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THE
TRANSACTIONS
OF THE
ROYAL IRISH ACADEMY.
VOL. XVII.
DUBLIN:
PRINTED BY P. DIXON HARDY, 3, CECILIA STREET,
PRINTER TO THE ROYAL IRISH ACADEMY.
1837.
Tue Acavemy desire it to be understood, that they are not answerable for any
opinion, representation of facts, or train of reasoning, that may appear in the fol-
lowing papers. The Authors of the several Essays are alone responsible for their
contents.
-
*
~
LIST OF THE
ROYAL IRISH ACADEMY.
MDCCCXXXVII.
Patron—THE KING.
VISITOR,
CHIEF GOVERNOR OF IRELAND.
PRESIDENT,
Rey. Bartholomew Lloyd, D.D., Provost of Trinity College.
VICE-PRESIDENTS,
His Grace the Most Rev. Richard Whateley, D.D. &c. Lord Archbishop of Dublin.
Rey. Franc Sadleir, D.D., S.F.T.C.D.
Samuel Litton, M.D.
Sir William Rowan Hamilton, Astronomer Royal of Ireland, and Andrews’ Professor
of Astronomy, Trinity College.
COUNCIL.
COMMITTEE OF SCIENCE.
Rey. Franc Sadleir, D.D., S.F.T.C.D.
Rey. Richard Mac Donnell, D.D., S.F.T.C.D.
Sir William Rowan Hamilton, Astronomer Royal of Ireland, &c.
Vili
Rey. Humphrey Lloyd, F.T.C.D., F.R.S., Professor of Natural and Experimental
Philosophy, Trinity College.
James Apjohn, M.D., Professor of Chemistry, Royal College of Surgeons.
Captain Portlock, R.E., F.G.S., &c.
James Mac Cullagh, Esq. F.T.C.D. Professd of Mathematics, Trinity College.
COMMITTEE OF POLITE LITERATURE.
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Dublin.
Rey. Joseph Henderson Singer, D.D., F.T.C.D.
Andrew Carmichael, Esq.
Samuel Litton, M.D.
Rey. William Hamilton Drummond, D.D.
Rey. Charles Richard Elrington, D.D. Regius Professor of Divinity, Trinity College.
William West, M.D.
COMMITTEE OF ANTIQUITIES.
Isaac D’Olier, LL.D.
Thomas Herbert Orpen, M.D.
Hugh Ferguson, M.D.
Sir William Betham, Ulster King of Arms, F.S.A. &c.
George Petrie, Esq.
Rey. Cesar Otway.
Very Rey. Henry R. Dawson, Dean of St. Patrick’s.
OFFICERS.
TreasurER—Thomas Herbert Orpen, M.D.
SroreTary To THE AcApEMy—Rey. Joseph Henderson Singer, D.D., F.T.C.D.
Secretary To Counciz—Rey. Richard Mac Donnell, D.D., S.F.T.C.D.
Secretary or Foreign CorresponpDENcE—Sir William Betham, F.S.A. &c.
Lisrarian—Rey. William Hamilton Drummond, D.D.
MEMBERS.
Those marked * are Members for Life.
A.
Sir James Eglinton Anderson, M.D., M.W.N.H.S.
James Apjohn, M.D. Professor of Chemistry, Royal College of Surgeons.
* John Ashburner, Esq.’
John Arabin, Esq.
* Viscount Adare, F.R.S. &c.
A. Armstrong, A.M.
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* Thomas Arthur, Esq.
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* William Ball, Esq. :
* Robert Ball, Esq.
* Rev. Charles Bardin.
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The Honorable Thomas Barnwell.
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* Captain F. Beaufort, R.N., F.R.S. &c.
* Charles Benson, M.D.
-
Thomas F. Bergin, Esq.
Sir William Betham, Ulster King of Arms, F.S.A. &c.
Samuel Black, M.D.
Marcus Blair, Esq.
Travers Blackley, Esq.
* Bindon Blood, Esq.
*
*
*
Rev. Sir Francis Lynch Blosse, Bart.
Rey. John Abraham Bolster.
William Edward Bolton, Esq.
Edward Borough, Esq.
Rey. Charles Boyton.
Maziere Brady, Esq.
Robert Brown, Esq.
Very Rev. Robert Burrowes, D.D., Dean of Cork.
Peter Burrowes, Esq.
Hon, Sir F. N. Burton.
Colonel J. F. Burgoyne.
Isaac Butt, Esq.
Earl of Caledon.
William W. Campbell, Esq.
Edward Cane, Esq.
Andrew Carmichael, Esq.
Richard Carmichael, Esq.
Joseph Carne, Esq.
George Carr, Esq.
Earl of Carysfort, F.R.S., F.S.A.
George Cash, Esq.
Hon. Henry Caulfield.
George Chamley, Esq.
Rey. John B. Chapman.
Earl of Charlemont.
xi
* Earl of Charleville.
* Earl of Clare.
Thomas Clarke, Esq.
Edward S. Clarke, Esq.
Lieutenant Colonel Colby, Royal Engineers, F.R.S.L. & E.
Owen Blayney Cole, Esq.
William C. Colville, Esq.
W. E. Conwell, M.D.
Frederick W. Conway, Esq.
* Marquis of Conyngham.
E. J. Cooper, Esq.
Jonathan Sisson Cooper, Esq.
* John Richard Colballis, Esq.
* Thomas Coulter, M.D.
Henry Courtney, Esq.
John Crampton, M.D., King’s Professor of Materia Medica.
Hon. Judge Crampton.
* Thomas Crofton Croker, Esq. F'.S.A.
* Charles Croker, M.D.
* William Cubitt, Esq. F.R.S.
Allan Cunningham, Esq.
Joshua Cunningham, Esq.
* James William Cusack, Esq. M.B.
*
* * *
*
John D’ Alton, Esq.
Frederick Darley, Esq.
Charles Davis, M.D.
Edmund Davy, Esq. F.R.S. Professor of Chemistry, Royal Dublin Society.
Very Rev. H. R. Dawson, Dean of St. Patrick’s.
Rey. Charles Dickinson, D.D.
Robert Dickenson, Esq.
Isaac D’Olier, LL.D.
Isaac M. D’Olier, Esq.
Michael Donovan, Esq.
Henry Grattan Douglas, M.D.
Charles Doyne, Esq.
*
*
*
*
*
*
*
xii
Rey. William Hamilton Drummond, D.D.
F. D. Dwyer, M.D.
E.
William Edington, Esq.
Rev. Charles Richard Elrington, D.D., Regius Professor of Divinity, T.C.D.
Richard T. Evanson, M.D.
William Farran, Esq.
Hugh Ferguson, M.D.
Samuel Ferguson, Esq.
Joseph M. Ferrall, Esq.
Right Hon. Maurice Fitzgerald.
Rey. Michael Fitzgerald.
Lord Fitzgerald and Vesci.
Valentine Flood, M.D.
Simon Foot, Esq.
Hon. Baron Foster.
R. Murray Frazer, Esq.
Robert French, Esq.
Right Hon. Lord Garvagh.
Arthur E. Gayer, Esq.
Rey. Edward Geoghegan.
Rey. Thomas Goff,
George Stephens Gough, Esq.
Robert J. Graves, M.D. King’s Professor of the Institutes of Medicine.
William Gregory, M.D., F.R.S,E.
* George A. Grierson, Esq.
*
*
Richard Griffith, Esq., F.R.S.E. &c.
Rey. Hosea Guinness, D.D.
John Haire, Esq.
* Daniel Halliday, M.D.
* Sir William Rowan Hamilton, Astronomer Royal of Ireland, and Andrews’ Pro-
fessor of Astronomy, T.C.D.
Arthur Hamilton, LL.D.
William Tighe Hamilton, Esq.
Charles William Hamilton, Esq.
James Hardiman, Esq.
*
*
Major General Hardwicke.
Philip Dixon Hardy, Esq.
* Robert Harrison, M.D.
John Hart, M.D.
Andrew S. Hart, F.T.C.D.
Rey. Henry H. Harte.
William Haygarth, Esq.
Henry A. Herbert, Esq.
Colonel Edward Hill.
Lord George A. Hill.
Rev. Thomas Hincks, D.D.
John Houston, M.D.
* Henry Hudson, M.D.
* Arthur Hume, Esq.
Robert Hutton, Esq. F.G.S.
Edward Hutton, M.D.
*x* * * *
ed
Arthur Jacob, M.D.
Sir John K. James, Bart.
Rey. John Jebb.
* Frederick Thomas Jessop, Esq.
Hon. Judge Johnson.
Rev. Edwin Johnson.
Right Hon. Henry Joy, Lord Chief Baron.
Henry Holmes Joy, Esq.
*
K,
Robert J. Kane, M.D. Professor of Natural Philosophy, Royal Dublin Society.
* D. H. Kelly, Esq.
Thomas F. Kelly, LL.D.
* Rev. James Kennedy, D.D.
VOL. XVII. c
x1V
George A. Kennedy, M.D.
Jeffries Kingsley, Esq.
*
George Kiernan, Esq.
Viscount Kingsborough.
Hon. Thomas Knox.
Rey. Thomas Knox.
George J. Knox, Esq.
Right Rev. Samuel Kyle, D.D. Lord Bishop of Cork and Ross.
William Cotter Kyle, LL.D.
* * * *
185
Charles Lambert, Esq.
Rey. Dionysius Lardner, LL.D. F.R.S. &c.
Thomas A. Larcom, Esq. R.E.
David Charles La Touche, Esq.
William Digges La Touche, Esq.
Earl of Leitrim.
James Lenigan, Esq.
Thomas Little, M.D.
* Samuel Litton, M.D. Professor of Botany, Royal Dublin Society.
* Rey. Bartholomew Lloyd, D.D. Provost of ‘Frinity College.
* Rey. H. Lloyd, F.R.S. Professor of Natural Philosophy, T C.D.
Rev. J. C. Lloyd.
Rey. H. F. C. Logan.
* Rey. Thomas Luby, F.T.C.D.
Acheson Lyle, Esq.
John Finnis Lynch, Esq.
* * * K *
M.
* James Macartney, M.D., F.R.S. &c. Professor of Anatomy and Surgery, T.C.D.
Viscount de Mac Carthy.
John M‘Caul, LL.D.
* James Mac Cullagh, Professor of Mathematics, T.C.D.
* Rev. Richard Mac Donnell, D.D., S.F.T.C.D.
* John Mac Donnell, M.D.
* J. M‘Neil, Esq.
* William M‘Guire, Esq.
*
xv
James Townsend Mackay, Esq.
Gerard Macklin, Esq.
Venerable Thomas Perceval Magee, Archdeacon of Killaloe.
George Magrath, M.D.
* Pierce Mahony, Esq.
Robert Mallet, Esq.
Right Rev. Richard Mant, D.D., Lord Bishop of Down and Connor.
Rey. Edward Marks.
Henry Marsh, M.D.
Rev. John C. Martin.
Alexander Marsden, Esq.
William Marsden, Esq.
Henry Joseph Monck Mason, LL.D.
William Shaw Mason, Esq.
Rey. J. W. Massie.
* Rey. Charles Mayne.
Anthony Meyler, M.D.
* Rev. George Miller, D.D.
* Rey. Marcus Monck.
William F. Montgomery, M.D.
William Morrison, Esq.
Richard Morrison, Esq.
William Murray, Esq.
*
* * * * *
N.
* Rey. William Nelson.
* Right Hon. Sir John Newport, Bart. a
Francis Nesbitt, Esq.
* Whitley Nicholl, M.D.
John Armitage Nicholson, M.D.
@.
Lucius O’Brien, Esq.
Edward Odell, Esq.
James O’Grady, Esq.
Michael Martin O’Grady, M.D.
* John Oldham, Esq.
Sir Samuel O’Malley, Bart.
Miles J. O’Reilly, Esq.
Thomas Herbert Orpen, M.D.
Rey. Cxsar Otway.
Cesar G. Otway, Esq.
Lord Oxmantown, F.A.S. &c.
* Paul Patrick, Esq.
* Robert Perceval, M.D.
* * *
*
* * * *
George Petrie, Esq. R.H.A.
James Pim, Jun. Esq.
Rey. Thomas H. Porter, D.D.
Captain Portlock, R.E. E.G.S. &c.
Henry H. Price, Esq.
Rev. Thomas Prior, D.D., S.F.T.C.D.
James Prior, Esq.
Right Hon. John Radcliffe, LL.D.
Robert Reid, Esq. M.D.
Rey. J. Seaton Reid, D.D.
W. Renny, Esq. R.E.
Rey. John Cramer Roberts.
Rey. Thomas Romney Robinson, D.D. Professor of Astronomy, Armagh.
Earl of Rosse.
Rey. A. B. Rowan.
John Ryan, M.D.
Francis Rynd, Esq.
S.
Rey. Franc Sadlier, D.D., 8.F.T.C.D.
Rey. William D. Sadlier, F.T.C.D.
Anthony Semple, Esq.
R. Sheehan, Esq.
Rey. Joseph Henderson Singer, D.D., F.T.C.D.
Rey. Joseph D’ Arcy Sirr.
Joseph Huband Smith, Esq.
*
XVil
Rev. George S. Smith, F.T.C.D.
Aquilla Smith, M.D.
Robert Sparks, M.D.
Hon. Alexander Stewart.
Whitley Stokes, M.D., Professor of Natural History, T.C.D.
William Stokes, M.D.
Rey. Charles Strong.
D. P. Thompson, Esq.
John Thwaites, M.D.
Rev. J. H. Todd, F.T.C.D.
Venerable John Torrens, D.D. Archdeacon of Dublin.
Rey. W. Trail, D.D.
* Right Hon. Lord Trimleston.
* William Turner, Esq.
V.
Crofton Moore Vandeleur, Esq.
Rey. C. Vignolles, D.D.
C. Vignolles, Esq. F.R.A.S.
N. A. Vigors, Esq. F.L.S., &c.
Ww.
Rev. Richard Wall.
William Baily Wallace, Jun. Esq.
* William Wallace, Esq.
*
* * * *
Francis Weldon Walsh, Esq.
R. C. Walker, Esq.
Right Hon. Colonel Robert Ward.
Thomas Weaver, Esq. F.G.S.
William Webb, Esq.
Isaac Weld, Esq.
William West, M.D.
Hon. H. R. Westenra.
Most Rey. Richard Whateley, D.D. &c. Lord Archbishop of Dublin.
*
*
*
* *
Robert Wigram, Esq.
James T. Wilkinson, Esq.
Thomas Williams, Esq.
Edward Wilmot, Esq.
Rey. James Wilson, D.D.
Benjamin Woodward, Esq.
Rey. Richard Woodward, D.D.
XViil
HONORARY MEMBERS.
His Royal Highness Augustus Frederick, Duke of Sussex, Pres. R. S. &c. &c.
General T. Abrahamson, Copenhagen.
George B. Airy, F.R.S. &c. Astronomer Royal.
Thomas Amyot, F.R.S. &c.
Charles Babbage, F.R.S. Lucasian Professor of Mathematics, Cambridge.
Francis Baily, V.P.R.S. F.A.S.
T. Berzelius, Professor of Chemistry, Upsal.
Nathaniel Bowditch, LL.D.
James Norris Brewer, Esq.
Sir David Brewster, LL.D., F.R.S. L. and E. &c.
Sir Thomas Mac Dougal Brisbane, Bart. F.R.S. L. and E. &c.
Chevalier Bronstead, Copenhagen.
Professor Brongniart.
Robert Brown, F.R.S., F.L.S., &c.
His Grace the Duke of Buckingham and Chandos.
George Combe, Esq.
C. P. Cooper, Esq. F.A.S., F.R.S.
W. R. Clanny, M.D., F.S.A.
John Dalton, L.LD. F.R.S. &c.
Charles Daubeny, M.D., F.R.S. &c. Professor of Chemistry, Oxford.
Baron De Donnop, Save Meiningen.
Charles Dupin, President of the Academy of Sciences, Paris.
Professor Antonmaria Vassali-Eandi.
Sir H. Ellis, K.E.H. F.R.S. Sec. S. A.
Rey. Josiah Forshall.
Davies Gilbert, Esq. V.P.R.S.
Robert Graham, M.D. F.R.S.E. Professor of Botany, Edinburgh.
R. K. Greville, LL.D.
Francis Hamilton, M.D.
Rey. W. Vernon Harcourt, F.R.S.
xx
G. C. Haughton, Esq.
Professor Heeren, Gottingen.
Sir John F. W. Herschel, F.R.S. &c.
Right Hon. Henry Hobhouse.
Sir William Jackson Hooker, LL.D. Professor of Botany, Glasgow
Thomas C. Hope, M.D., F.R.S.E., Professor of Chemistry, Edinburgh.
Baron Jacquin, Professor of Botany, Vienna.
Robert Jameson, F.R.S.L.& E. &c. Regius Professor of Natural History, Edinburgh
Charles Kénig, K.H.
Doctor Lagasca.
Aylmer Bourke Lambert, Esq. V. P. L. S.
Sinor Jacquine Jose de Costa de Macedo, Secretary to the Royal Academy of
Sciences at Lisbon.
Sir Frederick Madden, K.H. F.R.S. F.S.A.
David M‘Laughlin, M.D.
Roderick I. Murchison, Esq. F.R.S. F. G. S. &e.
Captain William Parry, R.N., F.R.S.
Sir John Phillipart, K.G.V.K.P.S.
John Prichard, M.D., F.R.S.
C. C. Rafn, Secretary of the Royal Antiquarian Society, Copenhagen.
George Rennie, F.R.S.
S. P. Rigaud, F.R.S., Savilian Professor of Anatomy, Oxford.
Baron Frederick Schreibers, Director of the Imperial Museum, Vienna.
Professor Schumacher, Copenhagen.
Rey. Adam Sedgwick, F'.R.S. &c. Woodwardian Professor of Geology, Cambridge.
Mrs. Somerville.
Sir James South, F.R.S. F.L.S.
Robert Southey, LL.D.
Rey. Edward Stanley, F.G.S.
Lieutenant Colonel Sykes, F.R.S. F.G.S.
Thomas Taylor, M.D.
Thomas Thompson, M.D. F.R.S. &c. Professor of Chemistry, Glasgow.
J. L. Villanueva, D.D.
Rey. Robert Walsh.
Rey. W. Whewell, F.R.S. &c.
1
II.
CONTENTS.
PART I.
SCIENCE.
Third Supplement to an Essay on the Theory of Systems of Rays.
By William R. Hamilton, A.B., M.R.I.A., MR. Ast. Soc. London,
M. G. Soc. Dublin, Hon. M. Soc. Arts Sor Scotland, Hon. M. Ports-
mouth Lit. and Phil. Soc., Member of the British Association Sor the
Advancement of Science, Fellow of the American Academy of Arts
and Sciences, Andrews’ Professor of Astronomy in the University of
Dublin, and Royal Astronomer of Ireland. Read January 23, 1832,
and October 22, DSN eon aes cosis ve caisnci<ewsinds ahs Omasatcent a ecee daca stncu icc
On the Phenomena presented by Light in its passage along the Axes
of Biaral Crystals. By the Rev. Humphrey Lloyd, A.M., M.R.LA.,
Fellow of Trinity College, and Professor of Natural and Eperimental
Philosophy in the University of Dublin. Read January 28, 1833
seseee
Page
145
—
CONTENTS.
PART II.
SCIENCE.
. An attempt to facilitate Observations of Terrestrial Magnetism. By
The Rev. Humphrey Lloyd, A.M., M.R.I.A., Fellow of Trinity
College, and Professor of Natural and Experimental dd Hole in
the University of Dublin. Read October 28, 1833.
. On a New Case of Interference of the Rays apebihte “By ie Rov
Humphrey Lloyd, A.M., M.R.LA., Fellow of Trinity College, and
Professor of Natural and Es it Sat ee baa in the ike
of Dublin. Read January 27, 1834...
. An Essay on the Climate - Ireland. By Teak Ms pe M.D.
In answer to the question proposed by the Royal Irish Academy.
* Whether we have reason to believe that a change has taken place in
the climate of Ireland, and if such change has occurred, through what
period can we trace it, and to what causes should we a it.’ Read
November 8, 1830.. : nceo poceatooc
. On Differences and Differ aan of ity unctions of oe By William
R. Hamilton, Royal Astronomer of Ireland. Read June 13, 1831
. On a Difficulty in the Theory of the Attraction of Spheroids. -
James M‘Cullagh, A.B. Read May 28, 1832...
. Geometrical Propositions applied to the Wave Theory oy Taal “By U
James M‘Cullagh, F.T.C.D. Read June 24, 1833..
VII. An account of a new Fulminating Silver, and tts apollo en as a
Test for Chlorine, §c. By Edmund Davy, F.R.S., M.R.LA., &c.,
Page
yill
mo of Chemistry to the Royal Dublin ee Read May 23,
VIII. On the cor 'y of aha Moist. bulb mee 3 James Apjohn, Esq.
M.D., M.R.I.A., Professor of Chemistry in the Royal College of
Surgeons. Read November 24, 1834..........-0.ssssscesssrsseccscrnccees
IX. Do. Continued.) Read April 27, 1835.. aeisielccaeere
X. Theory of Conjugate Functions, or ialaabi aic niles eb a
Preliminary and Elementary Essay on Algebra as the Science of Pure
Time. By William Rowan Hamilton, M. R. 1. A., F. R. A. S., Hon.
M.R.S. Ed. and Dub., Fellow of the American Academy of Arts
and Sciences, and of the Royal Northern Antiquarian Society at
Copenhagen, Andrews’ Professor of Astronomy in the University of
Dublin, and Royal Astronomer of Ireland. Read November 4, 1833,
ANd JUNEHS USST:s cssiccwsceccesee voeeescectueeclcemteeee setters eceae vive cterlelsp
265
275
283
293
Il.
Ill.
Le
Il.
CONTENTS.
PART III.
SCIENCE.
Researches on the Action of Ammonia upon the Chlorides and Oxides
of Mercury. By Robert Kane, M.D. M.R.1.A. Professor to the Royal
Dublin Society and the Apothecaries’ Hall, Corresponding Member of
the Society of Pharmacy of Paris, and of the Society of Medical
Chemistry in the same City. Read November 30th and December
28th, 1835.
Further development of a method of observing the Dip and the Magnetic
Intensity at the same time, and with the same Instrument. By The Rev.
Humphrey Lloyd, M.A., F.R.S., M.R.LA., Fellow of Trinity College,
and Professor of Natural and Experimental Philosophy in the Uni-
versity of Dublin. Read December 28th, 1835. ........s0seeeeeeee
On the Laws of the Double Refraction of Quartz. By James Mac
Cullagh, Fellow of Trinity College, Dublin. Read February 22, 1836.
An Investigation of the Principles upon which a new Self- Registering
Barometer may be constructed. By John Stevelly, Esq. Professor of
Natural He al in the Belfast Institution. Read November 30,
1835. . “no pAb CATES 7AB SCS CoO CHOCO OCOOASHED EDO RES
ANTIQUITIES.
On an Astronomical Instrument of the Ancient Irish. By Sir William
Betham, M.R.LA., F.A.S., Ulster rage 4, Arms, me a: Read
May 23, 1836. . ScbocnoconeccconnuocSeenpsnGcHooDachocoodeaEcosnonadnos
On the Ring Money of the Celte, and their System of Weights, which
appears to have been what is now called Troy Weight. By Sir
William Betham, M.R.I.A., Ulster a ee Arms, ak ae Read
May 23 and June 27, 1836... Rescisiasies
VOL. XVII. a
Page
449
461
471
ill.
IV.
V.
VI.
li
The Affinity of the Phenician and Celtic Languages, illustrated by the
Geographical Names in Ptolemy and the Periplous of Arrian. By
Sir William Betham, M.R.I.A., Ulster a a Arms, ae ee Read
May 23 and June 27, 1S36i doce =Satecroc
Notes on the Statistics and Natural History of the Island of Rathlin,
off the Northern Coast of Ireland. By James Drummond Marshall,
M.D., Secretary to the Natural History Society of Belfast, §C........+
On the Affinity of the Hiberno-Celtic and Phenician Languages.
By Sir William Betham, F.S.A., M.R.I1.A., Ulster . ae Arms, ee
§c. Read November 28, 1836. . nets cehac tence i Be
On the Ring Money of the Celte. By Sir William Betham, F.S.A.,
M.R.LA., Ulster King of Arms, a a Read November 28, 1836,
and January 9, 1837. Fodgaseneve scaneeenteeeeccansaeme cacess cesinsog>
21
37
73
Third Supplement to an Essay on the Theory of Systems of Rays. By WILLIAM
R. HAMILTON, A. B., M. R. I. A., WR. Ast. Soc. London, M. G. Soc.
Dublin, Hon. M. Soc. Arts for Scotland, Hon. M. Portsmouth Lit. and Phil.
Soc., Member of the British Association for the Advancement of Science, Fellow
of the American Academy of Arts and Sciences, Andrews’ Professor of Astro-
nomy in the University of Dublin, and Royal Astronomer of Ireland.
Read January 23, 1832, and October 22, 1832.
ERRATA.
Page 60, for Partridge, (Perdix coturniz,) read (Perdix cinerea.)
Page 67, for pittock, read piltock.
Page 68, for Insecta, read Crustacea.
Page 69, for accidentally, read incidentally.
Of these the theory of external and internal conical refraction, deduced by my
general methods from the principles of Fresnex, will probably be thought the least
undeserving of attention. It is right, therefore, to state that this theory had been de-
duced, and was communicated to a general meeting-of the Royal Irish Academy, not
at the earlier, but at the later of the two dates prefixed to the present Supplement.
After making this communication to the Academy, in October, 1832, I requested
Professor Luoyp to examine the question experimentally, and to try whether he could
perceive any such phenomena in biaxal crystals, as my theory of conical refraction had
led me to expect. The experiments of Professor Luoyp, confirming my theoretical
expectations, have been published by him in the numbers of the London and Edin-
VOL. XVII, a
yi
burgh Philosophical Magazine, for the months of February and March, 1833 ;
and they will be found with fuller details in the present Volume of the Irish Trans-
actions.
I am informed that James Mac Cutxacu, Esq. F.T.C.D. who published in the last
preceding Volume of these Transactions a series of elegant Geometrical Illustrations
of Fresnev’s theory, has, since he heard of the experiments of Professor Luoyp, em-
ployed his own geometrical methods to confirm my results respecting the existence of
those conoidal cusps and circles of contact on FREsNEL’s wave, from which I had been
led to the expectation of conical refraction. And on my lately mentioning to him
that I had connected these cusps and circles on FRESNEL’s wave, with circles and cusps
of the same kind on a certain other surface discovered by M. Caucuy, by a general
theory of reciprocal surfaces, which I stated last year at a general meeting of the Royal
Irish Academy, Mr. Mac Cuxtacu said that he had arrived independently at similar
results, and put into my hands a paper on the subject, which I have not yet been able
to examine, but which will (I hope) be soon presented to the Academy, and published
in their Transactions.
{ ought also to mention, that on my writing in last November to Professor Airy,
and communicating to him my results respecting the cusps and circles on FREsNEL’s
wave, and my expectation of conical refraction which had not then been verified, Pro-
fessor Arry replied that he had long been aware of the existence of the conoidal cusps,
which indeed it is surprising that Fresvex did not perceive. Professor Arry, how-
ever, had not perceived the existence of the circles of contact, nor had he drawn from
either cusps or circles any theory of conical refraction.
This latter theory was deduced, by my general methods, from the hypothesis of
transversal vibrations in a luminous ether, which hypothesis seems to have been first
proposed by Dr. Youne, but to have been independently framed and far more per-
fectly developed by Fresnex ; and from Fresnev’s other principle, of the existence of
three rectangular axes of elasticity within a biaxal crystallised medium. The verifi-
cation, therefore, of this theory of conical refraction, by the experiments of Professor
Lioyp, must be considered as affording a new and important probability in favour of
FRESNEL’s views: that is, a new encouragement to reason from those views, in com-
bining and predicting appearances.
vil
The length to which the present Supplement has already extended, obliges me to
reserve, for a future communication, many other results deduced by my general
methods from the principle of the characteristic function: and especially a general
theory of the focal lengths and aberrations of optical instruments of revolution.
WILLIAM R. HAMILTON.
OssERvATORY, June, 1833.
G
ia
12.
13.
14
. Fundamental Formula of Mathematical Optics. Design of the present Supplement
. Fundamental problem of Mathematical Optics, and solution by the fundamental formula. Partial
CONTENTS OF THE THIRD SUPPLEMENT.
differential equations, respecting the Characteristic Function V, and common to all optical com-
binations. Deductions of the medium-functions Q, v, from this characteristic function V. Re-
marks on the new symbols a, 7, v.--+++.+--
. Connexion of the characteristic function V, with the: nies i letersi ms re general
equations of a curved ray, ordinary or extraordinary ..........seseessseeeseeeees
. Transformations of the fundamental formula. New view ae the ransshinn faye W: new auxi-
liary function 7. Deductions of the characteristic and auxiliary functions, V, W, 7, each from
each. General theorem of maxima and minima, which includes all the details of such deduc-
tions. Remarks on the respective advantages of the characteristic and auxiliary functions ......
. General Transformations, by the auxiliary functions W, 7, of the partial differential equations in
V. Other partial differential equations in VY, for extreme uniform media. dias of these
equations, by the functions IV, 7...
. General deductions and Stare ee of the differential cad, aed ee of a “eet or
straight ray, ordinary or extraordinary, by the auxiliary functions W, 7. . Soprnccn
- General remarks on the connexions between the partial differential Sais ob the Ee cee
of the functions V, W, 7. General method of investigating those connexions. Deductions of
the coefficients of V from those of 7, when the final medium is uniform..
. Deductions of the coefficients of W from those of V. Homogeneous ae eruations eenereee ea:
Deductions of the coefficients of T from those of W, and reciprocally .. ta soasedenraone
General remarks, and cautions with respect to the foregoing deductions. rs of. a S85 cache
medium. Connexions between the coefficients of the functions v, Q, v, for any single medium.
General formula for Reflection or Refraction, ordinary or extraordinary. Changes of V, W, 7.
The difference A V is = 0; A W=A T= a homogeneous function of the first dimension
of the differences A o, Az, A v, depending on the shape and position of the reflecting or re-
fracting surface. Theorem of maxima and minima, for the elimination of the incident variables.
Combinations of reflectors or refractors. Compound and component combinations...........
Changes of the coefficients of the second order, of V, W, T, produced by reflection or Seen
Changes produced by transformation of co-ordinates. Nearly all the foregoing results may be ex-
tended to oblique co-ordinates. The fundamental formula may be presented so as to extend even
to polar or to any other marks of position, and new auxiliary functions may then be found, ana-
logous to, and including, the functions W, 7’: together with new and general differential and
integral equations for curved and polygon rays, ordinary and extraordinary..........1006seeeeeereees
General Geometrical relations of infinitely near rays. Classification of twenty-four independent
coefficients, which enter into the algebraical expressions of these general relations. Division
of the general discussion into four principal problems...........cssseeseeeeseseeceeceeceeteeeescesenseaeees
VOL. XVII. A
60
16.
17.
20.
21.
- Calculation of the elements of arrangement, for arbitrary axes of co- soraihetee” c 5
25. The general linear expressions for the arrangement of near rays, fail at a point of vergency. ane.
vill
P
. Discussion of the four problems. Elements of arrangement of near luminous paths Axis and
constant of chromatic dispersion. Axis of curvature of ray. Guiding paraboloid, and constant
of deviation. Guiding planes, and conjugate guiding axes..........s.ceeeeeeeeeeeeeeeneee soneeeees
Application of the elements of arrangement. Connexion of the igs final pettericted eid ae of
vergency, and guiding lines, with the two curvatures and planes of curvature of the guiding surface,
and with the constant of deviation. The planes of curvature are the planes of extreme projection
Ofsthe fittal Fay-UNESs.iosnccucs wweescers veces tdsbonee svoncsnasctentoncessenclsts covedtueatebamencienee cess chest ee:
Second application of the elements. Arrangement of the near final ray-lines from an oblique plane.
Generalisation of the theory of the guiding paraboloid and constant of deviation. General theory
of deflexures of surfaces. Circles and axes of deflexure. Rectangular planes and axes of ex-
treme deflexure. Deflected .ines, passing through these axes, and having the centres of deflexure
for their corresponding foci by projection. Conjugate planes of deflexure, and indicating cylinder
OP GEMEKION <2. s.cnseceevenissncoclovsacevscsctanescesaceeesetectenv ener seapedesacdsusetese sav aacaesis sacs ee neeNer
. Construction of the new auxiliary paraboloid, (or of an osculating hyperboloid,) and of the new
constant of deviation, for ray-lines from an oblique plawe, by the former elements of arrangement.
- Condition of intersection of two near final ray-lines. Conical locus of the near final points in a
variable medium which satisfy this condition. Investigations of Matus. Illustration of the con-
dition of intersection, by the theory of the auxiliary paraboloid, for ray-lines from an tacaatd
plane: 83, .00.29% He, FES aoe
Other geamictiieal aiinatraGons of thes conditied ‘of ifitersedtions niet of ie clined of arrangement.
Composition of partial deviations. Rotation round the axis of curvature of a final ray............
Relations between the elements of arrangement, depending only on the extreme points, directions,
and colour of a given luminous path, and on the extreme media. In a final uniform medium,
ordinary or extraordinary, the two planes of vergency are conjugate planes of deflexure of any
surface of a certain class determined by the final medium; and also of a certain analogous surface
determined by the whole combination. Relations between the visible magnitudes and distor-
tions of any two small objects viewed from each other through any optical combination. Inter-
changeable eye-axes and object-axes of distortion. Planes of no distortion..
termination of these points, and of their loci the caustic surfaces, in a straight or curved system,
by the methods of the present Supplement ..
- Connexion of the conditions of initial and final initeodec tier of 4 “ey near acenie of Aree’ Berea or
curved, with the maxima or minima of the time or action-function V+ V,= 3 fvds. Separating
planes, transition planes, and transition points, suggested by these maxima and minima. The
separating planes divide the near points of less from those of greater action, and they contain the
directions of osculation or intersection of the surfaces for which V and V, are constant; the
transition planes touch the caustic pencils, and the transition points are on the caustic curves.
Extreme osculating waves, or action surfaces. Law of osculation. Analogous theorems for
sudden reflection or refraction .......... sdeuccenesceevscerecneesuseeses
25. Principal rays and principal foci for strat lie or ited syeieuts General craked? of aetaathing
the arrangement and aberrations of the rays, near a principal focus, or other point of vergency...
26. Combination of the foregoing view of optics with the undulatory theory of light. The quantities
o, T, UV, or
ov 8V SV
Sz? by 2 Se?
that is, the partial differential co-efficients of the first order of the characteristic function V,
taken with respect to the final co-ordinates, are, in the undulatory theory of light, the compo-
nents of normal slowness of propagation of a wave. The fundamental formula (A) may easily
be explained and proved by the principles of the same theory.......ccscescecceseeseeereereeenseneenens
age
71
79
83
90
97
101
..108
Roll tit
116
123
27
28
30
31
ix
Page
. Theory of Fresne. New formule, founded on that theory, for the velocities and polarisations of
a plane wave or wave-element. New method of deducing the equation of FresneL’s curved
wave, propagated from a point in a uniform medium with three unequal elasticities. Lines of
single ray-velocity, and of single normal-velocity, discovered by FrEsNEL...... is/dtetniciatetelatote 128
. New properties of Fresneu’s wave. This wave has four conoidal cusps, at the ends of the
lines of single ray-velocity ; it has also four circles of contact, of which each is contained on a
touching plane of single normal-velocity. The lines of single ray-velocity may therefore be
called cusp-rays; and the lines of single normal-velocity may be called normals of circular
CONTACE occ cccislsiniewecddeiesicicccveccscccecuacccaseessedes Mis vateietusela's ste avelnereraters dite 6192
. New consequences of FresNeEv's principles. It follows from those principles that crystals of suffi-
cient biaxal energy ought to exhibit two kinds of conical refraction, an external and an internal ;
a cusp-ray giving an external cone of rays, and a normal of circular contact being connected
WILLMAN LC PNARGCONE s\sieiaiatelo(a(elslsrt=/e/elaisslaisieis\sslais.c c'e'sicleiei= =/<isa'e's sipiareleinlelsievetelsiatets pelo aie 's 134
. Theory of conical polarisation. Lines of vibration. These lines, on FresnEL’s wave, are the in-
tersections of a certain series of concentric and co-axal ellipsoids .........22eeeeeeeeeeeeeee 138
. In any uniform medium, the curved wave propagated from a point is connected with a certain
other surface, which may be called the surface of components, by relations discovered by M.
Caucuy, and by some new relations connected with a general theorem of reciprocity. This new
theorem of reciprocity gives a new construction for the wave in any undulatory theory of light :
and it connects the conoidal cusps and circles of contact on FRESNEL’s wave, with circles and
cusps of the same kind upon the surface of components ....+.seeeeeeeeseseeeseeeetes
cenit owed. eet. Sorat yl pene oor at a
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seb oak Fyiav- a1, aiende Yo cont T att
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hsp aicene moter “dade =
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Tf ee er Se eee ee
fats he Seas, ada :
sth arent ence
THIRD SUPPLEMENT.
Fundamental Formula of Mathematical Optics. Design of the present Supplement.
1. Wuen light is considered as propagated, according to that known general law
which is called the law of least action, or of swiftest propagation, along any curved
or polygon ray, ordinary or extraordinary, describing each element of that ray
ds= ¥ (da + dy’ + dz*) with a molecular velocity or undulatory slowness v, which is
supposed to depend, in the most general case, on the nature of the medium, the posi-
tion and direction of the element, and the colour of the light, having only a finite
number of values when these are given, and being therefore a function of the three
rectangular co-ordinates, or marks of position, 7, y, z, the three differential ratios or
cosines of direction,
di dz
o= Bap Y= as?
and a chromatic index or measure of colour, x, the form of which function v depends
on and characterises the medium; then if we denote as follows the variation of this
function,
Bo se by +e Pae4 Pea BoB += ay +e dx,
and if, by the help of the relation a* + 6° + y°=1, we determine
By Bo Be
Sa? yey 8 va
so as to satisfy the condition
namely, by making v homogeneous of the first dimension with respect to a, B, y; it
has been shown, in my First Supplement, that the variation of the definite integral
V =f vds, considered as a function, which I haye called the Characteristic Function
B
2 Professor Hamitton’s Third Supplement
of the final and initial co-ordinates, that is, the variation of the action, or the time,
expended by light of any one colour, in going from one variable point to another, is
sV=(sfvds = eae = ba" + a = oy’ + = be Be (A)
the accented being the initial quantities. This general equation, (4), which I have
called the Equation of the Characteristic Function, involves very various and exten-
sive consequences, and appears to me to include the whole of mathematical optics. I
propose, in the present Supplement, to offer some additional remarks and methods,
connected with the characteristic function 7, and the fundamental formula (A) ; and
in particular to point out a new view of the auxiliary function 7’, introduced in my
former memoirs, and a new auxiliary function 7, which may be employed with advan-
tage in many optical researches : I shall also give some other general transformations
and applications of the fundamental formula, and shall speak of the connection of my
view of optics with the undulatory theory of light.
Fundamental Problem of Mathematical Optics, and Solution by the Fundamental
Formula. Partial Differential Equations, respecting the Characteristic Function
FV’, and common to all optical combinations. Deduction of the Medium Functions
Q, v, from this Characteristic Function V. Remarks on the new symbols o, 7, v.
2. It may be considered as a fundamental problem in Mathematical Optics, to
which all others are reducible, to determine, for any proposed combination of media,
the law of dependence of the two extreme directions of a curved or polygon ray,
ordinary or extraordinary, on the positions of the two extreme points which are
visually connected by that ray, and on the colour of the light: that is, m our present
notation, to determine the law of dependence of the extreme direction-cosines a 3 y
a By, on the extreme co-ordinates x y z v' y' 2, and on the chromatic dex yx.
This fundamental problem is resolved by our fundamental formula (4); or by the
six following equations into which (4) resolves itself, and which express the law of
dependence required :
SV fv OVA See OF _ ob
3c ~ Sa? y 7 5B? de — By?
SV Bey A eee. meta
—3,/= 5,9 — 3y' = 3p" ; Tay ate) ¢
These equations appear to require, for their application to any proposed combination,
not only the knowledge of the form of the Characteristic Function V’, that is,
the law of dependence of the action or time on the extreme positions and on the
colour, but also the knowledge of the forms of the functions v, v’, that is, the optical
On Systems of Rays. 3
properties of the final and initial media ; but these final and initial mediwm-functions
v, vw, may themselves be deduced from the one characteristic function ’, by reasonings
of the following kind.
Whatever be the nature of the final medium, that is, whatever be the law of
dependence of v on the position, direction, and colour, we have supposed, in deducing
the general formula (4), that the expression of this dependence has been so prepared
as to make the medium-function v homogeneous of the first dimension relatively to
the direction-cosines a, 3, y ; the partial differential co-efficients
ov oe Ou
8a’ 3B” dy’
of this homogeneous function, are therefore themselves homogeneous, but of the
dimension zero ; that is, they are functions of the two ratios
a
ry?
involving also, in general, the co-ordinates « y z, and the chromatic index y: if then
we conceive the two ratios
to be eliminated between the three first of the equations (B), and if, in like manner,
we conceive
to be eliminated between the three last equations (B), we see that such eliminations
would give two partial differential equations of the first order, between the character-
istic function V’ and the co-ordinates and colour, of the form
V&V BV
= « ue Se De wi
O= Q’ eV ov OV , , , (C)
= (spp gpe ge oes x )»
which both conduct to the following general equation, of the second order and third
degree, common to all optical combinations,
BY OE og ui ip an FV apie Ye ilo AK hase
oxen §dydy' = 8z82’ i ardy’ dydz Bzer’ t ra? 3 yea’ dézéy’
< _ Oe CIR Oe eV 8V SV SV SF &V
—Ssde’ Syd Sade! | Med ByeT Bede ted Gye’ Busy"
If now we put, for abridgment,
Dil SIE UBT
ww —<9; ay —T; % =U,
ras Lei ae Le 8 Rie
(D)
4 Professor Hamuton’s Third Supplement
and if between the three first of these equations () we eliminate two of the three initial
co-ordinates wv’ 7/' 2’, it is easy to perceive, by (C) or (D), that in every optical com-
bination the third co-ordinate will disappear ; and similarly that between the three
last equations (EZ) we can eliminate all the three final co-ordinates, by eliminating
any two of them ; and that these eliminations will conduct to the relations (C) under
the form
O=Q (G, Ts Vy Ly Ys 2s X)s
0= a) (a, T,U,2'5 y; z, Xs t (F)
which can thus be obtained, by differentiation and elimination, from the characteristic
function V” alone: and which, as we are about to see, determine the forms of », v’,
that is, the properties of the extreme media. Comparing the differentials of the rela-
tions (F’), with the following, that is, with the conditions of homogeneity of v, v', pre-
pared by the definitions s ) a a ch relations (B),
Pa Bt ay = as + Br + yp, ' é)
Y= “+B a+ py ee cde
and with their nen that is with
ov sv, ov bv
aba + Bor ae ews) +582 +3 OX
/ i} H
ado’ + for’ oF you’ = = Sh ad aE ae
we find
a_o2 P_&Q y_ ea
yoo? 0 oF’ v dv?
a’ _ 6a’ p __ 60! 7 _ 6a’ t (1)
vy a! ? os Se wv SSN
and also
—le&_e2 -1&_e&@ —lov_82 —1_ oa
ode Se? voy dy? vdz dz? vdy dy’
=13¥ Bo! 1 Be 8a 1 ae a" 1 be _ 80
Dir ea Od es ennai: iG ay? v df 8”? 3x aon
ac a 0 aman 4 (L)
a ets |)
which can be done by putting those relations under the form
0 = (co? +7? 2) 2 —
Ca) 1 en (M)
O= (6? +7? +2)? w —-1=0';
On Systems of Rays. 5
in which w, w’, that is, (6% + 7° + v’) - and (6% +7r%+v")~ a are to be expressed
ie ¢ ee =o
as functions respectively of «(6° +7°+v°)*, r(e@++u) *, u(r t+) *a,y,
— , "9 U " —+4 ' "2 2 19 —3 , , ,
ZX and of o (6? +77 +") cm tT (o? +7" +v") a v (eo? +77 +v") po 3 Uae oe
After this preparation the partial differential coefficients
sQ 8a 8
oF Nes, Qe
or” ou
are homogeneous of dimension zero relatively to o, 7, v; and in like manner
za’ 30 20
So’ > a > Su’ ’
are homogeneous of dimension zero relatively to o’, 7’, v’; if, therefore, between the
three first equations (J), we eliminate any two of the three final quantities o, 7, v, the
third will disappear ; and similarly all the three initial quantities o’, 7’, v’', can be elimi-
nated together, between the three last of the equations (J): and by these eliminations
we shall be conducted to two relations of the form
QD
oo
o=¥ (5, E 7, Ds %X)s
ope ee (N)
o= Ds E a x's ¥/, Z,x)s
which determine the forms of the final and initial medium-functions v, v’; so that these
forms can be deduced from the form of the characteristic function 1”. We can there-
fore reduce to the study of this one function /’, that general problem of mathematical
optics which has been already mentioned.
The partial differential coefficients of the characteristic function 7, taken with
respect to the co-ordinates 2, ¥, z, are of continual occurrence in the optical methods
of my present and former memoirs; I have therefore thought it useful to denote
them in this Supplement by separate symbols, o, r, v, and I shall show in a future
number their meanings in the undulatory theory: namely, that they denote, in it, the
components of normal slowness of propagation of a wave.
Connection of the Characteristic Function V7, with the Formation and Integration
of the General Equations of a Curved Ray, Ordinary or Extraordinary.
3. It may be considered as a particular case of the foregoing general problem, to
determine general forms for the differential equations of a curved ray, ordinary or ex-
traordmary ; that is, to connect the general changes of direction with those of posi-
tion, in the passage of light through a variable medium. The following forms,
ov ov ov ov ov ov
ay Bs a
Sar ae sp a by 7 ay a ee ©
c
6 Professor Hamitton’s Third Supplement
(which are of the second order, because a, es y, @, By, are defined by the equations
vida =-
= ds P ae a ee
dz’ ijn te (P)
= ik jy? a= a “e i
a=
the symbol d referring, throughout the present . to motion along a ray,
while 8 refers to arbitrary infinitesimal changes of position, direction, and colour, and
ds’ being the initial element of the ray,) were deduced, in the First Supplement, by the
Calculus of Variations, from the law of least action. The same forms (0), which are
equivalent to but two distinct equations, may be deduced from the fundamental
formula (4), by the properties of the characteristic function /. For, if we differ-
entiate the first equation (C), (which involves the coefficients of this function V,
and was deduced from the formula (.4),) with reference to each of the three co-ordi-
nates, x, y, z, considered as three independent variables, and with reference to the
index of colour y, we find, by the foregoing number,
OV Or eV _ dv
aye + Pacey t 1 Bebe Se?
eV ae ct eV kv
* Scat P aye + 7 Syde— ay’
lla Pitas SONG 2 att 2 (Q)
se yes PY Bsa
eV Cada eV ov
” 8x8 ae Bydy | Yd dy?
and the three first of these equations (@Q), by the help of the general relations (B),
which were themselves deduced from (4), and by the meanings (P) of a, B, y, may
easily be transformed to (0). The differential equations (QO) may also be regarded
as the limits of the ere
=) e-fe@)u-vs@), = ®
oy
= saad
are obtained by differentiating ~ considered as a function of the seven variables
®, Y, % Ax, Ay, Az, x, if Av=ax—a', Ay=y—y', Az=z—Z; the variation of V,
when so considered, being by (4), and by the definitions (#7),
aT &V OV
bys =(o—o Jor +(r —T ‘ey+(u —v Na+ (EE ane + (se )aay + (are te 8x, (S)
in which
in which
~ sick z ee )=* i ==" (1)
On Systems of Rays. |
If we differentiate the first equation (C) relatively to 2’, 7’, 2’, we find, by the fore-
going number,
Dy aaah eV
ye “Ovba" +.B ier bye’ + Y Sede?
Freep SV eV
° Bray 7 B Syay’ | Y Sedy'’ (U)
0 OV eV Vv
mt OL
scar + Bayar + Y gar?
of which, in virtue of (D), any two include the third, and which may be put by (P)
under the form
V V
oad <>; 0nd ;0=d 55 5 (V)
and these differential equations (7) of the first order, in which the initial co-ordi-
nates and the colour are constant, belong to the ray, and may be regarded as integrals
of (OQ). They have, themselves, for integrals,
a= const. , = 4 a = const. , (W)
the constants being, by (B), the values of the initial quantities
i By
a> 8B * ky’ |
In like manner, by differentiating the last equation (C’), we find the following
equations, which are analogous to (@Q) and (UV),
= const.
we Po Der ae aN
: aad 77 ale 97 em
Sr SES Oi NB
a suey +O gat leayatin aay (x)
, 3V BV a
* 3yo7 * Payer FY oom a BA?
ef ala. gr ee Ne
* Boy = ce a
and
mmowea
- ia + Bs by ue a
37
Y
rn oe Pay” Y Syazt i (®)
Axe a pale
952 sae iPS s +7 Seg
The second members of the three first equations (X) vanish when the initial medium
is uniform, and those of the three first equations (@) when the final medium is so ;
and in this latter case, of a final uniform medium, the final portion of the ray is
8 Professor Hamitton’s Third Supplement
straight, and in its whole extent we have not only the equations (J¥”) but also the
following,
1 C) ov
= const. , las const. , ee const. , (Z)
the constants being by (B) those functions of the final direction-cosines and of the
colour which we have denoted by
cv dv ov
ga’? 88” by’
and which are here independent of the co-ordinates. In general, if we consider the
final co-ordinates and the colour as constant, the relations (Z) between the initial co-
ordinates are forms for the equations of a ray. And though we have hitherto consi-
dered rectangular co-ordinates only, yet we shall show in a future number that there
are analogous results for oblique and even for polar co-ordinates.
Transformations of the Fundamental Formula. New View of the Auxiliary
Function WV ; New Auxiliary Function T. Deductions of the Characteristic
and Auxiliary Functions, V, W, T, each from each. General Theorem of
Maxima and Minima, which includes all the details of such deductions. Remarks
on the respective advantages of the Characteristic and Auxiliary Functions.
4. The fundamental equation (7) may be put under the form
v
SV = ob4 — oéu' + roy — roy + vog— vee + = 8x3 (A )
employing the definitions (#7), and introducing the variation of colour ; it admits
also of the two following general transformations,
V
SW = ado + ydr + xu + o'da! + 78y' +82" ~~ 8x, (B’)
and
, , , , ne OV U
8 T= x80 — x80 + yor — or + zdv —2'bv' — =~ oy, (C)
XxX
in which
W=—-—V +40 + Yt + Zu, (D’)
and
T=W—a's -y'7 — 2. (E’)
In the two foregoing Supplements, the quantity 7” was introduced, and was consi-
dered as a function of the final direction-cosines a, B, y, the final medium being
regarded as uniform, and the luminous origin and colour as given ; we shall now take
another and a more general view of this auxiliary function 7”, and shall consider it
On Systems of Rays. 9
as depending, by (2), for all optical combinations, on the seven quantities a7 vy z x.
In like manner, we shall consider the new auxiliary function JZ’ as depending, by the
new transformation (C’), on the seven quantities rv o'r vy. The forms of these
auxiliary functions, V7, 7, are connected with each other, and with the characteristic
function V7, by relations of which the knowledge is important, in the theory of
optical systems. Let us therefore consider how the form of each of the three func-
tions, V, WW, T, can be deduced from the form of either of the other two.
These deductions may all be effected by suitable applications of the three forms
(41) (B) (C), of our fundamental equation (4), together with the definitions (D’)
(E), as we shall soon see more in detail, by means of the following remarks.
When the form of the characteristic function /” is known, and it is required to
deduce the form of the auxiliary function //’, we are to eliminate the three final
co-ordinates, x, y, z, between the equation (J) and the three first of the equations
(E£); and similarly when it is required to deduce the form of ZT from that of V,
we are to eliminate the six final and initial co-ordinates x y z x yz between the six
equations (#), (which are all included in the formula (4’),) and the following,
T=—-V+a0-270 tyr —yr +zu—Zu : (F)
and if it be required to deduce the form of ZY from that of WV, we are to eliminate
the three initial co-ordinates 2’ yz’, between the equation (’) and the three follow-
ing general equations,
7
Se ae : (G’)
But when it is required to deduce reciprocally V from T or from /V, or W from T,
we must distinguish between the cases of variable and of uniform media; because we
must then use the equations into which (B’) and (C’) resolve themselves, and this
resolution, when the extreme media are not both variable, requires the consideration
of the connexion that then exists between the quantities oz vo' 7 vy: which circum-
stance also, of a connexion between these variable quantities, leaves a partial indeter-
minateness in the forms of Zand JV as deduced from V7, and in the form of J’ as
deduced from WV, for the case of uniform media.
When the final medium is variable, then o,z,v, x, may in general vary indepen-
dently, and the equation (B’) gives
oo (H’)
and, in this case, Y can in general be deduced from WV by eliminating o, 7, v, between
the equation (D’), and the three first equations (H’). But if the final medium be uni-
form, then o, 7, v, xX» are connected by the first of the relations (#’), from which, in this
case, the final co-ordinates disappear ; and instead of the four equations (H’) we have
the three following
VOL. XVII. D
10 Professor Hamitton’s Third Supplement
ww OW LAE LA
8 cin Stacia ag
ae ealy ere 1’)
do or du 8x
by means of the two first of which, combined with the relation already mentioned,
namely,
0=Q (6, 7 v, x) (K’)
which depends on, and characterises, the nature of the final uniform medium, we can
eliminate o, 7, v, from the equation (D’), and so deduce V’ from VY.
In like manner, if both the extreme media be variable, then the seven quantities
otvo rv x may in general vary independently, and the equation (C’) resolves itself
into the seven following,
oT oT oT éT eV 8T OL, MO : ;
3 =O 5, SU lig, Pig = ig age aemepiy rlesetaey ieh 2 (L)
by the three first and three last of which we can eliminate or v o'r’ v' from (F"), and
so deduce /” from 7. And in the same case, or even in the case when only the
initial medium is variable, the three last of the equations (L’) are true, and suffice to
eliminate o', 7’, v', from (H’), and so to deduce VY from T.
But if the final medium be uniform, the initial being still variable, then o, 7, v, x;
are connected’ by the relation (A), while o'r’ v’ remain independent ; and instead of
the four first equations (L’) we have the three following,
a7 ae = a a tay 87
Se ae” ay ay ;
Besa = wean = Serhan 7 ener” atop)
ie oF SS ax
by the two first of which, combined with the relation (’), and with the three
last equations (Z’), we can eliminate o, 7, v, ¢, 7, v, from (F"), and so deduce /7
from 7.
If both the extreme media be uniform, we have then not only the relation (A‘’) for
the final medium, but also an analogous relation
0=Q' (o, T Vy x) (N’)
for the initial ; and instead of the seven equations (L’), we have the two first of the
equations (J/’), and the two following,
Spe SPS 8 .-
Sekt Skee ee |
Saree nyt) wren? (0)
30” 8r’ re
On Systems of Rays. 11
in which A is the common yalue of the three first equated quantities in (J/’), and X’ is
the common value of the three equated quantities in(O’). And in this case, by
means of the two equations (O'), and the two that remain of (JZ), combined with
the two relations (J<') (NV’), we can eliminate o, 7, v, o,7,v, from (F"), and so deduce
PV from T: while, in the same case, or even if the initial medium alone be uniform, we
are to deduce /V from T, by eliminating o’, x’, v', between the equations (7’)(N’)( 0’).
When all the media of the combination are not only uniform, but bounded by
plane surfaces, which happens in investigations respecting prisms, ordinary or extra-
ordinary, then of the seven quantities o, r, v, «,7',v, x, only three are independent ;
two other relations existing besides (KX) and (V’), which may be thus denoted,
0=o7 (o, Ty Uy Oy T5 Vy x)» ;
0=07 (o, 7, v, OyT,0U5 x) > t (Q)
because, in this case, the initial direction, and the colour, determine the final direc-
tion. In this case, we may still treat the variations of «, 7, v, o', 7, v, x, as indepen-
dent, in 87, by introducing the variations of the four conditions (A’) (N’) (Q),
multiplied by factors A, d’, \”, \’, that is by putting
8 T= 280 —2'8o' +ydr — yor + 280 — 2/80 — > x
+ ASQ + N'8Q' +2"8Q" +2"80": CR’)
an equation which decomposes itself into the seven following,
sT so, 80 on 82"
eel ad cee iit a ee
8T Sanewsa” 30”
SN aAY RA iN”
8T 8a » 0D Oe
ai 2 = AS +X Sig +2 Su :
oT fi len OQ “ 6Q” m 6a” /
3° 12 = Par ix Ww +2 Be? (S)
SPM OW RS Sarva MBry wea”
ge YSN G IN GE AG
Sth ce eel Syn OO, SNe
5 7 z= r 3 7 +X SV + r Te >
T V ¢ “u" vu
7 oh ees \ re) x oQ 17 02" 7 2"
By eT TT By TO?
between the six first of: which, and the five equations marked (F’) (K’) (N’) (Q’),
we can eliminate the ten quantities o, 7, v, 0,7, v,A, A’, X’, ”, and thus deduce the
relation between V, x, y, z, x’, y', 7, x, from that between 7; o, 7, v, 6, 7, v’, x. It
is easy to extend this method to other cases, in which there exists a mutual depend-
ence, expressed by any number of equations, betwen the seven quantities o, 7, v,
Ty T,V5 Xe
And all the foregoing details respecting the mutual deductions of the functions
12 Professor Hamitton’s Third Supplement
V, W, T, may be summed up in this one rule or theorem: that each of these three
functions may be deduced from either of the other two, by using one of the three
equations (D’) (Z’) (F") and by making the sought function a maximum or minimum
with respect to the variables that are to be eliminated. For example we may deduce
T from V’, by making the expression (#") a maximum or minimum with respect to
the initial and final co-ordinates.
An optical combination is more perfectly characterised by the original function /’,
than by either of the two connected and auxiliary functions 7, 7’; because
enables us to determine the properties of the extreme media, which /V and J’ do not;
but there is an advantage in using these latter functions when the extreme media are
uniform and known, because the known relations which in this case exist, of the forms
(K’) and (N’), (together with the other relations (@Q") which arise when the combi-
nation is prismatic,) leave fewer independent variables in the auxiliary than in the
original function. At the same time, as has been already remarked, and will be after-
wards more fully shown, the existence of relations between the variables produces a
partial indeterminateness in the forms of the auxiliary functions, from which the
characteristic function 7 is free, but which is rather advantageous than the contrary,
because it permits us to introduce suppositions and transformations, that contribute
to elegance or simplicity.
General Transformations, by the Auxiliary Functions IV, T, of the Partial Dif-
ferential Equations in V. Other Partial Differential Equations in V, for
Extreme Uniform Media. Integration of these Equations, by the Functions W, T.
5. Another advantage of the auxiliary functions W, 7, is that they serve to
transform, and in the case of extreme uniform media to integrate, the partial differ-
ential equations (C), which the characteristic function /” must satisfy. In fact, if the
final medium be variable, the first of the two partial differential equations (C) may
be put by the foregoing number under the two following forms,
SW 8W 8W
OSD (erry es Ee)
da * or” ou T
0=2( ee: ce)
= Oy Ts Vy Sau?) eae 32% 5)
and if the initial medium be variable, the second of the two partial differential equa-
tions (C’) may be put under these two forms,
ao QW WW
o=o'(—, ay” 3” a,y',2,X); U’
Sg aloe oy) W
>
So Se a
Obsa; (¢, r, Vv, ai
On Systems of Rays. 13
of which indeed the first is general. But if the final medium be uniform, then
W remains an arbitrary function of the four variables o, 7, v, y, which are in this
case connected with each other by the relation (A); and the two equations (D’)
(K’), together with the two first of those marked (J'), compose a system, which is a
form for the integral of the partial differential equation
OV OV BV
ae? oy 5) Oe ’ x)
to which the first equation (C) in this case reduces itself. In like manner, if both
the extreme media be uniform, in which case the second equation (C’) reduces itself
0=2 ( (V)
to the form
Sega eS
the system of the partial a equations (7) (JV) has for integral the system
composed of the equations (F”) (4’) (V’) (0'), and the two first equations (JZ’), in
which 7’ is considered an arbitrary function of «, 7, v, o, 7, v, x. It will be found
that these integrals are extensively useful, in the study of optical combinations.
The two partial differential equations, (/”’) (/V’’), of the first order, are them-
selves integrals of the two following, of the second order,
SV ET BY BV EV BY _
- dy? 82% | Sady Sydz dzdxr
2 SV (&V SV (eV >
or ae ‘ti oy? Or ae (X)
and
SV SV SV SV SV &V
ae by? B22 T ws ay Bye B78e'
eV % yr
au? Ce oF aa) seen mo @)
which are obtained by elimination from (@Q) and (X’), after making
ee 0, oo =i; a = (0);
ox oy oz 7
OH Be Cae ( )
The system of the three first of these six equations (Z’), or the partial differential
equation of the second order (X’), or its integral of the first order (V’"), expresses
that the final medium is uniform ; and the uniformity of the initial medium is, in like
manner, expressed by the three last equations (Z’), or by the partial differential equa-
tion ( Y"), or by its integral of the first order (/”’).. The integral systems of equa-
tions, also, which we have already assigned, express properties peculiar to optical
combinations that have one or both of the extreme media uniform.
VOL. XVII. E
14 Professor Hamitton’s Third Supplement
The first equation (U') has for transformation the second equation (U’), when the
initial medium is variable; and it has for integral, when the initial medium is uni-
form, the system (E’) (N’) (0'), by which, in that case, WV is deduced from the
arbitrary function 7’: while, in the same case, of an initial uniform medium, the
first equation (U’) becomes of the form
,foW SW sw
0=2 Gas i? a? XD (A®)
and is an integral of the following equation of the second order, analogous to ( Y’),
SHE OU Ua iy Paar gg BN
e a oy? Bt aa Syse Wea —
ev SLANG LLL :
a5 oe +5 by? = al Tay wap (BY)
When the final medium is variable, the function /V” satisfies the following partial
differential equation, analogous to the general equation (D),
Sn SW, Wing, Sel, CU Mie, By PY
Sade’ brdy' Sudz' © dady’ Broz’ Svdx’ dadz’ drdu’ dudy’
WSU Ce CHE a Sl SW SW i Pale ow OW p Cc)
~ Suz" Srdy/_ dodz2' + 3y 3rdz' Sodz" ' dvdz’ S782" dady’ ” (
and when both the extreme media are variable, the function 7’ satisfies the following
analogous equation,
OT OT aT OT oT OT eT BT ET
ba8c' Srér’ Sudu’ Sadr’ Srdu' Budo’ Sadv’ Orda’ Oudr’
LET MG BE go SL ST oT, MT ST AT
~ Suda’ Srér! dadu’ + Sar! ord! Seda" * Suv! oréa’ dadr’
(D*)
General Deductions and Transformations of the Differential and Integral Equa-
tions of a Curved or Straight Ray, Ordinary or Extraordinary, by the Auxiliary
Functions W, T.
6. The auxiliary functions JV, 7, give new equations for the initial and final por-
tions of a curved or polygon ray. Thus the function V7 gives generally the following
equations, between the final quantities c, +, v, analogous to the equations (VY),
ed =const., = const., = const., (E*)
in which 2’ »/ 2’ are the co-ordinates of some fixed point on the initial portion, and
the constants are, by (G'), the corresponding values of the initial quantities o', 7’, v’.
The equations (H*) have for differentials the following,
On Systems of Rays. 15
ow ow ow
OF ah ae aed
ov. SW, SW
i a (F)
WwW
jee Pe a gargs.
dadz" ordz’ Ovdz
d still referring to motion along a ray : and if we combine these with the following,
wVew wesw kv ew
Se Bode! * By Bria” * 82 Bde’?
wesw wesw wv sew
O= Be Say! * ay Bray’ + Be Bay"?
we wWeW ww
= Se Sole’ * dy Srdz’ + Sz Bude’ ’
which are obtained by differentiating the first equation (7"’) relatively to the initial
co-ordinates 2’ y' z’, and by attending to the relations (K ), we see that for a curved
ray the differentials do, dr, dv, are proportional to
dv du bu
we > ay 5) sz ?
and from this proportionality, combined with the relation
o=
(G*)
ads + Pdr + ydv=(a ant Bet 7) as, (H®)
which results from (7) and (P), we can easily infer the equations (0): these differ-
ential equations (Q) for the final portion of a curved ray, which can be extended to
the initial portion by merely accenting the symbols, may therefore be deduced from
the consideration of the auxiliary function JV. The equations (0) for a curved ray,
may also be deduced from the function VY, by combining the differentials d of the
three first equations (#7'), with the partial differentials of the first equation ( 7”),
taken with respect to o, 7, v.
The same auxiliary function /V gives for the final straight portion of a polygon
ray, the two first equations (I’), which may be thus written,
Bec geal 3 aw
a 3 ae er eo)
these equations may also be put under the form
if in virtue of (JX’), we consider o, 7, v, as functions, each, of y, and of two other
independent variables denoted by 0, ¢, and consider /V as a function of the six
16 Professor Hamitton’s Third Supplement
independent variables 6, $5 x» x’, y', 2. We may choose o,7, for the independent
variables 0, , considering v as, by (C’), a function of o, 7, x, such that by (#7),
du a ov B 8 _1 &
Se Se eg i L’
oo y? & vy’ 8x 8x’ ry)
and considering JV as a function of the six independent variables o, r, x, 2’, y', 2’ 5
and then the equations (J*) or (*), for the final straight portion of a polygon ray,
ordinary or extraordinary, will take these simpler forms, which we shall have frequent
occasion to employ,
ow
or *
The other auxiliary function, Z, gives the following equations between «, r, v, for
the final portion, straight or curved, when the initial medium is variable,
(M’)
pte NG ee oe
oY, i
dH oT
—— = const., yr = const. z =const., (N?)
So > Ru"
in which o’, 7’, v’, belong to some point on the initial portion, and in which the con-
stants are, by (L’), the negatives of the co-ordinates of that point; it gives, in like
manner, for the initial portion, when the final medium is variable, the following equa-
tions between o’, 7’, v’,
oT
oT 8T A
= const., 3 = const., > = const., (O°)
cs, t, v, belonging to some point upon the final portion, and the constants being the
co-ordinates of that pomt: and from these equations we might deduce the differential
equations (QO), by processes analogous to those already mentioned. When both the
extreme media are uniform, and therefore both the extreme portions straight, we
have, for these straight portions, the following equations, deduced from (1/’) ( 0‘) (7),
1 or 1 oT I oT
5 alii gn) = Ca
1 hab ho ih 1 ; Od 1 Petey
“(2 + J=ply +s a7 Piso
which may be thus transformed,
(P)
2 he ee a SY
60 CO” ~80 86 80”
eS a > aa
op op 8p Op” ;
028 Sree ene SE ne
oy’ 3 36 | 8H’
ona Hy yy gO
oY" oe by’ 8h"?
On Systems of Rays. 17
if, as before, by virtue of (J’), we consider o, r, v, as functions, each, of x and of two
other independent variables 0, , considering similarly o’, 7’, v', as functions, each, by
(N'), of three independent variables 0,¢,x 3 and 7 as a function of the five inde-
pendent variables 0, ¢, 6’, ¢> x: If we choose the independent variables 0, 4, so as to
coincide with o, 7, and if in like manner we take o’, 7’, for the independent variables
0, ¢, making, by (HZ),
ov a ov’ i Ope 1 &’ :
eA MRE Bay ©
and considering 7’ as a function of the five independent variables o, +, 0’, 7’, x, we
have the following transformed equations for the extreme straight portions of a
polygon ray, ordinary or extraordinary,
_ = 7 . — B. oT
O=u2 Pid A ha a Se .
Oa ee. o= i B py oes ( )
=: y Se’? =y 7 @ BES
which are analogous to the equations (J/*) and, like them, will often be found useful.
It may be remarked here, that from the differential equations (0) of a curved ray,
ordinary or extraordinary, to which, in the present and former numbers, we have
been conducted by so many processes, the following may be deduced,
(T*)
dVv= dT=(«2+ yt 2) ds=0da + ydr +2du
We may also remark, that when the final medium is uniform, and when therefore
the quantities «, +, v, x, are connected by a relation (’), the quantity
W (6 +r+v°)?
may, in general, by means of this relation, be expressed as a function of
oc T
, , ,
=5 =o 2 z
Re ae 9Ys2%5X>
and that ZT’ (o+7°+ v7? may, in like manner, be expressed as a function of
o Le , , ‘
ME O5T9VU9 X35
and that therefore JV, 7, may both be made homogeneous functions, of any assumed
dimension n, relatively to «, 7, v, so as to satisfy the following conditions
W
Be tae I
Seite” gen SB .
sr. OT Or (Us)
t— tu— =n,
Beedle to
VOL. XVII. F
18 Professor Hamiiton’s Third Supplement
With this preparation, the two first equations (J’), and the two first equations (mM"),
which belong to the straight final portion of the ray, may be transformed by (1) to
the following,
6Q oT oQ
x Tiss (ox +1y +02)=2—n nS - 3o —aTe s
6Q a éT 6Q
y “ee (ox +1 +w2)=o— nW = Slag nT — , (V*)
Ss, = (ow + ry tve)ae—n ee _— nT.
If then we make »=1, that is if we make JV homogeneous of the first dimension
relatively to o, 7, v, and if we attend to the relation (D’), we see that the equations
of this straight final portion may be thus written,
a W _ ow
t= +V ay =D oF yee
—& or
of which any two include the third, and which we shall often hereafter employ, on
account of their symmetry.
In like manner, when the initial medium is uniform, and therefore the initial por-
tion straight, the equations ( 0’) of this straight portion may be put under the form,
Lee ye = (We?)
; iy ai i Fj
Xu =e (o2' +ry oe ene Bes
; “ er pit dtc gee a ;
y¥-3 (ov try tuz)= — Se +1To, (&)
ae ae ee oT 1p!
2 a (ca +ry t+uz)= a +uT >_>
by making 7 homogeneous of dimension 7’ relatively to o', r, v, so as to have
of or + or +uv , oT
Cat bv’
If both the extreme media be uniform, and if we make 7=0, m =O, that is if we
express /V as a function of
=<0T. (X)
o T t ' 1
mY aL U,Y,2;, X
and 7’ as a function of
, ,
r
Cy ear, ara ©
Oe OMe x
we find the following forms for the equations of the extreme straight portions of a
a polygon ray, ordinary or extraordinary, less simple than (S*), but more sym-
metric,
On Systems of Rays. 19
6a SW _8T
&— — (t+ yt+uz)= =>
da é
SQ'F 5. BO VAR
es a, (ox +TYy +uz)= Se
or or’
8 a ayes
z- > (o@ + ry +z) = <§ a2 i
fee aa): oe oily A aa cy
Bie (ow +TY +vuZ = 3y"?
a SOE Mp ii pee te eee
y— <7 (6@ try tuzZ) = =37,
/ a 4 Se. if of. 6T
Ai (ox trefiud) Sis:
ou v
The case of prismatic combinations may be treated as in the fourth number.
General Remarks on the Connexions between the Partial Differential Coefficients of
the Second Order of the Functions V, IV, T. General Method of investigating
those Connexions. Deductions of the Coefficients of V’ from those of IV,
when the Final Medium is uniform.
7. It is easy to see, from the manner in which the equations of a ray involve the
partial differential coefficients of the first order, of the functions V7, /V, 7’, that the
partial differential coefficients of the second order, of the same three functions, must
present themselves in investigations respecting the geometrical relations between infi-
nitely near rays of a system; and that therefore it must be useful to know the gene-
ral connexions between these coefficients of the second order. Connexions of this
kind, between the coefficients of the second order of the characteristic function /’,
taken with respect to the final co-ordinates, and those of the auxiliary function JV,
considered as belonging to a final system of straight rays of a given colour, which
issued originally from a given luminous point, were investigated in the First Supple-
ment; but these connexions will now be considered in a more general manner, and
will be extended to the new auxiliary function 7, which was not introduced before :
the new investigations will differ also from the former, by making JV” depend on the
quantities o, 7, v, rather than on a, p, y.
The general problem of investigating these connexions may be decomposed into
many particular problems, according to the way in which we pair the functions, and
according as we suppose the extreme media to be uniform or variable ; but all these
particular problems may be resolved by attending to the following general principle,
that the connexions between the partial differential coefficients of the three functions,
whether of the second or of higher orders, are to be obtained by differentiating and
20 Professor Hamitton’s Third Supplement
comparing the equations which connect the three functions themselves: that is, by dif-
ferentiating and comparing the three forms (4') (B’) (C’) of the fundamental equa-
tion (4), and the equations into which these forms (4’) (B) (C’) resolve themselves.
Thus, to deduce the twenty-eight partial differential coefficients of the second
order, of the characteristic function 7, taken with respect to the extreme co-ordinates
and the colour, from the coefficients of the same order of the auxiliary function /V,
or 7, we are to differentiate the equations into which (B’) or (C’) resolves itself,
together with the relations between the variables on which JV or 7’ depends, if any
such relations exist; and then by elimination to deduce the variations of the first
order of the seven coefficients of the variation (4’) as linear functions of the seven
variations of the first order of the extreme co-ordinates and the colour: these seven
linear functions will have forty-nine coefficients, of which, however, only twenty-
eight will be distinct, and these will be the coefficients sought.
More particularly, if the final medium be variable, and if it be required to deduce
the coefficients of the second order of V from those of V7, we first obtain expres-
sions for 8s, 87, dv, as linear functions of 8, dy, dz, 82’, dy’, 62’, dy, from the differen-
tials of the three first equations (H’), deduced from (B’), expressions which will
necessarily satisfy the first condition (H7); we then substitute these expressions for
éc, ér, dv, in the differentials of the three equations (G’), deduced from (B'), so as to
get analogous expressions for 80’, or, 6v, which must satisfy the second condition
(#7); and substituting the same expressions for 6s, dr, dv, in the differential of the
last equation (/7’), also deduced from (B’), we get an expression of the same kind for
ee 5 - : :
6 wa after which, we have only to compare the expressions so obtained, with the
following, that is, with the differentials of the equations into which the formula (4’)
has ats
8 OF aig eee spp ng ae ety oO aati
oe Sedy Y * Sade" Trae” TB a YT arae * * 323, °%
a ap SV oo 1 piORLA
or= sae ¥ le ae ‘aa aa La may + Bay OX?
eV i“ 4 &V
du= ane + oy + Y 32+ + oT a0! ‘aa + —— 62’ + yay 8X
ee es a ic V eR ed o? Ma : =
om, 27 ev 52 V eV SV, ‘e
ae Say aoe &Y + 37 & tara +o =u nia Bioyeeat
ee cake eV eV S°V SV, yo*7
—w =a es 7 oy + ser ae @ +y797 et vay +o gor + 37 by.
4 o°2V 2! SELL eee gL
eh ee? om y+ be + pte + aa Y by + ay oz tea 8x
On Systems of Rays. 21
But if the final medium be uniform, then o, z, v, x, are not independent, but related
by (<'); and the formula (#) resolves itself, not into the seven equations (4G) and
(H') but into the six equations (G) and (J), the differentials of which are to be
combined with the differential of the relation (J<’), so as to give the expressions for
80, Or, Ou, da, Ory dv, 8 x , which are to be compared with (4%) as before. And in
this case, of a final uniform medium, we may employ, instead of the two first equa-
tions (/’), any of the transformations of those equations in the foregoing number ;
or we may employ the following transformations of (J’),
L+z ov OW. +2 ov peo: Zz ov ee OW | B’
Sr ee re a i ee
in which, WV is considered as a function of the six independent variables o, 7, y,
x,y’, 2, obtained by substituting for v its value as a function of o, 7, x; the form of
which function v depends on and characterises the properties of the final medium,
and is deduced from the relation (AK). It may be useful here to go through the
process last indicated, both to explain its nature more fully, and to have its results
ready for future researches.
Differentiating therefore the two first a (B*), we obtain
bv WwW (ew ew
bu +s oz — os + Zz corey os (-? =) 8a + (—- zo” )ar, Ae
3 es 7 =(<" Bs) 4 ee gos ()
Be ein | sale taee sapere acasnW Se 187 2
in which we have put for abridgment
OW ew iy ow Ow , ow
S eSeaeer ee? ote Saag vt’ Sag :
5g URE Sie STs SSW, ys STE Am)
See Lye aaa ae ee
8 referring only to the variations of the initial co-ordinates and of the colour: and
if we put
ow
i ‘(a = = ey i (egy (E°)
the equations (C*) give, by elimination,
w" so = v2 (ax +223 gor 255°)
bs mao ter 25x):
w" be = — 2) (ay + Be oo #55 8x) .*
2 es Bee ee a +a Pedy );
VOL. XVII. G
99
wm
Professor Hamitton’s Third Supplement
and hence by (.4*) we can deduce already, without any farther differentiation,
SV 1 ew LP ae by BY by BF
st ow? Cpe 7 * 5D? Sede 8 Oe Oe Sey?
a Su SV tv SV ou &&V
Say = 7 wAaode —255,)3 Syoz 8a Sady | or oy? (G*)
37 1 Sw See a dy
By? wo” Vace Boe J? Be Be Orde Or dyes
observing, in deducing the sixth of these equations (G*), that by the definitions
(EZ), and by the a of v on, 7, x, we have
age du .oY ov
gen =(=) 232 Fp ay it 5 ox (H’)
The equations (4*) (F"*) (1°) give also
OV as SW ew o°u 1 &w Ww o2u
vox’ ie eee =e w” Sod’ Crs tee Ds
eV aa: Lae REZ LOA
arey wv" ay Sadr Sadr? wo” Body’ De See
BY _ SW (SW Bey 1 SW SW ey,
Sziz) w” SroZ “Sadr * Bade] we” dade We * Be}
oy Sein ciao Ne ite Se i Se 8
byde’ ~~ w" sadx" \dadr Oodr a w Srdai “So? “ie a)
SV 1 &W ew o2u 1 &W sew ov
SySy wm” Sody Sede * Sa) wo Bye Baad =) ; (P)
BY gs) SG PIE | WEe yas LO EG Bi
dydz’ — w" Sodz' \8a8r dadr7 ow” Sr dz’ gry
SV b&w RK SV
dzdv’ 8a Orda’ Sr Syd2"’
SV ke SK be
ody’ oo Oxdy' Br: -Bydy' ”
SV we we eV.
dzd2’ ~ 8a Sxdz' Sr Sydz! ”? J
and
Se awe ly ow ow Saas 2a
ar Om ta) (< = 2ex)- Age - =i Ge
eV 1 ey ew ew ew bu Ke
yi w Seay 7 7 = Ge a wes - eee * * 52 ; ( )
SV vey wey ow
32x da Bzdy | Or(Syax Sx’
We have therefore found expressions (G*) (I*) (A*), for eighteen out of the twenty-
eight partial differential coefficients of VV of the second order; and with respect to
On Systems of Rays. 23
os OV i i
nine of the remaining ten, namely all except 52? We may obtain expressions for
x
these by differentiating the three equations (G’), and comparing the differentials with
(4°) ; for thus we find,
Sv BW EW eV SW OV,
Sc ~*~ —Sad2’ Sxdz' Fro’ Sydx"’
SY SW eW eV ew SV
S77 Sy? Sadi Sxdy drdy Syd?
Sv SW ewer Ew RV
S22 8e* Bade! Bde BrdeY BySY ? :
SF BW OW BV MW (L’)
Sx/dy’— Ox’Sy’ Sad’ Sxdx/ Sex" Syd’ ?
SV BW Ww BV ew eV
3/82 —«By/Sz’— Baby’ Sxd82 Sd Bydz’”
OV ¥ OW SW 82V CW 8V :
32/82’ —Ss'Sx" Sade’ Broa’ drdz’ Syda’ ”
and
SV BW. Bw BY Bw eV
Sx'dy SO aw'By Sada’ Ardy —Srdx’ Bydy ”
A lL ol Al z
By 8x = ByBx Baky” Rady — BrBy” ByBy” ae)
yy Pe ke On ae. eo Reel
d2'dx ~ 82'8x ~ 8082’ oudy ~ 8482! aydx -
the equations (G’) give also
SV BW BW BV BW BV
&x'8y——C'Sy—Sady’ Bxd2’ Bry’: By/Sz’ ; |
OV a OCW SW 8V OW §V 5
dy7d2'——«Sy'S2—Sadz’ Bx8y/ Srdz’ Sydy' ” (N°)
o2V CW SW bW OoW BV
378n'870x' Sada’ Sxdz" Stn’ yds?
but these three expressions (V*) agree with the corresponding expressions (L’),
because, by (J*),
ew ey SW ey ew ey ew eV
Badu’ Sudy/ * Sedu Sydy’ — Sady/ Sede! * Srdy” Bye"?
OW SV OE CRUE ORG IZ OW SV
Body’ Sede’ * Srdy/ Sydz’ Sade’ Sedy’ | Sede’ dydy’?
Deige OA Van Ong 10202 OV ORV 987 Wa Oi
SaBz’ Sede’ | SS? SySe’ ~ Sade’ Sude’ * Srda dy8z'*
(0*)
2
Finally, with respect to the twenty-eighth coefficient , this may be obtained by
differentiating the third equation (B*), which gives
24 Professor Hamriton’s Third Supplement
oy oy Ww & 8M) eV Sy 8W_y eV :
ay? - xe dy? ( ) oxvdyx ( orey orox ) oyex : hee
And if we would generalize the twenty-eight expressions (G*) (I*) (¢*) (L*) CZ")
(P*), so as to render them independent of the particular supposition, that //” has
“B0dx dadx
been made, by a previous elimination of v, a function involving only the six indepen-
dent variables «, 7, x, vy, #, we may do so by suitably generalising fifteen out of
the twenty-one coefficients of JV, of the second order, which result from the
foregoing suppositions ; that is by leaving unchanged the six that are formed by dif-
ferentiating only with aeapect to x’, y', 2, but changing , » &c. to the following more
= &e. 3
general expressions ee
a2
Eo. as Pench be ee dv? OV &v | 7}
~ 8cdu oc dv? =) du daz ”
eas) 24 a i Cow oy ow One Hess OW & “U S
O72 me Ore oréu. or éu? or pun 8 2
ar) 4 ow 129 oW bu oe oW &v ‘
ox” Pee oe oxou ex +3, ov dx? y
Es a OV ee Ow ou MV Se eW dv bu OW &u é
= C= ae oréu Se 2 Ne Sun One sue Oo: io 8a6r ’
ia |= be ad eure SW bu eee L eur SW &u ,
b08x Ss Perms ae, dy T Sy ba 5x ts oabx 4
[ 217 ]= SW a CW Ba, SW bu ee Se ou OV &u
drdy 4 — dxdu or ESS x bu? or By eee drdy 7 (Q’)
(ee eee BW by eH 4 _ 8h BW bv
Sobel Bebe + par ao Doar l=aer ther &?
[= |= BS 74 és OW bu, 1s a OW OW bu.
Body’ I Scby’ |) Buby 30° LBrdy I~ Srby * Buby! 8?
OW |= 1 yee =; CW are ee CW SW bu
Sie ess 7 Soave Sreea | Suber?
eee ev ew -
Wel Te Tee
a aa
5x5 1 yay * Bvdy’ By?
ee j= os
bx oz =P 5x62" a dudz' by" J
obtained by differentiating the three corresponding expressions of the first order,
ov ov oWw ou. - OW bu_ ol es Oe év ,
ails: Suan? = “|= Ine Ses [5 l= Bese? ae
which are to be substituted in (B*), in as of
SV OW BW
pee ine
On Systems of Rays. 25
Deduction of the Coefficients of WV from those of V. Homogeneous Trans-
formations.
8. Reciprocally, if it be required to deduce the partial differential coefficients of
IV, of the second order, from those of /”, in the case of a final variable medium,
we have only to compare the expressions for
rr
v7 /
dx, dy, dz, 8a, dr, dv, —S—,
as linear functions of 8c, dr, dv, da’, dy’, dz’, dx, deduced from the equations (4°), with
those that are obtained by differentiating the seven equations (G@') (#'), into which
(B) resolves itself: that is with the developed expressions for the variations of
aw aw 3W 3 aw aw aw
So” Sr ? by.” b2'’ BY ” 82” Sy
But if the final medium be uniform, then (B’) no longer furnishes the seven equa-
tions (G) (H’), nor can da, dy, 6z, themselves, but only certain combinations of
them, be deduced from (4*); and the auxiliary function JV” is no longer completely
determined in form, by the mere knowledge of the form of the characteristic function
PV’, with which it is connected ; because, in this case, the seven variables on which
W depends, are not independent of each other, four of them being connected by
the relation (/X"), by means of which relation the dependence of JV” on the seven
may be changed in an infinite variety of ways, while the dependence of /” on its
seven variables, and the properties of the optical combination, remain unaltered.
Accordingly this indeterminateness of JV”, as deduced from /’, in the case of a final
uniform medium, produces an indeterminateness, in the same case, in the partial
differential coefficients of 7 ; and whereas V7, considered as a function of seven
variables, has thirty-five partial differential coefficients of the first and second orders,
we have only twenty-seven relations between these thirty-five coefficients, unless we
make some particular supposition respecting the form of JV; such as the supposition,
already mentioned, that one of the related variables, for example v, has been removed
by a previous elimination, which gives the eight conditions,
a7 _ 9, BH _
SV. eW_. eh SW EW ow
Su” Sedu
se ee
x Pa eae! Sues Sane
This last supposition removes the indeterminateness of JV” itself, and therefore of its
partial differential coefficients; of which, for the two first orders, eight vanish by
VOL. XVII. H
26 Professor Hamitton’s Third Supplement
(S*), and the remaining twenty-seven are determined, (when the variables and
-coefficients of VY” are known,) by the six equations (G’), (B*), the three lefthand
equations (G*), the six first (7°), the two first (A*), and the ten (L*) (1Z*) (P*); in
resolving which equations it is useful to observe, that by (7°) and (G*),
wo? 8a
Ligure) owe (aa sig (T")
= ovdy
And the twenty-seven expressions thus found for the coefficients of 7 of the two
first orders, on the supposition of a previous elimination of one of the seven related
variables, may be generalised, by (@Q*) and (#*), into the twenty-seven relations
already mentioned as existing between the thirty-five coefficients on any other suppo-
sition ; which supposition, if it be sufficient to determine the form of JV’, will give
the eight remaining conditions analogous to the conditions (S*), that are necessary
to determine the coefficients sought.
If, for example, we determine JV” by supposing it made homogeneous of the first
dimension with respect to a, 7, v, we shall have the eight following conditions,
ow ow ow
ae ? 3
Pao bite this oS W, (U*)
and
PH BIA 6S Blane
© 352 Ru abr She 4
BB aly BM gin OW ails
7 dadr 3 or? y Strout”
BW ge, Wag ey M, ang
Solus, ein
ow Ow ow ow
TGS | Se ke ee cv")
Sw yw SW _3w
Seay Sy ge ae
ev ew EW 8w
* Bode! * 7 Boe ’ BBe" BE"?
Jou | ew | ew _ aw
dady ordx dvdx dx’
to be combined with the twenty-seven which are independent of the form of JV, and
are deduced by the general method already mentioned. But this supposition of
homogeneity appears to deserve a separate investigation, on account of the symmetry
of the processes and results to which it leads.
(cS)
~
On Systems of Rays.
Let us therefore resume the equations
aw * Spd weak SOAR» Sea Se) SM, ae) 7
Go + Vee sae! Sao cw)
which were deduced in the sixth number from the homogeneous form that we now
assign to JV’, and which are to be combined with the following
Rei, Be 6a r3
(0) = By Tr = + V ax ’ (W )
and with the general equations of the fourth number,
Sa 3, (S3w) 4.80 ,
ee al =? v S“ineet (G’)
and let us eliminate
Qu, Sy, 32, 80’, &', Sv’, 32,
Ce)
by (4°), from the differentials of these seven equations, (J/”*) (G') (JV’*), that is
from the seven following,”
soe: Fe po er.
oo oa oo
sw 8a oad
85 + V8 =~ BP,
ee V7 322 5, 25 V, (X°)
ou ou ou
ae te Seay OE
da = 60, 8 Ma Mae”
ow 6Q SV 8a
This elimination gives
OU Yel ow 6a OV ow eQ
X® BQ = — es + 8 yom (° - SBME sa COnass Yee)
:
ox* ardy |
eee vss),
wm aas ere 4 (ai 4 vat) + (a + ae) |
2 (2 4 78), |
A 3Q= —4F + (38 4 vas) +o = yee)
7 7%, |
28 Professor Hamitton’s Third Supplement
no aaaaee ago (38s vem) t a (asm + ae) !
+a + Vasa) nk
® 30.= oer + eee +5 (Bet 8a) Sg (a +738)
‘hy Ber ve aay,
yoaanaey 498 ST (3, vam) se ee ~+V3e
“Eee. Vass
nmansden + Va 8s +e OS + Se tan Ca +t £2,
“bau, vee |
if we put for abridgment
ay ae da” + aay oy + ae ~ 82 + = dy, ]
oF Fl , OF OB a SV See |
ay ~ aan © + ay Y 2 ayer + aye,
ye Hore ar + a ay + Sas be + = dy
Ny aa ou’ + wy oy + ae oz) + wi 8x; (Z)
y men oa + aa oy + a dz’ + = by;
i) ae oa" Oe. by/ ton 2 +E by,
FB oy)” aydy
using 8 as in the notation (D*); and if we observe that the partial differential equa-
tion of the fifth number,
=o (¥ VV WV =):
dc’ dy? ds’ (vo)
gives
On Systems of Rays.
29
o-2087 )saay | soe
os ox" or owdy = du: Swdz.’
_ 80 &F 60 &V 6Q &V
~ ba dvdy Sr by? ou dydz’
on MBY MEY | EY
~ 86 dvdz br Bydz bu Bz?’
oO eV 6 &V Ps) V
oF = Svd2’ * or Byda" a = =e 7 (4)
387 3087 Baer
~ 8a dwdy’ 3 dydy |) Sy Seay”
oe 80 &V 82 &V 02 &V
~ 8c dxde/ Sr Sydz’ bu S282’
80. 3087 (ASF AST
J
We have introduced, in the equations ( Y*), the terms A%8Q,...A78Q, that we may
treat as independent the variations és, ér, dv, 6x, which are connected by the condition
62 =0.
To determine the multipliers \",...4%, we are to observe that in deducing the
"By 8s Bzdy er Sydy Sy Seay |
foregoing equations, the relation © =O between the four variables o, 7, v, x, has been
supposed to have been so expressed, by the method mentioned in the second number,
that the function Q when increased by unity becomes homogeneous of the first dimen-
sion with respect to «, 7, v; in such a manner that we have identically, for all values
of the four variables o, 7, v, x,
so 8 ‘
os tty Bivigs —=2 115 (B*)
and therefore,
S07 [00 f ao adj
Aaa aa Tee
Bre) ‘ 0) x rQ _
¢ Oaédr ¢ or? ¥ Sreu” (C’)
Siu cone Katee c
so 2a | Sa _ 20
q oodx +e oro Sudy ox”
Hence, and from the conditions (/”*), relative to the homogeneity of the func-
tion WV, it is easy to infer that the multipliers have the following values ;
' , OV
AM= —o; A= —73 AM = —v; AMHe's AMHe’s AMHo's AMHR LD:
(D*‘)
VOL. XVII. I
30 Professor Hamitton’s Third Supplement
attending to (G’) and (/V’*).
If we substitute these values of the multipliers, in the
seven equations ( Y*), we may decompose each of those equations into seven others,
by treating the seven variations 8o, dr, dv, d4’, &y/, dz’, 8x, as independent; and thus
obtain forty-nine equations of the first degree, of which however only twenty-eight
are distinct, for the determination of the twenty-eight partial differential coefficients
of the second order, of JV considered as a function of o, 7, v, a’, y', 2, x, which
relatively to «, 7, v, is homogeneous of the first dimension: the corresponding coeff-
cients of the first order being determined by the seven equations (G@) (W*) (W*).
Instead of calculating in this manner the coefficients of /V of the second order,
by eliminating between the equations into which the system ( Y*) may be decomposed,
it is simpler to eliminate between the equations ( Y*) themselves, and thus to obtain
expressions for the variations
of the coefficients of the first order, from which expressions the coefficients of the
second order will then immediately result.
first equations ( Y*), in order to get expressions for the three variations
ow ow OW
Se oe
we find, after some symmetric reductions,
8
Eliminating, therefore, between the three
a V8 tr rhage) 4 (FE) (FE) F
O97 + a(t SA c (8 5r) =» (Be en |
(es vow) 4 + (%-8F)-o(%-eZ)ts
= vas (veo Ee) fo (2) - + (w-2Z)b
837 + (ve se) {2 (85 )—» (2 vz) @)
oy? aoa, — ie) An Be )— <8)
se = - vees— («oe - or) Sv ( —¥5r)— + (m— ar) t
Seville wo) f « ( ~ 85) — » (@- 95) b
+7 (0 ge team) Lr 8E)—o (88) fo
in which,
On Systems of Rays. 31
pil ae Kl) 17 8. \ ie RE SV \? &yv &v eV \?
ig ~i& oy (ay) Tay a (Se) ee ae ae =, . Ke
4 sty 30) ”
- vw \ oo or ou
» having the same meaning as before: @ also referring, as before, to the variations of
ay zy alone, and V'" having the same meaning as in the First Supplement. In effect-
ing this elimination, we have attended to the forms of the functions JV, Q, which give
Ww Ww eae
o (dat Von) +r (d+ Vde)ty(I+V2 e)=-97;
oo oa
we have also employed the Pace (4°), which give, by Gi aH
SV eV Le oe a OVE BV EV Ps , OO oO
ne a aC) = bate 30) 3 Srdy Sze x2 Gydz Vat? gpg 8
ay &V Sa i aT ee ee a thea g OND: ;
a =e wo 3) Wiis ia ee | OD
Breer iyPr Oa, Sy ey oF RF _ 8080
3x2 Gap as 0G =A Szdx Sydz 822 Sxdy ie aeree
Having thus obtained expressions (# *) for the three variations
537 er
80 ” é&r ” bu”
it only remains to substitute these expressions in the four last equations ( Y*), and so
to deduce, without any new elimination, the four other variations
SL LL
oe?) Oye Oz dx ”
after which, we shall have immediately the twenty-eight coefficients of JV, of the
r) f) 3
second order. ‘The six coefficients, for example, of this order, which are formed by
differentiating /V” with respect to o, r, v, are expressed by the six following equations,
deduced from (E*) ;
SV &Q 1 ,oV Ov Ss | }
Sao Vien tap (“sm t+ yy ce 5a) ?
eV = ey le a 2y
See li Vs + mp” "Se +o Srey Que ae
Sew goV OV
a ar Ce = Bor a) :
PW BOL) oul ae Sp OV. eh ae oe
Sadr ae SANT” bcbg? Bebe Syde Be?
a ee hg Pe a EV ey:
Scie 1 Mie eee © bytest ety} ee ee?
oo sa algal cua clay
—_ = —— ee = — :
dude dude * wey” ( ss bzdx cis by6z ap bxdy a ey dy?
32 Professor Hamitton’s Third Supplement
which may be shown to agree with the less simple equations of the same kind in the
First Supplement, and may be thus summed up,
VE 2 2 + pi : S (180—v8r)? ar 3: (weds — adv ) (6dr — rs)
a == = (ote _ sou)” + on x, (ar — ros ) (réu — vér)
lay ES —rd0)? + 9° (rdu—ver ) (vdc—adv), (K*)
=
the mark of variation &” 1 a only to the variables «, 7, v, as & referred only to
Bs Ys Zo X-
And the whole system of the twenty-eight expressions for the twenty-eight coeffi-
cients of JV’, of the second order, may be summed up in this one formula :
vV" BW + V8O+ B80 480) = EF Ae 8) or mor
Fe Att 8) 8 EES om 8) 0 20 ro) 2
Eafe tor DY nt sie)
Oe og) 8 Zt (w- 8 )-+(a-eZ)t
+e" (FES -Cst PLE «(ant B)— (ne Py}
in which the symbols &, 8", are easily understood by what precedes, and in which the
seven variations 80, Sr, du, da’, dy’, 82, 8x, may be treated as independent of each
other.
The formula (/¢*) has an inverse, deduced from (X°*), namely
ele eo iA ve 2
vey” =(Ga =? G = ay)
ol 6Q oan = O 82 2)
@ Wess =
+( a 2a. @ oy as © ar)
pa@har ye 0 5, 9 y) (2 Sr “a®)
+2(Se 40 veo) es wa) (Say Se © ar)
sa@her la) yee) Bee) a0
On Systems of Rays. 33
in which 8” refers to 2, y, Zz, and in which V” may be deduced from JV by the
relation
2 SV _ 80, sw Pare) ew eQ \?
<== a ( oo” Wid a = us Poe) (sar ¥ VS)
& ow & O° &
(Ge 40 ae) (ae + ae) ~ (Gant Y ie)
oe ow Or ow SO \?
Cs ay) (set? sé) ~ (5st =) (N’)
and the more extensive formula (Z*) has an inverse also, namely,
pas (8V + V9'04+ BV 3049 IP) =
(SF+7 22) 20 (3,— 9% _ py) Oy — 9 vee)
} 0 } OW vo
(et V ar) {a (et 785) 3 (85 eR) Le
(Get ae) fa (w- 8er Pay) ele 8 Pe) Y
6 OW ce x9) ow 6
oe ae a Ege iar pepe aid
sie Tate) U2 gH py y®D)G Y_20(0, _ gM _p-g 2D)
Pye) OW 3 3 37 vo
coe 2) = (a —&5—--V8 ay ees s)
Se ee Be heey gk) —2 (ar 9-3 2)
5 ow x0) 5 ow 6a
ec: ee ee eM Ee
Se eee a tee, rs.
é retaining its recent meaning, so that, as Q does not contain 2’, 7’, 2’, we have, in the
last formula,
(P*)
2 _ FQ. 6Q_ 0 , 02 _ 8&0
is ~ Sad Be es ~ brdx OG ~ 8udx 8x:
If we do not choose to suppose //” homogeneous of the first dimension with
respect to o, 7, v, and if we put for abridgment
VOL. XVII. K
34 Professor Hamitton’s Third Supplement
ow ow ow
ed (Q')
and denote by 8/V,, 8°77, the expressions already found on this particular supposi-
tion, for the variations of JV”, of the two first orders, so that, for the first order, by
(G) CP) (WPS
V
8, =1286 + yor +Zou+o00u + Toy, +u0z 5 ox —F 8Q, (R‘)
and, for the second order, 8°//7,= the value of 8°/V assigned by the formula (L*‘) ;
we may generalise these particular values 6/7, °/V,, by the following relations,
SV, =8 WV —w ed,
PIP, =P WV —w 8 Q — ew SQ
ow wy Owe 5
+(o3 Te T ee a Wise ) 80%,
(8)
in which 8/7, &W, are general expressions, independent of the condition of homo-
geneity w,=0, and of every other particular supposition respecting the form of W.
It is, however, here understood that the final medium is uniform, and that in forming
the variations of the function W, the quantities o, 7, v, x, 2’, y, 2, on which it
depends, are treated as if they were seven independent variables.
And if we would deduce expressions, 8W,,, 8° W,,, for the variations of W, of the
two first orders, on the supposition that W is made, before differentiation, homogene-
ous of any dimension 7, with respect to o, 7, v, we may put
aw, 3W Ww
Batt Higgs hg —nWe=w,, (T’)
o
and we shall have the following relations
SW, =sW—-w, 8,
2 W, =" W —w, 82 — 28w, 8Q (Uy
wy Own ow,
=— +, — NW, )8Q’,
5 Mighs TP +, — NW, )bQ
+6
which include the relations (S*). The general analysis of these homogeneous trans-
formations is interesting, but we cannot dwell upon it here.
Deductions of the Coefficients of T from those of W, and reciprocally.
9. The general principles of investigation, respecting the connexions between the
partial differential coefficients of the second order, of the characteristic and auxiliary
On Systems of Rays. 55
functions, having been sufficiently explained by the remarks made at the beginning of
the seventh number, and by the details into which we have since entered ; we shall
confine ourselves, in the remaining research of such connexions, for the new auxiliary
function 7; to the case of extreme uniform media. And haying already treated of
the mutual connexions between the coefficients of the two functions / and W, it will
be sufficient now to connect the coefficients of either of these two, for example, the
coefficients of W, with those of 7, of the first and second orders: since the connex-
ions between the coefficients of all three functions will thus be sufficiently known.
We shall also suppose that W has been made, before differentiation, homogeneous of
the first dimension with respect to o, 7, v, that our results may be the more easily
combined with the symmetric expressions already deduced from ‘this supposition,
expressions which can be generalised in the manner that has been explained: and
similarly we shall suppose that Z’ is made homogeneous of the first dimension with
respect to o, 7, v, and also with respect too’, 7,v. Let us then seek to express the
partial differential coefficients of the two first orders, of 7, by means of those of
W, both functions being thus symmetrically prepared.
In this inquiry, we have, as before, the conditions of homogeneity (U*) (/”*),
relative to the function W, and analogous conditions relative to ZT, namely, for the
first order,
Mie oar
Le ; Te + vy =F ;
(V*)
ol ,Or pols
monk OR So Wane
and, for the second order,
dk oT eT er A Mi oT
O=c55 Pr ey te OO tet eat Yas |
PRONE Niet wet D nT wee BE
Oadr or orev ” ~~ — 8a'dr’ or® or'du’ ”
pee Se OC one ae aed ae 5
Oadu orév oe” ~ — 8o'8u’ Or'dv’ ou’? ”
Oe OE) oT oT OLE fOr Le Ole ,or A
in’ ° dota’ +" Bede! * * ByBel } Bo ° Bode * * Babe *Y Boar? FOV
RS. ap FT My. ROT gM ET
or’ Badr’ t Oar i “Sousa ton Sroc'ie" abr Or’ 7 S780"?
OF 1 ONeee er oT OLS Neel One noe &
Su 7 BoB) Sede” Bue 2 By Bude! Bue” Be?
RU uth ped ld oT OL ore OL OE
[a yan ey iy Ose Mero eye? J
36 Professor Hamitton’s Third Supplement
together with the conditions relative to Q, Q’, namely (B*), (C*), and the follow-
ing,
6Q! 6Q/ 6Q’ "
' ‘ i = Oeil
o 30’ 5 ace Sy’ uv Su! me ’
ea’ ea’ SQ’
‘ , ae , pe Ge = 0.
2 oo” i do’ dr’ Ae 6a du’ ;
eo! 82/ SQ! £
gta , ’ —— = (6) Xx‘
< 3a0u) Tin oe » Bev 3 ( )
Ps: ss:
85 OU Spay id? eae ot
OQ! OQ! OQ! 8Q! |
© 08x ws or’ Ox tig ou'dx ae ;
we have also the general equations
OE iy 2S ON TOWRA, '
SM Hedaya (G’)
by combining which with the foregoing conditions and with the partial differential
equation (4°), we find the following, analogous to (4‘),
so’ WV 8a’ BW 8a SW
30 Sa® SY Seb * i Bwer’
80’ SW 8a SWB SW
Jor Be'sy 1 Br B® | Ov by'dz””
80’ SV 8a SW 8a BW
“Se Sve | Or By de By Be2 ?
3a. 80° SW 8a SW SIV j
ee = ae mm)
sa 8a’ BW 8 Sw 8a OW
Sr 7 8e Braz! SBD Soy’ | Ou erBe ?
Sa 80° SW (a SWS OW
dv 80" Suda" SF Sud | Sy’ Bude"?
a _30' 30'S __ 30's ba’ BW
Sx Sy 80" Syeu Be’ Byd/ TO dydF
we
and if we combine the conditions of homogeneity of the two functions W, 7, with
the fundamental relation (7’) between these two functions, and with the properties
of Q, 2%, and attend to (G'), we find the following expressions for the partial differ-
ential coefficients of J, of the first order,
On Systems of Rays. 37
oT ; 60’
S557 Os, (PS ae gt 2 Ws
ar_ aw 78M BT hy gy BD,
5 eg peg 2 Pe (Z*)
ee af e S76 aig
5 = +(T- wy, | Sas WT ay >
oF _ or
1 pr OQ!
oF eT W)S + Tae
Differentiating the expressions (Z*), a eliminating 62’, dy’, dz’, by means of the
differentials of the general equations (G'), we obtain, by ( Y*), the following system,
analogous to the system ( Y*);
yy 87 ca’) kW 7.87 79 oQk) 88W 7.87 ON}
M8042 80' = 8-8) + QO - Hg) te gs F aa
ala
aah a7 te ;
ee ae SPAY 8a’) oe >. BO
ASQ + 2’, 8Q =waypOx— ¥ 8x) oy? Ge = a) + tear slay Cer
We,
ae +67 5 |
ge OW So’ AYE er sr 80/
Ns6Q +N's80' = eos —Ws tae as -wae +e hoe — Woh)
ow
—8 +éu ;
epee 60’ Pe 4 9 62 fen
ASO +2130 = —— o+(8e5 ~Wa)+ = a as po) tae ee / ws) x
g og 8
ie —T )+ Was 5
ow 8a’ Gi! AWW 00’
X82 + po= 2 2 a; Wae)+ se Gx; -W35,)
ye: a pre ee
xo N20 (ee ee ee a yee SW oe)
Suda’ BY oa ov
ee-T2 », wae |
ow aa e |
A,8Q +280’ = Se uf ser mi): : se wh = -; (2 a seo
ea 8a!
ae ay, Dean,
VOL. XVII.
38
Professor Hamitton’s Third Supplement
in which é, refers only to the four variations és, ér, dv, 8y, and in which we may treat
the seven variations, dc, 6r, dv, dy, dc, &’, dv’, as independent, if we assign to the
fourteen multipliers \,,...
N';, the following values ;
‘ SOE Ve oaeose oT SW SRS
SES 8x 8y/ "Sy" Belbe Sx" ]
X WOT OW OnnO Wee OL O21 Can
Be! 82°Sy' * Sr Sy’ Su! 8y'82' Sy!”
\ _ 87 ew éT br 67 ow OW
“3 ol Seba!) Sr ody See oul, Sean Bet?
\ OL ow oT OW oT ew T so ;
Ae ~ be! 8a82' 67 ba6y/ bu! da62" oe da ”
aT 5W STEW BOW 9,80. (B)
Shy BOL OM Say ON! Orde” is a
or ew ar ew | ar ew 80.
<0 Se) Subel | SieGudy cio, SUSE ee
WOW OLA oT Sw oT ow 6, Ww 6Q' ;
“7 8e' 8y 8a! " Sr Bydy' bu Sy8z2' by ’
Ay =o 5A — Tae |
W Ww er 4 . oh 8
yee we SNe We, OW aXe We: |
8x 8x
the values of ),...A; may also be thus expressed,
éu Suet 62
Ae ae a MS BD (Qs? Ve |
Sw! é 6Q
= F° = (W=T)~
: : 20 -.
w' w
tS a ergs Genin Bye |
ow! 6Q
aa OP 5 J
if we put for abridgment
Ph gles Aue .
Wi, & de! to y 3y- a 9 (D*)
and consider w’, like W, as a function of o, 7, v, x, 2’, y', 2, which, relatively to
3, 7, v, is homogeneous of the first dimension. The four last equations (4°) give,
by addition, after multiplying them respectively, by 80, 8r, dv, dx,
On Systems of Rays. 39
eT =(T- W) 80+ Ws +(W—T) 80"
+ (8T'—8,w') 82 +(8, W= Wao) ae +°W
= 3 OF ABCs 3 = eh
- (2, =e) (8-- Wee)
-(3,F 0) (2 _ wey,
8 still referring only to the variations of o, 7, v, x; and the three first equations
(4°) give, by elimination,
st wee £0" (3, W— W30) + $720
] , OW , OW OW ‘a =
nf Om fate 47 Gears de aay —* 5
1 Os 4 5 sw
ee (4 y* = ae (8 Sr) — o¢ Hay
awe Cae OF) CF Oh,
38f Lee 0
sah (¢ Bw 0 (4 av) —u (a S|
oe a J Oe Av (3 -e) 6 (8 = wy |
se (eRh EE) (OR 1-102),
a ~ W333 w— Wie) +2730 ut
ae e a ay Ve (3203) eee $4!
tape («Sponge )fv (ag) - 6 (85s 3
1 , OoW ,o2WV ; ow . ‘ ow
ape Fangs say fe (ax —&)-7 (85> oe) -J
in which
oW SW Sew \? WW SW OCW \? SW ew OoW 2 ;
“= — (——_— — x’)
ou’? byt Gay) z' (Se ) F 3c? Sa ae ) ellen
and
4.0 Professor Hamitton’s Third Supplement
1 psa"? 730%? /80'y2 :
SSS a A 2 (H")
» having the same meaning as in the second number. In effecting the last elimina-
tion, we have attended to the relations ( ¥*), which give
oW &Ww (= )'= W's (Se )
dy? 822 \ By’82"
SW ew (wy? TOT Ba. fo Oe
S22 oe ( Oz 6a" Ne = Ge )
WV SW ew \? pe Mo
a2 Dy -(=.;) =” ea ii
or (OY éx'dy (1°)
ow ow wighs W ew = Wy” < 6ar .
S2'8y' 82/8x! 8x"? By'Sz! — bu’ ”
BH Se FW a W8Q' 80%
Sy'sz! Sa’dy’ Sy” Bax! aoe = bo! ”
DH OY SMOEK, De mea! sa"
32'S2" ? d/o, = 72 82 dy! ee v Sel pe
And combining (2°) (F"), we obtain the following formula for #7’, analogous to the
formula (Z*), which completes the solution of our present problem, because it is
equivalent to twenty-eight expressions for the twenty-eight partial differential coeffi-
cients of 7, of the second order, deduced from the coefficients of W ;
O=0" IV" fe T+ (W-T)8Q-VV 80-28, W .32'-87W + W280! + y'8r' + z'bv'Y8Q y
te ‘(oa 27) — vhf eae) U
ety ( 7 Me (ae ah yy
(ee vy es 5 ( bo" 5% ye
eA | a ’ (sea? w) ti ( Sy! ag BN «(2 ay
+25 v! oe As a ~ ye) (eZ yh
m)
,
ha ‘ : OW Bi) TEs Qo as
+25 = ae or — ae ( 30’ a8 eka (8v' ey er —v (x — a) h (K°)
And if we denote by 8°7',, the value of the second difféventinl &T assigned by the
formula (JC *), and determined on the supposition that Z’ has been made, before dif-
ferentiation, homogeneous of the first dimension with respect to 6, 7, v, and also with
On Systems of Rays. 41
respect to o’, 7, v, and denote by 87},, the corresponding value of 87, determined
by the coefficients (Z*), we may generalise these values by means of the following
relations, analogous to (S") ;
87. 1 =8T—80.y.T 80.9 7;
P71 = 8 T—80.7, T-80'.—) T
(L*)
— 280.8y, T— 280'.87) T
+ 60?.Wi(vi +1) T+ 202.00! qiyi T+ 802.91 (yi +1) 7:
Vv Vis being here characteristics of operation, defined by the following symbolic
equations,
) ) }
Rt Se, ta en Yeh 15
(M*)
— ks 1, o /o 1
eC ae pat a
More generally, if we denote by 7,,,,, the function deduced from 7 by the homo-
geneous preparation mentioned in the sixth number, which coincides with 7" when the
variables ¢ tv o' rv’ x are connected by the relations Q=0, Q’=0, and which is, for
arbitrary valnes of those variables, homogeneous of the dimension 7 with respect to
o, r, v, and of the dimension 7’ with respect to o’, 7’, v’, we have the following expres-
sions, analogous to (U*),
Of, §=89 1 —80.y, 1 — 0.9L;
2 T= 8 T— 80.9, T— 80.9 T — B2.89n T — 280.89’, T (N’*)
+ 607.Yn (Vn +1) T+ 280.82'.Ya Vn! L822. 7 WT wt +1) T:
defining the characteristics V7, V’,', as follows,
3 3 ah nk
Va=oy tre tus ms Vu=esstr sts an. (0°)
Reciprocally to deduce the coefficients of JV, of the second order, from
those of 7, on the same suppositions of homogeneity, and with the same dimensions
n=1, n =1, we are to eliminate 80, 6’, Sv, between the differentials of (G’) and
(Z"*), and we find the following system,
VOL. XVII. M
42
N32= (=
N',8Q= ee x
X"36Q= eu =
SQ = (Ss
r'8Q2 = (-=
dN’ SQ= (- r
0a OF
Professor Hamitton’s Third Supplement
So! \.3W ST 8:0/ \ 3, (&T 820/ \. 8
sla El 5a a) oy! at) ar
\ aoe
+35 W?, 7 +88 57 8;
30’ \.3W (8T S:0'\ 3W (87 S20’ \.8W
aaa ae IF a Pay * ear gar
+38 or 8, +a Sa;
So \ 3W (8T 320’ \.3W /33T so’ \ sw
ay) sand Pale a We jay
+8 Wa, a + +32 ow,
ar a0) ow (G2 a7: iy: aw (87 _ aT 80) .3W
TAR SS 8x" NaAC a Oraaon + (say ov =) 62!
nes
ar Se aw pal BT iy BE (a2 sT aa ow
& ror Or’ Sy | Nore’ bu’ oy
3, 322 (WT) 32+ OW-3,T);
aT 20) ow (ia AE 7 90).53 aw (ez a7 aa) aw
a 3a’ dud’ ou Sa Om Ot: o2"
WwW
— 3 3 4W- fe aes
30’ Lor iy 538 ow ae 820" apubly sw
do°3x oo OX or Ox dr'dx &r by by
ns 8°! ——) ow
ou'dy ma ou'dx du ox hyed
~ 8+ aS +(W- T) ate =< W387) - Ve Ss
6, still referring su to Re variations of s, 7, v, x, and the values of _ ne
being,
" , a 6Q
MS = 25 Ma = 5 Ties ]
” , » _8W_ 780
(Q")
” J, ” ow 62
pee Rr oe
ee 3Ww sa
ray
On Systems of Rays. 43
Hence may be deduced, by reasonings analogous to those already employed, the fol-
lowing formula for ©’ JV, which is equivalent to twenty-eight separate expressions for
the partial differential coefficients of W, of the second order, considered as deduced
from the coefficients of 7’, on the foregoing suppositions of homogeneity :
ate ll ee 2 ae (T— WSO + 1730 4 8W (8,0! ~80) + 23,177.80
ow"
* Ge a) D+ 8 (ons Sy; Fen) D'D!
a sr)" +2(R5; -W T2,)D'D
Ct. = "aC; = WV ae )DD ; (R’)
in which we have put for ratte
p-* G (av +20043 2 Wao 7) — 20" ( ay’ +980 43, =),
D=%
O (ar +0100 +352 — wa) 88 ( ary sara _ ws, my (s’)
By
ra Oy ty a) 3a’ Bai acon Ban
Ais +9304 3,27 7 — We) = << ( ar +a'80+8 5 - Was),
and in which W" can be deduced from 7} by the relation
o?47' +0" ee we) (= = we) & ‘a = ee
vw” — \de8 Sait \B72 372 Sadr’ Sale’
er ea! or fa! oer 820! \2
seem asso arate Monee «) i; Gael areas
er S0'\ (327 80! er So! \? .
+ (ge - Waa) Ge Wie) - (ag - Wines)
General Remarks and Cautions, with respect to the foregoing deductions. Case of
a Single Uniform Medium. Connexions between the Coefficients of the Function
v, Q, v, for any Single Medium.
10. We are then able, by combining the formule of the three preceding numbers,
to deduce the partial differential coefficients of the two first orders, of any one of the
three functions 7, W, 7, from those of either of the other two, when the extreme
media are uniform and known: since we have expressed the coefficients of Y by
those of JV, and the coefficients of W by those of 7, and reciprocally, for this case
4A Professor Hamiiton’s Third Supplement
of uniform media. And if the extreme media be not uniform, but variable, that is,
if they be atmospheres, ordinary or extraordinary, we can still connect the partial
differential coefficients of the three functions, by the general method mentioned at
the beginning of the seventh number: which method extends to orders higher than
the second, without much additional difficulty of elimination, but with results of
greater complexity, and of less interesting application.
This general method consists, as has been said, in differentiating and comparing the
equations into which the general expressions (4') (B’) (C’) for the variations of the
three functions resolve themselves : and in making this preliminary resolution of the
general expressions (A’) (B’) (C’), it is necessary to attend with care to the rela-
tions between the variables oc, 7, v, 6, 7, v, x, or between o, 7, v, v, 7’, 2, x, when
any such relations exist. The investigations into which we have entered im the three
last numbers, for the case of extreme uniform media, suppose that the variables are
connected only by the relations Q=0, Q'=0, which arise from and express the optical
properties of these media; and other but analogous processes must be deduced from
the general method, when any additional relations Q" =0, Q" =0,... between the
variables of the question, arise from the particular nature of a combination which we
wish to study. In the very simple case, for instance, of a single uniform medium,
we have the three relations
o=0, T=T, v=, (U’)
which are to be combined with the relation Q=0; and with this combination of rela-
tions, the general expression (C’) for the variation of 7’ will no longer admit of being
resolved in the same way as when more of the quantities on which 7” depends could
vary independently of each other.
In the case last mentioned, of a single uniform medium, the characteristic function
V’ involves the co-ordinates x, y, 2, 2’, y/, #, only by involving their differences 7 —2",
y—y, 2-2, and is, with respect to these differences, homogeneous of the first
dimension, being determined by an equation of the form
Oh = i > fee ? s, X ); G¥®)
which results from the equation (IV) for the medium function v, by first suppressing
in that equation the co-ordinates on account of the supposed uniformity, and then
making
a. 0-2. Bs Ws op. 2S2! 5
= (W")
The relation (V°) may also be deduced from the relation Q=0, by eliminating the
ratios of o, r, v, between the three following equations,
On Systems of Rays. 4S
7 tee
We have also, in this case of a single uniform medium,
Vao(e—2')+ry—y)+v@-2), — (¥)
and therefore, by (D') (#') (U°),
ANG +rYy +21 ' (2)
the last of which results may be verified by observing that the general expression for
the auxiliary function 7’ may be put under the form
3V ar SSS er ae a
DS Crag tt Wig at 2 axe Ht Dee 7 A, oy +2 aan VU, (AS)
so that 7’ vanishes when V is homogeneous of the first dimension with respect to the
six extreme co-ordinates. The formul of the last number, for the partial differen-
tial coefficients of 7, all fail in this case of a single uniform medium, for the reason
already assigned ; but we may consider all these coefficients of J’ as vanishing, like
T itself: we may however give any other values to these coefficients which when
combined with the relations betwen the variables will make the variations of TZ’ vanish.
The coefficients of W may be obtained by differentiating the expression (Z°), which
is of the homogeneous form that we have already found it convenient to adopt ; they
are, for the first two orders, included in the two following formule,
OV =27 80 + Yer +2/8u + c6u' + Toy + vez’, Be
& W = Badr’ + Q87dy/ aig Wvd-z', ( )
and they vanish for orders higher than the second. And the coefficients of V7, of the
two first orders, may be deduced from those of JV by the formule of the eighth
number, which are not vitiated by the existence of the relations (U°), because those
relations do not affect the variables that enter into the composition of VY and JV.
The variation of V, of the first order, is
oV’ =o(d4 —82") + + (dy—8y’) + (82 —82') — ye 7 8x3 (C*)
and that of the second order is given by the following eqiitiond disthced from (0*),
(W"), (B),
Cre) FQ 0) +8 ue FQ ‘a 10) y FQ FO =-y <<
302 3s Oaor 2 Sue éréu + 32 da" Ge o+r+u
627 . ; 3
9005, ag _p2% een ras
VOL. XVII.
46 Professor Hamitton’s Third Supplement
sa £20 8) 807, gy, 82) Y?
ood 5 (arte 82) P(e 8S)
2 § 6Q f cQy, 8a i
= $2(y-¥y ~ 739) (553018 oe
exe) x 6a ie) , , 62
so oS Gage ees, 5) (br -82— 8 =e
dae o¢ 9 '
™ (-2( yay vse (=22( a2 —ar— 77 52)
8a sumer 2 8a sa
vote | ~ epi, 45 = i aaa:
~(se—20 79 2)) (SC ar—av' — V8)
sa , 6a sa , 6Q
9 ea Boe ots =, (82-82 — Ve ;
+ 2 : D’)
ads ) 80 Sy JOR se 8a iS ante oa,
: —5 (ar—a - V3) -5,(y-W-735)
in which the symbol 8’ has the same meaning as before, so that as 2’ y' 2’ do not enter
into the composition of the function Q, & refers here to the variation of colour only.
This equation (D°) may be put under the following simpler form,
+2
= (8 + 782 + 28V8Q)
="! (ara Vy
+5 y-y Vee)
(wv 9 BY
+255 (e- Awe 2) (y- a a any
2 Y- a — 792) (82-8 V3 2)
a" (a8 $9) (ar—ar 792), (E*)
if we attend to the equations already established, in the second number,
a_ sO B_& y_8Q 1a _ 8a
ooo? 0 fers 0 Oeeee OBE Tey”
On Systems of Rays. 47
and to the relations which result from these, by differentiation and elimination. For
thus we obtain
a yoQ SOQ .&w SO wv SQ .w
ra” a 7 dc? © be * bss a.
oor ire y (F’)
5 te 12 ea 3 we A@) oe 0) 5 ov
v dadu te * 3:55 “sp 2 by’
aie ey _ vy 62_ FO po fee 3 2 ole tty
v ay. 7 ax ~dadx ca Ordx of dvdx dy
in which v is considered as a homogeneous function of the first dimension of a, f, y,
involving also the colour y ; and in which, although the three variations éa, 63, ey,
are connected by the relation a&a+(03+ysy=0, yet we may treat these variations
as independent; because, if we introduced indeterminate multipliers of ada + Pe + yey,
in (Ff), to allow for the relation, we should find that these multipliers vanish, on
account of the conditions of homogeneity of v. And if we put for abridgment
“oe ae oQ >? FO FQ oO eee a
— So? door rr Ss ot sa) +5 2 (2), (G )
wv
the equations (F"°) give the following formula for &v, that is, for the second variation
of v, taken as if a B y x were four weet variables,
oy — — 08 =) a
vw
a+r tu
a er) =
a 208 = 9 (aa—o 2m
a
, 82 F8 = ao Ge vy =) 38 —v8" wet
(Sv + 08"Q + 28v8Q) =
ov:
a = (38—o8 &) 20 (2)—vy 2) fea vy?) — (m—09 5) t
“uate (a) =) au — 09") ESP (a- ee -w)t
-arn $8 2 (aa—v 9 Se) 35 (28-08 se) (a) 29a)
which justifies the passage from (D") to (H°), and expresses the law of dependence
of the partial differential coefficients of the second order of the function v on those
of Q, for the case of a uniform medium.
If the medium be not uniform, and if we would still express the law of this depend-
ence, we have onlp to change &, in the four equations (F"°) to a new characteristic 8,,
48 Professor Hamitton’s Third Supplement
referring to the variations of « y z y, and to combine the four thus altered with the
three following,
Cagis é ly sQ_ SQ .o SQ .o&v FQ ww
v
l
"®a = So8e” Bat Sede 3B * Sede By?
v SQ SQ .& SO .&v SQ .& 3
Jou, stay ee a | Ory $38 + Say ° By’ (1)
1 bv 82 82 .& FO .&v FQ .&
2 ee 813 Babs ° ba t Brdz 5G t Bude ° By’
in which ¢,, is the same new characteristic, and which are deduced from the equations
already established for variable media,
1o 00
de or?
and we are conducted to a formula for 8v, which no otherwise differs from (H°)
than by having 6) instead of © throughout.
And if, reciprocally, we would express the law of dependance oF the coefficients of
Q of the second order, on those of v, we may do so by the following general formula,
v'v'(v8Q +820 +:28,080) = ot v (=, 2 ai (aa) b
+ oe ( eg 2, &) — » (20— a5) Y
feck 3) Bee: 3,53) ¢
+25 » (a=, 5)—+ (a-2,5 =) ES «( (8- 3) — ee. a5) }
Pee x= a, =) — » ( do—8, = haa Be 8,) — o ( ar— -8,53)}
+ oF 4 (8,2) = 6(%=8 “tye (8r— 8, + (8-3, 2) b; (KH
in which 6, refers still to the variations x, y, z, x, and in which v” has the same mean-
ing as in the First Supplement, namely
Sy oy? ve oer
»_ ov eal So \? sv Sv sv_\? Susy sv \*_ ;
= 59 3p ~ (Sax) +5pr 3° ~ (Say) top ae — 4) ee)
this quantity v” is also connected with the w” of (G*) (7°), by the relation
2 2 2
vy" On = ott tu . (M’)
vt
The formula (A) is equivalent to twenty-eight separate expressions for the partial
differential coefficients of Q, of the second order, which extend to variable as well as
On Systems of Rays. 19
to uniform media: the formula gives, for example, the six following general expres-
sions, which enable us to introduce the coefficients of the function v, of the second
order, instead of those of Q, if it be thought desirable so to do, in many of the gene-
ral equations of the present memoir, as the expressions contained in (#°) would
enable us to introduce Q instead of v, in many of those of the First Supplement :
ca 1 ov Ov Peo ) ;
3° — ya U ay + Spe _ aos 3
Qe S27, 02), Zn
= =F ( eg hay" = rae one =) 3
SOx yl ( 3 Sep 1s ov ov )
— = =O aay ;
vw of3? da? oa B R (N°)
ea _ 1 ( 2 OV ev ov OL ) f
aay | v'G \ ! 3asG . SySu)” BGR), op
&2Q 1 200 ov en ov
Srou vv ( ov opoy ae dacp va oyea i éa® ) z
ea I 2 Ov oy ov ov
Suda | vee (=. Syoa 1 Say +,” Sap 7 BB ):
To make more complete this theory of the coefficients of the function Q, which
determines the nature of the final uniform or variable medium by the manner of its
dependence on the seven variables « + v x y z y, and is supposed to have been so pre-
pared that Q+1 is homogeneous of the first dimension relatively to «7 v, let us
investigate the connexion of these coefficients of Q with those of the simpler though
less symmetric function v, considered as depending on the six other variables o 7 x y
2 x by the relation Q=0. For this purpose we are to combine the differentials of
that relation with the conditions of homogeneity (B*) (C'*), and with the following
other conditions of the same kind, which are only useful in variable media,
Fa
Ee
2
0a
+
eQ
epee:
orox
ea
80 _ 80
YSese . one?
oOF veo)
* Sady 1S ordy oe ouey Savy
0) ie
© Sadz
eQ
ca ca
aa
Oroz
ria Sede S23
In this manner we find, for the first order,
(0°)
éu éu éu U =
sQ= No — F bo — or — Rae — ty — ae ax) (Ps)
that is
ie 5) OL Ble bv 82).
so 8c? or a2”
Soy lp" OF WOR ae thedy > Sale eSqw8OK = «A Bo (Q’)
ap Omen a or we Set ye Oy”
VOL. XVII.
O
50 Professor Hamitton’s Zhird Supplement
py
\ being a multiplier introduced for the purpose of treating the variations of oruryzy
as independent ; and to determine the value of this multiplier we have, by the condi-
tion of homogeneity (B'),
Bu
(vo — 2 )=041=1: (R*)
the coefficients of Q of the first order are therefore known, and we have for example,
1
ao SA .
3 a= ee 6 (S°)
ra ou
Again, for the second order,
8
p= 3.5 + d, (8v © 2, — &e.); |
p= 8AS + a(S e— &e); |
6Q éu
BS = Bd + As (Rv —5* 8 — &e);
6Q ou ou
8 = — BAT + A (30-5 bo — &e.); (T’)
6Q 38
a a Xs (80 =* 6 — &e.);
gS? = sam + 2,(8 Sas — ae); |
20 oh tak, |
O35 =- 85 + Xi (80 — a — &e.) J
in which, by (C'*) (0°) (Q°), the multipliers \,...; have the following values,
Mar (ods sd). r®,
meA(oetre).ar®; |
waa (seer e)a; |
MEA (ee tre):a eee: (U")
ANIA (of+rg) ago’,
EA (oo 4-2).a2 ue,
m= A( oes re) ARN
On Systems of Rays. 51
i, like v, being here treated as a function of o, 7, 7, y, z, x: and if we put, as usual,
ye) 62 re
S0=808 + ara & — ae au & — oy ara 2 + apm + 8208 iw + ee , (V*)
and oe
= 303 a ae + aro e+ aye + 8 + ea (W")
we find
OS OE)
Se bu ue to a ee
+ 28. (80 — gor —5 gor — 5 by 5 be Fx)
2 8 76
— 8 (oe + Sor ge + 52) (We te Be d» &)
in which
bof ae: Pig ce 5
sd=r? (a +32 aoe pte Ox aah to GED
and which is equivalent to twenty-eight expressions for the partial differential coefti-
cients of Q of the second order: it gives, for example,
oOU Q eu 20U
sO 302 oT 5 Tete -
Bet fe cial se Ag Se ony
oc 8
And since the forms of the connected functions Q, v, v, of which each expresses
the optical properties of the final medium, may be deduced, by the method of the
second number, from the form of the characteristic function /’, it evident that their
partial differential coefficients also, of all orders, are not only related to each other,
but may be deduced from the coefficients of that one characteristic function.
General Formula for Reflection or Refraction, Ordinary or Extraordinary. Changes
of V,W,T. The Difference AV is =0; AFV=AT= a Homogeneous Func-
tion of the First Dimension of the Differences Ao, Ar, Av, depending on the
Shape and Position of the Reflecting or Refracting Surface. Theorem of Mas-
ma and Minima, for the Elimination of the Incident Variables. Combinations
of Reflectors or Refractors. Compound and Component Combinations.
11. Let us now endeavour to improve our theory of the characteristic and related
functions, by applying the methods of the present memoir to improve the determina-
52 Professor Hamitton’s Third Supplement
tion given in the First Supplement, of the sudden changes produced in these func-
tions and in their coefficients, by reflexion or refraction, ordinary or extraordinary.
The general formula of such changes, which easily results from the nature of the
characteristic function V, is
0O=AV=V,—V,; (A’)
V,, V., being the two successive forms of the function V, before and after the refiex-
ion or refraction ; and the final co-ordinates x, y, z, im these forms, being connected
by the equation
O=u (a; Ys Z) (B’)
of the reflecting or refracting surface. The formula 4’) may be differentiated any
number of times with reference to the final and initial co-ordinates and the colour,
attending to the relation (") ; and such differentiation, combined with the properties
of the final uniform or variable media, conducts to the general laws of reflexion and
refraction, and to all the conditions necessary for determining the changes of the
coefficients of 7, and therefore also of the connected coefficients of JV and T, as
well as to the laws of change of the functions V, JV, 7, themselves.
Thus, for the first order, we have the general formula
8V.—8V,=sAV =dou, €@)
which, on account of the multiplier A, and the definitions (#7), resolves itself into the
seven following,
ou ou ou
Ao= AT 3 ATS 5 Av= r= 5
. av Ge
Ag =Oigh Ar =10.5) Av =05 Ay =0:
xX
the symbol A referring, as in (4’), to the finite changes produced at the surface (B’),
so that Ac, Ar, Av, denote the differences 6,—0,, t2.—7), v.—v,, between the new and
the old values of o, 7, v, that is of the partial differential coefficients of the first order,
of the characteristic function V, taken with respect to the final co-ordinates. The
three first of the equations (D") contain the general laws of the sudden reflexion or
refraction of a straight or curved ray, ordinary or extraordinary ; because, when com-
bined with the equation of the form (F’),
O=Qr (on, To, Uo5 Ly Ys 2; x)» (E’)
which expresses the nature of the final medium, they suffice, in general, when that
final medium is known, to determine, or at least to restrict to a finite variety, the
new values o,, 72, v2, of the quantities c, 7, v, on which the direction of the reflected
or refracted ray depends, if we know the old values o, 7, v,, which depend on the
direction of the incident ray and on the properties of the medium containing it, and
On Systems of Rays. 53
fu du bu
dz’ Sy? 82’
of incidence, and the normal to the reflecting or refracting surface at that point. A
remarkable case of indeterminateness, however, or rather two such cases, will appear,
when we come to treat, in a future number, of external and internal conical refraction.
With respect to the new form V’, of the characteristic function V’, it is to be deter-
mined by the two following conditions ; first, by the condition of satisfying, at the sur-
face (B"), the equation in finite differences (4’), that is, by the condition of becoming
equal to the value of the old form V,, when the final co-ordinates x, y, z, are con-
nected by the relation w=0; and secondly by the condition of satisfying, when the
if we knowalso y, 2, y, Z, and the ratios of that is the colour, the point
final co-ordinates are considered as arbitrary, the partial differential equation of the
form (C),
en SaMe ae ale «ae 7
020, (EB swnx),
if the final medium be variable, or the simpler partial differential equation of the
form (/”’), if that final medium be uniform. And as it has been already shown that
the partial differential equations relative to the characteristic function V, may be
transformed, and in the case of uniform media integrated, by the help of the auxiliary
functions JV, J, it is useful to consider here the changes of those auxiliary func-
tions, which are also otherwise interesting.
It easily follows from the definitions of W, JT, that the increments of these two
functions, acquired in reflexion or refraction, are equal to each other, and may be
thus expressed,
AW=AT=zZAc +yAr +ZzAv. (G’)
And because the differences Ac, Ar, Av, are, by the general equations of reflexion or
ou du du
refraction (D"), proportional to Be? By? 32” we may consider these differences as
equal to the projections, on the rectangular axes of the co-ordinates x, y, z, of a
straight line = v(Ao*+Ar*+Av*), perpendicular to the reflecting or refracting sur-
face at the point of incidence, and making with the axes of co-ordinates angles of
which the cosines may be called 7,, ,, 2, ; in such a manner that we shall have
Ao=n, J (Ac* + Ar* + Av?) ;
Ar=n, J (Ao? + Ar + Av’);
Av=n, / (Ao? + Ar? + Av’);
AW=AT=(an, +yn, + 2N,) / (Ao* + Ar? + Av’).
(H’)
Now the quantity an, +yn,+zn, is equal, abstracting from sign, to the perpendicular
let fall from the origin of co-ordinates on the plane which touches the reflecting or
refracting surface at the point of incidence ; it is therefore constant if that surface be
VOL. XVII. P
54 Professor Hamitron’s Third Supplement
plane, and in general it may be considered as a function of the ratios of Ac, Ar, Av,
because when those ratios are given we know the direction of the normal, and there-
fore, if the surface be curved and given, we know the point of incidence, or at least
can in general restrict that point to a finite number of positions: we have therefore
in general
AW=AT=f (Ae, Ar, Av), (1)
the function f being homogeneous of the first dimension, and depending for its form
on the shape and position of the reflecting or refracting surface, from the equation
(B’) of which surface it is to be deduced, by eliminating 2 y 2 A between the equa-
tions (B’) (G") and the three first of those marked (D"). We have also
dite (22 = Acie _ 82, Ar_ _ 82.
Av Av’ Av’? Av, 82’ Av — oy’
OZ oz z oz
eaten
the form therefore of the homogeneous function f may easily be deduced from the enna,
8
(K)
tion of the surface (B) by so wine eparing that equation as to express 2 DP ra Sie, ay
as a function ¢ of — a? Bac , which function » reduces itself to a wi Re when
the surface is plane : and we hae a simple expression for the variation of the homo-
geneous function /; namely
of= rec + yoAr + zdAv, (L’)
which, when the.reflecting or refracting surface is curved, resolves itself into the fol-
lowing remarkable expressions for the co-ordinates of the point of incidence,
so that these co-ordinates, which, for a curved surface, we knew before to be functions
of the ratios Ac, Ar, Av, are now seen to be, for such a surface, the partial differential
coefficients of the homogeneous function f When the surface (3’) is plane, the
differences Ac, Ar, Av, are no longer independent, since their ratios are then given ;
and although the expression (L’) for 6f still holds, it no longer resolves itself into the
three equations (JZ”).
Having thus studied some of the chief properties of the common increment f,
which the functions J”, 7, receive, in the act of reflexion or refraction, we are pre-
pared to investigate the new forms JV, T., of these functions V, T, considered as
depending on the new quantities 62, 2, v2, instead of the old o, 7, vu. For this pur-
pose we have first the equations
W.= Wit f(a—a, 12-71, w—u1), (nN)
T=; +f(m—n1, TMi vv);
On Systems of Rays. 55
by which /V7,, 7;, at the reflecting or refracting surface, are expressed as explicit
functions of o; 7; 1; 62 7 2; the expression of JV, involving also 2’ ¥y 2% x, and the
expression of J), involving o’ 7’ v y: and to eliminate from these expressions the
incident quantities o, 7; v, we have, if the surface be curved, the following equations,
in which the symbol 6,,, ,,, ,, refers to the variations of those incident quantities,
851, Ty Vy f= — Loo, — yer, — 20,
=—8,,, Ts v, Wi=—3,,, ny v°13 (0)
< - _ .
and *.* Sn, TI) v, ‘V2=9; 8 o,, T19 Vy -T,=0;
we are therefore to disengage the incident quantities from the expressions for JV”, T.,
by making each of those expressions a maximum or minimum with respect to those
quantities, attending to the relation Q,=0, between them; the phrase maximum or
minimum being employed with the usual latitude. For the case of a plane surface
this method of elimination fails, the form of f becoming indeterminate, on account
of the constant ratios which then exist, by (J%") or (D"), between Ac, Ar, Av; but
these very ratios, combined with the relation Q,=0, between the quantities o, 7: v5
enable us in this case to eliminate those quantities from J/,, 7;. And when we
have thus determined the new forms /V,, T,, of the functions JV, 7’, for the points
of the reflecting or refracting surface, we may extend these forms to the other points
of the final medium, if that medium be uniform, because then the final rays are
straight, and for any one such ray the quantities 6, 7, v, J, T, are constant; but if
the final medium be variable, then the final rays are curved, and the general forms of
IV, T,, for arbitrary points of the medium, are to be determined by combinations
of partial differential equations and equations in finite differences, analogous to the
combinations of such equations for /”,, and easily deduced from the principles already
laid down.
It is easy to extend the foregoing remarks to any combination of reflexions or re-
fractions, and to show, for example, that in the case of any combination of uniform
media, producing any system of polygon rays, ordinary or extraordinary, the auxiliary
function 7” is equal to the following expression,
== f (As, Ar, Av), (4)
that is, to the sum of all the homogeneous functions f of the differences of the quan-
tities o, r, v, obtained by considering the successive reflecting or refracting surfaces :
from which expression the intermediate quantities of the form o, 7, v, are to be elimi-
nated by making the expression a maximum or minimum with respect to those inter-
mediate quantities, attending to the relations between them which result from the
properties of the media, and using, for plane surfaces, the other method of elimina-
tion, founded on the ratios of Ac, Ar, Av. And when the function 7’ is known, we
56 Professor Hamitton’s Third Supplement
can deduce from it, by the methods of the fourth number, the other auxiliary func-
tion JV, and the characteristic function V.
In general for all optical combinations, whether with uniform or with variable
media, we have, by the definitions of the functions /, VY, 7, and by the results of
former numbers, the following expressions,
i ov ov ;
V=f? vds; T= f"'(#5+95,+ 2) ds ;
a4 ER i) s ov ov _ ov ;
We=x0+y7 +z +f (*Et 957 z=) ds :
(Q’)
ds being, as before, the element of the curved or polygon ray ; and hence it follows
that if we consider any total combination, of m+—1 media, whether uniform or
variable, as resulting from two partial combinations, of m and of x media respectively,
combined so that the last medium of the one partial combination (7) is the first of the
other partial combination (7), and so that the final rays of the one partial combination
are the initial rays of the other, then the functions V7, 7, (but not in general JV) for
the total combination, are the sums of the corresponding functions for the partial com-
binations : it follows also from the general expressions for the variations of these func-
tions, that the intermediate variables, belonging to the last medium of the first partial
combination, or to the first medium of the second, are to be eliminated from the sum,
by the condition of making that sum a maximum or minimum with respect to them,
Analogous remarks apply to compound combinations, composed of more than two
component combinations. These properties of the functions V, T, for total or result-
ant combinations, will be found useful in the theory of double and triple object-glasses,
and other compound optical instruments.
Changes of the Coefficients of the Second Order, of V, WW, T, produced by
Reflexion or Refraction.
12. With respect to the changes produced by reflexion or refraction in the coefti-
cients of the second order, of the characteristic function /”, and therefore also of the |
connected functions W, T, they may be deduced from the following formula, analo-
gous to (C”),
CAV =8. UW =dANU + VWACU 5 (R’)
u, \, having the same meanings as in (B’) (C’); and the multiplier \, which was
introduced also in the First Supplement, and was there regarded as a function of the
final co-ordinates x, ¥, %, beg now considered as involving also the initial co-ordi-
nates ’, y', 2, and the chromatic index x- The seven variations dr, dy, dz, da’, dy’,
On Systems of Rays. 57
82’, dy, may be treated as independent in (/?’), if we assign a proper value to od, as
a linear function of these seven variations ; so that we may deduce from (£') the
seven following equations,
As oF 03 HO aus an ]
AS Be =A? get gy ut BMS
as =28 ae au +o d; (S?)
AS was ; AB =a ; nae = du;
Ad aes
of which each may again be decomposed into seven others. But of the forty-nine
expressions thus obtained for the changes of the twenty-eight coefficients of V of the
second order, only twenty-eight expressions are distinct ; and these involve seven
multipliers as yet unknown, namely, the seven partial differential coefficients of 2 :
however we can determine these seven multipliers, and the twenty-eight coefficients
of FV, of the second order, by introducing the seven additional equations obtained by
differentiating the partial differential equation (#”), with respect to cy z 2’ y/ 2 x.
The differential of the equation (#”), is
6Q2 éQ, OQ
2 Ve 2 Vy 2 V,
_ 60, 38 2, OQs 3° 2 , 6Q, 9° 2 , 6 on + oy + ohid Geeks (T’)
Dee SG ON etree ine atnisien' Bie By cone
and this, when combined with the three first equations (S*), conducts to the following
formula,
om 3 a ian +o ee + an ay + be + Se by
th (Geet Oy tae 2)
tu (asin By Th Be)
+0. (@ a ete =) 5 (U’)
which resolves itself into seven separate equations, sufficient to determine the seven
multipliers
AB A A A A
dz’ dy’ 827 Be ? B/ ? S’” dy°
VOL. XVII. Q
58 Professor Hamitron’s Third Supplement
Three of these seven equations into which (U") resolves itself, give, by a proper
combination, a value for the trinomial
s Sox Nae Sr. Sy Ou bz’
which enables us to eliminate that trinomial from (U") and so to deduce a value for
aA, which being combined with (FR) gives,
2 Ao 80, SQ. / &V, Su
Gs } Ge New a ae = on i daa)
= 80, 80, / 87, S
atcok es rar) * 25, eee)
2 \2 (8M j dQ, 80, /& & ky
(= ) (= - ee) 24 Oe: = (Set =)
80, 80, 20, 80; | 80, 30)
as a a Sa FEE
SQ. du SQ, du Qs, du
io) ot
(eesone a Sy | Bus 5)
6Q, oy ot\ dQe oF; ou\ Qe 8"; ou
Sz ( . oe ine =e as ( 8 oy +A8 a) ( )
OQ 6Qs . Qs =
+
bax
Laas yrs oz +
=(8V,—8V —d&'u)
3X
(2. &u ne ou 8D. li (v")
Ora dy ‘ 6u2 oz
a formula that is equivalent to twenty-eight separate expressions for the twenty-eight
coefficients of 772, of the second order. This formula supposes the rays to be reflected
or refracted into a variable medium; but it can be adapted to the simpler supposition
of reflexion or refraction into an uniform medium, by merely making the quantities
éQ, 60, rere
ox ‘ oy : en
formula (V") gives,
; vanish, Whether the last medium be variable or uniform, the
8°, =8°V, ; (W’)
8 referring, as in former numbers of this Supplement, to the variations of 2’, 7/, 2’, x,
alone, that is, to the variations of the initial co-ordinates and of the colour ; and the
final co-ordinates x y 2 being those of any point on the reflecting or refracting sur-
face. Thus the ten differential coefficients, of the second order, of the characteristic
function V, like the four of the first erder, taken with respect to the initial co-ordi-
nates and the colour, undergo no sudden change by reflexion or refraction ; but the
differential coefficients of both orders, which involve the final co-ordinates, take sud-
denly new values which we have shown how to determine : and from these new coefti-
On Systems of Rays. 59
cients of V7, we can deduce those of JV and T, by the methods of the foregoing
numbers. ‘The coefficients thus found, of /V, and 7, remain unchanged through
the whole extent of the last reflected or refracted portion of the ray, when this
last portion is straight, the final medium being uniform ; but the coefficients of /”,, of
the second order, change gradually in passing from one point to another, even of this
straight portion, according to laws deducible from their connexion, already explained,
with the constant coefficients of /7,.
The coefficients of W, and J, of the second and higher orders, may also be cal-
culated, whether the last medium be uniform or variable, by differentiating the expres-
sions (V"), and eliminating the variations of o, 7, v, by the help of the conditions
already mentioned, of maximum or minimum.
Another method of calculating the changes produced in the partial differential
coefficients of V7 of the second order, by reflexion or refraction, ordinary or extraor-
dinary, into a medium uniform or variable, is to develope the second differential of
the general formula (4"), considering AV” as a function of the seven variables «x, y, 2,
r, y, 2, x, and considering x, y, z, as themselves functions of two independent vari-
ables ; for example, considering z as a function of 2, y, of which the form is deter-
mined by the equation of the reflecting or refracting surface. In this manner we
obtain, besides the formula (W 7), which is equivalent to ten equations, the eleven
following ;
bar |. Bay be, Bay (&)* Bar Be
ox* ~ Sxbz Ox O22 ox oz Ot’
ae: OAV Ae SAV bz m CAV (2) peek oz
oy" dyoz dy be Ny dz 8y* ”
Gigs CAV dz SAV & q SAL oz dz Al oz
oxoy oxdz Oy byez «Ox oz? fx by bz) Garay’ |
SAV SAV & SAV SAV & =
OS Bei? Bade’ be = Qe igysen tsar Gy? Ce)
SOA; Py ad a Lhe ADAM: Se
~ Baby" Bedy? Be? > — Bydy * Sady’ dy? |
on TAM RAV Be Ay | Bay &
oxdz! zz Oa” ~ 8ydz’ | bzdz by”
jadaY FAV Be Bar Bay & |
oxdx ozdx or g uw oyex dzoy oy c
which may be put under the form
SV eV bz = 2 — Gtz
ee :
o=4 Sx i ~ brbz co +-(=) = ? \ (¥")
SC
60 Professor Hamitton’s Third Supplement
and are deduced by differentiation from the analogous equations of the first order
} Slate oV oz ove 8V &
Past f
OS ACS oye 3 23 Cs + be wy" @
And the eleven equations thus deduced, when combined with the ten given by (W‘),
and with the seven into which (7) resolves itself, suffice, in general, to determine
the twenty-eight coefficients of V7, of the second order.
Changes produced by Transformation of Co-ordinates. Nearly all the foregoing
Results may be extended to Oblique Co-ordinates. The Fundamental Formula
may be presented so as to extend even to Polar or any other marks of position ; and
new Auxiliary Functions may then be found, analogous to, and including, the
Functions W, T': together with New and General Differential and Integral
Equations for Curved and Polygon Rays, Ordinary or Extraordinary.
13. In all the foregoing investigations, it has been supposed that the final and
initial co-ordinates, x, y, 2, wv’, y', 2, were referred to one common set of rectangular
axes. But since it may be often convenient to change the mode of marking the final
and initial positions, let us now express the old rectangular co-ordinates as linear
functions of new and more general co-ordinates w,, ¥, 2,5 and w/, y/, 2’, which may
or may not be rectangular, and may or may not be referred to one common set of
final or initial axes. For this purpose we may employ the following formule,
= Z, ar Ux, v, ais vy, Y, a Uz, Z, > )
Y=Yo + Ye, U,+Yy, Y, +Yz, 2,5
o + Se, x, + Zy Y +22, 2,3
(A*)
aaa +0'e @) + ay Y +g! 23
y =y,+y'z! z, +Y'yY, +¥Ye) B53
hee) £5 ee wae , GR
2=Z,+ 22) UL tZy Y, +2, z3 J
in which each of the eighteen coefficients of the form 2, is the cosine of the angle
between the directions of the two corresponding semiaxes, so that these coefficients are
connected by the six following relations, on account of the rectangularity of the old
co-ordinates,
2 ° t
Kx, + Yx, “=p Sx, 2— 1 5 x,” +Y a? + Z of hl 5
2
Gy, + Uy +4 2H13 ys tye? +2y2=1;3 (B*)
3 2 2
Tepe +Yz," +Zz = 1 5 ' x! +y'2/ ar Ze! elle
On Systems of Rays. 61
Let us also establish, according to the analogy of our former notation, the following
definitions similar to (P),
dx, oe dy, es
i LRA a Aptana aa a cs
ee 8! dy; dz} (C’)
=== =— = —+
, itn ales de”? ds’ ”
and the following, similar to (),
OOK 8K Eincl
Lie? Ts Bah leah, 8%) 2
y Y F (D‘)
’ oV epee Oe.
o, 5 i ) 1 oy; ee 32’
we shall then have
ata vy +B, Vy Fy) Uz 5 |
B=a, Ye, +2, Yy, +, Yz,3
Y=4, 72, +B, zy, HY, %2,3 (E’)
a = a, Lig! a B; Ly! as Y, X's! >
U ‘ ' a , é ‘
B =a; Yo +B, Wy +7, Yes
y = a’ Zn! a5 ‘ow By! ate y, Bie! >
and
/ Ul , | Pas | (i }
o,=o08z, + TY x, SF Vex, 5 6, =oX x! +TY 2 tue x5
T,=OLy, +TYy, + ey, 5 T= omy’ +7 y'y/ + vey! 3 (EY)
v=o, +TYz, tues, 3 v, =o 02" + TY 2/ 5 UZ a! -
And if, by substituting in the former homogeneous medium-functions, v, v', the ex-
pressions (Z*) for a, 8, y, a, 8’, y, we obtain v under a new form, as a homogeneous
function of a,, 3, y,, of the first dimension, and v' as a homogeneous function of the
same dimension of a‘, 8’, y/, and then differentiate these new forms of v, v’, with
reference to their new variables, we find, by (4°), the following relations between the
new and the old coefficients,
ov Ov ou ov
Sf be + 3p a2 By 3
ov ov ov ov
‘Yeh wna Y, + + By 3
ov bv ou
(G*)
ee ee ae af aides 2! re
§B/ SH Y, ae yew Yy/ + by ¥,>
ov eu’ ie wr, Fa Soe,
Sear be baa Yet Hy 8S
oy oa” 8G Y 2; eae
VOL. XVII.
|
DeGmOu A, By ou’ |
R
62 Professor Hamitton’s Third Supplement
from which relations, combined with (D*) (#’*), and with the equations (B) (#), of
the second number, we obtain the following generalisations of the equations (B),
ov ob or Vv — Vo
Sz, ~8a,> &y, — §8,> &, ~ &,’ WI
sv _ ou vr <A ee: ~
tae Se eee oe
and therefore the following aval ea of the fundamental formula (4),
év bu’ ron Sy? bv og bu > :
sV= 5 cna pi? = 3a, Fi ox, sf ap, % oy, — a +5 ae aed re 3y/ 8z, ’ (1°)
which is thus shown to extend to oblique co-ordinates, < not even to require that
the initial should coincide with the final axes.
We may adapt nearly all the foregoing reasonings and results, of the present Sup-
plement, to this more general view. We have, for example, partial differential equa-
tions of the first order in V, analogous to the equations (C’), and of the form
OL OV Oa
0=0 SE AS DSR arc AU. ?
/ ( Se, Sy een? rentice x) (K*)
1/80 = SF BA oF
0=Q (-S, aye Sz’? Se RAED x)»
which conduct to a partial differential equation of the second order, analogous to
(D): and if we put the equations (A*) under the form
0=2, (s,, OP) BY,» > x)
’ , ‘ , , , ‘ (L')
0=Q) (c;; COE) 29 Y> Zs x)s
and suppose them so prepared, by the method indicated in the second number, that
the function Q +1 shall be homogeneous of the first dimension with respect to o,,7,,v,,
and that Q,/ +1 shall be homogeneous of the same dimension with respect to o/’, 7/, v,,
we shall have
eo (wr)
with many other relations, analogous to those of the second number. The differen-
tial equations of a curved ray, ordinary or extraordinary, in the third number, may
be generalised as follows,
d §& & d & =e, d & _w&
ds Sa, dx,* ds 3B, dy, ° ds dy, 82,’
(N*)
and their integrals may be extended to ee co-ordinates, under the form,
On Systems of Rays. 63
8V 8V
Saf = const. ; ay const. ; ier const. : (O*)
while, if the final portion of the ray be straight, we have also, for that final portion,
av av Bul ;
Salat const. ; ae const. ; ate const. CE)
The formula (4’) of reflexion or refraction, ordinary or extraordinary, namely,
AV=0,
extends to oblique co-ordinates ; and if we introduce new auxiliary functions W,, 7,,
analogous to /V, 7, and defined by the new equations
Via = Vite, OY) TZ, Ui,
(Q")
analogous to the definitions (D’) (£"), and attend to the meanings and properties of
the symbols o, 7, v, ¢/ z/ v/, we shall obtain the following expressions for the varia-
tions of VY, W, T.,
V
SUL w= o On, — o/ 8x; 35 TOY, = 7/éy, + voz, —v/dz; 55 = ox 5 ]
‘ rd ‘ ‘ fd Ve
SW = 480, +n! +y Sr, +7'8y' +2804 0/82! — = ay; (R')
= (ee ne res
f= =W -2, oF WY, a, FES; Us
3 qs = 2 80,— x8; +Y or, —y, or, =f zou, = Zev; = = ox 3
which resemble the expressions (4‘) (B’) (C’), and lead to analogous results. Thus,
the partial differential coefficients of the new auxiliary functions JY7,, T,, may be
deduced, by methods similar to those already employed, from the new coefficients of
the characteristic function V, which may themselves be deduced from the old coeffi-
cients of that function, by the following general formula,
QE Oe Oe)
( Xe, = + Ys, Ln Za! a ( tees + Yr + 2! =)
(S*)
( Ly, + Yy, = + %, =)" ( ays a + yy! + + Zy! i
( Fy = + Ys, 3, + 2 =) ( at 5? ~ - ies + 2a
and the equations of a straight final ray may be put under the forms,
1 8wW,. 1 ANE OW,
hie ED AB. (y- Sr, =; (, ~ Sy, ) ic
Loewe ears a ee ar oe
a, (2- os, ap, @- or, )=— (, a ou, )»
while those of a straight initial ray may be put under these other forms,
64 Professor Hamitton’s Third Supplement
7 Ges Ga mans
these new equations (7'*) (U*) being analogous to (J*) and (P”). It is evident that
the arbitrary constants introduced by these transformations of co-ordinates must often
assist to simplify the solution of optical problems. In the comparison, for example,
of a given polygon ray, ordinary or extraordinary, of any given system, with other
near rays of the same system, it will often be found convenient to choose the final
portion of the given polygon ray for the axis of z,, and the initial portion for the
the axis of z/, a choice which will make a, 8, a/ 3’ and many of the new partial dif-
ferential coefficients vanish, without producing, by this simplification, any real loss of
generality.
We may even carry these transformations farther, and introduce polar co-ordinates,
or any other marks of initial and final position, and still obtain results having much
analogy to the foregoing. For if we suppose that the final co-ordinates 2, y, 2 are
functions of any three quantities o, 0, ¢, and that in like manner the initial co-ordinates
x’, y’, z' are functions of any other three quantities o’, 0’, ¢’, so that
87,
z,
ar a5 tp +5 +5 80 + 8, dx= = ee +o dé c dd,
_ oy oy oy
oy = se 8p +55 00+ Z 36, dy= 3p dp + 350 +3 L dp,
pum, + 59 i d x = a0 += ad
7 = 3? "30 mae Bs 25 Oats ap OP “
Bd ee Selg ag a 7 alae Borg ela preaclany:+ | oe
on ~ 80! +o + > ae 0! Pt Ay 39" P »
ete seb DEV ne penny A 12 By
oy = 30 do + sq 80! + = dy The dp + aqr 28 + 3g" dp’,
heise ale) Nabtes slay Vg
oz Saar + 397 8 + =e dz'= aaa + 30 do! + 9 dp,
we ee consider V as a function of o 0 : 0 ¢' x, obtained by ee for vy Z
x y 2 their values; and if we substitute also the values of dx, dy, dz, in the differ-
ential dV’, or vds, which was before a homogeneous function of the first dimension of
dx, dy, dz, such that by our fundamental formula
dV duds_ ov _ 8V 7
édzx ddx ba
8dV _bds_ dv _0V \
ddy ~ ddy 8 by” Se
odV _ dvds _ dv _8V
Sdz ~ 8dz by bz’
On Systems of Rays. 65
we may consider this differential dV =vds as becoming now a homogeneous function
of do, dd, dp, of the first dimension, such that
dvds _ 8dV Vda 8Vey WV dz WV
Sdp Sdp dx Op Ty dp oz dp op’
dvds odV SV & ok sy OV oz WV
$40 —3d0 du 361 Sy 30° Sz 3090”
é.vds _ odV Bog or mle oy OV &z mor
odo on ox op ty 3p | 82 op op’
(X*)
the symbol d referring still to motion along a ray. In like manner we may consider
the initial differential element of /’, namely v'ds', as a homogeneous function of the
first dimension of do’, d6’, dg’, and then we shall find that the partial differential coeffi-
cients of the first order of this function, are equal respectively to
ov OV ov
> Bp! ean 50! ras ye. >
we may therefore generalise the fundamental formula (4) as follows
* = sat wads d.cds
sV= — + 70 80 ayy op
}.' f: 1, owds' ; ae v ‘ds!
om Sdp! rar 57 ia oo! — ny oo! oe an = By. @®)
And the auxiliary functions JY, 7, correspond to the TiN more general func-
tions,
8V a Van BO ae Oe 2 OK be
maak Ria: +oE era -P+pe +05 39 + P35 +8 grt 8 ag + $ <5
of which the first may be regarded as a function of
a Sv ow 0,
> 39° Se p ’ ° > ¢ aK
and the second as a function z
ay av yaya
Spo? S07.-86.2), Bp eae! ? Ba a x
It is easy also to establish the following general differential equations of a curved ray,
ordinary or extraordinary, and the following general integrals analogous to and in-
cluding those already assigned for rectangular and oblique co-ordinates,
odV _odV odV See sd BRAS
Bgamritisp ?) “SSa00 medd ado 3
Z
ae (2)
ap! = const. ; 5
VOL. XVII. S
OV
= const. ; no = const.
66 Professor Hamitton’s Zhird Supplement
General geometrical Relations of infinitely near Rays. Classification of twenty-
four independent Coefficients, which enter into the algebraical Expressions of these
general Relations. Division of the general Discussion into four principal
Problems.
14. It is an important general problem of mathematical optics, included in that
fundamental problem which was stated in the second number, to investigate the gene-
ral relations of infinitely near rays, or paths of light ; and especially to examine how
the extreme directions change, for any infinitely small changes of the extreme points,
and of the colour: that is, in the notation of this Supplement, to examine the gene-
ral dependence of the variations da, 9B, dy, da’, §8', dy’, on da, dy, Sz, da’, dy', dz’, dy.
This important case of our fundamental problem is easily resolved by the application
of our general methods, and by the partial differential coefficients, of the two first
orders, of the characteristic and related functions: it may also be resolved by the
partial differentials of the three first orders, of the characteristic function V alone.
For from these we can in general deduce six linear expressions for da, 83, dy, da, 8B",
dy’, in terms of dz, dy, dz, da", dy’, d2', dy, involving forty-two coefficients, of which
however only twenty-four are independent, because they are connected by fourteen re-
lations included in the formule asa + B88 + y8y=0, a'8a' + B'9B' + y'dy'=0, and by four
more included in the conditions that the final direction does not change when the
initial point takes any new position on the given luminous path, nor the initial direc-
tion when the final point is removed to any new point on that given path.
Thus, if we employ the characteristic function 7, and the final and initial medium-
functions v, v', we have, by (B), the following general relations :
OV ov OV bv OV bv
ra Sat ae. Age ie aye A’)
pat ote Nae Sh ae OS or aay
ae) a a One Pes. 8y/' = yeu eae Sela 3y’ ~
in which, by the last number, we are at liberty to assign different origins and different
and oblique directions to the axes of the final and initial co-ordinates, if we assign
new and corresponding values to the marks of final and initial direction, a, [, y,
a’, ', y', so as to have still the equations her
==, oy oi dz!
=a Say ae ds' mie a aa
ds being still the final, and ds' the initial element of the curved or polygon path. We
may suppose, for example, that both sets of co-ordinates are rectangular, but that the
origins of the final and initial co-ordinates are respectively the final and initial points
On Systems of Rays. 67
of a given ordinary or extraordinary path, and that the positive semiaxes of 2, 2’,
coincide with the final and initial directions, so as to give
B= 0/0570) a=0N oO. — 5 ov 0),
/ - , i U i , : i (B’)
2 =0, 7 =0,2=0, d=0,8.=0, y =1, sy =0;
and then the six equations (.4°), of which only four are distinct, reduce themselves
to the four ae
OV
o2v i OV F OV ev
Sano Ve 3B pian seem Oo (ae ee x }
eV 8 SV Sy do |
¥ i sel po (erage ape (=- Sais) bea
Sv sue {ay 4 OV ’ OV ow 5
sap +3 P= gar & + Har Y + (Ram apH) & |
eV ov eV &v cv dt
o eemarey ar+ (Se sma) y+ (5 — saz) 23 2 er mes,
a Sv! Lee sy OV OV coy
oar 8a'sp 2. Sede! + 5y8. ' oy + (a7 Sy oa'd x |
SF Sv! eV Soy, Se! Be
a i a oe oa! + ( ba'dy! + ae ar ee da'dz! a) a
Se! So! Se SV SF fy
ay ft 1 BY
Sap" ~ spr °F = pay®* t Hay Yt Leste see) x a
OV ov! } SF Su! o oe
=F (soak ae ou! + (52 + Spay’ a} sy — = —sg52) ©
they give therefore, by easy eliminations, expressions for éa, 03, ea’, @f', of the form
6 O o Sov & owe Qaee edte
ea= 5, ety + et ee + Hi BN }
38 = 2 ar + ze Cys oz + 2 aa! + ey wc
Z (D’)
ee ere ar + y + = By, |
apa a + Fy + Bee En E ees: J
which involye twenty-four coefficients, and enable us to determine the general
geometrical relations between the final and initial tangents to the near luminous paths.
If the extreme media be ordinary, that is, if the functions v, v', be independent of
the directions of the rays, we have
V=pV/ (a” + ice + y)s v = im Jf (a’* + p° ata ys (E’)
68 Professor Hamittron’s Third Supplement
u, » being functions of the colour y, of which » involves also the final co-ordinates,
and ,' the initial co-ordinates, when the extreme media are atmospheres: and then the
equations (C°) reduce themselves at once to the following expressions of the form
da ae ov + ae oy + ee éz + ae ov + =e oy + = é Ne }
Aye) =" (rs Sr + oe oy + oz + ae ou’ + ae oy + os 8x) 3
éa = sae ou + aay ¥ = oe oz’ + a ou + a sy + a 8x)»
so= 2 sal ay ou + a4 oy se z+ = ou + - gy + = 8x) +
In general we see that the twenty-four coefficients of the expressions (D*) can easily
be deduced, by (C°), from the partial differentials of the two first orders of the cha-
racteristic function 7, and of the extreme medium-functions v, v': we have for ex-
ample
Oni Wl oe Se _ & je Oh (aa _ oe )
Seo” Ss \3e Sade/ ov” Sad \oxdy Soe
8a a ea ov ee Sv (a _&e )
dy v 8B \evdy sady/ vw” SadB\dy2 —- day
(G’)
Spa es a Sake Va On
eet a ay spas) a dae[3 Oz" ee)
i ou 1
uo” a — 5Bay) ~ See 3a8 ap?
iN
Ox
vw
v' having the same ee as in the tenth number. The same twenty-four coefh-
cients of (D") may also be deduced (as we have said) from the partial differentials of
the two first orders of the other related and auxiliary functions: or eyen from the
partial differentials of the three first orders of the characteristic function VY” alone.
Let us therefore suppose that these twenty-four coefficients of the expressions (D")
are known, and let us consider their geometrical meanings and uses: that is, their
connexions with questions respecting the infinitely small variations of the extreme
directions or tangents of a luminous path, arising from variations of the extreme
points and of the colour.
In discussing these connexions, it is evidently permitted, by the linear form of the
differential expressions (D°), to consider separately and successively the influence of
the seven variations 6x, dy, dz, a’, dy’, d2', dy, of the extreme co-ordinates and the
colour, or the influence of any groupes of these seven variations, on the four varia-
tions da, 63, da’, o', of the extreme small cosines of direction. Thus, if it be required
On Systems of Rays. 69
to compare the extreme directions of a given path of ordinary or extraordinary light
of the colour x, from a given initial point 4 to a given final point B, which path we
shall denote as follows,
(A, B), bd (H*)
with the extreme directions of an infinitely near path of infinitely near colour xX + 0x
from an infinitely near initial pomt 4’ to an infinitely near final point B’, which near
path we shall in like manner denote thus
(A, B+ ox (1)
we may do so by comparing separately the extreme directions of the given path
(A, B), with those of the three following other infinitely near paths ;
Astin CAMB), sho; 32h CE BY, S. 8d. C4 3B), : (K*)
which are obtained by changing, successively and separately, the colour y, the final
point B, and the initial point 4. We are therefore led, by this consideration, to
examine separately and successively the meanings and uses of the three following
groupes, out of the twenty-four coefficients of (D*) :
sa SB &a Op’
1st groupe i a ioe A
2 da da da 8B OB OB sai oo =P BB’ 9
2d groupe we? ay’ 82? Sx? oy? See Ria e Sy a? dy > (L’)
da da OB OB a!’ bo’ OB’ OB’ OB
82" t) by ] 8a" ’ by ’ 8a’ ’ by > 82’ > 82’ >’ 8y/ > 82’
3d groupe
But we may simplify and improve the plan of our investigation, by means of the fol-
lowing considerations.
Of the three comparisons, of the given path (/Z°) with the three near paths (K°),
the third is evidently of the same kind with the second, and need not be treated as
distinct ; because, of the two extreme points of a luminous path, it is indifferent
which we consider as initial and which as final. We may therefore omit the third
comparison (J<’), and confine ourselves to the first and second, that is, we may omit
the consideration of the third groupe (L”), in forming a theory of the general rela-
tions of infinitely near rays. For a similar reason we may omit the consideration of
the two last coefficients of the first groupe (L’), and so may reduce the study of the
whole twenty-four to the study of half that number.
On the other hand, the second comparison (A°) may conveniently be decomposed
into two: for instead of the arbitrary infinitesimal line BB’, connecting the given
final point B with the near point B’, we may conveniently consider the two projec-
tions of this line, on the final element or tangent of the given luminous path, and on
the plane perpendicular to this element : that is, we may put
VOL. XVII. a
70 Professor Hamitton’s Third Supplement
BB - BB, + BB,, (M)
BBz being the projection on the element, and BB, the projection on the perpendi-
cular plane, and we may consider separately the two near points Ba, B,, upon this
element and plane, and the two corresponding paths,
yr BRE Cee Beet (N*)
instead of considering the more general near point B’, and the near path (4, B),.
In this manner we are led to consider separately, as one subordinate class or set, sug-
gested by the path (4, Bz ),, the system of the two coefficients . ; - ; distinguish-
ing these from the eight other coefficients of the second groupe (L*), which corres-
pond to the other near path (4, B,), ; and these eight may again be conveniently
divided into two distinct classes, according as we consider the changes of final or
of initial direction.
We are then led to arrange the twelve retained coefficients of the expressions (D’),
in four new sets or classes, suggesting four separate problems :
First set & 5 eB Second, =, 2,
x” dx oz” «OZ (0°)
i da Sa 8B sp. aa ba’ ba’ op’ op’
Third, Sz? By? Sn? By > Fourth, EU? yp eae easy
In each of these four problems, the initial point is considered as given, and may be
supposed to be a luminous origin, common to all the infinitely near paths of which we
compare the extreme directions. In the first problem, the final point also is given,
but the colour y is variable ; and we study the final chromatic dispersion of the dif-
ferent near paths of heterogeneous light, connecting the given final point with the
given luminous origin : whereas, in the three remaining problems, the light is consi-
dered as homogeneous, but the luminous path varies by the variation of its final point.
In the second problem, the new final point By, is on the original path, or on that path
prolonged ; and we examine whether and in what manner the final direction varies,
on account of the final curvature of that original path. In the third problem, the
new final point B, is on an infinitely small line
v= BB, 2)
which is drawn from the given final point of the original path, perpendicular to the
given final element of that path, namely to the element
ds= BB: ; (Q’)
and we inquire into the mutual arrangement and relations of the final system of right
lines which coincide with and mark the final directions of the near luminous paths,
On Systems of Rays. 71
at the several near points 4, where they meet the given final plane perpendicular to
the given element ds. In the fourth problem, we consider the initial system of right
lines, which mark, at the luminous origin, the initial directions of the same near paths
of homogeneous light ; and we compare these initial directions with the positions of
the points B;. Let us now consider separately these four principal problems, respect-
ing the geometrical relations of infinitely near rays.
Discussion of the Four Problems. Elements of Arrangement of near Luminous
Paths. Axis and Constant of Chromatic Dispersion. Axis of Curvature of
Ray. Guiding Paraboloid, and Constant of Deviation. Guiding Planes, and
Conjugate Guiding Axes.
15. The first of these four problems, namely that in which it is required to deter-
mine the final chromatic dispersion, by means of the two coefficients — , eB, is very
0) /
easily resolved: since we have the following equations for the magnitude and plane of
this dispersion,
Final angle of chromatic dispersion = &x ; =! + (Py 2
ox
: , : da op
Final plane of dispersion..........0000. Y — =U.
inal plane of disp Vaseig its
(R*)
We may geometrically construct the effect of this dispersion, by making the given
final line of direction of the original luminous path revolve through the small angle
&éy, in which € may be called the constant of final chromatic dispersion, round the
following line which may be called the axis of final chromatic dispersion,
KAN ead 6
x Ra z=0. (S*)
The second problem, which relates to the final curvature of the given luminous
path, is resolved by the analogous equations,
Final curvature of ray =f Oe);
(T)
Plane of curvature ......+. Y en Re?
we have also the following equations for the axis of curvature, that is, for the axis of
the circle of curvature, or of the final osculating circle to the given luminous path,
ont ye il, 20s (U*)
72 Professor Hamitton’s Third Supplement
and in all these equations of curvature we may, consistently with hes ction of the
e F a because
they relate to motion along a given luminous path It is evident that these coeff-
: ca 0
present Supplement, express the coefficients = ; =e by the symbols “2
cients vanish, when the final portion of this path is straight. But when this final
portion is curved, we may geometrically construct the effect of the curvature on the
final direction, by making the final element ds revolve through an infinitely small
angle round the final axis of curvature.
The two remaining problems are more complicated, because each involves two
independent variations dr, ey, namely the two rectangular co-ordinates of the near
point B, on the final plane of 2y, which point is considered as the final point of a
near luminous path. The equations of the right line, which is the final portion or
final tangent of this near path, are,
G=00 + 2(= ou ie ay ) )
ox oy
Ge)
y=yteZ ie dict +Py)s
and the equations of the right line which is the initial portion or the initial tangent
of the same near path, are
Be (a+ ay ) 5 au
y= 2 (Lars 2 x) il
a Tk
Our third problem is to investigate the geometrical relations of the system of right
lines (V”"), which we shall call final ray-lines, with each other, and with the co-ordi-
nates 6x, dy; and our fourth problem is to investigate the connexion of the same
co-ordinates or variations with the right lines of the system (//”*), which may be
called initial ray-lines.
The third problem may be considered as resolved, if we can assign any surface to
which the final ray-lines (/”°) are normals, or with which they are determinately con-
nected by any other known geometrical relation. Let us therefore examine whether
the ray-lines of the system (/”*) are normals to any common surface, which passes
through the given final point of the original luminous path. If so, this surface may be
considered, in our present order of approximation, as perpendicular to the final rays
themselves. Now, in general, when rays of a given colour diverge from a given lumi-
nous point, and undergo any number of ordinary or extraordinary and gradual or
sudden reflexions or refractions, they are, or are not, perpendicular in their final state
to a common surface, according as the following differential equation
ad + Boy +82 = (X°)
On Systems of Rays. 73
is or is not integrable ; and if there be any one surface perpendicular to all the final
rays, there is also a series of such surfaces, represented by the integral of this equa-
tion. Hence, in the present question, the normal surface sought is such, if it exist
at all, as to satisfy the conditions 6z=0, and
oz + dada + dBdy = 0 ; CX?)
that is, if it exist, it must touch the given final plane of ay, and must have contact of
the second order with the following paraboloid, which may therefore in our present
order of approximation be employed instead of it,
oa 2 éa 6B . 6p 2 — Th
an +(S+e ley ts, 7 =0- (2)
The normals to this paraboloid, near its summit, that is, near the final point of the
given luminous path, or the origin of the final co-ordinates, have for their approx-
imate equations,
Qe4+
8a éa
= br +2(> ox +5 ¥) + 2ney, “
y= +2(3 ox +3 ay ) —Znen ,
if we put for abridgment
ab Gh ot», 10
n=4 (5 -5)3 (B")
they coincide therefore with the ray-lines (/”*) when the following condition is satis-
fied,
Joes
oc by’
which is in fact the condition of integrability of the differential equation (X°), because
we have made a B vanish by our choice of the axis of z. The condition (CC) is
satisfied, by (#°), when the final medium is ordinary ; and in fact the final rays:whe-
ther straight or curved are then perpendicular to the series of surfaces represented by
the equation
(C*)
V =const. : ¢D*)
which is, for ordinary rays, the integral of the equation (X°), and gives, as an
approximate equation of the normal surface at the origin, the following,
Sl eV i a
Ogee 120! Uouztae Ge ta Yt hie Te (E")
agreeing, by (#”), with the equation of the paraboloid (Z°). In general, the condi-
tion (C') for the existence of a normal surface, may be put, by (@*), under the form
VOL. XVII. U
74 Professor Hamitton’s Third Supplement
Sv (ae ov ) ov Ge Sv )
Sa? \axdy sBdx/ Sad3\de* Sadr
_sv (eV sv ov (eV cv). é
Se ey eee ae Sh NS ce")
and it is not satisfied by extraordinary rays, except in particular cases. We may how-
ever always consider the paraboloid (Z°) as an auxiliary surface, with which the final
ray-lines of the proposed system (V°) are connected by a remarkable and simple
relation. For if we take the rectangular planes of curvature of this paraboloid for
the co-ordinate planes of az, yz, and denote the two curvatures corresponding by
7, t, so as to have the following form for the equation of the paraboloid
z=thre’+tty’, (G")
we shall satisfy the condition
éa 6B = 10
ay +32 = (H")
and may employ the following expressions for the four coefficients of our problem,
oa oa 6B _ op _ aris 10
Sac esky ae ae sy be (1 )
the ray-lines of our system (V*) may therefore be thus represented
L=8E—2 (rex + ney), 7
y = by—2(ty—nda), J
while the normals to the paraboloid are represented by these equations
(K")
T=or—zrex, y=sy—z2tdy ; (L”)
from which it follows that the angle év between a ray-line (A) and the correspond-
ing normal (L"°) may be thus expressed
év=nel, in which V= / 82" + dy’, (M”)
él being the same small line BB, as before; and that the plane of this angle 8, or
in other words, the plane containing the ray-line and the normal, has for equa-
tion
wou + ydy = dl? — 2(rda* + bey”): (N")
this plane therefore contains also the right line having for equations
von +ysy =0, z= (O”)
rbae+ toy? 2
that is, the axis of the osculating circle of curvature of the normal or diametral section
On Systems of Rays. 75
of the paraboloid, of which the line 8/ is anelement ; and the normal may be brought
to coincide with the ray-line by being made to revolve round the element él, through
an angle & proportional to él, and equal to that element multiplied by the constant n :
the direction of the rotation depending on the sign of the constant. On account of
this simple law of deviation of the final ray-lines from the normals of the paraboloid,
we shall call this paraboloid the guiding surface : and the constant n, we shall call
the constant of deviation. And we may consider this theory, of the guiding parabo-
loid and the constant. of deviation, as containing an adequate solution of our third ge-
neral problem, in the discussion of the geometrical relations of infinitely near rays :
since this theory shows adequately the general arrangement of the final system of ray-
lines (7°), and the geometrical meanings of the third set of coefficients ( 0"), namely,
du da 38 8B
bz” dy’ 82’ Sy
The geometrical construction suggested by this theory may be still farther simpli-
fied by observing that the infinitely near normals to the guiding surface, all pass
through two rectangular lines, namely, the axes of the two principal circles of curva-
ture of the surface ; it is therefore sufficient to draw through any proposed point B,
two planes containing respectively these two given axes of curvature, and then to
make the line of intersection of these two planes revolve round the proposed small
line 3/ or BB, through the same small angle nel as before, in order to obtain the
sought final ray-line for the proposed final point.
Finally, to compare, as required in the fourth problem, the initial system of ray-
lines (/7”*) with the corresponding final points B, on the given final plane, we may
denote these initial ray-lines by the equations
x =2'80'. cos. 4, y' =2'd0'. sin. ¢', (P”)
if we put
? Sa’ = 60’. cos. ¢, 33’ =80'.sin. ¢' : (Q”)
and if in like manner we put
sv =8l.cos.¢, oy=dl.sin. >, (R”)
we shall have the following relations, between 4, ¢, ol, 80’, and the fourth set of par-
tial differential coefficients ( O°),
80. cos. ¢' = (= cos. @ + = sin. > )a,
; : (S")
30’. sin. ¢’ = (2 cos. p + 2 sil. al.
These relations give
76 Professor Hamitton’s Therd Supplement
tan. ¢ = 375, ; (ES)
they enable us therefore to determine, for any given value of 4, that is, for any pro-
posed direction of the small final line &/, or BB;, the corresponding value of 9’, that
is, the direction of the initial plane of ray-lmes, having for equation
y =a tan. ¢. (U")
Thus the final line / and initial plane ¢’ revolve together, but not in general with
equal rapidity ; and arbitrary rectangular directions of the one do not in general give
rectangular directions of the other, because the conditions
“ Ui PY ui Ui
oe tan. ¢, ere tan. >,
tang fee pois ita eee poe nla
OD, oa oF ‘
ox ty ang ox by meter WV)
T ; f T
hh ty» H=ei+z, |
(in which x is the semicircumference to the radius unity,) give the following formula
for the angle ¢,,
Q (2 # + =) cotan. 26, =
Gee enna
which is not in general satisfied by arbitrary values of that angle. There are how-
ever in general two rectangular final directions determined by this formula, which
correspond to two rectangular initial planes; and if we take these rectangular direc-
tions and planes respectively for the directions of 2, y, and for the planes of w' 2’,
yz, we shall have
oa Situs ie
oy ah Scie (x )
We may alse in general satisfy, at the same time, by a proper choice of the semiaxes
of co-ordinates, the following other conditions,
ay 7 Z Se 7 By” Or)
By this choice of co-ordinates, the relations (S$) are simplified, and become
On Systems of Rays. 77
30’. cos. #' =< .2l. cos. ’3
a (Z”")
6’. sin. ¢ =~—.&l. sin. ¢ :
while the equations (JV”*) of the initial ter reduce themselves to the following,
{oo aA oa’ a ipa BAN 0B" ll
&L=2 Se Or 3 Y =e By. oY (A )
If, then, these initial ray-lines form a circular cone haying for equation
wet y? = 280", (B")
the corresponding locus of the final point B,, on the final plane of xy, will not in
general be a circle, but an ellipse, having for its equation
(Sy aa + (Py. Sy? = 30°, (C")
of which, by ( Y"), the axis of a coincides with the least and the axis of y with the
greatest axis ; and reciprocally if the final locus be a circle having for equation
82° + oy = 81’, (D")
the initial cone of ray-lines will have for equation
e@ aS ae, ay
so that its perpendicular sections are ellipses, having their greater axes in the plane of
x 2, and their lesser axes in the plane of y/ 2’. It is evident that a circle equal to
the final circle (D") may be obtained from the elliptic cone (#"), by cutting that
elliptic cone by any one of the four following planes,
; Sa \— e\
dae (E) ey (ES) (GF) 9
and in like manner the four elliptic sections of the circular cone (B"), made by the
same four planes, are all equal and similar to the final ellipse (C'"). In general it is
easy to prove by the equations of the initial ray-lines (4"), that whatever final locus
we take for the point B,, represented by the equation
ey =f (6x); (G")
the corresponding initial cone
VOL. XVII. x
78 Professor Hamitton’s Third Supplement
will have four sections equal and similar to this final locus, namely, the sections by the
four planes (F'""). We may therefore consider these as four guiding planes for the
initial ray, since each contains for any proposed final curve or locus (G") of the final
point B;, an equal and similar guiding curve or locus, which is a section of the
sought initial cone, and by which therefore that cone may be determined. If, then,
we know these four guiding planes, or any one of them, and the corresponding sys-
tem of final and initial rectangular directions, or conjugate guiding axes, of which
two are determined by a guiding plane, we shall be able to construct the imitial ray-
line or ray-cone corresponding to any final position or locus of the point B,. The
fourth and last general problem of those proposed above, may therefore be considered
as resolved, by this theory of the guiding planes and guiding axes.
We see then that in order to compare completely the extreme directions of any
two near luminous paths
(A, B),.s (A, BAYS, &3
in which 4 is the initial and B the final point of a given path, and 4’, B’, are any
other initial and final points infinitely near to these, the following geometrical elements
of arrangement, or some data equivalent to them, are necessary and sufficient to be
known.
First. The final axis, and the initial axis, of chromatic dispersion ; and the corres-
ponding final and initial constants & &, with their proper signs, to indicate the direc-
tions, as well as the quantities of dispersion.
Second. The final axis, and the initial axis, of curvature of the given path.
Third. The final pair, and the initial pair, of axes of curvature of the guiding
paraboloids, at the ends of this given path; and the final and initial constants of
deyiation 7, 7’.
Fourth. A guiding plane for the initial ray-lines, and a guiding plane for the final
ray-lines ; together with the final system and the initial system of rectangular direc-
tions, or conjugate guiding axes, connected with these guiding planes.
When these different elements of arrangement of the extreme ray-lines are known,
we can deduce from them the dependence of &a, 88, da’, 88’, and more generally of
da, 83, dy, ba’, 88, dy’, on aa, dy, Sz, dx’, dy’, dz’, dy ; and reciprocally when this latter
dependence has been deduced from the partial differential coefficients of the charac-
teristic or related functions, we can deduce from it the geometrical elements above
mentioned.
On Systems of Rays. 79
Application of the Elements of Arrangement. Connexion of the two final Ver-
gencies, and Planes of Vergency, and Guiding Lines, with the two principal
Curvatures and Planes of Curvature of the Guiding Paraboloid, and with the
Constant of Deviation. The Planes of Curvature are the Planes of Extreme
Projection of the final Ray-Lines.
16. To give now an example of the application of these geometrical elements of
arrangement, let us employ them to determine the conditions of intersection of two
near final ray-lines, corresponding to a given colour and to a given luminous origin ;
and let us suppose, for simplicity, that one of these two straight ray-lines being the
final portion or final tangent of a given luminous path (4d, B),, the other corres-
ponds (as in the third of the foregoing problems) to a final point B, on the given
final plane perpendicular to this given path at B. Then if the constant 7 of devia-
tion vanishes, so that the final ray-lines are normals to the guiding paraboloid, the
condition of intersection requires evidently that the near point B, should be in one
of the two principal diametral planes, that is, on one of the two rectangular tangents
to the lines of curvature on this surface ; and the corresponding point of intersection
must be one of the two centres of curvature. But when m does not vanish, the
deviation of the ray-lines obliges us to alter this result. The intersection of the near
ray-line with the given ray-line will not now take place for the directions of the lines
of curvature ; but for those other directions, if any, for which the angular deviation
nel of the ray-line from the normal is equal and contrary to the angular deviation of
the normal from the corresponding plane of normal section, that is, from the corres-
ponding diametral plane of the guiding paraboloid. This latter deviation, abstract-
ing from sign, is, by the general properties of normals, equal to the semidifference of
curvatures multiplied by the element of the normal section &/, and by the sign of
twice the inclination of this element to either of the lines of curvature; it cannot
therefore destroy the deviation of the ray-line from the normal, unless the semi-
difference of the two principal curvatures of the paraboloid is greater, or at least not
less, abstracting from sign, than the constant of deviation 7; this then is a necessary
condition for the possibility of the intersection sought. But when the semidifference
of curvatures is greater (abstracting from sign) than n, then there are two distinct
directions P,, P,, of the normal or diametral plane of section, symmetrically placed
with respect to the two principal planes of curvature, and such that if the element of
section o/ be contained in either of these two planes, P,, P,, (but not if the element
8 be in any other normal plane,) the corresponding ray-line from the extremity of
that element will be contained in the same normal plane P, or P,, and will intersect
the given ray-line as required ; and the point of intersection of these two near ray-
80 Professor Hamiiton’s Third Supplement
lines will be the centre of curvature of the corresponding normal section. We may
therefore call the curvatures of these two diametral sections the ¢2vo vergencies of the
final ray-lines ; and the two corresponding planes P, P, we may call the two planes
of vergency.
The same conclusions may be deduced algebraically from the equations (A), which
give the following conditions of intersection of a near ray-line with the given ray-
line or axis of 2,
O=(27'—r)dx—ndy; O=(z-'—#) dy + nda; (1")
2 being the sought ordinate of intersection, and therefore z—' the vergency: for thus
we find by elimination the following quadratic to determine the ratio of 8a, dy, that
is the direction of 6/,
(t—1) dady =n (dy? + 82"), (K")
which may be put under the form
sin. apa, (L")
the angle ¢ being, as in (F"), the inclination of 8 to the axis of x, that is, to one of
the tangents of the lines of curvature, while 7, ¢, are the two curvatures themselves,
of the guiding paraboloid ; there are therefore two real directions of 8/, or one, or
none, corresponding to the mtersection supposed, according as we have
9
t—r7\?
( ) PP OMS jos < 7B (M")
so that we are thus conducted anew to the same conditions of reality, and to the same
symmetric directions of the two planes of vergency, which we obtained before by a
reasoning of a more geometrical kind. The same conditions may also be obtained by
considering the quadratic for the vergency itself, namely
(z2-'—r) (e¢-'—f)+n7=0, (N")
which results from the equations (J") and shows that the sum and product of the two
vergencies may be thus expressed, by means of the curvatures 7, ¢, and the constant
of deviation 7,
ate arté; 2cetsrt+n. (O")
The equations (J'') give also, by elimination of 7,
z'=r cos. ¢' +# sin. 97; (P")
we see, therefore, as before, that the two vergencies, when real, of the final ray-lines,
are the curvatures of the two corresponding sections of the guiding paraboloid. In
general the centre of curvature of any section of this surface, made by a normal plane
drawn through the given final ray-line, is the common focus by projection of all the
=
On Systems of Rays. 81
near ray-lines from the points of that section; that is, the projections of these near
ray-lines on this plane, all pass through this centre of curvature. The two rectangu-
lar planes of curvature, or principal diametral planes, of the guiding paraboloid,
may therefore be called the planes of extreme projection ; under which view they
were considered in the First Supplement, for the case of an uniform medium, and
were proposed as a pair of natural co-ordinate planes passing through any given
straight ray. The two planes of vergency, for the case of straight final rays, were
also considered in that First Supplement, in connexion with the two developable pencils
or ray-surfaces which pass through a given straight ray, and of which the two tangent
planes contain rays infinitely near, and therefore coincide with the two planes of
vergency.
When the planes of vergency are real and distinct, then, whether the final rays are
straight or curved, there exist two guiding lines perpendicular to the given final ray-
line, which are both intersected by all the near final ray-lines from the points B, on
the given final plane of wy, and which therefore suffice to determine the geometrical
arrangement and relations of that system of final ray-lines. ‘To prove the existence
and determine the positions of these two guiding lines, let us examine what conditions
are necessary and sufficient, in order that a right line having for equations
Y= 0 talie Ds, 2 — LZ, (Q")
should be intersected by all the near final ray-lines of the system (A). These con-
ditions are
Z=r-+n cotan. ® = t—n tan. ©; (R")
they give
2 2n
sin. 2@= sar (S")
and
(Z4 —r) (2-1) +m=0: CES)
when therefore
(t a ry >4n’, (U")
that is, when there are two real vergencies there are also two real guiding lines of the
kind explained above ; and these two guiding lines are contained in the two planes of
vergency, and cross the final ray-line in the two corresponding points in which it is
crossed by other ray-lines of the same system: the intersection of each guiding line with
the given final ray-line being the point of convergence or divergence of the near ray-
lines contained in that plane of vergency which contains the other guiding line. When
the constant of deviation vanishes, these guiding lines are necessarily real, and are
the axes of the two principal circles of curvature of the guiding paraboloid. And when
the final rays are straight, then, whether 7 vanishes or not, the two guiding lines (if
VOL, XVII. us
82 Professor Hamitton’s Third Supplement
real) are tangents to the two caustic surfaces ; that is, to the two surfaces which are
touched by the final rays, and are the loci of the two points of vergency. If the
guiding lines are imaginary then the points of vergency are so too, and the final rays
are not all tangents to any common surface. We shall have occasion to resume here-
after the theory of the caustic and developable surfaces.
If it happen that
; t—r=+2n, (v")
without ¢—r and separately vanishing, then the two planes of vergency close up
into one plane, bisecting one pair of the right angles formed by the two principal
planes of curvature of the guiding paraboloid ; the two vergencies reduce themselves
to a single vergency, corresponding to this single plane, and equal to the semisum of
the two curvatures of the same surface: and the two guiding lines reduce themselves
to a single guiding line, passing through the corresponding point of convergence or
divergence, and having still the property of being intersected by all the near final
ray-lines, although this property is not now sufficient to determine this system of ray-
lines.
But if the two members of (/”") vanish separately, that is, if the difference of
curvatures and the constant of deviation are separately equal to zero, then the guiding
paraboloid is a surface of revolution, having its summit at the given final point B, and
all the near final ray-lines are normals to this paraboloid of revolution, and (with the
same order of approximation) to the osculating sphere at its summit, and they all pass
through the centre of this sphere. Reciprocally, if there be any one point 0, 0, Z,
through which all the final ray-lines pass, the equations (JC) give
=O t=T= 4 CW)
and the more general equations (7°), in which the rectangular axes of x and y are
arbitrary, give
da _ $B _ —1. 8a _ = a 5 vil
may ee 5 Aig OF vagreee ee)
that is, by (G°), or (C*),
Py ee eee ee
3a? Sa? Sadr.”
ov Z- Og Or epee z
Sriy |” dadB Sady SGdz° @)
eV Se ADtly ik Desa
ay + 2 BB — Spay’
When the final rays are straight, and satisfy these last conditions (¥"), which then
reduce themselves to the following,
On Systems of Rays. 83
ov 1, ow i ov OV ov IS
eee Oe 0 zn
Sal AG. nhOak TC aRy ACD aie DyA ced ow3B® \ ? eo)
the given final ray becomes one of those which we have called principal rays in for-
mer memoirs, and the point of convergence or divergence 0, 0, Z, is what we have
called a principal focus.
Second Application of the Elements. Arrangement of the Near Final Ray-lines
from an Oblique Plane. Generalisation of the Theory of the Guiding Parabo-
loid and Constant of Deviation. General Theory of Deflexures of Surfaces.
Circles and Axes of Deflecure. Rectangular Planes and Axes of Extreme De-
flexure. Deflected Lines passing through these Axes, and having the Centres of
Deflexure for their respective Foci by Projection. Conjugate Planes of Deflexure,
and Indicating Cylinder of Deflexion.
17. The foregoing theorems respecting the mutual relations of the final ray-lines,
suppose that the near final point B, is on the given plane which is perpendicular to
the given luminous path (4, B), at its given final point B: but analogous theorems
can be found for the more general case where the near final point B’ is not in this
given perpendicular plane, by combining the solutions of the second and third of the
four problems lately discussed ; that is, by considering jointly the second and third
sets of coefficients (O°), and therefore by employing the following equations for a
final ray-line,
da. oa oa
ox Peery ears az) :
y=yte2 (@ ov +2 oy +2 ez ) -
we =de +2(
(A”)
If, in these equations, we establish no relation between dv, dy, dz, then the system of
these final ray-lines (4) is what has been called (in my Theory of Systems of Rays)
a System of the Third Class, because the equations of a ray-line in this system in-
volve three arbitrary elements of position, namely, the co-ordinates ea, dy, éz, of the
near point JB’; but to study more conveniently the properties of this total system of
the third class, we may decompose it into partial systems of the second class, that is,
systems with only two arbitrary elements of position, by assuming some relation, with
an arbitrary parameter, between the three co-ordinates dr, 4y, 8z, or, in other words,
by assuming some arbitrary and variable surface, as a locus for the near point B.
For example we may assume, as this locus, an oblique plane passing through the given
point B, and having for equation
oz = pea + goy, (B”)
S84 Professor Hamitton’s Third Supplement
in which one of the two parameters p, g, is arbitrary, and the other depends on it by
some assumed law; and then, for every such assumed plane locus (B"), we shall have
to consider a partial system of the second class, deduced from and included in the
total system of the third class (4%); namely, a system in which the equations of a
ray-line are follows,
ratete(S + pe) dere (S49) ays
(c’)
avd eee GerDe 5
Let us therefore consider the geometrical arrangement and properties of this system
of final ray-lines (C"), corresponding to the oblique plane locus (B") of the final
point B’.
The system (C'"), of ray-lines from the arbitrary oblique plane (B™), includes, as
a particular case, the system of ray-lines from the plane of no obliquity : that is, the
system (/”*), considered in a former number. And as the ray-lines of that particular
system (/”*) were found to have a remarkable connexion with the guiding paraboloid
(Z*), which touched the given perpendicular plane locus of the near final point B;,
and which satisfied the differential condition of the second order (Y°): so, the ray-
lines of the more general system (€”) may be shown to be connected in an analogous
manner with the following more general paraboloid, which satisfies the same differen-
tial condition ( Y°), and touches the more general oblique plane locus (B") at the
given final point B,
2=px+qytthratsry+hty; (D”)
in which p, g, retain their recent meanings, and the coeflicients 7, s, ¢ have the follow-
ing values,
3 é ©
r=— (2+ pS); t=— (Bap rE) :
é
4(E +5 3 +3+ z = t 93 a:
But in order to develope this more general connexion, between the ray-lines ( C”), and
the paraboloid (D”), it will be useful previously to establish some general theorems
respecting the deflexures of curved surfaces, which include some of the known theo-
rems respecting their curvatures and planes of curvature.
Let us then consider the paraboloid (D"), or any other curved surface which has,
at the origin of co-ordinates, a complete contact of the second order therewith, and
which is therefore approximately represented by the same equation: that is, (on account
of the arbitrary position of the origin, and arbitrary values of the coefficients p, 9,7,5,¢,)
any surface of continuous curvature, near any assumed point upon this surface. The
tangent plane at this arbitrary point or origin, has for equation
(E")
s= —
On Systems of Rays. 85
ZSpU+Yy 5 a)
and the deflexion from this tangent plane, measured in the direction of the arbitrary
axis of z, which we shall call the axis of deflexion, or in any direction infinitely near
to this, is, for any point B’ infinitely near to the point of contact B,
Deflexion=3.8z=h réa? + sdudy + $ td’. (G”)
This deflexion depends therefore on the perpendicular distance é/ of the near point
B from the axis of deflexion, and on the direction of the plane containing this point
and axis; in such a manner that if we put, as in (/?"),
dr=dl. cos. ¢, Sy=el. sin. ¢,
and give the name of deflexure (after the analogy of the known name curvature) to
ha ’
pendicular distance from the axis of deflexion, we shall have the following law of
dependence of this deflexwre, which we shall denote by f; on the angle ¢,
the quotient that is, to the double deflexion divided by the square of the per-
Deflexure =f= < =r cos. ¢?+2s cos. ¢ sin. p+ésin. ¢”. (aks)
There are, therefore, wo rectangular planes of extreme deflexwre, corresponding to
angles $,, ¢., determined by the following formula,
2s
on — ——- 12
tan. 2p = 3 (es)
and if we take these for the co-ordinate planes of az, yz, and denote the two extreme
deflexures corresponding by fi, f2, we have
P= 80, o—fas (K”)
and the general formula for the deflexure becomes
SHfi cos. ¢ +fz sin. ¢? : (L”)
which is analogous to, and includes, the known formula for the curvature of a normal
section. And as it is usual to consider a system of circles of curvature, for any given
point of a curved surface, namely, the osculating circles of the normal sections of
that surface, so we may now more generally consider a system of circles of deflexure :
namely, in each plane of deflexure ¢, a circle passing through the given point of the
surface, and having its centre on the given axis of deflexion, and its curvature equal
to the deflexure f; so that the radius of this circle, or the ordinate of its centre,
, ; Sua) Fi :
which we may call the radius of deflexure, is 7? and so that the equations of the circle
of deflexure are,
y=autang, @t+y+2?=—. (M”)
VOL. XVII. Z
LP
86 Professor Hamitton’s Third Supplement
We may also give the name of ais of deflexure, to the axis of this circle, that is, to
the right line haying for equations
y= —x cotan. 9, — : (N")
and we easily see that there are two principal circles of deflexure, analogous to the
two principal circles of curvature, namely, the two circles having for equations
Firs, y=0, a+ 2° aah
wed (O”)
Second a=0, y+ 2=—>3
fr
and two principal rectangular axes of deflexure, namely,
First @=0; 2 == ; Second y=0, z= - : (Gis)
1
These principal axes of deflexure are analogous to the principal axes of curvature,
that is, to the axes of the two principal osculating circles of the normal sections, in
the less general theory of normals. And as, in that theory, the near normals all pass
through the two principal axes of curvature, so we may now consider a more general
system of right lines, which we shall call the deflected lines, all near the arbitrary axis
of deflexion, and all passing through the two corresponding principal axes of deflexure,
and therefore haying for equations,
wade de, y=y—-fiy, — (Q”)
when the co-ordinates are chosen as before. These deflected lines are normals,
in the present order of approximation, to the locus of the circles of deflexure (7 2),
that is, to the surface of the fourth degree
: a 22 (a7) '
2 Dita ud 12
P+yPr+e= Fitifey ? (R”)
and they might be defined by this condition, or by the condition that they are nor-
mals, in the same order of approximation, to the following paraboloid,
c=i(fietfiy’)» (s")
which osculates to the locus (#"), and has the property that its ordinates measure
the deflexions (G") of the given surface.
A deflected line of the system (@”) is in the corresponding plane of deflexure
YOu = LOY, Ci)
if that plane coincide with either of those two principal rectangular planes of deflex-
ure, which we have taken for co-ordinate planes; but otherwise the deflected line
makes with the plane of deflexure an infinitesimal angle oy, expressed as follows,
&=2( fi-—fr) ol. sin. 2¢: (U")
On Systems of Rays. 87
this angle, therefore, is equal to the semidifference of the extreme deflexures multi-
plied by the infinitesimal perpendicular distance from the axis of deflexion, and by the
sine of twice the inclination @ of this perpendicular (or of the plane of deflexure
containing it) to one of the two rectangular planes of extreme deflexure. In this
general case, the deflected line ((*) does not intersect the given axis of deflexion,
which we have made the axis of z; but the deflected line (Q") always intersects its
own axis of deflexure (VV), in a point of which the co-ordinates may be thus ex-
pressed
CS “2. SL a cosig, z= i (Vv")
the symbols f, ¢, and ey, retaining their recent meanings. It is easy also to see that
if a near deflected line be projected on the corresponding plane of deflexure, the pro-
jection will cross the axis of deflexion in the centre of the circle of deflexure ; and
therefore that this centre of deflexure may be considered as a focus by projection, and
that the planes of extreme deflexure are planes of extreme projection.
The foregoing results respecting the deflexures and deflected lines of a eurved
surface, near any given point upon that surface, and for any given axis of deflexion,
may easily be expressed by general formule extending to an arbitrary origin and arbi-
trary axes of co-ordinates. If, for simplicity, we still suppose the co-ordinates rectan-
gular, and still take the given point upon the surface for origin, and the given axis of
deflexion for axis of z, but leave the rectangular co-ordinate planes of xz and yz
arbitrary, so that the coefficient s in the equation of the surface shall not in general
vanish, then the equations of a deflected line become
va=du—z(rdx+sdy), y=sy—z (sdx + toy) ; (W”)
since the equation of the paraboloid (S”), to which they are nearly normals, and of
which the ordinates measure the deflexions (G"*) of the given surface, becomes
z=trrtsary + hty’. (X”)
The deflexure for any plane ¢ is expressed by the general formula (H™) ; and in like
manner the general formule (J/") (NV) determine still the circle and axis of deflex-
ure. ‘The two principal planes of deflexure, ¢,, ¢2, are still determined by the for-
mula (Z"), while the corresponding extreme deflexures, fi, f:, are the roots of the
following quadratic
SJ’ -f(r+O+rt-s=0: (Y”)
and the angular deviation éy of a deflected line from the corresponding plane of
deflexure, is thus expressed,
=k (fi—fa). sin. (24 — 4). = (> -sin, 2p — s. cos, 2g) &i. (Z")
88 Professor Hamitton’s Third Supplement
Before we proceed to apply these general remarks on the deflexures of surfaces to
the optical question proposed in the present number, that is, to the study of the con-
nexion of the ray-lines (C'”) with the paraboloid (D”), we may remark that the
theory which M. Durty has given, in his excellent Développements de Géométrie, of
the indicating curves and conjugate tangents of a surface, may be extended from cur-
vatures to deflexures. For if we consider the deflexion (402=4f0l*) in the given
arbitrary direction of z as equal to any given infinitesimal quantity of the second
order, that is, if we cut the given surface by a plane ~
2—px —qy =4,%z=deflexion = const. , (A®)
parallel and infinitely near to the given tangent plane (#""), we obtain in general a
plane curve of section which may be considered as of the second degree, namely, the
indicating curve considered by M. Durty, of which the axes by their directions and
values indicate the shape of the given surface near the given point, by indicating its
curvatures and planes of curvature. This indicating curve is on the following cylin-
der of the second degree, which has for its indefinite axis the axis of deflexion, and
which we shall call the indicating cylinder of deflexion,
ra? + Qsay + ty? =z = const. ; (B")
and it is easy to see that the two principal planes of deflexure, ¢,, ¢., are the princi-
pal diametral planes of this indicating cylinder, and that the two principal deflexures
Sis fz» positive or negative, are equal respectively to the given double deflexion
ez divided by the squares of the real or imaginary principal semidiameters or
semiaxes of the cylinder, perpendicular to its indefinite axis. In general, the
positive or negative deflexure f; corresponding to any plane of deflexure ¢, is equal
to the given double deflexion 6’z divided by the square of the real or imaginary semi-
diameter of the cylinder, contained in this plane of deflexure, and perpendicular to
the axis of deflexion, that is, to the indefinite axis of the cylinder. Hence it follows,
that if we consider any two conjugate diametral planes ¢, ¢,, which we shall call con-
sugate planes of deflecure, and which are connected by the relation
O=r+s(tan. ¢+tan. ¢) +¢. tan. » tan. ¢,, (€*)
the sum of the two corresponding conjugate radii of deflexure, ; + Z , is constant, and
equal to the sum of the two extreme or principal radii : that is, we have
UN eal 1 1
at fhe |
a relation which might also have been deduced from the general expression for
the deflexure, without its being necessary to employ the indicating cylinder. We may
(D")
we
On Systems of Rays. 89
remark that any two conjugate planes of deflexure, connected by the relation (C),
intersect the tangent plane of the surface in two conjugate tangents of the kind
considered by M. Durty.
Let us now resume the system of ray-lines (C), of which the equations may be
put by (2) under the form
=r —z (r0u + sby)—znsy, )
(E*)
y =ty—2 (sox + ty) +2ndx, J
if we make
$6 ‘8a 98.08 s
a 2 (Se am Pie — 155)? ee
and let us compare these ray-lines with the deflected lines from the auxiliary parabo-
loid (D"), which have for equations
vadr—2z (row + soy), y= sy —= (sda + ty). Cw")
We easily see, by this comparison, that the infinitesimal angle of deviation év of a
ray-line (EZ) from the corresponding deflected line (J/”™), is still determined by the
same formula (JZ"°)
dv=nel,
as in the simpler theory of the guiding paraboloid explained in the fifteenth number ;
that is, this angular deviation év is still equal to the perpendicular distance o/ of the
near final point from the given final ray-line, multiplied by a constant of deviation 7.
The plane of this angle &v, that is, the plane contaiming the ray-lme (4) and the
deflected line (JV), has for equation
vou + y8y = dl? — z (rea* + Qsdxdy + ty"), (G*)
and therefore contains the right line having for equations
Bsns eS i
Pan On rex? + Qsdxby + 18y?? o
that is, the axis of deflexure (N'”): results which are analogous to those of the
fifteenth number, expressed by the equations (V™) (O"). And we may construct
the final ray-line (Z7") by a process of rotation analogous to that already employed,
namely, by making the deflected line (7), which passes through the two rectan-
gular axes of deflexure of the auxiliary paraboloid (D"), revolve round the perpen-
dicular 8/, through the infinitesimal angle 6y, proportional to that perpendicular. The
theory, therefore, of the guiding paraboloid and constant of deviation, which was
given in the fifteenth number, for the ray-lines from the near points B, on the final
perpendicular plane, extends with little modification to the ray-lines from the points
B on any final oblique plane locus passing through the given final point: namely,
VOL. XVII. 2A
90 Professor Hamitton’s Third Supplement
by employing a more general auxiliary paraboloid, and by considering deflexures and
deflected lines, instead of curvatures and normals. And we may transfer to this more
general auxiliary paraboloid, and to its connected constant of deviation, the reason-
ings of the sixteenth number, respecting the system of final ray-lines ; for example,
the reasonings respecting the foci by projection, and those respecting the condition of
intersection of such ray-lmes. And since for any given values of p, g, that is, for
any given position of the oblique plane ("), we can construct the new auxiliary
paraboloid (D"), and its new constant of deviation (#'"), by the coefficients
Sa 38 3a 38 & 3B
dz’ 82? By” dy” Sz’ Se’
that is, by means of the former guiding paraboloid (Z°) and the former constant of
deviation (B"), and by the magnitude and plane of curvature (7"°) of the final ray,
we may be considered as having reduced the theory of the geometrical arrangement
and relations of the system of final ray-lines (C™), from an oblique plane (B"), to
the theory of the elements of arrangement, which was given in the fifteenth number.
Construction of the New Auxiliary Paraboloid, (or of an Osculating Hyperboloid, )
and of the New Constant of Deviation, for Ray-lines from an Oblique Plane,
by the former Elements of Arrangement.
18. To construct the new auxiliary paraboloid (D") by the former elements of
arrangement, we may observe that this new paraboloid not only touches the given
oblique plane (6) at the given final point B of the original luminous path, but
osculates in all directions at that given point to a certain hyperboloid, represented by
the following equation,
2=pr+qyttret+sjaythty—tz ( we y =) 5 ae)
in which 7, s, ¢, are the particular values
peo Coe boy 28 #0 8B 13
Urs ox’ $= 2 ate t,= by’ (K )
of the coefficients r s ¢, deduced from the general expressions (#*) by making
D=O;9=9; (L*)
that is, by passing to the case of no obliquity ; so that the equation (Z*) of the guiding
paraboloid may be put under the form
z=trgets.acyttty’s (M”)
which includes the form (G"). Reciprocally, the sought paraboloid (D") is the only
paraboloid which has its indefinite axis parallel to the given final ray-line, and oscu-
On Systems of Rays. 91
lates in all directions at the given final point to the hyperboloid (7): it is therefore
sufficient to construct this osculatng hyperboloid, in order to deduce the sought para-
boloid (D”). We might even employ the hyperboloid as a new guiding surface for
the ray-lines from the oblique plane, instead of employing the paraboloid, since thes
two osculating surfaces have the same deflexures and deflected lines, near their given
point of osculation.
Now to construct the osculating hyperboloid (J"*), by the oblique plane (B") or
(F'”), and by the former elements of arrangement, that is, by the guiding paraboloid
: ry : : 3
(11), and by the coefficients = - ee , which determine the magnitude and plane of
curvature of the final ray, we may compare the sought hyperboloid (J") with the
following new paraboloid
S=pL tq tare +s.ty + 3toy', (N*)
which may be called the guiding paraboloid removed, since it is equal and similar to
the guiding paraboloid (17°), and may be obtained by transporting that guiding para-
boloid without rotation to a new position such that it touches the given oblique plane
at the given point. ‘The intersection of the hyperboloid (7) and paraboloid (Vv),
consists in general of an ellipse or hyperbola in the given plane
z=0, (O")
perpendicular to the given final ray, and of a parabola in the plane
6 é
x+y P=0, GP?)
which contains the given final ray-line or ray-tangent, and is perpendicular to the
final plane of curvature of the ray. If then, we make this final plane of curvature
the plane of xz, so that its equation shall be
y=9, (Q")
and so that, by ( 7”),
2 =0; (R”)
we shall have the following equations for the two curves of intersection ; first, for the
ellipse or hyperbola,
, 2=0, pa+qy tyre +sry + 3ty'=035 (S*)
and Sc for the parabola,
a=0, z=qytaty’: ei)
and these two curves may be considered as known, since they are the intersections of
two known planes with the known guiding paraboloid removed to a known position.
To examine now how far a surface of the second degree is restricted by the condition
92 Professor Hamitton’s Third Supplement
of containing these two known curves, and what other conditions are necessary, in
order to oblige this surface to be the hyperboloid sought, let us employ the following
general form for the equation of a surface of the second degree,
Av’ + By + C2 + Day + Hyz+ Fzur+ Ga+ Hy +Iz+K=0, (Oke)
and let us seek the relations which restrict the coefficients of this equation when the
surface is obliged to contain the two known curves. The condition of containing the
parabola (Z""), gives
K=0, H= —ig, H=0,, C20, b= —+ i; (Vv")
so that, by this condition alone, the general equation (U™) is reduced to the follow-
ing form,
z=qythty—— (G+ Fe+Dy+ Az). Ww)
qYy +4ty —F ,
In order that this less general surface of the second degree, (JV), should contain
the ellipse or hyperbola (S'"), it is necessary and sufficient that we should have the
relations,
G=-Ip, D=-Is,, A4=—-tf,: (X*)
the general equation, therefore, of all those surfaces of the second degree which con-
tain at once the two known curves (S”) (Z""), involves only one arbitrary coeffi-
cient, and may be put under the form
s=pxtqyttriv+sayt+hty +rrz. Os,
This general equation, with the arbitrary coefficient X, belongs to the guiding parabo-
loid removed, that is, to the surface (VV), when we suppose
\=0; (2)
and the same general equation belongs by (/?") to the sought hyperboloid (7"*), when
ent ze (A)
To put this last condition under a geometrical form, let us, as we have already consi-
dered the intersections of the hyperboloid with the two rectangular co-ordinate planes
of ay and yz, consider now its intersection with the third co-ordinate plane of xz,
that is, with the plane of curvature (Q") of the given final ray. This intersection is
the following hyperbola, .
iNT
y=0, c= +3r0'—4 5 22, (B")
and the corresponding intersection for the surface ( Y"™) is
y=0, 2=pr+ dre +022; (C4)
the condition (4") is therefore equivalent to an expression of the coincidence of these
two intersections; and if we oblige the surface of the second degree (U") to contain
On Systems of Rays. 93
the three curves (S"*) (7""’) (5"), in the three rectangular co-ordinate planes, we
shall thereby oblige it to become the sought hyperboloid (2). It is not necessary,
however, though it is sufficient, to assign the hyperbola (6"), as a third curve upon
this hyperboloid. or, in general, if we know the intersections of a surface of the
second degree with two known planes, there remains only one unknown quantity in
the equation of that surface, and the intersection with a-third known plane is more
than sufficient to determine it. Thus, in the present question, if the intersection
(C") be distinct from the following parabola
y=0, z=p2+4r2, (D")
that is, if the surface (Y"™), containing the two known curves (S™) (7""), be dis-
tinct from the known guiding paraboloid removed, which also contains the same two
curves, the intersection (C'') with the plane of curvature of the ray is in general a
hyperbola, which touches the known parabola (D") at the known origin of co-
ordinates, and meets this parabola again in another known point on the axis of 2,
that is on the radius of curvature of the known final ray, namely, in the point
2p
r= — q=O,42—0); (Ee)
Tr. ’
the hyperbola (€") has also one asymptote parallel to the known final ray-line or axis
of z, namely, the asymptote having for equations
1
os y=9, ce”)
and it will be entirely determined, if, in addition to the foregoing properties, we know
also a line parallel to its other asymptote, namely, to that which has for equations
Ap 2s 1 2p
w= —2(—)2-5->, P=VE (GS)
it will therefore be obliged to coincide with the hyperbola (B"), if only we oblige its
second asymptote (G@") to be parallel to the following known right line,
= te, 22 4
Diet igen aa es
in which the coefficient
ope yew COTTER ORIOLE TENE (ry
r. dz deflecure of guiding paraboloid ”
°
the plane of the deflexure 7, being the plane of curvature of the ray. We see, then,
that this last condition, respecting the direction of the second asymptote (G"*) of the
hyperbolic section (C™), is sufficient, when combined with the conditions of contain-
ing the two known curves (S") (Z"), to determine completely the sought hyperbo-
loid (J). Even the conditions of containing the two curves (S*) (7"") are not
perfectly distinct and independent ; nor would their coexistence be possible, in the
VOL. XVII. 2B
94 Professor Hamitton’s Third Supplement
determination of a surface of the second degree, if the two points in which the para-
bola( 7") is intersected by the axis of y, that is, by the intersection-line of the planes
of the two curves, namely, the origin and the point
o—0; —— 3=0p (K")
were not also contained on the ellipse or hyperbola (S'°). But we may confine our-
selves to the last chosen conditions, of having these two known curves as the inter-
sections of the hyperboloid with two known planes, and of having known directions
for the asymptotes of its hyperbolic curve of intersection with a third known plane,
as adequate and sufficiently simple conditions for the construction of the sought
hyperboloid, and thereby of the auxiliary paraboloid (D"), to which that hyperboloid
osculates. And with respect to the new constant of deviation m, connected with this
auxiliary paraboloid, we may put its general yalue (#"") under the form
n=n,+hpe — 49%, (L")
n, being the particular value
op 8a
=e 14
eee (3. ay) an
for the plane of no obliquity, that is, the value (6") connected with the guiding para-
boloid (Z*) in the theory of the elements of arrangement which was given in a former
number : we may therefore construct the new constant 7, as the ordinate z of a plane
Z=px+qytn,, (N™)
which is parallel to the given oblique plane (B”), and contains the point
z=0, y=0, z=n,, (O")
so that it intersects the axis of z at a distance from the origin =the old constant of
deviation m,. The other co-ordinates x, y, to which the ordinate z = 7 corresponds,
are
e=42, y=-3e (P")
so that the corresponding line / 2° +7’ is equal to half the curvature of the ray, and
is perpendicular to the radius of that curvature.
The details of the present number have been given, in order to illustrate the sub-
ject, by combining it more closely with geometrical conceptions; but the new auxiliary
paraboloid, and the new constant of deviation, might have been considered as suffi-
ciently defined by their former algebraical expressions.
On Systems of Rays. 95
Condition of Intersection of Two Near Final Ray-lines. Conical Locus of the
Near Final Points, in a variable medium, which satisfy this condition. Investi-
gations of Matus. Jllustration of the Condition of Intersection, by the. Theory
of the Auxiliary Paraboloid, for Ray-lines from an Oblique. Plane.
19. Returning now to the system of final ray-lines (C™) from an oblique plane
(B"), let us consider the condition necessary in order that one of these near final
ray-lines (C’) may intersect the given final ray-line or axis of z. This condition
may be at once obtained by making x and y vanish in the equations (C'”), and then
eliminating z; it may therefore be thus expressed,
8 By. (8, B\s
an. (2 an fe 57) oe 45 5 + q swt
oa oa oa oa
=yy.4 (E+ pZ)w+ (E+ aut, (Q*)
or more concisely thus, on account of the equation of the oblique plane (B"),
8B... Bs. Bo,
dw, (Sax +5. ay +5. 8)
=ty. (Baw + Say +% 82) 3 (R*)
that is,
da 88 = dy 8a; (S"*)
it is therefore necessary and sufficient, for the intersection sought, that the near final
point B’ should be on a certain conical locus of the second degree, determined by
the equation (#"), between the co-ordinates dx, dy, dz. A conical locus of this kind,
appears to have been first discovered by Matus. That excellent mathematician and
observer had occasion, in his Zraité D’ Optique, to make some remarks on the gene-
ral properties of a system of right-lines in space, represented by equations of the
form
a—xv_y-y _z—2
i me
>
in which m, m, 0, are any given functions of the co-ordinates 2’, y/, z', of a point
through which the line is supposed to pass, and by which it is supposed to be deter-
mined ; and he remarked that the condition of intersection of a line thus determined,
with the corresponding near line from a point infinitely near, was expressed by an
equation of the second degree between the differentials of the co-ordinates 2’, y', 2’,
which might be considered as the equation of a conical locus of the second degree for
the infinitely near point. The theory of systems of rays which was given by Matus,
differs much, in form and in extent, from that proposed in the present Supplement ;
96 Professor Hamitton’s Third Supplement
especially because, in the former theory, the coefficients which mark the direction of a
ray were left as independent and unconnected functions, whereas, in the latter, they
are shown to be connected with each other, and to be deducible by uniform methods
from one characteristic function. But the mere consideration of the existence of some
functional laws, whether connected or arbitrary, of dependence of the coefficients m7 0
on the co-ordinates wv 7’ 2’, or of a B yon x y z, conducts easily, as we have seen, to a
conical locus of the kind (#"). This result may however be illustrated by the theory
which we have given of the geometrical relations of the near final ray-lines from an
oblique plane with the deflected lines of a certain auxiliary paraboloid, and with a
certain law and constant of deviation.
For, according to the theory of these relations, the ray-line from a near final point
B’ on a given oblique plane drawn through the given point B, will or will not inter-
sect the given final ray-line from B, according as its deviation Sv from its own
deflected line does or does not compensate for the deviation dy of that deflected line
from the corresponding plane of deflexure, by these two deviations being equal in
magnitude but opposite in direction ; the condition of intersection may therefore be
thus expressed,
dv + dv=0; (ie)
or, by the values of the deviations dv, dy, established in the seventeenth number,
nN = =. sin. 2 + 5. cos. 2, (U")
that is,
n (da° + by’) = (t—1) da dy +s (Ox? — by’): (V*)
and the condition of intersection thus obtained, by the consideration of two equal and
opposite deviations, is, on account of the meanings (#") (F"") of n, 7, s, ¢, equivalent
to (Q"), and therefore to the equation (2) of the cone of the second degree. In
this manner, then, as well as by the former less geometrical process, we might perceive
that the two planes of vergency for the ray-lines from an oblique plane, (determined
by (U™) or (V+), and analogous to the two less general planes of vergency consi-
dered in the sixteenth number,) intersect the oblique plane in the same two lines in
which that plane intersects a certain cone of the second degree, through the centre of
which cone it passes ; and that the planes of vergency are imaginary when the oblique
plane does not intersect this cone. We may remark that the intersection of the
oblique plane with the cone, or of a near final ray-line from the oblique plane with
the given final ray-line, is impossible, when the constant of deviation corresponding
to the oblique plane is greater (abstracting from its sign) than the semidifference of
the extreme deflexures of the auxiliary paraboloid: for then the compensation of the
two deviations év, dy, is impossible, the near ray-line always deviating more from the
corresponding deflected line of the auxiliary paraboloid, than this deflected line from
On Systems of Rays. 97
the corresponding plane of deflexure. And when the compensation and therefore
the intersection becomes possible, by the constant of deviation being less than the
semidifference of the two extreme deflexures, then the two real planes of vergency of
the near final ray-lines from the oblique plane are symmetrically situated with respect
to the two rectangular planes of extreme deflexure : which latter planes may also, for
a reason already alluded to, be called the planes of extreme projection of the final
ray-lines.
Other Geometrical Illustrations of the Condition of Intersection, and of the Elements
Arrangement. Composition of Partial Deviations. Rotation round the Axis of
Curvature of a Final Ray.
20. The condition of intersection of two near final ray-lines may also be illustrated,
and might have obtained, by other geometrical considerations, on which we shall
dwell a little, because they will help to illustrate and improve the theory of the
elements of arrangement.
It was remarked, in the fourteenth number, that the general comparison of a given
luminous path (4, B), with a near path (4’', B’), +, might be decomposed into
several particular comparisons, such as the comparisons with the less general near paths
(A, B,),, (A, B;),, and others, on account of the linear form of the expressions (D")
for the variations Sa, 83, 8a’, 88’, of the extreme small cosines of direction, which form
permits us to consider separately and successively the influence of the variations of
the extreme co-ordinates and colour, or the influence of any groupes of these varia-
tions. Accordingly, by an Analysis founded on this remark, we decomposed the
general discussion of the geometrical relations of infinitely near rays into four less
general problems, which were treated of, in the fifteenth number. The applications,
in the sixteenth number, to questions respecting the mutual intersections of the final
ray-lines from the final perpendicular plane, may be considered as only illustrations
and corollaries of the third of those four problems: but the questions since discussed,
respecting the ray-lines from an oblique plane, require a combination of the solutions
of the second and third of the four problems, and furnish therefore, an example of
the Synthesis of those elements of arrangement of near rays, to which the former
Analysis had conducted. This synthesis, however, has in the foregoing numbers been
itself algebraically performed, (namely, by the algebraical addition of certain partial
variations, ) although many of the results were enunciated geometrically, and com-
bined with geometrical conceptions: but a geometrical idea and method, of the Syn-
thesis of the Elements of Arrangement, may be obtained by considering, in a general
manner, the geometrical composition of partial deviations.
VOL. XVII. 2G
98 Professor Hamitton’s Third Supplement
To understand more fully the occasion of such composition, let us remember that
our theory of the Elements of Arrangement enables us to pass from the extreme
directions of a given luminous path (4, B),, to the four following sets of near
extreme directions, by the solution of the four problems considered in the fifteenth
number.
First. The extreme directions of the near path (4, B),,;,, which has the same
extreme points 4, B, but differs by chromatic dispersion.
Second. The final direction of (4, B,),, that is, of the original path prolonged at
the end, and the initial direction of (4,, B),, that is, of the same path prolonged at
the beginning ; these near extreme directions being in general affected by curvature.
Third. The final direction of the path (4, B,),, and the initial direction of
(A,, B),; the small lines 24,, BB,, being perpendicular to the given path at its
extremities.
Fourth. The initial direction of (4, B,),, and the final direction of (4,, B),.
We saw also that the initial direction of (4, B,), and the final direction of (4,, B),
do not differ from the corresponding extreme directions of the original luminous
path.
If then we would apply this theory to determine the final direction of an arbitrary
near path (4’, B’),+3,, we have to consider and compound, algebraically or geome-
trically, the following partial deviations from the given final direction of the given
path (4, B),: first, the chromatic deviation of the final direction of the near path
(A, B),+;, from that given final direction; second, the deviation of curvature of the
final direction of (4, B.),; third, the final deviation of the path (4, B,),, to be
determined by the theory of the final guiding paraboloid ; and fourth, the deviation
of the final direction of (4,, B),, to be found by the theory of the guiding planes
and conjugate guiding axes. A similar composition of four partial deviations is
required for the determination of the initial direction of the same arbitrary near path
(4, B tax:
Now to compound in a geometrical manner the four preceding partial deviations
of the final ray-line, we may proceed as follows. We may construct each partial
deviation, by drawing the deviated final ray-line corresponding, or a line parallel
thereto, through the given final point B; the line thus drawn will differ little in
direction from the given final ray-line or axis of z, and if we take its length equal to
unity, then its small projection on the given final plane of xy, to which it is nearly
perpendicular, will measure the magnitude and will indicate the direction of the —
deviation : and if we compound all these projections according to the usual geometri-
cal rule of composition of forces, the result will be the projection of the equal line
which represents in direction the resultant or total deviation. And similarly we
may compound the four partial deviations of a near initial ray-line.
On Systems of Rays. 99
The geometrical synthesis of the partial deviations may also be performed in other
ways. or example, we may consider each partial deviation as arising from a partial
or component rotation, and we may compound these several rotations by the geo-
metrical methods proper for such composition.
In particular, we may compound the final deviation of curvature with any of the
other partial deviations, by making the deviated ray-line, obtained without considering
the final curvature of the ray, revolve through an infinitely small angle round the
axis of final curvature, that is, round the axis of the final osculating circle of the
given final ray. By this rotation, the projection B, of a near final point B’ on the
final perpendicular plane, will be brought into the position B’; and, by the same
rotation, the near final ray-line, which had been obtained by abstracting from the final
curvature, and by considering B, as the final point, will be brought, at the same time,
into the position of the sought ray-line, which corresponds to a final point at B’.
Applying now these general principles to the particular question respecting the
condition of intersection of two near final ray-lines, from two near final points B, PB,
(the colour x and the initial point 4 being considered as common and given,) we see
that if the projection B, of B’ be given, the small projecting perpendicular B’B, or
éz and therefore also the near point B’ itself may in general be determined s0 as to
satisfy the condition of intersection: for the final ray-line from B, may in general
be brought to intersect the given final ray-line, by revolving through an infinitesimal
angle round the axis of curvature of the given final ray. We see also that the angular
quantity of rotation and therefore the length z= B,B’ depends on the position of
the projection B,, that is, on the co-ordinates dr, 8y ; and therefore that there must
be some determined surface as the locus of the near final point B’, when the final
ray-line from that point is supposed to intersect the given final ray-line.
To investigate the form of this locus, by the help of the foregoing geometrical
conceptions, we may observe that the only point, on the near ray-line from B;, which
is brought by the supposed rotation to meet the given final ray-line, is the point con-
tained in the final plane of curvature of the given final ray ; and that if we call this
point where the ray-line from B, intersects the given plane of curvature the point P,
the angle of rotation required is the angle between the line BP and the given final
ray-line ; because the same infinitesimal rotation which brings the near ray-line from
B,, that is, the line B,P, into a new position in which it intersects the given final
ray-line, brings also the line BP into the position of the given final ray-line itself.
Translating now these geometrical results into algebraical language, and taking the
given final plane of curvature for the plane of xz, so as to satisfy the condition (R"),
we find the following co-ordinates of the point P of intersection of this plane of cur-
_vature with the ray-line (V”°) from B,,
100 Professor Hamitton’s Third Supplement
oy. S ox + gy ) —ey
Se ee eg!
= Ou Bats, oy oe 7 <S oy
so that the angle between the line BP which connects this with the origin of
co-ordinates, and the given final ray-line or axis of 2, is
Bee ey ee ee ws
a= (~L=) gyde. (be + ay) (3 an + =y) (X")
and this being equal to the infinitesimal angle of rotation, that is, to the small line 8z
or BB’ multiplied by = or by the final curvature of the given ray taken with
its proper sign, we have the following equation for the locus of the near point B’,
when the condition of intersection is to be satisfied,
ot 5. a5 se (2 ar +P y) — (Se += Say): (¥")
oz ox
which is, accordingly, the equation of the former conical locus (/?"), only simplified
by the condition (#"), arising from a choice of co-ordinates. Without making that
choice, we might easily have deduced in a similar manner the equation (#2), under
the form
3 *8 8
a aw (2 aw +22 ay) — ay(S ax +5 ay ) a
aH op ;
in which each member is an expression for the infinitesimal angle of rotation divided
by the curvature of the ray.
Another way of applying the foregoing geometrical pritieiplan to investigate the
condition of intersection of two near final ray-lines, is to consider the infinitesimal
angle by which the ray-line from B, deviates from the plane containing the given final
ray-line and the near point B,. This angular deviation is expressed by the numera-
tor of the fraction (Z"*), divided by 8, that is, divided by the small line BB,; and
the denominator of the same fraction (Z"), divided also by 8/, is equal to the final
curvature of the ray multiplied by the sine of the inclination of the line & to the
radius of this final curvature: and hence it is easy to see, by geometrical considera-
tions, that the fraction in the second number of (Z") is equal to the infinitesimal angle
of rotation required for destroying the last mentioned deviation, divided by the curva-
ture of the ray, and therefore equal to the ordinate dz of the sought locus of the near
point B’, as expressed by the first member. We might therefore easily have obtained,
by calculations founded on this other geometrical view, the same condition of inter-
section as before, and the same conical locus.
On Systems of Rays. 101
Relations between the Elements of Arrangement, depending only on the Extreme
Points, Directions, and Colour, of a Given Luminous Path, and on the Extreme
Media. Ina Final Uniform Medium, Ordinary or Extraordinary, the two Planes
of Vergency are Conjugate Planes of Deflexure of any Surface of a certain class,
determined by the Final Medium ; and also of a certain Analogous Surface,
determined by the whole combination. Relations between the Visible Magnitudes
and Distortions of any two small objects, viewed from each other through any
Optical Combination. Interchangeable Eye-axes and Olyect-axves of Distortion.
Planes of No Distortion.
21. It was shown in the fourteenth number, and the result has since been developed
in detail, that the general geometrical relations between the extreme directions of
infinitely near rays are determined by the co-efficients of the linear variations éa, 83, dy,
da’, 68’, dy’, of the six marks of extreme direction, considered as functions of the six
extreme co-ordinates and of the colour; and that, between the forty-two general
coefficients of these six linear variations, there exist eighteen general relations, leaving
only twenty-four coefficients arbitrary, if we suppose for simplicity that the final and
initial co-ordinates are referred to rectangular axes. But besides these eighteen
general relations which are common to all optical combinations, there arise certain
other relations between the coefficients, when the extreme media are considered as
given, and when the extreme points, directions, and colour, of any one Juminous path,
are also supposed to be known. For, if we then employ the general equations (4°),
we may consider the extreme medium functions v, v’, and their partial differentials,
as known, and may deduce general expressions for the coefficients before mentioned
of the linear variations of the extreme cosines of direction, involving only, as
unknown quantities, twenty-seven partial differentials of the second order of the cha-
racteristic function 7, namely, all of this order, which are not relative to the variation
of colour only ; but these twenty-seven are connected by the fourteen general rela-
tions (Q) (U) (X) (Y), deduced in the third number, of which however only thir-
teen are distinct, because the two systems (U) (Y) conduct both to one common
equation (D) ; there remain, therefore, as independent quantities, only fourteen of
the partial differentials of V7, in the general expressions of those twenty-four coeffi-
cients of the linear variations of the extreme direction-cosines, which had before
been considered as independent, when the extreme medium-functions v, v’ were sup-
posed unknown and arbitrary : and if we eliminate the fourteen independent differen-
tials of V” between the expressions of these twenty-four coefficients, we shall obtain
ten general relations, between the elements of arrangement of infinitely near rays,
VOL. XVII. 2D
102 Professor Hamitton’s Third Supplement
involving only the extreme points, directions, and colour, of the given luminous path,
and the properties of the extreme media.
The simplest manner of obtaining these ten general relations, is to eliminate the
fourteen differentials of 7 which enter into the twenty-four expressions, deducible
from (C°), from the twenty-four coefficients (D"). The ten relations thus obtained,
may be arranged in three different groupes: the first groupe containing the two
following
A®
cv da ov 6B Sv _ wv Sr
SadpB Sz 1 9p? dz | 8882 —
and two others similar to these, but with accented or initial symbols ; the second
groupe containing the final relation
Sv da dv eB ev sv oa Sv OB ov (B")
Sa2 Sy + Sad8 dy | Sady dadp Oe Se 8x” §BSx’
and a similar initial relation ; and the third groupe comprising the four following,
Sv da Sv OB oul da yc 6p
Sa Sc’ * Sad de’ | Sa? Ox "eB ae
Sv Ba, 8 8B Sal Se BB _
Sa? By’ | SaB dy * Sa'dp’ Ox | S82 Ba is
Bo Ba, 7 8B, Bi BY BW 8B cc)
Sad Sx" | 32 Oe | Sa? Sy | Sade’ dy
cv 6a nee ep set Sv! oa’ mi Sv’ op’ =
3adB dy’ * 3G" dy’ | dads’ dy | SB dy
The two first relations of the first groupe, namely, the equations (4), are equi-
valent to the two first differential equations (0) of a curved ray, and express that the
magnitude and plane of final curvature of a luminous path, in a final variable medium,
are determined, in general, by the properties of that medium, the colour of the light,
the position of the final point, and the direction of the final tangent. And the two
other relations of the same groupe express, in like manner, a dependence of the
initial magnitude and plane of curvature of a luminous path, on the initial medium,
colour, point, and tangent.
The equation (B"), belonging to the second groupe, is a relation between the four
coefficients 7 ; = e = , and therefore a relation between the guiding paraboloid
and constant of deviation Pi the final ray-lines, depending on the final medium,
colour, point, and tangent. And similarly the other equation of the second sroune
expresses an analogous relation for the initial medium.
On Systems of Rays. 103
In the extensive case of a final uniform medium, the equation (B’) reduces itself
to the following,
oi ov pa ev ge a) ev 6B ‘
~ oa? dy cach \dy de 6p’ dx’
and, in the same case, the general conical locus of the second degree (#"'), connected
with the condition of intersection of the final ray-lines, reduces itself to two real or
(D")
imaginary planes of vergency, represented by the quadratic
= “ oy” + (=) oudy aE Sa’, (E*)
and coinciding with the two planes of vergency considered in the sixteenth number :
attending therefore to (C™), the relation (D”) may be geometrically enunciated by
saying, that in a final uniform medium the two planes of vergency are conjugate
planes of deflexure of any surface of a certain class determined by the nature of
the medium, namely, that class for which, at the origin of co-ordinates,
oz 28 Ow Oe COP: O82 BS
ae tet = € 15
Oa : da” dxoy . Sad” dy? a SB ” Le
and therefore nearly, for points near to this origin,
r ov Oy o2u 2
z=per+yrts (2 +2ay sap tS im (G")
ape
the given final ray or axis of z being taken as the axis of deflexion, and the constants
Ps 4, », being arbitrary. ‘This relation may be still farther simplified, by choosing the
arbitrary constants as follows,
; 1 ov af 1 &v aoe! *
a ie a0 Soe (H")
Z being any constant ordinate ; for then, (by the theory of the characteristic function
V’ for a single uniform medium, which was given in the tenth number,) the surface
(G°) acquires a simple optical property, and becomes, in the final uniform medium,
the approximate locus of the points a, y, z, for which
P, = fvds= Ve =const., ad")
the integral 1” = /vds being taken here, in the positive direction, along the variable
line , from the fixed point 0, 0, Z, to the variable point x, y, z, or from the latter to
the former, according as Z is negative or positive. And though the equation (4G'”)
is only an approximate representation of the medium-surface (J'’), which was called
in the First Supplement a spheroid of constant action, and which is in the undulatory
theory a curved wave propagated from or to a point in the final medium, yet since
the equation (G’) gives a correct development of the ordinate z of this surface as far
as terms of the second dimension inclusive, when the constants are determined by
104 Professor Hamitton’s Third Supplement
(H"), the conclusion respecting the deflexures applies rigorously to the surface (7°);
and the two planes of vergency (HE), in a final uniform medium, are conjugate
planes of deflexure of the spheroid or wave (I). We shall soon resume this result,
and endeavour to illustrate and extend it. In the mean time we may remark that the
same planes of vergency (H") are also conjugate planes of deflexure of a certain
analogous surface, determined by the whole combination, and not merely by the final
uniform medium, namely, the surface (D"), for which
JSvds (= V )= const., (K")
the integral being here extended to the whole luminous path, and being therefore
equal to the characteristic function V” of the whole optical combination ; an additional
property of the planes of vergency, which is proved by the following relation, analo-
gous to (D"), and deducible from (C*) or (G@*),
BF 3 EY (38 _Ba) _ BY 8
= 3a oy | Sxdy Sy Sy de'
Sy be ce
Finally, with respect to the four remaining equations, of the third groupe (C), it
is evident that they express certain general relations depending on the extreme media,
between the coefficients which determine the guiding planes and conjugate guiding
axes, for the final and initial ray-lines. In the extensive case of extreme ordinary
media, they reduce themselves to the four following, which may also be deduced
from (F'*),
(M")
nu, « being the indices of the media; and they conduct to some simple conclusions,
respecting the general relations between the visible magnitudes and distortions of a
small plane object, placed alternately at each end of any given luminous path, and
viewed from the other end, through any ordinary or extraordinary combination : at
least so far as we suppose these distortions and magnitudes to be measured by the
shape and size of the initial and final ray-cones. For then the conjugate guiding
axes, initial and final, perpendicular to the given path at its extremities, and deter-
mined in the fifteenth number, may be called the eye-axes and olject-axes of distor-
tion, for a small object placed in the final perpendicular plane, and viewed from the
initial point ; and if we take these for the axes of initial and final co-ordinates, so
as to have, by (X") (Y™),
ba’ 6B" 83" da’ 68’
= 0, =O, > 0, eg s
>
On Systems of Rays. 105
we shall then have also, by (Z'°), (the extreme media being supposed ordinary, and
their indices p, »’ positive, )
Se. Gg hBe) 9. 288: Bee 0 OE, qs
W) hames gay = ae = ay te ND
that is, in this case, the guiding axes for the initial ray-lines are also the guiding
axes of the same kind for the final ray-lines measured backward ; which is already
a remarkable relation, and may be enunciated by saying that the eye-axes and olyect-
axes of distortion are interchangeable, when the extreme media are ordinary : that
is, for such extreme media, the eye-axes of distortion become olject-axes, and the
object-axes become eye-axes, when the object is removed from the final to the initial
perpendicular plane, and is viewed from the final instead of the initial point. And
while the equations of the fifteenth number,
Jaf SOs ry OB u
Uae ee ov, y=z sy oY; (A")
represent the initial visual ray-line corresponding to a final visible point B’ which has
for co-ordinates da, dy, dz, the following other equations,
Rete eae, Crete ein a Is
aie eS ov, y=—2z. Rope? @)
will represent by (J/") the final visual ray-line corresponding to an initial visible
point 4’ which has for co-ordinates da’, dy, dz’; the initial visual ray-cone corres-
ponding to any small object
oy =f (8x) (G")
in the final perpendicular plane is therefore represented by the equation
6@=-EE).
and the final visual ray-cone corresponding to any smal] object
ef =f (82) (P")
in the initial perpendicular plane is represented by the following analogous equation
CFATTOT 6) in fae May GRAB cal Raa Viral 8 15
are al =ah (= we (= ) ) a (Q")
if therefore these two small objects, (G") (P"), at the ends of a given luminous
path, be equal and similar and similarly placed with respect to the conjugate axes of
distortion, that is, if the final and initial functions ff’ be the same, and if we cut the
two ray-cones (H") (@Q"°) respectively by perpendicular planes having for equations
i Z=wh, 2=—pRh, Ci)
in which # is any constant length, while », »’ are the same constant indices as before
VOL. XVII. 2E
106 Professor Hamiiton’s Third Supplement
of the extreme ordinary media, the two perpendicular sections thus obtained will be
equal and similar to each other ; and if, besides, we put, by ( Y"),
See = cos. G, (S*) \
( & being by (#"") the inclination of an initial guiding plane to the plane perpendicular
to the given initial ray-line,) and determine also the arbitrary quantity # as follows,
wl 8a’ —1_—-!1 oa —]
a aay Ge) = i lecaliovi2 (T”)
the perpendicular sections of the initial and final ray-cones may then be represented
as follows,
, : , ci :
U2, ChyiG@y = = G) - (U0)
and
Y éa\ —1 15
y=Ccos, Gafi();) 2 = (3) : CV?)
the visible distortions therefore, depending on the inclination G, are the same for any
two small equal objects, thus perpendicularly and similarly placed at the ends of any
given luminous path, and viewed from each other along that path, through any
optical combination.
The distortion here considered will in general change, if the object at either end of
the given luminous path be made to revolve in the perpendicular plane at that end, so
as to change its position with respect to the axes of distortion, For example, if the
object be a small right-angled triangle in the final perpendicular plane, having the
summit of the right angle at the given final pomt B of the path, we know, by the
theory given in the fifteenth number, that the right angle will appear right to an eye
placed at the initial point 4, when the rectangular directions of its sides $4, $25
coincide with those of the final guiding axes, or object axes of distortion ; but that
otherwise the right angle ¢’,—¢', will appear acute or obtuse, its apparent magnitude
¢2— 4 being determined by the formula
T sl 7 .
= tan. | ¢:—¢ —-) = —_—__ Sin: 264, wae) .
CaaS ae
ba Oy
which may, by (S"), be reduced to the following,
— tan. ( po— bi =) = dsin. G. tan. G. sin. 2 ¢). (X*)
The law of change of the distortion, corresponding to a rotation in the final perpen-
dicular plane, may also be deduced from the theory of the guiding planes, explained
in the fifteenth number.
“ww
On Systems of Rays. 107
The distortion will also change, if the small plane object be removed into an oblique
instead of a perpendicular plane. In this case we may still employ the equations
(A") (0) for the initial and final ray-lines, and may still represent the initial and
final ray-cones by the equations (47") (Q") ; but we are now to consider the equa-
tions (@") (P"), for the final and initial objects, as representing the projections of
those objects on the extreme perpendicular planes ; or rather the projecting cylinders,
which contain the objects, and which determine their visible magnitudes and distor-
tions, by determining the connected ray-cones. For example, the equation (C"') may
be considered as representing a final elliptic cylinder, of which any section near the
final point B of the given luminous path will correspond to an initial circular ray-
cone (B"), and will therefore appear a circle to an eye placed at the initial point 4 ;
while on the other hand we may regard the equation (D") as respecting a final
circular cylinder, such that any section of this cylinder, near the final point B, will
give an initial elliptic ray-cone ("'), and will appear an ellipse at 4. And as
the elliptic ray-cone (#") conducted, by its circular sections, to the guiding planes
(F") for the initial ray-lines, so, for small plane final objects, the planes
z=+z2 tan. G, CY?)
namely, by (S"), the planes of circular section of the elliptic cylinder (C™), are
planes of no distortion ; in such a manner that not only, by what has been said, the
circular sections themselves in these two planes appear each circular, but every other
small final object in either of the same two planes appears with its proper shape to an
_eye placed at the initial point A of the given luminous path ; the angular magnitude
of the final object thus placed, being the same as if it were viewed perpendicularly by
straight rays, bik ae refracting or reflecting surface or medium interposed, from
ay ) —1_ In like manner, the planes
a final distance = (
Zee. tan. Cr (Z')
which are the planes of circular section of an analogous initial elliptic cylinder, are
initial planes of no distortion, of the same kind as the final planes (Y'); since any
small initial object, placed in either of these two initial planes (Z"), and viewed from
the final point B of the given luminous path, will appear with its proper shape, and
with the same angular magnitude as if it were viewed directly from an initial distance
i] SBE nl efoo y= 1
es SUE ae
This theory of the planes of no distortion gives a simple determination of the
visible shape and size of any small object placed in any manner near either end of a
given luminous path ; since we have only to project the object on one of the two
planes of no distortion at that end, by lines parallel to the corresponding extreme
108 Professor Hamitton’s Third Supplement
direction of the path, and then to suppose this projection viewed directly from a final
or initial distance determined as above. We might, for example, deduce from this
theory the property of the guiding planes, the circular and elliptic appearances (B")
(E") of the ellipse and circle (C'") (D"), and the acute or obtuse appearance (X'"’)
of a right angle in the final perpendicular plane, when the directions of the sides of
this angle are different from those of the object-axes of distortion. And the relations
(M”) for extreme ordinary media may be expressed by the following theorems:
first, that the angle (2G) between the final pair of planes of no distortion (Y"’),
is equal to that between the initial pair (Z"); second, the visible angular mag-
nitudes of any small and equal linear objects in final and initial planes of no dis-
tortion, are proportional to the indices of the final and initial media, when the
objects are viewed along a given luminous path, from the initial and final points ; and
third, the two intersection-lines of the two pairs of planes of no distortion coincide
each with the visible direction of the other, when viewed along the path.
Calculation of the Elements of Arrangement, for Arbitrary Axes of Co-ordinates.
22. In the foregoing formule for the elements of arrangement of near rays, we
have chosen for simplicity the final and initial points of a given luminous path, as the
respective origins of two sets of rectangular co-ordinates, final and initial, and we
have made the final and initial ray-lines, or tangents to the given path, the axes of
z and 2’; a choice of co-ordinates which had the convenience of reducing to zero
eighteen of the forty-two general coefficients in the expressions of da, 8B, dy, da’, 88,87,
as linear functions of 8x, dy, dz, da’, dy’, 62’, 8x. The twenty-four remaining coeffi-
cients (D*) may however be easily deduced, by the methods already established, and
by the partial differential coefficients of the characteristic and related funtions, from
other systems of final and initial co-ordinates, for example, from any other rectangular
sets of final and initial axes.
In effecting this deduction, it will be useful to distinguish by lower accents the par-
ticular co-ordinates and cosines of direction, which enter into the expressions (D*),
and are referred to particular axes of the kind already described ; and then we may
connect these particular co-ordinates and cosines with the more general analogous
quantities « y za’ yz aB ya fy, by the formule of transformation given in the
thirteenth number, which may easily be shown to extend to the case of two distinct
rectangular sets of given or unaccented co-ordinates. In this manner the axes of
z, and 2’, considered in the thirteenth number, become the final and initial ray-lines,
and we have, by (4°),
On Systems of Rays. 109
OL =NXe, 84, +4y, Sy, +a 82,
SY =Yr, 8%, +Yy, Sy, +B 8z,,
Sz=z, da, +2 dy,+y (A")
Sa =2'n/ 8a) +a'y: Sy’ +a 82",
by =y'r) 8; +4; By, +B 8z/,
82 =2'z' 8a) +2y) Sy) +7 8z;,
because
Bz =a, Yx, =P, 2s, =75 Be
we=a, y'x=P, Z2;=7 5 oe
we have also
a —On0,—0; La ig °Y, =O} (c%)
a, =0, B,=0, y,=1, Sy, =0,
and therefore, by (£’*),
Sa =az 8a,+2y, 88,3 8a =a'n/ 8a) +2'y/ 98/5
SB=yr, 8a, +yy, 88,5 BB =y'x; ba) +r; 83/5 (D")
Sy=22, 8a,+2y, 88,3; Sy =2'r/ 8a, +2y/ 8B;
and substituting these values (4") (D") for the twelve variations 62, dy, dz, 6x’, dy’, dz,
8a, 8, Sy, 8a’, 38, Sy’, in the general linear relations (4°) between these twelve varia-
tions and the variation of colour 8y, or in any other linear relations of the same kind,
deduced from the characteristic and related functions, and referred to arbitrary rec-
tangular co-ordinates, we shall easily discover the particular dependence, of the form
(D*), of 8a,, 88,, on dx, dy,, 8z,, da’, dy/, 8x, and of da/, §3/, on dx, dy,, dx, dy/,
oz), ox.
We seem, by this transformation, to introduce twelve arbitrary cosines or coefii-
cients, namely,
*’ LA s / , ‘y , ‘ ’ / se
Ur,5 Yr,» Fx,5 Vy,» Yy,> %y,» Vas Y v/s Fay Vy/s Yy/s Zy/ 5
but these twelve coefficients are connected by ten relations, arising from the rectangu-
larity of each of the four sets of co-ordinates, and from the given directions of the
semiaxes of z and z’; so that there remain only two arbitrary quantities, correspond-
ing to the arbitrary planes of x, z,, x) z/, of which planes we often, lately, disposed
at pleasure, so as to make them coincide with certain given planes of curvature, or
otherwise to simplify the recent geometrical discussions. Thus, although we may
assign to the semiaxis of xv, any position in the given final plane perpendicular to the
VOL. XVII. 2F
110 Professor HamiLtTon’s Third Supplement
luminous path, and therefore may assign to its cosines of direction, %z,, yz,, Z,, any
values consistent with the first equation (B*), namely,
2," +Yzx,” ar @7,°= 1,
and with the following
ats, + Bye, +2, =0, (E")
yet when the axis of xz, has been so assumed, the perpendicular axis of y in the final
perpendicular plane is determined, and we have
Ty, = +(p Zr, —VYz, )s
Yu, = + (y Tz, — 42x, ); (F")
%y, = +(ayr, —B2,),
the upper or lower signs being here obliged to accompany each other : and similarly
for the initial axes of x/ and y’.
The characteristic and related functions give immediately, by their partial differen-
tials of the first order, the dependence of the quantities which we have denoted by
6, T) vy) ©, 7, v, rather than that of a, B, y, a, 8, 7, on the extreme co-ordinates and
the colour; and therefore the same functions give immediately, by their partial
differentials of the second order, the variations 8, 87, dv, 0, é, dv, rather than
Sa, 8B, dy, Sa, 88, Sy, in terms of dx, dy, dz, 6a’, dy’, dz, dy. But we can easily
deduce the variations of a B ya (3 y from those of or vo'r' v' and of ayza' yz x
by differentiating the relations
ee ee
Nh re ap 7 = yy?
ov Aete EL ' ou’
a. Ul
o= 3a , T= ip" = Sy ’
which have often been employed already in the present Supplement ; for thus we
obtain
= “4 + 55 2B + oe ES Se
ht 2B + Sp oy ee id
oe * BF share :
on ba + Sr OB + ze fat U ee.
ony © + 595 + on aan,
On Systems of Rays. 111
5, ; referring; as in former numbers, to the variations of x, y, z, x, and 8’ to those of
a’, y', %, x: and hence we have, by some symmetric eliminations,
Ray Mer Es ae oy oe e-( B85)
v'da = (2 yy
op ~& Sal ~ Sad “ §B/ ~ dy8a “Sy
v'§8 = Gr +e) ( er —8, 55) eat sy—8, ys Lay 28,2"),
vty = (Sr 4R) (H- =8,5) yp ( 8,2) aap (ar-2 a
vids (Fee + Se) ( 86-8 57) ~arag( B= 2 ia) “1
was (Sa + Fe) ( BB 5) ap B85) grag Be —8 S),
vy = ot + 5gu (2-2 oe )- — sg (et —8 oY a say (8! =3) =),
v' having the iy (L*), and v"" ine erplogaus meaning
Sv! 2 So Se’ ow’ 2
Sea ORE RONAN | (aa ys
= ha? = a + 3p" = <(ans my) + a we Gerd OY)
We ae also deduce the variations of a B y a’ B’ y' from those of ctuad ru Hy z
x’ y' z x, by differentiating the equations (J) of the second number, and by employ-
ing the functions Q, ©’, instead of v, v
(H"’)
The general Linear Expressions for the Arrangement of Near Rays, fail at a
Point of Vergency. Determination of these Points, and of their Loci, the
Caustic Surfaces, in a Straight or Curved System, by the Methods of the present
Supplement.
23. We have hitherto supposed that the infinitesimal or limiting expressions of the
variations of the extreme cosines of direction of a luminous path, are linear functions
of the variations of the extreme co-ordinates and colour. But although this supposi-
tion is in general true, it admits of an important and extensive exception ; for the
linear form becomes inapplicable when the given luminous path (4, B),, with which
other near paths are to be compared, is intersected in its initial and final points 4, B,
by another path infinitely near, and having the same colour y: since then the extreme
directions may undergo certain infinitesimal variations, while the extreme positions
A, B, and the colour x, remain unaltered. It is therefore an important general
problem of mathematical optics, to determine, for any proposed optical combination,
the relations between the extreme co-ordinates and the colour of a luminous path
112 Professor Hamitton’s Third Supplement
which is intersected in its extreme points by another infinitely near path of the same
colour. This general problem, of which the solution includes the general theory of
the caustic surfaces touched by the straight or curved rays of any proposed optical
system, may easily be resolved by the methods of the present Supplement.
In applying these methods to the present question, we are to differentiate the
general equations which connect the extreme directions with the extreme positions
and colour, by the partial differential coefficients of the first order of the character-
istic and related functions, and then to suppress the variations of xy za’ yz y.
And of the partial differential coefficients of the second order, introduced by such
differentiation, it is easy to see by (4°) that those of the characteristic function V,
or at least some of them, are infinite in the present research: it is therefore
advantageous here to employ one of the auxiliary functions 7, T, combined if
necessary with the functions v, v’, or Q, ©’, which express by their form the properties
of the extreme media.
Thus, when the final medium is uniform, and therefore the final rays straight, we
may conveniently employ the following equations, which involve the coefficients of
the functions V7, ©, and were established in the sixth number,
3W Pye} sw FY] sw rye) ”
taytV yy arth yo tH atl ss: (W*)
Differentiating these equations with respect to «7 v as the only variables, and sup-
pressing the var iation of the first order of V7, as well as those of yz a’ y' z' x, we
obtain
OW cQ ow 7 80
o=(5, + rae Sa ao +Ge+ Ve) Su, 7
ow yee E Pay
0= (55,7 V 3,5.) 8+ _ “) 2 (K")
ow OW aw W
iss =(ae + Vo) ie + Se Sho. Vo 20. or +
and hence, by a symmetric Sanat 6 te and ies the forms of /V, Q,
rr + en) Ge = ee ee Lae
a aM may 2)
V su
Ww yko 1
¢ Geter 2a.) Co. zay_ -Ge+r oe ee
which is a form of the condition required, for the final and initial intersections of two
near luminous paths, of any common colour, the final medium being uniform. The
condition (L") is quadratic with respect to /’, and determines, for any final system of
+
On Systems of Rays. 113
straight rays, corresponding to any given luminous or initial point 4, and to any
given colour x, two real or imaginary points of vergency B,, B,, on any one straight
final ray, that is, two points in which this ray is intersected by infinitely near rays of
the same final system ; and the joint equation in w y z, (involving also a’ 7/ 2’ y as
parameters, ) of the two caustic surfaces which are touched by all the final rays and
are the loci of the points of vergency, may be obtained by eliminating o + v between
the equations (//”*) and the quadratic ("): which quadratic, by the homogeneity
of the functions JV and Q +1, may be put under the following simpler form,
(ou =) (ee ae) Z ae xO ne A
get! xe) oe + ge) —\eae +” sc5 (M")
and admits of several other transformations. When V” has either of the two values
determined by this quadratic, that is, when the final point B of the luminous path has
any position B, or B, on either of the two caustic surfaces, then the equations
deduced from (/V’*) by differentiating with respect to x y 2 as well as oz v, namely,
ae poe Sern ae) U4
or — > (ode +7dy +v0z) = 8 3, + V3~ ;
Saigae wie ow 8a
ey =o (coax + 7réy + vdz) = ae Be ais ’ (N")
Bye) : AY Aye)
8z Se (oot + rey +vdz) = os + Vs~ ;
conduct to a linear relation between da, 8y, 82, which may be put under several forms,
for example under the following,
1 6Q 1 6Q
X fee -F tater rita) bar dy ae (ode + rly +82) b
=" fe = (oda + dy +82) } > (O*)
in which we may assign to d d' X” any of the following systems of values,
ow So 6, SV aso 5, Sw “0
LES Ss “A Aa ek er a rE
ao ee Pp BO ye Oy i
Second A= Sear t Ve » A=, + V se? * =z3, + V ss, a CE)
eM ie: 8M a BO BM PO
fi SEPT aati i 7 8rieatad odelat. T Balae So
and it is easy to see that the linear relation thus deduced, between 6a, dy, dz, is the
differential equation, or equation of the tangent plane, of the caustic surface at the
point of vergency x y 2. The same linear equation represents also the plane of
VOL. XVII. 26
114 Professor Hamitton’s Third Supplement
vergency, or the tangent plane to the developable pencil of straight rays, correspond-
ing to the other or conjugate point of vergency on the given final ray.
When the final medium is variable, the three first equations (#7'), namely,
37 ww
aT I ee Sr Paar out?
are to be differentiated with respect to o, 7, v; and thus we obtain
ow 3 ew ow
ee ti a —— a dadu v0,
SV. SW. SW. +
we eee (Q
ow ow ow
Saou oo + Srdu aa due a
and consequently, by elimination,
ew ew ew, kW ew ew _ ew ey ew ea ewe Hy Re
Sai BF Sul << Batr-Orlu bods Se Sey) Wa aba? 1 but Gear? = ae)
this equation, therefore, (which may be put under other forms,) takes the place, when
the final medium is variable, of the quadratic (Z") for a final uniform medium ; and
if we eliminate from it « z v by (47), it will give, for any proposed initial point and
colour, the equation of the single or multiple caustic surface, touched by the curved
rays of the corresponding final system.
The auxiliary function Z’ may also be employed for the case of curved rays, but it
is chiefly useful when both the extreme media are uniform. In that case the extreme
portions of a luminous path are straight, and we may employ for these extreme
straight portions the equations (S*) under the form
oS OSE, os ; oS ae
=.= gos i Game 8 Sago. ane snl (S")
in which we haye put, for abridgment,
S= T-2zv+ev, (Ee)
and in which we consider v as a function of o, 7, x3 v as a function of o’,7, ¥; Tas
a function of o, 7, 0,7, <3 and S as a function of 2, 2’, o, 7, o', 7, x. Differen-
tiating these equations (S") with respect to o, 7, o', r', we find that if the extreme
straight portions, ordinary or extraordinary, of two infinitely near paths of light of
the same colour, intersect in an initial point 2’ y' 2', and in a final point x y 2, the
final and initial variations 86, $r, 8’, 8r', and the final and initial ordinates of inter-
section 2, 2’, must satisfy the four following conditions,
On Systems of Rays.
115
eS. ss BOS | oS Best,
OF 54) tise ebindabe! ads Deer?
25 2,8)
Lee aes yg 7 So + oe or’,
Sadr or? Orda Orér (U")
os O28 SS. nce OS 4,
OS Sit oak Oa Dear
es . | 8S sal a) gigs
OF ya Oo tas Or t+ arate + oe OT 3
which give, by eliminating between the two first,
2 2 2 325
NE gaan te Jerk ew nay aL
dada’ Srér’ dadr’ brdo’ Sadr dcdr’ da" Or or’ Ore Oadr Oadr orér v")
(ee &S oS n= Ce es Laie ee eS 8S BS). (
Oooo oror’ oabr’ broa’ ck da* Ordo’ dedr Sado" Ooor Se 32) =) 2
and therefore, by substituting these values of 80’, dr’
, in the two last,
ona {28 (25.88, BS 9S), BS (ATES _ BS BS)
30°? \Sa8r daédr’ O02 Srér’ + 3y8r' 60% Srda’ dadr ae
os oS Fs es omy
"Sado" \Sado’ Srdr’ Sadr’ Srdo"
gc ee es &S &S &S es 28 8S _ &s ee
S02 \ Or? Badr’ Sadr Srdr’ adr’ \daér Sréc’ = Sr? Sarda’
Ses (S&S &S Ss os
tay Geoas Bae naa ae)
onan f PS (88 BS ESBS) BS (BS ES BS BS)
oor \dcdr Sadr 8a brér’ 8r% \8c2 Sr8a" Sadr Sada’
Ses (2S S&S 8&8 &s
me as ba ae oa
eer os (= es &s ay tiie (eee a)
30’Sr’ \Sr2 Sadr’ Sadr Srdr’ Or’? \8cdr Srdc’ br? Sado’
O28 / & Ss es & 3
Ted pee
so that by a new elimination we obtain, between the
final and initial ordinates 2, 2’,
the following equation, which, by the form of S, is quadratic with respect to each
ordinate separately, and involves the product of their squares :
= (55 5 - (ES) ) (Ee ge (Se) ) +
328 es BS SS Sg Ss
2
-orle Ge) - 2 ee?
6r
5.2
2
Oadr’ orda
be Bs
dae’ Stor’
828 yt ;
6067’
(
(
)
116 Professor Hamitton’s Third Supplement
2 os es SS FS oS e oS os =.) oS 62s a
2 S78r Vo" Std’ Sy8v Sad \Sode’ Srv * Sadr Sr80) TO Sede" Sadr’
os es (ay es SS ss a os (23 .
or? 60° \drdo" Sadr dadc’ Srdc’ = Sr?_—«\Sado" ‘
(X")
When the point of intersection of the infinitely near initial rays removes to an infinite
distance, this equation reduces itself to the following,
es Sg \2
US Se BSS
ST sy /8 s eT Sy \?
-@f--8) 07 -B)-GE-)
and when in like manner the two infinitely near final rays become parallel it gives the
following quadratic to determine the two corresponding positions of the point of initial
intersection, :
es ss os 2
Be or ea ieee)
CoO oD eE
The caustic surfaces of straight systems, ordinary or extraordinary, were determined
in the First Supplement: but it seemed useful to resume the subject in a more general
manner here, and to treat it by the new methods of the present memoir.
Connexion of the Conditions of Initial and Final Intersection of two Near Paths
of Light, Polygon or Curved, with the Maxima or Minima of the Time or Action-
Function V+V,= 3 fvds. Separating Planes, Transition Planes, and Transi-
tion Points, suggested by these Maxima and Minima. The Separating Planes
divide the Near Points of less from those of greater Action, and they contain the
Directions of Osculation or Intersection of the Surfaces for which V and FV, are
constant ; the Transition Planes toysh the Caustic Pencils, and the Transition
Points are on the Caustic Curves. Extreme Osculating WVaves, or Action-
Surfaces: Law of Osculation. Analogous Theorems for Sudden Reflexion or
Refraction.
24. The conditions of initial and final intersection of two near luminous paths,
have a remarkable connexion with the maxima and minima of the integral in the law
of least action, that is, with those of the characteristic function 74, or rather with
those of the sum of two such integrals or functions, which may be investigated in
the following manner.
On Systems of Rays. 117
Let A, B, C; be three successive points, at finite intervals, on one common lumi-
nous path. Let the rectangular co-ordinates of these three points be a’, 7/, 2 for 4;
x, y,2for B; ands, y, z,for C. Let V (A, B) denote the integral /vds taken
from the first point 4 to the second point B; let Y (B,C) denote the same integral,
taken from the second point B to the third point C; and similarly, let Y(4, C) be
the integral from 4 to C, which is evidently equal to the sum of the two former,
V (A, C)=V (A, B)+V (B, C), CA)
so that, if we put for abridgment
Mes, BV, V(b, ©)= Vi; (B”)
we shall have, by the continuity of the integral,
Vi (A, C)E=V+V,. (C”)
If we do not suppose that the intermediate point B is a point of sudden reflexion or
refraction, the final direction of the part (4, B) will coimcide with the initial direc-
tion of the part (B, C), and the final direction-cosines a B y of the one part will be
equal to the initial direction-cosines of the other ; considering V therefore, as usual,
as a function of wy z x’ y' z' xy, and V,as a function of x,y, 2,“ y z x, we have, by
our fundamental formula (4),
pipemediee el, 18K 280 ot OV ie Leo a) OP ey
2 Bee Se a. ge a Sele 7
Sean Gus ace? gy SG ay, ees Pope ee cp")
that is, we have
8V+8V,=0, (E")
for any infinitesimal variations of the co-ordinates x y z, and therefore, to the accuracy
of the first order,
V (A, BY+V(B, C)=V (A, B)+V(B, C)=V (4,6) (FO
B’ being any new intermediate point infinitely near to B, and the path (B’, C) being
not in general a continuation of the path (4, B’). If therefore we regard the
extreme points 4, C, as fixed, but consider the intermediate point B as variable and
as not necessarily situated on the path (4, C’), the function V+V_, or 2 fvds,
composed of the two partial and now not necessarily continuous integrals (B”), will
acquire what may be called a stationary value, when the paths (4, B) (B, C) become
continuous, that is, when the intermediate point B takes any position on the path
(A, C) from one given extreme point to the other: since then the change of this
function will be infinitely small of the second order, for any infinitely small alteration
BB, of the first order, in the position of the point B. The stationary value thus
determined, namely, / (A, C), might be called, by that customary latitude of expres-
VOL. XVII. 2H
118 Professor Hamitton’s Third Supplement
sion which leads to the received phrase of least action, a maximum or minimum of
the function V+ VY; but in order that this value should really be greater than all
the neighbouring values, or less than all, a new condition is necessary. To find this
new condition, we oh observe that the relations
eid lle dal al Soi atone
* 3 = + sca ee = ~ (4 get Bart Veal) >
BF 8 eros ALA LA a
® sy t Poy Se re eae andy + 8 o T eayse ‘)s (G")
payee OV Se OF eV, gals
* Srdz t Baraat ae 737 (4 ovdz + BS a 822 ay J
which result from the third number, give
eV SY, as \?, (VP, Seen.
8 +80 = (+ ) (se —£ as ) = Soe: 82)
5 ay ana Be nN, "
+2 (seay t gaa) (82-5 = “ az) (8 mp 2) 5 (H")
the condition of existence of a maximum or minimum, properly so called, of the
Function V + V,, is therefore,
20 dhe = Get a) yw) (et mE) a
When we have on the contrary
Q<0, (K")
the variation of the second order ®V+&V, admits of changing sign, in passing from
one set of values of dx, dy, dz to another, that is, in passing from one near point B’
to another ; and since, to the accuracy of the second order,
V (A, B’) + V (B, C)-—V (A, C)=4 (8V+EV), (L”)
we shall have the one or the other of the two following opposite inequalities
V (A, B’)+ V (B, C)> or < V (4, C), (M")
according as the near point B’ is in one or the other pair of opposite diedrate angles
formed by two separating planes P’ P” determined by the following equation
SV+8V,=0, (N"”)
which is, by (#7"’), quadratic with respect to the ratio
-_
On Systems of Rays. 119
These two separating planes P’ P” contain each the ray-line or element of the
path (A, B, C) at B ; and they divide the near points of less from those of greater
action, or those of shorter from those of longer time, when the continuous integral
V+V,=V (A, C) ts not greater than all, or less than all, the adjacent values
of the sum = fvds. ‘The directions of these planes depend on the positions of the
points 4, B, C; so that if we consider 4 and B as fixed, but suppose C to move
along the prolongation (8, C’) of the path (4, B), the separating planes P’, P’,
will in general revolve about the ray-line at B. They will even become imaginary,
when by this motion of C' the quantity @ becomes > instead of < 0, so as to satisfy
the condition of existence of a maximum or minimum of the function V+ V”,; and
in this transition from the real to the imaginary state the two separating planes P’ P”
will close up into one real transition-plane P, determined by either of the two follow-
ing equations,
eV oe ot B
oS ho ge) (8x — 82) + “Gag oo bz),
2 4 (O”)
OeiGeapih =r (ee 2 8z) + Gor
Ny
8),
while the corresponding position of the und C, which we may call by analogy a
transition-point, will satisfy the ee
a
Q=0, that is, Geet
gl OnVeona
1 yy = Goat oe =
We are now prepared to us te a remarkable connexion between the transition-
planes and transition-points to which we have been thus conducted by the considera-
tion of the maxima and the minima of the function V+ V,, and the condition of
final and initial intersection of two near luminous paths. For these conditions of
intersection may be obtained by supposing that not only the point B, having for co-
ordinates x y z, is on a given path (4, C’), so as to satisfy the equations (D"), but
that also an infinitely near pot B’, having for co-ordinates x +éx, y +éy, x +8z, is
on another path of the same colour connecting the same extreme points 4 and C, so
as to give the differential equations
ov 3V.. . 8V SV, ;
= 2 f — ete ee (Q”)
and since these last equations may be reduced, by the relations (G@"), to the forms
(0"7), we see that when the conditions of initial and final intersection of a given path
(A, B, C) with a near path (4, B’,C) are satisfied, and when we consider the initial
point A as fixed, the near intermediate point B’ must be in a transition-plane P of the
form (O"), and the final point of intersection C must be a transition-point of the form
(P"). Continuing therefore to regard the initial point A as the fixed origin of a system
120 Professor Hamitton’s Third Supplement
of luminous paths, polygon or curved, of any common colour, which undergo any num-
ber of refractions or reflexions, ordinary or extraordinary, and gradual or sudden, it
is easy to see that we may consider these paths as touching a certain set of caustic
curves, in the final state of the system, and therefore as grouped into certain sets of
consecutively intersecting paths, and as having for their loci certain corresponding
sets of ray-surfaces, which may be called caustic pencils: and that these caustic
pencils are touched by the transition planes (O"), while the transition-points
(P") are on the caustic curves, and therefore on their loci the caustic surfaces.
The transition-points are also evidently the points of consecutive intersection, or of
vergency, of the luminous paths from 4, in the final state of the system. And it is
manifest, from the foregoing remarks, that these final points of intersection are also
transition-points in the following other sense, that when the point C, in moving along
the prolongation of the path (4, 6), arrives at any one of these positions of inter-
section, the condition of existence of maximum or minimum of the function V+ V,
begins or ceases to be satisfied.
The separating planes P’ P”, have, when real, another remarkable property,
namely, that of containing the directions of mutual osculation, at the point B, of the
two action-surfaces or waves determined by the equations
V=const., V,=const.; (R”)
for these equations may be put approximately under the following forms, (when we
choose the point B for origin and the final direction of the path (4, B) for the
positive semiaxis of z, so as to have a=0, B=0, y=1,)
z=petqy +4hra’+sxy +h ty’, (s")
Z=pe+qy thre tsay they,
in which the coefficients have the following relations,
P,=P, 7%
_ 1/8¥, sv
ne ee Gases
an & Ve eV (GES)
Si aay (say ae, :
1 SV, 37
4—t= = Gr +50)
and therefore the planes
O=(r, —1) 2° +2(s,-s)ayt+(4—-by’, (U"’)
which pass through the given ray-line at the point B, and contain the directions of
osculation of the second order of the two touching surfaces (2) or (S"), are the
On Systems of Rays. 121
separating planes (VV). We might also characterise these separating planes, or
planes of osculation, as containing the directions of mutual intersection of the same
two touching surfaces for which V and VY’, are constant; or as the planes in which
the deflexures of these two surfaces are equal, the ray-line at B being made the axis
of deflexion.
The comparison of the same two waves or action-surfaces (J?) gives a new pro-
perty of the planes and points of transition; for the equations which determine a
plane and point of this kind may be put under the form
(r—r)xt+(s—s) y=, (s—s) «+(t—t)y=0, or, Sp,=Sp, oy =: (V")
they express, therefore, that when Cis a transition point, the two surfaces (#2) touch
one another not only at the point B, but in the whole extent of an infinitely small
arc contained in the transition-plane.
The point C may be called the focus of the second wave or action-surface V_,
since all the corresponding paths of light (B’, C’) are supposed to meet in it; and in
like manner the point 4 may be called the focus of the first surface V of the same
kind, since all the paths (4, B’) are supposed to diverge from A. The focus A and
the point of osculation B remaining fixed, we may change the focus C, and thereby
the directions of osculation; but there are, in general, certain extreme or limiting
positions for the osculating focus C, corresponding to extreme osculating waves or
action-surfaces V, and it is easy to show that these extreme osculating foci coincide
with the transition-points or points of vergency: and that the transition-planes or
tangent-planes of the caustic pencils contain the directions of such extreme or limiting
osculation.
These theorems of intersection and osculation include several less general theorems
of the same kind, assigned in former memoirs. It is easy also to see that they extend
to the case when the order of the points 4 B C on a luminous path is different, so
that B is not intermediate between A and C, and so that the paths (A, B) (A, B),
which go from A to the points B and B, coincide at those points with the paths
(C, B) (C, B), and not with the opposite paths (B, C) (B’, C), that is, tend from
the point C, not to it; observing only that we must then employ the difference instead
of the sum of the two integrals /vds, or of the two functions V and V,.
When the point C is on a given straight ray in a given uniform medium, we can
easily prove, by the theory of the partial differential coefficients of the second order
of the characteristic and related functions which was explained in former numbers,
that the equation (P) becomes quadratic with respect to z, or V, and assigns, in
general, two or real imaginary positions C,, C,, for the transition-point, or point of
vergency ; and that the equations (0") assign two corresponding real or imaginary
transition-planes P, P., or tangent planes of caustic pencils. And when, besides,
VOL. XVII. 21
122 Professor Haminton’s Third Supplement
the points B, C, are both in one common uniform medium, so that the paths
(B, C) (B, C) are straight, then each of the caustic pencils, or ray-surfaces,
composed of such straight paths consecutively intersecting each other and touching
one caustic curve, becomes a developable pencil, and its tangent plane becomes a
plane of vergency, of the kind considered in the sixteenth number. The relations
also between the two planes of vergency in a final uniform medium, which were
pointed out in the twenty-first number, may easily be deduced from the present more
general view and from the recent theorems of osculation ; for thus we are led to con-
sider a series of waves or action-surfaces /”,, similar and similarly placed, and deter-
mined in shape but not in size or focus by the uniform medium, and then to seek the
extreme or limiting surfaces of this set which osculate to the given surface ” at the
given point B; and since it can be shown that im general among any series of sur-
faces, similar and similarly placed, but having arbitrary magnitudes, and osculating
to a given surface at a given point, there are two extreme osculating surfaces, real
or imaginary, and that the tangents which mark the two corresponding directions of
osculation are conjugate tangents (of the kind discovered by M. Dupry) on each sur-
Sace of the osculating series, and also on the given surface, it follows as before that
the conjugate planes of vergency in a final uniform medium are conjugate planes of
deflexure of each medium-surface /”, and also of the surface /” determined by the
whole combination. When the final medium is ordinary as well as uniform, then the
osculating surfaces /”, are spheres, and the directions of extreme osculation are the
rectangular directions of the lines of curvature on the surface /, which is now per-
pendicular to the rays ; in this case, therefore, and more generally when a given final
ray in a final uniform medium corresponds to an umbilical point or point of spheric
curvature on the medium-surface /”,, the planes of vergency cut that surface, and the
surface Y” to which it osculates, in two rectangular directions, because two conjugate
tangents at an umbilical point are always perpendicular to each other: and, in like
manner, the planes of vergency being conjugate planes of deflexure will (by the
seventeenth number) be themselves rectangular, if the final ray whether ordinary or
extraordimary be such that taking it for the axis of deflexion of the medium-surface
V the indicating cylinder of deflexion is circular.
The foregoing principles give also the law of osculation of the variable medium-
surface /”, between its extreme positions, in a final uniform medium, namely, that the
distances of the variable osculating focus from the two points of vergency, are pre-
portional to the squares of the sines of the inclinations of the variable plane of oscu-
lation to the two planes of vergency, multiplied respectively by certain constant
Jactors. A formula expressing this law was deduced in the First Supplement ; but
the constant and in general unequal factors, (in the formula Z and 1,) for the squares
of the sines of the inclinations, were inadvertently omitted in the enunciation, Our
present methods would enable us to investigate without difficulty the law for the more
On Systems of Rays. 123
complicated case, when the osculating focus C’ being still in a uniform medium, the
point of osculation B is in another uniform medium, or even in an atmosphere ordi-
nary or extraordinary.
We might extend the reasonings of the present number to the case of sudden
reflexion or refraction, ordinary or extraordinary, and obtain analogous results, which
would include, in like manner, the results of former memoirs. In this case we should
find a certain analogous condition for the existence of a maximum or minimum of the
function = fvds ; and when this condition is not satisfied, we should have to consider
two pairs of separating planes, which cross the tangent plane of the reflecting or
refracting surface in one common pair of separating lines: the two pairs of planes
passing together from the real to the imaginary state, and in this passage closing up
into two transition-planes, which touch the caustic pencil before and after the sudden
reflexion or refraction, and intersect in one common ftransition-line, on the tangent
plane of the reflector or refractor, connected with a transition-point upon the caustic
curve of the pencil, and with certain extreme osculating waves or action-surfaces and
focal reflectors or refractors, of a kind easily discovered from the analogy of the
foregoing results.
Formule for the Principal Foci and Principal Rays of a Straight or Curved System,
Ordinary or Extraordinary. General method of investigating the Arrangement
and Aberrations of the Rays, near a Principal Focus, or other point of vergency.
25. Among the various points of consecutive intersection of the rays of an optical
system, there are in general certain eminent points of vergency, in which certain
particular luminous paths are intersected each by all the infinitely near paths of the
system. These eminent points and paths have been pointed out in my former
memoirs, and have been called principal foci, and principal rays. They may be
determined for straight final systems, by the characteristic function V7, and by any
three of the six following equations,
&Vv 1 dv OV 1 Sy
a2 t+ Roe — Saby | R Sadp— ;
eV | sy ov 1 82v
aye Rpt Bydz TR Hay ihe
OFA EAU OLS ov 1 38%
a2 TRO 7° Bact Roe
x, y, 2 being the co-ordinates of any point on a principal ray, and a +ah, y+PR,
z+yR being the co-ordinates of the principal focus; they may a!so be deduced from
124 Professor Hamitton’s Third Supplement
the auxiliary function 77, when made homogeneous of the first dimension with
respect to o, 7, v, by the equations
ow 2 ow ea
gor tee On 55, t Mig =:
ow SO ow oO *
or? iin ore S ordu e sy x3 (x )
ew P@) ew Bre)
du? 4 ou 0; dude i Suda e
of which only three are distinct, and in which V corresponds to the focus: or from
the function 7’, when expressed as depending on o, 7, ¢, 7’, x, by the following,
ia ce A le gts 88 Ss | 3S (is yt
80°? oo* \érd0" dar dada" Srd0 Sr? \Oada"
i eee
oo dr" Oa° Orda Srér’ Sadr \ Sade" Srdzr | Badr’ Srdo" or? dade’ Sadr’
_ (8S \-1 68S (BS \? SS SS Ss os / 8s \?
# hia ) aes (sy) a 2 aor Sadr’ S367’ iv 67” ( Sab’ ) }
m025) oes (ce 2
50 ae — (a8) »
in which as before, S= J’—zv+2'v'.
When the final medium is variable, we may employ the following equations,
ow ow ow ow ow ow
do? or ov? Sadr Oréu dvd
(a) "(ay)" (apa ae
oa or ou do dr or vu ov So
= (22 dQ 62 a) ae
dc 8c or Cy) ku Oz i
L SW: Sica tel SO. if OM dy Sik We,
or, — =—
a? Sc B? or y® Sv’ af Sadr _ By Srov _ ya Suda |
CD
(Z")
Sy ov bv ov\ —1
= (a at Bs + 1x) >
of which only three are distinct, but which are sufficient to determine the principal
foci and principal rays of a curved system, ordinary or extraordinary, by the aux-
iliary function /V’, considered as depending on o, 7, v, 2’, y’, 2’, x, in conformity to
the new view of that function, proposed in the present Supplement. ‘The new func-
tion Z’ might also be employed for the same purpose, but with somewhat less facility.
It was remarked, in a former number, that at a point of vergency the general
linear expressions for the relations of near rays fail; but the more complex expres-
On Systems of Rays. 125
sions by which these linear forms must be replaced at a principal focus or other point of
vergency, and generally when it is proposed to determine the aberrational corrections
of the first approximate or limiting relations, can always be obtained without difficulty
by developing to the required order of accuracy the general and rigorous equations
which we have given for a luminous path. An example of such deduction will occur,
when we come to consider the theory of instruments of revolution, which on account
of its extent and importance must be reserved for a future occasion.
Combination of the foregoing Lap of Optics with the Undulatory Theory of Light.
we mS By , that is, the Partial Differential Coeffi-
cients of the First Order of the Vis tees Function V, taken with respect to
the Final Co-ordinates, are, in the Undulatory Theory of Light, the Components
of Normal Slowness of Propagation of a Wave. The Fundamental Formula
(A) may easily be explained and proved by the principles of the same theory.
The quantities o, r, v, or
26. It remains, for the execution of the design announced at the beginning of this
Supplement, to illustrate the mathematical view of optics proposed in this and in for-
mer memoirs, by connecting it more closely with the undulatory theory of light.
For this purpose we shall begin by examining the undulatory meanings of the
symbols o, 7, v, of which, in the present Supplement, we have made so frequent a use,
and which we have defined by the equations (2),
her 8 oF
~~ Sn? ay? Paps ee
V being the undulatory time of propagation of light of some given colour, from
some origin v, 7,2, toa point x, y, z, through any combination of media. It is
evident that these quantities , 7, v are proportional to the direction-cosines of the
normal to the wave for which the time V’ is constant, and which has for its differen-
tial equation
= oon + rey + vez ; (A*®)
and if, as in the second number, we denote (0? + 7° + v°)~? by w, these direction-cosines
themselves will be ow, rw, vw ; and w will be the normal velocity, because the infinite-
simal time 8V, during which the wave propagates itself in the direction of its own
normal through the infinitesimal line 8/, from the point , y, z, to the point 7 +ow.0/,
Yt7w.dl, 2+vw.0l, is
8V =0.0w.81 +7.7w.8l + v.vw.0l= 5 al: (B")
VOL. XVII. 2K
126 Professor Hamitton’s Third Supplement
we may therefore call the quantities o, r, v, the components of normal slowness,
because they are equal to the reciprocal of the normal velocity, that is, to the normal
slowness, multiplied respectively by the direction-cosines of the normal, that is, by the
cosines of the angles which it makes with the rectangular axes of co-ordinates.
Such then may be said to be the optical meaning of our quantities o, 7, v, in the
theory of the propagation of light by waves. And we might easily deduce from this
meaning, and from the first principles of the undulatory theory, the general expres-
sion (4) for the variation of the characteristic function 7, which has been proposed
in the present and former memoirs, as fundamental in mathematical optics. For it is
an immediate consequence of the dynamical ideas of the undulatory theory of light,
that for a plane wave of a given direction and colour, in a given uniform medium, the
normal velocity of propagation is determined, or at least restricted to a finite variety
of values ; so that this normal velocity may be considered as a function of its cosines
of direction, volving also the colour, and depending for its form on the nature of
‘the uniform medium, and on the positions of the axes of co-ordinates, to which the
angles of direction are referred: and if the medium be variable instead of uniform,
and the wave curved instead of plane, we must suppose that the normal velocity w is
still a function of its direction-cosines o(6* +77 +v")~4, r(o +72 +u")-4, v(o? +72 +u")-4,
and of the colour x, involving also, in this more general case, the co-ordinates &, y, z.
In this manner we are conducted, by the principles of the undulatory theory, to a
relation between o, r, v, 2, y, 2, x, of the kind already often employed in the present
Supplement, namely,
0=Q=(6 +72 +v)to—1, (M)
Q+1 being a homogeneous function of «, 7, v, of the first dimension, which satisfies
therefore the condition
8Q 0G 6Q
SriGee ae wo Hl
ig, hoe SO
and which involves also in general the co-ordinates 2, y, z, and the colour y, and
depends for its form on the optical properties of the medium in which the pot x y 2
is placed. To connect now, for any given point and colour, the velocity and direction
of the ray with the direction of the normal of the wave, we may suppose, at first, that
the medium is uniform, and that the wave is plane. The two positions of this plane
wave, at the time J’, and at the time V+ AV, may be denoted by the equations
om
First ot +ty+uz=V + IV, (Cc)
Second cAaw+ TAy +vAz= AP,
in which o, 7, v, JV, are constants; and by the principles of the same undulatory
theory, if the point +A, ¥+Ay, 2+ Az, on the second plane wave, corresponding
On Systems of Rays. 127
to the time 7 + AV, be upon the ray that passes through the point w y z of the first
plane wave, it will be also on all the other infinitely near plane waves which corres-
pond to the same time V +AV, these other waves having passed through the point
wy z at the time VY, and having made infinitely small angles with the first plane
wave ; we are therefore to find the co-ordinaets # + Az, y + Ay, = + Az, of the second
point upon the ray, by seeking the intersection of the second wave (C’) with all those
other waves which are obtained from it by assigning to o, 7, v, any infinitely small
variations consistent with the relation
0= 50 ie oo + = oT Tae vu;
and thus we find
a_Av _80 B _Ay _ 80 dies Az _6Q (D")
v AV 8’ v AV &’? v AV wv’
as in the second number of this Supplement, and therefore
v =ac +PBr +y,
O = ado + Ber + you,
dv = oda +708 + vdy,
and finally
ov ov ov
ia Oy Fc a Ty; Sina (E*)
if we denote by v the reciprocal of the undulatory velocity with which the light is
propagated along the ray, and by a, , y, the cosines of the angles which the ray makes
with the axes of co-ordinates. We see, therefore, by the foregoing reasoning, which
it is easy to extend to the case of curved waves and of variable media, that the com-
ponents c,7, v, of normal slowness of a wave, or the partial differential coefficients of
the first order of the time- pene V, are equal to the partial differential coefficients
dv dv
of the first order, = aoe yy? of the undulatory slowness v of propagation along
the ray, when this latter sowineks is expressed as a homogeneous function of the
first dimension of the direction-cosines a B y of the ray : which is the general theorem
of mathematical optics, expressed by our fundamental formula (4).
That general theorem does not appear to have been perceived by other writers ;
nor do they seem to have distinctly thought of the components of normal slowness,
nor of the function of which these components are partial differential coefficients, that
is, the time V of propagation of light from one variable point to another, through
any combination of uniform or variable media, considered as depending on the final
and initial co-ordinates and on the colour: much less do those who have hitherto
written upon light, appear to have thought of this time-function V as a CHARACTER-
128 Professor Hamiiton’s Third Supplement
ISTIC FUNCTION, to the study of which may be reduced all the problems of mathema-
tical optics. But the problem of connecting by general equations the direction and
velocity of a ray with the direction and with the law of normal velocity of a wave,
has been elegantly resolved by M. Caucuy, in the 50th Livraison of the Evercices
de Mathématiques : and the formule which have been there deduced by considering
the normal velocity as a homogeneous function of the first dimension of its three
cosines of direction, may easily be shown to agree with the equations (D").
Theory of Fresnet, New Formule, founded on that theory, for the Velocities
and Polarisations of a Plane WVave, or WVave-Element. New method of deducing
the Equation of Fresnew’s Curved JV ave propagated from a Point in a Uniform
Medium with. Three Unequal Elasticities. Lines of Single Ray-Velocity, and
of Single Normal-Velocity, discovered by Fresne..
27. Let us now consider more particularly the undulatory theory of Fresnet.
In that theory, the small displacements of the vibrating etherial points are confined
to the surface of the wave, the ether being supposed to be sensibly incompressible,
and so to resist and prevent any sensible normal vibration: and the tangential forces,
which regulate the tangential or transversal vibrations, result in general from the
elasticity of the ether, combined with this normal resistance. It is also supposed that
the etherial medium has in general three principal unequal elasticities, corresponding
to displacements in the directions of three rectangular aves of elasticity ; in such a
manner that if we take these for the axes of co-ordinates, any small component dis-
placements dx, ay, dz parallel to these three axes will produce elastic forces —a%éz,
—b°ey, —c*éz parallel to the same axes, and equal to the displacements taken with
contrary signs and multiplied by certain constant positive factors a’, b*, c’: and any
small resultant displacement, 8/, in any other direction, having 62, ey, dz for its com-
ponents or projections, will produce a corresponding elastic force — Hél, of which
the components are —a*dx, —6°8y, —c’dz, and which has not in general the same
direction as the displacement &, nor a direction exactly opposite to that. Light,
polarised in any plane P, is supposed to correspond to vibrations perpendicular to
that plane, and propagated without change of direction ; and in order that a vibration
should thus preserve its: direction unchanged, while the plane wave or wave-element
to which it belongs is propagated through the uniform medium with a normal velocity
w, it is necessary and sufficient that the elastic force —H3/, when combined with a
normal resistance arising from the incompressibility of the ether, should produce a
tangential force —w°é/, in the direction opposite to the displacement 8/, and equal to
this displacement taken with a.contrary sign, and multiplied by the square of the nor-
On Systems of Rays. 129
mal velocity of propagation, so that its components are —w’da, —wy, —wdz: that
is, we must have the equations
: (w* —a’) da = Z (w —b*) dy = : (w* —c”) 82, (E"*)
v
in which o, 7, v, are, as before, the components of normal slowness, so that the equa-
tion of the wave-element containing the transversal vibration is
cou + Toy +voz=0. (A"”)
These equations (A) (F'") suffice in general to determine, on FRESNEL’s princi-
ples, the velocities of propagation and the planes of polarisation for any given wave-
element in any known crystallised medium.
Thus, eliminating the components of displacement 8x, dy, 8z, between the equations
(4) (F'*), we find the following law of the normal velocity w, considered as depend-
ing on the normal direction, that is, on the ratios of «, 7, v,
= + tote =0. (G")
wae to
To deduce hence the direction and velocity of a ray, for any given normal direction
and normal velocity, compatible with the foregoing law, that is, for any given values
of the components of normal slowness o, 7, v, compatible with the relation (G"), we
are to make, by (JZ),
Pie (Q ar 1 DE 18
ee o+72+u% ” CH )
and we then find, by (7), or by (D"), the following expressions for the components
of the velocity of the ray,
if we put for abridgment
Ps eee
e 2 2 2
(Gem)? (gem) tae)
And to deduce the law of the velocity + of the ray, considered as depending on its own
2
(K")
direction, that is, on the cosines a 3 y of its inclinations to the semiaxes a 6 c of elas-
ticity, we are to eliminate (according to the general method of the second number)
VOL. XVII. 2L
130 Professor Hamitton’s Third Supplement
the ratios of « + v between the three expressions (Z'*), and so to deduce the relation
between the three components of velocity ~ 5 = : ; now the equations (J) give
evidently, by (K"),
aca? be Cty?
A2—a? a A2—B + AL—e?
=05 (i)
they give also, when we attend to (@"),
Mee B 2 oa a5 se M's
Ne) shee oleae’ «2 ED
A therefore is the velocity of the ray, or the radius vector of the curved wnit-wave,
propagated in all directions from the origin of co-ordinates during the unit of time ;
and the equation of the wave in rectangular co-ordinates x y z, parallel to the axes of
elasticity, is
2
acre b2y2 et?
Pipe + eye t aye =o OY
or, when freed from fractions,
(a? +4? +27) (aa? + By? +072") + 0B?
HCV+ )P +0 (C+a)Yt+eaes+h)2. (O*)
This method of determining the equation of Fresnet’s ave, will perhaps be thought
simpler than that which was employed by the illustrious discoverer, and than others
which have since been proposed.
Reciprocally to determine by our general methods the normal direction and yelo-
city, or the components of normal slowness o, 7, v, for any proposed direction and
velocity of a ray compatible with this form of the wave, that is, for any values of
a B y X compatible with the relation (L"), we are to substitute for the ray-velocity A
in that relation its value (J/"*), and we find, by (#*),
su_a 1—a%
6a 0° =a?
_ ov Lins, We bay 18
ae aie Sok.
Bay Oe he
‘. sy v0” Hc?
if we put for abridgment
Ga) + Ge) + ES)
On Systems of Rays. 131
It is easy to see that the value of v thus determined is the normal slowness, or reci-
procal of w, because the expressions (P") give, by (L**),
os +77 ze v= vy? 5 (Chey
and since the same expressions give also evidently, by (Q"*),
3 7 :
v
a PS Lae Eee
0, (S'*)
we easily deduce the law (G@") of dependence of the normal velocity on the normal
direction, from the form of FREsNEL’s wave, as we had deduced the latter from the
former.
The equations (L"*) (MM) which gave us the equation of the wave in rectangular
co-ordinates, give also the following polar equation for the reciprocal of its radius-
vector, that is, for the slowness v of the ray,
O=v'—v* fab + 6%) + B(e*# +0") + y7(a?* + 8}
+ (a” + ice J yY) (a°b~ c + [Care= an ae y a- b=); Cr)
and therefore the following double expression for the square of this slowness,
veh +o) (@+f+7)
ee ee 18
+$(c%-a){ 4A" +t V0 +R +y—A*? Vat+fPh't+y —A}, (U")
if we put for abridgment
rae aad
oes Sh Ap ae
= eas a pee t= :
ca
ee
supposing therefore a’ > 0’ > c’, the polar equation of Fe wave may be put under
the form
ep =h(c* +a) +h (c?—@~) cos. ( (99) + (ye") ), cw")
e being the radius-vector or velocity, and (p9') (ep") being the angles which this radius
e makes with two constant radii o', p”, determined by the following cosines of their
inclinations to the semiaxes of w y z, or of abc,
‘ 7] (je , aoa’ 18
Pe a= eas Bip p— Os, Oc — =P c= / ae . (X")
The expression (/7"*), for the reciprocal of the square of the re of a ray, has
been assigned by Fresnex, who has also remarked that it gives always two unequal
velocities unless the direction p of the ray coincide with some one of the four direc-
tions +9, +o”, which are opposite two by two, and situated in the plane a c of the
132 Professor Hamitton’s Third Supplement
‘pp
extreme axes of elasticity. Fresnex has shown in like manner that any given nor-
mal direction corresponds to two unequal normal velocities, except four particular
directions, which we may call tw’, +w’, and which are determined by the following
cosines of direction,
2 §2 2
= —o g= ia 5 Os = Oz, = ON, jaw Ee E = (OG)
ce
and in fact it is easy to establish the following expression for the double value of the
square of the normal velocity, analogous to the expression (JV),
w=4(a’? +0) +4 (a —c’) cos. ( (ww) +(ww") ), (Z"*)
which cannot reduce itself to a single value, unless the sine of (ww) or of (ww’)
vanishes. FresNEL has given the name of optic axes sometimes to the one and some-
times to the other of the two sets of directions (X') (Y"*); but to prevent the con-
fusion which might arise from this double use of a term, we shall, for the present, call
the set +o, +e", by the longer but more expressive name of the directions or lines
of single ray-velocity : and similarly we shall call the set +w', +w’, the directions or
lines of single normal velocity.
New Properties of Fresnev’s ave. This Wave has Four Conoidal Cusps, at the
Einds of the Lines of Single Ray-Velocity : it has also Four Circles of Con-
tact, of which each is contained on a Touching Plane of Single Normal-V elocity.
The Lines of Single Ray-Velocity may therefore be called Cusp-Rays ; and the
Lines of Single Normal-Velocity may be called Normals of Circular Contact.
28. The reasonings of the foregoing number suppose that the axes of co-ordinates
coincide with the axes of elasticity ; but it is easy to extend the results thus obtained,
to any other axes of co-ordinates, by the formule of transformation which were given
in the thirteenth number. We shall content ourselves at present with considering two
remarkable transformations of this kind, suggested by the two foregoing sets of lines
of single velocity, which.conduct to some new properties of FRESNEL’ S wave, and to
some new consequences of his theory.
The polar equation (JV) of the wave may be put under the form
=4(c? +a) +4 (ce? —a) Srv" t VP Vert, (A”)
if we put for abridgment
r=A'p=20, +20 cs r =A'p=40, +20" 5 (B”)
so that r’, r", are the projections of the radius-vector » on the directions 9’, p’, of
On Systems of Rays. 133
single ray-velocity ; and if we take new rectangular co-ordinates x,y, z,, such that
the plane of «x, z, is still the plane a c of the extreme axes of elasticity, but that the
positive semi-axis of z, coincides with the line p’, we ney employ the following for-
mule of a eon
=X +2005 Y=Yip a —2,pat+Z,Pes (C*)
which give
Par tyi +27, =z, 7r”=7, sin. (p p') +2, COs. (¢' 9’), (D")
and change the equation (4) of the wave to the form
1=)7 27? +4h2,4 (c?—a@”) sin. ('p’) +4 (0? +.) (7 +”)
+4 (c°%-a”) Va? +y? V(,sin. (e'p )—2,cos.(op") PP +y”s (E")
This equation enables us easily to examine the shape of the wave near the end of the
radius 9’, that is, near the point having for its new co-ordinates
2 =Ony 012, =); (F”)
for it takes, near that point, the following approximate form,
z,=b— 4B Vo? —b— b- Vb — a an (4, + Vx +Y, ay (G")
which shows that at the point (#"") the wave has a conoidal cusp, and is touched not
by one determined tangent plane but by a tangent cone of the second degree, repre-
sented rigorously by the equation (G@"). FresneL does not appear to have been
aware of the existence of this tangent cone to his wave ; he seems to have thought
that at the end of a radius 9’ of single ray-velocity, the wave was touched only by two
right lines, contained in the plane of ac, namely, by the tangents to a certain circle
and ellipse, the intersections of the wave with that plane: but it is evident from the
foregoing transformation that every other section of the wave, made by a plane con-
taining the radius-vector 9’, is touched, at the end of that radius, by two tangent lines,
contained on the cone (G"). It is evident also that there are four such conoidal
cusps, at the ends of the four lines of single ray-velocity, +9, +”. They are deter-
mined by the following co-ordinates, when referred to the axes of elasticity,
ze: a_h pa a, b2_c2
t= sgt ei 4 Qe at RV (eae (H")
and they are the four intersections of FresNet’s circle and ellipse, in the plane of ac,
which have for their equations in that plane
Gate, Ot -- oz OC. Ge)
Again, if we employ the following new formule of transformation,
‘ ‘
DHL Wet Way Y=Y,» Z=—Lweth,wWes (K")
VOL. XVII. 2M
134 _ Professor Hamitton’s Third Supplement
so as to pass to a new system of rectangular co-ordinates such that the plane of x, =,
coincides with the plane of ac, and the positive semiaxis of z, with the line w’ of sin-
gle normal velocity, we find a new transformed equation of the wave, which may be
thus written,
(wit y 2+ 4,2, 07° V@—8 VB—eyY= Q A-—27b-), «L”)
if we put for abridgment .
Q=(a° +e?) pe + (at—e*) rr’ —a’e? (1 +2267); (M”)
and hence it is easy to prove that the plane
a b, (N*®)
which is perpendicular to the line w' at its extremity, touches the wave in the whole
extent of a circle ; the equation of this circle of contact being, in its own plane,
eity?+ 2,07 vVae—k Ve—Ce=0. (O”)
It is evident that there are four such circles of plane contact at the ends of the four
lines +o’, +w’, of single normal-velocity. They are all ae to each other, and the
common magnitude of their diameters is 0—' Va?—0? Vb?—c%. The same conclu-
sions may be drawn from FReEsNEL’s equation of the wave in co-ordinates @ y z
referred to the axes of elasticity: the equations of the fowr planes of circular con-
tact being, in these co-ordinates,
eVP—¢ +taVG@—8= tbVa—e, (P”)
FresNeEL however does not appear himself to have ‘ 3 the existence of these
circles of contact, nor do they seem to have been since perceived by any other person.
We shall find that the circles and cusps, pointed out in the present number, conduct
to some remarkable theoretical conclusions respecting the laws of refraction in biaxal
crystals.
New Consequences of Frusnew’s Principles. It follows from those Principles, that
Crystals of sufficient Biaxal Energy ought to exhibit two kinds of Conical
Refraction, an External and an Internal: a Cusp-Ray giving an External
Cone of Rays, and a Normal of Circular Contact being connected with an
Internal Cone.
29. The general formule for reflexion or refraction, ordinary or extraordinary,
which we haye deduced from the nature of the characteristic function VY, become
simply
On Systems of Rays. 135
Ac=0, Ar=0, (Q”)
when we take for the plane of x y the tangent plane to the refiecting or refracting
surface ; they show therefore that the components of normal slowness parallel to this
tangent plane are not changed, which is a new and general form for the laws of
reflexion and refraction. It is easy to combine this general theorem with F'Rresne1’s
law of velocity, and-so to deduce new consequences from that law with respect to
biaxal crystals.
For this deduction, our theorem may be expressed as follows,
ov ov a
a fee ee nes 19
o=A (a, s+ bag tox)» (R")
in which v is the undulatory slowness of a ray considered as a homogeneous function
of the first dimension of the cosines a B y of its inclinations to any three rectangular
semiaxes a b c, while A refers to the changes produced by reflexion or refraction, the
unaltered trinomial to which it is prefixed being the component of normal slowness in
the direction of any line ¢ on the tangent plane of the reflecting or refracting surface,
and a, b, c, being the cosines of the inclinations of this line to the semiaxes a bc: and
in order to combine this theorem with the principles of 'rrsneL, we have only to
suppose that the rectangular semiaxes a 6 c in each medium are the semiaxes of elas-
ticity of that medium, and that the form of the function v is determined as in the
twenty-seventh number.
Thus, to calculate the refraction of light on entering from a vacuum into a biaxal
crystal a 6 c bounded by a plane face F, we may denote by a, B, y, the cosines of
the inclinations of the external or incident ray to two rectangular lines s, ¢ upon the
face F, and to the inward normal, and we shall have the two equations following,
q5=0, a shel gig (=ca, +1), +vc,),
oa (3 oy
(S")
ov év én
po=a > + bse + C, 3y (=ca,+7b, +v0,),
which contain the required connexions between a, 3, y, anda y, that is, between
the external and internal directions. In this manner we find in general two incident
rays for one refracted, and two refracted for one incident ; because a given system of
values of a B y, that is, a given direction of the internal ray, corresponds in general
to two systems of values of the internal components of normal slowness o 7 v, and
therefore to two systems of values of a, 3, y,, that is, to two external directions ;
while, reciprocally, a given system of two linear relations between o, tr, v, deduced by
(S") from a given external direction, corresponds in general to two directions of the
internal ray. But there are two remarkable exceptions, connected with the two sets
of lines of single velocity, and with the conoidal cusps and circles of contact on Fres-
NEL’s wave.
136 Professor Hamitton’s Third Supplement
For we hava’@en that at a conoidal cusp the tangent plane to the wave is indeter-
minate ; it is evident therefore that a cusp-ray must correspond to an infinite variety
of systems of ponents of normal slowness o, 7, v, within the biaxal crystal, and
therefore also to an infinite variety of systems of direction-cosines a, 8, y, of the
external ray ; so that this one internal cusp-ray must correspond to an external cone
of rays, according to a new theoretical law of light, which may be called ExtTerna
ContcaL REFRActION.
And again, at a circle of contact, the wave has one common tangent plane for all
“the points of that circle, and therefore the infinite variety of internal rays which cor-
respond to these different points have all one common wave-normal, which may be
called a normal of circular contact, and all these internal rays have one common
system of components of normal slowness o 7 » within the crystal, and consequently
correspond to one common external ray : so that this one external ray is connected
with an internal cone of rays, according to another new theoretical law of light,
which may be called Inrernat Contcau Rerraction.
To develope, somewhat more fully, these two new consequences from FRESNEL’s prin-
ciples, let us begin by considering external conical refraction : and let us seek the
equation of the external cone of rays, corresponding to the internal cusp-ray p’. The
approximate equation (G") of the wave, near the end of this cusp-ray, in the trans-
formed co-ordinates x 7, 2,, gives the following approximate expression for the undu-
latory slowness v of a near ray, considered as a homogeneous function of the first
dimension of the cosines a, 3, y, of its inclinations to the positive semiaxes of these
co-ordinates 2, ¥, Z,,
4 =b- iets (a,+ Va? + 2?) Ci?)
in which
=4b Vo_—b? Vb?—a; 2 @z)
it gives therefore by our general method, the following components of normal slow-
ness parallel to the same semiaxes of wy, 2,,
i —¢@= oe tacts Be
gi aL oa eke PES)
m= = we =hee OB ie A) ise (v")
op, Vaz+Be {
2 , ov
# il hace Papua
the expressions for «, r, becoming itidefinitely more accurate as a, 8, diminish, that is,
as the near internal ray approaches to the cusp-ray ,, and the expression for v’ being
On Systems of Rays. 137
rigorous: the relations between the components of normal slowness o rv of the cusp-
ray o are therefore
.
(oo'. - Up a)” jr Beir (9'. - U0 a)s 0 4 + v0. =e e (w"’)
and the equation (ina, RB.) of the external cone of rays corresponding to the one
internal cusp-ray ois to be found by eliminating these three internal components orv
between the two relations (IV) and the two equations of refraction (S").
For example, if the internal cusp-ray p’ coincide with the inward normal to the
refracting face F' of the crystal, we may take, for the semiaxes s, ¢ upon that face, the
projection of a, and the semiaxis 6 of elasticity ; and then the equations of refraction
(S”) becoming
aap up yapi=rs (X")
we have, by (JV "), the following polar equation of the external cone of rays,
a2+P.2=27, a, 3 Ye)
or, in rectangular co-ordinates, an equation of the fourth degree,
(y+ y2) =473%a *(et+y, +2, ) (Z")
This cone is nearly circular in all the known biaxal crystals, because the coefficient 7,
is small, by (U'*), when the biaxal energy is weak, that is, when the semiaxes of elas-
ticity a 6 c are nearly equal to each other : and rigorously the external cone (Z")
meets the concentric sphere of radius unity in a curve contained on a circular cylin-
der of radius=r,, one side of this cylinder coinciding with a ray of the cone.
With respect to the internal conical refraction, the equation of the internal cone
of rays corresponding to the internal wave-normal w', or normal of circular contact,
is always, by (V") (0”),
wit+y?+9r ec 2 =0, if r,=hb?Vae—b Voc, (A®)
when referred to the rectangular co-ordinates w, yz, by the transformation (™) ;
and in the simpler rectangular co-ordinates 7 y z which are parallel to the axes of
elasticity the equation of this cone is
(tw,—z0 2 +y° +27, (tw, — Zw z) (tw, +2zw'.)=0, (B”)
in which we may change the co-ordinates x yz to the direction-cosines a (3 y of an
internal ray of the cone: while the one external ray corresponding is determined by
the following direction-cosines
abu, Balu); (Cc)
or by the ordinary law of proportional sines, since the internal wave-normal of circular
contact w’, which is one ray of the internal cone, is connected with the external
VOL. XVII. 2N
138 Professor Hamitton’s Third Supplement
ray by this ordinary law, if we take as the refracting index of the crystal the
reciprocal 5>' of the mean semiaxis of elasticity. It is evident hence that if the inter-
nal cone emerge at a new plane face, it will emerge a cylinder, whether the two faces
be parallel or inclined, that is, whether the crystal be a plate or a prism.
Theory of Conical Polarisation. Lines of Vibration. These Lines, on Fres-
NEL’s /Vave, are the Intersections of Two Series of Concentric and Co-axal
Ellipsoids.
30. A given direction of a wave-normal in a biaxal crystal corresponds in general
to two directions of vibration, and therefore to two planes of polarisation, determined
by the equations (#""), namely one for each of the two values «,’, ,” of the square
of the normal velocity deduced by (G") from the given system of ratios of o, 7, v;
and these two directions of vibration, or the two planes of polarisation, that is, the
two normal planes of the wave perpendicular to these vibrations, are perpendicular to
each other, since we can easily deduce from (G™) the following relation between
wr) W's
o® E v?
7
rerio + Wwe + Gaa@ee =9? PMY
which general rectangularity of the two vibrations on any one plane wave has been
otherwise established by Fresnet, and is an important result of his theory. But |
besides this general double polarisation connected with the general double refraction
in biaxal crystals, we may consider two other kinds which may be called conical polar-
isation, connected with the two kinds of conical refraction, which were pointed out in
the foregoing number.
To examine the law of the conical polarisation connected with the internal conical
refraction, and therefore with the planes of circular contact, we may employ the
co-ordinates 7, y,, 2,, defined by (AX), and thus transform the general equations of
polarisation (4°) (#"") into the following equally general,
, 7 Q
W. OL, +0 a oz,
aT tN OY, — 0,02, +0’, OF
7 ; *—a@)= =4 (W* —8) = —* 4 | (w*-—¢
WF, +W GV, & ) By (w ) — Wao, +0, (w ce), (E*)
o 82, oF 7,Y, + v8z,,= 0) 5
which give, for the projection of a vibration on the plane x, y, of single normal
velocity, the rigorous formula
oY, ae (w* —a’) (w? —c?) oi,
oar” Lesa v, Vae—B Vc to, (w?t+l?—at—ct) ”
(F*)
On Systems of Rays. 139
and for any plane wave slightly inclined to this plane of wy, the following approx-
imate relation between the components of normal slowness,
U,, = ie a Li (s, = jie o,. at 7, )s (G”)
retaining the meaning (4”™) of r,,; and if we attend to the general connexions,
established in this Supplement, between the direction-cosines of a ray and the compo-
nents of normal slowness of a wave, we easily deduce from (G*), by differentiation,
the following other relations,
a,, ou, on B,, ou, iia Ty 2
eee ), Bee 8 Fate pn
oy, = Ty, By 4 a?)
This formula contains the theory of the conical polarisation connected with internal
conical refraction. It shows that the vibrations at the circle of contact on FRESNEL’s
wave, are in the chords of that circle drawn from the extremity of the normal w of
single velocity ; and therefore that the corresponding planes of polarisation all pass
through another parallel normal at the opposite point of the circle. The plane of
polarisation, therefore, in passing from one position to another, revolves only half as
rapidly as the revolving radius, so that the angle between any two planes of polarisa-
tion is only half the angle between the two corresponding radii of this circle on
Fresnew’s wave. And if we suppose that the direction of the external incident ray
coincides with the wave-normal w’, and therefore also with the normal to the refracting
face of the crystal, then the small internal components of normal slowness, o,r,,,
parallel to this refracting face, are equal (by our general theorem of refraction) to the
small external direction-cosines a, 3, of the inclinations of a near incident ray to the
semiaxes of x, and y,; from which it follows, by (J), that the plane of external
incidence containing this near incident ray revolves twice as rapidly as the corres-
ponding plane of refraction.
For the other kind of conical polarisation, connected with the external conical
refraction, and therefore with the conoidal cusps on FREsNEL’s wave, we find by a
similar process,
= >
8z, o, aE Va? a] ah
our = By B, (K”)
and
oz,= —2br dr, (L”)
r having the meaning (U"). The formula (A) shows thatthe normal plane to the
140 Professor Hamitton’s Third Supplement
wave, containing any vibration near the cusp, contains either the cusp-ray itself, or a
line parallel to this ray ; so that the direction of any near vibration coincides with or
is parallel to the projection of the cusp-ray on the corresponding tangent plane of the
wave, or of the cone which touches it at the cusp: and the formula (L”) shows that
all these near vibrations are parallel to one common plane, which is easily seen to be
perpendicular to the plane of ac, and to contain the tangent at the cusp to the
elliptic section (I') of the wave, made by this latter plane ; so that all the planes of
polarisation near the cusp, contain, or are parallel to, the normal of this elliptic
section. And the direction of any near vibration on the wave, or on its tangent cone,
may be obtained by cutting the corresponding tangent plane of this wave or cone by
a plane perpendicular to this elliptic normal.
If the cusp-ray be incident perpendicularly on a refracting face of the crystal, then
the internal components o, 7, are equal to the direction-cosines a, 8, of the corres-
ponding ray of the emerging external cone ; and therefore, by (A), the plane of
refraction of this external ray contains the internal vibration, and therefore also, by
FresNnev’s principles, the external vibration corresponding: so that, i the external
conical polarisation, produced by the perpendicular internal incidence of a cusp-ray,
the plane of polarisation of an external ray is perpendicular to its plane of refrac-
tion ; and therefore revolves about half as rapidly as the plane containing, this
emergent ray and passing through the approximate axis of the nearly circular
emergent cone, when the biaxal energy is small. We see also, by (AX™), that the
plane containing the cusp-ray and containing or parallel to a near internal ray,
revolves with double the rapidity of the plane containing the cusp-ray and parallel to
the near waye-normal ; and therefore, in the case of perpendicular incidence of the
cusp-ray, the plane of incidence of a near internal ray revolves with double the
rapidity of the plane of external refraction, which, as we have seen, contains here the
external vibrations.
In general, the equations of polarisation (#""*), which we have deduced from Fres-
NEL’s principles, conduct, by (J) (L"*), to the following simple formula
aradx + b°Pdy + c’ydz =0, (M”)
dx, dy, z being still the components of displacement parallel to the semiaxis a, 6, ¢,
and a, (3, y being still the cosines of the inclinations of the ray to the same semiaxes of
elasticity : and this formula (J17*°), when combined with the equation of transversal
vibrations,
8V=0, or, od4+ roy + +vdz=0, (A*®)
determines easily the direction of vibration for any given direction and velocity of a
ray, that is, for any point of Fresnev’s curved wave propagated from a luminous origin
On Systems of Rays. 141
within a biaxal crystal. And we easily see that on any wave in a biaxral crystal, whe-
ther propagated from within or from without, the differential equation (17) deter-
mines a series of lines of vibration, having the property that at any point of such a
line the vibration is in the direction of the line itself. ‘To find these lines on Fres-
NEL’s wave (0°), we may change af y to wy z in the differential equation (17), and
we then find, by integration,
ee +ytese, (N”)
« being an arbitrary constant; and since this integral, when combined with the equa-
tion (O") of the wave itself, gives
ate ae bi+e a? +- c + ¢! 2 “e+ +c*) é—a he Cz oO”
Ly
we see that the lines of vibration on FRESNEL’s wave, propagated from a point in a
biawal crystal, are the intersections of two series (N®)(O*) of concentric and co-axal
ellipsoids.
By this general integration, extending to the whole wave, or by integrating the
approximate equations for vibrations near the conoidal cusps and circles of contact,
obtained from (A”) (J*) by changing the direction-cosines of a ray to the propor-
tional co-ordinates of the wave, we find that near a cusp the lines of vibration coincide
nearly with small parabolic arcs on the tangent cone of the wave, in planes perpendi-
cular to the elliptic normal already mentioned ; and that in crossing a circle of contact
the course of each line of vibration is directed towards that point of the circle which is
the end of the corresponding waye-normal of single velocity, that is, towards the foot
of the perpendicular let fall from the centre of the wave on the plane of circular
contact.
In any Uniform Medium, the Curved TVave propagated from a point is connected
with a certain other surface, which may be called the surface of components, by
relations discovered by M. Caucuy, and by some new relations connected with a
General Theorem of Reciprocity. This new Theorem of Reciprocity gives a new
construction for the WVave, in any Undulatory Theory of Light : and it connects
the Cusps and Circles of Contact on Fresnev’s WVave, with Circles and Cusps of
the same kind on the Surface of Components.
31. The theory of the wave propagated from a point in any uniform medium, may
be much illustrated by comparing this wave with a certain other surface which appears
to have been first discovered by M. Caucuy, who has pointed out some of its proper-
ties in the Livraison already referred to. In that Livraison, M. Caucny has treated
VOL, XVII. 20
2
142 Professor Hamitton’s Third Supplement
of the propagation of plane waves in a system of mutually attracting or repelling par-
ticles ; and has been conducted to a relation between the normal velocity of propaga-
tion, which he calls s, and the cosines of its inclinations to the positive semiaxes of
#, Ys %, which cosines he denotes by a, 6, c. ‘The relation thus found being expressed
by equating to zero a certain homogeneous function (of the sixth dimension) of s,a, b,c,
it has suggested to M. Caucuy the consideration of s as a homogeneous function of
the first dimension of the cosines a, 6, ce, whereas we have preferred to treat the normal
velocity (denoted in this Supplement by w) as a homogeneous function of its cosines
of direction of the dimension zero ; a difference m method which makes no real differ-
ence in the results, because the relation existing between the cosines (namely, that
the sum of their squares is unity,) permits us to transform in an infinite variety of
ways any equation into which they enter. M. Caucuy deduces from his view of the
relation between the normal velocity and cosines of normal direction, the following
equations between the time ¢ and the co-ordinates 2 y z of a ray from the origin of
co-ordinates,
ds Wiens,
Ka
rma Oe coe
which were alluded to in the twenty-sixth number of the present Supplement, as sub-
stantially equivalent to our equations (D"). He deduces also an equation of the
form
GQ. OG
F(5.595)=%
x Fs ty 7 pl wc ial E P
which he constructs by a surface having prise ag for its co-ordinates. Our methods
suggest immediately the same surface, as the construction of the same equation under
the form
OQ (6, 75 v)=0,
which has been so frequently employed in this Supplement ; and from the optical
meanings that we have pointed out for the co-ordinates o, r, v, of this surface Q=0,
we shall call it the surface of components of normal slowness, or simply the surface
of components. M. Caucuy shows that this surface is connected with the curved
wave propagated from the origin of co-ordinates in the unit of time, (which we have
called the wnit-wave and may denote by the equation
I ioresh 5).
by two remarkable relations, which can easily be deduced fron: our formule, and
may be thus enunciated: first, the swm of the products of their corresponding co-
ordinates, or, in other words, the product of any two corresponding radii multiplied
hy the cosine of the included angle, is unity ; and secondly, the wave is the enveloppe
On Systems of Rays. 143
of the planes which cut perpendicularly the radii of the surface of components at
distances from the centre equal to the reciprocals of those radit.
To these two relations, discovered by M. Caucny, we may add a third, not less
remarkable, which he does not seem to have perceived: namely, that the surface of
components is the enveloppe of the planes which cut perpendicularly the radii of the
wave at distances from its centre equal to the reciprocals of those radii, that is, equal
to the slownesses of the rays. For it is a general theorem of reciprocity between sur-
faces, which can easily be deduced from the evident coexistence of the three equations
,
ax +yy +227 =1,
udu’ + ysy' + 282/ =0, (rE?)
vou tycy +2dz=0,
that if one surface B be deduced from another A by drawing radii vectores to the
latter from an arbitrary origin O, and altering the lengths of these radii to their
reciprocals without charging their directions, and seeking the enveloppe B of the
planes perpendicular at the extremities to these altered radii of A, then reciprocally,
the surface A may be deduced from B by a repetition of the same construction,
employing the same origin O, and the same arbitrary unit of length. For example,
if the surface 4 be formed by the revolution of an ellipse about its greater axis, and
if we place the arbitrary origin O at one focus of this ellipsoid 4, and take the arbi-
trary unit equal to the semiaxis minor, the enveloped surface B will be a sphere,
having its diameter equal to the axis major of the ellipsoid, and its centre on that axis
major, the interval between the centres of the two surfaces being bisected by the origin
O ; and if from this excentric origin we draw radii to the sphere B, and change these
unequal radii to their reciprocals, and draw perpendicular planes at the extremities of
these new radii, the enveloppe of the planes so drawn will be the ellipsoid 4. Another
particular case of this general theory of reciprocal surfaces, namely, the case of two
concentric and co-axal ellipsoids, referred to their centre as origin, and having the
semiaxes of one equal to the reciprocals of those of the other, has been perceived by
Mr. MacCuxraen, and elegantly proved by him, in the Second Part of the Sixteenth
Volume of the Transactions of the Royal Irish Academy.
This general theorem of reciprocity, when applied to the unit-wave and surface of
components, gives a new construction for the unit-wave in any uniform medium, and
Sor any law of velocity : namely, that the wave is the locus of the points obtained by
letting fall perpendiculars from the centre on the tangent planes of the surface of
components, and then altering the lengths of these perpendiculars to their reciprocals,
without altering their directions.
It follows also from this general theory of reciprocal surfaces, that a conoidal cusp
on any surface A corresponds in general to a curve of plane contact on the reciprocal
144 Professor Hamiiton’s Third Supplement on Systems of Rays.
surface B, and reciprocally ; and, accordingly the cusps and circles on FREsNEL’s wave
are connected with circles and cusps on the corresponding surface of components, which
latter surface is indeed deducible from the former by merely changing the semiaxes of
elasticity abc to their reciprocals. And it was in fact by this general theorem that I
was led to discover the four circles of contact on FrrsNnew’s wave, by concluding that
this wave must touch four planes in curves instead of points of contact, as soon as I
had perceived the existence of four conoidal cusps on the surface of components, by
obtaining (in some investigations respecting the aberrations of biaxal lenses) the
formula (G@*), which is the approximate equation of such a cusp. I easily found also
that there were only four such cusps on each of the two reciprocal surfaces, and there-
fore concluded that there were only four curves of plane contact on each. I may
mention that though I have taken care to attribute to M. Caucuy the discovery of
the surface of components, yet before I met the Hwercices de Mathématiques, I was
familiar, in my own investigations, with the existence and with the foregoing properties
of this surface: it is indeed immediately suggested by the first principles of my view
of optics, since it constructs the fundamental partial differential equation
8V OV OV
(Seay? & )=°
which my characteristic function V must satisfy in a final uniform medium.
The surface of components possesses many other interesting properties, for exam-
ple the following, that in a final uniform medium any two conjugate planes of ver-
gency (") are perpendicular to two conjugate tangents on it: which is analogous
to the less simple relations considered in the twenty-first number. But the length to
which this Supplement has extended, confines me here to remarking, that the general
equations of reflexion or refraction,
Nise 10, Ac =O; (Q”)
may be thus enunciated; the corresponding points (c, 7, v, and « + Ac, 7 + Ar, v + Av)
upon the surface or surfaces of components (0 = 2, 0 = 2+AQ,) before and
after any reflexion or refraction ordinary or extraordinary, are situated on one
common perpendicular to the plane which touches the reflecting or refracting
surface at the point of reflexion or refraction ; a new geometrical relation, which
gives a new and general construction to determine a reflected or refracted ray, simpler
in many cases than the construction proposed by Huycuens.
R. AvC LAO SN
eek: 1°
CO NegisCua: RE ERA CY Lom ;
J Kirkwood Se. Dublin.
On the Phenomena presented by Light in its passage along the Axes of Biaxal
Crystals. By the Rev. HUMPHREY LLOYD, A.M., M.R.I. A., Fellow of
Trinity College, and Professor of Natural and Experimental Philosophy in the
University of Dublin.
Read January 28, 1833.
Ir is well known that when a ray of light is incident upon certain crystals, such as
Iceland spar and quartz, it is in general divided into two pencils, of which one is
refracted according to the known law of the sines, while the direction of the other is
determined by a new and extraordinary law, first assigned by Huycuens.
These laws were long supposed to apply to all doubly refracting substances ; and it
was not until the subject was examined by the ablest advocate of the undulatory _
theory, that the problem of double refraction was solved in all its generality. Setting
out from the hypothesis, that the elasticity of the vibrating medium within the crystal
is unequal in three rectangular directions, FRresNEL has shown that the surface of the
wave is not, in general, either a sphere or spheroid, as in the Huyghenian law, but a
surface of the fourth order, consisting of two sheets; and that the directions of the
two refracted rays are determined by tangent planes drawn to these surfaces under
known conditions. Such crystals have, in general, two optic axes, and are thence
denominated biaral. When the elasticity of the medium is the same in two of the
three directions, the equation of the waye-surface is resolvable into two, which repre-
sent the sphere and spheroid of the Huyghenian law. ‘The two optic axes in this case
coincide ; and the law of Huycuens is thus proved to be a case of a more general
law, and shown to belong to wniawal crystals only. Finally, when the elasticity is the
same in all the three directions, the wave-surface becomes a sphere; and the refraction
is single, and takes place according to the ordinary law of the sines. This case com-
prises a few of the crystallized, and most uncrystallized substances.
There are two remarkable cases, however, in this elegant and profound theory,
which its author seems to have overlooked, if not to have misapprehended. In a com-
munication made to the Academy at its last meeting, Professor Hamiiron has
supplied these omissions in the theory of Fresnex, and has been thus led to results in
the highest degree novel and remarkable.
VOL. XVII. 2P
146 Mr. Luioyp on Conical Refraction.
To understand these conclusions, it may be useful to revert for a moment to the
original theory of Fresnet. The general form of the wave-surface is determined by
the equation
(a cos.'a + b° cos." + c*cos.* y) r*
—[a'(B + c*)cos.°a + b*(a* + c*) cos.7B + ¢°(a® + b*)cos.*y] r°
+ab'?=0;
in which a, 3, y, denote the angles made by the radius-vector with the three axes, and
a’, 6°, c’, the elasticities of the medium in these directions. If now we make cos. 8 =0,
in this equation, so as to obtain the section of the surface made by the plane of xz, the
result is reducible to the form
(7° — b?) [(a@’cos.’a + e’sina) 7?—a’c?] =0.
So that the surface intersects the plane of xz in a circle and ellipse, whose equations
are
r=b, (a’cos”a+e'sin.” a) =a’ Cc’.
Now 4, the radius of the circle, being intermediate between a and c, the semiaxes of
the ellipse, it is obvious that the two curves must intersect in four points, or cusps, as
represented in (fig. 1); and the angle which the radius-vector OP, drawn to the cusp,
makes with the axis of z, is found by eliminating r between the two equations, by
which means we obtain
s a Bc?
sin.a= 2 Of Tere
At each of the points thus determined there will be two tangents to the plane section ;
and consequently the ray OP, proceeding within the crystal to one of these points,
might be supposed to be divided at emergence into two, whose directions are deter-
mined by those of the tangents.
Such seems to have been Fresnet’s conception of this case. Professor HamiLTon
has shown, however, that there is a cusp at each of these points, not only in this parti-
cular section, but in every section of the wave surface passing through the line OP ;
or, in other words, that there is a conoidal cusp on the general waye-surface at the
four points of intersection of the circle and ellipse ; so that there must be an infinite
number of tangent planes at each of these points, and consequently a single ray, such
as O P, proceeding from a point within the crystal to one of these points, must be divided
into an infinite number of emergent rays, constituting a conical surface.
It is evident further, that the circle and ellipse will have four common tangents,
such as MN (fig. 1.) The planes passing through these tangents, and parallel to the
third or mean axis, are parallel to the circular sections of the surface of elasticity of
Fresnex’s theory, or perpendicular to the optic axes. FResneL seems to have con-
Mr. Luoyp on Conical Refraction. 147
cluded that these planes touched the wave-surface only in the two points just men-
tioned ; and, consequently, that a single ray, incident upon a biaxal crystal in such a
manner that one of the refracted rays should coincide with an optic axis, would be
divided into two, determined by the points of contact. ‘This result, if verified by
experience, would place a remarkable distinction between the phenomena of uniaxal
and biaxal crystals ; but though the case was examined by M. Bior, no corresponding
appearances were observed.
Professor Hamiiron has shown that the four planes of which we have spoken touch
the wave-surface, not in two points only, but in an infinite number of points, consti-
tuting each a small circle of contact, whose plane is parallel to one of the two circular
sections of the surface of elasticity ; and that, consequently, a single ray of common
light, incident externally in the above-mentioned direction, should be divided within
the crystal into an infinite number of refracted rays, constituting a conical surface.
Here, then, are two singular and unexpected consequences of the undulatory theory,
not only unsupported by any facts hitherto observed, but even opposed to all the
analogies derived from experience. If confirmed by experiment, they would furnish
new and almost convincing proofs of the truth of that theory ; and if disproved, on the
other hand, it is evident that the theory must be abandoned or modified.
Being naturally anxious to submit the undulatory hypothesis to this delicate test, and
to establish or disprove these new results of theory, Professor HamiLron requested me
to institute a series of experiments with that view. I accordingly applied myself to this
interesting research with all the attention which the subject so well merited, and have
fortunately succeeded in verifying both cases of conical refraction. The substance |
employed in these experiments was arragonite, which is well known to be a biaxal
crystal, whose axes are inclined at an angle of nearly 20°. I selected it partly on
account of the magnitude of its biaxal energy, and partly also because the optical
elements of this mineral have been determined, apparently with great care, by Professor
Rupsere ; and therefore the results of theory could be applied to it at once without
further examination. ‘The specimen I used was one of considerable size dnd purity,
procured for me by Mr. Dotonp, and cut with its parallel faces perpendicular to the
line bisecting the optic axes.
The first-mentioned species of conical refraction, it has been observed, takes place
mm air, when a ray of common light is transmitted within the crystal in the direction
of the line joining two opposite cusps of the wave. If we suppose such a ray to pass
in both directions out of the crystal, it is evident that it must emerge similarly at both
surfaces ; consequently, the rays which are transmitted along this line within the
crystal, and form a diverging cone at emergence at the second surface, must be inci-
dent in a converging cone at the first. Having therefore nearly ascertained the required
direction by means of the system of rings in polarized light, I placed a lens of short
148 Mr. Lioypd on Conical Refraction.
focus at its focal distance from the first surface, and in such a position that the central
part of the pencil might have an incidence nearly corresponding to the cusp-ray within.
Then looking through the erystal at the light of a lamp placed at a considerable
distance, I observed in the expected direction a point more luminous than the space
immediately about it, and surrounded by something resembling a stellar radiation.
Fearing that this singular appearance might have arisen from some imperfection in the
crystal, I transmitted the light in the same manner through several different parts of
its substance, and always with the same result. The connexion of the phenomenon
with the optic axis was proved by the system of rings which appeared in the same direc-
tion when the light was examined with a polarizing and analyzing plate.
This result is of some interest in itself, independently of its connexion with theory.
It has been hitherto supposed that the only method of determining experimentally the
direction of the optic axes, in most doubly refracting substances, consisted in observing
the system of coloured rings, which appear around them when the incident and emer-
gent light is polarized. Here, however, we find that common, or unpolarized light,
undergoes such modifications in the neighbourhood of one of the optic axes, that the
apparent direction of that axis may be at once determined, and with the aid of the
simplest contrivance.*
But to examine the emergent cone, it was necessary to exclude the light which passed
through the crystal in all but one direction. For this purpose, a plate of thin metal,
having a minute aperture, was placed on the surface of the crystal next the eye, and
the position of the aperture so adjusted, that the line connecting it with the luminous
point on the first surface might be, as nearly as possible, in the direction of the cusp-
ray. The exact adjustment to this direction was made by subsequent trial. The
phenomenon which presented itself when this disposition was complete, was in the
highest degree curious. There appeared at first a luminous circle, with a small dark
space in the centre; and in this dark central space were two bright points, separated
by a narrow and well-defined dark line. ‘These appearances are represented in (figures
a and 6.) When the aperture in the plate was slightly shifted, the phenomena rapidly
changed, assuming in succession the forms represented in (figs. c, d, e.) In the first stage
of its change, the central dark space became greatly enlarged, and a double sector
appeared in the centre. The circle was reduced to about a quadrant, and was sepa-
rated by a dark interval from the sector just mentioned. This is represented in (fig. ¢.)
The remote sector then disappeared, and the circular arch diminished, as in (fig. d) ;
* This fact is here mentioned, rather as a matter of curiosity than as one likely to be of practical value
in determining the optical elements of crystals. It is to be observed, moreover, that the direction thus
determined is that of the normal to the circular section of the ellipsoid of Fresne’s theory; while the
rings (there is strong reason to believe) are related to the normals to the circular sections of the surface
of elasticity.
Mr. Luioyp on Conical Refraction. 149
and as the inclination of the internal ray to the cusp-ray was further increased, these
two luminous portions merged gradually into the two pencils, into which a single ray
is divided in the other parts of the crystal. ‘This change is represented in (fig. e.)
Similar observations were made without the lens, by bringing the flame of the lamp
near the first surface of the crystal, and forming the converging cone by covering that
surface also with a thin metallic plate, perforated with a minute aperture. In this case
the line connecting the two minute apertures was adjusted as before, and the pheno-
mena were the same as in the former instance, the rays which passed along this line
within the crystal forming a diverging cone at emergence.
In all these experiments the emergent rays were received directly by the eye placed
close to the aperture on the second surface. It was obviously desirable, however, to
receive them on a screen, and thus to observe the section of the cone at different
distances from its summit. After some trials, I effected this with the sun’s light, the
light of a lamp being too weak for the purpose. The emergent cone being made to
fall on a screen of roughened glass, I was enabled to observe its sections at various
distances, and therefore with all the advantages of enlargement. ‘The light was suffi-
ciently bright, and the appearance distinct, when the diameter of the section was
between one and two inches.
On examining the emergent cone with a tourmaline plate, I was surprised to observe
that one radius only of the circular section* vanished in a given position of the axis of
the tourmaline, and that the ray which disappeared ranged through 360° as the tour-
maline plate was turned through 180°. Thus it appeared that all the rays of the cone
are polarized in different planes.
On examining this curious phenomenon more attentively, I discovered the remark-
able law, “that the angle between the planes of polarization of any two rays of the
cone is half the angle between the planes containing the rays themselves and the
axis.”
Having assured myself of the near truth of this law by experiment, I was naturally
led to inquire how far it was in accordance with theory ; and on examining Fresnet’s
theory with this view, I was gratified to find that it led to the very same result.
According to the known rule, the plane of polarization of any one ray of the
emergent cone must bisect the angle contained by the planes passing through the cor-
responding normal to the front of the wave and the two optic axes. Now, it can be
easily shown that the normals to the wave, at the cusp, surround one of the optic
axes, and are inclined to it all round at small angles. For the tangent of the angle
* These sections are not mathematically circular, the line being, in fact, one of the fourth
order.
150 Mr. Luoyp on Conical Refraction.
which the normals to the circle and ellipse in the plane of wz make with one
another is
Va—b VB
ac 2
and it can be easily shown that the tangent of the angle which the optic axis makes
with the normal to the circle, or the cusp-ray, is
VE=B Ee
nace
Now, this is about half the former, since b?=ac, nearly ; and consequently the optic
axis nearly bisects the angle contained by the extreme normals in the plane of zz.
Hence if 4 and B be the intersections of the two optic axes with the sphere whose
centre is at the cusp, and JV the intersection of one of the normals at that point
with the same (fig. 2), the angle 4 C ranges through every magnitude between
O and 360°, the arch V A being all the time very small. Let the angle NA C be
denoted by a, and NP C by w, NP being the arch bisecting the angle V; then
in the triangle 4 P N, we have
cos. w=cos. 4 N. sin. a.sin. } N+cos. a.cos.} NV;
or, since 4 N is very small, and therefore cos. 4 N=1, nearly,
cos. w=cos, (a—4$ NV), and w=a—} N, nearly.
But, when any side of a spherical triangle is very small in comparison with the other
two, the adjacent angles are together equal to 180° q. p. Consequently,
N=a, and w=4a, nearly.
From this it appears that the angle which the plane of polarization of any ray makes
with the plane of the optic axes, is half the angle which the plane passing through the
normal and the near axis makes with the same plane. But this latter angle, it may be
easily shown, is very nearly the same as that which the plane passing through the
emergent ray and the axis of the cone makes with the plane of the optic axes. Con-
sequently, the angle which the plane of polarization of any ray of the emergent cone
makes with the plane of the optic axes is half of that which the plane containing that
ray and the axis of the cone forms with the same plane.
The general phenomena being observed, it remained to examine the magnitude and
position of the emergent cone, and to compare the results with those furnished by
theory. For this purpose I viewed the aperture in the second plate through a small
telescope, which was moved ina plane nearly perpendicular to the axis of the emer-
gent cone ; and by noting the points at which the light failed, I obtained the magni-
tude of the section of the cone made by that plane. ‘Lhe distance of this section from
Mr. Luioyp on Conical fefraction. 151
the crystal being then measured, the angle of the cone was obtained from the trigo-
nometrical tables ; and was found to be very nearly 6°. I then placed the flame of a
wax taper at the centre of the section, and removing the plate from the second surface
of the crystal, found the direction of the ray reflected from the surface. A well
defined mark was then placed on this line, at a considerable distance, and the angular
distance between the centre of the flame and the mark measured by a sextant, whose
centre was brought exactly to the place of the crystal. This angle was found to be
31° 56;; and consequently the angle of emergence corresponding to the central rays
of the cone was 15° 58’.
Now to compare these results with those of theory.—lIt is a well-known principle
of the theory of waves, that the direction of a ray incident upon, or emergent from,
a crystal, and the normal to the front of the wave, are always in the same plane per-
pendicular to the surface of incidence or emergence ; and the angles which these two
lines make with the perpendicular to the surface, are connected by the known law of
the sines; the index of refraction being the reciprocal of the normal velocity of the
wave, or of the perpendicular upon the tangent plane. Now, at the cusp, there are
an infinite number of normals to the wave, and consequently an infinite number of
corresponding emergent rays. Of these the two rays in the plane of the optic axes
form the greatest angle, and their directions are determined by those of the normals
to the circle and ellipse, which constitute the section of the wave-surface in that plane.
If then » and p’ denote the angles of emergence of these rays, «the angle which
the normal to the circle, or cusp-ray, makes with the perpendicular to the surface,
athe angle contained by the normals to the circle and ellipse, and p the perpendi-
cular from the centre on the tangent to the ellipse at the cusp, we have
c ae : bars
sin. p = > sin. ct, sin. p! = ji sin. («—a) ;
é
Tn which
1 2 0262" ¢— 8? ./82—e2
Il Vatt+e Li Wg ia b b?—e
Pp ac ac
Now in Arragonite, according to the determination of M. Rupsere,
1
'_ 1.5326, } = 1.6863, - = 1.6908 ;
a c
And substituting these values we find
1
i = 1,68708, a = 1°.44).48".
These values being introduced in the first two equations, ’ and pe" will be deter-
mined for any given surface of emergence. In this manner Professor HamiLron has
found that when «=O, or the surface of.emergence perpendicular to the cusp-ray,
pe! =0, and p'= 2°.56'-51". And when. = 9°. 56'-27", or the surface perpendicular
152 Mr. Luioyp on Conical Refraction.
to the line bisecting the optic axes, p = 16° 55' 27", and ep” = 13° 54/49". Ac-
cordingly, the difference of these angles, p —¢’, which is the extreme angle of the
emergent cone, is in the former case 2° 56 51",* and in the latter 3° 0' 38". Also,
half the sum of these angles, which is the angle of emergence corresponding to the
axis of the cone, is 15° 25' 8”.
Comparing these with the results of observation, it will be seen that they agree
nearly with respect to the mean angle of emergence, the difference amounting only to
33' ; whereas the angle of the cone determined by experiment is about double of that
furnished by calculation.
Talso measured the angle of the cone by tracing the outline of its section on a
screen of roughened glass, when the sun’s light was employed instead of that of a
lamp. The mean diameter of this section being then accurately ascertained, and the
distance of the screen from the aperture measured, the angle was given by the tables.
Measurements taken in this manner gave for the value of the angle, 6° 24’, 5° 56',
6° 22', respectively ; and the mean of these is 6° 14’, which, like the former mea-
surement, differs very little from the double of the calculated angle.
The results of observation thus appeared to be at variance with those of theory in
two important particulars. In the first place, the emergent rays appeared to form a
solid cone, instead of a conical surface ; and in the next, the magnitude of this cone
was about double of the expected magnitude. Conceiving that these discrepancies
might probably be owing to the rays which are inclined to the cusp-ray at small
angles, aud which pass by the edge of the aperture, I determined to ascertain the
fact by trying the effects of apertures of various sizes.
I found accordingly that when the aperture was at all considerable, such as that
formed by a large-sized pin, two concentric circles were seen to surround the axis, the
interior of which had about double the brightness of the exterior annulus. And it
was remarkable that the light of the interior circle was unpolarized, while that of the
surrounding annulus was polarized according to the law already explained. When
smaller apertures were used, the inner circle contracted, the breadth of the exterior
annulus remaining nearly the same; until the former was finally reduced to a point
in the centre of a fainter circle. When the aperture was still further diminished, a
dark space sprung up in the centre, enlarging as the aperture decreased ; until finally,
with a very minute aperture, the breadth of this central space increased to about
3ths of the entire diameter. ;
The phenomena exhibited in these cases assumed the forms represented in figures
* It is easily shown that the sine of the angle of the cone, in this case, is generally expressed by the
Va_h? VRP
abe
formula
.
Mr. Luioyp on Conical Refraction. 153
(f) and (g). (Fig. i) represents the appearance of the section when the line con-
necting the aperture with the luminous point on the first surface was slightly inclined
to the cusp-ray.
It is easy to render an account of these various appearances. When the aperture
mn, (fig. 3.) is at all considerable, the rays cm, cn, proceeding to its circumference
from a point on the first surface, will be sensibly inclined to the cusp-ray, which
we shall suppose to be the line ¢ 0, connecting the point on the first surface with the
centre of the aperture. Consequently the interior refracted rays, mq, m7, as well as
the exterior, mp, 2s, will be inclined outwards ; and it is obvious that there will be
a central bright space, limited by the lines mq, 27, each point of which will be illu-
minated by one interior and one exterior ray. The light in this space, therefore, will
have double the intensity of that of the surrounding space ; and as the rays which
combine to form it are polarized in planes at right-angles to one another, the result-
ing light will be unpolarized. When the aperture is diminished, the inclination of
the rays mq, n7, to one another is lessened, until finally they are reduced to
parallelism, and the central bright space contracts to a point. ‘This is represented in
(fig. 4.) When the aperture is still further diminished, the rays m q, nr, become in-
clined inwards, and cross (fig. 5.) It is obvious that beyond the point of intersection
there will be a dark space illumined by no ray whatever; and as in the surrounding
annulus there is no meeting of rays oppositely polarized, the whole of the light will
be polarized, and according to the law already explaimed. With a yet diminished
aperture, the rays mq, 2 1r, approach to parallelism with the exterior rays, ” s, im p ;
and the central dark space enlarges, and approaches to equality with the outer and
limiting cone. Thus the annulus of light in any section is diminished indefinitely in
breadth, and the cone approaches to a mathematical surface.
Now if we assume that the divergence of the two refracted rays in this plane, cor-
responding severally to the rays ¢ m, ¢ 0, ¢n, is the same, as must be nearly the case,
it will follow that the angle of the true cone, which would arise from the single ray
co, is half the sum of the angles of the exterior and interior surfaces of the conical
annulus ; and that when a bright circle appears in the centre, as is the case with a
considerable aperture, the dark space must be considered as negative, and the true
angle is half the difference of the observed angles.
From this it follows that when the central bright space is reduced to a point, the
true angle is just half the observed. Now this was very nearly the case in the expe-
riments from which the measures were taken ; consequently the corrected angle, de-
duced from these measures, coincides very nearly with that assigned by theory.
Two other measurements, taken since with a more direct reference to this correc-
tion, were as follows :—
VOL, XVII. 2Q
154 Mr. Luioyp on Conical Refraction.
1. Distance of screen from the aperture on the second surface of the crystal = 19.3
half inches. Mean diameter of section of exterior cone = 1.27. Mean diameter of
interior = 0.55. Corrected angle of cone thence computed = 2° 44.
2, Distance of screen = 11.9. Mean diameter of section of exterior cone = 0.93.
Mean diameter of interior = 0.41. Computed angle of cone = 3° 14’.
The mean of these two measurements is 2° 59’.
Inasmuch as the cusp-ray, within the crystal, corresponds to a cone of rays without,
it is evident that there must be a converging cone incident on the first surface, equal
to that which diverges from the second. With a view to determine its magnitude, J
placed a kind of rough micrometer, consisting of two moveable metallic plates, imme-
diately before the lens; and closed the plates until, on looking through the aperture
on the second surface, I could see them touching the circumference of the annular
section. The diameters of the interior and exterior circumferences of this section,
at the distance of the lens, being thus ascertained, and the focal length of the lens
measured, the corrected angle of the cone was found. ‘The mean of three measure-
ments taken in this manner gave for this angle 3° 47’. But the methods by which
this last result was obtained, do not seem susceptible of much accuracy.
Before I conclude this part of the subject, I may observe that an interesting yaria-
tion in the phenomena is obtained by substituting a narrow linear aperture for the
small circular one, in the plate which covers the first surface of the crystal—that
surface being close to the lamp. ‘The linear aperture is to be so fixed, that the plane
passing through it and the aperture in the plate next the eye, shall be the plane of
the optic axes. In this case, according to the received theory, all the rays trans-
mitted through the two apertures should be refracted doubly in the plane of the optic
axes, so that no part of the line should appear enlarged in breadth on looking through
the second aperture ; whereas, according to Professor Hamitroyn’s beautiful conclu-
sion from the same theory, the cusp-ray should be refracted in every possible azimuth.
I found accordingly that the luminous line was undilated, except in the direction cor-
responding to that of the cusp-ray ; and that in the neighbourhood of this direction
its boundaries were no longer rectilinear, but swelled out in the form of an oval
curve (fig. 7.)
When a very minute aperture was used on the surface next the eye, in this experi-
ment, the phenomenon was rendered much more remarkable. The swelling curves
in this case were separated by a considerable dark interval, and the luminous line was
prolonged into this dark space, terminating abruptly near its centre. ‘This appear-
ance is represented in (fig. k.) When the plate next the eye was slightly shifted, so
that the plane passing through the two apertures no longer coincided accurately with
the plane of the optic axes, the curves rapidly changed, preserving, however, in all
;
Mr. Luioyp on Conical Refraction. 155
cases, the form of the conchoid, whose pole was the projection of the axis of the
emergent cone, and asymptot the line on the first surface—(figs. /, m.) It is easy to
show that these results are in accordance with theory.
The second kind of conical refraction, whose existence has been anticipated by
Professor Hamitron, depends (it will be remembered) on the mathematical fact, that
the wave-surface is touched in an infinite number of points, constituting a small
circle of contact, by a single plane parallel to one of the circular sections of the surface
of elasticity. It takes place when a single external ray falls upon a biaxal crystal in
such a manner, that one refracted ray may coincide with an optic axis. When this is
the case, there will be a cone of rays within the crystal, determined by lines con-
necting the centre of the wave with the points of the periphery of the circle of con-
tact. The angle of this cone is equal to
tang. —1 Van Ne as 5
bz
and its numerical value in the case of Arragonite is 1° 55’, assuming the values of the
three indices as determined for the ray 7 by Professor RupBeEre ; (see page 151.)
As the rays constituting this cone will be refracted at emergence in a direction
parallel to the incident ray, they will form a small cylinder of rays in air. This
cylinder, it will be seen, is in all cases extremely small; for the diameter of its section
made by the surface of emergence: subtends an angle of 1° 55’ only, at a distance
equal to the thickness of the crystal. Hence the experiments required to detect its
existence and measure its magnitude, demand more care and precision than those
already described. The incident light was that of a lamp placed at some distance ;
and in order to reduce as much as possible the breadth of the incident beam, it was
constrained to pass through two small apertures, the first of which was in a screen
placed near the flame, and the second perforated in a thin metallic plate adjoining to
the first surface of the crystal. Under ordinary circumstances, it is obvious, the inci-
dent ray will be divided into two within the crystal, and these will emerge parallel
from the second surface. I was able to distinguish these two rays by the aid of a
lens ; and turning the crystal slowly, so as to vary the incidence gradually, I at length
observed that there was a position in which the two rays changed their relative places
rapidly, on any slight change of incidence, and appeared at times to revolve round
one another, as the incidence was altered. Being convinced that the ray was now
near the critical incidence, I changed the position of the crystal, with respect to the
incident ray, very slowly ; and after much care in the adjustment, I at last saw the
two rays spread into a continuous circle, whose diameter was apparently equal to
their former interval.
This phenomenon was exceedingly striking. It looked like a small ring of gold
viewed upon a dark ground ; and the sudden and almost magical change of the ap-
156 Mr. Luoyp on Conical Refraction.
pearance from two luminous points to a perfect luminous ring, contributed not a little
to enhance the interest.
The emergent light, in this experiment, being too faint to be reflected from a screen,
I repeated the experiment with the sun’s light, and received the emergent cylinder
upon a small piece of silver-paper. I could detect no sensible difference in the mag-
nitude of the circular sections at different distances from the crystal.
When the adjustment was perfect, the light of the entire annulus was white, and
of equal intensity throughout. But when there was a very slight deviation from the
exact position, two opposite quadrants of the circle appeared more faint than the other
two, and the two pairs were of complementary colours.* The light of the circle was
polarized, according to the law which I had before observed in the other case of
conical refraction. In this instance, however, the law was anticipated from theory
by Professor Hammron.
I measured the angle of incidence by a method similar to that already employed
for the emergent ray in the former case ; and found it to be 15° 40’. ‘This determi-
nation is, for many reasons, capable of much greater accuracy than the other ; and
was probably, in this instance, very near the truth.
In order to compare it with the result of theory, it is to be observed that the optic
axis isa normal to the wave-surface, and therefore the corresponding incident ray
will be given by the ordinary law of the sines, the index of refraction being the
mean index of the crystal. Now the angle which the normal to the circular
section of the surface of elasticity, or the optic axis, makes with the axis of x, or the
2 Hz
perpendicular to the surface, is equal to feo ya j ; andits numerical value
a
in the case of Arragonite, is 9° 1’. We have then
sin. c = 1.6863, sin. (9° 1’) ;
from which we finds = 15° 19’. The difference between this and the observed angle
SI
In order to measure the angle of the cone, I was compelled to employ a method
somewhat indirect, but (I think) susceptible of considerable accuracy. As the
aperture on the first surface of the crystal must have some physical magnitude, it is
obvious that instead of a cone of mathematical rays within the crystal, there will be
in all cases a cone of cylindrical pencils, overlapping one another near the point of
divergence ; and that the diameter of these pencils will be equal to the diameter of
the aperture. Now I tried a number of apertures, until I found one with which
these cylindrical pencils just separated at the second surface of the crystal. It is evi-
dent that, in this case, the interval between the axes of the cylinders at the surface of
* This part of the phenomenon appears to be explained by the non-coincidence of the optic axes for
the rays of different colours.
Mr. Luioyp on Conical Refraction. 157
emergence, is Just equal to the diameter of one of them, or to the diameter of the aper-
ture. 1 had then only to measure the aperture itself. ‘This was effected by the aid of
a micrometer divided to the 1-500th of an inch, placed along with the aperture before
a compound microscope ; and it was found to be .016 of an inch. ‘This therefore
was the diameter of the oblique section of the cone made by the surface of emer-
gence ; and the diameter of the circular section at the same distance was .016 cos. 9°,
since the axis of the cone makes an angle of 9° with the normal to the faces of the
crystal. ‘The perpendicular thickness of the crystal was .49 of an inch; and there-
: : : eee : 49
fore the thickness estimated in the direction of the axis of the cone was 95°
From these data the angle of the cone was calculated by the tables, and found to be
1° 50’; a result which differs from the theoretical angle by 5’ only.
SCIENCE.
an
LLL NT eee
An attempt to facilitate Observations of Terrestrial Magnetism.
By the Rev. Humeurey Lioyp, A. M., M.R.IA. Fellow of Trinity College, and
Professor of Natural and Experimental Philosophy in the University of Dublin.
Read October 28, 1838.
Since the important researches of Baron Humsotpr, Captain Sapine, and Professor
HanstTeen, much interest has been awakened on the subject of Terrestrial Magnet-
ism. The intensity of the magnetic force, and the dip, have been determined at
various parts of the earth’s surface, and are found to follow laws agreeing re-
markably with the results of theory. But the experimental data necessary for the
complete solution of this important problem are, as yet, far from adequate; and ac-
cordingly, the multiplication of observations in remote quarters of the globe, has
engaged the zeal of many a scientific individual, and even called forth the resources
of more than one state of Europe.
The imperfections of our instruments of observation have long presented a for-
midable obstacle to the advancement of these inquiries. Many of these have indeed
been guarded against by the skill of the observer, and by the application of suitable
corrections. ‘There are still, however, some inherent sources of fallacy in the instru-
ments themselves, which seem almost inseparable from them in their ordinary form.
The defects of the common dipping needle have been long acknowledged, and still
continue to embarrass observers. The chief of these defects arises from the nature
of the suspension. ‘The directive force, it is well known, diminishes with the angular
distance from the true line of the dip, varying as the sine of that distance. Hence
when the needle is near the position which it would assume under the influence of the
terrestrial action, if free to move, the directive tendency may be so small as to be
more than balanced by the friction of the axle against its supports ; and in this case
the needle will rest out of the position in which the earth’s force would place it. This
evil will of course be increased by any irregularity in the form of the axle itself, and
such irregularities can never be wholly removed. To this is added another source of
error, arising from the non-coincidence of the axle with the centre of gravity of the
VOL. XVII. 2k
160 Attempt to facilitate Observations of
needle. Where this fault exists, gravity unites with magnetism in determining the
position of equilibrium of the needle, and it will assume a direction different from that
of the resultant of the terrestrial magnetic forces. ‘The error arising from this cause,
it is true, is in a great degree corrected by the process of reversing the poles of
the needle, in the usual mode of observation ; but still it cannot be said to be wholly
removed, nor are the methods of correction altogether unobjectionable.
These and other difficulties in the direct determination of the dip, have led to the
occasional employment of indirect methods. ‘Thus, if the total magnetic intensity be
known, together with its vertical or its horizontal component, the dip is readily found
by elimination, A method founded on this principle was suggested by Lariace, and
practised by Humsotpr and Captain Sabine. The total intensity was determined by
noting the time of vibration of a dipping needle, oscillating in the plane of the mag-
netic meridian ; and the vertical component was similarly obtained from the time of
vibration of the same needle in the plane at right angles to the former: the ratio of
these two forces is equal to the sine of the dip. Captain Sazinr has also employed
another method derived from the same principle. The horizontal component of the
terrestrial magnetic force was found by observing the time of vibration of the needle
suspended horizontally : and this component, divided by the total intensity, is equal to
the cosine of the dip. The dipping needle, however, seems to be badly adapted to
observations of vibration. Owing to the friction of the axis, the needle is soon
brought to rest, unless the initial arc of vibration be considerable ; and, when this is
the case, the observer is involved in many sources of error, against which it seems dif-
ficult to guard.
A method similar to these in principle, but free from the objections just stated, was
adopted many years since by Coutoms. In this method, the elements obtained by ob-
servation were the vertical and horizontal components of the intensity, the ratio of
which is equal to the tangent of the dip. The horizontal component was determined,
as in the last case, by observing the number of oscillations performed in a given time
by the needle suspended horizontally ; while the vertical component was found by
determining the weight of a counterpoise, placed at a given distance on the southern
arm of the needle, and sufficient to bring it to the horizontal position. ‘The moment
of this counterpoise, or its weight multiplied by its distance from the axle, is ob-
viously equal to the vertical component of the magnetic intensity. M. Bror considers
this as the most accurate method that has been ever applied to the determination of
the dip. It seems, however, open to very serious objections, which have been pointed
out by M. Pourtier. In order to determine the horizontal component from the time
of vibration, so that it may be comparable with the other component determined sta-
tically, it is necessary that the moment of inertia of the needle should be known a
priori; and that this should be, the needle must have a regular mathematical form.
Terrestrial Magnetism. 161
That used by Coutomp was of uniform and very small dimensions throughout its en-
tire length, and in the calculation was treated as a mathematical line. ‘There are
other difficulties connected with the practice of this method, on which it is unnecessary
to dwell: so that, notwithstanding its high sanction, it has not, as far as [ am aware,
been adopted by any other observer.
I have alluded to the method of Coutoms in this place, because it suggested that
which I am now about to propose. It occurred to me that the components of the
magnetic intensity might both be determined statically, and by one and the same pro-
cess. Thus all the objections, to which Coutoms’s method is liable, would be avoided,
and the process itself rendered much simpler of application, as well as more accurate
in its results.
Let us conceive a magnetic needle suspended as the ordinary dipping needle, and
placed in the magnetic meridian ; and let us calculate the directive effect of the ter-
restrial magnetic force upon it in any position.
If ¢ denote the quantity of the magnetic fluid in any point of the needle, and »
the terrestrial magnetic force acting on the unit of quantity, ¢ q will be the force ex-
erted on the point in question. Let é denote the angle which the direction of this
force makes with the horizon, or the dip, and let @ be the angle formed by the needle
itself with the horizon : then é—@ is the angle which the direction of the force makes
with the needle ; and consequently, the moment of the force tending to turn the
needle round its axle is ¢ q7 sin (©—@); r being the distance of the point in question
from the axle. Now, to obtain the total effect of these forces acting on all the points
of the needle, we have only to multiply this moment by the element of the mass, dm,
and integrate the result. The total directive effect of the earth upon the needle is
therefore
S ¢ qrdm. sin (8—6) = ¢sin (6-0) f grdm.
The integral /grdm, in this expression, will depend on the form of the needle, and
on the law of the magnetic distribution in it. Let its value, taken within the limits
of the dimensions of the needle, be denoted by «: then the moment of the terrestrial
magnetic force, tending to turn the needle round its axle, is expressed by
po sin (6-86).
Now, if a weight be placed on the southern arm of the needle, its effect to turn the
needle in the opposite way is
pm Cos 8;
n being the moment of the mass, or the weight multiplied by its distance from the
axle. Accordingly, in the case of equilibrium, we have » cos 0 = ¢o sin (¢—8)
Or, u = ¢o (sin é—cos é tan 6) (1)
162 Attempt to facilitate Observations of
Now, let us suppose two weights to be placed in succession on the southern arm of
the needle, and let their moments, »’ and »”, be adjusted so as to bring the needle into
the positions forming half a right angle with the horizon, below and above; the corres-
ponding values of @ then are +45° and —45°, and substituting, we have the two equa-
tions following
nu = o (sindé—cos 0), nw’ =@ a (sin 8 + Cos 0).
Adding and substracting these equations, we have
uw +p =2oosind, pb’ —p =296 C08 6.
from which we obtain,
‘ tan 8H po V2= yn te (2)
wu
Accordingly, to determine the intensity of the terrestrial magnetic force, as well as
the dip, we have only to find the moments of two counterpoises which will bring the
needle into the two positions above mentioned ; the sum of these divided by their dif-
ference is equal to the tangent of the dip, and the square root of the sum of their
squares furnishes a measure of the intensity. Another set of results might be obtained
by observing also the moment of the counterpoise which brings the needle to the
horizontal position ; for it will be easily seen that this moment is equal to the arithme-
tical mean between the two preceding, or equal to ¢ o sin 6.
In the practice of this method, it is obvious that a divided circle is not required; it is
only necessary to place four lines on the frame of the instrument corresponding to the
inclinations of 45°. It is requisite, however, to have some means of determining the
moments of the counterpoises with facility. For this purpose I procured a needle fur-
nished with a small sliding weight, and having a scale on its southern arm, divided to
the 1-50th of aninch. I had then only to observe the division on the scale coinciding
with a fixed mark on the sliding weight, when the needle was brought to the two
required positions. 2
Simple as this method appears, a few trials were sufficient to convince me that it
would prove a most inconvenient one in practice. The adjustment of the counter-
poise, so as to bring the needle to the required inclinations, was found to be a tedious
and delicate operation, requiring a frequent removal of the needle from its supports ;
and the repetition of this process, in the various positions of the needle and instrument
in which an observation of dip is usually taken, multiplies, of course, the difficulty.
For these, and other reasons, it appeared far preferable to observe the positions
which the needle assumes with fixed counterpoises, rather than the counterpoises which
bring it into fixed positions. It will be easily seen, that two such observations are
sufficient to determine the dip and the magnetic intensity.
Terrestrial Magnetism. 163
Let »/ and »” be the moments of the two fixed counterpoises, and @ and 6" the
corresponding inclinations of the needle to the horizon: then, substituting in
equation (1),
nw =o (sin —cos o tan), pw = o (sin 6—cos 6 tan @”).
We have thus two equations containing % and 8, from which these quantities may be
obtained by elimination. To effect this, let the former be divided by the latter, and
we find
me tan 6—tan 4 ! a pw tan = tan 6’ :
nw” tand—tan 6 wap
y
And denoting the constant factors ar and gee by v' and v’,
we have finally,
tan =v’ tan 0” —v” tan 0’. (3)
If we eliminate 8 between this equation and either of the two equations given above, °
we readily obtain an expression for @ in terms of /, 1’, and @, 0”; but as the result-
ing expression is somewhat complicated, it will be much simpler in practice to obtain
the dip, in the first instance, from the equation last deduced, and to substitute its
value in the formula
pcos 0
Ke o sin (0 Liiy
The quantity o, in this formula, depending on the law of distribution of the magnetic
fluid in the needle employed, is unknown ; and, consequently, the absolute intensity
of the terrestrial magnetic force cannot be determined without some other artifices.
In general, however, the ratio of this force at different parts of the earth’s surface is
alone required ; and for this purpose the force at some given place (usually the mag-
netic equator) is taken as unity, and the force at other places determined by compa-
rison. Let ¢, be the force at the given place, and 6, @ the corresponding values of
8 0, then
_ cos 8,
‘~gsin(8,—0) ”
and dividing
i ere vee
¢, sin (8—6) cos 0,” (4)
164 Attempt to facilitate Observations of
The counterpoises I employed, in the practice of this method, were small pieces
of brass wire inserted in holes which were drilled at different distances along the axis
of the needle. Finding that the weights which were sufficient to balance the magnetic
force, at a moderate distance from the axle, were unmanageably small, I used two coun-
terpoises, one of which was a weight of one grain, placed on the southern arm, at the
distance 1.38 inches from the axle ; the other was a weight of .7 of a grain, placed on the
northern arm, at the distances 1.1 and 1.38 inches successively. ‘Thus the moments
of the actual counterpoises were the differences of those on the two arms, added to the
difference of the moments of the arms themselves.*
It was now necessary to determine with accuracy the ratio of these moments ; and,
as this could hardly be effected with the requisite precision by any direct measure-
ments, I determined to use for the purpose the means suggested by the proposed
method of observation itself. This method, it has been shown, determines the dip by
means of the two positions of the counterpoised needle, when the ratio of the moments
is known: therefore, conversely, if the dip be otherwise found, the same formule may
be used to calculate the ratio of the moments. The following observation was accord-
ingly taken for this purpose :
Killiney, Sept. 11. Needle I.+ Killiney, Sept. 12. Needle II.+
eo E marked side w hepa Er marked side w
22) —— —~ fees, oa
Seen s N s Sea N s N s
E
| Tee [Lo 7s Oa ‘ zr] — 18° 45'— 18° 37'| —19. 0’—18° 52’
wi70 45 70 45/70 50 70 50 w|—20 25 —20 15) =21° 0 =21%0
fs 71 10 71 20|71 30 71 45 nf +23 10 +23 20/ +25 40 +25 55
wi70 45 70 40/71 10.71 5 wl +22 30 +22 85| +25 20 +25 25
Mean |70°56.8 71°O 71° 11.3 71° 17.5|Mean | 6 = —19°44 O" = + 94°14"
Final Mean=71° 6'.4.
Substituting the values of ¢, 6’ and 6”, thus obtained, in the equation,
’
w _ tan o—tan
nu” tan §_tan 6” ’
we find the ratio of the connterpoises to be 1.3274.
* The moments of the two arms of the needle were rendered unequal by the weights which were
unsymmetrically abstracted in the process of drilling the holes.
+ Needle I. is the common needle—Needle II. the counterpoised needle.
Terrestrial Magnetism. 165
It now remained to take a series of observations, with the common as well as the
counterpoised needle, and to compare their results. The following tables contain the
results of two such series. The first series of observations was taken at places differing
in dip: the second series at the same place. The poles of the counterpoised needle
were not reversed, as such a process would be incompatible with one of the objects of
the needle—namely, the determination of the variations of terrestrial intensity ; and it
fortunately happens, that this troublesome operation is, in the present method, wholly
unnecessary. The use of reversing the poles in the common needle, it is well known,
is to correct for the moment of the needle itself. If the centre of gravity of the
needle deviates at all from the centre of motion, in the direction of its length,
the moment of the needle will increase or diminish its inclination to the horizon,
according as the deviation is towards the north or south pole; and this error
can only be corrected by reversing the poles of the needle, so that its moment may
act in opposite directions in the two cases.* But, in the needle proposed in these
pages, the centre of gravity of the whole is made intentionally to deviate from the
axle, and the dip determined from the altered inclinations. In this case, accordingly,
the moment of the needle, if any, simply increases or diminishes the moments of the
actual counterpoises ; and, as these are determined a posteriori, the moment of the
needle itself will of course be included in the determination.
-
* It may be readily shown, that the tangent of the true dip, is the arithmetical mean between the
tangents of the observed inclinations of the needle, when the same end is a north and south pole. But
when the difference of the observed inclinations is small, we may, without much error, suppose the same
relation to hold amongst the angles themselves.
166
Attempt to facilitate Observations of
TABLE I.— Containing Observations of the positions of the Common and Coun-
terpoised Needle, at places differing from one another in Dip.
Observatory, Armagh, Sept. 16.
|
Dundonnell, County of Down, Sept. 23.
3 3 - E marked side WwW E marked side WwW
8 35 aS SS SSS SS —“—_
2 N s Ain s N s N s
cn aale \" Tee We aI Ga il bar (BO YE Oe (aS aoe
3 bad 71 ile aa A ae aii eirsi la SOM cpa)
a sf 270. “Yo. 2 Ts” 2 “ee yi 5 7 OF” ONeeeee
3S |
Z tefl bas iLO | 71 710) ey 5 71 20 71 #15 | TL 22, a es
Mean | 71 43 WL (3¥.2)|ial ale 7 oeeon| 71 avo V1) 36:5 | (#1, ATS 7 40
Final mean = 71° 39’.7 Final mean = 71° 39’
§ 2.2 E marked side WwW E marked side WwW
eI aie (as SS (SS SSS) SS
oS own s N s N s N S
=) gE |~21° 20 =21° 15 | 22° 10°—22° 7 |~21°10 -—21° | —22° 50’—22° 50
salt > i =21 25 —21 15 |—22 5-21.55 |-22 0 —21 55 | —22 52-22 45
a io
a pn |+22 380 +22 30 | +25 40425 35 |+22 5 +22 5 | +25 15425 20
a ny | +22 0 +22 5 | +23 45423 45 }+22 7 422 7 | +24 40424 40
Mean |. & = — 21° 41.5 Yo 4008" ese) \@ 922° 4 = 423722!
Calculated dip = 71° 26’ Calculated dip = 71° 38’
Museum, Belfast, Sept. 28. Garden, Trinity College, Oct. 21.
3 mate E marked side Ww E marked side Ww
ES BE caicadnnina paws ss SS)
SA N s N Ss N Ss N s
st ra 72 20 We (22 | 72 15 72 415 71 16 epi Ns) | 71 LO 7 Lae
A “Une 71 30 71 25 | W380" eI 2s8 70 43 70 35 |70 33 70 30
a 5F 72 #415 72 20 | 72 LOY W742 15 7A UG) 7 201 18) ieee
“Vy 7 3 7 37 | 71 30 71 20 | 70 36 70 34 | 70 35 70 32
Mean 71 do 71 56 | 71 51.2 71 49.5 | 70 57.5 70 58 | 70 54 70 56.2
Final mean = 71° 52’.9 Final.mean = 70° 56.4
A 2.9 E marked side WwW E marked side Ww
50.8 SS aw) Tae meD. a ae
gy Ns S N s N s N s
5 E |—21° 10'—21° O | —21° 45'21° 35’ |—23° 27' _23° 30° | —23° 42/—23° 41!
a :
5 : se = 22.7 92 AF Vi 82 87-92 Sols 16 =/94) 17. | —2515 a7 ee oneee
S|
2 gE |+22 40 +22 40 | +25 30+25 35 |420 27 +20 27 | +21 40+421 43 |
uf +22 52 422 45 | +25 30+25 40 |+19 52 +19 52 | +22 37422 32
Mean | %= ~ 21° 51’ o” = +24 9 d'= + 24° 91.2 6’=+21° 81.8
Calculated dip = 71° 49 Calculated dip=71° 12’
|
Terrestrial Magnetism. 165
TABLE I1—Containing Observations of the positions of the Common and Coun-
terpoised Needle, taken in the Philosophy School of Trinity College.
i October 22. Therm. =56}° October 23. Therm.=57°
3 io E marked side W E marked side Ww
go4 SSS PS a a ae
She | N Ss N Ss N Ss N s
71° 20 «712 20 «| 71° 18 71° 20 «{ 71° 1s 71° 7 «| 71° 1s 71° 22
E
Sf | a0 70 30 | 70 25 70 30 |70 8 70 30 |70 35 70 30 |
a i Re 22 | 71 15 71 ee ee) ee
£/° wl 70 35 70 30 | 70 50 70 50 |70 35 70 25 |70 30 70 15 |
Mean | 70 57.5 70 55.5 | 70 56.2 71 0.5 /70 55 70 50.5 | 70 53.8 70 53
Final mean=70° 57'.4 Final mean=70° 53'.1 ip
a v9 E marked side Ww E marked side Ww
5 25 Se ee SS aes = ee ee
3 (= e5)| eis s N s N s N s
=| = |—27° 52° ~27° 52’ | —29° 20'29° 29’ |_12° 30” 12° 29° | 14° 15'—14° 30’
ie oe —28 45 —28 45 | —28 52-28 52 |-13 387 —13 37 | —-15 55-15 50
a ,|F16ulS +16 15 | +20 15420 25 426 7 +26 15 | 429 35429 45
a uf +15 15 +15 15 | +19 0+19 0 |+26 0 +26 0 | +30 10+30 10
Mean Qe 25 An OT = ao ADS | 6'= —14° 4.5 6’ = +28° 0.2
Calculated dip = 71° 22'.5 | Calculated dip = 71° 6’
(1) October 24. Therm. =58}° (2) October 24, Therm.=59°
3 3A E marked side Ww E marked side w
a 5-5) ——A~A—~ oN ae aa ait ae
id ee s N s N s N s
pijZede 71222, [7197 71° ger | 71° OF 7° IF | 71° lo” 7io oR
“ i 70 40 70 30 | 70 30 70 22 |70 45 70 45 | 70 30 70
8 FW AAG PO NAL 9) Fy The (| 7120) 71.40 (71, % 71 do.”
S ie 70 35. 70 25 {70 30 70 30 |70 35 70 30 | 70 35 70 30
“\Mean | 70 56.2 70 54.2 | 70 48.5 70 52.2|70 55 71 2.5 | 70 50.5 70 56,2
Final mean=70° 52’.8 Final mean=70° 56’
3 %.0 E marked side Ww E marked side Ww
EZ: fi A a a
mis} Ae) oN s N s N s N s
+ g |—35° 30 —85° 45° | —38° 10 —38° 15 |~10° 457 —10° 40° | —12° 457—12° 45°
5 a —38 7 —38 0 |—39 55—39 55 |-12 45 -12 45 |-14 0-14 0
a p|+6 35 +6 37 [+8 45+ 850 |+26 30 +26 30 | +30 30+30 30 |
nf lea 15 +4 20 |+ 6 304 630/427 0 +27 0 | +30 30430 30
Mean, = — 87? 57 o" = + 6 32/8 = — 12° 33 On = 428° 27.5 ||
Calculated dip = 70° 39’ | Calculated dip = 70° 56’ !
VOL. XVII. 2s
164 <lttempt to fucititate Observations of
From the first of these Tables we see that the values of the dip, obtained by the
common and counterpoised needle, do not differ from one another by 16’, at any of
the four stations ; while at two of the stations the difference is considerably within the
limits of error of a single observation of dip, taken in the usual manner.
The general agreement of the results of the two methods being thus established, the
second series of observations, given in Table I, enables us to compare the methods
with respect to accuracy ; for these observations being all taken at the same place, the
consistency of the results will afford the means of estimating the probable limit of
error in each method.
Comparing then, the results of Table IT, it will be seen in the first place, that the
mean of the four observations made with Needle I, gives 70° 55’.1 for the dip, in the
Philosophy school of Trinity College ; while the mean of the observations taken
with Needle H, is 71° 0.9, differing from the former by less than 6’.
The results of Needle LI, are, however, by no means as consistent among themselves
as those of Needle I. The greatest difference between any one result and the mean,
with the common needle, is only 24’; while, with the counterpoised needle, the cor-
responding difference amounts to 22.’ This, however, can scarcely be regarded
as decisive against the accuracy of the method. The common needle was one of re-
markable nicety of construction ; while that used with the counterpoises, was obviously
erroneous. It was necessary for the consistency of the results that the points of appli-
cation of the counterpoises, should coincide exactly with the magnetic axis, or cen-
tral line of the needle. Now, no yery precise means were adopted to insure this coin-
cidence, in drilling the holes for the counterpoises; and that the coincidence was in fact
not effected, will readily appear on looking over the results. It will be seen that, when
the second counterpoise is employed, there is a difference amounting sometimes to 4°
between the angles read off with the face of the needle in opposite positions; and
this plainly indicates an error in the place of the second hole on the northern
arm of the needle. It is to be observed, further, that the magnetic state of the nee-
dle was widely different in the several observations, having been altered before each
trial by a pair of bar magnets ; so that the method was subjected to a more severe
test than any to which it could be exposed in practice. Even with these disadvan-
tages, however, the values of the dip obtained with this needle do not appear to
differ from one another more than is common in such observations ; so that the result
of the trial cannot be regarded as unfavorable.
During the progress of the observations recorded in Table I, I made a cotempora-
neous series of observations on the rate of vibration of Needle II, suspended horizon-
tally ; with the view of ascertaining how far the results obtained with this needle could
he relied on for the determination of the force. For this purpose the magnetism of
the needle was altered by a pair of bar magnets, and its rate of vibration ascertained
in the usual manner, after each change in its magnetic condition, by a good chrono-
Terrestrial Magnetism. 165
meter. The observation of the angles of position of the same needle, when counter-
poised, was taken immediately before or after, and, as nearly as possible, at the same
temperature. But unfortunately the moment of inertia of the needle was unguard-
edly altered during the observations, so that no reliance could be placed on the results.
Since the preceding trials were made, it appeared to me desirable to examine theo-
retically the accuracy of the counterpoised needle in its different positions, and to as-
certain, if possible, the most eligible. It will appear from the foregoing observations,
that between two distant places, the range, or change of position of the counter-
poised needle, is considerably greater than the corresponding change in the dip. This
I had been previously led to expect from theory, and had found that the change of
position, corresponding to a given change in the dip, would be greatest when
tan20=1tand. Whend=70", this formula gives, for the angle of maximum range,
6=17° 15’; and the ratio of the range itself to the corresponding variation in the dip
was found, in this case, to amount to 3.275. I was thus led to expect that changes
of dip, which would be inappreciable by the common needle, might be detected with
this. It appeared, however, on trial, that the limits of error were likewise increased
in this needle ; so that it was necessary to examine it in another point of view.
Resuming the original equation
4 cos 0=0 (6—8),
and differentiating and dividing by the equation itself, considering 0, 3 and ¢ as all va-
riable, we find
cos 9 sin (6—6) “ +cos @ cos (6—6@) dé—cos 8 dd=0.
This equation will give the error in the dip, or in the force, corresponding to a
given error in the position of the counterpoised needle ; for, making dg and dé, suc-
cessively, equal to nothing, there is
ae cos 6 dé dp _ cos 6 dé
me = cos 0 cos (8—0)° @ cos @sin (6—@) *
(5)
Hence, supposing the error in the position of the needle, d0, to be given, we can find
the directions of the needle, in which the resulting error in the determination of the
dip, or of the force, shall be a minimum. It is easily seen that dé is a minimum, when
Ip . 5 3
6=49; and that pis is least, when 9=48—45°. Hence, in our latitudes, for the de-
termination of the dip and of the force, the counterpoises should be such as to bring
the needle into the positions forming the angles +35° and —10°, respectively, with
the horizon.
o
166 Attempt to facilitate Observations of
Such then should be the inclinations of the needle, in order that any constant error
of position, dé, may have the smallest influence on the calculated dip and inten-
sity. Of such constant errors the most obvious is the error to which the observer is
subject in reading the angle on the limb, arising from the smallness or imperfection of
the divisions. But the error of reading is not the only, or even the most important
error, to which we are liable in determining the position of the needle. It has been
already stated that, owing to the friction of the axle, the needle is often brought to
rest out of its true direction: now the error of position arising from this cause is,
in struments of the usual size, of greater magnitude than the error of reading, and
that magnitude is different in the different positions of the needle.
In order to determine the amount of this error, it will be necessary to consider the
directive force, by which the needle is urged to its position of equilibrium. This
force is obviously the difference between the magnetic moment, ¢o sin (6—4), and
the moment of the counterpoise, » cos 6; so that, if its magnitude be denoted by F,
F= $s sin (8—@) —p cos 9.
But, if @, be the position -of equilibrium, there: is ¢ «sin (8—0)—y cos 6=0; and
substituting the value of «, obtained from this equation, in the preceding formula,
and observing that
sin (S—@) cos @ —sin (6-6) cos @=cos é (sin @, cos @—cos @ sin #) =cos 6 sin (0,—6),
we have finally
cos }
F=$¢o6 cos 0, sin (0 —6).
Hence the directive force varies as the sine of the angular distance from the position
of equilibrium. Accordingly, when that angular distance is reduced to a certain limit,
the force becomes equal to the friction, and is balanced by it. Let « be the magnitude
of the angle, #, — 8, when the directive force becomes equal to the friction, f; then
cosé.. cos}
S=¢ooe eee sm <=po0 ——€ 3
since < is small; and therefore
cos 0
feos 0
E Tema : (6)
The angle «, thus found, is obviously the limit of error to which we are liable in
determining the position of the needle, arising from friction. Its value depends, as
we see, upon the force of friction, the intensity of the terrestrial magnetic force, the
magnetic moment of the needle, and its position with relation to the dip. In order to
determine the resulting error which it will produce, in the determination of the dip
and intensity, we have only to substitute its value for 0, in the equations (5). We
find, in this manner,
Terrestrial Magnetism. 167
da bie or Bopabihb ph 200 bt
: ~ pa cos (8—8) ’ ¢ posin(o—8) ’
or, denoting the limit of error arising from friction in the ordinary position of the
needle by « ,
d3 = ¢ sec (3-6) , tae cosec (3—0). (7)
We learn, then, that the error in the determination of the dip, arising from friction,
is least when 8—0=0; and that the smallest value of d8 is ¢.* The corresponding
error in the determination of the force will be a minimum, when §—6@=90° ; in which
d 3 A ‘
case ee fee being expressed in parts of radius.
As far then, as friction is concerned, it would appear to be the most advantageous
modification of the method suggested in the preceding pages, to observe the position
of the needle, in the first instance, without any counterpoise, and, afterwards, with a
counterpoise which will bring it into a position nearly perpendicular to the line of the
dip. ‘The former of these angles is the dip itself; and the two angles, when substi-
tuted in formula (4), furnish the measure of the intensity. But, in order to avoid
the error arising from the non-coincidence of the centre of gravity with the axle, I
think it would be far better to use a small counterpoise in the first instance; or
even to consider the moment of the needle itself, (or its weight multiplied by the
distance of its centre of gravity from the axle,) as a counterpoise acting with or
against the magnetic moment. The ratio of this moment to that of the other coun-
terpoise should, of course, be determined by‘ the indirect method which has been
already explained.t
We may now form an estimate of the accuracy of this method, as compared with
the usual one, in the determination of the magnetic intensity. In the received me-
thod, it is well known, the horizontal component of the force is determined by ob-
serving the time of vibration of the needle suspended horizontally. Now let us sup-
pose this portion of the force to be completely determined, and inquire how far the
probable error in the dip will affect the total intensity, thence deduced. If h denote
the horizontal component, we have
h=9ocos 8;
* We have here considered the effect of friction on the result, so far as it depends upon a single reading.
The eight readings usually taken may undoubtedly diminish still further the resulting error of dip; but
as these readings are taken in order to correct other errors, we have disregarded their effect here. If, how-
ever, the needle be so perfect in its construction, that the errors arising from the non-coincidence of the
centre of gravity with the axle, and the deviation of the magnetic axis from the axis of the needle, &c.
are less than the error of friction, then the multiplication of readings will have the effect of reducing
the latter in the ultimate result.
f See page 6.
168 Attempt to facilitate Observations of
and differencing and dividing, /, being considered as constant,
i
P83 tan 3.
?
Loe
Now, confining our consideration to the error arising from friction, dd=— = ¢; so
o
that the /imit of error in the determination of the force, in the usual method, is <’ tan é ;
while, in that now proposed, it may be reduced to. ‘The limit of error in the
common method, therefore, is to that of the method now proposed, in the ratio of
the tangent of the dip to unity ; that is, in our latitude, as 2.75 to 1, nearly.
In the instrument in my possession, constructed by Mr. Roginson, of Devon-
shire-street, London, I consider the limit of error arising from friction, in the ordi-
nary position of the needle, not to exceed 5’. With this instrument, therefore,
¢ =5'=.0015, nearly, radius being unity; so that the error in the determination of
the force does not exceed the .0015 part of the entire quantity.
In order to satisfy myself more fully as to the accuracy of this method, the follow-
ing series of observations was taken. The position of the needle (Needle I of the
preceding experiments) was observed, first without any counterpoise, and secondly
with a counterpoise which brought it into a position nearly perpendicular to the line
of the dip. ‘The observations were taken in the same spot of the room, and as nearly
as possible at the same temperature ; the magnetism of the needle was not interfered
with during the observations.
TABLE IIL. Containing the results of Four Observations taken with Needle I. in
the Philosophy School of Trinity College. Therm. = 61°.5.
!
| oe | I. E marked side Ww | II. E marked side Ww
ple ————oe ————— a eo
<4 ls s N s | oN s N s
(Ef eM% OF WUABofy TI 1B FI Wi pomislo MMO) 7A oMoweT
é-
lw! 70 35 70 30 | 70 25 70 22) 70 30. 70 22 | 70 80 70 28
#510 -sdvsavelso opis pasts inlay oleliy! (7 paws Ise
6
= —19 40 —19 37 | —18 15 —18 5|—19 40¥—J0 .45"|)'=1s8" 16°=18
[Mean | $= 70° 51.8, | 9=—17° 37'.4 8 = 70° 50; 0=—17° 32’
Liee.3 ill. E marked side WwW DV; E marked side Ww
Sue — a SSS aa
}<H | oN S N Ss N iS N s
| — .
8g To O° “MOM eis” Tie Sor me a! 7 IS) zi 1 ee
| tw) 70 30 70 25 | 70 30. 70.25) 70 30 70 22 | 70 30 70 30
| 52-16 65°=17...00 | =15 15 +15 -17 22-17 35 | —15 20 —15 20
0
Uwi-l9 30 —19 30 | —17 45 —17 45|-19 45 —19 45 |-17 55 —17 55
Mean | § = 70° 50'.6, @—-17° 20.9 $= 70° 50.2, @=-17° 371
Terrestrial Magnetism. 169
The mean values of 8 and @, as deduced from these four observations, are
=//02. 00) 7 ed — Ure o.6 3
and if we take the force corresponding to these mean values as unity, the numerical
values of the force, derived from each separate observation, will be obtained by sub-
stituting the corresponding values of 6 and 6, given by the preceding table, in the
formula
log ¢=log cos 6—log cos 6 + log sin (8, —0,) —log sin (¢—6).
We find in this manner
$1=-9994, $2=1.0000, g:=1.0011, ¢,=.9995.
The greatest deviation of these values from the mean is .0011; so that the limit of
error in the determination of the force, would appear from these observations to be
little more than the one-thousandth part of the entire quantity.
It appears then from the preceding, that the Terrestrial Magnetic Intensity may be
determined, together with the Dip, with the aid of a single instrument, and by a pro-
cess even somewhat less troublesome than that by which the dip alone is usually de-
termined.* To any one who considers the numerous precautions required, in the
common method of determining the magnetic intensity, the saving of time and labour
thus effected will be abundantly obvious. But it is an advantage of much greater
moment, that the results of the proposed method, so far as the intensity is con-
cerned, will be less liable to error than those obtained in the usual manner, as long as
the dip exceeds 45°; and that, in our latitudes, the accuracy of the new method is
nearly three-fold that of the old.
* The same number of readings is taken in the two cases, while in the proposed method the process of
reversing the poles is dispensed with. <—
—-~
Ory
L Stages
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On a New Case of Interference of the Rays of Light. By the Rev. Humpurery
Luoyp, A. M., M.R.I.A. Fellow of Trinity College, and Professor of Natural
and Experimental Philosophy in the University of Dublin.
Read January 27, 1834.
Tue experiment of Fresnex, on the interference of the lights proceeding from the
same origin, and reflected by two mirrors inclined at a very obtuse angle, has been
justly regarded as one of the most important in the whole range of physical optics.
The principle of interference itself had, indeed, been stated broadly by Youne, and
supported by the evidence of phenomena, which, to the unbiassed inquirer, left little
to desire. All these phenomena, however, admitted of other possible explanations ;
and the advocates of the corpuscular theory of light had recourse to these, rather
than admit the truth of a law which afforded such strong support to the undulatory
theory. In most of these phenomena, the light was in part intercepted by an ob-
stacle, and it was conceived that, in passing by the edge, the molecular action, which
might be supposed to exist between the particles of the body and those of light, was
sufficient to account for the facts observed. But, in Fresnet’s experiment, the two
lights which interfere are regularly reflected by the surfaces of the mirrors, according
to the ordinary laws, and are divested of every extraneous circumstance which could,
by possibility, be supposed to influence the result. This experiment, accordingly, has
materially changed the character of the controversy respecting the nature of light ;
and the advocates of the Newtonian theory, of the present day, are forced to admit
the principle thus rigidly established, and labour only to show how the theory and that
principle may be reconciled.
While examining this important experiment—the adjustment of which is a matter
of some delicacy—it occurred to me that the fact of direct interference might be
shown in a yet simpler manner, by the mutual action of direct and reflected light.
An interference of this kind was assumed by Youne to account for some of the phe-
nomena of diffraction; but Fresne~ showed that the explanation was incomplete,
and that the phenomena in question were caused merely by the interference of the
secondary waves, reflexion playing no part in their production. Under these circum-
VOL. XVII. 2T
172 On a New Case of Interference
stances it is somewhat strange that the fact of the interference of direct and reflected
lights should not have been itself submitted to the test of experiment ; especially as
the character of this interference, if it were found to exist, might be expected to throw
some light upon the laws of reflexion itself.
The theory of such interference is easily deduced from the general principles. Let.
light proceeding from a single luminous origin fall upon a reflecting surface, at an
incidence of nearly 90°: a screen placed at the other side of the reflector will be
illuminated, throughout a certain extent, by both direct and reflected lights; and, if
the difference of the paths traversed by these lights amount only to a small multiple
of the length of an undulation, the two lights will form fringes by their interference.
Let the intensities of the direct and reflected lights be denoted by a’ and a*, and
that of the resulting light by 4’; then, by the theory of the composition of coexisting
vibrations, we have
5 r o—8
J=7> 2 I,
A=a +2 aa’ cos Ix ( x ) +a :
3 and 8 denoting the lengths of the paths traversed by the two waves, from their origin
to any given point, and ) the length of an undulation.
The intensity of the resulting light will be a maximum, and equal to (a +a’), at
those points for which
cos 2m (2—*) = + as or 8—d= ans ;
It will be a minimum, and equal to (a—da' ), when
cos 2x (2) = 1, or 8—8= (@n+19
D >
n being any number of the natural series 0, 1, 2, 38, &e. Bright fringes, therefore,
will be formed at all the points included in the former equation, and dark ones at the
points corresponding to the latter.
i
A
Let op be the reflector, om the screen placed in contact with it, and perpen-
dicular to its plane; and let a be the luminous origin, and a’ its reflected image at
an equal distance below the line os ; then if m be any point, whose illumination is
required, 8=am, 6 =a'M.
Of the Rays of Light. 173
Now if aB be denoted by p, Bo by d, and om by 2, it is obvious that
e=P+(p—-ryY, S=ae+(p+ax).
T’rom these equations we have approximately
es p—r\?* i ‘aa\el a
aad 4i+4( = Jet sadfi+3( sty se
oS = =2 tana a,
and therefore
denotipg the angle aos by a. Hence the general expression of the intensity’of the light,
at any point M, is
A’ =a? + 2ad' cos (4 tana x) +a?.
Again, substituting for & —8 its value just found, we see that the successive fringes
will be formed at the distances given by the formula
x= md cotan a;
in which m is any number of the natural series, its even values giving the places
of the bright fringes, and its odd values those of the dark ones. Accordingly, the
bright fringes are formed at the distances 0, 2/, 4/, &c., and the dark ones at the
distances intermediate, /, 3/, 51, &c., 1 being equal to + cotan a: the successive
fringes, therefore, are equidistant. It is obvious that the angle a must be very small,
or the incidence very oblique, in order that the fringes should have any sensible breadth.
We have hitherto assumed that the light has undergone no change by reflexion, ex-
cepting the change of direction. Let us now suppose that the phase of the vibration
is accelerated, and let us examine the effect produced in the position of the fringes.
Let the amount of this acceleration be denoted by the angle pz ; then the difference
of the phases*will be
s—8 cm &—8—Lyr
2n(2—*) = pr = 2x (—*).
So that the successive fringes will be formed at the points for which
0 —s—$prA = km),
m being any number of the natural series. But we have already found that
&—8§=2 tan a «; so that the points in question are given by the formula,
a=} (m+) d cotan a;
the even values of m corresponding to the bright fringes, and the odd values to the
dark ones. It is evident from this that the magnitude of the fringes will be unal-
174 On a New Case of Interference
tered ; and that the only effect of the acceleration is to push the entire system from
the edge, the amount of the shifting being equal to } A cotan a.
In order to submit these results to the test of trial, I employed the apparatus
consisting of two moveable metallic plates, which is of so much use in experiments of
interference. The plates being closed, so as to form a narrow horizontal aperture, the
flame of a lamp was placed behind ; and the light thus diverging from the aperture
was received, at the distance of about three feet, on a piece of black glass truly
polished, and also horizontal. This reflector was then adjusted, so that its plane
might pass a little below the aperture ; or, in other words, that the light might be incident
upon it at an angle of nearly 90°. It is evident, then, that the light thus obliquely re-
flected will meet the direct light diverging from the aperture under a very small angle,
and with a difference in the lengths of their paths which is capable of indefinite dimi- -
nution. The two lights, therefore, are in a condition to interfere; and I found, ac-
cordingly, that when they were received upon an eyepiece, placed at a short distance
from the reflector, a very beautiful system of bands was visible, in every respect simi-
lar to one-half of the system formed by the two mirrors, in FresNeL’s experiment.
The first band was a bright one, and colourless. This was succeeded by a very
sharply defined black band; then followed a coloured bright band, and so on alter-
nately. Under favorable circumstances I could easily count seven alternations ; the
breadth of the bands being, as far as the eye could judge, the same throughout’ the
series, and increasing with the obliquity of the reflected beam. The first dark band
was of intense blackness; but the darkness of the succeeding bands was less intense,
as they were of higher orders; and after three or four orders, they were completely
obliterated by the closing in of the bright bands. At the same time the coloration of
the bright bands increased with the order of the band ; until, after six or seven alter-
nations, the colours of different orders became superimposed, and the bands were thus
lost in a diffused light of nearly uniform intensity. All these circumstances are simi-
lar to those observed in FresNet’s experiment, and correspond exactly with the results
of theory.
These bands are most perfectly defined when the eyepiece is close to the reflector.
Their breadth and coloration increased with the distance of the eyepiece, but re-
mained of a finite and very sensible magnitude, when the latter was brought into
actual contact with the edge—a circumstance whith distinguishes them altogether
from the diffracted fringes formed on the boundary of the shadow.
These fringes appear to me to possess some interest in a theoretical point of view,
independently of that which attaches to them as illustrations of an important general
law. Depending on the interference of two lights, one of which proceeds directly
from the luminous origin, while the other has undergone reflexion, they would seem
to afford the means of detecting any difference which might exist in their condition
when they meet, and therefore of tracing the modifications produced by reflexion.
Of the Rays of Light. 175
There are two circumstances which chiefly demand our attention in the case of re-
flected light—namely: Ist, the amplitude of the vibration, on which the intensity of
the light depends ; and 2dly, the phase. ‘The facts before us seem, toa certain extent,
to bear on both these points.
The reasonings of Fresnex with respect to the intensity of reflected light, are partly
of an analogical nature, and very far indeed from being strictly demonstrative. Still,
however, they have led to conclusions fully borne out by experience, and of the most
interesting kind; and we can hardly refuse our assent to doctrines which bear with
them such characters of truth. The formula which Fresnex has obtained for the in-
tensity of reflected light has not received any direct confirmation from experiment,
except in the case of a few observations made by M. Araco. It results from this for-
mula that the intensity of the reflected light must be equal to that of the incident, or
the whole of the light reflected, at the limiting incidence of 90°. Fresnev himself
notices this consequence, and adds that we should doubtless find it to be experimentally
true, if we could reach this limit. Now the present experiment affords the means of
examining this conclusion, and seems fully to establish it. We have already alluded to
the intense blackness of the first dark bar, in the phenomena now described. As far as
the eye can judge, the intensity of the light is absolutely nothing, at the points cor-
responding to this bar ; and as the intensity of the light in the dark bands is generally
expressed by the formula (a—a’)’, we are forced to admit that a=a’, or that the
intensities of the direct and reflected lights are equal at this extreme incidence.
With respect to the effect of reflexion upon the phase of vibration, there seems to
be some uncertainty in the theory. The phenomena of thin plates compel us to ad-
mit that half an undulation is either lost or gained, by the wave reflected from the
first or second surface; so that half an undulation must be added to, or subtracted
from, the difference in the lengths of the paths traversed by the two waves. That
such an effect should take place is in the highest degree probable from theoretical con-
siderations. ‘The light in the one case is reflected from the surface of a denser, in the
other from that of ararer medium; and the mechanical laws, on which Fresnet has
founded the doctrine of reflexion, lead us to the conclusion that the displacements of
the ethereal particles, in the moment after reflexion, must be of opposite signs in the
two cases. This difference in the phase of the vibration is equivalent to a difference of
half an undulation in the length of the path.
But it does not seem to be clearly understood to which surface we are to attribute
this physical change in the condition of the’'ray. Dr. Youne, indeed, who was the
first to state this law, says expressly that where “light has been reflected at the sur-
face of the rarer medium, it must be supposed to be retarded one-half of the appro-
priate interval.” I cannot avoid thinking, however, that the very analogy by which
he himself illustrates this point, and still more the reasonings of Fresnex on the sub-
176 On a New Case of Interference
ject, lead to an opposite conclusion, and tend to ascribe the effect which is found to
take place to reflexion at the surface of the denser medium. In fact, it would ap-
pear from Fresnex’s conclusions, that the sign of the vibratory movement is in all
cases changed by reflexion at the surface of the denser medium, the angle of incidence
exceeding the polarizing angle ; and it can readily be shown that this change of sign
. 1s equivalent to the addition of + 7 to the phase.
The present case of interference seems to support this view. It follows, as we have
seen, from theory, that if the light undergoes no change of phase by reflexion, the
distances of the successive dark fringes from the edge of the shadow will be as the odd
numbers 1, 3, 5, &c.; so that the distance of the first dark band from the edge will be
half the interval between each succeeding pair of dark bands. But it appears, on the
contrary, from the phenomena, that the distance is, as far as the eye can judge, exactly
equal to the succeeding intervals ; or that the bands are all shifted from the edge by
the amount of half an interval. The phenomena, therefore, require us to suppose
that the phase of the reflected wave is accelerated, and that the amount of this accele-
ration is exactly half a phase, or z. For the general expression for the shifting
of the bands is {A cotan a; and as this is found to be equal to 1A cotan a, it follows
that »=1, or the acceleration equal to 7. It appears then that when light is reflected
at the surface of a denser medium, the wave—at the limiting incidence at least—gains
half an undulation at the instant of reflexion.
In order to satisfy myself more fully of the effects of reflexion upon the phase, I
repeated the experiment with polarized light. The light was polarized, before it
reached the aperture in the screen, by transmission through a good tourmaline; and
the fringes were observed in various positions of the plane of polarization with respect
to the plane of reflexion. I could detect, however, no sensible difference in the posi-
tion of the fringes under all these changes of circumstance ; and, in particular, the
distance of the first dark band from the edge of the shadow seemed, as before, to be
precisely equal to the intervals of the succeeding bands, whether the light was polarized
in the plane of incidence, or the plane perpendicular to it.
This result seems to be just what might be expected from Fresnex’s theory of re-
flexion. From this theory it appears that if + a@ be the coefficient of the displacement
of the incident ray, or the amplitude of the vibration, and 7 and? the angles of inci-
dence and refraction, the coefficients of displacement of the reflected ray will be
sin (7—7’) tan(7—7’)
sin(Z+7) ’ oF 9 tani 7)?
according as the plane of polarization coincides with the plane of reflexion, or is per-
pendicular to it. Now the former quantity is always negative, so long as @ is greater
than 7, or the ray incident on the surface of the denser medium. Under the same
circumstances, the latter quantity is positive or negative, according as i+? is less or
Of the Rays of Light. 177
greater than 90°, or the angle of incidence below or above the polarizing angle. For
very oblique reflexion, then, both displacements are negative ; and, therefore, whether
the plane of polarization coincides with, or is perpendicular to, the plane of reflexion,
the wave will undergo a change of half a phase at the instant of reflexion.
From Sir Davip Brewsrter’s important researches on the nature of metallic reflexion,
it appears that a plane-polarized ray, which is incident upon a metallic reflector, be-
comes elliptically-polarized after reflexion ; a result which indicates a difference in the
phases of the two resolved vibrations. But it appears further, from the same researches,
that this difference of phase varies with the incidence, and vanishes altogether at the
extreme incidences; so that at the limiting incidence of 90°, there is either no alteration
in the phase of vibration, whether parallel or perpendicular to the plane of reflexion,
or that alteration is the same for the two vibrations. From some observation of the
fringes produced by the interference of direct light with that reflected from speculum
metal, I conclude that the former is the case.
An Essay on the Climate of Ireland. By Joserpn M‘Swerny, M. D. In
answer to the question proposed by the Royal Irish Academy.—* WHETHER WE
HAVE REASON TO BELIEVE THAT A CHANGE HAS TAKEN PLACE IN THE CLIMATE OF
IRELAND, AND IF SUCH CHANGE HAS OCCURRED, THROUGH WHAT PERIOD CAN WE
TRACE IT, AND TO WHAT CAUSES SHOULD WE ASSIGN IT.”
Read November 8th, 1830.
Sepe etiam steriles incendere profuit agros
Atque levem stipulam crepitantibus urere flammis.
VireiL. Geor. Lis. 1.
Dirricutt would be the task of treating of the climate of Ireland at different
periods, and of drawing conclusions from data so scanty as we possess, but that
general considerations of the immutable laws of nature aid us in the investigation,
and we derive no small assistance from the more accurate observations on the climate
of England, where the history of the weather has been preserved with more care,
than in Ireland; analogy in this case lends us its powerful aid.
The comparison of the nature of the vegetation in the island in ancient and modern
times, and the comparison also of the animal kingdom at different periods, give us
general views as to the climate. ‘The clothing of the ancient inhabitants, the nature
_ of their habitations, the casual allusions to the weather made by historians in describ-
‘ing sieges and battles; all these matters must engage the attention of the person,
who undertakes to treat of the climate of Ireland at remote periods, when no regular
accounts of the weather were kept to be handed down to posterity.
If a stranger were to direct his attention to the subject of the climate of Ireland,
he would be led to suppose from the situation of the island in a temperate zone, and
from its situation with respect to the vast Atlantic, to the vapours of which it must be
exposed, that the climate ought to be mild and moist; but on ascertaining that the
island contained no mountains of extraordinary height, and on tracing on the map the
number of rivers by which it is intersected, and amongst them such a river as the
VOL. XvU. 2u
180 M‘Sweeny, on the Climate of Ireland.
Shannon, and knowing the effect of insular situation on climate, he could have no
doubt on the subject.
Mr. Daniel, in one of his meteorological essays, compares the Caspian sea, which has
no outlet for the rivers which it receives, with the lakes of another continent, North
America, which send an immense volume of water to tle ccean; he considers them
as hygrometers on a large scale, by which we may judge of the state of saturation of
the two atmospheres. Jn Ireland we have also a hygrometer on a large scale, we
have the Shannon, an index of the large quantity of rain that annually falls. The
fact of this fine river being in so small an island, “spreading like a sea,” as the poet
Spencer described it, is alone quite sufficient to prove the great humidity of the cli-
mate of Ireland.
As the causes of climate here glanced at, are of a permanent nature, and as the
laws of nature are immutable, it is just to, suppose that mildness and humidity were
the characteristics of this climate always, when compared with the climate of other
countries of Europe. The effects of draining and cultivation on climate, have been
noticed in every quarter of the globe ; it would be strange if draining and cultivation,
which have gone on rapidly with an increasing population in this island, have not pro-
duced some effect on its climate. We will have to investigate in the course of this
essay, whether Ireland be an exception to a general rule in this particular. Before
entering on the task of tracing the history of the climate, it may be right to make
some observations on climate in general, and to examine the assertion so boldly made,
that the climate of Europe has greatly changed since the time of the Roman dominion.
Our knowledge of different agents, such as heat, light, and electricity, is yet limited,
their mode of operation is not sufficiently understood, to enable us to speak decidedly
on several meteorological points. The nature of the sun itself is only surmised, the
question of its being a habitable globe, surrounded by a phosphorescent atmosphere,
or of its being a body of fire, is one that admits of discussion. amples are
undecided about the nature of the solar spots; the question of their producing": any
effects on the weather is yet involved in doubt.
To Franklin we are indebted for the knowledge of the agency of electricity in the
clouds. Scheele pointed out the action of solar light in blackening the nitrate of
silver, but to modern research we owe the knowledge of the connexion between
light, heat, electricity, and magnetism. The curious effect produced by light on a
mixture of chlorine and hydrogen, excited attention, and the discovery of the pola-
risation of light by Malus, directed the first philosophers of Europe to the subject of
light. The action of the violet rays in exciting the magnetic influence, noticed by
Morichini, and the connexion between electricity and magnetism, discovered by
(Ersted, have opened a new field for inquiry, which has been cultivated with
assiduity and success.
The connexion between heat, light, electricity, and magnetism, fortunately, is en-
M‘Sweeny, on the Climate of Ireland. 181
s
gagiig the attention of the first philosophers of the day. According to Baumgart-
ner of Vienna, Ann. de Chimie, 1826, the direct white light of the sun, may be
made to produce magnetism more rapidly, than the process of Morichini, or wf Mrs.
Somerville. The researches of De La Rive, of Cirsted, and of Ampere, the creator
of the branch of physics, called electro dynamics, and of Faraday, on the movements
of continued rotation discovered by him; and of Becquerel, and of other eminent
philesophers, may help, when compared with the meteorological observations of such
men as Humboldt, Daniel, Howard, and Flaugergues, to give insight into intricate
points of meteorology. Much yet is to be done in the way of simultaneous obser-
vation, in different parts of the world, by philosophers capable of undertaking the
task, and aided by the most improved instruments, before we will be warranted to
speak dogmatically on several points.
A more intimate knowledge than we at present possess of the laws that govern
light, heat, electricity, and magnetism, may be absolutely necessary for the explana-
tion of several meteorological phenomena. It has been said that a second Newton is
yet to come, whose comprehensive mind will be capable of arranging the multitude of
facts and observations, and of deducing the general laws that govern climate. It is
only necessary to study attentively, the most approved of works on meteorology, to
know what little at present is understood, of the causes that make the winds to de-
viate at different times, and that influence temperature in different countries, and in
different years.
The doctrine of central heat in our earth, has latterly been brought forward on
the continent, to explain the difference of temperature in countries, where a similarity
of temperature might be expected. The Neptunian theory of Werner, which had
obtained ascendency in public opinion, cannot be well reconciled with the mass of
evidence now before the scientific world, with regard to the increase of temperature,
as we descend from the surface of the earth into mines. In the Mssai sur la tem-
_ perature de Vinteriéur de la Terre, par M. Cordier, will be found a deal of
information on this curious subject.
If the opinion were admitted of our earth being a planet partially cooled, as Des
Cartes, and Leibnitz supposed, and having its centre in a state of fluidity, there would
still be a difficulty in reconciling the accounts of the very severe cold of Europe in
former days, with its climate at present, and a cooling process constantly going on, by
_ means of the conducting power of the crust of the earth. ;
Humboldt, perhaps the first opinion in the world on the subject of temperature,
inclines to the belief of a central heat. After alluding to the observations of Arago
on the temperature of water brought up from deep borings, he remarks, that these
important observations show, that in the earliest state of our planet, tropical tempe-
rature, and tropical vegetation, could arise under every zone, and continue until the
radiation of heat from the hardened surface of the earth, could cause it to cool. If
182 M‘Sweeny, on the Climate of Ireland.
«
this be admitted, it follows as a consequence, that Ireland must have experienced, at
one period, a climate of a high temperature. In Jameson’s Edinburgh Philoso-
phical Journal, for 1828, vol. xviii., an extract of a lecture on climate, delivered by
Humboldt, is given, in which are to be found many valuable observations. He ob-
serves, that the time is passed, when persons were satisfied with some undefined views
of the difference of climates, and when all the modifications of temperature were
ascribed either to the shelter afforded by ridges of mountains, or to the various eleva-
tions of the earth. He remarks, that remarkable differences of climates are perceived
in large tracts of country under the same latitude, and on the same level above the
surface of the sea, which do not arise from the trifling influence of individual localities ;
but are subject to general laws determined by the form of the continents in general,
by their outlines, by the state of their swrface, but particularly by their respective
positions, and the proportion of their size to the neighbouring seas. [rom the pro-
portion of the size of Ireland to the vast Atlantic, no wonder that our winters should
be mild, and that our summers should not be very warm.
The influence of the sea on climate is a matter that admits of no dispute ; but other
matters relating to temperature are involved in obscurity—for instance, the occa-
sional visitation of very severe winters. Are the supporters of the doctrine of central
heat, prepared to maintain, that its agency in some winters is less than in others ? Cor-
dier supposes the thickness of the crust of the earth to vary in different places, and
explains on the principle of central heat, the difference of climate in countries in the
same latitude. This curious subject is engaging the attention of the scientific world,
but it must be acknowledged, that the occasional return of very severe winters, and
of extremely hot summers, does not at present admit of any satisfactory explanation.
Many writers, by selecting from the works of the ancients, passages which treat of
severe cold, have led to an almost general belief, that the climate of Europe has very
much changed: but a person residing in a distant country would form a very erro-
neous notion of the climate of England, if he were only to read a number of accounts
of the Thames having been frozen over at different times, and the extraordinary years
are these that are most likely to be recorded. Doctor Patterson had the courage, in his
work on the climate of Ireland, to deny the asserted change in the climate of Italy.
It is a matter of importance in this inquiry to investigate, if any change of conse-
quence has taken place in the climate of the continent; if it could be shown that an
important change has taken place there, it would be but reasonable to suppose a
change also in the climate of Ireland. But let the climate of Europe in general be
what it may, at any time, it may be safely asserted, that Ireland, from not having
mountains of great height when compared with those of other countries from its insular
situation in a temperate zone, from its being exposed to the vapours of the Atlantic,
must from the most remote periods have possessed a climate, mild and moist, when
compared with that of the rest of Europe.
M‘Sweeny, on the Climate of Ireland. 183
A large work has been written by M. Schow, professor of Botany in the University
of Copenhagen, which treats of the climate of the earth during the existence of man
on its surface; it treats of the climate of the antediluvian world, as far as can be
ascertained by fossils.
A paper by Professor Schow on the supposed changes in the meteorological con-
stitution of the different parts of the earth, during the historical period, was read
before the Royal Society of Copenhagen, and has been translated and published in
the Sth volume of Brewster’s Edinburgh Journal of Science. He investigates the
accounts given by the ancients, of animals and plants in different countries. He ob-
serves that the most rigorous criticism is required in such an inquiry, that persons
should not be led into error, as the ancients are not very careful in their description
of plants and animals, and matters considered essential in determining the species,
were unknown to them ; besides, their descriptions are not free from fabulous admix-
tures. He remarks that great caution must be observed in drawing conclusions as to
climate, from animals and plants ; for instance, it is not a higher temperature which
has driven the beaver from the greater part of Europe, and which in North America
compels it, more and more, to retire into the interior, but an increasing population.
With respect to the cultivation of plants, it is not enough to know that a plant was
not cultivated by the ancients, but it should be ascertained that they attempted its
cultivation in vain.
With regard to the freezing of the sea, a great difference must be made between
that which is usual, and that which is extraordinary ; and great allowance must be
made for the weakness of human memory, which recollects much better the remark-
able exceptions than the general rule of things.
He begins the investigation with Palestine, on the authority of the Bible, and
treats of the existence of the date tree, and of the vine, about which there can be no
question in ancient or modern times. The date tree was abundant, and principally
in the southern part of the country.
Jerico was noted for palm trees; the people had palm branches in their hands,
Pliny mentions the palm tree as abounding in Judea. ‘Tacitus and Josephus, speak
of woods of palm, as well as Strabo, Diodorus Siculus, and Theophrastus.—
Among the Hebrew coins, those with date trees are by no means rare, and the tree is
recognised as it is figured with its fruit. The vine was one of the plants most cul-
tivated in Palestine. In many places vineyards and wine are spoken of. Strabo and
Diodorus speak of the cultivation of the vine in Palestine. Both dates and grapes,
together, are symbols on Hebrew coins.
Professor Schow, argues—that as the date tree, in order to bring its fruit to per-
fection, required a mean temperature of 21° centigrade, that the country about Je-
rusalem could not have a lower mean temperature than 21° centigrade.
He observes, that in Barbary the vine succeeds only on the coast, and even there,
184 M‘Sweeny, on the Climate of Ireland.
the north sides of the hills are chosen for its cultivation. The mean temperature of
Algiers is 21° centigrade. In Egypt, the cultivation of the vine is insignificant ;
Cairo has a mean temperature of 22° centigrade. At Abusheer, in Persia, they are
obliged to plant vines in ditches, to protect them from the heat of the sun.
From the successful cultivation of the date and vine in Palestine, its mean tem-
perature could not have been above 22,, probably not above 21° centigrade, and he
concludes—“ if there has been any difference between the mean temperature of Jeru-
salem, in ancient and modern times, it can hardly amount to one degree.”
The time of harvest formerly in Palestine, compared with the time of harvest de-
scribed by modern travellers, he shows, favours the same conclusions.
It follows from the observations of Theophrastus and Pliny, that as the olive tree
was cultivated in Upper Egypt, the climate could not have been more warm, because
the olive tree does not bear a great heat. Professor Schow thinks from comparing
the accounts of the ancients with these of the French observers, that the rise of the
Nile happens at the same period of the year as formerly, showing that the rainy sea-
son began in the tropical part of Africa at the same period that it does now.
The ancients spoke of the central part of Africa, as unimhabitable on account of
the heat, but not from their own observation it is to be remarked. 5
From the careful study of the writings of the ancients, and from observation of
the vegetation of Italy at the present day, he is against the idea of any change of
consequence having taken place in the climate of Italy. He says that the passages of
Virgil are taken from his description of pastoral life suited to the mountains, since
in the lower plains, there is not sufficient grass on account of the heat. Myrtle
and bay have grown near Rome since the earliest times ; myrtle branches were made
use of in the peace between the Romans and Sabines; and bay crowns were used in
the time of their kings; he says that the climate could not be much colder, since
myrtle and bay grew there.
That the climate of the south of Europe has not been more warm, is proved by
the account which Theophrastus gives about the date tree in Persia, which, when
brought to Greece, did not ripen the fruit. Schow carefully compares the times of
the corn, and wine harvests, in ancient and modern times, and thinks that the climate
of Greece and Italy, like that of Palestine and Egypt, has undergone no important
change ; but if on account of somewhat later harvests, and the possible growth of
beech trees in the Roman plains, we might be led to the opinion, that formerly the
climate had been a little colder than at present, the difference will hardly come up to
one or two degrees, and will not be greater than might be occasioned by the im-
provement and cultivation of the north of Europe in modern times.
From Greece and Italy he passes to the countries on the Black and Caspian sea,
here it has been pretended that the change of climate has been extraordinary. The
Abbé Mann, who collected the accounts of the ancient writers, says that they concur
M‘Sweeny, on the Climate of Ireland. 185
in asserting that the climate there was such as is now hardly to be found in Lapland
or Siberia. At present there grow, according to the accounts of travellers, olive
trees, fig trees, bay trees, and most of these which grow in the south of Europe.
Schow maintains, that a severe criticism will do away with a deal of these pretended
changes. Herodotus says, that the Bosphorus freezes, over which the Scythians led
their armies, and their waggons. Strabo speaks of the freezing of the sound, and
adds, that it was reported thet Neoptolemus fought a battle with cavalry in winter,
where in summer he had engaged in a sea-fight.
But Pallas, who in modern times has described those regions, informs us that the
Bosphorus, even in moderately severe winters, is covered with ice, principally drift
ice from the river Don; and that in severe winters, loaded waggons are carried over
it. It is thus at present as it was in former times. To these remarks, adduced by
professor Schow, may be added the severity of the winter on the Danube, which inter-
fered with the operations of the Russian army in the late war; and the London
papers at the time, gave us accounts from Odessa, of the 3d of January, 1829, which
stated, that the sea, as far as the eye could reach, was frozen, and that vessels were pre-
vented from going out or coming in.
Schow enumerates the fruits of these regions, described by the ancients, to prove
that no material change has taken place.
In treating of the climate of France, he observes, that we are informed by Strabo,
that in Gallia Narbonnensis, the same fruits are found as in Italy, but that in going
farther north, the olive tree, and fig tree disappear. In comparing Decandolle’s
map to his Flore Francaise, the limit assigned by Strabo holds good, and it proves
that the climate had not been colder formerly.
The high authority of Professor Schow is here adduced, to prove that no change
of great consequence has taken place in the climate of the continent of Europe, in
the historical period ; it is, therefore, just to suppose, that during the same time, no
very great change has taken place in the climate of Ireland ; but it is also reason-
able to suppose, that it has been modified in some degree by draining and cultivation.
In endeavouring to arrive at general conclusions, with regard to the climate of
Ireland by analogical reasoning, the subject of the climate of England naturally sug-
gests itself, for our investigation. It may be right here not to confine our attention
to the historical period, but to endeavour to obtain some idea of the nature of the
climate of England in antediluvian times.
The fossil remains in England enable us to form an opinion on the subject ; not
only in England but even in the northern parts of Europe, the remains of animals
have been found, which prove that at some remote period, animals existed there in
great numbers, which are now only to be met with in warm regions. In different
parts of Europe also have been found the impressions of plants, so well defined, that
they are easily ascertained to have been the produce of a very warm climate. Im-
186 M‘Sweeny, on the Climate of Ireland.
pressions of vegetables have been found in countries, the climate of which now could
not produce such vegetable productions. It does not come within the plan of this
Essay, to treat of the geological changes of our earth. A work has been written by
Doctor Ure, of Glasgow, showing that the revolutions of our earth, and animated
nature, can be reconciled to modern science and sacred history.
In this work he alludes to the futile attempts of Voltaire to explain the impressions
of fishes, found on mountains. Doctor Ure, in the introduction to his work, truly
observes: “As the stream of civilization advances towards the general diffusion of
knowledge, truth, and piety, over the earth, new chambers of nature are unlocked,
new scenes of instruction are disclosed, and new means and motives of intellectual
and moral excellence, are presented to our view.”
In treating of the subject of vegetation in Europe in antediluvian times, he ob-
serves: ‘* There is no doubt, however, that palms with fan-shaped leaves, covered
Europe with their lofty vegetation at this remote period, in regions where no species
of these plants could now grow. The opinion of some writers that these vegetables
may have been transported from remote climates into the places where they are
actually deposited, appears at variance with every fact hitherto observed, and possesses
in reality no solid foundation.” Ure’s Geol. p. 452.
The fossil vegetables found at Newhaven, in England, agree with those found in
the Paris basin; one was the fruit of the palm tree, an instance of the produce of a
warm climate. In Doctor Ure’s work on geology, there is given the figure of an
impression of a vegetable in slate clay, from Lancashire, considered to be the pro-
duction of a tropical climate.
At Kirkdale, have been discovered the remains of hyzenas, and even of the hippo-
potamus, inhabitants of warm regions. Doctor Buckland in quoting Cuvier, to prove
the dispersion of the remains of elephants over every country in Europe, combats
the opinion, that the remains found in England, were of elephants imported by the
Roman armies. He shows that the fossil elephant belongs to an extinct species of
this genus. He observes, that the idea of their being drifted by the diluvian waters
from the tropical regions must be abandoned, on the evidence afforded at the den of
Kirkdale ; and he adds, it remains only to admit that they must have inhabited the
countries in which their bones are found.
If it be admitted, that the climate of England at any one period was capable of the
growth of such vegetable productions as palms, and served as the abode of the
elephant, hippopotamus, and hyzna, it follows as a natural consequence, that Ireland
from its proximity, must have possessed somewhat of a similar climate at the same
time. The evidence afforded by the den at Kirkdale, cannot be explained away.
Doctor Ure remarks, that “ there are few physical properties established on a larger
or sounder induction than that the Kirkdale and Torquay caves having been dens oc-
cupied by hyzenas in antediluvian times.”— Ure’s Geol. p. 574.
i
f
Z
;
M‘Sweeny, on the Climate of Ireland. 187
We shall now have to direct attention to the climate of England in the historical
period. According to Doctor Halley, Cwsar landed in England in the latter end of
August; an attempt has been made on this computation, to prove that the harvests
were earlier at that time than at present, from a passage in his commentaries, where
it is stated that the corn was all reaped except in one place. But by reading Czsar
attentively, it is easy to see that he alludes to the country near the camp, and that this
sweeping conclusion cannot be admitted. .
The Britons, who after the first battle had agreed to submit, no sooner learned that
the Roman fleet had been damaged, than they resolved to break off negotiations, and
to starve the Romans, ‘‘frwmento commeatuque nostros prohibere.” ‘They hoped by
preventing the return of the Romans, that no one after, would attempt to pass into
Britain for the purpose of waging war.
Under such circumstances, it is not to be supposed, that Cesar would be very parti-
cular in waiting until the very last day for the ripening of the corn; on the con-
trary, he would be willing to lay his hands on any thing that might support his troops.
“ dt Cesar etsi nondum eorum consilia cognoverat, tamen et ex eventu naviwmn
suarum, et ex eo quod obsides dare intermiserant, fore id quod accidit suspicabatur.
Itaque ad omnes casus subsidia comparabat. Nam et frumentum ex agris quotidie
in castra conferebat.”
He also repaired his damaged fleet ; and while these matters were going on, the
soldiers stationed before the camp, informed him that an unusual dust was to be seen
in the direction in which the legion had gone, which, according to custom, had been
sent out to forage. He hastened to their assistance, and found them engaged with
the Britons, who had formed an ambuscade for them in a place where the corn had
remained uncut, and who attacked them while engaged in reaping——‘‘ Nam quod
omni, ex reliquis partibus demesso frumento pars una erat reliqua suspicati hostes
huc nostros esse venturos.” Surely it cannot be contended for with any reason, from
this passage, that all the corn in Britain was reaped except in this place. The passage
_has reference only to the neighbourhood of the camp; all the corn on the ground
within sight of the camp, was probably cut down by the Romans themselves, except in
the spot where the ambuscade was laid for them : Cvesar informs us that they were daily
employed about it. Its being not quite ripe would not prevent them, situated as they
were ; besides the number of days which elapsed from the time of their fleet being
injured, until this engagement, is not mentioned.
We may infer from the dust seen from the camp, that the weather was at the time,
dry. The account of tlie weather after the engagement, coincides with the variability
of the climate of England in modern times—“ Secute sunt, continuos complures
dies, tempestates que et nostros in castris continerent, et hostem a pugna prohiberent.”
Thus this passage in the 4th Book of Czsar, is far from proving that a change has
taken place in the climate of England.
VOL. XVII. 2x
188 M‘Sweeny, on the Climate of Ireland.
The general character which Cesar gives of the climate of Britain, holds good to
this day. He describes it as being more temperate than that of Gaul, the cold being
less severe——
“ Loca sunt temperatiora quam in Gallia remissioribus frigoribus.”
Cesar de Bello Gal. Lib. y.
The cold in the north of-France in the winter, is much more severe than in Eng-
land. Persons from England, who reside during the winter at Paris, are surprised at
the cold of the weather.
Tacitus describes the British climate as foul, with frequent showers and clouds.
“Colum erebris imbribus ac nebulis foedum, asperitas frigorum abest.”
Vita Agricole.
This passage in Tacitus, is a difficulty not to be surmounted by those who maintain,
that the great humidity of England is of recent origin.
The account which Tacitus gives us of the vegetation of Britain, answers perfectly
at the present day.
“ Solum preter oleam vitemque et cetera, calidioribus terris oriri sueta patiens frugum fecundum.”
Vita Agricole.
If it should be maintained, that the climate of England at one period was well fitted
for the cultivation of the vine, and for the production of wine ; the supporters of
this doctrine, to get rid of the quotation from Tacitus, ought to be able to prove that
the climate ameliorated to the time of William of Malmesbury, and that subsequently
it again grew cold.
It was probably a succession of favourable seasons that led the Romans to encourage
attempts at cultivating the vine in Britain. Nothing was more likely to be fostered
than the vine by religious establishments, after the introduction of Christianity.
Bede speaks of the vine growing in some places in England—
“Vineas quibusdam in locis.”
The same may be said now.
Camden, speaking of Gloucestershire, says: ‘‘ The west part beyond the Severn,
is covered with woods. But .I need not spend much time on this head, William of
Malmesbury, will save me this trouble, who is lavish of his praises and description of
this country ; take, therefore, his words: ‘’The country is called from its principal
city, the vale of Gloucester, productive throughout of corn and fruits, either by the
sole bounty of nature or the industry of art, so that it invites the most indolent per-
sons to labour, when the product will return a hundred fold ; you may see the high roads
bedecked with fruit trees not planted by art, but natives of the soil. The ground
M‘Sweeny, on the Climate of Ireland. 189
spontaneously produces fruit in taste and colour far exceeding others, many of which
will keep the year round, so as to serve their owners till others come in again, No
county in England has more or richer vineyards, or which yield greater plenty of
grapes, or of a more agreeable flavour. The wine has not a disagreeable sharpness to
the taste, as it is little inferior to that of France in sweetness.
Camden comments on this passage thus: ‘“ What he says of the hundred-fold
999
increase of the land, is a mistake. Not that I am of the opinion of those peevish
lazy husbandmen, whom Columella complains of, that the soil is worn out and ren-
dered barren by its excessive ancient plenty. But on this account, not to mention
others, we need not wonder that so many places in this county were called vineyards
from their vines, since wine was one of the productions of this county ; and certainly
it seems more owing to the indolence of the inhabitants than to the alteration of the
climate, that it now yields none.”— Gough’s Camden, Vol. I. p. 379.
Here we have the opinion of Camden, that no alteration had taken place in the
climate from the time of William of Malmesbury to his time ; and the experience of
latter days shows, that in favourable years, wine may be produced in England, which
may be mistaken for continental wine by good judges, instead of being inferior, as
William of Malmesbury described the wine in his time.
Mr. Williams,* who advocates the opinion of a change of climate, of course has not
passed by the celebrated passage in Malmesbury ; he thinks with Camden on the sub-
ject of the hundred-fold produce from the land, that the learned monk may have drawn
rather too flattering a picture; if so, it can be immediately retorted, that the learned
monk may have also drawn too flattering a picture of the-fruit trees and vineyards.
The archives of the Church of Ely, give us positive information of the making of
wine from a vineyard for some years, and we learn that in an unfavourable year no
wine but verjuice was made.
Speechly, in his treatise on the culture of the vine, mentions a controversy between
the Rey. Mr. Pegg, and another, on the subject of vineyards in England, formerly.
The Rey. Mr. Pegg, after stating the evidence, observes, that it appears plainly, that
at Ely grapes would sometimes ripen, and the convent made wine of them, and some-
times not, and then they were converted into verjuice.
Speechly gives an account of many successful attempts at raising vines in England,
for the purpose of making wine at subsequent periods. He describes the vineyards
at Pain’s Hill, which belonged to the Hon. Charles Hamilton. This gentleman pro-
duced wine, which was supposed by good judges, to be superior to any champagne they
ever drank, and which was sold for fifty guineas a hogshead ; one merchant purchased
£500 worth.— Speechly on the Vine, p. 213, 3d edition.
* Williams on the Climate of Great Britain, p. 125.
190 M‘Sweeny, on the Climate of Ireland.
Speechly says : “‘ From the foregoing accounts it is evident, that good wine may be
made in this country in a propitious season.” ‘There can be little doubt but that the
opinion of Speechly would have been verified, with great profit to any individual, so for-
tunate as to possess a vineyard, in a favourable soil in England in the year 1826. But
where the seasons are so uncertain, such a speculation would be hazardous indeed.
Evelyn, in 1655, writes thus: ‘I went to see Colonel Blount’s subterranean warren,
and drank of the wine of his vineyard, which was good for little.” Phillips in his Po-
marium Britannicum, has the following very sensible remark on the subject of
English wine. . ‘‘ We may conclude that as our intercourse increased with the conti-
nent, it was found more advantageous to import wine, than to depend on the product
of our own crop, which must have been an uncertain one from the variableness of our
climate.” ;
When the English army assembled at York in the year 1327, to repel an inyasion of
the Scots, Froissart informs us that “ good wines from Gascony, Alsace, and the Rhine,
were in abundance, and reasonable.” —Froissart’s Chronicles, by Johnes, Vol. I. p. 55.
The fact of wines from such distant places being conveyed to the centre of Eng-
land, and sold at reasonable prices there, proves that the climate then was not suited
to the cultivation of the vine.
The statement of the production of wine in England at a distant period, can be met
by similar statements in modern times, and the description of the vale of Gloucester,
by Malmesbury, bears with it the marks of being an exaggerated statement.
Where the climate is so variable, it would not be an easy matter to attempt to draw
precise conclusions by means of a calendar of Flora ; in some years vegetation is more
forward than in others ; and it has been remarked by close observers, that after a suc-
cession of favourable years, many plants acquire, as if by habit, the power of blossom-
ing somewhat earlier for some time.
Lord Bacon, in his celebrated essays, gives us an account of gardening, and of the
time several plants come into flower, near London; probably the description was
drawn up from the state of the gardens in the year the essay was written ; we shall
compare some of his accounts with those of modern times.
“There followeth for the latter part of Janu- ** The mezereon sometimes blossoms as early
ary and February, the mezereon tree which then as the end of January or beginning of Febru-
blossoms. ’’— Bacon. ary.”—Phillips's Sylva Florifera.
“Por March there come violets.” —Bacon. “ Violet—This favourite flower is a native of
Europe, flowering in March and April.’’—Mil-
ler’s Gardener's Dictionary by Martyn.
M‘Sweeny, on the Climate of Ireland. 191
“Tn April—the wall flower.”—Bacon. : Wild wall flower, “its bright golden flowers
are very ornamentalin April and May.’’—Sow-
erby’s English Botany.
In April, the cowslip.”"—Bacon, Cowslip—April is commonly assigned as the
month of flowering for all; but the primrose ap-
pears in March, and the cowslip in April.”—
Miller's Dictionary by Martyn.
“Tn May and June, apple tree in blossom.” “When about the end of May it is covered
—Bacon. with bloom, few if any shrubs surpass the crab
in beauty.” —Sowerby's English Botany.
“In July—the lime tree in blossom.’’.—Bacon. Lime tree, “the flowers begin to open by the
middle of May, but are not in their full beauty
before the middle of July.”—Phillips'’s Sylva
Florifera.
From the mildness of some winters in England, furze is met with in blossom, some-
times about Christmas. In the calendar of Flora, in White’s Natural History of
Selborne, we find the primrose in flower on the 10th of November, and furze in blossom
on the 2ist of December.
It shows that where the years differ so much, and where the seasons are proverbially
variable, complete uniformity cannot be expected in the accounts of the flowering of
plants.
Phillips, in treating of the mulberry, observes: ‘‘ The mulberry tree is stated to
have been introduced into this Country (England) in the year 1548, and it is said that
it was first planted at Sion House, where the original trees still thrive, and which we
have seen since the first part of this work has been put to press.”—Phillips’s Poma-
rium Britannicum, p. 239, London 1823,
Although years may differ, yet on an average of a great number of years, the fact
as told by Phillips of these mulberry trees standing the weather so long, tends to show
that no great alteration has taken place in the climate of England since the time of
their being planted.
There is every reason to think, that variability of seasons is not of modern date.
Lord Bacon, in his essay on the vicissitude of things, says: ‘“ There is a toy which I
have heard, and would not have it given over, but waited upon a little. They say it is
observed in the Low Countries, I know not in what part, that every five and thirty years,
the same kind and suit of years and weathers comes about again ; as great frosts, great
wet, great droughts, warm winters, swmmers with little heat, and the like, and they
call it the prime : it is athing I do the rather mention, because, computing backwards,
I have found some concurrence.”
We find White, in 1774, in his Natural History of Selborne, complaining of a run
of wet seasons, and observing that there was no use in newspapers inflaming the public
192 M‘Sweeny, on the Climate of Ireland.
mind about combination, that plenty was not to be expected until Providence would
send more favourable seasons.
Howard, in his treatise on the climate of London, treating of the popular opinion
with regard to St. Swithin’s-day, observes: ‘* To do justice to popular observation, I
may now state, that in a majority of our summers, a showery period which, with
some latitude as to time and local circumstances, may be admitted to constitute rain
for forty days, does come on about the time indicated by this tradition, not that any long
space before is often so dry as to mark distinctly its commencement ; the tradition is
so far valuable, as it proves that the summers in this southern part of our island, were
subject a thousand years ago to occasional heavy rains in the same way as at present.”
Howard’s Climate of London, Vol. Il. p. 198.
When the English army were searching for the army of the Scots, to bring them to
an engagement, near New Castle upon Tyne, in July 1327, the English army suffered
severely from rain. Froissart describes their situation thus: ‘‘ To add to their un-
pleasant situation, it had rained all the week, by which their saddles and girths were
rotted, and the greater part of their cavalry were worn down. They had not where-
withal to shoe their horses that wanted it, nor had they any thing to clothe themselves
or preserve them from the rain and cold, but their jerkins or armour, and the green
huts.”—Froissart’s Chronicles by Johnes, Vol. I. p. 54.
This quotation from Froissart’ is valuable, as it corroborates the statement of
Howard. All the ancient accounts we have, tend to prove that the climate of Eng-
land has not materially changed ; the dress of the Britons, as described by Cesar, is
well fitted for a humid clime— :
« Pellibusque sunt yestiti.”
Cesar de Bello Gal. Lib. V.
Skins were well adapted for keeping out rain and preserving the animal heat.
The description of the climate by Tacitus, would hold good for some of the worst
years that are now experienced in England. It would be very difficult for Mr. Wil-
liams to get over the quotation from Tacitus. It may be objected to him at every
turn. This gentleman thinks that the humidity of the summers in England, has
greatly increased, owing to a change on the surface of the island from the increase of
hedge rows, from the planting of trees, and from the extension of green crops shading
the ground, and preventing its being parched up. There can be little doubt but that
the state of the surface of the island, must have some effect on its temperature ; but
when it is remembered that hot and cold seasons arise from general causes, and that the
vast Atlantic is the grand source of moisture ; it is to be supposed that Mr. Williams
attributed too much to modern improvements. .
The very observation which he has quoted of a very old gentleman of Worcester,
on the subject of drought, tells against him—* Never fear, I have often known Eng-
M‘Sweeny, on the Climate of Ireland. 193
land to suffer from too much cold and wet, but never from too much heat.”—JV7il-
liams’s Climate of Great Britain, p. 227.
Howard, in the preface to the second volume of his Climate of London, published
in 1820, says: ‘“ The result of my experience is, on the whole, unfavourable to the
opinion of a permanent change having taken place of latter times, either for the better
or the worse, in the climate of this country ; our recollection of the weather, even at
the distance of a few years, being very imperfect, we are apt to suppose that the sea-
sons are not what they formerly were ; while, in fact, they are only going through a
series of changes such as we may have heretofore already witnessed and forgotten.”
Howard also thinks that in its great leading features the climate differs little from what
» it was at a remote period.
Doctor Rutty, in his Natural History of the County of Dublin, instituted a compa-
rison between the climate of London and the climate of Dublin, by means of re-
gistries kept in both cities, from which he concluded that the winters in Dublin were
warmer and moister, than in London.
We have the decided opinion of Howard, that no change has taken place in the
climate of England, a man, who in knowledge of meteorological details, stands un-
| rivalled.
The comparison which Caesar made between the climate of Britain, and of Gaul,
is what a foreigner would be apt to make at the present day.
The description of the climate and vegetation of Britain, by Tacitus, holds good at
the present day.
From the proximity of Ireland to England, we may conclude that no great change
has taken place in the climate of Ireland since the time of Cesar.
That the temperature of Europe in antediluvian times, was greater than it is at the
present day, is a subject that admits of no doubt. Attempts have been made, to ex-
plain away the fact of the bones of the elephant being found in cold countries; but
the finding of thé numerous impressions of plants, that now only thrive in tropical
| regions, sets the matter at rest. .
\ The subject of fossil plants has been studied with great care by M. Brogniart.
Not only in England have been found the remains of the elephant, but also in
Scotland.— Buckland’s Reliquie Diluviane, p.179.
From the bones and vegetable impressions found in England, we would from analogy
be fully warranted in concluding, that the temperature of Ireland was also great in
antediluvian times; but the bones of the elephant have been found in Ireland also.
In Grierson’s edition of Boate’s and Molyneux’s Natural History of Ireland, there
. is an account from the Philosophical Transactions, of the remains of the elephant
) found in Ireland. ‘The writer who describes the finding of them, is sadly at a loss to
account for their being found in such a climate ; he thinks that no ship of the ancient
194 M‘Sweeny, on the Climate of Ireland.
inhabitants could be capable of importing the elephant, and leaves the reader to ima-
gine how the animal got there before the flood.
Molyneux, in commenting on this subject, thinks, that the elephant under consider-
ation was not brought to Ireland by any industry of man. He supposes that the
globe in the early ages of the world, before all records, differed materially from
its present geography, as to the distribution of ocean, dry land, islands, and continents,
so_as to allow this beast a free and open passage from the continent. But a change of
climate must be supposed too ; the evidence of the Kirkdale cave leaves no doubt but
that the climate of the British isles was once suited to animals, that are now only to be
met with in warm regions. We may therefore conclude that the elephant in Ireland
did not find its way there by mere chance, but that it inhabited the country, and that»
the climate of Ireland was suited to its existence and habits in antediluvian times.
The remains of the moose deer have been found in different places in Ireland ; the
bone, when treated with muriatic acid, has been rendered flexible. Dub. Phil. Journal,
Vol. I. p. 484. The bones discovered at Kirkdale, when treated in the same manner,
were made flexible ; a proof that the gelatine had not been destroyed by time.
The great temperature of Europe at this period has been explained on the suppo-
sition of central heat in our’globe: to this cause the writer of this essay attributes
the temperature which once rendered Ireland a fit abode for. the elephant.
We now come to the historical period, in endeavouring to trace the climate: the
researches of Professor Schow, prove that there is no decisive evidence of a material
change of climate on the continent of Europe, by records, during the historical period.
Although it is maintained in this essay, that the general character of the climate of
Treland, has been the same from a very early date ; yet it is contended for, that the
weather has been modified from local causes. The state of the surface of the island
has been different at different times, at one time abounding with timber, at another
time denuded ; at present the surface is furrowed from the potatoe culture in every
direction, An ancient name of Ireland was, the woody island.
Endeavours™have been made to trace back the history of Ireland to remote anti-
quity, but on account of some evident fable mixed up with the accounts given, many
think themselves warranted in rejecting the entire history of very remote periods
altogether.
There is one point (to which if any credence be given) that would be decisive
evidence of a change of climate in Ireland at a very distant period, namely, the ap-
pearance of rivers that did not exist before, and the formation of new lakes.
This would be evidence of the highest description to prove an increase of humidity
n the climate at the time.
Keating, in his History of Ireland, says: “ In the time of Partholanus, seven lakes
broke out in the island;” again he informs us that ‘ Partholanus found but nine
rivers and three lakes in the island.”
M‘Sweeny, on the Climate of Ireland. 195
In the Annals of the Four Masters we have the following accounts :
«« /Xtas mundi 1mpxxxir eruptio lacus con et lacus Techet anno hoc.
« /Etas mundi impr eruptio novem fluminum.
« ANtas mundi 11DLxxx1 eruptio novem lacuum.
“« Doctor O’ Connor’s Version of the Four Masters.”
If these accounts are to be credited, there can be no question as to the increase of
humidity of the climate. If we suppose them to be true, we may attempt to explain
the previous aridity by the theory of central heat. M. Cordier thinks that the thick-
ness of the crust of the earth varies in different places, and he explains on the prin-
ciple of central heat, the difference of climate in countries in the same latitude. We
know from the fact of volcanoes, that internal fire in some places is not far distant
from the surface of our earth, and that its operation sometimes is more active than it
is at other times.
We have only to suppose a cessation of activity in the operation of the internal
heat in that part of the globe on which Ireland is situated, to account for the con-
densation of vapours into rain, which previously might be dissolved by the heated air.
Indeed there is one point which appears to corroborate this explanation, it is the ac-
count in the Irish Annals of the formation of not only new rivers, but also of new
lakes.
An increased quantity of rain might cause old rivers to be flooded, or new rivers
to be formed, but it would not cause the formation of new lakes, unless the level of
the ground had been disturbed by its sinking in some places, or by its elevation in
other parts from the operation of an earthquake, a visitation universally attributed to
the agency of internal fire.
This matter rests entirely on the authority of the old Irish records; where they
assume more the shape of historical narrative, they give reason to believe that the
climate in Ireland did not materially differ from the climate of the present day, and
that remarkably wet, dry, cold, and mild seasons happened occasionally as at the pre-
_ sent time.
We have a very early account of a mild climate in Ireland, in the Annals of the
Four Masters; we have also an early account of snow. “ A‘tas mundi mipcccLxvit
Erat floribus estivis ornatus omnis campus in Hibernia tempore Fiachi.”— Annales
IV. Magistrorum, Dr. O’ Connor’s Version.
Keating tells us in his History of Ireland, that Fionnachta, the son of Ollamh
Fodhla, obtained the name by which he was distinguished, on account of the quantity
of snow that fell upon the island in his reign. We have in the Annals of the Four
Masters, an account of a mild winter at an early period.
« /Btas mundi m111cLx Regnante Conario—oberrabant armenta absque custode in
Hibernia in ejus regimine propter abundantiam pacis et concordiz, non fuit tonitrale
vel procellosum ejus regnum. Nam non fluxu afficiebat ventus asper, armenta a medio
VOL. XVII. 2Y
196 M‘Sweeny, on the Climate of Ireland.
autumno ad medium veris. Videbantur sylve pendentes pre pondere suorum fruc-
tuum ejus tempore.”— 0’ Connor’s Version.
Here we have a description of a mild winter ; this and the account of the great
fall of snow from which Fionnachta obtained his name, tend to prove the occasional
return of mild and severe winters at a remote period.
In Jocelin’s Life of St. Patrick, frost and snow are mentioned. ‘The Annals of
Ulster show us the recurrence of wet swmmers, of droughts, and of severe winters.
Years, remarkable for the abundance of nuts, are mentioned; and we find the fre-
quent recurrence of bowel complaints, which coincides with subsequent statements.
Subjoined is a list of some of the most remarkable years, from Doctor O’Connor’s
version of the Annals of Ulster :
“634 Nix magna occidit multos in campo Breg.
684 Ventus magnus terre motus in insula.
713 Siccitas magna.
719 “Estas pluvialis.
747 Nix insolite magnitudinis ita ut pene pecora deleta sunt totius Hibernie et postea insolita sic-
citate mundus exarsit.
758 Estas pluvialis,
761 Nix magna.
763 Nix magna tribus fere mensibus—siccitas magna ultra modum, fluxus sanguinis in tota Hibernia.
772 Insolita siccitas.
773 Eugan Mac Colmain a fluxu sanguinis moritur, et multi alii ex isto dolore mortui sunt.
776 Diluvia yentosa in estate, i.e. inundatio magna imbrium et ventus magnus.
778 e, ventris profluvies per Hiberniam totam.
817
Gelu mirandum, et nix magna permanserunt a natali quasi usque ad Quinquagesimam.
Transitus paludum pedibus siccis, et plurima flumina eodem modo gelata ac lacus.—Plurima
materia ad construendas domos transyecta trans lacum Eirne e regionibus Connaciz ‘in re-
gionem posterorum Crimthani.
821 Gelu mirabile gelavit mare et lacus et flumina ita ut conducerentur armenta equorum et greges
et vectigalia ultra citraque.
855 Nix et gelu magnum.
911 Pluvialis tenebrosus annus.
912 Pluvialis tenebrosus annus.
916 Nixet frigus valde magnum et gelu mirabile hoc anno ita ut transgredirentur principaliores lacus
et amnes Hiberniz.
944 Gelu magnum mirabile, ita ut transgredirentur lacus et flumina.
1011 Mortalitas, fluxus sanguinis hoe anno in Ardmacha occidit plurimos.
1026 Exercitus cum Flahertaco O'Neill, abstulit obsides et profectus est supra glaciem, in insulam
Mochtei et eam vastavit.
1047 Nix magna hoe anno a festo Maria ad festum Patricii, eujus non visa est similis ita ut inde mor-
talitas hominum plurimorum et armentorum et ferarum innumerabilium et volucrum.
1094 Inundationes ingentes in Hibernia tota.
1095 Nix magna ceeidit die primi jejunii (i. e. die Mercurii) post Kalendas hujus anni, ita ut occiderit
plurimos homines et volucres et armenta.
1107 Imbrium inundationes ingentes cum gelu et nive a xv Kal. Jan. ad xv Kal. Martii vel paulo plus,
ita ut mortalitas esset volucrum et armentorum et hominum.”
M‘Sweeny, on the Climate of Ireland. 197
In the year 821, which, according to the computation in these Annals, agrees with
the year 822 of the Christian era, it is recorded that the sea was frozen ; but we find
that in the year 822, the principal rivers of Europe, such as the Danube, the Elbe,
and the Seine, were frozen so hard as to bear heavy waggons for a month. It may
be boldly asserted, that Ireland, from its insular situation, suffered less by cold that
year than the continent of Europe. But it is to be believed, that the state of the
surface of the island, unimproved as it was, in comparison with its present state, must
have greatly aggravated the cold of a severe winter, when it happened from a general
cause. Swamps, and bogs, and pools, are soon frozen ; at the present, draining, cul-
tivation, and reclaiming, have made great progress.
By the aid of chemistry it is easy to prove that Ireland never could have just
claims to the appellation of Glacialis Ierne :
“ Tlla ego sum Graiis olim Glacialis Ierne
Dicta, et Jasonie puppis bene cognita nautis.”
The water of the ocean mitigates cold in this way ; the upper particles of the
water, when cooled by the air, sink, and allow a warmer stratum of water to come in
contact with the atmosphere ; this process goes on, until the water, by long exposure
to the cold, acquires its maximum of density.
The vast body of water of the Atlantic must, therefore, at all times, have rendered
the winters in Ireland, mild, when compared with the winters in other countries.
The appellation of “ Glacialis lerne,” might have arisen on account of navigators
from the Mediterranean, having been in the island during a severe winter, in for-
mer times. ‘The crew of a vessel from a warm climate in the Mediterranean, would
be apt now to form an erroneous opinion ae the climate of Ireland, if they had
been here in the winter of 1812.
The island described by Diodorus Siculus, is supposed by some to be Ireland, from
the description ; the soil fruitful, the country diversified with mountain and plain,
watered by navigable rivers, abounding in woods, and orchards, and all the island
watered by streams, and the summer season fitted for pleasure and amusement—
Diod. Sic. vers. Wesseling, t. 1, p. 344.
Whether this island was the one described by him, or not, is a matter of surmise ; how-
eyer, this description may be received as a true one of Ireland, at the present day. The
island inhabited by the Hyperborei, he tells us, was so fruitful, and the climate so tem-
perate, that they mowed twice in the year.—Diod. Sic. t. 1, p. 158.
Festus Avienus, writing of that which was called the sacred island, says :
“Hee inter undas, multum cespitem jacit
Eamque late, gens Hibernorum colit
Propinqua, rursus, insula Albionum patet.’”—De Oris Marit.
The expression ‘‘ multum cespitem” may be received now, as applicable to Ireland, in
reference to a peaty soil, or to its grassy sward,
198 M‘Sweeny, on the Climate of Ireland.
Much confidence is not to be placed in poetic description, with regard to a matter
of philosophical inquiry.
We find Claudian also giving Ireland the title of ‘* Glacialis lerne”—
* Scotorum cumulos flevit Glacialis Ierne.”
It has been shown that Ireland never could have been remarkable for cold, in com-
parison with other countries.
Tacitus described the climate of Britain as being foul, with frequent showers and
clouds, and stated the absence of severe cold; and he described the climate of Ire-
land as not differing much from it—
“ Solumque ccelum et ingenia cultusque hominum haud, multum a Britannia differunt.”
Cesar previously had described the cold of Britain as less than that of Gaul.
Professor Schow has proved that there is no evidence to establish a material change in
the climate of France ; we here connect Ireland with the chain of his reasoning.
The comparison which Doctor Rutty made between the climate of Dublin, and the
climate of London, shows, that in the grand leading features, there is a similarity in
the climate of both places: thus his statement corroborates that of Tacitus.
In a work supposed to be written by Athicus, the climate is described as superior
to that of Britain. Orosius repeats this statement— Hee propior Britannia, spatio
terrarum angustior, sed cali, solisque temperie magis utilis.”— Orosius, Lib. 1, Hist.
cap. 2.
Isidorus says that Ireland is more fertile than Britain.
The venerable Bede gives a decided preference to the climate of Ireland. ‘“ Hiber-
nia autem salubritate ac serenitate aerum multum Britannie prestat.”
The observations of Doctor Rutty show that the winters in Ireland are milder than
in England.”—Rutty’s Natural Hist. of the County of Dublin, Vol, I. p. 466.
The description of the island by Donatus, has been often given :
“Insula dives opum, gemmarum, vestis et auri
Commoda corporibus, aere, sole, solo
Melle fluit pulchris et lacteis Scotia campis
Vestibus atque armis, frugibus, arte, viris
Ursorum rabies nulla est ibi seva leonum
Semina nee unquam Scotica terra tulit
Nulla venena nocent nec serpens serpit in herba
Nec conquesta, canit, garrula rana lacu.”
The picture which Cambrensis has drawn of the climate of Ireland, bears with it
the marks of having been highly coloured ; in the first place the style is poetical—
“Terra autem terrarum temperatissima nec Cancri calor exeestuans compellit ad um-
M‘Sweeny, on the Climate of Ireland. 199
bras, nec ad focos Capricorni rigor invitat, aeris amznitate temperieque tempora fere
cuncta tepescunt.”— Topog. Hib. dist. 1, cap. 25.
Again, in another part he observes—‘ Aeris clementia tanta est ut nec nebula in-
ficiens, nec spiritus hic pestilens nec aura corrumpens, medicorum opera parum indiget
insula ; morbidos enim homines preter moribundos paucos invenies.”— Topog. Hib.
Dist. 1, cap. 27.
The Abbé Ma-Geoghegan, in commenting on this description by Cambrensis, in his
History of Ireland, remarks thus—‘‘ Cependant le temoignage de Cambrensis me
paroit un peu suspect, parce qu’ il est outré. En effet les pluies, les neiges et les ge-
lées y sont assez frequentes en hyver.” He might have shown by the Irish Annals,
previous to the invasion, and by Ware’s Annals, at a subsequent period, that cold
winters have been often recorded. It is manifest, by these documents, that the inha-
bitants of this country were not entirely so free from disease, as Cambrensis described
them to be.
In the Irisk Annals, we find that bowel complaints were not unfrequent. In fact,
distempers of this nature were called by the general name of the country disease.
Dermot Mac Murrough, the cause of the invasion in the time of Cambrensis, did
not die of old age, but of disease. The soldiers of the English army were affected
by sickness.
It is probable that the armour of the English adventurers, particularly of the chiefs,
afforded protection, not only against the weapons of the natives, but also protected
them in some measure, from the drenching rains of the island.
The expression of Cambrensis is very vague—‘‘ Morbidos enim homines preter
moribundos paucos invenies.”” It might as well imply that the faculty in those days,
made quick work with their patients.
Although the account by Cambrensis, of the climate, is exaggerated, still it may be
received in evidence, as to the general mildness of the weather in Ireland.
Kirwan in his work* on the temperature of different latitudes, thinks that ‘‘the as-
tronomical source of heat is permanent.” If this be the fact on an average of years,
and if it has held good in former times,t it must follow that less inconvenience was
felt from heat in summer, at the time of Cambrensis, in Ireland, than at a subsequent
period, when the woods were cut down. When woods abounded in Ireland, of course
a great portion of the surface of the island was sheltered from the rays of the sun ;
therefore, moisture on the ground in the woods, could not be rapidly dried up.
Evaporation causes a depression of temperature ; the constant evaporation from ex-
tensive woods must, independently of the shade afforded, have tended to keep the
surface of the island cool in summer, at the time of Cambrensis.
* Kirwan, page 107.
+ There is every reason to think that it has, from the investigation of Professor Schow of Copenhagen.
200 M‘Sweeny, on the Climate of Ireland.
A good deal of information relating to the weather in Ireland, may be collected
from Ware’s Annals. ‘The following are extracts :
** A.D.1171 This winter the English soldiers, by the scarcity of provision, and change
of air and diet, contracted several distempers, and many died.
“©1172 A very tempestuous winter, the king having stayed three months in Dublin.
‘*€1192 This likewise may seem worth the remembering, that this year there were
so great tempests in Desmond, that many houses and churches were beaten down, and
much cattle and men destroyed.
‘1209 The city of Dublin, by reason of some great mortality, being waste and
desolate, the inhabitants of Bristol flocked thither to mhabit.
«©1247 The same year, saith Florilegus, there was a marvellous and strange earth-
quake over England, but saith Feleon, over Ireland, and all the west of the world ;
and there followed immediately a continual intemperature of the air, with a filthy
skurf, the winter stormy, cold, and wet, which continued until the 11th of July, and
put the gardeners, fruiterers, and husbandmen, void of all hope, insomuch that they
complained that winter was turned to summer, and summer to winter, and that they
were like to lose all, and be undone.
‘©1326 The earth received fruitfulness, the air temperature, and the sea calmness.
‘©1348 This year there was great mortality in all places.
“1361 About Easter, began a great mortality of men, but few women in England
and Ireland.
*©1370 There was a third pestilence in Ireland.
“1383 The fourth great pestilence was in Ireland.
“1486 March, there happened so great a storm of wind and rain, that trees were
pulled up by the roots, and many houses, and some churches, were blown down to the
ground.
‘*€1489 This summer proving very pestilent and feverish, many people died.
“©1491 This year was commonly called by the natives, the dismal year, by reason
of the continual fall of rain all the summer and autumn, which caused great scarcity
of all sorts of grain throughout Ireland.
«< About the latter end of December, after the appearance of a blazing star, which
shone for some days, a certain grievous and pestilential sickness, commonly called the
English sweat, began first to afflict this nation.
‘1492 There was so great a drought this summer, throughout Ireland, that many
rivers were almost dried up, the cattle dying every where with thirst ; also soon after
the pestilence began to rage.
«1500 This year from the middle of September, till the end of winter, Ireland en-
dured continual rains, and many tempests.
“1504 This year the pestilence swept away many people, almost every where, but
especially in Ulster.
M‘Sweeny, on the climate of Ireland. 201
“1505 The plague not yet ceasing, did even this year also, grievously afflict Ire-
land, a great dearth of corn following it by reason of the continual rains that fell in
summer and harvest.
“©1510 This year, in the month of April, did happen great ¢nwndations, which
overturned trees, houses, and bridges.
“1517 In this year was a very hard winter, so that the ice of the rivers did not
only for a long season bear up men upon it, but also loaded carts.
‘©1522 The city of Limerick was sadly visited with the plague.
“1523 There was great scarcity of corn this year in Ireland, by reason of the
continual rains in summer.
‘©1525 The pestilence was rife all this autumn, especially at Dublin.
“1528 This year a certain grievous and pestilential disease, commonly called the
English sweat, did overspread a great part of Ireland.
“©1534 An earthquake happened at Dublin, which accident is so rare in Ireland,
that when it falls out so, it is esteemed as a prodigy.
©1535 A raging pestilence did this year sweep away many, especially in Ulster.
“1539 This summer so great a drought was in Ireland, that many rivers were
almost dried up. The autumn also was very sickly, fevers and bloody fluxes, being
rife every where, whereof many died. An extreme hard winter followed, insomuch
that store of cattle perished in many places.
“1548 February, there happened such a strange violent tempest, or rather hurri-
cane, in most parts of Ireland, that by the force of it, trees were rooted up, and
churches and other edifices, quite blown down.
“1552 In this year there was such a scarcity of corn in Ireland, that a peck of
wheat (which contains four bushels of English measure) was sold in Dublin for twenty-
four shillings ; but the following year carried such plenty with it, that a peck of pure
wheat was sold for five shillings.
‘©1554 This year there was a very sad winter, especially from the 21st of December,
to the end of the following spring, either perpetual rain, hail, or tempest.
“1574 This summer the plague raged in Dublin for several months.
1599 The Lord Lieutenant, Earl of Essex ‘ towards the end of July, returned to
Dublin, his army being much diminished in number, fatigued, and in a sickly con-
dition.’ ”
These annals show the occasional occurrence of very dry summers, of very severe
winters, and of seasons so wet as to cause a scarcity of corn in Ireland. It does not
follow by any means, that all the remarkable years are included in these annals ; for
instance, Ware had to collect the accounts of the weather from books, written
without any particular view to meteorology.
Necessity is justly called the mother of invention; where the mere whim of
202 M‘Sweeny, on the Climate of Ireland.
fashion does not influence, we may discover in the dress of particular nations, some
indications of the nature of the climate.
The large shading hats of the Spaniards, bespeak a sunny clime.
The conical caps of the ancient Irish, were admirably adapted for protecting the
head against rain, and may be received as collateral evidence of a moist climate.
Cambrensis describes the Irish as dressed in woollens, and the mantle as a protection
against the rain, is mentioned by Spencer.
The account of the Irish in the reign of James I. as given by Morryson, is scarcely
worthy of notice ; he says—‘‘ In the remote parts where the English laws and man-
ners are unknown, the very chiefs of the Irish, as well men as women, go naked in
the winter time.” Dr. Leland gives no credit to this account of Morryson ; he
remarks—‘ The fact is totally incredible, the climate must at all times have forced the
most barbarous to some covering in their retired chambers.” Walker, in his essay on
the dress of the ancient Irish, agrees with Leland on this point ; and it is worthy of
observation that Morryson speaks of the remote parts, with which of course he was
the least acquainted ; besides, and what has not been remarked by Leland or Walker,
Morryson had been present at the celebrated siege of Kingsale, at the time of the
Spanish invasion, and therefore was a witness of the severe weather that prevailed
during that siege.
In his history of Ireland, L. Abbé Ma Geoghegan, thus writes of the dress of the
Irish—‘‘ Les manufactures de toiles d’ etoffes, de tout ce qui etoit necessaire pour les
couvrir et garantir de l intemperie del’ air etoient connues aux anciens Irlandois.”
The abundance of timber in former times, must have led the inhabitants, in Ireland,
as it does in America at present, to construct habitations of that material. In Han-
mer’s Chronicle, we have the reasons assigned by Mac Mahon, an Irish chieftain, for
not residing in a castle. Hanmer informs us, that Mac Mahon levelled two castles
bestowed on him by Sir John De Courcy, a short time after the coming of the Eng-
lish. He said that he had promised not to hold stones, but land, and that it was
contrary to his nature to couch himself within cold stones, the woods being so nigh.
The desire of the ancient Irish to reside in woods, no doubt arose from the shelter
afforded against the winds, from the proximity of timber for the construction of habi-
tations, and for fuel, and probably from the facility of enclosing, by means of stakes
between the trees, during the night, their cattle, always a desirable prey amongst a
pastoral people, divided into a number of septs, frequently in a state of hostility with
each other.
Cambrensis tells us that woods were inhabited as places of defence : ‘* Hibernicus
enim populus castella non curat, sylvis namque pro castris, paludibus utitur pro fos-
satis.” — Top. ib. Dist. 3, c. 37.
In the Dublin Philosophical Journal, there is an account of the finding of the body
of a man preserved in a peat bog, and dressed in a singular costume. ‘The dress was
M‘Sweeny, on the Climate of Ireland. 203
composed of the skin of some animal, laced in front with thongs of the same ma-
terial, and having the hairy side inwards. The writer who describes it, thinks that it
belonged to a period antecedent to Cambrensis, as he described the Irish dressed in
woollen garments. However this may be, the dress is well adapted for keeping out
rain, and may be received in evidence of a rainy climate when the wearer lived.
The woods in winter afforded protection to the inhabitants against high winds, and
in warm summers yielded a pleasant shade; but military defence was probably the
great inducement for their choosing such places of residence,
In the Speculum Regale, a treatise written in the twelfth century, the inhabitants
of Ireland are described as being well clothed in winter and summer.— Antiquarian
Repertory, Vol. II. page 336,
The loose coat, a garment so much worn by our peasantry, is supposed to be a
remnant of the old Irish mantle.
This garment, the great coat, worn in winter and summer, is often valuable
in affording protection against rain in a variable climate.
However mild the climate of Ireland is to persons who can have the shelter of a
~ house when necessary, still to troops in the field, obliged to march at all hours, and,
of course, exposed to wet, it must be any thing but agreeable.
Therefore, it is not strange that we find, in the history of Irish warfare, complaints
of the weather, and of sickness amongst the troops, particularly amongst those newly
arrived.
It is extraordinary how well the Irish peasantry bear the drenching rains of this
climate—they travel, and frequently work, in weather that would prove destructive to
strangers, or even to men from the cities or towns in Ireland.
The sufferings of different armies, at different periods, tend to prove, therefore,
that the general character of the climate has been the same.
The troops of Henry the Second were affected with sickness.
The army of Bruce, when he invaded Ireland, suffered from the weather.
The army of the Earl of Essex, in the reign of Elizabeth, was diminished by sick-
ness ; indeed the English troops in Ireland, in her reign, suffered dreadfully from the
climate.
The sufferings of the soldiers of Cromwell are well known. It would appear that
officers of rank, at this period, had sometimes recourse to oil cloth, as a protection
against the weather. Ludlow, in his memoirs, says, ‘‘ I clothed myself as warm as I
could, putting on a fur coat over my buff, and an oiled one over that, by which means
I prevented the farther increase of my distemper.”
Who has not heard of the sufferings of the army of William the Third in Ireland
from the climate ?
In the Pacata Hibernia we have a good deal of detail given of the weather in
Ireland, in the reign of Elizabeth. ‘
VOL. XVII. Q2
204: M‘Sweeny, on the Climate of Ireland.
June is stated to be a convenient time to be in the camp. ‘‘ Whereas, if the service
should be deferred until winter, difficulties should they find in the foulness of the
weather, and deepness of the way.” In July, the Lord President left Limerick, to
relieve a place in Kerry, “and set forward the three and twentieth of July ; but,
whereas by reason of continual rain, that had lately fallen in great abundance, it was
thought that the mountain of Sleulogher was impassable for carriages, was con-
stramed to take the way of Thomond.” In January, the Lord Dunboyne forced
Redmond Burke’s forces into the Nore, where seventy of his men were drowned,
*‘ the river Nore being at that time very high. The ninth of August Sir Francis and
his troops lodged at Alphine, in the County Roscommon; the morning following was
dark and misty.”
(September.) And no sooner could there a ship appear upon the coast, but
presently it was supposed to be a Spaniard, but there none appeared before the seven-
teenth of the same month, which the Lord President perceiving, and that the
winds still were contrary, and the weather very stormy and tempestuous.”
(October.) It is stated that in this month some ships with provisions were detained
in Waterford, “ enforced to stay there, the wind being southerly.”
The early part of the month is described as so wet, that it was unfit for the army
to take the field. .
A short time after the commencement of the siege of Kingsale, the weather is
described ‘falling out very foul.” And again, “ We attended all that day for the
landing of the artillery, and perfected the entrenchment about the army, which was
left unperfected the day before, through the extreme foulness of the weather.”
(November.) ‘‘ For the mountain Slewphelim, which, in summer time, is a good
ground to pass over, was, by reason of great rains, so wet and boggy, as that no
carriage or horse could pass over.” ‘The writer of the Pacata Hibernia next speaks
of the frost that enabled O’Donnell’s army to cross this mountain, on their way to
assist the Spaniards: ‘There happened a great frost, the like whereof hath been
seldom seen in Ireland.”
An account is given, that ships, with supplies from England to the siege of
Kingsale, were driven to the “southermost part of Ireland” by’ the foulness of the -
weather. The besiegers were prevented on the 17th from attempting any thing, on
account of the weather, but at night, “when the storm was somewhat appeased,”
they caused some officers to view the ground of Castle Ny Parke. Next, the extreme
frost is spoken of asa difficulty in making approaches: “Continued to work all
night, and although the ground was extremely hard, and the night very light, yet
they brought the work to very good perfection.”
The variable nature of the climate is well marked by the next quotation—‘ The
enemy sallied about eight of the clock in the night, being extreme dark and rainy,
with about two thousand men.”
MG:
\
M‘Sweeny, on the Climate of Ireland. 205
(December.) Sir Richard Levison returned into the harbour of Kingsale, and
reported to the Lord Deputy the damage done to the Spanish fleet at Castlehaven :
“the seventh of December, the wind being extremely at south-east, he rode still at
Castlehaven, the night following, with wind at west-south-west, he warped out with
the ships.” While Sir Richard was at Castlehaven he was exposed for some time to
the fire of cannon from the shore, “being by no industry able to avoid it until some
calmer weather came.”
« The thirteenth, the weather fell out to be extreme foul and stormy. The four-
teenth, foul weather, wherein nothing was performed. The seventeenth, foul and
stormy weather. The nineteenth, by reason of stormy and foul weather, nothing on
either side was performed. This morning the ordnance played oftener.” By the
context it appears to have been the twentieth. ‘The next morning that work was
brought to great perfection, though the night fell out stormy, with great abundance
of thunder and lightning, to the wonder of all men, considering the season of the
year.” The latter end of December is described as being extremely tempestuous,
cold, and wet, at the time an attack was expected from O’Neal’s army.
In February Captain Flower was obliged to put back, in an attempt to reach the
castle of Dunboy, “by reason of foul weather and contrary winds.”
In a letter, dated the 15th of February, 1601, O. S. the wind is described so
westerly, as to prevent the arrival of shipping to carry away the Spaniards that sur-
rendered.
«The eighth of March, Don Juan being at Kingsale, hourly expecting a wind to
be gone, and, finding a flattering gale, went aboard, but, for want of a fair wind,
departed not from Kingsale until the sixteenth of the same month.”
(May.) The army is described as on its way to besiege the Castle of Dunboy.
“ The fifth and sixth the weather was so tempestuous, that we could not stir out of
quarters. The thirteenth, unseasonable weather. From the seventeenth to the six
and twentieth nothing happened worthy of notice, only we were detained in our
camp with contrary winds, and with strange, unseasonable, and tempestuous weather.
The six and twentieth the wind turned fair, and the shipping drew forth, but imme-
diately the weather proved so tempestuous, they were constrained to return to their
former road. The seven and twentieth, the eight and twentieth, the nine and
twentieth, and the thirtieth, we were detained with like contrary winds, and unseason-
able, foul, and stormy weather. The one and thirtieth the weather grew fair, and we
took advantage thereof.”
(June.) ‘ The sixth being Sunday, a foul and stormy morning.” It is to be
supposed that the rest of this month was favourable, as no complaint is made of the
weather during the siege of the castle of Dunboy.
(July.) We find, by the Pacata Hibernia, that in this month Sir Charles Willmot
was sent into Kerry, to remove all the inhabitants, with ‘e goods and cattle, into
206 M‘Sweeny, on the Climate of Ireland.
the County Limerick, and to destroy such corn as could not be presently reaped.
‘« But in effecting hereof, the governor found great difficulty, for the harvest, by
reason of the winter-like summer, was very backward.”
It ought to be remarked here, that barley was very much cultivated at this period
in Ireland, and in treating of the vegetation of the island, it can be shown that
barley in modern times has been reaped very early in Kerry.
(October) ‘ Easterly winds are so seldom upon this coast, as it would ask a long
time to transport victuals and munitions by sea.”
(January) ‘* The sharpness of this winter journey, did exceedingly weaken our
companies, for the mountains of Beare, being at that time quite covered with snow,
tasted the strong bodies, whereby many returned sick; and some, unable to endure
the extremity, died standing sentinel.”
Snow on the mountains, at this time of the year, is not of uncommon occurrence
at present ; and we have evidence from the same work, the Pacata Hibernia, to ren-
der it probable that the cold was not so severe as to freeze the rivers ; for, on the 5th
of January, in an account of a fight between the troops of Captain Taffe, and those
of Owen Mac Eggan, the troops of the latter were driven into the river Bandon:
“Jeaped into the river Bandon, hoping by that means to escape ; but that little availed
them, for they all for the most part, were either killed or drowned in the river.
(March) “After the Lord Deputy departed, by reason of easterly winds, the
President was stayed about three weeks in Dublin, during which time, every day, posts
were employed between them.”
Here we have an account of the prevalence of easterly wiuds in the spring, which is
well known to hold good at present. This prevalence of easterly winds, in the spring,
was also remarked at the time of Doctor Boate, in Ireland. Doctor Rutty informs
us, that the easterly winds in spring, are nearly double to what they are in autumn
and winter, and that the North East wind in spring, is double to what it is in autumn
and winter.—Natural History of the County of Dublin, Vol. II. p. 457.
Indeed, taking in general, the details of the weather in the reign of Elizabeth, as
they are to be gathered from the Pacata Hibernia, we find a similarity between them
and the observations of Doctor Rutty. June is stated to be a convenient time to be
in camp, in the above cited work: by Rutty’s statements, there would be a good
chance of fair weather in this month. By the Pacata Hibernia, it appears that
abundance of rain fell in July. During the space of forty-three years in Dublin, in
seven years only was the month of July fair and dry.—Natural History of the
County of Dublin, Vol. II. p. 462.
We have it expressly stated in the Pacata Hibernia, under the head of October,
that easterly winds were seldom on the south coast of Ireland.
There is reason to think from the context, that the frost, which was of unusual
severity, at the time of the siege of Kingsale, was ushered in by a north wind, because
the ships, with supplies from England, were driven to the southermost part,of Ire-
— Saye em
M‘Sweeny, on the Climate of Ireland. 207
land. Rutty observes, that the great frost in 1739-1740, was attended with an
unusual suspension of our trade winds of the west and south west.
The frost at the siege was put to an end, most likely by a change of the wind to the
south. ‘The enemy are stated to have sallied out on a dark, and rainy night ; soon
after, we find by a quotation, the wind to be at south east.
Campion, in his History of Ireland, says—‘ The soil is low and waterish, and in-
eludeth divers little islands, surrounded with bogs and marshes—highest hills have
standing pools on their top. Inhabitants, especially new come, are subject to distilla-
tions, rhums, and flixes, for remedy whereof they use an ordinary drink of aqua vite,
so qualified in the making, that it drieth more and inflameth less than other hot con-
fections. The air is wholesome, not altogether so ciear and subtle as ours of Eng-
land ; of bees, good store—no vineyards, contrary to the opinion of some writers,
who both in this and other errors, touching the land, may be easily excused, as those
that wrote of hearsay. Cambrensis, in his time, complaineth that Ireland had excess
of wood, and very little champaign ground, but now the English pale is too naked.”
This observation of Campion, respecting vineyards, may be considered as an an-
swer to Bede’s statement of the vine being found in Ireland. Indeed the expression
of Bede is not very strong; he says “nec vinearum expers.’? ‘The same may be
said of Ireland now, where the vine is cultivated for ornament.
Spencer describes the island adorned with woods, the heavens as most mild and tem-
perate, though somewhat moist ; but in another passage, in discussing the origin of the
Irish mantle, and in maintaining that it was introduced by invaders ; he says—‘* And
coming lastly into Ireland, they found there more special use thereof, by reason of the
raw cold climate.’”? On this passage it may be observed, that it is well known to philo-
sophers, that our sensations from cold are not always in proportion to the degree indi-
cated by the thermometer; a certain degree of cold, combined with moisture, will
produce on our frames very chilling effects. There is nothing in those observations
of Spencer, that ought to induce us to think that any great change has taken place in
the seasons, since his time. Moisture, combined with a certain degree of cold, is a
sufficient inducement to the use of warm clothing. In describing the various uses of
the mantle, he adds— When it raineth it is his pent house, when it bloweth it is his
tent, when it freezeth it is his tabernacle.”
In Spencer’s account of his plan for putting an end to the disturbances in Ireland,
we have some insight into the nature of the winters in his time ; he observes—“ It is
not with Ireland as it is with other countries, where the wars flame most in summer,
and the helmets glitter brightest in the fairest sunshine. But in Ireland the winter
yieldeth the best services, for then the trees are bare and naked, which use both to
clothe and house kern; the ground is cold and wet, which useth to be his bedding ;
the air is sharp and bitter to blow through his naked sides and legs ; the kyne are
barren and without milk, which useth to be his only food; neither if he kill them,
Ry 3}
208 M‘Sweeny, on the Climate of Ireland.
will they yield him flesh; nor if he keep them will they give him food: besides being
all with calf for the most part, they will, through much chasing and driving, cast all
their calves, and lose their milk which should relieve him next summer.”
It ought here to be particularly remarked, that no allusion is made to severe frosts, as
a hard frozen state of the ground would be of the greatest consequence to the pursuers,
in enabling them to follow through bogs and marshes, in the most direct line, those
who in other circumstances might escape by their knowledge of the country. It is,
therefore, just to infer, that severe frosts were not of frequent occurrence in Ireland
at the time. We have the ground described as cold and wet, such as it is commonly
with us in winter.
It does not follow but that severe winters occasionally occurred as they doin mo-
dern times.
In Sir W. Betham’s Antiquarian Researches, Life of O’Donnel, the frost is men-
tioned by which young O’Donnel lost some of his toes, in escaping from Dublin. In
like manner, an unusually severe frost is described in the Pacata Hibernia, by which
the same O’Donnel, at a subsequent period, was enabled to cross a swampy mountain
with his army, on his way to assist the Spaniards, besieged in Kingsale. But, ifa hard
frozen state of the ground was of common occurrence in winter, it is difficult to sup-
pose that the acute Spencer would not have alluded to it.
The remarks of Spencer are worthy of every attention; from his long residence in
Ireland, he had ample opportunities of observing the general state of the island, and of
making a comparison between its climate, and that of England. He, no doubt, con-
trasts the Irish climate with the English, when he says that it is most temperate,
though somewhat moist.
In modern times, another Englishman, Arthur Young, came to the same conclusion
from his own observation.
We shall now have to direct attention to the character of the climate, as given by
Sir James Ware ; he says—* Pomponius Mela affirms that the temperature of Ire-
land is unfit to bring seeds to maturity. But more particularly, Giraldus Cambrensis,
says: § Thus corn promises much in the grass, more in the straw, but least in the ear ;
for the grains of wheat are so small, that they can scarce be cleansed by the help of a
fan.’ Let us hear now what others of the ancients have written to the contrary. Thus,
therefore, Orosius: ‘It lies nearer,’ says he, ‘to Britain; is less in extent, but of a
more temperate air and profitable soil.’ Likewise, Isidore: ‘The next island to Bri-
tain, less in extent of land, but more fertile ;’ and Bede : ‘ Ireland,’ says he, ‘ both in
healthfulness, and also serenity of the air, much excels Britain. But to speak my opi-
nion: if these comparisons relate to the south part of Britain, which we call England,
they are not to be allowed, yet we grant that Ireland is of so temperate an air, that
we see the fields green and flourishing in the midst of winter, and cattle put daily to
grazing, unless in time of snow, which is rarely of two or three days continuance.
M‘Sweeny, on the Climate of Ireland. 209
Many boggy and fennish places being also now drained, the temperature of the air
has been much improved. As to the grains of corn, they are not generally so small
as Giraldus and his followers say ; for in very few of the neighbouring countries,
fairer or larger corn is to be found, than in Ireland. Nor can we allow of the opinion
of Raphael Maffeus Volateranus, that Ireland produces nothing but corn and horses,
The error likewise of Ranulphus Higden, that Ireland has no pheasants, partridges,
deer, nor hedgehogs, is to be corrected. We might observe many things that are fa-
bulously delivered by Giraldus Cambrensis, concerning Ireland ; and the reader is to
take notice that Giraldus’s Topography is to be read with caution, as Giraldus
himself in a manner acknowledges in the apology which he makes in his preface to his
book of the conquest of Ireland.”—/Vare’s Antiquities of Ireland, chap. 23.
Here we have the testimony of Sir James Ware, that in his time, the fields were
green in the midst of winter, and that cattle were not prevented from grazing, except
in case of snow, which rarely lasted two or three days.
In the Speculum Regale, a work supposed to be written in the twelfth century ; we
are informed that oxen and sheep were continually fed out of doors in Ireland.—
Antiquarian Repertory, Vol. I. p. 336.
Petrus Lombardus, in his book de De Regno Hibernize Sanctorum Insula, stated
that the habitants neglected to make hay. “ Hic plerique negligunt resecare foenum
ob summam temperiem aeris.” The mildness of the climate is here given by him as
the cause ; but though this neglect of saving hay might have been very common in his
time, yet the word ‘‘ plerique” shows that it was not universal.
Patterson thinks it may be accounted for, by the plenty of ground they had in pro-
portion to the stock of cattle.
The evidence of Lombard and of Sir James Ware, may be put in opposition to the
statement of Hamilton, who thinks that the great mildness of our winters is of recent
date. He says in his memoir on the Climate of Ireland— Winter has likewise felt
the general influence of this Atlantic temperature, our grasses scarcely droop beneath
the frost.” When he penned this he certainly could not have recollected, that Boate,
in his work on Ireland, had also mentioned, that in his time, cattle fed out in the fields,
day and night in winter, and were seldom troubled with great frost. Thus the state-
ment of Hamilton himself may be used now to show that no great change has taken
place in the climate.
Some particulars relating to the climate, may be gleaned from the history of the
civil war, at the time of Charles I. Sir J. Temple describes the weather as very severe
on the breaking out of the rebellion in 1641—* Most bitter cold and frosty.” He
describes it as the severest year in the memory of man. Among other reasons for
sending an army of Scots into Ireland, one was, “ that their bodies would better sort
with the climate."—Sir J. Temple’s History of the Rebellion of 1641.
210 M‘Sweeny, on the Cimate of Ireland.
About the middle of March, 1643, the Marquis of Ormonde’s army at the siege of
Ross, suffered from ‘ continual rains."—TVarner’s Civil Wars, Vol. I. p. 252.
In the middle of June, 1643, some cavalry under Lord Castlehaven, “being fa-
voured by the rain,” succeeded in a charge, in routing the troops under Sir Charles
Vavasour.— Varner, Vol. I. p. 271.
The forces of the council of Kilkenny, in 1645, laid siege to the fort of Dun-
cannon “in January, and in extreme bad weather.”—/Varner.
When O’Neil and Preston, advanced in the winter of 1646, to Dublin, to besiege
it, the bad weather and a flood in the Liffey, which carried away some bridges, in-
terfered with their operations.— Varner.
In September, 1649, the English fleet, with an army, and Cromwell aboard, were
put into Dublin by a strong gale from the south.— Ludlow’s Memoirs.
The English army, shortly after their arrival. were affected with flux. Ludlow.
Cromwell laid siege to Wexford, in October 1649, and took it after a short time.
The Marquis of Ormonde was greatly disappointed, for he had flattered himself that
it “would hold Cromwell long enough in play until his forces, which were unused to
the climate of Ireland, would be so considerably reduced by the fatigues of a siege at
such a season.” —/Varner, Vol. II. p. 188.
*« Though the siege of Wexford had been very short, yet Cromwell’s army were not
all pleased with a winter campaign, and complaining of great hardships, began to
mutiny.”—TIbid, p. 189.
A. D. 1650 “ The English army were much wasted with sickness and hard duty, as
well as the plague, and the greatest part of those he (Cromwell) had brought
with him, had perished; but the fatal revolt of the Munster forces, had recruited him
with men, habituated to the climate, and inured to the hardships of an Irish war.”—
Ibid, p. 208.
A.D. 1651, Siege of Limerick—* Ireton lost many men by hard service, change
of food, and alteration of the climate.”?—Tbid, p. 243.
Cromwell’s army being attacked with flux soon after their arrival, coincides with the
account of the climate which had been given by Campion.
We can judge of the general character of the climate of Ireland, on an average of
years from the work of Doctor Boate. He observes—“ So that the Irish air is greatly
defectuous in this part, and too much subject to wet and rainy weather, wherein if it
were of somewhat a better temperature, and as free from too much wet, as it is from
excessive cold, it would be one of the sweetest and pleasantest in the whole world ;
and very few countries could be named that might be compared with Ireland for
agreeable temperature. Although it is unlikely that any revolution of times will pro-
duce any considerable alteration in this, (the which indeed in some other countries,
hath caused wonderful changes) because that those who, many years ago, have written
of this island, do witness the self same things of it in this particular, as we do find
‘
M‘Sweeny, on the Climate of Ireland. 211
~
in our time: there is, nevertheless, great probability, that this defect may in part,
be amended by the industry of man, if the country, being once inhabited throughout
by a civil nation, care were taken every where to diminish and take away the super-
fluous and excessive wetness of the ground, in all the watery and boggy places,
whereby this too great moistness of the air is greatly increased, and also occasioned.
*« This opinion is not grounded upon some uncertain speculation, but upon assured
experience, for several knowing and credible persons have affirmed to me, that already
some years since, good beginnings have been seen of it, and that in some parts of the
land, well inhabited with English, and where great extents of bogs have been drained
and reduced to dry land, it hath been found by the observation of some years, one
after another, that they have had a drier air, and much less troubled with rain than in
former times.”
The number of rivers and brooks in Ireland is the best proof of its great moisture.
Boate says: ‘‘ No country in the world is fuller of brooks than Ireland, where the
same be numberless, and water all the parts of the land on all sides.”
On the subject of cold, he remarks. there are commouly three or four frosts in one
winter, but they are very short, seldom lasting longer than three or four days together,
and withal at their very worst, nothing near so violent as in most other countries.
«There hath been,” he observes, ‘‘ some winters wherein it hath frozen ten or twelve
days together, so as the Liffie, and other the like rivers, were quite frozen, and might
be gone upon by man and beast ; but those are altogether extraordinary, and do come
very seldom, hardly once in the space of ten or twelve years.’
Here we have some evidence to show, that the extension of cultivation had, up to
the time of Doctor Rutty, some effect in mitigating the cold of severe winters when
they did happen. We do not find the Liffey, on an average of years, so often frozen
over, that it might be gone upon by man and beast, as was described by Boate in his
time.
Kirwan’s observations on the weather, correspond in the general features with the
accounts handed down by Rutty.
Rutty’s descriptions are, as Kirwan remarks, merely popular ; they therefore can-
not be accurately compared with more precise accounts in latter days ; but, on the
other hand, these haye been made at different periods, in different places, and by dif-
ferent persons.
They lose much of their value, if the opinion maintained by many, be correct,
namely, that the seasons go through a cycle ; therefore it would be necessary, that ob-
servations made in any one place, should be continued for a very long time, before we
would be warranted in attempting to draw very precise conclusions.
The unconnected accounts we have, answer, however, to show the general nature of
the climate, which agrees in its principal points with ancient accounts, and with those
of Rutty and Kirwan. We have mostly a prevalence of south and south west winds
VOL, XVII, 3A
212 M‘Sweeny, on the Climate of Ireland.
in the winter, and of north and north east winds in the spring, the wetness of winter,
the humidity of the summer, particularly about the time of July, and the variability
of different years, when closely compared with each other. On the supposition of
a cycle, variability of years ought to be expected, and a variation is manifest in the
accounts we have.
When we have a severe winter in Ireland, it is the effect of a general cause, acting
with greater effect on the continent. Ancient and modern accounts agree as to the
usual mildness of our winters.
Boate observes—“ For the most part there falleth no great store of snow in Ireland,
and some years, none at all, especially in the plain countries. In the mountains,
there is commonly greater plenty of snow than in other parts, so that all kinds of
cattle do, all winter long, remain there abroad, being seldom troubled with very great
frost or snow, and do feed in the fields, night and day.”
The very attempt of Hamilton, in modern times, to prove a change of climate, by
describing its great mildness, only corroborates Boate’s statement, and must inevitably
lead the impartial reader, who compares the two accounts, to conclude that no great
change has taken place.
Boate describes the heat of summer thus—‘‘ The which is seldom so great, even
in the hottest times of the year, as to be greatly troublesome. And it falleth out
often enough in the very summer months, that the weather is more inclinable to cold,
than to heat, so as one may very well endure to come near a good fire. And this
cometh to pass only during the wet weather, for else, and whilst it is fair, it is very
warm all summer long, albeit seldom over hot.”
There is a strong similarity in the description of the spring of the year, given by
Boate, to that given by Rutty in his time. Rutty showed that the winds from the
rainy points were not prevalent in the spring. Doctor Boate, says—‘* And it raineth
there very much all the year long, in the summer as well as in the winter ; commonly
in the spring of the year it is very fair weather, with clear sunshine from morning till
night, for the space of five or six weeks together, with very little or no interruption,
which fair weather beginneth commonly in the month of March, some years in the begin-
ning, other years in the midst, and sometimes in the latter end of it. But the same
being once passed, it raineth afterwards very much all: the summer long, so as it is a
rare thing to see a whole week pass without it, and many swnmers ti is never dry
weather two or three days together. Which inconstancy of the weather, is not only —
troublesome to men, but also hurtful to all things growing out+of the earth for man’s
behoof.”
What will the advocates for an increase of humidity in summer, in modern times,
say to this ? ‘
The above quotation from Boate, is enough of itself, to upset the doctrine main-
tained in Hamilton’s Memoir on the Climate of Ireland.
ae
M‘Sweeny, on the Climate of Ireland. 213
That close observer, Doctor Rutty, tells us, that “a series of hot and dry weather,
evergin summer, is what the farmer ought not to expect, but to provide for the con-
trary.”’—Natural History of the County of Dublin, Vol. II. p. 281.
In the Introduction to Cox’s History of Ireland, the comparison is made between
the climate of England and Ireland. The summers are stated to be warmer, and the
winters colder in England, than in Ireland. He adds, thus—‘“ It may be expected, that
as the bogs are drained, and the country grows populous, the Trish air will meliorate,
since it is already brought to that pass, that fluxes and dysenteries, which are the
country diseases, are neither so rife, nor so mortal, as they have been heretofore.”
In the reign of William III. the potatoe culture was extending itself in some degree
in Ireland, the country was denuded of timber, and was therefore less shaded in sum-
mer, than it was in the time of Cambrensis. By Cox’s statement, bowel complaints
had become less prevalent than before.
The trenches in the potatoe culture, were admirably adapted for quickly removing
superfluous water, as the best mode of forming them is in the direction of the summit
of a range of hills at right angles with it. It was, therefore, no wonder, that in the
places where the culture of the potatoe was commenced, some improvement should
be experienced in the time of Cox, in the reign of TVilliam the Third.
As the country was denuded of trees, and as the surface was not shaded in every
direction by luxuriant stalks of potatoes in summer, as it is at present, when a warm
summer occurred, the inhabitants must have experienced some inconvenience from
heat. Leland says of the garrison, during the celebrated siege of Derry—‘ The
heats of summer proved even pestilential to men fatigued and confined ; and their
scanty, and unwholesome diet, inflamed their disorders.”
The history of the war at this period, in Ireland, affords ample testimony of the
general moist character of the climate.
Some extracts from Leland may answer better than a long commentary. Speaking
of the sufferings of Schomberg’s army, he says—‘‘ His men had already experienced
the hardships of their present service, wasted by a fatiguing march in rain and tem-
pest, in cold and hunger, through a country, dispiriting by its aspect, and by the in-
clemency of the season rendered still more dreary and distressing.”
When Schomberg halted, waiting for the arrival of artillery and provisions, the si-
tuation of the army is thus described by Leland—* His soldiers in a confined and
unwholesome situation, in the midst of damps and winter showers, without sufficient
food, fuel, or covering, an unfriendly climate and inclement season, soon weakened
‘the whole army by fluxes, and a burning fever was caught from the garrison of Derry.
The English, unaccustomed to severities, confined to a low and moist situation,
drenched with perpetual showers, without the means of health, or the relief necessary
in sickness, died daily, in great numbers.”
When strong efforts were made for the removal of the sick of the army, to places
Q14 M‘Sweeny, on the Climate of Ireland.
of safety, and of shelter, Leland informs us, that the general, at the age of four
score, afflicted with the scene of wretchedness, exposed to the violence of a dgeary
and tempestuous season, stood for hours at the bridge of Dundalk, directing every
means for alleviating the miseries of his men.
James’s council of officers, before the battle of the Boyne, advised him to decline an
engagement with the army of William, and to maintain a defensive war, as the one
most likely to destroy men in an unfriendly climate, in want of provisions, and suc-
eours.—Leland’s History of Ireland.
The sickness of the English army, corroborates the statement of Campion who,
when writing of the island at a previous period, said that persons newly arrived, were
particularly liable to be affected with bowel complaints.
The comparison which Cox made between the winters and summers of England and
Ireland, agrees with that which has been given to us by Doctor Rutty.— Natural His-
tory of the County of Dublin, Vol. LI. p. 469.
In the Pacata Hibernia, we find by a letter, dated August 1602, the Lord Presi-
dent giving his opinion against ever undertaking a winter siege in Ireland, “for
Kingsale was bought at so dear a rate, as while I live, | will protest against a winter
siege, if it may be avoided.”—Pacat. Hiber. p. 631.
In the reign of William HI. at the siege of Kinsale, as it is now called, the army
suffered from the weather. The garrison surrendered upon conditions which would
not have been granted, but that the weather was very bad, provisions scarce, and the
army very sickly.—Smith’s History of the County of Cork, Vol. II. p. 210.
It is difficult to ascertain the precise condition of the weather in distant periods,
the invention of the thermometer, by Sanctorio, being comparatively of modern date ;
and a long time elapsed before the instrument was reduced to a correct standard by
Fahrenheit. Great allowance must be made for the accounts in old chronicles, and it
is possible that extraordinary years happened, accounts of which have not been handed
down to posterity in Irish, or continental annals.
In the Philosophical Magazine for 1820, Vol LV. is given a list of extraordinary
years for a long period, chiefly from a work by Pilgram, in the German language,
published at Vienna, in 1788.
This list of years is valuable, as it shows the occasional return of very severe win-
ters in modern times, and may be used as an answer to those persons who, would at-
tempt to prove, by quotations from the classics, the greater cold of Europe in the
time of the Roman dominion ; it is also valuable, as it proves that one of the coldest
years recorded in the Annals of Ulster, arose from a general cause, by which the
great rivers of Europe were frozen so hard as to bear waggons for a month.
A. D. 401. The Black Sea was entirely frozen over.
462 ‘The Danube was frozen, so that an aoa marched over the ice.
545 The cold was intense.
ee SO, oe
¢,.
M‘Sweeny, on the Climate of Ireland. Ws
763!1 The Black Sea, and the Straits of the Dardanelles, were frozen over.
800 The winter was intensely cold.
822!! The great rivers of Europe such as the Danube, the Elbe, and the Seine,
were frozen so hard as to bear heavy waggons for a month.
860 The Adriatic was frozen.
874 Snow from the beginning of November to the end of March.
In 991 The vines were killed by the frost; and again in 993, cattle perished in
their stalls.
1067 The cold was so intense that travellers in Germany were frozen to death on
the roads.
1124 The winter was uncommonly severe.
1133!! In Italy, the Po was frozen from Cremona to the sea, the snow p adated
the roads impassable, and wine casks were burst by the frost.
1179 In Austria the snow lay on the ground until Easter, and the crops failed.
1216!! The Po was frozen fifteen ells deep, and wine burst the casks.
1234!! The Po was again frozen, and loaded waggons crossed the Adriatic to
Venice.
1236 The Danube was frozen to the bottom.
1261 The frost was intense in Scotland, and the Cattegat at Jutland, was frozen
over.
1281 A vast quantity of snow fell in Austria.
1292 The Rhine was frozen over at Brisach, and bore loaded waggons.
1323 The winter was so severe that both horse and foot passengers crossed on the
ice from Denmark to Dantzic.
1344!! All the rivers in Italy were frozen over.
1392 The vineyards and orchards were destroyed by frost and the trees torn to
pieces.
1408 One of the coldest winters ever remembered. Not only the Datube was
frozen over, but the sea between Norway and Denmark, so that wolves driven from
their forests, came over the ice into Jutland. In France the vineyards and orchards
were destroyed.
1423 Travellers passed on foot from Lubec to Dantzic, on the ice.
The successive winters of 1432-1433-1434 were uncommonly severe. All the
rivers in Germany were frozen over.
1460 Horse and foot passengers crossed the ice from Denmark to Sweden, and the
vineyards in Germany were destroyed.
1468 The winter was so severe in Flanders, that wine was cut in pieces with
hatchets.
1544 The same thing happened again, the wine being frozen into solid lumps.
1548 Between Denmark and Rostock, sledges drawn by horses, travelled over the
ice.
216 M‘Sweeny, on the Climate of Ireland.
In 1564 and in 1565, the Scheld was frozen so as to support loaded waggons.
1571!! All the rivers in France were covered with ice, and fruit trees, even in Lan-
guedoc, were killed by the frost.
1594 The Rhine and the Scheld were frozen, and even the sea at Venice.
1608 The snow lay of an immense depth even at Padua.
In 1621 and 1622!! All the rivers of Europe were frozen, and even the Zuyder
Zee, a sheet of ice covered the Hellespont, and the Venetian fleet was choaked up in
the Adriatic.
1655 The winter was very severe.
The winters of 1658, 1659, 1660!! were intensely cold. The rivers in Italy bore
heavy carriages, and so much snow had not fallen at Rome for several centuries. It
was in 1658, Charles X. of Sweden, crossed the Little Belt, over the ice into Den-
mark, with his whole army.
1670 In Denmark, both the Little and Great Belt were frozen over.
1684 The winter was excessively cold, even oak trees in England were split by the
frost, and coaches were driven along the Thames.
1691 The cold was so excessive, that the wolves entered Vienna, and attacked the
men and cattle.
1695 The frost in Germany began in October, and continued until April ; many
persons were frozen to death.
The years 1697 and 1699, were nearly as cold.
In 1709!!! occurred the famous winter called by distinction the cold winter. In
the south of France the olive plantations were almost entirely destroyed. The Adriatic
was quite frozen over, and the citron and orange groves suffered in the finest parts of
Italy.
1716 On the Thames booths were erected, and fairs held.
1726 People travelled from Copenhagen to Scania, in Sweden.
1729 Much injury done by the frost in Scotland, multitudes of cattle buried in the
snow.
.
The successive winters of 1731—1732 were extremely cold.
1740!! The cold was scarcely inferior to that-of 1709, the snow lay on the
ground eight or ten feet deep in Spain and Portugal, and the Zuder Zee was frozen
over.
1744 The winter was again very cold, the Mayne was covered with ice seven
weeks.
The winters during the five successive years 1745-1746-1747-1748-1749, were all
of them very cold.
In 1754 and again in 1755, the winters were particularly cold.
In England strong ale exposed to the air, in a quarter of an hour, was covered with
a film of ice.
M‘Sweeny, on the Climate of Ireland. 217
The winters of 1766-1767-1768, were very cold all over Europe.
In France the thermometer fell six degrees below the zero of Fahrenheit’s scale.
The thermometer laid on the snew at Glasgow, fell two degrees below zero.
1771 The Elbe was frozen to the bottom.
1776 The Danube was frozen that it had ice five feet thick below Vienna. Wine
was frozen in the.cellars in France.
The successive winters of 1784 and 1785, were so severe, that the Little Belt was
frozen over. i
In 1789 The cold was excessive, and again in 1795, when the republican armies
overran Holland.
The successive winters of 1799 and 1800, were both very cold.
In 1809, and again in 1812, the winters were remarkably cold.
The following list is of years, the summers of which were remarkable for being hot
and dry, from the same work, (the Philosophical Magazine, Vol. 55 :)
‘ASD. FAs | AsDs A.D. A.D. A.D. ASD" A.D, A.D.
763—1000—1171— —1473) —1556 —1646—1718
860—1022—1232—1293 1333—1474 —1615 arma:
9932 1130—1260—1294§ 1393) 1538 ) —1616§ —1660—1724
994§ 1159—1276) 1303 13946 1539
—12774 1304 1540
1541
1745) —1754—1763—1779
1746§ —1760) 1774—1788
1748 —1761§ 1778—1811
As a succession of severe winters and of hot summers, have occasionally occurred, it
is no wonder that an opinion of change of climate should have prevailed at different
times.
In the time of the Hon. Robert Boyle, it was supposed, that the climate of Russia
had changed. Boyle says in his Treatise on Cold—‘ The Czar’s physician tells me
by letter, that the winter he spent at Vologda, proved much less severe than usual, for
as it happened, they had not three days of what they there call winter weather. He
adds, that the cold which is thought to be excessive, hath been rare of late years, for
some English who have lived upon the spot thirty years declare, that during their
time, winters are become so mild, that the extreme cold which used to freeze people
on the road in several postures, hath not been felt as formerly.”—Shaw’s Edition of
Boyle's Works, Vol. I. p. 661.
Bonaparte, if he were alive, would not be very much inclined to subscribe to the
doctrine of a change of climate in Russia.
In Lowthorp’s Abridgment of the Philosophical Transactions, Vol. II. p. 42, we
have a paper on the alteration of the climate of Ireland. It tends to show that the
idea of change of climate has not been confined to modern days, and that a succession
of favourable years has led to the belief of a change. The writer observes, “ every
218 M‘Sweeny, on the Climate of Ireland.
one almost begins to take notice that this country Boing every year more and more
temperate.”
“It was not unusual to have frosts and deep snows of a fortnight and three weeks
continuance, and that twice or thrice, sometimes oftener in a winter ; nay, we have had
great rivers and lakes frozen all over, whereas of late, especially these two or three
years last past, we have had scarce any frost or snow at all. Neither.can | impute this
extraordinary alteration to any fortuitous circumstances, because it is manifest that it
hath succeeded gradually, every year becoming more temperate than the year preced-
ing.——-This winter 1675, I have kept an exact account of wind and weather. To
transcribe my journal here, would be too tedious, let it suffice therefore to tell, that it
hath been a very fair and warm, or rather no winter at all.”
The language used in 1675 is very like that of Hamilton in his Memoir on the
Climate of Ireland.
In Rutty’s time, complaints of the inversion of the seasons were common—“ wicked
exclamations we hear against the inclemency of the climate, our changeable, and par-
ticularly our moist and windy weather, an inversion of the seasons, &c., which are
owing to a want of due attention to this branch of natural history, for those changes
are common to us.”—Natural History of the County of Dublin, Vol. LI. p. 281.
The writer in 1675, described the prevalence of south and west winds, and stated
that persons sometimes had to wait three months for a fair wind to come to Ireland.
He gives the usual height of the barometer in Ireland as at “ 29 inches 4 tenths.”
Though our climate is variable, yet in its chief leading features its history shows it
to be the same. It is well known to us all, that September and October are generally
agreeable months ; Rutty, in describing this time of the year, calls it owr little summer.”
Ibid, p. 465. ¢
Doctor Boate says—“ In the latter end of autumn, weather is commonly fair again,
for some weeks together, in the same manner as in the spring, but not so long, which
as it doth serve for to dry up and to get in the corn, hay which till then hath remained
in the fields, the too much wet having hindered it from being brought away sooner ;
so it giveth the opportunity of ploughing the ground, and sowing the winter corn, the
which otherwise would very hardly be done. For that season being once passed, you
have very little dry weather the rest of the autumn, and during all the winter.”
Boate speaks of thunder being of rare occurrence in Ireland, and says when it does
happen, it is in the summer season. ;
in the reign of Elizabeth, at the siege of Kinsale, there was thunder in December
‘to the wonder of all men, considering the season of the year.”—Pacat. Hiber.
In speaking of dry summers, Boate observes—‘“‘ But as winters cruelly cold, so like-
wise over dry summers do in this island hardly come once in an age, and it is a com-
mon saying in Ireland, that the very driest summers never hurt the land : for al-
though the corn and grass upon the high and dry grounds may get harm, nevertheless
M‘Sweeny, on the Climate of Ireland. 219
the country in general gets more good, than hurt by it ; and when any dearths fall out
to be in Ireland, they are not caused through immoderate heat and drought as in most
other countries, but through too much wet and excessive rain.”
In 1826 a famine was apprehended in Ireland, yet the potatoe crop turned out much
better than could be expected after so very dry a summer.
In the Phil. Trans. No. 220, there is an account of a substance resembling butter,
noticed in Ireland in November, 1695. A similar substance was remarked on grass
in the autumn of 1826. The country people employed it to grease the wheels of
carts.
Boate remarked, that often in Ireland after days of rain, the nights were fair and
clear ; the writer of this essay has frequently made a similar observation.
The researches of Doctor Wells explain the formation of dew; the cloudy sky of
Ireland interferes with radiation, and is one of the chief causes of the mild tempera-
ture of our nights. Boate fell into an error, when he stated that there was as much
dew in Ireland as there was in hotter and drier countries ; but he may be refuted by his
own words—“ It is found ordinarily, that in a clear night following rain, the which is
very ordinary, the dew cometh as liberally, as if it had not rained the day before.”
But the nights in Ireland are not in general so clear as the nights in drier countries ;
and Boate remarked that before rain, little or no dew was to be found, and he described
the climate as subject to rain, therefore the formation of dew must have been fre-
quently interfered with. His words are—‘‘ When it is towards any great rain, little
or no dew doth fall.” In another part he observes—‘‘ We have showcth how much
Ireland is subject to rain, and it is likewise to dark weather and overcasting of the air, .
even when it raineth not, which continueth sometimes many days together, especially
in winter time.”
In Kirwan’s work on the Temperature of Different Latitudes, under the head of
Stockholm, we are informed that Wargentin, in examining a series of 39 years, could
not find that any one year resembled another... When a person examines the details
of the years such as they are registered in Ireland, he sees such a variation, that he
_ must be convinced of the folly of attempting to draw precise conclusions, unless he
had a series of accurate observations made in the same place for a very long period.
Thus if the seasons go through a cycle of fifty-four years, he should have observations
for 108 years, to be able to compare two cycles.
Mr. Howard, in treating of the mean temperature of London, observes: ‘ The
mean temperature of the year is found to vary in different years to the extent of full
four and a half degrees, and this variation is periodical. The extent of the periods for
want of a sufficient number of years of accurate observations cannot at present be
fully determined, but they have the appearance of being completed in seventeen
years.”
He ventured to make predictions of some succeeding years, and he has failed.
VOL. XVII. 3B
220 M‘Sweeny, on the Climate of Ireland.
Let us hear a prediction from him—* The year 1816 which was the coldest of a
cycle, appears to have had its parallels in 1799 and 1782, and there is every reason
to conclude from present appearances, that the warm temperature of 1806 will re-
appear in 1823.”—Howard on the Climate of London, Vol. II. p. 289.
Let us now see the character of the year 1828 in the Philosophical Magazine, Vol.
LXIU, p. 77—‘* The mean annual temperature fully confirms what has been before
advanced, that wet summers are generally cold. The whole of the monthly means,
with the exception of May and December, are unusually low, indeed the actual de-
ficiency as to the annual amount exceeds 25 degrees.” Howard predicted also that
the year 1821 would prove an extremely dry one.— Climate of London, Vol. I. p.
294.
Dr. Burney describes the ground in 1821 to be in a very moist state.-—Phil. Mag.
Vol. LIX. p. 278.
The failure of such a man as Howard in predicting a year, is the best possible proof
of the variableness of the climate of the British isles.
Kirwan endeavoured to form rules of prognostication from the observations of
Rutty. In describing the year 1792, in the 5th Vol. of the Transactions of the Royal
Irish Academy, he admits that the autumn turned out wet, the least probable event.
The autumn of 1794 turning out wet, he admits to be contrary to the rules of prog-
nostication.— Trans. Roy. Irish Acad. Vol. VI. p. 171.
It would not be an easy matter to draw up rules, or to talk dogmatically on the mi-
nute details of the weather from the scanty materials we have in Ireland. In the
diary of the weather for the year 1802 in the Transactions of the Dublin Society for
1803, we are informed that ‘the thermometer is noted at 8 morning, 12 noon, still
the month of May. ‘Then it is noted at 8 morning, 12 noon, and 4 afternoon, the
remainder of the year. In some instances it is noted at 8 morning, 12 noon, 4 after-
noon, 8 at night, and at other hours.” In the diary of the weather for the year 1806,
we are informed in the same work that ‘‘ the thermometer is noted at noon, and four
o’clock, and occasionally at other times.” Can any thing be more vague than this ?
In noticing some difference in the mean temperature of some years in Dublin, or in
other parts of Ireland, we are not to conclude that a change of climate has taken
place ; the mean temperature of London, according to Howard, varies to the extent
of four and a half degrees, therefore the mean temperature in Ireland ought to vary
also. The same reasoning will hold good with regard to the quantity of rain in dif-
ferent years.
Doctor Patterson’s work on the climate of Ireland,* is evidently for the purpose ot
combating the opinion of Hamilton in the Transactions of the Royal Irish Academy,
that trees fail now in situations where they once flourished, owing to a change of
* Observations on the Climate of Ireland, by William Patterson, M.D. Dublin, 1804.
M‘Sweeny, on the Climate of Ireland. 221
climate. But as it has been well shown, allowance has not been made for the shelter
afforded by large trees.
In a long course of time, young trees may gradually extend up the sides of moun-
tains, protected from the wind by the older and higher trees, until at length they may
crown the very summits.
Patterson might have also easily refuted the opinion of Hamilton, by showing from
Irish history, that in former times mountains were not the places remarkable for the
growth of timber.
To establish this point, is a matter of some importance ; if it can be satisfactorily
proved, it will tend to overthrow the doctrine of Hamilton, with regard to a change
of climate.
Jocelin, in his life of St. Patrick, makes the distinction between woods and mountains,
“ For he abode in the mountains, and in the woods.”
Swift's Jocelin, Chap. xtt-
Cambrensis describes the mountains for the pasturing of cattle :
“ Frugibus arya pecori montes.”
Spencer informs us of the Irish holding meetings on the mountains. He says, speak-
ing of the Irish—“ There is one use amongst them, to keep their cattle and to live
themselves the most part of the year in boolies, pasturing upon the mountain and
waste wild places, and removing still to fresh land as they have depastured the former.”
In another place he observes thus—‘“ Yet it is very behoofeful in this country of
Ireland, where there are great mountains and waste deserts full of grass, that the same
should be eaten down and nourish many thousands of cattle for the good of the whole
realm.”
The distinction between woods and mountains, is of frequent occurrence in accounts
of Irish warfare.
In Ware’s Annals, we have an account given of a disaster, which befel the forces of
Lord Grey in the county of Wicklow, in the reign of Elizabeth—* Marched with a
good force to attack, and ordered his foot to enter into the woods, whilst he with the
horse remained on the mountains hard by.”—/Vare’s Annals.
Here the distinction between woods and mountains, is well marked’; the mountains
on which cavalry could mancuvre, agree with the account Spencer has‘ given of the
mountains fittedfor the pasturing of cattle.
Sir John Davis, in his book to explain the reason why Ireland was never entirely
subdued until the reign of James I. says that the English settlers erected their castles
and habitations in the plains and open countries, and forced the Irish into the woods
and mountains. Again he observes =
“The over large grants of land and liberties to the English, the plantations made
222 -M‘Sweeny, on the Climate of Ireland.
by the English in the plains and open countries, leaving the woods and mountains
to the Irish, were great defects in the civil policy, and hindered the perfection of the
conquest very much.”—Sir J. Davis, Quarto Edition, p. 36.
In 1586, when Sir Richard Bingham marched to put. down an insurrection of the
Burks, the distinction is made between the mountains and the woods, thus—‘‘ he im-
mediately marched to the Abbey of Balintubber, from whence he sent his foot and
kerns into the mountains and woods.”—/Vare’s Annals.
In the Pacata Hibernia, Desmond is described as being a desolate country —‘ the
whole country being nothing else but mountains, woods, and bogs.”— Pacat. Hiber.
p- 538. ;
Patterson has given plenty of instances of the growth of trees in exposed and high
situations in modern times. Templeton, in the 8th Vol. Transactions of the Royal
Trish Academy, says—“ The Laurustinus is one of those plants that were introduced
to Ireland before green houses were known, consequently planted in the open
ground, and experience shows that it is seldom hurt by frost.”
In the same volume he also states, that at Fair Head, the northern extremity of
Ireland, the mountain ash, birch, and oak, grow luxuriantly within fifteen or
twenty yards of high water mark.
In the 4th number of the Dublin Philosophical Journal, there is an account given
of a number of plants naturalized under the climate of Ireland; by James Townsend
Mackay. It would take up too much space to enumerate them, but the paper shows
the great mildness of our climate, and proves that it is not becoming more ungenial,
as a person might be led to think by reading Hamilton’s Memoir on the Climate of
Ireland. It is not to be supposed that the great mildness of our winters is of any
recent origin, although it has been promoted by draining and cultivation.
Patterson, in describing the celebrated Arbutus at Mount Kennedy, states that in
1773, its age then exceeded one hundred years.
Some suppose that the arbutus which grows in such abundance at Killarney, was in-
troduced by the Spaniards in the reign of Elizabeth. It was probably introduced by
the monks'at a much earlier period... Smith, in describing Innisfallen in the Lake of
Killarney, says—‘‘ There are besides timber trees, the remains of several fruit, trees,
as plums, pears, &c. which have outlived the desolation that hath seized on the cells of
those recluses who first planted them,”
There can. be little, doubt but that apple trees, were cultivated in Ireland, before
the time of Henry I]. An apple tree is mentioned in the life of, St. Columba. , The
story of St. Kevin andthe apples may be cited; but. one of the most authentic docu-
ments we have relating to Ireland, St.. Bernard’s Life of; Malachy, proves that there
were apples in Ireland—
“ Accelera inquit fer illi tria poma.”
Vita Malac. cap. 23.
M‘Sweeny, on the Climate of Ireland. 292
The account which Mela has given of the vegetation in the island, although exag-
gerated, yet it has a tendency to prove the moist and mild nature of the climate in his
time. He states that the climate is unfit to bring grain to maturity, and that cattle, if
not restrained from feeding, would be in danger of bursting from the luxuriant her-
bage.— Mela, Lib. 3. c. 6.
Some agriculturists maintain now, that corn is liable to degenerate in this moist cli-
mate, and they advise the importation of seed corn from a more congenial country.
Cattle have been often injured by feeding on clover. Spencer says, speaking of corn
in Ireland—*‘as for corn, it isnothing natural, save only for barley and oats, and some
places for rye.” Arthur Young, m modern times, gives a decided preference to Eng-
lish grain in comparison to Irish.
Petrus Lombardus imagined that the vine could be cultivated with success in Ireland ;
this ought to be looked on as a speculation, probably encouraged by a succession of
favourable seasons about the period in which he wrote. He was an ecclesiastic who
had spent a good deal of his time on the continent. Let us hear Camden on the
subject of Ireland. ‘It has also vines, but more for shade than fruit, for when the
sun quits Leo, cool breezes ensue in this our climate, and the afternoon heats in
autumn, are too weak and short both here, and in Britain, to bring grapes to per-
fection.” — Gough’s Camden.
In an Irish almanack of the fourteenth century, the time of gathering grapes and
of drinking new made wine, is pointed out—Anthol. Hiber. Vol. I. p. 130.
This ought to be looked on in the light of a modern gardener’s book. Although
directions may be given in such a book how to cultivate the fig tree, we would not be
led to suppose that the climate was fitted for it ; yet in 1826, in the south of Ireland,
figs in some favourable situations, came to perfection in the open air.
There is an ancient canon which imposes penalties on the owners of hens that
damage vines.—Dacherii Spicil, tom. ix. p. 46.
Ecclesiastical communities might have raised vines for shade and ornament as at
present, or for making verjuice.
It has been supposed that yew trees did not abound in Ireland in the middle ages,
from an act being passed to oblige merchants to import bows. This may be accounted
for; the Irish probably destroyed the yew tree wherever they met it, for two reasons,
first, because it was poisonous to their cattle, secondly, because it afforded their ene-
mies a destructive weapon.
Although Patterson has, in his treatise on the Climate of Ireland, refuted Hamilton,
yet he has propagated an error relating to the quantity of rain in Ireland. He sup-
posed that more rain fell in England, than in Ireland, and he has led others into the
same mistake. We find in the Encyclopedia Metropolitana article, Ireland, the fol-
lowing :
“It is probable that the quantity of rain which falls annually in Ireland, is less than
224 M‘Sweeny, on the Climate of Ireland.
that which falls in England ; but it is evidently impossible to arrive at certain results
on this question, from the partial observations hitherto made on local climates.” The
writer of this essay has no hesitation im saying, that Patterson and his followers are
wrong. The great, quantity of rain that falls at Kendal, from its peculiar locality, de-
ceived him as to the average quantity of rain in England and Scotland.
Dr. Campbell of Lancaster, observes, that the influence of hills in attracting clouds
is no where more conspicuous, than at Kendal; that one third more rain falls at Ken-
dal, than at Lancaster, a distance of only twenty miles, and that it is by no means un-
usual, to, see from, the church-yard, of Lancaster, the hills about Kendal envolved in
thick clouds, while the sky at the Lancaster side of Farlton Knott, appears perfectly
clear.—Memoirs of the Lit. and Phil. Soc. of Manchester, Vol. IV. part 2, p. 635.
Dr. Campbell informs us in the same work, that the clouds from the South and
South-west, are attracted by the hills which divide Yorkshire from Westmorland, and
that while the western side of these hills is deluged with rain, frequently on the York-
shire side, the weather is dry. Doctor Garnett says—‘‘ The summer of 1792 was
remarkably dry in Yorkshire, and all the eastern side of the English Appenine was
burnt up for want of rain, while on the western they had plenty of rain and abundant
crops of grass.’—Tbid. p. 634. :
Doctor Patterson should have had observations made on the western side of the
high grounds in the centre of Ireland, or at a remarkably rainy spot, such as Killar-
ney, to institute a comparison with the very moist part of England.
No person, a priori, would suppose that more rain could fall in England, than in
Ireland. In the. first place, Ireland is, nearer to the. Atlantic, and in the second place,
it has. more mountains, than, England to attract clouds.
Long since, Boate. remarked, that no country in the world was fuller of brooks than
Ireland. The. number of rivers, is the best proof of greater humidity. The high
zrounds in the centre of the island, arrest the clouds loaded with moisture, which did’
not deposit, their burthen on the. western coast.; hence the magnificent Siannon,
swelled by tributary streams, rolls its vast volume of water to the ocean.
Any person may point to the: Shannon, and: laugh, at meteorological registries ;
here is the hygrometer of nature. which does not err; in pointing out the greater hu-
midity of Ireland. We haye in the Derry Survey as follows :
« Taking the annual quantity of rain. that falls in. the east of England, which rarely
is less than 18 inches, and the max, of the west of that country, the average will
exceed 51 inches; and we cannot suppose that Scotland would produce a lower
result.”
A comparison between the quantities of rain at Derry and Edinburgh, will show.
that Patterson was wrong.
M‘Sweeny, on the Climate of Ireland. 225
From the Derry Survey, - From Brewster’s Encyclop. article Scotland.
year inches inches
1795 - 32-861 - 35-7
1796 - 25-718394 - 19-4
bee 30-821272 : 25-9
1798 - 33-2310176 - 23-9
WE ers 34-7709468 - 25-9
In one year the quantities were nearly alike in both places ; in the other years, the
rain at Derry far exceeded that of Edinburgh. The quantity of rain that fell at
Glasgow, on an average of thirty years, is marked at 29 inches in the same work.—
Brewster’s Eneyclopedia.
The average quantity of 51 inches, which Patterson attributed to England, is en-
tirely too much.* Howard, in his work on the climate of London, states that the
greatest quantity of rain during twenty-three years, fell in 1816 at London, and he
gives the amotint as being 32 inches. He gives the general average for a period of
twenty years at 25-179 inches, and the means taken on the ground.— Howard, Vol.
IT. p. 185.
Wakefield quotes Doctor Young, to show, that the average quantity of rain for
England and Wales is 31 inches.
The average quantity during eighteen years at Liverpool, which is not a very
great distance from Kendal, was 34 inches.—Memoirs of the Lit. and Phil. Society
of Manchester, Vol. LV. part 2, p. 575.
Arthur Young observes—“ I have known gentlemen in Ireland deny their climate
being moister than England ; but if they have eyes let them open them, and see the
verdure that clothes theiy rocks, and compare it with ours in England, where the
rocky soils are of a russet brown, however sweet the feed for the sheep.”
In another place he remarks—* If as much rain fell upon the clays of England, as
falls upon the rocks of her sister island, those lands could not be cultivated.’”— Tour
in Ireland, Vol. 11. p. '74.
The prevalence of winds which waft vapours to the island, is from an early date.
The prevailing winds in the time of Camden, were the same in the time of Cam-
brensis. Camden observes, that Giraldus said, not without reason=nature beheld the
realm of zephyr, with an uncommonly favourable eye.
Solinus described the Irish sea as being stormy‘‘ Mare quod Britanniam et Hi-
berniam interluit, undosum et inquietum toto in anno, non nisi aestivis pauculis diebus
est navigabile, navigant autem vimineis alveis quos circumdant ambitione tergorum
bubulinorum— Solinus, c. 35.
* Williams quotes Hales, who estimated the annual quantity of rain in England at 22 inches:i—
Williams's Climate of Great Britain, p. 79.
226 M‘Sweeny, on the Climate of Ireland.
It may be said that it would be dangerous to cross the sea, in such craft at present.
He qualified the description by the words “ navigant autem.”
Boate gives a passage from Giraldus on the Trish sea, but he does not give his mean-
ing correctly in the translation—‘ Hibernicum mare, concurrentibus fluctibus undo-
sissimum fere semper est, inquietum ita, ut vix etiam aestivo tempore, paucis diebus se
navigantibus tranquillum prcebeat.”
Boate translates this passage thus—‘‘ The Irishsea being very boisterous through °
the concourse of the waves, is almost always restless, so as even in the summer time,
it is hardly for a few days quiet enough to be sailed on.”
Surely this is not the meaning of Cambrensis. His meaning in this passage is, that
scarcely in the summer time, is it calm for a few days.
A perfect calm is not a frequent occurrence at the present day in the Irish sea.
Boate, speaking of the want of east wind to bring ships from England, to Ireland,
obseryes—‘ But in the summer time, and chiefly in the spring, and in the months of
March, April, and May, one is not so much subject to that incommodity, as in the
other times of the year.” .
Boate’s observations agree with these of Rutty.
It would be difficult from the natural history of freland, at least during the histo-
rical period, to prove a change of climate; wolves have been exterminated, the em-
ployment of guns has tended to banish birds from countries more than any change of
climate.
In the list of birds by Cambrensis, are to be found cygni. Smith informs us
that wild swans were common in the north of Ireland, but were only observed in the
south in the great frost of 1739.
Wild swans were shot in the south of Ireland, in the winter of 1829.
It is allowed that magpies were driven here by a storm, at a period subsequent to
Cambrensis, but many birds might have escaped his observation. The increase of
population, and the use of fire-arms, no doubt, banished storks. They were in Ireland
in the reign of Henry II. ‘* We have seen,” says the Irish king Dermot, in a letter
preservel by Cambrensis, “the storks and the swallows. ‘The birds of the spring
have paid us their annual visit, and at the warning of the blast, have departed to
other climes. But our best friend has hitherto disappointed our hopes. Neither the
breezes of the summer, nor the storms of winter, have conducted him to these
shores.” —Lingard’s History of England.
Ireland, in ancient and modern times, is similar in being free from serpents
Spencer described the Irish as being tormented by gnats, in the woods. Arthur
Young says that the number of flies which devour one in a wood, prove the great hu-
midity of Ireland.”— Young’s Tour in Ireland, Vol. II. p. 77.
in every period of the history of the climate of Ireland, we find evidence of its
mildness,
M‘Sweeny, on the Climate of Ireland. 227
Sir John Davis in his time, stated as follows :—‘* During the time of my service in
Ireland, I have visited all the provinces of that kingdom, in sundry journies and cir-
- cuits, wherein I have observed the good temperature of the air, and the fruitfulness of
the soil.”
We find its character at different periods to agree with its character of the present
day, if we except the title of ‘glacialis Ierne,’ given in poetic description ; to this it
has been shown, it never could have just claims.
Cambrensis says—“ Pascuis tamen quam frugibus, gramine quam grano foecundior
est insula.”
Wakefield observes, that in the south of Ireland, the value of the mountains of
Tipperary, Cork, and Kerry, was frequently mentioned to him, as the climate allowed
them to be grazed throughout the whole year ; a statement which agrees with the fol-
lowing—* sicut aestivo, sic hiemali tempore, herbosa virescunt pascua, unde nec ad
pabula feena secari nec armentis unquam stabula parari solent, aeris amoenitate tempe-
rieque, tempora fere cuncta tepescunt.”
The woods have been cut down since the time of Cambrensis, with the exception
of the woods ; this description of the country may be received as applicable at the
present day.
* Hibernia quidem terra inaequalis est, mollis, et aquosa, sylvestris, et paludosa.”
The names of the letters in the Irish alphabet, show that a vegetation familiar to
the present generation, was known to the inhabitants of the island, at a distant period.
» Ledwich, on the authority of Lombard, says :—‘ About 1632, artichokes, colly-
flowers, pompions, and hops, seem to have been first introduced and grew very well.”
These vegetables growing in Ireland at that time, do not prove any change of cli-
mate, if they had been introduced at an earlier period, there is little doubt but that
they would have succeeded as well. The same reasoning holds good for the tobacco,
cultivated at present to such an extent in the county of Wexford.
In the Preface to the Translation of Dandolo, on the silk-worm, the writer states,
that ‘‘ during the last century, some French refugees in the south of Ireland, made
considerable plantations of the mulberry, and had begun the cultivation of silk with
every appearance of success; but since their removal, the trees have been cut down.”
An attempt has been made of late years in the south of Ireland, to produce silk on
the estate of the Earl of Kingston.
The following queries were transmitted to a person in the neighbourhood.
Have the mulberry plants thriven or died ?
Have the silk worms died ; the cause ?
Are any attempts continued to rear plants or worms ?
| To which the following was received :—‘ The plants did not thrive, the silk worms
. died, the climate did not appear congenial. No attempts are now made, the ground
has been let to tenants. Lord Kingston went to much expense, in this attempt to es-
| VOL. XVII. 3c
228 M‘Sweeny, on the Climate of Ireland.
tablish a silk factory ; the first season it appeared to have gone on well, and it was
imagined that it would have been successful ; however, the following season came wet,
and the worms perished.” ;
A great deal depends on favourable seasons. In 1768, a bounty of twenty guineas
was given to a Mrs. Gregg, for having raised a considerable quantity of silk in the
county Clare.— Transactions of Dublin Society for 1799.
In the Pacata Hibernia, it is stated, that they were prevented from reaping in
Kerry in the month of July, the crops being backward on account of an unfavourable
season. Barley was much cultivated in Ireland at that time, the country was but
thinly inhabited, and of course the best lands were selected for cultivation, and the
computation of time was according to the old style. Arthur Young, in his tour in
Ireland, describing the rotation of crops in the Mahagree islands, near Tralee, in the
County Kerry, observes—“ All grain is remarkably early, they have sown English
barley, and made bread of the crop in six weeks. I was assured, that in these islands,
they have known two crops of barley gained from the same land, in one year, and the
second better than the first. ‘They sowed the first in April, and reaped the middle of
May, and immediately sowed a second, which they reaped the end of August.”— Vol.
I. p. 472.
It would be a desirable thing, if we had an exact account of the weather in the
south of Ireland for a long time. There is no registry of the weather made at the
Royal Cork Institution, on Sundays. When the writer of this essay visited that es-
tablishment, to compare the statements in Smith’s History of the County Cork, with
a considerable series of years in modern times ; the officers of that establishment could
not tell what was become of the registry, previous to the year 1825.
In Smith’s time the rain was as follows in Cork :—In 1738, 54 inches 5 tenths—
the same nearly in 1739—in 1740, but 21 inches 5 tenths—in 1741, 33 inches 6
tenths—in 1742, 38 inches 1 tenth—in 1743, 39 inches 3 tenths—in 1744 33 inches
6 tenths—in 1745, 48 inches 4 tenths—in 1746, 30 inches ; the same nearly in 1747—
and in 1748, 37 inches 4 tenths.—Smith’s Cork, Vol. II. p. 404.
The quantity of rain at the Royal Cork Institution, was in round numbers as
follows :
years inches
1825 - - 32
1826 - . 28
hy ae = ar
1828 - . 40
1829 - - 39
Hamilton gives the mean temperature of different parts of the City of Cork in
1788, at 52—5 to 53—5.— Transactions Royal Irish Academy, Vol..II..
The mean temperature at the Royal Cork Institution on the average of five years
le
M‘Sweeny, on the Climate of Ireland. 229
is about 55. This does not prove a change of climate ; in the first place the average
is swelled by the warm year of 1826, and the thermometer is kept in the centre of the
city, in a situation surrounded by high buildings, where it must be affected by ra-
diation. Howard has shown, that the mean temperature of London yaries to the
extent of 44 degrees in different years; therefore Hamilton was not warranted, in
deducing the mean temperature of Ireland from the observations of a few years.
Smith’s account of the winds in his day, agrees well enough with the average of
five years in modern times in Cork. No precise conclusions can be drawn from com-
parisons between broken fragments of cycles of the weather.
Kirwan remarks—‘‘ Among all the years observed by Dr. Rutty from 1725 to
1765, there occurs but one similar to 1792, the year 1755, in that the three seasons,
spring, summer, and autumn, were wet.”— Transactions ef Royal Irish Academy,
Vol. V. p. 240.
It is a generally received opinion, that within the memory of our old peasants, the
winters have become milder, and the summers less warm ; in this essay it is contended
for, that the general character of the climate has from a very early period been the
same; yet it is certain that there is some good reason for this popular opinion, on
account of a modification of the climate, from the general extension of the potatoe
culture.
That the old people should imagine that the summers were warmer formerly, ought
not to surprise us, as the buoyancy of youth, and warm blood in young days, cause a
warm glow from moderate exercise; but, on account of the greater liability of old
people, to be affected by cold, they ought to feel more severely now, winters of the
same temperature. If no change has taken place, we should expect to hear from them
complaints of the cold; but on the contrary, the old peasants maintain firmly, that
the winters formerly were colder.
Popular and general opinion is not to be slightly passed over. The peasantry of
France, obtained a signal triumph over the philosophers of their country, with regard
to the fact of the fall of, meteoric stones.
The potatoe culture has extended in almost every direction, even up to the tops of
mountains in some places. The paring and burning of rough ground, makes it
smooth, and allows the sun to exert its full influence along the surface in winter. On
rough mountain ground, where there are inequalities, and tufts of heath, and furrows
worn by the rain, snow in such places is liable to remain a long time undissolved.
Boate remarked in his time, that there was a greater plenty of snow on the moun-
tains, than in other parts.
In summer the shade of the stalks of potatoes protect the ground from the sun, and
the trenches, which might serve in winter as fit receptacles for keeping snow undis-
solyed a considerable time, are obliterated by the digging out of the potatoe crop, on
the approach of severe weather. If the trenches remained during the winter, it is
=,
230. M‘Sweeny, on the Climate of Ireland.
evident that after a fall of snow, some of it would lie a considerable time undis-
solyed in these trenches.
The appearance of frost is a signal for the Irish peasant to dig out his potatoes,
and consequently to obliterate the trenches. The level dark-coloured ground, which
remains after the potatoe crop, is well adapted for melting snow. The sun exerts its
influence to great advantage on a dark surface, according to the experiments of
Franklin ; and at night, fallow ground of this description, is, according to the expe-
riments of Dr. Wells, particularly unfavourable for the production of hoar frost.
Thus modern science is in favour of popular opinion.
The potatoe culture shades the ground in summer also, in the following manner :
A number of the hills of Ireland lie east and west ; the potatoe trenches, which are
so many drains, run at right angles to the tops of these hills, for the purpose of con-
veying off superfluous water ; the rays of the sun from east to west do not therefore
traverse directly these trenches, and thus the beds, independently of the stalks, cause
ashade. In this way a great portion of the surface of Ireland is shaded in summer,
but particularly by the luxuriant stalks of potatoes, that meet the eye of the traveller
in every direction.
The potatoe culture has also wonderfully increased the number of enclosures, and
hedge rows, and has consequently added to the shading of the ground in summer, in
every direction, even up to the tops of mountains. Hedge rows also afford shelter to
cattle in winter. Smith, in his History of Kerry, stated that cattle in his time some-
times perished on the mountains in severe winters.
Before the general cultivation of the potatoe for the food of the people, (in the re-
collection of many, oaten bread constituted a considerable portion of their diet,) large
tracts of pasture ground denuded of timber, and not intersected by hedges, must have
been liable to be parched in summer. ‘The bed and trench plan of: culture, the fa-
vourite system with our peasantry, is admirably adapted for draining this moist island,
and for mixing clay with a peaty surface.
In winter, bogs and shallow pools, were easily frozen at night, and served as reser-
voirs of cold on the following day. According to the experiments of Doctor Wells,
grass is particularly liable to be covered with dew and hoar frost. ‘The extensive
system of pasture formerly followed in Ireland, must have often presented a large sur-
face of hoar frost to the action of the morning’s sun.
A considerable portion of the heat of the morning’s sun must have been therefore
expended, in thawing the ice on shallow pools, and in bogs, and in melting the hoar
frost, formed on the grass during the night.
If, notwithstanding the draining of swamps, the reclaiming of bogs, and the ame-
lioration of the soil by manures, and by more judicious cultivation, it’should be con-
tended, that on an average of years, the winters at present are exactly as cold as they
were previous to the general cultivation of the potatoe; it would imply, that the
F
:
M‘Sweeny, on the Climate of Ireland. 231
power of the sun was then greater than it is at present, as it had then more obstacles
to overcome in warming the surface of the island.
The vast quantity of manured fallow ground, of a colour dark in proportion as it
is not exhausted by severe cropping, now materially aids the sun, to warm the surface
of Ireland in the winter. It is well known, that the exhausting of ground by repeated
corn crops, causes its colour to become lighter. This injudicious system, was much
more practised formerly, than at present. Landlords everywhere endeavour to pre-
vent it. The old country people are positive in asserting, that the snow lay longer on™
the ground when they were young, than it now is observed to remain.
If, notwithstanding luxuriant crops from an improved soil, and the shading of the
surface by the general cultivation of the potatoe, and by the number of hedge rows,
it should be contended, that on an average of years, the summers now are ewactly as
warm as they were formerly, it would imply, that the power of the sun’s rays is greater
for warming the island now, as its surface is better shaded than it was in the period
subsequent to the destruction of the woods.
The greater power in the rays of the sun, on an average of years cannot be admitted,
as there is no evidence to prove it ; and the occasional return of very hot summers, and
of very severe winters, is attributed to causes at present not perfectly understood.
If this reasoning be allowed, it must be admitted, that the modification of climate
must have kept pace with agricultural improvement, on an average of years, and it
explains, and corroborates popular opinion on the subject.
The vast increase of the potatoe culture, and the general use of this vegetable as
the entire dependence of the peasantry, have been within the memory of the old per-
sons of the present generation; therefore it is just to believe, that a modification of
climate from local causes, has taken place within their recollection.
The general character of the climate has been the same from a very early period ;
hot summers, and cold winters arise from géneral, not from local causes; but when
they do happen, the temperature must be influenced by the state of the surface in
some degree.
The testimony of the peasantry, that the snow does not remain now so long on the
ground as formerly, must be received. The experience of old sportsmen, who had
heen in the habit of traversing tracts of country now reclaimed, corroborates the eyi-
dence.
In Rutty’s time, the cultivation of the potatoe was making progress on the rough
grounds, in the county Dublin; he admits that his account of frost and snow in the
city of Dublin, was too little when compared with the accounts of country parts.
On account of the scanty state of data on the weather in Ireland, it is impossible to
put popular opinion to a severe test, by scientific records.
In fact if it be true, as was supposed by Lord Bacon in his time, and as is imagined
by Howard and others, that the seasons go through a cyele, it is evident that we
232 M‘Sweeny, on the Climate of Ireland.
should have the weather accurately observed during two complete cycles, one at a late,
another at a distant-period, so as to be able to compare them, before we would be war-
ranted in attempting to draw precise conclusions.
The potatoe culture is well fitted for draining the moist surface of the island, the
trenches run from the summits of the hills to carry off superfluous water.
Not only have bogs been reclaimed, but in some districts they have been absolutely
removed, and the peat which they afforded consumed as fuel. In some places, a bog is
the most valuable part of an estate, where fuel is dear. Great masses, therefore, of
this vegetable sponge, retentive of moisture, and liable to be quickly frozen, have been
removed or reclaimed, and mixed with clay in the recollection of our peasantry. These
places must have been fertile sources of vapour; and in time of frost, when once
frozen, they must haye been magazines of cold for a considerable time.
We have the testimony of Sir James Ware, of Boate, and of Cox, that good ef-
fects on the climate from cultivation, were experienced in Ireland when they lived ;
then why should we reject the testimony of the old peasants who are yet alive, parti-
cularly when it is consistent with science ?
Smith, in his History of the County of Kerry, predicted that the culture of potatoes
would render the country more wholesome, and stated that enclosures sheltered the
land, and improved it, and kept it warm in winter.—Smith’s Kerry, p. 159.
The process of adding calcareous, vegetable, and animal manures to the soil is con-
stantly going on in Ireland, year after year. Sir H. Davy ascertained by experiment,
that a dark-coloured soil, containing animal or vegetable matter, if heated within the
common limits of solar heat, will cool more slowly, than a wet pale soil, composed
entirely of earthy matter. Therefore it is what ought to be expected, that snow
should not remain on the ground now, so long as formerly.— Agricul. Chemis. p. 156.
The few observations made with instruments in Ireland, have been made in towns,
not in the country parts. Towns are not the fit places for observations ; the heat from
fires, the number of inhabitants and of domestic animals crowded together, the friction.
of vehicles and of machinery, the dark colour of the streets from animal and vegeta-
ble manures, the process of fermentation going on, all these matters tend to raise
the temperature of towns.
Registries of the weather, kept for a very long period, and in country parts, only
could disprove popular opinion on the subject of snow remaining on the ground.
The observations should be made with great care and for an extended series of
years.
We have the authority of Howard for thinking, that implicit reliance is not to be
placed on the registry, even of the Royal Society of London.—Howard on the cli-
mate of London, Vol. IT. p. 190.
The name of mountain ground is frequently given in Ireland, to rough grounds
producing heath, and such kind of vegetation of little yalue. In such places, not
M‘Sweeny, on the Climate of Ireland. 233
only in the time of Boate, but also in the recollection of our own country people,
snow, when it fell, was apt to remain a considerable time. ‘The inequalities on such
ground, protect the snow from the rays of the sun.
By paring and burning, lands of this description are every year brought into culti-
vation.
The history of the weather in Ireland shows its general mild and moist nature,
what might be expected from the island’s locality, in regard to the Atlantic. The ya-
pours of this ocean produce frequent rain ; this produces rivers and verdure.
Well may Ireland be called—
Land of brooks, and murmuring streams
Of rivers wide, and verdant plains.
Sir James Ware, in his Antiquities of Ireland, quotes the following character of
the island from Alexander Nechamus.
“ Fluminibus magnis letatur Hibernia.”
Spencer, in his Fairy Queen, has described our rivers.
“There was the Liffie rolling down the lea,
The sandy Slane, the stony Au-brian
The spacious Shenan spreading like a sea,
The pleasant Boyne, the fishy fruitful Ban,”
The general character of humidity and mildness of our climate, cannot be disproved
by details of portions of cycles. ‘The same causes always produced the same effects.
Ireland in every age excelled other countries in mildness of climate and in yerdure.
On Differences and Differentials of Functions of Zero. By WILLIAM R.
HAMILTON, Royal Astronomer of Ireland.
Read June 13, 1831.
Tue first important researches onthe differences of powers of zero, appear to be
those which Dr. Brinxtey published in the Philosophical Transactions for the year
1807. The subject was resumed by Mr. Herscuer in the Philosophical Transactions
for 1816; and in a collection of Examples on the Calculus of Finite Differences,
published a few years afterwards at Cambridge. In the latter work, a remarkable
theorem is given, for the development of any function of a neperian exponential, by
means of differences of powers of zero. In meditating upon this theorem of Mr.
Herscuet, I have been-led to one more general, which is now submitted to the
Academy. It contains three arbitrary functions, by making one of which a power
and another a neperian exponential, the theorem of Mr. Herscuet may be obtained.
Mr. Herscuer’s Theorem is the following :
S(EJ=fAttf tA) o + 5 FU+A)e +&e. (A)
JF (1 +4) denoting any function which admits of being developed according to posi-
tive integer powers of A, and every product of the form A™ o” being interpreted,
asin Dr. Brinkiey’s notation, as a difference of a power of zero.
The theorem which I offer as a more general one may be thus written :
¢A+4) f¥O)=fA+4) oA+4) (0) "5 (B)
or thus
F(D) ft (V=fL+4) FD) (0) )”. (C)
In these equations, f, ¢, F, y, are arbitrary functions, such however that f(1 +A’),
¢(1 +A), (D), can be developed according to positive integer powers of A’ A D ;
and after this development, A’ A are considered as marks of differencing, referred to
the variables o’ 0, which vanish after the operations, and D as a mark of derivation
by differentials, referred to the variable 0 . And if in the form ( C) we particularise
VOL, XVII. 3D
236 On Differences and Differentials of Functions of Zero.
the functions #’, ¥, by making F’ a power, and ya neperian exponential, we deduce
the following corollory :
TEGCO=af AS A )Be =fK$ CEPA) 0" ;
A : e
that is, the coefficient of
1.2... 2
in the development of f(e') may be represented by
Ff (1 +A) o* ; which is the theorem (4) of Mr. Herscuet.
June 13, 1831.
ADDITION.
The two forms (B) ( C) may be included in the following :
VF¥O) =fA+AV (eC) Y. (D)
To explain and prove this equation, I observe that in MacLaurin’s series,
a PFO). 2 Df(0)
fn (Gt) = 5(O) et oat eps ae
we may put « =(1+A) 2° and therefore may put the series itself under the form
S(e)=f(0) +72. 1+ ayer +P.
or more concisely thus
(1+A)’ x +&c.
Fi (a) = fA): (E)
which latter expression is true even when Mactaurin’s series fails, and which
gives, by considering » as a function ¥ of a new variable o’ and performing any
operation vy’ with reference to the latter variable,
Vt (CO) =VFA+4) HO) Y- (F)
If now the operation y' consist in any combination of differencings and differ-
entiatings, as in the equations (B)and (C), and generally if we may transpose the
symbols of operation vy’ and f(1 +A), which happens for an infinite variety of forms
of VY’, we obtain the theorem (D). It is evident that this theorem may be extended
to functions of several variables.
June 20, 1831.
eos
:
On a difficulty in the Theory of the Attraction of Spheroids.
By James M‘Cutxacn, A.B.
Read May 28, 1882.
AN approximate theorem, discovered by Laprace, and relating to the attraction of
a solid slightly differing from a sphere, on a point placed at its surface, has given rise
to many disputes among mathematicians.* I hope the question will be set in a clear
light by the following remarks.
Left us consider the function which expresses the sum of every element of a solid
divided by its distance from a fixed point, and let us denote it, as Laptace has done,
by the letter V. It is necessary to find the value of V for a pyramid of indefinitely
small angle, the fixed point being at its vertex. Calling ¢ the small solid angle of the
pyramid (or the area which it intercepts on the surface of a sphere whose radius is
unity and centre at the vertex), it is manifest that the element of the pyramid at the
distance 7 from the vertex is gr*dr ; dividing therefore by 7, and integrating, we have
4@r°, or » multiplied into half the square of the length, for the value of V.
Again, supposing the force to vary inversely as the square of the distance—the
only hypothesis that can be of use in the present inquiry—the attraction of the same
pyramid on a point at its vertex, and in the direction of its length, is manifestly equal
to 9r. :
_ Let us now consider a solid of any shape, regular or irregular, terminated at one
end by a plane to which the straight line PQ (Fig. 1,) is perpendicular at
ie the point P ; and let there be a sphere of any magnitude, whose diameter P’ Q’ is
parallel to PQ. Let P” be a fixed point, and from the points P, P’, P”, draw three
parallel straight lines Pp, P’p', Pp", the first two terminated by the surfaces of the
solid and of the sphere, the third Pp’ in the same direction with them and equal to
their difference, without regarding which of them is the greater, and suppose all the
.
_* See Pontécoulant, Théorie analytique du systéme du monde, Tome II. p. 880; with the references
there given.
238 On a difficulty in the Theory of
points p’, taken according to the same law, to trace the surface of a third solid. Let
Pp, Pp’, P'p’, be edges of three small pyramids with their other edges proceeding
from P, P’, P", parallel, and having of course the same solid angle which we shall
call ¢, denoting by 7, 7, 7”, their respective lengths, and by V, V’, V’", the values of
the function V for each of them. Drawing pF perpendicular to PQ, the attraction
of the pyramid Pp in the direction of PQ will be equal to ¢x PR; call this attrac-
tion A, and let a be the radius of the sphere.
Since 7" is the difference of r and 7’, we have 72+ r?—7r"?=2 rr’ =2 PRx P'Q', = 4
and multiplying by 4 @ we find $77 +4 gr’*—$ ¢2”*=2a¢ x PR, that is V+ V'—V"
=2aA. The same thing is true for any other three pyramids similarly related to
each other, throughout the whole extent of the three solids which are exhausted by
them atthe same time; and hence, if we now denote by /, V’, V", the whole values of
the function V for the three solids, and by 4 the whole attraction of the first of them
parallel to PQ on a point at P, we shall still have V+ V'’—V" =2aA.
To express this general theorem in the notation of LapLacr, we have merely to
observe that the attraction 4 is synonymous with — (7) , and that the quantity V"
for the sphere is equal to 47a’. Substituting these values, we find
V+ 2a(3) = — wat"; Lo 2 ft
an exact equation, differing from the approximate one of Lapxace only in containing
the quantity W", and totally independent of the nature of the surface or of the mag-
nitude of the sphere ; the only things supposed being that all the lines drawn from P
meet the surface again but once, and that no part of it passes beyond a plane through
P at right angles to PQ.
With respect to the limit of the quantity VY", it is obvious that if a hemisphere be
described from P” as a centre with a radius equal to the greatest difference 6 between
the lines Pp, P’p’, the solid Pp” will lie wholly within this hemisphere, and ‘con-
sequently V" will be less than the value of V for the hemisphere, that is, less than {
76°; for here all the little pyramids from the centre have the same length 8, and their
bases are spread over the hemispherical surface ; wherefore V’=27 x }o:=7e. All
this is independent of any thing but the suppositions just mentioned.
If now PQ be supposed to be a spheroid of any sort, slightly differing from the
sphere P’Q’, and such that the line PQ, perpendicular to the surface at P, passes
nearly through the centre, than all the differences, of which é is the greatest, being of
the first order, the quantity V", which is less than 7é, will be of the second order ;
and therefore neglecting, as LarLace has done, the quantities of that order, we get the
theorem in question.
the Attraction of Spheroids. 239
__ It may be well to apply the general theorem to the simple case in which the first
solid is a sphere of the radius a’, because both Lacrance and Ivory have used this
case to show that the reasonings of Lapxace are incorrect. In this instance, then, the
surface described by the point p” is that of a sphere whose radius is the difference
between a and a’; and the values of V, 1”, V'", and A, are jra*, 4ra®, 4n(a’—a):
and 37a’, respectively.
Substituting these values in the equation 7+ V’—PV’"=2aA, and omitting the
common factor jr, thg resulting equation
az +a°—(a@ —a)*=2aa’
ought to be identical ;—and so it manifestly is.
November, 1831.
; Geometrical Propositions applied to the WVave Theory of Light,
. By James M‘Cuxxacu, F.T.C.D.
¢ : *
Read June 24, 1833.
Part I.—GeromerricaL Propositions.
1. Turorem I. Conceive a curved surface B to be generated from a given curved sur-
~~ face Jin the following manner : having assumed a fixed origin O, apply a tangent plane
at any point Q of the given surface, and perpendicular to this plane draw a right line
OPR cutting the plane in P, and terminated in RF, so that OP and OR may be re-
ciprocally proportional to’each other, their rectangle being equal tp a constant quan-
tity k°, and let all the points & taken according to this law generate the second sur-
face B. Then the relation between these two surfaces, and between the points Q and
_ R, will be reciprocral ; that is to say, if,a tangent plane be applied at the point F of
the second surface, a perpendicular ON to this plane will pass through the point Q of
the first surface, and ON and OQ will be reciprocally proportional to each other, the
rectangle under them being also equal to k*.
2. To prove this theorem, take a point g, in the tangent plane of the surface 4,
_ and near the point of contact Q. (Hig. 2.) Throughg let several other planes be
drawn touching the surface A in points Q’, Q", Q’, &c.. and draw the perpendi-
culars OP'R’, OP"R", OP" R”, &e. according to the same law asOPR. The
points R, R’, R", RK”, &c., will thus be upon the second surface B, and they will
moreover be all in the same plane ; for from any one of them F' let Fn be drawn
perpendicular to the right line Og and meeting Og in 7; then on account of the si-
ilar right-angled triangles OP'q and On’, the rectangle Oq will be equal to the
rectangle 2’ OP’, or to the constant quantity k*, so that the point x, or the foot
_ of the perpendicular let fall upon Og, will be the same for all the points R,R'.R’,R”,
_ &c., and consequently all these points will lie in a plane cutting the right line Ogn
perpendicularly in , so as to make the rectangle n Oq equal to k*. Now while the
point Q remains fixed, let the point g approach to it without limit in the tangent
242 Geometrical Propositions
plane at Q ; and the points R’, R”, R”’, &c. will in like manner approach without
limit to the fixed point R; the plane which contains all those neighbouring points
having for its limiting position the tangent plane at #. Also the point m will ulti-
mately coincide with V. It follows therefore that the tangent plane at # cuts the
right line OQ perpendicularly in N, so as to make the rectangle VOQ equal to k’.
3. Corollary. If any point Q upon the surface 4 should bea point of intersection,
where the surface admits an infinite number of tangent planes, the perpendiculars
from O upon these planes will form a conical surface having O for its vertex. In
O@ take, as before, a point N, so that ON x OQ =k*, and let a plane passing through
NV at right angles to OQ cut the conical surface. The intersection will be a certain
curve. From the preceding demonstration it is evident that every point of this curve
belongs to the surface B, and that thé plane which touches this surface at any point of
the curve cuts OQ perpendicularly in VV; or, in other words, that the same plane
touches the surface B through the whole extent of the curve.
4. Two surfaces related to each other like 4 and B in the preceding theorem may
be called reciprocal surfaces, and points like Q and -# reciprocal points ; the radiz
OQ and OR may likewise be termed reciprocal. A familiar example of such sur-
faces is afforded, as I have shown on a former occasion*, by two ellipsoids having a
common centre at the point O, and their semi-axes coincident in direction, and con-
nected by the relation aa’ =bb'=cc’'=k’ ; where a, b,c, are the semi-axes of one el-
lipsoid in the order of their magnitude, a being the greatest ; and a’, 6’, c’, those of
the other ellipsoid, a’ being the least. The mean semi-axes 6 and b’ coincide, and the
circular sections of both ellipsoids pass through the common direction of 6 and 6’.
5. It has also been shown with regard to those ellipsoids, that if Q and # be reci-
procal points on the surfaces of abc and a’b’c’ respectively, and if a right line Ogr,
perpendicular to the plane QO, cut the first ellipsoid in g and the second in 7, the
lines OQ and Oq will be the semi4fxes of the section made in the ellipsoid abc by a
plane passing through them; and the lines O# and Qr, in like manner, will be the
semi-axes of the section made in the other ellipsoid a@’b’c’ by the plane in which
they lie.
6. It may further be remarked, that if the radius OQ in one of the reciprocal el-
lipsoids describe a plane, the corresponding radius OF will describe another plane.
For the planes touching the ellipsoid abc in the points Q will all be parallel to a cer-
tain right line, and therefore the perpendiculars O# to these tangent planes will all
lie in a plane perpendicular to that right line. “These two planes, containing the re-
ciprocal radii, may, for brevity, be called reciprocal planes.
When two reciprocal radii lie in a principal plane, at right angles to a semi-axis of
-* Transactions of the Royal Irish Academy, Vol. XVI. Part II. pp. 67, 68.
applied to the Wave Theory of Light. 243
the ellipsoids, it is evident that two planes intersecting in this semi-axis and passing
through the reciprocal radii, are reciprocal planes.
7. Tueoren II. If three right lines at right angles to each other pass through a
: fixed point O, so that two of them are confined to given planes; the plane of these
two, in all its positions, touches the surface of a cone whose sections, parallel to the
: given planes, are parabolas ; while the third right line describes another cone, whose
sections parallel to the given planes are circles.
Let the plane of the figure, (/¥%g. 3) supposed parallel to one of the given planes,
be intersected by the other given plane in the right line J/N; and let OQ be per-
pendicular to the latter plane, while OP is perpendicular to the former and to the
plane of the figure, so that PQ being joined will meet WN at right angles in R.. Let
OA, OB, OC, be the three perpendicular lines, of which OA is parallel to the plane
of the figure ; this plane will be intersected by the plane of OA and OB in a right
line BT parallel to O_4, and therefore perpendicular to both OL and OP, and to
the plane BOP, and to theline BP. ‘Thus the angle PBT is always a right angle,
and therefore 6T' always touches the parabola whose focus is P and vertex R; or,
which comes to the same thing, the plane 4OBT always touches the cone which has
0 for its vertex, and the parabola for its section.
Again,,since OB, OP, OC, are all at right angles to O.4, they are in the same
plane, and therefore the points B, P, C, are in the same straight line; and as BOC
is a right angle, the rectangle under BP and PC is equal to the square of the per-
pendicular OP ; but QO# is also a right angle, and therefore QP x PR=OP’;
whence BP x PC=QP x PR, and therefore the points B, R, C,Q, are in the cir-
cumference of a circle, so that the angle at Cis a right angle, being in the same seg-
| _ ment with the angle at R. Thus the point C describes the circle whose = is
_ PQ, and OC describes the cone of which this circle is the section.
8. Of the two right lines OP and OQ perpendicular to the given planes, one is
_ also perpendicular to the plane of the section. That one is OP. Its extremity P
isthe focus of the parabola. The extremities of both are the extremities of the
diameter PQ of the circle. The vertex of the parabola is the point R where the
_ diameter of the circle intersects that given plane to which the plane of section is not
parallel.
_ 9. Turorem III. In a straight line at right angles to any diametral section QOq
z of an ellipsoid abc whose centre is O, let OT and OV be taken respectively equal to
OQ and O¢ the semi-axes of the section, and imagine the double surface which is
the locus of all the points T and V; then if OS be perpendicular to the plane which
Bit ouches the surface in 7',and OP to the plane which touches the ellipsoid in Q, the
f lines OP and OS will be equal and perpendicular to each other, and the four straight
. lines OP, OQ, OS, OT, will lie in the same plane at right angles to Og.
Pee vOL,. XVII, : 3E
QWs Geometrical Propositions
10. This theorem is taken from a former communication to the Academy*. The
surface to which it relates, being the wave surface of FResNEt, is one of frequent occur-
rence in optical inquiries, and it is therefore desirable to give it a distinctive name not
derived from any physical hypothesis. I shall call it a biaral surface, from the cir-
cumstance implied in its construction, and adopted as the definition on which the pre-
ceding theorem is founded ;—namely, that any pair of its coincident diameters are
equal to the two axes of a central section made in the generating ellipsoid abc, by a
plane perpendicular to the common direction of the two diameters. ‘The name, per-
haps, may appear the more appropriate, as it reminds us of the place which the surface
holds in the optical theory of biaxal crystals.
11. Turorem IV. The biaxal surfaces generated by two reciprocal ellipsoids are
themselves reciprocal.
For if Q and R (Fig. 4.) be reciprocal points on the two ellipsoids, abe and aé’c’,
a tangent plane at Q will cut OR perpendicularly in P ; a tangent plane at R will
cut OQ perpendicularly in V7; and the rectangles ROP and NOQ will be equal to
each other and to A? (4rt. 4). Also if the straight line Ogr, at right angles to the
plane of the figure, cut the first ellipsoid in g and the second in 7, then (5) the elliptic
section QOq will have OQ and Og for its semi-axes, and the lines O# and Or will be
the semi-axes of the other section ROr. Draw therefore, in the plane of the figure,
the right lines OTL and OSM perpendicular to the right lines OQN and OPR,
making OT, OL, OS, OM, equal to OQ, ON, OP, OR, respectively ; the angles at
Sand L being of course right angles. Then it is evident that the point Z’is on the
biaxal surface generated by the ellipsoid abc, because OT" is perpendicular to the
plane of the ellipse QOg and equal to the semi-axis OQ ; and by Theorem III. it ap-
pears that OS is perpendicular to the tangent plane at 7. In like manner, the point
Mis on the biaxal surface generated by the other ellipsoid a’b’c’, and OL is perpen-
dicular to the tangent plane at M. Moreover, the rectangles MOS and LOT, being
equal to the rectangles ROP and NOQ, are each equal to k?, Hence the proposi-
tion is manifest.
12. As the ellipsoid whose semi-axes are a,5,c, may be called the ellipsoid abc, so
the biaxal surface generated by this ellipsoid may be called the biaxal abc; and that
which is generated by the ellipsoid a’b‘c’ may be called the biaxal a’b’c’.
13. Prorostrion V. To find what properties of biaxal surfaces are indicated by
the cases wherein one of the two sections QOq, ROr, in the preceding theorem,
is a circle.
Case 1. When QO is a circular section of the ellipsoid abc, the points T' and V,
(9) in the description of the biaxal surface abc, coincide in a single point m. At this
* Transactions of the Royal Irish Academy, Vol. XVI. Part II. pp. 67, 68.
applied to the Wave Theory of Light. Q45
point there are an infinite number of tangent planes; because the semi-axes of the
circular section QOq being indeterminate, any two perpendicular radii of the circle
may take the place of OQ, Og, in the general construction. The point 7 is there-
fore a point of intersection (3), where the two biaxal sheets cross each other, and it
may be called a nodal point, or simply anode. As OQ always lies in the plane of the
circle QOq, the line OR, which is reciprocal to OQ, must lie (6) in a given plane
reciprocal to the plane of the circle. And as Qg lies in the plane of the circle, we
have three right lines OR, Og, OS, which are at right angles to each other, and of
which the first two are confined to given planes. Therefore by Theorem If. the third
line OS describes a cone whose sections parallel to the given planes are circles. Now
T'S—or in the present case »S—is parallel to the fixed plane which contains OR,
and therefore the point § describes a circle; or, in other words, the feet of the per-
pendiculars OS, let fall from O on the.nodal tangent planes, eg) the ee
ence of a circle passing (8) through the nodal point.
14, Parallel to the plane of the circle and to its reciprocal plane, conceive two
planes passing through the node, and call them the principal tangent planes at n.
The plane of the circle and its reciprocal plane are intersected in the right lines Og,
OR, by the plane gO which is parallel to a tangent plane at m. | Consequently this
tangent plane at ” intersects the two principal tangent planes in lines that are paral-
lel to Og, OR ; and as Og, OR are perpendicular to each other, it follows that every
nodal tangent plane intersects the two principal tangent planes in lines that are at
right angles.
Hence again, the nodal tangent planes touch (7) the surface of a cone whose sec-
tions, parallel to the principal tangent planes, are parabolas. As this cone touches
the biaxal surface all round the point 7, it may be called the modal tangent cone.
15. Case 2. When ROr is a circular section of the ellipsoid a’‘b’c’, any two per-
pendicular radii of the circle may be taken for OR, Or: and because OR =’, and
OR x OP =Kk?=b1'", we have OP or OS equal to b, the mean semiaxis of the ellip-
_ soid abe. Hence OS is given both in position and length; for it is perpendicular
_ to the fixed plane ROr, and it is equal to 6. Now a plane cutting OS perpendicularly
at S,is a tangent plane to the biaxal abc ; and we have just seen that this tangent
plane remains the same, whatever pair of rectangular radii are taken for OR, Or.
But the point of contact J’ is variable, for the plane ROS in which it lies changes
with OR. Therefore as OR revolves, the point J’ describes a curve of contact on
. a the tangent plane of the biaxal abc.
The lines OR, Or, are in the fixed plane ROr ; and as OQ is reciprocal to OR,
it lies in a fixed plane reciprocal to the plane Ror (6). Therefore the first two of
the three perpendicular right lines Or, OQ, OT, are confined to fixed planes. Hence
the third line OT describes a cone, whose sections parallel to these planes are circles.
But the tangent plane is parallel to the fixed plane ROr, and its intersection with
246 Geometrical Propositions
OT describes the curve of contact. Therefore the curve of coptact is a cirele J
0 Lat VecleT
passing (8) through the point S. pty
16. We have examined the two cases of circular section with peewee only to the
biaxal abc. If we examine the same cases with regard to the second biaxal wb'c’, we
shall find that their indications are reversed ; the supposition which gives a node
upon one biaxal, giving a circle of contact on the other: and that the node and the
circle, thus corresponding, are so related, that a line drawn from O to the node
passes through the circumference of the circle, cutting the plane of the circle perpen-
dicularly ; whilst every line drawn from O through the circumference of the circle is
perpendicular to some nodal tangent plage.
These things are evident on looking at the figure. For when Or is a circle,
it is plain that the point M is a node of the biaxal ab’c', since OM is perpen-
dicular to the plane of the circle # Or and equal to its radius O#. But we have
already seen (15) that when ROr is a circle, the other biaxal abe has a circle of con-
tact, whose plane is perpendicular to OM at the point S of its circumference. The
line OTL is perpendicular, in general (11), to a tangent plane at WM, and therefore
perpendicular, in the present case, to a nodal tangent plane; whilst the point 7’,
through which it passes, is on the circle of contact. It is also evident that OT’ x OL
sire
We have here an example of the general remark in the corollary of Theorem I.
17. The section made in a biaxal surface abe, by any of the principal planes of its
generating ellipsoid, consists of an ellipse and a circle.
For let the plane Q Og pass through one of the semiaxes a, and let it revolve round
this semiaxis, while the right line OZV (9), perpendicular to the plane QQq, re-
volves about O in the plane of the semiaxes b,c. Then the semiaxis a of the ellipsoid
will always be one of the semiaxes of the ellipse QQq ; and if O 7’ be equal to this
semiaxis, the point 7’ will describe a circle with the radius a about the centre O.
The other semiaxis of the ellipse QOq is that semidiameter of the principal
ellipse be which lies in the intersection of the plane bc with the plane @Oq ; and as
OV is equal and perpendicular to this semidiameter, the point V describes an ellipse
equal to be, but turned round through a right angle, so that the greater axis of the
ellipse described by V coincides in direction with the less axis of the ellipse be. As
the radius a of the circle is greater (4) than both the semiaxes b,c, of the ellipse, the
circle will lie wholly without the ellipse.
In like manner, the section made in the biaxal surface by the plane aé consists of
a circle with the radius c, and an ellipse with the semiaxes a,b ; and as the radius of
the circle is less than both the semiaxes of the ellipse, the circle lies wholly within the
ellipse.
18. But when the section lies in the plane of the greatest and least semiaxes a,c, _
the circle and ellipse, of which it is composed, intersect each other. For the radius
6 of the circle is less than one semiaxis of the ellipse ac and greater than the other.
-_—
applied to the Wave Theory of Light. Q47
Leaving the ellipse ac in the position which it has as a section of the ellipsoid abe, if
we describe the circle b with the centre O and radius 4, the ellipse and the circle will
cut each other in four points at the extremities of two diameters ; and planes, passing
through these diameters and through the semiaxis 4 of the ellipsoid, will evidently be
the planes of the two circular sections of the ellipsoid. Now turning the ellipse ac
round through a right angle (17), the circle and the ellipse in its new position will
constitute the section of the biaxal surface, and will cut each other (F% 1g. 5.) in four
points » at the extremities of two diameters n On, nOn, which are perpendicular to
the two former diameters, and therefore perpendicular to the planes of the two cir-
cular sections. Consequently, the biaxal surface has four nodes at the four points 7.
These nodes, it is manifest, are alike in all their properties ; and they are the only
points common to the two biaxal sheets, since the points Z’and V (9), in the deserip-
tion of the biaxal surface, cannot coincide unless the section Q Og, perpendicular to
OTF, be a circle.
19. The plane of the greatest and least semiaxes a, c, of the generating ellipsoid,
may be called the plane of the nodes; and the two diameters n On, n On, passing
through the nodes, may be called the nodal diameters.
At one of the nodes n (Fig. 5) draw tangents nf, nk, to the ellipse and the circle
that compose the biaxal section ; and through O draw Op perpendicular to On, cutting
the circle inp. Then as On is perpendicular to the plane of a circular section of the
ellipsoid abc, this circular section will have Op for its radius, and its circumference
will cross that of the ellipse ac (belonging to the ellipsoid) in the point p. A line
touching the ellipse ac at p will be parallel to every plane that touches the ellipsoid
in a point of the circular section, and will therefore (6) be perpendicular to the plane
which is reciprocal to the plane of the circular section. But the tangent at p is per-
pendicular to the tangent mf, since the two tangents would coincide if the ellipse ac
were turned round (18) through a right angle, the point p then falling upon n.
Hence the circular section and its reciprocal plane are parallel to the tangents nk, nf;
_ and thereforegwo planes perpendicular to the plane of the figure and passing through
_ these tangents, are the planes that we have called (14) the principal tangent planes
at 2.
20. Produce Op to meet vf in v, aad conceive a parabola having its focus at Q, its
vertex at v (8), and its plane perpendicular to the plane of the figure. A cone, with
its yertex at ” and this parabola for its section, is (14) the nodal tangent cone.
Draw Of perpendicular to nf at f, and meeting nk ink. ‘The perpendiculars let
fall from O upon the nodal tangent planes form a cone, of which the circles described
in planes perpendicular to the figure upon the diameters nf; nk, are sections (8). On
the other biaxal surface a’b’c’ there is (16) a circle of contact whose plane is perpendi-
cular to On. This circle of contact is (16) another section of the cone last mentioned.
21. To the circle b and to the principal section ae of the ellipsoid abc conceive a
7
248 Geometrical Propositions
common tangent di’ to be drawn, in a quadrant adjacent to that which contains the
node ”, and let it touch the circle in d’ and the ellipse acin7’. A radius Od", drawn
through the point d’ to meet the ellipsoid a’d’c’ in the point d”, will be reciprocal to the
radius O07’, because it is perpendicular toa tangent ut 2’, and it will be equal in length to
b', because Od" x Od' =k’ = bb’, and Od'=6 ; whence Od'=0'. Therefore Od" isin a cir-
cular section of the ellipsoid ab’c’ Two planes perpendicular to the plane of the
figure, and passing through the radlikocal radi Od’, O7’, are (6) reciprocal planes
we have seen that the first of them makes a circular section in the ellipsoid a‘b’c’. They
are therefore (15) the fixed planes in the second case of Prop, V.
22. Now draw di a common tangent to the circle b and ellipse ac composing the
biaxal section, and let it touch the circle in d and the ellipse m 7. The lines Od, Oi, are
of course perpendicular to the lines Od’, Oz’, and therefore perpendicular to the fixed
planes just mentioned. Hence the me Od and the point d are the same as the fixed
line OS and the point S in the second case of Prop. V. The plane of the circle of
contact is therefore perpendicular to Od at the point d (15); and the points d and 7,
where its plane intersects the right lines Od, O7, perpendicular to the fixed planes, are
(8) the extremities of a diameter.
These things agree with the obvious remark, that the points of contact d and 7
must be points of the circle of contact; and that d? must be a diameter, because the
plane of the circle is perpendicular to the plane of the figure, and this latter plane
divides the biaxal surface symmetrically.
As the circle and ellipse may have a common tangent opposite to each node, there
are four circles of contact in planes perpendicular to the plane of the nodes.*
23. The biaxal surface belongs to a class that may be called apsidal surfaces, from
the manner in which they are conceived to be generated.
Let G be agiven surface, and O a fixed origin or pole. If a plane passing through.
O cut the surface G, the curve of intersection will in general have several apsides
A, A’, A’, &c., where the lines OA, OA’, OA", &c. are perpendicular to the curve.
Through the point O conceive a right line perpendicular to the plane of the curve, and
on this perpendicular take from O the distances Oa, Oa’, Oa’, &c. respectively equal
to the apsidal distances, O04, OA’, OA”, &c. Imagine a similar construction to be
made in every possible position of the intersecting plane passing through O, and the
points a, a’, a’, &c. will describe the different sheets of an apsidal surface.
*The curves of contact on biaxal surfaces, and the conical intersections or nodes, were lately disco-
yered by Professor Hamilton, who deduced from these properties a theory of conical refraction, which
has been confirmed by the experiments of Professor Lloyd. See Transactions of the Royal Irish Aca-
demy, Vol. XVII. Part. I, pp. 132, 145; and the present paper, Art. 55—58.
The indeterminate cases of circular section—at least the case of the nodes—had cogurred to me long
ago; but haying neglected to examine the matter attentively, I did not perceive the properties involved
in it (13). April 2, 1834.
-s
applied to the Wave Theory of Light. 24.9
The apsidal surface has a centre at the point O, because the lengths Qa,
Oa’, Oa’ ,&c. may be measured on the perpendicular at either side of the intersecting
plane.
Referring* to the demonstration of Theorem III. it will be seen to depend only on
the supposition that the point Q is an apsis of the section made by the plane Q Oq; or,
which is the same thing, that OQ is a position wherein the radius rector from O to the
curve of section is a maximum or a minimum. Hence we have the following general
theorem :—
24. Prop. VI. Tueorem. If tangent planes be applied at corresponding points |
A, a,on the surface G and the apsidal surface which it generates ; these tangent
planes will be perpendicular to each other and to the plane of the points O, A, a.
This is equivalent to saying that perpendiculars from Oon the tangent planes are
equal to each other, and lie in the plane of the lines OA, Oa.
25. If Q and R be reciprocal points on two reciprocal surfaces of which O is the
fixed origin or pole, the tangent plane at Q will be (1) perpendicular to OF and to
the plane QOR. Leta plane also perpendicular to the plane Q OF pass through OQ,
cutting the surface to which the point Q belongs in a certain curve, and the tangent
plane at Q in a tangent to this curve. The tangent is evidently perpendicular to Og,
and therefore the point Q is an apsis of the curve.
In like manner, the point £ is an apsis of the section made in the other surface by
a plane passing through OF and perpendicular to the plane QOR.
26. From these observations, and from Prop. VI., it appears that if the points
Q, R, in the figure of Theorem IV., be reciprocal points on any two reciprocal sur-
faces, and if the same construction be supposed to remain, the points 7’ and M will
be points on the apsidal surfaces generated by these reciprocal surfaces, and the tangent
planes at Z’and MM will be perpendicular to the lines OM and OT respectively. Also
the rectangles LO Z' and MOS will be equal to &°. Hence we have another general
theorem :—-
Prop. VII: Turorem. The apsidal surfaces generated by two reciprocal surfaces
_ are themselves reciprocal.
27. A very simple example of apsidal surfaces, with nodes and circles of contact,
may be had by supposing the generatrix G@ to be a sphere, and the pole O to be within
the sphere, between the surface and the centre C.
It is evident that the apsidal surface in this case will be one of revolution round the
right line O Cas an axis. Therefore taking for the plane of the figure (Fg. 6.) a plane
passing through OC'and cutting the sphere in a great circle of which the radius is C'S,
let a plane at right angles to the figure revolve about O, cutting the circle C'S in the
points 4, 4’. The section of the sphere made by the ~ailiving plane will have only
* Transactions of the Royal Irish Academy, Vol. XVI. Part II. p. 68.
250 Geometrical Propositions
two apsides 4, 4’, with respect to the point O, except when the plane is perpendicular
to OC. Hence if we draw the right line Oaa’ perpendicular to 404’, taking Ga, Oa’,
always equal to OA, OA’, the points a, a’, will describe a section of the apsidal surface.
This section will evidently consist of two circles C’S'; C" S”, equal to the circle CS,
and having their centres C’, C”, on the opposite sides of O in aright line C'OC" per-
pendicular to OC; the distances OC, OC’, OC" being equal. The circles C'S’, C" 8”,
intersect in two points m, ’, on the line OC and have two common tangents di, d’7’,
which are bisected at right angles by OC in the points c,c’.
28. Now let the circles C'S’, C"'S”, with their common tangents, or only one of
the circles with the half tangents, revolve about the axis OC, and we shall have the
apsidal surface with nodes at 7,n’, and with circles of contact described by the radii
ed, cd.
The section of the sphere by a plane passing through O at right angles to On, is a
circle of which Ois the centre. If therefore we suppose that the point » answers to
ain Prop. VL, the apsis A corresponding to 7 will be indeterminate, and the position
of the tangent plane at 7 will also be indeterminate, which ought to be the case at a
node. 2
The surface reciprocal to the sphere, the pole being at O, is evidently a surface of
revolution about the axis OC (it is easily shown to be a spheroid haying a focus at O) ;
and the section of this reciprocal surface, by a plane perpendicular to the axis at Q, is
a circle of which Ois the centre. This circumstance indicates (15) that on the
apsidal surface there is a curve of contact, whose plane is parallel to the plane of cir-
cular section; which agrees with what we have already seen.
29. When the point O is without the sphere, the axis OC will pass between the cir-
cles C'S’, CO" S", without intersecting either of them. The apsidal surface, described
by the revolution of one of these circles about OC, will be a circular rng. The nodes
have disappeared ; but the circles of contact still exist, as is evident.
Part Il.—On tHe Wave Tueory or Licurt.
30. Some of the foregoing propositions lead to a simple transformation of the
wave theory of light.
In this theory, the surface of waves, or the wave surface, is a geometrical surface
used to determine the directions and velocities of refracted or reflected rays; being
the surface of a sphere in a singly refracting medium ; a double surface, or a surface
of two sheets, in a doubly refracting medium; a surface of three sheets on the sup-
position of triple refraction 5 and having always a centre O round which it is sym-
metrical. The radii of the wave surface, drawn from its centre O in different di-
rections, represent the velocities of rays to which they are parallel.
applied to the Wave Theory of Light. 251
31. We shall consider particularly the case of a doubly refracting crystal, with two
plane faces parallel to each other, and surrounded by a medium of the common kind”
wherein the constant velocity is.V; supposing, for the sake of clearness, that the
crystal refracts more powerfully than the surrounding medium, so that the velocities
in the crystal are less than the velocity V.
A ray S'O, falling on the first surface of the crystal at the point O, is partly re-
flected according to the common law of reflection, and partly refracted. The two re-
fracted rays pass on to the second surface, where each of them is divided by internal
reflection into a pair, the two reflected pairs being parallel to each other; while the
two emergent rays—one from each refracted ray—are parallel to each other and to
the incident ray S’O. ‘The directions of the rays within the crystal are usually found
by the following construction. j
32. Describe a wave surface of the crystal, having its centre at O the point of in-
cidence. By the nature of the wave surface, a right line O 7'U, drawn from the point
O, will in general cut this surface in two points 7} U, on the same side of O; and a ray
passing through the crystal in a direction parallel to OT'U willhave one of the two
velocities represented by the radii O7;, OU, taking a line of a certain length / to repre-
sent the uniform velocity V inthe external medium, With the centre O and a radius
OS equal to this line & describe a sphere. As the velocities in the crystal are sup-
posed to be less than V, the wave surface will lie wholly within this sphere. Let the
plane of the figure (Fg. 7) be the plane of incidence, perpendicular to the parallel
faces of the crystal, and intersecting the first face in the right line #A. Through the
point S, where the incident ray S’O, produced through the crystal, cuts the surface of
the sphere, draw SJ at right angles to OS and meeting F'A in the point ZA right
line perpendicular to the plane of the figure, and passing through this point J, we shall
call the right line Z.
33. Through the right line J draw two planes touching the two sheets cf the wave
surface, on the side remote from the incident light, in the points T, 7”, which will lie
within the sphere (32); then the incident plane wave, perpendicular to OS, will be re-
fracted into two plane waves parallel to these two tangent planes; and the linesO T, OT",
will be the directions of the refracted rays along which the refracted waves are pro-
pagated. The lengths 07, O7", represent the velocities with which the light moves
along the rays ; and of course the normal velocities, which are the velocities of the
S refracted waves, are represented by the perpendiculars OG, OJ, let fall from O on
the two tangent planes at 7,7”. These two perpendiculars OG, OH, evidently lie
in the plane of the figure; but the points T, 7”, in general, do not lie in this plane.
34. Again, through the right line J draw two other planes touching the wave surface, at
the side of the incident light, in the points ¢,¢’. The rays OT,0 7”, arriving at the second
surface of the crystal, will each be divided by internal reflection into two rays parallel (
to Ot,Ot'; and these four reflected rays, arriving at the first surface, will each be divided,
VOL. XVII. 3F
252 Geometrical Propositions
by a new reflection, into two rays parallel to O 7,07"; and so on, for any number of
reflections. Any of the rays emerging at the first surface after internal reflections, is
parallel to the ray Os produced by ordinary reflection at the point of incidence ; and
any ray emerging at the second surface is parallel to the incident ray S'OS.
35. This construction may be changed into another that will be found more con-
venient both in theory and practice.
Through S draw SR perpendicular to OJ, and meeting O G,OH, produced, in the
points P,M. Then as the angles at Gand R are right angles, the points J, R, G, P,
are in the circumference of a circle, and therefore OP x OG@= OI x OR=OS=Kk
and similarly, OMx OH=k'. If then wetake O for the fixed origin, or pole, and k*
for the constant rectangle (Theorem I.), and describe the surface which is reciprocal
to the wave surface, it is evident that the points P and M will be points of the sur-
face so described, and that O7,O7", will coincide in direction with perpendiculars
let fall from O on planes touching the surface at P and M, and will be inversely pro-
portional to these perpendiculars. It follows in the very same manner, that if per-
pendiculars Og,Qh, let fall from O on the tangent planes at ¢,t’, be produced to meet
SR in the points p,m, these points will also be on the surface reciprocal to the wave
surface.
In the present case, it is manifest that this reciprocal surface lies wholly without the
sphere OS.
86. The surface reciprocal to oy wave surface, the pole bing at os we shall call
the surface of refraction. {fy Ly f ie ede Kearse het tet / VA 4
It is hardly necessary to observe that the surface of refraction has a centre at the
point O, round which it is symmetrical; that it is a sphere in a singly refracting me-
dium, a double surface in a doubly refracting medium, and a surface of three sheets if
we suppose a case of triple refraction.
37. In the case that we are considering, let the figure (Fig. 8.) representa section
made in the double surface of refraction and its attendant sphere by the plane of in-
cidence. Through the point S, where the incident ray S'O prolonged cuts the cir-
cular section of the sphere, draw SR perpendicular to the face of the crystal, or to
FA; and let SR produced cut the circle again in the point s. Then Os is the di- —
rection of the ray given by ordinary reflection at the first surface of the crystal.
Produce the right line Ss both ways, to cut the surface of refraction in the points
P,M, behind the crystal, and in the points p,m, before it; and conceive planes to
touch the surface of refraction at the points P,M/, p,m. Suppose also that perpendi-—
culars OP', OM’,Op',Om’, are let fall from O upon these tangent planes, and that
they intersect the planes in the points P’,J7, p',m’, respectively. ;
Then from the preceding observations (33, 34, 35), it is manifest that OP’,OM’,
are the directions of the rays into which S’O is divided by refraction ; that each of
these refracted rays, on arriving at the second surface of the crystal, is divided by in- —
;
applied to the WVave Theory of Light. 253
ternal reflection into two rays parallel to Op',Om’; and that each of the four reflected
rays, on arriving at the first surface, is again divided by reflection into two rays paral-
lel to OP’,OM’; and so on. In general, every ray going into the crystal from the
first surface, whether after refraction or after any even number of internal reflections,
is parallel either toOP’ or to OM’; and every ray returning from the second surface
of the crystal after any odd number of internal reflections, is parallel either to Op’ or
to Om'. Thus the direction of every ray in the interior of the crystal is the same as
the direction of some one of the four lines OP',OM'’, Op',Om’; and the velocity of
the ray is inversely as the length of this line ; so that the velocity of the ray OM’, for
example, or of any ray parallel to OM’, is to the velocity V asOSis to OM’. The
little plane waves that, keeping always parallel to themselves, move along these rays,
are respectively perpendicular to the lines OP,OM, Op,Om; and the lengths of these
lines are inversely as the velocities of the waves estimated in directions perpendicular
to their planes ; so that the velocity of the wave which moves along the ray OM’, or
along any parallel ray, is to the velocity V as OS is to OM.
38. The ray OP’ and all the rays parallel to it are perpendicular to the plane which
touches at P the surface of refraction ; and the waves which move along these rays
are perpendicular to the right line OP. Any ray of this set may be called a ray P,
and any of the waves a wave P. In like manner, the rays M,p,m, are rays that are
perpendicular to the tangent planes at the points 1Z,p,m, respectively ; and the waves
M,p,m, are the waves that belong to these rays, and that have their planes respectively
perpendicular to the right lines OM,Op,Om. The rays P,M, all come from the first
surface of the crystal; the rays p,m, from the second.
As the ordinates RP, Rp, are greater than the ordinates RM, Rm, so the rays P,p,
are more refracted or more reflected than the rays M,m. The former rays may
therefore be said to be plus refracted, or plus reflected, and the latter to be minus re-
Fracted, or minus reflected. Or,—for the convenience of naming,—the rays P,p, may
be called plus rays ; and the rays M,m, minus rays. The waves P,p, in like manner,
may be termed plus waves, and the waves Mm, minus waves.
For a medium of the common kind, ora singly refracting medium, we may use the
letters Sands. Thus the incident ray S'OS, or any ray emerging parallel to OS
from the second surface of the crystal, may be marked by the letter S; while the ray
Os produced by common reflection, or any ray emerging parallel to Os from the first
surface, may be denoted by the letter s.
39. The course of a ray through the crystal may now be easily expressed. A ray
SMps, for example, is a ray (8) incident on the crystal, undergoing minus refraction
(M) at the first surface, plus reflection (p) at the second, and emerging (s) from the
first surface in a direction parallel to Os. Of this ray the part within the crystal is
Mp. A ray SPS isa ray plus refracted, and then emerging ina direction parallel to
that of incidence. A ray SPpMS isa ray plus refracted at the first surface, then
254 Geometrical Propositions
————
plusreflected at the second surface, then minus reflected at the first surface, and
finally emerging from the second surface in a direction parallel to that of incidence.
Its path within the crystal is PpM.
These examples indicate the general method of expressing the path of a ray.
40. Suppose light to be moving in the same direction and with the same velocity
along two proximate parallel rays, so that it is at the point A in one ray when it is at
the point B in the other; and through the points 4 and B conceive two planes per-
pendicular to the common direction of the rays. These planes are either coincident,
or maintain a constant distance. In the first case, the rays are said to be in complete
accordance. In the second case, the constant distance between the planes is called
the interval between the portions of light composing the rays, or the interval between
the waves that move along the rays.
We proceed to find the lengths of these intervals in the case of rays emerging
parallel to each other, at either side of the crystal that we have been hitherto con-
sidering.
41. Let the tangent planes at P, M, p, m, intersect the plane of the figure
(Fig. 8) inthe right lines PP, MI, pp,, mm,, which of course are tangents to the
section of the surface of refraction represented in the figure ; let a perpendicular at O
to the face of the crystal cut these tangents in the points P, 11, p,, m,; and let
the lines OP”, OM", Op", Om", respectively parallel to PP, MM, pp,, mm,, cut
the line S#s in the points P”, 1”, p”, m’.
The length of the path which a ray P describes within the crystal, is equal to the
thickness © of the crystal divided by the cosine of the angle P’ OP, which the path
of the ray makes with a perpendicular to the faces of the crystal; and the velocity of
: Os dere é
P isequalto V x op (37) dividing therefore the length of the path by the velocity,
By iy: - ; @x OP’
we find that the time in which a ray P crosses the crystal is equal to Fx 08x Cos POP
: : ORS
But as OP’ is perpendicular to the tangent plane at P, we have Gos POR,
=OP =PP". Therefore the time is equal to A ask
which rays M, p,m, pass from one surface of the crystal to the other, are equal to
OxMM” expp” Oxmm’ ;
x08? Vx08? Pos” respectively.
42. Now suppose the path of a ray P to be be projected perpendicularly on a right
line having any proposed direction in space. Through O conceive a right line OL
parallel to the proposed direction, and meeting in L the tangent plane at P. ‘The
length of the projection is equal to the length of the path multiplied by the cosine of —
the angle P’OL which the ray P makes with OL ; that is, the projection is equal to
Cos POL | ,
Similarly, the times in
But because OP’ is perpendicular to the tangent plane at P, we have
Cos P'OP, ' 1 2p
5 F OF : OP' OP Cos POL PP"
Cos P'OL= OL” and 08 Oe OP,= PP” ; therefore Cos POP, = OL Henee
the projection is equal to on
&
<t
H
5
applied to the FVave Theory of Light. 255
If the path of a ray P be projected on the incident ray O'S, then producing OS to
meet PP, in /, we see, by what has just been proved, that the length of the projec-
Ni s
tion is equal to 0 o =0 0) aby similar triangles. In like manner, the projections
of the paths of rays MM, p, m, on the direction of the incident ray OS, are equal to
ee Xe Sp" ‘s) Sm* , respectively.
Os Os
43. Let each rectilinear path be measured in the direction in which the light moves
along it; and according as the direction so measured makes an acute or an obtuse
angle with the direction OS, measured from O to S, let the projection of the path
on OS be reckoned positive or negative. Then if SPmMpMS be any ray entering
the crystal at O, and emerging from its second surface at H, and if a perpendicular
EI be let fall from LZ upon OS, meeting OS in I; the distance OF, from O to
to the foot of this perpendicular, will evidently be equal to the algebraic sum of the
projections of the paths P, m, M, p, M, contained within the crystal ; taking each
projection with its proper sign. It is obvious that the projections of the P and M
rays are always positive. And asthe lines Op’,Om’',—the directions of the rays p, m,—
lie in planes which are respectively perpendicular to pp,, mm,, or to Op", Om", it
is easy to see that these directions make acute or obtuse angles with O.S, according as
the points p”, m”, lie below the point S or above it ; that is, the projections are posi-
tive or negative according as the points p”, im”, lie without the circle OS towards
SR is within the circle. ‘Therefore the distance OJ, in the case of the figure, is
equal to — ax 2 (SP"—Sm" + )SM"—Sp"+SM"). - Lorre
44. If the paths of rays P, M, p, m, be projected on we direction Os of the or-
dinarily reflected ray, thelengths of their projections will be ate os? eyes = , 9 2 ,9 a ;
respectively. The projections upon Os of the rays p, m, will be nee ays positive ; and
the projections of the rays P, M, will be positive or negative according as the points
P", M", lie above the point s or below it ; that is, according as the points P”, WM”,
lie without the circle OS towards p and m, or within the circle. So that if SPmMps
be a ray entering the crystal at O and emerging from the first surface at e, and if a
perpendicular e7 be let fall from e upon Os, the distance Oz from the point O to the
foot of this perpendicular, or the algebraic sum of the projections of the paths P,m,
M, p, contained within the crystal, will be equal to = (—sP" + sm"—sM" + sp"), in
the case of the figure.
45. Let us imagine that the light in the incident ray S'O, instead of being inter-
rupted at O by the crystal, had continued to move with the same velocity V in the
same right line OS, leaving the point O at the moment when the refracted light
enters the crystal at O. Comparing the light in this imaginary ray with that in a ray
emerging parallel to it from the second surface of the crystal, after an even number
of internal reflections, we shall find that the emergent is behind the imaginary ray, and
256 Geometrical Propositions
that the interval between them (40),—or the retardation of the former,—may be de-
rived very easily from the letters that designate that ray. Let SPmMpMS be any
such ray. The sum of the distances of the point S from each of the points marked
by the letters (PmM/pM™) that denote (39) the part of the ray contained within the
crystal, is proportional to the interval of retardation; that interval being equal to
oy (SP+ Sm+SM+Sp+SM). Lorn
For if from the point , where the last internal ray M emerges from the second
surface of the crystal, a perpendicular HJ be let fall upon OS, meeting OS in J, the
time of describing OJ with the velocity V would (43) bey (SP "— Sm" +SM"
—Sp'+SM"). But (41) the actual time of describing the broken path PmMpM
e (PP” + mm" + MM" + pp" + MM"); and, on inspecting the figure, this time
ag VxOs
is seen to be greater than the time of describing OJ, by Toa (SP + Sm +SM+ Sp
+SM), or by the time in which the line a5 (SP+ Sm +SM+ Sp + SM)would
be described with the velocity V. Consequently, at the moment when the light in the
ray SPmMp MS emerges at the point EH from the second surface of the crystal, the
light in the imaginary uninterrupted ray OS will have passed the point J by an interval
equal to the line just mentioned ; and as the two rays afterwards have the same velo-
city and parallel directions, this interval is the retardation of the emergent ray.
46. The rays emerging from the first surface after any odd number of internal re-
flections are to be compared with the ordinarily reflected ray Os to which they are
parallel ; the light in Os, which moves with the velocity V, being supposed to leave
O at the moment when the refracted light enters the crystal at O. The mode of
proceeding in this case is exactly similar to that in the last, and the interval is de-
termined in the same way, using sin place of S; the retardation of the ray SPmMps,
for example, of which the part PmMp is contained within the crystal, being equal
to Ae (sP +sm+sMM+ sp).*
47. It is remarkable that the preceding demonstration nowise depends upon the sup-
position that the planes perpendicular to the rays P,M,p,m, are tangent planes to the
surface of refraction at the pots P,M,p,m. If we had supposed any planes—dif-
ferent from the plane of the figure— to pass through the points P,M,p,m, and the
rays to coincide in direction with perpendiculars let fall from O upon these planes,
and to have velocities inversely proportional to the lengths of the perpendiculars, the
intervals of retardation would haye remained unchanged. Hence the retardations are
the same as if the lines OP,OM, Op,Om, were the directions of the rays in passing
* The change of phase, which may take place at a surface of the crystal, is not here considered as af-
fecting the intervals. ’
applied to the Wave Theory of Light. ‘ 257
through the crystal ; as will appear by conceiving the planes that we have spoken of
to be perpendicular to these lines.
If the incident ray S'O were refracted in the ordinary way with an index equal
o it would take the direction OP; if it were refracted, in like manner, with
the index , it would take the direction OM; and if the two rays, thus ordinarily
oy)
Os
refracted, were to emerge from the second surface of the crystal in directions parallel
to OS, it is evident from what has been said, that they would be in complete accord-
ance, respectively, with the rays SPS and SMS.
If the surface of refraction should happen to have a node JN, which is a point of in-
tersection where it admits an infinite number of tangent planes (3), let the direction
of the incident ray S’ OS be chosen, so that the right line HS perpendicular to the
face of the crystal, being produced below S, may pass through J, and we shall have
a cone of refracted rays formed by the perpendiculars let fall from O upon the tangent
planes at NV; all of which rays, on emerging parallel to OS from the second surface
of the crystal, will be in complete accordance with one another. For we have just seen
that if the ray S'OS were supposed to emerge after being refracted in the ordinary
: ; WY : :
way with an index equal to 55 , it would be in complete accordance with any ray of
the cone.
48. The interval between any two rays emerging at the same side of the crystal is
the difference of their retardations. In taking the difference, the letters that are
common to the names of the two rays may be left out. Thus the ray SPmMS is
behind the ray SPS by the interval oy (Sm + SM)= <M. m. The line ~ Pp is
the interval between the rays SMS and SMpPS, or between the reflected ray Os
and the ray SPps ; and so on.
49. The retardations of the two refracted rays SPS and SMS, emerging without
vernal reflection, are SP and 5ySM respectively. The difference of these is
98 Os
face in directions parallel to the incident ray, the light in the plus emergent ray is
eal® Or, in
other words, the incident plane wave, perpendicular to OS, produces two emergent
waves parallel to each other and to the incident wave, moving along the emergent
PM, i
2 8 S between their planes,
the minus wave being foremost. If OS, the radius of the sphere, be taken for unity,
PM will be a number,—generally a very small fraction,—and the interval will be the
thickness of the crystal multiplied by this number.
50, Suppose the right line PVR, remaining always perpendicular to the face of the
crystal, to describe a cylindrical surface, with the condition that the part PM, inter-
Consequently, when the two refracted rays have emerged from the second sur-
behind the light in the minus emergent ray by an interval equal to
rays with equal velocities V, and preserving the distance
258 Geometrical Propositions
cepted between the two sheets of the surface of refraction, shall remain of a constant
length ; the point £ will then describe, on the surface of the crystal, a curve whose
radii O& are the sines’ (to the radius OS) of the angles of incidence of a cone of
rays; and every ray S’O of this cone, when refracted by the crystal, will afford two
emergent rays, or two waves, having the same given interval between them. Panes
drawn from the eye parallel to the sides of this cone are the emergent rays belonging
to a ring, when rings are made to appear, in any of the usual ways, on tenance
polarised light through the plate of crystal. In nominal conformity to this, we see
that the line PM describes a ring of constant breadth between the two sheets of the
surface of refraction. The ring described by supposing pm to remain constant cor-
responds to the interval between two rays p and m reflected at the same point of the
second surface of the crystal, and then emerging at the first. The other intercepts
Pp, Mm, Pm, Mp, ave proportional (48) to intervals like those in Newton’s-rings ;
to the intervals, namely, between the reflected ray Os and the rays SPps, SJMdms,
SPms, SMps, emerging at the first surface after one reflection within the crystal ; or
to the intervals between rays that are twice reflected in the crystal and the rays trans-
mitted without reflection.
51. The general investigation of the figure of a geometrical ring does not distin-
guish between the different intercepts, and will therefore include all the rings PW,
pm, Pp, Mm, Pm, Mp ; so that it will be sufficient to contemplate any one of them,
as PM, of which the breadth P.M is equal to a given line J.
The points P and M describe, in general, similar and equal curves of double cur-
vature, which may be called ring-edges, as being the edges of the ring ; and if we
imagine the surface of refraction, carrying these curves along with it, to be shifted
either way, in a direction parallel to PM, through a distance equal to J, it is clear
that the new position of one of the ring-edges will exactly coincide with the first po-
sition of the other, and that therefore the curve of the latter ring-edge will be given
by the intersection of the two equal surfaces in these two positions. Let U=0,—
where 0 is a function of x, y, z, and given quantities—be the equation of the sur-
face of refraction in its original position ; and, the axes of coordinates being fixed,
suppose that by the shifting of the surface the coordinates of a point assumed on it are
diminished by the given lines f, g, h, which are the projections of the given line J
on the axes of x, y, z, respectively. Then the equation of the surface in its new po-
sition will be had by substituting +f, y+g, 2+h, for x, y, z, in the equation U=0,
which will thus become U + V=0, where V is the increment of U produced by the sub-
stitution. These two equations combined are equivalent to the equations U=0,
V=0, which are therefore the equations of one of the ring-edges. If the surface
had been shifted the opposite way, in a direction parallel to PM, the intersection
would have been the other ring-edge, whose equations are therefore deducible from
those already found, by changing the signs of f, g, h.
PY
applied to the Wave Theory of Light. 259
52. If the equation of the surface of refraction be transformed, so that the plane of
xy may coincide with the face of the crystal, and the axis of z be perpendicular to it,
the origin of coordinates being at the centre O, no change will be produced in & or in
y by the motion of the surface, because P.M, the direction of the motion, is now pa-
rallel to the axis of z ; but z will be diminished or increased by 7; and accordingly,
if U'=0 be the equation of the surface in its first position, when the centre is at O,
and if U’ become U' + V"’ when z becomes 2 +2,—the equation of the surface in its
second position, when the centre has moved through a distance equal to J along the
axis of z, will be U'+ V'=0; and these two equations combined will give G=0)
V’=0, for the equations of one of the ring-edges. ‘The equations of the other ring-
edge are deduced from these by changing the sign of J.
The projection of each of the ring-edges on the plane of «y is the curve traced by
the point # on the surface of the crystal (50). This curve may be called a ring-
trace. Its equation is obtained by eliminating z between the equations of a ring-edge ;
and as the result must be the same whether J be taken positive or negative, the equa-
tion of the ring-trace, when found by this general method, will contain only even
powers of J. The radii drawn from O to the points # of the ring-trace, are (50)
the sines (to the radius OS,) of the angles of incidence or emergence of the rays that,
form an optical ring ; the rays that come from this ring to the eye being parallel to
the sides of the cone described by the right line S'OS while the point # describes the
ring-trace.
53. It is evident that tangents to the ring-edges, at the points P and M, are paral-
lel to each other, and therefore parallel to the intersection of two planes touching
the surface of refraction at P and MM, because these tangent planes pass through the
tangents. But the directions OP’, OM’, are perpendicular to the tangent planes,
and therefore the plane P OM, containing the two rays, is perpendicular to the inter-
section of the tangent planes, and of course perpendicular to the parallel tangents.
Hence the plane P’'OM intersects the face of the crystal in a right line perpendicular
to the projection of the parallel tangents on the face of the crystal. As this projec-
tion is a tangent to the curve described by R, it follows that the normal to the ring- |
trace at the point # is parallel to the line joining the points in which the two refracted
rays cut the second surface of the crystal.
In like manner, taking any two consecutive rays (P and m) having a common extre-
mity on one surface of the crystal, the line joining the points where these rays cut the
other surface, is parallel to the normal at the point # of the ring-trace which is de-
scribed when the intercept (Pm) between the letters that mark the rays is supposed to
remain constant.
54. In all that precedes we have made no supposition about the surface of refrac-
tion except that it isa surface of two sheets ; andif we supposed it to have three sheets,
the conclusions would be easily extended to this hypothesis.
VOL. XVII. 3.6
260 Geometrical Propositions
In the theory of Fresnex, the wave surface is* a biaxal whose generating ellipsoid
has its centre at the point O, and its semiaxes parallel to the three principal directions
of the crystal, the length of each semiaxis being equal to OS divided by one of the
principal indices of refraction. The surface of refraction is reciprocal to the wave
surface, aud is (11) theretore another biaxal generated by an ellipsoid reciprocal to
the former, having its centre at the same point O, and the directions of its semiaxes
the same as before, the rectangle under each coincident pair of semiaxes being equal
to k* or OS*.. Hence the semiaxes of the ellipsoid which generates the biaxal surface
of refraction are equal in length to OS multiplied by each of the three principal in-
dices. ‘This biaxal surface is of course to be substituted for the surface of refraction
in the preceding observations.
55. When the line #S, produced below S, passes through a node WN of the biaxal
surface of refraction, the points P, M, coincide in the point N, and the interval PM
vanishes. At the point N there are an infinite number of tangent planes, and the
perpendiculars from O on these tangent planes give a cone of refracted rays whose
sections we have already shown how to determine (20). All the rays in this cone, on
arriving at the second surface of the crystal, emerge parallel to the incident ray OS ;
‘and if the rays in the emergent cylinder be cut by a plane perpendicular to their com-
mon direction, they will all arrive at this plane at the same instant, because the inter-
val PM vanishes. See art. 47.
56. Suppose fig. 5 to be a section of the wave surface. The right line Od will pass
through V ; and the circle of contact, described on the diameter di in a plane per-
pendicular to the right liné OdN, will be a section of the refracted cone. Now it will be
recollected that, in general, the vibrations of a ray OT, which goes to any point 7’ of
the wave surface, are parallel to the line which joins the point 7' with the foot of the
perpendicular let fall from O on the tangent plane at 7’. “In the present case, the per-
pendicular is the same for all the rays of the refracted cone, and its extremity coin-
cides with the point d : so that the line d 7; drawn from d to any point T of the circle
of contact, is parallel to the vibrations of the ray O7' which passes through 7°
Conceive, therefore, a plane perpendicular to ON at the nodal pomt N. This
plane will cut the refracted cone in a circle whose circumference will pass through WN ;
and a line NT", drawn from the node to any other point 7" of the circumference, _
will be the direction of the vibrations in aray O7” which crosses the circle at this
point. The plane of polarisation is perpendicular to the direction of the vibrations.
57. The transverse section of the emergent cylinder is always a very small ellipse,
affording a hollow pencil of parallel rays in complete accordance (55). If the crystal
be thin, this ellipse will be of evanescent magnitude. Hence the line OS will be the
direction of a line drawn from the eye to the centre of the rings commonly observed
* Trans. R. I. A. Vol. XVI. p. 76., + Ibid.
A)
applied to the FVave Theory of Light. 261
(50) withypolarised light or it will be what is called the apparent direction of one of
the optic axes. The diameter passing through XV will be the direction of the optic
‘axis within the crystal. ‘There are therefore two optic axes, parallel to the two nodal
diameters (19) of the surface of refraction.
As ON is equal to the mean semiaxis of the generating ellipsoid, or to the mean in-
dex of refraction, when OS is unity, it follows that the apparent direction of an optic
axis is the direction of an incident ray, which, if refracted in the‘ordinary way, with
an index equal to the mean index of refraction, would pass along a nodal diameter of
the surface of refraction. ~~
58. We have seen (15) that there is a circle of contact on the biaxal surface of re-
fraction. If an incident ray S’OS be taken, cutting the sphere in S, so that the line
RS produced may pass through the circumference of this circle, it is manifest that the
direction of the refracted ray will be the same through whatever point II of the cir-
cumference the line #S may pass, because that direction is perpendicular to the tan-
gent plane at II, which is in fact the plane of the circle itself. If, therefore, the line
RS move parallel to itself along the circumference of the circle, cutting the sphere
in a series of points S, every incident ray SOS which passes through a point S so de-
termined, will be refracted into two rays of which one will have a fixed direction in the
crystal, being perpendicular to the plane of the circle of contact, and therefore coin-
ciding (16) with 2 On, one of the nodal diameters of the wave surface. But though
the direction On of the refracted ray is fixed, its polarisation changes with the
meident ray from which it is derived ; for if Ibe the point in which the line FS, cor-
responding to any position of the incident ray, crosses the circle of contact, the vibra-
tions of the refracted ray On will be-eontained in the plane of the lines On, OT, and
will be perpendicular to OII. Conceive a circle described on the diameter nf in a
plane perpendicular to the figure (Fig. 5). This cirele, and the circle of contact on
the surface of refraction, are (20) sections of the same cone. Let TI’ therefore be the
point at which OI, in any position of the incident ray, crosses the circumference of
the circle nf ; and the line Il’n, drawn to the node of the wave surface, will be the
corresponding direction of the vibrations in the ray On.
E 59. With regard to the general law of polarisation in tlie theory of Fresner, it
¢ _ may be observed, that if the ellipsoid abc which generates the biaxal surface of re-
fraction be cut by a plane perpendicular to OP, the vibrations of the ray P will be
. parallel to the greater axis of the section, and therefore the plane of polarisation will
pass through OP and the less axis ; whence it is easy to show that the plane of pola-
risation of a ray P bisects one of the angles made by two planes intersecting in OP
plane of polarisation of the ray p is found in like manner. But for the rays M, m,
the angle to be bisected is that which contains within it the greatest semiaxis a.
j, , 4
0
h / J
¥ ite 4 / /
As In hy m ye a
and passing through the nodal ‘diameters of the surface of refraction ; the bisected
angle being that which contains the least semiaxis c of the generating ellipsoid. The ~
262 Geometrical Propositions
If OP" be perpendicular to a tangent plane at P, the vibrations of the ray P will
be perpendicular to @P-and will-ie-in the plane POP’. A similar remark applies
to the rays M, p, m.
60. When two semiaxes a,b, of the ellipsoid abc become equal, it changes into a
spheroid aac described by the revolution of the ellipse ac about the semiaxis ¢ ; and
the biaxal aac, generated by this spheroid, is * composed of a sphere whose radius is a,
and a concentric spheroid acc described by the revolution of the ellipse ac about the
semiaxis a; so that, the diameter of the sphere being equal to the axis of revolution
of the spheroid, the two surfaces touch at the extremities of the axis. This combina-
tion of a sphere anda spheroid is the surface of refraction for uniaxal crystals. Jn
these crystals, therefore, the refracted ray whose direction is determined by the inter-
section of the right line #S with the surface of the sphere follows the ordinary law
of a constant ratio of the sines, and is called the ordinary ray ; whilst the other,
whose variable refraction is regulated by the intersection of FS with the spheroid, is
ealled the extraordinary ray: And hence uniaxal crystals are usually divided into the
two classes of positive and negative, according to the character of the extraordinary
ray ; being called positive when it is the plus ray, and negative when it is the minus
ray. The first case evidently happens when the spheroid is oblate, and therefore lies
without the sphere described on its axis ; the second, when the spheroid is prolate,
and therefore lies within the sphere. The second case, (which is that of Iceland
spar,) may be supposed to be represented in the figure (Fig. 8), where the elliptic
section of the spheroid, made by a plane of incidence oblique to the axis, lies within
the circular section of the sphere, and the minus ray is of course the extraordinary
one.
61. Let PM, preserving a constant length J, move parallel to itself between the
surfaces of the uniaxal sphere and spheroid, so as to form a ring (50). Then sup-
posing the spheroid, with the ring-edge described on it by. the point JZ, to remain
fixed, imagine the sphere, carrying the ring-edge P along with it, to move parallel to
PM, from P towards M, through a distance equal to J, and the two ring-edges will
exactly coincide. iy
Hence the uniaxal riag-edge is the intersection of a sphere anda spheroid, the
diameter of the sphere being equal to the axis of revolution of the spheroid, and the
line joining their centres being perpendicular to the faces of the crystal and equal to
the breadth J of the ring. And the projection of this intersection, on a plane perpen-
dicular to the line joining the centres of the sphere and the spheroid, is the uniaxal
ring-trace.
62. The biavxal ring-edge is (51) the intersection of two equal biaxal surfaces
similarly posited, the line joining their centres being perpendicular to the faces of the
* Trans. R. I. A. Vol. XVI. p. 77.
applied to the Wave Theory of Light. 263
erystal and equal to the breadth of the ring. And the projection of this intersec-
tion, on a plane perpendicular to the line joining the centres of the surfaces, is the
biawal ring-trace.*
* In applying the general theory (51, 52) to biaxal rings, it is necessary to know the equation of a
biaxal surface, which may be found in the following manner. Let 7, 7’, 7’, be three rectangular radii of
the generating ellipsoid abe, the two latter being the semiaxes of the section made by a plane passing
through them ; so that if from the centre O two distances OT, OV, equal to 7’, 7’, be taken on the di-
rection of 7, thepoints Tand V will belong (9) to the biaxal surface ; and let a plane parallel to the plane
of 7, x”, and touching the ellipsoid, cut the direction of r at the distance p from the centre. Then if 7
make the angles a, 8, 7, with the semiaxes a, b, c, we shall have, by the nature of the ellipsoid.
1 costa cos’ costy
a a oe as
>
» po=atcos?a+l*cos*h + c*costy.
Now since the sum of the squares of the reciprocals of three rectangular radii of an ellipsoid is con.
stant, as well as the parallelopiped described on three conjugate semidiameters, we haye the equations
1 1 1 1 1 1
a ae
pT tea c;
Or,
1 1 1 1 1 cos*a cos? cos*
a he = +5 ( + Bea ") = M,
we At” Bt” Ge
2 2
a &? ce
1 a®cos?a + 5°cos*B + c*cos?y
= =N-
rere ace
Whence it appears that 7’, rv’, are the yalues of p in the equation
-——4+ = 0,
in which p denotes indifferently either semidiameter, OT or OV, of the biaxal surface. Therefore
putting for Wand J their values, and writing z ¥ 2, instead of cos a, cos PB, cos y, and 22+y*+2°
instead of p?, we obtain, for the equation of the biaxal surface,
(a* +y° +2°)(a%a* +b? +e°2z*) + a2(b?+¢°)a*~— h(a +e*)y?—e(a? +62) z* + a*b%e° =0.
‘This is the equation of the surface of refraction for a biaxal crystal in which a, 6, ec, are (54) the three
principal indices of refraction, taking O.S' the radius of the sphere to be unity. The left-hand member
of the equation is therefore the expression supplied by the theory of FRESNEL for the function UJ in art.
51. :
When the faces of the crystal are parallel to any of the principal planes of the ellipsoid, —to the plane
of ay for example,—the nature of the ring-trace may be found very easily. For if the difference of the
two values of z, deduced from the preceding equation of the surface of refraction, be put equal to a con-
“stant quantity JZ, the result, when cleared of radicals, will be an equation of the fourth degree in x and y,
which will be the equation of the corresponding ring-trace. This is a case that occurs frequently in prac-
tice; the crystal being often cut with its faces perpendicular to the axis of x or of z, because these lines
ect the angles made by the optic axes. :
VOL. XVII. ; 3H
ot, loupe diag, Ses
tei
ys and :
ie et
7 :
UnagGh ine *
hoe DER Sal hs ws
~
An account of anew Fulminating Silver, and its application as a Test for Chlo-
rine, Sc. By EDMUND DAVY, F.R.S., M.R.LA., &c., Professor of
Chemistry to the Royal Dublin Society.
Read May 23, 1831.
I varecy had the honor of sending to the Royal Irish Academy, a paper of mine, “ On
a new Acid and its Combinations,” published by the Royal Dublin Society. Whilst
it was printing, I found that some of the compounds therein described, sponta-
neously exploded when brought in contact with chlorine gas. Such unexpected re-
sults appeared sufficiently interesting to merit further inquiry: accordingly, I insti-
tuted a series of experiments on the subject ; and as they have led to a number of
new facts which may admit of useful applications, I venture to submit some of them
to the consideration of the Academy. In the present communication, | purpose to
give a brief account of a new fulminating silver I have obtained, and of the principal
experiments which illustrate its efficacy as a test for chlorine.
1, Modes of making the new Fulminating Silver ; its properties and composition.
This compound may be readily prepared from Howard’s well-known fulminating
mercury by the following simple processes :—Put any quantity (suppose from ten to
fifty grains or more,) of Howard’s compound, either in a dry or moist state, into a
phial, with about half an ounce of pure water, and about twenty grains of zine filings,
or granulated zinc, for every ten grains of Howard’s compound used. Cork the
phial and occasionally agitate its contents, for about twenty or thirty minutes. Less
time will answer if the water is moderately warm. ‘The fluid is fulminate of zinc ;
filter and treat it with nitrate of silver, the white precipitate which occurs is the new
fulminating silver. This substance may also be easily made, by adding nitrate of sil-
ver to an aqueous solution of any of the soluble fulminates I have described in the
paper already alluded to, or from Howard’s fulminating silver by the means recom-
mended aboye, in the case of the fulminating mercury ; and after being collected on
266 Professor Davy’s account of
a filter, washed and dried, either in the open air or at a heat not exceeding 212° Faht.
its properties appear to be the same.
The new fulminating silver, in drying on the filter, breaks into small lumps, which
are easily separated from the paper, and reduced by the gentle pressure of a platina
spatula, or of a card, to an impalpable powder. Its colour, when recently prepared,
is white ; and if dried in the dark, or in a weak light, it retains this colour for some
time. On being exposed to a strong light or the sun’s rays in a moist state, it soon
undergoes progressive changes of colour ; from greyish white to yellowish brown, brick
red, blackish brown, and even black. These changes are facilitated by the presence
of water, and they appear to be connected with the partial decomposition of the com-
pound ; for on exposing a little of it in water for some time to the action of the solar
rays, I observed, by means of a magnifying glass, minute specks of metallic silver on
the surface of the water, apparently carried there by small globules of gas, which were
occasionally evolved from the compound.
The new fulminating silver, when heated to about 350° Faht. assumes a darkish
tint and explodes, producing a large yellowish flame, and a loud report. It also ex-
plodes by percussion, when struck with a hammer on an anvil ; and by friction, when
it is rubbed between two hard surfaces; hence, it must be handled with caution.
It is insoluble, or very nearly so, in cold water. I tried in vain to dissolve a single
grain of the dry compound in about eleven fluid ounces of pure water ; after some
days, the greater part was undissolyed. The water, however, became slightly turbid,
acquired a metallic taste, and on evaporating some of it to dryness, a very minute por-
tion of the compound remained. Boiling water dissolves a certain limited quantity
of the compound; but as the fluid cools, the greater part separates in minute crystals,
which are principally long needles, intersecting each other.
The new fulminating silver, in a dry state and even whilst it yet retains moisture,
instantly explodes when brought in contact with pure chlorine gas; and also when
this gas is mixed with most other gases, as will be presently stated.
Muriatic acid decomposes the new fulminating silver, readily, converting it into
chloride of silver, whilst hydrocyanic acid is evolved; and if the experiment is
continued to dryness, sal-ammoniac rises in vapour, and the pure chloride of silver
remains.
Strong nitric acid gradually decomposes the new fulminating silver with evolution
of gas ; but if diluted, fulminic acid is slowly disengaged, and nitrate of silver formed.
Strong sulphuric acid readily explodes the new compound ; but when diluted, the
fulminic acid is gradually evolved, and sulphate of silver produced.
The new compound is soluble, to a certain extent, in liquid ammonia, and as the
alkali evaporates, minute crystals are deposited, which explode by heat, but not in
chlorine gas. Solutions of the fixed alkalies dissolve a portion of the new compound,
and form crystallized fulminating compounds, which I have not examined.
A new Fulminating Silver. 267
The new fulminating silver readily explodes by the electric spark, producing a loud
report, and a reddish flame. I repeatedly succeeded in the experiment, by simply
placing a little of it on the prime conductor, bringing a brass ball near it, and putting
the machine in motion. I found that the very feeble charge remaining on the prime
conductor, after giving the machine a few turns, was quite sufficient to explode it.
The new fulminating silver is decomposed by a number of metals, as zinc, iron,
copper, &c. in cases when it is put into water, and these metals severally introduced ;
new fulminating compounds of each metal are in a little time produced.
As the new fulminating silver may be formed from Howard’s fulminating silver,
and may be readily converted into chloride of silver, (as has been stated ;) little diffi-
culty was anticipated in ascertaining its composition ; and yet from the precautions to
be observed, I made several experiments, (using both methods,) before I obtained
any satisfactory results. In converting Howard’s fulminating silver into the new
compound, a given weight of the former, well dried, was very cautiously put into a
small phial, (nearly filled with pure water,) with about twice its bulk of fine zinc
filings. The contents of the phial were occasionally agitated gently, and after some
hours the fluid, (fulminating zinc, ) was filtered, and carefully treated with solution of
crystallized nitrate of silver, until no farther precipitate took place. The new com-
pound thus produced was then thrown on a filter, washed, dried at about 212° Faht.
and weighed. In one experiment thus conducted, 1.41 grain of Howard’s fulminat-
ing silver, afforded 0.87 grain of the new fulminating silyer ; but a portion could not
be separated from the filter, which being dried, broken in pieces and heated, afforded
successive explosions. The loss thus sustained, may be estimated at 0.03 grain, which
being added to the 0.87 make 0.90. Making these results the basis of calculation,
100 grains of Howard’s compound would afford about 63.8 grains of the new com-
pound; for 1.41 : 0.90 :: 100: 63.8 nearly. Now, according to my analysis,*
100 grains of Howard’s compound, contain 26.25 grains of fulminic acid, and
63.8 : 26.25 :: 100 : 41.14 nearly. Hence, 100 grains of the new compound would
consist of
58.86 oxide of silver
41.14 fulminic acid
100.00
and taking from my experiments the proportional number of fulminic acid as 42.
hydrogene being unity : the new compound would consist of two proportions of ful-
minic acid 2 x 42=84 and one proportion of oxide of silver 118, for
58.86 : 41.14 :: 118 : 83.7
In a second experiment, conducted like the first, 5.45 grains of Howard’s compound,
* On a new acid and its combinations. rans. Royal Dublin Society, 1829.
268 Professor Davy’s Account of
were converted into the new compound ; but only 3.02 grains of this substance could
be collected from the filter, instead of about 3.48 grains: the loss on the filter, how-
ever, was nearly equivalent to the deficiency, as was ascertained by comparing the
weight of the dried filter, with another, equal in every respect, and placed under
sunilar circumstances.*
The preceding experiments afford the nearest approximations I could obtain, to the
composition of the new compound, by that mode of operating. The difficulty of
gaining uniform results, by converting Howard’s compound into the new compound,
arises partly from the facility with which the new compound, when first formed, is
redissolved by an excess of nitrate of silver ; and partly from its being to a certain
limited extent, either dissolved by water, or partially decomposed by it.
In converting the new compound into chloride of silver, a given weight of it, (well
dried, ) was put into a platina crucible ; a little pure water was added, and then some
pure muriatic acid of moderate strength ; a considerable action took place, hydrocy-
anic acid appeared to be evolved, and after digestion for a short time, water was put
into a crucible, the chloride was thrown on a filter, washed, dried, collected and fused
in a platina capsule, previously counterpoised in a very delicate balance. Operating in
this manner, I found, that on fusing the chloride in the capsule it was of a dark colour,
having some small specks in it like charcoal. On exposing the capsule to a full red heat,
the dark coloured chloride seemed to undergo ebullition ; it evolved gas, and gradually
assumed a yellowish white colour. It now lost no weight on being heated to redness,
and no part of it appeared to be reduced. In one experiment, thus conducted, 6
grains of the new compound afforded 4.70 grains, and in another experiment, 6 grains
yielded 4.82 grains of fused chloride of silver. Now, if the mean of these experi-
ments is taken, 6 grains will afford 4.76 grains, and 100 grains of the new com-
pound, will yield 79.33 grains of chloride, equivalent to 54,73 grains of metallic
silver, or 58.44 grains of oxide of silver. These results, so nearly correspond with
those derived from a different method of examination already noticed, that [ venture
to regard the new fulminating silver, as a compound of one proportion of oxide of
silver 118. and two proportions of fulminic acid 84, or of
58.42 oxide of silver
41.58 fulminic acid
100.00
I made a number of comparative experiments on the two fulminating silvers, all of
* I have since found, that in decomposing Howard’s compound, by Zine, &c. another compound of sil-
ver is formed. It is of a dull white colour, but changes on exposure to light. It does not explode by heat,
but ignites and burns for an instant, leaving a brown substance, which by a strong heat is converted
into silver. When decomposed over mercury, the products were oxide of silver, carbonic acid gas, and
carbonate of ammonia. This substance may partly account far the loss in the above experiments ;
but its quantity seems too minute to affect the accuracy of the statements made.
A new Fulminating Silver. 269
which tended to prove that they are different compounds. ‘Thus, | found that the
new compound, exploded when heated from about 350° to 370° of Faht.; whilst
Howard’s compound required an increase of 100° or from about 450° to 470° Faht.
to explode it. Howard’s compound readily dissolves in boiling water, as M. Liebig
observed, but the white silky crystals are rapidly deposited as the solution cools, and
very little remains in solution at the temperature of the air, The new compound
is nearly insoluble in boiling water; I could not dissolve half agrain of it in several
ounces of water, kept boiling for some time in a platina crucible. Even the small
portion that dissolves is found to have acquired new properties, for on being collected
and dried, it will not explode in chlorine, and is probably by a loss of acid, converted
into Howard’s compound.
Howard’s compound, however carefully prepared and dried ; whether in crystals,
or an impalpable powder, does not spontaneously explode in chlorine gas, or in mix-
tures of this gas with other gases. | Whereas, the new compound, whether dried at
60° or at 212° and even before it becomes pulverulent from loss of moisture, readily
explodes under such circumstances.
According to my experiments, Howard’s compound contains one proportion of
fulminic acid, and one proportion of oxide of silver; and it has been called fulminate
of silver: as the new compound appears to contain two proportions of the acid, and
one of the oxide, it is properly a bi-fulminate of silver.
5. Experiments on the application of the new Fulminating Silver, as a test for
Chlorine, §c.
The chlorine gas employed in the following experiments, was generally made in the
usual way, by adding diluted sulphuric acid to a mixture of common salt, and black
oxide of manganese. Occasionally, it was procured from muriatic acid and the same
oxide. Sometimes the gas was received over water, at other times it was collected in dry
bottles furnished with ground stoppers. ‘The new fulminating silver, or test, as I shall
now, for convenience, call it, was made by decomposing fulminating zinc by nitrate of
silver, as has been stated. It was sometimes dried on a sand bath at a temperature not
exceeding 212° Faht., and sometimes in the open air. It was always, however, suffi-
ciently dry to be pulverulent. It was commonly used in very minute quantities ; a simple
grain serving, on an average, for upwards of fifty,and occasionally for about one hun-
dred separate experiments. It indicated the presence of chlorine, by instantly ex-
ploding when brought in contact with this gas, and also with other gaseous mixtures
in general, which contained chlorine.
1. The test readily and repeatedly exploded when put into bottles of very impure
chlorine gas, made a long time and haying no vestige of colour; and also into bottles
270 Professor Davy’s Account of
of the gas containing about one-tenth their bulk of water, which had been exposed
last summer, during several weeks, in the open air, to light and the sun’s rays.
2. Water recently saturated with chlorine in one bottle, was partly transferred to
another bottle. On agitating the fluids in both bottles, and introducing the test, it
instantly exploded. Similar experiments were made with a saturated solution of the
gas which had been excluded from light for a month, and the results were precisely
similar.
3. Mixtures, both of pure and impure chlorine and hydrogene, being treated with
the test, instant explosions took place, and the production of muriatic acid gas.
4. On adding the tenth of a cubic inch of nitric acid to an equal bulk of muriatic
acid, chlorine gas was presently evolved ; and the test, on being applied, repeatedly
exploded. In cases where those acids were strong, and had acted for a short time on
each other, it was only necessary to put the test on a slip of platina, and bring it to
the mouth of the glass containing the acids, when an explosion took place, and the effect
was produced several times. I found, indeed, that two drops of strong nitric, and
one of muriatic acid, put into the same glass, produced a sufficient quantity of
chlorine to explode the test several times.
5. Nitric acid being added to common salt, both in its usual state of dryness and
also after being fused, the gas evolved readily, and repeatedly exploded the test.
The results were precisely similar, when nitric acid was added to a number of dry
chlorides, as those of potassium, lime, iron, &c. and the test applied.
6. Strong sulphuric acid was added to chloride of lime made above twelve months
since; chlorine gas appeared to be evolved and the test exploded. Fluid chloride of
magnesia made some time before, being treated with sulphuric acid, gas was disen-
gaged which exploded the test. :
7. A number of experiments were made to ascertain whether the power of the test
would be injured or destroyed by diluting chlorine with other gases. Thus nitro-
gene, nitrous oxide, nitrous gas and carbonic oxide gases were mixed in different pro-
portions with chlorine, but in every instance the test exploded with great facility.
When olefiant and carburetted hydrogene gases, were separately mixed with certain
proportions of chlorine, the test simply exploded when brought in contact with the
mixtures ; but when the same gases were severally mixed with chlorine in the pro-
portions of about equal volumes of each, and the test applied, the mixed gases in-
flamed the instant the test exploded, and the interior of the tube was completely
blackened from the deposition of carbon.
8. Different acid gases, as muriatic acid gas, nitrous acid gas, and carbonic acid gas,
were separately mixed with chlorine gas, without any regard to proportions; but this
circumstance neither prevented, or retarded in any degree, the action of the test,
which instantly and repeatedly exploded on being dropped into the respective mix-
tures. Even gases which are known to be rapidly acted on by chlorine, as sulphu-
A new Fulminating Silver. Q71
retted by hydrogene and sulphureous acid gas, did not hinder the action of the test
provided there was the slightest excess of chlorine present.
9. Melted sulphur, and phosphorus were put into bottles of chlorine, and after the
respective chlorides were formed, the test being repeatedly introduced into the bottles,
in every instance, it exploded.
10. The protoxide of chlorine was collected in an open tube, and a bit of the test
introduced ; a double explosion took place; first of the test and then of the gas.
The test being now added, it exploded as in a mixture of chlorine and oxygene. The
peroxide of chlorine being treated with the test, a much louder explosion took place
than in the foregoing experiment.
11. A number of experiments were made to ascertain whether the vapours of dif-
ferent fluids diffused through chlorine gas would prevent the usual action of the test.
The chlorine gas was agitated in contact with water heated nearly to the boiling point,
and whilst the hot vapour issued from the bottle, the test was several times applied,
and in every trial it instantly exploded.
12. Sulphuric ether was put into a bottle of chlorine gas, and agitated ; a bit of
the test being now added, it exploded; inflammation took place, and carbonaceous
matter was deposited on the sides of the bottle. A similar experiment being made
with alcohol, the test exploded several times.
13. A few drops of oil of turpentine and of naphtha were separately put into
phials of chlorine gas, a rapid action of course took place, and much heat was pro-
duced. ‘The test, on being instantly applied, exploded in both phials; but if a short
interval was suffered to elapse, and the fluids were agitated, the chlorine was all ab-
sorbed and no effect was produced on the test.
14. Strong muriatic, nitric, acetic, and hydrocyanic acids, were separately put into
phials of chlorine gas and agitated ; the test being repeatedly applied to all the phials,
exploded in every instance. ‘The test did not explode in any of those acids, or their
Tespective vapours: nor in aqua regia, but when the deep orange aqua regia was agi-
tated, a compound of chlorine was evolved from it which readily exploded the test.
15. Well stopped bottles containing Thomson’s chloro-chromic acid, and chloride
of sulphur, which had been made a considerable time, were opened, and the test being
applied, it exploded repeatedly before it reached the fluids. In both fluids there
seemed to be a partial decomposition. ‘The chloro-chromic acid had a very strong
odour of chlorine, and the gas in the bottle continued for some time to explode the
test. On opening the bottle of chloride of sulphur, a quantity of the vapour and
some of the fluid were forcibly expelled.
16. Being desirous of ascertaining to what extent common air might be mixed with
chlorine gas, without impairing the action of the test, I made a number of experi-
ments, using different proportions of chlorine and common air ; in all of which the
test readily exploded. I then collected in a cubic-inch measure divided into 100
VOL. XVII. 31
272 Professor Davy’s Account of
parts, 4, of chlorine gas, which, by absorption was reduced to about 5, the remain-
2 of the measure was now filled with common air, and being mixed, a bit of the
INS so
test was dropped into the tube, when it readily exploded.
The foregoing experiments seem to prove that the new fulminating silver is a very
delicate test of the presence of chlorine gas, nor does its delicacy appear to be im-
paired by exposure to the air, the light, or the sun’s rays. Thus, some of the com-
pound made last spring, was exposed on the sand bath during the summer months ;
part of it became of a dark brown, and part of a black colour. Some of the com-
pound was exposed to the direct agencies of the sun’s rays for some time, but the
changes of colour it thus underwent, did not prevent it from instantly exploding in
all cases in which it was put into chlorine gas, or into mixtures of this gas with other
gases. I may also remark, that on exposing some of the test to the action of boiling
water for some time, and then drying it, it exploded under the same circumstances as
before.
The properties which appear to be regarded as most characteristic of chlorine, are its
colour and odour. Though chlorine is easily recognised by its yellowish green colour ;
in cases when it is pure or nearly so, or when it forms the greater part of a gaseous
mixture on which it does not act; yet it may, as is well known, be present in conside-
rable quantity without exhibiting the least vestige of colour. Thus, in the common
modes of making the gas, a considerable quantity must be generated, before any co-
lour is apparent. And the purest chlorine, when mixed with a certain portion of
common air, or other gases, on which it exerts no immediate action, is no longer dis-
tinguishable by its colour. Whereas the new test readily detects chlorine in the first
bottle of air that comes over in the usual modes of making the gas; in cases when
the gas is mixed with 0.98 or 0.99 of common air, and also, when even a solution
of the gas in water is transferred from one bottle to another.
The odowr of chlorine, though perhaps sufficiently characteristic, when the gas is
mixed with other gases on which it exerts no action, or which have no powerful odour ;
yet it ceases to be so, when certain pungent gases or vapours are diffused through it.
Thus, when a portion of chlorine was mingled with muriatic acid, or nitrous acid gas,
or with the strongest liquid muriatic or nitric acid, the chlorine could not be satisfac-
torily distinguished ; but in every instance of the kind the test exploded with flame an
indefinite number of times. From a number of experiments I have made, I am dis-
posed to regard what is commonly called the odour of chlorine, as a vague, and by no
means a discriminative character, and that this odour exists in cases where we have no
evidence of the presence of chlorine, or where, according to received opinions, it
caunot exist. Thus, after exposing solutions of chlorine in water for several weeks
or even months to the action of the sun’s rays in summer, they are found still to have
a strong odour which has been, I think, erroneously referred to chlorine.
I have hitherto said little concerning the specific action of chlorine gas, on the new
A new Fulminating Silver. ; 273
fulminating silver. The experiments I have made throw some light on the subject,
while they afford additional proofs of the extreme delicacy of the test.
I filled a long narrow-necked matrass (smaller than a florence flask), with chlo-
rine gas ; placed it upright on a table, and successively dropped into it small portions
of the new fulminating silver, until the number of separate explosions exceeded six
hundred; when some fragments of the test on the table exploded, and the odour of
chlorine was perceived. On examining the matrass, it was found sufficiently cracked
.to admit the gas to escape. In the course of this experiment, the explosions were
uniformly accompanied with flame, and the appearance of a small dense white cloud.
These phenomena, at first, occurred in the neck of the matrass, and part of the cloud
sunk into the matrass, whilst the remainder rose into the air; but they took place,
lower and still lower in the glass, as the number of explosions increased. The in-
terior of the matrass was found covered with a finely divided dark purple substance,
which readily dissolved in ammonia ; and the solution treated with pure nitric acid in
slight excess, gave a white precipitate, which melted at a dull red heat, and was chlo-
ride of silver.
In another experiment, a dry half ounce phial, having a narrow mouth, was filled
with chlorine gas; small portions of the test were introduced, until one hundred
and ninety-nine explosions had taken place. A little pungent vapour arose from the
phial, and a peculiar odour was emitted, resembling that of chlorine in a state of
great dilution. ‘The vapour in the phial still possessed the property of bleaching ;
for it soon rendered moist litmus paper, white. A dark dove coloured substance re-
mained principally at the bottom of the phial. It was chloride of silver, and
there was a very minute portion of a crystallized substance attached to the upper
part of the phial, which exhibited the properties of sal-ammoniac.
In a third experiment, a long dry tube of about the capacity of two cubic inches,
was filled with chlorine gas, and the test was added until it ceased to explode. There
was now distinctly perceived an odour precisely similar to that of the compound
which is formed when fulminate of zinc is agitated in contact with chlorine gas. This
compound is a yellow, oily, volatile fluid, resembling azotane in appearance, but having
none of its explosive properties. Its odour is so acrid and peculiar that it can scarcely
be mistaken. Its taste is sweetish, and astringent, with a certain degree of pungency,
which remains for some time on the palate. It is apparently insoluble in water, but
readily forms a sort of saponaceous compound with ammonia. It does not imme-
diately redden litmus paper, but acquires this property after a short time. It is, I
presume, a compound of fulminic acid and chlorine. There appears to be another
compound of the same substances, but in different proportions. I obtained it by dis-
tillation after exposing the fulminating silver, and other analogous compounds, either
diffused in water or dissolved in it to the action of chlorine gas. It is a colourless, trans-
parent, and volatile fluid, haying a peculiar and disagreeable smell and a taste at first
274. Professor Davy’s Account of a new Fulminating Silver.
sweet, but which presently becomes sharp and enduring, somewhat resembling that of
cayenne pepper. It is soluble in water, has no bleaching, but some acid properties,
and may deserve farther examination.
It would seem from the foregoing results, that when the new fulminating silver is
exposed to the action of chlorine gas, it is decompounded, chloride of silver is formed ;
one part of the fulminic acid combines with that gas to form the peculiar compound
just referred to, whilst the other part is decomposed, and affords by the reunion of its
elements, sal-ammoniac. It seems probable, too, that carbonic acid gas and nitrogene,
are at the same time evolved.
The action of chlorine gas on the new fulminating silver, is uniformly accompanied
with flame ; this circumstance and the formation of ammonia above noticed, seem to
favour the opinion I have advanced in the papers already referred to, that hydrogene
enters into the composition of the fulminic acid.
The new fulminating silver, besides its use as a test for chlorine, might I think be
employed with advantage as a substitute for Howard’s fulminating mercury in the
caps for percussion locks, which are now so much approved, and getting into such ge-
neral use as threaten to supersede the common lock. The strong springs required in
the percussion locks in which Howard’s fulminating compound is used, are objection-
able. The new fulminating silver requires much less percussive force to explode it
than Howard's fulminating mercury ; nor is the effect of the explosion of the former,
accompanied with that loud, and almost deafening report, of the latter compound.
From the known analogies existing between chlorine and bromine, the vapour of
the latter might be expected to explode the test, as well as the former ; and this I find
is the case. Thus, on putting a few drops of bromine into a small stoppered bottle,
and dropping in a bit of the test; it immediately exploded in the vapour, and the ex-
periment was repeatedly tried at different intervals in the same bottle, with the same
result. The test does not explode when brought in contact with iodine, either at
the common temperature of the air, or when it is raised in vapour by heat.
On the Theory of the Moist-bulb Hygrometer. By JAMES APJOHN, Ese. M.D.
M.R.I.A., Professor of Chemistry in the Royal College of Surgeons.
Read November 24, 1834.
In the number of the Edinburgh Philosophical Journal just published, and which was
sent me by a friend, in consequence of its containing a report of the proceedings of
the late meeting of the British Association for the promotion of science, I find an
anonymous paper, entitled ‘‘ Observations on the Hygrometer,’”? which induces me
to take the earliest opportunity of submitting to the Academy the following remarks.
It is well known to Meteorologists that the Hygrometers of Saussure and De Luc,
and all others whose indications depend upon variations in the volume or weight of
the hygroscopic substance employed, have, as philosophical instruments, been in a
great measure discarded. The direct determination of the Dew-point, as it is techni-
cally called, is now almost universally practised, either according to the original
method of Dalton, or by means of the elegant instrument of Professor Daniell. The
method of the former, however, is much too tedious for practical purposes, and the
instrument of the latter, though in principle rigorously correct, requires, in order
to accuracy of result, a quickness of sight, and an adroitness in observation, which
few can boast of possessing. For the great purposes, therefore, of Meteorology, the
condensation Hygrometer must be considered as having, as yet, but imperfectly ful-
filled the high expectations which were entertained of it.
The Dew-point process was preceded by one of an analogous description, which is
said to have been first practised by Hutton the Geologist. If the bulb of a thermo-
meter be kept covered with a thin film of water, its temperature will sink beneath that
of the atmosphere, (this latter being supposed unsaturated with moisture, ) the maxi-
mum depression being attained when the heat received in a given time by the film
from the contiguous air is exactly equal to the caloric of elasticity of the water eva-
VOL. XVII. 3k
276 Professor Apsoun on the Theory
porated ; and it is obvious that the amount of this depression will bear some direct
ratio to the degree of dryness of the air at the time of the experiment. Further than
this, however, the wet-bulb Hygrometer does not go. It affords us, for example, no
information as to the exact quantity of moisture present in the surrounding atmo-
sphere at the time of observation, because it does not indicate the position of the Dew-
point. That there existed, however, between this latter and its indications, some
necessary connexion, so that the one might be inferred by calculation from the other,
must have been very early suspected; but Sir John Leslie, after having converted
his differential thermometer into an hygrometer, was the first who attempted to point
it out. In this attempt, I believe I may venture to say, he was but partially success-
ful. I have never seen any very explicit statement of the principle on which he pro-
ceeded ; but the table which he published in 1820, in his “ Description of Meteoro-
logical Instruments,” is undoubtedly erroneous.
In a very able article on Hygrometry in Brewster’s Encyclopedia, a formula for
the solution of this question is elaborately investigated ; but as it is in greater part
tentative, assumes, contrary to the fact, that the amount of moisture which air is
capable of taking up is influenced by the pressure; and, as lastly, it does not very
well accord with observations—at least some which I have made—it has been
adopted, I believe, but by few Meteorologists. A satisfactory solution of this problem
was still viewed as a desideratum, and of this no better proof can be given than that
the “theory of the moist-bulb Hygrometer” is found among the questions submitted
by the first meeting of the British Association, held at York, to the renewed consider-
ation of philosophers.
About four months since, on turning the matter in my mind, it occurred to me
that the relation between the Dew-point and the temperature of a thermometer with
moistened bulb might be made matter of calculation, and deduced from the theory of
mixed gases and vapours—a theory which the labours of Dalton and G. Lussac have
rendered as complete as any other in Physics. It was not, however, until the latter
end of August that I was enabled to return to the subject, when I succeeded in
arriving, by a direct rout, at a formula derived exclusively from experimental data,
and which represents with unexpected accuracy, the best observations with which I
have been able to compare it. To this formula (to explain, at length, the object of
this hurried communication,) I am anxious to draw, without further delay, the atten-
tion of men of science, as there is a brief notice of one for accomplishing the same
object in the article of Professor Jamieson’s Journal already alluded to. Of this
latter, which is said to belong to Mr. Ivory, I beg to say I was altogether ignorant ;
nor haye I yet been able to refer to the number of the Philosophical Magazine in
which it is stated to have been originally published.
of the Moist-bulb Hygrometer. 277
Having disposed of these preliminary observations, I shall now proceed to explain
the principle of my method of investigation ; but before doing so, I wish to observe
that I would have submitted my formula to the Committee of the British Association
in Edinburgh, charged with the subject of Meteorology, but that, having fallen upon
it during the vacation, while sojourning in the south of Ireland, I was not able, prior
to the meeting, to institute test observations myself, nor had I access to books, so as
to compare it with the recorded experiments of others, on the temperature of the
moist-bulb Hygrometer and the corresponding Dew-point.
When in the moist-bulb Hygrometer the stationary temperature is attained, the
caloric which vaporizes the water is necessarily exactly equal to that which the air
imparts in descending from the temperature of the atmosphere to that of the moistened
bulb; and the air which has undergone this reduction becomes saturated with
moisture. Now from these facts, and the known specific heat of air, we can calculate
the weight of water m which would be converted into vapour by the heat which a
given weight of air would evolve in cooling from ¢ the temperature of the atmosphere
to ¢ that of the moistened bulb; and we can also calculate the total quantity of
moisture m which the same weight of air would contain at ¢ if saturated. This being
accomplished, if f’ be the tension of vapour at the temperature ¢ (1 —=) f= Fe
the tension of aqueous vapour at ¢” the Dew-point. Hence, by looking in Dalton’s
table for f”, the Dew-point is found in the opposite column.
The value of the’ expression (1 _= ) f’ may be found in the following manner :—
1 being the specific heat of water, .267 (De la Roche and Berard) is that of air. Also
967° being the caloric of elasticity of steam at 212°, 212—50+967=1129° will be
its caloric of elasticity at 50°, assuming, as is generally done, that the sum of the
sensible and latent heats of vapour is the same at every temperature. One grain of
air, therefore, in cooling through any number of degrees d, will raise the temperature
of .267 grains of water through the same number, and will consequently be adequate
: 267d d : ey os
to vaporize a quantity of water represented by Sy55= 4195 grains ; or, multiplying by
the denominator, 4195 grains of air, in cooling through d degrees, give out the
exact quantity of heat which constitutes the caloric of elasticity of d grains of
vapour. But the volume of this weight of air, at 60°, and under a pressure of
30, is 13754 cubic inches; and, at the temperature ¢ and pressure p, 13754
Baste 30. 4484¢ ... 44847 Ee
508° 5 =810 cubic inches. Hence Cin, x (10,583 x MASTe
* This expression is obviously deduced from the fact of the tension, of vapour at a given temperature
and under a given volume, being proportional to the quantity or specific gravity.
278 Professor Apsoun on the Theory
x, 805)* = 2613.88 xt =the quantity of moisture which the air contains when satu-
ried at ¢. We will therefore have, on the principle already explained (1 a 4
) x J =f'- ee =f" the tension of vapour at the dew-point. If p = 30,
ri = <= =f".
I shall now proceed to state, and subsequently observe upon, the objections which
may be made to the method of investigation I have pursued. It may be said—
1°.. That the air which is cooled by contact with the moistened bulb at the sta-
tionary temperature, is assumed, without proof, to be saturated with moisture.
2°. That the caloric of elasticity of steam is 1129 only at 50°.
3°. That the specific heat of air is .267 only under a pressure of 30.
4°. That the medium which is cooled from ¢ to ¢ is not pure air, but a mixed at-
mosphere of air and vapour; and
5°. That the caloric, which at the temperature ¢ converts the water into vapour,
is not derived exclusively from the air by contact, but partly also by radiation from
surrounding bodies.
With respect to the first objection, I have only to observe, that air is an extremely
bad conductor of heat, and that it is, therefore, very unlikely that the reduction of
temperature which it experiences in the experiment in question can be effected in‘any
other way than by actual contact with the moistened bulb. But, if such contact be
established in the case of every indefinitely thin aerial shell, there can, I conceive, be
no doubt but that each becomes charged with the full amount of moisture which be-
longs to its reduced temperature.
In reference to the second objection, it must of course be admitted that the caloric
of elasticity of vapour varies with the temperature, and that it is represented by the
number 1129 only at the temperature of 50°, a point chosen by me as being nearly
the mean temperature of Dublin. In strictness, the number employed should be
967 +212—#', but it would be easy to shew that the uniform use of 1129 cannot give
rise to any material error.
The third objection is usually considered as one of considerable weight. The spe-
cific heat of air varies with the pressure, and in order to accuracy of result, a proper
correction must undoubtedly be made for this variation. But what is the law which
it observes? Upon this point, different opinions would appear to be entertained.
According, however, to De la Roche and Berard, (whose views, if not rigorously
exact, are at least sufficiently so for my present purpose,) for small variations of
* 10,583 x ro x ,305=the weight of a cubic inch of vapour whose tension is f’ and temperature ¢.
of the Moist-bulb Hygrometer. 279
pressure, such as occur to the natural atmosphere, the differences of specific heats
under a constant volume, are proportional to the differences of pressure. And the
same philosophers have shewn, that for pressures in the ratio of 1 to 1.3583, the cor-
responding ee are 1 and 1.2396. Hence, as
3583 : .2396 : & —1 2 —1, ¢ being the specific heat under a constant vo-
lume at 30, and x that at p,—a proportion from which we deduce
a = (.0223 p + 8312) c.
But the specific heats under a constant volume, divided by the densities, give the
specific heats of equal weights. And as the densities vary as the pressures directly,
and as the temperatures + 448 inversely, and are, therefore, to each other in the
30 -
present case as 77577Gy to ee , we will have
aan M8 te
: (.0223 p + .8312)c¢ cb ee
So that 2’, or the specific heat of air at temperature ¢ and pressure p, =
ex (.0228 p + .3812) x .267.
508
The erie therefore, of f' already given, when subjected to this correction, will
d A48+¢ |
become f" — 5, x —5g- x (.0223 p + 3312).
The equation for this correction, given by the writer in the Edinburgh Philosophi-
cal Journal, is exclusively a function of p; but if the method here explained be cor-
rect, and I believe it will be found so, the temperature ¢ of the Hygrometer has a
still greater influence on its amount. They both, however, affect it in the same
direction ; i.e. as they rise, it increases; and as they fall, it diminishes: so that if
the one should augment as the other diminishes, they will counteract, to some
extent, each other’s effects. When ¢’=50, and p= 29,
ape =f —£ x.958,
i.e. the value of the subtractive quantity is diminished by its ath part; but if ¢ being
still 50, p be supposed 31,
Ti ar x 1.008,
or the subtractive term is augmented we by its anth part,—facts from which the ge-
neral conclusion may be drawn, that when 60—¢ =30—p, the latter difference being
measured in tenths of an inch, and that they have opposite signs, the correction may
be altogether neglected.
The theoretical justness of the fourth objection must also be conceded. The
medium which is in contact with the bulb of the Hygrometer is not dry air, but air
280 Professor Arpsoun on the Theory
charged with the amount of vapour which belongs to the existing dew-point ; and as
the specific heats of air and vapour are different, this mixed atmosphere, in cooling
through @ —?¢°, will evidently not give out the same quantity of caloric, and can
therefore not convert into vapour the same quantity of water that would be cooled
and vaporized by the same weight of dry air alone. In fact, for .267, the specific
heat of air, we should in strictness use the specific heat of the mixture of air and
vapour ; or, what will answer the same purpose, multiply by the ratio of these, the
value of the quantity to be deducted from f’, already obtained. Now, to determine
the specific heat of the mixture, the relative weights of its constituents should be mul-
tiplied by their respective capacities, and the sum of the products divided by the sum
of the weights. But the weights, being obviously as the specific gravities, are to each
wr
other as 1 : .625 ce Also, the specific heat of air being .267, and that of vapour
.847, the former is to the latter as 1: 3.172. Hence, according to the rule given
above, we will have
14 6254" 3.172
(AS ees
1 + 625
P
for the specific heat of the mixture of air and vapour referred to that of dry air taken
as unity ; and, applying the correction as already explained, we will have an equation
in which f” is the only unknown quantity, and from which, therefore, its value may
be found. ‘The equation, however, being a quadratic, and the unknown quantity in
its first dimension having a coefficient of three terms, its solution would involve
tedious arithmetical operations, and can therefore not be recommended as a ready
means of making the correction in question. Nor is such course at all necessary, for
the same object may be achieved, and with a sufficient precision, by either assigning
to f” an average value, or by deducing approximately the tension of vapour at the
dew-point by the formula f"”= f’ — & x = (.0223 p + .3312), and using the
value of f” thus obtained, in order to determine that of
1+ .625 Sx 3.172
P
1 + 625 £
P
the specific heat of the mixture of air and vapour. The latter method is decidedly
the best ; and though not mathematically accurate, will not I believe exhibit a devia-
tion from the truth until the calculation is pushed to the seventh or eighth decimal
place.
of the Moist-bulb Hygrometer. 281
I have now to notice the last circumstance which, as far as I understand the sub-
ject, can have any influence upon the accuracy of my determination of the Dew-
point.
When the wet-bulb Hygrometer has attained its stationary temperature, the caloric
which it loses and gains in a given time are perfectly equal. This requires no demon-
stration. ‘The caloric lost also is entirely employed in converting the water into
vapour ; but the whole of the acquired caloric is not necessarily derived, although such is
assumed to be the case, from the air cooled by contact with the bulb of the instrument.
In fact, the Hygrometer is in the predicament of a cool body placed in a warm
medium, and it must consequently receive from surrounding bodies, by radiation, a
greater amount of caloric than it imparts to them in virtue of the same process. ‘To
the d grains, therefore, of moisture converted into vapour by the heat given out by
4195 grains of air, in cooling through d degrees, we should add the additional quan-
tity vaporized by the heat which the bulb has in the same time received by radiation.
When ¢—? is small, this quantity may probably be safely neglected; but it will
sometimes, I make no doubt, be of sufficient magnitude to exercise an appreciable
influence. I regret my inability to assign any means of determining its amount ;
and shall merely add, that the neglect of this correction will always tend to render the
calculated Dew-point somewhat higher than the true.
Having disposed of the theory of my method, I shall now conclude by subjecting
the results which it affords to the test of experiment. I shall not at present refer to
my own observations, though I have of late amassed a considerable number on the
Hygrometer and Dew-point. As a more unimpeachable criterion, I shall compare
my formula with the observations of others, and shall select for this purpose, it being
the nearest at hand, a table published in the last number of the Edinburgh Journal.
The differences, it will be seen, between the corresponding numbers of the fourth and
fifth* columns of this table are so small, that we may consider them as almost entirely
due to errors of observation. I may add, that as in the original table there is no
notice taken of the barometer, the formula, in its most complete form, could not be
applied; so that a perfect coincidence between calculation and observation was not in
this instance to be expected.
* The numbers in the fourth column are the observed, and in the fifth the calculated dew-points.
Professor Apyoun on the Theory, Sc.
Eo ge
SS SS = OES SSS ee
t’
Observed.
57.25
53.25
54.5
55.75
56.75
56.25
55.25
57.25
56.5
58.25
56.25
54.75
54.5
56.25
57.25
57.5
57.5
54,5
54,25
t!
Calculated,
ue
Observed,
53.75
50
53.75
52.25
49,25
51.25
53.25
53.25
53.75
55.75
58.25
Calculated.
t’
54,4
49,9
53.1
52.5
50.5
51.25
52.75
53.25
54
57.1
58.5
53.2
Ds
53.3
50.33
50.33
50.25
53.66
59
54.74
On the Theory of the Moist-bulb Hygrometer. By JAMES APJOHN, Esa. M.D.
M.R.I.A., Professor of Chemistry in the Royal College of Surgeons.
(Continued. )
Read April 27, 1335.
Ar the meeting of the Academy held in November last, I was permitted to read a
short memoir on the subject of a formula, at which I had a considerable time pre-
viously arrived, for inferring the Dew-point from the indications of the Moist-bulb
Hygrometer. This formula was deduced altogether from general considerations ;
and, though satisfied, from some hasty observations of my own, that it represented
facts with considerable accuracy, I was not, at the time, in possession of evidence
which could be considered as establishing this important point in an unequivocal
manner. ‘The table which is subjoined to my paper undoubtedly shows, that, within
certain limits, my formula is in accordance with experiment ; but the observed de-
pressions in the‘table are, generally speaking, so small, that a formula in itself incor-
rect might, it must be admitted, yield results which would deviate from the observed
dew-points by quantities not exceeding the possible errors of observation. Berzelius,
for example, states (Traité de Chimie, tom. viii. p. 254,) that, from the experiments
of August, Bohnenberger, and others, it appeared that the temperature of a thermo-
meter with moistened bulb was an arithmetic mean between that of the air and the
dew-point ; and this rule, which would make ¢’=2# —?, though utterly erroneous,
would apply to the table appended to my paper, nearly as well as the formula I have
deduced. The validity, therefore, of my method required to be more rigorously
tested ; and having been for some time engaged in experimental researches, instituted
with this object, which have led to interesting, and to me most satisfactory results, I
am anxious to submit them, with as little delay as possible, to the judgment of the
Academy.
VOL. XVII. oi
284 Professor Arsoun on the Theory
The equation which, as I believe, comprehends the theory of the wet-bulb hygro-
meter, is as follows :-—
f=. —mdx
in which f” is the tension of steam at the dew-point, /” its tension at the temperature
of the hygrometer, d the depression or difference between the temperature of the hy-
grometer and air, p the existing, and 30 the mean pressure, and m a coefficient de-
pending upon the specific heat of air and the caloric of elasticity of its peeuded
vapour, its arithmetical value being .01149, or the equivalent vulgar fr action 2
In the paper to which I have already referred, corrections are given for the reflnesec
on the specific heat of air, of the fluctuations of the barometer, and the moisture
present in the atmosphere. These corrections are, I believe, deduced from correct
principles, and should be resorted to when extreme precision is desirable. Expe-
rience, however, and a careful consideration of the subject, have satisfied me that they
are, generally speaking, in their effects, much too insignificant to be objects of atten-
tion to the practical meteorologist.
The first and most obvious method of verification which presented itself to my
mind, was the comparison of my formula with recorded cotemporaneous observations
on the temperature of air, that shown by a moist-bulb hygrometer, and the actual
dew-point. I have, however, unfortunately been able to meet but few at all suited
to my purpose. Those in which ¢—?' is small, and this is generally the case in the
few registers to which I have had access, cannot, as we have already seen, serve for
deciding the value of any formula. In the first report, indeed, of the British Asso-
ciation for the Promotion of Science, page 50, mention is made of a register of obser-
vations kept in the East Indies, which, as belonging to high temperatures, would
necessarily exhibit great depressions, and would therefore be valuable as a standard
of comparison ; but I have in vain searched for the Calcutta Journal, ‘* Gleanings in
Science,” in which they are said to be contained. In fact, the only observations I
have been able to procure, adapted to my purpose, and made, apparently at least,
with the necessary precision, are those adduced in the article ‘ Hygrometry’ of Sir
_ David Brewster’s Encyclopedia, and there made, by the author of the article, the
basis of a calculation for investigating the constants of a tentative formula for con-
necting the indications of the wet-bulb hygrometer with the dew-point. They are but
two in number, and are comprehended in the following table, in which the numbers
in the first column represent the temperatures of air, those in the second the corres-
ponding indications of the hygrometer, those in the third the depressions, those in the
of the Moist-bulb Hygrometer. 285
fourth the pressures, and those in the fifth the dew-points experimentally determined
by the method of Dalton :—
(1) (2) (3) (4) (5) (6)
t t Pp d p t’ ob. t’ cale.
G72 << Oe eemlOie weeded Ot moDal ss. JOOkO
56.4555. NAS oe Ss C.VNIK. feS0.02) ANSeS .2i09 4254
The numbers in column (6) are the dew-points calculated by my formula; and
while there is an almost exact correspondence between the first and the result of ex-
periment, the second, it will be seen, is higher than the observed temperature of
deposition by nearly three degrees. There is here, however, obviously some mistake.
It is impossible that, with the recorded temperatures of air and hygrometer, the dew-
point could have been so low ; and this conclusion I do Pot at present draw from my
theoretical views, for that would be to subject myself to the imputation of arguing in
a circle ; but from the following observation, made by me with great care on the 22d
of March :—
d t’ ob.
56 50 6 44
Here the temperatures ¢ and ¢ differ from those taken from the Encyclopedia only
by about half a degree ; and nevertheless the observed dew-point 44 is higher than
39.5 by 4.5 degrees. [rom these observations, therefore, I am, I conceive, entitled
to conclude—Ist, that the series in which the depression amounts to 15°.2, being
in exact accordance with my formula, lends it some degree of support; and 2dly, that
my method cannot be considered as impugned by the other’series, inasmuch as this is
in some particular manifestly incorrect. But it is time to enter upon the experimen-
tal tests to which I have resorted.
If air, in reference to which ¢, ¢ and ¢” have been accurately noted, be raised to
any elevated temperature, and the observation be repeated in the heated air, as far as
respects ¢ and ¢’, we will have two* separate sets of observations, from which to cal-
culate the point of deposition ; and as the amount of moisture in the air is not altered
by the augmentation of temperature it has experienced, both calculations, provided
our formula be correct, should give precisely the same result; 7. e. the dew-point in
the first instance determined by observation. Such is the principle of the test expe-
riments which I first performed. ‘The air- was heated by urging it in a continued
stream, by means of a double bellows, through the worm of a small still—such as are
for sale in the Opticians’ shops—the worm-tub being filled with water of the desired
temperature ; and, in order to the necessary observations, in a glass tube connected
by a cork with the upper extremity of the worm, a couple of small thermometers were
* Any number of observations, having reference to the same dew-point, may, it is obvious, be thus obtained.
286 Professor AryJoun on the Theory
placed, their bulbs being separated by about a quarter of an inch, and that of the in-
strument occupying the higher position being invested with a tunic of muslin kept
constantly moist with water. The blast was steadily maintained until the thermome- .
ters ceased to rise, and the temperature of each was then accurately noted, the eye
being assisted by a lens. Tables 1, 2, 3, and 4 exhibit the results of four distinct
series of experiments thus conducted :—
TABLE 1. TaBLe 2,
February 8, 1835, 1] o'clock, a.m. February 9, 1835: 11 o'clock, a.m.
Bi eels din | Bp t’ ob. | t” cale. Diff. t wv d P t’ ob. | t” cale. | Diff.
1 | 49.6 | 44.7} 4.9] 29.6 | 40 | 39.02 | —.98 | 1 | 47.2] 42.5| 4.7] 30.02] 38 | 36.58 | —1.42)
2 | 88.5 | 62 | 26.5] 29.6 | 40 |*39.18 | —.82 ] 2 | 76 57.5 | 18.5 | 30.02} 38 | 40.44 | +2.44)
3 | 80.5 | 59 21.5] 29.6 | 40 | 39.27 | —.73
Taste 3. ¢ TABLE 4.
March 4, 1835: 11 o’clock, a.m. March 25, 1835: 11 o’clock, a.m.
t tv d P t’ ob. | ¢” cale. Diff. t t d Pp t’ ob. | t' cale.| Diff.
48.3 | 43 5.3 | 29.76 | 37.5 | 36.41 | —1.09] 1 | 51.3] 45.5] 5.8] 30.7 | 38.5-\ 38.61] +.11 |
96 64 | 32 29.76 | 37.5 | 36.37 | —1.13 2 82 59 23 30.7 | 38.5 | 36.7 | —1.8 |
91 | 62.5| 28.5 | 29.76 | 37.5 | 37.66 | + .16
75 |56 |18 | 29.76 | 37.5 | 38.26 | + .76
The results exhibited in the preceding tables will, I believe, be considered by many
as going far towards establishing the accuracy of my theoretical views. Although the
depressions vary from 4°.7 to 28°.5, the differences between the observed dew-points and
Rone
those deduced from the formula, are certainly not greater than what may fairly be
ascribed to unavoidable inaccuracy of observation. But for the purpose of putting
this matter in a still clearer point of view, I have calculated a number of values of m,
the constant of our formula, from the preceding observations. This was easily done ;
for as all the observations in the same table refer to air in the same Beer a state,
each series should give. the same dew-point, and the pan i f'-md x = must
have in reference to them a constant value. f’ —md xz = o for one must therefore be
qeval to F’ —mDxk for any other—an equation hea which we deduce m=
D = nies The si icaiita of this method gives us the following values of m :—
TABLE I. TABLE 2.
(land 2) (J and 3) (2and 3) (1 and 2)
m = ,01155 ak -01185 mae -01075 m = .01489
TABLE 3.
(land 2) (land3) (land 4) (2 and 3) (2and4) (Sand 4)
m= O1879, om O88 7%...» GOL809s...., 00825), ... .00976 ..., .01045
TaBLe 4.
(1 and 2)
m = .00967
of the Moist-bulb Hygrometer. 287
If the mean of all these values of m be taken, it will be found to be .01122, or the
1
equivalent vulgar fraction = an approximation to the coefficient = employed in the
formula, which, under all the circumstances, cannot but be considered as remarkably
close. Indeed the difference, which is less than 3 in the fourth place of decimals, is
so small, that°they may be substituted indiscriminately for each other without the
occurrence, at least in ordinary cases, of sensible error. Had values of m been cal-
culated from the comparison alone of the first series of observations in each table with
the subsequent ones, the mean, it is worthy of remark, would be .01156, or almost
exactly a3 and as, for such observations, F” —f’ and D—d are necessarily greatest,
they are best calculated to afford correct results, since any error of experiment would
obviously, in their case, exercise the least influence.
The next test experiments performed were suggested by the formula itself. If
f'=f'-Z x = and f” be supposed equal to 0, a condition which can only be
fulfilled in perfectly dry air, f’= a a an equation from which we deduce
a 37 fx rh Hence, by determining experimentally the depression of the hygro-
meter in perfectly dry air, we will be able to pronounce upon the validity of the
general method under discussion.
The first attempts for determining values of d experimentally, consisted in suspend-
ing a pair of thermometers, one of which had its bulb moistened, in a close corked
bottle, the bottom of which was covered with a stratum of oil of vitriol; but this
method was soon abandoned, as the depressions it afforded were, on an average, one-
fifth less than they should be according to the formula. In fact the extreme depres-
sion could not be expected here, for it is obvious that the air, in contact with the bulb
of the moist thermometer, is never perfectly dry except at the very commencement of
the experiment.
The next contrivance to which I resorted was as follows. A bag of India rubber
cloth, furnished with a cap and stop-cock, was inflated by a bellows, and then con-
nected, by means of a caoutchane collar, to a glass tube, traversing a cork fitted to
the tubulure of the lower bottle of a Noothe’s apparatus. The middle bottle of the
apparatus was next filled, 2 rds, with oil of vitriol, and the pair of thermometers last
described being introduced into the axis of a small tube, perforating a cgrk fitted to
the upper opening of this bottle, a stream of air was forced by pressing on the caout-
chouc bag, through the oil of vitriol, and, of course, over the thermometers ; and as
soon as the instrument with moistened bulb ceased to fall, the temperatures of both
were noted. ‘The following table comprehends the results of five experiments thus
performed :—
288 Professor Apsoun on the Theory
t t Pp d ob. d cale. Diff.
March 11 | 48.5 | 31.5 | 29.37 | 17 17.4 — A
15 | 50.5 | 33.5 | 30.00 | 17 18.2 —1.2
20 | 54.5 | 35.5 | 30.25 | 19 19.4 — .4
21 | 57.5 | 37.5 | 30.27 | 20 20.8 — 8
22 | 54.5 | 36 30.35 | 18.5 19.7 —1.2
Now, as in all these instances, the observed depression differs from ‘the true ; this
difference, though small, being always on the same side, must be ascribed either to the
co-efficient m being assumed too great, or to the method of experiment employed not
being calculated to afford the extreme depression. That this latter was the real
cause of the discrepancy I was disposed to believe, from having observed that
when the hygrometer, in the course of an experiment, became stationary, it could be
made to sink a little further by pressing with great force upon the bag of air. In
fact, this observation rendered it probable that the tube, between the lower and middle
bottle of the Nooth, did not afford sufficient air-way; and that, therefore, there was
not a sufficient current from behind to propel forward, and immediately remove, from
contact with the moistened bulb, the air which had become saturated with its humi-
dity. To bring this conjecture to the criterion of experiment, it was obviously
necessary to operate so, that while the air underwent perfect desiccation, it was, at
the same time, made to pass over the thermometers in a strong and continuous
current ; and, after some trials, I found that both objects were secured by substi-
tuting for the Nooth a series of three Wolfe’s bottles, containing oil of vitriol, and
connected, as in the process for preparing the water of ammonia, by glass tubes and
caoutchouc collars, the bag of air being attached to a tube passing to the bottom of
the first bottle, and the thermometers being placed in the axis of a tube perforating
a cork inserted into one of the tubulures of the last bottle. ‘The experiments re-
corded in the following table were made with this apparatus.
t t P d ob. d cale. | Diff.
March 26 | 51 33.5 | 30.55 | 17.5 | 17.94 | +.44
27 | 53 34.5 | 80.385 | 18.5 | 17.73 | —.77
28 | 52 34 80.21 | 18 17.62 | —.38
29 | 51 33 30.05 | 18 17.97 | —.03
30 | 52 33.4 | 29.75 | 18.6 | 18.387 | —.23
31 | 53 84.3 | 29.50 | 18.7 | 19.14 | +.44
April 1 | 56.5| 35.8 | 29.70 | 20.7 | 20.04 | —.66
2 | 58 387 29.72 | 21 20.88 | —.12
° 3 | 58.2| 37 29.77 | 21.2 | 20.84 | —.36
4 | 58 37 30.03 | 21 20.68 | —.32
5 | 58 37 30.15 | 21 20.59 | —.41
6 | 59 37.5 | 30.25 | 21.5 | 20.88 | —.62
7 | 59 38 30.26 | 21 21.24 | +.24
8 | 61 38.7 | 30.21 | 22.3 | 21.80 | —.60
10 | 58.3| 387.7 | 30.85 | 20.6 | 20.96 | ++.36|
1l | 58 37.5 | 30.45 | 20.5 | 20.75 | +.35
12 | 56.3] 36.5 | 30.30 | 19.8 | 20.12 | +.32|
USF S725) enor 30.20 | 20.5 | 20.55 | +.05
14 | 57.5 | 87 30.15 | 20.5 | 20.59 | +.09
of the Moist-bulb Hygrometer. 289
Of the nineteen observations of depression in dry air registered in the preceding
table, eleven are greater, and eight less than the calculated results. ‘The mean of the
plus errors of the formula is, .28, and of the minus errors, .4 of a degree; so that
.28—.40= —.12 of a degree is the mean difference deducible from the whole be-
tween experiment and calculation. A closer approximation between them than this
could not, I think, be anticipated, even upon the hypothesis of the strict accuracy of
the formula. I may also observe that if by means of the equation f’ =m d x x which,
as we have already seen, belongs to perfectly dry air, we deduce from the preceding
tables 19 values of m, the mean of all will be found almost accurately equal to
= a result the more entitled to confidence inasmuch as the mean pressure for the
19 experiments being but very little over 30, and the air being perfectly dry, neither
of the corrections which I investigated in my former paper require to be applied.
If from the experiments already detailed I were to draw the conclusion that the
1 ‘ : :
equation f= f — a x 5 will afford the dew-point with a degree of accuracy far
surpassing ordinary hygrometrical observations, I would, probably, have the concur-
rence of most of my readers. The evidence adduced in support of the formula
appears, at least to me, ample and satisfactory. For the purpose, however, of dis-
pelling any doubts of its accuracy which may exist in the minds of others, I under-
took another series of test experiments, to the description of which I shall now
proceed.
The most direct method of testing our formula consists, as has been already ob-
served, in comparing its results with dew-points experimentally determined. In
order, however, that this criterion be decisive, it is not only necessary that the de-
pressions be considerable in amount, but also, as is obvious, that the dew-points be
accurately known. Now the registers to which I have had access do not perfectly
satisfy either of these conditions, the depressions being generally small, and the
observations made with an instrument—Daniell’s hygrometer, the difficulty of
observing with which is universally admitted. In reflecting on this matter it occurred
to me that both difficulties might be evaded in the following simple manner. Let
air, saturated with moisture, and whose temperature is, therefore, necessarily its dew-
point, be heated, and let the temperature of the heated air be taken, as also that shewn
by a moist bulb hygrometer, subjected to the action of a current of it. Let, then,
by the application of the formula, the dew-point, belonging to the two latter observa-
tions, be calculated, and from a comparison of it with the original temperature of the
air when saturated with humidity, we will be enabled to pronounce with confidence
upon the yalue of our method.
290 Professor Apsoun on the Theory
In the experiments which I performed on this plan the air was saturated with
moisture, by forcing it from a bellows through a succession of four Wolfe’s bottles, con-
nected in the usual way so as to cause the air to pass in each bottle through about two
inches of water, and the air thus saturated was heated by being made to pass through
a coil of copper tubing, immersed in a tub of warm water, the thermometer and
hygrometer being placed with their bulbs within a quarter of an inch of each other
in a narrow glass tube, attached to the farther extremity of the copper worm. The
following are the results thus obtained :-—
t’ ob. | t’ cale. Diff.
t t d
7s | 62.2 | 15.8 | 3030 | 51.3 | 50.47 | —. 83
WWenmaes 76 | 61.5 | 14.5 | 30.30 | 51.3 | 50.26 | —1.04
Agel | 73 | 60.3 | 12.7 | 30.30 | 51.3 | 51.58 | + .28
Meat 12 60 | 12 | 30.30 | 52.3 | 50.81 | — .49 |
69 | 58.6 | 10.4 | 30.30 | 51.3 | 50.40 | — .90
f 90.5 | 67 23.5 Be 50.8 | 50.17 | — .63
, 82.2 | 64.3 | 17.9 | 30.15 | 50.9 | 51 S50
eee | 79 | 62 | 16.4 | 30.15 | 50.9 | 50.23 | — .67
| 24M.) 71.7 | 60 | 11.7 | 30.15 | 51.2 | 50.66 | — .54
69 | 58.9 | 10.1 | 30.15 | 51.5 | 50.70 | — .80 |
(92 | 69 | 23 3942 | 541 | 54.40 | + .30 |
| April 20, 1835, | 83 | 65.8 | 17.2 | 30.42 | 54.5 | 54.36 | — .14
11 o'clock, A.M. | 76 | 63.3 | 12.7 | 30.42 | 54.9 | 54.54 | — .36
| 6s | 60.3 | 7.7 | 30.42 | 55 | 54.74 | — .26
98.5 | 71.5 | 27 | 30.36 | 55.5 | 55.51 | + .O1
April 21, 1835, } 84.6 | 67 | 17.6 | 30.36 | 56 | 55.79 | — .21 |
11 o'clock, A.M. ] 77.5 | 64.5 | 13 | 30.36 | 56.3 | 55.97 | — .33 |
s1_ | 62.2| 8.8 | 30.36 | 56.5 | 56.18 | — 32 |
83 | 66.5 | 16.5 | 30.51 | 56.8 | 55.87 | — .93 |
pert ia Hone a | 77 |65 | 12 | 30.51 | 57.2 | 57.23 | + .03 |
epte cata atl 70 Sa 6S 8.3 a B50) o74dt 0s |
Hee 91.8 | 68.6 | 23.2 | 30.51 | 54.1 | 53.70 | — .40 |
| Rte ee | 75.2 | 63.2! 12 | 30.51 | 55 | 54.94 | — .06 |
7AM. | 72 | 62 | 10 | 30.51 | 55.1 | 54.98 | — .12 |
| | — -35=mean.|
By a glance at the preceding table, which includes twenty-four distinct observations,
we will perceive, Ist. that in the case of seven of them, the observed and calculated
dew-points are almost coincident ; 2d. that the difference in no instance exceeds,
and in but a single instance reaches, one degree; and 3d. that the mean difference,
deducible from the whole, is but .35, or about one-third of a degree Fahrenheit. It
will also be noted that the difference is negative, or that the mean calculated dew-
point is lower than the observed, and not vice versa. If we were justified in consi-
dering this latter result as any thing more than accidental, it might certainly be
urged as an argument against the strict accuracy either of our experiments, or our
theoretical views ; for the corrections for the influence of pressure and aqueous
vapour on the specific heat of air being neglected in the preceding calculations, the
of the Moist-bulb Hygrometer. 291
caiculated dew-points, instead of being lower, should be higher than the truth. In
order, in fact, to account for the discrepancy in question, supposing it to be well
established, it would be necessary to conclude either that m, the co-efficient of our
hygrometric formula, is assumed somewhat too great, or that the observed depressions
are a little too small. The first, I believe, to be the true solution, and am, at pre-
sent disposed to consider m as more correctly represented by the fraction = than
= This point, however, I have not as yet been able fully to satisfy myself upon,
nor can the more exact determination of the value of the constant be considered a
matter of much practical importance, since the formula, in its present state, conducts,
as we have seen, to results which harmonize admirably with each other and with
observation.
I shall conclude by su}joining a couple of tables, by the aid of which the applica-
™ nt to ‘ . . s =
tion of my formula f”=f ~ a Xa to the determination of the dew-point, is
greatly facilitated. Table (A), which I have taken from the Edinburgh Encyclo-
pedia, article hygrometry, gives the elastic force of the vapour of water for every
: ne : d 1
degree Fahrenheit between 0° and 100" inclusive. Table (B) gives ea for every
( U
value of d between | and 10. This quotient, as is obvious from a glancez at the
formula, is, in calculating an observation, to be multiplied by p the existing pressure,
and the product, when deducted from J’, as given by table (A), will afford f”, or the
tension of vapour, at the dew-point. Should the depression exceed 10° the value of
d : ona : 1
a7xgo May still be got from table (B) by addition. Thus if d= 13°, s7e30=
-00383 + .00114= .00497.
Taste (A.)
+
~
~>
~
.
Ss
”
if ap Ff
-06121 | 21 | -13408 | 41 | .27376 | 61 | .54089 | 81 | 1.03350}
-06359 | 22 | -13906 | 42 | .28346 | 62 | .55913 | 82 | 1.06656 |
06605 | 23 | .14421 | 43 | .29848 | 63 | .57795 | 83 | 1.10058 |
06861 | 24 | .14954 | 44 | .30384 | 64 | .59735 | 84 | 1.13559
.07126 | 25 | .15506 | 45 | .31453 | 65 | .61734 | 85 | 1.17161
07401 | 26 | .16076 | 46 | .52557 | 66 | .63795 | 86 | 1.20867 |
07685 | 27 | .16667 | 47 | .33684 | 67 | .65919 | 87 | 1.24680
.07980 | 28 | -17277 | 48 | -34875 | 68 | .68108 | 88 | 1.28602
| .08286 | 29 | -17908 | 49 | -36090 | 69 | .70364 | 89 | 1.32636 |
| .08603 | 20 | -18561 | 50 | -37345 | 70 | .72688 | 90 | 1.36785 |
10 | .c8931 | 31 | -19237 | 51 | -38640 | 71 | .75083 | 91 | 1.41059
11 | .09270 | 32 | -19934 | 52 | -39977 | 72 | .77551 | 92 | 1.45438 |
12 | .09622 | 33 | -20658 | 53 | -41356 | 73 | .80092 | 93 | 1.49948 |
3 | .09987 | 34 | -21404 | 54 | -42779 | 74] .82710 | 94 | 1.54585
14 | .10364 | 35 | -22175 | 55 | -44249 | 75 | .85407 | 95 | 1.59352
15 | .10755 | 36 | -22972 | 56 | 45764] 76 | .88184 | 96 | 1.64251!
16 | -11160 | 37 | -23796 | 57 | 47328 | 77 | .91042 | 97 | 1.69286,
17 | .11579 | 38 | -24647 | 58 | *48940 | 78 | .93987 | 98 | 1.74461
18 | .12013 | 39 | -25527 | 59 | *50604 | 79 | .97017 | 99 | 1.79778
19 | .12462 | 40 | -26436 | 60 | °52320 | 80 | 1.00137 | 100 | 1.85241
20 | .12927 |
VOL. XVII. 3M
SoOortnankwworeo
(9)
Professor Apsoun on the Theory, &c.
TaBLe (B.)
d d fd eee d d d a ae
87 x 30 | 87 X 30 | 87x30 87 x 30 |
.1 | .00003 | 2.1 | .00080 | 4.1 | 00157 | 6.1 0033 8.1
.2 | 00007 | 2.2 .00084 | 4.2 | 00160 | 6.2 | .00237 } 8.2
| 3 | 00011 | 2.3 | .00087 | 4.3 | 00164 | 6.3 | .00241 | 83
| .4 | 00015 | 2.4 .00091 | 4.4 | 00168 | 6.4 | .00245 } 8.4
| 5 | 00019 { 2.5, .00095 | 4.5 | 00172 | 6.5 | .00248 | 8.5
| .6 | .00022 | 2.6 | .00099 | 4.6 | 00176 | 6.6 | .00252 | 8.6
| .7 | .00026 | 27) .00103 | 4.7 | 00180 | 6.7 | -00256 | 8.7
.8 | .00030 | 2.8 .00107 | 4.8 | 00183 | 6.8 | .00260 | 8.8
9 | .00034 } 2.9 | .00111 | 4.9 | 00187 | 6.9 | .00264 | 8.9
1 | .00038 | 8 | .00114 | 5 | 00191 | 7 | -00268]| 9
1.1 | .00042 | 3.1} .00118 } 5.1 | 00195 | 7.1 | -00271 | 9.1
| 1.2 | .00045 | 3.2 | .00122 | 5.2 | 00199 | 7.2 | 00275 | 9.2
1.3 | .00049 | 3.3) .00126 }.5.3 . 00202 | 7.3 | -00279 | 9.3 |
1.4 | .00053 | 3.4! .00130 | 5.4 | 00206 | 7.4 | -00283 | 9.4 |
15 | .00057 | 3.5 | .00134 | 5.5 , 00210 | 7.5 | -00287 | 9.5
16 | .00061 | 3.6! .00137 |] 5.6 00214 | 7.6 | -00291 | 9.6
17 | .00065 | 3.7 | .00141 | 5.7 | 00218 | 7.7 | .00294 | 9.7
18 | .00068 | 3.8 | .00145 | 5.8 00222 | 7.8 | .00.98 | 9.8 |
1,9 | .00072 | 3.9} .00149 } 59 00225 | 7.9 | .00302 | 9.9
2 | 00076 | 4 | .00153 | 6 00229 { 8 !| .00306 | 10
Theory of Conjugate Functions, or Algebraic Couples ; with a Preliminary and
Elementary Essay on Algebra as the Science of Pure Time.
By WILLIAM ROWAN HAMILTON,
MRI. A., F.R.A.S., Hon. M. R. S. Ed. and Dub., Fellow of the American Academy of Arts and
Sciences, and of the Royal Northern Antiquarian Society at Copenhagen, Andrews’ Professor of
Astronomy in the University of Dublin, and Royal Astronomer of Ireland.
Read November 4th, 1833, and June Ist, 1835.
General Introductory Remarks.
Tue Study of Algebra may be pursued in three very different schools, the Practical, the Philological, or
the Theoretical, according as Algebra itself is accounted an Instrument, or a Language, or a Contempla-
tion ; according as ease of operation, or symmetry of expression, or clearness of thought, (the agere, the
fari, or the sapere,) is eminently prized and sought for. The Practical person seeks a Rule which he
may apply, the Philological person seeks a Formula which he may write, the Theoretical person seeks a
Theorem on which he may meditate. The felt imperfections of Algebra are of three answering kinds.
The Practical Algebraist complains of imperfection when he finds his Instrument limited in power; when
a rule, which he could happily apply to many cases, can be hardly or not at all applied by him to some
new case; when it fails to enable him to do or to discover something else, in some other Art, or in some
other Science, to which Algebra with him was but subordinate, and for the sake of which and not for its
own sake, he studied Algebra. The Philological Algebraist complains of imperfection, when his Language
presents him with an Anomaly; when he finds an Exception disturb the simplicity of his Notation, or the
symmetrical structure of his Syntax; when a Formula must be written with precaution, and a Symbolism
isnot universal. The Theoretical Algebraist complains of imperfection, when the clearness of his Con-
templation is obscured; when the Reasonings of his Science seem anywhere to oppose each other, or
become in any part too complex or too little valid for his belief to rest firmly upon them; or when,
though trial may have taught him that a rule is useful, or that a formula gives true results, he cannot
prove that rule, nor understand that formula: when he cannot rise to intuition from induction, or cannot
look beyond the signs to the things signified.
294 Professor Hamitton on Conjugate Functions,
It is not here asserted that every or any Algebraist belongs exclusively to any one of these three
schools, so as to be only Practical, or only Philological, or only Theoretical. Language and Thought
react, and Theory and Practice help eachother. No man can be so merely practical as to use frequently
the rules of Algebra, and never to admire the beauty of the language which expresses those rules, nor
care to know the reasoning which deduces them. No man can be so merely philological an Algebraist
but that things or thoughts will at some times intrude upon signs; and occupied as he may habitually be
with the logical building up of his expressions, he will feel sometimes a desire to know what they mean,
or to apply them. And no man can be so merely theoretical or so exclusively devoted to thoughts, and
to the contemplation of theorems in Algebra, as not to feel an interest in its notation and language, its
symmetrical system of signs, and the logical forms of their combinations ; or not to prize those practical
aids, and especially those methods of research, which the discoveries and contemplations of Algebra have
given to other sciences. But, distinguishing without dividing, it is perhaps correct to say that every Alge-
braical Student and every Algebraical Composition may be referred upon the whole to one or other of
these three schools, according as one or other of these three views habitually actuates the man, and emi-
nently marks the work.
These remarks have been premised, that the reader may more easily and distinctly perceive what the
design of the following communication is, and what the Author hopes or at least desires to accomplish.
That design is Theoretical, in the sense already explained, as distinguished from what is Practical on the
one hand, and from what is Philological upon the other. The thing aimed at, is to improve the Science,
not the Art nor the Language of Algebra. The imperfections sought to be removed, are confusions of
thought, and obscurities or errors of reasoning; not difficulties of application of an instrument, nor
failures of symmetry in expression. And that confusions of thought, and errors of reasoning, still darken
the beginnings of Algebra, is the earnest and just complaint of sober and thoughtful men, who in a spirit
of love and honour have studied Algebraic Science, admiring, extending, and applying what has been al-
ready brought to light, and feeling all the beauty and consistence of many a remote deduction, from
principles which yet remain obscure, and doubtful.
For it has not fared with the principles of Algebra as with the principles of Geometry. No candid
and intelligent person can doubt the truth of the chief properties of Parallel Lines, as set forth by
Eucuip in his Elements, two thousand years ago; though he may well desire to see them treated in a
clearer and better method. The doctrine involves no obscurity nor confusion of thought, and leaves in
the mind no reasonable ground for doubt, although ingenuity may usefully be exercised in improving the
plan of the argument. But it requires no peculiar scepticism to doubt, or even to disbelieve, the doctrine
of Negatives and Imaginaries, when set forth (as it has commonly been) with principles like these: that a
greater magnitude may be subtracted from a less, and that the remainder is less than nothing ; that two
negative numbers, or numbers denoting magnitudes each less than nothing, may be multiplied the one by
the other, and that the product will be a positive number, or a number denoting a magnitude greater than
nothing ; and that although the square of a number, or the product obtained by multiplying that number
by itself, is therefore always positive, whether the number be positive or negative, yet that numbers, called
imaginary, can be found or conceived or determined, and operated on by all the rules of positive and
negative numbers, as if they were subject to those rules, although they have negative squares, and must
therefore be supposed to be themselves neither positive nor negative, uor yet null numbers, so that the
magnitudes which they are supposed to denote can neither be greater than nothing, nor less than nothing,
nor even equal to nothing. It must be hard to found a Screncz on such grounds as these, though the
and on Algebra as the Science of Pure Time. 295
forms of logic may build up from them a symmetrical system of expressions, and a practical art may be
learned of rightly applying useful rules which seem to depend upon them.
So useful are those rules, so symmetrical those expressions, and yet so unsatisfactory those principles
from which they are supposed to be derived, that a growing tendency may be perceived to the rejection
of that view which regarded Algebra as a SciENcE, in some sense analogous to Geometry, and to the
adoption of one or other of those two different views, which regard Algebra as an Art, or as a Language :
as a System of Rules, or else as a System of Expressions, but not as a System of Z’ruths, or Results having
any other validity than what they may derive from their practical usefulness, or their logical or philological
coherence. Opinions thus are tending to substitute for the Theoretical question—“Is a Theorem of
Algebra true?” the Practical question,—‘* Can it be applied as an Instrument, to do or to discover some-
thing else, in some research which is not Algebraical?” or else the Philological question,— Does its
expression harmonise, according to the Laws of Language, with other Algebraical expressions ?”
Yet a natural regret might be felt, if such were the destiny of Algebra; if a study, which is conti-
nually engaging mathematicians more and more, and has almost superseded the Study of Geometrical
Science, were found at last to be not, in any strict and proper sense, the Study of a Science at all: and
if, in thus exchanging the ancient for the modern Mathesis, there were a gain only of Skill or Ele-
gance, at the expense of Contemplation and Intuition. Indulgence, therefore, may be hoped for, by any
one who would inquire, whether existing Algebra, in the state to which it has been already unfolded by
the masters of its rules and of its language, offers indeed no rudiment which may encourage a hope of
developing a Science of Algebra: a Science properly so called ; strict, pure, and independent; deduced
by valid reasonings from its own intuitive principles ; and thus not less an object of priori contempla-
tion than Geometry, nor less distinct, in its own essence, from the Rules which it may teach or use, and
from the Signs by which it may express its meaning.
The Author of this paper has been led to the belief, that the Intuition of Time is such a rudiment.
This belief involves the three following as components: First, that the notion of Time is connected
with existing Algebra ; Second, that this notion orintuition of Time may be unfolded into an independent
Pure Science ; and Third, that the Science of Pure Time, thus unfolded, is co-extensive and identical with
Algebra, so far as Algebra itself is a Science. The first component judgment is the result of an induction;
the second of a deduction; the third is a joint result of the deductive and inductive processes.
I. The argument for the conclusion that the notion of Time is connected with existing Algebra, is an
induction of the following kind. The History of Algebraic Science shows that the most remarkable disco-
veriesin it have been made, either expressly through the medium of that notion of Time, or through the
closely connected (and in some sort coincident) notion of Continuous Progression. It is the genius
of Algebra to consider what it reasons on as flowing, as it was the genius of Geometry to consider what
it reasoned on as fixed. Evcxip* defined a tangent to a circle, APoLLoniust conceived a tangent to an
ellipse, as an indefinite straight line which had only one point in common with the curve ; they looked
upon the line and curve not as nascent or growing, but as already constructed and existing in space ; they
studied them as formed and fixed, they compared the one with the other, and the proved exclusion of any
second common point was to them the essential property, the constitutive character of the tangent.
The Newtonian Method of Tangents rests on another principle; it regards the curve and line not as
* EdSia xdxrov iparreaSar Aéyerau, Aris awroutyn Tov xixAou xat ExCadrouiyn ow reaver Toy xixdore-— EUCLID, Book III. Def. 2.
Oxford Edition, 1703.
ft Edy ty xdvou rom amd ris xopupis ris romms ayS7 eieia wapx rerayutvus xzrnyucyny Exrds weosira Ths rosstis.—txros Spa
megsiray diomwep tQanreras thi To4Hi.—APPOLLONIUS, Book 1. Prop. 17. Oxford Edition, 1710.
296 Professor Hamitton on Conjugate Functions,
already formed and fixed, but rather as nascent, or in process of generation: and employs, as its pri-
mary conception, the thought of a flowing point. And, generally, the revolution which NEwron* made in
the higher parts of both pure and applied Algebra, was founded mainly on the notion of fluwion, which
involves the notion of Time.
Before the age of Newron, another great revolution, in Algebra as well as in Arithmetic, had been
made by the invention of Logarithms ; and the “ Canon Mirificus” attests that Naprert deduced that in-
vention, not (as it is commonly said) from the arithmetical properties of powers of numbers, but from
the contemplation of a Continuous Progression ; in describing which, he speaks expresssly of Fluxions,
Velocities and Times.
In a more modern age, Lacranes, in the Philological spirit, sought to reduce the Theory of Fluxions
to a system of operations upon symbols, analogous to the earliest symbolic operations of Algebra, and
professed to reject the notion of time as foreign to such a system; yet admitted} that fluxions might be
considered only as the velocities with which magnitudes vary, and that in so considering them, abstrac-
tion might be made of every mechanical idea. And in one of his own most important researches in pure
Algebra, (the investigation of limits between which the sum of any number of terms in TayLor’s Series
is comprised,) Lackance|| employs the conception of continuous progression to show that a certain vari-
able quantity may be made as small as can be desired. And not to dwell on the beautiful discoveries
made by the same great mathematician, in the theory of singular primitives of equations, and in the al-
gebraical dynamics of the heavens, through an extension of the conception of variability, (that is, in
fact, of flowingness,) to quantities which had before been viewed as fixed or constant, it may suffice for
the present to observe that LacranGE considered Algebra to be the Science of Functions§, and that it is
not easy to conceive a clearer or juster idea of a Function in this Science, than by regarding its essence
as consisting in a Law connecting Change with Change. But where Change and Progression are, there is
Time. The notion of Time is, therefore, inductively found to be connected with existing Algebra.
If. The argument for the conclusion that the notion of time may be unfolded into an independent
Pure Science, or that a Science of Pure Time is possible, vests chiefly on the existence of certain priori
* Considerando igitur quod quantitates zqualibus temporibus crescentes et crescendo genite, pro velocitate majori vel mi-
nori qua crescunt ac generantur evadunt majores vel minores ; methodum quierebam determinandi quantitates ex velocitati-
bus motuum yel incrementorum quibus generantur; et has motuum vel incrementorum yelocitates nominando Fluxiones,
et quantitates genitas nominando Fluences, incidi paulatim annis 1665 et 1666 in Methodum Fluxionum qua hie usus sum in
Quadratura Curvarum—Traetatus de Quad. Curv., Introd., published at the end of Sir I. Newton’s Opticks, London 1704.
+ Logarithmus ergo cujusque sinus, est numerus quam proximé definiens lineam, que xqualiter crevit intered dum sinus
totius linea proportionaliter in sinum illum decrevit, existente utroque motu synchrono, atque initio equiveloce. Baron
Napier’s Mirifici Logarithmorum Canonis Descriptio, Def. 6, Edinburgh 1614.—Also in the explanation of Def. 1, the
words flucu and fluat occur.
¢ Calcul des Fonctions, Legon Premiere, page 2. Paris 1806.
|| Done puisque V devient nul lorsque i devient nul, il est clair qu’ en faisant croitre i par degrés insensibles depuis
zéro, la valeur de V croitra aussi insensiblement depuis zéro, soit en plus ou en moins, jusqu’ a un certain point, aprés
quoi elle pourra diminuer.— Calcul des Fonctions, Legon Neuviéme, page 90. Paris 1806. An instance still more strong
may be found in the First Note to Lagrange’s Equations Numeriques. Paris, 1808.
§ On doit regarder l'algébre comme la science des fonetions.—Cale. des Fonct., Legon Premiere.
© The word “ Algebra” is used throughout this whole paper, in the sense which is commonly but improperly given
by modern mathematical writers to the name “ Analysis,” and not with that narrow signification to which the unphilosophi-
eal use of the latter term (Analysis) has caused the former term (Algebra) to be too commonly confined. The author
confesses that he has often deserved the censure which he has here so freely expressed.
and on Algebra as the Science of Pure Time. 297
intuitions, connected with that notion of time, and fitted to become the sources of a pure Science; and on
the actual deduction of such a Science from those principles, which the author conceives that he has
begun. Whether he has at all succeeded in actually effecting this deduction, will be judged after the
Essay has been read; but that such a deduction is possible, may be concluded in an easier way, by an
appeal to those intuitions to which allusion has been made. That a moment of time respecting which
we inquire, as compared with a moment which we know, must either coincide with or precede or follow
it, is an intuitive truth, as certain, as clear, and as unempirical as this, that no two straight lines can
comprehend an area. The notion or intuition of OrpER 1n Time is not less but more deep-seated in the
human mind, than the notion or intuition of OrDER IN SpAcE; and a mathematical Science may be founded
on the former, as pure and as demonstrative as the science founded on the \atter. There is something
mysterious and transcendent involved in the idea of Time ; but there is also something definite and clear:
and while Metaphysicians meditate on the one, Mathematicians may reason from the other.
Ill. That the Mathematical Science of Time, when sufficiently unfolded, and distinguished on the
one hand from all actual Outward Chronology (or collections of recorded eyents and phenomenal marks
and measures), and on the other hand from all Dynamical Science (or reasonings and results from the
notion of cause and effect), will ultimately be found to be co-extensive and identical with Algebra, so far
as Algebra itselfis a Science: is a conclusion to which the author has been led by all his attempts, whe-
ther to analyse what is Scientific in Algebra, or to construct a Science of Pure Time. It is a joint result of the
inductive and deductive processes, and the grounds on which it rests cou!d not be stated in a few general
remarks. The author hopes to explain them more fully in a future paper; meanwhile he refers
to the present one, as removing (in his opinion) the difficulties of the usual theory of Negative and Ima-
ginary Quantities, or rather substituting a new Theory of Contrapositives and Couples, which he considers
free from those old difficulties, and which is deduced from the Intuition or Original Mental Form of Time :
the opposition of the (so-called) Negatives and Positives being referred by him, not to the oppos'tion of
the operations of increasing and diminishing a magnitude, but to the simpler and more extensive contrast
between the relations of Before and A fter,* or between the directions of Forward and Backward ; and
Pairs of Moments being used to suggest a Theory of Conjugate Functions,+ which gives reality and
meaning to conceptions that were before Imaginary,{ Impossible, or Contradictory, because Mathemati-
cians had derived them from that bounded notion of Magnitude, instead of the original and comprehensive
thought of OrpER 1N PRroGreEssion.
“It is, indeed, very common, in Elementary works upon Algebra, to allude to past and future time, as one among many
illustrations of the doctrine of negative quantities ; but this avails little for Science, so long as magnitude instead of PROGRES-
Sion is attempted to be made the basis of the doctrine.
+ The author was conducted to this Theory many years ago, in reflecting on the important symbolic results of Mr.
Graves respecting Imaginary Logarithms, and in attempting to explain to himself the theoretical meaning of those remark-
able symbolisms. The Preliminary and Elementary Essay on Algebra as the Science of Pure Time, is a much more
recent developement of an Idea against which the author struggled long, and which he still longer forbore to make public, on
account of its departing so far from views now commonly received. The novelty, however, is in the view and method, not
inthe results and details: in which the reader is warned to expect little addition, if any, to what is already known.
¢ The author acknowledges with pleasure that he agrees with M. Caucny, in considering every (so-called) Imaginary
Equation as a symbolic representation of two separate Real Equations: but he differs from that excellent mathematician in
_ his method generally, and especially in not introducing the sign »/—1 until he has provided for it, by his Theory of
Couples, a possible and real meaning, as a symbol of the couple (0.1).
298 Professor Hamitton on Conjugate Functions,
CONTENTS OF THE PRELIMINARY AND ELEMENTARY ESSAY ON ALGEBRA
AS THE SCIENCE OF PURE TIME.
Articles,
Comparison of any two moments with respect to identity or diversity, subsequence or precedence
Comparison of two pairs of moments, with respect to their analogy or non-analogy.......-..-+--+-+-+- Hea 3.
Combinations of two different analogies, or non-analogies, of pairs of moments, with each other.........sssseeeeeeeeee 4
On continued analogies, or equidistant series Of MOMENES... ...cesssceseeeecsseeeeesctseeeeteee teen setae PEO ort: ced 5,6,7,3
On steps in the progression of time: their application direct or inverse to moments, so as to generate other moments ;
and their combination with other steps, in the way of composition or decomposition........---.++sees++eeuwee sree0 9,10,11,12
On the multiples of a given base, or unit step ; and on the algebraic addition, subtraction, multiplication, and division,
of their determining or multipling whole numbers, whether positive or contra-positive, or null.. ............:24 13,14,15
On the submultiples and fractions of any given step in the progression of time ; on the algebraic addition, subtraction,
multiplication, and division, of reciprocal and fractional numbers, positive and contra-positive ; and on the impos-
sible or indeterminate act of submultipling or dividing by ZeT0...--...sscseseeeeneeseseceeeesneeeeeeesteresaneres -16,17,18,19,20
On the comparison of any one effective step with any other, in the way of ratio, and the generation of any one such
step from any other, in the way of multiplication; and on the addition, subtraction, multiplication, and division of
algebraic numbers in general, considered thus as ratios or as multipliers of stepS......---sss01 creseserees: ceeeeeee/D1,22,23
On the insertion of a mean proportional between two steps ; and on impossible, ambiguous, and incommensurable
square roots Of ratiOS...........scee---eeeee seers paodccosomonoscoabessaeoser wsoanesn fone. do tsans==n-ne cecscsrndoncereass PS -caccciod 24,25
More formal proof of the general existence of a determined positive square-root, commensurable or incommensurable,
for every determined positive ratio; continuity of eee of the square, and principles connected with this
continuity.......--.+ Ss rete Waters stee iecavarcsansreaeecesopeesenscenenssacseasene-suasaneacserssenessrauccc! ube cubinwsinmnsbieigsas «40g= apie 26,27
On continued analogies or series of siaconlinial steps ; va on powers and roots and logarithms of ratios...28,29,30,31 32,33
Remarks on the notation of this Essay and on some modifications by which it may be made more like the notation
commonly employed ........... oc CerECEe Sioteateait cence oes aavaaeaues Unahe than cons sense pmactunentaarsthawnetpasanssmesenas er tod: + 34,29,36
CONTENTS OF THE THEORY OF CONJUGATE FUNCTIONS, OR ALGEBRAIC COUPLES.
= Articles.
On couples of moments, and of steps, in time..........--.seesecseeescssseeeeseteetnes cocrenersersesserrsesassnaseesenannanearscansnes
On the composition and decomposition of step-couples.........
On the multiplication of a step-couple by a number
On the multiplication of a step-couple by a number-couple; and on the ratio of one step-couple to another............-+
On the addition, subtraction, multiplication, and division, of number-couples, as combined with each other..
On the powering of a number-couple by a single whole number............ sdinevuetidaunaceseqashtccentssshsu=-saHsssa5 0 aeigeeeMEan
On a particular class of exponential and logarithmic function-couples, connected with a particular series of integer
chencAaasreccoeeoos pope Ebo coe co 2800-5 noses cheba Meera rocaanaraceca ccoscocopeacadsoecobo cL MIS
On the powering of any number-couple tea any single number or number-couple
powers of number-couples.
On exponential and logarithmic function-couples in general.
and on Algebra as the Science of Pure Time. 299
PRELIMINARY AND ELEMENTARY ESSAY
ON ALGEBRA AS THE SCIENCE oF PURE TIME.
Comparison of any two moments with respect to identity or diversity, subsequence or
precedence.
1. If we have formed the thought of any one moment of time, we may afterwards
either repeat that thought, or else think of a different moment. And if any two
spoken or written names, such as the letters a and B, be dates, or answers to the
question /Vhen, denoting each a known moment of time#they must either be names
of one and the same known moment, or else: of two different moments. In each
case, we may speak of the pair of dates as denoting a pair of moments ; but in the
first case, the two moments are coincident, while in the second case they are distinct
from each other. ‘To express concisely the former case of relation, that is, the case
of identity between the moment named B and the moment named a, or of equivalence
between the date B and the date a, it is usual to write
B=A; (1.)
a written sentence or assertion, which is commonly called an equation : and to express
concisely the latter case of relation, that is, the case of diversity between the two mo-
ments, or of non-equivalence between the two dates, we may write
BA; (2.)
VOL. XVII. 3N
300 Professor Haminton on Conjugate Functions,
annexing, here and afterwards, to these concise written expressions, the side-marks
(1.) (2.), &c., merely to facilitate the subsequent reference in this essay to any such
assertion or result, whenever such reference may become necessary or convenient.
The latter case of relation, namely, the case (2.) of diversity between two moments»
or of non-equivalence between two dates, subdivides itself into the two cases of subse-
quence and of precedence, according as the moment 8 is later or earlier than a. To
express concisely the former sort of diversity, in which the moment B is ater than a,
we may write .
B>A; (3.)
and the latter sort of diversity, in which the moment B is earlier than A, may be ex-
pressed concisely in this other way,
B<A. (4.)
It is evident that
if B=A, then a=B; (5.)
if BA, then as ; (6.)
if B>a, then A<B; (7.)
if B<a, then a>B. (8.)
Comparison of two pairs of moments, with respect to their analogy or non-analogy.
2. Considering now any two other dates c and p, we perceive that they may and
must represent either the same pair of moments as that denoted by the former pair
of dates a and B, or else a different pair, according as the two conditions,
C=A, and p=B, (9.)
are, or are not, both satisfied. If the new pair of moments be the same with the old,
then the connecting relation of identity or diversity between the moments of the one
pair is necessarily the same with the relation which connects in like manner the mo-
ments of the other pair, because the pairs themselves are the same. But even if the
pairs be different, the relations may still be the same ; that is, the moments c and p,
even if not both respectively coincident with the moments a and 3, may still be re-
lated to each other exactly as those moments, (p to c as B to A;) and thus the two
pairs, A, B and c, D may be analogous, even if they be not coincident with each other.
An analogy of this sort (whether between coincident or different pairs) may be ex-
pressed in writing as follows,
and on Algebra as the Science of Pure Time. 801
D-—C=B—A, or, B—~A=D—C; (10.)
the interposed mark =, which before denoted identity of moments, denoting now
identity of relations : and the written assertion of this identity being called (as before)
an equation. The conditions of this exact identity between the relation of the mo-
ment p to c, and that of B to a, may be stated more fully as follows: that if the mo-
ment B be identical with a, then p must be identical with c ; if B be later than a,
then p must be later than c, and exactly so much later; and if B be earlier than
A, then p must be earlier than c, and exactly so much earlier. It is evident, that what-
ever the moments A B and c may be, there is always one, and only one, connected mo-
ment D, which is thus related to c, exactly as B is to A; and it is not difficult to per-
ceive that the same moment p is also related to B, exactly as c is to A: since, in the
case of coincident pairs, pD is identical with 8, and c with 4; while, in the case of pairs
analogous but not coincident, the moment p is later or earlier than B, according as c
is later or earlier than a, and exactly so much later or so much earlier. If then the
pairs A, B, and c, p, be analogous, the pairs ac and BD, which may be said to be
alternate to the former, are also analogous pairs ; that is,
if D—c=B-A, then D—B=c—a; (11.)
a change of statement of the relation between these four moments a B c D, which
may be called alternation of an analogy. It is still more easy to perceive, that if any
two pairs aB and cp be analogous, then the ¢nverse pairs BA and vc are analogous
also, and therefore that
if D—c— p— AS then cp —A—B; (12.)
a change in the manner of expressing the relation between the four moments
ABCD, which may be called znverscon of an analogy, Combining inversion with
alternation, we find that
if p—c=B—A, then B—D=A—C; (13.)
and thus that all the eight following written sentences express only one and the same
relation between the four moments a Bc D:
D— C=E— A, B—A—D —C,)}
CAG an ase—e, (14.)
CA BA — Cl Ds
B—D=A—C, A—C=B—D;
any one of these eight written sentences or equations being equivalent to any other.
302 Professor Hamitton on Conjugate Functions,
5. When the foregoing relation between four moments aBcpD does not exist,
that is, when the pairs aB and cp are not analogous pairs, we may mark this non-
analogy by writing
D—C=-B— A; (15.)
and the two possible cases into which this general conception of non-analogy or di-
versity of relation subdivides itself, namely, the case when the analogy fails on ac-
count of the moment pD being too /ate, and the case when it fails because that moment
p is too early, may be denoted, respectively, by writing in the first case,
D—C>B—A, - (16.)
and in the second case,
D—C<B—A; (17.)
while the two cases themselves may be called, respectively, a non-analogy of subse-
quence, and a non-analogy of precedence. We may also say that the relation of
p to c, as compared with that of B to 4, is in the first case a relation of comparative
lateness, and in the second case a relation of comparative earliness.
Alternations and inversions may be applied to these expressions of non-analogy, ead
the case of p too late may be expressed in any one of the eight following ways, which
are all equivalent to each other,
D—C>B—aA, B—A<D—C,}
D—E>C—A, saseoas (18.)
c—D<a—B, A—B>C—D,
B—D<a—C, A—cC>oB—D;
while the other case, when the analogy fails because the moment p is too early, may
be expressed at pleasure in any of the eight ways following,
D—c<B—a, B—A>D—CE,
pD—B<C —a, C_a>D—B,
C—D>A—B, A__B<C—D,
B—D>A—C, A—C<C¢B—D.
(19.)
In general, if we have any analogy or non-analogy between two pairs of moments,
4 B and c D, of which we may call the first and fourth mentioned moments, a and p,
the extremes, and the second and third mentioned moments, namely, 8 and c, the
and on Algebra as the Science of Pure Time. 808
means, and may call a and c the antecedents, and 8 and v the consequents ; we do
not disturb this analogy or non-analogy by interchanging the means among them.
selves, or the extremes among themselves ; or by altering equally, in direction and in
degree, the two consequents, or the two antecedents, of the analogy or of the non-
analogy, or the two moments of either pair ; or, finally, by altering oppositely in di-
rection, and equally in degree, the two extremes, or the two means. In an analogy,
we may also put, by inversion, extremes for means, and means for extremes; but if a
non-analogy be thus inverted, it must afterwards be changed in kind, from subse-
quence to precedence, or from precedence to subsequence.
Combinations of two different analogies, or non-analogies, of pairs of moments,
with each other.
4, From the remarks last made, it is manifest that
if D—C=B—A,
and p'—p=B'—B, (20.)
then pD’—c=B’—Aa;
because the second of these three analogies shews, that in passing from the first to the
third, we have either made no change, or only altered equally in direction and in
degree the two consequent moments B and p of the first analogy. In like manner,
if D—c=B-—aA,
an Cr— Gl Acts (21.)
then Dp —c’'=B —A;
because now, in passing from the first to the third analogy, the second analogy shews
that we have either made no change, or else have only altered equally, in direction
and degree, the antecedents a and c. Again,
ne 3) =) Se
chival Ipc 1b) ==(oy aor (22°)
then p’—c’=B —A;
because here we have only altered equally, if at all, the two moments c and p of one
common pair, in passing from the first analogy to the third. Again,
if oD — CB —A,
and c—c’=B —3, (23.)
then p—c'=B'—a;
304 Professor Haminton on Conjugate Functions,
because now we either do not alter the means B and c at all, or else alter them oppo-
sitely in direction and equally in degree. And similarly,
Rip ee Dy SSCS 3h
and Dp’ =peha ah (24:.)
then p’'—c=B—a’,
because here we only alter equally, if at all, in degree, and oppositely in direction,
the extremes, a and p, of the first analogy. It is still more evident that if two pairs
be analogous to the same third pair, they are analogous to each other ; that is
fee — co —seAe
and B—A=D—C, (25.)
then D—c=p'—c.
And each of the foregoing conclusions will still be true, if we change the first supposed
analogy p—c=B—A, to anon-analogy of subsequence D — c> B — A, or to anon-analogy
of precedence p —c <B—A, provided that we change, in like manner, the last or con-
cluded analogy to a non-analogy of subsequence in the one case, or of precedence
in the other.
It is easy also to see, that if we still suppose the first analogy p—c=B—A to
remain, we cannot conclude the third analogy, and are not even at liberty to suppose
that it exists, in any one of the foregoing combinations, unless we suppose the second
also to remain: that is, if two analogies have the same antecedents, they must have
analogous consequents ; if the consequents be the same in two analogies, the antece-
dents must themselves form two analogous pairs ; if the extremes of one analogy be
the same with the extremes of another, the means of either may be combined as
extremes with the means of the other as means, to form a new analogy ; if the means
of one analogy be the same with the means of another, then the extremes of either may
be combined as means with the extremes of the other as extremes, and the resulting
analogy will be true ; from which the principle of inversion enables us farther to
infer, that if the extremes of one analogy be the same with the means of another,
then the means of the former may be combined as means with the extremes of the
latter as extremes, and will thus generate another true analogy.
On continued Analogies, or Equidistant Series of Moments,
5, It is clear from the foregoing remarks, that in any analogy
B'— A’ =B—A, (26.)
the two moments of either pair A B or A’ B’ cannot coincide, and so reduce themselves
P ’
and on Algebra as the Science of Pure Time. 805
to one single moment, without the two moments of the other pair 4’ »’ or 4 B being
also identical with each other ; nor can the two antecedents a a’ coincide, without the
two consequents B B’ coinciding also, nor can the consequents without the antecedents.
The only way, therefore, in which two of the four moments AB a’B of an analogy
can coincide, without the two others coinciding alse, that is, the only way in which an
analogy can be constructed with three distinct moments of time, is either by the two
extremes a B’ coinciding, or else by the two means B a’ coinciding; and the principle
of inversion permits us to reduce the former of these two cases to the latter. We
may then take as a sufficient type of every analogy which can be constructed with
three distinct moments, the following : .
B—B=B—A; (27.)
that is, the case when an extreme moment B’ is related to a mean moment B, as that
mean moment B is related to another extreme moment a; in which case we shall say
that the three moments a B B’ compose a continued analogy. In such an analogy,
it is manifest that the three moments a B B’ compose also an equidistant series, B
being exactly so much later or so much earlier than 8, as B is later or earlier than a.
The moment B is evidently, in this case, exactly intermediate between the two other
moments 4 and B’, and may be therefore called the middle moment, or the bisector,
of the interval of time between them. It is clear that whatever two distinct moments
a and pg’ may be, there is always one and only one such bisector moment B; and that
thus a continued analogy between three moments can always be constructed in one
but in only one way, by inserting a mean, when the extremes are given. And it is
still more evident, from what was shewn before, that the middle moment B, along with
either of the extremes, determines the other extreme, so that it is always possible to
complete the analogy in one but in only one way, when an extreme and the middle
are given.
6. If, besides the continued analogy (27.) between the three moments A B B’, we
have also a continued analogy between the two last B B’ of these three and a fourth
moment B”, then the fowr moments a BR’ B” may themselves also be said to form
another continued analogy, and an equidistant series, and we may express their rela-
tions as follows:
B’— B’=B'—B=B—A. (28.)
In this case, the interval between the two extreme moments a and B” is ¢trisected by
the two intermediate moments kz and B’, and we may call B the first trisector, and 1
the second trisector of that interval. If the first extreme momenta and the first
306 Professor Hamitton on Conjugate Functions,
trisector moment B be given, it is evidently possible to complete the continued analogy
or equidistant series in one and in only one way, by supplying the second trisector
s’ and the second extreme B” ; and it is not much less easy to perceive that any two
of the four moments being given, (together with their names of position in the series,
as such particular extremes, or such particular trisectors,) the two other moments can
be determined, as necessarily connected with the given ones. Thus, if the extremes
be given, we must conceive their interval as capable of being trisected by two means,
in one and in only one way ; if the first extreme and second trisector be given, we
can bisect the interval between them, and so determine (in thought) the first trisector,
and afterwards the second extreme; if the two trisectors be given, we can continue
their interval equally in opposite directions, and thus determine (in thought) the two
extremes ; and if either of these two trisectors along with the last extreme be given,
we can determine, by processes of the same kind, the two other moments of the
series,
7. In general, we can imagine a continued analogy and an equidistant series, com-
prising any number of moments, and having the interval between the extreme moments
of the series divided into the next lesser number of portions equal to each other, by a
number of intermediate moments which is itself the next less number to the number of
those equal portions of the whole interval. For example, we may imagine an equi-
distant series of five moments, with the interval between the two extremes divided
into four partial and mutually equal intervals, by three intermediate moments, which
may be called the first, second, and third guadrisectors or quarterers of the total .
interval. And it is easy to perceive, that when any two moments of an equidistant
series are given, (as such or such known moments of time, ) together with their places in
that series, (as such particular extremes, or such particular intermediate moments, ) the
other moments of the series can then be all determined ; and farther, that the series
itself may be continued forward and backward, so as to include an unlimited number
of new moments, without losing its character of equidistance. Thus, if we know the
first extreme moment a, and the third quadrisector B’ of the total interval (from a
to B’”) in any equidistant series of five moments, a BB’ B’ B”, we can determine by
trisection the two first quadrisectors B and B’, and afterwards the last extreme moment
8”; and may then continue the series, forward and backward, so as to embrace other
moments B’, B*, &c., beyond the fifth of those originally conceived, and others also
such as E, E £’, &c., behind the first of the original five moments, that is, preceding
it in the order of progression of the series ; these new moments forming with the old
an equidistant series of moments, (which comprehends as a part of itself the original
series of five,) namely, the following unlimited series,
SEE ht Ra
and on Algebra as the Science of Pure Time. 307
Sh. bh ne BB! Be BB (29.)
constructed so as to satisfy the conditions of a continued analogy,
...BY— BY =B"—B’ =B" —B’ =B’ —B =B’—B=B—A=A—E=E—E=E—E... (30.)
8. By thus constructing and continuing an equidistant series, of which any two
moments are given, we can arrive at other moments, as far from those two, and as near
to each other, as we desire. For no moment 8 can be so distant from a given moment
A, (on either side of it, whether as later or as earlier,) that we cannot find others
still more distant, (and on the same side of a, still later or still earlier,) by continuing
(in both directions) any given analogy, or given equidistant series; and, therefore,
no two given moments, c and p, if not entirely coincident, can possibly be so near
to each other, that we cannot find two moments still more near by treating any twe
given distinct moments (A and B), whatever, as extremes of an equidistant series of
moments sufficiently many, and by inserting the appropriate means, or intermediate
moments, between those two given extremes. Since, however far it may be necessary
to continue the equidistant series c p...p’, with c and p for its two first moments, in
order to arrive at a moment pb" more distant from c than g is from a, it is only ne-
cessary to insert as many intermediate moments between a and B as between c and
p’, in order to generate a new equidistant series of moments, each nearer to the one
next it than p toc. ‘Three or more moments a Bc &c. may be said to be wniserial
with each other, when they all belong to one common continued analogy, or equi-
distant series; and though we have not proved (and shall find it not to be true) that
-any three moments whatever are thus uniserial moments, yet we see that if any two
moments be given, such as A and g, we can always find a third moment B’ uniserial
with these two, and differing (in either given direction) by less than any interval pro-
posed from any given third moment c, whatever that may be. This possibility of
indefinitely appproaching (on either side) to any given moment c, by moments
“ uniserial with any two given ones a and B, increases greatly the importance which
I would otherwise belong to the theory of continued analogies, or equidistant series
of moments. Thus if any two given dates, c and p, denote two distinct moments of
~ time, (c-£ b,) however near to each other they may be, we can always conceive their
diversity detected by inserting means sufficiently numerous between any two other
given distinct moments a and 8, as the extremes of an equidistant series, and then, if
necessary, extending this series in both directions beyond those given extremes, until
some one of the moments B’ of the equidistant series thus generated is found to fall
between the two near moments c and p, being later than the earlier, and earlier than
VOL. XVII. 30
308 Professor Hamitton on Conjugate Functions,
the later of those two. And, therefore, reciprocally, if in any case of two given
dates € and b,. we can prove that no moment B" whatever, of all that can be imagined
as uniserial with two given distinct moments a and s, falls thus between the momenta
c and p, we shall then have a sufficient proof that those two moments c and D are
identical, or, in other words, that the two dates c and p represent only one common
moment of time, (c=p,) and not two different moments, however little asunder.
And even in those cases in which we have not yet succeeded. in discovering
a rigorous proof of this sort, identifying a sought moment with a known one, or dis-
tinguishing the former from the latter, the conception of continued analogies offers
always a method of research, and of nomenclature, for investigating and. expressing,
or, at least, conceiving as investigated and expressed, with any proposed degree of
approximation if not with perfect accuracy, the situation of the sought moment in
the general progvession of time, by its relation to-a known equidistant series of
moments sufficiently close. ‘This might, perhaps, be a proper place, in a complete
treatise on the Science of Pure Time, to introduce a regular system of integer ordi-
nals, such as the words first, second, third, &c., with the written marks 1, 2, 3, &c.,
which answer both to them and to the cardinal or quotitative numbers, one, two,
three, &c.; but it is permitted and required, by the plan of the present essay, that we
should treat these spoken and written names of the integer ordinals and cardinals,
together with the elementary laws of their combinations, as already known and -
familiar, It is the more admissible in point of method to suppose this previous
acquaintance with the chief properties of integer numbers, as set forth in elementary
arithmetic, because these properties, although belonging to the Science of Pure Time;
as involving the conception of succession, may all be deduced fromthe unfolding of
that mere conception of succession, (among things or thoughts as counted,) without
requiring any notion of measurable intervals, equal or unequal, between successive
moments of time. Arithmetic, or the science of counting, is, therefore, a part, indeed,
of the Science of Pure Time, but a part so simple and familiar that it may be pre-
sumed to have been previously and separately studied, to some extent, by any one
who is entering on Algebra.
On steps in the progression of time ; their application (direct or inverse) ta moments,
so as to generate other moments ; and their combination with other steps, in the
way of composition or decomposition.
9. The foregoing remarks may have sufficiently shewn the importance, in the
general study of pure time, of the conception of a continued analogy or equidistant
and on Algebra as the Science of Pure Time. 309
series of moments. ‘This conception involves and depends on the conception of the
repeated transference of one common ordinal relation, or the continued application of
one common mental step, by which we pass, in thought, from any moment of such a
series to the moment immediately following. For this, and for other reasons, it is
desirable to study, generally, the properties and laws of the transference, or applica-
tion, direct or inverse, and of the composition or decomposition, of ordinal relations
between moments, or of steps in the progression of time; and to form a convenient
system of written signs, for concisely expressing and reasoning on such applications and
such combinations of steps.
In the foregoing articles, we have denoted, by the complex symbol p—a, the ordinal
relation of the moment B.to the moment A, whether that relation were one of identity
or of diversity ; and if of diversity, then whether it were one of subsequence or of
precedence, and‘in whatever degree. ‘Thus, having previously interposed the mark
= between two equivalent signs for one-common moment of time, we came to inter-
pose the same sign of equivalence between any two marks of one ordinal relation,
and to write
D—C=B-—A,
when we designed to express that the relations of p to c and of g to a were coin-
cident, being both relations of identity, or both relations of diversity; and if the
Jatter, then both relations of subsequence, or both relations of precedence, and both
in the same degree. In like manner, having agreed to interpose the mark + be-
tween the two signs of two moments essentially different from each other, we wrote
p—c+B—A,
when we wished to express that the ordinal relation of p to c (as identical, or sub-
_ sequent, or precedent) did no¢ coincide with the ordinal relation of the moment B to
a; and, more particularly, when we desired to distinguish between the two principal
cases of this non-coincidence of relations, namely the case when the relation of p to
¢ (as compared with that of s to a) was comparatively a relation of lateness, and the
‘case when the same relation (of p to c) was comparatively a relation of earliness,
_ we wrote, in the first case,
bp—c>B—a,
and in the second case,
D—c<B—AaA,
having previously agreed to write
BOA
310 Professor Haminton on Conjugate Functions,
if the moment B were later than the moment a, or
B¢A
if B were earlier than 4.
Now, without yet altering at all the foregoing conception of B—a, as the symbol
of an ordinal relation discovered by the comparison of two moments, we may in some
degree abridge and so far simplify all these foregoing expressions, by using a simpler
symbol of relation, such as a single letter 4 or b» &c. or in some cases the character
0, or other simple signs, instead of a complex symbol such as p—a, or p—c, &c.
Thus, if we agree to use the symbol 0 to denote the relation of identity between two
moments, writing
A Os (31.)
we may express the equivalence of any two dates B and a, by writing
B—A=0, (32.)
and may express the non-equivalence of two dates by writing
B-A+O0; (33.)
distinguishing the two cases when the moment B is later and when it is earlier than a,
by writing, in the first case,
B—A>O, (34..)
and in the second case,
B—aA<0, (35.)
to express, that as compared with the relation of identity 0, the relation B—A is im
the one case a relation of comparative lateness, and in the other case a relation of
comparative earliness : or, more concisely, by writing, in these four last cases re- —
spectively, which were the cases before marked (1.) (2.) (3.) and (4.)
a= 0; (36.)
a0, (37-)
a>O, (38.)
a<O, (89.)
if we put, for abridgement,
B—A=a, (40.)
Again, if we put, in like manner, for abridgement,
D—c=b, (41.)
and on Algebra as the Science of Pure Time. 311
the analogy (10.) namely,
D—C=B—A,
may be concisely expressed as follows,
b=a; (42.)
while the general non-analogy (15.),
pD—c=tB—A,
boEa 5 (43.)
may be expressed thus,
and the written expressions of its two cases (16.) and (17.), namely,
D—C>B—A
and p—c<B—A,
may be abridged in the following manner,
b>a, (44.)
and b<a. (45.)
Again, to denote a relation which shall be exactly the inverse or opposite of any
proposed ordinal relation a or b, we may agree to employ a complex symbol such
as 9 a or © b, formed by prefixing the mark ©, (namely, the initial letter O of the
Latin word Oppositio, distinguished by a bar across it, from the same letter used for
other purposes,) to the mark « or b of the proposed ordinal relation ; that is, we may
agree to use @ a to denote the ordinal relation of the moment a to B, or Ob to
denote the ordinal relation of ¢ to p, when the symbol a has been already chosen to
denote the relation of B to a, or b to denote that of p to c: considering the two
assertions
B—A=a, and a—B=0 a, (46.)
as equivalent each to the other, and in like manner the two assertions
DC =\b; and c—p=0 b, (47.)
and similarly in other cases. In this notation, the theorems (5.) (6.) (7.) (8.)
‘may be thus respectively written :
6 a=0, if a=0; (48.)
© xb0, if =O; (49.)
0 a <0, if a2>0; (50.)
© a>0, if a<0; (51.)
312 Professor Hamitton on Conjugate Functions,
and the theorem of inversion (12.) may be written thus :
0 b=Oa, if b=a, (52.)
The corresponding rules for inverting a non-analogy shew that, in general,
© +02, if ba; (53.)
and more particularly, that
Ol '<Os, iPS a (54.
and Ob>@a, if b< a, (55.)
It is evident also that ,
if Y=@a, then a=0 #; (56.)
that is, the opposite of the opposite of any proposed relation ais that proposed
relation itself; a theorem which may be concisely expressed as follows :
8 (8 a) —=2)5 (57.)
for, as a general rule of notation, when a complex symbol (as here © @) is substituted
in any written sentence (such as here the sentence »=© 2’) instead of a simple
symbol (which the symbol «', notwithstanding its accent, is here considered to be),
it is expedient, and in most cases necessary, for distinctness, to record and mark this
using of a complex as a simple symbol, by some such written warning as the enclosing
of the complex symbol in parentheses, or in brackets, or the drawing of a bar across
it. However, in the present case, no confusion would be likely to ensue from the
omission of such a warning ; and we might write at pleasure
9(Oa)=a, 8 fOa=a, Of[Oa] =a, O Oa= a, orsimplyO@O2=8. (58.)
10. For the purpose of expressing, in a somewhat similar notation, the properties
of alternations and combinations of analogies, set forth in the foregoing articles, with
some other connected results, and generally for the illustration and development of
the conception of ordinal relations between moments, it is advantageous to intro-
duce that other connected conception, already alluded to, of steps in the progression
of time ; and to establish this other symbolic definition, or conventional manner of
writing, namely,
B=(sp—A)+A4, or B=a+a if B—A=ajz (59.)
this notation 2+a, or (B—A)+a, corresponding to the above-mentioned concep-
tion of a certain mental step or act of transition, which is determined in direction
and degree by the ordinal relation a or gp—a, and may, therefore, be called ‘the
and on Algebra as the Science of Pure Time. 313
step 3,” or the step p—A, and which is such that by making this mental step, or
performing this act of transition, we pass, in thought, from the moment a to the
moment B, and thus suggest or generate (in thought) the latter from the former, as
a mental product or result B of the act « and of the object a. We may also express
the same relation between 8 and a by writing
A=(9 4)+B, or more simply a=0 2+8, (60.)
if we agree to write the sign 4 without parentheses, as if it were a simple or single
symbol, because there is no danger of causing confusion thereby ; and if we observe
that the notation A=© +B corresponds to the conception of another step, or
mental act of transition, 0 4, exactly opposite to the former step a, and such that by
it we may return (in thought) from the moment B to the moment a, and thus may
generate A as a result of the act 94 and of the object 8. The mark +, in this sort of
notation, is interposed, as a mark of combination, between the signs of the act and
the object, so as to form a complex sign of the result ; or, in other words, between
the sign of the transition (2 or @ ) and the sign of the moment (a or B) from which
that transition is made, so as to express, by a complex sign, (recording the suggestion
or generation of the thought, ) that other moment (8 or a) to which this mental tran-
sition conducts. And in any transition of this sort, such as that expressed by the
equation B= a + a, we may call (as before) the moment a, from which we pass, the
antecedent, and the moment 8, to which we pass, the consequent, of the ordinal rela-
tion a, or B—a, which suggests and determines the transition. In the particular
case when this ordinal relation is one of identity, (a=0,) the mental transition or
act (@ or 0) makes no change in the object of that act, namely in the moment a, but
only leads us to repeat the thought of that antecedent moment a, perhaps with a
new name B; in this case, therefore, the transition may be said to be null, or a null
step, as producing no real-alteration in the moment from which it is made. A step
not null, (»+.0,) corresponds to a relation of diversity, and may be called, by con-
trast, an effective step, because it is an act of thought which really alters its object,
namely the moment to which it is applied: An effective step a must be either a late-
making or an early-making step, according as the resultant moment «+a is later or
earlier than a; but even a null step 0 may be regarded as relatively late-making,
when compared with an early-making step 4, (0+A>a+<A, if 2<0,) or as relatively
early-making if compared. with a late-making step »; (O+a<b +a, if b>0;) and,
in like manner, of two unequal early-making steps, the lesser may be regarded as
relatively late-making, while of two unequal late-making steps the lesser step may be
considered as relatively early-making.. With these conceptions of the relative effecis
314 Professor Hamitton on Conjugate Functions,
of any two steps a and b, we may enunciate in words the non-analogy (44.), (b> a,
that is, b+A >a +a,) by saying that the step b as compared with the step 4 is relatively
late-making ; and the opposite non-analogy (45.), (> <a, that is, b+a <a-+4s,) by
saying that the step b as compared with a is relatively early-making.
11. After haying made any one step a from a proposed moment a to a resulting
moment represented (as before) by a+, we may conceive that we next make from
this new moment 4+ a new step b, and may denote the new result by the new com-
plex symbol b)+(a +a); enclosing in parentheses the sign 2+ of the olject of this
new act of mental transition, or (in other words) the sign of the new antecedent
moment, to mark that it isa complex used as a simple symbol; so that, in this
notation,
if B—a =a, and c—B=b, then c=b +(a +a). (61.)
It is evident that the total change or total step, effective or null, from the first
moment A to the last moment c, in this successive transition from a to B and from
B to c, may be considered as compounded of the two successive or partial steps @
and b, namely the step a from a tos, and the step > from Bs to c; and that the
ultimate ordinal relation of c to 4 may likewise be cénsidered as compounded of the
two intermediate (or suggesting) ordinal relations b and a, namely, the relation » of
¢ to B, and the relation 4 of B to a; a composition of steps or of relations which may
conveniently be denoted, by interposing, as a mark of combination, between the signs
of the component steps or of the component ordinal relations, the same mark +
which was before employed to combine an act of transition with its object, or an
ordinal relation with its antecedent. We shall therefore denote the compound trans-
ition from A to ©, or the compound relation of ¢ to a, by the complex symbol b + a,
writing,
C—A=b +a, if B—a=a, and c—B=b, (62.)
that is,
e=b+a, if p=a +A, C=b +B, Cpe +A. (63.)
For example, the case of coincidence between the moments A and c, that is, the
case when the resulting relation of c to a is the relation of identity, and when there-
fore the total or compound transition from a to © is null, because the two component
or successive steps a and » have been exactly opposite to each other, conducts to the
relations, ’
Gata=0; b+0b=0. (64.)
and on Algebra as the Science of Pure Time. 315
In general, the establishment of this new complex mark > +, for the compound
mental transition from A through B to c, permits us to regard the two written asser-
tions or equations
c=(>+4)+a and c=b +(a+a), (65.)
as expressing the same thing, or as each involving the other ; for which reason we are
at liberty to omit the parentheses, and may write, more simply, without fear of causing
confusion,
c=b+a+a, ifc=b+B, and B= a+; (66.)
because the complex symbol » + 2+ denotes only the one determined moment c,
whether it be interpreted by first applying the step 4 to the moment 4, so as to ge-
nerate another moment denoted by the complex mark « + a, and afterwards applying
to this moment the step denoted by >, or by first combining the steps a and > into
one compound step > + @, and afterwards applying this compound step to the original
moment A.
In like manner, if three successive steps abe have conducted successively (in
thought) from a to B, from gto c, and from c to p, and therefore ultimately and
upon the whole from a to p, we may consider this total transition from a to D as
compounded of the three steps a b ¢; we may also regard the resulting ordinal relation
of p to A as compounded of the three relations ¢, b, 2, namely of the relation ¢ of
p toc, the relation » of c to B, and the relation « of B to a; and may denote
this compound step or compound relation by the complex symbol « +» +, and the
last resulting moment p by the connected symbol ¢ + » +a +a; in such a manner
that
S
p—A=c+b+a, andp=c+b+a +a, (67.)
if B—A=a, c—gp=b, andp—c=c. :
_ For example,
e+@Oa+are, c+b+Ob=c,
Ob+b+a=a,e +Oc+a =a, (68.)
Remarks of the same kind apply to the composition of more successive steps than
three. And we see that in any complex symbol suggested by this sort of composition,
such as © + b +a +a, we are at liberty to enclose any two or more successive compo-
nent symbols, such as ¢ or b or a or A, in parentheses, with their proper combining
VOL. XVII. 3P
316 Professor Hamiiron on Conjugate Functions,
marks +, and to treat the enclosed set as if they formed only one single symbol ;
thus,
(69.)
e+b+a+A=>e4+b+(a+a)He+(b+a)+a
=(e+b+a)+a, &.,
the notation ¢ +(b+a)+A, for example, directing us to begin by combining (in
thought) the two steps 4 and > into one compound step » +, and then to apply
successively this compound step and the remaining step ¢ to the original moment A ;
while the notation (¢ +» +a)+A suggests a previous composition (in thought) of
all the three proposed steps 4, b, ©, into one compound step ¢ +» +a, and then the »
application of this one step to the same original moment. It is clear that all these
different processes must conduct to one common result ; and generally, that as, by the
very meaning and conception of a compound step, it may be applied to any moment
by applying in their proper order its component steps successively, so also may these
components be compounded successively with any other step, as a mode of com-
pounding with that other step the whole original compound.
We may also consider decomposition as well as composition of steps, and may pro-
pose to deduce either of two components a and > from the other component and from
the compound » +a. For this purpose, it appears from (68.) that we have the re-.
lations
a=Ob+c, and /=c +Oa, if c=b +a; (70.)
observing that a problem of decomposition is plainly a determinate problem, in the:
sense that if any one component step, such as here the step denoted by 0» +, or
that denoted by ¢ +9, has been found to conduct to a given compound ¢, when
combined in a given order with a given component > or a, then no other component
a or », essentially different from the one thus found, can conduct by the same process-
of composition to the same given compound step. We see then that each of the two
components a and » may be deduced from the other, and from the compound ¢, by
compounding with that given compound the opposite of the given component, in a
suitable order of composition, which order itself we shall shortly find to be indifferent.
Meanwhile it is important to observe, that though we have agreed, for the sake of
conciseness, to omit the parentheses about a complex symbol of the kind ©, when
combined with other written signs by the interposed mark +, yet it is in general ne-
cessary, if we would avoid confusion, to retain the parentheses, or some such con-
necting mark or marks, for any complex symbol of a step, when we wish to form, by —
prefixing the mark of opposition ©, a symbol for the opposite of that step: for
and on Algebra as the Science of Pure Time. ( gir
example, the opposite of a compound step » + must be denoted in some such
manner as 9 (> + 2), and not merely by writing 9> +. Attending to this remark,
we may write
@(>+2)=O2 +Ob, (71.)
because, in order to destroy or undo the effect of the compound step » + «, it is suf-
ficient first to apply the step 0 which destroys the effect of the last component step»,
and afterwards to destroy the effect of the first component step a by applying its op-
posite 92, whatever the two steps denoted by « and » may be. In like manner,
O@(¢+b+a)=O0a +O0b+0c; (72.)
and similarly for more steps than three.
12. We can now express, in the language of steps, several other general theorems,
for the most part contained under a different form in the early articles of this Essay.
Thus, the propositions (20.) and (21.), with their reciprocals, may be expressed by
saying that if equivalent steps be similarly combined with equivalent steps, whether in
the way of composition or of decomposition, they generate equivalent steps ; an asser-
tion which may be written thus :
if a=a, then’ b = a=b fa, a ba b,
b+Oa=b +O0a, Oa'+ b=Oa +b, (73.)
Ob+a=Ob +a, a+ Ob=a+Ob, &.
The proposition (25.) may be considered as expressing, that if two steps be equiva-
lent to the same third step, they are also equivalent to anh other ; or, that
if a'=a' and a’=a, then a’=a. ; (74.) Beg:
The theorem of alternation of an analogy (11.) may be included in the assertion
that in the composition of any two steps, the order of those two components may “5
changed, without altering the compound step ; or that
atb= ba, (75.)
For, whatever the four moments ABcpD may be, which construct any proposed ana-
logy or non-analogy, we may denote the step from a to B by a symbol such as 4, and
the step from B to p by another symbol », denoting also the step from a to c by \,
and that from c to p by #5 in such a manner that
B—A=4, D—B=b,c—A=V, D—C=8'; (76.)
318 Professor Hamitton on Conjugate Functions,
and then the total step from a to p may be denoted either by » +4 or by *+¥, ac-
cording as we conceive the transition performed by passing through 8 or through c ;
we have therefore the relation
Aye i ea (77.)
which becomes
a+b=b+a, (78.)
when we establish the analogy
D—c=B—A, that is, 2 = 5; (79.)
we see then that if the theorem (75.) be true, we cannot have the analogy (79.) with-
out haying also its alternate analogy, namely
b=, or D—B=C—A: (80.)
because the compound steps 2+ and 2+», with the common second component =,
could not be equivalent, if the first components b’ and » were not also equivalent
to each other. The theorem (75.) includes, therefore, the theorem of alternation.
Reciprocally, from the theorem of alternation considered as known, we can infer
the theorem (75.), namely, the indifference of the order of any two successive compo-
nents, », of a compound step: for, whatever those two component steps and >
may be, we can always apply them successively to any one moment A, so as to gene-
rate two other moments B and c, and may again apply the step 4 to c so as to gene-
rate a fourth moment p, the moments thus suggested having the properties
B=« +A, C=b-+ A; D=a +¢, (81.)
and being therefore such that
D—A=a+b, D—C=a=B—A; (82.)
by alternation of which last analogy, between the two pairs of moments 4 B and c D,
we find this other analogy,
D—B=c—A=b, D=b+B=b+2a+A, (83.)
and finally,
b+sa=D—A=an+d. (84.)
and on Algebra as the Science of Pure Time. 819
The propositions (22.) (23.) (24.), respecting certain combinations of analogies,
are included in the same assertion (75.) ; which may also, by (71.), be thus expressed,
at+b=0 (Oa + Ob), or, b+ a=O0 (8 b+6 a); (85.)
that is, by saying that it comes to the same thing, whether we compound any two
steps a and b themselves, or first compound their opposites © a, © b, into one com-
pound step 9 b+6 a, and then take the opposite of this. Under this form, the
theorem of the possibility of reversing the order of composition may be regarded as
evident, whatever the number of the component steps may be ; for example, in the
case of any three component steps a, b, c, we may regard it as evident that by apply-
ing these three steps successively to any moment a, and generating thus three
moments B, C, D, we generate moments related to A as a itself is related to those
three other moments B’, c’, p’, which are generated from it by applying successively,
in the same order, the three respectively opposite steps, © a, Ob, O ¢; that is, if
B=a+a, B=Oa+ A,
C=b+ 3B, c= Ob +8, (86.)
Dic +c) Dp) =\/Ole 1c’,
then the sets B’ AB, c’ Ac, D' AD, containing each three moments, form so many
continued analogies or equidistant series, such that
B-A=A-—B
c—-A=A-—C (87.)
D—A=A—D’
and therefore not only b + 2 =© (Ob +a), as before, but also
c+b+a=0 (0c+Ob+9O 4), (88.)
that is, by (72.) and (57.),
ctbtaxtatbt+c; (89.)
and similarly for more steps than three.
The theorem (89.) was contained, indeed, in the reciprocal of the proposition
(24..), namely, in the assertion that
if D—c=B-—A,
and Dp —c=B—A, (90.)
then D’—p=a—a’,
320 Professor Hamitton on Conjugate Functions,
and, therefore, by alternation,
D’—a=p—A’; (91.)
for, whatever the three steps abe may be, we may always conceive them applied suc-
cessively to any moment 4, so as to generate three other moments B, c, pv’, such that
B=a+A, C=bd+B, D=c +6, (92.)
and may also conceive two other moments a’ and p such that Bc D may be succes-
sively generated from a’ by applying the same three steps in the order c, b, a, so that
B=c+A, C=b+B, D=a+C3 (93.)
and then the two first analogies of the combination (90.) will hold, and, therefore,
also the last, together with its alternate (91.) ; that is, the step from a to D’, com-
pounded of the three steps abc, is equivalent to the step from a’ to p, compounded of
the same three steps in the reverse order cba,
Since we may thus reverse the order of any three successive steps, and also the
order of any two which immediately follow each other, it is easy to see that we may
interchange in any manner the order of three successive steps ; thus
pitt, et de (94.)
=atb+crlDatet+bob+t+atea
We might also have proved this theorem (94.), without previously establishing the ©
less general proposition (S9.), and in a manner which would extend to any number of
component steps ; namely, by observing that when any arrangement of component
steps is proposed, we may always reserve the first (and by still stronger reason any
other) of those steps to be applied the last, and leave the order of the remaining steps
unchanged, without altering the whole compound step; because the components
which followed, in the proposed arrangement, that one which we now reserve for the
last, may be conceived as themselves previously combined into one compound step,
and this then interchanged in place with the reserved one, by the theorem respecting
the arbitrary order of any two successive steps. In like manner, we might reserve
any other step to be the last but one, and any other to be the last but two, and so on ;
by pursuing which reasoning it becomes manifest that when any number of component
steps are applied to any original moment, or compounded with any primary step, their
order may be altered at pleasure, without altering the resultant moment, or the whole
compounded step: which is, perhaps, the most important and extensive property of
the composition of ordinal relations, or steps in the progression of time.
and on Algebra as the Science of Pure Time. 321
On the Multiples of a given base, or unit-step ; and on the Algebraic Addition,
Subtraction, Multiplication, and Division, of their determining or multipling
Whole Numbers, whether positive, or contra-positive, or null.
13. Let us now apply this general theory of successive and compound steps, from
any one moment to any others, or of component and compound ordinal relations
between the moments of any arbitrary set, to the case of an equidistant series of
moments,
EGGERE: AUBMB, Busi (29.)
constructed so as to satisfy the conditions of a continued analogy,
... B —B = B’—B=RB—A=A—E=E—E =E—E’, &e.; (30.)
and first, for distinctness of conception and of language, let some one moment a of
this series be selected as a standard with which all the others are to be compared, and
let it be called the zero-moment ; while the moments B, B’, &c. which follow it, in the
order of progression of the series, may be distinguished from those other moments
E, E, &c., which precede it in that order of progression, by some two contrasted
epithets, such as the words positive and contra-positive: the moment B being called
the positive fi'st, or the first moment of the series on the positive side of the zero ;
while in the same plan of nomenclature the moment B' is the positive second, B’ the
positive third, © the contra-positive first, © the contra-positive second, and so forth.
By the nature of the series, as composed of equi-distant moments, or by the condi-
tions (30.), all the positive or succeeding moments B B’ &c. may be conceived as
generated from the zero-moment a, by the continual and successive application of
one common step a, and all the contra-positive or preceding moments E £’ &c. may be
conceived as generated from the same zero-moment a, by the continual and successive
application of the opposite step © a, so that we may write
Bo=a+A, Bo=a+B, B’=a+B, &c., (95.)
and.
E=Oa +A, F’'=Oa +E, E =—Oa +E, &C.; (96.)
while the standard or zero-moment a itself may be denoted by the complex symbol
© + a, because it may be conceived as generated from itself by applying the null step
322 Professor Haminton on Conjugate Functions,
0. Hence, by the theory of compound steps, we have expressions of the following
sort for all the several moments of the equi-distant series (29.) :
U
E’—@a+Oa+@Qa+A,
E =Oa+6a+t+A,
= 0 e-vA5 (97.)
=a. As
SSC Ot eR
SS Gage am oss
with corresponding expressions for their several ordinal relations to the one standard
moment A, or for the acts of transition which are made in passing from a to them,
namely :
E’—A=0a+ O0a+ Qa,
E —aA=O0a+Oa,
E —Aa=QOa,
A —A=0; (98.)
B —A=a,
B —A>a+t+a,
Bi —A=a -atia;
&e.
'
The simple or compound step, a, or a +a, &c., from the zero-moment a to any
positive moment 8B or B’ &c. of the series, may be called a positive step ; and the
opposite simple or compound step, © a, or 92+6a, &c., from the same zero-
moment A to any contra-positive moment E or rE’, &c., of the series, may be called a
contra-positive step ; while the null step 0, from the zero-moment a to itself, may be
called, by analogy of language, the zero-step. The original step a is supposed to
be an effective step, and not a null one, since otherwise the whole series of moments
(97.) would reduce themselves to the one original moment a; but it may be either a
late-making or an early-making step, according as the (mental) order of progression of
that series is from earlier to later, or from later to earlier moments. And the whole
series or system of steps (98.), simple or compound, positive or contra-positive, effec-
tive or null, which serve to generate the several moments of the equi-distant series
(29.) or (97.) from the original or standard moment a, may be regarded as a system of
steps generated from the original step a, by a system of acts of generation which are
all of one common kind; each step haying therefore a certain relation of its own to
lige at ma,
| ~o tei <p re
and on Algebravas the Science of Pure Time. 323
that original step, and these relations having all a general resemblance to each other, so
that they may be conceived as composing a certain system of relations, having all one
common character. ‘To mark this common generation of the system of steps (98.)
from the one original’ step a, and their common relation thereto, we may call them all
by the common name of multiples of that original step, and may say that they are or
may be (mentally) formed by multipling that common base, or unit-step, a; distin-
guishing, however, these several multiples among themselves by peculiar or special
names, which shall serve to mark the peculiar relation of any one multiple to the
base, or the special act of multipling by which it may be conceived to be generated
therefrom.
Thus, the null step, or zero-step, 0, which conducts to the zero-moment a, may be
called, according to this way of conceiving it, the zero multiple of the original step a; and
the positive (effective) steps, simple or compound, a, ata, a+ a+ a, &¢., may
be called by thé general name of positive multiples of 2, and may be distinguished by
the special ordinal names of first, second, third, &c., so that the original step is, in
this view, its own first positive multiple; and finally, the contra-positive (but effective )
steps, simple or compound, namely, 0 a, 9a + 0, Oa + Oa + Oa, &e., may be
called the first contra-positive multiple of a, the second contra-positive multiple of
the same original step a, and so forth. Some particular multiples have particular
and familiar names ; for example, the second positive multiple of a step may also be
called the double of that step, and the third positive multiple may be called familiarly
the triple. In general, the original step a may be called (as we just now agreed) the
common base (or unit) of all these several multiples; and the ordinal name or
number, (such as zero, or positive first, or contra-positive second,) which serves as a
special mark to distinguish some one of these multiples from every other, in the
-general series of such multiples (98.), may be called the determining ordinal: so
that any one multiple step is sufficiently described, when we mention its base and its
determining ordinal. In conformity with this conception of the series of steps (98.,)
as a series of multiples of the base 2, we may denote them by the following series of
written symbols,
nielostete Oa, -2' Ola, 1 Olas? Ola Oa 2iay) Sia, .- (99.)
and may denote the moments themselves of the cqui-distant series (29.) or (97.) by
_. the symbols,
VOL. XVII. 3 Q
324 Professor Hamitton on Conjugate Functions,
arenes
B'=30a +A,
E =20a + A,
E =1@a-+a,
Ko Olay As (100.)
Boal Sate,
B= 2a +A,
B= 3a +4,
&e. ; J
in which
Ora=0; (101.)
and
lia = sa, I@pa— Ola,
Qara+a 2e6a=O0a+Oa,
3 teat aia gorzerson | | (102.)
&C. 5 &e.
The written sign O in Oa is here equivalent to the spoken name zero, as the
determining ordinal of the null step from a to a, which step was itself also denoted
before by the same character 0, and is here considered as the zero-multiple of the
base a; while the written signs 1, 2, 3, &c., in the symbols of the positive multiples
la, 2a, 3a, &c., correspond to and denote the determining positive ordinals, or the
spoken names first positive, second positive, third positive, &c.; and, finally, the re-
maining written signs 1 ©, 2 0, 3 @, &c., which are combined with the written sign of
the base a, in the symbols of the contra-positive multiples 1 @a, 2 Oa, 30a, &c., cor-
respond to and denote the determining ordinal names of those contra-positive multiples,
that is, they correspond to the spoken names, first contra-positive, second contra-posi-
tive, third contra-positive, &c.: so that the series of signs of multiple steps (99.), is
formed by combining the symbol of the base a with the following series of ordinal
symbols,
conse BESS) Os al a Gs ae (103.)
We may also conceive this last series of signs as equivalent, not to ordinal names,
such as the numeral word first, but to cardinal names, such as the numeral word
one ; or more fully, positive cardinals, contra-positive cardinals, and the null cardinal
(or number none); namely, the system of all possible answers to the following com-
plex question: ‘*‘ Have any effective steps (equivalent or opposite to the given
base a) been made (from the standard moment a), and if any, then How many, and
In which direction 2”? In this view, 3 © is a written sign of the cardinal name or
and on Algebra as the Science of Pure Time. 825
number contra-positive three, as a possible answer to the foregoing general question ;
and it implies, when prefixed to the sign of the base a, in the complex written sign
3 © a of the corresponding multiple step, that this multiple step has been formed,
(as already shown in the equations (102.), ) by making three steps equal to the base
a in length, but in the direction opposite thereto. Again, the mark 1 may be re-
garded as a written sign of the cardinal number positive one, and 1 a denotes (in
this view) the step formed by making one such step as a, and in the same direction,
that is, (as before,) the original step a itself ; and O denotes the cardinal number
none, so that O a is (as before) a symbol for the null step from a to a, which step we
haye also marked before by the simple symbol 0, and which is here considered as
formed by making no effective step like a. In general, this view of the numeral
signs (103.), as denoting cardinal numbers, conducts to the same ultimate interpre-
tations of the symbols (99.), for the steps of the series (98.), as the former view,
which regarded those signs (103.) as denoting ordinal numbers. ;
If we adopt the latter view of those numeral signs (103.), which we shall call by the
common name of whole (or integer) numbers, (as distinguished from certain broken
or fractional numbers to be considered afterwards,) we may conveniently continue to
use the word mudtiple (occasionally) as a verb active, and may speak of the several
multiple steps of the series (98.), or (99.), as formed from the base a, by mudtipling
that base by the several whole (cardinal) numbers: because every multiple step may be
conceived as generated (in thought) from the base, by a certain mental act, of which
the cardinal number is the mark. Thus we may describe the multiple step 3 0 a,
(which is, in the ordinal view, the third contra-positive multiple of a,) as formed from
the base a by multipling it by contra-positive three. Some particular acts of multi-
pling have familiar and special names, and we may speak (for instance) of doubling
or tripling a step, instead of describing that step as being multipled by positive two,
or by positive three. In general, to distinguish more clearly, in the written symbol
of a multiple step, between the base and the determining number (ordinal or cardi-
nal), and to indicate more fully the performance of that mental act (directed by the
number) which generates the multiple from the base, the mark x may be inserted
between the sign of the base, and the sign of the number; and thus we may
denote the series of multiple steps (99.) by the following fuller symbols,
SLOmas we OLE ia, UO Xsa) OLX a, ell X. a, eX B08 Xmen OCC ee LOd, )
and which 1 x a (for example) denotes the original step a itself, and 2 x a represents
the double of that original step.
326 Professor Hamitron on Conjugate Functions,
It is manifest that in this notation
nmOxa=in xOa=O(n ee (105.)
and m x a=nOxO@a=O(nOxa)=O( nm xa),
if 7» denote any one of the positive numbers 1, 2, 3, &c. and if m © denote the cor-
responding contra-positive number, 1 0, 20, 30, &c.; for example, the equation
“2x a= 2x @a is true, because it expresses that the second contra-positive multiple
of the base a is the same step as the second positive multiple of the opposite base or
step © a, the latter multiple being derived from this opposite base by merely doubling
its length without reversing its direction, while the former is derived from the original
base a itself by both reversing it in direction and doubling it in length, so that both
processes conduct to the one common compound step, 02+ Oa. In like manner
the equation 2 x a= 2 © x Oa is true, because by first reversing the direction of the
original step a, and then taking the reversed step Oa as a new base, and forming the
second contra-positive multiple of it, which is done by reversing and doubling, and
which is the process of generation expressed by the symbol 20x Oa, we form in
the end the same compound step, a +a, asif we had merely doubled a. We may
also conveniently annex the mark of opposition 0, at the left hand, to the symbol of
any whole number, 7 or n 9 or 0, in order to form a symbol of its opposite number,
nO, n, or 0; and thus may write
0 n=n 9, O(n O)=n, 8 O=0; (106.)
if we still denote by n any positive whole number, and if we call two whole numbérs
opposites of each other, when they are the determining or multipling numbers of
two opposite multiple steps.
14. Two or more multiples such as » x a, v x a, & x a, of the same base a, may
be compounded as successive steps with each other, and the resulting or compound
step will manifestly be itself some multiple, such as w x a, of the same common base
a; the signs n, v, &, denoting here any arbitrary whole numbers, whether positive, or
contra-positive, or null, and w denoting another whole number, namely the deter-
mining number of the compound multiple step, which must evidently depend on the
determining numbers » v € of the component multiple steps, and on those alone,
according to some general law of dependence. This law may conveniently be de-
noted, in writing, by the same mark of combination + which has been employed
already to form the complex symbol of the compound step itself, considered as de-
pending on the component steps ; that is, we may agree to write
aod
and on Algebra as the Science of Pure Time. 327
w—=v+p, when w x a=(v X a)+(m x a), (107.)
and
wr=E+vd4-p, when w x a= (e x a)+ x a) +(u x a), (108.)
together with other similar expressions for the case of more component steps than
three. In this notation, ;
(Exa)t@ xa) tx a)aE+vtu) xe, (109.)
&e.
(v x a)+(m x a)=(vtn) x a, \
whatever the whole numbers » v & may be; equations which are to be regarded here
as true by definition, and as only serving to explain the meaning attributed to such
complex signs as v+p, or €+v+p, when wv & are any symbols of whole nun-
bers: although when we farther assert that the equations (109.) are true inde-
pendently of the base or unit-step a, so that symbols of the form v + » or §+v +p,
denote whole numbers independent of that base, we express in a new way a theorem
which we had before assumed to be evidently true, as an axiom and not a definition,
respecting the composition of multiple steps.
In the particular case when the whole numbers denoted by , v & are positive, the
law of composition of those numbers expressed by the notation v + or E+v+ 4,
as explained by the equations (109.), is easily seen to be the law called addition of
numbers (that is of quotities) in elementary arithmetic ; and the quotity of the com-
pound or resulting whole number is the arithmetical swm of the quotities of the com-
ponent numbers, this arithmetical swm being the answer to the question, How many
things or thoughts does a total group contain, if it be composed of partial groups
of which the quotities are given, namely the numbers to be arithmetically added.
For example, since (3 x a) +(2 x a) is the symbol for the total or compound mul-
tiple step composed of the double and the triple of the base a, it must denote the
quintuple or fifth positive multiple of that base, namely 5 x a; and since we have
agreed to write
(8 x a)+(2xa)=(3 + 2) xa,
we must interpret the complex symbol 3 + 2 as equivalent to the simple symbol 5;
in seeking for which latter number five, we added, in the arithmetical sense, the given
component numbers fwo and three together, that is, we formed their arithmetical
sum, by considering how many steps are contained in a total group of steps, if the
component or partial groups contain two steps and three steps respectively. In like
328 Professor Hamitron on Conjugate Functions,
manner, if we admit in arithmetic the idea of the cardinal number none, as one of the
possible answers to the fundamental question How many, the rules of the arith-
metical addition of this number to others, and of others to it, and the properties
of the arithmetical sums thus composed, agree with the rules and properties
of such combinations as 0 +p, &+v+0, explained by the equations (109.),
when the whole numbers, p», v, £ are positive; we shall, therefore, not clash
in our enlarged phraseology with the language of elementary arithmetic, respecting
the addition of numbers regarded as answers to the question How many, if we now
establish, as a definition, in the more extensive Science of Pure Time, that any com-
bination of whole numbers up v &, of the form v + p, or E+v+ p, interpreted so as to
satisfy the equations (109.), is the swm of those whole numbers, and is composed by
adding them together, whether they be positive, or contra-positive, or null, But as a
mark that these words swm and adding are used in AuGEpra (as the general Science
of Pure Time), in a more extensive sense than that in which -drithmetic (as the
science of counting) employs them, we may, more fully, call v + u the algebraic sum
of the whole numbers » and v, and say that it is formed by the operation of algebrai-
cally adding them together, v to p.
In general, we may extend the arithmetical names of sum and addition to every
algebraical combination of the class marked by the sign +, and may give to that
combining sign the arithmetical name of Plus ; although in Algebra the idea of
more, (originally implied by plus,) is only occasionally and accidentally myolved in
the conception of such combinations. or example, the written symbol b +a, by
which we haye already denoted the compound step formed by compounding the step b
as a successive step with the step a, may be expressed in words by the phrase
‘*a plus b,”’ (such written algebraic expressions as these being read from right to left, )
or ‘the algebraic sum of the steps a and b ;”? and this algebraic sum or compound
step b + a may be said to be formed by “algebraically adding » to a :” although this
compound step is only occasionally and accidentally greater in length than its com-
ponents, being necessarily shorter than one of them, when they are both effective
steps with directions opposite to each other. Even the application of a step a to a
moment A, so as to generate another moment a + A, may not improperly be called
(by the same analogy of language) the algebraic addition of the step to the moment,
and the moment generated thereby may be called their algebraic sum, or ‘the original
moment plus the step ;” though in this sort of combination the moment and the step
to be combined are not even homogeneous with each other.
With respect to the process of calculation of an algebraic sum of whole numbers,
the following rules are evident consequences of what has been already shown respect-
and on Algebra as the Science of Pure Time. 329
ing the composition of steps. In the first place, the numbers to be added may be
added in any arbitrary order ; that is,
vtpoputy,
E+vtyr=ut+éEt+v= &e., (110.)
&e. ;
we may therefore collect the positive numbers into one algebraical sum, and the con-
tra-positive into another, and then add these two partial sums to find the total sum,
omitting (if it anywhere occur) the number None or Zero, as not capable of altering
the result. In the next place, positive numbers are algebraically added to each other,
by arithmetically adding the corresponding arithmetical numbers or quotities, and
considering the result as a positive number ; thus positive two and positive three,
when added, give positive five: and contra-positive numbers, in like manner, are al-
gebraically added to each other, by arithmetically adding their quotities, and consi-
dering the result as a contra-positive number ; thus,contra-positive two and contra-po-
sitive three have contra-positive five for their algebraic sum. In the third place, a
positive number and a contra-positive, when the quotity of the positive exceeds that
of the contra-positive, give a positive algebraic sum, in which the quotity is equal to
that excess ; thus positive five added to contra-positive three, gives positive two for
the algebraic sum: and similarly, a positive number and a contra-positive number, if
the quotity of the contra-positive exceed that of the positive, give a contra-positive
algebraic sum, with a quotity equal to the excess; for example, if we add positive
three to contra-positive five, we get contra-positive two for the result. Tinally, a posi-
tive number and a contra-positive, with equal quotities, (such as positive three and
contra-positive three,) destroy each other by addition ; that is, they generate as their
algebraic sum the number None or Zero.
It is unnecessary to dwell on the algebraical operation of decomposition of multiple
steps, and consequently of whole or multipling numbers, which corresponds to and
includes the operation of arithmetical subtraction ; since it follows manifestly from
the foregoing articles of this Essay, that the decomposition of numbers (like that of
steps) can always be performed by compounding with the given compound number
‘(that is, by algebraically adding thereto) the opposite or opposites of the given com-
ponent or components: the number or numbers proposed to be subtracted are there-
fore either to be neglected if they be null, since in that case they have no effect, or
else to be changed from positive to contra-positive, or from contra-positive to positive,
(their quotities being preserved,) and then added (algebraically) in this altered state.
Thus, positive five is subtracted algebraically from positive two by adding contra-posi-
330 Professor Hamiiton on Conjugate Functions,
tive five, and the result is contra-positive three ; that is, the given step 2 x a or 2a
x
g
may be decomposed into two others, of which the given component step 5 x a is one,
and the sought compo: en’ step 3 Oa is the other.
15. Any multiple step «a may be treated as a new base, or new unit-step; and
thus we may generate from it a new system of multiple steps. It is evident that these »
multiples of a multiple of a step are themselves also multiples of that step; that is, if
we first multiple a given base or unit-step a by any whole number », and then again
multiple the result « x a by any other whole number »v, the final result v x (u x a)
will necessary be of the form w x a, w being another whole number. It is easy also
to see that the new multipling number, such as w, of the new or derived multiple, must
pend on the old or given multipling numbers, such as » and », and on those alone ;
and the law ef its dependence on them may be conveniently expressed by the same
mark of combination x which we have already used to combine any multipling
number with its base ; so that we may agree to write
w =v Xp, when» x a=v x (ux a). eyo)
With this definition of the effect of the combining sign x, when interposed between
the signs of two whole numbers, we may write
yx (ue Xa) = (Xp) Kia x 25 (112.)
omitting the parentheses as unnecessary ; because, although their absence permits us
to interpret the complex symbol v x « x a either as v x (u x a) or as (vxp) Xa,
yet both the processes of combination thus denoted conduct to one common result, or
ultimate multiple step. . (Compare article 11.)
When x and v are positive numbers, the law of combination expressed by the nota-
tion » x #, as above explained, is easily seen to be that which is called Multiplication
in elementary Arithmetic, namely, the arithmetical addition of a given number v of
equal quotities » ; and the resulting quotity v x m is the arithmetical product of the
numbers to be combined, or the product of « multiplied by v: thus we must, by the
2
definition (112.), interpret 3 x 2 as denoting the positive number 6, because
3x (2x a)=6 x a, the triple of the double of any step a being the sextuple of that
step; and the quotity 6 is, for the same reason, the arithmetical product of 2 multi-
plied by 3, in the sense of being the answer to the question, How many things or
thoughts (in this case, steps) are contained in a total group, if that total group be
composed of 3 partial groups, and if 2 such things or thoughts be contained in_
each of these? From this analogy to arithmetic, we may in general call v x » the
and on Algebra as the Science of Pure Tune. 331
product, or (more fully) the algebraic product, of the whole numbers p» and v, whe-
ther these, which we may call the fuctors of the product, be positive, or contra-
positive, or null; and may speak of the process of combination of those numbers, as
the multipling, or (more fully) the algebraic multipling of » by v: reserving still the
more familiar arithmetical word ‘multiplying’? to be used in algebra in a more
general sense, which includes the operation of multipling, and which there will soon
be occasion to explain.
In like manner, three or more whole numbers, », v, &, may be used successively to
multiple a given step or one another, and so to generate a new derived multiple of the
original step or number ; thus, we may write
Ex fu x (ux a)}=é x {(v x #) x w= (Oa neat, (113.)
the symbol ~ x v x » denoting here a new whole number, which may be called the
algebraic product of the three whole numbers p, v, & those numbers themselves being
called the factors of this product. With respect to the actual processes of such
multipling, or the rules for forming such algebraic products of whole numbers,
(whether positive, or contra-positive, or null,) it is sufficient to observe that the pro-
duct is evidently null if any one of the factors be null, but that otherwise the product
is contra-positive or positive, according as there is or is not an odd number (such as
one, or three, or five, &c.) of contra-positive factors, because the direction of a step
is not changed, or is restored, when it is either not reversed at all, or reversed an
even number of times; and that, in every case, the quotity of the algebraic product
is the arithmetical product of the quotities of the factors. Hence, by the properties
of arithmetical products, or by the principles of the present essay, we see that in
forming an algebraical product the order of the factors may be altered in any manner
without altering the result, so that
yXpopxr, ExvxpaspxeExv=Ke., &e. 5 (114)
and that any one of the factors may be decomposed in any manner into algebraical
parts or component whole numbers, according to the rules of algebraic addition and
subtraction of whole numbers, and each part separately combined as a factor with the
other factors to form a partial product, and then these partial products algebraically
added together, and that the result will be the total product ; that is,
(' #y) x n=O! xn) + (vxn), t (115.)
yx (we t+p=( xv')4+ xp), &e.
VOL. XVII. Sue
332 Professor Hamitton on Conjugate Functions,
Again, we saw that if a factor » be null, the product is then null also,
52 Oi 0) 2 (116.)
because the multiples of a null multiple step are all themselves null steps. But if, in
a product of two whole numbers, v x «, the first factor » (with which by (114.)
the second factor v may be interchanged) be given, and effective, that is, if it be
any given positive or contra-positive whole number, («=E0,) then its several multi-
ples, or the products of the form v x », form an indefinite series of whole numbers,
sem OlO Xius 2 One lV OPXiis) Onxcursnil -Xqus 2S. Us OOS lyases (117.)
such that any proposed whole number w, whatever, must be cither a number of this
series, or else included between two successive numbers of it, such as » x » and
(1+v) x », being on the positive side of one of them, and on the contra-positive side
of the other, in the complete series of whole numbers (103.). In the one case, we can
satisfy the equation
wv Xun, or, 9 Xn) +w=0, (118.)
by a suitable choice of the whole number v; in the other case, we cannot indeed do
this, but we can choose a whole number v, such that
w=pt+(vxp), Or, O(vxp) +o=P~, (119.)
p being a whole number which lies between 0 and » in the general series of whole
numbers (103.), and which therefore has a quotity less than the quotity of that given
first factor », and is positive or contra-positive according as « is positive or con-
tra-positive. In each case, we may be said (by analogy to arithmetical division) to
have algebraically divided (or rather measured), accurately or approximately, the
whole number w by the whole number p, and to have found a whole number v which
is either the accurate quotient (or measure), as in the case (118.), or else the next
preceding integer, as in the other case (119.); in which last case the whole number p
may be called the remainder of the division (or of the measuring). In this second
case, namely, when it is impossible to perform the division, or the measuring, exactly,
in whole numbers, because the proposed dividend, or mensurand, w, is not contained
among the series (117.) of multiples of the proposed divisor, or measurer, », we may
choose to consider as the approximate integer quotient, or measure, the next suc-
ceeding whole number 1 + v, instead of the next preceding whole number v; and then
we shall have a different remainder, 94+ p, such that
w=(On+0)+(+v x p), (120.)
and on Algebra as the Science of Pure Time. 333
which new remainder © » + has still a quotity less than that of », but lies between
0 and 6x, instead of lying (like ») between 0 and », in the general series of whole
numbers (103.), and is therefore contra-positive if « be positive, or positive if « be
contra-positive. With respect to the actual process of calculation, for discovering
whether a proposed algebraical division (or measuring), of one whole number by
another, conducts to an accurate integer quotient, or only to two approximate integer
quotients, a next preceding and a next succeeding, with positive and contra-positive
remainders ; and for actually finding the names of these several quotients and re-
mainders, or their several special places in the general series of whole numbers : this
algebraical process differs only by some slight and obvious modifications (on which it
is unnecessary here to dwell,) from the elementary arithmetical operation of di-
viding one quotity by another ; that is, the operation of determining what multiple
the one is of the other, or between what two successive multiples it is contaimed.
Thus, having decomposed by arithmetical division the quotity 8 into the arithmetical
sum of 1 x 5 and 3, and having found that it falls short by 2 of the arithmetical pro-
duct 2 x 5, we may easily infer from hence that the algebraic whole number contra-
positive eight can be only approximately measured (in whole numbers), as a mensur-
and, by the measurer positive five ;
?
the next succeeding integer quotient or measure
being contra-positive one, with contra-positive three for remainder, and the next pre-
ceding integer quotient or measure being contra-positive two, with positive two as the
remainder. It is easy also to see that this algebraic measuring of one whole number
by another, corresponds to the accurate or approximate measuring of one step by
another. And in like manner may all other arithmetical operations and reasonings
upon quotities be generalised in Algebra, by the consideration of multiple steps, and
of their connected positive and contra-positive and null whole numbers.
On the Sub-multiples and Fractions of any given Step in the Progression of Time ;
on the Algebraic Addition, Subtraction, Multiplication, and Division, of Re-
' ciprocal and Fractional Numbers, positive and contra-positive ; and on the
impossible or indeterminate act of sub-multipling or dividing by zero.
16. We have seen that from the thought of any one step a, as a base or unit-
step, we can pass to the thought of a series or system of multiples of that base,
namely, the series (98.) or (99.) or (104.), having each a certain relation of its own
334 Professor Hamiiron on Conjugate Functions,
to the base, as such or such a particular multiple thereof, or as mentally generated
from that base by such or such a particular act of multipling; and that every such
particular relation, and every such particular act of multipling, may be distinguished
from all such other relations, and from all such other acts, in the entire series or
system of these relations, and in the entire system of these acts of multipling, by its
own special or determining whole number, whether ordinal or cardinal, and whether
positive, or contra-positive, or null. Now ‘every such relation or act must be con-
ceived to have a certain inverse or reciprocal, by which we may, in thought, connect
the base with the multiple, and return to the former from the latter: and, generally,
the conception of passing (in thought) from a base or unit-step to any one of its
multiples, or of returning from the multiple to the base, is included in the more com-
prehensive conception of passing from any one such multiple to any other ; that is,
from any one step to any other step commensurable therewith, two steps being said to
be commensurable with each other when they are multiples of one common base or
unit-step, because they have then that common base or unit for their common mea-
surers The base, when thus compared with one of its own multiples, may be called
a sub-multiple thereof ; and, more particularly, we may call it the ‘second positive
sub-multiple” of its own second positive multiple, the “first contra-positive sub-
multiple”? of its own first contra-positive multiple, and so forth; retaining always, to
distinguish any one sub-multiple, the determining ordinal of the multiple to which it
corresponds: and the act of returning from a multiple to the base, may be called an
act of swb-multipling or (more fully) of sub-multipling by the same determining
cardinal number by which the base had been multipled before; for example, we may
return to the base from its second contra-positive multiple, by an act of thought
which may be called sub-multipling by contra-positive two. Some particular sub-
multiples, and acts of sub-multipling, have particular and familiar names; thus, the
second positive sub-multiple of any given step, and the act of sub-multipling a given
step by positive two, may be more familiarly described as the half of that given step,
and as the act of halving it. And the more comprehensive conception above men-
tioned, of the act of passing from any one step b to any other step ¢ commensurable
therewith, or from any one to any other multiple of one common measure, or base,
or unit-step a, may evidently be resolved into the foregoing conceptions of the acts
of multipling and sub-multipling; since we can always pass first by an act of sub-
multipling from the given step b, considered as a multiple of the base a, to that
base a itself, as an auxiliary or intermediate thought, and then proceed, by an act of
multipling, from this auxiliary thought or step, to its other multiple c. Any one
step ¢ may therefore be considered as a multiple of a sub-multiple of any other
and on Algebra as the Science of Pure Time. 335
step b, if those two steps be commensurable ; and the act of passing from the one to
the other is an act compounded of sub-multipling and multipling.
Now, all acts thus compounded, besides the acts of multipling and sub-multipling
themselves, (and other acts, to be considered afterwards, which may be regarded as
of the same kind with these, being connected with them by certain intimate relations,
and by one common character,) may be classed in algebra under the general name of
multiplying acts, or acts of algebraic multiplication ; the object on which any such
act operates being called the muléiplicand, and the result being called the product ;
while the distinctive thought or sign of such an act is called the algebraic multiplier,
or multiplying number : whatever this distinctive thought or sign may be, that is, what-
ever conceived, or spoken, or written specific rule it may involve, for specifying one
particular act of multiplication, and for distinguishing it from every other. ‘The
relation of an algebraic product to its algebraic multiplicand may be called, in general,
ratio, or algebraic ratio ; but the particular ratio of any one particular product to
its own particular multiplicand, depends on the particular act of multiplication by
which the one may be generated from the other: the mwmber which specifies the act
of multiplication, serves therefore also to specify the resulting ratio, and every
number may be viewed either as the mark of a ratio, or as the mark of a multiplica-
tion, according as we conceive ourselves to be analytically examining a product
already formed, or synthetically generating that product.
We have already considered that series or system of algebraic integers, or whole
numbers, (positive, contra-positive, or null,) which mark the several possible ratios of
all multiple steps to their base, and the several acts of multiplication by which the
former may be generated from the latter; namely all those several acts which we
haye included under the common head of multipling. The inverse or reciprocal acts
of sub-multipling, which we must now consider, and which we have agreed to regard
as comprehended under the more general head of multiplication, conduct to a new
class of multiplying numbers, which we may call reciprocals of whole numbers, or,
more concisely, reciprocal numbers ; and to a corresponding class of ratios, which we
may call reciprocals of integer ratios. And the more comprehensive conception of
the act of passing from one to another of any two commensurable steps, conducts to
a correspondingly extensive class of multiplying acts, and therefore also of multiplying
numbers, and of ratios, which we may’ call acts of fractioning, and fractional
numbers, or fractional ratios ; while the product of any such ‘act of fractioning, or
of multiplying by any such fractional number, that is, the generated step which is any
multiple of any sub-multiple of any proposed step or multiplicand, may be called a
Fraction of that step, or of that multiplicand. A fractional number may therefore
836 Professor Hiamitton on Conjugate Functions,
always be determined, in thought and in expression, by two whole numbers, namely
the sub-multipling number, called also the denominator, and the multipling number,
called also the nwmerator, (of the fraction or fractional number,) which mark the two
successive or component acts that make up the complex act of fractioning. Hence
also the reciprocal number, or reciprocal of any proposed whole number, which marks
the act of multiplication conceived to be equivalent to the act of sub-multipling by
that whole number, coincides with the fractional number which has the same whole
number for its denominator, and the number 1 for its numerator, because a step is not
altered when it is multipled by positive one. And any whole number itself, consi-
dered as the mark of any special act of multipling, may be changed to a fractional
number with positive one for its denominator, and with the proposed whole number for
its numerator ; since such a fractional number, considered as the mark of a special
act of multiplication, is only the complex mark of a complex act of thought equi-
‘valent to the simpler act of multipling by the numerator of the fraction ; because the
other component act, of sub-multipling by positive one, produces no real alteration.
Thus, the conceptions of whole numbers, and of reciprocal numbers, are included in
the more comprehensive conception of fractional numbers; and a complete theory of
the latter would contain all the properties of the former.
17. To form now a notation of fractions, we may agree to denote a fractional num-
ber by writing the numerator over the denominator, with a bar between ; that is, we
may write
ce = ~a, or more fully, ¢ = oe (121.)
in mn
when we wish to express that two commensurable steps, b and c, (which we shall, for
the present, suppose to be both effective steps,) may be severally formed from some
one common base or unit-step 2, by multipling that base by the two (positive or
contra-positive) whole numbers « and v, so that
bomwxa, C= vxKa. (122.)
[We shall suppose throughout the whole of this and of the two next following arti-
cles, that all the steps are effective, and that all the numerators and denominatcrs are
positive or contra-positive, excluding for the present the consideration of null steps,
and of null numerators or null denominators. |
Under these conditions, the step ¢ is a fraction of b, and bears to that step b the
fractional ratio ~, called also ‘the ratio of v to u;” and e may be deduced or gene-
p
rated as a product from b by a corresponding act of fractioning, namely, by the act of
and on Algebra as the Science of Pure Time. 337
multiplying b as a multiplicand by the fractional number 5 as a multiplier, or finally
by the complex act of first submultipling b by the denominator », and then multi-
pling the result 2 by the numerator v. Under the same conditions, it is evident that
we may return from c¢ to b by an inverse or reciprocal act of fractioning, namely, by
that new complex act which is composed of submultipling instead of multipling by »,
and then multipling instead of submultipling by »; so that
b="x ec, whene=~xbd: (123.)
v B
on which account we may write
=!@x(2x»b), ande=~x ("x ce), (124.)
v liad B v
whatever (effective) steps may be denoted by » and c, and whatever (positive or con-
tra-positive) whole numbers may be denoted by » and v. The two acts of fractioning,
marked by the two fractional numbers ~ and =. are therefore opposite or reciprocal
acts, of which each destroys or undoes the effect of the other; and the fractional
numbers themselves may be called reciprocal fractional numbers, or, for shortness,
reciprocal fractions : to mark which reciprocity we may use a new symbol 4, (namely,
the initial letter of the word Reciprocatio, distinguished from the other uses of the
same letter by being written in an inverted position,) that is, we may write
=u
Lu
; (125.)
v
Be?
whatever positive or contra-positive whole numbers may be marked by » andy. In
this notation,
v v wh v
cf hile = tt) tS ae 126.
Ysu(ut) sab at, (126.)
or, to express the same thing in words, the reciprocal of the reciprocal of any frac-
tional number is that fractional number itself. (Compare equation (57.) ).
It is evident also, that
1 ‘
ecto andb=4 xo, ifb=p xa; (127.)
338 Professor Hamitton on Conjugate Functions,
that is, the whole number », regarded as a multiplier, or as a ratio, may be put under
the fractional form ©
7? 80 that we may write
eI
7/3 (128.)
and the reciprocal of this whole number, or the connected reciprocal number up» to
multiply by which is equivalent to submultipling by p, coincides with the reciprocal frac-
tional number 2 so that
results which were indeed anticipated in the remarks made at the close of the fore-
going article, respecting the extent of the conception of fractional numbers, as inelud-
ing whole numbers and their reciprocals. As an example of these results, the double
of any step a may be denoted by the symbol 7 x a as well as by 2 x a, and the half
of that step a may be denoted either by the symbol 5 x a, orbyu2xa. The sym-
bol u 1 is evidently equivalent to 1, the number positive one being its own reciprocal ;
and the opposite number, contra-positive one, has the same property, because to re-
verse the direction of a step is an act which destroys itself by repetition, leaving the
last resulting step the same as the original; we have therefore the equations,
wr ONZSOW: (180.)
By the definition of a fraction, as a multiple of a submultiple, we may now express
it as follows :
v
Y bau x(2 xb) = vx (up xd). (131.)
ph
Besides, under the conditions (122.), we have, by (112.) and (114.), that is, by the
principle of the indifference of the order in which any two successive multiplings are
performed,
wxc=px (vx a)=(u xv) Sa) <M) Se ee DO (4X a)=v xb; (182.)
so that a fractional product « = xb may be derived from the multiplicand b, by
first multipling by the numerator v and then submultipling by the denominator p, in-
stead of first submultipling by the latter and afterwards multipling by the former ;
that is, in any act of fractioning, we may change the order of the two successive and
,
and on Algebra as the Science of Pure Time. 339
component acts of submultipling and multipling, without altering the final result, and
may write
p= = x(t x Gx hb): (183.)
Iwi
In general it may easily be shown, by pursuing a reasoning of the same sort, that in
any set of acts of multipling and submultipling, to be performed successively on any
one original step, the order of succession of those acts may be altered in any arbitrary
manner, without altering the final result. We may therefore compound any proposed
set of successive acts of fractioning, by compounding first the several acts of submul-
tipling by the several denominators into the one act of submultipling by the product
of those denominators, and then the several acts of multipling by the several numerators
into the one act of multipling by the product of those numerators, and finally the two
acts thus derived into one last resultant act of fractioning ; that is, we have the relations,
v vy xv
(= xb ear
u CF ) wx pe d 9
" U a yf. 134.
{ex Exh = ee ys, Satie
K le ro wx wx fe
C. J
We may also introduce or remove any positive or contra-positive whole number as a
factor in both the numerator and the denominator of any fraction, without making
any real alteration ; that is, the following relation holds good :
= - (185.)
whatever positive or contra-positive whole numbers may be denoted by pvw; a
theorem which may often enable us to put a proposed fraction under a form more simple
in itself, or more convenient for comparison with others. As particular cases of this
theorem, corresponding to the case when the common factor w is contra-positive one,
we have
vy Ov Ov v
that is, the denominator of any fraction may be changed from contra-positive to posi-
tive, or from positive to contra-positive, without making any real change, provided
that the numerator is also changed to its own opposite whole number. Two frac-
tional numbers, such as 2” and P , may be said to be opposites, (though not recipro-
fh
VOL. XVII. Sie
340 Professor Hamiiton on Conjugate Functions,
cals), when (though oz themselves the marks of opposite acts), they generate opposite
Or ° os :
steps, such as thesteps —” x » and - x b; and to mark this opposition we may write
7
== = (3) =. (137.)
Hence every fractional number, with any positive or contra-positive whole numbers j
and » for its denominator and numerator, may be put under one or other of the two
following forms :
n
Ist, =, or IInd O-, (138.)
m
(m and n denoting positive whole numbers,) according as the proposed whole numbers
u and v agree or differ in respect of being positive or contra-positive ; and in the
Ist case we may say that the fractional number itself is positive, but in the IInd
case that it is contra-positive : definitions which agree with and include the former
conceptions of positive and contra-positive whole numbers, when we consider these as
equivalent to fractional numbers in which the numerator is a multiple of the denomi-
nator ; and lead us to regard the reciprocal of any positive or contra-positive whole
number (and more generally the reciprocal of any positive or contra-positive frac-
tional number) as positive or contra-positive like it; a fractional number being
equivalent to the reciprocal of a whole number, when the denominator is a multiple
of the numerator. A fraction of a late-making step b is itself a late-making or an
early-making step, according as the multiplying fractional number is positive or
contra-positive ; and as we have agreed to write b > 0 when »b is a late-making step,
so we may now agree to write
7 > 0, when a <b> Omandehs om Os (139.)
. Uae Ong . .
that is, when — is a positive fractional number, and to write, on the contrary,
bh
v
i <o when ; xb<0O and v> 0, (140.)
that is, when - is a contra-positive fractional number. More generally, we shall
write
f . v v
, if =x b>-x b, b>9O, (141.)
u lu
and
» if Dx bcix, b> Os (142.)
and on Algebra as the Science of Pure Time. 341
and shall enunciate these two cases respectively, by saying that in the first case the
. y . rd . . . .
fractional number — is on the positive side, and that in the second case it is on the
ord . . v . v .
contra-positive side, of the other fractional number —; or that in the first case —, _fol-
. . v . .
lows and that in the second it precedes —, in the general progression of numbers,
ia
from contra-positive to positive: definitions which may easily be shown to be con-
sistent with each other, and which extend to whole numbers and their reciprocals, as
included in fractional numbers, and to the number zero itself as compared with any of
these. Thus, every positive number is on the positive side of zero and of every
contra-positive number; while zero is on the positive side of all contra-positive
numbers, but on the contra-positive side of all positive numbers: for example,
2>0,2>03,03<0,08<2,0>03,0<2 (143.)
Of two unequal positive whole numbers, the one which has the greater quotity is on
the positive side, but among contra-positive numbers the reverse is the case; for
example,
3>2,08<02: (144)
and in general a relation of subsequence or precedence between any two whole or
fractional numbers is changed to the opposite relation of precedence or subsequence,
by altering those numbers to their opposites, though a relation of equality or coinci-
dence remains unaltered after such a change. Among reciprocals of positive whole
numbers, the reciprocal of that which has the lesser quotity is on the positive side of
the other, while reciprocals of contra-positive numbers are related by the opposite
rule; thus
! 1 1 :
57 3? 62 ~*63’ that is, u2>u3, uO2<193z (145.)
In general, to determine the ordinal relation of any one fractional number ~ to
v . . - .
another ie as subsequent, or coincident, or precedent, in the general progression
of numbers, it is sufficient to prepare them by the principle (185.) so that their deno-
minators may be equal and positive, and then to compare their numerators; for
which reason it is always sufficient to compare the two whole numbers p xp xy xv’
and »’ x xu xv, and we have
t
>
, according asuxpxpe xv =n xp xpxv: (146.)
<
All Vv
Rs
ee
> .
the abridged notation = implying the same thing as if we had written more fully
<
342 Professor Hamitton on Conjugate Functions,
“> or = or <.” If it had been merely required to prepare two fractional numbers
so as to make them have a common denominator, without obliging that denominator
to be positive, we might have done so ina simpler manner by the formula (135.),
namely by multipling the numerator and denominator of each fraction by the deno-
minator of the other fraction, that is, by employing the following expressions,
te onthe: BES (147.)
pte 7
aX mo XE
y
uw
a process which may be still farther simplified when the original denominators have
any whole number (other than positive or contra-positive one) for a common factor,
since it is sufficient then to multiple by the factors which are not thus common, that
is, to employ the expressions,
v xy v si vxp
~ Ny (148.)
ox wxexpe WX WXMXE
A similar process of preparation applies to more fractions than two.
18. This reduction of different fractional numbers to a common denominator is
chiefly useful in combining them by certain operations which may be called algebraical
addition and subtraction of fractions, (from their analogy to the algebraical addition
and subtraction of whole numbers, considered in the 14th article, and to the arith-
metical operations of addition and subtraction of quotities,) and which present them-
selves in considering the composition and decomposition of fractional steps. For
we compound, as successive steps, any two or more fractions ” x b, — x b, &c., of
. . ine .
any one effective step b, and generate thereby a new effective step, this resultant step
will evidently be itself a fraction of the step b, which we may agree to denote as
follows :
(3 9) +(
* x,
(3 xo) + (5 xo) +(-
and the resultant fractional number ~,+~ or 4+ Gare &c. may be called the
Heo)
fe (149.)
oe) et) 3 &e.;
w
v v
, Bes v
algebraical swm of the proposed fractional numbers -, —, >, &c. and may
. Hu He Oe .
be said to be formed by algebraically adding them together ; definitions which agree
with those established in the 14th article, when the fractional numbers reduce them-
selves to whole numbers. If the denominators of the proposed fractions be the same,
and on Algebra as the Science of Pure Time. 343
it is sufficient to add the numerators, because then the proposed fractional steps are all
multiples of one common sub-multiple of the common unit-step b, namely of that sub-
multiple which is determined by the common denominator ; it is therefore sufficient, in
other cases, to prepare the fractions so as to satisfy this condition of haying a common
denominator, and afterwards to add the numerators so prepared, and to combine their
sum as the new or resulting numerator of the resulting fractional sum, with the
common denominator of the added fractions as the denominator of the same frac-
tional sum ; which may, however, be sometimes simplified by the omission of common
factors, according to the principle (135.). ‘Thus
v (’ x pw) + (uw xv) p y Bs vat py 5
== er 5 .0r more concisely = -> = = =— , &e.; (150.)
MX pe lM he
Be
i=
for, as a general rule of algebraic notation, we may omit at pleasure the mark of
tmaultiplication between any two simple symbols of factors, (except the arithmetical
signs 1, 2, 3, &c.,) without causing any confusion ; and when a product thus denoted,
by the mere juxta-position of its factors, (without the mark x ,) is to be combined
with other symbols in the way of addition, by the mark +, it is not necessary to en-
close that symbol of a product in parentheses : although in this Elementary Essay we
have often used, and shall often use again, these combining and enclosing marks, for
greater clearness and fulness. It is evident that the addition of fractions may be
performed in any arbitrary order, because the order of composition of the fractional
steps is arbitrary.
The algebraical subtraction of one given fractional number — from another un-
. . . fe “7°
equal fractional number ~, is an operation suggested by the decomposition of a
. . v . . .
given compound fractional step — xb into a given component fractional step —, x b
” Bh
J
and a sought component fractional step “x b, (these three steps being here sup-
posed to be all effective :) and it may be performed by compounding the opposite of
the given component step with the given compound step, or by algebraically adding
the opposite O~ of the given fractional number ~ to the other given fractional
le
number ~, according to the rule (150.). When we thus subtract one fractional
number from another with which it does not coincide, the result is positive or contra-
positive according as the fraction from which we subtract is on the positive or contra-
positive side of the other; and thus we have another general method, besides the
tule (146.), for examining the ordinal relation of any two unequal fractions, in the
general progression of numbers. This ordinal relation between any two fractional
8344 Professor Hamitron on Conjugate Functions,
(or whole) numbers a and #3, is not altered by adding any fractional (or whole) num-
ber y to both, nor by subtracting it from both; so that
> >
y ee +a, and 0 y+P=0y + a, according as B=a. (151.)
< < <
19. Again, the composition and decomposition of swecessive acts of fractioning
(instead of successive fractional steps) conduct to algebraical operations of multzpli-
cation and division of fractional numbers, which are analogous to the arithmetical
operations of multiplication and division of quotities. For if we first multiply a
given step b by a given fractional number 4 that is, if we first perform on b the
act of fractioning denoted by this number, aa so form the fractional step me x b,
we may then perform on the result another act of fractioning denoted by aaather
fractional number +, and so deduce another fractional step ~ x ( a ») ,which
will evidently be itself a fraction of the original step b, and might therefore have
been deduced from » by one compound act of fractioning ; and thus we may proceed
to other and other fractions of that step, and to other compound acts of fractioning,
which may be thus denoted,
, Uy
v v v v 7
5x (= x b)=(4 x") x,
Me BE ee
" vihiog (152.)
v v v
mx dx(t xs) ba(5x = ) x bs Scen,
B it # le
, ah vy
and the resultant fractional numbers ? x : , 4x4x", &c., which thus express
fe
the resultant acts of fractioning, derived on the proposed component acts marked
by the fractional numbers 5 , =, 2 -,, &c., may be called the algebraic products
of those proposed fractional caumbees and may be said to be formed by algebraically
multiplying them as fractional factors together ; definitions which agree with the
definitions of product and multiplication already established for whole numbers.
The same definitions shew that every fraction may be regarded as the product of the
numerator (as one factor) and the reciprocal of the denominator (as another); and
give, in general, by (134.), the following rule for the calculation of a fractional
product
&e. (153.)
and on Algebra as the Scienee of Pure Time. 345
The properties (114.) and (115.) of algebraic products of whole numbers extend to
products of fractional numbers also ; that is, we may change in any manner the order
of the fractional factors ; and if we resolve any one of those factors into two or
more algebraic parts by the rules of algebraic addition and subtraction, we may com-
bine each part separately as a partial factor with the other factors proposed, so as to
form by algebraic multiplication a partial fractional product, and then add together
those partial products algebraically to obtain the total product: or, in written
symbols,
aKa Sx Gy ey (154.)
and
zx (2+5)=€ «5)4 (E* 2): &e., (155.)
because
x(t v)=(Zxw)+(2xv), (156.)
whatever steps may be denoted by wv and vw” and whatever fractional (or whole)
number by ~ . We may also remark that
be
> =
y x B=y xa, according as B = a, if y> 0, (157.)
< <
but that
< ‘ a .
y x B=y xa, according as B = a, if y <0, (158.)
= <
a B y denoting any three fractional (or whole) numbers.
The deduction of one of two fractional factors from the other and from the product,
_ may be called (by analogy to arithmetic) the algebraic division of the given fractional
product as a dividend, by the given fractional factor as a divisor ; and the result,
which may be called the quotient, may always be found by algebraically multiplying
the proposed dividend by the reciprocal of the proposed divisor. This more general
conception of quotient, agrees with the process of the 15th article, for the division of
one whole number by another, when that process gives an accurate quotient in whole
numbers ; and when no such integral and accurate quotient can be found, we may
still, by our present extended definitions, conceive the numerator of any fraction to
be divided by the denominator, and the quotient of this division will be the fractional
number itself. In this last case, the fractional number is not exactly equal to any
346 Professor HamiLton on Conjugate Functions,
whole number, but lies between two successive whole numbers, a next preceding and
a next succeeding, in the general progression of numbers ; and _ these may be dis-
covered by the process of approximate division above mentioned, while each of the
two remainders of that approximate division is the numerator of a new fraction,
which retains the proposed denominator, and must be added algebraically as a cor-
rection to the corresponding approximate integer quotient, in order to express, by the
help of it, the quotient of the accurate division. For example,
82 e8
Sa ae ee
Or} oe
AS) PS
ap (S) Ue
Or|
trl bo
®
w
5)
In general, a fractional number may be called a mixed number, when it is thus ex-
pressed as the algebraic sum of a whole number and a proper fraction, this last name
being given to a fractional number which lies between zero and positive or contra-
positive one. We may remark that an ordinal relation between two fractional
numbers is not altered by dividing them both by one common positive divisor ;_ but if
the divisor be contra-positive, it changes a relation of subsequence to one of pre-
cedence, and conversely, without disturbing a relation of coincidence.
20. In all the formulz of the three last articles, we have supposed that all the
numerators and all the denominators of those formule are positive or contra-positive
whole numbers, excluding the number zero. However, the general conception of a
fraction as a multiple of a sub-multiple, permits us: to suppose that the multipling
number or numerator is zero, and shows us that then the fractional step itself is null,
if the denominator be different from zero; that is,
o xb =0if wk 0. (159.)
Thus, although we supposed, in the composition (149.) of successive fractional steps,
(with positive or contra-positive numerators and denominators,) that the resultant
step was effective, yet we might have removed this limitation, and have presented the
formule (150.) for fractional sums as extending even to the case when the resultant
step is null, if we had observed that in every such case the resultant numerator of the
formula is zero, while the resultant denominator is different from zero, and therefore
that the formula rightly expresses that the resultant fraction or sum is null. For
cee : at Ov-.
example, the addition of any two opposite fractional numbers, such as ~ ‘anid! |= an
be
which » and v are different from zero, conducts to a null sum, under the form
sa , in which the numerator © v+v is zero, while the denominator is different
from zero.
and on Algebra as the Science of Pure Time. 347
But it is not so immediately clear what ought to be regarded as the meaning of a
fractional sign, in the case when the denominator is null, and when therefore the act
of fractioning prescribed by the notation involves a sub-multipling by zero. ‘To
discuss this case, we must remember that to sub-multiple a step b by a whole number
u, is, by its definition, to find another step a, which, when multipled by that whole
number p, shall produce the proposed step b; but, whatever step a may be, the theory
of multiple steps (explained im the 13th article) shows that it necessarily produces
the null step 0, when it is multipled by the null number zero ; that is, the equation
Ova 0) (1€0.)
is true independently of a, and consequently we have always
Oxa+bd, ifo+o. (161.)
It is, therefore, impossible to find any step a, in the whole progression of time, which
shall satisfy the equation
Mab = AOL. Ora. — b's (162.)
=
if the given step b be effective ; or, in other words, it is impossible to sub-multiple an
effective step by zero. The fractional sign 6 denotes therefore an ¢mpossible act, if it
be applied to an effective step: and the zero-submultiple of an effective step is a phrase
which involves a contradiction. On the other hand, if the given step » be null, it is
not only possible to choose some one step a which shall satisfy the equations (162.),
but every conceivable step possesses the same proposed property ; in this case, there-
fore, the proposed conditions lay no restriction on the result, but at the same time,
and for the same reason, they fail to give any information respecting it: and the act
of sub-multipling a null step by zero, is indeed a possible, but it is also an indeter-
minate act, or an act with an indeterminate result ; so that the zero-submultiple of a
: 1 : ; :
null step, and the written symbol 9 x O% are spoken or written signs which do not
specify any thing, although they do not involve a contradiction. We see then that
while a fractional number is in general the sign of a possible and determinate act of
fractioning, it loses one or other of those two essential characters whenever its deno-
minator is zero ; for which reason it becomes comparatively unfit, or at least inconve-
nient, in this case, for the purposes of mathematical reasoning. And to prevent the
confusion which might arise from the mixture of such cases with others, it is conve-
nient to lay down this general rule, to which we shall henceforth adhere: that all
VOL. XVII. 3T
348 Professor Hamitton on Conjugate Functions,
denominators and divisors are to be supposed different from zero unless the contrary
be mentioned expressly ; or that we shall never suwb-multiple nor divide by a null num-
ber without expressly recording that we do so.
On the Comparison of any one effective Step with any other, in the way of Ratio,
and the Generation of any one such step from any other, in the way of Multipli-
cation ; and on the Addition, Subtraction, Multiplication, and Division of Alge-
braic Numbers in general, considered thus as Ratios or as Multipliers of Steps.
é
21. The foregoing remarks upon fractions lead naturally to the more general con-
ception of algebraic ratio, as a complex relation of any one effective step to any
other, determined by their relative largeness and relative direction ; and to a simi-
larly extended conception of algebraic multiplication, as an act (of thought) which
enlarges, or preserves, or diminishes the magnitude, while it preserves or reverses the
direction, of any effective step proposed. In conformity with these conceptions, and
by analogy to our former notations, if we denote by a and b any two effective steps,
of which a may be called the antecedent or the multipiicand, and » the consequent
or the product, we may employ the symbol » to denote the ratio of the consequent »
to the antecedent a, or the algebraic number or multiplier by which we are to mul-
tiply a as a multiplicand in order to generate » as a product: and if we still employ
the mark of multiplication x, we may now write, in general,
b
b= =
- a
San: (163.)
or, more concisely,
pe ae ry ee a, (164.)
that is, if we employ, for abridgement, a simple symbol, such as the italic letter a, to
denote the same ratio or multiplier which is more fully denoted by the complex
symbol _
It is an immediate consequence of these conceptions and definitions, that the fol-
lowing relation holds good,
aes Ai geees Teh (165.)
a denoting any effective step, and » and v denoting any positive or contra-positive
and on Algebra as the Science of Pure Time. 34.9
whole numbers; since the fractional ratio denoted by the symbol ~ is the ratio of
le
the multiple step » x a to the multiple step » x a. In like manner it follows, from
the same conceptions and definitions, that
a iD) aad reciprocally v = : x bif x = =; (166.)
and more generally, that
and reciprocally,
sae e ft Fs (168.)
a
whatever effective steps may be denoted by a, b, c,d, and whatever fraction by ~.
We may also conceive combinations of ratios with each other, by operations which
we may call Addition, Subtraction, Multiplication, and Division of Ratios, or of
general algebraic numbers, from the analogy of these operations to those which we
have already called by the same names, in the theories of whole numbers and of
fractions. And as we wrote, in treating of whole numbers,
w=v+m when w x a= (vx a)t+(u Xa), (107.)
and
w=vX mu when w x a=v X (u Xa); (111.)
and, in the theory of fractions,
wr v Us J
a= pe when “xp=(=x b) +(% x b), (149.)
eye lu le B
and
n v’ un v’
~=+ x? when vxva=%x(% x), (152.)
jee ot a ft NE
with other similar expressions ; so we shall now write, in the more general theory of
ratios,
a =e when axe=(3xe)t+(2x e), (169.)
and
bei ob b’ vb! (bd
ae eae when ax esa x(>x e): (170.)
350 Professor HamiLton on Conjugate Functions,
and shall suppose that similar definitions are established for the algebraical sums and
products of more than two ratios, or general algebraic numbers. It follows that
b! b_ bi+b
ats 4
b! bb! bb __ bbb 171.
a a renee i Cm
&e.
and that
b’ b_bi
bts ase
b’! bio op bl! (172.)
tel a ole
A ratio between any two effective steps may be said to be positive or contra-positive,
according as those two steps are co-directional or contra-directional, that is, according
as their directions agree or differ; and then the product of any two or more positive
or contra-positive ratios will evidently be contra-positive or positive according as there
are or are not an odd number of contra-positive ratios, as factors of this product ;
because the direction of a step is not altered or is restored, if it either be not reversed
at all, or be reversed an even number of times. ,
Again, we may say, as in the case of fractions, that we subiract a ratio when we
add its opposite, and that we divide by a ratio when we multiply by its reciprocal, if we
agree to say that two ratios or numbers are opposites when they generate opposite
steps by multiplication from one common step as a multiplicand, and if we call them
reciprocals when their corresponding acts of multiplication are opposite acts, which
destroy, each, the effect of the other ; and we may mark such opposites and reci-
procals, by writing, as in the notation of fractions,
aes eb when 2x = e(2x c) ; (173.)
and
bY
a!
fae when © x (2x c) Se: (174.)
definitions from which it follows that
and that
and on Algebra as the Science of Pure Time. 351
And as, by our conceptions and notations respecting the ordinal relation of one
fractional number to another, (as subsequent, or coincident, or precedent, in the
general progression of such numbers from contra-positive to positive,) we had the
relations,
>
,
v Vv v
=, when = x= — x a,,8 > 05
woe lH < He
so we may now establish, by analogous conceptions and notations respecting ratios,
the relations,
fe ity ” >
by ee aee b 2 iy di
ae a) NOS nt aie iP app O:: (177-)
that is, more fully,
BS pa
= > ywhen (2 xa) +a>(Yxa) +a, (178.)
HW” , “a 4 hs
a = a when (3, x a) +a=(Fxe)+a, (179.)
and
b’ b’ b” b’
mw <g> When (5 xe) +a<(Zxa) +a; (180.)
the symbol a denoting any moment of time, and a any late-making step. ‘The rela-
tion (179.) is indeed an immediate consequence of the first conceptions of steps and
ratios; but it is inserted here along with the relations (178.) and (180.), to show
more distinctly in what manner the comparison and arrangement of the moments
,
(x s)+a (Gx e)+a, &. (181.)
which are suggested and determined by the ratios or numbers = ; = , &c., (in combi
nation with a standard moment a and with a late-making step a,) enable us to com-
pare and arrange those ratios or numbers themselves, and to conceive an indefinite
progression of ratio from contra-positive to positive, including the indefinite pro-
gression of whole numbers (103.), and the more comprehensive progression of frac-
tional numbers considered in the 17th article: for it will soon be shown, that though
every fractional number is a ratio, yet there are many ratios which cannot be ex-
pressed under the form of fractional numbers. Meanwhile we may observe, that the
theorems (151.) (157.) (158.) respecting the ordinal relations of fractions in the
general progression of number, are true, even when the symbols a B y denote ratios
which are not reducible to the fractional form; and that this indefinite progression
352 Professor Hamitton on Conjugate Functions,
of number, or of ratio, from contra-positive to positive, corresponds in all respects to
the thought from which it was deduced, of the progression of time itself, from mo-
ments indefinitely early to moments indefinitely late.
22. It is manifest, on a little attention, that the ratio of one effective step b to
another a, is a relation which is entirely determined when those steps are given, but
which is not altered by multiplying both those steps by any common multiplier,
whether positive or contra-positive ; for the relative largeness of the two steps is not
altered by doubling or halving both, or by enlarging or diminishing the magnitudes of
both in any other common ratio of magnitude, that is, by multiplying both by any
common positive multiplier: nor is their relative direction altered, by reversing the
directions of both. We have then, generally,
pS Ses (182.)
= (183.)
Hence, by (167.), the two steps= x b and © x eare related in one common ratio,
namely the ratio = to the common step ce, and therefore are equivalent to each
other ; that is, we have the equation,
c b
abe acs (184.)
whatever three effective steps may be denoted by a b ec.
In general, when any four effective steps a b ¢ @ are connected by the relation
d b
= = as (185.)
that is, when the ratio of the step a to c is the same as the ratio of the step b to a,
these two pairs of steps a, b and c, a may be said to be analogous or proportional
pairs ; the steps a and ¢ being called the antecedents of the analogy, (or of the
proportion) and the steps b and a being called the consequents, while a and d are the
extremes and b and ¢ the means. And since the last of these four steps, or the
second consequent 4, may, by (168.), be expressed by the symbol 2 x ec, we see, by
(184.), that it bears to the first consequent b the ratio < of the second antecedent
¢ to the first antecedent a; that is,
.
and on Algebra as the Science of Pure Time. 353
Seay Perea ot (186.)
a theorem which shows that we may transform the expression of an analogy (or pro-
portion) between two pairs of effective steps in a manner which may be called alter-
nation. (Compare the theorem (11.).)
It is still more easy to perceive that we may invert an analogy or proportion
between any two pairs of effective steps; or that the following theorem is true,
c By je
a, Syaput
= mae (187.)
Combining inversion with alternation, we see that
bree
d c
it et (188.)
(Compare the theorems (12.) and (13.).)
In general, if any two pairs of effective steps a, b and c, a be analogous, we do
not disturb this analogy by interchanging the extremes among themselves, or the
means among themselves, or by substituting extremes for means and means for ex-
tremes ; or by altering proportionally, that is, altering in one common ratio, or mul-
tiplying by one common multiplier, whether positive or contra-positive, the two con-
sequents, or the two antecedents, or the two steps of either pair: or, finally, by
altering in inverse proportion, that is, multiplying respectively by any two reciprocal
multipliers, the two extremes, or the two means. The analogy (185.) may therefore
be expressed, not only in the ways (186.), (187.), (188.), but also in the following :
axd axb d b axa b
en Te Ser: ae 189.
c a oa xee DUST Frias oh ae ( 9.)
uwaxd eee, eee axbd
Bac ae uae alain cis = sane (190.)
a denoting any ratio of one effective step to another, and u a denoting the reciprocal
ratio, of the latter step to the former.
23. We may also consider it as evident that if any effective step ¢ be com-
pounded of any others a and », this relation of compound and components will not
be disturbed by altering the magnitudes of all in any common ratio of magnitude,
that is by doubling or halving it, or multiplying all by any common positive multi-
plier ; and we saw, in the 12th article, that the same relation of compound and com-
ponents is not disturbed by reversing the directions of all: we may therefore mul-
354 Professor Hamitton on Conjugate Functions,
tiply all by any common multiplier a, whether positive or contra-positive, and may
establish the theorem,
axe=(axb)+(a xa), if c=b+a; (191.)
which gives, by the definitions (169.) (170.) for the sum and product of two ratios,
this other important relation,
ax (b+b)=(ax 8) +(ax B), (192.)
if b, b', and 6'+4, denote any three positive or contra-positive numbers, connected
with each other by the definition (169.), or by the following condition,
(6 +b) xa=(b'x a)+(b xa), (193.)
in which a denotes any arbitrary effective step. The definitions of the sum and
product of two ratios, or algebraic numbers, give still more simply the theorem,
(b +b) x a=(0 x a)+(b x a). (194:.)
The definition (169.) of a sum of two ratios, when combined with the theorem
(75.) respecting the arbitrary order of composition of two successive steps, gives the *
following similar theorem respecting the addition of two ratios,
b+a=a+b. (195.)
And if the definition (170.) of a product of two ratios or multipliers be combined
with the theorem (186.) of alternation of an analogy between two pairs of steps, in
the same way as the definition of a compound step was combined in the 12th article
with the theorem of alternation of an analogy between two pairs of moments, it
shows that as any two steps a, b, may be applied to any moment, or compounded
with each other, either in one or in the opposite order, (b +a=a+b,) so any two
ratios a and 6 may be applied as multipliers to any step, or combined as factors of a
product with each other, in an equally arbitrary order; that is, we have the relation,
DO GGs—NCEXE0s (196.)
It is easy to infer, from the thorems (195.) (196.), that the opposite of a sum of two
ratios is the sum of the opposites of those ratios, and that the reciprocal of the pro-
duct of two ratios is the product of their two reciprocals ; that is,
6 (b+a)=906+0a, (197.)
and
u (bxa)=ubxua. (198.)
and on Algebra as the Science of Pure Time. 355
And all the theorems of this article, respecting pairs of ratios or of steps, may easily
be extended to the comparison and combination of more ratios or steps than two.
In particular, when any number of ratios are to be added or multiplied together, we
may arrange them in any arbitrary order; and in any multiplication of ratios, we
may treat any one factor as the algebraic sum of any number of other ratios, or
partial factors, and substitute each of these separately and successively for it, and the
sum of the partial products thus obtained will be the total product sought. As ar
example of the multiplication of ratios, considered thus as sums, it is plain from the
foregoing principles that
(d+ce)x(b+a)= fdx(b+a)} + {ex(b+a}}
= (dx b) + (dxa) + (cx b) +(exa)
=db+da+cb+ea, (199.)
and that
(b+a)x(b+a)=(bxb) + (2xbxa) + (axa)
—bb+2ba+aa, (200.)
whatever positive or contra-positive ratios may be denoted by a bc d.
And though we have only considered effective steps, and positive or contra-
positive ratios, (or algebraic numbers,) in the few last articles of this Essay, yet the
results extend to null steps, and to null ratios, also; provided that for the reasons
given in the 20th article we treat all such null steps as consequents only and not as
antecedents of ratios, admitting null ratios themselves but not their reciprocals into our
formule, or employing null numbers as multipliers only but not as divisors, in order
to avoid the introduction of symbols which suggest either impossible or indeterminate
operations.
‘On the insertion of a Mean Proportional between two steps ; and on Impossible,
Ambiguous, and Incommensurable Square-Roots of Ratios.
24. Three effective steps ab b’ may be said to form a continued analogy or con-
tinued proportion, when the ratio of v to b is the same as that of b to a, that is,
when
b 9g
= =; (201.)
VOL, XVII, 2 Beh Agy
356 Professor HamiLton on Conjugate Functions,
a and b’ being then the extremes, and » the mean, or the mean proportional between
a and b’, in this continued analogy ; in which w is also the third proportional to a
and pv, and a is at the same time the third proportional to v and b, because the
analogy may be inverted thus,
a b
eee (202.)
When the condition (201.) is satisfied, we may express b! as follows,
b
Ss Seg (203.)
a
that is, if we denote by a the ratio of b to a, we shall have the relations
Di=texga', De=—teeee bi == e (204.)
and reciprocally when these relations exist, we can conclude the existence of the con-
tinued analogy (201.). It is clear that whatever effective steps may be denoted by
a and b, we can always determine, (or conceive determined,) in this manner, one
third proportional b’ and only one; that is, we can complete the continued analogy
(201.) in one, but in only one way, when an extreme a and the mean b are given:
and it is important to observe that whether the ratio a of the given mean b to the
given extreme a be positive or contra-positive, that is, whether the two given steps
a and b be co-directional or contra-directional steps, the product a x a will necessarily
be a positive ratio, and therefore the deduced extreme step v’ will necessarily be
co-directional with the given extreme step a. In fact, without recurring to the
theorem of the 21st article respecting the cases in which a product of contra-positive —
factors is positive, it is plain that the continued analogy requires, by its conception,
that the step b’ should be co-directional to b, if b» be co-directional to a, and that b’
should be contra-directional to » if b be contra-directional to a; so that in every
possible case the extremes themselves are co-directional, as both agreeing with the
mean or both differing from the mean in direction. Jt ts, therefore, impossible to
insert a mean proportional between two contra-directional steps ; but for the same —
reason we may insert either of two opposite steps as a mean proportional between
two given co-directional steps ; namely, either a step which agrees with each, or a
step which differs from each in direction, while the common magnitude of these two
opposite steps is exactly intermediate in the way of ratio between the magnitudes of
the two given extremes. (We here assume, as it seems reasonable to do, the con-
ception of the general existence of such an exactly intermediate magnitude, although
the nature and necessity of this conception will soon be more fully considered.) For ,
and on Algebra as the Science of Pure Time. — 357
example, it is impossible to insert a mean proportional between the two contra-
directional (effective) steps a and © 9 a, that is, it is impossible to find any step b
which shall satisfy the conditions of the continued analogy
: (205.)
or any number or ratio a which shall satisfy the equation
axa=09: (206.)
whereas it is possible to insert in two different ways a mean proportional b between
the two co-directional (effective) steps a and 9 a, or to satisfy by two different steps
b (namely, by the step 3 a, and also by the opposite step © 3 a) the conditions of the
continued analogy
ee (207.)
and it is possible to satisfy by two different ratios a the equation
axa=9, (208.)
namely, either by the ratio 3 or by the opposite ratio 6 8. In general, we may
agree to express the two opposite ratios a which satisfy the equation
axa=b (> 0), (209.)
by the two symbols
/b (> 0) and 0 vb (<0), (210.)
b and vb being positive ratios, but © vb being contra-positive ; for example,
/9=3, OvV9=08. (211.)
With this notation we may represent the two opposite steps of which each is a mean
proportional between two given co-directional (effective) steps a and v, by the
symbols
Wena and O = x a3 (212.)
and shall have for each the equation of a continued analogy,
b! rw yp =;
Wf Be Peak, Dey apa, (213.)
358 Professor Haminton on Conjugate Functions,
We may also call the numbers vb and @ yb by the common name of roots, or
(more fully) sqguare-roots of the positive number}; distinguishing them from each
other by the separate names of the positive square-root and the contra-positive
square-root of that number b, which may be called their common square: though we
may sometimes speak simply of the square-root of a (positive) number, meaning then
the positive root, which is simpler and more important than the other.
25. The idea of the contizxwity of the progression from moment to moment in
time involves the idea of a similarly continuous progression in magnitude from any
one effective step or interval between two different moments, to any other unequal
effective step or other unequal interval ; and also the idea of a continuous progres-
sion in ratio, from any one degree of inequality, in the way of relative largeness or
smallness, as a relation between two steps, to any other degree. Pursuing this train
of thought, we find ourselves compelled to conceive the existence (assumed in the
last article) of a determined magnitude », exactly intermediate im the way of ratio
between any two given unequal magnitudes a and ”’, that is, larger or smaller than
the one, in exactly the same proportion in which it is smaller or larger than the
other ; and therefore also the existence of a determined number or ratio a which is
the exact square-root of any proposed (positive) number or ratio 6. ‘To show this
more fully, let A B D be any three given distinct moments, connected by the relations
De
B—A
= i, iS ib (214.)
which require that the moment 8B should be situated between a and p ; and let c be
any fourth moment, lying between B and p, but capable of being chosen as near to B
or as near to D as we may desire, in the continuous progression of time. Then the
two ratios
D—A
Cas
and
Besa C—A
will both be positive ratios, and both will be ratios of largeness, (that is, each will be
a relation of a larger to a smaller step,) which we may denote for abridgement as
follows,
—A D—A
c =—
Sy Sax (215.)
B-—A cC—*&
but by choosing the moment c sufficiently near to B we may make the ratio x ap-
proach as near as we desire to the ratio of equality denoted by 1, while the ratio y
and on Algebra as the Science of Pure Time. 359
will tend to the given ratio of largeness denoted by ) ; results which we may express
by the following written sentence,
if Lc=sB, then L x=1 and Ly=d, (216.)
prefixing the symbol 4, (namely the initial letter L of the Latin word Limes, distin-
guished by a bar drawn under it,) to the respective marks of the variable moment c
and variable ratios’ 4, y, in order to denote the respective limits to which those
variables tend, while we vary the selection of one of them, and therefore also of the
rest. Again, we may choose the moment c nearer and nearer to p, and then the
ratio x will tend to the given ratio of largeness denoted by b, while the ratio y will
tend to the ratio of equality ; that is,
if Lc=p, then L w=), Ly=1; C217.)
and if we conceive a continuous progression of moments c from-B to p, we shall also
have a continuous progression of ratios x, determining higher and higher degrees of
relative largeness (of the increasing step c—a as compared with the fixed step B—)
from the ratio of equality 1 to the given ratio of largeness >, together with another
continuous but opposite progression of ratios y, determining lower and lower degrees
of relative largeness (of the fixed step D—A as compared with the increasing step
c—A) from the same given ratio of largeness ) down to the ratio of equality 1; so
that we cannot avoid conceiving the existence of some one determined state of the
progression of the moment c, for which the two progressions of ratio meet, and for
which they give .
D—A _C—A
(iy. i
aaxb=y=a, that is : (218.)
having given at first y > #, and giving afterwards y < #. And since, in general,
D—A Cc—A D—A
C—sA) B-—A B—A
, that is, (a a x b).x x=), (219.)
we can and must by (218.) and (214.), conceive the existence of a positive ratio a
which shall satisfy the condition (209.), ax a=), if b > 1, that is, we must conceive
the existence of a positive square-root of 6, if 6 denote any positive ratio of large-
ness. A reasoning of an entirely similar kind would prove that we must conceive the
existence of a positive square-root of 6, when ) denotes any positive ratio of small-
ness, (b < 13) andif b denote the positive ratio of equality, (J=1,) then it evi-
dently has that ratio of equality itself for a positive- square-root. We see then by
360 Professor Hamiiton on Conjugate Functions,
this more full examination what we before assumed to be true, that every positive
number or ratio } has a positive (and therefore also a contra-positive) square-root.
And hence we can easily prove another important property of ratios, which has
been already mentioned without proof; namely that several ratios can and must be
conceived to exist, which are incapable of being expressed under the form of whole
or fractional numbers ; or, in other words, that every effective step a has other steps
incommensurable with it ; and therefore that when any two distinct moments a and
B are given, it is possible to assign (in various ways) a third moment c which shall
not be wniserial with these two, in the sense of the Sth article, that is, shall not
belong in common with them to any one equi-distant series of moments, comprising
all the three. For example, the positive square-root of 2, which is evidently inter-
mediate between 1 and 2 in the general progression of numbers, and which therefore
is not a whole number, cannot be expressed as a fractional number either ; since if it
could be put under the fractional form = , so that
Ope deen (220.)
we should then have
@=~x7=7** (221.)
m ™m mxm
that is,
MXn=2Qxmxm; (222.)
but the arithmetical properties of quotities are sufficient to prove that this last equa-
tion is impossible, whatever positive whole numbers may be denoted by m and n.
And hence, if we imagine that
b= /2xX a, a >O, (223.)
the step b which is a mean proportional between the two effective and co-directional
steps a and 2a (of which the latter is double the former) will be zncommensurable
with the step a (and therefore also with the double step 2); that is, we cannot
find nor conceive any other step c which shall be a common measurer of the steps
a andb, so as to satisfy the conditions
a=Mc, b=Mc, (224)
whatever positive or contra-positive whole numbers we may denote m and n ; be-
cause, if we could do this, we should then have the relations,
2 n
bimini, « AA Sms (225.)
and on Algebra as the Science of Pure Time. 361
of which the latter has been shown to be impossible. Hence finally, if a and B be
any two distinct moments, and if we choose a third moment c such that
the moment c will not be uniserial with a and B, that is, no one equi-distant series of
moments can be imagined, comprising all the three. And all that has here been
sliown respecting the square-root of two, extends to the square-root of three, and
may be illustrated and applied in an infinite variety of other examples. We must
then admit the existence of pairs of steps which have no common measurer ; and
may call the ratio between any two such steps an incommensurable ratio, or incom-
mensurable number.
More formal proof of the general existence of a determined positive square-root,
commensurable or incommensurable, for every determined positive ratio: conti-
nuity of progression of the square, and principles connected with this continuity.
26. The existence of these incommensurables, (or the necessity of conceiving
them to exist,) is so curious and remarkable a result, that it may be usefully con-
firmed by an additional proof of the general existence of square-roots of positive
ratios, which will also offer an opportunity of considering some other important prin-
ciples.
The existence of a positive square-root a= /}, of any proposed ratio of largeness
b> 1, was proved in the foregoing article, by the comparison of the two opposite
progressions of the two ratios w and u ax, from the states r=1, u ax 6=5, for
which u «xb >a, to the states r=b, uxxb=1, for which uxxb<w; for this
comparison obliged us to conceive the existence of an intermediate state or ratio «
between the limits 1 and 4, as a common state or state of meeting of these two oppo-
site progressions, corresponding to the conception of a moment at which the de-
creasing ratio u # x 6 becomes exactly equal to the increasing ratio w, having been
previously a greater ratio (or a ratio of greater relative largeness between steps), and
becoming afterwards a lesser ratio (or a ratio of less relative largeness). And it
was remarked that an exactly similar comparison of two other inverse progressions
would prove the existence of a positive square-root / of any proposed positive
362 Professor Hamitton on Conjugate Functions,
ratio 5 of smallness, <1, 56> 0. But instead of thus comparing, with the progression
of the positive ratio 2, the connected but opposite progression of the connected posi-
tive ratio ux, and showing that these progressions meet each other in a certain
intermediate state or positive ratio a, we might have compared the two connected
and not opposite progressions of the two connected positive ratios 7 and 2x2, of
which the latter is the square of the former ; and might have shown that the square
(=a x v= x) increases constantly and continuously with the root (=x), from the
state zero, so as to pass successively through every state of positive ratio 4. To
develope this last conception, and to draw from it a more formal (if not a more con-
vineing) proof than that already given, of the necessary existence of a conceivable
positive square-root for every conceivable positive number, we shall here lay downa
fow Lemmas, or preliminary and auxiliary propositions.
Lemma I. If 2’ =a, and x>0, a'>0, then aa’
ax; (227.)
<<
Allv
that is, the square a’x’ of any one positive number or ratio 2’, is greater than, or
equal to, or less than the square wz of any other positive number or ratio xz, ac-
cording as the number ’ itself is greater than, or equal to, or less than the number
x 3; one number 2’ being said to be greater or less than another number x, when it is
on the positive or on the contra-positive side of that other, in the general progression
of numbers considered in the 21st article. This Lemma may be easily proved from
the conceptions of ratios and of squares ; it follows also without difficulty from the
theorem of multiplication (200.). And hence we may obviously deduce as a corollary
of the foregoing Lemma, this converse proposition :
if 2’
ALLY
va, and x>0, a>0, then 2’
ALLY
z5 (228.)
that is, if any two proposed positive numbers have positive square-roots, the root of
the one number is greater than, or equal to, or less than the root of the other
number, according as the former proposed number itself is greater than, or equal to,
or less than the latter proposed number.
The foregoing Lemma shows that the square constantly increases with the root,
from zero up to states indefinitely greater and greater. But to show that this in-
crease is continuous as well as constant, and to make more distinct the conception of
such continuous increase, these other Lemmas may be added.
Lemma Il. If @ and a” be any two unequal ratios, we can and must conceive the
and on Algebra as the Science of Pure Time. 363
existence of some intermediate ratio a; that is, we can always choose a or conceive
it chosen so that
>A G Gi lh eh (229.)
For then we have the following relation of subsequence between moments,
a’ (B—A)+A >a (B—A) +A, if B> a, (280.)
by the very meaning of the relation of subsequence between ratios, a’ > a’, as defined
in article 21.; and between any two distinct moments it is manifestly possible to
insert an intermediate moment, indeed as many such as we may desire: it is, there-
fore, possible to insert a moment c between the two non-coincident moments
ad (B—A)+4 and a” (B—a) +A,
such that
c>a@ (B—a)+a, c<a’ (B—A)+A, if B> A, a’ >a’; (231.)
and then if we put, for abridgement,
; (232.)
denoting by a the ratio of the step or interval c—a to the step or interval B—4,
we shall have
a (B—A)+A>a'(B—A) +A,
a (B—A)+a < a’(B—a) +A,
C=a (B—A)+A, B>A,
‘i (233.)
and therefore finally,
aS Gh, Gh Git
as was asserted in the Lemma. We see, too, that the ratio a is not determined by
the conditions of that Lemma, but that an indefinite variety of ratios may be chosen,
which shall all satisfy those conditions.
Corollary. It is possible to choose, or conceive chosen, a ratio a, which shall
satisfy all the following conditions,
a>a, a=b, a>c, ‘cy
ee iaed cg N (234:.)
if the least (or hindmost) of the ratios a’, 6", c’, ... be greater (or farther advanced
in the general progression of ratio from contra-positive to positive) than the greatest
(or foremost in that general progression) of the ratios a’, b’, c’, &c.
VOL. XVII. 3x
364 Professor Hamitton on Conjugate Functions,
For if c’ (for example) be the least or hindmost of the ratios a”, 6", c’,
. so
that
<=
et, OU SboesSw en. (235.)
and if J (for example) be the greatest or foremost of the ratios a’, b’, c’, ... so that
EGLO COO LI (236.)
= denoting what might be more fully written thus, ‘‘ < or =”
and the other abridged sign = denoting in like manner “> or =”,) then the con-
ditions (234.) of the Corollary will all be satisfied, if we can satisfy these two condi-
tions,
(the abridged sign
a> OU, we 5 (237.)
and this, by the Lemma, it is possible to do, if we have the relation
Coma (238.)
which relation the enunciation of the Corollary supposes to exist.
Remark.—lf the ratios a’ b'c... a” bc’... be all actually given, and therefore
limited in number ; or if, more generally, the least of the ratios a” b” ce’... and the
greatest of the ratios a’ b’ c’... be actually given and determined, so that we have
only to choose a ratio a intermediate between two given unequal ratios ; we can then
make this choice in an indefinite variety of ways, even if it should be farther required
that a should be a fractional number ”, since we saw, in the 8th article, that be-
mu
tween any two distinct moments, such as a’ (B—a)+a and a” (B—A) +A, it is pos-
sible to insert an indefinite variety of others, such as ~ (s—a) +, wniserial with the
two moments a and xB, and giving therefore fractions such as ”, intermediate (by
pe
the 21st article) between the ratios a’ and a’. But if, instead of actually knowing
the ratios a’ b' c’... a’ b' c’... themselves, in (234.), we only know a law by which we
may assign such ratios without end, this law may lead us to conceive new conditions
of the form (234.), incompatible with some (and perhaps ultimately with all) of these
selections of fractional ratios ~, although they can never exclude all ratios a what-
ever, unless they be incompatible with each other, that is, unless they fail to possess
the relation mentioned in the Corollary. The force of this remark will soon be felt
more fully.
Lemma Ill. If b denote any given positive ratio, whether it be or be not the
and on Algebra as the Science of Pure Time. 365
square of any whole or of any fractional number, it is possible to find, or to conceive
as found, one positive ratio a, and only one, which shall satisfy all the conditions of
the following forms :
eae ha hs (239.)
mn’ m' n" denoting here any positive whole numbers whatever, which can be chosen
so as to satisfy these relations,
geen ee
nn n'n
b
By Ti v
ve om
———s 19
m' m (240.)
For if the proposed ratio 4 be not the square of any whole or fractional number, then
the existence of such a ratio a may be proved from the two preceding Lemmas, or
from their Corollaries, by observing that the relations (240.) give
min’ nn
n!
—=— >. -_ alla, therefore —. >.—-
mm” mim? ~ m ~ mm’?
(241.)
so that no two conditions of the forms (239.) are incompatible with each other, and there
must be at least one positive ratio a which satisfies them all. And to prove in the same —
case that there is ovly one such ratio, or that if any one positive ratio a satisfy all the
conditions (239.), no greater ratio ¢ (> a@) can possibly satisfy all those conditions, we
may observe that however little may be the excess 0 a+ c of the ratio c over a, this
excess may be multiplied by a positive whole number m’ so large that the product
shall be greater than unity, in such a manner that
m (8 a+c)>1, (242.)
and therefore
1 1
@ate>=—, and ¢c>—+a; (243.)
m m
and that then another positive (or null) whole number 7’ can be so chosen that
,
n'n
Ll+n’ 1l+n’
7 = — b, x 5 z
UL Li m m
> b, (244)
with which selection we shall have, by (239.) (240.) (243.),
sve > : 2
iri ea sy Cra ere * (245.)
whereas, if c satisfied the conditions (239.) it ought to be less than this fraction
aa , because the square of this positive fraction iS greater by (244.) than the pro-
366 Professor Haminton on Conjugate Functions,
posed ratio 4. In like manner it may be proved that in the other case, when @ is the
square of a positive fractional or positive whole number a; one positive ratio @ and
only one, namely the number = itself, will satisfy all the conditions (239.) ; in both
cases, therefore, the Lemma is true: and the consideration of the latter case shows,
that, under the conditions (239.),
eh ae eee SO: (246.)
m mm me
In no case do the conditions (239.) exclude all ratios a whatever ; but except in the
case (246.) they exclude all fractional ratios : for it will soon be shown that the one
ratio a which they do not exclude has its square always =b, and must, therefore, be
an incommensurable number when # is not the square of any integer or fraction.
(Compare the Remark annexed to the Corollary of the IInd Lemma.)
Lemma IV. If b' and 6" be any two unequal positive ratios, it is always possible
to insert between them an intermediate fractional ratio which shall be itself the square
. ere = .
of another fractional ratio a that is, we can always find, or conceive found, two
positive whole numbers m and » which shall satisfy the two conditions,
nan
ey ag a moe if 6 > b, 6>0. (247.)
For, by the theorem of multiplication (200.), the square of the fraction : _- may
be expressed as follows,
a ! fo) ' 7 ok
Ua Cee Pe ee” Pade (248.)
m m mm mem mm
Sout? As Sc aa | ON epee
that is, its excess over the square of the fraction — 1s —_ + , which is less than
9 m mm mm
2 1+ : ; S40 ;
Sie og a , and constantly increases with the positive whole number 7’ when the
m m
positive whole number m remains unaltered ; so that the 1+ squares of fractions
with the common denominator m, in the following series,
1 1 2 2 3} 3 n' n! lin 1lxn
== x= Dee Sl Set a z (249.)
m m m m m me mm m m Ww
5 5 . o 5 I 2 1 + n d I f
increase by increasing differences which are each less than a} = therefore
Lieve ; : ae
than i? if we choose m and m’ so as to satisfy the conditions
m=2Zik, 1+n'=im, (250.)
and on Algebra as the Science of Pure Time. 367
é and & being any two positive whole numbers assumed at pleasure : with this choice,
nn
therefore, of the numbers m and 7’, some one (at least) such as —, among the
squares of fractions (249.), that is, some one at least among the following squares of
fractions,
1 1 Q Q 3 3 Qitk | @iik
Sek Dik Lik Pe tk Dik ek OTe Qik
of which the last is =72, must lie between any two proposed unequal positive ratios
6 and 6”, of which the greater 6” does not exceed that last square 77, and of which
i 1 aj) ‘
the difference 9 6'+6" is not less than BP and positive whole numbers i and & can
always be so chosen as to satisfy these last conditions, however great the proposed
ratio 6" may be, and however little may be its excess © b'+b" over the other pro-
posed ratio U’.
27. With these preparations it is easy to prove, in a new and formal way, the ex-
istence of one determined positive square root 6 for every proposed positive ratio
6, whether that ratio 6 be or be not the square of any whole or of any fractional
number ; for we can now prove this Theorem :
The square aa of the determined positive ratio a, of which ratio the existence
was shown in the IIId. Lemma, is equal to the proposed positive ratio } in the same
Lemma; that is,
‘ / t
: n n'n |
if @> —, whenever ——; < B,
mM mm
n'! n' ni! 259.
and a < all whenever ailonl. b, ¢ )
then aa=b, a= Vb, j
mn’ m' n' being any positive whole numbers which satisfy the conditions here men-
tioned, and } being any determined positive ratio.
For if the square a a of the positive ratio a, determined by these conditions, could
be greater than the proposed positive ratio 4, it would be possible, by the [Vth Lemma,
to insert between them some positive fraction which would be the square of another
positive fraction = ; that is, we could choose m and 7 so that
ah, pee <a * (253.)
mm” ? mm
368 Professor Hamitton on Conjugate Functions,
and then, by the Corollary to the Ist Lemma, and by the conditions (252.), we should
be conducted to the two following incompatible relations,
= a 254
m SiGe 4 mm (254.)
A similar absurdity would result, if we were to suppose aa less than b; a a must
therefore be equal to 4, that is, the theorem is true. It has, indeed, been here as-
sumed as evident, that every determined positive ratio a has a determined positive
square a a3 which is included in this more general but equally evident principle, that
any two determined positive ratios or numbers have a determined positive product.
We find it, therefore, proved, by the most minute and rigorous examination, that
if we conceive any positive ratio 2 or a to increase constantly and continuously from
0, we must conceive its square a x or aa to increase constantly and continuously with
it, so as to pass successively but only once through every state of positive ratio 6:
and therefore that every determined positive ratio 6 has one determined positive
square root yb, which will be commensurable or incommensurable, according as 6 can
or cannot be expressed as the square of a fraction. When ) cannot be so expressed,
it is still possible to approximate in fractions to the incommensurable square root v },
by choosing successively larger and larger positive denominators, and then seeking
for every such denominator m’ the corresponding positive numerator m’ which satisfies
the two conditions (244.); for although every fraction thus found will be less than
the sought root v 6, yet the error, or the positive correction which must be added to
it in order to produce the accurate root v 6, is less than the reciprocal of the deno-
minator m’, and therefore may be made as little different as we please from 0, (though
it can never be made exactly = 0,) by choosing that denominator large enough.
This process of approximation to an incommensurable root v 6 is capable, therefore,
of an indefinitely great, though never of a perfect accuracy ; and using the notation
already given for limits, we may write
P rn ., nn }+n l+n’ ;
SE RN yh ELAR SN (255.)
mn mim m m
and may think of the incommensurable root as the limit of the varying fractional
number.
The only additional remark which need be made, at present, on the subject of the
progression of the square x .r, or aa, as depending on the progression of the root x,
and on Algebra as the Science of Pure Time. 369
or a, is that since (by the 24th article) the square remains positive and unchanged
when the root is changed from positive to contra-positive, in such a manner that
Ola, xO "4% a> 256.)
the square aa must be conceived as first constantly and continuously decreasing or
retrograding towards 0, and afterwards constantly and continuously increasing or
advancing from 0, if the root a be conceived as constantly and continuously increas-
ing or advancing, in the general progression of ratio, from states indefinitely far from
0 on the contra-positive side, to other states indefinitely far from 0, but on the posi-
tive side in the progression.
On Continued Analogies, or Series of Proportional Steps; and on Powers, and
Roots, and Logarithms of Ratios.
28. Four effective steps ab v’ b’ may be said to form a continued analogy or conti-
nued proportion, a and b” being the extremes, and b and b’ the means, when they are
connected by one common ratio in the following manner :
as
6S = (257.)
and if we denote for abridgement this common ratio by a, we may write
Be eX Bs Dt =e XG) X, Big Di) a MAL Xone (258.)
Reciprocally, when » v b’ can be thus expressed, the four steps ab b’ b’ compose a
continued analogy ; and it is clear that if the first extreme step 2 and the common
ratio a be given, the other steps can be deduced by the multiplications (258.) It is
easy also to perceive, that if the two extremes a and bv’ be given, the two means b and
bv may be conceived to be determined (as necessarily connected with these) in one and
in only one way ; and thus that the insertion of two mean proportionals between two
given effective steps, is never impossible nor ambiguous, like the insertion of a single
mean proportional, In fact, it follows from the theorems of multiplication that the
product a x a x a, which may be called the cube of the number or ratio a, is not
obliged (like the square a x a) to be always a positive ratio, but is positive or contra-
positive according as a itself (which may be called the cube-root of this product
370 Professor Hamitton on Conjugate Functions,
a x a X a) is positive or contra-positive ; and on examining the law of its progression,
(as we lately examined the law of the progression of the square,) we find that the cube
a x axa increases constantly and continuously with its cube-root a from states inde-
finitely far from zero, on the contra-positive side, to states indefinitely far advanced
on the positive side of zero, in the general progression of ratio, so as to pass succes-
sively but only once through every state of contra-positive or positive ratio, instead of
first decreasing or retrograding, and afterwards increasing or advancing, like the
square. Thus every ratio has one and only one cube-root, (commensurable or in-
commensurable,) although a ratio has sometimes two square-roots and sometimes none,
according as it is positive or contra-positive ; and when the two extreme effective steps
a and b’ of the continued analogy (257.) are given, we can always conceive the cube-
root a of their ratio ~ determined, and hence the two mean steps or mean propor-
tionals of the analogy, b and v’.
29. In general, as we conceived a continued analogy or series of equi-distant mo-
ments, generated from a single standard moment a, by the repetition of a forward
step a and of a backward step Oa; so we may now conceive, as another sort of conti-
nued analogy, a series of proportional steps, generated from a single standard (effec-
tive) step a, by the repetition of the act of multiplication which corresponds to and is
determined by some one multiplier or ratio a (+E 0), and of the inverse or reciprocal
act of multiplication determined by the reciprocal multiplier or ratio ua: namely, the
following series of proportional steps,
~--Uaxuaxutaxa,taxuda X ay aX a, a,aX a,aA XaAXa,a XAX AX ayaee
(259.)
which may also be thus denoted,
. u (a aa) x a,u(a a) x a,ua@xa,lxa, @x a, @axa, Aaaxa,... (260.)
and in which we may consider the system or series of ratios or multipliers,
-» U(a@aa), U(aa), Ua, 1, a, aa, aaa,... (261.)
to be a system generated from the original ratio or multiplier a, by a system of acts
of generation having all one common character : as we before considered the system of
multiple steps (98.),
-o- Ola 11Ola + Ola, Oal+ Oa, On, 0; a, ata, at ata,
to be a system of steps generated from the original step a by a system of acts of ge-
neration to which we gave the common name of acts of multiplying.
by tT
and on Algebra as the Science of Pure Time. 371
In conformity with this conception, we may call the original ratio a the base of the
system of ratios (261.) and may call those ratios by the common name of powers of
that common base, and say that they are (or may be) formed by acts of powering
it. And to distinguish any one such power, or one such act of powering, from all
the other powers in the system, or from all the other acts of powering, we may
employ the aid of determining numbers, ordinal or cardinal, in a manner analogous
to that explained in the 13th article for a system of multiple steps. Thus, we may
call the ratios a, wa, aaa, ... by the common name of positive powers of the base
a, and may distinguish them by the special ordinal names first, second, third, &c. ; so
that the ratio a is, in this view, its own first positive power; the second positive
power is the square aa, and the third positive power is the cube. Again, we may
call the ratio 1, which immediately precedes these positive powers in the series, the
zero-power of the base a, by analogy to the zero-multiple in the series of multiple
steps, which immediately preceded in that series the system of positive multiples ;
and the ratios ua, u (aa), u (aaa), ... which precede this zero-power 1 in the
series of powers (261.), may be called, by the same analogy, from their order of
' position, contra-positive powers of a, so that the reciprocal u a of any ratio a is the
Jirst contra-positive power of that ratio, the reciprocal u (aa) of its square is its
second contra-positive power, and so on. We may also distinguish the several cor-
responding acts of powering by the corresponding cardinal numbers, positive, or
contra-positive, or null, and may say (for example) that the third positive power aaa
is formed from the base a by the act of powering by positive three ; that the second
contra-positive power u (aa) is formed from the same base a by powering by contra-
positive two ; and that the zero-power 1 is (or may be) formed from a by powering
that base by the null cardinal or number none. In written symbols, answering to
these thoughts and names, we may denote the series of powers (261.), and the series
of proportional steps (260.), as follows,
ee A ars Bos a a ae (262.)
and
meet KBs PE aes 20 eX hs aX as gh Xone ne oe, a X..8 dan AC sue)
in which
a°=1, (264.)
and
(5 2 alec
aa 2d SH (aa), 26
7 Ap (265.)
a°=aaa, a ua (aaa),
&ce. &e.
VOL. XVII, OM
372 Professor Hamitton on Conjugate Functions,
And we may give the name of exponents or logarithms to the determining numbers,
ordinal or cardinal,
SEO S402, Ole Otel et Sone, (266.)
which answer the question “ which in order is the Power?” or this other question
‘« Have any (effective) acts of multiplication, equivalent or reciprocal to the original
act of multiplying by the given ratio a, been combined to produce the act of multi-
plying by the Power ; and if any, then How many, and Jn which direction, that is,
whether are they similar or opposite in effect, (as enlarging or diminishing the step on
which they are performed,) to that original act?” Thus 2 is the logarithm of the
square or second power aa, when compared with the base a; 3 is the logarithm of
the cube aaa, 1 is the logarithm of the base a itself, © 1 is the logarithm of the
reciprocal u a, and O is the logarithm of the ratio 1 considered as the zero-power
of a.
With these conceptions and notations of powers and logarithms, we can easily
prove the relation
a’ x a“=za rte, (267.)
for any integer logarithms » and v, whether positive, or contra-positive, or null ; and
this other connected relation
b’ =a’ X+ if b=ae; 268.)
which may be thus expressed in words: ‘‘ Any two powers of any common base may
be multiplied together by adding their logarithms,’ and ‘‘ Any proposed power may
be powered by any proposed whole number, by multiplying its logarithm by that
number,” if the sum of the two proposed logarithms in the first case, or the multiple
of the proposed logarithm in the second case, be employed as a new logarithm, to
form a new power of the original base or ratio; the logarithms here considered being:
all whole numbers.
30. The act of passing from a base to a power, is connected with an inverse or
reciprocal act of returning from the power to the base ; and the conceptions of both
these acts are included in the more comprehensive conception of the act of passing
from any one to any other of the ratios of the series (261.) or (262.). This act of
passing from any one power a“ to any other power a” of a common base a, may be
still called in general an act of powering ; and more particularly, (keeping up the
analogy to the language already employed in the theory of multiple steps,) it may be
called the act of powering by the fractional number ae By the same analogy of
I
and on Algebra as the Science of Pure Time. 373
definition, this fractional number may be called the logarithm of the resulting power,
and the power itself may be denoted in written symbols as follows,
(a )“=a", (269.)
or thus,
cob, if b=a*, c=a’. (270.)
(The analogous formula (121.) ought to have been printed « = : b, and not ¢ = o :
when b=uxXx a, c=VvxX a.)
In the particular case when the numerator v is 1, and when, therefore, we have to
1
power by the reciprocal of a whole number, we may call the result (a “) x», that is
a', = a, a7root or more fully the p’th root of the power or ratio a“; and we may
call the corresponding act of powering, an extraction of the wth root, or a rooting by
the (whole) number . Thus, to power any proposed ratio by the reciprocal
number 5 8 3 is to extract the second or the third root, that is, (by what has been
already shown,) the square-root or the cube-root, of J, or to root the proposed ratio
b by the number 2 or 3; and in conformity with this last mode of expression, the
following notation may be employed,
1
a='/b when b=a", a=b+:; (271)
so that a square-root may also be denoted by the symbol 2/6, and the cube-root
of b may be denoted by 3/4. And whereas we saw, in considering square-roots that
a contra-positive ratio )<0O has no square-root, and that a positive ratio )>O has two
square-roots, one positive = // and the other contra-positive =9 / 6, of which each
has its square =) ; we may consider the new sign b? or x as denoting indifferently
either of these two roots, reserving the old sign v6 to denote specially that one of
them which is positive, and’the other old sign © 8 to denote specially that one of
them which is contra-positive. Thus yb and OvJ shall still remain determinate
signs, implying each a determinate ratio, (when 6>0,) while 7s and 4* shall be used
as ambiguous signs, susceptible each of two different meanings. But is a deter-
minate sign, because a ratio has only one cube-root. In general, an even root, such
as the second, fourth, or sixth, of a proposed ratio b, is ambiguous if that ratio be
positive, and impossible if 6 be contra-positive ; because an even power, or a power
with an even integer for its logarithm, is always a positive ratio, whether the base be
positive or contra-positive : but an odd root, such as the third or fifth, is always pos-
sible and determinate.
31. It may, however, be useful to show more distinctly, by a method analogous to that
374: Professor Hamitton on Conjugate Functions,
of the 26th and 27th articles, that for any proposed positive ratio 6 whatever, and
for any positive whole number m, it is possible to determine, or conceive determined,
one positive ratio a, and only one, which shall have its m’th power =2; and for this
purpose to show that the power a” increases constantly and continuously from zero
with a, so as to pass successively, but only once, through every state of positive
ratio 5. On examining the proof already given of this property, in the particular
case of the power a’, we see that in order to extend that proof to the more general
case of the power a”, we have only to generalise, as follows, the Ist, IJId, and [Vth
Lemmas, and the Corollary of the Ist, with the Theorem resulting from all four,
retaining the IInd Lemma.
Vth Lemma: (generalised from Ist.)
> >
iW y=; and 2>0; y>0; then y™ = 2. (272.)
> <
When m=1, this Lemma is evident, because the first powers y’ and x* coincide with
the ratios y and 2. When m > 1, the Lemma may be easily deduced from the con-
ceptions of ratios, and of powers with positive integer exponents; it may also be
proved by observing that the difference 0 x” +y”, between the powers x” and y”,
in which the symbol © a” denotes the same thing as if we had written more fully
© (#”™), and which may be obtained in one way by the subtraction of 2” from y”,
may also be obtained in another way by multiplication from the difference 6 a +y
as follows :
Oa™+y"™=(Oaty) x(Foltry? +a my tt. Hay ort 1a Yel tm), (B78:)
and is, therefore, positive, or contra-positive, or null, according as the difference
© «+y of the positive ratios 2 and y themselves is positive, or contra-positive, or
null, because the other factor of the product (273.) is positive. For example,
Oawi+yi=(Ort+y) x(e*+ayty’); (274.)
and, therefore, when a and y and consequently #*+2y+y?* are positive, the dif-
ference © «*+y* and the difference © 2+y are positive, or contra-positive, or null
together.
As a Corollary of this Lemma, we see that, conversely,
> >
if y"=a”", and 4 >0, y> 0, then y =. (275.)
<
and on Algebra as the Science of Pure Time. 375
Thus, the power 2” and the root a increase constantly together, when both are
positive ratios.
The logic of this last deduction, of the Corollary (275.) from the Lemma (272.),
must not be confounded with that erroneous form of argument which infers the truth
of the antecedent of a true hypothetical proposition from the truth of the conse-
quent ; that is, with the too common sophism: If a be true then B is true; but B is
true, therefore a is true. The Lemma (272.) asserts three hypothetical propositions,
which are tacitly supposed to be each transformed, or logically converted, according
to this valid principle, that the falsehood of the consequent of a true hypothetical
proposition infers the falsehood of the antecedent ; or according to this just formula :
If a were true then B would be true ; but 2 is false, therefore a is not true. Ap-
plying this just principle to each of the three hypothetical propositions of the Lemma,
we are entitled to infer, by the general principles of Logic, these three converse
hypothetical propositions :
i ener unease as
ify” > 2, then » > @;
i (276.)
if y™ ¢ v”, then y £7;
x and y being here any positive ratios, and m any positive whole number, and the
signs > <¢ denoting respectively “not >” and ‘‘not <” as the sign + denotes
“not =”. And if, to the propositions (276.), we join this principle of intuition in
Algebra, as the Science of Pure Time, that a variable moment B must either follow,
or coincide with, or precede a given or variable moment a, but cannot do two of
these three things at once, and therefore (by the 21st article) that a variable ratio
must also bear one but only one of these three ordinal relations to a given or yariable
ratio 2, which shows that
when .y” > 2”, then:y "= 2.” andy." ¢ a",
when 1% = 2", then 44 a" and y= > 2™,
when y”™ < a”; then y™ } @™ and y™ + 2”,
and that
when y ¢ @ and y } @, then y = 2,
when y= 2@ and y <¢ 2, then y >a,
(278.)
when y > @ and ya, then y < 2,
_ we find finally that the Corollary (275.) is true. The same logic was tacitly em-
ployed in deducing the Corollary of the Ist Lemma, in the hope that it would be
mentally supplied by the attentive reader. It has now been stated expressly, lest any
376 Professor Hamitton on Conjugate Functions,
should confound it with that dangerous and common fallacy, of inferring, in Pure
Science, the necessary truth of a premiss in an argument, from the known truth of
the conclusion.
Resuming the more mathematical part of the research, we may next establish
this
Vith Lemma (generalised from IIId): There exists one positive ratio a, and only
one, which satisfies all the following conditions,
a> z whenever er <4,
n" n"\™ (279.)
a<aa whenever S&S >b;
6 being any given positive ratio, and m any given positive whole number, while
mn’ m' n' are also positive but variable whole numbers. The proof of this Lemma
is so like that of the IIId, that it need not be written here; and it shows that in the
particular case when the given ratio 4 is the m‘* power of a positive fraction
n, $ 4 A : ;
cae then a is that fraction itself. In general, it will soon be shown that under the
]
conditions of this Lemma the mt" power of a is 4.
VIIth Lemma (generalised from IVth). It is always possible to find, or to con-
ceive as found, two positive whole numbers m, and 7,, which shall satisfy the two
conditions
(x) ">e, (“)" <o', if oh >, o>0, Se
m, }
m being any given positive whole number ; that is, we can insert between any two
unequal positive ratios 2’ and 2” an intermediate fractional ratio which is itself the
m+ power of a fraction.
For, when m=1, this Lemma reduces itself to the IInd; and when m > 1, the
m
theorem (273.) shows that the excess of =) over Ge) may be expressed as
follows :
e) (ear + (242)" == xp, * (281.)
mM, m,
n e@l-+m n o2+m n e@3+m 1 n a
dei al ppongs olin ere ea eee
™, mM, m, m, mM,
2
etsy) Gas e ere Fa ee El it (282.)
m, m,\ m, m,
in which
and on Algebra as the Science of Pure Time. 377
1 n 3 n \ 3 3
zs ) over the cube ‘= ) ,is
{
i
for example, when m=3, the excess of the cube (
n> Man\?_1 ie Dp, The byiyss ere ae
e(>) +( m, } = 4 = | nay m, eh) m, ) 2 (285.)
In general, the number of the terms (or added parts) in the expression (282.), 1
ol+m : :
m, and they are all unequal, the least being we ey) , and the greatest being
: el+m y 4
| ; their sum, therefore, is less than the hes multiple of this greatest term,
that is,
el+m
pa a ee (284.)
7
and therefore the excess (281.) is subject to the corresponding condition
1 1 el-+m 2
o(- oy 44th <5 =*) ; (285.)
aC m, ) + +C Oy ™, 4g =f) : (286.)
However this excess (281.) increases constantly with x, when m, remains unaltered,
because p so increases; so that the 1 +7 fractions of the series
) m Sym a
i Ce Ge Co (287.)
increase by increasing differences, (or advance by increasing intervals,) which are each
for example,
oel+m .
less than — c= , and therefore than z? if we choose m, and 7 so as
™, 7] C
to satisfy the conditions
r _o9l+m kmin
l+tn=im, m=kmxi ———
. (288.)
i and & being any two positive whole numbers assumed at pleasure ; with this choice,
therefore, of the numbers m,and 7, some one (at least), such as es jd of the
series of powers of fractions (287.), of which the last is =7”, will fall between any
two proposed unequal positive ratios 3' and 4”, if the greater 4” does not exceed that
last power 7”, and if the difference © J’ + 4” is not less than >; and these condi-
k
tions can be always satisfied by a suitable choice of the whole numbers 7 and k, how-
378 Professor HamiLTon on Conjugate Functions,
ever large may be the given greater positive ratio 4’, and however little may be its
given excess over the lesser positive ratio 2’.
Hence, finally, this Theorem :
n' n'!
Ifa>—,anda< —,, ]
Y m!
n' m n” ™m 28
whenever (|) < 4, (-) aE b (289.)
1
then a" = 0, a="/b =5™;
5 denoting any given positive ratio, and m any given positive whole number,
while m' 2’ m" n" are any arbitrary positive whole numbers which satisfy these
conditions, and a is another positive ratio which the VIth Lemma shows to be
determined.
For if a” could be >4, we could, by the VIIth Lemma, insert between them a
™
positive fraction of the form (=) > such that
Ci) o> (ZB) "<a", (290)
and then by the Corollary of the Vth Lemma, and by the conditions (289.), we
should deduce the two incompatible relations
a SG axm, (291.)
which would be absurd. A similar absurdity would follow from supposing that a”
could be less than 4; a” must therefore be =4, that is, the Theorem is true. It
has, indeed, been all along assumed as evident that every determined positive ratio a
has a determined positive mt power a”, when m is a positive whole number ; which
is included in this more general but also évident principle, that any m determined
positive ratios or numbers have a determined positive product.
Every positive ratio 6 has therefore one, and only one, positive ratio a for its
m‘ root, which is commensurable or incommensurable, according as 4 can or
cannot be put under the form (=) ; but which, when incommensurable, may be
theoretically conceived as the accurate limit of a variable fraction,
aaah ye if (By CRED TES (292.)
m
SS es a
and on Algebra as the Science of Pure Time. 379
and may be practically approached to, by determining such fractions — , with larger
and larger whole numbers m’ and 7’ for their denominators and numerators. And
whether m be odd or even, we see that the power a” increases continuously (as well
as constantly) with its positive root or base a, from zero up to states indefinitely
greater and greater. But if this root, or base, or ratio a be conceived to advance
constantly and continuously from states indefinitely far from zero on the contra-
positive side to states indefinitely far upon the positive side, then the power a” will
either advance constantly and continuously likewise, though not with the same quick-
ness, from contra-positive to positive states, or else will first constantly and continu-
ously retrograde to zero, and afterwards advance from zero, remaining always posi-
tive, according as the positive exponent or logarithm m is an odd or an eyen integer.
It is understood that for any such positive exponent m,
0" =0, (293.)
the powers of 0 with positive integer exponents being considered as all themselves
equal to 0, because the repeated multiplication by this null ratio generates from any
one effective step a the series of proportional steps,
aeOnxe tn = Os OOD ai Owe n> (294.)
which may be continued indefinitely iz one direction, and in which all steps after the
first are null ; although we were obliged to exclude the consideration of such null
ratios in forming the series (259.) because we wished to continue that series of steps
indefinitely in two opposite directions.
32. We are now prepared to discuss completely the meaning, or meanings, if any,
which ought to be assigned to any proposed symbol of the class bu, b denoting any
proposed ratio, and m and y any proposed whole numbers. By the 30th article,
the symbol b« denotes generally the v’th power of a ratio a of which 6 is the »’th
power ; or, in other words, the »th power of a nth root of 6; so that the mental
operation of passing from the ratio 4 to the ratio 4 m, is compounded, (when it can
be performed at all,) of the two operations of first rooting by the one whole
number ,», and then powering by the other whole number v: and we may write,
br = (4) = (0) (295.)
The ratio 4, and the whole numbers u and y, may each be either positive, or contra-
positive, or null; and thus there arise many cases, which may be still farther sub-
yOL. XVII, 32
380 Professor Hami.ton on Conjugate Functions,
divided, by distinguishing between odd and even values of the positive or contra-
positive whole numbers. or, if we suppose that B denotes a positive ratio, and that
m and n denote positive whole numbers, we may then suppose
b=8, or 6=0, or 6=0 8,
w=M, or n=O, or p=O m, (296.)
v=n, or w=0; or p= O0in;
and thus shall obtain the twenty-seven cases following,
” 2 On
B™ B*”, Bm
ze, g pa
Be 5 Be ’ B o (297.)
n mos on
BeOm, BOm, Bom
On 5 O m ; Om
n oO on
Oe Dan Ur ois (298.)
n o on
Qem, O om, O om,
2 ws ous.
(9 B) my (O B) my (9 B) my \
n o on
(9B)*, (OB)*, (OB) -, (299.)
ne Qe en
(9 B) on, (9 B) om, (9 B) em, J
which we may still farther sub-divide by putting m and m under the forms
m= 2 21, \or M=0 1 +.2.2,
n=2k, or 1=O0 142k, (300.)
_in which @ and & themselves denote positive whole numbers. But, various as these
- cases are, the only difficulty in discussing them arises from the occurrence, in some,
of the ratio or number 0; and to remove this difficulty, we may lay down the fol-
lowing rules, deduced from the foregoing principles.
To power the ratio 0 by any positive whole number m, gives, by (293.), the ratio
O as the result. This ratio 0 is, therefore, at least one m’th root of 0; and since
no positive or contra-positive ratio can thus give 0 when powered by any positive
whole number, we see that the on/y m’th root of O is O itself. Thus,
1
0m=~0, (301.)
acne tw
Pa
4
°
and on Algebra as the Science of Pure Time. 381
and generally,
Oe, (302.)
To power any positive ratio a, whether positive, or contra-positive, or null, by the
number or logarithm 0, may be considered to give 1 as the result; because we can
always construct at least this series of proportional steps, beginning with any one
effective step a, and proceeding indefinitely in one direction :
Ib eit CHS, Cp NSS OES Ce BD BEE (3083. )
and we may still call the ratio 1 the zero-power, and the ratios a, axa, ... the
positive powers of the ratio a, even when we cannot continue this series of proportional
steps (303.) backward, like the series (259.), so as to determine any contra-positive
powers of a; namely, in that particular case when a=0. We may, therefore, con-
sider the equation (264.), @’=1, as including even this particular case a=0; and
may write
OZ (304.)
and, therefore, by (301.) and (295.)
On=1: (305.)
we are also conducted to consider the symbols
0°", Om, (306.)
as absurd, the ratio 0 having no contra-positive powers.
From the generality which we have been led to attribute to the equation a’=1, it
follows that the symbol
oy
1°, and more generally Tree (307.)
is indeterminate, or that it is equally fit to denote all ratios whatever ; but that the
symbol
1
be, or be, if 61, (308.)
is absurd, or that it cannot properly denote any ratio. In particular, the symbols
fis jh) Or; (309.)
382 Professor Hamitton on Conjugate Functions,
are absurd, or denote no ratios whatever. In like manner the symbol
1 v
0®™, and more generally Oo», (310.)
is absurd, or denotes no ratio, because no ratio a can satisfy the equation
a°™=0. (311.)
We have thus discussed all the nine cases (298.), of powers in which the base is 0,
and haye found them all to be impossible, except the two first, in which the exponents
are ~, and ~, and in which the resulting powers are respectively 0 and 1. We.
have also obtained sufficient elements for discussing all the other cases (297.) and
(299.), with their sub-divisions (300.), as follows.
lst. B™ is determined and positive, unless m is even, and x odd; in which case
: e1+2k 3 : 4 Z .
it becomes of the form B —g:—, and is ambiguous, being capable of denoting either
of two opposite ratios, a positive or a contra-positive. To distinguish these among
themselves, we may denote the positive one by the symbol
eo1+42k
Bist cs (312.)
and the contra-positive one by the symbol
0 142k
OB ar 3 (313.)
for example, the two values of the square-root /B or B*, may be denoted for dis-
tinction by the two separate symbols
i oy
B?=/8B, OB®=0,//B. (314.)
°
The other three cases of the notation B”, namely, the symbols
olpzk _2k 2k
polt2i ,pelfai py Bi (315.)
>
denote determined positive ratios.
2d. The three cases
1+0 (2k) 0 (2k) © (2 k)
p OM? pelei , 8t) (316.)
on
of the notation B ™, are symbols of determined positive ratios; but the case
and on Algebra as the Science of Pure Time. 383
1+0@k)., , ; : : - ie é
B — 9; is ambiguous, this symbol denoting either a determined positive ratio or a
determined contra-positive ratio, which may be thus respectively marked, when we
wish to distinguish them from each other,
140 (2h) 1-0 (2 A) :
Stat orp O BiunrowE (317.)
In general, we may write,
Bo™ = 4u(B”), (318.)
the latter of these two symbols having the same meaning or meanings as the
former.
3d. The symbols
e142hk 2k Qk
B l+e Go) pire? ) pee s) (319.)
included in the form B°™, denote determined positive ratios ; but the other symbol
Olf+2k , = = fs A 5
B e@»), included in the same form B®”, is ambiguous, denoting either a deter-
mined positive or a determined contra-positive ratio,
e142k o142k
B (23) , or OB @(@). (320.)
In general, we may write
= 2 :
Bem=4 (B”). (321.)
4th. In like manner, we may write,
nm
Bem—B”, (322.
the former symbol having always the same meaning or meanings as the latter. The
cases
1+0 (2h) 0 (2k) 9 (2k)
Bree) Bae, Bp? @)
102k) | ,
are symbols of determined positive ratios; but the case B ¢@0 is ambiguous, and
includes two opposite ratios, which may be thus respectively denoted,
140 (2k) 140 (24)
zg °@i) ,On ¢2) . (324.)
384 Professor Hamizton on Conjugate Functions,
In general, we shall denote by the symbol
Bz, or bs, if b> 0, ~-E0, v20, (325.)
that positive ratio which is either the only value, or at least one of the values of the
symbol pu or 6»; and it is important to observe that this positive ratio is not
changed, when the form of the fractional logarithm S is changed, as if it were a
fractional multiplier, by the rule (135.), to the form rm or (as it may be more
concisely written) {’ 3 that is,
ov
Bo#=B": (326.)
a theorem which is easily proved by means of the relation (268.), combined with the
determinateness (already proved) of that positive ratio which results from powering
or rooting any proposed positive ratio by any positive or contra-positive whole
number.
5th. With respect to the five remaining notations of the group (297.), namely,
those in which O occurs, we have
o o
B@=1; Bem=15 (327.)
also the symbols
n
o
B°, B
on
0
: (328.)
are each indeterminate when B=1, and absurd in the contrary case ; and, finally, the
symbol
o
)
B
(329.)
is absurd when eB = 1, but determined and =1, when p=1.
6th. Proceeding to the group (299.), the symbols
(0»)7, (@5)%, (8)>, (330.)
are absurd; the symbols
(o2)", (OR)om, (331.)
are determined and each =1, if m be odd, but otherwise, they are absurd; and the
four remaining symbols
(Ox)*, a) aa (Cis)e>s (OB) on, (332.)
and on Algebra as the Science of Pure Time. 385
are absurd if m be even, but denote determined ratios when m is odd, which ratios
are positive if m be even, but contra-positive if m be odd.
It must be remembered that all the foregoing discussion of the cases of the general
notation bi, for powers with fractional logarithms, is founded on the definition laid
down in the 30th article, that b« denotes the v’th power of a ,’th root of 6, or in
other words, the v’th power of a ratio a of which 4 is the y’th power. When no such
ratio a can be found, consistently with the previous conception of powers with integer
logarithms, the symbol b+ is pronounced to be absurd, or to be incapable of denoting
any ratio consistently with its general definition ; and when two or more such ratios
a can be found, each having its »’th power = b, we have pronounced that the frac-
tional power bz is ambiguous or indeterminate, except in those cases in which the
second component act of powering by the numerator v has happened to destroy the inde-
terminateness. And with respect to powers with integer exponents, it is to be remem-
bered that we always define them by a reference to a series of proportional steps, of which
at least the original step (corresponding to the zero-power) is supposed to be an effective
step, and which can always be continued indefinitely, at least in the positive direction,
that is, in the way of repeated multiplication by the ratio proposed as the base, al-
though in the particular case of a null ratio, we cannot continue the series backward
by division, so as to find any contra-positive powers. These definitions appear to be
the most natural ; but others might have been assumed, and then other results would
have followed. In general, the definitions of mathematical science are not altogether
arbitrary, but a certain discretion is allowed in the selection of them, although when
once selected, they must then be consistently reasoned from.
33. The foregoing article enables us to assign one determined positive ratio, and
only one, as denoted by the symbol br, when @ is any determined positive ratio, and
a any fractional number with a numerator and a denominator each different from O :
it shows also that this ratio b does not change when we transform the expression of
the fractional logarithm a by introducing or suppressing any whole number w as a factor
common to both numerator and denominator ; and permits us to write
be2 =u (b*), (333.)
8 a being the opposite of the fraction a in the sense of the 17th article. More gene-
rally, by the meaning of the notation ba, and by the determinateness of those positive
ratios which result from the powering or rooting of determined positive ratios by de-
386 Professor Hamitton on Conjugate Functions,
termined integer numbers, (setting aside the impossible or indeterminate case of root-
ing by the number 0,) we have the relation
(i ileal Jaleo (334.)
which is analogous to (267.); and the relation
cf — exe ifc = DF, (335.)
analogous to (268.): a and f denoting here any two commensurable numbers. And
it is easy to see that while the fractional exponent or logarithm a increases, advancing
successively through all fractional states in the progression from contra-positive to
positive, the positive ratio b “ increases constantly if 5 >1, or else decreases constantly
if b<1, ()>0,) or remains constantly = 1 if 6= 1. But to show that this increase
or decrease of the power with the exponent is continuous as well as constant, we must
establish principles for the interpretation of the symbol be when a is not a fraction.
When a is incommensurable, but 6 still positive, it may be proved that we shall still
have these last relations (334.) and (335.), if we interpret the symbol be to denote
that determined positive ratio ¢ which satisfies the following conditions :
whenever a >=,
m whenever a < x, f (836.)
if b>1; J
c = bs < b® whenever a >=,
(337.)
c = b* > bw’ whenever a <=,
if b< 1, b>0;
or finally this equation,
c= h=1,if 6-1. (338.)
The reader will soon perceive the reasonableness of these interpretations ; but he may
desire to see it proved that the conditions (336.) or (337.) can always be satisfied by
one positive ratio c, and only one, whatever determined ratio may be denoted by a,
and whatever positive ratio (different from 1) by 4. ‘That at least one such positive
and on Algebra as the Science of Pure Time. 387
ratio ¢ = b can be found, whatever incommensurable number the exponent a may
be, is easily proved from the circumstance that none of the conditions (336.) are in-
compatible with one another if ) > 1, and that none of the conditions (337.) are in-
compatible with each other in the contrary case, by reason of the constant increase or
constant decrease of the fractional power Bin for constantly increasing values of the
fractional exponent *. And that only one such positive ratio ¢ = b can be found,
or that no two different positive ratios c, c’, can both satisfy all these conditions may
be proved for the case » > 1 by the following process, which can without difficulty be
adapted to the other case.
The fractional powers of comprised in the series
1. 3) dh im tim
bn, bm, bm, ... bm, bom, (339.)
° °
increase (when )>1) by increasing differences, of which the last is
im l+m
m
Ob™ + bw =b (01 + bn); ($40.)
this last difference, therefore, and by still stronger reason each of the others which
precede it, will be less than = if
l>kb (841.)
and
O1+bn<7: (342.)
. and this last condition will be satisfied, if
| m>1(01 +b), (343.)
J and m (like 7 and &) denoting any positive whole numbers ; for then we shall have
i So, (344.)
and by still stronger reason
(Lx 7)" >B, 1+ 7> Bry (345.)
observing that
(1 +4) ">1+4%, ifm>1, (346.)
VOL. XVII. 4A
388 Professor HamItton on Conjugate Functions,
because, by the theorem of multiplication (273.), or (281.),
m
] ] | ee: 1. S9l+m ;
Ol + Cts) =p {l+atp+aep t+. F04}) - (347.)
If then c c¢’ be any two proposed unequal positive ratios, of which we may suppose
that c’ is the greater,
rae ono 5 (348.)
we may choose two positive whole numbers 7, k, so large that
bi>c,2<Octe, (34 9.)
and two other positive whole numbers J, m, large enough to satisfy the conditions
(341.) (343.) ; and then we shall be sure that some one at least, such as 4”, of the
fractional powers of 4 comprised in the series (339.) will fall between the two proposed
unequal ratios ¢ c’, so that
ce<bm,c >in, (350.)
If then the one ratio c satisfy all the conditions (336.), the incommensurable number
a must be <},, and therefore, by the 2nd relation (350.), the other ratio c’ cannot also
satisfy all the conditions of the same form, since it is > 8 m, although a<;. In like
manner, if the greater ratio c’ satisfy all the conditions of the form (336.) the lesser ratio c
cannot also satisfy them all, because in this case the incommensurable number a will be > ~,
No two unequal positive ratios, therefore, can satisfy all those conditions : they are there-
fore satisfied by one positive ratio and only one, and the symbol bs (interpreted by.
them) denotes a determined positive ratio when b > 1. For a similar reason the same
symbol 6", interpreted by the conditions (337.), denotes a determined positive ratio
when 3 < 1, 6 > 0; and for the remaining case of a positive base, 5 = 1, the symbol
b* denotes still, by (338.) a determined positive ratio, namely, the ratio 1. The ex-
ponent or logarithm a has, in these late investigations, been supposed to be incom-
mensurable ; when that exponent a is commensurable, the base é being still positive,
we saw that the symbol % can be interpreted more easily, as a power of a root, and
that it always denotes a determined positive ratio.
Reciprocally, in the equation
B= lio (351.)
°
if the power c he any determined positive ratio, and if the exponent a be any deter-
and on Algebra as the Science of Pure Time. 389
mined ratio, positive or contra-positive, we can deduce the positive base or ratio 4, by
calculating the inverse or reciprocal power
6 =lcR (352.)
as appears from the relation (335.) which extends, as was above announced, together
with the relation (334.), even to the case of incommensurable exponents. The proof
of the important extension last alluded to, will easily suggest itself to those who have
studied the foregoing demonstrations ; and they will perceive that with the foregoing
rules for the interpretation of the symbol 4", for the case of an incommensurable ex-
ponent, the power 4* increases (as was said ‘shiote) continuously as well as constantly
with the exponent a if the base 4 be > 1, or else decreases continuously and con-
stantly if that positive base be <1, but remains constantly = 1 if} = 1. It is
therefore possible to find one determined exponent or logarithm a, and only one,
which shall satisfy the equation (351.), when the power c and the base 4 are any given
positive ratios, except in the impossible or indeterminate case when this base 4 is the
particular ratio 1; and the number a thus determined, whether positive or contra-
positive or null, may be called ‘‘ the logarithm of c to the base 4,” and may be denoted
by the symbol
a= log,.c. (353.)
It is still more easy to perceive, finally, that when this logarithm a is given, (even
if it be incommensurable,) the power c¢ increases constantly and continuously from
zero with the base 4, if a > 0, or else decreases constantly and continuously towards
zero ifa <0, or remains constant and = 1, ifa = 0.
Remarks on the Notation of this Essay, and on some modifications by which it may
be made more like the Notation commonly employed.
34. In the foregoing articles we have constantly denoted moments, or indivisible
points of time, by small capital letters, a, B, a’, B’, &c.; and steps, or transitions from
one such moment to others, by small Roman letters, a, b, a, b', &c. The mark —
has been interposed between the marks of two moments, to express the ordinal rela-
tion of those two moments, or the step which must be made in order to pass from
one to the other ; and the mark + has been inserted between the marks of a step
and a moment, or between the marks of two steps, to denote the application of the
390 Professor Hamitton on Conjugate Functions,
step to the moment, or the composition of the two steps with each other. For the
decomposition of a step into others, we have used no special mark; but employed
the theorem that such decomposition can be performed by compounding with the
given compound step the opposites of the given component steps, and a special nota-
tion for such opposite steps, namely, the mark* © prefixed; so that we have written
© a to denote the step opposite to the step a, and consequently 6a + b to denote the
algebraical excess of the step b over the step a, because this excess may be conceived
as a step compounded of b and Oa. However, we might have agreed to write
(») +4)—(ata)=b—a, (354:.)
denoting the step from the moment a+a to the moment b+ a, for conciseness by
b — a; and then b —a would have been another symbol for the algebraical excess of
the step b over the step a, and we should have had the equation
b—a=Oadb. (355.)
We might thus have been led to interpose the mark — between the marks of a com-
pound step b and a component step a, in order to denote the other component step,
or the algebraical remainder which results from the algebraical subtraction of the
component from the compound.
Again, we have used the Greek letters » v & p w, with or without accents, to denote
integer numbers in general, and the italic letters ik 1mm to denote positive whole
numbers in particular ; using also the earlier letters a B y aed to denote any
ratios whatever, commensurable or incommensurable, and in one recent investigation
the capital letter B to denote any positive ratio: and employed, in the combi-
nation of these symbols of numbers, or of ratios, the same marks of addition and of
opposition, + and ©, which had been already employed for steps, and the mark of
multiplication x , without any special mark for subtraction, We might, however,
have agreed to write, in general,
(6 x a)—(a x a) =(b—a) xK ay (356.)
as we wrote
(b x a)+(a x a)=(b+a) x a;
and then the symbol 4—a would have denoted the algebraical excess of the number
* This mark has been printed, for want of the proper type, like a Greek Theta in this Essay: it was
designed to be printed thus G,
and on Algebra as the Science of Pure Time. 391
b over the number a, or the remainder obtained by the algebraical subtraction of the
latter number from the former ; and we should have had the equation,
b6-—a=Oa+4, (357.)
which is, with respect to nwmbers, or ratios, what the equation (355.) is, with respect
to steps. And when such symbols of remainders, ») — a or 5—a, are to be combined
with other symbols in the way of algebraical addition, it results, from principles
already explained, that they need not be enclosed in parentheses; for example, we
may write simply ¢ + b —a instead of ¢ +(» —a), because the sum denoted by this
last notation is equivalent to the remainder (ce +b) —a. But the parentheses (or
some other combining mark) must be used, when a remainder is to be subtracted ;
thus the symbol ¢ — » — a is to be interpreted as (ec — b) — a, and not as ec — (b — a),
which latter symbol is equivalent to (¢—b) +a, or e—b+a.
35. With this way of denoting the algebraical subtraction of steps, and that of
numbers, we have the formula,
O—a=O9a, O-—a=0 a, (358.)
O denoting in the one a null step, and in the other a null number. And if we farther
agree to suppress (for abridgement) this symbol O when it occurs in such combina-
nations as the following, 0 + a, O—a, 0+a, O—a, writing, in the case of steps,
O+a=>+a, O—a=—a, (359.)
and similarly, in the case of numbers,
0+a= +a, O—a=—a, (360.)
and, in like manner,
oe (361.)
O+a+b=+a+tb, O-at+b=-atb,
we shall then have the formula
+asa, —a=QOa, (362,)
and
+a=a, —a=04, (363.)
of which the one refers to steps and the other to numbers. With these conventions,
392 Professor Hamitton on Conjugate Functions,
the prefixing of the mark + to an isolated symbol of a step or of a number, does
not change the meaning of the symbol; but the prefixing of the mark — converts
that symbol into another, which denotes the opposite of the original step, or the
opposite of the original number ; so that the series of whole numbers (103.) or (266.)
may be written as follows :
Q
—3, —2, —1, 0, +1, +2, +3, ... (364.)
Also, in this notation,
bt(+a)=bta, D+(—a)=bFa, (365.)
b+(t+ta)=b+a, b+(-a=bFa. .
36. Finally, as we wrote, for the case of commensurable steps,
Vv aay,
a aay 7c
» and vy being here whole numbers, so we may agree to write, in general,
2B Bed eek (366.)
eee AG
whatever ratios a and 6 may be; and then this symbol » will denote, in general, the
algebraic quotient obtained by dividing the number or ratio } by the number or ratio
a; whereas we had before no general way of denoting such a quotient, except by the
mark u prefixed to the symbol of the divisor a, so as to form a symbol of the reci-
procal number u a, to multiply by which latter number is equivalent to dividing by
the former. Comparing the two notations, we have the formula,
7 = 144, (367.)
and generally
b
SSG <i SS Oe Ue (368.)
a
These two marks © and u haye been the only new marks introduced in this Ele-
mentary Essay ; although the notation employed for powers differs a little from the
common notation: especially the symbol Be; suggested by those researches of Mr.
Graves respecting the general expression of powers and logarithms, which were the
first occasion of the conception of that Theory of Conjugate Functions to which we
now proceed.
END OF THE PRELIMINARY AND ELEMENTARY ESSAY.
and on Algebra as the Science of Pure Time. 393
THEORY OF CONJUGATE FUNCTIONS,
OR ALGEBRAIC COUPLES.
On Couples of Moments, and of Steps, in Time.
1. When we have imagined any one moment of time A,, which we may call a
primary moment, we may again imagine a moment of time 4,, and may call this a
secondary moment, without regarding whether it follows, or coincides with, or pre-
cedes the primary, in the common progression of time; we may also speak of this
primary and this secondary moment as forming a couple of moments, or a moment-
couple, which may be denoted thus, (A,, A,). Again, we may imagine any other two
moments, a primary and a secondary, B, and B,, distinct from or coincident with each
other, and forming another moment-couple, (B,, By); and we may compare the latter
couple of moments with the former, moment with moment, primary with primary,
and secondary with secondary, examining how B, is ordinally related to a,, and how
B, is ordinally related to a,, in the progression of time, as coincident, or subsequent,
or precedent ; and thus may obtain a couple of ordinal relations, which may be thus
separately denoted B,—A,, B,—A., or thus collectively, as a relation-couple,
(8; a iT Ap).
This couple of ordinal relations between moments may also be conceived as consti-
tuting a complex relation of one moment-couple to another ; and in conformity with
this conception it may be thus denoted,
(Bi, B,) — (A, Aa)s
394. Professor Hamiiton om Conjugate Functions,
so that, comparing this with the former way of representing it, we may establish the -
written equation,
(B, By) — (Ay Ay) =(Bi— Aj B,— Ay). (1.)
instead of conceiving thus a couple of ordinal relations between moments, or a
relation between two couples of moments, discovered by the (analytic) comparison of
one such couple of moments with another, we may conceive a couple of steps in the
progression of time, from moment to moment respectively, or a single complex step
which we may call a step-couple from one moment-couple to another, serving to
generate (synthetically) one of these moment-couples from the other; and if we
denote the two separate steps by a,, a2, (a, being the step from a, to B,, and a, being
the step from a, to B,,) so that in the notation of the Preliminary Essay,
Bj=a,;+A), By= ag+ Ay,
B,=(B)—A;) +41, B= (By— Ag) + As,
we may now establish this analogous notation for couples,
(8, B») =(a, + Aly a + A») }
=e ay) +(A, Ay) i (2.)
),
= $ (Bi, Bo) — (Ai, Ao)} +(A1, Ay
the symbol (#,, B,) —(a,, 4.) corresponding now to the conception of the step-couple by
which we may pass from the moment-couple (A,, A) to the moment-couple (B,, Bs), and
the equivalent symbol (a), a) or (Bj —A,, B:—A,) corresponding now to the conception
of the couple of steps a,, », from the two moments Aj, A,, to the two moments B,, B,,
respectively. The step ,, or B,—A, may be called the primary step of the couple
(a1, a), and the step a, or B,—A, may be called the secondary step.
A step-couple may be said to be effective when it actually changes the moment-
couple to which it is applied; that is, when one at least of its two coupled steps is
effective : and in the contrary case, that is, when both those coupled steps are sepa-
rately null, the step-couple itself may be said to be null also. And effective step-couples
may be distinguished into singly effective and doubly effective step-couples, according
as they alter one or both of the two moments of the moment-couples to which they
are applied. Finally, a singly effective step-couple may be called a pure primary or
pure secondary step-couple, according as only its primary or only its secondary step
is effective, that is, according as it alters only the primary or only the secondary
moment. Thus (0, 0) is a null step-couple, (a,, ») is a doubly effective step-couple,
a
2 ead
and on Algebra as the Science of Pure Time. 395
and (a,, 0) (0, a) are singly effective step-couples, the former (a, 0) being a pure
primary, and the latter (0, a,) being a pure secondary, if 0 denote a null step,
and a, a ellective steps.
On the Composition and Decomposition of Step- Couples,
2. Having stepped from one couple of moments (A,, A») to another couple of
moments (5, B,) by one step-couple (a,, a), we may afterwards step to a third
couple of moments (C,, C:) by a second step-couple (b,, b,), so as to have
(a, C.)=(b1, by.) + (Bi; B)), t (3.)
(Bi, By) =(a1 ap) +(A1p Ag) 3
and then we may consider ourselves as having made upon the whole a compound
couple of steps, or a compound step-couple, from the first moment-couple (A,, A,) to
the third moment-couple (c,, c,), and may agree to call this compound step-couple
the swm of the two component step-couples (@,, 4), (1, b), or to say that is formed &
by adding them, and to denote as follows,
(ci, C2) —(A; A») =( bay by) a ( 45 a,) 3 (4.)
as, in the language of the Preliminary Essay, the two separate compound steps, from
A, to c, and from a, to c, are the swms of the component steps, and are denoted by
the symbols »,+ 4, and b,+ 4, respectively. With these notations, we have evi:
dently the equation
(>, bo) + (41, a2)=(b, + 41, bo + a2); (5.)
that is, the swm of two step-couples may be formed by coupling the two sum-steps.
Hence, also,
(>,,b.)+( a1, a)=(), ao) +(>,, b»), (6.)
that is, the order of any two component step-couples may be changed without altering
the result ; and
(#1, 42)=(41, 0) + (0; 4), (7)
that is, every doubly effective step-couple is the sum of a pure primary and a pure
VOL. XVII, 43
396 Professor Hamitron on Conjugate Functions,
secondary. In like manner, we can conceive sums of more than two step-couples,
and may establish, for such sums, relations analogous to those marked (5.) and (6.) ;
thus,
(ey, %)+(by, bay +(ay, as)—Cer+ bit a1, oot byt a4), (s
= (an, ay) +(d,, db») +(e, 20) &e. t )
We may also consider the decomposition of a step-couple, or the subtraction of one
such step-couple from another, and may write,
(b,, bo) —( ay, ay)=(b,— a,, by— a4), (9.)
(b,, b,)—(a,, a.) being that sought step-couple which must be compounded with or
added to the given component step-couple (;, 4,) in order to produce the given
compound step-couple (»,, b»). And if we agree to suppress the symbol of a null
step-couple, when it is combined with others or others with it in the way of addition
or subtraction, we may write
(41 a,)=(0, 0) +( a1 ay)= +(ay ao),
(-— 1, —a,)=(0, 0)—( a1, ayy= —(a,, ay), Os)
employing a notation analogous to that explained for single steps in the 35th article
of the Preliminary Essay. Thus +(,, 2.) is another way of denoting the step-
couple (@,, 2.) itself; and —(,, ®,) isa way of denoting the opposite step-couple
(— 41, — #2).
On the Multiplication of a Step- Couple by a Number.
3. From any proposed moment-couple (a;, 42), and any proposed step-couple
(a, a), we may generate a series of other moment-couples
So (£1, E>»), (4), Ey); (Ai Aa); (Bi, Bp); (Bi, B’2) eee (11.)
by repeatedly applying this step-couple (a), a), itself, and the opposite step-couple
—(a, ), or (—a,, — a), ina way analogous to the process of the 13th article of
the Preliminary Essay, as follows :
and a series of multiple step-couples, namely
and on Algebra as the Science of Pure Time.
eee eeneee
(: E,)=(—a, —ay) +(-a4, —ay) + (Ai, A)s
(], &)=(—a1, —a,)+(A; 41),
(A, A:)=(A, 42),
@,, B )=(a,, ay) Si (A, Ay),
(8), Bo) = (a, ao) zt (a ’ ay) ar (A; As),
(©);
(4,
(A,
(2,
(3',
ee eeeeeees
E,)— (A, As)=(—a1, —a2)
A.) — (A, A2) = (0, 0),
B,)—(A, A) =(a, aa)»
B,)y—(A, A)=(a, ao) +(a,, as),
which may be thus more concisely denoted,
and
a
nig eT we
(f, Eo) —(Ai, As) = — 2 (a1
(1, Ey )—(Ay 42)= —1(@,
«,
(2,
(A,
(2,
(’ 5]
E>) =— a (a, a2) + (A, Ay),
E,)=
=
B;) =
Es) =
(Ay A, )—(Ay i
(B, B.)—(A, 4)= +1,
(3), Bs) — (Ay A,)=
&e
—1{(a,
0,
+1,
+2(a,
O (a1
+ 2(a,
ao) +(a, Ao),
ao) oF (A, Ay)s
ao) + (A; As),
ao) + (A, Ad)>
a,)= —2x (a,
Ay) = —1x (a1,
a)= Ox(a,
ay) = +1x(,
an) i= +2x(a,
|
J
a a
e 2)»
Ue a)
a 2)>
a a)
(12.)
(14.)
|
|
J
397
_ We may also conceive step-couples which shall be swb-multiples and fractions of a
; given step-couple, and may write
§
(ci, co) — =
a
v
x (b,, by) = Poles bs),
(16.)
398 Professor HamintTon on Conjugate Functions,
when the two step-couples (b,, b:) (c:, «) are related as multiples to one common
step-couple (a,, a.) as follows :
(bi, Ne) = tees (Gyn ai); (e1, Cn p= yd (Canstan)5 (17.)
» and » being any two proposed whole numbers. And if we suppose the fractional
multiplier ~ in (16.) to tend to any incommensurable limit a, we may denote by
@ x (bj, b,) the corresponding limit of the fractional product, and may consider this
latter limit as the product obtained by multiplying the step-couple (b,, b,) by the in-
commensurable muitiplier or number a; so that we may write,
(ep eo) =a x (b1, bo) = a(b, bo),
‘i us)
if (cj, cs) = L (“(b, be)) anda=L~,
Bh B
using L as the mark of a limit, as in the notation of the Preliminary Essay. It follows
from these conceptions of the multiplication of a step-couple by a number, that gene-
rally
aX (aieaa) = (aa, a as)s (19.)
whatever steps may be denoted by a,, a, and whatever number (commensurable or
imcommensurable, and positive or contra-positive or null) may be denoted by a.
Hence also we may write
ue ts ? aS (20.)
and may consider the number a as expressing the ratio of the step-couple (a 9), @ s,)
to the step-coup!e (a1, 22).
On the Multiplication of a Step-Couple by a Number-Couple ; and on the Ratio of —
one Step-Couple to another.
4. The formula (20.) enables us, in an infinite variety of cases, to assign a single
number a as the ratio of one proposed step-couple (b,, b») to another (a), a2);
namely, in all those cases in which the primary and secondary steps of the one couple
are proportional to those of the other : but it fails to assign such a ratio, in all those
and on Algebra as the Science of Pure Time. 399
other cases in which this condition is not satisfied. ‘The spirit of the present Theory
of Couples leads us, however, to conceive that the ratio of any one effective step-
couple to any other may perhaps be expressed in general by a number-couple, or
couple of numbers, a primary and a secondary ; and that with reference to this more
general view of such ratio, the relation (20.) might be more fully written thus,
(4) aw a, 44) = (a, 0), (21.)
(a, ay)
and the relation (19.) as follows,
(a), 0) x (ay ao) = (a1, 0) (a1 a,) = (a, aj, a, a»)5 (22.)
the single number a, being changed to the couple (a, 0), which may be called a pure
primary number-couple. The spirit of this theory of primaries and secondaries leads
us also to conceive that the ratio of any step-couple (b,, b») to any pure primary
step-couple (a), 0), may be expressed by coupling the two ratios ti, ?2. which the
ae el
two steps b!, b, bear to the effective primary step a,; so that we may write :
(>, ba ee b, bo (a ayy Cy a;) = z
(a. 0) ie ee re Aly 0) a (a, a,); (23.)
and in like manner, by the weneral connexion of multiplication with ratio
d fo} ’
(@, a.) x (a, 0) = (A, G2) (ay 0) = (G 24, @ a,). (24.)
From the relations (22.) (24.), it follows by (5.) that
. (6, +a, 0) (a,, a2)=(b,, O) (Cai, a2) + (G1, O) (ay, a2), (25.)
and that
(Gy, G2) (b, + a,, O) = (Gi, a2) (by, 0) + (GQ, Az) Cai, 0); (26.)
and the spirit of the present extension of reasonings and operations on single mo-
ments, steps, and numbers, to moment-couples, step-couples, and number-couples,
leads us to determine (if we can) what remains yet undetermined in the conception of
a number-couple, as a multiplier or as a ratio, so as to satisfy the two following more
general conditions,
(+m, b:+a:) (41, a,)=(b, b.) (a1 a) +(@, 2) (91, 82)5 (27.)
and
(a, a») (>, + a), bot a,)=(M, a; ) (>, bo) +(M, a2) (41, a»), (28.)
4.00 Professor HamiLtron on Conjugate Functions,
whatever numbers may be denoted by a, Oy b, b,, and whatever steps by a, a, by by
With these conditions we have
(a, ds) (4, a)=(a, O) (4, ay +(0, az) (4, ay), (29.)
(0, ) (4, a.) = (0, a) (4, 0) + (0, a2) (0, a), (30.)
and, therefore, by (22.) and (24.), and by the formula for sums,
(Mm, ay) (41, a)=(a4 Ao, Gy 44) +(0, ay a) +(0, a») (0, ay)
=@ 4, QA %+Q, a,)+(0, a2) (0, a), (31.)
in which the product (0, a.) (0, @,) remains still undetermined. It must, however,
by the spirit of the present theory, be supposed to be some step-couple,
(0, as) (0, a )y=(¢, oy) 5 (32.)
and these two steps ¢, ¢. must each vary proportionally to the product a, a,, since
otherwise it could be proved that the foregoing conditions, (27.) and (28. 4 would not
be satisfied ; we are, therefore, to suppose
(= ¥J1 Ay % “2= Yo Az Ag, (33.)
that is,
(0, @2) (0, #2)=(yr Ge M5 Ya Ga %)s (34.)
yi ys being some two constant numbers, independent of a, and a, but otherwise
capable of being chosen at pleasure. Thus, the general formula for the product of a
step-couple («,, @,) multiplied by a number-couple (a, a), is, by (31.) (34.) and by
the theorem for sums,
(a, Q) (41, 2)= (a, a, a 4+, 4) + (yy Gs 9% Y2 A a,)
= (M1 + Yr Ay My Gy A + Ay + Y2 Ge %): (35.)
and accordingly this formula satisfies the conditions (27.) and (28.), and includes the
relations (22.) and (24.), whatever arbitrary numbers we choose for y,, and y, ; pro-
vided that after once choosing these numbers, which we may call the constants of mul-
tiplication, we retain thenceforth unaltered, and treat them as independent of both
the multiplier and the multiplicand. It is clear, however, that the simplicity and
elegance of our future operations and results must mainly depend on our making a
simple and suitable choice of these two constants of multiplication ; and that in making
and on Algebra as the Science of Pure Time. 401
this choice, we ought to take care to satisfy, if possible, the essential condition that
there shall be always one determined number-couple to express the ratio of any one
determined step-couple to any other, at least when the latter is not null : since this was
the very object, to accomplish which we were led to introduce the conception of these
number-couples. It is easy to show that no choice simpler than the following,
yi——I1, y= 0, (36.)
would satisfy this essential condition : and for that reason we shall now select these
two numbers, contra-positive one and zero, for the two constants of multiplication,
and shall establish finally this formula for the multiplication of any step-couple (4, 42)
by any number-couple (a,, 0),
(4; %) (1, By) = (4 a, —@, a5, @, a, + @ ay), (37.)
5. In fact, whatever constants of multiplication y, y, we may select, if we denote
by (»,,>,) the product of the step-couple (4,, 4,) by the number-couple (4, @,), so
that
(>, b) = (4, ay) x (4, ay), (38.)
we have by (35.) the following expressions for the two coupled steps b,, »., of the
product,
Wat CRRA ea & Ay Any
a SV | 9 eae Was \ (39.)
= @% a, + Gy a, + Ya % Foy
and therefore
Bi =a, a, +Y1 U2 ay 5 t (40.)
Py = A, a2 + Az a, +2 Ay a2 y
if a, a, (3; 2. denote respectively the ratios of the four steps 4, 4, b, b, to one effective
step c, so that
8a) %, 4 =a °¢, {41.)
and
B= Gite ans (42.)
from which it follows that 2
a, fa, (a; + y2 a) —y, a2} =P: (a +2 a2) — Pa ys as t (43.)
Ay fa; (a! + y2 a2) — yi a2 $ =P: at— Py a2;
in order therefore that the numbers a, a, should always be determined by the equa-
402 Professor Hamitton on Conjugate Functions,
tion (38.), when a, and a, are not both null steps, it is necessary and sufficient that |
the factor
a, (a, + Ya ay) a ay =(a, + by ay)? — (yn ots 4y,*) a” (4.4.)
should never become null, when a, and a, are not both null numbers; and this con-
dition will be satisfied if we so choose the constants of multiplication y, y, as to make
yn t+iy’<0, (45.)
but not otherwise. Whatever constants y, y2 we choose, we have, by the foregoing
principles,
(ec, 0) (0, ¢) (O, ¢)
Se lO) 2. = (Oeil) ae (0) ia (46,
AO) » 93 (ec, 0) (O, 1); (ORS 20); )
and finally
25.0) 8( sae, »)., (47.)
(O, c) yi ¥y ie)
because, when we make, in (43.),
¢,=0, a=1, B,=1, B’=0, (48.)
we find
Gras et ; (49.)
yi yi
so that although the ratio of the pure primary step-couple (c, 0) to the pure second-
ary step-couple (0, ¢) can never be expressed as a pure primary number-couple, it
may be expressed as a pure secondary number-couple, namely (0, aed if we choose
O, as in (36.), for the value’ of the secondary constant y,, but not otherwise : this
choice y,=0 is therefore required by simplicity. And since by the condition (45.),
the primary constant y, must be contrapositive, the simplest way of determining it’is
to make it contrapositive one, y,= —1, as announced in (36.). We have there-
fore justified that selection (36.) of the two constants of multiplication ; and find,
with that selection,
Ce, ee 50
(20) (0, -1), (50.)
and generally, for the ratio of any one step-couple to any other, the formula
( b,, b.)_ (Bi c, Pr c) = (Biss =F By ay ‘ B, a —B, ay J (51.)
CA 2) Care ape)
2 2
ay +a, a; +a;
and on Algebra as the Science of Pure Time. 403
On the Addition, Subtraction, Multiplication, and Division, of Number- Couples,
as combined with each other.
6. Proceeding to operations upon number-couples, considered in combination with
each other, it is easy now to see the reasonableness of the following definitions, and
even their necessity, if we would preserve in the simplest way, the analogy of the the-
ory of couples to the theory of singles :
(1, b)+ (Gy, G)=(i+a, b+); (52.)
(A, by —-(4,, a,jy=(h —, b,—a.) B (53.)
(, ’ b,) (a, ’ ar=(b, > by) x (Q ) a)=(b, (oh Si), Gs; b, aq,t+ b, as) : 6 1.)
(ho) (ba +h, boa —b i;
“ (Q 5 a2) ( Gy Fas OPN a?-Fa? ) (55.)
«
Were these definitions even altogether arbitrary, they would at least not contradict
each other, nor the earlier principles of Algebra, and it would be possible to draw
legitimate conclusions, by rigorous mathematical reasoning, from premises thus arbi-
trarily assumed : but the persons who have read with attention the foregoing remarks
of this theory, and have compared them with the Preliminary Essay, will see that
these definitions are really not arbitrarily chosen, and that though others might have
been assumed, no others would be equally proper.
With these definitions, addition and subtraction of number-couples are mutually
inyerse operations, and so are multiplication and division ; and we have the relations,
(1, 5) + (2), a=(a, , %) + (21, Fy), (56.)
(2,5 2) x (4, %)=(Q, 4) x (4, be)s (57.)
(A> i) f(a a’) + (4, a,)} = (4, b,) (a, @) + (1, b,) (@,, @): (58.)
we may, therefore, extend to number-couples all those results respecting numbers,
which have been deduced from principles corresponding to these last relations. For
example,
{(4,, by) +(a, a2)} x {(h, by) + (a, &)}=
(h, b2) (i, b,) +2(6, b,) (M4, a2) + (a, a2) (M, 1), (59.)
VOL. XVII. 4 c¢
404 Professor Hamitton on Conjugate Functions,
in which
2(h, bs) (a; a.) =(2, 0) (di, b:) (a; a.) =(b,, bz) (iy G2) + (h,, b.) (Gis Ga) 5 (60.)
for, in general, we may mia the signs of numbers with those of number-couples, if we
consider every single number a as equivalent to a pure primary number-couple,
=O) (61.)
When the pure primary couple (1, 0) is thus considered as equivalent to the number
1, it may be called, for shortness, the primary unit ; and the pure secondary couple
(0, 1) may be called in like manner the secondary unit.
We may also agree to write, by analogy to notations already explained,
(0, 0) +(a; a2) =+ (Q; a),
(62.)
(O, 0) —(a, a) = (a, a’) 3
and then + (a, @,) will be another symbol for the number-couple (a, a) itself, and
—(a, a) will be asymbol for the opposite number-couple (—a, —a,). The reciprocal
of a number-couple (a, “) is this othe#number-couple,
1 =o -( a =a )=o=@) (63.)
— nl 2 Pe 2 2 ag "| nae
(Qj, Gs) (%, 4%) ay + 4% ay +4, ay + az
It need scarcely be mentioned that the insertion of the sign of coincidence =
between any two number-couples implies that those two couples coincide, number with
number, primary with primary, and secondary with secondary ; so that an equation
between number-couples is equivalent to a couple of equations between numbers.
On the Powering of a Number-couple by Single WV hole Number.
7. Any number-couple (a, a.) may be used as a base to generate a series of
powers, with integer exponents, or logarithms, namely, the series
o'er, (a, ay)~*, (a, Q)—"; (a, a), (a, as)', (a, as)", eee (64.)
in which the first positive power (a, a)' is the base itself, and all the others are
formed from it by repeated multiplication or division by that base, according as they
follow or precede it in the series ; thus,
(a, a2)’ =(1, 0), (65.)
and on Algebra as the Science of Pure Time. 405
and
(1, 0)
(4, a)'= (4, 4); (4, (as a,)
se 2 (SF)
(4, GY = (4, a) (%, &), (a, a2) Gi 2) (dy 2) |
ec &e. J
To power the couple (4, a.) by any positive whole number m, is, therefore, to
multiply, m times successively, the primary unit, or the couple (1, 0), by the proposed
couple (4, @); and to power (4, a) by any contra-positive whole number —m,
is to divide (1, 0) by the same couple (4, a), m times successively : but to power by
O produces always (1, 0.). Hence, generally, for any whole numbers p, v,
(4, a) u (4, a2) v— (4; ay) uty ae
CG; 4)" )”=(Can %)*”- : (07.)
8. In the theory of single numbers,
(a+b)” _ a” ¥e He b a ans? BP
1x2x3...xm ~1x2x3...xm 1x2x3x...(m—I) 1 1x2x8x...(m—2) 1x2 a
a ii um :
# x FTX aKa xm? (68.)
1 1x2x8x...(m—1)
and similarly in the theory of number-couples,
{(a, ay) ar (A, b,)}” = (4; a,)” (4, ay" Cr Os)
1X2 XIX 5m 1x2x3x...m UX2X3 xin (n—1) Te
(4, CDN (4, b,)? +
+ 1x2x8x...(m—2) 1x2 ws
ph GT EE rake Gi (69.)
1x2x3x...(m—1) 1x2x3x...m’
m being in both these formula a positive whole number, but @ 4 @, @, 4, 4, being any
numbers whatever. The latter formula, which includes the former, may easily be
proved by considering the product of m unequal factor sums,
(a, 2) F (6, bx"); (a, a>) + Ge; sd) ee (a, a2) as (™, 5, ) > (70.)
for, in this product, when developed by the rules of multiplication, the power
(4; a,)"~" is multiplied by the sum of all the products of m factor couples
£06 Professor Hamitton on Conjugate Functions,
such as (0,%, B,) (6,%, 5,)...(6,, b) ; and the number of such products is the
number of combinations of m things, taken » by n, that is,
1x2x3x...«xm
}x2x3~x ...(m—n)x 1x2x3x...0? (71.)
while these products themselyes become each =(4,, 2,)", when we return to the case of
equal factors.
The formula (69.) enables us to determine separately the primary and secondary
numbers of the power or couple (a, a,)”, by treating the base (a,, a2) as the sum of
a pure primary couple (@, 0) and a pure secondary (0, a,), and by observing that the
powering of these latter number-couples may be performed by multiplying the powers
of the numbers a, a, by the powers of the primary and secondary units, (1, 0) and
(0, 1); for, whatever whole number 7 may be,
(a,, 0)'=a;' (1, 07,
(Fa; 2=ar(Ors)
We have also the following expressions for the powers of these two units,
aCe, OO} SGROy
(0; 1945 SCOR):
(0, 4) = 1; 0), (73.),
(1)! S00. = 1),
(0, 1) “* =(1,0); J
that is, the powers of the primary unit are all themselves equal to that primary unit ;
but the first, second, third, and fourth powers of the secondary unit are re-
spectively
(72.)
(0, 1) (—1, 0), (0, —1), (1, 0),
and the higher powers are formed by merely repeating this period. In like manner
we find that the equation
(4, %)"=(b,, be), (74)
is equivalent to the two following,
b, =¢, 22 (m—1) G Arnie 4m (n—1) (m—2) (m —3) a,"—"a,'— &¢e
Sao 1x2x3x4 i i
(75.)
m= m (m—1) (m—2)
b.=m a, (Ca ORa aaa Ga Oa tC.
For example, the square and cube of a couple, that is, the second and third positive
powers of it, may be developed thus,
(a1; a)?={Cam, 0) +0, a) P=(a2— a’, 2a a), * (76.)
ae
and on Algebra as the Science of Pure Time. 407
and
( a, a2)®= {(ai, 0) + (0, a:)}°=( a2 —3 a1 a, 3a, a,— a,°). (77.)
9. In general, if
(a, a) (ay a'2)=(a"s, a’2), (78.)
then, by the theorem or rule of multiplication (54.)
a") = a, 2 — a, Ay A =, 1 + Ay (79.)
and therefore
ay t+ a'y=(a"+ a?) (a+ a7); (80.)
and in like manner it may be proved that
mo
” af (a, a> ) (a, as) (a1, a's) =I (ala taia\s ' (81.)
then (a+ a"s)=( a+ ar) (a+ a?) (a’24+ a’),
and so on, for any number m of factors. Hence, in particular, when all these m fac-
tors are equal, so that the product becomes a power, and the equation (74.) is satisfied,
the two numbers 0, ), of the power-couple must be connected with the two numbers
a, a» of the base-couple by the relation
bP +bP=( a. + a7)". (82.)
For example, in the cases of the square and cube, this relation holds good under the
forms
(a?— a®l+(2a w)’=(a’?+ a)’, (83.)
and
(a2—3 a, a?)+ (8a: a— a*)=(a?+ a2). (84.)
The relation (82.) is true even for powers with contra-positive exponents —7, that is,
bt +bt=(at+ az)” if (by b)=(a» a@)~"5 (85.)
for in general
at
(41, a2) (a1, a») (@'1,4"2) «.
UCT, Os) =i, cali eis da) (Cis. eal ;
ay B2)— (ai? + as?) (a)? + a2?) (a's? + a"2?)...
then (bh; 3 De ee ara (1? + eo?) (e'1? + co”)...
408 Professor Hamitton on Conjugate Functions,
On a particular Class of Exponential and Logarithmic Function- Couples, connected
with a particular Series of Integer Powers of Number-Couples.
10. The theorem (69.) shows, that if we employ the symbols r,, ( a, a2) and F,, (4), b2)
to denote concisely two number-couples, which depend in the following way on the
couples (ai, a)) and (4, 2),
a> 2) aoe Ay) Qo)” =
F, Cas a) = d, iy ee : +6 Smo ihe eee ee (87.)
25 (A, b,)' a(n by) (4, b,)™
Fe (6 6) = (1,0) +5 Teena gi Sr Yt pero SC a (88.)
and if we denote in like manner by the symbol
Pn (@, a.) ar (2, b.)) = Bn (4, ao by a, at: by) (89.)
the couple which depends in the same way on the sum (4, @,) + (4,, 2), or on the
couple (4, +2, ¢,+4,), and develope by the rule (69.) the powers of this latter sum,
we shall have the relation
{r,, (4, ay) X En (4, b,)% Fh (a; a2) + (G1, b2)) =
(4, @)" Le ee CI ear eG ad t
1 2
1x2x3x_..m 2x3 xm
(io 20 FGA et Gu BEND
1x2x3x...(m—1) ISDN i tee linge acon cen
+
(2%, %)' (5, &)”
1 Ix 23K.) (90.)
This expression may be farther developed, by the rule for the multiplication of a sun,
into the sum of several terms or couples, (¢,, ¢:), of which the number is
14+24+8+4...4m=0 (4D, (91.)
and which are of the form
t b- b,)*
Cc : os (4, a.) ( ly 2 2.
(, C1) = eeu eS Xa (92.)
and on Algebra as the Science of Pure Time. 409
i and k being positive integers, such that
itm, kbm, i+k>m; (93.)
and if we put for abridgment c
Ja) +4,=a, VJ 6, + 6,2 =, (94.)
and
a’ pe ~
Bs Awe 95.
Y = TxOx3x dx Ix2x3xak? (95.)
we shall have, by principles lately explained,
ores = y (96.)
and therefore
Cc; } + Ys Orde => C2 > Ys Cr dg yrs (97.)
if then the entire sum (90.) of all these couples (¢,, c.) be put under the form
= (4, e)=(2%, = ©), (98.)
the letter = being used as a mark of summation, we shall have the corresponding li-
mitations
Za pdy, Bat — zy, :
Ze bly, q+ —Zy,
(99.)
m (m+1)
2
= being the positive sum of the such terms as that marked (95.). This
latter sum depends on the positive whole number m, and on the positive numbers
a, (2; but whatever these two latter numbers may be, it is easy to show that by taking
the former number sufficiently great, we can make the positive sum = y become
smaller, that is nearer to 0, than any positive number é previously assigned, however
small that number 6 may be. For if we use the symbols F,, (a), F,, (8)5 Fn (a+ 3), to
denote positive numbers connected with the positive numbers a, 3, a+, by relations
analogous to those marked (87.) and (88.), so that
7 Ge a”
F = —~+t— +... + —.~_,__—_
m(a)= 1+ 7+ 4+ + TXaxax um
(100.)
it is easy to prove, by (68.), that the product r,, (a) x r,,() exceeds the number
410 Professor Hamitton on Conjugate Functions,
F,, (a+) by = y, but falls short of the number F,,,(a+), that is of the following
number
oa ~ 14. (¢+8)' , (+B)? (actBy Ga
lt) ee ae pene COREE (101.)
so that
Ly=C;,, (a) x F,(B)) —Fn (a +B), (102.)
and
Ly< hk, (a+B)—F, (a+): (103.)
if then we choose a positive integer 7, so as to satisfy the condition
a
+
eed (104. }
nm+1>2(a+)5), that is
and take m >, we shall have
(a +f)” < 1 (a +)" , and therefore < 8, (105.)
1x2x3x...m ~ 2-2" TxX2xX3x...2n
however small the positive number 8 may be, and however large a + B may be, if we
take m large enough ; but also
Fo,(a+) —F,,(a+B)= Sey | and therefore <éxn, (106.)
in which
at Gae ge)! (a +)” (107.)
~ mxl (m+i)(m+2) igs Gat 1) (m+2) x ...(2m)”
and, therefore,
nigh, (108.)
because
a+p 1 (a+f) 1 (a+)” 1 F
m+1 2 (m+1) (m+2) Sepa es (m+1) (m+2) x ...(2 m) S om} (109.)
therefore, combining the inequalities (103.) (106.) (108.), we find finally
Sy <no: (110.)
And hence, by (99.), the two sums = ¢,, © @, may both be made to approach as near
as we desire to 0, by taking m sufficiently large; so that, in the notation of limits
already employed,
LZy=0, £ 3¢,=0, & 36,=0, (111.)
: and on Algebra as the Science of Pure Time. 411
and, therefore,
LPF, (4) Fx (8)— Fp (a +8)} =O, (112.)
LL Fn (Ai @) Fn (bis 0.) — Fn ((ar, a2) + (bis b,)) } = (0, 0). (113.)
In the foregoing investigation, a and 8 denoted positive numbers; but the theorem
(113.) shows that the formula (112.) holds good, whatever numbers may be denoted
by a and 23, if we still interpret the symbol me by the rule (100.).
11. Ifa still retain the signification (94.), it results, from the foregoing rea-
sonings, that the primary and secondary numbers of the couple
Frntm’ (iy Qa) —Fm (diy Ae) (114.)
are each
+ Fram (a)—F,, (a), and ¢ F(a) —Frim (a); (115.)
and, therefore, may each be made nearer to O (on the positive or on the contra-
positive side) than any proposed positive number é by choosing m large enough,
however large m’ and a may be, and however small 8 may be: because in the ex-
pression
a” a a a”
Fn tm (#) — Fn (a) = 1x2x3x.. amet mE ey tt eee} (116.)
m
Sales a Q pie :
the positive factor eel be made <6, chat is, as near as we please to 0,
: 1 1 1
and also the other factor, as being < A ileus + 7, and therefore SESE y aps if
m+1>na. Pursuing this train of reasoning, we find that as m Spe te greater
and greater without end, the couple F,, (a,, a;) tends to a determinate /imit-couple,
which depends on the couple (%, a,), and may be denoted by the symbol r,(a,, a),
or simply F(a, a),
F (M4, Q2)=F_ (dh, G2) =" Fm (Ay, Az) 5 C7)
and similarly, that for any determinate number a, whether positive or not, the
number F,,(a) tends to a determinate limit-number, which depends on the number a,
and may be denoted thus,
F (a) =F, (a) = L Fn(a). (118.)
It is easy also to prove, by (112.), that this function, or dependent number, ¥ (a);
must always satisfy the conditions
F(a) x¥ (8) =F (a +8), (119.)
VOL. XVII. 4D
412 Professor HAmILTon on Conjugate Functions,
and that it increases constantly and continuously from positive states indefinitely near
to 0 to positive states indefinitely far from 0, while a increases or advances constantly,
and continuously, and indefinitely in the progression from contra-positive to positive ;
so that, for every positive number /, there is some determined number a which satis-
fies the condition
B=r (@), (120.)
and which may be thus denoted,
a=F-' (). (121.)
It may also be easily proved that we have always the relations,
F (a) =e", F—'(B) =log,. B, (122.3
if we put, for abridgement,
F (1) =e, (123.)
and employ the notation of powers and logarithms explained in the Preliminary
Essay. A power b when considered as depending on its exponent, is called an
exponential function thereof; its most general and essential properties are those
expressed by the formule,
rele sweaty (Sti (124.3
of which the first is independent of the base }, while the second specifies that base ;
and since, by (113.), the function-couple F (a, @,) satisfies the analogous condition,
F (@, a2) x F (b,, 6 =P ((, 2) + (A, 0.) JHE (a, +5, a. + dy), (125.)
(whatever numbers a, a, b, b, may be,) we may say by analogy that this function-couple
F (a), a) is an exponential function-couple, and that its base-couple is
1 (HS MEG; O) 2 (126.)
and because the exponent a of a power &, when considered as depending on that
power, is called a logarithmic function thereof, we may say by analogy that the
couple (a), a.) is a logarithmic function, or function-couple, of the couple F(a, @,),
and may denote it thus,
(a, a.) =Fa'G, 4), if ©, 4)=r@, a,). (127.)
and on Algebra as the Science of Pure Time. 413
In general, if we can discover any law of dependence of one couple ® (4, @,), upon
another (4, a,), such that fer all values of the numbers a, a, 4, b, the condition
© (4, 4) ® (%, b= (a, + A, b, + b) (128.)
is satisfied, then, whether this function-couple © (@%, 4) be or be not coincident with
the particular function-couple F (4, 4), we may call it (by the same analogy of defi-
nition) an eaponential functien-couple, calling the particular couple (1, 0) its base, or
dase-couple ; and may call the couple (@,, ¢,), when considered as depending inversely
on ® (4, @,), a logarithmic function, or function-covple, which we may thus denote,
(a, 7) =b—'@,, 4), if ©, 4,)=o (a, a). (129.)
12. We have shown that the particular exponential function-couple (4,, 4.) =
F(a, a) is always possible and determinate, whatever determinate couple (a, a2)
may be ; let us now consider whether, inversely, the particular logarithmic function-
couple (a, a.) =F7' (4,, 6.) is always possible and determinate, for every determined
couple (4, &.). By the exponential properties of the function r, we have
(4, &J=FCa, a)=E(am, 0) F(O, w)=EF( a) F(O, a)
(130.)
: sin a»),
=(e"'cos ay, e*
if we define the functions cosa and sina, or more fully the cosine and sine of any
number a, to be the primary and secondary numbers of the couple F (0, a), or the
numbers which satisfy the couple-equation,
F (0, a) =(cos a, sina). {131.)
Vrom this definition, or from these two others which it includes, namely from the fol-
lowing expressions of the functions cosine and sine as limits of the sums of series,
which are already familiar to mathematicians,
Ph aa ai 2. a &c
= 1x2 1x2x3x4 i
(132.)
: a a &
$1 = _ Se
ae ¢ iKaKS 1 LKDKONANS ¢
it is possible to deduce all the other known properties of these two functions ; and
especially that they are periodical functions, in such a manner that while the variable
414 Professor Hamitron on Conjugate Functions,
number a increases constantly and continuously from 0 to a certain constant positive
number = (7 being a certain number between 3 and 4,) the function sin a increases
with it (constantly and continuously) from 0 to 1, but cosa decreases (constantly and
continuously) from 1 to 0; while a continues to increase from = to m, sina decreases
from 1 to 0, and cos a from 0 to —1; while a increases from 7 to om sin a decreases
from 0 to —1, but cos a increases from —1 to 0; while a still increases from Sr to Qr,
sin a increases from —t to 0, and cosa from 0 to 1, the sum of the squares of the
cosine and sine remaining always = 1; and that then the same changes recur in the
same order, having also occurred before for contra-positive values of a, according to
this law of periodicity, that
cos (a+2iz)=cosa, sin (a+2i7)=sina, (132.)
? denoting here (as elsewhere in the present paper) any positive whole number. But
because the proof of these well known properties may be deduced from the equations
(132.), without any special reference to the theory of couples, it is not necessary, and
it might not be proper, to dwell upon it here.
It is, however, important to observe here, that by these properties we can always
find (or conceive found) an indefinite variety of numbers a, differing from each other by
multiples of the constant number 27, and yet each having its cosine equal to any one
proposed number 3,, and its sine equal to any other proposed number f,, provided
that the sum of the squares of these two proposed numbers (3,, 3,, is = 1; and reci-
procally, that if two different numbers a both satisfy the conditions
cosa = B,, sina = f,, (134. }
P, and , being two given numbers, such that /3,’ + B,?=1, then the difference of these
two numbers a is necessarily a multiple of 27. Among all these numbers a, there will
always be one which will satisfy these other conditions
a> —T, a; 7, (135.)
t
and this particular number a may be called the principal solution of the equations
134.), because it is always nearer to 0 than any other number a which satisfies the
same equations, except in the particular case when B,= —1, 3,=03; and because, in
this particular case, though the two numbers 7 and —7 are equally near to 0, and
both satisfy the equations (134.), yet still the principal solution 7, assigned by the
conditions (135.), is simpler than the other solution —7, which is rejected by those
last conditions. It is therefore always possible to find not only one, but infinitely
many number-couples (m, a), differing from eaeh other by multiples of the constant
and on Algebra as the Science of Pure Time. 415
couple (0, 27), but satisfying each the equation (130.), and therefore each entitled to
be represented by, or included in the meaning of, the general symbol F~'(4,, 42),
whatever proposed effective couple (4,, ,) may be. For we have only to satisfy, by
(130.), the two separate equations
: (136.
e™ cosa, = by, ee"! sina, = by} ea
which are equivalent to the three following,
e N= J bf + b> S77)
and
by : bo
COS a. =~ ——_., SM a = ~—>— 3 (138.)
J by + bo" VY by + bs
and if a be the principal solution of these two last equations, we shall have as their
most general solution
a,—a+2wn7, (139.)
while the formula (137.) gives
a—log../ b+ be: (140.)
the couple (a, a.) admits therefore of all the following values, consistently with the
conditions (130.) or (136.),
(a1) as) =F —" (51; b= Uog J be +b, at+2wn), (141.)
in which w is any whole number, and a isa number > —7z, but not >, which has its
cosine and sine respectively equal to the proposed numbers 5,, 5, divided each by the
square-root of the sum of their squares. ‘To specify any one value of (a,, a,), or
F—'(5,, 5,), corresponding to any one particular whole number w, we may use the
symbol r—'(f,, 4.) ; and then the symbol i (41, 42) will denote what may be called
the principal value of the inverse or logarithmic function-couple r—' (4, 4,), because
it corresponds to the principal value of the number a, as determined by the condi-
tions (138.).
416 Professor Haminron on Conjugate Functions,
On the Powering of any Number- Couple by any Single Number or Number- Couple.
13. Resuming now the problem of powering a number-couple by a number, we
may employ this property of the exponential function F,
(F (a, a:))"=F(u di, 1 a), (142.)
» being any whole number whether positive or contrapositive or null; which easily
follows from (125.), and gives this expression for the »’th power, or power-couple, of
any effective number-couple,
(8, 6,)*=F(n FCG, , b,)). (143.)
Reciprocally if (a, a2) be an mth root, or root-couple, of a proposed couple (0, , bs),
so that the equation (74.) is satisfied, then
1 1
(a, a) =(, bo) =F (— F(h,, b)). (144.)
This last expression admits of many values, when the positive whole number m is > 1,
on account of the indeterminateness of the inverse or logarithmic function r~'; and
to specify any one of these values of the root-couple, corresponding to any one value
¥-! of that inverse function, which value of the root we may call the wth value of that
root, we may employ the notation
1
6, m= F (EG, %); (145.)
o
we may also call the particular value
Bs 1
(, &)m = F(— FG, 4)), (146.)
the principal value of the root-couple, or the principal m’th root of the couple @,, 4).
In this notation,
2
wT 3
m
Oh y (0, (147.)
Gitge @,,8)% Gils Oyns (148.)
so that generally, the w*th value of the mth root of any number-couple ts equal to
the principal value of that root multiplied by the wth value of the mth root of the
and on Algebra as the Science of Pure Time. 417
primary unit (1,0). The mth root of any couple has therefore m distinct values,
and no more, because the mth root of the primary unit (1, 0) has m distinct values,
and no more, since it may be thus expressed, by (147.) and (131.),
Z 2ur . 207
a, O)m= ( cos. —-— 5 sin. a) ' (149.)
so that, by the law of periodicity (133.), for any different whole number w,
(1, 0) m = 0)", (150.)
and therefore generally, -
(b, b) = h, ae : (151.)
if ; é :
w —w+im, (152.)
but not otherwise. For example, the cube-root of the primary unit (1, 0) has three
distinct values, and no more, namely
a1 ENS}
a,OF = 4,0); a,07=(—3, 2);a,or=("5,-); a3,
27 2
60 that each of these three couples, but no other, has its cube =(1, 0). Again the
couple (—1, 0) has two distinct square-reots, and no more, namely
1 1
(—1, 0) =" (O05 (-1, 0)? =(0, —1). (154.)
In general we may agree to denote the principal square-root of a couple (b,, b:) by
the symbol
Jb, &) = @, %)7; (155.)
and then we shall have the particular equation
V(—1, 0)=©, 1); (156.)
which may, by the principle (61.), be concisely denoted as follows,
¥—1=(0, 1). (157.)
In the THEORY OF sINGLE NUMBERS, the symbol /—1 is absurd, and denotes an
IMPOSSIBLE EXTRACTION, or a merely IMAGINARY NUMBER; but in the THEORY OF
couptes, the same symbol ¥ —1 is significant, and denotes a POSSIBLE EXTRACTION,
418 Professor Hamitron on Conjugate Functions,
or a REAL COUPLE, namely (as we have just now seen) the principal square-root of
the couple (—1, 0). In the latter theory, therefore, though not in the former, this
signy —1 may properly be employed ; and we may write, if we choose, for any cou-
ple (a, az) whatever,
(a, @)=a,+a,V—}, (158.)
interpreting the symbols q; and @,, in the expression @, + @, / —1, as denoting the pure
primary couples (@, 0) (4, 0), according to the law of mixture (61.) of numbers with
number-couples, and interpreting the symbol /—1, in the same expression, as de-
noting the secondary unit or pure secondary couple (0, 1), according ,to the formula
(157.). However, the notation (a, @,) appears to be sufficiently simple.
14. In like manner, if we write, by analogy to the notation of fractional powers
of numbers, -
(a, Cy) = @, b)e, (159.)
whenever the two couples (%,, 4,) and (¢, ¢:) are both related as integer powers to one
common base couple (4%, @,) as follows,
(G, b.) = (a, ds)"; (4a, C2) = (a, a)’, (160.)
(u and v being any two whole numbers, of which » at least is different from 0,) we
can easily prove that this fractional power-couple (c¢,, ¢), or this result of powering
the couple (d,, 5.) by the fractional number a has in general many values, which
are all expressed by the formula
(cy )=(by be = F ("e 1b, b,))s 161.)
and of which any one may be distinguished from the others by the notation
(by byt = F Bap ' (b,, b,)) (162.)
We may call the couple thus denoted the w’th value of the fractional power, and in
particular we may call
(6, b:) r= r (= F-"(b, b,)) (163.)
the principal value. The w’th value may be formed from the principal value, by
multiplying it by the w’th value of the corresponding fractional power of the primary
unit, that is, by the following couple,
Sead
and on Algebra as the Science of Pure Time. 419
anor =( cos S24, sia Om); (164.)
and therefore the number of distinct values of any fractional power of a couple, is
equal to the number m of units which remain in the denominator, when the fraction -
has been reduced to its simplest possible expression, by the rejection of common
factors.
15. Thus, the powering of any couple (b,, b:) by any commensurable number «
may be effected by the formula,
(5;, 5)? =F (x F—' (5,, by); (165.)
or by these more specific expressions,
(b, 6,’ =¥ (wF-'(b,, b,))
=i, Bs) (15.0), (166.)
im which
(1, 0)'=(cos2wan, sinQwarn): (167.)
and it is natural to extend the same formule by definition, for reasons of analogy and
continuity, even to the case when the exponent or number w is tncommensurable, in
which latter case the variety of values of the power is infinite, though no confusion
can arise, if each be distinguished from the others by its specific ordinal number, or
determining integer w.
And since the spirit of the present theory leads us to extend all operations with
single numbers to: operations with number-couples, we shall further define (being
authorised by this analogy to do so) that the powering of any one number-couple
(b;, bs) hy any other number-couple (2, x») is the calculation of a third number-couple
(c , ¢2), such that
(4; C2) = (41, by) (1 ,) =F (@, X,) XPS! (b,, b)); (168.)
or more specifically of any one of the infinitely many couples corresponding to the
infinite variety of specific ordinals or determining integers w, according to this for-
mula,
(4:5 by) (1 ) =F (in, ) XBT (6,5)
= (h,, by) (x1; X>) (1, 0) (Xi, Xs), (169.)
in which the factor (4, 52) (» 2) may be called the principal value of the general
VOL, XVII. : 4,5
4.20 Professor Haminton on Conjugate Functions,
power- couple, and in which the other factor may be calculated by the following ex-
pression,
(1, 0) (ts 2)» (Cay, ay) x (0, 2 w 7)
=F(-207 &, 2w7 2X)
=e —2em (cos Quart, sin 2Qwr%). (170.)
For example,
(1, 0) (t» %)— (1, 0), (171.)
and
(6, 0) (#1 7) = (ERD) R (172.)
also
(¢, 0) (i %) p(w, a) x (1, 207) (173.)
On Exponential and Logarithmic Function- Couples in general.
16. It is easy now to discover this general expression for an exponential function-
couple :
® (2, 2) =F (CX, X2) x (Gis G2) 5 (174)
in which (a,, a2) is any constant couple, independent of (a, a). This general expo-
nential function © includes the particular function r, and satisfies (as it ought) the
condition of the form (128.),
® (21, a) ®D (YM, Yr) =P (a, +s Hop) +Yo) 5 (175.)
its base, or base-couple, which may be denoted for conciseness by (4, 52), is, by the
11th article, the couple
(Gib) =o Oe 0) "Gi a3); (176.)
and if we determine that integer number w which satisfies the conditions
Q,-2w7 > —7, —L2wT PT, (177.)
we shall have the general transformation
® (@y 24) = (by by) Fo ™), (178.)
and on Algebra as the Science of Pure Time. 421
And the general inverse exponential or logarithmic function-couple, which may, by
(129.), be thus denoted,
(2, r,)=0' CY Yr), if (YW Y2) =P (X,, 22), (179.)
may also, by (174.) and (176.), be thus expressed :
ig =F Yom),
® CY ==, b)? (180.)
it involves, therefore, two arbitrary integer numbers, when only the couple (¥,, y;)
and the base (b,, b,) are given, and it may be thus more fully written,
wo" a ee Cy ’ Yo)
®"'(Yis Yo) = log wy my + (Yo Y2) =F :
Fa( b,, b,) (181.)
For example, the general expression for the logarithms of the primary unit (1, 0) to
the base (e, 0), is
(0, 20’ z) (2 w' 7.0)
ea ely er OE Gira mea co aay 3
lo}
wo
(182.)
or, if we choose to introduce the symbol “—1, as explained in the 13th article,
that is, as denoting the couple (0, 1) according to the law of mixture of numbers with
number-couples, then
2 war JT 20 7
OS e* Ser Li eee (183.)
In general,
nf EY Yo) + (0, Qo! a 7
198 orm)» (I W=EI CH, In) + 0, 20) oe
The integer number w may be called the first specific ordinal, or simply the orpEr,
and the other integer number w’ may be called the second specific ordinal, or simply
the rank, of the particular logarithmic function, or logarithm-couple, which is deter-
mined by these two integer numbers. ‘This existence of two arbitrary and inde-
pendent integers in the general expression of a logarithm, was discovered in the year
1826, by Mr. Graves, who published a Memoir upon the subject in the Philosophical
Transactions for 1829, and has since made another communication upon the same
-subject to the British Association for the Advancement of Science, during the meeting
of that Association at Edinburgh, in 1834: and it was he who propesed these names
of Orders and Ranks of Logarithms. But because Mr. Graves employed, in his
422 Professor Hamitton on Conjugate Functions, &c.
reasoning, the usual principles respecting Imaginary Quantities, and was content to
prove the symbolical necessity without showing the interpretation, or inner meaning,
of his formule, the present Theory of Couples is published to make manifest that
hidden meaning: and to show, by this remarkable instance, that expressions which
seem according to common views to be merely symbolical, and quite incapable of
being interpreted, may pass into the world of thoughts, and acquire reality and signi-
ficance, if Algebra be viewed as not a mere Art or Language, but as the Science of
Pure Time. The author hopes to publish hereafter many other applications of this
view ; especially to Equations and Integrals, and to a Theory of Triplets and Sets of
Moments, Steps, and Numbers, which includes this Theory of Couples.
THE END.
Pray
Researches on the Action of Ammonia upon the Chlorides and O.xides of Mercury.
By Roserr Kaye, M.D. M.R.1.A., Professor to the Royal Dublin Society and
the Apothecaries’ Hall, Corresponding Member of the Society of Pharmacy of
Paris, and of the Society of Medical Chemistry in the same City.
Read November 30th and December 28th 1835.
Secrion I.— Of the Action of Ammonia upon the Bichloride of Mercury.
Tue action of ammonia upon corrosive sublimate is of so remarkable a nature,
as to have attracted attention from a very early period. There have been, therefore,
very many facts elicited; but the discordance in the quantitative results of the most
eminent chemists who have applied themselves to its elucidation, renders an attentive
and fundamental re-examination of the subject necessary before any conclusion can be
justly drawn. In the elementary works on chemistry, a theory is given as an ascer-
tained truth, which we shall find to be totally devoid of foundation ; and almost every
analyst that has written, has brought forward a different hypothesis of his own.
Ammonia may be made to act upon corrosive sublimate in either of two ways—first.
by passing the gas directly over dry deuto-chloride ; or, secondly, by adding solution
of ammonia to that of corrosive sublimate. The reaction of the first case has only been
examined by Grouvelle and Rose,*—that of the second has obtained an amount of
study bestowed on few bodies. The memoirs of Fourcroy,t of Hennell,+ Guibourt,$
Soubeiran,|| and Mitcherlich,§ have shown the interest attached by chemists to
the investigation of the nature of these bodies ; but, by the absolute discordance of
the results, and by the accidental misinterpretation of some of them, have demon-
strated the existence of some latent error, and the necessity for new investigations, in
which it should be sought for, detected, and avoided.
* Poggendorff, Annalen der Physik, vol. xci.
+ Journal de l'Ecole Polytechnique, vol. vi. p. 312.
+ Quarterly Journal of Science, vol. xviii. p. 291.
§ Journal de Pharmacie, vol. vi.
|| Soubeiran, Journal de Pharmacie, vol. xii. 24; Annales de Chimie, vol. xxxvi. p. 220.
q Mitcherlich, Poggendorff, vol. Ixxxv. p, 410.
VOL. XVII. 4F
424. Professor Kane on the Action of Ammonia
The white precipitate, produced by adding water of ammonia to a solution of cor-
rosive sublimate, is the substance with which we shall commence, as a knowledge of
its history will be found to facilitate very much the study of those remaining.
§ 1.—Of the White Precipitate of Mercury.
It is well known that on adding water of ammonia to a solution of corrosive sub-
limate, there is obtained a milk-white precipitate, insoluble in water, and possessing
properties which, as they are made the foundation of many experiments hereafter to
be described, I shall briefly mention. This precipitate, when first produced, is milk-
white, very bulky, depositing but slowly, and rather aluminous-looking. If very
hot water be used in preparing it, or if it be washed very much, it loses its pure white
colour, and acquires a yellow tinge; if boiled for a few minutes in the liquor, it is
completely decomposed, and then results a canary-yellow powder, very heavy and
granular. This white precipitate is perfectly insoluble, as such, in water. An occa-
sional appearance of solution results from its being decomposed, and one or other of
its elements entering into new combinations. When heated in a glass tube, sealed at
one end, it is completely decomposed below a red heat; there is disengaged a mixture
of ammonia and azote, some water, and calomel sublimes, generally darkened by some
ammonia, from which, however, it can readily be freed.
White precipitate dissolves readily in nitric or in muriatic acids ; when mixed with
an alcali, as potash or soda, or with lime or baryta, there is ammonia disengaged, and
the mass becomes yellowish, but the decomposition is never complete. ‘The real nature
of the products we shall hereafter study ; but it may be remarked, that no excess of
aleali can expel all the ammonia of the body.
If we add to white precipitate a solution of iodide of potassium, there is deposited
a red powder, and much ammonia is disengaged. ‘The powder is biniodide of mer-
cury, and the liquor contains free potash ; all the ammonia is liberated. Sulphuret .
of barium in solution produces a similar reaction—all the ammonia being disengaged,
and all the quicksilver deposited as bisulphuret.
In order to obtain white precipitate of sufficient purity for examination, some precau-
tions must be observed, the neglect of which has produced much of the confusion in re-
sults ; different chemists having analyzed heterogeneous products. Toa cold solution of
bichloride of mercury, there is to be added water of ammonia in very slight excess ; the
whole is to be thrown on a filter, and as much as possible of the liquor allowed to
come away before any washing is attempted. It may then be cautiously washed with
as much distilled water as may suffice to remove the original liquor from the mass,
but over excess avoided, as, even by cold water, some portions are decomposed, and
the milk-white colour of the powder lost.
on the Haloid Compounds of Mercury. 425
Having, by careful repetition of the above method, obtained a product such as I
might consider pure, I entered upon its analysis. I think it advisable, however, before
commencing the details of my own experiments, or noticing the results to which they
have led, to bring forward a tabular statement of the analytical results of other writers,
in order to show in what obscurity the subject was involved. ‘The results are taken,
divested of the theories to which they have respectively conducted their authors, and
are exhibited in the real quantities of chlorine, quicksilver, and ammonia, which were
obtained as the products of the experimental trials.
There were obtained from 100 parts :
Authors. Mercury. Chlorine. Ammonia.
Fourcroy,* 74,1 13,2 6,03
Hennell,t 74:,24 13,14 6,31
Mitcherlich,t 76,37 13,82 vial!
Guibourt,§ feck) 13,3il 4,45
Soubeiran, || 82,1 7,8 5,0
A glance is sufficient to show the impossibility of deducing from such a chaos any
general expression for the composition of white precipitate. In attempting to find
some principle by which it could be set in order, I rested my probabilities of success
on the number of trials I would make ; for, conscious that the chance of incorrectness in
any one analysis should be still greater with me than with any one of those chemists
whose ill success I have exhibited above, I trusted to diminishing the errors by taking
averages of numerous results, and obtaining the results themselves by processes dif-
fering in principle, so as to render it very unlikely that any one error could pervade
all. I shall therefore describe in order the results of each different method of
analysis.
A.—When a solution of corrosive sublimate is precipitated by ammonia, the
liquor contains no mercury, but much chlorine. All the mercury of the sublimate
is contained in the white precipitate, and a portion of the chlorine is removed, and
remains in the solution as sal-ammoniac. It is evident that on this principle may be
founded a method of determining the amount of mercury and chlorine in white pre-
cipitate ; and this was accordingly that first made use of.
One hundred grains of corrosive sublimate were dissolved in cold water, the solution
decomposed by a slight excess of ammonia, and the precipitate thrown on a weighed
filter, and washed with cold water. The precipitate was carefully dried and weighed ;
the liquor added to the washings was acidulated by nitric acid, and precipitated by solu-
* By result given by Leopold Gmelin, Handbuch.
+ Quarterly Journal of Science.
$ Poggendorff, vol. Ixxxv.
§ Result given by Thenard, Traité Elementaire.
|| Journal de Pharmacie, vol. xii.
4.26 Proressor Kane on the Action of Ammonia
tion of nitrate of silver ; the chloride produced was collected on a weighed filter, and its
guantity determined. In this way were obtained the results given in the following table:
No. of Experiments. Precipitate. Chlorine in liquor.
1 91,3 12,9
2 92,4 13,3
3 92,9 1S lis
4 95,4 dEAY
£5) 93,4 12,95
Mean 93,1 13,00
But 100 grains of sublimate contain,
Mercury 74,09
Chlorine 25,91
The quantity of chlorine contained in the liquor was, therefore, evidently one-half of
that in the corrosive sublimate, and we have by this method, in 93,1 of white precipitate,
Mercury 74,09
Chlorine 12,91
\
Or in one hundred parts,
Mercury 79,57
Chlorine 13,87
B.—When white precipitate is heated, there is obtained, besides gaseous matter and
watery vapour, the whole of the mercury and chlorine united as calomel. This fact,
which confirms the relation between the quantities of mercury and chlorine found in
the preceding results, affords a means of determining the actual amount, which was next
put in practice. The operation was thus performed. A small tube retort was taken,
and sometimes a straight tube having a strong bulb blown at one extremity, and it
was accurately tared ; there was next introduced a amount of white precipitate suf-
ficient to about half-fill the bulb, and the whole weighed ; the increase indicated the
quantity of white precipitate employed. The bulb was now heated, and the tube it-
self so warmed, as perfectly to get rid of any water that might tend to deposit itself
in the throat of the apparatus, but guarding against the loss of any calomel. When
the latter has been completely sublimated, it is generally dark-coloured, owing to the
contact of free ammonia ; but, by allowing the tube to cool, and the atmospheric air
to gain admission, and then again heating the calomel, it is finally obtained quite white.
The apparatus was then again weighed ; the loss from the gross weight gave the amount
of volatile ingredients ; the excess above tare gave the weight of the calomel, from which
the quantities of quicksilver and chlorine may be calculated. The following table
contains the results of this method.
No. of Ex. Matter used. Calomel. Calomel per cent.
1 20,42 18,95 92,80
2 19,42 18,07 92,53
3 12,14 11,28 92,91
4 14,71 13,79 93,68
on the Haloid Compounds of Mercury. 427
These results, which agree very closely with each other, give us as a mean from
one hundred of white precipitate 92,98 of calomel, containing
Mercury 79,14.
Chlorine 13,84.
C.—To obtain a value for the quicksilver by reduction, the usual method was pur-
sued. ‘The white precipitate was dissolved in muriatic acid, and precipitated by solu-
tion of protochloride of tin. The quicksilver was dried in the cold, and there was
obtained from one hundred grains of white precipitate, in one experiment, 77,3, and
in a second 78,1 of mercury, giving a mean of 77,7 per cent.
D.—105,40 of white precipitate were dissolved in muriatic acid, and the liquor
diluted with four times its volume of water. Sulphuretted hydrogen was then passed
through until the resulting precipitate became perfectly black, and deposited readily.
The whole was then thrown on a weighed filter, and washed with distilled water.
The washings and liquor were cautiously evaporated to dryness, and the residue
weighed. ‘There were thus obta ined :
Filter and sulphuret 143,58 grs.
Filter - - 48,35
Bisulphuret of mercury " 95,23
Consisting of Mercury - rink, 82,17.
Sulphur — - - 13,06
And Sal-ammoniac - - 23,59 grs.
Consisting of Muriatic acid - 16,04
Ammonia - - [eco
Or from 100 of white precipitate :
Mercury - 77,96
Ammonia - : 7,16
E.—For the determination of the ammonia of white precipitate, potash or lime
cannot be employed, as only one-half of the ammonia is thus separated. In addition
to the mode applied in (D), which is perhaps the best, the following means were had
recourse to :
a. 100 grains of white precipitate were introduced into a flask, and a solution of
sulphuret of barium poured on it; from the flask a bent tube conducted to a tall jar
containing dilute muriatic acid. By the application of heat, the ammonia and much
water were driven over ; the liquor in the jar was then evaporated to dryness. There
remained sal-ammoniac 21,57 grs. consisting of—
Muriatic acid 14,85
Ammonia 6,72
4.28 Professor Kane on the Action of Ammonia
B. 100 grains of white precipitate, treated with iodide of potassium by the same
method, gave sal-ammoniac 19,83, consisting of—
Muriatic acid 13,50
Ammonia 6,33
F.—In all theories of the composition of white precipitate hitherto advanced, oxygen
is enumerated as a constituent to a very considerable amount, generally so much as
to peroxidize the whole of the quicksilver. The results hitherto obtained in my ex-
periments would not appear to leave room for so much oxygen, and I therefore en-
deavoured to obtain an estimate of the amount of that element by direct experiment.
The principle of which I made use, is the following. When white precipitate is heated
there are obtained ammonia and azote, water and calomel, but no free oxygen; there-
fore all the oxygen of the substance has formed water at the expense of the ammonia.
I resolved upon obtaining this water, and determining from its weight how much
oxygen the white precipitate contained.
A small retort was blown of strong glass, and of a capacity of from 0,2 to 0,3 of
a cubic inch, the neck being about two inches long. ‘To this was tightly connected a
small tube, containing sometimes dry lime, and at others fused potash, and communi-
cating by a narrow tube with the mercurial pneumatic trough. The retort was carefully
weighed, and the white prévipitate introduced ; the whole then weighed, and thus the
quantity of materiel determined ; the desiccating tube was also carefully weighed ; and
the apparatus having been connected, heat was applied to the retort until the chlorine
and mercury had all sublimed as calomel. The water was driven completely out of
the neck of the retort, and passed into the tube with lime or potash, where it re-
mained. ‘The azote and ammonia were collected in a jar over the mercury.
After the operation, the retort was weighed: the residue was calomel ; the loss,
water, azote and ammonia: the desiccating tube was weighed ; the increase gave the
quantity of water formed. The mixture of gases was corrected for pressure and
temperature, and analyzed by water.
Expt. 1st.—Retort and material 99,93
Retort = Tae
White precipitate 22,21
Tube with lime before = 269,00
Tube with lime after = 269,22
Water condensed 0,22
In this experiment, owing to an accident, the estimates for the calomel and gaseous
matters were rendered unavailable.
Expt. 2d.—Retort and material 85,25
Retort = 64,78
White precipitate 20,47
on the Halotd Compounds of Mercury. 429
Retort and sublimate residue Ss 84,22
Retort = : 64,78
Sublimate residue 19,44
Tube with lime before : 268,72
Tube with lime after = 268,86.
Water condensed 0,14.
The gaseous mixture, reduced to mean temperature and pressure, amounted to 4,24
cubic inches, of which 2,67 were absorbed by water. Of the remainder, 0,23 were
the atmospheric air of the apparatus; there were obtained therefore—
Cub. Inch.
Ammonia 2,67 weighing 0,488 grs.
Azote 1,34 0,404
But the sublimed residue of calomel was dark-coloured from having absorbed
ammonia; it must, therefore, be corrected by subtracting from it that quantity, and
adding it to the ammonia actually obtained. Now it has been shown that 100 of
white precipitate give 92,28 of calomel; therefore 20,47 should have given 19,135,
which had absorbed 0,305 of ammonia.
Summing up these results, we obtain—
Calomel 19,135
Azote 0,4.04
Ammonia 0,793
Water 0,140
It is unnecessary to give all the details of other experiments similarly made ; the
results will suffice.
Expt. 3d gave for 12,14 white precipitate—
Calomel 11,280)
peak eee 12,14
Water 0,080
Experiment 4th furnished the calomel result, No. 2 of the table, page 426. But
as some gas was lost in shifting a jar, the azote or ammonia could not be determined ;
20,472
the desiccating tube had not sensibly increased in weight.
It is not necessary to use the above results for the chlorine or mercury constituents,
as there have been already gotten numbers by simpler processes, and consequently
less exposed to error. The difference is, however, very trivial. The product of
water gives the the following :—
Expt. Material. Water. Water per cent.
1 22,21 0,22 0,990
2 20,47 O,14 0,684:
3 12,14 0,08 0,658
4 19,42 0,00 0,000
430 Professor Kane on the Action of Ammonia
Giving as the average of water per cent. 0,583 ; but of this a portion probably con-
sists of moisture, hygrometric, or arising from imperfect desiccation of the precipitate.
To obtain the ammonia, we must convert the azote into ammonia, and add it to that
actually gotten; we thus find—
Expt. 2d gives 1,282, or 6,26 per cent.
3d 0,845, or 6,96.
Giving an average of 6,61 per cent of ammonia.
Summing up all the analytical results for white precipitate, we have—
Process. Mercury. Chlorine. Ammonia. Water.
A 79,57 13,87
B 79,14 13,84
c 77,70
D 77,96 7,16
E 6,53
F 6,61 0,583
And we obtain the final mean of
. Mercury 78,60
Chlorine 13,85
Ammonia 6,77
Water 0,58
Loss 0,20
100,00
The theory of white precipitate given by the majority of chemists supposes it to
consist of an atom of peroxide of mercury, united to an atom of sal-ammoniac. This
view, which is founded on the experiments of Fourcroy and Hennell, gives the fol-
lowing numerical results :—
1 atom mercury 202,8 or 74,46
2 oxygen 16,0 5,88
1 muriatic acid 36,42 6,29
1 Ammonia VS 13537,
272,37 100,00
The fallacy of this theoryis completely proved by the great difference in the de-
termination of the mercury. ‘The result of Fourcroy was obtained so long since, that
it necessarily participates in the imperfections of the analytic methods of that time ;
and Hennell’s results are evidently of a very inaccurate description. He states that
he got a quantity of mercury very near the atomic proportion, but does not give the
number actually gotten; hence there cannot be placed implicit confidence in the
theory he supports. In addition, the existence of a quantity of oxygen, so large
per cent. (5,88) in white precipitate, is completely impossible. My researches, di-
rected expressly to that object, sufficiently disprove such an idea.
on the Haloid Compounds of Mercury. 431
Another circumstance which demonstrates the incorrectness of the popular theory
of this body, is the fact that 100 of sublimate yield only 93,1 of product ; on Hen-
nell’s view, there should be 99,5 of white precipitate for
1 atom of mercury 202,80 give 1 atom mercury 202,80
2 of chlorine 70,84 2 oxygen 16,00
1 muriatic acid 36,42
273,64 1 ammonia iy pas
272,37
The source of error in such analyses, evidently consisted in not having dried the
white precipitate until it ceased to lose weight.
In the Memoir already quoted, George Mitcherlich adopts the hypothesis just
described, and his doing so has created great confusion. He gives the name
Chlor-Wasserstoff Siure to the hypothetic dry muriatic acid; and his formula
(NH + M) + He is so constructed. ‘This is shown by the numbers for muriatic acid
and ammonia, 10,7 and 7,1, which, he states, form sal-ammoniac, (dry). In order to
get his value for chlorine, given in the table of results, p. 425, I had to add to his
muriatic acid, half the oxygen which he gives to the oxide of mercury. In fact, his
analysis, correctly interpreted, overturns the very hypothesis which it has been sup-
posed to be in accordance with, for—
Mercury 76,37
Dry muriatic acid 10,7 :
Oxygen p 3,12 Chlorine 13,82
Ammonia 7,10
97.29
leaves only a vacancy of 2,71 per cent for the oxygen to
oxidize the whole of the quicksilver.
Not having access to Guibourt’s Memoir, I can only speak of his opinions and re-
sults, by the references made to him by Thenard,* and other writers. He considers
white precipitate to be composed of corrosive sublimate, peroxide of mercury and
ammonia in proportions, giving the formula—
fs (2Ch + Hg) + enurt + 43 lig+2 wut
which gives as the per centage result—
Mercury 78,71
Chlorine 13,78
Oxygen 3,10
Ammonia 4,41
100,00
* Traité Elémentaire de Chimie.
VOL. XVII. 4G
432 Professor Kane on the Action of Ammonia
Here the mercury and chlorine agree pretty nearly with my determination, but the
quantity of ammonia is only about one-half. I think it probable, that Guibourt deter-
mined the ammonia by means of potash, thinking the decomposition perfect, as it is
generally described, and as I myself at first considered it to be. In order to show
the probability of this being the source of Guibourt’s error, I shall describe the fol-
lowing, one of the earliest experiments I made in the matter.
100 grains of white precipitate were mixed with a very strong solution of potash,
in a flask from which a bent tube passed to a jar containing dilute muriatic acid. The
flask was heated until most of the water in it had distilled over. ‘The liquor in the
jar was then evaporated to dryness, and gave 11,5 grs. of sal-ammoniac, consisting of
Muriatic acid 7,84
Ammonia 3,66
Now, supposing the loss to be oxygen, which was the method generally pursued in
analyzing this substance hitherto, we should have the result—
Mercury 78,60
Chlorine 13,85
Ammonia 3,66
Oxygen 8,89
These results agree so closely as to point out the manner in which Guibourt’s
analysis, so correct in the chlorine and quicksilver constituents, became erroneous by
reducing the ammonia to nearly one-half the actual amount.
100,00
I shall return to the causes of error in Soubeiran’s analysis; at present it is not
necessary to advert to them, as he evidently analyzed a body quite different from white
precipitate.
The simplest view to take of the existence of the chlorine in this substance, is to
suppose it united with half the mercury as corrosive sublimate ; it is almost the only
view possible. Then, in what state is the remainder of the mercury? We may sup-
pose it peroxidized, and the oxide united with the ammonia giving the formula
(2Ch + Hg) + (Hg +2NH’) and the following numerical arrangements :—
2atoms mercury 405,60 or 77,00
2 oxygen 16,00 3,04
2 chlorine 70,84 13,45
Q ammonia 34,30 6,51
526,74 100,00
This agrees closely with Mitcherlich’s, and also with some of my own analyses. It
differs, nevertheless, from the mean of my results in the quantities of mercury and
chlorine, and particularly in the quantity of oxygen. This hypothesis supposes the
existence of 3,04 per cent. of oxygen, a body of which I could not determine the
on the Haloid Compounds of Mercury. 433
existence as a constituent at all, and which the relative quantities of the other ele-
ments would appear to exclude. It is therefore necessary to examine whether any
other method of arrangement is more suitable.
From Dumas’ researches on oxamide, benzamide, &c. it follows, that by the ac-
tion of ammonia on an oxide, there may be formed water anda compound of the body
NH’, with the basis of the oxide. If we consider this to have taken place in white
precipitate, we should have the formula (2 Ch + Hg) + (2 NH’ + Hg), giving—
Hg 202,80 or 79,73
Ch 35,42 13,93
NH? 16,15 6,34
254,37 100,00
And this compound should give by decomposition 6,73 per cent. of ammonia.
I should not wish to adopt too positively the opinion that white precipitate is a
compound of deuto-chloride and deutamide of mercury, although the non-existence
by experiment of oxygen as a constituent, renders it extremely probable. In addition,
the decomposition of white precipitate by iodide of potassium appears to afford a pre-
sumption that the mercury is not oxydized, as red oxide of mercury does not de-
compose iodide of potassium ; on the other theory, the reaction is at once explained.
Thus—
4 f1+K} + 4 (@Ch+ Ha) +(2NH'+He)t ie
a21+Hg)+2(Ch+K)+2(NH?+K).
and 2(NH?+K)+2H = 2K+2NH’.
The question whether ammonia, in acting on metallic oxides, forms water and me-
tallic amides, is one of the most interesting now beginning to be examined ; but,
notwithstanding the bearing which the results just described, have on the question, I
do not wish to come to any positive conclusion, until a more extensive basis for in-
duction is obtained. The atomic weight of mercury is so large, and preponderates
so much on the other constituents, as to make small differences in their quantities fall
within the limits of necessary error; and hence, I shall leave the two explanations
until, by experiments on the compounds of a metal of a smaller combining number, I
shall have an opportunity of seeing them diverge at a greater angle.
To conclude, the white precipitate yields, as the mean results of my analyses,
Mercury 78,60
Chlorine 13,85
Ammonia 6,77 ¢ 100,00
Hygrometric water, ial 0,78
and oxygen
434 Professor Kane on the Action of Ammonia
and there are two formule which give results approaching closely to those, viz.—
1 a
(2Ch + Hg) + (2NH’ + Hg) (2Ch + Hg) + (2NH* + Hg)
which gives which gives
Hg 79,73 Hg 77,00
Ch_ 13,93 Ch , 13,45
NH? 6,34 NH?. 6,51
Ox 3,04
100,00
It is evident, that although neither is so far removed as to be beyond admissibility,
the balance of probability is inclined much to the side of the former.
$2.—Of the Powder formed by the Action of Water on HFhite Precipitate.
It is generally stated by chemical writers, that by the action of much boiling water,
white precipitate is decomposed completely, red oxide of mercury being left behind.
I never could succeed in effecting this ; but the reaction that did take place appear-
ing to me perfectly definite, and identical in its results at different times, I was in-
duced to examine it in detail.
When white precipitate is boiled in water, it is changed into a heavy canary-yellow
powder, subsiding rapidly, and very easily dried, when it appears granular. This
powder is not quite insoluble in water ; when heated, it gives out ammonia, azote,
water, and there sublimes a mixture of calomel and metallic mercury : it dissolves
readily in muriatic or nitric acids. Alcalies appear to have scarcely any action upon
it, except slightly altering its colour; when digested with iodide of potassium, there
is ammonia disengaged, and a brown powder formed. To this reaction I shall hereafter
recur. In order to determine the composition of this yellow powder, the following
experiments were made :—
A.—100 parts of corrosive sublimate were dissolved in water, and ammonia added
in excess. The mass, in place of being filtered cold, was boiled until the light-white
precipitate was changed into the clear yellow heavy powder ; it was then filtered, and
the quantity of product determined. ‘The liquor and washings were acidulated by
nitric acid, and precipitated by nitrate of silver, and the chloride abstracted from the
sublimate thus determined; the liquor contained a very small trace of mercury.
Several experiments were made on this plan, the result of which are exhibited in
the following table :
100 parts of sublimate gaye—
on the Haloid Compounds of Mercury. 435
Expt. Yellow powder. Chlorine in liquor.
1 83,5 19,25
2 83,3 18,50
3 84,7 18,90
mean 83,83 18,89
Now 100 of sublimate, contain
Mercury 74,09
Chlorine 25,91
Hence we see that there have been abstracted from the sublimate three-fourths of
its chlorine ; the remaining fourth, and all the mercury, existing in the yellow powder.
We have therefore in 83,83 parts of it—
Mercury 74.09
Chlorine 71,02
Or in one hundred parts,
Mercury 88,381
Chlorine 8,374
B.—When white precipitate already prepared is boiled with water, there is ob-
tained a similar yellow powder, and the supernatant liquor is found to contain only
sal-ammoniac. As we know, within very strict limits, the composition of the white
powder, we can make use of this reaction to illustrate the nature of the yellow
product :
100 parts of white precipitate were boiled with water until completely converted
into the yellow powder ; the liquor, which was quite neutral, was acidulated ; and the
chlorine dissolved precipitated as chloride of silver, from which its quantity was ob-
tained by calculation. The following table gives the results of experiments conducted
in this manner :—
100 of white precipitate gave—
Expt. Yellow powder. Chlorine in liquor.
1 90,00 5,93
2 88,50 6,50
3 90,30 6,40
mean 89,60 6,29
But 100 of white precicipate contain
Mercury —_78,60
Chlorine 13,85
436 Professor Kane on the Action of Ammonia
Therefore 89,60 of the yellow powder contain
Mercury 78,60
Chlorine 7,06
and 100 contain
Mercury 87,95
Chlorine 8,44
C.—100 grains of white precipitate were boiled with water until completely de-
composed ; the resulting yellow powder weighed 91,15 grains. ‘The liquor was cau-
tiously evaporated to dryness, and gave 10,23 of sal-ammoniac, consisting of
Chlorine 6,76
Hydrogen ,19 > 10,23
Ammonia 3,28
Therefore there are obtained, by this experiment, for the constituents of yellow powder,
Mercury 86,23
Chlorine 7,77
Ammonia 3,83
D.—It has been already stated, that when this powder is heated, it is resolved into
ammonia, azote, water, calomel, and quicksilver. Having found that, by performing
this operation in a very small retort, the water and gases could be dissipated without
any remarkable loss of the other constituents, I made some trials in this way to de-
termine the amount of the chlorine and quicksilver. For this purpose a higher tem-
perature is required than for the corresponding analysis of white precipitate, and the
condensation of the mercurial vapour must be very carefully effected. In other
respects, the manipulation was the same, and the following table contains the results :
Exot Quantity of Subliemd Sublimed Residue
cB Material. Residue. from 100 parts.
] 14,30 13,37 93,50
2 19,65 18,53 94,30
3 93502 29,35 94,22
Per cent. mean 94,01
From this result we can easily calculate the quantities of chlorine and mercury
the residue contains, for—
Let m = the residue = 94,01
= the quantity of chlorine
the quantity of mercury
atomic weight of chlorine = 35,42
atomic weight of mercury = 202,8
cae s
{I
on the Haloid Compounds of Mercury. 437
This is (1) c=m—y
and (2) = = = by other processes
. Remains
Then pels = ps . 24m = (at+2b) y
y 2b
2bm
nas fares
We thus find 100 of yellow powder, to contain
Mercury 86,46
Chlorine (aa)
E.—105,28 grains of yellow powder were dissolved in muriatic acid, and the solu-
tion having been somewhat diluted, was decomposed by a current of sulphuretted
hydrogen gas. The perfectly black sulphuret was collected on a weighed filter, and
the liquor evaporated to dryness, and the residual sal-ammoniac, weighed :
The filter and sulphuret 126,71
Filter - - 23,00
Sulphuret of mercury 103,71, consisting of
Sulphur - . 14,22
Mercury - - 89,49
The sal-ammoniac weighed 12,86 grs. consisting of
Chlorine - = 8,50
Hydrogen - »24
Ammonia = 4,12
Therefore, the yellow powder consisted of
in 105,28 parts— in 100 parts—
Mercury 89,49 Mercury 85,00
Ammonia 4,12 Ammonia 3,91
Summing up these different results, we haye—
Process. Mercury. Chlorine. Ammonia.
A 88,381 8,374
B 87,95 8,44
C 86,23 7,77 8,83
D 86,46 7,55
E 85,00 3,91
And taking the mean result of all, we get, for the composition of this yellow
powder,
438 Proressor Kane on the Action of Ammonia
Mercury 86,80
Chlorine 8,03
Ammonia 3,87 100,00
Oxygen and loss 1,30
In the processes, A and B, a small quantity of the yellow powder was lost, in con-
sequence of its being not perfectly insoluble in water. This quantity, I have reason
to believe, varied from one to two per cent. and by the means of calculation we em-
ployed, the mercury and chlorine constituents are given above what is correct in that
proportion. As far as I can judge, by considering the circumstances of the ex-
periments, I conceive the mean to be consequently too high; and I believe the
analysis C. by itself, to approach closer to the truth. By it we have in 100 of the
yellow powder—
Mercury 86,23)
Chlorine 777
Ammonia 3,83 100,00
Oxygen and loss 2,17
This yellow powder is generated evidently by the reaction of water on white pre-
cipitate, in which one-half of the chlorine and ammonia are converted into sal-
ammoniac, a corresponding portion of the mercury being oxydized. I shall compare
the results of this reaction with each of the formule for white precipitate that I have
previously given.
24 (ech + Hg) + (2 Nur + fie) t 4 2H =
4 (2Ch + Hg) + (8 He + ont) t + 2ChNH!
Here two atoms of white precipitate and two of water, mutually reacting, give two
atoms of sal-ammoniac and one of the powder. On this idea, the yellow powder
should consist of
Mercury 84,1 ‘|
Chlorine 7,36
Ammonia 3,56 100,00
Oxygen 4,96 J
These numbers, with the exception of oxygen, fall below the lowest experimental
result, and probabilities are therefore rather against this formula being true. Let us
next try the result of reaction with that formula for white precipitate which does not
include oxygen, and supposes the ammonia to exist as amidogene.
on the Haloid Compounds of Mercury. 439
Q {(@Ch + Hg) + (2 NH? + Hg) +4 H=
f(2 Ch + Hg) +2Hig +(@2NH? + Hg) +2Ch NH
There are here equally produced by the two atoms of white powder and two of water,
two of sal-ammoniac, and one of the yellow powder, of which the composition should be,
Mercury 85,72
Chlorine 7,48
Amidogene 3,42 skies
Oxygen 3,38
and it should yield in analysis 3,63 per cent. of ammonia.
The definite composition of this yellow powder is thus evident, and the decom-
position by which it is formed perfectly explained. We see that all the results tend
to show that in these bodies the ammonia is not united with oxide of mercury, but
rather the metal with amidogene. The perfect demonstration of this principle, how-
ever, must be sought for in the other metals.
§3.— Of the products of the Action of Alcalies in Excess on WV hite Precipitate.
Grouvelle and other chemists have stated that by the action of an excess of alcali
on a sublimate solution, there is produced the ammoniuret of mercury which was dis-
covered by Fourcroy and examined by Guibourt, and to which I shall hereafter
speedily recur ; and even Dumas states, that ‘the same compound (the ammoniuret)
is obtained by pouring ammonia into a solution of corrosive sublimate, and then adding
caustic-potash in excess.”” My anxiety to obtain pure ammoniuret of mercury, joined
to the interest of the preceding investigations, led me to examine the nature of the
products thus obtained; and the results, as correcting an error very generally fallen
into, are worthy of being described.
When corrosive sublimate is decomposed by ammonia, the quantity of alcali in
excess does not appear to interfere much with the reaction before described. If the
liquors be cold, there is obtained white precipitate; and if it be boiled, the heavy
yellowish powder is produced. The liquor retaining in the former, one-half, in the
latter, three-fourths of the chlorine of the sublimate. Again if white precipitate be
boiled in water, rendered strongly alcaline by ammonia, we obtain the yellowish
powder, and half the chlorine and half the ammonia of the precipitate are disengaged.
Thus, water of ammonia acts on white precipitate only as water itself does, the nature
of the reaction being the same in both instances. Again when white precipitate
VOL. XVII. 4H
440 Professor Kang on the Action of Ammonia
was treated with potash for analysis, as in p. 432. it has been seen that the ammonia
disengaged was but one-half what it contained, the formation of the yellowish powder
being the limit at which the decomposition stops. In these cases, however, the
powder product is not so bright in colour as that produced by the action of mere
water. It does not appear to be quite so pure, but in its properties it manifests com-
plete identity.
Nevertheless, in order to leave no room for doubt upon the matter, I decomposed
corrosive sublimate by a great excess of ammonia, added a strong solution of potash,
and boiled for a considerable time. ‘The yellowish white powder produced, was se-
parated by the filter and washed until the liquors ceased to effect turmeric paper :
dried carefully. It weighed from 100 of sublimate, 85 grains. When heated it gave
out water, ammonia, and azote, and calomel with metallic mercury sublimed. When
suddenly heated, it puffed up, more so than the pure yellow powders, which was pro-
bably the reason of its having been confounded with the ammoniuret which possesses
a very slight detonating property.
To analyze it, 66,83 grains were dissolved in muriatic acid, and having been diluted
were decomposed by a current of sulphuretted hydrogen gas. The black sulphuret
was collected on a weighed filter, and having been carefully dried, weighed 67,70
grains, consisting of
Mercury 58,42
Sulphur 9,28
The liquor evaporated, gave sal-ammoniac 6,58 grains, consisting of
Chlorine 4,35
Hydrogen 912
Ammonia 2,11
There were thus obtained from 66,83 of this powder—
Mercury 58,42)
Chlorine 7 hl be
Ammonia 2,11 66,83
Oxygen and loss 1,935 J
But the yellow powder by water, should have given from the formula :
(2 Ch + Hg) +2 Hg + (2 NH? + Hg)
Mercury 57,30
Chlorine 5,00
Amidogene 2,28
Oxygen 2,25
This result proves the identity of the effect in the two cases of the action of water
and of an alkali.
66,83
on the Haloid Compounds of Mercury. 441
Rose and Grouvelle have already shown that when dry ammonia is passed over
melted sublimate, an atom of the former is absorbed by one of the latter, and a white
mass formed. The history of its properties has been so well given in Rose’s Memoir,
that it is unnecessary for me to do more than mention my results as having been con-
firmatory of his. This compound is decomposed by water giving white precipitate and
sal-alembroth, as may be at once seen :
4 (2 Cl+Hg)+4 NH =
§ (2Cl+ Hz) + (@NH?+ Hg) +2 fe Cl +Hg) +(Cl + NH)?
Section II.— Of the Action of Ammonia on the Proto- Chloride of Mercury.
§1.—Aection of Liquid Ammonia upon Calomel.
The decomposition resulting from the action of water of ammonia upon the proto-
chloride of mercury, does not appear to have attracted particular attention, as all
writers who speak at all upon the subject, mention ammonia, along with potash and
soda, as decomposing calomel into black oxide of mercury. Hennell in particular,
states expressly, that calomel decomposed by excess of ammonia, yields a black powder
containing in 100 parts, 96 of mercury and four of oxygen. I was therefore rather
surprised when experiment showed me that a reaction of a totally different nature
takes place, giving rise to a compound possessed of very remarkable properties.
When water of ammonia is poured on calomel, whether sublimed or precipited, the
the mass immediately becomes black, and the appearance is not altered by boiling the
mixture for a long time. While yet wet the powder remains almost black, but it
becomes much lighter on drying, so that when quite dry it is of a dark-grey. This
powder is not altered by exposure to air, or to a moderate heat ; a portion of it was
exposed in a platinum crucible on a sand bath for several hours to a temperature of
180° Fahrenheit, without being altered in weight or colour. When moistened it
becomes nearly as dark as when first generated, but it again loses its black colour on
being dried: boiled with water it does not appear altered in its composition. When
this powder is heated in a tube sealed at one end, it first gives a trace of water, with
much azote and ammonia; then there sublimes calomel mixed with metallic mer-
442 Professor Kane on the Action of Ammonia
cury, the decomposition being accompanied with that sort of effervescence which
appears in the heating of so many of the substances under examination.
For the examination of this body, an order of analysis similar to that adopted for
white precipitate was pursued.
A.—148, 15 grains of precipitated calomel were boiled for some minutes with a great
excess of water of ammonia, and the whole thrown on a filter. ‘The black powder
thus obtained weighed 141,92 grains corresponding to 95,79 grains from 100 of
calomel.
The liquor that had been filtered off was acidulated by nitric acid and nitrate of
silver added in excess ; the chloride of silver precipitated was collected and dried : it
weighed 44,44 grains corresponding to 30,0 from 100 of calomel; and the 30,0
grains of chloride of silver containing 7,401. But calomel consists in 100 parts of
Mercury 85,117
Chlorine 14,883
Therefore we have by this experiment, the black powder composed of
Mercury 85,117 and 88,85
Chlorine 7,482 7,76
Other matters $8,191 : 3,39
95,790 100,00
No. 2.—153,36 grains of calomel were boiled with water of ammonia for a few
minutes, and filtered. The dry dark-grey powder weighed 146,71 grains, correspond-
ing to 95,66 per cent.
The liquor treated with nitrate of silver gave 44,03 of chloride of silver, corre-
sponding to 28,71 of chloride per cent. and which contains 7,08 of chlorine.
Thus we obtain,
Mercury 85,117 or 88,98
Chlorine 7,803 8,15
Other matters 2,774.0 2,87
95,660 100,00
The mean of these experiments gives,
Mercury 88,91
Chlorine 7,95
Other matters 3,14
100,00
on the Haloid Compounds of Mercury. 443
B.—As the above method necessarily throws the chlorine and mercury estimate
rather too high, the following experiment was made, in which the necessary loss pro-
duces an opposite effect :
101,37 of the powder were boiled with strong muriatic acid, and an acid solution
of proto-chloride of tin added. The reduction of the quicksilver took place readily,
and large well-formed globules appeared; the metal collected and carefully dried,
weighed 89,39 grains, or 100 of the powder had given 88,18.
C.—51,42 of the grey powder were dissolved in dilute aqua regia, and a current of
sulphuretted hydrogen in excess passed through the liquor. It was found, that owing
to free chlorine, the sulphur precipitated invalidated the result. The whole was
therefore mixed with nitric acid, and boiled until the sulphuret of mercury was com-
pletely decomposed ; the liquor was then freed from the particles of pure sulphur and
evaporated until all free nitric acid and chlorine were completely dissipated. Being
then treated by sulphuretted hydrogen, it yielded a sulphuret, pure and jet black,
which collected and dried, weighed 52,39 grains, consisting of
Sulphur 7,19
Mercury 45,20
The 51,42 grains therefore contained 45,20 of mercury
or 100,00 - - - 87,90
In this experiment so much ammonia was lost by the treatment with nitric acid, that
its quantity could not be determined,
D.—As in none of these former analyses had the ammonia constituent been de-
termined, the following experiments were made for the purpose of ascertaining its
52,39
precise quantity :
Ist. 66,43 grains were boiled with an excess of solution of iodide of potassium,
and the flask being connected with a bent tube dipping into dilute muriatic acid, the
heat was kept up until all the ammonia and about half the water had passed over.
The liquor was then evaporated to dryness, and yielded a residue of 6,96 grs. of sal-
ammoniac, consisting of
Muriatic acid 4,73
Ammonia 2533
or 100 of powder gives 3,36 of ammonia.
The action of potash on the grey powder liberates ammonia likewise ; but it was
found so difficult to obtain complete decomposition that the method was abandoned.
Another process tried, consisted in repeatedly distilling strong muriatic acid off the
powder, in order to convert it into metallic mercury, corrosive sublimate, and sal-
ammoniac, and thus obtain a quantative result ; but this method also was found of so
imperfect action, that it could not be well applied.
444 Professor Kane on the Action of Ammonia
Summing up the results of the analyses above recorded, we have for 100 parts of
the powder :
Process. Mercury. Chlorine. Ammonia.
A 88,91 7:95
B 88,18
C 87,90
D 3,36
Or the mean result is—
Mercury 88,33
Chlorine 7,95
Ammonia 3,36
Loss, &c. 0,36
100,00
It is evident that we have here a body precisely corresponding to white precipitate ;
the mercury, however, being in proto-combination. Water of ammonia acting on
calomel, abstracts half the chlorine, which is replaced by a corresponding quantity of
ammonia in some form of combination. We can accordingly construct two formule
corresponding to white precipitate ; in the first half, the mercury being conserved as
protoxidized and combined with an atom of ammonia : in the second, that half of the
mercury being directly united to amidogene. The former theory gives from the formula
(Ch + Hg) + (Hg + NH’).
Mercury 87,00
Chlorine 7,59
Oxygen 1,73
E Ammonia 3,68
and 100 of calomel should yield 97,84 of product; whilst the second, from the
formula (Ch + Hg) + (NH’ + Hg) gives,
Mercury _— 88,72
Chlorine vast 100,00
Amidogene 3,54
and 100 of calomel should yield 95,95 of product, which is almost precisely the quan-
tity obtained in experiment.
We here find the evidence in favour of the existence of amidogene in combination
to be almost insuperable. I shall nevertheless retain all through this paper the two
methods of expression, until by examining the compounds of the other metals, the
differences may become so much larger, as to completely prevent their falling within
possible limits of error of observation.
on the Haloid Compounds of Mercury. 445
Section III.—Of the Action of Ammonia upon Peroxide of Mercury.
The accurate examination of the action of ammonia upon peroxide of mercury is
of very great importance, as the compound resulting ; the ammoniuret of mercury is
one of a very remarkable class of bodies, viz. the fulminating compounds containing
ammonia; and in addition, the experiments of Guibourt, the only chemist I believe
who has made analyses of it, would appear to demonstrate in it, a relation between
the number of atoms of ammonia and oxygen, which must influence the ammoniacal
theories to a very great extent. These circumstances made me trace out the proper-
ties of this body with more exactness than should have been otherwise required.
I have not been able to prepare a substance possessing the external characters of
the ammoniuret of mercury described by Fourcroy and Thenard. I have varied in
every manner I could imagine, the method of obtaining it ; but, although I got asub-
stance constantly the same in its properties and composition, it differed much in
appearance from that described by the French chemists. They state, that by digesting
liquid ammonia on red oxide of mercury during eight or ten days, the oxide gra-
dually covers itself with a yellowish-white powder which generally passes to a very
fine white. I have never obtained it of a pure white, but always with a tinge of
yellow, possessing an appearance and affording on analysis, results always the same.
The constancy of its properties justifies me, I should think, in considering it as pure,
notwithstanding its not exactly agreeing with their result. Unfortunately they did
not publish any quantative analysis of their product; the only one known to me is
that in Guibourt’s thesis.
In order to prepare ammoniuret of mercury, I precipitated a solution of sublimate
by potash, and the precipitate having been well washed from all excess of alcali, was
put into a bottle of water of ammonia and left for some days; its colour became
much lighter, but never completely white. Other portions of recently precipitated
peroxide were boiled in water of ammonia for a few minutes, until the colour ceased
to undergo any change: the reaction was very much accelerated by heat. ‘These dif-
ferent portions of product had all the same colour, and were indifferently, but with-
out mixture, used in the following examination without any difference of properties
becoming observable.
When this ammonuret is heated it gives off much ammonia, and azote ; a consider-
able quantity of water collects in the tube, and the matter remaining becomes dark-
red, like peroxide ; but if it be allowed to cool, it reassumes its whitish colour, and is
evidently still unaltered ammoniuret. ‘The reaction evidently does not consist in a
separation of the ammoniuret into ammonia and peroxide; but, from the commence-
ment to the termination, there are disengaged water, ammonia, azote, oxygen, and
446 Professor KANE on the Action of Ammonia
metallic mercury. The ammoniuret, like many other mercurial compounds, is dark-
red when hot, but of a whitish colour when cold. When a quantity of the ammo-
nuret is suddenly thrown on ignited coals it explodes very feebly, and far inferiorly to
fulminating gold with which its discoverers have compared it : it dissolves readily in
nitric or muriatic acid.
To analyse this compound, processes of a simple nature were sufficient.
A.—72,07 grains of ammoniuret were dissolved in muriatic acid, and the liquor
having been diluted was decomposed by sulphuretted hydrogen. The resulting sul-
phuret dried and weighed, amounted to '70,08 grains, consisting of
Sulphur 9,61
Mercury 60,47
The liquor and washings evaporated to dryness, gave sal-ammoniac, 9,21 grains,
consisting of
Muriatic acid 6,28
Ammonia 2,93
Hence, supposing the mercury to exist as peroxide, we have as the result of the
analysis :
Mercury 60,47)
Oxygen 4,78
Ammonia 2,93 12,07
Water and loss 3,89
or in one hundred parts—
Mercury 83,90
Oxygen 6,63
Ammonia 4,07
Water and loss 5,40
2.—The following analysis was made on a portion of ammoniuret prepared at a
different time and in another manner than that used in the former experiment.
67,57 grains were dissolved in muriatic acid and decomposed by a stream of sul-
phuretted hydrogen. The precipitated sulphuret weighed 65,37 grains, consisting of
Sulphur 8,96 :
Mercury 56,41 t 65,37
The liquor evaporated to dryness gave 8,15 grains of sal-ammoniac, consisting of
Muriatic acid 5,54
Ammonia 2,61
we have therefore the result—
Mercury 56,41
Oxygen 4,46
Ammonia 2.61 67,57
Water and loss 4,09
on the Haloid Compounds of Mercury. 44:7
or in 100 parts—
Mercury 83,48)
Oxygen 6,59
Ammonia 3,86
Water and loss 6,07 J
a result almost identical with the former.
B.—52,22 grains were dissolved in muriatic acid and decomposed by chloride of
tin. There were obtained 43,74 of mercury corresponding to 83,76 per cent.
C.—As the constancy of the amount of mercury and ammonia in the preceding
results, proved completely that the loss did not arise from error, but probably from
water present, the following experiment was made to ascertain whether water ex-
isted in such quantity: A small green-glass retort was blown, with a pretty long
neck ; to it was attached a tube containing potash ; and the ammoniuret in the retort
having been decomposed by a red heat, its gaseous elements were allowed to escape,
the mercury condensed in the neck of the retort and the water in the potash-tube ; the
result, though not absolutely true, is sufficiently accurate for the determination of the
point required.
Weighth of retort and material - - 75,38
Weight of retort - = = - 63,00
Ammoniuret used 12,38 grains
Weight of retort and mercury-residue - 73,35
Weight of retort - a = = 63,00
Mercury remaining 10,35
Weight of potash-tube before = - 278,28
Weight of potash-tube after = - 278,95
Water absorbed - 0,67
We thus obtain as results—
Mercury 10,35 = 83,62
Water 367 = 5,39
Gases and loss 1,36 = 10,99
But the gases consist of oxygen and ammonia, the former being such as to perox-
idize the mercury; and assuming the remainder to be ammonia without loss, we
have,
Mercury “6.60 |
Oxygen 6,60
renee 4,39 100,00
Water 5,39 J
These results summed up, give—
Process. Mercury. Oxygen. Ammonia. Water.
A, No.1. 83,90 6,63 4,07 5,40
— No.2. 83,48 6,59 3,86 6,07
B 83,76 6,60
€ 83,62 6,60 4,39 5,39
VOL. XVII. 41
448 Professor Kane on the Action of Ammonia, &c.
Giving a mean result of
Mercury 83,68 |
Oxygen 6,60
Ammonia 4,10 roe
Water 5,62
on abstracting the water, we have—
Mercury 88,67
Oxygen 6,99 + 100,00
Ammonia 4,34
The only analysis of this substance that I am aware of having been published, is that
of Guibourt, already quoted, and he considers it to be a compound of oxide of mer-
cury and ammonia in such proportion that the hydrogen of the ammonia could con-
vert the oxygen of the oxide of mercury into water, consequently his formula is the
following (3 Hg + 2NH®) and the per centage result :
Mercury 88,08
Oxygen 6,95 $ 100,00
Ammonia 4,97
with which my analyses may be considered as completely agreeing. In the abstracts
of Guibourt’s paper that I have seen, there is not any notice taken of the water
present ; but yet its constant value shows it to be a chemical ingredient, and we have
its atomic proportion, thus—
2Hg _ 405,6 _ 83,68 jy 83:68
SH wee Nin Soteati. 5:
The compound (3 He +2NH +4H) gives us in per cent. composition, the following :
Mercury 83,72)
Oxygen 6,60 |
Ammonia 4,72 pe
Water 4,96 J
a result agreeing very closely with that of experiment.
Admitting that the azotic element is engaged in the combination as amidogene,
and not as ammonia, the above formula converts itself into
(2 Hg +(2 NH?+ Hg) +6H)
a method of arrangement which we have already met with as an element of the yellow
powder, formed by water on white precipitate.
a
Further development of a method of observing the Dip and the Magnetic Intensity at
the same time, and with the same Instrument. By the Rev. Humpurey Lioyp,
M.A., F.R.S., M.R.LA. Fellow of Trinity College, and Professor of Natural
and Experimental Philosophy in the University of Dublin.
Read December 28, 1835.
Ow a former occasion I had the honour of submitting to the Academy a new me-
thod of observing, as applied to Terrestrial Magnetism, in which the dip and the
intensity of the magnetic force were determined with the same instrument, and by one
observation. As this method has fully realized the expectations which I ventured
at that time to entertain respecting it, I feel it my duty to enter somewhat more mi-
nutely into its details, and to explain the modifications which experience has led me
to adopt in the practice of it. The ordinary dipping needle is supported on an axle
which is supposed to pass precisely through its centre of gravity ; and, consequently,
the position which it assumes, when placed in the magnetic meridian, is the direction
of the magnetic force. But if one of the arms of the needle be loaded with a weight,
the needle will no longer rest in the line of the dip, but will assume a new position of
equilibrium under the combined influence of magnetism and gravity ;—the inclination
of the needle to the horizon being connected with the dip, the magnetic force, and the
moment of the added weight, by a very simple relation. This is the simple principle
of the method which has been already laid before the Academy. In order to apply
it, let us supose two small weights to be attached in succession to the southern arm of
the needle, at fixed distances from its centre ; and let the statical moments of these
weights be » and vy, and the corresponding inclinations of the needle to the horizon
~ and @; then, it has been shown* that
uw cos Z=go sin (6—Z), (1)
v cos 0=¢o sin (8—8) ; (2)
in which 6 denotes the dip, ¢ the earth’s magnetic force, and « a constant depending
on the distribution of magnetism in the needle itself. If therefore the angles Z and @
* Page 161.
450 Rey. Mr. Luoyp on a New Method of Observation
be observed in the usual manner, and if the ratio of the moments u and v be pre-
viously ascertained, these equations will give the dip, and the relative force, at the
several places of observation.
The degree of accuracy with which these elements are thus determined is, however,
not independent of the moments of the added weights ; and, for a given amount of
friction of the axle on its supports, the errors of the final results will vary with the
position of equilibrium of the needle. It has been already shown* from theoretical
considerations that the probable error in the determination of the dip, arising from
the friction of the axle, will be least when the needle is entirely unloaded, and of
course in the line of the dip ;—while the probable error in the determination of the
force is least, when the needle is at right angles to the same line. Hence the most
advantageous mode of applying the preceding method consists in observing the po-
sition of the needle,—first, when unloaded,—and, secondly, when loaded with a weight
sufficient to bring it into a position nearly perpendicular to the line of the dip.
It is obvious that if »= 0,¢=6; or the first of the observed inclinations becomes equal
to the dip, when there is no weight whatever acting with or against the directive force.
This condition, however, is never perfectly attained in practice. Owing to the want of
perfect coincidence of the centre of gravity of the needle with the axle, the weight of
the needle itself has a certain moment, which must deflect it from the true line of the
dip. But, as this deflexion is, in all cases, small, it will be convenient to consider the
angle Z as the approximate value of the dip, and to seek the correction necessary to
reduce it to its exact value. For this purpose, let equation (1) be divided by (2), and
let the ratio of the moments, “, be denoted by p; then
v
ceeGte omnis 4)
P cos8 ~ sin (S—6) *
Now making
d= Ute, . (3)
the 2d member of the preceding equation becomes aa 9) UP since «is avery
small quantity, and we haye
COS
cos St (¢—6). (4)
The dip is therefore determined by means of the two equations (3) and (4); and the
correction due to the want of perfect balance of the needle is inferred from the two
observed angles, without the reversal of the poles.
The value of the constant coefficient, p, in equation (4), will be given by the
formula
sin s= p
_ cos § sin (6—Z)
~ cos Z sin (8—0)
* Page 167.
applied to Terrestrial Magnetism. 451
when the corresponding values of the angles 8, ¢ and 0, are accurately known at some
one station. A series of cotemporaneous observations were made for this purpose, in
the Philosophy School of Trinity College, with a dipping needle of the ordinary form,
and with two needles which have been constructed for the observation of the dip and
the force by the the present method. The following are the results :
Need. I. Need. III. Need. IV.
70° 51.2 70m
19| 70 49.0 70 . 48-1 p=. 48.8
Mr Osh +) 7. bo |= fe oSl.Sy | 70 . 42 | a8 pl0.6
Mean | 70 50.5 | 71 34! — 7 169 | 70 4821-8 27
Substituting the mean results in the formula, we find
Needle III. log p=2. 06670, p= — .01166.
Needle IV. log p=3. 31175, p= + .00205.
When the value of p is small, as in the case of Needle IV, the variations in the
value of « arising from moderate changes in the angles on which it depends, are incon-
siderable ; and, accordingly, where the district over which the observations extend is
limited, the correction may be regarded as constant. Thus the computed difference
in the value of this correction at London and Dublin is only 0.’2 for Needle IV.
But as a very slight abrasion, or oxidation of the surface, will affect in a very sensi-
ble manner the position of equilibrium of the needle, it is probable that the correc-
tion will undergo some change in the course of time. ‘The present value of the cor-
rection of Needle IV in Dublin appears to be + 1’.5.*
The dip being known, the relative values of the magnetic intensity at different
stations will be given by equation (2). But before we can apply this equation, it is
necessary to examine, a little more particularly, the coefficients « and v which enter
it.
The former of these quantities is the statical moment of the free magnetism of the
needle, or the value of the integral fgrdm, taken throughout its extent,—dm being
the element of the mass, g the quantity of free magnetism which belongs to it, and 7
* It has been here assumed that all constant errors are removed by reversal in the mean of the various
readings taken with the ordinary needle, or that the deviations of the results from the absolute dip are
equally probable on the positive and on the negative side. This however, it has been elsewhere shown,
is not the case, and a correction seems to be required even in needles whose poles are reversed. This
correction in the case of Needle I appears to be +10’, so that the true correction of Needle IV in Dublin
is + 11.5
452 Rey. Mr. Lioyp on a New Method of Observation
its distance from the centre. Now this quantity (which we may denominate the
magnetic moment) varies with the temperature, and this variation must be taken into
account before we can make any accurate inference from the formula. Let 7 denote
the temperature of observation, 7’ a certain standard temperature, and o' the cor-
responding value of c. ‘Then, assuming the changes of the magnetic moment to be
proportional to the changes of temperature, we have
a=o' [ l—a(r—r) ]5 (9)
in which a is a constant whose value is to be determined by observation.
In order to obtain the value of this constant, the Needles III and IV were sus-
pended horizontally by a few filaments of silkworm’s thread, and vibrated in a large
glass bell, the air of which was heated from beneath by means of a spirit lamp. The
time of 100 vibrations was observed at the artificial temperature, and at the ordinary
temperature of the room before and after. ‘The following are the results :
Needle III. Needle IV.
Hour Time ‘Temp. Hour ‘Lime ‘Temp.
10" 28" 222.64 | 52°2 12° 28° | 238".00 | 58°2
10 46 222.50 58.5 12 49 238.00 58.5
1 42 222.80 63.5 3 39 238.45 58.9
Hg 27 222.84 63.8 3 50 238.56 58.6
Mean 222.70 61.0 Mean 238.25 58.5
LLORES 222.97 90.3 2 4 238.80 78.8
ES iN) 222.84 91.1 emis) 238.76 79.5
12 29 223.28 92.3 Mean 238.78 79.2
Mean 223.03 OND |
Hence we have for Needle Ill
A= 299070 sk — Ae Ol, SOee rt OO e
and substituting in the formula
—(so—«c) sy 2 (TT)
Hira) MG)
there isa= 00010. For Needle IV
{MSOs Oops MNS MSOs 4 7—7 —20°.7 ;
and a=.00021. But as these two needles were made at the same time by the same
artist, and are therefore probably similar in temper, as they are in material and size,
it is natural to suppose that the effects of temperature will be the same on both, and
that the difference here observed is due to the uncertainties attending observations of
this nature. Taking then the mean of the preceding results as the most probable
value of the coefficient for both needles, we have
a— .00016.,
a=
applied to Terrestrial Magnetism. 458
The quantity v in formula (2) is the sum (or difference) of the moments of the
weight of the needle and of the added weight. Accordingly, if m denote the mass
of the added weight, and a the distance of its point of application from the axle,
v-ph= mag;
or, since n=p v,
Now the force of gravity, g, varies with the latitude ; and the variation is expressed
by the formula
£=g (l1—e cos 2),
in which g" is the force at the latitude of 45°, \ the latitude of the place of ohsery-
ation, and e a constant whose numerical value is .002588. Accordingly, substituting
this value of g in the preceding expression, and employing the symbol v’ to denote
-, mag
the value of » corresponding to the latitude of 45°, or the quantity gyre have
v=v 1—e cos 2X). (6)
The equations (2) (5) (6) contain all that is requisite for the comparison of the
magnetic force at different places of the earth’s surface, and under different circum-
stances as to temperatnre. The expression for the force, obtained from them by sub-
stitution, is
v' cos 8 1—e cos 2d
ia o’sin(d—8@) * feweor) J
(7)
This expression is peculiarly adapted to logarithmic computation : for, since e cos 2d
and a (r—r) are very small fractions whose squares and higher powers may be ne-
glected,
log (l—e cos 21) = — Mecos2d,
log[1—a (r—7') ] = — Ma (7-7);
M being the modulus of the common system, whose numerical yalue is .43429. Hence,
if we take
A = log cos 6 — log sin (8—0@) + Ma (+—7') — Me cos 2X,
4 = log cos 0, — log sin(8,—6,) + Ma(z,—7) — Me cos 2X,, (8)
in which 8, 6, &c. denote the values of 8, 0, &c. at the place for which the force is
taken as unit, we have
log ¢= log v’—log o' + A,
O= log v —loga' + A;
454 Rev. Mr. Lroyp on a New Method of Observation
and, subtracting,
og o= A-A,, (9)
The last term in equations (8), or the correction for the variation of gravity, may
be omitted when the places at which the magnetic force is compared do not differ
considerably in latitude ; for in that case the quantity Me (cos 2,—cos 2X), which
enters the logarithmic formula for the force, may from its smallness be disregarded.
In applying this method of observation to the determination of the changes which
the intensity of the magnetic force undergoes at the same place, it will be convenient
to substitute an approximate formula for the preceding. To obtain this, we have
only to differentiate the equations (8) and (9), and we find
d
i —tan 6 dé—cot (3—6) (d3—d0) + adr.
in which we are to substitute for tan 0, cot (6—6), and a, their values belonging to
the particular place and needle. If the needle be loaded so that 5—6 is very nearly
equal to 90°, the term multiplied by cot (8—6) may be neglected, and we have
= — — tan 0 d0+<adr.
This seems to offer a very simple means of observing the diurnal and menstrual ya-
riations of the total intensity, and of ascertaining the law of a phenomenon of which
nothing certain is as yet known.
I shall now adduce, in exemplication of this method, the results which I have ob-
tained on the direction and intensity of the magnetic force in Dublin ; and shall com-
pare these results with those of the received method, the observations being made, for
the most part, at the same time. The needle employed (Needle IV) is one of those
made expressly for the practice of this method ; and care has been therefore taken
to preserve its magnetic state undisturbed. In each arm are drilled three small holes,
close to one another, coinciding, as accurately as could be effected, with the axis of
form of the needle, and distant from its centre by about two-thirds of the length of
the arm. The weight is a small piece of brass wire, which is introduced into one of
the holes on the southern arm,—the diameter of the wire corresponding accurately to
that of the hole; the magnitude of the weight is such as to bring the needle into a
position nearly perpendicular to the line of the dip. The length of the needle is 44
inches.
The following are the observations, and the results computed by the formule (3)
(8) and (9); they constitute three distinct comparisons of Dublin and London, the
first of which was made in the year 1834, and the other two in 1835.
applied to Terrestrial Magnetism. 455
I. August and September 1834.
Dublin |
London. Dublin.
t= 69° 7.4, c= +15 t= 70° 53.6, «= 41/5
s8= 69 8.9. € oO —/ Ooo
6=-—12 8.9, log cos = 9.99016 Oo Os log cos = 9.99579
8-—0= 81 17.8, logsin = 9.99497 | 8-@= 78 53.0, log sin = 9.99177
diff. = — .00481 diff. — + .00402
tr—rTr=+8°.7, corr, = +.00060 | r—7 = +3°.0, corr. = +.00021
A, = —.00421 A = +.00423
log g=A—A, =+.00844, ¢=1.0196.
IT. September 1835.
Place : : (8)
Dublin Sept. 4 1°56 71-8 70° 43.6 | — 12° 29.4
5 2519 65.5 70 52.8;—13 15.4
7 4 8 70.0 70 02.2} —13 20.8
15 i Os) 62.0 70 55.0}— 13 10.5
Mean 67.3 70 50.9|}—13 4.0
London Sept. 19 1 10 68.0 69 74|/—16 54.2
22 | 11 32 70.8 69 124|/—17 163
Preity tena 70.0 69 13.6|—16 49.4
Mean 69.6 (oo) | ET | i OX)
London. Dublin.
es 000g tell, a= +15 C=) 7 Ora50-95 e= F125
8= 69 12.6. o= 5 2/0 524.
6=-17 0.0, log cos = 9.98060 6=—-13 4.0, log cos = 9.98861
8—0=86 12.6, log sin = 9.99905 8—0=83 56.4, log sin = 9.99756
diff. = — .01845 diff. = —.00895
t—r=+ 9.°6, corr. = +.00066 r—-r=+7.° 38, corr. = +.00050
A, = —.01779 A = — .00845
log ¢= A—A,= +.00934, = 1.0217.
* The coefficient of r—r'in the correction, or the value of Ma, is .000069.
VOL. XVII. 4K
456 Rey. Mr. Luoyp on a New Method of Observation
III. October and November 1835.
Place Date Hour ‘Temp. (Z) ; (6)
London Oct. 23 Tess 50°5 69° 106 |— 16° 32/0
— See 51.6 69 2.2)}—16 35.4
24 1 16 53.8 69 6.0 |—16 44.4
Mean 52.0 69 63 |}—16 37.3
Dublin Noy. 5 i2> 522 56.2 70 49.6 |—12 54.6
— QO. f32 52.8 70 45.8 |—12 54.5
6 TE Pcie) 49.0 70 53.9 |—12 36.6
Mean 52.7 12 48.6
London, Dublin.
Z=) 60° 36.3) 6 ere t= 70° 40'.8, «= +15
Js Wolth ygick oa /0 51.3.
0=—16 37.3, log cos = 9.98146 6=—12 48.6, log cos = 9.98905
$-0= 85 45.1, log sin = 9.99881 8—0= 83 39.9, log sin = 9.99734
diff. = —.0O1735 diff. = — .00829
fess SEP OP corr. = —.00055 r—t=— 7. 3, corr. =— .00050
A, = -.01790 A = — .00879
log ¢=A—A,= +.00911, $= 1.0212.
In order to reduce these results to the same epoch, the values of the dip obtained
in the autumn of 1834 must be diminished by 3’, that being nearly the amount of the
annual decrease of dip in this part of the globe at the present time. We have then,
on summing up,
(8,) (8) (9)
I. 69° 5.9 70° 52.1 1.0196
II. 69 12.6 70 52.4 1.0217
69 7.8 70 51.3 1.0212
69 8.8 70 51.9 1.0208
We may now compare the preceding determinations of the Intensity with those
obtained by the received method. We have for this purpose three comparisons of the
horizontal part of the magnetic force at Dublin and London, made in the summer and
autumn of last year; the latter two having been cotemporaneous with (II) and (III)
of the preceding table. The results of these comparisons are given in the annexed
table, which contains 1. the place, and 2. the date of the observation ; 3. the name of
needle employed ; 4. the number of observations ; 5. the mean time of 100 vibrations,
corrected for temperature and for the rate of the chronometer; and 6. the computed
ratio of the horizontal intensities. The observations were made in the manner adopted
by Professor Hansteen, viz. by allowing the needle to make 360 vibrations com-
Da,
arp,
applied to Terrestrial Magnetism. 457
mencing with the arc of 20°, and noting the time of completion of every 10th vibra-
tion during the interval by a chronometer. The needles are short cylinders, 24
inches long, and .13 of an inch in diameter; such being the form and size recom-
mended by the same skilful and indefatigable observer.
lace Date Cyl. | No. | ‘Lime Intensity
London | July 8-19 | R(c)| 19 | 441."59 ;
Aug. 30,31 6 | 441.46
Mean 25 441.53 1.0000
454.06 9456
438.57
* Dublin | Aug. 16 — | 38
) s
6 4:39.07
14
3
London | July 19,20| R (d)
Aug. 28,29 | ——
Mean —— 438.82 1.0000
Dublin | Aug. 14. | —— 452.11 9421
London | Sept.19-22; L(1)| 3 | 235.98 1.0000
Dublin 12-15) —— | 4] 243.52 .9390
CII) | London | Sept.19-22) L(2)| 3 | 284.01 1.0000
Dublin 12-15, —— | 3] 293.31 .9376
London | Oct. 23,24} L(1)| 3 | 235.42 1.0000
London | Oct. 23,24) L(2)| 4 | 283.11 1.0000
Dublin | Nov. 5,6 | ——| 3] 293.27 .9319
The mean of these three determinations gives the horizontal component of the
magnetic force in Dublin equal to .9380, that in London being unity. To deduce
from this the ratio of the total force at the two places, we must know the amount
of the dip at each. The following are the results of the observations made with the
needles (1) and (IV), in the autumn of 1835. In taking the mean, only half the
weight has been allowed to each observation with the latter,—the number of readings
from which the dip is deduced in the statical method being half of that taken in the
ordinary process.
* Place | Date Needle No. Dip
Dublin Sept. 4-15 I 6 70° 53.5
IV 6 70 52.0
Mean 12 70 53.0
London Sept. 19-25 I 9 69 6.3
Sept.—Oct. IV a 69 9.8
Mean 16 HOF 725
* The observations of comparison (I) were made by Captain James Ross.
458 Rey. Mr. Luoyp on a New Method of Observation
Hence, if » denote, as before, the total intensity of the magnetic force at Dublin,
that at London being unity, we have
om
$= .9380 x ee = 1.0208.*
The mean results of the two methods, then, agree ina very remarkable manner ; the
agreement extending to the fourth place of decimals inclusive. But the differences
between the partial results and the mean (by which we are accustomed to judge of the
value of observations) are very different in the two cases. The greatest of these dif-
ferences, in the method which forms the subject of this paper, is only .0012; while the
greatest difference, in the three comparisons of the horizontal intensity, amounts to
.0061, and the corresponding difference in the value of the total force is .0066.
The difference .006, though it does not appear to be greater than that commonly
met with in different comparisons of the horizontal force at two places, is yet much
beyond the limits of the errors of observation ; and, to account for it, we must sup-
pose the horizontal force to have varied at one or both of the places of observation.
The existence of such variations seems to be well established. Besides the regular pe-
riodical changes dependent on the hour and on the season, the horizontal force appears
to be liable also to accidental fluctuations, or irregular oscillations round its mean
state ; and the variations of the latter kind (like those of the barometer in our cli-
mates) are probably more considerable than those that are periodic and regular. These
variations are, in all probability, the effects of changes both in the intensity and direc-
tion of the magnetic force; but the latter appear to be (in these high magnetic lati-
tudes) the predominating cause. The relation which subsists among these changes is
obtained by differentiating, the equation h—@ cos 8, considering h, ¢, and 8, as all
variable ; dividing the result by the equation itself, we find
. = = @ — tan 8 sin Vd;
the change of dip, dé, being expressed in minutes. When the dip is 71°, the last term
of this equation becomes — .00084 x d8; so that considering the change of dip as the
sole cause of the effect observed, a variation of .006 in the amount of the forizontal
force will be produced by a variation of 7' in the dip ; and this is, probably, within the
limits of the irregular changes to which that element is subject.
* The values of the dip employed in the preceding calculation are the apparent values, reduced to
Needle I, as the standard. When the correction due to the latter needle (see note, p. 451) is applied,
there will be a small alteration, amounting to +.0007, in the computed value of the relative intensity.
applied to Terrestrial Magnetism. 459
If the preceding views be correct, it will follow that, where the dip is considerable,
we cannot hope to determine with any accuracy the relative values of the total in-
tensity, by the observation of its horizontal component, unless the dip be observed at
the same time at each place. Still, however, the final values of the intensity will be
affected, to a large amount, by the errors of observation to which the results obtained
with the dipping needle are liable. The chief of these errors is that due to the fric-
tion of the axle on its supports ; and it has been already shown* that if the amount of
that error in the natural position of the needle be denoted (in parts of radius) by e,
the induced error in the determination of the force by the receiyed method, will be
e tan 0;
while the error, arising from the same cause, in the needle, is
e
; sin (6—@) ~
Accordingly, when 6—6=90°, or when the position of the needle when loaded is
perpendicular to the line of the dip, the error in the determination of the force is
reduced to e ; and is less than the corresponding error in the common method in the
ratio of unity to the tangent of the dip.
* p. 167—8.
VOL. XVII. 41
On the Laws of the Double Refraction of Quartz. By James MacCutvacu,
Fellow of Trinity College, Dublin.
Read February 22, 1836.
The singular laws of the double refraction of quartz, which have been discovered
by the successive researches of Arago, Biot, Fresnel, and Airy, are known merely as
so many independent facts ; they have not been connected by a theory of any kind.
I propose, therefore, to show how these laws may be explained hypothetically, by
introducing differential coefficients of the third order into the equations of vibratory
motion.
Suppose a plane wave of light to be propagated within a crystal of quartz. Let the
coordinates x, y, z, of a vibrating molecule be rectangular, and take the axis of z per-
pendicular to the plane of the wave, and the axis of y perpendicular to the axis of the
crystal. Let us admit that the vibrations are accurately in the plane of the wave, and
of course parallel to the plane of wy. Then, using § and 7 to denote, at any time ¢,
the displacements parallel to the axes of x and y respectively, we shall assume the two
following equations for explaining the laws of quartz :—
a ae a
ene Soden a1)
dn <_ d°n BE f
ee ae ame (2)
The peculiar properties of this crystal depend on the constant C. When C=0, the
third differentials disappear, and the equations are reduced to the ordinary form, in
which state they ought to agree with the common equations for uniaxal crystals.
Hence, putting a for the reciprocal of the ordinary index, b for the reciprocal of the
extraordinary, and ¢ for the angle made by the axis of z with the axis of the crystal,
we must have
A=a, « B=a’—(a’—2’) sin’ ¢, - (3.)
supposing the velocity of propagation in air to be unity.
462 Mr. MacCuttacu on the Laws of
Now, from the nature of equations (1.) and (2.), the vibrations must be elliptical.
In fact, if we put
b=p cos} Zist-2b, n= sin} ot-2)t, (4.)
where p, 9, s, J, are constant quantities, the differential equations will be satisfied by
assigning proper values to s and to the ratio 1. For, after substituting in equations (1.)
and (2.) the values of the partial differential coefficients obtained by differentiating
formule (4.), we shall find that every term of each equation will have the same sine or
cosine for a factor ; omitting, therefore, the common factors, and making ies we
shall get the two following equations of condition : ;
F=A— ae Ck, (5 )
2 27, C
s=B-—"y-. (6.)
Subtracting these, we have
27C / 1
i Bt (= ~k) =0, (7.)
which, by formule (3.), becomes
U :
ke — al a*—0* ) sin’ 4. ha (8.)
Let us now interpret these results. It is obvious, from formule (4.), that s is the
velocity of propagation for a wave whose length is /, and that each vibrating molecule
describes a little ellipse whose semiaxes p and q are parallel to the directions of w and y.
But the number k, expressing the ratio of the semiaxes, has two values, one of which
is the negative reciprocal of the other, as appears by equation (8.); and each value of
Kk has a corresponding value of s determined by equation (5.) or (6.) Hence there
will be two waves elliptically polarized, and moving with different velocities, the ratio
of the axes being the same in both ellipses ; but the greater axis of the one will coin-
cide with the less axis of the other. The difference of sign in the two values of &,
shows that if the vibration be from left to right in one wave, it will be from right to
left in the other. These laws were discovered by Mr. Airy.
The law by which the ellipticity of the vibrations depends on the inclination ¢,
and on the colour of the light, is contained in equation (8.). The value of the
constant C will be determined presently. In the mean time we may observe, that
C denotes a line, whose length is very small, compared with the length of a wave.
a
the Double Refraction of Quartz. 463
When ¢=0, the light passes along the axis of the crystal. In this case we have
"je =1, and k= +1; which shows that there are two rays, circularly polarized in oppo-
site directions. The value of s for each ray may be had from equation (5.) or (6.),
by putting +1 and —1 successively for &. Calling these values s’ and s”, we find
‘ C ' aC
s*=a'—2r7, YSe ere (9.)
C aC
2 9 SP Bay en .
s=a+2n—, s"=a (1 ay) (10.)
Suppose a plate of quartz to have two parallel faces perpendicular to the axis, and
conceive a ray of light, polarized in a given plane, to fall perpendicularly on it. The
incident rectilinear vibration may be resolved into two opposite circular vibrations,
which will pass through the crystal with different velocities ; and which, after their
emergence into air, will again compound a rectilinear vibration, whose direction wil]
make a certain angle p with that of the incident vibration : so that the plane of polari-
zation will appear to have been turned round through an angle equal to p, called the
- angle of rotation. This angle may be determined by means of the preceding formule.
Putting @ for the thickness of the crystalline plate, the circularly polarized wave whose
velocity is s’, will pass through it in the time
68 ( aC ) :
al
and the wave whose velocity is s", in the time
—
s a
ees salt) aC
7 ( ead ).
Therefore, if ° be the difference of the times, we have
QrCd
P s= =T" (11.)
But, since the velocity of propagation in air is supposed to be unity, the time and
the space described are represented by the same quantity ; and therefore 0, which is
evidently a line, denotes the distance between the fronts of the two circularly polarized
waves, when they emerge into air. ‘The waves being at this distance from each other,
if we conceive, at the same depth in each of them, a molecule performing its circular
vibration, and carrying a radius of its circle along with it, the two radii will revolve
in contrary directions, and will always cross each other in a position parallel to the
incident rectilinear vibration. Now let two series of such waves be superposed, so as
to agitate every molecule by their compound effect, and it is evident, that, when
the radius vector of one component vibration attains the position just mentioned, the
radius vector of the other will be separated from it by an angle equal to =, where
VOL. XVII. es 4M :
me 4y/ |
4 L4 a
464 Mr. MacCutracu on the Laws of
\ is the length of a wave in air. The resultant rectilinear vibration will bisect this
; j 8 «Wie
angle; and therefore p, the angle of rotation, will be equal to = Hence, substituting
for 8 its value, and observing that /, the length of a wave in quartz, is equal to a,
we find
2n°CO
rd Seat 12.
a‘? Med
which gives the experimental law of M. Biot, that the angle of rotation is directly as
the thickness of the crystal, and inversely as the square of the length of a wave for any
particular colour. By changing the sign of C, we should have an equal rotation in
the opposite direction. And here we may remark, that C may be made negative in
all the preceding equations, its magnitude remaining. There are two kinds of quartz,
the right-handed and left-handed, distinguished by the sign of C.
The angle of rotation, for a given colour and thickness, is known from M. Biot’s
experiments. We can therefore find the value of C by means of the last formula ;
and substituting this value in equation (8.), we shall be able to compute & when ¢ and
lare given. Now it happens that Mr. Airy *, by a very ingenious method of observ-
ation, has determined the values of & in red light for two different values of ¢; and
of course we must compare these observed values of & with the independent results of
theory. As Mr. Airy’s experiments were made upon red light, we shall select, for
the object of our calculations, the red ray which is marked by the letter C in the
spectrum of Fraunhofer. For this ray, Fraunhofer has given the length \, which,
- expressed in parts of an English inch, is equal to .0000258; and M. Rudberg has
found a= .64859, b= .64481. Moreover, from the experiments of M. Biot, we may
collect, that the arc of rotation, produced by the thickness of a millimetre, is some-
thing more than 19 degrees for the ray we have chosen; so that the fraction } may
be taken to express nearly the length of that arc in a circle whose radius is unity.
We have, therefore, 8=.03937 inch, and p=.333, Substituting these values in the
formula
derived from (12.), we find
l
from which it appears that C is about twenty thousand times less than the millionth
part of an inch.
* Transactions of the Cambridge Philosophical Society, vol. iv. p. 205.
——
the Double Refraction of Quartz. 465
Again, since a?—b’=.00489, we have
Lipa
——_ (a? —b”) = 258,
ae
so that equation (8.) becomes
k? — 258 sin’ ¢. k=1. (13.)
The results of this formula are compared with Mr. Airy’s experiments in the fol-
lowing table, in which the less root is taken for /, and its sign is neglected.
Values k. |
Values of ¢. -
Observed. “| Galcalated. |
6° 15’ | tan 16° 38'=.2987 | .2980
8° 54 tan 8° 56 =.1572 1579
The angles 4, in the first column, are deduced from the observed inclinations of the
rays in air to the axis of the crystal; and as &/ was observed to be somewhat different
for the ordinary and extraordinary rays, its mean values are given in the second
column. ‘The exact coincidence between these and the calculated values is, perhaps,
in some degree accidental ; but a less perfect agreement would be sufficient to confirm
the theory.
The magnitude of k varies considerably with the colour of the light, increasing
from the red to the violet, while the coefficient of sin? ¢. k, in formula (18.) diminishes.
If we take the violet ray H, for example, this coefficient will be about 159. But
it would be useless to make any more calculations, as we haye no experiments with
which they might be compared.
The figure of the wave surface yet remains to be examined.
Eliminating k between formulz (5.) and (6.), we obtain the equation
(#— A)(—B) =4n? <, 14.)
which expresses the nature of the surface, s being a perpendicular from the origin on
a tangent plane. From this equation it follows that the two values of s can never
become equal in quartz, as they do in other crystals ; for if we solve the equation for
s’, and put the radical equal to zero, we shall get the condition
(A-B)'+160"S =0,
which cannot be fulfilled, since the quantity which ought to vanish is the sum of two
squares. The two sheets, or nappes, of the wave surface, are therefore absolutely
separated.
4.66 Mr. MacCutiacu on the Laws of
It is commonly assumed that one of the rays is refracted according to the ordinary
law; but this is not the case, since neither of the values of s is constant. However,
the ray which has the greater velocity, (a being greater than b,) may still, for conve-
nience, be called the ordinary ray. Of the two roots of equation (8), the one &,,
whose numerical value (supposing @ not to vanish) is less than unity, corresponds to
this ray. When C is positive, &, is negative ; and when C is negative, i, is positive :
therefore in both kinds of quartz, by formuleze (5) and (6), we have s2> A, and s2<B;
denoting by s, and s. the respective velocities of propagation of the ordinary and ex-
traordinary waves. Hence, if we conceive a sphere of the radius a, with its centre
at the origin, and a concentric prolate spheroid, whose semiaxis of revolution is also
equal to a, and parallel to the axis of the crystal, while the radius of its equator is
equal to b, the ordinary nappe of the wave surface will fall entirely without the sphere,
_and the extraordinary nappe entirely within the spheroid, whether the crystal be right-
handed or left-handed. With respect to the little ellipse in which the vibrations are
performed, and of which the semiaxes parallel to x and y are represented by p and q
respectively, it is evident that p>q for the ordinary wave, since k,<1 ; and that p<q
for the extraordinary wave. When C vanishes, the minor axis of each ellipse also
vanishes, and the rays become plane-polarized, the ordinary vibrations being then pa-
rallel to the direction of x, and the extraordinary parallel to that of y. This is ex-
actly what ought to happen on the supposition that the vibrations of a plane-polarized
ray* are parallel to its plane of polarization ; a supposition which was kept in view
in framing the fundamental equations (1.) and (2.).
To show, with precision, how the two kinds of quartz are to be distinguished by
the sign of C, we must give definite directions to the axes of coordinates. To this
end, let us imagine the plane of xy to be horizontal, and a circle to be described in
it with the origin O for its centre ; and let the north, east, and south points of this
circle be marked respectively with the letters N, E, S. Let the direction of +z be
eastward, from O to E; that of +y northward, from O to N; and that of +2 ver-
tically downwards; the progress of the light through the crystal being also down-
wards, and the plane of the wave moving parallel, as before, to the plane of zy.
Then the crystal will be right handed or left handed, according as C is positive or
* On this point there are two very different opinions. Fresnel supposed, as is well known, that the
vibrations of a plane-polarized ray are perpendicular to its plane of polarization; whereas, according to M.
Cauchy, whom I have followed, they are parallel to that plane. I am induced to adopt the latter sup-
position, because I have succeeded, by means of hypotheses which are grounded on it, in discovering the
laws of reflexion from crystallized surfaces ; laws which include, as a particular case, those discovered by
Fresnel for ordinary media. The hypotheses alluded to, along with some of their results, are published
in the London and Edinburgh Philosophical Magazine, vol. viii. p. 103, in a letter to Sir David Brewster.
See also vol. vii. p. 295, of the same Journal. I hope soon to offer the Academy a detailed account of my
researches on this subject.
ee
iia
cao
the Double Refraction of Quartz. 4.67
negative. For, if C be positive, &, will be negative, and formule (4) will become,
by exhibiting the sign of k,,
=p cos} Fiet—2yh, n= —k, psin§ F(st-)t, (15.)
for the ordinary vibration ; and
7)
b= F0 cos f Fst —2)t, n=q sin} Fost—2) b, (16.)
- : : : 2 :
for the extraordinary vibration. Now if we suppose the are 7 (st =z) either to va-
nish, or to be a multiple of the circumference, the molecule will be at the east point
of its vibration; and upon increasing the time a little, the value of » will become
negative in (15.), and positive in (16.),so that the movement will be towards the south
in the first case, and towards the north in the second. Therefore, when C is positive,
the ordinary vibration takes place in the direction NES, or from left to right, and the
extraordinary in the direction SEN, or from right to left, supposing a spectator to
look in the direction of the progress of the light. It may be shown, in like manner,
that, when C is negative, the ordinary and extraordinary vibrations are in the direc-
tions SEN and NES, or from right to left and from left to right respectively. Now
if a plane-polarized ray be transmitted along the axis of the crystal, the plane of po-
larization will be turned in the direction of the ordinary vibration, because this vibra-
tion, being propagated more quickly, will be in advance of the other, upon emerging
from the crystal. Hence, the rotation is from left to right when C is positive, and
from right to left when C is negative; and the crystal is called right-handed in the
first case, and left-handed in the second.
We have all along supposed that C is a constant quantity, and the agreement of our
results with experiment proves that this supposition is at least very nearly true in the
neighbourhood of the axis. It is probable, however, not only that C varies with 9,
but that it becomes different in equations (1.) and (2.) ; that is to say, it is probable
that the following equations
PE —Aavé +C dn
d@ dz dz (17)
dn —-péa oc @é UG
de dz de
in which C’ is a little different from C, would be more correct than those which we
have assumed. Indeed Mr. Airy’s experiments seem to indicate that C’ is greater
than C; for he found, as we haye already said, that the ratio of the axes of the little
VOL. XVII. 40N
468 Mr. MacCuttacu on the Laws of
ellipse described by a vibrating molecule is somewhat different for the two rays,
being more nearly a ratio of equality for the ordinary than for the extraordinary ray.
Now if we set out from equations (17.), instead of (1.) and (2.), and proceed in all
respects as before, we shall arrive at the formula
altar ae Cc’
ke — sal @—0 )sin o k=% (18.)
instead of formula (8.). The quantity = will be greater than unity, if C’ be greater
than C, and the value of &, will be greater than before. This seems to be the expla-
nation of the difference between the ratios observed by Mr. Airy.
It may be proper to state briefly the the considerations which led to the foregoing
theory. Beginning with the simple case of a ray passing along the axis, the first thing
to be explained was the law of M. Biot, that the angle of rotation varies inversely as
the square of / or of \. Now it was remarked by Fresnel, who first resolved the
phenomena of rotation into the interference of two circularly polarized waves, that
the interval § between these waves, at their emergence from the crystal, must be in-
versely as J, if the angle of rotation be inversely as the square of /. This remark sug-
gested* to me the idea of adding, to the equations of the common theory, terms con-
taining the third differential coefficients of the displacements ; for it was evident that
such additional terms would give, in the value of s*, a part inversely proportional to /.
It was also evident, that the third differential coefficient of € should be combined with
the second differential coefficients of », and the third of » with the second of &, in
order that, after substitutions such as we have indicated in deducing formule (5.)
and (6.), the sines or cosines might disappear by division, and that thus the value
of s’ might be independent of the time, as it ought to be. This kind of reasoning
led me to assume the equations
Ce ao Ce aN dn
— = —" or — 9
dt? dz dz
dn =, dn D ae
dt? dz” dz
(19.)
, (20.)
for the case of a ray passing along the axis of quartz ; and then, substituting in these
equations the values of the differential coefficients obtained by differentiating the
formule
ep cos} Fost—2) b, n= + psin f2r(s-2)h,
*«<The singular relation between the interval of retardation [6] and the length of the wave [/] seems
to afford the only clue to the unravelling of this difficulty."—Report on Physical Optics, by Professor
Lloyd; (Fourth Report of the British Association, p. 409). It was in reading this Report, that
Fresnel’s remark, about the relation between 6 and J, first came to my knowledge.
the Double Refraction of Quartz. 469
which express a circular vibration, (from right to left, or from left to right, according
to the sign of the second p,) the result was
S=a= +c
from (19.), and
es=a'+ 2D
from (20.); which showed that D-=- —C, since the values of s, corresponding to the
same circular vibration, ought to be equal. The transition from this simple case to that
of a ray inclined at a given angle ¢ to the axis, was easily made, by taking into ac-
count the doubly refracting structure of the crystal. This was done by supposing
€ and » parallel to the principal directions in the plane of the wave, and by chang-
ing a’, in equation (20.), into a’—(a*—b*) sin *¢; and thus the fundamental equa-
tions (1.) and (2.) were obtained.
“
>
dow; dba ws Foire “onli ahd iger : me
’ Wy ees eared ess am (a
i |
%, re 4d
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Bobs: rai es
hf
Raa} vad ah
i a fal ay
“
. lo on
; U ligt
+
yy at
a it
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eh po adres 2h ham eit MBL ae Ee tee eas eet 2)
ge {it pete 9 ge “ams Sent
Lig godiieeirs one: to
ee a May ay hatieg “
fe. te! , SF Me
f 4 Lo ’ Mag. * ih.”
pewitien. eiw sige
el ee a‘ < << uae y =f
i> he +b
‘ f = oo
2 r
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if :
ITAA i
i Para Ss co i
PROFESSOR STRVRLLY'S.
Set Registering Barometer
DESCRIPTION OF THE ENGRAVING
The Barometer tube tirmly tived in its place dipping at bottom into
the listen supported by
the Frame resting upon
the Stem or Pilar of
the Hydrometer floating a the tlaid im
a Vessel firmly tired in tts place
the Pncal attached w the trame F shewing the height of the Barometer
upon the ruled sheet of paper at 9% A.M.
1 2 3 rd 5 6 7
.
a
| |
ie era
<i
An Investigation of the Principles upon which a new Self: Registering Barometer
may be constructed. By Joun Srevetty, Esq. Professor of Natural Philosophy
in the Belfast Institution.
Read Noy. 30, 1835.
My attention having been of late much directed to subjects connected with Me-
teorology, I could scarcely fail to remark how much time was consumed in making
and recording observations, and how very limited, as to their extent, were our best
observations, when considered in relation to the multiplicity of changes which are
taking place at times when no person is present to observe or record. It is, there-
fore, an object of much interest to endeavour to construct instruments of every
description, as far as is practicable, in such a way as that they may register their own
indications, not only at stated hours of the day, but if possible at all times, and may
place as clearly as possible the general results before the mind.
The barometer and the rain gauge are unquestionably two very important instru-
ments in this science. The following’ observations will, I trust, clearly show how a
self-registering barometer may be constructed ; and a method of constructing a rain
gauge capable of registering its own indications, may be so readily deduced from this,
that it would be a waste of time to allude to it more particularly.
It is very obvious that if a pencil can be caused to rise and fall through equal dis-
tances, in such a way as to correspond with equal elevations and depressions of the
barometer, this pencil may be made to press against a sheet of paper divided by
twenty-four equidistant vertical lines, which would represent the twenty-four hours
of the day; while the sheet may be carried by clock-work across the pencil laterally
or in a direction at right angles to the vertical lines. The pencil would thus trace
out on the sheet of paper, a curve which would present to the eye a correct view of
the actual oscillations of the barometer during the several parts of that day; the
height at each hour, or intervening portion of time, being readily and distinctly trace-
able. Avsecond portion of this same sheet, or a second sheet, may then have the
curve for the next day traced upon it similarly ; and so on from day to day; these
sheets being then dated, and arranged in consecutive order, would afford a correct re-
gister of the barometer, and thus we should haye within our reach a means of ascer-
VOL. XVII. 40
472 Professor STEVELLY on a new
taining the height at which it had ranged at any instant of past time that our re-
searches may render desirable.
But farther, the same sheet of paper being used on many successive days, for a
great length of time, it is obyious that the points at which the point of the pencil
arrived oftenest would become darker than those points at which it was only occa-
sionally found at the same hour, and therefore the succession of these darker points
would at length trace out the true curve of mean diurnal oscillation. In this case,
and perhaps indeed always, it would be best to have the sheet of paper strained upon
the surface of a cylinder, which the clock-work should cause to turn round once in
twenty-four hours. In this way it is obvious that nearly all the labour and tedium of
observing, recording, and afterwards reducing actual observations to mean results,
may be avoided.
Most, if not all of these advantages, I conceive, may be attained by the use of an
instrument, one modification of which I had the honour of hastily describing to the
subsection of useful arts at the late meeting of the British Association in Dublin.
I shall now give a more full and a more general description of the instrument,
together with the formule, which will in all ordinary circumstances of variety of
construction, give the connection of its scale with the scale of the common
barometer. I shall also briefly point out the manner in which its indications are af-
fected by changes of temperature, at least so far as may be required for making the ne-
cessary corrections.
During the oscillations of the common barometer, when it falls, a certain quantity
of mercury is added to that already in the cistern, which of course increases its weight
by so much. On the contrary, when the barometer rises, mercury retires from the
cistern into the tube ; the cistern thus becoming by so much lighter than it was before.
If then the tube of the barometer be firmly sustained in its place, but the cistern be
suspended by any mechanical means, so as to descend by arithmetical distances for
equal additions to its weight, and to rise similarly when its weight has been similarly
diminished ; an index carried by the cistern may be made either to point to a fixed
scale placed beside the instrument, as in the common barometer, or to mark on a
sheet of paper a variety of positions corresponding with the synchronous variations
of the height of the barometer, while, as will appear just now, the range of the scale
of this new barometer may be made to bear any proportion, that may be desired, to
the three-inch scale of the common barometer.
Although various mechanical means of suspending the cistern may be readily de-
vised, either on the principle of the lever, or by certain curved surfaces made to turn
freely on an axis, or even by simple counterpoise, the alteration of the buoyant force
of a metallic cylinder as it is drawn more or less out of a fluid being made the means
of restoring the equilibrium, yet for reasons which J shall not now stop to detail, I
should prefer to any of these, the method of suspending it, derived from the buoyant
Seif- Registering Barometer. 473
force of a hydrometer ; and I also conceive that a hydrometer floating in mercury is
to be preferred, although almost any other fluid may be made to answer. ‘This me-
thod of suspending, or, to speak more correctly, of weighing the cistern, was suggested
to me by having long since observed that a most simple, cheap, and sensible instru-
ment for weighing, may be constructed on the hydrometrical principle, by using an
index, or the wire of a microscope attached to the vessel in which the hydrometer
floats, for noting the position of the mark upon the stem of the hydrometer, instead
of using the surface of the fluid for that purpose, as is usually done.
The following description of a barometer constructed upon these principles will, I
trust, be readily understood.
Let B be a barometer tube, which may be of iron, turned accurately cylindrical,
internally, where the upper surface of the mercury rises and falls , externally, where it
dips into the cistern ; its shape and dimensions, in other respects, is of no consequence.
This tube must be firmly fixed in its place. Let C be the cistern which should be
cylindrical near the surface of the mercury, suspended by the frame F' from the cylin-
drical pillar or stem S of a hydrometer H, which floats in a vessel 4 cylindrical near-
its upper part, firmly fixed in its place, and filled to a proper height with mercury, oil,
water, or any other proper fluid ; it is now obvious, that if the barometer should fall,
mercury would descend from B into C, add to the weight of C, and cause it to sink
until the stem S should have displaced as much additional fluid as would be equal in
weight to the weight added to C’; on the contrary, if the barometer should rise, mer-
cury would ascend from C into B, and the hydrometer would rise as far as was
necessary for re-establishing the equilibrium by the emerging of part of S.A scale,
it is also obvious, may be placed beside an index attached to any moveable part of
instrument, which scale shall bear a certain proportion to the scale of the common ba-
rometer. Let us now endeavour to discover what that proportion is.
Let h denote the height of the common barometer at any instant (say when it
stands at 30 inches).
Let sh denote a change in that height caused by a change of atmospheric pressure,
Let 8h’ denote the vertical alteration of level of the surface of the mercury in the
tube B, caused by that change.
Let dh” denote the vertical descent or ascent of the cistern and hydrometer in con-
sequence of that change; it is then obvious that eh” is the part of the scale of the
new barometer which corresponds with éh upon the scale of the common barometer.
Let sh’ denote the rise or fall of the surface of the mercury in the cistern.
Let 8° denote the rise or fall of the surface of the mercury in the vessel 4, in
consequence of the sinking or rising of the stem of the hydrometer.
Let s denote the internal cross section of the tube B at its upper part; s’ the
external cross section of B where it dips into the cistern ; s” the excess of the cross
section of the cistern, at the surface of the mercury, above s’ ; s’” the cross section of
AT 4 Professor STEVELLY on a@ new
the pillar or stem of the hydrometer; s* the excess of the cross section of the vessel
A, at the surface of the fluid it contains, above s’’. All these quantities are sup-
posed to have one inch for their linear unit. Let w denote the weight of one cubic
inch of mercury, and if the hydrometer float in any other fluid than mercury, let w’
denote the weight of one cubic inch of that other fluid.
Then 8h” —dh’” is the part of the lower portion of the tube B which emerges
from the mercury in the cistern, or becomes more immersed in it, in consequence
of the oscillations of the instrument. And 6h’+ 0h" is the portion of the stem or
pillar of the hydrometer which becomes immersed in the fluid in the vessel 4, or
emerges from it in consequence of the same oscillation.
Again dh.’ s. w—(sh —oh”) s. w is the alteration of weight of the cistern, and
(8h’ + 8h") s.” w' is the alteration of the buoyant force of the hydrometer, and that
the equilibrium may continue these must be equal ; hence
m ww
dh.'s —(8h" —8h'") s' = (8h + 8h ) s. (D)
Also because the perpendicular distance of the surface of the mercury in the tube B
from the surface of that in the cistern C is at all times equal to the height of the
common barometer, therefore
oh = bh! — bh" + oh'" . (1)
And from considering the cause of the rising or falling of the surface of the mercury
in the cistern, it will appear that .
dh'.s — (8h —8h'") s'=8h" 5", hence
oh's—oh''s!
Ai = fo)
oh a (2)
and by elimination from (1) and (2) we get
_ 8h C's!) —$h"' (s!—s') +8's—8h''s!
oh _ ee ence
cS ‘
yf BAG SD TEMS” (3)
ae co
and by substitution of this in (2) we get -
onl! — dh s+ 6h's—dh's (4)
s+s'—s
Also by considering the cause of the rising or falling of the surface of the fluid in
the vessel 4 it will be seen that
(8h" + 8h’) s\’ =8h.” 8’.) Hence—
3h! sl"
SY +3"
(5)
w=
oa
pe
Self-Registering Barometer. ATS
mt
Then by substituting in D the respective values for dh,’ éh'’ and éh* found in
(3) (4) and (5) we obtain
dh (s'—s')s + 8h’. 8".s — 8h". (8 +8'—s8) + 0h.s.9 + Oh'.8.8' —Oh'.s'.s
8+s"—s' Pape re
fei tn td EES, f
w
wo
s¥—gl
Hence concinnating both sides, we have
h.s'.s+ 0h" .s" (s—s’) Ons 8” wl
s+s"—s! sv¥—s" w
Hence we deduce the general equation
3h = 8h" pelaese) eae tty
ss (s’—s") w s (E)
In which equation it is obvious that the multiplier of 64” 1s a constant quantity ;
and since that multiplier is capable of receiving any value by altering the dimensions
of the cross sections, of the upper and lower part of the tube B, of the cistern, of the
pillar of the hydrometer, and of the vessel A, we have it in our power to establish any
relation we may desire between oh and o/” that is between the scale of the common
barometer and of this new barometer.
When the fluid used in the vessel A is mercury, which for many reasons is to be
preferred, w'=w and equation / becomes
3°.8""(s+s"—s') s—s’
sha sh" rary -— (G)
Equation (#) is more general; but when (G) is used, a much less ball will be re-
quired for the hydrometer, the other dimensions being the same. In these equations
the effects of diversified atmospheric pressure alone, is considered; no account being
as yet taken of a variation of temperature.
In the modification of this instrument, which I described popularly to the subsec-
tion of useful arts of the British Association, I supposed the cross sections of the
cistern and of the vessel 4 to be so large as that the level of the surfaces of the mer-
cury in them was not materially altered by the oscillations of the instrument; also,
in order to diminish or remove its oscillations, I supposed the cross section of the part
_of the tube B which dips into the cistern, to be very small. Now, if in the equation
( G) we suppose s” and s” to be very large and s’ to be very small, it assumes the form
aaah" (=)
in which case if s’” the cross section of the stem of the hydrometer, were equal to s the
cross section of the upper part of the tube B, the multiplier of 8h” would be cipher and
ch” would be infinite for any finite 8h, or the new instrument would require an infi-
VOL. XVII. é‘ 4p
476 Professor STEVELLY 07 a new
nitely long scale, and would therefore be useless ; but if s” were double of s, the mul-
tiplier of sh’ would be unity and the scale of the instrument would be precisely the
same as that of the common barometer; between these, any desired length of scale
may be obtained by proportioning the pillar of the hydrometer to the cross section of
the upper part of the tube B. But of course the formula affords the more correct
rule for constructing the instrument, so as to suit the use for which it is intended.
I shall, however, give one other example of a particular construction of it, before I
proceed to examine the effects of changes of temperature upon the indications of
this barometer.
If the internal cross section of the upper part of the tube B be made equal to the
external cross section of that part of it which dips into the cistern, that is if s=s’;
and if the cross section of the vessel 4 near the surface of the fluid it contains be so
large that s” shall be nearly equal to s’—s’”; then will (@) become
sh= oh" ~ cin ant Yap aago2y7
that is, the scale of this instrument will bear to the scale of the common barometer, the
ratio of the internal cross section of the upper part of the tube B to the cross section
of the stem or pillar of the hydrometer. Now, these conditions may be obtained
while the passage for the mercury through the lower part of the tube B, shall be as
small as is necessary for getting rid of the irregular oscillations, before alluded to ;
and since the cross section of the vessel 4, must be always pretty large, this rule will
be sufficiently exact to enable a workman to guess at the size that the instrument will
be when constructed, as also to know before hand, pretty exactly what length of scale
an instrument of which he had only made the tube B and the hydrometer stem would
require ; or, vice versa, what relative magnitudes he should give these, in order to pro-
duce an instrument with a required length of scale; all these considerations also will
enable an intelligent person before hand to calculate with sufficient exactness the cost
of a rerequired instrument constructed on this principle.
Let us now suppose the pressure of the atmosphere to remain unvaried, but the
temperature to change, and let us endeavour briefly to investigate the manner in which
the indications of the instrument are affected, and the allowances to be made, correc-
tions to be applied, or the precautions to be adopted in consequence of that change.
Any person who has attended to the subject of compensating pendulums, will rea-
dily perceive, that there is a certain position in the frame which supports the cistern,
depending upon the materials of which the hydrometer is constructed, and upon its
dimensions, as also upon the frame, and scale, or whatever supports the sheet of paper
on which the curve of barometric oscillation is to be delineated ; at which position, if
the index or tracing pencil be fixed, any alteration of temperature will move it as
much upwards, in consequence of the expansion or contraction of one part, as down-
Self Registering Barometer. 477
‘wards, in consequence of the corresponding expansion or contraction of other parts,
so that the position of the index will remain unchanged, notwithstanding any change
of temperature experienced by the frame work of the instrument. I shall accordingly
suppose the index or pencil to be so placed, and then we need not take into account
the effect of any expansion or contraction of the mere frame work of the instrument.
But there are three other causes, in a barometer of this particular construction,
affecting its indications by changes of temperature, which I shall now notice, and en-
deavour to trace their effects. The first is, that when the mercury in the tube expands,
by an increase of temperature, part of it will be required to descend into the cistern,
namely, the pillar which would have been otherwise raised upon the expansion of its
cross section ; but since the material of the tube (suppose it to be iron) expands like-
wise, the quantity of mercury that will truly have to descend into the cistern, is only
the column corresponding to the excess of the expansion of the mercury above the
expansion of the tube ; the effect is, of course, reversed when contraction takes place
by a fall of temperature. It is then, upon the excess of the expansion of the bulk of
mercury in the tube above the expansion of the capacity of the tube, and e contra,
that this effect depends. It is likewise to be observed, that a mere vertical lengthening
or shortening of the barometric column, by changes of temperature, since it leaves
the weight of the column unchanged, has therefore, no effect upon the weight of the
cistern. The second cause is, that the effect of the capacity of the part of the tube
which is plunged into the mercury in the cistern upon its weight, alters on three ac-
counts—first, because the capacity of that part of the tube itself alters; secondly,
because the level of the surface of the mercury in the cistern is changed; also,
thirdly, the specific gravity of the mercury which it displaces, is changed; and on
these accounts, the weight of mercury in the cistern, is virtually changed, and the in-
dication of the instrument affected. The third cause is, that the buoyant force of the
hydrometer is affected: first, by its own change of bulk; secondly, by the change of
the weight of the bulk of mercury, whose place it occupies; thirdly, by the alteration
of the portion of its pillar or stem which is immersed in the fluid, partly by the sink-
ing or rising of the hydrometer itself; partly by the alteration of the level of the sur-
face of the mercury in which it floats. _Upon these principles we can, in the follow-
ing way investigate the effect of changes of temperature upon the indications of the
instrument.
Let us suppose the materials of which the parts of the instrument are constructed,
to be cast iron and mercury.
Let c = capacity of the mercury in the barometer tube B at 30 inches pressure
and 60° temperature.
Let c’ = bulk of same tube B which is immersed in fluid in cistern.
ce” = bulk of fluid in the cistern.
e” = bulk of fluid in vessel 4.
c’ = capacity of part of hydrometer immersed in fluid in vessel A.
478 Professor STEVELLY 07 a new ~
Let 8m, denote the variation of cubic unit of mercury for a change of tempe-
ot cast iron rature of 1° Fahrenheit.
dt = the total change of temperature from 60°,
dh = the sinking of the index of the instrument caused by that change,
dh’ = the elevation of the surface of the fluid in the cistern, so caused.
dii' = the elevation of the surface of the fluid in vessel A.
Then will ¢ (8m—8i) dé = the column of mercury which descends from B.
c” (8m—é) dt = change of volume of mercury in cistern.
ce & dt = change of bulk of the part of B immersed in cistern.
dh'—dh = total alteration of immersion of B in fluid in cistern.
Then by considering the causes of alteration of level of surface in the cistern, it
will be seen that
a! at fon 98 ae
(ahi —ah) * ¢ ae e c sal ot) dt i ce (8m 3 8) dt mona) male
ane (c+c") (8m— SHS +c di dt—s' dh : nd
+s
dki=+dk) = (c+c") (8m—&) di+c' di dt—s" dh ; pipee
s'—s
c+e’) (d8m—s2) dt+c!' di dt—s'' dh : weight virtually added to the cistern, in conse-
asia M Sela dia heros » SwW= je of change of part of tube B immersed in
SoHE
Ss 8 cistern.
30 dt __ § weight lost by cistern in consequence of the decrease of specific gravity of fluid dis-
EPL UR ON at laced by part of B immersed in cistern.
P YP
c(sm—v)di.w = { weight of mercury which descends from tube B into the cistern.
(cs! +0’s') (8m bi) dt-+e's (8m + di) dt~c's" dm dt—s's' dh
ee eee
v w= $ tot weight gained by the cistern.
—s'
rising of surface in vessel A, in consequence of total sinking of the pillar
: sll!
Again, (dh + dh’) i= sai the hydrometer.
(cc) (dm—er) dt _ § rising of surface in vessel A, in consequence of change of volume of mercury,
vessel A, and hydrometer.
s’
hence,
Th+dh’)s" OF — oi) de
ch” = Seen’ te 2) Gm Bat , from whence we deduce
Py Ww pv om
an Soh ee ; and from hence
n _ S*dh+(c"—c" —6i) dt Apes :
dh +dh’ = ALES Ra Gide) ee sinking of pillar of hydrometer ; from
sv —sg
whence we infer
(2 dh+ (c"~c") (om—di) dt ea = cat of buoyant force caused by the sinking of the
" g¥—eg" P hydrometer.
Self Registering Barometer. 479
also
4 : _ § diminution of buoyant force caused by the increased volume of the mercury
c” (om — 8) dt. w= se the hydrometer.
hence by subtraction we deduce
s¥ 8!" dh+(cl" s!"—c'v") (8m—8) dt
—— ,W= 4 total increase of the buoyant force of the hydrometer.
sv—g!!’
hence by equating the total gain of weight of the cistern to the total gain of buoyant
force of the hydrometer, and dividing both sides by w, we deduce the equation
(cs e's’) (8m—di) dt+c's' (6m+8t) dt—c's” dm dt—s's" dh i
Ee
ss" dh+(e"s'" —c's") (8m~—8i) dt
a ee eee
vs”
from whence by the ordinary processes of concinnation we deduce
Sah
{es +e"s!) (g's!) — (6's! — c's") (s"' ~81)] (Sm—8i)+e's! (sv 8") (8m+4 61) —C's" (8*— 81) dm
Vall
Sra (siRHs))i--sist (v6)
x dt
(K)
being the value, for the entire effect of change of temperature equal to the number of
degrees of Fahrenheit, df. upon the indication of the instrument.
From equation (4), when the dimensions of the several parts of a barometer on
this principle are known, we might, were it desirable, calculate the rising or falling of
the instrument, consequent upon a change of temperature, for we know from
the mean of very accurate observations by several eminent philosophers, that
8m = .0183345 gp, and that 6 =.003367. gp. But any dependance upon such a cal-
culation would in practice be by no means desirable, as the sources of error would be
too many to admit of much confidence in the result. The better practical method,
however, is readily deduced from a careful consideration of the foregoing equation, as
follows.
In equation (KX) it is obvious that s, s', s,s’, s’ dm, and 8 are all independant of the
amount of atmospheric pressure, but that ¢, c’, c’,c” and c’ are all dependant upon that
pressure, and are each functions of 8h” of the form y+ +‘ dh” ; y and y being certain
constant quantities. The entire equation () therefore, if such values were substi-
tuted for each of these equations, would assume the form
dh=(0 +1" 8h") dt (L)
In which T and I’ are constants, the values of which are very easily discoverable by
two simple experiments, in which the effects of two known changes of temperature
upon a given instrument, while the pressure remained unvaried, should be accurately
noted, or more correctly should be deduced from means of many pairs of observations,
VOL. XVII. 4a
480 Professor STEVELLY on a new Self-Registering Barometer.
the pressure being maintained unchanged during each pair. Should the fluid in which
the hydrometer is sustained, be any other than mercury, the division of both sides of
the equation from which we deduce (_) by w, cannot be resorted to, as the w which
multiplies the second member, is the weight of a cubic inch of that other fluid ; but
practically this would be-of little consequence, as the form of the equation (L) would
not be affected by this difference.
As I stated in the beginning of this paper, it is obvious that a self-registering rain
gauge may be constructed on a similar principle, by conducting the rain from a
proper funnel into a vessel like the cistern sustained by a hydrometer, the descent of
which would become an index of the quantity of rain received by the funnel, and a
pencil might be made to trace at the several portions of the entire day, the height at
which the gauge stood at each instant by means of a sheet of paper carried across it
by clock work.
VOL. XVII.
ANTIQUITIES,
A
A
On an Astronomical Instrument of the Ancient Irish. By Sin Witxi1AM Betuam,
Ulster King of Arms, M.R.1.A., F.A.S., §c. &e.
Read 23d of May, 1836.
We learn from Cesar that the Druids of Gaul were the philosophers and men of
science, as well as the priests, of the Celti.
Yirst—As priests, they were occupied in expounding religion, and ordering and
directing sacrifices.
Secondly—The education of youth was solely committed to them.
Thirdly—<As judges they settled all disputes and controversies.
Fourthly—That their learning and science was derived from the British Islands,
where all persons who wished to obtain a perfect education, went to study, and to
become adepts required attention for twenty years.
Vifthly—-That they were a literate people, and in their common concerns of life,
both public and private, they used, in writing, a character similar to the Greek.
Sixthly—They taught the doctrine of the metempsychosis.
Lastly —* They also taught the youth many points touching the motion of the
heavenly bodies, the magnitude of the earth, the nature of the world and of things,
and the dignity and power of the gods.”
It thus appears that the Celtz were learned in the sciences of their day. Their
youth were taught astronomy and other alistruse and difficult subjects, the doctrines
of Pythagoras, and the learning of the Egyptians. In fact, they were highly in-
structed, and knew as much as was then known by any people.
Hibernia being inhabited by the same race of people as Gaul, was, in fact, as one
of the British Islands, the chief seat of Druidic learning, for Cisar distinctly states,
that all who wished to perfect themselves in the sciences and learning of the Druids,
went to the British Islands for instruction. I trust, the time has arrived when
the subject of Antient Irish History can be logically discussed, with patient candour,
free, as well from the sneers of those who condemn, because they do not understand,
as from the injudicious zeal of those who, by crediting every wild notion, have injured
the cause they have endeavoured to promote.
I beg to call the attention of the Academy to a very singular instrument lately
found in Ireland, which has fortunately been preserved from the melting pot by
4 Astronomical Instrument of the Ancient Irish.
our excellent friend and zealous antiquary, the Dean of St. Patrick’s. If I am not
greatly mistaken, it is a Celtic astronomical instrument, invented to exhibit to the
pupil a diagram of the earth’s polar inclination, and the phenomena of the phases of
the moon. It is certainly a rude effort, but it displays considerable ingenuity, and a
progress, or, at least an attempt, towards demonstration. It is very nearly the diagram
now made use of, in elementary works on astronomy, for the same purpose.
People possessing so correct a notion of the heavenly bodies, and skill to form
an instrument so nearly representing their motions and real state, were, at all events,
a discriminating race. It required a long period of study, a succession of ages of
learning, with a state of quietude and undisturbed repose, to have arrived at such a.
point of scientific knowledge.
It is of Celtic brass, and evidently was in common use ; for though parts of many
have been found, no other, as far as I have seen, was perfect. It consists of a circle, the
outside edge of which represents the moon’s orbit, having on it eight rings _represent-
ing the different phases of the planet. In the inside of this circle, is another fixed
on an axis, in the line of the inclination of the poles, on which this, which represents
the earth, traverses,
FIRST QUARTER,
NEW
jy MOON.
SUPPOSED
POSITION OF
THE SUN.
LAST QUARTER,
eee oe
Astronomical Instrument of the Ancient Irish. 5
It is not possible, I think, to imagine any other use for this instrument, or to deny
that assigned to it.
On the upper ring was another loose one, apparently for the purpose of hanging it
up, which was broken in the pocket of the Dean as he brought it to me; but it is
preserved, and exactly fits the broken part, so as to indicate its true position, as re-
presented in the drawings.
The outside edge is the equator or ecliptic, with the phases of the moon. The
interior moveable circle exhibits the inclination of the earth’s axis to the equator.
{t will be observed, that the learning and doctrines of the Druids, as related by
Cesar, exactly tally with those of the Phenician philosopher Pythagoras, which, as far
as astronomy is concerned, is shortly summed up in the following passage in a work
published by Mr. Mason Good, and Doctor Olynthus Gregory :
“In astronomy his (Pythagoras) inventions were many and great. It is reported,
that he discovered and maintained the true system of the world, which places the sun
in the centre, and makes all the planets revolve about him ; from him it is to this day
called the Old or Pythagorean System, and is the same as that revived by Coper-
nicus. He discovered that Lucifer and Hesperus were but one and the same, being
the planet Venus, though formerly thought to be two different stars. The invention
of the obliquity of the Zodiac, is also ascribed to him. He first gave the world the
name of J<osmos, from the order and beauty of all things comprehended in it, assert-
ing that it was made according to musical proportion ; for he held that the sun (by
him and his followers termed the fiery globe of unity) was seated in the midst of the
universe, and the earth and planets moving around him ; so he held that the seven
planets had an harmonious motion, and their distances from the sun corresponded to
the musical intervals or divisions of the monochord.
** Pythagoras and his followers held the transmigration of souls, making them suc-
cessively occupy one body after another.”
Let us see how exactly this agrees with what Cesar says of the Druids :—
“‘Imprimis hoc volunt persuadere non interire animas, sed ab aliis post mortem
transire ad alios : atque hos maxime ad virtutem exercitari putant metu mortis neglecto.
Multa preterea de sideribus atque eorum motu, de mundi, ac terrarum magnitudine,
de rerum natura de deorum immortalium viac potestate disputant, et juventuti trans-
dunt.”—Lib. VI. 13.
In this short passage is condensed the precise philosophy of Pythagoras which is
declared to have been taught to the youth by the Druid philosophers of the Celt.
Pythagoras was born at Sidon, in Phenicia, about the 47th Olympiad, or 590 years
before our era ; but the Greeks say his father was a Greek merchant of Samos, where
he was brought when young ; but his thirst for knowledge not being satisfied with the
ignorance which preyailed at that place, at eighteen years of age he travelled, first to
VOL. XVII. BB
6 Astronomical Instrument of the Ancient Irish.
Syros, tosee Pherecydes the philosopher, then to Miletus, to Thales, then to Sidon, in
Phenicia, where he remained some time, and then to Egypt, where Solon and Thales
had been before him, and stayed there twenty-five years. He also went to Chaldea, to
visit Babylon.
I am aware that Pythagoras is claimed by the Greeks, but they admit his birth and
education to have been Phenician. The Greeks were furtive in this respect. Accord-
ing to them, Hercules was a Greek, and at the same time, they admit the Tyrian Her-
cules was the most ancient who bore that name. There is little doubt, that had the
Tyrian Hercules not existed, we should never have heard of the Greek hero. The
fact is, they borrowed their science, learning, gods, heroes, and philosophers, all from
the Phenicians; and there are strong grounds for doubting every Greek relation
which has reference to their national vanity.
Pythagoras taught what he learned from his masters at Sidon—the science, learn-
ing, and philosophy, long known and taught in Phenicia ; which, however, being newv
to the Greeks when promulgated by Pythagoras, were by them attributed to him as
his own discoveries, and the works of his great mind.
Here we have a glimpse of the reality on which is grounded the tradition of the
ancient learning of the ancestors of the Irish people—that they were a colony of
Phenicians. If not clearly established, every advance in the inquiry seems to confirm
that opinion.
On the Ring Money of the Celta, and their System of IV eights, which appears to
have been what is now called Troy Weight. By Sim Wituiam Bernam,
M.R.1.A., Ulster King of Arms, §c¢. Sc.
Read 23d of May, and 27th of June, 1836.
That the Britons at Cesar’s invasion, were considerably advanced in civilization,
fully appears from his statements. They had large ships, and much trade and com-
mercial intercourse with Gaul, Spain, and Germany. ‘The island was populous ; they
had good houses; great abundance of cattle ; and, among other proofs of a polished
and cultivated people, they possessed a well regulated and graduated metallic circu-
lating medium. “ Hominum est infinita multitudo, creberrimaque edificia, feré Gallicis
consimilia, pecoris magnus numerus, utuntur autem nummo aureo aut annulis ferreis
ad certum pondus examinatus pro nummo. Nascitur ibi plumbum album in mediter-
raneis regionibus, in maritimis ferrum, sed ejus exigua est copia, wre utuntur im-
portato.” Many circumstances are stated in this short passage which only could refer
to a people somewhat advanced in civilized life. ‘These observations are equally ap-
plicable to the Celtz of Hibernia, which the remains daily found under her surface
fully substantiate.
Suetonius says, (in Czsare, cap. 54,) “In Gallia fana templaque Deum donis re-
ferta expilavit, urbes diruit, sepius ob praedam quam ob delictum, unde factum est ut
auro abundaret ternisque millibus nummum in libras promercale, per Italiam provin-
ciasque divenderet.”
The great abundance of gold found by Cesar in the cities and temples of Gaul,
absolutely diminished its value in Italy.
It appears also, from Czsar and Diodorus Siculus, that there was a great abundance
of gold among the Celtz ; the people wore the richest and most ponderous ornaments
of that metal, as torques, chains, breastplates, and even had the frontlets of their hel-
mets covered with plates of gold. Many specimens of all these have been found of
great Weight and value in Gaul and Britain ; but, in recent times, those found in Ire-
land greatly exceed them, both in number and value. Several are still in the Dean of
St. Patrick’s and other museums.
Many reasons may be given for the more frequent occurrence of these remains in
Ireland than in either Britain or Gaul The former has ever been more a grazing
8 On the Ring Money of the Celte.
than an agricultural country, and therefore its surface has been much less disturbed
than the other parts of Celtica; the plough and the spade have been less active, there-
fore the remains of the magnificence of former ages have there remained undiscovered.
The cultivation of the potato has been, however, of late years, greatly instru-
mental to the discovery of these antiquities: most of those found without the bogs,
have been brought to light by the potato-planter’s spade.
The existence of bogs, or peat moss, of such immense extent in Ireland, has also
greatly operated in preserving the remains of antiquity which have been there de-
posited, either by accident or design—I say design, for no doubt many valuable
articles have been purposely hidden on sudden alarms.
They were secure places of secretion of the precious metals from the incursions of
an enemy or a sudden emergency ; this description of property could be deposited
without any external indication to lead the plunderer to the spot; and a token or
clue could easily and safely be fixed to guide the owner, when the danger had passed,
to the certain recovery of his property. In many instances, no doubt, the possessors
of the secret fell by the swords of their enemies, and thus the treasure remained, uutil
accident, in recent times, brought it to light.
The bog is also of such a nature as sometimes to ovyergrow the level of the ground
adjoining, and overflow it, and thus cover even the habitations of men, and with them
all their valuables. An instance of this having taken place has lately been disco-
vered by Captain Mudge, R.N., while employed in making a survey of the coast of
Ireland. In cutting out a bog he discovered a series of wooden buildings of a very
rude character ; and the stone axes with which the timbers had been hewed into
shape were found on the spot, which were, no doubt, covered by the moving bog,
while in the very act of construction, and so remained until discovered by Captain
Mudge. These must have been built of a period previous to the Celtic invasion—as
stone axes were the implements of the Zuath de daone, or Northern People.
The aggregate amount of the articles of manufactured gold found in the course of
twelve months in the bogs and fields of Ireland, is truly surprising—most of them of
exquisite and elaborate workmanship, particularly torques, helmets, breastplates, brace-
lets, rings, and ring-money, with many implements, the use of which it is difficult
even to suggest. Besides those which come under the notice of the antiquary or the
curious investigator, immense quantities are silently broken up and sold by the finders
as old gold, lest the owners of the soil should make their claim, and deprive the
finders of the fruits of their good fortune.
Ancient silver articles are, however, of much rarer occurrence, at least those which
may be considered Celtic. It may safely be said that there are found a thousand ar-
ticles in gold to one of silver. This may possibly be attributed to the ease with which
gold was collected, compared with the exertion necessary to obtain silver—the latter
requiring all the labour and skill of mining and refining operations, while gold is
On the Ring Money of the Celte. 9
found frequently, if not generally, in a pure state in the soil washed down by the
mountain streams.
Cesar tells us, that the Gauls “ use for money, gold and iron rings, by certain
weight.” The latter have perished by oxidation, but the two former are found, in
great abundance, in the fields and bogs in every part of Ireland. These curious re-
mains are so perfectly anolagous to the accounts given of the Britons and Gauls, by
Greek and Roman writers, that they of themselves afford the most powerful testimony
of the identity of the origin of the ancient Irish and that people. To gold and iron,
may be added silver and brass rings of a graduated weight.
There are also great quantities of rings of jet, coal, or ebony, found in our bogs,
which may possibly have passed as a circulating medium : but there is not, as far as J
have discovered, any authority beyond conjecture that they were so used ; althougl
it is well known that such substances have, in other countries, been used as a circu-
lating medium.
It has often been objected against the Irish pretensions to early civilization, that
no very ancient coins, or medals, of the early Irish monarchs have been found ; and,
certainly, the absence of any indication of a metallic circulating medium, would sup-
ply a fair inference of a low state of commercial intercourse ; but, on the other hand,
the appearance of a well-regulated, convenient, and graduated circulating medium of
the precious metals, demonstrates an advanced progress in civilization. There
have, however, been found in Ireland many specimens of very ancient silver coins,
the legends on which have never been deciphered, or appropriated to any monarchs
or people, and yet remain unexplained.
These were, certainly, very early attempts at coinage ; and as exactly similar coins
are often found in Britain, it is a fair conclusion, that they were the production of
the ancient Celtic inhabitants of each country.
But it is not necessary to rest the pretensions of the Celtz, to the possession of a
metallic circulating medium, on these rude specimens of early coinage; the period of
their fabrication is recent, when compared with the circulating medium which, we have
now irrefragible testimony, existed for ages previous to our era, in all parts of Celtica.
It is very remarkable, that rings of gold and iron being used as money among the
Britons, appears to have been an idea new to Cesar ; there is no remark that any such
medium was known at the time to have existed among any nation with whom he was
acquainted.
The kings of Lydia, we are told by Herodotus, were the first who coined metallic
money, about 600 years A.C. and the practice soon after obtained universally among
the nations with whom they had intercourse. The discontinuance, and a lapse of 560
years, in Casar’s day, had obliterated the recollection that the ancient currency of
rings had ever existed. Recent investigations have exhibited evidence, that the most an-
cient of nations used rings of gold and silyer for money, or a metallic circulating medium.
VOL, XVII. oe
10 On the Ring Money of the Celta.
So extensively a commercial people as the Phenicians, of whom the Celtz were
unquestionably colonists, could not long carry on their affairs of trade by the means
of barter and exchange. ‘They would soon feel the necessity for something defined to
represent property, and the precious metals would be naturally suggested as the rea-
diest mean, and weight would be adopted as the measure. ‘They were, in all proba-
bility, the inventors of ring-money, for they were certainly the first people who car-
ried on an extensive commerce.
Gold and silver wire, cut into equal lengths, were most likely the first attempt at
money, because the pieces could more easily be made of the required weight and
value. Straight pieces are inconvenient of carriage, would wear a purse, or bag, and
escape from small apertures ; this inconvenience naturally suggested twisting the wire
into the form of a ring, such were the gold ring-money of the Gael of Gaul, Bri-
tain, and Ireland. ‘The brass rings were cast, first, like those of gold, afterwards in a
perfect ring ; and both are every day found in Ireland in great numbers.
Vast quantities of articles in gold and brass are also found, the use of which have
not a little puzzled the learned antiquaries. Vallancey calls them paterz, but as pa-
tere they are of the most inconvenient shape; they will not stand, so as to hold a li-
quid ; and, having two cups, one would discharge itself while a person was drinking
from the other. Vallancey supposes them sacrificial cups, and that they were used to
pour forth oblations to the gods ; others have fancied them to be used to cover the
breasts of the dea mater. All these speculations, I conceive, are untenable, as some
of them have flat surfaces, instead of cups, consequently could not be used as pateree,
as they would contain.no liquid whatever.
The objections against their being fibule, are equally cogent ; a buckle, or fastening
of gold, of fifty-six ounces weight, appears absurd ; besides, undoubted fibulz of the
precious metals and brass are found in Ireland, in great quantities, of convenient and
palpable shapes.
Their peculiar form appear to render them incapable of application to any active
operative purpose ; and the only conclusion which appears satisfactory as to their use
is, that they were ingots of gold, or the larger species of the circulating medium, and
but a variety of the ring-money.
The following specimens will illustrate my notice of the transition from the straight
wire to the ring, and from them to the larger ingots or gold money.
‘This is the most common form of the smaller gold ring-money found in Ireland. They
are made of pieces of gold wire formed into the required thickness, cut into lengths of
equal weights, and then bent round into the shape above represented. I have seen
—
On the Ring Money of the Celte. 11
counterfeits which have been occasionally found, made of cast brass, exactly of the
same shape and size, and so neatly covered over with a coating of gold plate, as to
defy detection unless weighed. There can be no doubt of the fraud being ancient, for
the brass is of copper and tin, the same as the brazen spear-heads and other Celtic
utensils of that metal.
The smallest I have seen, weighed 12 grains, or half a pennyweight, which seems to
have been the unit by which the larger were graduated, for all have been found to
bear a relative value to it ; taking this for the wnt, the rest are its multiples. Among
those in the possession of the Dean of St. Patrick’s, and G. Petrie, Esq. are the following :
oz, dwt. grs.
No. 1 *1, weighing O20 1a
2 1. Dean of St. Patrick’s, O 1 12 equalto 3 of No. 1*
3 3. do. 0; .2al2 = ie Ofte do,
4 2, do. 0.3 Mito ts Oe
5 1 do. Ona O = 10
6 ie do. O}das 50 = 22
7 1. do. oll 8 = 2h
8 1. George Petrie, Esq. Ont SeaO = Ue
9 1. Mr. Stewart, Goldsmith, 120 0 = 480
10 1. Alderman West, apo LD) St!
It will be observed, that there are but Nos. 4 and 7 in all these which do not con-
tain in weight the exact value of their multiple No. 1*: thus No. 2 is equal to
3 of No. 1—No. 5 equal to 10—No. 6 to 22—Nos. 4 and 7, these may have been
wasted by use or fraud. They, however, contain each two fractional thirds of No. 1.;
and it is possible they may have so graduated for convenience of exchange, as our
half-crown is contrived to represent 2s. 6d., or as the old quarter of a guinea repre-
sented 5s. 3d.
This is a specimen of the simple cast brass ring, and is found in immense quan-
tities, from the weight of one pennyweight to that of several ounces. Many are
12 On the Ring Money of the Celte.
In the Memoir lately published by the Ordnance Surveyors, mention is made, from
the Irish Annals, of many rings having been presented by the princes of Ireland to
the corb or successor of St. Columbkill, In 1151, Cooly O’Flynn presented one
weighing two ounces, and another in 1153, of one ounce. In 1004, Brian Bor-
roihme presented to the altar at Armagh, a ring of gold weighing 20 ounces. It is
obseryed that all these are described of equal weight in ownces. Such passages are of
yery frequent occurrence in the Irish Annals.
This is of brass, and is the first variation or change from the perfect circle ; it is
larger than the largest of the perfect brass rings: it was cast, and they are found in
very great quantities together in bogs ; the points are flat.
This is also of brass, but larger than the last specimen which it much resembles,
but the points are larger and at a greater distance, with flat surfaces. It is obviously
a progressive and farther deviation from the perfect circle, as the weight and value
increased. Its diameter is somewhat more than two and a-half inches. An immense
number of these, many thousands, as many as loaded a large cart, were found in a
kind of tumulus, in the county of Monaghan, a few years since. They were sold for
old brass, which is the general fate of the brazen articles found in Ireland. One spe-
cimen, of which the above is a representation, is preserved in the museum of George
Petrie, Esq.
This specimen is of gold, and weighed 4 0z. 16 dwts.; it was found, about two
years since, with ten others, under a large stone in the County of Mayo. They were
sold to Alderman West of Dublin, who sent them to me for inspection and to have
drawings made. ‘This has more of the circle than the last specimen, but varies in
having the points of a concaye or cup-like shape.
On the Ring Money of the Celte. 13
This was found with the last, which it resembles, but its points are more distant ;
it is of gold, and weighed 9 oz.; it has deep cups at the points, which originally,
perhaps, were made to regulate its weight.
These are very attenuated specimens. The first only weighed 16 dwts, 12 grs., and is
in the Museum of the Royal Dublin Society ; the other two ounces. The cups are
large and very thin ; the transition is gradual from the last.
This is also of gold, and weighed twelve ounces; the cups are very large. Ano-
ther weighed 30z. 12dwts. Ihave seen many other specimens of greater weight ;
all, however, of equal penny-weights.
This is also of gold, and weighed 19 oz.: it was found in a stone chest in the
church-yard of Ballymoney, in the County of Antrim. I had one nearly, if not pre-
cisely, of the same figure, which weighed 36 oz.; and Vallancey mentions one which
weighed 56 oz. ‘The cups of this are five inches in dianfeter. ‘Two others are men-
tioned by Vallancey, one weighing 15 oz., the other 1 oz. 12grs. It should be
observed, that one or two have been found of a weight two or three grains less than
the even twelve grains, which may have been lost by attrition, but they are very few—
I have met with but two or three, and they may be considered exceptions to the
rule.
VOL. XVII. DD
14 On the Ring Money of the Celte.
This specimen is singular in its shape, from its very broad, thin, and flat points,
which are of equal thickness, except just at the junction with the stem. They are
found of various sizes and weights, some of the plates are two inches in diameter, but
all very thin. One specimen, in the collection of Alderman West, has, on the back
of one of the plates, a small loop, through which a cord of the size of a pack-thread
might pass.
It is very difficult to decide on this specimen; it bears a strong resemblance in
some respects to its predecessors, and yet it differs so much that it might be consi-
dered a different article altogether.
To look at the last five specimens, unconnected with the others, no one would
imagine them ring-money ; but, seeing the gradual variations, I think they may
fairly be considered as the same. ‘The necessity of stringing small money is obvious
to preserve them from loss, but for the larger and more valuable this precaution is
not required.
These ring coins of the Gael suggest a very early period of civilization, before the
Phenicians struck medallic coins on flat plates, with the effigies of a sovereign, or
emblem of a people, and a legend or inscription. The Phenicians were, at a very
early period, acquainted with the art of coining money ; and as there are very few, if
any, instances of Phenician coins found in Ireland, the period of their intercourse
must have been of very remote antiquity. I have seen but one brass coin which was
thought to be Phenician, and that of doubtful character—their intercourse most likely
ceased with Ireland before they coined money on plates.
SILVER RINGS.
Since the foregoing was written, the Dean of Saint Patrick’s has placed in my hands
six rings of silver, of very rude workmanship. I have never seen any similar, and,
at first sight, could scarcely believe they could be money; but further investigation,
and their graduated weight, led me to the affirmative conclusion. I have weighed
On the Ring Money of the Celte. 15
them, and find they also, like gold rings, are all multiples of the half penny-weights,
as follows :
oz. dwts. rs.
No. 1 weighs 2 10 12
2 Leia 12
3 — 1 7 12
4——1 2 0
5 — 0 19 12
*6 ORL 50
These also are so nicely adjusted to their weight that they nearly balance the scales.
BRASS RINGS.
Mr. Petrie has also placed in my hands three cast brass rings, which fit most cu-
* No. 6 is exactly like the others in shape. Mr. Alderman Brady has lately purchased several other
silver rings exactly like these, and of the same graduated weight.
16 On the Ring Money of the Celte.
riously one within another ; they have been considered money, and are perforated
laterally through in such a manner as to admit a thong of leather or cord to pass
through them. They are of the old Celtic brass. Their weight also is of equal
penny-weights.
02. dwts.
No. 1, the outside, weighs 7 9
2, the second, weighs 3 18
3, the centre, weighs 1 13
Having been perforated, they are not so exactly balanced as the gold and silver,
though they are very nearly of equal penny-weights, but just turning the scale.
The smaller specimens, however, of the brass ring-money which are perfect, are
quite as accurately balanced as the gold and silver ; of these I lay before the Academy
eight specimens, and it is singular that, with one exception, they exhihit odd multiples
of the unit of twelve grains.
dwts. grs.
No. 1, or the largest is 19 12
ON. oy Ue
Sy. -agasoC BASSCM Riveniclaclts cope al
4, Selassie coy IY
Ds RoncdS iy 10)
6, treble rings ...... 8 12
7, double rings...... Fe he
8, dittokcasceseeess By al
No. 5 alone consists of even penny-weights. I have weighed a great many of
these rings, and found them, without a single exception, multiples of half a penny-
weight. It would, indeed, be difficult to persuade ourselves that this circumstance
could be accidental. The ring money, gold, silver, and brass, as Czsar tells us, was
‘ad certum pondus ;” and that weight, all our specimens show, was formed on the
same scale, or, perhaps, was derived from the same original as the Troy weight. A
pound Troy of gold thus formed 480 rings, weighing each half a penny-weight, 40 of
which were equal to an ounce. It is not easy to say whether their system was duo-
decimal or decimal—from these specimens it might have been either. If we could to
a certainty say their penny-weight consisted of 24 grains, we might conclude it was
duodecimal ; and the coincidence of all the other fractional parts being so accurately
Troy weight, and several of the specimens being of duodecimal multiples of the
unit, and all consistent with with it, seem to lead to the conclusion, that it was the
same ancient system from which the Troy weight was derived.
To what remote period of antiquity do these singular facts carry us back! To
many ages before the time of Cesar, or even Herodotus. The latter speaks, as I have
observed, of the Lydians as the first who coined medallic money, at least six centuries
On the Ring Money of the Celta. 17
before our era. ‘These are no visionary speculations : we have here the remains and
imperishable reliques of these early times to verify the whole ; and recent investigations
and discoveries, in a most singularly convincing manner, come to our aid by showing
that the fresco paintings in the tombs of Egypt exhibit people bringing, as tribute,
to the foot of the throne of Pharoah, bags of gold and silver rings, at a period
before the Exodus of the Israelites.
It is right to observe that many of these weights were taken by me before I had an
idea that these things were money.
The Troy weight is said to have been brought to Europe from Palestine and Egypt
by the Crusaders, and obtained its present name from the town of Troyes, in France,
where it was first used at the great fair held there. ‘There is reason to believe that it
is the old Phenician mercantile standard weight which once prevailed all over the
East, and that, like most other commercial improvements, originated with that great
commercial people, as they must have first felt the necessity for such a means of ad-
justment of those commodities which were disposed of by weight.
The old Celtic (raye) unsha was the exact ounce Troy—it is a compound word
aon, one, and ye, siath, or the one-sixth part of a given weight, containing the quantity
of our half pound ‘Troy ; the name of this weight I have not yet been able to ascer-
tain. The weight of twelve ounces, now called a pound, having eventually prevailed
in general computation of larger quantities, the word wnsha, as the twelfth part of a
pound, became the Celtic, and from them the Latin, word signifying the twelfth part
of anything, even of time.
I submit the foregoing remarks, if not with reluctance, yet with diffidence ; for
although I feel satisfied that the evidence sustains all the conclusions I have drawn, I
am more anxious for truth than for any hypothesis ; and shall be glad if the discussion
produces a correct result, even if it should be contrary to what I myself have fondly
fancied to be irrefragibly established.
YOL. XVII. EE
NOTES.
In the interesting work published by Mr. Wilkinson on the Thebaid and Egypt, in which it appears,
that the most ancient money of that country, even before the Exodus of the Israelites, was gold and silver
rings of a graduated weight, as the following extracts will show:
“In the second line black chiefs of Cush or Ethiopia bring presents of gold rings, copper, skins, fans,
or umbrellas, of feather-work, and an ox bearing on its horns an artificial garden and a lake of fish.” —
Wilkinson's Thebaid, p. 136.
“A continuation of these presents follows in the third line, where, besides rings* of gold, and bags of
precious stones, are the cameleopard, &c.”
Thothmes III. (1495 B.C. p. 154.) ‘The money used at that epoch was, as I have already observed,
of gold and silver rings.”
“On the right hand are some very elegant vases of what has been called the Greek style, but common
in the oldest tombs of Thebes. They are ornamented as usual with arabasques and other devices. Indeed,
all these forms of vases, the Tuscan border, and the greater part of the painted ornaments, which exist
on Greek remainss, are found on Egyptian monuments of the earliest epoch, even before the Exodus of
the Israelites, and plainly removes all doubts as to their original invention. Above these are, chariot
makers and other artisans. Others are employed in weighing gold and silver rings, the property of the
deceased ; their weights are an entire calf—the head of an ox (the half weight), and small oval balls (the
quarter weights.) They have a very ingenious mode of preventing the scale from sinking when the ob-
ject they have weighed is taken out by means of a ring upon the beam,” p. 151—vide Genesis xliii. 21,
“ our money in full weight.”
Pomponius Mela thus speaks of the Phenicians: “ Phcenicen illustravere Phcenices solers hominum
genus, et ad belli pacisque munia eximium literas et literarum operas aliasque etiam artes maria navibus
adire classe confligere, imperitare gentibus regnum preliamque commenti.’ — Viait A.D. I. under Claudius,
De Situ orbis.
They were the shrewdest and most acute of mankind—skilled eminently in the arts of peace and
war, and by their skill and valour, kept the empire of the seas, and governed nations—skilled in sci-
ence, literature, and the arts—addicted to navigation and commerce.
When Pliny and other ancient authors declares the Peni to have been the inventors of navigation and
astronomy, he intended the Phenicians and not the Carthagenians. It was under the conduct of the
Phenicians, the fleets of Solomon sailed to Ophir and Tarshish, from the ports of Eilath and Esiongeber,
in the Red Sea. Ophir was the general name of the eastern coasts of Africa, which was always a great
market for gold-dust. Tharshish was also a general name for all distant countries.
The Greeks, in their lists of the nations who have been masters of the Mediterranean, give the seventh
place to the Phenicians, and the eighth to the Egyptians; but they were always reproached by the
Egyptians, as novices in antiquities, as they really were. The Egyptians do not appear to have ever been
a naval power, except by their allies, the Phenicians.
* The money of the Ethiopians and Egyptians was in rings of gold and silver, like those still in use about Sennaar. I had
interpreted the hieroglyphic signifying silver, ‘ wrought-gold,”” but the white colour of the rings placed opposite to others
painted yellow, (in another tomb at Thebes) decides the question in fayour of the word silyex,
Arbuthnot on WPeights and Measures. 4to. London, 1754.
“The Romans used the /ibra, which they divided into twelve uncia; and the later Greeks had their
litra, which they divided into the same measure.”—p. 47.
“The ratel, or litra, used all over Egypt, is of different quantities in several places, and in the same
place for several goods, but always divided into twelve parts, which are their ounces.”’—p. 55.
According to Appian—“ As to Gaul, Cesar exacted from it yearly quadringenties, £322,916. 13s. 4d.”
—p. 194,
“Lipsius is of opinion that quatermillies should be read for guadringenties, which would make the
sum ten times bigger, viz. £3,229,166. 13s. 4d. But it is not probable that Gaul would be able to pay
such asum yearly. However Velleius Paterculus affirms that Gaul was reckoned on the same footing
with Egypt as to taxes.—p. 194.
“The commerce of the Phceenicians lying more towards the west than that of the Egyptians, was the
occasion of their being celebrated as the inventors of astronomy and navigation. When Pliny names the
Peni, as inventors of navigation, it must not be understood of the Carthagenians, but the Phcenicians,
from whom they were descended. They navigated into the ocean by the straights of Gibraltar—estab-
lished many colonies—Thebes in Beotia, Cadiz, and Carthage itself, which was built fifty years before the
destruction of Troy. It was under the conduct of the Pheenicians that Solomon’s fleet sailed to Ophir
and Tharsis, from the ports of Ailath and Esiongeber on the Red Sea. Ophir was the general name of
the eastern coast of Africa—and Tharsis, that of the western coast, both of Africa and Spain. This
commerce, Jehosaphat, king of Judah, endeavoured to restore; but his enterprise was blasted by the
destruction of his vessels in the harbour.’ —p. 218.
“It is past all doubt that the Cape of Good Hope was doubled in those early times, and that the Por-
tuguese were not the first discoverers of that navigation.” —p. 219.
“The Pheenicians were much older sailers than the Greeks; the naval expedition of their Hercules
mentioned by Sanchoniathan, under the name of Malcanthus, being 300 years before that of Jason.”
“The Corinthians were said to (have been) be the inventors of weights and measures, though both
their sea craft and arithmetic came originally from the Pheenicians.”—p. 248.
« Strabo relates, that the commodities of Britain were corn, cattle, gold, silver, iron, skins, leather, and
hunting dogs; and speaking of the Cassiterides, tin and lead. Tacitus adds pearls.” —p. 253.
“ Tin and lead were used in the time of the Trojan war.’—p. 254.
“ Arabia was a country of great commerce in the time of the Romans. Aden had in its harbour, ships
from all parts of the world. The Gerrheans and the Mineans, ancient inhabitants of Arabia, carried their
spices, by land, to the frontiers of Palestine.’—p. 364.
Aden was afterwards called Portus Romanus.
The Affinity of the Phenician and Celtic Languages, illustrated by the Geographical
names in Ptolemy and the Periplous of Arrian. By Str Witusam Beruam,
S.R.LA., Ulster King of Arms, &c. §c.
Read 23d of May, and 27th of June, 1836.
Many years since, I ventured to suggest, that the inhabitants of Celtic Gaul, Bri-
tain and Ireland, at Czsar’s invasion, spoke the language now called Irish or Gaelic,
and that they were the only true Celtz. I have since, being satisfied that the ancient
Caledonians were an exception, they being of Cimbric or Gothic origin, and the an-
cestors of the Picts and Welsh.
With this exception, farther investigation have convinced me of the accuracy of my
opinions; and I have had the satisfaction of the approval of many most competent
individuals who have given the subject consideration, and flatter myself, that the time
is not far distant when there will be no doubts on the subject.
The more difficult question of—from what branch of the great family of mankind
the Celtz proceeded—! have endeavoured ‘to answer by showing the cognate cha-
racter of the languages of the Celte and the Phenicians, if they were not precisely
the same, and that the former being a colony of the latter, the Celtic language de-
monstrated the signification of the names of Greco-Phenician geography, as well as
those of the deities of the Greek and Roman theology.
“« Language, (says the writer in the Quarterly Review*) under the guidance of the
extensive research and philosophic spirit of modern philology, has been the safest clue
to the application of remote races.”
“The grounds from. which we may infer the affiliation or the relationship of the
different races of mankind and similarity—
Ist. Of languages, including written characters.
2d. Religion.
3d. Civil Institutions.
4th. Manners.
* Sept. 1885, p. 433. Micali on the ancient peoples of Italy.
VOL. XVII. FE
22 On the Affinity of the Phenician and Celtic Languages.
5th. Of Arts, and perhaps
6th. Physical Form.
But these points of similarity must be so marked and peculiar, as not to be resolved
into the common habits and usages of mankind in a similar period of civilization.”
The surest test, indeed, of the affinity of nations, is the community of language:
The Phenicians, no doubt, were the earliest civilized maritime visitors of Western
Europe, and conferred names on countries and places when they discovered and
visited them for the first time; and though their language may be said to have been
lost, or hitherto unknown, yet it remains, in a degree, in the names they conferred.
The names given to places by maritime discoverers, are usually, if not universally,
significant ; and, as it is well known that Ptolemy borrowed the names for his geo-
graphy from the Phenician mariners, if we can discover an existing tongue in which
the names to be found in Ptolemy are significant, and accurately describe the peculiar
features of the places and things which bear them, we may fairly conclude that lan-
guage to be a branch of the Phenician, and the people speaking it, the descendants of
their colonists.
That language is found in the Gaelic or Irish and Scottish Celtic.
By these names we are able to trace the proceedings and voyages of these primitive
mariners in every portion of the world, known to the ancients, and their wanderings
and maritime enterprise, at a period, compared with which, the Greek and Roman
was, are occurrences of yesterday. It may, without any great stretch of imagination,
be surmised that even Egypt was greatly indebted to the illustrious Homerite, or
Arabian Phenicians, as well for the first dawn of her commerce and consequent ci-
vilization, as for her knowledge of alphabetic writing.
In my volume, on the Gael and Cymbri, I have treated largely on the names of that
part of Europe, inhabited by Phenician Celtze, which appear clearly significant in the
Gaelic tongue, and fully bear me out in presuming the fact of the Celt being Phe-
nician colonies.
A test has lately presented itself to my mind, which at once appeared likely to
settle the question satisfactorily and conclusively.
I felt strongly, if my hypothesis be true, that the Gaelic and Phenician were
originally the same tongue, the names of places in the Eastern or Indian Seas, also
avowedly borrowed from the Phenicians by Ptolemy, ought also to be as significant as
those of Europe, which he had from the same source. I now.propose to give the re-
sult of my investigation into this proposition.
It was my intention to have made a few brief remarks on the Etrurians, and on
taly generally, but on dipping into that question, so much did it interest me, and led
me so far, that I found that I should do injustice to a subject which deserves a more
critical and serious investigation. I haye discovered enough to induce me to think
a
nan
On the Affinity of the Phenician and Celtic Languages. 23
that the solving the mystery of the origin of the Pelasgi and the Etruscans is within
my reach: I shall, therefore, reserve that subject for a future paper.
Heeren’s Researches has rendered services to historic literature, surpassing all other
writers, and too much praise can scarcely be given, or any eulogium, perhaps, be too
laudatory, for the soundness of his reasoning and the general accuracy of his deduc-
tions; it is, therefore, with great diffidence and reluctance I venture to promulgate
opinions differing, or even slightly varying from his, and but for the irresistible force
of the evidence I have to adduce, as it appears to my mind, I would not venture on so
great a responsibility.
* Heeren supposes, and argues on the supposition, that Sidon, Tyre and Aradus,
were the first seats of Phenician greatness—that they were a Syrian people, a@ Canaan-
itish tribe, which settled on the coast, the descendants of Ham. “ Zhe oldest of these,
the first-born of Canaan, according to the Mosaic record, was Sidon,” the foundress
of the trade and navigation of the Phenicians.’ It is true, as far as the western
greatness of the Phenicians was concerned, that Sidon was their first seat, but erro-
neous as to the eastern world and their origin; they were no originally Syrians, or
descendants of Ham, but from Shem; they were Arabians, from Yemen, or Arabia
Felix, and originally Chaldeans.
It is singular that Heeren should have considered so lightly, and worthy of so little
weight, what Herodotus states so distinctly and clearly, that the Phenicians were
a colony of the Homerite (which name he says, means the same as Phenician
in their language) a great commercial people, who inhabited the southern coast
of Arabia Felix; and Dyonisius declares they came from Chaldea, to the Persian
Gulph, and eventually settled on the southern coast of Arabia.” There, from
their ports of Aden, Hargia, Cana, &c. they carried on their commerce with all parts
of the Erythrean ocean for ages before they knew of the existence of the Mediter-
ranean. They first possessed the island of Tyrus (now Tylos) and Aradus, in the
Persian Gulph, from which it is said, and truly, I think, the Tyrian cities bor-
rowed their names. The inhabitants of those islands, claimed to be the parent
country of their Syrian namesakes.
Heeren observes—‘ The principal direction in which the Phenician race extended
itself by colonization, was towards the west, because, from their situation, their sea
trade could take no other. But notwithstanding this, so soon as their land trade
through Asia had reached the coasts of the Indian ocean, the want of settlements
there must naturally have been felt. Traces of them, though certainly only doubtful
traces, are found both on the Persian and Arabian Gulphs. The names of two islands
in the midst of the Persian Gulph, named Tyrus, or Tylos, and Aradus, bear striking
marks of Phenician origin; and in these have lately been discovered vestiges of Phe-
nician workmanship and buildings.”
Again, in the chapter on Babylonish commerce, he says—“ First, however, there
24 On the Affinity of the Phenician and Celtic Languages.
remains an investigation, as obscure as it is important, concerning some islands si-
tuated near this coast, which, as they are said to have been eminent trading places,
must not be passed over in silence. Jn the Greek geographers, for instance, we read
of two islands named Tyrus, or Tylos, and Aradus, which boasted that they were the
mother country of the Phenicians, and exhibited relics of Phenician temples.”
Pliny and Strabo are the principal authorities, yet they are both indebted to more
ancient authors. ‘On sailing farther south from Gerrha, (says Strabo) we came to
two islands, where there are to be seen Phenician temples; and the inhabitants assure
us that the cities of Phenicia are colonies from them.’
It appears very extraordinary, that with this testimony before him, Heeren should
still speak of the Phenicians as a Syrian tribe, and of their commerce, as originating
from the Syrian cities, and that we should meet with such passages as the following :
** Having thus shown the direction and extent of the trade and navigation of the
Phenicians, towards the west, let us now bend our course eastwards, and trace their
progress on the two great south-western gulphs of Asia, the Arabian and Persian. In
these, it has already been stated, they had partially settled ; and thus gained secure
harbours, from which to set forth on their still more distant enterprises.
*«It must, however, be at once perceived, that their navigation could not have a
like undisturbed continuance with that of the Mediterranean : as the proper domi-
nions of the Phenicians never stretched so far as either of these gulphs. It depended
upon their political relations, how far they could make use of the harbours they pos-
sessed there. For even though the way might be open to caravans, the dominant
nations of inner Asia might not be always willing to allow foreign colonies on their
coasts.
Their navigation upon the Arabian gulph, arose out of their connexion with the
Jews, and the extension of dominions of the latter under David.” So far Heeren.
Had the learned Professor allowed due weight to the evidence of Herodotus, Strabo,
and Pliny, one would conceive it impossible he could not have observed, that the
operations in the east must have been carried on by a people possessing territory, ci-
ties, and harbours, on those seas. The whole tenor of the evidence is so strong, that
it appears really astonishing, so acute and able a reasoner as Heeren, should not have
seen that the Syrian cities were, and must have been, but colonies of the Homerite
or Asiatic Phenicians, as they are stated to be by Herodotus, Strabo, and Pliny.
We must, therefore, look to a much more remote period for the origin and history
of the Phenicians, to a time when the sites of their Syrian cities of Tyre, Sidon, and
Aradus, were to be marked out. Herodotus says, Tyre was 2400 years before his
time ; and the prophet Ezekicl speaks of it about the same time, as being of the
“antiquity of ancient days.”
Arabia Felix, or the kingdom of Yemen, may safely be considered as the previous
country of the Phenicians, where, under the names. of Homeriti, and Sabeans, they
On the Affinity of the Phenician and Celtic Languages. 25
established and carried on an extensive commerce with India and all the coasts of the
Erythrean or Indian ocean. Their ships visited the eastern coasts of Africa, and sup-
plied Nubia and the countries about the upper regions of the Nile, with the rich ma-
nufactures and products of India, and obtained gold, ivory, and other products of that
country, in return. Their ports of Cana, Aden, Saba, and Macala, situated in Ara-
bia Felix, at the entrance of the Arabian Gulph, were admirably circumstanced as to
locality for the purposes of commerce, both with Asia and Africa.
The direct testimony of Herodotus would be sufficient, in my mind, to settle the
question, that the Sabeans or Homerite were the same people as the Phenicians ; but
the evidence I shall soon lay before the Academy, comes with irresistible force, in
corroboration of the statements of the father of history, and shows forth his fidelity
and accuracy in a most extraordinary, if not an unexpected manner.
We are now contemplating a period of remote antiquity—a time long before any
written history, previous to the foundation of those most ancient cities, Sidon and
Tyre, of whom the prophet Ezekiel says, “ whose antiquity is of ancient days ;” and
the former is said to have been the first seat of Phenician commerce on the Mediter-
ranean. It is not, perhaps, venturing too much, if we suggest, that through the
African ports of Sabe, Avalites Emporium, and Malao, commerce and consequent
civilization laid the foundation of Egypt’s glory and greatness ; and that the Sabean
and Homeritz were the first to “lighten up the flame of commerce and consequent
humanity’’ among that most ancient and magnificent people.
With respect to the extent of the commercial and geographical discovery of the
Homeritz, or Arabian Phenicians, history is silent, or nearly so: it has not come
down to us but in imperfect glimpses, confused and unsatisfactory : the queen of the
Sabeans, who visited Solomon, appears as sovereign of a highly cultivated nation, and
supplies a few hints. Dyonisius (Periegetes) declares, they were originally from Chal-
dea; but little, indeed, do we know of their acts or the extent of their power and
commerce. The extent of their maritime discovery, however, for the first time, if [
have not deceived myself, are within our reach ; but before we enter into the detail,
that the subject may be better understood, it may be as well to glance briefly at the
origin and early history of the Arabians.
The old patriarchal government and history of the Arabians, as detailed in the
sacred writings, prove them to have been a very ancient people, and trace them back
to ages near the deluge. They are divided into classes—the primitive Arabians, and
the descendants of Ishmael, from whom the present Arabians are descended,
The primitive Arabians are generally derived in descent, from Jokten the son of
Eber, or Heber, of the line of Shem, whose son Jarab, or Yarab, is said (after the
confusion of Babel) to have founded the kingdom of Yemen, and his brother Jorham,
that of Hejaz.
The kingdom Yemen was governed by princes of the tribe Hamyar, son of Saber,
vol. XVI. GG
26 On the Affinity of the Phenician and Celtic Languages.
great grandson of Joktan, but at length passed to the descendants of his brother
Cohlan, who retained the title of king of the Harngarites, (by the Greeks called
Homerites ). It is said to have continued in existence 2020 years, when the inunda-
tion of Aram, soon after the time of Alexander the Great, dislodged many of the
tribes who emigrated to other countries.
Such is briefly the early received history of Arabia. It rests mainly on tradition,
and, as was usual, especially with the Greeks, a personage is constructed to give name
to a people. Hamyar is made the patriarch and ancestor of the Homerite. Hero-
dotus, however, tells us the name of Homerite was significant, and had the same
meaning as Phenicians, being evidently conferred, in consequence of the profession
and habits of the people, that is, a mariner or navigator of the sea. This statement
of the father of history is more rational and satisfactory than any tradition, especially
when dependant on so slender a foundation.
The Homerite may, therefore, be considered the most ancient and primitive inha-
bitants of Arabia Felix, who flourished for ages, as the greatest, and perhaps, the only
maritime commercial people of antiquity. They were conquered, and probably exter-
minated long after the foundation of Tyre, by the warlike descendants of Ishmael ;
and their commerce and mercantile settlements, or colonies, if they formed any, were
transferred to their Syrian colonies who, in their turn, possessed of the empire of the
seas, and after a long possession of the commerce of the world, and great glory, they
eventually fell under the sword of a conqueror, whose sagacity and enlarged mind
discovered the cause of their greatness, and ever assiduously promoted the commerce
of his own subjects. Alexander acquired thereby, the name of Great, more deserv-
edly, than from the deeds of arms which made his name terrible, as a scourge of the
human race. ‘
After the fall of Tyre, Carthage, her most illustrious western colony, succeeded to
her commerce and consequent wealth, power, and dignity, in the western world. In
her turn, she fell a victim to the jealous rivalry of Rome; and with Carthage, the
Phenician race, as rulers, ceased altogether. The savage decree of delenda est Car-
thago, was extended to her records, muniments and monuments, which were so sedu-
lously destroyed, that scarcely a vestige remains of this once great and illustrious
people. The Greeks, indeed, acknowledge that they owe to the Phenician Cadmus
their alphabet ; but the Romans would not tolerate the idea, and did all they could to
efface the recollection and remembrance of a greater and more illustrious nation than
themselyes—a people who did so much to promote the cause of humanity and the ci-
vilization of mankind ; who conquered but the bad passions of mankind, and by com-
merce, taught man that it was the interest of all that each should be prosperous and
happy.
A dark and almost impenetrable cloud has since obscured the Phenician story ; even
the language was apparently annihilated; and thus was removed the only certain
On the Affinity of the Phenician and Celtic Languages. 27
means, and key, or clue for ascertaining the extent of their commerce, geographical
discoveries, and extended empire, in the names given by them to the countries, rivers,
places, and people, they visited for commercial purposes. The course of events placed
most of their European colonies under the power of the Romans, whose policy led
them to make their newly conquered provinces essentially Jtoman, in customs and
language ; and thus the Punic tongue soon ceased to be colloquial in the countries
conquered by the Romans.
In one solitary, separated corner of the remote west, a colony of Phenicians es-
caped the overwhelming influence of the Roman sword, and kept their language and
traditions pure and unmixed. Ireland was never visited by a Roman, at least we
have no historical notice: the Romanised Britons probably visited the island for com-
mercial purposes, but never with a view to conquest,
The Irish language, although losing ground every day, but before it was entirely
extinguished, the zeal of modern investigation has discovered its identity with the
Phenician, demonstrating the fact almost beyond a question; and the discovery has
been productive of the most important results to history, geography, and philology, and
opens a view into events which occurred long before the existence of written his-
tory.
Whenever the names of places, as they appear in Ptolemy’s Geography, are sig-
nificant, and explain their peculiarity of character in the Gaelic or Celtic tongue, we
may conclude they were conferred by the Phenicians. By these names we may now
define and mark out the extent of the voyages and discoveries of those adventurous
and bold mariners, with almost unerring accuracy. They have, in fact, put a label
upon each which tells its story, if not so fully and clearly as certainly as if we pos-
sessed it written by Sanconiathon, or in the records of Tyre and Sidon, or their more
ancient sisters of Saba, Aden, Avalites, Semaa and Corana.
The geography of the ancient known world, which has hitherto been a jargon of
unmeaning names, without sense or import, becomes intelligible, clear, and significant.
The character of many nations, and tribes who inhabited regions, of which no written
or even traditional history is extant, is strikingly exhibited, and the reasons for
the names they bear at this day, made clear, explicit, and satisfactory.
The deadly and unwholesome malaria of the Deltas of the Ganges, Surawady, and
other great rivers of the east, is pointed out with alarming epithets, and the savage
cruelty of the people, and the dangerous rocks and unhospitable coasts, are made
known to voyagers, by warnings contained in their names; while places of secure re-
treat and supply, are expressed by such inviting demonstrations, as would not fail to
induce the mariners to visit a port, whose name indicated secure and refreshing repose
after his labours, sufferings, and deprivations.
The extent and importance of the early commerce of-the Homeritz, Sabeans, or
Phenicians of Arabia, may be traced by their names along the coasts of the eastern
28 On the Affinity of the Phenician and Celtic Languages.
seas, where they established, chiefly wpon islands, at convenient points, fortified depots
or entrepots for their goods and merchandize, which answered, as in modern times»
the double purpose of protecting warehouses and supply to their vessels in long voy-
ages.
Herodotus, as before stated, tells us that the Homerite, who inhabited the southern
coast of Arabia, were also called by the Greeks, Phenicians, and that the name meant
in Arabic, the same thing as Phenician. By Arabic, must be meant the ancient lan-
guage of Arabia Felix—that of the Homerite, not what is now spoken by the de-
scendants of Ishmael.
Herodotus in this, as in all other cases, is faithfully correct—Phenicia is the coun-
try of the ploughers of the sea—Homerite, that of mariners—fein, a ploughman—
ojee, of the sea—I)a, country ;—ua, the country—mapajoe, of mariners. The old
Arabian and the Phenician were the same language.
The Arabian Gulph was called the Red Sea, by the Romans and Greeks, from
Erythorus, the son of Perseus and Andromeda. ‘This is one of those ingenious
fictions, by which, as Sammes observes, ‘‘ the Greeks endeavoured to bring down the
origin of every thing to their own pitiful era,’’ and was invented to disguise and ob-
scure its real origin, or ignorantly for want of a better. The name of the Erythrean
ocean, probably arose from its position—o1p, east—icyoy, headland. ‘The Phenicians
of Tyre and Sidon called the sea which washed the shores of Arabia Felix the coun-
try of their ancestors ; muyp oy) ttor—the sea beyond the eastern headland, alluding
to what is now called Cape Gardefan, the north-eastern point of Africa. The Greeks
copying the sound, made the sea, Hrythrean, and the people, Hrythreans.
Erythrus was but a personification of the Hrythreans, Homerite, or Phenicians,
the same people, and first mariners and ship-builders of the human race, who first
brought ship-building and navigation, for commercial purposes, to any perfection, in
the same manner as the Tyrians were personified by Hercules.
From their ports of Cana, Aden, Saba, Sanaa, and Corana, they visited, for the first
time, the coasts of the whole Indian ocean, as far as the Straits of Malacca ; and as
they discovered new countries, islands, rivers, promontories, estuaries, coasts, or peo-
ple, they gave them appropriate names, expressive of their respective products, rela-
tive positions, appearance, qualities, or other palpable and obviously striking cireum-
stances.
This conclusion is so natural, that it would be insulting to the understanding, to
use arguments to establish so evident a proposition. ‘Thus we find in Ptolemy and
other other ancient geographers, as among modern discoveries, places distinguished
by names, which, in the Celtic, indicate—the round hil, the good market, the swampy
marshy inlet, the happy tribe, the welcome, the island of gentle showers, the fruitful
hill, the pleasant town on the sea, the promontory of turtles, the brilliant principality
on the sea, the farthest torrent or great river, the eastern island fruitful in corn, the
On the Affinity of the Phenician and Celtic Languages. 29
the bounteous islands, the island of rich earth, or fruitful soil, the land of love, the
good harbour for ships, the harbour of refuge, the fortified depot for goods, the high-
land tribe, the narrow district. Mariners are warned against other places by fearful
denominations, as—the coast of death and evil, the gulph of the power of death, the
gulph of cruel pirates, the deceitful invitation, or false bay, the land of robbers, the
unhealthy country, the weedy river, the muddy stream, the quarrelsome people, the
shipwreck rocks, the inhospitable coast ; and many others, all of which are appro-
priate and descriptive of the places, too palpably to be mistaken, and too obviously
Gaelic, to admit of question.
It is now proposed to take a coasting investigation of the ancient Ptolemaic names
of M. D’Anville’s Map of the world known to the ancients, commencing at the north-
eastern point of the Arabian Gulph, or Red Sea, at Elana and Ezion Geber, a port
mentioned in the Sacred Writings ; thence down to the Erythrean, or Indian Ocean
by the Straits of Babelmandel, along the coasts of Arabia to the Persian Gulph,
the Gulphs of Cutch and Cambay, and the Malabar coast to Cape Cormorin and the
island of Ceylon ; then up the Coromandel to the Ganges ; and again, southward, on
the coast of the Birman empire, to the Straits of Malacca, and passing Sincapore to
northward, up to the Gulph of Siam, which appears to have been the farthest limit of
Phenician navigation in that direction. We shall then proceed down the eastern
coast of Africa, from Cape Gardefan to Zanguebar, an island a few degrees south of
the equator, beyond which the names do not, as far as I have investigated, indicate
Phenician origin :—
Elana, a rumous town on the north-eastern branch of the Red Sea, in Arabia
Petrea ; aed, the eye ; leana, a swampy plain. The eye or inlet of the swampy plain.
Exion Geber—aron, bad ; zabac, dangerous. The bad or dangerous harbour.
“ The ships of Tarshish were destroyed by a storm at Ezion Geber.”
Sina Mountain—yin, round ; «, hill, or mount. The round hill, or mount.
Madion, a town in Arabia Petrea, on the Red Sea ; mad, a field, or plain ; aojne,
fasting. The unfruitful plain, or hungry plain.
Pharan—promontory. The cape or head land of Arabia which divides the two
bays or gulphs—one ending at Elana, the other at Suez; yayan, a turtle. The pro-
montory of turtles.
Phenicon, a town in the Gulph of Suez; rei, a plougher ; ojce, of the sea. Ma-
riner’s town.
Raunat, a town on the Red Sea in Arabia, in about lat. 26° N.; panad, a market.
Place of commerce.
Leucecome, atown in Arabia, lat. 25° N. on the Red Sea ; leoz, a marsh or swamp ;
coym, a cover. The covered marsh.
Latrippa, now Medina—ta, the place ; tyreaba, of the tribes. The place of meeting
of the tribes.
VOL. XVII. HH
380 On the Affinity of the Phenician and Celtic Languages.
Betuis, river ; beic, double, probably a double river like the Guadalquiver, or Betis,
in Spain. This river discharges itself about lat, 23° north.
Chersonesus.—This name occurs about lat. 24° N. on the coast of Arabia, in the
Red Sea; but I have sought in vain for a promontory or peninsula answering to the
meaning of the Greek. I am, therefore, inclined to think, that it is a corruption of a
Phenician term having a similar sound: probably cojp-ronay, deceitful happiness, like
our false bay, deceitful promise, treacherous port, of which we have numerous in-
stances in our own geography.
Badeo, a town in Arabia, about lat. 22° N.; bare, gratitude, friendship, kindness,
Sacacia, a town about lat. 21° N.; rare, plenty, abundance ; ace, a leading to.
The town in a fruitful neighbourhood.
eli, now Heli, a town in Arabia Felix, lat. 18° N., on the coast : perhaps ajle, per-
Fume, odour, smell.
Gasandi, now Ghezan, a town in lat. 14° N.; ze1r, promise ; aon, country. The
town or land of promise, or invitation.
Orine, an island near the coast of Ethiopia, lat. 16° N. This was probably a market
for gold, and the Ophir of the Scriptures. Its contiguity to the African gold country,
and its name indicates such a circumstance; oj, golden ; in1y, island. ‘The golden
island, or market for gold. Ophir is indicative of the same idea; oip, golden ; reap,
man. Men inhabiting the country of gold. Men who dealt in gold—as we call them
goldsmiths, refiners, or workers.
Sava, a town in Arabia, lat. 14° N. This town may have acquired the appellation
from quarrel or fight having occurred there ; rabav, a skirmish ; or from yabar, sorrel,
which may be had in abundance there.
Musa, a town in Arabia Felix, in lat. 12° N.; mear, fruit ; a, hill, or eminence.
The fruitful hill.
Ocelis, a town or place in Arabia Felix, near the Straits of Babelmandel ; oc, the
sea ; aleay, a pleasant place. A pleasant or agreeable residence on the sea.
Having now examined most of the names mentioned in Ptolemy’s Geography in the
Arabian Gulph, I now proceed to the south and south-east coast of Arabia, on the
Erythrean, or Indian Ocean, a country, as before stated, which was the habitation of
Sabeans and Homerite, or Arabian Phenicians, before their discovery of the Medi-
terranean sea, and the building of Sidon and Tyre.
Lamala, the wet or oozy hills; caom, swampy; al, rocks. This name indicates
an unwholesome position.
Minea, the hilly country on the western shore of the Hargiah river: this is un-
doubtedly manac, abounding in ore—the mine country. The Phenicians were the
first great miners,
Abisama, now called Cape Aden, a peninsula in Arabia Felix—the abode of delight ;
abad, abode ; yam, pleasure or delight.
On the Affinity of the Phenician and Celtic Languages. 31
Cana Emporium—the Hargiah river—the chief market ; caai, the head, or chief
market—the metropolis of the Homerit, or Arabian Phenicians, or mariners.
Corana, a town on the west branch of the Hargiah river, situated where the ruins
of Dhafen now appear: the site or city of a fair, or annual market; con, a place,
district, or neighbourhood ; aonaic, of a fair or market.
Sanaa, the capital of Arabia Felix—the river or stream of ships; ya, a stream ;
naj, ships.
Mariaba, now called Mareb, a city in the mountainous country to the east of the
Hargiah river—perhaps the boast or glory of seamen ; mapayde, of the seamen ; ajbery,
boast or glory—pronounced Maraba.
Saphor, a place on the east bank of the Hargiah ; a, the river ; ror, protection,
a defence, or fortress, on a stream.
Catabanum, a place in the mountains to the east of Mareb—the worship of the
woman goddess, or female deity ; caca, the worship ; bean, the woman. This was,
no doubt, a temple to Onvana, the same deity who was worshipped by the Celte—the
Diana, or Ardurena, whose temple was erected in the ards, or high mountains.
Chatramotide, a people living on this fruitful coast to the east of the Hargiah—a
name which well expresses their position ; cat, a tribe ; tpog, the shore ; mocac, fruitful.
Sochor, a country about one hundred miles east of the Hargiah, on the coast now
called Doan, or Dofar ; roiceapn, bountiful, plentiful, liberal.
Prion river.—lIf the water of this river be foul, dirty, unwholesome, or fetid, it is
from the Celtic word bxean, which means all that. Thus the Severn, or Sabrina, was
so called, as the dirty stream.
Savalite, a people residing to the east of the mountains, now called Lous Kebir.
The people rich, or wealthy, in precious stones ; yordbip, rich, opulent, wealthy ; Ur,
precious stones.
Corte, island—now called Maziera ; perhaps caot, showery.
Syagros, promontory—now called Ras al Had, or Rosalgate, the eastern point of
Arabia. The farther promontory ; ia, farther ; zpor, nose, or promontory.
Moscha, now Muscat ; mor, drought, exhaustion ; caota, shower. The place of
few showers, or little rain.
PERSIAN GULPH.
Heeren tells us that there are indications of the Phenicians in the Persian Gulph ;
and Herodotus distinctly says, they came originally from Chaldea through it, before
they settled on the southern coast of Arabia. I have before mentioned the islands
of Aradus and Tylos, or Tyre, and therefore need not again advert to them.
Indus, or Sind—pin> is also the old name for the Shannon river—the old river ;
rind, old ; aban, river.
32 On the Affinity of the Phenician and Celtic Languages.
MALABAR COAST, FROM THE GULPH OF CAMBAY, TO CAPE CORMORIN AND THE ISLAND
OF CEYLON.
Dachanabades, the country of misfortune and plunderers, or robbers; da)z, plun-
derer ; ana, ill-luck, or misfortune ; bad, district.
Pandionis regio, from 10 to 12° N. lat.—The mountaineers ; pannal, a crew or
body of men, a tribe ; >on, surface, top, summit.
Baragazenus, Sinus. The Gulph of Cambay—the gulph of the small fruitful
stream ; banac, fruitful ; zaiyin, smaller torrent, or stream.
Baragaza, now the town of Baroach. ‘The town on that stream.
Calliana, now called Bombay. The small or narrow district ; caoylamnz, the nar-
row district, the country between the Ghaut Mountains and the sea.
Tyndis, now called Choule, or in that neighbourhood, neat and clean ; dejn, neat ;
vejy, clean.
Suppora, a place about lat.18° N. The pleasant town; yuba, pleasure, delight ;
apna, country.
Musiris, in 18° N, lat. ; muy, pleasant ; spay, variegated, pretty place, near Severn-
droog.
Limyrica—Rampara, in lat. 17° N. The place of trade ; tai, a hand, or adroit-
ness ; a, for; yeic, trade. Limeric in Ireland, may be from the same root ; though
it is generally considered to be derived from laympuz, a ford, or pier. The latter may
be also the origin of this name.
Barace, now Goa.—The lowland swampy district, with aquatic plants growing in
it, from bappoz, aquatic plants.
Nelcynda. The first cloud, probably the monsoon first makes its appearance here ;
neal, a oloud ; cyonbva, first.
Sisecreine island. ‘The protection of trade or merchandize ; yeayda, defence, pro-
tection ; cpeana, the merchant’s business or merchandize. The coast to the north is
marked Prrara, by D’Anville. This island was a fortified depot for merchandize. I
had written thus far, when referring to a modern map, I found that it now goes by the
name of the Fortified Island, from the remains of ancient earth-works which cover
the whole extent of its shores.*
Cotonara, now called Canara. The shore of the country on the sea ; za0t, the
sea ; an, of the ; apa, country.
Elancin Port—ejlayn, an island or peninsula ; cea, a head. The port of the
headland.
Au Cottiara. The tribe living on the country on the sea; a), ¢ribe ; zaot, the
sed ; ana, country ; now called Cranganore.
* Captain Eatwell of the Indian Navy, who knows it well, assured me that the island is nearly covered
by antient earth-works,
On the Affinity of the Phenician and Celtic Languages. 33
NAMES ON THE ISLAND OF CEYLON.
Taprobana, the old name of Ceylon. The bright sparkling principality of the
ocean ; cab, the ocean ; puad, lordship or principality ; ban, white or bright. Ceylon
produces greater variety of precious stones than any other country.
Comaria, now called Cape Cormorin, the south point of Hindostan ; comap, the
point: nose, promontory ; 14, a country.
Dagana, aplace in the south point of Ceylon, now called Matura ; az, good ; ana,
Suir weather, or climate.
Sindocan, now Columbo—rjon, stormy weather ; vocan, harm, wyury. The place
of storms.
Anuro Grannan, a place on the Arippo river ; anuayp, once, at one time, formerly ;
zjuanan, the royal residence,
Ganges, now called the Montiganga river, discharges itself at Trincomalee—the
further torrent, or most northern in the island.
NAMES ON THE COAST OF COROMANDEL, BEGINNING AT CAPE CORMORIN,
Sosicure, a town in lat. 8° N. The mountain town; ro, this; pyozaipe, moun-
taineer. A place built on, or near, the mountain.
Calligicum—cala, a port ; xeocam, of wanderers, or voyagers. One of the mouths
of the river.
Nigama, now Negapatam, nezam, an indenture in the coast ; a bay, an inlet.
Chaberris, now Caverypatnam ; cabap, a conjunction or confluence with ; yay, a
river that joins others.
Maliarphia, now Meliapour, the land of merchants ; malja, a merchant ; ia, land
or country.
Malanga, a town on the river, about 13. N. lat. The good harbour for ships ; ma,
good ; longa, for ships. Here we have the Celtic longa, for ships, again.
Tyna, river in lat. 14° N. The same as the Tyne, in England; ceign, hasty,
rapid river.
Palura, a town in lat. 16° N. The town of strife or contention ; bal, a town ; una,
strife or contention.
Mesolia, now Masulipatam. The cliff of fishing; .meay, fish ; asl, cliff.
Calingon (Portus). The harbour for ships; ca, house, refuge ; longa, of ships.
Here is longa again.
Tyndis, river—now Guadaveri. ‘The rapid but shallow river ; ceizn, rapid ;
ojorc, shallow or dry.
Calinga, now Calingapatan ; ea, harbour ; lonza, of ships ; longa, again.
Cosamba, now Ganjam. The river which runs through a ravine, or high bank ;
cor, a ravine ; aman, a river.
VOL, XVII. ri
34 On the Affinity of the Phenician and Celtic Languages.
Cambysum, now called the Subunreeka river. The crooked silk river; cam,
crooked or winding ; bioyap, silk.
Tilogrannum.—The Delta land at the mouth of the Ganges, now called the Sun-
derbunds, a place of jungle and disease. ‘The land of deformity, ugliness, or danger ;
calam, land, earth, district ; zpanead, or zpaneam, ugliness, abomination, baseness :
a most appropriate name.
NAMES ON THE EASTERN COAST OF THE BAY OF BENGAL BEGINNING AT THE MOUTH OF
THE CHITTAGONG RIVER.
Catabeda river, now called the Chittagong. The river of friendly boatmen ; cata,
Friendship ; bavac, having boats.
Baracura river, now called the Sunkar. ‘The weedy river, or the marshy river
with weeds ; baypoz, plants that grow in the water ; cuppac, a marsh, or fen.
Triglyphon, the river of Aracan—the dirty river with three branches ; cya, three ;
zlajb, dirty water ; aban, river.
Tecosanna—the house of dissolution or death ; teac, house ; 0, of; yanad, dis-
solution, or dying. The mouth of the river of Aracan: perhaps one of the most
unhealthy and pestiferous places on the globe.
Sabva, a town on the coast, a little to the south of the above; yab, death ; ua, coun-
try. The unhealthy country, or district of death.
Mareura, one of the mouths of the river Ava, the Surawaddy: the sea of strife ;
mapa, sea; uta, contention, strife.
Barabenna, a place at one of the mouths of the Ava river: the head or promon-
tory of strife ; bana, strife, anger ; 4, of ; bei, head.
Tamala, The S.-W. promontory of the Delta, formed by the mouths of the Ava
river, now called Cape Negrais : the place or habitation of the plague, or disease from
Malaria ; caom, the plague or pestilence arising from unwholesome exhalations ; ayll,
a place or locality ; caom, also means ooze, or swampy low land: it also means death.
Sabaricus sinus—the Gulph of Martaban or Pegu: the gulph into which the Su-
rawaddy and other rivers empty themselyes—the gulph of the power of death ; yab,
death ; anacar, power. In this bay is the entrance to the port of Rangoon.
Leste Daone—The people of that part of the coast of the Burman Empire which
lies on the Bay of Bengal, between 9 and 17° N. lat. This appellation is very re-
markable ; its literal meaning in Gaelic is, the contentious or quarrelsome people ;
loy>a, contentious or quarrelsome ; >a0ne, people: the pronunciation is precisely
Lestee Daonse, as given by D’ Anville.
Berobe, now called Mergi, a place on the coast, in lat. 14° N. ; be, visage ; peab,
crafty—the people of dissimulation, or fraud.
Ligor, a place on the peninsula of Malacca, about 8° N. lat; bazon, a tongue ;
which exactly answers its character.
—=
<ee
sens
On the Affinity of the Phenician and Celtic Languages. 35
Malencolen ; mool, bald, naked ; an, of the; colam, body. The land of naked
people.
Zabe, a town on the point of the peninsula of Malacca ; yab, death.
ISLANDS ON THE SAME COAST.
Bazacata, in about 18° N., now called Cheduba ; bayod, slaying ; ceadac, striking :
the island which strikes with disease and death.
Andaman, in lat. 14° N.—the evil island; andac, evil ; man, sin: the very abo-
minable islands: they are called the islands of the cannibals, or anthropophagi.
Maniole, the little Andaman island, about 12° N. ; man, sin, wickedness ; jola,
loss, destruction: the island of wickedness and destruction ; the people here are also
called anthropophagi.
Baruss@, now Nicobar ; bapj, death or disease ; apacd, power : the islands of death
or disease.
Sabadibe, in lat. 6 N.; yabad, quarrel! ; jbe, people: the quarrelsome people.
Jabaidii, now Sumatra ; 1s, country ; bard, of love : the rich and lovely country.
Perimulicus sinns—the Straits of Malacca. The gulph of cruel or savage pirates ;
plonard, a pirate, or sea robber ; mjolac, cruel, savage, beastly: pronounced Peri-
mulac. This name is appropriate, even to this day.
Malacca is probably derived from this word’ mjolac, cruel, savage ; 14, country.
The cruel country.
NAMES ON THE EASTERN COAST OF THE PENINSULA OF MALACCA.
Thagora, in lat. 4° north; tazay, fight, battle : the place of skirmish.
Balonga, now called Patara, in lat. 9° north: this is the same name as that of
Barlonga, on the coast of Portugal; bapy, death, destruction; lonza, of ships. No
doubt given for the same reason—the dangerous rocky islands on this coast, where
many ships have been wrecked.
Cotiaris river, the river of Cambojia—the river of boats ; coreojn, a boat builder.
It is a singular fact, that the houses are built in this river, upon stakes, or piles, driven
into the bottom of the shores of the river. The inhabitants communicate by boats.
Cattigara, a little to the north of Camboja river. Whether this place had its name
from cactazad, temptation, it is impossible to say ; but it is probable, from the pro-
nunciation, it is the same.
D’Anville traces the coast very little farther northward, and gives no names but
Sinhou, which he calls Sinarum Metropolis, or the chief city of the Chinese.
SOUTHERN COAST OF ARABIA AND ISLANDS.
Dioscoridis insula, now called Socotora, an island in lat. 15° north, about five leagues
from the north-east cape of Africa or Gardefan, This name indicates its fruitful-
36 On the Affinity of the Phenician and Celtic Languages.
ness ; djanac, abounding in fruit, fertile ; oi, eastern ; 3, high land: the eastern
island of high land, abounding in fruits and corn.
Sacalite sinus et insul, islands on the south-east of Arabia; rac, plenty, abun-
dance ; \cead, overflowing, fruitful, bounteous ; islands of abundance.
Availites sinus et emporium, on the coast of Africa, near Babelmandel—the wel-
come ; a, the ; rajlce, welcome: the desired port.
Aronata, Promontorium, the north-east Cape of Africa, now called Cape Gardefan,
or Gardefoy. The commencement or beginning of the broad sea ; apeanac.
Barbaria or Asania—the country from Cape Gardefan to the river of Patta. Both
these names very significantly describe this district, which is most wretched, unhealthy,
unhospitable, and barren ; there is not a river, or creek, for several hundred miles ;
banban, deadly ; 1a, country ; ayan, evil ; 10, country.
Rapta, the capital of Barbaria, about lat. 3° south, on the banks of the river: per-
haps paobta, form : separated by convulsion.
Menuthias Insule, now Zanguebar, an island situated in about lat. 65° south ; mayn,
riches, abundance, goods, wealth ; uat, earth or mould, ‘The rich island of fertile
soil. This is a very accurate description of Zanguebar, Captain Owen, who lately
surveyed this coast, says this island is eminently fruitful, and produces sugar, grain,
and fruit, in the greatest abundance.
I now leave this etymological examination to the scrutinizing criticism of the most
incredulous and sceptical, as well as to the candid. . I feel convinced of the utter im-
possibility of my conclusions being otherwise than correct on the whole. I may have
fallen into some inaccuracies, but that so many coincidences of accurate descrip-
tion, and precise terms of meaning and import, should, be accidental, appear to me
quite impossible. A few names and sounds might happen perchance ; but that all
the names’ on lines of coasts, of some thousand miles extent, should be significant, and
in correct and applicable terms, of the nature and properties of the places described
in any language but that of the people who conferred these names, cannot I think be
seriously asserted. It may also be observed, that where two names are given by
Ptolemy, each are found equally expressive of the nature and qualities of the place.
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Notes on the Statistics and Natural History of the Island of Rathlin, off the North-
ern Coast of Ireland. By James Drummonp Marsuatt, M.D., Secretary to the
Natural History Society of Belfast, §c.
Rathlin forms one of that vast multitude of islands which are every where scattered
round the shores of Ireland. Upwards of six hundred of these have been enumerated,
of which one hundred and forty are inhabited—the others, consisting chiefly of islets
and holms, are tenanted only by cattle or birds.
This island is situated off the northern coast of the county of Antrim, in north
latitude 55°15. It isa part of the county, a rectory of the diocese of Connor, and
a distinct parish, having been separated from that of Ballintoy in 1723.
Doctor Hamilton justly observes, that “the name of this island has suffered so
many variations in its orthography, as to render it now very difficult to determine what
may be the most proper. It is called Ricnia, by Pliny; Ficina, by Ptolemy ; Riduna,
by Antoninus ; Raclinda, by Buchanan, the Scotch historian, (who classes it among the
Ebudz or Western Isles of Scotland); Raghlin, by Sir James Ware; Rathlin, by
Sir William Petty, and most of the modern map-makers.”* Mr, J.O’Donovan, to whom
the student of Irish antiquities is largely indebted, farther informs us, that “ it was called
Reépayyo and Raclame by Tigernach ; Rachra, by John Lesley ; Rachrine, by John
Fordun, in his Scoti-chronicon ; and Reacpamy, by Dudley M‘Firbis.”+
The nearest point of Rathlin lies about three miles from the promontory on the
mainland, called Fairhead, but from Ballycastle it is nearly 55. The usual point of
disembarkation in Rathlin, is Church bay, which lies at the distance of 7} miles from
* In anote he subjoins, that “ Raghery, as pronounced in Ireland, corresponds with the spelling and sound
of the name in use at this day. If one were inclined to speculate in the dangerous field of etymology,
perhaps Ragh Erin, or the Fort of Erin, might appear to be somewhat in the midst of these various
sounds; and the command of the Irish coast, which must have attended the possessors of this island in
early ages, might make it not unaptly be styled the Fortress of Ireland.”
+ Sce note to a translation “from the Autograph of the Four Irish Masters, in the Library of the Royal
Irish Academy,” by J. O'D. in the Dublin Penny Journal, vol. I. p. 315. In the “ Ancient Topography
of Ireland,” contained in the 11th Number of the Collectanea, (p. 411), it is stated, that “ all these words
are derived from Fach, Ridh, Rudh, Riada, and Reuda, a tribe or habitation ; and ean or lean, water ;
whence the habitation in the water—the present isle of Rathlin.”’
VOL. XVII. KK
38 J. D. Marsuaut on the Statistics and
Ballycastle ; this, therefore, may be considered the mean distance of the island from
the mainland.
The form of the island has been compared, like Italy, to that of a boot,* the toe
pointing to the coal-works of Ballycastle—the heel, where Bruce’s Castle is situated, to
Cantire—and the top to the great western ocean. Towards the middle, which lies op-
posite Ballycastle, it is bent in an angle, and thus is formed Church bay, almost the
only good harbour in Rathlin.
The length of the island from the Bull or western point, to Bruce’s Castle on the
extreme east, is 54 English miles. From Rue-point, the most southerly, to Altacarry,
at the north-east extremity, the distance is upwards of four miles. The greatest
breadth of the island at any part is 1} miles, and the narrowest, half a mile.
The highest point of Rathlin is 447 feet above the level of the sea, it is in North
Kenramer,t at the north-western extremity of the island. The other highest points
are:
Kenramer Head, 445 feet. Sliebh-an-all, - 347 feet.
Farganlack Point, 347 do. Bull Point, = 295 do.
Lough Cleggan, 350 do. Slack-na-Calye + 240 do.
Altahuile = /S00rdb. Broagh-mor-na-Hoosid 237 do.
Kintroan Head 318 do.
So precipitous are the cliffs, that from the vicinity of Bruce’s Castle, round the whole
northern shore, by the Bull-point to the church in Church bay, the lowest pomt is
180 feet above the level of the sea, and the mean height may be said to be 300 feet.
From the striking similitude existing between the Island of Rathlin and the
adjoining continent, it is the general opinion that this island had, at one period,
formed a part of the county of Antrim, from which it has been separated by some
violent convulsion of nature. All geologists who have made this the subject of inquiry,
have stated, that in geological structure the island and adjacent continent are accu-
rately thesame, the principal strata in both being limestone? and basalt. Along the north-
eastern coast of Ireland, for a space of at least sixty miles, these strata every where
present themselyes—in one place the limestone rises to a considerable height above the
level of the sea, and in another gives place to the basalt. On the range of cliffs
running westward, and forming the northern boundary of Church bay in Rathlin, we
see the limestone rising abruptly from the ocean, and forming a line of coast fantasti-
cally beautiful. At Kenbaan-head, on the mainland, corresponding in situation to
these cliffs, we have the limestone appearing in a similar manner. In both, the lime-
* «Sir William Petty says it resembles an Irish stochin, the toe of which pointeth to the main lande.”’—
J. O'D.
+ Ceai Ratan, the thick head, or promontory. The south end was anciently called Ceai) Caol, or
slender head, being a small point of land, pointing to the north coast of Antrim.”—J, O'D.
+ The term limestone must be considered synonymous with the English chalk.
——
Natural History of the Island of Rathlin. 39
stone is overlaid by the basalt. To the east of Bruce’s Castle, the limestone disap-
pears ; and at Doon Point, which is situated exactly opposite to Fair Head, that singu-
larly beautiful arrangement of basaltic pillars, which is very imperfectly delineated in
the sketch, may be observed.
Sandstone, coals, iron, ore, &c. the substances which form the eastern side of Bally-
eastle-bay, and which appear different from the common mineral productions of the
country, may also be traced directly opposite, running under Rathlin, which, in con-
nection with other circumstances, would tend materially to confirm the opinion of their
being a continuation of the same general strata.
Doctor Hamilton entertained the idea, that this island, standing as it were in the
midst between this and the Scottish coast, may be the surviving fragment of a large
tract of country, which at some period of time has been buried in the deep, and may
have formerly united Staffa and the Giant’s Causeway.
I have already mentioned, that the limestone traverses the island from west to east.
It is stated, that this chalk or white limestone, when crossed by a basaltic dyke, often
undergoes a remarkable alteration near the point of contact, the limestone becoming
granular marble, highly phosphorescent when subjected to heat. On the west side of
Church bay, within a distance of 100 feet, the chalk is thus intersected by three
basaltic dykes, and converted in each instance into granular marble. Doctor Hamilton
supposes, from the sandy texture of this marble, called by him phosphorescent cal-
careous sandstone, that it might be considered to have been originally formed of water-
worn grains cemented together ; but the gradual change of the white limestone into
this substance, together with the fact of marine exuvie being sometimes contained in
it, seems to preclude this opinion, and renders it more probable that some local cir-
cumstances may have converted the original limestone into this state.
“This sandstone,” Doctor Hamilton remarks, “occurs near Larne, and in Island
Magee ; but in point of phosphorescent qualities, that found in Raghery is much su-
perior.”
I was some time ago favoured by Mrs. Mant, of Down and Connor House, with
some specimens of minerals and fossils, which were collected in Rathlin by the late
Rey. Andrew O’Beirne, while curate of the island. In consequence of no catalogue
having been furnished with these, it is uncertain whether they are indigenous to the
island, or procured from boulders that may have been washed across from the oppo-
site shore. They all appeared, however, to be similar to those found on the Antrim
shore, containing some beautiful specimens of the zeolite family, with calcareous spar,
hornblende, &c.
My friend, Mr. M‘Adam, M.G.S. examined the fossils, which seemed all to belong
to the lias formation : the only species he could at the time determine, were the lima
antiquata and the avicula inequivalvis ; a number of obscure terebratule could also be
4.0 J. D. Marsuat on the Statistics and
discerned. The specimen of fossil coal was similar to that found at the Giant’s Cause-
way and elsewhere.
When strolling along the base of the white cliffs in Church bay, I was frequently
struck with the appearance of nodules of dark flint which occurred amid the substance
of the limestone, and which have been before noticed by Dr. Hamilton. When
broken, they were of a dark grey colour, and imperfect transparency.
Zeolite and calcareous spar I found in great abundance in detached pieces in dif-
ferent parts of the island, but more particularly on the southern side of Church bay,
where the basalt is strewn about in fragments of every shape and size. On breaking
some of these, beautiful crystals of quartz and zeolite appeared in the cells and
cavities.
A mineral somewhat resembling the puozzolana of Italy has been frequently found
along the shores of Raghery. As described by Dr. Hamilton, it is “ very cellular,
sharp and cutting in its feel, of a specific gravity little superior to that of water, and
of the character of a basaltic cinder broken down.” ‘The specimens which I pro-
cured were not exactly analogous in appearance to that just mentioned ; but as it is
said to exist in the state of a gritty powder before its contact with water, and those
pieces which I obtained had, I should think, been submitted to the action of that fluid,
it is scarcely correct to institute a comparison.
Mr. Gage mentioned to me, that specimens of this puozzolana had been forwarded
for experiment, and that it was supposed that it might be serviceable for the same
important purposes as those volcanic products found at Naples and in the Canary
Islands.
In the island of Rathlin there are several systems of basaltic pillars. At Kenramer,
or the western extremity, many may be observed, the pillars standing in a vertical
position, although none of them are very regular. At one of these headlands, called
Croch-an-Teriy, 1 noticed a peculiar arrangement, which I do not believe has been
mentioned by any writer on Rathlin. The pillars were arranged like the spokes of a
wheel, each running out from a common centre. The headland is between three and
four hundred feet high, and the columns near its summit, against the face of the cliff,
being in some places fractured and interspersed with grass, present an aspect more
interesting than if they had been arranged in unbroken regularity.
At Thivigh, or the side point, there is a headland which slopes down into the sea and
is covered with grass; but the section, sideways, exhibits two assemblages of square
pillars not unlike those of Fairhead.
Near Ushet, on the south-east side of a hill, named in Irish Broagh-mor-na-
Hoosid, there is an elegant causeway, which runs to the extent of between four and
five hundred yards. ‘he pillars are pentagonal and hexagonal, a few only having
seven sides.
Natural History of the Island of Rathlin. 41
But the most remarkable disposition of columns, I believe, at present known, occurs
at Doon Point, on the south-eastern side of the island. In Dr. Hamilton’s words,
“the base of this little promontory isa natural pier or mole ; above this is a collection
of columns, many of which are curved, apparently assumed in conformity with the
surface on which they rest, and inducing a belief that they were so moulded when in
a state of softness ; and above this arrangement there is a variety of differently dis-
posed columns, partaking of every position and form in which basalt has yet been dis-
covered.”
I took the sketch of Doon Point, from which I made the drawing, from a headland,
a short distance to the south ; and on examining the seat which I occupied, I found it
to be a basaltic column, one of a series which projected quite horizontally through
the face of the headland, thus exhibiting their ends to the view of any one looking
from below. ‘The arrangement of the basalt on this spot was so varied that it would
haye required hours to note the different positions of the columns.
Caves are very numerous in Raghery, particularly on the northern shore of the
island, where the heavy sea which almost constantly beats against its base, has formed
many excavations in both the limestone and basalt. In sailing round the shore, these
caves can, for the most part, in calm weather, be entered by a boat; they vary much
both in extent and form, some retiring only a few feet from the edge of the cliff, and
others extending as far as the eye can reach.
The finest of the caves which I had an opportunity of seeing, is Bruce’s Cave, si-
tuated a short distance north-west of the castle. It rises at its entrance, as well as I
could conjecture, sixty or seventy feet, and appeared from forty to fifty in depth,
formed of noble arches of dark basalt, resting in layers behind one another. Bruce’s
Cave faces the Northern Ocean ; and, as I have remarked, the sea which sets in on this
part of the coast is tremendous. Although the day I obtained a view of it was
almost perfectly calm, yet the swell was too great to permit us to approach its en-
trance, and I was reluctantly compelled to pass it after a very imperfect inspection.
The other caves on this part of the island were all smaller than that just mentioned.
Some of these we entered, but others were so situated as to prevent the boat ap-
proaching them. ‘The echoes excited by the firing of our guns in some of these were
awfully grand, and the rocks rung for many seconds. We never failed to disturb
the corvorant from his apparently dismal seat in the most retired part of the cave, and
his hard croak sounded in unison with the appearance of his habitation.
The scenery on this shore far exceeded in grandeur any I had before witnessed.
The rocks are not only of very considerable height, but from their rising abruptly
from the surface of the water, appear much more so than they really are. The line
of the coast is not straight, but intersected into different amphitheatres, the bases of
which are composed of huge masses of limestone, converted by the incessant rolling
of the waves into the most fantastical shapes. Aboye the limestone, the grass and
42 J. D. Marsuaut on the Statistics and
earth appear, and still higher the basalt, arranged in many places in regular and beau-
tiful columns, similar to those at Fairhead, or the Causeway, and exhibiting, on one
of the cliffs, the singular appearance of which I have already spoken.
On the southern side of Church-bay, on the headland called Cloch-a-doos, there
are four caves situated a short distance from the water’s edge, but considerably above
that elevation. In one of those which I entered, the mouth of the cave was at least
thirty or forty feet above the level of the sea; but the floor of the cave gradually
descended towards its extremity, which was too retired for inspection except by torch-
light. Doctor Berger notices an interesting geological fact about some of these cayes :
** Although excavated in the basaltic rock, and at a point remote from any calcareous
formation, they are nevertheless invested with calcareous stalactites depending from
their roofs, and by their droppings on the floor, depositing a crust of about an inch
in thickness.” Doctor Berger thinks this circumstance worthy of attention, since
‘calcareous matter seems evidently, from the situation of the caverns, to have been
derived from that which enters as a chemical ingredient into the composition of the
basaltic rock, separated from the mass, and deposited in its present situation by the
percolation of water, which the rains or springs must have furnished.”
In those which I visited, I did not observe any calcareous depositions ; but I ob-
tained from Mr. M‘Donnell a stalactite of very considerable size, which he had pro-
cured from one of these caves some time before the period of my visit. I may add,
that these are the caves in which the bones of different animals were found by Doctor
Andrews during the past summer, a notice of which was laid before the Meeting of
the British Association in Edinburgh ; and from the deposition of marine exuviz, it
was conjectured that the respective levels of the sea and land had there undergone
some material change.
Rathlin is, generally speaking, fertile, and in those parts where the land is well
cultivated, good crops are produced. It has been already mentioned that a bed of
limestone traverses the island from east to west, and lime is in consequence abun-
dantly procured for manure.
The crops usually grown, are barley, potatoes, and oats, with occasionally some
wheat and flax. The wheat would grow well in many parts of the island, were the
attempt made to procure its introduction. At present it grows only on the farms
belonging to Mr Gage and Mr. M‘Donnell ; but on these, the crop was fully equal
to what I had seen on the mainland ; and the opinion universally expressed was, that
wheat might be grown as abundantly as any of the other grains.
Of barley, besides the quantity necessary for home consumption, the inhabitants
are enabled annually to export upwards of 90 tons to Scotland.
The oats, wheat, and flax which are grown, are also made use of on the island ; and
potatoes are exported to Ballycastle and other places, whence formerly they were im-
ported into Rathlin.
Natural History of the Island of Rathlin. 43
The reason for this increased attention to agricultural pursuits, appears obvious,
when we consider that kelp now obtains in the market scarcely half the price it did
twenty years ago, and hence the inhabitants have of late years bestowed their labour
on the improvement of their farms; and although these now occupy the greater por-
tion of their time, they find leisure to make a small quantity of kelp.
The most fertile parts of the island are the valleys; the hills are generally rocky
and sterile—some so entirely covered with broken masses of rock, as to render it a
matter of much difficulty to force a passage among them. ‘The stone is made use
of for building houses and fences, and so extensive has been its use, that no less than
thirty quarries are to be found on the island; but many of these are not now wrought.
Rathlin is liberally supplied with fresh water, both by springs and lakes. Of the
former there are above thirty, the most remarkable of which is one situated about a
quarter of a mile north-west of Bruce’s Castle. The water in this well rises and
falls with the tide, although it is about one hundred feet above the level of the sea.
Its rise and fall vary, it is said, from four to six inches, and at spring and neap tides
there is a proportional variation. ‘There are two lakes: one, called Lough Cleggan,
situated on the north-western side of the island, is three hundred and fifty feet above
the level of the sea. It extends over the space of ten or twelve acres. The other,
Lough Runaolin, covers upwards of thirty English acres, and is situated at the Ushet
end. ‘The streams of water which flow into these are very small, so insignificant,
indeed, as not to deserve a name.
Besides these lakes, there are fifty or sixty sheets of water of different sizes, but the
principal of these are in marshy districts. ‘The largest marsh is towards Doon Point,
in a very wild and uninhabited district. On its surface I found, in great luxuriance,
the white and yellow water-lilies, (mymphea alba and nuphar lutea) and also the com-
mon reed.
The leaves and flowers of the lilies covered the greater part of the water’s surface,
and the beautiful variety of their white and yellow petals gave an air of beauty and
interest to the marsh which it would not otherwise have possessed. This was the only
part on the island where these plants were to be found.
The soil in Rathlin is in most places light and dry, but in parts the clay is firm. and
in one of the valleys about the centre of the island, it is well adapted for making
bricks. As I have just mentioned, marshes are very abundant, and in most of the
valleys, peat is dug and used by the natives as fuel ; but the supply thus furnished is
not adequate to the demand, and it consequently forms one of their imports from
Ballycastle. It is, however, principally imported by the inhabitants of the southern
end of the island; for towards the Kenramer, or western end, fuel is much more abun-
dant.
The quality of the soil is, on the whole, good ; for every where I saw the appear-
ance of flourishing crops. The potatoes were planted principally in ridges, and each
44 J. D. Marsuatu on the Statistics and
house had its potato garden attached. This vegetable appeared to thrive well
throughout the island; and I was very much surprised, when sailing along the base
of the white cliffs in Church bay, to see little plats of potatoes appearing here and
there among the enormous masses of rock which had at one time been dislodged from
the adjoining precipices. The only way of getting access to these potato-fields is, by
descending the steep declivities by which they are on every side surrounded; and I
conceived that the value of all the vegetables grown on these sequestered spots, would
not repay the trouble necessary for their culture. I afterwards learned, that the seed
was planted every second year only—that is, that as many potatoes were allowed to
remain in the ground during the winter, as would produce a sufficiently large crop
the following summer, and by this means the labour was materially diminished.
The pasturage ground in Rathlin is very extensive, as might be supposed in so
rocky an island. On all the headlands large flocks of sheep may be seen picking
out the tufts of grass from amid the rocky enclosures ; and no place furnishes a finer
breed than Raghery, Each family has a flock, however few in number, and they
are enabled to kill a sheep occasionally for home consumption, unless an unusually
unproductive season should force them to sell these useful animals. The sheep and
goat will thrive well in situations where scarcely any other domestic animal could gain
subsistence, and hence, to the inhabitants of islands, they are invaluable. The trouble
attendant on rearing them is very trifling—they are turned out to graze in com-
mon, each with its owner’s mark upon it, and they roam over the rocky eminences of
Raghery, free and undisturbed,
The climate of Rathlin is but little different from that of the mainland, the varia-
tions of temperature being nearly the same. In winter there is no snow of any con-
sequence, the weather usually continuing very mild. Fogs are, however, very preva-
lent, particularly in spring and autumn, and they are frequently so dense as to prevent
the island being seen even at a short distance. Hence many vessels are exposed to
danger in rough weather in approaching this rock-bound isle, and shipwrecks fre-
quently take place on its shores, from which no one survives to tell the tale. Nota
winter elapses, during which at least one vessel is not wrecked ; and at the Bull Point,
fragments are frequently driven on shore, as notifications to the islanders that some
unfortunate crew have sunk to rise no more.
In March 1833, two vessels were wrecked on the eastern side of the island, the re-
mains of which were washed on the rocks near Doon Point. The night was dark and
stormy, the crew, in all probability, ignorant of the coast, and a strong easterly wind
setting in on the shore. On such a shore, no hope of rescue could for a moment be
entertained : the islanders were doomed to hear the shrieks of their expiring fellow-
creatures, without having it in their power to render the slightest assistance. The
vessels beat to pieces near Doon, and in the morning their timbers lay here and there
on the rocky points of the shore.
Natural History of the Island of Rathlin. 45
The mariner, in coasting by Rathlin, has every thing to contend with. Even if the
weather be calm, the tides are so irregular, and set down the channel with such a force,
that considerable risk is incurred. If a storm arise and the wind should be blowing on
the shore of the island nearest the vessel, but little chance of escape can be offered.
The shores consist of one entire range of rocks; and these so steep and rugged, as to
prevent any aid being afforded by those on shore, should even a pause in the fury of
the elements enable her cries to be heard by the islanders.
Surrounded as it is by a wild and turbulent sea—from the precipitous nature of its
shores, and the number of caves every where to be found, Rathlin has been in former
days a favourite resort of smugglers. In Queen’s Anne’s reign, a French privateer
had made Church bay her head quarters ; and there is not the smallest doubt that were
there any persons on the island favourable to illicit trade, the system might be pursued
to a great extent. Any one who would take the trouble of walking round Raghery,
must be struck with the impossibility of a body of men preventing boats or vessels from
having intercourse with the island, unless by keeping a sentinel and guard on each of
the headlands. Although many vessels can be boarded coming through the channel
in fine weather, it would be out of the question attempting it during the gales of wind
which, for many months in the year, sweep round its shores.
The channel between Rathlin and the mainland has, it is said, a strong resemblance
to the Straits of Reggio between Sicily and the coast of Calabria, particularly in the
indenting of its shores, the velocity of its tides, and the vortices produced by counter
currents. Like it, the water is frequently agitated and thrown into ridges and whirl-
ings by the violence of the current, the particular direction of certain winds, and the
irregular conformation of the coasts. At times it likewise happens, as I have observed,
_ that a very dense vapour is accumulated over the waters of the channel. If the atmos-
phere be highly impregnated with these vapours and dense exhalations not previously
dispersed by the action of the winds or waves, or rarefied by the sun, it then happens
that in this vapour, as in a curtain extended along the channel for some height above
the sea, the extraordinary phenomenon called the Fata Morgana may be observed.
In one instance, many years ago, a gentleman of undoubted veracity, the comman-
der of a corps of yeomen, being at some distance from the shore, with a party in his
pleasure-boat, distinctly saw a body of armed men going through their exercise on the
beach; and so complete was the deception, that he supposed it had been a field-day
which he had forgotten. A woman also, ata time when an alarm of French invasion pre-
vailed, very early on a summer’s morning, saw a numerous fleet of French vessels ad-
yancing in full sail up the channel. She withdrew in amazement to call her friends to
witness the spectacle, but on her return the whole had vanished!
On the evening I first crossed the channel to Raghery, the boatmen were pointing
out the most interesting objects to my notice, and enlivening the time by the relation
of any little occurrences of more than ordinary note which they had met with. One
VOL. XVII. LL
46 J. D. Marsuart on the Statistics and
of them spoke of the dense fog which they frequently encountered in crossing to and
fro ; and after mentioning the deceptions they occasionally experienced, he stated, that
one morning very lately they had been crossing to Ballycastle during a fog, when sud-
denly they perceived a brig in full sail bearing down on them. The illusion was so
great, that the crew used every exertion to escape being run down, as they momen-
tarily anticipated ; and they had just accomplished their object, when the vessel totally
disappeared. 3
In connexion with this subject, I may mention, that a belief was formerly prevalent
among the inhabitants, that a green island rises, every seventh year, out of the sea
between Bengore and Rathlin. Many individuals, they say, have distinctly seen it
adorned with woods and lawns, and crowded with people selling yarn, and engaged in
the common occupations of a fair. Could this have been the Fata Morgana 2
Here oft, ’tis said, Morgana’s fairy train
Sport with the senses of the wondering swain :
Spread on the eastern haze a rainbow light,
And charm with visions fair th’ enchanted sight.
At first a beauteous island scene behold,
Like that Hy Brasail found, by swains of old,
In ocean’s depths ;—and then a rustic throng,
With booths and tents the forest glades among ;
Next, warrior bands in scarlet files arise,
Chariots and steeds, and towers that reach the skies :
But soon they flit, and bounding in the breeze,
Embattled navies plough the azure seas;
Sail crowds on sail, the boiling wake grows hoar,
And whitening surges climb each sculptured prore.
Thus shifts each pageant, like the scenes that fall
Through lens, or lantern, pictured on the wall
Of chamber dark—till all dissolves away,
As filmy vapour in the noon-tide ray.
Drummond's Poem on the Giant's Causeway.
The tides of Rathlin are most remarkable ; and to navigate the channel between the
island and the opposite coast, requires more than ordinary skill and caution. Against
the north-west point, the Bull, the great body of water which flows from the ocean
during flood-tide to supply the northern part of the Irish channel, is first interrupted
and broken in its course, and counter-tides are here created. Thus, along part of the
coasts of Antrim, Derry, and Donegal, the flood-tide appears to flow nine hours, and
the ebb only three. In Church bay, in Rathlin, the same is the case ; but at Archill
bay, south of Bruce’s Castle, the ebb runs nine hours, and the flood only three. So
prevalent are these currents round the island, that I have frequently observed that
one tide or current will be setting round part of the coast for a short distance from
the shore, while another will be running in the very opposite direction at the distance
Natural History of the Island of Rathlin. 47
of some perches. A person navigating the channel, must not leave either shore with-
out reference to the state of the tides, otherwise, the passage which might be both safe
and expeditious, may become the very reverse. In leaving Ballycastle for Church
bay, the time usually chosen is, at the last of ebb-water, when the current is setting
down towards the north, and when the boat will be carried to the north-western side
of Church bay ; and at the time this is reached, the flood-tide sets down the channel
towards Fairhead, and carries the boat in nearly the opposite direction to that in which
she started, viz. into the heart of the bay. If, howeyer, there be a smart breeze, and
the wind fair, a good boat will make the passage in a direct line across the channel
from Church bay to Ballycastle, without paying that strict attention to the tides,
which, under other circumstances, is absolutely necessary.
Lying at so considerable a distance from the mainland, the depth of water every
where round the shores of Rathlin is considerable. There are but few adjacent banks—
one, a cod bank called Skirnaw, lies between it and Isla, in Scotland. While I re-
mained on the island, I had opportunities of ascertaining the depth of water at differ-
ent parts along the coast, and although it varied much, it generally exceeded ten
fathoms within a very short distance of the shore. In many places a fifty-fathom line
did not reach the bottom ; and I was much struck with the accuracy of the fishermen
in pointing out most minutely the particularly deep or shallow parts, and calculating
correctly the depth of water at those places where we found it desirable to gain the
information.
The greatest depth of the channel between Rathlin and Ballycastle has been ascer-
tained to be fifty-three fathoms, and between the north-east of Ireland and the west of
Scotland, ninety fathoms
There is good anchorage in Church and Archill bays, in ten to twenty fathom
water ; and also at Ushet, where a small rude pier or quay is formed for the accommo-
dation of the boats in that part of the island, and where an occasional collier or mer-
chantman may be seen at anchor.
The traditions of the island of Rathlin do not reach beyond the commencement of
the fifth century, when we learn that it was well inhabited, and was garrisoned by a
small army. At the period alluded to, St. Comgall, a religious individual, landed
here, but was instantly seized and driven out of the island, After him came St. Co-
lumba, the celebrated missionary of the north, who founded in Raghery a religious
establishment, and placed over it Colman, the son of Roi. Here it flourished for the
space of nearly 300 years, in peace and quietness, until the latter end of the eighth
century, when (to use the words of Doctor Hamilton) “ the northern storm filling at
once the whole horizon, and bursting impetuously from the ocean, overwhelmed the
island, burying in blind and brutal destruction the inoffensive ministers of the Chris-
tian religion, in the very moment when they were cultivating the olive branch, and
preaching peace and good will among men.”
48 J. D. Marswatu on the Statistics and
In 790, the monastery established by St. Columba was ravaged and destroyed by
the Danes; and in 973 by a second visitation of these freebooters, who put the abbot
of the island to death.
Tts vicinity to Ireland rendering it an object of importance to an invading enemy,
it became a scene of contention between the inhabitants of the opposite coasts of Scot-
land and Ireland. The memory of a dreadful massacre perpetrated by the Campbells,
a Highland clan, is still preserved, and a place called Sloc-na-Calleach, perpetuates a
tradition of the destruction, by precipitation over the rocks, of all the women in ad-
vanced life, then resident on the island. Doctor Hamilton remarks, ‘that the re-
membrance of this horrid deed remains so strongly impressed on the minds of the
present inhabitants, that no person of the name of Campbell is allowed to settle on the
island.” ‘This feeling, however, seems now to have subsided; for during my visit to
Raghery in the summer of 1834, no such enmity to the name of Campbell was mani-
fested.
During the civil wars which devastated Scotland after the appointment of Baliol to
the throne of that kingdom, Robert Bruce was driven out and obliged to seek shelter
in the isle of Raghery, in a fortress whose ruined walls still retain the name of the illus-
trious fugitive. His enemies, however, pursued him even to this remote spot, and forced
him to embark in a little skiff and seek refuge on the ocean. ‘The ruins of Bruce’s
Castle are situated on a bold headland at the extreme eastern part of the island, im-
mediately fronting Scotland. Although apparently very lofty, the height of the rock on
which the castle stood, is marked, according to the late survey, between seventy and
eighty feet only above the level of the sea.
It rises perpendicularly from the water’s edge; and about forty or fifty feet from
the eastern extremity, a deep chasm traverses the ground, insulating, as it were, the
huge mass on which the outer part of the fortress has been situated. On this, the
ruins now standing, consist only of part of a wall fronting the west, entirely destitute
of all ornament and style of architecture. About eighty or one hundred feet on the
western side of the chasm, the remains of another part of the building are still visible,
from which we may fairly infer, that the castle had originally been of very consider-
able extent. In the face of the rock fronting the south, and immediately under the
wall, there is the appearance of a small cave, in which, it is said by some, that Bruce
concealed himself, the castle not having been built at the time of his residence there.
Considerable difference of opinion has, however, been manifested concerning the
exact date of the erection of this building. Its antiquity is dated at least five hundred
years back; and it is supposed to be considerably older, as the time which Bruce
spent on the island was not sufficient to erect it. One of the strongest proofs
alleged of its antiquity, is, that the lime with which it is built has been burned with
sea-coal, the cinders of which are still visible, and bear so strong a resemblance to the
cinder of the Ballycastle coal, as makes it extremely probable that at some period so
LIL
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Natural History of the Isiand of Rathlin. 49
early as the year 1500, sea-coal had been used as fuel in Raghery. On this subject
Doctor Hamilton remarks—« It might be imagined that the coals were brought from
« Britain, but a little reflection will show that supposition to be extremely improbable
“even so late as the time of Robert Bruce. It was but just then that the English
‘themselves had discovered the use of sea-coal as fuel ; and we find that in the time
“of Edward I. after being tried in London, they were immediately prohibited on
“a hasty opinion that the vapour was noxious to the health of the inhabitants. It is
‘not, therefore, to be readily believed, that at this early period, England could have
‘‘had any extensive export trade in coals; or, if so, it must have been to some po-
*‘pulous and civilized country—to some safe harbour—to a great and commercial
‘town ; but, at the time we speak of, the British charts do not lay down a single
“village on all this line of coast. Further reflection on the subject,” continues he,
“might lead me to suppose, that the building of this castle was of much more ancient
“date, because, in the time of Edward I. the kingdom of Ireland was an almost unin-
“terrupted forest, so that the abundance of more convenient fuel would then have
“anticipated all necessity for searching for fossil coal; indeed, for several ages sub-
“sequent to the year 1171, at which time the English invaders found Ireland to be a
“country overrun with wood, instructions may frequently be found among the annual
‘orders of government, to have successive portions of forest cleared away, for the
*‘ purpose of rendering the country more accessible to the English forces; and it was
‘not until four hundred years after, about the latter end of Elizabeth’s reign, that
‘‘any considerable progress was made in this work of devastation.”
In 1550, Rathlin was invaded by the English, who were repulsed with the loss of a
vessel and several men. We are told in the Annals of the Four Masters, as translated
by Mr. J. O'Donovan, that in 1551, ‘the Lord Chief Justice (Anthony St. Leger)
marched at the head of an army into Ulster, and dispatched the crew of four ships to
the island of Reac pyny, to plunder it. James and Colla, the two sons of Mac Donnell
of Scotland, were on the island to defend it. A battle ensued, which ended in the
total defeat of the English; not one of whom survived the battle, excepting the lieu-
tenant who commanded them on this excursion, whom the Albanians ( Scots ) kept as
a prisoner, until they got in his stead their own brother, Sorley Boy M‘Donnell, who
had been imprisoned in Dublin a year before that time, besides other ransoms.”’
In 1558, the Scots took possession of the island ; but were soon expelled with dread-
ful slaughter by the Lord Deputy Sussex, who seized upon it for the English crown,
and left in it a garrison for its defence. In 1575, General Morris landed here with
a body of men from Carrickfergus, and having killed two hundred and forty of the
islanders, seized upon the castle. In consequence of successive barbarities committed
on the peaceable inhabitants of Raghery by various savage invaders during the unset-
tled ages of Ireland, this little island became at length totally uninhabited, in which
state it is represented in a manuscript of the country, so late as 1580, now in the
50 J. D. Marswatt on the Statistics and
hands of the Macdonalds; and it is further mentioned, that some Highlanders who
fled to it for safety, were forced to feed on colt’s flesh for want of other provisions.
Subsequent to this date, but little is related concerning the island. Doctor Francis
Hutchinson, Bishop of Down and Connor, who published an Irish Almanac and other
works, procured for the inhabitants of Raghery a translation of the Church Catechism
into Irish, with the English annexed. It was published in Belfast, in the Roman letter;
but so scarce has it now become, that it is a matter of uncertainty whether or not a
copy of the Raghery catechism is at present extant.
About the year 1740, the island became the property of Mr Gage, in whose family
it still continues. Mr. Gage is both the rector and magistrate of Rathlin; and from
the interest he feels, and the attention he bestows on the inhabitants, is much respected
by all. His residence is situated in Church bay, within a short distance of the church ;
and in no house is the virtue of hospitality more freely exercised. To Mrs. Gage I
am particularly indebted for many facts noticed m my paper, and for opportunities
afforded me of observing whatever was rare or interesting on the island. To Mr. and
Mrs. Macdonnell I must also stand indebted for many acts of kindness,
Dr. Hamilton, in his “ Letters on the County of Antrim,” and Doctor Drummond
in his “ Giant’s Causeway,”’ are almost the only writers who have noticed any of the
particulars of the island. Doctor Hamilton’s remarks have now been written many
years ; and although comprising much valuable information, are, nevertheless, inter-
spersed with many observations, which, after the lapse of fifty years of civilization and
improvement, would now admit of considerable modification. ‘These, where differing
from those made during my visit, I have noticed under their respective heads, paying,
at the same time, all due deference to an authority so highly respected.
Rathlin contains about 2000 Irish plantation acres, or 3239 English. The popula-
tion is stated by Dr. Hamilton at 1200, or 130 families ; but there are now not more
than 1050 inhabitants. Of this number, upwards of 900 are Roman Catholics, and
the remainder belonging to the Established Church. From this statement, it will
appear that the population has latterly much decreased ; a great number of the young
men go to Glasgow and Greenock, or to some of the ports in Ireland, to learn differ-
ent trades, particularly that of ship-carpenters; and emigration to America has, of
late years, taken away considerable numbers. In one of the years immediately pre-
ceding my visit, upwards of forty had left the island for America; and during the
summer of 1834, sixteen had emigrated.
The inhabitants are simple, laborious, honest, and exceedingly attached to their
native soil. Doctor Hamilton says, ‘that in conversation they always speak of Ire-
land as a foreign kingdom—and a common and heavy curse among them is, ‘ may
Ireland be your latter end.’” In my intercourse with them, however, I did not ob-
serve that repugnance to visit the mainland, experienced by Doctor Hamilton. ‘This,
no doubt, has arisen from the more frequent intercourse of late years, kept up between
Natural History of the Island of Rathlin. 51
the islanders and their brethren across the water ; and the distinction made by him
between the inhabitants of the Kenramer or western end, and the Ushet or southern
extremity of the island, however correct in his time, has now in a great degree ceased
to exist. In consequence of the Kenramer end being of a much more precipitous
character, and not being well situated for intercourse with Ballycastle, its inhabitants
retain more of their primitive manners than those residing further to the south, who,
from having more frequent communication with the mainland, have acquired a more
intimate acquaintance with English manners and customs.
The houses in Rathlin, with the exception of those of Mr Gage, Mrs, M‘Donnell,
and a very few others, are of the very poorest description. I was particularly struck
with this circumstance during my excursion through the island, and one house might
almost be taken as a fair sample of all the others: they are chiefly of one story, and
built of stone—many without a window or chimney, the inhabitants being obliged to
be content with the light admitted by the door, and the hole made in the roof, by
which part only, and that a small portion of the smoke is carried off. To each dwell-
ing-house there is commonly attached a small shed for cattle.
The land is divided into smal] farms; each house has a plot for vegetables, while
the remainder is occupied by potatoes and barley. ‘The only ¢rees on the island, are
those planted in Mr. Gage’s garden, and in the immediate vicinity of his house ; the
hedges number but two or three, the fields being divided by stone walls and fences,
That trees once abounded in Raghery, is beyond a doubt. One of its ancient names
is, as has been observed, employed in consequence of the abundance of wood with which
it was furnished ; and although now scarcely a tree is to be seen, we must not infer
that this has always been the case. In fact, we have indisputable evidence in the wood
which is occasionally dug up in some of the bogs. Hazel is frequently found in con-
siderable quantities—oak is also found, but of small dimensions. Were sufficient care
bestowed in the selection and planting of trees in situations similar to Raghery, they
would frequently succeed, where, at the present day, they fail.
The average number of a family in Rathlin does not exceed six or seven, a few
only contain so many as ten. This is generally attributed to the want of early medical
assistance and other causes. ‘Their food consists of potatoes, oaten and barley-bread,
and fish ; and their own mutton, pork, and beef, in winter, unless when the price of
meat is high, or the season unusually unproductive, when they prefer exporting their
cattle.
The inhabitants employ themselves in tilling the ground, in fishing, and in making
kelp. Towards the latter end of the last century, when kelp gave a good price, it
furnished to the industrious people of Rathlin a rich source of wealth. At the period
I allude to, one hundred tons were annually exported, and the kelp alone frequently
paid the entire rent of the island. Now, however, the demand and price have both
declined, and thirty tons constitute the annual exportation. The kelp is usually ex-
52 J. D. Marswatu on the Statistics and
ported to Scotland in a vessel which pays an annual visit to the island ; and as it would
be very inconvenient to the islanders to bring their kelp to Church bay, just at the
time when the vessel would be anchored there, Mr. Gage has kindly fitted up a small
store-house, where the kelp is allowed to remain till the time of its exportation.
In Rathlin all the larger fuci are indiscriminately used in the manufacture of kelp;
but the fucus nodosus, or bladder-wrack, and the fucus serratus, are the species most
employed. Besides these, they use the chorda filum, or long tangle which they call rrog ;
this, and the /aminaria digitata, as well as the other species, are furnished in great abun-
dance round all the shores, and many a kelp-kiln bears testimony to their profusion.
At low water the women and children walk out on the ledges of the rocks, and
with old knives or reaping hooks, cut off the sea-weed, which is then borne to the
shore, and spread out before the sun to dry: in the evening it is made into little
heaps or hillocks, and in the morning again shaken out, just in the manner in which
they make hay. This process is continued till the weeds are dry enough to be burnt.
An excavation of about five feet long, two or three broad, and two deep, is then made
in the ground, and lined with large stones; and in this, which is called the kelp-kiln, the
dried weeds are burned, the fire being kept up by constantly throwing them on the
flame. During this process, the alkali, and every thing not capable of being dissipated
by the heat, accumulate in the bottom of the kiln ; and when liquid, like molten metal,
are stirred about with an iron rod till they form the hard bluish mass called kelp. I
know nothing which produces a more beautiful and picturesque effect, than when saun-
tering along the sea-shore on a calm summer’s evening, we see the numerous kelp-kilns
sending forth their dense wreaths of smoke. The effect is more striking when the kiln
is viewed from the sea, or from a point of land where the yellow smoke can be con-
trasted with the dark-green of a verdant back-ground. At night, if the kiln be kept
burning, the figures of the persons engaged in the occupation are thrown finely for-
ward by the fire; and this, aided by the reflection from their faces, gives such an air
of wildness to the scene, as to lead the spectator to imagine that he is witnessing a noc-
turnal sacrifice, or the infernal cauldron of Macbeth’s witches :
“What clouds of smoke in azure curls aspire
From many an altar’s dark and smouldering fire ?
What shadowy forms dim gleam upon the sight,
Now hid in fume—now clear with sudden light?
Do stern Crom-cruach’s priests to life return,
And here, once more, their fires unholy burn.?
Or Albyn’s friends their beacon lights relume,
To guide the spoiler through the midnight gloom ?
Or those dread sisters who, on blasted plain,
Are wont to meet in thunder, lightning, rain;
Their cauldron boil, and round it as they go,
Their cursed enchantments in the hell-broth throw ?
ee . —. _ hee
er
Natural History of the Island of Rathlin. 53
Ah! no, a race inured to toil severe,
Of manners simple, and of heart sincere,
Sons of the rock and nurslings of the surge,
Around the kiln their daily labours urge;
O’er the dried weed the smoky volume coils,
And deep beneath, the precious kali boils.”
Drummond.
The customs and manners of the islanders differ in some points from those of their
fellow-men on this side of the channel. ‘The island is divided into townlands ; and
when one of the natives dies, even an infant, the inhabitants of his townland will do
no work that can possibly be avoided: whether they deem it irreligious, or merely
a fit opportunity for self-indulgence, I cannot determine. They observe all their
holidays and fast-days very strictly ; and hence, between these and the funeral holidays,
they idle no inconsiderable portion of time. They possess, in an eminent degree, the
virtue of hospitality. I found them remarkably kind and obliging—ever ready to
point out to my attention any object particularly worthy of notice, or to give me any
information I required.
The only ¢radesmen in Rathlin are, a smith, two or three tailors and shoemakers,
and a few boat-builders. The natives manufacture the chief part of their woollen and
linen clothing from their own wool and flax, grown and spun on the island ; they
also plait straw hats, and knit their own stockings. A public-house has been estab-
lished within these few years, and a shop for groceries and a few medicines.
There are three public schools—one male and two female, to which competent
teachers are appointed, and which are under the superintendence of Mr. Gage’s fa-
mily, who devote much attention to the children. A mixture of Irish and Gaelic
forms the native dialect ; and though the English has made some progress, by many it
is neither spoken nor understood. On entering some of the cottages, I not unfre-
quently found the only house-keepers to be two or three children under seven or eight
years of age. On putting a question to them, I was answered by a wild stare, evine-
ing their total ignorance of the language in which I addressed them. In different
parts of the island, the women particularly, knew nothing of the English tongue.
To the taste which some at least of the islanders possess for music, I can bear wit-
ness ; and I shall make an extract from the journal which I kept, as it contains a
short account of the first eyening I spent in Raghery :
“I reached Ballycastle on Friday the 27th June, 1834, at '7 P. M. My first in-
quiry was for a boat to Rathlin, but to my disappointment I was informed that one
had started about an hour before my arrival. I found one, however, which had just
reached Ballycastle quay, laden with potatoes from the island ; and although the crew
did not purpose returning till morning, I induced them to leave for the island this
evening, which they did at 8 o’clock. Although already sufficiently impatient to visit
VOL, XVII. MM
54 J. D. Marsuatt on the Statistics and
Raghery, an additional stimulus was afforded me in the mildness of the evening. I
could not have chosen a more delightful one—the sun was sinking in the waves, and
he had already dipped so low as to throw the island into deep shade; the water was
clear and calm—not a ripple disturbed its surface ; and every thing around me wore
a pleasing aspect.
«< Aftera most agreeable sail, I landed on the island, and was escorted to the public-
house in Church bay, where I was informed by my boatmen I should get good accom-
modation. Although not exactly corresponding with the character bestowed on it, I
endeavoured to make myself comfortable. The boatmen, after their pull across the
channel, had no objection to drink my health, and they accordingly retired to the large
room above that in which I was seated. The joke and laugh now circulated, and
being musically inclined, a song was commenced. ‘The peculiarity of the tune at-
tracted my notice, and one of the men having kindly requested my presence, if I
wished to hear a Raghery song, I gladly joined the party. A new song was now
begun—it was a duet, to which a chorus was attached, sung by the whole party, The
two principal performers took hold of each other by the right hand, and kept time
with the tune by striking their hands thus entwined, on the table. The song lasted at
least fifteen minutes, and was sung in their native language, with greater spirit and
warmth of feeling than is usually displayed by more fashionable vocalists. 1 retired
to my room, highly pleased with this my first introduction to the inhabitants of
Rathlin.”
The diseases most prevalent in Raghery are, asthma, rheumatism, and diseases of
the eye. The asthma and ophthalmic diseases are principally occasioned by residing in
ill-constructed and ill-yentilated houses, in an atmosphere of smoke, by which the
lungs and eyes must inevitably suffer. In the turf of the island, sulphur abounds, as
I experienced by inhaling its smoke for some time; and the lungs of the inhabitants
being thus daily exposed for many hours to the fumes of sulphur, may lay the foun-
dation of asthma. I not unfrequently found the junior members of families afflicted
with this disease, although under other circumstances it affects adults only.
One of the greatest desiderata of the islanders, is a physician, the want of which is
severely felt. In summer and in calm weather, they can, in cases of emergency,
command one at Ballycastle ; but in winter, when the waters of the channel are fretted
by the fury of the winds, and the wide expanse of water exhibits but one sheet of
foam, it were in vain to seek for aid from the mainland. One, two, or three weeks,
not unfrequently elapse without any communication with the opposite shore ; and a
physician has been detained for many days on the island, waiting the return of milder
weather. Should, therefore, any accident occur, or symptoms arise requiring imme-
diate surgical or medical aid, the poor inhabitants are doomed, perhaps, to see their
nearest and dearest friends cut off for want of timely assistance.
On my first passage across the channel, the boatmen, my Kenramer friends, were
Natural History of the Island of Rathlin. 55
very anxious to discover the purport of my visit to Raghery. Though they all con-
versed in their native dialect, I saw from their looks, and manners, the anxiety they
were unable to suppress. At length, one of them ventured to inquire, if I was “ one
of the gentlemen who looked after the schools?” J answered I was not. ‘* Maybe
you would be one of the government gentlemen going to take observations ?” said
another. Here again they were unsuccessful, and I was at length obliged to inform
them of my profession. A physician so seldom visited their island, that they looked
upon their present voyager as a kind of godsend ; and I observed, that during the
remainder of the passage, I was treated with much increased deference.
Questions on the treatment of different diseases now flowed on me; and we had
become uncommonly good friends when we reached Raghery. They invited me to
visit them when in their part of the island, which I did about a week afterwards. I
was proceeding along the banks of Lough Cleggan, in pursuit of a ring-plover, and in
the course of the chase, I was led over a small hill forming the southern boundary of
the lake. From the summit of this, I overheard my Kenramer friends busied in
cutting turf; they immediately recognised me, and they and their companions form-
ing a party of about twenty in number, gathered round me, and, for at least half an
hour, did all in their power to aid my search for the ring-plover’s nest. They pointed
out to me their flock of sheep and herd of cattle grazing in the valley in which their
cottages were situated ; the animals, though belonging to eight or ten families, were
all kept together, each having its owner’s mark. ‘The inhabitants of this and other
townlands gain a livelihood by cultivating their land and carrying the overplus of
their crops to Ballycastle ; and from their residing three hundred feet above the level
of the sea, fishing and kelp-making are not pursued, as along the southern shores.
One day, while sauntering on the road at Church bay, an old man overtook and
accosted me. ‘‘I understand, sir,” said he, ‘‘that you are a doctor.” ‘* You are
right,” I replied. ‘If it was not taking too great a liberty, then, might I ask what
you would recommend for my leg which has been hurt.” I examined the wound, and
prescribed what I thought necessary. He now wished to pay me my fee for advice,
although his dress bespoke any thing but wealth in its wearer. Of course, I expressed
my obligation for his good intention, but declined the proffered kindness. What then
could he do to oblige me? he would carry my gun—my shot-belt—my botanical
box, or any thing else I had; something he must do to evince his gratitude for my
attention, and I was at length obliged to send him on some errand to one of the ad-
joining houses. I met him frequently afterwards in different parts of the island, and
he never failed to inform all the invalids in the townland of my approach ; and when
I had reached the houses, he would present himself with a train of followers, all de-
sirous of medical advice. In many places he acted as interpreter ; and some days
after I had first met him, I was greatly amused by the dexterity he displayed in disco-
56 J. D. Marswauu on the Statistics and
vering the invalids’ complaints, and making me acquainted with them. Had I wanted
practice in my profession, Raghery offered me every facility.
ZOOLOGY.
The Zoology of Rathlin does not, so far as I am acquainted, offer anything novel, or
afford any species not hitherto known as natives of Ireland. Placed at so short a
distance from the mainland, it cannot be expected that the animals frequenting Rathlin
should be found differing in any considerable degree from those found in the other
parts of the country ; and the few remarks I have to offer on this part of the subject,
will rather serve to mark the habitats of some of the species, than throw light on
that interesting, though, hitherto, too neglected subject—the Natural History of
Treland.
The only quadrupeds on the island, with the exception of those which are domesti-
cated, are the
Common Hare. (Lepus timidus.)
Norway Rat. (Mus decwmanus.)
Common Mouse. (Mus musculus.)
Surew Mouse. (Sorex araneus.)
The Hare (Lepus timidus) is very rarely seen in Rathlin; and I was unable to
procure a specimen, although anxious to examine whether it belonged to the British
species, or to the variety lately discovered to be peculiar to Ireland. The total want
of underwood, such as whins, broom, briar, or heath, must account for the scarcity of
this animal. Dr. Barry states that the same reason, he is satisfied, operates in the Orkney
Islands where the hare is now literally extinct, although in former times it must have
been abundant, as it formed an object of the chase to the ancient Earls of Orkney.
The Brown or Norwecian Rat. As Dr. Hamilton remarks, we might have ex-
pected that the native black rat would have found an asylum in Raghery, although
driven from the other parts of Ireland ; but here the brown rat only is found, having
entirely usurped the place of its predecessor, It infests all the dwelling-houses, barns,
and store-houses, on the island, and is here, as elsewhere, universally detested.
The Mouse (Mus musculus) is found in and near all the houses on the island. I
was unable to learn whether the field mouse is seen here, but
The Surew Mouse (Sorex araneus) is found occasionally in the fields.
LAND BIRDS.
Crnzrzous Eacir. (Falco albicilla.) This eagle generally frequents the island
during summer, and on some of the lofty headlands makes its eyrie. It is, however,
—— Oe
——
Natural History of the Island of Rathlin. 57
but rarely seen. It feeds on young lambs, sickly sheep, fish, and, when pressed by
hunger, on carrion. A friend on the island informed me that he one day saw a sea
eagle strike down a raven, and afterwards soar away apparently regardless of his fallen
victim.
Prerecrine Farcon. (Falco peregrinus.) This falcon, which is found on many
of the headlands in the County of Antrim, may be also seen on the precipices on the
northern shore of Rathlin, where it annually rears its young. During my visit to
the island the situation of one of their nests was pointed out, and a man descended
the cliff and procured two of the young, which were brought to me for sale.
Sparrow Hawk (Falco nisus) is occasionally seen hovering over the corn fields
in search of mice and small birds, on which it subsists.
Kestret. (Falco tinnunculus.) The kestrel is seen in different parts of the
island ; and on the cliffs on the southern shore I had frequent opportunities of ob-
serving it. The young starlings, which were, at the time I allude to, just fledged and
frequenting the island in large flocks, offered to this and the other birds of prey ample
means of subsistence.
Wuire Owx. (Strix flammea.) This is the only species of owl found in Rathlin,
and is but very rarely seen.
Raven. (Corvus corax.) A few pairs annually breed on some of the precipitous
cliffs on the island, but it is a bird only occasionally seen. It is destructive here as
elsewhere to young lambs and sickly sheep, which it attacks and commonly destroys.
Chickens and ducks not unfrequently fall victims to its voracity.
Hoopep Crow. (Corvus corniz.) ‘This wary bird is found in Rathlin through-
out the year. They associate mostly in pairs, frequenting the inland parts of the
island, and occasionally resorting to the shore in search of shell-fish, and other marine
substances, on which they sometimes feed. Like the raven, they breed in the most
inaccessible parts of the rocks.
Macriz. (Corvus pica.) From the total want of wood on the island, the magpie
is very rare, one or two pairs only being seen.
Cornisu-Cuoven. (Pyrrhocorax graculus.) This is called by the islanders
the Jackdaw, and is by far the most numerous species on the island. In the month
of July I found them everywhere, associated in large flocks, at one place frequenting
inland situations, and at another congregated on the sea-shore. They had just col-
lected together their different families, now fully fledged, and were picking up their
food (which consisted chiefly of insects) either on the shore, in the crevices of rocks,
or in the pasture fields. Mr. Selby mentions that the chough will not alight on the
turf if it can possibly avoid it, always preferring gravel, stones, or walls. On Rathlin
its choice of situation seemed to be but sparingly exhibited, as I found it frequenting
the corn and pasture fields in even greater numbers than along the shores. This may
have arisen from the shores being in general high and rocky, not affording them the
58 J. D. MarsHaut on the Statistics and
same opportunities of seeking their food as a low, sandy, or gravelly beach. They
breed on the lofty cliffs overhanging the sea; the eggs are of a whitish colour,
speckled at the larger end with brown. ‘The chough is of a restless active disposition,
hopping or flying about from place to place; it is also very shy, and can be with diffi-
culty approached. ‘Temminck says that the legs of this bird, before the first moult,
are of a dark colour, while Montagu affirms that they are orange-coloured from the
first. ‘The young which I examined were about six weeks old, and m them the bills
were of a brownish orange ; not of that brilliant colour which marks the adult plumage,
but certainly exhibiting enough of the orange to lead us to conjecture that they would
become completely of that colour after the moult. The legs could not be called
“ orange-coloured,” for although there was a tinge of that colour, yet the brown pre-
dominated. I should, therefore, agree with Temminck in stating the legs and feet to
be ‘‘dark-coloured” in the young birds.
Sranyine. (Sturnus vulgaris.) This is one of the most common birds in Rathlin.
It is found over the greater part of the island, but principally about Church bay,
where the houses are more numerous, and where there are a few trees and shrubs.
In July they were assembled in flocks of one or two hundred, dispersing themselves
over the fields and along the sea-shores. They frequented the more rocky parts of the
pasture fields, and seemed busy picking among the loose stones, both here and on the
shore, for insects of various kinds. Mr. Low, in his ‘‘ Fauna Orcadensis,” says, that
in the Orkney Islands they feed, during the severity of winter, on the sea-louse,
(oniscus marinus,) which they obtain by turning over the small stones on the beach
with their bills. It is more than probable that this insect constitutes part of their
food in Rathlin, when the small and inefficient supply of berries and grain has been
consumed ; but I regret that I did not make the matter certain, by examining the
contents of the stomach in my specimens. ‘The cry uttered by this bird was a kind
of chatter, which, when coming from a number assembled together, was rather harsh
and grating to the ear. They build among the rocks; the young are of a brown
colour, lighter on the belly and throat.
Cuckoo. (Cuculus canorus.) ‘This well-known bird annually visits Rathlin for
two or three weeks in May, when her pleasing note is occasionally heard. Her eggs
must be deposited here, as in other places, in the nest of the tit-lark, or some other of
the smaller birds, but, so far as I could learn, it has not yet been discovered.
Cummney Swatitow. (Hirundo rustica.) This bird is not very common on the
island, being only occasionally seen in the vicinity of the houses in Church bay.
Martin. (Hirundo urbica.) This swallow is very generally distributed, being
found in all parts of the island, as well inland as along the cliffs which overhang the
sea. It is the most numerous of the genus in Rathlin. The situations it selects for
the purposes of incubation are the out-houses in different parts of the island, and the
lofty precipices near the sea. The latter it does not select without regard to eligibility
Natural History of the Island of Rathlin. 59
of situation, for I invariably found it frequenting the range of white cliffs running
along the north-western side of Church bay. ‘These have a southerly aspect ; and
being formed almost entirely of limestone, become in summer good reflectors of heat,
and, consequently, render the situation well adapted to the nests of the martin. These
are placed against projecting masses of the rock, and built of mud and the usual
materials. One of these birds which | shot had its mouth completely filled with in-
sects, among which were a large dragon-fly, and one of the tipule. ‘The former was
alive and apparently unhurt. How it had procured this insect, struck me at the time
as singular, for the bird was killed on one of the rocks at least a mile distant from
any marsh or pond, the usual residence of the dragon-fly.
Swirr. (Cypselus murarius.) This bird I found in different parts of the island,
in populous districts, and near barren and rocky cliffs. Although I saw it so often
flying about the headlands on the sea-shore, and conjectured that it had its nest formed
either in the earth at the summits of the cliffs, or in holes ready made in the rocks,
I was unable to ascertain the fact, from the inaccessible height at which it usually re-
mained. Mr. Selby thinks, that the cry made by swifts, when flying in small parties
“near houses, steeples, &c. to be the consequence of irritability, excited by the highly
electrical state of the atmosphere ; and not, as Mr. White supposed, a serenade to
their respective families. From what Mr. Selby says, it would be imagined that the
cry was only uttered in very sultry close weather with approaching thunder-storms, as
these are the periods at which he mentions it as being heard. From the opinions of
these justly celebrated naturalists, I should be inclined to differ, and suppose the
scream to be uttered by the swift when in the enjoyment of the fine weather in which
it delights, and when an abundant store of insects is afforded it, for at these times is its
cry most audible. I have very frequently heard a flock screaming when no thunder-
storm portended, and when the weather was particularly mild, though cool; and in
some places where they abound, I have never seen them without hearing their scream.
Mr. Selby’s remark may be, generally speaking, correct ; but in the north of Ireland,
so far as my observation has extended, the opinion I have formed appears correct.
TurusH. ( Furdus musicus.) One or two pairs are found on the island.
Bracxsirp. (Zurdus merula.) About Mr. Gage’s house one or two are occa-
sionally seen; they breed in the garden.
Repgreast. (Sylvia rubecula.) Very rarely seen.
Wren. (Troglodytes Europeus.) Very rare.
Wueatear. (Saxicola enanthe.) This bird is very numerous on the island, and
may be seen in almost every direction, flitting from stone to stone, or fence to fence.
All situations seemed equally well adapted to it—the barren moor, the gravelly beach,
or the rocky headland. It arrives on the island about April, and after performing the
process of incubation, departs for a warmer region. Its nest is placed under a large
stone, or in the crevices and interstices of rocks and fences. In July, the young
60 J. D. MarsuHatu on the Statistics and
birds are very numerous, and were readily distinguished from the adult by the brown
on the upper parts of the plumage, which in the old birds are bluish-grey. It is
called by the islanders, the stonechat, or stonecheck.
StronecHatT. (Saxicola rubicola.) ‘This lively little bird is common in Rathlin,
although not so frequently met with as the foregoing.
Prep Wactait. (Motacilla alba.) This well-known bird is as numerous in
Rathlin as on the mainland. It prefers the neighbourhood of the houses, and resorts
to the moist watery places which abound on the island, and where a constant supply
of insects may be procured.
Rock Pirtr. (Anthus aquaticus.) This is a constant resident, and is found
round all the coasts, the situation being very rocky, and consequently well adapted to
its habits. It flits from stone to stone, along the shore, and may always be seen, ge-
nerally in search of the smaller marine insects, its usual food.
Sxy-Larx. (Alauda arvensis.) The island is enlivened by the sweet notes of this
delightful songster ; it is a constant resident. It builds in the corn and pasture-fields ;
and in winter, assembles in large flocks in the same way as on the mainland.
YeLttow Buntinc. (Hmberiza citrinella.) Not very numerous.
Reep Buntinc. (Hmberiza scheniculus.) 1 saw a pair of these birds on a
swamp in the southern part of the island where the common reed (arundo phragmites)
grew in considerable quantities, and where they had formed their nest. This was the
only situation in which I found them.
House Sparrow. (Fringilla domestica.) Very common.
Common Linner. (Lringilla cannabina.) This little bird was common through-
out the island; and at the time of my visit was in its full summer dress, having the
bright red markings on the head and breast.
Rocx Dove. (Columba livia.) This bird, better known by the name of wild
pigeon, is not unfrequently seen in Rathlin. It frequents the rocky precipices over-
hanging the sea, and is found in all districts in the island. I found it in flocks of ten
to twenty in the corn fields and open moors. They were very wild, and could not be
approached within gun-shot. They breed in the caves along the shore ; but from the
inaccessible situations they occupy, their nests are seldom disturbed.
Partripce. (Perdix coturnix.) But very rarely met with.
Quai. (Perdix coturnix.) I heard this little bird occasionally in the corn
fields.
WATER BIRDS.
Common Heron. (Ardea cinerea.) There are but few places in the British isles,
where this common, though not uninteresting, bird is not met with ; and in the island
of Rathlin a few pairs have taken up their abode. The numerous ponds and marshes
Tr
ips
Natural History of the Island of Rathlin. 61
with which Rathlin is furnished, render it a desirable situation for a bird like the heron,
_ whose food consists of small fish; for eels abound on the island, and afford it an easily
obtained and plentiful supply. The ponds and marshes are for the most part totally
devoid of planting or other shelter, and surrounded with bleak barren hills, on which
scarcely a trace of vegetation is to be seen—yet here, on these dreary solitudes, the
heron may be seen standing in its apparently listless attitude, on a stone at the edge of
the water, watching for its prey. The heron is gifted with extreme wariness ; and the
situation it selects here, precludes the possibility of approaching within gun-shot, for it
can see and be seen from a great distance. On one of the marshes at the southern
extremity of the island, the water covers the ground to any depth, only after heavy
rains. Here the marsh is almost entirely covered by the nymphea alba, nuphar lutea,
and arundo phragmites, the latter of which grows to a considerable height. This is
the only spot on the island where their nests could be placed, there being no trees,
their usual resort, during the season of incubation. This affords us another instance
of the facility with which birds can adapt themselves to situations different from those
usually selected ; for it is a well known fact, that herons almost invariably place their
nests at the summit of some lofty trees; yet in such a spot as Rathlin, where no trees
are to be had, they choose such a situation as I have described. Here the reeds are
sufficiently tall to enable them successfully to conceal their lurking places ; and the
ground is so soft as to prevent any disturbance from persons inclined to annoy them.
A boy told me, that in the summer of 1833, he ran a heron down ; it had, in all pro-
bability, gorged itself with food, and was unable to make its escape from among the
reeds and grass. :
Common Curtew. (Numenius arquata.) The curlew is occasionally seen during
the summer season, although by no means numerous. In winter, however, they as-
semble in large flocks; and in severe weather, approach the houses and cultivated
grounds, in search of food.
Repsuank Sanppiper. (Totanus calidris.) Isaw only one pair of these birds
on the island, although I expected to find them much more numerous, the marshes
and bleak hills being favourite situations.
Common Sanvpiver. (Totanus hypoleucos.) This bird, like the preceding, I was
surprised at finding so rarely, having seen only two or three round the shores, and
never having met with it about the ponds or marshes.
Common Snipe. (Scolopax gallinago.) On inquiry, I was told that this bird was
seen only very rarely in winter, and was, therefore, quite unprepared for finding it in
July ; which, however, I did on the marsh at the southern end of the island. I was
watching the motions of another bird, when I suddenly heard what I thought was the
bleating of a goat in the neighbourhood. I looked round—the sound was now before
me, at one time very loud, and, at the next moment, apparently at the further extre-
mity of the marsh. After looking frequently around me in vain, I at last heard it so
VOL. XVII. NN
62 J. D. Marsuwatu on the Statistics and
distinctly above me, that I looked up, and there the snipe was hovering, sweeping
down repeatedly towards the marsh, and again wheeling aloft, all the time uttering its
peculiar bleating ery. Its young, I conjectured, were at the time in the marsh ; and
its anxiety was severely tried, while my companion and I remained in the vicinity.
Meapvow or Corn Crake. ( Crex pratensis.) The rail is plenty in the corn fields in
all parts of the island, and forms her nest in similar situations to those on the op-
posite coast of Antrim.
Common GaLuinuLe. (Gallinula chloropus.) This bird inhabits the marsh
which I have before alluded to in the southern extremity of the island, and makes its
nest among the reeds and water-lilies covering its surface.
Crestep or Green Lapwine. (Vanellus cristatus.) 1 found this plover on
some of the high grounds on the island ; but at this season, they are by no means so
numerous as in winter.
Riycep Prover. (Charadrius hiaticula.) This handsome little plover is not
unfrequently seen in summer, on the high, retired, and stony parts of Rathlin, where
it forms its nest. I found it also on the gravelly beach of Church bay, where, near
high-water mark, its eggs have been frequently found. In the month of July they
were running about, accompanied by their young, for whom they evinced the greatest
attachment.
Bean or Witp Goose. (Anser ferus.) Seen in small flocks in winter.
Brent Goose. (Anser brenta.) Occasionally met with in Church bay.
Wuistiine Swan. (Cygnus ferus.) Seen sometimes in hard winters.
Common Witp Duck. (Anas boschas.) Occasionally shot in winter.
Common Tra. (Querquedula crecca.) In small flocks on the marshes and ponds
in winter.
Common Wicron. (Mareca penelope.) Rarely seen in winter. ;
Norruern Diver. (Colymbus glacialis.) Frequents the bays and shores of
Raghery in winter and spring, and has been seen both im adult and immature
plumage.
Footish Guittemor. (Uria troile.) These birds were congregated in very con-
siderable numbers on the north-western extremity of the island, where the high and
precipitous rocks afford them facilities for incubation. They were not, however, so
plentiful as either the razor-bill auks or puffins, but they frequented the same rocks
indiscriminately. This guillemot lays one large egg on the bare rock, to which it is
secured as it were by its peculiarly conical shape, being very large at one end, and
diminishing rapidly towards the other. It is thus prevented rolling off the rock ; but
it was in former times supposed to be retained on the rock by some glutinous sub-
stance applied to one side by the bird. The young guillemots I had frequent oppor-
tunities of examining ; they were, when excluded from the shell, covered with a dark
grey down, of a whitish colour underneath.
Natural History of the Island of Rathlin. 63
Buack Guittemor. (Uria grylie.) This bird frequents the southern or Ushet
extremity of the island—a place totally devoid of any other sea-fowl—and the shores
which immediately front Ballycastle, where I found them in number about thirty,
flying backwards and forwards among the rocks, where they had established them-
selves. I saw only one pair on the northern shores, and could not ascertain whether
they bred there or not. At their breeding haunts on the southern shore they were
very wary, and could scarcely be approached; but the day I visited the immediate
vicinity of the spot I allude to, was so stormy, and the sea ran so high, that I dared
not keep the boat closer to the rocks, in order to examine their breeding places more
particularly. The black guillemots were easily distinguished from all the others, by
the dark plumage and the white spots on the wings.
Razor-sitt Aux. (Alca torda.) This auk was found associated with the foolish
guillemot in countless numbers on the northern shores of Rathlin. It was, however,
much more plentiful than the guillemot, but so much resembling it in general appear-
ance, that by the boatmen they were invariably confounded, and, while sitting on the
rocks, regarded as belonging to the same species. The cry of the razor-bill auk is a
kind of croak, harsh and disagreeable; and by an imitation of it, the birds, securing
themselves behind the ledges of rock, are drawn out from their lurking places by the
fowlers. The egg is similar in size and markings to that of the guillemot ; the young
were covered with dark grey down, the bill slightly hooked at the tip, but not pre-
senting the peculiar marks which characterize that of the adult.
Common Purrix. (F'ratercula arctica.) These little birds breed in great num-
bers at the Bull Point, and the headlands adjoining, where the rocks are based with
mould, and intersected and covered here and there with patches of grass ; thus afford-
ing them facilities for scooping out their nests. These we found wherever the earth
appeared among the rocks. Here they excavate or burrow in the mould to the depth
of two or three feet; and, at the extremity of the excavation, the egg, which is white
and about the size of a hen’s, is deposited on the bare earth. From being surrounded
by the damp mould, it appears, when taken from the hole, of a dirty brown, but, on
being washed, it acquires its natural colour. The puffins seemed equally numerous
as the razor-bill auks ; they took possession of the earthy parts, while the latter sat
close beside them on all those bare ledges of rock not otherwise occupied. These
birds, with a few guillemots, were met with in considerable numbers along the range
of white cliffs facing the south, and forming the northern boundary of Church bay ;
they were not, however, by any means so numerous as on the northern side of the
Bull Point. The opinion prevails here, as well as elsewhere, that the puffins feed
their young with sorrel, when they become, as it is stated, too fat to allow them to
make their escape from their burrowed nests. This idea I conceived might have ori-
ginated in consequence of the quantity of the plant not unfrequently found growing,
as in Rathlin, in the vicinity of their nests.
64 J. D. Marsuatt on the Statistics and
Common Corvorant. (Phalacrocorax carbo.) ‘This bird breeds in the caves
round the northern coast.
Crestep Corvorant. (Phalacrocorax cristatus.) We found this corvorant in
pairs, frequenting the numerous caves with which the northern and western shores of
Rathlin are indented. They formed their nests on the high ledges of rock, almost
touching the summit of the caves; the nest was composed of fuci of various kinds»
matted and plastered together ; the eggs were of a bluish-green colour. We some-
times, by good management, entered the caves ere the coryorants had left, and at such
times we found them sitting, with the neck and head thrust over the ledge of rock,
looking down on the boat as it made its way to the inner extremity of the cave. On
firing our guns, they would drop into the water as if they had been shot, and, with
great expertness, dive under the boat, and make their way out to sea. This species
seemed much more numerous than the preceding.
Soran Ganner. (Sula bassana.) Occasionally seen fishing in the channel. The
first time I crossed to Rathlin, two or three pairs were very busily engaged at the fry,
then passing down along the coast.
Common Gui. (Larus canus.) ‘This species occupied one of the large natural
amphitheatres formed on the north-western side of the island, and which seemed to be
occupied by no other species. ‘Their nests were placed towards the summits of the
cliffs in situations equally inaccessible from above or below; and, when disturbed, the
birds would soar away at such a distance as to leave them free and undisturbed by any
intruder.
Kirtiwake. (Larus rissa.) This is by far the most common species of gull in
Rathlin. On all the precipitous headlands north of the Bull, with the exception of a
few, these birds take up their summer residence ; and they were, during my visit, in
such countless multitudes, as to darken the air above our heads. I have never wit-
nessed so great a congregation of birds as along the headlands of Raghery. Every
pinnacle and ledge of rock was tenanted by the razor-bill, puffin, or kittiwake gull ;
and, numerous as the others were, the latter far outstripped them innumber. The nests
were formed of dried grass, sea-weed, &c. ; and the eggs, usually two in number, are
of a grey colour, blotched and dotted with brown and purple. When I looked down
from a height on these nests, it appeared wonderful how the birds found room to sit
and hatch their eggs or tend their young, for five or six nests were placed on a shelf
of rock so close to each other that the birds sat in contact, and, if not peaceably
inclined, would have thrown the whole into confusion, and prevented each other
from fulfilling the process of incubation. Yet they all seemed to live in harmony ;
and, except when one unintentionally occupied a nest not its own, (which very rarely
happened, ) they never attempted to disturb one another. The young, when first ex-
cluded from the shell, are covered with a greyish down, intermixed with white. Their
food consisted chiefly of fry. For two or three miles along the base of these cliffs the
Natural History of the Island of Rathlin. 65
rocks were covered with eggs, from which the young had been liberated—young birds
which had been precipitated from the rocks, and with the excrement and feathers of
the adult birds.
Herrine Gui. (Larus argentatus.) This gull occupied the summits of the
cliffs, tenanted below by the foregoing species; but their nests, like those of the com-
mon gull, were placed far beyond reach, except by lowering a man by a rope. Besides
being found on the northern side of the island, these birds occupied the range of white
cliffs on the northern side of Church bay ; here they remained quite secure, for they
scarcely ever ventured lower than the middle of the precipices, and could in this
manner effectually escape the gun of the fowler, either from the summit or base.
This species I also found in pairs on the eastern coast, although on this part of the
island it was rare.
The cry of the herring gull is very similar to that of the common gull, and the two
were not unfrequently confounded with each other, when soaring towards the summits
of their respective cliffs.
Among the Ampuisia, I may briefly mention the Common Sea. (Phoca vitulina.)
This animal frequents the numerous caves in Rathlin, particularly during the winter
season, when it is seen in very considerable numbers in Church bay and other parts
of the island. It varies in size from three to five feet. It is seldom or never seen
nearer to the shore than high-water mark, and it generally prefers the rocks at the
mouth of its cave, where it will lie for hours, basking in the sun. It is generally
taken by the natives with the gun, but may also be secured by throwing a net across
the mouth of the cave, and driving the seal out from the interior extremity, to which,
when alarmed, it retires. The skin and oil are made use of; the former in the
manufacture of shoes, caps, &c. and the latter, for burning.
FISHES.
Although classed among the Mammalia, I shall in the present instance, place the
Ceracea at the head of the fishes.
Common Wuate. (Balena mysticetus.) This monster of the deep is occasion-
ally seen in the channel between Rathlin and Ballycastle, though of late years very
rarely. Part of the skeleton of one may be seen on the shore of Church bay; but
whether belonging to one of these animals which had been killed by the natives, or
one accidentally thrown on shore, I could not ascertain.
Porrotse. (Delphinus phocena) is frequently seen in large shoals or herds in the
channel and round the coast during summer ; but in the more inclement season, they
desert these shores.
Grampus. (Delphinus orca). This voracious fish is also met with in great num-
66 J. D. Marsuauu on the Statistics and
bers, during the summer months; it is said to be very mischievous, and not unfre-
quently to endanger boats.
Froc-risH, ANGLER OR SEA-DEVIL. (Lophius piscatorius.) ‘This fish has been taken
on the coast of Raghery.
Skate. (Raia batis.) The skate is taken in deep water, in considerable numbers
round the shores; and were it not for the prejudice entertained towards it by the
islanders, might afford them, as in Scotland, Shetland, and other places, a nutritive
article of food; but here they will eat the skate, only when nothing better can be had.
When salted and well dried, it will keep for upwards of a twelvemonth.
Lesser Doc-Fisu (Squalus catulus) is often caught on the long line, during sum-
mer, and when captured, is valued by the natives for the oil which it affords.
Concer-Eet. (Murena conger.) Conger eels are very abundant round all the
coasts, and often take the bait on the long line. They furnish a small quantity of oil,
but are never used as food in Raghery.
Launce. (Ammodytes tobianus.) This little fish, commonly known by the name
of sand-eel, is yery abundant round the island, and furnishes a favourite food to the
different sea-fowl frequenting Raghery. Almost every sea-fowl I had an opportunity
of examining, had the mouth and stomach filled with the fry of this fish ; and from
the innumerable flocks of birds which reside here during summer, the quantity of fry
devoured at this period must be quite incalculable.
Cop-Fisu. (Gadus morhua.) This valuable fish was formerly very abundant
round the shores of Rathlin, but of late years it has been but occasionally caught.
The only cod bank near the island is called Skirnaw, and lies between Rathlin and
Isla. The red cod-fish is much esteemed. :
Coat-Fisu. (Gadus carbonarius.) This fish, known through all its stages, by the
names of pickoc, blockan, glashan, and grey lord, was in former times a most abun-
dant species in Rathlin, and furnished a cheap, wholesome, and nutritious food. In
latter years, however, they have become scarce; and on an average, one fish may now
be caught, where at least twenty were captured before. So numerous were they for-
merly, that they could be taken by a common boat hook, or a pole armed with iron ;
and in Church bay, they not unfrequently loaded a boat in this manner, and in a very
short space of time. In Orkney, they were equally abundant, as Mr. Barry mentions
that they were caught in myriads, and valued not only as articles of food, but for the
quantity of oil furnished by the livers.
Round the coast of the county of Antrim, they were taken in the following manner.
At the ebb and flow of tide, two men rowed against the current, so that the boat con-
tinued nearly stationary, the impulse of the oars counteracting the force of the stream.
The hook was coarsely dressed with a goose feather thrown on the water, and greedily
caught by the fishes, which were often so plentiful as literally to cover the surface of
the water.
Natural History of the Isiand of Rathlin. 67
From Rathlin, the coal-fish after being salted, was exported in very considerable
numbers; and the quantity of oil collected during the summer season, served the in-
habitants for lighting their lamps during winter.
At Drainsbay, near Larne, in 1810, 456 fishes of this species, supposed to weigh
upwards of five tons, were captured by a single boat in one night. The coal-fish, in
Scotland is, according to my friend Dr. Neill, called the sillock, till it attains the
length of five inches; and the pittock, when it measures twelve or fourteen inches.
Lyrue or Potrack. (Gadus pollachus.) The lythe is caught in summer in the
deep pools, which, from the craggy nature of the coasts, are very numerous. They
are sometimes caught so large, as to weigh seven pounds. The bait used in Raghery
is crab.
Line ( Gadus molva) is not an uncommon visitor of Rathlin ; but few are captured
in comparison to the number frequenting the coasts.
Fatner-Lasner. (Cottus scorpius.) This fish is found in the small pools regu-
larly left by the ebbing tide. :
Pratse (Pleuronectes platessa) is generally caught on the long line, particularly on
the eastern coast towards the great cod-bank already alluded to.
‘Sore. (Pleuronectes solea.) Occasionally caught with the plaise.
Tursot. (Pleuronectes maximus.) This delicate fish is not unfrequently taken
near the island; and specimens have been obtained, weighing twenty pounds.
Wrasse. (Labrus tinca,) and the Battan-wrasse. (Labrus ballanus.) Both
indiscriminately called by the islanders, murrans, are caught in considerable numbers
in Raghery. The capture of these fish occupies a great proportion of the boys on
the island—as on fine days, almost every projecting point of rock in some parts of the
island contains one or two fishers. The bait principally used is crab. I could scarcely
persuade some of them that the hooks were too large in comparison with the size of
the fishes’ mouths ; and their bait was nibbled away almost as fast as it was put on, in
consequence of the disproportion of the hook to the fish’s mouth.
The wrasse is easily caught, but not much esteemed, their flesh being soft and
watery, and the bones small and numerous. ‘They are found on all the rocky parts of
the coast in the county of Antrim, frequenting deep wracky holes, where, from the
brilliancy of their colour, on a clear day they may be seen to a considerable depth.
Suort-sPINED STICKLEBACK. (Gasterosteus brachycentrus.) This species of
stickleback, first added to the British fauna by William Thompson, Esq. Vice-Presi-
dent of the Belfast Natural History Society, and described and figured in “ Yarrell’s
British Fishes,” I found the only species of three-spined stickleback inhabiting Rathlin.
It must be noticed as a remarkable fact, that, of all the species of three-spined stickle-
back, common to England and Scotland, none should be found in Treland, while in
their place we have a truly continental species, the only inhabitant of our Irish lakes
and rivers. The specimens which 1 obtained in Raghery were all smaller than those
68 J. D. Marsuaru on the Statistics and
frequenting the mainland ; and although nearly agreeing with the ‘ brachycentrus,”’
had the appearance of the lateral plates extending towards the tail. Mr. Thompson
examined those specimens which I procured on the island, and came to the conclusion,
that although exhibiting an apparent difference, they belonged to the same species. I
found them in all the ponds and small streams in Rathlin.
FirreEN-SPINED STICKLEBACK. (Gasterosteus spinachia.) ‘This species does not,
like the other, frequent lakes or rivers, but prefers the rocky pools of salt water which
occur round the coasts. I found it in such situations in Rathlin.
Grey Gurnarp. (Trigla gurnardus.) During the months of June, July, and
August, this fish is very abundant round the shores, and is caught in great numbers
by the natives, who hang and dry what they do not want for immediate
use. The gurnard is easily captured ; by a bit of fish-skin, or similar substance, tied
firmly on the hook, many hundreds may be caught without ever changing the bait.
In Larne, and other places on the coast of the County of Antrim, this fish occasion-
ally furnishes a rich harvest to the industrious fisherman, When the shoals of gur-
nard make their appearance on the coast, the gulls congregate in innumerable flocks
near the fishing stations, by which means the boatmen are directed to their prey.
They leave the shore for these fishing grounds about four in the morning, so as to
reach their destination about six. From this till nine or ten they continue fishing ;
and a boat will frequently take, in a morning’s fishing, from four hundred to seven
hundred gurnards. It is affirmed by some, that, when brought into the boat, the gur
nard utters a croaking noise ; but the accuracy of this I have never had attested.
Herrinc. (Clupea heringus.) ‘This valuable fish is seldom met with between
Lough Swilly and the Point of Tor, on the Coast of Antrim ; and in the vicinity of
Rathlin it rarely appears. This may, perhaps, be occasioned by the very powerful
currents which sweep round the shores of the island; and to avoid contending with
these streams, the herring may keep more towards the middle of the channel, between
Rathlin and Scotland.
Satmon. (Salmo salar.) In the immediate vicinity of the island it is seldom seen,
but, on the opposite coast, at Ballycastle and Carrick-a-rede, it occurs in great
numbers.
Wuirte-Trout or Sea-rrout. (Salmo trutta.) Occasionally caught.
Among the Insgcra, I may briefly allude to the following :
Common Cras or Partin. (Cancer pagurus.) These are very abundant round
all the shores, and are sought after by the boys for bait to the wrasse or murran.
Lozsster. (Cancer gammarus.) Around many parts of the coast, these are
caught in considerable numbers in summer. A large fishing smack anchors at some
favourable situation, as off Stroanadergan point, on the western side of the island.
The men are provided with lobster baskets, made of a conical shape, with a hole at
Natural History of the Island of Rathlin. 69
the top, sufficiently large to allow the animals to crawl in, but prevent their leaving
their prison when once within its walls. Each basket is furnished with a piece of fish,
flesh, or other similar bait to attract the lobsters ; and left down for some hours. They
are then drawn up, the lobsters taken out and again sunk, until they have thus taken
the requisite number. When caught, their claws are tied together to prevent them
injuring each other; and they are put into other baskets which are suspended in the
water over the vessel’s side, that by this means they may be kept alive till enough have
been caught for the market.
One of these fishing smacks will be furnished with twenty or thirty baskets, or
lobster-pots, as they are termed, and will take in a morning, not unfrequently, several
lobsters from each. To prevent the pots sinking too far, or going astray, each is
provided with a large piece of cork-wood, which floats on the surface, and points out
the situation. When the vessel has remained here for two or three days, the pots are
lifted, the anchor weighed, and the lobsters carried from the coast of Rathlin into
Liverpool, Dublin, or some other port, where they bring a handsome price.
BOTANY.
The following are the names of a few plants which were observed on the island ; but
as the writer’s attention was not directed to the Botany of Rathlin, they must be
considered merely as a list of those accidentally noticed in his excursions through
the Island.
The species marked thus (*), were found on the island by the late J. Templeton, Esq.
Ranunculus acris, Linn. Upright meadow crowfoot. Frequent.
Ranunculus sceleratus, L. Celery-leaved crowfoot. In pools, &c.
Ranunculus repens, L. Creeping crowfoot. Common.
Caltha palustris, L. Common marsh-marigold. Marshes and ditches.
Nymphea alba, L. Great white water-lily. Covering one of the marshes in the
southern extremity of the island,
Nuphar Jutea, L. Common yellow water-lily. Growing with the white water-
lily.
Fumaria officinalis, L. Common fumitory. Frequent.
Cochlearia officinalis, L. Common scurvy-grass. Rocks on the shore.
* Cochlearia coronopus, L. Common wart-cress. On the waste grounds, common.
* Crambe maritima, L. Sea-kale. _ Gravelly shore of Church bay.
Viola tricolor, L. Pansy or heart’s-ease. Fields, rather frequent.
Polygala vulgaris, L. Common milk-wort. Hills, common.
Malva sylvestris, L. Common mallow. On one of the hills on the northern shore
of the island.
VOL. XVIL 00
70 J. D. MarsHatt on the Statistics and
Silene maritima, With. Sea campion. Not common.
Silene armeria, L. Common catch-fly. Not frequent.
Geranium robertianum, L. Stinking crane’s-bill. Common on waste grounds.
* Rhodiola rosea, L. Rose-root. On rocks at the north end in great abundance.
* Ulex nanus, Forst. Dwarf-furze. Sown in Rathlin by Mr. Gage, in 1790.
Anthyllis vuineraria, L. Common kidney vetch. Common.
Trifolium pratense, L. Common purple trefoil or red clover. In great abundance
in front of Mr. Gage’s house.
* Trifolium agrariwm 2?
Potentilla anserina, L. Silver-weed. Road-side and meadows, frequent.
Rosa spinosissima, L. Burnet-leayed rose. Frequent in many parts of the
island.
Crategus oxyacantha, L. Hawthorn. Very rare.
Daucus carota, L. Wild carrot. Borders of fields.
Hydrocotyle vulgaris, L. Common white-rot. Marshy grounds..
Galium verum, L. Yellow bed-straw. Common.
Galium palustre, L. White water bed-straw. Ditches and marshes.
Jasione montana, L. Annual sheep’s-bit. On the pasture grounds..
Senecio vulgaris, L. Common groundsel. Not common.
Bellis perennis, L. Common daisy. Frequent.
Arctium lappa, L. Common burdock. Road-sides, common.
Carduus acanthoides, L. Welted thistle. Common.
Carduus tenwiflorus, Curt. Slender-flowered thistle. Road-sides.
Leontodon tarazacum, L. Dandelion. Not common.
Myosotis arvensis, Hoffm. Field scorpion grass. Common.
Myosotis palustris, Roth. Forget me not. Ditches and marshy ground.
Plantago coronopus, L. Buck’s-horn plantain. Waste grounds.
Statice armeria, L. Sea gilliflower. Rocks on sea-side, and on the hills.
Erythraa centaurium, Pers. Common centaury. Not frequent.
Anagallis arvensis, L. Common pimpernel. Corn fields and hills, common.
* Anagallis cerulea, Schreb. Blue pimpernel.
* Anagallis tenella, L. Bog pimpernel.
Veronica beccabunga, L. Brooklime. Ditches and marshy ground.
Veronica chamedrys, L. Germander speedwell. Waste grounds, common.
Euphrasia officinalis, L. Eye-bright. Hills, abundant.
Rumex crispus, L. Curled dock. Common.
Rumex acetosa, L. Common sorrel. Pasture grounds, and by the sides of the
marshes and ponds.
Polygonum amphibium, L. Amphibious persicaria. Marshy grounds.
Urtica dioica, L. Great nettle. Common.
Natural History of the Island of Rathlin. 71
* Ceratophyllum demersum, L. Common horn-wort. Large lake (Lough Ru-
naolin) in southern end of Rathlin.
Lemna minor, L. Lesser duck-weed. Marshes, common.
Poa pratensis, L. Smooth-stalked meadow-grass. Frequent.
Poa maritima, Huds. Creeping sea meadow-grass. Sea shore.
Poa annua, L. Annual meadow-grass. Road-side.
Festuca ovina, L. Sheep’s fescue-grass. Abundant.
Arundo phragmites, L. Common reed. Covering great part of the marsh in the
southern part of the island.
* Triticum loliaceum, L. Dwarf sea wheat-grass. On dry ground.
* Lolium arvense, L. Short-awned annual darnel. In dry places, as on hills
above Church bay.
Aspidium jfiliz mas, L. Abundant at the entrance to one of the caves, south of
Church bay ; growing also on borders of the marshes.
Chara vulgaris, L. Common chara. Ditches and riyulets.
i
DA
*
. }
,
‘
.
*. Vv
On the Affinity of the Hiberno- Celtic and Phenician Languages. By Sir WitL1am
Berna, F.S.A. M.R.I.A. Secretary of Foreign Correspondence, Member. of
the Royal Academy of Sciences of Lisbon, &c. §c.
Read 28th of November, 1836.
In my former paper, read the 23d May and 27th June, I have stated that the
names of places in Ptolemy’s Geography are significant of their local position, cir-
cumstances, or peculiar qualities, in the Hiberno-Celtic language ; from which it may
fairly be inferred, that the Celta must have been an early colony of Phenicians, as all
those names were avowedly borrowed from the Phenician mariners.
Before I proceed to lay before the Academy the results of my more recent inyesti-
gations, I wish to say something in answer to the observation—which, as it has been
made by many, requires, perhaps, some preliminary remarks—viz. That my theory
derives the Celte, and all their early learning and science, from Ireland and the
Trish.
This is altogether an erroneous notion. I claim for Ireland itself no pre-eminence
in science, learning, or the arts, above the other branches of the Celt ; all I de-
mand for Ireland is— That her people, being a branch of the great colonizing people
of antiquity, enjoyed an equal portion of civilization with the mother country, im the
ratio which colonies usually possess.
It is no part of my theory that the other colonies of the Phenicians, or branches
of the Celt, derived anything from Ireland, or the British islands, further than what
Cesar asserts, that the chief seat of Druidic science and learning was from thence.
Were I to assert that the early Greeks and Romans borrowed their learning,
science, and civilization, from the Irish, I should receive and deserve the ridicule due
to such an assertion. I shall not, however, fear it, when I assert that they derived
those blessings from the Phenician ancestors of the Irish Celta. Were I to assert
that the Etruscans and Pelasgi were descended from the Irish, I should receive the
derision such a declaration would justly call for ; but I do not fear it when I assert
that they were colonies of the same great people.
VOL. XVII. PP
74 On the Affinity of the Hiberno- Celtic and Phenician Languages.
Fortunately, the Phenician language has been preserved by their Irish descendants,
and by it we are enabled to unravel difficulties, solve problems, and elucidate facts
which, without an acquaintance with that tongue, must have remained for ever unex-
plained and inexplicable mysteries.
ON ITALY AND ITS ANTIENT INHABITANTS,
Previously to the building of Rome, the history of the various antient people of
Italy is involved in the deepest obscurity, and of the inhabitants of the more northern
and western portions of Europe we absolutely know nothing whatever. We learn,
indeed, from Pliny, (3 c. 5) Strabo, (5) Plutarch, (in Romulo) and Mela, (2 c. 4)
that the Etruscans, or Tuscans, occupied the countries west of the Tiber, between that
river and the Tyrhenian sea; and that they were divided into twelve tribes or dis-
tricts, having each a chief, monarch, or leader, called a Lucoman.
I have long been satisfied that the Etruscans were an early Phenician colony, and
of the same race as the Pelasgi. They are both represented as civilized polished
people; and the remains of the former, lately brought to light by the excavations of
the Prince of Canino, as well as those which have been long known, exhibit a progress
and perfection in the arts which moderns are happy to copy, seldom equal, and never
excel,
It may be asked, what other civilized people of that period, except the Phenicians,
possessed a local habitation or a certain country? The Pelasgi are said to have
spread abroad and settled colonies, but no historian has ever given them an original
country. Were they not the Phenicians, disguised under another specific name ? as
the Phenicians who inhabited the southern coasts of Arabia were called Homerite,
which, Herodotus tells us, meant the same as Phenician, each indicating a seaman or
mariner.
The researches of Micali on the antient peoples of Italy, has thrown considerable
light on this most interesting subject. He says, ‘In the religion of the Etruscans
there is rather a general resemblance to the great oriental systems than to that which
is purely and exclusively Egyptian—monuments of Phenician and other eastern super-
stitions appear mingled with those of an Egyptian character.”
A writer in the Quarterly Review for September, 1835, says, in the critique on
Micali’s work, ‘“‘ The Etruscan language stands alone a problem and a mystery, not
merely allied to none of the older dialects of Italy, but bearing no resemblance to any
language to which it has been compared.
The means of explaining and unravelling this difficulty has, hitherto, been wanting,
but I shall endeavour to show that at least some of the Etruscan words and names
are significant in an existing tongue, and indicate a resemblance so striking and pal-
On the Affinity of the Hiberno- Celtic and Phenician Languages. 75
pable, that it will be difficult, if not impossible, to avoid recognizing them as cognate
tongues.
Exclusive of proper names, but few Etruscan words have come down to us through
the Latin writers. One most remarkable word, however, is mentioned by Suetonius,
in the life of Augustus, which is 4sar. He relates, that the electric fluid haying
struck the statue of Augustus at Rome, melted the C from the name Czsar on its
pedestal. The augurs declared that C denoting one hundred, and God being called
4isar in the Etruscan language, the emperor had but one hundred days to remain on
earth, when he would have his apotheosis, and be taken to the gods.* The death of
Augustus took place shortly after, and apparently verified the prediction.
Moran is the (Irish) Celtic word for God; and not only is the word itself to be
found in the Irish Dictionary, but its roots, or the two words of which it is a com-
pound—aor, age, and ap, ruling, guiding, dividing, judging, controlling, i. e. God,
as the guider, ruler, and controller of ages, the eternal ruler.
I have collated the names of the Etrurian nations with the Celto-Phenician, and
the following is the result :—
Veientes. The people living in an undulating or sloping country, from yaoyn, un-
dulating or sloping.
Clusini. Cluarm, a pot, or porringer, of brass or other metal; probably from the
manufacture of such articles being carried on in the place.
Cortonensis. Caone, sheep; oun, a hill. A hilly country, favourable to the
feeding of sheep.
Arretani. The arable or agricultural country : sojpead, a ploughed field or land ;
cana, country.
Vetuloni. ¥eacal, a cup or vase of earthenware; ana, rich or productive of. This
name is very interesting on account of the recent discoveries on the site of the old
city or town of Vetulonia by the Prince of Canino. Featalon means the manufactory
of earthen vessels, vases, cups, &c.
Volaterrani. ¥olava, cattle ; yp, land; aya, rich. The country rich in cattle.
Russellani. Ruy, a wood ; alajyeact, beautiful. The weli-wooded or beautiful
country of groves.
Volcinit. ¥4ol, wild, fierce, bold ; ¥cjan, a knife or dagger.
* «“ Mors quoque ejus, de qua de hinc dicam, divinitasque post mortem, evidentissimis ostentis praecog-
nita est. Cum lustrum in campo Martio magna populi frequentia conderet aquila eum szpius circum-
volavit; transgressaque in vicinam «dem, super nomen Agrippe ad primam literam sedit: quo animad-
verso, vota, que in proximum lustrum suscipi mos est, collegam suum Tiberium nuncupare jussit: nam
se quanquam conscriptis paratisque jam tabulis negavit suscepturum que non esset soliturus. Sub idem
tempus ictu fulminis ex inscriptione statue ejus prima nominis litera effluwit; Responsum est, centum solos
dies post hac victurum, quem numerum C litera notaret ; futurumque, ut inter Deos referretur quod
ZEsanr, id est, reliqua pars e Cesaris nomine, Etrusca lingua Duvs vocaretur.” (Suet. in Aug. 97.)
76 On the Affinity of the Hiberno- Celtic and Phenician Languages.
Tarquinii. Tapconar, a ferry or passage over a river. These people resided in a
town situated on the river Arone.
Falisci. ¥ajl, a precipice or cliff ; wyrse, water. The people residing on or near a
cataract or fall of water.
Coretani. The pastoral or sheep feeding tribe or people. Caon, sheep ; cana,
country.
Lucamo, or Lucamon. The title of the hereditary chiefs of the Etrurian nation,
derived from lejc, a precious stone, and caoman, a noble, chief, or prince. One orna-
mented with or wearing a diadem, as a mark of dignity or sovereignty.
The name of the country Tuscia is probably derived from the circumstance of its
being the first settlement made by the Phenicians on the coasts of Italy. From corac,
a beginning or commencement.
The Irish have many figurative refinements of language, by which they denominate
the Deity : as aoran, the eternal ruler ; corac 3an corac, the beginning without a be-
ginning, or the first cause. Their common name is va, which also means a day ; so
a sory, the antient of days, may be the true origin of deus. How little foundation is
there for the assertion that the Irish language is without terms of art or refined ideas ;
it would be made only in total ignorance or relying on the ignorant decla-
ration of others. The language is rich in such refinements as could only have origi-
nated among a thinking and reflecting people.
The Greeks called Etruria ruppnvia, and the sea the Tyrhenian sea. This is proba-
bly from ap, land; epsea, of necessity, force, compulsion, violence, or conquest,
from being driven upon it first by stress of weather or other necessity.
THE GREEKS.
The first civilized people who are said to have visited Greece were the Pelasgi,
a most mysterious nation. We know little of them but their name, and their cha-
racter as a civilized people ; but from whence they came, or of their original country,
not a word. They are said to have settled colonies in Argolis, in Peloponesus, Thes-
saly, and Epirus, while others extend them over all Greece, and even into Thrace ;
and, as usual, their name, for want of correct knowledge, is derived: by the Greeks
from an individual called Pelasgus, who is said to have been their first king. But this
derivation is unworthy of a moment’s consideration.
Two theories have been promulgated respecting their origin. One bringing them
from the barbarous hordes residing on the Caspian and Euxine sea; the other makes
them autochthones, or aborigines; both unsupported by evidence. An admirable
German writer, Conrad Mannert, of Nuremberg, (in his Geografie der Gtriechen,
1792-5) gives a more probable hypothesis, while he rejects both the former. He says,
they every where met, on their arrival, with races of men less civilized than them-
On the Affinity of the Hiberno- Celtic and Phenician Languages. 77
selves ; some living in forests, and others just formed into civil societies. ‘The name
Pelasgi was never assumed by them, but was given them by the Greeks. The name
was more antiently written MeAapyoi, and was applied to them in familiar language by
the Greeks, from the resemblance they bore to storks and other birds of passage,
when they first became known to the Greeks; for it seems before they fixed them-
selves permanently in Greece they would appear and disappear from the coasts at
almost stated and regular intervals.’ How exactly, I may say, this notion agrees
with the habits of mariners in their trading voyages. And it may be asked, who were
the commercial mariners of antiquity? The Phenicians alone.
Again, Mannert says, ‘ All the Pelasgic colonies which established themselves
among the early Greeks, brought with them the elements of civilization and the arts.
IF hence did they obtain them 2’? Certainly not among the ignorant barbarians of
the Caspian and Euxine.
Again, ‘“‘ The Pelasgi are acknowledged, by the concurrent voice of antiquity, to
have brought with them into Greece a peculiar and distinct system of religion. They
are acknowledged, moreover, to have been the founders of Grecian theology. They
established an oracle at Dodona, instituted the mysteries of the Cabiri, and there is
reason to believe that those of Eleusis were of similar origin.”
Few will venture, I think, to question the truth of these observations. Now if we
can discover an antient language, in which all the names of the Greek divinities and
heroes are significant of their peculiar attributes, we may justly conclude it to be a
cognate language with that of the Pelasgi; and if that language also be proved to
be the same as the Phenicians and Etrurians, it follows that they were a branch of the
same people as the Phenicians and Etrurians.
“ Profound night (says Mamnert) rests on this portion of history ; a single gleam
of light pierces the darkness which involves it. On the one side of the Pelasgi many
tribes of Illyrians practised navigation ; as, for example, the Phoeacians of the island
of Scheria, afterwards Corcyra. At the head of the Adriatic existed long established
commercial cities, and artificial canals were seen at an early period. Every thing
seems to indicate that at an early period the shores of the Adriatic were inhabited by
civilized communities.” Thus far Mannert. We may add that the name of the city
of Venice itself would indicate a Phenician origin ; and their position, and the charac-
ter of the people, strongly corroborates the same idea. Phenice and Venice are very
similar in sound, and both suggest a common origin or meaning ; that is, of a ma-
riner or plougher of the seas. Mannert, indeed, supposes the name to be derived
from the Sclavonic wenden, to rove about, and that they were a northern race ; but
he was not acquainted with the Celtic language, or he would not have made so impro-
bable a conjecture.
Herodotus states that letters were introduced into Greece by Cadmus, but Dio-
dorus claims a previous possession of written characters; and Pausanius mentions an
78 On the Affinity of the Hiberno- Celtic and Phenician Languages.
inscription on a monument earlier than the time of Cadmus. The learned editor
of the last edition of Lempriere’s Classical Dictionary meets this with a
very conclusive answer. ‘How came,” says he, ‘the alphabet used by the
Greek nation to bear so close a resemblance in the names, order, and very form of
the letters, to the alphabets of the nations which belonged to the Shemetic race ;
namely, to those of the Phenicians, Samaritans, and Jews; or, to speak more cor-
rectly, to that of the Phenicians, for those and the Jews, until the time of Cyrus,
used the same characters? One of two suppositions must be the answer to this ques-
tion. Either the Phenicians introduced an alphabet into Greece so far superior to the
old Pelasgic as to be adopted in its stead, or the alphabet of Cadmus and that of the
Pelasgi were the same.
“* The first supposition will be found extremely difficult to support. It takes for
granted what few, if any, will be willing to allow, that there existed in those early
ages a sufficient degree of mental activity and refinement, on the part of the rude
inhabitants of Greece, to induce them to discriminate between the comparative ad-
vantages of two rival systems of alphabetic writing.” The most rational conclusion
is, that the Pelasgic and Cadmean alphabets were the same, and both were Phe-
nician.
The history of Greece, previous to the period when Cadmus taught them the use
of an alphabet, is nearly a blank, and involved in dark fable for near 800 years after.
Rome was founded about the year 704, A.C. But both these periods are, however,
recent when compared with the glorious era of Tyre and Sidon ; and it will not be
denied that the Greeks, when first visited by the Pelasgi, were nearly as ignorant
and illiterate barbarians as the South-Sea islanders were, on their first discovery, to the
English.
Their learning, science, arts, and the whole of their mythology, with its ap-
pendages, were all borrowed from their schoolmasters. They really have nothing
antient of their own. It must have appeared to every scholar how absurd, far-
fetched, and puerile, are most of the attempts made to derive Greek names from their
own language.
The Greek mythology seems to have been a disfigured and corrupted paraphrase
of the Phenician system: each mythos appears a confused representation of some-
thing they had learned without being acquainted with its precise or defined meaning ;
and every story being involved in a mist, is exaggerated and distorted by being viewed
through it.
Most of the names of their divinities and heroes have no meaning in the Greek
language, but appear mere barbarous and unmeaning epithets. This is equally true
with the Romans, who, indeed, invented some new divinities, and gave them names
indicating their supposed attributes, but the names of their old deities are equally
without meaning in the Latin.
On the Affinity of the Hiberno- Celtic and Phenician Languages. 79
It is, however, very remarkable, that although many of the Greek names for the
same divinity differ from the Roman, yet both being derived from the same Phenician
source, by different channels, each are significant in that language, and express the
same, or common, attributes of the given deity, so as to mark their identity with ex-
traordinary precision. ‘Thus Vutcan means the adroit, or able and intelligent, smith.
Hepuaistos, one skilled in the effects of heat or fire. Venus, the woman of the com-
munity, or the courtesan. Mererrix, violent lust or lechery. Adpotire, the produce
of froth. Mercury, swift cap, or the man with wings on his cap. Hermes, the ray
of the sun. Baccnus, the lame or staggering drunkard. Avvvooc, the drunken
man, &c.
It would be inexpedient, even if time would permit, to fatigue the Academy with
a collation with the Celto-Phenician of every name to be found in the Greek and
Roman mythology ; but it is necessary to state that they are found, almost without an
exception, significant in too remarkable a manner to be mistaken.
I shall now proceed to state the meaning, in the Celtic, of some of the most re-
markable of the divinities and heroes of the Greeks and Romans, and commence with
the most antient.
Uranus. The most antient of the gods. This name is derived from the sun’s sup-
posed orbit round the earth. up, the sun; ayy, circle. The ideaof the motion of the
sun round the earth, his benificent warmth, vivifying and generating powers, naturally
became the object of early and grateful devotion; and the Greeks worshipped the
sun, (the Phenician or Tyrian Hercules,) under the name of Ovpavoc, which is nothing
more than the Phenician una Hellenised.
Celus. The same deity as Uranus, under another name. From Ceyl, a husband,
generator, father, progenitor, creator ; soy, of ages.
Thea, or Tithea—she is represented as the wife of Calus—is merely 4, plenty,
abundance, riches. Thus*the generator or creator produced all things from his
abundant power or capability of production. This goddess is also called Zitea, which
is a compound of a, power, government, rule, design, intention, contemplation, and
aja, divinity or god. Thus Ceil, the creator, by his power, wisdom, prudence, and
fore-knowledge, produced Saturn, Chronos, or Time, heaven, earth, and all things.
Her name of Ops may be from ob, or op, force, or violence, of parturition. Eheu,
from pe, the moon.
Saturn, or Chronos. The former of these names is palpably Phenician—it literally
means tle Lord, ye cjapna; the latter equally so, Time, Cpon.
The Titans, brothers of Saturn, meant nothing more than the Princes. T), the ;
Ga, Princes,
80 On the Affinity of the Hiberno- Celtic and Phenician Languages.
INFERNAL DEITIES, RIVERS, &c.
Pluto. The god of hell, or the infernal regions. Blocac, is one who'dwells in a
cave, or under the earth ; from bloc, a cave or mine. From wealth being found in a
mine Plutus, the god of wealth, has his name, as also the word Iovroc, wealth.
Acheron. A river in hell. The river of bitterness, severity. Ucap, bitier, severe ;
aban, Tiver.
Abastor. One of Pluto’s horses, so called because he barked like a dog. 2borzpac,
barking like a dog.
Ades or Hades. Pluto ; the infernal regions. jo, cold ; aj, death.
Hecate. The same as Proserpine, Luna, and Diana. The eye of death. cc,
death ; ser, eye.
Cerberus. The dog of hell. Geapboyne, the worryer, slaughterer ; or cean, death,
burial ; vepac, barking, the incessant barker.
Charon. The ferryman over the Styx. The river of interment. Cean, burial,
interment ; aban, river.
Barathrum. The infernal regions. The change of death. Bapp, death ; acpwysynp
I change.
Gorgones. Frightful women. ops, cruel, fierce, frightful ; sean, women.
Nemesis. The goddess of revenge. WNemayre, terrible, cruel, revengeful.
Yartarus. The region of hell, where the most impious and guilty were punished.
The place of scorn, reproach, and contempt. Tan, contempt, reproach, scorn ;
canpac, horrid, terrible, fearful, appalling.
Tisiphone. One of the furies—represented with a whip in her hand. Tear, hot ;
tebe, @ whip or scourge ; yeanaym, to flay, scourge, excoriate. ‘These words, com-
pounded, give the exact sound of the Greek, or Latin name, with its equally correct
meaning.
Lyphon. ‘A famous giant, son of Tariarus and Terra, who had a hundred
heads, like those of a serpent or dragon. Flames of fire were darted from his eyes
and mouth, and he uttered horrid yells, like the dissonant shrieks of different animals.
He was no sooner born than he made war against heaven and frightened the gods.”’
Such is the description of this portentous being, and very accurately does the perso-
nification of the monster describe the subterranean fires and volcanoes of Etna, which
its name indicate. Ty, burning, or fire ; yon, under, or below, the earth. The hundred
heads are the peaks of the mountain, the mouth and eyes are the craters which vomit
forth fire, and the horrid yells are the thunder-like voices and hissings. The making
war against heaven are the missiles projected from the craters. ‘‘ Jupiter is said to
have put Typhon to flight and crushed him below Etna.”
The Egyptians reckoned Typhon to be the cause of all evil. Tj also signifies judg-
On the Affinity of the Hiberno- Celtic and Phenician Languages. 81
ment ; yon, below, in which sense it meant the infernal regions, and without doubt
was the origin of Zartarus, or Hell. He is represented as the father of Geryon,
Cerberus, and Orthos. Geryon—the rugged, dreadful river, with the rocky bed.
Cerberus—the dog of hell: ceap, death ; vepac, incessant talking, barking, shouting.
The noises of Etna were supposed to proceed from the barking of Cerberus: our
word bark was probably from the Phenician berac. Orthos—op, noise, sound ; zor,
first. The vigilant dog who gave the first bark.
Bacchus. The god of revels and drinking. Bac, loving ; ay, drink, liquor, wine ;
or bacaé, lame, staggering, from drink.
Dyonusus (Awvucec). The Greek name of Bacchus. aoneay, manliness ; ar,
drink. The drunken manly hero.
Eleleus. A name of Bacchus, from the word ede\ev, which the Bacchanals loudly
repeated during the festivals. This is the cry of the Irish, of uwlulu, or pillulu, at
funerals and when melancholy drunk.
Euhyus. A name of Bacchus. eu, dying, or dead ; ay, drink—i. e. dead drunk.
facchus. A surname of Bacchus. jac, a yell, scream, shout ; ay, drink. From
the noise the Bacchanals made at his feasts. Iayew, shout ; waxyatw, to shout, revel,
to be drunk. ‘This word is also, no doubt, from the Phenician root.
Menades. The Bacchantes, priestesses of Bacchus. This word is generally
derived from pawoua, to be furious; but I would rather say from the Phenician
neanao, gaping, yawning, vomiting.
Orgia. Festivals in honour of Bacchus. Opsjor, cheer, entertainment, revelry.
Orphica. A name by which the orgies of Bacchus were called—because they were
introduced by Orpheus? On, sound, voice, music ; joc, concert, combined, united,
a choir. ‘Thus singing in a choir isso called, and not from Orpheus.
Diana. ya, divinity ; ya, the—ya, is the feminine of the article, an, the. The
goddess. Diana was called the goddess, in consideration of her eminence.
“ Great is Diana of the Ephesians.”
Echo. A daughter of Air and Tellus. eccors, a model, shape, likeness, repetition,
simulance.
Folus. @otar, knowledge, science, philosophy, art ; king of the winds, a great
astronomer and inventer of sails. How much more palpable is this derivation than
the Greek atodoc, varius.
Fortuna... The goddess of fortune. This was a very antient deity. She
has been represented standing on the prow of a ship and holding the rudder in
her hands, which is probably one of her most antient representations, ‘The import of
the name Fortuna is, yon, protection, defence ; cuyn, from the waves, which is well ex-
pressed in the above figure.
Janus. His temples were closed in time of peace and open in war ; jan, the blade
of a sword.
VOL. XVII. QQ
82 On the Affinity of the Hiberno- Celtic and Phenician Languages.
Tris. One of the Oceanides, messenger of Juno and the gods. wp, the sun ; aer,
a shower. The rainbow. ‘The production of the sun and a shower.
Lares. Gods who presided over houses, families, and grounds, or estates. tap, the
sround, floor, hearth.
Lucina. The goddess who presided over the birth of children. Her mother
brought her forth without pain, from whence her name. ta, litéle ; cj, lamentation ;
na, of the.
Luna. Diana, the moon. tuay, moon; na, the feminine article the. Literally,
the moon.
Nape. Certain divinities who presided over hills and woods. It is said from vazn,
a grove, rather from, naob, a holy person, a nymph. The Irish call the fairies the
good people ; naoyi, the nymphs—see nymphs.
Naiades. Inferior deities who presided over rivers, springs, wells, and fountains.
They are said to resort to the woods near the stream or fountains over which they
presided, and hence their names, (says Lempriere) vaew, to fow. The true derivation
is from nao, sacred or holy, or a holy person; oja, divinity, or a person made a divi-
nity—a demi-deity pronounced Naidia.
Nemesis. The goddess of revenge. Neamajre, terrible, cruel, revengeful. She is
also called Adrastia, from (says Lempriere) the temple of Adrastus, king of Argus,
but rather from gyopar; the Phenician and Celtic name for a fury or infernal deity.
Neptune. The god of the sea or waves. Waob, a holy person, a saint or divinity ;
con, the waves.
Nereus. A deity of the sea. wyapdear, the south-west ; pronounced Nereus.
Orthia, A name of Diana at Sparta—whence boys were whipped at her sacrifices.
On, voice, shouting, exclamation, crying ; 214, goddess. The goddess of crying,
shouting, or exclamation.
Orthus. The dog of Geryon. On, noise ; cor, first,
Orus. ‘The sun, skilled in medicine, the benefactor of the human race. An
Egyptian deity. up, the sun ; air, a shower, or water. Thus the united influence of
these two agents acquired for this god the name of benefactor of man and divine
honours.
Osiris. A deity of the Egyptians. Oy, above, superior ; un, the sun ; ayy, water
or showers. The deity who commanded and directed the influence of the sun and
the rain. The supreme director of all things. God above, the sun, the rain, and
genial influence.
Pan. The god of shepherds. pean, a reed ; so called from playing on a musical
instrument made of reeds, which shepherds played on.
Parce. The fates. Bapp, death ; ca, a road ; the road or course to death.
Atropos. One of the Parce or Fates. She outs the thread of life. 2, the ; mpoc,
On the Affinity of the Hiberno- Celtic and Phenician Languages. 83
dark ; van, death. The common derivation of a non (rpexw,) muto ; is not so pal-
pable, although she be inexorable.
Clotho. The youngest of the Parce. She presided over the moment of birth,
held the distaff, and spun the thread of life; whence her name xray, to spin ;
vloécar, separating, loosing, spinning.
Lachesis. One of the Parcze, whose name is said to be derived from Aayav, to mea-
sure out by lot ; rather from lac, reckoning ; aoyr, of ages. She is represented spinning
the thread of life; gojy, is repeated to signify the plural; the Phenician word
would stand thus, Lac-aoyr-aoyrp—which is pronounced exactly as the Greek or Latin
name.
Priapus. The phallus. The godof generation. pyyob4yo, privacy, secrecy.
Sancus. A deity of the Sabines. san, holy ; coy, foot. The holy foot.
Sibille. The sybyls or fortune-telling women. sya, a fairy or witch ; beala,
mouth. The women who foretold events.
Sicheus, Shicharbus, or Acerbas. The husband of Dido, put to death by Pyg-
malion. sSjozaé, 2mdolent, inactive. The latter name from his death. Sjcajpe, a
motive, occasion, reason. Bay, for death ; acan, sour, bitter, or acayp, poison ; bar:
death.
Taautes. A Phenician deity, the same as the Saturn of the Latins, and probably
the Thaut, or Thoth, the Mercury of the Egyptians. (Cic. de Nat. Deorum, 3 c. 22.
Varro.) Teutates. ja, the god, Taat or Tait, the Celtic or Phenician god of
trade, one of the deified heroes or patriarchs of the Phenician Gael. (See Gael and
Cymbre, 225, &c.) Ta, pronounced tha, or thor, is being, God. Literally, am, I
am, existence, “ Iam, has sent me unto you ;”’ cay, is merciful, clement, beneficent,
compassionate. Thus we have Thortais, or Taautes, the merciful God.
Triton. A sea deity, the son of Neptune ; powerful among the sea deities. Gpjaz,
lord, sovereign, or king ; com, of the waves or billows. King or lord or prince of the
billows.
Venus. ‘The goddess of beauty, the mother of love, the queen of laughter, the
mistress of the graces, and patroness of courtezans. Some mention more than one.
Plato mentions two: Venus Urania, the daughter of Uranus ; and Venus Popularia, the
daughter of Jupiter and Dione. Cicero speaks of four, a daughter of Coelus and
Light ; one sprung from the froth of the sea, the third daughter of Jupiter and Dione ;
and the fourth born at Tyre, the Astarte of the Phenicians. Of these the Venus
sprung from the froth of the sea, after the mutilated parts of the body of Uranus
had been thrown there by Saturn is most known. The name of this goddess is very
apposite ; teay, the woman; aor, of the people or community ; pronounced /“anus,
the prostitute, courtezan, or woman of the town.
Vertumnus. The god who presided over spring, orchards, fruits, and village
84 On the Affinity of the Hiberno- Celtic and Phenician Languages.
pursuits. ean, @ man; cuajm, @ village or farmers house ; yey, active, vigorous,
attentive. The good husbandman.
Vesta. The goddess ‘of chastity and female virtue. wary, gentleness ; ca, exist-
ence, gentleness personified.
Vulcan. The god of the antients who presided over fire, and was patron of all
artists who worked in iron and metals. ols, clever, active, intelligent, adroit ; satan,
smith, pronounced folgaun, or vulgaun. ‘The clever smith. Tubal Cain ?
Alecto. (A, Anyw, non desino, not laid aside,) one of the furies. la, a@ wound ;
ece, death ; co, feminine. The woman who inflicts wounds and death.
Harpyle or Harpies. Winged monsters with the faces of women and bodies of
vultures ; three innumber. There names—Aello, Ocypete, and Celeno. %n, plague ;
bojleac, altogether, or entirely. They emitted an infectious smell. ello, ajle, smell.
Occipete. Occ, with ; cj, lamentation ; prey, music.
Celeno. Cy, lamentation ; Yon, a snare or net.
Medusa. One of the Gorgons, whose hair was composed of serpents, and who
changed into stones, or killed, whoever they looked upon. aycear, necromancy,
sorcery, magic.
Narcissus. A beautiful youth, who, seeing his own image reflected in a stream,
became enamoured of it, thinking it the nymph of the water. ‘Naob, a nymph ; cear,
sight, vision; ary, a stream, or waterfall—waocea yay, the sight of a nymph in the
stream. .
Orpheus. On, sound, voice, music ; yoy, Skill, knowledge, science.
Pandora. Beay, the woman; svoppac or roppda, harsh, rough, fierce, cruel,
austere, unpleasant. The woman of mischief. ,
Pythia. The priestess of Apollo at Delphos. She always delivered the oracles in
hexameter verses, and with musical intonation. pPjceas, music; from whence the
name.
Titii. The priests of Apollo at Rome. Tycm, the sun.
Tityus. The giant son of Terra or Jupiter. Tyz, the earth ; cu, bulky, large,
gigantic.
Parthenope. One of the syrens. Bann, death ; cem, approaching ; aojve,
civility, politeness, deceitful invitation.
Pasiphae. The wife of Minos enamoured of a bull. Ba, a cow ; yabar, taste, relish.
The propensity, fancy, or disposition of a cow.
Pygmalion. King of Tyre, son of Belus, or Baal. Beas, little ; maoitnyyn, mule.
The Little mule or person of a low stature and obstinate disposition.
Pythagoras. The philosopher. He was most probably a Phenician nobleman,
or—Pejs, musician ; sjazun, nobleman. The noble musician. He was a great poet
and musician. .
On the Affinity of the Hiberno- Celtic and Phenician Languages. 85
Sanconiathon. The Phenician historian born at Berytus. san, holy ; con, under-
standing, sense, or wise man ; yor, real; cayn, Of the country. The sacred writer,
or wise recorder of the events of his country.
Sesostris. The great king of Egypt, and conqueror of Asia, &c. His father
ordered all the male children born on the same day with him, to be educated at the
public expense. Sey, pleasure, delight, happiness, fortune, success ; yor, knowledge,
science, learning ; sperres force, strength, power. His Egyptian name was Ramesis
the Great.
Tages. A son of Genius, who taught the Etrurians augury and divination. He
is said to have been found under a clod by a Tuscan ploughman, and assumed a
human shape to instruct this nation, so celebrated for their knowledge of incantations.
Tazbayy, chance, fortune, hope ; pronounced Tages.
Talus. The inventor of the saw, compasses, and other instruments. Taylm, 27-
struments, or tools ; cal, an adze.
Tantalus. King of Lydia, son of Jupiter or Pluto. Represented as punished
in hell with insatiable thirst, in the midst of a pool of water which recedes as he
brings his mouth to it; a bough hangs over his head, loaded with delicious fruit,
which, as he attempts to take it, is removed from his grasp by a gust of wind, Others
say, alarge rock or stone was suspended over his head, ready to crush him to pieces ;
all the same idea of a state if constant excitement, fear, and trembling disappoint-
ment. Tajn, water, dropping, or falling ; cal, receding ; ayy, back, water receding
backwards.
Tlepolemus. Son of Hercules. Tlaz, soft; peollayo, a skin without hair ; may;
comely. The handsome man with a fine soft skin, without hair on it. The union of
the 77, a peculiarity of the Irish language is singularly exhibited in this word.
Triptolemus. Son of Oceanus and Terra. Gpeat, tilling, ploughing ; calam,
the ground ; aor, fo a people, or community. Ceres gave him her chariot, and he
travelled in it over the world, distributing corn, and teaching people agriculture.
This name is a very singular confirmation of the identity of the Phenician and Celtic
tongues. The three words express the correct idea, and is pronounced exactly like
the Greek name.
Haruspex. A soothsayer who drew omens from the resistance, throes, pangs
in dying, and the inspection of the entrails of beasts sacrificed, The usual derivation
of this word is ab aris inspiciendis, which is but a guess, although not so improbable
as most of the classical derivations. The true derivation is, an, yudging, conducting,
deciding, guiding ; uypoy3, from a throe, pang, gasp, heave, agony. This superstition
had its origin with the Chaldeans, and their Phenician descendants communicated it to
the Greeks and the Romans.
VOL. XVII. RR
86 On the Affinity of the Hiberno- Celtic and Phenician Languages.
ANTIENT GEOGRAPHIC NAMES,
Abyssinia. Literally, the country of rain. 4, the ; bao, rain; yead, extension,
progress, dilatation. The country or district where, heavy, or intense, rain or showers
fall. ‘The water which supplies the Nile, falls in a great measure in Abyssinia, and it
had its name from the Phenicians, who were the first civilized people who penetrated
into it.
Aigyptus. ‘The civilized valley. 2, learned, cizilized, cultivated ; syy, valley, or
glyn.
Aithiopia. The civilized country of springs, or wells. j, civilized 3 sjoban, Of
wells ; ja, country.
Aitna. The holy hill. jz, a hill, or mountain ; yaory, holy.
Africa. The desert country. a, the ; ynac, barren, desert, bleak ; 4a, country,
From its barren plains and districts, (the name given it by ourselves is the Desert, )
which eventually became the denomination of the whole continent.
Adriaticum Mare. The Phoceans first made the Greeks acquainted with sea;
but it had its name from the Phenicians, who, as was their custom, endeavoured to
prevent other nations from trading to those places they found a source of wealth;
they, therefore, called this the sea of enchantment. U, the; opaoveacd, enchanted ;
oyce, Sea. The Greeks and Romans added, THedayoc, Pontus, and Mare, because they
knew not the intent of its name, the last syllable of which means sea. Therefore,
the additions were surplusage.
Luxinus Pontus. Here a similar unnecessary addition has been made. The first
word means the little sea. Ojce, sea ; 4n, little.
Parnassus. One of the highest mountains in Greece. Bap, chief; principal,
highest ; year, hill; ar, the sign of the comparative. The highest hill in its
neighbourhood.
Samothrace. An island in the Egean sea, where the Eleusinian mysteries were
carried to the highest perfection. It was an early settlement of the Phenicians. The
name is from the antient god Satay, heaven, or Celus ; and cperre, power, strength.
The residence of the gods.
Scylla. ‘The rocks in the straits of Messina in Sicily, many rocks in Ireland are
called Sceligs, from yeatlac, bald, bare, naked, without verdure.
Tamyras. A river in Phenicia, between Tyre and Sidon. Ta, the; mean, quick,
rapid ; ay, water, or cataract.
Yanas. A river in Numidia. TGanar, a spirit, apparition, ghost. The antients
always affixed a genius or spirit to a river. Or ceji, swift, hasty ; ay, stream.
Tentyra. A place in Thrace opposite Samothrace. Tean, rugged ; sym; land.
_
On the Affinity of the Hiberno- Celtic and Phenician Languages. 87
Thule. The island which the antients considered the farthest point of the
inhabited world, called Ultima thule, supposed to be Iceland. Tul, pronounced
Thule, is rest, sleep, repose. ‘The last place of rest.
Fatican. Ahillat Rome, covered with stagnated waters, long disregarded on that
account by the Romans. Feaz, a fen or bog ; a, the ; cean, head, or top; the hill
Covered with bog or swamp. Heliogabalus first drained and cleared it.
NYMPHS.
Nymph. The word Nymph (wwugn) is derived from the Phenician. Naor, holy,
sacred, sanctified, set apart, and was used by the Greeks to designate a bride, as
well asa female divinity ; yaorj, which in pronunciation sounds nearly the same, is used
by the Irish at this day in the same sense, it means a holy person, or saint, and a
bride. It is spelled by the Highland Scottish Gael, Naomh.
The derivation from the word Lympha, and that from rov da véac galvecba, their perpe-
tual youthful appearance it is unnecessary to refute. They were the imaginary divini-
ties, or spirits, who were supposed to inhabit the seas, rivers, fountains, springs, woods,
trees, mountains, &c. &c. from which our modern fairy elves are derived, as well as
the genii of the east.
The Dryades, are probably derived from opaojdeac, witchcraft, sorcery, magic,
elfish craft, which has ever been supposed to be performed by the agency of the
gen, fairies, demons, or evil spirits; or under the Greek and Roman mythology,
the demi-deity, who presided over fountains, woods, &c. &c.
The Hamadryades were formed by the addition of the word onmna, a large tree, or
timber ; the supposed residence of the spiritual being, for the sake of euphony the
Celto-Phenicians often dropped a consonant which interfered with the agreeable sound
of a word.
The number of nymphs according to Hesiod, amounted to 3000; it would, there-
fore, be a laborious task to collate their names, so as to ascertain their respective
meanings. I shall, therefore, merely take the Neriedes in the order their names
stand in Lempriere’s Classical Dictionary, which will afford ample testimony, that they
were all derived from Phenician roots: their signification is not only striking, but
highly poetical.
Sao. The light, flitting, aerial nymph. saob, light, Sc.
Amphitrite. ‘The nymph of the sea-weed or fuci. %m, the deep sea, or ocean ;
yitpleac, sea-weed, fuct.
Proto. ‘The arrogant, conceited nymph. Ppov, or boo, arrogance, conceit.
Galatea. ‘The bright, fair, or beautiful nymph of the wave or froth. galla, fuir-
ness, brightness ; cea3, froth, delusive vision or appearance.
88 On the Affinity of the Hiberno- Celtic and Phenician Languages.
Thoe. Thesilent nymph. Toa, silence, quietude.
Eucrate. @a, the; cneac, swan. 4
Eudora. The leading, early, first, or prominent nymph. @a, the ; copa, leader.
Gallena. The nymph of the rock. 3allan, a rock.
Glauce. ‘The sea-green nymph. Slay, sea-green.
Thetis. The nymph of the deep water. Ges, smooth, pronounced Thee ; jy,
under.
Spio. ‘The reproachful nymph. sjjoo, reproach.
Cymothoe. The nymph of the gentle quiet wave. Cyma, a small wave ; coa, silent.
Melita. The nymph who sports on the wave. 2ealav, sporting, enjoying.
Thalia. The nymph of the calm. Talat, rocking to sleep, composing.
Agave. The nymph of the boisterous wave. a, the; satac, dangerous, perilous,
boisterous.
Eulimine. The nymph who leaps or dances on the waves, or from wave to wave.
@a, the ; mac, leaper, or dancer.
Erato. ‘The nymph who runs on the waves. @a, the ; paca, running.
Pasithea. The gentle nymph who deviously sports in the waves. Ba, good ; tycea>,
bending, twisting, revolving.
Doto. The female sea-snail or nautilus. ao, the sea-snail ; co, female.
Eunice. The nimble nymph. @a, the ; yey, nimble.
Neasa. The generous, friendly nymph. Wearya, generous, friendly.
Dynamene. The nymph who appears and disappears rapidly. eynmne, swift,
active, nimble, supple, hasty, rapid.
Ferusa. The nymph who sports in devious contortions. ¥yapay, crooked, twisted,
bent.
Protomelia. The nymph who sports in the phosphoric spark of the wave. Bnoz,
Jire ; co, feminine, or female ; mealad, enjoying, sporting.
Actea. The nymph of despair. e@j5, a cry ; cao5, frenzy, passion.
Panope. The nymph of the estuary. Bean, the woman; aba, an estuary.
Doris. The invisible nymph. o-ajpeac, invisible.
Cymatolege. The nymph of the wet rock or breaker. Cyjm, a drop, water, or
wave ; acolejc, on a rock or stone.
Hippothoe. The silent, happy nymph. %ojb, joyous, pleasant ; 0, from co, silent,
Cymadoce. The nymph of the white foamy wave. Cjm, a wave ; 0,,from 3; vor,
scum, froth of the wave.
Neso. The nimble nymph. Wey, quick, nimble, agile.
Eupompe. The nymph who sounds the conc or musical shell. e@us, music; veut,
to sound, strike.
Pronoe. The complaining or bewailing nymph. Bpojneas, complaining.
Mae
On the Affinity of the Hiberno- Celtic and Phenician Languages. 89
Themiste. The nymph of the flowing tide. Taom, to pour out, overflow, throw
water out of a vessel ; arse, out of her.
Glauconome. The nymph with long waving hair. 5leacdac, with waving hair ;
naon, a nymph.
Halimede. The walking nymph of the rocks. 2, @ rock; jmeaéo, walking or
pacing.
Pontoporia. The nymph of the wells, or spring, district. Bean, @ woman ; coban,
a spring, or well; ja, a district, or country.
Evagora. The kind and smiling nymph. ojb, civil, courteous, pleasant ; 4, the ;
zon, laughter, or smiling.
Liagora. The nymphof the pleasant countenance. ja, the face, or countenance ;
zon, laughter, cheerfulness.
Polynome. The nymph of the cavern. poll, a hole, pit, or cavern ; ynaom, a
nymph.
Laomedia. The mid-day. taomeavon, the mid-day or noon.
Lysianassa. The nymph of the light of the promontory. Lear, light ; na, of the ;
near, promontory, headland.
Antonoe. The nymph of the spark of the wave. ay, a spark of fire; con, of the
wave ; naom, nymph.
Menippe. The sipping nymph. jon, little; jb, drinker.
Evarne. The nymph of the marble rocks. jtean, marble ; na, of the.
Psamathe. The nymph of the summer breeze. Sam, the swmmer ; aca, breeze.
Nemertes. The nymph of the boat. Waom, nymph; ancaé, boat.
The most incredulous must admit, that all those names are extremely apposite ;
and some, if not most of them, palpably derived from the same language as the
Hiberno-Celtic. It is scarcely possible so great, so universal a coincidence in name
and import can have been accidental. If they stood alone, they would amount almost
to demonstration, but supported as they are by other evidence and analogies, they
appear to me to be irresistible.
The further prosecution of this investigation is likely to dispel much of the mist,
and unravel many of the difficulties in which antient history is involved. Every step
we advance opens a new vista and field of view, and, at the same time, points out the
means of further progress.
It was my intention to have said something of the Cabiri, but I think it a subject
worthy separate consideration, and a future paper.
VOL. XVII. ss
nities
* 4
sk
n ta
F ae |
fy~of
On the Ring Money of the Celte. By Str Wititam Beruam, M.R.I. A.
B.S.A. Secretary for Foreign Correspondence, Member of the Royal Academy
of Sciences of Lisbon, &c.
Read 28th of November, 1836, and 9th of January, 1837.
In a letter from my friend, Richard Sainthill of Cork, dated the 5th (Novy. 1836)
instant, is the following paragraph :—
“ Cork, 5th November, 1836.
‘«* T am anxiously expecting your paper on the Ring Money of Ireland, which Mr.
Orpen will at any time send me. A vessel going to Africa, to trade with the natives,
was wrecked last summer on the coast here; among the articles on board for barter,
were some boxes of cast iron pieces, extremely like one species of gold articles found
in Ireland, annexed is a rude outline, one of which was saved from the iron foundry,
where they were sold and melted. I understand they pass in barter at about the value
of one halfpenny, but the similarity in shape to the old Irish articles is curious.”
Another letter from the same gentleman, dated 12th November, instant, he says—
“Thad the pleasure to receive your letters and paper on the Ring Money of Ire-
land last night, and the present is merely to request you to accept the enclosed speci-
men of the Anglo-African Iron Ring Money. I will procure the information you
require, and send it to you as soon as possible.”
I have now the pleasure of placing before the Academy specimens of the Irish
bronze, the iron found on board the wrecked vessel, and two other specimens, both
found in Italy, one said to be found in Herculaneum, but this I should doubt. This
latter is much corroded, and is also very singularly incrusted with what appears to me
to be crystals of carbonate of lime.
I yesterday received a letter from my friend T. C. Croker, dated 25th November,
(1836,) of which the following is an extract :—
“‘ T have got some curious information for you respecting the ring money now
current in Africa, which goes completely to establish your theory. Not a shadow of
doubt can now exist on the subject. My informants are our friend Sainthill and an
Egyptian traveller, Mr. Bonomi. Their letters to me are not immediately at hand,
or I should send them over to you.”
92 On the Ring Money of the Celte.
These are most unexpected confirmations of the theory I had ventured to pro-
pound, respecting the larger description of ring money ; with respect to the smaller,
no one could have any doubt, I mean those of mere rings.
9th of January, 1837.
On the 28th of November, 1836, I had the honour to place before you specimens
of certain articles of cast iron, found on board a vessel wrecked on the coast of Cork,
so exactly similar im size and shape to those found of Celtic brass in Ireland, that it is
quite impossible to refuse assent to the notion of their being of the same use and in-
tent. I have since obtained the most satisfactory information on the subject, which I
have now the pleasure of placing before the Academy.
Although very unwilling to intrude on the time of the Academy more than is ab-
solutely necessary to convey the information, yet, rather than not be sufficiently ex-
plicit, I think it best to give the information in the order I received it, and then pro-
ceed to make a few observations.
In a letter from my friend Mr, Sainthill of Cork, dated 7th of December last, he
says—
“The gentleman who collected the African trading bracelet money, is named Mr.
Abraham Abell, a Quaker, to whom I applied for particulars which he promised, and
I have no doubt intends to give. Iam weary with waiting. I can only inform you,
that the schooner Magnificent, the property (I believe) of Sir John Tobin of Liver-
pool, on her voyage to Africa, was wrecked at Ballycotten Bay, about a year ago. A
nephew of Sir John Tobin’s is working the gunpowder mills at Ballincollig, near Cork,
for lis uncle, but since your letter | have not met him.”
In another letter, dated the 10th of December, 1836, Mr. Sainthill says—‘‘T this
day met Mr. Tobin of Ballincollig gunpowder mills, I spoke to him respecting the
African money, he promised most kindly to procure me every information. He in-
formed me that the articles are manufactured at Birmingham, and are a composition
of brass and copper, they are called manillas, and are worn as ornaments, and pass as
the representative of money in Africa. They send out about forty chests annually,
and, according to the number of vessels on the coast, a greater or lesser number re-
present a bar, which is a certain ideal standard of value, like our pound; and as our
pound, according to the rate of exchange, say with France, for instance, would to-day
represent twenty-four franks, to-morrow twenty-six, next day twenty-two; so the
bar one day represents so many manillas, and to-morrow, if a dozen vessels arrived,
so many less.”’
Having a slight acquaintance with Sir John Tobin, and knowing his gentlemanly
courtesy, I wrote to request all the information he could give me on the subject, and
I had the pleasure of receiving a most satisfactory letter in answer to my queries, of
which the following is an extract of so much as refers to the subject :-—
On the Ring Money of the Celte. 93
**On the subject of the schooner Magnificent, which was lost some where near
Cork, some time since ; she was bound to the river Bonney, or New Calabar, which is
not far from the kingdom of Benin. The trade to these rivers for palm oil and ivory,
is cotton goods of a great variety, gunpowder, muskets, and an extensive number of
other articles and manillas, both of iron, and copper mixed, which is the money that
the people of the Eboe and Brass country, and all the natives in that neighbourhood,
go to market with. On Wednesday next, I will send you a manilla of each kind.
This vessel did not belong to me, but the master of her was named Tobin, and had
been.a master of one of my vessels.
‘If at any time I can give you any information of the West Coast of Africa and
its inhabitants, and their customs and habits, I shall have great pleasure, having in
early life seen a good deal of these people.
** T am, dear Sir, yours, &c.
“J. Topix.”
On the 28th of December, 1836, the day after I received Sir John Tobin’s letter,
which I have just read, I received the parcel containing the books which I had the
pleasure of presenting just now, from my learned and valued friend Dr. Hibbert, of
York, (late of Edinburgh,) in which, among other things, he was kind enough to send
me the following :—
“ Extract of a letter from Edward Jones, Esq. Captain of the Ist Regiment of
Royal Lancashire Militia, to Dr. Hibbert, dated 9, Bridge-street, Manchester, 16th
November, 1836—
«<The annexed two sketches are taken from a cast of the species of money now,
at the present day, passing current among the Africans. It so strongly resembles
what we saw in Ireland, that I thought you might be interested in a copy of it. A
Mr. Dyson, who was for some years surgeon on board an African merchantman,
brought it with him; and the first opportunity I shall make inquiries respecting this
coin, and other sorts in use among the natives.
««< J am told that in the country they are made of solid gold as in Ireland, but that
they are now counterfeited in England, and sent out to Africa in large chests, espe-
cially from Liverpool.
** «Do you think this will throw any light on the antient Irish relics, &c. &c.’
“I wrote to Captain Jones, in reply, that I believed Sir William Betham had pub-
lished some remarks on the gold relics found in Ireland, and that I would communi-
cate to him the information given me.
“S. Hippert.”
Thus has corroborating information and evidence crowded upon me, in a very re-
markable manner, from different quarters and in appropriate season.
Mr. J. Bonomi, who travelled with Lord Prudhoe in Egypt, Nubia, and Sennaar,
writes to my friend T. C. Croker as follows :
94 On the Ring Money of the Celte.
«‘ You ask me for a note on the ring money of Africa—here it is. So little has the
interior of that country changed, in that particular, since the days of the Pharaohs,
that to this day, among the inhabitants of Sennaar, pieces of gold, in the form of
rings, pass current. ‘The rings of gold have a cut in them,
O
for the convenience of keeping them together, the gold being so pure you easily bind
them and unite them in the manner of a chain. The money is weighed, as in the
days of Joseph. ;
«‘T shall soon be able to show you a cast from an Egyptian basso-relievo, where
occurs an example of the money in the form of a chain; and you may see, in
Heskyn’s work, plenty of rings of gold.
“‘ Yours, very truly,
«J. Bonomi.”
These gold rings are identically the same, as to shape and character, with those
found in Ireland ; the sketch of one most accurately represents the other.
The manillas are still more interesting, in as much as they give the name by which
they are known among the Africans; a name, no doubt, the same which they
bore when they were first introduced among these people so many centuries, aye, it
may be said, thousands of years ago, by the Phenicians, the same commercial people
who introduced them into Ireland. So little has the customs of the negro race of
Western Africa, like those of Sennaar, changed since the period of their intercourse
with the Phenicians, that these manillas pass current as money among them as they
did two or three thousand years ago. Africa seems to have stood still, while the rest
of the world progressed in civilization ; her deadly unwholesome climate forbade in-
tercourse beyond the exchange of manufactured goods for her raw materials.
This name maniila is in itself a powerful testimony ; it is no doubt the name the
articles bore in Phenicia, and by which they were known when first introduced to the
knowledge of the African negro nations, who have preserved it to our day. In the
Celto-Phenician it literally means the value or representation of property. ayn,
riches, patrimony, goods, value, and eallac, cattle, or any description of property ;
the word chattles rightly expresses it. Thus it appears that as pecunia had its name
from pecus, cattle, because flocks and herds were the first riches, and a number of cattle
were the standard of value before money existed, and where it was not to be had; so
manilla means literally the value of cattle or goods, or the representative of the value
of cattle, or any chattle property.
Money was so scarce in Ireland, in the fourteenth and fifteenth century, that the
fines and amerciaments, mentioned on the rolls as imposed by the courts, were pigs,
sheep, and cattle ; but it is not necessary to use arguments in support of so self-
evident a proposition.
On the Ring Money of the Celte. 95
Benin and Calabar are situated on the Gulph of Guinea, in latitude from 7 to 10
N.; longitude, 5 to 10 east of London. It would appear from those facts that the
Phenicians had penetrated to the Gulph of Guinea, and were acquainted with the
whole of this coast, probably beyond the line. We know they circumnavigated Africa,
by order of Pharaoh Necho, king of Egypt; but it now appears that they traded
regularly to the coasts of Guinea, and there introduced a money which still bears a
Phenician name, and is still as much in estimation as it was when the merchant princes
of Tyre supplied them with manillas in exchange for their gold, ivory, and palm oil.
The Tobins, and other English merchants, equally eminent and illustrious with those
of Tyre, now occupy the Tyrians’ position, while the negroe’s is not a whit more
elevated than it was two or three thousand years since.
The Romans knew nothing on the west coast of Africa beyond the port of Sala,
now Sallee, in latitude 34 N. and a very narrow slip on the north coast, not even so
far as the Great Desert ; except, perhaps, they may be said to have been acquainted
with the existence of the Desert, but had no intercourse with any people beyond it.
The coast of Guinea, within ten degrees of the equator, was far beyond their ken ;
consequently, during their sway, the people of that district having no intercourse
with any great commercial people, their customs and habits of commerce received
no impetus likely to produce any change of their antient mode of traffic, and the me-
tallic currency they learned from the Phenicians remains unchanged to the present
day. The English finding the manillas current, naturally availed themselves of the
facilities which they possessed of fabricating them, it can scarcely be justly called
counterfeiting, because they bear no impress or mark of authority.
The Carthaginians may have carried on the trade with these coasts after the de-
struction of Tyre, but there is no evidence that they or the Romans ever visited
them. The intercourse of the English, French, Spaniards, Portuguese, and Dutch,
is of a very recent date.
When I first ventured to assert that these things were money, the smile of some-
thing very like pity, if not of contempt, met me on all sides; it was considered so
wild a proposition, that sage individuals almost considered it a proof of a disordered
imagination, and some laughed outright, it appeared so preposterous and improbable.
I have never doubted the truth of my opinion from the moment I first saw the speci-
men of the Monaghan manilla in the museum of my friend Mr. George Petrie; and
it is no small gratification to me to have been able to collect such irrefragable evidence
in its support, and to conduct the inquiry to so triumphant and satisfactory a conclusion.
This matter of the ring money has arisen out of the investigation into the question
of who were the Scoti? and the tracing them to be the Celtz, led to the consequent
question of who were the Celte? which being answered that they were Phenicians,
have led to further questions and inquiries respecting that great people, as well as to
who were the Pelasgi and Etruscans? and in short, into the investigation of the
96 On the Ring Money of the Celte.
history of the proceedings of the illustrious nation, who so abundantly sowed the seed
of civilization in the world. As we proceed new lights break in upon us, the inves-
tigation and elucidation of one question opens a vista to our sight into farther mys-
_ teries, and eventually enables us to force a passage into the remotest penetralia of an-
tiquity. If I have not deceived myself, I think I see my way still farther, and that I
shall be able on a future occasion to produce evidence to illustrate the progress of the
nations of the most remote antiquity, and to throw some light on the channel by which
civilization poured to the west.
[ have now the pleasure of laying before you the specimens of the two descriptions
of manillas alluded to in Sir John Tobin’s letter, which | have since received from that
gentleman. The price at first cost is as under :
The copper manilla is £105. per ton.
The cast iron manilla £22. per ton.
The copper manilla weighs somewhat more than two and a half ounces avoirdupois,
or about six to the pound, and would be about twopence each; but they, no doubt,
pass for much more in Africa, I believe about fourpence.
The iron manilla weighs rather more than an ounce and a half, and gives ten to
the pound, or about one farthing each; I believe they pass in Africa for about one
halfpenny.
The weight of the ancient Celtic brass manilla found in Monaghan, is somewhat
more than three ounces avoirdupois.
As nothing tends so much to the right understanding of the subject as figures of
the articles in juxta-position, I have placed the wood-cuts of all the different speci-
mens in one point of view.
GOLD RING MONEY, OR MANILLAS, FOUND IN IRELAND.
0000 ©
WEIGHTS OF THE ABOVE SPECIMENS.
Now Verte esce 12 grs. No. 9, 5 dwt. No. 16, 9 oz.
2, 1 dwt. 12 grs. 10, 10 dwt. 17, 16 dwt. 12 grs.
3, 2 dwt. 12 grs, 11, 11 dwt. ! 18, 2 oz.
4, 2dwt. 12 grs. 12, 8 dwt. 19, 3 oz. 12 dwt:
5, 2 dwt. 12 grs, 13, 11 dwt. 12 grs. 20, 19 oz. to 56 oz.
6, 3 dwt. 16 ers. 14, Counterfeit, plated with 21, 1 oz.
7, 3 dwt. 16 grs. gold over brass. 22, seeese 12 dwt.
8, 6 dwt. 15, 4 oz. 16 dwt. QOS cdaece 2 dwt.
VOL. XV. oN
BRASS MANILLAS.
26
Manilla in Copper and Cast Iron, fabricated in England, and now passing current as
money in Africa.
ERRATA.
Page
321, last line, for 0 + read 0 +
323, line 17, for + © read + Oa,
330, line 11, for pend read depend
336, equations (121.) for 4 read b
342, line 18, for For read For if
347, 6th line from foot, for thorems read theorems
360, last line but 2, for denote read denote by
383, last line, for B read B
384, line 7, for = read Ta
387, equation (340.) for 1 + m read 1 + im
391, before (358.) for formula read formule
391, before (862.) for formula read formule
391, last line, for one read one set
394, line 10, for A, to By read A, to B,
395, before (4.) for denote read denote it
399, in (21.) for (a, a, 4 ) read (a, % , a a )
399, in (23.) for (> =) read 2 he )
2 2
ay ay
“I
400, line 6, for (a 4, a, 4 ) read (a, 4 , a 4 )
400, last but 3, for retain read retain them
401, before (37.) for (a,,0) read (a, a2 )
401, 2d formula (39.) for b, read by
401, in (42.) for Bz read By
402, end of (44.) for a, read ay”
402, in (48.) for B® read By
404, in (62.) for a? read az
404, in (63.) for a,” +a) read a? + a”
405, after (69.) for formula read formule
407, in (78.) for (a",, a) read (a",, a’, )
407, in (79.) for a, a’, read a ay
407, in (83.) for a,* —a® read a\* — ay?
408, in (90.) for (a2, a2 )"—! read (a, , a2 )*—!
410, in (104.) for nx1 read n+1
410, in (107.) for mx1 read m+1
417, in (151.) for (1, By) read (by , bs Jn
421, in (182.) for (2 w' 7.0) read (2 w' mr, 0)
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