the fresh- water organisms distributed over the entire
temperate region of Europe, while the usually high summer
temperature, due to the shallowness of our lakes, is undoubtedly
one of the main factors to which we must ascribe the extremely
rich organic life, both as to the number of species and of indi-
viduals, characteristic of our own as well as most of the lakes in
the northern part of the Central European plain. We shall now
consider the vegetable and animal life in the Danish and Scottish
lakes respectively, according to the three main regions that may
be recognised in every lake, viz., the Littoral region, the Pelagic
region, and the Abyssal region. As far as possible, we shall
endeavour to indicate how the different characters of the lakes in
the two countries have produced great differences in their
associations of animals and plants.

A. The Danish Lakes,

1. The Littoral Region. — Owing to the gently sloping shores,
the smooth wash of the waves, the sandy beaches, often covered
with decaying vegetable matter, and the high summer temperature
of the coastal waters, most of our lakes are bordered by dense and
luxurious bands of vegetation, which in shallow bays may attain
a considerable width, merging imperceptibly into the vegetation
of the adjoining land. Thus our lakes are often in certain parts
bordered by humid meadows, which in winter and spring are
covered by ice or water, while in hot summers they may be quite
dry, so that it is frequently difficult to say where the land ends
and the lake begins.

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1904-6.] ^vdy of the Lakes of Scotland and Denmark. 413

As the depth of the lakes increases very regularly from the
shore outward, and as the different plants are on the whole limited
to certain depths, the vegetation arranges itself in zones (see fig. 2).
For details I may refer to the excellent work of Professor Warming
(1895),* and will here restrict myself to the following remarks.
In most of our larger lakes we have a narrower or wider shore-
zone, mainly characterised by Scirpvs lacustris and Phragmites
communis. Further out we shall find zones of Potaniogefon lucens
and perfoliatus and some other plants, especially Batrachium,
Mynophyllum, and Ceratophyllum, Still further out, by dredging
on the bottom, we find a zone formed of Characea and some Fonti-
nalis, which extend to a depth of 8 or 9 metres (25 or 30 feet),
and beyond this limit we usually find no higher plants. With
the exception perhaps of the outer border of the Characea zone,
all these zones of vegetation die off in winter, leaving only their
resting organs, their rhizomes, etc., on the bottom. The higher
plants are in summer nearly always covered by a very rich
epiphytic vegetation of blue- green AlgSB, Diatoms, and green
Algse. On the windward side of the lakes the vegetation is, of
course, less abundant, and here we often find beaches of stones
and gravel, without any higher plants. The stones themselves in
all our lakes are in winter covered with a rich brown coating of
Diatoms, which in summer often disappears, but in several lakes
its place is taken by a crust of greyish lime deposited from the
blue-green Algse, as in many of the Swiss lakes.

The plentiful vegetation is the home of an abundant and re-
markable animal life : of the higher invertebrate groups we
specially notice many larvte of insects,— of Diptera, Phryganidse,
Ephemerid», Libellulidae, certain Coleoptera, and a few Neuroptera
(Sialts) ; of the Crustacea there are Amphipoda {Oammams piUex^
Pallasiella quadrispinosa), Asellus, Daphnids and Copepods in
great abundance; besides many Rhabdocoela, a few Dendrocoela
and Oligochffita, very many Rotif era, a very rich Protozoan fauna,
and a great many snails and mussels. Beneath the stones we also
find numerous organisms, especially Phryganidse, Ephemeridse, and
Planaria, and on the upper sides of the stones snails are nearly
everywhere found.

• This work will shortly appear in English, translated by Professor Balfour.

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414 Proeeeding9 of Royal Society of Edinburgh, [snt.

A stranger unacquainted with our lakes on reading these lines
might form the impression that the shores of our lakes were for
the most part inhabited by the common fresh-water fauna to be
found in every shallow pond with rich yegetation. This impres-
sion would be incorrect, for a closer examination would certainly
show that, while many species are common to ponds and to the
vegetation zone of the lakes, still it would appear that most of the
PhryganidsB, Libellulidse, EphemeridaB, some of the Crustacea,
many Planaria, some OligochsBta and Rotifera are quite peculiar
to the lake-shores, and rarely appear in ponds. Further, it would
seem that several species of snails common to the ponds and the
shores of the larger lakes are represented in the lakes by special
forms differing from those found in ponds. I cannot in this short
paper discuss this point in greater detail, but will content myself
by remarking that the fauna of the littoral zone of our lakes is
on the whole very different from that of our ditches and ponds.

In winter the greater part of this rich fauna disappears. In
November and December many of the organisms, especially snails
and some insect larvae, migrate into deeper water before the shores
are covered with ice; other organisms, for instance many insect
larvse, go ashore and burrow holes in the ground, while a great
many other species, especially Daphnids and Rotifers, make resting
organs * and, by means of them, survive the freezing in the ice.
Still, there are numerous organisms which appear to live in winter
beneath the ice as they do in summer in water having a tempera-
ture of about 25* C. (77° F.) ; for example, Planaria, Phryganidae,
Amphipoda, NepfieliSy etc.

2. Tfie Pelagic Region, — With regard to the plankton, I may
refer to my Plankton studies (1904), and restrict myself in this
place to the following brief remarks. Our lakes are nearly always
extremely rich in plankton, so much so that throughout the greater
part of the year — from April to December — it affects the colour
and transparency of the water, and is doubtless one of the main
factors in determining the varying amounts of oxygen and carbonic
acid dissolved in the water. It will thus be understood that the
plankton of our lakes — its composition and its abundance — must
necessarily greatly influence the other organisms in the lakes.
* Hibernating buds, ephippia, or <

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1904-5.] Study of the Lakes of Scotland and Denmark, 415

With regard to the fresh-water plankton of the world, two r^
markable characteristics should be noted. Firstly, that generally
speaking it seems to be very homogeneous from pole to pole. The
plankton of the Greenland lakes is similar to that of the North
African lakes, only certain groups of plankton-Algae being apparently
rare, or perhaps entirely absent, near the pole. From this general
rule we know only a few exceptions, especially as regards some
Crustacea. Very many species are common to the fresh-waters of
Iceland and those of North Italy. Secondly, that the central
domain for the full development of all fresh-water plankton is
apparently in the temperate zone, and not in the tropics. If these
characteristics hold good, the fresh-water plankton differs essentially
in both these respects from all other associations of organisms in
the sea or on the land. These two points cannot, however, be
held as proved until the tropical fresh-water plankton has been
fully explored ; and I consider it extremely desirable that one of
the great nations having possessions in the tropics should despatch
an expedition with the main object of investigating the tropical
fresh- water plankton.

The plankton of our lakes does not differ, on the whole, from
that to be found in any of the larger lakes in the northern parts
of the Central European plateau, but, as Forel justly remarks, all
these lakes scarcely merit the name. In most of these compara-
tively shallow lakes the plankton is characterised by a great
development of Melosira and blue-green Algae, by the presence of
Bosmina coregoni, and perhaps by the occurrence of the only two
common species of the Copepod genus DtaptomnSj D, gracilis and
graciloid&t. The Cydotdla and Oscillatoria, so characteristic of
alpine lakes, are usually rare, and often entirely absent, while
certain species of Diaptomus and some peculiar species of
Chlorophycea, common in southern alpine lakes, have never been
found in the Central European plateau.

The plankton of the Danish lakes differs somewhat perhaps
from that of the lakes in the surrounding lowland countries in the
rich development of the Diatom genus StephanodisfmSy of the blue-
green Alga genus Lynghya^ and of the Conferva Tnbonema
hombydnum. As our lakes are usually shallow and the littoral
zone very extensive, it will be readily understood that many

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416 ProcudiTigB of Boyal Society of Edinburgh, [

organisms from the Littoral region find their way to the central
parts of the Pelagic region, and that many of the forms peculiar to
the central parts of the larger ponds, especially many Chlorophycea,
may be carried by the rivers into the Pelagic region of the lakes ;
still, the mcyority of these organisms never play a prominent part
in the composition of the plankton.

Out of about 150 plankton organisms which have been
recognised in the Danish lakes, very few appear in such vast
quantities as to give the plankton a monotonous character, or to
influence the life-conditions of the lake during the greater part of
the year. Among these are Melosira crenulata and granulata^
AiterioneUa graciUimOy Aphanizomenon flos aqtut^ Ceratium hirun-
dinella^ the species of Diaptomua^ Daphndla hrackyurOj
Hi/dloilaphnia cucuUcUa^ Botmiina coregoni, and Leptodara
kindtti. From April to December there are in almost every
lake, besides the above-mentioned species, others which may pre-
dominate during a shorter period. Among these I would mention
Fragilaria crotonewds, and other Diatoms, Coelospkan'ium kutzm-
gianum, Polycystis^ and a few other Cyanophycea, a very few
Chlorophycea and Protozoa, some Kotifera, and of Crustacea
especially Cyclops oithonoides, Bosmina longirostrisy and Daphnia
hyalina. Besides those organisms whose home is in the littoral
zone, or in the central parts of ponds, which are always rare in
the Pelagic region of the lakes, there are other rare forms found
in this region that only appear in the summer months. These
organisms, as far as I know, have apparently reached or nearly
reached their northern limit with us; this applies especially to
some Rotifers, Cyanophycea, etc.

Though the life-conditions in our lakes do not vary very much,
still there is a good deal of difference in the plankton of the
different lakes: this refers mostly to the Diatoms and Cyano-
phycea, those two great groups of organisms which, in my opinion,
affect more than any other the common life-conditions of the lake.
As a general rule, we may say that these two groups rarely attain
their maximum development in the same lake or simultaneously.
Most of the fresh-water Diatoms reach their highest development
at a relatively low temperature (below 12* or 10* C. = 54* or 50' F.)
and in the colder of our lakes; on the other hand, the Cyano-

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1904-5.] Study of the Lakes of Scotland and Denmark, 417

phycea— except Osdllatoria — usually reach their greatest develop-
ment at the highest summer temperature (between 19* and %S'* C.
= 66' and 73* F.) and in the warmer lakes. Accordingly, we find
a great development of Diatoms in the cold northern lakes as well
as in the southern alpine lakes, and an almost complete absence of
the Cyanophycea in both these localities, the only exception being
the OsciUatoi'ia and partly AnabcBna flos aquos^ which are both
common in the alpine lakes of Switzerland. In our colder lakes
a great development of Diatoms occurs in the last days of April,
when the lakes are ice-free, and continues till June ; then a great
development of Ceraiium hirundinella sets in, and in September
a second development of Diatoms appears. On the other hand,
in our shallower and warmer lakes the great development of
Diatoms is discontinued a little earlier, then the Cyanophycea
appear, and often predominate throughout the rest of the year;
still, in these lakes also the development of Ceraiium and a second
development of Diatoms occur, but rarely to such an extent as in
the deeper and colder lakes.

The deep cold lakes rarely present the phenomenon of " wasser-
bliithe " ; and if it appear, it is only for a short time in June, caused
by Anabcenafloa agiue. As the chromatophores of the Diatoms, as
well as those of Ceraiium hirundinella, are a yellowish-green, the
colour of the water in nearly all our colder lakes is also yellow-
green. The colour of the water in the shallower and warmer
lakes is in spring also yellow-green, owing to the first great
development of Diatoms ; but when the maximum development of
the Cyanophycea sets in, the colour becomes more bluish-green,
owing to the blue-green colour of the Cyanophycea cells, and the
surface of the water on calm days is covered by a thick layer of
" wasserblUthe " : in August and September, when the great
development of Cyanophycea is intermixed with that of Ceraiium
hirundinella and the second development of Diatoms, the water in
these lakes clianges somewhat towards yellow-green.

As a rule, we may say that the colour of the water in our lakes
throughout the greater part of the year is determined by the
colour of the plankton-organisms, especially by that of the
chromatophores of the Diatoms and of the Cyanophycea. Only in
April, immediately after the breaking up of the ice, is the quantity

PROC. BOY. SOC. EDIN. — VOL. XXV. 27

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418 Proceedings of Moyal Society of Ediriburgh, [i

of plankton so insignificant that one may decide as to the original
colour of our fresh-waters : to determine the colour of the water at
any other season it would be necessary to filter it This probably
applies to all the lakes of the Central European plain, but, as far as
I am aware, the colour of the water in all these lakes has never
been determined from filtered samples; and if so, it must be
remembered that such determinations may have been greatly
influenced by a foreign factor, viz., the colour of the chromato-
phores of the plankton-organisms in greatest profusion at the time.
Until the colour of the water has been determined from filtered
samples, we cannot, in my opinion, directly compare the colour of
the water in these lakes with that of the water in the alpine lakes,
in which the amount of plankton, especially in the surface layers
of water, is altogether insignificant as compared with our lakes.

In winter a great many plankton-organisms totally disappear
from the water: this is the case with certain species which in
more southern latitudes occur all the year round (Ceraiium
htrundiriella), but with us they produce their resting organs in
autumn and disappear. I think it is very probable that those
resting organs which, before winter sets in, are precipitated to the
bottom in the deepest parts of the lakes, never rise to the surface
again, but sooner or later die off, not finding the necessary
conditions for germination. In my opinion, the plankton-organisms
of the following year are mostly derived from those resting organs
which were deposited in shallower water nearer the shore, where
the waves during the spring gales sweep the bottom, carrying
away the resting organs and scattering them over the lake. In
our lakes the resting organs of the different plankton-organisms
are most plentiful in April and May, after the heavy storms ; and
I have shown in my Plankton paper that many plankton-organisms
are in May most abundant near shore, and that their distribution
over the whole lake does not take place till later in the year.

I may here remark that very probably — though direct observa-
tion is very difficult — various plankton-organisms, especially
certain Diatoms (TahellaHa fenestratOy Diatoma elongatum), may
have alternately a fixed littoral stage and a free-swimming or free-
floating pelagic stage, and these two stages may be restricted to
certain seasons, the shape of the colonies in the littoral stage

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1904-5.] Stvdy of the Lakes of Scotland aiid Denmark. 419

(chains) being different from that of the colonies in the pelagic stage
(stars). These remarks may prove of some importance, inasmuch as
future investigations may show how littoral organisms become trans-
formed into pelagic organisms, and as they support the hypothesis,
now commonly adopted, that the fresh-water plankton is derived
from the common microscopical littoral and bottom fauna and
flora, very few organisms having immigrated directly from the sea.

As a character common to all our plankton, I may add that the
seasonal variations of the organisms are very conspicuous, and
more especially those of Daphnia (Hyalodaphnia) etunUlata,
Bosmina coregoniy Asplanchna priodonta, Ceratium hirundinella^
AgterioneUa gracillima, Melosira crenvlaiOy FragilaHa crotonensis,
Pediastrumy etc. I shall return to the investigations on this point
after treating of the plankton of the Scottish lakes.

I may point out that the vivid red colour characteristic of many
Crustacea in other countries is not with us very conspicuous;
several Copepoda do, as a rule, in winter, change from yellowish-
white into a deep red colour.

With regard to the vertical distribution of the plankton, I only
venture to remark that the greatest profusion of plankton is to
be found in the upper layers of water. Like most of the
naturalists who have studied the plankton in the lakes of the
northern part of Central Europe, I have not been able to dis-
tinguish any vertical wanderings at different hours of the day;
I venture to think that such wandeiings are rather incouFpicuous
with us, but further investigations with improved appliances will
be necessary to decide this question.

3. The Abyssal Region. — In my paper on the bottom-exploration
of the Danish lakes (1901), I have pointed out that there are
reasons for fixing the limit between the Littoral region and the
Abyssal region at about 9 or 10 metres (30 or 35 feet). Li speaking
of our shallow lakes we cannot, of course, strictly use the term
** abyssal region"; the principal conditions laid down by Forel
regarding this region, especially the uniformity of all the life-
conditions, are never fully realised in the Danish lakes. Still, it
may be maintained that we can speak of an abyssal fauna,
inasmuch as this is quite different from the littoral fauna, and
apparently similar to the abyssal faima in deep alpine lakes.

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420 Proceedings of Royal Society of Edinburgh, [skss.

Outside the 9metre (30-feet) contour we find no plants except
certain species of OscUlatoria and bottom Diatoms; all higher
vegetation is limited within this contour, and the slight trans-
parency of the water is probably the main factor in determining
this distribution. The majority of the snails also are limited
by this contour, only Valvata piscinalis extending a little beyond ;
the pulmonary snails never cross this boundary, the abyssal
LimncBa known from the Lake of (Geneva being entirely absent
from our lakes. The same contour also marks the boundary of
nearly all the insect larvje, only Sidlts penetrating so far.

The deep bottom of our lakes is chiefly inhabited by Pisidium^
the larvffi of Chironomus and Tanypus, the OligochaBte
Psammorydes fossor, Ostracoda {Limnicythere relicta and some
species of Candona), a few Planaria {Plagio$tmna lemani), etc.
The Daphnidae and the very minute forms of animal life, such
as Protozoa, have not been studied. On the whole, I think I may
say that our abyssal fauna, though imperfectly known, is still
undoubtedly very like the abyssal fauna of the Swiss lakes.

B. Tfie ScoUish Lakes.

In comparing the associations of fresh-water organisms in the
Scottish lakes with those in the Danish lakes, we shall find in
nearly every particular the greatest contrast.

1. TJie Littoral Region, — With regard to this region we may,
in the first place, point out that the belt of vegetation which
nearly always surrounds our lakes is often entirely abeent from
the larger Scottish alpine lakes, due to the precipitous or stone-
eovered shores, devoid of deposits of sand or decaying vegetable
matter : even river deltas and other sandy flats are often almost
bare of vegetation, partly, I suppose, because of the powerful
erosion of the waves, and partly because the sudden changes in
the level of the lakes is destructive to the amphibial plants. In
the smaller and shallower lakes, for instance Loch Oich, in
which we find some higher vegetation along the shore, this
vegetation is not arranged in those elegant zones so characteristic
of the Danish lakes.

As far as I am aware, the stones have never been found
clothed with blue-green Algae ; but when I had the opportunity

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1904-5.] Stvdy of the Lakes of Scotland and Denmark. 421

of examining them, they were always covered with coatings,
often thick, of Diatoms. I found such coatings at the height of
summer, at a time when they never occur in our coimtry, owing
to the high temperature of the water; and from what I have
observed in Danish lakes, I suppose they may possibly also occur
in the Scottish lakes in winter. I visited Scotland at an
extremely dry season of the year, when the rivers were only
moderately supplied with water and the level of the water in the
lakes singularly low ; on the stony shores and precipitous
mountain sides I often found a more or less distinct whitish
band, which on closer examination proved to be due to dried
Diatoms and other plants, the upper stripe being identical with
high-water mark. We find a similar band on the stones in our
lakes in May, but later on the Diatoms are often covered over
by blue-green Algsp..

The animal life in the littoral region of the larger Highland
lochs seemed to me, compared with the Danish lakes, to be
extremely poor, but it must be kept in mind that I only
examined the lakes during the season when the animal life of
the littoral zone is almost everywhere at a minimum ; most of
those insects which, as larvsB, live in the littoral zone, disappear
in summer as full-grown insects, though they may possibly
have been numerous at an earlier season. Still, the animal
life whose home is in the vegetation zone, living or resting
on the vegetation, is rare compared with our lakes. When
I had occasion to examine the vegetation, for example in Loch
Oich, I always found it extremely void of the epiphytic organisms
so characteristic of most of our submerged fresh-water plants ; stiU,
in rapid streams the leaves of Potamogeton natans often constitute
a support for a great many larvae of Chironorrms^ Phryganea, and
of the family Hydroptilidae (I suppose ffydroptila maclachlani),
as well as for Stylaria proboscidea and Sida crystallina. Along
the shores of the lakes I observed very little of the extremely
rich winged insect life, consisting of swarms of images of all
those insects which as larvse abound in the water, and which
both in bright simshine and on calm moonlight nights are
characteristic of our lakes, and highly attractive to the student.
Beneath the stones I only found a few Planarians and one or

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422 Proceedings of Bayed Society of Edivburgh, [i

two species of EphemeridsB and Phryganidse. In the small bays
of the lakes, where the bottom may be seen well covered with
vegetation, for example Littordla, MyriophyUum, etc., we often
find a comparatively rich fauna of insect larvae, Cladocera, and
Rotifera; in such localities the fauna in these respects does not
seem to be piuch inferior to that found in the Danish lakes.

Between the littoral fauna of the Highland lakes as compared
with that of the Danish lakes, the main difference appears to be in
the MoUusca, which play a very prominent part in our lakes, but
are extremely rare in the Highland lakes. Along the shores of
Loch Ness and the other lochs of the Caledonian Canal I never
found a single mollusc shell, and on exploring the shores only a
few living specimens of Limnoea ovata and Planorhis coniorius
were to be found. Still, I expect that a closer examination by a
malacologist would reveal more species, and that in the shallow
water, in depths of 15 or 20 feet, species of Valvata, Bithynia,
etc. would be found, but all the larger species of Planorhis and
Limivea seem to be entirely wanting. At any rate the moUuscan
life in the Highland lakes generally is so extremely poor that it
cannot possibly influence the general conditions of life in the zone
in which it is principally found.

This special difference between the Scottish and Danish lakes I
consider to be due to the large amount of humic acid in the water
of the Scottish lakes, to the total absence of lime in the water and
on the floor of these lakes, to the absence of all lime-secreting
AlgflB and of lime-encrusted blue-green Algae covering the stones, of
Characea, etc., on which the snails in our lakes principally feed,
and to the, generally speaking, extremely poor vegetation. That
the first-mentioned is the principal cause is evident from the fact
that even in lakes rich in vegetation the molluscan life is greatly
inferior to that in the Danish lakes.

2. Tfie Pelagic Begion, — The investigations of Mr James
Murray,* assistant on the Lake Survey staff, of Messrs West, and
my own cursory examinations, have shown that there is a great
resemblance, and at the same time a great difference, between the
plankton of the Scottish and of the Danish lakes. Nearly all the
common plankton-organisms of the Scottish lakes also occur in the
* I desire to express my thanks to Mr Murray for information supplied to me.

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1904-5.] Stvdy of the Lakes of Scotland and Denmark. 423

Danish lakes, while, on the other hand, many forms found in the
Danish lakes have not hitherto heen ohserved in the Scottish
lakes. I may here give a short account of the commoner
plankton-forms, hased on the investigations above referred to.

The Cyanophycea play an altogether inferior part in the com-
position of the plankton in the larger Highland lakes, the only
rather common forms being Anabcenafloa aqtuB and Codosphcertum
ncegdianum. With regard to Lynghya and Oscillaioria further
explorations may give information, but as Mr Murray often speaks
of " filamentous Algae in abundance " they are probably common.

Of the Diatoms, it may be pointed out that Melosira, as in
many other mountain lakes, seems to be relatively rare, and
never forms those huge masses of plankton found in tlie Danish
and other lowland lakes. Stephanodiscus adrma has not yet been
observed as a plankton-organism ; and Cyclotdla, which has often
been considered as characteristic of alpine lakes, was not so common
as might have been expected, yet I suppose that closer examina-
tion at other seasons may prove that it is abundant ; Fragilaria
erotonensis also seems to be rare in the Scottish lakes. The
commonest forms are : — Asterionella gracilHma, Tabellaria fenes-
irata, var. asteriortellotdes, T, flocculosa (in chains), and a remark-
ably large number of bottom and shore Diatoms (Naviculoidese and
Surirelloidefie).

With regard to the Chlorophycea, Chodat has observed that
nearly all the small forms belonging to the Euchlorophycea are
warm-water plants, having their home in small ponds, the water
of which is rich in disintegrated organic matter ; in the Pelagic
region of the greater lakes they are nearly all rare, and must be
considered as merely chance visitors, introduced by streams and
rivers, soon finding their graves in the Pelagic region of the lakes :
to this rule we find only a few, but very peculiar, exceptions.
A study of the Chlorophycea in the Danish lakes has shown this
view of Chodat's to be quite correct : as all our lakes are shallow,
and the water in summer very warm, they should, according to
Chodat, be extremely rich in Chlorophycea, and this is exactly
the case. With regard to the Scottish lakes, we find some very
remarkable features. All the Euchlorophycea seem, from my own
observations, to be rare, and Messrs West (1904, p. 554) have

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424 Proceedings of Royal Society of Edinburgh, [smb.

lilso pointed out the very "remarkable scarcity of many of the
free-swimming Protococcoideae." Still, it must be remembered
that these organisms, judging from Chodat's investigations, eould
by no means be expected, all these plants, except Sphoerocystis and
a few others, being extremely rare in lakes : the numerous species
recorded by Lemmermann, Bruno Schroder, and others, all inhabit
the shallower and warmer lakes (see West, p. 564).

On the other hand, the explorations of Messrs West have proved
that the Desmidiacea play a most prominent and remarkable part
in the Pelagic region of a considerable number of the larger lakes.
The authors state that the Scottish phytoplankton " is unique in
the abundance of its Desraids. No known plankton can compare
with it in the richness and diversity of the Desmid flora." In the
present state of our knowledge, I consider the presence of these
numerous Desmids to be one of the most peculiar traits in the
composition of the plankton of the Scottish lakes. As far as I
know, very few of them have hitherto been recorded in the Pelagic
region of any of the greater European lakes, and their common
occurrence is quite the reverse of what might have been expected
from Chodat^s and my own observations. In the other European
lakes only two species, viz., Staurastrum gracHe and S, paradoxum^
are common. ITie manner in which, I think, we may endeavour
to account for their frequent occurrence will be referred to after
the plankton groups have been treated of.

As the Flagellata, Heliozoa, and Infusoria have not hitherto
been specially studied, and I myself have had no opportunity of
visiting the lakes in the season during which many of the Flagellata
and Infusoria are generally most abundant, I do not venture to
deal with these groups in detail, but restrict myself to the follow-
ing remarks. I have found Diiwhryum in all the lakes explored,
and in a few instances also species of the genera Mallomonas and
Gymnodinium, Geratium hirundinella seems to be common, and
the frequent occurrence of Clathndina is very remarkable — usually
empty shells, for only once, I think, did I see a living animal
As far as one may judge from the investigations of Mr James
Murray, it seems that the plankton Rotifers are quite similar to
those in other countries, but the absence of Madigocerra capudna
is remarkable.

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1904-5.] Stvdy of the Lakes of Scotland and Denmark. 425

With regard to the geographical distribution, none of the
plankton-organisms present points of so much interest as the
Crustacea. It has been mentioned that the plankton-organisms
have an extremely wide distribution, and may be regarded as
cosmopolitan ; most of the exceptions belong to the Copepoda
and Cladocera. Steuer (1901) was the first to draw attention to
the fact that the Diaptomidae and some of the plankton Cladocera
seem to have well-marked areas of distribution. Steuer's views
have been corroborated and modified or enlarged by the excellent
investigations of Ekman (1904) in the northern part of Sweden;
Ekman's results fully accord in all the main points with my observa-
tions in the Danish lakes (1904). Having referred to these
papers, I shall here restrict myself to those points having special
reference to the fauna of the Scottish and Danish lakes.

It may be regarded as a fact that there exists a peculiar associa-
tion of Arctic plankton Crustacea, mainly restricted to the Arctic or
North European lakes. This association is characterised by the
common occurrence of Holopedium gihherum^ Daphnia hyalina, Bo$-
mina obtusirostris, Bythotrephes longimanuSy Diaptomus lactniatfiSy
the genus Ileterocope (perhaps), and certain other species of
Copepoda. Bosmina coregoni, as well as B, longirostris and
BycUodaphnia cucidlata^ are almost entirely absent ; these are the
particular forms, besides several others, especially Diaptomus
gracilis and D. graciloides, Daphnella brachyura, I^ptodora kindtii,
which constitute the huge masses of zooplankton in the Central
European plains. Of the sub-arctic association, some of the species,
especially Diaptomus laciniatus, are also common in the alpine
lakes of Switzerland and other lakes in the Central European alpine
zone, but most of them {Holopedium giblteruvi, Bosmina obttud-
rostris) are never, or only exceptionally, found there. It seems
to me that these southern alpine lakes are mostly inhabited by
the same species which are characteristic of the Central European
plains, and that the arctic elements are on the whole subordinate.

The following facts may be briefly stated, from the explorations
of Mr James Murray, and the exceUent papers of Mr Scourfield
and Mr Scott quoted in the Bibliography, as well as from my
own investigations : —

Holopedium gibberum is very common, and frequently "so

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426 Proceedings of Boycd Society of Edinbwrgh. [sbsb.

abuudant that it chokes up the nets in a short time, and makes
it impossible to get a fair proportion of the other animals present"
(Murray, 1904a, p. 42); it may be added that the animals are
extremely large.

Of the genus Daphuia the common species is Daphnia hycdina
in different varieties {lacustrts, galeatOj etc.). D. {Hydlodaphrda)
cucuUcda is very rare, and only found in one locality (a lowland
lake). Bosmina arregrmi is almost entirely absent, and it seems
as though the genus Bosmina were only represented by one species,
B, obtusiivstris. BythotrepJies Longimanua occurs generally in
the Highland lakes, and is extremely large. Leptodora kindtii and
Daphndla bracJiyura are common in nearly all the lakes. Of the
Copepoda, the Diaptomidae are represented by D, gracilis, the com-
monest species, as well as by D, laciniattis, D. laticeps, and the
peculiar D. toiei'zejskii ; of the Cyclops, C. strenuus is the main form.

As will readily be seen, the common occurrence of Leptodora
kindtii and Daphnella brachyura is the only feature that gives
the otherwise almost entirely sub-arctic association of Scottish
plankton Crustacea a more southern facies. Otherwise we may
point to a very close connection between the associations of
plankton Crustacea in the Scottish and the sub-arctic lakes — a con-
nection much closer than that between the plankton Crustacea of
the Scottish lakes and of the lakes of the Central European plain
and of Switzerland. This result is only what might have been
expected, considering the situation of the Scottish lakes and the
geological structure of the country, but still it seems to me not
without interest.

With regard to the other plankton-organisms, I shall only point
out that Cortfhra plumicomis has been found by Mr James
Murray as a plankton-organism in Loch Oich, and that different
species of Hydrachnids are common in most of the lakes.

As regards the quantity of plankton in the Highland lakes, it
can only be regarded as extremely poor when compared with that
in the Danish lakes. It appears that the plankton in the larger
Highland lakes affects the transparency or colour of the water only
to a very slight extent, therefore the plankton can only slightly
influence the general conditions of life for all the other organisms
in these lakes. Only in small lakes has Mr James Murray

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1904-6.] StvAy of the Lakes of Scotland and Denmark. 427

observed the transparency and the colour of the water to be
inflaenced by the plankton. Further, I should think it is
exceptional to find in the Highland lakes a single plankton -
organism giving the entire plankton the unifoi-m monotonous
character frequently observed in our lakes due to Melosira^
Aphanizomenon, and others. And it will be easily understood
thiEit the marked changes which almost invariably take place in
our lakes when the great development of Diatoms ceases and the
maximum development of the Cyanophycea sets in are never so
conspicuous in the Scottish lakes. Finally, I am inclined to
think that many of the plankton-organisms in the Scottish lakes
show a less marked maximum and minimum development than
is the case in our lakes ; and should further explorations confirm
this supposition, the fact must be ascribed to the much lesser
amplitude in the annual variation of temperature in the Highland
lakes, where the water never attains those very low or very high
temperatures at which life in an active form, owing to the
structure of the organisms, becomes impossible ; the organisms
may therefore not be forced to form resting organs, but may
remain in the layers of water as free swimmers.

According to the observations of Mr James Murray and myself,
the seasonal variations of the plankton-organisms are never so
conspicuous in the Scottish as in the Danish lakes. I have
pointed out (1900) that in several very diflferent plankton-
organisms the longitudinal axis is simultaneously lengthened
during summer and shortened during winter, and that the
formation of all the various structures (spines, floating apparatus,
etc.) considered necessary to enable the organism to float are most
distinctly visible in summer-forms and summer-individuals. I
also pointed out that the explanation must be looked for in the
varying external conditions, which, so to speak, compel the
organisms to vary regularly in accordance therewith. I ascribed
these variations mainly to the annual changes in the specific
gravity of the water, occasioned by the regular annual fluctuations
in the temperature, starting from the supposition that if the
velocity of the falling motion of the plankton-organisms be not
the same at all seasons, the organisms must, in order to exist as
such during the season when the velocity of the falling motion is

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428 Proceedings of Royal Society of Edinburgh. [

invariably greatest, of necessity be capable of developing properties
tending to reduce the velocity of the falling motion. Knowing, as
we now do, that the spherical form in all bodies has the quickest
falling velocity, and seeing that so many organisms, with the
increasing temperature and decreasing specific gravity of the water,
often obviously became lengthened in form, the thought struck me
that very probably the seasonal variations in the specific gravity
of the water were the main factor in determining the seasonal
variations in the shape of the organisms. Subsequently Ostwald
(1903) pointed out that the lengthening of the longitudinal
axis with increase of temperature, and the shortening of the
longitudinal axis with decrease of temperature, cannot be
attributed solely to the variations in the specific gravity of the
water consequent upon the rising temperature in spring and falling
temperature in autumn ; he draws attention to the fact that the
oscillations in the specific gravity of the water with a temperature
varying from 0' to 24' C. (32' to 75" F.) are too slight to account
for these great seasonal variations in the form of the organisms.
He agrees with me in taking it for granted that these seasonal
variations in so many very different plankton-organisms can only
be due to variations in the external conditions, but he believes
them to be due to the varying viscosity of the water, which, like
the specific gravity, is dependent on the oscillations in the
temperature of the water, while the variations in viscosity are far
more perceptible than the variations in specific gravity. I think
that Ostwald's modification of my views is quite correct.

The conclusions arrived at by Ostwald and myself have been
greatly strengthened by recent observations. It is evident that if
tlie seasonal variations are occasioned by variations in the external
conditions, in accordance with the variations in the temperature of
the water, these seasonal variations must be most conspicuous in
those lakes having the most pronounced annual variations in
temperature. It has now been shown that the seasonal variations
are very conspicuous in a great many lakes in Denmark, South
Sweden, and North Germany, and many interesting facts regarding
these seasonal variations, the sinking of short-spined individuals
during the early summer months, etc. (Max Voigt, 1904, p. 113),
have been brought to light by the explorers in these countries.

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1904-5.] Study of the Lakes of Scotland and Denmark, 429

with their shallow and, in summer, warm lakes. On the other
hand, from Ekman's explorations in the northern alpine lakes in
the Sarek, we know that the seasonal variations are by no means
so conspicuous there as in the more southerly parts of Sweden.
Brehm (1902) arrives at a similar result as regards the Daphnids
in the Achensee, North Tyrol. From my own observations in the
Icelandic lakes (which will be published shortly), I know that the
seasonal variations are there extremely inconspicuous, and now
the investigation of the Scottish lakes has given the same result.
From these facts, and in accordance with the observations of
Ostwald and myself, we may conclude that the seasonal variations
are of slight importance in arctic and cold alpine lakes, while, as
might have been expected, they are conspicuous in the lakes of
the Central European plain, characterised by the great annual
variations in the temperature of the water. In this connection it
will be seen how interesting a thorough exploration of the great
tropical lakes would prove to be.

According to the published papers by the investigators of the
alpine lakes and the lakes of the European plains, it may be con-
sidered as a general rule that many animals always display more
vivid colours in the cold alpine lakes than in the warm lakes of
the plains, and that the animals retain their bright colours in the
alpine lakes throughout the year, whereas in the lakes of the
plains the vivid colouring is only observed in winter when the
temperature is low. Brehm supposes that the red colouring of
alpine organisms is a means of protection against the cold, and
gives good reasons for this supposition. The examination of the
plankton in the Scottish lakes has now shown that the Crustacea,
for instance Daphnia hyalina^ Diaptomus gracilis, Gydops strenuus,
as in other alpine lakes, are frequently in summer of a deep red
or deep blue colour ; in my own country I have only seen these
vivid colours in winter, and never in the summer months.

With regard to the vertical distribution of the plankton-organisms
iu the Scottish lakes we know very little, and further observations
on this point are necessary. It is an interesting fact that what
is known of the vertical movements of the plankton shows that
these movements are very conspicuoiis in the alpine lakes, but
inconspicuous, and often hardly traceable, in the lakes of the

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430 Proceedings of Royal Society of Edinburgh, [sbss.

plains. Seeing, however, that no thorough investigations have as
yet been carried out on this point in the lakes of the plains, or
the facts have not been sufficiently elucidated, I consider any
discussion on this subject as rather premature. Mr James Murray
has told me that at night a very great accumulation of plankton
takes place in the surface waters of the Highland lakes, and we
may therefore conclude that very conspicuous movements occur
at different times of the day and night ; in this particular also the
plankton of the Highland lakes agrees with that of other alpine lakes.
Before leaving the plankton of the Scottish lakes I wish to draw
attention to a very peculiar feature. The singular abundance of
Desmids has been already mentioned, and needs an explanation.
To suppose that the Scottish lakes should be the only known home
of an entire plankton-flora of Desmids seems to me, at first sight,
from my knowledge of fresh-water planktons, on the whole an odd
idea. I presume that the occurrence of the Desmids in the
plankton must be regarded in connection with the appearance of
a good many other organisms in the Pelagic region of the lakes ;
for instance, Polyphemun pedicidus, Sida crystcUlinOy Chydorus
iphcericuB, ClathrulinOj several Rotifers, and very many Diatoms
of the sub-orders Naviculoidece and SurirelloidecB. All these
organisms may be considered as littoral forms, washed out by the
waves from the precipitous hillsides, blown out by the wind from
the few shallow bays, and carried out into the deeper part of the
lakes by rivers and currents. Knowing that the original home of
the Desmids is in peat-moors, and that the sloping sides of the
hills in Scotland are almost everywhere covered with mosses,
which are quite moist for the greater part of the year, and in many
places all the year round, the thought immediately struck me that
the plankton Desmids must have been originally derived from the
hillsides, or from tarns and moors on the hilltops, and, associated
with the littoral species above named, have been carried by the
rivers out into the centre of the lakes. Later on, when I read
the most interesting paper of Messrs West, I observed that,
according to these gentlemen, the plankton Desmids of the
Scottish lakes "are also known to us from the bogs and rocky
pools of north-west Scotland and the Outer Hebrides" (1904,
p. 553). Further, the authors report the very interesting fact

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1904-5.] Stvdy of the Lakes of Scotland and Denmark, 431

that "the majority of the species of Staurastrum and Arthro-
desmwf which occur iu the plankton are remarkable for their long
spines, or long processes with spinate apices. Even those species
which are normally long-spined increase the length of their spines
when in the plankton" (1904, p. 554).

From the results of these thorough explorations I think we may
conclude, on the one hand, that the home of the plankton Desmids
is in fact in the pools and moss-covered sides of the hills, from
which the plankton-flora of the lakes is nowadays recruited, and,
on the other hand, that some of those forms which, according to
their primeval structure, were best adapted to plankton-life, are
now in fact, under the new conditions, about to develop those
processes (spines, etc.), common to very many exclusively plankton-
organisms, that we always regard as a floating apparatus. The
adoption of a pelagic life by the Desmids — a process really going
on as regards so many species in the Scottish lakes — may be more
easily understood when we remember that these lakes, unlike most
other large lakes, offer one of those great life-conditions which so
many of the Desmids seem to require, viz., peaty water rich in
humic acid. What I have here set forth is, of course, only a
theory, but one which may perhaps prove a starting-point for
further investigations.

3. Ths Abyssal Region, —Our knowledge of the abyssal fauna of
the Highland lakes is at the present time very deficient. Before
my arrival in Scotland, Mr James Murray had been drclging a
good deal, especially in Loch Ness. As mentioned in the Intro-
duction, opportunities were afforded me for dredging in Loch
Lochy, Loch Oich, and Loch Ness, and from a good steamer I
used all the various apparatus employed in deep-sea trawling. I
thus, of course, obtained some idea of the abyssal fauna in the
lakes mentioned, but still I consider my impressions to be altogether
insufficient, and the results at which I have arrived need in a
great measure to be tested and corrected by further explorations.

The distance from shore at which the alluvial deposits settle on
the bottom depends in the first instance, of course, upon the
declivity of the shore. As the shores of Loch Lochy and Loch
Ness are very precipitous, with depths of 300 to 500 feet only a
few hundred yards from shore, I suppose that the alluvial deposits

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432 Proceedings of Royal Society of Bdiniurgh, [sea

settle on the bottom only at remarkably great depths. It is very
difficult to dredge upon these almost vertical planes ; and in the
few instances where a dredging gave any result, I never got any
finer alluvial deposits, but only stones and gravel, upon which I
never found any sign of animal life. At the present moment we
have no knowledge of the animal life on the precipitous sides of
the lochs from 100 to about 300 feet, but I expect that further
investigations will show that it is extremely poor. Mr James
Murray has shown me samples from 300 feet in Loch Ness, con-
taining many insect larvsB, especially Perlidee, Coleoptera, and
EphemeridflB, as well as many Daphnidce and Rotifera. In the
dredgings in Loch Ness I never found these animals, and I
conclude that, especially during the spring, they will be found to
accumulate in the abyssal region. These forms must certainly be
regarded as having fallen down the precipitous sides of the
bordering hills, washed out by the waves, and carried out into
deep water. Further, I think it quite probable that the rivers,
especially after heavy rains, may be able to sweep away the
river-fauna from the rocks and carry it out into the lakes so far
from shore that it does not subside until depths exceeding 200 or
300 feet have been reached. Further observations may show
whether this littoral fauna of the great depths will be starved out,
or will be able to reach its primary home again.

I had hoped to find in the lakes of the Caledonian Canal traces
of the fauna of rehct animals, first discovered by I«oven in the
great Swedish lakes, subsequently observed in Finland, Norway,
Iceland, and North America, and in recent years also in Germany
and Denmark (1902). I had expected to find both the relicts
common in all these countries (Mysis relicta, PaUasiella qttadri-
spinosaj Pontoporeia afflnis\ and also those whose home is in very
deep and very cold water (Idothea eniomon and Gamtnarus
loricatus), hitherto recorded only from the Swedish lakes and
Lake Ladoga. It is most extraordinary that the deep fauna of
the great Swedish lakes has never been investigated since Loven
drew the attention of the entire scientific world to the existence
of marine animals in their great depths. I thought that the
sources of knowledge regarding this peculiar fauna could not have
been exhausted with Loven's discoveries, and that modem

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1904-6.] Stvdy of the Lakes of Scotland and Denmark, 433

appliances would have brought to light quite new fresh- water
organisms. I hoped, further, that the explorations might reveal
some of those species found by Forel in the deep water of the
Lake of Geneva — Niphargvs fordi, Asdlvs fordi^ Limncea
profunda and abysaicola^ etc. It will thus be understood that I
began the deep bottom dredgings with great expectations, which
were, of course, nourished by Mr James Murray's discoveries,
larvae of PerKdse and EphemeridsB never having previously been
found in the abyssal region. All my expectations, however, fell
short of realisation. While I think it necessary to emphasise the
fact that the explorations hitherto carried on have been quite
fragmentary, yet I consider it most extraordinary that with our
excellent apparatus we were unable to procure one specimen either
of the relict fauna or of the deep-water fauna taken by Forel in
the Lake of Geneva. I may add, that in the exploration of Loch
Ness I used the very same net with which I have taken the relict
fauna in our Danish lakes.

The genuine abyssal fauna of the Highland lakes appears to be
poor, consisting mainly of Chironomus larvsB, a very few species of
OligochsBta, Ostracoda, and Pisidium (probably Plagiostoma
lemani was found in Loch Ness), and the number of individuals
seemed to me inconsiderable. The microscopic abyssal fauna is im-
perfectly known; but seeing that many Rhizopods are most common
in peaty water, I think it probable that further investigations will
reveal a great many species as inhabitants of the abyssal region of
the Scottish lakes. As probably pointing to the cause of the
apparently extreme poverty of organic life in the abyssal region of
the Scottish lakes, I would draw attention to a fact well known
in our country, viz., that in all our peat-moors the animal life at
the bottom of the moors is extremely poor ; we find only a few
snails, larvae of Chironomidse, while the Oligochaeta are often
almost entirely absent, and only the Khizopods are numerous.
For my part I have always thought that this must be due to the
large amount of humic acid, which acts as poison to many
animals; and if this be the true explanation, it may indicate the
principal reason why the abyssal region of the Scottish lakes is so
thinly populated, the peaty water being a hindrance to the
development of life.

PROC. BOY. 800. EDIN. — VOL. XXV. 28

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434 Proceedings of Royal Society of JBdinburgh. [gns.

Once more calling to mind the mist- wrapped, moes-covered
Scottish hills, with their peaty moors and precipitous sides, I
think we must seek the main cause of the general extreme poverty
of animal and vegetable life in the Highland lakes in the general
geographical conditions of the country itself.

From this sketch of the organic life in the Danish and Scottish
lakes it will appear that the differences are extremely great I
suppose that what has been said with regard to the life in the
Danish lakes will hold good also as to the lakes of the northern
part of the Central European plain. On the other hand, the very
imperfect sketch I have given of the Highland lakes can by no
means be taken as applicable also to alpine lakes in general. It
would indeed have been fortunate could we have drawn a com-
parison between the Highland lakes of Scotland, their nature and
their organic life, and the Norwegian alpine lakes, many of which
are similar in some respects; but this is impossible, since the
Norwegian lakes have been very insufficiently explored, and we
can only compare the Scottish lakes with the southern alpine
lakes, especially the well-explored Swiss lakes. I may refer to the
admirable works of Forel (1892-1902), Zschokke (1900), and
others, relating to the fauna and flora of the Swiss lakes. Any-
one who has read these, and knows something of the life in the
Scottish lakes, will be aware that in every respect life is much
richer in the Swiss lakes than in the Scottish lakes.

III.

The Influence op the Organic Life upon the Lakes
themselves and their surroundings.

A. The Danish Lakes.

It stands to reason that the organic life will always exercise the
greatest influence upon the surrounding medium where the
organisms are in excess, both as regards the number of species
and the number of individuals. When we remember that
Denmark is built up of friable soil, while Scotland, on the other
hand, consists for the greater part of hard rocks, it will be
evident that the influence of organic life is far more intense, and

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1904-6.] Study of the Lakes of Scotland and Denmark. 435

consequently more conspicuous, in Denmark than in Scotland.
Every year the wide zone of vegetation which surrounds our
lakes decays in October and November, is broken up by the
waves, partly pulverised on the shore, and, as detritus, carried
out over the whole lake; the vegetation which withstood the
force of the autumn gales is frozen in the ice, and in spring,
when the ice breaks up, is scattered over the lake as leaves and
stems. The lime-crusts, derived from the blue-green Algss
covering the stones, are peeled off by the action of the ice, and as
powder carried out from the shore. As stated by Forel ( 1 89 2-1 902),
Kirchner (1896), myself (1901), and others, the blue-green Algss
uid the fauna living in the Algsd-crusts corrode the stones, so that
the stones become brittle, decay, and are pulverised. Every spring,
after the first heavy storms, we find the shores strewn with
thousands of dying snails or empty shells, which are broken up,
polverised, and as a tine lime powder, colouring the water in calm
bays a whitish-grey, are scattered over the lake ; the lime incrusta-
tions on Potamogeton and other plants will, especially in spring
and autumn, share the same fate. During these seasons the
waves reach the bottom in depths of 10 to 15 feet, and the great
Characea growths, which often cover the bottom, are uprooted,
cast on to the beach, and undergo the same process of pulverisa-
tion. The pulverised material remains in suspension in the water
for a long time, and as detritus affects the transparency of the
water, — the amount of detritus, especially in spring after heavy
gales, being very considerable. It may be added, that by no means
all the material thrown up on the beach is subjected to pulverisa-
tion, for a larger or smaller proportion is deposited in shallow bays,
and forming peat, fills them up, and thus diminishes the size of
the lake.

The huge masses of plankton will also in the course of time
reach the bottom. I have shown (1900) that we can often
detect beneath the layers of living plankton — I think below the
" sprungschicht " — layers of dead plankton, which three or four
weeks previously had been living plankton in the upper layers of
water. This dead plankton mostly consists of skeletons, and by
means of vertical hauls I have followed it on its way to the lake-
bottom. I have shown, further, that nearly all the protoplasm of

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436 Proceedings of Royal Society of Edinburgh. [sna.

the cells in the plankton is eaten away by Phycomycetes befoie
reaching the bottom : my observations prove that an organism in
the latter part of the period of maximum development may very
often be infected by Phycomycetes, which feed upon the proto
plasm and kill it, leaving the skeleton intact.

All the decayed matter derived from the plankton or from the
littoral organisms, on settling upon the bottom, will be mixed with
the inorganic material washed out by the waves from the shores
or carried by the rivers out into the lakes. In our country this
material consists mainly of lime and clay, but as yet the inorganic
constituents of our lake-bottoms have not been thoroughly studied.
The percentage of lime in our deeper lake-deposits is very variable,
but in most cases it is extremely high, often 15 to 25 per cent.,
and in the Fureso 35*30 per cent., while in other lakes it may
rise to 46*98, and even 59*44 per cent, (see my bottom explora-
tions, 1901, p. 93). We have no chemical analyses of the water
of the greater lakes, and therefore cannot speak of any deposits
due to chemical precipitations from the water of the lakes.

The rich bottom-fauna, consisting mainly of Chtronomus^
Oligochseta, Ostracoda, and Pisidium, obtains its nutriment from
the rain of organic and inorganic matter which drops down
through the water and reaches the bottom. I have studied the
Hfe of this fauna in aquaria at the fresh-water biological laboratory
at Fureso. If we take the mud from the greatest depths of our
lakes and place it in aquaria, we shall observe, after the lapse of
some days, upon the surface of the mud, elevations consisting of
granules, as well as some jelly tubes covered with mud, and sur-
rounded by similar granules. Beneath the elevations and in the
tubes we find respectively Oligochaeta and Ghironomus larvae ; we
can detect the granules being pushed out, and we know them to
be excrementa. If we take some mud from the deep lake-bottoms
and sift it through a very fine sieve we shall find enormous
quantities of these granules, and if we allow the mud to remain
sufficiently long in the aquaria the whole surface becomes con-
verted into granules, that is, into excrements. From these
observations we conclude that the upper layers of the deeper lake-
bottoms become, consequent upon the digestive action of the
fauna, converted into layers of excrements.

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1904-5.] Study of the Lakes of Scotland and Denvnark. 437

As far back as 1862 these layers were termed **gytje*' in an
admirable paper by the eminent Swedish naturalist H. v. Post, and
this term is very much used in North European and Danish
literature. V. Post distinguishes different forms of " gytje," but
we shall here only deal with the so-called " Lake-gytje," which is
formed principally in clear, limpid water. In my paper (1901) I
have pointed out that the main condition for the formation of this
" gytje " appears to be, that no greater quantities of organic matter
be precipitated than the bottom-fauna and the bacteria coi^jointly
may be capable of digesting. If the supply of organic matter be
superabundant, black fetid mud-formations (river-deltas, common
sewers, etc.) result, while, on the other hand, where the organic
matter, owing to the presence of humic acid, remains undecayed
and is preserved, peat is formed. Owing to the digestive processes,
the excrements are generally of a lighter colour than that of the
lake-bottom itself. This might be accounted for by supposing that
the animals of the upper layers feed mainly on the organic dark-
coloured debris, allowing the inorganic matter, which in our lakes
consists especially of lime and clay, to pass through their ali-
mentary canals. By means of bore samples from shallow lakes I
have shown that the colour of the lake-bottom grows lighter the
deeper we go down ; it may be greyish-white 4 feet beneath a surface
which is often quite black. I am of opinion that layers of almost
pure lime or clay — so-called coprogenic lime and clay layers —
may result from the digestive action of the bottom fauna and flora.

With regard to the process of formation, these layers are not
identical with those layers of clay which, during and immediately
after the Ice Age, were formed on the primary sandy bottoms of
our lakes, and were one of the first conditions for the development
of a higher and more specialised organic life in the lakes. Nowa-
days, in all our lakes, and probably in many of the lakes of the
Central European plains, the precipitation of organic matter —
debris from the littoral zone as well as plankton — is very copious.
In all our deeper lakes it is mainly the plankton which determines
the composition of the lake-gytje ; and as the plankton varies in
the different lakes, it will be understood that the lake-gytjes
consequently also differ from each other.

In our lakes I have been able to distinguish three different

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438 Proceedings of Royal Society of Edinburgh. [ant.

forms of lake-gytje, viz., Diatom-gytje, Cyanophyceagytje, and
Chitin-gytje. The firat-named, which occurs mostly in the colder
lakes, contains enormous quantities of plankton Diatom frustoleB,
and may consist almost exclusively of these ; skeletons of bottom
Diatoms are very rare. From gytjes of this composition the
Diatom clay may arise. According to Forel it seems that Uie
Diatom skeletons in deeper lake-bottoms may be dissolved and
disappear, but this is not the case in our shallower lakes. The
Cyanophycea-gytje is a black, fetid substance, consisting of decaying
plankton Cyanophycea, and mostly occurs in warm shallow lakes.
The Chitin-gytje contains enormous quantities of the valves of
Daphnids, and is generally formed in small lakes devoid of
Cyanophycea. Lately, Holmboe (1903) has found Diatom-gytje as
well as Chitin-gytje fossil in Norwegian peat-moors.

The constituents of the lake-gytje are not the same all over the
lake-floor, notable differences being recognisable on the two sides
of the 30-feet contour-line. Outside this contour we hardly ever
find stems, shells, and Mollusca (except Pisidtum), and veiy
seldom leaves, the deposit nearly always consisting of fine mud.
On the other hand, inside the 30-feet contour we often find the
whole bottom strewn with shells ; leaves and stems are common, and
the deposit is much coarser in texture, often containing considerable
quantities of sand and gravel, which are rarely found outside the
30-feet contour. As already stated, the material inside the 30-feet
contour is either deposited, and forms, for example, peat, or is, sooner
or later, pulverised by the action of the waves dashing it against
the stones and sandy bays of the beach ; hand in hand with this
mechanical action a chemical process goes on, especially as regards
the lime deposits. A close study of the mollusc shells from the
shore and shallow water shows a very conspicuous corrosion, caused
by different factors. On this point I may refer to my bottom
explorations (1901, p. 152), and would here only observe that
hitherto the corrosion of the shells of living animals has been
studied chiefly as a conchological curiosity, witliout full appreciation
of the fact that the corroding influences are nature's principal
instruments in the pulverisation and dissolution of lime secreted
by organisms The process of pulverisation and dissolution of all
the waste material inside the 30-feet contour is greatly accelerated

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1904-5.] Stvdy of the Lakes of Scotland and Denmark. 439

by the operations of the abundant littoral fauna, which feeds alike
on the living vegetation and on the decayed matter ; a large part
of these passes through the alimentary canals of animals, and is
transformed into excrementa. The animals which cause this
transformation are not the same as those found in deeper water,
but consist mostly of insect larvaB and molluscs; very often we
find the bottom covered with long greyish-white excrements of
snails, especially lAmnaea aurictdariOy ampla, and ovata.

In our lakes the space between the 16-feet and 30-feet contours
is marked by a remarkable and often very conspicuous elevation
of the bottom. Explorations show that in two of the lakes at
some distance from shore a series of banks occur, consisting chiefly
of mollusc shells embedded in a bluish-grey lake-marl. There is
no doubt that the molluscs here act as reef -forming factors, and
it will be understood that in our lakes the molluscs must act as
such. In the Danish lakes molluscan life (except Pisidium) does
not extend beyond the 30-feet contour. The shells in the vegeta-
tion zone are in great measure dissolved or pulverised by the
powerful action of the various erosive agencies of this zone. In
the zone occupying the space between the vegetation zone and the
outer limit of molluscan life on the lake-floor the erosive power
of these agencies is much diminished, and in the deeper part of
the zone almost niL In the tranquil water here the accumulation
of shells may go on undisturbed by the grinding and dissolving
forces, and thus banks of mollusc shells are formed. These banks
consist of the shells of those mollusca which can live outside the
vegetation zone, especially VcUvata piscinalis, Bithynia^ Anodonta,
and Unto, but only to a slight extent of the shells of Limna-a
and PlanorfnSj which live mostly in the vegetation zone. The
accumulation of shells in the " shell-zone " is often enormous, and
apparently there is a striking disproportion between the large
amount of empty shells and the relatively few specimens of
living molluscs ; yet it must be remembered that vast accumula-
tions of shells may result from a slow process of deposition during
long periods of time, as from a more rapid deposition during a
shorter period.

Inside the shell-zone and closer to the shore we often find more
local and very peculiar formations, among which may be mentioned

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440 Proceedings of Royal Society of Edinburgh. [:

the great lime-deposits, consisting solely of lime-incrustations
formed by the Characea, composed of very conspicuous broken
stems and leaves. These lime-deposits, in which the percentage
of lime may amount to 88*50, are dug out of the lakes by
machinery and used as manure on the fields.

In other locaKties within the 30-feet contour a high percentage
of lime is found, but very often it is impossible to discover from
what source the lime originates. In our lakes we often find lime-
incrustations upon other plants besides Characea, especially
Potamogeton, Elodea, etc. In studying these lime-incrustations
(1901) I arrived at the following result : — In clear, calm weather
the lime accumulates in thick flakes on the leaves and stems of
Potamogeton^ etc. ; in stormy weather it is swept off by wave
action. The precipitation of lime upon the leaves probably goes
on unceasingly during assimilation ; and the leaves not being able
to carry the full weight of the lime, broken particles are con-
tinually dropping off", which sink to the bottom at a greater or less
distance from the plant. In order to show, as far as practicable,
that the precipitations of lime from Potamogeton and Eiodea play
a prominent part in the formation of lake-lime, two bottom-
samples were taken in the Furesci ; one from a bed of Potamogeton
lucensy the other from a depth of 100 feet, the former containing
72*41 per cent., the latter 35*30 per cent, of lime. On separately
weighing the dried leaves of P, lucens and their coatings, it
appeared that a leaf often carried more lime than its own weight ;
one leaf weighing 0*35 gram carried no less than 4*1 grams of
lime. As one plant has often some thirty leaves, it will be easily
understood that the percentage of lime on the lake-floor beneath
the dense growths of Potamogeton may be considerably raised by
means of the constant rain of lime-powder dropping down from
the leaves.

Other local formations are the often extensive layers of peat
arising from the decaying- vegetation along the protected shores
and in the shallow bays, often bordered by abundant growths of
Phragmites and Scirptts, In the shell-zone lime-deposits likewise
occur, abounding in mollusc shells ; and in certain lakes these
shells are transformed into limonite, so that considerable layers of
"bohnenerz" have been formed; on this point I may refer to

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1904-5.] Study of the Lakes of Scotland and Denmark. 441

my bottom explorations, where such transformations are figured
(1901, p. 159, tab. iii.).

The preceding pages will have shown to what a large extent the
organic life of a lake may influence the lake itself and its environs.
We observe the vegetation of the littoral zone being transformed
into peat, or in other localities being pulverised, and as detritus
scattered over the lake, reducing the transparency of the water,
and ultimately find it on the deeper lake-floor, constituting a part
of the general precipitation. We see the blue-green Algae of the
shore corroding the stones, reducing them in size, and the Algse-
crusts in turn broken off and pulverised by the ice. We are able
to follow the accumulation, as well as the pulverisation, of shells
near the shore, and to see the white powder colouring the water
a greyish-white. We observe whole layers of lime (often several
feet thick) arising from the precipitated stems and leaves of
Characea, and are also able to show that the percentage of lime
on the bottom is raised by the lime dropping down from the great
leaves of Potamogeton, We see the huge plankton masses deter-
mining the colour of the water, affecting the quality of the air
contained in the water, causing accumulations of gases unfit for
the respiration of animals, and greatly reducing the transparency
of the water. We are able to recognise the once-living plankton
as skeletons in the deeper layers of water, and to show how the
nature of the lake-bottom is mainly determined by the character
of the plankton, and, furthermore, that whole layers are derived
from the accumulation of Diatom skeletons. We also note how
the different precipitations are eaten by the bottom-fauna and
converted into excrementa, and that the excremental processes
result in layers having a lesser amount of organic matter and a
greater amount of inorganic matter than if the precipitations had
not been subjected to the digestive action of the bottom-fauna.

B. The Scottish Lakes.

As the result of my investigations on the Danish lakes, I have
dwelt at some length upon the manner in which the fauna and
flora influence and react upon the general character of the lakes
themselves, thereby transforming the conditions of life common
to all organisms in the lakes and their surroundings. From the

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442 Proceedings of Royal Society of Edinburgh. [sbbs.

impressions I formed of the Scottish lakes, I shall next endeavow
to show how the organic life here also influences the lakes and
their environs. I have, of course, seen too little of Scotland to
be able to do so as satisfactorily and exhaustively as I should wish.
From what I did see, I gathered that, owing to the extreme
paucity of organic life and the hardness of the soil, as well as the
lesser amplitude of the variations in the temperature of the
water, the intensity of all those processes due to the influence of
organic life is much less marked than in the Danish lakes.

As a zone of higher vegetation in the larger Highland lakes is
almost entirely wanting, peat formation along the shores is almost
out of the question ; only a very small amount of organic material
from the shores is scattered over the lakes, in the form of
detritus, diminishing the transparency of the water. The stones,
as far as I am aware, are never covered with lime-incrustations
derived from blue-green Algse ; the Potamogetons, etc., are never
seen covered with lime-crusts ; and the shells of mussels or snails
never abound in such quantities on the beach that their pulverised
fragments, in the shape of lime-powder, are scattered over the
lakes, or influence the percentage of lime in the water or in the
deposits on the lake-floor. The amount of plankton in the larger
Highland lakes is never or very rarely so great as to aflect the
colour of the water in any notable degree ; most probably it may
aflect, to a relatively slight extent, the transparency, and the
amount and quality of the air in the water.

From my studies of the deposits in Loch Ness, Loch Oich, and
Loch Lochy, I suppose that the precipitation of decayed or
decaying matter derived from the plankton is very insignificant
Of course, I never found any great quantities of blue-green mud
derived from blue-green plankton AlgsB, but even the chitinous
valves of Daphnids are rare. Li Loch Lochy, at a depth of 500
feet, I most frequently found the carapaces with long antennae of
Bosmina. A most remarkable and interesting thing is that the
frustules of Diatoms, as in the Swiss lakes, are comparatively
rare, and the skeletons that do occur are, to my knowledge, only
those of bottom and shore Diatoms, the plankton Diatoms being
almost entirely absent. It has long been an enigma to me why
the skeletons of the plankton Diatoms accumulate on the bottom in

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1904-6.] Stvdy of the Lakes of Scotland and Denmark, 443

OUT lakes at 120 feet, while in the certainly much deeper alpine
lakes they always appear to he dissolved before reaching the
bottom. I can hardly imagine that the solution in the alpine
lakes is solely due to the greater depth, because of which the
deposition would occupy a longer period of time. On becoming
acquainted with the plankton Diatoms of the Scottish lakes, it
struck me that the Diatoms in nearly all alpine lakes are the
thin-shelled Gyclotella, AsterioneUOy and FragUaria, Of these
the two last-mentioned are also common in our lakes, but there
also their skeletons never produce Diatom-ooze ; in many hundreds
of samples I have observed very few frustules of these forms, and
I suppose that in the Danish lakes also they are dissolved before
sedimentation. The Diatom-ooze in our lakes is produced by
thick-shelled plankton Diatoms (Mdonra, Stephanodiscus attrcea^
etc.), species which are rare in the plankton of the alpine lakes, but
still occurring in the littoral zone. Provisionally, I am inclined
to believe that the formation of plankton Diatom-ooze in our
lakes may perhaps be explained by the presence of thick-shelled
Diatoms in the plankton. The circulation of silicates in the
lakes is a study of the greatest interest, and one regarding which
we know very little.

I think it very probable that a future more exhaustive explora-
tion wiU only further prove that the precipitation of organic
matter derived from the littoral zone and plankton in the Scottish
lakes is only relatively small. The greater part of the organic
matter ultimately reaching the bottom in a more or less pulverised
state is, as far as I can make out, derived from the tops and sides
of the mountains, carried into the lakes by the rivers. In the
preceding page^ I have made it my object to point out that,
according to my view, the organic life in the Scottish lake«, both as
regards the littoral faunct^ the bottom fauna^ and the plankton^ to a
very considerable extent likeioise originally belonged to the adjoining
country^ and not to the lake itself. Regarding the deposits on the
lake-floor^ we shall arrive at a similar conclusion. With us it is a
common rule that the deposits already at about 50 feet mainly
consist of fine mud, mingled with very few stems, shells, or leaves.
When dredging at 300 feet in Loch Ness I was greatly surprised
to find the bottom mainly consisting of very coarse material, mixed

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444 Proceedings of Royal Society of Edinburgh, [skss.

with large stems, leaves, etc. ; it was only at about 500 feet that I
found the deposits to be as finely pulverised as at about 50 feet in
the Danish lakes. This phenomenon is easily accounted for — in
our lakes everything in the shallow water between the shore and
the 30-feet contour is pulverised by the dash of the waves,
whereas in the Scottish lakes, owing to the precipitous hillsides,
everything is carried away from shore by the rivers and waves,
and subsides in depths of 200 to 300 feet, without being exposed
to the eroding force of the waves on a shallow coast.

From all the bottom-samples I have seen it appears that the
deposition of organic matter is not nearly so abundant as in the
Danish lakes, the deposits consisting principally of inorganic
materials; there is further a total absence of lime — ^a very con-
spicuous difference between the lake-bottoms in the two countries.
Further observations may show in what manner the bottom fauna
deals with the deposited material, and the changes to which this
material is, in consequence, subjected; £ cannot but think that
here also layers of " gytje " are being formed.

I suppose that most of the observations on the influence of
organic life upon the general conditions of the lakes and their
surroundings in our own country will hold good also with regard
to most of the lakes in the southern part of Sweden and in the
northern part of Germany ; my investigations of the Danish lake-
gytjes are in accordance with v. Post*s explorations of Swedish
gytjes, and most of my observations with regard to the lime-
deposits have been corroborated by Passarge. The explorations
of Halbfass among the lakes of Pomerania show that the natural
conditions of those lakes are very similar to those of our own.

My Visit to thb Lowland Lakes.

Subsequent to my examination of the Highland lakes, I visited
some lakes in the Lowlands, as well as some smaller lakes near
Edinburgh, including Loch Leven — famous for its excellent trout.
These lakes presented many points of similarity with those of our
own country. 1 found in Loch Leven the same gently sloping
shores, a very slight transparency of the water, and a considerable
amount of detritus ; the mud was very fine, and the large amount

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1904-5.] Study of the Lakes of Scotland and Denmark. 445

of organic matter, on the whole, very similar to that at the hottom
of our lakes. The organic life also has some resemblance to that of
the Danish lakes, but still I noticed some very striking differences.
The band of vegetation visible above water was narrow, but the
evenly sloping sandy shores, especially along the north-east coast,
were covered with dense growths of Characea : strangely enough,
in the deepest parts of the lake, in depths of about 80 feet, I
found the mud covered with long filaments of blue -green Algse.
From the explorations of the Lake Survey (1901a, p. 124) we
know that the mud contains no carbonate of lime. The animal
life has been studied by Mr T. Scott, to whose paper I refer.
The molluscan life I found to be much richer than that in
the Highland lakes, but still by no means so rich as with us.
Limntjea and Planorbis were, both as regards species and in-
dividuals, relatively few in number ; only in the Characea-growths
were there great quantities of Valvataj and in the bottom-mud
Sphoeriumy Pisidium^ and Anodonta abounded. The Crustacea,
especially the Cladocera, were represented by numerous species,
and in the Characea-growths the animal life was extremely rich.

The quantity of plankton was enormous : I do not remember to
have seen, even in our lakes, such huge masses of Leptodora, The
plankton, at the time I visited the lake, consisted chiefly of this
Daphnid, with Cyclops gtrenuus and other Entomostraca. The
phytoplankton was less conspicuous, Anabamaflos aqtue being the
most predominant, and it might have formed " wasserbliithe."

General Conclusions.

It will easily be understood that where the alluvial deposits in
shallow lakes are as copious as in our country, the lakes will in
the course of time become silted up and overgrown, and will
finally disappear. When looking at old maps and when studying
nature we meet with traces of numerous former lakes. Many have
been drained by man and converted into arable land, but yet in
such cases man has only forestalled what nature would have
accomplished in a relatively short period of time. All our lakes
were formerly much larger, and their form and coast-lines far more
irregular, the bays having in many cases been silted up, and at the

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446 Proceedings of Boyal Society of Edinburgh, [s

end of the more elongated lakes we generally find more or lese
extensive marshy ground. Many of our existing lakes are
apparently doomed, and it is difficult to imagine how in our
country new lakes could be formed.

There can be little doubt that in Scotland the coast-lines of the
lakes have altered very little during thousands of years, and that
the lakes themselves will remain through long ages.

List of Literature.

1902. Brehm, v., " Zusammensetzung, Verteilung und Periodi-
citat des Zooplankton im Achensee," Zeitschr. d, Ferdinandeums,
Bd. 46, p. 1.

1904. Ekman, S., "Die PhyUopoden, Gladoceren und frei-
lebenden Copepoden der nord-schwedischen Hochgebirge," 2jOoL
Jahrb., Bd. 21, Abth. Syst, p. 1.

1892-1902. Forel, F. A., "Le L^man," Monographie linmo-
logif^f t. 1-3, Lausanne.

1901. Forel, F. A., "Etude thermique des lacs du Nord de
TEurope," Arch, des set. phys. et natur,^ s^r. 4, t. 12, p. 35.

1901. Geikie, a.. The Scenery of Scotland. London.

1901. Halbpass, W., " Beitrage zur Kenntniss der Pommerschen
Seen," Petermann^s MitteUungen^ Erganzungsheft Nr. 136, Gotha.

1903. HoLMBOB, J., " Planterester i norske torvmyrer," Videns-
kab, Selsk. Skri/ter Ghriatianiay 1903, Math, naturv. Klasse,
No. 2.

1896. Kirchner, D., in Schroter und Kirchner, "Die Vegeta-
tion des Bodensees," Schriften d, Vereins fur GreschicfUe des
Bodensees und seiner Umgebung, Lindau, I. 1896, II. 1902.

1900. Murray, Sir John, and Pullar, F. P., "A Bathymetrical
Survey of the Fresh- water Lochs of Scotland," Part I., Oeogr,
Joum,j vol. XV. p. 309.

1901a. Murray, Sir John, and Pullar, F. P., ibid., Part II.,
ibid., vol. xvii, p. 273.

1901b. Murray, Sir John, and Pullar, F. P., ibid., Part III.,
No. 1. ibid., p. 289.

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1904-6.] Study of the Lakes of Scotland and Denmark. 447

1903a. Murray, Sir John, and Pullar, L., ibid.. Part III.,
Nob. 2-6, ibid., vol. xxii. p. 237.

1903b. Murray, Sir John, and Pullar, L., ibid.. Part III.,
No6. 7-9, ibid., p. 621.

1904a. Murray, Sir John, and Pullar, L., ibid., Part III.,
No. 10, ibid.y vol. xxiii. p. 32.

1904b. Murray, Sir John, and Pullar, L., ibid., Part IV.,
ibid., p. 444.

1903. Ostwald, W., " tjber eine neue theoretische Betrach-
tungsweise in der Plank tologie," Forschungsber. aus der biolog.
Station zu Pl<m, T. x. p. 1.

1862. Post, H. v., ** Studier ofver Nutidens koprogena Jord-
bildningar : Oyttja, Dy, Torf och Mylla," KongL Svenska Vetena-
kaps-ako'I., Handl. N.F., Bd. 4, No. 1.

1890-1899. Scott, Th., "The Invertebrate Fauna of the
Inland Waters of Scotland," Parts I.-IX. Annual Reports of
the Fishery Board for Scotland.

1893. Scott, Th., " On some Entomostraca from Castlemilk,
near Kutherglen," Trans. Nat. Hist. Sac. Glasgoio, vol. iv. (N.S.)
p. 69.

1892-94. Scott, Th., *-The Land and Fresh-water Crustacea
of the District around Edinburgh : I. Amphipoda, Isopoda ; II.
Ostracoda and Copepoda; III. Cladocera," Proc. Roy. Phya.
Soc. Edinimrghf vol. xi. p. 73 ; vol. xii. pp. 45, 362.

1899. Scott, Th., " Some Notes on the Fresh- water Entomos-
traca of Aberdeenshire," Annah of Scottish Nat. Hist., 1899, p. 216.

1901-2. Scott, Th., "Notes on some Fresh- and Brackish-
water Entomostraca found in Aberdeenshire," ibid., 1901, p. 157 ;
1902, p. 21.

1903. Scott, Th., "Some Observations on British Fresh-water
Harpactids,'* Ann. Mag. Nat. Hist., ser. 7, vol. xi. p. 185.

1903a. Scourpibld, D. J., " Synopsis of the Known Species of
British Fresh-water Entomostraca. Part I. Cladocera," Joum.
Quekett Micr. Club, ser. 2, vol. viii. p. 431.

1903b. Scourpikld, D. J., ibid.. Part II. Copepoda, ibid., ser. 2,
vol. viii. p. 531.

1904. Scourpibld, D. J., ibid.. Part III. Ostracoda, Phyllopoda^
and Branchiura, ibid., ser. 2, vol. ix. p. 29.

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448 ProceediTigs of Royal Society of JSdinburgh. [sess.

1901. Steuer, A., "Die Entomostracenfauiia der *alteii Donau'
bei Wien," Zoolog. Jahrb., Bd. 15, Abth. Syst, p. 1.

1904. UssiNG, N., ** Danmarks Geologi i almenfatteligt Om-
rids," Danmarks geologuke Undersdgdse, III. R. Nr. 2, Kjbbenhavn.

1904. VoiGT, M., ** Die Rotatorien und Gastrotrichen der
Umgebimg von Plon," Fffrschungsber. aus der biolog, StcUion zu
Plan, T. 11, p. 1.

1895. Warming, E., *• Plantesamfund," Grundirosk af den
okoloffiske Plantegeograji, Kjbbenhavn.

1904. West, W., and West, G. S., "Scottish Fresh-water
Plankton, No. I.," Joum. Linn, Soc, Botany, vol. xxxv. p. 519.

1900. Wbsbnberg-Lund, C, "Von dem Abhangigkeitsver-
haltnis zwischen dem Bau der Planktonorganismen und dem
epezifischen Gewicht des Siisswassers," Biolog. Centralbl., Bd. 20,
pp. 606, 644.

1901. Wesenbbrg-Lund, C., " Studier over Sbkalk, Bbnnemalm
og Sbgytje i danske Indsoer," with summary of contents, Meddd.
fra dansk geol. Foren, Kjbbenhavn, Bd. 7, p. 1.

1902. Wesenbbrg-Lund, C., " Sur Texistence d'une faune relicte
dans le lac de Furesb," Kong, Danske Videnskab. Sdsk. Forhand-
linger, 1902, p. 257.

1904. Wesenbbrg-Lund, C, "Plankton Investigations of the
Danish Lakes," Danish Fresh-water Biological Laboratory, Op. 5,
Copenhagen.

1900. ZscHOKKB, F., "Die Tierwelt der Hochgebirgseen,"
Neue Denkschr. d, Schweh, naturf. Oes., Ziirich, Bd. 37, p. 1.

{Ismed separately March 8, 1906.)

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Proc Roy, Soc. of Min . ] [ V o L. X X V .

Plate I.

Fig. 1. — Ice erosion on the shores of the Fureso.
(Photo by Dr C. Wesenberg-Lund.)

Fir.. 2. — Furesii with its zones of PJir<y(fiiiffrft and Srirpua.
(Plioto, V>y Dr C. Wcsen berg- Lund.)

Dii VVe8p:nbek(;-Lund

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Proc. Roil. Sue. ofEdin.] [Vol. XXV.

Plate II.

Fig. 3.— Loch Xess from Borluni, looking north-east.
(Photo, by MrG. West.)

Flo. 4.— Loch Killiii (near Loch Ness), looking north, showing steep
escarpment on the \vest<^rn shore,
(Photo, by MrG. West.)
Dr WesenbkkgLund.

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1904-5.] Crystallisation of Potassium Hydrogen Siuxinaie. 449

Variations in the Crystallisation of Potassium Hydrogen
Succinate due to the presence of other metctUic
compounds in the Solution. {Preliminary Notice,) By
Alexander T. Cameron, M.A. Communicated by Dr
Hugh Marshall, F.R.S.

(MS. received January 9, 1905. Read January 23, 1905.)

In the summer of 1902, while working as a student in the
Chemistry Department of Edinburgh University, I prepared a
quantity of potassium chromoxalate (Gregory's salt) as an ordinary
exercise, and this led me to attempt the preparation of a similar
derivative of succinic acid, since such derivatives apparently had
not been obtained.

For this purpose a solution of potassium hydrogen succinate
(prepared by half - neutralising succinic acid with potassium
carbonate) was boiled' for some time with freshly precipitated
chromic hydroxide (prepared by adding ammonia to a boiling
solution of chrome alum, filtering, and washing thoroughly). The
undissolved hydroxide was filtered off, and the filtrate subjected
to the same treatment with fresh chromic hydroxide ; the whole
process was repeated two or three times, the final filtrate being
dark green in colour. A portion of this solution was evaporated
to small bulk by boiling ; on cooling, potassium hydrogen succinate
first crystallised out, and then a green crystalline powder was
obtained. The remainder of the solution was concentrated only
to about half its volume and allowed to stand for three days ; at
the end of that time dark green crystals were deposited. These
showed the striking pecuHarity of being bounded only by curved
surfaces, plane faces being entirely absent ; from their shape they
might be described as obhque elliptical double cones, possessing
monoclinic symmetry (plane of symmetry with digonal axis normal
to it). A perfect cleavage, yielding highly lustrous faces, was
observed parallel to the plane of symmetry, and the parallelogram

PROC. ROY. 800. EDIN. — VOL. XXV. 29

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460 Proceedings of Boyal Society of Edinbv/rgh, [i

formed by the outline of the cleavage face had an obtuse angle
of about 135'.

Until recently I was unable to continue the investigation, but,
owing to the publication of a paper by Werner on " The Behaviour
of Chromic Hydroxide towards Oxalic Acid and certain other
Organic Acids" (7. Ch&nu Soc,, 1904, 85, p. 1438), I have con-
sidered it desirable to publish a preliminary note, although the
results so far obtained can only be stated generally.

Several preparations have been made similar to that described
above, and the crystalline products analysed for chromium. The
percentage varies considerably in the different preparations, and as
yet it is impossible to state what is the maximum, but specimens
hitherto analysed show considerably less than 1 per cent. Since
potassium hydrogen succinate crystallises in monoclinic crystals
possessing a plane of symmetry and showing perfect cleavage
faces parallel to it, the small amount and the fluctuation in that
amount of chromium present in these crystals lead to the
assumption that they are potassium hydrogen succinate, the
external surfaces being modified by the presence of some chromium
compound, possibly in solid solution.

Attempts have also been made to dissolve other hydroxides and
certain carbonates in potassium hydrogen succinate solution.

When copper carbonate was taken a precipitate of copper
succinate was first produced ; the filtrate from this was coloured
slightly blue, and after standing for some time deposited crystals
of the acid succinate. These showed six-sided prism faces, and
also, superimposed on these, curved faces similar to those observed
with chromic hydroxide.

Crystals showing traces of these curved faces have been obtained
from solutions in which aluminium hydroxide had been dissolved.

Equal quantities of fairly concentrated solutions of potassium
hydrogen succinate and ferric chloride (containing a few drops of
hydrochloric acid) were mixed, and in a few minutes a brick-red
precipitate appeared. The solution and precipitate were boiled
with another equal quantity of potassium hydrogen succinate, the
precipitate filtered off, and the filtrate, which was slightly yellow
in colour, set aside to crystallise. At the end of three weeks pale
yellow cryst^s were found at the bottom of the crystallising

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1004-6.] CrystaUisatifm of Potassium H^i/droffen Siuxinate. 461

dish, elliptical in form, and growing in towards the centre. Their
appearance was that of truncated cones. They were removed, and
a month later a single elliptical biconical crystal was obtained ; it
was brownish-yellow in colour, and resembled those obtained with
chromic hydroxide, but was much more perfect in form.

Chromic hydroxide dissolves in potassium hydrogen malate
much more readily than in the corresponding succinate, and gives
finally a very dark green solution, from which crystals similar to
those already described have been obtained.

I am continuing the investigation, and hope to be able to publish
a detailed examination of these crystals at an early date.

Chemical Laboratort,
SuBOBONs' Hall, Edinburgh*

{Issued separately February 4, 1905.)

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452 Proceedings of Royal Society of Edinburgh, [i

A Laboratory Apparatus for Measuring the Lateral
Strains in Tension and Compression Members, with
some Applications to the Measurement of the
Elastic Constants of Metals. By E. G. Coker, M.A.
(Cantab.), D.Sc. (Edin.), F.R.S.E., Professor of Mechanical
Engineering and Applied Mathematics, City and Guilds
Technical College, Finsbury, London. (With a Plate.)

(MS. received October 26, 1904. Read November 21, 1904.)

The recognition of the imi)ortance of lateral strain in the theory
of elasticity, as now taught in most engineering coUeges, makes
it very desirable that students should make experiments upon the
Ikteral contraction of tension specimens and the lateral extension
of compression pieces with the same facility that they now deter-
mine the values of Young's modulus and the modulus of shear.

With this purpose in view, the author designed an instrument
which has been very thoroughly tested by student-use for the
past two years in the testing laboratory of M'Gill University.

For the object in view it was necessary to make an apparatus
of simple construction, easily operated and understood, and
capable of standing a considerable amount of wear and tear without
injury, while at the same time it must be capable of measuring
with accuracy linear strains of the order of ^^y.^nnr ^^ *^ itic^i.

After some minor alterations, an apparatus was constructed
which fulfilled these requirements.

The instrument is shown in sectional elevation by fig. 1, and
in part sectional plan by fig. 2, and it consbts essentially of
a pair of tubular arms A^ connected by a flexible steel plate B^
which forms the fulcrum. This plate is very thin, in order to
allow the arms to turn in the plane passing through their axes,
and is very deep, to give the necessary rigidity perpendicular to the
plane of motion, and thereby ensure that the arms have no other
motion. The plate is gripped by a pair of collars (7, mounted on the
arms A^ and provided with grooved ends and tightening screws.

The instrument is attached to the specimen by a pair of screws
Z>,. threaded through nuts formed on the arms and provided

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1904-5.] Apparatus for Measuring Zaieral Strains.

iSZ

'with lock nuts, and the pressure of the screw points on the
specimen is regulated by a spring threaded over a hollow spindle
^ pivoted to one arm, this spindle being guided by a second, F,
pivoted to the other arm: the compression of the spring is
regulated by a nut G upon the outer spindle.

FHiyu^re 1

The free ends of the tubes are prolonged beyond the screw
grips, and one of them is fitted with an ebony finger H, having
a thin steel plate 1 secured to its outer end, which presses against
a double knife-edge J, seated in a shallow V-notch cut in the end
of the other arm.

This knife-edge carries a mirror K pivoted upon a vertical
spindle, and capable of adjustment about an horizontal axis alsa
An adjusting screw Z, secured in one of the collars, bears against
the specimen, and keeps the instrument from swinging round on
the points of the screws.

With this arrangement any alteration in the diameter of the
specimen between the screw points causes a movement of the
outer end of one arm relatively to the other, and a proportional
rotation of the knife-edge and its attached mirror is obtained.
This rotation is observed by a telescope and scale placed at a
Convenient distance away, and a measure of the change in the

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454 ' Proceedings of Royal Society of Edinburgh, [sim.

diameter of the specimen is thus obtained. A photograph of
the apparatus is shown in fig. 3 mounted upon a tension specimen.

The value of a unit of the scale was obtained by calibrating the
instrument bj a Whitworth measuring machine, and it was found
that, with the scale 24*8 inches distant from the mirror, one
division of the scale corresponded to one-millionth of an inch.

At first some minor difficulties were experienced owing to the
longer branches of the tubes being insufficiently rigid, and they
were therefore trussed, with good effect, and afterwards pieces of
hard wood, of square section, were forced down the tubes;
this overcame, the difficulty completely. As the instrument was
wholly of brass, some difficulty was experienced owing to small
changes of temperature in the laboratory, which sometimes altered
the zero of the instrument during a test ; this error was guarded
against by lagging with chamois leather.

The instrument, when used in conjunction with an apparatus
for measuring longitudinal strain, gives a measure of Poisson's

ratio — if the material fulfils the conditions assumed by the theory
m

of elasticity ; and knowing the value of Young's modulus £, we
can easily calculate the modulus of shear C and the bulk modulus
D from the formulflB

C = l-?^ E
2m+l

3 m-2
As an example of this we may quote a test of a piece of
machinery steel in tension, when the lateral extensometer above
described and a Ewing extensometer were secured to the specimen.
The experiment gave the following results : —
Steel specimen I'Ol inches in diameter.
Length under test 8*00 inches.

Ewing Extensometer, one division = 77.^17 of an inch.
Lateral Extensometer, one division = T,insh,wuT5 ^^ ®^ ^^^•
The accompanying table of observations (page 455) shows that
the mean longitudinal strain per unit of length is '0000825 inches,
and the mean lateral strain -0000206, corresponding to a value
for m of 4*01, and the value of E, obtained in the usual manner, is
30,250,000, the units being pounds and inches.

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1904-6.] Apparatutfor Measuring Lateral Strains.

455

LoDgitadinal

Strain.

Lateral Strain.

Load
Poands.

Reading.

Reading. A |

1,000

0 ,

-84

-20 '

8,000

-82

-22 ,

6,000

»3

-21

7,000

-83

-22

9.000

132

-33

-21

11,000

165

106 '

0 i

The values of C and D are respectively 12,378,000 and
20,603,000, with the same units.

As a further example we may quote the case of a wrought-iron
bar in tension, having a diameter of 1 inch, the test being similar
to the one previously described. The readings obtained were —

Load
Pounds.

1,000
8,000
5,000
7,000
9,000
11,000
9,000
7,000
6,000
8,000
1,000

Longitudinal Strain.

Lateral Strain.

Reading. A

-87

-25

-85

-25

-34

-25

106

-87

-24

143

-36

-25

179

124

-86

-22

148

102

-86

-24 '

108

-36

-24

-85

-25

-37

-25

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456 Proceedings of Roycd Society of Edinburgh, [sess.

And from these readings we derive the following values : —

m = 3-64
E = 28,450,000
0=11,160,000
D = 21,048,000

Other metals were also experimented upon, and in some cases
under compression, when the longitudinal strain was measured by
a compressometer of Professor Swing's design. It will be suflB-
cient to quote the results of these experiments without the detailed
observations, which present no peculiarity except that, in the
cases of cast-iron, brass, and copper, the stress strain curve for a
complete cycle of stress was a very narrow loop. In these cases
the mean value of the strains for the whole range of stress was
taken for calculating the values of the constants. The results,
including the tests above cited, were as follows : —

Tension Experiments,

Specimen.

E '

D !

Machinery Steel, .

4-01

30,260,000

12,378,000

20,608,000

Wrought- Iron,

3-64

28,450,000

11,160,000

21,048,000 1

Rolled- Brass,

3-10

14,700,000

6,667,000

18,809,000

Rolled-Copper,

3 02

10.100,000

8,794,000

9,640,000 1

Compression Experiment

Specimen.

Machinery Steel, .

4-09

29,600,000

11,891,000

19.310,000

Wrought- Iron,

3-58

28,100,000

11,000,000

21,200.000

Rolled- Brass,

3-12

14,820,000

6.620,000

13,760,000

Cast-Iron, .

4-07

14,900,000

6,960,000

9,750,000

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Proc. Roy, Soctj. of Edin.]

[V(»L. XXV.

Pkofkssou E. G. Coker.

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1904-5.] Apparattbs for Measuinng Lateral Strains. 457

These results correspond with those obtained by Bauschinger,*
Stromeyer,t Morrow J and others.

It should be noted, in conclusion, that the specimens of material
for the tension and compression specimens were not identical,
but they were taken from the same consignments.

* Der Civilingenieur, vol. xxv., 1879.

t " Ezperimental DetennlnatioD of PoiBson's Ratio/' Proc R,S.t 1894.
t " On an Instrument for Measuring the Lateral Contraction of the Bars,
and on the Determination of Poisson's Ratio," Phil, Mag,, 1903.

{I89iied separately March 3, 1905.)

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458 Proceedings of Royal Society of Edinburgh, [i

On Astronomical Seeing. By Dr J. Halm,

Lecturer in Astronomy in the University of Edinburgh.

(Read May 6, 1904. MS. received October 14, 1904.)

In the Annual Report of the Smithsonian Institution for 1902
Prof. Langley has published an important note on " Good Seeing,"
in which he describes some experiments undertaken with the view
of improving the definition of telescopic images, so far as it depends
on the conditions of the air in the vicinity of the instrument. Up
to now the belief has prevailed among astronomers that in order
to obtaiu good definitions the air inside the telescope-tubes should
be kept as much as possible not only at a uniform temperature but
also in a state of perfect tranquillity. Langley, however, shows that
this view is not quite correct, and that maintaining constant and
uniform temperature inside the tube, while preventing circulation
between the air inside and outside the instrument, is not sufficient
to produce satisfactory telescopic images. Particularly, this method
does not diminish the troublesome boiling which in solar observa-
tions proves so often to be a source of grave inconvenience to the
observer. But he shows that if the air inside and near the
telescope-tube is agitated by stirring, the definition becomes at
once markedly better. The improvement has in all cases been so
decided that the reality of this beneficial effect of stirring cannot
well be doubted.

This result has led me to investigate the question as to whether
a similar conclusion may perhaps be drawn with regard to the
great mass of atmosphere which is traversed by the luminous rays
of the celestial object before they reach our telescopes. Is there
any reason for assuming that stirring this mass of air would
improve the definition, sharpness, and steadiness of the star
images ? The question, I think, has not been asked before ; and

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1904-5.] Dr J. Halm on Astronomical Suing, 469

I should like, therefore, to discuss it here in a few words, especially
as the answer to it seems to be simple and conclusive.

Let us first get an insight into the cause of the blurrings
of telescopic images, so far as atmospheric circumstances are
responsible for it. We feel no hesitation to look for this cause
in the incessant motions of our atmosphere, in the spontaneous,
fitful, and ever varying displacements of air from one place to
another, in consequence of local changes of temperature and
pressure. Now, the motion itself can have no direct effect on the
definition. The cause of the blurring must be looked for in sudden
changes of the index of refraction of the air resulthig from its
internal motions. If, for instance, a volume of heated air rises
from the surface of the soil to a higher layer, and arrives there
with a temperature higher or lower than that of the layer itself,
the temperature and density of that particular point of the
atmosphere, and thus its index of refraction, will be momentarily
altered. Hence the direction of a ray of light passing through this
point must suffer a corresponding change ; the consequence being,
that among the rays which, under undisturbed and perfectly ideal
conditions, would all reach the object-glass in parallel directions,
those passing through the affected area will be thrown into slightly
different paths, and will therefore be focussed at different points of
the field of view.

Now, we may ask: If the definition of telescopic images
depends on these fitful changes of the index of refraction which
are caused by the unavoidable movements and displacements of
air in the atmosphere, are there conditions under which these
movements have a minimum disturbing effect ? It is well known
that there is indeed one particular state of the atmosphere in
which these conditions seem to be present, viz., the so-called state
of adiabatic equilibrium. In this state a volume of air carried
from one layer to another will arrive at its new position with
exactly the S€ane temperature and density which were previously
possessed by the mass of air whose place it has taken. Hence
motion of air, in whatever direction it may take place, is not
accompanied by change of the index of refraction. We may
compare the atmosphere in this particular state to a liquid in
which bodies are suspended, of any size and shape, but of the

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460 Proceedings of Royal Society of Edinburgh, [i

same transparency and refrangibility as the liquid itself. What-
ever may be the motions of these bodies within the liquid, they
can have no disturbing effect on the course of the rays passing
through the medium, which will behave as an homogeneous
substance.

This reasoning leads us to expect the most perfect telescopic
images whenever the atmosphere traversed by the light of the
star is in the state of adiabatic equilibrium. Now, it is a well-
known fact that this state is reached, or at least approached, when
air is agitated by convection. It is for this reason that Lord
Kelvin long ago proposed to call this equilibrium * convective,'
instead of * adiabatic' or * indiflFerent.* Hence we conclude that
seeing should be most favourable when the air has been previously
stirred by convection-currents. With regard to the general
atmosphere, we reach therefore the same conclusion at which
Langley has arrived by his experiments where he considered the
comparatively small mass of air in the immediate vicinity of the
instrument

Several facts may be mentioned which seem to corroborate this
explanation, and in some measure to bear out its validity. For
instance, we know that on clear summer days, especially at
continental stations, convection between the upper and lower
layers of the atmosphere takes place during the daytime, being
most energetic in the afternoon. Hence we infer that convective
equilibrium is most nearly attained in the early evening, and
consequently that the definition of stellar images should be best
during the first hours of the night. In the later hours the seeing
must become worse, because, in consequence of nocturnal radiation,
the vertical distribution of temperature changes gradually so as
to become incompatible with the conditions of adiabatic equilibrium.
Towards the morning hours conditions become, therefore, more
and more prevalent under which spontaneous displacements of
masses of air must be accompanied by fitful changes of its re-
frangibility. My experience as an observer at Strasbourg is in
perfect accordance with these conclusions. As a rule, the seeing
in the early summer evenings at the time of sunset was excellent,
while after two o'clock in the morning the images had usually
become so bad that the observations had to be discontinued.

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1904-5.] Dr J. Halm on Astronomical Seeing. 461

The worst definition was commonly experienced shortly before
sunrise. Professor Copeland tells me that at Parsenstown the
seeing was specially good during a gale, and my own experience
here in Edinburgh confirms this statement.

The superiority of the definition in summer over that in winter
which is very marked at continental observatories is readily
explained by the fact that convection is much more energetic in
the former season. Indeed, at continental stations the atmo-
sphere in winter is on the whole very far from the condition of
adiabatic equilibrium, the temperature-gradient being much too
small, and often even reversed.

The question is doubtless of practical importance, and should
receive attention when sites for new observatories are selected.
The erection of observatories on or near mountains may be
advocated from this point of view, because horizontal movements
of the atmosphere are deflected at the mountain sides into more
vertical directions, thus enhancing that ** stirring " of the atmo-
sphere above the station which leads to the establishment of con-
vective equilibrium. The atmosphere on mountains, besides being
more transparent, must also be steadier, in an optical sense, not
from the absence of motions, but because these motions, by taking
place under adiabatic conditions, exert little or no disturbing
influence on the normal refrangibility of the air.

Meteorologists may perhaps give us definite and practical hints
as to the more or less favourable conditions under which convection
takes place in our atmosphere. Astronomers should be guided by
these advices in the selection of localities for their observatories.
Clearly, we have no means to prevent the incessant general and
local movements of the vast gaseous ocean above us. But knowing
that under one certain condition these uncontrollable motions,
otherwise so much inclined to impair our vision, may be rendered
optically ineflFective, we must avail ourselves of every possible
chance by which this ideal condition may be approached, — on the
one hand, by taking full advantage of favourable topographic and
climatic features, and on the other, by designing mechanical devices
for inducing convection in the neighbourhood of our instruments.

It would be interesting to hear the opinion of practical
astronomers on this question, and to see how far their experiences

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462 Proceedings of Boyqi Society of JSdinburgh, [sssft.

confirm my conclusions. I also wish to induce observers to take
regular notes of the conditions of seeing, and to enter into their
notebooks such remarks on the meteorological conditions prevail-
ing at the time of observation as may enable us to test the views
here expressed.

{Issued separately March 8, 1905.)

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1904-6.] OraptolUe^aring Rocks of the South Orkneys, 463

On the Ghraptolite-bearing Bocks of the South Orkneys.
By J. H. Harvey Pirie, B.Sc, M.B., Ch.B. Communi-
cated by Dr Horne, F.R.S. With a Note by Dr Peach on
Specimens from the South Orkneys.

(MS. receired February 7, 1905. Read February 20, 1905.)

The South Orkneys are a small group of islands situated in the
Southern Ocean, in about 62' S. lat. and 46' W. long., roughly
800 miles S.£. of Cape Horn. A single landing was made from
the " Scotia " on Saddle Island, a small island on the north side
of the group, and another on Coronation Island, the largest and
most westerly. With these two exceptions all the rock specimens
were obtained on Laurie Island, the most easterly of the group.

The rock got on Coronation Island is a coarse conglomerate, in
which the bedding is well marked, the individual beds averaging
about 2 feet in thickness, and dipping at about 30' in a north-
easterly direction. The rock is composed of a mixture of rounded
water worn pebbles and of angular fragments of dark-coloured shale
and mica-schist. Whether this rock belongs to the same series as
the Laurie Island beds or not I do not know, but the strike is
approximately the same.

Saddle Island is composed of a massive greenish grey wacke, very
similar to the Laurie Island rocks. The typical rock of Laurie
Island is a fine-grained grey wacke of a blue-grey or greenish-grey
colour. To the naked eye it appears almost homogeneous: the
only constituents that can be recognised are some minute rounded
quartz grains, small black shaly particles, and a few specks of
pyrites. Thin quartz and calcite veins traverse the rock irregu-
larly. A microscopic section shows that the derived constituents
consist of angular and sub-angular grains, with a mean diameter
of about 0*2 mm. The great majority of these are quartz, originally
of plutonic origin ; there are also a goodly number of small
crystals of plagioclase, wonderfully fresh, some grains of both
sphene and zircon, and a few minute flakes of biotite. The

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464 Proceedings of Royal Society of Edinburgh, [i

cemeuting material is very largely obscured by a dusty-grey or
brown amorphous substance and by black carbonaceous matter.
Where the grains are fairly large and well packed this forms a
sort of network, in the meshes of which lie the quartz grains.
Where the grains are not so close it is more distinct, and under
crossed Nicols has a crypto-crystalline appearance, practically
identical with that of chalcedony. A few chlorite flakes occur
in it here and there. Small veins traverse the section, some con-
taining calcite, others a fine quartz mosaic. Bedding is not seen
in hand specimens, and in many places in the field it cannot
be made out either, the rocks having a massive character, but
much traversed by cracks and faults, shattering them into irregular
masses.

In other places, again, the bedding is distinct, or even marked.
Where this is the case the individual beds vary in thickness
from a few inches to several feet. Very often the bedding has
a contorted, or rather, wavy character, more conspicuous when
viewed from some distance off.

On some of the cliffs faulting is very marked, which has
probably given rise to the general shattered condition of the
rocks. Most of the faults noted are strike-faults. When the
faults are not so much in evidence, the rock shows in places well-
marked jointing, often very difficult to distinguish from bedding
planes.

Varieties of the greywacke occur. These are of very local
occurrence, and are not usually sharply defined, but shade off
imperceptibly into the common type. The following are the
principal varieties : —

1. Greywacke conglomerate. Contains rounded quartz pebbles,
not usually larger than \ in. in diameter, and pieces of dark slate
or shale, rounded or flattened angular laminse, up to f in. in length.
This is an extremely hard, tough rock, intimately pervaded by the
siliceous matrix, so that the grains seem to fade into each other and
into the cementing material, instead of having sharp outlines. When
fractured, the component pebbles break across, but on natural
weathered surface the matrix gives way sooner, leaving the indi-
vidual pebbles sticking out as in a conglomerate. A microscopic
section shows that the allothigenic or derived materials are practically

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Ill

To prevent delay, authors residing abroad should appoint some one
residing in this country to correct their proofs.

5. Brief Abstracts op Transactions Papers will be published in
the Proceedings, provided they are sent along with the original paper.

The author receives fifty of these reprints free, and may have any
reasonable number of additional copies at a fixed scale of prices which
will be furnished by the printer, who will charge him with the cost.
To prevent disappointment, especially if the paper contains plates,
the author should, immediately after receiving his first proof, notify
to the printer the number of additional copies required.

MODEL INDEX.

Schafer, E. A.— On the Existence within the Liver Cells of Channels which can
be directly injected from the Blood-vessels. Proc. Roy. See. Ed in., voL ,
1902, pp.
Cells, Liver, — Intra-cellular Canaliculi in.

E. A. Schafer. Proc. Roy. Soc. Edin., vol. , 1902, pp.
Liver, — Injection within Cells of.

E. A. Schafer. Proc. Roy. Soc Edin., vol. , 1902, pp.

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IV CONTENTS.

PAGE

A La>K)ratory Apparatus for Measuring the lateral Strains
in Tension and Compression Members, with some
Applications to the Measui-ement of the Elastic
Constants of Metals. By E. G. Coker, M.A. (Cantab.),
D.Sc. (Edin.), F.R.S.E., Professor of Mechanical
Engineering and Applied Mathematics, City and
Guilds Technical College, Finsbury, London. (With
a Plate), . . . . . 452

{Issued separately March 3, 1905.)

On Astronomical Seeing. By Dr J. Halm, Lecturer in

Astronomy in the University of Edinburgh, . . 45^

{Issu-ed separate! p March 3, 1905.)

On the Graptolite-bearing Rocks of the South Orkneys.
By J. H. Harvey Pirie, B.Sc, M.B., Ch.B. {Com-
muntcated by Dr Horne, F.R.S.) With a Note by
Dr Peach on Specimens from the South Orkneys, . ' 463

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PROCEEDINGS

OF THE

ROYAL SOCIETY OF EDINBURGH.

SESSION 1904-5.

No. VII.] VOL. XXV. [Pp. 466-692.

CONTENTS.

PAOE

A Possible Explanation of the Formation of the Moon.

By George Romanes, C.E., . . .471

{Issued separately March 30, 1905.)

On Pennella : a Crustacean parasitic on the Finner Whale
(Batxjwptei'a musculus), {Ahdract.) By Sir "William
Turner, K.C.B., LL.D., . . . .480

(Issued separately March 30, 1905.)

The Diameters of Twisted Threads, with an Account of
the History of the Mathematical Setting of Cloths.
By Thomas Oliver, B.Sc. (Lond. & Edin.). {Com-
municated by Dr C. G. Knott), . .481
{Issued separately April 8, 1905.)

[CoTUinned <*n page iv of Cover,

^EDINBURGH:
PUBLISHBD BY ROBERT GRANT & SON, 107 Princes Strbbt, and
WILLIAMS k NORGATE, 14 Hknkietta. Street, Covent Garden, London.

MDCCCCV.
Price Six Shillings and Sixpence.

0 -^^v,-S

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KEGULATIONS KEGAKDING THE PUBLICATION OF
PAPERS IN THE PEOCEEDINGS AND TRANS-
ACTIONS OF THE SOCIETY.

The Council beg to direct the attention of authors of communications to
tlie Society to the following Regulations, which have been drawn up io
order to accelerate the publication of the Proceedings and Transactions,
and to utilise as widely and as fairly as possible the funds which the
Society devotes to the publication of Scientific and Literary Researches.

2. Illustrations.— All illustrations must be drawn in a form im-
mediately suitable for reproduction; and such illustrations as can be
reproduced by photographic processes should, so far as possible, be
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fine white bristol board, free from folds or creases ; smooth, clean lines
or sharp dots, but no washes or colours should be used. If the drawings
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lettering or other legend must be on a corresponding scale.

If an author finds it inconvenient to furnish such drawings, the Society
will have the figures re-drawn at his expense ; but this will cause delay.

All proofs must, if possible, be returned within one week, addressed to
The Secretary, Royal Society^ Mound, Edinhargh, and not to the printer.

[Continued on page m of Cover,

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1904-5.] GrraptolUe-bearing Bocks of the SotUh Orkneys. 465

the same as in the normal greywacke, ue, pebbles of quartz and
chalcedony, pieces of shale, small crystals of plagioclase, a few
grains of sphene and zircon, and biotite flakes. Of the larger
quartz pebbles, some at least are typical plutonic quartz, with lines
of fluid inclusions, but showing strain shadows : the majority
seem to be derived from some metamorphic rock — pebbles which
in ordinary transmitted light appear quite uniform, between
crossed Nicols are seen to "be composed of a mosaic of different
crystallographic individuals. The cementing material is not so
obvious as one would expect from a naked-eye examination, as
the interstices between the larger pebbles are filled up by smaller
fragments, chiefly of quartz. It has the same chalcedonic
appearance as in the typical greywacke, but green chloritic flakes
are more abundant. There are numerous small veins of both
calcite and quartz : one of the latter, about 0 03 mm. in width, was
observed running right through some of the large quartz pebbles.

2. Greywacke-slate. Has a fine laminar structure parallel to the
places of deposition, is of a lighter grey colour, and splits up readily
into thin laminse. There is no true slaty cleavage developed,
however.

3. Greywacke, showing gneissic banding and folding. This was
only got in one patch of very limited extent.

Shaly rocks also occur. In one situation only were regular beds
of shale found alternating with layers of greywacke. Commonly
the shale occurs simply as patches in the greywacke, seemingly
irregularly mixed up with it, or with ill-defined borders shading
oflf into the greywacke. The shale is much cleaved and broken,
the individual'pieces being bent and curved, and showing numerous
slickensided faces, the result of the crushing and faulting to which
it has been subjected. Microscopically it shows much brownish-
grey amorphous material and black carbonaceous matter in the
lines of stratification — forming a sort of network in the silica
matrix. Interstratified lenticular-shaped patches occur, which are
much freer horn amorphous matter. With crossed Nicols these
resolve themselves into a crypto-crystalline chalcedony, identical in
character with the cementing material of the greywackes.

The largest development of the shale occurs on a small islet off
the south coast of Laurie Island, near Cape Dundas — its eastern

PROC. ROY. SOC. EDIN. — VOL. XXV. 30

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466 Proceedings of BoycU Society of Edinburgh. [i

end — and which has been called Graptolite Island. Here three
fossils were got. One of these is a graptolite, which has been
examined by Miss Elles, who considers it to be part of a Pleuro-
graptus. This would make the bed correspond in age with the
Hartfell shales — almost the uppermost beds of the Ordovician
system. The others have been kindly examined for me by Dr
Peach. As is seen in his Note, he considers them to be parts of a
Phyllocarid crustacean, probably nearly allied to Discinocaris^ a
form typical in this country of the Lower Birkhill shales, at the
base of the Upper Silurian.

If this is the case, then there is here an association in one bed of
two forms which, in the South of Scotland, are characteristic of
two different but at the same time closely contiguous zones.

As regards the structure of the island as a whole, it is un-
fortunate that the data regarding the dip and strike of the rocks
are rather meagre. This is due partly to the fact that so much of
the area is covered by ice, and partly because in so many places
the dip could not be made out. The most common strike of the
rocks is north-westerly, varying from N.N.W. to W.N:W.,
the dip being in most cases at a high angle north-easterly or
south-westerly. One definite anticlinal axis was observed, running
in a N.N.W. and S.S.E. direction. In a few localities other
directions of strike were noted, but these were nowhere of large
extent, and are probably only local contortions.

Laurie Island itself, although its greatest length is in an E.N.K
and W.S.W. direction, consists of a series of peninsulas and hill
ridges, running in a general N.W. and S.E. direction, with deep
bays between adjacent peninsulas, and usually low cols crossing the
island from the head of a bay on the north side to the head of
another on the south side.

The same structure is repeated in the group as a whole, which,
though it extends furthest in an east and west direction, is cut up
by two large straits, which cross it in about a N.N.W, and S.S.E.
direction.

These two sets of facts — the strike of the rocks and the general
alignment of the hill ridges — lead one to believe we have here to
deal with a series of plications whose axes run in a general N. W.
and S.E. direction — probably rather nearer N.N.W. and S.S.K

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1904-5.] GraptoHte-bearirig Bocks of the South Orkneys, 467

In the only previous reference to the structure of these islands
that I have been able to find, viz., the reports of M. Dumont
D'Urville's voyage,* their only landing seems to have been on a
small islet about half a mile from Saddle Island, where they report
greyish limestone and phyllitic shales, with a K.N.W. and S.S.E.
strike, and inclined at over 60"*.

Although geographically situated nearer the South Shetlands
and Graham Land, the strike of the rocks leads one to consider
whether these islands are not more intimately connected with

2000 fothom line.

Alternative line. Position doubtful.

No soundings.

South America. In this connection it is important to consider
some geological facts from areas further afield. In the Falkland
Islands the Silurian or Devonian rocks there are folded along an
east and west axis. South Georgia, composed entirely of clay
elates, in which one fossil shell has been found — of Upper
Palaeozoic or Lower Mesozoic Age, according to Professor Koken —
is stated t by Dr Andersson, of the recent Swedish Antarctic

* " Voyage au Pole Sud, sous le conunandement de M. Dumont D'TXrviUe,'*
OiologiCf par M. J. Grange,
t Andersson, Oeog. Jour,^ Oct. 1902.

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468 Proceedings of Royal Society of Edinburgh. [sEse.

Expedition, to consist of a series of folds along an axis nearly
parallel to the long axis of the island, ue, a north-west and south-
east axis. Then the soundings taken by the " Scotia " indicate
that the deep water between Cape Horn and the South Shetlands
narrows as we go eastwards into a trough-like depression of over
2000 fathoms, passing north of the South Orkneys, then probably
turning south-eastwards, to become continuous with the deep area
of the Weddell Sea.

It may be, therefore, that the Andean axis, already turning east-
wards in Southern Patagonia and Tierra del Fuego, is continued in
this direction south of the Burdwood bank, and then curves south-
eastwards between the South Orkneys and South Georgia.

If this is the case, then there is a relationship established
between these Silurian rocks of the South Orkneys and the Silurian
rocks occurring on both sides of the main Andean chain in Bolivia
and Northern Argentina,* and in the province of Buenos Aires, in
the Sierra Tandil and Sierra de la Ventana.

More soundings in the area between the South Orkneys, Cape
Horn, and South Georgia would probably shed further light
on this problem ; and they are also much to be desired between
the South Orkneys and Graham Land, where rocks of an entirely
diflferent type occur, viz., plutonic and metamorphic rocks on the
Pacific side, and on the eastern side Lower Tertiary rocks, similar
to those of Patagonia.

At all events, the presence of isolated islands such as the South
Orkneys and South Georgia, composed of sedimentary rocks, mostly
inclined at high angles, and surrounded by deep water, proves a
former much greater extension of land in this area. If they formed
part of the Tertiary Antarctica postulated by Professor H. F.
Osborn and many others, t to explain the floral and faunal relation-
ships of S. America, S. Africa, and Australia, it is evident from the
recent soundings X that the changes of level in sea and land in this
region have been very considerable: it would now require an
elevation of nearer 20,000 feet than the 10,000 assumed by
Professor Osborn as necessary to unite S. America with Antarctica.

* Cf. Suess, La Face de la Terre, vol. i. pp. 684-686.
t H.F. Osborn, Science, 1900, vol. xi p. 666.

X Andersson, loe, cU,, and *' Second Voyage of * SooUtL,^^ Scot. Geog. Mag.,
Jan. 1904.

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1904-5.] Grraptolite-bearing Bocks of the South Orkneys. 469

Note by Dr Peach on Specimens from the South Orkneys.

Two specimens of black shale, Nos. 014 and 015, from the South
Orkneys, have been submitted to me by Dr Pirie for examination.
No. 015. — In addition to some stipes of graptolite, determined
by Miss Elles to belong to the genus Pleurograpius, there occurs
a fragment of another organism, showing a web of dark carbon-
aceous matter, with a succession of sub-parallel ridges which appears
to belong to a Phyllocarid crustacean, probably nearly allied to
Discinoearis.

No. 014 shows the remains of what appears to have been
another form of Phyllocarid crustacean, preserved in a dark shining
anthracitic substance. What seems to be the carapace is broad
and smooth, with faint indications of raised lines directed outwards
and forwards on the left side. Where the supposed carapace
has broken away in splitting the shale, a succession of bands about
^ inch broad, and numbering six within about the same breadth
backwards, may be observed. These are each ornamented with
sub-parallel lines and with broadened posterior margins. Both the
carapace and the apparent body segments are abruptly truncated
posteriorly in the breaking of the shale.

A wide experience of the black graptolite-shales of the Southern
Uplands of Scotland and North Wales, of all horizons, from the
Lowest Arenig up to the Wenlock and Ludlow rocks, has shown
that, with the exception of a few small hingeless brachiopods and
some glass-rope sponges, only the tests of chitinous Phyllocarid
crustaceans have been met with. Of these, the genus Caryocaris
characterises the Arenig, Pinnocaris the Lowest Hartfell shales
(Caradoc), Diseinocaris and Peltocaris the Lower Birkhill shales
(Lower Llandovery), and Aptychopsis and Ceratiocaris (the Wenlock)
dark graptolitic shales.

The general style of ornament found in the test of most of the
above genera is that of the sub-parallel raised lines, which may be
arranged on the carapaces almost concentrically to rudely simulate
lines of growth in some forms ; but in Ceratiocaris they run longi-
tudinally backwards. On specimen 015 there appears to be a slight
curve in the raised lines similar to what occurs in Diseinocaris
gigaSy Jones and Woodward, and figured in their monograph.

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470 Proceedings of Boyal Society of Edinburgh. [i

This form has only been found in the BiikhiU shales (Llandovery)
of Moffat, while the graptolite Pleurograptus found on specimen
015 shows that this specimen belongs to a lower horizon (Caradoc).
Pleurograptus linearis, Carruthers, is the zonal form of the Upper-
most zone of the Lower Hartfell shales (Caradoc) of Moffat. I do
not, therefore, consider that any of the specimens could be deter-
mined either specifically or generically ; but if these organic remains
belong, as they appear to do, to Phyllocarid crustaceans, their
occurrence along with graptolites in black shales in both the
northern and southern hemispheres would signify more than a
near coincidence.

{Ismed separately March 80, 1905.)

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1904-5.] Mr Eomanes on the Formation of the Moon. 471

A Possible ExplaDation of the Formation of the Moon.
By Gteorge Romanes, C.K

(Read November 21, 1904.)

The subject of the moon's development has been dealt with by
Professor G. H. Darwin by means of a highly abstruse mathe-
matical analysis, which the present writer cannot pretend to be
able to discuss. He wishes to point out, however, that Professor
Darwin's theory requires the assumption that earth and moon
formed, at one time, a single highly -heated fluid mass ; the theory
being that the moon was thrown off by centrifugal force aided by
the sun's tidal influence and synchronous vibratory motion of the
fluid mass.

There is another possible explanation of the formation of the
moon, that gets over many difficulties in explaining its features.

It is to suppose that earth and moon were separately formed out
of different parts of the same nebula, or crowd of small parts which
were at one time circulating round their common centre of mass
at great varieties of distances, in every plane and with every
degree of eccentricity, the whole having a balance of moment in
the plane and direction in which earth and moon are now revolv-
ing. The portions near the centre would tend to collect there to
fonn the earth, while the outer portions gradually collected into
larger and larger masses to form the moon, and in doing so built up
its mass in such a way as to leave a record, which it is the purpose
of this paper to endeavour to interpret.

Before considering the markings on the moon's surface, the
writer wishes to show, as clearly as he can, how such a result as
the building up of the moon in this way is possible. All bodies
circulating round the earth are subject not only to the influence of
the earth, but also that of the sun and of each other ; which must
have caused great irregularities in their motions, and increased the
chances of collisions among each other, and thus gradually reduced

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472 Proceedings of Roycd Society of Edivburgh, [

the number and increased the average size ; and the largest body,
the moon, would capture most matter in this way.

There is a very important difference between the collisions of
bodies moving in the same direction of revolution and of those
moving in opposite directions, which must be kept in view. The
former are caused principally by bodies attracting each other ; they
are not destructive ; and while they cause the mean distances of
the orbits to be diminished, they tend to make these orbits less
eccentric. The latter occur at high speeds ; they are highly
destructive, and cause the orbits to become Tnore eccentric The
moon's moment of momentum round the earth proves that it has
been built up principally of bodies having the same direction of
revolution.

The several portions which now form the moon must have long
had independent orbits round the earth, and many may have
grown to a considerable size before being caught by the moon. The
moon's mass is now an eighty-first part of that of the earth, and at
distances of 23,800 miles (more or less, according to circumstances)
from the moon its influence is equal to that of the earth. Hence,
when a small body having an independent orbit round the earth
came near the moon, it would be drawn into a subsidiary orbit with
the moon's centre as focus, which, with reference to the moon,
would be a hyperbola ; and the body might strike or graze the
moon's surface, or escape and keep on an orbit round the earth,
much modified by the encounter, till some other close approach,
when it might be captured.

With regard to bodies being captured by the earth, if two equal
masses circulating at the same mean distance in opposite directions
were to collide, their moments of momentum would be mutually
destroyed, they would be highly heated and driven to pieces, and
they would fall direct to the earth. So exact a balance as this is
against all probability, and the most usual result of such collisions
would be to render the resultant orbits more eccentric, and thiis
give increased chances of further collisions, because they would
cross other orbits to a greater extent. Finally, many orbits would
be rendered so eccentric as to cause the bodies to graze the earth's
atmosphere at each revolution, which would thus reduce the orbit
till the earth captured the whole in small pieces, this effect being

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1904-5.] Mr Romanes on the Formation of the Moon, 473

aided by the disintegrating influence of the atmosphere and of the
earth's tidal attraction. The earth's atmosphere would thus be the
first and principal recipient of the heat caused in this way.
Direct impacts on the earth would be rare, and their marks would
in time be effaced by the various geological influences.

Impacts on the moon, of bodies having independent orbits round
the earth, would be of a very diflferent nature ; these would often
be very direct, and the bodies themselves might be of considerable
size, possibly up to 20 miles or more in diameter. Such bodies
being built up of many parts loosely held together by their own
feeble gravity, would be more like masses of sand and dust than
solid stone ; hence a grazing impact of such a body on the moon
would be like a sand-blast which would liquefy the rock and plough
out a straight groove. The utmost velocity the moon can produce
by its attraction is 1 '476 mile per second, and bodies having orbits
round the earth at the same mean distance in the opposite
direction would, if they collided, strike it with the velocity of 1 946
mile per second, and it would be struck by bodies having orbits
within its own, as well as by others beyond it ; thus velocities of
impact might range from 1 '4 mile to even 2 miles per second on
rare occasions. These velocities represent energies capable of.
raising the temperature of the bodies striking by 5200** Fahr. to
10600* Fahr., or rather of raising the temperature not only of the
bodies themselves, but also of much of the moon's surface, to an
extent sufficient to liquefy them ; while the mechanical force of
the impact would cause much of the surrounding surface to be
forced up into irregular mountain ranges all round, and cause
great splashings of liquid rock from the hollows thus formed, and
great surgings to and fro of the liquid rock within them ; and no
doubt gases would be formed and fly off, till the liquid rock had
time to cooL

Besides being struck by single bodies, the moon may often have
been struck and grazed by nebulie — that is to say, swarms of small
bodies which had sufficient moments of momentum about their
centres of mass to keep them from aggregating more closely.

Impacts of large bodies having independent orbits round the sun
would be very rare, and it is doubtful if any would leave marks
large enough to be seen from the earth.

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474 Proceedings of Royal Society of Edinburgh, [

The writer has been referred to Professor N. S. Shaler's great
essay in the Smithsonian Contribuiiofu to Knowledge to study his
views, and to discoss them herein.

Professor Shaler does not discuss the manner of the moon's growth
except by a reference to Professor G. H. Darwin's theory, a modified
form of which he apparently accepts (page 3 of his essay), and he
makes the following assumption on pp. 31-32: "The most
reasonable view of the interior condition of the moon when its
Yulcanoids (craters) were in activity is that it was in a state of
essential fluidity with a relatively thin crust." This is making
use of a popular idea that the moon, like all other cosmic bodies,
must at one time have been so hot as to be fluid. This is not a
scientific view, as no proof of it is possible. Professor Shaler
makes no attempt to show how the moon became so hot as to be
fluid, and on page 48, under " Adjustments of the Surface to Con-
traction," he gives the following strong evidence that leads to a con-
trary inference : " On the earth he (the geologist) sees in the ample
folds of the sea-basins and of the continents, as well as in many
folded mountain chains, what he takes to be evidence of a long-
continued accommodation of an anciently cooled crust to a central
mass which is ever losing heat. On the moon he finds what, in
proportion to the size of that sphere, is surely not the hundredth

part of such action What then is the meaning of this

startling diversity in the orogenic history of the two spheres?"
Also on page 4 Professor Shaler states the relative densities of the
moon and earth as six to ten ; but he does not draw the inference
that the moon has been less subjected to gravitational compression,
and therefore has had less internal heating than the earth ; indeed,
the influence of mass in causing the heating of cosmic bodies
seems not to have been sufficiently present to his mind. Professor
Shaler requires the presence of fluid lava a short way below the
surface of the moon to explain the formation of vulcanoids (craters)
by a rise and fall of liquid lava through holes in the crust, which
he supposes have been formed by the help of gases like slow
boiling; and he accounts for the formation of terraces on the
inside of the surrounding walls of the vidcanoids by the different
levels at which the lava successively stood. These terraces are
very irregular, and by no means continuously horizontal, and they

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1904-6.] Mr Eomanes on the Formation of the Moon, 475

occur outside the crater walls as well as inside; but the writer
does not find that Professor Shaler accounts for those on the out-
side. In discussing G. K. Gilbert's hypothesis, that the craters
were due to impacts, he rejects it, because, he says in a footnote
(page 12), the masses or bolides would have struck with velocities
that would have raised their temperature more than 150,000
degrees (scale not stated). He (probably after Gilbert) is thinking
of velocities of 7 J miles per second or more. The possibility of
such masses (bolides) having always been in company with the
earth and moon has not occurred to him ; and he objects (page 12)
that such impacts would have caused much cracking of the moon's
surface — thinking, no doubt, of hard masses striking stone, but
not considering that the bodies striking might have been more
like heaps of loose material moving generally with the velocity of
only 1 J mile per second.

Again, Professor Shaler thinks that the maria must have been
formed, each by the impact of one or more bolides with planetary
velocities (page 1 7), and he considers the great amount of melting
of rock they could produce ; but he does not sufficiently consider
that a mass moving at such velocity, instead of melting a great
quantity of rock, would melt only a moderate quantity, and spend
much of its energy in driving the melted rock right away from
the moon in a great splash.

Professor Shaler has taken an immense amount of care, and
given many years of labour to accumulate facts as to the moon,
and he has stated those facts with great impartiality for the
benefit of science; but in explaining the causes at work in pro-
ducing them, the writer thinks he has started from wrong
premises, and found difficulties that disappear when the true
causes become known.

The writer will now state his views as to the cause of some of
the principal lunar formations. He thinks that the circular or
slightly elliptical craters have been formed by the impacts of bodies
belonging to the earth's system, of all sizes up to 20 miles or more
in diameter. The floors of these craters are in general much
depressed below the surrounding surface, and the crater walls are
sometimes of such great elevations as 17,000 feet or more above
the floors, while the diameter of the craters varies from the

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476 Proceedings of Royal Society of Ediiiburgh, [sbs.

smallest size that can be seen up to at least 1 40 miles. Some of
the crater floors are above the general level, such as Gassendi close
to Mare Humorum, and some more nearly at the same level, when
thej are situated in or near the maria. The general characteristic,
however, of those that are not near the maria is to have their floors
much depressed, even to the extent of thousands of feet in some
cases. The forms of these craters can be fairly well imitated by
firing bullets into a mass of lead. The cavity thus formed has
always a raised burr round it, is much larger in diameter than the

bullet, and is generally fairly round even when the bullet strikes
obliquely, if not so obliquely as to glance off altogether. There is
always a small cone left in the cavity, and the surface of the whole
cavity can be seen to glow red-hot immediately after the shot is
fired. In the case here illustrated the bullets were elongated
leaden ones -22 inch in diameter, and the cavities were '44 inch
diameter. The three shots down the middle were fired perpen-
dicular and the others obliquely, but not at measured angles ;
however, the mark on the right of the centre was roughly estimated
to be at about 45*.

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1904-5.] Mr Eomanes on the Formation of the Moon, 477

The velocity of impacfc in these experiments is not known, but
might have been about 1200 feet per second; and, no doubt,
bullets fired at higher velocities would form cavities wider in
proportion to the bullet. A velocity of 1200 feet per second is
from one-sixth to one-ninth of the velocities we are dealing with
in the case of the moon, and the body striking is a compact one ;
whereas, as has been shown above, the bodies striking the moon
were by no means compact ; and the circumstances are so different
that the analogy between the bullet marks and the lunar craters
will not be very close. However, the experiments make it clear
that cavities so formed on the moon's surface may be expected to
be greatly larger in diameter than the body that caused them, and
generally fairly round.

The great radial streaks, notably those from Tycho, are probably
caused by splashes of liquid rock comminuted and blown out by
the gas formed at the same time. Their great brilliancy at full
moon is probably due to the surface being rough — that is, covered
with small particles, and not appearing vitrified like the rest of
the moon's surface. As no shadows can be seen at full moon,
rough surfaces must then appear brighter than under indirect
illumination. Although these streaks extend to great distances,
such as 1000 miles, it is obvious that the initial. velocity, neces-
sary to project them from their source to any other part of the
moon's surface, is much less than the moon caused by its attraction
on the bodies that produced them; and therefore this cause of
them is quite within the limits of possibility.

The irregular terraces or wrinkles, seen on both the inner and
the outer slopes of the circular moimtain rings, and particularly
well seen in Copernicus, are probably caused by the powerful side
thrust that raised them up.

The cones inside the craters are evidence that part of the body
striking was unmelted, and was piled up in a heap or heaps near
the centre, and cemented together by the liquid rock surging to
and fro. The absence of cones in some craters shows that the
whole has been melted, either at first or by lava from other sources,
such as molten lava being thrown in by the violent surgings of
the maria when they were formed.

The Valley of the Alps has all the appearance of having been

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478 Proceedings of Boyal Society of JSdinburgh, [i

ploughed out by the grazing impact of a moonlet. Its floor is
level with Mare Imbrium, and its sides are nearly vertical ; hence
it may have been scoured through by white-hot lava from the
Mare Imbrium when that mare was violently suiging on its for-
mation. There are numerous features of the nature of the Valley
of the Alps on the moon's surface, notably in the region of craters
Albategnius and Ptolemaeus, and also in the region south of Mare
Serenitatis. These are arranged in series of parallel lines, and
may be due to the grazing impact of swarms.

A large portion of the moon's surface is covered with the maria,
some of which have a roughly circular outline, such as Mare
Imbrium, Mare Serenitatis, and Mare Crisium, which seems to
indicate that each is the result of some single great catastrophe.
These may have been formed by the impact of a nebula or swarm
of bodies ; and the mountain ranges bordering them, such as the
Alps and Apennines bordering Mare Imbrium, may have been the
result of the same catastrophe which formed the sea they are
associated with. These mountain ranges have all the appearance
of masses of matter thrown down in a sidelong heap and splashed
over with liquid rock. There is much appearance on the sur&ce
of the maria of their having been in commotion, and indeed they
must have been in violent commotion when they were formed.
Many long ridges on their surfaces show that they have not quite
come to a level surface tiD they were too viscous to do so. These
ridges seem to indicate a creeping together of the lava from
opposite sides when it was nearly solid. The surfaces of the maria
are generally darker than the rest of the moon's surface, owing, no
doubt, to their comparative smoothness rather than to any differ-
ence in the kind of rock ; obviously, a polished surface would look
black at full moon, if not at the centre of its disc.

A very interesting feature, that may be noticed more or less on
all parts of the moon's surface, is the immense number of old
craters and mountain ranges that have been overwhelmed by the
lava of the maria, or battered down by more recent formations ;
which shows that the formation of those craters and maria is no
casual occurrence, depending on the chance meeting of meteors
from outer space, but the natural process by which the moon's
mass has been built up.

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1904-6.] Mr Romanes on the Formation of the Moon, 479

It may finally be suggested that the sadden accession of large
quantities of matter, such as that of a mare, to the moon's surface,
might slightly alter its balance, and cause it to turn a somewhat
different face to the earth. The frequent occurrence of such
changes would be in favour of its assuming the true form of
equilibrium even although it has never been fluid; and all in-
fluences to which it has been subjected would have the same
tendency.

The writer has heard, since this paper was read, that former
attempts have been made to illustrate the formation of lunar
craters by firing bullets ; but he has heard of no former attempt
to explain the whole formation of the moon's mass as due to
impacts of bodies which have always been part of the earth's
system, in the manner explained above.

He wishes to state that he is greatly indebted to Mr Heath of
the Koyal Observatory for help of every kind in gaining informa-
tion, and for the slides which were shown in illustration of this
paper.

{Issued separately March 30, 1905.)

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480 Proceedings of Boyal Society of JEdiriburgh. [:

On Pennella : a Crustacecm paxasitic on the Finner Whale
(Balsmoptera musculus). By Sir William Turner, K.C.B.,
LL.D.

In this memoir the author described a Pennella found attached
to the back of a BdUtnoptera musculus, specimens of which were
given to him in 1903 by Mr Chr. Castberg. The specimens were
of the same species as the Pennella baUmoptera described by Keren
and Danielssen in 1857, and found infesting B. rostrata. The
species is a giant Copepod, and the longest examples measured
about 12^ inches.

The description included a short historical introduction to the
genus, an account of the external characters and internal anatomy
of the species, its comparison with other species, and the attach-
ment to one of the specimens of Conchoderma VirgcUa. The
memoir, with illustrations, will appear in the Transactions of the
Society.

(Issued separately March 80, 1905.)

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1904-5.] Mr T. Oliver on Diameters of Twisted Threads, 481

The Diameters of Twisted Threads, with an Aocount of
the History of the MathematioeJ Setting of Cloths.
By Thomas Oliver, B.Sc. (Lond. & Edin.). Communicated
by Dr C. G. Knott.

(MS. received January 27, 1906. Read March 20, 1905.)

During the last generation the idea of reducing the ** setting "
of cloths to mathematical accuracy has heen gradually taking hold
of the minds of thinking men in the various textile trades. That
this end is perfectly attainable is perhaps an open question, but
there can be no doubt that the investigation of such problems
most lead to a more satisfactory knowledge of the factors which
determine the construction of fabrics.

The base from which these " setting " theories begin is natur-
ally the diameter of the thread, since the "set" of a cloth, i.e.
the number of threads in some unit distance, usually the inch,
made in any one weave or scheme of interlacing, is inversely
proportional to the diameter of the thread employed in the
construction of the cloth. Clearly, then, the first step in this
investigation must be the determination of the diameters of the
numerous " counts " or numbers of yams in the various materials
which are in use in the textile industries. But this is by no
means such an easy task as it may seem at first sight. The
diameter of a thread is neither easily measured at any one section,
nor a constant quantity throughout its length. Especially is this
the case with woollen yarns, in which the fibres projecting from
the body of the thread in every conceivable direction renders the
averaging up of the section a tedious and often unsatisfactory
operation.

The history of the mathematical setting of cloths is, however
not confined to the last generation. The earliest record of a
systematic attempt to attain this end is preserved in the British
Museum in a copy of Maihematical Sleaing Tables, calculated
by Mr Joseph Beaumont, a wiiter on the Irish linen trade in
1712. He recognised that the setting of cloths should be based

PROC. ROY. SOC. EDIN. — VOL. XXV. 31

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482 Proceedings of Roycd Society of EdvnJtmrgh. [i

on the diameter of the thread, although he erroneously applied
this term, not to the actual diameter, but to what was really the
pitch of the threads in the warp, i.e. the diameter of the thread
plus the space between the threads. We find another stepping-
stone in the evolution of this subject in comparative setting or
caaming tables included in Murphy's classical Art of Weaning^
published about the beginning of last century. It is, hotvever, not
too much to say that " rule of thumb " held practically undisputed
sway in this field until thirty years ago.

About 1875 the late Mr Robert Johnstone, of Gralashiels, a
shrewd Scotch designer, possessed of remarkable powers of obser-
vation, put out a little work entitled Designer's Handbook^ in
which he gave a rule to set webs in the reed. After stating the
rule, he appends the following note : — " I have often been asked
why the square root of the size weight of a yam multiplied by the
numbers stated in this rule gives the number of the reed which
should be used. I answer the question in this way : \ of an inch
divided by the square root of any weight of yam is equal to the
diameter of it. Now if that is so, the diameter of 1 cut yarn will
be ^ of an inch, and that of 25 cut will be ^ of an inch." The
yarns were numbered on the Gralashiels system. The above state-
ment, though rather loosely worded, is the first instance, so far as
the present writer is aware, in which the diameter of a yam was
employed in its proper sense as a basis on which the " set " for
a given yam might b^ determined. The conclusions arrived at are
all the more remarkable since Mr Johnstone must have deduced
them by observation on cloths alone, as he had no means of
making micro-measurements. Besides, neither he nor his fellow-
workmen could have been burdened with much education, nor had
he the advantage of consulting literature on the subject, since there
was none. Johnstone's rule is held in high repute amongst Scotch
designers, and it is safe to say that it gives very good results for
the average Scotch woollen cloths, for which the rule was intended.

The great epoch in this subject, however, occurred in 1880, when
the late Mr Thos. B. Ashenhurst, then head of the textile de-
partment of Bradford Teclmical College, gave out the results of
his experiments and deductions to the textile public. Mr
Ashenhurst's experiments consisted of measuring the diameters of

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1904-5.] Mr T. Oliver an Diameters of Twisted Threads. 483

a large number of threads in different sizes and materials with a
micrometer, and taking the average for each yam number. These
numbers are tabulated in his work on Textile Calculations,
published in 1884.

Subsequently he found that the following empirical formula
gave results closely approaching to the number tabulated from
his experiments. The diameter, expressed as a fraction of an
inch, is equal to the reciprocal of the square root of the number of
yards per lb., with a deduction of 10 per cent, from the square
root for worsted, cotton, linen, and silk yams, while for woollen
yams a deduction of 16 per cent, should be made. This deduction
is sometimes spoken of as the allowance for surface fibre, which
is, however, quite erroneous, as the surface fibre is far too variable
a quantity to be reckoned as proportional to the diameter or any
•ther attribute of the thread. It has really no physical meaning
whatever. The reason that there should be a deduction is purely
a mathematical one, i.e. to make one number correspond with
another. Ashenhurst was helped towards the explanation of his
diameter rule by Mr T. F. Bell, of Belfast, in 1889. The full
correspondence on this matter will be found in the Textile
Educator, February 1889, of which Mr Ashenhurst was the
editor. There is little doubt that it is a very useful formula,
and gives very good results when applied, in conjunction with his
other setting formulsB dealing with variations in weave (a subject,
however, outside the scope of this paper), to the average mn of
cloths made in Yorkshire, where the practice is to set cloths
much closer than is customary in the Scotch trade. There has
been very little done in this field of research since the time of
Mr Ashenhurst's experiments. The statements enunciated by him
have been repeated by lecturers, and have figured in text-books
and examination papers for over twenty years, until textile
students are beginning to consider these statements as absolute
as the inverse square law of gravitation, while practical men
rock over to the other extreme, treating the whole matter as
theoretical humbug, and people generally do not trouble to in-
vestigate the subject further. This course is clearly not in
accordance with the scientific spirit of inquiry permeating other
branches of industry at the present time. While all honour is

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484 Proceedings ofltoyal Society of Edinburgh, [sess.

due to the memory of Mr Ashenhurst in connection with his
pioneer labour in this field of research, to recognise that it was
only a forward step in the evolution of a difficult subject in no
way detracts from that honour. Textile students would do well
to consider the foundation on which Ashenhurst's assumptions
rest, and to investigate the limitations to which they are subject,
as set forth in his own words in the second section of his TextUe
GcUculcUions ; so that by the aid of experiment and reasoning
the next twenty years may be more fruitful in results than the
same period which has just passed.

As the author's experiments on the absolute diameters of threads
do not admit of generalisation at the present stage, we shall pass
on to consider what is the main subject of this paper, viz., the
diameter of a twisted thread compared with the diameter of its
component singles. The subject is admittedly a difficult one
both on the analytical and experimental sides, which may
doubtless have deterred textile writers from discussing it. But
it is, nevertheless, a logical consequence of Ashenhurst's teaching.

Single threads for purposes of calculation may be assumed to
be flexible cylinders if not subjected to lateral stress, since to this
form single threads approximate according as they approach per-
fection in structure. Writers on textile calculation have always
tacitly reckoned twisted threads to have the same form also, in
order to avoid the mathematical difficulties which more complex
forms must introduce. If the thread is twofold, i.c. consists of
two threads twisted together, then its diameter is considered to
be the same as the diameter of a single thread of twice the weight
and volume per unit length, or twice the sectional area. A little
consideration, however, will show that this is an erroneous idea,
and sufficient in many cases to vitiate the results arrived at. It
is very evident from fig. 1 that a twofold twist consists of two
spirab interlocking each other, a form differing very markedly
from that of the cylindrical single thread.

The dimension of a thread which is of practical importance in
the theory of cloth-setting is its horizontal projection, since in all
ordinary cases cloth is constructed by the interlacing of two series
of threads which cross each other at right angles. The series
which is stretched lengthways in the loom is called the "warp,"

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1904-5.] Mr T. Oliver on Diameters of Tvnsted Threads, 485

while the other series, which interlaces the warp transversely
according to some definite scheme or weave, is called the '^ weft."
Therefore the number of threads which can be crowded into a
given distance in a horizontal plane, i.e. into cloth, must be
dependent upon the horizontal dimensions of the threads. If a
single thread is stretched horizontally, it is evident that its

^ c 3
Fig. 1.— Horizontal Plan of Thread.

horizontal projection is a rectangle if perfectly even spun, but in
the case of a twofold twist the outline of the projection consists
of two overlapping curves, each of which will be readily recog-
nised as a curve of sines.

At section A of fig. 1 the maximum width = two diameters
of the single thread; at section B, the minimum width = one
diameter only ; while between A and B the projection width
assumes every value from two diameters to one diameter as we
pass from A to B.

Fio. 2, — Section A.

Fig. 3.— Section B.

In passing beyond B on to D it is evident that the same values
will be reached, but in the reverse order, until at D the projection
width is again %1, where d = the diameter of the single thread.
The next part of the problem is to find the average horizontal
projection, because if we warp a large number of threads or weave
a large number of picks (as the weft threads are technically
termed) side by side, the probability is that the broad parts of

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486

Proceedings of Royal Society of Edinburgh, [

some of the threads will come against the narrow parts of others
in such a way that they will average up and fill the same space as
an equal number of threads of a hypothetical yam uniform
throughout its length and with a diameter equal to the mean
projection width of the real yam. To find this mean value we
may proceed in one or other of two ways. (1) The most ex-
peditious method is to employ the integral calculus. We may
consider, for purposes of calculation, that the twist is generated by
keeping one thread stationary and rotating the other about the
axis of the first as centre. Proceeding from section A to section
B, the angle of twist grows from 0* to 90°, %,e, through \ turn of
twist.

Fig. 4.— Section C.

If we call the angle of twist 6 and consider any intermediate
section C, the horizontal projection is AD or AB + CD + BC,
but AB + CD => e2, the diameter of the single thread,
andBO = rf
.-. BC = <icos^
.-. AD = ci(l+co8^)
And the sum of all the sections = c2 / (1 + cos 0)dd between the

limits ^ = 0 and 6 = 90* or ^ radians,

or cz|5 (1 + cos e)de = c/l"^ + sin ^1^ = d(l + 1)

Integral |+1 .
. *. the mean width of projection = ^ = rf = M -|- ? \<f

^ 2

= 1-63W

A graphical method of solving the problem, — The following
graphical method will be intelligible to those who are not familiar

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1904-5.] Mr T. Oliver m Diameters of Twisted Threads. 487

with the calculus. Plot to a large scale on squared paper the
values of ^ as ahecissae and the corresponding values of AD as
ordinates, and draw a curve through the tops of the ordinates in
the usual way. The values of AD may be found by drawing
figures for the ten values of 6, i.e, 0\ 10', 20" ... . 90', and

4. — —

Fio. 5,

^ynMxk/tu

measure off the lengths for each case, or the values of cos 6 may be
taken from a four-figure table of cosines.

The area inclosed by the base line, the curve, and the two end
ordinates may be found by the planimeter, or any of the rules for
summing areas in mensuration. Of the latter, the mid-ordinate
rule, being the simplest and sufficiently accurate, might be used.

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488 Proceedinffs of Itoyal Society of Edinburgh, [ana.

The mean projection width = the mean value of the mid-
ordinates of the nine strips into which the diagram is convenientlj
divided.

'~(l-996 + 1-966+ 1-906+ 1-819 + 1-707+ 1-574 + 1-423

+ 1-259 + 1-087)
= l-637d.

Now, if the twist had been taken as equivalent to a single thread
of twice the sectional area of one of the component singles, the
conclusion would have been arrived at that the projection width
= J2d or l-414t?. Thus an error of about 14 per cent, would
have been made, following the usual assumption.

Fio. 6.

In practice, however, it will be found that the discrepancy is not
so great as shown above, because, for the sake of simplicity in intro-
ducing the subject, a hypothetical case has been considered which
would never arise in practice, t,e. an unstretched thread. When
yarn is formed into a warp it is necessary that it should be sub-
jected to a relatively large longitudinal stress in order to secure
uniformity in weaving. The result of this is that the spirals in
the twist tend to become straight, and consequently each single
thread exerts a transverse pressure on the other along the spiral
line of contact : in practice, contact takes place along, not a line,
but a surface, the extent of which depends upon the compressi-
bility of the material of which the thread is composed. A thread
also presents this deformation to a lesser degree, even when not
subjected to longitudinal stress. Because, in the process of form-

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1904-6.] Mr T. Oliver on Diameters of Twisted Threads. 489

ing the twist on the throstle frame, the threads are under
considerable tension, which strains the cylindrical singles. When
the stress is relieved, after the thread passes away from the
throstle, the friction between the rough surfaces of the singles
prevent to some extent the natural elasticity of the material from
bringing the thread back to its original form. The single threads
no longer present a circular cross section, but elliptical, with the
minor axes of the ellipses everywhere at right angles to the line
or surface of contact. The mean projection width is now more
difficult to find, since the integral is of a higher order. Section
C is now as shown in fig. 6.

Let BE = a, BF = &, BA = r.

The polar equation to the ellipse when 0 is the angle GEO or
angle of twist is

1 sins^^cos^^

r =

a2 62
ab

cos^^ + ^sin^^

V'-(.-3

siD^e^

,_ where e^ = i _

N/l-e2sin2^ a2

Therefore AB or CD = ,

Jl - e2 8in2 0

and BC = BO cos tf = 2 fe cos ^

But the projection width = AD = AB + CD + BC

= 2r+26cos 6

\ Jl -e^ 8m2 0 /

n/1
Then the sum of all the sections between the limits ^ = 0 and

e= 90' or % radians = 26 [^ / + cos ^V^

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490 Proceedings of Royal Society of JBdiriburgh. [

The first integral is evidently a complete elliptic function of the
first order, and therefore not expressible in terms of elementary
transcendents. For convenience this function will be referred to,
as is usual, by F^ and its value taken from tables or determined
by quadrature for any value of e (or e^ preferably).

.'. the sum of all the sections = 26(Fj + 1)

and the mean projection width = ^r = — (F^H-l)

2
Instead of using tables of elliptic functions, it is instructive to
use approximate methods of solution.

(1) Expanding the radical -— -, -^-^ by the Binomial

Theorem, the series 1 + Je^ sin* ^ + f c* sin* 0+ .... is obtained
which is uniformly convergent from ^ = 0 to 0 = ^ radians, since
e^<l. Integrating this series term by term and using the formula

f^sin'^ 0 ^^^(n-l)(n-3) 1^

' 0 7» (n - 2) . . . . 2 2

the value of the function is obtained as

Vl-e-^sin*^"

which can be easily evaluated for all values of e* and to any
degree of approximation by taking sufficient terms of the series.
From the nature of the problem, it is, however, not only unneces-
sary but misleading to use more than three or four significant
figures.

(2) The graphical solution, — Calculate the value of the expres-
sion 2h( ,--==-+ cos ^) for 10 values of $, viz., 0*, 10",

xvl-e^sm^^ /

20** ... . 90*, keeping e^ constant, say '1. Plot these values as
ordinates and $ as abscissae. Draw a curve through the plotted
points. The mean height of the diagram gives, as before, the
mean projection width for e'^—'l. Plot out the results on the
same sheet for e^ _ -2, -3 . . . . and the different curves on the
same diagram will render evident to the eye at a glance how the

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1904-5.] Mr T. Oliver on Diameters of Ttoisted Threads. 491

projection width varies with the square of the eccentricity of the
elliptical section.

These curves are shown in fig. 7.

The comparison of these results with that obtained by con-
sidering the thread in its unstrained condition is beset with
difficulties. The volume of the thread must necessarily be less

Fio. 7.

in the strained than in the unstrained condition, because (1) the
yarn will stretch and thus decrease its sectional area; (2) each
single thread is subjected to lateral compression. The latter cause,
however, will not greatly affect the volume unless the twist is
hard, as the fibres are free to a considerable extent to move
away from the surface of compression. The amount of this com-
pression cannot be arrived at by a priori reasoning, but must be

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492 Proceedings of Royal Society of Edinburgh. [sbss.

the subject of experiment. The results of the author's experi-
ments give reasonable ground for the belief that the law of
compression is such that a + 6 = (f to a first approximation if e^
is not > '6, where a and h are the semi-major and semi-minor axes
respectively of the elliptical section, and d the original diameter
of the unstrained single thread; In any case it is instructive to
work out the results for this hypothetical case. This is practi-
cally equivalent to reckoning the perimeter constant if e is not
large.

Proof, — The perimeter of an ellipse = 4a/* ^(i -e'sin^B)dO

which is a complete elliptic function of the second order, values
of which may be obtained from tables for values of e and 6> But
as the compressibility of the material is not known exactly, it is
unnecessary to work with exact values.

Vl-c2 8in2^=l-Je2 gin2 0-.^ gin* ^ .... (by Binomial
Theorem).
Integrating term by term between the limits ^ = 0 -I- ^ = !^ radians.

The perimeter = 2 ira (1 - \e^ - ^^ . . . .)

Neglecting all powers of e of the fourth and higher degree

= Tra + Tra I

= 7r(a + 6) •.• b^^ajl^^

/>2

= a n - ^ j approx. when e

is small, and if a + b = d

then TT (a-|-^) = 7r(/ a constant, viz., the original circumference of

the single thread.

Substituting for a in a -\-b = d

o = .; ,- — - a . a = - —r

.*. mean projection width of strained thread = -(F| + 1)6

= *(^^+i)r^^'^

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1904-6.] Mr T. Oliver on Diameters of Twisted Threads, 493

F,+l

Mean projection width
BectiomJ area of thd.

Tables of Functions.

•1 -i -3 -4

1671

1^612

1-660

1-713

1-777

1^64

1^960

2-076

2-267

2571

2-612

2660

2713

2777

2-864

2^950

3-076

3267

8-274

S-826

3-388

3465

3-637

3-634

3-767

3-917

4148

-9

-8

•7

•6

•4

-8

•2

•9487

•8944

•8367

7746

•7071

•6826

•6477

•4472

1-9487

1-8944

18367

1^7746

17071

1-6826

1-5477

1-4472

•6

•4867

•4722

'4665

•4364

•4142

•3876

•8638

•3001

lM7d

l-619d

l-600d

l-674d

1646d

1605d

l-466d

1887d

l-282d

S-799

3-687

3-875

3-150

2^914

2-662

2-397

2-094

•2600

•2498

•2493

•2479

•2469

•2427

•2376

•2285

•2186

•2500

•2498

•2493

•2479

•2469

•2427

•2376

2286

•2186

The sectional area of thread = vab =

1-62

^.d^ ^'-_±

jY^^ (1 + vr^>

Table showing the variation in the width of projection from
e^ = 0 to a^ = '6 through \ turn of twist (when a+h^d).

Values of ^2

Valufifi of 0

•1

•2

•3

•4

•5

'« 1

0-*

2-OOOci

l-947rf

l-889<i

l-822(i

l-746(i

Vmd

Vhhdd

10''

l-985rf

l-984rf

1-877(3?

l-816rf

i-nu

l-652(«

Vb^hd

20'

l-940rf

l-894d

I'^AU

vmd

in^d

l-633ti

1-532(3?

SO**

l-866d

l-830rf

l-786(i

vmd

l-676(i

1-604(3?

1-512(3?

40''

1 -766(3?

l-740rf

V1\0d

1 -671(3?

\mid

1-565(3?

1-486(2

50"

l-643rf

1629(i

reisc?

1-590(3?

l-659(i

Vb\M

l-460d

60''

l-500rf

1*499(£

l-497(i

l-490(i

1-480(3?

l-468rf

1-432(3?

70**

r842d

1-851(3?

l-862rf

I'ZIU

l-384(i

1-398(3?

1 -395(2

80'*

1 -174(3?

l-194rf

1-217(3?

1-241(3?

l-268(i

1 •299(i

1-338(2

90''

1 -OOOc?

l-026ci

l-055(i

l-091(i

\\21d

ll72rf
l-505(i

1-225(2
1-456(2

Mean Valaes

l-637(i

\'^\U

l-600(^
1-79

\-f>l\d
1-67

1 -546(^
1-55

Maximum Value

2-00

1-90

1-41

1-26

Minimum Value

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494 Proceedings of Royal Society of Edinburgh. [:

The experimental work of this subject has been greatly facilitated
by accessories invented and added to the microscope by Mr George
R. Smith, of Bradford, about three years ago. The complete

Ha
u

Fig. 8.

Carve A shows the mftximum values of the pFojectlon width as ^ changes.
„ B „ minimum „ „ „

I, C ,, mean „ „ „

,, D ,, ratio of the maximum to the minimum.

apparatus is shown in fig. 9. A frame is fixed in grooves under
the stage of the microscope, and it can be moved to and fro by
a rack and pinion. One end of the frame carries a bell crank lever
neatly pivoted, the upright arm of which carries a jaw for securing

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1904-6.] Mr T. Oliver on Diametei^s of Twisted Threads, 495

one end of the thread, while the other consists of a notched lever
on which a weight can be moved along to produce the required
tension. The other end of the frame carries a sliding jaw, which
can also be rotated by a handle, and the rotations indicated by
a counter. Any length of thread from half an inch to four inches
can be operated on, the sliding jaw being drawn back to any of
the numbers on the base under the stage. The number of turns

Fio. 9.

of twist is indicated by the counter when all the twist is taken
out by turning the sliding jaw. The twist can also be varied at
will by the same arrangement. The diameter of the thread is
measured by means of an eye-piece micrometer, which is much
better for this purpose than a stage micrometer, as with the latter
it is impossible to bring the image of the widest part of the
thread to coincide with the image of the scale if the thread is
moderately thick. Another advantage of this instrument is that

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496 Proceedings of Royal Society of Edvnhurgh. [sbbb.

the whole length of thread may be moved across the field of view
of the microscope by the rack and pinion underneath the stage.

The following tables show the results of micro-measurements
on three representative yams selected from a large collection, the
general tendency of which is to confirm the theory discussed in
this paper. The numbers are in micrometer divisions, each of
which = -00618 inch. But as the subject is only relative, i.e.
the comparison of a twist thread with a single thread, it is
unnecessary to translate the readings into absolute measure. The
three yarns selected are, (1) a 2/368 worsted with 16 turns per
inch, (2) a 50-cut 2-ply woollen yarn with 9 turns per inch, (3)
a 2/ 20s cotton with 9 turns per inch.

(1) 2/368 Worsted

(2) 50-cut 2-ply Woollen.

Minimum

Uaximum

width of ,

width of

Projection. 1

Projection.

1-98

2-92

2 12

3-02

2-08

310

2-15

3-25

2-20

3 '36

216

3-30

2-10

3 25

210

3-25

2-15

3-26

218

815

10)21-16

10)31-84

212

3 18

average

(3) 2/208 Cotton.

Minimum

width of

Projection.

1-54
1-88
2-05
2-00
1-96
1-72
1-86
1 82
1-72
1-65

lo)l8"

1-82
averaj^

Maximnm

width of

Projection.

2-61
2-68
2-68
2-60
2-55
250
2*55
2-56
2-50
2-60

10 26 62

2-6«
averagp

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1904-6.] Mr T. Oliver on Diameters of TvnsUd Threads. 497

(I)

(2)

(3)

From
experiments.

From
graphs

of
fig. 8.

From
experiments.

From
graphs

of
fig. 8.

vnd

I'lid

From
experiments.

From
graphs

of
fig. 8.

Maarimnm .
Minimum

Maximum

Minimum

Avera|^ diameter
of single .

1 '48 divisions

l-58rf
\'2\d

»18 = l-60
2-12

1-88

1-88 divisions

1-52

1*52 divisions

l-66rf
1*1 7rf

The author is indebted to the Camegie Trust for the Universi-
ties of Scotland for a grant to meet the expenses of this research.

{Isnied separately April 8, 1905.)

PKOC. ROY. SOC. BDIN. — VOL. XXV.

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498 Proceedings of Koyal Society of Edinburgh, [i

A Study of Three Vegetarian Diets. By D. Noel.Paton
and J. C. Dunlop.

( From the Research Lahoraiory of the Royal College of
Physicians^ Edinburgh,)

(MS. receiyed February 24th, 1905. Read March 6th, 1905.)

The recent publication of Prof. Chittenden's PhysiologiccU
Economy in Nutrition tends to establish a new standard of
dietary requirements, if not for the labouring classes, at least for
men, middle-aged and young, who are not undergoing continued
and sustained muscular work.

He records a prolonged series of observations upon himself and
on his colleagues, representing professional men, on soldiers and
upon student athletes. In the first class, health and undiminished
working capacity were sustained for 7 to 9 months on a diet con-
taining only about 46 grms. of proteid per diem, and yielding only
from 1550 to 2530 Calories of energy. In the group of soldiers,
44 to 50 grms. of proteid and from 2500 to 2800 Calories of
energy were sufficient to maintain their working power ; and in the
case of the students 55 grms. of proteid and under 3000 Calories
of energy were found to be sufficient to meet the dietary require-
ments of men in training.

From the fact that most of the diets of those able to select
their food contain at least 100 grms. of proteid, it has been,
perhaps too readily, assumed that this amount of proteid is
essential for the maintenance of health and a good state of
muscular activity. Chittenden has certainly shown that adult
men not subjected to sustained muscular exertion can maintain
themselves in a state of good muscular development on less than
half this amount. He does not, however, touch the question of
whether, in growing children, pregnant women, and labouring
men, it is advantageous or, indeed, possible to reduce the proportion
of proteids in the diet to anything like this extent.

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1904-5.] A Study of Three Vegetarian Diets. 499

It is not our purpose here to consider this aspect of the question,
but we think that the new light thrown upon dietetics by
Chittenden's book makes the study of what might be considered
atypical diets of considerable interest.

In the diets recorded by him, vegetables, as might be expected,
figure very largely, and while in all of them the amount of
animal food is lower than is usual, in some of the diets vegetables
almost entirely replace animal products.

As a result of the publication of our Dietary Studies of the
Labouring Classes in Edinburgh in 1898, the opportunity has
been presented to us of studying three very atypical vegetarian
diets, which had been selected by their consumers for what
appeared to them reasons of health and economy, and they seem
to us to present features of sufficient interest to warrant their
publication.

The first illustrates the danger of a refusal to accept the very
evident fact that the food must supply the necessary energy for
work ; the second records what, in the light of Chittenden's work,
might be considered a very liberal diet, but illustrates one of the
difficulties of vegetarianism ; while the third reveals the diet of
a vegetarian glutton, and shows how the res angusta domi have
produced a reformation.

Study L

The subject of this study was a retired professional man. His
theory is that most men overeat themselves, and that the less
a man eats the better and the stronger he is. His physical
condition does not support his theory. He is in a state of emacia-
tion, and his appearance is more that of a man suffering from
some wasting disease than that of a man in robust health. His
height is 5 feet \0\ inches; his weight at the commencement of
the week's observation was only 52 kilos. — about 40 per cent, less
than the normal for his height.

The food which he selected for himself during the period of
observation, as suitable for the maintenance of health, was banana
and hot water. The quantity of banana he consumed during the
five days was 9| lbs.; on four of the observation days he ate one
pound of the bananas at about 8.30 a.m., and a second pound at

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500 Proceedings of Royal Society of Edinburgh. [i

about 3 p.m., taking a little hot water twice or thrice in the 24
hours. During the observation period he reported that he was
feeling well and satisfied, but on the last day allowed that he had
not slept well, that he was feeling hungry, and that he would
appreciate a change to a diet containing some bread and butter.
After five days of the banana diet his weight was 50 kilos. — a
loss of 2 kilos.

The food- value of his diet amounted for the five days to:
proteid, 37*5 grammes ; fat, 3*5 grammes; and carbohydrates, 999
grammes, the equivalent per man per day being : —

Proteids .7*5 grms.

Fats .... 0-7 „

Carbohydrates .... 199-8 „
Calories . . .856

His excretions were carefully analysed during the period, and
the results of the analyses are shown in the following table : —

Urini.

Quantity, c.c

Specific gravity
Reaction, on each day alkaline
Total nitrogen, grammes
Urea nitrogen, grammes
Ammonia nitrogen, grammes
Uric acid, grammes
Non-urea nitrogen, grammes .
Phosphoric acid, grammes

Dry weight, grammes
ToUl nitrogen, grammes

Intake.

Food

1st

2nd

8rd

Day.

1040

960

500

1014

1020

3-92

4-26

2^96

330

3-68

2-26

•112

•095

•061

•308

•804

•345

•62

•67

•70

•88

•80

4th
Day.

5th
Day.

Average.

460

860

762

1022

1012

285

2^€8

323

2-35

2-68

2-88»

•084

-095

•09

833

•828

■82t

•78

•76

1-00

•84

•88

I Per

cent, of I
Total N.l

25
80

F.fi01CS.

I 28-2
I -81

29-2
102

29 1 29-5
156 112

30-5
1-21

22-5
114

NiTRooBM Balance.

1-21

Output.

Urine
F»ces

328
1*14

4-87

Food Analtsis.
Bananas, Proteid, 0*87 ; Fat, 0*06; Carhohydrate, 28*17 per eent

* Average urea, 6 2.

t Or 0-106 grm. N.

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1904-6.] A Stiidy of Three Vegetarian Diets. 501

The more uoteworthy points about this study were : —

1. The extreme smaUness of the diet The caloric value of it
was only about one-fourth of the normal diet for moderate labour,
and the proteid value was only about one-twentieth of the normal.

2. The urine was alkaline throughout the entire period. It
was more strongly so on the third, fourth, and fifth days than on
the first two days. This alkalinity was due to the food being
purely vegetable.

3. The excretion of nitrogen was very similar to that found in
total starvation. Of the total nitrogen, only 80 per cent, was
excreted as urea, a proportion less than the normal. The total
amount of non-urea nitrogen was less than the normal, but was
relatively not so much reduced as was the excretion of nitrogen
in urea.

4. The excretion of preformed ammonia was very small. This
may be ascribed to the presence of excess of alkali and to the
comparative absence of organic sulphur in the food.

5. The nitrogen balance was decidedly negative, and indicated
a daily average loss of 19*8 grms. of tissue proteid, or about 100
grms. of flesh.

Study IL

The subject was a woman aged forty-two, a typist, who had for
a long time been a modified vegetarian. The study was made at
the same time and in the same way as our studies of the diets of
the labouring classes of Edinburgh. She stated that she was
strong and well, and able for a large amount of exercise, that she
habitually bicycled and walked long distances. She always sat
with the window of her room open, and did not feel cold. The
study extended over a period of one week.

The food she used during the period was as follows : — Butter,
20 oz. ; milk, 60 oz. ; eggs, 8 ; cream, 10 oz. ; cheese, 3 oz. ;
bread, 32 oz. ; brown bread, 22 oz. ; cakes and pastry, 50 oz. ;
chocolate cream, 2 oz. ; sugar, 13 oz. ; jam 11 oz. ; potatoes, 46
oz. ; fresh vegetables, 24 oz. ; prunes, 16 oz. ; bananas, 21 oz. ;
oranges, 29 oz. ; and apples, 8 oz.

The food principles in such a diet are estimated by us to
amount per week to: proteid, 406*5 grammes; fat, 896*1

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502 Proceedings of Roycd Society of Edinburgh. [se

grammes; carbohydrates,

2923

grammes.

The equivalent (

per man per day is: —

Proteids

73*6 grms.

Fats .

. 160-0 „

Carbohydrates .

. 5220 „

Calories

. 3926

Here a fair energy value is yielded by a large supply of
fats and carbohydrates, while the proportion of proteids is un-
usually low.

The point of special interest in this diet is the very large
amount of fat and carbohydrate food taken to get the necessary
energy, an amount which many persons would find it difficult to
digest.

The cost of the week's diet was 12s. 4d., or equivalent to
14s. lOd. per man per week, or 25 J pence per day. The ordinary
labourer's family in Edinburgh gets a larger supply of proteid
and a fair supply of energy for about 7d. per day.

Study II L

Along with a cutting concerning our Dietary Studies from the
South-Eastern Advertiser of 24th February 1900, we received a
letter from a Mr H., of which the following is an extract : —

\Uh October 1900.
Dear Sir, — After reading the above, it occurred to me that I
might as well send you a copy of my half-year's expenditure.
.... I cannot possibly be called a typical person ; but there
are so few people who do keep exact records of what they eat,
drink, and spend, that I suppose scientific men are glad to get
such records from almost anybody."

With this letter was a very full and detailed budget of C. H.'s
income and expenditure, and a detailed statement of the food
consumed during the six months from Ist April to 30th September
1900. It is unnecessary to publish this at length. The following
list contains the articles of importance, and the quantity of each
used, the quantities being expressed in kilogrammes : — Apples, 1 5-88 :
cherries, 0*45 ; bilberries, 045 ; strawberries, 0*45 ; melons, 8'00 ;
red currants, I'OO; gooseberries, 2*50; oranges, 20*00; lemons,

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1904-5.] A Study of Three Vegetarian Diets. 503

4-00; tomatoes, 3'37 ; monkey nuts, 94 00; hovis bread, 84-00;
other breads, 1000; nucoline, 9'50 ; quaker oats, 20*00; sugar,
4100; nut butter, 1*00; jam, 15*00; golden syrup, 0*90;
cocoa, 0*50; coffee, 0*15; peas, 1000; lentils, 4*70; onions,
3-20; carrots, 0-20; radishes, 4*00; rhubarb, 4*00; biscuits,
3'00; chocolate, 110; peppermints, 010; eggs, 0*30; condensed
milk, 1*50; lemon squash, TOO; nutta, 0*50; plasmon, 0*20;
yeast, 0*07 ; bananas (dried), 0*40.

The food-value of such a diet has been estimated by us, and
it is found that its value is per man per diem : —

Proteids 2303 grms.

Fats .... 275-3 „
Carbohydrates . 7342 „

Calories . .6514

Of the total energy, 8 '5 per cent, is derived from animal food,
and 91*5 from vegetable food. The cost of the diet for the six
months was £8, 18s. lid., which is equal to Gs. lOd. per man
per week, or 11 -7^ per day.

Even supposing that this diet is over-estimated by 10 per cent.,
it is still Gargantuan, yielding over 200 grs. of proteid and 5800
Calories of energy. From the observations of Avsititkiski, of
Dunlop upon prisoners, and of Noel Paton on dogs, it is almost
certain that a great part of this enormous diet was not digested
and absorbed, and was therefore not available.

When putting together our results, we wrote to Mr H. as to
the enormous amount of food consumed, and he writes, under
date 5th February 1904 :—

"I must own, however, that I am a larger eater than most,
indeed, a glutton. Everyone has his own physical vice, and I
make up for abstinence from alcohol, tobacco, tea, coffee, meat,
and breakfasts, and for devotion to the morning cold tub, by
overeating myself three or four evenings a week. I always read
at meals, and this tends to make one go on feeding mechanically."

With this letter he sends details of the diet of himself and
of his wife and three children from 1st April to 30th September
1903. He says : — " I do not think I eat quite so much
now as in 1900. I cannot say I have ever suffered much in

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504 Proceedings of Royal Society of Edinburgh. [i

health from overfeeding (though I suppose all physical sins
must be paid for in the long run), but a growing family and
growing debts exert on me a highly beneficial pressure. I enclose
a list of the food I ate April-September '03. The table was
made primarily with a view to cost, but weights can be deduced
from it within probably 5 per cent, of the truth.'*

The diet here recorded is a much more normal one, and con-
sidering the non-availability of the proteid in many v^etable
foods, and the fact that many of the vegetables used contain a
large proportion of non-proteid nitrogen which is here recorded as
proteid, the food consumption is by no means above the average.
The growing family and growing debts have certainly been bene-
ficial so far as his diet is concerned.

The food consumed during this second six-months period was
of essentially the same kind as during the first period, but differed
from the latter in quantity. He had reduced his six-monthly
consumption of monkey nuts from 94 kilogrammes to 131, of
ho vis bread from 84 kilogrammes to 30; but had increased his
supplies of other, more ordinary, breads from 10 to 65. Another
notable change was that he had much increased his supply of
fresh vegetables, using no loss than 22 kilogrammes of carrots,
while during the first period he only used 0*02 of that vegetable.
Here is a list of the food used during the second period,
expressed in kilos: — Monkey nuts, 13-10; roasted peanuts,
0*90; apples, 9*80; oranges, 1*50; lemons, 2*00; tomatoes,
1*40; melons, 5 00; red currants, I'OO; cucumbers, 1*50;
stoned raisins, 180; hovis bread, 30*613; whole - meal
bread, 44*73; malt bread, 16*00; white bread, 5*00; biscuits,
4-00; bannocks, 12*00; cake, 0*30; quaker oats, 0*90
force, 1-80; sugar, 10*5; carrots, 22*20; onions, 3*00
scallions, 1*20; turnips, 8*00; green peas, 2 00; rhubarb, 14*00
radishes, 1*50; jam, 6*30; honey, 0*40; syrup, 9*00; nucoline,
6*30 ; walnut butter, 0*40 ; peanut butter, 0*40 ; cow («tc)
butter, 7*7; cocoa, 0*2; coffee, 0*2; chocolate, 4*0; sweets,
0*6; eggs, 1*40; Briggs' food, 0*40 ; orange wine ; plasmon, 0*4;
Maggi's soup powder, 0*1.

The total food principles in these six months' rations, as estimated
by us, amount to: proteid, 19,054*4 grammes; fat, 18,981*9;

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1904-5.] A Study of Three Vegetarian Diets. 505

Ksarbohydrates, 93, 689 '0 ; and from that estimate the diet per
-day per man is found to be : —

Proteids
Fats .

Carbohydrates
Calories

104*1 grms.
103-7 „
512-3 „
3497

The cost of the diet for six months was X7, 12s. Id. ; the
•equivalent cost per man per week was Ss. lOd., or lOd. per day.

Considered in the light of the older standards, the diet is here
a very liberal one, while in the light of Chittenden's observations
it may be considered as still excessive.

The diet of this man's wife and children for the period of six
months included the following, quantities being expressed as kilo-
grammes:—Flour, 126; butter, 23*1; 236 eggs; sugar, 271;
potatoes, 144; milk, 204; lentils, 10; bacon, 9*5; and smaller
quantities of bread, lard, cheese, onions, peas, turnips, cabbages,
carrots, rhubarb, radishes, tea, coffee, cocoa, oranges, lemons,
tomatoes, cucumbers, apples, bananas, plums, currants, monkey
nuts, quaker oats, cake, biscuits, jam, sweets, peel, corn-flour,
nut butter, meat, ham, sausages, sardines, and tinned salmon.
The food principles in the six months' rations amount to : proteid,
35,324-6 ; fat, 34,568 9 ; carbohydrates, 172,506-0. Using Atwater's
estimate of the proportional requirements of a man, and of women
and children, we estimate that the requirements of his wife and
family, three children, aged six, four, and two, would amount to
2-1 times that of a man. On that basis we estimate that the diet
submitted is equivalent to a diet per man per diem : —

Proteids . .65*7 grms.

Fats .... 90*2 „
Carbohydrates 444-1 „

Calories .... 2929

The cost of the diet was for the six months JB12, 19s. lOd. ; this
is equal to a cost of 4s. 9d. per man per week, or 8* Id. per day.

It is a diet, largely vegetarian, which meets the requirements
laid down by Chittenden, but which by an older standard would
be considered deficient in proteid and in energy.

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506 Proceedings of Boycd Society of Edinburgh. [i

Conclusions.

On two of the diets studied the suhject was ahle to maintain
health and muscular vigour because the amount of proteid and
energy yielded was sufficient, but in both the cost was considerably
in excess of that for which the labouring classes in town or
country are able to procure an equally satisfactory diet. They are
both essentially wasteful diets, and are not to be recommended for
general adoption.

The study of the ordinary diets of the labouring classes in all
countries seems to show that whenever possible a diet is secured
which will yield something over 3000 Calories of energy and over
100 grms. of proteids per man per diem. It is improbable that so
many different races should have made the same mistakes in the
essential elements of their very varied diets, and we think that the
evidence afforded by these diets cannot be set aside even by so
careful a set of experiments as those conducted by Chittenden.

{Issued separately April 8, 1905.)

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1904-5.] Continuants whose Main Diagonal is Univarial. 507

Continiiants whose Main Dickgoncd is Univarial.
By Thomas Muir. LLD.

(MS. receiyed December 12, 1904. Read January 23, 1905.)

(I) In a recently written paper* dealing with a continuant first
considered by Cayley, it was pointed out that the function in
question owed its complicated law of development to peculiarities
of specialisation, there being a much more general continuant
governed by a simpler law. The theorem enunciated regarding
the latter was : — If A, be written for the sum of all the r-ary pro-
ducts formed from \, bg , . . . . with tJie restriction that no two
conseciUive b'a shall appear in any single product, then

0
-1

- 1

= ^ + Ai^-HA2^-*+ . .

For example, when n = 6 the expansion is

+ b^b^ + bzbA

(2) The curious fact has now to be noted that this theorem
itself can be generalised with a minimum of change in the mode of
expression by altering the 2nd, 4th, 6th, .... diagonal-elements
on the left into <^ and writing 0<f> for $^ on the right, the resulting
theorem being then formulated as follows : —

0 L

-1

= {$<!>)"' + A^{$it>)^- ' + A^{0<l>)^-' +

■.oi^{e<i>r-'-^A,{Oi>r

(I)

n
when w=2m,

when n = 2m - 1 .

* See Messenger of Math, ^ xxxiv. p. 126.

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508

Proceedings of Royal Society of Edinburgh, [

That ^ is a factor in the latter case is evident from a consideration
of the fundamental identity

which shows that if, as is easily seen to be the case, the continuant
of the 3rd order has 0 for a factor, so also must the continuant of
the 5th order, and therefore also the continuant of the 7 th order,
and so on.

(3) The fact that the change from a univarial to a bivarial
diagonal necessitates no change in the coefficients on the right-
hand side of the identity prepares one for an analogous widening
of other theorems in which continuants with a univarial diagonal
are involved. Thus, denoting the continuant in (I) by *„ we have
the important condensation theorem —

*^ =

0<l> + b^

Oi> + b^^b,

0<t^ + b^ + b^

(II)

^«^ + ^Jm-i+^Im-l

^r.-v^e\ 04>+b,-^b^

04>-¥b,^b, b,

bi H-^b^^-b^

I (rr)

> + /'«m-S+^-«l

Dividing *„ as it appears in (II) by the cofactor of its first element
we obtain a continued fraction, and dividing the equivalent con-
tinuant in (I) by the cofactor of Us first element we obtain another
continued fraction : and as, when n is even, the two divisors
differ only by the factor ^, the two continued fractions differ to
the same extent. We thus have

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1904-6.] Continuants whose Main Diagonal is Univarial. 509

. + 6i-

b,b.

¥i.

'+h+^-e4,+b;+b,-

=i,\e+''j b, \

"^•^^TV^.fc,

+ V

1 +

Consideration of the case where n is odd leads to the same result,
— a result given, probably for the first time, by Heilermnan.*

(4) Similarly we have the theorem

n-l <l> 2 .

I . 71-2 0 3

w-3 <l>

(Ill)

= (0<l> - 22) (Oi> - 42) (d<^ - 62) when n is even,

and = 0{$<l> - 12) {O4, - 3-) (^<^ - 52) when n is odd,

— a theorem which degenerates into Sylvester's {Nouv. Ann, de
Math.y xiii. p. 305) when ^ is put equal to 0, It has to be noted,
however, that the mode of proof followed in the case of Sylvester's
theorem, viz., removing the linear factors separately, is now unsuit-
able. A mode of removing the quadratic factors will be found in
the Proc. Roy. Soc, Edin,, xxiv. pp. 105-112.

(5) Thirdly, if we denote the preceding generalisation of
Sylvester's continuant by <r„ we obtain

4> 2

a?-l e 3
. a:-2 if>

n{n-\)
= <r„ 7) {x - n + l)<r,._.

(IV>

n(n-l)(n-2)(n-3)
2-4

{x-n-\r\)(x-n-\- 2)(r„_^

n(n-l)...(n-5),

rjTjTg (ic-n+l)(a;-«+2)

• See Zeitsehriftf, Math, u, Phya., v. (1860), pp. 862-363 ; also Gunther'a
Darstellung der NdherungBwerthe der KeUenbriicfietif p. 75, Leipzig, 1878.

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510 Proceedings of jRoycU Society of Edinburgh. [

— a theorem which degenerates into Cay ley's (Quart. Joum. of
Math,, ii. pp. 163-166) when <f> is put equal to $.

(6) Fourthly, all the theorems given in the paper referred to in
§ 3 are capable of the same extension as Sylvester's. Only one of
them need be quoted in its generalised form, viz. : — If in the con-
tinuant of the n** order

Pn-

h
*

(V)

the difference between the element following any $ and the element
preceding the same he constant, equal to \ Sfiy ; and the correspond-
ing difference in the rows containing ^ be also constant, equal to ^^
say ; then

is a factor of the continuant, the cofador being the similar con-
tinuant of the (n - 2)** order whose minor diagonals are got from
those of the original by striking otit \ , \ from the one and ^ , j8,
from the other.

(7) Fifthly, with the notation of § 4 we find

B ar-n + 2

.r <f> x-n + S

X- 1 ar-n + 4 .

x-2 <l>

+ (^2^V(«-n+l)(a;+l).(aj-n + 2)a:-<

(VI)

where the putting of <f>^0 gives a theorem first published in the
paper referred to in § 1.

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1904-6.] Continuants whose Main Diagonal is Univarial. 511

(8) Proofs of the foregoing six theorems have been purposely
omitted, because the modes of procedure followed in the case of
the original ungeneralised theorems are applicable without altera-
tion to the new theorems. In only one instance, that of (II), does
previous work stand markedly in need of being supplemented.
The first part of it, viz., where n is even, is best dealt with as
follows : —

<>ex*' = ,^ h

-1 4> b.

-10 63 . .
. -1 <^ 64 .
. . -I 0 h

0<l> + b^

-1

-1 4> ;

-1 Oif> + b^ + b^
-1

|<^ 1 . . . .

1 . . . .

. . 1 . .

. . -6, <^ 1

. . . . 1

^
*

^3
-1

-V4

1 + 64 + 65

4> ,

= ^s

*« =

^<^ + 6i
-1

0<l> + b^

-V2
^^^ + 62 + 63

-1
^45^ + 62 + 68

-V4
^ + 64 + 65

0<l> + b^ + b^
Applying the same treatment to * when of odd order we obtain

*7 =

$<l> + b^

^ + 63 + 63

^^^ + 64 + 65

-^<^,

$<l>+h.

— a result interesting in itself, although not the form desired.
Increasing each column by the column which immediately follows
it, we have

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512 Froeudingi of Riryal Society of Edinburgh, [i

^7 = ^ + 61 + 62 ^ + fti + ft, + fts \ K*

i b^ ^ + 62 + «», + ft4 ^ + 2>8 + ^ + ^ft *5 I

and now diminishing the second column by the first, the third
by the new second, and so on, we obtain

► + ^1 + ^2

^ + ^8 + *4

H + h + ^6

. \^<t>.

given in § 2.

» + ^1 + ^2 *s

63 ^ + 63 + 64

^ + 65 + 5^

{Issued ssparaUly April 8, 1905.)

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1904-5.] On Prof. Seeliger's TTieory of Temporary Stars. 513

On Professor Seeliger's Theory of Temporary Stars.
By J. Halm, Ph.D., Lecturer on Astronomy in the Uni-
versity of Edinburgh, and Assistant Astronomer at the Royal
Observatory.

(Read November 7, 1»04. MS. received November 28, 1904.)

Professor Becker's paper " On the Spectrum of Nova Persei and
the Structure of its Bands," recently published in the Transactions
of this Society, contains an interesting confirmation of some results
already pointed out by Messrs Campbell and Wright of the Lick
Observatory,* which seem to be of considerable importance for the
theory of temporary stars. By most careful micrometric measure-
ments of the positions of the bright and dark bands in the photo-
graphic spectrum of Nova Persei, Professor Becker arrives at the
conclusion that all the bands are similar in type, and that the
distances of corresponding maxima and minima from the centres
of the bands are proportional to the wave-lengths. The results
derived from the Lick photographs point to exactly the same
conclusion. It appears, therefore, from these two carefully and
independently executed series of observations, that the chemical
nature of the elements, whose light-vibrations gave rise to the
selective radiations and absorptions noticed in Nova Persei, had
no influence on the appearance of the bands. . According to the
Lick observers, there is no evidence that the structure and char-
acter of these bands were affected by other considerations than that
of wave-length.

This important result appears to necessitate now the exclusion
from our view of those theories in which chemical or physical
properties of the incandescent gases and vapours figure as deter-
mining factors. It seems, for instance, incompatible with the high-
pressure theory advocated by Professor Wilsing of Potsdam, because
those effects of pressure on the displacements of spectral lines
which form the basis of Wilsing's theory are by no means the
* Eighth Bulletin of the Lick Observatory.

PROC. ROY. SOC. EDIN. — VOL. XXV. 33

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514 Proceedings of Royal Society of Edivburgh, [i

same for all gases and vapours.* The identical behaviour of lines
in the spectrum of Nova Persei pertaining to different chemical
elements must be considered to contradict this explanation. For
the same and also some other reasons we cannot perhaps accept
a new theory advanced by Dr Ebert of Munich, t in which
abnormal refraction is claimed as the principal cause of the
peculiar duplex character of the Nova lines. Dr Ebert's con-
siderations are based on the fact that in a medium, the spectrum
of which shows distinct absorption maxima, the index of refrac-
tion changes abruptly in the immediate neighbourhood of such a
maximum, being greater on the less refrangible, and smaller on
the more refrangible side. In his opinion, the light-radiation
perceived in the bands of the Nova is not due to the radiative
energy of the gas itself, but to light originally emanating from Uie
incandescent surface of the star, which is abnormally refracted in
the gaseous envelopes outside in such a manner that bright and
dark bands are formed lying on the red and violet edges of
the lines peculiar to the traversed gases. Three objections may be
raised against this theory. Firstly, Dr Ebert's theoretical intensity-
curve of the bands, as we shall see, differs materially from that
observed in Nova Persei. Secondly^ his theory gives insufficient
account of the presence of the bright bands after the continuous
spectrum had disappeared. For obviously, if the continuous
spectrum is, in Dr Ebert's opinion, the condiHo sine qua nan for
the bands, we are at a loss to explain how these bands could have
possibly outlived, as they have actually done, the incandescence of
the starts surface. Thirdly^ the effect of abnormal dispersion is
by no means the same for all gaseous media. According to the
electro-magnetic theory, it depends on the elastic resistance of
the ions, a force which cannot be supposed to be the same for
different atoms. According to observation, even lines of one and
the same metallic vapour, e.g, sodium, behave quite differently.
Hence the theory of abnormal dispersion seems to offer no
explanation of that similarity in structure and character of the
Nova bands, which, according to the observations, appears to be
a fundamental feature of the Nova spectrum.

* See H. Kayser, Handbiteh der Spectroscopies ii. p. 825.
t Ueher die Spectren der neuen Steme^ K,l^. 8917.

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1904-5.] On Prof, Seeliger's Theory of Temporary Stars. 515

If the results of the Glasgow and Mount Hamilton observations
thus limit our field of search by excluding those explanations in
which the internal properties of the vibrating atoms must be
considered to play an important part, there remains, in my opinion,
only one explanation which is in a priori agreement with the
observed facts, viz., that based on Doppler's principle. Indeed, if
motions in the line of sight are the cause of the peculiar emission
and absorption bands in the spectra of the Novae, the similarity
of their structure, independent of the chemical nature of the
elements, and the strict proportionality of all the displacements
to the wave-length are necessary desiderata. The crucial point
seems to be, therefore, this: On the one hand, observation has
demonstrated that the structure of the bands is governed by no
other conditions than that of wave-length ; on the other hand, of
all explanations, only that based on Doppler's principle accounts
for this fact : hence motions of matter in the line of sight must be
considered as the probable cause of the remarkable spectrum of
temporary stars.

This conclusion has increased my confidence in some theoretical
views published two years ago, by which I attempted to explain
the Nova spectrum. The new facts brought to light by subse-
quent observations, especially by those referred to, make it now
desirable to again publish these tentative speculations in a some-
what modified form, and at the same time to compare the results
of theory with our present empirical knowledge.

Before entering upon the subject, I beg to pass a few general
remarks of an historical character. In the earlier days of star
spectroscopy explanations of the Nova spectrum were pre-eminently
based on Doppler's principle. Theoretical views focussed more or
less round the one conception that motions of radiating and
absorbing matter must be responsible for the observed displace-
ments of the spectral lines. Later, however, doubts began to be
felt as to the correctness of this view. Apart from the fact that the
velocities of matter in the Novae exceeded by far the average
motions in the line of sight commonly dealt with in stellar spectro-
scopy, a serious objection to this view was thought to be found in
the striking similarities between the spectra of all the Novae
hitherto accessible to spectroscopic investigation. A universal

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516 Proceedings of Royal Society of Edinburgh, [aitt,

feature of all these spectra is the appearance of broad bright lines,
bordered on their more refrangible edges by diffuse absorption-
bands. Now, assuming that these composite bands were caused
by two bodies moving in different directions, why should the bright
bands invariably appear on the less refrangible and the dark
bands on tlie more refrangible side? In other words, why
should the body or bodies giving the bright line spectrum
always move from us, while those showing the absorption-
lines should be invariably directed towards us? It was the
seeming inexplicability, on the ground of Doppler's principle, of
this universal phenomenon which led astrophysicists to search
for other explanations, such as high pressure and abnormal
dispersion.

£ut if we look more closely into the question, the reasons for
abandoning Doppler's principle seem by no means so convincing
as they were thought to be. The position was abandoned before
the field was thoroughly reconnoitred. Indeed, I shall endeavour
to show in the subsequent remarks that an explanation of the
extremely complicated spectrum of new stars based on Doppler's
principle is not only possible, but also sufficiently probable, on
account of the simplicity of the underlying hypotheses on the one
hand, and the satisfactory agreement between theory and observa-
tion on the other.

The leading idea upon which these considerations are based is
well known to astronomers through Professor Seeliger's ingenious
investigations: my present contribution is indeed merely an
extension of the celebrated theory which we owe to this dis-
tinguished astronomer. Seeliger's hypothesis, which ascribes the
outburst of a new star to the collision between a dark solid body
and matter of a nebular constituency, has so far not been worked
out in detail, so that no definite conclusions have been formed as
to the motions of the matter involved in the catastrophe. In a
general way, however. Professor Seeliger draws attention to the
important r6le performed by the star's gravitational attraction on
the approaching nebulous matter, a consideration we often find
seriously neglected in subsequent investigations. He remarks
that, OS the star approaches the nebulous cloud, the latter, through
the action of gravitation, will extend out to meet it. The attracted

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1904-6.] On Prof, Sediger's Theory of Temporary Stars, 517

cloud particles, in obedience to gravitational laws, describe hyper-
bolic orbits round the star's centre as focus, in exactly the same
manner as do those meteoric swarms of our own system which
have been launched upon us from the remote recesses of space.
The idea occasionally met with in papers on this subject, that the
star penetrates into the cloud as a bullet pierces the air, is
quite erroneous. Its fallacy is so obvious that I need not dwell
upon it.

The hyperbolic paths described by the attracted particles are of
course extremely different in shape and position, forming a chaos
of motions which to unravel seems at first sight a hopeless task.
But, fortunately, at least one definite conclusion may be drawn
which is of vital importance for our problem. We know that the
character of the conic section described by a body round a centre
of attraction is perfectly defined by its velocity V at any point of
the orbit. The body describes

an ellipse, ifV2<?^
r

a parabola, ifV2 = ?^ fi^k^ (M + m)
r

a hyperbola, if V2>?^

where r is the radius vector at that point (expressed in units of
the mean distance O - 5 )> ^ ^1<^ Gaussian constant, and M, m
the masses of the attracting and attracted body in units of the
solar mass. Now, in our case a collision between the star and a
meteoric particle must occur in all instances where the perihelion
distances are less than the radius of the star. Such particles will
impinge upon the surface. But impact means loss of energy of
motion (molar energy), which is converted into kinetic (molecular)
energy, i.e. heat. Hence V, the orbital velocity, must be smaller
after the impact than it had been before. In other words, the
impact-friction on and near the star's surface, by converting a more
or less considerable portion of energy of motion into energy of heat,
acts as a resisting medium, with the effect that in many cases Y

becomes less than — , i.e. that the hyperbolae are transformed
into ellipses.

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518

Proceedings of JRoyal Society of Edhiburgh, [sbss.

An example may illustrate this simple reasoning. Suppose a
swarm of meteors approaches the star 0 from A in hyperbolic
orbits. The perihelion distance of the inner particles is assumed
to be less than the star's radius R. These particles must im-
pinge upon the star's surface, where their further career will be

checked ; i.e, V^, which was greater than -^ immediately before

the impact, will be zero after the catastrophe, supposing that the
whole orbital motion has been transformed into heat. On the
other hand, the orbital velocities of particles grazing the sur-
face, though impeded by surface friction, will undergo much

smaller reductions, while bodies sufficiently removed from the
star may pursue their hyperbolic paths practically undisturbed.
Hence we notice a gradual transition in the values of V from zero
to hyperbolic velocities, so that the swarm, although arriving at
the star with practically uniform velocity, exhibits after the impact
the most heterogeneous motions of its individual members. These
new motions determine the character of the orbits described by
those particles which are at all capable of escaping the star after
impact. Since V may have all possible values, the new orbits
contain all possible conic sections, from the circle to the hyperbola.
The important point is, that many of these new orbits are closed,
the particles becoming permanently attached to the system of the

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1901-5.] On Prof, Seeliger*s Theory of Temporary Stars, 519

star. In other words, as a consequence of the collision, the star
l)ecomes permanently surrounded with a ring of luminous meteoric
matter, revolving in ellipses with eccentricities probably ranging
from zero to unity. The transformation of molar into molecular
energy must lead to incandescence, which will be in proportion to
the amount of converted energy. But this latter is evidently
greatest in the case of circular orbits, because here the reduction
of V from its original hyperbolic value is most considerable. Hence
the brightest parts of the ring are composed of particles moving
round the star in ellipses of small eccentricities.

Now, we cannot avoid the conclusion that the kind of collision
here described must occur in the case of a new star, provided that
Seeliger's fundamental assumptions be true. I can imagine only
one exceptional instance to which the above reasoning would seem
inapplicable, viz., that the cloud particles move towards the star
exactly in the direction of its centre, but I think the scarcity of
such a phenomenon will at once be admitted. The most probable
assumption is that of a more or less one-sided collision, such as is
represented in fig. 1. Granting the reasoning so far, we conclude
that after the catastrophe the star is surrounded by radiating
nebular (meteoric) matter revolving in closed elliptical paths
round the star's centre as focus, the brightest nebular particles
describing orbits of small eccentricities.

The result in this general form is sufficient to assist us later on
in the interpretation of the Nova spectrum. With regard to the
constituency of the luminous ring, the most general assumption
is that it consists of a mixture of bodies in all three states of
aggregation — solid, liquid, and gaseous. But owing to their high
power of radiation, the liquids and solids will cool down much
sooner than the gases, so that in a more advanced state the spectral
appearance of the ring will be that of an incandescent gaseous body
emitting a line spectrum.

The problem, in its main principle, is seen to be closely related
to Encke's celebrated theory of a resisting medium. A force
acting near perihelion in the direction of the tangent against the
orbital motion of a body causes a progressive (secular) diminution
of the major axis and eccentricity of the orbit, and therefore tends
to incorporate the body into the system of the attracting centre.

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520 Proceedings of Royal Society of Ediriburgh. [sbss.

which otherwise might not he capahle of hringing the cosmic
invader under its permanent gravitational sway.

Still another consequence of importance, however, must he
drawn from those fundamental considerations hy which Professor
Seeliger was guided in the framing of his theory. We admit that
the development of heat at the surface of the star must be
enormous, even granting the most unfavourable assumptions as to
the tenuity of the impinging cloud. We may safely assume that
the amount of heat developed during the bombardment may have
exceeded many times that expended by the sun during a corre-
sponding time. This fact seems to warrant the conclusion, not
only that the surface of the star is rapidly liquefied, but also thai
from this surface of molten lava an incessant escape takes place
of molecules with extremely high velocities, leading to the
formation of an expanding incandescent atmosphere of vapours
and gases. This unquestionable fact of an expanding atmosphere
has, so far, not been considered in theories of temporary stars.
Is there reason for neglecting the influence of its motions on the
appearance of the lines in the Nova spectrum ? There can be little
doubt that the gaseous molecules escaping from the liquid surface
of the star would tend towards a state of eqiiilibrium such as is
presented in the gaseous envelopes surrounding the photospheres
of ordinary stars. The height of this * atmosphere ' is determined
by gravitation on the one hand, and by the surface temperature
on the other. If, for instance, we assume the mass and radius of
the star to be equal to that of the sun, and its surface temperature
to that of the solar photosphere, then the atmosphere would most
probably assume the dimensions of the solar chromosphere, pro-
vided that it contains the same gaseous materials. If, however,
the surface temperature of the new star equals that of the photo-
spheres of so-called * white ' stars, which, as we know, possess very
extensive atmospheres, its gaseous molecules would tend to form
an envelope of similar dimensions. Now, I have shown in a
paper in the Astronomische Nachrichien^ Nos. 3822-3, that the
extension of stellar atmospheres must be supposed to increase very
rapidly if the surface temperature is raised. On the other hand,
as shown in the same paper, our assumptions as to the tempera-
ture of new stars immediately after the collision are practically

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1904-5.] On Prof. Seeligers Theory of Temporary Stars, 521

unlimited. I have mentioned already that the surface tempera-
ture of a Nova may exceed many times that of the solar photo-
sphere.*^ Hence there is no reason to contradict the assertion that
the atmosphere of a new star, after the catastrophe, may assume
dimensions surpassing considerably even those presented in the
white stars. Indeed, this atmosphere may even extend infinitely,
for it is well known that when the temperature of the surface
exceeds a certain critical value, the height of the atmosphere
above the surface must become infinite, i,e, gravitation then
proves insufficient to counteract the continuous dissipation of the
gases into space. As is shown in the paper referred to, this state
of matters may happen already at a comparatively low tempera-
ture, exceeding not many times that of stars of the Sirian class
(/.c, 118). Now, in this peculiar case of infinite expansion, the
initial velocities of the gaseous molecules at the surface must have
been greater than the so-called critical velocity of the star (t.6.
610 km. per second if the sun's mass and dimensions be assumed).

* Some estimate of the amount of heat develojied by the impact may be
gained from the following consideration. Supitose the materials of a cosmic
cloud to fall from infinity upon onr sun. The velocity V with which the
cloud particles arrive at the sun's surface is hyperbolic, and therefore
greater than 600 km. per second. Now we know that 1 kgr. matter moving
with a velocity of V metres per second, if completely stopped, develops a

quantity of heat which equals -j^* «" calories. If, then, a quantity of

8330

cosmic matter weighing 1 kgr. at the surface of the earth would impinge

upon the sun with parabolic velocity (Hbout 600,000 metres per second), ca.

45 millions of calories would be developed by the collision. Suppose that

during every second 1 kgr. matter impinges upon the area of 1 square metre,

then the heat developed would be about 2400 times the amount of heat

actually radiated by our sun during the same time. Now it is easy to see

that this kgr. of matter is distributed within a parallelopipedon whose basis

is 1 square metre and whose height is 600 km., because when the first

particle of the kilogram arrives at the surface, the last particle which

impinges exactly one second later will be, rou)(hly speaking, at a distance of

600,000 metres from the surface. But the density of such a cloud is only

about 1 : 800,000 of the density of air at ordinary temperature and pressnre.

Hence we conclude that an all-round impact of cosmic matter whose density

is only the 1 : 2,000,000,000th part of that of our atmosphere would still

produce an amount of heat equivalent to the energy radiated into space

during the same time by our sun under normal circumstances. This rough

calculation appears to justify the remark in the text, that the amount of

heat supplied by the collision may indeed be assumed to be practically

unlimited.

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522 Proceedings of Royal Society of Edinburgh, [sess.

Such velocities are of the order of magnitude which would corre-
spond to the displacements in the spectra of new stars if thej
were to he explained on the hasis of Doppler's principle. Hence,
if the expansion of the Nova atmosphere is associated with
enormous surface temperature, the velocities involved must have
a most profound hearing on the structure of the spectrum. This
conclusion necessitates an examination of the spectral character of
lines emitted hy gases which form a rapidly expanding atmosphere
round the incandescent nucleus of the star. It may he well to
rememher at this stage that we are hy no means unfamiliar with
the phenomenon of rapid gaseous expansion at the surfaces of
celestial hodies. Notahle instances are afforded in the sidar
eruptions, where the motions sometimes recorded fall little ahori
of the sun's critical velocity. My opinion on these phenomena,
more fully expressed in the paper referred to, is that they are
the inevitahle consequences of local changes of temperature in
the interior layers of the sun. If, for some reason or other, upon
which I will not enter at present, the temperature of the photo-
sphere at a certain locality should he raised to that, say, of a
Sirian star, the conditions of equilihrium over this particular
area would require that the hydrogen atmosphere of the son
should expand to the dimensions of the hydrogen atmosphere of
Sirius. What we perceive in a solar eruption is therefore, accord-
ing to this view, the violent transition from a state of atmospheric
equilihrium at solar temperature to that corresponding to the
higher Sirian temperature. If the sun were suddenly homharded
by a shower of meteors, raising the temperature of the photosphere,
an inevitahle consequence would be the rapid development of a
protuberance over the place of impact, simply because the atmo-
sphere would tend to assume that form of thermal and mechanical
equilibrium which corresponds to the higher temperature of the
layers underneath. I conclude that the expansion in solar erup-
tions and that of the atmospheres on new stars are analogous
phenomena, in both cases due to the tendency on the part of
the gaseous molecules to assume that state of equilibrium which
corresponds to the temperature at the surface. If this analogy be
accepted, and if we take note of the high velocities so often re-
vealed in the solar gases, we see probably no further difficulty in

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1904-6.] On Prof. Sediger's Theory of Temporary Stars, 523

admittiDg enormous atmospheric expansion in temporary stars.
The correctness of this view will be more fully evidenced if we
now investigate the effects of such a rapidly expanding gaseous
envelope on the appearance of the spectrum. Almost at a glance
we notice that a satisfactory explanation of one of the most
enigmatic features of the Nova spectrum is here offered.

To show this in a few words, let us consider the star immediately
after the coHision, when its surface is in a state of high incandes-
cence, and when the gaseous matter evaporating from the surface
expands in radial directions outwards. Let the circle A A' repre-

S^2

sent the boundary of the star nucleus in a plane passing through
the observer, 0 E being the line of sight. We suppose the outside
boundary of the expanding atmosphere at this particular moment
to be at B C C B' D' D. We may also assume the star to be so
far removed that the light of its photosphere («= incandescent
star surface) and of the surrounding atmosphere is tlirown simul-
taneously upon the slit of the spectroscope. Now, obviously,
all the rays leaving the photosphere in the direction OE, i.e.
towards us, have to pass through that part of the atmosphere
which lies within the area ADD' A'. The natural assumption
being that the gases of the outside layers at D D', in consequence

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524 Proceedings of Boyod Society of Edinbvnrgh, [i

of cooling by expansion, are at a lower temperature than the
photosphere, absorption-lines, characteristic of the subetances of
the atmosphere, appear in the otherwise continuous photoepheric
spectrum. But since all these atmospheric particles move towards
us, their lines must be displaced towards the more refrangible eide
of the spectrum, in accordance with Doppler's principle. Now,
it will be noticed that, in whatever direction we may look at
the star, i.e. in whatever part of space the observer may be
stationed, the phenomenon will always be the same. The dis-
placement of the absorption-lines towards the more refrangible
side of the spectrum is therefore a general feature peculiar to all
stars possessing expanding atmospheres.

There are two reasons why these absorption-lines, instead of
being narrow and sharply defined, as in normal star spectra,
should be broad and hazy. Firstly, the motions of the gaseous
particles towards us are not uniform. We may take it for certain
that considerable differences must exist in the amount and
direction of these motions which would tend to broaden the
lines. Secondly, the density of the atmosphere near the surface
may be considerable, especially during the first stages of the
star's development. We know that from this cause, too, a
broadening of the lines may be expected. Considering the
doubtless violent character of the catastrophe, we may also safely
conclude that the broadening due to the causes mentioned must
have been considerable.

We now turn our attention to the radiations emanating
from those parts of the expanding atmosphere lying inside the
segments DBG and D' B' C. Obviously the spectrum produced
by these radiations must show bright lines, characteristic of the
same substances which cause the absorption spectrum in front of
the star. But since in this case there are as many motions toward*
OS from us, tlie centres of these lines — which are also broad and
hazy, owing to the effects of density and divergence of directions
— must appear in their normal positions.

In consequence of the great distance of the star, the two spectra
are superimposed upon one another in the spectroscope. We see,
therefore, a double spectrum, consisting of broad bright lines in
approximately normal positions, edged on their more refrangible

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1904-6.] On Prof, Sedigei*'8 Theory of Temporary Stars. 525

sides by broad and hazy absorption-lines, these duplex lines being
projected on the continuous photospheric spectrum.

This consideration of the conditions prevailing on a star
whose atmosphere is rapidly expanding leads already to con-
clusions with regard to the character of its spectrum which are in
satisfactory agreement with the principal and most important
feature of the Nova spectrum. Our conclusions require only some
further modification through the existence of the rotating ring of
luminous matter we have considered before. Great significance
must be attached to the fact that the same type of spectrum must
appear under all circumstances, whatever may be the relative
positions of star and observer. Hence the remarkable uniformity
of the spectra of all the Novae appears to be capable of a simple
explanation.

According to the views here expressed, the described phenomena
should occur in a certain sequence which deserves careful attention.
The immediate effect of the collision being incandescence of the
star's surface, the spectrum of the star, at the moment of the
catastrophe, will be purely continuous. Subsequent to this
stage, which is probably of short duration, we have the
development of the expanding atmosphere, which impresses its
existence on the spectrum only after the expanding gases have
cooled below the temperature of the surface. At this stage broad
and diffuse dark lines, strongly displaced towards the violet, make
their appearance. Some further time will elapse, however, before
the atmospheric halo round the star has sufficiently expanded
to render its bright lines visible against the luminous background
of the continuous spectrum. Now this order of events deduced from
the theory seems to be confirmed by certain observations. In the
case of Nova Persei, thanks to its timely discovery by Dr Anderson,
we were fortunately permitted to watch the celestial catastrophe
almost from its very commencement. When the spectrum
was first viewed here in Edinburgh by the Astronomer- Royal for
Scotland on the early evening of the 22nd February 1901, it
certainly appeared to be purely continuous. A few hours later,
however, I noticed distinctly faint dark bands, one of which agreed
in position with the absorption-band afterwards noted on the
violet edge of the bright F-line. On that night emission-bands

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526 Proceedings of Boyal Society of Edinburgh, [sess.

were not conspicuous. A few days afterwards these bright bands
appeared strongly developed, and were then the most promi-
nent feature of the spectrum.

I liave already mentioned that the rotating ring of luminous
nebular matter modi ties to a certain extent the appearance of the
spectral lines. Its effect will be to produce two additional maxima
of brightness, the one displaced towards the red, the other
towards the violet. For in whatever direction we may view the

star— unless the line of sight be at right angles to the plane of
rotation — we will always have some substance of the ring moving
towards us on the one side of the star, and matter moving from us
on the other. Between these maxima a more or less hazy absorp-
tion-line appears at approximately normal wave-length (leaving, of
course, out of consideration the relative motion of the whole
system : star + ring). This absorption-line is due to the gaseous
particles of the ring travelling in front of the star, as seen from the
standpoint of the observer. (Compare the sketch given in fig. 3.)
The complete structure of the bands is determined by these

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1904-5.] On Prof, Seeliger's Theory of Temporary Stars. 527

considerations. We obtain an at least approximate idea of the
appearance of such a band, so far as it is due to the radiations of
the photosphere and the expanding atmosphere, by combining the
intensity-curves of (1) the continuous spectrum in the neighbour-
hood of the line, (2) the absorption -band displaced towards the
violet, and (3) the emission-band at normal wave-length. A com-
bined band of this character is schematically represented in fig. 4,
A A representing the normal position of the special line.

On the other hand, the radiations contributed to the band by
the luminous ring may be roughly represented by the intensity-
curve in fig. 5. If, in combining the two curves 4 and 5, we apply
the simple additive rule, we obtain the total intensity-curve of the
band in fig. 6. Since our assumptions as to the relative shift and
intensity of the various components must of necessity be vague,
there are, of course, many ways of drawing these curves, and the
diagrams therefore represent only one special case out of a great
number of possible combinations. But in constructing the curves I
have aimed at adapting their relative dimensions to the phenomena
actually observed in one particular case of new stars, viz., in Nova

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528

Proceedings of Royal Society of Edinburgh. [sebr.

Aurigae. Afterwards I shall have occasion to exhibit other cnrres
representative of the conditions present in Nova Persei, and I shall
then be in a position to show how the observed dififerences in the
character of the bands of these two stars can be accounted for.
Fixing our attention for the present on fig. 6, we notice a broad
bright band strongly displaced towards the red, and a broad diffuse
absorption-band considerably shifted towards the violet. A re-
markable feature is the distinct appearance of seeming ' reversals '
in both the emission- and absorption -bands. Now, this same
phenomenon was noticed in the observed spectrum. I may quote
the following remark from Scheiner- Frost's Astronomical Spec-
troscopy^ pp. 288-9 : — " The microscopical examination of the
photographic spectra showed the individual lines, both dark
and bright, to be quite complex. A fine bright line could be seen
extending down through the middle of many of the dark lines,
and many of the bright lines had two or more points of maximum

intensity From measurements on nine plates obtained

at Potsdam between 14th February and 4th March (1892) Vogel
deduces the following results, to which are added those calculated
by Vogel from CampbelPs measurements on six plates taken between
8th February and 6th March, and those published by Belopolsky from
measurements on six plates taken from 24th February to 3rd March
1892. The velocities have been corrected for the motion of the
earth, and are therefore relative to the sun. A + velocity denotes
recession from the sun, a - velocity approach toward the sun.

Displacements in Tenth-Metres,

Line employed.

Bright line
within dark.

First Max.
of intensity.

Second Max.
of intensity.

Third Max.
of intensity.

H5
H
K
Vogel Mean

-10-3

- 9-8
-101

- 9-4

- 9-8

-0-1
-0-8
-21
-1-7
-1-2

+ 8-2
+ 6-4
+ 36
+ 8-3
+ 6-1

F
Campbell Mean

-10-4

- 9-3

- 7-

- 8-9

...
...

+ 8-8
+ 7-2
+ 6-

+ 7-2

+ 14-4

Belopolhky Wy

- 9-8

-0-7

+ 8-6

+ 16-7

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1904-5.] On Prof, Seeliger's Theory of Temporary Sta7\s. 529

The bright bands showed two maxima, except Hy, where
three were noticed. We know, however, from the observations of
the spectrum of Nova Persei, that the Hy band has a specially
complex structure, being overlapped by another band on the less
refrangible side. The third maximum noticed in Nova Aurigae
may therefore not belong to the Hy radiations at all. Accepting
this not improbable supposition, we conclude that in Nova Aurigae
the bright bands showed two maxima with displacement of - 0*6
and +6*3 tenth-metres, whereas the absorption-band exhibited a
maximum in form of a bright line at - 9*6 tenth -metres. If we
assume a common motion of the whole system of about - 1*6 t.m.,
the reduced motions of the maxima of the bright band would be
+ 1*0 and +7*9 t.m., and that of the maximum in the absorption-
band - 8*0 t.m. Now if we draw the observed intensity-curve
corresponding to the arbitrary scale of fig. 6, the positions of the
three maxima will be indicated by the three arrows above the
curve. We notice, therefore, that the relative distances between
these maxima are very closely represented by the theoretical curve
in our diagram. At this stage we may submit the theory to a
further test, which in my opinion goes far to show its probability.
Doubtless the presence of the continuous spectrum has a decisive
influence on the appearance of the bands, whose character on the
more refrangible side is mainly determined by the absorption-band
which is caused by gaseous matter moving between the in-
candescent star and the observer. But we know from observations
that in Nova Aurigae, as well as in Nova Persei, the continuous
spectrum has gradually faded away in such a degree that tlio
star in its last stages of luminosity was almost entirely reduced to
its gaseous emissions. Seeliger's hypothesis explains this course
of events quite naturally. We have only to consider that an
incandescent solid or liquid radiates heat more freely than a gas,
and also that the brilliance of the star nucleus is confined to a
shallow surface layer whose energy will be rapidly dissipated.
We may ask : What becomes of the band shown in fig. 6 after the
continuous spectrum has disappeared? Considering that the
former absorptions Mrill now have become radiations, the combined
spectrum of the bands will be represented by an intensity-curve
such as is shown in fig. 7. Here, again, our want of knowledge

PROC. ROY. SOC. BDIN. — VOL. XXV. 34

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530 Proceedings of Royai Society of Edinbavgh, [i

of the actual intensities of all the components implicated in the
formation of the band renders it impossible to select from the
infinite number of possible cases the one which corresponds to the
actual conditions. But the fact I want to point out here will be
shovm under all circumstances. It becomes at once apparent if
we compare the bright band of fig. 7 with that of fig. 6.
While the continuous spectrum was present, the band appeared
shifted towards the red (fig, 6) ; after the continuous spectrum
had vanished, the band appears in approximately normal
position, but the maximum of light lies on the violet side (fig. 7).

cE^ Y^-v^t s

It is readily noticed that the excess of brightness on the more
refrangible side is due to the expanding atmosphere between star
and observer. Since the density of this atmosphere must be
supposed to diminish in the course of time, the same quantity of
gas occupying more and more extended spaces on its outward
journey, and since at the same time its temperature will be
reduced by expansion, its contribution to the light of the bands
will gradually lessen, and we may finally imagine a state in which
ihe light of the star is mainly due to the gaseous radiations of the
ring, whose temperature may be maintained for a more consider-
able time by the doubtless frequent collisions between its in-
dividual meteoric members. At this stage the intensity-curve of

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19J4-5.] On Prof, Seeliger*s Theory of Temp&rary Stars. 531

the band will somewhat resemble that given in fig. 8, the light
being now distributed symmetrically to the normal position. The
sequence of phenomena, as theory would require it, may therefore
be described as follows. While the continuous spectrum was bril-
liant, the observer must have noticed a strong displacement of the
bright bands towards the red. We eaw already that this conclusion
is borne out by the observations (fig. 6). After the vanishing of
the continuous spectrum, however, the same bands must have
appeared shifted towards the violet, since then the maximum on
which the observer would make his mettsurements, on account of
the breadth and indistinctness of the band, lies on the violet side
(fig. 7). The observer would therefore gain the impression that
the star's motion in the line of sight had been considerably
changed during the interval between his two observations.

This apparent shift towards the more refrangible side would
gradually lessen, and finally the bright bands would appear in their
normal positions (provided that the common motion of the whole
system has been accounted for). Now, the student of the spectro-
scopic evolution of Nova Aurigae will at once recognise an agreement
between these theoretical conclusions and the facts actually observed.
The agreement is sufficiently demonstrated by the following data.
It is well known that the spectrum of Nova Aurigae during its last
stages of luminosity, from August 1902 to the end of 1903, wa&
almost purely gaseous, and resembled that of a planetary nebula.
There is also the possibly strongest evidence that Clie hydrogen-
lines were represented in this later spectrum as well as in that of
the former period when continuous radiation was powerful. In
the following table I give the measured wave-lengths of these lines,
in both cases,* and also their normal wave-lengths : —

First Period (Feb. 1892). Secon«l Period (An^. 1892).
Continuoas Spectrum, Contiuuous Spectrura,

strong. feeble or absent (?). Normal wave-length

Two Maxima in Maxima of of H-lines.

bright H-bands. bright H-baiids.

Hr «JJ:8f 4336 43««

4098 4101-9

Of. Scheiner-Frost, pp. 287 and 291.

HS 4108 (

4102

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532 Proceedings of Royal Society of Edinburgh, [sess.

While we notice a displacement of the centres of the lines
towards the red in the first period, we also see clearly their shiit
towardh the violet in the second period, therefore confirming the
conclusions drawn from figs. 6 and 7. Now, Professor Campbell
of the. Lick Observatory has given special attention to this shift
of the bands during the second period, and has found evidence
of a tendency of these bands to approach their normal positions.
His measurements were made on the chief nebular line X= 5007 15
(normal). They are exhibited in the following table taken from
p. 293 of Scheiner-Frost's Asironomiecd Spectroscopy.

1892. Aug.

20-30.

X = 5003-3

AX =

- 3-8 t.m.

Sept.

3-22.

5002-2

-4-9

Oct.

12-19.

5003-7

-3-4

Nov.

2-24.

5004-6

-2-5

1893. Feb,

10-27.

50060

-1-1

Mar.

26- May 9.

5005-3

-1-8

Aug. 6-Oct. 10. 5005-9 -1-2

The decrease in AX in quite apparent. In 1892 the displacement
amounted to about - 4 t.m., a value which agrees very well with
that of the H-lines of the preceding table. In 1893, on the other
hand, the displacement was only about - 1*5 t.m. Since we had
found before from other coiisideiations that this was probably in
amount and sign the common displacement of the whole system,
we conclude 'that at this stage the in tensity -curve of the bands
must have approximately agreed with that given in fig. 8.

Thus the theory here discussed seems to offer a simple and
probable explanation of an otherwise extremely puzzling pheno-
menon, viz., of the enormous shift of the bands from red towards
violet during the time between the first and second period of the
history of Nova Aurigae.

So far, our attempt to explain the character of the spectrum of
Novae on the basis of Seeliger's theory has been of a purely
qualitative character. It is important to show now its possibility
also from the quantitative point of view. In order to explain the
displacements of the absorption-bands towards the violet by
motions in the line of sight, we have to assume average velocities
of the expanding gases amounting to about 600 km. per second.

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1904-5.] On Prof. Seeliger*$ Theory of Temporary Stars, 533

This figure exceeds the average velocity of the outrushing gases in
solar eruptions, but it may nevertheless be assumed to be at least
of the order of these velocities. If, therefore, we grant the
reality of the motions in the case of the sun, we should find no
difficulty in accepting the explanation of the absorption-bands in
new stars which I have proposed in this paper. I fully admit
that a physical explanation of such exorbitant velocities in
expanding gases has still to be framed, and that our present
thermodynamical views, by accepting Boyle's law, offer no clue
whatever. But these views, it must be remembered, are based
on conceptions of molecular matter which we now admit to be
imperfect. The new physics of the molecule and atom is quite
dififerent from that which led formerly to the kinetic explanation
of Boyle's law. We are no longer permitted to conceive of the
motions of gaseous molecules at solar temperatures as being
exclusively governed by the frequency and intensity of their
mutual impacts, and uninfluenced by any other forces acting
between them. A number of facts point to the conclusion that
gases emitting line-spectra are ionised. If this view is accepted,
we have to take into consideration the electric agencies which are
brought into play in cases of moving electric charges, and which,
as Professor J. J. Thomson and others have shown, influence
profoundly our conceptions of mechanical mass and energy. The
kinetic theory of an ionised gas is therefore diflerent from that of
an electrically neutral gas, because in the former internal forces,
viz. electric agencies, are operating which are not present in the
latter. But the existence of these electric forces demands the
introduction of an internal virial in Clausius' fundamental equa-
tion, which means, in other words, that Boyle's law is inapplicable,
since the definition of a so-called * perfect * gas excludes the
presence of an internal virial. Now, our difficulty in under-
standing the greatness of motions in solar eruptions arises mainly
from the fact that we have hitherto considered the gases on the
sun as being in this * perfect ' state, and therefore have accepted
Boyle's law as the basis from which we formed our opinions of the
greatest possible speeds in expanding gases at solar temperatures.
We have reasoned in the following way : The velocity with which
a certain disturbance of equilibrium within the gas can be propa-

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534 Proceedings of B<yyal Society of Bdinhargk, [i

gated by internal mechanical agencies can under no ciicumstances
exceed that with which sound would travel through the gas. Now,
we have a fairly warranted estimate of the surface temperature of
the sun, and with this ^temperature we can compute the speed of
sound in the solar chromosphere. But we find that the computed
velocity falls considerably short of that usually noticed in solar
jwominences. Hence we argue that the phenomenon of solar
eruptions cannot be explained on the basis of thermodynamical
reasoning. The argumentation seems strong enough, only we must
not forget that our computation of the velocity of sound is essen-
tially founded on Boyle's law. If, for instance, we assumed
that between the molecules of the solar gases powerful repulsive
forces were acting, the computed speed of sound would become
considerably greater, and hence our conclusion as to the maximum
speed of propagation would also differ from that we hold at present.
We are, I think, clearly placed before the alternative : either
Boyle's law is correct, then it is difficult to see how solar eruptions
can be real displacements of matter; or Boyle's law does not
express the true kinetic conditions existing in solar gases, then the
high velocities in solar eruptions become conceivable if we assume
powerful repulsive forces acting between the molecules. As I
mentioned before, there are reasons which seem to favour the
second alternative. If, for instance, we accept the modem view
that radiation is due to motions of the electrons within the atom
or molecule, are we not bound to look upon the molecules of an
incandescent gas as moving electric currents ? And suppose, under
this condition, two molecules to approach each other, will not the
induced electric force tend to drive them apart, i.e. act as a
repulsive force ? We are quite certain that this will happen in
the case of ordinary currents and conductors, such as we are able
to produce in laboratories : why not also in currents of molecular
dimensions? What difference is there between a current produced
by electrons moving along a conducting wire, and one caused by
electrons moving round the nucleus of the atom or molecule ? I
think questions of this kind may at least shake the hitherto
implicit confidence in the so-called ' perfect ' state of incandescent
gases, and may also make us aware that the kinetic theory of
matter endowed with distinct inherent electric properties must

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1904-6.] On Prof. Seeliger*8 Theory of Temporary Stars. 535

differ essentially from that with which we are at present
familiar.

The object of this short transgression upon the field of molecular
physics is to show that the reality of enormous velocities in
expanding gases, such as we see in solar eruptions, cannot well be
refuted by a reasoning based on conceptions of molecular matter
in which electric agencies are ignored. There is no cogent reason,
either on the part of theory or observation, which would force us
to pronounce \he displacements of gaseous matter on the sun as
'appearances' only. Hence, from the point of view here
advocated, the stupendous rate of expansion of the ' atmosphere '
of a new star may also be brought within the range of mental
comprehension. It must be considered as a decided advantage of
this theory that the asserted displacements of the spectral lines by
motions of expanding gases in the line of sight are phenomena
clearly noticeable in solar spectroscopic observations, whereas
we have no recorded instance in cosmic evolution which might
support, in a similar convincing manner, the assumption of
exorbitant pressure or of abnormal refraction.

We will now turn to the quantitative test of the displacements
caused by the rotating ring. The motions in the line of sight are
here, according to the theory, of the order of the orbital velocities
of bodies revolving roimd the nucleus of the attracting star near
its surface. If we assume the star of the mass and dimensions of
our sun, and if we remember that the brightest part of the ring
is formed by substance revolving in circular orbits whose radius
is practically that of the star, we find displacements of the two
maxima in the bands corresponding to approximately 4-500 km.
per second, which would be equivalent to a distance of about
11-14 tenth-metres between the two maxima. These figures are
in close agreement with the observations which showed a distance
of about 15 tenth-metres. Hence there is no difficulty in com-
prehending these displacements, and therefore also the enormous
breadth of the Nova lines, on the assumption that they are caused
by the orbital motions of particles revolving in the immediate
neighbourhood of the star's surface.

It will doubtless be noticed that the theory requires no assump-
tion as regards the magnitude of the original relative motion

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536 Proceedings of Boyal Society of Edinburgh, [i

of star and nebula. Certainly the two objects must have approached
each other, otherwise a collision would, of course, have been
impossible. But the rate of approach is here a question of no
relevancy. In this point the theory may be clearly distinguishes 1
from the earlier attempts, in which two or more bodies were
assumed to move through space in dififerent directions, with speeds
far transcending the average proper motions of celestial bodies.

In illustrating these views on the physical processes connected
with the phenomenon of temporary stars, I have discussed some
of the more important facts brought to hght by the observations
of Nova Aurigae. I beg now to enter upon a brief discussion of
the observational records of Nova Persei. Broadly speaking, the
spectral phenomena noticed in this specially remarkable new star
were in fair accordance with those of its predecessor of 1892.
There are, however, some peculiar differences in the structure of
the bands which seem to require an explanation. Most noticeable
among these is the fact that during the time when the continuous
spectrum was strong, the bright band, which in Nova Aurigae was
strongly displaced towards the red (fig. 6), appeared in its normal
position in Nova Persei. Fortunately, our theory is sufficiently
flexible to explain this peculiar difference. We have seen before
that the absorption-band on the violet side is caused by the rapid
development of an expanding atmosphere at the moment of the
collision. Now, obviously, the rate of expansion will depend on
the temperature developed during the impact. If, therefore, on
account of greater density of the impinging cloud, we suppose the
catastrophe of Nova Persei to have been considerably more violent
than that of Nova Aurigae — an assumption which is perhaps
supported by the relative brightness of the two stars — then the
displacement of the absorption-band would also be more consider-
able in Nova Persei. On the other hand, if the masses of the two
stars have been nearly the same, the two maxima of the bright
bands which are due to gravitational effects would appear in the
same positions. Thus, while the curve in fig. 4 would have to be
extended in the horizontal direction (fig. 9), fig. 5 would remain
unaltered (fig. 10). By combining the two curves in the same
way as before we obtain the intensity-curve in fig 11. Hence, as
the total effect of the combined radiations and absorptions, we find

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1904-5.] On Prof, Seeliger's Theory of Temporary Stars. 537

in this case a broad bright line with two maxima, the centre of
which lies at the normal position, and a hazy absorption- band
on the violet side of the bright band. We notice that the
assumption of a more energetic expansion at once explains why
the bright Perseus-lines should have been found in normal positions,
contrary to what had been seen in the former fainter Novae,
where these bands were displaced towards the red.

The fact that the bright bands in Nova Persei were not dis-

placed renders it difficult, on the other hand, to accept either the
high-pressure or the abnormal-refraction theory. According to
the former, we must expect, under all circumstances, displace-
ment of the bright bands towards the red, while the absorption-
bands should appear in normal positions. The observations show
that in Nova Persei just the opposite phenomenon occurred. The
a priori improbable assumption which might save the theory, viz.,
that the star may have possessed an enormous proper motion towards
us, is clearly contradicted by other observed facts. In the case of
the refraction theory the same difficulty is experienced, even in

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538 Proceedings of Boyai Society of Editibv/rgh. [i

a more pronounced form, because it is inconceivable thai any
photospheric light can have been abnormallj refracted on wave-
lengths shorter than the normal. The whole of the bright band
should have developed on the less refrangible side. This is clearly
shown by the theoretical intensity-curve in Dr Eberf s paper.

I have mentioned before that the density of the expanding
atmosphere, which may have been considerable at the moment of
impact, must be assumed to decrease in course of time^ and I have
pointed out the effect which this must have on the appearance of the
absorption-bands. These bands, being very broad and hazy at first.

qL^ 12

will gradually shrink into narrow lines. Suppose fig. 1 2 to represent
a central section through the star and its atmosphere, A B indicat-
ing the line of sight. It is clear that atmospheric particles, at
one time distributed along the arc A A, will, by radial expansion,
in course of time be distributed over the greater arc B' B'. Now
all the particles within A A have contributed to the absorption-
band at the first moment, but of these only those lying within the
arc B B will absorb the photospheric light at the second moment
Hence the total number of gaseous molecules passed through by
photospheric rays in the direction of the line of sight will be less
at the second moment. Now, since the breadth and haziness of

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1904-5.] On Prof, Seeliger's Theory of Temporary Stars. 539

the spectral lines, as experiments show, uicrease with the density,
and perhaps also with the temperature, of the emitting gas ; and
since hoth density and temperature are more considerable at A
than at B, we must conclude that the broad and hazy absorption-
band is gradually reduced to a narrow line, and finally fades away
altogether.

This peculiar shrinkage of the absorption-bands has indeed
been noticed during the spectral evolution of Nova Persei. But,
curiously, the band resolved into ttDO lines instead of one. To
explain this duplicity we have to make a further assumption, but
fortunately one which seems not improbable. We have indeed
only to suppose that in this special case the dark body was a double
«tar. We are quite familiar with double-star systems in which
one of the components is invisible (stars of the Algol type). There
is, however, no reason that might debar us from assuming double
4tars in which the surfaces of both components have cooled below
the range of visibility. Now, in such a case it is very unlikely
that both stars should have the same mass. But if the masses are
different, then the gravitational effects on the cloud particles should
also be different, and hence the heat-development at the surges
4uid the orbital velocities of the encircling rings. In other words,
we should then obtain an intensity-curve of the bands which is
found by combining two curves of the shape of fig. 1 1 drawn on
<lifferent scales. The resultant curve is shown in fig. 13, which is
indeed typical of the first stage of development in Nova Persei.
The following stage is characterised by fig. 14, where the broad
hazy absorption-band has already been resolved into two compara-
tively distinct absorption-lines. At a still further stage, when the
density of the expanding atmosphere has become extremely small,
the absorption has practically disappeared, and there remains only
the radiation of the two rings, giving rise to a bright band with
four more or less pronounced maxima, its centre lying at normal
wave-length (fig. 15). All these conclusions are well borne out
by the observed facts.

1 may be allowed here to quote the folloMring remark from a
paper by Father Sidgreaves on the spectrum of Nova Persei in
Monthly NoticeSy vol. xii. p. 141, descriptive of the gradual
changes in the dark hydrogen-bands : — *' At the beginning these

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540 Proceedings of Royal Society of Edinburgh. [i

dark lines appeared to grow in strength between 28th February
and 8th March .... But after 8th March their decline was regu-
lar and uninterrupted ; they slowly disappeared, together with the
bright calcium line K. On 12th March they had lost their centres
and appeared as well-defined double lines, separated by a thin clear
reversal. The more refracted components were much the weaker,
and were the first to disappear. They had lost much on the 16th,
and were quite extinct on the 20th, when the red side components

formed the series of sharp thin lines, which were seen for the last
time on the 21st."

It appears from this quotation that in Nova Persei the two
absorption-lines in fig. 14 have been of different intensity, the one
less refracted being decidedly the stronger of the two. The con-
sequence was that this line outlived its more refracted feebler
neighbour, and that there was a stage when the intensity-curve
of the bands showed the structure exhibited in fig. 16. Suppose
now this state of matters to have lasted for some time, during
which the continuous spectrum has more and more decreased in
brightness. Under these circumstances the absorption-line would

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1904-5.] On Prof. Seeliger's Theory of Temporary Stars, 541

gradually become an emissioD-line, and, as such, might enhance the
intensity of the violet edge of the emission-band. We should then
notice those peculiar finger-post structures (fig. 17) which are
so prominent features in the later spectrograms of the Lick
Observatory (8eeZ».0. Bulletin, No. 8).

There is a good reason for the longer persistence of the less re-
fracted absorption-line. The more rapidly the atmosphere expands,
the more quickly will the absorption-band thin out and disappear.
But since the more refracted band is due to the more rapidly ex-
panding atmosphere, we may naturally infer that its existence

must be of shorter duration than that of its neighbour, which is
caused by the less expanding gases.

The double-star hypothesis, which apparently explains in a
satisfactory way some of the peculiar spectral features of Nova
Persei, may also assist us in understanding more fully the peculiar
variability of the star's light, specially noticed during the first
stages of development. In an earlier paper {Astronomische
NachricJUen, Nos. 3822-3) I have attempted to show that the
principal features of this variability may be explained by a rotation
of the star round an axis. I have there emphasised the fact that
by the more or less one-sided collision the star's superficial layers
must be melted unequally, the liquefaction reaching down into

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542 Proceedings of Royal Society of Edinburgh, [i

lower levels at the place of maximum impact A central sectioD
through the star immediately after the catastrophe may therefore
be represented by No. 1 of fig. 18, the ring ABA'B' showing the
incandescent surface layers, and A being the locality of maximum
impact. After the collision has passed over, the surface begins ta
cool, and the star will gradiially arrive at the stage No. 2, where
the surface at B has cooled down to darkness, while the surface
at A, through more vigorous conduction, aud perhaps convection
of heat from the interior, may still be in a state of incandescence*
Some time afterwards the stage No. 3 will be reached, where the
incandescence is now limited to a small lenticular segment at A.
In this way the star would gradually pass from a state of all-
round incandescence to total obscurity.* If, now, we suppose the

\n^ uu

star to possess a rotatory motion, by which the points A and B are
successively brought into the line of sight, we would notice the
following features of variability : At No. 1 a uniform gradual
decrease of brightness ; at No. 2 the same, but in addition a periodic
recurrence of pronounced maxima and minima, the former being
much extended and covering the greater part of the period, the
latter being indicated by abrupt and short inflections of the light
curve ; at No. 3, protracted minima covering the greater part of the
period and maxima of short duration, hence the reverse of No. 2.
These three theoretical light-curves are also represented in fig. 18.
They are in fair accordance with the observed phenomena.

This assumption of an axial rotation advocated in my former
paper is by no means improbable, since the impacts will doubtless
impart a certain moment of momentum to the star nucleus. But
it may perhaps seem unnecessary in the case of a double star,

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1904-5.] On Prof, Seeliger's Tlieory of Tmvporary Stars, 543

where the observed phenomena may as well be explained by the
revolution of the two stars round their common centre of gravity.
In the introductory remarks to this paper I have laid consider-
able stress on the fact that the observed displacements of the
spectral lines are proportional to their wave-lengths, and independ-
ent of the chemical nature of the emitting gas. I pointed out that
this remarkable fact supports the view that the displacements are
due to motions in the line of sight. Indeed, if an incandescent
gaseous body is moved with a velocity y, its lines are displaced by
an amount ± d\^ so that

where V is the velocity of light ( = 300,000 km. per second) and A.
the wave-length of the line. This equation holds also if the body
consists of a mixture of gases moving in the line of sight with a
common velocity v. Hence any line of the spectrum emitted by
these various gases wiU be displaced by an amount

±d\ — const. X A,
t.e. the displacement depends solely on the wave-length. Professor
Becker's elaborate measurements confirm this statement in every
respect. I should like here to supplement his important con-
clusions, which bear out so admirably the theoretical results of
this communication, by a few similar measurements published by
Messrs Campbell and Wright in the 8th Bulletin of the Lick
Observatory. I begin with the displacements of the absorption-
bands. The following table contains in the first column the lines
measured and the elements to which they belong, in the second
and third columns the observed and computed displacements
towards the violet. The values of the third column have been
computed from the formula : —

-ciA. = 00046xX

Displacement

g i Calcium |

Observed. Computed.
- 17 t.m. - 18

16 -18

H8 I ( -19 -19

Hy \ Hydrogen < - 19 - 19

HJ8) ( -24 -22

D Sodium -27 -27

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544 Proceedings of Boyal Society of Edinburgh. [sbss.

The agreement between observation and computation is so ex-
traordinary that the observers felt justified to remark : " There
is, then, no evidence that the position of the band is affected by
other considerations than that of wave-length."

A similar result is obtained from the investigation of the
bright bands. According to theory, the enormous width of these
bands, as well as the appearance of maxima within them, are also
to be explained by motions of gaseous matter in the line of sight
Hence we conclude that the width and the displacements of corre-
sponding maxima should be linear functions of the wave-lengths,
but independent of the chemical nature of the emitting substances.
The correctness of this conclusion is shown in the following table.
Here the measurements given in the second column refer to the chief
(violet) maximum of the bright bands, while the displacements iu
the third column have been computed from the formula

-(fX = 0-00212xX.

The fourth column contains the observed widths of the bright
bands. Naturally these measurements are far less reliable, but
neverthelejis the alleged proportionality to the wave-length is quite
evident.

Displacement of chief maximum
of bright bands

Wave-lengtli.

Obs.

Comp.

Width,

3868-9 t.m.

- 8-7 t.m.

- 8-2 t.m.

3967-6

-8 6

-8-4

4101-9 H

-8-2

-8-7

4340-6 H

-9-4

-9-1

4363-3 Neb.

-9-7

-9-2

4471-6 He

-9-6

-9-5

4643-

-11

-10

...

4685-9 Neb.

-10

4713-2 He

-10

4861-5 H

-9

-10

4959-0 Neb.

-11

5007-0 Neb

-11

5752- Neb.

-11

-12

5875-9 He

-13

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1904-5.] On Prof, Seeliger^s Theory of Temporai-y Stars. 545

I cannot believe that these> results, combined with the cor-
roborating evidence of Professor Becker's observations, leave any
doubt as to the fact that the displacements in the spectra of new
stars depend exclusively on the wave-length, and are not caused
by agencies which depend on the atomic structure of the emitting
substances. This fact must be considered as the touch-stone
of theories on temporary stars; so much so, indeed, that we
may at once dismiss any explanation, however plausible in other
respects, which is not in entire accordance with it.

We are now in a position to form, step by step, a mental picture
of the evolution of a new star, and to compare our deductive
conclusions with the observed facts. The more important events
in the star's history as a radiating body may be thus summarised : —

(1) The immediate consequence of the impact between star and
cosmic cloud is a more or less one-sided incandescence of the star's
surface, causing a purely continuous spectrum. This stage was
noticed here in Edinburgh about sixteen hours after the outburst.

(2) In consequence of the sudden and enormous heating a gaseous
envelope is formed, which expands very rapidly in radial directions.
The velocity of expansion may be assumed to exceed that noticed
in solar eruptions. The expanding gases now begin to influence
the spectrum. At first absorption predominates, and is shown by
broad absorption-lines, displaced towards the more refrangible side.
The lines must be broad and hazy, on account of the density and
the divergent motions of the gases. This stage was observed
here in Edinburgh about twenty hours after the discovery, when
the visual spectrum was strongly continuous, but interrupted at
various places by faint broad absorption-bands. As the density
must have decreased while the atmosphere was more and more
expanding, the broad and hazy absorption-bands in course of
time reduced to sharp dark lines, which ultimately thinned out
and faded away. This peculiar feature, too, has been noticed by
observers (see F. Sidgreaves' note quoted above). At the same
time the star's atmosphere becoming more extensive, its radiation
outside the star's disc grows more and more prominent, giving rise
to broad emission-bands in normal positions. Hence, after a time,
the spectrum shows bright bands, bordered on their violet edges
by absorption-bands. This constitutes the typical new star

PROC. ROY. SOC. EDIN. — VOL. XXV. 35

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546 Proceedings of Rayed Soctety of Edinburgh. [§■».

spectrum. It may be specially noticed that, from the theoretical
point of view, the absorption-line must under aU circunutanees be
on the violet side of the emission-line.

(3) The expanding atmosphere, formed from the volatilised sub-
stances of the star matter, and being at temperatures comparable to
those prevalent in star-atmospheres, will spectroscopically resemble
the chromosphere. This conclusion is confirmed by the table on
pp. 286-7 of Scheiner-Frost's Spectroscopy^ in which a comparison
is made between the lines seen in Nova Aurigae and those most
frequently and most intensely noticed in the solar chromosphere.

(4) Besides the expanding atmosphere, account must be taken
of the revolving ring of nebular matter which, after the collision,
has been brought under the permanent gravitational sway of the
star. The presence of motions of this character explains uot only
the enormous width of the bright bands, but also the appearance
of symmetrically grouped maxima within them. We are further
enabled to understand the strong displacement of the bright band
towards the red in one case (Nova Aurigae), and the absence of
such a shift in another (Nova Persei).

(5) When once this gyrating ring of matter has been established,
further direct impact of meteoric matter upon the star will be im-
peded, since a considerable number of nebular particles may collide
already inside the ring without reaching the surface of the star.
This enhances, on the one hand, the luminosity of the ring, and
reduces, on the other hand, the incandescence of the nucleus.
Consequently we notice a decrease of the continuous spectrum
coupled with an increase of those gaseous radiations which are caused
by the incandescence or luminosity of the gyrating nebular matter.
The expanding atmosphere having gradually faded away, the
chromospheric spectrum has also disappeared, and has been super-
seded by those lines which are peculiar to the spectrum of gaseous
nebulee.* This is, briefly, the course of events which theory would
lead us to expect. At the same time, it is also in many respects the
sequence of phenomena shown by observation. The gradual dis-
appearance of the continuous spectrum together with the lines
which belong to the chromospheric radiations, and the simultaneous
intensification of the nebular lines, — the peculiar process of

* See note at end of paper.

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1904-5.] On Prof, Seeliger's Theory of Temporary Stars. 547

" backwardation," as Sir Norman Lockyer appropriately calls it,
because it reveals a sequence of phenomena exactly opposite to
what we comprehend as the natural evolution of a cosmic body, —
are indeed features well known to students of this problem.

We notice, then; that Seeliger's ingenious hypothesis can be made
to respond to a number of observed facts if the circumstances are
duly considered under which the supposed collision between a dark
body and a cosmic cloud may occur. An effort has been made
in this communication to emphasise the important r6le played by
the star's gravitational force, and to. show that the motions of
incandescent matter generated by the star's attraction are probably
sufficient, from a qualitative as well as a quantitative point of view,
to explain the peculiarities of the Nova spectrum, and also to
account for the extraordinary process of evolution noticed in
temporary stars.

One of the conclusions reached in this paper is that, as an effect
of one-sided collision, the cosmic body may become surrounded
by a revolving ring of nebular matter. Before the collision,
neither the star nor the nebula were supposed to possess a rotational
momentum. But the mere fact of a meteoric swarm impinging upon
the star leads to the conclusion that a permanent ring of meteoric
matter may be formed, the constituents of which revolve with
orbital velocities round the star nucleus. May not this conclusion
perhaps assist us in explaining the origin of the rotation of our
own solar system 1 It is well known that Laplace, in his
celebrated hypothesis, assumed rotation as a pre-existing quality
of the solar nebula. He clearly recognised, what had escaped the
less mathematical genius of Kant, that rotation could not have
been generated by the internal motions of the contracting matter ;
that only an external agency could have introduced it into our
system. Laplace made no attempt to define this agency : he boldly
assumed its primeval operation, and started his hypothesis from
the moment when rotation had been impressed upon the vast
cosmic cloud from which our present system has gradually been
formed. No doubt, our attempts to grasp the evolution of the
natural world can only begin from a certain stage ; unconceivable
creation stands at the beginning of the cosmos. Laplace's assump-
tion of original rotation is therefore certainly justified, and must be

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548 Proceedings of Roycd Society of Edinburgh. [sess.

preferred to the Kantian attempt to explain this quality of solar
matter from an untenable mathematical and physical point of view.
If the Laplaceian hypothesis would otherwise satisfactorily account
for the development of our system, we might well grant his assiirop-
tion that rotation was due to an external impulse beyond the
grasp of our intelligence. But a recent criticism of the hypothesis
has shown that our minds cannot readily accept all the con-
clusions drawn in this great poem of cosmic evolution. In a paper
contributed to the Astrophysical Journal^ vol. xi., Mr Moulton,
a mathematical astronomer of high repute, attacks the hypothesis
from various mathematical and physical points of view. His
negative conclusions appear in many respects suflficiently sound
and vigorous to convey the impression that the evolution of
our system must have differed very largely from the ideal picture
of Laplace. Since a brief review of ISIr Moulton's arguments
seems necessary in order to understand more clearly the bearing
of our own hypothesis upon solar evolution, I beg to quote
a few passages from his work which may give an idea of the
nature and extent of the difficulties encountered in the nebular
hypothesis.

(P. 104.) " The methods of testing the theory will be divided
into three categories : — (i.) Comparison of observed phenomena
with those which result from the expressed or implied conditions
maintained by the hjrpothesis ; (ii.) Answers to the question
whether the supposed initial conditions could have developed into
the existing system ; (iii.) Comparison of those properties of the
supposed initial system with the one now existing, which are
invariant under all changes resulting from the action of internal
forces."

(P. 129.) "Under the methods of the first category certain
phenomena are enumerated which contradict the hypothesis so
flatly that candid minds must admit that its validity in the form
considered is open to serious question. In less exact sciences
such objections would overthrow a theory or lead to its reconstruc-
tion. The objections are, that the planes of the planets' orbits
present considerable deviations, while four satellites revolve in
planes making practically right angles with the average of the
system ; that the distribution of mass in the planets is unaccount-

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1904-5.] On Prof, Seeliger's Theory of Tempo7*ary Stars. 549

ably and suspiciously irregular ; and that there is an unexplainable
anomaly in the motion of the inner ring of Saturn.

*' Under the methods of the second category, it is shown that
the development of a system of planets and satellites from an
extended nebula is by no means a simple matter, and that in the
system under consideration the conclusions which it was possible
to make were invariably adverse to the theory. In subjects where
perfectly rigorous mathematical processes cannot be employed,
such a uniform agreement of conclusions, when so various methods
of attack are employed, is sufficient to establish a proposition.
The objections are, that the lighter elements would have escaped ;
that matter would have been left off continually, instead of in rings
at rare intervals ; that if a ring were all contracted into a planet
except an infinitesimal remainder distributed in its path, the
process of aggregation could not complete itself ; that the gravita-
tion between the masses occurring in the rare media would be so
feeble that they would seldom come in contact, and that Roche's
limit and a similar new criterion show that fluid masses of the
density which must have existed would be disintegrated by the
disturbing action of the sun.

" The one objection which is advanced in the methods of the
third category * is of great simplicity, and leads to certain conclu-
sions. It is of such a character, and the numerical discrepancies
are so great, that it seems to render the nebular hypothesis, in the
simple form in which it has usually been accepted, absolutely
untenable, unless some fundamental postulates, now generally

* (P. 126.) ** It is known from the elementary principles of dynamics that
the moment of momentum of a system which is subject to no external forces
is constant." Mr Moulton demonstrates, however, that when the solar
nebula extended to Neptune's orbit, the moment of momentum was 32*176,
while in the system at present it is only 0'151. Hence, ''instead of being a
constant, the moment of momentum is found to vary in a remarkable
manner. ... It follows from these figures that if the mass of the solar
system filled a spheroid exten«ling to Neptune's orbit, and rotated with a
velocity sufficient to make its moment of momentum equal to that of the
present system, and if it then contracted .... the centrifugal force would
not equal the centripetal until it had shrunk far within Mercury's orbit.
Such an enormous difference cannot be ascribed to uncertainties in the law
of density, or to the approximations in the mechanical quadratures ; but it
points to a mode of development quite different from, and much more com-
plicated than, that postulated in the nebular theory under discussion."

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550 Proceedings of Rayed Society of Ediiibwryh, \^r^^.

accepted, are radically erroneous. It seems a necessary inference from
the results of tlio discussion that the solar nebula was heterogeneous
to a degree not lieretofore considered as being probable. . . ."

Now, it seems to me that some of these difficulties arc avoided
if we ascribe the formation of the planets to the rotating ring
engendered by the collision between the solar body and a
dense cosmic cloud. I would still assume the original solar body
to have been formed from a nebula by the process of contraction.
But this nebula had no inherent property of rotation. Conse-
quently the resulting liquid body had neither a tendency to
rotate nor was it surrounded by a revolving planetary system.
Now let us suppose this body, on its journey through space, to
approach a cosmic cloud of considerable density. As a consequence
of the collision, which in all probability will be one-sided, not only
a revolving ring of matter will become permanently attacheil to
the star, but also those particles which impinge upon the IkkIv
will impart a rotation to it in the same direction as that of the
ring. The result is a slowly rotating central nucleus surrounded
by a ring of quickly revolving matter. I have pointed out that
the orbits of the ring particles, immediately after the catastrophe,
have all possible eccentricities ranging between zero and unity,
those near the star describing circles, those farther removed
elongated ellipses. But this aspect will gradually change. On
each return to periastron the particles will encounter fresh colli-
sions, by which the major axes and the eccentricities of their
orbits are lessened, the ring thereby becoming denser, and at the
same time more and more circular. For we must keep in mind
that, in consequence of the enormous heat communicated to the
star by the impacts, there will be a dense and extensive atmosphere
around it, through which the ring particles have to force their way
every time they return to periastron. The tendency would there-
fore be to establish a circular ring. The density of matter within
this ring may be quite heterogeneous. It is indeed to be expected
that matter may be more concentrated in some of its parts than in
others. From the beginning distinct nuclei may be present,
around which matter is more or less densely grouped. These
nuclei would form centres of attraction, and, as such, would mark
the initial steps towards the formation of planets. From this

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1904-5.] On Prof, Seeliger's Theory of Temporal^ Stars, 551

point of view, however crude it may appear in its present form,*
the difficulties of Mr Moulton's first and third category are at once
removed, and those of the second certainly reduced. We under-
stand better why the distribution of matter in the solar system
should be so heterogeneous, and why there is not that constancy
of the moment of momentum which would have to be expected if
the Laplaceian hypothesis were correct. Besides, we are here for
the first time confronted with a possible explanation of how
rotation may have been introduced into the solar system. In the
problem of cosmic evolution, this question has always proved an
insurmountable difficulty to those philosophers who attempted to
trace the natural development of our world from the primordial
chaos. That matter endowed with gravitational force may have
contracted from nebulse into spherical bodies, and that these latter
may have originally been impressed with chance motions through
space — such conclusions are quite compatible with our conception
of the chaos where chance has ruled supreme. But how, from this
anarchy of forces and directions, a system of cosmic bodies could
have been moulded, in which one particular tendency of motion
prevails to the exclusion of all others — this question has so far been
considered as pertaining to the domain of metaphysics rather than
of natural philosophy. The difficulty seems now to be somewhat
lessened, inasmuch as it can be shown that the chance approach of
a star towards a nebular or meteoric agglomeration of matter may
entail the formation of a rotating ring surrounding the star, and also
the impression of an equally directed moment of momentum upon
the body itself. It seems not unlikely, therefore, that in the pheno-
menon of a new star we notice the initial stage of the fabric of a
solar system, and that. Nature presents here to our eyes — although,
perhaps, on a less gigantic scale — a sequence of events which had
taken place in our own system in the remote past.

Note added on Slst January 1905. — It has been pointed out to
me that I do not explain the noteworthy fact that the nebular
lines have appeared a considerable time after the outburst, and
were not present during the initial stages, whereas the theory
demands the existence of nebular matter round the star from the
• See my paper, ** Some Suggestions on the Nebular Hypothesis."

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552 Proceedings of Royal Society of Edinburgh, [sesb.

very beginning of the catastrophe. I admit that my exposition
contains no direct allusion to this point, which, however, seemed to
me too obvious to require a special explanation. We must grant,
I tlnnk without hesitation, that the appearance of the nebular
spectrum presupposes not only the presence of nebulous matter,
but also those special conditions of temperature under which
alone this matter can emit the peculiar lines of gaseous nebulae.
Nobody denies now that the materials of which the stars are
composed once formed nebular clouds, and that under such condi-
tions they emitted the typical nebular spectrum, of which at
present, with a few exceptions, we see no traces in their atmo-
spheres. It is one of the great achievements of modern spectroscopy
to have shown that the same substance emits essentially different
spectra under different conditions (e,g. the spectrum of hydrogen
at low and high tem{>erature). Hence we are clearly not permitted
to think that nebular matter — an infinitely more complex structure
than the simple hydrogen atom — will betray its existence by one
and the same typical spectrum under all circumstances. ITie
spectrum of nebular matter at a high temperature will most likely
be essentially different from that at a low temperature. If our
ideas of cosmic evolution be correct, the former must resemble
that of incandescent cosmic matter in the star atmospheres, i.e.
it must be chromospheric, while the latter is typical of the condi-
tions in nebulae which our modern views suppose to be at very
low temperatures, and luminous rather than incandescent. Doubt-
less the nebular matter round a temporary star is under the former
conditions immediately after the outburst. It is only after the
subsidence of impacts that the star and the nebulous matter round
it gradually cool down and approach those conditions of low
temperature which finally lead to the appearance of the typical
nebular spectrum. In the ordinary process of evolution, therefore,
cosmic matter begins its spectroscopic existence by showing the
low temperature nebular spectnmi, and thence develops its high
temperature or chromospheric character ; in temporary stars we
notice the inverse process — so to speak, a negative evolution.
These remarks will suffice to explain why the nebular spectrum
should be absent at first, and should gradually develop with the
cooling of the star.

{Issued separcUely April 15, 1905.)

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1904-6.] Dr J. Halm on the Nehdar Hypothesis. 553

Some Suggestions on the Nebular Hypothesis.
By J. Halm, Ph.D.

(MS. received March 6, 1905. Read March 20, 1905.)

The hypothesis of Laplace on the genesis of the solar system
from an extensive nebula presents difficulties of so serious a
character that important modifications appear to be required in
order to make it conformable with the laws of dynamics. The
objection most frequently brought forward refers to the mode in
which Laplace assumes the separation of the planets from the
contracting nebula to have taken place. It is urged that the
intermittent shedding-oflf of rings is a somewhat unintelligible
process considering the physical constitution of the nebula ; that
we should rather expect a contintu>u8 separation of particles at the
equator, where the centripetal force is overbalanced by the centri-
fugal force, and hence that no fissure of a large ring from the
main bulk is to be expected. Much hope is now entertained that
the brilliant researches of M. Poincar^ and Professor Darwin on the
form of equilibrium of rotating fluids may eventually remove this
difficulty, and teach us something about the evolution of the solar
fluid when its axial rotation was quickening through contraction.
It is conceivable that even in a heterogeneous body, as the solar
nebula doubtless was, a course of events might take place which
would lead from the sphere through the series of spheroids
and Jacobi ellipsoids to Poincare^s well-known pear-shape; and
ultimately, by increasing constriction of the waist of the pear, to
the division of the body into two or more. But even granting
such a possibility, some difficulty is felt in approaching an
explanation of solar evolution from this groove of thought,
because, as Mr Moulton has pointed out, the solar nebula has not
fulfilled the law of constant moment of momentum. There
can be no doubt that the present sum of rotary moments is
considerably less than it should be if the planets had been

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554 Proceedings of Royal Society of Edinburgh, [sns.

formed by the contraction of a rotating nebula. But the re-
duction of rotary momentum in a system cannot be explained
in absence of external forces ; and since such forces, acting with
sufficient power, cannot be claimed, we must conclude that the
formation of the planets must have been due to a cause different
from that assumed by Laplace. In the following communication,
which is of a merely suggestive character and is based on some
conclusions arrived at in my previous paper on temporary stars,
I have tried to avoid this difficulty by proposing a possible
mode of development, in some respects different from Laplace's
view, but ultimately leading to the same conclusions. I assume
that the conditions necessary for the formation of planets were
introduced after the solar body had condensed from a lum-rotating
nebula into a spherical body of a diameter probably less than the
distance of Mercury. I suppose that at this stage the solar body,
on its course through space, had approached a cosmic cloud of
meteoric constitution, and had passed through a series of cTents
such as have been described in my previous paper, leading — as
was shown there — to the formation of a ring of meteors rotating
with orbital velocities round the solar nucleus. The question to
be discussed is whether we may explain the formation of planets
and their rotation round an axis simply from the heterogeneity of
the ring and the mutual perturbing action of its constituents.
The conclusion, although reached by somewhat general and ad-
mittedly crude considerations, seems yet to be that these perturba-
tions would introduce motions in the particles round a point of the
ring where matter was denser than on the average, such as would
impart a rotation to the condensing planet in the required direction.
There seems also reason to suppose that Professor Darwin's in-
genious conception of fluid-pressure in a meteoric swarm would
sufficiently account for a gradual evacuation of the ring by the
gravitational action of the planets. Lastly, I propose to show
that the suggested view offers an explanation of the origin of
comets compatible with observed facts, and may thus perhaps
supplement the nebular hypothesis with regard to a point as to
which Laplace's theory gives no satisfactory account.

In my paper, "On Professor Seeliger's Theory of Temporary
Stars," an attempt was formerly made to explain the genesis of

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1904-5.] Dr J. Halm 071 the Nebular Hypothesis. 555

rotation in the solar system by one-sided impacts of a meteoric
cloud upon the solar nucleus. We must admit, on dynamical
grounds, that the partial destruction of the orbital velocities of the
meteors involves the generation of closed orbits round the star as
focus, and also that one-sidedness of the impacts leads to the pre-
ponderance of a distinct direction of rotation. It may, however,
seem difficult at first sight to understand how a system, in which
the outer orbits must have possessed large eccentricities, should
have developed into one in which all the bodies move now sensibly
in circles. But on closer examination this difficulty seems to be
lessened. Professor Darwin, in his essay on "The Mechanical
Conditions of a Swarm of Meteorites and on Theories of Cos-
mogony," in the Transactions and Proceedings of the Royal Society
for 1888, has proposed an ingenious thermodynamical theory of
meteoric matter based on the laws of the kinetic theory as
ordinarily applied to gases. One point of his investigation refers
to the viscosity of such an agglomeration of meteoric substance,
which he finds to be remarkably great. His conclusion suggests
that friction must have largely influenced the orbital motions of
the ring-particles. The passage in Professor Darwin's paper
which has a direct bearing on this point may here be quoted : —

" The very essence of the nebular hypothesis is the conception
of fluid-pressure, since without it the idea of a figure of equilibrium
becomes inapplicable. Now, at first sight, the meteoric condition
of matter seems absolutely inconsistent with a fluid-pressure
exercised by one part of the system on another. We thus seem
driven either to the absolute rejection of the nebular hypothesis,
or to deny that the meteoric condition was the immediate ante-
cedent of the sun and the planets. The object of this paper \Proc
Roy, SoCy vol. 45, p. 4] is to point out that by a certain interpreta-
tion of the meteoric theory we may obtain a reconciliation of these
two orders of ideas, and may hold that the origin of stellar and
planetary systems is meteoric, whilst retaining the conception of
fluid-pressure. According to the kinetic theory of gases^ fluid-
pressure is the average result of the impacts of molecules. If we
imagine the molecules magnified until of the size of meteorites,
their impacts will stiU, on a coarser scale, give a quasi-fluid-
pressure. I suggest, then, that the fluid-pressure essential to the

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556 Proceedings of Eoyal Society of Edinburgh, [j

nebular hypothesis is in fact the resultant of countless impacts of
meteorites."

In applying this idea of a kinetic theory of meteors to the
present problem, we have to consider the conditions prevailing
in a system which consists of a non-rotating nucleus surrounded
^y a gyrating ring of meteoric substance. The conception of
fluid-pressure, as proposed by Professor Darwin, involves the
assumption of friction between star and ring. The star's surface
being continually bombarded by neighbouring ring-particles, rotary
momentum is imparted to the star, and is consequently lost by
the ring. The motion of the inner ring is thus gradually reduced,
in much the same way as that of an air-current passing along
the earth's surface. The friction being propagated throughout the
whole ring in accordance with laws similar to those of the internal
friction in gaseous media, the materials of the ring will be con-
stantly submitted to resisting forces acting in the direction of their
motion. Hence, in course of time, the eccentricity of the ring,
as a whole, must be lessened, and the system will tend towards
a figure of equilibrium consistent with fluid-pressure.

This reasoning has brought us to the state of matters from
which Laplace started his hypothesis. We see now some possi-
bility, at least, how, by accepting this new hypothesis, we may ex-
plain the introduction of rotation into our system without abandon-
ing any of the Laplaceian conclusions. So far the present view
may therefore be considered merely as an extension of Laplace's
cosmogonic conceptions. But in consideration of the grave objec-
tions raised against the nebular hypothesis in its present form, it
may seem advisable to trace also the further development of the
rotating fluid, and to see whether the difficulties expressed by
Mr Moulton and others are indeed so insurmountable as they
appear. One of the most serious objections refers to the formation
of the planetary rings. The intermittent shedding-ofi* of annidar
aggregates, which Laplace assumes, is a process not easily adapt-
able to our conception of the physical properties of meteoric
matter. But I think this hiatus may be avoided. Little doubt
can be felt regarding the assumption that the original ring must
have been heterogeneous. Granting this, we must admit the
existence of nuclei of condensation within the ring attracting the

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1904-5.] Dr J. Halm an the Nebular Hypothesis. 557

smaller particles around them. Now let us consider the effect of
these mutual attractions. All the particles in front of the nucleus
(counting in the direction of the rotation of the ring) are pulled
towards the condensation by forces, the tangential components of
which are acting against their orbital motions round the central
star. It is evident that these particles must fall toicards the sun ;
they acquire radial velocities in the inward direction. Exactly
the opposite course of events must happen with particles in the
rear. Here the tangential pull is in the direction of orbital
motion ; they must move from the sun, and hence acquire radial
velocities in the outward direction. On the other hand, the
attraction of the particles on the nucleus acting equally in all
directions, the latter suffers no deflection from its original motion.
Now it is easy to picture what will happen when the attracted
particles coalesce with the nucleus. The conclusion is that
neither the front nor the rear particles fall directly towards the
centre of the condensation : all the front particles must show a
tendency to swing round on the inner side, i,e, between nucleus
and sun, and all the rear particles on the outer side. Hence the
accreting meteors must impart a rotary motion to the condensing
nucleus, and the direction of this rotation must necessarily be
that of the ring itself. Here, then, we have the conditions of
rotary motions actually existing in our system. Whereas Laplace
explains planetary rotation by the difference of speed between *
the outer and inner parts of the ring, which he must therefore
assume to rotate with uniform angular velocity, we find now
that the detachment of a Laplaceian ring and its subsequent
coalescence into a planet is not necessarily required to account
for the rotary motions of the planets.

The next point I desire to illustrate may be inferred from the
following consideration. Let us imagine two bodies of equal
masses to revolve in the same circle round tlie sun. Suppose, also,
that the distance between the two bodies is sufficiently great to
permit us to neglect their mutual attractions. Obviously the
time of their revolution will also be exactly the same, and hence
their distance from each other will remain unaltered. But let us
assume their masses to be unequal. The period of revolution of
the heavier body being shorter than that of the lighter body, the

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658 Proceedings of Royal Society of Ediiiburgh, [ass.

former must gradually overtake the latter. If, for instance, one
of the bodies possesses the mass' of Jupiter, while the other body
has only half this mass, their periods would be in the ratio
1 : 1 000237 ; and hence if the bodies had been 180° apart in the
beginning, they would be at identical points of the common orbit
in about 14,000 years. This reasoning shows clearly that the
planet, after having attained a portion of its mass through
accretion, must gradually bring under its gravitational influence
the smaller masses revolving in its orbit. The planet would
therefore evacuate its own ring. But if we accept Professor
Darwin's conception of fluid-pressure, the idea of a vacuum cannot
be maintained. The gap round the planet would be constantly
filled up by meteors rushing into the planet's orbit from the
outer and inner parts of the ring. The planet would act some-
what like a powerful air-pump, sucking in the meteoric molecules
thrown into its sphere of gravitational attraction by the outside
collisions. We may also gather from the mode in which the
planet acts on the particles in its front and rear that the motions
of those meteors which escape amalgamation with the attracting
nucleus are deflected either towards or from the sun. This, no
doubt, must increase the chance of collisions with the inner and
outer portions of the ring. It is also understood that the in-
creasing diversity of motions of the smaller meteors may assist
the planets in their function of incorporating the small fragments
thrown into their paths. There should be a gradual approach
towards conditions such as we notice at the present moment when
we find meteors crossing the earth's orbit in all possible directions.
I am far from saying, however, that all the present meteors should
be considered in this way as the last remnant of the original ring.

A point of extreme difficulty in Laplace's hypothesis is the
explanation of the present slow rotation of the sun. Mr Moulton *
has demonstrated that if the solar nebula had contracted in the way
Laplace assumed, the moment of momentum of the solar system
should be more than two hundred times what it actually is ; but,
on the other hand, if the nebula had always possessed its present

*F. R. Moulton, "An Attempt to Test the Nebular Hypothesis by an
Appeal to the Laws of DynBm\c%^^ AstropkysicaZ Joximal, vol. xl, 1900,
p. 108.

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1904-6.] Dr J. Halm on the Nebular Hypothesis. 559

moment of momentum, the centrifugal force could not have over-
balanced the centripetal force until the solar body had shrunk
far within the orbit of Mercury. This argument against the
Laplaceian view seems to me unanswerable, and I agree with
Mr Moulton when he contends that it "points to a mode of
development quite different from, and much more complicated
than, that postulated in the nebular theory." The present view is
not exposed to this difficulty ; on the contrary, the slow rotation
of the sun follows of necessity from the mode in which rotation
is supposed to have been brought into the system.

A further point in favour of the hypothesis seems to be the reason-
ing by which the existence of comets and the peculiarities of their
orbits may be explained. Laplace, as is well known, considered the
comets as bodies not belonging to our system. He arrived at this
conclusion by investigating the question what form of cometary
orbits should be the most probable if they are bodies launched
upon us from outside space. He found that the most probable
orbit must be the parabola ; and since this is indeed the typical
form of cometary orbits, he concluded that his supposition on
their origin was correct. Subsequently, however, Schiaparelli has
proved that Laplace committed an error in his analysis, and that
the result to be expected from Laplace's supposition should be
exactly opposite to his conjecture. Schiaparelli showed that the
parabola is in fact the least probable curve in which a foreign
body may intrude upon our system, and that under Laplace's
supposition the great majority of orbits should be hyperbolic.
His researches leave scarcely any doubt that the comets are
members of our own system ; that at practically infinite distance
there exists a cosmic cloud travelling with our sun through space
with practically the same speed and in the same direction ; and
that all the comets originate from this mysterious appendage. To
explain these facts by the Laplaceian hypothesis seems to me
extremely difficult ; but they are rendered almost obvious by the
present theory. I have shown in my paper on temporary stars
that through the catastrophe the star becomes sun'ounded with
an expanding atmosphere of gases and vapours. We have the
strongest possible evidence of the presence of this atmosphere in
the absorption-bands of the spectra of new stars, which by their

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560 Proceedings of Royal Society of Edinburgh. [i

displacements towards the violet indicate that the ejected gases
move with enormous velocities, the greatest exceeding the critical
velocity of a star such as our sun. Matter moving with such speeds
is most probably for ever lost But we may admit that there
must be a considerable range of velocities among the various
parts of this expanding cloud. Some move quicker, others
more slowly, and all those particles whose initial velocities were
less than the critical will sooner or later come to a point of
rest whence they begin their return journey towards the star.
Obviously the slowest particles must have returned soonest ; they
have either impinged upon the solar body long ago, or, in
consequence of perturbations, have been drawn into elliptic
orbits. They may have formed systems of periodic comets in the
past, which now, through the continued disintegrating action of
the sun and planets, have degenerated into meteoric swarms.
Some of the present ^leriodic swarms probably had this origin.
If this view is correct, then the comets falling upon our system
at the present moment must move in ellipses not distinguishable
from parabolae, since their return points must have been at
practically infinite distances from the sun. That the fall of
these bodies is not central, may be explained by the doubtless
inevitable perturbations experienced, not only during their outward
journey, but perhaps also at the outer limit, where the cloud may
at times have been under the gravitational influence of neigh-
bouring stars.* Accepting this view, we understand why comets
describe parabolic and elliptic orbits, why all inclinations are
possible, and why there is the well-known physical resemblance
between the members of this cosmic family. The expanding
atmosphere of a new star would thus be a cometary cloud in statu
nascendi.

The assumption of previous solar condensation, which is clearly
necessary in this theory, may appear as a disadvantage, because it
involves the creation and expenditure of solar energy before the
planets were formed, and thereby seems to limit the time at our
disposal for explaining the evolution of the planetary system. But,
on the one hand, we must keep in mind that the generation of heat
by contraction is at first a slow process. Indeed, the amount of
* This is also Schiaparelli*8 view.

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1904-5.] Dv J.KolmontJie Nehilar Rypothesis, 561

caloric energy produced by the sun through contraction from infinity
to the orbit of Mercury is little more than one per cent, of what he
acquired afterwards through shrinkage to the present diameter.
On the other hand, this loss may have been fully compensated by
the impact of the meteoric cloud. Considering the enormous rise
of temperature when this happened, it is not unlikely that the ring
which probably first developed near the sun's surface, where the
destruction of orbital motion was greatest, through the heat de-
veloped by the collisions, expanded and afterwards filled the whole
space of the planetary system.

But these are perhaps futile speculations which I will not pursue
further, fearing that in this general outline already the hypothesis
has stretched too far into the regions of uncontrolled imagination.
Considered by itself, the theory would be of little value. But the
fact, acknowledged by common consent, that collisions between
stars and nebulae occur even now before our eyes in temporary
stars, and that they are accompanied by phenomena which, judging
from the spectroscopic evidence, point to the genesis of a rotating
ring of nebular matter round the attracting body, is so suggestive
of a similar course of events having been the cause of the rotation
in our system, that I could not resist the temptation to venture
upon speculative ground. Certainly no extraordinary gift of
imagination is required to picture to ourselves the spectrum of the
solar system under the initial conditions here assumed, with its
expanding atmosphere of embryonal comets and the luminous ring
of meteoric substance, the protoplasm of the future planets, and
then to realise that this spectrum must have appeared to a distant
observer in space as the typical spectrum of a new star. Tlie
** experimental " proof of the theory, afforded by the preceding
examination of the spectroscopic evidence of temporary stars, is
therefore encouraging, whatever may otherwise be urged against
the superficial and highly incomplete treatment of so important a
question in this communication.

{Isftued separately April 15, 1905. )

PROC. ROY. SOC. BDIN. — VOL. XXV. 36

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562 Proceedings of Saycd Society of Edinburgh, [i

Deep Water Ship-Wavee.* {Continued from Proc. R.S.K,
June 20th, 1904.) By Lord Kelvin.

(MS. received January 28, 1905. Read same date. )

S 32-64. Canal Ship-Waves.

§ 32. To avoid the somewhat cumhrous title "Two-dimensional,"
I now use the designation " Canal t Waves " to denote waves in
a canal with horizontal bottom and vertical sides, which, if
not two-dimensional in their source, become more and more
approximately two-dimensional at greater and greater distances
from the source. In the present communication the source is
such as to render the motion two-dimensional throughout; the
two dimensions being respectively perpendicular to the bottom,
and parallel to the length of the canal : the canal being straight.

§ 33. The word " deep " in the present communication and
its two predecessors (g 1-31) is used for brevity to mean
infinitely deep; or so deep that the motion does not differ
sensibly from what it would be if the water, being incompressible,
were infinitely deep. This condition is practically fulfilled in
water of finite depth if the distance between every crest (point
of maximum elevation), and neighbouring crest on either side, is
more than two or three times its distance from the bottom.

§ 34. By " ship- waves " I mean any waves produced in open
sea or in a canal by a moving generator; and for simplicity I
r..tr»ru^ao tiiA nirttion of the generator to be rectilineal and uniform.

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1904-5.] Lord Kelvin on Deep Water Ship-Waves, 563

The generator may be a ship floating on the water, or a submarine
ship or a flsh moving at uniform speed below the surface; or,
as suggested by Bayleigh, an electrified body moving above the
surface. For canal ship-waves, if the motion of the water close
to the source is to be two-dimensional, the ship or submarine
must be a pontoon having its sides (or a submerged bar having
its ends) plane and fitting to the sides of the canal, with
freedom to move horizontally. The submerged surface must be
cylindric with generating lines perpendicular to the sides.

§ 35. The case of a circular cylindric bar of diameter small com-
pared with its depth below the surface, moving horizontally at a
constant speed, is a mathematical problem which presents interest-
ing difficulties, worthy of serious work for anyone who may care
to undertake it. The case of a floating pontoon is much more
difficult, because of the discontinuity between free surface of
water and water-surface pressed by a rigid body of given shape,
displacing the water.

§ 36. Choosing a much easier problem than either of those, I
take as wave generator a forcive * consisting of a given continuous
distribution of pressure at the surface, travelling over the surface
at a given speed. To understand the relation of this to the
pontoon problem, imagine the rigid surface of the pontoon to
become flexible ; and imagine applied to it, a given distribution n
of pressure, everywhere perpendicular to it. Take 0, any point at
a distance h above the undisturbed water-level, draw O X parallel
to the length of the canal and OZ vertically downwards. Let
^, { be the displacement-components of any particle of the water
whose undisturbed position is (a;, z). We suppose the disturbance
infinitesimal; by which we mean that the change of distance
between any two particles of water is infinitely small in comparison
with their undisturbed distance ; and that the line joining them
experiences changes of direction which are infinitely small in
comparison with the radian. For liberal interpretation o^ this
condition see § 61 below. Water being assumed frictionless, its
motion, started primarily from rest by pressure applied to the

* '* Forcive" is a very useful word introduced, after careful consultation
with literary authorities, by my brother the late Prof. James Thomson, to
denote any system of force.

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564 Proceedings of Royal Society of EdiTiburgh. [sbssl

free surface^ is essentially irrotational. But we need not assume
this at present : we see immediately that it is proved by onr
equations of motion, when in them we suppose the motion to be
infinitesimal. The equations of motion, when the density of the
liquid is taken as unity, are : —

df- ^dx ^dz dx

dfi ax dz dz

(59).

where g denotes the force of gravity and p the pressure at (a:, 2, t).
Assuming now the liquid to be incompressible, we have

i4-« <«»■

§ 37. The motion being assumed to be infinitesimal, the second
and third terms of the first members of (59) are n^ligible, and
the equations of motion become : —

^^ _^dp\
dt^ dx

dt'^ -^ dz]

(61).

This, by taking the difference of two differentiations, gives : —

(62),

d/d^ dt\
dt\dz ^ dx)

which shows that if at any time the motion is zero or irrotational,
it remains irrotational for ever.

§ 38. If at any time there is rotational motion in any part of
the liquid, it is interesting to know what becomes of it. Leaving
for a moment our present restriction to canal waves, imagine our-
selves on a very smooth sea in a ship, kept moving uniformly at
a good speed by a tow-rope above the water. Looking over the
8hip'& side we see a layer of disturbed motion, showing by dimples
in the surface innumerable little whirl p6ol8. The thickness of
this layer increases from nothing perceptible near the bow to
perhaps 10 or 20 cms. near the stem; more or less according
to the length and speed of the ship. If now the water suddenly
loses viscosity and becomes a perfect fluid, the dynamics of vortex

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1904-5.] Lord Kelvia on Deep JVater Ship- Waves, 565

motion tells us that the rotationally moving water gets left behind
by the ship, and spreads out in the more and more distant wake
and becomes lost;* without, however, losing its kinetic energy,
which becomes reduced to infinitely small velocities in an
infinitely large portion of liquid. The ship now goes on through
the calm sea without producing any more eddies along its sides
and stern, but leaving within an acute angle on each side of its
wake, smooth ship-waves with no eddies or turbulence of any
kind. The ideal annulment of the water's viscosity diminishes
considerably the tension of the tow-rope, but by no means annuls
it; it has still work to do on an ever increasing assemblage of
regular waves extending farther and farther right astern, and

over an area of 19* 28' (tan ~^. /t;) on each side of mid-wake, as

we shall see in about § 80 below. Returning now to two-dimen-
sional motion and canal waves : we, in virtue of (62), put

where ^ denotes what is commonly called the "velocity-
potential"; which, when convenient, we shall write in full
^(a:, 2, i). With this notation (6 1 ) gives by integration with
respect to x and 2,

^--y+iK'+C) m-

And (60) gives

Following Fourier's method, take now

<^(ar, 2, 0 = - li^-"^ sin m{x^vt) , . . . (66),

* It now seems to me certain that if any moliou be given within a finite
portion of an infinite incompressible liquid originally at rest, its fate is
necessarily dissipation to infinite distances with infinitely small velocities
evtrywhere; while the total kinetic energy remains constant. After
many years of failure to prove that the motion in the ordinary Helmholtz
circular ring is stable, I came to the conclusion that it is essentially unstable,
and that its fate must be to become dissipated as now described. I came
to this conclusion by extensions not hitherto published of the considerations
described in a short paper entitled : ** On the stability of .steady and periodic
fluid motion/' in the Fhil, Mag, for Mny 1887.

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566 Proceedbxgs of Royal Society of Bdiriburgh. [siss.

which satisfies (65) and expresses a sinusoidal wave-disturbance,
of wave-length 2ir/m, travelling ar- wards with velocity v,

§ 39. To find the boundary-pressure II, which must act on the
water-surface to get the motion represented by (66), when m, r, A:
are given, we must apply (64) to the boundary. Let z==0 be the
undisturbed surface ; and let d denote its depression, at (a-, o, t),
below undisturbed level ; that is to say,

<l = ^(ic, o, t) = --<f>{x, z, t%^^ink sin m(x-vt) . (67),
az

whence by integration with respect to t,

d=- cos m(x-vt) (68).

To apply (64) to the surface, we must, in gz^ put z — d; and in
dff>/dt we may put z =» 0, because d, k, are infinitely small quantities
of the first order, and their product is neglected in our problem of
infinitesimal displacements. Hence with (66) and (68), and
with n taken to denote surface-pressure, (64) becomes

kmv cos yn{x - r/) = ^k cos m{x - vt) -Il-{-gG , (69) ;

whence, with the arbitrary constant C taken = 0 ,

11 = Ai7^ -- - mj cos m{x -vt) (70) ;

and, eliminating k by (68), we have finally,

n = (i/-mt;2)d (71).

Thus we see that if r = n/^//w, we have n = 0, and therefore we
have a train of free sinusoidal waves having wave-length equal to
27r/m, This is the well-known law of relation between velocity
and length of free deep-sea waves. But if r is not equal to ,Jg/m ,
we have forced waves with a surface-pressure (^-mr*)d which
is directed with or against the displacement according as
v< or >^g/m.

§ 40. Let now our problem be : — given n, a sum of sinusoidal
functions, instead of a single one, as in (70); — required d the
resulting displacement of the water-surface. We have by (71)
and (70), with properly altered notation.

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1904-6.] Lord Kelvin on Deep Water Ship- Waves. 567

U = 'SBcoam{x-vt + fi) (72),

d^':i, ^-^-^008 m(X'-vt + 8) + A cosi.{x-vt + y) . (73),

where B, m, fi are given constants having different values in the
different terms of the sums ; and t; is a given constant velocity.
The last term of (73) expresses, with two arbitrary constants
(A, y), a train of free waves which we may superimpose on any
solution of our problem.

§ 41. It is very interesting and instructive in respect to the
dynamics of water-waves, to apply (72) to a particular case of
Fourier's expansion of periodic arbitrary functions such as a dis-
tribution of alternate constant pressures, and zeros, on equal
successive spaces, travelling with velocity v. But this must be
left undone for the present, to let us get on with ship- waves ; and
for this purpose we may take as a case of (72), (73),

n = ^c{^-Heco8^-He«cos2^ + etc.)«yc^_|^^J^^^^ ^^ (74),

'-^'{^

+ .l^costf-H^i^cos2tf + etc. I . . .(75);
J — 1 J — 2 3

where

Q^'^JLi^^vt^fl) (76);

•"-'h^-l-B <">^

and e may be any numeric < 1. Remark that when t? = 0 , J = oo ,
and we have by (75) and (74), d = n/^, which explains our unit
of pressure,

§ 42. To understand the dynamical conditions thus prescribed,
and the resulting motion: — remark first that (74), with (76),
represents a space-periodic distribution of pressure on the surface,
travelling with velocity » ; and (75) represents the displacement
of the water-surface in the resulting motion, when space-periodic
of the same space-period as the surface-pressure. Any motion
whatever; consequent on any initial disturbance and no subse-
quent application of surface-pressure ; may be superimposed on the
solution represented by (75), to constitute the complete solution

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568 Proceedings of Bat/al Society of Edinburgh. [siaB,

of the problem of finding the motion in which the sorface-pressore
is that given in (74).

§ 43. To understand thoroughly the constitution of the forcive-
datum (74) for n, it is helpful to know that, n denoting any
positive or negative integer, we have

2ir(i + ecos^ + e«co82^ + etc.)= 2

n— OD l;^-¥{x-na)^

ft = ^log(l/e)
2ir

(78),

(79).

This we find by applying § 15 above to the periodic function
represented by the second member of (78).

The equality of the two members of (78) is illustrated by fig. 11 ;

O 'I l 'I 4 -5 -6 -r •« •<} C'

Fio. 11 ; e=*6,

in which ; for the case c= '5 and consequently, by (79), 6/a= -1103;
the heavy curve represents the first member, and the two light
curves represent two terms of the second member ; which are as

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1901-5.] Lord Kelvin on Deep WcUer Skip- Waves.

669

many as the scale of the diagram allows to be seen on it. There
is a somewhat close agreement between each of the light curves,
and the part of the heavy curve between a maximum and the
minimum on each side of it. Thus we see that even with e so
small as '5, we have a not very roicgh approximation to equality

i 'f f

4—4-

Fio. 12 ; «=-9.

between successive half periods of the first member of (78) and a
single term of its second member. If e is <1 by an infinitely
small difference this approximation is infinitely nearly perfect.
It is so nearly perfect for 6=*9 that fig. 12 cannot show any
deviation from it, on a scale of ordinates 1/10 of that of fig. 11.
The tendency to agreement between the first member of (78) and
a single term of its second member with values of e approaching to
1, is well shown by the following modification of the last member
of (74) :—

^(l-e«) ^(1-e^)

""•^^l-2eco8^ + e2-^^(l-6)2 + 4eain2J^ ' ' ^^^)-

Thus we see that if e = 1 , 11 is very great when 6 is very small ;
and n is very small urUesa $ is very small (or very nearly = 2i7r).
Thus when e = 1, we have

1(1 -«^)

(81);

which means expressing n approximately by a single term of the
second member of (78).

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570 Proceedings of Royal Society of Edinburgh, [i

§ 44. Return to our dynamical solution (75) ; and remark that
if J is an integer, one term of (75) is infinite, of which the
dynamical meaning is clear in (70). Hence to have every term
of (75) finite we must have J -J + 8, where ; is an integer and 8 is
< 1 ; and we may conveniently write (75) as follows :

l = c(8+i){i8|

e COS d e* cos 20 ^ cosjtf

• + ^ , • 1 + s~T7 o + . . . . + K

e/^^cosO>l)^ ^"^70^^^)^-adinf,] (82);

d=<§^+c/ (83),

where y and J denote finite and infinite series shown in (82).

§ 45. We are going to make $ = | ; and in this case J can be
summed, in finite terms, as follows. First midtiply each term hy
^•+1 e-;-«' and we find

c/= -c(8+^y+«[j-j'^cos(i+l)^+ 2*:^cosO' + 2)^+etc.]
= -c(8+y)e^+« /^(/d e-«cos(y+l)tf+ei-«cos(y + 2)tf + etc. 1
= - c(8 +jV+* [de e-«{RS}^+i(l + eg + eV + ©tc.) ;

where g denotes €^ ; and, as in § 3 above, {RS} denotes
realisation by taking half sum for ±t. Summing the infinite
series, and performing /Se , for the case 8 = |, we find

c/=-cO' + J)e»+*{RS}2*+Uogi±>^ (84),

^ ^ I i^i 1- ^ecosj^-t^esm^^

^=tan-»/7^^^,.. f = tan-i ^^^i-^ij . . (86),
l + /v/ecos^^ l-/^€cosJ^

where

and therefore

1 -e •

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1904-5.] Lord Kelvin on Deep Water SMp-Waves^ 571

Hence finally

+«iiy + i)» l.ii-'?4^iil» I . (86).
1 -e J

For our present case, of 8 = J , (82) gives
^-«'0>i){ij^+^+^-^'+.... + ?^'} (87).

With c/ and <^ thus expressed, (83) gives the solution of our
problem.

§ 46. In all the calculations of §§ 46-61 I have taken e= '9, as
suggested for hydrokinetic illustrations in Lecture X. of my
Baltimore Lectures, pp. 113, 114, from which fig. 12, and part of
fig. 11 above, are taken. Results calculated from (83), (86), (87),
are represented in figs. 13-16, all for the same forcive, (74) with
€"'9, and for the four different velocities of its travel, which
correspond to the values 20, 9, 4, 0, of /. The wave-lengths
of free waves having these velocities are [(77) above] 2a/41,
2a/19, 2a/ 9, and 2a. The velocities are inversely proportional
to ^41, ^19, ^9, J2, Each diagram shows the forcive by one
curve, a repetition of fig. 12; and shows by another curve the
depression, d, of the water-surface produced by it, when travelling
at one or other of the four speeds.

§ 47. Taking first the last, being the highest, of those speeds,
we see by fig. 16 that the forcive travelling at that speed produces
maximum displacement uptcards where the dovmward pressure is
greatest ; and maximum dovmward displacement where the pressure
(everywhere downward) is least. Judging dynamically it is easy
to see that greater and greater speeds of the forcive would still
give displacements above the mean level where the downward
pressure of the forcive is greatest, and below the mean level where
it is least; but with diminishing magnitudes down to zero for
infinite speed.

And in (75) we have, for all positive values of J<1, a series
always convergent, (though sluggishly when e=l,) by which the
displacement can be exactly calculated for every value of $,

§48. Take nextffig. 15, for which J=4J, and therefore, by

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572 Proceedings of Royal Society of JSdinburgh. , [i

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1904-6.] Lord Kelvin on Deep Water Ship-Wave^, olZ

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674 Proceedings of Royal Society of Edinburgh, \\

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1904^.] Lord Kelvin on Deep Water Ship^ Waves. 575

o
ii

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576 Proceedings of Royal Society of JEdirihurgh. [i

(77), V = Jgaj^w, and X = a/4-5. Remark that the scale of
ordinates is, in fig. 15, only 1/2*5 of the scale in fig. 16 ; and see how
enormously great is the water-disturhance now in comparison with
what we had with the same forcive, but three times greater speed
and nine times greater wave-length (v = Jga/v^ X = 2a). Within
the space-period of fig. 1 5 we see four complete waves, very approxi-
mately sinusoidal, between M, M, two maximums of depression
which are almost exactly (but very slightly less than) quarter
wave-lengths between C and C. Imagine the curve to be exactly
sinusoidal throughout, and continued sinusoidally to cut the zero
line at CC.

We should thus have in C C a train of 4 J sinusoidal waves ;
and if the same is continued throughout the infinite procession
.... CCC .... we have a discontinuous periodic curve
made up of continuous portions each 4^ periods of sinusoidal
curve beginning and ending with zero. The change at each point
of discontinuity C is merely a half-period change of phase. A
slight alteration of this discontinuous curve within 60* on each
side of each C, converts it into the continuous wavy curve of fig. 15,
which represents the water-surface due to motion at speed fjga/9v
of the pressural forcive represented by the other continuous curve
of fig. 15.

§ 49. Every word of § 48 is applicable to figs. 14 and 13 except
references to ^eed of the forcive, which is Jga/ldir for fig. 14
and Jga/^lir for fig. 13; and other statements requiring modifica-
tion as follows : —

For 4^ "periods" or ** waves," in respect to fig. 15; substitute
9i in respect to fig. 14, and 20 J in respect to fig. 13.

For "depression" in defining M M in respect to figs. 15, 14;
substitute elevation in the case of fig. 13. ,

§ 50. How do we know that, as said in § 48, the formula
{(83), (86), (87)} gives for a wide range of about 120' on each
side of ^=180',

dW==(-l)^d(180').sinO' + i)^ . . . (88),
which is merely §§ 48, 49 in symbols ? it being understood that j
is any integer not < 4 ; and that e is '9^ or any numeric between
'9 and 1 ? I wish I could give a short answer to this question

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1904-5.] Lord Kelvin on Deep Water Ship- Waves. 577

without help of hydrokinetic ideas ! Here is the only answer I
can give at present.

§ 51. Look at figs. 12-16, and see how, in the forcive de-
fined by c='9, the pressure is almost wholly confined to the
spaces ^<60* on each side of each of its maximums, and is very
Ttearly null from ^=60* to ^=300*. It is obvious that if the
j -essure were perfectly annulled in these last-mentioned spaces,
viiile in the spaces within 60* on each side of each maximum
the pressure is that expressed by (74), the resulting motion would
be sensibly the same as if the pressure were throughout the whole
space C C (^ = 0* to ^= 360"), exactly that given by (74). Hence
we must expect to find through nearly the whole space of 240*,
from 60* to 300*, an almost exactly sinusoidal displacement of
water-surface, having the wave-length 360*/(y-!-J) due to the
translational speed of the forcive.

§ 52. I confess that I did 7ioi expect so small a difference from
sinusoidality through the whole 240*, as calculation by {(83), (86),
(87)} has proved; and as is shown in figs. 18, 19, 20, by the
D-curve on the right-hand side of C, which represents in each
case the value of

D(^) = d(^) - ( - 1)M(180*). sin U + i)^ . . . (89),

being the difference of d(^) from one continuous sinusoidal curve.
The exceeding smallness of this difference for distances from
C exceeding 20* or 30*, and therefore through a range between
C C of 320*, or 300*, is very remarkable in each case.

§ 53. The dynamical interpretation of (88), and figs. 18, 19, 20,
is this: — Superimpose on the solution {(83), (86), (87)} a "free
-wave " solution according to (73), taken as

-(-l)>d(180*). sin(i-hi)d .... (90).

This approximately annuls the approximately sinusoidal portion
between C and C shown in figs. (13), (14), (15); and approxi-
mately doubles the approximately sinusoidal displacement in the
corresponding portions of the spaces CC, and CC on the two
sides of C C. This is a very interesting solution of our problem
§ 41 ; and, though it is curiously artificial, it leads direct and
short to the determinate solution of the following general problem
of canal ship- waves : —

PROC. ROY. SOC. EDIN. — VOL. XXV. 37

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678 Proceedings of Royal Society of Edinburgh,

[SBSS.

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1904-5.] Lord Kelvin on Deep Water Ship-Waves, 579

«N

t«

•^4

•9

0»

^ Cr

» « Ml

-><»*

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580 Proceedings of Royal Society of Edinburgh. [t

§ 54. Given, as forcive, the isolated distribution of pressure
defined in fig. 12, travelling at a given constant speed; required
the steady distribution of displacement of the water in the place
of the forcive, and before it and behind it ; which becomes estab-
lished after the motion of the forcive has been kept steady for
a sufficiently long time. Pure synthesis of the special solution
given in g 1-10 above, solves not only the problem now proposed,
but gives the whole motion from the instant of the application
of the moving forcive. This synthesis, though easily put into
formula, is not easily worked out to any practical conclusion. On
the other hand, here is my present short but complete solution of
the problem of steady motion for which we have been preparing,
and working out illustrations in §§ 32-53.

Continue leftward, indefinitely, as a curve of sines, the D curve
of each of figs. 18, 19, 20; leaving the forcive curve, F, isolated,
as shown already in these diagrams. Or, analytically stated : —
in (89) calculate the equal values of d(0) for equal positive and
negative values of $ from 0* to 40* or 50* by {(83), (86), (87)} ;
and for all larger values of 6 take

d(^)=(-iyd(180')sin0* + i)^ (91),

where d(180') is calculated by {(83), (86), (87)}. This used in
(89), makes D(^)=0 for all positive values of 0 greater than 40*
or 50* ; and makes it the double of (91) for all negative values of
e beyond - 40* or - 50*.

^ 55, 56. Rigid Covers or Pontoons^ introduced to apply the given
forcive (pressure on the water-surf ace),

§ 55. In any one of our diagrams showing a water-surface
imagine a rigid cover to be fixed, fitting close to the whole water-
surface. Now look at the forcive curve, F, on the same diagram,
and wherever it shows no sensible pressure remove the cover.
The motion (non-motion in some parts) of the whole water remains
unchanged. Thus, for example, in figs. 13, 14, 15, 16, let the
water be covered by stiff covers fitting it to 60* on each side of
each C ; and let the surface be free from 60* to 300* in each of
the spaces between these covers. The motion remains unchanged

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1904-5.] Lord Kelvin on Deep Watei* Ship- Waves, 581

under the covers, and under the free portions of the surface. The
pressure n constituting the given forcive, and represented by the
F curve in each case, is now automatically applied by the covera.

§ 56. Do the same in figs. 18, 19, 20 Mrith reference to the
isolated forcives which they show. Thus we have three different
cases in which a single rigid cover, which we may construct as the
bottom of a floating pontoon, kept moving at a stated velocity rela-
tively to the still water before it, leaves a train of sinusoidal waves
in its rear. The D curve represents the bottom of the pontoon in
each case. The arrow shows the direction of the motion of the
pontoon. The F curve shows the pressure on the bottom of the
pontoon. In fig. 20 this pressure is so small at - 2q that the
pontoon may be supposed to end there; and it will leave the
water with free surface almost exactly sinusoidal to an indefinite
distance behind it (infinite distance if the motion has been
uniform for an infinite time). The F curve shows that in fig. 19
the water wants guidance as far back as - 3^, and in fig. 18 as far
back as - 8g to keep it sinusoidal when left free ; q being in each
case the quarter wave-length.

§5 57-60. Shapes for Waoeless Pontoons, and their Forcives.

S 57. Taking any case such as those represented in figs. 18, 19,
20 ; we see obviously that if any two equal and similar forcives
are applied, with a distance ^X between corresponding points, and
if the forcive thus constituted is caused to travel at speed equal to
iJgXI^TTy being, according to (77) above, the velocity of free waves
of length X, the water will be left waveless (at rest) behind the
travelling forcive.

§ 58. Taking for example the forcives and speeds of figs. 18, 19,
20, and duplicating each forcive in the manner defined in § 57, we
find, (by proper additions of two numbers, taken from our tables
of numbers calculated for figs. 18, 19, 20,) the numbers which give
the depressions of the water in the three corresponding waveless
motions. These results are shown graphically in fig. 21, on scales
arranged for a common velocity. The free wave-length for this
velocity is shown as iq in the diagram.

§ 59. The three forcives, and the. three waveless water-shapes

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582 Proceedings of Boyal Society of Edinburgh. [sess.

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1904-5.] Lord Kelvin on Deep Waier Ship-Waves, 583

produced by them, are shown in figs. 22, 23, 24 on different scales,
of wave-length, and pressure, chosen for the convenience of each
case.

§ 60. As most interesting of the three cases take that derived
fromy = 9 of our original investigation. By looking at fig. 23 we
see that a pontoon having its bottom shaped according to the
D curve from -3g to +3g, \\ free wave-lengths, will leave the
water sensibly fiat and at j*est if it moves along the canal at the
velocity for which the free-wave-length is 4g. And the pressure
of the water on the bottom of the pontoon is that represented
hydrostatically by the F curve.

§ 61. Imagine the scale of abscissas in each of the four diagrams,
figs. 21-24, to be enlarged tenfold. The greatest steepnesses of the
D curve in each case are rendered sufficiently moderate to allow it
to fairly represent a real water-surface under the given forcive.
The same may be said of figs. 15, 16, 18, 19, 20 ; and of figs. 13,
14 with abscissas enlarged twentyfold. In respect to mathematical
hydrokinetics generally; it is interesting to remark that a very
liberal interpretation of the condition of infinitesimality (§ 36
above) is practically allowable. Inclinations to the horizon of as
much as 1/10 of a radian (5* '7 ; or, say, 6*), in any real case of
water-waves or disturbances, will not seriously vitiate the mathe-
matical result.

§ 62. Fig. 17 represents the calculations of d(0*) and
(-iyd(180*) for twenty-nine integral values of j; 0, 1, 2, 3,
.... 19, 20, 30, 40, ... . 90, 100; from the following
formulas, found by putting ^ = 0* and ^=180°; and with e=*9
in each case, and c = 1

dXl80') = (- 1)'(2;-+ l)e'[ie«tan-'^-^^+ 1 " y + y'+ • • •

The asymptote of d(0*) shown in the diagram is explained by
remarking that when j is infinitely great, the travelling velocity of

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584 Proceedings of Royal Society of Edinburgh. [sbps.

the forcive is infinitely small; and therefore, by end of §41, the
depression is that hydrostatically due to the forcive pressure. Thijs
at ^ = 0*, is equal to

^ 1-c 2

§ 63. The interpretation of the curves of fig. 17 for points
between those corresponding to integral values of j is exceedingly
interesting. We shall be led by it into an investigation of the
disturbance produced by the motion of a single forcive, expressed

n-jg, (94);

but this must be left for a future communication, when it will be
taken up as a preliminary to sea ship-waves,

§ 64. The plan of solving by aid of periodic functions the
two-dimensional ship-wave problem for infinitely deep water,
adopted in the present communication, was given in Part IV.
of a series of papers on Stationary Waves in Flowing Water,
published in the Philosophical Magazine, October 1886 to January
1887, with analytical methods suited for water of finite depths.
The annidment of sinusoidal waves in front of the source of
disturbance (a bar across the bottom of the canal), by the super-
position of a train of free sinusoidal waves which double the
sinusoidal waves in the rear, was illustrated (December 1886) by
a diagram on a scale too small to show the residual disturbance
of the water in front, described in § 53 above, and represented
in figs. 18, 19, 20.

In conclusion, I desire to thank Mr J. de Graaff Hunter for
his interested and zealous co-operation with me in all the work of
the present communication, and for the great labour he has given
in the calculation of results, and their representation by diagrams.

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1904-5.] Lord Kelvin on Deep Water Ship- Waves.

585

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586 Proceedings of Royal Society of EdvnJbvrgh, [i

>oi

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1904-5.] Lord Kelvin on Deep Water Ship -Waves,

587

•3

{Issued separately April 18, 1905.)

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588 Proceedings of Royal Society of Edinhirgh, [sisa.

On Two Liquid States of Sulphur Sx and S^ and their

Transition Point. By Alexander Smith.

(MS. received Febroaiy 17, 1905. Read March 20, 1905.)

{Ahsti'oct,)

It is well known that melted sulphur when heated beror**-
suddenly dark brown and viscous in the neighbourhood of • •" ^
170*. These and other facts rendered it probable that th^n* v .^
a transition point in liquid sulphur, and that two Uquid ^•.: •
could be proved to exist, one of them being stable belov
transition point, and the other above it. According to the phase
rule, a single substance can exist in three phases (two liquid and
one vapour phase) only as a non- variant system at a single tempera-
ture and pressure. Thus, if the two liquid forms were not
completely miscible, the lower one might form the greater part of
the material until, as the temperature rose, it became saturated
with the upper one and a new phase separated out This
phenomenon would mark the transition point, and the smallest
further rise in temperature would cause the complete disappearance
of the first phase. The substance would then contain a small
proportion of the lower form in solution in the upper, and this
proportion would diminish with rising temperature. No case of
an exactly parallel nature is known ; but the transition from
*' liquid crystals " to an isotropic liquid in the case of certain
organic compounds is to a certain extent analogous.

For the discovery of a transition point of this kind the study of
the progressive change in any physical property as the temperature
rises is available. The author examined successively the change
in viscosity, the change in solubility, the variation in the co-
efficient of dilatation, and the rather marked absorption of heat
which is observed to accompany the sudden onset of viscosity in
the fluid. The results were as follows : —

1. In melted sulphur, easily perceptible viscosity appears first at
159*5*. At 160** the viscosity is already very great.

2. When sulphur is held at 162*5* or any higher temperature a
sudden absorption of heat and simultaneous sudden access of vis-
cosity occur, and the temperature falls to l62*. The transition
point is therefore not above the latter temperature.

3. Distilled sulphur does not show either of these phenomena

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1904-5.] Prof. Smith on Two Liquid States of Sulphur. 589

so sharply as does recrystaUised sulphur, and seems to be subject
to superheating.

4. It was shown in a previous paper {Proc, Roy, Soc, Edin.,
vol. xxiv. p. 342) that ordinary sulphur owes to the presence of
sulphur dioxide its tendency to give amorphous sulphur when
chilled, and that sulphur which while melted has been treated
with ammonia, gives when quenched nothing but soluble crystal-
line sulphur. The phenomena described in 1 and 2 above take
place in the same way and at precisely the same temperatures,
whether the sulphur concerned is such as by chilling gives in-
soluble sulphur, or, having been treated with ammonia, does not.

5. The existence of two independent curves of solubility for
the two kinds of liquid sulphur in triphenylmethane and other
solvents is demonstrated. The solubility of yellow mobile
sulphur (Sx) increases, that of brown viscous sulphur (S^) de-
creases, with rise in temperature.

6. The expansion of yellow mobile sulphur (Sa) diminishes
rapidly from 154** to 160°; that of brown viscous sulphur (S^)
increases rapidly from 160" upwards. The statement under 4
holds in this case also.

7. The dilatometric method gives no evidence of the existence
of Frankenheim's transition point (250-260**).

8. It is shown that the point of minimum dilatation is displaced
upwards when triphenylmethane is dissolved in the sulphur. The
displacement averages 2*8° for 1 per cent, of this foreign body.

9. The production of the new phase is easily to be seen when
strongly heated brown viscous sulphur is allowed to cool in a test-
tube. The radiation from the greater surface at the bottom
causes the formation of the mobile yellow liquid first in that
region. The interface between the two varieties is quite distinct,
and recedes slowly up the tube as the transition proceeds. Owing,
however, to the progress of the change mthin the upper brown
layer, the interface gradually becomes indistinct.

10. It is thus shown conclusively that there are two liquid
states of sulphur, which are partially, but only partially, miscible.
These are S^, which predominates from the melting point to 160%
and S^, which prevails above 160*. As the temperature ascends,
saturation of the former with the latter determines the separation
of the new phase, and conversely when the temperature falls.

(Issiied separately April 18, 1905.)

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590 Proceedings of Royal Society of JEdiriburgh,

The Nature of Amorphous Sulphur, and ContributionB
to the Study of the Influence of Foreign Bodies *
the Phenomena of Superoooling observed -wl
Melted Sulphur is suddenly Chilled. By Aiezano**
Smith.

(MS. received February 17, 1905. Read March 20, 1905.)

(Abstract)

1. The hardening of plastic sulphur was investigated, and it was
found that partial reversion to soluble sulphur prevents the securing
in quasi-solid form of the whole of the amorphous sulphur present.
It was discovered, however, that sulphur formed by precipitation
in presence of concentrated acids does yield 100 per cent, of
insoluble sulphur, and that only the impossibility of realising the
requisite condition of very fine subdivision is therefore responsible
for the smaller yields from melted sulphur which has reached the
highest temperatures previous to being chilled.

2. A new series of measurements of the proportions of insoluble
sulphur formed when common sulphur is chilled from various
temperatures was made. The amounts vary from 4*2 per cent, at
130" to 34 per cent, at 448°. In this, and in all other cases
described below, only the insoluble sulphur which remains after
the viscous material has completely hardened was estimated.

3. It was found that when sulphur was subjected to prolonged
heating at 448'', or was heated for a shorter time in vacuo, or was
used immediately after recrystallisation, or was washed with water
before being heated, the amount of insoluble sulphur obtainable
by chilling was greatly reduced. The eflfects of these modes
of treatment seemed to be to remove a trace of sulphuric acid
which sulphur acquires by exposure to the air.

4. It appeared that gases like carbon dioxide, and particularly
ammonia and hydrogen sulphide, when led through melted sulphur,

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1904-5.] Professor Smith on Amorphovs Sulphur. 591

destroyed the ahiHty to give insoluble sulphur. Their use did not
however, affect the viscosity above 160°.

5. It was shown that air and sulphur dioxide restored the ability
to give insoluble sulphur. The halogens, the halogen hydrides,
and even dry phosphoric acid, had the same effect.

6. It was found that sulphur which had been treated with
ammonia while melted, and had afterwards been recrystallised, if
used at once, froze at 119*17° and contained no insoluble sulphur.
In a previous investigation {Proc. R,S,E,, vol. xxiv. p. 299) it had
been shown that insoluble sulphur, when present, depressed the
freezing point, in accordance with Raoult's law.

7. Sulphur containing iodine (100:2) gave when heated and
chilled large amounts of insoluble sulphur. These ranged from
4 per cent, at 110" to 62-7 per cent, at 448%

8. The amount of insoluble sulphur obtained at 150° was pro-
portional to the quantity of iodine present when the quantity of
the latter was 1 per cent, or more.

9. Sulphur prepared by distilling the element and quenching
the burning stream in ice-water gave 51 per cent, of insoluble
sulphur. Chilling boiling sulphur in ether gave 44*1 per cent, of
the insoluble form.

10. It was demonstrated, by identity in boiling points under
ordinary and reduced pressures, and by identity in specific gravities,
that sulphur which will give the insoluble form when chilled is
identical in constitution near the boiling point with that which
will not.

11. It was shown by identity in solubility between 120° and
160° that the two kinds of sulphur mentioned in 10 are identical
in constitution also below the transition point of S^ to S;^ (160*).

12. The facts referred to in 10 and 11, together with the con-
clusions of the preceding paper showing the identity of the two
kinds of sulphur at the transition point (160°) itself, demonstrate
that the insoluble form is present in all specimens of melted
sulphur in proportions depending upon the temperature alone,
whether, by treatment with ammonia or otherwise, they have lost
the capacity to give insoluble sulphur by chilling or not.

13. The conclusion is reached that amorphous sulphur is
supercooled S/*, — the form stable above 160°.

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592 ProceediTigs of Royal Society of Edinburgh. [i

14. With pure sulphur, freed from sulphur dioxide by recrystal-
lisation or by treatment with carbon dioxide above 310*, or by
treatment with ammonia or hydrogen sulphide at any temperature
at which it is fluid, the S^ reverts so rapidly to the soluble form
that it cannot be supercooled. When traces of sulphur dioxide,
iodine, and other substances are present, S^ is more or less co"!-
pletely supercooled and gives amorphous sulphur. The way ■
which the latter class of foreign substances produces this eff

is still being investigated.

15. There is a close analogy of these phenomena to thi ^
observed in the cooling of cast-iron and steel.

{l8S%ted separaiehj April 18, 1906.)

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Ill

To prevent delay, authors residing abroad sliould appoint some one
residing in this country to correct their proofs.

4. Additions to a Papek after it has been finally handed in for
publication, if accepted* by the Council, will be treated and dated as
separate communications, and may, or may not, be printed immediately
after the original paper.

5. Brief Abstracts op Transactions Papers will be published in
the Proceedings, provided they are sent along with the original paper.

The author receives fifty of these reprints free, and may have any
reasonable number of additional copies at a fixed scale of prices which
will be furnished by the printer, who will charge him with the cost.
To prevent disappointment, especially if the paper contains plates,
the author should, immediately after receiving his first proof, notify
to the jpi-inier the number of additional copies required.

7. Index Slips. — In order to facilitate the compilation of Subject
Indices, and to secure that due attention to the Important points in a
paper shall bo given in General Catalogues of Scientific Literature and
in Abstracts by Periodicals, every author is requested to return to the
Secretnry along with his final proof a brief index (on the model given
below), of the points in it which he considers new or important. These
indices will be edited by the Secretary, and incorporated in Separate
Index Slips, to be issued with each part of the Proceedings and
Transactions.

IklODEL INDEX.

Schafer, E. A. — On the Existence within the Liver Cells of Channels which can
be directly injected from the Blood-vessels. Proc. Roy. Soc. Edin., vol. ,
1902, pp.
Cells, Liver, — Intra-cellular Canaliculi in.

E. A. Schafer. Proc. Roy. Soc Edin., vol. , 1902, pp.
Liver, — Injection within Cells of.

E. A. Schafer. Proc. Roy. Soc Edin., vol. , 1902, pp.

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IV CONTENTS.

PAGE

A Study of Three Vegetarian Diets. By J). Noel Patox
and J. C. Dun LOP. {From the Research Lafxtrafory
of the Royal College of Phydciam, Eilinlnir(jh\ . 498

{Issued seixtraiely ApHl 8, 1905.)

Continuants whose Main Diagonal is Univarial. By

Thomas Muir, LI^D., .... 507

{Issued separately April 8, 1905.)

On Professor Seeliger's Theory of Temporary Stars. By
J. Halm, Ph.D., Lecturer on Astronomy in the
University of Edinburgh, and Assistant Astronomei
at the Royal Observatory, . . . .513

{Issued separately April 15, 1905.)

Some Suggestions on the Nebular Hypothesis. By

J. Halm, Ph.D., . . . . .553

{Issued separately April 15, 1905.)

Deep Water Ship-Waves. {Goiitimied from Proc. R.S.E.,

.Tune 20th, 1904.) By T^rd Kelvin, . . 562

{Issued separately April 18, 1905.)

On Two Liquid States of Sulphur Sa and S^ and^^their

Transition Point. By Alexander Smith. (Abstract), 588
{Issued separately April 18, 1905.)

The Nature of Amorphous Sulphur, and Contributions to
the Study of the Influence of Foreign Bodies^ on the
Phenomena of Supercooling observed when Melted
Sulphur is suddenly Chilled. By Albxander Smith.
{Abstract), ...... 590

{Issued separately April 18, 1905.)

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