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Proceedings of the 

Royal Society of Edinburgh 



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 



33 



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 



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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. 


1 


Andreas Diurhuas, 
Jacob Jacobeen, . 


65 


^Z 


brown 


2 


68 


brown 


8 


Andreas Jacobsen, 


44 


blue 


brown 


4 


Jacob MikkelsoD, . 


86 


blue 


fair 


5 


Christian Christiansen, 


42 


blue 


fair 


6 


Ole Hansen, 


40 


blue 


brown 


7 


Rasmns Andreassen, . 


46 


blue 


fair 


8 


Paul Nichodemussen, . 


65 


blue 


fair 


9 


Joen Gjoueraa, 


44 


blue 


fair 


10 


Paul Hansen, 


40 


blue 


dark 


11 


Tomas Yule Nichalsen, 


87 


blue 


red 


12 


William Paulsen, . 


55 


bine 


brown 


18 


Daniel Samuelsen, 


81 


blue 


fair 


14 


Peter Hana SiSrensen, . 


59 


dark brown 


fair 


15 


Andreas Olsen, . 


24 


dark 


16 


Peter Haraldsen, . 


63 


bine 


brown 


17 


Hans Mikkelsen, . 


52 


dark grey 


black 


18 


NilsJoensen, 


27 


blue 


fair 


19 


Djone Isaksen, 


54 


blue 


brown 


20 


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 




J) 

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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16 



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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30 



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 


12 


D13P12V9. 


5§ times 


10 


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^ 


8" 


m 


17 cm. 


12 D14V9. 


11 


344" 


rr 


m 


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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46 



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 
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 

1 


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), . 


13 

12-6 

13 

13-6 

13 

12 

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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52 



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 



2' 



3! 



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. 


% 


^ 


A 


tV 


mol. 


1-005 .. 


. 1-005 .. 


-0 


i 


»> • 




1012 .. 


. 1011 .. 


. - -001 


i 


M 




1-024 .. 


. 1-022 . 


. --002 




mol. 




1-0 16 .. 


. 1-046 . 




2 


»» 




1-089 .. 


. 1092 . 


'. +-*()03 






Acetamide, 






Concentration. 


% 


^ 


A 


1 mol. 


1-013 .. 


. 1-014 .. 


. +-001 


■ 


f$ * 


1-028 .. 


. 1-028 .. 


. 


[ 


»i • 


T067 .. 


. 1-057 .. 


. 




mol. 


1-117 .. 


. 1-118 .. 


*. +-001 


2 


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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56 



Proceedings of Roycd Society of Edinburgh, 



[sBsa. 



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

O 

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 


A 


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 



60 



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 = 

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

If \ bgCad^ I = 0, then \ agCgd^ | • | biCgd^ | = - | aib2C,d4 1 • | c^d J . 
This, it is readily seen, is not a very obscure foreshadowing of 
Jacobi's identity 

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.. + ... 


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 
(V o - 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 

1 

(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 

1 

xfo+' xfii+' — «f!!r'^' 

in expressione 

1 

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 



«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 



A 



B 



^1^2- 



ac generalius 



1 



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 , . . . . 






Po 


fPi* 




Y, 
" A 


Y, 
'A '•••• 






S: 


>'h, 




- n, 


r, , . . . . 






^0 » A > 




= «i, 


«j 










A 


= A, 






X 


1 


•X, 


rn+ij 

X^ 


I 


■ • *»•»+' 


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 






1 



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, - , 



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 






1 



^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^ ^)hxM t-' &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^ = 



^^a^ + rn^b^ -J- c^ = | 
itts + mj6g + Cg = 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 = , 
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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76 



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 



X 



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„ 



I; 



* See Sylvester's paper of 1840. 



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1903-4.] 



Dr Muir on General Detei^minants. 



77 



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



(3) 






K» 



A, 



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) 









K 



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. 



79 



(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 .... 


X 

x' 

• 



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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80 



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 





X 


y 


z 




1 


a 


P 


y 




v 


1 
a 


^ 


y 




i 


It 
a 


pr 


n 

y 





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 



81 



-1^7' • • 


1 lay--- 


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


1^/... 


1 -lay"... 


1 |«)8"...| .... 


-\H--- 


1 ky... 


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



X 


y 


z .... 


«1 


Oj 


Og .... 


*1 


h 


\ .... 


<=! 


H 


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 , 


B, 


....,A', 


A= ff 


y • • . 




r 


y • • • 


A' = ± 


i8" 


y" • • • 




r 


y" • • • 



be given by the equations 



B = 



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' +.... = 


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


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


Ba + W'a +.... = 
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, . . . 


, . . 


... ,1 


,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. 



83 



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



wh 


K . . 

Ok.. 

1 


* • 


. . . 

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




ich is equivalent to 


K 







l/*-* .... 






or 


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 



"l 


«2 




«i 


«8 


«1 


«4 


«2 

= 1 


«S 


«2 


«4 




«» 


«4 


*1 


6, 




''. 


^8 


^1 


h 


'^2 


b» 


t>. 


h 




^3 


b. 



= 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. 



85 



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 ^ 


L 






» 




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'] 



y 

z' 



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 


1 


1 




1 

1 


-1 

1 


-1 

1 


-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, 



91 



"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. 



b 



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. 




^ 



o 



I ^ 



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Ptoc. Roy. Socy. of Edin.] 



[Vol. XXV. 



2i 



J 



< 

2u 



5b 



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Proc. Roy. Sory. of Edin,] 



[Vol. XXV. 




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Proc, Ritij. Sijry. of Edin.] 



[Vo\. XXV. 



r^ 



> 

'A 

< 

5U 






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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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. 



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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 



Ai 


1 











1 


A, 


1 











1 


As 


1 











1 


A4 


1 











1 


A5 



* 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 


1 


A, 1 


> 


Ai 1 


1 A, 




1 Aj 1 




1 A, 1 




1 A, 




1 A, 1 










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... 



D. 



viz. N,.D,.,-N<_,D<=1. 



For if we construct the matrix which for greater simplicity 
I limit to five lines and columns, 



A 


1 











1 


B 


1 











1 


C 


1 











1 


D 


1 











1 


E 



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 




= 


55 


^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 


= 


11 




''2 *$ *4 


ft' 




= 






ftj ft, ft^ 


ft* 




- 


1 B 
1 


1 

C 1 "" 


1 
B 1 













1 D 


1 C 1 












1 


1 D 


1, 




= 


1 X 


1 = 1, 










as was to be 


proved. And the demonstration is 


evidently 




general in its nature." 






In 


regard 


to this there has to be noted, first, the use of 










«s 


a$ «4 










h 


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 

3 



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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 



A 


B 


C" 








1 


A' 


B" 


C" 








1 


A" 


B'" 


C"" 








1 


A" 


B"" 











1 


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 



1 

A- 


i-i 




-k^^ 


7 - 


A 1 





...00 




1 B 


1 


...00 




1 


c 


...00 










...Ml 










...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 

1 





1 

c 



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 




-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 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 



= 



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 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 

A, 



Nouv, 



X. 1 


X 


1 







X 


1 








1 X. 


2 


X 


2 




3 


X 


2 










1 


A., 








2 



X 

1 


3 

X, 


X 


1 



















4 


X 


2 



















3 


X 


3 



















2 


X 


4 



















1 


X, 


. < 


. • 


, , 





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 




yo 










- "hyo 


+ yi 








- Vo 


-0^1 + 


2/s 








- ^tUx - 


«»!'« + 


ys 





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 



-\ 
= 
= 



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 



a+ 



+ 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 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 = 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, 



or 



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 



N, 



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 











1 


1 













1 




1 
1 



i.e. + 1 ; 



(2) that the cofactor* of (-a^)( -a,)(— a,) 
the a's are consecutive is 



when two of 



1 1 

1 1 






1 1 
1 
1 





I 1 






1 



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 
1 1 

1 1 
1 
1 1 

1 1 
1 
1 1 



155 
when no 



1 1 
1 



t.e. 



1 1 
1 



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 



1 



1 



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 1 

1 

- a*+i 1 1 



1 1 



1 I 

1 



'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 , tt o , . . . , a„) 
(ttg, . . . , a„) ' 



an 



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 ^^^ 



a 



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) |, 



W) 



= 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, 

Kk 

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 

Kk 

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 



*1 


a'* 


*8 


«11 


«1J 


«l. 


«M 


«!2 


<hi 


«B1 


Si 


«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- 



W 



1 



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 

A 



A 



•"21 

A 



A 

^28 

A 



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 



or 



( 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 


^2 


^8 


«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- 



Gi 



Go 



G« 



The repetition of the substitution upon^this equation, and the 
application of the same process to the equation 



1 



* 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 



a. 



«18 



a22 - a: a, 



81 



*82 



28 
«83-^ 



0, 



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 ) 

I 

' ^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. 


IW 


81 


12 


81 


68-2 


6 


4.40 „ 5.80 p.ni. 


210 


82 


14 '9 


82 


41 -4 


9 


5.30 „ 6.15 p.ni. 


212 


88 


19 -9 


82 


89 -5 


10 


6.10 „ 7.80 a.m. 


218 


84 


7 7 


82 


45-8 


10 


5.35 „ 6.80 p.m. 


219 


84 


28 -9 


38 


20-9 


12 


6.80 „ 7.10 a.m. 


220 


88 


38 *9 


88 


15 -8 


12 


8.10 „ 4.15 p.m. 


222 


82 


64-1 


83 


14 -5 


13 


6.10 „ 7.15 a.m. 


228 


38 


19-6 


38 


88 


13 


6 „ 6.45 p.m. 


226 


84 


7-8 


38 


47 -3 


14 


6.15 „ 7.30 a.m. 


226 


84 


62-6 


83 


47-6 


14 


6 ,, 6.46 p.m. 


228 


88 


21 -6 


84 




15 


6.10 „ 7.80 a.m. 


229 


34 


28 -6 


84 


6-7 


16 


3.15 „ 4.20 p.m. 


281 


38 


57-7 


84 


10-5 


16 


6. 5 „ 6.50 a.m. 


282 


83 


46 -1 


84 


85-7 


16 


1. 6 „ 2 p.m. 


285 


84 


8 -5 


34 


43 


21 


5.55 „ 6.15 a.m. 


248 


88 


17 


85 


29 -6 


26 


6.45 „ 7.20 a.m. 


260 


88 


2-6 


85 


51 


26 


2. 6 „ 2.80 p.m. 


262 


32 


60-2 


85 


57 -2 


27 


7.15 „ 9.45 a.m. 


268 


82 


7-4 


85 


40 


27 


4. 2 „ 6. 5 p.m. 


257 


31 


29 -1 


34 


82 -1 


28 


2.10 „ 6 p.m. 


269 


31 


6-5 


86 


27 1 


29 


6.10 „ 6.55 a.m. 


260 


31 


21 -7 


86 


8-9 


29 


2.10 „ 6 p.m. 


262 


30 


40 '9 


36 


10-4 


80 


6.30 „ 7. 5 a.m. 


264 


30 


19-8 


86 


5 -2 


80 


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. 




















*-■ _ 





2 


5 


10 


20 


30 


50 


70 


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 


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 


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 


17-9' 


16-8 


235 


27-0 


26-9 


26-8' 


27-0 


257 


21-3 


19-0 


17-8' 


167' 


238 


27-4 


27-4 


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 


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. 




















— ^>__ 





2 


5 


10 


20 


80 


50 


70 


100 


Station. 


1 


•7 


•8 




1 




1-8 







191-188 


1-4 


•8 




218-212 


•5 


•8 


•8 


•4 





•1 


•3 


"•8 


•5 


220-219 


•4 


•8 


•8 


•2 


•2 


•2 


•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 


•1 


226-225 


•3 


•1 


•1 


•2 


•1 


•6 


•4 


•6 


•6 


226-228 


•4 


-•2 


-•2 





- -1 


- -2 


•2 








229-228 


•2 








•2 


•2 


-1-7 


-1^7 


-1-2 


-•6 


229-281 


1-2 


11 


•8 


7 


•4 


•4 


•4 


- 1 


-•1 


232-281 


1 


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 


•2 


•8 


14 


- 8 


- -6 


-•4 


250-262 


-1 


-•2 


-•3 


-•6 


•1 


- -6 


- -8 


- -8 


-•6 


258-252 


•1 





-•2 


-•5 


- -7 


-2^0 


-1-2 


- -7 


-•4 


257-259 


•2 


•5 


•5 


•5 


•7 


2-0 


•8 


•4 


•2 


260-269 


•9 


•9 


•9 


11 


Vi 


•5 


•7 


•8 


•4 


260-262 





-•8 


-•8 


— '2 


•8 


- •o 


-1 


- -9 


-•2 


264-262 


•4 


•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 


2 


0-87 


±0065 


5 


0-29 


±0-062 


10 


0-22 


±0-088 


20 


0-09 


±012 


80 


-0-08 




50 


-0-09 


... 


70 


-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 = 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 


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^ 



i 






('), 



(8). 



where = 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 ^ = 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 ^ = 
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 < = 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* = 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. 6. 


Col. 7. 


X. 


^/^ 


1 
II 


98 




0||8 




P 


I* 

1^0000 


•sis *** 

14142 


T 




+ 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 




o 

& 



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192 Proceedings of ItoycU Society of Edinburgh. [i 




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residing in this country to correct their proofs. 

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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. 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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IV 



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 



5- 



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 



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the Society to the following Regulations, which have been drawn up in 
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and to utilise as widely and as fairly as possible the funds which the 
Society devotes bo the publication of Scientific and Literary Researches. 

1. Manuscript op Papers. — 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 
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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 
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When the ilhistrations 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. Proofs. — In general, a first proof and a revise of each paper will 
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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 
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. 

1 

It 

•5490 


1 Col. 5. 

1 


Col. 6. 


1 Col. 7. 


X. 


P 


II 


s 


€P^"- 


i|ir 


4-0 


•6965 


•7882 


•4866 


•2298 


•07771 


4-41 






... 





... 





4-6 


•6588 


•7798 


•6189 


- ^0944 


•3088 


-•01917 


6 


•6262 


•7788 


•4848 


- 6684 


•8823 


- 1386 


5-6 


, •5981 


•76-78 


•4592 


- -8457 


'4498 


-•2273 


60 


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 


•4980 


-7494 


•8734 


- •7053 


•6808 


- -2892 


9-0 


•4699 


•7461 


•8601 


- ^4289 


•7872 


- -1486 


10 


1 -4461 


•7416 


•3308 


- -1679 


•6846 


- -04768 


10-62 
















11 


, -4255 


•7885 


•3U2 


•05698 


•81*47 


•01975 


12 


1 -4076 


•7369 


•2999 


•2428 


•8416 


•08876 


13 


•8916 


•7889 


-2874 


•8940 


•8644 


•1834 


14 


•8775 


•7318 


•2762 


•5176 


•8808 


•1721 


15 


•3648 


•7302 


•2663 


•6168 


•8954 


•2018 


16 


•3683 


•7286 


•2574 


•6953 


•9082 


•2281 


18 


•3331 


•7266 


•2419 


•8098 


•9260 


•2498 


20 


•8160 


•7256 


•2290 


•8881 


•9396 


•2622 


22 


•3014 


•7230 


•2179 


•9818 


•9497 


•2666 


24 


•2885 


•7216 


•2082 


•9627 


-9679 


•2661 


26 


•2772 


•7206 


•1997 


•9816 


•9638 


•2622 


28 


•2672 


•7193 


•1923 


•9916 


•9685 


•2566 


80 


•2681 


•7187 


•1856 


•9979 


•9727 


•2505 


82 


•2500 


•7181 


•1795 


•9999 


•9759 


•2489 


84 


•2425 


7173 


•1740 


•9993 


•9786 


•2371 


88 


•2294 


•7163 


•1648 


•9938 


•9828 


•2240 


42 


•2182 


•7155 


•1661 


•9840 


•9847 


•2188 


46 


•2084 


•7147 


•1490 


•9734 


•9883 


•2005 


50 


•2000 


•7141 


•1429 


•9623 


•9902 


•1905 


55 


•1906 


•7136 


•1360 


•9486 , 


•9917 


•1794 


60 


•1826 


•7129 


•1802 


•9361 


•9931 


•1697 


70 


•1690 


•7120 


•1208 


•9125 


•9949 


•1535 


80 


•1581 


•7114 


•1125 


•8931 


•9961 


•1407 


100 


•1415 


•7108 


•1005 


•8626 


•9977 


•1217 


00 





•7071 




1 


•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 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 < = 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 






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 




t 



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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 








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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 

8t 

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 0-640 501 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. 


0- 


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 

i 



• 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** 


3 


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. 


0' 


3 


15'' 


1 
2 
3 


25* 


1 

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. 


0* 


1 
2 
3 


1-0160 
1-0327 
1-0489 


1-000 


16 


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. 



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 




(T 



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 


1 




65-4 




2 




631 




3 




62-4 


KBr 


1 




68-3 




2 




67-6 




3 




65-8 


KI 


1 




700 




2 




69-5 




3 




68-0 


HCl 


1 




187-0 




2 




165-7 




3 




143-5 


HBr 


1 




203-0 




2 




175-0 




3 




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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1903-4.] Date of Upheaval of Raised Searches in Scotland. 245 

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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1903-4.] Date of Upheaval of liaised Beaches in Scotland. 247 

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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1903-4.] DcUe of Upheaved of Raised Beaches in Scotland, 251 

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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1903-4.] Date of Upheaval of Raised Beaches in Scotland. 253 

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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1903-4.] DcUe of Upheaval of liaised Beaches in Scotland. 257 

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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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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KEGULATIONS 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 Papers. — ^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 
ot 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. 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 
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, 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 = 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+2 1 

^*""[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 

(«>,).= -„ = (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 



E= 



[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 = , 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 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. 


15 


1-18566 

1 


1-18566 


20 


1-18416 


1-18418 


26 


1-18269 


1 18266 


80 


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 

A 

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 


15 


1-18468 


•84415 


•83740 


-h -00675 


J 


20 


1-18208 


•84596 


•83815 


-h -00781 


1? 


26 


1-17891 


•84823 


•83935 


-f -00888 


t y 


30 


1 17676 


-84988 


•84031 


+ -00967 


6-0785 


15 


1-06884 


•93559 


•94003 


- -00444 


)) 


20 


1-06699 


-93721 


•94087 


- 00366 




26 


1-06452 


•93938 


•94222 


- -00284 


^ 


30 


1-06294 


•94078 


•94829 


-•00261 


3-1805 


15 


1-03532 


-96589 


•96954 


- -00866 




20 


1 03373 


•96737 


•97041 


- -00804 


J 


26 


1-03180 


■96918 


-97179 


- -00261 




30 


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 



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 



1 



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. 


15 


1-00363 


t"C.(Vc.c.). 
-99639 


tion (V ca> 




-32744 


-99759 


- -00120 


)) 


20 


1-00263 


-99738 


99848 


- -00110 




26 


1-00114 


•99886 


•99992 


- 00106 


)) 


30 


•99996 


1 -00004 


1-00105 


- 00101 


12162 


15 


1-00072 


•99923 


-99965 


- -00042 


)) 


20 


•99971 


1 00029 


1-00055 


- -00026 


1) 


26 


•99831 


1-00169 


1-00197 


- -00028 


»» 


30 


•99708 


1-00293 


1-00313 


- -00020 


•02354 


15 


-99946 


1-00054 


100063 


- -00009 


jj 


20 


•99849 


1-00151 


1-00153 


- -00002 


)) 


26 


•99700 


1 -00806 


1-00296 


+ -00010 


»» 


30 


•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 



U 






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 


10 


20 


29 


+ 130 


+ 980 


-210 


18 


21 


26 


+ 148 


+ 630 


- 28 


14 


20 


27 



51 
86 
88 



84 
28 
31 



Strontium Hydrate. 



8274 


+ 180 


-^340 


+ 50 


18 


21 


27 


1216 


+ 220 


-410 


+ 380 


22 


21 


26 


•0235 


+ 165 


+ 600 


- 12 


16 


22 


29 



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 

« 
1 

T 



*«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 


40 


201 


350 


5? 


250 ! 

J 







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 ^ = 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 

yW 

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 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 

A 



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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 = 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 ^ = 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 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 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(^, = 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 ; 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, - 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 , + 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, = ( - 1)'P(0. - ( - 1)'D(0, ± . . + D(* - 1, (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 







I 



55 

OQ 



^ 

^ 



« 

^ 






FROC. HOY. SOC. KDIN. — VOL. XXV. 



21 



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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(^, + Q( - ', = '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 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 ^ = 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 






1 

2 
8 

4 

5 

6 

7 

8 

9 

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 


X 


<»(«,0) 


D(-l,0)=-D(o,() 


Xa:,0) 


34 




D(x,u) 


•8155 


•1740 


•0026 


•2942 


35 


•1714 


•0025 


•1698 


36 


•1689 


•0023 


•0962 


37 


•1666 


•0028 


•0647 


38 


-1643 


•0022 


•0468 


39 1 


•1621 


•0021 


•0357 


40 , 


•1600 


-0020 


•0284 


41 1 


•1580 


•0019 


•0232 


42 


•1561 


•0019 


•0194 


48 ' 


•1642 


•0018 


•0166 


44 


-1624 


•0017 


•0143 


45 


•1507 


•0017 


•0125 


46 


•1490 


•0016 


•0111 


47 


■1474 


•0016 


•0100 


48 


-1468 


•0016 


•0089 


49 


•1443 


-0016 


-0081 


50 


•1428 


•0014 


•0073 


51 


•1414 


-0014 


•0068 


52 


•1400 


•0014 


•0062 


63 


•1886 


•0013 


•0058 


54 


•1373 


•0018 


•0053 


56 


•1860 


•0012 


•0050 


56 


•1348 


-0012 


•0047 


57 


-1336 


•0912 


•0043 


58 


-1324 


•0011 


•0041 


59 


•1318 


•0011 


•0039 


60 


■1302 


•0011 


-0036 


61 


•1291 


•0011 


•0035 


62 


-1280 


-0010 


-0033 


63 


-1270 


-0010 


•0031 


64 


•1260 


•0010 


-0029 


65 


•1250 


•0010 


-0028 


66 


•1240 


-0009 


•0027 


67 


•1231 


0009 



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1903-4.] Lord Kelvin 07i a Free Procession of Waves, 327 



Table II. 



t = 25t ; T = ^TT ; x = ^^ 



15 

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, 
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 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 



aw 



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) = 

c=2.3, 4.5, 6.7, i.e. 6, 20, 42, . . (12) 

The roots of @(c, 1) = and @(c, 1) = 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) 
= c = 2-77 .... , and for the corresponding root of (S(c, 1) = 
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 =• , 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 




Fio. 2. 



If q, Cj, Cj iv .... be the real positive roots 

taken in order of magnitude of the equations @(c, 1) = 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 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 A and 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) = . . (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?, D' = iVy then we may represent the portions A B, B D, 
I) 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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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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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 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 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^ | 


-■ 



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' 


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 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 



1 



1 

Co 



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 



of 



/ 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 


1 


1 


1 ... 


Cs 


Cj 


<^i 


1 ... 


\ 


*2 


h 


1 ... 


<h 


a, 


«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. 



I 



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) 

.J 



X»»-i~ 



a 
-1 



-be 

a -be 
-1 a 



^-2 



a 
-1 



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 <^^*) 

m 



-™.,„-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 



+1 



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" . + 



1 



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 = ; 

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 .-. . . 


n 



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 

.A 






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 : — 



I 



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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m 

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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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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. 

I. 

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. 


A 


Reading. A | 

1 


1,000 







, 






-84 


-20 ' 


8,000 


84 




20 






-82 


-22 , 


6,000 


66 




42 






»3 


-21 


7,000 


99 




68 






-83 


-22 


9.000 


132 




85 






-33 


-21 


11,000 


165 




106 ' 










i 

1 



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 


Reading. A 








-87 


-25 


37 


25 


-85 


-25 


72 


60 


-34 


-25 


106 


75 


-87 


-24 


143 


99 


-36 


-25 


179 


124 


-86 


-22 


148 


102 


-86 


-24 ' 


108 


78 


-36 


-24 


72 


54 


-85 


-25 


37 


29 


-37 


-25 





4 



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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. 


m 


E ' 


C 


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. 


m 


E 





D 


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. 



1 



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 

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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. 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. 



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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 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 ^ = 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 is the angle GEO or 
angle of twist is 

1 sins^^cos^^ 



r = 



a2 62 
ab 



V. 



cos^^ + ^sin^^ 



V'-(.-3 



siD^e^ 



,_ where e^ = i _ 

N/l-e2sin2^ a2 

b 



Therefore AB or CD = , 

Jl - e2 8in2 

and BC = BO cos tf = 2 fe cos ^ 

But the projection width = AD = AB + CD + BC 

= 2r+26cos 6 



\ Jl -e^ 8m2 / 



n/1 
Then the sum of all the sections between the limits ^ = 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 ^ = to = ^ radians, since 
e^<l. Integrating this series term by term and using the formula 

f^sin'^ ^^^(n-l) (n-3) 1^ 

' 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 ^ = -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 


1- 


-9 


-8 


•7 


•6 


•6 


•4 


-8 


•2 


1- 


•9487 


•8944 


•8367 


7746 


•7071 


•6826 


•6477 


•4472 


2- 


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 


4- 


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^ = to a^ = '6 through \ turn of twist (when a+h^d). 





Values of ^2 


Valufifi of 









•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 


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 



18 



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 

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. 



32 



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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. 


Day. 


Day. 


1040 


960 


500 


1014 


1014 


1020 


3-92 


4-26 


2^96 


330 


3-68 


2-26 


•112 


•095 


•061 


•308 


•804 


•345 


•62 


•67 


•70 


•88 


•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 


70 


1-00 


•84 


•88 



I 

I Per 

cent, of I 
Total N.l 



I 



25 
80 

20 



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 




-1 



- 1 



^2 

6 



= ^ + 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 : — 

L 



-1 



i> 



1 





-1 



^8 



= {$<!>)"' + A^{$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 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 

6. 



. + 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 

^1 

n-l <l> 2 . 

I . 71-2 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 



1 

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 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>. 



H 



^0 



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 Nov